(* Title: HOL/Sexp
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
S-expressions, general binary trees for defining recursive data structures
*)
open Sexp;
(** sexp_case **)
Goalw [sexp_case_def] "sexp_case c d e (Leaf a) = c(a)";
by (Blast_tac 1);
qed "sexp_case_Leaf";
Goalw [sexp_case_def] "sexp_case c d e (Numb k) = d(k)";
by (Blast_tac 1);
qed "sexp_case_Numb";
Goalw [sexp_case_def] "sexp_case c d e (Scons M N) = e M N";
by (Blast_tac 1);
qed "sexp_case_Scons";
Addsimps [sexp_case_Leaf, sexp_case_Numb, sexp_case_Scons];
(** Introduction rules for sexp constructors **)
val [prem] = goalw Sexp.thy [In0_def] "M: sexp ==> In0(M) : sexp";
by (rtac (prem RS (sexp.NumbI RS sexp.SconsI)) 1);
qed "sexp_In0I";
val [prem] = goalw Sexp.thy [In1_def] "M: sexp ==> In1(M) : sexp";
by (rtac (prem RS (sexp.NumbI RS sexp.SconsI)) 1);
qed "sexp_In1I";
AddIs sexp.intrs;
Goal "range(Leaf) <= sexp";
by (Blast_tac 1);
qed "range_Leaf_subset_sexp";
val [major] = goal Sexp.thy "Scons M N : sexp ==> M: sexp & N: sexp";
by (rtac (major RS setup_induction) 1);
by (etac sexp.induct 1);
by (ALLGOALS Blast_tac);
qed "Scons_D";
(** Introduction rules for 'pred_sexp' **)
Goalw [pred_sexp_def] "pred_sexp <= sexp Times sexp";
by (Blast_tac 1);
qed "pred_sexp_subset_Sigma";
(* (a,b) : pred_sexp^+ ==> a : sexp *)
val trancl_pred_sexpD1 =
pred_sexp_subset_Sigma RS trancl_subset_Sigma RS subsetD RS SigmaD1
and trancl_pred_sexpD2 =
pred_sexp_subset_Sigma RS trancl_subset_Sigma RS subsetD RS SigmaD2;
Goalw [pred_sexp_def]
"!!M. [| M: sexp; N: sexp |] ==> (M, Scons M N) : pred_sexp";
by (Blast_tac 1);
qed "pred_sexpI1";
Goalw [pred_sexp_def]
"!!M. [| M: sexp; N: sexp |] ==> (N, Scons M N) : pred_sexp";
by (Blast_tac 1);
qed "pred_sexpI2";
(*Combinations involving transitivity and the rules above*)
val pred_sexp_t1 = pred_sexpI1 RS r_into_trancl
and pred_sexp_t2 = pred_sexpI2 RS r_into_trancl;
val pred_sexp_trans1 = pred_sexp_t1 RSN (2, trans_trancl RS transD)
and pred_sexp_trans2 = pred_sexp_t2 RSN (2, trans_trancl RS transD);
(*Proves goals of the form (M,N):pred_sexp^+ provided M,N:sexp*)
Addsimps (sexp.intrs @ [pred_sexp_t1, pred_sexp_t2,
pred_sexp_trans1, pred_sexp_trans2, cut_apply]);
val major::prems = goalw Sexp.thy [pred_sexp_def]
"[| p : pred_sexp; \
\ !!M N. [| p = (M, Scons M N); M: sexp; N: sexp |] ==> R; \
\ !!M N. [| p = (N, Scons M N); M: sexp; N: sexp |] ==> R \
\ |] ==> R";
by (cut_facts_tac [major] 1);
by (REPEAT (eresolve_tac ([asm_rl,emptyE,insertE,UN_E]@prems) 1));
qed "pred_sexpE";
Goal "wf(pred_sexp)";
by (rtac (pred_sexp_subset_Sigma RS wfI) 1);
by (etac sexp.induct 1);
by (ALLGOALS (blast_tac (claset() addSEs [allE, pred_sexpE])));
qed "wf_pred_sexp";
(*** sexp_rec -- by wf recursion on pred_sexp ***)
Goal "(%M. sexp_rec M c d e) = wfrec pred_sexp \
\ (%g. sexp_case c d (%N1 N2. e N1 N2 (g N1) (g N2)))";
by (simp_tac (HOL_ss addsimps [sexp_rec_def]) 1);
(* sexp_rec a c d e =
sexp_case c d
(%N1 N2.
e N1 N2 (cut (%M. sexp_rec M c d e) pred_sexp a N1)
(cut (%M. sexp_rec M c d e) pred_sexp a N2)) a
*)
bind_thm("sexp_rec_unfold",
[result() RS eq_reflection, wf_pred_sexp] MRS def_wfrec);
(** conversion rules **)
Goal "sexp_rec (Leaf a) c d h = c(a)";
by (stac sexp_rec_unfold 1);
by (rtac sexp_case_Leaf 1);
qed "sexp_rec_Leaf";
Goal "sexp_rec (Numb k) c d h = d(k)";
by (stac sexp_rec_unfold 1);
by (rtac sexp_case_Numb 1);
qed "sexp_rec_Numb";
Goal "[| M: sexp; N: sexp |] ==> \
\ sexp_rec (Scons M N) c d h = h M N (sexp_rec M c d h) (sexp_rec N c d h)";
by (rtac (sexp_rec_unfold RS trans) 1);
by (asm_simp_tac (simpset() addsimps [pred_sexpI1, pred_sexpI2]) 1);
qed "sexp_rec_Scons";