src/HOL/Induct/LList.ML
author paulson
Wed Jul 15 10:15:13 1998 +0200 (1998-07-15)
changeset 5143 b94cd208f073
parent 5102 8c782c25a11e
child 5148 74919e8f221c
permissions -rw-r--r--
Removal of leading "\!\!..." from most Goal commands
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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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(** 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. [| 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 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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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 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 [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 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 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_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
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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_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
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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;
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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 (res_inst_tac [("n", "n")] natE 1);
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by (Asm_simp_tac 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]) 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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    "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 "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 (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 [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. [| 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. [| 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 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 "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 
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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 
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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;
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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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(*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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(*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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   299
by (Simp_tac 1);
paulson@3120
   300
qed "ntrunc_CONS";
paulson@3120
   301
paulson@4521
   302
Addsimps [ntrunc_one_CONS, ntrunc_CONS];
paulson@4521
   303
paulson@4521
   304
paulson@3120
   305
val [prem1,prem2] = goal LList.thy
wenzelm@3842
   306
 "[| !!x. h1(x) = sum_case (%u. NIL) (split(%z w. CONS z (h1 w))) (f x);  \
wenzelm@3842
   307
\    !!x. h2(x) = sum_case (%u. NIL) (split(%z w. CONS z (h2 w))) (f x) |]\
paulson@3120
   308
\ ==> h1=h2";
paulson@3120
   309
by (rtac (ntrunc_equality RS ext) 1);
paulson@3120
   310
by (rename_tac "x k" 1);
paulson@3120
   311
by (res_inst_tac [("x", "x")] spec 1);
paulson@3120
   312
by (res_inst_tac [("n", "k")] less_induct 1);
paulson@3120
   313
by (rename_tac "n" 1);
paulson@3120
   314
by (rtac allI 1);
paulson@3120
   315
by (rename_tac "y" 1);
paulson@3120
   316
by (stac prem1 1);
paulson@3120
   317
by (stac prem2 1);
nipkow@4831
   318
by (Simp_tac 1);
paulson@3120
   319
by (strip_tac 1);
paulson@3120
   320
by (res_inst_tac [("n", "n")] natE 1);
paulson@3120
   321
by (rename_tac "m" 2);
paulson@3120
   322
by (res_inst_tac [("n", "m")] natE 2);
paulson@4521
   323
by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq])));
paulson@3120
   324
result();
paulson@3120
   325
paulson@3120
   326
paulson@3120
   327
(*** Lconst -- defined directly using lfp, but equivalent to a LList_corec ***)
paulson@3120
   328
wenzelm@5069
   329
Goal "mono(CONS(M))";
paulson@3120
   330
by (REPEAT (ares_tac [monoI, subset_refl, CONS_mono] 1));
paulson@3120
   331
qed "Lconst_fun_mono";
paulson@3120
   332
