diff -r b93cc55cb7ab -r df6b3bd14dcb ex/SList.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/ex/SList.ML	Fri Dec 02 16:09:49 1994 +0100
@@ -0,0 +1,397 @@
+(*  Title: 	HOL/ex/SList.ML
+    ID:         $Id$
+    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1993  University of Cambridge
+
+Definition of type 'a list by a least fixed point
+*)
+
+open SList;
+
+val list_con_defs = [NIL_def, CONS_def];
+
+goal SList.thy "list(A) = {Numb(0)} <+> (A <*> list(A))";
+let val rew = rewrite_rule list_con_defs in  
+by (fast_tac (univ_cs addSIs (equalityI :: map rew list.intrs)
+                      addEs [rew list.elim]) 1)
+end;
+qed "list_unfold";
+
+(*This justifies using list in other recursive type definitions*)
+goalw SList.thy list.defs "!!A B. A<=B ==> list(A) <= list(B)";
+by (rtac lfp_mono 1);
+by (REPEAT (ares_tac basic_monos 1));
+qed "list_mono";
+
+(*Type checking -- list creates well-founded sets*)
+goalw SList.thy (list_con_defs @ list.defs) "list(sexp) <= sexp";
+by (rtac lfp_lowerbound 1);
+by (fast_tac (univ_cs addIs sexp.intrs@[sexp_In0I,sexp_In1I]) 1);
+qed "list_sexp";
+
+(* A <= sexp ==> list(A) <= sexp *)
+val list_subset_sexp = standard ([list_mono, list_sexp] MRS subset_trans);
+
+(*Induction for the type 'a list *)
+val prems = goalw SList.thy [Nil_def,Cons_def]
+    "[| P(Nil);   \
+\       !!x xs. P(xs) ==> P(x # xs) |]  ==> P(l)";
+by (rtac (Rep_list_inverse RS subst) 1);   (*types force good instantiation*)
+by (rtac (Rep_list RS list.induct) 1);
+by (REPEAT (ares_tac prems 1
+     ORELSE eresolve_tac [rangeE, ssubst, Abs_list_inverse RS subst] 1));
+qed "list_induct";
+
+(*Perform induction on xs. *)
+fun list_ind_tac a M = 
+    EVERY [res_inst_tac [("l",a)] list_induct M,
+	   rename_last_tac a ["1"] (M+1)];
+
+(*** Isomorphisms ***)
+
+goal SList.thy "inj(Rep_list)";
+by (rtac inj_inverseI 1);
+by (rtac Rep_list_inverse 1);
+qed "inj_Rep_list";
+
+goal SList.thy "inj_onto(Abs_list,list(range(Leaf)))";
+by (rtac inj_onto_inverseI 1);
+by (etac Abs_list_inverse 1);
+qed "inj_onto_Abs_list";
+
+(** Distinctness of constructors **)
+
+goalw SList.thy list_con_defs "CONS(M,N) ~= NIL";
+by (rtac In1_not_In0 1);
+qed "CONS_not_NIL";
+val NIL_not_CONS = standard (CONS_not_NIL RS not_sym);
+
+val CONS_neq_NIL = standard (CONS_not_NIL RS notE);
+val NIL_neq_CONS = sym RS CONS_neq_NIL;
+
+goalw SList.thy [Nil_def,Cons_def] "x # xs ~= Nil";
+by (rtac (CONS_not_NIL RS (inj_onto_Abs_list RS inj_onto_contraD)) 1);
+by (REPEAT (resolve_tac (list.intrs @ [rangeI, Rep_list]) 1));
+qed "Cons_not_Nil";
