author | lcp |
Thu, 13 Apr 1995 15:08:39 +0200 | |
changeset 1046 | 9d2261a3e682 |
parent 972 | e61b058d58d2 |
child 1266 | 3ae9fe3c0f68 |
permissions | -rw-r--r-- |
969 | 1 |
(* Title: HOL/llist |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1993 University of Cambridge |
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SHOULD LListD_Fun_CONS_I, etc., be equations (for rewriting)? |
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*) |
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open LList; |
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(** Simplification **) |
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val llist_ss = univ_ss addcongs [split_weak_cong, sum_case_weak_cong] |
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setloop split_tac [expand_split, expand_sum_case]; |
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(*For adding _eqI rules to a simpset; we must remove Pair_eq because |
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it may turn an instance of reflexivity into a conjunction!*) |
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fun add_eqI ss = ss addsimps [range_eqI, image_eqI] |
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delsimps [Pair_eq]; |
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(*This justifies using llist in other recursive type definitions*) |
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goalw LList.thy llist.defs "!!A B. A<=B ==> llist(A) <= llist(B)"; |
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by (rtac gfp_mono 1); |
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by (REPEAT (ares_tac basic_monos 1)); |
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qed "llist_mono"; |
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goal LList.thy "llist(A) = {Numb(0)} <+> (A <*> llist(A))"; |
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let val rew = rewrite_rule [NIL_def, CONS_def] in |
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by (fast_tac (univ_cs addSIs (equalityI :: map rew llist.intrs) |
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addEs [rew llist.elim]) 1) |
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end; |
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qed "llist_unfold"; |
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(*** Type checking by coinduction, using list_Fun |
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THE COINDUCTIVE DEFINITION PACKAGE COULD DO THIS! |
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***) |
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goalw LList.thy [list_Fun_def] |
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"!!M. [| M : X; X <= list_Fun A (X Un llist(A)) |] ==> M : llist(A)"; |
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by (etac llist.coinduct 1); |
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by (etac (subsetD RS CollectD) 1); |
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by (assume_tac 1); |
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qed "llist_coinduct"; |
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goalw LList.thy [list_Fun_def, NIL_def] "NIL: list_Fun A X"; |
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by (fast_tac set_cs 1); |
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qed "list_Fun_NIL_I"; |
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goalw LList.thy [list_Fun_def,CONS_def] |
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"!!M N. [| M: A; N: X |] ==> CONS M N : list_Fun A X"; |
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by (fast_tac set_cs 1); |
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qed "list_Fun_CONS_I"; |
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(*Utilise the "strong" part, i.e. gfp(f)*) |
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goalw LList.thy (llist.defs @ [list_Fun_def]) |
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"!!M N. M: llist(A) ==> M : list_Fun A (X Un llist(A))"; |
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by (etac (llist.mono RS gfp_fun_UnI2) 1); |
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qed "list_Fun_llist_I"; |
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(*** LList_corec satisfies the desired recurion equation ***) |
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(*A continuity result?*) |
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goalw LList.thy [CONS_def] "CONS M (UN x.f(x)) = (UN x. CONS M (f x))"; |
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by (simp_tac (univ_ss addsimps [In1_UN1, Scons_UN1_y]) 1); |
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qed "CONS_UN1"; |
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(*UNUSED; obsolete? |
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goal Prod.thy "split p (%x y.UN z.f x y z) = (UN z. split p (%x y.f x y z))"; |
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by (simp_tac (prod_ss setloop (split_tac [expand_split])) 1); |
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qed "split_UN1"; |
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goal Sum.thy "sum_case s f (%y.UN z.g y z) = (UN z.sum_case s f (%y.g y z))"; |
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by (simp_tac (sum_ss setloop (split_tac [expand_sum_case])) 1); |
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qed "sum_case2_UN1"; |
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*) |
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val prems = goalw LList.thy [CONS_def] |
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"[| M<=M'; N<=N' |] ==> CONS M N <= CONS M' N'"; |
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by (REPEAT (resolve_tac ([In1_mono,Scons_mono]@prems) 1)); |
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qed "CONS_mono"; |
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val corec_fun_simps = [LList_corec_fun_def RS def_nat_rec_0, |
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LList_corec_fun_def RS def_nat_rec_Suc]; |
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val corec_fun_ss = llist_ss addsimps corec_fun_simps; |
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(** The directions of the equality are proved separately **) |
