src/HOL/List.ML
author nipkow
Mon Aug 10 11:30:12 1998 +0200 (1998-08-10)
changeset 5283 0027ddfbc831
parent 5281 f4d16517b360
child 5296 bdef7d349d27
permissions -rw-r--r--
More lemmas about lex.
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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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(** @ - 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] addss simpset()
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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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(** mem **)
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section "mem";
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Goal "x mem (xs@ys) = (x mem xs | x mem ys)";
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by (induct_tac "xs" 1);
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by (Auto_tac);
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qed "mem_append";
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Addsimps[mem_append];
clasohm@923
   351
nipkow@4935
   352
Goal "x mem [x:xs. P(x)] = (x mem xs & P(x))";
nipkow@3040
   353
by (induct_tac "xs" 1);
nipkow@5129
   354
by (Auto_tac);
clasohm@923
   355
qed "mem_filter";
nipkow@2512
   356
Addsimps[mem_filter];
clasohm@923
   357
nipkow@3465
   358
(** set **)
paulson@1812
   359
nipkow@3467
   360
section "set";
nipkow@3467
   361
nipkow@4935
   362
Goal "set (xs@ys) = (set xs Un set ys)";
nipkow@3040
   363
by (induct_tac "xs" 1);
nipkow@5129
   364
by (Auto_tac);
paulson@3647
   365
qed "set_append";
paulson@3647
   366
Addsimps[set_append];
paulson@1812
   367
nipkow@4935
   368
Goal "(x mem xs) = (x: set xs)";
nipkow@3040
   369
by (induct_tac "xs" 1);
nipkow@5129
   370
by (Auto_tac);
paulson@3647
   371
qed "set_mem_eq";
paulson@1812
   372
nipkow@4935
   373
Goal "set l <= set (x#l)";
nipkow@5129
   374
by (Auto_tac);
paulson@3647
   375
qed "set_subset_Cons";
paulson@1936
   376
nipkow@4935
   377
Goal "(set xs = {}) = (xs = [])";
paulson@3457
   378
by (induct_tac "xs" 1);
nipkow@5129
   379
by (Auto_tac);
paulson@3647
   380
qed "set_empty";
paulson@3647
   381
Addsimps [set_empty];
nipkow@2608
   382
nipkow@4935
   383
Goal "set(rev xs) = set(xs)";
paulson@3457
   384
by (induct_tac "xs" 1);
nipkow@5129
   385
by (Auto_tac);
paulson@3647
   386
qed "set_rev";
paulson@3647
   387
Addsimps [set_rev];
nipkow@2608
   388
nipkow@4935
   389
Goal "set(map f xs) = f``(set xs)";
paulson@3457
   390
by (induct_tac "xs" 1);
nipkow@5129
   391
by (Auto_tac);
paulson@3647
   392
qed "set_map";
paulson@3647
   393
Addsimps [set_map];
nipkow@2608
   394
nipkow@4935
   395
Goal "(x : set(filter P xs)) = (x : set xs & P x)";
nipkow@4605
   396
by (induct_tac "xs" 1);
nipkow@5129
   397
by (Auto_tac);
nipkow@4605
   398
qed "in_set_filter";
nipkow@4605
   399
Addsimps [in_set_filter];
nipkow@4605
   400
nipkow@5272
   401
Goal "(x : set xs) = (? ys zs. xs = ys@x#zs)";
nipkow@5272
   402
by(induct_tac "xs" 1);
nipkow@5272
   403
 by(Simp_tac 1);
nipkow@5272
   404
by(Asm_simp_tac 1);
nipkow@5272
   405
br iffI 1;
nipkow@5272
   406
by(blast_tac (claset() addIs [eq_Nil_appendI,Cons_eq_appendI]) 1);
nipkow@5272
   407
by(REPEAT(etac exE 1));
nipkow@5272
   408
by(exhaust_tac "ys" 1);
nipkow@5272
   409
by(Auto_tac);
nipkow@5272
   410
qed "in_set_conv_decomp";
nipkow@5272
   411
nipkow@5272
   412
(* eliminate `lists' in favour of `set' *)
nipkow@5272
   413
nipkow@5272
   414
Goal "(xs : lists A) = (!x : set xs. x : A)";
nipkow@5272
   415
by(induct_tac "xs" 1);
nipkow@5272
   416
by(Auto_tac);
nipkow@5272
   417
qed "in_lists_conv_set";
nipkow@5272
   418
nipkow@5272
