Documentation

Mathlib.Algebra.BigOperators.Intervals

Results about big operators over intervals #

We prove results about big operators over intervals.

theorem Finset.add_sum_Ico_eq_sum_Icc {α : Type u_1} {M : Type u_2} [PartialOrder α] [AddCommMonoid M] {f : αM} {a : α} {b : α} [LocallyFiniteOrder α] (h : a b) :
(f b + Finset.sum (Finset.Ico a b) fun (x : α) => f x) = Finset.sum (Finset.Icc a b) fun (x : α) => f x
theorem Finset.mul_prod_Ico_eq_prod_Icc {α : Type u_1} {M : Type u_2} [PartialOrder α] [CommMonoid M] {f : αM} {a : α} {b : α} [LocallyFiniteOrder α] (h : a b) :
(f b * Finset.prod (Finset.Ico a b) fun (x : α) => f x) = Finset.prod (Finset.Icc a b) fun (x : α) => f x
theorem Finset.sum_Ico_add_eq_sum_Icc {α : Type u_1} {M : Type u_2} [PartialOrder α] [AddCommMonoid M] {f : αM} {a : α} {b : α} [LocallyFiniteOrder α] (h : a b) :
(Finset.sum (Finset.Ico a b) fun (x : α) => f x) + f b = Finset.sum (Finset.Icc a b) fun (x : α) => f x
theorem Finset.prod_Ico_mul_eq_prod_Icc {α : Type u_1} {M : Type u_2} [PartialOrder α] [CommMonoid M] {f : αM} {a : α} {b : α} [LocallyFiniteOrder α] (h : a b) :
(Finset.prod (Finset.Ico a b) fun (x : α) => f x) * f b = Finset.prod (Finset.Icc a b) fun (x : α) => f x
theorem Finset.add_sum_Ioc_eq_sum_Icc {α : Type u_1} {M : Type u_2} [PartialOrder α] [AddCommMonoid M] {f : αM} {a : α} {b : α} [LocallyFiniteOrder α] (h : a b) :
(f a + Finset.sum (Finset.Ioc a b) fun (x : α) => f x) = Finset.sum (Finset.Icc a b) fun (x : α) => f x
theorem Finset.mul_prod_Ioc_eq_prod_Icc {α : Type u_1} {M : Type u_2} [PartialOrder α] [CommMonoid M] {f : αM} {a : α} {b : α} [LocallyFiniteOrder α] (h : a b) :
(f a * Finset.prod (Finset.Ioc a b) fun (x : α) => f x) = Finset.prod (Finset.Icc a b) fun (x : α) => f x
theorem Finset.sum_Ioc_add_eq_sum_Icc {α : Type u_1} {M : Type u_2} [PartialOrder α] [AddCommMonoid M] {f : αM} {a : α} {b : α} [LocallyFiniteOrder α] (h : a b) :
(Finset.sum (Finset.Ioc a b) fun (x : α) => f x) + f a = Finset.sum (Finset.Icc a b) fun (x : α) => f x
theorem Finset.prod_Ioc_mul_eq_prod_Icc {α : Type u_1} {M : Type u_2} [PartialOrder α] [CommMonoid M] {f : αM} {a : α} {b : α} [LocallyFiniteOrder α] (h : a b) :
(Finset.prod (Finset.Ioc a b) fun (x : α) => f x) * f a = Finset.prod (Finset.Icc a b) fun (x : α) => f x
theorem Finset.add_sum_Ioi_eq_sum_Ici {α : Type u_1} {M : Type u_2} [PartialOrder α] [AddCommMonoid M] {f : αM} [LocallyFiniteOrderTop α] (a : α) :
(f a + Finset.sum (Finset.Ioi a) fun (x : α) => f x) = Finset.sum (Finset.Ici a) fun (x : α) => f x
theorem Finset.mul_prod_Ioi_eq_prod_Ici {α : Type u_1} {M : Type u_2} [PartialOrder α] [CommMonoid M] {f : αM} [LocallyFiniteOrderTop α] (a : α) :
(f a * Finset.prod (Finset.Ioi a) fun (x : α) => f x) = Finset.prod (Finset.Ici a) fun (x : α) => f x
theorem Finset.sum_Ioi_add_eq_sum_Ici {α : Type u_1} {M : Type u_2} [PartialOrder α] [AddCommMonoid M] {f : αM} [LocallyFiniteOrderTop α] (a : α) :
(Finset.sum (Finset.Ioi a) fun (x : α) => f x) + f a = Finset.sum (Finset.Ici a) fun (x : α) => f x
theorem Finset.prod_Ioi_mul_eq_prod_Ici {α : Type u_1} {M : Type u_2} [PartialOrder α] [CommMonoid M] {f : αM} [LocallyFiniteOrderTop α] (a : α) :
(Finset.prod (Finset.Ioi a) fun (x : α) => f x) * f a = Finset.prod (Finset.Ici a) fun (x : α) => f x
theorem Finset.add_sum_Iio_eq_sum_Iic {α : Type u_1} {M : Type u_2} [PartialOrder α] [AddCommMonoid M] {f : αM} [LocallyFiniteOrderBot α] (a : α) :
