src/HOL/List.ML
author paulson
Thu, 01 Jun 2000 13:28:00 +0200
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simplified the proof of nth_upt
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(*  Title:      HOL/List
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    Copyright   1994 TU Muenchen
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List lemmas
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*)
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Goal "!x. xs ~= x#xs";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed_spec_mp "not_Cons_self";
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bind_thm("not_Cons_self2",not_Cons_self RS not_sym);
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Addsimps [not_Cons_self,not_Cons_self2];
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Goal "(xs ~= []) = (? y ys. xs = y#ys)";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "neq_Nil_conv";
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(* Induction over the length of a list: *)
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val [prem] = Goal
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  "(!!xs. (!ys. length ys < length xs --> P ys) ==> P xs) ==> P(xs)";
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by (rtac measure_induct 1 THEN etac prem 1);
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qed "length_induct";
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(** "lists": the list-forming operator over sets **)
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Goalw lists.defs "A<=B ==> lists A <= lists B";
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by (rtac lfp_mono 1);
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by (REPEAT (ares_tac basic_monos 1));
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qed "lists_mono";
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val listsE = lists.mk_cases "x#l : lists A";
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AddSEs [listsE];
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AddSIs lists.intrs;
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Goal "l: lists A ==> l: lists B --> l: lists (A Int B)";
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by (etac lists.induct 1);
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by (ALLGOALS Blast_tac);
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qed_spec_mp "lists_IntI";
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Goal "lists (A Int B) = lists A Int lists B";
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by (rtac (mono_Int RS equalityI) 1);
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by (simp_tac (simpset() addsimps [mono_def, lists_mono]) 1);
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by (blast_tac (claset() addSIs [lists_IntI]) 1);
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qed "lists_Int_eq";
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Addsimps [lists_Int_eq];
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(**  Case analysis **)
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section "Case analysis";
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val prems = Goal "[| P([]); !!x xs. P(x#xs) |] ==> P(xs)";
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by (induct_tac "xs" 1);
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by (REPEAT(resolve_tac prems 1));
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qed "list_cases";
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Goal "(xs=[] --> P([])) & (!y ys. xs=y#ys --> P(y#ys)) --> P(xs)";
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by (induct_tac "xs" 1);
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by (Blast_tac 1);
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by (Blast_tac 1);
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bind_thm("list_eq_cases",
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  impI RSN (2,allI RSN (2,allI RSN (2,impI RS (conjI RS (result() RS mp))))));
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(** length **)
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(* needs to come before "@" because of thm append_eq_append_conv *)
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section "length";
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Goal "length(xs@ys) = length(xs)+length(ys)";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed"length_append";
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Addsimps [length_append];
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Goal "length (map f xs) = length xs";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "length_map";
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Addsimps [length_map];
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Goal "length(rev xs) = length(xs)";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "length_rev";
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Addsimps [length_rev];
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Goal "length(tl xs) = (length xs) - 1";
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by (case_tac "xs" 1);
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by Auto_tac;
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qed "length_tl";
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Addsimps [length_tl];
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Goal "(length xs = 0) = (xs = [])";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "length_0_conv";
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AddIffs [length_0_conv];
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Goal "(0 = length xs) = (xs = [])";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "zero_length_conv";
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AddIffs [zero_length_conv];
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Goal "(0 < length xs) = (xs ~= [])";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "length_greater_0_conv";
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AddIffs [length_greater_0_conv];
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Goal "(length xs = Suc n) = (? y ys. xs = y#ys & length ys = n)";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "length_Suc_conv";
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(** @ - append **)
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section "@ - append";
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Goal "(xs@ys)@zs = xs@(ys@zs)";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "append_assoc";
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Addsimps [append_assoc];
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Goal "xs @ [] = xs";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "append_Nil2";
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Addsimps [append_Nil2];
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Goal "(xs@ys = []) = (xs=[] & ys=[])";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "append_is_Nil_conv";
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AddIffs [append_is_Nil_conv];
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Goal "([] = xs@ys) = (xs=[] & ys=[])";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "Nil_is_append_conv";
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AddIffs [Nil_is_append_conv];
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Goal "(xs @ ys = xs) = (ys=[])";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "append_self_conv";
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Goal "(xs = xs @ ys) = (ys=[])";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "self_append_conv";
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AddIffs [append_self_conv,self_append_conv];
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Goal "!ys. length xs = length ys | length us = length vs \
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\              --> (xs@us = ys@vs) = (xs=ys & us=vs)";
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by (induct_tac "xs" 1);
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 by (rtac allI 1);
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 by (case_tac "ys" 1);
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  by (Asm_simp_tac 1);
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 by (Force_tac 1);
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by (rtac allI 1);
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by (case_tac "ys" 1);
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by (Force_tac 1);
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by (Asm_simp_tac 1);
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qed_spec_mp "append_eq_append_conv";
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Addsimps [append_eq_append_conv];
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Goal "(xs @ ys = xs @ zs) = (ys=zs)";
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by (Simp_tac 1);
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qed "same_append_eq";
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Goal "(xs @ [x] = ys @ [y]) = (xs = ys & x = y)"; 
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by (Simp_tac 1);
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qed "append1_eq_conv";
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Goal "(ys @ xs = zs @ xs) = (ys=zs)";
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by (Simp_tac 1);
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qed "append_same_eq";
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AddIffs [same_append_eq, append1_eq_conv, append_same_eq];
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Goal "(xs @ ys = ys) = (xs=[])";
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by (cut_inst_tac [("zs","[]")] append_same_eq 1);
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by Auto_tac;
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qed "append_self_conv2";
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Goal "(ys = xs @ ys) = (xs=[])";
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by (simp_tac (simpset() addsimps
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     [simplify (simpset()) (read_instantiate[("ys","[]")]append_same_eq)]) 1);
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by (Blast_tac 1);
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qed "self_append_conv2";
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AddIffs [append_self_conv2,self_append_conv2];
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Goal "xs ~= [] --> hd xs # tl xs = xs";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed_spec_mp "hd_Cons_tl";
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Addsimps [hd_Cons_tl];
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Goal "hd(xs@ys) = (if xs=[] then hd ys else hd xs)";
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by (induct_tac "xs" 1);
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by Auto_tac;
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qed "hd_append";
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Goal "xs ~= [] ==> hd(xs @ ys) = hd xs";
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by (asm_simp_tac (simpset() addsimps [hd_append]
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                           addsplits [list.split]) 1);
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qed "hd_append2";
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Addsimps [hd_append2];
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Goal "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)";
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by (simp_tac (simpset() addsplits [list.split]) 1);
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qed "tl_append";
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Goal "xs ~= [] ==> tl(xs @ ys) = (tl xs) @ ys";
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by (asm_simp_tac (simpset() addsimps [tl_append]
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                           addsplits [list.split]) 1);
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qed "tl_append2";
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Addsimps [tl_append2];
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(* trivial rules for solving @-equations automatically *)
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Goal "xs = ys ==> xs = [] @ ys";
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by (Asm_simp_tac 1);
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qed "eq_Nil_appendI";
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Goal "[| x#xs1 = ys; xs = xs1 @ zs |] ==> x#xs = ys@zs";
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by (dtac sym 1);
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by (Asm_simp_tac 1);
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qed "Cons_eq_appendI";
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Goal "[| xs@xs1 = zs; ys = xs1 @ us |] ==> xs@ys = zs@us";
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by (dtac sym 1);
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by (Asm_simp_tac 1);
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qed "append_eq_appendI";
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5427
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(***
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Simplification procedure for all list equalities.
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Currently only tries to rearranges @ to see if
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- both lists end in a singleton list,
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- or both lists end in the same list.