   419
bind_thm("in_listsD",in_lists_conv_set RS iffD1);
nipkow@5272
   420
AddSDs [in_listsD];
nipkow@5272
   421
bind_thm("in_listsI",in_lists_conv_set RS iffD2);
nipkow@5272
   422
AddSIs [in_listsI];
paulson@1812
   423
clasohm@923
   424
(** list_all **)
clasohm@923
   425
nipkow@3467
   426
section "list_all";
nipkow@3467
   427
nipkow@4935
   428
Goal "list_all (%x. True) xs = True";
nipkow@3040
   429
by (induct_tac "xs" 1);
nipkow@5129
   430
by (Auto_tac);
clasohm@923
   431
qed "list_all_True";
nipkow@2512
   432
Addsimps [list_all_True];
clasohm@923
   433
nipkow@4935
   434
Goal "list_all p (xs@ys) = (list_all p xs & list_all p ys)";
nipkow@3040
   435
by (induct_tac "xs" 1);
nipkow@5129
   436
by (Auto_tac);
nipkow@2512
   437
qed "list_all_append";
nipkow@2512
   438
Addsimps [list_all_append];
clasohm@923
   439
nipkow@4935
   440
Goal "list_all P xs = (!x. x mem xs --> P(x))";
nipkow@3040
   441
by (induct_tac "xs" 1);
nipkow@5129
   442
by (Auto_tac);
clasohm@923
   443
qed "list_all_mem_conv";
clasohm@923
   444
clasohm@923
   445
nipkow@2608
   446
(** filter **)
clasohm@923
   447
nipkow@3467
   448
section "filter";
nipkow@3467
   449
nipkow@4935
   450
Goal "filter P (xs@ys) = filter P xs @ filter P ys";
paulson@3457
   451
by (induct_tac "xs" 1);
nipkow@5129
   452
by (Auto_tac);
nipkow@2608
   453
qed "filter_append";
nipkow@2608
   454
Addsimps [filter_append];
nipkow@2608
   455
nipkow@4935
   456
Goal "filter (%x. True) xs = xs";
nipkow@4605
   457
by (induct_tac "xs" 1);
nipkow@5129
   458
by (Auto_tac);
nipkow@4605
   459
qed "filter_True";
nipkow@4605
   460
Addsimps [filter_True];
nipkow@4605
   461
nipkow@4935
   462
Goal "filter (%x. False) xs = []";
nipkow@4605
   463
by (induct_tac "xs" 1);
nipkow@5129
   464
by (Auto_tac);
nipkow@4605
   465
qed "filter_False";
nipkow@4605
   466
Addsimps [filter_False];
nipkow@4605
   467
nipkow@4935
   468
Goal "length (filter P xs) <= length xs";
paulson@3457
   469
by (induct_tac "xs" 1);
nipkow@5129
   470
by (Auto_tac);
nipkow@4605
   471
qed "length_filter";
paulson@3383
   472
nipkow@2608
   473
nipkow@2608
   474
(** concat **)
nipkow@2608
   475
nipkow@3467
   476
section "concat";
nipkow@3467
   477
nipkow@4935
   478
Goal  "concat(xs@ys) = concat(xs)@concat(ys)";
nipkow@3040
   479
by (induct_tac "xs" 1);
nipkow@5129
   480
by (Auto_tac);
nipkow@2608
   481
qed"concat_append";
nipkow@2608
   482
Addsimps [concat_append];
nipkow@2512
   483
nipkow@4935
   484
Goal "(concat xss = []) = (!xs:set xss. xs=[])";
wenzelm@4423
   485
by (induct_tac "xss" 1);
nipkow@5129
   486
by (Auto_tac);
nipkow@3896
   487
qed "concat_eq_Nil_conv";
nipkow@3896
   488
AddIffs [concat_eq_Nil_conv];
nipkow@3896
   489
nipkow@4935
   490
Goal "([] = concat xss) = (!xs:set xss. xs=[])";
wenzelm@4423
   491
by (induct_tac "xss" 1);
nipkow@5129
   492
by (Auto_tac);
nipkow@3896
   493
qed "Nil_eq_concat_conv";
nipkow@3896
   494
AddIffs [Nil_eq_concat_conv];
nipkow@3896
   495
nipkow@4935
   496
Goal  "set(concat xs) = Union(set `` set xs)";
nipkow@3467
   497
by (induct_tac "xs" 1);
nipkow@5129
   498
by (Auto_tac);
paulson@3647
   499
qed"set_concat";
paulson@3647
   500
Addsimps [set_concat];
nipkow@3467
   501
nipkow@4935
   502
Goal "map f (concat xs) = concat (map (map f) xs)"; 
nipkow@3467
   503
by (induct_tac "xs" 1);
nipkow@5129
   504
by (Auto_tac);
nipkow@3467
   505
qed "map_concat";
nipkow@3467
   506
nipkow@4935
   507
Goal "filter p (concat xs) = concat (map (filter p) xs)"; 
nipkow@3467
   508
by (induct_tac "xs" 1);
nipkow@5129
   509
by (Auto_tac);
nipkow@3467
   510
qed"filter_concat"; 
nipkow@3467
   511
nipkow@4935
   512
Goal "rev(concat xs) = concat (map rev (rev xs))";
