src/HOL/Induct/LList.ML
author nipkow
Fri Oct 17 15:25:12 1997 +0200 (1997-10-17)
changeset 3919 c036caebfc75
parent 3842 b55686a7b22c
child 4089 96fba19bcbe2
permissions -rw-r--r--
setloop split_tac -> addsplits
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(*  Title:      HOL/ex/LList
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1993  University of Cambridge
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SHOULD LListD_Fun_CONS_I, etc., be equations (for rewriting)?
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*)
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open LList;
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(** Simplification **)
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simpset := !simpset addsplits [expand_split, expand_sum_case];
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(*For adding _eqI rules to a simpset; we must remove Pair_eq because
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  it may turn an instance of reflexivity into a conjunction!*)
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fun add_eqI ss = ss addsimps [range_eqI, image_eqI] 
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                    delsimps [Pair_eq];
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(*This justifies using llist in other recursive type definitions*)
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goalw LList.thy llist.defs "!!A B. A<=B ==> llist(A) <= llist(B)";
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by (rtac gfp_mono 1);
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by (REPEAT (ares_tac basic_monos 1));
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qed "llist_mono";
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goal LList.thy "llist(A) = {Numb(0)} <+> (A <*> llist(A))";
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let val rew = rewrite_rule [NIL_def, CONS_def] in  
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by (fast_tac (!claset addSIs (map rew llist.intrs)
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                      addEs [rew llist.elim]) 1)
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end;
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qed "llist_unfold";
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(*** Type checking by coinduction, using list_Fun 
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     THE COINDUCTIVE DEFINITION PACKAGE COULD DO THIS!
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***)
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goalw LList.thy [list_Fun_def]
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    "!!M. [| M : X;  X <= list_Fun A (X Un llist(A)) |] ==>  M : llist(A)";
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by (etac llist.coinduct 1);
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by (etac (subsetD RS CollectD) 1);
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by (assume_tac 1);
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qed "llist_coinduct";
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goalw LList.thy [list_Fun_def, NIL_def] "NIL: list_Fun A X";
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by (Fast_tac 1);
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qed "list_Fun_NIL_I";
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goalw LList.thy [list_Fun_def,CONS_def]
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    "!!M N. [| M: A;  N: X |] ==> CONS M N : list_Fun A X";
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by (Fast_tac 1);
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qed "list_Fun_CONS_I";
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(*Utilise the "strong" part, i.e. gfp(f)*)
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goalw LList.thy (llist.defs @ [list_Fun_def])
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    "!!M N. M: llist(A) ==> M : list_Fun A (X Un llist(A))";
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by (etac (llist.mono RS gfp_fun_UnI2) 1);
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qed "list_Fun_llist_I";
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(*** LList_corec satisfies the desired recurion equation ***)
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(*A continuity result?*)
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goalw LList.thy [CONS_def] "CONS M (UN x. f(x)) = (UN x. CONS M (f x))";
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by (simp_tac (!simpset addsimps [In1_UN1, Scons_UN1_y]) 1);
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qed "CONS_UN1";
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(*UNUSED; obsolete?
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goal Prod.thy "split p (%x y. UN z. f x y z) = (UN z. split p (%x y. f x y z))";
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by (simp_tac (!simpset addsplits [expand_split]) 1);
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qed "split_UN1";
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goal Sum.thy "sum_case s f (%y. UN z. g y z) = (UN z. sum_case s f (%y. g y z))";
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by (simp_tac (!simpset addsplits [expand_sum_case]) 1);
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qed "sum_case2_UN1";
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*)
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val prems = goalw LList.thy [CONS_def]
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    "[| M<=M';  N<=N' |] ==> CONS M N <= CONS M' N'";
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by (REPEAT (resolve_tac ([In1_mono,Scons_mono]@prems) 1));
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qed "CONS_mono";
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Addsimps [LList_corec_fun_def RS def_nat_rec_0,
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          LList_corec_fun_def RS def_nat_rec_Suc];
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(** The directions of the equality are proved separately **)
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goalw LList.thy [LList_corec_def]
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    "LList_corec a f <= sum_case (%u. NIL) \
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\                          (split(%z w. CONS z (LList_corec w f))) (f a)";
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by (rtac UN1_least 1);
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by (res_inst_tac [("n","k")] natE 1);
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by (ALLGOALS (Asm_simp_tac));
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by (REPEAT (resolve_tac [allI, impI, subset_refl RS CONS_mono, UN1_upper] 1));
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qed "LList_corec_subset1";
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goalw LList.thy [LList_corec_def]
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    "sum_case (%u. NIL) (split(%z w. CONS z (LList_corec w f))) (f a) <= \
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\    LList_corec a f";
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by (simp_tac (!simpset addsimps [CONS_UN1]) 1);
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by (safe_tac (!claset));
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by (ALLGOALS (res_inst_tac [("x","Suc(?k)")] UN1_I THEN' Asm_simp_tac));
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qed "LList_corec_subset2";
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(*the recursion equation for LList_corec -- NOT SUITABLE FOR REWRITING!*)
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goal LList.thy
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    "LList_corec a f = sum_case (%u. NIL) \
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\                           (split(%z w. CONS z (LList_corec w f))) (f a)";
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by (REPEAT (resolve_tac [equalityI, LList_corec_subset1, 
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                         LList_corec_subset2] 1));
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qed "LList_corec";
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(*definitional version of same*)
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val [rew] = goal LList.thy
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    "[| !!x. h(x) == LList_corec x f |] ==>     \
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\    h(a) = sum_case (%u. NIL) (split(%z w. CONS z (h w))) (f a)";
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by (rewtac rew);
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by (rtac LList_corec 1);
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qed "def_LList_corec";
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(*A typical use of co-induction to show membership in the gfp. 
