src/CCL/Trancl.thy
author wenzelm
Sat May 15 22:15:57 2010 +0200 (2010-05-15)
changeset 36948 d2cdad45fd14
parent 32153 a0e57fb1b930
child 42156 df219e736a5d
permissions -rw-r--r--
renamed Outer_Parse to Parse (in Scala);
     1 (*  Title:      CCL/Trancl.thy
     2     Author:     Martin Coen, Cambridge University Computer Laboratory
     3     Copyright   1993  University of Cambridge
     4 *)
     5 
     6 header {* Transitive closure of a relation *}
     7 
     8 theory Trancl
     9 imports CCL
    10 begin
    11 
    12 consts
    13   trans   :: "i set => o"                   (*transitivity predicate*)
    14   id      :: "i set"
    15   rtrancl :: "i set => i set"               ("(_^*)" [100] 100)
    16   trancl  :: "i set => i set"               ("(_^+)" [100] 100)
    17   relcomp :: "[i set,i set] => i set"       (infixr "O" 60)
    18 
    19 axioms
    20   trans_def:       "trans(r) == (ALL x y z. <x,y>:r --> <y,z>:r --> <x,z>:r)"
    21   relcomp_def:     (*composition of relations*)
    22                    "r O s == {xz. EX x y z. xz = <x,z> & <x,y>:s & <y,z>:r}"
    23   id_def:          (*the identity relation*)
    24                    "id == {p. EX x. p = <x,x>}"
    25   rtrancl_def:     "r^* == lfp(%s. id Un (r O s))"
    26   trancl_def:      "r^+ == r O rtrancl(r)"
    27 
    28 
    29 subsection {* Natural deduction for @{text "trans(r)"} *}
    30 
    31 lemma transI:
    32   "(!! x y z. [| <x,y>:r;  <y,z>:r |] ==> <x,z>:r) ==> trans(r)"
    33   unfolding trans_def by blast
    34 
    35 lemma transD: "[| trans(r);  <a,b>:r;  <b,c>:r |] ==> <a,c>:r"
    36   unfolding trans_def by blast
    37 
    38 
    39 subsection {* Identity relation *}
    40 
    41 lemma idI: "<a,a> : id"
    42   apply (unfold id_def)
    43   apply (rule CollectI)
    44   apply (rule exI)
    45   apply (rule refl)
    46   done
    47 
    48 lemma idE:
    49     "[| p: id;  !!x.[| p = <x,x> |] ==> P |] ==>  P"
    50   apply (unfold id_def)
    51   apply (erule CollectE)
    52   apply blast
    53   done
    54 
    55 
    56 subsection {* Composition of two relations *}
    57 
    58 lemma compI: "[| <a,b>:s; <b,c>:r |] ==> <a,c> : r O s"
    59   unfolding relcomp_def by blast
    60 
    61 (*proof requires higher-level assumptions or a delaying of hyp_subst_tac*)
    62 lemma compE:
    63     "[| xz : r O s;
    64         !!x y z. [| xz = <x,z>;  <x,y>:s;  <y,z>:r |] ==> P
    65      |] ==> P"
    66   unfolding relcomp_def by blast
    67 
    68 lemma compEpair:
    69   "[| <a,c> : r O s;
    70       !!y. [| <a,y>:s;  <y,c>:r |] ==> P
    71    |] ==> P"
    72   apply (erule compE)
    73   apply (simp add: pair_inject)
    74   done
    75 
    76 lemmas [intro] = compI idI
    77   and [elim] = compE idE
    78   and [elim!] = pair_inject
    79 
    80 lemma comp_mono: "[| r'<=r; s'<=s |] ==> (r' O s') <= (r O s)"
    81   by blast
    82 
    83 
    84 subsection {* The relation rtrancl *}
    85 
    86 lemma rtrancl_fun_mono: "mono(%s. id Un (r O s))"
    87   apply (rule monoI)
    88   apply (rule monoI subset_refl comp_mono Un_mono)+
    89   apply assumption
    90   done
    91 
    92 lemma rtrancl_unfold: "r^* = id Un (r O r^*)"
    93   by (rule rtrancl_fun_mono [THEN rtrancl_def [THEN def_lfp_Tarski]])
    94 
    95 (*Reflexivity of rtrancl*)
    96 lemma rtrancl_refl: "<a,a> : r^*"
    97   apply (subst rtrancl_unfold)
    98   apply blast
    99   done
   100 
   101 (*Closure under composition with r*)
   102 lemma rtrancl_into_rtrancl: "[| <a,b> : r^*;  <b,c> : r |] ==> <a,c> : r^*"
   103   apply (subst rtrancl_unfold)
   104   apply blast
   105   done
   106 
   107 (*rtrancl of r contains r*)
   108 lemma r_into_rtrancl: "[| <a,b> : r |] ==> <a,b> : r^*"
   109   apply (rule rtrancl_refl [THEN rtrancl_into_rtrancl])
