Theory Typechecking

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theory Typechecking
imports CTT
begin

(*  Title:      CTT/ex/Typechecking.thy
    ID:         $Id: Typechecking.thy,v 1.1 2006/06/02 16:15:38 wenzelm Exp $
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1991  University of Cambridge
*)

header "Easy examples: type checking and type deduction"

theory Typechecking
imports CTT
begin

subsection {* Single-step proofs: verifying that a type is well-formed *}

lemma "?A type"
apply (rule form_rls)
done

lemma "?A type"
apply (rule form_rls)
back
apply (rule form_rls)
apply (rule form_rls)
done

lemma "PROD z:?A . N + ?B(z) type"
apply (rule form_rls)
apply (rule form_rls)
apply (rule form_rls)
apply (rule form_rls)
apply (rule form_rls)
done


subsection {* Multi-step proofs: Type inference *}

lemma "PROD w:N. N + N type"
apply (tactic form_tac)
done

lemma "<0, succ(0)> : ?A"
apply (tactic "intr_tac []")
done

lemma "PROD w:N . Eq(?A,w,w) type"
apply (tactic "typechk_tac []")
done

lemma "PROD x:N . PROD y:N . Eq(?A,x,y) type"
apply (tactic "typechk_tac []")
done

text "typechecking an application of fst"
lemma "(lam u. split(u, %v w. v)) ` <0, succ(0)> : ?A"
apply (tactic "typechk_tac []")
done

text "typechecking the predecessor function"
lemma "lam n. rec(n, 0, %x y. x) : ?A"
apply (tactic "typechk_tac []")
done

text "typechecking the addition function"
lemma "lam n. lam m. rec(n, m, %x y. succ(y)) : ?A"
apply (tactic "typechk_tac []")
done

(*Proofs involving arbitrary types.
  For concreteness, every type variable left over is forced to be N*)
ML {* val N_tac = TRYALL (rtac (thm "NF")) *}

lemma "lam w. <w,w> : ?A"
apply (tactic "typechk_tac []")
apply (tactic N_tac)
done

lemma "lam x. lam y. x : ?A"
apply (tactic "typechk_tac []")
apply (tactic N_tac)
done

text "typechecking fst (as a function object)"
lemma "lam i. split(i, %j k. j) : ?A"
apply (tactic "typechk_tac []")
apply (tactic N_tac)
done

end

Single-step proofs: verifying that a type is well-formed

lemma

  N type

lemma

  N --> N type

lemma

  N --> N + N type

Multi-step proofs: Type inference

lemma

  N --> N + N type

lemma

  <0,succ(0)> ∈ N × N

lemma

  Π w∈N. Eq(N, w, w) type

lemma

  Π x∈N. Prod(N, Eq(N, x)) type

lemma

  (λλu. split(u, λv w. v)) ` <0,succ(0)> ∈ N

lemma

  λλn. rec(n, 0, λx y. x) ∈ N --> N

lemma

  λλn m. rec(n, m, λx. succ) ∈ N --> N --> N

lemma

  λλw. <w,w> ∈ N --> N × N

lemma

  λλx y. x ∈ N --> N --> N

lemma

  λλi. split(i, λj k. j) ∈ N × N --> N