Real exponentiation, part I

Analysis I, Section 5.6: Real exponentiation, part I

I have attempted to make the translation as faithful a paraphrasing as possible of the original text. When there is a choice between a more idiomatic Lean solution and a more faithful translation, I have generally chosen the latter. In particular, there will be places where the Lean code could be "golfed" to be more elegant and idiomatic, but I have consciously avoided doing so.

Main constructions and results of this section:

  • Exponentiating reals to natural numbers and integers.

  • nth roots.

  • Raising a real to a rational number.

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namespace Chapter5

Definition 5.6.1 (Exponentiating a real by a natural number). Here we use the Mathlib definition coming from Monoid.{u} (M : Type u) : Type uMonoid.

lemma Real.pow_zero (x: Real) : x ^ 0 = 1 := rfl
lemma Real.pow_succ (x: Real) (n:ℕ) : x ^ (n+1) = (x ^ n) * x := rfllemma Real.pow_of_coe (q: ℚ) (n:ℕ) : (q:Real) ^ n = (q ^ n:ℚ) := q:ℚn:ℕ⊢ ↑q ^ n = ↑(q ^ n) q:ℚ⊢ ↑q ^ 0 = ↑(q ^ 0)q:ℚn:ℕhn:↑q ^ n = ↑(q ^ n)⊢ ↑q ^ (n + 1) = ↑(q ^ (n + 1)) q:ℚ⊢ ↑q ^ 0 = ↑(q ^ 0)q:ℚn:ℕhn:↑q ^ n = ↑(q ^ n)⊢ ↑q ^ (n + 1) = ↑(q ^ (n + 1)) All goals completed! 🐙
/- The claims below can be handled easily by existing Mathlib API (as `Real` already is known to be a `Field`), but the spirit of the exercises is to adapt the proofs of Proposition 4.3.10 that you previously established. -/

Analogue of Proposition 4.3.10(a)

theorem declaration uses `sorry`Real.pow_add (x:Real) (m n:ℕ) : x^n * x^m = x^(n+m) := x:Realm:ℕn:ℕ⊢ x ^ n * x ^ m = x ^ (n + m) All goals completed! 🐙

Analogue of Proposition 4.3.10(a)

theorem declaration uses `sorry`Real.pow_mul (x:Real) (m n:ℕ) : (x^n)^m = x^(n*m) := x:Realm:ℕn:ℕ⊢ (x ^ n) ^ m = x ^ (n * m) All goals completed! 🐙

Analogue of Proposition 4.3.10(a)

theorem declaration uses `sorry`Real.mul_pow (x y:Real) (n:ℕ) : (x*y)^n = x^n * y^n := x:Realy:Realn:ℕ⊢ (x * y) ^ n = x ^ n * y ^ n All goals completed! 🐙

Analogue of Proposition 4.3.10(b)

theorem declaration uses `sorry`Real.pow_eq_zero (x:Real) (n:ℕ) (hn : 0 < n) : x^n = 0 ↔ x = 0 := x:Realn:ℕhn:0 < n⊢ x ^ n = 0 ↔ x = 0 All goals completed! 🐙

Analogue of Proposition 4.3.10(c)

theorem declaration uses `sorry`Real.pow_nonneg {x:Real} (n:ℕ) (hx: x ≥ 0) : x^n ≥ 0 := x:Realn:ℕhx:x ≥ 0⊢ x ^ n ≥ 0 All goals completed! 🐙

Analogue of Proposition 4.3.10(c)

theorem declaration uses `sorry`Real.pow_pos {x:Real} (n:ℕ) (hx: x > 0) : x^n > 0 := x:Realn:ℕhx:x > 0⊢ x ^ n > 0 All goals completed! 🐙

Analogue of Proposition 4.3.10(c)

theorem declaration uses `sorry`Real.pow_ge_pow (x y:Real) (n:ℕ) (hxy: x ≥ y) (hy: y ≥ 0) : x^n ≥ y^n := x:Realy:Realn:ℕhxy:x ≥ yhy:y ≥ 0⊢ x ^ n ≥ y ^ n All goals completed! 🐙

Analogue of Proposition 4.3.10(c)

theorem declaration uses `sorry`Real.pow_gt_pow (x y:Real) (n:ℕ) (hxy: x > y) (hy: y ≥ 0) (hn: n > 0) : x^n > y^n := x:Realy:Realn:ℕhxy:x > yhy:y ≥ 0hn:n > 0⊢ x ^ n > y ^ n All goals completed! 🐙

Analogue of Proposition 4.3.10(d)

theorem declaration uses `sorry`Real.pow_abs (x:Real) (n:ℕ) : |x|^n = |x^n| := x:Realn:ℕ⊢ |x| ^ n = |x ^ n| All goals completed! 🐙

Definition 5.6.2 (Exponentiating a real by an integer). Here we use the Mathlib definition coming from DivInvMonoid.{u} (G : Type u) : Type uDivInvMonoid.

lemma Real.pow_eq_pow (x: Real) (n:ℕ): x ^ (n:ℤ) = x ^ n := x:Realn:ℕ⊢ x ^ ↑n = x ^ n All goals completed! 🐙
@[simp] lemma Real.zpow_zero (x: Real) : x ^ (0:ℤ) = 1 := x:Real⊢ x ^ 0 = 1 All goals completed! 🐙lemma Real.zpow_neg {x:Real} (n:ℕ) : x^(-n:ℤ) = 1 / (x^n) := x:Realn:ℕ⊢ x ^ (-↑n) = 1 / x ^ n All goals completed! 🐙

Analogue of Proposition 4.3.12(a)

theorem declaration uses `sorry`Real.zpow_add (x:Real) (n m:ℤ) (hx: x ≠ 0): x^n * x^m = x^(n+m) := x:Realn:ℤm:ℤhx:x ≠ 0⊢ x ^ n * x ^ m = x ^ (n + m) All goals completed! 🐙

