src/HOL/Lambda/ParRed.thy
 author berghofe Wed Jul 11 11:23:24 2007 +0200 (2007-07-11) changeset 23750 a1db5f819d00 parent 22271 51a80e238b29 child 25972 94b15338da8d permissions -rw-r--r--
- Renamed inductive2 to inductive
- Renamed some theorems about transitive closure for predicates
```     1 (*  Title:      HOL/Lambda/ParRed.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Tobias Nipkow
```
```     4     Copyright   1995 TU Muenchen
```
```     5
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```     6 Properties of => and "cd", in particular the diamond property of => and
```
```     7 confluence of beta.
```
```     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 inductive par_beta :: "[dB, dB] => bool"  (infixl "=>" 50)
```
```    18   where
```
```    19     var [simp, intro!]: "Var n => Var n"
```
```    20   | abs [simp, intro!]: "s => t ==> Abs s => Abs t"
```
```    21   | app [simp, intro!]: "[| s => s'; t => t' |] ==> s \<degree> t => s' \<degree> t'"
```
```    22   | beta [simp, intro!]: "[| s => s'; t => t' |] ==> (Abs s) \<degree> t => s'[t'/0]"
```
```    23
```
```    24 inductive_cases par_beta_cases [elim!]:
```
```    25   "Var n => t"
```
```    26   "Abs s => Abs t"
```
```    27   "(Abs s) \<degree> t => u"
```
```    28   "s \<degree> t => u"
```
```    29   "Abs s => t"
```
```    30
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```    31
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```    32 subsection {* Inclusions *}
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```    33
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```    34 text {* @{text "beta \<subseteq> par_beta \<subseteq> beta^*"} \medskip *}
```
```    35
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```    36 lemma par_beta_varL [simp]:
```
```    37     "(Var n => t) = (t = Var n)"
```
```    38   by blast
```
```    39
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```    40 lemma par_beta_refl [simp]: "t => t"  (* par_beta_refl [intro!] causes search to blow up *)
```
```    41   by (induct t) simp_all
```
```    42
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```    43 lemma beta_subset_par_beta: "beta <= par_beta"
```
```    44   apply (rule predicate2I)
```
```    45   apply (erule beta.induct)
```
```    46      apply (blast intro!: par_beta_refl)+
```
```    47   done
```
```    48
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```    49 lemma par_beta_subset_beta: "par_beta <= beta^**"
```
```    50   apply (rule predicate2I)
```
```    51   apply (erule par_beta.induct)
```
```    52      apply blast
```
```    53     apply (blast del: rtranclp.rtrancl_refl intro: rtranclp.rtrancl_into_rtrancl)+
```
```    54       -- {* @{thm[source] rtrancl_refl} complicates the proof by increasing the branching factor *}
```
```    55   done
```
```    56
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```    57
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```    58 subsection {* Misc properties of par-beta *}
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```    59
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```    60 lemma par_beta_lift [simp]:
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```    61     "t => t' \<Longrightarrow> lift t n => lift t' n"
```
```    62   by (induct t arbitrary: t' n) fastsimp+
```
```    63
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```    64 lemma par_beta_subst:
```
```    65     "s => s' \<Longrightarrow> t => t' \<Longrightarrow> t[s/n] => t'[s'/n]"
```
```    66   apply (induct t arbitrary: s s' t' n)
```
```    67     apply (simp add: subst_Var)
```
```    68    apply (erule par_beta_cases)
```
```    69     apply simp
```
```    70    apply (simp add: subst_subst [symmetric])
```
```    71    apply (fastsimp intro!: par_beta_lift)
```
```    72   apply fastsimp
```
```    73   done
```
```    74
```
```    75
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```    76 subsection {* Confluence (directly) *}
```
```    77
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```    78 lemma diamond_par_beta: "diamond par_beta"
```
```    79   apply (unfold diamond_def commute_def square_def)
```
```    80   apply (rule impI [THEN allI [THEN allI]])
```
```    81   apply (erule par_beta.induct)
```
```    82      apply (blast intro!: par_beta_subst)+
```
```    83   done
```
```    84
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```    85
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```    86 subsection {* Complete developments *}
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```    87
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```    88 consts
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```    89   "cd" :: "dB => dB"
```
```    90 recdef "cd" "measure size"
```
```    91   "cd (Var n) = Var n"
```
```    92   "cd (Var n \<degree> t) = Var n \<degree> cd t"
```
```    93   "cd ((s1 \<degree> s2) \<degree> t) = cd (s1 \<degree> s2) \<degree> cd t"
```
```    94   "cd (Abs u \<degree> t) = (cd u)[cd t/0]"
```
```    95   "cd (Abs s) = Abs (cd s)"
```
```    96
```
```    97 lemma par_beta_cd: "s => t \<Longrightarrow> t => cd s"
```
```    98   apply (induct s arbitrary: t rule: cd.induct)
```
```    99       apply auto
```
```   100   apply (fast intro!: par_beta_subst)
```
```   101   done
```
```   102
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```   103
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```   104 subsection {* Confluence (via complete developments) *}
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```   105
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```   106 lemma diamond_par_beta2: "diamond par_beta"
```
```   107   apply (unfold diamond_def commute_def square_def)
```
```   108   apply (blast intro: par_beta_cd)
```
```   109   done
```
```   110
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```   111 theorem beta_confluent: "confluent beta"
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```   112   apply (rule diamond_par_beta2 diamond_to_confluence
```
```   113     par_beta_subset_beta beta_subset_par_beta)+
```
```   114   done
```
```   115
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```   116 end
```