diff options
Diffstat (limited to 'theories/Logic')
| -rw-r--r-- | theories/Logic/ChoiceFacts.v | 712 | ||||
| -rw-r--r-- | theories/Logic/ClassicalChoice.v | 38 | ||||
| -rw-r--r-- | theories/Logic/ClassicalDescription.v | 124 | ||||
| -rw-r--r-- | theories/Logic/ClassicalEpsilon.v | 90 | ||||
| -rw-r--r-- | theories/Logic/ClassicalFacts.v | 3 | ||||
| -rw-r--r-- | theories/Logic/ClassicalUniqueChoice.v | 79 | ||||
| -rw-r--r-- | theories/Logic/Classical_Prop.v | 11 | ||||
| -rw-r--r-- | theories/Logic/DecidableType.v | 156 | ||||
| -rw-r--r-- | theories/Logic/DecidableTypeEx.v | 50 | ||||
| -rw-r--r-- | theories/Logic/Diaconescu.v | 222 | ||||
| -rw-r--r-- | theories/Logic/RelationalChoice.v | 15 |
11 files changed, 1289 insertions, 211 deletions
diff --git a/theories/Logic/ChoiceFacts.v b/theories/Logic/ChoiceFacts.v index bc892ca9..e0be9ed3 100644 --- a/theories/Logic/ChoiceFacts.v +++ b/theories/Logic/ChoiceFacts.v @@ -1,3 +1,4 @@ +(* -*- coding: utf-8 -*- *) (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) (* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) @@ -6,155 +7,453 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: ChoiceFacts.v 8132 2006-03-05 10:59:47Z herbelin $ i*) +(*i $Id: ChoiceFacts.v 8892 2006-06-04 17:59:53Z herbelin $ i*) -(** We show that the functional formulation of the axiom of Choice - (usual formulation in type theory) is equivalent to its relational - formulation (only formulation of set theory) + the axiom of - (parametric) definite description (aka axiom of unique choice) *) +(** ** Some facts and definitions concerning choice and description in + intuitionistic logic. -(** This shows that the axiom of choice can be assumed (under its - relational formulation) without known inconsistency with classical logic, - though definite description conflicts with classical logic *) +We investigate the relations between the following choice and +description principles + +- AC_rel = relational form of the (non extensional) axiom of choice + (a "set-theoretic" axiom of choice) +- AC_fun = functional form of the (non extensional) axiom of choice + (a "type-theoretic" axiom of choice) +- AC! = functional relation reification + (known as axiom of unique choice in topos theory, + sometimes called principle of definite description in + the context of constructive type theory) + +- GAC_rel = guarded relational form of the (non extensional) axiom of choice +- GAC_fun = guarded functional form of the (non extensional) axiom of choice +- GAC! = guarded functional relation reification + +- OAC_rel = "omniscient" relational form of the (non extensional) axiom of choice +- OAC_fun = "omniscient" functional form of the (non extensional) axiom of choice + (called AC* in Bell [Bell]) +- OAC! + +- ID_iota = intuitionistic definite description +- ID_epsilon = intuitionistic indefinite description + +- D_iota = (weakly classical) definite description principle +- D_epsilon = (weakly classical) indefinite description principle + +- PI = proof irrelevance +- IGP = independence of general premises + (an unconstrained generalisation of the constructive principle of + independence of premises) +- Drinker = drinker's paradox (small form) + (called Ex in Bell [Bell]) + +We let also + +IPL_2^2 = 2nd-order impredicative, 2nd-order functional minimal predicate logic +IPL_2 = 2nd-order impredicative minimal predicate logic +IPL^2 = 2nd-order functional minimal predicate logic (with ex. quant.) + +Table of contents + +A. Definitions + +B. IPL_2^2 |- AC_rel + AC! = AC_fun + +C. 1. AC_rel + PI -> GAC_rel and PL_2 |- AC_rel + IGP -> GAC_rel and GAC_rel = OAC_rel + +C. 2. IPL^2 |- AC_fun + IGP = GAC_fun = OAC_fun = AC_fun + Drinker + +D. Derivability of choice for decidable relations with well-ordered codomain + +E. Equivalence of choices on dependent or non dependent functional types + +F. Non contradiction of constructive descriptions wrt functional choices + +G. Definite description transports classical logic to the computational world -Section ChoiceEquivalences. +References: + +[Bell] John L. Bell, Choice principles in intuitionistic set theory, +unpublished. + +[Bell93] John L. Bell, Hilbert's Epsilon Operator in Intuitionistic +Type Theories, Mathematical Logic Quarterly, volume 39, 1993. + +[Carlstrøm05] Jesper Carlstrøm, Interpreting descriptions in +intentional type theory, Journal of Symbolic Logic 70(2):488-514, 2005. +*) + +Set Implicit Arguments. + +Notation Local "'inhabited' A" := A (at level 10, only parsing). + +(**********************************************************************) +(** *** A. Definitions *) + +(** Choice, reification and description schemes *) + +Section ChoiceSchemes. Variables A B :Type. -Definition RelationalChoice := - forall (R:A -> B -> Prop), - (forall x:A, exists y : B, R x y) -> - exists R' : A -> B -> Prop, - (forall x:A, - exists y : B, R x y /\ R' x y /\ (forall y':B, R' x y' -> y = y')). +Variables P:A->Prop. + +Variables R:A->B->Prop. + +(** **** Constructive choice and description *) + +(** AC_rel *) + +Definition RelationalChoice_on := + forall R:A->B->Prop, + (forall x : A, exists y : B, R x y) -> + (exists R' : A->B->Prop, subrelation R' R /\ forall x, exists! y, R' x y). + +(** AC_fun *) + +Definition FunctionalChoice_on := + forall R:A->B->Prop, + (forall x : A, exists y : B, R x y) -> + (exists f : A->B, forall x : A, R x (f x)). + +(** AC! or Functional Relation Reification (known as Axiom of Unique Choice + in topos theory; also called principle of definite description *) + +Definition FunctionalRelReification_on := + forall R:A->B->Prop, + (forall x : A, exists! y : B, R x y) -> + (exists f : A->B, forall x : A, R x (f x)). + +(** ID_epsilon (constructive version of indefinite description; + combined with proof-irrelevance, it may be connected to + Carlstrøm's type theory with a constructive indefinite description + operator) *) + +Definition ConstructiveIndefiniteDescription_on := + forall P:A->Prop, + (exists x, P x) -> { x:A | P x }. + +(** ID_iota (constructive version of definite description; combined + with proof-irrelevance, it may be connected to Carlstrøm's and + Stenlund's type theory with a constructive definite description + operator) *) + +Definition ConstructiveDefiniteDescription_on := + forall P:A->Prop, + (exists! x, P x) -> { x:A | P x }. + +(** **** Weakly classical choice and description *) -Definition FunctionalChoice := - forall (R:A -> B -> Prop), - (forall x:A, exists y : B, R x y) -> - exists f : A -> B, (forall x:A, R x (f x)). +(** GAC_rel *) -Definition ParamDefiniteDescription := - forall (R:A -> B -> Prop), - (forall x:A, exists y : B, R x y /\ (forall y':B, R x y' -> y = y')) -> - exists f : A -> B, (forall x:A, R x (f x)). +Definition GuardedRelationalChoice_on := + forall P : A->Prop, forall R : A->B->Prop, + (forall x : A, P x -> exists y : B, R x y) -> + (exists R' : A->B->Prop, + subrelation R' R /\ forall x, P x -> exists! y, R' x y). + +(** GAC_fun *) + +Definition GuardedFunctionalChoice_on := + forall P : A->Prop, forall R : A->B->Prop, + inhabited B -> + (forall x : A, P x -> exists y : B, R x y) -> + (exists f : A->B, forall x, P x -> R x (f x)). + +(** GFR_fun *) + +Definition GuardedFunctionalRelReification_on := + forall P : A->Prop, forall R : A->B->Prop, + inhabited B -> + (forall x : A, P x -> exists! y : B, R x y) -> + (exists f : A->B, forall x : A, P x -> R x (f x)). + +(** OAC_rel *) + +Definition OmniscientRelationalChoice_on := + forall R : A->B->Prop, + exists R' : A->B->Prop, + subrelation R' R /\ forall x : A, (exists y : B, R x y) -> exists! y, R' x y. + +(** OAC_fun *) + +Definition OmniscientFunctionalChoice_on := + forall R : A->B->Prop, + inhabited B -> + exists f : A->B, forall x : A, (exists y : B, R x y) -> R x (f x). + +(** D_epsilon *) + +Definition ClassicalIndefiniteDescription := + forall P:A->Prop, + A -> { x:A | (exists x, P x) -> P x }. + +(** D_iota *) + +Definition ClassicalDefiniteDescription := + forall P:A->Prop, + A -> { x:A | (exists! x, P x) -> P x }. + +End ChoiceSchemes. + +(** Generalized schemes *) + +Notation RelationalChoice := + (forall A B, RelationalChoice_on A B). +Notation FunctionalChoice := + (forall A B, FunctionalChoice_on A B). +Notation FunctionalChoiceOnInhabitedSet := + (forall A B, inhabited B -> FunctionalChoice_on A B). +Notation FunctionalRelReification := + (forall A B, FunctionalRelReification_on A B). + +Notation GuardedRelationalChoice := + (forall A B, GuardedRelationalChoice_on A B). +Notation GuardedFunctionalChoice := + (forall A B, GuardedFunctionalChoice_on A B). +Notation GuardedFunctionalRelReification := + (forall A B, GuardedFunctionalRelReification_on A B). + +Notation OmniscientRelationalChoice := + (forall A B, OmniscientRelationalChoice_on A B). +Notation OmniscientFunctionalChoice := + (forall A B, OmniscientFunctionalChoice_on A B). + +Notation ConstructiveDefiniteDescription := + (forall A, ConstructiveDefiniteDescription_on A). +Notation ConstructiveIndefiniteDescription := + (forall A, ConstructiveIndefiniteDescription_on A). + +(** Subclassical schemes *) + +Definition ProofIrrelevance := + forall (A:Prop) (a1 a2:A), a1 = a2. + +Definition IndependenceOfGeneralPremises := + forall (A:Type) (P:A -> Prop) (Q:Prop), + inhabited A -> + (Q -> exists x, P x) -> exists x, Q -> P x. + +Definition SmallDrinker'sParadox := + forall (A:Type) (P:A -> Prop), inhabited A -> + exists x, (exists x, P x) -> P x. + +(**********************************************************************) +(** *** B. AC_rel + PDP = AC_fun + + We show that the functional formulation of the axiom of Choice + (usual formulation in type theory) is equivalent to its relational + formulation (only formulation of set theory) + the axiom of + (parametric) definite description (aka axiom of unique choice) *) + +(** This shows that the axiom of choice can be assumed (under its + relational formulation) without known inconsistency with classical logic, + though definite description conflicts with classical logic *) Lemma description_rel_choice_imp_funct_choice : - ParamDefiniteDescription -> RelationalChoice -> FunctionalChoice. -intros Descr RelCh. -red in |- *; intros R H. -destruct (RelCh R H) as [R' H0]. -destruct (Descr R') as [f H1]. -intro x. -elim (H0 x); intros y [H2 [H3 H4]]; exists y; split; [ exact H3 | exact H4 ]. + forall A B : Type, + FunctionalRelReification_on A B -> RelationalChoice_on A B -> FunctionalChoice_on A B. +Proof. +intros A B Descr RelCh R H. +destruct (RelCh R H) as (R',(HR'R,H0)). +destruct (Descr R') as (f,Hf). +firstorder. exists f; intro x. -elim (H0 x); intros y [H2 [H3 H4]]. -rewrite <- (H4 (f x) (H1 x)). -exact H2. +destruct (H0 x) as (y,(HR'xy,Huniq)). +rewrite <- (Huniq (f x) (Hf x)). +apply HR'R; assumption. Qed. -Lemma funct_choice_imp_rel_choice : FunctionalChoice -> RelationalChoice. -intros FunCh. -red in |- *; intros R H. -destruct (FunCh R H) as [f H0]. -exists (fun x y => y = f x). -intro x; exists (f x); split; - [ apply H0 - | split; [ reflexivity | intros y H1; symmetry in |- *; exact H1 ] ]. +Lemma funct_choice_imp_rel_choice : + forall A B, FunctionalChoice_on A B -> RelationalChoice_on A B. +Proof. +intros A B FunCh R H. +destruct (FunCh R H) as (f,H0). +exists (fun x y => f x = y). +split. + intros x y Heq; rewrite <- Heq; trivial. + intro x; exists (f x); split. + reflexivity. + trivial. Qed. -Lemma funct_choice_imp_description : - FunctionalChoice -> ParamDefiniteDescription. -intros FunCh. -red in |- *; intros R H. +Lemma funct_choice_imp_description : + forall A B, FunctionalChoice_on A B -> FunctionalRelReification_on A B. +Proof. +intros A B FunCh R H. destruct (FunCh R) as [f H0]. (* 1 *) intro x. -elim (H x); intros y [H0 H1]. -exists y; exact H0. +destruct (H x) as (y,(HRxy,_)). +exists y; exact HRxy. (* 2 *) exists f; exact H0. Qed. Theorem FunChoice_Equiv_RelChoice_and_ParamDefinDescr : - FunctionalChoice <-> RelationalChoice /\ ParamDefiniteDescription. -split. + forall A B, FunctionalChoice_on A B <-> + RelationalChoice_on A B /\ FunctionalRelReification_on A B. +Proof. +intros A B; split. intro H; split; [ exact (funct_choice_imp_rel_choice H) | exact (funct_choice_imp_description H) ]. intros [H H0]; exact (description_rel_choice_imp_funct_choice H0 H). Qed. -End ChoiceEquivalences. +(**********************************************************************) +(** *** C. Connection between the guarded, non guarded and descriptive choices and *) (** We show that the guarded relational formulation of the axiom of Choice comes from the non guarded formulation in presence either of the independance of premises or proof-irrelevance *) -Definition GuardedRelationalChoice (A B:Type) := - forall (P:A -> Prop) (R:A -> B -> Prop), - (forall x:A, P x -> exists y : B, R x y) -> - exists R' : A -> B -> Prop, - (forall x:A, - P x -> - exists y : B, R x y /\ R' x y /\ (forall y':B, R' x y' -> y = y')). - -Definition ProofIrrelevance := forall (A:Prop) (a1 a2:A), a1 = a2. +(**********************************************************************) +(** **** C. 1. AC_rel + PI -> GAC_rel and AC_rel + IGP -> GAC_rel and GAC_rel = OAC_rel *) Lemma rel_choice_and_proof_irrel_imp_guarded_rel_choice : - (forall A B, RelationalChoice A B) - -> ProofIrrelevance -> (forall A B, GuardedRelationalChoice A B). + RelationalChoice -> ProofIrrelevance -> GuardedRelationalChoice. Proof. intros rel_choice proof_irrel. red in |- *; intros A B P R H. -destruct (rel_choice _ _ (fun (x:sigT P) (y:B) => R (projT1 x) y)) as [R' H0]. -intros [x HPx]. -destruct (H x HPx) as [y HRxy]. +destruct (rel_choice _ _ (fun (x:sigT P) (y:B) => R (projT1 x) y)) as (R',(HR'R,H0)). +intros (x,HPx). +destruct (H x HPx) as (y,HRxy). exists y; exact HRxy. set (R'' := fun (x:A) (y:B) => exists H : P x, R' (existT P x H) y). -exists R''; intros x HPx. -destruct (H0 (existT P x HPx)) as [y [HRxy [HR'xy Huniq]]]. -exists y. split. - exact HRxy. - split. - red in |- *; exists HPx; exact HR'xy. - intros y' HR''xy'. +exists R''; split. + intros x y (HPx,HR'xy). + change x with (projT1 (existT P x HPx)); apply HR'R; exact HR'xy. + intros x HPx. + destruct (H0 (existT P x HPx)) as (y,(HR'xy,Huniq)). + exists y; split. exists HPx; exact HR'xy. + intros y' (H'Px,HR'xy'). apply Huniq. - unfold