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  Bisimulation is  range(%x. LList_corec x f) *)
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goal LList.thy "LList_corec a f : llist({u. True})";
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by (res_inst_tac [("X", "range(%x. LList_corec x ?g)")] llist_coinduct 1);
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by (rtac rangeI 1);
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by (safe_tac (!claset));
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by (stac LList_corec 1);
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by (simp_tac (!simpset addsimps [list_Fun_NIL_I, list_Fun_CONS_I, CollectI]
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                       |> add_eqI) 1);
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qed "LList_corec_type";
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(*Lemma for the proof of llist_corec*)
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goal LList.thy
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   "LList_corec a (%z. sum_case Inl (split(%v w. Inr((Leaf(v),w)))) (f z)) : \
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\   llist(range Leaf)";
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by (res_inst_tac [("X", "range(%x. LList_corec x ?g)")] llist_coinduct 1);
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by (rtac rangeI 1);
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by (safe_tac (!claset));
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by (stac LList_corec 1);
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by (asm_simp_tac (!simpset addsimps [list_Fun_NIL_I]) 1);
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by (fast_tac (!claset addSIs [list_Fun_CONS_I]) 1);
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qed "LList_corec_type2";
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(**** llist equality as a gfp; the bisimulation principle ****)
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(*This theorem is actually used, unlike the many similar ones in ZF*)
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goal LList.thy "LListD(r) = diag({Numb(0)}) <++> (r <**> LListD(r))";
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let val rew = rewrite_rule [NIL_def, CONS_def] in  
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by (fast_tac (!claset addSIs (map rew LListD.intrs)
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                      addEs [rew LListD.elim]) 1)
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end;
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qed "LListD_unfold";
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goal LList.thy "!M N. (M,N) : LListD(diag(A)) --> ntrunc k M = ntrunc k N";
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by (res_inst_tac [("n", "k")] less_induct 1);
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by (safe_tac ((claset_of "Fun") delrules [equalityI]));
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by (etac LListD.elim 1);
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by (safe_tac (claset_of "Prod" delrules [equalityI] addSEs [diagE]));
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by (res_inst_tac [("n", "n")] natE 1);
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by (asm_simp_tac (!simpset addsimps [ntrunc_0]) 1);
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by (rename_tac "n'" 1);
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by (res_inst_tac [("n", "n'")] natE 1);
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by (asm_simp_tac (!simpset addsimps [CONS_def, ntrunc_one_In1]) 1);
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by (asm_simp_tac (!simpset addsimps [CONS_def, ntrunc_In1, ntrunc_Scons, less_Suc_eq]) 1);
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qed "LListD_implies_ntrunc_equality";
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(*The domain of the LListD relation*)
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goalw LList.thy (llist.defs @ [NIL_def, CONS_def])
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    "fst``LListD(diag(A)) <= llist(A)";
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by (rtac gfp_upperbound 1);
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(*avoids unfolding LListD on the rhs*)
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by (res_inst_tac [("P", "%x. fst``x <= ?B")] (LListD_unfold RS ssubst) 1);
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by (Simp_tac 1);
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by (Fast_tac 1);
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qed "fst_image_LListD";
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(*This inclusion justifies the use of coinduction to show M=N*)
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goal LList.thy "LListD(diag(A)) <= diag(llist(A))";
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by (rtac subsetI 1);
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by (res_inst_tac [("p","x")] PairE 1);
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by (safe_tac (!claset));
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by (rtac diag_eqI 1);
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by (rtac (LListD_implies_ntrunc_equality RS spec RS spec RS mp RS 
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          ntrunc_equality) 1);
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by (assume_tac 1);
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by (etac (fst_imageI RS (fst_image_LListD RS subsetD)) 1);
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qed "LListD_subset_diag";
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(** Coinduction, using LListD_Fun
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    THE COINDUCTIVE DEFINITION PACKAGE COULD DO THIS!
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 **)
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goalw thy [LListD_Fun_def] "!!A B. 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 LList.thy [LListD_Fun_def]
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    "!!M. [| M : X;  X <= LListD_Fun r (X Un LListD(r)) |] ==>  M : LListD(r)";
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by (etac LListD.coinduct 1);
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by (etac (subsetD RS CollectD) 1);
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by (assume_tac 1);
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qed "LListD_coinduct";
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goalw LList.thy [LListD_Fun_def,NIL_def] "(NIL,NIL) : LListD_Fun r s";
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by (Fast_tac 1);
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qed "LListD_Fun_NIL_I";
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goalw LList.thy [LListD_Fun_def,CONS_def]
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 "!!x. [| x:A;  (M,N):s |] ==> (CONS x M, CONS x N) : LListD_Fun (diag A) s";
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by (Fast_tac 1);
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qed "LListD_Fun_CONS_I";
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(*Utilise the "strong" part, i.e. gfp(f)*)
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goalw LList.thy (LListD.defs @ [LListD_Fun_def])
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    "!!M N. M: LListD(r) ==> M : LListD_Fun r (X Un LListD(r))";
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by (etac (LListD.mono RS gfp_fun_UnI2) 1);
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qed "LListD_Fun_LListD_I";
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(*This converse inclusion helps to strengthen LList_equalityI*)
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goal LList.thy "diag(llist(A)) <= LListD(diag(A))";
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by (rtac subsetI 1);
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by (etac LListD_coinduct 1);
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by (rtac subsetI 1);
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by (etac diagE 1);
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by (etac ssubst 1);
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by (eresolve_tac [llist.elim] 1);
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by (ALLGOALS
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    (asm_simp_tac (!simpset addsimps [diagI, LListD_Fun_NIL_I,
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                                      LListD_Fun_CONS_I])));
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qed "diag_subset_LListD";
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goal LList.thy "LListD(diag(A)) = diag(llist(A))";
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by (REPEAT (resolve_tac [equalityI, LListD_subset_diag, 
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                         diag_subset_LListD] 1));
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qed "LListD_eq_diag";
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goal LList.thy 
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    "!!M N. M: llist(A) ==> (M,M) : LListD_Fun (diag A) (X Un diag(llist(A)))";
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by (rtac (LListD_eq_diag RS subst) 1);
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by (rtac LListD_Fun_LListD_I 1);
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by (asm_simp_tac (!simpset addsimps [LListD_eq_diag, diagI]) 1);
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qed "LListD_Fun_diag_I";
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(** To show two LLists are equal, exhibit a bisimulation! 
