--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/BNF/Examples/Stream_Processor.thy Fri Jan 10 11:47:10 2014 +0100
@@ -0,0 +1,187 @@
+(* Title: HOL/BNF/Examples/Stream_Processor.thy
+ Author: Dmitriy Traytel, TU Muenchen
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2014
+
+Stream processors---a syntactic representation of continuous functions on streams
+*)
+
+header {* Stream Processors *}
+
+theory Stream_Processor
+imports Stream
+begin
+
+section {* Continuous Functions on Streams *}
+
+datatype_new ('a, 'b, 'c) sp\<^sub>\<mu> = Get "'a \<Rightarrow> ('a, 'b, 'c) sp\<^sub>\<mu>" | Put "'b" "'c"
+codatatype ('a, 'b) sp\<^sub>\<nu> = In (out: "('a, 'b, ('a, 'b) sp\<^sub>\<nu>) sp\<^sub>\<mu>")
+
+primrec_new run\<^sub>\<mu> :: "('a, 'b, 'c) sp\<^sub>\<mu> \<Rightarrow> 'a stream \<Rightarrow> ('b \<times> 'c) \<times> 'a stream" where
+ "run\<^sub>\<mu> (Get f) s = run\<^sub>\<mu> (f (shd s)) (stl s)"
+| "run\<^sub>\<mu> (Put b sp) s = ((b, sp), s)"
+
+primcorec run\<^sub>\<nu> :: "('a, 'b) sp\<^sub>\<nu> \<Rightarrow> 'a stream \<Rightarrow> 'b stream" where
+ "run\<^sub>\<nu> sp s = (let ((h, sp), s) = run\<^sub>\<mu> (out sp) s in h ## run\<^sub>\<nu> sp s)"
+
+primcorec copy :: "('a, 'a) sp\<^sub>\<nu>" where
+ "copy = In (Get (\<lambda>a. Put a copy))"
+
+lemma run\<^sub>\<nu>_copy: "run\<^sub>\<nu> copy s = s"
+ by (coinduction arbitrary: s) simp
+
+text {*
+To use the function package for the definition of composition the
+wellfoundedness of the subtree relation needs to be proved first.
+*}
+
+definition "sub \<equiv> {(f a, Get f) | a f. True}"
+
+lemma subI[intro]: "(f a, Get f) \<in> sub"
+ unfolding sub_def by blast
+
+lemma wf_sub[simp, intro]: "wf sub"
+proof (rule wfUNIVI)
+ fix P :: "('a, 'b, 'c) sp\<^sub>\<mu> \<Rightarrow> bool" and x
+ assume "\<forall>x. (\<forall>y. (y, x) \<in> sub \<longrightarrow> P y) \<longrightarrow> P x"
+ hence I: "\<And>x. (\<forall>y. (\<exists>a f. y = f a \<and> x = Get f) \<longrightarrow> P y) \<Longrightarrow> P x" unfolding sub_def by blast
+ show "P x" by (induct x) (auto intro: I)
+qed
+
+function
+ sp\<^sub>\<mu>_comp :: "('a, 'b, 'c) sp\<^sub>\<mu> \<Rightarrow> ('d, 'a, ('d, 'a) sp\<^sub>\<nu>) sp\<^sub>\<mu> \<Rightarrow> ('d, 'b, 'c \<times> ('d, 'a) sp\<^sub>\<nu>) sp\<^sub>\<mu>"
+ (infixl "o\<^sub>\<mu>" 65)
+where
+ "Put b sp o\<^sub>\<mu> fsp = Put b (sp, In fsp)"
+| "Get f o\<^sub>\<mu> Put b sp = f b o\<^sub>\<mu> out sp"
+| "Get f o\<^sub>\<mu> Get g = Get (\<lambda>a. Get f o\<^sub>\<mu> g a)"
+by pat_completeness auto
+termination by (relation "lex_prod sub sub") auto
+
+primcorec sp\<^sub>\<nu>_comp (infixl "o\<^sub>\<nu>" 65) where
+ "out (sp o\<^sub>\<nu> sp') = map_sp\<^sub>\<mu> id (\<lambda>(sp, sp'). sp o\<^sub>\<nu> sp') (out sp o\<^sub>\<mu> out sp')"
+
