src/HOL/Library/While_Combinator.thy
 author krauss Mon Jun 05 14:22:58 2006 +0200 (2006-06-05) changeset 19769 c40ce2de2020 parent 19736 d8d0f8f51d69 child 20807 bd3b60f9a343 permissions -rw-r--r--
Added [simp]-lemmas "in_inv_image" and "in_lex_prod" in the spirit of "in_measure".
This simplifies some proofs.
```     1 (*  Title:      HOL/Library/While.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Tobias Nipkow
```
```     4     Copyright   2000 TU Muenchen
```
```     5 *)
```
```     6
```
```     7 header {* A general ``while'' combinator *}
```
```     8
```
```     9 theory While_Combinator
```
```    10 imports Main
```
```    11 begin
```
```    12
```
```    13 text {*
```
```    14  We define a while-combinator @{term while} and prove: (a) an
```
```    15  unrestricted unfolding law (even if while diverges!)  (I got this
```
```    16  idea from Wolfgang Goerigk), and (b) the invariant rule for reasoning
```
```    17  about @{term while}.
```
```    18 *}
```
```    19
```
```    20 consts while_aux :: "('a => bool) \<times> ('a => 'a) \<times> 'a => 'a"
```
```    21 recdef (permissive) while_aux
```
```    22   "same_fst (\<lambda>b. True) (\<lambda>b. same_fst (\<lambda>c. True) (\<lambda>c.
```
```    23       {(t, s).  b s \<and> c s = t \<and>
```
```    24         \<not> (\<exists>f. f (0::nat) = s \<and> (\<forall>i. b (f i) \<and> c (f i) = f (i + 1)))}))"
```
```    25   "while_aux (b, c, s) =
```
```    26     (if (\<exists>f. f (0::nat) = s \<and> (\<forall>i. b (f i) \<and> c (f i) = f (i + 1)))
```
```    27       then arbitrary
```
```    28       else if b s then while_aux (b, c, c s)
```
```    29       else s)"
```
```    30
```
```    31 recdef_tc while_aux_tc: while_aux
```
```    32   apply (rule wf_same_fst)
```
```    33   apply (rule wf_same_fst)
```
```    34   apply (simp add: wf_iff_no_infinite_down_chain)
```
```    35   apply blast
```
```    36   done
```
```    37
```
```    38 definition
```
```    39   while :: "('a => bool) => ('a => 'a) => 'a => 'a"
```
```    40   "while b c s == while_aux (b, c, s)"
```
```    41
```
```    42 lemma while_aux_unfold:
```
```    43   "while_aux (b, c, s) =
```
```    44     (if \<exists>f. f (0::nat) = s \<and> (\<forall>i. b (f i) \<and> c (f i) = f (i + 1))
```
```    45       then arbitrary
```
```    46       else if b s then while_aux (b, c, c s)
```
```    47       else s)"
```
```    48   apply (rule while_aux_tc [THEN while_aux.simps [THEN trans]])
```
```    49   apply (rule refl)
```
```    50   done
```
```    51
```
```    52 text {*
```
```    53  The recursion equation for @{term while}: directly executable!
```
```    54 *}
```
```    55
```
```    56 theorem while_unfold [code]:
```
```    57     "while b c s = (if b s then while b c (c s) else s)"
```
```    58   apply (unfold while_def)
```
```    59   apply (rule while_aux_unfold [THEN trans])
```
```    60   apply auto
```
```    61   apply (subst while_aux_unfold)
```
```    62   apply simp
```
```    63   apply clarify
```
```    64   apply (erule_tac x = "\<lambda>i. f (Suc i)" in allE)
```
```    65   apply blast
```
```    66   done
```
```    67
```
```    68 hide const while_aux
```
```    69
```
```    70 lemma def_while_unfold:
```
```    71   assumes fdef: "f == while test do"
```
```    72   shows "f x = (if test x then f(do x) else x)"
```
```    73 proof -
```
```    74   have "f x = while test do x" using fdef by simp
```
```    75   also have "\<dots> = (if test x then while test do (do x) else x)"
```
```    76     by(rule while_unfold)
```
```    77   also have "\<dots> = (if test x then f(do x) else x)" by(simp add:fdef[symmetric])
```
```    78   finally show ?thesis .
```
```    79 qed
```
```    80
```
```    81
```
```    82 text {*
```
```    83  The proof rule for @{term while}, where @{term P} is the invariant.