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goalw LList.thy [LList_corec_def] |
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"LList_corec a f <= sum_case (%u.NIL) \ |
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\ (split(%z w. CONS z (LList_corec w f))) (f a)"; |
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by (rtac UN1_least 1); |
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by (res_inst_tac [("n","k")] natE 1); |
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by (ALLGOALS (asm_simp_tac corec_fun_ss)); |
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by (REPEAT (resolve_tac [allI, impI, subset_refl RS CONS_mono, UN1_upper] 1)); |
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qed "LList_corec_subset1"; |
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goalw LList.thy [LList_corec_def] |
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"sum_case (%u.NIL) (split(%z w. CONS z (LList_corec w f))) (f a) <= \ |
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\ LList_corec a f"; |
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by (simp_tac (corec_fun_ss addsimps [CONS_UN1]) 1); |
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by (safe_tac set_cs); |
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by (ALLGOALS (res_inst_tac [("x","Suc(?k)")] UN1_I THEN' |
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asm_simp_tac corec_fun_ss)); |
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qed "LList_corec_subset2"; |
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(*the recursion equation for LList_corec -- NOT SUITABLE FOR REWRITING!*) |
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goal LList.thy |
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"LList_corec a f = sum_case (%u. NIL) \ |
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\ (split(%z w. CONS z (LList_corec w f))) (f a)"; |
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by (REPEAT (resolve_tac [equalityI, LList_corec_subset1, |
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LList_corec_subset2] 1)); |
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qed "LList_corec"; |
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(*definitional version of same*) |
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val [rew] = goal LList.thy |
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"[| !!x. h(x) == LList_corec x f |] ==> \ |
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\ h(a) = sum_case (%u.NIL) (split(%z w. CONS z (h w))) (f a)"; |
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by (rewtac rew); |
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by (rtac LList_corec 1); |
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qed "def_LList_corec"; |
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(*A typical use of co-induction to show membership in the gfp. |
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Bisimulation is range(%x. LList_corec x f) *) |
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goal LList.thy "LList_corec a f : llist({u.True})"; |
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by (res_inst_tac [("X", "range(%x.LList_corec x ?g)")] llist_coinduct 1); |
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by (rtac rangeI 1); |
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by (safe_tac set_cs); |
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by (stac LList_corec 1); |
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by (simp_tac (llist_ss addsimps [list_Fun_NIL_I, list_Fun_CONS_I, CollectI] |
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|> add_eqI) 1); |
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qed "LList_corec_type"; |
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(*Lemma for the proof of llist_corec*) |
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goal LList.thy |
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"LList_corec a (%z.sum_case Inl (split(%v w.Inr((Leaf(v),w)))) (f z)) : \ |
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\ llist(range(Leaf))"; |
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by (res_inst_tac [("X", "range(%x.LList_corec x ?g)")] llist_coinduct 1); |
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by (rtac rangeI 1); |
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by (safe_tac set_cs); |
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by (stac LList_corec 1); |
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by (asm_simp_tac (llist_ss addsimps [list_Fun_NIL_I]) 1); |
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by (fast_tac (set_cs addSIs [list_Fun_CONS_I]) 1); |
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qed "LList_corec_type2"; |
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(**** llist equality as a gfp; the bisimulation principle ****) |
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(*This theorem is actually used, unlike the many similar ones in ZF*) |
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goal LList.thy "LListD(r) = diag({Numb(0)}) <++> (r <**> LListD(r))"; |
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let val rew = rewrite_rule [NIL_def, CONS_def] in |
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by (fast_tac (univ_cs addSIs (equalityI :: map rew LListD.intrs) |
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addEs [rew LListD.elim]) 1) |
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end; |
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qed "LListD_unfold"; |
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goal LList.thy "!M N. (M,N) : LListD(diag(A)) --> ntrunc k M = ntrunc k N"; |
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by (res_inst_tac [("n", "k")] less_induct 1); |
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by (safe_tac set_cs); |
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by (etac LListD.elim 1); |
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by (safe_tac (prod_cs addSEs [diagE])); |
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by (res_inst_tac [("n", "n")] natE 1); |
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by (asm_simp_tac (univ_ss addsimps [ntrunc_0]) 1); |
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by (rename_tac "n'" 1); |
