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feat(library/data/list/perm): add perm_cross_product theorem

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Leonardo de Moura 2015-04-11 19:13:34 -07:00
parent 41ddc97e0d
commit 4c827293a8
2 changed files with 26 additions and 1 deletions
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2
library/data/list/comb.lean
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@ -275,7 +275,7 @@ definition cross_product : list A → list B → list (A × B)
| [] l₂ := []
| (a::l₁) l₂ := map (λ b, (a, b)) l₂ ++ cross_product l₁ l₂
theorem nil_cross_product_nil (l : list B) : cross_product (@nil A) l = []
theorem nil_cross_product (l : list B) : cross_product (@nil A) l = []
theorem cross_product_cons (a : A) (l₁ : list A) (l₂ : list B)
: cross_product (a::l₁) l₂ = map (λ b, (a, b)) l₂ ++ cross_product l₁ l₂

25
library/data/list/perm.lean
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@ -726,4 +726,29 @@ theorem perm_ext : ∀ {l₁ l₂ : list A}, nodup l₁ → nodup l₂ → (∀a
... ~ s₁++(a₁::s₂) : !perm_middle
... = a₂::t₂ : by rewrite t₂_eq
end ext
/- cross_product -/
section cross_product
theorem perm_cross_product_left {l₁ l₂ : list A} (t₁ : list B) : l₁ ~ l₂ → (cross_product l₁ t₁) ~ (cross_product l₂ t₁) :=
assume p : l₁ ~ l₂, perm.induction_on p
!perm.refl
(λ x l₁ l₂ p r, perm_app !perm.refl r)
(λ x y l,
let m₁ := map (λ b, (x, b)) t₁ in
let m₂ := map (λ b, (y, b)) t₁ in
let c := cross_product l t₁ in
calc m₂ ++ (m₁ ++ c) = (m₂ ++ m₁) ++ c : by rewrite append.assoc
... ~ (m₁ ++ m₂) ++ c : perm_app !perm_app_comm !perm.refl
... = m₁ ++ (m₂ ++ c) : by rewrite append.assoc)
(λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, trans r₁ r₂)
theorem perm_cross_product_right (l : list A) {t₁ t₂ : list B} : t₁ ~ t₂ → (cross_product l t₁) ~ (cross_product l t₂) :=
list.induction_on l
(λ p, by rewrite [*nil_cross_product]; exact !perm.refl)
(λ a t r p,
perm_app (perm_map _ p) (r p))
theorem perm_cross_product {l₁ l₂ t₁ t₂ : list A} : l₁ ~ l₂ → t₁ ~ t₂ → (cross_product l₁ t₁) ~ (cross_product l₂ t₂) :=
assume p₁ p₂, trans (perm_cross_product_left t₁ p₁) (perm_cross_product_right l₂ p₂)
end cross_product
end perm