Results about big operators over intervals

We prove results about big operators over intervals.

theorem Finset.mul_prod_Ico_eq_prod_Icc {α : Type u_1} {M : Type u_2} [PartialOrder α] [CommMonoid M] {f : α → M} {a : α} {b : α} [LocallyFiniteOrder α] (h : a ≤ b) : f b * ∏ x ∈ Finset.Ico a b, f x = ∏ x ∈ Finset.Icc a b, f x

theorem Finset.add_sum_Ico_eq_sum_Icc {α : Type u_1} {M : Type u_2} [PartialOrder α] [AddCommMonoid M] {f : α → M} {a : α} {b : α} [LocallyFiniteOrder α] (h : a ≤ b) : f b + ∑ x ∈ Finset.Ico a b, f x = ∑ x ∈ Finset.Icc a b, f x

theorem Finset.prod_Ico_mul_eq_prod_Icc {α : Type u_1} {M : Type u_2} [PartialOrder α] [CommMonoid M] {f : α → M} {a : α} {b : α} [LocallyFiniteOrder α] (h : a ≤ b) : (∏ x ∈ Finset.Ico a b, f x) * f b = ∏ x ∈ Finset.Icc a b, f x

theorem Finset.sum_Ico_add_eq_sum_Icc {α : Type u_1} {M : Type u_2} [PartialOrder α] [AddCommMonoid M] {f : α → M} {a : α} {b : α} [LocallyFiniteOrder α] (h : a ≤ b) : ∑ x ∈ Finset.Ico a b, f x + f b = ∑ x ∈ Finset.Icc a b, f x

theorem Finset.mul_prod_Ioc_eq_prod_Icc {α : Type u_1} {M : Type u_2} [PartialOrder α] [CommMonoid M] {f : α → M} {a : α} {b : α} [LocallyFiniteOrder α] (h : a ≤ b) : f a * ∏ x ∈ Finset.Ioc a b, f x = ∏ x ∈ Finset.Icc a b, f x

theorem Finset.add_sum_Ioc_eq_sum_Icc {α : Type u_1} {M : Type u_2} [PartialOrder α] [AddCommMonoid M] {f : α → M} {a : α} {b : α} [LocallyFiniteOrder α] (h : a ≤ b) : f a + ∑ x ∈ Finset.Ioc a b, f x = ∑ x ∈ Finset.Icc a b, f x

theorem Finset.prod_Ioc_mul_eq_prod_Icc {α : Type u_1} {M : Type u_2} [PartialOrder α] [CommMonoid M] {f : α → M} {a : α} {b : α} [LocallyFiniteOrder α] (h : a ≤ b) : (∏ x ∈ Finset.Ioc a b, f x) * f a = ∏ x ∈ Finset.Icc a b, f x

theorem Finset.sum_Ioc_add_eq_sum_Icc {α : Type u_1} {M : Type u_2} [PartialOrder α] [AddCommMonoid M] {f : α → M} {a : α} {b : α} [LocallyFiniteOrder α] (h : a ≤ b) : ∑ x ∈ Finset.Ioc a b, f x + f a = ∑ x ∈ Finset.Icc a b, f x

theorem Finset.mul_prod_Ioi_eq_prod_Ici {α : Type u_1} {M : Type u_2} [PartialOrder α] [CommMonoid M] {f : α → M} [LocallyFiniteOrderTop α] (a : α) : f a * ∏ x ∈ Finset.Ioi a, f x = ∏ x ∈ Finset.Ici a, f x

theorem Finset.add_sum_Ioi_eq_sum_Ici {α : Type u_1} {M : Type u_2} [PartialOrder α] [AddCommMonoid M] {f : α → M} [LocallyFiniteOrderTop α] (a : α) : f a + ∑ x ∈ Finset.Ioi a, f x = ∑ x ∈ Finset.Ici a, f x

theorem Finset.prod_Ioi_mul_eq_prod_Ici {α : Type u_1} {M : Type u_2} [PartialOrder α] [CommMonoid M] {f : α → M} [LocallyFiniteOrderTop α] (a : α) : (∏ x ∈ Finset.Ioi a, f x) * f a = ∏ x ∈ Finset.Ici a, f x

theorem Finset.sum_Ioi_add_eq_sum_Ici {α : Type u_1} {M : Type u_2} [PartialOrder α] [AddCommMonoid M] {f : α → M} [LocallyFiniteOrderTop α] (a : α) : ∑ x ∈ Finset.Ioi a, f x + f a = ∑ x ∈ Finset.Ici a, f x

