src/HOL/List.ML
author berghofe
Fri Jul 24 13:03:20 1998 +0200 (1998-07-24)
changeset 5183 89f162de39cf
parent 5162 53e505c6019c
child 5200 a23c23af335f
permissions -rw-r--r--
Adapted to new datatype package.
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(*  Title:      HOL/List
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    Copyright   1994 TU Muenchen
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List lemmas
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*)
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Goal "!x. xs ~= x#xs";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed_spec_mp "not_Cons_self";
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bind_thm("not_Cons_self2",not_Cons_self RS not_sym);
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Addsimps [not_Cons_self,not_Cons_self2];
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Goal "(xs ~= []) = (? y ys. xs = y#ys)";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "neq_Nil_conv";
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(* Induction over the length of a list: *)
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val [prem] = Goal
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  "(!!xs. (!ys. length ys < length xs --> P ys) ==> P xs) ==> P(xs)";
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by (rtac measure_induct 1 THEN etac prem 1);
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qed "length_induct";
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(** "lists": the list-forming operator over sets **)
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Goalw lists.defs "A<=B ==> lists A <= lists B";
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by (rtac lfp_mono 1);
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by (REPEAT (ares_tac basic_monos 1));
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qed "lists_mono";
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val listsE = lists.mk_cases list.simps  "x#l : lists A";
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AddSEs [listsE];
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AddSIs lists.intrs;
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Goal "l: lists A ==> l: lists B --> l: lists (A Int B)";
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by (etac lists.induct 1);
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by (ALLGOALS Blast_tac);
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qed_spec_mp "lists_IntI";
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Goal "lists (A Int B) = lists A Int lists B";
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by (rtac (mono_Int RS equalityI) 1);
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by (simp_tac (simpset() addsimps [mono_def, lists_mono]) 1);
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by (blast_tac (claset() addSIs [lists_IntI]) 1);
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qed "lists_Int_eq";
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Addsimps [lists_Int_eq];
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(**  Case analysis **)
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section "Case analysis";
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val prems = Goal "[| P([]); !!x xs. P(x#xs) |] ==> P(xs)";
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by (induct_tac "xs" 1);
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by (REPEAT(resolve_tac prems 1));
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qed "list_cases";
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Goal "(xs=[] --> P([])) & (!y ys. xs=y#ys --> P(y#ys)) --> P(xs)";
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by (induct_tac "xs" 1);
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by (Blast_tac 1);
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by (Blast_tac 1);
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bind_thm("list_eq_cases",
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  impI RSN (2,allI RSN (2,allI RSN (2,impI RS (conjI RS (result() RS mp))))));
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(** length **)
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(* needs to come before "@" because of thm append_eq_append_conv *)
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section "length";
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Goal "length(xs@ys) = length(xs)+length(ys)";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed"length_append";
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Addsimps [length_append];
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Goal "length (map f xs) = length xs";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "length_map";
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Addsimps [length_map];
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Goal "length(rev xs) = length(xs)";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "length_rev";
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Addsimps [length_rev];
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Goal "xs ~= [] ==> length(tl xs) = (length xs) - 1";
