Section ChoiceSchemes.
Variables A B :Type.
Variable P:A->Prop.
(** ** Constructive choice and description *)
(** 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 = 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)).
(** 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 = 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! = 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_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 formulation of choice); Note also a typo in its intended formulation in [[Werner97]]. *)
Definition RepresentativeFunctionalChoice_on :=
forall R:A->A->Prop,
(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 = functional form of the (so-called extensional) axiom of choice from setoids *)
Definition SetoidFunctionalChoice_on :=
forall R : A -> A -> Prop,
forall T : A -> B -> Prop,
Equivalence R ->
(forall x x' y, R x x' -> T x y -> T x' y) ->
(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 = 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]]. *)
Definition GeneralizedSetoidFunctionalChoice_on :=
forall R : A -> A -> Prop,
forall S : B -> B -> Prop,
forall T : A -> B -> Prop,
Equivalence R ->
Equivalence S ->
(forall x x' y y', R x x' -> S y y' -> T x y -> T x' y') ->
(forall x, exists y, T x y) ->
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 = 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,
forall T : A -> B -> Prop,
Equivalence R ->
(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 *)
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) *)
Definition ConstructiveDefiniteDescription_on :=
forall P:A->Prop,
(exists! x, P x) -> { x:A | P x }.
(** ** Weakly classical choice and description *)
(** GAC_rel = guarded relational form of the (non extensional) axiom of choice *)
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 = guarded functional form of the (non extensional) axiom of choice *)
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)).
(** GAC! = guarded functional relation reification *)
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 = "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 = "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 = (weakly classical) indefinite description principle *)
Definition EpsilonStatement_on :=
forall P:A->Prop,
inhabited A -> { x:A | (exists x, P x) -> P x }.
(** D_iota = (weakly classical) definite description principle *)
Definition IotaStatement_on :=
forall P:A->Prop,
inhabited A -> { x:A | (exists! x, P x) -> P x }.
End ChoiceSchemes.
(** Generalized schemes *)
Notation RelationalChoice :=
(forall A B : Type, RelationalChoice_on A B).
Notation FunctionalChoice :=
(forall A B : Type, FunctionalChoice_on A B).
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).
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 :=
(forall A  B: Type, SetoidFunctionalChoice_on A B).
Notation GeneralizedSetoidFunctionalChoice :=
(forall A B : Type, GeneralizedSetoidFunctionalChoice_on A B).
Notation SimpleSetoidFunctionalChoice :=
(forall A B : Type, SimpleSetoidFunctionalChoice_on A B).
Notation GuardedRelationalChoice :=
(forall A B : Type, GuardedRelationalChoice_on A B).
Notation GuardedFunctionalChoice :=
(forall A B : Type, GuardedFunctionalChoice_on A B).
Notation GuardedFunctionalRelReification :=
(forall A B : Type, GuardedFunctionalRelReification_on A B).
Notation OmniscientRelationalChoice :=
(forall A B : Type, OmniscientRelationalChoice_on A B).
Notation OmniscientFunctionalChoice :=
(forall A B : Type, OmniscientFunctionalChoice_on A B).
Notation ConstructiveDefiniteDescription :=
(forall A : Type, ConstructiveDefiniteDescription_on A).
Notation ConstructiveIndefiniteDescription :=
(forall A : Type, ConstructiveIndefiniteDescription_on A).
Notation IotaStatement :=
(forall A : Type, IotaStatement_on A).
Notation EpsilonStatement :=
(forall A : Type, EpsilonStatement_on A).
(** 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 *)
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(** * 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 *)
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(** * AC_rel + AC! = 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) + functional relation reification (aka axiom of unique choice, or, principle of (parametric) definite descriptions) *)
(** This shows that the axiom of choice can be assumed (under its relational formulation) without known inconsistency with classical logic, though functional relation reification conflicts with classical logic *)
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(** * Connection between the guarded, non guarded and omniscient choices *)
(** 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 independence of general premises or subset types (themselves derivable from subtypes thanks to proof- irrelevance) *)
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(** ** AC_rel + PI -> GAC_rel and AC_rel + IGP -> GAC_rel and GAC_rel = OAC_rel *)
(** OAC_rel = GAC_rel *)
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(** ** AC_fun + IGP = GAC_fun = OAC_fun = AC_fun + Drinker *)
(** AC_fun + IGP = GAC_fun *)
(** AC_fun + Drinker = OAC_fun *)
(** This was already observed by Bell [[Bell]] *)
(** OAC_fun = GAC_fun *)
(** This is derivable from the intuitionistic equivalence between IGP and Drinker but we give a direct proof *)
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(** ** D_iota -> ID_iota  and  D_epsilon <-> ID_epsilon + Drinker *)
(** D_iota -> ID_iota *)
(** ID_epsilon + Drinker <-> D_epsilon *)
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(** * 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 functional relation reification and the functional form of the axiom of choice are equivalent on decidable relation with [nat] as codomain *)
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(** * AC_fun = AC_fun_dep = AC_trunc *)
(** ** Choice on dependent and non dependent function types are equivalent *)
(** The easy part *)
(** 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.
(** ** Functional choice and truncation choice are equivalent *)
(** ** Reification of dependent and non dependent functional relation  are equivalent *)
(** The easy part *)
(** 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 *)
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(** * Non contradiction of constructive descriptions wrt functional axioms of choice *)
(** ** Non contradiction of indefinite description *)
(** ** Non contradiction of definite description *)
(** Remark, the following corollaries morally hold:  Definition In_propositional_context (A:Type) := forall C:Prop, (A -> C) -> C.  Corollary constructive_definite_descr_in_prop_context_iff_fun_reification : In_propositional_context ConstructiveIndefiniteDescription <-> FunctionalChoice.  Corollary constructive_definite_descr_in_prop_context_iff_fun_reification : In_propositional_context ConstructiveDefiniteDescription <-> FunctionalRelReification.  but expecting [FunctionalChoice] (resp. [FunctionalRelReification]) to be applied on the same Type universes on both sides of the first (resp. second) equivalence breaks the stratification of universes. *)
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(** * 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.  *)
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(** * Choice => Dependent choice => Countable choice *)
(* The implications below are standard *)
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(** * About the axiom of choice over setoids                          *)
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(** ** Consequences of the choice of a representative in an equivalence class *)
(** *** This is a variant of Diaconescu and Goodman-Myhill theorems *)
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(** ** AC_fun_setoid = AC_fun_setoid_gen = AC_fun_setoid_simple       *)
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(** ** AC_fun_setoid = AC! + AC_fun_repr                              *)
(** Note: What characterization to give of RepresentativeFunctionalChoice? A formulation of it as a functional relation would certainly be equivalent to the formulation of SetoidFunctionalChoice as a functional relation, but in their functional forms, SetoidFunctionalChoice seems strictly stronger *)
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(** * AC_fun_setoid = AC_fun + Ext_fun_repr + EM                      *)
(** ** This is the main theorem in [[Carlström04]] *)
(** Note: all ingredients have a computational meaning when taken in separation. However, to compute with the functional choice, existential quantification has to be thought as a strong existential, which is incompatible with the computational content of excluded-middle *)

Theorem setoid_functional_choice_first_characterization :
FunctionalChoice /\ ExtensionalFunctionRepresentative /\ ExcludedMiddle <-> SetoidFunctionalChoice.