author | lcp |
Fri, 12 Aug 1994 12:51:34 +0200 | |
changeset 516 | 1957113f0d7d |
parent 484 | 70b789956bd3 |
child 525 | e62519a8497d |
permissions | -rw-r--r-- |
435 | 1 |
(* Title: ZF/List.ML |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1993 University of Cambridge |
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Datatype definition of Lists |
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*) |
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open List; |
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(*** Aspects of the datatype definition ***) |
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(*An elimination rule, for type-checking*) |
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val ConsE = list.mk_cases list.con_defs "Cons(a,l) : list(A)"; |
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(*Proving freeness results*) |
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val Cons_iff = list.mk_free "Cons(a,l)=Cons(a',l') <-> a=a' & l=l'"; |
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val Nil_Cons_iff = list.mk_free "~ Nil=Cons(a,l)"; |
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(*Perform induction on l, then prove the major premise using prems. *) |
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fun list_ind_tac a prems i = |
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EVERY [res_inst_tac [("x",a)] list.induct i, |
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rename_last_tac a ["1"] (i+2), |
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ares_tac prems i]; |
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goal List.thy "list(A) = {0} + (A * list(A))"; |
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by (rtac (list.unfold RS trans) 1); |
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bws list.con_defs; |
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br equalityI 1; |
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by (fast_tac sum_cs 1); |
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by (fast_tac (sum_cs addIs datatype_intrs |
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addDs [list.dom_subset RS subsetD]) 1); |
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val list_unfold = result(); |
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(** Lemmas to justify using "list" in other recursive type definitions **) |
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goalw List.thy list.defs "!!A B. A<=B ==> list(A) <= list(B)"; |
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by (rtac lfp_mono 1); |
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by (REPEAT (rtac list.bnd_mono 1)); |
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by (REPEAT (ares_tac (univ_mono::basic_monos) 1)); |
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val list_mono = result(); |
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(*There is a similar proof by list induction.*) |
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goalw List.thy (list.defs@list.con_defs) "list(univ(A)) <= univ(A)"; |
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by (rtac lfp_lowerbound 1); |
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by (rtac (A_subset_univ RS univ_mono) 2); |
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by (fast_tac (ZF_cs addSIs [zero_in_univ, Inl_in_univ, Inr_in_univ, |
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Pair_in_univ]) 1); |
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val list_univ = result(); |
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val list_subset_univ = standard ([list_mono, list_univ] MRS subset_trans); |
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goal List.thy "!!l A B. [| l: list(A); A <= univ(B) |] ==> l: univ(B)"; |
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by (REPEAT (ares_tac [list_subset_univ RS subsetD] 1)); |
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val list_into_univ = result(); |
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val major::prems = goal List.thy |
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"[| l: list(A); \ |
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15
6c6d2f6e3185
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
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\ c: C(Nil); \ |
6c6d2f6e3185
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
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\ !!x y. [| x: A; y: list(A) |] ==> h(x,y): C(Cons(x,y)) \ |
6c6d2f6e3185
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
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\ |] ==> list_case(c,h,l) : C(l)"; |
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by (rtac (major RS list.induct) 1); |
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by (ALLGOALS (asm_simp_tac (ZF_ss addsimps (list.case_eqns @ prems)))); |
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val list_case_type = result(); |
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(** For recursion **) |
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goalw List.thy list.con_defs "rank(a) < rank(Cons(a,l))"; |
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8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
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changeset
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by (simp_tac rank_ss 1); |
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val rank_Cons1 = result(); |
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goalw List.thy list.con_defs "rank(l) < rank(Cons(a,l))"; |
6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset
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by (simp_tac rank_ss 1); |
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val rank_Cons2 = result(); |
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(*** List functions ***) |
