Documentation

Mathlib.Data.Finset.MulAntidiagonal

Multiplication antidiagonal as a Finset. #

We construct the Finset of all pairs of an element in s and an element in t that multiply to a, given that s and t are well-ordered.

theorem Set.IsPWO.add {α : Type u_1} {s : Set α} {t : Set α} [OrderedCancelAddCommMonoid α] (hs : s.IsPWO) (ht : t.IsPWO) :
(s + t).IsPWO
theorem Set.IsPWO.mul {α : Type u_1} {s : Set α} {t : Set α} [OrderedCancelCommMonoid α] (hs : s.IsPWO) (ht : t.IsPWO) :
(s * t).IsPWO
theorem Set.IsWF.add {α : Type u_1} {s : Set α} {t : Set α} [LinearOrderedCancelAddCommMonoid α] (hs : s.IsWF) (ht : t.IsWF) :
(s + t).IsWF
theorem Set.IsWF.mul {α : Type u_1} {s : Set α} {t : Set α} [LinearOrderedCancelCommMonoid α] (hs : s.IsWF) (ht : t.IsWF) :
(s * t).IsWF
theorem Set.IsWF.min_add {α : Type u_1} {s : Set α} {t : Set α} [LinearOrderedCancelAddCommMonoid α] (hs : s.IsWF) (ht : t.IsWF) (hsn : s.Nonempty) (htn : t.Nonempty) :
.min = hs.min hsn + ht.min htn
theorem Set.IsWF.min_mul {α : Type u_1} {s : Set α} {t : Set α} [LinearOrderedCancelCommMonoid α] (hs : s.IsWF) (ht : t.IsWF) (hsn : s.Nonempty) (htn : t.Nonempty) :
.min = hs.min hsn * ht.min htn
noncomputable def Finset.addAntidiagonal {α : Type u_1} [OrderedCancelAddCommMonoid α] {s : Set α} {t : Set α} (hs : s.IsPWO) (ht : t.IsPWO) (a : α) :
Finset (α × α)

Finset.addAntidiagonal hs ht a is the set of all pairs of an element in s and an element in t that add to a, but its construction requires proofs that s and t are well-ordered.

Equations
Instances For
    theorem Finset.addAntidiagonal.proof_1 {α : Type u_1} [OrderedCancelAddCommMonoid α] {s : Set α} {t : Set α} (hs : s.IsPWO) (ht : t.IsPWO) (a : α) :
    (s.addAntidiagonal t a).Finite
    noncomputable def Finset.mulAntidiagonal {α : Type u_1} [OrderedCancelCommMonoid α] {s : Set α} {t : Set α} (hs : s.IsPWO) (ht : t.IsPWO) (a : α) :
    Finset (α × α)

    Finset.mulAntidiagonal hs ht a is the set of all pairs of an element in s and an element in t that multiply to a, but its construction requires proofs that s and t are well-ordered.

    Equations
    Instances For
      @[simp]
      theorem Finset.mem_addAntidiagonal {α : Type u_1} [OrderedCancelAddCommMonoid α] {s : Set α} {t : Set α} {hs : s.IsPWO} {ht : t.IsPWO} {a : α} {x : α × α} :
      x Finset.addAntidiagonal hs ht a x.1 s x.2 t x.1 + x.2 = a
      @[simp]
      theorem Finset.mem_mulAntidiagonal {α : Type u_1} [OrderedCancelCommMonoid α] {s : Set α} {t : Set α} {hs : s.IsPWO} {ht : t.IsPWO} {a : α} {x : α × α} :
      x Finset.mulAntidiagonal hs ht a x.1 s x.2 t x.1 * x.2 = a
      theorem Finset.addAntidiagonal_mono_left {α : Type u_1} [OrderedCancelAddCommMonoid α] {s : Set α} {t : Set α} {hs : s.IsPWO} {ht : t.IsPWO} {a : α} {u : Set α} {hu : u.IsPWO} (h : u s) :
      theorem Finset.mulAntidiagonal_mono_left {α : Type u_1} [OrderedCancelCommMonoid α] {s : Set α} {t : Set α} {hs : s.IsPWO} {ht : t.IsPWO} {a : α} {u : Set α} {hu : u.IsPWO} (h : u s) :
      theorem Finset.addAntidiagonal_mono_right {α : Type u_1} [OrderedCancelAddCommMonoid α] {s : Set α} {t : Set α} {hs : s.IsPWO} {ht : t.IsPWO} {a : α} {u : Set α} {hu : u.IsPWO} (h : u t) :
      theorem Finset.mulAntidiagonal_mono_right {α : Type u_1} [OrderedCancelCommMonoid α] {s : Set α} {t : Set α} {hs : s.IsPWO} {ht : t.IsPWO} {a : α} {u : Set α} {hu : u.IsPWO} (h : u t) :
      theorem Finset.swap_mem_addAntidiagonal {α : Type u_1} [OrderedCancelAddCommMonoid α] {s : Set α} {t : Set α} {hs : s.IsPWO} {ht : t.IsPWO} {a : α} {x : α × α} :
      theorem Finset.swap_mem_mulAntidiagonal {α : Type u_1} [OrderedCancelCommMonoid α] {s : Set α} {t : Set α} {hs : s.IsPWO} {ht : t.IsPWO} {a : α} {x : α × α} :
      theorem Finset.support_addAntidiagonal_subset_add {α : Type u_1} [OrderedCancelAddCommMonoid α] {s : Set α} {t : Set α} {hs : s.IsPWO} {ht : t.IsPWO} :
      {a : α | (Finset.addAntidiagonal hs ht a).Nonempty} s + t
      theorem Finset.support_mulAntidiagonal_subset_mul {α : Type u_1} [OrderedCancelCommMonoid α] {s : Set α} {t : Set α} {hs : s.IsPWO} {ht : t.IsPWO} :
      {a : α | (Finset.mulAntidiagonal hs ht a).Nonempty} s * t
      theorem Finset.isPWO_support_addAntidiagonal {α : Type u_1} [OrderedCancelAddCommMonoid α] {s : Set α} {t : Set α} {hs : s.IsPWO} {ht : t.IsPWO} :
      {a : α | (Finset.addAntidiagonal hs ht a).Nonempty}.IsPWO
      theorem Finset.isPWO_support_mulAntidiagonal {α : Type u_1} [OrderedCancelCommMonoid α] {s : Set α} {t : Set α} {hs : s.IsPWO} {ht : t.IsPWO} :
      {a : α | (Finset.mulAntidiagonal hs ht a).Nonempty}.IsPWO
      theorem Finset.addAntidiagonal_min_add_min {α : Type u_2} [LinearOrderedCancelAddCommMonoid α] {s : Set α} {t : Set α} (hs : s.IsWF) (ht : t.IsWF) (hns : s.Nonempty) (hnt : t.Nonempty) :
      Finset.addAntidiagonal (hs.min hns + ht.min hnt) = {(hs.min hns, ht.min hnt)}
      theorem Finset.mulAntidiagonal_min_mul_min {α : Type u_2} [LinearOrderedCancelCommMonoid α] {s : Set α} {t : Set α} (hs : s.IsWF) (ht : t.IsWF) (hns : s.Nonempty) (hnt : t.Nonempty) :
      Finset.mulAntidiagonal (hs.min hns * ht.min hnt) = {(hs.min hns, ht.min hnt)}