src/HOL/List.ML
author paulson
Tue, 19 Jan 1999 11:18:11 +0100
changeset 6141 a6922171b396
parent 6073 fba734ba6894
child 6162 484adda70b65
permissions -rw-r--r--
removal of the (thm list) argument of mk_cases
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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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5427
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(***
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   246
Simplification procedure for all list equalities.
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   247
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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   255
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   256
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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   261
fun list1 (Const("List.list.op #",_) $ _ $ Const("List.list.[]",_)) = true
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  | list1 _ = false;
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   263
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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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   283
            handle ERROR =>
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   284
            error("The error(s) above occurred while trying to prove " ^
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   285
                  string_of_cterm ct)
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      in Some((conv RS (thm RS trans)) RS eq_reflection) end
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   287
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   288
  in if list1 lastl andalso list1 lastr
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   289
     then rearr append1_eq_conv
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   290
     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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   294
  end;
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   295
in
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val list_eq_simproc = mk_simproc "list_eq" [list_eq_pattern] list_eq;
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   297
end;
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   298
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Addsimprocs [list_eq_simproc];
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(** map **)
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   303
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section "map";
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   305
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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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   310
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Goal "map (%x. x) = (%xs. xs)";
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450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   312
by (rtac ext 1);
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   313
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   314
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   315
qed "map_ident";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   316
Addsimps[map_ident];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   317
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   318
Goal "map f (xs@ys) = map f xs @ map f ys";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   319
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   320
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   321
qed "map_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   322
Addsimps[map_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   323
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   324
Goalw [o_def] "map (f o g) xs = map f (map g xs)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   325
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   326
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   327
qed "map_compose";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   328
Addsimps[map_compose];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   329
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   330
Goal "rev(map f xs) = map f (rev xs)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   331
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   332
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   333
qed "rev_map";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   334
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   335
(* a congruence rule for map: *)
5278
a903b66822e2 even more tidying of Goal commands
paulson
parents: 5272
diff changeset
   336
Goal "(xs=ys) --> (!x. x : set ys --> f x = g x) --> map f xs = map g ys";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   337
by (rtac impI 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   338
by (hyp_subst_tac 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   339
by (induct_tac "ys" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   340
by Auto_tac;
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   341
val lemma = result();
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   342
bind_thm("map_cong",impI RSN (2,allI RSN (2,lemma RS mp RS mp)));
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   343
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   344
Goal "(map f xs = []) = (xs = [])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   345
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   346
by Auto_tac;
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   347
qed "map_is_Nil_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   348
AddIffs [map_is_Nil_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   349
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   350
Goal "([] = map f xs) = (xs = [])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   351
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   352
by Auto_tac;
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   353
qed "Nil_is_map_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   354
AddIffs [Nil_is_map_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   355
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   356
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   357
(** rev **)
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   358
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   359
section "rev";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   360
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   361
Goal "rev(xs@ys) = rev(ys) @ rev(xs)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   362
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   363
by Auto_tac;
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   364
qed "rev_append";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   365
Addsimps[rev_append];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   366
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   367
Goal "rev(rev l) = l";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   368
by (induct_tac "l" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   369
by Auto_tac;
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   370
qed "rev_rev_ident";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   371
Addsimps[rev_rev_ident];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   372
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   373
Goal "(rev xs = []) = (xs = [])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   374
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   375
by Auto_tac;
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   376
qed "rev_is_Nil_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   377
AddIffs [rev_is_Nil_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   378
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   379
Goal "([] = rev xs) = (xs = [])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   380
