(* Title: CCL/ex/Stream.thy
Author: Martin Coen, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
*)
section \<open>Programs defined over streams\<close>
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 \<open>Map of composition is composition of maps\<close>
lemma map_comp:
assumes 1: "l:Lists(A)"
shows "map(f \ g,l) = map(f,map(g,l))"
apply (eq_coinduct3 "{p. EX x y. p= \ (EX l:Lists (A) .x=map (f \ g,l) \ y=map (f,map (g,l)))}")
apply (blast intro: 1)
apply safe
apply (drule ListsXH [THEN iffD1])
apply EQgen
apply fastforce
done
(*** Mapping the identity function leaves a list unchanged ***)
lemma map_id:
assumes 1: "l:Lists(A)"
shows "map(\x. x, l) = l"
apply (eq_coinduct3 "{p. EX x y. p= \ (EX l:Lists (A) .x=map (\x. x,l) \ y=l) }")
apply (blast intro: 1)
apply safe
apply (drule ListsXH [THEN iffD1])
apply EQgen
apply blast
done
subsection \<open>Mapping distributes over append\<close>
lemma map_append:
assumes "l:Lists(A)"
and "m:Lists(A)"
shows "map(f,l@m) = map(f,l) @ map(f,m)"
apply (eq_coinduct3
"{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))}")
apply (blast intro: assms)
apply safe
apply (drule ListsXH [THEN iffD1])
apply EQgen
apply (drule ListsXH [THEN iffD1])
apply EQgen
apply blast
done
subsection \<open>Append is associative\<close>
lemma append_assoc:
assumes "k:Lists(A)"
and "l:Lists(A)"
and "m:Lists(A)"
shows "k @ l @ m = (k @ l) @ m"
apply (eq_coinduct3
"{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) }")
apply (blast intro: assms)
apply safe
apply (drule ListsXH [THEN iffD1])
apply EQgen
prefer 2
apply blast
apply (tactic \<open>DEPTH_SOLVE (eresolve_tac \<^context> [XH_to_E @{thm ListsXH}] 1
THEN EQgen_tac \<^context> [] 1)\<close>)
done
subsection \<open>Appending anything to an infinite list doesn't alter it\<close>
lemma ilist_append:
assumes "l:ILists(A)"
shows "l @ m = l"
apply (eq_coinduct3 "{p. EX x y. p= \ (EX l:ILists (A) .EX m. x=l@m \ y=l)}")
apply (blast intro: assms)
apply safe
apply (drule IListsXH [THEN iffD1])
apply EQgen
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" for lhs rhs in iter2B [THEN ssubst])
apply (simp add: nmapBcons)
done
lemma iter1_iter2_eq: "iter1(f,a) = iter2(f,a)"
apply (eq_coinduct3
"{p. EX x y. p= \ (EX n:Nat. x=iter1 (f,f^n`a) \ y=map (f) ^n`iter2 (f,a))}")
apply (fast intro!: napplyBzero [symmetric] napplyBzero [symmetric, THEN arg_cong])
apply (EQgen iter1B iter2Blemma)
apply (subst napply_f, assumption)
apply (rule_tac f1 = f in napplyBsucc [THEN subst])
apply blast
done
end
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