Documentation

Init.Data.Int.Order

Results about the order properties of the integers, and the integers as an ordered ring. #

Order properties of the integers #

theorem Int.nonneg_def {a : Int} :
a.NonNeg ∃ (n : Nat), a = n
theorem Int.NonNeg.elim {a : Int} :
a.NonNeg∃ (n : Nat), a = n
theorem Int.nonneg_or_nonneg_neg (a : Int) :
a.NonNeg (-a).NonNeg
theorem Int.le_def {a : Int} {b : Int} :
a b (b - a).NonNeg
theorem Int.lt_iff_add_one_le {a : Int} {b : Int} :
a < b a + 1 b
theorem Int.le.intro_sub {a : Int} {b : Int} (n : Nat) (h : b - a = n) :
a b
theorem Int.le.intro {a : Int} {b : Int} (n : Nat) (h : a + n = b) :
a b
theorem Int.le.dest_sub {a : Int} {b : Int} (h : a b) :
∃ (n : Nat), b - a = n
theorem Int.le.dest {a : Int} {b : Int} (h : a b) :
∃ (n : Nat), a + n = b
theorem Int.le_total (a : Int) (b : Int) :
a b b a
@[simp]
theorem Int.ofNat_le {m : Nat} {n : Nat} :
m n m n
theorem Int.ofNat_zero_le (n : Nat) :
0 n
theorem Int.eq_ofNat_of_zero_le {a : Int} (h : 0 a) :
∃ (n : Nat), a = n
theorem Int.eq_succ_of_zero_lt {a : Int} (h : 0 < a) :
∃ (n : Nat), a = n.succ
theorem Int.lt_add_succ (a : Int) (n : Nat) :
a < a + n.succ
theorem Int.lt.intro {a : Int} {b : Int} {n : Nat} (h : a + n.succ = b) :
a < b
theorem Int.lt.dest {a : Int} {b : Int} (h : a < b) :
∃ (n : Nat), a + n.succ = b
@[simp]
theorem Int.ofNat_lt {n : Nat} {m : Nat} :
n < m n < m
@[simp]
theorem Int.ofNat_pos {n : Nat} :
0 < n 0 < n
theorem Int.ofNat_nonneg (n : Nat) :
0 n
theorem Int.ofNat_succ_pos (n : Nat) :
0 < n.succ
@[simp]
theorem Int.le_refl (a : Int) :
a a
theorem Int.le_trans {a : Int} {b : Int} {c : Int} (h₁ : a b) (h₂ : b c) :
a c
theorem Int.le_antisymm {a : Int} {b : Int} (h₁ : a b) (h₂ : b a) :
a = b
@[simp]
theorem Int.lt_irrefl (a : Int) :
¬a < a
theorem Int.ne_of_lt {a : Int} {b : Int} (h : a < b) :
a b
theorem Int.ne_of_gt {a : Int} {b : Int} (h : b < a) :
a b
theorem Int.le_of_lt {a : Int} {b : Int} (h : a < b) :
a b
theorem Int.lt_iff_le_and_ne {a : Int} {b : Int} :
a < b a b a b
theorem Int.lt_succ (a : Int) :
a < a + 1
theorem Int.lt_iff_le_not_le {a : Int} {b : Int} :
a < b a b ¬b a
theorem Int.lt_of_not_ge {a : Int} {b : Int} (h : ¬a b) :
b < a
theorem Int.not_le_of_gt {a : Int} {b : Int} (h : b < a) :
¬a b
theorem Int.not_le {a : Int} {b : Int} :
¬a b b < a
theorem Int.not_lt {a : Int} {b : Int} :
¬a < b b a
theorem Int.lt_trichotomy (a : Int) (b : Int) :
a < b a = b b < a
theorem Int.ne_iff_lt_or_gt {a : Int} {b : Int} :
a b a < b b < a
theorem Int.lt_or_gt_of_ne {a : Int} {b : Int} :
a ba < b b < a
theorem Int.eq_iff_le_and_ge {x : Int} {y : Int} :
x = y x y y x
theorem Int.lt_of_le_of_lt {a : Int} {b : Int} {c : Int} (h₁ : a b) (h₂ : b < c) :
a < c
