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

• Finset.addAntidiagonal hs ht a = ⋯.toFinset

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

• Finset.mulAntidiagonal hs ht a = ⋯.toFinset

@[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) : Finset.addAntidiagonal hu ht a ⊆ Finset.addAntidiagonal hs ht a

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) : Finset.mulAntidiagonal hu ht a ⊆ Finset.mulAntidiagonal hs ht a

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) : Finset.addAntidiagonal hs hu a ⊆ Finset.addAntidiagonal hs ht a

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) : Finset.mulAntidiagonal hs hu a ⊆ Finset.mulAntidiagonal hs ht a

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} : x.swap ∈ Finset.addAntidiagonal hs ht a ↔ x ∈ Finset.addAntidiagonal ht hs a

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} : x.swap ∈ Finset.mulAntidiagonal hs ht a ↔ x ∈ Finset.mulAntidiagonal ht hs a

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

abbrev Finset.support_addAntidiagonal_subset_add.match_1 {α : Type u_1} [OrderedCancelAddCommMonoid α] {s : Set α} {t : Set α} {hs : s.IsPWO} {ht : t.IsPWO} (a : α) (motive : a ∈ {a : α | (Finset.addAntidiagonal hs ht a).Nonempty} → Prop) : ∀ (x : a ∈ {a : α | (Finset.addAntidiagonal hs ht a).Nonempty}), (∀ (b : α × α) (hb : b ∈ Finset.addAntidiagonal hs ht a), motive ⋯) → motive x

Equations

• ⋯ = ⋯

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)}