lean2/library/logic/instances.lean

/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Jeremy Avigad

Class instances for iff and eq.
-/

import logic.connectives algebra.relation

namespace relation

/- logical equivalence relations -/

definition is_equivalence_eq [instance] (T : Type) : relation.is_equivalence (@eq T) :=
relation.is_equivalence.mk (@eq.refl T) (@eq.symm T) (@eq.trans T)

definition is_equivalence_iff [instance] : relation.is_equivalence iff :=
relation.is_equivalence.mk @iff.refl @iff.symm @iff.trans

/- congruences for logic operations -/

definition is_congruence_not : is_congruence iff iff not :=
is_congruence.mk @congr_not

definition is_congruence_and : is_congruence2 iff iff iff and :=
is_congruence2.mk @congr_and

definition is_congruence_or : is_congruence2 iff iff iff or :=
is_congruence2.mk @congr_or

definition is_congruence_imp : is_congruence2 iff iff iff imp :=
is_congruence2.mk @congr_imp

definition is_congruence_iff : is_congruence2 iff iff iff iff :=
is_congruence2.mk @congr_iff

definition is_congruence_not_compose [instance] := is_congruence.compose is_congruence_not
definition is_congruence_and_compose [instance] := is_congruence.compose21 is_congruence_and
definition is_congruence_or_compose [instance] := is_congruence.compose21 is_congruence_or
definition is_congruence_implies_compose [instance] := is_congruence.compose21 is_congruence_imp
definition is_congruence_iff_compose [instance] := is_congruence.compose21 is_congruence_iff

/- a general substitution operation with respect to an arbitrary congruence -/

namespace general_subst
  theorem subst {T : Type} (R : T → T → Prop) ⦃P : T → Prop⦄ [C : is_congruence R iff P]
    {a b : T} (H : R a b) (H1 : P a) : P b := iff.elim_left (is_congruence.app C H) H1
end general_subst

/- iff can be coerced to implication -/

definition mp_like_iff [instance] : relation.mp_like iff :=
relation.mp_like.mk @iff.mp

/- support for calculations with iff -/

namespace iff
  theorem subst {P : Prop → Prop} [C : is_congruence iff iff P] {a b : Prop}
    (H : a ↔ b) (H1 : P a) : P b :=
  @general_subst.subst Prop iff P C a b H H1
end iff

attribute iff.subst [subst]

namespace iff_ops
  notation H ⁻¹ := iff.symm H
  notation H1 ⬝ H2 := iff.trans H1 H2
  notation H1 ▸ H2 := iff.subst H1 H2
  definition refl  := iff.refl
  definition symm  := @iff.symm
  definition trans := @iff.trans
  definition subst := @iff.subst
  definition mp    := @iff.mp
end iff_ops

end relation
refactor(library/algebra/relation, library/logic/instances): revise equivalence relations and congruences to use structure command 2014-11-28 15:20:52 +00:00			`/-`
			`Copyright (c) 2014 Microsoft Corporation. All rights reserved.`
			`Released under Apache 2.0 license as described in the file LICENSE.`
			`Author: Jeremy Avigad`

			`Class instances for iff and eq.`
			`-/`
feat(library/standard): add classes for relations 2014-08-05 00:07:59 +00:00
refactor(library/init/logic): move theorems to library/logic 2014-12-12 21:20:27 +00:00			`import logic.connectives algebra.relation`
feat(library/standard): add classes for relations 2014-08-05 00:07:59 +00:00
feat(library/standard): port int, and reorganize a lot 2014-08-20 02:32:44 +00:00			`namespace relation`

refactor(library/algebra/relation, library/logic/instances): revise equivalence relations and congruences to use structure command 2014-11-28 15:20:52 +00:00			`/- logical equivalence relations -/`

feaf(library): make sure basic standard library can be compiled with option "--to_axiom" We use this option to erase proofs when generating the javascript version. The proofs are erased to minimize the size of the file that must be downloaded by users 2015-07-29 23:11:23 +00:00			`definition is_equivalence_eq [instance] (T : Type) : relation.is_equivalence (@eq T) :=`
refactor(library/algebra/relation, library/logic/instances): revise equivalence relations and congruences to use structure command 2014-11-28 15:20:52 +00:00			`relation.is_equivalence.mk (@eq.refl T) (@eq.symm T) (@eq.trans T)`

feaf(library): make sure basic standard library can be compiled with option "--to_axiom" We use this option to erase proofs when generating the javascript version. The proofs are erased to minimize the size of the file that must be downloaded by users 2015-07-29 23:11:23 +00:00			`definition is_equivalence_iff [instance] : relation.is_equivalence iff :=`
refactor(library/algebra/relation, library/logic/instances): revise equivalence relations and congruences to use structure command 2014-11-28 15:20:52 +00:00			`relation.is_equivalence.mk @iff.refl @iff.symm @iff.trans`