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by (exhaust_tac "xs" 1);
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by (Auto_tac);
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qed "length_tl";
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Addsimps [length_tl];
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Goal "(length xs = 0) = (xs = [])";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "length_0_conv";
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AddIffs [length_0_conv];
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Goal "(0 = length xs) = (xs = [])";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "zero_length_conv";
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AddIffs [zero_length_conv];
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Goal "(0 < length xs) = (xs ~= [])";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "length_greater_0_conv";
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AddIffs [length_greater_0_conv];
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(** @ - append **)
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section "@ - append";
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Goal "(xs@ys)@zs = xs@(ys@zs)";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "append_assoc";
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Addsimps [append_assoc];
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Goal "xs @ [] = xs";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "append_Nil2";
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Addsimps [append_Nil2];
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Goal "(xs@ys = []) = (xs=[] & ys=[])";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "append_is_Nil_conv";
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AddIffs [append_is_Nil_conv];
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Goal "([] = xs@ys) = (xs=[] & ys=[])";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "Nil_is_append_conv";
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AddIffs [Nil_is_append_conv];
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Goal "(xs @ ys = xs) = (ys=[])";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "append_self_conv";
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Goal "(xs = xs @ ys) = (ys=[])";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "self_append_conv";
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AddIffs [append_self_conv,self_append_conv];
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Goal "!ys. length xs = length ys | length us = length vs \
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\              --> (xs@us = ys@vs) = (xs=ys & us=vs)";
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by (induct_tac "xs" 1);
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 by (rtac allI 1);
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 by (exhaust_tac "ys" 1);
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  by (Asm_simp_tac 1);
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 by (fast_tac (claset() addIs [less_add_Suc2] addss simpset()
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                      addEs [less_not_refl2 RSN (2,rev_notE)]) 1);
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by (rtac allI 1);
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by (exhaust_tac "ys" 1);
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 by (fast_tac (claset() addIs [less_add_Suc2] addss simpset()
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                      addEs [(less_not_refl2 RS not_sym) RSN (2,rev_notE)]) 1);
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by (Asm_simp_tac 1);
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qed_spec_mp "append_eq_append_conv";
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Addsimps [append_eq_append_conv];
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Goal "(xs @ ys = xs @ zs) = (ys=zs)";
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by (Simp_tac 1);
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qed "same_append_eq";
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Goal "(xs @ [x] = ys @ [y]) = (xs = ys & x = y)"; 
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by (Simp_tac 1);
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qed "append1_eq_conv";
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Goal "(ys @ xs = zs @ xs) = (ys=zs)";
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by (Simp_tac 1);
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qed "append_same_eq";
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AddSIs
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 [same_append_eq RS iffD2, append1_eq_conv RS iffD2, append_same_eq RS iffD2];
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AddSDs
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 [same_append_eq RS iffD1, append1_eq_conv RS iffD1, append_same_eq RS iffD1];
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Goal "(xs @ ys = ys) = (xs=[])";
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by (cut_inst_tac [("zs","[]")] append_same_eq 1);