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      [also admits true equality]
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   Replace "A" by some particular set, like {x.True}??? *)
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goal LList.thy 
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    "!!r. [| (M,N) : r;  r <= LListD_Fun (diag A) (r Un diag(llist(A))) \
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\         |] ==>  M=N";
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by (rtac (LListD_subset_diag RS subsetD RS diagE) 1);
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by (etac LListD_coinduct 1);
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by (asm_simp_tac (!simpset addsimps [LListD_eq_diag]) 1);
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by (safe_tac (!claset));
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qed "LList_equalityI";
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(*** Finality of llist(A): Uniqueness of functions defined by corecursion ***)
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(*abstract proof using a bisimulation*)
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val [prem1,prem2] = goal LList.thy
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 "[| !!x. h1(x) = sum_case (%u. NIL) (split(%z w. CONS z (h1 w))) (f x);  \
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\    !!x. h2(x) = sum_case (%u. NIL) (split(%z w. CONS z (h2 w))) (f x) |]\
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\ ==> h1=h2";
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by (rtac ext 1);
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(*next step avoids an unknown (and flexflex pair) in simplification*)
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by (res_inst_tac [("A", "{u. True}"),
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                  ("r", "range(%u. (h1(u),h2(u)))")] LList_equalityI 1);
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by (rtac rangeI 1);
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by (safe_tac (!claset));
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by (stac prem1 1);
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by (stac prem2 1);
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by (simp_tac (!simpset addsimps [LListD_Fun_NIL_I,
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                                 CollectI RS LListD_Fun_CONS_I]
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                       |> add_eqI) 1);
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qed "LList_corec_unique";
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val [prem] = goal LList.thy
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 "[| !!x. h(x) = sum_case (%u. NIL) (split(%z w. CONS z (h w))) (f x) |] \
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\ ==> h = (%x. LList_corec x f)";
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by (rtac (LList_corec RS (prem RS LList_corec_unique)) 1);
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qed "equals_LList_corec";
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(** Obsolete LList_corec_unique proof: complete induction, not coinduction **)
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goalw LList.thy [CONS_def] "ntrunc (Suc 0) (CONS M N) = {}";
paulson@3120
   292
by (rtac ntrunc_one_In1 1);
paulson@3120
   293
qed "ntrunc_one_CONS";
paulson@3120
   294
paulson@3120
   295
goalw LList.thy [CONS_def]
paulson@3120
   296
    "ntrunc (Suc(Suc(k))) (CONS M N) = CONS (ntrunc k M) (ntrunc k N)";
paulson@3120
   297
by (simp_tac (!simpset addsimps [ntrunc_Scons,ntrunc_In1]) 1);
paulson@3120
   298
qed "ntrunc_CONS";
paulson@3120
   299
paulson@3120
   300
val [prem1,prem2] = goal LList.thy
wenzelm@3842
   301
 "[| !!x. h1(x) = sum_case (%u. NIL) (split(%z w. CONS z (h1 w))) (f x);  \
wenzelm@3842
   302
\    !!x. h2(x) = sum_case (%u. NIL) (split(%z w. CONS z (h2 w))) (f x) |]\
paulson@3120
   303
\ ==> h1=h2";
paulson@3120
   304
by (rtac (ntrunc_equality RS ext) 1);
paulson@3120
   305
by (rename_tac "x k" 1);
paulson@3120
   306
by (res_inst_tac [("x", "x")] spec 1);
paulson@3120
   307
by (res_inst_tac [("n", "k")] less_induct 1);
paulson@3120
   308
by (rename_tac "n" 1);
paulson@3120
   309
by (rtac allI 1);
paulson@3120
   310
by (rename_tac "y" 1);
paulson@3120
   311
by (stac prem1 1);
paulson@3120
   312
by (stac prem2 1);
nipkow@3919
   313
by (simp_tac (!simpset addsplits [expand_sum_case]) 1);
paulson@3120
   314
by (strip_tac 1);
paulson@3120
   315
by (res_inst_tac [("n", "n")] natE 1);
paulson@3120
   316
by (rename_tac "m" 2);
paulson@3120
   317
by (res_inst_tac [("n", "m")] natE 2);
paulson@3120
   318
by (ALLGOALS(asm_simp_tac(!simpset addsimps
paulson@3120
   319
            [ntrunc_0,ntrunc_one_CONS,ntrunc_CONS, less_Suc_eq])));
paulson@3120
   320
result();
paulson@3120
   321
paulson@3120
   322
paulson@3120
   323
(*** Lconst -- defined directly using lfp, but equivalent to a LList_corec ***)
paulson@3120
   324
paulson@3120
   325
goal LList.thy "mono(CONS(M))";
paulson@3120
   326
by (REPEAT (ares_tac [monoI, subset_refl, CONS_mono] 1));
paulson@3120
   327
qed "Lconst_fun_mono";
paulson@3120
   328
paulson@3120
   329
(* Lconst(M) = CONS M (Lconst M) *)
paulson@3120
   330
bind_thm ("Lconst", (Lconst_fun_mono RS (Lconst_def RS def_lfp_Tarski)));
paulson@3120
   331
paulson@3120
   332
(*A typical use of co-induction to show membership in the gfp.
paulson@3120
   333
  The containing set is simply the singleton {Lconst(M)}. *)
paulson@3120
   334