theorem Int.lt_of_lt_of_le {a : Int} {b : Int} {c : Int} (h₁ : a < b) (h₂ : b c) :
a < c
theorem Int.lt_trans {a : Int} {b : Int} {c : Int} (h₁ : a < b) (h₂ : b < c) :
a < c
instance Int.instTransLe :
Trans (fun (x1 x2 : Int) => x1 x2) (fun (x1 x2 : Int) => x1 x2) fun (x1 x2 : Int) => x1 x2
Equations
instance Int.instTransLtLe :
Trans (fun (x1 x2 : Int) => x1 < x2) (fun (x1 x2 : Int) => x1 x2) fun (x1 x2 : Int) => x1 < x2
Equations
instance Int.instTransLeLt :
Trans (fun (x1 x2 : Int) => x1 x2) (fun (x1 x2 : Int) => x1 < x2) fun (x1 x2 : Int) => x1 < x2
Equations
instance Int.instTransLt :
Trans (fun (x1 x2 : Int) => x1 < x2) (fun (x1 x2 : Int) => x1 < x2) fun (x1 x2 : Int) => x1 < x2
Equations
theorem Int.min_def (n : Int) (m : Int) :
min n m = if n m then n else m
theorem Int.max_def (n : Int) (m : Int) :
max n m = if n m then m else n
theorem Int.min_comm (a : Int) (b : Int) :
min a b = min b a
theorem Int.min_le_right (a : Int) (b : Int) :
min a b b
theorem Int.min_le_left (a : Int) (b : Int) :
min a b a
theorem Int.min_eq_left {a : Int} {b : Int} (h : a b) :
min a b = a
theorem Int.min_eq_right {a : Int} {b : Int} (h : b a) :
min a b = b
theorem Int.le_min {a : Int} {b : Int} {c : Int} :
a min b c a b a c
theorem Int.max_comm (a : Int) (b : Int) :
max a b = max b a
theorem Int.le_max_left (a : Int) (b : Int) :
a max a b
theorem Int.le_max_right (a : Int) (b : Int) :
b max a b
theorem Int.max_le {a : Int} {b : Int} {c : Int} :
max a b c a c b c
theorem Int.max_eq_right {a : Int} {b : Int} (h : a b) :
max a b = b
theorem Int.max_eq_left {a : Int} {b : Int} (h : b a) :
max a b = a
theorem Int.eq_natAbs_of_zero_le {a : Int} (h : 0 a) :
a = a.natAbs
theorem Int.le_natAbs {a : Int} :
a a.natAbs
@[simp]
theorem Int.ofNat_max_zero (n : Nat) :
max (↑n) 0 = n
@[simp]
theorem Int.zero_max_ofNat (n : Nat) :
max 0 n = n
@[simp]
theorem Int.negSucc_max_zero (n : Nat) :
max (Int.negSucc n) 0 = 0
@[simp]
theorem Int.zero_max_negSucc (n : Nat) :
max 0 (Int.negSucc n) = 0
theorem Int.add_le_add_left {a : Int} {b : Int} (h : a b) (c : Int) :
c + a c + b
theorem Int.add_lt_add_left {a : Int} {b : Int} (h : a < b) (c : Int) :
c + a < c + b
theorem Int.add_le_add_right {a : Int} {b : Int} (h : a b) (c : Int) :
a + c b + c
theorem Int.add_lt_add_right {a : Int} {b : Int} (h : a < b) (c : Int) :
a + c < b + c
theorem Int.le_of_add_le_add_left {a : Int} {b : Int} {c : Int} (h : a + b a + c) :
b c
theorem Int.le_of_add_le_add_right {a : Int} {b : Int} {c : Int} (h : a + b c + b) :
a c
theorem Int.add_le_add_iff_left {b : Int} {c : Int} (a : Int) :
a + b a + c b c
theorem Int.add_le_add_iff_right {a : Int} {b : Int} (c : Int) :
a + c b + c a b
theorem Int.add_le_add {a : Int} {b : Int} {c : Int} {d : Int} (h₁ : a b) (h₂ : c d) :
