src/HOL/List.ML
author oheimb
Wed Aug 12 15:29:34 1998 +0200 (1998-08-12)
changeset 5296 bdef7d349d27
parent 5283 0027ddfbc831
child 5316 7a8975451a89
permissions -rw-r--r--
added length_Suc_conv, finite_set
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(*  Title:      HOL/List
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    Copyright   1994 TU Muenchen
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List lemmas
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*)
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Goal "!x. xs ~= x#xs";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed_spec_mp "not_Cons_self";
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bind_thm("not_Cons_self2",not_Cons_self RS not_sym);
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Addsimps [not_Cons_self,not_Cons_self2];
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Goal "(xs ~= []) = (? y ys. xs = y#ys)";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "neq_Nil_conv";
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(* Induction over the length of a list: *)
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val [prem] = Goal
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  "(!!xs. (!ys. length ys < length xs --> P ys) ==> P xs) ==> P(xs)";
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by (rtac measure_induct 1 THEN etac prem 1);
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qed "length_induct";
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(** "lists": the list-forming operator over sets **)
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Goalw lists.defs "A<=B ==> lists A <= lists B";
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by (rtac lfp_mono 1);
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by (REPEAT (ares_tac basic_monos 1));
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qed "lists_mono";
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val listsE = lists.mk_cases list.simps  "x#l : lists A";
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AddSEs [listsE];
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AddSIs lists.intrs;
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Goal "l: lists A ==> l: lists B --> l: lists (A Int B)";
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by (etac lists.induct 1);
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by (ALLGOALS Blast_tac);
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qed_spec_mp "lists_IntI";
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Goal "lists (A Int B) = lists A Int lists B";
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by (rtac (mono_Int RS equalityI) 1);
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by (simp_tac (simpset() addsimps [mono_def, lists_mono]) 1);
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by (blast_tac (claset() addSIs [lists_IntI]) 1);
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qed "lists_Int_eq";
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Addsimps [lists_Int_eq];
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(**  Case analysis **)
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section "Case analysis";
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val prems = Goal "[| P([]); !!x xs. P(x#xs) |] ==> P(xs)";
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by (induct_tac "xs" 1);
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by (REPEAT(resolve_tac prems 1));
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qed "list_cases";
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Goal "(xs=[] --> P([])) & (!y ys. xs=y#ys --> P(y#ys)) --> P(xs)";
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by (induct_tac "xs" 1);
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by (Blast_tac 1);
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by (Blast_tac 1);
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bind_thm("list_eq_cases",
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  impI RSN (2,allI RSN (2,allI RSN (2,impI RS (conjI RS (result() RS mp))))));
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(** length **)
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(* needs to come before "@" because of thm append_eq_append_conv *)
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section "length";
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Goal "length(xs@ys) = length(xs)+length(ys)";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed"length_append";
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Addsimps [length_append];
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Goal "length (map f xs) = length xs";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "length_map";
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Addsimps [length_map];
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Goal "length(rev xs) = length(xs)";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "length_rev";
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Addsimps [length_rev];
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Goal "xs ~= [] ==> length(tl xs) = (length xs) - 1";
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by (exhaust_tac "xs" 1);
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by (Auto_tac);
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qed "length_tl";
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Addsimps [length_tl];
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Goal "(length xs = 0) = (xs = [])";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "length_0_conv";