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by (res_inst_tac [("n", "n'")] natE 1); |
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by (asm_simp_tac (univ_ss addsimps [CONS_def, ntrunc_one_In1]) 1); |
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by (asm_simp_tac (univ_ss addsimps [CONS_def, ntrunc_In1, ntrunc_Scons]) 1); |
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qed "LListD_implies_ntrunc_equality"; |
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(*The domain of the LListD relation*) |
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goalw LList.thy (llist.defs @ [NIL_def, CONS_def]) |
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"fst``LListD(diag(A)) <= llist(A)"; |
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by (rtac gfp_upperbound 1); |
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(*avoids unfolding LListD on the rhs*) |
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by (res_inst_tac [("P", "%x. fst``x <= ?B")] (LListD_unfold RS ssubst) 1); |
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by (simp_tac fst_image_ss 1); |
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by (fast_tac univ_cs 1); |
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qed "fst_image_LListD"; |
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(*This inclusion justifies the use of coinduction to show M=N*) |
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goal LList.thy "LListD(diag(A)) <= diag(llist(A))"; |
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by (rtac subsetI 1); |
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by (res_inst_tac [("p","x")] PairE 1); |
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by (safe_tac HOL_cs); |
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by (rtac diag_eqI 1); |
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by (rtac (LListD_implies_ntrunc_equality RS spec RS spec RS mp RS |
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ntrunc_equality) 1); |
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by (assume_tac 1); |
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by (etac (fst_imageI RS (fst_image_LListD RS subsetD)) 1); |
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qed "LListD_subset_diag"; |
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(** Coinduction, using LListD_Fun |
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THE COINDUCTIVE DEFINITION PACKAGE COULD DO THIS! |
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**) |
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goalw LList.thy [LListD_Fun_def] |
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"!!M. [| M : X; X <= LListD_Fun r (X Un LListD(r)) |] ==> M : LListD(r)"; |
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by (etac LListD.coinduct 1); |
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by (etac (subsetD RS CollectD) 1); |
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by (assume_tac 1); |
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qed "LListD_coinduct"; |
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goalw LList.thy [LListD_Fun_def,NIL_def] "(NIL,NIL) : LListD_Fun r s"; |
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by (fast_tac set_cs 1); |
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qed "LListD_Fun_NIL_I"; |
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goalw LList.thy [LListD_Fun_def,CONS_def] |
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"!!x. [| x:A; (M,N):s |] ==> (CONS x M, CONS x N) : LListD_Fun (diag A) s"; |
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by (fast_tac univ_cs 1); |
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qed "LListD_Fun_CONS_I"; |
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(*Utilise the "strong" part, i.e. gfp(f)*) |
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goalw LList.thy (LListD.defs @ [LListD_Fun_def]) |
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"!!M N. M: LListD(r) ==> M : LListD_Fun r (X Un LListD(r))"; |
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by (etac (LListD.mono RS gfp_fun_UnI2) 1); |
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qed "LListD_Fun_LListD_I"; |
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(*This converse inclusion helps to strengthen LList_equalityI*) |
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goal LList.thy "diag(llist(A)) <= LListD(diag(A))"; |
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by (rtac subsetI 1); |
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by (etac LListD_coinduct 1); |
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by (rtac subsetI 1); |
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by (etac diagE 1); |
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by (etac ssubst 1); |
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by (eresolve_tac [llist.elim] 1); |
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by (ALLGOALS |
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(asm_simp_tac (llist_ss addsimps [diagI, LListD_Fun_NIL_I, |
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LListD_Fun_CONS_I]))); |
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qed "diag_subset_LListD"; |
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goal LList.thy "LListD(diag(A)) = diag(llist(A))"; |
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by (REPEAT (resolve_tac [equalityI, LListD_subset_diag, |
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diag_subset_LListD] 1)); |
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qed "LListD_eq_diag"; |
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goal LList.thy |
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"!!M N. M: llist(A) ==> (M,M) : LListD_Fun (diag A) (X Un diag(llist(A)))"; |
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by (rtac (LListD_eq_diag RS subst) 1); |
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by (rtac LListD_Fun_LListD_I 1); |
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by (asm_simp_tac (HOL_ss addsimps [LListD_eq_diag, diagI]) 1); |
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qed "LListD_Fun_diag_I"; |
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(** To show two LLists are equal, exhibit a bisimulation! |