a + c b + d
theorem Int.le_add_of_nonneg_right {a : Int} {b : Int} (h : 0 b) :
a a + b
theorem Int.le_add_of_nonneg_left {a : Int} {b : Int} (h : 0 b) :
a b + a
theorem Int.neg_le_neg {a : Int} {b : Int} (h : a b) :
-b -a
theorem Int.le_of_neg_le_neg {a : Int} {b : Int} (h : -b -a) :
a b
theorem Int.neg_nonpos_of_nonneg {a : Int} (h : 0 a) :
-a 0
theorem Int.neg_nonneg_of_nonpos {a : Int} (h : a 0) :
0 -a
theorem Int.neg_lt_neg {a : Int} {b : Int} (h : a < b) :
-b < -a
theorem Int.neg_neg_of_pos {a : Int} (h : 0 < a) :
-a < 0
theorem Int.neg_pos_of_neg {a : Int} (h : a < 0) :
0 < -a
theorem Int.sub_nonneg_of_le {a : Int} {b : Int} (h : b a) :
0 a - b
theorem Int.le_of_sub_nonneg {a : Int} {b : Int} (h : 0 a - b) :
b a
theorem Int.sub_pos_of_lt {a : Int} {b : Int} (h : b < a) :
0 < a - b
theorem Int.lt_of_sub_pos {a : Int} {b : Int} (h : 0 < a - b) :
b < a
theorem Int.sub_left_le_of_le_add {a : Int} {b : Int} {c : Int} (h : a b + c) :
a - b c
theorem Int.sub_le_self (a : Int) {b : Int} (h : 0 b) :
a - b a
theorem Int.sub_lt_self (a : Int) {b : Int} (h : 0 < b) :
a - b < a
theorem Int.add_one_le_of_lt {a : Int} {b : Int} (H : a < b) :
a + 1 b
theorem Int.mul_nonneg {a : Int} {b : Int} (ha : 0 a) (hb : 0 b) :
0 a * b
theorem Int.mul_pos {a : Int} {b : Int} (ha : 0 < a) (hb : 0 < b) :
0 < a * b
theorem Int.mul_lt_mul_of_pos_left {a : Int} {b : Int} {c : Int} (h₁ : a < b) (h₂ : 0 < c) :
c * a < c * b
theorem Int.mul_lt_mul_of_pos_right {a : Int} {b : Int} {c : Int} (h₁ : a < b) (h₂ : 0 < c) :
a * c < b * c
theorem Int.mul_le_mul_of_nonneg_left {a : Int} {b : Int} {c : Int} (h₁ : a b) (h₂ : 0 c) :
c * a c * b
theorem Int.mul_le_mul_of_nonneg_right {a : Int} {b : Int} {c : Int} (h₁ : a b) (h₂ : 0 c) :
a * c b * c
theorem Int.mul_le_mul {a : Int} {b : Int} {c : Int} {d : Int} (hac : a c) (hbd : b d) (nn_b : 0 b) (nn_c : 0 c) :
a * b c * d
theorem Int.mul_nonpos_of_nonneg_of_nonpos {a : Int} {b : Int} (ha : 0 a) (hb : b 0) :
a * b 0
theorem Int.mul_nonpos_of_nonpos_of_nonneg {a : Int} {b : Int} (ha : a 0) (hb : 0 b) :
a * b 0
theorem Int.mul_le_mul_of_nonpos_right {a : Int} {b : Int} {c : Int} (h : b a) (hc : c 0) :
a * c b * c
theorem Int.mul_le_mul_of_nonpos_left {a : Int} {b : Int} {c : Int} (ha : a 0) (h : c b) :
a * b a * c
@[simp]
theorem Int.natAbs_ofNat (n : Nat) :
(↑n).natAbs = n
@[simp]
theorem Int.natAbs_negSucc (n : Nat) :
(Int.negSucc n).natAbs = n.succ
@[simp]
@[simp]
@[simp]
theorem Int.natAbs_eq_zero {a : Int} :
a.natAbs = 0 a = 0
theorem Int.natAbs_pos {a : Int} :
0 < a.natAbs a 0
@[simp]
theorem Int.natAbs_neg (a : Int) :
(-a).natAbs = a.natAbs
theorem Int.natAbs_eq (a : Int) :
a = a.natAbs a = -a.natAbs
