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Mathlib.Data.Finset.NatAntidiagonal

Antidiagonals in ℕ × ℕ as finsets #

This file defines the antidiagonals of ℕ × ℕ as finsets: the n-th antidiagonal is the finset of pairs (i, j) such that i + j = n. This is useful for polynomial multiplication and more generally for sums going from 0 to n.

Notes #

This refines files Data.List.NatAntidiagonal and Data.Multiset.NatAntidiagonal, providing an instance enabling Finset.antidiagonal on Nat.

The antidiagonal of a natural number n is the finset of pairs (i, j) such that i + j = n.

Equations
  • One or more equations did not get rendered due to their size.
theorem Finset.Nat.antidiagonal_eq_map (n : ) :
Finset.antidiagonal n = Finset.map { toFun := fun (i : ) => (i, n - i), inj' := (_ : ∀ (x x_1 : ), (fun (i : ) => (i, n - i)) x = (fun (i : ) => (i, n - i)) x_1((fun (i : ) => (i, n - i)) x).1 = ((fun (i : ) => (i, n - i)) x_1).1) } (Finset.range (n + 1))
theorem Finset.Nat.antidiagonal_eq_map' (n : ) :
Finset.antidiagonal n = Finset.map { toFun := fun (i : ) => (n - i, i), inj' := (_ : ∀ (x x_1 : ), (fun (i : ) => (n - i, i)) x = (fun (i : ) => (n - i, i)) x_1((fun (i : ) => (n - i, i)) x).2 = ((fun (i : ) => (n - i, i)) x_1).2) } (Finset.range (n + 1))
@[simp]

The cardinality of the antidiagonal of n is n + 1.

@[simp]

The antidiagonal of 0 is the list [(0, 0)]

theorem Finset.Nat.antidiagonal.fst_lt {n : } {kl : × } (hlk : kl Finset.antidiagonal n) :
kl.1 < n + 1
theorem Finset.Nat.antidiagonal.snd_lt {n : } {kl : × } (hlk : kl Finset.antidiagonal n) :
kl.2 < n + 1
@[simp]
theorem Finset.Nat.antidiagonal_filter_snd_le_of_le {n : } {k : } (h : k n) :
Finset.filter (fun (a : × ) => a.2 k) (Finset.antidiagonal n) = Finset.map (Function.Embedding.prodMap { toFun := fun (x : ) => x + (n - k), inj' := (_ : Function.Injective fun (x : ) => x + (n - k)) } (Function.Embedding.refl )) (Finset.antidiagonal k)
@[simp]
theorem Finset.Nat.antidiagonal_filter_fst_le_of_le {n : } {k : } (h : k n) :
Finset.filter (fun (a : × ) => a.1 k) (Finset.antidiagonal n) = Finset.map (Function.Embedding.prodMap (Function.Embedding.refl ) { toFun := fun (x : ) => x + (n - k), inj' := (_ : Function.Injective fun (x : ) => x + (n - k)) }) (Finset.antidiagonal k)
@[simp]
theorem Finset.Nat.antidiagonal_filter_le_fst_of_le {n : } {k : } (h : k n) :
Finset.filter (fun (a : × ) => k a.1) (Finset.antidiagonal n) = Finset.map (Function.Embedding.prodMap { toFun := fun (x : ) => x + k, inj' := (_ : Function.Injective fun (x : ) => x + k) } (Function.Embedding.refl )) (Finset.antidiagonal (n - k))
@[simp]
theorem Finset.Nat.antidiagonal_filter_le_snd_of_le {n : } {k : } (h : k n) :
Finset.filter (fun (a : × ) => k a.2) (Finset.antidiagonal n) = Finset.map (Function.Embedding.prodMap (Function.Embedding.refl ) { toFun := fun (x : ) => x + k, inj' := (_ : Function.Injective fun (x : ) => x + k) }) (Finset.antidiagonal (n - k))
@[simp]
theorem Finset.Nat.antidiagonalEquivFin_symm_apply_coe (n : ) :
∀ (x : Fin (n + 1)), ((Finset.Nat.antidiagonalEquivFin n).symm x) = (x, n - x)

The set antidiagonal n is equivalent to Fin (n+1), via the first projection. -

Equations
  • One or more equations did not get rendered due to their size.
Instances For