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[also admits true equality] |
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Replace "A" by some particular set, like {x.True}??? *) |
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goal LList.thy |
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changeset
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"!!r. [| (M,N) : r; r <= LListD_Fun (diag A) (r Un diag(llist(A))) \ |
969 | 252 |
\ |] ==> M=N"; |
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by (rtac (LListD_subset_diag RS subsetD RS diagE) 1); |
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by (etac LListD_coinduct 1); |
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by (asm_simp_tac (HOL_ss addsimps [LListD_eq_diag]) 1); |
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by (safe_tac prod_cs); |
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qed "LList_equalityI"; |
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(*** Finality of llist(A): Uniqueness of functions defined by corecursion ***) |
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(*abstract proof using a bisimulation*) |
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val [prem1,prem2] = goal LList.thy |
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"[| !!x. h1(x) = sum_case (%u.NIL) (split(%z w. CONS z (h1 w))) (f x); \ |
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\ !!x. h2(x) = sum_case (%u.NIL) (split(%z w. CONS z (h2 w))) (f x) |]\ |
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\ ==> h1=h2"; |
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by (rtac ext 1); |
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(*next step avoids an unknown (and flexflex pair) in simplification*) |
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by (res_inst_tac [("A", "{u.True}"), |
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("r", "range(%u. (h1(u),h2(u)))")] LList_equalityI 1); |
969 | 271 |
by (rtac rangeI 1); |
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by (safe_tac set_cs); |
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by (stac prem1 1); |
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by (stac prem2 1); |
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by (simp_tac (llist_ss addsimps [LListD_Fun_NIL_I, |
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CollectI RS LListD_Fun_CONS_I] |
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|> add_eqI) 1); |
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qed "LList_corec_unique"; |
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val [prem] = goal LList.thy |
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"[| !!x. h(x) = sum_case (%u.NIL) (split(%z w. CONS z (h w))) (f x) |] \ |
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\ ==> h = (%x.LList_corec x f)"; |
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by (rtac (LList_corec RS (prem RS LList_corec_unique)) 1); |
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qed "equals_LList_corec"; |
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285 |
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(** Obsolete LList_corec_unique proof: complete induction, not coinduction **) |
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goalw LList.thy [CONS_def] "ntrunc (Suc 0) (CONS M N) = {}"; |
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by (rtac ntrunc_one_In1 1); |
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qed "ntrunc_one_CONS"; |
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292 |
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goalw LList.thy [CONS_def] |
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"ntrunc (Suc(Suc(k))) (CONS M N) = CONS (ntrunc k M) (ntrunc k N)"; |
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by (simp_tac (HOL_ss addsimps [ntrunc_Scons,ntrunc_In1]) 1); |
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qed "ntrunc_CONS"; |
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val [prem1,prem2] = goal LList.thy |
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"[| !!x. h1(x) = sum_case (%u.NIL) (split(%z w. CONS z (h1 w))) (f x); \ |
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\ !!x. h2(x) = sum_case (%u.NIL) (split(%z w. CONS z (h2 w))) (f x) |]\ |
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\ ==> h1=h2"; |
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by (rtac (ntrunc_equality RS ext) 1); |
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by (res_inst_tac [("x", "x")] spec 1); |
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by (res_inst_tac [("n", "k")] less_induct 1); |
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by (rtac allI 1); |
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by (stac prem1 1); |
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by (stac prem2 1); |
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by (simp_tac (sum_ss setloop (split_tac [expand_split,expand_sum_case])) 1); |
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by (strip_tac 1); |
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by (res_inst_tac [("n", "n")] natE 1); |
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by (res_inst_tac [("n", "xc")] natE 2); |
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by (ALLGOALS(asm_simp_tac(nat_ss addsimps |
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[ntrunc_0,ntrunc_one_CONS,ntrunc_CONS]))); |
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result(); |
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316 |
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(*** Lconst -- defined directly using lfp, but equivalent to a LList_corec ***) |
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318 |
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goal LList.thy "mono(CONS(M))"; |
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by (REPEAT (ares_tac [monoI, subset_refl, CONS_mono] 1)); |
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qed "Lconst_fun_mono"; |
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(* Lconst(M) = CONS M (Lconst M) *) |
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bind_thm ("Lconst", (Lconst_fun_mono RS (Lconst_def RS def_lfp_Tarski))); |
