src/HOL/List.ML
author nipkow
Thu Aug 06 12:46:18 1998 +0200 (1998-08-06)
changeset 5272 95cfd872fe66
parent 5200 a23c23af335f
child 5278 a903b66822e2
permissions -rw-r--r--
New lemmas in List and Lambda in IsaMakefile
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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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(* trivial rules for solving @-equations automatically *)
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Goal "xs = ys ==> xs = [] @ ys";
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by(Asm_simp_tac 1);
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qed "eq_Nil_appendI";
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Goal "[| x#xs1 = ys; xs = xs1 @ zs |] ==> x#xs = ys@zs";
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bd sym 1;
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by(Asm_simp_tac 1);
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qed "Cons_eq_appendI";
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Goal "[| xs@xs1 = zs; ys = xs1 @ us |] ==> xs@ys = zs@us";
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bd sym 1;
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by(Asm_simp_tac 1);
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qed "append_eq_appendI";
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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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fun rev_induct_tac xs = res_inst_tac [("xs",xs)] 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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   351
qed "mem_append";
nipkow@2512
   352
Addsimps[mem_append];
clasohm@923
   353
nipkow@4935
   354
Goal "x mem [x:xs. P(x)] = (x mem xs & P(x))";
nipkow@3040
   355
by (induct_tac "xs" 1);
nipkow@5129
   356
by (Auto_tac);
clasohm@923
   357
qed "mem_filter";
nipkow@2512
   358
Addsimps[mem_filter];
clasohm@923
   359
nipkow@3465
   360
(** set **)
paulson@1812
   361
nipkow@3467
   362
section "set";
nipkow@3467
   363
nipkow@4935
   364
Goal "set (xs@ys) = (set xs Un set ys)";
nipkow@3040
   365
by (induct_tac "xs" 1);
nipkow@5129
   366
by (Auto_tac);
paulson@3647
   367
qed "set_append";
paulson@3647
   368
Addsimps[set_append];
paulson@1812
   369
nipkow@4935
   370
Goal "(x mem xs) = (x: set xs)";
nipkow@3040
   371
by (induct_tac "xs" 1);
nipkow@5129
   372
by (Auto_tac);
paulson@3647
   373
qed "set_mem_eq";
paulson@1812
   374
nipkow@4935
   375
Goal "set l <= set (x#l)";
nipkow@5129
   376
by (Auto_tac);
paulson@3647
   377
qed "set_subset_Cons";
paulson@1936
   378
nipkow@4935
   379
Goal "(set xs = {}) = (xs = [])";
paulson@3457
   380
by (induct_tac "xs" 1);
nipkow@5129
   381
by (Auto_tac);
paulson@3647
   382
qed "set_empty";
paulson@3647
   383
Addsimps [set_empty];
nipkow@2608
   384
nipkow@4935
   385
Goal "set(rev xs) = set(xs)";
paulson@3457
   386
by (induct_tac "xs" 1);
nipkow@5129
   387
by (Auto_tac);
paulson@3647
   388
qed "set_rev";
paulson@3647
   389
Addsimps [set_rev];
nipkow@2608
   390
nipkow@4935
   391
Goal "set(map f xs) = f``(set xs)";
paulson@3457
   392
by (induct_tac "xs" 1);
nipkow@5129
   393