theorem Int.natAbs_negOfNat (n : Nat) :
(Int.negOfNat n).natAbs = n
theorem Int.natAbs_mul (a : Int) (b : Int) :
(a * b).natAbs = a.natAbs * b.natAbs
theorem Int.natAbs_eq_natAbs_iff {a : Int} {b : Int} :
a.natAbs = b.natAbs a = b a = -b
theorem Int.natAbs_of_nonneg {a : Int} (H : 0 a) :
a.natAbs = a
theorem Int.ofNat_natAbs_of_nonpos {a : Int} (H : a 0) :
a.natAbs = -a

toNat #

theorem Int.toNat_eq_max (a : Int) :
a.toNat = max a 0
@[simp]
@[simp]
theorem Int.toNat_of_nonneg {a : Int} (h : 0 a) :
a.toNat = a
@[simp]
theorem Int.toNat_ofNat (n : Nat) :
(↑n).toNat = n
@[simp]
theorem Int.toNat_negSucc (n : Nat) :
(Int.negSucc n).toNat = 0
@[simp]
theorem Int.toNat_ofNat_add_one {n : Nat} :
(n + 1).toNat = n + 1
@[simp]
theorem Int.ofNat_toNat (a : Int) :
a.toNat = max a 0
theorem Int.self_le_toNat (a : Int) :
a a.toNat
@[simp]
theorem Int.le_toNat {n : Nat} {z : Int} (h : 0 z) :
n z.toNat n z
@[simp]
theorem Int.toNat_lt {n : Nat} {z : Int} (h : 0 z) :
z.toNat < n z < n
theorem Int.toNat_add {a : Int} {b : Int} (ha : 0 a) (hb : 0 b) :
(a + b).toNat = a.toNat + b.toNat
theorem Int.toNat_add_nat {a : Int} (ha : 0 a) (n : Nat) :
(a + n).toNat = a.toNat + n
@[simp]
theorem Int.pred_toNat (i : Int) :
(i - 1).toNat = i.toNat - 1
theorem Int.toNat_sub_toNat_neg (n : Int) :
n.toNat - (-n).toNat = n
@[simp]
theorem Int.toNat_add_toNat_neg_eq_natAbs (n : Int) :
n.toNat + (-n).toNat = n.natAbs
@[simp]
theorem Int.toNat_neg_nat (n : Nat) :
(-n).toNat = 0

toNat' #

theorem Int.mem_toNat' {a : Int} {n : Nat} :
a.toNat' = some n a = n

Order properties of the integers #

theorem Int.le_of_not_le {a : Int} {b : Int} :
¬a bb a
@[simp]
theorem Int.eq_negSucc_of_lt_zero {a : Int} :
a < 0∃ (n : Nat), a = Int.negSucc n
theorem Int.lt_of_add_lt_add_left {a : Int} {b : Int} {c : Int} (h : a + b < a + c) :
b < c
theorem Int.lt_of_add_lt_add_right {a : Int} {b : Int} {c : Int} (h : a + b < c + b) :
a < c
theorem Int.add_lt_add_iff_left {b : Int} {c : Int} (a : Int) :
a + b < a + c b < c
theorem Int.add_lt_add_iff_right {a : Int} {b : Int} (c : Int) :
a + c < b + c a < b
theorem Int.add_lt_add {a : Int} {b : Int} {c : Int} {d : Int} (h₁ : a < b) (h₂ : c < d) :
a + c < b + d
theorem Int.add_lt_add_of_le_of_lt {a : Int} {b : Int} {c : Int} {d : Int} (h₁ : a b) (h₂ : c < d) :
a + c < b + d
theorem Int.add_lt_add_of_lt_of_le {a : Int} {b : Int} {c : Int} {d : Int} (h₁ : a < b) (h₂ : c d) :
a + c < b + d
theorem Int.lt_add_of_pos_right (a : Int) {b : Int} (h : 0 < b) :
a < a + b
theorem Int.lt_add_of_pos_left (a : Int) {b : Int} (h : 0 < b) :
a < b + a
theorem Int.add_nonneg {a : Int} {b : Int} (ha : 0 a) (hb : 0 b) :
0 a + b
theorem Int.add_pos {a : Int} {b : Int} (ha : 0 < a) (hb : 0 < b) :
0 < a + b
