src/HOL/List.ML
author paulson
Tue Oct 13 11:08:28 1998 +0200 (1998-10-13)
changeset 5641 5266f09db46c
parent 5537 c2bd39a2c0ee
child 5644 85fd64148873
permissions -rw-r--r--
length_Suc_conv is no longer given to AddIffs
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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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Goal "(length xs = Suc n) = (? y ys. xs = y#ys & length ys = n)";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "length_Suc_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 (Force_tac 1);
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by (rtac allI 1);
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by (exhaust_tac "ys" 1);
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by (Force_tac 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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by (dtac 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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by (dtac sym 1);
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by (Asm_simp_tac 1);
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qed "append_eq_appendI";
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(***
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Simplification procedure for all list equalities.
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Currently only tries to rearranges @ to see if
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- both lists end in a singleton list,
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- or both lists end in the same list.
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***)
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local
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val list_eq_pattern =
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  read_cterm (sign_of List.thy) ("(xs::'a list) = ys",HOLogic.boolT);
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fun last (cons as Const("List.list.op #",_) $ _ $ xs) =
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      (case xs of Const("List.list.[]",_) => cons | _ => last xs)
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  | last (Const("List.op @",_) $ _ $ ys) = last ys
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  | last t = t;
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fun list1 (Const("List.list.op #",_) $ _ $ Const("List.list.[]",_)) = true
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  | list1 _ = false;
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fun butlast ((cons as Const("List.list.op #",_) $ x) $ xs) =
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      (case xs of Const("List.list.[]",_) => xs | _ => cons $ butlast xs)
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  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
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  | butlast xs = Const("List.list.[]",fastype_of xs);
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val rearr_tac =
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  simp_tac (HOL_basic_ss addsimps [append_assoc,append_Nil,append_Cons]);
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fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
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  let
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    val lastl = last lhs and lastr = last rhs
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    fun rearr conv =
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      let val lhs1 = butlast lhs and rhs1 = butlast rhs
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          val Type(_,listT::_) = eqT
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          val appT = [listT,listT] ---> listT
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          val app = Const("List.op @",appT)
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          val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
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          val ct = cterm_of sg (HOLogic.mk_Trueprop(HOLogic.mk_eq(F,F2)))
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          val thm = prove_goalw_cterm [] ct (K [rearr_tac 1])
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            handle ERROR =>