R'' in HR''xy'. - destruct HR''xy' as [H'Px HR'xy']. - rewrite proof_irrel with (a1 := HPx) (a2 := H'Px). - exact HR'xy'. + rewrite proof_irrel with (a1 := HPx) (a2 := H'Px); exact HR'xy'. Qed. -Definition IndependenceOfGeneralPremises := - forall (A:Type) (P:A -> Prop) (Q:Prop), - (Q -> exists x, P x) -> exists x, Q -> P x. - Lemma rel_choice_indep_of_general_premises_imp_guarded_rel_choice : - forall A B, RelationalChoice A B -> - IndependenceOfGeneralPremises -> GuardedRelationalChoice A B. -Proof. -intros A B RelCh IndPrem. -red in |- *; intros P R H. -destruct (RelCh (fun x y => P x -> R x y)) as [R' H0]. - intro x. apply IndPrem. - apply H. - exists R'. - intros x HPx. - destruct (H0 x) as [y [H1 H2]]. - exists y. split. - apply (H1 HPx). - exact H2. + forall A B, inhabited B -> RelationalChoice_on A B -> + IndependenceOfGeneralPremises -> GuardedRelationalChoice_on A B. +Proof. +intros A B Inh AC_rel IndPrem P R H. +destruct (AC_rel (fun x y => P x -> R x y)) as (R',(HR'R,H0)). + intro x. apply IndPrem. exact Inh. intro Hx. + apply H; assumption. + exists (fun x y => P x /\ R' x y). + firstorder. +Qed. + +Lemma guarded_rel_choice_imp_rel_choice : + forall A B, GuardedRelationalChoice_on A B -> RelationalChoice_on A B. +Proof. +intros A B GAC_rel R H. +destruct (GAC_rel (fun _ => True) R) as (R',(HR'R,H0)). + firstorder. +exists R'; firstorder. Qed. +(** OAC_rel = GAC_rel *) + +Lemma guarded_iff_omniscient_rel_choice : + GuardedRelationalChoice <-> OmniscientRelationalChoice. +Proof. +split. + intros GAC_rel A B R. + apply (GAC_rel A B (fun x => exists y, R x y) R); auto. + intros OAC_rel A B P R H. + destruct (OAC_rel A B R) as (f,Hf); exists f; firstorder. +Qed. + +(**********************************************************************) +(** **** C. 2. AC_fun + IGP = GAC_fun = OAC_fun = AC_fun + Drinker *) + +(** AC_fun + IGP = GAC_fun *) + +Lemma guarded_fun_choice_imp_indep_of_general_premises : + GuardedFunctionalChoice -> IndependenceOfGeneralPremises. +Proof. +intros GAC_fun A P Q Inh H. +destruct (GAC_fun unit A (fun _ => Q) (fun _ => P) Inh) as (f,Hf). +tauto. +exists (f tt); auto. +Qed. + +Lemma guarded_fun_choice_imp_fun_choice : + GuardedFunctionalChoice -> FunctionalChoiceOnInhabitedSet. +Proof. +intros GAC_fun A B Inh R H. +destruct (GAC_fun A B (fun _ => True) R Inh) as (f,Hf). +firstorder. +exists f; auto. +Qed. + +Lemma fun_choice_and_indep_general_prem_imp_guarded_fun_choice : + FunctionalChoiceOnInhabitedSet -> IndependenceOfGeneralPremises + -> GuardedFunctionalChoice. +Proof. +intros AC_fun IndPrem A B P R Inh H. +apply (AC_fun A B Inh (fun x y => P x -> R x y)). +intro x; apply IndPrem; eauto. +Qed. + +(** AC_fun + Drinker = OAC_fun *) + +(** This was already observed by Bell [Bell] *) + +Lemma omniscient_fun_choice_imp_small_drinker : + OmniscientFunctionalChoice -> SmallDrinker'sParadox. +Proof. +intros OAC_fun A P Inh. +destruct (OAC_fun unit A (fun _ => P)) as (f,Hf). +auto. +exists (f tt); firstorder. +Qed. + +Lemma omniscient_fun_choice_imp_fun_choice : + OmniscientFunctionalChoice -> FunctionalChoiceOnInhabitedSet. +Proof. +intros OAC_fun A B Inh R H. +destruct (OAC_fun A B R Inh) as (f,Hf). +exists f; firstorder. +Qed. + +Lemma fun_choice_and_small_drinker_imp_omniscient_fun_choice : + FunctionalChoiceOnInhabitedSet -> SmallDrinker'sParadox + -> OmniscientFunctionalChoice. +Proof. +intros AC_fun Drinker A B R Inh. +destruct (AC_fun A B Inh (fun x y => (exists y, R x y) -> R x y)) as (f,Hf). + intro x; apply (Drinker B (R x) Inh). + exists f; assumption. +Qed. + +(** OAC_fun = GAC_fun *) + +(** This is derivable from the intuitionistic equivalence between IGP and Drinker +but we give a direct proof *) + +Lemma guarded_iff_omniscient_fun_choice : + GuardedFunctionalChoice <-> OmniscientFunctionalChoice. +Proof. +split. + intros GAC_fun A B R Inh. + apply (GAC_fun A B (fun x => exists y, R x y) R); auto. + intros OAC_fun A B P R Inh H. + destruct (OAC_fun A B R Inh) as (f,Hf). + exists f; firstorder. +Qed. + +(**********************************************************************) +(** *** D. Derivability of choice for decidable relations with well-ordered codomain *) (** Countable codomains, such as [nat], can be equipped with a well-order, which implies the existence of a least element on inhabited decidable subsets. As a consequence, the relational form of the axiom of choice is derivable on [nat] for decidable relations. - We show instead that definite description and the functional form - of the axiom of choice are equivalent on decidable relation with [nat] - as codomain + We show instead that functional relation reification and the + functional form of the axiom of choice are equivalent on decidable + relation with [nat] as codomain *) Require Import Wf_nat. @@ -163,12 +462,11 @@ Require Import Decidable. Require Import Arith. Definition has_unique_least_element (A:Type) (R:A->A->Prop) (P:A->Prop) := - (exists x, (P x /\ forall x', P x' -> R x x') - /\ forall x', P x' /\ (forall x'', P x'' -> R x' x'') -> x=x'). + exists! x, P x /\ forall x', P x' -> R x x'. Lemma dec_inh_nat_subset_has_unique_least_element : forall P:nat->Prop, (forall n, P n \/ ~ P n) -> - (exists n, P n) -> has_unique_least_element nat le P. + (exists n, P n) -> has_unique_least_element le P. Proof. intros P Pdec (n0,HPn0). assert @@ -194,30 +492,228 @@ assert assumption. destruct H0. rewrite Heqn; assumption. -destruct (H n0) as [(n,(Hltn,(Hmin,Huniqn)))|]; [exists n | exists n0]; - repeat split; + destruct (H n0) as [(n,(Hltn,(Hmin,Huniqn)))|]; [exists n | exists n0]; + repeat split; assumption || intros n' (HPn',Hminn'); apply le_antisym; auto. Qed. -Definition FunctionalChoice_on (A B:Type) (R:A->B->Prop) := - (forall x:A, exists y : B, R x y) -> - exists f : A -> B, (forall x:A, R x (f x)). +Definition FunctionalChoice_on_rel (A B:Type) (R:A->B->Prop) := + (forall x:A, exists y : B, R x y) -> + exists f : A -> B, (forall x:A, R x (f x)). Lemma classical_denumerable_description_imp_fun_choice : forall A:Type, - ParamDefiniteDescription A nat -> - forall R, (forall x y, decidable (R x y)) -> FunctionalChoice_on A nat R. + FunctionalRelReification_on A nat -> + forall R:A->nat->Prop, + (forall x y, decidable (R x y)) -> FunctionalChoice_on_rel R. Proof. intros A Descr. red in |- *; intros R Rdec H. set (R':= fun x y => R x y /\ forall y', R x y' -> y <= y'). -destruct (Descr R') as [f Hf]. +destruct (Descr R') as (f,Hf). intro x. apply (dec_inh_nat_subset_has_unique_least_element (R x)). apply Rdec. apply (H x). exists f. intros x. -destruct (Hf x) as [Hfx _]. +destruct (Hf x) as (Hfx,_). +assumption. +Qed. + +(**********************************************************************) +(** *** E. Choice on dependent and non dependent function types are equivalent *) + +(** **** E. 1. Choice on dependent and non dependent function types are equivalent *) + +Definition DependentFunctionalChoice_on (A:Type) (B:A -> Type) := + forall R:forall x:A, B x -> Prop, + (forall x:A, exists y : B x, R x y) -> + (exists f : (forall x:A, B x), forall x:A, R x (f x)). + +Notation DependentFunctionalChoice := + (forall A (B:A->Type), DependentFunctionalChoice_on B). + +(** The easy part *) + +Theorem dep_non_dep_functional_choice : + DependentFunctionalChoice -> FunctionalChoice. +Proof. +intros AC_depfun A B R H. + destruct (AC_depfun A (fun _ => B) R H) as (f,Hf). + exists f; trivial. +Qed. + +(** Deriving choice on product types requires some computation on + singleton propositional types, so we need computational + conjunction projections and dependent elimination of conjunction + and equality *) + +Scheme and_indd := Induction for and Sort Prop. +Scheme eq_indd := Induction for eq Sort Prop. + +Definition proj1_inf (A B:Prop) (p : A/\B) := + let (a,b) := p in a. + +Theorem non_dep_dep_functional_choice : + FunctionalChoice -> DependentFunctionalChoice. +Proof. +intros AC_fun A B R H. +pose (B' := { x:A & B x }). +pose (R' := fun (x:A) (y:B') => projT1 y = x /\ R (projT1 y) (projT2 y)). +destruct (AC_fun A B' R') as (f,Hf). +intros x. destruct (H x) as (y,Hy). +exists (existT (fun x => B x) x y). split; trivial. +exists (fun x => eq_rect _ _ (projT2 (f x)) _ (proj1_inf (Hf x))). +intro x; destruct (Hf x) as (Heq,HR) using and_indd. +destruct (f x); simpl in *. +destruct Heq using eq_indd; trivial. +Qed. + +(** **** E. 2. Reification of dependent and non dependent functional relation are equivalent *) + +Definition DependentFunctionalRelReification_on (A:Type) (B:A -> Type) := + forall (R:forall x:A, B x -> Prop), + (forall x:A, exists! y : B x, R x y) -> + (exists f : (forall x:A, B x), forall x:A, R x (f x)). + +Notation DependentFunctionalRelReification := + (forall A (B:A->Type), DependentFunctionalRelReification_on B). + +(** The easy part *) + +Theorem dep_non_dep_functional_rel_reification : + DependentFunctionalRelReification -> FunctionalRelReification. +Proof. +intros DepFunReify A B R H. + destruct (DepFunReify A (fun _ => B) R H) as (f,Hf). + exists f; trivial. +Qed. + +(** Deriving choice on product types requires some computation on + singleton propositional types, so we need computational + conjunction projections and dependent elimination of conjunction + and equality *) + +Theorem non_dep_dep_functional_rel_reification : + FunctionalRelReification -> DependentFunctionalRelReification. +Proof. +intros AC_fun A B R H. +pose (B' := { x:A & B x }). +pose (R' := fun (x:A) (y:B') => projT1 y = x /\ R (projT1 y) (projT2 y)). +destruct (AC_fun A B' R') as (f,Hf). +intros x. destruct (H x) as (y,(Hy,Huni)). + exists (existT (fun x => B x) x y). repeat split; trivial. + intros (x',y') (Heqx',Hy'). + simpl in *. + destruct Heqx'. + rewrite (Huni y'); trivial. +exists (fun x => eq_rect _ _ (projT2 (f x)) _ (proj1_inf (Hf x))). +intro x; destruct (Hf x) as (Heq,HR) using and_indd. +destruct (f x); simpl in *. +destruct Heq using eq_indd; trivial. +Qed. + +(**********************************************************************) +(** *** F. Non contradiction of constructive descriptions wrt functional axioms of choice *) + +(** **** F. 1. Non contradiction of indefinite description *) + +Lemma relative_non_contradiction_of_indefinite_desc : + (ConstructiveIndefiniteDescription -> False) + -> (FunctionalChoice -> False). +Proof. +intros H AC_fun. +assert (AC_depfun := non_dep_dep_functional_choice AC_fun). +pose (A0 := { A:Type & { P:A->Prop & exists x, P x }}). +pose (B0 := fun x:A0 => projT1 x). +pose (R0 := fun x:A0 => fun y:B0 x => projT1 (projT2 x) y). +pose (H0 := fun x:A0 => projT2 (projT2 x)). +destruct (AC_depfun A0 B0 R0 H0) as (f, Hf). +apply H. +intros A P H'. +exists (f (existT (fun _ => sigT _) A + (existT (fun P => exists x, P x) P H'))). +pose (Hf' := + Hf (existT (fun _ => sigT _) A + (existT (fun P => exists x, P x) P H'))). assumption. Qed. + +Lemma constructive_indefinite_descr_fun_choice : + ConstructiveIndefiniteDescription -> FunctionalChoice. +Proof. +intros IndefDescr A B R H. +exists (fun x => proj1_sig (IndefDescr B (R x) (H x))). +intro x. +apply (proj2_sig (IndefDescr B (R x) (H x))). +Qed. + +(** **** F. 2. Non contradiction of definite description *) + +Lemma relative_non_contradiction_of_definite_descr : + (ConstructiveDefiniteDescription -> False) + -> (FunctionalRelReification -> False). +Proof. +intros H FunReify. +assert (DepFunReify := non_dep_dep_functional_rel_reification FunReify). +pose (A0 := { A:Type & { P:A->Prop & exists! x, P x }}). +pose (B0 := fun x:A0 => projT1 x). +pose (R0 := fun x:A0 => fun y:B0 x => projT1 (projT2 x) y). +pose (H0 := fun x:A0 => projT2 (projT2 x)). +destruct (DepFunReify A0 B0 R0 H0) as (f, Hf). +apply H. +intros A P H'. +exists (f (existT (fun _ => sigT _) A + (existT (fun P => exists! x, P x) P H'))). +pose (Hf' := + Hf (existT (fun _ => sigT _) A + (existT (fun P => exists! x, P x) P H'))). +assumption. +Qed. + +Lemma constructive_definite_descr_fun_reification : + ConstructiveDefiniteDescription -> FunctionalRelReification. +Proof. +intros DefDescr A B R H. +exists (fun x => proj1_sig (DefDescr B (R x) (H x))). +intro x. +apply (proj2_sig (DefDescr B (R x) (H x))). +Qed. + +(**********************************************************************) +(** *** G. Excluded-middle + definite description => computational excluded-middle *) + +(** The idea for the following proof comes from [ChicliPottierSimpson02] *) + +(** Classical logic and axiom of unique choice (i.e. functional + relation reification), as shown in [ChicliPottierSimpson02], + implies the double-negation of excluded-middle in [Set] (which is + incompatible with the impredicativity of [Set]). + + We adapt the proof to show that constructive definite description + transports excluded-middle from [Prop] to [Set]. + + [ChicliPottierSimpson02] Laurent Chicli, Loïc Pottier, Carlos + Simpson, Mathematical Quotients and Quotient Types in Coq, + Proceedings of TYPES 2002, Lecture Notes in Computer Science 2646, + Springer Verlag. *) + +Require Import Setoid. + +Theorem constructive_definite_descr_excluded_middle : + ConstructiveDefiniteDescription -> + (forall P:Prop, P \/ ~ P) -> (forall P:Prop, {P} + {~ P}). +Proof. +intros Descr EM P. +pose (select := fun b:bool => if b then P else ~P). +assert { b:bool | select b } as ([|],HP). + apply Descr. + rewrite <- unique_existence; split. + destruct (EM P). + exists true; trivial. + exists false; trivial. + intros [|] [|] H1 H2; simpl in *; reflexivity || contradiction. +left; trivial. +right; trivial. +Qed. diff --git a/theories/Logic/ClassicalChoice.v b/theories/Logic/ClassicalChoice.v index 5a633f84..bb8186ae 100644 --- a/theories/Logic/ClassicalChoice.v +++ b/theories/Logic/ClassicalChoice.v @@ -6,28 +6,40 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: ClassicalChoice.v 6401 2004-12-05 16:44:57Z herbelin $ i*) +(*i $Id: ClassicalChoice.v 8892 2006-06-04 17:59:53Z herbelin $ i*) -(** This file provides classical logic and functional choice *) +(** This file provides classical logic, and functional choice *) -(** This file extends ClassicalDescription.v with the axiom of choice. - As ClassicalDescription.v, it implies the double-negation of - excluded-middle in Set and implies a strongly classical - world. Especially it conflicts with impredicativity of Set, knowing - that true<>false in Set. -*) +(** This file extends ClassicalUniqueChoice.v with the axiom of choice. + As ClassicalUniqueChoice.v, it implies the double-negation of + excluded-middle in [Set] and leads to a classical world populated + with non computable functions. Especially it conflicts with the + impredicativity of [Set], knowing that [true<>false] in [Set]. *) -Require Export ClassicalDescription. +Require Export ClassicalUniqueChoice. Require Export RelationalChoice. Require Import ChoiceFacts. +Set Implicit Arguments. + +Definition subset (U:Type) (P Q:U->Prop) : Prop := forall x, P x -> Q x. + +Theorem singleton_choice : + forall (A : Type) (P : A->Prop), + (exists x : A, P x) -> exists P' : A->Prop, subset P' P /\ exists! x, P' x. +Proof. +intros A P H. +destruct (relational_choice unit A (fun _ => P) (fun _ => H)) as (R',(Hsub,HR')). +exists (R' tt); firstorder. +Qed. + Theorem choice : - forall (A B:Type) (R:A -> B -> Prop), - (forall x:A, exists y : B, R x y) -> - exists f : A -> B, (forall x:A, R x (f x)). + forall (A B : Type) (R : A->B->Prop), + (forall x : A, exists y : B, R x y) -> + exists f : A->B, (forall x : A, R x (f x)). Proof. intros A B. apply description_rel_choice_imp_funct_choice. -exact (description A B). +exact (unique_choice A B). exact (relational_choice A B). Qed. diff --git a/theories/Logic/ClassicalDescription.v b/theories/Logic/ClassicalDescription.v index ce3e279c..7053266a 100644 --- a/theories/Logic/ClassicalDescription.v +++ b/theories/Logic/ClassicalDescription.v @@ -6,73 +6,95 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: ClassicalDescription.v 5920 2004-07-16 20:01:26Z herbelin $ i*) +(*i $Id: ClassicalDescription.v 8892 2006-06-04 17:59:53Z herbelin $ i*) (** This file provides classical logic and definite description *) -(** Classical logic and definite description, as shown in [1], - implies the double-negation of excluded-middle in Set, hence it - implies a strongly classical world. Especially it conflicts with - impredicativity of Set, knowing that true<>false in Set. +(** Classical definite description operator (i.e. iota) implies + excluded-middle in [Set] and leads to a classical world populated + with non computable functions. It conflicts with the + impredicativity of [Set] *) - [1] Laurent Chicli, Loïc Pottier, Carlos Simpson, Mathematical - Quotients and Quotient Types in Coq, Proceedings of TYPES 2002, - Lecture Notes in Computer Science 2646, Springer Verlag. -*) +Set Implicit Arguments. Require Export Classical. +Require Import ChoiceFacts. -Axiom - dependent_description : - forall (A:Type) (B:A -> Type) (R:forall x:A, B x -> Prop), - (forall x:A, - exists y : B x, R x y /\ (forall y':B x, R x y' -> y = y')) -> - exists f : forall x:A, B x, (forall x:A, R x (f x)). +Notation Local "'inhabited' A" := A (at level 200, only parsing). + +Axiom constructive_definite_description : + forall (A : Type) (P : A->Prop), (exists! x : A, P x) -> { x : A | P x }. + +(** The idea for the following proof comes from [ChicliPottierSimpson02] *) + +Theorem excluded_middle_informative : forall P:Prop, {P} + {~ P}. +Proof. +apply + (constructive_definite_descr_excluded_middle + constructive_definite_description classic). +Qed. + +Theorem classical_definite_description : + forall (A : Type) (P : A->Prop), inhabited A -> + { x : A | (exists! x : A, P x) -> P x }. +Proof. +intros A P i. +destruct (excluded_middle_informative (exists! x, P x)) as [Hex|HnonP]. + apply constructive_definite_description with (P:= fun x => (exists! x : A, P x) -> P x). + destruct Hex as (x,(Hx,Huni)). + exists x; split. + intros _; exact Hx. + firstorder. +exists i; tauto. +Qed. + +(** Church's iota operator *) -(** Principle of definite descriptions (aka axiom of unique choice) *) +Definition iota (A : Type) (i:inhabited A) (P : A->Prop) : A + := proj1_sig (classical_definite_description P i). + +Definition iota_spec (A : Type) (i:inhabited A) (P : A->Prop) : + (exists! x:A, P x) -> P (iota i P) + := proj2_sig (classical_definite_description P i). + +(** Weaker lemmas (compatibility lemmas) *) + +Unset Implicit Arguments. + +Lemma dependent_description : + forall (A:Type) (B:A -> Type) (R:forall x:A, B x -> Prop), + (forall x:A, exists! y : B x, R x y) -> + (exists f : (forall x:A, B x), forall x:A, R x (f x)). +Proof. +intros A B R H. +assert (Hexuni:forall x, exists! y, R x y). + intro x. apply H. +exists (fun x => proj1_sig (constructive_definite_description (R x) (Hexuni x))). +intro x. +apply (proj2_sig (constructive_definite_description (R x) (Hexuni x))). +Qed. Theorem description : forall (A B:Type) (R:A -> B -> Prop), - (forall x:A, exists y : B, R x y /\ (forall y':B, R x y' -> y = y')) -> - exists f : A -> B, (forall x:A, R x (f x)). + (forall x : A, exists! y : B, R x y) -> + (exists f : A->B, forall x:A, R x (f x)). Proof. intros A B. apply (dependent_description A (fun _ => B)). Qed. -(** The followig proof comes from [1] *) +(** Axiom of unique "choice" (functional reification of functional relations) *) + +Set Implicit Arguments. -Theorem classic_set : ((forall P:Prop, {P} + {~ P}) -> False) -> False. +Require Import Setoid. + +Theorem unique_choice : + forall (A B:Type) (R:A -> B -> Prop), + (forall x:A, exists! y : B, R x y) -> + (exists f : A -> B, forall x:A, R x (f x)). Proof. -intro HnotEM. -set (R := fun A b => A /\ true = b \/ ~ A /\ false = b). -assert (H : exists f : Prop -> bool, (forall A:Prop, R A (f A))). -apply description. -intro A. -destruct (classic A) as [Ha| Hnota]. - exists true; split. - left; split; [ assumption | reflexivity ]. - intros y [[_ Hy]| [Hna _]]. - assumption. - contradiction. - exists false; split. - right; split; [ assumption | reflexivity ]. - intros y [[Ha _]| [_ Hy]]. - contradiction. - assumption. -destruct H as [f Hf]. -apply HnotEM. -intro P. -assert (HfP := Hf P). -(* Elimination from Hf to Set is not allowed but from f to Set yes ! *) -destruct (f P). - left. - destruct HfP as [[Ha _]| [_ Hfalse]]. - assumption. - discriminate. - right. - destruct HfP as [[_ Hfalse]| [Hna _]]. - discriminate. - assumption. +intros A B R H. +apply (description A B). +intro x. apply H. Qed. - diff --git a/theories/Logic/ClassicalEpsilon.v b/theories/Logic/ClassicalEpsilon.v new file mode 100644 index 00000000..b7293bec --- /dev/null +++ b/theories/Logic/ClassicalEpsilon.v @@ -0,0 +1,90 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: ClassicalEpsilon.v 8933 2006-06-09 14:08:38Z herbelin $ i*) + +(** *** This file provides classical logic and indefinite description + (Hilbert's epsilon operator) *) + +(** Classical epsilon's operator (i.e. indefinite description) implies + excluded-middle in [Set] and leads to a classical world populated + with non computable functions. It conflicts with the + impredicativity of [Set] *) + +Require Export Classical. +Require Import ChoiceFacts. + +Set Implicit Arguments. + +Notation Local "'inhabited' A" := A (at level 200, only parsing). + +Axiom constructive_indefinite_description : + forall (A : Type) (P : A->Prop), + (ex P) -> { x : A | P x }. + +Lemma constructive_definite_description : + forall (A : Type) (P : A->Prop), + (exists! x : A, P x) -> { x : A | P x }. +Proof. +intros; apply constructive_indefinite_description; firstorder. +Qed. + +Theorem excluded_middle_informative : forall P:Prop, {P} + {~ P}. +Proof. +apply + (constructive_definite_descr_excluded_middle + constructive_definite_description classic). +Qed. + +Theorem classical_indefinite_description : + forall (A : Type) (P : A->Prop), inhabited A -> + { x : A | ex P -> P x }. +Proof. +intros A P i. +destruct (excluded_middle_informative (exists x, P x)) as [Hex|HnonP]. + apply constructive_indefinite_description with (P:= fun x => ex P -> P x). + destruct Hex as (x,Hx). + exists x; intros _; exact Hx. + firstorder. +Qed. + +(** Hilbert's epsilon operator *) + +Definition epsilon (A : Type) (i:inhabited A) (P : A->Prop) : A + := proj1_sig (classical_indefinite_description P i). + +Definition epsilon_spec (A : Type) (i:inhabited A) (P : A->Prop) : + (ex P) -> P (epsilon i P) + := proj2_sig (classical_indefinite_description P i). + +Opaque epsilon. + +(** Open question: is classical_indefinite_description constructively + provable from [relational_choice] and + [constructive_definite_description] (at least, using the fact that + [functional_choice] is provable from [relational_choice] and + [unique_choice], we know that the double negation of + [classical_indefinite_description] is provable (see + [relative_non_contradiction_of_indefinite_desc]). *) + +(** Remark: we use [ex P] rather than [exists x, P x] (which is [ex + (fun x => P x)] to ease unification *) + +(** *** Weaker lemmas (compatibility lemmas) *) + +Theorem choice : + forall (A B : Type) (R : A->B->Prop), + (forall x : A, exists y : B, R x y) -> + (exists f : A->B, forall x : A, R x (f x)). +Proof. +intros A B R H. +exists (fun x => proj1_sig (constructive_indefinite_description (H x))). +intro x. +apply (proj2_sig (constructive_indefinite_description (H x))). +Qed. + diff --git a/theories/Logic/ClassicalFacts.v b/theories/Logic/ClassicalFacts.v index 91056250..70da74d3 100644 --- a/theories/Logic/ClassicalFacts.v +++ b/theories/Logic/ClassicalFacts.v @@ -1,3 +1,4 @@ +(* -*- coding: utf-8 -*- *) (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) (* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) @@ -6,7 +7,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: ClassicalFacts.v 8136 2006-03-05 21:57:47Z herbelin $ i*) +(*i $Id: ClassicalFacts.v 8892 2006-06-04 17:59:53Z herbelin $ i*) (** ** Some facts and definitions about classical logic diff --git a/theories/Logic/ClassicalUniqueChoice.v b/theories/Logic/ClassicalUniqueChoice.v new file mode 100644 index 00000000..79bef2af --- /dev/null +++ b/theories/Logic/ClassicalUniqueChoice.v @@ -0,0 +1,79 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: ClassicalUniqueChoice.v 8893 2006-06-04 18:04:53Z herbelin $ i*) + +(** This file provides classical logic and unique choice *) + +(** Classical logic and unique choice, as shown in + [ChicliPottierSimpson02], implies the double-negation of + excluded-middle in [Set], hence it implies a strongly classical + world. Especially it conflicts with the impredicativity of [Set]. + + [ChicliPottierSimpson02] Laurent Chicli, Loïc Pottier, Carlos + Simpson, Mathematical Quotients and Quotient Types in Coq, + Proceedings of TYPES 2002, Lecture Notes in Computer Science 2646, + Springer Verlag. *) + +Require Export Classical. + +Axiom + dependent_unique_choice : + forall (A:Type) (B:A -> Type) (R:forall x:A, B x -> Prop), + (forall x : A, exists! y : B x, R x y) -> + (exists f : (forall x:A, B x), forall x:A, R x (f x)). + +(** Unique choice reifies functional relations into functions *) + +Theorem unique_choice : + forall (A B:Type) (R:A -> B -> Prop), + (forall x:A, exists! y : B, R x y) -> + (exists f:A->B, forall x:A, R x (f x)). +Proof. +intros A B. +apply (dependent_unique_choice A (fun _ => B)). +Qed. + +(** The followig proof comes from [ChicliPottierSimpson02] *) + +Require Import Setoid. + +Theorem classic_set : ((forall P:Prop, {P} + {~ P}) -> False) -> False. +Proof. +intro HnotEM. +set (R := fun A b => A /\ true = b \/ ~ A /\ false = b). +assert (H : exists f : Prop -> bool, (forall A:Prop, R A (f A))). +apply unique_choice. +intro A. +destruct (classic A) as [Ha| Hnota]. + exists true; split. + left; split; [ assumption | reflexivity ]. + intros y [[_ Hy]| [Hna _]]. + assumption. + contradiction. + exists false; split. + right; split; [ assumption | reflexivity ]. + intros y [[Ha _]| [_ Hy]]. + contradiction. + assumption. +destruct H as [f Hf]. +apply HnotEM. +intro P. +assert (HfP := Hf P). +(* Elimination from Hf to Set is not allowed but from f to Set yes ! *) +destruct (f P). + left. + destruct HfP as [[Ha _]| [_ Hfalse]]. + assumption. + discriminate. + right. + destruct HfP as [[_ Hfalse]| [Hna _]]. + discriminate. + assumption. +Qed. + diff --git a/theories/Logic/Classical_Prop.v b/theories/Logic/Classical_Prop.v index f8b0e65b..ce3e84a7 100644 --- a/theories/Logic/Classical_Prop.v +++ b/theories/Logic/Classical_Prop.v @@ -6,7 +6,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Classical_Prop.v 8642 2006-03-17 10:09:02Z notin $ i*) +(*i $Id: Classical_Prop.v 8892 2006-06-04 17:59:53Z herbelin $ i*) (** Classical Propositional Logic *) @@ -22,6 +22,15 @@ unfold not in |- *; intros; elim (classic p); auto. intro NP; elim (H NP). Qed. +(** Peirce's law states [forall P Q:Prop, ((P -> Q) -> P) -> P]. + Thanks to [forall P, False -> P], it is equivalent to the + following form *) + +Lemma Peirce : forall P:Prop, ((P -> False) -> P) -> P. +Proof. +intros P H; destruct (classic P); auto. +Qed. + Lemma not_imply_elim : forall P Q:Prop, ~ (P -> Q) -> P. Proof. intros; apply NNPP; red in |- *. diff --git a/theories/Logic/DecidableType.v b/theories/Logic/DecidableType.v new file mode 100644 index 00000000..a38b111f --- /dev/null +++ b/theories/Logic/DecidableType.v @@ -0,0 +1,156 @@ +(***********************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) +(* \VV/ *************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(***********************************************************************) + +(* $Id: DecidableType.v 8933 2006-06-09 14:08:38Z herbelin $ *) + +Require Export SetoidList. +Set Implicit Arguments. +Unset Strict Implicit. + +(** * Types with decidable Equalities (but no ordering) *) + +Module Type DecidableType. + + Parameter t : Set. + + Parameter eq : t -> t -> Prop. + + Axiom eq_refl : forall x : t, eq x x. + Axiom eq_sym : forall x y : t, eq x y -> eq y x. + Axiom eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z. + + Parameter eq_dec : forall x y : t, { eq x y } + { ~ eq x y }. + + Hint Immediate eq_sym. + Hint Resolve eq_refl eq_trans. + +End DecidableType. + +(** * Additional notions about keys and datas used in FMap *) + +Module KeyDecidableType(D:DecidableType). + Import D. + + Section Elt. + Variable elt : Set. + Notation key:=t. + + Definition eqk (p p':key*elt) := eq (fst p) (fst p'). + Definition eqke (p p':key*elt) := + eq (fst p) (fst p') /\ (snd p) = (snd p'). + + Hint Unfold eqk eqke. + Hint Extern 2 (eqke ?a ?b) => split. + + (* eqke