+
+val Nil_not_Cons = standard (Cons_not_Nil RS not_sym);
+
+val Cons_neq_Nil = standard (Cons_not_Nil RS notE);
+val Nil_neq_Cons = sym RS Cons_neq_Nil;
+
+(** Injectiveness of CONS and Cons **)
+
+goalw SList.thy [CONS_def] "(CONS(K,M)=CONS(L,N)) = (K=L & M=N)";
+by (fast_tac (HOL_cs addSEs [Scons_inject, make_elim In1_inject]) 1);
+qed "CONS_CONS_eq";
+
+val CONS_inject = standard (CONS_CONS_eq RS iffD1 RS conjE);
+
+(*For reasoning about abstract list constructors*)
+val list_cs = set_cs addIs [Rep_list] @ list.intrs
+	             addSEs [CONS_neq_NIL,NIL_neq_CONS,CONS_inject]
+		     addSDs [inj_onto_Abs_list RS inj_ontoD,
+			     inj_Rep_list RS injD, Leaf_inject];
+
+goalw SList.thy [Cons_def] "(x#xs=y#ys) = (x=y & xs=ys)";
+by (fast_tac list_cs 1);
+qed "Cons_Cons_eq";
+val Cons_inject = standard (Cons_Cons_eq RS iffD1 RS conjE);
+
+val [major] = goal SList.thy "CONS(M,N): list(A) ==> M: A & N: list(A)";
+by (rtac (major RS setup_induction) 1);
+by (etac list.induct 1);
+by (ALLGOALS (fast_tac list_cs));
+qed "CONS_D";
+
+val prems = goalw SList.thy [CONS_def,In1_def]
+    "CONS(M,N): sexp ==> M: sexp & N: sexp";
+by (cut_facts_tac prems 1);
+by (fast_tac (set_cs addSDs [Scons_D]) 1);
+qed "sexp_CONS_D";
+
+
+(*Basic ss with constructors and their freeness*)
+val list_free_simps = [Cons_not_Nil, Nil_not_Cons, Cons_Cons_eq,
+		       CONS_not_NIL, NIL_not_CONS, CONS_CONS_eq]
+                      @ list.intrs;
+val list_free_ss = HOL_ss  addsimps  list_free_simps;
+
+goal SList.thy "!!N. N: list(A) ==> !M. N ~= CONS(M,N)";
+by (etac list.induct 1);
+by (ALLGOALS (asm_simp_tac list_free_ss));
+qed "not_CONS_self";
+
+goal SList.thy "!x. l ~= x#l";
+by (list_ind_tac "l" 1);
+by (ALLGOALS (asm_simp_tac list_free_ss));
+qed "not_Cons_self";
+
+
+goal SList.thy "(xs ~= []) = (? y ys. xs = y#ys)";
+by(list_ind_tac "xs" 1);
+by(simp_tac list_free_ss 1);
+by(asm_simp_tac list_free_ss 1);
+by(REPEAT(resolve_tac [exI,refl,conjI] 1));
+qed "neq_Nil_conv";
+
+(** Conversion rules for List_case: case analysis operator **)
+
+goalw SList.thy [List_case_def,NIL_def] "List_case(c, h, NIL) = c";
+by (rtac Case_In0 1);
+qed "List_case_NIL";
+
+goalw SList.thy [List_case_def,CONS_def]  "List_case(c, h, CONS(M,N)) = h(M,N)";
+by (simp_tac (HOL_ss addsimps [Split,Case_In1]) 1);
+qed "List_case_CONS";
+
+(*** List_rec -- by wf recursion on pred_sexp ***)
+
+(* The trancl(pred_sexp) is essential because pred_sexp_CONS_I1,2 would not
+   hold if pred_sexp^+ were changed to pred_sexp. *)
+
+val List_rec_unfold = [List_rec_def, wf_pred_sexp RS wf_trancl] MRS def_wfrec
