src/HOL/List.ML
author nipkow
Thu Nov 11 11:43:14 1999 +0100 (1999-11-11)
changeset 8009 29a7a79ee7f4
parent 7570 a9391550eea1
child 8064 357652a08ee0
permissions -rw-r--r--
Imported Conny's lemmas from MicroJava
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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 "length(tl xs) = (length xs) - 1";
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by (exhaust_tac "xs" 1);
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by Auto_tac;
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qed "length_tl";
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Addsimps [length_tl];
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Goal "(length xs = 0) = (xs = [])";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "length_0_conv";
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AddIffs [length_0_conv];
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Goal "(0 = length xs) = (xs = [])";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "zero_length_conv";
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AddIffs [zero_length_conv];
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Goal "(0 < length xs) = (xs ~= [])";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "length_greater_0_conv";
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AddIffs [length_greater_0_conv];
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Goal "(length xs = Suc n) = (? y ys. xs = y#ys & length ys = n)";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "length_Suc_conv";
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(** @ - append **)
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section "@ - append";
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Goal "(xs@ys)@zs = xs@(ys@zs)";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "append_assoc";
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Addsimps [append_assoc];
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Goal "xs @ [] = xs";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "append_Nil2";
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Addsimps [append_Nil2];
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Goal "(xs@ys = []) = (xs=[] & ys=[])";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "append_is_Nil_conv";
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AddIffs [append_is_Nil_conv];
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Goal "([] = xs@ys) = (xs=[] & ys=[])";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "Nil_is_append_conv";
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AddIffs [Nil_is_append_conv];
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Goal "(xs @ ys = xs) = (ys=[])";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "append_self_conv";
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Goal "(xs = xs @ ys) = (ys=[])";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "self_append_conv";
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AddIffs [append_self_conv,self_append_conv];
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Goal "!ys. length xs = length ys | length us = length vs \
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\              --> (xs@us = ys@vs) = (xs=ys & us=vs)";
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by (induct_tac "xs" 1);
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 by (rtac allI 1);
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 by (exhaust_tac "ys" 1);
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  by (Asm_simp_tac 1);
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 by (Force_tac 1);
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by (rtac allI 1);
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by (exhaust_tac "ys" 1);
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by (Force_tac 1);
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by (Asm_simp_tac 1);
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qed_spec_mp "append_eq_append_conv";
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Addsimps [append_eq_append_conv];
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Goal "(xs @ ys = xs @ zs) = (ys=zs)";
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by (Simp_tac 1);
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qed "same_append_eq";
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Goal "(xs @ [x] = ys @ [y]) = (xs = ys & x = y)"; 
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by (Simp_tac 1);
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qed "append1_eq_conv";
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Goal "(ys @ xs = zs @ xs) = (ys=zs)";
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by (Simp_tac 1);
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qed "append_same_eq";
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AddSIs
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 [same_append_eq RS iffD2, append1_eq_conv RS iffD2, append_same_eq RS iffD2];
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AddSDs
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 [same_append_eq RS iffD1, append1_eq_conv RS iffD1, append_same_eq RS iffD1];
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Goal "(xs @ ys = ys) = (xs=[])";
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by (cut_inst_tac [("zs","[]")] append_same_eq 1);
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by Auto_tac;
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qed "append_self_conv2";
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Goal "(ys = xs @ ys) = (xs=[])";
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by (simp_tac (simpset() addsimps
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     [simplify (simpset()) (read_instantiate[("ys","[]")]append_same_eq)]) 1);
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by (Blast_tac 1);
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qed "self_append_conv2";
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AddIffs [append_self_conv2,self_append_conv2];
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Goal "xs ~= [] --> hd xs # tl xs = xs";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed_spec_mp "hd_Cons_tl";
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Addsimps [hd_Cons_tl];
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Goal "hd(xs@ys) = (if xs=[] then hd ys else hd xs)";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "hd_append";
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Goal "xs ~= [] ==> hd(xs @ ys) = hd xs";
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by (asm_simp_tac (simpset() addsimps [hd_append]
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                           addsplits [list.split]) 1);
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qed "hd_append2";
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Addsimps [hd_append2];
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Goal "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)";
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by (simp_tac (simpset() addsplits [list.split]) 1);
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qed "tl_append";
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Goal "xs ~= [] ==> tl(xs @ ys) = (tl xs) @ ys";
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by (asm_simp_tac (simpset() addsimps [tl_append]