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AddIffs [length_0_conv];
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Goal "(0 = length xs) = (xs = [])";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "zero_length_conv";
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AddIffs [zero_length_conv];
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Goal "(0 < length xs) = (xs ~= [])";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "length_greater_0_conv";
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AddIffs [length_greater_0_conv];
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Goal "(length xs = Suc n) = (? y ys. xs = y#ys & length ys = n)";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "length_Suc_conv";
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AddIffs [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 (fast_tac (claset() addIs [less_add_Suc2] addss simpset()
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                      addEs [less_not_refl2 RSN (2,rev_notE)]) 1);
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by (rtac allI 1);
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by (exhaust_tac "ys" 1);
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by (fast_tac (claset() addIs [less_add_Suc2] 
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		       addss (simpset() delsimps [length_Suc_conv])
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                       addEs [(less_not_refl2 RS not_sym) RSN (2,rev_notE)]) 1);
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by (Asm_simp_tac 1);
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qed_spec_mp "append_eq_append_conv";
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Addsimps [append_eq_append_conv];
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Goal "(xs @ ys = xs @ zs) = (ys=zs)";
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by (Simp_tac 1);
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qed "same_append_eq";
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Goal "(xs @ [x] = ys @ [y]) = (xs = ys & x = y)"; 
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by (Simp_tac 1);
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qed "append1_eq_conv";
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Goal "(ys @ xs = zs @ xs) = (ys=zs)";
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by (Simp_tac 1);
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qed "append_same_eq";
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AddSIs
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 [same_append_eq RS iffD2, append1_eq_conv RS iffD2, append_same_eq RS iffD2];
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AddSDs
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 [same_append_eq RS iffD1, append1_eq_conv RS iffD1, append_same_eq RS iffD1];
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Goal "(xs @ ys = ys) = (xs=[])";
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by (cut_inst_tac [("zs","[]")] append_same_eq 1);
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by (Auto_tac);
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qed "append_self_conv2";
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Goal "(ys = xs @ ys) = (xs=[])";
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by (simp_tac (simpset() addsimps
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     [simplify (simpset()) (read_instantiate[("ys","[]")]append_same_eq)]) 1);
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by (Blast_tac 1);
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qed "self_append_conv2";
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AddIffs [append_self_conv2,self_append_conv2];
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Goal "xs ~= [] --> hd xs # tl xs = xs";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed_spec_mp "hd_Cons_tl";
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Addsimps [hd_Cons_tl];
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Goal "hd(xs@ys) = (if xs=[] then hd ys else hd xs)";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "hd_append";
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Goal "xs ~= [] ==> hd(xs @ ys) = hd xs";
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by (asm_simp_tac (simpset() addsimps [hd_append]
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                           addsplits [list.split]) 1);
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qed "hd_append2";
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Addsimps [hd_append2];
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Goal "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)";
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by (simp_tac (simpset() addsplits [list.split]) 1);
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qed "tl_append";
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Goal "xs ~= [] ==> tl(xs @ ys) = (tl xs) @ ys";
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by (asm_simp_tac (simpset() addsimps [tl_append]
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                           addsplits [list.split]) 1);
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qed "tl_append2";
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Addsimps [tl_append2];
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(* trivial rules for solving @-equations automatically *)
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Goal "xs = ys ==> xs = [] @ ys";
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by(Asm_simp_tac 1);
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qed "eq_Nil_appendI";