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***)
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local
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val list_eq_pattern =
6394
3d9fd50fcc43 Theory.sign_of;
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  Thm.read_cterm (Theory.sign_of List.thy) ("(xs::'a list) = ys",HOLogic.boolT);
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7224
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fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
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      (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
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  | last (Const("List.op @",_) $ _ $ ys) = last ys
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  | last t = t;
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fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
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  | list1 _ = false;
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fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
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      (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
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  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
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  | butlast xs = Const("List.list.Nil",fastype_of xs);
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val rearr_tac =
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  simp_tac (HOL_basic_ss addsimps [append_assoc,append_Nil,append_Cons]);
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fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
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  let
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    val lastl = last lhs and lastr = last rhs
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    fun rearr conv =
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      let val lhs1 = butlast lhs and rhs1 = butlast rhs
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          val Type(_,listT::_) = eqT
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          val appT = [listT,listT] ---> listT
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          val app = Const("List.op @",appT)
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          val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
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          val ct = cterm_of sg (HOLogic.mk_Trueprop(HOLogic.mk_eq(F,F2)))
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          val thm = prove_goalw_cterm [] ct (K [rearr_tac 1])
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   280
            handle ERROR =>
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            error("The error(s) above occurred while trying to prove " ^
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                  string_of_cterm ct)
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      in Some((conv RS (thm RS trans)) RS eq_reflection) end
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  in if list1 lastl andalso list1 lastr
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     then rearr append1_eq_conv
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     else
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     if lastl aconv lastr
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     then rearr append_same_eq
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     else None
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   291
  end;
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   292
in
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val list_eq_simproc = mk_simproc "list_eq" [list_eq_pattern] list_eq;
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   294
end;
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Addsimprocs [list_eq_simproc];
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(** map **)
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section "map";
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   302
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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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   307
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Goal "map (%x. x) = (%xs. xs)";
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by (rtac ext 1);
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7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
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by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   311
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   312
qed "map_ident";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   313
Addsimps[map_ident];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   314
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   315
Goal "map f (xs@ys) = map f xs @ map f ys";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   316
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   317
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   318
qed "map_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   319
Addsimps[map_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   320
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   321
Goalw [o_def] "map (f o g) xs = map f (map g xs)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   322
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   323
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   324
qed "map_compose";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   325
Addsimps[map_compose];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   326
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   327
Goal "rev(map f xs) = map f (rev xs)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   328
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   329
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   330
qed "rev_map";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   331
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   332
(* a congruence rule for map: *)
6451
paulson
parents: 6433
diff changeset
   333
Goal "xs=ys ==> (!x. x : set ys --> f x = g x) --> map f xs = map g ys";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   334
by (hyp_subst_tac 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   335
by (induct_tac "ys" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   336
by Auto_tac;
6451
paulson
parents: 6433
diff changeset
   337
bind_thm("map_cong", impI RSN (2,allI RSN (2, result() RS mp)));
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   338
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   339
Goal "(map f xs = []) = (xs = [])";
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   340
by (case_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   341
by Auto_tac;
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   342
qed "map_is_Nil_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   343
AddIffs [map_is_Nil_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   344
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   345
Goal "([] = map f xs) = (xs = [])";
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   346
by (case_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   347
by Auto_tac;
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   348
qed "Nil_is_map_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   349
AddIffs [Nil_is_map_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   350
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   351
Goal "(map f xs = y#ys) = (? x xs'. xs = x#xs' & f x = y & map f xs' = ys)";
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   352
by (case_tac "xs" 1);
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   353
by (ALLGOALS Asm_simp_tac);
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   354
qed "map_eq_Cons";
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   355
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   356
Goal "!xs. map f xs = map f ys --> (!x y. f x = f y --> x=y) --> xs=ys";
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   357
by (induct_tac "ys" 1);
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   358
 by (Asm_simp_tac 1);
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   359
by (fast_tac (claset() addss (simpset() addsimps [map_eq_Cons])) 1);
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   360
qed_spec_mp "map_injective";
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   361
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   362
Goal "inj f ==> inj (map f)";
8064
357652a08ee0 expandshort
paulson
parents: 8009
diff changeset
   363
by (blast_tac (claset() addDs [map_injective,injD] addIs [injI]) 1);
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   364
qed "inj_mapI";
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   365
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   366
Goalw [inj_on_def] "inj (map f) ==> inj f";
8064
357652a08ee0 expandshort
paulson
parents: 8009
diff changeset
   367
by (Clarify_tac 1);
357652a08ee0 expandshort
paulson
parents: 8009
diff changeset
   368
by (eres_inst_tac [("x","[x]")] ballE 1);
357652a08ee0 expandshort
paulson
parents: 8009
diff changeset
   369
 by (eres_inst_tac [("x","[y]")] ballE 1);
357652a08ee0 expandshort
paulson
parents: 8009
diff changeset
   370
  by (Asm_full_simp_tac 1);
357652a08ee0 expandshort
paulson
parents: 8009
diff changeset
   371
 by (Blast_tac 1);
357652a08ee0 expandshort
paulson
parents: 8009
diff changeset
   372
by (Blast_tac 1);
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   373
qed "inj_mapD";
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   374
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   375
Goal "inj (map f) = inj f";
8064
357652a08ee0 expandshort
paulson
parents: 8009
diff changeset
   376
by (blast_tac (claset() addDs [inj_mapD] addIs [inj_mapI]) 1);
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   377
qed "inj_map";
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   378
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   379
(** rev **)
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   380
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   381
section "rev";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   382
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   383
Goal "rev(xs@ys) = rev(ys) @ rev(xs)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   384
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   385
by Auto_tac;
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   386
qed "rev_append";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   387
Addsimps[rev_append];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   388
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   389
Goal "rev(rev l) = l";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   390
by (induct_tac "l" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   391
by Auto_tac;
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   392
qed "rev_rev_ident";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   393
Addsimps[rev_rev_ident];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   394
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   395
Goal "(rev xs = []) = (xs = [])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   396
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   397
by Auto_tac;
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   398
qed "rev_is_Nil_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   399
AddIffs [rev_is_Nil_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   400
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   401
Goal "([] = rev xs) = (xs = [])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   402
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   403
by Auto_tac;
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   404
qed "Nil_is_rev_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   405
AddIffs [Nil_is_rev_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   406
6820
41d9b7bbf968 rev=rev lemma.
nipkow
parents: 6813
diff changeset
   407
Goal "!ys. (rev xs = rev ys) = (xs = ys)";
6831
799859f2e657 expandshort
paulson
parents: 6820
diff changeset
   408
by (induct_tac "xs" 1);
6820
41d9b7bbf968 rev=rev lemma.
nipkow
parents: 6813
diff changeset
   409
 by (Force_tac 1);
6831
799859f2e657 expandshort
paulson
parents: 6820
diff changeset
   410
by (rtac allI 1);
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   411
by (case_tac "ys" 1);
6820
41d9b7bbf968 rev=rev lemma.
nipkow
parents: 6813
diff changeset
   412
 by (Asm_simp_tac 1);
41d9b7bbf968 rev=rev lemma.
nipkow
parents: 6813
diff changeset
   413
by (Force_tac 1);
41d9b7bbf968 rev=rev lemma.
nipkow
parents: 6813
diff changeset
   414
qed_spec_mp "rev_is_rev_conv";
41d9b7bbf968 rev=rev lemma.
nipkow
parents: 6813
diff changeset
   415
AddIffs [rev_is_rev_conv];
41d9b7bbf968 rev=rev lemma.
nipkow
parents: 6813
diff changeset
   416
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   417
val prems = Goal "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P xs";
5132
24f992a25adc isatool expandshort;
wenzelm
parents: 5129
diff changeset
   418
by (stac (rev_rev_ident RS sym) 1);
6162
484adda70b65 expandshort
paulson
parents: 6141
diff changeset
   419
by (res_inst_tac [("list", "rev xs")] list.induct 1);
5132
24f992a25adc isatool expandshort;
wenzelm
parents: 5129
diff changeset
   420
by (ALLGOALS Simp_tac);
24f992a25adc isatool expandshort;
wenzelm
parents: 5129
diff changeset
   421
by (resolve_tac prems 1);
24f992a25adc isatool expandshort;
wenzelm
parents: 5129
diff changeset
   422
by (eresolve_tac prems 1);
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   423
qed "rev_induct";
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   424
5272
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   425
fun rev_induct_tac xs = res_inst_tac [("xs",xs)] rev_induct;
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   426
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   427
Goal  "(xs = [] --> P) -->  (!ys y. xs = ys@[y] --> P) --> P";
5132
24f992a25adc isatool expandshort;
wenzelm
parents: 5129
diff changeset
   428
by (res_inst_tac [("xs","xs")] rev_induct 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   429
by Auto_tac;
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   430
bind_thm ("rev_exhaust",
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   431
  impI RSN (2,allI RSN (2,allI RSN (2,impI RS (result() RS mp RS mp)))));
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   432
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   433
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   434
(** set **)
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   435
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   436
section "set";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   437
7032
d6efb3b8e669 NatBin: binary arithmetic for the naturals
paulson
parents: 7028
diff changeset
   438
Goal "finite (set xs)";
d6efb3b8e669 NatBin: binary arithmetic for the naturals
paulson
parents: 7028
diff changeset
   439
by (induct_tac "xs" 1);
d6efb3b8e669 NatBin: binary arithmetic for the naturals
paulson
parents: 7028
diff changeset
   440
by Auto_tac;
d6efb3b8e669 NatBin: binary arithmetic for the naturals
paulson
parents: 7028
diff changeset
   441
qed "finite_set";
d6efb3b8e669 NatBin: binary arithmetic for the naturals
paulson
parents: 7028
diff changeset
   442
AddIffs [finite_set];
5296
bdef7d349d27 added length_Suc_conv, finite_set
oheimb
parents: 5283
diff changeset
   443
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   444
Goal "set (xs@ys) = (set xs Un set ys)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   445
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   446
by Auto_tac;
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   447
qed "set_append";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   448
Addsimps[set_append];
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   449
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   450
Goal "set l <= set (x#l)";
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   451
by Auto_tac;
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   452
qed "set_subset_Cons";
1936
979e8b4f5fa5 Proved set_of_list_subset_Cons
paulson
parents: 1908
diff changeset
   453
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   454
Goal "(set xs = {}) = (xs = [])";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   455
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   456
by Auto_tac;
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   457
qed "set_empty";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   458
Addsimps [set_empty];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   459
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   460
Goal "set(rev xs) = set(xs)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   461
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   462
by Auto_tac;
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   463
qed "set_rev";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   464
Addsimps [set_rev];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   465
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   466
Goal "set(map f xs) = f``(set xs)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   467
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   468
by Auto_tac;
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   469
qed "set_map";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   470
Addsimps [set_map];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   471
6433
228237ec56e5 Added new thms.
