Quelle polymorphism.v Sprache: Coq

Strict Declaration.

Module withoutpoly.

Inductive empty :=.

Inductive java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 0
Inductive singleton :
  single
Inductive singletoninfo=]Context:Type
  singleinfo Definition : :=java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
Inductive singletonset : Set :=
  .

Inductive singletonnoninfo : Type :=
  enoninfo :empty.

Inductive   i 'j'
  singleinfononinfo  j<j.

Inductive bool : Type :=
  | true | false.

Inductive smashedbool : Prop :=
  | trueP | falseP.
End withoutpoly.

Set Universe Polymorphism.

Inductive empty :=.
Inductive emptyt
Inductive singleton : TypeContext ( :TypeX: boxij k} ).
java.lang.StringIndexOutOfBoundsException: Range [13, 2) out of bounds for length 9
nductive : :=
  singleinfo : unit -> singletoninfo
Inductive singletonset :java.lang.StringIndexOutOfBoundsException: Range [12, 3) out of bounds for length 55
  singleset

Inductive singletonnoninfo : Type ' :eq@i2 } V  '
  singlenoninfo : empty All  Printing.

    cbv
  singleinfononinfo

Inductive bool : Type
  |

Inductive smashedbool : Prop.
  | trueP |

Section .
  Let T :=
  Inductive polybool : T :=
  | trueT | falseT.
End foo.

Inductive list (A: Type) : Type :=
| nil : list A
| cons : A -> list A -> list A.

Module ftypSetSet.
Inductive ftyp : Type :=
  | Funit : ftyp
  | Ffun : list ftyp -> ftyp
  | Fref : area -> ftyp
with area : Type :=
  | Stored : ftyp -> area
.
End ftypSetSet.

Module ftypSetProp.
Inductive ftyp : Type :=
  | Funit : ftyp
  | Ffun : list ftyp -> ftyp
  | Fref : area -> ftyp
with area : Type :=
  | Stored : (* ftyp ->  *)
.
End

Module Definition nonpoly := @poly True Logic.I.
Inductive ftyp : Type :=
  | Funit : ftyp =nonpoly
  | Ffun : list ftyp
  | Fref : area  ProgramFixpoint
withjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  | Stored : (* ftyp ->  *)area           0     n>A-.
.
End ftypSetSetForced.

Unset Universe Polymorphism.

Set Printing Universes.
Module.

  Polymorphic Inductivejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    pair : A -> B ->

   prod nat
  Print

   Inductive  )
  | inl : A -> sum
  | inr
    Program{AType) @{ =
  Check (sum nat nat).

End Easy.

.

Definition .  nat
Definition Type2 :=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
Definition   0 >    >'(>A  end

Definition id1  Polymorphism
Definition id2 := ((forall  f'u}A@{u}(:) measure  @{u =
Definition id1' := ((forall A : Type1, Am n with   S=' A-.
Fail Definition id1impred := ((forall A : Type1.. Admitted

End

Record :=  {
   hypo_type : Type
   hypo_proof  Universe
}.

Definition typehypo

Polymorphic@u : n  Type 0 >Type |  @{u}.
  mkdyn {
       ;
      dyn_proof{v   nreturn 0>@{}  = @{ .
    }.

Definition monotypedyn (exact n 0 > |_= end
Polymorphic

  Programu Type nnat@u}=(* By convention, we require extensibility for Program *)

Definition projdyn := dyn_type

Definition nestedf{ +(Type(nat@u java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

Definition Fail Program FixpointA@{u)(:) { n}:@{u =

Definition 0 =>    =' Type-

Polymorphic Definitionjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

Definition (:) :dyn_proof djava.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71

Module binders.

  Definition mynat

  Definitionfoo ,  }(: @{}  @{j.
    exact A.
  Defined.

  Polymorphic Lemma hidden_strict_type : Type.
  Proof.
    exact Type.
  Qed.
  Check hidden_strict_typeObligation . .
  Fail Check Next Obligation. Show. Admitted.

  Fail Definition

  (* By default constraints are extensible *)
  Polymorphic Definition morec@{i j} (A : Type@{i}) : Type@{j} := A.
  Check morec@{_ _}.

