# Theory Stream

```(*  Title:      CCL/ex/Stream.thy
Author:     Martin Coen, Cambridge University Computer Laboratory
*)

section ‹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 (eq_coinduct3 "{p. EX x y. p=<x,y> ∧ (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(λx. x, l) = l"
apply (eq_coinduct3 "{p. EX x y. p=<x,y> ∧ (EX l:Lists (A) .x=map (λx. x,l) ∧ y=l) }")
apply (blast intro: 1)
apply safe
apply (drule ListsXH [THEN iffD1])
apply EQgen
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 (eq_coinduct3
"{p. EX x y. p=<x,y> ∧ (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 ‹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 (eq_coinduct3
"{p. EX x y. p=<x,y> ∧ (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 ‹DEPTH_SOLVE (eresolve_tac \<^context> [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 (eq_coinduct3 "{p. EX x y. p=<x,y> ∧ (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 = "λx. lhs(x) = rhs" for lhs rhs in iter2B [THEN ssubst])
done

lemma iter1_iter2_eq: "iter1(f,a) = iter2(f,a)"
apply (eq_coinduct3
"{p. EX x y. p=<x,y> ∧ (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
```