author | paulson |
Mon, 02 Nov 1998 15:31:29 +0100 | |
changeset 5788 | e3a98a7c0634 |
parent 5278 | a903b66822e2 |
child 5977 | 9f0c8869cf71 |
permissions | -rw-r--r-- |
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(* Title: HOL/Induct/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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bind_thm ("UN1_I", UNIV_I RS UN_I); |
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||
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(** Simplification **) |
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Addsplits [split_split, split_sum_case]; |
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(*This justifies using llist in other recursive type definitions*) |
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Goalw llist.defs "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(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 (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 [list_Fun_def] |
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"[| 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 [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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AddIffs [list_Fun_NIL_I]; |
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Goalw [list_Fun_def,CONS_def] |
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"[| 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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Addsimps [list_Fun_CONS_I]; |
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AddSIs [list_Fun_CONS_I]; |
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(*Utilise the "strong" part, i.e. gfp(f)*) |
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Goalw (llist.defs @ [list_Fun_def]) |
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"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 [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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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_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 UN_least 1); |
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by (exhaust_tac "k" 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, |
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UNIV_I RS UN_upper] 1)); |
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qed "LList_corec_subset1"; |
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Goalw [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; |
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by (ALLGOALS (res_inst_tac [("a","Suc(?k)")] UN_I)); |
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by (ALLGOALS 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_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_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; |
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by (stac LList_corec 1); |
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by (Simp_tac 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_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; |
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by (stac LList_corec 1); |
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by (Asm_simp_tac 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 "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 (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 "!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.thy) delrules [equalityI])); |
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by (etac LListD.elim 1); |
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by (safe_tac (claset_of Prod.thy delrules [equalityI] addSEs [diagE])); |
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by (exhaust_tac "n" 1); |
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by (Asm_simp_tac 1); |
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by (rename_tac "n'" 1); |
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by (exhaust_tac "n'" 1); |
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by (asm_simp_tac (simpset() addsimps [CONS_def]) 1); |
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by (asm_simp_tac (simpset() addsimps [CONS_def, 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.defs @ [NIL_def, CONS_def]) |
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"Domain (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. Domain 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 "Domain_LListD"; |
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|
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(*This inclusion justifies the use of coinduction to show M=N*) |
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Goal "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; |
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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 (DomainI RS (Domain_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 [LListD_Fun_def] "A<=B ==> LListD_Fun r A <= LListD_Fun r B"; |
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by (REPEAT (ares_tac basic_monos 1)); |
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qed "LListD_Fun_mono"; |
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Goalw [LListD_Fun_def] |
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"[| 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 [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 [LListD_Fun_def,CONS_def] |
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"[| 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 (LListD.defs @ [LListD_Fun_def]) |
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"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 "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 "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 "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 "[| (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; |
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qed "LList_equalityI"; |
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|
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(*** Finality of llist(A): Uniqueness of functions defined by corecursion ***) |
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(*We must remove Pair_eq because it may turn an instance of reflexivity |
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(h1 b, h2 b) = (h1 ?x17, h2 ?x17) into a conjunction! |
|
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(or strengthen the Solver?) |
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*) |
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Delsimps [Pair_eq]; |
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||
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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; |
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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]) 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 [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 [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 1); |
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qed "ntrunc_CONS"; |
