src/HOL/Relation_Power.thy
author paulson
Tue Jun 28 15:27:45 2005 +0200 (2005-06-28)
changeset 16587 b34c8aa657a5
parent 15410 18914688a5fd
child 18049 156bba334c12
permissions -rw-r--r--
Constant "If" is now local
     1 (*  Title:      HOL/Relation_Power.thy
     2     ID:         $Id$
     3     Author:     Tobias Nipkow
     4     Copyright   1996  TU Muenchen
     5 *)
     6 
     7 header{*Powers of Relations and Functions*}
     8 
     9 theory Relation_Power
    10 imports Nat
    11 begin
    12 
    13 instance
    14   set :: (type) power ..  
    15       --{* only type @{typ "('a * 'a) set"} should be in class @{text power}!*}
    16 
    17 (*R^n = R O ... O R, the n-fold composition of R*)
    18 primrec (relpow)
    19   "R^0 = Id"
    20   "R^(Suc n) = R O (R^n)"
    21 
    22 
    23 instance
    24   fun :: (type, type) power ..
    25       --{* only type @{typ "'a => 'a"} should be in class @{text power}!*}
    26 
    27 (*f^n = f o ... o f, the n-fold composition of f*)
    28 primrec (funpow)
    29   "f^0 = id"
    30   "f^(Suc n) = f o (f^n)"
    31 
    32 text{*WARNING: due to the limits of Isabelle's type classes, exponentiation on
    33 functions and relations has too general a domain, namely @{typ "('a * 'b)set"}
    34 and @{typ "'a => 'b"}.  Explicit type constraints may therefore be necessary.
    35 For example, @{term "range(f^n) = A"} and @{term "Range(R^n) = B"} need
    36 constraints.*}
    37 
    38 lemma funpow_add: "f ^ (m+n) = f^m o f^n"
    39 by(induct m) simp_all
    40 
    41 lemma rel_pow_1: "!!R:: ('a*'a)set. R^1 = R"
    42 by simp
    43 declare rel_pow_1 [simp]
    44 
    45 lemma rel_pow_0_I: "(x,x) : R^0"
    46 by simp
    47 
    48 lemma rel_pow_Suc_I: "[| (x,y) : R^n; (y,z):R |] ==> (x,z):R^(Suc n)"
    49 apply (auto ); 
    50 done
    51 
    52 lemma rel_pow_Suc_I2 [rule_format]:
    53      "\<forall>z. (x,y) : R --> (y,z):R^n -->  (x,z):R^(Suc n)"
    54 apply (induct_tac "n", simp_all)
    55 apply blast
    56 done
    57 
    58 lemma rel_pow_0_E: "[| (x,y) : R^0; x=y ==> P |] ==> P"
    59 by simp
    60 
    61 lemma rel_pow_Suc_E: 
    62      "[| (x,z) : R^(Suc n);  !!y. [| (x,y) : R^n; (y,z) : R |] ==> P |] ==> P"
    63 by auto
    64 
    65 lemma rel_pow_E: 
    66     "[| (x,z) : R^n;  [| n=0; x = z |] ==> P;         
    67         !!y m. [| n = Suc m; (x,y) : R^m; (y,z) : R |] ==> P   
    68      |] ==> P"
    69 by (case_tac "n", auto)
    70 
    71 lemma rel_pow_Suc_D2 [rule_format]:
    72      "\<forall>x z. (x,z):R^(Suc n) --> (\<exists>y. (x,y):R & (y,z):R^n)"
    73 apply (induct_tac "n")
    74 apply (blast intro: rel_pow_0_I elim: rel_pow_0_E rel_pow_Suc_E)
    75 apply (blast intro: rel_pow_Suc_I elim: rel_pow_0_E rel_pow_Suc_E)
    76 done
    77 
    78 
    79 lemma rel_pow_Suc_D2':
    80      "\<forall>x y z. (x,y) : R^n & (y,z) : R --> (\<exists>w. (x,w) : R & (w,z) : R^n)"
    81 by (induct_tac "n", simp_all, blast)
    82 
    83 lemma rel_pow_E2: 
    84     "[| (x,z) : R^n;  [| n=0; x = z |] ==> P;         
    85         !!y m. [| n = Suc m; (x,y) : R; (y,z) : R^m |] ==> P   
    86      |] ==> P"
    87 apply (case_tac "n", simp) 
    88 apply (cut_tac n=nat and R=R in rel_pow_Suc_D2', simp, blast) 
    89 done
    90 
    91 lemma rtrancl_imp_UN_rel_pow: "!!p. p:R^* ==> p : (UN n. R^n)"
    92 apply (simp only: split_tupled_all)
    93 apply (erule rtrancl_induct)
    94 apply (blast intro: rel_pow_0_I rel_pow_Suc_I)+
    95 done
    96 
    97 lemma rel_pow_imp_rtrancl: "!!p. p:R^n ==> p:R^*"
    98 apply (simp only: split_tupled_all)
    99 apply (induct "n")
   100 apply (blast intro: rtrancl_refl elim: rel_pow_0_E)
   101 apply (blast elim: rel_pow_Suc_E intro: rtrancl_into_rtrancl)
   102 done
   103 
   104 lemma rtrancl_is_UN_rel_pow: "R^* = (UN n. R^n)"
   105 by (blast intro: rtrancl_imp_UN_rel_pow rel_pow_imp_rtrancl)
   106 
   107 
   108 lemma single_valued_rel_pow [rule_format]:
   109      "!!r::('a * 'a)set. single_valued r ==> single_valued (r^n)"
   110 apply (rule single_valuedI)
   111 apply (induct_tac "n", simp)
   112 apply (fast dest: single_valuedD elim: rel_pow_Suc_E)
   113 done
   114 
   115 ML
   116 {*
   117 val funpow_add = thm "funpow_add";
   118 val rel_pow_1 = thm "rel_pow_1";
   119 val rel_pow_0_I = thm "rel_pow_0_I";
   120 val rel_pow_Suc_I = thm "rel_pow_Suc_I";
   121 val rel_pow_Suc_I2 = thm "rel_pow_Suc_I2";
   122 val rel_pow_0_E = thm "rel_pow_0_E";
   123 val rel_pow_Suc_E = thm "rel_pow_Suc_E";
   124 val rel_pow_E = thm "rel_pow_E";
   125 val rel_pow_Suc_D2 = thm "rel_pow_Suc_D2";
   126 val rel_pow_Suc_D2 = thm "rel_pow_Suc_D2";
   127 val rel_pow_E2 = thm "rel_pow_E2";
   128 val rtrancl_imp_UN_rel_pow = thm "rtrancl_imp_UN_rel_pow";
   129 val rel_pow_imp_rtrancl = thm "rel_pow_imp_rtrancl";
   130 val rtrancl_is_UN_rel_pow = thm "rtrancl_is_UN_rel_pow";
   131 val single_valued_rel_pow = thm "single_valued_rel_pow";
   132 *}
   133 
   134 end