src/HOL/Library/Old_Recdef.thy
author krauss
Tue Aug 02 10:36:50 2011 +0200 (2011-08-02)
changeset 44013 5cfc1c36ae97
parent 39302 src/HOL/Recdef.thy@d7728f65b353
child 44014 88bd7d74a2c1
permissions -rw-r--r--
moved recdef package to HOL/Library/Old_Recdef.thy
     1 (*  Title:      HOL/Library/Old_Recdef.thy
     2     Author:     Konrad Slind and Markus Wenzel, TU Muenchen
     3 *)
     4 
     5 header {* TFL: recursive function definitions *}
     6 
     7 theory Old_Recdef
     8 imports Main
     9 uses
    10   ("~~/src/HOL/Tools/TFL/casesplit.ML")
    11   ("~~/src/HOL/Tools/TFL/utils.ML")
    12   ("~~/src/HOL/Tools/TFL/usyntax.ML")
    13   ("~~/src/HOL/Tools/TFL/dcterm.ML")
    14   ("~~/src/HOL/Tools/TFL/thms.ML")
    15   ("~~/src/HOL/Tools/TFL/rules.ML")
    16   ("~~/src/HOL/Tools/TFL/thry.ML")
    17   ("~~/src/HOL/Tools/TFL/tfl.ML")
    18   ("~~/src/HOL/Tools/TFL/post.ML")
    19   ("~~/src/HOL/Tools/recdef.ML")
    20 begin
    21 
    22 inductive
    23   wfrec_rel :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b => bool"
    24   for R :: "('a * 'a) set"
    25   and F :: "('a => 'b) => 'a => 'b"
    26 where
    27   wfrecI: "ALL z. (z, x) : R --> wfrec_rel R F z (g z) ==>
    28             wfrec_rel R F x (F g x)"
    29 
    30 definition
    31   cut        :: "('a => 'b) => ('a * 'a)set => 'a => 'a => 'b" where
    32   "cut f r x == (%y. if (y,x):r then f y else undefined)"
    33 
    34 definition
    35   adm_wf :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => bool" where
    36   "adm_wf R F == ALL f g x.
    37      (ALL z. (z, x) : R --> f z = g z) --> F f x = F g x"
    38 
    39 definition
    40   wfrec :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b" where
    41   "wfrec R F == %x. THE y. wfrec_rel R (%f x. F (cut f R x) x) x y"
    42 
    43 subsection{*Well-Founded Recursion*}
    44 
    45 text{*cut*}
    46 
    47 lemma cuts_eq: "(cut f r x = cut g r x) = (ALL y. (y,x):r --> f(y)=g(y))"
    48 by (simp add: fun_eq_iff cut_def)
    49 
    50 lemma cut_apply: "(x,a):r ==> (cut f r a)(x) = f(x)"
    51 by (simp add: cut_def)
    52 
    53 text{*Inductive characterization of wfrec combinator; for details see:
    54 John Harrison, "Inductive definitions: automation and application"*}
    55 
    56 lemma wfrec_unique: "[| adm_wf R F; wf R |] ==> EX! y. wfrec_rel R F x y"
    57 apply (simp add: adm_wf_def)
    58 apply (erule_tac a=x in wf_induct)
    59 apply (rule ex1I)
    60 apply (rule_tac g = "%x. THE y. wfrec_rel R F x y" in wfrec_rel.wfrecI)
    61 apply (fast dest!: theI')
    62 apply (erule wfrec_rel.cases, simp)
    63 apply (erule allE, erule allE, erule allE, erule mp)
    64 apply (fast intro: the_equality [symmetric])
    65 done
    66 
    67 lemma adm_lemma: "adm_wf R (%f x. F (cut f R x) x)"
    68 apply (simp add: adm_wf_def)
    69 apply (intro strip)
    70 apply (rule cuts_eq [THEN iffD2, THEN subst], assumption)
    71 apply (rule refl)
    72 done
    73 
    74 lemma wfrec: "wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a"
    75 apply (simp add: wfrec_def)
    76 apply (rule adm_lemma [THEN wfrec_unique, THEN the1_equality], assumption)
    77 apply (rule wfrec_rel.wfrecI)
    78 apply (intro strip)
    79 apply (erule adm_lemma [THEN wfrec_unique, THEN theI'])
    80 done
    81 
    82 
    83 text{** This form avoids giant explosions in proofs.  NOTE USE OF ==*}
    84 lemma def_wfrec: "[| f==wfrec r H;  wf(r) |] ==> f(a) = H (cut f r a) a"
    85 apply auto
    86 apply (blast intro: wfrec)
    87 done
    88 
    89 
    90 subsection {* Nitpick setup *}
    91 
    92 axiomatization wf_wfrec :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
    93 
    94 definition wf_wfrec' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
    95 [nitpick_simp]: "wf_wfrec' R F x = F (cut (wf_wfrec R F) R x) x"
