lean2/library/logic/core/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.
-- Author: Jeremy Avigad
-- logic.core.instances
-- ====================
import logic.core.connectives algebra.relation
namespace relation
open relation
-- Congruences for logic
-- ---------------------
theorem congruence_not : congruence iff iff not :=
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 congruence_and : congruence2 iff iff iff and :=
congruence2.mk
(take a1 b1 a2 b2,
assume H1 : a1 ↔ b1, assume H2 : a2 ↔ b2,
iff.intro
(assume H3 : a1 ∧ a2, and.imp_and H3 (iff.elim_left H1) (iff.elim_left H2))
(assume H3 : b1 ∧ b2, and.imp_and H3 (iff.elim_right H1) (iff.elim_right H2)))
theorem congruence_or : congruence2 iff iff iff or :=
congruence2.mk
(take a1 b1 a2 b2,
assume H1 : a1 ↔ b1, assume H2 : a2 ↔ b2,
iff.intro
(assume H3 : a1 a2, or.imp_or H3 (iff.elim_left H1) (iff.elim_left H2))
(assume H3 : b1 b2, or.imp_or H3 (iff.elim_right H1) (iff.elim_right H2)))
theorem congruence_imp : congruence2 iff iff iff imp :=
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 congruence_iff : congruence2 iff iff iff iff :=
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))))
-- theorem congruence_const_iff [instance] := congruence.const iff iff.refl
definition congruence_not_compose [instance] := congruence.compose congruence_not
definition congruence_and_compose [instance] := congruence.compose21 congruence_and
definition congruence_or_compose [instance] := congruence.compose21 congruence_or
definition congruence_implies_compose [instance] := congruence.compose21 congruence_imp
definition congruence_iff_compose [instance] := congruence.compose21 congruence_iff
-- Generalized substitution
-- ------------------------
-- TODO: note that the target has to be "iff". Otherwise, there is not enough
-- information to infer an mp-like relation.
namespace general_operations
theorem subst {T : Type} (R : T → T → Prop) ⦃P : T → Prop⦄ {C : congruence R iff P}
{a b : T} (H : R a b) (H1 : P a) : P b := iff.elim_left (congruence.app C H) H1
end general_operations
-- = is an equivalence relation
-- ----------------------------
theorem is_reflexive_eq [instance] (T : Type) : relation.is_reflexive (@eq T) :=
relation.is_reflexive.mk (@eq.refl T)
theorem is_symmetric_eq [instance] (T : Type) : relation.is_symmetric (@eq T) :=
relation.is_symmetric.mk (@eq.symm T)
theorem is_transitive_eq [instance] (T : Type) : relation.is_transitive (@eq T) :=
relation.is_transitive.mk (@eq.trans T)
-- TODO: this is only temporary, needed to inform Lean that is_equivalence is a class
theorem is_equivalence_eq [instance] (T : Type) : relation.is_equivalence (@eq T) :=
relation.is_equivalence.mk _ _ _
-- iff is an equivalence relation
-- ------------------------------
theorem is_reflexive_iff [instance] : relation.is_reflexive iff :=
relation.is_reflexive.mk (@iff.refl)
theorem is_symmetric_iff [instance] : relation.is_symmetric iff :=
relation.is_symmetric.mk (@iff.symm)
theorem is_transitive_iff [instance] : relation.is_transitive iff :=
relation.is_transitive.mk (@iff.trans)
-- Mp-like for iff
-- ---------------
theorem mp_like_iff [instance] (a b : Prop) (H : a ↔ b) : @relation.mp_like iff a b H :=
relation.mp_like.mk (iff.elim_left H)
-- Substition for iff
-- ------------------
namespace iff
theorem subst {P : Prop → Prop} {C : congruence iff iff P} {a b : Prop} (H : a ↔ b) (H1 : P a) :
P b :=
@general_operations.subst Prop iff P C a b H H1
end iff
-- Support for calculations with iff
-- ----------------
calc_subst iff.subst
namespace iff_ops
postfix `⁻¹`:100 := iff.symm
infixr `⬝`:75 := iff.trans
infixr `▸`:75 := iff.subst
abbreviation refl := iff.refl
abbreviation symm := @iff.symm
abbreviation trans := @iff.trans
abbreviation subst := @iff.subst
abbreviation mp := @iff.mp
end iff_ops
end relation