nipkow
parents: 6408
diff changeset
   472
Goal "set(filter P xs) = {x. x : set xs & P x}";
6813
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
   473
by (induct_tac "xs" 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
   474
by Auto_tac;
6433
228237ec56e5 Added new thms.
nipkow
parents: 6408
diff changeset
   475
qed "set_filter";
228237ec56e5 Added new thms.
nipkow
parents: 6408
diff changeset
   476
Addsimps [set_filter];
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   477
6433
228237ec56e5 Added new thms.
nipkow
parents: 6408
diff changeset
   478
Goal "set[i..j(] = {k. i <= k & k < j}";
6813
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
   479
by (induct_tac "j" 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
   480
by Auto_tac;
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
   481
by (arith_tac 1);
6433
228237ec56e5 Added new thms.
nipkow
parents: 6408
diff changeset
   482
qed "set_upt";
228237ec56e5 Added new thms.
nipkow
parents: 6408
diff changeset
   483
Addsimps [set_upt];
228237ec56e5 Added new thms.
nipkow
parents: 6408
diff changeset
   484
5272
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   485
Goal "(x : set xs) = (? ys zs. xs = ys@x#zs)";
5318
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   486
by (induct_tac "xs" 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   487
 by (Simp_tac 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   488
by (Asm_simp_tac 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   489
by (rtac iffI 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   490
by (blast_tac (claset() addIs [eq_Nil_appendI,Cons_eq_appendI]) 1);
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   491
by (REPEAT(etac exE 1));
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   492
by (case_tac "ys" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   493
by Auto_tac;
5272
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   494
qed "in_set_conv_decomp";
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   495
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   496
5272
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   497
(* eliminate `lists' in favour of `set' *)
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   498
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   499
Goal "(xs : lists A) = (!x : set xs. x : A)";
5318
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
   500
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   501
by Auto_tac;
5272
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   502
qed "in_lists_conv_set";
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   503
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   504
bind_thm("in_listsD",in_lists_conv_set RS iffD1);
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   505
AddSDs [in_listsD];
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   506
bind_thm("in_listsI",in_lists_conv_set RS iffD2);
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
   507
AddSIs [in_listsI];
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   508
5518
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   509
(** mem **)
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   510
 
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   511
section "mem";
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   512
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   513
Goal "(x mem xs) = (x: set xs)";
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   514
by (induct_tac "xs" 1);
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   515
by Auto_tac;
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   516
qed "set_mem_eq";
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   517
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   518
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   519
(** list_all **)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   520
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   521
section "list_all";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   522
5518
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   523
Goal "list_all P xs = (!x:set xs. P x)";
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   524
by (induct_tac "xs" 1);
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   525
by Auto_tac;
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   526
qed "list_all_conv";
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   527
5443
e2459d18ff47 changed constants mem and list_all to mere translations
oheimb
parents: 5427
diff changeset
   528
Goal "list_all P (xs@ys) = (list_all P xs & list_all P ys)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   529
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   530
by Auto_tac;
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   531
qed "list_all_append";
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   532
Addsimps [list_all_append];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   533
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   534
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   535
(** filter **)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   536
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   537
section "filter";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   538
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   539
Goal "filter P (xs@ys) = filter P xs @ filter P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   540
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   541
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   542
qed "filter_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   543
Addsimps [filter_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   544
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   545
Goal "filter (%x. True) xs = xs";
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   546
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   547
by Auto_tac;
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   548
qed "filter_True";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   549
Addsimps [filter_True];
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   550
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   551
Goal "filter (%x. False) xs = []";
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   552
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   553
by Auto_tac;
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   554
qed "filter_False";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   555
Addsimps [filter_False];
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   556
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   557
Goal "length (filter P xs) <= length xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   558
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   559
by Auto_tac;
8741
61bc5ed22b62 removal of less_SucI, le_SucI from default simpset
paulson
parents: 8442
diff changeset
   560
by (asm_simp_tac (simpset() addsimps [le_SucI]) 1);
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   561
qed "length_filter";
5443
e2459d18ff47 changed constants mem and list_all to mere translations
oheimb
parents: 5427
diff changeset
   562
Addsimps[length_filter];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   563
5443
e2459d18ff47 changed constants mem and list_all to mere translations
oheimb
parents: 5427
diff changeset
   564
Goal "set (filter P xs) <= set xs";
e2459d18ff47 changed constants mem and list_all to mere translations
oheimb
parents: 5427
diff changeset
   565
by Auto_tac;
e2459d18ff47 changed constants mem and list_all to mere translations
oheimb
parents: 5427
diff changeset
   566
qed "filter_is_subset";
e2459d18ff47 changed constants mem and list_all to mere translations
oheimb
parents: 5427
diff changeset
   567
Addsimps [filter_is_subset];
e2459d18ff47 changed constants mem and list_all to mere translations
oheimb
parents: 5427
diff changeset
   568
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   569
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   570
section "concat";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   571
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   572
Goal  "concat(xs@ys) = concat(xs)@concat(ys)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   573
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   574
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   575
qed"concat_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   576
Addsimps [concat_append];
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   577
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   578
Goal "(concat xss = []) = (!xs:set xss. xs=[])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   579
by (induct_tac "xss" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   580
by Auto_tac;
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   581
qed "concat_eq_Nil_conv";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   582
AddIffs [concat_eq_Nil_conv];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   583
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   584
Goal "([] = concat xss) = (!xs:set xss. xs=[])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   585
by (induct_tac "xss" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   586
by Auto_tac;
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   587
qed "Nil_eq_concat_conv";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   588
AddIffs [Nil_eq_concat_conv];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   589
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   590
Goal  "set(concat xs) = Union(set `` set xs)";
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   591
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   592
by Auto_tac;
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   593
qed"set_concat";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   594
Addsimps [set_concat];
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   595
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   596
Goal "map f (concat xs) = concat (map (map f) xs)"; 
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   597
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   598
by Auto_tac;
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   599
qed "map_concat";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   600
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   601
Goal "filter p (concat xs) = concat (map (filter p) xs)"; 
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   602
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   603
by Auto_tac;
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   604
qed"filter_concat"; 
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   605
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   606
Goal "rev(concat xs) = concat (map rev (rev xs))";
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   607
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   608
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   609
qed "rev_concat";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   610
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   611
(** nth **)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   612
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   613
section "nth";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   614
6408
5b443d6331ed new definition for nth.
pusch
parents: 6394
diff changeset
   615
Goal "(x#xs)!0 = x";
5b443d6331ed new definition for nth.
pusch
parents: 6394
diff changeset
   616
by Auto_tac;
5b443d6331ed new definition for nth.
pusch
parents: 6394
diff changeset
   617
qed "nth_Cons_0";
5b443d6331ed new definition for nth.
pusch
parents: 6394
diff changeset
   618
Addsimps [nth_Cons_0];
5644
85fd64148873 Nat: added zero_neq_conv
nipkow
parents: 5641
diff changeset
   619
6408
5b443d6331ed new definition for nth.
pusch
parents: 6394
diff changeset
   620
Goal "(x#xs)!(Suc n) = xs!n";
5b443d6331ed new definition for nth.
pusch
parents: 6394
diff changeset
   621
by Auto_tac;
5b443d6331ed new definition for nth.
pusch
parents: 6394
diff changeset
   622
qed "nth_Cons_Suc";
5b443d6331ed new definition for nth.
pusch
parents: 6394
diff changeset
   623
Addsimps [nth_Cons_Suc];
5b443d6331ed new definition for nth.
pusch
parents: 6394
diff changeset
   624
5b443d6331ed new definition for nth.
pusch
parents: 6394
diff changeset
   625
Delsimps (thms "nth.simps");
5b443d6331ed new definition for nth.
pusch
parents: 6394
diff changeset
   626
5b443d6331ed new definition for nth.
