src/HOL/Induct/SList.ML
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(*  Title:      HOL/ex/SList.ML
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1998  University of Cambridge
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Definition of type 'a list by a least fixed point
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*)
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open SList;
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val list_con_defs = [NIL_def, CONS_def];
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Goal "list(A) = {Numb(0)} <+> (A <*> list(A))";
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let val rew = rewrite_rule list_con_defs in  
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by (fast_tac (claset() addSIs (equalityI :: map rew list.intrs)
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                      addEs [rew list.elim]) 1)
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end;
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qed "list_unfold";
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(*This justifies using list in other recursive type definitions*)
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Goalw list.defs "!!A B. A<=B ==> list(A) <= list(B)";
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by (rtac lfp_mono 1);
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by (REPEAT (ares_tac basic_monos 1));
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qed "list_mono";
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(*Type checking -- list creates well-founded sets*)
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Goalw (list_con_defs @ list.defs) "list(sexp) <= sexp";
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by (rtac lfp_lowerbound 1);
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by (fast_tac (claset() addIs sexp.intrs@[sexp_In0I,sexp_In1I]) 1);
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qed "list_sexp";
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(* A <= sexp ==> list(A) <= sexp *)
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bind_thm ("list_subset_sexp", ([list_mono, list_sexp] MRS subset_trans));
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(*Induction for the type 'a list *)
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val prems = goalw SList.thy [Nil_def,Cons_def]
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    "[| P(Nil);   \
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\       !!x xs. P(xs) ==> P(x # xs) |]  ==> P(l)";
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by (rtac (Rep_list_inverse RS subst) 1);   (*types force good instantiation*)
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by (rtac (Rep_list RS list.induct) 1);
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by (REPEAT (ares_tac prems 1
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     ORELSE eresolve_tac [rangeE, ssubst, Abs_list_inverse RS subst] 1));
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qed "list_induct2";
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(*Perform induction on xs. *)
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fun list_ind_tac a M = 
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    EVERY [res_inst_tac [("l",a)] list_induct2 M,
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           rename_last_tac a ["1"] (M+1)];
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(*** Isomorphisms ***)
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Goal "inj(Rep_list)";
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by (rtac inj_inverseI 1);
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by (rtac Rep_list_inverse 1);
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qed "inj_Rep_list";
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Goal "inj_on Abs_list (list(range Leaf))";
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by (rtac inj_on_inverseI 1);
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by (etac Abs_list_inverse 1);
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qed "inj_on_Abs_list";
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(** Distinctness of constructors **)
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Goalw list_con_defs "CONS M N ~= NIL";
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by (rtac In1_not_In0 1);
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qed "CONS_not_NIL";
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bind_thm ("NIL_not_CONS", (CONS_not_NIL RS not_sym));
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bind_thm ("CONS_neq_NIL", (CONS_not_NIL RS notE));
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val NIL_neq_CONS = sym RS CONS_neq_NIL;
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Goalw [Nil_def,Cons_def] "x # xs ~= Nil";
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by (rtac (CONS_not_NIL RS (inj_on_Abs_list RS inj_on_contraD)) 1);
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by (REPEAT (resolve_tac (list.intrs @ [rangeI, Rep_list]) 1));
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qed "Cons_not_Nil";
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bind_thm ("Nil_not_Cons", Cons_not_Nil RS not_sym);
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(** Injectiveness of CONS and Cons **)
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Goalw [CONS_def] "(CONS K M=CONS L N) = (K=L & M=N)";
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by (fast_tac (claset() addSEs [Scons_inject, make_elim In1_inject]) 1);
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qed "CONS_CONS_eq";
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(*For reasoning about abstract list constructors*)
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AddIs ([Rep_list] @ list.intrs);
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AddIffs [CONS_not_NIL, NIL_not_CONS, CONS_CONS_eq];
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AddSDs [inj_on_Abs_list RS inj_onD,
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        inj_Rep_list RS injD, Leaf_inject];
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Goalw [Cons_def] "(x#xs=y#ys) = (x=y & xs=ys)";