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            error("The error(s) above occurred while trying to prove " ^
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                  string_of_cterm ct)
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      in Some((conv RS (thm RS trans)) RS eq_reflection) end
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  in if list1 lastl andalso list1 lastr
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     then rearr append1_eq_conv
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     else
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     if lastl aconv lastr
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     then rearr append_same_eq
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     else None
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  end;
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in
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val list_eq_simproc = mk_simproc "list_eq" [list_eq_pattern] list_eq;
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end;
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Addsimprocs [list_eq_simproc];
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(** map **)
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section "map";
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Goal "(!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 "(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";
nipkow@3860
   348
AddIffs [map_is_Nil_conv];
nipkow@3860
   349
nipkow@4935
   350
Goal "([] = map f xs) = (xs = [])";
wenzelm@4423
   351
by (induct_tac "xs" 1);
paulson@5316
   352
by Auto_tac;
nipkow@3860
   353
qed "Nil_is_map_conv";
nipkow@3860
   354
AddIffs [Nil_is_map_conv];
nipkow@3860
   355
nipkow@3860
   356
lcp@1169
   357
(** rev **)
lcp@1169
   358
nipkow@3467
   359
section "rev";
nipkow@3467
   360
nipkow@4935
   361
Goal "rev(xs@ys) = rev(ys) @ rev(xs)";
nipkow@3040
   362
by (induct_tac "xs" 1);
paulson@5316
   363
by Auto_tac;
lcp@1169
   364
qed "rev_append";
nipkow@2512
   365
Addsimps[rev_append];
lcp@1169
   366
nipkow@4935
   367
Goal "rev(rev l) = l";
nipkow@3040
   368
by (induct_tac "l" 1);
paulson@5316
   369
by Auto_tac;
lcp@1169
   370
qed "rev_rev_ident";
nipkow@2512
   371
Addsimps[rev_rev_ident];
lcp@1169
   372
nipkow@4935
   373
Goal "(rev xs = []) = (xs = [])";
wenzelm@4423
   374
by (induct_tac "xs" 1);
paulson@5316
   375
by Auto_tac;
nipkow@3860
   376
qed "rev_is_Nil_conv";
nipkow@3860
   377
AddIffs [rev_is_Nil_conv];
nipkow@3860
   378
nipkow@4935
   379
Goal "([] = rev xs) = (xs = [])";
wenzelm@4423
   380
by (induct_tac "xs" 1);
paulson@5316
   381
by Auto_tac;
nipkow@3860
   382
qed "Nil_is_rev_conv";
nipkow@3860
   383
AddIffs [Nil_is_rev_conv];
nipkow@3860
   384
nipkow@4935
   385
val prems = Goal "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P xs";
wenzelm@5132
   386
by (stac (rev_rev_ident RS sym) 1);
nipkow@4935
   387
br(read_instantiate [("P","%xs. ?P(rev xs)")]list.induct)1;
wenzelm@5132
   388
by (ALLGOALS Simp_tac);
wenzelm@5132
   389
by (resolve_tac prems 1);
wenzelm@5132
   390
by (eresolve_tac prems 1);
nipkow@4935
   391
qed "rev_induct";
nipkow@4935
   392
nipkow@5272
   393
fun rev_induct_tac xs = res_inst_tac [("xs",xs)] rev_induct;
nipkow@5272
   394
nipkow@4935
   395
Goal  "(xs = [] --> P) -->  (!ys y. xs = ys@[y] --> P) --> P";
wenzelm@5132
   396
by (res_inst_tac [("xs","xs")] rev_induct 1);
paulson@5316
   397
by Auto_tac;
nipkow@4935
   398
bind_thm ("rev_exhaust",
nipkow@4935
   399
  impI RSN (2,allI RSN (2,allI RSN (2,impI RS (result() RS mp RS mp)))));
nipkow@4935
   400
nipkow@2608
   401
nipkow@3465
   402
(** set **)
paulson@1812
   403
nipkow@3467
   404
section "set";
nipkow@3467
   405
oheimb@5296
   406
qed_goal "finite_set" thy "finite (set xs)" 
oheimb@5296
   407
	(K [induct_tac "xs" 1, Auto_tac]);
oheimb@5296
   408
Addsimps[finite_set];
oheimb@5296
   409
AddSIs[finite_set];
oheimb@5296
   410
nipkow@4935
   411
Goal "set (xs@ys) = (set xs Un set ys)";
nipkow@3040
   412
by (induct_tac "xs" 1);
paulson@5316
   413
by Auto_tac;
paulson@3647
   414
qed "set_append";
paulson@3647
   415
Addsimps[set_append];
paulson@1812
   416
nipkow@4935
   417
Goal "set l <= set (x#l)";
paulson@5316
   418
by Auto_tac;
paulson@3647
   419
