101 lines
3.7 KiB
Text
101 lines
3.7 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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Class instances for iff and eq.
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-/
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import logic.connectives algebra.relation
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namespace relation
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/- logical equivalence relations -/
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theorem is_equivalence_eq [instance] (T : Type) : relation.is_equivalence (@eq T) :=
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relation.is_equivalence.mk (@eq.refl T) (@eq.symm T) (@eq.trans T)
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theorem is_equivalence_iff [instance] : relation.is_equivalence iff :=
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relation.is_equivalence.mk @iff.refl @iff.symm @iff.trans
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/- congruences for logic operations -/
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theorem is_congruence_not : is_congruence iff iff not :=
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is_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 is_congruence_and : is_congruence2 iff iff iff and :=
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is_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_of_and_of_imp_of_imp H3 (iff.elim_left H1) (iff.elim_left H2))
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(assume H3 : b1 ∧ b2, and_of_and_of_imp_of_imp H3 (iff.elim_right H1) (iff.elim_right H2)))
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theorem is_congruence_or : is_congruence2 iff iff iff or :=
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is_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_of_or_of_imp_of_imp H3 (iff.elim_left H1) (iff.elim_left H2))
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(assume H3 : b1 ∨ b2, or_of_or_of_imp_of_imp H3 (iff.elim_right H1) (iff.elim_right H2)))
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theorem is_congruence_imp : is_congruence2 iff iff iff imp :=
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is_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 is_congruence_iff : is_congruence2 iff iff iff iff :=
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is_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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definition is_congruence_not_compose [instance] := is_congruence.compose is_congruence_not
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definition is_congruence_and_compose [instance] := is_congruence.compose21 is_congruence_and
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definition is_congruence_or_compose [instance] := is_congruence.compose21 is_congruence_or
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definition is_congruence_implies_compose [instance] := is_congruence.compose21 is_congruence_imp
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definition is_congruence_iff_compose [instance] := is_congruence.compose21 is_congruence_iff
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/- a general substitution operation with respect to an arbitrary congruence -/
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namespace general_subst
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theorem subst {T : Type} (R : T → T → Prop) ⦃P : T → Prop⦄ [C : is_congruence R iff P]
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{a b : T} (H : R a b) (H1 : P a) : P b := iff.elim_left (is_congruence.app C H) H1
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end general_subst
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/- iff can be coerced to implication -/
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definition mp_like_iff [instance] : relation.mp_like iff :=
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relation.mp_like.mk (λa b (H : a ↔ b), iff.elim_left H)
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/- support for calculations with iff -/
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namespace iff
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theorem subst {P : Prop → Prop} [C : is_congruence iff iff P] {a b : Prop}
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(H : a ↔ b) (H1 : P a) : P b :=
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@general_subst.subst Prop iff P C a b H H1
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end iff
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attribute iff.subst [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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