(* Title: HOLCF/FOCUS/Fstream.thy
Author: David von Oheimb, TU Muenchen
FOCUS streams (with lifted elements).
TODO: integrate Fstreams.thy
*)
header {* FOCUS flat streams *}
theory Fstream
imports "../ex/Stream"
begin
default_sort type
types 'a fstream = "'a lift stream"
definition
fscons :: "'a \<Rightarrow> 'a fstream \<rightarrow> 'a fstream" where
"fscons a = (\<Lambda> s. Def a && s)"
definition
fsfilter :: "'a set \<Rightarrow> 'a fstream \<rightarrow> 'a fstream" where
"fsfilter A = (sfilter\<cdot>(flift2 (\<lambda>x. x\<in>A)))"
abbreviation
emptystream :: "'a fstream" ("<>") where
"<> == \<bottom>"
abbreviation
fscons' :: "'a \<Rightarrow> 'a fstream \<Rightarrow> 'a fstream" ("(_~>_)" [66,65] 65) where
"a~>s == fscons a\<cdot>s"
abbreviation
fsfilter' :: "'a set \<Rightarrow> 'a fstream \<Rightarrow> 'a fstream" ("(_'(C')_)" [64,63] 63) where
"A(C)s == fsfilter A\<cdot>s"
notation (xsymbols)
fscons' ("(_\<leadsto>_)" [66,65] 65) and
fsfilter' ("(_\<copyright>_)" [64,63] 63)
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.nchotomy)
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
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 (cases t)
apply (cut_tac fscons_not_empty)
apply (fast dest: eq_UU_iff [THEN iffD2])
apply (simp add: fscons_def2)
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"
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 [thm "DefE"]) 1 *})
apply (tactic {* step_tac (HOL_cs addSEs [thm "DefE"]) 1 *})
apply (tactic {* step_tac (HOL_cs addSEs [thm "DefE"]) 1 *})
apply (tactic {* step_tac (HOL_cs addSEs [thm "DefE"]) 1 *})
apply (tactic {* step_tac (HOL_cs addSEs [thm "DefE"]) 1 *})
apply (tactic {* step_tac (HOL_cs addSEs [thm "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.induct)
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 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; (\<Squnion>i. Y i) = 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_mono)
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; (\<Squnion>i. Y i) = 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) & (\<Squnion>i. X i) = 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 i="j" and j="i+j" in chain_mono)
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