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(** hd and tl **) |
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goalw List.thy [hd_def] "hd(Cons(a,l)) = a"; |
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by (resolve_tac list.case_eqns 1); |
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val hd_Cons = result(); |
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goalw List.thy [tl_def] "tl(Nil) = Nil"; |
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by (resolve_tac list.case_eqns 1); |
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val tl_Nil = result(); |
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goalw List.thy [tl_def] "tl(Cons(a,l)) = l"; |
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by (resolve_tac list.case_eqns 1); |
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val tl_Cons = result(); |
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goal List.thy "!!l. l: list(A) ==> tl(l) : list(A)"; |
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by (etac list.elim 1); |
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by (ALLGOALS (asm_simp_tac (ZF_ss addsimps (list.intrs @ [tl_Nil,tl_Cons])))); |
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val tl_type = result(); |
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(** drop **) |
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goalw List.thy [drop_def] "drop(0, l) = l"; |
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by (rtac rec_0 1); |
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val drop_0 = result(); |
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goalw List.thy [drop_def] "!!i. i:nat ==> drop(i, Nil) = Nil"; |
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by (etac nat_induct 1); |
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by (ALLGOALS (asm_simp_tac (nat_ss addsimps [tl_Nil]))); |
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val drop_Nil = result(); |
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goalw List.thy [drop_def] |
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"!!i. i:nat ==> drop(succ(i), Cons(a,l)) = drop(i,l)"; |
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by (etac nat_induct 1); |
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by (ALLGOALS (asm_simp_tac (nat_ss addsimps [tl_Cons]))); |
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val drop_succ_Cons = result(); |
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goalw List.thy [drop_def] |
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"!!i l. [| i:nat; l: list(A) |] ==> drop(i,l) : list(A)"; |
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by (etac nat_induct 1); |
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by (ALLGOALS (asm_simp_tac (nat_ss addsimps [tl_type]))); |
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val drop_type = result(); |
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(** list_rec -- by Vset recursion **) |
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goal List.thy "list_rec(Nil,c,h) = c"; |
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by (rtac (list_rec_def RS def_Vrec RS trans) 1); |
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by (simp_tac (ZF_ss addsimps list.case_eqns) 1); |
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val list_rec_Nil = result(); |
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goal List.thy "list_rec(Cons(a,l), c, h) = h(a, l, list_rec(l,c,h))"; |
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by (rtac (list_rec_def RS def_Vrec RS trans) 1); |
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by (simp_tac (rank_ss addsimps (rank_Cons2::list.case_eqns)) 1); |
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val list_rec_Cons = result(); |
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(*Type checking -- proved by induction, as usual*) |
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val prems = goal List.thy |
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"[| l: list(A); \ |
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\ c: C(Nil); \ |
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\ !!x y r. [| x:A; y: list(A); r: C(y) |] ==> h(x,y,r): C(Cons(x,y)) \ |
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\ |] ==> list_rec(l,c,h) : C(l)"; |
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by (list_ind_tac "l" prems 1); |
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by (ALLGOALS (asm_simp_tac |
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(ZF_ss addsimps (prems@[list_rec_Nil,list_rec_Cons])))); |
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val list_rec_type = result(); |
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(** Versions for use with definitions **) |
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val [rew] = goal List.thy |
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"[| !!l. j(l)==list_rec(l, c, h) |] ==> j(Nil) = c"; |
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by (rewtac rew); |
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by (rtac list_rec_Nil 1); |
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val def_list_rec_Nil = result(); |
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val [rew] = goal List.thy |
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"[| !!l. j(l)==list_rec(l, c, h) |] ==> j(Cons(a,l)) = h(a,l,j(l))"; |
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by (rewtac rew); |
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by (rtac list_rec_Cons 1); |
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val def_list_rec_Cons = result(); |
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fun list_recs def = map standard |
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([def] RL [def_list_rec_Nil, def_list_rec_Cons]); |
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(** map **) |
