src/HOL/Library/While_Combinator.thy

 
 theory While_Combinator
 imports Main
 begin
 
-subsection {* Option result *}
+subsection {* Partial version *}
 
 definition while_option :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a option" where
 "while_option b c s = (if (\<exists>k. ~ b ((c ^^ k) s))
    then Some ((c ^^ (LEAST k. ~ b ((c ^^ k) s))) s)
    else None)"

       by (induct i) (auto simp: init step 1) }
   thus "P t" by (auto simp: t)
 qed
 
 
-subsection {* Totalized version *}
+subsection {* Total version *}
 
 definition while :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
 where "while b c s = the (while_option b c s)"
 
 lemma while_unfold:

    apply blast
   apply (erule wf_subset)
   apply blast
   done
 
-text {*
- \medskip An application: computation of the @{term lfp} on finite
- sets via iteration.
-*}
-
-theorem lfp_conv_while:
-  "[| mono f; finite U; f U = U |] ==>
-    lfp f = fst (while (\<lambda>(A, fA). A \<noteq> fA) (\<lambda>(A, fA). (fA, f fA)) ({}, f {}))"
-apply (rule_tac P = "\<lambda>(A, B). (A \<subseteq> U \<and> B = f A \<and> A \<subseteq> B \<and> B \<subseteq> lfp f)" and
-                r = "((Pow U \<times> UNIV) \<times> (Pow U \<times> UNIV)) \<inter>
-                     inv_image finite_psubset (op - U o fst)" in while_rule)
-   apply (subst lfp_unfold)
-    apply assumption
-   apply (simp add: monoD)
-  apply (subst lfp_unfold)
-   apply assumption
-  apply clarsimp
-  apply (blast dest: monoD)
- apply (fastsimp intro!: lfp_lowerbound)
- apply (blast intro: wf_finite_psubset Int_lower2 [THEN [2] wf_subset])
-apply (clarsimp simp add: finite_psubset_def order_less_le)
-apply (blast intro!: finite_Diff dest: monoD)
-done
-
-
-subsection {* Example *}
-
-text{* Cannot use @{thm[source]set_eq_subset} because it leads to
-looping because the antisymmetry simproc turns the subset relationship
-back into equality. *}
-
-theorem "P (lfp (\<lambda>N::int set. {0} \<union> {(n + 2) mod 6 | n. n \<in> N})) =
-  P {0, 4, 2}"
-proof -
-  have seteq: "!!A B. (A = B) = ((!a : A. a:B) & (!b:B. b:A))"
-    by blast
-  have aux: "!!f A B. {f n | n. A n \<or> B n} = {f n | n. A n} \<union> {f n | n. B n}"
-    apply blast
-    done
-  show ?thesis
-    apply (subst lfp_conv_while [where ?U = "{0, 1, 2, 3, 4, 5}"])
-       apply (rule monoI)
-      apply blast
-     apply simp
-    apply (simp add: aux set_eq_subset)
-    txt {* The fixpoint computation is performed purely by rewriting: *}
-    apply (simp add: while_unfold aux seteq del: subset_empty)
-    done
-qed
 
 end