--- a/List.ML Fri Dec 02 16:09:49 1994 +0100
+++ b/List.ML Fri Dec 02 16:13:34 1994 +0100
@@ -1,296 +1,38 @@
(* Title: HOL/List
ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1993 University of Cambridge
+ Author: Tobias Nipkow
+ Copyright 1994 TU Muenchen
-Definition of type 'a list by a least fixed point
+List lemmas
*)
open List;
-val list_con_defs = [NIL_def, CONS_def];
-
-goal List.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 List.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 List.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);
+val [Nil_not_Cons,Cons_not_Nil] = list.distinct;
-(*Induction for the type 'a list *)
-val prems = goalw List.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 List.thy "inj(Rep_list)";
-by (rtac inj_inverseI 1);
-by (rtac Rep_list_inverse 1);
-qed "inj_Rep_list";
-
-goal List.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 List.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 = store_thm("Cons_neq_Nil", standard (Cons_not_Nil RS notE));
+val Nil_neq_Cons = store_thm("Nil_neq_Cons", sym RS Cons_neq_Nil);
-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] "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 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);
-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];
+val Cons_inject =
+ store_thm("Cons_inject", standard ((hd list.inject) RS iffD1 RS conjE));
-goalw List.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 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));
-qed "CONS_D";
+val list_ss = HOL_ss addsimps list.simps;
-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);
-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 List.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 List.thy "!x. l ~= x#l";
-by (list_ind_tac "l" 1);
-by (ALLGOALS (asm_simp_tac list_free_ss));
+goal List.thy "!x. xs ~= x#xs";
+by (list.induct_tac "xs" 1);
+by (ALLGOALS (asm_simp_tac list_ss));
qed "not_Cons_self";
-
goal List.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 (list.induct_tac "xs" 1);
+by(simp_tac list_ss 1);
+by(asm_simp_tac list_ss 1);
by(REPEAT(resolve_tac [exI,refl,conjI] 1));
qed "neq_Nil_conv";
-(** Conversion rules for List_case: case analysis operator **)
-
-goalw List.thy [List_case_def,NIL_def] "List_case(c, h, NIL) = c";
-by (rtac Case_In0 1);
-qed "List_case_NIL";
-
-goalw List.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 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);
-qed "pred_sexp_CONS_I1";
-
-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);
-qed "pred_sexp_CONS_I2";
-
-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));
-qed "pred_sexp_CONS_D";
-
-(** Conversion rules for List_rec **)
-
-goal List.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 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 (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 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(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 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 List.thy [Rep_map_def] "Rep_map(f,Nil) = NIL";
-by (rtac list_rec_Nil 1);
-qed "Rep_map_Nil";
-
-goalw List.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 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));
-qed "Rep_map_type";
-
-goalw List.thy [Abs_map_def] "Abs_map(g,NIL) = Nil";
-by (rtac List_rec_NIL 1);
-qed "Abs_map_NIL";
-
-val prems = goalw List.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 List.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 List.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,
+val list_ss = arith_ss addsimps list.simps @
+ [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,
@@ -300,68 +42,52 @@
(** @ - append **)
goal List.thy "(xs@ys)@zs = xs@(ys@zs)";