nipkow@3467
   513
by (induct_tac "xs" 1);
nipkow@5129
   514
by (Auto_tac);
nipkow@2608
   515
qed "rev_concat";
clasohm@923
   516
clasohm@923
   517
(** nth **)
clasohm@923
   518
nipkow@3467
   519
section "nth";
nipkow@3467
   520
paulson@5278
   521
Goal "!xs. (xs@ys)!n = (if n < length xs then xs!n else ys!(n - length xs))";
berghofe@5183
   522
by (induct_tac "n" 1);
paulson@3457
   523
 by (Asm_simp_tac 1);
paulson@3457
   524
 by (rtac allI 1);
paulson@3457
   525
 by (exhaust_tac "xs" 1);
nipkow@5129
   526
  by (Auto_tac);
nipkow@2608
   527
qed_spec_mp "nth_append";
nipkow@2608
   528
nipkow@4935
   529
Goal "!n. n < length xs --> (map f xs)!n = f(xs!n)";
nipkow@3040
   530
by (induct_tac "xs" 1);
nipkow@1301
   531
(* case [] *)
nipkow@1301
   532
by (Asm_full_simp_tac 1);
nipkow@1301
   533
(* case x#xl *)
nipkow@1301
   534
by (rtac allI 1);
berghofe@5183
   535
by (induct_tac "n" 1);
nipkow@5129
   536
by (Auto_tac);
nipkow@1485
   537
qed_spec_mp "nth_map";
nipkow@1301
   538
Addsimps [nth_map];
nipkow@1301
   539
nipkow@4935
   540
Goal "!n. n < length xs --> list_all P xs --> P(xs!n)";
nipkow@3040
   541
by (induct_tac "xs" 1);
nipkow@1301
   542
(* case [] *)
nipkow@1301
   543
by (Simp_tac 1);
nipkow@1301
   544
(* case x#xl *)
nipkow@1301
   545
by (rtac allI 1);
berghofe@5183
   546
by (induct_tac "n" 1);
nipkow@5129
   547
by (Auto_tac);
nipkow@1485
   548
qed_spec_mp "list_all_nth";
nipkow@1301
   549
nipkow@4935
   550
Goal "!n. n < length xs --> xs!n mem xs";
nipkow@3040
   551
by (induct_tac "xs" 1);
nipkow@1301
   552
(* case [] *)
nipkow@1301
   553
by (Simp_tac 1);
nipkow@1301
   554
(* case x#xl *)
nipkow@1301
   555
by (rtac allI 1);
berghofe@5183
   556
by (induct_tac "n" 1);
nipkow@1301
   557
(* case 0 *)
nipkow@1301
   558
by (Asm_full_simp_tac 1);
nipkow@1301
   559
(* case Suc x *)
nipkow@4686
   560
by (Asm_full_simp_tac 1);
nipkow@1485
   561
qed_spec_mp "nth_mem";
nipkow@1301
   562
Addsimps [nth_mem];
nipkow@1301
   563
nipkow@5077
   564
(** list update **)
nipkow@5077
   565
nipkow@5077
   566
section "list update";
nipkow@5077
   567
nipkow@5077
   568
Goal "!i. length(xs[i:=x]) = length xs";
nipkow@5077
   569
by (induct_tac "xs" 1);
nipkow@5077
   570
by (Simp_tac 1);
berghofe@5183
   571
by (asm_full_simp_tac (simpset() addsplits [nat.split]) 1);
nipkow@5077
   572
qed_spec_mp "length_list_update";
nipkow@5077
   573
Addsimps [length_list_update];
nipkow@5077
   574
nipkow@5077
   575
nipkow@3896
   576
(** last & butlast **)
nipkow@1327
   577
nipkow@4935
   578
Goal "last(xs@[x]) = x";
wenzelm@4423
   579
by (induct_tac "xs" 1);
nipkow@5129
   580
by (Auto_tac);
nipkow@3896
   581
qed "last_snoc";
nipkow@3896
   582
Addsimps [last_snoc];
nipkow@3896
   583
nipkow@4935
   584
Goal "butlast(xs@[x]) = xs";
wenzelm@4423
   585
by (induct_tac "xs" 1);
nipkow@5129
   586
by (Auto_tac);
nipkow@3896
   587
qed "butlast_snoc";
nipkow@3896
   588
Addsimps [butlast_snoc];
nipkow@3896
   589
nipkow@4935
   590
Goal "length(butlast xs) = length xs - 1";
nipkow@4935
   591
by (res_inst_tac [("xs","xs")] rev_induct 1);
nipkow@5129
   592
by (Auto_tac);
nipkow@4643
   593
qed "length_butlast";
nipkow@4643
   594
Addsimps [length_butlast];
nipkow@4643
   595
paulson@5278
   596
Goal "!ys. butlast (xs@ys) = (if ys=[] then butlast xs else xs@butlast ys)";
wenzelm@4423
   597
by (induct_tac "xs" 1);
nipkow@5129
   598
by (Auto_tac);
nipkow@3896
   599
qed_spec_mp "butlast_append";
nipkow@3896
   600
nipkow@4935
   601
Goal "x:set(butlast xs) --> x:set xs";
wenzelm@4423
   602
by (induct_tac "xs" 1);
nipkow@5129
   603
by (Auto_tac);
nipkow@3896
   604
qed_spec_mp "in_set_butlastD";