pusch
parents: 6394
diff changeset
   627
Goal "!n. (xs@ys)!n = (if n < length xs then xs!n else ys!(n - length xs))";
5b443d6331ed new definition for nth.
pusch
parents: 6394
diff changeset
   628
by (induct_tac "xs" 1);
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   629
 by (Asm_simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   630
 by (rtac allI 1);
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   631
 by (case_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   632
  by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   633
qed_spec_mp "nth_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   634
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   635
Goal "!n. n < length xs --> (map f xs)!n = f(xs!n)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   636
by (induct_tac "xs" 1);
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   637
 by (Asm_full_simp_tac 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   638
by (rtac allI 1);
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   639
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   640
by Auto_tac;
1485
240cc98b94a7 Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents: 1465
diff changeset
   641
qed_spec_mp "nth_map";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   642
Addsimps [nth_map];
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   643
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   644
Goal "set xs = {xs!i |i. i < length xs}";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   645
by (induct_tac "xs" 1);
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   646
 by (Simp_tac 1);
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
   647
by (Asm_simp_tac 1);
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
   648
by Safe_tac;
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
   649
  by (res_inst_tac [("x","0")] exI 1);
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   650
  by (Simp_tac 1);
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
   651
 by (res_inst_tac [("x","Suc i")] exI 1);
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
   652
 by (Asm_simp_tac 1);
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   653
by (case_tac "i" 1);
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
   654
 by (Asm_full_simp_tac 1);
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
   655
by (rename_tac "j" 1);
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
   656
 by (res_inst_tac [("x","j")] exI 1);
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
   657
by (Asm_simp_tac 1);
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   658
qed "set_conv_nth";
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   659
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   660
Goal "n < length xs ==> Ball (set xs) P --> P(xs!n)";
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   661
by (simp_tac (simpset() addsimps [set_conv_nth]) 1);
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
   662
by (Blast_tac 1);
5518
654ead0ba4f7 re-added mem and list_all
oheimb
parents: 5448
diff changeset
   663
qed_spec_mp "list_ball_nth";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   664
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   665
Goal "n < length xs ==> xs!n : set xs";
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   666
by (simp_tac (simpset() addsimps [set_conv_nth]) 1);
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
   667
by (Blast_tac 1);
1485
240cc98b94a7 Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents: 1465
diff changeset
   668
qed_spec_mp "nth_mem";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   669
Addsimps [nth_mem];
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   670
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   671
Goal "(!i. i < length xs --> P(xs!i)) --> (!x : set xs. P x)";
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   672
by (simp_tac (simpset() addsimps [set_conv_nth]) 1);
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
   673
by (Blast_tac 1);
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   674
qed_spec_mp "all_nth_imp_all_set";
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   675
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   676
Goal "(!x : set xs. P x) = (!i. i<length xs --> P (xs ! i))";
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   677
by (simp_tac (simpset() addsimps [set_conv_nth]) 1);
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
   678
by (Blast_tac 1);
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   679
qed_spec_mp "all_set_conv_all_nth";
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   680
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   681
5077
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   682
(** list update **)
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   683
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   684
section "list update";
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   685
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   686
Goal "!i. length(xs[i:=x]) = length xs";
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   687
by (induct_tac "xs" 1);
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   688
by (Simp_tac 1);
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   689
by (asm_full_simp_tac (simpset() addsplits [nat.split]) 1);
5077
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   690
qed_spec_mp "length_list_update";
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   691
Addsimps [length_list_update];
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   692
5644
85fd64148873 Nat: added zero_neq_conv
nipkow
parents: 5641
diff changeset
   693
Goal "!i j. i < length xs  --> (xs[i:=x])!j = (if i=j then x else xs!j)";
6162
484adda70b65 expandshort
paulson
parents: 6141
diff changeset
   694
by (induct_tac "xs" 1);
484adda70b65 expandshort
paulson
parents: 6141
diff changeset
   695
 by (Simp_tac 1);
484adda70b65 expandshort
paulson
parents: 6141
diff changeset
   696
by (auto_tac (claset(), simpset() addsimps [nth_Cons] addsplits [nat.split]));
5644
85fd64148873 Nat: added zero_neq_conv
nipkow
parents: 5641
diff changeset
   697
qed_spec_mp "nth_list_update";
85fd64148873 Nat: added zero_neq_conv
nipkow
parents: 5641
diff changeset
   698
8144
c4b5cbfb90dd optimized xs[i:=x]!j lemmas.
nipkow
parents: 8118
diff changeset
   699
Goal "i < length xs  ==> (xs[i:=x])!i = x";
c4b5cbfb90dd optimized xs[i:=x]!j lemmas.
nipkow
parents: 8118
diff changeset
   700
by (asm_simp_tac (simpset() addsimps [nth_list_update]) 1);
c4b5cbfb90dd optimized xs[i:=x]!j lemmas.
nipkow
parents: 8118
diff changeset
   701
qed "nth_list_update_eq";
c4b5cbfb90dd optimized xs[i:=x]!j lemmas.
nipkow
parents: 8118
diff changeset
   702
Addsimps [nth_list_update_eq];
c4b5cbfb90dd optimized xs[i:=x]!j lemmas.
nipkow
parents: 8118
diff changeset
   703
c4b5cbfb90dd optimized xs[i:=x]!j lemmas.
nipkow
parents: 8118
diff changeset
   704
Goal "!i j. i ~= j --> xs[i:=x]!j = xs!j";
c4b5cbfb90dd optimized xs[i:=x]!j lemmas.
nipkow
parents: 8118
diff changeset
   705
by (induct_tac "xs" 1);
c4b5cbfb90dd optimized xs[i:=x]!j lemmas.
nipkow
parents: 8118
diff changeset
   706
 by (Simp_tac 1);
c4b5cbfb90dd optimized xs[i:=x]!j lemmas.
nipkow
parents: 8118
diff changeset
   707
by (auto_tac (claset(), simpset() addsimps [nth_Cons] addsplits [nat.split]));
c4b5cbfb90dd optimized xs[i:=x]!j lemmas.
nipkow
parents: 8118
diff changeset
   708
qed_spec_mp "nth_list_update_neq";
c4b5cbfb90dd optimized xs[i:=x]!j lemmas.
nipkow
parents: 8118
diff changeset
   709
Addsimps [nth_list_update_neq];
c4b5cbfb90dd optimized xs[i:=x]!j lemmas.
nipkow
parents: 8118
diff changeset
   710
6433
228237ec56e5 Added new thms.
nipkow
parents: 6408
diff changeset
   711
Goal "!i. i < size xs --> xs[i:=x, i:=y] = xs[i:=y]";
6813
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
   712
by (induct_tac "xs" 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
   713
 by (Simp_tac 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
   714
by (asm_simp_tac (simpset() addsplits [nat.split]) 1);
6433
228237ec56e5 Added new thms.
nipkow
parents: 6408
diff changeset
   715
qed_spec_mp "list_update_overwrite";
228237ec56e5 Added new thms.
nipkow
parents: 6408
diff changeset
   716
Addsimps [list_update_overwrite];
228237ec56e5 Added new thms.
nipkow
parents: 6408
diff changeset
   717
228237ec56e5 Added new thms.
nipkow
parents: 6408
diff changeset
   718
Goal "!i < length xs. (xs[i := x] = xs) = (xs!i = x)";
6813
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
   719
by (induct_tac "xs" 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
   720
 by (Simp_tac 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
   721
by (simp_tac (simpset() addsplits [nat.split]) 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
   722
by (Blast_tac 1);
6433
228237ec56e5 Added new thms.
nipkow
parents: 6408
diff changeset
   723
qed_spec_mp "list_update_same_conv";
228237ec56e5 Added new thms.
nipkow
parents: 6408
diff changeset
   724
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   725
Goal "!i xy xs. length xs = length ys --> \
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   726
\     (zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])";
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   727
by (induct_tac "ys" 1);
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   728
 by Auto_tac;
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   729
by (case_tac "xs" 1);
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   730
 by (auto_tac (claset(), simpset() addsplits [nat.split]));
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   731
qed_spec_mp "update_zip";
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   732
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   733
Goal "!i. set(xs[i:=x]) <= insert x (set xs)";
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   734
by (induct_tac "xs" 1);
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   735
 by (asm_full_simp_tac (simpset() addsimps []) 1);
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   736
by (asm_full_simp_tac (simpset() addsplits [nat.split]) 1);
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   737
by (Fast_tac  1);
8287
42911a6bb13f Added and renamed a lemma.
nipkow
parents: 8254
diff changeset
   738
qed_spec_mp "set_update_subset_insert";
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   739
8287
42911a6bb13f Added and renamed a lemma.
nipkow
parents: 8254
diff changeset
   740
Goal "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A";
42911a6bb13f Added and renamed a lemma.
