341 lines
11 KiB
Text
341 lines
11 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
|
|
-/
|
|
prelude
|
|
import init.datatypes init.reserved_notation
|
|
|
|
definition not (a : Type) := a → empty
|
|
prefix `¬` := not
|
|
|
|
definition absurd {a : Type} {b : Type} (H₁ : a) (H₂ : ¬a) : b :=
|
|
empty.rec (λ e, b) (H₂ H₁)
|
|
|
|
theorem mt {a b : Type} (H₁ : a → b) (H₂ : ¬b) : ¬a :=
|
|
assume Ha : a, absurd (H₁ Ha) H₂
|
|
|
|
-- not
|
|
-- ---
|
|
|
|
theorem not_empty : ¬ empty :=
|
|
assume H : empty, H
|
|
|
|
theorem not_not_intro {a : Type} (Ha : a) : ¬¬a :=
|
|
assume Hna : ¬a, absurd Ha Hna
|
|
|
|
theorem not_intro {a : Type} (H : a → empty) : ¬a := H
|
|
|
|
theorem not_elim {a : Type} (H₁ : ¬a) (H₂ : a) : empty := H₁ H₂
|
|
|
|
theorem not_implies_left {a b : Type} (H : ¬(a → b)) : ¬¬a :=
|
|
assume Hna : ¬a, absurd (assume Ha : a, absurd Ha Hna) H
|
|
|
|
theorem not_implies_right {a b : Type} (H : ¬(a → b)) : ¬b :=
|
|
assume Hb : b, absurd (assume Ha : a, Hb) H
|
|
|
|
-- eq
|
|
-- --
|
|
|
|
notation a = b := eq a b
|
|
definition rfl {A : Type} {a : A} := eq.refl a
|
|
|
|
namespace eq
|
|
variables {A : Type}
|
|
variables {a b c a': A}
|
|
|
|
theorem subst {P : A → Type} (H₁ : a = b) (H₂ : P a) : P b :=
|
|
rec H₂ H₁
|
|
|
|
theorem trans (H₁ : a = b) (H₂ : b = c) : a = c :=
|
|
subst H₂ H₁
|
|
|
|
definition symm (H : a = b) : b = a :=
|
|
subst H (refl a)
|
|
|
|
namespace ops
|
|
notation H `⁻¹` := symm H --input with \sy or \-1 or \inv
|
|
notation H1 ⬝ H2 := trans H1 H2
|
|
notation H1 ▸ H2 := subst H1 H2
|
|
end ops
|
|
end eq
|
|
|
|
calc_subst eq.subst
|
|
calc_refl eq.refl
|
|
calc_trans eq.trans
|
|
calc_symm eq.symm
|
|
|
|
-- ne
|
|
-- --
|
|
|
|
definition ne {A : Type} (a b : A) := ¬(a = b)
|
|
notation a ≠ b := ne a b
|
|
|
|
namespace ne
|
|
open eq.ops
|
|
variable {A : Type}
|
|
variables {a b : A}
|
|
|
|
theorem intro : (a = b → empty) → a ≠ b :=
|
|
assume H, H
|
|
|
|
theorem elim : a ≠ b → a = b → empty :=
|
|
assume H₁ H₂, H₁ H₂
|
|
|
|
theorem irrefl : a ≠ a → empty :=
|
|
assume H, H rfl
|
|
|
|
theorem symm : a ≠ b → b ≠ a :=
|
|
assume (H : a ≠ b) (H₁ : b = a), H (H₁⁻¹)
|
|
end ne
|
|
|
|
section
|
|
open eq.ops
|
|
variables {A : Type} {a b c : A}
|
|
|
|
theorem false.of_ne : a ≠ a → empty :=
|
|
assume H, H rfl
|
|
|
|
theorem ne.of_eq_of_ne : a = b → b ≠ c → a ≠ c :=
|
|
assume H₁ H₂, H₁⁻¹ ▸ H₂
|
|
|
|
theorem ne.of_ne_of_eq : a ≠ b → b = c → a ≠ c :=
|
|
assume H₁ H₂, H₂ ▸ H₁
|
|
end
|
|
|
|
calc_trans ne.of_eq_of_ne
|
|
calc_trans ne.of_ne_of_eq
|
|
|
|
-- iff
|
|
-- ---
|
|
|
|
definition iff (a b : Type) := prod (a → b) (b → a)
|
|
|
|
notation a <-> b := iff a b
|
|
notation a ↔ b := iff a b
|
|
|
|
namespace iff
|
|
