author | lcp |
Thu, 06 Apr 1995 11:49:42 +0200 | |
changeset 246 | 0f9230a24164 |
parent 234 | 80f45ad991cb |
permissions | -rw-r--r-- |
113 | 1 |
(* Title: HOL/List |
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ID: $Id$ |
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Author: Tobias Nipkow |
61620d959717
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parents:
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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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val list_ss = HOL_ss addsimps list.simps; |
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goal List.thy "!x. xs ~= x#xs"; |
61620d959717
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by (list.induct_tac "xs" 1); |
61620d959717
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by (ALLGOALS (asm_simp_tac list_ss)); |
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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); |
61620d959717
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nipkow
parents:
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by(simp_tac list_ss 1); |
61620d959717
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nipkow
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by(asm_simp_tac list_ss 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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val list_ss = arith_ss addsimps list.simps @ |
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[null_Nil, null_Cons, hd_Cons, tl_Cons, ttl_Nil, ttl_Cons, |
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mem_Nil, mem_Cons, |
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append_Nil, append_Cons, |
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map_Nil, map_Cons, |
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flat_Nil, flat_Cons, |
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list_all_Nil, list_all_Cons, |
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filter_Nil, filter_Cons, |
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foldl_Nil, foldl_Cons, |
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length_Nil, length_Cons]; |
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(** @ - append **) |
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goal List.thy "(xs@ys)@zs = xs@(ys@zs)"; |
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parents:
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by (list.induct_tac "xs" 1); |
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by(ALLGOALS(asm_simp_tac list_ss)); |
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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 list_ss)); |
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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 list_ss)); |
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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 list_ss)); |
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qed "same_append_eq"; |
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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 (list_ss 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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61620d959717
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nipkow
parents:
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diff
changeset
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by (list.induct_tac "xs" 1); |
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by(ALLGOALS(asm_simp_tac (list_ss setloop (split_tac [expand_if])))); |
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qed "mem_filter"; |
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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 list_ss)); |
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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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nipkow
parents:
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by (list.induct_tac "xs" 1); |
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by(ALLGOALS(asm_simp_tac list_ss)); |
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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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61620d959717
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nipkow
parents:
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diff
changeset
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by (list.induct_tac "xs" 1); |
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by(ALLGOALS(asm_simp_tac (list_ss setloop (split_tac [expand_if])))); |
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by(fast_tac HOL_cs 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)) & \ |
48
21291189b51e
changed "." to "$" and Cons to infix "#" to eliminate ambiguity
clasohm
parents:
44
diff
changeset
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\ (!y ys. xs=y#ys --> P(f(y,ys))))"; |
196
61620d959717
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nipkow
parents:
171
diff
changeset
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by (list.induct_tac "xs" 1); |
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by(ALLGOALS(asm_simp_tac list_ss)); |
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by(fast_tac HOL_cs 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 HOL_cs 1); |
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by(fast_tac HOL_cs 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 (list_ss 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 list_ss)); |
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qed"length_append"; |
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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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(** Additional mapping lemmas **) |
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goal List.thy "map(%x.x, xs) = xs"; |
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196
61620d959717
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nipkow
parents:
171
diff
changeset
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by (list.induct_tac "xs" 1); |
61620d959717
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nipkow
parents:
171
diff
changeset
|
139 |
by (ALLGOALS (asm_simp_tac list_ss)); |
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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); |
61620d959717
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nipkow
parents:
171
diff
changeset
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by (ALLGOALS (asm_simp_tac list_ss)); |
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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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61620d959717
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nipkow
parents:
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diff
changeset
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by (list.induct_tac "xs" 1); |
61620d959717
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nipkow
parents:
171
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changeset
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by (ALLGOALS (asm_simp_tac list_ss)); |
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qed "map_compose"; |
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val list_ss = list_ss 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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map_ident, map_append, map_compose, |
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flat_append, length_append, list_all_True, list_all_conj, nth_0, nth_Suc]; |
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