nipkow
parents: 8254
diff changeset
   741
by(fast_tac (claset() addSDs [set_update_subset_insert RS subsetD]) 1);
42911a6bb13f Added and renamed a lemma.
nipkow
parents: 8254
diff changeset
   742
qed "set_update_subsetI";
5077
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 5043
diff changeset
   743
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   744
(** last & butlast **)
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   745
5644
85fd64148873 Nat: added zero_neq_conv
nipkow
parents: 5641
diff changeset
   746
section "last / butlast";
85fd64148873 Nat: added zero_neq_conv
nipkow
parents: 5641
diff changeset
   747
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   748
Goal "last(xs@[x]) = x";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   749
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   750
by Auto_tac;
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   751
qed "last_snoc";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   752
Addsimps [last_snoc];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   753
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   754
Goal "butlast(xs@[x]) = xs";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   755
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   756
by Auto_tac;
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   757
qed "butlast_snoc";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   758
Addsimps [butlast_snoc];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   759
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   760
Goal "length(butlast xs) = length xs - 1";
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   761
by (res_inst_tac [("xs","xs")] rev_induct 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   762
by Auto_tac;
4643
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   763
qed "length_butlast";
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   764
Addsimps [length_butlast];
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   765
5278
a903b66822e2 even more tidying of Goal commands
paulson
parents: 5272
diff changeset
   766
Goal "!ys. butlast (xs@ys) = (if ys=[] then butlast xs else xs@butlast ys)";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   767
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   768
by Auto_tac;
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   769
qed_spec_mp "butlast_append";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   770
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   771
Goal "xs ~= [] --> butlast xs @ [last xs] = xs";
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
   772
by (induct_tac "xs" 1);
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
   773
by (ALLGOALS Asm_simp_tac);
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   774
qed_spec_mp "append_butlast_last_id";
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   775
Addsimps [append_butlast_last_id];
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   776
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   777
Goal "x:set(butlast xs) --> x:set xs";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   778
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   779
by Auto_tac;
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   780
qed_spec_mp "in_set_butlastD";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   781
5448
40a09282ba14 in_set_butlast_appendI supersedes in_set_butlast_appendI1,2
paulson
parents: 5443
diff changeset
   782
Goal "x:set(butlast xs) | x:set(butlast ys) ==> x:set(butlast(xs@ys))";
40a09282ba14 in_set_butlast_appendI supersedes in_set_butlast_appendI1,2
paulson
parents: 5443
diff changeset
   783
by (auto_tac (claset() addDs [in_set_butlastD],
40a09282ba14 in_set_butlast_appendI supersedes in_set_butlast_appendI1,2
paulson
parents: 5443
diff changeset
   784
	      simpset() addsimps [butlast_append]));
40a09282ba14 in_set_butlast_appendI supersedes in_set_butlast_appendI1,2
paulson
parents: 5443
diff changeset
   785
qed "in_set_butlast_appendI";
3902
265a5d8ab88f Removed comment.
nipkow
parents: 3896
diff changeset
   786
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   787
(** take  & drop **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   788
section "take & drop";
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   789
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   790
Goal "take 0 xs = []";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   791
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   792
by Auto_tac;
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   793
qed "take_0";
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   794
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   795
Goal "drop 0 xs = xs";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   796
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   797
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   798
qed "drop_0";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   799
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   800
Goal "take (Suc n) (x#xs) = x # take n xs";
1552
6f71b5d46700 Ran expandshort
paulson
parents: 1485
diff changeset
   801
by (Simp_tac 1);
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   802
qed "take_Suc_Cons";
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   803
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   804
Goal "drop (Suc n) (x#xs) = drop n xs";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   805
by (Simp_tac 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   806
qed "drop_Suc_Cons";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   807
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   808
Delsimps [take_Cons,drop_Cons];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   809
Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   810
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   811
Goal "!xs. length(take n xs) = min (length xs) n";
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   812
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   813
 by Auto_tac;
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   814
by (case_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   815
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   816
qed_spec_mp "length_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   817
Addsimps [length_take];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   818
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   819
Goal "!xs. length(drop n xs) = (length xs - n)";
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   820
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   821
 by Auto_tac;
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   822
by (case_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   823
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   824
qed_spec_mp "length_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   825
Addsimps [length_drop];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   826
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   827
Goal "!xs. length xs <= n --> take n xs = xs";
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   828
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   829
 by Auto_tac;
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   830
by (case_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   831
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   832
qed_spec_mp "take_all";
7246
33058867d6eb Added take_all and drop_all to simpset.
nipkow
parents: 7224
diff changeset
   833
Addsimps [take_all];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   834
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   835
Goal "!xs. length xs <= n --> drop n xs = []";
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   836
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   837
 by Auto_tac;
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   838
by (case_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   839
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   840
qed_spec_mp "drop_all";
7246
33058867d6eb Added take_all and drop_all to simpset.
nipkow
parents: 7224
diff changeset
   841
Addsimps [drop_all];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   842
5278
a903b66822e2 even more tidying of Goal commands
paulson
parents: 5272
diff changeset
   843
Goal "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)";
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   844
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   845
 by Auto_tac;
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   846
by (case_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   847
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   848
qed_spec_mp "take_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   849
Addsimps [take_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   850
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   851
Goal "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; 
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   852
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   853
 by Auto_tac;
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   854
by (case_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   855
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   856
qed_spec_mp "drop_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   857
Addsimps [drop_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   858
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   859
Goal "!xs n. take n (take m xs) = take (min n m) xs"; 
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   860
by (induct_tac "m" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   861
 by Auto_tac;
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   862
by (case_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   863
 by Auto_tac;
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   864
by (case_tac "na" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   865
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   866
qed_spec_mp "take_take";
7570
a9391550eea1 Mod because of new solver interface.
nipkow
parents: 7246
diff changeset
   867
Addsimps [take_take];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   868
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   869
Goal "!xs. drop n (drop m xs) = drop (n + m) xs"; 
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   870
by (induct_tac "m" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   871
 by Auto_tac;
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   872
by (case_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   873
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   874
qed_spec_mp "drop_drop";
7570
a9391550eea1 Mod because of new solver interface.