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by (Fast_tac 1);
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qed "Cons_Cons_eq";
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bind_thm ("Cons_inject2", (Cons_Cons_eq RS iffD1 RS conjE));
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val [major] = goal SList.thy "CONS M N: list(A) ==> M: A & N: list(A)";
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by (rtac (major RS setup_induction) 1);
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by (etac list.induct 1);
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by (ALLGOALS (Fast_tac));
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qed "CONS_D";
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val prems = goalw SList.thy [CONS_def,In1_def]
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    "CONS M N: sexp ==> M: sexp & N: sexp";
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by (cut_facts_tac prems 1);
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by (fast_tac (claset() addSDs [Scons_D]) 1);
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qed "sexp_CONS_D";
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(*Reasoning about constructors and their freeness*)
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Addsimps list.intrs;
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AddIffs [Cons_not_Nil, Nil_not_Cons, Cons_Cons_eq];
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Goal "!!N. N: list(A) ==> !M. N ~= CONS M N";
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by (etac list.induct 1);
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by (ALLGOALS Asm_simp_tac);
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qed "not_CONS_self";
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Goal "!x. l ~= x#l";
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by (list_ind_tac "l" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "not_Cons_self2";
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Goal "(xs ~= []) = (? y ys. xs = y#ys)";
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by (list_ind_tac "xs" 1);
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by (Simp_tac 1);
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by (Asm_simp_tac 1);
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by (REPEAT(resolve_tac [exI,refl,conjI] 1));
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qed "neq_Nil_conv2";
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(** Conversion rules for List_case: case analysis operator **)
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Goalw [List_case_def,NIL_def] "List_case c h NIL = c";
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by (rtac Case_In0 1);
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qed "List_case_NIL";
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Goalw [List_case_def,CONS_def]  "List_case c h (CONS M N) = h M N";
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by (Simp_tac 1);
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qed "List_case_CONS";
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Addsimps [List_case_NIL, List_case_CONS];
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(*** List_rec -- by wf recursion on pred_sexp ***)
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(* The trancl(pred_sexp) is essential because pred_sexp_CONS_I1,2 would not
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   hold if pred_sexp^+ were changed to pred_sexp. *)
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Goal
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   "(%M. List_rec M c d) = wfrec (trancl pred_sexp) \
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                           \     (%g. List_case c (%x y. d x y (g y)))";
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by (simp_tac (HOL_ss addsimps [List_rec_def]) 1);
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val List_rec_unfold = standard 
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  ((wf_pred_sexp RS wf_trancl) RS ((result() RS eq_reflection) RS def_wfrec));
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(*---------------------------------------------------------------------------
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 * Old:
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 * val List_rec_unfold = [List_rec_def,wf_pred_sexp RS wf_trancl] MRS def_wfrec
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 *                     |> standard;
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 *---------------------------------------------------------------------------*)
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(** pred_sexp lemmas **)
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Goalw [CONS_def,In1_def]
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    "!!M. [| M: sexp;  N: sexp |] ==> (M, CONS M N) : pred_sexp^+";
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by (Asm_simp_tac 1);
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qed "pred_sexp_CONS_I1";
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Goalw [CONS_def,In1_def]
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    "!!M. [| M: sexp;  N: sexp |] ==> (N, CONS M N) : pred_sexp^+";
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by (Asm_simp_tac 1);
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qed "pred_sexp_CONS_I2";
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val [prem] = goal SList.thy
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    "(CONS M1 M2, N) : pred_sexp^+ ==> \
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\    (M1,N) : pred_sexp^+ & (M2,N) : pred_sexp^+";
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by (rtac (prem RS (pred_sexp_subset_Sigma RS trancl_subset_Sigma RS 
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                   subsetD RS SigmaE2)) 1);
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by (etac (sexp_CONS_D RS conjE) 1);
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by (REPEAT (ares_tac [conjI, pred_sexp_CONS_I1, pred_sexp_CONS_I2,
