author | berghofe |
Fri, 21 Jun 1996 12:18:50 +0200 | |
changeset 1820 | e381e1c51689 |
parent 1673 | d22110ddd0af |
child 1824 | 44254696843a |
permissions | -rw-r--r-- |
1465 | 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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simpset := !simpset 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 (!claset 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 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 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 (!simpset 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 (!simpset 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 (!simpset 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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Addsimps [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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(** 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)); |
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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 (!simpset addsimps [CONS_UN1]) 1); |
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by (safe_tac (!claset)); |
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by (ALLGOALS (res_inst_tac [("x","Suc(?k)")] UN1_I THEN' Asm_simp_tac)); |
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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 (!claset)); |
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by (stac LList_corec 1); |
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by (simp_tac (!simpset 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 (!claset)); |
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by (stac LList_corec 1); |
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by (asm_simp_tac (!simpset addsimps [list_Fun_NIL_I]) 1); |
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by (fast_tac (!claset 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 (!claset 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 ((claset_of "Fun") delrules [equalityI])); |
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by (etac LListD.elim 1); |
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by (safe_tac (((claset_of "Prod") delrules [equalityI]) addSEs [diagE])); |
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by (res_inst_tac [("n", "n")] natE 1); |
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by (asm_simp_tac (!simpset 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 (!simpset addsimps [CONS_def, ntrunc_one_In1]) 1); |
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by (asm_simp_tac (!simpset addsimps [CONS_def, ntrunc_In1, ntrunc_Scons, less_Suc_eq]) 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 1); |
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by (Fast_tac 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 (!claset)); |
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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 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 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 (!simpset 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 (!simpset 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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"!!r. [| (M,N) : r; r <= LListD_Fun (diag A) (r Un diag(llist(A))) \ |
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\ |] ==> 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 (!simpset addsimps [LListD_eq_diag]) 1); |
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by (safe_tac (!claset)); |
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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); |