+                      |> standard;
+
+(** pred_sexp lemmas **)
+
+goalw SList.thy [CONS_def,In1_def]
+    "!!M. [| M: sexp;  N: sexp |] ==> <M, CONS(M,N)> : pred_sexp^+";
+by (asm_simp_tac pred_sexp_ss 1);
+qed "pred_sexp_CONS_I1";
+
+goalw SList.thy [CONS_def,In1_def]
+    "!!M. [| M: sexp;  N: sexp |] ==> <N, CONS(M,N)> : pred_sexp^+";
+by (asm_simp_tac pred_sexp_ss 1);
+qed "pred_sexp_CONS_I2";
+
+val [prem] = goal SList.thy
+    "<CONS(M1,M2), N> : pred_sexp^+ ==> \
+\    <M1,N> : pred_sexp^+ & <M2,N> : pred_sexp^+";
+by (rtac (prem RS (pred_sexp_subset_Sigma RS trancl_subset_Sigma RS 
+		   subsetD RS SigmaE2)) 1);
+by (etac (sexp_CONS_D RS conjE) 1);
+by (REPEAT (ares_tac [conjI, pred_sexp_CONS_I1, pred_sexp_CONS_I2,
+		      prem RSN (2, trans_trancl RS transD)] 1));
+qed "pred_sexp_CONS_D";
+
+(** Conversion rules for List_rec **)
+
+goal SList.thy "List_rec(NIL,c,h) = c";
+by (rtac (List_rec_unfold RS trans) 1);
+by (simp_tac (HOL_ss addsimps [List_case_NIL]) 1);
+qed "List_rec_NIL";
+
+goal SList.thy "!!M. [| M: sexp;  N: sexp |] ==> \
+\    List_rec(CONS(M,N), c, h) = h(M, N, List_rec(N,c,h))";
+by (rtac (List_rec_unfold RS trans) 1);
+by (asm_simp_tac
+    (HOL_ss addsimps [List_case_CONS, list.CONS_I, pred_sexp_CONS_I2, 
+		      cut_apply])1);
+qed "List_rec_CONS";
+
+(*** list_rec -- by List_rec ***)
+
+val Rep_list_in_sexp =
+    [range_Leaf_subset_sexp RS list_subset_sexp, Rep_list] MRS subsetD;
+
+local
+  val list_rec_simps = list_free_simps @
+	          [List_rec_NIL, List_rec_CONS, 
+		   Abs_list_inverse, Rep_list_inverse,
+		   Rep_list, rangeI, inj_Leaf, Inv_f_f,
+		   sexp.LeafI, Rep_list_in_sexp]
+in
+  val list_rec_Nil = prove_goalw SList.thy [list_rec_def, Nil_def]
+      "list_rec(Nil,c,h) = c"
+   (fn _=> [simp_tac (HOL_ss addsimps list_rec_simps) 1]);
+
+  val list_rec_Cons = prove_goalw SList.thy [list_rec_def, Cons_def]
+      "list_rec(a#l, c, h) = h(a, l, list_rec(l,c,h))"
+   (fn _=> [simp_tac (HOL_ss addsimps list_rec_simps) 1]);
+end;
+
+val list_simps = [List_rec_NIL, List_rec_CONS,
+		  list_rec_Nil, list_rec_Cons];
+val list_ss = list_free_ss addsimps list_simps;
+
+
+(*Type checking.  Useful?*)
+val major::A_subset_sexp::prems = goal SList.thy
+    "[| M: list(A);    	\
+\       A<=sexp;      	\
+\       c: C(NIL);      \
+\       !!x y r. [| x: A;  y: list(A);  r: C(y) |] ==> h(x,y,r): C(CONS(x,y)) \
+\    |] ==> List_rec(M,c,h) : C(M :: 'a item)";
+val sexp_ListA_I = A_subset_sexp RS list_subset_sexp RS subsetD;
+val sexp_A_I = A_subset_sexp RS subsetD;