variables {a b c : Type}
|
|
|
|
definition def : (a ↔ b) = (prod (a → b) (b → a)) :=
|
|
rfl
|
|
|
|
definition intro (H₁ : a → b) (H₂ : b → a) : a ↔ b :=
|
|
prod.mk H₁ H₂
|
|
|
|
definition elim (H₁ : (a → b) → (b → a) → c) (H₂ : a ↔ b) : c :=
|
|
prod.rec H₁ H₂
|
|
|
|
definition elim_left (H : a ↔ b) : a → b :=
|
|
elim (assume H₁ H₂, H₁) H
|
|
|
|
definition mp := @elim_left
|
|
|
|
definition elim_right (H : a ↔ b) : b → a :=
|
|
elim (assume H₁ H₂, H₂) H
|
|
|
|
definition flip_sign (H₁ : a ↔ b) : ¬a ↔ ¬b :=
|
|
intro
|
|
(assume Hna, mt (elim_right H₁) Hna)
|
|
(assume Hnb, mt (elim_left H₁) Hnb)
|
|
|
|
definition refl (a : Type) : a ↔ a :=
|
|
intro (assume H, H) (assume H, H)
|
|
|
|
definition rfl {a : Type} : a ↔ a :=
|
|
refl a
|
|
|
|
theorem trans (H₁ : a ↔ b) (H₂ : b ↔ c) : a ↔ c :=
|
|
intro
|
|
(assume Ha, elim_left H₂ (elim_left H₁ Ha))
|
|
(assume Hc, elim_right H₁ (elim_right H₂ Hc))
|
|
|
|
theorem symm (H : a ↔ b) : b ↔ a :=
|
|
intro
|
|
(assume Hb, elim_right H Hb)
|
|
(assume Ha, elim_left H Ha)
|
|
|
|
theorem true_elim (H : a ↔ unit) : a :=
|
|
mp (symm H) unit.star
|
|
|
|
theorem false_elim (H : a ↔ empty) : ¬a :=
|
|
assume Ha : a, mp H Ha
|
|
|
|
open eq.ops
|
|
theorem of_eq {a b : Type} (H : a = b) : a ↔ b :=
|
|
iff.intro (λ Ha, H ▸ Ha) (λ Hb, H⁻¹ ▸ Hb)
|
|
end iff
|
|
|
|
calc_refl iff.refl
|
|
calc_trans iff.trans
|
|
|
|
-- inhabited
|
|
-- ---------
|
|
|
|
inductive inhabited [class] (A : Type) : Type :=
|
|
mk : A → inhabited A
|
|
|
|
namespace inhabited
|
|
|
|
protected definition destruct {A : Type} {B : Type} (H1 : inhabited A) (H2 : A → B) : B :=
|
|
inhabited.rec H2 H1
|
|
|
|
definition fun_inhabited [instance] (A : Type) {B : Type} (H : inhabited B) : inhabited (A → B) :=
|
|
destruct H (λb, mk (λa, b))
|
|
|
|
definition dfun_inhabited [instance] (A : Type) {B : A → Type} (H : Πx, inhabited (B x)) :
|
|
inhabited (Πx, B x) :=
|
|
mk (λa, destruct (H a) (λb, b))
|
|
|
|
definition default (A : Type) [H : inhabited A] : A := destruct H (take a, a)
|
|
|
|
end inhabited
|
|
|
|
-- decidable
|
|
-- ---------
|
|
|
|
inductive decidable [class] (p : Type) : Type :=
|
|
inl : p → decidable p,
|
|
inr : ¬p → decidable p
|
|
|
|
namespace decidable
|
|
variables {p q : Type}
|
|
|
|
definition pos_witness [C : decidable p] (H : p) : p :=
|
|
rec_on C (λ Hp, Hp) (λ Hnp, absurd H Hnp)
|
|
|
|
definition neg_witness [C : decidable p] (H : ¬ p) : ¬ p :=
|
|
rec_on C (λ Hp, absurd Hp H) (λ Hnp, Hnp)
|
|
|
|
definition by_cases {q : Type} [C : decidable p] (Hpq : p → q) (Hnpq : ¬p → q) : q :=
|
|
rec_on C (assume Hp, Hpq Hp) (assume Hnp, Hnpq Hnp)
|
|
|
|
theorem em (p : Type) [H : decidable p] : sum p ¬p :=
|
|
by_cases (λ Hp, sum.inl Hp) (λ Hnp, sum.inr Hnp)
|
|
