49 lines
1.7 KiB
Text
49 lines
1.7 KiB
Text
/-
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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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Module init.relation
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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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-- 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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universe variables u v
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variables {A : Type u} {B : Type v} (R : B → B → Prop)
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local infix `≺`:50 := R
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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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definition equivalence := reflexive R ∧ symmetric R ∧ transitive R
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definition total := ∀ x y, x ≺ y ∨ y ≺ x
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definition mk_equivalence (r : reflexive R) (s : symmetric R) (t : transitive R) : equivalence R :=
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⟨r, s, t⟩
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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 := λ a₁ 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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λ a₁ 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 u} (R : A → A → Prop) : A → A → Prop
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| base : ∀a b, R a b → tc a b
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| trans : ∀a b c, tc a b → tc b c → tc a c
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