Theory Buffer

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theory Buffer
imports FOCUS

(*  Title:      HOLCF/FOCUS/Buffer.thy
Author: David von Oheimb, TU Muenchen

Formalization of section 4 of

@inproceedings {broy_mod94,
author = {Manfred Broy},
title = {{Specification and Refinement of a Buffer of Length One}},
booktitle = {Deductive Program Design},
year = {1994},
editor = {Manfred Broy},
volume = {152},
series = {ASI Series, Series F: Computer and System Sciences},
pages = {273 -- 304},
publisher = {Springer}
}

Slides available from http://ddvo.net/talks/1-Buffer.ps.gz

*)


theory Buffer
imports FOCUS
begin


typedecl D

datatype

M = Md D | Mreq ("•")


datatype

State = Sd D | Snil ("¤")


types

SPF11 = "M fstream -> D fstream"
SPEC11 = "SPF11 set"
SPSF11 = "State => SPF11"
SPECS11 = "SPSF11 set"


definition
BufEq_F :: "SPEC11 => SPEC11" where
"BufEq_F B = {f. ∀d. f·(Md d\<leadsto><>) = <> ∧
(∀x. ∃ff∈B. f·(Md d\<leadsto>•\<leadsto>x) = d\<leadsto>ff·x)}"


definition
BufEq :: "SPEC11" where
"BufEq = gfp BufEq_F"


definition
BufEq_alt :: "SPEC11" where
"BufEq_alt = gfp (λB. {f. ∀d. f·(Md d\<leadsto><> ) = <> ∧
(∃ff∈B. (∀x. f·(Md d\<leadsto>•\<leadsto>x) = d\<leadsto>ff·x))})"


definition
BufAC_Asm_F :: " (M fstream set) => (M fstream set)" where
"BufAC_Asm_F A = {s. s = <> ∨
(∃d x. s = Md d\<leadsto>x ∧ (x = <> ∨ (ft·x = Def • ∧ (rt·x)∈A)))}"


definition
BufAC_Asm :: " (M fstream set)" where
"BufAC_Asm = gfp BufAC_Asm_F"


definition
BufAC_Cmt_F :: "((M fstream * D fstream) set) =>
((M fstream * D fstream) set)"
where
"BufAC_Cmt_F C = {(s,t). ∀d x.
(s = <> --> t = <> ) ∧
(s = Md d\<leadsto><> --> t = <> ) ∧
(s = Md d\<leadsto>•\<leadsto>x --> (ft·t = Def d ∧ (x,rt·t)∈C))}"


definition
BufAC_Cmt :: "((M fstream * D fstream) set)" where
"BufAC_Cmt = gfp BufAC_Cmt_F"


definition
BufAC :: "SPEC11" where
"BufAC = {f. ∀x. x∈BufAC_Asm --> (x,f·x)∈BufAC_Cmt}"


definition
BufSt_F :: "SPECS11 => SPECS11" where
"BufSt_F H = {h. ∀s . h s ·<> = <> ∧
(∀d x. h ¤ ·(Md d\<leadsto>x) = h (Sd d)·x ∧
(∃hh∈H. h (Sd d)·(• \<leadsto>x) = d\<leadsto>(hh ¤·x)))}"


definition
BufSt_P :: "SPECS11" where
"BufSt_P = gfp BufSt_F"


definition
BufSt :: "SPEC11" where
"BufSt = {f. ∃h∈BufSt_P. f = h ¤}"



lemma set_cong: "!!X. A = B ==> (x:A) = (x:B)"
by (erule subst, rule refl)


(**** BufEq *******************************************************************)

lemma mono_BufEq_F: "mono BufEq_F"
by (unfold mono_def BufEq_F_def, fast)

lemmas BufEq_fix = mono_BufEq_F [THEN BufEq_def [THEN eq_reflection, THEN def_gfp_unfold]]

lemma BufEq_unfold: "(f:BufEq) = (!d. f·(Md d\<leadsto><>) = <> &
(!x. ? ff:BufEq. f·(Md d\<leadsto>•\<leadsto>x) = d\<leadsto>(ff·x)))"

apply (subst BufEq_fix [THEN set_cong])
apply (unfold BufEq_F_def)
apply (simp)
done

lemma Buf_f_empty: "f:BufEq ==> f·<> = <>"
by (drule BufEq_unfold [THEN iffD1], auto)

lemma Buf_f_d: "f:BufEq ==> f·(Md d\<leadsto><>) = <>"
by (drule BufEq_unfold [THEN iffD1], auto)

lemma Buf_f_d_req:
"f:BufEq ==> ∃ff. ff:BufEq ∧ f·(Md d\<leadsto>•\<leadsto>x) = d\<leadsto>ff·x"

by (drule BufEq_unfold [THEN iffD1], auto)


