diff options
Diffstat (limited to 'theories/Logic/ChoiceFacts.v')
| -rw-r--r-- | theories/Logic/ChoiceFacts.v | 170 |
1 files changed, 115 insertions, 55 deletions
diff --git a/theories/Logic/ChoiceFacts.v b/theories/Logic/ChoiceFacts.v index 3d434b37..b2c4a049 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 *) @@ -6,7 +7,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: ChoiceFacts.v 10756 2008-04-04 17:10:45Z herbelin $ i*) +(*i $Id: ChoiceFacts.v 12363 2009-09-28 15:04:07Z letouzey $ i*) (** Some facts and definitions concerning choice and description in intuitionistic logic. @@ -18,9 +19,11 @@ description principles (a "set-theoretic" axiom of choice) - AC_fun = functional form of the (non extensional) axiom of choice (a "type-theoretic" axiom of choice) +- DC_fun = functional form of the dependent axiom of choice +- ACw_fun = functional form of the countable axiom of choice - AC! = functional relation reification (known as axiom of unique choice in topos theory, - sometimes called principle of definite description in + 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 @@ -47,9 +50,9 @@ description principles We let also -IPL_2 = 2nd-order impredicative minimal predicate logic (with ex. quant.) -IPL^2 = 2nd-order functional minimal predicate logic (with ex. quant.) -IPL_2^2 = 2nd-order impredicative, 2nd-order functional minimal pred. logic (with ex. quant.) +- IPL_2 = 2nd-order impredicative minimal predicate logic (with ex. quant.) +- IPL^2 = 2nd-order functional minimal predicate logic (with ex. quant.) +- IPL_2^2 = 2nd-order impredicative, 2nd-order functional minimal pred. logic (with ex. quant.) with no prerequisite on the non-emptyness of domains @@ -73,6 +76,8 @@ Table of contents 7. Definite description transports classical logic to the computational world +8. Choice -> Dependent choice -> Countable choice + References: [[Bell]] John L. Bell, Choice principles in intuitionistic set theory, @@ -81,7 +86,7 @@ unpublished.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 +[[Carlström05]] Jesper Carlström, Interpreting descriptions in intentional type theory, Journal of Symbolic Logic 70(2):488-514, 2005. *) @@ -116,6 +121,20 @@ Definition FunctionalChoice_on := (forall x : A, exists y : B, R x y) -> (exists f : A->B, forall x : A, R x (f x)). +(** DC_fun *) + +Definition FunctionalDependentChoice_on := + forall (R:A->A->Prop), + (forall x, exists y, R x y) -> forall x0, + (exists f : nat -> A, f 0 = x0 /\ forall n, R (f n) (f (S n))). + +(** ACw_fun *) + +Definition FunctionalCountableChoice_on := + forall (R:nat->A->Prop), + (forall n, exists y, R n y) -> + (exists f : nat -> A, forall n, R n (f n)). + (** AC! or Functional Relation Reification (known as Axiom of Unique Choice in topos theory; also called principle of definite description *) @@ -126,7 +145,7 @@ Definition FunctionalRelReification_on := (** 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 + Carlström's type theory with a constructive indefinite description operator) *) Definition ConstructiveIndefiniteDescription_on := @@ -134,7 +153,7 @@ Definition ConstructiveIndefiniteDescription_on := (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 + with proof-irrelevance, it may be connected to Carlström's and Stenlund's type theory with a constructive definite description operator) *) @@ -146,16 +165,16 @@ Definition ConstructiveDefiniteDescription_on := (** GAC_rel *) -Definition GuardedRelationalChoice_on := +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, + (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, +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)). @@ -163,34 +182,34 @@ Definition GuardedFunctionalChoice_on := (** GFR_fun *) Definition GuardedFunctionalRelReification_on := - forall P : A->Prop, forall R : A->B->Prop, + 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 := +Definition OmniscientRelationalChoice_on := forall R : A->B->Prop, - exists 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, +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 EpsilonStatement_on := +Definition EpsilonStatement_on := forall P:A->Prop, inhabited A -> { x:A | (exists x, P x) -> P x }. (** D_iota *) -Definition IotaStatement_on := +Definition