by (Auto_tac);
paulson@3647
   394
qed "set_map";
paulson@3647
   395
Addsimps [set_map];
nipkow@2608
   396
nipkow@4935
   397
Goal "(x : set(filter P xs)) = (x : set xs & P x)";
nipkow@4605
   398
by (induct_tac "xs" 1);
nipkow@5129
   399
by (Auto_tac);
nipkow@4605
   400
qed "in_set_filter";
nipkow@4605
   401
Addsimps [in_set_filter];
nipkow@4605
   402
nipkow@5272
   403
Goal "(x : set xs) = (? ys zs. xs = ys@x#zs)";
nipkow@5272
   404
by(induct_tac "xs" 1);
nipkow@5272
   405
 by(Simp_tac 1);
nipkow@5272
   406
by(Asm_simp_tac 1);
nipkow@5272
   407
br iffI 1;
nipkow@5272
   408
by(blast_tac (claset() addIs [eq_Nil_appendI,Cons_eq_appendI]) 1);
nipkow@5272
   409
by(REPEAT(etac exE 1));
nipkow@5272
   410
by(exhaust_tac "ys" 1);
nipkow@5272
   411
by(Auto_tac);
nipkow@5272
   412
qed "in_set_conv_decomp";
nipkow@5272
   413
nipkow@5272
   414
(* eliminate `lists' in favour of `set' *)
nipkow@5272
   415
nipkow@5272
   416
Goal "(xs : lists A) = (!x : set xs. x : A)";
nipkow@5272
   417
by(induct_tac "xs" 1);
nipkow@5272
   418
by(Auto_tac);
nipkow@5272
   419
qed "in_lists_conv_set";
nipkow@5272
   420
nipkow@5272
   421
bind_thm("in_listsD",in_lists_conv_set RS iffD1);
nipkow@5272
   422
AddSDs [in_listsD];
nipkow@5272
   423
bind_thm("in_listsI",in_lists_conv_set RS iffD2);
nipkow@5272
   424
AddSIs [in_listsI];
paulson@1812
   425
clasohm@923
   426
(** list_all **)
clasohm@923
   427
nipkow@3467
   428
section "list_all";
nipkow@3467
   429
nipkow@4935
   430
Goal "list_all (%x. True) xs = True";
nipkow@3040
   431
by (induct_tac "xs" 1);
nipkow@5129
   432
by (Auto_tac);
clasohm@923
   433
qed "list_all_True";
nipkow@2512
   434
Addsimps [list_all_True];
clasohm@923
   435
nipkow@4935
   436
Goal "list_all p (xs@ys) = (list_all p xs & list_all p ys)";
nipkow@3040
   437
by (induct_tac "xs" 1);
nipkow@5129
   438
by (Auto_tac);
nipkow@2512
   439
qed "list_all_append";
nipkow@2512
   440
Addsimps [list_all_append];
clasohm@923
   441
nipkow@4935
   442
Goal "list_all P xs = (!x. x mem xs --> P(x))";
nipkow@3040
   443
by (induct_tac "xs" 1);
nipkow@5129
   444
by (Auto_tac);
clasohm@923
   445
qed "list_all_mem_conv";
clasohm@923
   446
clasohm@923
   447
nipkow@2608
   448
(** filter **)
clasohm@923
   449
nipkow@3467
   450
section "filter";
nipkow@3467
   451
nipkow@4935
   452
Goal "filter P (xs@ys) = filter P xs @ filter P ys";
paulson@3457
   453
by (induct_tac "xs" 1);
nipkow@5129
   454
by (Auto_tac);
nipkow@2608
   455
qed "filter_append";
nipkow@2608
   456
Addsimps [filter_append];
nipkow@2608
   457
nipkow@4935
   458
Goal "filter (%x. True) xs = xs";
nipkow@4605
   459
by (induct_tac "xs" 1);
nipkow@5129
   460
by (Auto_tac);
nipkow@4605
   461
qed "filter_True";
nipkow@4605
   462
Addsimps [filter_True];
nipkow@4605
   463
nipkow@4935
   464
Goal "filter (%x. False) xs = []";
nipkow@4605
   465
by (induct_tac "xs" 1);
nipkow@5129
   466
by (Auto_tac);
nipkow@4605
   467
qed "filter_False";
nipkow@4605
   468
Addsimps [filter_False];
nipkow@4605
   469
nipkow@4935
   470
Goal "length (filter P xs) <= length xs";
paulson@3457
   471
by (induct_tac "xs" 1);
nipkow@5129
   472
by (Auto_tac);
nipkow@4605
   473