is stricter than eqk *) + + Lemma eqke_eqk : forall x x', eqke x x' -> eqk x x'. + Proof. + unfold eqk, eqke; intuition. + Qed. + + (* eqk, eqke are equalities *) + + Lemma eqk_refl : forall e, eqk e e. + Proof. auto. Qed. + + Lemma eqke_refl : forall e, eqke e e. + Proof. auto. Qed. + + Lemma eqk_sym : forall e e', eqk e e' -> eqk e' e. + Proof. auto. Qed. + + Lemma eqke_sym : forall e e', eqke e e' -> eqke e' e. + Proof. unfold eqke; intuition. Qed. + + Lemma eqk_trans : forall e e' e'', eqk e e' -> eqk e' e'' -> eqk e e''. + Proof. eauto. Qed. + + Lemma eqke_trans : forall e e' e'', eqke e e' -> eqke e' e'' -> eqke e e''. + Proof. + unfold eqke; intuition; [ eauto | congruence ]. + Qed. + + Hint Resolve eqk_trans eqke_trans eqk_refl eqke_refl. + Hint Immediate eqk_sym eqke_sym. + + Lemma InA_eqke_eqk : + forall x m, InA eqke x m -> InA eqk x m. + Proof. + unfold eqke; induction 1; intuition. + Qed. + Hint Resolve InA_eqke_eqk. + + Lemma InA_eqk : forall p q m, eqk p q -> InA eqk p m -> InA eqk q m. + Proof. + intros; apply InA_eqA with p; auto; apply eqk_trans; auto. + Qed. + + Definition MapsTo (k:key)(e:elt):= InA eqke (k,e). + Definition In k m := exists e:elt, MapsTo k e m. + + Hint Unfold MapsTo In. + + (* An alternative formulation for [In k l] is [exists e, InA eqk (k,e) l] *) + + Lemma In_alt : forall k l, In k l <-> exists e, InA eqk (k,e) l. + Proof. + firstorder. + exists x; auto. + induction H. + destruct y. + exists e; auto. + destruct IHInA as [e H0]. + exists e; auto. + Qed. + + Lemma MapsTo_eq : forall l x y e, eq x y -> MapsTo x e l -> MapsTo y e l. + Proof. + intros; unfold MapsTo in *; apply InA_eqA with (x,e); eauto. + Qed. + + Lemma In_eq : forall l x y, eq x y -> In x l -> In y l. + Proof. + destruct 2 as (e,E); exists e; eapply MapsTo_eq; eauto. + Qed. + + Lemma In_inv : forall k k' e l, In k ((k',e) :: l) -> eq k k' \/ In k l. + Proof. + inversion 1. + inversion_clear H0; eauto. + destruct H1; simpl in *; intuition. + Qed. + + Lemma In_inv_2 : forall k k' e e' l, + InA eqk (k, e) ((k', e') :: l) -> ~ eq k k' -> InA eqk (k, e) l. + Proof. + inversion_clear 1; compute in H0; intuition. + Qed. + + Lemma In_inv_3 : forall x x' l, + InA eqke x (x' :: l) -> ~ eqk x x' -> InA eqke x l. + Proof. + inversion_clear 1; compute in H0; intuition. + Qed. + + End Elt. + + Hint Unfold eqk eqke. + Hint Extern 2 (eqke ?a ?b) => split. + Hint Resolve eqk_trans eqke_trans eqk_refl eqke_refl. + Hint Immediate eqk_sym eqke_sym. + Hint Resolve InA_eqke_eqk. + Hint Unfold MapsTo In. + Hint Resolve In_inv_2 In_inv_3. + +End KeyDecidableType. + + + + + diff --git a/theories/Logic/DecidableTypeEx.v b/theories/Logic/DecidableTypeEx.v new file mode 100644 index 00000000..a4f99de2 --- /dev/null +++ b/theories/Logic/DecidableTypeEx.v @@ -0,0 +1,50 @@ +(***********************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) +(* \VV/ *************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(***********************************************************************) + +(* $Id: DecidableTypeEx.v 8933 2006-06-09 14:08:38Z herbelin $ *) + +Require Import DecidableType OrderedType OrderedTypeEx. +Set Implicit Arguments. +Unset Strict Implicit. + +(** * Examples of Decidable Type structures. *) + +(** A particular case of [DecidableType] where + the equality is the usual one of Coq. *) + +Module Type UsualDecidableType. + Parameter t : Set. + Definition eq := @eq t. + Definition eq_refl := @refl_equal t. + Definition eq_sym := @sym_eq t. + Definition eq_trans := @trans_eq t. + Parameter eq_dec : forall x y, { eq x y }+{~eq x y }. +End UsualDecidableType. + +(** a [UsualDecidableType] is in particular an [DecidableType]. *) + +Module UDT_to_DT (U:UsualDecidableType) <: DecidableType := U. + +(** An OrderedType can be seen as a DecidableType *) + +Module OT_as_DT (O:OrderedType) <: DecidableType. + Module OF := OrderedTypeFacts O. + Definition t := O.t. + Definition eq := O.eq. + Definition eq_refl := O.eq_refl. + Definition eq_sym := O.eq_sym. + Definition eq_trans := O.eq_trans. + Definition eq_dec := OF.eq_dec. +End OT_as_DT. + +(** (Usual) Decidable Type for [nat], [positive], [N], [Z] *) + +Module Nat_as_DT <: UsualDecidableType := OT_as_DT (Nat_as_OT). +Module Positive_as_DT <: UsualDecidableType := OT_as_DT (Positive_as_OT). +Module N_as_DT <: UsualDecidableType := OT_as_DT (N_as_OT). +Module Z_as_DT <: UsualDecidableType := OT_as_DT (Z_as_OT). diff --git a/theories/Logic/Diaconescu.v b/theories/Logic/Diaconescu.v index 3e94deda..19d5d7ec 100644 --- a/theories/Logic/Diaconescu.v +++ b/theories/Logic/Diaconescu.v @@ -1,3 +1,4 @@ +(* -*- coding: utf-8 -*- *) (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) (* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) @@ -6,26 +7,46 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Diaconescu.v 6401 2004-12-05 16:44:57Z herbelin $ i*) +(*i $Id: Diaconescu.v 8892 2006-06-04 17:59:53Z herbelin $ i*) -(** R. Diaconescu [Diaconescu] showed that the Axiom of Choice in Set Theory - entails Excluded-Middle; S. Lacas and B. Werner [LacasWerner] - adapted the proof to show that the axiom of choice in equivalence - classes entails Excluded-Middle in Type Theory. +(** Diaconescu showed that the Axiom of Choice entails Excluded-Middle + in topoi [Diaconescu75]. Lacas and Werner adapted the proof to show + that the axiom of choice in equivalence classes entails + Excluded-Middle in Type Theory [LacasWerner99]. - This is an adaptatation of the proof by Hugo Herbelin to show that - the relational form of the Axiom of Choice + Extensionality for - predicates entails Excluded-Middle + Three variants of Diaconescu's result in type theory are shown below. - [Diaconescu] R. Diaconescu, Axiom of Choice and Complementation, in - Proceedings of AMS, vol 51, pp 176-178, 1975. + A. A proof that the relational form of the Axiom of Choice + + Extensionality for Predicates entails Excluded-Middle (by Hugo + Herbelin) - [LacasWerner] S. Lacas, B Werner, Which Choices imply the excluded middle?, - preprint, 1999. + B. A proof that the relational form of the Axiom of Choice + Proof + Irrelevance entails Excluded-Middle for Equality Statements (by + Benjamin Werner) + C. A proof that extensional Hilbert epsilon's description operator + entails excluded-middle (taken from Bell [Bell93]) + + See also [Carlström] for a discussion of the connection between the + Extensional Axiom of Choice and Excluded-Middle + + [Diaconescu75] Radu Diaconescu, Axiom of Choice and Complementation, + in Proceedings of AMS, vol 51, pp 176-178, 1975. + + [LacasWerner99] Samuel Lacas, Benjamin Werner, Which Choices imply + the excluded middle?, preprint, 1999. + + [Bell93] John L. Bell, Hilbert's epsilon operator and classical + logic, Journal of Philosophical Logic, 22: 1-18, 1993 + + [Carlström04] Jesper Carlström, EM + Ext_ + AC_int <-> AC_ext, + Mathematical Logic Quaterly, vol 50(3), pp 236-240, 2004. *) -Section PredExt_GuardRelChoice_imp_EM. +(**********************************************************************) +(** *** A. Pred. Ext. + Rel. Axiom of Choice -> Excluded-Middle *) + +Section PredExt_RelChoice_imp_EM. (** The