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