qed "length_filter";
paulson@3383
   474
nipkow@2608
   475
nipkow@2608
   476
(** concat **)
nipkow@2608
   477
nipkow@3467
   478
section "concat";
nipkow@3467
   479
nipkow@4935
   480
Goal  "concat(xs@ys) = concat(xs)@concat(ys)";
nipkow@3040
   481
by (induct_tac "xs" 1);
nipkow@5129
   482
by (Auto_tac);
nipkow@2608
   483
qed"concat_append";
nipkow@2608
   484
Addsimps [concat_append];
nipkow@2512
   485
nipkow@4935
   486
Goal "(concat xss = []) = (!xs:set xss. xs=[])";
wenzelm@4423
   487
by (induct_tac "xss" 1);
nipkow@5129
   488
by (Auto_tac);
nipkow@3896
   489
qed "concat_eq_Nil_conv";
nipkow@3896
   490
AddIffs [concat_eq_Nil_conv];
nipkow@3896
   491
nipkow@4935
   492
Goal "([] = concat xss) = (!xs:set xss. xs=[])";
wenzelm@4423
   493
by (induct_tac "xss" 1);
nipkow@5129
   494
by (Auto_tac);
nipkow@3896
   495
qed "Nil_eq_concat_conv";
nipkow@3896
   496
AddIffs [Nil_eq_concat_conv];
nipkow@3896
   497
nipkow@4935
   498
Goal  "set(concat xs) = Union(set `` set xs)";
nipkow@3467
   499
by (induct_tac "xs" 1);
nipkow@5129
   500
by (Auto_tac);
paulson@3647
   501
qed"set_concat";
paulson@3647
   502
Addsimps [set_concat];
nipkow@3467
   503
nipkow@4935
   504
Goal "map f (concat xs) = concat (map (map f) xs)"; 
nipkow@3467
   505
by (induct_tac "xs" 1);
nipkow@5129
   506
by (Auto_tac);
nipkow@3467
   507
qed "map_concat";
nipkow@3467
   508
nipkow@4935
   509
Goal "filter p (concat xs) = concat (map (filter p) xs)"; 
nipkow@3467
   510
by (induct_tac "xs" 1);
nipkow@5129
   511
by (Auto_tac);
nipkow@3467
   512
qed"filter_concat"; 
nipkow@3467
   513
nipkow@4935
   514
Goal "rev(concat xs) = concat (map rev (rev xs))";
nipkow@3467
   515
by (induct_tac "xs" 1);
nipkow@5129
   516
by (Auto_tac);
nipkow@2608
   517
qed "rev_concat";
clasohm@923
   518
clasohm@923
   519
(** nth **)
clasohm@923
   520
nipkow@3467
   521
section "nth";
nipkow@3467
   522
nipkow@4935
   523
Goal
nipkow@4502
   524
  "!xs. (xs@ys)!n = (if n < length xs then xs!n else ys!(n - length xs))";
berghofe@5183
   525
by (induct_tac "n" 1);
paulson@3457
   526
 by (Asm_simp_tac 1);
paulson@3457
   527
 by (rtac allI 1);
paulson@3457
   528
 by (exhaust_tac "xs" 1);
nipkow@5129
   529
  by (Auto_tac);
nipkow@2608
   530
qed_spec_mp "nth_append";
nipkow@2608
   531
nipkow@4935
   532
Goal "!n. n < length xs --> (map f xs)!n = f(xs!n)";
nipkow@3040
   533
by (induct_tac "xs" 1);
nipkow@1301
   534
(* case [] *)
nipkow@1301
   535
by (Asm_full_simp_tac 1);
nipkow@1301
   536
(* case x#xl *)
nipkow@1301
   537
by (rtac allI 1);
berghofe@5183
   538
by (induct_tac "n" 1);
nipkow@5129
   539
by (Auto_tac);
nipkow@1485
   540
qed_spec_mp "nth_map";
nipkow@1301
   541
Addsimps [nth_map];
nipkow@1301
   542
nipkow@4935
   543
Goal "!n. n < length xs --> list_all P xs --> P(xs!n)";
nipkow@3040
   544
by (induct_tac "xs" 1);
nipkow@1301
   545
(* case [] *)
nipkow@1301
   546
by (Simp_tac 1);
nipkow@1301
   547
(* case x#xl *)
nipkow@1301
   548
by (rtac allI 1);
berghofe@5183
   549
by (induct_tac "n" 1);
nipkow@5129
   550
by (Auto_tac);
nipkow@1485
   551
qed_spec_mp "list_all_nth";
nipkow@1301
   552
nipkow@4935
   553
Goal "!n. n < length xs --> xs!n mem xs";