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   381
by Auto_tac;
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   382
qed "Nil_is_rev_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   383
AddIffs [Nil_is_rev_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   384
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   385
val prems = Goal "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P xs";
5132
24f992a25adc isatool expandshort;
wenzelm
parents: 5129
diff changeset
   386
by (stac (rev_rev_ident RS sym) 1);
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   387
br(read_instantiate [("P","%xs. ?P(rev xs)")]list.induct)1;
5132
24f992a25adc isatool expandshort;
wenzelm
parents: 5129
diff changeset
   388
by (ALLGOALS Simp_tac);
24f992a25adc isatool expandshort;
wenzelm
parents: 5129
diff changeset
   389
by (resolve_tac prems 1);
24f992a25adc isatool expandshort;
wenzelm
parents: 5129
diff changeset
   390
by (eresolve_tac prems 1);
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   391
qed "rev_induct";
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   392
5272
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   393
fun rev_induct_tac xs = res_inst_tac [("xs",xs)] rev_induct;
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   394
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   395
Goal  "(xs = [] --> P) -->  (!ys y. xs = ys@[y] --> P) --> P";
5132
24f992a25adc isatool expandshort;
wenzelm
parents: 5129
diff changeset
   396
by (res_inst_tac [("xs","xs")] rev_induct 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   397
by Auto_tac;
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   398
bind_thm ("rev_exhaust",
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   399
  impI RSN (2,allI RSN (2,allI RSN (2,impI RS (result() RS mp RS mp)))));
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   400
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   401
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   402
(** set **)
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   403
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   404
section "set";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   405
5296
bdef7d349d27 added length_Suc_conv, finite_set
oheimb
parents: 5283
diff changeset
   406
qed_goal "finite_set" thy "finite (set xs)" 
bdef7d349d27 added length_Suc_conv, finite_set
oheimb
parents: 5283
diff changeset
   407
	(K [induct_tac "xs" 1, Auto_tac]);
bdef7d349d27 added length_Suc_conv, finite_set
oheimb
parents: 5283
diff changeset
   408
Addsimps[finite_set];
bdef7d349d27 added length_Suc_conv, finite_set
oheimb
parents: 5283
diff changeset
   409
AddSIs[finite_set];
bdef7d349d27 added length_Suc_conv, finite_set
oheimb
parents: 5283
diff changeset
   410
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   411
Goal "set (xs@ys) = (set xs Un set ys)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   412
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   413
by Auto_tac;
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   414
qed "set_append";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   415
Addsimps[set_append];
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   416
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   417
Goal "set l <= set (x#l)";
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   418
by Auto_tac;
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   419
qed "set_subset_Cons";
1936
979e8b4f5fa5 Proved set_of_list_subset_Cons
paulson
parents: 1908
diff changeset
   420
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   421
Goal "(set xs = {}) = (xs = [])";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   422
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   423
by Auto_tac;
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   424
qed "set_empty";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   425
Addsimps [set_empty];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   426
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   427
Goal "set(rev xs) = set(xs)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   428
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   429
by Auto_tac;
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   430
qed "set_rev";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   431
Addsimps [set_rev];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   432
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   433
Goal "set(map f xs) = f``(set xs)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   434
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   435
by Auto_tac;
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   436
qed "set_map";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   437
Addsimps [set_map];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   438
5443
e2459d18ff47 changed constants mem and list_all to mere translations
oheimb
parents: 5427
diff changeset
   439
Goal "(x : set (filter P xs)) = (x : set xs & P x)";
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   440
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   441
by Auto_tac;
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   442
qed "in_set_filter";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   443
Addsimps [in_set_filter];
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   444
5272
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   445
Goal "(x : set xs) = (? ys zs. xs = ys@x#zs)";
5318
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   446
by (induct_tac "xs" 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   447
 by (Simp_tac 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   448
by (Asm_simp_tac 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   449
by (rtac iffI 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   450
by (blast_tac (claset() addIs [eq_Nil_appendI,Cons_eq_appendI]) 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   451
by (REPEAT(etac exE 1));
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   452
by (exhaust_tac "ys" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   453
by Auto_tac;
5272
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   454
qed "in_set_conv_decomp";
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   455
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   456
(* eliminate `lists' in favour of `set' *)
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   457
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   458
Goal "(xs : lists A) = (!x : set xs. x : A)";
5318
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   459
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   460
by Auto_tac;
5272
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   461
qed "in_lists_conv_set";
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   462
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   463
bind_thm("in_listsD",in_lists_conv_set RS iffD1);
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   464
AddSDs [in_listsD];
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   465
bind_thm("in_listsI",in_lists_conv_set RS iffD2);
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   466
AddSIs [in_listsI];
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   467
5518
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   468
(** mem **)
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   469
 
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   470
section "mem";
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   471
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   472
Goal "(x mem xs) = (x: set xs)";
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   473
by (induct_tac "xs" 1);
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   474
by Auto_tac;
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   475
qed "set_mem_eq";
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   476
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   477
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   478