theorem Int.add_pos_of_pos_of_nonneg {a : Int} {b : Int} (ha : 0 < a) (hb : 0 b) :
0 < a + b
theorem Int.add_pos_of_nonneg_of_pos {a : Int} {b : Int} (ha : 0 a) (hb : 0 < b) :
0 < a + b
theorem Int.add_nonpos {a : Int} {b : Int} (ha : a 0) (hb : b 0) :
a + b 0
theorem Int.add_neg {a : Int} {b : Int} (ha : a < 0) (hb : b < 0) :
a + b < 0
theorem Int.add_neg_of_neg_of_nonpos {a : Int} {b : Int} (ha : a < 0) (hb : b 0) :
a + b < 0
theorem Int.add_neg_of_nonpos_of_neg {a : Int} {b : Int} (ha : a 0) (hb : b < 0) :
a + b < 0
theorem Int.lt_add_of_le_of_pos {a : Int} {b : Int} {c : Int} (hbc : b c) (ha : 0 < a) :
b < c + a
theorem Int.add_one_le_iff {a : Int} {b : Int} :
a + 1 b a < b
theorem Int.lt_add_one_iff {a : Int} {b : Int} :
a < b + 1 a b
@[simp]
theorem Int.succ_ofNat_pos (n : Nat) :
0 < n + 1
theorem Int.not_ofNat_neg (n : Nat) :
¬n < 0
theorem Int.le_add_one {a : Int} {b : Int} (h : a b) :
a b + 1
theorem Int.nonneg_of_neg_nonpos {a : Int} (h : -a 0) :
0 a
theorem Int.nonpos_of_neg_nonneg {a : Int} (h : 0 -a) :
a 0
theorem Int.lt_of_neg_lt_neg {a : Int} {b : Int} (h : -b < -a) :
a < b
theorem Int.pos_of_neg_neg {a : Int} (h : -a < 0) :
0 < a
theorem Int.neg_of_neg_pos {a : Int} (h : 0 < -a) :
a < 0
theorem Int.le_neg_of_le_neg {a : Int} {b : Int} (h : a -b) :
b -a
theorem Int.neg_le_of_neg_le {a : Int} {b : Int} (h : -a b) :
-b a
theorem Int.lt_neg_of_lt_neg {a : Int} {b : Int} (h : a < -b) :
b < -a
theorem Int.neg_lt_of_neg_lt {a : Int} {b : Int} (h : -a < b) :
-b < a
theorem Int.sub_nonpos_of_le {a : Int} {b : Int} (h : a b) :
a - b 0
theorem Int.le_of_sub_nonpos {a : Int} {b : Int} (h : a - b 0) :
a b
theorem Int.sub_neg_of_lt {a : Int} {b : Int} (h : a < b) :
a - b < 0
theorem Int.lt_of_sub_neg {a : Int} {b : Int} (h : a - b < 0) :
a < b
theorem Int.add_le_of_le_neg_add {a : Int} {b : Int} {c : Int} (h : b -a + c) :
a + b c
theorem Int.le_neg_add_of_add_le {a : Int} {b : Int} {c : Int} (h : a + b c) :
b -a + c
theorem Int.add_le_of_le_sub_left {a : Int} {b : Int} {c : Int} (h : b c - a) :
a + b c
theorem Int.le_sub_left_of_add_le {a : Int} {b : Int} {c : Int} (h : a + b c) :
b c - a
theorem Int.add_le_of_le_sub_right {a : Int} {b : Int} {c : Int} (h : a c - b) :
a + b c
theorem Int.le_sub_right_of_add_le {a : Int} {b : Int} {c : Int} (h : a + b c) :
a c - b
theorem Int.le_add_of_neg_add_le {a : Int} {b : Int} {c : Int} (h : -b + a c) :
a b + c
theorem Int.neg_add_le_of_le_add {a : Int} {b : Int} {c : Int} (h : a b + c) :
-b + a c
theorem Int.le_add_of_sub_left_le {a : Int} {b : Int} {c : Int} (h : a - b c) :
a b + c
theorem Int.le_add_of_sub_right_le {a : Int} {b : Int} {c : Int} (h : a - c b) :
a b + c
theorem Int.sub_right_le_of_le_add {a : Int} {b : Int} {c : Int} (h : a b + c) :