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Addsimps [ntrunc_one_CONS, 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 (rename_tac "x k" 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 (rename_tac "n" 1); |
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by (rtac allI 1); |
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by (rename_tac "y" 1); |
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by (stac prem1 1); |
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by (stac prem2 1); |
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by (Simp_tac 1); |
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by (strip_tac 1); |
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by (exhaust_tac "n" 1); |
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by (rename_tac "m" 2); |
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by (exhaust_tac "m" 2); |
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by (ALLGOALS (asm_simp_tac (simpset() addsimps [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 "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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|
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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 "M:A ==> Lconst(M): llist(A)"; |
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by (rtac (singletonI RS llist_coinduct) 1); |
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by Safe_tac; |
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by (res_inst_tac [("P", "%u. u: ?A")] (Lconst RS ssubst) 1); |
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by (REPEAT (ares_tac [list_Fun_CONS_I, singletonI, UnI1] 1)); |
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qed "Lconst_type"; |
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Goal "Lconst(M) = LList_corec M (%x. Inr((x,x)))"; |
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by (rtac (equals_LList_corec RS fun_cong) 1); |
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by (Simp_tac 1); |
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by (rtac Lconst 1); |
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qed "Lconst_eq_LList_corec"; |
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336 |
|
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(*Thus we could have used gfp in the definition of Lconst*) |
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Goal "gfp(%N. CONS M N) = LList_corec M (%x. Inr((x,x)))"; |
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by (rtac (equals_LList_corec RS fun_cong) 1); |
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by (Simp_tac 1); |
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by (rtac (Lconst_fun_mono RS gfp_Tarski) 1); |
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qed "gfp_Lconst_eq_LList_corec"; |
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(*** Isomorphisms ***) |
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Goal "inj(Rep_llist)"; |
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by (rtac inj_inverseI 1); |
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by (rtac Rep_llist_inverse 1); |
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qed "inj_Rep_llist"; |
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Goal "inj_on Abs_llist (llist(range Leaf))"; |
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by (rtac inj_on_inverseI 1); |
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by (etac Abs_llist_inverse 1); |
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qed "inj_on_Abs_llist"; |
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(** Distinctness of constructors **) |
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Goalw [LNil_def,LCons_def] "~ LCons x xs = LNil"; |
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by (rtac (CONS_not_NIL RS (inj_on_Abs_llist RS inj_on_contraD)) 1); |
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by (REPEAT (resolve_tac (llist.intrs @ [rangeI, Rep_llist]) 1)); |
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qed "LCons_not_LNil"; |
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bind_thm ("LNil_not_LCons", LCons_not_LNil RS not_sym); |
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AddIffs [LCons_not_LNil, LNil_not_LCons]; |
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(** llist constructors **) |
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Goalw [LNil_def] |
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"Rep_llist(LNil) = NIL"; |
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by (rtac (llist.NIL_I RS Abs_llist_inverse) 1); |
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qed "Rep_llist_LNil"; |
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Goalw [LCons_def] |
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"Rep_llist(LCons x l) = CONS (Leaf x) (Rep_llist l)"; |
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by (REPEAT (resolve_tac [llist.CONS_I RS Abs_llist_inverse, |
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rangeI, Rep_llist] 1)); |
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qed "Rep_llist_LCons"; |
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381 |
|
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(** Injectiveness of CONS and LCons **) |
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Goalw [CONS_def] "(CONS M N=CONS M' N') = (M=M' & N=N')"; |
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by (fast_tac (claset() addSEs [Scons_inject]) 1); |
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qed "CONS_CONS_eq2"; |
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bind_thm ("CONS_inject", (CONS_CONS_eq RS iffD1 RS conjE)); |
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(*For reasoning about abstract llist constructors*) |
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AddIs ([Rep_llist]@llist.intrs); |
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AddSDs [inj_on_Abs_llist RS inj_onD, |
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inj_Rep_llist RS injD, Leaf_inject]; |
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Goalw [LCons_def] "(LCons x xs=LCons y ys) = (x=y & xs=ys)"; |
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by (Fast_tac 1); |
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qed "LCons_LCons_eq"; |
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AddIffs [LCons_LCons_eq]; |
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val [major] = goal LList.thy "CONS M N: llist(A) ==> M: A & N: llist(A)"; |
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by (rtac (major RS llist.elim) 1); |
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by (etac CONS_neq_NIL 1); |
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by (Fast_tac 1); |
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qed "CONS_D2"; |
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408 |
|
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409 |
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(****** Reasoning about llist(A) ******) |
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Addsimps [List_case_NIL, List_case_CONS]; |
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|
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(*A special case of list_equality for functions over lazy lists*) |