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val [map_Nil,map_Cons] = list_recs map_def; |
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val prems = goalw List.thy [map_def] |
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"[| l: list(A); !!x. x: A ==> h(x): B |] ==> map(h,l) : list(B)"; |
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by (REPEAT (ares_tac (prems @ list.intrs @ [list_rec_type]) 1)); |
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val map_type = result(); |
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val [major] = goal List.thy "l: list(A) ==> map(h,l) : list({h(u). u:A})"; |
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by (rtac (major RS map_type) 1); |
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by (etac RepFunI 1); |
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val map_type2 = result(); |
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(** length **) |
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val [length_Nil,length_Cons] = list_recs length_def; |
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goalw List.thy [length_def] |
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"!!l. l: list(A) ==> length(l) : nat"; |
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by (REPEAT (ares_tac [list_rec_type, nat_0I, nat_succI] 1)); |
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val length_type = result(); |
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(** app **) |
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val [app_Nil,app_Cons] = list_recs app_def; |
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goalw List.thy [app_def] |
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"!!xs ys. [| xs: list(A); ys: list(A) |] ==> xs@ys : list(A)"; |
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by (REPEAT (ares_tac [list_rec_type, list.Cons_I] 1)); |
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val app_type = result(); |
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(** rev **) |
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val [rev_Nil,rev_Cons] = list_recs rev_def; |
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goalw List.thy [rev_def] |
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"!!xs. xs: list(A) ==> rev(xs) : list(A)"; |
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by (REPEAT (ares_tac (list.intrs @ [list_rec_type, app_type]) 1)); |
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val rev_type = result(); |
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(** flat **) |
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val [flat_Nil,flat_Cons] = list_recs flat_def; |
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goalw List.thy [flat_def] |
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"!!ls. ls: list(list(A)) ==> flat(ls) : list(A)"; |
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by (REPEAT (ares_tac (list.intrs @ [list_rec_type, app_type]) 1)); |
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val flat_type = result(); |
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(** list_add **) |
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val [list_add_Nil,list_add_Cons] = list_recs list_add_def; |
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goalw List.thy [list_add_def] |
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"!!xs. xs: list(nat) ==> list_add(xs) : nat"; |
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by (REPEAT (ares_tac [list_rec_type, nat_0I, add_type] 1)); |
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val list_add_type = result(); |
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(** List simplification **) |
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val list_typechecks = |
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list.intrs @ |
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[list_rec_type, map_type, map_type2, app_type, length_type, |
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rev_type, flat_type, list_add_type]; |
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val list_ss = arith_ss |
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addsimps list.case_eqns |
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addsimps [list_rec_Nil, list_rec_Cons, |
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map_Nil, map_Cons, app_Nil, app_Cons, |
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length_Nil, length_Cons, rev_Nil, rev_Cons, |
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flat_Nil, flat_Cons, list_add_Nil, list_add_Cons] |
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setsolver (type_auto_tac list_typechecks); |
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(*** theorems about map ***) |
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val prems = goal List.thy |
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"l: list(A) ==> map(%u.u, l) = l"; |
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by (list_ind_tac "l" prems 1); |
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by (ALLGOALS (asm_simp_tac list_ss)); |
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val map_ident = result(); |
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val prems = goal List.thy |
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"l: list(A) ==> map(h, map(j,l)) = map(%u.h(j(u)), l)"; |
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by (list_ind_tac "l" prems 1); |
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by (ALLGOALS (asm_simp_tac list_ss)); |
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val map_compose = result(); |
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val prems = goal List.thy |
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"xs: list(A) ==> map(h, xs@ys) = map(h,xs) @ map(h,ys)"; |
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by (list_ind_tac "xs" prems 1); |