+lemma run\<^sub>\<nu>_sp\<^sub>\<nu>_comp: "run\<^sub>\<nu> (sp o\<^sub>\<nu> sp') = run\<^sub>\<nu> sp o run\<^sub>\<nu> sp'"
+proof (rule ext, unfold o_apply)
+ fix s
+ show "run\<^sub>\<nu> (sp o\<^sub>\<nu> sp') s = run\<^sub>\<nu> sp (run\<^sub>\<nu> sp' s)"
+ proof (coinduction arbitrary: sp sp' s)
+ case Eq_stream
+ show ?case
+ proof (induct "out sp" "out sp'" arbitrary: sp sp' s rule: sp\<^sub>\<mu>_comp.induct)
+ case (1 b sp'')
+ show ?case by (auto simp add: 1[symmetric])
+ next
+ case (2 f b sp'')
+ from 2(1)[of "In (f b)" sp''] show ?case by (simp add: 2(2,3)[symmetric])
+ next
+ case (3 f h)
+ from 3(1)[of _ "shd s" "In (h (shd s))", OF 3(2)] show ?case by (simp add: 3(2,3)[symmetric])
+ qed
+ qed
+qed
+
+text {* Alternative definition of composition using primrec_new instead of function *}
+
+primrec_new sp\<^sub>\<mu>_comp2R where
+ "sp\<^sub>\<mu>_comp2R f (Put b sp) = f b (out sp)"
+| "sp\<^sub>\<mu>_comp2R f (Get h) = Get (sp\<^sub>\<mu>_comp2R f o h)"
+
+primrec_new sp\<^sub>\<mu>_comp2 (infixl "o\<^sup>*\<^sub>\<mu>" 65) where
+ "Put b sp o\<^sup>*\<^sub>\<mu> fsp = Put b (sp, In fsp)"
+| "Get f o\<^sup>*\<^sub>\<mu> fsp = sp\<^sub>\<mu>_comp2R (op o\<^sup>*\<^sub>\<mu> o f) fsp"
+
+primcorec sp\<^sub>\<nu>_comp2 (infixl "o\<^sup>*\<^sub>\<nu>" 65) where
+ "out (sp o\<^sup>*\<^sub>\<nu> sp') = map_sp\<^sub>\<mu> id (\<lambda>(sp, sp'). sp o\<^sup>*\<^sub>\<nu> sp') (out sp o\<^sup>*\<^sub>\<mu> out sp')"
+
+lemma run\<^sub>\<nu>_sp\<^sub>\<nu>_comp2: "run\<^sub>\<nu> (sp o\<^sup>*\<^sub>\<nu> sp') = run\<^sub>\<nu> sp o run\<^sub>\<nu> sp'"
+proof (rule ext, unfold o_apply)
+ fix s
+ show "run\<^sub>\<nu> (sp o\<^sup>*\<^sub>\<nu> sp') s = run\<^sub>\<nu> sp (run\<^sub>\<nu> sp' s)"
+ proof (coinduction arbitrary: sp sp' s)
+ case Eq_stream
+ show ?case
+ proof (induct "out sp" arbitrary: sp sp' s)
+ case (Put b sp'')
+ show ?case by (auto simp add: Put[symmetric])
+ next
+ case (Get f)
+ then show ?case
+ proof (induct "out sp'" arbitrary: sp sp' s)
+ case (Put b sp'')
+ from Put(2)[of "In (f b)" sp''] show ?case by (simp add: Put(1,3)[symmetric])
+ next
+ case (Get h)
+ from Get(1)[OF _ Get(3,4), of "In (h (shd s))"] show ?case by (simp add: Get(2,4)[symmetric])
+ qed
+ qed
+ qed
+qed
+
+text {* The two definitions are equivalent *}
+
+lemma sp\<^sub>\<mu>_comp_sp\<^sub>\<mu>_comp2[simp]: "sp o\<^sub>\<mu> sp' = sp o\<^sup>*\<^sub>\<mu> sp'"
+ by (induct sp sp' rule: sp\<^sub>\<mu>_comp.induct) auto
+
+(*will be provided by the package*)
+lemma sp\<^sub>\<mu>_rel_map_map[unfolded vimage2p_def, simp]:
+ "rel_sp\<^sub>\<mu> R1 R2 (map_sp\<^sub>\<mu> f1 f2 sp) (map_sp\<^sub>\<mu> g1 g2 sp') =
+ rel_sp\<^sub>\<mu> (BNF_Def.vimage2p f1 g1 R1) (BNF_Def.vimage2p f2 g2 R2) sp sp'"
+by (tactic {*
+ let val ks = 1 upto 2;
+ in