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val [Mlist,gMlist,NILcase,CONScase] = goal LList.thy |
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"[| M: llist(A); g(NIL): llist(A); \ |
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\ f(NIL)=g(NIL); \ |
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\ !!x l. [| x:A; l: llist(A) |] ==> \ |
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\ (f(CONS x l),g(CONS x l)) : \ |
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\ LListD_Fun (diag A) ((%u.(f(u),g(u)))``llist(A) Un \ |
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\ diag(llist(A))) \ |
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\ |] ==> f(M) = g(M)"; |
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by (rtac LList_equalityI 1); |
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by (rtac (Mlist RS imageI) 1); |
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by (rtac image_subsetI 1); |
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by (etac llist.elim 1); |
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by (etac ssubst 1); |
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by (stac NILcase 1); |
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by (rtac (gMlist RS LListD_Fun_diag_I) 1); |
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by (etac ssubst 1); |
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by (REPEAT (ares_tac [CONScase] 1)); |
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qed "LList_fun_equalityI"; |
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434 |
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435 |
(*** The functional "Lmap" ***) |
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436 |
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5069 | 437 |
Goal "Lmap f NIL = NIL"; |
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by (rtac (Lmap_def RS def_LList_corec RS trans) 1); |
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by (Simp_tac 1); |
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qed "Lmap_NIL"; |
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441 |
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Goal "Lmap f (CONS M N) = CONS (f M) (Lmap f N)"; |
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by (rtac (Lmap_def RS def_LList_corec RS trans) 1); |
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by (Simp_tac 1); |
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qed "Lmap_CONS"; |
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446 |
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Addsimps [Lmap_NIL, Lmap_CONS]; |
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||
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(*Another type-checking proof by coinduction*) |
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val [major,minor] = goal LList.thy |
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"[| M: llist(A); !!x. x:A ==> f(x):B |] ==> Lmap f M: llist(B)"; |
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by (rtac (major RS imageI RS llist_coinduct) 1); |
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by Safe_tac; |
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by (etac llist.elim 1); |
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by (ALLGOALS Asm_simp_tac); |
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456 |
by (REPEAT (ares_tac [list_Fun_NIL_I, list_Fun_CONS_I, |
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|
457 |
minor, imageI, UnI1] 1)); |
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|
458 |
qed "Lmap_type"; |
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paulson
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diff
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|
459 |
|
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paulson
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|
460 |
(*This type checking rule synthesises a sufficiently large set for f*) |
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|
461 |
val [major] = goal LList.thy "M: llist(A) ==> Lmap f M: llist(f``A)"; |
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paulson
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|
462 |
by (rtac (major RS Lmap_type) 1); |
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paulson
parents:
diff
changeset
|
463 |
by (etac imageI 1); |
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paulson
parents:
diff
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|
464 |
qed "Lmap_type2"; |
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paulson
parents:
diff
changeset
|
465 |
|
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|
466 |
(** Two easy results about Lmap **) |
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paulson
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diff
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|
467 |
|
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|
468 |
val [prem] = goalw LList.thy [o_def] |
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|
469 |
"M: llist(A) ==> Lmap (f o g) M = Lmap f (Lmap g M)"; |
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|
470 |
by (rtac (prem RS imageI RS LList_equalityI) 1); |
4160 | 471 |
by Safe_tac; |
3120
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|
472 |
by (etac llist.elim 1); |
4521 | 473 |
by (ALLGOALS Asm_simp_tac); |
3120
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|
474 |
by (REPEAT (ares_tac [LListD_Fun_NIL_I, imageI, UnI1, |
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|
475 |
rangeI RS LListD_Fun_CONS_I] 1)); |
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paulson
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|
476 |
qed "Lmap_compose"; |
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|
477 |
|
3842 | 478 |
val [prem] = goal LList.thy "M: llist(A) ==> Lmap (%x. x) M = M"; |
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|
479 |
by (rtac (prem RS imageI RS LList_equalityI) 1); |
4160 | 480 |
by Safe_tac; |
3120
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|
481 |
by (etac llist.elim 1); |
4521 | 482 |
by (ALLGOALS Asm_simp_tac); |
3120
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|
483 |
by (REPEAT (ares_tac [LListD_Fun_NIL_I, imageI RS UnI1, |
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paulson
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|
484 |
rangeI RS LListD_Fun_CONS_I] 1)); |
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|
485 |
qed "Lmap_ident"; |
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|
486 |
|
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paulson
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|
487 |
|
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|
488 |
(*** Lappend -- its two arguments cause some complications! ***) |
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|
489 |
|
5069 | 490 |
Goalw [Lappend_def] "Lappend NIL NIL = NIL"; |
3120
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|
491 |
by (rtac (LList_corec RS trans) 1); |
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|
492 |
by (Simp_tac 1); |
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|
