src/HOL/Sexp.ML
author clasohm
Tue Jan 30 15:24:36 1996 +0100 (1996-01-30)
changeset 1465 5d7a7e439cec
parent 1264 3eb91524b938
child 1475 7f5a4cd08209
permissions -rw-r--r--
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     1 (*  Title:      HOL/Sexp
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1994  University of Cambridge
     5 
     6 S-expressions, general binary trees for defining recursive data structures
     7 *)
     8 
     9 open Sexp;
    10 
    11 (** sexp_case **)
    12 
    13 val sexp_free_cs = 
    14     set_cs addSDs [Leaf_inject, Numb_inject, Scons_inject] 
    15            addSEs [Leaf_neq_Scons, Leaf_neq_Numb,
    16                    Numb_neq_Scons, Numb_neq_Leaf,
    17                    Scons_neq_Leaf, Scons_neq_Numb];
    18 
    19 goalw Sexp.thy [sexp_case_def] "sexp_case c d e (Leaf a) = c(a)";
    20 by (rtac select_equality 1);
    21 by (ALLGOALS (fast_tac sexp_free_cs));
    22 qed "sexp_case_Leaf";
    23 
    24 goalw Sexp.thy [sexp_case_def] "sexp_case c d e (Numb k) = d(k)";
    25 by (rtac select_equality 1);
    26 by (ALLGOALS (fast_tac sexp_free_cs));
    27 qed "sexp_case_Numb";
    28 
    29 goalw Sexp.thy [sexp_case_def] "sexp_case c d e (M$N) = e M N";
    30 by (rtac select_equality 1);
    31 by (ALLGOALS (fast_tac sexp_free_cs));
    32 qed "sexp_case_Scons";
    33 
    34 
    35 (** Introduction rules for sexp constructors **)
    36 
    37 val [prem] = goalw Sexp.thy [In0_def] 
    38     "M: sexp ==> In0(M) : sexp";
    39 by (rtac (prem RS (sexp.NumbI RS sexp.SconsI)) 1);
    40 qed "sexp_In0I";
    41 
    42 val [prem] = goalw Sexp.thy [In1_def] 
    43     "M: sexp ==> In1(M) : sexp";
    44 by (rtac (prem RS (sexp.NumbI RS sexp.SconsI)) 1);
    45 qed "sexp_In1I";
    46 
    47 val sexp_cs = set_cs addIs sexp.intrs@[SigmaI, uprodI];
    48 
    49 goal Sexp.thy "range(Leaf) <= sexp";
    50 by (fast_tac sexp_cs 1);
    51 qed "range_Leaf_subset_sexp";
    52 
    53 val [major] = goal Sexp.thy "M$N : sexp ==> M: sexp & N: sexp";
    54 by (rtac (major RS setup_induction) 1);
    55 by (etac sexp.induct 1);
    56 by (ALLGOALS 
    57     (fast_tac (set_cs addSEs [Scons_neq_Leaf,Scons_neq_Numb,Scons_inject])));
    58 qed "Scons_D";
    59 
    60 (** Introduction rules for 'pred_sexp' **)
    61 
    62 goalw Sexp.thy [pred_sexp_def] "pred_sexp <= Sigma sexp (%u.sexp)";
    63 by (fast_tac sexp_cs 1);
    64 qed "pred_sexp_subset_Sigma";
    65 
    66 (* (a,b) : pred_sexp^+ ==> a : sexp *)
    67 val trancl_pred_sexpD1 = 
    68     pred_sexp_subset_Sigma RS trancl_subset_Sigma RS subsetD RS SigmaD1
    69 and trancl_pred_sexpD2 = 
    70     pred_sexp_subset_Sigma RS trancl_subset_Sigma RS subsetD RS SigmaD2;
    71 
    72 val prems = goalw Sexp.thy [pred_sexp_def]
    73     "[| M: sexp;  N: sexp |] ==> (M, M$N) : pred_sexp";
    74 by (fast_tac (set_cs addIs prems) 1);
    75 qed "pred_sexpI1";
    76 
    77 val prems = goalw Sexp.thy [pred_sexp_def]
    78     "[| M: sexp;  N: sexp |] ==> (N, M$N) : pred_sexp";
    79 by (fast_tac (set_cs addIs prems) 1);
    80 qed "pred_sexpI2";
    81 
    82 (*Combinations involving transitivity and the rules above*)
    83 val pred_sexp_t1 = pred_sexpI1 RS r_into_trancl
    84 and pred_sexp_t2 = pred_sexpI2 RS r_into_trancl;
    85 
    86 val pred_sexp_trans1 = pred_sexp_t1 RSN (2, trans_trancl RS transD)
    87 and pred_sexp_trans2 = pred_sexp_t2 RSN (2, trans_trancl RS transD);
    88 
    89 (*Proves goals of the form (M,N):pred_sexp^+ provided M,N:sexp*)
    90 Addsimps (sexp.intrs @ [pred_sexp_t1, pred_sexp_t2,
    91                         pred_sexp_trans1, pred_sexp_trans2, cut_apply]);
    92 
    93 val major::prems = goalw Sexp.thy [pred_sexp_def]
    94     "[| p : pred_sexp;  \
    95 \       !!M N. [| p = (M, M$N);  M: sexp;  N: sexp |] ==> R; \
    96 \       !!M N. [| p = (N, M$N);  M: sexp;  N: sexp |] ==> R  \
    97 \    |] ==> R";
    98 by (cut_facts_tac [major] 1);
    99 by (REPEAT (eresolve_tac ([asm_rl,emptyE,insertE,UN_E]@prems) 1));
   100 qed "pred_sexpE";
   101 
   102 goal Sexp.thy "wf(pred_sexp)";
   103 by (rtac (pred_sexp_subset_Sigma RS wfI) 1);
   104 by (etac sexp.induct 1);
   105 by (fast_tac (HOL_cs addSEs [mp, pred_sexpE, Pair_inject, Scons_inject]) 3);
   106 by (fast_tac (HOL_cs addSEs [mp, pred_sexpE, Pair_inject, Numb_neq_Scons]) 2);
   107 by (fast_tac (HOL_cs addSEs [mp, pred_sexpE, Pair_inject, Leaf_neq_Scons]) 1);
   108 qed "wf_pred_sexp";
   109 
   110 (*** sexp_rec -- by wf recursion on pred_sexp ***)
   111 
   112 (** conversion rules **)
   113 
   114 val sexp_rec_unfold = wf_pred_sexp RS (sexp_rec_def RS def_wfrec);
   115 
   116 
   117 goal Sexp.thy "sexp_rec (Leaf a) c d h = c(a)";
   118 by (stac sexp_rec_unfold 1);
   119 by (rtac sexp_case_Leaf 1);
   120 qed "sexp_rec_Leaf";
   121 
   122 goal Sexp.thy "sexp_rec (Numb k) c d h = d(k)";
   123 by (stac sexp_rec_unfold 1);
   124 by (rtac sexp_case_Numb 1);
   125 qed "sexp_rec_Numb";
   126 
   127 goal Sexp.thy "!!M. [| M: sexp;  N: sexp |] ==> \
   128 \    sexp_rec (M$N) c d h = h M N (sexp_rec M c d h) (sexp_rec N c d h)";
   129 by (rtac (sexp_rec_unfold RS trans) 1);
   130 by (asm_simp_tac (!simpset addsimps [sexp_case_Scons,pred_sexpI1,pred_sexpI2])
   131     1);
   132 qed "sexp_rec_Scons";