(* Title: HOL/HOLCF/IOA/Seq.thy
Author: Olaf Müller
*)
section \<open>Partial, Finite and Infinite Sequences (lazy lists), modeled as domain\<close>
theory Seq
imports HOLCF
begin
default_sort pcpo
domain (unsafe) 'a seq = nil ("nil") | cons (HD :: 'a) (lazy TL :: "'a seq") (infixr "##" 65)
inductive Finite :: "'a seq \ bool"
where
sfinite_0: "Finite nil"
| sfinite_n: "Finite tr \ a \ UU \ Finite (a ## tr)"
declare Finite.intros [simp]
definition Partial :: "'a seq \ bool"
where "Partial x \ seq_finite x \ \ Finite x"
definition Infinite :: "'a seq \ bool"
where "Infinite x \ \ seq_finite x"
subsection \<open>Recursive equations of operators\<close>
subsubsection \<open>\<open>smap\<close>\<close>
fixrec smap :: "('a \ 'b) \ 'a seq \ 'b seq"
where
smap_nil: "smap \ f \ nil = nil"
| smap_cons: "x \ UU \ smap \ f \ (x ## xs) = (f \ x) ## smap \ f \ xs"
lemma smap_UU [simp]: "smap \ f \ UU = UU"
by fixrec_simp
subsubsection \<open>\<open>sfilter\<close>\<close>
fixrec sfilter :: "('a \ tr) \ 'a seq \ 'a seq"
where
sfilter_nil: "sfilter \ P \ nil = nil"
| sfilter_cons:
"x \ UU \
sfilter \<cdot> P \<cdot> (x ## xs) =
(If P \<cdot> x then x ## (sfilter \<cdot> P \<cdot> xs) else sfilter \<cdot> P \<cdot> xs)"
lemma sfilter_UU [simp]: "sfilter \ P \ UU = UU"
by fixrec_simp
subsubsection \<open>\<open>sforall2\<close>\<close>
fixrec sforall2 :: "('a \ tr) \ 'a seq \ tr"
where
sforall2_nil: "sforall2 \ P \ nil = TT"
| sforall2_cons: "x \ UU \ sforall2 \ P \ (x ## xs) = ((P \ x) andalso sforall2 \ P \ xs)"
lemma sforall2_UU [simp]: "sforall2 \ P \ UU = UU"
by fixrec_simp
definition "sforall P t \ sforall2 \ P \ t \ FF"
subsubsection \<open>\<open>stakewhile\<close>\<close>
fixrec stakewhile :: "('a \ tr) \ 'a seq \ 'a seq"
where
stakewhile_nil: "stakewhile \ P \ nil = nil"
| stakewhile_cons:
"x \ UU \ stakewhile \ P \ (x ## xs) = (If P \ x then x ## (stakewhile \ P \ xs) else nil)"
lemma stakewhile_UU [simp]: "stakewhile \ P \ UU = UU"
by fixrec_simp
subsubsection \<open>\<open>sdropwhile\<close>\<close>
fixrec sdropwhile :: "('a \ tr) \ 'a seq \ 'a seq"
where
sdropwhile_nil: "sdropwhile \ P \ nil = nil"
| sdropwhile_cons:
"x \ UU \ sdropwhile \ P \ (x ## xs) = (If P \ x then sdropwhile \ P \ xs else x ## xs)"
lemma sdropwhile_UU [simp]: "sdropwhile \ P \ UU = UU"
by fixrec_simp
subsubsection \<open>\<open>slast\<close>\<close>
fixrec slast :: "'a seq \ 'a"
where
slast_nil: "slast \ nil = UU"
| slast_cons: "x \ UU \ slast \ (x ## xs) = (If is_nil \ xs then x else slast \ xs)"
lemma slast_UU [simp]: "slast \ UU = UU"
by fixrec_simp
subsubsection \<open>\<open>sconc\<close>\<close>
fixrec sconc :: "'a seq \ 'a seq \ 'a seq"
where
sconc_nil: "sconc \ nil \ y = y"
| sconc_cons': "x \ UU \ sconc \ (x ## xs) \ y = x ## (sconc \ xs \ y)"
abbreviation sconc_syn :: "'a seq \ 'a seq \ 'a seq" (infixr "@@" 65)
where "xs @@ ys \ sconc \ xs \ ys"
lemma sconc_UU [simp]: "UU @@ y = UU"
by fixrec_simp
lemma sconc_cons [simp]: "(x ## xs) @@ y = x ## (xs @@ y)"
by (cases "x = UU") simp_all
declare sconc_cons' [simp del]
subsubsection \<open>\<open>sflat\<close>\<close>
fixrec sflat :: "'a seq seq \ 'a seq"
where
sflat_nil: "sflat \ nil = nil"
| sflat_cons': "x \ UU \ sflat \ (x ## xs) = x @@ (sflat \ xs)"
lemma sflat_UU [simp]: "sflat \ UU = UU"
by fixrec_simp
lemma sflat_cons [simp]: "sflat \ (x ## xs) = x @@ (sflat \ xs)"
