--- a/src/ZF/ListFn.ML Sat Apr 05 16:18:58 2003 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,366 +0,0 @@
-(* Title: ZF/list-fn.ML
- ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1992 University of Cambridge
-
-For list-fn.thy. Lists in Zermelo-Fraenkel Set Theory
-*)
-
-open ListFn;
-
-(** hd and tl **)
-
-goalw ListFn.thy [hd_def] "hd(Cons(a,l)) = a";
-by (resolve_tac List.case_eqns 1);
-val hd_Cons = result();
-
-goalw ListFn.thy [tl_def] "tl(Nil) = Nil";
-by (resolve_tac List.case_eqns 1);
-val tl_Nil = result();
-
-goalw ListFn.thy [tl_def] "tl(Cons(a,l)) = l";
-by (resolve_tac List.case_eqns 1);
-val tl_Cons = result();
-
-goal ListFn.thy "!!l. l: list(A) ==> tl(l) : list(A)";
-by (etac List.elim 1);
-by (ALLGOALS (asm_simp_tac (ZF_ss addsimps (List.intrs @ [tl_Nil,tl_Cons]))));
-val tl_type = result();
-
-(** drop **)
-
-goalw ListFn.thy [drop_def] "drop(0, l) = l";
-by (rtac rec_0 1);
-val drop_0 = result();
-
-goalw ListFn.thy [drop_def] "!!i. i:nat ==> drop(i, Nil) = Nil";
-by (etac nat_induct 1);
-by (ALLGOALS (asm_simp_tac (nat_ss addsimps [tl_Nil])));
-val drop_Nil = result();
-
-goalw ListFn.thy [drop_def]
- "!!i. i:nat ==> drop(succ(i), Cons(a,l)) = drop(i,l)";
-by (etac nat_induct 1);
-by (ALLGOALS (asm_simp_tac (nat_ss addsimps [tl_Cons])));
-val drop_succ_Cons = result();
-
-goalw ListFn.thy [drop_def]
- "!!i l. [| i:nat; l: list(A) |] ==> drop(i,l) : list(A)";
-by (etac nat_induct 1);
-by (ALLGOALS (asm_simp_tac (nat_ss addsimps [tl_type])));
-val drop_type = result();
-
-(** list_rec -- by Vset recursion **)
-
-goal ListFn.thy "list_rec(Nil,c,h) = c";
-by (rtac (list_rec_def RS def_Vrec RS trans) 1);
-by (simp_tac (ZF_ss addsimps List.case_eqns) 1);
-val list_rec_Nil = result();
-
-goal ListFn.thy "list_rec(Cons(a,l), c, h) = h(a, l, list_rec(l,c,h))";
-by (rtac (list_rec_def RS def_Vrec RS trans) 1);
-by (simp_tac (rank_ss addsimps (rank_Cons2::List.case_eqns)) 1);
-val list_rec_Cons = result();
-
-(*Type checking -- proved by induction, as usual*)
-val prems = goal ListFn.thy
- "[| l: list(A); \
-\ c: C(Nil); \
-\ !!x y r. [| x:A; y: list(A); r: C(y) |] ==> h(x,y,r): C(Cons(x,y)) \
-\ |] ==> list_rec(l,c,h) : C(l)";
-by (list_ind_tac "l" prems 1);
-by (ALLGOALS (asm_simp_tac
- (ZF_ss addsimps (prems@[list_rec_Nil,list_rec_Cons]))));
-val list_rec_type = result();
-
-(** Versions for use with definitions **)
-
-val [rew] = goal ListFn.thy
- "[| !!l. j(l)==list_rec(l, c, h) |] ==> j(Nil) = c";
-by (rewtac rew);
-by (rtac list_rec_Nil 1);
-val def_list_rec_Nil = result();
-
-val [rew] = goal ListFn.thy
- "[| !!l. j(l)==list_rec(l, c, h) |] ==> j(Cons(a,l)) = h(a,l,j(l))";
-by (rewtac rew);
-by (rtac list_rec_Cons 1);
-val def_list_rec_Cons = result();
-
-fun list_recs def = map standard
- ([def] RL [def_list_rec_Nil, def_list_rec_Cons]);
-
-(** map **)
-
-val [map_Nil,map_Cons] = list_recs map_def;
-