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by (rtac rangeI 1); |
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by (safe_tac (!claset)); |
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by (stac prem1 1); |
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by (stac prem2 1); |
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by (simp_tac (!simpset 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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(** 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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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 (!simpset 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 (!simpset setloop (split_tac [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", "xb")] natE 2); |
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by (ALLGOALS(asm_simp_tac(!simpset addsimps |
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[ntrunc_0,ntrunc_one_CONS,ntrunc_CONS, less_Suc_eq]))); |
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result(); |
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(*** Lconst -- defined directly using lfp, but equivalent to a LList_corec ***) |
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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)}. *) |
|
325 |
goal LList.thy "!!M A. M:A ==> Lconst(M): llist(A)"; |
|
326 |
by (rtac (singletonI RS llist_coinduct) 1); |
|
1820 | 327 |
by (safe_tac (!claset)); |
969 | 328 |
by (res_inst_tac [("P", "%u. u: ?A")] (Lconst RS ssubst) 1); |
329 |
by (REPEAT (ares_tac [list_Fun_CONS_I, singletonI, UnI1] 1)); |
|
330 |
qed "Lconst_type"; |
|
331 |
||
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diff
changeset
|
332 |
goal LList.thy "Lconst(M) = LList_corec M (%x.Inr((x,x)))"; |
969 | 333 |
by (rtac (equals_LList_corec RS fun_cong) 1); |
1266 | 334 |
by (Simp_tac 1); |
969 | 335 |
by (rtac Lconst 1); |
336 |
qed "Lconst_eq_LList_corec"; |
|
337 |
||
338 |
(*Thus we could have used gfp in the definition of Lconst*) |
|
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969
diff
changeset
|
339 |
goal LList.thy "gfp(%N. CONS M N) = LList_corec M (%x.Inr((x,x)))"; |
969 | 340 |
by (rtac (equals_LList_corec RS fun_cong) 1); |
1266 | 341 |
by (Simp_tac 1); |
969 | 342 |
by (rtac (Lconst_fun_mono RS gfp_Tarski) 1); |
343 |
qed "gfp_Lconst_eq_LList_corec"; |
|
344 |
||
345 |
||
346 |
(*** Isomorphisms ***) |
|
347 |
||
348 |
goal LList.thy "inj(Rep_llist)"; |
|
349 |
by (rtac inj_inverseI 1); |
|
350 |
by (rtac Rep_llist_inverse 1); |
|
351 |
qed "inj_Rep_llist"; |
|
352 |
||
1642 | 353 |
goal LList.thy "inj_onto Abs_llist (llist(range Leaf))"; |
969 | 354 |
by (rtac inj_onto_inverseI 1); |
355 |
by (etac Abs_llist_inverse 1); |
|
356 |
qed "inj_onto_Abs_llist"; |
|
357 |
||
358 |
(** Distinctness of constructors **) |
|
359 |
||
360 |
goalw LList.thy [LNil_def,LCons_def] "~ LCons x xs = LNil"; |
|
361 |
by (rtac (CONS_not_NIL RS (inj_onto_Abs_llist RS inj_onto_contraD)) 1); |
|
362 |
by (REPEAT (resolve_tac (llist.intrs @ [rangeI, Rep_llist]) 1)); |
|
363 |
qed "LCons_not_LNil"; |
|
364 |
||
365 |
bind_thm ("LNil_not_LCons", (LCons_not_LNil RS not_sym)); |
|
366 |
||
367 |
bind_thm ("LCons_neq_LNil", (LCons_not_LNil RS notE)); |
|
368 |
val LNil_neq_LCons = sym RS LCons_neq_LNil; |
|
369 |
||
370 |
(** llist constructors **) |
|
371 |
||
372 |
goalw LList.thy [LNil_def] |
|
373 |
"Rep_llist(LNil) = NIL"; |
|
374 |
by (rtac (llist.NIL_I RS Abs_llist_inverse) 1); |
|
375 |
qed "Rep_llist_LNil"; |
|
376 |
||
377 |
goalw LList.thy [LCons_def] |
|
378 |
"Rep_llist(LCons x l) = CONS (Leaf x) (Rep_llist l)"; |
|
379 |
by (REPEAT (resolve_tac [llist.CONS_I RS Abs_llist_inverse, |
|
1465 | 380 |
