src/HOL/List.ML
author paulson
Tue Apr 20 14:32:48 1999 +0200 (1999-04-20)
changeset 6451 bc943acc5fda
parent 6433 228237ec56e5
child 6794 ac367328b875
permissions -rw-r--r--
tidied
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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 "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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  Thm.read_cterm (Theory.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 (hyp_subst_tac 1);
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by (induct_tac "ys" 1);
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by Auto_tac;
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bind_thm("map_cong", impI RSN (2,allI RSN (2, result() 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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   348
Goal "([] = map f xs) = (xs = [])";
wenzelm@4423
   349
by (induct_tac "xs" 1);
paulson@5316
   350
by Auto_tac;
nipkow@3860
   351
qed "Nil_is_map_conv";
nipkow@3860
   352
AddIffs [Nil_is_map_conv];
nipkow@3860
   353
nipkow@3860
   354
lcp@1169
   355
(** rev **)
lcp@1169
   356
nipkow@3467
   357
section "rev";
nipkow@3467
   358
nipkow@4935
   359
Goal "rev(xs@ys) = rev(ys) @ rev(xs)";
nipkow@3040
   360
by (induct_tac "xs" 1);
paulson@5316
   361
by Auto_tac;
lcp@1169
   362
qed "rev_append";
nipkow@2512
   363
Addsimps[rev_append];
lcp@1169
   364
nipkow@4935
   365
Goal "rev(rev l) = l";
nipkow@3040
   366
by (induct_tac "l" 1);
paulson@5316
   367
by Auto_tac;
lcp@1169
   368
qed "rev_rev_ident";
nipkow@2512
   369
Addsimps[rev_rev_ident];
lcp@1169
   370
nipkow@4935
   371
Goal "(rev xs = []) = (xs = [])";
wenzelm@4423
   372
by (induct_tac "xs" 1);
paulson@5316
   373
by Auto_tac;
nipkow@3860
   374
qed "rev_is_Nil_conv";
nipkow@3860
   375
AddIffs [rev_is_Nil_conv];
nipkow@3860
   376
nipkow@4935
   377
Goal "([] = rev xs) = (xs = [])";
wenzelm@4423
   378
by (induct_tac "xs" 1);
paulson@5316
   379
by Auto_tac;
nipkow@3860
   380
qed "Nil_is_rev_conv";
nipkow@3860
   381
AddIffs [Nil_is_rev_conv];
nipkow@3860
   382
nipkow@4935
   383
val prems = Goal "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P xs";
wenzelm@5132
   384
by (stac (rev_rev_ident RS sym) 1);
paulson@6162
   385
by (res_inst_tac [("list", "rev xs")] list.induct 1);
wenzelm@5132
   386
by (ALLGOALS Simp_tac);
wenzelm@5132
   387
by (resolve_tac prems 1);
wenzelm@5132
   388
by (eresolve_tac prems 1);
nipkow@4935
   389
qed "rev_induct";
nipkow@4935
   390
nipkow@5272
   391
fun rev_induct_tac xs = res_inst_tac [("xs",xs)] rev_induct;
nipkow@5272
   392
nipkow@4935
   393
Goal  "(xs = [] --> P) -->  (!ys y. xs = ys@[y] --> P) --> P";
wenzelm@5132
   394
by (res_inst_tac [("xs","xs")] rev_induct 1);
paulson@5316
   395
by Auto_tac;
nipkow@4935
   396
bind_thm ("rev_exhaust",
nipkow@4935
   397
  impI RSN (2,allI RSN (2,allI RSN (2,impI RS (result() RS mp RS mp)))));
nipkow@4935
   398
nipkow@2608
   399
nipkow@3465
   400
(** set **)
paulson@1812
   401
nipkow@3467
   402
section "set";
nipkow@3467
   403
oheimb@5296
   404
qed_goal "finite_set" thy "finite (set xs)" 
oheimb@5296
   405
	(K [induct_tac "xs" 1, Auto_tac]);
oheimb@5296
   406
Addsimps[finite_set];
oheimb@5296
   407
AddSIs[finite_set];
oheimb@5296
   408
nipkow@4935
   409
Goal "set (xs@ys) = (set xs Un set ys)";
nipkow@3040
   410
by (induct_tac "xs" 1);
paulson@5316
   411
by Auto_tac;
paulson@3647
   412
qed "set_append";
paulson@3647
   413
Addsimps[set_append];
paulson@1812
   414
nipkow@4935
   415
Goal "set l <= set (x#l)";
paulson@5316
   416
by Auto_tac;
paulson@3647
   417
qed "set_subset_Cons";
paulson@1936
   418
nipkow@4935
   419
Goal "(set xs = {}) = (xs = [])";
paulson@3457
   420
by (induct_tac "xs" 1);
paulson@5316
   421
by Auto_tac;
paulson@3647
   422
qed "set_empty";
paulson@3647
   423
Addsimps [set_empty];
nipkow@2608
   424
nipkow@4935
   425
Goal "set(rev xs) = set(xs)";
paulson@3457
   426
by (induct_tac "xs" 1);
paulson@5316
   427
by Auto_tac;
paulson@3647
   428
qed "set_rev";
paulson@3647
   429
Addsimps [set_rev];
nipkow@2608
   430
nipkow@4935
   431
Goal "set(map f xs) = f``(set xs)";
paulson@3457