(**** BufAC_Asm ***************************************************************)

lemma mono_BufAC_Asm_F: "mono BufAC_Asm_F"
by (unfold mono_def BufAC_Asm_F_def, fast)

lemmas BufAC_Asm_fix =
mono_BufAC_Asm_F [THEN BufAC_Asm_def [THEN eq_reflection, THEN def_gfp_unfold]]


lemma BufAC_Asm_unfold: "(s:BufAC_Asm) = (s = <> | (? d x.
s = Md d\<leadsto>x & (x = <> | (ft·x = Def • & (rt·x):BufAC_Asm))))"

apply (subst BufAC_Asm_fix [THEN set_cong])
apply (unfold BufAC_Asm_F_def)
apply (simp)
done

lemma BufAC_Asm_empty: "<> :BufAC_Asm"
by (rule BufAC_Asm_unfold [THEN iffD2], auto)

lemma BufAC_Asm_d: "Md d\<leadsto><>:BufAC_Asm"
by (rule BufAC_Asm_unfold [THEN iffD2], auto)
lemma BufAC_Asm_d_req: "x:BufAC_Asm ==> Md d\<leadsto>•\<leadsto>x:BufAC_Asm"
by (rule BufAC_Asm_unfold [THEN iffD2], auto)
lemma BufAC_Asm_prefix2: "a\<leadsto>b\<leadsto>s:BufAC_Asm ==> s:BufAC_Asm"
by (drule BufAC_Asm_unfold [THEN iffD1], auto)


(**** BBufAC_Cmt **************************************************************)

lemma mono_BufAC_Cmt_F: "mono BufAC_Cmt_F"
by (unfold mono_def BufAC_Cmt_F_def, fast)

lemmas BufAC_Cmt_fix =
mono_BufAC_Cmt_F [THEN BufAC_Cmt_def [THEN eq_reflection, THEN def_gfp_unfold]]


lemma BufAC_Cmt_unfold: "((s,t):BufAC_Cmt) = (!d x.
(s = <> --> t = <>) &
(s = Md d\<leadsto><> --> t = <>) &
(s = Md d\<leadsto>•\<leadsto>x --> ft·t = Def d & (x, rt·t):BufAC_Cmt))"

apply (subst BufAC_Cmt_fix [THEN set_cong])
apply (unfold BufAC_Cmt_F_def)
apply (simp)
done

lemma BufAC_Cmt_empty: "f:BufEq ==> (<>, f·<>):BufAC_Cmt"
by (rule BufAC_Cmt_unfold [THEN iffD2], auto simp add: Buf_f_empty)

lemma BufAC_Cmt_d: "f:BufEq ==> (a\<leadsto>⊥, f·(a\<leadsto>⊥)):BufAC_Cmt"
by (rule BufAC_Cmt_unfold [THEN iffD2], auto simp add: Buf_f_d)

lemma BufAC_Cmt_d2:
"(Md d\<leadsto>⊥, f·(Md d\<leadsto>⊥)):BufAC_Cmt ==> f·(Md d\<leadsto>⊥) = ⊥"

by (drule BufAC_Cmt_unfold [THEN iffD1], auto)

lemma BufAC_Cmt_d3:
"(Md d\<leadsto>•\<leadsto>x, f·(Md d\<leadsto>•\<leadsto>x)):BufAC_Cmt ==> (x, rt·(f·(Md d\<leadsto>•\<leadsto>x))):BufAC_Cmt"

by (drule BufAC_Cmt_unfold [THEN iffD1], auto)

lemma BufAC_Cmt_d32:
"(Md d\<leadsto>•\<leadsto>x, f·(Md d\<leadsto>•\<leadsto>x)):BufAC_Cmt ==> ft·(f·(Md d\<leadsto>•\<leadsto>x)) = Def d"

by (drule BufAC_Cmt_unfold [THEN iffD1], auto)

