src/HOL/Lambda/ParRed.thy
 author wenzelm Thu Feb 16 21:12:00 2006 +0100 (2006-02-16) changeset 19086 1b3780be6cc2 parent 18241 afdba6b3e383 child 19363 667b5ea637dd permissions -rw-r--r--
new-style definitions/abbreviations;
```     1 (*  Title:      HOL/Lambda/ParRed.thy
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```     2     ID:         \$Id\$
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```     3     Author:     Tobias Nipkow
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```     4     Copyright   1995 TU Muenchen
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```     5
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```     6 Properties of => and "cd", in particular the diamond property of => and
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```     7 confluence of beta.
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```     8 *)
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```     9
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```    10 header {* Parallel reduction and a complete developments *}
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```    11
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```    12 theory ParRed imports Lambda Commutation begin
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```    13
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```    14
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```    15 subsection {* Parallel reduction *}
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```    16
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```    17 consts
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```    18   par_beta :: "(dB \<times> dB) set"
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```    19
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```    20 abbreviation (output)
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```    21   par_beta_red :: "[dB, dB] => bool"  (infixl "=>" 50)
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```    22   "(s => t) = ((s, t) \<in> par_beta)"
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```    23
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```    24 inductive par_beta
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```    25   intros
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```    26     var [simp, intro!]: "Var n => Var n"
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```    27     abs [simp, intro!]: "s => t ==> Abs s => Abs t"
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```    28     app [simp, intro!]: "[| s => s'; t => t' |] ==> s \<degree> t => s' \<degree> t'"
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```    29     beta [simp, intro!]: "[| s => s'; t => t' |] ==> (Abs s) \<degree> t => s'[t'/0]"
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```    30
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```    31 inductive_cases par_beta_cases [elim!]:
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```    32   "Var n => t"
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```    33   "Abs s => Abs t"
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```    34   "(Abs s) \<degree> t => u"
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```    35   "s \<degree> t => u"
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```    36   "Abs s => t"
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```    37
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```    38
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```    39 subsection {* Inclusions *}
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```    40
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```    41 text {* @{text "beta \<subseteq> par_beta \<subseteq> beta^*"} \medskip *}
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```    42
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```    43 lemma par_beta_varL [simp]:
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```    44     "(Var n => t) = (t = Var n)"
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```    45   by blast
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```    46
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```    47 lemma par_beta_refl [simp]: "t => t"  (* par_beta_refl [intro!] causes search to blow up *)
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```    48   by (induct t) simp_all
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```    49
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```    50 lemma beta_subset_par_beta: "beta <= par_beta"
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```    51   apply (rule subsetI)
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```    52   apply clarify
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```    53   apply (erule beta.induct)
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```    54      apply (blast intro!: par_beta_refl)+
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```    55   done
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```    56
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```    57 lemma par_beta_subset_beta: "par_beta <= beta^*"
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```    58   apply (rule subsetI)
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```    59   apply clarify
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```    60   apply (erule par_beta.induct)
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```    61      apply blast
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```    62     apply (blast del: rtrancl_refl intro: rtrancl_into_rtrancl)+
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```    63       -- {* @{thm[source] rtrancl_refl} complicates the proof by increasing the branching factor *}
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```    64   done
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```    65
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```    66
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```    67 subsection {* Misc properties of par-beta *}
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```    68
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```    69 lemma par_beta_lift [simp]:
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```    70     "t => t' \<Longrightarrow> lift t n => lift t' n"
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```    71   by (induct t fixing: t' n) fastsimp+
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```    72
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```    73 lemma par_beta_subst:
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```    74     "s => s' \<Longrightarrow> t => t' \<Longrightarrow> t[s/n] => t'[s'/n]"
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```    75   apply (induct t fixing: s s' t' n)
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```    76     apply (simp add: subst_Var)
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```    77    apply (erule par_beta_cases)
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```    78     apply simp
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```    79    apply (simp add: subst_subst [symmetric])
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```    80    apply (fastsimp intro!: par_beta_lift)
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```    81   apply fastsimp
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```    82   done
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```    83
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```    84
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```    85 subsection {* Confluence (directly) *}
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```    86
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```    87 lemma diamond_par_beta: "diamond par_beta"
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```    88   apply (unfold diamond_def commute_def square_def)
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```    89   apply (rule impI [THEN allI [THEN allI]])
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```    90   apply (erule par_beta.induct)
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```    91      apply (blast intro!: par_beta_subst)+
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```    92   done
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```    93
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```    94
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```    95 subsection {* Complete developments *}
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```    96
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```    97 consts
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```    98   "cd" :: "dB => dB"
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```    99 recdef "cd" "measure size"
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```   100   "cd (Var n) = Var n"
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```   101   "cd (Var n \<degree> t) = Var n \<degree> cd t"
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```   102   "cd ((s1 \<degree> s2) \<degree> t) = cd (s1 \<degree> s2) \<degree> cd t"
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```   103   "cd (Abs u \<degree> t) = (cd u)[cd t/0]"
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```   104   "cd (Abs s) = Abs (cd s)"
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```   105
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```   106 lemma par_beta_cd: "s => t \<Longrightarrow> t => cd s"
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```   107   apply (induct s fixing: t rule: cd.induct)
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```   108       apply auto
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```   109   apply (fast intro!: par_beta_subst)
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```   110   done
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```   111
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```   112
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```   113 subsection {* Confluence (via complete developments) *}
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```   114
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```   115 lemma diamond_par_beta2: "diamond par_beta"
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```   116   apply (unfold diamond_def commute_def square_def)
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```   117   apply (blast intro: par_beta_cd)
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```   118   done
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```   119
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```   120 theorem beta_confluent: "confluent beta"
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```   121   apply (rule diamond_par_beta2 diamond_to_confluence
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```   122     par_beta_subset_beta beta_subset_par_beta)+
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```   123   done
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```   124
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```   125 end
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