nipkow
parents: 7246
diff changeset
   875
Addsimps [drop_drop];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   876
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   877
Goal "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; 
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   878
by (induct_tac "m" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   879
 by Auto_tac;
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   880
by (case_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   881
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   882
qed_spec_mp "take_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   883
6813
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
   884
Goal "!xs. take n xs @ drop n xs = xs";
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
   885
by (induct_tac "n" 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
   886
 by Auto_tac;
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   887
by (case_tac "xs" 1);
6813
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
   888
 by Auto_tac;
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
   889
qed_spec_mp "append_take_drop_id";
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   890
Addsimps [append_take_drop_id];
6813
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
   891
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   892
Goal "!xs. take n (map f xs) = map f (take n xs)"; 
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   893
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   894
 by Auto_tac;
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   895
by (case_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   896
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   897
qed_spec_mp "take_map"; 
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   898
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   899
Goal "!xs. drop n (map f xs) = map f (drop n xs)"; 
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   900
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   901
 by Auto_tac;
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   902
by (case_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   903
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   904
qed_spec_mp "drop_map";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   905
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   906
Goal "!n i. i < n --> (take n xs)!i = xs!i";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   907
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   908
 by Auto_tac;
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   909
by (case_tac "n" 1);
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   910
 by (Blast_tac 1);
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   911
by (case_tac "i" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   912
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   913
qed_spec_mp "nth_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   914
Addsimps [nth_take];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   915
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   916
Goal  "!xs i. n + i <= length xs --> (drop n xs)!i = xs!(n+i)";
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 5162
diff changeset
   917
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   918
 by Auto_tac;
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   919
by (case_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   920
 by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   921
qed_spec_mp "nth_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   922
Addsimps [nth_drop];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   923
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   924
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   925
Goal
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   926
 "!zs. (xs@ys = zs) = (xs = take (length xs) zs & ys = drop (length xs) zs)";
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
   927
by (induct_tac "xs" 1);
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
   928
 by (Simp_tac 1);
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
   929
by (Asm_full_simp_tac 1);
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
   930
by (Clarify_tac 1);
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   931
by (case_tac "zs" 1);
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
   932
by (Auto_tac);
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   933
qed_spec_mp "append_eq_conv_conj";
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   934
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   935
(** takeWhile & dropWhile **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   936
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   937
section "takeWhile & dropWhile";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   938
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   939
Goal "takeWhile P xs @ dropWhile P xs = xs";
3586
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   940
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   941
by Auto_tac;
3586
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   942
qed "takeWhile_dropWhile_id";
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   943
Addsimps [takeWhile_dropWhile_id];
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   944
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   945
Goal  "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   946
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   947
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   948
bind_thm("takeWhile_append1", conjI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   949
Addsimps [takeWhile_append1];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   950
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   951
Goal "(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   952
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   953
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   954
bind_thm("takeWhile_append2", ballI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   955
Addsimps [takeWhile_append2];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   956
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   957
Goal "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   958
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   959
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   960
bind_thm("dropWhile_append1", conjI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   961
Addsimps [dropWhile_append1];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   962
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   963
Goal "(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   964
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   965
by Auto_tac;
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   966
bind_thm("dropWhile_append2", ballI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   967
Addsimps [dropWhile_append2];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   968
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
   969
Goal "x:set(takeWhile P xs) --> x:set xs & P x";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   970
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
   971
by Auto_tac;
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   972
qed_spec_mp"set_take_whileD";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   973
6306
81e7fbf61db2 modified zip
nipkow
parents: 6162
diff changeset
   974
(** zip **)
81e7fbf61db2 modified zip
nipkow
parents: 6162
diff changeset
   975
section "zip";
81e7fbf61db2 modified zip
nipkow
parents: 6162
diff changeset
   976
81e7fbf61db2 modified zip
nipkow
parents: 6162
diff changeset
   977
Goal "zip [] ys = []";
6813
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
   978
by (induct_tac "ys" 1);
6306
81e7fbf61db2 modified zip
nipkow
parents: 6162
diff changeset
   979
by Auto_tac;
81e7fbf61db2 modified zip
nipkow
parents: 6162
diff changeset
   980
qed "zip_Nil";
81e7fbf61db2 modified zip
nipkow
parents: 6162
diff changeset
   981
Addsimps [zip_Nil];
81e7fbf61db2 modified zip
nipkow
parents: 6162
diff changeset
   982
81e7fbf61db2 modified zip
nipkow
parents: 6162
diff changeset
   983
Goal "zip (x#xs) (y#ys) = (x,y)#zip xs ys";
6813
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
   984
by (Simp_tac 1);
6306
81e7fbf61db2 modified zip
nipkow
parents: 6162
diff changeset
   985
qed "zip_Cons_Cons";
81e7fbf61db2 modified zip
nipkow
parents: 6162
diff changeset
   986
Addsimps [zip_Cons_Cons];
81e7fbf61db2 modified zip
nipkow
parents: 6162
diff changeset
   987
81e7fbf61db2 modified zip
nipkow
parents: 6162
diff changeset
   988
Delsimps(tl (thms"zip.simps"));
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   989
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
   990
Goal "!xs. length (zip xs ys) = min (length xs) (length ys)";
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   991
by (induct_tac "ys" 1);
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   992
 by (Simp_tac 1);
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   993
by (Clarify_tac 1);
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
   994
by (case_tac "xs" 1);
8064
357652a08ee0 expandshort
paulson
parents: 8009
diff changeset
   995
 by (Auto_tac);
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   996
qed_spec_mp "length_zip";
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   997
Addsimps [length_zip];
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   998
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
   999
Goal
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1000
 "!xs. zip (xs@ys) zs = \
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1001
\      zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)";
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1002
by (induct_tac "zs" 1);
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1003
 by (Simp_tac 1);
8064
357652a08ee0 expandshort
paulson
parents: 8009
diff changeset
  1004
by (Clarify_tac 1);
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
  1005
by (case_tac "xs" 1);
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1006
 by (Asm_simp_tac 1);
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1007
by (Asm_simp_tac 1);
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1008
qed_spec_mp "zip_append1";
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1009
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1010
Goal
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1011
 "!ys. zip xs (ys@zs) = \
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1012
\      zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs";
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1013
by (induct_tac "xs" 1);
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1014
 by (Simp_tac 1);
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1015
by (Clarify_tac 1);
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
  1016
by (case_tac "ys" 1);
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1017
 by (Asm_simp_tac 1);
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1018
by (Asm_simp_tac 1);
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1019
qed_spec_mp "zip_append2";
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1020
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1021
Goal
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1022
 "[| length xs = length us; length ys = length vs |] ==> \
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1023
\ zip (xs@ys) (us@vs) = zip xs us @ zip ys vs";
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1024
by (asm_simp_tac (simpset() addsimps [zip_append1]) 1);
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1025
qed_spec_mp "zip_append";
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1026
Addsimps [zip_append];
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1027
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1028
Goal "!xs. length xs = length ys --> zip (rev xs) (rev ys) = rev (zip xs ys)";
8064
357652a08ee0 expandshort
paulson
parents: 8009
diff changeset
  1029
by (induct_tac "ys" 1);
357652a08ee0 expandshort
paulson
parents: 8009
diff changeset
  1030
 by (Asm_full_simp_tac 1);
357652a08ee0 expandshort
paulson
parents: 8009
diff changeset
  1031
by (Asm_full_simp_tac 1);
357652a08ee0 expandshort
paulson
parents: 8009
diff changeset
  1032
by (Clarify_tac 1);
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
  1033
by (case_tac "xs" 1);
8064
357652a08ee0 expandshort
paulson
parents: 8009
diff changeset
  1034
 by (Auto_tac);
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1035
qed_spec_mp "zip_rev";
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1036
8115
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1037
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1038
Goal
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1039
"!i xs. i < length xs --> i < length ys --> (zip xs ys)!i = (xs!i, ys!i)";
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1040
by (induct_tac "ys" 1);
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1041
 by (Simp_tac 1);
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1042
by (Clarify_tac 1);
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
  1043
by (case_tac "xs" 1);
8064
357652a08ee0 expandshort
paulson
parents: 8009
diff changeset
  1044
 by (Auto_tac);
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1045
by (asm_full_simp_tac (simpset() addsimps (thms"nth.simps") addsplits [nat.split]) 1);
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1046
qed_spec_mp "nth_zip";
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1047
Addsimps [nth_zip];
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1048
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1049
Goal "set(zip xs ys) = {(xs!i,ys!i) |i. i < min (length xs) (length ys)}";
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1050
by (simp_tac (simpset() addsimps [set_conv_nth]addcongs [rev_conj_cong]) 1);
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1051
qed_spec_mp "set_zip";
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1052
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1053
Goal
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1054
 "length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]";
8064
357652a08ee0 expandshort
paulson
parents: 8009
diff changeset
  1055
by (rtac sym 1);
357652a08ee0 expandshort
paulson
parents: 8009
diff changeset
  1056
by (asm_simp_tac (simpset() addsimps [update_zip]) 1);
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1057
qed_spec_mp "zip_update";
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1058
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1059
Goal "!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)";
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1060
by (induct_tac "i" 1);
8064
357652a08ee0 expandshort
paulson
parents: 8009
diff changeset
  1061
 by (Auto_tac);
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
  1062
by (case_tac "j" 1);
8064
357652a08ee0 expandshort
paulson
parents: 8009
diff changeset
  1063
 by (Auto_tac);
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1064
qed "zip_replicate";
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1065
Addsimps [zip_replicate];
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1066
8115
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1067
(** list_all2 **)
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1068
section "list_all2";
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1069
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1070
Goalw [list_all2_def] "list_all2 P xs ys ==> length xs = length ys";
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1071
by (Asm_simp_tac 1);
8115
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1072
qed "list_all2_lengthD";
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1073
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1074
Goalw [list_all2_def] "list_all2 P [] ys = (ys=[])";
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1075
by (Simp_tac 1);
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1076
qed "list_all2_Nil";
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1077
AddIffs [list_all2_Nil];
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1078
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1079
Goalw [list_all2_def] "list_all2 P xs [] = (xs=[])";
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1080
by (Simp_tac 1);
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1081
qed "list_all2_Nil2";
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1082
AddIffs [list_all2_Nil2];
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1083
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1084
Goalw [list_all2_def]
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1085
 "list_all2 P (x#xs) (y#ys) = (P x y & list_all2 P xs ys)";
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1086
by (Auto_tac);
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1087
qed "list_all2_Cons";
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1088
AddIffs[list_all2_Cons];
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1089
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1090
Goalw [list_all2_def]
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1091
 "list_all2 P (x#xs) ys = (? z zs. ys = z#zs & P x z & list_all2 P xs zs)";
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
  1092
by (case_tac "ys" 1);
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1093
by (Auto_tac);
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1094
qed "list_all2_Cons1";
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1095
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1096
Goalw [list_all2_def]
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1097
 "list_all2 P xs (y#ys) = (? z zs. xs = z#zs & P z y & list_all2 P zs ys)";
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
  1098
by (case_tac "xs" 1);
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1099
by (Auto_tac);
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1100
qed "list_all2_Cons2";
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1101
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1102
Goalw [list_all2_def]
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1103
 "list_all2 P (xs@ys) zs = \
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1104
\ (EX us vs. zs = us@vs & length us = length xs & length vs = length ys & \
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1105
\            list_all2 P xs us & list_all2 P ys vs)";
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1106
by (simp_tac (simpset() addsimps [zip_append1]) 1);
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1107
by (rtac iffI 1);
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1108
 by (res_inst_tac [("x","take (length xs) zs")] exI 1);
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1109
 by (res_inst_tac [("x","drop (length xs) zs")] exI 1);
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1110
 by (asm_full_simp_tac (simpset() addsimps [min_def,eq_sym_conv]) 1);
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1111
by (Clarify_tac 1);
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1112
by (asm_full_simp_tac (simpset() addsimps [ball_Un]) 1);
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1113
qed "list_all2_append1";
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1114
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1115
Goalw [list_all2_def]
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1116
 "list_all2 P xs (ys@zs) = \
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1117
\ (EX us vs. xs = us@vs & length us = length ys & length vs = length zs & \
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1118
\            list_all2 P us ys & list_all2 P vs zs)";
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1119
by (simp_tac (simpset() addsimps [zip_append2]) 1);
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1120
by (rtac iffI 1);
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1121
 by (res_inst_tac [("x","take (length ys) xs")] exI 1);
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1122
 by (res_inst_tac [("x","drop (length ys) xs")] exI 1);
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1123
 by (asm_full_simp_tac (simpset() addsimps [min_def,eq_sym_conv]) 1);
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1124
by (Clarify_tac 1);
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1125
by (asm_full_simp_tac (simpset() addsimps [ball_Un]) 1);
8118
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1126
qed "list_all2_append2";
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1127
746c5cf09bde More lemmas.