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                           addsplits [list.split]) 1);
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qed "tl_append2";
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Addsimps [tl_append2];
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(* trivial rules for solving @-equations automatically *)
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Goal "xs = ys ==> xs = [] @ ys";
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by (Asm_simp_tac 1);
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qed "eq_Nil_appendI";
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Goal "[| x#xs1 = ys; xs = xs1 @ zs |] ==> x#xs = ys@zs";
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by (dtac sym 1);
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by (Asm_simp_tac 1);
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qed "Cons_eq_appendI";
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Goal "[| xs@xs1 = zs; ys = xs1 @ us |] ==> xs@ys = zs@us";
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by (dtac sym 1);
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by (Asm_simp_tac 1);
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qed "append_eq_appendI";
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(***
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Simplification procedure for all list equalities.
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Currently only tries to rearranges @ to see if
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- both lists end in a singleton list,
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- or both lists end in the same list.
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***)
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local
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val list_eq_pattern =
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  Thm.read_cterm (Theory.sign_of List.thy) ("(xs::'a list) = ys",HOLogic.boolT);
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fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
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      (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
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  | last (Const("List.op @",_) $ _ $ ys) = last ys
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  | last t = t;
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fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
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  | list1 _ = false;
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fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
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      (case xs of Const("List.list.Nil",_) => 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.Nil",fastype_of xs);
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val rearr_tac =
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  simp_tac (HOL_basic_ss addsimps [append_assoc,append_Nil,append_Cons]);
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fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
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  let
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    val lastl = last lhs and lastr = last rhs
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    fun rearr conv =
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      let val lhs1 = butlast lhs and rhs1 = butlast rhs
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          val Type(_,listT::_) = eqT
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          val appT = [listT,listT] ---> listT
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          val app = Const("List.op @",appT)
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          val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
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          val ct = cterm_of sg (HOLogic.mk_Trueprop(HOLogic.mk_eq(F,F2)))
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          val thm = prove_goalw_cterm [] ct (K [rearr_tac 1])
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            handle ERROR =>
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            error("The error(s) above occurred while trying to prove " ^
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                  string_of_cterm ct)
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      in Some((conv RS (thm RS trans)) RS eq_reflection) end
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  in if list1 lastl andalso list1 lastr
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     then rearr append1_eq_conv
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     else
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     if lastl aconv lastr
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     then rearr append_same_eq
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     else None
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  end;
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in
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val list_eq_simproc = mk_simproc "list_eq" [list_eq_pattern] list_eq;
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end;
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Addsimprocs [list_eq_simproc];
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(** map **)
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section "map";
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Goal "(!x. x : set xs --> f x = g x) --> map f xs = map g xs";
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by (induct_tac "xs" 1);
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by Auto_tac;
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bind_thm("map_ext", impI RS (allI RS (result() RS mp)));
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Goal "map (%x. x) = (%xs. xs)";
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by (rtac ext 1);
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "map_ident";
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Addsimps[map_ident];
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Goal "map f (xs@ys) = map f xs @ map f ys";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "map_append";
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Addsimps[map_append];
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Goalw [o_def] "map (f o g) xs = map f (map g xs)";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "map_compose";
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Addsimps[map_compose];
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Goal "rev(map f xs) = map f (rev xs)";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "rev_map";
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(* a congruence rule for map: *)
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Goal "xs=ys ==> (!x. x : set ys --> f x = g x) --> map f xs = map g ys";
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by (hyp_subst_tac 1);
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by (induct_tac "ys" 1);
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by Auto_tac;
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bind_thm("map_cong", impI RSN (2,allI RSN (2, result() RS mp)));