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(*A typical use of co-induction to show membership in the gfp. |
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The containing set is simply the singleton {Lconst(M)}. *) |
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goal LList.thy "!!M A. M:A ==> Lconst(M): llist(A)"; |
|
329 |
by (rtac (singletonI RS llist_coinduct) 1); |
|
330 |
by (safe_tac set_cs); |
|
331 |
by (res_inst_tac [("P", "%u. u: ?A")] (Lconst RS ssubst) 1); |
|
332 |
by (REPEAT (ares_tac [list_Fun_CONS_I, singletonI, UnI1] 1)); |
|
333 |
qed "Lconst_type"; |
|
334 |
||
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
969
diff
changeset
|
335 |
goal LList.thy "Lconst(M) = LList_corec M (%x.Inr((x,x)))"; |
969 | 336 |
by (rtac (equals_LList_corec RS fun_cong) 1); |
337 |
by (simp_tac sum_ss 1); |
|
338 |
by (rtac Lconst 1); |
|
339 |
qed "Lconst_eq_LList_corec"; |
|
340 |
||
341 |
(*Thus we could have used gfp in the definition of Lconst*) |
|
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
969
diff
changeset
|
342 |
goal LList.thy "gfp(%N. CONS M N) = LList_corec M (%x.Inr((x,x)))"; |
969 | 343 |
by (rtac (equals_LList_corec RS fun_cong) 1); |
344 |
by (simp_tac sum_ss 1); |
|
345 |
by (rtac (Lconst_fun_mono RS gfp_Tarski) 1); |
|
346 |
qed "gfp_Lconst_eq_LList_corec"; |
|
347 |
||
348 |
||
349 |
(*** Isomorphisms ***) |
|
350 |
||
351 |
goal LList.thy "inj(Rep_llist)"; |
|
352 |
by (rtac inj_inverseI 1); |
|
353 |
by (rtac Rep_llist_inverse 1); |
|
354 |
qed "inj_Rep_llist"; |
|
355 |
||
356 |
goal LList.thy "inj_onto Abs_llist (llist(range(Leaf)))"; |
|
357 |
by (rtac inj_onto_inverseI 1); |
|
358 |
by (etac Abs_llist_inverse 1); |
|
359 |
qed "inj_onto_Abs_llist"; |
|
360 |
||
361 |
(** Distinctness of constructors **) |
|
362 |
||
363 |
goalw LList.thy [LNil_def,LCons_def] "~ LCons x xs = LNil"; |
|
364 |
by (rtac (CONS_not_NIL RS (inj_onto_Abs_llist RS inj_onto_contraD)) 1); |
|
365 |
by (REPEAT (resolve_tac (llist.intrs @ [rangeI, Rep_llist]) 1)); |
|
366 |
qed "LCons_not_LNil"; |
|
367 |
||
368 |
bind_thm ("LNil_not_LCons", (LCons_not_LNil RS not_sym)); |
|
369 |
||
370 |
bind_thm ("LCons_neq_LNil", (LCons_not_LNil RS notE)); |
|
371 |
val LNil_neq_LCons = sym RS LCons_neq_LNil; |
|
372 |
||
373 |
(** llist constructors **) |
|
374 |
||
375 |
goalw LList.thy [LNil_def] |
|
376 |
"Rep_llist(LNil) = NIL"; |
|
377 |
by (rtac (llist.NIL_I RS Abs_llist_inverse) 1); |
|
378 |
qed "Rep_llist_LNil"; |
|
379 |
||
380 |
goalw LList.thy [LCons_def] |
|
381 |
"Rep_llist(LCons x l) = CONS (Leaf x) (Rep_llist l)"; |
|
382 |
by (REPEAT (resolve_tac [llist.CONS_I RS Abs_llist_inverse, |
|
383 |
rangeI, Rep_llist] 1)); |
|
384 |
qed "Rep_llist_LCons"; |
|
385 |
||
386 |
(** Injectiveness of CONS and LCons **) |
|
387 |
||
388 |
goalw LList.thy [CONS_def] "(CONS M N=CONS M' N') = (M=M' & N=N')"; |
|
389 |
by (fast_tac (HOL_cs addSEs [Scons_inject, make_elim In1_inject]) 1); |
|
390 |
qed "CONS_CONS_eq"; |
|
391 |
||
392 |
bind_thm ("CONS_inject", (CONS_CONS_eq RS iffD1 RS conjE)); |
|
393 |
||
394 |
||
395 |
(*For reasoning about abstract llist constructors*) |
|
396 |
val llist_cs = set_cs addIs [Rep_llist]@llist.intrs |
|
397 |
addSEs [CONS_neq_NIL,NIL_neq_CONS,CONS_inject] |
|
398 |
addSDs [inj_onto_Abs_llist RS inj_ontoD, |
|
399 |
inj_Rep_llist RS injD, Leaf_inject]; |
|
400 |
||
401 |
goalw LList.thy [LCons_def] "(LCons x xs=LCons y ys) = (x=y & xs=ys)"; |
|
402 |
by (fast_tac llist_cs 1); |
|
403 |
qed "LCons_LCons_eq"; |
|
404 |
bind_thm ("LCons_inject", (LCons_LCons_eq RS iffD1 RS conjE)); |
|
405 |
||
406 |
val [major] = goal LList.thy "CONS M N: llist(A) ==> M: A & N: llist(A)"; |
|
407 |
by (rtac (major RS llist.elim) 1); |
|
408 |
by (etac CONS_neq_NIL 1); |
|
409 |
by (fast_tac llist_cs 1); |
|
410 |
qed "CONS_D"; |
|
411 |
||
412 |
||
413 |
(****** Reasoning about llist(A) ******) |
|
414 |
||
415 |
(*Don't use llist_ss, as it does case splits!*) |
|
416 |
val List_case_ss = univ_ss addsimps [List_case_NIL, List_case_CONS]; |
|
417 |
||
418 |
(*A special case of list_equality for functions over lazy lists*) |
|
419 |
val [Mlist,gMlist,NILcase,CONScase] = goal LList.thy |
|
420 |
"[| M: llist(A); g(NIL): llist(A); \ |
|
421 |
\ f(NIL)=g(NIL); \ |
|
422 |
\ !!x l. [| x:A; l: llist(A) |] ==> \ |
|
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
969
diff
changeset
|
423 |
\ (f(CONS x l),g(CONS x l)) : \ |
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
969
diff
changeset
|
424 |
\ LListD_Fun (diag A) ((%u.(f(u),g(u)))``llist(A) Un \ |
969 | 425 |
\ diag(llist(A))) \ |
426 |
\ |] ==> f(M) = g(M)"; |
|
427 |
by (rtac LList_equalityI 1); |
|
1046 | 428 |
by (rtac (Mlist RS imageI) 1); |
969 | 429 |
by (rtac subsetI 1); |
430 |
by (etac imageE 1); |
|
431 |
by (etac ssubst 1); |
|
432 |
by (etac llist.elim 1); |
|
433 |
by (etac ssubst 1); |
|
434 |
by (stac NILcase 1); |
|
1046 | 435 |
by (rtac (gMlist RS LListD_Fun_diag_I) 1); |
969 | 436 |
by (etac ssubst 1); |
437 |
by (REPEAT (ares_tac [CONScase] 1)); |
|
438 |
qed "LList_fun_equalityI"; |
|
439 |
||
440 |
||
441 |
(*** The functional "Lmap" ***) |
|
442 |
||
443 |
goal LList.thy "Lmap f NIL = NIL"; |
|
444 |
by (rtac (Lmap_def RS def_LList_corec RS trans) 1); |
|
445 |
by (simp_tac List_case_ss 1); |
|
446 |
qed "Lmap_NIL"; |
|
447 |
||
448 |
goal LList.thy "Lmap f (CONS M N) = CONS (f M) (Lmap f N)"; |
|
449 |
by (rtac (Lmap_def RS def_LList_corec RS trans) 1); |
|
450 |
by (simp_tac List_case_ss 1); |
|
451 |
qed "Lmap_CONS"; |
|
452 |
||
453 |
(*Another type-checking proof by coinduction*) |
|
454 |
val [major,minor] = goal LList.thy |
|
455 |
"[| M: llist(A); !!x. x:A ==> f(x):B |] ==> Lmap f M: llist(B)"; |
|
456 |
by (rtac (major RS imageI RS llist_coinduct) 1); |
|
457 |
by (safe_tac set_cs); |
|
458 |
by (etac llist.elim 1); |
|
459 |
by (ALLGOALS (asm_simp_tac (HOL_ss addsimps [Lmap_NIL,Lmap_CONS]))); |
|
460 |
by (REPEAT (ares_tac [list_Fun_NIL_I, list_Fun_CONS_I, |
|
461 |
minor, imageI, UnI1] 1)); |
|
462 |
qed "Lmap_type"; |
|
463 |
||
464 |
(*This type checking rule synthesises a sufficiently large set for f*) |
|
465 |
val [major] = goal LList.thy "M: llist(A) ==> Lmap f M: llist(f``A)"; |
|
466 |
by (rtac (major RS Lmap_type) 1); |
|
467 |
by (etac imageI 1); |
|
468 |
qed "Lmap_type2"; |
|
469 |
||
470 |
(** Two easy results about Lmap **) |
|
471 |
||
472 |
