andrejbauer

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andrejbauer

(* Post's theorem in computability theory states that a subset S ⊆ ℕ is
   decidable if both S and its complement ℕ \ S are semidecidable.
   Just how much classical logic is needed to prove the theorem, though?
   In this note we show that Markov's principle suffices.

   Rather than working with subsets of ℕ we work synthetically, with
   semidecidable and decidable truth values. (This is more general than
   the traditional Post's theorem.)
*)

andrejbauer

(*
   A formalization of first-order logic, theories and their models.
   As an example we formalize the language of ZF set theory and a a small fragment of ZF.
*)

(* A signature is given by function symbols, relation symbols, and their arities. *)
Structure Signature :=
  { funSym : Type (* names of function symbols *)
  ; funArity : funSym -> Type (* arities of function symbols *)
  ; relSym : Type (* names of relation symbols *)

andrejbauer

open import Data.Nat
open import Data.Product

module example where

  data SimpleTree (A : Set) : Set where
    empty : SimpleTree A
    leaf : A -> SimpleTree A
    node : A -> SimpleTree A -> SimpleTree A -> SimpleTree A

andrejbauer

{-
  Weak excluded middle states that for every propostiion P, either ¬¬P or ¬P holds.

  We give a proof of weak excluded middle, relying just on basic facts about real numbers,
  which we carefully state, in order to make the dependence on them transparent.

  Some people might doubt the formalization. To convince yourself that there is no cheating, you should:

  * Carefully read the facts listed in the RealFacts below to see that these are all
    just familiar facts about the real numbers.

andrejbauer

{- Here is a short demonstration on how to use Agda's solvers that automatically
   prove equations. It seems quite impossible to find complete worked examples
   of how Agda solvers are used.

   Thanks go to @anormalform who helped out with these on Twitter, see
   https://twitter.com/andrejbauer/status/1549693506922364929?s=20&t=7cb1TbB2cspIgktKXU8rng

   I welcome improvements and additions to this file.

   This file works with Agda 2.6.2.2 and standard-library-1.7.1

andrejbauer

Require Import Setoid.

Definition coerce {A B : Type} : A = B -> A -> B.
Proof.
  intros [] a.
  exact a.
Defined.

Lemma coerce_coerce {A B : Type} (e : A = B) (b : B) :
  coerce e (coerce (eq_sym e) b) = b.

andrejbauer

open import Level
open import Relation.Binary
open import Data.Product
open import Data.Unit
open import Agda.Builtin.Sigma
open import Relation.Binary.PropositionalEquality renaming (refl to ≡-refl; trans to ≡-trans; sym to ≡-sym; cong to ≡-cong)
open import Relation.Binary.PropositionalEquality.Properties
open import Function.Equality hiding (setoid)

module Epimorphism where

andrejbauer

open import Data.Nat
open import Data.Maybe
open import Data.Product

-- An implementation of infinite queues fashioned after the von Neuman ordinals

module Queue where

   infixl 5 _⊕_

andrejbauer

open import Data.Nat
open import Data.Fin as Fin

module Primrec where

  -- The datatype of primitive recursive function
  data PR : ∀ (k : ℕ) → Set where
    pr-zero : PR 0 -- zero
    pr-succ : PR 1 -- successor
    pr-proj : ∀ {k} (i : Fin k) → PR k -- projection