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by (ALLGOALS (asm_simp_tac list_ss)); |
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val map_app_distrib = result(); |
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val prems = goal List.thy |
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"ls: list(list(A)) ==> map(h, flat(ls)) = flat(map(map(h),ls))"; |
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by (list_ind_tac "ls" prems 1); |
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by (ALLGOALS (asm_simp_tac (list_ss addsimps [map_app_distrib]))); |
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val map_flat = result(); |
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val prems = goal List.thy |
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"l: list(A) ==> \ |
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\ list_rec(map(h,l), c, d) = \ |
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\ list_rec(l, c, %x xs r. d(h(x), map(h,xs), r))"; |
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by (list_ind_tac "l" prems 1); |
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by (ALLGOALS (asm_simp_tac list_ss)); |
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val list_rec_map = result(); |
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(** theorems about list(Collect(A,P)) -- used in ex/term.ML **) |
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(* c : list(Collect(B,P)) ==> c : list(B) *) |
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val list_CollectD = standard (Collect_subset RS list_mono RS subsetD); |
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val prems = goal List.thy |
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"l: list({x:A. h(x)=j(x)}) ==> map(h,l) = map(j,l)"; |
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by (list_ind_tac "l" prems 1); |
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by (ALLGOALS (asm_simp_tac list_ss)); |
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val map_list_Collect = result(); |
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(*** theorems about length ***) |
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val prems = goal List.thy |
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"xs: list(A) ==> length(map(h,xs)) = length(xs)"; |
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by (list_ind_tac "xs" prems 1); |
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by (ALLGOALS (asm_simp_tac list_ss)); |
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val length_map = result(); |
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val prems = goal List.thy |
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"xs: list(A) ==> length(xs@ys) = length(xs) #+ length(ys)"; |
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by (list_ind_tac "xs" prems 1); |
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by (ALLGOALS (asm_simp_tac list_ss)); |
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val length_app = result(); |
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(* [| m: nat; n: nat |] ==> m #+ succ(n) = succ(n) #+ m |
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Used for rewriting below*) |
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val add_commute_succ = nat_succI RSN (2,add_commute); |
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val prems = goal List.thy |
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"xs: list(A) ==> length(rev(xs)) = length(xs)"; |
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by (list_ind_tac "xs" prems 1); |
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by (ALLGOALS (asm_simp_tac (list_ss addsimps [length_app, add_commute_succ]))); |
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val length_rev = result(); |
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val prems = goal List.thy |
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"ls: list(list(A)) ==> length(flat(ls)) = list_add(map(length,ls))"; |
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by (list_ind_tac "ls" prems 1); |
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by (ALLGOALS (asm_simp_tac (list_ss addsimps [length_app]))); |
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val length_flat = result(); |
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(** Length and drop **) |
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(*Lemma for the inductive step of drop_length*) |
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goal List.thy |
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"!!xs. xs: list(A) ==> \ |
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\ ALL x. EX z zs. drop(length(xs), Cons(x,xs)) = Cons(z,zs)"; |
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by (etac list.induct 1); |
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by (ALLGOALS (asm_simp_tac (list_ss addsimps [drop_0,drop_succ_Cons]))); |
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by (fast_tac ZF_cs 1); |
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val drop_length_Cons_lemma = result(); |
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val drop_length_Cons = standard (drop_length_Cons_lemma RS spec); |
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goal List.thy |
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"!!l. l: list(A) ==> ALL i: length(l). EX z zs. drop(i,l) = Cons(z,zs)"; |
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by (etac list.induct 1); |
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by (ALLGOALS (asm_simp_tac (list_ss addsimps bquant_simps))); |
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by (rtac conjI 1); |
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by (etac drop_length_Cons 1); |
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by (rtac ballI 1); |