(f a + Finset.sum (Finset.Iio a) fun (x : α) => f x) = Finset.sum (Finset.Iic a) fun (x : α) => f x
theorem Finset.mul_prod_Iio_eq_prod_Iic {α : Type u_1} {M : Type u_2} [PartialOrder α] [CommMonoid M] {f : αM} [LocallyFiniteOrderBot α] (a : α) :
(f a * Finset.prod (Finset.Iio a) fun (x : α) => f x) = Finset.prod (Finset.Iic a) fun (x : α) => f x
theorem Finset.sum_Iio_add_eq_sum_Iic {α : Type u_1} {M : Type u_2} [PartialOrder α] [AddCommMonoid M] {f : αM} [LocallyFiniteOrderBot α] (a : α) :
(Finset.sum (Finset.Iio a) fun (x : α) => f x) + f a = Finset.sum (Finset.Iic a) fun (x : α) => f x
theorem Finset.prod_Iio_mul_eq_prod_Iic {α : Type u_1} {M : Type u_2} [PartialOrder α] [CommMonoid M] {f : αM} [LocallyFiniteOrderBot α] (a : α) :
(Finset.prod (Finset.Iio a) fun (x : α) => f x) * f a = Finset.prod (Finset.Iic a) fun (x : α) => f x
theorem Finset.sum_Ico_add' {α : Type u_1} {M : Type u_2} [AddCommMonoid M] [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (f : αM) (a : α) (b : α) (c : α) :
(Finset.sum (Finset.Ico a b) fun (x : α) => f (x + c)) = Finset.sum (Finset.Ico (a + c) (b + c)) fun (x : α) => f x
theorem Finset.prod_Ico_add' {α : Type u_1} {M : Type u_2} [CommMonoid M] [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (f : αM) (a : α) (b : α) (c : α) :
(Finset.prod (Finset.Ico a b) fun (x : α) => f (x + c)) = Finset.prod (Finset.Ico (a + c) (b + c)) fun (x : α) => f x
theorem Finset.sum_Ico_add {α : Type u_1} {M : Type u_2} [AddCommMonoid M] [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (f : αM) (a : α) (b : α) (c : α) :
(Finset.sum (Finset.Ico a b) fun (x : α) => f (c + x)) = Finset.sum (Finset.Ico (a + c) (b + c)) fun (x : α) => f x
theorem Finset.prod_Ico_add {α : Type u_1} {M : Type u_2} [CommMonoid M] [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (f : αM) (a : α) (b : α) (c : α) :
(Finset.prod (Finset.Ico a b) fun (x : α) => f (c + x)) = Finset.prod (Finset.Ico (a + c) (b + c)) fun (x : α) => f x
theorem Finset.sum_Ico_succ_top {M : Type u_2} [AddCommMonoid M] {a : } {b : } (hab : a b) (f : M) :
(Finset.sum (Finset.Ico a (b + 1)) fun (k : ) => f k) = (Finset.sum (Finset.Ico a b) fun (k : ) => f k) + f b
theorem Finset.prod_Ico_succ_top {M : Type u_2} [CommMonoid M] {a : } {b : } (hab : a b) (f : M) :
(Finset.prod (Finset.Ico a (b + 1)) fun (k : ) => f k) = (Finset.prod (Finset.Ico a b) fun (k : ) => f k) * f b
theorem Finset.sum_eq_sum_Ico_succ_bot {M : Type u_2} [AddCommMonoid M] {a : } {b : } (hab : a < b) (f : M) :
(Finset.sum (Finset.Ico a b) fun (k : ) => f k) = f a + Finset.sum (Finset.Ico (a + 1) b) fun (k : ) => f k
theorem Finset.prod_eq_prod_Ico_succ_bot {M : Type u_2} [CommMonoid M] {a : } {b : } (hab : a < b) (f : M) :
(Finset.prod (Finset.Ico a b) fun (k : ) => f k) = f a * Finset.prod (Finset.Ico (a + 1) b) fun (k : ) => f k
theorem Finset.sum_Ico_consecutive {M : Type u_2} [AddCommMonoid M] (f : M) {m : } {n : } {k : } (hmn : m n) (hnk : n k) :
((Finset.sum (Finset.Ico m n) fun (i : ) => f i) + Finset.sum (Finset.Ico n k) fun (i : ) => f i) = Finset.sum (Finset.Ico m k) fun (i : ) => f i
theorem Finset.prod_Ico_consecutive {M : Type u_2} [CommMonoid M] (f : M) {m : } {n : } {k : } (hmn : m n) (hnk : n k) :
((Finset.prod (Finset.Ico m n) fun (i : ) => f i) * Finset.prod (Finset.Ico n k) fun (i : ) => f i) = Finset.prod (Finset.Ico m k) fun (i : ) => f i
