209 lines
8 KiB
Text
209 lines
8 KiB
Text
-- 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, Floris van Doorn
|
|
|
|
-- logic.connectives.eq
|
|
-- ====================
|
|
|
|
-- Equality.
|
|
|
|
import logic.prop
|
|
|
|
-- eq
|
|
-- --
|
|
|
|
inductive eq {A : Type} (a : A) : A → Prop :=
|
|
refl : eq a a
|
|
|
|
infix `=` := eq
|
|
definition rfl {A : Type} {a : A} := eq.refl a
|
|
|
|
-- proof irrelevance is built in
|
|
theorem proof_irrel {a : Prop} {H1 H2 : a} : H1 = H2 := rfl
|
|
|
|
namespace eq
|
|
theorem id_refl {A : Type} {a : A} (H1 : a = a) : H1 = (eq.refl a) :=
|
|
proof_irrel
|
|
|
|
theorem irrel {A : Type} {a b : A} (H1 H2 : a = b) : H1 = H2 :=
|
|
proof_irrel
|
|
|
|
theorem subst {A : Type} {a b : A} {P : A → Prop} (H1 : a = b) (H2 : P a) : P b :=
|
|
rec H2 H1
|
|
|
|
theorem trans {A : Type} {a b c : A} (H1 : a = b) (H2 : b = c) : a = c :=
|
|
subst H2 H1
|
|
|
|
theorem symm {A : Type} {a b : A} (H : a = b) : b = a :=
|
|
subst H (refl a)
|
|
end eq
|
|
|
|
calc_subst eq.subst
|
|
calc_refl eq.refl
|
|
calc_trans eq.trans
|
|
|
|
namespace eq_ops
|
|
postfix `⁻¹` := eq.symm
|
|
reserve infixr `⬝`:75 infixr `⬝` := eq.trans
|
|
infixr `▸` := eq.subst
|
|
end eq_ops
|
|
open eq_ops
|
|
|
|
namespace eq
|
|
-- eq_rec with arguments swapped, for transporting an element of a dependent type
|
|
|
|
-- definition rec_on {A : Type} {a1 a2 : A} {B : A → Type} (H1 : a1 = a2) (H2 : B a1) : B a2 :=
|
|
-- eq.rec H2 H1
|
|
|
|
definition drec_on {A : Type} {a a' : A} {B : Πa' : A, a = a' → Type} (H1 : a = a') (H2 : B a (refl a)) : B a' H1 :=
|
|
eq.rec (λH1 : a = a, show B a H1, from H2) H1 H1
|
|
|
|
theorem drec_on_id {A : Type} {a : A} {B : Πa' : A, a = a' → Type} (H : a = a) (b : B a H) : drec_on H b = b :=
|
|
refl (drec_on rfl b)
|
|
|
|
theorem drec_on_constant {A : Type} {a a' : A} {B : Type} (H : a = a') (b : B) : drec_on H b = b :=
|
|
drec_on H (λ(H' : a = a), drec_on_id H' b) H
|
|
|
|
theorem drec_on_constant2 {A : Type} {a₁ a₂ a₃ a₄ : A} {B : Type} (H₁ : a₁ = a₂) (H₂ : a₃ = a₄) (b : B) : drec_on H₁ b = drec_on H₂ b :=
|
|
drec_on_constant H₁ b ⬝ drec_on_constant H₂ b ⁻¹
|
|
|
|
theorem drec_on_irrel {A B : Type} {a a' : A} {f : A → B} {D : B → Type} (H : a = a') (H' : f a = f a') (b : D (f a)) : drec_on H b = drec_on H' b :=
|
|
drec_on H (λ(H : a = a) (H' : f a = f a), drec_on_id H b ⬝ drec_on_id H' b⁻¹) H H'
|
|
|
|
theorem rec_id {A : Type} {a : A} {B : A → Type} (H : a = a) (b : B a) : rec b H = b :=
|
|
id_refl H⁻¹ ▸ refl (eq.rec b (refl a))
