lean2/library/logic/connectives/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
import logic.connectives.basic logic.connectives.eq struc.relation
namespace relation
using relation
-- Congruences for logic
-- ---------------------
theorem congr_not : congr iff iff not :=
congr_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 congr_and : congr2 iff iff iff and :=
congr2_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 congr_or : congr2 iff iff iff or :=
congr2_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 congr_imp : congr2 iff iff iff imp :=
congr2_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 congr_iff : congr2 iff iff iff iff :=
congr2_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 congr_const_iff [instance] := congr.const iff iff_refl
definition congr_not_compose [instance] := congr.compose congr_not
definition congr_and_compose [instance] := congr.compose21 congr_and
definition congr_or_compose [instance] := congr.compose21 congr_or
definition congr_implies_compose [instance] := congr.compose21 congr_imp
definition congr_iff_compose [instance] := congr.compose21 congr_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 : congr R iff P}
{a b : T} (H : R a b) (H1 : P a) : P b := iff_elim_left (congr.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 (@refl T)
theorem is_symmetric_eq [instance] (T : Type) : relation.is_symmetric (@eq T) :=
relation.is_symmetric_mk (@symm T)
theorem is_transitive_eq [instance] (T : Type) : relation.is_transitive (@eq T) :=
relation.is_transitive_mk (@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
-- ------------------
theorem subst_iff {P : Prop → Prop} {C : congr 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
-- Support for calculations with iff
-- ----------------
calc_subst subst_iff
namespace iff_ops
postfix `⁻¹`:100 := iff_symm
infixr `⬝`:75 := iff_trans
infixr `▸`:75 := subst_iff
abbreviation refl := iff_refl
abbreviation symm := @iff_symm
abbreviation trans := @iff_trans
abbreviation subst := @subst_iff
abbreviation mp := @iff_mp
end iff_ops
end relation