|
|
theorem by_contradiction [Hp : decidable p] (H : ¬p → empty) : p :=
|
|
by_cases
|
|
(assume H₁ : p, H₁)
|
|
(assume H₁ : ¬p, empty.rec (λ e, p) (H H₁))
|
|
|
|
definition decidable_iff_equiv (Hp : decidable p) (H : p ↔ q) : decidable q :=
|
|
rec_on Hp
|
|
(assume Hp : p, inl (iff.elim_left H Hp))
|
|
(assume Hnp : ¬p, inr (iff.elim_left (iff.flip_sign H) Hnp))
|
|
|
|
definition decidable_eq_equiv.{l} {p q : Type.{l}} (Hp : decidable p) (H : p = q) : decidable q :=
|
|
decidable_iff_equiv Hp (iff.of_eq H)
|
|
end decidable
|
|
|
|
section
|
|
variables {p q : Type}
|
|
open decidable (rec_on inl inr)
|
|
|
|
definition unit.decidable [instance] : decidable unit :=
|
|
inl unit.star
|
|
|
|
definition empty.decidable [instance] : decidable empty :=
|
|
inr not_empty
|
|
|
|
definition prod.decidable [instance] (Hp : decidable p) (Hq : decidable q) : decidable (prod p q) :=
|
|
rec_on Hp
|
|
(assume Hp : p, rec_on Hq
|
|
(assume Hq : q, inl (prod.mk Hp Hq))
|
|
(assume Hnq : ¬q, inr (λ H : prod p q, prod.rec_on H (λ Hp Hq, absurd Hq Hnq))))
|
|
(assume Hnp : ¬p, inr (λ H : prod p q, prod.rec_on H (λ Hp Hq, absurd Hp Hnp)))
|
|
|
|
definition sum.decidable [instance] (Hp : decidable p) (Hq : decidable q) : decidable (sum p q) :=
|
|
rec_on Hp
|
|
(assume Hp : p, inl (sum.inl Hp))
|
|
(assume Hnp : ¬p, rec_on Hq
|
|
(assume Hq : q, inl (sum.inr Hq))
|
|
(assume Hnq : ¬q, inr (λ H : sum p q, sum.rec_on H (λ Hp, absurd Hp Hnp) (λ Hq, absurd Hq Hnq))))
|
|
|
|
definition not.decidable [instance] (Hp : decidable p) : decidable (¬p) :=
|
|
rec_on Hp
|
|
(assume Hp, inr (not_not_intro Hp))
|
|
(assume Hnp, inl Hnp)
|
|
|
|
definition implies.decidable [instance] (Hp : decidable p) (Hq : decidable q) : decidable (p → q) :=
|
|
rec_on Hp
|
|
(assume Hp : p, rec_on Hq
|
|
(assume Hq : q, inl (assume H, Hq))
|
|
(assume Hnq : ¬q, inr (assume H : p → q, absurd (H Hp) Hnq)))
|
|
(assume Hnp : ¬p, inl (assume Hp, absurd Hp Hnp))
|
|
|
|
definition iff.decidable [instance] (Hp : decidable p) (Hq : decidable q) : decidable (p ↔ q) := _
|
|
end
|
|
|
|
definition decidable_pred {A : Type} (R : A → Type) := Π (a : A), decidable (R a)
|
|
definition decidable_rel {A : Type} (R : A → A → Type) := Π (a b : A), decidable (R a b)
|
|
definition decidable_eq (A : Type) := decidable_rel (@eq A)
|
|
|
|
definition ite (c : Type) [H : decidable c] {A : Type} (t e : A) : A :=
|
|
decidable.rec_on H (λ Hc, t) (λ Hnc, e)
|
|
|
|
definition if_pos {c : Type} [H : decidable c] (Hc : c) {A : Type} {t e : A} : (if c then t else e) = t :=
|
|
decidable.rec
|
|
(λ Hc : c, eq.refl (@ite c (decidable.inl Hc) A t e))
|
|
(λ Hnc : ¬c, absurd Hc Hnc)
|
|
H
|
|
|
|
definition if_neg {c : Type} [H : decidable c] (Hnc : ¬c) {A : Type} {t e : A} : (if c then t else e) = e :=