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Goal "[| x#xs1 = ys; xs = xs1 @ zs |] ==> x#xs = ys@zs";
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bd sym 1;
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by(Asm_simp_tac 1);
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qed "Cons_eq_appendI";
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Goal "[| xs@xs1 = zs; ys = xs1 @ us |] ==> xs@ys = zs@us";
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bd sym 1;
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by(Asm_simp_tac 1);
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qed "append_eq_appendI";
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(** map **)
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section "map";
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Goal "(!x. x : set xs --> f x = g x) --> map f xs = map g xs";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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bind_thm("map_ext", impI RS (allI RS (result() RS mp)));
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Goal "map (%x. x) = (%xs. xs)";
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by (rtac ext 1);
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "map_ident";
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Addsimps[map_ident];
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Goal "map f (xs@ys) = map f xs @ map f ys";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "map_append";
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Addsimps[map_append];
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Goalw [o_def] "map (f o g) xs = map f (map g xs)";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "map_compose";
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Addsimps[map_compose];
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Goal "rev(map f xs) = map f (rev xs)";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "rev_map";
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(* a congruence rule for map: *)
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Goal "(xs=ys) --> (!x. x : set ys --> f x = g x) --> map f xs = map g ys";
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by (rtac impI 1);
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by (hyp_subst_tac 1);
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by (induct_tac "ys" 1);
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by (Auto_tac);
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val lemma = result();
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bind_thm("map_cong",impI RSN (2,allI RSN (2,lemma RS mp RS mp)));
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Goal "(map f xs = []) = (xs = [])";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "map_is_Nil_conv";
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AddIffs [map_is_Nil_conv];
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Goal "([] = map f xs) = (xs = [])";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "Nil_is_map_conv";
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AddIffs [Nil_is_map_conv];
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(** rev **)
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section "rev";
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Goal "rev(xs@ys) = rev(ys) @ rev(xs)";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "rev_append";
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Addsimps[rev_append];
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Goal "rev(rev l) = l";
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by (induct_tac "l" 1);
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by (Auto_tac);
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qed "rev_rev_ident";
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Addsimps[rev_rev_ident];
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Goal "(rev xs = []) = (xs = [])";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "rev_is_Nil_conv";
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AddIffs [rev_is_Nil_conv];
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Goal "([] = rev xs) = (xs = [])";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "Nil_is_rev_conv";
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AddIffs [Nil_is_rev_conv];
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val prems = Goal "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P xs";
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by (stac (rev_rev_ident RS sym) 1);
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br(read_instantiate [("P","%xs. ?P(rev xs)")]list.induct)1;
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by (ALLGOALS Simp_tac);
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by (resolve_tac prems 1);
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by (eresolve_tac prems 1);
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qed "rev_induct";
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fun rev_induct_tac xs = res_inst_tac [("xs",xs)] rev_induct;
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Goal  "(xs = [] --> P) -->  (!ys y. xs = ys@[y] --> P) --> P";
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by (res_inst_tac [("xs","xs")] rev_induct 1);
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by (Auto_tac);
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bind_thm ("rev_exhaust",
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  impI RSN (2,allI RSN (2,allI RSN (2,impI RS (result() RS mp RS mp)))));
