(* Title: HOL/list
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
For list.thy.
*)
open List;
(** the list functional **)
goalw List.thy [List_Fun_def] "mono(List_Fun(A))";
by (REPEAT (ares_tac [monoI, subset_refl, usum_mono, uprod_mono] 1));
val List_Fun_mono = result();
goalw List.thy [List_Fun_def]
"!!A B. A<=B ==> List_Fun(A,Z) <= List_Fun(B,Z)";
by (REPEAT (ares_tac [subset_refl, usum_mono, uprod_mono] 1));
val List_Fun_mono2 = result();
(*This justifies using List in other recursive type definitions*)
goalw List.thy [List_def] "!!A B. A<=B ==> List(A) <= List(B)";
by (rtac lfp_mono 1);
by (etac List_Fun_mono2 1);
val List_mono = result();
(** Type checking rules -- List creates well-founded sets **)
val prems = goalw List.thy [List_def,List_Fun_def] "List(Sexp) <= Sexp";
by (rtac lfp_lowerbound 1);
by (fast_tac (univ_cs addIs [Sexp_NumbI,Sexp_In0I,Sexp_In1I,Sexp_SconsI]) 1);
val List_Sexp = result();
(* A <= Sexp ==> List(A) <= Sexp *)
val List_subset_Sexp = standard
(List_mono RS (List_Sexp RSN (2,subset_trans)));
(** Induction **)
(*Induction for the set List(A) *)
val major::prems = goalw List.thy [NIL_def,CONS_def]
"[| M: List(A); P(NIL); \
\ !!M N. [| M: A; N: List(A); P(N) |] ==> P(CONS(M,N)) |] \
\ ==> P(M)";
by (rtac (major RS (List_def RS def_induct)) 1);
by (rtac List_Fun_mono 1);
by (rewtac List_Fun_def);
by (fast_tac (set_cs addIs prems addEs [usumE,uprodE]) 1);
val List_induct = result();
(*Induction for the type 'a list *)
val prems = goalw List.thy [Nil_def,Cons_def]
"[| P(Nil); \
\ !!x xs. P(xs) ==> P(Cons(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));
val list_induct = result();
(*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)];
(** Introduction rules for List constructors **)
val List_unfold = rewrite_rule [List_Fun_def]
(List_Fun_mono RS (List_def RS def_lfp_Tarski));
(* c : {Numb(0)} <+> A <*> List(A) ==> c : List(A) *)
val ListI = List_unfold RS equalityD2 RS subsetD;
(* NIL is a List -- this also justifies the type definition*)
goalw List.thy [NIL_def] "NIL: List(A)";
by (rtac (singletonI RS usum_In0I RS ListI) 1);
val NIL_I = result();
goalw List.thy [CONS_def]
"!!a A M. [| a: A; M: List(A) |] ==> CONS(a,M) : List(A)";
by (REPEAT (ares_tac [uprodI RS usum_In1I RS ListI] 1));
val CONS_I = result();
(*** Isomorphisms ***)
goal List.thy "inj(Rep_List)";
by (rtac inj_inverseI 1);
by (rtac Rep_List_inverse 1);
val inj_Rep_List = result();
goal List.thy "inj_onto(Abs_List,List(range(Leaf)))";
by (rtac inj_onto_inverseI 1);
by (etac Abs_List_inverse 1);
val inj_onto_Abs_List = result();
(** Distinctness of constructors **)
goalw List.thy [NIL_def,CONS_def] "CONS(M,N) ~= NIL";
by (rtac In1_not_In0 1);
val CONS_not_NIL = result();
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 List.thy [Nil_def,Cons_def] "Cons(x,xs) ~= Nil";
by (rtac (CONS_not_NIL RS (inj_onto_Abs_List RS inj_onto_contraD)) 1);
by (REPEAT (resolve_tac [rangeI, NIL_I, CONS_I, Rep_List] 1));
val Cons_not_Nil = result();
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 List.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);
val CONS_CONS_eq = result();
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, NIL_I, CONS_I]
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 List.thy [Cons_def] "(Cons(x,xs)=Cons(y,ys)) = (x=y & xs=ys)";
by (fast_tac List_cs 1);
val Cons_Cons_eq = result();
val Cons_inject = standard (Cons_Cons_eq RS iffD1 RS conjE);
val [major] = goal List.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));
val CONS_D = result();
val prems = goalw List.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);
val Sexp_CONS_D = result();
(*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,
NIL_I, CONS_I];
val list_free_ss = HOL_ss addsimps list_free_simps;
goal List.thy "!!N. N: List(A) ==> !M. N ~= CONS(M,N)";
by (etac List_induct 1);
by (ALLGOALS (asm_simp_tac list_free_ss));
val not_CONS_self = result();
goal List.thy "!x. l ~= Cons(x,l)";
by (list_ind_tac "l" 1);
by (ALLGOALS (asm_simp_tac list_free_ss));
val not_Cons_self = result();
(** Conversion rules for List_case: case analysis operator **)
goalw List.thy [List_case_def,NIL_def] "List_case(NIL,c,h) = c";
by (rtac Case_In0 1);
val List_case_NIL = result();
goalw List.thy [List_case_def,CONS_def] "List_case(CONS(M,N), c, h) = h(M,N)";
by (simp_tac (HOL_ss addsimps [Split,Case_In1]) 1);
val List_case_CONS = result();
