lean2/library/data/prod.lean

/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.

Module: data.prod
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] [h₁ : inhabited A] [h₂ : inhabited B] : inhabited (prod A B) :=
  inhabited.mk (default A, default B)

  protected definition has_decidable_eq [instance] [h₁ : decidable_eq A] [h₂ : decidable_eq B] : decidable_eq (A × B) :=
  take (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_of_decidable_of_iff _ (iff.symm H₃)
end prod
-												refactor(library/data): remove folders with a single file

											
										
										
											2014-12-23 20:35:06 +00:00
+								/-
 								Copyright (c) 2014 Microsoft Corporation. All rights reserved.
 								Released under Apache 2.0 license as described in the file LICENSE.
 								Module: data.prod
 								Author: Leonardo de Moura, Jeremy Avigad
 								-/
-												refactor(library): add 'init' folder

											
										
										
											2014-12-01 04:34:12 +00:00
+								import logic.eq
-												refactor(library/data/prod): cleanup

											
										
										
											2014-10-05 20:38:08 +00:00
+								open inhabited decidable eq.ops
-												feat(library/standard): port int, and reorganize a lot

											
										
										
											2014-08-20 02:32:44 +00:00
-												feat(library): add definition of subsingleton and some other minor changes

											
										
										
											2014-10-09 01:41:18 +00:00
+								namespace prod
-												refactor(data.prod): move theorems about products from data.quotient.util to data.prod.thms

											
										
										
											2014-11-21 04:21:45 +00:00
+								  variables {A B : Type} {a₁ a₂ : A} {b₁ b₂ : B} {u : A × B}
-												refactor(library): add 'eq' and 'ne' namespaces

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>

											
										
										
											2014-09-05 01:41:06 +00:00
-												refactor(library/data/prod): cleanup

											
										
										
											2014-10-05 20:38:08 +00:00
+								  theorem pair_eq : a₁ = a₂ → b₁ = b₂ → (a₁, b₁) = (a₂, b₂) :=
 								  assume H1 H2, H1 ▸ H2 ▸ rfl
-												feat(library/standard/data/quotient): import quotient from lean 0.1

											
										
										
											2014-08-17 21:41:23 +00:00
-												refactor(library/data/prod): cleanup

											
										
										
											2014-10-05 20:38:08 +00:00
+								  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))
-												feat(library/standard): add sigma types and subtypes, make inhabited constructive

											
										
										
											2014-08-15 03:12:54 +00:00
-												refactor(library): use `[]` binder annotation when declaring instances

											
										
										
											2015-02-24 23:25:02 +00:00
+								  protected definition is_inhabited [instance] [h₁ : inhabited A] [h₂ : inhabited B] : inhabited (prod A B) :=
 								  inhabited.mk (default A, default B)
-												feat(library/standard): add sigma types and subtypes, make inhabited constructive

											
										
										
											2014-08-15 03:12:54 +00:00
-												refactor(library): use `[]` binder annotation when declaring instances

											
										
										
											2015-02-24 23:25:02 +00:00
+								  protected definition has_decidable_eq [instance] [h₁ : decidable_eq A] [h₂ : decidable_eq B] : decidable_eq (A × B) :=
 								  take (u v : A × B),
-												refactor(library/data/prod): cleanup

											
										
										
											2014-10-05 20:38:08 +00:00
+								    have H₃ : u = v ↔ (pr₁ u = pr₁ v) ∧ (pr₂ u = pr₂ v), from
-												refactor(library): add namespaces 'or', 'and' and 'iff'

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>

											
										
										
											2014-09-05 04:25:21 +00:00
+								      iff.intro
-												refactor(library): add 'eq' and 'ne' namespaces

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>

											
										
										
											2014-09-05 01:41:06 +00:00
+								        (assume H, H ▸ and.intro rfl rfl)
-												refactor(library/data/prod): cleanup

											
										
										
											2014-10-05 20:38:08 +00:00
+								        (assume H, and.elim H (assume H₄ H₅, equal H₄ H₅)),
-												refactor(library): rename theorems "iff.flip_sign -> not_iff_not_of_iff" and "decidable_iff_equiv -> decidable_of_decidable_of_iff"

											
										
										
											2014-12-16 03:17:51 +00:00
+								    decidable_of_decidable_of_iff _ (iff.symm H₃)
-												feat(library/standard): port int, and reorganize a lot

											
										
										
											2014-08-20 02:32:44 +00:00
+								end prod