src/HOL/UNITY/Detects.thy
 author wenzelm Thu Jul 23 22:13:42 2015 +0200 (2015-07-23) changeset 60773 d09c66a0ea10 parent 58889 5b7a9633cfa8 child 61635 c657ee4f59b7 permissions -rw-r--r--
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```     1 (*  Title:      HOL/UNITY/Detects.thy
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```     2     Author:     Tanja Vos, Cambridge University Computer Laboratory
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```     3     Copyright   2000  University of Cambridge
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```     4
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```     5 Detects definition (Section 3.8 of Chandy & Misra) using LeadsTo
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```     6 *)
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```     7
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```     8 section{*The Detects Relation*}
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```     9
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```    10 theory Detects imports FP SubstAx begin
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```    11
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```    12 definition Detects :: "['a set, 'a set] => 'a program set"  (infixl "Detects" 60)
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```    13   where "A Detects B = (Always (-A \<union> B)) \<inter> (B LeadsTo A)"
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```    14
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```    15 definition Equality :: "['a set, 'a set] => 'a set"  (infixl "<==>" 60)
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```    16   where "A <==> B = (-A \<union> B) \<inter> (A \<union> -B)"
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```    17
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```    18
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```    19 (* Corollary from Sectiom 3.6.4 *)
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```    20
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```    21 lemma Always_at_FP:
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```    22      "[|F \<in> A LeadsTo B; all_total F|] ==> F \<in> Always (-((FP F) \<inter> A \<inter> -B))"
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```    23 apply (rule LeadsTo_empty)
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```    24 apply (subgoal_tac "F \<in> (FP F \<inter> A \<inter> - B) LeadsTo (B \<inter> (FP F \<inter> -B))")
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```    25 apply (subgoal_tac [2] " (FP F \<inter> A \<inter> - B) = (A \<inter> (FP F \<inter> -B))")
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```    26 apply (subgoal_tac "(B \<inter> (FP F \<inter> -B)) = {}")
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```    27 apply auto
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```    28 apply (blast intro: PSP_Stable stable_imp_Stable stable_FP_Int)
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```    29 done
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```    30
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```    31
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```    32 lemma Detects_Trans:
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```    33      "[| F \<in> A Detects B; F \<in> B Detects C |] ==> F \<in> A Detects C"
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```    34 apply (unfold Detects_def Int_def)
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```    35 apply (simp (no_asm))
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```    36 apply safe
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```    37 apply (rule_tac [2] LeadsTo_Trans, auto)
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```    38 apply (subgoal_tac "F \<in> Always ((-A \<union> B) \<inter> (-B \<union> C))")
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```    39  apply (blast intro: Always_weaken)
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```    40 apply (simp add: Always_Int_distrib)
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```    41 done
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```    42
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```    43 lemma Detects_refl: "F \<in> A Detects A"
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```    44 apply (unfold Detects_def)
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```    45 apply (simp (no_asm) add: Un_commute Compl_partition subset_imp_LeadsTo)
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```    46 done
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```    47
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```    48 lemma Detects_eq_Un: "(A<==>B) = (A \<inter> B) \<union> (-A \<inter> -B)"
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```    49 by (unfold Equality_def, blast)
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```    50
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```    51 (*Not quite antisymmetry: sets A and B agree in all reachable states *)
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```    52 lemma Detects_antisym:
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```    53      "[| F \<in> A Detects B;  F \<in> B Detects A|] ==> F \<in> Always (A <==> B)"
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```    54 apply (unfold Detects_def Equality_def)
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```    55 apply (simp add: Always_Int_I Un_commute)
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```    56 done
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```    57
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```    58
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```    59 (* Theorem from Section 3.8 *)
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```    60
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```    61 lemma Detects_Always:
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```    62      "[|F \<in> A Detects B; all_total F|] ==> F \<in> Always (-(FP F) \<union> (A <==> B))"
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```    63 apply (unfold Detects_def Equality_def)
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```    64 apply (simp add: Un_Int_distrib Always_Int_distrib)
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```    65 apply (blast dest: Always_at_FP intro: Always_weaken)
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```    66 done
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```    67
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```    68 (* Theorem from exercise 11.1 Section 11.3.1 *)
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```    69
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```    70 lemma Detects_Imp_LeadstoEQ:
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```    71      "F \<in> A Detects B ==> F \<in> UNIV LeadsTo (A <==> B)"
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```    72 apply (unfold Detects_def Equality_def)
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```    73 apply (rule_tac B = B in LeadsTo_Diff)
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```    74  apply (blast intro: Always_LeadsToI subset_imp_LeadsTo)
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```    75 apply (blast intro: Always_LeadsTo_weaken)
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```    76 done
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```    77
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```    78
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```    79 end
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```    80
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