src/HOL/Library/While_Combinator.thy

--- a/src/HOL/Library/While_Combinator.thy	Fri Jan 26 15:50:28 2001 +0100
+++ b/src/HOL/Library/While_Combinator.thy	Fri Jan 26 15:50:52 2001 +0100
@@ -67,14 +67,16 @@
   apply blast
   done
 
+hide const while_aux
+
 text {*
  The proof rule for @{term while}, where @{term P} is the invariant.
 *}
 
 theorem while_rule_lemma[rule_format]:
-  "(!!s. P s ==> b s ==> P (c s)) ==>
-    (!!s. P s ==> \<not> b s ==> Q s) ==>
-    wf {(t, s). P s \<and> b s \<and> t = c s} ==>
+  "[| !!s. P s ==> b s ==> P (c s);
+      !!s. P s ==> \<not> b s ==> Q s;
+      wf {(t, s). P s \<and> b s \<and> t = c s} |] ==>
     P s --> Q (while b c s)"
 proof -
   case antecedent
@@ -89,10 +91,12 @@
 qed
 
 theorem while_rule:
-  "[| P s; !!s. [| P s; b s  |] ==> P (c s);
-    !!s. [| P s; \<not> b s  |] ==> Q s;
-    wf r;  !!s. [| P s; b s  |] ==> (c s, s) \<in> r |] ==>
-    Q (while b c s)"
+  "[| P s;
+      !!s. [| P s; b s  |] ==> P (c s);
+      !!s. [| P s; \<not> b s  |] ==> Q s;
+      wf r; 
+      !!s. [| P s; b s  |] ==> (c s, s) \<in> r |] ==>
+   Q (while b c s)"
 apply (rule while_rule_lemma)
 prefer 4 apply assumption
 apply blast
@@ -101,6 +105,50 @@
 apply blast
 done
 
-hide const while_aux
+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 <*> UNIV) <*> (Pow U <*> 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: inv_image_def finite_psubset_def order_less_le)
+apply (blast intro!: finite_Diff dest: monoD)
+done
+
+
+(*
+text {*
+ An example of using the @{term while} combinator.
+*}
+
+lemma aux: "{f n | n. A n \<or> B n} = {f n | n. A n} \<union> {f n | n. B n}"
+  apply blast
+  done
+
+theorem "P (lfp (\<lambda>N::int set. {#0} \<union> {(n + #2) mod #6 | n. n \<in> N})) =
+    P {#0, #4, #2}"
+  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 set_eq_subset del: subset_empty)
+  done
+*)
 
 end