diff options
Diffstat (limited to 'theories')
| -rw-r--r-- | theories/Logic/ChoiceFacts.v | 283 |
1 files changed, 131 insertions, 152 deletions
diff --git a/theories/Logic/ChoiceFacts.v b/theories/Logic/ChoiceFacts.v index f1f20606b..116897f4c 100644 --- a/theories/Logic/ChoiceFacts.v +++ b/theories/Logic/ChoiceFacts.v @@ -8,94 +8,9 @@ (************************************************************************) (** Some facts and definitions concerning choice and description in - intuitionistic 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) -- 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 - the context of constructive type theory, sometimes - called axiom of no choice) - -- AC_fun_repr = functional choice of a representative in an equivalence class -- AC_fun_setoid_gen = functional form of the general form of the (so-called - extensional) axiom of choice over setoids -- AC_fun_setoid = functional form of the (so-called extensional) axiom of - choice from setoids -- AC_fun_setoid_simple = functional form of the (so-called extensional) axiom of - choice from setoids on locally compatible relations - -- 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]]) -- EM = excluded-middle - -- Ext_pred_repr = choice of a representative among extensional predicates -- Ext_pred = extensionality of predicates -- Ext_fun_prop_repr = choice of a representative among extensional functions to Prop - -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.) - -with no prerequisite on the non-emptiness of domains - -Table of contents - -1. Definitions - -2. IPL_2^2 |- AC_rel + AC! = AC_fun - -3.1. typed IPL_2 + Sigma-types + PI |- AC_rel = GAC_rel and IPL_2 |- AC_rel + IGP -> GAC_rel and IPL_2 |- GAC_rel = OAC_rel - -3.2. IPL^2 |- AC_fun + IGP = GAC_fun = OAC_fun = AC_fun + Drinker - -3.3. D_iota -> ID_iota and D_epsilon <-> ID_epsilon + Drinker - -4. Derivability of choice for decidable relations with well-ordered codomain - -5. Equivalence of choices on dependent or non dependent functional types - -6. Non contradiction of constructive descriptions wrt functional choices - -7. Definite description transports classical logic to the computational world - -8. Choice -> Dependent choice -> Countable choice - -9.1. AC_fun_ext = AC_fun + Ext_fun_repr + EM - -9.2. AC_fun_ext = AC_fun + Ext_prop_fun_repr + PI - -References: - + intuitionistic logic. *) +(** * References: *) +(** [[Bell]] John L. Bell, Choice principles in intuitionistic set theory, unpublished. @@ -133,47 +48,75 @@ Variable P:A->Prop. (** ** Constructive choice and description *) -(** AC_rel *) +(** AC_rel = relational form of the (non extensional) axiom of choice + (a "set-theoretic" axiom of choice) *) 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 *) +(** AC_fun = functional form of the (non extensional) axiom of choice + (a "type-theoretic" axiom of choice) *) (* Note: This is called Type-Theoretic Description Axiom (TTDA) in [[Werner97]] (using a non-standard meaning of "description"). This is called intensional axiom of choice (AC_int) in [[Carlström04]] *) +Definition FunctionalChoice_on_rel (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 := 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)). -(** DC_fun *) +(** AC_fun_dep = functional form of the (non extensional) axiom of + choice, with dependent functions *) +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)). + +(** AC_trunc = axiom of choice for propositional truncations + (truncation and quantification commute) *) +Definition InhabitedForallCommute_on (A : Type) (B : A -> Type) := + (forall x, inhabited (B x)) -> inhabited (forall x, B x). + +(** DC_fun = functional form of the dependent axiom of choice *) 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 *) +(** ACw_fun = functional form of the countable axiom of choice *) 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 *) +(** 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, sometimes + called axiom of no choice) *) 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)). -(** AC_fun_repr *) +(** AC_dep! = functional relation reification, with dependent functions + see AC! *) +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)). + +(** AC_fun_repr = functional choice of a representative in an equivalence class *) (* Note: This is called Type-Theoretic Choice Axiom (TTCA) in [[Werner97]] (by reference to the extensional set-theoretic @@ -187,7 +130,8 @@ Definition RepresentativeFunctionalChoice_on := (Equivalence R) -> (exists f : A->A, forall x : A, (R x (f x)) /\ forall x', R x x' -> f x = f x'). -(** AC_fun_setoid *) +(** AC_fun_setoid = functional form of the (so-called extensional) axiom of + choice from setoids *) Definition SetoidFunctionalChoice_on := forall R : A -> A -> Prop, @@ -197,7 +141,8 @@ Definition SetoidFunctionalChoice_on := (forall x, exists y, T x y) -> exists f : A -> B, forall x : A, T x (f x) /\ (forall x' : A, R x x' -> f x = f x'). -(** AC_fun_setoid_gen *) +(** AC_fun_setoid_gen = functional form of the general form of the (so-called + extensional) axiom of choice over setoids *) (* Note: This is called extensional axiom of choice (AC_ext) in [[Carlström04]]. *) @@ -213,7 +158,8 @@ Definition GeneralizedSetoidFunctionalChoice_on := exists f : A -> B, forall x : A, T x (f x) /\ (forall x' : A, R x x' -> S (f x) (f x')). -(** AC_fun_setoid_simple *) +(** AC_fun_setoid_simple = functional form of the (so-called extensional) axiom of + choice from setoids on locally compatible relations *) Definition SimpleSetoidFunctionalChoice_on A B := forall R : A -> A -> Prop, @@ -222,19 +168,19 @@ Definition SimpleSetoidFunctionalChoice_on A B := (forall x, exists y, forall x', R x x' -> T x' y) -> exists f : A -> B, forall x : A, T x (f x) /\ (forall x' : A, R x x' -> f x = f x'). -(** ID_epsilon (constructive version of indefinite description; - combined with proof-irrelevance, it may be connected to - Carlström's type theory with a constructive indefinite description - operator) *) +(** ID_epsilon = constructive version of indefinite description; + combined with proof-irrelevance, it may be connected to + Carlström'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 Carlström's and - Stenlund's type theory with a constructive definite description - operator) *) +(** ID_iota = constructive version of definite description; + combined with proof-irrelevance, it may be connected to + Carlström's and Stenlund's type theory with a + constructive definite description operator) *) Definition ConstructiveDefiniteDescription_on := forall P:A->Prop, @@ -242,7 +188,7 @@ Definition ConstructiveDefiniteDescription_on := (** ** Weakly classical choice and description *) -(** GAC_rel *) +(** GAC_rel = guarded relational form of the (non extensional) axiom of choice *) Definition GuardedRelationalChoice_on := forall P : A->Prop, forall R : A->B->Prop, @@ -250,7 +196,7 @@ Definition GuardedRelationalChoice_on := (exists R' : A->B->Prop, subrelation R' R /\ forall x, P x -> exists! y, R' x y). -(** GAC_fun *) +(** GAC_fun = guarded functional form of the (non extensional) axiom of choice *) Definition GuardedFunctionalChoice_on := forall P : A->Prop, forall R : A->B->Prop, @@ -258,7 +204,7 @@ Definition GuardedFunctionalChoice_on := (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 *) +(** GAC! = guarded functional relation reification *) Definition GuardedFunctionalRelReification_on := forall P : A->Prop, forall R : A->B->Prop, @@ -266,27 +212,28 @@ Definition