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                      prem RSN (2, trans_trancl RS transD)] 1));
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qed "pred_sexp_CONS_D";
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(** Conversion rules for List_rec **)
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Goal "List_rec NIL c h = c";
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by (rtac (List_rec_unfold RS trans) 1);
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by (Simp_tac 1);
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qed "List_rec_NIL";
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Goal "!!M. [| M: sexp;  N: sexp |] ==> \
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\    List_rec (CONS M N) c h = h M N (List_rec N c h)";
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by (rtac (List_rec_unfold RS trans) 1);
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by (asm_simp_tac (simpset() addsimps [pred_sexp_CONS_I2]) 1);
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qed "List_rec_CONS";
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Addsimps [List_rec_NIL, List_rec_CONS];
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(*** list_rec -- by List_rec ***)
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val Rep_list_in_sexp =
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    [range_Leaf_subset_sexp RS list_subset_sexp, Rep_list] MRS subsetD;
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local
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  val list_rec_simps = [Abs_list_inverse, Rep_list_inverse,
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                        Rep_list, rangeI, inj_Leaf, inv_f_f,
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                        sexp.LeafI, Rep_list_in_sexp]
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in
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  val list_rec_Nil = prove_goalw SList.thy [list_rec_def, Nil_def]
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      "list_rec Nil c h = c"
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   (fn _=> [simp_tac (simpset() addsimps list_rec_simps) 1]);
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  val list_rec_Cons = prove_goalw SList.thy [list_rec_def, Cons_def]
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      "list_rec (a#l) c h = h a l (list_rec l c h)"
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   (fn _=> [simp_tac (simpset() addsimps list_rec_simps) 1]);
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end;
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Addsimps [List_rec_NIL, List_rec_CONS, list_rec_Nil, list_rec_Cons];
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(*Type checking.  Useful?*)
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val major::A_subset_sexp::prems = goal SList.thy
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    "[| M: list(A);     \
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\       A<=sexp;        \
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\       c: C(NIL);      \
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\       !!x y r. [| x: A;  y: list(A);  r: C(y) |] ==> h x y r: C(CONS x y) \
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\    |] ==> List_rec M c h : C(M :: 'a item)";
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val sexp_ListA_I = A_subset_sexp RS list_subset_sexp RS subsetD;
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val sexp_A_I = A_subset_sexp RS subsetD;
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by (rtac (major RS list.induct) 1);
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by (ALLGOALS(asm_simp_tac (simpset() addsimps ([sexp_A_I,sexp_ListA_I]@prems))));
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qed "List_rec_type";
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(** Generalized map functionals **)
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Goalw [Rep_map_def] "Rep_map f Nil = NIL";
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by (rtac list_rec_Nil 1);
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qed "Rep_map_Nil";
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Goalw [Rep_map_def]
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    "Rep_map f (x#xs) = CONS (f x) (Rep_map f xs)";
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by (rtac list_rec_Cons 1);
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qed "Rep_map_Cons";
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Goalw [Rep_map_def] "!!f. (!!x. f(x): A) ==> Rep_map f xs: list(A)";
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by (rtac list_induct2 1);
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by (ALLGOALS Asm_simp_tac);
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qed "Rep_map_type";
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Goalw [Abs_map_def] "Abs_map g NIL = Nil";
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by (rtac List_rec_NIL 1);
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qed "Abs_map_NIL";
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val prems = goalw SList.thy [Abs_map_def]
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    "[| M: sexp;  N: sexp |] ==> \
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\    Abs_map g (CONS M N) = g(M) # Abs_map g N";
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by (REPEAT (resolve_tac (List_rec_CONS::prems) 1));
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qed "Abs_map_CONS";
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(*These 2 rules ease the use of primitive recursion.  NOTE USE OF == *)
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val [rew] = goal SList.thy
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    "[| !!xs. f(xs) == list_rec xs c h |] ==> f([]) = c";
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by (rewtac rew);
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by (rtac list_rec_Nil 1);
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qed "def_list_rec_Nil";
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val [rew] = goal SList.thy