(**** BufAC *******************************************************************)

lemma BufAC_f_d: "f ∈ BufAC ==> f·(Md d\<leadsto>⊥) = ⊥"
apply (unfold BufAC_def)
apply (fast intro: BufAC_Cmt_d2 BufAC_Asm_d)
done

lemma ex_elim_lemma: "(? ff:B. (!x. f·(a\<leadsto>b\<leadsto>x) = d\<leadsto>ff·x)) =
((!x. ft·(f·(a\<leadsto>b\<leadsto>x)) = Def d) & (LAM x. rt·(f·(a\<leadsto>b\<leadsto>x))):B)"

(* this is an instance (though unification cannot handle this) of
lemma "(? ff:B. (!x. f·x = d\<leadsto>ff·x)) = \
\((!x. ft·(f·x) = Def d) & (LAM x. rt·(f·x)):B)"*)

apply safe
apply ( rule_tac [2] P="(%x. x:B)" in ssubst)
prefer 3
apply ( assumption)
apply ( rule_tac [2] ext_cfun)
apply ( drule_tac [2] spec)
apply ( drule_tac [2] f="rt" in cfun_arg_cong)
prefer 2
apply ( simp)
prefer 2
apply ( simp)
apply (rule_tac bexI)
apply auto
apply (drule spec)
apply (erule exE)
apply (erule ssubst)
apply (simp)
done

lemma BufAC_f_d_req: "f∈BufAC ==> ∃ff∈BufAC. ∀x. f·(Md d\<leadsto>•\<leadsto>x) = d\<leadsto>ff·x"
apply (unfold BufAC_def)
apply (rule ex_elim_lemma [THEN iffD2])
apply safe
apply (fast intro: BufAC_Cmt_d32 [THEN Def_maximal]
monofun_cfun_arg BufAC_Asm_empty [THEN BufAC_Asm_d_req])

apply (auto intro: BufAC_Cmt_d3 BufAC_Asm_d_req)
done


(**** BufSt *******************************************************************)

lemma mono_BufSt_F: "mono BufSt_F"
by (unfold mono_def BufSt_F_def, fast)

lemmas BufSt_P_fix =
mono_BufSt_F [THEN BufSt_P_def [THEN eq_reflection, THEN def_gfp_unfold]]


lemma BufSt_P_unfold: "(h:BufSt_P) = (!s. h s·<> = <> &
(!d x. h ¤ ·(Md d\<leadsto>x) = h (Sd d)·x &
(? hh:BufSt_P. h (Sd d)·(•\<leadsto>x) = d\<leadsto>(hh ¤ ·x))))"

apply (subst BufSt_P_fix [THEN set_cong])
apply (unfold BufSt_F_def)
apply (simp)
done

lemma BufSt_P_empty: "h:BufSt_P ==> h s · <> = <>"
by (drule BufSt_P_unfold [THEN iffD1], auto)
lemma BufSt_P_d: "h:BufSt_P ==> h ¤ ·(Md d\<leadsto>x) = h (Sd d)·x"
by (drule BufSt_P_unfold [THEN iffD1], auto)
lemma BufSt_P_d_req: "h:BufSt_P ==> ∃hh∈BufSt_P.
h (Sd d)·(• \<leadsto>x) = d\<leadsto>(hh ¤ ·x)"

by (drule BufSt_P_unfold [THEN iffD1], auto)


(**** Buf_AC_imp_Eq ***********************************************************)

lemma Buf_AC_imp_Eq: "BufAC ⊆ BufEq"
apply (unfold BufEq_def)
apply (rule gfp_upperbound)
apply (unfold BufEq_F_def)
apply safe
apply (erule BufAC_f_d)
apply (drule BufAC_f_d_req)
apply (fast)
done