goal LList.thy "!!M A. M:A ==> Lconst(M): llist(A)";
paulson@3120
   335
by (rtac (singletonI RS llist_coinduct) 1);
paulson@3120
   336
by (safe_tac (!claset));
paulson@3120
   337
by (res_inst_tac [("P", "%u. u: ?A")] (Lconst RS ssubst) 1);
paulson@3120
   338
by (REPEAT (ares_tac [list_Fun_CONS_I, singletonI, UnI1] 1));
paulson@3120
   339
qed "Lconst_type";
paulson@3120
   340
wenzelm@3842
   341
goal LList.thy "Lconst(M) = LList_corec M (%x. Inr((x,x)))";
paulson@3120
   342
by (rtac (equals_LList_corec RS fun_cong) 1);
paulson@3120
   343
by (Simp_tac 1);
paulson@3120
   344
by (rtac Lconst 1);
paulson@3120
   345
qed "Lconst_eq_LList_corec";
paulson@3120
   346
paulson@3120
   347
(*Thus we could have used gfp in the definition of Lconst*)
wenzelm@3842
   348
goal LList.thy "gfp(%N. CONS M N) = LList_corec M (%x. Inr((x,x)))";
paulson@3120
   349
by (rtac (equals_LList_corec RS fun_cong) 1);
paulson@3120
   350
by (Simp_tac 1);
paulson@3120
   351
by (rtac (Lconst_fun_mono RS gfp_Tarski) 1);
paulson@3120
   352
qed "gfp_Lconst_eq_LList_corec";
paulson@3120
   353
paulson@3120
   354
paulson@3120
   355
(*** Isomorphisms ***)
paulson@3120
   356
paulson@3120
   357
goal LList.thy "inj(Rep_llist)";
paulson@3120
   358
by (rtac inj_inverseI 1);
paulson@3120
   359
by (rtac Rep_llist_inverse 1);
paulson@3120
   360
qed "inj_Rep_llist";
paulson@3120
   361
paulson@3120
   362
goal LList.thy "inj_onto Abs_llist (llist(range Leaf))";
paulson@3120
   363
by (rtac inj_onto_inverseI 1);
paulson@3120
   364
by (etac Abs_llist_inverse 1);
paulson@3120
   365
qed "inj_onto_Abs_llist";
paulson@3120
   366
paulson@3120
   367
(** Distinctness of constructors **)
paulson@3120
   368
paulson@3120
   369
goalw LList.thy [LNil_def,LCons_def] "~ LCons x xs = LNil";
paulson@3120
   370
by (rtac (CONS_not_NIL RS (inj_onto_Abs_llist RS inj_onto_contraD)) 1);
paulson@3120
   371
by (REPEAT (resolve_tac (llist.intrs @ [rangeI, Rep_llist]) 1));
paulson@3120
   372
qed "LCons_not_LNil";
paulson@3120
   373
paulson@3120
   374
bind_thm ("LNil_not_LCons", LCons_not_LNil RS not_sym);
paulson@3120
   375
paulson@3120
   376
AddIffs [LCons_not_LNil, LNil_not_LCons];
paulson@3120
   377
paulson@3120
   378
paulson@3120
   379
(** llist constructors **)
paulson@3120
   380
paulson@3120
   381
goalw LList.thy [LNil_def]
paulson@3120
   382
    "Rep_llist(LNil) = NIL";
paulson@3120
   383
by (rtac (llist.NIL_I RS Abs_llist_inverse) 1);
paulson@3120
   384
qed "Rep_llist_LNil";
paulson@3120
   385
paulson@3120
   386
goalw LList.thy [LCons_def]
paulson@3120
   387
    "Rep_llist(LCons x l) = CONS (Leaf x) (Rep_llist l)";
paulson@3120
   388
by (REPEAT (resolve_tac [llist.CONS_I RS Abs_llist_inverse,
paulson@3120
   389
                         rangeI, Rep_llist] 1));
paulson@3120
   390
qed "Rep_llist_LCons";
paulson@3120
   391
paulson@3120
   392
(** Injectiveness of CONS and LCons **)
paulson@3120
   393
paulson@3120
   394
goalw LList.thy [CONS_def] "(CONS M N=CONS M' N') = (M=M' & N=N')";
paulson@3427
   395
by (fast_tac (!claset addSEs [Scons_inject]) 1);
paulson@3120
   396
qed "CONS_CONS_eq2";
paulson@3120
   397
paulson@3120
   398
bind_thm ("CONS_inject", (CONS_CONS_eq RS iffD1 RS conjE));
paulson@3120
   399
paulson@3120
   400
paulson@3120
   401
(*For reasoning about abstract llist constructors*)
paulson@3120
   402
paulson@3120
   403
AddIs ([Rep_llist]@llist.intrs);
paulson@3120
   404
AddSDs [inj_onto_Abs_llist RS inj_ontoD,
paulson@3120
   405
        inj_Rep_llist RS injD, Leaf_inject];
paulson@3120
   406
paulson@3120
   407
goalw LList.thy [LCons_def] "(LCons x xs=LCons y ys) = (x=y & xs=ys)";
paulson@3120
   408
by (Fast_tac 1);
paulson@3120
   409
qed "LCons_LCons_eq";
paulson@3120
   410
paulson@3120
   411
AddIffs [LCons_LCons_eq];
paulson@3120
   412
paulson@3120
   413
val [major] = goal LList.thy "CONS M N: llist(A) ==> M: A & N: llist(A)";
paulson@3120
   414
by (rtac (major RS llist.elim) 1);
paulson@3120
   415
by (etac CONS_neq_NIL 1);
paulson@3120
   416
by (Fast_tac 1);
paulson@3120
   417
qed "CONS_D2";
paulson@3120
   418
paulson@3120
   419
paulson@3120
   420
(****** Reasoning about llist(A) ******)
paulson@3120
   421
paulson@3120
   422
Addsimps [List_case_NIL, List_case_CONS];
paulson@3120
   423
paulson@3120
   424
(*A special case of list_equality for functions over lazy lists*)
paulson@3120
   425
val [Mlist,gMlist,NILcase,CONScase] = goal LList.thy
paulson@3120
   426
 "[| M: llist(A); g(NIL): llist(A);                             \
paulson@3120
   427
\    f(NIL)=g(NIL);                                             \
paulson@3120
   428
\    !!x l. [| x:A;  l: llist(A) |] ==>                         \
paulson@3120
   429
\           (f(CONS x l),g(CONS x l)) :                         \
paulson@3120
   430
\               LListD_Fun (diag A) ((%u.(f(u),g(u)))``llist(A) Un  \
paulson@3120
   431
\                                   diag(llist(A)))             \
paulson@3120
   432
\ |] ==> f(M) = g(M)";
paulson@3120
   433
by (rtac LList_equalityI 1);
paulson@3120
   434
by (rtac (Mlist RS imageI) 1);
paulson@3120
   435
by (rtac subsetI 1);
paulson@3120
   436
by (etac imageE 1);
paulson@3120
   437
by (etac ssubst 1);
paulson@3120
   438
by (etac llist.elim 1);
paulson@3120
   439
by (etac ssubst 1);
paulson@3120
   440
by (stac NILcase 1);
paulson@3120
   441
by (rtac (gMlist RS LListD_Fun_diag_I) 1);
paulson@3120
   442
by (etac ssubst 1);
paulson@3120
   443
by (REPEAT (ares_tac [CONScase] 1));
paulson@3120
   444
qed "LList_fun_equalityI";
paulson@3120
   445
paulson@3120
   446
paulson@3120
   447
(*** The functional "Lmap" ***)
paulson@3120
   448