493 |
qed "Lappend_NIL_NIL"; |
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|
494 |
|
5069 | 495 |
Goalw [Lappend_def] |
3120
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|
496 |
"Lappend NIL (CONS N N') = CONS N (Lappend NIL N')"; |
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paulson
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|
497 |
by (rtac (LList_corec RS trans) 1); |
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paulson
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|
498 |
by (Simp_tac 1); |
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paulson
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|
499 |
qed "Lappend_NIL_CONS"; |
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|
500 |
|
5069 | 501 |
Goalw [Lappend_def] |
3120
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|
502 |
"Lappend (CONS M M') N = CONS M (Lappend M' N)"; |
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paulson
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changeset
|
503 |
by (rtac (LList_corec RS trans) 1); |
c58423c20740
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paulson
parents:
diff
changeset
|
504 |
by (Simp_tac 1); |
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paulson
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|
505 |
qed "Lappend_CONS"; |
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paulson
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|
506 |
|
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paulson
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|
507 |
Addsimps [llist.NIL_I, Lappend_NIL_NIL, Lappend_NIL_CONS, |
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paulson
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|
508 |
Lappend_CONS, LListD_Fun_CONS_I, range_eqI, image_eqI]; |
4521 | 509 |
|
3120
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|
510 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
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5102
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|
511 |
Goal "M: llist(A) ==> Lappend NIL M = M"; |
3120
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paulson
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|
512 |
by (etac LList_fun_equalityI 1); |
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paulson
parents:
diff
changeset
|
513 |
by (ALLGOALS Asm_simp_tac); |
c58423c20740
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paulson
parents:
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|
514 |
qed "Lappend_NIL"; |
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paulson
parents:
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|
515 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
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5102
diff
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|
516 |
Goal "M: llist(A) ==> Lappend M NIL = M"; |
3120
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|
517 |
by (etac LList_fun_equalityI 1); |
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paulson
parents:
diff
changeset
|
518 |
by (ALLGOALS Asm_simp_tac); |
c58423c20740
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paulson
parents:
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changeset
|
519 |
qed "Lappend_NIL2"; |
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paulson
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|
520 |
|
4521 | 521 |
Addsimps [Lappend_NIL, Lappend_NIL2]; |
522 |
||
523 |
||
3120
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|
524 |
(** Alternative type-checking proofs for Lappend **) |
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|
525 |
|
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paulson
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|
526 |
(*weak co-induction: bisimulation and case analysis on both variables*) |
5278 | 527 |
Goal "[| M: llist(A); N: llist(A) |] ==> Lappend M N: llist(A)"; |
3120
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|
528 |
by (res_inst_tac |
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paulson
parents:
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|
529 |
[("X", "UN u:llist(A). UN v: llist(A). {Lappend u v}")] llist_coinduct 1); |
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paulson
parents:
diff
changeset
|
530 |
by (Fast_tac 1); |
4160 | 531 |
by Safe_tac; |
5102 | 532 |
by (eres_inst_tac [("aa", "u")] llist.elim 1); |
533 |
by (eres_inst_tac [("aa", "v")] llist.elim 1); |
|
4521 | 534 |
by (ALLGOALS Asm_simp_tac); |
535 |
by (Blast_tac 1); |
|
3120
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|
536 |
qed "Lappend_type"; |
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paulson
parents:
diff
changeset
|
537 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
538 |
(*strong co-induction: bisimulation and case analysis on one variable*) |
5278 | 539 |
Goal "[| M: llist(A); N: llist(A) |] ==> Lappend M N: llist(A)"; |
3842 | 540 |
by (res_inst_tac [("X", "(%u. Lappend u N)``llist(A)")] llist_coinduct 1); |
3120
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paulson
parents:
diff
changeset
|
541 |
by (etac imageI 1); |
4521 | 542 |
by (rtac image_subsetI 1); |
5102 | 543 |
by (eres_inst_tac [("aa", "x")] llist.elim 1); |
4521 | 544 |
by (asm_simp_tac (simpset() addsimps [list_Fun_llist_I]) 1); |
3120
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paulson
parents:
diff
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|
545 |
by (Asm_simp_tac 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
546 |
qed "Lappend_type"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
547 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
548 |
(**** Lazy lists as the type 'a llist -- strongly typed versions of above ****) |
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paulson
parents:
diff
changeset
|
549 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
550 |
(** llist_case: case analysis for 'a llist **) |
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paulson
parents:
diff
changeset
|
551 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
552 |
Addsimps ([Abs_llist_inverse, Rep_llist_inverse, |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
553 |
Rep_llist, rangeI, inj_Leaf, inv_f_f] @ llist.intrs); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
554 |
|
5069 | 555 |
Goalw [llist_case_def,LNil_def] "llist_case c d LNil = c"; |
3120
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paulson
parents:
diff
changeset
|
556 |
by (Simp_tac 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
557 |
qed "llist_case_LNil"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
558 |
|
5069 | 559 |
Goalw [llist_case_def,LCons_def] |
3120
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|
560 |
"llist_case c d (LCons M N) = d M N"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
561 |
by (Simp_tac 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
562 |
qed "llist_case_LCons"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
563 |
|
c58423c20740
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paulson
parents:
diff
changeset
|
564 |
(*Elimination is case analysis, not induction.*) |