   432
by (induct_tac "xs" 1);
paulson@5316
   433
by Auto_tac;
paulson@3647
   434
qed "set_map";
paulson@3647
   435
Addsimps [set_map];
nipkow@2608
   436
nipkow@6433
   437
Goal "set(filter P xs) = {x. x : set xs & P x}";
nipkow@6433
   438
by(induct_tac "xs" 1);
nipkow@6433
   439
by(Auto_tac);
nipkow@6433
   440
qed "set_filter";
nipkow@6433
   441
Addsimps [set_filter];
nipkow@6433
   442
(*
oheimb@5443
   443
Goal "(x : set (filter P xs)) = (x : set xs & P x)";
nipkow@4605
   444
by (induct_tac "xs" 1);
paulson@5316
   445
by Auto_tac;
nipkow@4605
   446
qed "in_set_filter";
nipkow@4605
   447
Addsimps [in_set_filter];
nipkow@6433
   448
*)
nipkow@6433
   449
Goal "set[i..j(] = {k. i <= k & k < j}";
nipkow@6433
   450
by(induct_tac "j" 1);
nipkow@6433
   451
by(Auto_tac);
nipkow@6433
   452
by(arith_tac 1);
nipkow@6433
   453
qed "set_upt";
nipkow@6433
   454
Addsimps [set_upt];
nipkow@6433
   455
nipkow@6433
   456
Goal "!i < size xs. set(xs[i := x]) <= insert x (set xs)";
nipkow@6433
   457
by(induct_tac "xs" 1);
nipkow@6433
   458
 by(Simp_tac 1);
nipkow@6433
   459
by(asm_simp_tac (simpset() addsplits [nat.split]) 1);
nipkow@6433
   460
by(Blast_tac 1);
nipkow@6433
   461
qed_spec_mp "set_list_update_subset";
nipkow@4605
   462
nipkow@5272
   463
Goal "(x : set xs) = (? ys zs. xs = ys@x#zs)";
paulson@5318
   464
by (induct_tac "xs" 1);
paulson@5318
   465
 by (Simp_tac 1);
paulson@5318
   466
by (Asm_simp_tac 1);
paulson@5318
   467
by (rtac iffI 1);
paulson@5318
   468
by (blast_tac (claset() addIs [eq_Nil_appendI,Cons_eq_appendI]) 1);
paulson@5318
   469
by (REPEAT(etac exE 1));
paulson@5318
   470
by (exhaust_tac "ys" 1);
paulson@5316
   471
by Auto_tac;
nipkow@5272
   472
qed "in_set_conv_decomp";
nipkow@5272
   473
nipkow@5272
   474
(* eliminate `lists' in favour of `set' *)
nipkow@5272
   475
nipkow@5272
   476
Goal "(xs : lists A) = (!x : set xs. x : A)";
paulson@5318
   477
by (induct_tac "xs" 1);
paulson@5316
   478
by Auto_tac;
nipkow@5272
   479
qed "in_lists_conv_set";
nipkow@5272
   480
nipkow@5272
   481
bind_thm("in_listsD",in_lists_conv_set RS iffD1);
nipkow@5272
   482
AddSDs [in_listsD];
nipkow@5272
   483
bind_thm("in_listsI",in_lists_conv_set RS iffD2);
nipkow@5272
   484
AddSIs [in_listsI];
paulson@1812
   485
oheimb@5518
   486
(** mem **)
oheimb@5518
   487
 
oheimb@5518
   488
section "mem";
oheimb@5518
   489
oheimb@5518
   490
Goal "(x mem xs) = (x: set xs)";
oheimb@5518
   491
by (induct_tac "xs" 1);
oheimb@5518
   492
by Auto_tac;
oheimb@5518
   493
qed "set_mem_eq";
oheimb@5518
   494
oheimb@5518
   495
clasohm@923
   496
(** list_all **)
clasohm@923
   497
nipkow@3467
   498
section "list_all";
nipkow@3467
   499
oheimb@5518
   500
Goal "list_all P xs = (!x:set xs. P x)";
oheimb@5518
   501
by (induct_tac "xs" 1);
oheimb@5518
   502
by Auto_tac;
oheimb@5518
   503
qed "list_all_conv";
oheimb@5518
   504
oheimb@5443
   505
Goal "list_all P (xs@ys) = (list_all P xs & list_all P ys)";
nipkow@3040
   506
by (induct_tac "xs" 1);
paulson@5316
   507
by Auto_tac;
nipkow@2512
   508
qed "list_all_append";
nipkow@2512
   509
Addsimps [list_all_append];
clasohm@923
   510
clasohm@923
   511
nipkow@2608
   512
(** filter **)
clasohm@923
   513
nipkow@3467
   514
section "filter";
nipkow@3467
   515
nipkow@4935
   516
Goal "filter P (xs@ys) = filter P xs @ filter P ys";
paulson@3457
   517
by (induct_tac "xs" 1);
paulson@5316
   518
by Auto_tac;
nipkow@2608
   519
qed "filter_append";
nipkow@2608
   520
Addsimps [filter_append];
nipkow@2608
   521
nipkow@4935
   522
Goal "filter (%x. True) xs = xs";
nipkow@4605
   523
by (induct_tac "xs" 1);
paulson@5316
   524
by Auto_tac;
nipkow@4605
   525
qed "filter_True";
nipkow@4605
   526
Addsimps [filter_True];
nipkow@4605
   527
nipkow@4935
   528
Goal "filter (%x. False) xs = []";
nipkow@4605
   529
by (induct_tac "xs" 1);
paulson@5316
   530
by Auto_tac;
nipkow@4605
   531
qed "filter_False";
nipkow@4605
   532
Addsimps [filter_False];
nipkow@4605
   533
nipkow@4935
   534
Goal "length (filter P xs) <= length xs";
paulson@3457
   535
by (induct_tac "xs" 1);
paulson@5316
   536
by Auto_tac;
nipkow@4605
   537
qed "length_filter";
oheimb@5443
   538