-by(list_ind_tac "xs" 1);
+by (list.induct_tac "xs" 1);
by(ALLGOALS(asm_simp_tac list_ss));
qed "append_assoc";
goal List.thy "xs @ [] = xs";
-by(list_ind_tac "xs" 1);
+by (list.induct_tac "xs" 1);
by(ALLGOALS(asm_simp_tac list_ss));
qed "append_Nil2";
(** mem **)
goal List.thy "x mem (xs@ys) = (x mem xs | x mem ys)";
-by(list_ind_tac "xs" 1);
+by (list.induct_tac "xs" 1);
by(ALLGOALS(asm_simp_tac (list_ss setloop (split_tac [expand_if]))));
qed "mem_append";
goal List.thy "x mem [x:xs.P(x)] = (x mem xs & P(x))";
-by(list_ind_tac "xs" 1);
+by (list.induct_tac "xs" 1);
by(ALLGOALS(asm_simp_tac (list_ss setloop (split_tac [expand_if]))));
qed "mem_filter";
(** list_all **)
goal List.thy "(Alls x:xs.True) = True";
-by(list_ind_tac "xs" 1);
+by (list.induct_tac "xs" 1);
by(ALLGOALS(asm_simp_tac list_ss));
qed "list_all_True";
goal List.thy "list_all(p,xs@ys) = (list_all(p,xs) & list_all(p,ys))";
-by(list_ind_tac "xs" 1);
+by (list.induct_tac "xs" 1);
by(ALLGOALS(asm_simp_tac list_ss));
qed "list_all_conj";
goal List.thy "(Alls x:xs.P(x)) = (!x. x mem xs --> P(x))";
-by(list_ind_tac "xs" 1);
+by (list.induct_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 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])));
-qed "Abs_map_inverse";
-
-(*Rep_map_inverse is obtained via Abs_Rep_map and map_ident*)
-
(** list_case **)
goal List.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 (list.induct_tac "xs" 1);
by(ALLGOALS(asm_simp_tac list_ss));
by(fast_tac HOL_cs 1);
qed "expand_list_case";
@@ -370,28 +96,22 @@
(** Additional mapping lemmas **)
goal List.thy "map(%x.x, xs) = xs";
-by (list_ind_tac "xs" 1);
-by (ALLGOALS (asm_simp_tac map_ss));
+by (list.induct_tac "xs" 1);
+by (ALLGOALS (asm_simp_tac list_ss));
qed "map_ident";
goal List.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])));
+by (list.induct_tac "xs" 1);
+by (ALLGOALS (asm_simp_tac list_ss));
qed "map_append";
goalw List.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));
+by (list.induct_tac "xs" 1);
+by (ALLGOALS (asm_simp_tac list_ss));
qed "map_compose";
-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])));
-qed "Abs_Rep_map";
-
val list_ss = list_ss addsimps
- [mem_append, mem_filter, append_assoc, append_Nil2, map_ident,
+ [not_Cons_self, append_assoc, append_Nil2, mem_append, mem_filter,
+ map_ident, map_append, map_compose,
list_all_True, list_all_conj];
--- a/List.thy Fri Dec 02 16:09:49 1994 +0100
+++ b/List.thy Fri Dec 02 16:13:34 1994 +0100
@@ -1,52 +1,32 @@
-(* Title: HOL/list
+(* Title: HOL/List.thy
ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1993 University of Cambridge
+ Author: Tobias Nipkow
+ Copyright 1994 TU Muenchen
-Definition of type 'a list by a least fixed point
+Definition of type 'a list as a datatype. This allows primrec to work.
-We use list(A) == lfp(%Z. {NUMB(0)} <+> A <*> Z)
-and not list == lfp(%Z. {NUMB(0)} <+> range(Leaf) <*> Z)
-so that list can serve as a "functor" for defining other recursive types
*)
-List = Sexp +
+List = Arith +
-types
- 'a list
-
-arities
- list :: (term) term
-
+datatype 'a list = "[]" ("[]") | "#"('a,'a list) (infixr 65)
consts
- list :: "'a item set => 'a item set"
- Rep_list :: "'a list => 'a item"
- Abs_list :: "'a item => 'a list"
- NIL :: "'a item"
- CONS :: "['a item, 'a item] => 'a item"
- Nil :: "'a list"
- "#" :: "['a, 'a list] => 'a list" (infixr 65)
- List_case :: "['b, ['a item, 'a item]=>'b, 'a item] => 'b"
- List_rec :: "['a item, 'b, ['a item, 'a item, 'b]=>'b] => 'b"
- list_case :: "['b, ['a, 'a list]=>'b, 'a list] => 'b"
- list_rec :: "['a list, 'b, ['a, 'a list, 'b]=>'b] => 'b"
- Rep_map :: "('b => 'a item) => ('b list => 'a item)"
- Abs_map :: "('a item => 'b) => 'a item => 'b list"
null :: "'a list => bool"
hd :: "'a list => 'a"
