Theory Examples

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theory Examples
imports Natural
begin

(*  Title:      HOL/IMP/Examples.thy
    ID:         $Id: Examples.thy,v 1.4 2005/12/08 19:15:50 wenzelm Exp $
    Author:     David von Oheimb, TUM
    Copyright   2000 TUM
*)

header "Examples"

theory Examples imports Natural begin

constdefs
  factorial :: "loc => loc => com"
  "factorial a b == b :== (%s. 1);
                    \<WHILE> (%s. s a ~= 0) \<DO>
                    (b :== (%s. s b * s a); a :== (%s. s a - 1))"

declare update_def [simp]

subsection "An example due to Tony Hoare"

lemma lemma1:
  assumes 1: "!x. P x --> Q x"
    and 2: "⟨w,s⟩ -->c t"
  shows "w = While P c ==> ⟨While Q c,t⟩ -->c u ==> ⟨While Q c,s⟩ -->c u"
  using 2 apply induct
  using 1 apply auto
  done

lemma lemma2 [rule_format (no_asm)]:
  "[| !x. P x --> Q x; ⟨w,s⟩ -->c u |] ==>
  !c. w = While Q c --> ⟨While P c; While Q c,s⟩ -->c u"
apply (erule evalc.induct)
apply (simp_all (no_asm_simp))
apply blast
apply (case_tac "P s")
apply auto
done

lemma Hoare_example: "!x. P x --> Q x ==>
  (⟨While P c; While Q c, s⟩ -->c t) = (⟨While Q c, s⟩ -->c t)"
  by (blast intro: lemma1 lemma2 dest: semi [THEN iffD1])


subsection "Factorial"

lemma factorial_3: "a~=b ==>
    ⟨factorial a b, Mem(a:=3)⟩ -->c Mem(b:=6, a:=0)"
  by (simp add: factorial_def)

text {* the same in single step mode: *}
lemmas [simp del] = evalc_cases
lemma  "a~=b ==> ⟨factorial a b, Mem(a:=3)⟩ -->c Mem(b:=6, a:=0)"
apply (unfold factorial_def)
apply (frule not_sym)
apply (rule evalc.intros)
apply  (rule evalc.intros)
apply simp
apply (rule evalc.intros)
apply   simp
apply  (rule evalc.intros)
apply   (rule evalc.intros)
apply  simp
apply  (rule evalc.intros)
apply simp
apply (rule evalc.intros)
apply   simp
apply  (rule evalc.intros)
apply   (rule evalc.intros)
apply  simp
apply  (rule evalc.intros)
apply simp
apply (rule evalc.intros)
apply   simp
apply  (rule evalc.intros)
apply   (rule evalc.intros)
apply  simp
apply  (rule evalc.intros)
apply simp
apply (rule evalc.intros)
apply simp
done

end

An example due to Tony Hoare

lemma lemma1:

  [| ∀x. P x --> Q x; w,s -->c t; w = WHILE P DO c; WHILE Q DO c,t -->c u |]
  ==> WHILE Q DO c,s -->c u

lemma lemma2:

  [| ∀x. P x --> Q x; w,s -->c u; w = WHILE Q DO c |]
  ==> WHILE P DO c; WHILE Q DO c,s -->c u

lemma Hoare_example:

  x. P x --> Q x
  ==> WHILE P DO c; WHILE Q DO c,s -->c t = WHILE Q DO c,s -->c t

Factorial

lemma factorial_3:

  a  b ==> factorial a b,Mem(a := 3) -->c Mem(b := 6, a := 0)

lemma

  SKIP,s -->c s' = (s' = s)
  x :== a ,s -->c s' = (s' = s[x ::= a s])
  b s ==> IF b THEN c0.0 ELSE c1.0,s -->c s' = c0.0,s -->c s'
  ¬ b s ==> IF b THEN c0.0 ELSE c1.0,s -->c s' = c1.0,s -->c s'
  ¬ b s ==> WHILE b DO c,s -->c s' = (s' = s)
  c0.0; c1.0,s -->c s' = (∃s''. c0.0,s -->c s''c1.0,s'' -->c s')
  b s ==> WHILE b DO c,s -->c s' =
          (∃s''. c,s -->c s''WHILE b DO c,s'' -->c s')

lemma

  a  b ==> factorial a b,Mem(a := 3) -->c Mem(b := 6, a := 0)