val [prem] = goalw LList.thy [o_def] |
|
473 |
"M: llist(A) ==> Lmap (f o g) M = Lmap f (Lmap g M)"; |
|
474 |
by (rtac (prem RS imageI RS LList_equalityI) 1); |
|
475 |
by (safe_tac set_cs); |
|
476 |
by (etac llist.elim 1); |
|
477 |
by (ALLGOALS (asm_simp_tac (HOL_ss addsimps [Lmap_NIL,Lmap_CONS]))); |
|
478 |
by (REPEAT (ares_tac [LListD_Fun_NIL_I, imageI, UnI1, |
|
479 |
rangeI RS LListD_Fun_CONS_I] 1)); |
|
480 |
qed "Lmap_compose"; |
|
481 |
||
482 |
val [prem] = goal LList.thy "M: llist(A) ==> Lmap (%x.x) M = M"; |
|
483 |
by (rtac (prem RS imageI RS LList_equalityI) 1); |
|
484 |
by (safe_tac set_cs); |
|
485 |
by (etac llist.elim 1); |
|
486 |
by (ALLGOALS (asm_simp_tac (HOL_ss addsimps [Lmap_NIL,Lmap_CONS]))); |
|
487 |
by (REPEAT (ares_tac [LListD_Fun_NIL_I, imageI RS UnI1, |
|
488 |
rangeI RS LListD_Fun_CONS_I] 1)); |
|
489 |
qed "Lmap_ident"; |
|
490 |
||
491 |
||
492 |
(*** Lappend -- its two arguments cause some complications! ***) |
|
493 |
||
494 |
goalw LList.thy [Lappend_def] "Lappend NIL NIL = NIL"; |
|
495 |
by (rtac (LList_corec RS trans) 1); |
|
496 |
by (simp_tac List_case_ss 1); |
|
497 |
qed "Lappend_NIL_NIL"; |
|
498 |
||
499 |
goalw LList.thy [Lappend_def] |
|
500 |
"Lappend NIL (CONS N N') = CONS N (Lappend NIL N')"; |
|
501 |
by (rtac (LList_corec RS trans) 1); |
|
502 |
by (simp_tac List_case_ss 1); |
|
503 |
qed "Lappend_NIL_CONS"; |
|
504 |
||
505 |
goalw LList.thy [Lappend_def] |
|
506 |
"Lappend (CONS M M') N = CONS M (Lappend M' N)"; |
|
507 |
by (rtac (LList_corec RS trans) 1); |
|
508 |
by (simp_tac List_case_ss 1); |
|
509 |
qed "Lappend_CONS"; |
|
510 |
||
511 |
val Lappend_ss = |
|
512 |
List_case_ss addsimps [llist.NIL_I, Lappend_NIL_NIL, Lappend_NIL_CONS, |
|
513 |
Lappend_CONS, LListD_Fun_CONS_I] |
|
514 |
|> add_eqI; |
|
515 |
||
516 |
goal LList.thy "!!M. M: llist(A) ==> Lappend NIL M = M"; |
|
517 |
by (etac LList_fun_equalityI 1); |
|
518 |
by (ALLGOALS (asm_simp_tac Lappend_ss)); |
|
519 |
qed "Lappend_NIL"; |
|
520 |
||
521 |
goal LList.thy "!!M. M: llist(A) ==> Lappend M NIL = M"; |
|
522 |
by (etac LList_fun_equalityI 1); |
|
523 |
by (ALLGOALS (asm_simp_tac Lappend_ss)); |
|
524 |
qed "Lappend_NIL2"; |
|
525 |
||
526 |
(** Alternative type-checking proofs for Lappend **) |
|
527 |
||
528 |
(*weak co-induction: bisimulation and case analysis on both variables*) |
|
529 |
goal LList.thy |
|
530 |
"!!M N. [| M: llist(A); N: llist(A) |] ==> Lappend M N: llist(A)"; |
|
531 |
by (res_inst_tac |
|
532 |
[("X", "UN u:llist(A). UN v: llist(A). {Lappend u v}")] llist_coinduct 1); |
|
533 |
by (fast_tac set_cs 1); |
|
534 |
by (safe_tac set_cs); |
|
535 |
by (eres_inst_tac [("a", "u")] llist.elim 1); |
|
536 |
by (eres_inst_tac [("a", "v")] llist.elim 1); |
|
537 |
by (ALLGOALS |
|
538 |
(asm_simp_tac Lappend_ss THEN' |
|
539 |
fast_tac (set_cs addSIs [llist.NIL_I, list_Fun_NIL_I, list_Fun_CONS_I]))); |
|
540 |
qed "Lappend_type"; |
|
541 |
||
542 |
(*strong co-induction: bisimulation and case analysis on one variable*) |
|
543 |
goal LList.thy |
|
544 |
"!!M N. [| M: llist(A); N: llist(A) |] ==> Lappend M N: llist(A)"; |
|
545 |
by (res_inst_tac [("X", "(%u.Lappend u N)``llist(A)")] llist_coinduct 1); |
|
1046 | 546 |
by (etac imageI 1); |
547 |
by (rtac subsetI 1); |
|
548 |
by (etac imageE 1); |
|
969 | 549 |
by (eres_inst_tac [("a", "u")] llist.elim 1); |
550 |
by (asm_simp_tac (Lappend_ss addsimps [Lappend_NIL, list_Fun_llist_I]) 1); |
|
551 |
by (asm_simp_tac Lappend_ss 1); |
|
552 |
by (fast_tac (set_cs addSIs [list_Fun_CONS_I]) 1); |
|
553 |
qed "Lappend_type"; |
|
554 |
||
555 |
(**** Lazy lists as the type 'a llist -- strongly typed versions of above ****) |
|
556 |
||
557 |
(** llist_case: case analysis for 'a llist **) |
|
558 |
||
559 |
val Rep_llist_simps = |
|
560 |
[List_case_NIL, List_case_CONS, |
|
561 |
Abs_llist_inverse, Rep_llist_inverse, |
|
562 |
Rep_llist, rangeI, inj_Leaf, Inv_f_f] |
|
563 |
@ llist.intrs; |
|
564 |
val Rep_llist_ss = llist_ss addsimps Rep_llist_simps; |
|
565 |
||
566 |
goalw LList.thy [llist_case_def,LNil_def] "llist_case c d LNil = c"; |
|
567 |
by (simp_tac Rep_llist_ss 1); |
|
568 |
qed "llist_case_LNil"; |
|
569 |
||
570 |
goalw LList.thy [llist_case_def,LCons_def] |
|
571 |
"llist_case c d (LCons M N) = d M N"; |
|
572 |
by (simp_tac Rep_llist_ss 1); |
|
573 |
qed "llist_case_LCons"; |
|
574 |
||
575 |
(*Elimination is case analysis, not induction.*) |
|
576 |
val [prem1,prem2] = goalw LList.thy [NIL_def,CONS_def] |
|
577 |
"[| l=LNil ==> P; !!x l'. l=LCons x l' ==> P \ |
|
578 |
\ |] ==> P"; |
|
579 |
by (rtac (Rep_llist RS llist.elim) 1); |
|
580 |
by (rtac (inj_Rep_llist RS injD RS prem1) 1); |
|
581 |
by (stac Rep_llist_LNil 1); |
|
582 |
by (assume_tac 1); |
|
583 |
by (etac rangeE 1); |
|
584 |
by (rtac (inj_Rep_llist RS injD RS prem2) 1); |
|
585 |
by (asm_simp_tac (HOL_ss addsimps [Rep_llist_LCons]) 1); |
|
586 |
by (etac (Abs_llist_inverse RS ssubst) 1); |
|
587 |
by (rtac refl 1); |
|
588 |
qed "llistE"; |
|
589 |
||
590 |
(** llist_corec: corecursion for 'a llist **) |
|
591 |
||
592 |
goalw LList.thy [llist_corec_def,LNil_def,LCons_def] |
|
593 |
"llist_corec a f = sum_case (%u. LNil) \ |
|
594 |
\ (split(%z w. LCons z (llist_corec w f))) (f a)"; |
|
595 |
by (stac LList_corec 1); |
|
596 |
by (res_inst_tac [("s","f(a)")] sumE 1); |
|
597 |
by (asm_simp_tac (llist_ss addsimps [LList_corec_type2,Abs_llist_inverse]) 1); |
|
598 |
by (res_inst_tac [("p","y")] PairE 1); |
|
599 |
by (asm_simp_tac (llist_ss addsimps [LList_corec_type2,Abs_llist_inverse]) 1); |
|
600 |
(*FIXME: correct case splits usd to be found automatically: |
|
601 |
by (ASM_SIMP_TAC(llist_ss addsimps [LList_corec_type2,Abs_llist_inverse]) 1);*) |
|
602 |
qed "llist_corec"; |
|
603 |
||
604 |
(*definitional version of same*) |
|
605 |
val [rew] = goal LList.thy |
|
606 |
"[| !!x. h(x) == llist_corec x f |] ==> \ |
|
607 |
\ h(a) = sum_case (%u.LNil) (split(%z w. LCons z (h w))) (f a)"; |
|
608 |
by (rewtac rew); |
|
609 |
by (rtac llist_corec 1); |
|
610 |
qed "def_llist_corec"; |
|
611 |
||
612 |
(**** Proofs about type 'a llist functions ****) |
|
613 |
||
614 |
(*** Deriving llist_equalityI -- llist equality is a bisimulation ***) |
|
615 |
||
616 |
goalw LList.thy [LListD_Fun_def] |
|
617 |
"!!r A. r <= Sigma (llist A) (%x.llist(A)) ==> \ |
|
618 |
\ LListD_Fun (diag A) r <= Sigma (llist A) (%x.llist(A))"; |