|
|
|
|
theorem drec_on_compose {A : Type} {a b c : A} {P : A → Type} (H1 : a = b) (H2 : b = c)
|
|
(u : P a) :
|
|
drec_on H2 (drec_on H1 u) = drec_on (trans H1 H2) u :=
|
|
(show ∀(H2 : b = c), drec_on H2 (drec_on H1 u) = drec_on (trans H1 H2) u,
|
|
from drec_on H2 (take (H2 : b = b), drec_on_id H2 _))
|
|
H2
|
|
end eq
|
|
|
|
open eq
|
|
|
|
theorem congr_fun {A : Type} {B : A → Type} {f g : Π x, B x} (H : f = g) (a : A) : f a = g a :=
|
|
H ▸ rfl
|
|
|
|
theorem congr_arg {A : Type} {B : Type} {a b : A} (f : A → B) (H : a = b) : f a = f b :=
|
|
H ▸ rfl
|
|
|
|
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 ▸ rfl
|
|
|
|
theorem congr_arg2 {A B C : Type} {a a' : A} {b b' : B} (f : A → B → C) (Ha : a = a') (Hb : b = b') : f a b = f a' b' :=
|
|
congr (congr_arg f Ha) Hb
|
|
|
|
theorem congr_arg3 {A B C D : Type} {a a' : A} {b b' : B} {c c' : C} (f : A → B → C → D) (Ha : a = a') (Hb : b = b') (Hc : c = c') : f a b c = f a' b' c' :=
|
|
congr (congr_arg2 f Ha Hb) Hc
|
|
|
|
theorem congr_arg4 {A B C D E : Type} {a a' : A} {b b' : B} {c c' : C} {d d' : D} (f : A → B → C → D → E) (Ha : a = a') (Hb : b = b') (Hc : c = c') (Hd : d = d') : f a b c d = f a' b' c' d' :=
|
|
congr (congr_arg3 f Ha Hb Hc) Hd
|
|
|
|
theorem congr_arg5 {A B C D E F : Type} {a a' : A} {b b' : B} {c c' : C} {d d' : D} {e e' : E} (f : A → B → C → D → E → F) (Ha : a = a') (Hb : b = b') (Hc : c = c') (Hd : d = d') (He : e = e') : f a b c d e = f a' b' c' d' e' :=
|
|
congr (congr_arg4 f Ha Hb Hc Hd) He
|
|
|
|
theorem congr2 {A B C : Type} {a a' : A} {b b' : B} (f f' : A → B → C) (Hf : f = f') (Ha : a = a') (Hb : b = b') : f a b = f' a' b' :=
|
|
Hf ▸ congr_arg2 f Ha Hb
|
|
|
|
theorem congr3 {A B C D : Type} {a a' : A} {b b' : B} {c c' : C} (f f' : A → B → C → D) (Hf : f = f') (Ha : a = a') (Hb : b = b') (Hc : c = c') : f a b c = f' a' b' c' :=
|
|
Hf ▸ congr_arg3 f Ha Hb Hc
|
|
|
|
theorem congr4 {A B C D E : Type} {a a' : A} {b b' : B} {c c' : C} {d d' : D} (f f' : A → B → C → D → E) (Hf : f = f') (Ha : a = a') (Hb : b = b') (Hc : c = c') (Hd : d = d') : f a b c d = f' a' b' c' d' :=
|
|
Hf ▸ congr_arg4 f Ha Hb Hc Hd
|
|
|
|
theorem congr5 {A B C D E F : Type} {a a' : A} {b b' : B} {c c' : C} {d d' : D} {e e' : E} (f f' : A → B → C → D → E → F) (Hf : f = f') (Ha : a = a') (Hb : b = b') (Hc : c = c') (Hd : d = d') (He : e = e') : f a b c d e = f' a' b' c' d' e' :=
|
|
Hf ▸ congr_arg5 f Ha Hb Hc Hd He
|
|
|
|
theorem congr_arg2_dep {A : Type} {B : A → Type} {C : Type} {a₁ a₂ : A}
|
|
