src/HOL/HOLCF/Tr.thy
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(*  Title:      HOL/HOLCF/Tr.thy
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    Author:     Franz Regensburger
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*)
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header {* The type of lifted booleans *}
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theory Tr
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imports Lift
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begin
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subsection {* Type definition and constructors *}
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type_synonym
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  tr = "bool lift"
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translations
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  (type) "tr" <= (type) "bool lift"
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definition
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  TT :: "tr" where
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  "TT = Def True"
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definition
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  FF :: "tr" where
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  "FF = Def False"
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text {* Exhaustion and Elimination for type @{typ tr} *}
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lemma Exh_tr: "t = \<bottom> \<or> t = TT \<or> t = FF"
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unfolding FF_def TT_def by (induct t) auto
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lemma trE [case_names bottom TT FF]:
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  "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; p = TT \<Longrightarrow> Q; p = FF \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
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unfolding FF_def TT_def by (induct p) auto
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lemma tr_induct [case_names bottom TT FF]:
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  "\<lbrakk>P \<bottom>; P TT; P FF\<rbrakk> \<Longrightarrow> P x"
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by (cases x rule: trE) simp_all
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text {* distinctness for type @{typ tr} *}
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lemma dist_below_tr [simp]:
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  "TT \<notsqsubseteq> \<bottom>" "FF \<notsqsubseteq> \<bottom>" "TT \<notsqsubseteq> FF" "FF \<notsqsubseteq> TT"
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lemma dist_eq_tr [simp]:
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  "TT \<noteq> \<bottom>" "FF \<noteq> \<bottom>" "TT \<noteq> FF" "\<bottom> \<noteq> TT" "\<bottom> \<noteq> FF" "FF \<noteq> TT"
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lemma TT_below_iff [simp]: "TT \<sqsubseteq> x \<longleftrightarrow> x = TT"
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by (induct x rule: tr_induct) simp_all
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lemma FF_below_iff [simp]: "FF \<sqsubseteq> x \<longleftrightarrow> x = FF"
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by (induct x rule: tr_induct) simp_all
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lemma not_below_TT_iff [simp]: "x \<notsqsubseteq> TT \<longleftrightarrow> x = FF"
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by (induct x rule: tr_induct) simp_all
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lemma not_below_FF_iff [simp]: "x \<notsqsubseteq> FF \<longleftrightarrow> x = TT"
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by (induct x rule: tr_induct) simp_all
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subsection {* Case analysis *}
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default_sort pcpo
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definition tr_case :: "'a \<rightarrow> 'a \<rightarrow> tr \<rightarrow> 'a" where
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  "tr_case = (\<Lambda> t e (Def b). if b then t else e)"
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abbreviation
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  cifte_syn :: "[tr, 'c, 'c] \<Rightarrow> 'c"  ("(If (_)/ then (_)/ else (_))" [0, 0, 60] 60)
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where
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  "If b then e1 else e2 == tr_case\<cdot>e1\<cdot>e2\<cdot>b"
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translations
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  "\<Lambda> (XCONST TT). t" == "CONST tr_case\<cdot>t\<cdot>\<bottom>"
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  "\<Lambda> (XCONST FF). t" == "CONST tr_case\<cdot>\<bottom>\<cdot>t"
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lemma ifte_thms [simp]:
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  "If \<bottom> then e1 else e2 = \<bottom>"
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  "If FF then e1 else e2 = e2"
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  "If TT then e1 else e2 = e1"
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by (simp_all add: tr_case_def TT_def FF_def)
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subsection {* Boolean connectives *}
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definition
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  trand :: "tr \<rightarrow> tr \<rightarrow> tr" where
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  andalso_def: "trand = (\<Lambda> x y. If x then y else FF)"
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abbreviation
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  andalso_syn :: "tr \<Rightarrow> tr \<Rightarrow> tr"  ("_ andalso _" [36,35] 35)  where
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  "x andalso y == trand\<cdot>x\<cdot>y"
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definition
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  tror :: "tr \<rightarrow> tr \<rightarrow> tr" where
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  orelse_def: "tror = (\<Lambda> x y. If x then TT else y)"
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abbreviation
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  orelse_syn :: "tr \<Rightarrow> tr \<Rightarrow> tr"  ("_ orelse _"  [31,30] 30)  where
