LList.ML
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(*  Title: 	HOL/llist
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    ID:         $Id$
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    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1993  University of Cambridge
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For llist.thy.  
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SHOULD LListD_Fun_CONS_I, etc., be equations (for rewriting)?
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*)
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open LList;
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(** Simplification **)
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val llist_simps = [case_Inl, case_Inr];
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val llist_ss = univ_ss delsimps [Pair_eq] 
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                       addsimps llist_simps
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                       addcongs [split_weak_cong, case_weak_cong]
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                       setloop (split_tac [expand_split, expand_case]);
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(** the llist functional **)
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val LList_unfold = rewrite_rule [List_Fun_def]
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	 (List_Fun_mono RS (LList_def RS def_gfp_Tarski));
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(*This justifies using LList in other recursive type definitions*)
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goalw LList.thy [LList_def] "!!A B. A<=B ==> LList(A) <= LList(B)";
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by (rtac gfp_mono 1);
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by (etac List_Fun_mono2 1);
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val LList_mono = result();
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(*Elimination is case analysis, not induction.*)
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val [major,prem1,prem2] = goalw LList.thy [NIL_def,CONS_def]
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    "[| L : LList(A);  \
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\       L=NIL ==> P; \
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\       !!M N. [| M:A;  N: LList(A);  L=CONS(M,N) |] ==> P \
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\    |] ==> P";
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by (rtac (major RS (LList_unfold RS equalityD1 RS subsetD RS usumE)) 1);
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by (etac uprodE 2);
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by (rtac prem2 2);
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by (rtac prem1 1);
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by (REPEAT (ares_tac [refl] 1
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     ORELSE eresolve_tac [singletonE,ssubst] 1));
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val LListE = result();
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(*** Type checking by co-induction, using List_Fun ***)
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val prems = goalw LList.thy [LList_def]
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    "[| M : X;  X <= List_Fun(A, X Un LList(A)) |] ==>  M : LList(A)";
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by (REPEAT (resolve_tac (prems@[List_Fun_mono RS coinduct]) 1));
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val LList_coinduct = result();
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(** Rules to prove the 2nd premise of LList_coinduct **)
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goalw LList.thy [List_Fun_def,NIL_def] "NIL: List_Fun(A,X)";
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by (resolve_tac [singletonI RS usum_In0I] 1);
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val List_Fun_NIL_I = result();
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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 (REPEAT (ares_tac [uprodI RS usum_In1I] 1));
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val List_Fun_CONS_I = result();
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(*Utilise the "strong" part, i.e. gfp(f)*)
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goalw LList.thy [LList_def]
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    "!!M N. M: LList(A) ==> M : List_Fun(A, X Un LList(A))";
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by (etac (List_Fun_mono RS gfp_fun_UnI2) 1);
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val List_Fun_LList_I = result();
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(*** LList_corec satisfies the desired recurion equation ***)
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(*A continuity result?*)
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goalw LList.thy [CONS_def] "CONS(M, UN x.f(x)) = (UN x. CONS(M, f(x)))";
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by (simp_tac (univ_ss addsimps [In1_UN1, Scons_UN1_y]) 1);
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val CONS_UN1 = result();
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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 (pair_ss setloop (split_tac [expand_split])) 1);
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val split_UN1 = result();
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goal Sum.thy "case(s, f, %y. UN z.g(y,z)) = (UN z. case(s, f, %y. g(y,z)))";
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by (simp_tac (sum_ss setloop (split_tac [expand_case])) 1);
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val case2_UN1 = result();
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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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val CONS_mono = result();
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val corec_fun_simps = [LList_corec_fun_def RS def_nat_rec_0,
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		       LList_corec_fun_def RS def_nat_rec_Suc];
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val corec_fun_ss = llist_ss addsimps corec_fun_simps;
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(** The directions of the equality are proved separately **)
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goalw LList.thy [LList_corec_def]
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    "LList_corec(a,f) <= case(f(a), %u.NIL, \
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\			   %v. split(v, %z w. CONS(z, LList_corec(w,f))))";
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by (rtac UN1_least 1);
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by (res_inst_tac [("n","k")] natE 1);
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by (ALLGOALS (asm_simp_tac corec_fun_ss));
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by (REPEAT (resolve_tac [allI, impI, subset_refl RS CONS_mono, UN1_upper] 1));
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val LList_corec_subset1 = result();
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goalw LList.thy [LList_corec_def]
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    "case(f(a), %u.NIL, %v. split(v, %z w. CONS(z, LList_corec(w,f)))) <= \
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\    LList_corec(a,f)";
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by (simp_tac (corec_fun_ss addsimps [CONS_UN1]) 1);
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by (safe_tac set_cs);
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by (ALLGOALS (res_inst_tac [("x","Suc(?k)")] UN1_I THEN' 
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	      asm_simp_tac corec_fun_ss));
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val LList_corec_subset2 = result();
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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) = case(f(a), %u. NIL, \
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\			     %v. split(v, %z w. CONS(z, LList_corec(w,f))))";
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by (REPEAT (resolve_tac [equalityI, LList_corec_subset1, 
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			 LList_corec_subset2] 1));
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val LList_corec = result();
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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) = case(f(a), %u.NIL, %v. split(v, %z w. CONS(z, h(w))))";
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by (rewtac rew);
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by (rtac LList_corec 1);
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val def_LList_corec = result();
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(*A typical use of co-induction to show membership in the gfp. 
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  Bisimulation is  range(%x. LList_corec(x,f)) *)
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goal LList.thy "LList_corec(a,f) : LList({u.True})";
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by (res_inst_tac [("X", "range(%x.LList_corec(x,?g))")] LList_coinduct 1);
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by (rtac rangeI 1);
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by (safe_tac set_cs);
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by (stac LList_corec 1);
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by (simp_tac (llist_ss addsimps [List_Fun_NIL_I,List_Fun_CONS_I,
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                                 CollectI, range_eqI]) 1);
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(* 6.7 vs 3.4 !!!