			`/- congruences for logic operations -/`
feat(library/standard): add classes for relations 2014-08-05 00:07:59 +00:00
feaf(library): make sure basic standard library can be compiled with option "--to_axiom" We use this option to erase proofs when generating the javascript version. The proofs are erased to minimize the size of the file that must be downloaded by users 2015-07-29 23:11:23 +00:00			`definition is_congruence_not : is_congruence iff iff not :=`
feat(library): clean up "sorry"s in library Breaking changes: pnat was redefined to use subtype instead of a custom inductive type, which affects the notation for pnat 2 and 3 2015-07-24 15:56:18 +00:00			`is_congruence.mk @congr_not`
feat(library/standard): add classes for relations 2014-08-05 00:07:59 +00:00
feaf(library): make sure basic standard library can be compiled with option "--to_axiom" We use this option to erase proofs when generating the javascript version. The proofs are erased to minimize the size of the file that must be downloaded by users 2015-07-29 23:11:23 +00:00			`definition is_congruence_and : is_congruence2 iff iff iff and :=`
feat(library): clean up "sorry"s in library Breaking changes: pnat was redefined to use subtype instead of a custom inductive type, which affects the notation for pnat 2 and 3 2015-07-24 15:56:18 +00:00			`is_congruence2.mk @congr_and`
feat(library/standard): add classes for relations 2014-08-05 00:07:59 +00:00
feaf(library): make sure basic standard library can be compiled with option "--to_axiom" We use this option to erase proofs when generating the javascript version. The proofs are erased to minimize the size of the file that must be downloaded by users 2015-07-29 23:11:23 +00:00			`definition is_congruence_or : is_congruence2 iff iff iff or :=`
feat(library): clean up "sorry"s in library Breaking changes: pnat was redefined to use subtype instead of a custom inductive type, which affects the notation for pnat 2 and 3 2015-07-24 15:56:18 +00:00			`is_congruence2.mk @congr_or`
feat(library/standard): add classes for relations 2014-08-05 00:07:59 +00:00
feaf(library): make sure basic standard library can be compiled with option "--to_axiom" We use this option to erase proofs when generating the javascript version. The proofs are erased to minimize the size of the file that must be downloaded by users 2015-07-29 23:11:23 +00:00			`definition is_congruence_imp : is_congruence2 iff iff iff imp :=`
feat(library): clean up "sorry"s in library Breaking changes: pnat was redefined to use subtype instead of a custom inductive type, which affects the notation for pnat 2 and 3 2015-07-24 15:56:18 +00:00			`is_congruence2.mk @congr_imp`
feat(library/standard): add classes for relations 2014-08-05 00:07:59 +00:00
feaf(library): make sure basic standard library can be compiled with option "--to_axiom" We use this option to erase proofs when generating the javascript version. The proofs are erased to minimize the size of the file that must be downloaded by users 2015-07-29 23:11:23 +00:00			`definition is_congruence_iff : is_congruence2 iff iff iff iff :=`
feat(library): clean up "sorry"s in library Breaking changes: pnat was redefined to use subtype instead of a custom inductive type, which affects the notation for pnat 2 and 3 2015-07-24 15:56:18 +00:00			`is_congruence2.mk @congr_iff`
feat(library/standard): add classes for relations 2014-08-05 00:07:59 +00:00
refactor(library/algebra/relation, library/logic/instances): revise equivalence relations and congruences to use structure command 2014-11-28 15:20:52 +00:00			`definition is_congruence_not_compose [instance] := is_congruence.compose is_congruence_not`
			`definition is_congruence_and_compose [instance] := is_congruence.compose21 is_congruence_and`
			`definition is_congruence_or_compose [instance] := is_congruence.compose21 is_congruence_or`
			`definition is_congruence_implies_compose [instance] := is_congruence.compose21 is_congruence_imp`
			`definition is_congruence_iff_compose [instance] := is_congruence.compose21 is_congruence_iff`
