165 lines
5.5 KiB
Text
165 lines
5.5 KiB
Text
----------------------------------------------------------------------------------------------------
|
||
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
||
-- Released under Apache 2.0 license as described in the file LICENSE.
|
||
-- Author: Leonardo de Moura
|
||
----------------------------------------------------------------------------------------------------
|
||
|
||
import logic.connectives.basic logic.connectives.quantifiers logic.connectives.cast
|
||
|
||
using eq_proofs
|
||
|
||
axiom prop_complete (a : Prop) : a = true ∨ a = false
|
||
|
||
theorem case (P : Prop → Prop) (H1 : P true) (H2 : P false) (a : Prop) : P a :=
|
||
or_elim (prop_complete a)
|
||
(assume Ht : a = true, Ht⁻¹ ▸ H1)
|
||
(assume Hf : a = false, Hf⁻¹ ▸ H2)
|
||
|
||
theorem em (a : Prop) : a ∨ ¬a :=
|
||
or_elim (prop_complete a)
|
||
(assume Ht : a = true, or_inl (eqt_elim Ht))
|
||
(assume Hf : a = false, or_inr (eqf_elim Hf))
|
||
|
||
theorem prop_complete_swapped (a : Prop) : a = false ∨ a = true :=
|
||
case (λ x, x = false ∨ x = true)
|
||
(or_inr (refl true))
|
||
(or_inl (refl false))
|
||
a
|
||
|
||
theorem not_true : (¬true) = false :=
|
||
have aux : (¬true) ≠ true, from
|
||
assume H : (¬true) = true,
|
||
absurd_not_true (H⁻¹ ▸ trivial),
|
||
resolve_right (prop_complete (¬true)) aux
|
||
|
||
theorem not_false : (¬false) = true :=
|
||
have aux : (¬false) ≠ false, from
|
||
assume H : (¬false) = false,
|
||
H ▸ not_false_trivial,
|
||
resolve_right (prop_complete_swapped (¬false)) aux
|
||
|
||
theorem not_not_eq (a : Prop) : (¬¬a) = a :=
|
||
case (λ x, (¬¬x) = x)
|
||
(calc (¬¬true) = (¬false) : {not_true}
|
||
... = true : not_false)
|
||
(calc (¬¬false) = (¬true) : {not_false}
|
||
... = false : not_true)
|
||
a
|
||
|
||
theorem not_not_elim {a : Prop} (H : ¬¬a) : a :=
|
||
(not_not_eq a) ◂ H
|
||
|
||
theorem propext {a b : Prop} (Hab : a → b) (Hba : b → a) : a = b :=
|
||
or_elim (prop_complete a)
|
||
(assume Hat, or_elim (prop_complete b)
|
||
(assume Hbt, Hat ⬝ Hbt⁻¹)
|
||
(assume Hbf, false_elim (a = b) (Hbf ▸ (Hab (eqt_elim Hat)))))
|
||
(assume Haf, or_elim (prop_complete b)
|
||
(assume Hbt, false_elim (a = b) (Haf ▸ (Hba (eqt_elim Hbt))))
|
||
(assume Hbf, Haf ⬝ Hbf⁻¹))
|
||
|
||
theorem iff_to_eq {a b : Prop} (H : a ↔ b) : a = b :=
|
||
iff_elim (assume H1 H2, propext H1 H2) H
|
||
|
||
theorem iff_eq_eq {a b : Prop} : (a ↔ b) = (a = b) :=
|
||
propext
|
||
(assume H, iff_to_eq H)
|
||
(assume H, eq_to_iff H)
|
||
|
||
theorem eqt_intro {a : Prop} (H : a) : a = true :=
|
||
propext
|
||
(assume H1 : a, trivial)
|
||
(assume H2 : true, H)
|
||
|
||
theorem eqf_intro {a : Prop} (H : ¬a) : a = false :=
|
||
propext
|
||
(assume H1 : a, absurd H1 H)
|
||
(assume H2 : false, false_elim a H2)
|
||
|
||
theorem by_contradiction {a : Prop} (H : ¬a → false) : a :=
|
||
or_elim (em a)
|
||
(assume H1 : a, H1)
|
||
(assume H1 : ¬a, false_elim a (H H1))
|
||
|
||