Analogue of Proposition 4.3.12(a)

theorem declaration uses `sorry`Real.zpow_mul (x:Real) (n m:ℤ) : (x^n)^m = x^(n*m) := x:Realn:ℤm:ℤ⊢ (x ^ n) ^ m = x ^ (n * m) All goals completed! 🐙

Analogue of Proposition 4.3.12(a)

theorem declaration uses `sorry`Real.mul_zpow (x y:Real) (n:ℤ) : (x*y)^n = x^n * y^n := x:Realy:Realn:ℤ⊢ (x * y) ^ n = x ^ n * y ^ n All goals completed! 🐙

Analogue of Proposition 4.3.12(b)

theorem declaration uses `sorry`Real.zpow_pos {x:Real} (n:ℤ) (hx: x > 0) : x^n > 0 := x:Realn:ℤhx:x > 0⊢ x ^ n > 0 All goals completed! 🐙

Analogue of Proposition 4.3.12(b)

theorem declaration uses `sorry`Real.zpow_ge_zpow {x y:Real} {n:ℤ} (hxy: x ≥ y) (hy: y > 0) (hn: n > 0): x^n ≥ y^n := x:Realy:Realn:ℤhxy:x ≥ yhy:y > 0hn:n > 0⊢ x ^ n ≥ y ^ n All goals completed! 🐙
theorem declaration uses `sorry`Real.zpow_ge_zpow_ofneg {x y:Real} {n:ℤ} (hxy: x ≥ y) (hy: y > 0) (hn: n < 0) : x^n ≤ y^n := x:Realy:Realn:ℤhxy:x ≥ yhy:y > 0hn:n < 0⊢ x ^ n ≤ y ^ n All goals completed! 🐙

Analogue of Proposition 4.3.12(c)

theorem declaration uses `sorry`Real.zpow_inj {x y:Real} {n:ℤ} (hx: x > 0) (hy : y > 0) (hn: n ≠ 0) (hxy: x^n = y^n) : x = y := x:Realy:Realn:ℤhx:x > 0hy:y > 0hn:n ≠ 0hxy:x ^ n = y ^ n⊢ x = y All goals completed! 🐙

Analogue of Proposition 4.3.12(d)

theorem declaration uses `sorry`Real.zpow_abs (x:Real) (n:ℤ) : |x|^n = |x^n| := x:Realn:ℤ⊢ |x| ^ n = |x ^ n| All goals completed! 🐙

Definition 5.6.2. We permit ``junk values'' when Unknown identifier `x`x is negative or Unknown identifier `n`n vanishes.

noncomputable abbrev Real.root (x:Real) (n:ℕ) : Real := sSup { y:Real | y ≥ 0 ∧ y^n ≤ x }
noncomputable abbrev Real.sqrt (x:Real) := x.root 2

Lemma 5.6.5 (Existence of n^th roots)

theorem declaration uses `sorry`Real.rootset_nonempty {x:Real} (hx: x ≥ 0) (n:ℕ) (hn: n ≥ 1) : { y:Real | y ≥ 0 ∧ y^n ≤ x }.Nonempty := x:Realhx:x ≥ 0n:ℕhn:n ≥ 1⊢ {y | y ≥ 0 ∧ y ^ n ≤ x}.Nonempty x:Realhx:x ≥ 0n:ℕhn:n ≥ 1⊢ 0 ∈ {y | y ≥ 0 ∧ y ^ n ≤ x} All goals completed! 🐙
theorem declaration uses `sorry`Real.rootset_bddAbove {x:Real} (n:ℕ) (hn: n ≥ 1) : BddAbove { y:Real | y ≥ 0 ∧ y^n ≤ x } := x:Realn:ℕhn:n ≥ 1⊢ BddAbove {y | y ≥ 0 ∧ y ^ n ≤ x} -- This proof is written to follow the structure of the original text. x:Realn:ℕhn:n ≥ 1⊢ ∃ x_1, ∀ y ∈ {y | y ≥ 0 ∧ y ^ n ≤ x}, y ≤ x_1 x:Realn:ℕhn:n ≥ 1h:x ≤ 1⊢ ∃ x_1, ∀ y ∈ {y | y ≥ 0 ∧ y ^ n ≤ x}, y ≤ x_1x:Realn:ℕhn:n ≥ 1h:1 < x⊢ ∃ x_1, ∀ y ∈ {y | y ≥ 0 ∧ y ^ n ≤ x}, y ≤ x_1 x:Realn:ℕhn:n ≥ 1h:x ≤ 1⊢ ∃ x_1, ∀ y ∈ {y | y ≥ 0 ∧ y ^ n ≤ x}, y ≤ x_1 x:Realn:ℕhn:n ≥ 1h:x ≤ 1⊢ ∀ y ∈ {y | y ≥ 0 ∧ y ^ n ≤ x}, y ≤ 1; intro y x:Realn:ℕhn:n ≥ 1h:x ≤ 1y:Realhy:y ∈ {y | y ≥ 0 ∧ y ^ n ≤ x}⊢ y ≤ 1; x:Realn:ℕhn:n ≥ 1h:x ≤ 1y:Realhy:0 ≤ y ∧ y ^ n ≤ x⊢ y ≤ 1 x:Realn:ℕhn:n ≥ 1h:x ≤ 1y:Realhy:0 ≤ y ∧ y ^ n ≤ xhy':1 < y⊢ False replace hy' : 1 < y^n := x:Realn:ℕhn:n ≥ 1⊢ BddAbove {y | y ≥ 0 ∧ y ^ n ≤ x} All goals completed! 🐙 All goals completed! 🐙 x:Realn:ℕhn:n ≥ 1h:1 < x⊢ ∀ y ∈ {y | y ≥ 0 ∧ y ^ n ≤ x}, y ≤ x; intro y x:Realn:ℕhn:n ≥ 1h:1 < xy:Realhy:y ∈ {y | y ≥ 0 ∧ y ^ n ≤ x}⊢ y ≤ x; x:Realn:ℕhn:n ≥ 1h:1 < xy:Realhy:0 ≤ y ∧ y ^ n ≤ x⊢ y ≤ x x:Realn:ℕhn:n ≥ 1h:1 < xy:Realhy:0 ≤ y ∧ y ^ n ≤ xhy':x < y⊢ False replace hy' : x < y^n := x:Realn:ℕhn:n ≥ 1⊢ BddAbove {y | y ≥ 0 ∧ y ^ n ≤ x} All goals completed! 🐙 All goals completed! 🐙