Addsimps[length_filter];
nipkow@2608
   539
oheimb@5443
   540
Goal "set (filter P xs) <= set xs";
oheimb@5443
   541
by Auto_tac;
oheimb@5443
   542
qed "filter_is_subset";
oheimb@5443
   543
Addsimps [filter_is_subset];
oheimb@5443
   544
nipkow@2608
   545
nipkow@3467
   546
section "concat";
nipkow@3467
   547
nipkow@4935
   548
Goal  "concat(xs@ys) = concat(xs)@concat(ys)";
nipkow@3040
   549
by (induct_tac "xs" 1);
paulson@5316
   550
by Auto_tac;
nipkow@2608
   551
qed"concat_append";
nipkow@2608
   552
Addsimps [concat_append];
nipkow@2512
   553
nipkow@4935
   554
Goal "(concat xss = []) = (!xs:set xss. xs=[])";
wenzelm@4423
   555
by (induct_tac "xss" 1);
paulson@5316
   556
by Auto_tac;
nipkow@3896
   557
qed "concat_eq_Nil_conv";
nipkow@3896
   558
AddIffs [concat_eq_Nil_conv];
nipkow@3896
   559
nipkow@4935
   560
Goal "([] = concat xss) = (!xs:set xss. xs=[])";
wenzelm@4423
   561
by (induct_tac "xss" 1);
paulson@5316
   562
by Auto_tac;
nipkow@3896
   563
qed "Nil_eq_concat_conv";
nipkow@3896
   564
AddIffs [Nil_eq_concat_conv];
nipkow@3896
   565
nipkow@4935
   566
Goal  "set(concat xs) = Union(set `` set xs)";
nipkow@3467
   567
by (induct_tac "xs" 1);
paulson@5316
   568
by Auto_tac;
paulson@3647
   569
qed"set_concat";
paulson@3647
   570
Addsimps [set_concat];
nipkow@3467
   571
nipkow@4935
   572
Goal "map f (concat xs) = concat (map (map f) xs)"; 
nipkow@3467
   573
by (induct_tac "xs" 1);
paulson@5316
   574
by Auto_tac;
nipkow@3467
   575
qed "map_concat";
nipkow@3467
   576
nipkow@4935
   577
Goal "filter p (concat xs) = concat (map (filter p) xs)"; 
nipkow@3467
   578
by (induct_tac "xs" 1);
paulson@5316
   579
by Auto_tac;
nipkow@3467
   580
qed"filter_concat"; 
nipkow@3467
   581
nipkow@4935
   582
Goal "rev(concat xs) = concat (map rev (rev xs))";
nipkow@3467
   583
by (induct_tac "xs" 1);
paulson@5316
   584
by Auto_tac;
nipkow@2608
   585
qed "rev_concat";
clasohm@923
   586
clasohm@923
   587
(** nth **)
clasohm@923
   588
nipkow@3467
   589
section "nth";
nipkow@3467
   590
pusch@6408
   591
Goal "(x#xs)!0 = x";
pusch@6408
   592
by Auto_tac;
pusch@6408
   593
qed "nth_Cons_0";
pusch@6408
   594
Addsimps [nth_Cons_0];
nipkow@5644
   595
pusch@6408
   596
Goal "(x#xs)!(Suc n) = xs!n";
pusch@6408
   597
by Auto_tac;
pusch@6408
   598
qed "nth_Cons_Suc";
pusch@6408
   599
Addsimps [nth_Cons_Suc];
pusch@6408
   600
pusch@6408
   601
Delsimps (thms "nth.simps");
pusch@6408
   602
pusch@6408
   603
Goal "!n. (xs@ys)!n = (if n < length xs then xs!n else ys!(n - length xs))";
pusch@6408
   604
by (induct_tac "xs" 1);
paulson@3457
   605
 by (Asm_simp_tac 1);
paulson@3457
   606
 by (rtac allI 1);
pusch@6408
   607
 by (exhaust_tac "n" 1);
paulson@5316
   608
  by Auto_tac;
nipkow@2608
   609
qed_spec_mp "nth_append";
nipkow@2608
   610
nipkow@4935
   611
Goal "!n. n < length xs --> (map f xs)!n = f(xs!n)";
nipkow@3040
   612
by (induct_tac "xs" 1);
nipkow@1301
   613
(* case [] *)
nipkow@1301
   614
by (Asm_full_simp_tac 1);
nipkow@1301
   615
(* case x#xl *)
nipkow@1301
   616
by (rtac allI 1);
berghofe@5183
   617
by (induct_tac "n" 1);
paulson@5316
   618
by Auto_tac;
nipkow@1485
   619
qed_spec_mp "nth_map";
nipkow@1301
   620
Addsimps [nth_map];
nipkow@1301
   621
oheimb@5518
   622
Goal "!n. n < length xs --> Ball (set xs) P --> P(xs!n)";
nipkow@3040
   623
by (induct_tac "xs" 1);
nipkow@1301
   624
(* case [] *)
nipkow@1301
   625
by (Simp_tac 1);
nipkow@1301
   626
(* case x#xl *)
nipkow@1301
   627
by (rtac allI 1);
berghofe@5183
   628
by (induct_tac "n" 1);
paulson@5316
   629
by Auto_tac;
oheimb@5518
   630
qed_spec_mp "list_ball_nth";
nipkow@1301
   631
oheimb@5518
   632
Goal "!n. n < length xs --> xs!n : set xs";
nipkow@3040
   633
by (induct_tac "xs" 1);
nipkow@1301
   634
(* case [] *)
nipkow@1301
   635
by (Simp_tac 1);
nipkow@1301
   636
(* case x#xl *)
nipkow@1301
   637
by (rtac allI 1);
berghofe@5183
   638
by (induct_tac "n" 1);
nipkow@1301
   639
(* case 0 *)
nipkow@1301
   640
by (Asm_full_simp_tac 1);
nipkow@1301
   641
(* case Suc x *)
nipkow@4686
   642
by (Asm_full_simp_tac 1);
nipkow@1485
   643
qed_spec_mp "nth_mem";
nipkow@1301
   644
Addsimps [nth_mem];
nipkow@1301