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by (ASM_SIMP_TAC (llist_ss addsimps [List_Fun_NIL_I,List_Fun_CONS_I,
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				    CollectI, rangeI]) 1);
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*)
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val LList_corec_type = result();
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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. case(f(z),Inl,%x. split(x,%v w. Inr(<Leaf(v),w>)))) : \
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\    LList(range(Leaf))";
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by (res_inst_tac [("X", "range(%x.LList_corec(x,?g))")] LList_coinduct 1);
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by (rtac rangeI 1);
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by (safe_tac set_cs);
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by (stac LList_corec 1);
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(*nested "case"; requires an explicit split*)
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by (res_inst_tac [("s", "f(xa)")] sumE 1);
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by (asm_simp_tac (univ_ss addsimps (llist_simps@[List_Fun_NIL_I])) 1);
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by (asm_simp_tac (univ_ss addsimps (llist_simps@[List_Fun_CONS_I, range_eqI])
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                     setloop (split_tac [expand_split])) 1);
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(* FIXME: can the selection of the case split be automated?
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by (ASM_SIMP_TAC (llist_ss addsimps [List_Fun_CONS_I, rangeI]) 1);*)
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val LList_corec_type2 = result();
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(**** LList equality as a gfp; the bisimulation principle ****)
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goalw LList.thy [LListD_Fun_def] "mono(LListD_Fun(r))";
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by (REPEAT (ares_tac [monoI, subset_refl, dsum_mono, dprod_mono] 1));
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val LListD_Fun_mono = result();
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val LListD_unfold = rewrite_rule [LListD_Fun_def]
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	 (LListD_Fun_mono RS (LListD_def RS def_gfp_Tarski));
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goal LList.thy "!M N. <M,N> : LListD(diag(A)) --> ntrunc(k,M) = ntrunc(k,N)";
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by (res_inst_tac [("n", "k")] less_induct 1);
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by (safe_tac set_cs);
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by (etac (LListD_unfold RS equalityD1 RS subsetD RS dsumE) 1);
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by (safe_tac (set_cs addSEs [Pair_inject, dprodE, diagE]));
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by (res_inst_tac [("n", "n")] natE 1);
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by (asm_simp_tac (univ_ss addsimps [ntrunc_0]) 1);
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by (res_inst_tac [("n", "xb")] natE 1);
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by (asm_simp_tac (univ_ss addsimps [ntrunc_one_In1]) 1);
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by (asm_simp_tac (univ_ss addsimps [ntrunc_In1, ntrunc_Scons]) 1);
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val LListD_implies_ntrunc_equality = result();
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goalw LList.thy [LList_def,List_Fun_def] "fst``LListD(diag(A)) <= LList(A)";
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by (rtac gfp_upperbound 1);
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by (res_inst_tac [("P", "%x. fst``x <= ?B")] (LListD_unfold RS ssubst) 1);
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by (simp_tac fst_image_ss 1);
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val fst_image_LListD = result();
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(*This inclusion justifies the use of coinduction to show M=N*)
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goal LList.thy "LListD(diag(A)) <= diag(LList(A))";
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by (rtac subsetI 1);
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by (res_inst_tac [("p","x")] PairE 1);
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by (safe_tac HOL_cs);
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by (res_inst_tac [("s","xa")] subst 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 (rtac diagI 1);
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by (etac (fst_imageI RS (fst_image_LListD RS subsetD)) 1);
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val LListD_subset_diag = result();
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(*This converse inclusion helps to strengthen LList_equalityI*)
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goalw LList.thy [LListD_def] "diag(LList(A)) <= LListD(diag(A))";
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by (rtac gfp_upperbound 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 (etac (LList_unfold RS equalityD1 RS subsetD RS usumE) 1);
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by (rewtac LListD_Fun_def);
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by (ALLGOALS (fast_tac (set_cs addIs [diagI,dsum_In0I,dsum_In1I,dprodI]
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                               addSEs [uprodE])));
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val diag_subset_LListD = result();
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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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val LListD_eq_diag = result();
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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 (rewrite_rule [LListD_def]
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           (LListD_subset_diag RS subsetD RS diagE)) 1);
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by (etac (LListD_Fun_mono RS coinduct) 1);
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by (etac (rewrite_rule [LListD_def] LListD_eq_diag RS ssubst) 1);
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by (safe_tac (set_cs addSEs [Pair_inject]));
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val LList_equalityI = result();
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(** Rules to prove the 2nd premise of LList_equalityI **)
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goalw LList.thy [LListD_Fun_def,NIL_def] "<NIL,NIL> : LListD_Fun(r,s)";
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by (rtac (singletonI RS diagI RS dsum_In0I) 1);
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val LListD_Fun_NIL_I = result();
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val prems = goalw LList.thy [LListD_Fun_def,CONS_def]
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    "[| x:A;  <M,N>:s |] ==> <CONS(x,M), CONS(x,N)> : LListD_Fun(diag(A),s)";
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by (rtac (dprodI RS dsum_In1I) 1);
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by (REPEAT (resolve_tac (diagI::prems) 1));
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val LListD_Fun_CONS_I = result();
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(*Utilise the "strong" part, i.e. gfp(f)*)
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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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br (rewrite_rule [LListD_def] LListD_eq_diag RS subst) 1;
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br (LListD_Fun_mono RS gfp_fun_UnI2) 1;
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br (rewrite_rule [LListD_def] LListD_eq_diag RS ssubst) 1;
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be diagI 1;
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val LListD_Fun_diag_I = result();
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0
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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) = case(f(x), %u.NIL, %v. split(v, %z w. CONS(z,h1(w))));   \
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\    !!x. h2(x) = case(f(x), %u.NIL, %v. split(v, %z w. CONS(z,h2(w)))) |] \
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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 set_cs);
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by (stac prem1 1);
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by (stac prem2 1);
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by (simp_tac (llist_ss addsimps [LListD_Fun_NIL_I, range_eqI,
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				 CollectI RS LListD_Fun_CONS_I]) 1);
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(* 9.5 vs 9.2/4.1/4.3
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by (ASM_SIMP_TAC (llist_ss addsimps [LListD_Fun_NIL_I, rangeI,
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				    CollectI RS LListD_Fun_CONS_I]) 1);*)
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val LList_corec_unique = result();
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val [prem] = goal LList.thy
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 "[| !!x. h(x) = case(f(x), %u.NIL, %v. split(v, %z w. CONS(z,h(w)))) |] \