+by (rtac (major RS list.induct) 1);
+by (ALLGOALS(asm_simp_tac (list_ss addsimps ([sexp_A_I,sexp_ListA_I]@prems))));
+qed "List_rec_type";
+
+(** Generalized map functionals **)
+
+goalw SList.thy [Rep_map_def] "Rep_map(f,Nil) = NIL";
+by (rtac list_rec_Nil 1);
+qed "Rep_map_Nil";
+
+goalw SList.thy [Rep_map_def]
+    "Rep_map(f, x#xs) = CONS(f(x), Rep_map(f,xs))";
+by (rtac list_rec_Cons 1);
+qed "Rep_map_Cons";
+
+goalw SList.thy [Rep_map_def] "!!f. (!!x. f(x): A) ==> Rep_map(f,xs): list(A)";
+by (rtac list_induct 1);
+by(ALLGOALS(asm_simp_tac list_ss));
+qed "Rep_map_type";
+
+goalw SList.thy [Abs_map_def] "Abs_map(g,NIL) = Nil";
+by (rtac List_rec_NIL 1);
+qed "Abs_map_NIL";
+
+val prems = goalw SList.thy [Abs_map_def]
+    "[| M: sexp;  N: sexp |] ==> \
+\    Abs_map(g, CONS(M,N)) = g(M) # Abs_map(g,N)";
+by (REPEAT (resolve_tac (List_rec_CONS::prems) 1));
+qed "Abs_map_CONS";
+
+(*These 2 rules ease the use of primitive recursion.  NOTE USE OF == *)
+val [rew] = goal SList.thy
+    "[| !!xs. f(xs) == list_rec(xs,c,h) |] ==> f([]) = c";
+by (rewtac rew);
+by (rtac list_rec_Nil 1);
+qed "def_list_rec_Nil";
+
+val [rew] = goal SList.thy
+    "[| !!xs. f(xs) == list_rec(xs,c,h) |] ==> f(x#xs) = h(x,xs,f(xs))";
+by (rewtac rew);
+by (rtac list_rec_Cons 1);
+qed "def_list_rec_Cons";
+
+fun list_recs def =
+      [standard (def RS def_list_rec_Nil),
+       standard (def RS def_list_rec_Cons)];
+
+(*** Unfolding the basic combinators ***)
+
+val [null_Nil,null_Cons] = list_recs null_def;
+val [_,hd_Cons] = list_recs hd_def;
+val [_,tl_Cons] = list_recs tl_def;
+val [ttl_Nil,ttl_Cons] = list_recs ttl_def;
+val [append_Nil,append_Cons] = list_recs append_def;
+val [mem_Nil, mem_Cons] = list_recs mem_def;
+val [map_Nil,map_Cons] = list_recs map_def;
+val [list_case_Nil,list_case_Cons] = list_recs list_case_def;
+val [filter_Nil,filter_Cons] = list_recs filter_def;
+val [list_all_Nil,list_all_Cons] = list_recs list_all_def;
+
+val list_ss = arith_ss addsimps
+  [Cons_not_Nil, Nil_not_Cons, Cons_Cons_eq,
+   list_rec_Nil, list_rec_Cons,
+   null_Nil, null_Cons, hd_Cons, tl_Cons, ttl_Nil, ttl_Cons,
+   mem_Nil, mem_Cons,
+   list_case_Nil, list_case_Cons,
+   append_Nil, append_Cons,
+   map_Nil, map_Cons,
+   list_all_Nil, list_all_Cons,
+   filter_Nil, filter_Cons];
+
+
+(** @ - append **)
+
+goal SList.thy "(xs@ys)@zs = xs@(ys@zs)";
+by(list_ind_tac "xs" 1);
+by(ALLGOALS(asm_simp_tac list_ss));
+qed "append_assoc";
+
+goal SList.thy "xs @ [] = xs";
+by(list_ind_tac "xs" 1);
+by(ALLGOALS(asm_simp_tac list_ss));
+qed "append_Nil2";