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nipkow@2608
   348
clasohm@923
   349
(** mem **)
clasohm@923
   350
nipkow@3467
   351
section "mem";
nipkow@3467
   352
nipkow@4935
   353
Goal "x mem (xs@ys) = (x mem xs | x mem ys)";
nipkow@3040
   354
by (induct_tac "xs" 1);
nipkow@5129
   355
by (Auto_tac);
clasohm@923
   356
qed "mem_append";
nipkow@2512
   357
Addsimps[mem_append];
clasohm@923
   358
nipkow@4935
   359
Goal "x mem [x:xs. P(x)] = (x mem xs & P(x))";
nipkow@3040
   360
by (induct_tac "xs" 1);
nipkow@5129
   361
by (Auto_tac);
clasohm@923
   362
qed "mem_filter";
nipkow@2512
   363
Addsimps[mem_filter];
clasohm@923
   364
nipkow@3465
   365
(** set **)
paulson@1812
   366
nipkow@3467
   367
section "set";
nipkow@3467
   368
oheimb@5296
   369
qed_goal "finite_set" thy "finite (set xs)" 
oheimb@5296
   370
	(K [induct_tac "xs" 1, Auto_tac]);
oheimb@5296
   371
Addsimps[finite_set];
oheimb@5296
   372
AddSIs[finite_set];
oheimb@5296
   373
nipkow@4935
   374
Goal "set (xs@ys) = (set xs Un set ys)";
nipkow@3040
   375
by (induct_tac "xs" 1);
nipkow@5129
   376
by (Auto_tac);
paulson@3647
   377
qed "set_append";
paulson@3647
   378
Addsimps[set_append];
paulson@1812
   379
nipkow@4935
   380
Goal "(x mem xs) = (x: set xs)";
nipkow@3040
   381
by (induct_tac "xs" 1);
nipkow@5129
   382
by (Auto_tac);
paulson@3647
   383
qed "set_mem_eq";
paulson@1812
   384
nipkow@4935
   385
Goal "set l <= set (x#l)";
nipkow@5129
   386
by (Auto_tac);
paulson@3647
   387
qed "set_subset_Cons";
paulson@1936
   388
nipkow@4935
   389
Goal "(set xs = {}) = (xs = [])";
paulson@3457
   390
by (induct_tac "xs" 1);
nipkow@5129
   391
by (Auto_tac);
paulson@3647
   392
qed "set_empty";
paulson@3647
   393
Addsimps [set_empty];
nipkow@2608
   394
nipkow@4935
   395
Goal "set(rev xs) = set(xs)";
paulson@3457
   396
by (induct_tac "xs" 1);
nipkow@5129
   397
by (Auto_tac);
paulson@3647
   398
qed "set_rev";
paulson@3647
   399
Addsimps [set_rev];
nipkow@2608
   400
nipkow@4935
   401
Goal "set(map f xs) = f``(set xs)";
paulson@3457
   402
by (induct_tac "xs" 1);
nipkow@5129
   403
by (Auto_tac);
paulson@3647
   404
qed "set_map";
paulson@3647
   405
Addsimps [set_map];
nipkow@2608
   406
nipkow@4935
   407
Goal "(x : set(filter P xs)) = (x : set xs & P x)";
nipkow@4605
   408
by (induct_tac "xs" 1);
nipkow@5129
   409
by (Auto_tac);
nipkow@4605
   410
qed "in_set_filter";
nipkow@4605
   411
Addsimps [in_set_filter];
nipkow@4605
   412
nipkow@5272
   413
Goal "(x : set xs) = (? ys zs. xs = ys@x#zs)";
nipkow@5272
   414
by(induct_tac "xs" 1);
nipkow@5272
   415
 by(Simp_tac 1);
nipkow@5272
   416
by(Asm_simp_tac 1);
nipkow@5272
   417
br iffI 1;
nipkow@5272
   418
by(blast_tac (claset() addIs [eq_Nil_appendI,Cons_eq_appendI]) 1);
nipkow@5272
   419
by(REPEAT(etac exE 1));
nipkow@5272
   420
by(exhaust_tac "ys" 1);
nipkow@5272
   421
by(Auto_tac);
nipkow@5272
   422
qed "in_set_conv_decomp";
nipkow@5272
   423
nipkow@5272
   424
(* eliminate `lists' in favour of `set' *)
nipkow@5272
   425
nipkow@5272
   426
Goal "(xs : lists A) = (!x : set xs. x : A)";
nipkow@5272
   427
by(induct_tac "xs" 1);
nipkow@5272
   428
by(Auto_tac);
nipkow@5272
   429
qed "in_lists_conv_set";
nipkow@5272
   430
nipkow@5272
   431
bind_thm("in_listsD",in_lists_conv_set RS iffD1);
nipkow@5272
   432
AddSDs [in_listsD];
nipkow@5272
   433
bind_thm("in_listsI",in_lists_conv_set RS iffD2);
nipkow@5272
   434
AddSIs [in_listsI];
paulson@1812
   435
clasohm@923
   436
(** list_all **)
clasohm@923
   437
nipkow@3467
   438
section "list_all";
nipkow@3467
   439
nipkow@4935
   440
Goal "list_all (%x. True) xs = True";
nipkow@3040
   441
by (induct_tac "xs" 1);
nipkow@5129
   442
by (Auto_tac);
clasohm@923
   443
qed "list_all_True";
nipkow@2512
   444
Addsimps [list_all_True];
clasohm@923
   445
nipkow@4935
   446
Goal "list_all p (xs@ys) = (list_all p xs & list_all p ys)";
nipkow@3040
   447
by (induct_tac "xs" 1);
nipkow@5129
   448
by (Auto_tac);
nipkow@2512
   449
qed "list_all_append";
nipkow@2512
   450
Addsimps [list_all_append];
clasohm@923
   451
nipkow@4935
   452
Goal "list_all P xs = (!x. x mem xs --> P(x))";
nipkow@3040
   453
by (induct_tac "xs" 1);
nipkow@5129
   454
by (Auto_tac);
clasohm@923
   455
qed "list_all_mem_conv";
clasohm@923
   456
clasohm@923
   457
nipkow@2608
   458
(** filter **)
clasohm@923
   459
nipkow@3467
   460
section "filter";
nipkow@3467
   461
nipkow@4935
   462
Goal "filter P (xs@ys) = filter P xs @ filter P ys";
paulson@3457
   463
by (induct_tac "xs" 1);
nipkow@5129
   464
by (Auto_tac);
nipkow@2608
   465
qed "filter_append";
nipkow@2608
   466
Addsimps [filter_append];
nipkow@2608
   467
nipkow@4935
   468
Goal "filter (%x. True) xs = xs";
nipkow@4605
   469
by (induct_tac "xs" 1);
nipkow@5129
   470
by (Auto_tac);
nipkow@4605
   471
qed "filter_True";
nipkow@4605
   472
Addsimps [filter_True];
nipkow@4605
   473
nipkow@4935
   474
Goal "filter (%x. False) xs = []";
nipkow@4605
   475
by (induct_tac "xs" 1);
nipkow@5129
   476
by (Auto_tac);
nipkow@4605
   477
qed "filter_False";
nipkow@4605
   478
Addsimps [filter_False];
nipkow@4605
   479
nipkow@4935
   480
Goal "length (filter P xs) <= length xs";
paulson@3457
   481
by (induct_tac "xs" 1);
nipkow@5129
   482
by (Auto_tac);
nipkow@4605
   483
qed "length_filter";
paulson@3383
   484
nipkow@2608
   485
nipkow@2608
   486
(** concat **)
nipkow@2608
   487
nipkow@3467
   488
section "concat";
nipkow@3467
   489
nipkow@4935
   490
Goal  "concat(xs@ys) = concat(xs)@concat(ys)";
nipkow@3040
   491
by (induct_tac "xs" 1);
nipkow@5129
   492
by (Auto_tac);
nipkow@2608
   493
qed"concat_append";
nipkow@2608
   494
Addsimps [concat_append];
nipkow@2512