nipkow@3896
   605
nipkow@5043
   606
Goal "x:set(butlast xs) ==> x:set(butlast(xs@ys))";
nipkow@4686
   607
by (asm_simp_tac (simpset() addsimps [butlast_append]) 1);
wenzelm@4423
   608
by (blast_tac (claset() addDs [in_set_butlastD]) 1);
nipkow@3896
   609
qed "in_set_butlast_appendI1";
nipkow@3896
   610
nipkow@5043
   611
Goal "x:set(butlast ys) ==> x:set(butlast(xs@ys))";
nipkow@4686
   612
by (asm_simp_tac (simpset() addsimps [butlast_append]) 1);
wenzelm@4423
   613
by (Clarify_tac 1);
wenzelm@4423
   614
by (Full_simp_tac 1);
nipkow@3896
   615
qed "in_set_butlast_appendI2";
nipkow@3902
   616
nipkow@2608
   617
(** take  & drop **)
nipkow@2608
   618
section "take & drop";
nipkow@1327
   619
nipkow@4935
   620
Goal "take 0 xs = []";
nipkow@3040
   621
by (induct_tac "xs" 1);
nipkow@5129
   622
by (Auto_tac);
nipkow@1327
   623
qed "take_0";
nipkow@1327
   624
nipkow@4935
   625
Goal "drop 0 xs = xs";
nipkow@3040
   626
by (induct_tac "xs" 1);
nipkow@5129
   627
by (Auto_tac);
nipkow@2608
   628
qed "drop_0";
nipkow@2608
   629
nipkow@4935
   630
Goal "take (Suc n) (x#xs) = x # take n xs";
paulson@1552
   631
by (Simp_tac 1);
nipkow@1419
   632
qed "take_Suc_Cons";
nipkow@1327
   633
nipkow@4935
   634
Goal "drop (Suc n) (x#xs) = drop n xs";
nipkow@2608
   635
by (Simp_tac 1);
nipkow@2608
   636
qed "drop_Suc_Cons";
nipkow@2608
   637
nipkow@2608
   638
Delsimps [take_Cons,drop_Cons];
nipkow@2608
   639
Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons];
nipkow@2608
   640
nipkow@4935
   641
Goal "!xs. length(take n xs) = min (length xs) n";
berghofe@5183
   642
by (induct_tac "n" 1);
nipkow@5129
   643
 by (Auto_tac);
paulson@3457
   644
by (exhaust_tac "xs" 1);
nipkow@5129
   645
 by (Auto_tac);
nipkow@2608
   646
qed_spec_mp "length_take";
nipkow@2608
   647
Addsimps [length_take];
clasohm@923
   648
nipkow@4935
   649
Goal "!xs. length(drop n xs) = (length xs - n)";
berghofe@5183
   650
by (induct_tac "n" 1);
nipkow@5129
   651
 by (Auto_tac);
paulson@3457
   652
by (exhaust_tac "xs" 1);
nipkow@5129
   653
 by (Auto_tac);
nipkow@2608
   654
qed_spec_mp "length_drop";
nipkow@2608
   655
Addsimps [length_drop];
nipkow@2608
   656
nipkow@4935
   657
Goal "!xs. length xs <= n --> take n xs = xs";
berghofe@5183
   658
by (induct_tac "n" 1);
nipkow@5129
   659
 by (Auto_tac);
paulson@3457
   660
by (exhaust_tac "xs" 1);
nipkow@5129
   661
 by (Auto_tac);
nipkow@2608
   662
qed_spec_mp "take_all";
clasohm@923
   663
nipkow@4935
   664
Goal "!xs. length xs <= n --> drop n xs = []";
berghofe@5183
   665
by (induct_tac "n" 1);
nipkow@5129
   666
 by (Auto_tac);
paulson@3457
   667
by (exhaust_tac "xs" 1);
nipkow@5129
   668
 by (Auto_tac);
nipkow@2608
   669
qed_spec_mp "drop_all";
nipkow@2608
   670
paulson@5278
   671
Goal "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)";
berghofe@5183
   672
by (induct_tac "n" 1);
nipkow@5129
   673
 by (Auto_tac);
paulson@3457
   674
by (exhaust_tac "xs" 1);
nipkow@5129
   675
 by (Auto_tac);
nipkow@2608
   676
qed_spec_mp "take_append";
nipkow@2608
   677
Addsimps [take_append];
nipkow@2608
   678
nipkow@4935
   679
Goal "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; 
berghofe@5183
   680
by (induct_tac "n" 1);
nipkow@5129
   681
 by (Auto_tac);
paulson@3457
   682
by (exhaust_tac "xs" 1);
nipkow@5129
   683
 by (Auto_tac);
nipkow@2608
   684
qed_spec_mp "drop_append";
nipkow@2608
   685
Addsimps [drop_append];
nipkow@2608
   686
nipkow@4935
   687
Goal "!xs n. take n (take m xs) = take (min n m) xs"; 
berghofe@5183
   688
by (induct_tac "m" 1);
nipkow@5129
   689
 by (Auto_tac);
paulson@3457
   690
by (exhaust_tac "xs" 1);
nipkow@5129
   691
 by (Auto_tac);