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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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val equals_LList_corec = result();
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(** Obsolete LList_corec_unique proof: complete induction, not coinduction **)
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goalw LList.thy [CONS_def] "ntrunc(Suc(0), CONS(M,N)) = {}";
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by (rtac ntrunc_one_In1 1);
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val ntrunc_one_CONS = result();
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goalw LList.thy [CONS_def]
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    "ntrunc(Suc(Suc(k)), CONS(M,N)) = CONS (ntrunc(k,M), ntrunc(k,N))";
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by (simp_tac (HOL_ss addsimps [ntrunc_Scons,ntrunc_In1]) 1);
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val ntrunc_CONS = result();
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val [prem1,prem2] = goal LList.thy
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 "[| !!x. h1(x) = case(f(x), %u.NIL, %v. split(v, %z w. CONS(z,h1(w))));  \
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\    !!x. h2(x) = case(f(x), %u.NIL, %v. split(v, %z w. CONS(z,h2(w)))) |] \
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\ ==> h1=h2";
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by (rtac (ntrunc_equality RS ext) 1);
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by (res_inst_tac [("x", "x")] spec 1);
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by (res_inst_tac [("n", "k")] less_induct 1);
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by (rtac allI 1);
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by (stac prem1 1);
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by (stac prem2 1);
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by (simp_tac (sum_ss setloop (split_tac [expand_split,expand_case])) 1);
0
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by (strip_tac 1);
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by (res_inst_tac [("n", "n")] natE 1);
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by (res_inst_tac [("n", "xc")] natE 2);
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by (ALLGOALS(asm_simp_tac(nat_ss addsimps
0
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            [ntrunc_0,ntrunc_one_CONS,ntrunc_CONS])));
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val LList_corec_unique = result();
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(*** Lconst -- defined directly using lfp, but equivalent to a LList_corec ***)
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goal LList.thy "mono(CONS(M))";
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parents:
diff changeset
   319
by (REPEAT (ares_tac [monoI, subset_refl, CONS_mono] 1));
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   320
val Lconst_fun_mono = result();
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   321
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parents:
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   322
(* Lconst(M) = CONS(M,Lconst(M)) *)
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   323
val Lconst = standard (Lconst_fun_mono RS (Lconst_def RS def_lfp_Tarski));
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parents:
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   324
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parents:
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   325
(*A typical use of co-induction to show membership in the gfp.
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parents:
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   326
  The containing set is simply the singleton {Lconst(M)}. *)
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parents:
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   327
goal LList.thy "!!M A. M:A ==> Lconst(M): LList(A)";
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clasohm
parents:
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   328
by (rtac (singletonI RS LList_coinduct) 1);
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clasohm
parents:
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   329
by (safe_tac set_cs);
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clasohm
parents:
diff changeset
   330
by (res_inst_tac [("P", "%u. u: ?A")] (Lconst RS ssubst) 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   331
by (REPEAT (ares_tac [List_Fun_CONS_I, singletonI, UnI1] 1));
0
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parents:
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   332
val Lconst_type = result();
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clasohm
parents:
diff changeset
   333
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clasohm
parents:
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   334
goal LList.thy "Lconst(M) = LList_corec(M, %x.Inr(<x,x>))";
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clasohm
parents:
diff changeset
   335
by (rtac (equals_LList_corec RS fun_cong) 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   336
by (simp_tac sum_ss 1);
0
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clasohm
parents:
diff changeset
   337
by (rtac Lconst 1);
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clasohm
parents:
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   338
val Lconst_eq_LList_corec = result();
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clasohm
parents:
diff changeset
   339
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clasohm
parents:
diff changeset
   340
(*Thus we could have used gfp in the definition of Lconst*)
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clasohm
parents:
diff changeset
   341
goal LList.thy "gfp(%N. CONS(M, N)) = LList_corec(M, %x.Inr(<x,x>))";
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clasohm
parents:
diff changeset
   342
by (rtac (equals_LList_corec RS fun_cong) 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   343
by (simp_tac sum_ss 1);
0
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clasohm
parents:
diff changeset
   344
by (rtac (Lconst_fun_mono RS gfp_Tarski) 1);
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clasohm
parents:
diff changeset
   345
val gfp_Lconst_eq_LList_corec = result();
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clasohm
parents:
diff changeset
   346
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clasohm
parents:
diff changeset
   347
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clasohm
parents:
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   348
(** Introduction rules for LList constructors **)
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   349
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parents:
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   350
(* c : {Numb(0)} <+> A <*> LList(A) ==> c : LList(A) *)
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parents:
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   351
val LListI = LList_unfold RS equalityD2 RS subsetD;
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clasohm
parents:
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   352
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clasohm
parents:
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   353
(*This justifies the type definition: LList(A) is nonempty.*)
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parents:
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   354
goalw LList.thy [NIL_def] "NIL: LList(A)";
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clasohm
parents:
diff changeset
   355
by (rtac (singletonI RS usum_In0I RS LListI) 1);
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clasohm
parents:
diff changeset
   356
val NIL_LListI = result();
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clasohm
parents:
diff changeset
   357
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clasohm
parents:
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   358
val prems = goalw LList.thy [CONS_def]
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clasohm
parents:
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   359
    "[| M: A;  N: LList(A) |] ==> CONS(M,N) : LList(A)";
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clasohm
parents:
diff changeset
   360
by (rtac (uprodI RS usum_In1I RS LListI) 1);
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clasohm
parents:
diff changeset
   361
by (REPEAT (resolve_tac prems 1));
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clasohm
parents:
diff changeset
   362
val CONS_LListI = result();
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clasohm
parents:
diff changeset
   363
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clasohm
parents:
diff changeset
   364
(*** Isomorphisms ***)
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clasohm
parents:
diff changeset
   365
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clasohm
parents:
diff changeset
   366
goal LList.thy "inj(Rep_LList)";
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clasohm
parents:
diff changeset
   367
by (rtac inj_inverseI 1);
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clasohm
parents:
diff changeset
   368
by (rtac Rep_LList_inverse 1);
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clasohm
parents:
diff changeset
   369
val inj_Rep_LList = result();
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clasohm
parents:
diff changeset
   370
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clasohm
parents:
diff changeset