   495
nipkow@4935
   496
Goal "(concat xss = []) = (!xs:set xss. xs=[])";
wenzelm@4423
   497
by (induct_tac "xss" 1);
nipkow@5129
   498
by (Auto_tac);
nipkow@3896
   499
qed "concat_eq_Nil_conv";
nipkow@3896
   500
AddIffs [concat_eq_Nil_conv];
nipkow@3896
   501
nipkow@4935
   502
Goal "([] = concat xss) = (!xs:set xss. xs=[])";
wenzelm@4423
   503
by (induct_tac "xss" 1);
nipkow@5129
   504
by (Auto_tac);
nipkow@3896
   505
qed "Nil_eq_concat_conv";
nipkow@3896
   506
AddIffs [Nil_eq_concat_conv];
nipkow@3896
   507
nipkow@4935
   508
Goal  "set(concat xs) = Union(set `` set xs)";
nipkow@3467
   509
by (induct_tac "xs" 1);
nipkow@5129
   510
by (Auto_tac);
paulson@3647
   511
qed"set_concat";
paulson@3647
   512
Addsimps [set_concat];
nipkow@3467
   513
nipkow@4935
   514
Goal "map f (concat xs) = concat (map (map f) xs)"; 
nipkow@3467
   515
by (induct_tac "xs" 1);
nipkow@5129
   516
by (Auto_tac);
nipkow@3467
   517
qed "map_concat";
nipkow@3467
   518
nipkow@4935
   519
Goal "filter p (concat xs) = concat (map (filter p) xs)"; 
nipkow@3467
   520
by (induct_tac "xs" 1);
nipkow@5129
   521
by (Auto_tac);
nipkow@3467
   522
qed"filter_concat"; 
nipkow@3467
   523
nipkow@4935
   524
Goal "rev(concat xs) = concat (map rev (rev xs))";
nipkow@3467
   525
by (induct_tac "xs" 1);
nipkow@5129
   526
by (Auto_tac);
nipkow@2608
   527
qed "rev_concat";
clasohm@923
   528
clasohm@923
   529
(** nth **)
clasohm@923
   530
nipkow@3467
   531
section "nth";
nipkow@3467
   532
paulson@5278
   533
Goal "!xs. (xs@ys)!n = (if n < length xs then xs!n else ys!(n - length xs))";
berghofe@5183
   534
by (induct_tac "n" 1);
paulson@3457
   535
 by (Asm_simp_tac 1);
paulson@3457
   536
 by (rtac allI 1);
paulson@3457
   537
 by (exhaust_tac "xs" 1);
nipkow@5129
   538
  by (Auto_tac);
nipkow@2608
   539
qed_spec_mp "nth_append";
nipkow@2608
   540
nipkow@4935
   541
Goal "!n. n < length xs --> (map f xs)!n = f(xs!n)";
nipkow@3040
   542
by (induct_tac "xs" 1);
nipkow@1301
   543
(* case [] *)
nipkow@1301
   544
by (Asm_full_simp_tac 1);
nipkow@1301
   545
(* case x#xl *)
nipkow@1301
   546
by (rtac allI 1);
berghofe@5183
   547
by (induct_tac "n" 1);
nipkow@5129
   548
by (Auto_tac);
nipkow@1485
   549
qed_spec_mp "nth_map";
nipkow@1301
   550
Addsimps [nth_map];
nipkow@1301
   551
nipkow@4935
   552
Goal "!n. n < length xs --> list_all P xs --> P(xs!n)";
nipkow@3040
   553
by (induct_tac "xs" 1);
nipkow@1301
   554
(* case [] *)
nipkow@1301
   555
by (Simp_tac 1);
nipkow@1301
   556
(* case x#xl *)
nipkow@1301
   557
by (rtac allI 1);
berghofe@5183
   558
by (induct_tac "n" 1);
nipkow@5129
   559
by (Auto_tac);
nipkow@1485
   560
qed_spec_mp "list_all_nth";
nipkow@1301
   561
nipkow@4935
   562
Goal "!n. n < length xs --> xs!n mem xs";
nipkow@3040
   563
by (induct_tac "xs" 1);
nipkow@1301
   564
(* case [] *)
nipkow@1301
   565
by (Simp_tac 1);
nipkow@1301
   566
(* case x#xl *)
nipkow@1301
   567
by (rtac allI 1);
berghofe@5183
   568
by (induct_tac "n" 1);
nipkow@1301
   569
(* case 0 *)
nipkow@1301
   570
by (Asm_full_simp_tac 1);
nipkow@1301
   571
(* case Suc x *)
nipkow@4686
   572
by (Asm_full_simp_tac 1);
nipkow@1485
   573
qed_spec_mp "nth_mem";
nipkow@1301
   574
Addsimps [nth_mem];
nipkow@1301
   575
nipkow@5077
   576
(** list update **)
nipkow@5077
   577
nipkow@5077
   578
section "list update";
nipkow@5077
   579
nipkow@5077
   580
Goal "!i. length(xs[i:=x]) = length xs";
nipkow@5077
   581
by (induct_tac "xs" 1);
nipkow@5077
   582
by (Simp_tac 1);
berghofe@5183
   583
by (asm_full_simp_tac (simpset() addsplits [nat.split]) 1);
nipkow@5077
   584
qed_spec_mp "length_list_update";
nipkow@5077
   585
Addsimps [length_list_update];
nipkow@5077
   586
nipkow@5077
   587
nipkow@3896
   588
(** last & butlast **)
nipkow@1327
   589
nipkow@4935
   590
Goal "last(xs@[x]) = x";
wenzelm@4423
   591
by (induct_tac "xs" 1);
nipkow@5129
   592
by (Auto_tac);
nipkow@3896
   593
qed "last_snoc";
nipkow@3896
   594
Addsimps [last_snoc];
nipkow@3896
   595
nipkow@4935
   596
Goal "butlast(xs@[x]) = xs";
wenzelm@4423
   597
by (induct_tac "xs" 1);
nipkow@5129
   598
by (Auto_tac);
nipkow@3896
   599
qed "butlast_snoc";
nipkow@3896
   600
Addsimps [butlast_snoc];
nipkow@3896
   601
nipkow@4935
   602
Goal "length(butlast xs) = length xs - 1";
nipkow@4935
   603
by (res_inst_tac [("xs","xs")] rev_induct 1);
nipkow@5129
   604
by (Auto_tac);
nipkow@4643
   605
qed "length_butlast";
nipkow@4643
   606
Addsimps [length_butlast];
nipkow@4643
   607
paulson@5278
   608
Goal "!ys. butlast (xs@ys) = (if ys=[] then butlast xs else xs@butlast ys)";
wenzelm@4423
   609
by (induct_tac "xs" 1);
nipkow@5129
   610
by (Auto_tac);
nipkow@3896
   611
qed_spec_mp "butlast_append";
nipkow@3896
   612
nipkow@4935
   613
Goal "x:set(butlast xs) --> x:set xs";
wenzelm@4423
   614
by (induct_tac "xs" 1);
nipkow@5129
   615
by (Auto_tac);
nipkow@3896
   616
qed_spec_mp "in_set_butlastD";
nipkow@3896
   617
nipkow@5043
   618
Goal "x:set(butlast xs) ==> x:set(butlast(xs@ys))";
nipkow@4686
   619
by (asm_simp_tac (simpset() addsimps [butlast_append]) 1);
wenzelm@4423
   620
by (blast_tac (claset() addDs [in_set_butlastD]) 1);
nipkow@3896
   621
qed "in_set_butlast_appendI1";
nipkow@3896
   622
nipkow@5043
   623
Goal "x:set(butlast ys) ==> x:set(butlast(xs@ys))";
nipkow@4686
   624
by (asm_simp_tac (simpset() addsimps [butlast_append]) 1);
wenzelm@4423
   625
by (Clarify_tac 1);
wenzelm@4423
   626
by (Full_simp_tac 1);
nipkow@3896
   627
qed "in_set_butlast_appendI2";
nipkow@3902
   628
nipkow@2608
   629
(** take  & drop **)
nipkow@2608
   630
section "take & drop";
nipkow@1327
   631
nipkow@4935
   632