Lemma 5.6.6 (ab) / Exercise 5.6.1

theorem declaration uses `sorry`Real.eq_root_iff_pow_eq {x y:Real} (hx: x ≥ 0) (hy: y ≥ 0) {n:ℕ} (hn: n ≥ 1) : y = x.root n ↔ y^n = x := x:Realy:Realhx:x ≥ 0hy:y ≥ 0n:ℕhn:n ≥ 1⊢ y = x.root n ↔ y ^ n = x All goals completed! 🐙

Lemma 5.6.6 (c) / Exercise 5.6.1

theorem declaration uses `sorry`Real.root_nonneg {x:Real} (hx: x ≥ 0) {n:ℕ} (hn: n ≥ 1) : x.root n ≥ 0 := x:Realhx:x ≥ 0n:ℕhn:n ≥ 1⊢ x.root n ≥ 0 All goals completed! 🐙

Lemma 5.6.6 (c) / Exercise 5.6.1

theorem declaration uses `sorry`Real.root_pos {x:Real} (hx: x ≥ 0) {n:ℕ} (hn: n ≥ 1) : x.root n > 0 ↔ x > 0 := x:Realhx:x ≥ 0n:ℕhn:n ≥ 1⊢ x.root n > 0 ↔ x > 0 All goals completed! 🐙
theorem declaration uses `sorry`Real.pow_of_root {x:Real} (hx: x ≥ 0) {n:ℕ} (hn: n ≥ 1) : (x.root n)^n = x := x:Realhx:x ≥ 0n:ℕhn:n ≥ 1⊢ x.root n ^ n = x All goals completed! 🐙theorem declaration uses `sorry`Real.root_of_pow {x:Real} (hx: x ≥ 0) {n:ℕ} (hn: n ≥ 1) : (x^n).root n = x := x:Realhx:x ≥ 0n:ℕhn:n ≥ 1⊢ (x ^ n).root n = x All goals completed! 🐙

Lemma 5.6.6 (d) / Exercise 5.6.1

theorem declaration uses `sorry`Real.root_mono {x y:Real} (hx: x ≥ 0) (hy: y ≥ 0) {n:ℕ} (hn: n ≥ 1) : x > y ↔ x.root n > y.root n := x:Realy:Realhx:x ≥ 0hy:y ≥ 0n:ℕhn:n ≥ 1⊢ x > y ↔ x.root n > y.root n All goals completed! 🐙

Lemma 5.6.6 (e) / Exercise 5.6.1

theorem declaration uses `sorry`Real.root_mono_of_gt_one {x : Real} (hx: x > 1) {k l: ℕ} (hkl: k > l) (hl: l ≥ 1) : x.root k < x.root l := x:Realhx:x > 1k:ℕl:ℕhkl:k > lhl:l ≥ 1⊢ x.root k < x.root l All goals completed! 🐙

Lemma 5.6.6 (e) / Exercise 5.6.1

theorem declaration uses `sorry`Real.root_mono_of_lt_one {x : Real} (hx0: 0 < x) (hx: x < 1) {k l: ℕ} (hkl: k > l) (hl: l ≥ 1) : x.root k > x.root l := x:Realhx0:0 < xhx:x < 1k:ℕl:ℕhkl:k > lhl:l ≥ 1⊢ x.root k > x.root l All goals completed! 🐙

Lemma 5.6.6 (e) / Exercise 5.6.1

theorem declaration uses `sorry`Real.root_of_one {k: ℕ} (hk: k ≥ 1): (1:Real).root k = 1 := k:ℕhk:k ≥ 1⊢ root 1 k = 1 All goals completed! 🐙

Lemma 5.6.6 (f) / Exercise 5.6.1

theorem declaration uses `sorry`Real.root_mul {x y:Real} (hx: x ≥ 0) (hy: y ≥ 0) {n:ℕ} (hn: n ≥ 1) : (x*y).root n = (x.root n) * (y.root n) := x:Realy:Realhx:x ≥ 0hy:y ≥ 0n:ℕhn:n ≥ 1⊢ (x * y).root n = x.root n * y.root n All goals completed! 🐙

