2014-12-01 04:34:12 +00:00
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/-
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Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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2014-12-15 21:43:42 +00:00
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Module init.relation
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2014-12-01 04:34:12 +00:00
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Authors: Leonardo de Moura
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-/
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prelude
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import init.logic
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2014-11-19 01:32:22 +00:00
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-- TODO(Leo): remove duplication between this file and algebra/relation.lean
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-- We need some of the following definitions asap when "initializing" Lean.
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variables {A B : Type} (R : B → B → Prop)
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2015-01-26 19:49:08 +00:00
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local infix `≺`:50 := R
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2014-11-19 01:32:22 +00:00
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definition reflexive := ∀x, x ≺ x
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definition symmetric := ∀⦃x y⦄, x ≺ y → y ≺ x
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definition transitive := ∀⦃x y z⦄, x ≺ y → y ≺ z → x ≺ z
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2015-04-01 02:56:05 +00:00
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definition equivalence := reflexive R ∧ symmetric R ∧ transitive R
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2014-11-19 01:32:22 +00:00
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definition irreflexive := ∀x, ¬ x ≺ x
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definition anti_symmetric := ∀⦃x y⦄, x ≺ y → y ≺ x → x = y
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definition empty_relation := λa₁ a₂ : A, false
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definition subrelation (Q R : B → B → Prop) := ∀⦃x y⦄, Q x y → R x y
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definition inv_image (f : A → B) : A → A → Prop :=
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λa₁ a₂, f a₁ ≺ f a₂
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theorem inv_image.trans (f : A → B) (H : transitive R) : transitive (inv_image R f) :=
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λ (a₁ a₂ a₃ : A) (H₁ : inv_image R f a₁ a₂) (H₂ : inv_image R f a₂ a₃), H H₁ H₂
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theorem inv_image.irreflexive (f : A → B) (H : irreflexive R) : irreflexive (inv_image R f) :=
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λ (a : A) (H₁ : inv_image R f a a), H (f a) H₁
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inductive tc {A : Type} (R : A → A → Prop) : A → A → Prop :=
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2015-02-26 01:00:10 +00:00
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| base : ∀a b, R a b → tc R a b
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| trans : ∀a b c, tc R a b → tc R b c → tc R a c
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