Goal "take 0 xs = []";
nipkow@3040
   633
by (induct_tac "xs" 1);
nipkow@5129
   634
by (Auto_tac);
nipkow@1327
   635
qed "take_0";
nipkow@1327
   636
nipkow@4935
   637
Goal "drop 0 xs = xs";
nipkow@3040
   638
by (induct_tac "xs" 1);
nipkow@5129
   639
by (Auto_tac);
nipkow@2608
   640
qed "drop_0";
nipkow@2608
   641
nipkow@4935
   642
Goal "take (Suc n) (x#xs) = x # take n xs";
paulson@1552
   643
by (Simp_tac 1);
nipkow@1419
   644
qed "take_Suc_Cons";
nipkow@1327
   645
nipkow@4935
   646
Goal "drop (Suc n) (x#xs) = drop n xs";
nipkow@2608
   647
by (Simp_tac 1);
nipkow@2608
   648
qed "drop_Suc_Cons";
nipkow@2608
   649
nipkow@2608
   650
Delsimps [take_Cons,drop_Cons];
nipkow@2608
   651
Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons];
nipkow@2608
   652
nipkow@4935
   653
Goal "!xs. length(take n xs) = min (length xs) n";
berghofe@5183
   654
by (induct_tac "n" 1);
nipkow@5129
   655
 by (Auto_tac);
paulson@3457
   656
by (exhaust_tac "xs" 1);
nipkow@5129
   657
 by (Auto_tac);
nipkow@2608
   658
qed_spec_mp "length_take";
nipkow@2608
   659
Addsimps [length_take];
clasohm@923
   660
nipkow@4935
   661
Goal "!xs. length(drop n xs) = (length xs - n)";
berghofe@5183
   662
by (induct_tac "n" 1);
nipkow@5129
   663
 by (Auto_tac);
paulson@3457
   664
by (exhaust_tac "xs" 1);
nipkow@5129
   665
 by (Auto_tac);
nipkow@2608
   666
qed_spec_mp "length_drop";
nipkow@2608
   667
Addsimps [length_drop];
nipkow@2608
   668
nipkow@4935
   669
Goal "!xs. length xs <= n --> take n xs = xs";
berghofe@5183
   670
by (induct_tac "n" 1);
nipkow@5129
   671
 by (Auto_tac);
paulson@3457
   672
by (exhaust_tac "xs" 1);
nipkow@5129
   673
 by (Auto_tac);
nipkow@2608
   674
qed_spec_mp "take_all";
clasohm@923
   675
nipkow@4935
   676
Goal "!xs. length xs <= n --> drop n xs = []";
berghofe@5183
   677
by (induct_tac "n" 1);
nipkow@5129
   678
 by (Auto_tac);
paulson@3457
   679
by (exhaust_tac "xs" 1);
nipkow@5129
   680
 by (Auto_tac);
nipkow@2608
   681
qed_spec_mp "drop_all";
nipkow@2608
   682
paulson@5278
   683
Goal "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)";
berghofe@5183
   684
by (induct_tac "n" 1);
nipkow@5129
   685
 by (Auto_tac);
paulson@3457
   686
by (exhaust_tac "xs" 1);
nipkow@5129
   687
 by (Auto_tac);
nipkow@2608
   688
qed_spec_mp "take_append";
nipkow@2608
   689
Addsimps [take_append];
nipkow@2608
   690
nipkow@4935
   691
Goal "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; 
berghofe@5183
   692
by (induct_tac "n" 1);
nipkow@5129
   693
 by (Auto_tac);
paulson@3457
   694
by (exhaust_tac "xs" 1);
nipkow@5129
   695
 by (Auto_tac);
nipkow@2608
   696
qed_spec_mp "drop_append";
nipkow@2608
   697
Addsimps [drop_append];
nipkow@2608
   698
nipkow@4935
   699
Goal "!xs n. take n (take m xs) = take (min n m) xs"; 
berghofe@5183
   700
by (induct_tac "m" 1);
nipkow@5129
   701
 by (Auto_tac);
paulson@3457
   702
by (exhaust_tac "xs" 1);
nipkow@5129
   703
 by (Auto_tac);
berghofe@5183
   704
by (exhaust_tac "na" 1);
nipkow@5129
   705
 by (Auto_tac);
nipkow@2608
   706
qed_spec_mp "take_take";
nipkow@2608
   707
nipkow@4935
   708
Goal "!xs. drop n (drop m xs) = drop (n + m) xs"; 
berghofe@5183
   709
by (induct_tac "m" 1);
nipkow@5129
   710
 by (Auto_tac);
paulson@3457
   711
by (exhaust_tac "xs" 1);
nipkow@5129
   712
 by (Auto_tac);
nipkow@2608
   713
qed_spec_mp "drop_drop";
clasohm@923
   714
nipkow@4935
   715
Goal "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; 
berghofe@5183
   716
by (induct_tac "m" 1);
nipkow@5129
   717
 by (Auto_tac);
paulson@3457
   718
by (exhaust_tac "xs" 1);
nipkow@5129
   719
 by (Auto_tac);
nipkow@2608
   720
qed_spec_mp "take_drop";
nipkow@2608
   721
nipkow@4935
   722
Goal "!xs. take n (map f xs) = map f (take n xs)"; 
berghofe@5183
   723
by (induct_tac "n" 1);
nipkow@5129
   724
 by (Auto_tac);
paulson@3457
   725
by (exhaust_tac "xs" 1);
nipkow@5129
   726
 by (Auto_tac);
nipkow@2608
   727
qed_spec_mp "take_map"; 
nipkow@2608
   728
nipkow@4935
   729
Goal "!xs. drop n (map f xs) = map f (drop n xs)"; 
berghofe@5183
   730
by (induct_tac "n" 1);
nipkow@5129
   731
 by (Auto_tac);
paulson@3457
   732
by (exhaust_tac "xs" 1);
nipkow@5129
   733
 by (Auto_tac);
nipkow@2608
   734
qed_spec_mp "drop_map";
nipkow@2608
   735
nipkow@4935
   736
Goal "!n i. i < n --> (take n xs)!i = xs!i";
paulson@3457
   737
by (induct_tac "xs" 1);
nipkow@5129
   738
 by (Auto_tac);
paulson@3457
   739
by (exhaust_tac "n" 1);
paulson@3457
   740
 by (Blast_tac 1);
paulson@3457
   741
by (exhaust_tac "i" 1);
nipkow@5129
   742
 by (Auto_tac);
nipkow@2608
   743
qed_spec_mp "nth_take";
nipkow@2608
   744
Addsimps [nth_take];
clasohm@923
   745
nipkow@4935
   746
Goal  "!xs i. n + i <= length xs --> (drop n xs)!i = xs!(n+i)";
berghofe@5183
   747
by (induct_tac "n" 1);
nipkow@5129
   748
 by (Auto_tac);
paulson@3457
   749
by (exhaust_tac "xs" 1);
nipkow@5129
   750
 by (Auto_tac);
nipkow@2608
   751
qed_spec_mp "nth_drop";
nipkow@2608
   752
Addsimps [nth_drop];
nipkow@2608
   753
nipkow@2608
   754
(** takeWhile & dropWhile **)
nipkow@2608
   755
nipkow@3467
   756
section "takeWhile & dropWhile";
nipkow@3467
   757
nipkow@4935
   758
Goal "takeWhile P xs @ dropWhile P xs = xs";
nipkow@3586
   759
by (induct_tac "xs" 1);
nipkow@5129
   760
by (Auto_tac);
nipkow@3586
   761
qed "takeWhile_dropWhile_id";
nipkow@3586
   762
Addsimps [takeWhile_dropWhile_id];
nipkow@3586
   763
nipkow@4935
   764
Goal  "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs";
paulson@3457
   765
by (induct_tac "xs" 1);
nipkow@5129
   766
by (Auto_tac);
nipkow@2608
   767
bind_thm("takeWhile_append1", conjI RS (result() RS mp));
nipkow@2608
   768
