(* Title: HOL/HOLCF/Lift.thy
Author: Olaf Mueller
*)
section \<open>Lifting types of class type to flat pcpo's\<close>
theory Lift
imports Discrete Up
begin
default_sort type
pcpodef 'a lift = "UNIV :: 'a discr u set"
by simp_all
lemmas inst_lift_pcpo = Abs_lift_strict [symmetric]
definition
Def :: "'a \<Rightarrow> 'a lift" where
"Def x = Abs_lift (up\<cdot>(Discr x))"
subsection \<open>Lift as a datatype\<close>
lemma lift_induct: "\<lbrakk>P \<bottom>; \<And>x. P (Def x)\<rbrakk> \<Longrightarrow> P y"
apply (induct y)
apply (rule_tac p=y in upE)
apply (simp add: Abs_lift_strict)
apply (case_tac x)
apply (simp add: Def_def)
done
old_rep_datatype "\<bottom>::'a lift" Def
by (erule lift_induct) (simp_all add: Def_def Abs_lift_inject inst_lift_pcpo)
text \<open>@{term bottom} and @{term Def}\<close>
lemma not_Undef_is_Def: "(x \<noteq> \<bottom>) = (\<exists>y. x = Def y)"
by (cases x) simp_all
lemma lift_definedE: "\<lbrakk>x \<noteq> \<bottom>; \<And>a. x = Def a \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
by (cases x) simp_all
text \<open>
For @{term "x ~= \<bottom>"} in assumptions \<open>defined\<close> replaces \<open>x\<close> by \<open>Def a\<close> in conclusion.\<close>
method_setup defined = \<open>
Scan.succeed (fn ctxt => SIMPLE_METHOD'
(eresolve_tac ctxt @{thms lift_definedE} THEN' asm_simp_tac ctxt))
\<close>
lemma DefE: "Def x = \<bottom> \<Longrightarrow> R"
by simp
lemma DefE2: "\<lbrakk>x = Def s; x = \<bottom>\<rbrakk> \<Longrightarrow> R"
by simp
lemma Def_below_Def: "Def x \<sqsubseteq> Def y \<longleftrightarrow> x = y"
by (simp add: below_lift_def Def_def Abs_lift_inverse)
lemma Def_below_iff [simp]: "Def x \<sqsubseteq> y \<longleftrightarrow> Def x = y"
by (induct y, simp, simp add: Def_below_Def)
subsection \<open>Lift is flat\<close>
instance lift :: (type) flat
proof
fix x y :: "'a lift"
assume "x \<sqsubseteq> y" thus "x = \<bottom> \<or> x = y"
by (induct x) auto
qed
subsection \<open>Continuity of @{const case_lift}\<close>
lemma case_lift_eq: "case_lift \<bottom> f x = fup\<cdot>(\<Lambda> y. f (undiscr y))\<cdot>(Rep_lift x)"
apply (induct x, unfold lift.case)
apply (simp add: Rep_lift_strict)
apply (simp add: Def_def Abs_lift_inverse)
done
lemma cont2cont_case_lift [simp]:
"\<lbrakk>\<And>y. cont (\<lambda>x. f x y); cont g\<rbrakk> \<Longrightarrow> cont (\<lambda>x. case_lift \<bottom> (f x) (g x))"
unfolding case_lift_eq by (simp add: cont_Rep_lift)
subsection \<open>Further operations\<close>
definition
flift1 :: "('a \<Rightarrow> 'b::pcpo) \<Rightarrow> ('a lift \<rightarrow> 'b)" (binder "FLIFT " 10) where
"flift1 = (\<lambda>f. (\<Lambda> x. case_lift \<bottom> f x))"
translations
"\<Lambda>(XCONST Def x). t" => "CONST flift1 (\<lambda>x. t)"
"\<Lambda>(CONST Def x). FLIFT y. t" <= "FLIFT x y. t"
"\<Lambda>(CONST Def x). t" <= "FLIFT x. t"
definition
flift2 :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a lift \<rightarrow> 'b lift)" where
"flift2 f = (FLIFT x. Def (f x))"
lemma flift1_Def [simp]: "flift1 f\<cdot>(Def x) = (f x)"
by (simp add: flift1_def)
lemma flift2_Def [simp]: "flift2 f\<cdot>(Def x) = Def (f x)"
by (simp add: flift2_def)
lemma flift1_strict [simp]: "flift1 f\<cdot>\<bottom> = \<bottom>"
by (simp add: flift1_def)
lemma flift2_strict [simp]: "flift2 f\<cdot>\<bottom> = \<bottom>"
by (simp add: flift2_def)
lemma flift2_defined [simp]: "x \<noteq> \<bottom> \<Longrightarrow> (flift2 f)\<cdot>x \<noteq> \<bottom>"
by (erule lift_definedE, simp)
lemma flift2_bottom_iff [simp]: "(flift2 f\<cdot>x = \<bottom>) = (x = \<bottom>)"
by (cases x, simp_all)
lemma FLIFT_mono:
"(\<And>x. f x \<sqsubseteq> g x) \<Longrightarrow> (FLIFT x. f x) \<sqsubseteq> (FLIFT x. g x)"
by (rule cfun_belowI, case_tac x, simp_all)
lemma cont2cont_flift1 [simp, cont2cont]:
"\<lbrakk>\<And>y. cont (\<lambda>x. f x y)\<rbrakk> \<Longrightarrow> cont (\<lambda>x. FLIFT y. f x y)"
by (simp add: flift1_def cont2cont_LAM)
end