272 lines
8.9 KiB
Text
272 lines
8.9 KiB
Text
/-
|
||
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
Authors: Jeremy Avigad, Leonardo de Moura
|
||
|
||
The propositional connectives. See also init.datatypes and init.logic.
|
||
-/
|
||
open eq.ops
|
||
|
||
variables {a b c d : Prop}
|
||
|
||
/- false -/
|
||
|
||
theorem false.elim {c : Prop} (H : false) : c :=
|
||
false.rec c H
|
||
|
||
/- implies -/
|
||
|
||
definition imp (a b : Prop) : Prop := a → b
|
||
|
||
theorem mt (H1 : a → b) (H2 : ¬b) : ¬a :=
|
||
assume Ha : a, absurd (H1 Ha) H2
|
||
|
||
theorem imp_true (a : Prop) : (a → true) ↔ true :=
|
||
iff.intro (assume H, trivial) (assume H H1, trivial)
|
||
|
||
theorem true_imp (a : Prop) : (true → a) ↔ a :=
|
||
iff.intro (assume H, H trivial) (assume H H1, H)
|
||
|
||
theorem imp_false (a : Prop) : (a → false) ↔ ¬ a := iff.rfl
|
||
|
||
theorem false_imp (a : Prop) : (false → a) ↔ true :=
|
||
iff.intro (assume H, trivial) (assume H H1, false.elim H1)
|
||
|
||
/- not -/
|
||
|
||
theorem not.elim (H1 : ¬a) (H2 : a) : false := H1 H2
|
||
|
||
theorem not.intro (H : a → false) : ¬a := H
|
||
|
||
theorem not_not_intro (Ha : a) : ¬¬a :=
|
||
assume Hna : ¬a, absurd Ha Hna
|
||
|
||
theorem not_not_of_not_implies (H : ¬(a → b)) : ¬¬a :=
|
||
assume Hna : ¬a, absurd (assume Ha : a, absurd Ha Hna) H
|
||
|
||
theorem not_of_not_implies (H : ¬(a → b)) : ¬b :=
|
||
assume Hb : b, absurd (assume Ha : a, Hb) H
|
||
|
||
theorem not_not_em : ¬¬(a ∨ ¬a) :=
|
||
assume not_em : ¬(a ∨ ¬a),
|
||
have Hnp : ¬a, from
|
||
assume Hp : a, absurd (or.inl Hp) not_em,
|
||
absurd (or.inr Hnp) not_em
|
||
|
||
theorem not_true : ¬ true ↔ false :=
|
||
iff.intro (assume H, H trivial) (assume H, false.elim H)
|
||
|
||
theorem not_false_iff : ¬ false ↔ true :=
|
||
iff.intro (assume H, trivial) (assume H H1, H1)
|
||
|
||
/- and -/
|
||
|
||
definition not_and_of_not_left (b : Prop) (Hna : ¬a) : ¬(a ∧ b) :=
|
||
assume H : a ∧ b, absurd (and.elim_left H) Hna
|
||
|
||
definition not_and_of_not_right (a : Prop) {b : Prop} (Hnb : ¬b) : ¬(a ∧ b) :=
|
||
assume H : a ∧ b, absurd (and.elim_right H) Hnb
|
||
|
||
theorem and.swap (H : a ∧ b) : b ∧ a :=
|
||
and.intro (and.elim_right H) (and.elim_left H)
|
||
|
||
theorem and_of_and_of_imp_of_imp (H₁ : a ∧ b) (H₂ : a → c) (H₃ : b → d) : c ∧ d :=
|
||
and.elim H₁ (assume Ha : a, assume Hb : b, and.intro (H₂ Ha) (H₃ Hb))
|
||
|
||
theorem and_of_and_of_imp_left (H₁ : a ∧ c) (H : a → b) : b ∧ c :=
|
||
and.elim H₁ (assume Ha : a, assume Hc : c, and.intro (H Ha) Hc)
|
||
|
||
theorem and_of_and_of_imp_right (H₁ : c ∧ a) (H : a → b) : c ∧ b :=
|
||
and.elim H₁ (assume Hc : c, assume Ha : a, and.intro Hc (H Ha))
|
||
|
||
theorem and.comm : a ∧ b ↔ b ∧ a :=
|
||
iff.intro (λH, and.swap H) (λH, and.swap H)
|
||
|
||
theorem and.assoc : (a ∧ b) ∧ c ↔ a ∧ (b ∧ c) :=
|
||
iff.intro
|
||
(assume H,
|
||
obtain [Ha Hb] Hc, from H,
|
||
and.intro Ha (and.intro Hb Hc))
|
||
(assume H,
|
||
obtain Ha Hb Hc, from H,
|
||
and.intro (and.intro Ha Hb) Hc)
|
||
|
