refactor(library/data/prod): move specialized theorems to quotient
This commit is contained in:
Jeremy Avigad
Leonardo de Moura
parent
3e92cd4922
commit
d48a332876
3 changed files with 91 additions and 88 deletions
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@ -326,7 +326,7 @@ or.elim (cases_of_nat_succ a)
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/- addition -/
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definition padd (p q : ℕ × ℕ) : ℕ × ℕ := map_pair2 nat.add p q
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definition padd (p q : ℕ × ℕ) : ℕ × ℕ := (pr1 p + pr1 q, pr2 p + pr2 q)
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theorem repr_add (a b : ℤ) : repr (add a b) ≡ padd (repr a) (repr b) :=
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cases_on a
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@ -28,86 +28,4 @@ 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_of_decidable_of_iff _ (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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@ -1,6 +1,10 @@
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-- Copyright (c) 2014 Floris van Doorn. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Author: Floris van Doorn
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/-
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Copyright (c) 2014 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: data.quotient.util
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Author: Floris van Doorn
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-/
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import logic ..prod algebra.relation
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import tools.fake_simplifier
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@ -10,8 +14,89 @@ open fake_simplifier
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namespace quotient
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-- auxliary facts about products
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-- -----------------------------
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/- auxiliary facts about products -/
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variables {A B : Type}
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/- flip -/
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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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-- 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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