(** list_all **)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   479
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   480
section "list_all";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   481
5518
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   482
Goal "list_all P xs = (!x:set xs. P x)";
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   483
by (induct_tac "xs" 1);
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   484
by Auto_tac;
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   485
qed "list_all_conv";
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   486
5443
e2459d18ff47 changed constants mem and list_all to mere translations
oheimb
parents: 5427
diff changeset
   487
Goal "list_all P (xs@ys) = (list_all P xs & list_all P ys)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   488
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   489
by Auto_tac;
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   490
qed "list_all_append";
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   491
Addsimps [list_all_append];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   492
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   493
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   494
(** filter **)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   495
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   496
section "filter";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   497
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   498
Goal "filter P (xs@ys) = filter P xs @ filter P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   499
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   500
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   501
qed "filter_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   502
Addsimps [filter_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   503
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   504
Goal "filter (%x. True) xs = xs";
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   505
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   506
by Auto_tac;
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   507
qed "filter_True";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   508
Addsimps [filter_True];
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   509
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   510
Goal "filter (%x. False) xs = []";
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   511
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   512
by Auto_tac;
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   513
qed "filter_False";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   514
Addsimps [filter_False];
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   515
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   516
Goal "length (filter P xs) <= length xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   517
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   518
by Auto_tac;
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   519
qed "length_filter";
5443
e2459d18ff47 changed constants mem and list_all to mere translations
oheimb
parents: 5427
diff changeset
   520
Addsimps[length_filter];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   521
5443
e2459d18ff47 changed constants mem and list_all to mere translations
oheimb
parents: 5427
diff changeset
   522
Goal "set (filter P xs) <= set xs";
e2459d18ff47 changed constants mem and list_all to mere translations
oheimb
parents: 5427
diff changeset
   523
by Auto_tac;
e2459d18ff47 changed constants mem and list_all to mere translations
oheimb
parents: 5427
diff changeset
   524
qed "filter_is_subset";
e2459d18ff47 changed constants mem and list_all to mere translations
oheimb
parents: 5427
diff changeset
   525
Addsimps [filter_is_subset];
e2459d18ff47 changed constants mem and list_all to mere translations
oheimb
parents: 5427
diff changeset
   526
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   527
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   528
section "concat";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   529
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   530
Goal  "concat(xs@ys) = concat(xs)@concat(ys)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   531
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   532
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   533
qed"concat_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   534
Addsimps [concat_append];
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   535
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   536
Goal "(concat xss = []) = (!xs:set xss. xs=[])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   537
by (induct_tac "xss" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   538
by Auto_tac;
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   539
qed "concat_eq_Nil_conv";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   540
AddIffs [concat_eq_Nil_conv];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   541
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   542
Goal "([] = concat xss) = (!xs:set xss. xs=[])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   543
by (induct_tac "xss" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   544
by Auto_tac;
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   545
qed "Nil_eq_concat_conv";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   546
AddIffs [Nil_eq_concat_conv];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   547
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   548
Goal  "set(concat xs) = Union(set `` set xs)";
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   549
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   550
by Auto_tac;
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   551
qed"set_concat";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   552
Addsimps [set_concat];
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   553
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   554
Goal "map f (concat xs) = concat (map (map f) xs)"; 
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   555
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   556
by Auto_tac;
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   557
qed "map_concat";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   558
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   559
Goal "filter p (concat xs) = concat (map (filter p) xs)"; 
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   560
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   561
by Auto_tac;
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   562
qed"filter_concat"; 
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   563
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   564
Goal "rev(concat xs) = concat (map rev (rev xs))";
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   565
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   566
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   567
qed "rev_concat";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   568
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   569
(** nth **)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   570
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   571
section "nth";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   572
5644
85fd64148873 Nat: added zero_neq_conv
nipkow
parents: 5641
diff changeset
   573
Goal "(x#xs)!n = (case n of 0 => x | Suc m => xs!m)";
85fd64148873 Nat: added zero_neq_conv
nipkow
parents: 5641
diff changeset
   574
by(simp_tac (simpset() addsplits [nat.split]) 1);
85fd64148873 Nat: added zero_neq_conv
nipkow
parents: 5641
diff changeset
   575
qed "nth_Cons";
85fd64148873 Nat: added zero_neq_conv
nipkow
parents: 5641
diff changeset
   576
5278
a903b66822e2 even more tidying of Goal commands
paulson
parents: 5272
diff changeset
   577
Goal "!xs. (xs@ys)!n = (if n < length xs then xs!n else ys!(n - length xs))";
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   578
by (induct_tac "n" 1);
