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From mathcomp Require Import ssreflect.
Require Import Nat.

Definition name := nat.

Inductive proc :=
  | Nil
  | Tau (P : proc)
  | Para (P Q : proc)
  | Sum (P Q : proc)

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From mathcomp Require Import ssreflect.
From Autosubst Require Import Autosubst.
Require Import Nat.

Inductive proc_ :=
  | Var (x : var)
  | Nil
  | Para (P Q : proc_)
  | Repl (P : proc_)
  | Send (M : proc_) (N : proc_) (P : proc_)

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  function Pow(n:nat, k:nat) : (r:nat)
    ensures n > 0 ==> r > 0     
    {
        if k == 0 then 1
        else if k == 1 then n
        else
            var p := k / 2;
            var np := Pow(n,p);
            if p*2 == k then np * np
            else

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function Pow(n:nat, k:nat) : (r:nat)
  // Following needed for some proofs
  ensures n > 0 ==> r > 0     
  {
      if k == 0 then 1
      else if k == 1 then n
      else
          var p := k / 2;
          var np := Pow(n,p);
          if p*2 == k then np * np

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include "evm-dafny/src/dafny/core/memory.dfy"
include "evm-dafny/src/dafny/util/int.dfy"
include "evm-dafny/src/dafny/util/bytes.dfy"
include "evm-dafny/src/dafny/bytecode.dfy"
include "evm-dafny/src/dafny/evm.dfy"
// include "evm-dafny/libs/DafnyCrypto/src/dafny/util/math.dfy"

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datatype Expr =
  | Num(nat)
  | Add(Expr, Expr)
  | Mul(Expr, Expr)

function eval (e : Expr) : nat
{
  match e
  case Num(n) => n
  case Add(e1, e2) => eval(e1) + eval(e2)

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From UniMath Require Export MoreFoundations.All.
From UniMath Require Export OrderTheory.Posets.
From UniMath Require Export OrderTheory.DCPOs.
From DomainTheory Require Export Pataraia.
From mathcomp Require Export ssreflect.

Open Scope dcpo.
Open Scope subtype.
Open Scope logic.

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Require Export UniMath.MoreFoundations.All.
Require Export UniMath.Algebra.Monoids.
Require Import UniMath.OrderTheory.Posets.Basics.
Require Import UniMath.OrderTheory.Posets.MonotoneFunctions.
Require Import UniMath.OrderTheory.DCPOs.Core.DirectedSets.
Require Import UniMath.OrderTheory.DCPOs.Core.Basics.
Require Import UniMath.OrderTheory.DCPOs.Examples.Products.
Require Import UniMath.OrderTheory.DCPOs.Examples.SubDCPO.
Require Import UniMath.OrderTheory.DCPOs.FixpointTheorems.Pataraia.

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Require Import Ensembles Classical_sets.
Require Import ssreflect.
From mathcomp Require Import finset eqtype ssrbool seq.
Require Import NonEmptyFintype.
Open Scope list_scope.

Unset Strict Implicit.
Unset Printing Implicit Defensive.

(* ================== Notation of finset ================== *)

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From mathcomp Require Export fintype.

Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

Module NonEmptyFintype.

  Section Def.