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by (Auto_tac);
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qed "append_self_conv2";
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Goal "(ys = xs @ ys) = (xs=[])";
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by (simp_tac (simpset() addsimps
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     [simplify (simpset()) (read_instantiate[("ys","[]")]append_same_eq)]) 1);
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by (Blast_tac 1);
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qed "self_append_conv2";
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AddIffs [append_self_conv2,self_append_conv2];
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Goal "xs ~= [] --> hd xs # tl xs = xs";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed_spec_mp "hd_Cons_tl";
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Addsimps [hd_Cons_tl];
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Goal "hd(xs@ys) = (if xs=[] then hd ys else hd xs)";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "hd_append";
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Goal "xs ~= [] ==> hd(xs @ ys) = hd xs";
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by (asm_simp_tac (simpset() addsimps [hd_append]
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                           addsplits [list.split]) 1);
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qed "hd_append2";
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Addsimps [hd_append2];
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Goal "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)";
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by (simp_tac (simpset() addsplits [list.split]) 1);
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qed "tl_append";
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Goal "xs ~= [] ==> tl(xs @ ys) = (tl xs) @ ys";
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by (asm_simp_tac (simpset() addsimps [tl_append]
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                           addsplits [list.split]) 1);
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qed "tl_append2";
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Addsimps [tl_append2];
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(** map **)
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section "map";
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Goal
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  "(!x. x : set xs --> f x = g x) --> map f xs = map g xs";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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bind_thm("map_ext", impI RS (allI RS (result() RS mp)));
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Goal "map (%x. x) = (%xs. xs)";
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by (rtac ext 1);
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "map_ident";
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Addsimps[map_ident];
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Goal "map f (xs@ys) = map f xs @ map f ys";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "map_append";
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Addsimps[map_append];
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Goalw [o_def] "map (f o g) xs = map f (map g xs)";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "map_compose";
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Addsimps[map_compose];
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Goal "rev(map f xs) = map f (rev xs)";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "rev_map";
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(* a congruence rule for map: *)
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Goal
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 "(xs=ys) --> (!x. x : set ys --> f x = g x) --> map f xs = map g ys";
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by (rtac impI 1);
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by (hyp_subst_tac 1);
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by (induct_tac "ys" 1);
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by (Auto_tac);
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val lemma = result();
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bind_thm("map_cong",impI RSN (2,allI RSN (2,lemma RS mp RS mp)));
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Goal "(map f xs = []) = (xs = [])";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "map_is_Nil_conv";
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AddIffs [map_is_Nil_conv];
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Goal "([] = map f xs) = (xs = [])";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "Nil_is_map_conv";
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AddIffs [Nil_is_map_conv];
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(** rev **)
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section "rev";
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Goal "rev(xs@ys) = rev(ys) @ rev(xs)";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "rev_append";