feat(library/standard): add classes for relations 2014-08-05 00:07:59 +00:00
refactor(library/algebra/relation, library/logic/instances): revise equivalence relations and congruences to use structure command 2014-11-28 15:20:52 +00:00			`/- a general substitution operation with respect to an arbitrary congruence -/`
feat(library/standard): add classes for relations 2014-08-05 00:07:59 +00:00
refactor(library/algebra/relation, library/logic/instances): revise equivalence relations and congruences to use structure command 2014-11-28 15:20:52 +00:00			`namespace general_subst`
			`theorem subst {T : Type} (R : T → T → Prop) ⦃P : T → Prop⦄ [C : is_congruence R iff P]`
			`{a b : T} (H : R a b) (H1 : P a) : P b := iff.elim_left (is_congruence.app C H) H1`
			`end general_subst`
feat(library/standard): add classes for relations 2014-08-05 00:07:59 +00:00
refactor(library/algebra/relation, library/logic/instances): revise equivalence relations and congruences to use structure command 2014-11-28 15:20:52 +00:00			`/- iff can be coerced to implication -/`
feat(library/standard): add classes for relations 2014-08-05 00:07:59 +00:00
refactor(library/algebra/relation, library/logic/instances): revise equivalence relations and congruences to use structure command 2014-11-28 15:20:52 +00:00			`definition mp_like_iff [instance] : relation.mp_like iff :=`
feat(library): clean up "sorry"s in library Breaking changes: pnat was redefined to use subtype instead of a custom inductive type, which affects the notation for pnat 2 and 3 2015-07-24 15:56:18 +00:00			`relation.mp_like.mk @iff.mp`
feat(library/standard): add classes for relations 2014-08-05 00:07:59 +00:00
refactor(library/algebra/relation, library/logic/instances): revise equivalence relations and congruences to use structure command 2014-11-28 15:20:52 +00:00			`/- support for calculations with iff -/`
feat(library/standard): add classes for relations 2014-08-05 00:07:59 +00:00
refactor(library): add namespaces 'or', 'and' and 'iff' Signed-off-by: Leonardo de Moura <leonardo@microsoft.com> 2014-09-05 04:25:21 +00:00			`namespace iff`
refactor(library/logic/connectives): rename theorems 2014-12-15 20:05:44 +00:00			`theorem subst {P : Prop → Prop} [C : is_congruence iff iff P] {a b : Prop}`
			`(H : a ↔ b) (H1 : P a) : P b :=`
refactor(library/algebra/relation, library/logic/instances): revise equivalence relations and congruences to use structure command 2014-11-28 15:20:52 +00:00			`@general_subst.subst Prop iff P C a b H H1`
refactor(library): add namespaces 'or', 'and' and 'iff' Signed-off-by: Leonardo de Moura <leonardo@microsoft.com> 2014-09-05 04:25:21 +00:00			`end iff`
fix(lean/library/standard): fix tests, more cleanup 2014-08-20 22:49:44 +00:00
refactor(library): replace 'calc_trans', 'calc_symm', 'calc_refl' and 'calc_subst' commands with attributes '[symm]', '[refl]', '[trans]' and '[subst]' These attributes are used by the calc command. They will also be used by tactics such as 'reflexivity', 'symmetry' and 'transitivity'. See issue #500 2015-05-02 22:15:35 +00:00			`attribute iff.subst [subst]`
feat(library/standard): port int, and reorganize a lot 2014-08-20 02:32:44 +00:00
			`namespace iff_ops`
refactor(library): use 'reserve' notation in the standard library 2014-10-21 21:08:07 +00:00			`notation H ⁻¹ := iff.symm H`
			`notation H1 ⬝ H2 := iff.trans H1 H2`
			`notation H1 ▸ H2 := iff.subst H1 H2`
feat(frontends/lean): definitions are opaque by default 2014-09-17 21:39:05 +00:00			`definition refl := iff.refl`
			`definition symm := @iff.symm`
			`definition trans := @iff.trans`
			`definition subst := @iff.subst`
			`definition mp := @iff.mp`
feat(library/standard): port int, and reorganize a lot 2014-08-20 02:32:44 +00:00			`end iff_ops`
refactor(library/algebra/relation, library/logic/instances): revise equivalence relations and congruences to use structure command 2014-11-28 15:20:52 +00:00
feat(library/standard): port int, and reorganize a lot 2014-08-20 02:32:44 +00:00			`end relation`