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    "[| !!xs. f(xs) == list_rec xs c h |] ==> f(x#xs) = h x xs (f xs)";
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by (rewtac rew);
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by (rtac list_rec_Cons 1);
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qed "def_list_rec_Cons";
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fun list_recs def =
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      [standard (def RS def_list_rec_Nil),
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       standard (def RS def_list_rec_Cons)];
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(*** Unfolding the basic combinators ***)
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val [null_Nil, null_Cons]               = list_recs null_def;
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val [_, hd_Cons]                        = list_recs hd_def;
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val [_, tl_Cons]                        = list_recs tl_def;
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val [ttl_Nil, ttl_Cons]                 = list_recs ttl_def;
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val [append_Nil3, append_Cons]          = list_recs append_def;
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val [mem_Nil, mem_Cons]                 = list_recs mem_def;
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val [set_Nil, set_Cons]                 = list_recs set_def;
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val [map_Nil, map_Cons]                 = list_recs map_def;
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val [list_case_Nil, list_case_Cons]     = list_recs list_case_def;
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val [filter_Nil, filter_Cons]           = list_recs filter_def;
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Addsimps
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  [null_Nil, ttl_Nil,
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   mem_Nil, mem_Cons,
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   list_case_Nil, list_case_Cons,
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   append_Nil3, append_Cons,
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   set_Nil, set_Cons, 
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   map_Nil, map_Cons,
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   filter_Nil, filter_Cons];
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(** @ - append **)
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Goal "(xs@ys)@zs = xs@(ys@zs)";
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by (list_ind_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "append_assoc2";
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Goal "xs @ [] = xs";
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by (list_ind_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "append_Nil4";
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(** mem **)
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Goal "x mem (xs@ys) = (x mem xs | x mem ys)";
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by (list_ind_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "mem_append2";
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Goal "x mem [x:xs. P(x)] = (x mem xs & P(x))";
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by (list_ind_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "mem_filter2";
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(** The functional "map" **)
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Addsimps [Rep_map_Nil, Rep_map_Cons, Abs_map_NIL, Abs_map_CONS];
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val [major,A_subset_sexp,minor] = goal SList.thy 
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    "[| M: list(A);  A<=sexp;  !!z. z: A ==> f(g(z)) = z |] \
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\    ==> Rep_map f (Abs_map g M) = M";
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by (rtac (major RS list.induct) 1);
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by (ALLGOALS (asm_simp_tac (simpset() addsimps [sexp_A_I,sexp_ListA_I,minor])));
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qed "Abs_map_inverse";
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(*Rep_map_inverse is obtained via Abs_Rep_map and map_ident*)
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(** list_case **)
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Goal
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 "P(list_case a f xs) = ((xs=[] --> P(a)) & \
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\                        (!y ys. xs=y#ys --> P(f y ys)))";
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by (list_ind_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "split_list_case2";
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(** Additional mapping lemmas **)
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Goal "map (%x. x) xs = xs";
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by (list_ind_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "map_ident2";
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Goal "map f (xs@ys) = map f xs @ map f ys";
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by (list_ind_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "map_append2";
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Goalw [o_def] "map (f o g) xs = map f (map g xs)";
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by (list_ind_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "map_compose2";
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Goal "!!f. (!!x. f(x): sexp) ==> \
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\       Abs_map g (Rep_map f xs) = map (%t. g(f(t))) xs";
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by (list_ind_tac "xs" 1);
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by (ALLGOALS (asm_simp_tac(simpset() addsimps
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			   [Rep_map_type, list_sexp RS subsetD])));
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qed "Abs_Rep_map";
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Addsimps [append_Nil4, map_ident2];