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Mathlib.Algebra.BigOperators.NatAntidiagonal

Big operators for NatAntidiagonal #

This file contains theorems relevant to big operators over Finset.NatAntidiagonal.

theorem Finset.Nat.prod_antidiagonal_succ {M : Type u_1} [CommMonoid M] {n : } {f : × M} :
pFinset.antidiagonal (n + 1), f p = f (0, n + 1) * pFinset.antidiagonal n, f (p.1 + 1, p.2)
theorem Finset.Nat.sum_antidiagonal_succ {N : Type u_2} [AddCommMonoid N] {n : } {f : × N} :
pFinset.antidiagonal (n + 1), f p = f (0, n + 1) + pFinset.antidiagonal n, f (p.1 + 1, p.2)
theorem Finset.Nat.sum_antidiagonal_swap {M : Type u_1} [AddCommMonoid M] {n : } {f : × M} :
pFinset.antidiagonal n, f p.swap = pFinset.antidiagonal n, f p
theorem Finset.Nat.prod_antidiagonal_swap {M : Type u_1} [CommMonoid M] {n : } {f : × M} :
pFinset.antidiagonal n, f p.swap = pFinset.antidiagonal n, f p
theorem Finset.Nat.prod_antidiagonal_succ' {M : Type u_1} [CommMonoid M] {n : } {f : × M} :
pFinset.antidiagonal (n + 1), f p = f (n + 1, 0) * pFinset.antidiagonal n, f (p.1, p.2 + 1)
theorem Finset.Nat.sum_antidiagonal_succ' {N : Type u_2} [AddCommMonoid N] {n : } {f : × N} :
pFinset.antidiagonal (n + 1), f p = f (n + 1, 0) + pFinset.antidiagonal n, f (p.1, p.2 + 1)
theorem Finset.Nat.sum_antidiagonal_subst {M : Type u_1} [AddCommMonoid M] {n : } {f : × M} :
pFinset.antidiagonal n, f p n = pFinset.antidiagonal n, f p (p.1 + p.2)
theorem Finset.Nat.prod_antidiagonal_subst {M : Type u_1} [CommMonoid M] {n : } {f : × M} :
pFinset.antidiagonal n, f p n = pFinset.antidiagonal n, f p (p.1 + p.2)
theorem Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk {M : Type u_3} [AddCommMonoid M] (f : × M) (n : ) :
ijFinset.antidiagonal n, f ij = kFinset.range n.succ, f (k, n - k)
theorem Finset.Nat.prod_antidiagonal_eq_prod_range_succ_mk {M : Type u_3} [CommMonoid M] (f : × M) (n : ) :
ijFinset.antidiagonal n, f ij = kFinset.range n.succ, f (k, n - k)
theorem Finset.Nat.sum_antidiagonal_eq_sum_range_succ {M : Type u_3} [AddCommMonoid M] (f : M) (n : ) :
ijFinset.antidiagonal n, f ij.1 ij.2 = kFinset.range n.succ, f k (n - k)

This lemma matches more generally than Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk when using rw ← .

theorem Finset.Nat.prod_antidiagonal_eq_prod_range_succ {M : Type u_3} [CommMonoid M] (f : M) (n : ) :
ijFinset.antidiagonal n, f ij.1 ij.2 = kFinset.range n.succ, f k (n - k)

This lemma matches more generally than Finset.Nat.prod_antidiagonal_eq_prod_range_succ_mk when using rw ← .