nipkow@3040
   554
by (induct_tac "xs" 1);
nipkow@1301
   555
(* case [] *)
nipkow@1301
   556
by (Simp_tac 1);
nipkow@1301
   557
(* case x#xl *)
nipkow@1301
   558
by (rtac allI 1);
berghofe@5183
   559
by (induct_tac "n" 1);
nipkow@1301
   560
(* case 0 *)
nipkow@1301
   561
by (Asm_full_simp_tac 1);
nipkow@1301
   562
(* case Suc x *)
nipkow@4686
   563
by (Asm_full_simp_tac 1);
nipkow@1485
   564
qed_spec_mp "nth_mem";
nipkow@1301
   565
Addsimps [nth_mem];
nipkow@1301
   566
nipkow@5077
   567
(** list update **)
nipkow@5077
   568
nipkow@5077
   569
section "list update";
nipkow@5077
   570
nipkow@5077
   571
Goal "!i. length(xs[i:=x]) = length xs";
nipkow@5077
   572
by (induct_tac "xs" 1);
nipkow@5077
   573
by (Simp_tac 1);
berghofe@5183
   574
by (asm_full_simp_tac (simpset() addsplits [nat.split]) 1);
nipkow@5077
   575
qed_spec_mp "length_list_update";
nipkow@5077
   576
Addsimps [length_list_update];
nipkow@5077
   577
nipkow@5077
   578
nipkow@3896
   579
(** last & butlast **)
nipkow@1327
   580
nipkow@4935
   581
Goal "last(xs@[x]) = x";
wenzelm@4423
   582
by (induct_tac "xs" 1);
nipkow@5129
   583
by (Auto_tac);
nipkow@3896
   584
qed "last_snoc";
nipkow@3896
   585
Addsimps [last_snoc];
nipkow@3896
   586
nipkow@4935
   587
Goal "butlast(xs@[x]) = xs";
wenzelm@4423
   588
by (induct_tac "xs" 1);
nipkow@5129
   589
by (Auto_tac);
nipkow@3896
   590
qed "butlast_snoc";
nipkow@3896
   591
Addsimps [butlast_snoc];
nipkow@3896
   592
nipkow@4935
   593
Goal "length(butlast xs) = length xs - 1";
nipkow@4935
   594
by (res_inst_tac [("xs","xs")] rev_induct 1);
nipkow@5129
   595
by (Auto_tac);
nipkow@4643
   596
qed "length_butlast";
nipkow@4643
   597
Addsimps [length_butlast];
nipkow@4643
   598
nipkow@4935
   599
Goal
nipkow@3896
   600
  "!ys. butlast (xs@ys) = (if ys=[] then butlast xs else xs@butlast ys)";
wenzelm@4423
   601
by (induct_tac "xs" 1);
nipkow@5129
   602
by (Auto_tac);
nipkow@3896
   603
qed_spec_mp "butlast_append";
nipkow@3896
   604
nipkow@4935
   605
Goal "x:set(butlast xs) --> x:set xs";
wenzelm@4423
   606
by (induct_tac "xs" 1);
nipkow@5129
   607
by (Auto_tac);
nipkow@3896
   608
qed_spec_mp "in_set_butlastD";
nipkow@3896
   609
nipkow@5043
   610
Goal "x:set(butlast xs) ==> x:set(butlast(xs@ys))";
nipkow@4686
   611
by (asm_simp_tac (simpset() addsimps [butlast_append]) 1);
wenzelm@4423
   612
by (blast_tac (claset() addDs [in_set_butlastD]) 1);
nipkow@3896
   613
qed "in_set_butlast_appendI1";
nipkow@3896
   614
nipkow@5043
   615
Goal "x:set(butlast ys) ==> x:set(butlast(xs@ys))";
nipkow@4686
   616
by (asm_simp_tac (simpset() addsimps [butlast_append]) 1);
wenzelm@4423
   617
by (Clarify_tac 1);
wenzelm@4423
   618
by (Full_simp_tac 1);
nipkow@3896
   619
qed "in_set_butlast_appendI2";
nipkow@3902
   620
nipkow@2608
   621
(** take  & drop **)
nipkow@2608
   622
section "take & drop";
nipkow@1327
   623
nipkow@4935
   624
Goal "take 0 xs = []";
nipkow@3040
   625
by (induct_tac "xs" 1);
nipkow@5129
   626
by (Auto_tac);
nipkow@1327
   627
qed "take_0";
nipkow@1327
   628
nipkow@4935
   629
Goal "drop 0 xs = xs";
nipkow@3040
   630
by (induct_tac "xs" 1);
nipkow@5129
   631
by (Auto_tac);