(**** Buf_Eq_imp_AC by coinduction ********************************************)

lemma BufAC_Asm_cong_lemma [rule_format]: "∀s f ff. f∈BufEq --> ff∈BufEq -->
s∈BufAC_Asm --> stream_take n·(f·s) = stream_take n·(ff·s)"

apply (induct_tac "n")
apply (simp)
apply (intro strip)
apply (drule BufAC_Asm_unfold [THEN iffD1])
apply safe
apply (simp add: Buf_f_empty)
apply (simp add: Buf_f_d)
apply (drule ft_eq [THEN iffD1])
apply (clarsimp)
apply (drule Buf_f_d_req)+
apply safe
apply (erule ssubst)+
apply (simp (no_asm))
apply (fast)
done

lemma BufAC_Asm_cong: "[|f ∈ BufEq; ff ∈ BufEq; s ∈ BufAC_Asm|] ==> f·s = ff·s"
apply (rule stream.take_lemma)
apply (erule (2) BufAC_Asm_cong_lemma)
done

lemma Buf_Eq_imp_AC_lemma: "[|f ∈ BufEq; x ∈ BufAC_Asm|] ==> (x, f·x) ∈ BufAC_Cmt"
apply (unfold BufAC_Cmt_def)
apply (rotate_tac)
apply (erule weak_coinduct_image)
apply (unfold BufAC_Cmt_F_def)
apply safe
apply (erule Buf_f_empty)
apply (erule Buf_f_d)
apply (drule Buf_f_d_req)
apply (clarsimp)
apply (erule exI)
apply (drule BufAC_Asm_prefix2)
apply (frule Buf_f_d_req)
apply (clarsimp)
apply (erule ssubst)
apply (simp)
apply (drule (2) BufAC_Asm_cong)
apply (erule subst)
apply (erule imageI)
done
lemma Buf_Eq_imp_AC: "BufEq ⊆ BufAC"
apply (unfold BufAC_def)
apply (clarify)
apply (erule (1) Buf_Eq_imp_AC_lemma)
done

(**** Buf_Eq_eq_AC ************************************************************)

lemmas Buf_Eq_eq_AC = Buf_AC_imp_Eq [THEN Buf_Eq_imp_AC [THEN subset_antisym]]


(**** alternative (not strictly) stronger version of Buf_Eq *******************)

lemma Buf_Eq_alt_imp_Eq: "BufEq_alt ⊆ BufEq"
apply (unfold BufEq_def BufEq_alt_def)
apply (rule gfp_mono)
apply (unfold BufEq_F_def)
apply (fast)
done

(* direct proof of "BufEq ⊆ BufEq_alt" seems impossible *)


lemma Buf_AC_imp_Eq_alt: "BufAC <= BufEq_alt"
apply (unfold BufEq_alt_def)
apply (rule gfp_upperbound)
apply (fast elim: BufAC_f_d BufAC_f_d_req)
done

lemmas Buf_Eq_imp_Eq_alt = subset_trans [OF Buf_Eq_imp_AC Buf_AC_imp_Eq_alt]

lemmas Buf_Eq_alt_eq = subset_antisym [OF Buf_Eq_alt_imp_Eq Buf_Eq_imp_Eq_alt]


(**** Buf_Eq_eq_St ************************************************************)

lemma Buf_St_imp_Eq: "BufSt <= BufEq"
apply (unfold BufSt_def BufEq_def)
apply (rule gfp_upperbound)
apply (unfold BufEq_F_def)
apply safe
apply ( simp add: BufSt_P_d BufSt_P_empty)
apply (simp add: BufSt_P_d)
apply (drule BufSt_P_d_req)
apply (force)
done

lemma Buf_Eq_imp_St: "BufEq <= BufSt"
apply (unfold BufSt_def BufSt_P_def)
apply safe
apply (rename_tac f)
apply (rule_tac x="λs. case s of Sd d => Λ x. f·(Md d\<leadsto>x)| ¤ => f" in bexI)
apply ( simp)
apply (erule weak_coinduct_image)
apply (unfold BufSt_F_def)
apply (simp)
apply safe
apply ( rename_tac "s")
apply ( induct_tac "s")
apply ( simp add: Buf_f_d)
apply ( simp add: Buf_f_empty)
apply ( simp)
apply (simp)
apply (rename_tac f d x)
apply (drule_tac d="d" and x="x" in Buf_f_d_req)
apply auto
done

lemmas Buf_Eq_eq_St = Buf_St_imp_Eq [THEN Buf_Eq_imp_St [THEN subset_antisym]]

end