142 lines
3.7 KiB
Text
142 lines
3.7 KiB
Text
-- 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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-- Authors: Leonardo de Moura, Jeremy Avigad
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-- logic.connectives.eq
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-- ====================
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-- Equality.
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import .prop
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-- eq
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-- --
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inductive eq {A : Type} (a : A) : A → Prop :=
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refl : eq a a
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infix `=` := eq
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notation `rfl` := eq.refl _
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namespace eq
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theorem id_refl {A : Type} {a : A} (H1 : a = a) : H1 = (eq.refl a) :=
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rfl
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theorem irrel {A : Type} {a b : A} (H1 H2 : a = b) : H1 = H2 :=
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rfl
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theorem subst {A : Type} {a b : A} {P : A → Prop} (H1 : a = b) (H2 : P a) : P b :=
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rec H2 H1
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theorem trans {A : Type} {a b c : A} (H1 : a = b) (H2 : b = c) : a = c :=
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subst H2 H1
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theorem symm {A : Type} {a b : A} (H : a = b) : b = a :=
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subst H (refl a)
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end eq
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calc_subst eq.subst
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calc_refl eq.refl
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calc_trans eq.trans
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namespace eq_ops
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postfix `⁻¹` := eq.symm
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infixr `⬝` := eq.trans
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infixr `▸` := eq.subst
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end eq_ops
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open eq_ops
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namespace eq
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-- eq_rec with arguments swapped, for transporting an element of a dependent type
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definition rec_on {A : Type} {a1 a2 : A} {B : A → Type} (H1 : a1 = a2) (H2 : B a1) : B a2 :=
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eq.rec H2 H1
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theorem rec_on_id {A : Type} {a : A} {B : A → Type} (H : a = a) (b : B a) : rec_on H b = b :=
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refl (rec_on rfl b)
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theorem rec_id {A : Type} {a : A} {B : A → Type} (H : a = a) (b : B a) : rec b H = b :=
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rec_on_id H b
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theorem rec_on_compose {A : Type} {a b c : A} {P : A → Type} (H1 : a = b) (H2 : b = c)
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(u : P a) :
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rec_on H2 (rec_on H1 u) = rec_on (trans H1 H2) u :=
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(show ∀(H2 : b = c), rec_on H2 (rec_on H1 u) = rec_on (trans H1 H2) u,
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from rec_on H2 (take (H2 : b = b), rec_on_id H2 _))
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H2
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end eq
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theorem congr_fun {A : Type} {B : A → Type} {f g : Π x, B x} (H : f = g) (a : A) : f a = g a :=
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H ▸ eq.refl (f a)
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theorem congr_arg {A : Type} {B : Type} {a b : A} (f : A → B) (H : a = b) : f a = f b :=
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H ▸ eq.refl (f a)
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theorem congr {A : Type} {B : Type} {f g : A → B} {a b : A} (H1 : f = g) (H2 : a = b) :
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f a = g b :=
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H1 ▸ H2 ▸ eq.refl (f a)
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theorem equal_f {A : Type} {B : A → Type} {f g : Π x, B x} (H : f = g) : ∀x, f x = g x :=
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take x, congr_fun H x
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theorem eqmp {a b : Prop} (H1 : a = b) (H2 : a) : b :=
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H1 ▸ H2
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theorem eqmpr {a b : Prop} (H1 : a = b) (H2 : b) : a :=
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H1⁻¹ ▸ H2
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theorem eq_true_elim {a : Prop} (H : a = true) : a :=
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H⁻¹ ▸ trivial
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theorem eq_false_elim {a : Prop} (H : a = false) : ¬a :=
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assume Ha : a, H ▸ Ha
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theorem imp_trans {a b c : Prop} (H1 : a → b) (H2 : b → c) : a → c :=
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assume Ha, H2 (H1 Ha)
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theorem imp_eq_trans {a b c : Prop} (H1 : a → b) (H2 : b = c) : a → c :=
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assume Ha, H2 ▸ (H1 Ha)
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theorem eq_imp_trans {a b c : Prop} (H1 : a = b) (H2 : b → c) : a → c :=
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assume Ha, H2 (H1 ▸ Ha)
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-- ne
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-- --
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definition ne [inline] {A : Type} (a b : A) := ¬(a = b)
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infix `≠` := ne
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namespace ne
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theorem intro {A : Type} {a b : A} (H : a = b → false) : a ≠ b :=
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H
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theorem elim {A : Type} {a b : A} (H1 : a ≠ b) (H2 : a = b) : false :=
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H1 H2
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theorem irrefl {A : Type} {a : A} (H : a ≠ a) : false :=
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H rfl
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theorem symm {A : Type} {a b : A} (H : a ≠ b) : b ≠ a :=
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assume H1 : b = a, H (H1⁻¹)
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end ne
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theorem a_neq_a_elim {A : Type} {a : A} (H : a ≠ a) : false :=
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H rfl
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theorem eq_ne_trans {A : Type} {a b c : A} (H1 : a = b) (H2 : b ≠ c) : a ≠ c :=
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H1⁻¹ ▸ H2
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theorem ne_eq_trans {A : Type} {a b c : A} (H1 : a ≠ b) (H2 : b = c) : a ≠ c :=
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H2 ▸ H1
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calc_trans eq_ne_trans
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calc_trans ne_eq_trans
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theorem p_ne_false {p : Prop} (Hp : p) : p ≠ false :=
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assume Heq : p = false, Heq ▸ Hp
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theorem p_ne_true {p : Prop} (Hnp : ¬p) : p ≠ true :=
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assume Heq : p = true, absurd trivial (Heq ▸ Hnp)
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theorem true_ne_false : ¬true = false :=
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assume H : true = false,
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H ▸ trivial
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