tl,ttl :: "'a list => 'a list"
- mem :: "['a, 'a list] => bool" (infixl 55)
+ mem :: "['a, 'a list] => bool" (infixl 55)
list_all :: "('a => bool) => ('a list => bool)"
map :: "('a=>'b) => ('a list => 'b list)"
- "@" :: "['a list, 'a list] => 'a list" (infixr 65)
+ "@" :: "['a list, 'a list] => 'a list" (infixr 65)
filter :: "['a => bool, 'a list] => 'a list"
+ foldl :: "[['b,'a] => 'b, 'b, 'a list] => 'b"
+ length :: "'a list => nat"
+syntax
(* list Enumeration *)
-
- "[]" :: "'a list" ("[]")
- "@list" :: "args => 'a list" ("[(_)]")
+ "@list" :: "args => 'a list" ("[(_)]")
(* Special syntax for list_all and filter *)
"@Alls" :: "[idt, 'a list, bool] => bool" ("(2Alls _:_./ _)" 10)
@@ -55,66 +35,42 @@
translations
"[x, xs]" == "x#[xs]"
"[x]" == "x#[]"
- "[]" == "Nil"
-
- "case xs of Nil => a | y#ys => b" == "list_case(a, %y ys.b, xs)"
"[x:xs . P]" == "filter(%x.P,xs)"
"Alls x:xs.P" == "list_all(%x.P,xs)"
-defs
- (* Defining the Concrete Constructors *)
- NIL_def "NIL == In0(Numb(0))"
- CONS_def "CONS(M, N) == In1(M $ N)"
-
-inductive "list(A)"
- intrs
- NIL_I "NIL: list(A)"
- CONS_I "[| a: A; M: list(A) |] ==> CONS(a,M) : list(A)"
-
-rules
- (* Faking a Type Definition ... *)
- Rep_list "Rep_list(xs): list(range(Leaf))"
- Rep_list_inverse "Abs_list(Rep_list(xs)) = xs"
- Abs_list_inverse "M: list(range(Leaf)) ==> Rep_list(Abs_list(M)) = M"
-
-
-defs
- (* Defining the Abstract Constructors *)
- Nil_def "Nil == Abs_list(NIL)"
- Cons_def "x#xs == Abs_list(CONS(Leaf(x), Rep_list(xs)))"
-
- List_case_def "List_case(c, d) == Case(%x.c, Split(d))"
-
- (* list Recursion -- the trancl is Essential; see list.ML *)
-
- List_rec_def
- "List_rec(M, c, d) == wfrec(trancl(pred_sexp), M, \
-\ List_case(%g.c, %x y g. d(x, y, g(y))))"
-
- list_rec_def
- "list_rec(l, c, d) == \
-\ List_rec(Rep_list(l), c, %x y r. d(Inv(Leaf, x), Abs_list(y), r))"
-
- (* Generalized Map Functionals *)
-
- Rep_map_def "Rep_map(f, xs) == list_rec(xs, NIL, %x l r. CONS(f(x), r))"
- Abs_map_def "Abs_map(g, M) == List_rec(M, Nil, %N L r. g(N)#r)"
-
- null_def "null(xs) == list_rec(xs, True, %x xs r.False)"
- hd_def "hd(xs) == list_rec(xs, @x.True, %x xs r.x)"
- tl_def "tl(xs) == list_rec(xs, @xs.True, %x xs r.xs)"
- (* a total version of tl: *)
- ttl_def "ttl(xs) == list_rec(xs, [], %x xs r.xs)"
-
- mem_def "x mem xs == \
-\ list_rec(xs, False, %y ys r. if(y=x, True, r))"
- list_all_def "list_all(P, xs) == list_rec(xs, True, %x l r. P(x) & r)"
- map_def "map(f, xs) == list_rec(xs, [], %x l r. f(x)#r)"
- append_def "xs@ys == list_rec(xs, ys, %x l r. x#r)"
- filter_def "filter(P,xs) == \
-\ list_rec(xs, [], %x xs r. if(P(x), x#r, r))"
-
- list_case_def "list_case(a, f, xs) == list_rec(xs, a, %x xs r.f(x, xs))"
-
+primrec null list
+ null_Nil "null([]) = True"
+ null_Cons "null(x#xs) = False"
+primrec hd list
+ hd_Nil "hd([]) = (@x.False)"
+ hd_Cons "hd(x#xs) = x"
+primrec tl list
+ tl_Nil "tl([]) = (@x.False)"
+ tl_Cons "tl(x#xs) = xs"
+primrec ttl list
+ (* a "total" version of tl: *)
+ ttl_Nil "ttl([]) = []"
+ ttl_Cons "ttl(x#xs) = xs"
+primrec "op mem" list
+ mem_Nil "x mem [] = False"
+ mem_Cons "x mem (y#ys) = if(y=x, True, x mem ys)"
+primrec list_all list
+ list_all_Nil "list_all(P,[]) = True"
+ list_all_Cons "list_all(P,x#xs) = (P(x) & list_all(P,xs))"
+primrec map list
+ map_Nil "map(f,[]) = []"
+ map_Cons "map(f,x#xs) = f(x)#map(f,xs)"
+primrec "op @" list
+ append_Nil "[] @ ys = ys"
+ append_Cons "(x#xs)@ys = x#(xs@ys)"
+primrec filter list
+ filter_Nil "filter(P,[]) = []"
+ filter_Cons "filter(P,x#xs) = if(P(x), x#filter(P,xs), filter(P,xs))"
+primrec foldl list
+ foldl_Nil "foldl(f,a,[]) = a"
+ foldl_Cons "foldl(f,a,x#xs) = foldl(f, f(a,x), xs)"
+primrec length list
+ length_Nil "length([]) = 0"
+ length_Cons "length(x#xs) = Suc(length(xs))"
end