qed "set_subset_Cons";
paulson@1936
   420
nipkow@4935
   421
Goal "(set xs = {}) = (xs = [])";
paulson@3457
   422
by (induct_tac "xs" 1);
paulson@5316
   423
by Auto_tac;
paulson@3647
   424
qed "set_empty";
paulson@3647
   425
Addsimps [set_empty];
nipkow@2608
   426
nipkow@4935
   427
Goal "set(rev xs) = set(xs)";
paulson@3457
   428
by (induct_tac "xs" 1);
paulson@5316
   429
by Auto_tac;
paulson@3647
   430
qed "set_rev";
paulson@3647
   431
Addsimps [set_rev];
nipkow@2608
   432
nipkow@4935
   433
Goal "set(map f xs) = f``(set xs)";
paulson@3457
   434
by (induct_tac "xs" 1);
paulson@5316
   435
by Auto_tac;
paulson@3647
   436
qed "set_map";
paulson@3647
   437
Addsimps [set_map];
nipkow@2608
   438
oheimb@5443
   439
Goal "(x : set (filter P xs)) = (x : set xs & P x)";
nipkow@4605
   440
by (induct_tac "xs" 1);
paulson@5316
   441
by Auto_tac;
nipkow@4605
   442
qed "in_set_filter";
nipkow@4605
   443
Addsimps [in_set_filter];
nipkow@4605
   444
nipkow@5272
   445
Goal "(x : set xs) = (? ys zs. xs = ys@x#zs)";
paulson@5318
   446
by (induct_tac "xs" 1);
paulson@5318
   447
 by (Simp_tac 1);
paulson@5318
   448
by (Asm_simp_tac 1);
paulson@5318
   449
by (rtac iffI 1);
paulson@5318
   450
by (blast_tac (claset() addIs [eq_Nil_appendI,Cons_eq_appendI]) 1);
paulson@5318
   451
by (REPEAT(etac exE 1));
paulson@5318
   452
by (exhaust_tac "ys" 1);
paulson@5316
   453
by Auto_tac;
nipkow@5272
   454
qed "in_set_conv_decomp";
nipkow@5272
   455
nipkow@5272
   456
(* eliminate `lists' in favour of `set' *)
nipkow@5272
   457
nipkow@5272
   458
Goal "(xs : lists A) = (!x : set xs. x : A)";
paulson@5318
   459
by (induct_tac "xs" 1);
paulson@5316
   460
by Auto_tac;
nipkow@5272
   461
qed "in_lists_conv_set";
nipkow@5272
   462
nipkow@5272
   463
bind_thm("in_listsD",in_lists_conv_set RS iffD1);
nipkow@5272
   464
AddSDs [in_listsD];
nipkow@5272
   465
bind_thm("in_listsI",in_lists_conv_set RS iffD2);
nipkow@5272
   466
AddSIs [in_listsI];
paulson@1812
   467
oheimb@5518
   468
(** mem **)
oheimb@5518
   469
 
oheimb@5518
   470
section "mem";
oheimb@5518
   471
oheimb@5518
   472
Goal "(x mem xs) = (x: set xs)";
oheimb@5518
   473
by (induct_tac "xs" 1);
oheimb@5518
   474
by Auto_tac;
oheimb@5518
   475
qed "set_mem_eq";
oheimb@5518
   476
oheimb@5518
   477
clasohm@923
   478
(** list_all **)
clasohm@923
   479
nipkow@3467
   480
section "list_all";
nipkow@3467
   481
oheimb@5518
   482
Goal "list_all P xs = (!x:set xs. P x)";
oheimb@5518
   483
by (induct_tac "xs" 1);
oheimb@5518
   484
by Auto_tac;
oheimb@5518
   485
qed "list_all_conv";
oheimb@5518
   486
oheimb@5443
   487
Goal "list_all P (xs@ys) = (list_all P xs & list_all P ys)";
nipkow@3040
   488
by (induct_tac "xs" 1);
paulson@5316
   489
by Auto_tac;
nipkow@2512
   490
qed "list_all_append";
nipkow@2512
   491
Addsimps [list_all_append];
clasohm@923
   492
clasohm@923
   493
nipkow@2608
   494
(** filter **)
clasohm@923
   495
nipkow@3467
   496
section "filter";
nipkow@3467
   497
nipkow@4935
   498
Goal "filter P (xs@ys) = filter P xs @ filter P ys";
paulson@3457
   499
by (induct_tac "xs" 1);
paulson@5316
   500
by Auto_tac;
nipkow@2608
   501
qed "filter_append";
nipkow@2608
   502
Addsimps [filter_append];
nipkow@2608
   503
nipkow@4935
   504
Goal "filter (%x. True) xs = xs";
nipkow@4605
   505
by (induct_tac "xs" 1);
paulson@5316
   506
by Auto_tac;
nipkow@4605
   507
qed "filter_True";
nipkow@4605
   508
Addsimps [filter_True];
nipkow@4605
   509
nipkow@4935
   510
Goal "filter (%x. False) xs = []";
nipkow@4605
   511
by (induct_tac "xs" 1);
paulson@5316
   512
by Auto_tac;
nipkow@4605
   513
qed "filter_False";
nipkow@4605
   514
Addsimps [filter_False];
nipkow@4605
   515
nipkow@4935
   516
Goal "length (filter P xs) <= length xs";
paulson@3457
   517
by (induct_tac "xs" 1);
paulson@5316
   518
by Auto_tac;
nipkow@4605
   519
qed "length_filter";
oheimb@5443
   520
Addsimps[length_filter];
nipkow@2608
   521
oheimb@5443
   522
Goal "set (filter P xs) <= set xs";
oheimb@5443
   523
by Auto_tac;
oheimb@5443
   524
qed "filter_is_subset";
oheimb@5443
   525
Addsimps [filter_is_subset];
oheimb@5443
   526
nipkow@2608
   527
nipkow@3467
   528
section "concat";
nipkow@3467
   529
nipkow@4935
   530
Goal  "concat(xs@ys) = concat(xs)@concat(ys)";
nipkow@3040
   531
by (induct_tac "xs" 1);
paulson@5316
   532
by Auto_tac;
