lean2/library/standard/logic/connectives/eq.lean

106 lines
3 KiB
Text
Raw Normal View History

----------------------------------------------------------------------------------------------------
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Authors: Leonardo de Moura, Jeremy Avigad
----------------------------------------------------------------------------------------------------
import .basic
-- eq
-- --
inductive eq {A : Type} (a : A) : A → Prop :=
| refl : eq a a
infix `=`:50 := eq
theorem subst {A : Type} {a b : A} {P : A → Prop} (H1 : a = b) (H2 : P a) : P b :=
eq_rec H2 H1
theorem trans {A : Type} {a b c : A} (H1 : a = b) (H2 : b = c) : a = c :=
subst H2 H1
calc_subst subst
calc_refl refl
calc_trans trans
theorem true_ne_false : ¬true = false :=
assume H : true = false,
subst H trivial
theorem symm {A : Type} {a b : A} (H : a = b) : b = a :=
subst H (refl a)
namespace eq_proofs
postfix `⁻¹`:100 := symm
infixr `⬝`:75 := trans
infixr `▸`:75 := subst
end
using eq_proofs
theorem congr1 {A : Type} {B : A → Type} {f g : Π x, B x} (H : f = g) (a : A) : f a = g a :=
H ▸ refl (f a)
theorem congr2 {A : Type} {B : Type} {a b : A} (f : A → B) (H : a = b) : f a = f b :=
H ▸ refl (f a)
theorem congr {A : Type} {B : Type} {f g : A → B} {a b : A} (H1 : f = g) (H2 : a = b) : f a = g b :=
H1 ▸ H2 ▸ refl (f a)
theorem equal_f {A : Type} {B : A → Type} {f g : Π x, B x} (H : f = g) : ∀x, f x = g x :=
take x, congr1 H x
theorem not_congr {a b : Prop} (H : a = b) : (¬a) = (¬b) :=
congr2 not H
theorem eqmp {a b : Prop} (H1 : a = b) (H2 : a) : b :=
H1 ▸ H2
infixl `<|`:100 := eqmp
infixl `◂`:100 := eqmp
theorem eqmpr {a b : Prop} (H1 : a = b) (H2 : b) : a :=
H1⁻¹ ◂ H2
theorem eqt_elim {a : Prop} (H : a = true) : a :=
H⁻¹ ◂ trivial
theorem eqf_elim {a : Prop} (H : a = false) : ¬a :=
assume Ha : a, H ◂ Ha
theorem imp_trans {a b c : Prop} (H1 : a → b) (H2 : b → c) : a → c :=
assume Ha, H2 (H1 Ha)
theorem imp_eq_trans {a b c : Prop} (H1 : a → b) (H2 : b = c) : a → c :=
assume Ha, H2 ◂ (H1 Ha)
theorem eq_imp_trans {a b c : Prop} (H1 : a = b) (H2 : b → c) : a → c :=
assume Ha, H2 (H1 ◂ Ha)
theorem eq_to_iff {a b : Prop} (H : a = b) : a ↔ b :=
iff_intro (λ Ha, H ▸ Ha) (λ Hb, H⁻¹ ▸ Hb)
-- ne
-- --
definition ne [inline] {A : Type} (a b : A) := ¬(a = b)
infix `≠`:50 := ne
theorem ne_intro {A : Type} {a b : A} (H : a = b → false) : a ≠ b := H
theorem ne_elim {A : Type} {a b : A} (H1 : a ≠ b) (H2 : a = b) : false := H1 H2
theorem a_neq_a_elim {A : Type} {a : A} (H : a ≠ a) : false := H (refl a)
theorem ne_irrefl {A : Type} {a : A} (H : a ≠ a) : false := H (refl a)
theorem ne_symm {A : Type} {a b : A} (H : a ≠ b) : b ≠ a :=
assume H1 : b = a, H (H1⁻¹)
theorem eq_ne_trans {A : Type} {a b c : A} (H1 : a = b) (H2 : b ≠ c) : a ≠ c := H1⁻¹ ▸ H2
theorem ne_eq_trans {A : Type} {a b c : A} (H1 : a ≠ b) (H2 : b = c) : a ≠ c := H2 ▸ H1
calc_trans eq_ne_trans
calc_trans ne_eq_trans