theorem Finset.sum_Ioc_consecutive {M : Type u_2} [AddCommMonoid M] (f : M) {m : } {n : } {k : } (hmn : m n) (hnk : n k) :
((Finset.sum (Finset.Ioc m n) fun (i : ) => f i) + Finset.sum (Finset.Ioc n k) fun (i : ) => f i) = Finset.sum (Finset.Ioc m k) fun (i : ) => f i
theorem Finset.prod_Ioc_consecutive {M : Type u_2} [CommMonoid M] (f : M) {m : } {n : } {k : } (hmn : m n) (hnk : n k) :
((Finset.prod (Finset.Ioc m n) fun (i : ) => f i) * Finset.prod (Finset.Ioc n k) fun (i : ) => f i) = Finset.prod (Finset.Ioc m k) fun (i : ) => f i
theorem Finset.sum_Ioc_succ_top {M : Type u_2} [AddCommMonoid M] {a : } {b : } (hab : a b) (f : M) :
(Finset.sum (Finset.Ioc a (b + 1)) fun (k : ) => f k) = (Finset.sum (Finset.Ioc a b) fun (k : ) => f k) + f (b + 1)
theorem Finset.prod_Ioc_succ_top {M : Type u_2} [CommMonoid M] {a : } {b : } (hab : a b) (f : M) :
(Finset.prod (Finset.Ioc a (b + 1)) fun (k : ) => f k) = (Finset.prod (Finset.Ioc a b) fun (k : ) => f k) * f (b + 1)
theorem Finset.sum_range_add_sum_Ico {M : Type u_2} [AddCommMonoid M] (f : M) {m : } {n : } (h : m n) :
((Finset.sum (Finset.range m) fun (k : ) => f k) + Finset.sum (Finset.Ico m n) fun (k : ) => f k) = Finset.sum (Finset.range n) fun (k : ) => f k
theorem Finset.prod_range_mul_prod_Ico {M : Type u_2} [CommMonoid M] (f : M) {m : } {n : } (h : m n) :
((Finset.prod (Finset.range m) fun (k : ) => f k) * Finset.prod (Finset.Ico m n) fun (k : ) => f k) = Finset.prod (Finset.range n) fun (k : ) => f k
theorem Finset.sum_Ico_eq_add_neg {δ : Type u_3} [AddCommGroup δ] (f : δ) {m : } {n : } (h : m n) :
(Finset.sum (Finset.Ico m n) fun (k : ) => f k) = (Finset.sum (Finset.range n) fun (k : ) => f k) + -Finset.sum (Finset.range m) fun (k : ) => f k
theorem Finset.prod_Ico_eq_mul_inv {δ : Type u_3} [CommGroup δ] (f : δ) {m : } {n : } (h : m n) :
(Finset.prod (Finset.Ico m n) fun (k : ) => f k) = (Finset.prod (Finset.range n) fun (k : ) => f k) * (Finset.prod (Finset.range m) fun (k : ) => f k)⁻¹
theorem Finset.sum_Ico_eq_sub {δ : Type u_3} [AddCommGroup δ] (f : δ) {m : } {n : } (h : m n) :
(Finset.sum (Finset.Ico m n) fun (k : ) => f k) = (Finset.sum (Finset.range n) fun (k : ) => f k) - Finset.sum (Finset.range m) fun (k : ) => f k
theorem Finset.prod_Ico_eq_div {δ : Type u_3} [CommGroup δ] (f : δ) {m : } {n : } (h : m n) :
(Finset.prod (Finset.Ico m n) fun (k : ) => f k) = (Finset.prod (Finset.range n) fun (k : ) => f k) / Finset.prod (Finset.range m) fun (k : ) => f k
theorem Finset.sum_range_sub_sum_range {α : Type u_3} [AddCommGroup α] {f : α} {n : } {m : } (hnm : n m) :
((Finset.sum (Finset.range m) fun (k : ) => f k) - Finset.sum (Finset.range n) fun (k : ) => f k) = Finset.sum (Finset.filter (fun (k : ) => n k) (Finset.range m)) fun (k : ) => f k
theorem Finset.prod_range_div_prod_range {α : Type u_3} [CommGroup α] {f : α} {n : } {m : } (hnm : n m) :
((Finset.prod (Finset.range m) fun (k : ) => f k) / Finset.prod (Finset.range n) fun (k : ) => f k) = Finset.prod (Finset.filter (fun (k : ) => n k) (Finset.range m)) fun (k : ) => f k
theorem Finset.sum_Ico_Ico_comm {M : Type u_3} [AddCommMonoid M] (a : ) (b : ) (f : M) :
(Finset.sum (Finset.Ico a b) fun (i : ) => Finset.sum (Finset.Ico i b) fun (j : ) => f i j) = Finset.sum (Finset.Ico a b) fun (j : ) => Finset.sum (Finset.Ico a (j + 1)) fun (i : ) => f i j