(*** 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 = wf_pred_Sexp RS wf_trancl RS (List_rec_def RS def_wfrec);
(** pred_Sexp lemmas **)
goalw List.thy [CONS_def,In1_def]
"!!M. [| M: Sexp; N: Sexp |] ==> <M, CONS(M,N)> : pred_Sexp^+";
by (asm_simp_tac pred_Sexp_ss 1);
val pred_Sexp_CONS_I1 = result();
goalw List.thy [CONS_def,In1_def]
"!!M. [| M: Sexp; N: Sexp |] ==> <N, CONS(M,N)> : pred_Sexp^+";
by (asm_simp_tac pred_Sexp_ss 1);
val pred_Sexp_CONS_I2 = result();
val [prem] = goal List.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));
val pred_Sexp_CONS_D = result();
(** Conversion rules for List_rec **)
goal List.thy "List_rec(NIL,c,h) = c";
by (rtac (List_rec_unfold RS trans) 1);
by (rtac List_case_NIL 1);
val List_rec_NIL = result();
goal List.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 (rtac (List_case_CONS RS trans) 1);
by(asm_simp_tac(HOL_ss addsimps [CONS_I, pred_Sexp_CONS_I2, cut_apply])1);
val List_rec_CONS = result();
(*** list_rec -- by List_rec ***)
val Rep_List_in_Sexp =
Rep_List RS (range_Leaf_subset_Sexp RS List_subset_Sexp RS 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 List.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 List.thy [list_rec_def, Cons_def]
"list_rec(Cons(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 List.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 node set)";
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))));
val List_rec_type = result();
(** Generalized map functionals **)
goalw List.thy [Rep_map_def] "Rep_map(f,Nil) = NIL";
by (rtac list_rec_Nil 1);
val Rep_map_Nil = result();
goalw List.thy [Rep_map_def]
"Rep_map(f, Cons(x,xs)) = CONS(f(x), Rep_map(f,xs))";
by (rtac list_rec_Cons 1);
val Rep_map_Cons = result();
goalw List.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));
val Rep_map_type = result();
goalw List.thy [Abs_map_def] "Abs_map(g,NIL) = Nil";
by (rtac List_rec_NIL 1);
val Abs_map_NIL = result();
val prems = goalw List.thy [Abs_map_def]
"[| M: Sexp; N: Sexp |] ==> \
\ Abs_map(g, CONS(M,N)) = Cons(g(M), Abs_map(g,N))";
by (REPEAT (resolve_tac (List_rec_CONS::prems) 1));
val Abs_map_CONS = result();
(** null, hd, tl, list_case **)
goalw List.thy [null_def] "null([]) = True";
by (rtac list_rec_Nil 1);
val null_Nil = result();
goalw List.thy [null_def] "null(Cons(x,xs)) = False";
by (rtac list_rec_Cons 1);
val null_Cons = result();
goalw List.thy [hd_def] "hd(Cons(x,xs)) = x";
by (rtac list_rec_Cons 1);
val hd_Cons = result();
goalw List.thy [tl_def] "tl(Cons(x,xs)) = xs";
by (rtac list_rec_Cons 1);
val tl_Cons = result();
goalw List.thy [list_case_def] "list_case([],a,f) = a";
by (rtac list_rec_Nil 1);
val list_case_Nil = result();
goalw List.thy [list_case_def] "list_case(Cons(x,xs),a,f) = f(x,xs)";
by (rtac list_rec_Cons 1);
val list_case_Cons = result();
(** @ - append **)
goalw List.thy [append_def] "[]@xs = xs";
by (rtac list_rec_Nil 1);
val append_Nil = result();
goalw List.thy [append_def] "Cons(x,xs)@ys = Cons(x,xs@ys)";
by (rtac list_rec_Cons 1);
val append_Cons = result();
goal List.thy "(xs@ys)@zs = xs@(ys@zs)";
by(res_inst_tac [("l","xs")] list_induct 1);
by(REPEAT(asm_simp_tac (list_ss addsimps [append_Nil,append_Cons]) 1));
val append_assoc = result();
(** The functional "map" **)
goalw List.thy [map_def] "map(f,Nil) = Nil";
by (rtac list_rec_Nil 1);
val map_Nil = result();
goalw List.thy [map_def] "map(f, Cons(x,xs)) = Cons(f(x), map(f,xs))";
by (rtac list_rec_Cons 1);
val map_Cons = result();
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 List.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])));
val Abs_map_inverse = result();
(*Rep_map_inverse is obtained via Abs_Rep_map and map_ident*)
(** The functional "list_all" -- creates predicates over lists **)
goalw List.thy [list_all_def] "list_all(P,Nil) = True";
by (rtac list_rec_Nil 1);
val list_all_Nil = result();
goalw List.thy [list_all_def]
"list_all(P, Cons(x,xs)) = (P(x) & list_all(P,xs))";
by (rtac list_rec_Cons 1);
val list_all_Cons = result();
(** Additional mapping lemmas **)
goal List.thy "map(%x.x, xs) = xs";
by (list_ind_tac "xs" 1);
by (ALLGOALS (asm_simp_tac map_ss));
val map_ident = result();
goal List.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])));
val Abs_Rep_map = result();
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,
list_case_Nil, list_case_Cons,
append_Nil, append_Cons, append_assoc,
map_Nil, map_Cons, map_ident,
list_all_Nil, list_all_Cons];