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paulson
parents:
diff
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|
565 |
val [prem1,prem2] = goalw LList.thy [NIL_def,CONS_def] |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
566 |
"[| l=LNil ==> P; !!x l'. l=LCons x l' ==> P \ |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
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|
567 |
\ |] ==> P"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
568 |
by (rtac (Rep_llist RS llist.elim) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
569 |
by (rtac (inj_Rep_llist RS injD RS prem1) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
570 |
by (stac Rep_llist_LNil 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
571 |
by (assume_tac 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
572 |
by (etac rangeE 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
573 |
by (rtac (inj_Rep_llist RS injD RS prem2) 1); |
4521 | 574 |
by (asm_simp_tac (simpset() delsimps [CONS_CONS_eq] |
575 |
addsimps [Rep_llist_LCons]) 1); |
|
3120
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paulson
parents:
diff
changeset
|
576 |
by (etac (Abs_llist_inverse RS ssubst) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
577 |
by (rtac refl 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
578 |
qed "llistE"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
579 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
580 |
(** llist_corec: corecursion for 'a llist **) |
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New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
581 |
|
5069 | 582 |
Goalw [llist_corec_def,LNil_def,LCons_def] |
3120
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|
583 |
"llist_corec a f = sum_case (%u. LNil) \ |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
584 |
\ (split(%z w. LCons z (llist_corec w f))) (f a)"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
585 |
by (stac LList_corec 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
586 |
by (res_inst_tac [("s","f(a)")] sumE 1); |
4089 | 587 |
by (asm_simp_tac (simpset() addsimps [LList_corec_type2]) 1); |
3120
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New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
588 |
by (res_inst_tac [("p","y")] PairE 1); |
4089 | 589 |
by (asm_simp_tac (simpset() addsimps [LList_corec_type2]) 1); |
3120
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paulson
parents:
diff
changeset
|
590 |
(*FIXME: correct case splits usd to be found automatically: |
4089 | 591 |
by (ASM_SIMP_TAC(simpset() addsimps [LList_corec_type2]) 1);*) |
3120
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paulson
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diff
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|
592 |
qed "llist_corec"; |
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New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
593 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
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|
594 |
(*definitional version of same*) |
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New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
595 |
val [rew] = goal LList.thy |
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paulson
parents:
diff
changeset
|
596 |
"[| !!x. h(x) == llist_corec x f |] ==> \ |
3842 | 597 |
\ h(a) = sum_case (%u. LNil) (split(%z w. LCons z (h w))) (f a)"; |
3120
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parents:
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|
598 |
by (rewtac rew); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
599 |
by (rtac llist_corec 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
600 |
qed "def_llist_corec"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
601 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
602 |
(**** Proofs about type 'a llist functions ****) |
c58423c20740
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paulson
parents:
diff
changeset
|
603 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
604 |
(*** Deriving llist_equalityI -- llist equality is a bisimulation ***) |
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New directory to contain examples of (co)inductive definitions
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|
605 |
|
5069 | 606 |
Goalw [LListD_Fun_def] |
5148
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More tidying and removal of "\!\!... from Goal commands
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parents:
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changeset
|
607 |
"r <= (llist A) Times (llist A) ==> \ |
3120
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parents:
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|
608 |
\ LListD_Fun (diag A) r <= (llist A) Times (llist A)"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
609 |
by (stac llist_unfold 1); |
4089 | 610 |
by (simp_tac (simpset() addsimps [NIL_def, CONS_def]) 1); |
3120
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New directory to contain examples of (co)inductive definitions
paulson
parents:
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|
611 |
by (Fast_tac 1); |
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paulson
parents:
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|
612 |
qed "LListD_Fun_subset_Sigma_llist"; |
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New directory to contain examples of (co)inductive definitions
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diff
changeset
|
613 |
|
5278 | 614 |
Goal "prod_fun Rep_llist Rep_llist `` r <= \ |
3120
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paulson
parents:
diff
changeset
|
615 |
\ (llist(range Leaf)) Times (llist(range Leaf))"; |
4521 | 616 |
by (fast_tac (claset() delrules [image_subsetI] |
617 |
addIs [Rep_llist]) 1); |
|
3120
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paulson
parents:
diff
changeset
|
618 |
qed "subset_Sigma_llist"; |
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paulson
parents:
diff
changeset
|
619 |
|
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|
620 |
val [prem] = goal LList.thy |
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New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
621 |
"r <= (llist(range Leaf)) Times (llist(range Leaf)) ==> \ |
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New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
622 |
\ prod_fun (Rep_llist o Abs_llist) (Rep_llist o Abs_llist) `` r <= r"; |
4160 | 623 |
by Safe_tac; |
3120
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New directory to contain examples of (co)inductive definitions
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parents:
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changeset
|
624 |
by (rtac (prem RS subsetD RS SigmaE2) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
625 |
by (assume_tac 1); |
4521 | 626 |
by (asm_simp_tac (simpset() addsimps [Abs_llist_inverse]) 1); |
3120
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New directory to contain examples of (co)inductive definitions
paulson
parents:
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|
627 |
qed "prod_fun_lemma"; |
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paulson
