Library Coq.Logic.ChoiceFacts


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)
  • 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 = 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
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
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.
[Carlström05] Jesper Carlström, Interpreting descriptions in intentional type theory, Journal of Symbolic Logic 70(2):488-514, 2005.

Set Implicit Arguments.

Definitions

Choice, reification and description schemes

Section ChoiceSchemes.

Variables A B :Type.

Variable P:A->Prop.

Variable 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)).

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

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 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

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 EpsilonStatement_on :=
  forall P:A->Prop,
    inhabited A -> { x:A | (exists x, P x) -> P x }.

D_iota

Definition IotaStatement_on :=
  forall P:A->Prop,
    inhabited A -> { x:A | (exists! x, P x) -> P x }.

End ChoiceSchemes.

Generalized schemes
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.

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

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 independance of general premises or subset types (themselves derivable from subtypes thanks to proof- irrelevance)

AC_rel + PI -> GAC_rel and AC_rel + IGP -> GAC_rel and GAC_rel = OAC_rel

OAC_rel = GAC_rel

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

D_iota -> ID_iota and D_epsilon <-> ID_epsilon + Drinker

D_iota -> ID_iota
ID_epsilon + Drinker <-> D_epsilon

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

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 ->
    forall R:A->nat->Prop,
      (forall x y, decidable (R x y)) -> FunctionalChoice_on_rel R.

Choice on dependent and non dependent function types are equivalent

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
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.

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
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

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.

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}).

Corollary fun_reification_descr_computational_excluded_middle_in_prop_context :
  FunctionalRelReification ->
  (forall P:Prop, P \/ ~ P) ->
  forall C:Prop, ((forall P:Prop, {P} + {~ P}) -> C) -> C.

Choice => Dependent choice => Countable choice