axiom of extensionality for predicates *) @@ -59,15 +80,9 @@ Qed. Require Import ChoiceFacts. -Variable rel_choice : forall A B:Type, RelationalChoice A B. +Variable rel_choice : RelationalChoice. -Lemma guarded_rel_choice : - forall (A B:Type) (P:A -> Prop) (R:A -> B -> Prop), - (forall x:A, P x -> exists y : B, R x y) -> - exists R' : A -> B -> Prop, - (forall x:A, - P x -> - exists y : B, R x y /\ R' x y /\ (forall y':B, R' x y' -> y = y')). +Lemma guarded_rel_choice : GuardedRelationalChoice. Proof. apply (rel_choice_and_proof_irrel_imp_guarded_rel_choice rel_choice proof_irrel). @@ -78,16 +93,19 @@ Qed. Require Import Bool. -Lemma AC : +Lemma AC_bool_subset_to_bool : exists R : (bool -> Prop) -> bool -> Prop, (forall P:bool -> Prop, (exists b : bool, P b) -> exists b : bool, P b /\ R P b /\ (forall b':bool, R P b' -> b = b')). Proof. - apply guarded_rel_choice with - (P := fun Q:bool -> Prop => exists y : _, Q y) - (R := fun (Q:bool -> Prop) (y:bool) => Q y). - exact (fun _ H => H). + destruct (guarded_rel_choice _ _ + (fun Q:bool -> Prop => exists y : _, Q y) + (fun (Q:bool -> Prop) (y:bool) => Q y)) as (R,(HRsub,HR)). + exact (fun _ H => H). + exists R; intros P HP. + destruct (HR P HP) as (y,(Hy,Huni)). + exists y; firstorder. Qed. (** The proof of the excluded middle *) @@ -98,7 +116,7 @@ Proof. intro P. (** first we exhibit the choice functional relation R *) -destruct AC as [R H]. +destruct AC_bool_subset_to_bool as [R H]. set (class_of_true := fun b => b = true \/ P). set (class_of_false := fun b => b = false \/ P). @@ -135,4 +153,152 @@ left; assumption. Qed. -End PredExt_GuardRelChoice_imp_EM. +End PredExt_RelChoice_imp_EM. + +(**********************************************************************) +(** *** B. Proof-Irrel. + Rel. Axiom of Choice -> Excl.-Middle for Equality *) + +(** This is an adaptation of Diaconescu's paradox exploiting that + proof-irrelevance is some form of extensionality *) + +Section ProofIrrel_RelChoice_imp_EqEM. + +Variable rel_choice : RelationalChoice. + +Variable proof_irrelevance : forall P:Prop , forall x y:P, x=y. + +(** Let [a1] and [a2] be two elements in some type [A] *) + +Variable A :Type. +Variables a1 a2 : A. + +(** We build the subset [A'] of [A] made of [a1] and [a2] *) + +Definition A' := sigT (fun x => x=a1 \/ x=a2). + +Definition a1':A'. +exists a1 ; auto. +Defined. + +Definition a2':A'. +exists a2 ; auto. +Defined. + +(** By proof-irrelevance, projection is a retraction *) + +Lemma projT1_injective : a1=a2 -> a1'=a2'. +Proof. + intro Heq ; unfold a1', a2', A'. + rewrite Heq. + replace (or_introl (a2=a2) (refl_equal a2)) + with (or_intror (a2=a2) (refl_equal a2)). + reflexivity. + apply proof_irrelevance. +Qed. + +(** But from the actual proofs of being in [A'], we can assert in the + proof-irrelevant world the existence of relevant boolean witnesses *) + +Lemma decide : forall x:A', exists y:bool , + (projT1 x = a1 /\ y = true ) \/ (projT1 x = a2 /\ y = false). +Proof. + intros [a [Ha1|Ha2]]; [exists true | exists false]; auto. +Qed. + +(** Thanks to the axiom of choice, the boolean witnesses move from the + propositional world to the relevant world *) + +Theorem proof_irrel_rel_choice_imp_eq_dec : a1=a2 \/ ~a1=a2. +Proof. + destruct + (rel_choice A' bool + (fun x y => projT1 x = a1 /\ y = true \/ projT1 x = a2 /\ y = false)) + as (R,(HRsub,HR)). + apply decide. + destruct (HR a1') as (b1,(Ha1'b1,_Huni1)). + destruct (HRsub a1' b1 Ha1'b1) as [(_, Hb1true)|(Ha1a2, _Hb1false)]. + destruct (HR a2') as (b2,(Ha2'b2,Huni2)). + destruct (HRsub a2' b2 Ha2'b2) as [(Ha2a1, _Hb2true)|(_, Hb2false)]. + left; symmetry; assumption. + right; intro H. + subst b1; subst b2. + rewrite (projT1_injective H) in Ha1'b1. + assert (false = true) by auto using Huni2. + discriminate. + left; assumption. +Qed. + +(** An alternative more concise proof can be done by directly using + the guarded relational choice *) + +Declare Implicit Tactic auto. + +Lemma proof_irrel_rel_choice_imp_eq_dec' : a1=a2 \/ ~a1=a2. +Proof. + assert (decide: forall x:A, x=a1 \/ x=a2 -> + exists y:bool, x=a1 /\ y=true \/ x=a2 /\ y=false). + intros a [Ha1|Ha2]; [exists true | exists false]; auto. + assert (guarded_rel_choice := + rel_choice_and_proof_irrel_imp_guarded_rel_choice + rel_choice + proof_irrelevance). + destruct + (guarded_rel_choice A bool + (fun x => x=a1 \/ x=a2) + (fun x y => x=a1 /\ y=true \/ x=a2 /\ y=false)) + as (R,(HRsub,HR)). + apply decide. + destruct (HR a1) as (b1,(Ha1b1,_Huni1)). left; reflexivity. + destruct (HRsub a1 b1 Ha1b1) as [(_, Hb1true)|(Ha1a2, _Hb1false)]. + destruct (HR a2) as (b2,(Ha2b2,Huni2)). right; reflexivity. + destruct (HRsub a2 b2 Ha2b2) as [(Ha2a1, _Hb2true)|(_, Hb2false)]. + left; symmetry; assumption. + right; intro H. + subst b1; subst b2; subst a1. + assert (false = true) by auto using Huni2, Ha1b1. + discriminate. + left; assumption. +Qed. + +End ProofIrrel_RelChoice_imp_EqEM. + +(**********************************************************************) +(** *** B. Extensional Hilbert's epsilon description operator -> Excluded-Middle *) + +(** Proof sketch from Bell [Bell93] (with thanks to P. Castéran) *) + +Notation Local "'inhabited' A" := A (at level 10, only parsing). + +Section ExtensionalEpsilon_imp_EM. + +Variable epsilon : forall A : Type, inhabited A -> (A -> Prop) -> A. + +Hypothesis epsilon_spec : + forall (A:Type) (i:inhabited A) (P:A->Prop), + (exists x, P x) -> P (epsilon A i P). + +Hypothesis epsilon_extensionality : + forall (A:Type) (i:inhabited A) (P Q:A->Prop), + (forall a, P a <-> Q a) -> epsilon A i P = epsilon A i Q. + +Notation Local eps := (epsilon bool true) (only parsing). + +Theorem extensional_epsilon_imp_EM : forall P:Prop, P \/ ~ P. +Proof. +intro P. +pose (B := fun y => y=false \/ P). +pose (C := fun y => y=true \/ P). +assert (B (eps B)) as [Hfalse|HP] + by (apply epsilon_spec; exists false; left; reflexivity). +assert (C (eps C)) as [Htrue|HP] + by (apply epsilon_spec; exists true; left; reflexivity). + right; intro HP. + assert (forall y, B y <-> C y) by (intro y; split; intro; right; assumption). + rewrite epsilon_extensionality with (1:=H) in Hfalse. + rewrite Htrue in Hfalse. + discriminate. +auto. +auto. +Qed. + +End ExtensionalEpsilon_imp_EM. diff --git a/theories/Logic/RelationalChoice.v b/theories/Logic/RelationalChoice.v index 11979057..ec168f09 100644 --- a/theories/Logic/RelationalChoice.v +++ b/theories/Logic/RelationalChoice.v @@ -6,15 +6,12 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: RelationalChoice.v 6001 2004-08-01 09:27:26Z herbelin $ i*) +(*i $Id: RelationalChoice.v 8892 2006-06-04 17:59:53Z herbelin $ i*) (** This file axiomatizes the relational form of the axiom of choice *) -Axiom - relational_choice : - forall (A B:Type) (R:A -> B -> Prop), - (forall x:A, exists y : B, R x y) -> - exists R' : A -> B -> Prop, - (forall x:A, - exists y : B, - R x y /\ R' x y /\ (forall y':B, R' x y' -> y = y')). +Axiom relational_choice : + forall (A B : Type) (R : A->B->Prop), + (forall x : A, exists y : B, R x y) -> + exists R' : A->B->Prop, + subrelation R' R /\ forall x : A, exists! y : B, R' x y. |