GuardedFunctionalRelReification_on := (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 *) +(** OAC_rel = "omniscient" relational form of the (non extensional) axiom of choice *) 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 *) +(** OAC_fun = "omniscient" functional form of the (non extensional) axiom of choice + (called AC* in Bell [[Bell]]) *) 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 *) +(** D_epsilon = (weakly classical) indefinite description principle *) Definition EpsilonStatement_on := forall P:A->Prop, inhabited A -> { x:A | (exists x, P x) -> P x }. -(** D_iota *) +(** D_iota = (weakly classical) definite description principle *) Definition IotaStatement_on := forall P:A->Prop, @@ -300,14 +247,20 @@ Notation RelationalChoice := (forall A B : Type, RelationalChoice_on A B). Notation FunctionalChoice := (forall A B : Type, FunctionalChoice_on A B). -Definition FunctionalDependentChoice := +Notation DependentFunctionalChoice := + (forall A (B:A->Type), DependentFunctionalChoice_on B). +Notation InhabitedForallCommute := + (forall A (B : A -> Type), InhabitedForallCommute_on B). +Notation FunctionalDependentChoice := (forall A : Type, FunctionalDependentChoice_on A). -Definition FunctionalCountableChoice := +Notation FunctionalCountableChoice := (forall A : Type, FunctionalCountableChoice_on A). Notation FunctionalChoiceOnInhabitedSet := (forall A B : Type, inhabited B -> FunctionalChoice_on A B). Notation FunctionalRelReification := (forall A B : Type, FunctionalRelReification_on A B). +Notation DependentFunctionalRelReification := + (forall A (B:A->Type), DependentFunctionalRelReification_on B). Notation RepresentativeFunctionalChoice := (forall A : Type, RepresentativeFunctionalChoice_on A). Notation SetoidFunctionalChoice := @@ -341,38 +294,87 @@ Notation EpsilonStatement := (** Subclassical schemes *) +(** PI = proof irrelevance *) Definition ProofIrrelevance := forall (A:Prop) (a1 a2:A), a1 = a2. +(** IGP = independence of general premises + (an unconstrained generalisation of the constructive principle of + independence of premises) *) Definition IndependenceOfGeneralPremises := forall (A:Type) (P:A -> Prop) (Q:Prop), inhabited A -> (Q -> exists x, P x) -> exists x, Q -> P x. +(** Drinker = drinker's paradox (small form) + (called Ex in Bell [[Bell]]) *) Definition SmallDrinker'sParadox := forall (A:Type) (P:A -> Prop), inhabited A -> exists x, (exists x, P x) -> P x. +(** EM = excluded-middle *) Definition ExcludedMiddle := forall P:Prop, P \/ ~ P. (** Extensional schemes *) +(** Ext_prop_repr = choice of a representative among extensional propositions *) Local Notation ExtensionalPropositionRepresentative := (forall (A:Type), exists h : Prop -> Prop, forall P : Prop, (P <-> h P) /\ forall Q, (P <-> Q) -> h P = h Q). +(** Ext_pred_repr = choice of a representative among extensional predicates *) Local Notation ExtensionalPredicateRepresentative := (forall (A:Type), exists h : (A->Prop) -> (A->Prop), forall (P : A -> Prop), (forall x, P x <-> h P x) /\ forall Q, (forall x, P x <-> Q x) -> h P = h Q). +(** Ext_fun_repr = choice of a representative among extensional functions *) Local Notation ExtensionalFunctionRepresentative := (forall (A B:Type), exists h : (A->B) -> (A->B), forall (f : A -> B), (forall x, f x = h f x) /\ forall g, (forall x, f x = g x) -> h f = h g). +(** 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.) + +with no prerequisite on the non-emptiness of domains +*) + +(**********************************************************************) +(** * Table of contents *) + +(* This is very fragile. *) +(** +1. Definitions + +2. IPL_2^2 |- AC_rel + AC! = AC_fun + +3.1. typed IPL_2 + Sigma-types + PI |- AC_rel = GAC_rel and IPL_2 |- AC_rel + IGP -> GAC_rel and IPL_2 |- GAC_rel = OAC_rel + +3.2. IPL^2 |- AC_fun + IGP = GAC_fun = OAC_fun = AC_fun + Drinker + +3.3. D_iota -> ID_iota and D_epsilon <-> ID_epsilon + Drinker + +4. Derivability of choice for decidable relations with well-ordered codomain + +5. AC_fun = AC_fun_dep = AC_trunc + +6. Non contradiction of constructive descriptions wrt functional choices + +7. Definite description transports classical logic to the computational world + +8. Choice -> Dependent choice -> Countable choice + +9.1. AC_fun_setoid = AC_fun + Ext_fun_repr + EM + +9.2. AC_fun_setoid = AC_fun + Ext_pred_repr + PI + *) + (**********************************************************************) (** * AC_rel + AC! = AC_fun @@ -400,9 +402,6 @@ Proof. apply HR'R; assumption. Qed. -Notation description_rel_choice_imp_funct_choice := - functional_rel_reification_and_rel_choice_imp_fun_choice (compat "8.6"). - Lemma fun_choice_imp_rel_choice : forall A B : Type, FunctionalChoice_on A B -> RelationalChoice_on A B. Proof. @@ -416,8 +415,6 @@ Proof. trivial. Qed. -Notation funct_choice_imp_rel_choice := fun_choice_imp_rel_choice (compat "8.6"). - Lemma fun_choice_imp_functional_rel_reification : forall A B : Type, FunctionalChoice_on A B -> FunctionalRelReification_on A B. Proof. @@ -431,8 +428,6 @@ Proof. exists f; exact H0. Qed. -Notation funct_choice_imp_description := fun_choice_imp_functional_rel_reification (compat "8.6"). - Corollary fun_choice_iff_rel_choice_and_functional_rel_reification : forall A B : Type, FunctionalChoice_on A B <-> RelationalChoice_on A B /\ FunctionalRelReification_on A B. @@ -444,8 +439,6 @@ Proof. intros [H H0]; exact (functional_rel_reification_and_rel_choice_imp_fun_choice H0 H). Qed. -Notation FunChoice_Equiv_RelChoice_and_ParamDefinDescr := - fun_choice_iff_rel_choice_and_functional_rel_reification (compat "8.6"). (**********************************************************************) (** * Connection between the guarded, non guarded and omniscient choices *) @@ -687,10 +680,6 @@ Qed. Require Import Wf_nat. Require Import Decidable. -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 -> @@ -712,18 +701,10 @@ Proof. Qed. (**********************************************************************) -(** * Choice on dependent and non dependent function types are equivalent *) +(** * AC_fun = AC_fun_dep = AC_trunc *) (** ** 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 : @@ -760,13 +741,7 @@ Proof. destruct Heq using eq_indd; trivial. Qed. -(** Functional choice can be reformulated as a property on [inhabited] *) - -Definition InhabitedForallCommute_on (A : Type) (B : A -> Type) := - (forall x, inhabited (B x)) -> inhabited (forall x, B x). - -Notation InhabitedForallCommute := - (forall A (B : A -> Type), InhabitedForallCommute_on B). +(** ** Functional choice and truncation choice are equivalent *) Theorem functional_choice_to_inhabited_forall_commute : FunctionalChoice -> InhabitedForallCommute. @@ -795,14 +770,6 @@ Qed. (** ** 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 : @@ -1337,3 +1304,15 @@ Proof. apply repr_fun_choice_imp_excluded_middle. now apply setoid_fun_choice_imp_repr_fun_choice. Qed. + +(**********************************************************************) +(** * Compatibility notations *) +Notation description_rel_choice_imp_funct_choice := + functional_rel_reification_and_rel_choice_imp_fun_choice (compat "8.6"). + +Notation funct_choice_imp_rel_choice := fun_choice_imp_rel_choice (compat "8.6"). + +Notation FunChoice_Equiv_RelChoice_and_ParamDefinDescr := + fun_choice_iff_rel_choice_and_functional_rel_reification (compat "8.6"). + +Notation funct_choice_imp_description := fun_choice_imp_functional_rel_reification (compat "8.6"). |