(* Title: CCL/ex/Stream.thy Author: Martin Coen, Cambridge University Computer Laboratory Copyright 1993 University of Cambridge *) header {* Programs defined over streams *} theory Stream imports List begin definition iter1 :: "[i=>i,i]=>i" where "iter1(f,a) == letrec iter x be x$iter(f(x)) in iter(a)" definition iter2 :: "[i=>i,i]=>i" where "iter2(f,a) == letrec iter x be x$map(f,iter(x)) in iter(a)" (* Proving properties about infinite lists using coinduction: Lists(A) is the set of all finite and infinite lists of elements of A. ILists(A) is the set of infinite lists of elements of A. *) subsection {* Map of composition is composition of maps *} lemma map_comp: assumes 1: "l:Lists(A)" shows "map(f \ g,l) = map(f,map(g,l))" apply (tactic {* eq_coinduct3_tac @{context} "{p. EX x y. p= & (EX l:Lists (A) .x=map (f \ g,l) & y=map (f,map (g,l)))}" 1 *}) apply (blast intro: 1) apply safe apply (drule ListsXH [THEN iffD1]) apply (tactic "EQgen_tac @{context} [] 1") apply fastsimp done (*** Mapping the identity function leaves a list unchanged ***) lemma map_id: assumes 1: "l:Lists(A)" shows "map(%x. x,l) = l" apply (tactic {* eq_coinduct3_tac @{context} "{p. EX x y. p= & (EX l:Lists (A) .x=map (%x. x,l) & y=l) }" 1 *}) apply (blast intro: 1) apply safe apply (drule ListsXH [THEN iffD1]) apply (tactic "EQgen_tac @{context} [] 1") apply blast done subsection {* Mapping distributes over append *} lemma map_append: assumes "l:Lists(A)" and "m:Lists(A)" shows "map(f,l@m) = map(f,l) @ map(f,m)" apply (tactic {* eq_coinduct3_tac @{context} "{p. EX x y. p= & (EX l:Lists (A). EX m:Lists (A). x=map (f,l@m) & y=map (f,l) @ map (f,m))}" 1 *}) apply (blast intro: assms) apply safe apply (drule ListsXH [THEN iffD1]) apply (tactic "EQgen_tac @{context} [] 1") apply (drule ListsXH [THEN iffD1]) apply (tactic "EQgen_tac @{context} [] 1") apply blast done subsection {* Append is associative *} lemma append_assoc: assumes "k:Lists(A)" and "l:Lists(A)" and "m:Lists(A)" shows "k @ l @ m = (k @ l) @ m" apply (tactic {* eq_coinduct3_tac @{context} "{p. EX x y. p= & (EX k:Lists (A). EX l:Lists (A). EX m:Lists (A). x=k @ l @ m & y= (k @ l) @ m) }" 1*}) apply (blast intro: assms) apply safe apply (drule ListsXH [THEN iffD1]) apply (tactic "EQgen_tac @{context} [] 1") prefer 2 apply blast apply (tactic {* DEPTH_SOLVE (etac (XH_to_E @{thm ListsXH}) 1 THEN EQgen_tac @{context} [] 1) *}) done subsection {* Appending anything to an infinite list doesn't alter it *} lemma ilist_append: assumes "l:ILists(A)" shows "l @ m = l" apply (tactic {* eq_coinduct3_tac @{context} "{p. EX x y. p= & (EX l:ILists (A) .EX m. x=l@m & y=l)}" 1 *}) apply (blast intro: assms) apply safe apply (drule IListsXH [THEN iffD1]) apply (tactic "EQgen_tac @{context} [] 1") apply blast done (*** The equivalance of two versions of an iteration function ***) (* *) (* fun iter1(f,a) = a$iter1(f,f(a)) *) (* fun iter2(f,a) = a$map(f,iter2(f,a)) *) lemma iter1B: "iter1(f,a) = a$iter1(f,f(a))" apply (unfold iter1_def) apply (rule letrecB [THEN trans]) apply simp done lemma iter2B: "iter2(f,a) = a $ map(f,iter2(f,a))" apply (unfold iter2_def) apply (rule letrecB [THEN trans]) apply (rule refl) done lemma iter2Blemma: "n:Nat ==> map(f) ^ n ` iter2(f,a) = (f ^ n ` a) $ (map(f) ^ n ` map(f,iter2(f,a)))" apply (rule_tac P = "%x. ?lhs (x) = ?rhs" in iter2B [THEN ssubst]) apply (simp add: nmapBcons) done lemma iter1_iter2_eq: "iter1(f,a) = iter2(f,a)" apply (tactic {* eq_coinduct3_tac @{context} "{p. EX x y. p= & (EX n:Nat. x=iter1 (f,f^n`a) & y=map (f) ^n`iter2 (f,a))}" 1*}) apply (fast intro!: napplyBzero [symmetric] napplyBzero [symmetric, THEN arg_cong]) apply (tactic {* EQgen_tac @{context} [@{thm iter1B}, @{thm iter2Blemma}] 1 *}) apply (subst napply_f, assumption) apply (rule_tac f1 = f in napplyBsucc [THEN subst]) apply blast done end