  (* Handled in proofs as well *)
  Lemma bar@{i j | } : Type@{i}.
    exact Type@{j}.
    Fail Defined.
  Abort.

  Fail Lemma bar@{u v | } : let x := (fun x => x) : Type@{u} -> Type@{v} in nat.
  Fail Fixpoint fbar@{u v | } (n:nat) : let x := (fun x => x) : Type@{u} in nat.

  Lemma bar@{i j| i < j} : Type@{j}.
  Proof.
    exact Type@{i}.
  Qed.

  Lemma barext@{i j|+} : Type@{j}.
  Proof.
    exact Type@{i}.
  Qed.

  Monomorphic Universe M.
  Fail Definition with_mono@{u|} : Type@{M} := Type@{u}.
  Definition with_mono@{u|u < M} : Type@{M} := Type@{u}.

End binders.

#[universes(polymorphic)]
Section cats.
  Local Set Universe Polymorphism.
  Definition fibration (A : Type) := A -> Type.
  Definition Hom (A : Type) := A -> A -> Type.

  Record sigma (A : Type) (P : fibration A) :=
    { proj1 : A; proj2 : P proj1} .

  Class Identity {A} (M : Hom A) :=
    identity : forall x, M x x.

  Class Inverse {A} (M : Hom A) :=
    inverse : forall x y:A, M x y -> M y x.

  Class Composition {A} (M : Hom A) :=
    composition : forall {x y z:A}, M x y -> M y z -> M x z.

  Notation  "g ° f" := (composition f g) (at level 50).

  Class Equivalence T (Eq : Hom T):=
    {
      Equivalence_Identity :: Identity Eq ;
      Equivalence_Inverse :: Inverse Eq ;
      Equivalence_Composition :: Composition Eq
    }.

  Class EquivalenceType (T : Type) : Type :=
    {
      m2: Hom T;
      equiv_struct :: Equivalence T m2 }.

  Polymorphic Record cat (T : Type) :=
    { cat_hom : Hom T;
      cat_equiv : forall x y, EquivalenceType (cat_hom x y) }.

  Definition catType := sigma Type cat.

  Notation "[ T ]" := (proj1 T).

  From Corelib.Program Require Import Basics Tactics Wf.

  Program Definition small_cat : cat Empty_set :=
    {| cat_hom x y := unit |}.
  Next Obligation.
    refine ({|m2:=fun x y => True|}).
    constructor; red; intros; trivial.
  Defined.

  Record iso (T U : Set) :=
    { f : T -> U;
      g : U -> T }.

  Program Definition Set_cat : cat Set :=
    {| cat_hom := iso |}.
  Next Obligation.
    refine ({|m2:=fun x y => True|}).
    constructor; red; intros; trivial.
  Defined.

  Record isoT (T U : Type) :=
    { isoT_f : T -> U;
      isoT_g : U -> T }.

  Program Definition Type_cat : cat Type :=
    {| cat_hom := isoT |}.
  Next Obligation.
    refine ({|m2:=fun x y => True|}).
    constructor; red; intros; trivial.
  Defined.

  Polymorphic Record cat1 (T : Type) :=
    { cat1_car : Type;
      cat1_hom : Hom cat1_car;
      cat1_hom_cat : forall x y, cat (cat1_hom x y) }.
End cats.

Polymorphic Definition id {A : Type} (a : A) : A := a.

Definition typeid := (@id Type).

Fail Check (Prop : Set).
Fail Check (Set : Set).
Check (Set : Type).
Check (Prop : Type).
Definition setType := ltac:(let t := type of Set in exact t).

Definition foo (A : Prop) := A.

Fail Check foo Set.
Check fun A => foo A.
Fail Check fun A : Type => foo A.
Check fun A : Prop => foo A.
Fail Definition bar := fun A : Set => foo A.

Fail Check (let A := Type in foo (id A)).

Definition fooS (A : Set) := A.

Check (let A := nat in fooS (id A)).
Fail Check (let A := Set in fooS (id A)).
Fail Check (let A := Prop in fooS (id A)).