   645
oheimb@5518
   646
nipkow@5077
   647
(** list update **)
nipkow@5077
   648
nipkow@5077
   649
section "list update";
nipkow@5077
   650
nipkow@5077
   651
Goal "!i. length(xs[i:=x]) = length xs";
nipkow@5077
   652
by (induct_tac "xs" 1);
nipkow@5077
   653
by (Simp_tac 1);
berghofe@5183
   654
by (asm_full_simp_tac (simpset() addsplits [nat.split]) 1);
nipkow@5077
   655
qed_spec_mp "length_list_update";
nipkow@5077
   656
Addsimps [length_list_update];
nipkow@5077
   657
nipkow@5644
   658
Goal "!i j. i < length xs  --> (xs[i:=x])!j = (if i=j then x else xs!j)";
paulson@6162
   659
by (induct_tac "xs" 1);
paulson@6162
   660
 by (Simp_tac 1);
paulson@6162
   661
by (auto_tac (claset(), simpset() addsimps [nth_Cons] addsplits [nat.split]));
nipkow@5644
   662
qed_spec_mp "nth_list_update";
nipkow@5644
   663
nipkow@6433
   664
Goal "!i. i < size xs --> xs[i:=x, i:=y] = xs[i:=y]";
nipkow@6433
   665
by(induct_tac "xs" 1);
nipkow@6433
   666
 by(Simp_tac 1);
nipkow@6433
   667
by(asm_simp_tac (simpset() addsplits [nat.split]) 1);
nipkow@6433
   668
qed_spec_mp "list_update_overwrite";
nipkow@6433
   669
Addsimps [list_update_overwrite];
nipkow@6433
   670
nipkow@6433
   671
Goal "!i < length xs. (xs[i := x] = xs) = (xs!i = x)";
nipkow@6433
   672
by(induct_tac "xs" 1);
nipkow@6433
   673
 by(Simp_tac 1);
nipkow@6433
   674
by(simp_tac (simpset() addsplits [nat.split]) 1);
nipkow@6433
   675
by(Blast_tac 1);
nipkow@6433
   676
qed_spec_mp "list_update_same_conv";
nipkow@6433
   677
nipkow@5077
   678
nipkow@3896
   679
(** last & butlast **)
nipkow@1327
   680
nipkow@5644
   681
section "last / butlast";
nipkow@5644
   682
nipkow@4935
   683
Goal "last(xs@[x]) = x";
wenzelm@4423
   684
by (induct_tac "xs" 1);
paulson@5316
   685
by Auto_tac;
nipkow@3896
   686
qed "last_snoc";
nipkow@3896
   687
Addsimps [last_snoc];
nipkow@3896
   688
nipkow@4935
   689
Goal "butlast(xs@[x]) = xs";
wenzelm@4423
   690
by (induct_tac "xs" 1);
paulson@5316
   691
by Auto_tac;
nipkow@3896
   692
qed "butlast_snoc";
nipkow@3896
   693
Addsimps [butlast_snoc];
nipkow@3896
   694
nipkow@4935
   695
Goal "length(butlast xs) = length xs - 1";
nipkow@4935
   696
by (res_inst_tac [("xs","xs")] rev_induct 1);
paulson@5316
   697
by Auto_tac;
nipkow@4643
   698
qed "length_butlast";
nipkow@4643
   699
Addsimps [length_butlast];
nipkow@4643
   700
paulson@5278
   701
Goal "!ys. butlast (xs@ys) = (if ys=[] then butlast xs else xs@butlast ys)";
wenzelm@4423
   702
by (induct_tac "xs" 1);
paulson@5316
   703
by Auto_tac;
nipkow@3896
   704
qed_spec_mp "butlast_append";
nipkow@3896
   705
nipkow@4935
   706
Goal "x:set(butlast xs) --> x:set xs";
wenzelm@4423
   707
by (induct_tac "xs" 1);
paulson@5316
   708
by Auto_tac;
nipkow@3896
   709
qed_spec_mp "in_set_butlastD";
nipkow@3896
   710
paulson@5448
   711
Goal "x:set(butlast xs) | x:set(butlast ys) ==> x:set(butlast(xs@ys))";
paulson@5448
   712
by (auto_tac (claset() addDs [in_set_butlastD],
paulson@5448
   713
	      simpset() addsimps [butlast_append]));
paulson@5448
   714
qed "in_set_butlast_appendI";
nipkow@3902
   715
nipkow@2608
   716
(** take  & drop **)
nipkow@2608
   717
section "take & drop";
nipkow@1327
   718
nipkow@4935
   719
Goal "take 0 xs = []";
nipkow@3040
   720
by (induct_tac "xs" 1);
paulson@5316
   721
by Auto_tac;
nipkow@1327
   722
qed "take_0";
nipkow@1327
   723
nipkow@4935
   724
Goal "drop 0 xs = xs";
nipkow@3040
   725
by (induct_tac "xs" 1);
paulson@5316
   726
by Auto_tac;
nipkow@2608
   727
qed "drop_0";
nipkow@2608
   728
nipkow@4935
   729
Goal "take (Suc n) (x#xs) = x # take n xs";
paulson@1552
   730
by (Simp_tac 1);
nipkow@1419
   731
qed "take_Suc_Cons";
nipkow@1327
   732
nipkow@4935
   733
Goal "drop (Suc n) (x#xs) = drop n xs";
nipkow@2608
   734
by (Simp_tac 1);
nipkow@2608
   735
qed "drop_Suc_Cons";
nipkow@2608
   736
nipkow@2608
   737
Delsimps [take_Cons,drop_Cons];
nipkow@2608
   738
Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons];
nipkow@2608
   739
nipkow@4935
   740
Goal "!xs. length(take n xs) = min (length xs) n";
berghofe@5183
   741
by (induct_tac "n" 1);
paulson@5316
   742
 by Auto_tac;
paulson@3457
   743
by (exhaust_tac "xs" 1);