+
+(** mem **)
+
+goal SList.thy "x mem (xs@ys) = (x mem xs | x mem ys)";
+by(list_ind_tac "xs" 1);
+by(ALLGOALS(asm_simp_tac (list_ss setloop (split_tac [expand_if]))));
+qed "mem_append";
+
+goal SList.thy "x mem [x:xs.P(x)] = (x mem xs & P(x))";
+by(list_ind_tac "xs" 1);
+by(ALLGOALS(asm_simp_tac (list_ss setloop (split_tac [expand_if]))));
+qed "mem_filter";
+
+(** list_all **)
+
+goal SList.thy "(Alls x:xs.True) = True";
+by(list_ind_tac "xs" 1);
+by(ALLGOALS(asm_simp_tac list_ss));
+qed "list_all_True";
+
+goal SList.thy "list_all(p,xs@ys) = (list_all(p,xs) & list_all(p,ys))";
+by(list_ind_tac "xs" 1);
+by(ALLGOALS(asm_simp_tac list_ss));
+qed "list_all_conj";
+
+goal SList.thy "(Alls x:xs.P(x)) = (!x. x mem xs --> P(x))";
+by(list_ind_tac "xs" 1);
+by(ALLGOALS(asm_simp_tac (list_ss setloop (split_tac [expand_if]))));
+by(fast_tac HOL_cs 1);
+qed "list_all_mem_conv";
+
+
+(** The functional "map" **)
+
+val map_simps = [Abs_map_NIL, Abs_map_CONS, 
+		 Rep_map_Nil, Rep_map_Cons, 
+		 map_Nil, map_Cons];
+val map_ss = list_free_ss addsimps map_simps;
+
+val [major,A_subset_sexp,minor] = goal SList.thy 
+    "[| M: list(A);  A<=sexp;  !!z. z: A ==> f(g(z)) = z |] \
+\    ==> Rep_map(f, Abs_map(g,M)) = M";
+by (rtac (major RS list.induct) 1);
+by (ALLGOALS (asm_simp_tac(map_ss addsimps [sexp_A_I,sexp_ListA_I,minor])));
+qed "Abs_map_inverse";
+
+(*Rep_map_inverse is obtained via Abs_Rep_map and map_ident*)
+
+(** list_case **)
+
+goal SList.thy
+ "P(list_case(a,f,xs)) = ((xs=[] --> P(a)) & \
+\                         (!y ys. xs=y#ys --> P(f(y,ys))))";
+by(list_ind_tac "xs" 1);
+by(ALLGOALS(asm_simp_tac list_ss));
+by(fast_tac HOL_cs 1);
+qed "expand_list_case";
+
+
+(** Additional mapping lemmas **)
+
+goal SList.thy "map(%x.x, xs) = xs";
+by (list_ind_tac "xs" 1);
+by (ALLGOALS (asm_simp_tac map_ss));
+qed "map_ident";
+
+goal SList.thy "map(f, xs@ys) = map(f,xs) @ map(f,ys)";
+by (list_ind_tac "xs" 1);
+by (ALLGOALS (asm_simp_tac (map_ss addsimps [append_Nil,append_Cons])));
+qed "map_append";
+
+goalw SList.thy [o_def] "map(f o g, xs) = map(f, map(g, xs))";
+by (list_ind_tac "xs" 1);
+by (ALLGOALS (asm_simp_tac map_ss));
+qed "map_compose";
+
+goal SList.thy "!!f. (!!x. f(x): sexp) ==> \
+\	Abs_map(g, Rep_map(f,xs)) = map(%t. g(f(t)), xs)";
+by (list_ind_tac "xs" 1);
+by(ALLGOALS(asm_simp_tac(map_ss addsimps
+       [Rep_map_type,list_sexp RS subsetD])));
+qed "Abs_Rep_map";
+
+val list_ss = list_ss addsimps
+  [mem_append, mem_filter, append_assoc, append_Nil2, map_ident,
+   list_all_True, list_all_conj];
+