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  "x orelse y == tror\<cdot>x\<cdot>y"
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definition
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  neg :: "tr \<rightarrow> tr" where
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  "neg = flift2 Not"
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definition
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  If2 :: "[tr, 'c, 'c] \<Rightarrow> 'c" where
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  "If2 Q x y = (If Q then x else y)"
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text {* tactic for tr-thms with case split *}
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lemmas tr_defs = andalso_def orelse_def neg_def tr_case_def TT_def FF_def
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text {* lemmas about andalso, orelse, neg and if *}
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lemma andalso_thms [simp]:
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  "(TT andalso y) = y"
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  "(FF andalso y) = FF"
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  "(\<bottom> andalso y) = \<bottom>"
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  "(y andalso TT) = y"
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  "(y andalso y) = y"
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apply (unfold andalso_def, simp_all)
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apply (cases y rule: trE, simp_all)
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apply (cases y rule: trE, simp_all)
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done
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lemma orelse_thms [simp]:
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  "(TT orelse y) = TT"
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  "(FF orelse y) = y"
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  "(\<bottom> orelse y) = \<bottom>"
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  "(y orelse FF) = y"
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  "(y orelse y) = y"
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apply (unfold orelse_def, simp_all)
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apply (cases y rule: trE, simp_all)
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apply (cases y rule: trE, simp_all)
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done
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lemma neg_thms [simp]:
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  "neg\<cdot>TT = FF"
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  "neg\<cdot>FF = TT"
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  "neg\<cdot>\<bottom> = \<bottom>"
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by (simp_all add: neg_def TT_def FF_def)
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text {* split-tac for If via If2 because the constant has to be a constant *}
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lemma split_If2:
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  "P (If2 Q x y) = ((Q = \<bottom> \<longrightarrow> P \<bottom>) \<and> (Q = TT \<longrightarrow> P x) \<and> (Q = FF \<longrightarrow> P y))"
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apply (unfold If2_def)
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apply (rule_tac p = "Q" in trE)
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apply (simp_all)
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done
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ML {*
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val split_If_tac =
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  simp_tac (HOL_basic_ss addsimps [@{thm If2_def} RS sym])
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    THEN' (split_tac [@{thm split_If2}])
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*}
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subsection "Rewriting of HOLCF operations to HOL functions"
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lemma andalso_or:
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  "t \<noteq> \<bottom> \<Longrightarrow> ((t andalso s) = FF) = (t = FF \<or> s = FF)"
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apply (rule_tac p = "t" in trE)
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apply simp_all
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done
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lemma andalso_and:
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  "t \<noteq> \<bottom> \<Longrightarrow> ((t andalso s) \<noteq> FF) = (t \<noteq> FF \<and> s \<noteq> FF)"
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apply (rule_tac p = "t" in trE)
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apply simp_all
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done
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lemma Def_bool1 [simp]: "(Def x \<noteq> FF) = x"
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by (simp add: FF_def)
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lemma Def_bool2 [simp]: "(Def x = FF) = (\<not> x)"
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by (simp add: FF_def)
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lemma Def_bool3 [simp]: "(Def x = TT) = x"
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by (simp add: TT_def)
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lemma Def_bool4 [simp]: "(Def x \<noteq> TT) = (\<not> x)"
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by (simp add: TT_def)
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lemma If_and_if:
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  "(If Def P then A else B) = (if P then A else B)"
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apply (rule_tac p = "Def P" in trE)
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apply (auto simp add: TT_def[symmetric] FF_def[symmetric])
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done
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subsection {* Compactness *}
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lemma compact_TT: "compact TT"
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by (rule compact_chfin)
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lemma compact_FF: "compact FF"
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by (rule compact_chfin)
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end