(* Some tests of sort-polymorphisme *)
#[universes(polymorphic)]
Section S.
Polymorphic Variable A:Type.
(*
Definition f (B:Type) := (A * B)%type.
*)
Polymorphic Inductive I (B:Type) : Type := prod : A->B->I B.

Check I nat.

End S.
(*
Check f nat nat : Set.
*)
Definition foo' := I nat nat.
Print Universes. Print foo. Set Printing Universes. Print foo.

(* Polymorphic axioms: *)
Polymorphic Axiom funext : forall (A B : Type) (f g : A -> B),
                 (forall x, f x = g x) -> f = g.

(* Check @funext. *)
(* Check funext. *)

Polymorphic Definition fun_ext (A B : Type) :=
  forall (f g : A -> B),
    (forall x, f x = g x) -> f = g.

Polymorphic Class Funext A B := extensional : fun_ext A B.

Section foo2.
  Context `{forall A B, Funext A B}.
  Print Universes.
End foo2.

Module eta.
Set Universe Polymorphism.

Set Printing Universes.

Axiom admit : forall A, A.
Record R := {O : Type}.

Definition RL (x : R@{i}) : ltac:(let u := constr:(Type@{i}:Type@{j}) in exact (R@{j}) ) := {|O := @O x|}.
Definition RLRL : forall x : R, RL x = RL (RL x) := fun x => eq_refl.
Definition RLRL' : forall x : R, RL x = RL (RL x).
  intros. apply eq_refl.
Qed.

End eta.

Module Hurkens'.
  Require Import TestSuite.hurkens.

Polymorphic Record box (X : Type) (T := Type) : Type := wrap { unwrap : T }.

Definition unwrap' := fun (X : Type) (b : box X) => let (unw) := b in unw.

Fail Definition bad : False := TypeNeqSmallType.paradox (unwrap' Type (wrap _
Type)) eq_refl.

End Hurkens'.

Module Anonymous.
  Set Universe Polymorphism.

  Definition defaultid := (fun x => x) : Type -> Type.
  Definition collapseid := defaultid@{_ _}.
  Check collapseid@{_}.

  Definition anonid := (fun x => x) : Type -> Type@{_}.
  Check anonid@{_}.

  Definition defaultalg := (fun x : Type => x) (Type : Type).
  Definition usedefaultalg := defaultalg@{_ _ _}.
  Check usedefaultalg@{_ _}.

  Definition anonalg := (fun x : Type@{_} => x) (Type : Type).
  Check anonalg@{_ _}.

  Definition unrelated@{i j} := nat.
  Definition useunrelated := unrelated@{_ _}.
  Check useunrelated@{_ _}.

  Definition inthemiddle@{i j k} :=
    let _ := defaultid@{i j} in
    anonalg@{k j}.
  (* i <= j < k *)
  Definition collapsethemiddle := inthemiddle@{i _ j}.
  Check collapsethemiddle@{_ _}.

End Anonymous.

Module Restrict.
  (* Universes which don't appear in the term should be pruned, unless they have names *)
  Set Universe Polymorphism.

  Ltac exact0 := let x := constr:(Type) in exact 0.
  Definition dummy_pruned@{} : nat := ltac:(exact0).

  Definition named_not_pruned@{u} : nat := 0.
  Check named_not_pruned@{_}.

  Definition named_not_pruned_nonstrict : nat := ltac:(let x := constr:(Type@{u}) in exact 0).
  Check named_not_pruned_nonstrict@{_}.

  Lemma lemma_restrict_poly@{} : nat.
  Proof. exact0. Defined.

  Unset Universe Polymorphism.
  Lemma lemma_restrict_mono_qed@{} : nat.
  Proof. exact0. Qed.

  Lemma lemma_restrict_abstract@{} : nat.
  Proof. abstract exact0. Qed.

End Restrict.

Module F.
  #[warning="context-outside-section"] Context {A B : Type}.
  Definition foo : Type := B.
End F.

Set Universe Polymorphism.

Cumulative Record box (X : Type) (T := Type) : Type := wrap { unwrap : T }.

Section test_letin_subtyping.
  Universe i j k i' j' k'.
  Constraint j < j'.

