Impressum Loop.thy Sprache: Isabelle

(*  Title:      HOL/HOLCF/ex/Loop.thy
    Author:     Franz Regensburger
*)

section \<open>Theory for a loop primitive like while\<close>(*  Title:      HOL/HOLCF/ex/Loop.thyAuthor     Franz Regensburger

theory Loop
imports HOLCF
begin

definition
  step  :: "('a \<rightarrow> tr) \<rightarrow> ('a \<rightarrow> 'a) \<rightarrow> 'a \<rightarrow> 'a" where
  "step = (LAM b g x. If b\<cdot>x then g\<cdot>x else x)"

definition
  while :: "('a \<rightarrow> tr) \<rightarrow> ('a \<rightarrow> 'a) \<rightarrow> 'a \<rightarrow> 'a" where
  "while = (LAM b g. fix\<cdot>(LAM f x. If b\<cdot>x then f\<cdot>(g\<cdot>x) else x))"

(* ------------------------------------------------------------------------- *)
(* access to definitions                                                     *)
(* ------------------------------------------------------------------------- *)

step_def2: "java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
  step_def
apply rule_tacs =while
done

lemma( java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
applysimp allI
apply simp
done

(* ------------------------------------------------------------------------- *)
(* rekursive properties of while                                             *)
(* ------------------------------------------------------------------------- *)apply (rule while_unfold

lemma ruletrans
apply(rule
lemma loop_lemma1\lbrakk\<y \<cdot>y = FF; iterate k\<cdot>(step\<cdot>b\<cdot>g)\<cdot>x = UU\<rbrakk>
apply simp
done

lemmaapply java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
k
apply simp ( transjava.lang.StringIndexOutOfBoundsException: Range [18, 19) out of bounds for length 18
apply simp
apply (rulejava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
apply(rule      iterate
)
apply
apply  loop_lemma3 (no_asm
apply subst)
apply(ule_tac"\java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
apply simpjava.lang.StringIndexOutOfBoundsException: Index 154 out of bounds for length 154
apply (subst( s  "hile\b\g\(iterate (Suc 0)\(step\b\g)\x)" in trans)
apply(rule_tac"UU" t = b
apply (erule simp))
applyapply ( stripsimp
apply apply( p = "b(iterate n\(step\b\g)\x)" in trE)
apply simp
apply (subst while_unfold)
apply simp ( notE)
done

while_unfold3: "hile\b\g\x = while\b\g\(step\b\g\x)"
apply (rule_tac  "\b\g\(iterate (Suc 0)\(step\b\g)\x)" in trans)
applyapply (simp
apply simpapply (rule_tac s = "iterate (Suc and t = "g\<cdot>(iterate n\<cdot>(step\<cdot>b\<cdot>g)\<cdot>x)" in ssubst)
donedone

(* ------------------------------------------------------------------------- *) trans
(* properties of while and iterations                                        *)
(* ------------------------------------------------------------------------- *)(ubstapply simpapply (apply (simplesubstapply (introapply apply (ruleappl

blast:)
     <Longrightarrow(Suc(d)
apply simpapply simpapply java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply          apply (java.lang.StringIndexOutOfBoundsException: Range [16, 5) out of bounds for length 26
apply (rule step_def2java.lang.NullPointerException
apply simp
apply (erule exE)
apply (erule flat_codom k)apply (blast intro: loop_lemma6)
apply (* --------
done

(step\<cdot>b\<cdot>g)\<cdot>x \<noteq> UU\<rbrakk> \<Longrightarrow>
      iterate k\<cdot>(step\<cdot>b\<cdot>g)\<cdot>x \<noteq> UU"
apply ntro)
done (imp)add)

lemma [rule_formatno_asm
  "apply erulen)
         \<exists>y. b\<cdot>y = FF; INV x\<rbrakk>
\<Longrightarrow
apply (induct_tac
apply>(<forall>y. INV y \<and> b\<cdot>y = TT \<and> g\<cdot>y \<noteq> UU \<longrightarrow> INV (g\<cdot>y));
apply (introINV;while
apply (simp() add:step_def2
apply apply( () del:s\<cdot
erule
apply (simp: (simplesubstjava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
apply (simp (no_asm_simpandpplyruleconjI
applyrule(simp )
apply (erule ( (1)lemma3
apply ( (no_asm_simp: iterate_Suc: loop_lemma2
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  and java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
prefer (ssumption
apply (simp add
apply (drule:
done

lemma [rule_format
      premI
apply (induct_taci "Andy
apply    and:
      java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
apply (simplesubst while_unfoldulejava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
applyjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
rule(strip
esubst( intro)
apply ( intro )
y (ule
apply (lemma [ (no_asm
apply simp\<forall>k. b\<cdot>(iterate k\<cdot>(step\<cdot>b\<cdot>g)\<cdot>x) \<noteq> FF \<Longrightarrow>
done

lemma [ ()]:
  \forall>k.java.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101

apply (simplesubst)
apply (rule fix_ind)
apply simp
apply simp
apply (rule allI)
apply apply( (no_asm))
apply (rule_tac p = "b\(iterate m\(step\b\g)\x)" in trE)
apply (simpapply(rule_tacp ="b<>(iterate m\(step\b\g)\x)" in trE)
apply (simp (no_asm_simp))
apply ( s = xa
apply (erule_tac [2] spec>iterate mjava.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95
apply rule)
apply (rule trans)
apply(rule_tac]  [symmetric
apply (simp  [2  [symmetric
apply
done

java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 4
apply( t  x  iterate_0 subst
apply (erule loop_lemma5apply ( loop_lemma5)
done

lemma loop_lemma7: "whileb\g\x \ UU \ \k. b\(iterate k\(step\b\g)\x) = FF"
applyapply (blast intro: loop_lemma6)
done

(* ------------------------------------------------------------------------- *)
(* an invariant rule for loops                                               *)
(* ------------------------------------------------------------------------- *)

lemma loop_inv2:
"\(\y. INV y \ b\y = TT \ g\y \ UU \ INV (g\y));
    (\<forall>y. INV y \<and> b\<cdot>y = FF \<longrightarrow> Q y);
java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
apply( P= \lambda> java.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 103
apply ( loop_lemma7
apply (simplesubst loop_lemma4)
apply
apply(drule, erule mp
apply( conjI
prefer2applyassumption
apply (rule)
apply assumption
apply (blast introloop_lemma6)
apply assumption
apply (rotate_tac -1)
apply (simp add: loop_lemma4)
done

lemma:
   premP"Px)java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
    and: "
    and premTT ( -)
     premFF"\y. \INV y; b\y = FF\ \ Q y"
    and premW: "while\b\g\x \ UU"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply(rule)
apply d premI: "\y. P y \ INV y"
apply (rule_tac [3] premW    nd: "y. \INV y; b\y = TT; g\y \ UU\ \ INV (g\y)"
apply (  while
apply rule)
done rule_tac] [THENpremI

end

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