paulson@3120
   333
(* Lconst(M) = CONS M (Lconst M) *)
paulson@3120
   334
bind_thm ("Lconst", (Lconst_fun_mono RS (Lconst_def RS def_lfp_Tarski)));
paulson@3120
   335
paulson@3120
   336
(*A typical use of co-induction to show membership in the gfp.
paulson@3120
   337
  The containing set is simply the singleton {Lconst(M)}. *)
paulson@5143
   338
Goal "M:A ==> Lconst(M): llist(A)";
paulson@3120
   339
by (rtac (singletonI RS llist_coinduct) 1);
paulson@4160
   340
by Safe_tac;
paulson@3120
   341
by (res_inst_tac [("P", "%u. u: ?A")] (Lconst RS ssubst) 1);
paulson@3120
   342
by (REPEAT (ares_tac [list_Fun_CONS_I, singletonI, UnI1] 1));
paulson@3120
   343
qed "Lconst_type";
paulson@3120
   344
wenzelm@5069
   345
Goal "Lconst(M) = LList_corec M (%x. Inr((x,x)))";
paulson@3120
   346
by (rtac (equals_LList_corec RS fun_cong) 1);
paulson@3120
   347
by (Simp_tac 1);
paulson@3120
   348
by (rtac Lconst 1);
paulson@3120
   349
qed "Lconst_eq_LList_corec";
paulson@3120
   350
paulson@3120
   351
(*Thus we could have used gfp in the definition of Lconst*)
wenzelm@5069
   352
Goal "gfp(%N. CONS M N) = LList_corec M (%x. Inr((x,x)))";
paulson@3120
   353
by (rtac (equals_LList_corec RS fun_cong) 1);
paulson@3120
   354
by (Simp_tac 1);
paulson@3120
   355
by (rtac (Lconst_fun_mono RS gfp_Tarski) 1);
paulson@3120
   356
qed "gfp_Lconst_eq_LList_corec";
paulson@3120
   357
paulson@3120
   358
paulson@3120
   359
(*** Isomorphisms ***)
paulson@3120
   360
wenzelm@5069
   361
Goal "inj(Rep_llist)";
paulson@3120
   362
by (rtac inj_inverseI 1);
paulson@3120
   363
by (rtac Rep_llist_inverse 1);
paulson@3120
   364
qed "inj_Rep_llist";
paulson@3120
   365
wenzelm@5069
   366
Goal "inj_on Abs_llist (llist(range Leaf))";
nipkow@4831
   367
by (rtac inj_on_inverseI 1);
paulson@3120
   368
by (etac Abs_llist_inverse 1);
nipkow@4831
   369
qed "inj_on_Abs_llist";
paulson@3120
   370
paulson@3120
   371
(** Distinctness of constructors **)
paulson@3120
   372
wenzelm@5069
   373
Goalw [LNil_def,LCons_def] "~ LCons x xs = LNil";
nipkow@4831
   374
by (rtac (CONS_not_NIL RS (inj_on_Abs_llist RS inj_on_contraD)) 1);
paulson@3120
   375
by (REPEAT (resolve_tac (llist.intrs @ [rangeI, Rep_llist]) 1));
paulson@3120
   376
qed "LCons_not_LNil";
paulson@3120
   377
paulson@3120
   378
bind_thm ("LNil_not_LCons", LCons_not_LNil RS not_sym);
paulson@3120
   379
paulson@3120
   380
AddIffs [LCons_not_LNil, LNil_not_LCons];
paulson@3120
   381
paulson@3120
   382
paulson@3120
   383
(** llist constructors **)
paulson@3120
   384
wenzelm@5069
   385
Goalw [LNil_def]
paulson@3120
   386
    "Rep_llist(LNil) = NIL";
paulson@3120
   387
by (rtac (llist.NIL_I RS Abs_llist_inverse) 1);
paulson@3120
   388
qed "Rep_llist_LNil";
paulson@3120
   389
wenzelm@5069
   390
Goalw [LCons_def]
paulson@3120
   391
    "Rep_llist(LCons x l) = CONS (Leaf x) (Rep_llist l)";
paulson@3120
   392
by (REPEAT (resolve_tac [llist.CONS_I RS Abs_llist_inverse,
paulson@3120
   393
                         rangeI, Rep_llist] 1));
paulson@3120
   394
qed "Rep_llist_LCons";
paulson@3120
   395
paulson@3120
   396
(** Injectiveness of CONS and LCons **)
paulson@3120
   397
wenzelm@5069
   398
Goalw [CONS_def] "(CONS M N=CONS M' N') = (M=M' & N=N')";
wenzelm@4089
   399
by (fast_tac (claset() addSEs [Scons_inject]) 1);
paulson@3120
   400
qed "CONS_CONS_eq2";
paulson@3120
   401
paulson@3120
   402
bind_thm ("CONS_inject", (CONS_CONS_eq RS iffD1 RS conjE));
paulson@3120
   403
paulson@3120
   404
paulson@3120
   405
(*For reasoning about abstract llist constructors*)
paulson@3120
   406
paulson@3120
   407
AddIs ([Rep_llist]@llist.intrs);
nipkow@4831
   408
AddSDs [inj_on_Abs_llist RS inj_onD,
paulson@3120
   409
        inj_Rep_llist RS injD, Leaf_inject];
paulson@3120
   410
wenzelm@5069
   411
Goalw [LCons_def] "(LCons x xs=LCons y ys) = (x=y & xs=ys)";
paulson@3120
   412
by (Fast_tac 1);
paulson@3120
   413
qed "LCons_LCons_eq";
paulson@3120
   414
paulson@3120
   415
AddIffs [LCons_LCons_eq];
paulson@3120
   416
paulson@3120
   417
val [major] = goal LList.thy "CONS M N: llist(A) ==> M: A & N: llist(A)";
paulson@3120
   418
by (rtac (major RS llist.elim) 1);
paulson@3120
   419
by (etac CONS_neq_NIL 1);