Lemma 5.6.6 (g) / Exercise 5.6.1

theorem declaration uses `sorry`Real.root_root {x:Real} (hx: x ≥ 0) {n m:ℕ} (hn: n ≥ 1) (hm: m ≥ 1): (x.root n).root m = x.root (n*m) := x:Realhx:x ≥ 0n:ℕm:ℕhn:n ≥ 1hm:m ≥ 1⊢ (x.root n).root m = x.root (n * m) All goals completed! 🐙
theorem declaration uses `sorry`Real.root_one {x:Real} (hx: x > 0): x.root 1 = x := x:Realhx:x > 0⊢ x.root 1 = x All goals completed! 🐙theorem declaration uses `sorry`Real.pow_cancel {y z:Real} (hy: y > 0) (hz: z > 0) {n:ℕ} (hn: n ≥ 1) (h: y^n = z^n) : y = z := y:Realz:Realhy:y > 0hz:z > 0n:ℕhn:n ≥ 1h:y ^ n = z ^ n⊢ y = z All goals completed! 🐙example : ¬(∀ (y:Real) (z:Real) (n:ℕ) (_: n ≥ 1) (_: y^n = z^n), y = z) := ⊢ ¬∀ (y z : Real), ∀ n ≥ 1, y ^ n = z ^ n → y = z ⊢ ∃ x x_1 x_2, 1 ≤ x_2 ∧ x ^ x_2 = x_1 ^ x_2 ∧ ¬x = x_1; ⊢ 1 ≤ 2⊢ (-3) ^ 2 = 3 ^ 2⊢ ¬-3 = 3 ⊢ 1 ≤ 2⊢ (-3) ^ 2 = 3 ^ 2⊢ ¬-3 = 3 All goals completed! 🐙

Definition 5.6.7

noncomputable abbrev Real.ratPow (x:Real) (q:ℚ) : Real := (x.root q.den)^(q.num)
noncomputable instance Real.instRatPow : Pow Real ℚ where pow x q := x.ratPow qtheorem Rat.eq_quot (q:ℚ) : ∃ a:ℤ, ∃ b:ℕ, b > 0 ∧ q = a / b := q:ℚ⊢ ∃ a, ∃ b > 0, q = ↑a / ↑b q:ℚ⊢ q.den > 0 ∧ q = ↑q.num / ↑q.den; q:ℚthis:q.den ≠ 0 := q.den_nz⊢ q.den > 0 ∧ q = ↑q.num / ↑q.den refine ⟨ q:ℚthis:q.den ≠ 0 := q.den_nz⊢ q.den > 0 All goals completed! 🐙, (Rat.num_div_den q).symm ⟩