nipkow@2608
   533
qed"concat_append";
nipkow@2608
   534
Addsimps [concat_append];
nipkow@2512
   535
nipkow@4935
   536
Goal "(concat xss = []) = (!xs:set xss. xs=[])";
wenzelm@4423
   537
by (induct_tac "xss" 1);
paulson@5316
   538
by Auto_tac;
nipkow@3896
   539
qed "concat_eq_Nil_conv";
nipkow@3896
   540
AddIffs [concat_eq_Nil_conv];
nipkow@3896
   541
nipkow@4935
   542
Goal "([] = concat xss) = (!xs:set xss. xs=[])";
wenzelm@4423
   543
by (induct_tac "xss" 1);
paulson@5316
   544
by Auto_tac;
nipkow@3896
   545
qed "Nil_eq_concat_conv";
nipkow@3896
   546
AddIffs [Nil_eq_concat_conv];
nipkow@3896
   547
nipkow@4935
   548
Goal  "set(concat xs) = Union(set `` set xs)";
nipkow@3467
   549
by (induct_tac "xs" 1);
paulson@5316
   550
by Auto_tac;
paulson@3647
   551
qed"set_concat";
paulson@3647
   552
Addsimps [set_concat];
nipkow@3467
   553
nipkow@4935
   554
Goal "map f (concat xs) = concat (map (map f) xs)"; 
nipkow@3467
   555
by (induct_tac "xs" 1);
paulson@5316
   556
by Auto_tac;
nipkow@3467
   557
qed "map_concat";
nipkow@3467
   558
nipkow@4935
   559
Goal "filter p (concat xs) = concat (map (filter p) xs)"; 
nipkow@3467
   560
by (induct_tac "xs" 1);
paulson@5316
   561
by Auto_tac;
nipkow@3467
   562
qed"filter_concat"; 
nipkow@3467
   563
nipkow@4935
   564
Goal "rev(concat xs) = concat (map rev (rev xs))";
nipkow@3467
   565
by (induct_tac "xs" 1);
paulson@5316
   566
by Auto_tac;
nipkow@2608
   567
qed "rev_concat";
clasohm@923
   568
clasohm@923
   569
(** nth **)
clasohm@923
   570
nipkow@3467
   571
section "nth";
nipkow@3467
   572
paulson@5278
   573
Goal "!xs. (xs@ys)!n = (if n < length xs then xs!n else ys!(n - length xs))";
berghofe@5183
   574
by (induct_tac "n" 1);
paulson@3457
   575
 by (Asm_simp_tac 1);
paulson@3457
   576
 by (rtac allI 1);
paulson@3457
   577
 by (exhaust_tac "xs" 1);
paulson@5316
   578
  by Auto_tac;
nipkow@2608
   579
qed_spec_mp "nth_append";
nipkow@2608
   580
nipkow@4935
   581
Goal "!n. n < length xs --> (map f xs)!n = f(xs!n)";
nipkow@3040
   582
by (induct_tac "xs" 1);
nipkow@1301
   583
(* case [] *)
nipkow@1301
   584
by (Asm_full_simp_tac 1);
nipkow@1301
   585
(* case x#xl *)
nipkow@1301
   586
by (rtac allI 1);
berghofe@5183
   587
by (induct_tac "n" 1);
paulson@5316
   588
by Auto_tac;
nipkow@1485
   589
qed_spec_mp "nth_map";
nipkow@1301
   590
Addsimps [nth_map];
nipkow@1301
   591
oheimb@5518
   592
Goal "!n. n < length xs --> Ball (set xs) P --> P(xs!n)";
nipkow@3040
   593
by (induct_tac "xs" 1);
nipkow@1301
   594
(* case [] *)
nipkow@1301
   595
by (Simp_tac 1);
nipkow@1301
   596
(* case x#xl *)
nipkow@1301
   597
by (rtac allI 1);
berghofe@5183
   598
by (induct_tac "n" 1);
paulson@5316
   599
by Auto_tac;
oheimb@5518
   600
qed_spec_mp "list_ball_nth";
nipkow@1301
   601
oheimb@5518
   602
Goal "!n. n < length xs --> xs!n : set xs";
nipkow@3040
   603
by (induct_tac "xs" 1);
nipkow@1301
   604
(* case [] *)
nipkow@1301
   605
by (Simp_tac 1);
nipkow@1301
   606
(* case x#xl *)
nipkow@1301
   607
by (rtac allI 1);
berghofe@5183
   608
by (induct_tac "n" 1);
nipkow@1301
   609
(* case 0 *)
nipkow@1301
   610
by (Asm_full_simp_tac 1);
nipkow@1301
   611
(* case Suc x *)
nipkow@4686
   612
by (Asm_full_simp_tac 1);
nipkow@1485
   613
qed_spec_mp "nth_mem";
nipkow@1301
   614
Addsimps [nth_mem];
nipkow@1301
   615
oheimb@5518
   616
nipkow@5077
   617
(** list update **)
nipkow@5077
   618
nipkow@5077
   619
section "list update";
nipkow@5077
   620
nipkow@5077
   621
Goal "!i. length(xs[i:=x]) = length xs";
nipkow@5077
   622
by (induct_tac "xs" 1);
nipkow@5077
   623
by (Simp_tac 1);
berghofe@5183
   624
by (asm_full_simp_tac (simpset() addsplits [nat.split]) 1);
nipkow@5077
   625
qed_spec_mp "length_list_update";
nipkow@5077
   626
Addsimps [length_list_update];
nipkow@5077
   627
nipkow@5077
   628
nipkow@3896
   629
(** last & butlast **)
nipkow@1327
   630
nipkow@4935
   631
Goal "last(xs@[x]) = x";
wenzelm@4423
   632
by (induct_tac "xs" 1);
paulson@5316
   633
by Auto_tac;
nipkow@3896
   634
qed "last_snoc";
nipkow@3896
   635
Addsimps [last_snoc];
nipkow@3896
   636
nipkow@4935
   637
Goal "butlast(xs@[x]) = xs";
wenzelm@4423
   638
by (induct_tac "xs" 1);
paulson@5316
   639
by Auto_tac;
nipkow@3896
   640
qed "butlast_snoc";
nipkow@3896
   641
Addsimps [butlast_snoc];
nipkow@3896
   642
nipkow@4935