rangeI, Rep_llist] 1)); |
969 | 381 |
qed "Rep_llist_LCons"; |
382 |
||
383 |
(** Injectiveness of CONS and LCons **) |
|
384 |
||
385 |
goalw LList.thy [CONS_def] "(CONS M N=CONS M' N') = (M=M' & N=N')"; |
|
1820 | 386 |
by (fast_tac (!claset addSEs [Scons_inject, make_elim In1_inject]) 1); |
1266 | 387 |
qed "CONS_CONS_eq2"; |
969 | 388 |
|
389 |
bind_thm ("CONS_inject", (CONS_CONS_eq RS iffD1 RS conjE)); |
|
390 |
||
391 |
||
392 |
(*For reasoning about abstract llist constructors*) |
|
1820 | 393 |
|
394 |
AddIs ([Rep_llist]@llist.intrs); |
|
395 |
AddSEs [CONS_neq_NIL,NIL_neq_CONS,CONS_inject]; |
|
396 |
AddSDs [inj_onto_Abs_llist RS inj_ontoD, |
|
1465 | 397 |
inj_Rep_llist RS injD, Leaf_inject]; |
969 | 398 |
|
399 |
goalw LList.thy [LCons_def] "(LCons x xs=LCons y ys) = (x=y & xs=ys)"; |
|
1820 | 400 |
by (Fast_tac 1); |
969 | 401 |
qed "LCons_LCons_eq"; |
402 |
bind_thm ("LCons_inject", (LCons_LCons_eq RS iffD1 RS conjE)); |
|
403 |
||
404 |
val [major] = goal LList.thy "CONS M N: llist(A) ==> M: A & N: llist(A)"; |
|
405 |
by (rtac (major RS llist.elim) 1); |
|
406 |
by (etac CONS_neq_NIL 1); |
|
1820 | 407 |
by (Fast_tac 1); |
1266 | 408 |
qed "CONS_D2"; |
969 | 409 |
|
410 |
||
411 |
(****** Reasoning about llist(A) ******) |
|
412 |
||
1266 | 413 |
Addsimps [List_case_NIL, List_case_CONS]; |
969 | 414 |
|
415 |
(*A special case of list_equality for functions over lazy lists*) |
|
416 |
val [Mlist,gMlist,NILcase,CONScase] = goal LList.thy |
|
1465 | 417 |
"[| M: llist(A); g(NIL): llist(A); \ |
418 |
\ f(NIL)=g(NIL); \ |
|
419 |
\ !!x l. [| x:A; l: llist(A) |] ==> \ |
|
420 |
\ (f(CONS x l),g(CONS x l)) : \ |
|
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969
diff
changeset
|
421 |
\ LListD_Fun (diag A) ((%u.(f(u),g(u)))``llist(A) Un \ |
1465 | 422 |
\ diag(llist(A))) \ |
969 | 423 |
\ |] ==> f(M) = g(M)"; |
424 |
by (rtac LList_equalityI 1); |
|
1046 | 425 |
by (rtac (Mlist RS imageI) 1); |
969 | 426 |
by (rtac subsetI 1); |
427 |
by (etac imageE 1); |
|
428 |
by (etac ssubst 1); |
|
429 |
by (etac llist.elim 1); |
|
430 |
by (etac ssubst 1); |
|
431 |
by (stac NILcase 1); |
|
1046 | 432 |
by (rtac (gMlist RS LListD_Fun_diag_I) 1); |
969 | 433 |
by (etac ssubst 1); |
434 |
by (REPEAT (ares_tac [CONScase] 1)); |
|
435 |
qed "LList_fun_equalityI"; |
|
436 |
||
437 |
||
438 |
(*** The functional "Lmap" ***) |
|
439 |
||
440 |
goal LList.thy "Lmap f NIL = NIL"; |
|
441 |
by (rtac (Lmap_def RS def_LList_corec RS trans) 1); |
|
1266 | 442 |
by (Simp_tac 1); |
969 | 443 |
qed "Lmap_NIL"; |
444 |
||
445 |
goal LList.thy "Lmap f (CONS M N) = CONS (f M) (Lmap f N)"; |
|
446 |
by (rtac (Lmap_def RS def_LList_corec RS trans) 1); |
|
1266 | 447 |
by (Simp_tac 1); |
969 | 448 |
qed "Lmap_CONS"; |
449 |
||
450 |
(*Another type-checking proof by coinduction*) |
|
451 |
val [major,minor] = goal LList.thy |
|
452 |
"[| M: llist(A); !!x. x:A ==> f(x):B |] ==> Lmap f M: llist(B)"; |
|
453 |
by (rtac (major RS imageI RS llist_coinduct) 1); |
|
1820 | 454 |
by (safe_tac (!claset)); |
969 | 455 |
by (etac llist.elim 1); |
1266 | 456 |
by (ALLGOALS (asm_simp_tac (!simpset addsimps [Lmap_NIL,Lmap_CONS]))); |
969 | 457 |
by (REPEAT (ares_tac [list_Fun_NIL_I, list_Fun_CONS_I, |
1465 | 458 |
minor, imageI, UnI1] 1)); |
969 | 459 |
qed "Lmap_type"; |
460 |
||
461 |
(*This type checking rule synthesises a sufficiently large set for f*) |
|
462 |
val [major] = goal LList.thy "M: llist(A) ==> Lmap f M: llist(f``A)"; |
|
463 |
by (rtac (major RS Lmap_type) 1); |
|
464 |
by (etac imageI 1); |
|
465 |
qed "Lmap_type2"; |
|
466 |
||
467 |
(** Two easy results about Lmap **) |
|
468 |
||
469 |
val [prem] = goalw LList.thy [o_def] |
|
470 |
"M: llist(A) ==> Lmap (f o g) M = Lmap f (Lmap g M)"; |