paulson@3120
   420
by (Fast_tac 1);
paulson@3120
   421
qed "CONS_D2";
paulson@3120
   422
paulson@3120
   423
paulson@3120
   424
(****** Reasoning about llist(A) ******)
paulson@3120
   425
paulson@3120
   426
Addsimps [List_case_NIL, List_case_CONS];
paulson@3120
   427
paulson@3120
   428
(*A special case of list_equality for functions over lazy lists*)
paulson@3120
   429
val [Mlist,gMlist,NILcase,CONScase] = goal LList.thy
paulson@3120
   430
 "[| M: llist(A); g(NIL): llist(A);                             \
paulson@3120
   431
\    f(NIL)=g(NIL);                                             \
paulson@3120
   432
\    !!x l. [| x:A;  l: llist(A) |] ==>                         \
paulson@3120
   433
\           (f(CONS x l),g(CONS x l)) :                         \
paulson@3120
   434
\               LListD_Fun (diag A) ((%u.(f(u),g(u)))``llist(A) Un  \
paulson@3120
   435
\                                   diag(llist(A)))             \
paulson@3120
   436
\ |] ==> f(M) = g(M)";
paulson@3120
   437
by (rtac LList_equalityI 1);
paulson@3120
   438
by (rtac (Mlist RS imageI) 1);
paulson@4521
   439
by (rtac image_subsetI 1);
paulson@3120
   440
by (etac llist.elim 1);
paulson@3120
   441
by (etac ssubst 1);
paulson@3120
   442
by (stac NILcase 1);
paulson@3120
   443
by (rtac (gMlist RS LListD_Fun_diag_I) 1);
paulson@3120
   444
by (etac ssubst 1);
paulson@3120
   445
by (REPEAT (ares_tac [CONScase] 1));
paulson@3120
   446
qed "LList_fun_equalityI";
paulson@3120
   447
paulson@3120
   448
paulson@3120
   449
(*** The functional "Lmap" ***)
paulson@3120
   450
wenzelm@5069
   451
Goal "Lmap f NIL = NIL";
paulson@3120
   452
by (rtac (Lmap_def RS def_LList_corec RS trans) 1);
paulson@3120
   453
by (Simp_tac 1);
paulson@3120
   454
qed "Lmap_NIL";
paulson@3120
   455
wenzelm@5069
   456
Goal "Lmap f (CONS M N) = CONS (f M) (Lmap f N)";
paulson@3120
   457
by (rtac (Lmap_def RS def_LList_corec RS trans) 1);
paulson@3120
   458
by (Simp_tac 1);
paulson@3120
   459
qed "Lmap_CONS";
paulson@3120
   460
paulson@4521
   461
Addsimps [Lmap_NIL, Lmap_CONS];
paulson@4521
   462
paulson@3120
   463
(*Another type-checking proof by coinduction*)
paulson@3120
   464
val [major,minor] = goal LList.thy
paulson@3120
   465
    "[| M: llist(A);  !!x. x:A ==> f(x):B |] ==> Lmap f M: llist(B)";
paulson@3120
   466
by (rtac (major RS imageI RS llist_coinduct) 1);
paulson@4160
   467
by Safe_tac;
paulson@3120
   468
by (etac llist.elim 1);
paulson@4521
   469
by (ALLGOALS Asm_simp_tac);
paulson@3120
   470
by (REPEAT (ares_tac [list_Fun_NIL_I, list_Fun_CONS_I, 
paulson@3120
   471
                      minor, imageI, UnI1] 1));
paulson@3120
   472
qed "Lmap_type";
paulson@3120
   473
paulson@3120
   474
(*This type checking rule synthesises a sufficiently large set for f*)
paulson@3120
   475
val [major] = goal LList.thy  "M: llist(A) ==> Lmap f M: llist(f``A)";
paulson@3120
   476
by (rtac (major RS Lmap_type) 1);
paulson@3120
   477
by (etac imageI 1);
paulson@3120
   478
qed "Lmap_type2";
paulson@3120
   479
paulson@3120
   480
(** Two easy results about Lmap **)
paulson@3120
   481
paulson@3120
   482
val [prem] = goalw LList.thy [o_def]
paulson@3120
   483
    "M: llist(A) ==> Lmap (f o g) M = Lmap f (Lmap g M)";
paulson@3120
   484
by (rtac (prem RS imageI RS LList_equalityI) 1);
paulson@4160
   485
by Safe_tac;
paulson@3120
   486
by (etac llist.elim 1);
paulson@4521
   487
by (ALLGOALS Asm_simp_tac);
paulson@3120
   488
by (REPEAT (ares_tac [LListD_Fun_NIL_I, imageI, UnI1,
paulson@3120
   489
                      rangeI RS LListD_Fun_CONS_I] 1));
paulson@3120
   490
qed "Lmap_compose";
paulson@3120
   491
wenzelm@3842
   492
val [prem] = goal LList.thy "M: llist(A) ==> Lmap (%x. x) M = M";
paulson@3120
   493
by (rtac (prem RS imageI RS LList_equalityI) 1);
paulson@4160
   494
by Safe_tac;
paulson@3120
   495
by (etac llist.elim 1);
paulson@4521
   496
by (ALLGOALS Asm_simp_tac);
paulson@3120
   497
by (REPEAT (ares_tac [LListD_Fun_NIL_I, imageI RS UnI1,
paulson@3120
   498
                      rangeI RS LListD_Fun_CONS_I] 1));
paulson@3120
   499
qed "Lmap_ident";
paulson@3120
   500
paulson@3120
   501
paulson@3120
   502