   643
Goal "length(butlast xs) = length xs - 1";
nipkow@4935
   644
by (res_inst_tac [("xs","xs")] rev_induct 1);
paulson@5316
   645
by Auto_tac;
nipkow@4643
   646
qed "length_butlast";
nipkow@4643
   647
Addsimps [length_butlast];
nipkow@4643
   648
paulson@5278
   649
Goal "!ys. butlast (xs@ys) = (if ys=[] then butlast xs else xs@butlast ys)";
wenzelm@4423
   650
by (induct_tac "xs" 1);
paulson@5316
   651
by Auto_tac;
nipkow@3896
   652
qed_spec_mp "butlast_append";
nipkow@3896
   653
nipkow@4935
   654
Goal "x:set(butlast xs) --> x:set xs";
wenzelm@4423
   655
by (induct_tac "xs" 1);
paulson@5316
   656
by Auto_tac;
nipkow@3896
   657
qed_spec_mp "in_set_butlastD";
nipkow@3896
   658
paulson@5448
   659
Goal "x:set(butlast xs) | x:set(butlast ys) ==> x:set(butlast(xs@ys))";
paulson@5448
   660
by (auto_tac (claset() addDs [in_set_butlastD],
paulson@5448
   661
	      simpset() addsimps [butlast_append]));
paulson@5448
   662
qed "in_set_butlast_appendI";
nipkow@3902
   663
nipkow@2608
   664
(** take  & drop **)
nipkow@2608
   665
section "take & drop";
nipkow@1327
   666
nipkow@4935
   667
Goal "take 0 xs = []";
nipkow@3040
   668
by (induct_tac "xs" 1);
paulson@5316
   669
by Auto_tac;
nipkow@1327
   670
qed "take_0";
nipkow@1327
   671
nipkow@4935
   672
Goal "drop 0 xs = xs";
nipkow@3040
   673
by (induct_tac "xs" 1);
paulson@5316
   674
by Auto_tac;
nipkow@2608
   675
qed "drop_0";
nipkow@2608
   676
nipkow@4935
   677
Goal "take (Suc n) (x#xs) = x # take n xs";
paulson@1552
   678
by (Simp_tac 1);
nipkow@1419
   679
qed "take_Suc_Cons";
nipkow@1327
   680
nipkow@4935
   681
Goal "drop (Suc n) (x#xs) = drop n xs";
nipkow@2608
   682
by (Simp_tac 1);
nipkow@2608
   683
qed "drop_Suc_Cons";
nipkow@2608
   684
nipkow@2608
   685
Delsimps [take_Cons,drop_Cons];
nipkow@2608
   686
Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons];
nipkow@2608
   687
nipkow@4935
   688
Goal "!xs. length(take n xs) = min (length xs) n";
berghofe@5183
   689
by (induct_tac "n" 1);
paulson@5316
   690
 by Auto_tac;
paulson@3457
   691
by (exhaust_tac "xs" 1);
paulson@5316
   692
 by Auto_tac;
nipkow@2608
   693
qed_spec_mp "length_take";
nipkow@2608
   694
Addsimps [length_take];
clasohm@923
   695
nipkow@4935
   696
Goal "!xs. length(drop n xs) = (length xs - n)";
berghofe@5183
   697
by (induct_tac "n" 1);
paulson@5316
   698
 by Auto_tac;
paulson@3457
   699
by (exhaust_tac "xs" 1);
paulson@5316
   700
 by Auto_tac;
nipkow@2608
   701
qed_spec_mp "length_drop";
nipkow@2608
   702
Addsimps [length_drop];
nipkow@2608
   703
nipkow@4935
   704
Goal "!xs. length xs <= n --> take n xs = xs";
berghofe@5183
   705
by (induct_tac "n" 1);
paulson@5316
   706
 by Auto_tac;
paulson@3457
   707
by (exhaust_tac "xs" 1);
paulson@5316
   708
 by Auto_tac;
nipkow@2608
   709
qed_spec_mp "take_all";
clasohm@923
   710
nipkow@4935
   711
Goal "!xs. length xs <= n --> drop n xs = []";
berghofe@5183
   712
by (induct_tac "n" 1);
paulson@5316
   713
 by Auto_tac;
paulson@3457
   714
by (exhaust_tac "xs" 1);
paulson@5316
   715
 by Auto_tac;
nipkow@2608
   716
qed_spec_mp "drop_all";
nipkow@2608
   717
paulson@5278
   718
Goal "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)";
berghofe@5183
   719
by (induct_tac "n" 1);
paulson@5316
   720
 by Auto_tac;
paulson@3457
   721
by (exhaust_tac "xs" 1);
paulson@5316
   722
 by Auto_tac;
nipkow@2608
   723
qed_spec_mp "take_append";
nipkow@2608
   724
Addsimps [take_append];
nipkow@2608
   725
nipkow@4935
   726
Goal "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; 
berghofe@5183
   727
by (induct_tac "n" 1);
paulson@5316
   728
 by Auto_tac;
paulson@3457
   729
by (exhaust_tac "xs" 1);
paulson@5316
   730
 by Auto_tac;
nipkow@2608
   731
qed_spec_mp "drop_append";
nipkow@2608
   732
Addsimps [drop_append];
nipkow@2608
   733
nipkow@4935
   734
Goal "!xs n. take n (take m xs) = take (min n m) xs"; 
berghofe@5183
   735
by (induct_tac "m" 1);
paulson@5316
   736
 by Auto_tac;
paulson@3457
   737
by (exhaust_tac "xs" 1);
paulson@5316
   738
 by Auto_tac;
berghofe@5183
   739
by (exhaust_tac "na" 1);
paulson@5316
   740
 by Auto_tac;
nipkow@2608
   741
qed_spec_mp "take_take";
nipkow@2608
   742
nipkow@4935
   743
Goal "!xs. drop n (drop m xs) = drop (n + m) xs"; 
berghofe@5183
   744
by (induct_tac "m" 1);
paulson@5316
   745
 by Auto_tac;
paulson@3457
   746