-val prems = goalw ListFn.thy [map_def]
- "[| l: list(A); !!x. x: A ==> h(x): B |] ==> map(h,l) : list(B)";
-by (REPEAT (ares_tac (prems@[list_rec_type, NilI, ConsI]) 1));
-val map_type = result();
-
-val [major] = goal ListFn.thy "l: list(A) ==> map(h,l) : list({h(u). u:A})";
-by (rtac (major RS map_type) 1);
-by (etac RepFunI 1);
-val map_type2 = result();
-
-(** length **)
-
-val [length_Nil,length_Cons] = list_recs length_def;
-
-goalw ListFn.thy [length_def]
- "!!l. l: list(A) ==> length(l) : nat";
-by (REPEAT (ares_tac [list_rec_type, nat_0I, nat_succI] 1));
-val length_type = result();
-
-(** app **)
-
-val [app_Nil,app_Cons] = list_recs app_def;
-
-goalw ListFn.thy [app_def]
- "!!xs ys. [| xs: list(A); ys: list(A) |] ==> xs@ys : list(A)";
-by (REPEAT (ares_tac [list_rec_type, ConsI] 1));
-val app_type = result();
-
-(** rev **)
-
-val [rev_Nil,rev_Cons] = list_recs rev_def;
-
-val prems = goalw ListFn.thy [rev_def]
- "xs: list(A) ==> rev(xs) : list(A)";
-by (REPEAT (ares_tac (prems @ [list_rec_type, NilI, ConsI, app_type]) 1));
-val rev_type = result();
-
-
-(** flat **)
-
-val [flat_Nil,flat_Cons] = list_recs flat_def;
-
-val prems = goalw ListFn.thy [flat_def]
- "ls: list(list(A)) ==> flat(ls) : list(A)";
-by (REPEAT (ares_tac (prems @ [list_rec_type, NilI, ConsI, app_type]) 1));
-val flat_type = result();
-
-
-(** list_add **)
-
-val [list_add_Nil,list_add_Cons] = list_recs list_add_def;
-
-val prems = goalw ListFn.thy [list_add_def]
- "xs: list(nat) ==> list_add(xs) : nat";
-by (REPEAT (ares_tac (prems @ [list_rec_type, nat_0I, add_type]) 1));
-val list_add_type = result();
-
-(** ListFn simplification **)
-
-val list_typechecks =
- [NilI, ConsI, list_rec_type,
- map_type, map_type2, app_type, length_type, rev_type, flat_type,
- list_add_type];
-
-val list_ss = arith_ss
- addsimps List.case_eqns
- addsimps [list_rec_Nil, list_rec_Cons,
- map_Nil, map_Cons,
- app_Nil, app_Cons,
- length_Nil, length_Cons,
- rev_Nil, rev_Cons,
- flat_Nil, flat_Cons,
- list_add_Nil, list_add_Cons]
- setsolver (type_auto_tac list_typechecks);
-(*Could also rewrite using the list_typechecks...*)
-
-(*** theorems about map ***)
-
-val prems = goal ListFn.thy
- "l: list(A) ==> map(%u.u, l) = l";
-by (list_ind_tac "l" prems 1);
-by (ALLGOALS (asm_simp_tac list_ss));
-val map_ident = result();
-
-val prems = goal ListFn.thy
- "l: list(A) ==> map(h, map(j,l)) = map(%u.h(j(u)), l)";
-by (list_ind_tac "l" prems 1);
-by (ALLGOALS (asm_simp_tac list_ss));
-val map_compose = result();
-
-val prems = goal ListFn.thy
- "xs: list(A) ==> map(h, xs@ys) = map(h,xs) @ map(h,ys)";
-by (list_ind_tac "xs" prems 1);
-by (ALLGOALS (asm_simp_tac list_ss));
-val map_app_distrib = result();
-
-val prems = goal ListFn.thy
- "ls: list(list(A)) ==> map(h, flat(ls)) = flat(map(map(h),ls))";
-by (list_ind_tac "ls" prems 1);
-by (ALLGOALS (asm_simp_tac (list_ss addsimps [map_app_distrib])));
-val map_flat = result();