theorem Finset.mul_prod_Iio_eq_prod_Iic {α : Type u_1} {M : Type u_2} [PartialOrder α] [CommMonoid M] {f : α → M} [LocallyFiniteOrderBot α] (a : α) : f a * ∏ x ∈ Finset.Iio a, f x = ∏ x ∈ Finset.Iic a, f x

theorem Finset.add_sum_Iio_eq_sum_Iic {α : Type u_1} {M : Type u_2} [PartialOrder α] [AddCommMonoid M] {f : α → M} [LocallyFiniteOrderBot α] (a : α) : f a + ∑ x ∈ Finset.Iio a, f x = ∑ x ∈ Finset.Iic a, f x

theorem Finset.prod_Iio_mul_eq_prod_Iic {α : Type u_1} {M : Type u_2} [PartialOrder α] [CommMonoid M] {f : α → M} [LocallyFiniteOrderBot α] (a : α) : (∏ x ∈ Finset.Iio a, f x) * f a = ∏ x ∈ Finset.Iic a, f x

theorem Finset.sum_Iio_add_eq_sum_Iic {α : Type u_1} {M : Type u_2} [PartialOrder α] [AddCommMonoid M] {f : α → M} [LocallyFiniteOrderBot α] (a : α) : ∑ x ∈ Finset.Iio a, f x + f a = ∑ x ∈ Finset.Iic a, f x

theorem Finset.prod_prod_Ioi_mul_eq_prod_prod_off_diag {α : Type u_1} {M : Type u_2} [Fintype α] [LinearOrder α] [LocallyFiniteOrderTop α] [LocallyFiniteOrderBot α] [CommMonoid M] (f : α → α → M) : ∏ i : α, ∏ j ∈ Finset.Ioi i, f j i * f i j = ∏ i : α, ∏ j ∈ {i}ᶜ, f j i

theorem Finset.sum_sum_Ioi_add_eq_sum_sum_off_diag {α : Type u_1} {M : Type u_2} [Fintype α] [LinearOrder α] [LocallyFiniteOrderTop α] [LocallyFiniteOrderBot α] [AddCommMonoid M] (f : α → α → M) : ∑ i : α, ∑ j ∈ Finset.Ioi i, (f j i + f i j) = ∑ i : α, ∑ j ∈ {i}ᶜ, f j i

theorem Finset.prod_Ico_add' {α : Type u_1} {M : Type u_2} [CommMonoid M] [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (f : α → M) (a : α) (b : α) (c : α) : ∏ x ∈ Finset.Ico a b, f (x + c) = ∏ x ∈ Finset.Ico (a + c) (b + c), f x

theorem Finset.sum_Ico_add' {α : Type u_1} {M : Type u_2} [AddCommMonoid M] [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (f : α → M) (a : α) (b : α) (c : α) : ∑ x ∈ Finset.Ico a b, f (x + c) = ∑ x ∈ Finset.Ico (a + c) (b + c), f x

theorem Finset.prod_Ico_add {α : Type u_1} {M : Type u_2} [CommMonoid M] [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (f : α → M) (a : α) (b : α) (c : α) : ∏ x ∈ Finset.Ico a b, f (c + x) = ∏ x ∈ Finset.Ico (a + c) (b + c), f x

theorem Finset.sum_Ico_add {α : Type u_1} {M : Type u_2} [AddCommMonoid M] [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (f : α → M) (a : α) (b : α) (c : α) : ∑ x ∈ Finset.Ico a b, f (c + x) = ∑ x ∈ Finset.Ico (a + c) (b + c), f x

@[simp]

theorem Finset.prod_Ico_add_right_sub_eq {α : Type u_1} {M : Type u_2} [CommMonoid M] {f : α → M} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] [Sub α] [OrderedSub α] (a : α) (b : α) (c : α) : ∏ x ∈ Finset.Ico (a + c) (b + c), f (x - c) = ∏ x ∈ Finset.Ico a b, f x