by (exhaust_tac "xs" 1);
paulson@5316
   747
 by Auto_tac;
nipkow@2608
   748
qed_spec_mp "drop_drop";
clasohm@923
   749
nipkow@4935
   750
Goal "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; 
berghofe@5183
   751
by (induct_tac "m" 1);
paulson@5316
   752
 by Auto_tac;
paulson@3457
   753
by (exhaust_tac "xs" 1);
paulson@5316
   754
 by Auto_tac;
nipkow@2608
   755
qed_spec_mp "take_drop";
nipkow@2608
   756
nipkow@4935
   757
Goal "!xs. take n (map f xs) = map f (take n xs)"; 
berghofe@5183
   758
by (induct_tac "n" 1);
paulson@5316
   759
 by Auto_tac;
paulson@3457
   760
by (exhaust_tac "xs" 1);
paulson@5316
   761
 by Auto_tac;
nipkow@2608
   762
qed_spec_mp "take_map"; 
nipkow@2608
   763
nipkow@4935
   764
Goal "!xs. drop n (map f xs) = map f (drop n xs)"; 
berghofe@5183
   765
by (induct_tac "n" 1);
paulson@5316
   766
 by Auto_tac;
paulson@3457
   767
by (exhaust_tac "xs" 1);
paulson@5316
   768
 by Auto_tac;
nipkow@2608
   769
qed_spec_mp "drop_map";
nipkow@2608
   770
nipkow@4935
   771
Goal "!n i. i < n --> (take n xs)!i = xs!i";
paulson@3457
   772
by (induct_tac "xs" 1);
paulson@5316
   773
 by Auto_tac;
paulson@3457
   774
by (exhaust_tac "n" 1);
paulson@3457
   775
 by (Blast_tac 1);
paulson@3457
   776
by (exhaust_tac "i" 1);
paulson@5316
   777
 by Auto_tac;
nipkow@2608
   778
qed_spec_mp "nth_take";
nipkow@2608
   779
Addsimps [nth_take];
clasohm@923
   780
nipkow@4935
   781
Goal  "!xs i. n + i <= length xs --> (drop n xs)!i = xs!(n+i)";
berghofe@5183
   782
by (induct_tac "n" 1);
paulson@5316
   783
 by Auto_tac;
paulson@3457
   784
by (exhaust_tac "xs" 1);
paulson@5316
   785
 by Auto_tac;
nipkow@2608
   786
qed_spec_mp "nth_drop";
nipkow@2608
   787
Addsimps [nth_drop];
nipkow@2608
   788
nipkow@2608
   789
(** takeWhile & dropWhile **)
nipkow@2608
   790
nipkow@3467
   791
section "takeWhile & dropWhile";
nipkow@3467
   792
nipkow@4935
   793
Goal "takeWhile P xs @ dropWhile P xs = xs";
nipkow@3586
   794
by (induct_tac "xs" 1);
paulson@5316
   795
by Auto_tac;
nipkow@3586
   796
qed "takeWhile_dropWhile_id";
nipkow@3586
   797
Addsimps [takeWhile_dropWhile_id];
nipkow@3586
   798
nipkow@4935
   799
Goal  "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs";
paulson@3457
   800
by (induct_tac "xs" 1);
paulson@5316
   801
by Auto_tac;
nipkow@2608
   802
bind_thm("takeWhile_append1", conjI RS (result() RS mp));
nipkow@2608
   803
Addsimps [takeWhile_append1];
clasohm@923
   804
nipkow@4935
   805
Goal "(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys";
paulson@3457
   806
by (induct_tac "xs" 1);
paulson@5316
   807
by Auto_tac;
nipkow@2608
   808
bind_thm("takeWhile_append2", ballI RS (result() RS mp));
nipkow@2608
   809
Addsimps [takeWhile_append2];
lcp@1169
   810
nipkow@4935
   811
Goal "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys";
paulson@3457
   812
by (induct_tac "xs" 1);
paulson@5316
   813
by Auto_tac;
nipkow@2608
   814
bind_thm("dropWhile_append1", conjI RS (result() RS mp));
nipkow@2608
   815
Addsimps [dropWhile_append1];
nipkow@2608
   816
nipkow@4935
   817
Goal "(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys";
paulson@3457
   818
by (induct_tac "xs" 1);
paulson@5316
   819
by Auto_tac;
nipkow@2608
   820
bind_thm("dropWhile_append2", ballI RS (result() RS mp));
nipkow@2608
   821
Addsimps [dropWhile_append2];
nipkow@2608
   822
nipkow@4935
   823
Goal "x:set(takeWhile P xs) --> x:set xs & P x";
paulson@3457
   824
by (induct_tac "xs" 1);
paulson@5316
   825
by Auto_tac;
paulson@3647
   826
qed_spec_mp"set_take_whileD";
nipkow@2608
   827
oheimb@4132
   828
qed_goal "zip_Nil_Nil"   thy "zip []     []     = []" (K [Simp_tac 1]);
oheimb@4132
   829
qed_goal "zip_Cons_Cons" thy "zip (x#xs) (y#ys) = (x,y)#zip xs ys" 
oheimb@4132
   830
						      (K [Simp_tac 1]);
nipkow@4605
   831
nipkow@5272
   832
nipkow@5272
   833
(** foldl **)
nipkow@5272
   834
section "foldl";
nipkow@5272
   835
nipkow@5272
   836
Goal "!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys";
paulson@5318
   837
by (induct_tac "xs" 1);
paulson@5316
   838
by Auto_tac;
nipkow@5272
   839
qed_spec_mp "foldl_append";
nipkow@5272
   840
Addsimps [foldl_append];
nipkow@5272
   841
nipkow@5272
   842
(* Note: `n <= foldl op+ n ns' looks simpler, but is more difficult to use
nipkow@5272
   843
   because it requires an additional transitivity step