paulson@5316
   744
 by Auto_tac;
nipkow@2608
   745
qed_spec_mp "length_take";
nipkow@2608
   746
Addsimps [length_take];
clasohm@923
   747
nipkow@4935
   748
Goal "!xs. length(drop n xs) = (length xs - n)";
berghofe@5183
   749
by (induct_tac "n" 1);
paulson@5316
   750
 by Auto_tac;
paulson@3457
   751
by (exhaust_tac "xs" 1);
paulson@5316
   752
 by Auto_tac;
nipkow@2608
   753
qed_spec_mp "length_drop";
nipkow@2608
   754
Addsimps [length_drop];
nipkow@2608
   755
nipkow@4935
   756
Goal "!xs. length xs <= n --> take n xs = xs";
berghofe@5183
   757
by (induct_tac "n" 1);
paulson@5316
   758
 by Auto_tac;
paulson@3457
   759
by (exhaust_tac "xs" 1);
paulson@5316
   760
 by Auto_tac;
nipkow@2608
   761
qed_spec_mp "take_all";
clasohm@923
   762
nipkow@4935
   763
Goal "!xs. length xs <= n --> drop n xs = []";
berghofe@5183
   764
by (induct_tac "n" 1);
paulson@5316
   765
 by Auto_tac;
paulson@3457
   766
by (exhaust_tac "xs" 1);
paulson@5316
   767
 by Auto_tac;
nipkow@2608
   768
qed_spec_mp "drop_all";
nipkow@2608
   769
paulson@5278
   770
Goal "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)";
berghofe@5183
   771
by (induct_tac "n" 1);
paulson@5316
   772
 by Auto_tac;
paulson@3457
   773
by (exhaust_tac "xs" 1);
paulson@5316
   774
 by Auto_tac;
nipkow@2608
   775
qed_spec_mp "take_append";
nipkow@2608
   776
Addsimps [take_append];
nipkow@2608
   777
nipkow@4935
   778
Goal "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; 
berghofe@5183
   779
by (induct_tac "n" 1);
paulson@5316
   780
 by Auto_tac;
paulson@3457
   781
by (exhaust_tac "xs" 1);
paulson@5316
   782
 by Auto_tac;
nipkow@2608
   783
qed_spec_mp "drop_append";
nipkow@2608
   784
Addsimps [drop_append];
nipkow@2608
   785
nipkow@4935
   786
Goal "!xs n. take n (take m xs) = take (min n m) xs"; 
berghofe@5183
   787
by (induct_tac "m" 1);
paulson@5316
   788
 by Auto_tac;
paulson@3457
   789
by (exhaust_tac "xs" 1);
paulson@5316
   790
 by Auto_tac;
berghofe@5183
   791
by (exhaust_tac "na" 1);
paulson@5316
   792
 by Auto_tac;
nipkow@2608
   793
qed_spec_mp "take_take";
nipkow@2608
   794
nipkow@4935
   795
Goal "!xs. drop n (drop m xs) = drop (n + m) xs"; 
berghofe@5183
   796
by (induct_tac "m" 1);
paulson@5316
   797
 by Auto_tac;
paulson@3457
   798
by (exhaust_tac "xs" 1);
paulson@5316
   799
 by Auto_tac;
nipkow@2608
   800
qed_spec_mp "drop_drop";
clasohm@923
   801
nipkow@4935
   802
Goal "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; 
berghofe@5183
   803
by (induct_tac "m" 1);
paulson@5316
   804
 by Auto_tac;
paulson@3457
   805
by (exhaust_tac "xs" 1);
paulson@5316
   806
 by Auto_tac;
nipkow@2608
   807
qed_spec_mp "take_drop";
nipkow@2608
   808
nipkow@4935
   809
Goal "!xs. take n (map f xs) = map f (take n xs)"; 
berghofe@5183
   810
by (induct_tac "n" 1);
paulson@5316
   811
 by Auto_tac;
paulson@3457
   812
by (exhaust_tac "xs" 1);
paulson@5316
   813
 by Auto_tac;
nipkow@2608
   814
qed_spec_mp "take_map"; 
nipkow@2608
   815
nipkow@4935
   816
Goal "!xs. drop n (map f xs) = map f (drop n xs)"; 
berghofe@5183
   817
by (induct_tac "n" 1);
paulson@5316
   818
 by Auto_tac;
paulson@3457
   819
by (exhaust_tac "xs" 1);
paulson@5316
   820
 by Auto_tac;
nipkow@2608
   821
qed_spec_mp "drop_map";
nipkow@2608
   822
nipkow@4935
   823
Goal "!n i. i < n --> (take n xs)!i = xs!i";
paulson@3457
   824
by (induct_tac "xs" 1);
paulson@5316
   825
 by Auto_tac;
paulson@3457
   826
by (exhaust_tac "n" 1);
paulson@3457
   827
 by (Blast_tac 1);
paulson@3457
   828
by (exhaust_tac "i" 1);
paulson@5316
   829
 by Auto_tac;
nipkow@2608
   830
qed_spec_mp "nth_take";
nipkow@2608
   831
Addsimps [nth_take];
clasohm@923
   832
nipkow@4935
   833
Goal  "!xs i. n + i <= length xs --> (drop n xs)!i = xs!(n+i)";
berghofe@5183
   834
by (induct_tac "n" 1);
paulson@5316
   835
 by Auto_tac;
paulson@3457
   836
by (exhaust_tac "xs" 1);
paulson@5316
   837
 by Auto_tac;
nipkow@2608
   838
qed_spec_mp "nth_drop";
nipkow@2608
   839
Addsimps [nth_drop];
nipkow@2608
   840
nipkow@2608
   841
(** takeWhile & dropWhile **)
nipkow@2608
   842
nipkow@3467
   843
section "takeWhile & dropWhile";
nipkow@3467
   844
nipkow@4935
   845