paulson@3120
   449
goal LList.thy "Lmap f NIL = NIL";
paulson@3120
   450
by (rtac (Lmap_def RS def_LList_corec RS trans) 1);
paulson@3120
   451
by (Simp_tac 1);
paulson@3120
   452
qed "Lmap_NIL";
paulson@3120
   453
paulson@3120
   454
goal LList.thy "Lmap f (CONS M N) = CONS (f M) (Lmap f N)";
paulson@3120
   455
by (rtac (Lmap_def RS def_LList_corec RS trans) 1);
paulson@3120
   456
by (Simp_tac 1);
paulson@3120
   457
qed "Lmap_CONS";
paulson@3120
   458
paulson@3120
   459
(*Another type-checking proof by coinduction*)
paulson@3120
   460
val [major,minor] = goal LList.thy
paulson@3120
   461
    "[| M: llist(A);  !!x. x:A ==> f(x):B |] ==> Lmap f M: llist(B)";
paulson@3120
   462
by (rtac (major RS imageI RS llist_coinduct) 1);
paulson@3120
   463
by (safe_tac (!claset));
paulson@3120
   464
by (etac llist.elim 1);
paulson@3120
   465
by (ALLGOALS (asm_simp_tac (!simpset addsimps [Lmap_NIL,Lmap_CONS])));
paulson@3120
   466
by (REPEAT (ares_tac [list_Fun_NIL_I, list_Fun_CONS_I, 
paulson@3120
   467
                      minor, imageI, UnI1] 1));
paulson@3120
   468
qed "Lmap_type";
paulson@3120
   469
paulson@3120
   470
(*This type checking rule synthesises a sufficiently large set for f*)
paulson@3120
   471
val [major] = goal LList.thy  "M: llist(A) ==> Lmap f M: llist(f``A)";
paulson@3120
   472
by (rtac (major RS Lmap_type) 1);
paulson@3120
   473
by (etac imageI 1);
paulson@3120
   474
qed "Lmap_type2";
paulson@3120
   475
paulson@3120
   476
(** Two easy results about Lmap **)
paulson@3120
   477
paulson@3120
   478
val [prem] = goalw LList.thy [o_def]
paulson@3120
   479
    "M: llist(A) ==> Lmap (f o g) M = Lmap f (Lmap g M)";
paulson@3120
   480
by (rtac (prem RS imageI RS LList_equalityI) 1);
paulson@3120
   481
by (safe_tac (!claset));
paulson@3120
   482
by (etac llist.elim 1);
paulson@3120
   483
by (ALLGOALS (asm_simp_tac (!simpset addsimps [Lmap_NIL,Lmap_CONS])));
paulson@3120
   484
by (REPEAT (ares_tac [LListD_Fun_NIL_I, imageI, UnI1,
paulson@3120
   485
                      rangeI RS LListD_Fun_CONS_I] 1));
paulson@3120
   486
qed "Lmap_compose";
paulson@3120
   487
wenzelm@3842
   488
val [prem] = goal LList.thy "M: llist(A) ==> Lmap (%x. x) M = M";
paulson@3120
   489
by (rtac (prem RS imageI RS LList_equalityI) 1);
paulson@3120
   490
by (safe_tac (!claset));
paulson@3120
   491
by (etac llist.elim 1);
paulson@3120
   492
by (ALLGOALS (asm_simp_tac (!simpset addsimps [Lmap_NIL,Lmap_CONS])));
paulson@3120
   493
by (REPEAT (ares_tac [LListD_Fun_NIL_I, imageI RS UnI1,
paulson@3120
   494
                      rangeI RS LListD_Fun_CONS_I] 1));
paulson@3120
   495
qed "Lmap_ident";
paulson@3120
   496
paulson@3120
   497
paulson@3120
   498
(*** Lappend -- its two arguments cause some complications! ***)
paulson@3120
   499
paulson@3120
   500
goalw LList.thy [Lappend_def] "Lappend NIL NIL = NIL";
paulson@3120
   501
by (rtac (LList_corec RS trans) 1);
paulson@3120
   502
by (Simp_tac 1);
paulson@3120
   503
qed "Lappend_NIL_NIL";
paulson@3120
   504
paulson@3120
   505
goalw LList.thy [Lappend_def]
paulson@3120
   506
    "Lappend NIL (CONS N N') = CONS N (Lappend NIL N')";
paulson@3120
   507
by (rtac (LList_corec RS trans) 1);
paulson@3120
   508
by (Simp_tac 1);
paulson@3120
   509
qed "Lappend_NIL_CONS";
paulson@3120
   510
paulson@3120
   511
goalw LList.thy [Lappend_def]
paulson@3120
   512
    "Lappend (CONS M M') N = CONS M (Lappend M' N)";
paulson@3120
   513
by (rtac (LList_corec RS trans) 1);
paulson@3120
   514
by (Simp_tac 1);
paulson@3120
   515
qed "Lappend_CONS";
paulson@3120
   516
paulson@3120
   517
Addsimps [llist.NIL_I, Lappend_NIL_NIL, Lappend_NIL_CONS,
paulson@3120
   518
          Lappend_CONS, LListD_Fun_CONS_I, range_eqI, image_eqI];
paulson@3120
   519
Delsimps [Pair_eq];
paulson@3120
   520
paulson@3120
   521
goal LList.thy "!!M. M: llist(A) ==> Lappend NIL M = M";
paulson@3120
   522
by (etac LList_fun_equalityI 1);
paulson@3120
   523
by (ALLGOALS Asm_simp_tac);
paulson@3120
   524
qed "Lappend_NIL";
paulson@3120
   525
paulson@3120
   526
goal LList.thy "!!M. M: llist(A) ==> Lappend M NIL = M";
paulson@3120
   527
by (etac LList_fun_equalityI 1);
paulson@3120
   528
by (ALLGOALS Asm_simp_tac);
paulson@3120
   529
qed "Lappend_NIL2";
paulson@3120
   530
paulson@3120
   531
(** Alternative type-checking proofs for Lappend **)
paulson@3120
   532
paulson@3120
   533
(*weak co-induction: bisimulation and case analysis on both variables*)
paulson@3120
   534
goal LList.thy
paulson@3120
   535
    "!!M N. [| M: llist(A); N: llist(A) |] ==> Lappend M N: llist(A)";
paulson@3120
   536
by (res_inst_tac
paulson@3120
   537
    [("X", "UN u:llist(A). UN v: llist(A). {Lappend u v}")] llist_coinduct 1);
paulson@3120
   538
by (Fast_tac 1);
paulson@3120
   539
by (safe_tac (!claset));
paulson@3120
   540
by (eres_inst_tac [("a", "u")] llist.elim 1);
paulson@3120
   541
by (eres_inst_tac [("a", "v")] llist.elim 1);
paulson@3120
   542
by (ALLGOALS
paulson@3120
   543
    (Asm_simp_tac THEN'
paulson@3120
   544
     fast_tac (!claset addSIs [llist.NIL_I, list_Fun_NIL_I, list_Fun_CONS_I])));
paulson@3120
   545
qed "Lappend_type";
paulson@3120
   546
paulson@3120
   547
(*strong co-induction: bisimulation and case analysis on one variable*)
paulson@3120
   548
goal LList.thy
paulson@3120
   549
    "!!M N. [| M: llist(A); N: llist(A) |] ==> Lappend M N: llist(A)";
wenzelm@3842
   550
by (res_inst_tac [("X", "(%u. Lappend u N)``llist(A)")] llist_coinduct 1);
paulson@3120
   551
by (etac imageI 1);
paulson@3120
   552
by (rtac subsetI 1);
paulson@3120
   553
by (etac imageE 1);
paulson@3120
   554
