src/HOL/Library/While_Combinator.thy

left_minus axiom;

(*  Title:      HOL/Library/While.thy
    ID:         $Id$
    Author:     Tobias Nipkow
    Copyright   2000 TU Muenchen
*)

header {*
 \title{A general ``while'' combinator}
 \author{Tobias Nipkow}
*}

theory While_Combinator = Main:

text {*
 We define a while-combinator @{term while} and prove: (a) an
 unrestricted unfolding law (even if while diverges!)  (I got this
 idea from Wolfgang Goerigk), and (b) the invariant rule for reasoning
 about @{term while}.
*}

consts while_aux :: "('a => bool) \<times> ('a => 'a) \<times> 'a => 'a"
recdef while_aux
  "same_fst (\<lambda>b. True) (\<lambda>b. same_fst (\<lambda>c. True) (\<lambda>c.
      {(t, s).  b s \<and> c s = t \<and>
        \<not> (\<exists>f. f 0 = s \<and> (\<forall>i. b (f i) \<and> c (f i) = f (i + 1)))}))"
  "while_aux (b, c, s) =
    (if (\<exists>f. f 0 = s \<and> (\<forall>i. b (f i) \<and> c (f i) = f (i + 1)))
      then arbitrary
      else if b s then while_aux (b, c, c s)
      else s)"

constdefs
  while :: "('a => bool) => ('a => 'a) => 'a => 'a"
  "while b c s == while_aux (b, c, s)"

ML_setup {*
  goalw_cterm [] (cterm_of (sign_of (the_context ()))
    (HOLogic.mk_Trueprop (hd (RecdefPackage.tcs_of (the_context ()) "while_aux"))));
  br wf_same_fstI 1;
  br wf_same_fstI 1;
  by (asm_full_simp_tac (simpset() addsimps [wf_iff_no_infinite_down_chain]) 1);
  by (Blast_tac 1);
  qed "while_aux_tc";
*} (* FIXME cannot access recdef tcs in Isar yet! *)

lemma while_aux_unfold:
  "while_aux (b, c, s) =
    (if \<exists>f. f 0 = s \<and> (\<forall>i. b (f i) \<and> c (f i) = f (i + 1))
      then arbitrary
      else if b s then while_aux (b, c, c s)
      else s)"
  apply (rule while_aux_tc [THEN while_aux.simps [THEN trans]])
   apply (simp add: same_fst_def)
  apply (rule refl)
  done

text {*
 The recursion equation for @{term while}: directly executable!
*}

theorem while_unfold:
    "while b c s = (if b s then while b c (c s) else s)"
  apply (unfold while_def)
  apply (rule while_aux_unfold [THEN trans])
  apply auto
  apply (subst while_aux_unfold)
  apply simp
  apply clarify
  apply (erule_tac x = "\<lambda>i. f (Suc i)" in allE)
  apply blast
  done

text {*
 The proof rule for @{term while}, where @{term P} is the invariant.
*}

theorem while_rule [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} ==>
    P s --> Q (while b c s)"
proof -
  case antecedent
  assume wf: "wf {(t, s). P s \<and> b s \<and> t = c s}"
  show ?thesis
    apply (induct s rule: wf [THEN wf_induct])
    apply simp
    apply clarify
    apply (subst while_unfold)
    apply (simp add: antecedent)
    done
qed

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)" in while_rule)
     apply (subst lfp_unfold)
      apply assumption
     apply clarsimp
     apply (blast dest: monoD)
    apply (fastsimp intro!: lfp_lowerbound)
   apply (rule_tac r = "((Pow U <*> UNIV) <*> (Pow U <*> UNIV)) \<inter>
       inv_image finite_psubset (op - U o fst)" in wf_subset)
    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)
  apply (subst lfp_unfold)
   apply assumption
  apply (simp add: monoD)
  done

end