author | clasohm |
Sun, 24 Apr 1994 11:27:38 +0200 | |
changeset 70 | 9459592608e2 |
parent 48 | 21291189b51e |
permissions | -rw-r--r-- |
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(* Title: HOL/sexp |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1992 University of Cambridge |
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For sexp.thy. S-expressions. |
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*) |
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open Sexp; |
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(** the sexp functional **) |
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goal Sexp.thy "mono(%Z. range(Leaf) Un range(Numb) Un Z<*>Z)"; |
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by (REPEAT (ares_tac [monoI, subset_refl, Un_mono, uprod_mono] 1)); |
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val Sexp_fun_mono = result(); |
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val Sexp_unfold = Sexp_fun_mono RS (Sexp_def RS def_lfp_Tarski); |
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(** Induction **) |
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val major::prems = goal Sexp.thy |
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"[| ii: Sexp; !!a. P(Leaf(a)); !!k. P(Numb(k)); \ |
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\ !!i j. [| i: Sexp; j: Sexp; P(i); P(j) |] ==> P(i$j) \ |
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\ |] ==> P(ii)"; |
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by (rtac (major RS (Sexp_def RS def_induct)) 1); |
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by (rtac Sexp_fun_mono 1); |
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by (fast_tac (set_cs addIs prems addSEs [uprodE]) 1); |
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val Sexp_induct = result(); |
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(** Sexp_case **) |
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goalw Sexp.thy [Sexp_case_def] "Sexp_case(Leaf(a),c,d,e) = c(a)"; |
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by (fast_tac (HOL_cs addIs [select_equality] |
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addSDs [Leaf_inject] |
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addSEs [Leaf_neq_Scons, Leaf_neq_Numb]) 1); |
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val Sexp_case_Leaf = result(); |
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goalw Sexp.thy [Sexp_case_def] "Sexp_case(Numb(k),c,d,e) = d(k)"; |
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by (fast_tac (HOL_cs addIs [select_equality] |
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addSDs [Numb_inject] |
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addSEs [Numb_neq_Scons, Numb_neq_Leaf]) 1); |
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val Sexp_case_Numb = result(); |
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goalw Sexp.thy [Sexp_case_def] "Sexp_case(M$N, c, d, e) = e(M,N)"; |
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by (fast_tac (HOL_cs addIs [select_equality] |
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addSDs [Scons_inject] |
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addSEs [Scons_neq_Leaf, Scons_neq_Numb]) 1); |
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val Sexp_case_Scons = result(); |
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(** Introduction rules for Sexp constructors **) |
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val SexpI = Sexp_unfold RS equalityD2 RS subsetD; |
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goal Sexp.thy "Leaf(a): Sexp"; |
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by (fast_tac (set_cs addIs [SexpI]) 1); |
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val Sexp_LeafI = result(); |
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goal Sexp.thy "Numb(a): Sexp"; |
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by (fast_tac (set_cs addIs [SexpI]) 1); |
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val Sexp_NumbI = result(); |
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val prems = goal Sexp.thy |
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"[| M: Sexp; N: Sexp |] ==> M$N : Sexp"; |
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by (fast_tac (set_cs addIs ([uprodI,SexpI]@prems)) 1); |
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val Sexp_SconsI = result(); |
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val [prem] = goalw Sexp.thy [In0_def] |
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"M: Sexp ==> In0(M) : Sexp"; |
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by (rtac (prem RS (Sexp_NumbI RS Sexp_SconsI)) 1); |
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val Sexp_In0I = result(); |
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val [prem] = goalw Sexp.thy [In1_def] |
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"M: Sexp ==> In1(M) : Sexp"; |
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by (rtac (prem RS (Sexp_NumbI RS Sexp_SconsI)) 1); |
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val Sexp_In1I = result(); |
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goal Sexp.thy "range(Leaf) <= Sexp"; |
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by (fast_tac (set_cs addIs [SexpI]) 1); |
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val range_Leaf_subset_Sexp = result(); |
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val [major] = goal Sexp.thy "M$N : Sexp ==> M: Sexp & N: Sexp"; |
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by (rtac (major RS setup_induction) 1); |
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by (etac Sexp_induct 1); |
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by (ALLGOALS |
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(fast_tac (set_cs addSEs [Scons_neq_Leaf,Scons_neq_Numb,Scons_inject]))); |
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val Scons_D = result(); |
