src/HOL/Library/While_Combinator.thy
 author huffman Thu Jun 11 09:03:24 2009 -0700 (2009-06-11) changeset 31563 ded2364d14d4 parent 30738 0842e906300c child 37757 dc78d2d9e90a permissions -rw-r--r--
cleaned up some proofs
```     1 (*  Title:      HOL/Library/While_Combinator.thy
```
```     2     Author:     Tobias Nipkow
```
```     3     Copyright   2000 TU Muenchen
```
```     4 *)
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```     5
```
```     6 header {* A general ``while'' combinator *}
```
```     7
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```     8 theory While_Combinator
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```     9 imports Main
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```    10 begin
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```    11
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```    12 text {*
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```    13   We define the while combinator as the "mother of all tail recursive functions".
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```    14 *}
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```    15
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```    16 function (tailrec) while :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
```
```    17 where
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```    18   while_unfold[simp del]: "while b c s = (if b s then while b c (c s) else s)"
```
```    19 by auto
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```    20
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```    21 declare while_unfold[code]
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```    22
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```    23 lemma def_while_unfold:
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```    24   assumes fdef: "f == while test do"
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```    25   shows "f x = (if test x then f(do x) else x)"
```
```    26 proof -
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```    27   have "f x = while test do x" using fdef by simp
```
```    28   also have "\<dots> = (if test x then while test do (do x) else x)"
```
```    29     by(rule while_unfold)
```
```    30   also have "\<dots> = (if test x then f(do x) else x)" by(simp add:fdef[symmetric])
```
```    31   finally show ?thesis .
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```    32 qed
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```    33
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```    34
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```    35 text {*
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```    36  The proof rule for @{term while}, where @{term P} is the invariant.
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```    37 *}
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```    38
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```    39 theorem while_rule_lemma:
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```    40   assumes invariant: "!!s. P s ==> b s ==> P (c s)"
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```    41     and terminate: "!!s. P s ==> \<not> b s ==> Q s"
```
```    42     and wf: "wf {(t, s). P s \<and> b s \<and> t = c s}"
```
```    43   shows "P s \<Longrightarrow> Q (while b c s)"
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```    44   using wf
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```    45   apply (induct s)
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```    46   apply simp
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```    47   apply (subst while_unfold)
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```    48   apply (simp add: invariant terminate)
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```    49   done
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```    50
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```    51 theorem while_rule:
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```    52   "[| P s;
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```    53       !!s. [| P s; b s  |] ==> P (c s);
```
```    54       !!s. [| P s; \<not> b s  |] ==> Q s;
```
```    55       wf r;
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```    56       !!s. [| P s; b s  |] ==> (c s, s) \<in> r |] ==>
```
```    57    Q (while b c s)"
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```    58   apply (rule while_rule_lemma)
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```    59      prefer 4 apply assumption
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```    60     apply blast
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```    61    apply blast
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```    62   apply (erule wf_subset)
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```    63   apply blast
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```    64   done
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```    65
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```    66 text {*
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```    67  \medskip An application: computation of the @{term lfp} on finite
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```    68  sets via iteration.
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```    69 *}
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```    70
```
```    71 theorem lfp_conv_while:
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```    72   "[| mono f; finite U; f U = U |] ==>
```
```    73     lfp f = fst (while (\<lambda>(A, fA). A \<noteq> fA) (\<lambda>(A, fA). (fA, f fA)) ({}, f {}))"
```
```    74 apply (rule_tac P = "\<lambda>(A, B). (A \<subseteq> U \<and> B = f A \<and> A \<subseteq> B \<and> B \<subseteq> lfp f)" and
```
```    75                 r = "((Pow U \<times> UNIV) \<times> (Pow U \<times> UNIV)) \<inter>
```
```    76                      inv_image finite_psubset (op - U o fst)" in while_rule)
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```    77    apply (subst lfp_unfold)
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```    78     apply assumption
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```    79    apply (simp add: monoD)
```
```    80   apply (subst lfp_unfold)
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```    81    apply assumption
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```    82   apply clarsimp
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```    83   apply (blast dest: monoD)
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```    84  apply (fastsimp intro!: lfp_lowerbound)
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```    85  apply (blast intro: wf_finite_psubset Int_lower2 [THEN  wf_subset])
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```    86 apply (clarsimp simp add: finite_psubset_def order_less_le)
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```    87 apply (blast intro!: finite_Diff dest: monoD)
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```    88 done
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```    89
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```    90
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```    91 text {*
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```    92  An example of using the @{term while} combinator.
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```    93 *}
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```    94
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```    95 text{* Cannot use @{thm[source]set_eq_subset} because it leads to
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```    96 looping because the antisymmetry simproc turns the subset relationship
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```    97 back into equality. *}
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```    98
```
```    99 theorem "P (lfp (\<lambda>N::int set. {0} \<union> {(n + 2) mod 6 | n. n \<in> N})) =
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```   100   P {0, 4, 2}"
```
```   101 proof -
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```   102   have seteq: "!!A B. (A = B) = ((!a : A. a:B) & (!b:B. b:A))"
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```   103     by blast
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```   104   have aux: "!!f A B. {f n | n. A n \<or> B n} = {f n | n. A n} \<union> {f n | n. B n}"
```
```   105     apply blast
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```   106     done
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```   107   show ?thesis
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```   108     apply (subst lfp_conv_while [where ?U = "{0, 1, 2, 3, 4, 5}"])
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```   109        apply (rule monoI)
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```   110       apply blast
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```   111      apply simp
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```   112     apply (simp add: aux set_eq_subset)
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```   113     txt {* The fixpoint computation is performed purely by rewriting: *}
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```   114     apply (simp add: while_unfold aux seteq del: subset_empty)
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```   115     done
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```   116 qed
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```   117
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```   118 end
```