Goal "takeWhile P xs @ dropWhile P xs = xs";
nipkow@3586
   846
by (induct_tac "xs" 1);
paulson@5316
   847
by Auto_tac;
nipkow@3586
   848
qed "takeWhile_dropWhile_id";
nipkow@3586
   849
Addsimps [takeWhile_dropWhile_id];
nipkow@3586
   850
nipkow@4935
   851
Goal  "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs";
paulson@3457
   852
by (induct_tac "xs" 1);
paulson@5316
   853
by Auto_tac;
nipkow@2608
   854
bind_thm("takeWhile_append1", conjI RS (result() RS mp));
nipkow@2608
   855
Addsimps [takeWhile_append1];
clasohm@923
   856
nipkow@4935
   857
Goal "(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys";
paulson@3457
   858
by (induct_tac "xs" 1);
paulson@5316
   859
by Auto_tac;
nipkow@2608
   860
bind_thm("takeWhile_append2", ballI RS (result() RS mp));
nipkow@2608
   861
Addsimps [takeWhile_append2];
lcp@1169
   862
nipkow@4935
   863
Goal "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys";
paulson@3457
   864
by (induct_tac "xs" 1);
paulson@5316
   865
by Auto_tac;
nipkow@2608
   866
bind_thm("dropWhile_append1", conjI RS (result() RS mp));
nipkow@2608
   867
Addsimps [dropWhile_append1];
nipkow@2608
   868
nipkow@4935
   869
Goal "(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys";
paulson@3457
   870
by (induct_tac "xs" 1);
paulson@5316
   871
by Auto_tac;
nipkow@2608
   872
bind_thm("dropWhile_append2", ballI RS (result() RS mp));
nipkow@2608
   873
Addsimps [dropWhile_append2];
nipkow@2608
   874
nipkow@4935
   875
Goal "x:set(takeWhile P xs) --> x:set xs & P x";
paulson@3457
   876
by (induct_tac "xs" 1);
paulson@5316
   877
by Auto_tac;
paulson@3647
   878
qed_spec_mp"set_take_whileD";
nipkow@2608
   879
nipkow@6306
   880
(** zip **)
nipkow@6306
   881
section "zip";
nipkow@6306
   882
nipkow@6306
   883
Goal "zip [] ys = []";
nipkow@6306
   884
by(induct_tac "ys" 1);
nipkow@6306
   885
by Auto_tac;
nipkow@6306
   886
qed "zip_Nil";
nipkow@6306
   887
Addsimps [zip_Nil];
nipkow@6306
   888
nipkow@6306
   889
Goal "zip (x#xs) (y#ys) = (x,y)#zip xs ys";
nipkow@6306
   890
by(Simp_tac 1);
nipkow@6306
   891
qed "zip_Cons_Cons";
nipkow@6306
   892
Addsimps [zip_Cons_Cons];
nipkow@6306
   893
nipkow@6306
   894
Delsimps(tl (thms"zip.simps"));
nipkow@4605
   895
nipkow@5272
   896
nipkow@5272
   897
(** foldl **)
nipkow@5272
   898
section "foldl";
nipkow@5272
   899
nipkow@5272
   900
Goal "!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys";
paulson@5318
   901
by (induct_tac "xs" 1);
paulson@5316
   902
by Auto_tac;
nipkow@5272
   903
qed_spec_mp "foldl_append";
nipkow@5272
   904
Addsimps [foldl_append];
nipkow@5272
   905
nipkow@5272
   906
(* Note: `n <= foldl op+ n ns' looks simpler, but is more difficult to use
nipkow@5272
   907
   because it requires an additional transitivity step
nipkow@5272
   908
*)
nipkow@5272
   909
Goal "!n::nat. m <= n --> m <= foldl op+ n ns";
paulson@5318
   910
by (induct_tac "ns" 1);
nipkow@6058
   911
by Auto_tac;
nipkow@5272
   912
qed_spec_mp "start_le_sum";
nipkow@5272
   913
nipkow@5272
   914
Goal "n : set ns ==> n <= foldl op+ 0 ns";
oheimb@5758
   915
by (force_tac (claset() addIs [start_le_sum],
oheimb@5758
   916
              simpset() addsimps [in_set_conv_decomp]) 1);
nipkow@5272
   917
qed "elem_le_sum";
nipkow@5272
   918
nipkow@5272
   919
Goal "!m. (foldl op+ m ns = 0) = (m=0 & (!n : set ns. n=0))";
paulson@5318
   920
by (induct_tac "ns" 1);
paulson@5316
   921
by Auto_tac;
nipkow@5272
   922
qed_spec_mp "sum_eq_0_conv";
nipkow@5272
   923
AddIffs [sum_eq_0_conv];
nipkow@5272
   924
nipkow@5425
   925
(** upto **)
nipkow@5425
   926
nipkow@5427
   927
(* Does not terminate! *)
nipkow@5427
   928
Goal "[i..j(] = (if i<j then i#[Suc i..j(] else [])";
paulson@6162
   929
by (induct_tac "j" 1);
nipkow@5427
   930
by Auto_tac;
nipkow@5427
   931
qed "upt_rec";
nipkow@5425
   932
nipkow@5427
   933
Goal "j<=i ==> [i..j(] = []";
paulson@6162
   934
by (stac upt_rec 1);
paulson@6162
   935
by (Asm_simp_tac 1);
nipkow@5427
   936
qed "upt_conv_Nil";
nipkow@5427
   937
Addsimps [upt_conv_Nil];
nipkow@5427
   938
nipkow@5427
   939
Goal "i<=j ==> [i..(Suc j)(] = [i..j(]@[j]";
nipkow@5427
   940
by (Asm_simp_tac 1);
nipkow@5427
   941