by (cases "x = UU") simp_all
declare sflat_cons' [simp del]
subsubsection \<open>\<open>szip\<close>\<close>
fixrec szip :: "'a seq \ 'b seq \ ('a \ 'b) seq"
where
szip_nil: "szip \ nil \ y = nil"
| szip_cons_nil: "x \ UU \ szip \ (x ## xs) \ nil = UU"
| szip_cons: "x \ UU \ y \ UU \ szip \ (x ## xs) \ (y ## ys) = (x, y) ## szip \ xs \ ys"
lemma szip_UU1 [simp]: "szip \ UU \ y = UU"
by fixrec_simp
lemma szip_UU2 [simp]: "x \ nil \ szip \ x \ UU = UU"
by (cases x) (simp_all, fixrec_simp)
subsection \<open>\<open>scons\<close>, \<open>nil\<close>\<close>
lemma scons_inject_eq: "x \ UU \ y \ UU \ x ## xs = y ## ys \ x = y \ xs = ys"
by simp
lemma nil_less_is_nil: "nil \ x \ nil = x"
by (cases x) simp_all
subsection \<open>\<open>sfilter\<close>, \<open>sforall\<close>, \<open>sconc\<close>\<close>
lemma if_and_sconc [simp]:
"(if b then tr1 else tr2) @@ tr = (if b then tr1 @@ tr else tr2 @@ tr)"
by simp
lemma sfiltersconc: "sfilter \ P \ (x @@ y) = (sfilter \ P \ x @@ sfilter \ P \ y)"
apply (induct x)
text \<open>adm\<close>
apply simp
text \<open>base cases\<close>
apply simp
apply simp
text \<open>main case\<close>
apply (rule_tac p = "P\a" in trE)
apply simp
apply simp
apply simp
done
lemma sforallPstakewhileP: "sforall P (stakewhile \ P \ x)"
apply (simp add: sforall_def)
apply (induct x)
text \<open>adm\<close>
apply simp
text \<open>base cases\<close>
apply simp
apply simp
text \<open>main case\<close>
apply (rule_tac p = "P\a" in trE)
apply simp
apply simp
apply simp
done
lemma forallPsfilterP: "sforall P (sfilter \ P \ x)"
apply (simp add: sforall_def)
apply (induct x)
text \<open>adm\<close>
apply simp
text \<open>base cases\<close>
apply simp
apply simp
text \<open>main case\<close>
apply (rule_tac p="P\a" in trE)
apply simp
apply simp
apply simp
done
subsection \<open>Finite\<close>
(*
Proofs of rewrite rules for Finite:
1. Finite nil (by definition)
2. \<not> Finite UU
3. a \<noteq> UU \<Longrightarrow> Finite (a ## x) = Finite x
*)
lemma Finite_UU_a: "Finite x \ x \ UU"
apply (rule impI)
apply (erule Finite.induct)
apply simp
apply simp
done
lemma Finite_UU [simp]: "\ Finite UU"
using Finite_UU_a [where x = UU] by fast
lemma Finite_cons_a: "Finite x \ a \ UU \ x = a ## xs \ Finite xs"
apply (intro strip)
apply (erule Finite.cases)
apply fastforce
apply simp
done
lemma Finite_cons: "a \ UU \ Finite (a##x) \ Finite x"
apply (rule iffI)
apply (erule (1) Finite_cons_a [rule_format])
apply fast
apply simp
done
lemma Finite_upward: "Finite x \ x \ y \ Finite y"
apply (induct arbitrary: y set: Finite)
apply (case_tac y, simp, simp, simp)
apply (case_tac y, simp, simp)
apply simp
done
lemma adm_Finite [simp]: "adm Finite"
by (rule adm_upward) (rule Finite_upward)
subsection \<open>Induction\<close>
text \<open>Extensions to Induction Theorems.\<close>
lemma seq_finite_ind_lemma:
assumes "\n. P (seq_take n \ s)"
shows "seq_finite s \ P s"
apply (unfold seq.finite_def)
apply (intro strip)
apply (erule exE)
apply (erule subst)
apply (rule assms)
done
lemma seq_finite_ind:
assumes "P UU"
and "P nil"
and "\x s1. x \ UU \ P s1 \ P (x ## s1)"
shows "seq_finite s \ P s"
apply (insert assms)
apply (rule seq_finite_ind_lemma)
apply (erule seq.finite_induct)
apply assumption
apply simp
done
end
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