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Addsimps[rev_append];
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Goal "rev(rev l) = l";
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by (induct_tac "l" 1);
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by (Auto_tac);
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qed "rev_rev_ident";
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Addsimps[rev_rev_ident];
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Goal "(rev xs = []) = (xs = [])";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "rev_is_Nil_conv";
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AddIffs [rev_is_Nil_conv];
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Goal "([] = rev xs) = (xs = [])";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "Nil_is_rev_conv";
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AddIffs [Nil_is_rev_conv];
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val prems = Goal "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P xs";
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by (stac (rev_rev_ident RS sym) 1);
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br(read_instantiate [("P","%xs. ?P(rev xs)")]list.induct)1;
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by (ALLGOALS Simp_tac);
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by (resolve_tac prems 1);
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by (eresolve_tac prems 1);
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qed "rev_induct";
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Goal  "(xs = [] --> P) -->  (!ys y. xs = ys@[y] --> P) --> P";
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by (res_inst_tac [("xs","xs")] rev_induct 1);
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by (Auto_tac);
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bind_thm ("rev_exhaust",
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  impI RSN (2,allI RSN (2,allI RSN (2,impI RS (result() RS mp RS mp)))));
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(** mem **)
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section "mem";
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Goal "x mem (xs@ys) = (x mem xs | x mem ys)";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "mem_append";
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Addsimps[mem_append];
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Goal "x mem [x:xs. P(x)] = (x mem xs & P(x))";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "mem_filter";
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Addsimps[mem_filter];
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(** set **)
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section "set";
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Goal "set (xs@ys) = (set xs Un set ys)";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "set_append";
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Addsimps[set_append];
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Goal "(x mem xs) = (x: set xs)";
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   353
by (induct_tac "xs" 1);
nipkow@5129
   354
by (Auto_tac);
paulson@3647
   355
qed "set_mem_eq";
paulson@1812
   356
nipkow@4935
   357
Goal "set l <= set (x#l)";
nipkow@5129
   358
by (Auto_tac);
paulson@3647
   359
qed "set_subset_Cons";
paulson@1936
   360
nipkow@4935
   361
Goal "(set xs = {}) = (xs = [])";
paulson@3457
   362
by (induct_tac "xs" 1);
nipkow@5129
   363
by (Auto_tac);
paulson@3647
   364
qed "set_empty";
paulson@3647
   365
Addsimps [set_empty];
nipkow@2608
   366
nipkow@4935
   367
Goal "set(rev xs) = set(xs)";
paulson@3457
   368
by (induct_tac "xs" 1);
nipkow@5129
   369
by (Auto_tac);
paulson@3647
   370
qed "set_rev";
paulson@3647
   371
Addsimps [set_rev];
nipkow@2608
   372
nipkow@4935
   373
Goal "set(map f xs) = f``(set xs)";
paulson@3457
   374
by (induct_tac "xs" 1);
nipkow@5129
   375
by (Auto_tac);
paulson@3647
   376
qed "set_map";
paulson@3647
   377
Addsimps [set_map];
nipkow@2608
   378
nipkow@4935
   379
Goal "(x : set(filter P xs)) = (x : set xs & P x)";
nipkow@4605
   380
by (induct_tac "xs" 1);
nipkow@5129
   381
by (Auto_tac);
nipkow@4605
   382
qed "in_set_filter";
nipkow@4605
   383
Addsimps [in_set_filter];
nipkow@4605
   384
paulson@1812
   385
clasohm@923
   386
(** list_all **)
clasohm@923
   387
nipkow@3467
   388
section "list_all";
nipkow@3467
   389
nipkow@4935
   390
Goal "list_all (%x. True) xs = True";
nipkow@3040
   391
by (induct_tac "xs" 1);
nipkow@5129
   392
by (Auto_tac);
clasohm@923
   393
qed "list_all_True";
nipkow@2512
   394
Addsimps [list_all_True];
clasohm@923
   395
nipkow@4935
   396
Goal "list_all p (xs@ys) = (list_all p xs & list_all p ys)";
nipkow@3040
   397
by (induct_tac "xs" 1);
nipkow@5129
   398
by (Auto_tac);
nipkow@2512
   399
qed "list_all_append";
nipkow@2512
   400
Addsimps [list_all_append];
clasohm@923
   401
nipkow@4935
   402
Goal "list_all P xs = (!x. x mem xs --> P(x))";
nipkow@3040
   403
by (induct_tac "xs" 1);
nipkow@5129
   404
by (Auto_tac);
clasohm@923
   405
qed "list_all_mem_conv";
clasohm@923
   406
clasohm@923
   407
nipkow@2608
   408