qed "upt_Suc";
nipkow@5427
   942
nipkow@5427
   943
Goal "i<j ==> [i..j(] = i#[Suc i..j(]";
paulson@6162
   944
by (rtac trans 1);
paulson@6162
   945
by (stac upt_rec 1);
paulson@6162
   946
by (rtac refl 2);
nipkow@5427
   947
by (Asm_simp_tac 1);
nipkow@5427
   948
qed "upt_conv_Cons";
nipkow@5427
   949
nipkow@5427
   950
Goal "length [i..j(] = j-i";
paulson@6162
   951
by (induct_tac "j" 1);
nipkow@5427
   952
 by (Simp_tac 1);
paulson@6162
   953
by (asm_simp_tac (simpset() addsimps [Suc_diff_le]) 1);
nipkow@5427
   954
qed "length_upt";
nipkow@5427
   955
Addsimps [length_upt];
nipkow@5425
   956
nipkow@5427
   957
Goal "i+k < j --> [i..j(] ! k = i+k";
paulson@6162
   958
by (induct_tac "j" 1);
paulson@6162
   959
 by (Simp_tac 1);
paulson@6162
   960
by (asm_simp_tac (simpset() addsimps [nth_append,less_diff_conv]@add_ac) 1);
paulson@6162
   961
by (Clarify_tac 1);
paulson@6162
   962
by (subgoal_tac "n=i+k" 1);
paulson@6162
   963
 by (Asm_simp_tac 2);
paulson@6162
   964
by (Asm_simp_tac 1);
nipkow@5427
   965
qed_spec_mp "nth_upt";
nipkow@5427
   966
Addsimps [nth_upt];
nipkow@5425
   967
nipkow@6433
   968
Goal "!i. i+m <= n --> take m [i..n(] = [i..i+m(]";
nipkow@6433
   969
by(induct_tac "m" 1);
nipkow@6433
   970
 by(Simp_tac 1);
nipkow@6433
   971
by(Clarify_tac 1);
nipkow@6433
   972
by(stac upt_rec 1);
nipkow@6433
   973
br sym 1;
nipkow@6433
   974
by(stac upt_rec 1);
nipkow@6433
   975
by(asm_simp_tac (simpset() delsimps (thms"upt.simps")) 1);
nipkow@6433
   976
qed_spec_mp "take_upt";
nipkow@6433
   977
Addsimps [take_upt];
nipkow@6433
   978
nipkow@6433
   979
Goal "!m i. i < n-m --> (map f [m..n(]) ! i = f(m+i)";
nipkow@6433
   980
by(induct_tac "n" 1);
nipkow@6433
   981
 by(Simp_tac 1);
nipkow@6433
   982
by(Clarify_tac 1);
nipkow@6433
   983
by(subgoal_tac "m < Suc n" 1);
nipkow@6433
   984
 by(arith_tac 2);
nipkow@6433
   985
by(stac upt_rec 1);
nipkow@6433
   986
by(asm_simp_tac (simpset() delsplits [split_if]) 1);
nipkow@6433
   987
by(split_tac [split_if] 1);
nipkow@6433
   988
br conjI 1;
nipkow@6433
   989
 by(simp_tac (simpset() addsimps [nth_Cons] addsplits [nat.split]) 1);
nipkow@6433
   990
 by(simp_tac (simpset() addsimps [nth_append] addsplits [nat.split]) 1);
nipkow@6433
   991
 by(Clarify_tac 1);
nipkow@6433
   992
 br conjI 1;
nipkow@6433
   993
  by(Clarify_tac 1);
nipkow@6433
   994
  by(subgoal_tac "Suc(m+nat) < n" 1);
nipkow@6433
   995
   by(arith_tac 2);
nipkow@6433
   996
  by(Asm_simp_tac 1);
nipkow@6433
   997
 by(Clarify_tac 1);
nipkow@6433
   998
 by(subgoal_tac "n = Suc(m+nat)" 1);
nipkow@6433
   999
  by(arith_tac 2);
nipkow@6433
  1000
 by(Asm_simp_tac 1);
nipkow@6433
  1001
by(simp_tac (simpset() addsimps [nth_Cons] addsplits [nat.split]) 1);
nipkow@6433
  1002
by(arith_tac 1);
nipkow@6433
  1003
qed_spec_mp "nth_map_upt";
nipkow@6433
  1004
nipkow@5272
  1005
nipkow@4605
  1006
(** nodups & remdups **)
nipkow@4605
  1007
section "nodups & remdups";
nipkow@4605
  1008
nipkow@4935
  1009
Goal "set(remdups xs) = set xs";
nipkow@4605
  1010
by (induct_tac "xs" 1);
nipkow@4605
  1011
 by (Simp_tac 1);
nipkow@4686
  1012
by (asm_full_simp_tac (simpset() addsimps [insert_absorb]) 1);
nipkow@4605
  1013
qed "set_remdups";
nipkow@4605
  1014
Addsimps [set_remdups];
nipkow@4605
  1015
nipkow@4935
  1016
Goal "nodups(remdups xs)";
nipkow@4605
  1017
by (induct_tac "xs" 1);
paulson@5316
  1018
by Auto_tac;
nipkow@4605
  1019
qed "nodups_remdups";
nipkow@4605
  1020
nipkow@4935
  1021
Goal "nodups xs --> nodups (filter P xs)";
nipkow@4605
  1022
by (induct_tac "xs" 1);
paulson@5316
  1023
by Auto_tac;
nipkow@4605
  1024
qed_spec_mp "nodups_filter";
nipkow@4605
  1025
nipkow@3589
  1026
(** replicate **)
nipkow@3589
  1027
section "replicate";
nipkow@3589
  1028
nipkow@4935
  1029
Goal "set(replicate (Suc n) x) = {x}";
wenzelm@4423
  1030
by (induct_tac "n" 1);
paulson@5316
  1031
by Auto_tac;
nipkow@3589
  1032
val lemma = result();
nipkow@3589
  1033
nipkow@5043
  1034
Goal "n ~= 0 ==> set(replicate n x) = {x}";
wenzelm@4423
  1035
by (fast_tac (claset() addSDs [not0_implies_Suc] addSIs [lemma]) 1);
nipkow@3589
  1036
qed "set_replicate";
nipkow@3589
  1037
Addsimps [set_replicate];