   371
goal LList.thy "inj_onto(Abs_LList,LList(range(Leaf)))";
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clasohm
parents:
diff changeset
   372
by (rtac inj_onto_inverseI 1);
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clasohm
parents:
diff changeset
   373
by (etac Abs_LList_inverse 1);
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clasohm
parents:
diff changeset
   374
val inj_onto_Abs_LList = result();
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clasohm
parents:
diff changeset
   375
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clasohm
parents:
diff changeset
   376
(** Distinctness of constructors **)
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clasohm
parents:
diff changeset
   377
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clasohm
parents:
diff changeset
   378
goalw LList.thy [LNil_def,LCons_def] "~ LCons(x,xs) = LNil";
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clasohm
parents:
diff changeset
   379
by (rtac (CONS_not_NIL RS (inj_onto_Abs_LList RS inj_onto_contraD)) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   380
by (REPEAT (resolve_tac [rangeI, NIL_LListI, CONS_LListI, Rep_LList] 1));
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   381
val LCons_not_LNil = result();
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clasohm
parents:
diff changeset
   382
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clasohm
parents:
diff changeset
   383
val LNil_not_LCons = standard (LCons_not_LNil RS not_sym);
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clasohm
parents:
diff changeset
   384
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clasohm
parents:
diff changeset
   385
val LCons_neq_LNil = standard (LCons_not_LNil RS notE);
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clasohm
parents:
diff changeset
   386
val LNil_neq_LCons = sym RS LCons_neq_LNil;
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clasohm
parents:
diff changeset
   387
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clasohm
parents:
diff changeset
   388
(** llist constructors **)
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clasohm
parents:
diff changeset
   389
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clasohm
parents:
diff changeset
   390
goalw LList.thy [LNil_def]
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clasohm
parents:
diff changeset
   391
    "Rep_LList(LNil) = NIL";
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clasohm
parents:
diff changeset
   392
by (rtac (NIL_LListI RS Abs_LList_inverse) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   393
val Rep_LList_LNil = result();
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clasohm
parents:
diff changeset
   394
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clasohm
parents:
diff changeset
   395
goalw LList.thy [LCons_def]
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clasohm
parents:
diff changeset
   396
    "Rep_LList(LCons(x,l)) = CONS(Leaf(x),Rep_LList(l))";
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clasohm
parents:
diff changeset
   397
by (REPEAT (resolve_tac [CONS_LListI RS Abs_LList_inverse,
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   398
			 rangeI, Rep_LList] 1));
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clasohm
parents:
diff changeset
   399
val Rep_LList_LCons = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   400
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   401
(** Injectiveness of CONS and LCons **)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   402
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   403
goalw LList.thy [CONS_def] "(CONS(M,N)=CONS(M',N')) = (M=M' & N=N')";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   404
by (fast_tac (HOL_cs addSEs [Scons_inject, make_elim In1_inject]) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   405
val CONS_CONS_eq = result();
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clasohm
parents:
diff changeset
   406
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   407
val CONS_inject = standard (CONS_CONS_eq RS iffD1 RS conjE);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   408
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   409
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   410
(*For reasoning about abstract llist constructors*)
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clasohm
parents:
diff changeset
   411
val LList_cs = set_cs addIs [Rep_LList, NIL_LListI, CONS_LListI]
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   412
	              addSEs [CONS_neq_NIL,NIL_neq_CONS,CONS_inject]
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   413
		      addSDs [inj_onto_Abs_LList RS inj_ontoD,
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   414
			      inj_Rep_LList RS injD, Leaf_inject];
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clasohm
parents:
diff changeset
   415
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   416
goalw LList.thy [LCons_def] "(LCons(x,xs)=LCons(y,ys)) = (x=y & xs=ys)";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   417
by (fast_tac LList_cs 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   418
val LCons_LCons_eq = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   419
val LCons_inject = standard (LCons_LCons_eq RS iffD1 RS conjE);
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clasohm
parents:
diff changeset
   420
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   421
val [major] = goal LList.thy "CONS(M,N): LList(A) ==> M: A & N: LList(A)";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   422
by (rtac (major RS LListE) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   423
by (etac CONS_neq_NIL 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   424
by (fast_tac LList_cs 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   425
val CONS_D = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   426
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   427
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   428
(****** Reasoning about LList(A) ******)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   429
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   430
val List_case_simps = [List_case_NIL, List_case_CONS];
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   431
val List_case_ss = llist_ss addsimps List_case_simps;
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   432
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   433
(*A special case of list_equality for functions over lazy lists*)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   434
val [MList,gMList,NILcase,CONScase] = goal LList.thy
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   435
 "[| M: LList(A); g(NIL): LList(A); 				\
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   436
\    f(NIL)=g(NIL);						\
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   437
\    !!x l. [| x:A;  l: LList(A) |] ==>				\
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   438
\	    <f(CONS(x,l)),g(CONS(x,l))> :			\
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   439
\               LListD_Fun(diag(A), (%u.<f(u),g(u)>)``LList(A) Un  \
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   440
\                                   diag(LList(A)))		\
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   441
\ |] ==> f(M) = g(M)";
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   442
by (rtac LList_equalityI 1);
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   443
br (MList RS imageI) 1;
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   444
by (rtac subsetI 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   445
by (etac imageE 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   446
by (etac ssubst 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   447
by (etac LListE 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   448
by (etac ssubst 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   449
by (stac NILcase 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   450
br (gMList RS LListD_Fun_diag_I) 1;
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   451
by (etac ssubst 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   452
by (REPEAT (ares_tac [CONScase] 1));
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   453
val LList_fun_equalityI = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   454
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   455
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   456
(*** The functional "Lmap" ***)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   457
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   458
goal LList.thy "Lmap(f,NIL) = NIL";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   459
by (rtac (Lmap_def RS def_LList_corec RS trans) 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   460
by (simp_tac List_case_ss 1);
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   461
val Lmap_NIL = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   462
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   463
goal LList.thy "Lmap(f, CONS(M,N)) = CONS(f(M), Lmap(f,N))";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   464
by (rtac (Lmap_def RS def_LList_corec RS trans) 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   465
by (simp_tac List_case_ss 1);
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   466
val Lmap_CONS = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   467
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   468