(*** Lappend -- its two arguments cause some complications! ***)
paulson@3120
   503
wenzelm@5069
   504
Goalw [Lappend_def] "Lappend NIL NIL = NIL";
paulson@3120
   505
by (rtac (LList_corec RS trans) 1);
paulson@3120
   506
by (Simp_tac 1);
paulson@3120
   507
qed "Lappend_NIL_NIL";
paulson@3120
   508
wenzelm@5069
   509
Goalw [Lappend_def]
paulson@3120
   510
    "Lappend NIL (CONS N N') = CONS N (Lappend NIL N')";
paulson@3120
   511
by (rtac (LList_corec RS trans) 1);
paulson@3120
   512
by (Simp_tac 1);
paulson@3120
   513
qed "Lappend_NIL_CONS";
paulson@3120
   514
wenzelm@5069
   515
Goalw [Lappend_def]
paulson@3120
   516
    "Lappend (CONS M M') N = CONS M (Lappend M' N)";
paulson@3120
   517
by (rtac (LList_corec RS trans) 1);
paulson@3120
   518
by (Simp_tac 1);
paulson@3120
   519
qed "Lappend_CONS";
paulson@3120
   520
paulson@3120
   521
Addsimps [llist.NIL_I, Lappend_NIL_NIL, Lappend_NIL_CONS,
paulson@3120
   522
          Lappend_CONS, LListD_Fun_CONS_I, range_eqI, image_eqI];
paulson@4521
   523
paulson@3120
   524
paulson@5143
   525
Goal "M: llist(A) ==> Lappend NIL M = M";
paulson@3120
   526
by (etac LList_fun_equalityI 1);
paulson@3120
   527
by (ALLGOALS Asm_simp_tac);
paulson@3120
   528
qed "Lappend_NIL";
paulson@3120
   529
paulson@5143
   530
Goal "M: llist(A) ==> Lappend M NIL = M";
paulson@3120
   531
by (etac LList_fun_equalityI 1);
paulson@3120
   532
by (ALLGOALS Asm_simp_tac);
paulson@3120
   533
qed "Lappend_NIL2";
paulson@3120
   534
paulson@4521
   535
Addsimps [Lappend_NIL, Lappend_NIL2];
paulson@4521
   536
paulson@4521
   537
paulson@3120
   538
(** Alternative type-checking proofs for Lappend **)
paulson@3120
   539
paulson@3120
   540
(*weak co-induction: bisimulation and case analysis on both variables*)
wenzelm@5069
   541
Goal
paulson@3120
   542
    "!!M N. [| M: llist(A); N: llist(A) |] ==> Lappend M N: llist(A)";
paulson@3120
   543
by (res_inst_tac
paulson@3120
   544
    [("X", "UN u:llist(A). UN v: llist(A). {Lappend u v}")] llist_coinduct 1);
paulson@3120
   545
by (Fast_tac 1);
paulson@4160
   546
by Safe_tac;
berghofe@5102
   547
by (eres_inst_tac [("aa", "u")] llist.elim 1);
berghofe@5102
   548
by (eres_inst_tac [("aa", "v")] llist.elim 1);
paulson@4521
   549
by (ALLGOALS Asm_simp_tac);
paulson@4521
   550
by (Blast_tac 1);
paulson@3120
   551
qed "Lappend_type";
paulson@3120
   552
paulson@3120
   553
(*strong co-induction: bisimulation and case analysis on one variable*)
wenzelm@5069
   554
Goal
paulson@3120
   555
    "!!M N. [| M: llist(A); N: llist(A) |] ==> Lappend M N: llist(A)";
wenzelm@3842
   556
by (res_inst_tac [("X", "(%u. Lappend u N)``llist(A)")] llist_coinduct 1);
paulson@3120
   557
by (etac imageI 1);
paulson@4521
   558
by (rtac image_subsetI 1);
berghofe@5102
   559
by (eres_inst_tac [("aa", "x")] llist.elim 1);
paulson@4521
   560
by (asm_simp_tac (simpset() addsimps [list_Fun_llist_I]) 1);
paulson@3120
   561
by (Asm_simp_tac 1);
paulson@3120
   562
qed "Lappend_type";
paulson@3120
   563
paulson@3120
   564
(**** Lazy lists as the type 'a llist -- strongly typed versions of above ****)
paulson@3120
   565
paulson@3120
   566
(** llist_case: case analysis for 'a llist **)
paulson@3120
   567
paulson@3120
   568
Addsimps ([Abs_llist_inverse, Rep_llist_inverse,
paulson@3120
   569
           Rep_llist, rangeI, inj_Leaf, inv_f_f] @ llist.intrs);
paulson@3120
   570
wenzelm@5069
   571
Goalw [llist_case_def,LNil_def]  "llist_case c d LNil = c";
paulson@3120
   572
by (Simp_tac 1);
paulson@3120
   573
qed "llist_case_LNil";
paulson@3120
   574
wenzelm@5069
   575
Goalw [llist_case_def,LCons_def]
paulson@3120
   576
    "llist_case c d (LCons M N) = d M N";
paulson@3120
   577
by (Simp_tac 1);
paulson@3120
   578
qed "llist_case_LCons";
paulson@3120
   579
paulson@3120
   580
(*Elimination is case analysis, not induction.*)
paulson@3120
   581
val [prem1,prem2] = goalw LList.thy [NIL_def,CONS_def]
paulson@3120
   582
    "[| l=LNil ==> P;  !!x l'. l=LCons x l' ==> P \
paulson@3120
   583
\    |] ==> P";
paulson@3120
   584
by (rtac (Rep_llist RS llist.elim) 1);
paulson@3120
   585
by (rtac (inj_Rep_llist RS injD RS prem1) 1);
paulson@3120
   586
by (stac Rep_llist_LNil 1);
paulson@3120