Addsimps [takeWhile_append1];
clasohm@923
   769
nipkow@4935
   770
Goal "(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys";
paulson@3457
   771
by (induct_tac "xs" 1);
nipkow@5129
   772
by (Auto_tac);
nipkow@2608
   773
bind_thm("takeWhile_append2", ballI RS (result() RS mp));
nipkow@2608
   774
Addsimps [takeWhile_append2];
lcp@1169
   775
nipkow@4935
   776
Goal "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys";
paulson@3457
   777
by (induct_tac "xs" 1);
nipkow@5129
   778
by (Auto_tac);
nipkow@2608
   779
bind_thm("dropWhile_append1", conjI RS (result() RS mp));
nipkow@2608
   780
Addsimps [dropWhile_append1];
nipkow@2608
   781
nipkow@4935
   782
Goal "(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys";
paulson@3457
   783
by (induct_tac "xs" 1);
nipkow@5129
   784
by (Auto_tac);
nipkow@2608
   785
bind_thm("dropWhile_append2", ballI RS (result() RS mp));
nipkow@2608
   786
Addsimps [dropWhile_append2];
nipkow@2608
   787
nipkow@4935
   788
Goal "x:set(takeWhile P xs) --> x:set xs & P x";
paulson@3457
   789
by (induct_tac "xs" 1);
nipkow@5129
   790
by (Auto_tac);
paulson@3647
   791
qed_spec_mp"set_take_whileD";
nipkow@2608
   792
oheimb@4132
   793
qed_goal "zip_Nil_Nil"   thy "zip []     []     = []" (K [Simp_tac 1]);
oheimb@4132
   794
qed_goal "zip_Cons_Cons" thy "zip (x#xs) (y#ys) = (x,y)#zip xs ys" 
oheimb@4132
   795
						      (K [Simp_tac 1]);
nipkow@4605
   796
nipkow@5272
   797
nipkow@5272
   798
(** foldl **)
nipkow@5272
   799
section "foldl";
nipkow@5272
   800
nipkow@5272
   801
Goal "!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys";
nipkow@5272
   802
by(induct_tac "xs" 1);
nipkow@5272
   803
by(Auto_tac);
nipkow@5272
   804
qed_spec_mp "foldl_append";
nipkow@5272
   805
Addsimps [foldl_append];
nipkow@5272
   806
nipkow@5272
   807
(* Note: `n <= foldl op+ n ns' looks simpler, but is more difficult to use
nipkow@5272
   808
   because it requires an additional transitivity step
nipkow@5272
   809
*)
nipkow@5272
   810
Goal "!n::nat. m <= n --> m <= foldl op+ n ns";
nipkow@5272
   811
by(induct_tac "ns" 1);
nipkow@5272
   812
 by(Simp_tac 1);
nipkow@5272
   813
by(Asm_full_simp_tac 1);
nipkow@5272
   814
by(blast_tac (claset() addIs [trans_le_add1]) 1);
nipkow@5272
   815
qed_spec_mp "start_le_sum";
nipkow@5272
   816
nipkow@5272
   817
Goal "n : set ns ==> n <= foldl op+ 0 ns";
nipkow@5272
   818
by(auto_tac (claset() addIs [start_le_sum],
nipkow@5272
   819
             simpset() addsimps [in_set_conv_decomp]));
nipkow@5272
   820
qed "elem_le_sum";
nipkow@5272
   821
nipkow@5272
   822
Goal "!m. (foldl op+ m ns = 0) = (m=0 & (!n : set ns. n=0))";
nipkow@5272
   823
by(induct_tac "ns" 1);
nipkow@5272
   824
by(Auto_tac);
nipkow@5272
   825
qed_spec_mp "sum_eq_0_conv";
nipkow@5272
   826
AddIffs [sum_eq_0_conv];
nipkow@5272
   827
nipkow@5272
   828
nipkow@4605
   829
(** nodups & remdups **)
nipkow@4605
   830
section "nodups & remdups";
nipkow@4605
   831
nipkow@4935
   832
Goal "set(remdups xs) = set xs";
nipkow@4605
   833
by (induct_tac "xs" 1);
nipkow@4605
   834
 by (Simp_tac 1);
nipkow@4686
   835
by (asm_full_simp_tac (simpset() addsimps [insert_absorb]) 1);
nipkow@4605
   836
qed "set_remdups";
nipkow@4605
   837
Addsimps [set_remdups];
nipkow@4605
   838
nipkow@4935
   839
Goal "nodups(remdups xs)";
nipkow@4605
   840
by (induct_tac "xs" 1);
nipkow@5129
   841
by (Auto_tac);
nipkow@4605
   842
qed "nodups_remdups";
nipkow@4605
   843
nipkow@4935
   844
Goal "nodups xs --> nodups (filter P xs)";
nipkow@4605
   845
by (induct_tac "xs" 1);
nipkow@5129
   846
by (Auto_tac);
nipkow@4605
   847
qed_spec_mp "nodups_filter";
nipkow@4605
   848
nipkow@3589
   849
(** replicate **)
nipkow@3589
   850
section "replicate";
nipkow@3589
   851
nipkow@4935
   852
Goal "set(replicate (Suc n) x) = {x}";
wenzelm@4423
   853
by (induct_tac "n" 1);
nipkow@5129
   854
by (Auto_tac);
nipkow@3589
   855
val lemma = result();
nipkow@3589
   856
nipkow@5043
   857
Goal "n ~= 0 ==> set(replicate n x) = {x}";
wenzelm@4423
   858
by (fast_tac (claset() addSDs [not0_implies_Suc] addSIs [lemma]) 1);
nipkow@3589
   859
qed "set_replicate";
nipkow@3589
   860
Addsimps [set_replicate];
nipkow@5162
   861
nipkow@5162
   862
nipkow@5281
   863
(*** Lexcicographic orderings on lists ***)
nipkow@5281
   864
section"Lexcicographic orderings on lists";
nipkow@5281
   865
nipkow@5281
   866
Goal "wf r ==> wf(lexn r n)";
nipkow@5281
   867
by(induct_tac "n" 1);
nipkow@5281
   868
by(Simp_tac 1);
nipkow@5281
   869
by(Simp_tac 1);
nipkow@5281
   870
br wf_subset 1;
nipkow@5281
   871
br Int_lower1 2;
nipkow@5281
   872
br wf_prod_fun_image 1;
nipkow@5281
   873
br injI 2;
nipkow@5281
   874
by(Auto_tac);
nipkow@5281
   875
qed "wf_lexn";
nipkow@5281
   876
nipkow@5281
   877
Goal "!xs ys. (xs,ys) : lexn r n --> length xs = n & length ys = n";
nipkow@5281
   878
by(induct_tac "n" 1);
nipkow@5281
   879
by(Auto_tac);
nipkow@5281
   880
qed_spec_mp "lexn_length";
nipkow@5281
   881
nipkow@5281
   882
Goalw [lex_def] "wf r ==> wf(lex r)";
nipkow@5281
   883
br wf_UN 1;
nipkow@5281
   884
by(blast_tac (claset() addIs [wf_lexn]) 1);
nipkow@5281
   885
by(Clarify_tac 1);
nipkow@5281
   886
by(rename_tac "m n" 1);
nipkow@5281
   887
by(subgoal_tac "m ~= n" 1);
nipkow@5281
   888
 by(Blast_tac 2);
nipkow@5281
   889
by(blast_tac (claset() addDs [lexn_length,not_sym]) 1);
nipkow@5281
   890
qed "wf_lex";
nipkow@5281
   891
AddSIs [wf_lex];
nipkow@5281
   892
nipkow@5281
   893
Goal
nipkow@5281
   894
 "lexn r n = \
nipkow@5281
   895
\ {(xs,ys). length xs = n & length ys = n & \
nipkow@5281
   896
\           (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}";