IotaStatement_on := forall P:A->Prop, inhabited A -> { x:A | (exists! x, P x) -> P x }. @@ -202,12 +221,16 @@ Notation RelationalChoice := (forall A B, RelationalChoice_on A B). Notation FunctionalChoice := (forall A B, FunctionalChoice_on A B). +Definition FunctionalDependentChoice := + (forall A, FunctionalDependentChoice_on A). +Definition FunctionalCountableChoice := + (forall A, FunctionalCountableChoice_on A). Notation FunctionalChoiceOnInhabitedSet := (forall A B, inhabited B -> FunctionalChoice_on A B). Notation FunctionalRelReification := (forall A B, FunctionalRelReification_on A B). -Notation GuardedRelationalChoice := +Notation GuardedRelationalChoice := (forall A B, GuardedRelationalChoice_on A B). Notation GuardedFunctionalChoice := (forall A B, GuardedFunctionalChoice_on A B). @@ -219,14 +242,14 @@ Notation OmniscientRelationalChoice := Notation OmniscientFunctionalChoice := (forall A B, OmniscientFunctionalChoice_on A B). -Notation ConstructiveDefiniteDescription := +Notation ConstructiveDefiniteDescription := (forall A, ConstructiveDefiniteDescription_on A). -Notation ConstructiveIndefiniteDescription := +Notation ConstructiveIndefiniteDescription := (forall A, ConstructiveIndefiniteDescription_on A). -Notation IotaStatement := +Notation IotaStatement := (forall A, IotaStatement_on A). -Notation EpsilonStatement := +Notation EpsilonStatement := (forall A, EpsilonStatement_on A). (** Subclassical schemes *) @@ -235,7 +258,7 @@ Definition ProofIrrelevance := forall (A:Prop) (a1 a2:A), a1 = a2. Definition IndependenceOfGeneralPremises := - forall (A:Type) (P:A -> Prop) (Q:Prop), + forall (A:Type) (P:A -> Prop) (Q:Prop), inhabited A -> (Q -> exists x, P x) -> exists x, Q -> P x. @@ -270,7 +293,7 @@ Proof. apply HR'R; assumption. Qed. -Lemma funct_choice_imp_rel_choice : +Lemma funct_choice_imp_rel_choice : forall A B, FunctionalChoice_on A B -> RelationalChoice_on A B. Proof. intros A B FunCh R H. @@ -283,7 +306,7 @@ Proof. trivial. Qed. -Lemma funct_choice_imp_description : +Lemma funct_choice_imp_description : forall A B, FunctionalChoice_on A B -> FunctionalRelReification_on A B. Proof. intros A B FunCh R H. @@ -297,7 +320,7 @@ Proof. Qed. Corollary FunChoice_Equiv_RelChoice_and_ParamDefinDescr : - forall A B, FunctionalChoice_on A B <-> + forall A B, FunctionalChoice_on A B <-> RelationalChoice_on A B /\ FunctionalRelReification_on A B. Proof. intros A B; split. @@ -312,7 +335,7 @@ Qed. (** We show that the guarded formulations of the axiom of choice are equivalent to their "omniscient" variant and comes from the non guarded - formulation in presence either of the independance of general premises + formulation in presence either of the independance of general premises or subset types (themselves derivable from subtypes thanks to proof- irrelevance) *) @@ -341,12 +364,12 @@ Proof. Qed. Lemma rel_choice_indep_of_general_premises_imp_guarded_rel_choice : - forall A B, inhabited B -> RelationalChoice_on A B -> + 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. + intro x. apply IndPrem. exact Inh. intro Hx. apply H; assumption. exists (fun x y => P x /\ R' x y). firstorder. @@ -385,7 +408,7 @@ Qed. (** ** 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. @@ -446,7 +469,7 @@ Proof. Qed. Lemma fun_choice_and_small_drinker_imp_omniscient_fun_choice : - FunctionalChoiceOnInhabitedSet -> SmallDrinker'sParadox + FunctionalChoiceOnInhabitedSet -> SmallDrinker'sParadox -> OmniscientFunctionalChoice. Proof. intros AC_fun Drinker A B R Inh. @@ -456,10 +479,10 @@ Proof. Qed. Corollary fun_choice_and_small_drinker_iff_omniscient_fun_choice : - FunctionalChoiceOnInhabitedSet /\ SmallDrinker'sParadox + FunctionalChoiceOnInhabitedSet /\ SmallDrinker'sParadox <-> OmniscientFunctionalChoice. Proof. - auto decomp using + auto decomp using omniscient_fun_choice_imp_small_drinker, omniscient_fun_choice_imp_fun_choice, fun_choice_and_small_drinker_imp_omniscient_fun_choice. @@ -510,7 +533,7 @@ Lemma constructive_indefinite_description_and_small_drinker_imp_epsilon : SmallDrinker'sParadox -> ConstructiveIndefiniteDescription -> EpsilonStatement. Proof. - intros Drinkers D_epsilon A P Inh; + intros Drinkers D_epsilon A P Inh; apply D_epsilon; apply Drinkers; assumption. Qed. @@ -542,7 +565,7 @@ Qed. 