|
471 |
by (rtac (prem RS imageI RS LList_equalityI) 1); |
|
1820 | 472 |
by (safe_tac (!claset)); |
969 | 473 |
by (etac llist.elim 1); |
1266 | 474 |
by (ALLGOALS (asm_simp_tac (!simpset addsimps [Lmap_NIL,Lmap_CONS]))); |
969 | 475 |
by (REPEAT (ares_tac [LListD_Fun_NIL_I, imageI, UnI1, |
1465 | 476 |
rangeI RS LListD_Fun_CONS_I] 1)); |
969 | 477 |
qed "Lmap_compose"; |
478 |
||
479 |
val [prem] = goal LList.thy "M: llist(A) ==> Lmap (%x.x) M = M"; |
|
480 |
by (rtac (prem RS imageI RS LList_equalityI) 1); |
|
1820 | 481 |
by (safe_tac (!claset)); |
969 | 482 |
by (etac llist.elim 1); |
1266 | 483 |
by (ALLGOALS (asm_simp_tac (!simpset addsimps [Lmap_NIL,Lmap_CONS]))); |
969 | 484 |
by (REPEAT (ares_tac [LListD_Fun_NIL_I, imageI RS UnI1, |
1465 | 485 |
rangeI RS LListD_Fun_CONS_I] 1)); |
969 | 486 |
qed "Lmap_ident"; |
487 |
||
488 |
||
489 |
(*** Lappend -- its two arguments cause some complications! ***) |
|
490 |
||
491 |
goalw LList.thy [Lappend_def] "Lappend NIL NIL = NIL"; |
|
492 |
by (rtac (LList_corec RS trans) 1); |
|
1266 | 493 |
by (Simp_tac 1); |
969 | 494 |
qed "Lappend_NIL_NIL"; |
495 |
||
496 |
goalw LList.thy [Lappend_def] |
|
497 |
"Lappend NIL (CONS N N') = CONS N (Lappend NIL N')"; |
|
498 |
by (rtac (LList_corec RS trans) 1); |
|
1266 | 499 |
by (Simp_tac 1); |
969 | 500 |
qed "Lappend_NIL_CONS"; |
501 |
||
502 |
goalw LList.thy [Lappend_def] |
|
503 |
"Lappend (CONS M M') N = CONS M (Lappend M' N)"; |
|
504 |
by (rtac (LList_corec RS trans) 1); |
|
1266 | 505 |
by (Simp_tac 1); |
969 | 506 |
qed "Lappend_CONS"; |
507 |
||
1266 | 508 |
Addsimps [llist.NIL_I, Lappend_NIL_NIL, Lappend_NIL_CONS, |
1465 | 509 |
Lappend_CONS, LListD_Fun_CONS_I, range_eqI, image_eqI]; |
1266 | 510 |
Delsimps [Pair_eq]; |
969 | 511 |
|
512 |
goal LList.thy "!!M. M: llist(A) ==> Lappend NIL M = M"; |
|
513 |
by (etac LList_fun_equalityI 1); |
|
1266 | 514 |
by (ALLGOALS Asm_simp_tac); |
969 | 515 |
qed "Lappend_NIL"; |
516 |
||
517 |
goal LList.thy "!!M. M: llist(A) ==> Lappend M NIL = M"; |
|
518 |
by (etac LList_fun_equalityI 1); |
|
1266 | 519 |
by (ALLGOALS Asm_simp_tac); |
969 | 520 |
qed "Lappend_NIL2"; |
521 |
||
522 |
(** Alternative type-checking proofs for Lappend **) |
|
523 |
||
524 |
(*weak co-induction: bisimulation and case analysis on both variables*) |
|
525 |
goal LList.thy |
|
526 |
"!!M N. [| M: llist(A); N: llist(A) |] ==> Lappend M N: llist(A)"; |
|
527 |
by (res_inst_tac |
|
528 |
[("X", "UN u:llist(A). UN v: llist(A). {Lappend u v}")] llist_coinduct 1); |
|
1820 | 529 |
by (Fast_tac 1); |
530 |
by (safe_tac (!claset)); |
|
969 | 531 |
by (eres_inst_tac [("a", "u")] llist.elim 1); |
532 |
by (eres_inst_tac [("a", "v")] llist.elim 1); |
|
533 |
by (ALLGOALS |
|
1266 | 534 |
(Asm_simp_tac THEN' |
1820 | 535 |
fast_tac (!claset addSIs [llist.NIL_I, list_Fun_NIL_I, list_Fun_CONS_I]))); |
969 | 536 |
qed "Lappend_type"; |
537 |
||
538 |
(*strong co-induction: bisimulation and case analysis on one variable*) |
|
539 |
goal LList.thy |
|
540 |
"!!M N. [| M: llist(A); N: llist(A) |] ==> Lappend M N: llist(A)"; |
|
541 |
by (res_inst_tac [("X", "(%u.Lappend u N)``llist(A)")] llist_coinduct 1); |
|
1046 | 542 |
by (etac imageI 1); |
543 |
by (rtac subsetI 1); |
|
544 |
by (etac imageE 1); |
|
969 | 545 |
by (eres_inst_tac [("a", "u")] llist.elim 1); |
1266 | 546 |
by (asm_simp_tac (!simpset addsimps [Lappend_NIL, list_Fun_llist_I]) 1); |
547 |
by (Asm_simp_tac 1); |
|
1820 | 548 |
by (fast_tac (!claset addSIs [list_Fun_CONS_I]) 1); |
969 | 549 |
qed "Lappend_type"; |
550 |
||
551 |
(**** Lazy lists as the type 'a llist -- strongly typed versions of above ****) |
|
552 |
||
553 |
(** llist_case: case analysis for 'a llist **) |
|
554 |
||
1266 | 555 |