(*Another type-checking proof by coinduction*)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   469
val [major,minor] = goal LList.thy
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   470
    "[| M: LList(A);  !!x. x:A ==> f(x):B |] ==> Lmap(f,M): LList(B)";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   471
by (rtac (major RS imageI RS LList_coinduct) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   472
by (safe_tac set_cs);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   473
by (etac LListE 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   474
by (ALLGOALS (asm_simp_tac (HOL_ss addsimps [Lmap_NIL,Lmap_CONS])));
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   475
by (REPEAT (ares_tac [List_Fun_NIL_I, List_Fun_CONS_I, 
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   476
		      minor, imageI, UnI1] 1));
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   477
val Lmap_type = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   478
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   479
(*This type checking rule synthesises a sufficiently large set for f*)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   480
val [major] = goal LList.thy  "M: LList(A) ==> Lmap(f,M): LList(f``A)";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   481
by (rtac (major RS Lmap_type) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   482
by (etac imageI 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   483
val Lmap_type2 = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   484
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   485
(** Two easy results about Lmap **)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   486
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   487
val [prem] = goal LList.thy
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   488
    "M: LList(A) ==> Lmap(f o g, M) = Lmap(f, Lmap(g, M))";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   489
by (rtac (prem RS imageI RS LList_equalityI) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   490
by (stac o_def 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   491
by (safe_tac set_cs);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   492
by (etac LListE 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   493
by (ALLGOALS (asm_simp_tac (HOL_ss addsimps [Lmap_NIL,Lmap_CONS])));
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   494
by (REPEAT (ares_tac [LListD_Fun_NIL_I, imageI, UnI1,
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   495
		      rangeI RS LListD_Fun_CONS_I] 1));
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   496
val Lmap_compose = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   497
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   498
val [prem] = goal LList.thy "M: LList(A) ==> Lmap(%x.x, M) = M";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   499
by (rtac (prem RS imageI RS LList_equalityI) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   500
by (safe_tac set_cs);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   501
by (etac LListE 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   502
by (ALLGOALS (asm_simp_tac (HOL_ss addsimps [Lmap_NIL,Lmap_CONS])));
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   503
by (REPEAT (ares_tac [LListD_Fun_NIL_I, imageI RS UnI1,
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   504
		      rangeI RS LListD_Fun_CONS_I] 1));
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   505
val Lmap_ident = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   506
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   507
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   508
(*** Lappend -- its two arguments cause some complications! ***)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   509
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   510
goalw LList.thy [Lappend_def] "Lappend(NIL,NIL) = NIL";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   511
by (rtac (LList_corec RS trans) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   512
(* takes 2.4(3.4 w NORM) vs 0.9 w/o NORM terms *)
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   513
by (simp_tac List_case_ss 1);
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   514
(*by (SIMP_TAC List_case_ss 1);*)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   515
val Lappend_NIL_NIL = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   516
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   517
goalw LList.thy [Lappend_def]
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   518
    "Lappend(NIL,CONS(N,N')) = CONS(N, Lappend(NIL,N'))";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   519
by (rtac (LList_corec RS trans) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   520
(* takes 5(7 w NORM) vs 2.1 w/o NORM terms *)
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   521
by (simp_tac List_case_ss 1);
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   522
(*by (SIMP_TAC List_case_ss 1);*)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   523
val Lappend_NIL_CONS = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   524
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   525
goalw LList.thy [Lappend_def]
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   526
    "Lappend(CONS(M,M'), N) = CONS(M, Lappend(M',N))";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   527
by (rtac (LList_corec RS trans) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   528
(* takes 4.9(6.7) vs 2.2 w/o NORM terms *)
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   529
by (simp_tac List_case_ss 1);
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   530
(*by (SIMP_TAC List_case_ss 1);*)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   531
val Lappend_CONS = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   532
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   533
val Lappend_ss = List_case_ss addsimps
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   534
                 [NIL_LListI, Lappend_NIL_NIL, Lappend_NIL_CONS,
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   535
	          Lappend_CONS, image_eqI, LListD_Fun_CONS_I];
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   536
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   537
goal LList.thy "!!M. M: LList(A) ==> Lappend(NIL,M) = M";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   538
by (etac LList_fun_equalityI 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   539
by (ALLGOALS (asm_simp_tac Lappend_ss));
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   540
val Lappend_NIL = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   541
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   542
goal LList.thy "!!M. M: LList(A) ==> Lappend(M,NIL) = M";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   543
by (etac LList_fun_equalityI 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   544
by (ALLGOALS (asm_simp_tac Lappend_ss));
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   545
val Lappend_NIL2 = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   546
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   547
(** Alternative type-checking proofs for Lappend **)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   548
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   549
(*weak co-induction: bisimulation and case analysis on both variables*)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   550
goal LList.thy
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   551
    "!!M N. [| M: LList(A); N: LList(A) |] ==> Lappend(M,N): LList(A)";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   552
by (res_inst_tac
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   553
    [("X", "UN u:LList(A). UN v: LList(A). {Lappend(u,v)}")] LList_coinduct 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   554
by (fast_tac set_cs 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   555
by (safe_tac set_cs);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   556
by (eres_inst_tac [("L", "u")] LListE 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   557
by (eres_inst_tac [("L", "v")] LListE 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   558
(* 7/12 vs 7.8/13.3/8.2/13.4 *)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   559
by (ALLGOALS
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   560
    (asm_simp_tac Lappend_ss THEN'
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   561
     fast_tac (set_cs addSIs [NIL_LListI,List_Fun_NIL_I,List_Fun_CONS_I]) ));
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   562
(*
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   563
by (REPEAT 
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   564
    (ASM_SIMP_TAC Lappend_ss 1 THEN
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   565
     fast_tac (set_cs addSIs [NIL_LListI,List_Fun_NIL_I,List_Fun_CONS_I])1));
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   566
*)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   567
val Lappend_type = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   568
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   569
(*strong co-induction: bisimulation and case analysis on one variable*)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   570
goal LList.thy
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   571
    "!!M N. [| M: LList(A); N: LList(A) |] ==> Lappend(M,N): LList(A)";
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   572
by (res_inst_tac [("X", "(%u.Lappend(u,N))``LList(A)")] LList_coinduct 1);
