src/HOLCF/FOCUS/Fstream.thy

   fscons_def:    "fscons a   \<equiv> \<Lambda> s. Def a && s"
   fsfilter_def:  "fsfilter A \<equiv> sfilter\<cdot>(flift2 (\<lambda>x. x\<in>A))"
 
 ML {* use_legacy_bindings (the_context ()) *}
 
+lemma Def_maximal: "a = Def d \<Longrightarrow> a\<sqsubseteq>b \<Longrightarrow> b = Def d"
+apply (rule flat_eq [THEN iffD1, symmetric])
+apply (rule Def_not_UU)
+apply (drule antisym_less_inverse)
+apply (erule (1) conjunct2 [THEN trans_less])
+done
+
+
+section "fscons"
+
+lemma fscons_def2: "a~>s = Def a && s"
+apply (unfold fscons_def)
+apply (simp)
+done
+
+lemma fstream_exhaust: "x = UU |  (? a y. x = a~> y)"
+apply (simp add: fscons_def2)
+apply (cut_tac stream.exhaust)
+apply (fast dest: not_Undef_is_Def [THEN iffD1])
+done
+
+lemma fstream_cases: "[| x = UU ==> P; !!a y. x = a~> y ==> P |] ==> P"
+apply (cut_tac fstream_exhaust)
+apply (erule disjE)
+apply fast
+apply fast
+done
+(*
+fun fstream_case_tac s i = res_inst_tac [("x",s)] fstream_cases i
+                          THEN hyp_subst_tac i THEN hyp_subst_tac (i+1);
+*)
+lemma fstream_exhaust_eq: "(x ~= UU) = (? a y. x = a~> y)"
+apply (simp add: fscons_def2 stream_exhaust_eq)
+apply (fast dest: not_Undef_is_Def [THEN iffD1] elim: DefE)
+done
+
+
+lemma fscons_not_empty [simp]: "a~> s ~= <>"
+by (simp add: fscons_def2)
+
+
+lemma fscons_inject [simp]: "(a~> s = b~> t) = (a = b &  s = t)"
+by (simp add: fscons_def2)
+
+lemma fstream_prefix: "a~> s << t ==> ? tt. t = a~> tt &  s << tt"
+apply (rule_tac x="t" in stream.casedist)
+apply (cut_tac fscons_not_empty)
+apply (fast dest: eq_UU_iff [THEN iffD2])
+apply (simp add: fscons_def2)
+apply (drule stream_flat_prefix)
+apply (rule Def_not_UU)
+apply (fast)
+done
+
+lemma fstream_prefix' [simp]:
+        "x << a~> z = (x = <> |  (? y. x = a~> y &  y << z))"
+apply (simp add: fscons_def2 Def_not_UU [THEN stream_prefix'])
+apply (safe)
+apply (erule_tac [!] contrapos_np)
+prefer 2 apply (fast elim: DefE)
+apply (rule Lift_cases)
+apply (erule (1) notE)
+apply (safe)
+apply (drule Def_inject_less_eq [THEN iffD1])
+apply fast
+done
+
+(* ------------------------------------------------------------------------- *)
+
+section "ft & rt"
+
+lemmas ft_empty = stream.sel_rews (1)
+lemma ft_fscons [simp]: "ft\<cdot>(m~> s) = Def m"
+by (simp add: fscons_def)
+
+lemmas rt_empty = stream.sel_rews (2)
+lemma rt_fscons [simp]: "rt\<cdot>(m~> s) = s"
+by (simp add: fscons_def)
+
+lemma ft_eq [simp]: "(ft\<cdot>s = Def a) = (? t. s = a~> t)"
+apply (unfold fscons_def)
+apply (simp)
+apply (safe)
+apply (erule subst)
+apply (rule exI)
+apply (rule surjectiv_scons [symmetric])
+apply (simp)
+done
+
+lemma surjective_fscons_lemma: "(d\<leadsto>y = x) = (ft\<cdot>x = Def d & rt\<cdot>x = y)"
+by auto
+
+lemma surjective_fscons: "ft\<cdot>x = Def d \<Longrightarrow> d\<leadsto>rt\<cdot>x = x"
+by (simp add: surjective_fscons_lemma)
+
+
+(* ------------------------------------------------------------------------- *)
+
+section "take"
+
+lemma fstream_take_Suc [simp]:
+        "stream_take (Suc n)\<cdot>(a~> s) = a~> stream_take n\<cdot>s"
+by (simp add: fscons_def)
+
+
+(* ------------------------------------------------------------------------- *)
+
+section "slen"
+
+(*bind_thm("slen_empty", slen_empty);*)
+
+lemma slen_fscons: "#(m~> s) = iSuc (#s)"
+by (simp add: fscons_def)
+
+lemma slen_fscons_eq:
+        "(Fin (Suc n) < #x) = (? a y. x = a~> y & Fin n < #y)"
+apply (simp add: fscons_def2 slen_scons_eq)