nipkow
parents: 8115
diff changeset
  1128
Goalw [list_all2_def]
8115
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1129
  "list_all2 P xs ys = \
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1130
\  (length xs = length ys & (!i<length xs. P (xs!i) (ys!i)))";
8254
84a5fe44520f expandshort
paulson
parents: 8144
diff changeset
  1131
by (force_tac (claset(), simpset() addsimps [set_zip]) 1);
8115
c802042066e8 Forgot to "call" MicroJava in makefile.
nipkow
parents: 8064
diff changeset
  1132
qed "list_all2_conv_all_nth";
5272
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
  1133
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
  1134
(** foldl **)
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
  1135
section "foldl";
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
  1136
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
  1137
Goal "!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys";
5318
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
  1138
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
  1139
by Auto_tac;
5272
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
  1140
qed_spec_mp "foldl_append";
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
  1141
Addsimps [foldl_append];
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
  1142
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
  1143
(* Note: `n <= foldl op+ n ns' looks simpler, but is more difficult to use
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
  1144
   because it requires an additional transitivity step
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
  1145
*)
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
  1146
Goal "!n::nat. m <= n --> m <= foldl op+ n ns";
5318
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
  1147
by (induct_tac "ns" 1);
6058
a9600c47ace3 Shortened a proof.
nipkow
parents: 6055
diff changeset
  1148
by Auto_tac;
5272
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
  1149
qed_spec_mp "start_le_sum";
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
  1150
8935
548901d05a0e added type constraint ::nat because 0 is now overloaded
paulson
parents: 8741
diff changeset
  1151
Goal "!!n::nat. n : set ns ==> n <= foldl op+ 0 ns";
5758
27a2b36efd95 corrected auto_tac (applications of unsafe wrappers)
oheimb
parents: 5644
diff changeset
  1152
by (force_tac (claset() addIs [start_le_sum],
27a2b36efd95 corrected auto_tac (applications of unsafe wrappers)
oheimb
parents: 5644
diff changeset
  1153
              simpset() addsimps [in_set_conv_decomp]) 1);
5272
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
  1154
qed "elem_le_sum";
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
  1155
8935
548901d05a0e added type constraint ::nat because 0 is now overloaded
paulson
parents: 8741
diff changeset
  1156
Goal "!m::nat. (foldl op+ m ns = 0) = (m=0 & (!n : set ns. n=0))";
5318
72bf8039b53f expandshort
paulson
parents: 5316
diff changeset
  1157
by (induct_tac "ns" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
  1158
by Auto_tac;
5272
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
  1159
qed_spec_mp "sum_eq_0_conv";
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
  1160
AddIffs [sum_eq_0_conv];
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
  1161
5425
157c6663dedd Added function upto to List.
nipkow
parents: 5355
diff changeset
  1162
(** upto **)
157c6663dedd Added function upto to List.
nipkow
parents: 5355
diff changeset
  1163
5427
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1164
(* Does not terminate! *)
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1165
Goal "[i..j(] = (if i<j then i#[Suc i..j(] else [])";
6162
484adda70b65 expandshort
paulson
parents: 6141
diff changeset
  1166
by (induct_tac "j" 1);
5427
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1167
by Auto_tac;
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1168
qed "upt_rec";
5425
157c6663dedd Added function upto to List.
nipkow
parents: 5355
diff changeset
  1169
5427
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1170
Goal "j<=i ==> [i..j(] = []";
6162
484adda70b65 expandshort
paulson
parents: 6141
diff changeset
  1171
by (stac upt_rec 1);
484adda70b65 expandshort
paulson
parents: 6141
diff changeset
  1172
by (Asm_simp_tac 1);
5427
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1173
qed "upt_conv_Nil";
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1174
Addsimps [upt_conv_Nil];
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1175
8982
4cb682fc083d renamed upt_Suc, since that name is needed for its primrec rule
paulson
parents: 8935
diff changeset
  1176
(*Only needed if upt_Suc is deleted from the simpset*)
5427
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1177
Goal "i<=j ==> [i..(Suc j)(] = [i..j(]@[j]";
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1178
by (Asm_simp_tac 1);
8982
4cb682fc083d renamed upt_Suc, since that name is needed for its primrec rule
paulson
parents: 8935
diff changeset
  1179
qed "upt_Suc_append";
5427
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1180
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1181
Goal "i<j ==> [i..j(] = i#[Suc i..j(]";
6162
484adda70b65 expandshort
paulson
parents: 6141
diff changeset
  1182
by (rtac trans 1);
484adda70b65 expandshort
paulson
parents: 6141
diff changeset
  1183
by (stac upt_rec 1);
484adda70b65 expandshort
paulson
parents: 6141
diff changeset
  1184
by (rtac refl 2);
5427
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1185
by (Asm_simp_tac 1);
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1186
qed "upt_conv_Cons";
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1187
9003
3747ec2aeb86 added new proofs and simplified an old one
paulson
parents: 8982
diff changeset
  1188
(*LOOPS as a simprule, since j<=j*)
3747ec2aeb86 added new proofs and simplified an old one
paulson
parents: 8982
diff changeset
  1189
Goal "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]";
3747ec2aeb86 added new proofs and simplified an old one
paulson
parents: 8982
diff changeset
  1190
by (induct_tac "k" 1);
3747ec2aeb86 added new proofs and simplified an old one
paulson
parents: 8982
diff changeset
  1191
by Auto_tac;
3747ec2aeb86 added new proofs and simplified an old one
paulson
parents: 8982
diff changeset
  1192
qed "upt_add_eq_append";
3747ec2aeb86 added new proofs and simplified an old one
paulson
parents: 8982
diff changeset
  1193
5427
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1194
Goal "length [i..j(] = j-i";
6162
484adda70b65 expandshort
paulson
parents: 6141
diff changeset
  1195
by (induct_tac "j" 1);
5427
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1196
 by (Simp_tac 1);
6162
484adda70b65 expandshort
paulson
parents: 6141
diff changeset
  1197
by (asm_simp_tac (simpset() addsimps [Suc_diff_le]) 1);
5427
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1198
qed "length_upt";
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1199
Addsimps [length_upt];
5425
157c6663dedd Added function upto to List.
nipkow
parents: 5355
diff changeset
  1200
5427
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1201
Goal "i+k < j --> [i..j(] ! k = i+k";
6162
484adda70b65 expandshort
paulson
parents: 6141
diff changeset
  1202
by (induct_tac "j" 1);
9014
4382883421ec simplified the proof of nth_upt
paulson
parents: 9003
diff changeset
  1203
 by (asm_simp_tac (simpset() addsimps [less_Suc_eq, nth_append] 
4382883421ec simplified the proof of nth_upt
paulson
parents: 9003
diff changeset
  1204
                             addsplits [nat_diff_split]) 2);
4382883421ec simplified the proof of nth_upt
paulson
parents: 9003
diff changeset
  1205
by (Simp_tac 1);
5427
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1206
qed_spec_mp "nth_upt";
26c9a7c0b36b Arith: less_diff_conv
nipkow
parents: 5425
diff changeset
  1207
Addsimps [nth_upt];
5425
157c6663dedd Added function upto to List.
nipkow
parents: 5355
diff changeset
  1208
6433
228237ec56e5 Added new thms.