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   573
be imageI 1;
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   574
br subsetI 1;
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   575
be imageE 1;
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   576
by (eres_inst_tac [("L", "u")] LListE 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   577
by (asm_simp_tac (Lappend_ss addsimps [Lappend_NIL, List_Fun_LList_I]) 1);
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   578
by (asm_simp_tac Lappend_ss 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   579
by (fast_tac (set_cs addSIs [List_Fun_CONS_I]) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   580
val Lappend_type = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   581
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   582
(**** Lazy lists as the type 'a llist -- strongly typed versions of above ****)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   583
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   584
(** llist_case: case analysis for 'a llist **)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   585
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   586
val Rep_LList_simps =
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   587
                [List_case_NIL, List_case_CONS, 
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   588
		 Abs_LList_inverse, Rep_LList_inverse, NIL_LListI, CONS_LListI,
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   589
		 Rep_LList, rangeI, inj_Leaf, Inv_f_f];
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   590
val Rep_LList_ss = llist_ss addsimps Rep_LList_simps;
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   591
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   592
goalw LList.thy [llist_case_def,LNil_def]  "llist_case(LNil, c, d) = c";
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   593
by (simp_tac Rep_LList_ss 1);
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   594
val llist_case_LNil = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   595
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   596
goalw LList.thy [llist_case_def,LCons_def]
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   597
    "llist_case(LCons(M,N), c, d) = d(M,N)";
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   598
by (simp_tac Rep_LList_ss 1);
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   599
val llist_case_LCons = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   600
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   601
(*Elimination is case analysis, not induction.*)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   602
val [prem1,prem2] = goalw LList.thy [NIL_def,CONS_def]
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   603
    "[| l=LNil ==> P;  !!x l'. l=LCons(x,l') ==> P \
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   604
\    |] ==> P";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   605
by (rtac (Rep_LList RS LListE) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   606
by (rtac (inj_Rep_LList RS injD RS prem1) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   607
by (stac Rep_LList_LNil 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   608
by (assume_tac 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   609
by (etac rangeE 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   610
by (rtac (inj_Rep_LList RS injD RS prem2) 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   611
by (asm_simp_tac (HOL_ss addsimps [Rep_LList_LCons]) 1);
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   612
by (etac (Abs_LList_inverse RS ssubst) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   613
by (rtac refl 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   614
val llistE = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   615
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   616
(** llist_corec: corecursion for 'a llist **)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   617
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   618
goalw LList.thy [llist_corec_def,LNil_def,LCons_def]
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   619
    "llist_corec(a,f) = case(f(a), %u. LNil, \
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   620
\			     %v. split(v, %z w. LCons(z, llist_corec(w,f))))";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   621
by (stac LList_corec 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   622
by (res_inst_tac [("s","f(a)")] sumE 1);
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   623
by (asm_simp_tac (llist_ss addsimps [LList_corec_type2,Abs_LList_inverse]) 1);
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   624
by (res_inst_tac [("p","y")] PairE 1);
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   625
by (asm_simp_tac (llist_ss addsimps [LList_corec_type2,Abs_LList_inverse]) 1);
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   626
(*FIXME: correct case splits usd to be found automatically:
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   627
by (ASM_SIMP_TAC(llist_ss addsimps [LList_corec_type2,Abs_LList_inverse]) 1);*)
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   628
val llist_corec = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   629
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   630
(*definitional version of same*)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   631
val [rew] = goal LList.thy
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   632
    "[| !!x. h(x) == llist_corec(x,f) |] ==>	\
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   633
\    h(a) = case(f(a), %u.LNil, %v. split(v, %z w. LCons(z, h(w))))";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   634
by (rewtac rew);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   635
by (rtac llist_corec 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   636
val def_llist_corec = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   637
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   638
(**** Proofs about type 'a llist functions ****)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   639
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   640
(*** Deriving llist_equalityI -- llist equality is a bisimulation ***)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   641
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   642
val prems = goalw LList.thy [LListD_Fun_def]
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   643
    "r <= Sigma(LList(A), %x.LList(A)) ==> \
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   644
\    LListD_Fun(diag(A),r) <= Sigma(LList(A), %x.LList(A))";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   645
by (stac LList_unfold 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   646
by (cut_facts_tac prems 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   647
by (fast_tac univ_cs 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   648
val LListD_Fun_subset_Sigma_LList = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   649
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   650
goal LList.thy
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   651
    "prod_fun(Rep_LList,Rep_LList) `` r <= \
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   652
\    Sigma(LList(range(Leaf)), %x.LList(range(Leaf)))";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   653
by (fast_tac (set_cs addSEs [prod_fun_imageE] addIs [SigmaI, Rep_LList]) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   654
val subset_Sigma_LList = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   655
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   656
val [prem] = goal LList.thy
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   657
    "r <= Sigma(LList(range(Leaf)), %x.LList(range(Leaf))) ==> \
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   658
\    prod_fun(Rep_LList o Abs_LList, Rep_LList o Abs_LList) `` r <= r";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   659
by (safe_tac (set_cs addSEs [prod_fun_imageE]));
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   660
by (rtac (prem RS subsetD RS SigmaE2) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   661
by (assume_tac 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   662
by (asm_simp_tac (HOL_ss addsimps [o_def,prod_fun,Abs_LList_inverse]) 1);
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   663
val prod_fun_lemma = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   664
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   665
goal LList.thy
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   666
    "prod_fun(Rep_LList, Rep_LList) `` range(%x. <x, x>) = \
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   667
\    diag(LList(range(Leaf)))";
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   668
br equalityI 1;
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   669
by (fast_tac (set_cs addIs  [diagI, Rep_LList]
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   670
		     addSEs [prod_fun_imageE, Pair_inject]) 1);
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   671
by (fast_tac (set_cs addIs  [prod_fun_imageI, rangeI]
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   672
		     addSEs [diagE, Abs_LList_inverse RS subst]) 1);
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   673
val prod_fun_range_eq_diag = result();
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   674
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   675
(** To show two llists are equal, exhibit a bisimulation! 