berghofe@5183
   692
by (exhaust_tac "na" 1);
nipkow@5129
   693
 by (Auto_tac);
nipkow@2608
   694
qed_spec_mp "take_take";
nipkow@2608
   695
nipkow@4935
   696
Goal "!xs. drop n (drop m xs) = drop (n + m) xs"; 
berghofe@5183
   697
by (induct_tac "m" 1);
nipkow@5129
   698
 by (Auto_tac);
paulson@3457
   699
by (exhaust_tac "xs" 1);
nipkow@5129
   700
 by (Auto_tac);
nipkow@2608
   701
qed_spec_mp "drop_drop";
clasohm@923
   702
nipkow@4935
   703
Goal "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; 
berghofe@5183
   704
by (induct_tac "m" 1);
nipkow@5129
   705
 by (Auto_tac);
paulson@3457
   706
by (exhaust_tac "xs" 1);
nipkow@5129
   707
 by (Auto_tac);
nipkow@2608
   708
qed_spec_mp "take_drop";
nipkow@2608
   709
nipkow@4935
   710
Goal "!xs. take n (map f xs) = map f (take n xs)"; 
berghofe@5183
   711
by (induct_tac "n" 1);
nipkow@5129
   712
 by (Auto_tac);
paulson@3457
   713
by (exhaust_tac "xs" 1);
nipkow@5129
   714
 by (Auto_tac);
nipkow@2608
   715
qed_spec_mp "take_map"; 
nipkow@2608
   716
nipkow@4935
   717
Goal "!xs. drop n (map f xs) = map f (drop n xs)"; 
berghofe@5183
   718
by (induct_tac "n" 1);
nipkow@5129
   719
 by (Auto_tac);
paulson@3457
   720
by (exhaust_tac "xs" 1);
nipkow@5129
   721
 by (Auto_tac);
nipkow@2608
   722
qed_spec_mp "drop_map";
nipkow@2608
   723
nipkow@4935
   724
Goal "!n i. i < n --> (take n xs)!i = xs!i";
paulson@3457
   725
by (induct_tac "xs" 1);
nipkow@5129
   726
 by (Auto_tac);
paulson@3457
   727
by (exhaust_tac "n" 1);
paulson@3457
   728
 by (Blast_tac 1);
paulson@3457
   729
by (exhaust_tac "i" 1);
nipkow@5129
   730
 by (Auto_tac);
nipkow@2608
   731
qed_spec_mp "nth_take";
nipkow@2608
   732
Addsimps [nth_take];
clasohm@923
   733
nipkow@4935
   734
Goal  "!xs i. n + i <= length xs --> (drop n xs)!i = xs!(n+i)";
berghofe@5183
   735
by (induct_tac "n" 1);
nipkow@5129
   736
 by (Auto_tac);
paulson@3457
   737
by (exhaust_tac "xs" 1);
nipkow@5129
   738
 by (Auto_tac);
nipkow@2608
   739
qed_spec_mp "nth_drop";
nipkow@2608
   740
Addsimps [nth_drop];
nipkow@2608
   741
nipkow@2608
   742
(** takeWhile & dropWhile **)
nipkow@2608
   743
nipkow@3467
   744
section "takeWhile & dropWhile";
nipkow@3467
   745
nipkow@4935
   746
Goal "takeWhile P xs @ dropWhile P xs = xs";
nipkow@3586
   747
by (induct_tac "xs" 1);
nipkow@5129
   748
by (Auto_tac);
nipkow@3586
   749
qed "takeWhile_dropWhile_id";
nipkow@3586
   750
Addsimps [takeWhile_dropWhile_id];
nipkow@3586
   751
nipkow@4935
   752
Goal  "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs";
paulson@3457
   753
by (induct_tac "xs" 1);
nipkow@5129
   754
by (Auto_tac);
nipkow@2608
   755
bind_thm("takeWhile_append1", conjI RS (result() RS mp));
nipkow@2608
   756
Addsimps [takeWhile_append1];
clasohm@923
   757
nipkow@4935
   758
Goal "(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys";
paulson@3457
   759
by (induct_tac "xs" 1);
nipkow@5129
   760
by (Auto_tac);
nipkow@2608
   761
bind_thm("takeWhile_append2", ballI RS (result() RS mp));
nipkow@2608
   762
Addsimps [takeWhile_append2];
lcp@1169
   763
nipkow@4935
   764
Goal "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys";
paulson@3457
   765
by (induct_tac "xs" 1);
nipkow@5129
   766
by (Auto_tac);
nipkow@2608
   767
bind_thm("dropWhile_append1", conjI RS (result() RS mp));
nipkow@2608
   768
Addsimps [dropWhile_append1];
nipkow@2608
   769
nipkow@4935
   770
Goal "(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys";
paulson@3457
   771
by (induct_tac "xs" 1);
nipkow@5129
   772
by (Auto_tac);
nipkow@2608
   773
bind_thm("dropWhile_append2", ballI RS (result() RS mp));
nipkow@2608
   774
Addsimps [dropWhile_append2];
nipkow@2608