    96 
    97 definition wfrec' ::  "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
    98 "wfrec' R F x \<equiv> if wf R then wf_wfrec' R F x
    99                 else THE y. wfrec_rel R (%f x. F (cut f R x) x) x y"
   100 
   101 setup {*
   102   Nitpick_HOL.register_ersatz_global
   103     [(@{const_name wf_wfrec}, @{const_name wf_wfrec'}),
   104      (@{const_name wfrec}, @{const_name wfrec'})]
   105 *}
   106 
   107 hide_const (open) wf_wfrec wf_wfrec' wfrec'
   108 hide_fact (open) wf_wfrec'_def wfrec'_def
   109 
   110 
   111 subsection {* Lemmas for TFL *}
   112 
   113 lemma tfl_wf_induct: "ALL R. wf R -->  
   114        (ALL P. (ALL x. (ALL y. (y,x):R --> P y) --> P x) --> (ALL x. P x))"
   115 apply clarify
   116 apply (rule_tac r = R and P = P and a = x in wf_induct, assumption, blast)
   117 done
   118 
   119 lemma tfl_cut_apply: "ALL f R. (x,a):R --> (cut f R a)(x) = f(x)"
   120 apply clarify
   121 apply (rule cut_apply, assumption)
   122 done
   123 
   124 lemma tfl_wfrec:
   125      "ALL M R f. (f=wfrec R M) --> wf R --> (ALL x. f x = M (cut f R x) x)"
   126 apply clarify
   127 apply (erule wfrec)
   128 done
   129 
   130 lemma tfl_eq_True: "(x = True) --> x"
   131   by blast
   132 
   133 lemma tfl_rev_eq_mp: "(x = y) --> y --> x";
   134   by blast
   135 
   136 lemma tfl_simp_thm: "(x --> y) --> (x = x') --> (x' --> y)"
   137   by blast
   138 
   139 lemma tfl_P_imp_P_iff_True: "P ==> P = True"
   140   by blast
   141 
   142 lemma tfl_imp_trans: "(A --> B) ==> (B --> C) ==> (A --> C)"
   143   by blast
   144 
   145 lemma tfl_disj_assoc: "(a \<or> b) \<or> c == a \<or> (b \<or> c)"
   146   by simp
   147 
   148 lemma tfl_disjE: "P \<or> Q ==> P --> R ==> Q --> R ==> R"
   149   by blast
   150 
   151 lemma tfl_exE: "\<exists>x. P x ==> \<forall>x. P x --> Q ==> Q"
   152   by blast
   153 
   154 use "~~/src/HOL/Tools/TFL/casesplit.ML"
   155 use "~~/src/HOL/Tools/TFL/utils.ML"
   156 use "~~/src/HOL/Tools/TFL/usyntax.ML"
   157 use "~~/src/HOL/Tools/TFL/dcterm.ML"
   158 use "~~/src/HOL/Tools/TFL/thms.ML"
   159 use "~~/src/HOL/Tools/TFL/rules.ML"
   160 use "~~/src/HOL/Tools/TFL/thry.ML"
   161 use "~~/src/HOL/Tools/TFL/tfl.ML"
   162 use "~~/src/HOL/Tools/TFL/post.ML"
   163 use "~~/src/HOL/Tools/recdef.ML"
   164 setup Recdef.setup
   165 
   166 text {*Wellfoundedness of @{text same_fst}*}
   167 
   168 definition
   169  same_fst :: "('a => bool) => ('a => ('b * 'b)set) => (('a*'b)*('a*'b))set"
   170 where
   171     "same_fst P R == {((x',y'),(x,y)) . x'=x & P x & (y',y) : R x}"
   172    --{*For @{text rec_def} declarations where the first n parameters
   173        stay unchanged in the recursive call. *}
   174 
   175 lemma same_fstI [intro!]:
   176      "[| P x; (y',y) : R x |] ==> ((x,y'),(x,y)) : same_fst P R"
   177 by (simp add: same_fst_def)
   178 
   179 lemma wf_same_fst:
   180   assumes prem: "(!!x. P x ==> wf(R x))"
   181   shows "wf(same_fst P R)"
   182 apply (simp cong del: imp_cong add: wf_def same_fst_def)
   183 apply (intro strip)
   184 apply (rename_tac a b)
   185 apply (case_tac "wf (R a)")
   186  apply (erule_tac a = b in wf_induct, blast)
   187 apply (blast intro: prem)
   188 done
   189 
   190 text {*Rule setup*}
   191 
   192 lemmas [recdef_simp] =
   193   inv_image_def
   194   measure_def
   195   lex_prod_def
   196   same_fst_def
   197   less_Suc_eq [THEN iffD2]
   198 
   199 lemmas [recdef_cong] =
   200   if_cong let_cong image_cong INT_cong UN_cong bex_cong ball_cong imp_cong
   201   map_cong filter_cong takeWhile_cong dropWhile_cong foldl_cong foldr_cong 
   202 
   203 lemmas [recdef_wf] =
   204   wf_trancl
   205   wf_less_than
   206   wf_lex_prod
   207   wf_inv_image
   208   wf_measure
   209   wf_measures
   210   wf_pred_nat
   211   wf_same_fst
   212   wf_empty
   213 
   214 end