Lemma 5.6.8

theorem declaration uses `sorry`Real.pow_root_eq_pow_root {a a':ℤ} {b b':ℕ} (hb: b > 0) (hb' : b' > 0) (hq : (a/b:ℚ) = (a'/b':ℚ)) {x:Real} (hx: x > 0) : (x.root b')^(a') = (x.root b)^(a) := a:ℤa':ℤb:ℕb':ℕhb:b > 0hb':b' > 0hq:↑a / ↑b = ↑a' / ↑b'x:Realhx:x > 0⊢ x.root b' ^ a' = x.root b ^ a -- This proof is written to follow the structure of the original text. a:ℤa':ℤb:ℕb':ℕhb:b > 0hb':b' > 0hq:↑a / ↑b = ↑a' / ↑b'x:Realhx:x > 0this:∀ {a a' : ℤ} {b b' : ℕ}, b > 0 → b' > 0 → ↑a / ↑b = ↑a' / ↑b' → a > 0 → x.root b' ^ a' = x.root b ^ aha:¬a > 0⊢ x.root b' ^ a' = x.root b ^ ax:Realhx:x > 0a:ℤa':ℤb:ℕb':ℕhb:b > 0hb':b' > 0hq:↑a / ↑b = ↑a' / ↑b'ha:a > 0⊢ x.root b' ^ a' = x.root b ^ a a:ℤa':ℤb:ℕb':ℕhb:b > 0hb':b' > 0hq:↑a / ↑b = ↑a' / ↑b'x:Realhx:x > 0this:∀ {a a' : ℤ} {b b' : ℕ}, b > 0 → b' > 0 → ↑a / ↑b = ↑a' / ↑b' → a > 0 → x.root b' ^ a' = x.root b ^ aha:¬a > 0⊢ x.root b' ^ a' = x.root b ^ a a:ℤa':ℤb:ℕb':ℕhb:b > 0hb':b' > 0hq:↑a / ↑b = ↑a' / ↑b'x:Realhx:x > 0this:∀ {a a' : ℤ} {b b' : ℕ}, b > 0 → b' > 0 → ↑a / ↑b = ↑a' / ↑b' → a > 0 → x.root b' ^ a' = x.root b ^ aha:a ≤ 0⊢ x.root b' ^ a' = x.root b ^ a a:ℤa':ℤb:ℕb':ℕhb:b > 0hb':b' > 0hq:↑a / ↑b = ↑a' / ↑b'x:Realhx:x > 0this:∀ {a a' : ℤ} {b b' : ℕ}, b > 0 → b' > 0 → ↑a / ↑b = ↑a' / ↑b' → a > 0 → x.root b' ^ a' = x.root b ^ aha✝:a ≤ 0ha:a < 0⊢ x.root b' ^ a' = x.root b ^ aa:ℤa':ℤb:ℕb':ℕhb:b > 0hb':b' > 0hq:↑a / ↑b = ↑a' / ↑b'x:Realhx:x > 0this:∀ {a a' : ℤ} {b b' : ℕ}, b > 0 → b' > 0 → ↑a / ↑b = ↑a' / ↑b' → a > 0 → x.root b' ^ a' = x.root b ^ aha✝:a ≤ 0ha:a = 0⊢ x.root b' ^ a' = x.root b ^ a a:ℤa':ℤb:ℕb':ℕhb:b > 0hb':b' > 0hq:↑a / ↑b = ↑a' / ↑b'x:Realhx:x > 0this:∀ {a a' : ℤ} {b b' : ℕ}, b > 0 → b' > 0 → ↑a / ↑b = ↑a' / ↑b' → a > 0 → x.root b' ^ a' = x.root b ^ aha✝:a ≤ 0ha:a < 0⊢ x.root b' ^ a' = x.root b ^ a replace hq : ((-a:ℤ)/b:ℚ) = ((-a':ℤ)/b':ℚ) := a:ℤa':ℤb:ℕb':ℕhb:b > 0hb':b' > 0hq:↑a / ↑b = ↑a' / ↑b'x:Realhx:x > 0⊢ x.root b' ^ a' = x.root b ^ a a:ℤa':ℤb:ℕb':ℕhb:b > 0hb':b' > 0hq:↑a / ↑b = ↑a' / ↑b'x:Realhx:x > 0this:∀ {a a' : ℤ} {b b' : ℕ}, b > 0 → b' > 0 → ↑a / ↑b = ↑a' / ↑b' → a > 0 → x.root b' ^ a' = x.root b ^ aha✝:a ≤ 0ha:a < 0⊢ -↑a / ↑b = -↑a' / ↑b'; a:ℤa':ℤb:ℕb':ℕhb:b > 0hb':b' > 0x:Realhx:x > 0ha✝:a ≤ 0ha:a < 0this:∀ {a a' : ℤ} {b b' : ℕ}, b > 0 → b' > 0 → ↑a * (↑b)⁻¹ = ↑a' * (↑b')⁻¹ → a > 0 → x.root b' ^ a' = x.root b ^ ahq:↑a * (↑b)⁻¹ = ↑a' * (↑b')⁻¹⊢ -(↑a * (↑b)⁻¹) = -(↑a' * (↑b')⁻¹); All goals completed! 🐙 specialize this hb hb' hq (a:ℤa':ℤb:ℕb':ℕhb:b > 0hb':b' > 0x:Realhx:x > 0this:∀ {a a' : ℤ} {b b' : ℕ}, b > 0 → b' > 0 → ↑a / ↑b = ↑a' / ↑b' → a > 0 → x.root b' ^ a' = x.root b ^ aha✝:a ≤ 0ha:a < 0hq:↑(-a) / ↑b = ↑(-a') / ↑b' := Eq.mpr (id (congr (congrArg Eq (congrFun' (congrArg HDiv.hDiv (Int.cast_neg a)) ↑b)) (congrFun' (congrArg HDiv.hDiv (Int.cast_neg a')) ↑b'))) (Eq.mpr (id (congr (congrArg Eq (Eq.trans (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Tactic.Ring.inv_mul (Eq.refl (↑b)⁻¹) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))))) (Mathlib.Tactic.Ring.mul_pf_right (↑b)⁻¹ (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b)⁻¹ (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * ((↑b)⁻¹ ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))) (Mathlib.Tactic.Ring.zero_mul ((↑b)⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * ((↑b)⁻¹ ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑a) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑a))) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (↑b)⁻¹) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (↑b)⁻¹))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑b)⁻¹ 1)) (congrArg Neg.neg (mul_one (↑b)⁻¹)))) (Mathlib.Tactic.RingNF.mul_neg (↑a) (↑b)⁻¹))) 0) (add_zero (-(↑a * (↑b)⁻¹)))))) (Eq.trans (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Tactic.Ring.inv_mul (Eq.refl (↑b')⁻¹) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))))) (Mathlib.Tactic.Ring.mul_pf_right (↑b')⁻¹ (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a') (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b')⁻¹ (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast))) (Mathlib.Tactic.Ring.mul_zero (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * ((↑b')⁻¹ ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))) (Mathlib.Tactic.Ring.zero_mul ((↑b')⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * ((↑b')⁻¹ ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑a') Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑a'))) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (↑b')⁻¹) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (↑b')⁻¹))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑b')⁻¹ 1)) (congrArg Neg.neg (mul_one (↑b')⁻¹)))) (Mathlib.Tactic.RingNF.mul_neg (↑a') (↑b')⁻¹))) 0) (add_zero (-(↑a' * (↑b')⁻¹))))))) (of_eq_true (Eq.trans (congrFun' (congrArg Eq (congrArg Neg.neg (Eq.mp (congr (congrArg Eq (Eq.trans (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Tactic.Ring.inv_mul (Eq.refl (↑b)⁻¹) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))))) (Mathlib.Tactic.Ring.mul_pf_right (↑b)⁻¹ (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b)⁻¹ (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * ((↑b)⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((↑b)⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * ((↑b)⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑a) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑a))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (↑b)⁻¹) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (↑b)⁻¹))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (↑b)⁻¹)))) 0) (add_zero (↑a * (↑b)⁻¹))))) (Eq.trans (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Tactic.Ring.inv_mul (Eq.refl (↑b')⁻¹) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))))) (Mathlib.Tactic.Ring.mul_pf_right (↑b')⁻¹ (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a') (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b')⁻¹ (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * ((↑b')⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((↑b')⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * ((↑b')⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑a') Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑a'))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (↑b')⁻¹) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (↑b')⁻¹))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (↑b')⁻¹)))) 0) (add_zero (↑a' * (↑b')⁻¹))))) _fvar.94286))) (-(↑a' * (↑b')⁻¹))) (eq_self (-(↑a' * (↑b')⁻¹))))))⊢ -a > 0 All goals completed! 🐙) All goals completed! 🐙 have : a' = 0 := a:ℤa':ℤb:ℕb':ℕhb:b > 0hb':b' > 0hq:↑a / ↑b = ↑a' / ↑b'x:Realhx:x > 0⊢ x.root b' ^ a' = x.root b ^ a All goals completed! 🐙 All goals completed! 🐙 have : a' > 0 := a:ℤa':ℤb:ℕb':ℕhb:b > 0hb':b' > 0hq:↑a / ↑b = ↑a' / ↑b'x:Realhx:x > 0⊢ x.root b' ^ a' = x.root b ^ a All goals completed! 🐙 x:Realhx:x > 0a:ℤa':ℤb:ℕb':ℕhb:b > 0hb':b' > 0ha:a > 0this:a' > 0 := sorryhq:↑a * ↑b' = ↑b * ↑a'⊢ x.root b' ^ a' = x.root b ^ a lift a to ℕ using x:Realhx:x > 0a:ℤa':ℤb:ℕb':ℕhb:b > 0hb':b' > 0ha:a > 0this:a' > 0 := sorryhq:↑a * ↑b' = ↑b * ↑a'⊢ 0 ≤ a All goals completed! 