(** filter **)
clasohm@923
   409
nipkow@3467
   410
section "filter";
nipkow@3467
   411
nipkow@4935
   412
Goal "filter P (xs@ys) = filter P xs @ filter P ys";
paulson@3457
   413
by (induct_tac "xs" 1);
nipkow@5129
   414
by (Auto_tac);
nipkow@2608
   415
qed "filter_append";
nipkow@2608
   416
Addsimps [filter_append];
nipkow@2608
   417
nipkow@4935
   418
Goal "filter (%x. True) xs = xs";
nipkow@4605
   419
by (induct_tac "xs" 1);
nipkow@5129
   420
by (Auto_tac);
nipkow@4605
   421
qed "filter_True";
nipkow@4605
   422
Addsimps [filter_True];
nipkow@4605
   423
nipkow@4935
   424
Goal "filter (%x. False) xs = []";
nipkow@4605
   425
by (induct_tac "xs" 1);
nipkow@5129
   426
by (Auto_tac);
nipkow@4605
   427
qed "filter_False";
nipkow@4605
   428
Addsimps [filter_False];
nipkow@4605
   429
nipkow@4935
   430
Goal "length (filter P xs) <= length xs";
paulson@3457
   431
by (induct_tac "xs" 1);
nipkow@5129
   432
by (Auto_tac);
nipkow@4605
   433
qed "length_filter";
paulson@3383
   434
nipkow@2608
   435
nipkow@2608
   436
(** concat **)
nipkow@2608
   437
nipkow@3467
   438
section "concat";
nipkow@3467
   439
nipkow@4935
   440
Goal  "concat(xs@ys) = concat(xs)@concat(ys)";
nipkow@3040
   441
by (induct_tac "xs" 1);
nipkow@5129
   442
by (Auto_tac);
nipkow@2608
   443
qed"concat_append";
nipkow@2608
   444
Addsimps [concat_append];
nipkow@2512
   445
nipkow@4935
   446
Goal "(concat xss = []) = (!xs:set xss. xs=[])";
wenzelm@4423
   447
by (induct_tac "xss" 1);
nipkow@5129
   448
by (Auto_tac);
nipkow@3896
   449
qed "concat_eq_Nil_conv";
nipkow@3896
   450
AddIffs [concat_eq_Nil_conv];
nipkow@3896
   451
nipkow@4935
   452
Goal "([] = concat xss) = (!xs:set xss. xs=[])";
wenzelm@4423
   453
by (induct_tac "xss" 1);
nipkow@5129
   454
by (Auto_tac);
nipkow@3896
   455
qed "Nil_eq_concat_conv";
nipkow@3896
   456
AddIffs [Nil_eq_concat_conv];
nipkow@3896
   457
nipkow@4935
   458
Goal  "set(concat xs) = Union(set `` set xs)";
nipkow@3467
   459
by (induct_tac "xs" 1);
nipkow@5129
   460
by (Auto_tac);
paulson@3647
   461
qed"set_concat";
paulson@3647
   462
Addsimps [set_concat];
nipkow@3467
   463
nipkow@4935
   464
Goal "map f (concat xs) = concat (map (map f) xs)"; 
nipkow@3467
   465
by (induct_tac "xs" 1);
nipkow@5129
   466
by (Auto_tac);
nipkow@3467
   467
qed "map_concat";
nipkow@3467
   468
nipkow@4935
   469
Goal "filter p (concat xs) = concat (map (filter p) xs)"; 
nipkow@3467
   470
by (induct_tac "xs" 1);
nipkow@5129
   471
by (Auto_tac);
nipkow@3467
   472
qed"filter_concat"; 
nipkow@3467
   473
nipkow@4935
   474
Goal "rev(concat xs) = concat (map rev (rev xs))";
nipkow@3467
   475
by (induct_tac "xs" 1);
nipkow@5129
   476
by (Auto_tac);
nipkow@2608
   477
qed "rev_concat";
clasohm@923
   478
clasohm@923
   479
(** nth **)
clasohm@923
   480
nipkow@3467
   481
section "nth";
nipkow@3467
   482
nipkow@4935
   483
Goal
nipkow@4502
   484
  "!xs. (xs@ys)!n = (if n < length xs then xs!n else ys!(n - length xs))";
berghofe@5183
   485
by (induct_tac "n" 1);
paulson@3457
   486
 by (Asm_simp_tac 1);
paulson@3457
   487
 by (rtac allI 1);
paulson@3457
   488
 by (exhaust_tac "xs" 1);
nipkow@5129
   489
  by (Auto_tac);
nipkow@2608
   490
qed_spec_mp "nth_append";
nipkow@2608
   491
nipkow@4935
   492
Goal "!n. n < length xs --> (map f xs)!n = f(xs!n)";
nipkow@3040
   493
by (induct_tac "xs" 1);
nipkow@1301
   494
(* case [] *)
nipkow@1301
   495
by (Asm_full_simp_tac 1);
nipkow@1301
   496
(* case x#xl *)
nipkow@1301
   497
by (rtac allI 1);
berghofe@5183
   498
by (induct_tac "n" 1);
nipkow@5129
   499
by (Auto_tac);
nipkow@1485
   500
qed_spec_mp "nth_map";
nipkow@1301
   501
Addsimps [nth_map];
nipkow@1301
   502
nipkow@4935
   503
Goal "!n. n < length xs --> list_all P xs --> P(xs!n)";
nipkow@3040
   504
by (induct_tac "xs" 1);
nipkow@1301
   505
(* case [] *)
nipkow@1301
   506
by (Simp_tac 1);
nipkow@1301
   507
(* case x#xl *)
nipkow@1301
   508
by (rtac allI 1);
berghofe@5183
   509
by (induct_tac "n" 1);
nipkow@5129
   510
by (Auto_tac);
nipkow@1485
   511
qed_spec_mp "list_all_nth";
nipkow@1301
   512
nipkow@4935
   513
Goal "!n. n < length xs --> xs!n mem xs";
nipkow@3040
   514
by (induct_tac "xs" 1);
nipkow@1301
   515
(* case [] *)
nipkow@1301
   516
by (Simp_tac 1);
nipkow@1301
   517
(* case x#xl *)
nipkow@1301
   518
by (rtac allI 1);
berghofe@5183
   519
by (induct_tac "n" 1);
nipkow@1301
   520
(* case 0 *)
nipkow@1301
   521
by (Asm_full_simp_tac 1);
nipkow@1301
   522
(* case Suc x *)
nipkow@4686
   523
by (Asm_full_simp_tac 1);
nipkow@1485
   524
qed_spec_mp "nth_mem";
nipkow@1301
   525
Addsimps [nth_mem];
nipkow@1301
   526
nipkow@5077
   527
(** list update **)
nipkow@5077
   528
nipkow@5077
   529
section "list update";
nipkow@5077
   530
nipkow@5077
   531
Goal "!i. length(xs[i:=x]) = length xs";
nipkow@5077
   532
by (induct_tac "xs" 1);
nipkow@5077
   533
by (Simp_tac 1);
berghofe@5183
   534
by (asm_full_simp_tac (simpset() addsplits [nat.split]) 1);
nipkow@5077
   535
qed_spec_mp "length_list_update";
nipkow@5077
   536
Addsimps [length_list_update];
nipkow@5077
   537
nipkow@5077
   538
nipkow@3896
   539
(** last & butlast **)
nipkow@1327
   540
nipkow@4935
   541
Goal "last(xs@[x]) = x";