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Goal "(map f xs = []) = (xs = [])";
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by (exhaust_tac "xs" 1);
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by Auto_tac;
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qed "map_is_Nil_conv";
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AddIffs [map_is_Nil_conv];
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nipkow@4935
   348
Goal "([] = map f xs) = (xs = [])";
nipkow@8009
   349
by (exhaust_tac "xs" 1);
paulson@5316
   350
by Auto_tac;
nipkow@3860
   351
qed "Nil_is_map_conv";
nipkow@3860
   352
AddIffs [Nil_is_map_conv];
nipkow@3860
   353
nipkow@8009
   354
Goal "(map f xs = y#ys) = (? x xs'. xs = x#xs' & f x = y & map f xs' = ys)";
nipkow@8009
   355
by (exhaust_tac "xs" 1);
nipkow@8009
   356
by (ALLGOALS Asm_simp_tac);
nipkow@8009
   357
qed "map_eq_Cons";
nipkow@8009
   358
nipkow@8009
   359
Goal "!xs. map f xs = map f ys --> (!x y. f x = f y --> x=y) --> xs=ys";
nipkow@8009
   360
by (induct_tac "ys" 1);
nipkow@8009
   361
 by (Asm_simp_tac 1);
nipkow@8009
   362
by (fast_tac (claset() addss (simpset() addsimps [map_eq_Cons])) 1);
nipkow@8009
   363
qed_spec_mp "map_injective";
nipkow@8009
   364
nipkow@8009
   365
Goal "inj f ==> inj (map f)";
nipkow@8009
   366
by(blast_tac (claset() addDs [map_injective,injD] addIs [injI]) 1);
nipkow@8009
   367
qed "inj_mapI";
nipkow@8009
   368
nipkow@8009
   369
Goalw [inj_on_def] "inj (map f) ==> inj f";
nipkow@8009
   370
by(Clarify_tac 1);
nipkow@8009
   371
by(eres_inst_tac [("x","[x]")] ballE 1);
nipkow@8009
   372
 by(eres_inst_tac [("x","[y]")] ballE 1);
nipkow@8009
   373
  by(Asm_full_simp_tac 1);
nipkow@8009
   374
 by(Blast_tac 1);
nipkow@8009
   375
by(Blast_tac 1);
nipkow@8009
   376
qed "inj_mapD";
nipkow@8009
   377
nipkow@8009
   378
Goal "inj (map f) = inj f";
nipkow@8009
   379
by(blast_tac (claset() addDs [inj_mapD] addIs [inj_mapI]) 1);
nipkow@8009
   380
qed "inj_map";
nipkow@3860
   381
lcp@1169
   382
(** rev **)
lcp@1169
   383
nipkow@3467
   384
section "rev";
nipkow@3467
   385
nipkow@4935
   386
Goal "rev(xs@ys) = rev(ys) @ rev(xs)";
nipkow@3040
   387
by (induct_tac "xs" 1);
paulson@5316
   388
by Auto_tac;
lcp@1169
   389
qed "rev_append";
nipkow@2512
   390
Addsimps[rev_append];
lcp@1169
   391
nipkow@4935
   392
Goal "rev(rev l) = l";
nipkow@3040
   393
by (induct_tac "l" 1);
paulson@5316
   394
by Auto_tac;
lcp@1169
   395
qed "rev_rev_ident";
nipkow@2512
   396
Addsimps[rev_rev_ident];
lcp@1169
   397
nipkow@4935
   398
Goal "(rev xs = []) = (xs = [])";
wenzelm@4423
   399
by (induct_tac "xs" 1);
paulson@5316
   400
by Auto_tac;
nipkow@3860
   401
qed "rev_is_Nil_conv";
nipkow@3860
   402
AddIffs [rev_is_Nil_conv];
nipkow@3860
   403
nipkow@4935
   404
Goal "([] = rev xs) = (xs = [])";
wenzelm@4423
   405
by (induct_tac "xs" 1);
paulson@5316
   406
by Auto_tac;
nipkow@3860
   407
qed "Nil_is_rev_conv";
nipkow@3860
   408
AddIffs [Nil_is_rev_conv];
nipkow@3860
   409
nipkow@6820
   410
Goal "!ys. (rev xs = rev ys) = (xs = ys)";
paulson@6831
   411
by (induct_tac "xs" 1);
nipkow@6820
   412
 by (Force_tac 1);
paulson@6831
   413
by (rtac allI 1);
paulson@6831
   414
by (exhaust_tac "ys" 1);
nipkow@6820
   415
 by (Asm_simp_tac 1);
nipkow@6820
   416
by (Force_tac 1);
nipkow@6820
   417
qed_spec_mp "rev_is_rev_conv";
nipkow@6820
   418
AddIffs [rev_is_rev_conv];
nipkow@6820
   419
nipkow@4935
   420
val prems = Goal "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P xs";
wenzelm@5132
   421
by (stac (rev_rev_ident RS sym) 1);
paulson@6162
   422
by (res_inst_tac [("list", "rev xs")] list.induct 1);
wenzelm@5132
   423
by (ALLGOALS Simp_tac);
wenzelm@5132
   424
by (resolve_tac prems 1);
wenzelm@5132
   425
by (eresolve_tac prems 1);
nipkow@4935
   426
qed "rev_induct";
nipkow@4935
   427
nipkow@5272
   428
fun rev_induct_tac xs = res_inst_tac [("xs",xs)] rev_induct;
nipkow@5272
   429
nipkow@4935
   430
Goal  "(xs = [] --> P) -->  (!ys y. xs = ys@[y] --> P) --> P";
wenzelm@5132
   431
by (res_inst_tac [("xs","xs")] rev_induct 1);
paulson@5316
   432
by Auto_tac;
nipkow@4935
   433
bind_thm ("rev_exhaust",
nipkow@4935
   434
  impI RSN (2,allI RSN (2,allI RSN (2,impI RS (result() RS mp RS mp)))));
nipkow@4935
   435
nipkow@2608
   436
nipkow@3465
   437
(** set **)
paulson@1812
   438
nipkow@3467
   439
section "set";
nipkow@3467
   440
paulson@7032
   441
Goal "finite (set xs)";
paulson@7032
   442
by (induct_tac "xs" 1);
paulson@7032
   443
by Auto_tac;
paulson@7032
   444
qed "finite_set";
paulson@7032
   445
AddIffs [finite_set];
oheimb@5296
   446
nipkow@4935
   447
Goal "set (xs@ys) = (set xs Un set ys)";
nipkow@3040
   448
by (induct_tac "xs" 1);
paulson@5316
   449
by Auto_tac;
paulson@3647
   450
qed "set_append";
paulson@3647
   451
Addsimps[set_append];
paulson@1812
   452
nipkow@4935
   453
Goal "set l <= set (x#l)";
paulson@5316
   454
by Auto_tac;
paulson@3647
   455
qed "set_subset_Cons";
paulson@1936
   456
nipkow@4935
   457
Goal "(set xs = {}) = (xs = [])";
paulson@3457
   458
by (induct_tac "xs" 1);
paulson@5316
   459
by Auto_tac;
paulson@3647
   460
qed "set_empty";
paulson@3647
   461
Addsimps [set_empty];
nipkow@2608
   462
nipkow@4935
   463
Goal "set(rev xs) = set(xs)";
paulson@3457
   464
by (induct_tac "xs" 1);
paulson@5316
   465
by Auto_tac;
paulson@3647
   466
qed "set_rev";
paulson@3647
   467
Addsimps [set_rev];
nipkow@2608
   468
nipkow@4935
   469
Goal "set(map f xs) = f``(set xs)";
paulson@3457
   470
by (induct_tac "xs" 1);
paulson@5316
   471
by Auto_tac;
paulson@3647
   472
qed "set_map";
paulson@3647
   473
Addsimps [set_map];
nipkow@2608
   474
nipkow@6433
   475
Goal "set(filter P xs) = {x. x : set xs & P x}";
paulson@6813
   476
by (induct_tac "xs" 1);
paulson@6813
   477
by Auto_tac;
nipkow@6433
   478
qed "set_filter";
nipkow@6433
   479
Addsimps [set_filter];
nipkow@8009
   480
nipkow@6433
   481
Goal "set[i..j(] = {k. i <= k & k < j}";
paulson@6813
   482
by (induct_tac "j" 1);
paulson@6813
   483
by Auto_tac;
paulson@6813
   484
by (arith_tac 1);
nipkow@6433
   485
qed "set_upt";
nipkow@6433
   486
Addsimps [set_upt];
nipkow@6433
   487
nipkow@6433
   488
Goal "!i < size xs. set(xs[i := x]) <= insert x (set xs)";
paulson@6813
   489
by (induct_tac "xs" 1);
paulson@6813
   490
 by (Simp_tac 1);
paulson@6813
   491
by (asm_simp_tac (simpset() addsplits [nat.split]) 1);
paulson@6813
   492
by (Blast_tac 1);
nipkow@6433
   493
qed_spec_mp "set_list_update_subset";
nipkow@4605
   494
nipkow@5272
   495
Goal "(x : set xs) = (? ys zs. xs = ys@x#zs)";
paulson@5318
   496
by (induct_tac "xs" 1);
paulson@5318
   497
 by (Simp_tac 1);
paulson@5318
   498
by (Asm_simp_tac 1);
paulson@5318
   499
by (rtac iffI 1);
paulson@5318
   500
by (blast_tac (claset() addIs [eq_Nil_appendI,Cons_eq_appendI]) 1);
paulson@5318
   501
by (REPEAT(etac exE 1));
paulson@5318
   502
by (exhaust_tac "ys" 1);
paulson@5316
   503
by Auto_tac;
nipkow@5272
   504
qed "in_set_conv_decomp";
nipkow@5272
   505
nipkow@8009
   506
nipkow@5272
   507
(* eliminate `lists' in favour of `set' *)
nipkow@5272
   508
nipkow@5272
   509
Goal "(xs : lists A) = (!x : set xs. x : A)";
paulson@5318
   510
by (induct_tac "xs" 1);
paulson@5316
   511
by Auto_tac;
nipkow@5272
   512
qed "in_lists_conv_set";
nipkow@5272
   513
nipkow@5272
   514
bind_thm("in_listsD",in_lists_conv_set RS iffD1);
nipkow@5272
   515
AddSDs [in_listsD];
nipkow@5272
   516
bind_thm("in_listsI",in_lists_conv_set RS iffD2);