by (eres_inst_tac [("a", "u")] llist.elim 1);
paulson@3120
   555
by (asm_simp_tac (!simpset addsimps [Lappend_NIL, list_Fun_llist_I]) 1);
paulson@3120
   556
by (Asm_simp_tac 1);
paulson@3120
   557
by (fast_tac (!claset addSIs [list_Fun_CONS_I]) 1);
paulson@3120
   558
qed "Lappend_type";
paulson@3120
   559
paulson@3120
   560
(**** Lazy lists as the type 'a llist -- strongly typed versions of above ****)
paulson@3120
   561
paulson@3120
   562
(** llist_case: case analysis for 'a llist **)
paulson@3120
   563
paulson@3120
   564
Addsimps ([Abs_llist_inverse, Rep_llist_inverse,
paulson@3120
   565
           Rep_llist, rangeI, inj_Leaf, inv_f_f] @ llist.intrs);
paulson@3120
   566
paulson@3120
   567
goalw LList.thy [llist_case_def,LNil_def]  "llist_case c d LNil = c";
paulson@3120
   568
by (Simp_tac 1);
paulson@3120
   569
qed "llist_case_LNil";
paulson@3120
   570
paulson@3120
   571
goalw LList.thy [llist_case_def,LCons_def]
paulson@3120
   572
    "llist_case c d (LCons M N) = d M N";
paulson@3120
   573
by (Simp_tac 1);
paulson@3120
   574
qed "llist_case_LCons";
paulson@3120
   575
paulson@3120
   576
(*Elimination is case analysis, not induction.*)
paulson@3120
   577
val [prem1,prem2] = goalw LList.thy [NIL_def,CONS_def]
paulson@3120
   578
    "[| l=LNil ==> P;  !!x l'. l=LCons x l' ==> P \
paulson@3120
   579
\    |] ==> P";
paulson@3120
   580
by (rtac (Rep_llist RS llist.elim) 1);
paulson@3120
   581
by (rtac (inj_Rep_llist RS injD RS prem1) 1);
paulson@3120
   582
by (stac Rep_llist_LNil 1);
paulson@3120
   583
by (assume_tac 1);
paulson@3120
   584
by (etac rangeE 1);
paulson@3120
   585
by (rtac (inj_Rep_llist RS injD RS prem2) 1);
paulson@3120
   586
by (asm_simp_tac (!simpset delsimps [CONS_CONS_eq] addsimps [Rep_llist_LCons]) 1);
paulson@3120
   587
by (etac (Abs_llist_inverse RS ssubst) 1);
paulson@3120
   588
by (rtac refl 1);
paulson@3120
   589
qed "llistE";
paulson@3120
   590
paulson@3120
   591
(** llist_corec: corecursion for 'a llist **)
paulson@3120
   592
paulson@3120
   593
goalw LList.thy [llist_corec_def,LNil_def,LCons_def]
paulson@3120
   594
    "llist_corec a f = sum_case (%u. LNil) \
paulson@3120
   595
\                           (split(%z w. LCons z (llist_corec w f))) (f a)";
paulson@3120
   596
by (stac LList_corec 1);
paulson@3120
   597
by (res_inst_tac [("s","f(a)")] sumE 1);
paulson@3120
   598
by (asm_simp_tac (!simpset addsimps [LList_corec_type2]) 1);
paulson@3120
   599
by (res_inst_tac [("p","y")] PairE 1);
paulson@3120
   600
by (asm_simp_tac (!simpset addsimps [LList_corec_type2]) 1);
paulson@3120
   601
(*FIXME: correct case splits usd to be found automatically:
paulson@3120
   602
by (ASM_SIMP_TAC(!simpset addsimps [LList_corec_type2]) 1);*)
paulson@3120
   603
qed "llist_corec";
paulson@3120
   604
paulson@3120
   605
(*definitional version of same*)
paulson@3120
   606
val [rew] = goal LList.thy
paulson@3120
   607
    "[| !!x. h(x) == llist_corec x f |] ==>     \
wenzelm@3842
   608
\    h(a) = sum_case (%u. LNil) (split(%z w. LCons z (h w))) (f a)";
paulson@3120
   609
by (rewtac rew);
paulson@3120
   610
by (rtac llist_corec 1);
paulson@3120
   611
qed "def_llist_corec";
paulson@3120
   612
paulson@3120
   613
(**** Proofs about type 'a llist functions ****)
paulson@3120
   614
paulson@3120
   615
(*** Deriving llist_equalityI -- llist equality is a bisimulation ***)
paulson@3120
   616
paulson@3120
   617
goalw LList.thy [LListD_Fun_def]
paulson@3120
   618
    "!!r A. r <= (llist A) Times (llist A) ==> \
paulson@3120
   619
\           LListD_Fun (diag A) r <= (llist A) Times (llist A)";
paulson@3120
   620
by (stac llist_unfold 1);
paulson@3120
   621
by (simp_tac (!simpset addsimps [NIL_def, CONS_def]) 1);
paulson@3120
   622
by (Fast_tac 1);
paulson@3120
   623
qed "LListD_Fun_subset_Sigma_llist";
paulson@3120
   624
paulson@3120
   625
goal LList.thy
paulson@3120
   626
    "prod_fun Rep_llist Rep_llist `` r <= \
paulson@3120
   627
\    (llist(range Leaf)) Times (llist(range Leaf))";
paulson@3120
   628
by (fast_tac (!claset addIs [Rep_llist]) 1);
paulson@3120
   629
qed "subset_Sigma_llist";
paulson@3120
   630
paulson@3120
   631
val [prem] = goal LList.thy
paulson@3120
   632
    "r <= (llist(range Leaf)) Times (llist(range Leaf)) ==> \
paulson@3120
   633
\    prod_fun (Rep_llist o Abs_llist) (Rep_llist o Abs_llist) `` r <= r";
paulson@3120
   634
by (safe_tac (!claset));
paulson@3120
   635
by (rtac (prem RS subsetD RS SigmaE2) 1);
paulson@3120
   636
by (assume_tac 1);
paulson@3120
   637
by (asm_simp_tac (!simpset addsimps [o_def,prod_fun,Abs_llist_inverse]) 1);
paulson@3120
   638
qed "prod_fun_lemma";
paulson@3120
   639
paulson@3120
   640
goal LList.thy
paulson@3120
   641
    "prod_fun Rep_llist  Rep_llist `` range(%x. (x, x)) = \
paulson@3120
   642
\    diag(llist(range Leaf))";
paulson@3120
   643
by (rtac equalityI 1);
paulson@3120
   644
by (fast_tac (!claset addIs [Rep_llist]) 1);
paulson@3120
   645
by (fast_tac (!claset addSEs [Abs_llist_inverse RS subst]) 1);
paulson@3120
   646
qed "prod_fun_range_eq_diag";
paulson@3120
   647
paulson@3120
   648
(*Surprisingly hard to prove.  Used with lfilter*)
paulson@3120
   649
goalw thy [llistD_Fun_def, prod_fun_def]
paulson@3120
   650
    "!!A B. A<=B ==> llistD_Fun A <= llistD_Fun B";
paulson@3120
   651
by (Auto_tac());
paulson@3120
   652
by (rtac image_eqI 1);
paulson@3120
   653
by (fast_tac (!claset addss (!simpset)) 1);
paulson@3120
   654
by (blast_tac (!claset addIs [impOfSubs LListD_Fun_mono]) 1);
paulson@3120
   655
qed "llistD_Fun_mono";
paulson@3120
   656
paulson@3120
   657
(** To show two llists are equal, exhibit a bisimulation! 