wenzelm@4423
   542
by (induct_tac "xs" 1);
nipkow@5129
   543
by (Auto_tac);
nipkow@3896
   544
qed "last_snoc";
nipkow@3896
   545
Addsimps [last_snoc];
nipkow@3896
   546
nipkow@4935
   547
Goal "butlast(xs@[x]) = xs";
wenzelm@4423
   548
by (induct_tac "xs" 1);
nipkow@5129
   549
by (Auto_tac);
nipkow@3896
   550
qed "butlast_snoc";
nipkow@3896
   551
Addsimps [butlast_snoc];
nipkow@3896
   552
nipkow@4935
   553
Goal "length(butlast xs) = length xs - 1";
nipkow@4935
   554
by (res_inst_tac [("xs","xs")] rev_induct 1);
nipkow@5129
   555
by (Auto_tac);
nipkow@4643
   556
qed "length_butlast";
nipkow@4643
   557
Addsimps [length_butlast];
nipkow@4643
   558
nipkow@4935
   559
Goal
nipkow@3896
   560
  "!ys. butlast (xs@ys) = (if ys=[] then butlast xs else xs@butlast ys)";
wenzelm@4423
   561
by (induct_tac "xs" 1);
nipkow@5129
   562
by (Auto_tac);
nipkow@3896
   563
qed_spec_mp "butlast_append";
nipkow@3896
   564
nipkow@4935
   565
Goal "x:set(butlast xs) --> x:set xs";
wenzelm@4423
   566
by (induct_tac "xs" 1);
nipkow@5129
   567
by (Auto_tac);
nipkow@3896
   568
qed_spec_mp "in_set_butlastD";
nipkow@3896
   569
nipkow@5043
   570
Goal "x:set(butlast xs) ==> x:set(butlast(xs@ys))";
nipkow@4686
   571
by (asm_simp_tac (simpset() addsimps [butlast_append]) 1);
wenzelm@4423
   572
by (blast_tac (claset() addDs [in_set_butlastD]) 1);
nipkow@3896
   573
qed "in_set_butlast_appendI1";
nipkow@3896
   574
nipkow@5043
   575
Goal "x:set(butlast ys) ==> x:set(butlast(xs@ys))";
nipkow@4686
   576
by (asm_simp_tac (simpset() addsimps [butlast_append]) 1);
wenzelm@4423
   577
by (Clarify_tac 1);
wenzelm@4423
   578
by (Full_simp_tac 1);
nipkow@3896
   579
qed "in_set_butlast_appendI2";
nipkow@3902
   580
nipkow@2608
   581
(** take  & drop **)
nipkow@2608
   582
section "take & drop";
nipkow@1327
   583
nipkow@4935
   584
Goal "take 0 xs = []";
nipkow@3040
   585
by (induct_tac "xs" 1);
nipkow@5129
   586
by (Auto_tac);
nipkow@1327
   587
qed "take_0";
nipkow@1327
   588
nipkow@4935
   589
Goal "drop 0 xs = xs";
nipkow@3040
   590
by (induct_tac "xs" 1);
nipkow@5129
   591
by (Auto_tac);
nipkow@2608
   592
qed "drop_0";
nipkow@2608
   593
nipkow@4935
   594
Goal "take (Suc n) (x#xs) = x # take n xs";
paulson@1552
   595
by (Simp_tac 1);
nipkow@1419
   596
qed "take_Suc_Cons";
nipkow@1327
   597
nipkow@4935
   598
Goal "drop (Suc n) (x#xs) = drop n xs";
nipkow@2608
   599
by (Simp_tac 1);
nipkow@2608
   600
qed "drop_Suc_Cons";
nipkow@2608
   601
nipkow@2608
   602
Delsimps [take_Cons,drop_Cons];
nipkow@2608
   603
Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons];
nipkow@2608
   604
nipkow@4935
   605
Goal "!xs. length(take n xs) = min (length xs) n";
berghofe@5183
   606
by (induct_tac "n" 1);
nipkow@5129
   607
 by (Auto_tac);
paulson@3457
   608
by (exhaust_tac "xs" 1);
nipkow@5129
   609
 by (Auto_tac);
nipkow@2608
   610
qed_spec_mp "length_take";
nipkow@2608
   611
Addsimps [length_take];
clasohm@923
   612
nipkow@4935
   613
Goal "!xs. length(drop n xs) = (length xs - n)";
berghofe@5183
   614
by (induct_tac "n" 1);
nipkow@5129
   615
 by (Auto_tac);
paulson@3457
   616
by (exhaust_tac "xs" 1);
nipkow@5129
   617
 by (Auto_tac);
nipkow@2608
   618
qed_spec_mp "length_drop";
nipkow@2608
   619
Addsimps [length_drop];
nipkow@2608
   620
nipkow@4935
   621
Goal "!xs. length xs <= n --> take n xs = xs";
berghofe@5183
   622
by (induct_tac "n" 1);
nipkow@5129
   623
 by (Auto_tac);
paulson@3457
   624
by (exhaust_tac "xs" 1);
nipkow@5129
   625
 by (Auto_tac);
nipkow@2608
   626
qed_spec_mp "take_all";
clasohm@923
   627
nipkow@4935
   628
Goal "!xs. length xs <= n --> drop n xs = []";
berghofe@5183
   629
by (induct_tac "n" 1);
nipkow@5129
   630
 by (Auto_tac);
paulson@3457
   631
by (exhaust_tac "xs" 1);
nipkow@5129
   632
 by (Auto_tac);
nipkow@2608
   633
qed_spec_mp "drop_all";
nipkow@2608
   634
nipkow@4935
   635
Goal 
nipkow@2608
   636
  "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)";
berghofe@5183
   637
by (induct_tac "n" 1);
nipkow@5129
   638
 by (Auto_tac);
paulson@3457
   639
by (exhaust_tac "xs" 1);
nipkow@5129
   640
 by (Auto_tac);
nipkow@2608
   641
qed_spec_mp "take_append";
nipkow@2608
   642
Addsimps [take_append];
nipkow@2608
   643
nipkow@4935
   644
Goal "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; 
berghofe@5183
   645
by (induct_tac "n" 1);
nipkow@5129
   646
 by (Auto_tac);
paulson@3457
   647
by (exhaust_tac "xs" 1);
nipkow@5129
   648
 by (Auto_tac);
nipkow@2608
   649
qed_spec_mp "drop_append";
nipkow@2608
   650
Addsimps [drop_append];
nipkow@2608
   651
nipkow@4935
   652
Goal "!xs n. take n (take m xs) = take (min n m) xs"; 
berghofe@5183
   653
by (induct_tac "m" 1);
nipkow@5129
   654
 by (Auto_tac);
paulson@3457
   655
by (exhaust_tac "xs" 1);
nipkow@5129
   656
 by (Auto_tac);
berghofe@5183
   657
by (exhaust_tac "na" 1);
nipkow@5129
   658
 by (Auto_tac);
nipkow@2608
   659
qed_spec_mp "take_take";
nipkow@2608
   660
nipkow@4935
   661
Goal "!xs. drop n (drop m xs) = drop (n + m) xs"; 
berghofe@5183
   662
by (induct_tac "m" 1);
nipkow@5129
   663
 by (Auto_tac);
paulson@3457
   664
by (exhaust_tac "xs" 1);
nipkow@5129
   665
 by (Auto_tac);
nipkow@2608
   666