nipkow@2608
   632
qed "drop_0";
nipkow@2608
   633
nipkow@4935
   634
Goal "take (Suc n) (x#xs) = x # take n xs";
paulson@1552
   635
by (Simp_tac 1);
nipkow@1419
   636
qed "take_Suc_Cons";
nipkow@1327
   637
nipkow@4935
   638
Goal "drop (Suc n) (x#xs) = drop n xs";
nipkow@2608
   639
by (Simp_tac 1);
nipkow@2608
   640
qed "drop_Suc_Cons";
nipkow@2608
   641
nipkow@2608
   642
Delsimps [take_Cons,drop_Cons];
nipkow@2608
   643
Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons];
nipkow@2608
   644
nipkow@4935
   645
Goal "!xs. length(take n xs) = min (length xs) n";
berghofe@5183
   646
by (induct_tac "n" 1);
nipkow@5129
   647
 by (Auto_tac);
paulson@3457
   648
by (exhaust_tac "xs" 1);
nipkow@5129
   649
 by (Auto_tac);
nipkow@2608
   650
qed_spec_mp "length_take";
nipkow@2608
   651
Addsimps [length_take];
clasohm@923
   652
nipkow@4935
   653
Goal "!xs. length(drop n xs) = (length xs - n)";
berghofe@5183
   654
by (induct_tac "n" 1);
nipkow@5129
   655
 by (Auto_tac);
paulson@3457
   656
by (exhaust_tac "xs" 1);
nipkow@5129
   657
 by (Auto_tac);
nipkow@2608
   658
qed_spec_mp "length_drop";
nipkow@2608
   659
Addsimps [length_drop];
nipkow@2608
   660
nipkow@4935
   661
Goal "!xs. length xs <= n --> take n xs = xs";
berghofe@5183
   662
by (induct_tac "n" 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 "take_all";
clasohm@923
   667
nipkow@4935
   668
Goal "!xs. length xs <= n --> drop n xs = []";
berghofe@5183
   669
by (induct_tac "n" 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 "drop_all";
nipkow@2608
   674
nipkow@4935
   675
Goal 
nipkow@2608
   676
  "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)";
berghofe@5183
   677
by (induct_tac "n" 1);
nipkow@5129
   678
 by (Auto_tac);
paulson@3457
   679
by (exhaust_tac "xs" 1);
nipkow@5129
   680
 by (Auto_tac);
nipkow@2608
   681
qed_spec_mp "take_append";
nipkow@2608
   682
Addsimps [take_append];
nipkow@2608
   683
nipkow@4935
   684
Goal "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; 
berghofe@5183
   685
by (induct_tac "n" 1);
nipkow@5129
   686
 by (Auto_tac);
paulson@3457
   687
by (exhaust_tac "xs" 1);
nipkow@5129
   688
 by (Auto_tac);
nipkow@2608
   689
qed_spec_mp "drop_append";
nipkow@2608
   690
Addsimps [drop_append];
nipkow@2608
   691
nipkow@4935
   692
Goal "!xs n. take n (take m xs) = take (min n m) xs"; 
berghofe@5183
   693
by (induct_tac "m" 1);
nipkow@5129
   694
 by (Auto_tac);
paulson@3457
   695
by (exhaust_tac "xs" 1);
nipkow@5129
   696
 by (Auto_tac);
berghofe@5183
   697
by (exhaust_tac "na" 1);
nipkow@5129
   698
 by (Auto_tac);
nipkow@2608
   699
qed_spec_mp "take_take";
nipkow@2608
   700
nipkow@4935
   701
Goal "!xs. drop n (drop m xs) = drop (n + m) xs"; 
berghofe@5183
   702
by (induct_tac "m" 1);
nipkow@5129
   703
 by (Auto_tac);
paulson@3457
   704
by (exhaust_tac "xs" 1);
nipkow@5129
   705
 by (Auto_tac);
nipkow@2608
   706
qed_spec_mp "drop_drop";
clasohm@923
   707
nipkow@4935
   708
Goal "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; 
berghofe@5183
   709
by (induct_tac "m" 1);
nipkow@5129
   710
 by (Auto_tac);
paulson@3457
   711
by (exhaust_tac "xs" 1);
nipkow@5129
   712
 by (Auto_tac);