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   579
 by (Asm_simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   580
 by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   581
 by (exhaust_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   582
  by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   583
qed_spec_mp "nth_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   584
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   585
Goal "!n. n < length xs --> (map f xs)!n = f(xs!n)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   586
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   587
(* case [] *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   588
by (Asm_full_simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   589
(* case x#xl *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   590
by (rtac allI 1);
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   591
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   592
by Auto_tac;
1485
240cc98b94a7 Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents: 1465
diff changeset
   593
qed_spec_mp "nth_map";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   594
Addsimps [nth_map];
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   595
5518
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   596
Goal "!n. n < length xs --> Ball (set xs) P --> P(xs!n)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   597
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   598
(* case [] *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   599
by (Simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   600
(* case x#xl *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   601
by (rtac allI 1);
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   602
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   603
by Auto_tac;
5518
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   604
qed_spec_mp "list_ball_nth";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   605
5518
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   606
Goal "!n. n < length xs --> xs!n : set xs";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   607
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   608
(* case [] *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   609
by (Simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   610
(* case x#xl *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   611
by (rtac allI 1);
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   612
by (induct_tac "n" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   613
(* case 0 *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   614
by (Asm_full_simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   615
(* case Suc x *)
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   616
by (Asm_full_simp_tac 1);
1485
240cc98b94a7 Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents: 1465
diff changeset
   617
qed_spec_mp "nth_mem";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   618
Addsimps [nth_mem];
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   619
5518
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   620
5077
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   621
(** list update **)
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   622
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   623
section "list update";
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   624
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   625
Goal "!i. length(xs[i:=x]) = length xs";
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   626
by (induct_tac "xs" 1);
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   627
by (Simp_tac 1);
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   628
by (asm_full_simp_tac (simpset() addsplits [nat.split]) 1);
5077
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   629
qed_spec_mp "length_list_update";
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   630
Addsimps [length_list_update];
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   631
5644
85fd64148873 Nat: added zero_neq_conv
nipkow
parents: 5641
diff changeset
   632
Goal "!i j. i < length xs  --> (xs[i:=x])!j = (if i=j then x else xs!j)";
85fd64148873 Nat: added zero_neq_conv
nipkow
parents: 5641
diff changeset
   633
by(induct_tac "xs" 1);
85fd64148873 Nat: added zero_neq_conv
nipkow
parents: 5641
diff changeset
   634
 by(Simp_tac 1);
85fd64148873 Nat: added zero_neq_conv
nipkow
parents: 5641
diff changeset
   635
by(auto_tac (claset(), simpset() addsimps [nth_Cons] addsplits [nat.split]));
85fd64148873 Nat: added zero_neq_conv
nipkow
parents: 5641
diff changeset
   636
qed_spec_mp "nth_list_update";
85fd64148873 Nat: added zero_neq_conv
nipkow
parents: 5641
diff changeset
   637
5077
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   638
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   639
(** last & butlast **)
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   640
5644
85fd64148873 Nat: added zero_neq_conv
nipkow
parents: 5641
diff changeset
   641
section "last / butlast";
85fd64148873 Nat: added zero_neq_conv
nipkow
parents: 5641
diff changeset
   642
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   643
Goal "last(xs@[x]) = x";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   644
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   645
by Auto_tac;
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   646
qed "last_snoc";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   647
Addsimps [last_snoc];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   648
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   649
Goal "butlast(xs@[x]) = xs";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   650
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   651
by Auto_tac;
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   652
qed "butlast_snoc";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   653
Addsimps [butlast_snoc];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   654
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   655
Goal "length(butlast xs) = length xs - 1";
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   656
by (res_inst_tac [("xs","xs")] rev_induct 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   657
by Auto_tac;
4643
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   658
qed "length_butlast";
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   659
Addsimps [length_butlast];
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   660
5278
a903b66822e2 even more tidying of Goal commands
paulson
parents: 5272
diff changeset
   661
Goal "!ys. butlast (xs@ys) = (if ys=[] then butlast xs else xs@butlast ys)";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   662
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   663
by Auto_tac;
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   664
qed_spec_mp "butlast_append";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   665
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   666
Goal "x:set(butlast xs) --> x:set xs";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   667
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   668
by Auto_tac;
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   669
qed_spec_mp "in_set_butlastD";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   670
5448
40a09282ba14 in_set_butlast_appendI supersedes in_set_butlast_appendI1,2
paulson
parents: 5443
diff changeset
   671
Goal "x:set(butlast xs) | x:set(butlast ys) ==> x:set(butlast(xs@ys))";
40a09282ba14 in_set_butlast_appendI supersedes in_set_butlast_appendI1,2
paulson
parents: 5443
diff changeset
   672
by (auto_tac (claset() addDs [in_set_butlastD],
40a09282ba14 in_set_butlast_appendI supersedes in_set_butlast_appendI1,2
paulson
parents: 5443
diff changeset
   673
	      simpset() addsimps [butlast_append]));
40a09282ba14 in_set_butlast_appendI supersedes in_set_butlast_appendI1,2
paulson
parents: 5443
diff changeset
   674
qed "in_set_butlast_appendI";
3902
265a5d8ab88f Removed comment.