Addsimps ([Abs_llist_inverse, Rep_llist_inverse, |
556 |
Rep_llist, rangeI, inj_Leaf, Inv_f_f] @ llist.intrs); |
|
969 | 557 |
|
558 |
goalw LList.thy [llist_case_def,LNil_def] "llist_case c d LNil = c"; |
|
1266 | 559 |
by (Simp_tac 1); |
969 | 560 |
qed "llist_case_LNil"; |
561 |
||
562 |
goalw LList.thy [llist_case_def,LCons_def] |
|
563 |
"llist_case c d (LCons M N) = d M N"; |
|
1266 | 564 |
by (Simp_tac 1); |
969 | 565 |
qed "llist_case_LCons"; |
566 |
||
567 |
(*Elimination is case analysis, not induction.*) |
|
568 |
val [prem1,prem2] = goalw LList.thy [NIL_def,CONS_def] |
|
569 |
"[| l=LNil ==> P; !!x l'. l=LCons x l' ==> P \ |
|
570 |
\ |] ==> P"; |
|
571 |
by (rtac (Rep_llist RS llist.elim) 1); |
|
572 |
by (rtac (inj_Rep_llist RS injD RS prem1) 1); |
|
573 |
by (stac Rep_llist_LNil 1); |
|
574 |
by (assume_tac 1); |
|
575 |
by (etac rangeE 1); |
|
576 |
by (rtac (inj_Rep_llist RS injD RS prem2) 1); |
|
1266 | 577 |
by (asm_simp_tac (!simpset delsimps [CONS_CONS_eq] addsimps [Rep_llist_LCons]) 1); |
969 | 578 |
by (etac (Abs_llist_inverse RS ssubst) 1); |
579 |
by (rtac refl 1); |
|
580 |
qed "llistE"; |
|
581 |
||
582 |
(** llist_corec: corecursion for 'a llist **) |
|
583 |
||
584 |
goalw LList.thy [llist_corec_def,LNil_def,LCons_def] |
|
585 |
"llist_corec a f = sum_case (%u. LNil) \ |
|
1465 | 586 |
\ (split(%z w. LCons z (llist_corec w f))) (f a)"; |
969 | 587 |
by (stac LList_corec 1); |
588 |
by (res_inst_tac [("s","f(a)")] sumE 1); |
|
1266 | 589 |
by (asm_simp_tac (!simpset addsimps [LList_corec_type2]) 1); |
969 | 590 |
by (res_inst_tac [("p","y")] PairE 1); |
1266 | 591 |
by (asm_simp_tac (!simpset addsimps [LList_corec_type2]) 1); |
969 | 592 |
(*FIXME: correct case splits usd to be found automatically: |
1266 | 593 |
by (ASM_SIMP_TAC(!simpset addsimps [LList_corec_type2]) 1);*) |
969 | 594 |
qed "llist_corec"; |
595 |
||
596 |
(*definitional version of same*) |
|
597 |
val [rew] = goal LList.thy |
|
1465 | 598 |
"[| !!x. h(x) == llist_corec x f |] ==> \ |
969 | 599 |
\ h(a) = sum_case (%u.LNil) (split(%z w. LCons z (h w))) (f a)"; |
600 |
by (rewtac rew); |
|
601 |
by (rtac llist_corec 1); |
|
602 |
qed "def_llist_corec"; |
|
603 |
||
604 |
(**** Proofs about type 'a llist functions ****) |
|
605 |
||
606 |
(*** Deriving llist_equalityI -- llist equality is a bisimulation ***) |
|
607 |
||
608 |
goalw LList.thy [LListD_Fun_def] |
|
1642 | 609 |
"!!r A. r <= (llist A) Times (llist A) ==> \ |
610 |
\ LListD_Fun (diag A) r <= (llist A) Times (llist A)"; |
|
969 | 611 |
by (stac llist_unfold 1); |
1266 | 612 |
by (simp_tac (!simpset addsimps [NIL_def, CONS_def]) 1); |
1820 | 613 |
by (Fast_tac 1); |
969 | 614 |
qed "LListD_Fun_subset_Sigma_llist"; |
615 |
||
616 |
goal LList.thy |
|
617 |
"prod_fun Rep_llist Rep_llist `` r <= \ |
|
1642 | 618 |
\ (llist(range Leaf)) Times (llist(range Leaf))"; |
1820 | 619 |
by (fast_tac (!claset addIs [Rep_llist]) 1); |
969 | 620 |
qed "subset_Sigma_llist"; |
621 |
||
622 |
val [prem] = goal LList.thy |
|
1642 | 623 |
"r <= (llist(range Leaf)) Times (llist(range Leaf)) ==> \ |
969 | 624 |
\ prod_fun (Rep_llist o Abs_llist) (Rep_llist o Abs_llist) `` r <= r"; |
1820 | 625 |
by (safe_tac (!claset)); |
969 | 626 |
by (rtac (prem RS subsetD RS SigmaE2) 1); |
627 |
by (assume_tac 1); |
|
1266 | 628 |
by (asm_simp_tac (!simpset addsimps [o_def,prod_fun,Abs_llist_inverse]) 1); |
969 | 629 |
qed "prod_fun_lemma"; |
630 |
||
631 |
goal LList.thy |
|
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
969
diff
changeset
|
632 |
"prod_fun Rep_llist Rep_llist `` range(%x. (x, x)) = \ |
1642 | 633 |
\ diag(llist(range Leaf))"; |
1046 | 634 |
by (rtac equalityI 1); |
1820 | 635 |
by (fast_tac (!claset addIs [Rep_llist]) 1); |
636 |