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 + relation with [nat] as codomain *) Require Import Wf_nat. @@ -552,10 +575,10 @@ 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, - FunctionalRelReification_on A nat -> - forall R:A->nat->Prop, +Lemma classical_denumerable_description_imp_fun_choice : + forall A:Type, + FunctionalRelReification_on A nat -> + forall R:A->nat->Prop, (forall x y, decidable (R x y)) -> FunctionalChoice_on_rel R. Proof. intros A Descr. @@ -563,7 +586,7 @@ Proof. set (R':= fun x y => R x y /\ forall y', R x y' -> y <= y'). destruct (Descr R') as (f,Hf). intro x. - apply (dec_inh_nat_subset_has_unique_least_element (R x)). + apply (dec_inh_nat_subset_has_unique_least_element (R x)). apply Rdec. apply (H x). exists f. @@ -582,12 +605,12 @@ Definition DependentFunctionalChoice_on (A:Type) (B:A -> Type) := (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 := +Notation DependentFunctionalChoice := (forall A (B:A->Type), DependentFunctionalChoice_on B). (** The easy part *) -Theorem dep_non_dep_functional_choice : +Theorem dep_non_dep_functional_choice : DependentFunctionalChoice -> FunctionalChoice. Proof. intros AC_depfun A B R H. @@ -606,12 +629,12 @@ 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 : +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)). + 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. @@ -633,7 +656,7 @@ Notation DependentFunctionalRelReification := (** The easy part *) -Theorem dep_non_dep_functional_rel_reification : +Theorem dep_non_dep_functional_rel_reification : DependentFunctionalRelReification -> FunctionalRelReification. Proof. intros DepFunReify A B R H. @@ -646,12 +669,12 @@ Qed. conjunction projections and dependent elimination of conjunction and equality *) -Theorem non_dep_dep_functional_rel_reification : +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)). + 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. @@ -665,7 +688,7 @@ Proof. destruct Heq using eq_indd; trivial. Qed. -Corollary dep_iff_non_dep_functional_rel_reification : +Corollary dep_iff_non_dep_functional_rel_reification : FunctionalRelReification <-> DependentFunctionalRelReification. Proof. auto decomp using @@ -764,7 +787,7 @@ be applied on the same Type universes on both sides of the first We adapt the proof to show that constructive definite description transports excluded-middle from [Prop] to [Set]. - [[ChicliPottierSimpson02]] Laurent Chicli, Loïc Pottier, Carlos + [[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. *) @@ -786,14 +809,51 @@ Proof. intros [|] [|] H1 H2; simpl in *; reflexivity || contradiction. left; trivial. right; trivial. -Qed. +Qed. Corollary fun_reification_descr_computational_excluded_middle_in_prop_context : FunctionalRelReification -> - (forall P:Prop, P \/ ~ P) -> + (forall P:Prop, P \/ ~ P) -> forall C:Prop, ((forall P:Prop, {P} + {~ P}) -> C) -> C. Proof. intros FunReify EM C; auto decomp using constructive_definite_descr_excluded_middle, (relative_non_contradiction_of_definite_descr (C:=C)). Qed. + +(**********************************************************************) +(** * Choice => Dependent choice => Countable choice *) + +(* The implications below are standard *) + +Require Import Arith. + +Theorem functional_choice_imp_functional_dependent_choice : + FunctionalChoice -> FunctionalDependentChoice. +Proof. + intros FunChoice A R HRfun x0. + apply FunChoice in HRfun as (g,Rg). + set (f:=fix f n := match n with 0 => x0 | S n' => g (f n') end). + exists f; firstorder. +Qed. + +Theorem functional_dependent_choice_imp_functional_countable_choice : + FunctionalDependentChoice -> FunctionalCountableChoice. +Proof. + intros H A R H0. + set (R' (p q:nat*A) := fst q = S (fst p) /\ R (fst p) (snd q)). + destruct (H0 0) as (y0,Hy0). + destruct H with (R:=R') (x0:=(0,y0)) as (f,(Hf0,HfS)). + intro x; destruct (H0 (fst x)) as (y,Hy). + exists (S (fst x),y). + red. auto. + assert (Heq:forall n, fst (f n) = n). + induction n. + rewrite Hf0; reflexivity. + specialize HfS with n; destruct HfS as (->,_); congruence. + exists (fun n => snd (f (S n))). + intro n'. specialize HfS with n'. + destruct HfS as (_,HR). + rewrite Heq in HR. + assumption. +Qed. |