nipkow
parents: 3896
diff changeset
   675
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   676
(** take  & drop **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   677
section "take & drop";
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   678
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   679
Goal "take 0 xs = []";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   680
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   681
by Auto_tac;
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   682
qed "take_0";
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   683
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   684
Goal "drop 0 xs = xs";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   685
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   686
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   687
qed "drop_0";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   688
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   689
Goal "take (Suc n) (x#xs) = x # take n xs";
1552
6f71b5d46700 Ran expandshort
paulson
parents: 1485
diff changeset
   690
by (Simp_tac 1);
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   691
qed "take_Suc_Cons";
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   692
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   693
Goal "drop (Suc n) (x#xs) = drop n xs";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   694
by (Simp_tac 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   695
qed "drop_Suc_Cons";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   696
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   697
Delsimps [take_Cons,drop_Cons];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   698
Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   699
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   700
Goal "!xs. length(take n xs) = min (length xs) n";
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   701
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   702
 by Auto_tac;
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   703
by (exhaust_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   704
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   705
qed_spec_mp "length_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   706
Addsimps [length_take];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   707
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   708
Goal "!xs. length(drop n xs) = (length xs - n)";
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   709
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   710
 by Auto_tac;
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   711
by (exhaust_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   712
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   713
qed_spec_mp "length_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   714
Addsimps [length_drop];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   715
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   716
Goal "!xs. length xs <= n --> take n xs = xs";
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   717
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   718
 by Auto_tac;
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   719
by (exhaust_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   720
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   721
qed_spec_mp "take_all";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   722
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   723
Goal "!xs. length xs <= n --> drop n xs = []";
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   724
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   725
 by Auto_tac;
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   726
by (exhaust_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   727
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   728
qed_spec_mp "drop_all";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   729
5278
a903b66822e2 even more tidying of Goal commands
paulson
parents: 5272
diff changeset
   730
Goal "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)";
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   731
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   732
 by Auto_tac;
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   733
by (exhaust_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   734
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   735
qed_spec_mp "take_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   736
Addsimps [take_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   737
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   738
Goal "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; 
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   739
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   740
 by Auto_tac;
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   741
by (exhaust_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   742
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   743
qed_spec_mp "drop_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   744
Addsimps [drop_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   745
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   746
Goal "!xs n. take n (take m xs) = take (min n m) xs"; 
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   747
by (induct_tac "m" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   748
 by Auto_tac;
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   749
by (exhaust_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   750
 by Auto_tac;
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   751
by (exhaust_tac "na" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   752
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   753
qed_spec_mp "take_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   754
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   755
Goal "!xs. drop n (drop m xs) = drop (n + m) xs"; 
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   756
by (induct_tac "m" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   757
 by Auto_tac;
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   758
by (exhaust_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   759
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   760
qed_spec_mp "drop_drop";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   761
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   762
Goal "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; 
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   763
by (induct_tac "m" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   764
 by Auto_tac;
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   765
by (exhaust_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   766
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   767
qed_spec_mp "take_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   768
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   769
Goal "!xs. take n (map f xs) = map f (take n xs)"; 
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   770
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   771
 by Auto_tac;
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   772
by (exhaust_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   773
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   774
qed_spec_mp "take_map"; 
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   775
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   776
Goal "!xs. drop n (map f xs) = map f (drop n xs)"; 
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   777
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   778
 by Auto_tac;
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   779
by (exhaust_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   780
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   781
qed_spec_mp "drop_map";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   782
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   783
Goal "!n i. i < n --> (take n xs)!i = xs!i";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   784
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   785
 by Auto_tac;
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   786
by (exhaust_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   787
 by (Blast_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   788
by (exhaust_tac "i" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   789
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   790
qed_spec_mp "nth_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   791
Addsimps [nth_take];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   792
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   793
Goal  "!xs i. n + i <= length xs --> (drop n xs)!i = xs!(n+i)";