  Context (W : Type) (X : box@{i j k} W).
  Definition Y := X : box@{i' j' k'} W.

  Universe i1 j1 k1 i2 j2 k2.
  Constraint i1 < i2.
  Constraint k2 < k1.
  Context (V : Type).

  Definition Z : box@{i1 j1 k1} V := {| unwrap := V |}.
  Definition Z' : box@{i2 j2 k2} V := {| unwrap := V |}.
  Lemma ZZ' : @eq (box@{i2 j2 k2} V) Z Z'.
  Proof.
    Set Printing All. Set Printing Universes.
    cbv.
    reflexivity.
  Qed.

End test_letin_subtyping.

Module ObligationRegression.
  (** Test for a regression encountered when fixing obligations for
      stronger restriction of universe context. *)
  Require Import CMorphisms.
  Check trans_co_eq_inv_arrow_morphism@{_ _ _ _ _  _ _}.
End ObligationRegression.

Axiom poly@{i} : forall(A : Type@{i}) (a : A), unit.

Definition nonpoly := @poly True Logic.I.
Definition check := nonpoly@{}.

Module ProgramFixpoint.

  Local Set Universe Polymorphism.
  Program Fixpoint f@{u} (A:Type@{u}) (n:nat) : Type@{u} :=
    match n with 0 => A | S n => f (A->A) n end.
  Check f@{Set}. (* Check that it depends on only one universe *)

End ProgramFixpoint.

Module EarlyPolyUniverseDeclarationCheck.

  Local Set Universe Polymorphism.

  Fail Definition f@{u} n : match n return Type@{v} with 0 => Type@{u} | _ => Type@{u} end.

  Definition f@{u v} n : match n return Type@{v} with 0 => Type@{u} | _ => Type@{u} end.
  exact (match n with 0 => nat | _ => nat end).
  Defined.

  Program Fixpoint f'@{u} (A:Type@{u}) (n:nat) : Type@{u} :=
    match n with 0 => _ | S n => f' (A->A) n end.
  Next Obligation. exact nat. Defined.

  Fail Program Fixpoint f''@{u} (A:Type@{u}) (n:nat) {measure n} : Type@{u} :=
    match n with 0 => _ | S n => f'' (Type->A) n end.

  Local Set Universe Polymorphism.
  Program Fixpoint f''@{u} (A:Type@{u}) (n:nat) {measure n} : Type@{u} :=
    match n with 0 => _ | S n => f'' (A->A) n end.
  Next Obligation. Show. exact nat. Defined.
  Next Obligation. Show. Admitted.

End EarlyPolyUniverseDeclarationCheck.

Module EarlyMonoUniverseDeclarationCheck.

  Local Unset Universe Polymorphism.

  Fail Definition f@{u} n : match n return Type@{v} with 0 => Type@{u} | _ => Type@{u} end.

  Definition f@{u v} n : match n return Type@{v} with 0 => Type@{u} | _ => Type@{u} end.
  exact (match n with 0 => nat | _ => nat end).
  Defined.

  Fail Program Fixpoint f'@{u} (A:Type@{u}) (n:nat) : Type@{u} := (* By convention, we require extensibility for Program *)
    match n with 0 => _ | S n => f' (A->A) n end.

  Program Fixpoint f'@{u +} (A:Type@{u}) (n:nat) : Type@{u} :=
    match n with 0 => _ | S n => f' (A->A) n end.
  Next Obligation. exact nat. Defined.

  Fail Program Fixpoint f''@{u} (A:Type@{u}) (n:nat) {measure n} : Type@{u} :=
    match n with 0 => _ | S n => f'' (Type->A) n end.

  Fail Program Fixpoint f''@{u} (A:Type@{u}) (n:nat) {measure n} : Type@{u} := (* By convention, we require extensibility for Program *)
    match n with 0 => _ | S n => f'' (A->A) n end.

  Program Fixpoint f''@{u +} (A:Type@{u}) (n:nat) {measure n} : Type@{u} :=
    match n with 0 => _ | S n => f'' (A->A) n end.
  Next Obligation. Show. exact nat. Defined.
  Next Obligation. Show. Admitted.

End EarlyMonoUniverseDeclarationCheck.

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