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diff
changeset
|
628 |
|
5278 | 629 |
Goal "prod_fun Rep_llist Rep_llist `` range(%x. (x, x)) = \ |
3120
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parents:
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changeset
|
630 |
\ diag(llist(range Leaf))"; |
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New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
631 |
by (rtac equalityI 1); |
4089 | 632 |
by (fast_tac (claset() addIs [Rep_llist]) 1); |
4818
90dab9f7d81e
split_all_tac is now added to claset() _before_ other safe tactics
oheimb
parents:
4521
diff
changeset
|
633 |
by (fast_tac (claset() delSWrapper "split_all_tac" |
90dab9f7d81e
split_all_tac is now added to claset() _before_ other safe tactics
oheimb
parents:
4521
diff
changeset
|
634 |
addSEs [Abs_llist_inverse RS subst]) 1); |
3120
c58423c20740
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parents:
diff
changeset
|
635 |
qed "prod_fun_range_eq_diag"; |
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paulson
parents:
diff
changeset
|
636 |
|
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paulson
parents:
diff
changeset
|
637 |
(*Surprisingly hard to prove. Used with lfilter*) |
5069 | 638 |
Goalw [llistD_Fun_def, prod_fun_def] |
5148
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More tidying and removal of "\!\!... from Goal commands
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parents:
5143
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changeset
|
639 |
"A<=B ==> llistD_Fun A <= llistD_Fun B"; |
4477
b3e5857d8d99
New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents:
4160
diff
changeset
|
640 |
by Auto_tac; |
3120
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New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
641 |
by (rtac image_eqI 1); |
4089 | 642 |
by (fast_tac (claset() addss (simpset())) 1); |
643 |
by (blast_tac (claset() addIs [impOfSubs LListD_Fun_mono]) 1); |
|
3120
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paulson
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|
644 |
qed "llistD_Fun_mono"; |
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paulson
parents:
diff
changeset
|
645 |
|
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paulson
parents:
diff
changeset
|
646 |
(** To show two llists are equal, exhibit a bisimulation! |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
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|
647 |
[also admits true equality] **) |
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paulson
parents:
diff
changeset
|
648 |
val [prem1,prem2] = goalw LList.thy [llistD_Fun_def] |
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New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
649 |
"[| (l1,l2) : r; r <= llistD_Fun(r Un range(%x.(x,x))) |] ==> l1=l2"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
650 |
by (rtac (inj_Rep_llist RS injD) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
651 |
by (res_inst_tac [("r", "prod_fun Rep_llist Rep_llist ``r"), |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
652 |
("A", "range(Leaf)")] |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
653 |
LList_equalityI 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
654 |
by (rtac (prem1 RS prod_fun_imageI) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
655 |
by (rtac (prem2 RS image_mono RS subset_trans) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
656 |
by (rtac (image_compose RS subst) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
657 |
by (rtac (prod_fun_compose RS subst) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
658 |
by (stac image_Un 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
659 |
by (stac prod_fun_range_eq_diag 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
660 |
by (rtac (LListD_Fun_subset_Sigma_llist RS prod_fun_lemma) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
661 |
by (rtac (subset_Sigma_llist RS Un_least) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
662 |
by (rtac diag_subset_Sigma 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
663 |
qed "llist_equalityI"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
664 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
665 |
(** Rules to prove the 2nd premise of llist_equalityI **) |
5069 | 666 |
Goalw [llistD_Fun_def,LNil_def] "(LNil,LNil) : llistD_Fun(r)"; |
3120
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New directory to contain examples of (co)inductive definitions
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parents:
diff
changeset
|
667 |
by (rtac (LListD_Fun_NIL_I RS prod_fun_imageI) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
668 |
qed "llistD_Fun_LNil_I"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
669 |
|
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New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
670 |
val [prem] = goalw LList.thy [llistD_Fun_def,LCons_def] |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
671 |
"(l1,l2):r ==> (LCons x l1, LCons x l2) : llistD_Fun(r)"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
672 |
by (rtac (rangeI RS LListD_Fun_CONS_I RS prod_fun_imageI) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
673 |
by (rtac (prem RS prod_fun_imageI) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
674 |
qed "llistD_Fun_LCons_I"; |
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New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
675 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
676 |
(*Utilise the "strong" part, i.e. gfp(f)*) |
5069 | 677 |
Goalw [llistD_Fun_def] |
5148
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More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5143
diff
changeset
|
678 |
"(l,l) : llistD_Fun(r Un range(%x.(x,x)))"; |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
679 |
by (rtac (Rep_llist_inverse RS subst) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
680 |
by (rtac prod_fun_imageI 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
681 |
by (stac image_Un 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
682 |
by (stac prod_fun_range_eq_diag 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
683 |
by (rtac (Rep_llist RS LListD_Fun_diag_I) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
684 |
qed "llistD_Fun_range_I"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
685 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
686 |
(*A special case of list_equality for functions over lazy lists*) |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
687 |
val [prem1,prem2] = goal LList.thy |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
688 |
"[| f(LNil)=g(LNil); \ |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
689 |
\ !!x l. (f(LCons x l),g(LCons x l)) : \ |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
690 |
\ llistD_Fun(range(%u. (f(u),g(u))) Un range(%v. (v,v))) \ |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
691 |
\ |] ==> f(l) = (g(l :: 'a llist) :: 'b llist)"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
692 |
by (res_inst_tac [("r", "range(%u. (f(u),g(u)))")] llist_equalityI 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