paulson@3120
   658
      [also admits true equality] **)
paulson@3120
   659
val [prem1,prem2] = goalw LList.thy [llistD_Fun_def]
paulson@3120
   660
    "[| (l1,l2) : r;  r <= llistD_Fun(r Un range(%x.(x,x))) |] ==> l1=l2";
paulson@3120
   661
by (rtac (inj_Rep_llist RS injD) 1);
paulson@3120
   662
by (res_inst_tac [("r", "prod_fun Rep_llist Rep_llist ``r"),
paulson@3120
   663
                  ("A", "range(Leaf)")] 
paulson@3120
   664
        LList_equalityI 1);
paulson@3120
   665
by (rtac (prem1 RS prod_fun_imageI) 1);
paulson@3120
   666
by (rtac (prem2 RS image_mono RS subset_trans) 1);
paulson@3120
   667
by (rtac (image_compose RS subst) 1);
paulson@3120
   668
by (rtac (prod_fun_compose RS subst) 1);
paulson@3120
   669
by (stac image_Un 1);
paulson@3120
   670
by (stac prod_fun_range_eq_diag 1);
paulson@3120
   671
by (rtac (LListD_Fun_subset_Sigma_llist RS prod_fun_lemma) 1);
paulson@3120
   672
by (rtac (subset_Sigma_llist RS Un_least) 1);
paulson@3120
   673
by (rtac diag_subset_Sigma 1);
paulson@3120
   674
qed "llist_equalityI";
paulson@3120
   675
paulson@3120
   676
(** Rules to prove the 2nd premise of llist_equalityI **)
paulson@3120
   677
goalw LList.thy [llistD_Fun_def,LNil_def] "(LNil,LNil) : llistD_Fun(r)";
paulson@3120
   678
by (rtac (LListD_Fun_NIL_I RS prod_fun_imageI) 1);
paulson@3120
   679
qed "llistD_Fun_LNil_I";
paulson@3120
   680
paulson@3120
   681
val [prem] = goalw LList.thy [llistD_Fun_def,LCons_def]
paulson@3120
   682
    "(l1,l2):r ==> (LCons x l1, LCons x l2) : llistD_Fun(r)";
paulson@3120
   683
by (rtac (rangeI RS LListD_Fun_CONS_I RS prod_fun_imageI) 1);
paulson@3120
   684
by (rtac (prem RS prod_fun_imageI) 1);
paulson@3120
   685
qed "llistD_Fun_LCons_I";
paulson@3120
   686
paulson@3120
   687
(*Utilise the "strong" part, i.e. gfp(f)*)
paulson@3120
   688
goalw LList.thy [llistD_Fun_def]
paulson@3120
   689
     "!!l. (l,l) : llistD_Fun(r Un range(%x.(x,x)))";
paulson@3120
   690
by (rtac (Rep_llist_inverse RS subst) 1);
paulson@3120
   691
by (rtac prod_fun_imageI 1);
paulson@3120
   692
by (stac image_Un 1);
paulson@3120
   693
by (stac prod_fun_range_eq_diag 1);
paulson@3120
   694
by (rtac (Rep_llist RS LListD_Fun_diag_I) 1);
paulson@3120
   695
qed "llistD_Fun_range_I";
paulson@3120
   696
paulson@3120
   697
(*A special case of list_equality for functions over lazy lists*)
paulson@3120
   698
val [prem1,prem2] = goal LList.thy
paulson@3120
   699
    "[| f(LNil)=g(LNil);                                                \
paulson@3120
   700
\       !!x l. (f(LCons x l),g(LCons x l)) :                            \
paulson@3120
   701
\              llistD_Fun(range(%u. (f(u),g(u))) Un range(%v. (v,v)))   \
paulson@3120
   702
\    |] ==> f(l) = (g(l :: 'a llist) :: 'b llist)";
paulson@3120
   703
by (res_inst_tac [("r", "range(%u. (f(u),g(u)))")] llist_equalityI 1);
paulson@3120
   704
by (rtac rangeI 1);
paulson@3120
   705
by (rtac subsetI 1);
paulson@3120
   706
by (etac rangeE 1);
paulson@3120
   707
by (etac ssubst 1);
paulson@3120
   708
by (res_inst_tac [("l", "u")] llistE 1);
paulson@3120
   709
by (etac ssubst 1);
paulson@3120
   710
by (stac prem1 1);
paulson@3120
   711
by (rtac llistD_Fun_range_I 1);
paulson@3120
   712
by (etac ssubst 1);
paulson@3120
   713
by (rtac prem2 1);
paulson@3120
   714
qed "llist_fun_equalityI";
paulson@3120
   715
paulson@3120
   716
(*simpset for llist bisimulations*)
paulson@3120
   717
Addsimps [llist_case_LNil, llist_case_LCons, 
paulson@3120
   718
          llistD_Fun_LNil_I, llistD_Fun_LCons_I];
paulson@3120
   719
paulson@3120
   720
paulson@3120
   721
(*** The functional "lmap" ***)
paulson@3120
   722
paulson@3120
   723
goal LList.thy "lmap f LNil = LNil";
paulson@3120
   724
by (rtac (lmap_def RS def_llist_corec RS trans) 1);
paulson@3120
   725
by (Simp_tac 1);
paulson@3120
   726
qed "lmap_LNil";
paulson@3120
   727
paulson@3120
   728
goal LList.thy "lmap f (LCons M N) = LCons (f M) (lmap f N)";
paulson@3120
   729
by (rtac (lmap_def RS def_llist_corec RS trans) 1);
paulson@3120
   730
by (Simp_tac 1);
paulson@3120
   731
qed "lmap_LCons";
paulson@3120
   732
paulson@3120
   733
Addsimps [lmap_LNil, lmap_LCons];
paulson@3120
   734
paulson@3120
   735
paulson@3120
   736
(** Two easy results about lmap **)
paulson@3120
   737
paulson@3120
   738
goal LList.thy "lmap (f o g) l = lmap f (lmap g l)";
paulson@3120
   739
by (res_inst_tac [("l","l")] llist_fun_equalityI 1);
paulson@3120
   740
by (ALLGOALS Simp_tac);
paulson@3120
   741
qed "lmap_compose";
paulson@3120
   742
wenzelm@3842
   743
goal LList.thy "lmap (%x. x) l = l";
paulson@3120
   744
by (res_inst_tac [("l","l")] llist_fun_equalityI 1);
paulson@3120
   745
by (ALLGOALS Simp_tac);
paulson@3120
   746
qed "lmap_ident";
paulson@3120
   747
paulson@3120
   748
paulson@3120
   749
(*** iterates -- llist_fun_equalityI cannot be used! ***)
paulson@3120
   750
paulson@3120
   751
goal LList.thy "iterates f x = LCons x (iterates f (f x))";
paulson@3120
   752
by (rtac (iterates_def RS def_llist_corec RS trans) 1);
paulson@3120
   753
by (Simp_tac 1);
paulson@3120
   754
qed "iterates";
paulson@3120
   755
paulson@3120
   756
goal LList.thy "lmap f (iterates f x) = iterates f (f x)";
paulson@3120
   757
by (res_inst_tac [("r", "range(%u.(lmap f (iterates f u),iterates f (f u)))")] 
paulson@3120
   758
    llist_equalityI 1);
paulson@3120
   759
by (rtac rangeI 1);
paulson@3120
   760
by (safe_tac (!claset));
paulson@3120
   761
by (res_inst_tac [("x1", "f(u)")] (iterates RS ssubst) 1);
paulson@3120
   762
by (res_inst_tac [("x1", "u")] (iterates RS ssubst) 1);
paulson@3120
   763
by (Simp_tac 1);
paulson@3120
   764
qed "lmap_iterates";
paulson@3120
   765
paulson@3120
   766
goal LList.thy "iterates f x = LCons x (lmap f (iterates f x))";
paulson@3120
   767
by (stac lmap_iterates 1);
paulson@3120
   768
by (rtac iterates 1);
paulson@3120
   769
qed "iterates_lmap";
paulson@3120
   770
paulson@3120
   771