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   794
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   795
 by Auto_tac;
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   796
by (exhaust_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   797
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   798
qed_spec_mp "nth_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   799
Addsimps [nth_drop];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   800
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   801
(** takeWhile & dropWhile **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   802
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   803
section "takeWhile & dropWhile";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   804
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   805
Goal "takeWhile P xs @ dropWhile P xs = xs";
3586
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   806
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   807
by Auto_tac;
3586
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   808
qed "takeWhile_dropWhile_id";
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   809
Addsimps [takeWhile_dropWhile_id];
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   810
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   811
Goal  "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   812
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   813
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   814
bind_thm("takeWhile_append1", conjI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   815
Addsimps [takeWhile_append1];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   816
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   817
Goal "(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   818
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   819
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   820
bind_thm("takeWhile_append2", ballI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   821
Addsimps [takeWhile_append2];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   822
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   823
Goal "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   824
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   825
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   826
bind_thm("dropWhile_append1", conjI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   827
Addsimps [dropWhile_append1];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   828
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   829
Goal "(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   830
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   831
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   832
bind_thm("dropWhile_append2", ballI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   833
Addsimps [dropWhile_append2];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   834
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   835
Goal "x:set(takeWhile P xs) --> x:set xs & P x";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   836
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   837
by Auto_tac;
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   838
qed_spec_mp"set_take_whileD";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   839
4132
daff3c9987cc added zip and nodup
oheimb
parents: 4089
diff changeset
   840
qed_goal "zip_Nil_Nil"   thy "zip []     []     = []" (K [Simp_tac 1]);
daff3c9987cc added zip and nodup
oheimb
parents: 4089
diff changeset
   841
qed_goal "zip_Cons_Cons" thy "zip (x#xs) (y#ys) = (x,y)#zip xs ys" 
daff3c9987cc added zip and nodup
oheimb
parents: 4089
diff changeset
   842
						      (K [Simp_tac 1]);
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   843
5272
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   844
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   845
(** foldl **)
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   846
section "foldl";
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   847
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   848
Goal "!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys";
5318
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   849
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   850
by Auto_tac;
5272
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   851
qed_spec_mp "foldl_append";
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   852
Addsimps [foldl_append];
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   853
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   854
(* Note: `n <= foldl op+ n ns' looks simpler, but is more difficult to use
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   855
   because it requires an additional transitivity step
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   856
*)
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   857
Goal "!n::nat. m <= n --> m <= foldl op+ n ns";
5318
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   858
by (induct_tac "ns" 1);
6058
a9600c47ace3 Shortened a proof.
nipkow
parents: 6055
diff changeset
   859
by Auto_tac;
5272
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   860
qed_spec_mp "start_le_sum";
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   861
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   862
Goal "n : set ns ==> n <= foldl op+ 0 ns";
5758
27a2b36efd95 corrected auto_tac (applications of unsafe wrappers)
oheimb
parents: 5644
diff changeset
   863
by (force_tac (claset() addIs [start_le_sum],
27a2b36efd95 corrected auto_tac (applications of unsafe wrappers)
oheimb
parents: 5644
diff changeset
   864
              simpset() addsimps [in_set_conv_decomp]) 1);
5272
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   865
qed "elem_le_sum";
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   866
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   867
Goal "!m. (foldl op+ m ns = 0) = (m=0 & (!n : set ns. n=0))";
5318
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   868
by (induct_tac "ns" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   869
by Auto_tac;
5272
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   870
qed_spec_mp "sum_eq_0_conv";
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   871
AddIffs [sum_eq_0_conv];
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   872
5425
157c6663dedd Added function upto to List.
nipkow
parents: 5355
diff changeset
   873
(** upto **)
157c6663dedd Added function upto to List.
nipkow
parents: 5355
diff changeset
   874
5427
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   875
(* Does not terminate! *)
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   876
Goal "[i..j(] = (if i<j then i#[Suc i..j(] else [])";
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   877
by(induct_tac "j" 1);
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   878
by Auto_tac;
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   879
qed "upt_rec";
5425
157c6663dedd Added function upto to List.
nipkow
parents: 5355
diff changeset
   880
5427
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   881
Goal "j<=i ==> [i..j(] = []";
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   882
by(stac upt_rec 1);
5983
79e301a6a51b At last: linear arithmetic for nat!
nipkow
parents: 5758
diff changeset
   883
by(Asm_simp_tac 1);
5427
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   884
qed "upt_conv_Nil";
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   885
Addsimps [upt_conv_Nil];
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   886
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   887
Goal "i<=j ==> [i..(Suc j)(] = [i..j(]@[j]";
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   888
by (Asm_simp_tac 1);
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   889
qed "upt_Suc";
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   890
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   891
Goal "i<j ==> [i..j(] = i#[Suc i..j(]";
5425
157c6663dedd Added function upto to List.
nipkow
parents: 5355
diff changeset
   892
br trans 1;
5427
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   893
by(stac upt_rec 1);
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   894
br refl 2;
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   895
by (Asm_simp_tac 1);
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   896
qed "upt_conv_Cons";
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   897
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   898
Goal "length [i..j(] = j-i";
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   899
by(induct_tac "j" 1);
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   900
 by (Simp_tac 1);
5983
79e301a6a51b At last: linear arithmetic for nat!
