NotBad4U

------------------------------ MODULE real ------------------------------- 

EXTENDS Naturals, Integers, Sequences, FiniteSets, FiniteSetsExt, NaturalsInduction

CONSTANT max

McNat == 0..max

(***********************************************************************)
(* Approximate sqrt(r)                                                  *)

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trait Category<'a> { // A
    type Y<A: 'a, B: 'a>; // B
    
    fn id<A: 'a>() -> Self::Y<A, A>;
    
    fn compose<A:'a,B: 'a, C:'a>(f: Self::Y<A, B>, g: Self::Y<B, C>) -> Self::Y<A, C>;
}

struct WrapperFunc;

NotBad4U

From Coq Require Import Bool.Bool.
From Coq Require Import Arith.Arith.
From Coq Require Import Arith.EqNat.
From Coq Require Import Arith.PeanoNat. Import Nat.
From Coq Require Import Lia.
Require Export Coq.Strings.String.
From Coq Require Import Logic.FunctionalExtensionality.
From Coq Require Import Lists.List.
Import ListNotations.

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// PRELUDE ##############
// Définitions intuitioniste nécessaires pour illustrer le problème dans Classic.

constant symbol Prop : TYPE;

injective symbol π : Prop → TYPE; // `p

constant symbol ∨ : Prop → Prop → Prop; notation ∨ infix left 6; // \/ or \vee

constant symbol ∨ᵢ₁ [p q] : π p → π (p ∨ q);

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require open Stdlib.FOL;
require open Stdlib.Prop;
require open Stdlib.Set;
require open Stdlib.Nat;
require open Stdlib.Bool;
require open Stdlib.Eq;
require open Stdlib.Z;
require open Stdlib.List;

// References use

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require open Stdlib.Set Stdlib.Prop Stdlib.FOL Stdlib.Eq Stdlib.Pos Stdlib.Bool;



inductive ℤ : TYPE ≔ // \BbbZ
| Z0 : ℤ
| Zpos : ℙ → ℤ
| Zneg : ℙ → ℤ;

constant symbol int : Set;

NotBad4U

require open Stdlib.Prop;
require open Stdlib.FOL;
require open Stdlib.Eq;
require open Stdlib.Nat;
require Stdlib.Bool;
require open Stdlib.Set;
require open Stdlib.List;

inductive prop: TYPE ≔
|pTrue: prop

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require open Stdlib.Prop;
require open Stdlib.FOL;
require open Stdlib.HOL;
require open Stdlib.Eq;
require open Stdlib.Set;
require open Stdlib.Z;
require open Stdlib.Pos;
require open Stdlib.Bool;
require open Stdlib.Comp;
require open Stdlib.Impred;

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require open Stdlib.Z Stdlib.Set Stdlib.Prop;

constant symbol R : TYPE;

symbol var : ℤ → ℤ → R;
/* semantics: [var k x] = k * x
note that the second argument could be any type that has a ring structure
(we then would take R : TYPE → TYPE) */
constant symbol cst:ℤ → R;
symbol opp:R → R;

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require open Stdlib.Nat Stdlib.Z Stdlib.Set Stdlib.Prop Stdlib.Bool Stdlib.Comp Stdlib.List;

constant symbol R : TYPE;

constant symbol cst:ℤ → R;

symbol var : ℤ → ℕ → R;
/* semantics: [var k x] = k * x
note that the second argument could be any type that has a ring structure
(we then would take R : TYPE → TYPE) */