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