nipkow@5281
   897
by(induct_tac "n" 1);
nipkow@5281
   898
 by(Simp_tac 1);
nipkow@5281
   899
 by(Blast_tac 1);
oheimb@5296
   900
by(asm_full_simp_tac (simpset() delsimps [length_Suc_conv] 
oheimb@5296
   901
				addsimps [lex_prod_def]) 1);
oheimb@5296
   902
by(auto_tac (claset(), simpset() delsimps [length_Suc_conv]));
nipkow@5281
   903
  by(Blast_tac 1);
nipkow@5281
   904
 by(rename_tac "a xys x xs' y ys'" 1);
nipkow@5281
   905
 by(res_inst_tac [("x","a#xys")] exI 1);
nipkow@5281
   906
 by(Simp_tac 1);
nipkow@5281
   907
by(exhaust_tac "xys" 1);
oheimb@5296
   908
 by(ALLGOALS (asm_full_simp_tac (simpset() delsimps [length_Suc_conv])));
nipkow@5281
   909
by(Blast_tac 1);
nipkow@5281
   910
qed "lexn_conv";
nipkow@5281
   911
nipkow@5281
   912
Goalw [lex_def]
nipkow@5281
   913
 "lex r = \
nipkow@5281
   914
\ {(xs,ys). length xs = length ys & \
nipkow@5281
   915
\           (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}";
oheimb@5296
   916
by(force_tac (claset(), simpset() delsimps [length_Suc_conv] addsimps [lexn_conv]) 1);
nipkow@5281
   917
qed "lex_conv";
nipkow@5281
   918
nipkow@5281
   919
Goalw [lexico_def] "wf r ==> wf(lexico r)";
nipkow@5281
   920
by(Blast_tac 1);
nipkow@5281
   921
qed "wf_lexico";
nipkow@5281
   922
AddSIs [wf_lexico];
nipkow@5281
   923
nipkow@5281
   924
Goalw
nipkow@5281
   925
 [lexico_def,diag_def,lex_prod_def,measure_def,inv_image_def]
nipkow@5281
   926
"lexico r = {(xs,ys). length xs < length ys | \
nipkow@5281
   927
\                     length xs = length ys & (xs,ys) : lex r}";
nipkow@5281
   928
by(Simp_tac 1);
nipkow@5281
   929
qed "lexico_conv";
nipkow@5281
   930
nipkow@5283
   931
Goal "([],ys) ~: lex r";
nipkow@5283
   932
by(simp_tac (simpset() addsimps [lex_conv]) 1);
nipkow@5283
   933
qed "Nil_notin_lex";
nipkow@5283
   934
nipkow@5283
   935
Goal "(xs,[]) ~: lex r";
nipkow@5283
   936
by(simp_tac (simpset() addsimps [lex_conv]) 1);
nipkow@5283
   937
qed "Nil2_notin_lex";
nipkow@5283
   938
nipkow@5283
   939
AddIffs [Nil_notin_lex,Nil2_notin_lex];
nipkow@5283
   940
nipkow@5283
   941
Goal "((x#xs,y#ys) : lex r) = \
nipkow@5283
   942
\     ((x,y) : r & length xs = length ys | x=y & (xs,ys) : lex r)";
nipkow@5283
   943
by(simp_tac (simpset() addsimps [lex_conv]) 1);
nipkow@5283
   944
br iffI 1;
nipkow@5283
   945
 by(blast_tac (claset() addIs [Cons_eq_appendI]) 2);
nipkow@5283
   946
by(REPEAT(eresolve_tac [conjE, exE] 1));
nipkow@5283
   947
by(exhaust_tac "xys" 1);
nipkow@5283
   948
by(Asm_full_simp_tac 1);
nipkow@5283
   949
by(Asm_full_simp_tac 1);
nipkow@5283
   950
by(Blast_tac 1);
nipkow@5283
   951
qed "Cons_in_lex";
nipkow@5283
   952
AddIffs [Cons_in_lex];
nipkow@5283
   953
nipkow@5283
   954
nipkow@5162
   955
(***
nipkow@5162
   956
Simplification procedure for all list equalities.
nipkow@5162
   957
Currently only tries to rearranges @ to see if
nipkow@5162
   958
- both lists end in a singleton list,
nipkow@5162
   959
- or both lists end in the same list.
nipkow@5162
   960
***)
nipkow@5162
   961
local
nipkow@5162
   962
nipkow@5162
   963
val list_eq_pattern =
nipkow@5162
   964
  read_cterm (sign_of List.thy) ("(xs::'a list) = ys",HOLogic.boolT);
nipkow@5162
   965
berghofe@5183
   966
fun last (cons as Const("List.list.op #",_) $ _ $ xs) =
berghofe@5183
   967
      (case xs of Const("List.list.[]",_) => cons | _ => last xs)
nipkow@5200
   968
  | last (Const("List.op @",_) $ _ $ ys) = last ys
nipkow@5162
   969
  | last t = t;
nipkow@5162
   970
berghofe@5183
   971
fun list1 (Const("List.list.op #",_) $ _ $ Const("List.list.[]",_)) = true
nipkow@5162
   972
  | list1 _ = false;
nipkow@5162
   973
berghofe@5183
   974
fun butlast ((cons as Const("List.list.op #",_) $ x) $ xs) =
berghofe@5183
   975
      (case xs of Const("List.list.[]",_) => xs | _ => cons $ butlast xs)
nipkow@5200
   976
  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
berghofe@5183
   977
  | butlast xs = Const("List.list.[]",fastype_of xs);
nipkow@5162
   978
nipkow@5162
   979
val rearr_tac =
nipkow@5162
   980
  simp_tac (HOL_basic_ss addsimps [append_assoc,append_Nil,append_Cons]);
nipkow@5162
   981
nipkow@5162
   982
fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
nipkow@5162
   983
  let
nipkow@5162
   984
    val lastl = last lhs and lastr = last rhs
nipkow@5162
   985
    fun rearr conv =
nipkow@5162
   986
      let val lhs1 = butlast lhs and rhs1 = butlast rhs
nipkow@5162
   987
          val Type(_,listT::_) = eqT
nipkow@5162
   988
          val appT = [listT,listT] ---> listT
nipkow@5200
   989
          val app = Const("List.op @",appT)
nipkow@5162
   990
          val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
nipkow@5162
   991
          val ct = cterm_of sg (HOLogic.mk_Trueprop(HOLogic.mk_eq(F,F2)))
nipkow@5162
   992
          val thm = prove_goalw_cterm [] ct (K [rearr_tac 1])
nipkow@5162
   993
            handle ERROR =>
nipkow@5162
   994
            error("The error(s) above occurred while trying to prove " ^
nipkow@5162
   995
                  string_of_cterm ct)
nipkow@5162
   996
      in Some((conv RS (thm RS trans)) RS eq_reflection) end
nipkow@5162
   997
nipkow@5162
   998
  in if list1 lastl andalso list1 lastr
nipkow@5162
   999
     then rearr append1_eq_conv
nipkow@5162
  1000
     else
nipkow@5162
  1001
     if lastl aconv lastr
nipkow@5162
  1002
     then rearr append_same_eq
nipkow@5162
  1003
     else None
nipkow@5162
  1004
  end;
nipkow@5162
  1005
in
nipkow@5162
  1006
val list_eq_simproc = mk_simproc "list_eq" [list_eq_pattern] list_eq;
nipkow@5162
  1007
end;
nipkow@5162
  1008
nipkow@5162
  1009
Addsimprocs [list_eq_simproc];