|
|
decidable.rec
|
|
(λ Hc : c, absurd Hc Hnc)
|
|
(λ Hnc : ¬c, eq.refl (@ite c (decidable.inr Hnc) A t e))
|
|
H
|
|
|
|
definition if_t_t (c : Type) [H : decidable c] {A : Type} (t : A) : (if c then t else t) = t :=
|
|
decidable.rec
|
|
(λ Hc : c, eq.refl (@ite c (decidable.inl Hc) A t t))
|
|
(λ Hnc : ¬c, eq.refl (@ite c (decidable.inr Hnc) A t t))
|
|
H
|
|
|
|
definition if_unit {A : Type} (t e : A) : (if unit then t else e) = t :=
|
|
if_pos unit.star
|
|
|
|
definition if_empty {A : Type} (t e : A) : (if empty then t else e) = e :=
|
|
if_neg not_empty
|
|
|
|
theorem if_cond_congr {c₁ c₂ : Type} [H₁ : decidable c₁] [H₂ : decidable c₂] (Heq : c₁ ↔ c₂) {A : Type} (t e : A)
|
|
: (if c₁ then t else e) = (if c₂ then t else e) :=
|
|
decidable.rec_on H₁
|
|
(λ Hc₁ : c₁, decidable.rec_on H₂
|
|
(λ Hc₂ : c₂, if_pos Hc₁ ⬝ (if_pos Hc₂)⁻¹)
|
|
(λ Hnc₂ : ¬c₂, absurd (iff.elim_left Heq Hc₁) Hnc₂))
|
|
(λ Hnc₁ : ¬c₁, decidable.rec_on H₂
|
|
(λ Hc₂ : c₂, absurd (iff.elim_right Heq Hc₂) Hnc₁)
|
|
(λ Hnc₂ : ¬c₂, if_neg Hnc₁ ⬝ (if_neg Hnc₂)⁻¹))
|
|
|
|
theorem if_congr_aux {c₁ c₂ : Type} [H₁ : decidable c₁] [H₂ : decidable c₂] {A : Type} {t₁ t₂ e₁ e₂ : A}
|
|
(Hc : c₁ ↔ c₂) (Ht : t₁ = t₂) (He : e₁ = e₂) :
|
|
(if c₁ then t₁ else e₁) = (if c₂ then t₂ else e₂) :=
|
|
Ht ▸ He ▸ (if_cond_congr Hc t₁ e₁)
|
|
|
|
theorem if_congr {c₁ c₂ : Type} [H₁ : decidable c₁] {A : Type} {t₁ t₂ e₁ e₂ : A} (Hc : c₁ ↔ c₂) (Ht : t₁ = t₂) (He : e₁ = e₂) :
|
|
(if c₁ then t₁ else e₁) = (@ite c₂ (decidable.decidable_iff_equiv H₁ Hc) A t₂ e₂) :=
|
|
have H2 [visible] : decidable c₂, from (decidable.decidable_iff_equiv H₁ Hc),
|
|
if_congr_aux Hc Ht He
|
|
|
|
|
|
-- We use "dependent" if-then-else to be able to communicate the if-then-else condition
|
|
-- to the branches
|
|
definition dite (c : Type) [H : decidable c] {A : Type} (t : c → A) (e : ¬ c → A) : A :=
|
|
decidable.rec_on H (λ Hc, t Hc) (λ Hnc, e Hnc)
|
|
|
|
definition dif_pos {c : Type} [H : decidable c] (Hc : c) {A : Type} {t : c → A} {e : ¬ c → A} : (if H : c then t H else e H) = t (decidable.pos_witness Hc) :=
|
|
decidable.rec
|
|
(λ Hc : c, eq.refl (@dite c (decidable.inl Hc) A t e))
|
|
(λ Hnc : ¬c, absurd Hc Hnc)
|
|
H
|
|
|
|
definition dif_neg {c : Type} [H : decidable c] (Hnc : ¬c) {A : Type} {t : c → A} {e : ¬ c → A} : (if H : c then t H else e H) = e (decidable.neg_witness Hnc) :=
|
|
decidable.rec
|
|
(λ Hc : c, absurd Hc Hnc)
|
|
(λ Hnc : ¬c, eq.refl (@dite c (decidable.inr Hnc) A t e))
|
|
H
|
|
|
|
-- Remark: dite and ite are "definitionally equal" when we ignore the proofs.
|
|
theorem dite_ite_eq (c : Type) [H : decidable c] {A : Type} (t : A) (e : A) : dite c (λh, t) (λh, e) = ite c t e :=
|
|
rfl
|