# Theory Stream

theory Stream
imports List
`(*  Title:      CCL/ex/Stream.thy    Author:     Martin Coen, Cambridge University Computer Laboratory    Copyright   1993  University of Cambridge*)header {* Programs defined over streams *}theory Streamimports Listbegindefinition 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 o g,l) = map(f,map(g,l))"  apply (tactic {* eq_coinduct3_tac @{context}    "{p. EX x y. p=<x,y> & (EX l:Lists (A) .x=map (f o 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 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 (tactic {* eq_coinduct3_tac @{context}    "{p. EX x y. p=<x,y> & (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  donesubsection {* 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=<x,y> & (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  donesubsection {* 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=<x,y> & (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) *})  donesubsection {* 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=<x,y> & (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  donelemma iter2B: "iter2(f,a) = a \$ map(f,iter2(f,a))"  apply (unfold iter2_def)  apply (rule letrecB [THEN trans])  apply (rule refl)  donelemma 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)  donelemma iter1_iter2_eq: "iter1(f,a) = iter2(f,a)"  apply (tactic {* eq_coinduct3_tac @{context}    "{p. EX x y. p=<x,y> & (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  doneend`