136 lines
4.5 KiB
Text
136 lines
4.5 KiB
Text
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Author: Jeremy Avigad
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-- logic.instances
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-- ====================
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import logic.connectives algebra.relation
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namespace relation
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open relation
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-- Congruences for logic
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-- ---------------------
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theorem congruence_not : congruence iff iff not :=
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congruence.mk
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(take a b,
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assume H : a ↔ b, iff.intro
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(assume H1 : ¬a, assume H2 : b, H1 (iff.elim_right H H2))
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(assume H1 : ¬b, assume H2 : a, H1 (iff.elim_left H H2)))
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theorem congruence_and : congruence2 iff iff iff and :=
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congruence2.mk
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(take a1 b1 a2 b2,
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assume H1 : a1 ↔ b1, assume H2 : a2 ↔ b2,
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iff.intro
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(assume H3 : a1 ∧ a2, and.imp_and H3 (iff.elim_left H1) (iff.elim_left H2))
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(assume H3 : b1 ∧ b2, and.imp_and H3 (iff.elim_right H1) (iff.elim_right H2)))
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theorem congruence_or : congruence2 iff iff iff or :=
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congruence2.mk
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(take a1 b1 a2 b2,
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assume H1 : a1 ↔ b1, assume H2 : a2 ↔ b2,
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iff.intro
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(assume H3 : a1 ∨ a2, or.imp_or H3 (iff.elim_left H1) (iff.elim_left H2))
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(assume H3 : b1 ∨ b2, or.imp_or H3 (iff.elim_right H1) (iff.elim_right H2)))
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theorem congruence_imp : congruence2 iff iff iff imp :=
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congruence2.mk
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(take a1 b1 a2 b2,
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assume H1 : a1 ↔ b1, assume H2 : a2 ↔ b2,
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iff.intro
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(assume H3 : a1 → a2, assume Hb1 : b1, iff.elim_left H2 (H3 ((iff.elim_right H1) Hb1)))
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(assume H3 : b1 → b2, assume Ha1 : a1, iff.elim_right H2 (H3 ((iff.elim_left H1) Ha1))))
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theorem congruence_iff : congruence2 iff iff iff iff :=
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congruence2.mk
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(take a1 b1 a2 b2,
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assume H1 : a1 ↔ b1, assume H2 : a2 ↔ b2,
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iff.intro
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(assume H3 : a1 ↔ a2, iff.trans (iff.symm H1) (iff.trans H3 H2))
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(assume H3 : b1 ↔ b2, iff.trans H1 (iff.trans H3 (iff.symm H2))))
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-- theorem congruence_const_iff [instance] := congruence.const iff iff.refl
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definition congruence_not_compose [instance] := congruence.compose congruence_not
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definition congruence_and_compose [instance] := congruence.compose21 congruence_and
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definition congruence_or_compose [instance] := congruence.compose21 congruence_or
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definition congruence_implies_compose [instance] := congruence.compose21 congruence_imp
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definition congruence_iff_compose [instance] := congruence.compose21 congruence_iff
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-- Generalized substitution
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-- ------------------------
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-- TODO: note that the target has to be "iff". Otherwise, there is not enough
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-- information to infer an mp-like relation.
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namespace general_operations
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theorem subst {T : Type} (R : T → T → Prop) ⦃P : T → Prop⦄ [C : congruence R iff P]
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{a b : T} (H : R a b) (H1 : P a) : P b := iff.elim_left (congruence.app C H) H1
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end general_operations
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-- = is an equivalence relation
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-- ----------------------------
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theorem is_reflexive_eq [instance] (T : Type) : relation.is_reflexive (@eq T) :=
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relation.is_reflexive.mk (@eq.refl T)
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theorem is_symmetric_eq [instance] (T : Type) : relation.is_symmetric (@eq T) :=
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relation.is_symmetric.mk (@eq.symm T)
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theorem is_transitive_eq [instance] (T : Type) : relation.is_transitive (@eq T) :=
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relation.is_transitive.mk (@eq.trans T)
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-- TODO: this is only temporary, needed to inform Lean that is_equivalence is a class
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theorem is_equivalence_eq [instance] (T : Type) : relation.is_equivalence (@eq T) :=
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relation.is_equivalence.mk _ _ _
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-- iff is an equivalence relation
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-- ------------------------------
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theorem is_reflexive_iff [instance] : relation.is_reflexive iff :=
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relation.is_reflexive.mk (@iff.refl)
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theorem is_symmetric_iff [instance] : relation.is_symmetric iff :=
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relation.is_symmetric.mk (@iff.symm)
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theorem is_transitive_iff [instance] : relation.is_transitive iff :=
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relation.is_transitive.mk (@iff.trans)
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-- Mp-like for iff
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-- ---------------
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theorem mp_like_iff [instance] (a b : Prop) (H : a ↔ b) : @relation.mp_like iff a b H :=
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relation.mp_like.mk (iff.elim_left H)
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-- Substition for iff
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-- ------------------
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namespace iff
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theorem subst {P : Prop → Prop} [C : congruence iff iff P] {a b : Prop} (H : a ↔ b) (H1 : P a) :
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P b :=
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@general_operations.subst Prop iff P C a b H H1
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end iff
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-- Support for calculations with iff
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-- ----------------
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calc_subst iff.subst
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namespace iff_ops
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notation H ⁻¹ := iff.symm H
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notation H1 ⬝ H2 := iff.trans H1 H2
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notation H1 ▸ H2 := iff.subst H1 H2
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definition refl := iff.refl
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definition symm := @iff.symm
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definition trans := @iff.trans
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definition subst := @iff.subst
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definition mp := @iff.mp
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end iff_ops
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end relation
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