🐙 lift a' to ℕ using x:Realhx:x > 0a':ℤb:ℕb':ℕhb:b > 0hb':b' > 0this:a' > 0 := sorrya:ℕha:↑a > 0hq:↑↑a * ↑b' = ↑b * ↑a'⊢ 0 ≤ a' All goals completed! 🐙 x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'⊢ x.root b' ^ a' = x.root b ^ a x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')⊢ x.root b' ^ a' = x.root b ^ a have h1 : y = (x.root b').root a := a:ℤa':ℤb:ℕb':ℕhb:b > 0hb':b' > 0hq:↑a / ↑b = ↑a' / ↑b'x:Realhx:x > 0⊢ x.root b' ^ a' = x.root b ^ a x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')⊢ x ≥ 0x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')⊢ b' ≥ 1x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')⊢ a ≥ 1 x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')⊢ x ≥ 0x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')⊢ b' ≥ 1x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')⊢ a ≥ 1 All goals completed! 🐙 have h2 : y = (x.root b).root a' := a:ℤa':ℤb:ℕb':ℕhb:b > 0hb':b' > 0hq:↑a / ↑b = ↑a' / ↑b'x:Realhx:x > 0⊢ x.root b' ^ a' = x.root b ^ a x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')h1:y = (x.root b').root a := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (mul_comm b' a))) (Eq.refl y))⊢ x ≥ 0x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')h1:y = (x.root b').root a := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (mul_comm b' a))) (Eq.refl y))⊢ b ≥ 1x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')h1:y = (x.root b').root a := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (mul_comm b' a))) (Eq.refl y))⊢ a' ≥ 1 x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')h1:y = (x.root b').root a := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (mul_comm b' a))) (Eq.refl y))⊢ x ≥ 0x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')h1:y = (x.root b').root a := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (mul_comm b' a))) (Eq.refl y))⊢ b ≥ 1x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')h1:y = (x.root b').root a := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (mul_comm b' a))) (Eq.refl y))⊢ a' ≥ 1 All goals completed! 🐙 have h3 : y^a = x.root b' := a:ℤa':ℤb:ℕb':ℕhb:b > 0hb':b' > 0hq:↑a / ↑b = ↑a' / ↑b'x:Realhx:x > 0⊢ x.root b' ^ a' = x.root b ^ a x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')h1:y = (x.root b').root a := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (mul_comm b' a))) (Eq.refl y))h2:y = (x.root b).root a' := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b)) hb))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b 1) (congrArg (LT.lt ↑b) Nat.cast_one)) a))))))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a')) this))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a' 1) (congrArg (LT.lt ↑a') Nat.cast_one)) a)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (Eq.symm hq))) (Eq.refl y))⊢ (x.root b').root a ^ a = x.root b'; x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')h1:y = (x.root b').root a := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (mul_comm b' a))) (Eq.refl y))h2:y = (x.root b).root a' := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b)) hb))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b 1) (congrArg (LT.lt ↑b) Nat.cast_one)) a))))))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a')) this))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a' 1) (congrArg (LT.lt ↑a') Nat.cast_one)) a)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (Eq.symm hq))) (Eq.refl y))⊢ a ≥ 1x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')h1:y = (x.root b').root a := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (mul_comm b' a))) (Eq.refl y))h2:y = (x.root b).root a' := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b)) hb))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b 1) (congrArg (LT.lt ↑b) Nat.cast_one)) a))))))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a')) this))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a' 1) (congrArg (LT.lt ↑a') Nat.cast_one)) a)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (Eq.symm hq))) (Eq.refl y))⊢ x ≥ 0x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')h1:y = (x.root b').root a := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (mul_comm b' a))) (Eq.refl y))h2:y = (x.root b).root a' := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b)) hb))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b 1) (congrArg (LT.lt ↑b) Nat.cast_one)) a))))))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a')) this))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a' 1) (congrArg (LT.lt ↑a') Nat.cast_one)) a)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (Eq.symm hq))) (Eq.refl y))⊢ b' ≥ 1 x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')h1:y = (x.root b').root a := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (mul_comm b' a))) (Eq.refl y))h2:y = (x.root b).root a' := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b)) hb))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b 1) (congrArg (LT.lt ↑b) Nat.cast_one)) a))))))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a')) this))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a' 1) (congrArg (LT.lt ↑a') Nat.cast_one)) a)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (Eq.symm hq))) (Eq.refl y))⊢ a ≥ 1x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')h1:y = (x.root b').root a := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (mul_comm b' a))) (Eq.refl y))h2:y = (x.root b).root a' := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b)) hb))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b 1) (congrArg (LT.lt ↑b) Nat.cast_one)) a))))))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a')) this))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a' 1) (congrArg (LT.lt ↑a') Nat.cast_one)) a)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (Eq.symm hq))) (Eq.refl y))⊢ x ≥ 0x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')h1:y = (x.root b').root a := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (mul_comm b' a))) (Eq.refl y))h2:y = (x.root b).root a' := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b)) hb))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b 1) (congrArg (LT.lt ↑b) Nat.cast_one)) a))))))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a')) this))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a' 1) (congrArg (LT.lt ↑a') Nat.cast_one)) a)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (Eq.symm hq))) (Eq.refl y))⊢ b' ≥ 1 All goals completed! 