by (fast_tac (!claset addSEs [Abs_llist_inverse RS subst]) 1); |
|
969 | 637 |
qed "prod_fun_range_eq_diag"; |
638 |
||
639 |
(** To show two llists are equal, exhibit a bisimulation! |
|
640 |
[also admits true equality] **) |
|
641 |
val [prem1,prem2] = goalw LList.thy [llistD_Fun_def] |
|
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
969
diff
changeset
|
642 |
"[| (l1,l2) : r; r <= llistD_Fun(r Un range(%x.(x,x))) |] ==> l1=l2"; |
969 | 643 |
by (rtac (inj_Rep_llist RS injD) 1); |
644 |
by (res_inst_tac [("r", "prod_fun Rep_llist Rep_llist ``r"), |
|
1465 | 645 |
("A", "range(Leaf)")] |
646 |
LList_equalityI 1); |
|
969 | 647 |
by (rtac (prem1 RS prod_fun_imageI) 1); |
648 |
by (rtac (prem2 RS image_mono RS subset_trans) 1); |
|
649 |
by (rtac (image_compose RS subst) 1); |
|
650 |
by (rtac (prod_fun_compose RS subst) 1); |
|
651 |
by (rtac (image_Un RS ssubst) 1); |
|
652 |
by (stac prod_fun_range_eq_diag 1); |
|
653 |
by (rtac (LListD_Fun_subset_Sigma_llist RS prod_fun_lemma) 1); |
|
654 |
by (rtac (subset_Sigma_llist RS Un_least) 1); |
|
655 |
by (rtac diag_subset_Sigma 1); |
|
656 |
qed "llist_equalityI"; |
|
657 |
||
658 |
(** Rules to prove the 2nd premise of llist_equalityI **) |
|
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
969
diff
changeset
|
659 |
goalw LList.thy [llistD_Fun_def,LNil_def] "(LNil,LNil) : llistD_Fun(r)"; |
969 | 660 |
by (rtac (LListD_Fun_NIL_I RS prod_fun_imageI) 1); |
661 |
qed "llistD_Fun_LNil_I"; |
|
662 |
||
663 |
val [prem] = goalw LList.thy [llistD_Fun_def,LCons_def] |
|
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
969
diff
changeset
|
664 |
"(l1,l2):r ==> (LCons x l1, LCons x l2) : llistD_Fun(r)"; |
969 | 665 |
by (rtac (rangeI RS LListD_Fun_CONS_I RS prod_fun_imageI) 1); |
666 |
by (rtac (prem RS prod_fun_imageI) 1); |
|
667 |
qed "llistD_Fun_LCons_I"; |
|
668 |
||
669 |
(*Utilise the "strong" part, i.e. gfp(f)*) |
|
670 |
goalw LList.thy [llistD_Fun_def] |
|
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
969
diff
changeset
|
671 |
"!!l. (l,l) : llistD_Fun(r Un range(%x.(x,x)))"; |
1046 | 672 |
by (rtac (Rep_llist_inverse RS subst) 1); |
673 |
by (rtac prod_fun_imageI 1); |
|
969 | 674 |
by (rtac (image_Un RS ssubst) 1); |
675 |
by (stac prod_fun_range_eq_diag 1); |
|
1046 | 676 |
by (rtac (Rep_llist RS LListD_Fun_diag_I) 1); |
969 | 677 |
qed "llistD_Fun_range_I"; |
678 |
||
679 |
(*A special case of list_equality for functions over lazy lists*) |
|
680 |
val [prem1,prem2] = goal LList.thy |
|
1465 | 681 |
"[| f(LNil)=g(LNil); \ |
682 |
\ !!x l. (f(LCons x l),g(LCons x l)) : \ |
|
683 |
\ llistD_Fun(range(%u. (f(u),g(u))) Un range(%v. (v,v))) \ |
|
684 |
\ |] ==> f(l) = (g(l :: 'a llist) :: 'b llist)"; |
|
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
969
diff
changeset
|
685 |
by (res_inst_tac [("r", "range(%u. (f(u),g(u)))")] llist_equalityI 1); |
969 | 686 |
by (rtac rangeI 1); |
687 |
by (rtac subsetI 1); |
|
688 |
by (etac rangeE 1); |
|
689 |
by (etac ssubst 1); |
|
690 |
by (res_inst_tac [("l", "u")] llistE 1); |
|
691 |
by (etac ssubst 1); |
|
692 |
by (stac prem1 1); |
|
693 |
by (rtac llistD_Fun_range_I 1); |
|
694 |
by (etac ssubst 1); |
|
695 |
by (rtac prem2 1); |
|
696 |
qed "llist_fun_equalityI"; |
|
697 |
||
698 |
(*simpset for llist bisimulations*) |
|
1266 | 699 |
Addsimps [llist_case_LNil, llist_case_LCons, |
1465 | 700 |
llistD_Fun_LNil_I, llistD_Fun_LCons_I]; |
969 | 701 |
|
702 |
||
703 |
(*** The functional "lmap" ***) |
|
704 |
||
705 |
goal LList.thy "lmap f LNil = LNil"; |
|
706 |
by (rtac (lmap_def RS def_llist_corec RS trans) 1); |
|
1266 | 707 |
by (Simp_tac 1); |
969 | 708 |
qed "lmap_LNil"; |
709 |
||
710 |
goal LList.thy "lmap f (LCons M N) = LCons (f M) (lmap f N)"; |
|
711 |