nipkow
parents: 6408
diff changeset
  1209
Goal "!i. i+m <= n --> take m [i..n(] = [i..i+m(]";
6813
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1210
by (induct_tac "m" 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1211
 by (Simp_tac 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1212
by (Clarify_tac 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1213
by (stac upt_rec 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1214
by (rtac sym 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1215
by (stac upt_rec 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1216
by (asm_simp_tac (simpset() delsimps (thms"upt.simps")) 1);
6433
228237ec56e5 Added new thms.
nipkow
parents: 6408
diff changeset
  1217
qed_spec_mp "take_upt";
228237ec56e5 Added new thms.
nipkow
parents: 6408
diff changeset
  1218
Addsimps [take_upt];
228237ec56e5 Added new thms.
nipkow
parents: 6408
diff changeset
  1219
9003
3747ec2aeb86 added new proofs and simplified an old one
paulson
parents: 8982
diff changeset
  1220
Goal "map Suc [m..n(] = [Suc m..n]";
6813
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1221
by (induct_tac "n" 1);
9003
3747ec2aeb86 added new proofs and simplified an old one
paulson
parents: 8982
diff changeset
  1222
by Auto_tac;
3747ec2aeb86 added new proofs and simplified an old one
paulson
parents: 8982
diff changeset
  1223
qed "map_Suc_upt";
3747ec2aeb86 added new proofs and simplified an old one
paulson
parents: 8982
diff changeset
  1224
3747ec2aeb86 added new proofs and simplified an old one
paulson
parents: 8982
diff changeset
  1225
Goal "ALL i. i < n-m --> (map f [m..n(]) ! i = f(m+i)";
3747ec2aeb86 added new proofs and simplified an old one
paulson
parents: 8982
diff changeset
  1226
by (res_inst_tac [("m","n"),("n","m")] diff_induct 1);
3747ec2aeb86 added new proofs and simplified an old one
paulson
parents: 8982
diff changeset
  1227
by (stac (map_Suc_upt RS sym) 3);
3747ec2aeb86 added new proofs and simplified an old one
paulson
parents: 8982
diff changeset
  1228
by (auto_tac (claset(), simpset() addsimps [less_diff_conv, nth_upt]));
6433
228237ec56e5 Added new thms.
nipkow
parents: 6408
diff changeset
  1229
qed_spec_mp "nth_map_upt";
228237ec56e5 Added new thms.
nipkow
parents: 6408
diff changeset
  1230
6813
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1231
Goal "ALL xs ys. k <= length xs --> k <= length ys -->  \
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1232
\        (ALL i. i < k --> xs!i = ys!i)  \
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1233
\     --> take k xs = take k ys";
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1234
by (induct_tac "k" 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1235
by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq_0_disj, 
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1236
						all_conj_distrib])));
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1237
by (Clarify_tac 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1238
(*Both lists must be non-empty*)
8442
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
  1239
by (case_tac "xs" 1);
96023903c2df case_tac now subsumes both boolean and datatype cases;
wenzelm
parents: 8423
diff changeset
  1240
by (case_tac "ys" 2);
6813
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1241
by (ALLGOALS Clarify_tac);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1242
(*prenexing's needed, not miniscoping*)
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1243
by (ALLGOALS (full_simp_tac (simpset() addsimps (all_simps RL [sym])  
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1244
                                       delsimps (all_simps))));
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1245
by (Blast_tac 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1246
qed_spec_mp "nth_take_lemma";
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1247
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1248
Goal "[| length xs = length ys;  \
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1249
\        ALL i. i < length xs --> xs!i = ys!i |]  \
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1250
\     ==> xs = ys";
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1251
by (forward_tac [[le_refl, eq_imp_le] MRS nth_take_lemma] 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1252
by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [take_all])));
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1253
qed_spec_mp "nth_equalityI";
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1254
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1255
(*The famous take-lemma*)
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1256
Goal "(ALL i. take i xs = take i ys) ==> xs = ys";
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1257
by (dres_inst_tac [("x", "max (length xs) (length ys)")] spec 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1258
by (full_simp_tac (simpset() addsimps [le_max_iff_disj, take_all]) 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1259
qed_spec_mp "take_equalityI";
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1260
5272
95cfd872fe66 New lemmas in List and Lambda in IsaMakefile
nipkow
parents: 5200
diff changeset
  1261
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
  1262
(** nodups & remdups **)
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
  1263
section "nodups & remdups";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
  1264
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
  1265
Goal "set(remdups xs) = set xs";
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
  1266
by (induct_tac "xs" 1);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
  1267
 by (Simp_tac 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
  1268
by (asm_full_simp_tac (simpset() addsimps [insert_absorb]) 1);
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
  1269
qed "set_remdups";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
  1270
Addsimps [set_remdups];
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
  1271
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
  1272
Goal "nodups(remdups xs)";
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
  1273
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
  1274
by Auto_tac;
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
  1275
qed "nodups_remdups";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
  1276
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
  1277
Goal "nodups xs --> nodups (filter P xs)";
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
  1278
by (induct_tac "xs" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
  1279
by Auto_tac;
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
  1280
qed_spec_mp "nodups_filter";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
  1281
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
  1282
(** replicate **)
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
  1283
section "replicate";
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
  1284
6794
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1285
Goal "length(replicate n x) = n";
6813
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1286
by (induct_tac "n" 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1287
by Auto_tac;
6794
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1288
qed "length_replicate";
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1289
Addsimps [length_replicate];
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1290
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1291
Goal "map f (replicate n x) = replicate n (f x)";
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1292
by (induct_tac "n" 1);
6813
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1293
by Auto_tac;
6794
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1294
qed "map_replicate";
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1295
Addsimps [map_replicate];
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1296
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1297
Goal "(replicate n x) @ (x#xs) = x # replicate n x @ xs";
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1298
by (induct_tac "n" 1);
6813
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1299
by Auto_tac;
6794
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1300
qed "replicate_app_Cons_same";
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1301
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1302
Goal "rev(replicate n x) = replicate n x";
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1303
by (induct_tac "n" 1);
6813
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1304
 by (Simp_tac 1);
6794
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1305
by (asm_simp_tac (simpset() addsimps [replicate_app_Cons_same]) 1);
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1306
qed "rev_replicate";
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1307
Addsimps [rev_replicate];
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1308
8009
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1309
Goal "replicate (n+m) x = replicate n x @ replicate m x";
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1310
by (induct_tac "n" 1);
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1311
by Auto_tac;
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1312
qed "replicate_add";
29a7a79ee7f4 Imported Conny's lemmas from MicroJava
nipkow
parents: 7570
diff changeset
  1313
6794
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1314
Goal"n ~= 0 --> hd(replicate n x) = x";
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1315
by (induct_tac "n" 1);
6813
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1316
by Auto_tac;
6794
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1317
qed_spec_mp "hd_replicate";
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1318
Addsimps [hd_replicate];
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1319
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1320
Goal "n ~= 0 --> tl(replicate n x) = replicate (n-1) x";
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1321
by (induct_tac "n" 1);
6813
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1322
by Auto_tac;
6794
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1323
qed_spec_mp "tl_replicate";
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1324
Addsimps [tl_replicate];
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1325
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1326
Goal "n ~= 0 --> last(replicate n x) = x";
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1327
by (induct_tac "n" 1);
6813
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1328
by Auto_tac;
6794
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1329
qed_spec_mp "last_replicate";
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1330
Addsimps [last_replicate];
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1331
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1332
Goal "!i. i<n --> (replicate n x)!i = x";
6813
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1333
by (induct_tac "n" 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1334
 by (Simp_tac 1);
bf90f86502b2 many new lemmas about take & drop, incl the famous take-lemma
paulson
parents: 6794
diff changeset
  1335
by (asm_simp_tac (simpset() addsimps [nth_Cons] addsplits [nat.split]) 1);
6794
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1336
qed_spec_mp "nth_replicate";
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1337
Addsimps [nth_replicate];
ac367328b875 Added lots of 'replicate' lemmas.
nipkow
parents: 6451
diff changeset
  1338
4935
1694e2daef8f Cleaned up and simplified etc.
nipkow
parents: 4911
diff changeset
  1339
Goal "set(replicate (Suc n) x) = {x}";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
  1340
by (induct_tac "n" 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5296
diff changeset
  1341
by Auto_tac;
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
  1342
val lemma = result();
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
  1343
5043
3fdc881e8ff9 goal -> Goal
nipkow
parents: 4935
diff changeset
  1344
Goal "n ~= 0 ==> set(replicate n x) = {x}";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
  1345
by (fast_tac (claset() addSDs [not0_implies_Suc] addSIs [lemma]) 1);
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
  1346
qed "set_replicate";
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
  1347
Addsimps [set_replicate];