nipkow@5162
  1038
nipkow@5162
  1039
nipkow@5281
  1040
(*** Lexcicographic orderings on lists ***)
nipkow@5281
  1041
section"Lexcicographic orderings on lists";
nipkow@5281
  1042
nipkow@5281
  1043
Goal "wf r ==> wf(lexn r n)";
paulson@5318
  1044
by (induct_tac "n" 1);
paulson@5318
  1045
by (Simp_tac 1);
paulson@5318
  1046
by (Simp_tac 1);
paulson@5318
  1047
by (rtac wf_subset 1);
paulson@5318
  1048
by (rtac Int_lower1 2);
paulson@5318
  1049
by (rtac wf_prod_fun_image 1);
paulson@5318
  1050
by (rtac injI 2);
paulson@5318
  1051
by (Auto_tac);
nipkow@5281
  1052
qed "wf_lexn";
nipkow@5281
  1053
nipkow@5281
  1054
Goal "!xs ys. (xs,ys) : lexn r n --> length xs = n & length ys = n";
paulson@5318
  1055
by (induct_tac "n" 1);
paulson@5318
  1056
by (Auto_tac);
nipkow@5281
  1057
qed_spec_mp "lexn_length";
nipkow@5281
  1058
nipkow@5281
  1059
Goalw [lex_def] "wf r ==> wf(lex r)";
paulson@5318
  1060
by (rtac wf_UN 1);
paulson@5318
  1061
by (blast_tac (claset() addIs [wf_lexn]) 1);
paulson@5318
  1062
by (Clarify_tac 1);
paulson@5318
  1063
by (rename_tac "m n" 1);
paulson@5318
  1064
by (subgoal_tac "m ~= n" 1);
paulson@5318
  1065
 by (Blast_tac 2);
paulson@5318
  1066
by (blast_tac (claset() addDs [lexn_length,not_sym]) 1);
nipkow@5281
  1067
qed "wf_lex";
nipkow@5281
  1068
AddSIs [wf_lex];
nipkow@5281
  1069
nipkow@5281
  1070
Goal
nipkow@5281
  1071
 "lexn r n = \
nipkow@5281
  1072
\ {(xs,ys). length xs = n & length ys = n & \
nipkow@5281
  1073
\           (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}";
paulson@5318
  1074
by (induct_tac "n" 1);
paulson@5318
  1075
 by (Simp_tac 1);
paulson@5318
  1076
 by (Blast_tac 1);
paulson@5641
  1077
by (asm_full_simp_tac (simpset() 
oheimb@5296
  1078
				addsimps [lex_prod_def]) 1);
paulson@5641
  1079
by (auto_tac (claset(), simpset()));
paulson@5318
  1080
  by (Blast_tac 1);
paulson@5318
  1081
 by (rename_tac "a xys x xs' y ys'" 1);
paulson@5318
  1082
 by (res_inst_tac [("x","a#xys")] exI 1);
paulson@5318
  1083
 by (Simp_tac 1);
paulson@5318
  1084
by (exhaust_tac "xys" 1);
paulson@5641
  1085
 by (ALLGOALS (asm_full_simp_tac (simpset())));
paulson@5318
  1086
by (Blast_tac 1);
nipkow@5281
  1087
qed "lexn_conv";
nipkow@5281
  1088
nipkow@5281
  1089
Goalw [lex_def]
nipkow@5281
  1090
 "lex r = \
nipkow@5281
  1091
\ {(xs,ys). length xs = length ys & \
nipkow@5281
  1092
\           (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}";
paulson@5641
  1093
by (force_tac (claset(), simpset() addsimps [lexn_conv]) 1);
nipkow@5281
  1094
qed "lex_conv";
nipkow@5281
  1095
nipkow@5281
  1096
Goalw [lexico_def] "wf r ==> wf(lexico r)";
paulson@5318
  1097
by (Blast_tac 1);
nipkow@5281
  1098
qed "wf_lexico";
nipkow@5281
  1099
AddSIs [wf_lexico];
nipkow@5281
  1100
nipkow@5281
  1101
Goalw
nipkow@5281
  1102
 [lexico_def,diag_def,lex_prod_def,measure_def,inv_image_def]
nipkow@5281
  1103
"lexico r = {(xs,ys). length xs < length ys | \
nipkow@5281
  1104
\                     length xs = length ys & (xs,ys) : lex r}";
paulson@5318
  1105
by (Simp_tac 1);
nipkow@5281
  1106
qed "lexico_conv";
nipkow@5281
  1107
nipkow@5283
  1108
Goal "([],ys) ~: lex r";
paulson@5318
  1109
by (simp_tac (simpset() addsimps [lex_conv]) 1);
nipkow@5283
  1110
qed "Nil_notin_lex";
nipkow@5283
  1111
nipkow@5283
  1112
Goal "(xs,[]) ~: lex r";
paulson@5318
  1113
by (simp_tac (simpset() addsimps [lex_conv]) 1);
nipkow@5283
  1114
qed "Nil2_notin_lex";
nipkow@5283
  1115
nipkow@5283
  1116
AddIffs [Nil_notin_lex,Nil2_notin_lex];
nipkow@5283
  1117
nipkow@5283
  1118
Goal "((x#xs,y#ys) : lex r) = \
nipkow@5283
  1119
\     ((x,y) : r & length xs = length ys | x=y & (xs,ys) : lex r)";
paulson@5318
  1120
by (simp_tac (simpset() addsimps [lex_conv]) 1);
paulson@5318
  1121
by (rtac iffI 1);
paulson@5318
  1122
 by (blast_tac (claset() addIs [Cons_eq_appendI]) 2);
paulson@5318
  1123
by (REPEAT(eresolve_tac [conjE, exE] 1));
paulson@5318
  1124
by (exhaust_tac "xys" 1);
paulson@5318
  1125
by (Asm_full_simp_tac 1);
paulson@5318
  1126
by (Asm_full_simp_tac 1);
paulson@5318
  1127
by (Blast_tac 1);
nipkow@5283
  1128
qed "Cons_in_lex";
nipkow@5283
  1129
AddIffs [Cons_in_lex];