lean2/library/standard/classical.lean

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-- 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
axiom boolcomplete (a : Bool) : a = true a = false
theorem case (P : Bool → Bool) (H1 : P true) (H2 : P false) (a : Bool) : P a
:= or_elim (boolcomplete a)
(assume Ht : a = true, subst (symm Ht) H1)
(assume Hf : a = false, subst (symm Hf) H2)
theorem em (a : Bool) : a ¬ a
:= or_elim (boolcomplete a)
(assume Ht : a = true, or_intro_left (¬ a) (eqt_elim Ht))
(assume Hf : a = false, or_intro_right a (eqf_elim Hf))
theorem boolcomplete_swapped (a : Bool) : a = false a = true
:= case (λ x, x = false x = true)
(or_intro_right (true = false) (refl true))
(or_intro_left (false = true) (refl false))
a
theorem not_true : (¬ true) = false
:= have aux : ¬ (¬ true) = true, from
not_intro (assume H : (¬ true) = true,
absurd_not_true (subst (symm H) trivial)),
resolve_right (boolcomplete (¬ true)) aux
theorem not_false : (¬ false) = true
:= have aux : ¬ (¬ false) = false, from
not_intro (assume H : (¬ false) = false,
subst H not_false_trivial),
resolve_right (boolcomplete_swapped (¬ false)) aux
theorem not_not_eq (a : Bool) : (¬ ¬ 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 : Bool} (H : ¬ ¬ a) : a
:= (not_not_eq a) ◂ H
theorem boolext {a b : Bool} (Hab : a → b) (Hba : b → a) : a = b
:= or_elim (boolcomplete a)
(λ Hat : a = true, or_elim (boolcomplete b)
(λ Hbt : b = true, trans Hat (symm Hbt))
(λ Hbf : b = false, false_elim (a = b) (subst Hbf (Hab (eqt_elim Hat)))))
(λ Haf : a = false, or_elim (boolcomplete b)
(λ Hbt : b = true, false_elim (a = b) (subst Haf (Hba (eqt_elim Hbt))))
(λ Hbf : b = false, trans Haf (symm Hbf)))
theorem iff_to_eq {a b : Bool} (H : a ↔ b) : a = b
:= iff_elim (assume H1 H2, boolext H1 H2) H
theorem iff_eq_eq {a b : Bool} : (a ↔ b) = (a = b)
:= boolext
(assume H, iff_to_eq H)
(assume H, eq_to_iff H)
theorem eqt_intro {a : Bool} (H : a) : a = true
:= boolext (assume H1 : a, trivial)
(assume H2 : true, H)
theorem eqf_intro {a : Bool} (H : ¬ a) : a = false
:= boolext (assume H1 : a, absurd H1 H)
(assume H2 : false, false_elim a H2)
theorem by_contradiction {a : Bool} (H : ¬ a → false) : a
:= or_elim (em a) (λ H1 : a, H1) (λ H1 : ¬ a, false_elim a (H H1))
theorem a_neq_a {A : Type} (a : A) : (a ≠ a) = false
:= boolext (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 : Bool) : (¬ (a b)) = (¬ a ∧ ¬ b)
:= boolext
(assume H, or_elim (em a)
(assume Ha, absurd_elim (¬ a ∧ ¬ b) (or_intro_left b Ha) H)
(assume Hna, or_elim (em b)
(assume Hb, absurd_elim (¬ a ∧ ¬ b) (or_intro_right a Hb) H)
(assume Hnb, and_intro Hna Hnb)))
(assume (H : ¬ a ∧ ¬ b), not_intro (assume (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 : Bool) : (¬ (a ∧ b)) = (¬ a ¬ b)
:= boolext
(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_intro_right (¬ a) Hnb))
(assume Hna, or_intro_left (¬ b) Hna))
(assume (H : ¬ a ¬ b), not_intro (assume (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 : Bool) : (a → b) = (¬ a b)
:= boolext
(assume H : a → b,
(or_elim (em a)
(λ Ha : a, or_intro_right (¬ a) (H Ha))
(λ Hna : ¬ a, or_intro_left b Hna)))
(assume H : ¬ a b,
assume Ha : a,
resolve_right H ((symm (not_not_eq a)) ◂ Ha))
theorem not_implies (a b : Bool) : (¬ (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 : Bool) : (a = ¬ a) = false
:= boolext
(assume H, or_elim (em a)
(assume Ha, absurd Ha (subst H Ha))
(assume Hna, absurd (subst (symm H) Hna) Hna))
(assume H, false_elim (a = ¬ a) H)
theorem true_eq_false : (true = false) = false
:= subst not_true (a_eq_not_a true)
theorem false_eq_true : (false = true) = false
:= subst not_false (a_eq_not_a false)
theorem a_eq_true (a : Bool) : (a = true) = a
:= boolext
(assume H, eqt_elim H)
(assume H, eqt_intro H)
theorem a_eq_false (a : Bool) : (a = false) = (¬ a)
:= boolext
(assume H, eqf_elim H)
(assume H, eqf_intro H)