-
-val prems = goal ListFn.thy
- "l: list(A) ==> \
-\ list_rec(map(h,l), c, d) = \
-\ list_rec(l, c, %x xs r. d(h(x), map(h,xs), r))";
-by (list_ind_tac "l" prems 1);
-by (ALLGOALS (asm_simp_tac list_ss));
-val list_rec_map = result();
-
-(** theorems about list(Collect(A,P)) -- used in ex/term.ML **)
-
-(* c : list(Collect(B,P)) ==> c : list(B) *)
-val list_CollectD = standard (Collect_subset RS list_mono RS subsetD);
-
-val prems = goal ListFn.thy
- "l: list({x:A. h(x)=j(x)}) ==> map(h,l) = map(j,l)";
-by (list_ind_tac "l" prems 1);
-by (ALLGOALS (asm_simp_tac list_ss));
-val map_list_Collect = result();
-
-(*** theorems about length ***)
-
-val prems = goal ListFn.thy
- "xs: list(A) ==> length(map(h,xs)) = length(xs)";
-by (list_ind_tac "xs" prems 1);
-by (ALLGOALS (asm_simp_tac list_ss));
-val length_map = result();
-
-val prems = goal ListFn.thy
- "xs: list(A) ==> length(xs@ys) = length(xs) #+ length(ys)";
-by (list_ind_tac "xs" prems 1);
-by (ALLGOALS (asm_simp_tac list_ss));
-val length_app = result();
-
-(* [| m: nat; n: nat |] ==> m #+ succ(n) = succ(n) #+ m
- Used for rewriting below*)
-val add_commute_succ = nat_succI RSN (2,add_commute);
-
-val prems = goal ListFn.thy
- "xs: list(A) ==> length(rev(xs)) = length(xs)";
-by (list_ind_tac "xs" prems 1);
-by (ALLGOALS (asm_simp_tac (list_ss addsimps [length_app, add_commute_succ])));
-val length_rev = result();
-
-val prems = goal ListFn.thy
- "ls: list(list(A)) ==> length(flat(ls)) = list_add(map(length,ls))";
-by (list_ind_tac "ls" prems 1);
-by (ALLGOALS (asm_simp_tac (list_ss addsimps [length_app])));
-val length_flat = result();
-
-(** Length and drop **)
-
-(*Lemma for the inductive step of drop_length*)
-goal ListFn.thy
- "!!xs. xs: list(A) ==> \
-\ ALL x. EX z zs. drop(length(xs), Cons(x,xs)) = Cons(z,zs)";
-by (etac List.induct 1);
-by (ALLGOALS (asm_simp_tac (list_ss addsimps [drop_0,drop_succ_Cons])));
-by (fast_tac ZF_cs 1);
-val drop_length_Cons_lemma = result();
-val drop_length_Cons = standard (drop_length_Cons_lemma RS spec);
-
-goal ListFn.thy
- "!!l. l: list(A) ==> ALL i: length(l). EX z zs. drop(i,l) = Cons(z,zs)";
-by (etac List.induct 1);
-by (ALLGOALS (asm_simp_tac (list_ss addsimps bquant_simps)));
-by (rtac conjI 1);
-by (etac drop_length_Cons 1);
-by (rtac ballI 1);
-by (rtac natE 1);
-by (etac ([asm_rl, length_type, Ord_nat] MRS Ord_trans) 1);
-by (assume_tac 1);
-by (asm_simp_tac (list_ss addsimps [drop_0]) 1);
-by (fast_tac ZF_cs 1);
-by (asm_simp_tac (list_ss addsimps [drop_succ_Cons]) 1);
-by (dtac bspec 1);
-by (fast_tac ZF_cs 2);
-by (fast_tac (ZF_cs addEs [succ_in_naturalD,length_type]) 1);
-val drop_length_lemma = result();
-val drop_length = standard (drop_length_lemma RS bspec);
-
-
-(*** theorems about app ***)
-
-val [major] = goal ListFn.thy "xs: list(A) ==> xs@Nil=xs";
-by (rtac (major RS List.induct) 1);
-by (ALLGOALS (asm_simp_tac list_ss));