nipkow@5272
   517
AddSIs [in_listsI];
paulson@1812
   518
oheimb@5518
   519
(** mem **)
oheimb@5518
   520
 
oheimb@5518
   521
section "mem";
oheimb@5518
   522
oheimb@5518
   523
Goal "(x mem xs) = (x: set xs)";
oheimb@5518
   524
by (induct_tac "xs" 1);
oheimb@5518
   525
by Auto_tac;
oheimb@5518
   526
qed "set_mem_eq";
oheimb@5518
   527
oheimb@5518
   528
clasohm@923
   529
(** list_all **)
clasohm@923
   530
nipkow@3467
   531
section "list_all";
nipkow@3467
   532
oheimb@5518
   533
Goal "list_all P xs = (!x:set xs. P x)";
oheimb@5518
   534
by (induct_tac "xs" 1);
oheimb@5518
   535
by Auto_tac;
oheimb@5518
   536
qed "list_all_conv";
oheimb@5518
   537
oheimb@5443
   538
Goal "list_all P (xs@ys) = (list_all P xs & list_all P ys)";
nipkow@3040
   539
by (induct_tac "xs" 1);
paulson@5316
   540
by Auto_tac;
nipkow@2512
   541
qed "list_all_append";
nipkow@2512
   542
Addsimps [list_all_append];
clasohm@923
   543
clasohm@923
   544
nipkow@2608
   545
(** filter **)
clasohm@923
   546
nipkow@3467
   547
section "filter";
nipkow@3467
   548
nipkow@4935
   549
Goal "filter P (xs@ys) = filter P xs @ filter P ys";
paulson@3457
   550
by (induct_tac "xs" 1);
paulson@5316
   551
by Auto_tac;
nipkow@2608
   552
qed "filter_append";
nipkow@2608
   553
Addsimps [filter_append];
nipkow@2608
   554
nipkow@4935
   555
Goal "filter (%x. True) xs = xs";
nipkow@4605
   556
by (induct_tac "xs" 1);
paulson@5316
   557
by Auto_tac;
nipkow@4605
   558
qed "filter_True";
nipkow@4605
   559
Addsimps [filter_True];
nipkow@4605
   560
nipkow@4935
   561
Goal "filter (%x. False) xs = []";
nipkow@4605
   562
by (induct_tac "xs" 1);
paulson@5316
   563
by Auto_tac;
nipkow@4605
   564
qed "filter_False";
nipkow@4605
   565
Addsimps [filter_False];
nipkow@4605
   566
nipkow@4935
   567
Goal "length (filter P xs) <= length xs";
paulson@3457
   568
by (induct_tac "xs" 1);
paulson@5316
   569
by Auto_tac;
nipkow@4605
   570
qed "length_filter";
oheimb@5443
   571
Addsimps[length_filter];
nipkow@2608
   572
oheimb@5443
   573
Goal "set (filter P xs) <= set xs";
oheimb@5443
   574
by Auto_tac;
oheimb@5443
   575
qed "filter_is_subset";
oheimb@5443
   576
Addsimps [filter_is_subset];
oheimb@5443
   577
nipkow@2608
   578
nipkow@3467
   579
section "concat";
nipkow@3467
   580
nipkow@4935
   581
Goal  "concat(xs@ys) = concat(xs)@concat(ys)";
nipkow@3040
   582
by (induct_tac "xs" 1);
paulson@5316
   583
by Auto_tac;
nipkow@2608
   584
qed"concat_append";
nipkow@2608
   585
Addsimps [concat_append];
nipkow@2512
   586
nipkow@4935
   587
Goal "(concat xss = []) = (!xs:set xss. xs=[])";
wenzelm@4423
   588
by (induct_tac "xss" 1);
paulson@5316
   589
by Auto_tac;
nipkow@3896
   590
qed "concat_eq_Nil_conv";
nipkow@3896
   591
AddIffs [concat_eq_Nil_conv];
nipkow@3896
   592
nipkow@4935
   593
Goal "([] = concat xss) = (!xs:set xss. xs=[])";
wenzelm@4423
   594
by (induct_tac "xss" 1);
paulson@5316
   595
by Auto_tac;
nipkow@3896
   596
qed "Nil_eq_concat_conv";
nipkow@3896
   597
AddIffs [Nil_eq_concat_conv];
nipkow@3896
   598
nipkow@4935
   599
Goal  "set(concat xs) = Union(set `` set xs)";
nipkow@3467
   600
by (induct_tac "xs" 1);
paulson@5316
   601
by Auto_tac;
paulson@3647
   602
qed"set_concat";
paulson@3647
   603
Addsimps [set_concat];
nipkow@3467
   604
nipkow@4935
   605
Goal "map f (concat xs) = concat (map (map f) xs)"; 
nipkow@3467
   606
by (induct_tac "xs" 1);
paulson@5316
   607
by Auto_tac;
nipkow@3467
   608
qed "map_concat";
nipkow@3467
   609
nipkow@4935
   610
Goal "filter p (concat xs) = concat (map (filter p) xs)"; 
nipkow@3467
   611
by (induct_tac "xs" 1);
paulson@5316
   612
by Auto_tac;
nipkow@3467
   613
qed"filter_concat"; 
nipkow@3467
   614
nipkow@4935
   615
Goal "rev(concat xs) = concat (map rev (rev xs))";
nipkow@3467
   616
by (induct_tac "xs" 1);
paulson@5316
   617
by Auto_tac;
nipkow@2608
   618
qed "rev_concat";
clasohm@923
   619
clasohm@923
   620
(** nth **)
clasohm@923
   621
nipkow@3467
   622
section "nth";
nipkow@3467
   623
pusch@6408
   624
Goal "(x#xs)!0 = x";
pusch@6408
   625
by Auto_tac;
pusch@6408
   626
qed "nth_Cons_0";
pusch@6408
   627
Addsimps [nth_Cons_0];
nipkow@5644
   628
pusch@6408
   629
Goal "(x#xs)!(Suc n) = xs!n";
pusch@6408
   630
by Auto_tac;
pusch@6408
   631
qed "nth_Cons_Suc";
pusch@6408
   632
Addsimps [nth_Cons_Suc];
pusch@6408
   633
pusch@6408
   634
Delsimps (thms "nth.simps");
pusch@6408
   635
pusch@6408
   636
Goal "!n. (xs@ys)!n = (if n < length xs then xs!n else ys!(n - length xs))";
pusch@6408
   637
by (induct_tac "xs" 1);
paulson@3457
   638
 by (Asm_simp_tac 1);
paulson@3457
   639
 by (rtac allI 1);
pusch@6408
   640
 by (exhaust_tac "n" 1);
paulson@5316
   641
  by Auto_tac;
nipkow@2608
   642
qed_spec_mp "nth_append";
nipkow@2608
   643
nipkow@4935
   644
Goal "!n. n < length xs --> (map f xs)!n = f(xs!n)";
nipkow@3040
   645
by (induct_tac "xs" 1);
nipkow@1301
   646
(* case [] *)
nipkow@1301
   647
by (Asm_full_simp_tac 1);
nipkow@1301
   648
(* case x#xl *)
nipkow@1301
   649
by (rtac allI 1);
berghofe@5183
   650
by (induct_tac "n" 1);
paulson@5316
   651
by Auto_tac;
nipkow@1485
   652
qed_spec_mp "nth_map";
nipkow@1301
   653
Addsimps [nth_map];
nipkow@1301
   654
oheimb@5518
   655
Goal "!n. n < length xs --> Ball (set xs) P --> P(xs!n)";
nipkow@3040
   656
by (induct_tac "xs" 1);
nipkow@1301
   657
(* case [] *)
nipkow@1301
   658
by (Simp_tac 1);
nipkow@1301
   659
(* case x#xl *)
nipkow@1301
   660
by (rtac allI 1);
berghofe@5183
   661
by (induct_tac "n" 1);
paulson@5316
   662
by Auto_tac;
oheimb@5518
   663
qed_spec_mp "list_ball_nth";
nipkow@1301
   664
oheimb@5518
   665
Goal "!n. n < length xs --> xs!n : set xs";
nipkow@3040
   666
by (induct_tac "xs" 1);
nipkow@8009
   667
 by (Simp_tac 1);
nipkow@1301
   668
by (rtac allI 1);
berghofe@5183
   669
by (induct_tac "n" 1);
nipkow@8009
   670
 by (Asm_full_simp_tac 1);
nipkow@4686
   671
by (Asm_full_simp_tac 1);
nipkow@1485
   672
qed_spec_mp "nth_mem";
nipkow@1301
   673
Addsimps [nth_mem];
nipkow@1301
   674
oheimb@5518
   675
nipkow@8009
   676
Goal "(!i. i < length xs --> P(xs!i)) --> (!x : set xs. P x)";
nipkow@8009
   677
by (induct_tac "xs" 1);
nipkow@8009
   678
 by (Asm_full_simp_tac 1);
nipkow@8009
   679
by (simp_tac (simpset() addsplits [nat.split] addsimps [nth_Cons]) 1);
nipkow@8009
   680
by (fast_tac (claset() addss simpset()) 1);
nipkow@8009
   681
qed_spec_mp "all_nth_imp_all_set";
nipkow@8009
   682
nipkow@8009
   683
Goal "(!x : set xs. P x) = (!i. i<length xs --> P (xs ! i))";
nipkow@8009
   684
br iffI 1;
nipkow@8009
   685
 by (Asm_full_simp_tac 1);
nipkow@8009
   686
be all_nth_imp_all_set 1;
nipkow@8009
   687
qed_spec_mp "all_set_conv_all_nth";
nipkow@8009
   688
nipkow@8009
   689
nipkow@5077
   690
(** list update **)
nipkow@5077
   691
nipkow@5077
   692
section "list update";
nipkow@5077
   693
nipkow@5077
   694
Goal "!i. length(xs[i:=x]) = length xs";
nipkow@5077
   695
by (induct_tac "xs" 1);
nipkow@5077
   696
by (Simp_tac 1);
berghofe@5183
   697
by (asm_full_simp_tac (simpset() addsplits [nat.split]) 1);
nipkow@5077
   698
qed_spec_mp "length_list_update";
nipkow@5077
   699
Addsimps [length_list_update];
nipkow@5077
   700
nipkow@5644
   701
Goal "!i j. i < length xs  --> (xs[i:=x])!j = (if i=j then x else xs!j)";
paulson@6162
   702
by (induct_tac "xs" 1);
paulson@6162
   703
 by (Simp_tac 1);
paulson@6162
   704
by (auto_tac (claset(), simpset() addsimps [nth_Cons] addsplits [nat.split]));