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   676
      [also admits true equality] **)
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   677
val [prem1,prem2] = goalw LList.thy [llistD_Fun_def]
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   678
    "[| <l1,l2> : r;  r <= llistD_Fun(r Un range(%x.<x,x>)) |] ==> l1=l2";
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   679
by (rtac (inj_Rep_LList RS injD) 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   680
by (res_inst_tac [("r", "prod_fun(Rep_LList,Rep_LList)``r"),
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   681
		  ("A", "range(Leaf)")] 
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   682
	LList_equalityI 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   683
by (rtac (prem1 RS prod_fun_imageI) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   684
by (rtac (prem2 RS image_mono RS subset_trans) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   685
by (rtac (image_compose RS subst) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   686
by (rtac (prod_fun_compose RS subst) 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   687
by (rtac (image_Un RS ssubst) 1);
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   688
by (stac prod_fun_range_eq_diag 1);
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   689
by (rtac (LListD_Fun_subset_Sigma_LList RS prod_fun_lemma) 1);
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   690
by (rtac (subset_Sigma_LList RS Un_least) 1);
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   691
by (rtac diag_subset_Sigma 1);
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   692
val llist_equalityI = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   693
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   694
(** Rules to prove the 2nd premise of llist_equalityI **)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   695
goalw LList.thy [llistD_Fun_def,LNil_def] "<LNil,LNil> : llistD_Fun(r)";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   696
by (rtac (LListD_Fun_NIL_I RS prod_fun_imageI) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   697
val llistD_Fun_LNil_I = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   698
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   699
val [prem] = goalw LList.thy [llistD_Fun_def,LCons_def]
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   700
    "<l1,l2>:r ==> <LCons(x,l1), LCons(x,l2)> : llistD_Fun(r)";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   701
by (rtac (rangeI RS LListD_Fun_CONS_I RS prod_fun_imageI) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   702
by (rtac (prem RS prod_fun_imageI) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   703
val llistD_Fun_LCons_I = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   704
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   705
(*Utilise the "strong" part, i.e. gfp(f)*)
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   706
goalw LList.thy [llistD_Fun_def]
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   707
     "!!l. <l,l> : llistD_Fun(r Un range(%x.<x,x>))";
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   708
br (Rep_LList_inverse RS subst) 1;
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   709
br prod_fun_imageI 1;
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   710
by (rtac (image_Un RS ssubst) 1);
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   711
by (stac prod_fun_range_eq_diag 1);
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   712
br (Rep_LList RS LListD_Fun_diag_I) 1;
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   713
val llistD_Fun_range_I = result();
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   714
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   715
(*A special case of list_equality for functions over lazy lists*)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   716
val [prem1,prem2] = goal LList.thy
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   717
    "[| f(LNil)=g(LNil);						\
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   718
\       !!x l. <f(LCons(x,l)),g(LCons(x,l))> :				\
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   719
\              llistD_Fun(range(%u. <f(u),g(u)>) Un range(%v. <v,v>))	\
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   720
\    |]	==> f(l) = g(l :: 'a llist) :: 'b llist";
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   721
by (res_inst_tac [("r", "range(%u. <f(u),g(u)>)")] llist_equalityI 1);
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   722
by (rtac rangeI 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   723
by (rtac subsetI 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   724
by (etac rangeE 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   725
by (etac ssubst 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   726
by (res_inst_tac [("l", "u")] llistE 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   727
by (etac ssubst 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   728
by (stac prem1 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   729
by (rtac llistD_Fun_range_I 1);
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   730
by (etac ssubst 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   731
by (rtac prem2 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   732
val llist_fun_equalityI = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   733
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   734
(*simpset for llist bisimulations*)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   735
val llistD_simps = [llist_case_LNil, llist_case_LCons, range_eqI,
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   736
		    llistD_Fun_LNil_I, llistD_Fun_LCons_I];
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   737
val llistD_ss = llist_ss addsimps llistD_simps;
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   738
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   739
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   740
(*** The functional "lmap" ***)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   741
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   742
goal LList.thy "lmap(f,LNil) = LNil";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   743
by (rtac (lmap_def RS def_llist_corec RS trans) 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   744
by (simp_tac llistD_ss 1);
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   745
val lmap_LNil = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   746
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   747
goal LList.thy "lmap(f, LCons(M,N)) = LCons(f(M), lmap(f,N))";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   748
by (rtac (lmap_def RS def_llist_corec RS trans) 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   749
by (simp_tac llistD_ss 1);
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   750
val lmap_LCons = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   751
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   752
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   753
(** Two easy results about lmap **)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   754
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   755
goal LList.thy "lmap(f o g, l) = lmap(f, lmap(g, l))";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   756
by (res_inst_tac [("l","l")] llist_fun_equalityI 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   757
by (ALLGOALS (simp_tac (llistD_ss addsimps [lmap_LNil, lmap_LCons])));
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   758
val lmap_compose = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   759
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   760
goal LList.thy "lmap(%x.x, l) = l";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   761
by (res_inst_tac [("l","l")] llist_fun_equalityI 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   762
by (ALLGOALS (simp_tac (llistD_ss addsimps [lmap_LNil, lmap_LCons])));
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   763
val lmap_ident = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   764
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   765
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   766
(*** iterates -- llist_fun_equalityI cannot be used! ***)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   767
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   768
goal LList.thy "iterates(f,x) = LCons(x, iterates(f,f(x)))";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   769
by (rtac (iterates_def RS def_llist_corec RS trans) 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   770
by (simp_tac sum_ss 1);
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   771
val iterates = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   772
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   773
goal LList.thy "lmap(f, iterates(f,x)) = iterates(f,f(x))";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   774
by (res_inst_tac [("r", "range(%u.<lmap(f,iterates(f,u)),iterates(f,f(u))>)")] 
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   775
    llist_equalityI 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   776
by (rtac rangeI 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   777
by (safe_tac set_cs);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   778
by (res_inst_tac [("x1", "f(u)")] (iterates RS ssubst) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   779
by (res_inst_tac [("x1", "u")] (iterates RS ssubst) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   780
by (simp_tac (llistD_ss addsimps [lmap_LCons]) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   781
val lmap_iterates = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   782
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   783
goal LList.thy "iterates(f,x) = LCons(x, lmap(f, iterates(f,x)))";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   784
br (lmap_iterates RS ssubst) 1;
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   785
br iterates 1;
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   786
val iterates_lmap = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   787
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   788
(*** A rather complex proof about iterates -- cf Andy Pitts ***)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   789
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   790
(** Two lemmas about natrec(n,x,%m.g), which is essentially (g^n)(x) **)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   791
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   792
goal LList.thy
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   793
    "nat_rec(n, LCons(b, l), %m. lmap(f)) =	\
