(* 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 *)
bind_thm ("list_subset_sexp", ([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_induct2";
(*Perform induction on xs. *)
fun list_ind_tac a M =
EVERY [res_inst_tac [("l",a)] list_induct2 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";
bind_thm ("NIL_not_CONS", (CONS_not_NIL RS not_sym));
bind_thm ("CONS_neq_NIL", (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";
bind_thm ("Nil_not_Cons", (Cons_not_Nil RS not_sym));
bind_thm ("Cons_neq_Nil2", (Cons_not_Nil RS notE));
val Nil_neq_Cons = sym RS Cons_neq_Nil2;
(** 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";
bind_thm ("CONS_inject", (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";
bind_thm ("Cons_inject2", (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*)
Addsimps ([Cons_not_Nil, Nil_not_Cons, Cons_Cons_eq, CONS_not_NIL,
NIL_not_CONS, CONS_CONS_eq] @ list.intrs);
goal SList.thy "!!N. N: list(A) ==> !M. N ~= CONS M N";
by (etac list.induct 1);
by (ALLGOALS Asm_simp_tac);
qed "not_CONS_self";
goal SList.thy "!x. l ~= x#l";
by (list_ind_tac "l" 1);
by (ALLGOALS Asm_simp_tac);
qed "not_Cons_self2";
goal SList.thy "(xs ~= []) = (? y ys. xs = y#ys)";
by(list_ind_tac "xs" 1);
by(Simp_tac 1);
by(Asm_simp_tac 1);
by(REPEAT(resolve_tac [exI,refl,conjI] 1));
qed "neq_Nil_conv2";
(** 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 (!simpset 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. *)
goal SList.thy
"(%M. List_rec M c d) = wfrec (trancl pred_sexp) \
\ (%g. List_case c (%x y. d x y (g y)))";
by (simp_tac (HOL_ss addsimps [List_rec_def]) 1);
val List_rec_unfold = standard
((wf_pred_sexp RS wf_trancl) RS ((result() RS eq_reflection) RS def_wfrec));
(*---------------------------------------------------------------------------
* Old:
* 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 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 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 (!simpset 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 (!simpset addsimps [List_case_CONS, pred_sexp_CONS_I2]) 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_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 (!simpset 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 (!simpset addsimps list_rec_simps) 1]);
end;
Addsimps [List_rec_NIL, List_rec_CONS, list_rec_Nil, list_rec_Cons];
(*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 (!simpset 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_induct2 1);
by(ALLGOALS Asm_simp_tac);
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_Nil3,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;
Addsimps
[null_Nil, ttl_Nil,
mem_Nil, mem_Cons,
list_case_Nil, list_case_Cons,
append_Nil3, append_Cons,
map_Nil, map_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);
qed "append_assoc2";
goal SList.thy "xs @ [] = xs";
by(list_ind_tac "xs" 1);
by(ALLGOALS Asm_simp_tac);
qed "append_Nil4";
(** 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 (!simpset setloop (split_tac [expand_if]))));
qed "mem_append2";
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 (!simpset setloop (split_tac [expand_if]))));
qed "mem_filter2";
(** The functional "map" **)
Addsimps [Rep_map_Nil, Rep_map_Cons, Abs_map_NIL, Abs_map_CONS];
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 (!simpset 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);
by(fast_tac HOL_cs 1);
qed "expand_list_case2";
(** Additional mapping lemmas **)
goal SList.thy "map (%x.x) xs = xs";
by (list_ind_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "map_ident2";
goal SList.thy "map f (xs@ys) = map f xs @ map f ys";
by (list_ind_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "map_append2";
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);
qed "map_compose2";
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(!simpset addsimps
[Rep_map_type,list_sexp RS subsetD])));
qed "Abs_Rep_map";
Addsimps [append_Nil4, map_ident2];