   110   apply assumption
   111   done
   112 
   113 
   114 subsection {* standard induction rule *}
   115 
   116 lemma rtrancl_full_induct:
   117   "[| <a,b> : r^*;
   118       !!x. P(<x,x>);
   119       !!x y z.[| P(<x,y>); <x,y>: r^*; <y,z>: r |]  ==>  P(<x,z>) |]
   120    ==>  P(<a,b>)"
   121   apply (erule def_induct [OF rtrancl_def])
   122    apply (rule rtrancl_fun_mono)
   123   apply blast
   124   done
   125 
   126 (*nice induction rule*)
   127 lemma rtrancl_induct:
   128   "[| <a,b> : r^*;
   129       P(a);
   130       !!y z.[| <a,y> : r^*;  <y,z> : r;  P(y) |] ==> P(z) |]
   131     ==> P(b)"
   132 (*by induction on this formula*)
   133   apply (subgoal_tac "ALL y. <a,b> = <a,y> --> P(y)")
   134 (*now solve first subgoal: this formula is sufficient*)
   135   apply blast
   136 (*now do the induction*)
   137   apply (erule rtrancl_full_induct)
   138    apply blast
   139   apply blast
   140   done
   141 
   142 (*transitivity of transitive closure!! -- by induction.*)
   143 lemma trans_rtrancl: "trans(r^*)"
   144   apply (rule transI)
   145   apply (rule_tac b = z in rtrancl_induct)
   146     apply (fast elim: rtrancl_into_rtrancl)+
   147   done
   148 
   149 (*elimination of rtrancl -- by induction on a special formula*)
   150 lemma rtranclE:
   151   "[| <a,b> : r^*;  (a = b) ==> P;
   152       !!y.[| <a,y> : r^*; <y,b> : r |] ==> P |]
   153    ==> P"
   154   apply (subgoal_tac "a = b | (EX y. <a,y> : r^* & <y,b> : r)")
   155    prefer 2
   156    apply (erule rtrancl_induct)
   157     apply blast
   158    apply blast
   159   apply blast
   160   done
   161 
   162 
   163 subsection {* The relation trancl *}
   164 
   165 subsubsection {* Conversions between trancl and rtrancl *}
   166 
   167 lemma trancl_into_rtrancl: "[| <a,b> : r^+ |] ==> <a,b> : r^*"
   168   apply (unfold trancl_def)
   169   apply (erule compEpair)
   170   apply (erule rtrancl_into_rtrancl)
   171   apply assumption
   172   done
   173 
   174 (*r^+ contains r*)
   175 lemma r_into_trancl: "[| <a,b> : r |] ==> <a,b> : r^+"
   176   unfolding trancl_def by (blast intro: rtrancl_refl)
   177 
   178 (*intro rule by definition: from rtrancl and r*)
   179 lemma rtrancl_into_trancl1: "[| <a,b> : r^*;  <b,c> : r |]   ==>  <a,c> : r^+"
   180   unfolding trancl_def by blast
   181 
   182 (*intro rule from r and rtrancl*)
   183 lemma rtrancl_into_trancl2: "[| <a,b> : r;  <b,c> : r^* |]   ==>  <a,c> : r^+"
   184   apply (erule rtranclE)
   185    apply (erule subst)
   186    apply (erule r_into_trancl)
   187   apply (rule trans_rtrancl [THEN transD, THEN rtrancl_into_trancl1])
   188     apply (assumption | rule r_into_rtrancl)+
   189   done
   190 
   191 (*elimination of r^+ -- NOT an induction rule*)
   192 lemma tranclE:
   193   "[| <a,b> : r^+;
   194       <a,b> : r ==> P;
   195       !!y.[| <a,y> : r^+;  <y,b> : r |] ==> P
   196    |] ==> P"
   197   apply (subgoal_tac "<a,b> : r | (EX y. <a,y> : r^+ & <y,b> : r)")
   198    apply blast
   199   apply (unfold trancl_def)
   200   apply (erule compEpair)
   201   apply (erule rtranclE)
   202    apply blast
   203   apply (blast intro!: rtrancl_into_trancl1)
   204   done
   205 
   206 (*Transitivity of r^+.
   207   Proved by unfolding since it uses transitivity of rtrancl. *)
   208 lemma trans_trancl: "trans(r^+)"
   209   apply (unfold trancl_def)
   210   apply (rule transI)
   211   apply (erule compEpair)+
   212   apply (erule rtrancl_into_rtrancl [THEN trans_rtrancl [THEN transD, THEN compI]])
   213     apply assumption+
   214   done
   215 
   216 lemma trancl_into_trancl2: "[| <a,b> : r;  <b,c> : r^+ |]   ==>  <a,c> : r^+"
   217   apply (rule r_into_trancl [THEN trans_trancl [THEN transD]])
   218    apply assumption+
   219   done
   220 
   221 end