src/HOL/List.ML
author paulson
Thu Aug 22 12:24:25 1996 +0200 (1996-08-22)
changeset 1936 979e8b4f5fa5
parent 1908 55d8e38262a8
child 1985 84cf16192e03
permissions -rw-r--r--
Proved set_of_list_subset_Cons
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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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open List;
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val [Nil_not_Cons,Cons_not_Nil] = list.distinct;
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bind_thm("Cons_neq_Nil", Cons_not_Nil RS notE);
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bind_thm("Nil_neq_Cons", sym RS Cons_neq_Nil);
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bind_thm("Cons_inject", (hd list.inject) RS iffD1 RS conjE);
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goal List.thy "!x. xs ~= x#xs";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "not_Cons_self";
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goal List.thy "(xs ~= []) = (? y ys. xs = y#ys)";
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by (list.induct_tac "xs" 1);
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by (Simp_tac 1);
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by (Asm_simp_tac 1);
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by (REPEAT(resolve_tac [exI,refl,conjI] 1));
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qed "neq_Nil_conv";
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(** @ - append **)
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goal List.thy "(xs@ys)@zs = xs@(ys@zs)";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "append_assoc";
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goal List.thy "xs @ [] = xs";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "append_Nil2";
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goal List.thy "(xs@ys = []) = (xs=[] & ys=[])";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "append_is_Nil";
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goal List.thy "(xs @ ys = xs @ zs) = (ys=zs)";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "same_append_eq";
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goal List.thy "hd(xs@ys) = (if xs=[] then hd ys else hd xs)";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "hd_append";
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(** rev **)
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goal List.thy "rev(xs@ys) = rev(ys) @ rev(xs)";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS (asm_simp_tac (!simpset addsimps [append_Nil2,append_assoc])));
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qed "rev_append";
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goal List.thy "rev(rev l) = l";
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by (list.induct_tac "l" 1);
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by (ALLGOALS (asm_simp_tac (!simpset addsimps [rev_append])));
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qed "rev_rev_ident";
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(** mem **)
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goal List.thy "x mem (xs@ys) = (x mem xs | x mem ys)";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
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qed "mem_append";
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goal List.thy "x mem [x:xs.P(x)] = (x mem xs & P(x))";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
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qed "mem_filter";
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(** set_of_list **)
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goal thy "set_of_list (xs@ys) = (set_of_list xs Un set_of_list ys)";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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by (Fast_tac 1);
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qed "set_of_list_append";
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goal thy "(x mem xs) = (x: set_of_list xs)";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
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by (Fast_tac 1);
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qed "set_of_list_mem_eq";
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goal List.thy "set_of_list l <= set_of_list (x#l)";
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by (Simp_tac 1);
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by (Fast_tac 1);
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qed "set_of_list_subset_Cons";
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(** list_all **)
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goal List.thy "(Alls x:xs.True) = True";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "list_all_True";
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goal List.thy "list_all p (xs@ys) = (list_all p xs & list_all p ys)";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "list_all_conj";
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goal List.thy "(Alls x:xs.P(x)) = (!x. x mem xs --> P(x))";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
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by (Fast_tac 1);
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qed "list_all_mem_conv";
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(** list_case **)
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goal List.thy
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 "P(list_case a f xs) = ((xs=[] --> P(a)) & \
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\                         (!y ys. xs=y#ys --> P(f y ys)))";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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by (Fast_tac 1);
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qed "expand_list_case";
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goal List.thy  "(xs=[] --> P([])) & (!y ys. xs=y#ys --> P(y#ys)) --> P(xs)";
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by (list.induct_tac "xs" 1);
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by (Fast_tac 1);
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by (Fast_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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(** flat **)
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goal List.thy  "flat(xs@ys) = flat(xs)@flat(ys)";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS (asm_simp_tac (!simpset addsimps [append_assoc])));
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qed"flat_append";
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(** length **)
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goal List.thy "length(xs@ys) = length(xs)+length(ys)";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed"length_append";
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Addsimps [length_append];
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goal List.thy "length (map f l) = length l";
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by (list.induct_tac "l" 1);
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by (ALLGOALS Simp_tac);
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qed "length_map";
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Addsimps [length_map];
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goal List.thy "length(rev xs) = length(xs)";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "length_rev";
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Addsimps [length_rev];
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(** nth **)
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val [nth_0,nth_Suc] = nat_recs nth_def; 
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store_thm("nth_0",nth_0);
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store_thm("nth_Suc",nth_Suc);
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Addsimps [nth_0,nth_Suc];
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goal List.thy "!n. n < length xs --> nth n (map f xs) = f (nth n xs)";
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by (list.induct_tac "xs" 1);
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(* case [] *)
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by (Asm_full_simp_tac 1);
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(* case x#xl *)
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by (rtac allI 1);
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by (nat_ind_tac "n" 1);
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by (ALLGOALS Asm_full_simp_tac);
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qed_spec_mp "nth_map";
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Addsimps [nth_map];
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goal List.thy "!n. n < length xs --> list_all P xs --> P(nth n xs)";
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by (list.induct_tac "xs" 1);
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(* case [] *)
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by (Simp_tac 1);
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(* case x#xl *)
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by (rtac allI 1);
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by (nat_ind_tac "n" 1);
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by (ALLGOALS Asm_full_simp_tac);
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qed_spec_mp "list_all_nth";
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goal List.thy "!n. n < length xs --> (nth n xs) mem xs";
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by (list.induct_tac "xs" 1);
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(* case [] *)
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by (Simp_tac 1);
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(* case x#xl *)
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by (rtac allI 1);
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by (nat_ind_tac "n" 1);
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(* case 0 *)
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by (Asm_full_simp_tac 1);
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(* case Suc x *)
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by (asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 1);
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qed_spec_mp "nth_mem";
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Addsimps [nth_mem];
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(** drop **)
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goal thy "drop 0 xs = xs";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "drop_0";
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goal thy "drop (Suc n) (x#xs) = drop n xs";
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by (Simp_tac 1);
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qed "drop_Suc_Cons";
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Delsimps [drop_Cons];
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Addsimps [drop_0,drop_Suc_Cons];
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(** take **)
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goal thy "take 0 xs = []";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "take_0";
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goal thy "take (Suc n) (x#xs) = x # take n xs";
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by (Simp_tac 1);
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qed "take_Suc_Cons";
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Delsimps [take_Cons];
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Addsimps [take_0,take_Suc_Cons];
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(** Additional mapping lemmas **)
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goal List.thy "map (%x.x) = (%xs.xs)";
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by (rtac ext 1);
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by (list.induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "map_ident";
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goal List.thy "map f (xs@ys) = map f xs @ map f ys";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "map_append";
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goalw List.thy [o_def] "map (f o g) xs = map f (map g xs)";
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by (list.induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "map_compose";
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goal List.thy "rev(map f l) = map f (rev l)";
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by (list.induct_tac "l" 1);
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by (ALLGOALS (asm_simp_tac (!simpset addsimps [map_append])));
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qed "rev_map_distrib";
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goal List.thy "rev(flat ls) = flat (map rev (rev ls))";
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by (list.induct_tac "ls" 1);
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by (ALLGOALS (asm_simp_tac (!simpset addsimps 
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       [map_append, flat_append, rev_append, append_Nil2])));
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qed "rev_flat";
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Addsimps
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  [not_Cons_self, append_assoc, append_Nil2, append_is_Nil, same_append_eq,
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   mem_append, mem_filter,
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   rev_append, rev_rev_ident,
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   map_ident, map_append, map_compose,
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   flat_append, list_all_True, list_all_conj];
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