   587
by (assume_tac 1);
paulson@3120
   588
by (etac rangeE 1);
paulson@3120
   589
by (rtac (inj_Rep_llist RS injD RS prem2) 1);
paulson@4521
   590
by (asm_simp_tac (simpset() delsimps [CONS_CONS_eq] 
paulson@4521
   591
		            addsimps [Rep_llist_LCons]) 1);
paulson@3120
   592
by (etac (Abs_llist_inverse RS ssubst) 1);
paulson@3120
   593
by (rtac refl 1);
paulson@3120
   594
qed "llistE";
paulson@3120
   595
paulson@3120
   596
(** llist_corec: corecursion for 'a llist **)
paulson@3120
   597
wenzelm@5069
   598
Goalw [llist_corec_def,LNil_def,LCons_def]
paulson@3120
   599
    "llist_corec a f = sum_case (%u. LNil) \
paulson@3120
   600
\                           (split(%z w. LCons z (llist_corec w f))) (f a)";
paulson@3120
   601
by (stac LList_corec 1);
paulson@3120
   602
by (res_inst_tac [("s","f(a)")] sumE 1);
wenzelm@4089
   603
by (asm_simp_tac (simpset() addsimps [LList_corec_type2]) 1);
paulson@3120
   604
by (res_inst_tac [("p","y")] PairE 1);
wenzelm@4089
   605
by (asm_simp_tac (simpset() addsimps [LList_corec_type2]) 1);
paulson@3120
   606
(*FIXME: correct case splits usd to be found automatically:
wenzelm@4089
   607
by (ASM_SIMP_TAC(simpset() addsimps [LList_corec_type2]) 1);*)
paulson@3120
   608
qed "llist_corec";
paulson@3120
   609
paulson@3120
   610
(*definitional version of same*)
paulson@3120
   611
val [rew] = goal LList.thy
paulson@3120
   612
    "[| !!x. h(x) == llist_corec x f |] ==>     \
wenzelm@3842
   613
\    h(a) = sum_case (%u. LNil) (split(%z w. LCons z (h w))) (f a)";
paulson@3120
   614
by (rewtac rew);
paulson@3120
   615
by (rtac llist_corec 1);
paulson@3120
   616
qed "def_llist_corec";
paulson@3120
   617
paulson@3120
   618
(**** Proofs about type 'a llist functions ****)
paulson@3120
   619
paulson@3120
   620
(*** Deriving llist_equalityI -- llist equality is a bisimulation ***)
paulson@3120
   621
wenzelm@5069
   622
Goalw [LListD_Fun_def]
paulson@3120
   623
    "!!r A. r <= (llist A) Times (llist A) ==> \
paulson@3120
   624
\           LListD_Fun (diag A) r <= (llist A) Times (llist A)";
paulson@3120
   625
by (stac llist_unfold 1);
wenzelm@4089
   626
by (simp_tac (simpset() addsimps [NIL_def, CONS_def]) 1);
paulson@3120
   627
by (Fast_tac 1);
paulson@3120
   628
qed "LListD_Fun_subset_Sigma_llist";
paulson@3120
   629
wenzelm@5069
   630
Goal
paulson@3120
   631
    "prod_fun Rep_llist Rep_llist `` r <= \
paulson@3120
   632
\    (llist(range Leaf)) Times (llist(range Leaf))";
paulson@4521
   633
by (fast_tac (claset() delrules [image_subsetI]
paulson@4521
   634
		       addIs [Rep_llist]) 1);
paulson@3120
   635
qed "subset_Sigma_llist";
paulson@3120
   636
paulson@3120
   637
val [prem] = goal LList.thy
paulson@3120
   638
    "r <= (llist(range Leaf)) Times (llist(range Leaf)) ==> \
paulson@3120
   639
\    prod_fun (Rep_llist o Abs_llist) (Rep_llist o Abs_llist) `` r <= r";
paulson@4160
   640
by Safe_tac;
paulson@3120
   641
by (rtac (prem RS subsetD RS SigmaE2) 1);
paulson@3120
   642
by (assume_tac 1);
paulson@4521
   643
by (asm_simp_tac (simpset() addsimps [Abs_llist_inverse]) 1);
paulson@3120
   644
qed "prod_fun_lemma";
paulson@3120
   645
wenzelm@5069
   646
Goal
paulson@3120
   647
    "prod_fun Rep_llist  Rep_llist `` range(%x. (x, x)) = \
paulson@3120
   648
\    diag(llist(range Leaf))";
paulson@3120
   649
by (rtac equalityI 1);
wenzelm@4089
   650
by (fast_tac (claset() addIs [Rep_llist]) 1);
oheimb@4818
   651
by (fast_tac (claset() delSWrapper "split_all_tac"
oheimb@4818
   652
		       addSEs [Abs_llist_inverse RS subst]) 1);
paulson@3120
   653
qed "prod_fun_range_eq_diag";
paulson@3120
   654
paulson@3120
   655
(*Surprisingly hard to prove.  Used with lfilter*)
wenzelm@5069
   656
Goalw [llistD_Fun_def, prod_fun_def]
paulson@3120
   657
    "!!A B. A<=B ==> llistD_Fun A <= llistD_Fun B";
paulson@4477
   658
by Auto_tac;
paulson@3120
   659
by (rtac image_eqI 1);
wenzelm@4089
   660
by (fast_tac (claset() addss (simpset())) 1);
wenzelm@4089
   661
by (blast_tac (claset() addIs [impOfSubs LListD_Fun_mono]) 1);
paulson@3120
   662
qed "llistD_Fun_mono";
paulson@3120
   663
paulson@3120
   664
(** To show two llists are equal, exhibit a bisimulation! 