|
619 |
by (stac llist_unfold 1); |
|
620 |
by (simp_tac (HOL_ss addsimps [NIL_def, CONS_def]) 1); |
|
621 |
by (fast_tac univ_cs 1); |
|
622 |
qed "LListD_Fun_subset_Sigma_llist"; |
|
623 |
||
624 |
goal LList.thy |
|
625 |
"prod_fun Rep_llist Rep_llist `` r <= \ |
|
626 |
\ Sigma (llist(range(Leaf))) (%x.llist(range(Leaf)))"; |
|
627 |
by (fast_tac (prod_cs addIs [Rep_llist]) 1); |
|
628 |
qed "subset_Sigma_llist"; |
|
629 |
||
630 |
val [prem] = goal LList.thy |
|
631 |
"r <= Sigma (llist(range Leaf)) (%x.llist(range Leaf)) ==> \ |
|
632 |
\ prod_fun (Rep_llist o Abs_llist) (Rep_llist o Abs_llist) `` r <= r"; |
|
633 |
by (safe_tac prod_cs); |
|
634 |
by (rtac (prem RS subsetD RS SigmaE2) 1); |
|
635 |
by (assume_tac 1); |
|
636 |
by (asm_simp_tac (HOL_ss addsimps [o_def,prod_fun,Abs_llist_inverse]) 1); |
|
637 |
qed "prod_fun_lemma"; |
|
638 |
||
639 |
goal LList.thy |
|
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
969
diff
changeset
|
640 |
"prod_fun Rep_llist Rep_llist `` range(%x. (x, x)) = \ |
969 | 641 |
\ diag(llist(range(Leaf)))"; |
1046 | 642 |
by (rtac equalityI 1); |
969 | 643 |
by (fast_tac (univ_cs addIs [Rep_llist]) 1); |
644 |
by (fast_tac (univ_cs addSEs [Abs_llist_inverse RS subst]) 1); |
|
645 |
qed "prod_fun_range_eq_diag"; |
|
646 |
||
647 |
(** To show two llists are equal, exhibit a bisimulation! |
|
648 |
[also admits true equality] **) |
|
649 |
val [prem1,prem2] = goalw LList.thy [llistD_Fun_def] |
|
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
969
diff
changeset
|
650 |
"[| (l1,l2) : r; r <= llistD_Fun(r Un range(%x.(x,x))) |] ==> l1=l2"; |
969 | 651 |
by (rtac (inj_Rep_llist RS injD) 1); |
652 |
by (res_inst_tac [("r", "prod_fun Rep_llist Rep_llist ``r"), |
|
653 |
("A", "range(Leaf)")] |
|
654 |
LList_equalityI 1); |
|
655 |
by (rtac (prem1 RS prod_fun_imageI) 1); |
|
656 |
by (rtac (prem2 RS image_mono RS subset_trans) 1); |
|
657 |
by (rtac (image_compose RS subst) 1); |
|
658 |
by (rtac (prod_fun_compose RS subst) 1); |
|
659 |
by (rtac (image_Un RS ssubst) 1); |
|
660 |
by (stac prod_fun_range_eq_diag 1); |
|
661 |
by (rtac (LListD_Fun_subset_Sigma_llist RS prod_fun_lemma) 1); |
|
662 |
by (rtac (subset_Sigma_llist RS Un_least) 1); |
|
663 |
by (rtac diag_subset_Sigma 1); |
|
664 |
qed "llist_equalityI"; |
|
665 |
||
666 |
(** Rules to prove the 2nd premise of llist_equalityI **) |
|
972
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changeset
|
667 |
goalw LList.thy [llistD_Fun_def,LNil_def] "(LNil,LNil) : llistD_Fun(r)"; |
969 | 668 |
by (rtac (LListD_Fun_NIL_I RS prod_fun_imageI) 1); |
669 |
qed "llistD_Fun_LNil_I"; |
|
670 |
||
671 |
val [prem] = goalw LList.thy [llistD_Fun_def,LCons_def] |
|
972
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diff
changeset
|
672 |
"(l1,l2):r ==> (LCons x l1, LCons x l2) : llistD_Fun(r)"; |
969 | 673 |
by (rtac (rangeI RS LListD_Fun_CONS_I RS prod_fun_imageI) 1); |
674 |
by (rtac (prem RS prod_fun_imageI) 1); |
|
675 |
qed "llistD_Fun_LCons_I"; |
|
676 |
||
677 |
(*Utilise the "strong" part, i.e. gfp(f)*) |
|
678 |
goalw LList.thy [llistD_Fun_def] |
|
972
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parents:
969
diff
changeset
|
679 |
"!!l. (l,l) : llistD_Fun(r Un range(%x.(x,x)))"; |
1046 | 680 |
by (rtac (Rep_llist_inverse RS subst) 1); |
681 |
by (rtac prod_fun_imageI 1); |
|
969 | 682 |
by (rtac (image_Un RS ssubst) 1); |
683 |
by (stac prod_fun_range_eq_diag 1); |
|
1046 | 684 |
by (rtac (Rep_llist RS LListD_Fun_diag_I) 1); |
969 | 685 |
qed "llistD_Fun_range_I"; |
686 |
||
687 |
(*A special case of list_equality for functions over lazy lists*) |
|
688 |
val [prem1,prem2] = goal LList.thy |
|
689 |
"[| f(LNil)=g(LNil); \ |
|
972
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diff
changeset
|
690 |
\ !!x l. (f(LCons x l),g(LCons x l)) : \ |
e61b058d58d2
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969
diff
changeset
|
691 |
\ llistD_Fun(range(%u. (f(u),g(u))) Un range(%v. (v,v))) \ |
969 | 692 |
\ |] ==> f(l) = (g(l :: 'a llist) :: 'b llist)"; |
972
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parents:
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diff
changeset
|
693 |
by (res_inst_tac [("r", "range(%u. (f(u),g(u)))")] llist_equalityI 1); |
969 | 694 |
by (rtac rangeI 1); |
695 |
by (rtac subsetI 1); |
|
696 |
by (etac rangeE 1); |
|
697 |
by (etac ssubst 1); |
|
698 |
by (res_inst_tac [("l", "u")] llistE 1); |
|
699 |
by (etac ssubst 1); |
|
700 |
by (stac prem1 1); |
|
701 |
by (rtac llistD_Fun_range_I 1); |
|
702 |
by (etac ssubst 1); |
|
703 |
by (rtac prem2 1); |
|
704 |
qed "llist_fun_equalityI"; |
|
705 |
||
706 |
(*simpset for llist bisimulations*) |
|
707 |
val llistD_simps = [llist_case_LNil, llist_case_LCons, |
|
708 |
llistD_Fun_LNil_I, llistD_Fun_LCons_I]; |
|
709 |
(*Don't use llist_ss, as it does case splits!*) |
|
710 |
val llistD_ss = univ_ss addsimps llistD_simps |> add_eqI; |
|
711 |
||
712 |
||
713 |
(*** The functional "lmap" ***) |
|
714 |
||
715 |
goal LList.thy "lmap f LNil = LNil"; |
|
716 |
by (rtac (lmap_def RS def_llist_corec RS trans) 1); |
|
717 |
by (simp_tac llistD_ss 1); |
|
718 |
qed "lmap_LNil"; |
|
719 |
||
720 |
goal LList.thy "lmap f (LCons M N) = LCons (f M) (lmap f N)"; |
|
721 |
by (rtac (lmap_def RS def_llist_corec RS trans) 1); |
|
722 |
by (simp_tac llistD_ss 1); |
|
723 |
qed "lmap_LCons"; |
|
724 |
||
725 |
||
726 |
(** Two easy results about lmap **) |
|
727 |
||
728 |
goal LList.thy "lmap (f o g) l = lmap f (lmap g l)"; |
|
729 |
by (res_inst_tac [("l","l")] llist_fun_equalityI 1); |
|
730 |
by (ALLGOALS (simp_tac (llistD_ss addsimps [lmap_LNil, lmap_LCons]))); |
|
731 |
qed "lmap_compose"; |
|
732 |
||
733 |
goal LList.thy "lmap (%x.x) l = l"; |
|
734 |
by (res_inst_tac [("l","l")] llist_fun_equalityI 1); |
|
735 |
by (ALLGOALS (simp_tac (llistD_ss addsimps [lmap_LNil, lmap_LCons]))); |
|
736 |
qed "lmap_ident"; |
|
737 |
||
738 |
||
739 |
(*** iterates -- llist_fun_equalityI cannot be used! ***) |
|
740 |
||
741 |
goal LList.thy "iterates f x = LCons x (iterates f (f x))"; |
|
742 |
by (rtac (iterates_def RS def_llist_corec RS trans) 1); |
|
743 |
by (simp_tac sum_ss 1); |
|
744 |
qed "iterates"; |
|
745 |
||
746 |
goal LList.thy "lmap f (iterates f x) = iterates f (f x)"; |
|
972
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clasohm