||
theorem and_true (a : Prop) : a ∧ true ↔ a :=
|
||
iff.intro (assume H, and.left H) (assume H, and.intro H trivial)
|
||
|
||
theorem true_and (a : Prop) : true ∧ a ↔ a :=
|
||
iff.intro (assume H, and.right H) (assume H, and.intro trivial H)
|
||
|
||
theorem and_false (a : Prop) : a ∧ false ↔ false :=
|
||
iff.intro (assume H, and.right H) (assume H, false.elim H)
|
||
|
||
theorem false_and (a : Prop) : false ∧ a ↔ false :=
|
||
iff.intro (assume H, and.left H) (assume H, false.elim H)
|
||
|
||
theorem and_self (a : Prop) : a ∧ a ↔ a :=
|
||
iff.intro (assume H, and.left H) (assume H, and.intro H H)
|
||
|
||
/- or -/
|
||
|
||
definition not_or (Hna : ¬a) (Hnb : ¬b) : ¬(a ∨ b) :=
|
||
assume H : a ∨ b, or.rec_on H
|
||
(assume Ha, absurd Ha Hna)
|
||
(assume Hb, absurd Hb Hnb)
|
||
|
||
theorem or_of_or_of_imp_of_imp (H₁ : a ∨ b) (H₂ : a → c) (H₃ : b → d) : c ∨ d :=
|
||
or.elim H₁
|
||
(assume Ha : a, or.inl (H₂ Ha))
|
||
(assume Hb : b, or.inr (H₃ Hb))
|
||
|
||
theorem or_of_or_of_imp_left (H₁ : a ∨ c) (H : a → b) : b ∨ c :=
|
||
or.elim H₁
|
||
(assume H₂ : a, or.inl (H H₂))
|
||
(assume H₂ : c, or.inr H₂)
|
||
|
||
theorem or_of_or_of_imp_right (H₁ : c ∨ a) (H : a → b) : c ∨ b :=
|
||
or.elim H₁
|
||
(assume H₂ : c, or.inl H₂)
|
||
(assume H₂ : a, or.inr (H H₂))
|
||
|
||
theorem or.elim3 (H : a ∨ b ∨ c) (Ha : a → d) (Hb : b → d) (Hc : c → d) : d :=
|
||
or.elim H Ha (assume H₂, or.elim H₂ Hb Hc)
|
||
|
||
theorem or_resolve_right (H₁ : a ∨ b) (H₂ : ¬a) : b :=
|
||
or.elim H₁ (assume Ha, absurd Ha H₂) (assume Hb, Hb)
|
||
|
||
theorem or_resolve_left (H₁ : a ∨ b) (H₂ : ¬b) : a :=
|
||
or.elim H₁ (assume Ha, Ha) (assume Hb, absurd Hb H₂)
|
||
|
||
theorem or.swap (H : a ∨ b) : b ∨ a :=
|
||
or.elim H (assume Ha, or.inr Ha) (assume Hb, or.inl Hb)
|
||
|
||
theorem or.comm : a ∨ b ↔ b ∨ a :=
|
||
iff.intro (λH, or.swap H) (λH, or.swap H)
|
||
|
||
theorem or.assoc : (a ∨ b) ∨ c ↔ a ∨ (b ∨ c) :=
|
||
iff.intro
|
||
(assume H, or.elim H
|
||
(assume H₁, or.elim H₁
|
||
(assume Ha, or.inl Ha)
|
||
(assume Hb, or.inr (or.inl Hb)))
|
||
(assume Hc, or.inr (or.inr Hc)))
|
||
(assume H, or.elim H
|
||
(assume Ha, (or.inl (or.inl Ha)))
|
||
(assume H₁, or.elim H₁
|
||
(assume Hb, or.inl (or.inr Hb))
|
||
(assume Hc, or.inr Hc)))
|
||
|
||
theorem or_true (a : Prop) : a ∨ true ↔ true :=
|
||
iff.intro (assume H, trivial) (assume H, or.inr H)
|
||
|
||
theorem true_or (a : Prop) : true ∨ a ↔ true :=
|
||
iff.intro (assume H, trivial) (assume H, or.inl H)
|
||
|
||
theorem or_false (a : Prop) : a ∨ false ↔ a :=
|
||
iff.intro
|
||
(assume H, or.elim H (assume H1 : a, H1) (assume H1 : false, false.elim H1))
|
||
(assume H, or.inl H)
|
||
|
||
theorem false_or (a : Prop) : false ∨ a ↔ a :=
|
||
iff.intro
|
||
(assume H, or.elim H (assume H1 : false, false.elim H1) (assume H1 : a, H1))
|
||
(assume H, or.inr H)
|
||
|
||
theorem or_self (a : Prop) : a ∨ a ↔ a :=
|
||
iff.intro
|
||
(assume H, or.elim H (assume H1, H1) (assume H1, H1))
|
||
(assume H, or.inl H)
|
||
|
||
/- distributivity -/