-val app_right_Nil = result();
-
-val prems = goal ListFn.thy "xs: list(A) ==> (xs@ys)@zs = xs@(ys@zs)";
-by (list_ind_tac "xs" prems 1);
-by (ALLGOALS (asm_simp_tac list_ss));
-val app_assoc = result();
-
-val prems = goal ListFn.thy
- "ls: list(list(A)) ==> flat(ls@ms) = flat(ls)@flat(ms)";
-by (list_ind_tac "ls" prems 1);
-by (ALLGOALS (asm_simp_tac (list_ss addsimps [app_assoc])));
-val flat_app_distrib = result();
-
-(*** theorems about rev ***)
-
-val prems = goal ListFn.thy "l: list(A) ==> rev(map(h,l)) = map(h,rev(l))";
-by (list_ind_tac "l" prems 1);
-by (ALLGOALS (asm_simp_tac (list_ss addsimps [map_app_distrib])));
-val rev_map_distrib = result();
-
-(*Simplifier needs the premises as assumptions because rewriting will not
- instantiate the variable ?A in the rules' typing conditions; note that
- rev_type does not instantiate ?A. Only the premises do.
-*)
-goal ListFn.thy
- "!!xs. [| xs: list(A); ys: list(A) |] ==> rev(xs@ys) = rev(ys)@rev(xs)";
-by (etac List.induct 1);
-by (ALLGOALS (asm_simp_tac (list_ss addsimps [app_right_Nil,app_assoc])));
-val rev_app_distrib = result();
-
-val prems = goal ListFn.thy "l: list(A) ==> rev(rev(l))=l";
-by (list_ind_tac "l" prems 1);
-by (ALLGOALS (asm_simp_tac (list_ss addsimps [rev_app_distrib])));
-val rev_rev_ident = result();
-
-val prems = goal ListFn.thy
- "ls: list(list(A)) ==> rev(flat(ls)) = flat(map(rev,rev(ls)))";
-by (list_ind_tac "ls" prems 1);
-by (ALLGOALS (asm_simp_tac (list_ss addsimps
- [map_app_distrib, flat_app_distrib, rev_app_distrib, app_right_Nil])));
-val rev_flat = result();
-
-
-(*** theorems about list_add ***)
-
-val prems = goal ListFn.thy
- "[| xs: list(nat); ys: list(nat) |] ==> \
-\ list_add(xs@ys) = list_add(ys) #+ list_add(xs)";
-by (cut_facts_tac prems 1);
-by (list_ind_tac "xs" prems 1);
-by (ALLGOALS
- (asm_simp_tac (list_ss addsimps [add_0_right, add_assoc RS sym])));
-by (rtac (add_commute RS subst_context) 1);
-by (REPEAT (ares_tac [refl, list_add_type] 1));
-val list_add_app = result();
-
-val prems = goal ListFn.thy
- "l: list(nat) ==> list_add(rev(l)) = list_add(l)";
-by (list_ind_tac "l" prems 1);
-by (ALLGOALS
- (asm_simp_tac (list_ss addsimps [list_add_app, add_0_right])));
-val list_add_rev = result();
-
-val prems = goal ListFn.thy
- "ls: list(list(nat)) ==> list_add(flat(ls)) = list_add(map(list_add,ls))";
-by (list_ind_tac "ls" prems 1);
-by (ALLGOALS (asm_simp_tac (list_ss addsimps [list_add_app])));
-by (REPEAT (ares_tac [refl, list_add_type, map_type, add_commute] 1));
-val list_add_flat = result();
-
-(** New induction rule **)
-
-val major::prems = goal ListFn.thy
- "[| l: list(A); \
-\ P(Nil); \
-\ !!x y. [| x: A; y: list(A); P(y) |] ==> P(y @ [x]) \
-\ |] ==> P(l)";
-by (rtac (major RS rev_rev_ident RS subst) 1);
-by (rtac (major RS rev_type RS List.induct) 1);
-by (ALLGOALS (asm_simp_tac (list_ss addsimps prems)));
-val list_append_induct = result();
-