nipkow@5644
   705
qed_spec_mp "nth_list_update";
nipkow@5644
   706
nipkow@6433
   707
Goal "!i. i < size xs --> xs[i:=x, i:=y] = xs[i:=y]";
paulson@6813
   708
by (induct_tac "xs" 1);
paulson@6813
   709
 by (Simp_tac 1);
paulson@6813
   710
by (asm_simp_tac (simpset() addsplits [nat.split]) 1);
nipkow@6433
   711
qed_spec_mp "list_update_overwrite";
nipkow@6433
   712
Addsimps [list_update_overwrite];
nipkow@6433
   713
nipkow@6433
   714
Goal "!i < length xs. (xs[i := x] = xs) = (xs!i = x)";
paulson@6813
   715
by (induct_tac "xs" 1);
paulson@6813
   716
 by (Simp_tac 1);
paulson@6813
   717
by (simp_tac (simpset() addsplits [nat.split]) 1);
paulson@6813
   718
by (Blast_tac 1);
nipkow@6433
   719
qed_spec_mp "list_update_same_conv";
nipkow@6433
   720
nipkow@8009
   721
Goal "!i xy xs. length xs = length ys --> \
nipkow@8009
   722
\     (zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])";
nipkow@8009
   723
by (induct_tac "ys" 1);
nipkow@8009
   724
 by Auto_tac;
nipkow@8009
   725
by (exhaust_tac "xs" 1);
nipkow@8009
   726
 by (auto_tac (claset(), simpset() addsplits [nat.split]));
nipkow@8009
   727
qed_spec_mp "update_zip";
nipkow@8009
   728
nipkow@8009
   729
Goal "!i. set(xs[i:=x]) <= insert x (set xs)";
nipkow@8009
   730
by (induct_tac "xs" 1);
nipkow@8009
   731
 by (asm_full_simp_tac (simpset() addsimps []) 1);
nipkow@8009
   732
by (asm_full_simp_tac (simpset() addsplits [nat.split]) 1);
nipkow@8009
   733
by (Fast_tac  1);
nipkow@8009
   734
qed_spec_mp "set_update_subset";
nipkow@8009
   735
nipkow@5077
   736
nipkow@3896
   737
(** last & butlast **)
nipkow@1327
   738
nipkow@5644
   739
section "last / butlast";
nipkow@5644
   740
nipkow@4935
   741
Goal "last(xs@[x]) = x";
wenzelm@4423
   742
by (induct_tac "xs" 1);
paulson@5316
   743
by Auto_tac;
nipkow@3896
   744
qed "last_snoc";
nipkow@3896
   745
Addsimps [last_snoc];
nipkow@3896
   746
nipkow@4935
   747
Goal "butlast(xs@[x]) = xs";
wenzelm@4423
   748
by (induct_tac "xs" 1);
paulson@5316
   749
by Auto_tac;
nipkow@3896
   750
qed "butlast_snoc";
nipkow@3896
   751
Addsimps [butlast_snoc];
nipkow@3896
   752
nipkow@4935
   753
Goal "length(butlast xs) = length xs - 1";
nipkow@4935
   754
by (res_inst_tac [("xs","xs")] rev_induct 1);
paulson@5316
   755
by Auto_tac;
nipkow@4643
   756
qed "length_butlast";
nipkow@4643
   757
Addsimps [length_butlast];
nipkow@4643
   758
paulson@5278
   759
Goal "!ys. butlast (xs@ys) = (if ys=[] then butlast xs else xs@butlast ys)";
wenzelm@4423
   760
by (induct_tac "xs" 1);
paulson@5316
   761
by Auto_tac;
nipkow@3896
   762
qed_spec_mp "butlast_append";
nipkow@3896
   763
nipkow@4935
   764
Goal "x:set(butlast xs) --> x:set xs";
wenzelm@4423
   765
by (induct_tac "xs" 1);
paulson@5316
   766
by Auto_tac;
nipkow@3896
   767
qed_spec_mp "in_set_butlastD";
nipkow@3896
   768
paulson@5448
   769
Goal "x:set(butlast xs) | x:set(butlast ys) ==> x:set(butlast(xs@ys))";
paulson@5448
   770
by (auto_tac (claset() addDs [in_set_butlastD],
paulson@5448
   771
	      simpset() addsimps [butlast_append]));
paulson@5448
   772
qed "in_set_butlast_appendI";
nipkow@3902
   773
nipkow@2608
   774
(** take  & drop **)
nipkow@2608
   775
section "take & drop";
nipkow@1327
   776
nipkow@4935
   777
Goal "take 0 xs = []";
nipkow@3040
   778
by (induct_tac "xs" 1);
paulson@5316
   779
by Auto_tac;
nipkow@1327
   780
qed "take_0";
nipkow@1327
   781
nipkow@4935
   782
Goal "drop 0 xs = xs";
nipkow@3040
   783
by (induct_tac "xs" 1);
paulson@5316
   784
by Auto_tac;
nipkow@2608
   785
qed "drop_0";
nipkow@2608
   786
nipkow@4935
   787
Goal "take (Suc n) (x#xs) = x # take n xs";
paulson@1552
   788
by (Simp_tac 1);
nipkow@1419
   789
qed "take_Suc_Cons";
nipkow@1327
   790
nipkow@4935
   791
Goal "drop (Suc n) (x#xs) = drop n xs";
nipkow@2608
   792
by (Simp_tac 1);
nipkow@2608
   793
qed "drop_Suc_Cons";
nipkow@2608
   794
nipkow@2608
   795
Delsimps [take_Cons,drop_Cons];
nipkow@2608
   796
Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons];
nipkow@2608
   797
nipkow@4935
   798
Goal "!xs. length(take n xs) = min (length xs) n";
berghofe@5183
   799
by (induct_tac "n" 1);
paulson@5316
   800
 by Auto_tac;
paulson@3457
   801
by (exhaust_tac "xs" 1);
paulson@5316
   802
 by Auto_tac;
nipkow@2608
   803
qed_spec_mp "length_take";
nipkow@2608
   804
Addsimps [length_take];
clasohm@923
   805
nipkow@4935
   806
Goal "!xs. length(drop n xs) = (length xs - n)";
berghofe@5183
   807
by (induct_tac "n" 1);
paulson@5316
   808
 by Auto_tac;
paulson@3457
   809
by (exhaust_tac "xs" 1);
paulson@5316
   810
 by Auto_tac;
nipkow@2608
   811
qed_spec_mp "length_drop";
nipkow@2608
   812
Addsimps [length_drop];
nipkow@2608
   813
nipkow@4935
   814
Goal "!xs. length xs <= n --> take n xs = xs";
berghofe@5183
   815
by (induct_tac "n" 1);
paulson@5316
   816
 by Auto_tac;
paulson@3457
   817
by (exhaust_tac "xs" 1);
paulson@5316
   818
 by Auto_tac;
nipkow@2608
   819
qed_spec_mp "take_all";
nipkow@7246
   820
Addsimps [take_all];
clasohm@923
   821
nipkow@4935
   822
Goal "!xs. length xs <= n --> drop n xs = []";
berghofe@5183
   823
by (induct_tac "n" 1);
paulson@5316
   824
 by Auto_tac;
paulson@3457
   825
by (exhaust_tac "xs" 1);
paulson@5316
   826
 by Auto_tac;
nipkow@2608
   827
qed_spec_mp "drop_all";
nipkow@7246
   828
Addsimps [drop_all];
nipkow@2608
   829
paulson@5278
   830
Goal "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)";
berghofe@5183
   831
by (induct_tac "n" 1);
paulson@5316
   832
 by Auto_tac;
paulson@3457
   833
by (exhaust_tac "xs" 1);
paulson@5316
   834
 by Auto_tac;
nipkow@2608
   835
qed_spec_mp "take_append";
nipkow@2608
   836
Addsimps [take_append];
nipkow@2608
   837
nipkow@4935
   838
Goal "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; 
berghofe@5183
   839
by (induct_tac "n" 1);
paulson@5316
   840
 by Auto_tac;
paulson@3457
   841
by (exhaust_tac "xs" 1);
paulson@5316
   842
 by Auto_tac;
nipkow@2608
   843
qed_spec_mp "drop_append";
nipkow@2608
   844
Addsimps [drop_append];
nipkow@2608
   845
nipkow@4935
   846
Goal "!xs n. take n (take m xs) = take (min n m) xs"; 
berghofe@5183
   847
by (induct_tac "m" 1);
paulson@5316
   848
 by Auto_tac;
paulson@3457
   849
by (exhaust_tac "xs" 1);
paulson@5316
   850
 by Auto_tac;
berghofe@5183
   851
by (exhaust_tac "na" 1);
paulson@5316
   852
 by Auto_tac;
nipkow@2608
   853
qed_spec_mp "take_take";
nipkow@7570
   854
Addsimps [take_take];
nipkow@2608
   855
nipkow@4935
   856
Goal "!xs. drop n (drop m xs) = drop (n + m) xs"; 
berghofe@5183
   857
by (induct_tac "m" 1);
paulson@5316
   858
 by Auto_tac;
paulson@3457
   859
by (exhaust_tac "xs" 1);
paulson@5316
   860
 by Auto_tac;
nipkow@2608
   861
qed_spec_mp "drop_drop";
nipkow@7570
   862
Addsimps [drop_drop];
clasohm@923
   863
nipkow@4935
   864
Goal "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; 
berghofe@5183
   865
by (induct_tac "m" 1);
paulson@5316
   866
 by Auto_tac;
paulson@3457
   867
by (exhaust_tac "xs" 1);
paulson@5316
   868
 by Auto_tac;
nipkow@2608
   869
qed_spec_mp "take_drop";
nipkow@2608
   870
paulson@6813
   871
Goal "!xs. take n xs @ drop n xs = xs";
paulson@6813
   872
by (induct_tac "n" 1);
paulson@6813
   873
 by Auto_tac;
paulson@6813
   874
by (exhaust_tac "xs" 1);
paulson@6813
   875
 by Auto_tac;
paulson@6813
   876
qed_spec_mp "append_take_drop_id";
paulson@6813
   877
nipkow@4935
   878
Goal "!xs. take n (map f xs) = map f (take n xs)"; 