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

theorem Finset.sum_Ico_eq_sum_range {M : Type u_2} [AddCommMonoid M] (f : M) (m : ) (n : ) :
(Finset.sum (Finset.Ico m n) fun (k : ) => f k) = Finset.sum (Finset.range (n - m)) fun (k : ) => f (m + k)
theorem Finset.prod_Ico_eq_prod_range {M : Type u_2} [CommMonoid M] (f : M) (m : ) (n : ) :
(Finset.prod (Finset.Ico m n) fun (k : ) => f k) = Finset.prod (Finset.range (n - m)) fun (k : ) => f (m + k)
theorem Finset.prod_Ico_reflect {M : Type u_2} [CommMonoid M] (f : M) (k : ) {m : } {n : } (h : m n + 1) :
(Finset.prod (Finset.Ico k m) fun (j : ) => f (n - j)) = Finset.prod (Finset.Ico (n + 1 - m) (n + 1 - k)) fun (j : ) => f j
theorem Finset.sum_Ico_reflect {δ : Type u_3} [AddCommMonoid δ] (f : δ) (k : ) {m : } {n : } (h : m n + 1) :
(Finset.sum (Finset.Ico k m) fun (j : ) => f (n - j)) = Finset.sum (Finset.Ico (n + 1 - m) (n + 1 - k)) fun (j : ) => f j
theorem Finset.prod_range_reflect {M : Type u_2} [CommMonoid M] (f : M) (n : ) :
(Finset.prod (Finset.range n) fun (j : ) => f (n - 1 - j)) = Finset.prod (Finset.range n) fun (j : ) => f j
theorem Finset.sum_range_reflect {δ : Type u_3} [AddCommMonoid δ] (f : δ) (n : ) :
(Finset.sum (Finset.range n) fun (j : ) => f (n - 1 - j)) = Finset.sum (Finset.range n) fun (j : ) => f j
@[simp]
theorem Finset.prod_Ico_id_eq_factorial (n : ) :
(Finset.prod (Finset.Ico 1 (n + 1)) fun (x : ) => x) = Nat.factorial n
theorem Finset.sum_range_id_mul_two (n : ) :
(Finset.sum (Finset.range n) fun (i : ) => i) * 2 = n * (n - 1)