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by (rtac natE 1); |
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by (etac ([asm_rl, length_type, Ord_nat] MRS Ord_trans) 1); |
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by (assume_tac 1); |
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by (asm_simp_tac (list_ss addsimps [drop_0]) 1); |
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by (fast_tac ZF_cs 1); |
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by (asm_simp_tac (list_ss addsimps [drop_succ_Cons]) 1); |
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by (dtac bspec 1); |
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by (fast_tac ZF_cs 2); |
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by (fast_tac (ZF_cs addEs [succ_in_naturalD,length_type]) 1); |
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val drop_length_lemma = result(); |
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val drop_length = standard (drop_length_lemma RS bspec); |
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344 |
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345 |
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(*** theorems about app ***) |
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347 |
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val [major] = goal List.thy "xs: list(A) ==> xs@Nil=xs"; |
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by (rtac (major RS list.induct) 1); |
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by (ALLGOALS (asm_simp_tac list_ss)); |
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val app_right_Nil = result(); |
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352 |
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353 |
val prems = goal List.thy "xs: list(A) ==> (xs@ys)@zs = xs@(ys@zs)"; |
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by (list_ind_tac "xs" prems 1); |
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by (ALLGOALS (asm_simp_tac list_ss)); |
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val app_assoc = result(); |
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357 |
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358 |
val prems = goal List.thy |
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359 |
"ls: list(list(A)) ==> flat(ls@ms) = flat(ls)@flat(ms)"; |
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by (list_ind_tac "ls" prems 1); |
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by (ALLGOALS (asm_simp_tac (list_ss addsimps [app_assoc]))); |
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362 |
val flat_app_distrib = result(); |
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363 |
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364 |
(*** theorems about rev ***) |
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365 |
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366 |
val prems = goal List.thy "l: list(A) ==> rev(map(h,l)) = map(h,rev(l))"; |
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367 |
by (list_ind_tac "l" prems 1); |
|
368 |
by (ALLGOALS (asm_simp_tac (list_ss addsimps [map_app_distrib]))); |
|
369 |
val rev_map_distrib = result(); |
|
370 |
||
371 |
(*Simplifier needs the premises as assumptions because rewriting will not |
|
372 |
instantiate the variable ?A in the rules' typing conditions; note that |
|
373 |
rev_type does not instantiate ?A. Only the premises do. |
|
374 |
*) |
|
375 |
goal List.thy |
|
376 |
"!!xs. [| xs: list(A); ys: list(A) |] ==> rev(xs@ys) = rev(ys)@rev(xs)"; |
|
377 |
by (etac list.induct 1); |
|
378 |
by (ALLGOALS (asm_simp_tac (list_ss addsimps [app_right_Nil,app_assoc]))); |
|
379 |
val rev_app_distrib = result(); |
|
380 |
||
381 |
val prems = goal List.thy "l: list(A) ==> rev(rev(l))=l"; |
|
382 |
by (list_ind_tac "l" prems 1); |
|
383 |
by (ALLGOALS (asm_simp_tac (list_ss addsimps [rev_app_distrib]))); |
|
384 |
val rev_rev_ident = result(); |
|
385 |
||
386 |
val prems = goal List.thy |
|
387 |
"ls: list(list(A)) ==> rev(flat(ls)) = flat(map(rev,rev(ls)))"; |
|
388 |
by (list_ind_tac "ls" prems 1); |
|
389 |
by (ALLGOALS (asm_simp_tac (list_ss addsimps |
|
390 |
[map_app_distrib, flat_app_distrib, rev_app_distrib, app_right_Nil]))); |
|
391 |
val rev_flat = result(); |
|
392 |
||
393 |
||
394 |
(*** theorems about list_add ***) |
|
395 |
||
396 |
val prems = goal List.thy |
|
397 |
"[| xs: list(nat); ys: list(nat) |] ==> \ |
|
398 |
\ list_add(xs@ys) = list_add(ys) #+ list_add(xs)"; |
|
399 |
by (cut_facts_tac prems 1); |
|
400 |
by (list_ind_tac "xs" prems 1); |
|
401 |
by (ALLGOALS |
|
402 |
(asm_simp_tac (list_ss addsimps [add_0_right, add_assoc RS sym]))); |
|
403 |
by (rtac (add_commute RS subst_context) 1); |
|
404 |
by (REPEAT (ares_tac [refl, list_add_type] 1)); |
|
405 |
val list_add_app = result(); |
|
406 |
||
407 |
val prems = goal List.thy |
|
408 |
"l: list(nat) ==> list_add(rev(l)) = list_add(l)"; |
|
409 |
by (list_ind_tac "l" prems 1); |
|
410 |
by (ALLGOALS |
|
411 |
(asm_simp_tac (list_ss addsimps [list_add_app, add_0_right]))); |
|
412 |
val list_add_rev = result(); |
|
413 |
||
414 |
val prems = goal List.thy |
|
415 |
"ls: list(list(nat)) ==> list_add(flat(ls)) = list_add(map(list_add,ls))"; |
|
416 |
by (list_ind_tac "ls" prems 1); |
|
417 |
by (ALLGOALS (asm_simp_tac (list_ss addsimps [list_add_app]))); |
|
418 |
by (REPEAT (ares_tac [refl, list_add_type, map_type, add_commute] 1)); |
|
419 |
val list_add_flat = result(); |
|
420 |
||
421 |
(** New induction rule **) |
|
422 |
||
423 |
val major::prems = goal List.thy |
|
424 |
"[| l: list(A); \ |
|
425 |
\ P(Nil); \ |
|
426 |
\ !!x y. [| x: A; y: list(A); P(y) |] ==> P(y @ [x]) \ |
|
427 |
\ |] ==> P(l)"; |
|
428 |
by (rtac (major RS rev_rev_ident RS subst) 1); |
|
429 |
by (rtac (major RS rev_type RS list.induct) 1); |
|
430 |
by (ALLGOALS (asm_simp_tac (list_ss addsimps prems))); |
|
431 |
val list_append_induct = result(); |
|
432 |