lean2/library/data/prod/thms.lean
2014-11-30 20:34:12 -08:00

109 lines
4.8 KiB
Text
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Leonardo de Moura, Jeremy Avigad
import logic.eq
open inhabited decidable eq.ops
namespace prod
variables {A B : Type} {a₁ a₂ : A} {b₁ b₂ : B} {u : A × B}
theorem pair_eq : a₁ = a₂ → b₁ = b₂ → (a₁, b₁) = (a₂, b₂) :=
assume H1 H2, H1 ▸ H2 ▸ rfl
protected theorem equal {p₁ p₂ : prod A B} : pr₁ p₁ = pr₁ p₂ → pr₂ p₁ = pr₂ p₂ → p₁ = p₂ :=
destruct p₁ (take a₁ b₁, destruct p₂ (take a₂ b₂ H₁ H₂, pair_eq H₁ H₂))
protected definition is_inhabited [instance] : inhabited A → inhabited B → inhabited (prod A B) :=
take (H₁ : inhabited A) (H₂ : inhabited B),
inhabited.destruct H₁ (λa, inhabited.destruct H₂ (λb, inhabited.mk (pair a b)))
protected definition has_decidable_eq [instance] : decidable_eq A → decidable_eq B → decidable_eq (A × B) :=
take (H₁ : decidable_eq A) (H₂ : decidable_eq B) (u v : A × B),
have H₃ : u = v ↔ (pr₁ u = pr₁ v) ∧ (pr₂ u = pr₂ v), from
iff.intro
(assume H, H ▸ and.intro rfl rfl)
(assume H, and.elim H (assume H₄ H₅, equal H₄ H₅)),
decidable_iff_equiv _ (iff.symm H₃)
-- ### flip operation
definition flip (a : A × B) : B × A := pair (pr2 a) (pr1 a)
theorem flip_def (a : A × B) : flip a = pair (pr2 a) (pr1 a) := rfl
theorem flip_pair (a : A) (b : B) : flip (pair a b) = pair b a := rfl
theorem flip_pr1 (a : A × B) : pr1 (flip a) = pr2 a := rfl
theorem flip_pr2 (a : A × B) : pr2 (flip a) = pr1 a := rfl
theorem flip_flip (a : A × B) : flip (flip a) = a :=
destruct a (take x y, rfl)
theorem P_flip {P : A → B → Prop} (a : A × B) (H : P (pr1 a) (pr2 a))
: P (pr2 (flip a)) (pr1 (flip a)) :=
(flip_pr1 a)⁻¹ ▸ (flip_pr2 a)⁻¹ ▸ H
theorem flip_inj {a b : A × B} (H : flip a = flip b) : a = b :=
have H2 : flip (flip a) = flip (flip b), from congr_arg flip H,
show a = b, from (flip_flip a) ▸ (flip_flip b) ▸ H2
-- ### coordinatewise unary maps
definition map_pair (f : A → B) (a : A × A) : B × B :=
pair (f (pr1 a)) (f (pr2 a))
theorem map_pair_def (f : A → B) (a : A × A)
: map_pair f a = pair (f (pr1 a)) (f (pr2 a)) :=
rfl
theorem map_pair_pair (f : A → B) (a a' : A)
: map_pair f (pair a a') = pair (f a) (f a') :=
(pr1.mk a a') ▸ (pr2.mk a a') ▸ rfl
theorem map_pair_pr1 (f : A → B) (a : A × A) : pr1 (map_pair f a) = f (pr1 a) :=
!pr1.mk
theorem map_pair_pr2 (f : A → B) (a : A × A) : pr2 (map_pair f a) = f (pr2 a) :=
!pr2.mk
-- ### coordinatewise binary maps
definition map_pair2 {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) : C × C :=
pair (f (pr1 a) (pr1 b)) (f (pr2 a) (pr2 b))
theorem map_pair2_def {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) :
map_pair2 f a b = pair (f (pr1 a) (pr1 b)) (f (pr2 a) (pr2 b)) := rfl
theorem map_pair2_pair {A B C : Type} (f : A → B → C) (a a' : A) (b b' : B) :
map_pair2 f (pair a a') (pair b b') = pair (f a b) (f a' b') :=
calc
map_pair2 f (pair a a') (pair b b')
= pair (f (pr1 (pair a a')) b) (f (pr2 (pair a a')) (pr2 (pair b b')))
: {pr1.mk b b'}
... = pair (f (pr1 (pair a a')) b) (f (pr2 (pair a a')) b') : {pr2.mk b b'}
... = pair (f (pr1 (pair a a')) b) (f a' b') : {pr2.mk a a'}
... = pair (f a b) (f a' b') : {pr1.mk a a'}
theorem map_pair2_pr1 {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) :
pr1 (map_pair2 f a b) = f (pr1 a) (pr1 b) := !pr1.mk
theorem map_pair2_pr2 {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) :
pr2 (map_pair2 f a b) = f (pr2 a) (pr2 b) := !pr2.mk
theorem map_pair2_flip {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) :
flip (map_pair2 f a b) = map_pair2 f (flip a) (flip b) :=
have Hx : pr1 (flip (map_pair2 f a b)) = pr1 (map_pair2 f (flip a) (flip b)), from
calc
pr1 (flip (map_pair2 f a b)) = pr2 (map_pair2 f a b) : flip_pr1 _
... = f (pr2 a) (pr2 b) : map_pair2_pr2 f a b
... = f (pr1 (flip a)) (pr2 b) : {(flip_pr1 a)⁻¹}
... = f (pr1 (flip a)) (pr1 (flip b)) : {(flip_pr1 b)⁻¹}
... = pr1 (map_pair2 f (flip a) (flip b)) : (map_pair2_pr1 f _ _)⁻¹,
have Hy : pr2 (flip (map_pair2 f a b)) = pr2 (map_pair2 f (flip a) (flip b)), from
calc
pr2 (flip (map_pair2 f a b)) = pr1 (map_pair2 f a b) : flip_pr2 _
... = f (pr1 a) (pr1 b) : map_pair2_pr1 f a b
... = f (pr2 (flip a)) (pr1 b) : {flip_pr2 a}
... = f (pr2 (flip a)) (pr2 (flip b)) : {flip_pr2 b}
... = pr2 (map_pair2 f (flip a) (flip b)) : (map_pair2_pr2 f _ _)⁻¹,
pair_eq Hx Hy
end prod