paulson@3120
   665
      [also admits true equality] **)
paulson@3120
   666
val [prem1,prem2] = goalw LList.thy [llistD_Fun_def]
paulson@3120
   667
    "[| (l1,l2) : r;  r <= llistD_Fun(r Un range(%x.(x,x))) |] ==> l1=l2";
paulson@3120
   668
by (rtac (inj_Rep_llist RS injD) 1);
paulson@3120
   669
by (res_inst_tac [("r", "prod_fun Rep_llist Rep_llist ``r"),
paulson@3120
   670
                  ("A", "range(Leaf)")] 
paulson@3120
   671
        LList_equalityI 1);
paulson@3120
   672
by (rtac (prem1 RS prod_fun_imageI) 1);
paulson@3120
   673
by (rtac (prem2 RS image_mono RS subset_trans) 1);
paulson@3120
   674
by (rtac (image_compose RS subst) 1);
paulson@3120
   675
by (rtac (prod_fun_compose RS subst) 1);
paulson@3120
   676
by (stac image_Un 1);
paulson@3120
   677
by (stac prod_fun_range_eq_diag 1);
paulson@3120
   678
by (rtac (LListD_Fun_subset_Sigma_llist RS prod_fun_lemma) 1);
paulson@3120
   679
by (rtac (subset_Sigma_llist RS Un_least) 1);
paulson@3120
   680
by (rtac diag_subset_Sigma 1);
paulson@3120
   681
qed "llist_equalityI";
paulson@3120
   682
paulson@3120
   683
(** Rules to prove the 2nd premise of llist_equalityI **)
wenzelm@5069
   684
Goalw [llistD_Fun_def,LNil_def] "(LNil,LNil) : llistD_Fun(r)";
paulson@3120
   685
by (rtac (LListD_Fun_NIL_I RS prod_fun_imageI) 1);
paulson@3120
   686
qed "llistD_Fun_LNil_I";
paulson@3120
   687
paulson@3120
   688
val [prem] = goalw LList.thy [llistD_Fun_def,LCons_def]
paulson@3120
   689
    "(l1,l2):r ==> (LCons x l1, LCons x l2) : llistD_Fun(r)";
paulson@3120
   690
by (rtac (rangeI RS LListD_Fun_CONS_I RS prod_fun_imageI) 1);
paulson@3120
   691
by (rtac (prem RS prod_fun_imageI) 1);
paulson@3120
   692
qed "llistD_Fun_LCons_I";
paulson@3120
   693
paulson@3120
   694
(*Utilise the "strong" part, i.e. gfp(f)*)
wenzelm@5069
   695
Goalw [llistD_Fun_def]
paulson@3120
   696
     "!!l. (l,l) : llistD_Fun(r Un range(%x.(x,x)))";
paulson@3120
   697
by (rtac (Rep_llist_inverse RS subst) 1);
paulson@3120
   698
by (rtac prod_fun_imageI 1);
paulson@3120
   699
by (stac image_Un 1);
paulson@3120
   700
by (stac prod_fun_range_eq_diag 1);
paulson@3120
   701
by (rtac (Rep_llist RS LListD_Fun_diag_I) 1);
paulson@3120
   702
qed "llistD_Fun_range_I";
paulson@3120
   703
paulson@3120
   704
(*A special case of list_equality for functions over lazy lists*)
paulson@3120
   705
val [prem1,prem2] = goal LList.thy
paulson@3120
   706
    "[| f(LNil)=g(LNil);                                                \
paulson@3120
   707
\       !!x l. (f(LCons x l),g(LCons x l)) :                            \
paulson@3120
   708
\              llistD_Fun(range(%u. (f(u),g(u))) Un range(%v. (v,v)))   \
paulson@3120
   709
\    |] ==> f(l) = (g(l :: 'a llist) :: 'b llist)";
paulson@3120
   710
by (res_inst_tac [("r", "range(%u. (f(u),g(u)))")] llist_equalityI 1);
paulson@3120
   711
by (rtac rangeI 1);
paulson@3120
   712
by (rtac subsetI 1);
paulson@3120
   713
by (etac rangeE 1);
paulson@3120
   714
by (etac ssubst 1);
paulson@3120
   715
by (res_inst_tac [("l", "u")] llistE 1);
paulson@3120
   716
by (etac ssubst 1);
paulson@3120
   717
by (stac prem1 1);
paulson@3120
   718
by (rtac llistD_Fun_range_I 1);
paulson@3120
   719
by (etac ssubst 1);
paulson@3120
   720
by (rtac prem2 1);
paulson@3120
   721
qed "llist_fun_equalityI";
paulson@3120
   722
paulson@3120
   723
(*simpset for llist bisimulations*)
paulson@3120
   724
Addsimps [llist_case_LNil, llist_case_LCons, 
paulson@3120
   725
          llistD_Fun_LNil_I, llistD_Fun_LCons_I];
paulson@3120
   726
paulson@3120
   727
paulson@3120
   728
(*** The functional "lmap" ***)
paulson@3120
   729
wenzelm@5069
   730
Goal "lmap f LNil = LNil";
paulson@3120
   731
by (rtac (lmap_def RS def_llist_corec RS trans) 1);
paulson@3120
   732
by (Simp_tac 1);
paulson@3120
   733
qed "lmap_LNil";
paulson@3120
   734
wenzelm@5069
   735
Goal "lmap f (LCons M N) = LCons (f M) (lmap f N)";