nipkow
parents: 5758
diff changeset
   901
by(asm_simp_tac (simpset() addsimps [Suc_diff_le]) 1);
5427
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   902
qed "length_upt";
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   903
Addsimps [length_upt];
5425
157c6663dedd Added function upto to List.
nipkow
parents: 5355
diff changeset
   904
5427
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   905
Goal "i+k < j --> [i..j(] ! k = i+k";
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   906
by(induct_tac "j" 1);
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   907
 by(Simp_tac 1);
5983
79e301a6a51b At last: linear arithmetic for nat!
nipkow
parents: 5758
diff changeset
   908
by(asm_simp_tac (simpset() addsimps [nth_append,less_diff_conv]@add_ac) 1);
5427
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   909
by(Clarify_tac 1);
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   910
by(subgoal_tac "n=i+k" 1);
6073
fba734ba6894 Refined arith tactic.
nipkow
parents: 6058
diff changeset
   911
 by(Asm_simp_tac 2);
fba734ba6894 Refined arith tactic.
nipkow
parents: 6058
diff changeset
   912
by(Asm_simp_tac 1);
5427
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   913
qed_spec_mp "nth_upt";
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
   914
Addsimps [nth_upt];
5425
157c6663dedd Added function upto to List.
nipkow
parents: 5355
diff changeset
   915
5272
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   916
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   917
(** nodups & remdups **)
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   918
section "nodups & remdups";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   919
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   920
Goal "set(remdups xs) = set xs";
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   921
by (induct_tac "xs" 1);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   922
 by (Simp_tac 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   923
by (asm_full_simp_tac (simpset() addsimps [insert_absorb]) 1);
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   924
qed "set_remdups";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   925
Addsimps [set_remdups];
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   926
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   927
Goal "nodups(remdups xs)";
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   928
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   929
by Auto_tac;
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   930
qed "nodups_remdups";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   931
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   932
Goal "nodups xs --> nodups (filter P xs)";
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   933
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   934
by Auto_tac;
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   935
qed_spec_mp "nodups_filter";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   936
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   937
(** replicate **)
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   938
section "replicate";
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   939
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   940
Goal "set(replicate (Suc n) x) = {x}";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   941
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   942
by Auto_tac;
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   943
val lemma = result();
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   944
5043
3fdc881e8ff9 goal -> Goal
nipkow
parents: 4935
diff changeset
   945
Goal "n ~= 0 ==> set(replicate n x) = {x}";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   946
by (fast_tac (claset() addSDs [not0_implies_Suc] addSIs [lemma]) 1);
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   947
qed "set_replicate";
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   948
Addsimps [set_replicate];
5162
53e505c6019c Added simproc list_eq.
nipkow
parents: 5132
diff changeset
   949
53e505c6019c Added simproc list_eq.
nipkow
parents: 5132
diff changeset
   950
5281
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
   951
(*** Lexcicographic orderings on lists ***)
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
   952
section"Lexcicographic orderings on lists";
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
   953
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
   954
Goal "wf r ==> wf(lexn r n)";
5318
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   955
by (induct_tac "n" 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   956
by (Simp_tac 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   957
by (Simp_tac 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   958
by (rtac wf_subset 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   959
by (rtac Int_lower1 2);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   960
by (rtac wf_prod_fun_image 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   961
by (rtac injI 2);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   962
by (Auto_tac);
5281
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
   963
qed "wf_lexn";
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
   964
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
   965
Goal "!xs ys. (xs,ys) : lexn r n --> length xs = n & length ys = n";
5318
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   966
by (induct_tac "n" 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   967
by (Auto_tac);
5281
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
   968
qed_spec_mp "lexn_length";
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
   969
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
   970
Goalw [lex_def] "wf r ==> wf(lex r)";
5318
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   971
by (rtac wf_UN 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   972
by (blast_tac (claset() addIs [wf_lexn]) 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   973
by (Clarify_tac 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   974
by (rename_tac "m n" 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   975