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(** Introduction rules for 'pred_Sexp' **) |
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val sexp_cs = set_cs addIs [SigmaI, uprodI, SexpI]; |
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goalw Sexp.thy [pred_Sexp_def] "pred_Sexp <= Sigma(Sexp, %u.Sexp)"; |
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by (fast_tac sexp_cs 1); |
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val pred_Sexp_subset_Sigma = result(); |
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(* <a,b> : pred_Sexp^+ ==> a : Sexp *) |
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val trancl_pred_SexpD1 = |
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pred_Sexp_subset_Sigma RS trancl_subset_Sigma RS subsetD RS SigmaD1 |
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and trancl_pred_SexpD2 = |
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pred_Sexp_subset_Sigma RS trancl_subset_Sigma RS subsetD RS SigmaD2; |
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val prems = goalw Sexp.thy [pred_Sexp_def] |
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"[| M: Sexp; N: Sexp |] ==> <M, M$N> : pred_Sexp"; |
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by (fast_tac (set_cs addIs prems) 1); |
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val pred_SexpI1 = result(); |
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val prems = goalw Sexp.thy [pred_Sexp_def] |
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"[| M: Sexp; N: Sexp |] ==> <N, M$N> : pred_Sexp"; |
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by (fast_tac (set_cs addIs prems) 1); |
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val pred_SexpI2 = result(); |
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(*Combinations involving transitivity and the rules above*) |
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val pred_Sexp_t1 = pred_SexpI1 RS r_into_trancl |
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and pred_Sexp_t2 = pred_SexpI2 RS r_into_trancl; |
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val pred_Sexp_trans1 = pred_Sexp_t1 RSN (2, trans_trancl RS transD) |
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and pred_Sexp_trans2 = pred_Sexp_t2 RSN (2, trans_trancl RS transD); |
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(*Proves goals of the form <M,N>:pred_Sexp^+ provided M,N:Sexp*) |
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val pred_Sexp_simps = |
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[Sexp_LeafI, Sexp_NumbI, Sexp_SconsI, |
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pred_Sexp_t1, pred_Sexp_t2, |
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pred_Sexp_trans1, pred_Sexp_trans2, cut_apply]; |
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val pred_Sexp_ss = HOL_ss addsimps pred_Sexp_simps; |
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val major::prems = goalw Sexp.thy [pred_Sexp_def] |
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"[| p : pred_Sexp; \ |
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\ !!M N. [| p = <M, M$N>; M: Sexp; N: Sexp |] ==> R; \ |
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\ !!M N. [| p = <N, M$N>; M: Sexp; N: Sexp |] ==> R \ |
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\ |] ==> R"; |
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by (cut_facts_tac [major] 1); |
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by (REPEAT (eresolve_tac ([asm_rl,emptyE,insertE,UN_E]@prems) 1)); |
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val pred_SexpE = result(); |
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goal Sexp.thy "wf(pred_Sexp)"; |
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by (rtac (pred_Sexp_subset_Sigma RS wfI) 1); |
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by (etac Sexp_induct 1); |
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by (fast_tac (HOL_cs addSEs [mp, pred_SexpE, Pair_inject, Scons_inject]) 3); |
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by (fast_tac (HOL_cs addSEs [mp, pred_SexpE, Pair_inject, Numb_neq_Scons]) 2); |
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by (fast_tac (HOL_cs addSEs [mp, pred_SexpE, Pair_inject, Leaf_neq_Scons]) 1); |
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val wf_pred_Sexp = result(); |
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(*** Sexp_rec -- by wf recursion on pred_Sexp ***) |
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(** conversion rules **) |
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val Sexp_rec_unfold = wf_pred_Sexp RS (Sexp_rec_def RS def_wfrec); |
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goal Sexp.thy "Sexp_rec(Leaf(a), c, d, h) = c(a)"; |
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by (stac Sexp_rec_unfold 1); |
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by (rtac Sexp_case_Leaf 1); |
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val Sexp_rec_Leaf = result(); |
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goal Sexp.thy "Sexp_rec(Numb(k), c, d, h) = d(k)"; |
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by (stac Sexp_rec_unfold 1); |
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by (rtac Sexp_case_Numb 1); |
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val Sexp_rec_Numb = result(); |
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goal Sexp.thy "!!M. [| M: Sexp; N: Sexp |] ==> \ |
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\ Sexp_rec(M$N, c, d, h) = h(M, N, Sexp_rec(M,c,d,h), Sexp_rec(N,c,d,h))"; |
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by (rtac (Sexp_rec_unfold RS trans) 1); |
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by (asm_simp_tac(HOL_ss addsimps |
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[Sexp_case_Scons,pred_SexpI1,pred_SexpI2,cut_apply])1); |
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val Sexp_rec_Scons = result(); |