by (rtac (lmap_def RS def_llist_corec RS trans) 1); |
|
1266 | 712 |
by (Simp_tac 1); |
969 | 713 |
qed "lmap_LCons"; |
714 |
||
715 |
||
716 |
(** Two easy results about lmap **) |
|
717 |
||
718 |
goal LList.thy "lmap (f o g) l = lmap f (lmap g l)"; |
|
719 |
by (res_inst_tac [("l","l")] llist_fun_equalityI 1); |
|
1266 | 720 |
by (ALLGOALS (simp_tac (!simpset addsimps [lmap_LNil, lmap_LCons]))); |
969 | 721 |
qed "lmap_compose"; |
722 |
||
723 |
goal LList.thy "lmap (%x.x) l = l"; |
|
724 |
by (res_inst_tac [("l","l")] llist_fun_equalityI 1); |
|
1266 | 725 |
by (ALLGOALS (simp_tac (!simpset addsimps [lmap_LNil, lmap_LCons]))); |
969 | 726 |
qed "lmap_ident"; |
727 |
||
728 |
||
729 |
(*** iterates -- llist_fun_equalityI cannot be used! ***) |
|
730 |
||
731 |
goal LList.thy "iterates f x = LCons x (iterates f (f x))"; |
|
732 |
by (rtac (iterates_def RS def_llist_corec RS trans) 1); |
|
1266 | 733 |
by (Simp_tac 1); |
969 | 734 |
qed "iterates"; |
735 |
||
736 |
goal LList.thy "lmap f (iterates f x) = iterates f (f x)"; |
|
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
969
diff
changeset
|
737 |
by (res_inst_tac [("r", "range(%u.(lmap f (iterates f u),iterates f (f u)))")] |
969 | 738 |
llist_equalityI 1); |
739 |
by (rtac rangeI 1); |
|
1820 | 740 |
by (safe_tac (!claset)); |
969 | 741 |
by (res_inst_tac [("x1", "f(u)")] (iterates RS ssubst) 1); |
742 |
by (res_inst_tac [("x1", "u")] (iterates RS ssubst) 1); |
|
1266 | 743 |
by (simp_tac (!simpset addsimps [lmap_LCons]) 1); |
969 | 744 |
qed "lmap_iterates"; |
745 |
||
746 |
goal LList.thy "iterates f x = LCons x (lmap f (iterates f x))"; |
|
1046 | 747 |
by (rtac (lmap_iterates RS ssubst) 1); |
748 |
by (rtac iterates 1); |
|
969 | 749 |
qed "iterates_lmap"; |
750 |
||
751 |
(*** A rather complex proof about iterates -- cf Andy Pitts ***) |
|
752 |
||
753 |
(** Two lemmas about natrec n x (%m.g), which is essentially (g^n)(x) **) |
|
754 |
||
755 |
goal LList.thy |
|
1465 | 756 |
"nat_rec n (LCons b l) (%m. lmap(f)) = \ |
969 | 757 |
\ LCons (nat_rec n b (%m. f)) (nat_rec n l (%m. lmap(f)))"; |
758 |
by (nat_ind_tac "n" 1); |
|
1266 | 759 |
by (ALLGOALS (asm_simp_tac (!simpset addsimps [lmap_LCons]))); |
969 | 760 |
qed "fun_power_lmap"; |
761 |
||
762 |
goal Nat.thy "nat_rec n (g x) (%m. g) = nat_rec (Suc n) x (%m. g)"; |
|
763 |
by (nat_ind_tac "n" 1); |
|
1266 | 764 |
by (ALLGOALS Asm_simp_tac); |
969 | 765 |
qed "fun_power_Suc"; |
766 |
||
767 |
val Pair_cong = read_instantiate_sg (sign_of Prod.thy) |
|
768 |
[("f","Pair")] (standard(refl RS cong RS cong)); |
|
769 |
||
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
969
diff
changeset
|
770 |
(*The bisimulation consists of {(lmap(f)^n (h(u)), lmap(f)^n (iterates(f,u)))} |
969 | 771 |
for all u and all n::nat.*) |
772 |
val [prem] = goal LList.thy |
|
773 |
"(!!x. h(x) = LCons x (lmap f (h x))) ==> h = iterates(f)"; |
|
1046 | 774 |
by (rtac ext 1); |
969 | 775 |
by (res_inst_tac [("r", |
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
969
diff
changeset
|
776 |
"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
|
777 |
\ nat_rec n (iterates f u) (%m y.lmap f y)))")] |
969 | 778 |
llist_equalityI 1); |
779 |
by (REPEAT (resolve_tac [UN1_I, range_eqI, Pair_cong, nat_rec_0 RS sym] 1)); |
|
1820 | 780 |
by (safe_tac (!claset)); |
969 | 781 |
by (stac iterates 1); |
782 |
by (stac prem 1); |
|
783 |
by (stac fun_power_lmap 1); |
|
784 |
by (stac fun_power_lmap 1); |
|
1046 | 785 |
by (rtac llistD_Fun_LCons_I 1); |
969 | 786 |
by (rtac (lmap_iterates RS subst) 1); |
787 |
by (stac fun_power_Suc 1); |
|
788 |
by (stac fun_power_Suc 1); |
|
1046 | 789 |
by (rtac (UN1_I RS UnI1) 1); |
790 |
by (rtac rangeI 1); |
|
969 | 791 |
qed "iterates_equality"; |