nipkow@2608
   713
qed_spec_mp "take_drop";
nipkow@2608
   714
nipkow@4935
   715
Goal "!xs. take n (map f xs) = map f (take n xs)"; 
berghofe@5183
   716
by (induct_tac "n" 1);
nipkow@5129
   717
 by (Auto_tac);
paulson@3457
   718
by (exhaust_tac "xs" 1);
nipkow@5129
   719
 by (Auto_tac);
nipkow@2608
   720
qed_spec_mp "take_map"; 
nipkow@2608
   721
nipkow@4935
   722
Goal "!xs. drop n (map f xs) = map f (drop n xs)"; 
berghofe@5183
   723
by (induct_tac "n" 1);
nipkow@5129
   724
 by (Auto_tac);
paulson@3457
   725
by (exhaust_tac "xs" 1);
nipkow@5129
   726
 by (Auto_tac);
nipkow@2608
   727
qed_spec_mp "drop_map";
nipkow@2608
   728
nipkow@4935
   729
Goal "!n i. i < n --> (take n xs)!i = xs!i";
paulson@3457
   730
by (induct_tac "xs" 1);
nipkow@5129
   731
 by (Auto_tac);
paulson@3457
   732
by (exhaust_tac "n" 1);
paulson@3457
   733
 by (Blast_tac 1);
paulson@3457
   734
by (exhaust_tac "i" 1);
nipkow@5129
   735
 by (Auto_tac);
nipkow@2608
   736
qed_spec_mp "nth_take";
nipkow@2608
   737
Addsimps [nth_take];
clasohm@923
   738
nipkow@4935
   739
Goal  "!xs i. n + i <= length xs --> (drop n xs)!i = xs!(n+i)";
berghofe@5183
   740
by (induct_tac "n" 1);
nipkow@5129
   741
 by (Auto_tac);
paulson@3457
   742
by (exhaust_tac "xs" 1);
nipkow@5129
   743
 by (Auto_tac);
nipkow@2608
   744
qed_spec_mp "nth_drop";
nipkow@2608
   745
Addsimps [nth_drop];
nipkow@2608
   746
nipkow@2608
   747
(** takeWhile & dropWhile **)
nipkow@2608
   748
nipkow@3467
   749
section "takeWhile & dropWhile";
nipkow@3467
   750
nipkow@4935
   751
Goal "takeWhile P xs @ dropWhile P xs = xs";
nipkow@3586
   752
by (induct_tac "xs" 1);
nipkow@5129
   753
by (Auto_tac);
nipkow@3586
   754
qed "takeWhile_dropWhile_id";
nipkow@3586
   755
Addsimps [takeWhile_dropWhile_id];
nipkow@3586
   756
nipkow@4935
   757
Goal  "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs";
paulson@3457
   758
by (induct_tac "xs" 1);
nipkow@5129
   759
by (Auto_tac);
nipkow@2608
   760
bind_thm("takeWhile_append1", conjI RS (result() RS mp));
nipkow@2608
   761
Addsimps [takeWhile_append1];
clasohm@923
   762
nipkow@4935
   763
Goal "(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys";
paulson@3457
   764
by (induct_tac "xs" 1);
nipkow@5129
   765
by (Auto_tac);
nipkow@2608
   766
bind_thm("takeWhile_append2", ballI RS (result() RS mp));
nipkow@2608
   767
Addsimps [takeWhile_append2];
lcp@1169
   768
nipkow@4935
   769
Goal "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys";
paulson@3457
   770
by (induct_tac "xs" 1);
nipkow@5129
   771
by (Auto_tac);
nipkow@2608
   772
bind_thm("dropWhile_append1", conjI RS (result() RS mp));
nipkow@2608
   773
Addsimps [dropWhile_append1];
nipkow@2608
   774
nipkow@4935
   775
Goal "(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys";
paulson@3457
   776
by (induct_tac "xs" 1);
nipkow@5129
   777
by (Auto_tac);
nipkow@2608
   778
bind_thm("dropWhile_append2", ballI RS (result() RS mp));
nipkow@2608
   779
Addsimps [dropWhile_append2];
nipkow@2608
   780
nipkow@4935
   781
Goal "x:set(takeWhile P xs) --> x:set xs & P x";
paulson@3457
   782
by (induct_tac "xs" 1);
nipkow@5129
   783
by (Auto_tac);
paulson@3647
   784