|
||
|
||
theorem and.distrib_left (a b c : Prop) : a ∧ (b ∨ c) ↔ (a ∧ b) ∨ (a ∧ c) :=
|
||
iff.intro
|
||
(assume H, or.elim (and.right H)
|
||
(assume Hb : b, or.inl (and.intro (and.left H) Hb))
|
||
(assume Hc : c, or.inr (and.intro (and.left H) Hc)))
|
||
(assume H, or.elim H
|
||
(assume Hab, and.intro (and.left Hab) (or.inl (and.right Hab)))
|
||
(assume Hac, and.intro (and.left Hac) (or.inr (and.right Hac))))
|
||
|
||
theorem and.distrib_right (a b c : Prop) : (a ∨ b) ∧ c ↔ (a ∧ c) ∨ (b ∧ c) :=
|
||
propext (!and.comm) ▸ propext (!and.comm) ▸ propext (!and.comm) ▸ !and.distrib_left
|
||
|
||
theorem or.distrib_left (a b c : Prop) : a ∨ (b ∧ c) ↔ (a ∨ b) ∧ (a ∨ c) :=
|
||
iff.intro
|
||
(assume H, or.elim H
|
||
(assume Ha, and.intro (or.inl Ha) (or.inl Ha))
|
||
(assume Hbc, and.intro (or.inr (and.left Hbc)) (or.inr (and.right Hbc))))
|
||
(assume H, or.elim (and.left H)
|
||
(assume Ha, or.inl Ha)
|
||
(assume Hb, or.elim (and.right H)
|
||
(assume Ha, or.inl Ha)
|
||
(assume Hc, or.inr (and.intro Hb Hc))))
|
||
|
||
theorem or.distrib_right (a b c : Prop) : (a ∧ b) ∨ c ↔ (a ∨ c) ∧ (b ∨ c) :=
|
||
propext (!or.comm) ▸ propext (!or.comm) ▸ propext (!or.comm) ▸ !or.distrib_left
|
||
|
||
/- iff -/
|
||
|
||
definition iff.def : (a ↔ b) = ((a → b) ∧ (b → a)) :=
|
||
!eq.refl
|
||
|
||
theorem iff_true (a : Prop) : (a ↔ true) ↔ a :=
|
||
iff.intro
|
||
(assume H, iff.mp' H trivial)
|
||
(assume H, iff.intro (assume H1, trivial) (assume H1, H))
|
||
|
||
theorem true_iff (a : Prop) : (true ↔ a) ↔ a :=
|
||
iff.intro
|
||
(assume H, iff.mp H trivial)
|
||
(assume H, iff.intro (assume H1, H) (assume H1, trivial))
|
||
|
||
theorem iff_false (a : Prop) : (a ↔ false) ↔ ¬ a :=
|
||
iff.intro
|
||
(assume H, and.left H)
|
||
(assume H, and.intro H (assume H1, false.elim H1))
|
||
|
||
theorem false_iff (a : Prop) : (false ↔ a) ↔ ¬ a :=
|
||
iff.intro
|
||
(assume H, and.right H)
|
||
(assume H, and.intro (assume H1, false.elim H1) H)
|
||
|
||
theorem iff_true_of_self (Ha : a) : a ↔ true :=
|
||
iff.intro (assume H, trivial) (assume H, Ha)
|
||
|
||
theorem iff_self (a : Prop) : (a ↔ a) ↔ true :=
|
||
iff_true_of_self !iff.refl
|
||
|
||
/- if-then-else -/
|
||
|
||
section
|
||
open eq.ops
|
||
|
||
variables {A : Type} {c₁ c₂ : Prop}
|
||
|
||
definition if_true (t e : A) : (if true then t else e) = t :=
|
||
if_pos trivial
|
||
|
||
definition if_false (t e : A) : (if false then t else e) = e :=
|
||
if_neg not_false
|
||
|
||
theorem if_congr_cond [H₁ : decidable c₁] [H₂ : decidable c₂] (Heq : c₁ ↔ c₂) (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 [H₁ : decidable c₁] [H₂ : decidable c₂] {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_congr_cond Hc t₁ e₁)
|
||
|
||
theorem if_congr [H₁ : decidable c₁] {t₁ t₂ e₁ e₂ : A} (Hc : c₁ ↔ c₂) (Ht : t₁ = t₂)
|
||
(He : e₁ = e₂) :
|
||
(if c₁ then t₁ else e₁) = (@ite c₂ (decidable_of_decidable_of_iff H₁ Hc) A t₂ e₂) :=
|
||
assert H2 : decidable c₂, from (decidable_of_decidable_of_iff H₁ Hc),
|
||
if_congr_aux Hc Ht He
|
||
end
|