167 lines
5 KiB
Text
167 lines
5 KiB
Text
----------------------------------------------------------------------------------------------------
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--- 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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----------------------------------------------------------------------------------------------------
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import logic.connectives.basic logic.connectives.eq struc.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.class 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 : congr.class2 iff iff iff and :=
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congr.mk2
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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 : congr.class2 iff iff iff or :=
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congr.mk2
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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 : congr.class2 iff iff iff imp :=
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congr.mk2
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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 : congr.class2 iff iff iff iff :=
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congr.mk2
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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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theorem congr_not_compose [instance] := congr.compose congr_not
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theorem congr_and_compose [instance] := congr.compose21 congr_and
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theorem congr_or_compose [instance] := congr.compose21 congr_or
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theorem congr_implies_compose [instance] := congr.compose21 congr_imp
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theorem congr_iff_compose [instance] := congr.compose21 congr_iff
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-- Generalized substitution
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-- ------------------------
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namespace gensubst
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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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theorem subst {T : Type} {R : T → T → Prop} {P : T → Prop} {C : congr.class 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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infixr `▸`:75 := subst
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end -- gensubst
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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.class (@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.class (@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.class (@eq T) :=
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relation.is_transitive.mk (@trans T)
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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.class iff :=
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relation.is_reflexive.mk (@iff_refl)
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theorem is_symmetric_iff [instance] : relation.is_symmetric.class iff :=
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relation.is_symmetric.mk (@iff_symm)
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theorem is_transitive_iff [instance] : relation.is_transitive.class 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.class H :=
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relation.mp_like.mk (iff_elim_left H)
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-- Tests
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-- -----
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namespace logic_instances_tests
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section
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using relation.operations
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theorem test1 (a b : Prop) (H : a ↔ b) (H1 : a) : b := mp H H1
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end
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section
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using gensubst
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theorem test2 (a b c d e : Prop) (H1 : a ↔ b) (H2 : a ∨ c → ¬(d → a)) : b ∨ c → ¬(d → b) :=
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subst H1 H2
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theorem test3 (a b c d e : Prop) (H1 : a ↔ b) (H2 : a ∨ c → ¬(d → a)) : b ∨ c → ¬(d → b) :=
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H1 ▸ H2
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end
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theorem test4 (a b c d e : Prop) (H1 : a ↔ b) : (a ∨ c → ¬(d → a)) ↔ (b ∨ c → ¬(d → b)) :=
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congr.infer iff iff (λa, (a ∨ c → ¬(d → a))) H1
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section
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using relation.symbols
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theorem test5 (T : Type) (a b c d : T) (H1 : a = b) (H2 : c = b) (H3 : c = d) : a = d :=
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H1 ⬝ H2⁻¹ ⬝ H3
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theorem test6 (a b c d : Prop) (H1 : a ↔ b) (H2 : c ↔ b) (H3 : c ↔ d) : a ↔ d :=
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H1 ⬝ (H2⁻¹ ⬝ H3)
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end
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end
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-- Boolean calculations
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-- --------------------
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-- TODO: move these to new file
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-- TODO: declare trans
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theorem or_right_comm (a b c : Prop) : (a ∨ b) ∨ c ↔ (a ∨ c) ∨ b :=
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calc
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(a ∨ b) ∨ c ↔ a ∨ (b ∨ c) : or_assoc _ _ _
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... ↔ a ∨ (c ∨ b) : congr.infer iff iff _ (or_comm b c)
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... ↔ (a ∨ c) ∨ b : iff_symm (or_assoc _ _ _)
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-- TODO: add or_left_comm, and_right_comm, and_left_comm
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