   775
nipkow@4935
   776
Goal "x:set(takeWhile P xs) --> x:set xs & P x";
paulson@3457
   777
by (induct_tac "xs" 1);
nipkow@5129
   778
by (Auto_tac);
paulson@3647
   779
qed_spec_mp"set_take_whileD";
nipkow@2608
   780
oheimb@4132
   781
qed_goal "zip_Nil_Nil"   thy "zip []     []     = []" (K [Simp_tac 1]);
oheimb@4132
   782
qed_goal "zip_Cons_Cons" thy "zip (x#xs) (y#ys) = (x,y)#zip xs ys" 
oheimb@4132
   783
						      (K [Simp_tac 1]);
nipkow@4605
   784
nipkow@5272
   785
nipkow@5272
   786
(** foldl **)
nipkow@5272
   787
section "foldl";
nipkow@5272
   788
nipkow@5272
   789
Goal "!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys";
nipkow@5272
   790
by(induct_tac "xs" 1);
nipkow@5272
   791
by(Auto_tac);
nipkow@5272
   792
qed_spec_mp "foldl_append";
nipkow@5272
   793
Addsimps [foldl_append];
nipkow@5272
   794
nipkow@5272
   795
(* Note: `n <= foldl op+ n ns' looks simpler, but is more difficult to use
nipkow@5272
   796
   because it requires an additional transitivity step
nipkow@5272
   797
*)
nipkow@5272
   798
Goal "!n::nat. m <= n --> m <= foldl op+ n ns";
nipkow@5272
   799
by(induct_tac "ns" 1);
nipkow@5272
   800
 by(Simp_tac 1);
nipkow@5272
   801
by(Asm_full_simp_tac 1);
nipkow@5272
   802
by(blast_tac (claset() addIs [trans_le_add1]) 1);
nipkow@5272
   803
qed_spec_mp "start_le_sum";
nipkow@5272
   804
nipkow@5272
   805
Goal "n : set ns ==> n <= foldl op+ 0 ns";
nipkow@5272
   806
by(auto_tac (claset() addIs [start_le_sum],
nipkow@5272
   807
             simpset() addsimps [in_set_conv_decomp]));
nipkow@5272
   808
qed "elem_le_sum";
nipkow@5272
   809
nipkow@5272
   810
Goal "!m. (foldl op+ m ns = 0) = (m=0 & (!n : set ns. n=0))";
nipkow@5272
   811
by(induct_tac "ns" 1);
nipkow@5272
   812
by(Auto_tac);
nipkow@5272
   813
qed_spec_mp "sum_eq_0_conv";
nipkow@5272
   814
AddIffs [sum_eq_0_conv];
nipkow@5272
   815
nipkow@5272
   816
nipkow@4605
   817
(** nodups & remdups **)
nipkow@4605
   818
section "nodups & remdups";
nipkow@4605
   819
nipkow@4935
   820
Goal "set(remdups xs) = set xs";
nipkow@4605
   821
by (induct_tac "xs" 1);
nipkow@4605
   822
 by (Simp_tac 1);
nipkow@4686
   823
by (asm_full_simp_tac (simpset() addsimps [insert_absorb]) 1);
nipkow@4605
   824
qed "set_remdups";
nipkow@4605
   825
Addsimps [set_remdups];
nipkow@4605
   826
nipkow@4935
   827
Goal "nodups(remdups xs)";
nipkow@4605
   828
by (induct_tac "xs" 1);
nipkow@5129
   829
by (Auto_tac);
nipkow@4605
   830
qed "nodups_remdups";
nipkow@4605
   831
nipkow@4935
   832
Goal "nodups xs --> nodups (filter P xs)";
nipkow@4605
   833
by (induct_tac "xs" 1);
nipkow@5129
   834
by (Auto_tac);
nipkow@4605
   835
qed_spec_mp "nodups_filter";
nipkow@4605
   836
nipkow@3589
   837
(** replicate **)
nipkow@3589
   838
section "replicate";
nipkow@3589
   839
nipkow@4935
   840
Goal "set(replicate (Suc n) x) = {x}";
wenzelm@4423
   841
by (induct_tac "n" 1);
nipkow@5129
   842
by (Auto_tac);
nipkow@3589
   843
val lemma = result();
nipkow@3589
   844
nipkow@5043
   845
Goal "n ~= 0 ==> set(replicate n x) = {x}";
wenzelm@4423
   846
by (fast_tac (claset() addSDs [not0_implies_Suc] addSIs [lemma]) 1);
nipkow@3589
   847
qed "set_replicate";
nipkow@3589
   848
Addsimps [set_replicate];
nipkow@5162
   849
nipkow@5162
   850
nipkow@5281
   851
(*** Lexcicographic orderings on lists ***)
nipkow@5281
   852
section"Lexcicographic orderings on lists";
nipkow@5281
   853
nipkow@5281
   854
Goal "wf r ==> wf(lexn r n)";
nipkow@5281
   855
by(induct_tac "n" 1);
nipkow@5281
   856
by(Simp_tac 1);
nipkow@5281
   857
by(Simp_tac 1);
nipkow@5281
   858
br wf_subset 1;
nipkow@5281
   859
br Int_lower1 2;
nipkow@5281
   860
br wf_prod_fun_image 1;
