Quelle bug_15048.v Sprache: Coq

Reserved Notation "x = y :> T"
(at level 70, y at next level, no associativity).
Reserved Notation "x = y" (at level 70, no associativity).

Reserved Notation "x + y" (at level 50, left associativity).

Reserved Notation "n .+1" (at level 2, left associativity, format "n .+1").
Reserved Notation "n .+2" (at level 2, left associativity, format "n .+2").
Reserved Notation "g 'o' f" (at level 40, left associativity).
Reserved Notation "! x" (at level 3, format "'!' x").
Declare Scope path_scope.
Delimit Scope type_scope with type.
Delimit Scope trunc_scope with trunc.

Global Open Scope trunc_scope.
Global Open Scope path_scope.
Global Open Scope type_scope.

Global Set Universe Polymorphism.

Notation "A -> B" := (forall (_ : A), B) : type_scope.

Create HintDb typeclass_instances discriminated.

Definition Relation (A : Type) := A -> A -> Type.

Class Symmetric {A} (R : Relation A) :=
  symmetry : forall x y, R x y -> R y x.

Notation Type0 := Set.

Cumulative Inductive paths {A : Type} (a : A) : A -> Type :=
  idpath : paths a a.

Arguments idpath {A a} , [A] a.

Notation "x = y :> A" := (@paths A x y) : type_scope.
Notation "x = y" := (x = y :>_) : type_scope.

Definition inverse {A : Type} {x y : A} (p : x = y) : y = x
  := match p with idpath => idpath end.

Global Instance symmetric_paths {A} : Symmetric (@paths A) | 0 := @inverse A.

Notation "1" := idpath : path_scope.

Class Contr_internal (A : Type) := Build_Contr {
  center : A ;
  contr : (forall y : A, center = y)
}.

Inductive trunc_index : Type :=
| minus_two : trunc_index
| trunc_S : trunc_index -> trunc_index.

Bind Scope trunc_scope with trunc_index.

Notation "n .+1" := (trunc_S n) : trunc_scope.
Notation "n .+2" := (n.+1.+1)%trunc : trunc_scope.

Fixpoint IsTrunc_internal (n : trunc_index) (A : Type) : Type :=
  match n with
    | minus_two => Contr_internal A
    | n'.+1 => forall (x y : A), IsTrunc_internal n' (x = y)
  end.

Class IsTrunc (n : trunc_index) (A : Type) : Type :=
  Trunc_is_trunc : IsTrunc_internal n A.

Notation Contr := (IsTrunc minus_two).
Notation IsHSet := (IsTrunc minus_two.+2).

Inductive Empty : Type0 := .

Inductive Unit : Type0 := tt : Unit.
Module Export Core.
Set Implicit Arguments.

Delimit Scope morphism_scope with morphism.
Delimit Scope category_scope with category.
Delimit Scope object_scope with object.

Record PreCategory :=
  Build_PreCategory' {
      object :> Type;
      morphism : object -> object -> Type;

      identity : forall x, morphism x x;
      compose : forall s d d',
                  morphism d d'
                  -> morphism s d
                  -> morphism s d'
                              where "f 'o' g" := (compose f g);

      associativity : forall x1 x2 x3 x4
                             (m1 : morphism x1 x2)
                             (m2 : morphism x2 x3)
                             (m3 : morphism x3 x4),
                        (m3 o m2) o m1 = m3 o (m2 o m1);

      associativity_sym : forall x1 x2 x3 x4
                                 (m1 : morphism x1 x2)
                                 (m2 : morphism x2 x3)
                                 (m3 : morphism x3 x4),
                            m3 o (m2 o m1) = (m3 o m2) o m1;

      left_identity : forall a b (f : morphism a b), identity b o f = f;
      right_identity : forall a b (f : morphism a b), f o identity a = f;

      identity_identity : forall x, identity x o identity x = identity x;

      trunc_morphism : forall s d, IsHSet (morphism s d)
    }.

Bind Scope category_scope with PreCategory.
Arguments identity {!C%_category} / x%_object : rename.
Arguments compose {!C%_category} / {s d d'}%_object (m1 m2)%_morphism : rename.

Infix "o" := compose : morphism_scope.

End Core.
Delimit Scope functor_scope with functor.

Local Open Scope morphism_scope.

Section Functor.
  Variables C D : PreCategory.

  Record Functor :=
    {
      object_of :> C -> D;
      morphism_of : forall s d, morphism C s d
                                -> morphism D (object_of s) (object_of d);
      composition_of : forall s d d'
                              (m1 : morphism C s d) (m2: morphism C d d'),
                         morphism_of _ _ (m2 o m1)
                         = (morphism_of _ _ m2) o (morphism_of _ _ m1);
      identity_of : forall x, morphism_of _ _ (identity x)
                              = identity (object_of x)
    }.
End Functor.