paulson@3120
   736
by (rtac (lmap_def RS def_llist_corec RS trans) 1);
paulson@3120
   737
by (Simp_tac 1);
paulson@3120
   738
qed "lmap_LCons";
paulson@3120
   739
paulson@3120
   740
Addsimps [lmap_LNil, lmap_LCons];
paulson@3120
   741
paulson@3120
   742
paulson@3120
   743
(** Two easy results about lmap **)
paulson@3120
   744
wenzelm@5069
   745
Goal "lmap (f o g) l = lmap f (lmap g l)";
paulson@3120
   746
by (res_inst_tac [("l","l")] llist_fun_equalityI 1);
paulson@3120
   747
by (ALLGOALS Simp_tac);
paulson@3120
   748
qed "lmap_compose";
paulson@3120
   749
wenzelm@5069
   750
Goal "lmap (%x. x) l = l";
paulson@3120
   751
by (res_inst_tac [("l","l")] llist_fun_equalityI 1);
paulson@3120
   752
by (ALLGOALS Simp_tac);
paulson@3120
   753
qed "lmap_ident";
paulson@3120
   754
paulson@3120
   755
paulson@3120
   756
(*** iterates -- llist_fun_equalityI cannot be used! ***)
paulson@3120
   757
wenzelm@5069
   758
Goal "iterates f x = LCons x (iterates f (f x))";
paulson@3120
   759
by (rtac (iterates_def RS def_llist_corec RS trans) 1);
paulson@3120
   760
by (Simp_tac 1);
paulson@3120
   761
qed "iterates";
paulson@3120
   762
wenzelm@5069
   763
Goal "lmap f (iterates f x) = iterates f (f x)";
paulson@3120
   764
by (res_inst_tac [("r", "range(%u.(lmap f (iterates f u),iterates f (f u)))")] 
paulson@3120
   765
    llist_equalityI 1);
paulson@3120
   766
by (rtac rangeI 1);
paulson@4160
   767
by Safe_tac;
paulson@3120
   768
by (res_inst_tac [("x1", "f(u)")] (iterates RS ssubst) 1);
paulson@3120
   769
by (res_inst_tac [("x1", "u")] (iterates RS ssubst) 1);
paulson@3120
   770
by (Simp_tac 1);
paulson@3120
   771
qed "lmap_iterates";
paulson@3120
   772
wenzelm@5069
   773
Goal "iterates f x = LCons x (lmap f (iterates f x))";
paulson@3120
   774
by (stac lmap_iterates 1);
paulson@3120
   775
by (rtac iterates 1);
paulson@3120
   776
qed "iterates_lmap";
paulson@3120
   777
paulson@3120
   778
(*** A rather complex proof about iterates -- cf Andy Pitts ***)
paulson@3120
   779
paulson@3120
   780
(** Two lemmas about natrec n x (%m.g), which is essentially (g^n)(x) **)
paulson@3120
   781
wenzelm@5069
   782
Goal
paulson@3120
   783
    "nat_rec (LCons b l) (%m. lmap(f)) n =      \
paulson@3120
   784
\    LCons (nat_rec b (%m. f) n) (nat_rec l (%m. lmap(f)) n)";
paulson@3120
   785
by (nat_ind_tac "n" 1);
paulson@3120
   786
by (ALLGOALS Asm_simp_tac);
paulson@3120
   787
qed "fun_power_lmap";
paulson@3120
   788
paulson@3120
   789
goal Nat.thy "nat_rec (g x) (%m. g) n = nat_rec x (%m. g) (Suc n)";
paulson@3120
   790
by (nat_ind_tac "n" 1);
paulson@3120
   791
by (ALLGOALS Asm_simp_tac);
paulson@3120
   792
qed "fun_power_Suc";
paulson@3120
   793
paulson@3120
   794
val Pair_cong = read_instantiate_sg (sign_of Prod.thy)
paulson@3120
   795
 [("f","Pair")] (standard(refl RS cong RS cong));
paulson@3120
   796
paulson@3120
   797
(*The bisimulation consists of {(lmap(f)^n (h(u)), lmap(f)^n (iterates(f,u)))}
paulson@3120
   798
  for all u and all n::nat.*)
paulson@3120
   799
val [prem] = goal LList.thy
paulson@3120
   800
    "(!!x. h(x) = LCons x (lmap f (h x))) ==> h = iterates(f)";
paulson@3120
   801
by (rtac ext 1);
paulson@3120
   802
by (res_inst_tac [("r", 
wenzelm@3842
   803
   "UN u. range(%n. (nat_rec (h u) (%m y. lmap f y) n, \
wenzelm@3842
   804
\                    nat_rec (iterates f u) (%m y. lmap f y) n))")] 
paulson@3120
   805
    llist_equalityI 1);
paulson@3120
   806
by (REPEAT (resolve_tac [UN1_I, range_eqI, Pair_cong, nat_rec_0 RS sym] 1));
paulson@4160
   807
by (Clarify_tac 1);
paulson@3120
   808
by (stac iterates 1);
paulson@3120
   809
by (stac prem 1);
paulson@3120
   810
by (stac fun_power_lmap 1);
paulson@3120
   811
by (stac fun_power_lmap 1);
paulson@3120
   812
by (rtac llistD_Fun_LCons_I 1);
paulson@3120
   813
by (rtac (lmap_iterates RS subst) 1);
paulson@3120
   814