nipkow@5281
   861
br injI 2;
nipkow@5281
   862
by(Auto_tac);
nipkow@5281
   863
qed "wf_lexn";
nipkow@5281
   864
nipkow@5281
   865
Goal "!xs ys. (xs,ys) : lexn r n --> length xs = n & length ys = n";
nipkow@5281
   866
by(induct_tac "n" 1);
nipkow@5281
   867
by(Auto_tac);
nipkow@5281
   868
qed_spec_mp "lexn_length";
nipkow@5281
   869
nipkow@5281
   870
Goalw [lex_def] "wf r ==> wf(lex r)";
nipkow@5281
   871
br wf_UN 1;
nipkow@5281
   872
by(blast_tac (claset() addIs [wf_lexn]) 1);
nipkow@5281
   873
by(Clarify_tac 1);
nipkow@5281
   874
by(rename_tac "m n" 1);
nipkow@5281
   875
by(subgoal_tac "m ~= n" 1);
nipkow@5281
   876
 by(Blast_tac 2);
nipkow@5281
   877
by(blast_tac (claset() addDs [lexn_length,not_sym]) 1);
nipkow@5281
   878
qed "wf_lex";
nipkow@5281
   879
AddSIs [wf_lex];
nipkow@5281
   880
nipkow@5281
   881
Goal
nipkow@5281
   882
 "lexn r n = \
nipkow@5281
   883
\ {(xs,ys). length xs = n & length ys = n & \
nipkow@5281
   884
\           (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}";
nipkow@5281
   885
by(induct_tac "n" 1);
nipkow@5281
   886
 by(Simp_tac 1);
nipkow@5281
   887
 by(Blast_tac 1);
nipkow@5281
   888
by(asm_full_simp_tac (simpset() addsimps [lex_prod_def]) 1);
nipkow@5281
   889
by(Auto_tac);
nipkow@5281
   890
  by(Blast_tac 1);
nipkow@5281
   891
 by(rename_tac "a xys x xs' y ys'" 1);
nipkow@5281
   892
 by(res_inst_tac [("x","a#xys")] exI 1);
nipkow@5281
   893
 by(Simp_tac 1);
nipkow@5281
   894
by(exhaust_tac "xys" 1);
nipkow@5281
   895
 by(ALLGOALS Asm_full_simp_tac);
nipkow@5281
   896
by(Blast_tac 1);
nipkow@5281
   897
qed "lexn_conv";
nipkow@5281
   898
nipkow@5281
   899
Goalw [lex_def]
nipkow@5281
   900
 "lex r = \
nipkow@5281
   901
\ {(xs,ys). length xs = length ys & \
nipkow@5281
   902
\           (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}";
nipkow@5281
   903
by(force_tac (claset(), simpset() addsimps [lexn_conv]) 1);
nipkow@5281
   904
qed "lex_conv";
nipkow@5281
   905
nipkow@5281
   906
Goalw [lexico_def] "wf r ==> wf(lexico r)";
nipkow@5281
   907
by(Blast_tac 1);
nipkow@5281
   908
qed "wf_lexico";
nipkow@5281
   909
AddSIs [wf_lexico];
nipkow@5281
   910
nipkow@5281
   911
Goalw
nipkow@5281
   912
 [lexico_def,diag_def,lex_prod_def,measure_def,inv_image_def]
nipkow@5281
   913
"lexico r = {(xs,ys). length xs < length ys | \
nipkow@5281
   914
\                     length xs = length ys & (xs,ys) : lex r}";
nipkow@5281
   915
by(Simp_tac 1);
nipkow@5281
   916
qed "lexico_conv";
nipkow@5281
   917
nipkow@5283
   918
Goal "([],ys) ~: lex r";
nipkow@5283
   919
by(simp_tac (simpset() addsimps [lex_conv]) 1);
nipkow@5283
   920
qed "Nil_notin_lex";
nipkow@5283
   921
nipkow@5283
   922
Goal "(xs,[]) ~: lex r";
nipkow@5283
   923
by(simp_tac (simpset() addsimps [lex_conv]) 1);
nipkow@5283
   924
qed "Nil2_notin_lex";
nipkow@5283
   925
nipkow@5283
   926
AddIffs [Nil_notin_lex,Nil2_notin_lex];
nipkow@5283
   927
nipkow@5283
   928
Goal "((x#xs,y#ys) : lex r) = \
nipkow@5283
   929
\     ((x,y) : r & length xs = length ys | x=y & (xs,ys) : lex r)";
nipkow@5283
   930
by(simp_tac (simpset() addsimps [lex_conv]) 1);
nipkow@5283
   931
br iffI 1;
nipkow@5283
   932
 by(blast_tac (claset() addIs [Cons_eq_appendI]) 2);
nipkow@5283
   933
by(REPEAT(eresolve_tac [conjE, exE] 1));
nipkow@5283
   934
by(exhaust_tac "xys" 1);
nipkow@5283
   935
by(Asm_full_simp_tac 1);
nipkow@5283
   936
by(Asm_full_simp_tac 1);
nipkow@5283
   937
by(Blast_tac 1);
nipkow@5283
   938
qed "Cons_in_lex";
nipkow@5283
   939
AddIffs [Cons_in_lex];
nipkow@5283
   940
nipkow@5283
   941
nipkow@5162
   942
(***
nipkow@5162
   943
Simplification procedure for all list equalities.