@[simp]

theorem Finset.sum_Ico_add_right_sub_eq {α : Type u_1} {M : Type u_2} [AddCommMonoid M] {f : α → M} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] [Sub α] [OrderedSub α] (a : α) (b : α) (c : α) : ∑ x ∈ Finset.Ico (a + c) (b + c), f (x - c) = ∑ x ∈ Finset.Ico a b, f x

theorem Finset.prod_Ico_succ_top {M : Type u_2} [CommMonoid M] {a : ℕ} {b : ℕ} (hab : a ≤ b) (f : ℕ → M) : ∏ k ∈ Finset.Ico a (b + 1), f k = (∏ k ∈ Finset.Ico a b, f k) * f b

theorem Finset.sum_Ico_succ_top {M : Type u_2} [AddCommMonoid M] {a : ℕ} {b : ℕ} (hab : a ≤ b) (f : ℕ → M) : ∑ k ∈ Finset.Ico a (b + 1), f k = ∑ k ∈ Finset.Ico a b, f k + f b

theorem Finset.prod_eq_prod_Ico_succ_bot {M : Type u_2} [CommMonoid M] {a : ℕ} {b : ℕ} (hab : a < b) (f : ℕ → M) : ∏ k ∈ Finset.Ico a b, f k = f a * ∏ k ∈ Finset.Ico (a + 1) b, f k

theorem Finset.sum_eq_sum_Ico_succ_bot {M : Type u_2} [AddCommMonoid M] {a : ℕ} {b : ℕ} (hab : a < b) (f : ℕ → M) : ∑ k ∈ Finset.Ico a b, f k = f a + ∑ k ∈ Finset.Ico (a + 1) b, f k

theorem Finset.prod_Ico_consecutive {M : Type u_2} [CommMonoid M] (f : ℕ → M) {m : ℕ} {n : ℕ} {k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) : (∏ i ∈ Finset.Ico m n, f i) * ∏ i ∈ Finset.Ico n k, f i = ∏ i ∈ Finset.Ico m k, f i

theorem Finset.sum_Ico_consecutive {M : Type u_2} [AddCommMonoid M] (f : ℕ → M) {m : ℕ} {n : ℕ} {k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) : ∑ i ∈ Finset.Ico m n, f i + ∑ i ∈ Finset.Ico n k, f i = ∑ i ∈ Finset.Ico m k, f i

theorem Finset.prod_Ioc_consecutive {M : Type u_2} [CommMonoid M] (f : ℕ → M) {m : ℕ} {n : ℕ} {k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) : (∏ i ∈ Finset.Ioc m n, f i) * ∏ i ∈ Finset.Ioc n k, f i = ∏ i ∈ Finset.Ioc m k, f i

theorem Finset.sum_Ioc_consecutive {M : Type u_2} [AddCommMonoid M] (f : ℕ → M) {m : ℕ} {n : ℕ} {k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) : ∑ i ∈ Finset.Ioc m n, f i + ∑ i ∈ Finset.Ioc n k, f i = ∑ i ∈ Finset.Ioc m k, f i

theorem Finset.prod_Ioc_succ_top {M : Type u_2} [CommMonoid M] {a : ℕ} {b : ℕ} (hab : a ≤ b) (f : ℕ → M) : ∏ k ∈ Finset.Ioc a (b + 1), f k = (∏ k ∈ Finset.Ioc a b, f k) * f (b + 1)

theorem Finset.sum_Ioc_succ_top {M : Type u_2} [AddCommMonoid M] {a : ℕ} {b : ℕ} (hab : a ≤ b) (f : ℕ → M) : ∑ k ∈ Finset.Ioc a (b + 1), f k = ∑ k ∈ Finset.Ioc a b, f k + f (b + 1)

theorem Finset.prod_Icc_succ_top {M : Type u_2} [CommMonoid M] {a : ℕ} {b : ℕ} (hab : a ≤ b + 1) (f : ℕ → M) : ∏ k ∈ Finset.Icc a (b + 1), f k = (∏ k ∈ Finset.Icc a b, f k) * f (b + 1)

theorem Finset.sum_Icc_succ_top {M : Type u_2} [AddCommMonoid M] {a : ℕ} {b : ℕ} (hab : a ≤ b + 1) (f : ℕ → M) : ∑ k ∈ Finset.Icc a (b + 1), f k = ∑ k ∈ Finset.Icc a b, f k + f (b + 1)

theorem Finset.prod_range_mul_prod_Ico {M : Type u_2} [CommMonoid M] (f : ℕ → M) {m : ℕ} {n : ℕ} (h : m ≤ n) : (∏ k ∈ Finset.range m, f k) * ∏ k ∈ Finset.Ico m n, f k = ∏ k ∈ Finset.range n, f k

theorem Finset.sum_range_add_sum_Ico {M : Type u_2} [AddCommMonoid M] (f : ℕ → M) {m : ℕ} {n : ℕ} (h : m ≤ n) : ∑ k ∈ Finset.range m, f k + ∑ k ∈ Finset.Ico m n, f k = ∑ k ∈ Finset.range n, f k