🐙 have h4 : y^a' = x.root b := a:ℤa':ℤb:ℕb':ℕhb:b > 0hb':b' > 0hq:↑a / ↑b = ↑a' / ↑b'x:Realhx:x > 0⊢ x.root b' ^ a' = x.root b ^ a x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')h1:y = (x.root b').root a := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (mul_comm b' a))) (Eq.refl y))h2:y = (x.root b).root a' := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b)) hb))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b 1) (congrArg (LT.lt ↑b) Nat.cast_one)) a))))))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a')) this))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a' 1) (congrArg (LT.lt ↑a') Nat.cast_one)) a)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (Eq.symm hq))) (Eq.refl y))h3:y ^ a = x.root b' := Eq.mpr (id (congrArg (fun _a => _a ^ a = x.root b') h1)) (pow_of_root (root_nonneg (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a)))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1))))))))⊢ (x.root b).root a' ^ a' = x.root b; x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')h1:y = (x.root b').root a := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (mul_comm b' a))) (Eq.refl y))h2:y = (x.root b).root a' := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b)) hb))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b 1) (congrArg (LT.lt ↑b) Nat.cast_one)) a))))))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a')) this))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a' 1) (congrArg (LT.lt ↑a') Nat.cast_one)) a)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (Eq.symm hq))) (Eq.refl y))h3:y ^ a = x.root b' := Eq.mpr (id (congrArg (fun _a => _a ^ a = x.root b') h1)) (pow_of_root (root_nonneg (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a)))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1))))))))⊢ a' ≥ 1x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')h1:y = (x.root b').root a := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (mul_comm b' a))) (Eq.refl y))h2:y = (x.root b).root a' := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b)) hb))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b 1) (congrArg (LT.lt ↑b) Nat.cast_one)) a))))))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a')) this))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a' 1) (congrArg (LT.lt ↑a') Nat.cast_one)) a)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (Eq.symm hq))) (Eq.refl y))h3:y ^ a = x.root b' := Eq.mpr (id (congrArg (fun _a => _a ^ a = x.root b') h1)) (pow_of_root (root_nonneg (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a)))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1))))))))⊢ x ≥ 0x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')h1:y = (x.root b').root a := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (mul_comm b' a))) (Eq.refl y))h2:y = (x.root b).root a' := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b)) hb))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b 1) (congrArg (LT.lt ↑b) Nat.cast_one)) a))))))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a')) this))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a' 1) (congrArg (LT.lt ↑a') Nat.cast_one)) a)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (Eq.symm hq))) (Eq.refl y))h3:y ^ a = x.root b' := Eq.mpr (id (congrArg (fun _a => _a ^ a = x.root b') h1)) (pow_of_root (root_nonneg (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a)))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1))))))))⊢ b ≥ 1 x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')h1:y = (x.root b').root a := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (mul_comm b' a))) (Eq.refl y))h2:y = (x.root b).root a' := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b)) hb))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b 1) (congrArg (LT.lt ↑b) Nat.cast_one)) a))))))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a')) this))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a' 1) (congrArg (LT.lt ↑a') Nat.cast_one)) a)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (Eq.symm hq))) (Eq.refl y))h3:y ^ a = x.root b' := Eq.mpr (id (congrArg (fun _a => _a ^ a = x.root b') h1)) (pow_of_root (root_nonneg (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a)))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1))))))))⊢ a' ≥ 1x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')h1:y = (x.root b').root a := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (mul_comm b' a))) (Eq.refl y))h2:y = (x.root b).root a' := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b)) hb))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b 1) (congrArg (LT.lt ↑b) Nat.cast_one)) a))))))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a')) this))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a' 1) (congrArg (LT.lt ↑a') Nat.cast_one)) a)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (Eq.symm hq))) (Eq.refl y))h3:y ^ a = x.root b' := Eq.mpr (id (congrArg (fun _a => _a ^ a = x.root b') h1)) (pow_of_root (root_nonneg (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a)))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1))))))))⊢ x ≥ 0x:Realb:ℕb':ℕhb:b > 0hb':b' > 0a:ℕa':ℕhx:x > 0ha:0 < athis:0 < a'hq:a * b' = b * a'y:Real := x.root (a * b')h1:y = (x.root b').root a := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (mul_comm b' a))) (Eq.refl y))h2:y = (x.root b).root a' := Eq.mpr (id (congrArg (fun _a => y = _a) (root_root (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b)) hb))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b 1) (congrArg (LT.lt ↑b) Nat.cast_one)) a))))))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a')) this))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a' 1) (congrArg (LT.lt ↑a') Nat.cast_one)) a)))))))))) (Eq.mpr (id (congrArg (fun _a => y = x.root _a) (Eq.symm hq))) (Eq.refl y))h3:y ^ a = x.root b' := Eq.mpr (id (congrArg (fun _a => _a ^ a = x.root b') h1)) (pow_of_root (root_nonneg (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul x (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (x ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf x) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (x ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero x (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw Real (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw Real 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat Real Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt hx) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b') (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b' ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 b') (congrFun' (congrArg LT.lt Nat.cast_zero) ↑b')) hb'))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 b' 1) (congrArg (LT.lt ↑b') Nat.cast_one)) a)))))))) (le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrFun' (congrArg LT.lt Nat.cast_zero) ↑a)) ha))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 a 1) (congrArg (LT.lt ↑a) Nat.cast_one)) a_1))))))))⊢ b ≥ 1 All goals completed! 🐙 All goals completed! 🐙
theorem Real.ratPow_def {x:Real} (hx: x > 0) (a:ℤ) {b:ℕ} (hb: b > 0) : x^(a/b:ℚ) = (x.root b)^a := x:Realhx:x > 0a:ℤb:ℕhb:b > 0⊢ x ^ (↑a / ↑b) = x.root b ^ a x:Realhx:x > 0a:ℤb:ℕhb:b > 0q:ℚ := ↑a / ↑b⊢ x ^ q = x.root b ^ a x:Realhx:x > 0a:ℤb:ℕhb:b > 0q:ℚ := ↑a / ↑b⊢ q.den > 0x:Realhx:x > 0a:ℤb:ℕhb:b > 0q:ℚ := ↑a / ↑b⊢ ↑a / ↑b = ↑q.num / ↑q.den x:Realhx:x > 0a:ℤb:ℕhb:b > 0q:ℚ := ↑a / ↑b⊢ q.den > 0 x:Realhx:x > 0a:ℤb:ℕhb:b > 0q:ℚ := ↑a / ↑bthis:q.den ≠ 0 := q.den_nz⊢ q.den > 0; All goals completed! 🐙 All goals completed! 🐙theorem declaration uses `sorry`Real.ratPow_eq_root {x:Real} (hx: x > 0) {n:ℕ} (hn: n ≥ 1) : x^(1/n:ℚ) = x.root n := x:Realhx:x > 0n:ℕhn:n ≥ 1⊢ x ^ (1 / ↑n) = x.root n All goals completed! 🐙theorem declaration uses `sorry`Real.ratPow_eq_pow {x:Real} (hx: x > 0) (n:ℤ) : x^(n:ℚ) = x^n := x:Realhx:x > 0n:ℤ⊢ x ^ ↑n = x ^ n All goals completed! 🐙