nipkow@5162
   944
Currently only tries to rearranges @ to see if
nipkow@5162
   945
- both lists end in a singleton list,
nipkow@5162
   946
- or both lists end in the same list.
nipkow@5162
   947
***)
nipkow@5162
   948
local
nipkow@5162
   949
nipkow@5162
   950
val list_eq_pattern =
nipkow@5162
   951
  read_cterm (sign_of List.thy) ("(xs::'a list) = ys",HOLogic.boolT);
nipkow@5162
   952
berghofe@5183
   953
fun last (cons as Const("List.list.op #",_) $ _ $ xs) =
berghofe@5183
   954
      (case xs of Const("List.list.[]",_) => cons | _ => last xs)
nipkow@5200
   955
  | last (Const("List.op @",_) $ _ $ ys) = last ys
nipkow@5162
   956
  | last t = t;
nipkow@5162
   957
berghofe@5183
   958
fun list1 (Const("List.list.op #",_) $ _ $ Const("List.list.[]",_)) = true
nipkow@5162
   959
  | list1 _ = false;
nipkow@5162
   960
berghofe@5183
   961
fun butlast ((cons as Const("List.list.op #",_) $ x) $ xs) =
berghofe@5183
   962
      (case xs of Const("List.list.[]",_) => xs | _ => cons $ butlast xs)
nipkow@5200
   963
  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
berghofe@5183
   964
  | butlast xs = Const("List.list.[]",fastype_of xs);
nipkow@5162
   965
nipkow@5162
   966
val rearr_tac =
nipkow@5162
   967
  simp_tac (HOL_basic_ss addsimps [append_assoc,append_Nil,append_Cons]);
nipkow@5162
   968
nipkow@5162
   969
fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
nipkow@5162
   970
  let
nipkow@5162
   971
    val lastl = last lhs and lastr = last rhs
nipkow@5162
   972
    fun rearr conv =
nipkow@5162
   973
      let val lhs1 = butlast lhs and rhs1 = butlast rhs
nipkow@5162
   974
          val Type(_,listT::_) = eqT
nipkow@5162
   975
          val appT = [listT,listT] ---> listT
nipkow@5200
   976
          val app = Const("List.op @",appT)
nipkow@5162
   977
          val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
nipkow@5162
   978
          val ct = cterm_of sg (HOLogic.mk_Trueprop(HOLogic.mk_eq(F,F2)))
nipkow@5162
   979
          val thm = prove_goalw_cterm [] ct (K [rearr_tac 1])
nipkow@5162
   980
            handle ERROR =>
nipkow@5162
   981
            error("The error(s) above occurred while trying to prove " ^
nipkow@5162
   982
                  string_of_cterm ct)
nipkow@5162
   983
      in Some((conv RS (thm RS trans)) RS eq_reflection) end
nipkow@5162
   984
nipkow@5162
   985
  in if list1 lastl andalso list1 lastr
nipkow@5162
   986
     then rearr append1_eq_conv
nipkow@5162
   987
     else
nipkow@5162
   988
     if lastl aconv lastr
nipkow@5162
   989
     then rearr append_same_eq
nipkow@5162
   990
     else None
nipkow@5162
   991
  end;
nipkow@5162
   992
in
nipkow@5162
   993
val list_eq_simproc = mk_simproc "list_eq" [list_eq_pattern] list_eq;
nipkow@5162
   994
end;
nipkow@5162
   995
nipkow@5162
   996
Addsimprocs [list_eq_simproc];