lean2/library/logic/core/identities.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.
-- Authors: Jeremy Avigad, Leonardo de Moura
-- logic.connectives.identities
-- ============================
-- Useful logical identities. In the absence of propositional extensionality, some of the
-- calculations use the type class support provided by logic.connectives.instances
import logic.core.instances logic.classes.decidable logic.core.quantifiers logic.core.cast
open relation decidable relation.iff_ops
theorem or_right_comm (a b c : Prop) : (a b) c ↔ (a c) b :=
calc
(a b) c ↔ a (b c) : or_assoc
... ↔ a (c b) : {or_comm}
... ↔ (a c) b : iff_symm or_assoc
theorem or_left_comm (a b c : Prop) : a (b c)↔ b (a c) :=
calc
a (b c) ↔ (a b) c : iff_symm or_assoc
... ↔ (b a) c : {or_comm}
... ↔ b (a c) : or_assoc
theorem and_right_comm (a b c : Prop) : (a ∧ b) ∧ c ↔ (a ∧ c) ∧ b :=
calc
(a ∧ b) ∧ c ↔ a ∧ (b ∧ c) : and_assoc
... ↔ a ∧ (c ∧ b) : {and_comm}
... ↔ (a ∧ c) ∧ b : iff_symm and_assoc
theorem and_left_comm (a b c : Prop) : a ∧ (b ∧ c)↔ b ∧ (a ∧ c) :=
calc
a ∧ (b ∧ c) ↔ (a ∧ b) ∧ c : iff_symm and_assoc
... ↔ (b ∧ a) ∧ c : {and_comm}
... ↔ b ∧ (a ∧ c) : and_assoc
theorem not_not_iff {a : Prop} {D : decidable a} : (¬¬a) ↔ a :=
iff_intro
(assume H : ¬¬a,
by_cases (assume H' : a, H') (assume H' : ¬a, absurd H' H))
(assume H : a, assume H', H' H)
theorem not_not_elim {a : Prop} {D : decidable a} (H : ¬¬a) : a :=
iff_mp not_not_iff H
theorem not_true : (¬true) ↔ false :=
iff_intro (assume H, H trivial) false_elim
theorem not_false : (¬false) ↔ true :=
iff_intro (assume H, trivial) (assume H H', H')
theorem not_or {a b : Prop} {Da : decidable a} {Db : decidable b} : (¬(a b)) ↔ (¬a ∧ ¬b) :=
iff_intro
(assume H, or_elim (em a)
(assume Ha, absurd (or_inl Ha) H)
(assume Hna, or_elim (em b)
(assume Hb, absurd (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} {Da : decidable a} {Db : decidable b} : (¬(a ∧ b)) ↔ (¬a ¬b) :=
iff_intro
(assume H, or_elim (em a)
(assume Ha, or_elim (em b)
(assume Hb, absurd (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} {Da : decidable a} : (a → b) ↔ (¬a b) :=
iff_intro
(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_iff⁻¹ ▸ Ha))
theorem not_implies {a b : Prop} {Da : decidable a} {Db : decidable b} : (¬(a → b)) ↔ (a ∧ ¬b) :=
calc (¬(a → b)) ↔ (¬(¬a b)) : {imp_or}
... ↔ (¬¬a ∧ ¬b) : not_or
... ↔ (a ∧ ¬b) : {not_not_iff}
theorem peirce {a b : Prop} {D : decidable a} : ((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)
theorem not_exists_forall {A : Type} {P : A → Prop} {D : ∀x, decidable (P x)}
(H : ¬∃x, P x) : ∀x, ¬P x :=
-- TODO: when type class instances can use quantifiers, we can use write em
take x, or_elim (@em _ (D x))
(assume Hp : P x, absurd (exists_intro x Hp) H)
(assume Hn : ¬P x, Hn)
theorem not_forall_exists {A : Type} {P : A → Prop} {D : ∀x, decidable (P x)}
{D' : decidable (∃x, ¬P x)} (H : ¬∀x, P x) :
∃x, ¬P x :=
@by_contradiction _ D' (assume H1 : ¬∃x, ¬P x,
have H2 : ∀x, ¬¬P x, from @not_exists_forall _ _ (take x, not_decidable (D x)) H1,
have H3 : ∀x, P x, from take x, @not_not_elim _ (D x) (H2 x),
absurd H3 H)
theorem iff_true_intro {a : Prop} (H : a) : a ↔ true :=
iff_intro
(assume H1 : a, trivial)
(assume H2 : true, H)
theorem iff_false_intro {a : Prop} (H : ¬a) : a ↔ false :=
iff_intro
(assume H1 : a, absurd H1 H)
(assume H2 : false, false_elim H2)
theorem a_neq_a {A : Type} (a : A) : (a ≠ a) ↔ false :=
iff_intro
(assume H, a_neq_a_elim H)
(assume H, false_elim H)
theorem eq_id {A : Type} (a : A) : (a = a) ↔ true :=
iff_true_intro (refl a)
theorem heq_id {A : Type} (a : A) : (a == a) ↔ true :=
iff_true_intro (hrefl a)
theorem a_iff_not_a (a : Prop) : (a ↔ ¬a) ↔ false :=
iff_intro
(assume H,
have H' : ¬a, from assume Ha, (H ▸ Ha) Ha,
H' (H⁻¹ ▸ H'))
(assume H, false_elim H)
theorem true_eq_false : (true ↔ false) ↔ false :=
not_true ▸ (a_iff_not_a true)
theorem false_eq_true : (false ↔ true) ↔ false :=
not_false ▸ (a_iff_not_a false)
theorem a_eq_true (a : Prop) : (a ↔ true) ↔ a :=
iff_intro (assume H, iff_true_elim H) (assume H, iff_true_intro H)
theorem a_eq_false (a : Prop) : (a ↔ false) ↔ ¬a :=
iff_intro (assume H, iff_false_elim H) (assume H, iff_false_intro H)