nipkow@5272
   844
*)
nipkow@5272
   845
Goal "!n::nat. m <= n --> m <= foldl op+ n ns";
paulson@5318
   846
by (induct_tac "ns" 1);
paulson@5318
   847
 by (Simp_tac 1);
paulson@5318
   848
by (Asm_full_simp_tac 1);
paulson@5318
   849
by (blast_tac (claset() addIs [trans_le_add1]) 1);
nipkow@5272
   850
qed_spec_mp "start_le_sum";
nipkow@5272
   851
nipkow@5272
   852
Goal "n : set ns ==> n <= foldl op+ 0 ns";
paulson@5318
   853
by (auto_tac (claset() addIs [start_le_sum],
nipkow@5272
   854
             simpset() addsimps [in_set_conv_decomp]));
nipkow@5272
   855
qed "elem_le_sum";
nipkow@5272
   856
nipkow@5272
   857
Goal "!m. (foldl op+ m ns = 0) = (m=0 & (!n : set ns. n=0))";
paulson@5318
   858
by (induct_tac "ns" 1);
paulson@5316
   859
by Auto_tac;
nipkow@5272
   860
qed_spec_mp "sum_eq_0_conv";
nipkow@5272
   861
AddIffs [sum_eq_0_conv];
nipkow@5272
   862
nipkow@5425
   863
(** upto **)
nipkow@5425
   864
nipkow@5427
   865
(* Does not terminate! *)
nipkow@5427
   866
Goal "[i..j(] = (if i<j then i#[Suc i..j(] else [])";
nipkow@5427
   867
by(induct_tac "j" 1);
nipkow@5427
   868
by Auto_tac;
nipkow@5427
   869
by(REPEAT(trans_tac 1));
nipkow@5427
   870
qed "upt_rec";
nipkow@5425
   871
nipkow@5427
   872
Goal "j<=i ==> [i..j(] = []";
nipkow@5427
   873
by(stac upt_rec 1);
nipkow@5427
   874
by(asm_simp_tac (simpset() addSolver cut_trans_tac) 1);
nipkow@5427
   875
qed "upt_conv_Nil";
nipkow@5427
   876
Addsimps [upt_conv_Nil];
nipkow@5427
   877
nipkow@5427
   878
Goal "i<=j ==> [i..(Suc j)(] = [i..j(]@[j]";
nipkow@5427
   879
by (Asm_simp_tac 1);
nipkow@5427
   880
qed "upt_Suc";
nipkow@5427
   881
nipkow@5427
   882
Goal "i<j ==> [i..j(] = i#[Suc i..j(]";
nipkow@5425
   883
br trans 1;
nipkow@5427
   884
by(stac upt_rec 1);
nipkow@5427
   885
br refl 2;
nipkow@5427
   886
by (Asm_simp_tac 1);
nipkow@5427
   887
qed "upt_conv_Cons";
nipkow@5427
   888
nipkow@5427
   889
Goal "length [i..j(] = j-i";
nipkow@5427
   890
by(induct_tac "j" 1);
nipkow@5427
   891
 by (Simp_tac 1);
nipkow@5427
   892
by(asm_simp_tac (simpset() addsimps [Suc_diff_le] addSolver cut_trans_tac) 1);
nipkow@5427
   893
qed "length_upt";
nipkow@5427
   894
Addsimps [length_upt];
nipkow@5425
   895
nipkow@5427
   896
Goal "i+k < j --> [i..j(] ! k = i+k";
nipkow@5427
   897
by(induct_tac "j" 1);
nipkow@5427
   898
 by(Simp_tac 1);
paulson@5537
   899
by(asm_simp_tac (simpset() addsimps [nth_append,less_diff_conv]@add_ac
nipkow@5427
   900
                           addSolver cut_trans_tac) 1);
nipkow@5427
   901
br conjI 1;
nipkow@5427
   902
 by(Clarify_tac 1);
nipkow@5427
   903
 bd add_lessD1 1;
nipkow@5427
   904
 by(trans_tac 1);
nipkow@5427
   905
by(Clarify_tac 1);
nipkow@5427
   906
br conjI 1;
nipkow@5427
   907
 by(Clarify_tac 1);
nipkow@5427
   908
 by(subgoal_tac "n=i+k" 1);
nipkow@5427
   909
  by(Asm_full_simp_tac 1);
nipkow@5427
   910
 by(trans_tac 1);
nipkow@5427
   911
by(Clarify_tac 1);
nipkow@5427
   912
by(subgoal_tac "n=i+k" 1);
nipkow@5427
   913
 by(Asm_full_simp_tac 1);
nipkow@5427
   914
by(trans_tac 1);
nipkow@5427
   915
qed_spec_mp "nth_upt";
nipkow@5427
   916
Addsimps [nth_upt];
nipkow@5425
   917
nipkow@5272
   918
nipkow@4605
   919
(** nodups & remdups **)
nipkow@4605
   920
section "nodups & remdups";
nipkow@4605
   921
nipkow@4935
   922
Goal "set(remdups xs) = set xs";
nipkow@4605
   923
by (induct_tac "xs" 1);
nipkow@4605
   924
 by (Simp_tac 1);
nipkow@4686
   925
by (asm_full_simp_tac (simpset() addsimps [insert_absorb]) 1);
nipkow@4605
   926
qed "set_remdups";
nipkow@4605
   927
Addsimps [set_remdups];
nipkow@4605
   928
nipkow@4935
   929
Goal "nodups(remdups xs)";
nipkow@4605
   930
by (induct_tac "xs" 1);
paulson@5316
   931
by Auto_tac;
nipkow@4605
   932
qed "nodups_remdups";
nipkow@4605
   933
nipkow@4935
   934
Goal "nodups xs --> nodups (filter P xs)";
nipkow@4605
   935
by (induct_tac "xs" 1);
paulson@5316
   936
by Auto_tac;
nipkow@4605
   937
qed_spec_mp "nodups_filter";
nipkow@4605
   938
nipkow@3589
   939
(** replicate **)
nipkow@3589
   940
section "replicate";
nipkow@3589
   941
nipkow@4935
   942
Goal "set(replicate (Suc n) x) = {x}";
wenzelm@4423
   943
by (induct_tac "n" 1);
paulson@5316
   944
by Auto_tac;
nipkow@3589
   945
val lemma = result();
nipkow@3589
   946
nipkow@5043
   947
Goal "n ~= 0 ==> set(replicate n x) = {x}";