by (subgoal_tac "m ~= n" 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   976
 by (Blast_tac 2);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   977
by (blast_tac (claset() addDs [lexn_length,not_sym]) 1);
5281
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
   978
qed "wf_lex";
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
   979
AddSIs [wf_lex];
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
   980
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
   981
Goal
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
   982
 "lexn r n = \
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
   983
\ {(xs,ys). length xs = n & length ys = n & \
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
   984
\           (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}";
5318
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   985
by (induct_tac "n" 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   986
 by (Simp_tac 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   987
 by (Blast_tac 1);
5641
5266f09db46c length_Suc_conv is no longer given to AddIffs
paulson
parents: 5537
diff changeset
   988
by (asm_full_simp_tac (simpset() 
5296
bdef7d349d27 added length_Suc_conv, finite_set
oheimb
parents: 5283
diff changeset
   989
				addsimps [lex_prod_def]) 1);
5641
5266f09db46c length_Suc_conv is no longer given to AddIffs
paulson
parents: 5537
diff changeset
   990
by (auto_tac (claset(), simpset()));
5318
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   991
  by (Blast_tac 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   992
 by (rename_tac "a xys x xs' y ys'" 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   993
 by (res_inst_tac [("x","a#xys")] exI 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   994
 by (Simp_tac 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   995
by (exhaust_tac "xys" 1);
5641
5266f09db46c length_Suc_conv is no longer given to AddIffs
paulson
parents: 5537
diff changeset
   996
 by (ALLGOALS (asm_full_simp_tac (simpset())));
5318
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   997
by (Blast_tac 1);
5281
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
   998
qed "lexn_conv";
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
   999
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
  1000
Goalw [lex_def]
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
  1001
 "lex r = \
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
  1002
\ {(xs,ys). length xs = length ys & \
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
  1003
\           (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}";
5641
5266f09db46c length_Suc_conv is no longer given to AddIffs
paulson
parents: 5537
diff changeset
  1004
by (force_tac (claset(), simpset() addsimps [lexn_conv]) 1);
5281
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
  1005
qed "lex_conv";
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
  1006
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
  1007
Goalw [lexico_def] "wf r ==> wf(lexico r)";
5318
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
  1008
by (Blast_tac 1);
5281
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
  1009
qed "wf_lexico";
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
  1010
AddSIs [wf_lexico];
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
  1011
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
  1012
Goalw
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
  1013
 [lexico_def,diag_def,lex_prod_def,measure_def,inv_image_def]
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
  1014
"lexico r = {(xs,ys). length xs < length ys | \
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
  1015
\                     length xs = length ys & (xs,ys) : lex r}";
5318
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
  1016
by (Simp_tac 1);
5281
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
  1017
qed "lexico_conv";
f4d16517b360 List now contains some lexicographic orderings.
nipkow
parents: 5278
diff changeset
  1018
5283
0027ddfbc831 More lemmas about lex.
nipkow
parents: 5281
diff changeset
  1019
Goal "([],ys) ~: lex r";
5318
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
  1020
by (simp_tac (simpset() addsimps [lex_conv]) 1);
5283
0027ddfbc831 More lemmas about lex.
nipkow
parents: 5281
diff changeset
  1021
qed "Nil_notin_lex";
0027ddfbc831 More lemmas about lex.
nipkow
parents: 5281
diff changeset
  1022
0027ddfbc831 More lemmas about lex.
nipkow
parents: 5281
diff changeset
  1023
Goal "(xs,[]) ~: lex r";
5318
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
  1024
by (simp_tac (simpset() addsimps [lex_conv]) 1);
5283
0027ddfbc831 More lemmas about lex.
nipkow
parents: 5281
diff changeset
  1025
qed "Nil2_notin_lex";
0027ddfbc831 More lemmas about lex.
nipkow
parents: 5281
diff changeset
  1026
0027ddfbc831 More lemmas about lex.
nipkow
parents: 5281
diff changeset
  1027
AddIffs [Nil_notin_lex,Nil2_notin_lex];
0027ddfbc831 More lemmas about lex.
nipkow
parents: 5281
diff changeset
  1028
0027ddfbc831 More lemmas about lex.
nipkow
parents: 5281
diff changeset
  1029
Goal "((x#xs,y#ys) : lex r) = \
0027ddfbc831 More lemmas about lex.
nipkow
parents: 5281
diff changeset
  1030
\     ((x,y) : r & length xs = length ys | x=y & (xs,ys) : lex r)";
5318
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
  1031
by (simp_tac (simpset() addsimps [lex_conv]) 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
  1032
by (rtac iffI 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
  1033
 by (blast_tac (claset() addIs [Cons_eq_appendI]) 2);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
  1034
by (REPEAT(eresolve_tac [conjE, exE] 1));
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
  1035
by (exhaust_tac "xys" 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
  1036
by (Asm_full_simp_tac 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
  1037
by (Asm_full_simp_tac 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
  1038
by (Blast_tac 1);
5283
0027ddfbc831 More lemmas about lex.
nipkow
parents: 5281
diff changeset
  1039
qed "Cons_in_lex";
0027ddfbc831 More lemmas about lex.
nipkow
parents: 5281
diff changeset
  1040
AddIffs [Cons_in_lex];