```
```    84 *}
```
```    85
```
```    86 theorem while_rule_lemma:
```
```    87   assumes invariant: "!!s. P s ==> b s ==> P (c s)"
```
```    88     and terminate: "!!s. P s ==> \<not> b s ==> Q s"
```
```    89     and wf: "wf {(t, s). P s \<and> b s \<and> t = c s}"
```
```    90   shows "P s \<Longrightarrow> Q (while b c s)"
```
```    91   using wf
```
```    92   apply (induct s)
```
```    93   apply simp
```
```    94   apply (subst while_unfold)
```
```    95   apply (simp add: invariant terminate)
```
```    96   done
```
```    97
```
```    98 theorem while_rule:
```
```    99   "[| P s;
```
```   100       !!s. [| P s; b s  |] ==> P (c s);
```
```   101       !!s. [| P s; \<not> b s  |] ==> Q s;
```
```   102       wf r;
```
```   103       !!s. [| P s; b s  |] ==> (c s, s) \<in> r |] ==>
```
```   104    Q (while b c s)"
```
```   105   apply (rule while_rule_lemma)
```
```   106      prefer 4 apply assumption
```
```   107     apply blast
```
```   108    apply blast
```
```   109   apply (erule wf_subset)
```
```   110   apply blast
```
```   111   done
```
```   112
```
```   113 text {*
```
```   114  \medskip An application: computation of the @{term lfp} on finite
```
```   115  sets via iteration.
```
```   116 *}
```
```   117
```
```   118 theorem lfp_conv_while:
```
```   119   "[| mono f; finite U; f U = U |] ==>
```
```   120     lfp f = fst (while (\<lambda>(A, fA). A \<noteq> fA) (\<lambda>(A, fA). (fA, f fA)) ({}, f {}))"
```
```   121 apply (rule_tac P = "\<lambda>(A, B). (A \<subseteq> U \<and> B = f A \<and> A \<subseteq> B \<and> B \<subseteq> lfp f)" and
```
```   122                 r = "((Pow U \<times> UNIV) \<times> (Pow U \<times> UNIV)) \<inter>
```
```   123                      inv_image finite_psubset (op - U o fst)" in while_rule)
```
```   124    apply (subst lfp_unfold)
```
```   125     apply assumption
```
```   126    apply (simp add: monoD)
```
```   127   apply (subst lfp_unfold)
```
```   128    apply assumption
```
```   129   apply clarsimp
```
```   130   apply (blast dest: monoD)
```
```   131  apply (fastsimp intro!: lfp_lowerbound)
```
```   132  apply (blast intro: wf_finite_psubset Int_lower2 [THEN  wf_subset])
```
```   133 apply (clarsimp simp add: finite_psubset_def order_less_le)
```
```   134 apply (blast intro!: finite_Diff dest: monoD)
```
```   135 done
```
```   136
```
```   137
```
```   138 text {*
```
```   139  An example of using the @{term while} combinator.
```
```   140 *}
```
```   141
```
```   142 text{* Cannot use @{thm[source]set_eq_subset} because it leads to
```
```   143 looping because the antisymmetry simproc turns the subset relationship
```
```   144 back into equality. *}
```
```   145
```
```   146 theorem "P (lfp (\<lambda>N::int set. {0} \<union> {(n + 2) mod 6 | n. n \<in> N})) =
```
```   147   P {0, 4, 2}"
```
```   148 proof -
```
```   149   have seteq: "!!A B. (A = B) = ((!a : A. a:B) & (!b:B. b:A))"
```
```   150     by blast
```
```   151   have aux: "!!f A B. {f n | n. A n \<or> B n} = {f n | n. A n} \<union> {f n | n. B n}"
```
```   152     apply blast
```
```   153     done
```
```   154   show ?thesis
```
```   155     apply (subst lfp_conv_while [where ?U = "{0, 1, 2, 3, 4, 5}"])
```
```   156        apply (rule monoI)
```
```   157       apply blast
```
```   158      apply simp
```
```   159     apply (simp add: aux set_eq_subset)
```
```   160     txt {* The fixpoint computation is performed purely by rewriting: *}
```
```   161     apply (simp add: while_unfold aux seteq del: subset_empty)
```
```   162     done
```
```   163 qed
```
```   164
```
```   165 end
```