Gauss' summation formula

theorem Finset.sum_range_id (n : ) :
(Finset.sum (Finset.range n) fun (i : ) => i) = n * (n - 1) / 2

Gauss' summation formula

theorem Finset.sum_range_diag_flip {M : Type u_2} [AddCommMonoid M] (n : ) (f : M) :
(Finset.sum (Finset.range n) fun (m : ) => Finset.sum (Finset.range (m + 1)) fun (k : ) => f k (m - k)) = Finset.sum (Finset.range n) fun (m : ) => Finset.sum (Finset.range (n - m)) fun (k : ) => f m k
theorem Finset.prod_range_diag_flip {M : Type u_2} [CommMonoid M] (n : ) (f : M) :
(Finset.prod (Finset.range n) fun (m : ) => Finset.prod (Finset.range (m + 1)) fun (k : ) => f k (m - k)) = Finset.prod (Finset.range n) fun (m : ) => Finset.prod (Finset.range (n - m)) fun (k : ) => f m k
theorem Finset.sum_range_succ_sub_sum {M : Type u_3} (f : M) {n : } [AddCommGroup M] :
((Finset.sum (Finset.range (n + 1)) fun (i : ) => f i) - Finset.sum (Finset.range n) fun (i : ) => f i) = f n
theorem Finset.prod_range_succ_div_prod {M : Type u_3} (f : M) {n : } [CommGroup M] :
((Finset.prod (Finset.range (n + 1)) fun (i : ) => f i) / Finset.prod (Finset.range n) fun (i : ) => f i) = f n
theorem Finset.sum_range_succ_sub_top {M : Type u_3} (f : M) {n : } [AddCommGroup M] :
(Finset.sum (Finset.range (n + 1)) fun (i : ) => f i) - f n = Finset.sum (Finset.range n) fun (i : ) => f i
theorem Finset.prod_range_succ_div_top {M : Type u_3} (f : M) {n : } [CommGroup M] :
(Finset.prod (Finset.range (n + 1)) fun (i : ) => f i) / f n = Finset.prod (Finset.range n) fun (i : ) => f i
theorem Finset.sum_Ico_sub_bot {M : Type u_3} (f : M) {m : } {n : } [AddCommGroup M] (hmn : m < n) :
(Finset.sum (Finset.Ico m n) fun (i : ) => f i) - f m = Finset.sum (Finset.Ico (m + 1) n) fun (i : ) => f i
theorem Finset.prod_Ico_div_bot {M : Type u_3} (f : M) {m : } {n : } [CommGroup M] (hmn : m < n) :
(Finset.prod (Finset.Ico m n) fun (i : ) => f i) / f m = Finset.prod (Finset.Ico (m + 1) n) fun (i : ) => f i
theorem Finset.sum_Ico_succ_sub_top {M : Type u_3} (f : M) {m : } {n : } [AddCommGroup M] (hmn : m n) :
(Finset.sum (Finset.Ico m (n + 1)) fun (i : ) => f i) - f n = Finset.sum (Finset.Ico m n) fun (i : ) => f i
theorem Finset.prod_Ico_succ_div_top {M : Type u_3} (f : M) {m : } {n : } [CommGroup M] (hmn : m n) :
(Finset.prod (Finset.Ico m (n + 1)) fun (i : ) => f i) / f n = Finset.prod (Finset.Ico m n) fun (i : ) => f i