+ BNF_Tactics.unfold_thms_tac @{context}
+ @{thms sp\<^sub>\<mu>.rel_compp sp\<^sub>\<mu>.rel_conversep sp\<^sub>\<mu>.rel_Grp vimage2p_Grp} THEN
+ HEADGOAL (EVERY' [rtac iffI, rtac @{thm relcomppI}, rtac @{thm GrpI}, rtac refl, rtac CollectI,
+ BNF_Util.CONJ_WRAP' (K (rtac @{thm subset_UNIV})) ks, rtac @{thm relcomppI}, atac,
+ rtac @{thm conversepI}, rtac @{thm GrpI}, rtac refl, rtac CollectI,
+ BNF_Util.CONJ_WRAP' (K (rtac @{thm subset_UNIV})) ks,
+ REPEAT_DETERM o eresolve_tac @{thms relcomppE conversepE GrpE},
+ hyp_subst_tac @{context}, atac])
+ end
+*})
+
+lemma sp\<^sub>\<mu>_rel_self: "\<lbrakk>op = \<le> R1; op = \<le> R2\<rbrakk> \<Longrightarrow> rel_sp\<^sub>\<mu> R1 R2 x x"
+ by (erule (1) predicate2D[OF sp\<^sub>\<mu>.rel_mono]) (simp only: sp\<^sub>\<mu>.rel_eq)
+
+lemma sp\<^sub>\<nu>_comp_sp\<^sub>\<nu>_comp2: "sp o\<^sub>\<nu> sp' = sp o\<^sup>*\<^sub>\<nu> sp'"
+ by (coinduction arbitrary: sp sp') (auto intro!: sp\<^sub>\<mu>_rel_self)
+
+
+section {* Generalization to an Arbitrary BNF as Codomain *}
+
+bnf_decl ('a, 'b) F (map: F)
+
+definition \<theta> :: "('p,'a) F * 'b \<Rightarrow> ('p,'a * 'b) F" where
+ "\<theta> xb = F id <id, \<lambda> a. (snd xb)> (fst xb)"
+
+(* The strength laws for \<theta>: *)
+lemma \<theta>_natural: "F id (map_pair f g) o \<theta> = \<theta> o map_pair (F id f) g"
+ unfolding \<theta>_def F.map_comp o_def id_apply convol_def map_pair_def split_beta fst_conv snd_conv ..
+
+definition assl :: "'a * ('b * 'c) \<Rightarrow> ('a * 'b) * 'c" where
+ "assl abc = ((fst abc, fst (snd abc)), snd (snd abc))"
+
+lemma \<theta>_rid: "F id fst o \<theta> = fst"
+ unfolding \<theta>_def F.map_comp F.map_id o_def id_apply convol_def fst_conv sym[OF id_def] ..
+
+lemma \<theta>_assl: "F id assl o \<theta> = \<theta> o map_pair \<theta> id o assl"
+ unfolding assl_def \<theta>_def F.map_comp o_def id_apply convol_def map_pair_def split fst_conv snd_conv ..
+
+datatype_new ('a, 'b, 'c) spF\<^sub>\<mu> = GetF "'a \<Rightarrow> ('a, 'b, 'c) spF\<^sub>\<mu>" | PutF "('b,'c) F"
+codatatype ('a, 'b) spF\<^sub>\<nu> = InF (outF: "('a, 'b, ('a, 'b) spF\<^sub>\<nu>) spF\<^sub>\<mu>")
+
+codatatype 'b JF = Ctor (dtor: "('b, 'b JF) F")
+
+(* Definition of run for an arbitrary final coalgebra as codomain: *)
+
+primrec_new
+ runF\<^sub>\<mu> :: "('a, 'b, ('a, 'b) spF\<^sub>\<nu>) spF\<^sub>\<mu> \<Rightarrow> 'a stream \<Rightarrow> (('b, ('a, 'b) spF\<^sub>\<nu>) F) \<times> 'a stream"
+where
+ "runF\<^sub>\<mu> (GetF f) s = (runF\<^sub>\<mu> o f) (shd s) (stl s)"
+| "runF\<^sub>\<mu> (PutF x) s = (x, s)"
+
+primcorec runF\<^sub>\<nu> :: "('a, 'b) spF\<^sub>\<nu> \<Rightarrow> 'a stream \<Rightarrow> 'b JF" where
+ "runF\<^sub>\<nu> sp s = (let (x, s) = runF\<^sub>\<mu> (outF sp) s in Ctor (F id (\<lambda> sp. runF\<^sub>\<nu> sp s) x))"
+
+end