theorem Finset.prod_range_eq_mul_Ico {M : Type u_2} [CommMonoid M] (f : ℕ → M) {n : ℕ} (hn : 0 < n) : ∏ x ∈ Finset.range n, f x = f 0 * ∏ x ∈ Finset.Ico 1 n, f x

theorem Finset.sum_range_eq_add_Ico {M : Type u_2} [AddCommMonoid M] (f : ℕ → M) {n : ℕ} (hn : 0 < n) : ∑ x ∈ Finset.range n, f x = f 0 + ∑ x ∈ Finset.Ico 1 n, f x

theorem Finset.prod_Ico_eq_mul_inv {δ : Type u_3} [CommGroup δ] (f : ℕ → δ) {m : ℕ} {n : ℕ} (h : m ≤ n) : ∏ k ∈ Finset.Ico m n, f k = (∏ k ∈ Finset.range n, f k) * (∏ k ∈ Finset.range m, f k)⁻¹

theorem Finset.sum_Ico_eq_add_neg {δ : Type u_3} [AddCommGroup δ] (f : ℕ → δ) {m : ℕ} {n : ℕ} (h : m ≤ n) : ∑ k ∈ Finset.Ico m n, f k = ∑ k ∈ Finset.range n, f k + -∑ k ∈ Finset.range m, f k

theorem Finset.prod_Ico_eq_div {δ : Type u_3} [CommGroup δ] (f : ℕ → δ) {m : ℕ} {n : ℕ} (h : m ≤ n) : ∏ k ∈ Finset.Ico m n, f k = (∏ k ∈ Finset.range n, f k) / ∏ k ∈ Finset.range m, f k

theorem Finset.sum_Ico_eq_sub {δ : Type u_3} [AddCommGroup δ] (f : ℕ → δ) {m : ℕ} {n : ℕ} (h : m ≤ n) : ∑ k ∈ Finset.Ico m n, f k = ∑ k ∈ Finset.range n, f k - ∑ k ∈ Finset.range m, f k

theorem Finset.prod_range_div_prod_range {α : Type u_3} [CommGroup α] {f : ℕ → α} {n : ℕ} {m : ℕ} (hnm : n ≤ m) : (∏ k ∈ Finset.range m, f k) / ∏ k ∈ Finset.range n, f k = ∏ k ∈ Finset.filter (fun (k : ℕ) => n ≤ k) (Finset.range m), f k

theorem Finset.sum_range_sub_sum_range {α : Type u_3} [AddCommGroup α] {f : ℕ → α} {n : ℕ} {m : ℕ} (hnm : n ≤ m) : ∑ k ∈ Finset.range m, f k - ∑ k ∈ Finset.range n, f k = ∑ k ∈ Finset.filter (fun (k : ℕ) => n ≤ k) (Finset.range m), f k

theorem Finset.sum_Ico_Ico_comm {M : Type u_3} [AddCommMonoid M] (a : ℕ) (b : ℕ) (f : ℕ → ℕ → M) : ∑ i ∈ Finset.Ico a b, ∑ j ∈ Finset.Ico i b, f i j = ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j

The two ways of summing over (i, j) in the range a ≤ i ≤ j < b are equal.

theorem Finset.sum_Ico_Ico_comm' {M : Type u_3} [AddCommMonoid M] (a : ℕ) (b : ℕ) (f : ℕ → ℕ → M) : ∑ i ∈ Finset.Ico a b, ∑ j ∈ Finset.Ico (i + 1) b, f i j = ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j

The two ways of summing over (i, j) in the range a ≤ i < j < b are equal.

theorem Finset.prod_Ico_eq_prod_range {M : Type u_2} [CommMonoid M] (f : ℕ → M) (m : ℕ) (n : ℕ) : ∏ k ∈ Finset.Ico m n, f k = ∏ k ∈ Finset.range (n - m), f (m + k)

theorem Finset.sum_Ico_eq_sum_range {M : Type u_2} [AddCommMonoid M] (f : ℕ → M) (m : ℕ) (n : ℕ) : ∑ k ∈ Finset.Ico m n, f k = ∑ k ∈ Finset.range (n - m), f (m + k)

theorem Finset.prod_Ico_reflect {M : Type u_2} [CommMonoid M] (f : ℕ → M) (k : ℕ) {m : ℕ} {n : ℕ} (h : m ≤ n + 1) : ∏ j ∈ Finset.Ico k m, f (n - j) = ∏ j ∈ Finset.Ico (n + 1 - m) (n + 1 - k), f j