qed_spec_mp "drop_drop";
clasohm@923
   667
nipkow@4935
   668
Goal "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; 
berghofe@5183
   669
by (induct_tac "m" 1);
nipkow@5129
   670
 by (Auto_tac);
paulson@3457
   671
by (exhaust_tac "xs" 1);
nipkow@5129
   672
 by (Auto_tac);
nipkow@2608
   673
qed_spec_mp "take_drop";
nipkow@2608
   674
nipkow@4935
   675
Goal "!xs. take n (map f xs) = map f (take n xs)"; 
berghofe@5183
   676
by (induct_tac "n" 1);
nipkow@5129
   677
 by (Auto_tac);
paulson@3457
   678
by (exhaust_tac "xs" 1);
nipkow@5129
   679
 by (Auto_tac);
nipkow@2608
   680
qed_spec_mp "take_map"; 
nipkow@2608
   681
nipkow@4935
   682
Goal "!xs. drop n (map f xs) = map f (drop n xs)"; 
berghofe@5183
   683
by (induct_tac "n" 1);
nipkow@5129
   684
 by (Auto_tac);
paulson@3457
   685
by (exhaust_tac "xs" 1);
nipkow@5129
   686
 by (Auto_tac);
nipkow@2608
   687
qed_spec_mp "drop_map";
nipkow@2608
   688
nipkow@4935
   689
Goal "!n i. i < n --> (take n xs)!i = xs!i";
paulson@3457
   690
by (induct_tac "xs" 1);
nipkow@5129
   691
 by (Auto_tac);
paulson@3457
   692
by (exhaust_tac "n" 1);
paulson@3457
   693
 by (Blast_tac 1);
paulson@3457
   694
by (exhaust_tac "i" 1);
nipkow@5129
   695
 by (Auto_tac);
nipkow@2608
   696
qed_spec_mp "nth_take";
nipkow@2608
   697
Addsimps [nth_take];
clasohm@923
   698
nipkow@4935
   699
Goal  "!xs i. n + i <= length xs --> (drop n xs)!i = xs!(n+i)";
berghofe@5183
   700
by (induct_tac "n" 1);
nipkow@5129
   701
 by (Auto_tac);
paulson@3457
   702
by (exhaust_tac "xs" 1);
nipkow@5129
   703
 by (Auto_tac);
nipkow@2608
   704
qed_spec_mp "nth_drop";
nipkow@2608
   705
Addsimps [nth_drop];
nipkow@2608
   706
nipkow@2608
   707
(** takeWhile & dropWhile **)
nipkow@2608
   708
nipkow@3467
   709
section "takeWhile & dropWhile";
nipkow@3467
   710
nipkow@4935
   711
Goal "takeWhile P xs @ dropWhile P xs = xs";
nipkow@3586
   712
by (induct_tac "xs" 1);
nipkow@5129
   713
by (Auto_tac);
nipkow@3586
   714
qed "takeWhile_dropWhile_id";
nipkow@3586
   715
Addsimps [takeWhile_dropWhile_id];
nipkow@3586
   716
nipkow@4935
   717
Goal  "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs";
paulson@3457
   718
by (induct_tac "xs" 1);
nipkow@5129
   719
by (Auto_tac);
nipkow@2608
   720
bind_thm("takeWhile_append1", conjI RS (result() RS mp));
nipkow@2608
   721
Addsimps [takeWhile_append1];
clasohm@923
   722
nipkow@4935
   723
Goal "(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys";
paulson@3457
   724
by (induct_tac "xs" 1);
nipkow@5129
   725
by (Auto_tac);
nipkow@2608
   726
bind_thm("takeWhile_append2", ballI RS (result() RS mp));
nipkow@2608
   727
Addsimps [takeWhile_append2];
lcp@1169
   728
nipkow@4935
   729
Goal "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys";
paulson@3457
   730
by (induct_tac "xs" 1);
nipkow@5129
   731
by (Auto_tac);
nipkow@2608
   732
bind_thm("dropWhile_append1", conjI RS (result() RS mp));
nipkow@2608
   733
Addsimps [dropWhile_append1];
nipkow@2608
   734
nipkow@4935
   735
Goal "(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys";
paulson@3457
   736
by (induct_tac "xs" 1);
nipkow@5129
   737
by (Auto_tac);
nipkow@2608
   738
bind_thm("dropWhile_append2", ballI RS (result() RS mp));
nipkow@2608
   739
Addsimps [dropWhile_append2];
nipkow@2608
   740
nipkow@4935
   741
Goal "x:set(takeWhile P xs) --> x:set xs & P x";
paulson@3457
   742
by (induct_tac "xs" 1);
nipkow@5129
   743
by (Auto_tac);
paulson@3647
   744
qed_spec_mp"set_take_whileD";
nipkow@2608
   745
oheimb@4132
   746
qed_goal "zip_Nil_Nil"   thy "zip []     []     = []" (K [Simp_tac 1]);
oheimb@4132
   747
qed_goal "zip_Cons_Cons" thy "zip (x#xs) (y#ys) = (x,y)#zip xs ys" 
oheimb@4132
   748
						      (K [Simp_tac 1]);
nipkow@4605
   749
nipkow@4605
   750
(** nodups & remdups **)
nipkow@4605
   751
section "nodups & remdups";
nipkow@4605
   752
nipkow@4935
   753
Goal "set(remdups xs) = set xs";
nipkow@4605
   754
by (induct_tac "xs" 1);
nipkow@4605
   755
 by (Simp_tac 1);
nipkow@4686
   756
by (asm_full_simp_tac (simpset() addsimps [insert_absorb]) 1);
nipkow@4605
   757
qed "set_remdups";
nipkow@4605
   758
Addsimps [set_remdups];
nipkow@4605
   759
nipkow@4935
   760
Goal "nodups(remdups xs)";
nipkow@4605
   761
by (induct_tac "xs" 1);
nipkow@5129
   762
by (Auto_tac);
nipkow@4605
   763
qed "nodups_remdups";
nipkow@4605
   764
nipkow@4935
   765
Goal "nodups xs --> nodups (filter P xs)";
nipkow@4605
   766
by (induct_tac "xs" 1);
nipkow@5129
   767
by (Auto_tac);
nipkow@4605
   768
qed_spec_mp "nodups_filter";
nipkow@4605
   769
nipkow@3589
   770
(** replicate **)
nipkow@3589
   771
section "replicate";
nipkow@3589
   772
nipkow@4935
   773
Goal "set(replicate (Suc n) x) = {x}";
wenzelm@4423
   774
by (induct_tac "n" 1);
nipkow@5129
   775
by (Auto_tac);
nipkow@3589
   776
val lemma = result();
nipkow@3589
   777
nipkow@5043
   778
Goal "n ~= 0 ==> set(replicate n x) = {x}";
wenzelm@4423
   779
by (fast_tac (claset() addSDs [not0_implies_Suc] addSIs [lemma]) 1);
nipkow@3589
   780
qed "set_replicate";
nipkow@3589
   781
Addsimps [set_replicate];
nipkow@5162
   782
nipkow@5162
   783
nipkow@5162
   784
(***
nipkow@5162
   785
Simplification procedure for all list equalities.