Lemma 5.6.9(a) / Exercise 5.6.2

theorem declaration uses `sorry`Real.ratPow_pos {x:Real} (hx: x > 0) (q:ℚ) : x^q > 0 := x:Realhx:x > 0q:ℚ⊢ x ^ q > 0 All goals completed! 🐙

Lemma 5.6.9(b) / Exercise 5.6.2

theorem declaration uses `sorry`Real.ratPow_add {x:Real} (hx: x > 0) (q r:ℚ) : x^(q+r) = x^q * x^r := x:Realhx:x > 0q:ℚr:ℚ⊢ x ^ (q + r) = x ^ q * x ^ r All goals completed! 🐙

Lemma 5.6.9(b) / Exercise 5.6.2

theorem declaration uses `sorry`Real.ratPow_ratPow {x:Real} (hx: x > 0) (q r:ℚ) : (x^q)^r = x^(q*r) := x:Realhx:x > 0q:ℚr:ℚ⊢ (x ^ q) ^ r = x ^ (q * r) All goals completed! 🐙

Lemma 5.6.9(c) / Exercise 5.6.2

theorem declaration uses `sorry`Real.ratPow_neg {x:Real} (hx: x > 0) (q:ℚ) : x^(-q) = 1 / x^q := x:Realhx:x > 0q:ℚ⊢ x ^ (-q) = 1 / x ^ q All goals completed! 🐙

Lemma 5.6.9(d) / Exercise 5.6.2

theorem declaration uses `sorry`Real.ratPow_mono {x y:Real} (hx: x > 0) (hy: y > 0) {q:ℚ} (h: q > 0) : x > y ↔ x^q > y^q := x:Realy:Realhx:x > 0hy:y > 0q:ℚh:q > 0⊢ x > y ↔ x ^ q > y ^ q All goals completed! 🐙

Lemma 5.6.9(e) / Exercise 5.6.2

theorem declaration uses `sorry`Real.ratPow_mono_of_gt_one {x:Real} (hx: x > 1) {q r:ℚ} : x^q > x^r ↔ q > r := x:Realhx:x > 1q:ℚr:ℚ⊢ x ^ q > x ^ r ↔ q > r All goals completed! 🐙

Lemma 5.6.9(e) / Exercise 5.6.2

theorem declaration uses `sorry`Real.ratPow_mono_of_lt_one {x:Real} (hx0: 0 < x) (hx: x < 1) {q r:ℚ} : x^q > x^r ↔ q < r := x:Realhx0:0 < xhx:x < 1q:ℚr:ℚ⊢ x ^ q > x ^ r ↔ q < r All goals completed! 🐙

Lemma 5.6.9(f) / Exercise 5.6.2

theorem declaration uses `sorry`Real.ratPow_mul {x y:Real} (hx: x > 0) (hy: y > 0) (q:ℚ) : (x*y)^q = x^q * y^q := x:Realy:Realhx:x > 0hy:y > 0q:ℚ⊢ (x * y) ^ q = x ^ q * y ^ q All goals completed! 🐙

Exercise 5.6.3

theorem declaration uses `sorry`Real.pow_even (x:Real) {n:ℕ} (hn: Even n) : x^n ≥ 0 := x:Realn:ℕhn:Even n⊢ x ^ n ≥ 0 All goals completed! 🐙

Exercise 5.6.5

theorem declaration uses `sorry`Real.max_ratPow {x y:Real} (hx: x > 0) (hy: y > 0) {q:ℚ} (hq: q > 0) : max (x^q) (y^q) = (max x y)^q := x:Realy:Realhx:x > 0hy:y > 0q:ℚhq:q > 0⊢ max (x ^ q) (y ^ q) = max x y ^ q All goals completed! 🐙

Exercise 5.6.5

theorem declaration uses `sorry`Real.min_ratPow {x y:Real} (hx: x > 0) (hy: y > 0) {q:ℚ} (hq: q > 0) : min (x^q) (y^q) = (min x y)^q := x:Realy:Realhx:x > 0hy:y > 0q:ℚhq:q > 0⊢ min (x ^ q) (y ^ q) = min x y ^ q All goals completed! 🐙
-- Final part of Exercise 5.6.5: state and prove versions of the above lemmas covering the case of negative q. end Chapter5