theorem Finset.sum_Ico_reflect {δ : Type u_3} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m : ℕ} {n : ℕ} (h : m ≤ n + 1) : ∑ j ∈ Finset.Ico k m, f (n - j) = ∑ j ∈ Finset.Ico (n + 1 - m) (n + 1 - k), f j

theorem Finset.prod_range_reflect {M : Type u_2} [CommMonoid M] (f : ℕ → M) (n : ℕ) : ∏ j ∈ Finset.range n, f (n - 1 - j) = ∏ j ∈ Finset.range n, f j

theorem Finset.sum_range_reflect {δ : Type u_3} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) : ∑ j ∈ Finset.range n, f (n - 1 - j) = ∑ j ∈ Finset.range n, f j

@[simp]

theorem Finset.prod_Ico_id_eq_factorial (n : ℕ) : ∏ x ∈ Finset.Ico 1 (n + 1), x = n.factorial

@[simp]

theorem Finset.prod_range_add_one_eq_factorial (n : ℕ) : ∏ x ∈ Finset.range n, (x + 1) = n.factorial

theorem Finset.sum_range_id_mul_two (n : ℕ) : (∑ i ∈ Finset.range n, i) * 2 = n * (n - 1)

Gauss' summation formula

theorem Finset.sum_range_id (n : ℕ) : ∑ i ∈ Finset.range n, i = n * (n - 1) / 2

Gauss' summation formula

theorem Finset.prod_range_diag_flip {M : Type u_2} [CommMonoid M] (n : ℕ) (f : ℕ → ℕ → M) : ∏ m ∈ Finset.range n, ∏ k ∈ Finset.range (m + 1), f k (m - k) = ∏ m ∈ Finset.range n, ∏ k ∈ Finset.range (n - m), f m k

theorem Finset.sum_range_diag_flip {M : Type u_2} [AddCommMonoid M] (n : ℕ) (f : ℕ → ℕ → M) : ∑ m ∈ Finset.range n, ∑ k ∈ Finset.range (m + 1), f k (m - k) = ∑ m ∈ Finset.range n, ∑ k ∈ Finset.range (n - m), f m k

theorem Finset.prod_range_succ_div_prod {M : Type u_3} (f : ℕ → M) {n : ℕ} [CommGroup M] : (∏ i ∈ Finset.range (n + 1), f i) / ∏ i ∈ Finset.range n, f i = f n

theorem Finset.sum_range_succ_sub_sum {M : Type u_3} (f : ℕ → M) {n : ℕ} [AddCommGroup M] : ∑ i ∈ Finset.range (n + 1), f i - ∑ i ∈ Finset.range n, f i = f n

theorem Finset.prod_range_succ_div_top {M : Type u_3} (f : ℕ → M) {n : ℕ} [CommGroup M] : (∏ i ∈ Finset.range (n + 1), f i) / f n = ∏ i ∈ Finset.range n, f i

theorem Finset.sum_range_succ_sub_top {M : Type u_3} (f : ℕ → M) {n : ℕ} [AddCommGroup M] : ∑ i ∈ Finset.range (n + 1), f i - f n = ∑ i ∈ Finset.range n, f i

theorem Finset.prod_Ico_div_bot {M : Type u_3} (f : ℕ → M) {m : ℕ} {n : ℕ} [CommGroup M] (hmn : m < n) : (∏ i ∈ Finset.Ico m n, f i) / f m = ∏ i ∈ Finset.Ico (m + 1) n, f i

theorem Finset.sum_Ico_sub_bot {M : Type u_3} (f : ℕ → M) {m : ℕ} {n : ℕ} [AddCommGroup M] (hmn : m < n) : ∑ i ∈ Finset.Ico m n, f i - f m = ∑ i ∈ Finset.Ico (m + 1) n, f i

theorem Finset.prod_Ico_succ_div_top {M : Type u_3} (f : ℕ → M) {m : ℕ} {n : ℕ} [CommGroup M] (hmn : m ≤ n) : (∏ i ∈ Finset.Ico m (n + 1), f i) / f n = ∏ i ∈ Finset.Ico m n, f i

theorem Finset.sum_Ico_succ_sub_top {M : Type u_3} (f : ℕ → M) {m : ℕ} {n : ℕ} [AddCommGroup M] (hmn : m ≤ n) : ∑ i ∈ Finset.Ico m (n + 1), f i - f n = ∑ i ∈ Finset.Ico m n, f i