src/HOL/Library/While_Combinator.thy
author nipkow
Wed, 13 Dec 2000 09:32:55 +0100
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(*  Title:      HOL/Library/While.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    Copyright   2000 TU Muenchen
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*)
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header {*
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 \title{A general ``while'' combinator}
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 \author{Tobias Nipkow}
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*}
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theory While_Combinator = Main:
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text {*
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 We define a while-combinator @{term while} and prove: (a) an
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 unrestricted unfolding law (even if while diverges!)  (I got this
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 idea from Wolfgang Goerigk), and (b) the invariant rule for reasoning
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 about @{term while}.
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*}
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consts while_aux :: "('a => bool) \<times> ('a => 'a) \<times> 'a => 'a"
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recdef while_aux
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  "same_fst (\<lambda>b. True) (\<lambda>b. same_fst (\<lambda>c. True) (\<lambda>c.
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      {(t, s).  b s \<and> c s = t \<and>
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        \<not> (\<exists>f. f 0 = s \<and> (\<forall>i. b (f i) \<and> c (f i) = f (i + 1)))}))"
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  "while_aux (b, c, s) =
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    (if (\<exists>f. f 0 = s \<and> (\<forall>i. b (f i) \<and> c (f i) = f (i + 1)))
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      then arbitrary
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      else if b s then while_aux (b, c, c s)
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      else s)"
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constdefs
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  while :: "('a => bool) => ('a => 'a) => 'a => 'a"
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  "while b c s == while_aux (b, c, s)"
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ML_setup {*
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  goalw_cterm [] (cterm_of (sign_of (the_context ()))
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    (HOLogic.mk_Trueprop (hd (RecdefPackage.tcs_of (the_context ()) "while_aux"))));
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  br wf_same_fst 1;
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  br wf_same_fst 1;
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  by (asm_full_simp_tac (simpset() addsimps [wf_iff_no_infinite_down_chain]) 1);
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  by (Blast_tac 1);
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  qed "while_aux_tc";
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*} (* FIXME cannot access recdef tcs in Isar yet! *)
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lemma while_aux_unfold:
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  "while_aux (b, c, s) =
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    (if \<exists>f. f 0 = s \<and> (\<forall>i. b (f i) \<and> c (f i) = f (i + 1))
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      then arbitrary
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      else if b s then while_aux (b, c, c s)
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      else s)"
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  apply (rule while_aux_tc [THEN while_aux.simps [THEN trans]])
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   apply (simp add: same_fst_def)
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  apply (rule refl)
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  done
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text {*
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 The recursion equation for @{term while}: directly executable!
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*}
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theorem while_unfold:
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    "while b c s = (if b s then while b c (c s) else s)"
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  apply (unfold while_def)
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  apply (rule while_aux_unfold [THEN trans])
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  apply auto
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  apply (subst while_aux_unfold)
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  apply simp
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  apply clarify
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  apply (erule_tac x = "\<lambda>i. f (Suc i)" in allE)
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  apply blast
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  done
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text {*
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 The proof rule for @{term while}, where @{term P} is the invariant.
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*}
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theorem while_rule_lemma[rule_format]:
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  "(!!s. P s ==> b s ==> P (c s)) ==>
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    (!!s. P s ==> \<not> b s ==> Q s) ==>
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    wf {(t, s). P s \<and> b s \<and> t = c s} ==>
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    P s --> Q (while b c s)"
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proof -
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  case antecedent
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  assume wf: "wf {(t, s). P s \<and> b s \<and> t = c s}"
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  show ?thesis
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    apply (induct s rule: wf [THEN wf_induct])
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    apply simp
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    apply clarify
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    apply (subst while_unfold)
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    apply (simp add: antecedent)
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    done
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qed
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theorem while_rule:
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  "\<lbrakk> P s; \<And>s. \<lbrakk> P s; b s \<rbrakk> \<Longrightarrow> P(c s);
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    \<And>s.\<lbrakk> P s; \<not> b s \<rbrakk> \<Longrightarrow> Q s;
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    wf r;  \<And>s. \<lbrakk> P s; b s \<rbrakk> \<Longrightarrow> (c s,s) \<in> r \<rbrakk> \<Longrightarrow>
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    Q (while b c s)"
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apply (rule while_rule_lemma)
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prefer 4 apply assumption
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apply blast
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apply blast
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apply(erule wf_subset)
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apply blast
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done
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hide const while_aux
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end