berghofe@5183
   879
by (induct_tac "n" 1);
paulson@5316
   880
 by Auto_tac;
paulson@3457
   881
by (exhaust_tac "xs" 1);
paulson@5316
   882
 by Auto_tac;
nipkow@2608
   883
qed_spec_mp "take_map"; 
nipkow@2608
   884
nipkow@4935
   885
Goal "!xs. drop n (map f xs) = map f (drop n xs)"; 
berghofe@5183
   886
by (induct_tac "n" 1);
paulson@5316
   887
 by Auto_tac;
paulson@3457
   888
by (exhaust_tac "xs" 1);
paulson@5316
   889
 by Auto_tac;
nipkow@2608
   890
qed_spec_mp "drop_map";
nipkow@2608
   891
nipkow@4935
   892
Goal "!n i. i < n --> (take n xs)!i = xs!i";
paulson@3457
   893
by (induct_tac "xs" 1);
paulson@5316
   894
 by Auto_tac;
paulson@3457
   895
by (exhaust_tac "n" 1);
paulson@3457
   896
 by (Blast_tac 1);
paulson@3457
   897
by (exhaust_tac "i" 1);
paulson@5316
   898
 by Auto_tac;
nipkow@2608
   899
qed_spec_mp "nth_take";
nipkow@2608
   900
Addsimps [nth_take];
clasohm@923
   901
nipkow@4935
   902
Goal  "!xs i. n + i <= length xs --> (drop n xs)!i = xs!(n+i)";
berghofe@5183
   903
by (induct_tac "n" 1);
paulson@5316
   904
 by Auto_tac;
paulson@3457
   905
by (exhaust_tac "xs" 1);
paulson@5316
   906
 by Auto_tac;
nipkow@2608
   907
qed_spec_mp "nth_drop";
nipkow@2608
   908
Addsimps [nth_drop];
nipkow@2608
   909
nipkow@2608
   910
(** takeWhile & dropWhile **)
nipkow@2608
   911
nipkow@3467
   912
section "takeWhile & dropWhile";
nipkow@3467
   913
nipkow@4935
   914
Goal "takeWhile P xs @ dropWhile P xs = xs";
nipkow@3586
   915
by (induct_tac "xs" 1);
paulson@5316
   916
by Auto_tac;
nipkow@3586
   917
qed "takeWhile_dropWhile_id";
nipkow@3586
   918
Addsimps [takeWhile_dropWhile_id];
nipkow@3586
   919
nipkow@4935
   920
Goal  "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs";
paulson@3457
   921
by (induct_tac "xs" 1);
paulson@5316
   922
by Auto_tac;
nipkow@2608
   923
bind_thm("takeWhile_append1", conjI RS (result() RS mp));
nipkow@2608
   924
Addsimps [takeWhile_append1];
clasohm@923
   925
nipkow@4935
   926
Goal "(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys";
paulson@3457
   927
by (induct_tac "xs" 1);
paulson@5316
   928
by Auto_tac;
nipkow@2608
   929
bind_thm("takeWhile_append2", ballI RS (result() RS mp));
nipkow@2608
   930
Addsimps [takeWhile_append2];
lcp@1169
   931
nipkow@4935
   932
Goal "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys";
paulson@3457
   933
by (induct_tac "xs" 1);
paulson@5316
   934
by Auto_tac;
nipkow@2608
   935
bind_thm("dropWhile_append1", conjI RS (result() RS mp));
nipkow@2608
   936
Addsimps [dropWhile_append1];
nipkow@2608
   937
nipkow@4935
   938
Goal "(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys";
paulson@3457
   939
by (induct_tac "xs" 1);
paulson@5316
   940
by Auto_tac;
nipkow@2608
   941
bind_thm("dropWhile_append2", ballI RS (result() RS mp));
nipkow@2608
   942
Addsimps [dropWhile_append2];
nipkow@2608
   943
nipkow@4935
   944
Goal "x:set(takeWhile P xs) --> x:set xs & P x";
paulson@3457
   945
by (induct_tac "xs" 1);
paulson@5316
   946
by Auto_tac;
paulson@3647
   947
qed_spec_mp"set_take_whileD";
nipkow@2608
   948
nipkow@6306
   949
(** zip **)
nipkow@6306
   950
section "zip";
nipkow@6306
   951
nipkow@6306
   952
Goal "zip [] ys = []";
paulson@6813
   953
by (induct_tac "ys" 1);
nipkow@6306
   954
by Auto_tac;
nipkow@6306
   955
qed "zip_Nil";
nipkow@6306
   956
Addsimps [zip_Nil];
nipkow@6306
   957
nipkow@6306
   958
Goal "zip (x#xs) (y#ys) = (x,y)#zip xs ys";
paulson@6813
   959
by (Simp_tac 1);
nipkow@6306
   960
qed "zip_Cons_Cons";
nipkow@6306
   961
Addsimps [zip_Cons_Cons];
nipkow@6306
   962
nipkow@6306
   963
Delsimps(tl (thms"zip.simps"));
nipkow@4605
   964
nipkow@8009
   965
Goal "!xs. length xs = length ys --> length (zip xs ys) = length ys";
nipkow@8009
   966
by (induct_tac "ys" 1);
nipkow@8009
   967
 by (Simp_tac 1);
nipkow@8009
   968
by (Clarify_tac 1);
nipkow@8009
   969
by (exhaust_tac "xs" 1);
nipkow@8009
   970
 by(Auto_tac);
nipkow@8009
   971
qed_spec_mp "length_zip";
nipkow@8009
   972
Addsimps [length_zip];
nipkow@8009
   973
nipkow@8009
   974
Goal
nipkow@8009
   975
"!xs. length xs = length us --> length ys = length vs --> \
nipkow@8009
   976
\     zip (xs@ys) (us@vs) = zip xs us @ zip ys vs";
nipkow@8009
   977
by(induct_tac "us" 1);
nipkow@8009
   978
 by(Asm_full_simp_tac 1);
nipkow@8009
   979
by(Asm_full_simp_tac 1);
nipkow@8009
   980
by(Clarify_tac 1);
nipkow@8009
   981
by(exhaust_tac "xs" 1);
nipkow@8009
   982
 by(Auto_tac);
nipkow@8009
   983
qed_spec_mp "zip_append";
nipkow@8009
   984
nipkow@8009
   985
Goal "!xs. length xs = length ys --> zip (rev xs) (rev ys) = rev (zip xs ys)";
nipkow@8009
   986
by(induct_tac "ys" 1);
nipkow@8009
   987
 by(Asm_full_simp_tac 1);
nipkow@8009
   988
by(Asm_full_simp_tac 1);
nipkow@8009
   989
by(Clarify_tac 1);
nipkow@8009
   990
by(exhaust_tac "xs" 1);
nipkow@8009
   991
 by(Auto_tac);
nipkow@8009
   992
by (asm_full_simp_tac (simpset() addsimps [zip_append]) 1);
nipkow@8009
   993
qed_spec_mp "zip_rev";
nipkow@8009
   994
nipkow@8009
   995
Goal
nipkow@8009
   996
"!i xs. i < length xs --> i < length ys --> (zip xs ys)!i = (xs!i, ys!i)";
nipkow@8009
   997
by (induct_tac "ys" 1);
nipkow@8009
   998
 by (Simp_tac 1);
nipkow@8009
   999
by (Clarify_tac 1);
nipkow@8009
  1000
by(exhaust_tac "xs" 1);
nipkow@8009
  1001
 by(Auto_tac);
nipkow@8009
  1002
by (asm_full_simp_tac (simpset() addsimps (thms"nth.simps") addsplits [nat.split]) 1);
nipkow@8009
  1003
qed_spec_mp "nth_zip";
nipkow@8009
  1004
Addsimps [nth_zip];
nipkow@8009
  1005
nipkow@8009
  1006
Goal
nipkow@8009
  1007
 "length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]";
nipkow@8009
  1008
by(rtac sym 1);
nipkow@8009
  1009
by(asm_simp_tac (simpset() addsimps [update_zip]) 1);
nipkow@8009
  1010
qed_spec_mp "zip_update";
nipkow@8009
  1011
nipkow@8009
  1012
Goal "!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)";
nipkow@8009
  1013
by (induct_tac "i" 1);
nipkow@8009
  1014
 by(Auto_tac);
nipkow@8009
  1015
by (exhaust_tac "j" 1);
nipkow@8009
  1016
 by(Auto_tac);
nipkow@8009
  1017
qed "zip_replicate";
nipkow@8009
  1018
Addsimps [zip_replicate];
nipkow@8009
  1019
nipkow@5272
  1020
nipkow@5272
  1021
(** foldl **)
nipkow@5272
  1022
section "foldl";
nipkow@5272
  1023
nipkow@5272
  1024
Goal "!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys";
paulson@5318
  1025
by (induct_tac "xs" 1);
paulson@5316
  1026
by Auto_tac;
nipkow@5272
  1027
qed_spec_mp "foldl_append";
nipkow@5272
  1028
Addsimps [foldl_append];
nipkow@5272
  1029
nipkow@5272
  1030
(* Note: `n <= foldl op+ n ns' looks simpler, but is more difficult to use
nipkow@5272
  1031
   because it requires an additional transitivity step
nipkow@5272
  1032
*)
nipkow@5272
  1033
Goal "!n::nat. m <= n --> m <= foldl op+ n ns";
paulson@5318
  1034
by (induct_tac "ns" 1);
nipkow@6058
  1035
by Auto_tac;
nipkow@5272
  1036
qed_spec_mp "start_le_sum";
nipkow@5272
  1037
nipkow@5272
  1038
Goal "n : set ns ==> n <= foldl op+ 0 ns";
oheimb@5758
  1039
by (force_tac (claset() addIs [start_le_sum],
oheimb@5758
  1040
              simpset() addsimps [in_set_conv_decomp]) 1);
nipkow@5272
  1041
qed "elem_le_sum";
nipkow@5272
  1042
nipkow@5272
  1043
Goal "!m. (foldl op+ m ns = 0) = (m=0 & (!n : set ns. n=0))";
paulson@5318
  1044
by (induct_tac "ns" 1);
paulson@5316
  1045
by Auto_tac;
nipkow@5272
  1046
qed_spec_mp "sum_eq_0_conv";
nipkow@5272
  1047
AddIffs [sum_eq_0_conv];
nipkow@5272
  1048
nipkow@5425
