src/HOL/UNITY/Comp/Progress.thy
 author wenzelm Sat Jun 14 23:19:51 2008 +0200 (2008-06-14) changeset 27208 5fe899199f85 parent 18556 dc39832e9280 child 28866 30cd9d89a0fb permissions -rw-r--r--
proper context for tactics derived from res_inst_tac;
```     1 (*  Title:      HOL/UNITY/Transformers
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```     2     ID:         \$Id\$
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```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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```     4     Copyright   2003  University of Cambridge
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```     5
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```     6 David Meier's thesis
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```     7 *)
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```     8
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```     9 header{*Progress Set Examples*}
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```    10
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```    11 theory Progress imports "../UNITY_Main" begin
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```    12
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```    13 subsection {*The Composition of Two Single-Assignment Programs*}
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```    14 text{*Thesis Section 4.4.2*}
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```    15
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```    16 constdefs
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```    17   FF :: "int program"
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```    18     "FF == mk_total_program (UNIV, {range (\<lambda>x. (x, x+1))}, UNIV)"
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```    19
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```    20   GG :: "int program"
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```    21     "GG == mk_total_program (UNIV, {range (\<lambda>x. (x, 2*x))}, UNIV)"
```
```    22
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```    23 subsubsection {*Calculating @{term "wens_set FF (atLeast k)"}*}
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```    24
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```    25 lemma Domain_actFF: "Domain (range (\<lambda>x::int. (x, x + 1))) = UNIV"
```
```    26 by force
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```    27
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```    28 lemma FF_eq:
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```    29       "FF = mk_program (UNIV, {range (\<lambda>x. (x, x+1))}, UNIV)"
```
```    30 by (simp add: FF_def mk_total_program_def totalize_def totalize_act_def
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```    31               program_equalityI Domain_actFF)
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```    32
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```    33 lemma wp_actFF:
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```    34      "wp (range (\<lambda>x::int. (x, x + 1))) (atLeast k) = atLeast (k - 1)"
```
```    35 by (force simp add: wp_def)
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```    36
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```    37 lemma wens_FF: "wens FF (range (\<lambda>x. (x, x+1))) (atLeast k) = atLeast (k - 1)"
```
```    38 by (force simp add: FF_eq wens_single_eq wp_actFF)
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```    39
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```    40 lemma single_valued_actFF: "single_valued (range (\<lambda>x::int. (x, x + 1)))"
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```    41 by (force simp add: single_valued_def)
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```    42
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```    43 lemma wens_single_finite_FF:
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```    44      "wens_single_finite (range (\<lambda>x. (x, x+1))) (atLeast k) n =
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```    45       atLeast (k - int n)"
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```    46 apply (induct n, simp)
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```    47 apply (force simp add: wens_FF
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```    48             def_wens_single_finite_Suc_eq_wens [OF FF_eq single_valued_actFF])
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```    49 done
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```    50
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```    51 lemma wens_single_FF_eq_UNIV:
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```    52      "wens_single (range (\<lambda>x::int. (x, x + 1))) (atLeast k) = UNIV"
```
```    53 apply (auto simp add: wens_single_eq_Union)
```
```    54 apply (rule_tac x="nat (k-x)" in exI)
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```    55 apply (simp add: wens_single_finite_FF)
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```    56 done
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```    57
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```    58 lemma wens_set_FF:
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```    59      "wens_set FF (atLeast k) = insert UNIV (atLeast ` atMost k)"
```
```    60 apply (auto simp add: wens_set_single_eq [OF FF_eq single_valued_actFF]
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```    61                       wens_single_FF_eq_UNIV wens_single_finite_FF)
```
```    62 apply (erule notE)
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```    63 apply (rule_tac x="nat (k-xa)" in range_eqI)
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```    64 apply (simp add: wens_single_finite_FF)
```
```    65 done
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```    66
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```    67 subsubsection {*Proving @{term "FF \<in> UNIV leadsTo atLeast (k::int)"}*}
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```    68
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```    69 lemma atLeast_ensures: "FF \<in> atLeast (k - 1) ensures atLeast (k::int)"
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```    70 apply (simp add: Progress.wens_FF [symmetric] wens_ensures)
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```    71 apply (simp add: wens_ensures FF_eq)
```
```    72 done
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```    73
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```    74 lemma atLeast_leadsTo: "FF \<in> atLeast (k - int n) leadsTo atLeast (k::int)"
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```    75 apply (induct n)
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```    76  apply (simp_all add: leadsTo_refl)
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```    77 apply (rule_tac A = "atLeast (k - int n - 1)" in leadsTo_weaken_L)
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```    78  apply (blast intro: leadsTo_Trans atLeast_ensures, force)
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```    79 done
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```    80
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```    81 lemma UN_atLeast_UNIV: "(\<Union>n. atLeast (k - int n)) = UNIV"
```
```    82 apply auto
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```    83 apply (rule_tac x = "nat (k - x)" in exI, simp)
```
```    84 done
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```    85
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```    86 lemma FF_leadsTo: "FF \<in> UNIV leadsTo atLeast (k::int)"
```
```    87 apply (subst UN_atLeast_UNIV [symmetric])
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```    88 apply (rule leadsTo_UN [OF atLeast_leadsTo])
```
```    89 done
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```    90
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```    91 text{*Result (4.39): Applying the leadsTo-Join Theorem*}
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```    92 theorem "FF\<squnion>GG \<in> atLeast 0 leadsTo atLeast (k::int)"
```
```    93 apply (subgoal_tac "FF\<squnion>GG \<in> (atLeast 0 \<inter> atLeast 0) leadsTo atLeast k")
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```    94  apply simp
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```    95 apply (rule_tac T = "atLeast 0" in leadsTo_Join)
```
```    96   apply (simp add: awp_iff FF_def, safety)
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```    97  apply (simp add: awp_iff GG_def wens_set_FF, safety)
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```    98 apply (rule leadsTo_weaken_L [OF FF_leadsTo], simp)
```
```    99 done
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```   100
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```   101 end
```