refactor(data.prod): move theorems about products from data.quotient.util to data.prod.thms
This commit is contained in:
Floris van Doorn
Leonardo de Moura
parent
24e35a9f2c
commit
7bfbe039d9
2 changed files with 83 additions and 85 deletions
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@ -6,7 +6,7 @@ import data.prod.decl logic.inhabited logic.eq logic.decidable
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open inhabited decidable eq.ops
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namespace prod
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variables {A B : Type} {a₁ a₂ : A} {b₁ b₂ : B}
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variables {A B : Type} {a₁ a₂ : A} {b₁ b₂ : B} {u : A × B}
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theorem pair_eq : a₁ = a₂ → b₁ = b₂ → (a₁, b₁) = (a₂, b₂) :=
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assume H1 H2, H1 ▸ H2 ▸ rfl
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@ -25,4 +25,86 @@ namespace prod
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(assume H, H ▸ and.intro rfl rfl)
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(assume H, and.elim H (assume H₄ H₅, equal H₄ H₅)),
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decidable_iff_equiv _ (iff.symm H₃)
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-- ### flip operation
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definition flip (a : A × B) : B × A := pair (pr2 a) (pr1 a)
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theorem flip_def (a : A × B) : flip a = pair (pr2 a) (pr1 a) := rfl
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theorem flip_pair (a : A) (b : B) : flip (pair a b) = pair b a := rfl
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theorem flip_pr1 (a : A × B) : pr1 (flip a) = pr2 a := rfl
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theorem flip_pr2 (a : A × B) : pr2 (flip a) = pr1 a := rfl
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theorem flip_flip (a : A × B) : flip (flip a) = a :=
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destruct a (take x y, rfl)
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theorem P_flip {P : A → B → Prop} (a : A × B) (H : P (pr1 a) (pr2 a))
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: P (pr2 (flip a)) (pr1 (flip a)) :=
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(flip_pr1 a)⁻¹ ▸ (flip_pr2 a)⁻¹ ▸ H
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theorem flip_inj {a b : A × B} (H : flip a = flip b) : a = b :=
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have H2 : flip (flip a) = flip (flip b), from congr_arg flip H,
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show a = b, from (flip_flip a) ▸ (flip_flip b) ▸ H2
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-- ### coordinatewise unary maps
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definition map_pair (f : A → B) (a : A × A) : B × B :=
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pair (f (pr1 a)) (f (pr2 a))
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theorem map_pair_def (f : A → B) (a : A × A)
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: map_pair f a = pair (f (pr1 a)) (f (pr2 a)) :=
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rfl
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theorem map_pair_pair (f : A → B) (a a' : A)
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: map_pair f (pair a a') = pair (f a) (f a') :=
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(pr1.mk a a') ▸ (pr2.mk a a') ▸ rfl
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theorem map_pair_pr1 (f : A → B) (a : A × A) : pr1 (map_pair f a) = f (pr1 a) :=
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!pr1.mk
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theorem map_pair_pr2 (f : A → B) (a : A × A) : pr2 (map_pair f a) = f (pr2 a) :=
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!pr2.mk
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-- ### coordinatewise binary maps
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definition map_pair2 {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) : C × C :=
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pair (f (pr1 a) (pr1 b)) (f (pr2 a) (pr2 b))
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theorem map_pair2_def {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) :
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map_pair2 f a b = pair (f (pr1 a) (pr1 b)) (f (pr2 a) (pr2 b)) := rfl
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theorem map_pair2_pair {A B C : Type} (f : A → B → C) (a a' : A) (b b' : B) :
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map_pair2 f (pair a a') (pair b b') = pair (f a b) (f a' b') :=
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calc
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map_pair2 f (pair a a') (pair b b')
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= pair (f (pr1 (pair a a')) b) (f (pr2 (pair a a')) (pr2 (pair b b')))
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: {pr1.mk b b'}
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... = pair (f (pr1 (pair a a')) b) (f (pr2 (pair a a')) b') : {pr2.mk b b'}
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... = pair (f (pr1 (pair a a')) b) (f a' b') : {pr2.mk a a'}
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... = pair (f a b) (f a' b') : {pr1.mk a a'}
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theorem map_pair2_pr1 {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) :
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pr1 (map_pair2 f a b) = f (pr1 a) (pr1 b) := !pr1.mk
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theorem map_pair2_pr2 {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) :
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pr2 (map_pair2 f a b) = f (pr2 a) (pr2 b) := !pr2.mk
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theorem map_pair2_flip {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) :
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flip (map_pair2 f a b) = map_pair2 f (flip a) (flip b) :=
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have Hx : pr1 (flip (map_pair2 f a b)) = pr1 (map_pair2 f (flip a) (flip b)), from
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calc
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pr1 (flip (map_pair2 f a b)) = pr2 (map_pair2 f a b) : flip_pr1 _
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... = f (pr2 a) (pr2 b) : map_pair2_pr2 f a b
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... = f (pr1 (flip a)) (pr2 b) : {(flip_pr1 a)⁻¹}
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... = f (pr1 (flip a)) (pr1 (flip b)) : {(flip_pr1 b)⁻¹}
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... = pr1 (map_pair2 f (flip a) (flip b)) : (map_pair2_pr1 f _ _)⁻¹,
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have Hy : pr2 (flip (map_pair2 f a b)) = pr2 (map_pair2 f (flip a) (flip b)), from
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calc
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pr2 (flip (map_pair2 f a b)) = pr1 (map_pair2 f a b) : flip_pr2 _
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... = f (pr1 a) (pr1 b) : map_pair2_pr1 f a b
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... = f (pr2 (flip a)) (pr1 b) : {flip_pr2 a}
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... = f (pr2 (flip a)) (pr2 (flip b)) : {flip_pr2 b}