  1049
(** upto **)
nipkow@5425
  1050
nipkow@5427
  1051
(* Does not terminate! *)
nipkow@5427
  1052
Goal "[i..j(] = (if i<j then i#[Suc i..j(] else [])";
paulson@6162
  1053
by (induct_tac "j" 1);
nipkow@5427
  1054
by Auto_tac;
nipkow@5427
  1055
qed "upt_rec";
nipkow@5425
  1056
nipkow@5427
  1057
Goal "j<=i ==> [i..j(] = []";
paulson@6162
  1058
by (stac upt_rec 1);
paulson@6162
  1059
by (Asm_simp_tac 1);
nipkow@5427
  1060
qed "upt_conv_Nil";
nipkow@5427
  1061
Addsimps [upt_conv_Nil];
nipkow@5427
  1062
nipkow@5427
  1063
Goal "i<=j ==> [i..(Suc j)(] = [i..j(]@[j]";
nipkow@5427
  1064
by (Asm_simp_tac 1);
nipkow@5427
  1065
qed "upt_Suc";
nipkow@5427
  1066
nipkow@5427
  1067
Goal "i<j ==> [i..j(] = i#[Suc i..j(]";
paulson@6162
  1068
by (rtac trans 1);
paulson@6162
  1069
by (stac upt_rec 1);
paulson@6162
  1070
by (rtac refl 2);
nipkow@5427
  1071
by (Asm_simp_tac 1);
nipkow@5427
  1072
qed "upt_conv_Cons";
nipkow@5427
  1073
nipkow@5427
  1074
Goal "length [i..j(] = j-i";
paulson@6162
  1075
by (induct_tac "j" 1);
nipkow@5427
  1076
 by (Simp_tac 1);
paulson@6162
  1077
by (asm_simp_tac (simpset() addsimps [Suc_diff_le]) 1);
nipkow@5427
  1078
qed "length_upt";
nipkow@5427
  1079
Addsimps [length_upt];
nipkow@5425
  1080
nipkow@5427
  1081
Goal "i+k < j --> [i..j(] ! k = i+k";
paulson@6162
  1082
by (induct_tac "j" 1);
paulson@6162
  1083
 by (Simp_tac 1);
paulson@6162
  1084
by (asm_simp_tac (simpset() addsimps [nth_append,less_diff_conv]@add_ac) 1);
paulson@6162
  1085
by (Clarify_tac 1);
paulson@6162
  1086
by (subgoal_tac "n=i+k" 1);
paulson@6162
  1087
 by (Asm_simp_tac 2);
paulson@6162
  1088
by (Asm_simp_tac 1);
nipkow@5427
  1089
qed_spec_mp "nth_upt";
nipkow@5427
  1090
Addsimps [nth_upt];
nipkow@5425
  1091
nipkow@6433
  1092
Goal "!i. i+m <= n --> take m [i..n(] = [i..i+m(]";
paulson@6813
  1093
by (induct_tac "m" 1);
paulson@6813
  1094
 by (Simp_tac 1);
paulson@6813
  1095
by (Clarify_tac 1);
paulson@6813
  1096
by (stac upt_rec 1);
paulson@6813
  1097
by (rtac sym 1);
paulson@6813
  1098
by (stac upt_rec 1);
paulson@6813
  1099
by (asm_simp_tac (simpset() delsimps (thms"upt.simps")) 1);
nipkow@6433
  1100
qed_spec_mp "take_upt";
nipkow@6433
  1101
Addsimps [take_upt];
nipkow@6433
  1102
nipkow@6433
  1103
Goal "!m i. i < n-m --> (map f [m..n(]) ! i = f(m+i)";
paulson@6813
  1104
by (induct_tac "n" 1);
paulson@6813
  1105
 by (Simp_tac 1);
paulson@6813
  1106
by (Clarify_tac 1);
paulson@6813
  1107
by (subgoal_tac "m < Suc n" 1);
paulson@6813
  1108
 by (arith_tac 2);
paulson@6813
  1109
by (stac upt_rec 1);
paulson@6813
  1110
by (asm_simp_tac (simpset() delsplits [split_if]) 1);
paulson@6813
  1111
by (split_tac [split_if] 1);
paulson@6813
  1112
by (rtac conjI 1);
paulson@6813
  1113
 by (simp_tac (simpset() addsimps [nth_Cons] addsplits [nat.split]) 1);
paulson@6813
  1114
 by (simp_tac (simpset() addsimps [nth_append] addsplits [nat.split]) 1);
paulson@6813
  1115
 by (Clarify_tac 1);
paulson@6813
  1116
 by (rtac conjI 1);
paulson@6813
  1117
  by (Clarify_tac 1);
paulson@6813
  1118
  by (subgoal_tac "Suc(m+nat) < n" 1);
paulson@6813
  1119
   by (arith_tac 2);
paulson@6813
  1120
  by (Asm_simp_tac 1);
paulson@6813
  1121
 by (Clarify_tac 1);
paulson@6813
  1122
 by (subgoal_tac "n = Suc(m+nat)" 1);
paulson@6813
  1123
  by (arith_tac 2);
paulson@6813
  1124
 by (Asm_simp_tac 1);
paulson@6813
  1125
by (simp_tac (simpset() addsimps [nth_Cons] addsplits [nat.split]) 1);
paulson@6813
  1126
by (arith_tac 1);
nipkow@6433
  1127
qed_spec_mp "nth_map_upt";
nipkow@6433
  1128
paulson@6813
  1129
Goal "ALL xs ys. k <= length xs --> k <= length ys -->  \
paulson@6813
  1130
\        (ALL i. i < k --> xs!i = ys!i)  \
paulson@6813
  1131
\     --> take k xs = take k ys";
paulson@6813
  1132
by (induct_tac "k" 1);
paulson@6813
  1133
by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq_0_disj, 
paulson@6813
  1134
						all_conj_distrib])));
paulson@6813
  1135
by (Clarify_tac 1);
paulson@6813
  1136
(*Both lists must be non-empty*)
paulson@6813
  1137
by (exhaust_tac "xs" 1);
paulson@6813
  1138
by (exhaust_tac "ys" 2);
paulson@6813
  1139
by (ALLGOALS Clarify_tac);
paulson@6813
  1140
(*prenexing's needed, not miniscoping*)
paulson@6813
  1141
by (ALLGOALS (full_simp_tac (simpset() addsimps (all_simps RL [sym])  
paulson@6813
  1142
                                       delsimps (all_simps))));
paulson@6813
  1143
by (Blast_tac 1);
paulson@6813
  1144
qed_spec_mp "nth_take_lemma";
paulson@6813
  1145
paulson@6813
  1146
Goal "[| length xs = length ys;  \
paulson@6813
  1147
\        ALL i. i < length xs --> xs!i = ys!i |]  \
paulson@6813
  1148
\     ==> xs = ys";
paulson@6813
  1149
by (forward_tac [[le_refl, eq_imp_le] MRS nth_take_lemma] 1);
paulson@6813
  1150
by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [take_all])));
paulson@6813
  1151
qed_spec_mp "nth_equalityI";
paulson@6813
  1152
paulson@6813
  1153
(*The famous take-lemma*)
paulson@6813
  1154
Goal "(ALL i. take i xs = take i ys) ==> xs = ys";
paulson@6813
  1155
by (dres_inst_tac [("x", "max (length xs) (length ys)")] spec 1);
paulson@6813
  1156
by (full_simp_tac (simpset() addsimps [le_max_iff_disj, take_all]) 1);
paulson@6813
  1157
qed_spec_mp "take_equalityI";
paulson@6813
  1158
nipkow@5272
  1159
nipkow@4605
  1160
(** nodups & remdups **)
nipkow@4605
  1161
section "nodups & remdups";
nipkow@4605
  1162
nipkow@4935
  1163
Goal "set(remdups xs) = set xs";
nipkow@4605
  1164
by (induct_tac "xs" 1);
nipkow@4605
  1165
 by (Simp_tac 1);
nipkow@4686
  1166
by (asm_full_simp_tac (simpset() addsimps [insert_absorb]) 1);
nipkow@4605
  1167
qed "set_remdups";
nipkow@4605
  1168
Addsimps [set_remdups];
nipkow@4605
  1169
nipkow@4935
  1170
Goal "nodups(remdups xs)";
nipkow@4605
  1171
by (induct_tac "xs" 1);
paulson@5316
  1172
by Auto_tac;
nipkow@4605
  1173
qed "nodups_remdups";
nipkow@4605
  1174
nipkow@4935
  1175
Goal "nodups xs --> nodups (filter P xs)";
nipkow@4605
  1176
by (induct_tac "xs" 1);
paulson@5316
  1177
by Auto_tac;
nipkow@4605
  1178
qed_spec_mp "nodups_filter";
nipkow@4605
  1179
nipkow@3589
  1180
(** replicate **)
nipkow@3589
  1181
section "replicate";
nipkow@3589
  1182
nipkow@6794
  1183
Goal "length(replicate n x) = n";
paulson@6813
  1184
by (induct_tac "n" 1);
paulson@6813
  1185
by Auto_tac;
nipkow@6794
  1186
qed "length_replicate";
nipkow@6794
  1187
Addsimps [length_replicate];
nipkow@6794
  1188
nipkow@6794
  1189
Goal "map f (replicate n x) = replicate n (f x)";
nipkow@6794
  1190
by (induct_tac "n" 1);
paulson@6813
  1191
by Auto_tac;
nipkow@6794
  1192
qed "map_replicate";
nipkow@6794
  1193
Addsimps [map_replicate];
nipkow@6794
  1194
nipkow@6794
  1195
Goal "(replicate n x) @ (x#xs) = x # replicate n x @ xs";
nipkow@6794
  1196
by (induct_tac "n" 1);
paulson@6813
  1197
by Auto_tac;
nipkow@6794
  1198
qed "replicate_app_Cons_same";
nipkow@6794
  1199
nipkow@6794
  1200
Goal "rev(replicate n x) = replicate n x";
nipkow@6794
  1201
by (induct_tac "n" 1);
paulson@6813
  1202
 by (Simp_tac 1);
nipkow@6794
  1203
by (asm_simp_tac (simpset() addsimps [replicate_app_Cons_same]) 1);
nipkow@6794
  1204
qed "rev_replicate";
nipkow@6794
  1205
Addsimps [rev_replicate];
nipkow@6794
  1206
nipkow@8009
  1207
Goal "replicate (n+m) x = replicate n x @ replicate m x";
nipkow@8009
  1208
by (induct_tac "n" 1);
nipkow@8009
  1209
by Auto_tac;
nipkow@8009
  1210
qed "replicate_add";
nipkow@8009
  1211
nipkow@6794
  1212