a - c b
theorem Int.le_add_of_neg_add_le_left {a : Int} {b : Int} {c : Int} (h : -b + a c) :
a b + c
theorem Int.neg_add_le_left_of_le_add {a : Int} {b : Int} {c : Int} (h : a b + c) :
-b + a c
theorem Int.le_add_of_neg_add_le_right {a : Int} {b : Int} {c : Int} (h : -c + a b) :
a b + c
theorem Int.neg_add_le_right_of_le_add {a : Int} {b : Int} {c : Int} (h : a b + c) :
-c + a b
theorem Int.le_add_of_neg_le_sub_left {a : Int} {b : Int} {c : Int} (h : -a b - c) :
c a + b
theorem Int.neg_le_sub_left_of_le_add {a : Int} {b : Int} {c : Int} (h : c a + b) :
-a b - c
theorem Int.le_add_of_neg_le_sub_right {a : Int} {b : Int} {c : Int} (h : -b a - c) :
c a + b
theorem Int.neg_le_sub_right_of_le_add {a : Int} {b : Int} {c : Int} (h : c a + b) :
-b a - c
theorem Int.sub_le_of_sub_le {a : Int} {b : Int} {c : Int} (h : a - b c) :
a - c b
theorem Int.sub_le_sub_left {a : Int} {b : Int} (h : a b) (c : Int) :
c - b c - a
theorem Int.sub_le_sub_right {a : Int} {b : Int} (h : a b) (c : Int) :
a - c b - c
theorem Int.sub_le_sub {a : Int} {b : Int} {c : Int} {d : Int} (hab : a b) (hcd : c d) :
a - d b - c
theorem Int.add_lt_of_lt_neg_add {a : Int} {b : Int} {c : Int} (h : b < -a + c) :
a + b < c
theorem Int.lt_neg_add_of_add_lt {a : Int} {b : Int} {c : Int} (h : a + b < c) :
b < -a + c
theorem Int.add_lt_of_lt_sub_left {a : Int} {b : Int} {c : Int} (h : b < c - a) :
a + b < c
theorem Int.lt_sub_left_of_add_lt {a : Int} {b : Int} {c : Int} (h : a + b < c) :
b < c - a
theorem Int.add_lt_of_lt_sub_right {a : Int} {b : Int} {c : Int} (h : a < c - b) :
a + b < c
theorem Int.lt_sub_right_of_add_lt {a : Int} {b : Int} {c : Int} (h : a + b < c) :
a < c - b
theorem Int.lt_add_of_neg_add_lt {a : Int} {b : Int} {c : Int} (h : -b + a < c) :
a < b + c
theorem Int.neg_add_lt_of_lt_add {a : Int} {b : Int} {c : Int} (h : a < b + c) :
-b + a < c
theorem Int.lt_add_of_sub_left_lt {a : Int} {b : Int} {c : Int} (h : a - b < c) :
a < b + c
theorem Int.sub_left_lt_of_lt_add {a : Int} {b : Int} {c : Int} (h : a < b + c) :
a - b < c
theorem Int.lt_add_of_sub_right_lt {a : Int} {b : Int} {c : Int} (h : a - c < b) :
a < b + c
theorem Int.sub_right_lt_of_lt_add {a : Int} {b : Int} {c : Int} (h : a < b + c) :
a - c < b
theorem Int.lt_add_of_neg_add_lt_left {a : Int} {b : Int} {c : Int} (h : -b + a < c) :
a < b + c
theorem Int.neg_add_lt_left_of_lt_add {a : Int} {b : Int} {c : Int} (h : a < b + c) :
-b + a < c
theorem Int.lt_add_of_neg_add_lt_right {a : Int} {b : Int} {c : Int} (h : -c + a < b) :
a < b + c
theorem Int.neg_add_lt_right_of_lt_add {a : Int} {b : Int} {c : Int} (h : a < b + c) :
-c + a < b
theorem Int.lt_add_of_neg_lt_sub_left {a : Int} {b : Int} {c : Int} (h : -a < b - c) :