+apply (fast dest: not_Undef_is_Def [THEN iffD1] elim: DefE)
+done
+
+lemma slen_fscons_eq_rev:
+        "(#x < Fin (Suc (Suc n))) = (!a y. x ~= a~> y | #y < Fin (Suc n))"
+apply (simp add: fscons_def2 slen_scons_eq_rev)
+apply (tactic "step_tac (HOL_cs addSEs [DefE]) 1")
+apply (tactic "step_tac (HOL_cs addSEs [DefE]) 1")
+apply (tactic "step_tac (HOL_cs addSEs [DefE]) 1")
+apply (tactic "step_tac (HOL_cs addSEs [DefE]) 1")
+apply (tactic "step_tac (HOL_cs addSEs [DefE]) 1")
+apply (tactic "step_tac (HOL_cs addSEs [DefE]) 1")
+apply (erule contrapos_np)
+apply (fast dest: not_Undef_is_Def [THEN iffD1] elim: DefE)
+done
+
+lemma slen_fscons_less_eq:
+        "(#(a~> y) < Fin (Suc (Suc n))) = (#y < Fin (Suc n))"
+apply (subst slen_fscons_eq_rev)
+apply (fast dest!: fscons_inject [THEN iffD1])
+done
+
+
+(* ------------------------------------------------------------------------- *)
+
+section "induction"
+
+lemma fstream_ind:
+        "[| adm P; P <>; !!a s. P s ==> P (a~> s) |] ==> P x"
+apply (erule stream.ind)
+apply (assumption)
+apply (unfold fscons_def2)
+apply (fast dest: not_Undef_is_Def [THEN iffD1])
+done
+
+lemma fstream_ind2:
+  "[| adm P; P UU; !!a. P (a~> UU); !!a b s. P s ==> P (a~> b~> s) |] ==> P x"
+apply (erule stream_ind2)
+apply (assumption)
+apply (unfold fscons_def2)
+apply (fast dest: not_Undef_is_Def [THEN iffD1])
+apply (fast dest: not_Undef_is_Def [THEN iffD1])
+done
+
+
+(* ------------------------------------------------------------------------- *)
+
+section "fsfilter"
+
+lemma fsfilter_empty: "A(C)UU = UU"
+apply (unfold fsfilter_def)
+apply (rule sfilter_empty)
+done
+
+lemma fsfilter_fscons:
+        "A(C)x~> xs = (if x:A then x~> (A(C)xs) else A(C)xs)"
+apply (unfold fsfilter_def)
+apply (simp add: fscons_def2 sfilter_scons If_and_if)
+done
+
+lemma fsfilter_emptys: "{}(C)x = UU"
+apply (rule_tac x="x" in fstream_ind)
+apply (simp)
+apply (rule fsfilter_empty)
+apply (simp add: fsfilter_fscons)
+done
+
+lemma fsfilter_insert: "(insert a A)(C)a~> x = a~> ((insert a A)(C)x)"
+by (simp add: fsfilter_fscons)
+
+lemma fsfilter_single_in: "{a}(C)a~> x = a~> ({a}(C)x)"
+by (rule fsfilter_insert)
+
+lemma fsfilter_single_out: "b ~= a ==> {a}(C)b~> x = ({a}(C)x)"
+by (simp add: fsfilter_fscons)
+
+lemma fstream_lub_lemma1:
+    "\<lbrakk>chain Y; lub (range Y) = a\<leadsto>s\<rbrakk> \<Longrightarrow> \<exists>j t. Y j = a\<leadsto>t"
+apply (case_tac "max_in_chain i Y")
+apply  (drule (1) lub_finch1 [THEN thelubI, THEN sym])
+apply  (force)
+apply (unfold max_in_chain_def)
+apply auto
+apply (frule (1) chain_mono3)
+apply (rule_tac x="Y j" in fstream_cases)
+apply  (force)
+apply (drule_tac x="j" in is_ub_thelub)
+apply (force)
+done
+
+lemma fstream_lub_lemma:
+      "\<lbrakk>chain Y; lub (range Y) = a\<leadsto>s\<rbrakk> \<Longrightarrow> (\<exists>j t. Y j = a\<leadsto>t) & (\<exists>X. chain X & (!i. ? j. Y j = a\<leadsto>X i) & lub (range X) = s)"
+apply (frule (1) fstream_lub_lemma1)
+apply (clarsimp)
+apply (rule_tac x="%i. rt\<cdot>(Y(i+j))" in exI)
+apply (rule conjI)
+apply  (erule chain_shift [THEN chain_monofun])
+apply safe
+apply  (drule_tac x="j" and y="i+j" in chain_mono3)
+apply   (simp)
+apply  (simp)
+apply  (rule_tac x="i+j" in exI)
+apply  (drule fstream_prefix)
+apply  (clarsimp)
+apply  (subst contlub_cfun [symmetric])
+apply   (rule chainI)
+apply   (fast)
+apply  (erule chain_shift)
+apply (subst lub_const [THEN thelubI])
+apply (subst lub_range_shift)
+apply  (assumption)
+apply (simp)
+done
+
 end