Generalizable Variables A B m n f.

Fixpoint trunc_index_inc (k : trunc_index) (n : nat)
  : trunc_index
  := match n with
      | O => k
      | S m => (trunc_index_inc k m).+1
    end.

Definition nat_to_trunc_index (n : nat) : trunc_index
  := (trunc_index_inc minus_two n).+2.

Definition int_to_trunc_index (v : Decimal.int) : option trunc_index
  := match v with
     | Decimal.Pos d => Some (nat_to_trunc_index (Nat.of_uint d))
     | Decimal.Neg d => match Nat.of_uint d with
                        | 2%nat => Some minus_two
                        | 1%nat => Some (minus_two.+1)
                        | 0 => Some (minus_two.+2)
                        | _ => None
                        end
     end.

Definition num_int_to_trunc_index v : option trunc_index :=
  match Nat.of_num_int v with
  | Some n => Some (nat_to_trunc_index n)
  | None => None
  end.

Fixpoint trunc_index_to_little_uint n acc :=
  match n with
  | minus_two => acc
  | minus_two.+1 => acc
  | minus_two.+2 => acc
  | trunc_S n => trunc_index_to_little_uint n (Decimal.Little.succ acc)
  end.

Definition trunc_index_to_int n :=
  match n with
  | minus_two => Decimal.Neg (Nat.to_uint 2)
  | minus_two.+1 => Decimal.Neg (Nat.to_uint 1)
  | n => Decimal.Pos (Decimal.rev (trunc_index_to_little_uint n Decimal.zero))
  end.

Definition trunc_index_to_num_int n :=
  Number.IntDecimal (trunc_index_to_int n).

Global Instance istrunc_succ `{IsTrunc n A}
  : IsTrunc n.+1 A | 1000.
Admitted.

Global Instance contr_unit : Contr Unit | 0 := let x := {|
  center := tt;
  contr := fun t : Unit => match t with tt => 1 end
|} in x.

Definition groupoid_category X `{IsTrunc (trunc_S (trunc_S (trunc_S minus_two))) X} : PreCategory.
Admitted.
  Definition discrete_category X `{IsHSet X} := groupoid_category X.
  Section indiscrete_category.

    Variable X : Type.

    Definition indiscrete_category : PreCategory
      := @Build_PreCategory' X
                             (fun _ _ => Unit)
                             (fun _ => tt)
                             (fun _ _ _ _ _ => tt)
                             (fun _ _ _ _ _ _ _ => idpath)
                             (fun _ _ _ _ _ _ _ => idpath)
                             (fun _ _ f => match f with tt => idpath end)
                             (fun _ _ f => match f with tt => idpath end)
                             (fun _ => idpath)
                             _.
  End indiscrete_category.
Local Open Scope nat_scope.

  Fixpoint CardinalityRepresentative (n : nat) : Type0 :=
    match n with
      | 0 => Empty
      | 1 => Unit
      | S n' => (CardinalityRepresentative n' + Unit)%type
    end.

  Coercion CardinalityRepresentative : nat >-> Sortclass.

  Global Instance trunc_cardinality_representative (n : nat)
  : IsHSet (CardinalityRepresentative n).
Admitted.

  Definition nat_category (n : nat) :=
    match n with
      | 0 => indiscrete_category 0
      | 1 => indiscrete_category 1
      | S (S n') => discrete_category (S (S n'))
    end.
    Notation "1" := (nat_category 1) : category_scope.
Notation terminal_category := (nat_category 1) (only parsing).

Class IsTerminalCategory (C : PreCategory)
      `{Contr (object C)}
      `{forall s d, Contr (morphism C s d)} := {}.
Global Instance: IsTerminalCategory 1 | 0 := {}.
Generalizable All Variables.

Section functors.
  Variable C : PreCategory.

  Definition from_terminal `{@IsTerminalCategory one Hone Hone'} (c : C)
  : Functor one C
    := Build_Functor
         one C
         (fun _ => c)
         (fun _ _ _ => identity c)
         (fun _ _ _ _ _ => symmetry _ _ (@identity_identity _ _))
         (fun _ => idpath).
End functors.

Local Notation "! x" := (@from_terminal _ terminal_category _ _ _ x) : functor_scope.

Definition CC_Functor' (C : PreCategory) (D : PreCategory) := Functor C D.

Coercion cc_functor_from_terminal' (C : PreCategory) (x : C) : CC_Functor' _ C
  := (!x)%functor.

quality96%

¤ Dauer der Verarbeitung: 0.7 Sekunden ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.