parents:
969
diff
changeset
|
747 |
by (res_inst_tac [("r", "range(%u.(lmap f (iterates f u),iterates f (f u)))")] |
969 | 748 |
llist_equalityI 1); |
749 |
by (rtac rangeI 1); |
|
750 |
by (safe_tac set_cs); |
|
751 |
by (res_inst_tac [("x1", "f(u)")] (iterates RS ssubst) 1); |
|
752 |
by (res_inst_tac [("x1", "u")] (iterates RS ssubst) 1); |
|
753 |
by (simp_tac (llistD_ss addsimps [lmap_LCons]) 1); |
|
754 |
qed "lmap_iterates"; |
|
755 |
||
756 |
goal LList.thy "iterates f x = LCons x (lmap f (iterates f x))"; |
|
1046 | 757 |
by (rtac (lmap_iterates RS ssubst) 1); |
758 |
by (rtac iterates 1); |
|
969 | 759 |
qed "iterates_lmap"; |
760 |
||
761 |
(*** A rather complex proof about iterates -- cf Andy Pitts ***) |
|
762 |
||
763 |
(** Two lemmas about natrec n x (%m.g), which is essentially (g^n)(x) **) |
|
764 |
||
765 |
goal LList.thy |
|
766 |
"nat_rec n (LCons b l) (%m. lmap(f)) = \ |
|
767 |
\ LCons (nat_rec n b (%m. f)) (nat_rec n l (%m. lmap(f)))"; |
|
768 |
by (nat_ind_tac "n" 1); |
|
769 |
by (ALLGOALS (asm_simp_tac (nat_ss addsimps [lmap_LCons]))); |
|
770 |
qed "fun_power_lmap"; |
|
771 |
||
772 |
goal Nat.thy "nat_rec n (g x) (%m. g) = nat_rec (Suc n) x (%m. g)"; |
|
773 |
by (nat_ind_tac "n" 1); |
|
774 |
by (ALLGOALS (asm_simp_tac nat_ss)); |
|
775 |
qed "fun_power_Suc"; |
|
776 |
||
777 |
val Pair_cong = read_instantiate_sg (sign_of Prod.thy) |
|
778 |
[("f","Pair")] (standard(refl RS cong RS cong)); |
|
779 |
||
972
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parents:
969
diff
changeset
|
780 |
(*The bisimulation consists of {(lmap(f)^n (h(u)), lmap(f)^n (iterates(f,u)))} |
969 | 781 |
for all u and all n::nat.*) |
782 |
val [prem] = goal LList.thy |
|
783 |
"(!!x. h(x) = LCons x (lmap f (h x))) ==> h = iterates(f)"; |
|
1046 | 784 |
by (rtac ext 1); |
969 | 785 |
by (res_inst_tac [("r", |
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
969
diff
changeset
|
786 |
"UN u. range(%n. (nat_rec n (h u) (%m y.lmap f y), \ |
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
969
diff
changeset
|
787 |
\ nat_rec n (iterates f u) (%m y.lmap f y)))")] |
969 | 788 |
llist_equalityI 1); |
789 |
by (REPEAT (resolve_tac [UN1_I, range_eqI, Pair_cong, nat_rec_0 RS sym] 1)); |
|
790 |
by (safe_tac set_cs); |
|
791 |
by (stac iterates 1); |
|
792 |
by (stac prem 1); |
|
793 |
by (stac fun_power_lmap 1); |
|
794 |
by (stac fun_power_lmap 1); |
|
1046 | 795 |
by (rtac llistD_Fun_LCons_I 1); |
969 | 796 |
by (rtac (lmap_iterates RS subst) 1); |
797 |
by (stac fun_power_Suc 1); |
|
798 |
by (stac fun_power_Suc 1); |
|
1046 | 799 |
by (rtac (UN1_I RS UnI1) 1); |
800 |
by (rtac rangeI 1); |
|
969 | 801 |
qed "iterates_equality"; |
802 |
||
803 |
||
804 |
(*** lappend -- its two arguments cause some complications! ***) |
|
805 |
||
806 |
goalw LList.thy [lappend_def] "lappend LNil LNil = LNil"; |
|
807 |
by (rtac (llist_corec RS trans) 1); |
|
808 |
by (simp_tac llistD_ss 1); |
|
809 |
qed "lappend_LNil_LNil"; |
|
810 |
||
811 |
goalw LList.thy [lappend_def] |
|
812 |
"lappend LNil (LCons l l') = LCons l (lappend LNil l')"; |
|
813 |
by (rtac (llist_corec RS trans) 1); |
|
814 |
by (simp_tac llistD_ss 1); |
|
815 |
qed "lappend_LNil_LCons"; |
|
816 |
||
817 |
goalw LList.thy [lappend_def] |
|
818 |
"lappend (LCons l l') N = LCons l (lappend l' N)"; |
|
819 |
by (rtac (llist_corec RS trans) 1); |
|
820 |
by (simp_tac llistD_ss 1); |
|
821 |
qed "lappend_LCons"; |
|
822 |
||
823 |
goal LList.thy "lappend LNil l = l"; |
|
824 |
by (res_inst_tac [("l","l")] llist_fun_equalityI 1); |
|
825 |
by (ALLGOALS |
|
826 |
(simp_tac (llistD_ss addsimps [lappend_LNil_LNil, lappend_LNil_LCons]))); |
|
827 |
qed "lappend_LNil"; |
|
828 |
||
829 |
goal LList.thy "lappend l LNil = l"; |
|
830 |
by (res_inst_tac [("l","l")] llist_fun_equalityI 1); |
|
831 |
by (ALLGOALS |
|
832 |
(simp_tac (llistD_ss addsimps [lappend_LNil_LNil, lappend_LCons]))); |
|
833 |
qed "lappend_LNil2"; |
|
834 |
||
835 |
(*The infinite first argument blocks the second*) |
|
836 |
goal LList.thy "lappend (iterates f x) N = iterates f x"; |
|
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
969
diff
changeset
|
837 |
by (res_inst_tac [("r", "range(%u.(lappend (iterates f u) N,iterates f u))")] |
969 | 838 |
llist_equalityI 1); |
839 |
by (rtac rangeI 1); |
|
840 |
by (safe_tac set_cs); |
|
841 |
by (stac iterates 1); |
|
842 |
by (simp_tac (llistD_ss addsimps [lappend_LCons]) 1); |
|
843 |
qed "lappend_iterates"; |
|
844 |
||
845 |
(** Two proofs that lmap distributes over lappend **) |
|
846 |
||
847 |
(*Long proof requiring case analysis on both both arguments*) |
|
848 |
goal LList.thy "lmap f (lappend l n) = lappend (lmap f l) (lmap f n)"; |
|
849 |
by (res_inst_tac |
|
850 |
[("r", |
|
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
969
diff
changeset
|
851 |
"UN n. range(%l.(lmap f (lappend l n),lappend (lmap f l) (lmap f n)))")] |
969 | 852 |
llist_equalityI 1); |
853 |
by (rtac UN1_I 1); |
|
854 |
by (rtac rangeI 1); |
|
855 |
by (safe_tac set_cs); |
|
856 |
by (res_inst_tac [("l", "l")] llistE 1); |
|
857 |
by (res_inst_tac [("l", "n")] llistE 1); |
|
858 |
by (ALLGOALS (asm_simp_tac (llistD_ss addsimps |
|
859 |
[lappend_LNil_LNil,lappend_LCons,lappend_LNil_LCons, |
|
860 |
lmap_LNil,lmap_LCons]))); |
|
861 |
by (REPEAT_SOME (ares_tac [llistD_Fun_LCons_I, UN1_I RS UnI1, rangeI])); |
|
862 |
by (rtac range_eqI 1); |
|
863 |
by (rtac (refl RS Pair_cong) 1); |
|
864 |
by (stac lmap_LNil 1); |
|
865 |
by (rtac refl 1); |
|
866 |
qed "lmap_lappend_distrib"; |
|
867 |
||
868 |
(*Shorter proof of theorem above using llist_equalityI as strong coinduction*) |
|
869 |
goal LList.thy "lmap f (lappend l n) = lappend (lmap f l) (lmap f n)"; |
|
870 |
by (res_inst_tac [("l","l")] llist_fun_equalityI 1); |
|
871 |
by (simp_tac (llistD_ss addsimps [lappend_LNil, lmap_LNil])1); |
|
872 |
by (simp_tac (llistD_ss addsimps [lappend_LCons, lmap_LCons]) 1); |
|
873 |
qed "lmap_lappend_distrib"; |
|
874 |
||
875 |
(*Without strong coinduction, three case analyses might be needed*) |
|
876 |
goal LList.thy "lappend (lappend l1 l2) l3 = lappend l1 (lappend l2 l3)"; |
|
877 |
by (res_inst_tac [("l","l1")] llist_fun_equalityI 1); |
|
878 |
by (simp_tac (llistD_ss addsimps [lappend_LNil])1); |
|
879 |
by (simp_tac (llistD_ss addsimps [lappend_LCons]) 1); |
|
880 |
qed "lappend_assoc"; |