c < a + b
theorem Int.neg_lt_sub_left_of_lt_add {a : Int} {b : Int} {c : Int} (h : c < a + b) :
-a < b - c
theorem Int.lt_add_of_neg_lt_sub_right {a : Int} {b : Int} {c : Int} (h : -b < a - c) :
c < a + b
theorem Int.neg_lt_sub_right_of_lt_add {a : Int} {b : Int} {c : Int} (h : c < a + b) :
-b < a - c
theorem Int.add_lt_iff {a : Int} {b : Int} {c : Int} :
a + b < c a < -b + c
theorem Int.sub_lt_iff {a : Int} {b : Int} {c : Int} :
a - b < c a < c + b
theorem Int.sub_lt_of_sub_lt {a : Int} {b : Int} {c : Int} (h : a - b < c) :
a - c < b
theorem Int.sub_lt_sub_left {a : Int} {b : Int} (h : a < b) (c : Int) :
c - b < c - a
theorem Int.sub_lt_sub_right {a : Int} {b : Int} (h : a < b) (c : Int) :
a - c < b - c
theorem Int.sub_lt_sub {a : Int} {b : Int} {c : Int} {d : Int} (hab : a < b) (hcd : c < d) :
a - d < b - c
theorem Int.lt_of_sub_lt_sub_left {a : Int} {b : Int} {c : Int} (h : c - a < c - b) :
b < a
theorem Int.lt_of_sub_lt_sub_right {a : Int} {b : Int} {c : Int} (h : a - c < b - c) :
a < b
@[simp]
theorem Int.sub_lt_sub_left_iff {a : Int} {b : Int} {c : Int} :
c - a < c - b b < a
@[simp]
theorem Int.sub_lt_sub_right_iff {a : Int} {b : Int} {c : Int} :
a - c < b - c a < b
theorem Int.sub_lt_sub_of_le_of_lt {a : Int} {b : Int} {c : Int} {d : Int} (hab : a b) (hcd : c < d) :
a - d < b - c
theorem Int.sub_lt_sub_of_lt_of_le {a : Int} {b : Int} {c : Int} {d : Int} (hab : a < b) (hcd : c d) :
a - d < b - c
theorem Int.add_le_add_three {a : Int} {b : Int} {c : Int} {d : Int} {e : Int} {f : Int} (h₁ : a d) (h₂ : b e) (h₃ : c f) :
a + b + c d + e + f
theorem Int.exists_eq_neg_ofNat {a : Int} (H : a 0) :
∃ (n : Nat), a = -n
theorem Int.lt_of_add_one_le {a : Int} {b : Int} (H : a + 1 b) :
a < b
theorem Int.lt_add_one_of_le {a : Int} {b : Int} (H : a b) :
a < b + 1
theorem Int.le_of_lt_add_one {a : Int} {b : Int} (H : a < b + 1) :
a b
theorem Int.sub_one_lt_of_le {a : Int} {b : Int} (H : a b) :
a - 1 < b
theorem Int.le_of_sub_one_lt {a : Int} {b : Int} (H : a - 1 < b) :
a b
theorem Int.le_sub_one_of_lt {a : Int} {b : Int} (H : a < b) :
a b - 1
theorem Int.lt_of_le_sub_one {a : Int} {b : Int} (H : a b - 1) :
a < b
theorem Int.mul_lt_mul {a : Int} {b : Int} {c : Int} {d : Int} (h₁ : a < c) (h₂ : b d) (h₃ : 0 < b) (h₄ : 0 c) :
a * b < c * d
theorem Int.mul_lt_mul' {a : Int} {b : Int} {c : Int} {d : Int} (h₁ : a c) (h₂ : b < d) (h₃ : 0 b) (h₄ : 0 < c) :
a * b < c * d
theorem Int.mul_neg_of_pos_of_neg {a : Int} {b : Int} (ha : 0 < a) (hb : b < 0) :
a * b < 0
theorem Int.mul_neg_of_neg_of_pos {a : Int} {b : Int} (ha : a < 0) (hb : 0 < b) :
a * b < 0
theorem Int.mul_nonneg_of_nonpos_of_nonpos {a : Int} {b : Int} (ha : a 0) (hb : b 0) :
0 a * b