nipkow@5162
   786
Currently only tries to rearranges @ to see if
nipkow@5162
   787
- both lists end in a singleton list,
nipkow@5162
   788
- or both lists end in the same list.
nipkow@5162
   789
***)
nipkow@5162
   790
local
nipkow@5162
   791
nipkow@5162
   792
val list_eq_pattern =
nipkow@5162
   793
  read_cterm (sign_of List.thy) ("(xs::'a list) = ys",HOLogic.boolT);
nipkow@5162
   794
berghofe@5183
   795
fun last (cons as Const("List.list.op #",_) $ _ $ xs) =
berghofe@5183
   796
      (case xs of Const("List.list.[]",_) => cons | _ => last xs)
berghofe@5183
   797
  | last (Const("List.list.op @",_) $ _ $ ys) = last ys
nipkow@5162
   798
  | last t = t;
nipkow@5162
   799
berghofe@5183
   800
fun list1 (Const("List.list.op #",_) $ _ $ Const("List.list.[]",_)) = true
nipkow@5162
   801
  | list1 _ = false;
nipkow@5162
   802
berghofe@5183
   803
fun butlast ((cons as Const("List.list.op #",_) $ x) $ xs) =
berghofe@5183
   804
      (case xs of Const("List.list.[]",_) => xs | _ => cons $ butlast xs)
berghofe@5183
   805
  | butlast ((app as Const("List.list.op @",_) $ xs) $ ys) = app $ butlast ys
berghofe@5183
   806
  | butlast xs = Const("List.list.[]",fastype_of xs);
nipkow@5162
   807
nipkow@5162
   808
val rearr_tac =
nipkow@5162
   809
  simp_tac (HOL_basic_ss addsimps [append_assoc,append_Nil,append_Cons]);
nipkow@5162
   810
nipkow@5162
   811
fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
nipkow@5162
   812
  let
nipkow@5162
   813
    val lastl = last lhs and lastr = last rhs
nipkow@5162
   814
    fun rearr conv =
nipkow@5162
   815
      let val lhs1 = butlast lhs and rhs1 = butlast rhs
nipkow@5162
   816
          val Type(_,listT::_) = eqT
nipkow@5162
   817
          val appT = [listT,listT] ---> listT
berghofe@5183
   818
          val app = Const("List.list.op @",appT)
nipkow@5162
   819
          val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
nipkow@5162
   820
          val ct = cterm_of sg (HOLogic.mk_Trueprop(HOLogic.mk_eq(F,F2)))
nipkow@5162
   821
          val thm = prove_goalw_cterm [] ct (K [rearr_tac 1])
nipkow@5162
   822
            handle ERROR =>
nipkow@5162
   823
            error("The error(s) above occurred while trying to prove " ^
nipkow@5162
   824
                  string_of_cterm ct)
nipkow@5162
   825
      in Some((conv RS (thm RS trans)) RS eq_reflection) end
nipkow@5162
   826
nipkow@5162
   827
  in if list1 lastl andalso list1 lastr
nipkow@5162
   828
     then rearr append1_eq_conv
nipkow@5162
   829
     else
nipkow@5162
   830
     if lastl aconv lastr
nipkow@5162
   831
     then rearr append_same_eq
nipkow@5162
   832
     else None
nipkow@5162
   833
  end;
nipkow@5162
   834
in
nipkow@5162
   835
val list_eq_simproc = mk_simproc "list_eq" [list_eq_pattern] list_eq;
nipkow@5162
   836
end;
nipkow@5162
   837
nipkow@5162
   838
Addsimprocs [list_eq_simproc];