theorem a_neq_a {A : Type} (a : A) : (a ≠ a) = false :=
|
||
propext
|
||
(assume H, a_neq_a_elim H)
|
||
(assume H, false_elim (a ≠ a) H)
|
||
|
||
theorem eq_id {A : Type} (a : A) : (a = a) = true :=
|
||
eqt_intro (refl a)
|
||
|
||
theorem heq_id {A : Type} (a : A) : (a == a) = true :=
|
||
eqt_intro (hrefl a)
|
||
|
||
theorem not_or (a b : Prop) : (¬(a ∨ b)) = (¬a ∧ ¬b) :=
|
||
propext
|
||
(assume H, or_elim (em a)
|
||
(assume Ha, absurd_elim (¬a ∧ ¬b) (or_inl Ha) H)
|
||
(assume Hna, or_elim (em b)
|
||
(assume Hb, absurd_elim (¬a ∧ ¬b) (or_inr Hb) H)
|
||
(assume Hnb, and_intro Hna Hnb)))
|
||
(assume (H : ¬a ∧ ¬b) (N : a ∨ b),
|
||
or_elim N
|
||
(assume Ha, absurd Ha (and_elim_left H))
|
||
(assume Hb, absurd Hb (and_elim_right H)))
|
||
|
||
theorem not_and (a b : Prop) : (¬(a ∧ b)) = (¬a ∨ ¬b) :=
|
||
propext
|
||
(assume H, or_elim (em a)
|
||
(assume Ha, or_elim (em b)
|
||
(assume Hb, absurd_elim (¬a ∨ ¬b) (and_intro Ha Hb) H)
|
||
(assume Hnb, or_inr Hnb))
|
||
(assume Hna, or_inl Hna))
|
||
(assume (H : ¬a ∨ ¬b) (N : a ∧ b),
|
||
or_elim H
|
||
(assume Hna, absurd (and_elim_left N) Hna)
|
||
(assume Hnb, absurd (and_elim_right N) Hnb))
|
||
|
||
theorem imp_or (a b : Prop) : (a → b) = (¬a ∨ b) :=
|
||
propext
|
||
(assume H : a → b, (or_elim (em a)
|
||
(assume Ha : a, or_inr (H Ha))
|
||
(assume Hna : ¬a, or_inl Hna)))
|
||
(assume (H : ¬a ∨ b) (Ha : a),
|
||
resolve_right H ((not_not_eq a)⁻¹ ◂ Ha))
|
||
|
||
theorem not_implies (a b : Prop) : (¬(a → b)) = (a ∧ ¬b) :=
|
||
calc (¬(a → b)) = (¬(¬a ∨ b)) : {imp_or a b}
|
||
... = (¬¬a ∧ ¬b) : not_or (¬a) b
|
||
... = (a ∧ ¬b) : {not_not_eq a}
|
||
|
||
theorem a_eq_not_a (a : Prop) : (a = ¬a) = false :=
|
||
propext
|
||
(assume H, or_elim (em a)
|
||
(assume Ha, absurd Ha (H ▸ Ha))
|
||
(assume Hna, absurd (H⁻¹ ▸ Hna) Hna))
|
||
(assume H, false_elim (a = ¬a) H)
|
||
|
||
theorem true_eq_false : (true = false) = false :=
|
||
not_true ▸ (a_eq_not_a true)
|
||
|
||
theorem false_eq_true : (false = true) = false :=
|
||
not_false ▸ (a_eq_not_a false)
|
||
|
||
theorem a_eq_true (a : Prop) : (a = true) = a :=
|
||
propext (assume H, eqt_elim H) (assume H, eqt_intro H)
|
||
|
||
theorem a_eq_false (a : Prop) : (a = false) = ¬a :=
|
||
propext (assume H, eqf_elim H) (assume H, eqf_intro H)
|
||
|
||
theorem not_exists_forall {A : Type} {P : A → Prop} (H : ¬∃x, P x) : ∀x, ¬P x :=
|
||
take x, or_elim (em (P x))
|
||
(assume Hp : P x, absurd_elim (¬P x) (exists_intro x Hp) H)
|
||
(assume Hn : ¬P x, Hn)
|
||
|
||
theorem not_forall_exists {A : Type} {P : A → Prop} (H : ¬∀x, P x) : ∃x, ¬P x :=
|
||
by_contradiction (assume H1 : ¬∃ x, ¬P x,
|
||
have H2 : ∀x, ¬¬P x, from not_exists_forall H1,
|
||
have H3 : ∀x, P x, from take x, not_not_elim (H2 x),
|
||
absurd H3 H)
|
||
|
||
theorem peirce (a b : Prop) : ((a → b) → a) → a :=
|
||
assume H, by_contradiction (assume Hna : ¬a,
|
||
have Hnna : ¬¬a, from not_implies_left (mt H Hna),
|
||
absurd (not_not_elim Hnna) Hna)
|