Goal"n ~= 0 --> hd(replicate n x) = x";
nipkow@6794
  1213
by (induct_tac "n" 1);
paulson@6813
  1214
by Auto_tac;
nipkow@6794
  1215
qed_spec_mp "hd_replicate";
nipkow@6794
  1216
Addsimps [hd_replicate];
nipkow@6794
  1217
nipkow@6794
  1218
Goal "n ~= 0 --> tl(replicate n x) = replicate (n-1) x";
nipkow@6794
  1219
by (induct_tac "n" 1);
paulson@6813
  1220
by Auto_tac;
nipkow@6794
  1221
qed_spec_mp "tl_replicate";
nipkow@6794
  1222
Addsimps [tl_replicate];
nipkow@6794
  1223
nipkow@6794
  1224
Goal "n ~= 0 --> last(replicate n x) = x";
nipkow@6794
  1225
by (induct_tac "n" 1);
paulson@6813
  1226
by Auto_tac;
nipkow@6794
  1227
qed_spec_mp "last_replicate";
nipkow@6794
  1228
Addsimps [last_replicate];
nipkow@6794
  1229
nipkow@6794
  1230
Goal "!i. i<n --> (replicate n x)!i = x";
paulson@6813
  1231
by (induct_tac "n" 1);
paulson@6813
  1232
 by (Simp_tac 1);
paulson@6813
  1233
by (asm_simp_tac (simpset() addsimps [nth_Cons] addsplits [nat.split]) 1);
nipkow@6794
  1234
qed_spec_mp "nth_replicate";
nipkow@6794
  1235
Addsimps [nth_replicate];
nipkow@6794
  1236
nipkow@4935
  1237
Goal "set(replicate (Suc n) x) = {x}";
wenzelm@4423
  1238
by (induct_tac "n" 1);
paulson@5316
  1239
by Auto_tac;
nipkow@3589
  1240
val lemma = result();
nipkow@3589
  1241
nipkow@5043
  1242
Goal "n ~= 0 ==> set(replicate n x) = {x}";
wenzelm@4423
  1243
by (fast_tac (claset() addSDs [not0_implies_Suc] addSIs [lemma]) 1);
nipkow@3589
  1244
qed "set_replicate";
nipkow@3589
  1245
Addsimps [set_replicate];
nipkow@5162
  1246
nipkow@8009
  1247
Goal "set(replicate n x) = (if n=0 then {} else {x})";
nipkow@8009
  1248
by(Auto_tac);
nipkow@8009
  1249
qed "set_replicate_conv_if";
nipkow@8009
  1250
nipkow@8009
  1251
Goal "x : set(replicate n y) --> x=y";
nipkow@8009
  1252
by(asm_simp_tac (simpset() addsimps [set_replicate_conv_if]) 1);
nipkow@8009
  1253
qed_spec_mp "in_set_replicateD";
nipkow@8009
  1254
nipkow@5162
  1255
nipkow@5281
  1256
(*** Lexcicographic orderings on lists ***)
nipkow@5281
  1257
section"Lexcicographic orderings on lists";
nipkow@5281
  1258
nipkow@5281
  1259
Goal "wf r ==> wf(lexn r n)";
paulson@5318
  1260
by (induct_tac "n" 1);
paulson@5318
  1261
by (Simp_tac 1);
paulson@5318
  1262
by (Simp_tac 1);
paulson@5318
  1263
by (rtac wf_subset 1);
paulson@5318
  1264
by (rtac Int_lower1 2);
paulson@5318
  1265
by (rtac wf_prod_fun_image 1);
paulson@5318
  1266
by (rtac injI 2);
paulson@6813
  1267
by Auto_tac;
nipkow@5281
  1268
qed "wf_lexn";
nipkow@5281
  1269
nipkow@5281
  1270
Goal "!xs ys. (xs,ys) : lexn r n --> length xs = n & length ys = n";
paulson@5318
  1271
by (induct_tac "n" 1);
paulson@6813
  1272
by Auto_tac;
nipkow@5281
  1273
qed_spec_mp "lexn_length";
nipkow@5281
  1274
nipkow@5281
  1275
Goalw [lex_def] "wf r ==> wf(lex r)";
paulson@5318
  1276
by (rtac wf_UN 1);
paulson@5318
  1277
by (blast_tac (claset() addIs [wf_lexn]) 1);
paulson@5318
  1278
by (Clarify_tac 1);
paulson@5318
  1279
by (rename_tac "m n" 1);
paulson@5318
  1280
by (subgoal_tac "m ~= n" 1);
paulson@5318
  1281
 by (Blast_tac 2);
paulson@5318
  1282
by (blast_tac (claset() addDs [lexn_length,not_sym]) 1);
nipkow@5281
  1283
qed "wf_lex";
nipkow@5281
  1284
AddSIs [wf_lex];
nipkow@5281
  1285
nipkow@5281
  1286
Goal
nipkow@5281
  1287
 "lexn r n = \
nipkow@5281
  1288
\ {(xs,ys). length xs = n & length ys = n & \
nipkow@5281
  1289
\           (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}";
paulson@5318
  1290
by (induct_tac "n" 1);
paulson@5318
  1291
 by (Simp_tac 1);
paulson@5318
  1292
 by (Blast_tac 1);
paulson@5641
  1293
by (asm_full_simp_tac (simpset() 
oheimb@5296
  1294
				addsimps [lex_prod_def]) 1);
paulson@5641
  1295
by (auto_tac (claset(), simpset()));
paulson@5318
  1296
  by (Blast_tac 1);
paulson@5318
  1297
 by (rename_tac "a xys x xs' y ys'" 1);
paulson@5318
  1298
 by (res_inst_tac [("x","a#xys")] exI 1);
paulson@5318
  1299
 by (Simp_tac 1);
paulson@5318
  1300
by (exhaust_tac "xys" 1);
paulson@5641
  1301
 by (ALLGOALS (asm_full_simp_tac (simpset())));
paulson@5318
  1302
by (Blast_tac 1);
nipkow@5281
  1303
qed "lexn_conv";
nipkow@5281
  1304
nipkow@5281
  1305
Goalw [lex_def]
nipkow@5281
  1306
 "lex r = \
nipkow@5281
  1307
\ {(xs,ys). length xs = length ys & \
nipkow@5281
  1308
\           (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}";
paulson@5641
  1309
by (force_tac (claset(), simpset() addsimps [lexn_conv]) 1);
nipkow@5281
  1310
qed "lex_conv";
nipkow@5281
  1311
nipkow@5281
  1312
Goalw [lexico_def] "wf r ==> wf(lexico r)";
paulson@5318
  1313
by (Blast_tac 1);
nipkow@5281
  1314
qed "wf_lexico";
nipkow@5281
  1315
AddSIs [wf_lexico];
nipkow@5281
  1316
nipkow@5281
  1317
Goalw
nipkow@5281
  1318
 [lexico_def,diag_def,lex_prod_def,measure_def,inv_image_def]
nipkow@5281
  1319
"lexico r = {(xs,ys). length xs < length ys | \
nipkow@5281
  1320
\                     length xs = length ys & (xs,ys) : lex r}";
paulson@5318
  1321
by (Simp_tac 1);
nipkow@5281
  1322
qed "lexico_conv";
nipkow@5281
  1323
nipkow@5283
  1324
Goal "([],ys) ~: lex r";
paulson@5318
  1325
by (simp_tac (simpset() addsimps [lex_conv]) 1);
nipkow@5283
  1326
qed "Nil_notin_lex";
nipkow@5283
  1327
nipkow@5283
  1328
Goal "(xs,[]) ~: lex r";
paulson@5318
  1329
by (simp_tac (simpset() addsimps [lex_conv]) 1);
nipkow@5283
  1330
qed "Nil2_notin_lex";
nipkow@5283
  1331
nipkow@5283
  1332
AddIffs [Nil_notin_lex,Nil2_notin_lex];
nipkow@5283
  1333
nipkow@5283
  1334
Goal "((x#xs,y#ys) : lex r) = \
nipkow@5283
  1335
\     ((x,y) : r & length xs = length ys | x=y & (xs,ys) : lex r)";
paulson@5318
  1336
by (simp_tac (simpset() addsimps [lex_conv]) 1);
paulson@5318
  1337
by (rtac iffI 1);
paulson@5318
  1338
 by (blast_tac (claset() addIs [Cons_eq_appendI]) 2);
paulson@5318
  1339
by (REPEAT(eresolve_tac [conjE, exE] 1));
paulson@5318
  1340
by (exhaust_tac "xys" 1);
paulson@5318
  1341
by (Asm_full_simp_tac 1);
paulson@5318
  1342
by (Asm_full_simp_tac 1);
paulson@5318
  1343
by (Blast_tac 1);
nipkow@5283
  1344
qed "Cons_in_lex";
nipkow@5283
  1345
AddIffs [Cons_in_lex];
paulson@7032
  1346
paulson@7032
  1347
paulson@7032
  1348
(*** Versions of some theorems above using binary numerals ***)
paulson@7032
  1349
paulson@7032
  1350
AddIffs (map (rename_numerals thy) 
paulson@7032
  1351
	  [length_0_conv, zero_length_conv, length_greater_0_conv,
paulson@7032
  1352
	   sum_eq_0_conv]);
paulson@7032
  1353
paulson@7032
  1354
Goal "take n (x#xs) = (if n = #0 then [] else x # take (n-#1) xs)";
paulson@7032
  1355
by (exhaust_tac "n" 1);
paulson@7032
  1356
by (ALLGOALS 
paulson@7032
  1357
    (asm_simp_tac (simpset() addsimps [numeral_0_eq_0, numeral_1_eq_1])));
paulson@7032
  1358
qed "take_Cons'";
paulson@7032
  1359
paulson@7032
  1360
Goal "drop n (x#xs) = (if n = #0 then x#xs else drop (n-#1) xs)";
paulson@7032
  1361
by (exhaust_tac "n" 1);
paulson@7032
  1362
by (ALLGOALS
paulson@7032
  1363
    (asm_simp_tac (simpset() addsimps [numeral_0_eq_0, numeral_1_eq_1])));
paulson@7032
  1364
qed "drop_Cons'";
paulson@7032
  1365
paulson@7032
  1366
Goal "(x#xs)!n = (if n = #0 then x else xs!(n-#1))";
paulson@7032
  1367
by (exhaust_tac "n" 1);
paulson@7032
  1368
by (ALLGOALS
paulson@7032
  1369
    (asm_simp_tac (simpset() addsimps [numeral_0_eq_0, numeral_1_eq_1])));
paulson@7032
  1370
qed "nth_Cons'";
paulson@7032
  1371
paulson@7032
  1372
Addsimps (map (inst "n" "number_of ?v") [take_Cons', drop_Cons', nth_Cons']);
paulson@7032
  1373