wenzelm@4423
   948
by (fast_tac (claset() addSDs [not0_implies_Suc] addSIs [lemma]) 1);
nipkow@3589
   949
qed "set_replicate";
nipkow@3589
   950
Addsimps [set_replicate];
nipkow@5162
   951
nipkow@5162
   952
nipkow@5281
   953
(*** Lexcicographic orderings on lists ***)
nipkow@5281
   954
section"Lexcicographic orderings on lists";
nipkow@5281
   955
nipkow@5281
   956
Goal "wf r ==> wf(lexn r n)";
paulson@5318
   957
by (induct_tac "n" 1);
paulson@5318
   958
by (Simp_tac 1);
paulson@5318
   959
by (Simp_tac 1);
paulson@5318
   960
by (rtac wf_subset 1);
paulson@5318
   961
by (rtac Int_lower1 2);
paulson@5318
   962
by (rtac wf_prod_fun_image 1);
paulson@5318
   963
by (rtac injI 2);
paulson@5318
   964
by (Auto_tac);
nipkow@5281
   965
qed "wf_lexn";
nipkow@5281
   966
nipkow@5281
   967
Goal "!xs ys. (xs,ys) : lexn r n --> length xs = n & length ys = n";
paulson@5318
   968
by (induct_tac "n" 1);
paulson@5318
   969
by (Auto_tac);
nipkow@5281
   970
qed_spec_mp "lexn_length";
nipkow@5281
   971
nipkow@5281
   972
Goalw [lex_def] "wf r ==> wf(lex r)";
paulson@5318
   973
by (rtac wf_UN 1);
paulson@5318
   974
by (blast_tac (claset() addIs [wf_lexn]) 1);
paulson@5318
   975
by (Clarify_tac 1);
paulson@5318
   976
by (rename_tac "m n" 1);
paulson@5318
   977
by (subgoal_tac "m ~= n" 1);
paulson@5318
   978
 by (Blast_tac 2);
paulson@5318
   979
by (blast_tac (claset() addDs [lexn_length,not_sym]) 1);
nipkow@5281
   980
qed "wf_lex";
nipkow@5281
   981
AddSIs [wf_lex];
nipkow@5281
   982
nipkow@5281
   983
Goal
nipkow@5281
   984
 "lexn r n = \
nipkow@5281
   985
\ {(xs,ys). length xs = n & length ys = n & \
nipkow@5281
   986
\           (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}";
paulson@5318
   987
by (induct_tac "n" 1);
paulson@5318
   988
 by (Simp_tac 1);
paulson@5318
   989
 by (Blast_tac 1);
paulson@5641
   990
by (asm_full_simp_tac (simpset() 
oheimb@5296
   991
				addsimps [lex_prod_def]) 1);
paulson@5641
   992
by (auto_tac (claset(), simpset()));
paulson@5318
   993
  by (Blast_tac 1);
paulson@5318
   994
 by (rename_tac "a xys x xs' y ys'" 1);
paulson@5318
   995
 by (res_inst_tac [("x","a#xys")] exI 1);
paulson@5318
   996
 by (Simp_tac 1);
paulson@5318
   997
by (exhaust_tac "xys" 1);
paulson@5641
   998
 by (ALLGOALS (asm_full_simp_tac (simpset())));
paulson@5318
   999
by (Blast_tac 1);
nipkow@5281
  1000
qed "lexn_conv";
nipkow@5281
  1001
nipkow@5281
  1002
Goalw [lex_def]
nipkow@5281
  1003
 "lex r = \
nipkow@5281
  1004
\ {(xs,ys). length xs = length ys & \
nipkow@5281
  1005
\           (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}";
paulson@5641
  1006
by (force_tac (claset(), simpset() addsimps [lexn_conv]) 1);
nipkow@5281
  1007
qed "lex_conv";
nipkow@5281
  1008
nipkow@5281
  1009
Goalw [lexico_def] "wf r ==> wf(lexico r)";
paulson@5318
  1010
by (Blast_tac 1);
nipkow@5281
  1011
qed "wf_lexico";
nipkow@5281
  1012
AddSIs [wf_lexico];
nipkow@5281
  1013
nipkow@5281
  1014
Goalw
nipkow@5281
  1015
 [lexico_def,diag_def,lex_prod_def,measure_def,inv_image_def]
nipkow@5281
  1016
"lexico r = {(xs,ys). length xs < length ys | \
nipkow@5281
  1017
\                     length xs = length ys & (xs,ys) : lex r}";
paulson@5318
  1018
by (Simp_tac 1);
nipkow@5281
  1019
qed "lexico_conv";
nipkow@5281
  1020
nipkow@5283
  1021
Goal "([],ys) ~: lex r";
paulson@5318
  1022
by (simp_tac (simpset() addsimps [lex_conv]) 1);
nipkow@5283
  1023
qed "Nil_notin_lex";
nipkow@5283
  1024
nipkow@5283
  1025
Goal "(xs,[]) ~: lex r";
paulson@5318
  1026
by (simp_tac (simpset() addsimps [lex_conv]) 1);
nipkow@5283
  1027
qed "Nil2_notin_lex";
nipkow@5283
  1028
nipkow@5283
  1029
AddIffs [Nil_notin_lex,Nil2_notin_lex];
nipkow@5283
  1030
nipkow@5283
  1031
Goal "((x#xs,y#ys) : lex r) = \
nipkow@5283
  1032
\     ((x,y) : r & length xs = length ys | x=y & (xs,ys) : lex r)";
paulson@5318
  1033
by (simp_tac (simpset() addsimps [lex_conv]) 1);
paulson@5318
  1034
by (rtac iffI 1);
paulson@5318
  1035
 by (blast_tac (claset() addIs [Cons_eq_appendI]) 2);
paulson@5318
  1036
by (REPEAT(eresolve_tac [conjE, exE] 1));
paulson@5318
  1037
by (exhaust_tac "xys" 1);
paulson@5318
  1038
by (Asm_full_simp_tac 1);
paulson@5318
  1039
by (Asm_full_simp_tac 1);
paulson@5318
  1040
by (Blast_tac 1);
nipkow@5283
  1041
qed "Cons_in_lex";
nipkow@5283
  1042
AddIffs [Cons_in_lex];