src/HOL/Library/While_Combinator.thy
 author wenzelm Wed Jan 03 21:24:29 2001 +0100 (2001-01-03) changeset 10774 4de3a0d3ae28 parent 10673 337c00fd385b child 10984 8f49dcbec859 permissions -rw-r--r--
recdef_tc;
```     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"
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```    22 recdef 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 = 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 = 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 = 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]])
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```    50    apply (simp add: same_fst_def)
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```    51   apply (rule refl)
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```    52   done
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```    53
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```    54 text {*
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```    55  The recursion equation for @{term while}: directly executable!
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```    56 *}
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```    57
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```    58 theorem while_unfold:
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```    59     "while b c s = (if b s then while b c (c s) else s)"
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```    60   apply (unfold while_def)
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```    61   apply (rule while_aux_unfold [THEN trans])
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```    62   apply auto
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```    63   apply (subst while_aux_unfold)
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```    64   apply simp
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```    65   apply clarify
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```    66   apply (erule_tac x = "\<lambda>i. f (Suc i)" in allE)
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```    67   apply blast
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```    68   done
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```    69
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```    70 text {*
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```    71  The proof rule for @{term while}, where @{term P} is the invariant.
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```    72 *}
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```    73
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```    74 theorem while_rule_lemma[rule_format]:
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```    75   "(!!s. P s ==> b s ==> P (c s)) ==>
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```    76     (!!s. P s ==> \<not> b s ==> Q s) ==>
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```    77     wf {(t, s). P s \<and> b s \<and> t = c s} ==>
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```    78     P s --> Q (while b c s)"
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```    79 proof -
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```    80   case antecedent
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```    81   assume wf: "wf {(t, s). P s \<and> b s \<and> t = c s}"
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```    82   show ?thesis
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```    83     apply (induct s rule: wf [THEN wf_induct])
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```    84     apply simp
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```    85     apply clarify
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```    86     apply (subst while_unfold)
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```    87     apply (simp add: antecedent)
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```    88     done
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```    89 qed
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```    90
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```    91 theorem while_rule:
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```    92   "[| P s; !!s. [| P s; b s  |] ==> P (c s);
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```    93     !!s. [| P s; \<not> b s  |] ==> Q s;
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```    94     wf r;  !!s. [| P s; b s  |] ==> (c s, s) \<in> r |] ==>
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```    95     Q (while b c s)"
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```    96 apply (rule while_rule_lemma)
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```    97 prefer 4 apply assumption
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```    98 apply blast
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```    99 apply blast
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```   100 apply(erule wf_subset)
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```   101 apply blast
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```   102 done
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```   103
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```   104 hide const while_aux
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```   105
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```   106 end
```