lean2/library/logic/instances.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.
Module: logic.instances
Author: Jeremy Avigad
Class instances for iff and eq.
-/
import logic.connectives algebra.relation
namespace relation
/- logical equivalence relations -/
theorem is_equivalence_eq [instance] (T : Type) : relation.is_equivalence (@eq T) :=
relation.is_equivalence.mk (@eq.refl T) (@eq.symm T) (@eq.trans T)
theorem is_equivalence_iff [instance] : relation.is_equivalence iff :=
relation.is_equivalence.mk @iff.refl @iff.symm @iff.trans
/- congruences for logic operations -/
theorem is_congruence_not : is_congruence iff iff not :=
is_congruence.mk
(take a b,
assume H : a ↔ b, iff.intro
(assume H1 : ¬a, assume H2 : b, H1 (iff.elim_right H H2))
(assume H1 : ¬b, assume H2 : a, H1 (iff.elim_left H H2)))
theorem is_congruence_and : is_congruence2 iff iff iff and :=
is_congruence2.mk
(take a1 b1 a2 b2,
assume H1 : a1 ↔ b1, assume H2 : a2 ↔ b2,
iff.intro
(assume H3 : a1 ∧ a2, and_of_and_of_imp_of_imp H3 (iff.elim_left H1) (iff.elim_left H2))
(assume H3 : b1 ∧ b2, and_of_and_of_imp_of_imp H3 (iff.elim_right H1) (iff.elim_right H2)))
theorem is_congruence_or : is_congruence2 iff iff iff or :=
is_congruence2.mk
(take a1 b1 a2 b2,
assume H1 : a1 ↔ b1, assume H2 : a2 ↔ b2,
iff.intro
(assume H3 : a1 a2, or_of_or_of_imp_of_imp H3 (iff.elim_left H1) (iff.elim_left H2))
(assume H3 : b1 b2, or_of_or_of_imp_of_imp H3 (iff.elim_right H1) (iff.elim_right H2)))
theorem is_congruence_imp : is_congruence2 iff iff iff imp :=
is_congruence2.mk
(take a1 b1 a2 b2,
assume H1 : a1 ↔ b1, assume H2 : a2 ↔ b2,
iff.intro
(assume H3 : a1 → a2, assume Hb1 : b1, iff.elim_left H2 (H3 ((iff.elim_right H1) Hb1)))
(assume H3 : b1 → b2, assume Ha1 : a1, iff.elim_right H2 (H3 ((iff.elim_left H1) Ha1))))
theorem is_congruence_iff : is_congruence2 iff iff iff iff :=
is_congruence2.mk
(take a1 b1 a2 b2,
assume H1 : a1 ↔ b1, assume H2 : a2 ↔ b2,
iff.intro
(assume H3 : a1 ↔ a2, iff.trans (iff.symm H1) (iff.trans H3 H2))
(assume H3 : b1 ↔ b2, iff.trans H1 (iff.trans H3 (iff.symm H2))))
definition is_congruence_not_compose [instance] := is_congruence.compose is_congruence_not
definition is_congruence_and_compose [instance] := is_congruence.compose21 is_congruence_and
definition is_congruence_or_compose [instance] := is_congruence.compose21 is_congruence_or
definition is_congruence_implies_compose [instance] := is_congruence.compose21 is_congruence_imp
definition is_congruence_iff_compose [instance] := is_congruence.compose21 is_congruence_iff
/- a general substitution operation with respect to an arbitrary congruence -/
namespace general_subst
theorem subst {T : Type} (R : T → T → Prop) ⦃P : T → Prop⦄ [C : is_congruence R iff P]
{a b : T} (H : R a b) (H1 : P a) : P b := iff.elim_left (is_congruence.app C H) H1
end general_subst
/- iff can be coerced to implication -/
definition mp_like_iff [instance] : relation.mp_like iff :=
relation.mp_like.mk (λa b (H : a ↔ b), iff.elim_left H)
/- support for calculations with iff -/
namespace iff
theorem subst {P : Prop → Prop} [C : is_congruence iff iff P] {a b : Prop}
(H : a ↔ b) (H1 : P a) : P b :=
@general_subst.subst Prop iff P C a b H H1
end iff
calc_subst iff.subst
namespace iff_ops
notation H ⁻¹ := iff.symm H
notation H1 ⬝ H2 := iff.trans H1 H2
notation H1 ▸ H2 := iff.subst H1 H2
definition refl := iff.refl
definition symm := @iff.symm
definition trans := @iff.trans
definition subst := @iff.subst
definition mp := @iff.mp
end iff_ops
end relation