693 |
by (rtac rangeI 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
694 |
by (rtac subsetI 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
695 |
by (etac rangeE 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
696 |
by (etac ssubst 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
697 |
by (res_inst_tac [("l", "u")] llistE 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
698 |
by (etac ssubst 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
699 |
by (stac prem1 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
700 |
by (rtac llistD_Fun_range_I 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
701 |
by (etac ssubst 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
702 |
by (rtac prem2 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
703 |
qed "llist_fun_equalityI"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
704 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
705 |
(*simpset for llist bisimulations*) |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
706 |
Addsimps [llist_case_LNil, llist_case_LCons, |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
707 |
llistD_Fun_LNil_I, llistD_Fun_LCons_I]; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
708 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
709 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
710 |
(*** The functional "lmap" ***) |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
711 |
|
5069 | 712 |
Goal "lmap f LNil = LNil"; |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
713 |
by (rtac (lmap_def RS def_llist_corec RS trans) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
714 |
by (Simp_tac 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
715 |
qed "lmap_LNil"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
716 |
|
5069 | 717 |
Goal "lmap f (LCons M N) = LCons (f M) (lmap f N)"; |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
718 |
by (rtac (lmap_def RS def_llist_corec RS trans) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
719 |
by (Simp_tac 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
720 |
qed "lmap_LCons"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
721 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
722 |
Addsimps [lmap_LNil, lmap_LCons]; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
723 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
724 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
725 |
(** Two easy results about lmap **) |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
726 |
|
5069 | 727 |
Goal "lmap (f o g) l = lmap f (lmap g l)"; |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
728 |
by (res_inst_tac [("l","l")] llist_fun_equalityI 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
729 |
by (ALLGOALS Simp_tac); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
730 |
qed "lmap_compose"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
731 |
|
5069 | 732 |
Goal "lmap (%x. x) l = l"; |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
733 |
by (res_inst_tac [("l","l")] llist_fun_equalityI 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
734 |
by (ALLGOALS Simp_tac); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
735 |
qed "lmap_ident"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
736 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
737 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
738 |
(*** iterates -- llist_fun_equalityI cannot be used! ***) |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
739 |
|
5069 | 740 |
Goal "iterates f x = LCons x (iterates f (f x))"; |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
741 |
by (rtac (iterates_def RS def_llist_corec RS trans) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
742 |
by (Simp_tac 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
743 |
qed "iterates"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
744 |
|
5069 | 745 |
Goal "lmap f (iterates f x) = iterates f (f x)"; |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
746 |
by (res_inst_tac [("r", "range(%u.(lmap f (iterates f u),iterates f (f u)))")] |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
747 |
llist_equalityI 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
748 |
by (rtac rangeI 1); |
4160 | 749 |
by Safe_tac; |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
750 |
by (res_inst_tac [("x1", "f(u)")] (iterates RS ssubst) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
751 |
by (res_inst_tac [("x1", "u")] (iterates RS ssubst) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
752 |
by (Simp_tac 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
753 |
qed "lmap_iterates"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
754 |
|
5069 | 755 |
Goal "iterates f x = LCons x (lmap f (iterates f x))"; |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
756 |
by (stac lmap_iterates 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
757 |
by (rtac iterates 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
758 |
qed "iterates_lmap"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
759 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
760 |
(*** A rather complex proof about iterates -- cf Andy Pitts ***) |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
761 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
762 |
(** Two lemmas about natrec n x (%m.g), which is essentially (g^n)(x) **) |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
763 |
|
5278 | 764 |
Goal "nat_rec (LCons b l) (%m. lmap(f)) n = \ |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
765 |
\ LCons (nat_rec b (%m. f) n) (nat_rec l (%m. lmap(f)) n)"; |
5184 | 766 |
by (induct_tac "n" 1); |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
767 |
by (ALLGOALS Asm_simp_tac); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
768 |
qed "fun_power_lmap"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
769 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
770 |
goal Nat.thy "nat_rec (g x) (%m. g) n = nat_rec x (%m. g) (Suc n)"; |
5184 | 771 |
by (induct_tac "n" 1); |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
772 |
by (ALLGOALS Asm_simp_tac); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
773 |
qed "fun_power_Suc"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
774 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
775 |
val Pair_cong = read_instantiate_sg (sign_of Prod.thy) |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
776 |
[("f","Pair")] (standard(refl RS cong RS cong)); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
777 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
778 |
(*The bisimulation consists of {(lmap(f)^n (h(u)), lmap(f)^n (iterates(f,u)))} |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
779 |
for all u and all n::nat.*) |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
780 |
val [prem] = goal LList.thy |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
781 |
"(!!x. h(x) = LCons x (lmap f (h x))) ==> h = iterates(f)"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
782 |
by (rtac ext 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
783 |
by (res_inst_tac [("r", |
3842 | 784 |
"UN u. range(%n. (nat_rec (h u) (%m y. lmap f y) n, \ |
785 |