792 |
||
793 |
||
794 |
(*** lappend -- its two arguments cause some complications! ***) |
|
795 |
||
796 |
goalw LList.thy [lappend_def] "lappend LNil LNil = LNil"; |
|
797 |
by (rtac (llist_corec RS trans) 1); |
|
1266 | 798 |
by (Simp_tac 1); |
969 | 799 |
qed "lappend_LNil_LNil"; |
800 |
||
801 |
goalw LList.thy [lappend_def] |
|
802 |
"lappend LNil (LCons l l') = LCons l (lappend LNil l')"; |
|
803 |
by (rtac (llist_corec RS trans) 1); |
|
1266 | 804 |
by (Simp_tac 1); |
969 | 805 |
qed "lappend_LNil_LCons"; |
806 |
||
807 |
goalw LList.thy [lappend_def] |
|
808 |
"lappend (LCons l l') N = LCons l (lappend l' N)"; |
|
809 |
by (rtac (llist_corec RS trans) 1); |
|
1266 | 810 |
by (Simp_tac 1); |
969 | 811 |
qed "lappend_LCons"; |
812 |
||
813 |
goal LList.thy "lappend LNil l = l"; |
|
814 |
by (res_inst_tac [("l","l")] llist_fun_equalityI 1); |
|
815 |
by (ALLGOALS |
|
1266 | 816 |
(simp_tac (!simpset addsimps [lappend_LNil_LNil, lappend_LNil_LCons]))); |
969 | 817 |
qed "lappend_LNil"; |
818 |
||
819 |
goal LList.thy "lappend l LNil = l"; |
|
820 |
by (res_inst_tac [("l","l")] llist_fun_equalityI 1); |
|
821 |
by (ALLGOALS |
|
1266 | 822 |
(simp_tac (!simpset addsimps [lappend_LNil_LNil, lappend_LCons]))); |
969 | 823 |
qed "lappend_LNil2"; |
824 |
||
825 |
(*The infinite first argument blocks the second*) |
|
826 |
goal LList.thy "lappend (iterates f x) N = iterates f x"; |
|
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
969
diff
changeset
|
827 |
by (res_inst_tac [("r", "range(%u.(lappend (iterates f u) N,iterates f u))")] |
969 | 828 |
llist_equalityI 1); |
829 |
by (rtac rangeI 1); |
|
1820 | 830 |
by (safe_tac (!claset)); |
969 | 831 |
by (stac iterates 1); |
1266 | 832 |
by (simp_tac (!simpset addsimps [lappend_LCons]) 1); |
969 | 833 |
qed "lappend_iterates"; |
834 |
||
835 |
(** Two proofs that lmap distributes over lappend **) |
|
836 |
||
837 |
(*Long proof requiring case analysis on both both arguments*) |
|
838 |
goal LList.thy "lmap f (lappend l n) = lappend (lmap f l) (lmap f n)"; |
|
839 |
by (res_inst_tac |
|
840 |
[("r", |
|
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
969
diff
changeset
|
841 |
"UN n. range(%l.(lmap f (lappend l n),lappend (lmap f l) (lmap f n)))")] |
969 | 842 |
llist_equalityI 1); |
843 |
by (rtac UN1_I 1); |
|
844 |
by (rtac rangeI 1); |
|
1820 | 845 |
by (safe_tac (!claset)); |
969 | 846 |
by (res_inst_tac [("l", "l")] llistE 1); |
847 |
by (res_inst_tac [("l", "n")] llistE 1); |
|
1266 | 848 |
by (ALLGOALS (asm_simp_tac (!simpset addsimps |
969 | 849 |
[lappend_LNil_LNil,lappend_LCons,lappend_LNil_LCons, |
850 |
lmap_LNil,lmap_LCons]))); |
|
851 |
by (REPEAT_SOME (ares_tac [llistD_Fun_LCons_I, UN1_I RS UnI1, rangeI])); |
|
852 |
by (rtac range_eqI 1); |
|
853 |
by (rtac (refl RS Pair_cong) 1); |
|
854 |
by (stac lmap_LNil 1); |
|
855 |
by (rtac refl 1); |
|
856 |
qed "lmap_lappend_distrib"; |
|
857 |
||
858 |
(*Shorter proof of theorem above using llist_equalityI as strong coinduction*) |
|
859 |
goal LList.thy "lmap f (lappend l n) = lappend (lmap f l) (lmap f n)"; |
|
860 |
by (res_inst_tac [("l","l")] llist_fun_equalityI 1); |
|
1266 | 861 |
by (simp_tac (!simpset addsimps [lappend_LNil, lmap_LNil])1); |
862 |
by (simp_tac (!simpset addsimps [lappend_LCons, lmap_LCons]) 1); |
|
969 | 863 |
qed "lmap_lappend_distrib"; |
864 |
||
865 |
(*Without strong coinduction, three case analyses might be needed*) |
|
866 |
goal LList.thy "lappend (lappend l1 l2) l3 = lappend l1 (lappend l2 l3)"; |
|
867 |
by (res_inst_tac [("l","l1")] llist_fun_equalityI 1); |
|
1266 | 868 |
by (simp_tac (!simpset addsimps [lappend_LNil])1); |
869 |
by (simp_tac (!simpset addsimps [lappend_LCons]) 1); |
|
969 | 870 |
qed "lappend_assoc"; |