by (stac fun_power_Suc 1);
paulson@3120
   815
by (stac fun_power_Suc 1);
paulson@3120
   816
by (rtac (UN1_I RS UnI1) 1);
paulson@3120
   817
by (rtac rangeI 1);
paulson@3120
   818
qed "iterates_equality";
paulson@3120
   819
paulson@3120
   820
paulson@3120
   821
(*** lappend -- its two arguments cause some complications! ***)
paulson@3120
   822
wenzelm@5069
   823
Goalw [lappend_def] "lappend LNil LNil = LNil";
paulson@3120
   824
by (rtac (llist_corec RS trans) 1);
paulson@3120
   825
by (Simp_tac 1);
paulson@3120
   826
qed "lappend_LNil_LNil";
paulson@3120
   827
wenzelm@5069
   828
Goalw [lappend_def]
paulson@3120
   829
    "lappend LNil (LCons l l') = LCons l (lappend LNil l')";
paulson@3120
   830
by (rtac (llist_corec RS trans) 1);
paulson@3120
   831
by (Simp_tac 1);
paulson@3120
   832
qed "lappend_LNil_LCons";
paulson@3120
   833
wenzelm@5069
   834
Goalw [lappend_def]
paulson@3120
   835
    "lappend (LCons l l') N = LCons l (lappend l' N)";
paulson@3120
   836
by (rtac (llist_corec RS trans) 1);
paulson@3120
   837
by (Simp_tac 1);
paulson@3120
   838
qed "lappend_LCons";
paulson@3120
   839
paulson@3120
   840
Addsimps [lappend_LNil_LNil, lappend_LNil_LCons, lappend_LCons];
paulson@3120
   841
wenzelm@5069
   842
Goal "lappend LNil l = l";
paulson@3120
   843
by (res_inst_tac [("l","l")] llist_fun_equalityI 1);
paulson@3120
   844
by (ALLGOALS Simp_tac);
paulson@3120
   845
qed "lappend_LNil";
paulson@3120
   846
wenzelm@5069
   847
Goal "lappend l LNil = l";
paulson@3120
   848
by (res_inst_tac [("l","l")] llist_fun_equalityI 1);
paulson@3120
   849
by (ALLGOALS Simp_tac);
paulson@3120
   850
qed "lappend_LNil2";
paulson@3120
   851
paulson@3120
   852
Addsimps [lappend_LNil, lappend_LNil2];
paulson@3120
   853
paulson@3120
   854
(*The infinite first argument blocks the second*)
wenzelm@5069
   855
Goal "lappend (iterates f x) N = iterates f x";
paulson@3120
   856
by (res_inst_tac [("r", "range(%u.(lappend (iterates f u) N,iterates f u))")] 
paulson@3120
   857
    llist_equalityI 1);
paulson@3120
   858
by (rtac rangeI 1);
paulson@4160
   859
by Safe_tac;
paulson@3120
   860
by (stac iterates 1);
paulson@3120
   861
by (Simp_tac 1);
paulson@3120
   862
qed "lappend_iterates";
paulson@3120
   863
paulson@3120
   864
(** Two proofs that lmap distributes over lappend **)
paulson@3120
   865
paulson@3120
   866
(*Long proof requiring case analysis on both both arguments*)
wenzelm@5069
   867
Goal "lmap f (lappend l n) = lappend (lmap f l) (lmap f n)";
paulson@3120
   868
by (res_inst_tac 
paulson@3120
   869
    [("r",  
paulson@3120
   870
      "UN n. range(%l.(lmap f (lappend l n),lappend (lmap f l) (lmap f n)))")] 
paulson@3120
   871
    llist_equalityI 1);
paulson@3120
   872
by (rtac UN1_I 1);
paulson@3120
   873
by (rtac rangeI 1);
paulson@4160
   874
by Safe_tac;
paulson@3120
   875
by (res_inst_tac [("l", "l")] llistE 1);
paulson@3120
   876
by (res_inst_tac [("l", "n")] llistE 1);
paulson@3120
   877
by (ALLGOALS Asm_simp_tac);
paulson@3120
   878
by (REPEAT_SOME (ares_tac [llistD_Fun_LCons_I, UN1_I RS UnI1, rangeI]));
paulson@3120
   879
qed "lmap_lappend_distrib";
paulson@3120
   880
paulson@3120
   881
(*Shorter proof of theorem above using llist_equalityI as strong coinduction*)
wenzelm@5069
   882
Goal "lmap f (lappend l n) = lappend (lmap f l) (lmap f n)";
paulson@3120
   883
by (res_inst_tac [("l","l")] llist_fun_equalityI 1);
paulson@3120
   884
by (Simp_tac 1);
paulson@3120
   885
by (Simp_tac 1);
paulson@3120
   886
qed "lmap_lappend_distrib";
paulson@3120
   887
paulson@3120
   888
(*Without strong coinduction, three case analyses might be needed*)
wenzelm@5069
   889
Goal "lappend (lappend l1 l2) l3 = lappend l1 (lappend l2 l3)";
paulson@3120
   890
by (res_inst_tac [("l","l1")] llist_fun_equalityI 1);
paulson@3120
   891
by (Simp_tac 1);
paulson@3120
   892
by (Simp_tac 1);
paulson@3120
   893
qed "lappend_assoc";