src/HOL/Library/While_Combinator.thy
author nipkow
Thu Dec 18 08:20:36 2003 +0100 (2003-12-18)
changeset 14300 bf8b8c9425c3
parent 12791 ccc0f45ad2c4
child 14589 feae7b5fd425
permissions -rw-r--r--
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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 (permissive) 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::nat) = 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::nat) = 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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recdef_tc while_aux_tc: while_aux
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  apply (rule wf_same_fst)
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  apply (rule wf_same_fst)
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  apply (simp add: wf_iff_no_infinite_down_chain)
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  apply blast
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  done
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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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lemma while_aux_unfold:
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  "while_aux (b, c, s) =
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    (if \<exists>f. f (0::nat) = 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 (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 [code]:
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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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hide const while_aux
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lemma def_while_unfold: assumes fdef: "f == while test do"
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      shows "f x = (if test x then f(do x) else x)"
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proof -
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  have "f x = while test do x" using fdef by simp
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  also have "\<dots> = (if test x then while test do (do x) else x)"
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    by(rule while_unfold)
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  also have "\<dots> = (if test x then f(do x) else x)" by(simp add:fdef[symmetric])
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  finally show ?thesis .
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qed
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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 rule_context
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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: rule_context)
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    done
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qed
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theorem while_rule:
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  "[| P s;
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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 r;
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      !!s. [| P s; b s  |] ==> (c s, s) \<in> r |] ==>
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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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text {*
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 \medskip An application: computation of the @{term lfp} on finite
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 sets via iteration.
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*}
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theorem lfp_conv_while:
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  "[| mono f; finite U; f U = U |] ==>
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    lfp f = fst (while (\<lambda>(A, fA). A \<noteq> fA) (\<lambda>(A, fA). (fA, f fA)) ({}, f {}))"
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apply (rule_tac P = "\<lambda>(A, B). (A \<subseteq> U \<and> B = f A \<and> A \<subseteq> B \<and> B \<subseteq> lfp f)" and
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                r = "((Pow U \<times> UNIV) \<times> (Pow U \<times> UNIV)) \<inter>
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                     inv_image finite_psubset (op - U o fst)" in while_rule)
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   apply (subst lfp_unfold)
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    apply assumption
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   apply (simp add: monoD)
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  apply (subst lfp_unfold)
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   apply assumption
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  apply clarsimp
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  apply (blast dest: monoD)
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 apply (fastsimp intro!: lfp_lowerbound)
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 apply (blast intro: wf_finite_psubset Int_lower2 [THEN [2] wf_subset])
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apply (clarsimp simp add: inv_image_def finite_psubset_def order_less_le)
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apply (blast intro!: finite_Diff dest: monoD)
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done
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text {*
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 An example of using the @{term while} combinator.\footnote{It is safe
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 to keep the example here, since there is no effect on the current
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 theory.}
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*}
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theorem "P (lfp (\<lambda>N::int set. {0} \<union> {(n + 2) mod 6 | n. n \<in> N})) = P {0, 4, 2}"
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proof -
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  have aux: "!!f A B. {f n | n. A n \<or> B n} = {f n | n. A n} \<union> {f n | n. B n}"
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    apply blast
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    done
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  show ?thesis
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    apply (subst lfp_conv_while [where ?U = "{0, 1, 2, 3, 4, 5}"])
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       apply (rule monoI)
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      apply blast
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     apply simp
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    apply (simp add: aux set_eq_subset)
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    txt {* The fixpoint computation is performed purely by rewriting: *}
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    apply (simp add: while_unfold aux set_eq_subset del: subset_empty)
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    done
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qed
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end