qed_spec_mp"set_take_whileD";
nipkow@2608
   785
oheimb@4132
   786
qed_goal "zip_Nil_Nil"   thy "zip []     []     = []" (K [Simp_tac 1]);
oheimb@4132
   787
qed_goal "zip_Cons_Cons" thy "zip (x#xs) (y#ys) = (x,y)#zip xs ys" 
oheimb@4132
   788
						      (K [Simp_tac 1]);
nipkow@4605
   789
nipkow@5272
   790
nipkow@5272
   791
(** foldl **)
nipkow@5272
   792
section "foldl";
nipkow@5272
   793
nipkow@5272
   794
Goal "!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys";
nipkow@5272
   795
by(induct_tac "xs" 1);
nipkow@5272
   796
by(Auto_tac);
nipkow@5272
   797
qed_spec_mp "foldl_append";
nipkow@5272
   798
Addsimps [foldl_append];
nipkow@5272
   799
nipkow@5272
   800
(* Note: `n <= foldl op+ n ns' looks simpler, but is more difficult to use
nipkow@5272
   801
   because it requires an additional transitivity step
nipkow@5272
   802
*)
nipkow@5272
   803
Goal "!n::nat. m <= n --> m <= foldl op+ n ns";
nipkow@5272
   804
by(induct_tac "ns" 1);
nipkow@5272
   805
 by(Simp_tac 1);
nipkow@5272
   806
by(Asm_full_simp_tac 1);
nipkow@5272
   807
by(blast_tac (claset() addIs [trans_le_add1]) 1);
nipkow@5272
   808
qed_spec_mp "start_le_sum";
nipkow@5272
   809
nipkow@5272
   810
Goal "n : set ns ==> n <= foldl op+ 0 ns";
nipkow@5272
   811
by(auto_tac (claset() addIs [start_le_sum],
nipkow@5272
   812
             simpset() addsimps [in_set_conv_decomp]));
nipkow@5272
   813
qed "elem_le_sum";
nipkow@5272
   814
nipkow@5272
   815
Goal "!m. (foldl op+ m ns = 0) = (m=0 & (!n : set ns. n=0))";
nipkow@5272
   816
by(induct_tac "ns" 1);
nipkow@5272
   817
by(Auto_tac);
nipkow@5272
   818
qed_spec_mp "sum_eq_0_conv";
nipkow@5272
   819
AddIffs [sum_eq_0_conv];
nipkow@5272
   820
nipkow@5272
   821
nipkow@4605
   822
(** nodups & remdups **)
nipkow@4605
   823
section "nodups & remdups";
nipkow@4605
   824
nipkow@4935
   825
Goal "set(remdups xs) = set xs";
nipkow@4605
   826
by (induct_tac "xs" 1);
nipkow@4605
   827
 by (Simp_tac 1);
nipkow@4686
   828
by (asm_full_simp_tac (simpset() addsimps [insert_absorb]) 1);
nipkow@4605
   829
qed "set_remdups";
nipkow@4605
   830
Addsimps [set_remdups];
nipkow@4605
   831
nipkow@4935
   832
Goal "nodups(remdups xs)";
nipkow@4605
   833
by (induct_tac "xs" 1);
nipkow@5129
   834
by (Auto_tac);
nipkow@4605
   835
qed "nodups_remdups";
nipkow@4605
   836
nipkow@4935
   837
Goal "nodups xs --> nodups (filter P xs)";
nipkow@4605
   838
by (induct_tac "xs" 1);
nipkow@5129
   839
by (Auto_tac);
nipkow@4605
   840
qed_spec_mp "nodups_filter";
nipkow@4605
   841
nipkow@3589
   842
(** replicate **)
nipkow@3589
   843
section "replicate";
nipkow@3589
   844
nipkow@4935
   845
Goal "set(replicate (Suc n) x) = {x}";
wenzelm@4423
   846
by (induct_tac "n" 1);
nipkow@5129
   847
by (Auto_tac);
nipkow@3589
   848
val lemma = result();
nipkow@3589
   849
nipkow@5043
   850
Goal "n ~= 0 ==> set(replicate n x) = {x}";
wenzelm@4423
   851
by (fast_tac (claset() addSDs [not0_implies_Suc] addSIs [lemma]) 1);
nipkow@3589
   852
qed "set_replicate";
nipkow@3589
   853
Addsimps [set_replicate];
nipkow@5162
   854
nipkow@5162
   855
nipkow@5162
   856
(***
nipkow@5162
   857
Simplification procedure for all list equalities.