theorem Int.mul_lt_mul_of_neg_left {a : Int} {b : Int} {c : Int} (h : b < a) (hc : c < 0) :
c * a < c * b
theorem Int.mul_lt_mul_of_neg_right {a : Int} {b : Int} {c : Int} (h : b < a) (hc : c < 0) :
a * c < b * c
theorem Int.mul_pos_of_neg_of_neg {a : Int} {b : Int} (ha : a < 0) (hb : b < 0) :
0 < a * b
theorem Int.mul_self_le_mul_self {a : Int} {b : Int} (h1 : 0 a) (h2 : a b) :
a * a b * b
theorem Int.mul_self_lt_mul_self {a : Int} {b : Int} (h1 : 0 a) (h2 : a < b) :
a * a < b * b
@[simp]
theorem Int.sign_zero :
@[simp]
theorem Int.sign_one :
theorem Int.sign_neg_one :
(-1).sign = -1
@[simp]
theorem Int.sign_of_add_one (x : Nat) :
(x + 1).sign = 1
@[simp]
theorem Int.sign_negSucc (x : Nat) :
(Int.negSucc x).sign = -1
theorem Int.natAbs_sign (z : Int) :
z.sign.natAbs = if z = 0 then 0 else 1
theorem Int.natAbs_sign_of_nonzero {z : Int} (hz : z 0) :
z.sign.natAbs = 1
theorem Int.sign_ofNat_of_nonzero {n : Nat} (hn : n 0) :
(↑n).sign = 1
@[simp]
theorem Int.sign_neg (z : Int) :
(-z).sign = -z.sign
theorem Int.sign_mul_natAbs (a : Int) :
a.sign * a.natAbs = a
@[simp]
theorem Int.sign_mul (a : Int) (b : Int) :
(a * b).sign = a.sign * b.sign
theorem Int.sign_eq_one_of_pos {a : Int} (h : 0 < a) :
a.sign = 1
theorem Int.sign_eq_neg_one_of_neg {a : Int} (h : a < 0) :
a.sign = -1
theorem Int.eq_zero_of_sign_eq_zero {a : Int} :
a.sign = 0a = 0
theorem Int.pos_of_sign_eq_one {a : Int} :
a.sign = 10 < a
theorem Int.neg_of_sign_eq_neg_one {a : Int} :
a.sign = -1a < 0
theorem Int.sign_eq_one_iff_pos {a : Int} :
a.sign = 1 0 < a
theorem Int.sign_eq_neg_one_iff_neg {a : Int} :
a.sign = -1 a < 0
@[simp]
theorem Int.sign_eq_zero_iff_zero {a : Int} :
a.sign = 0 a = 0
@[simp]
theorem Int.sign_sign {x : Int} :
x.sign.sign = x.sign
@[simp]
theorem Int.sign_nonneg {x : Int} :
0 x.sign 0 x
theorem Int.mul_sign (i : Int) :
i * i.sign = i.natAbs
theorem Int.natAbs_ne_zero {a : Int} :
a.natAbs 0 a 0
theorem Int.natAbs_mul_self {a : Int} :
(a.natAbs * a.natAbs) = a * a
theorem Int.eq_nat_or_neg (a : Int) :
∃ (n : Nat), a = n a = -n
theorem Int.natAbs_mul_natAbs_eq {a : Int} {b : Int} {c : Nat} (h : a * b = c) :
a.natAbs * b.natAbs = c
@[simp]
theorem Int.natAbs_mul_self' (a : Int) :
a.natAbs * a.natAbs = a * a
theorem Int.natAbs_eq_iff {a : Int} {n : Nat} :
a.natAbs = n a = n a = -n
theorem Int.natAbs_add_le (a : Int) (b : Int) :
(a + b).natAbs a.natAbs + b.natAbs
theorem Int.natAbs_sub_le (a : Int) (b : Int) :
(a - b).natAbs a.natAbs + b.natAbs
theorem Int.negSucc_eq' (m : Nat) :
Int.negSucc m = -m - 1
theorem Int.natAbs_lt_natAbs_of_nonneg_of_lt {a : Int} {b : Int} (w₁ : 0 a) (w₂ : a < b) :
a.natAbs < b.natAbs
theorem Int.eq_natAbs_iff_mul_eq_zero {a : Int} {n : Nat} :
a.natAbs = n (a - n) * (a + n) = 0