\ nat_rec (iterates f u) (%m y. lmap f y) n))")] |
|
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
786 |
llist_equalityI 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
787 |
by (REPEAT (resolve_tac [UN1_I, range_eqI, Pair_cong, nat_rec_0 RS sym] 1)); |
4160 | 788 |
by (Clarify_tac 1); |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
789 |
by (stac iterates 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
790 |
by (stac prem 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
791 |
by (stac fun_power_lmap 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
792 |
by (stac fun_power_lmap 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
793 |
by (rtac llistD_Fun_LCons_I 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
794 |
by (rtac (lmap_iterates RS subst) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
795 |
by (stac fun_power_Suc 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
796 |
by (stac fun_power_Suc 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
797 |
by (rtac (UN1_I RS UnI1) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
798 |
by (rtac rangeI 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
799 |
qed "iterates_equality"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
800 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
801 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
802 |
(*** lappend -- its two arguments cause some complications! ***) |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
803 |
|
5069 | 804 |
Goalw [lappend_def] "lappend LNil LNil = LNil"; |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
805 |
by (rtac (llist_corec RS trans) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
806 |
by (Simp_tac 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
807 |
qed "lappend_LNil_LNil"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
808 |
|
5069 | 809 |
Goalw [lappend_def] |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
810 |
"lappend LNil (LCons l l') = LCons l (lappend LNil l')"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
811 |
by (rtac (llist_corec RS trans) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
812 |
by (Simp_tac 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
813 |
qed "lappend_LNil_LCons"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
814 |
|
5069 | 815 |
Goalw [lappend_def] |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
816 |
"lappend (LCons l l') N = LCons l (lappend l' N)"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
817 |
by (rtac (llist_corec RS trans) 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
818 |
by (Simp_tac 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
819 |
qed "lappend_LCons"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
820 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
821 |
Addsimps [lappend_LNil_LNil, lappend_LNil_LCons, lappend_LCons]; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
822 |
|
5069 | 823 |
Goal "lappend LNil l = l"; |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
824 |
by (res_inst_tac [("l","l")] llist_fun_equalityI 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
825 |
by (ALLGOALS Simp_tac); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
826 |
qed "lappend_LNil"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
827 |
|
5069 | 828 |
Goal "lappend l LNil = l"; |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
829 |
by (res_inst_tac [("l","l")] llist_fun_equalityI 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
830 |
by (ALLGOALS Simp_tac); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
831 |
qed "lappend_LNil2"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
832 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
833 |
Addsimps [lappend_LNil, lappend_LNil2]; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
834 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
835 |
(*The infinite first argument blocks the second*) |
5069 | 836 |
Goal "lappend (iterates f x) N = iterates f x"; |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
837 |
by (res_inst_tac [("r", "range(%u.(lappend (iterates f u) N,iterates f u))")] |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
838 |
llist_equalityI 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
839 |
by (rtac rangeI 1); |
4160 | 840 |
by Safe_tac; |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
841 |
by (stac iterates 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
842 |
by (Simp_tac 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
843 |
qed "lappend_iterates"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
844 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
845 |
(** Two proofs that lmap distributes over lappend **) |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
846 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
847 |
(*Long proof requiring case analysis on both both arguments*) |
5069 | 848 |
Goal "lmap f (lappend l n) = lappend (lmap f l) (lmap f n)"; |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
849 |
by (res_inst_tac |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
850 |
[("r", |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
851 |
"UN n. range(%l.(lmap f (lappend l n),lappend (lmap f l) (lmap f n)))")] |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
852 |
llist_equalityI 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
853 |
by (rtac UN1_I 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
854 |
by (rtac rangeI 1); |
4160 | 855 |
by Safe_tac; |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
856 |
by (res_inst_tac [("l", "l")] llistE 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
857 |
by (res_inst_tac [("l", "n")] llistE 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
858 |
by (ALLGOALS Asm_simp_tac); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
859 |
by (REPEAT_SOME (ares_tac [llistD_Fun_LCons_I, UN1_I RS UnI1, rangeI])); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
860 |
qed "lmap_lappend_distrib"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
861 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
862 |
(*Shorter proof of theorem above using llist_equalityI as strong coinduction*) |
5069 | 863 |
Goal "lmap f (lappend l n) = lappend (lmap f l) (lmap f n)"; |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
864 |
by (res_inst_tac [("l","l")] llist_fun_equalityI 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
865 |
by (Simp_tac 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
866 |
by (Simp_tac 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
867 |
qed "lmap_lappend_distrib"; |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
868 |
|
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
869 |
(*Without strong coinduction, three case analyses might be needed*) |
5069 | 870 |
Goal "lappend (lappend l1 l2) l3 = lappend l1 (lappend l2 l3)"; |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
871 |
by (res_inst_tac [("l","l1")] llist_fun_equalityI 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
872 |
by (Simp_tac 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
873 |
by (Simp_tac 1); |
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
874 |
qed "lappend_assoc"; |