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... = pr2 (map_pair2 f (flip a) (flip b)) : (map_pair2_pr2 f _ _)⁻¹,
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pair_eq Hx Hy
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end prod
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@ -13,90 +13,6 @@ namespace quotient
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-- auxliary facts about products
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-- -----------------------------
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-- ### flip
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definition flip {A B : Type} (a : A × B) : B × A := pair (pr2 a) (pr1 a)
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theorem flip_def {A B : Type} (a : A × B) : flip a = pair (pr2 a) (pr1 a) := eq.refl (flip a)
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theorem flip_pair {A B : Type} (a : A) (b : B) : flip (pair a b) = pair b a := rfl
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theorem flip_pr1 {A B : Type} (a : A × B) : pr1 (flip a) = pr2 a := rfl
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theorem flip_pr2 {A B : Type} (a : A × B) : pr2 (flip a) = pr1 a := rfl
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theorem flip_flip {A B : Type} (a : A × B) : flip (flip a) = a :=
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prod.destruct a (take x y, rfl)
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theorem P_flip {A B : Type} {P : A → B → Prop} (a : A × B) (H : P (pr1 a) (pr2 a))
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: P (pr2 (flip a)) (pr1 (flip a)) :=
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(flip_pr1 a)⁻¹ ▸ (flip_pr2 a)⁻¹ ▸ H
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theorem flip_inj {A B : Type} {a b : A × B} (H : flip a = flip b) : a = b :=
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have H2 : flip (flip a) = flip (flip b), from congr_arg flip H,
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show a = b, from (flip_flip a) ▸ (flip_flip b) ▸ H2
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-- ### coordinatewise unary maps
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definition map_pair {A B : Type} (f : A → B) (a : A × A) : B × B :=
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pair (f (pr1 a)) (f (pr2 a))
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theorem map_pair_def {A B : Type} (f : A → B) (a : A × A)
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: map_pair f a = pair (f (pr1 a)) (f (pr2 a)) :=
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rfl
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theorem map_pair_pair {A B : Type} (f : A → B) (a a' : A)
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: map_pair f (pair a a') = pair (f a) (f a') :=
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(pr1.mk a a') ▸ (pr2.mk a a') ▸ rfl
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theorem map_pair_pr1 {A B : Type} (f : A → B) (a : A × A) : pr1 (map_pair f a) = f (pr1 a) :=
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!pr1.mk
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theorem map_pair_pr2 {A B : Type} (f : A → B) (a : A × A) : pr2 (map_pair f a) = f (pr2 a) :=
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!pr2.mk
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-- ### coordinatewise binary maps
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definition map_pair2 {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) : C × C :=
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pair (f (pr1 a) (pr1 b)) (f (pr2 a) (pr2 b))
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theorem map_pair2_def {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) :
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map_pair2 f a b = pair (f (pr1 a) (pr1 b)) (f (pr2 a) (pr2 b)) := rfl
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theorem map_pair2_pair {A B C : Type} (f : A → B → C) (a a' : A) (b b' : B) :
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map_pair2 f (pair a a') (pair b b') = pair (f a b) (f a' b') :=
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calc
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map_pair2 f (pair a a') (pair b b')
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= pair (f (pr1 (pair a a')) b) (f (pr2 (pair a a')) (pr2 (pair b b')))
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: {pr1.mk b b'}
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... = pair (f (pr1 (pair a a')) b) (f (pr2 (pair a a')) b') : {pr2.mk b b'}
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... = pair (f (pr1 (pair a a')) b) (f a' b') : {pr2.mk a a'}
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... = pair (f a b) (f a' b') : {pr1.mk a a'}
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theorem map_pair2_pr1 {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) :
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pr1 (map_pair2 f a b) = f (pr1 a) (pr1 b) := !pr1.mk
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theorem map_pair2_pr2 {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) :
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pr2 (map_pair2 f a b) = f (pr2 a) (pr2 b) := !pr2.mk
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theorem map_pair2_flip {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) :
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flip (map_pair2 f a b) = map_pair2 f (flip a) (flip b) :=
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have Hx : pr1 (flip (map_pair2 f a b)) = pr1 (map_pair2 f (flip a) (flip b)), from
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calc
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pr1 (flip (map_pair2 f a b)) = pr2 (map_pair2 f a b) : flip_pr1 _
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... = f (pr2 a) (pr2 b) : map_pair2_pr2 f a b
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... = f (pr1 (flip a)) (pr2 b) : {(flip_pr1 a)⁻¹}
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... = f (pr1 (flip a)) (pr1 (flip b)) : {(flip_pr1 b)⁻¹}
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... = pr1 (map_pair2 f (flip a) (flip b)) : (map_pair2_pr1 f _ _)⁻¹,
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have Hy : pr2 (flip (map_pair2 f a b)) = pr2 (map_pair2 f (flip a) (flip b)), from
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calc
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pr2 (flip (map_pair2 f a b)) = pr1 (map_pair2 f a b) : flip_pr2 _
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... = f (pr1 a) (pr1 b) : map_pair2_pr1 f a b
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... = f (pr2 (flip a)) (pr1 b) : {flip_pr2 a}
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... = f (pr2 (flip a)) (pr2 (flip b)) : {flip_pr2 b}
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... = pr2 (map_pair2 f (flip a) (flip b)) : (map_pair2_pr2 f _ _)⁻¹,
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pair_eq Hx Hy
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-- add_rewrite flip_pr1 flip_pr2 flip_pair
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-- add_rewrite map_pair_pr1 map_pair_pr2 map_pair_pair
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-- add_rewrite map_pair2_pr1 map_pair2_pr2 map_pair2_pair
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