(*** A rather complex proof about iterates -- cf Andy Pitts ***)
paulson@3120
   772
paulson@3120
   773
(** Two lemmas about natrec n x (%m.g), which is essentially (g^n)(x) **)
paulson@3120
   774
paulson@3120
   775
goal LList.thy
paulson@3120
   776
    "nat_rec (LCons b l) (%m. lmap(f)) n =      \
paulson@3120
   777
\    LCons (nat_rec b (%m. f) n) (nat_rec l (%m. lmap(f)) n)";
paulson@3120
   778
by (nat_ind_tac "n" 1);
paulson@3120
   779
by (ALLGOALS Asm_simp_tac);
paulson@3120
   780
qed "fun_power_lmap";
paulson@3120
   781
paulson@3120
   782
goal Nat.thy "nat_rec (g x) (%m. g) n = nat_rec x (%m. g) (Suc n)";
paulson@3120
   783
by (nat_ind_tac "n" 1);
paulson@3120
   784
by (ALLGOALS Asm_simp_tac);
paulson@3120
   785
qed "fun_power_Suc";
paulson@3120
   786
paulson@3120
   787
val Pair_cong = read_instantiate_sg (sign_of Prod.thy)
paulson@3120
   788
 [("f","Pair")] (standard(refl RS cong RS cong));
paulson@3120
   789
paulson@3120
   790
(*The bisimulation consists of {(lmap(f)^n (h(u)), lmap(f)^n (iterates(f,u)))}
paulson@3120
   791
  for all u and all n::nat.*)
paulson@3120
   792
val [prem] = goal LList.thy
paulson@3120
   793
    "(!!x. h(x) = LCons x (lmap f (h x))) ==> h = iterates(f)";
paulson@3120
   794
by (rtac ext 1);
paulson@3120
   795
by (res_inst_tac [("r", 
wenzelm@3842
   796
   "UN u. range(%n. (nat_rec (h u) (%m y. lmap f y) n, \
wenzelm@3842
   797
\                    nat_rec (iterates f u) (%m y. lmap f y) n))")] 
paulson@3120
   798
    llist_equalityI 1);
paulson@3120
   799
by (REPEAT (resolve_tac [UN1_I, range_eqI, Pair_cong, nat_rec_0 RS sym] 1));
paulson@3120
   800
by (safe_tac (!claset));
paulson@3120
   801
by (stac iterates 1);
paulson@3120
   802
by (stac prem 1);
paulson@3120
   803
by (stac fun_power_lmap 1);
paulson@3120
   804
by (stac fun_power_lmap 1);
paulson@3120
   805
by (rtac llistD_Fun_LCons_I 1);
paulson@3120
   806
by (rtac (lmap_iterates RS subst) 1);
paulson@3120
   807
by (stac fun_power_Suc 1);
paulson@3120
   808
by (stac fun_power_Suc 1);
paulson@3120
   809
by (rtac (UN1_I RS UnI1) 1);
paulson@3120
   810
by (rtac rangeI 1);
paulson@3120
   811
qed "iterates_equality";
paulson@3120
   812
paulson@3120
   813
paulson@3120
   814
(*** lappend -- its two arguments cause some complications! ***)
paulson@3120
   815
paulson@3120
   816
goalw LList.thy [lappend_def] "lappend LNil LNil = LNil";
paulson@3120
   817
by (rtac (llist_corec RS trans) 1);
paulson@3120
   818
by (Simp_tac 1);
paulson@3120
   819
qed "lappend_LNil_LNil";
paulson@3120
   820
paulson@3120
   821
goalw LList.thy [lappend_def]
paulson@3120
   822
    "lappend LNil (LCons l l') = LCons l (lappend LNil l')";
paulson@3120
   823
by (rtac (llist_corec RS trans) 1);
paulson@3120
   824
by (Simp_tac 1);
paulson@3120
   825
qed "lappend_LNil_LCons";
paulson@3120
   826
paulson@3120
   827
goalw LList.thy [lappend_def]
paulson@3120
   828
    "lappend (LCons l l') N = LCons l (lappend l' N)";
paulson@3120
   829
by (rtac (llist_corec RS trans) 1);
paulson@3120
   830
by (Simp_tac 1);
paulson@3120
   831
qed "lappend_LCons";
paulson@3120
   832
paulson@3120
   833
Addsimps [lappend_LNil_LNil, lappend_LNil_LCons, lappend_LCons];
paulson@3120
   834
paulson@3120
   835
goal LList.thy "lappend LNil l = l";
paulson@3120
   836
by (res_inst_tac [("l","l")] llist_fun_equalityI 1);
paulson@3120
   837
by (ALLGOALS Simp_tac);
paulson@3120
   838
qed "lappend_LNil";
paulson@3120
   839
paulson@3120
   840
goal LList.thy "lappend l LNil = l";
paulson@3120
   841
by (res_inst_tac [("l","l")] llist_fun_equalityI 1);
paulson@3120
   842
by (ALLGOALS Simp_tac);
paulson@3120
   843
qed "lappend_LNil2";
paulson@3120
   844
paulson@3120
   845
Addsimps [lappend_LNil, lappend_LNil2];
paulson@3120
   846
paulson@3120
   847
(*The infinite first argument blocks the second*)
paulson@3120
   848
goal LList.thy "lappend (iterates f x) N = iterates f x";
paulson@3120
   849
by (res_inst_tac [("r", "range(%u.(lappend (iterates f u) N,iterates f u))")] 
paulson@3120
   850
    llist_equalityI 1);
paulson@3120
   851
by (rtac rangeI 1);
paulson@3120
   852
by (safe_tac (!claset));
paulson@3120
   853
by (stac iterates 1);
paulson@3120
   854
by (Simp_tac 1);
paulson@3120
   855
qed "lappend_iterates";
paulson@3120
   856
paulson@3120
   857
(** Two proofs that lmap distributes over lappend **)
paulson@3120
   858
paulson@3120
   859
(*Long proof requiring case analysis on both both arguments*)
paulson@3120
   860
goal LList.thy "lmap f (lappend l n) = lappend (lmap f l) (lmap f n)";
paulson@3120
   861
by (res_inst_tac 
paulson@3120
   862
    [("r",  
paulson@3120
   863
      "UN n. range(%l.(lmap f (lappend l n),lappend (lmap f l) (lmap f n)))")] 
paulson@3120
   864
    llist_equalityI 1);
paulson@3120
   865
by (rtac UN1_I 1);
paulson@3120
   866
by (rtac rangeI 1);
paulson@3120
   867
by (safe_tac (!claset));
paulson@3120
   868
by (res_inst_tac [("l", "l")] llistE 1);
paulson@3120
   869
by (res_inst_tac [("l", "n")] llistE 1);
paulson@3120
   870
by (ALLGOALS Asm_simp_tac);
paulson@3120
   871
by (REPEAT_SOME (ares_tac [llistD_Fun_LCons_I, UN1_I RS UnI1, rangeI]));
paulson@3120
   872
qed "lmap_lappend_distrib";
paulson@3120
   873
paulson@3120
   874
(*Shorter proof of theorem above using llist_equalityI as strong coinduction*)
paulson@3120
   875
goal LList.thy "lmap f (lappend l n) = lappend (lmap f l) (lmap f n)";
paulson@3120
   876
by (res_inst_tac [("l","l")] llist_fun_equalityI 1);
paulson@3120
   877
by (Simp_tac 1);
paulson@3120
   878
by (Simp_tac 1);
paulson@3120
   879
qed "lmap_lappend_distrib";
paulson@3120
   880
paulson@3120
   881
(*Without strong coinduction, three case analyses might be needed*)
paulson@3120
   882
goal LList.thy "lappend (lappend l1 l2) l3 = lappend l1 (lappend l2 l3)";
paulson@3120
   883
by (res_inst_tac [("l","l1")] llist_fun_equalityI 1);
paulson@3120
   884
by (Simp_tac 1);
paulson@3120
   885
by (Simp_tac 1);
paulson@3120
   886
qed "lappend_assoc";