{b₁ : B a₁} {b₂ : B a₂} (f : Πa, B a → C) (H₁ : a₁ = a₂) (H₂ : eq.drec_on H₁ b₁ = b₂) :
|
|
f a₁ b₁ = f a₂ b₂ :=
|
|
eq.drec_on H₁
|
|
(λ (b₂ : B a₁) (H₁ : a₁ = a₁) (H₂ : eq.drec_on H₁ b₁ = b₂),
|
|
calc
|
|
f a₁ b₁ = f a₁ (eq.drec_on H₁ b₁) : {(eq.drec_on_id H₁ b₁)⁻¹}
|
|
... = f a₁ b₂ : {H₂})
|
|
b₂ H₁ H₂
|
|
|
|
theorem congr_arg3_dep {A : Type} {B : A → Type} {C : Πa, B a → Type} {D : Type} {a₁ a₂ : A} {b₁ : B a₁} {b₂ : B a₂} {c₁ : C a₁ b₁} {c₂ : C a₂ b₂} (f : Πa b, C a b → D)
|
|
(H₁ : a₁ = a₂) (H₂ : eq.drec_on H₁ b₁ = b₂) (H₃ : eq.drec_on (congr_arg2_dep C H₁ H₂) c₁ = c₂) :
|
|
f a₁ b₁ c₁ = f a₂ b₂ c₂ :=
|
|
eq.drec_on H₁
|
|
(λ (b₂ : B a₁) (H₂ : b₁ = b₂) (c₂ : C a₁ b₂) (H₃ : _ = c₂),
|
|
have H₃' : eq.drec_on H₂ c₁ = c₂,
|
|
from (drec_on_irrel H₂ (congr_arg2_dep C (refl a₁) H₂) c₁⁻¹) ▸ H₃,
|
|
congr_arg2_dep (f a₁) H₂ H₃')
|
|
b₂ H₂ c₂ H₃
|
|
|
|
theorem congr_arg3_ndep_dep {A B : Type} {C : A → B → Type} {D : Type} {a₁ a₂ : A} {b₁ b₂ : B} {c₁ : C a₁ b₁} {c₂ : C a₂ b₂} (f : Πa b, C a b → D)
|
|
(H₁ : a₁ = a₂) (H₂ : b₁ = b₂) (H₃ : eq.drec_on (congr_arg2 C H₁ H₂) c₁ = c₂) :
|
|
f a₁ b₁ c₁ = f a₂ b₂ c₂ :=
|
|
congr_arg3_dep f H₁ (drec_on_constant H₁ b₁ ⬝ H₂) H₃
|
|
|
|
theorem equal_f {A : Type} {B : A → Type} {f g : Π x, B x} (H : f = g) : ∀x, f x = g x :=
|
|
take x, congr_fun H x
|
|
|
|
theorem eqmp {a b : Prop} (H1 : a = b) (H2 : a) : b :=
|
|
H1 ▸ H2
|
|
|
|
theorem eqmpr {a b : Prop} (H1 : a = b) (H2 : b) : a :=
|
|
H1⁻¹ ▸ H2
|
|
|
|
theorem eq_true_elim {a : Prop} (H : a = true) : a :=
|
|
H⁻¹ ▸ trivial
|
|
|
|
theorem eq_false_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)
|
|
|
|
-- ne
|
|
-- --
|
|
|
|
definition ne {A : Type} (a b : A) := ¬(a = b)
|
|
infix `≠` := ne
|
|
|
|
namespace ne
|
|
theorem intro {A : Type} {a b : A} (H : a = b → false) : a ≠ b :=
|
|
H
|
|
|
|
theorem elim {A : Type} {a b : A} (H1 : a ≠ b) (H2 : a = b) : false :=
|
|
H1 H2
|
|
|
|
theorem irrefl {A : Type} {a : A} (H : a ≠ a) : false :=
|
|
H rfl
|
|
|
|
theorem symm {A : Type} {a b : A} (H : a ≠ b) : b ≠ a :=
|
|
assume H1 : b = a, H (H1⁻¹)
|
|
end ne
|
|
|
|
theorem a_neq_a_elim {A : Type} {a : A} (H : a ≠ a) : false :=
|
|
H rfl
|
|
|
|
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
|
|
|
|
theorem p_ne_false {p : Prop} (Hp : p) : p ≠ false :=
|
|
assume Heq : p = false, Heq ▸ Hp
|
|
|
|
theorem p_ne_true {p : Prop} (Hnp : ¬p) : p ≠ true :=
|
|
assume Heq : p = true, absurd trivial (Heq ▸ Hnp)
|
|
|
|
theorem true_ne_false : ¬true = false :=
|
|
assume H : true = false,
|
|
H ▸ trivial
|