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   794
\    LCons(nat_rec(n, b, %m. f), nat_rec(n, l, %m. lmap(f)))";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   795
by (nat_ind_tac "n" 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   796
by (ALLGOALS (asm_simp_tac (nat_ss addsimps [lmap_LCons])));
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   797
val fun_power_lmap = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   798
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   799
goal Nat.thy "nat_rec(n, g(x), %m. g) = nat_rec(Suc(n), x, %m. g)";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   800
by (nat_ind_tac "n" 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   801
by (ALLGOALS (asm_simp_tac nat_ss));
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   802
val fun_power_Suc = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   803
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   804
val Pair_cong = read_instantiate_sg (sign_of Prod.thy)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   805
 [("f","Pair")] (standard(refl RS cong RS cong));
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   806
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   807
(*The bisimulation consists of {<lmap(f)^n (h(u)), lmap(f)^n (iterates(f,u))>}
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   808
  for all u and all n::nat.*)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   809
val [prem] = goal LList.thy
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   810
    "(!!x. h(x) = LCons(x, lmap(f,h(x)))) ==> h = iterates(f)";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   811
br ext 1;
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   812
by (res_inst_tac [("r", 
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   813
   "UN u. range(%n. <nat_rec(n, h(u), %m y.lmap(f,y)), \
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   814
\                  nat_rec(n, iterates(f,u), %m y.lmap(f,y))>)")] 
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   815
    llist_equalityI 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   816
by (REPEAT (resolve_tac [UN1_I, range_eqI, Pair_cong, nat_rec_0 RS sym] 1));
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   817
by (safe_tac set_cs);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   818
by (stac iterates 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   819
by (stac prem 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   820
by (stac fun_power_lmap 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   821
by (stac fun_power_lmap 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   822
br llistD_Fun_LCons_I 1;
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   823
by (rtac (lmap_iterates RS subst) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   824
by (stac fun_power_Suc 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   825
by (stac fun_power_Suc 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   826
br (UN1_I RS UnI1) 1;
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   827
br rangeI 1;
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   828
val iterates_equality = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   829
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   830
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   831
(*** lappend -- its two arguments cause some complications! ***)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   832
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   833
goalw LList.thy [lappend_def] "lappend(LNil,LNil) = LNil";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   834
by (rtac (llist_corec RS trans) 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   835
by (simp_tac llistD_ss 1);
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   836
val lappend_LNil_LNil = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   837
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   838
goalw LList.thy [lappend_def]
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   839
    "lappend(LNil,LCons(l,l')) = LCons(l, lappend(LNil,l'))";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   840
by (rtac (llist_corec RS trans) 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   841
by (simp_tac llistD_ss 1);
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   842
(* 3.3(5.7) vs 1.3 !by (SIMP_TAC llistD_ss 1);*)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   843
val lappend_LNil_LCons = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   844
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   845
goalw LList.thy [lappend_def]
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   846
    "lappend(LCons(l,l'), N) = LCons(l, lappend(l',N))";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   847
by (rtac (llist_corec RS trans) 1);
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   848
by (simp_tac llistD_ss 1);
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   849
(* 5(5.5) vs 1.3 !by (SIMP_TAC llistD_ss 1);*)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   850
val lappend_LCons = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   851
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   852
goal LList.thy "lappend(LNil,l) = l";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   853
by (res_inst_tac [("l","l")] llist_fun_equalityI 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   854
by (ALLGOALS (simp_tac (llistD_ss addsimps [lappend_LNil_LNil,
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   855
                                            lappend_LNil_LCons])));
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   856
val lappend_LNil = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   857
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   858
goal LList.thy "lappend(l,LNil) = l";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   859
by (res_inst_tac [("l","l")] llist_fun_equalityI 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   860
by (ALLGOALS (simp_tac(llistD_ss addsimps [lappend_LNil_LNil,lappend_LCons])));
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   861
val lappend_LNil2 = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   862
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   863
(*The infinite first argument blocks the second*)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   864
goal LList.thy "lappend(iterates(f,x), N) = iterates(f,x)";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   865
by (res_inst_tac [("r", "range(%u.<lappend(iterates(f,u),N),iterates(f,u)>)")] 
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   866
    llist_equalityI 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   867
by (rtac rangeI 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   868
by (safe_tac set_cs);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   869
by (stac iterates 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   870
by (simp_tac (llistD_ss addsimps [lappend_LCons]) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   871
val lappend_iterates = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   872
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   873
(** Two proofs that lmap distributes over lappend **)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   874
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   875
(*Long proof requiring case analysis on both both arguments*)
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   876
goal LList.thy "lmap(f, lappend(l,n)) = lappend(lmap(f,l), lmap(f,n))";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   877
by (res_inst_tac 
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   878
    [("r",  
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   879
      "UN n. range(%l.<lmap(f,lappend(l,n)), lappend(lmap(f,l),lmap(f,n))>)")] 
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   880
    llist_equalityI 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   881
by (rtac UN1_I 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   882
by (rtac rangeI 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   883
by (safe_tac set_cs);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   884
by (res_inst_tac [("l", "l")] llistE 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   885
by (res_inst_tac [("l", "n")] llistE 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   886
by (ALLGOALS (asm_simp_tac (llistD_ss addsimps
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   887
      [lappend_LNil_LNil,lappend_LCons,lappend_LNil_LCons,
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   888
       lmap_LNil,lmap_LCons])));
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   889
by (REPEAT_SOME (ares_tac [llistD_Fun_LCons_I, UN1_I RS UnI1, rangeI]));
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   890
by (rtac range_eqI 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   891
by (rtac (refl RS Pair_cong) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   892
by (stac lmap_LNil 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   893
by (rtac refl 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   894
val lmap_lappend_distrib = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   895
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   896
(*Shorter proof of theorem above using llist_equalityI as strong coinduction*)
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   897
goal LList.thy "lmap(f, lappend(l,n)) = lappend(lmap(f,l), lmap(f,n))";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   898
by (res_inst_tac [("l","l")] llist_fun_equalityI 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   899
by (simp_tac (llistD_ss addsimps [lappend_LNil, lmap_LNil])1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   900
by (simp_tac (llistD_ss addsimps [lappend_LCons, lmap_LCons]) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   901
val lmap_lappend_distrib = result();
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   902
2
befa4e9f7c90 Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents: 0
diff changeset
   903
(*Without strong coinduction, three case analyses might be needed*)
0
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   904
goal LList.thy "lappend(lappend(l1,l2) ,l3) = lappend(l1, lappend(l2,l3))";
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   905
by (res_inst_tac [("l","l1")] llist_fun_equalityI 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   906
by (simp_tac (llistD_ss addsimps [lappend_LNil])1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   907
by (simp_tac (llistD_ss addsimps [lappend_LCons]) 1);
7949f97df77a Initial revision
clasohm
parents:
diff changeset
   908
val lappend_assoc = result();