nipkow@5162
   858
Currently only tries to rearranges @ to see if
nipkow@5162
   859
- both lists end in a singleton list,
nipkow@5162
   860
- or both lists end in the same list.
nipkow@5162
   861
***)
nipkow@5162
   862
local
nipkow@5162
   863
nipkow@5162
   864
val list_eq_pattern =
nipkow@5162
   865
  read_cterm (sign_of List.thy) ("(xs::'a list) = ys",HOLogic.boolT);
nipkow@5162
   866
berghofe@5183
   867
fun last (cons as Const("List.list.op #",_) $ _ $ xs) =
berghofe@5183
   868
      (case xs of Const("List.list.[]",_) => cons | _ => last xs)
nipkow@5200
   869
  | last (Const("List.op @",_) $ _ $ ys) = last ys
nipkow@5162
   870
  | last t = t;
nipkow@5162
   871
berghofe@5183
   872
fun list1 (Const("List.list.op #",_) $ _ $ Const("List.list.[]",_)) = true
nipkow@5162
   873
  | list1 _ = false;
nipkow@5162
   874
berghofe@5183
   875
fun butlast ((cons as Const("List.list.op #",_) $ x) $ xs) =
berghofe@5183
   876
      (case xs of Const("List.list.[]",_) => xs | _ => cons $ butlast xs)
nipkow@5200
   877
  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
berghofe@5183
   878
  | butlast xs = Const("List.list.[]",fastype_of xs);
nipkow@5162
   879
nipkow@5162
   880
val rearr_tac =
nipkow@5162
   881
  simp_tac (HOL_basic_ss addsimps [append_assoc,append_Nil,append_Cons]);
nipkow@5162
   882
nipkow@5162
   883
fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
nipkow@5162
   884
  let
nipkow@5162
   885
    val lastl = last lhs and lastr = last rhs
nipkow@5162
   886
    fun rearr conv =
nipkow@5162
   887
      let val lhs1 = butlast lhs and rhs1 = butlast rhs
nipkow@5162
   888
          val Type(_,listT::_) = eqT
nipkow@5162
   889
          val appT = [listT,listT] ---> listT
nipkow@5200
   890
          val app = Const("List.op @",appT)
nipkow@5162
   891
          val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
nipkow@5162
   892
          val ct = cterm_of sg (HOLogic.mk_Trueprop(HOLogic.mk_eq(F,F2)))
nipkow@5162
   893
          val thm = prove_goalw_cterm [] ct (K [rearr_tac 1])
nipkow@5162
   894
            handle ERROR =>
nipkow@5162
   895
            error("The error(s) above occurred while trying to prove " ^
nipkow@5162
   896
                  string_of_cterm ct)
nipkow@5162
   897
      in Some((conv RS (thm RS trans)) RS eq_reflection) end
nipkow@5162
   898
nipkow@5162
   899
  in if list1 lastl andalso list1 lastr
nipkow@5162
   900
     then rearr append1_eq_conv
nipkow@5162
   901
     else
nipkow@5162
   902
     if lastl aconv lastr
nipkow@5162
   903
     then rearr append_same_eq
nipkow@5162
   904
     else None
nipkow@5162
   905
  end;
nipkow@5162
   906
in
nipkow@5162
   907
val list_eq_simproc = mk_simproc "list_eq" [list_eq_pattern] list_eq;
nipkow@5162
   908
end;
nipkow@5162
   909
nipkow@5162
   910
Addsimprocs [list_eq_simproc];