# Theory Tr

theory Tr
imports Lift
```(*  Title:      HOL/HOLCF/Tr.thy
Author:     Franz Regensburger
*)

section ‹The type of lifted booleans›

theory Tr
imports Lift
begin

subsection ‹Type definition and constructors›

type_synonym
tr = "bool lift"

translations
(type) "tr" <= (type) "bool lift"

definition
TT :: "tr" where
"TT = Def True"

definition
FF :: "tr" where
"FF = Def False"

text ‹Exhaustion and Elimination for type @{typ tr}›

lemma Exh_tr: "t = ⊥ ∨ t = TT ∨ t = FF"
unfolding FF_def TT_def by (induct t) auto

lemma trE [case_names bottom TT FF, cases type: tr]:
"⟦p = ⊥ ⟹ Q; p = TT ⟹ Q; p = FF ⟹ Q⟧ ⟹ Q"
unfolding FF_def TT_def by (induct p) auto

lemma tr_induct [case_names bottom TT FF, induct type: tr]:
"⟦P ⊥; P TT; P FF⟧ ⟹ P x"
by (cases x) simp_all

text ‹distinctness for type @{typ tr}›

lemma dist_below_tr [simp]:
"TT \<notsqsubseteq> ⊥" "FF \<notsqsubseteq> ⊥" "TT \<notsqsubseteq> FF" "FF \<notsqsubseteq> TT"
unfolding TT_def FF_def by simp_all

lemma dist_eq_tr [simp]:
"TT ≠ ⊥" "FF ≠ ⊥" "TT ≠ FF" "⊥ ≠ TT" "⊥ ≠ FF" "FF ≠ TT"
unfolding TT_def FF_def by simp_all

lemma TT_below_iff [simp]: "TT ⊑ x ⟷ x = TT"
by (induct x) simp_all

lemma FF_below_iff [simp]: "FF ⊑ x ⟷ x = FF"
by (induct x) simp_all

lemma not_below_TT_iff [simp]: "x \<notsqsubseteq> TT ⟷ x = FF"
by (induct x) simp_all

lemma not_below_FF_iff [simp]: "x \<notsqsubseteq> FF ⟷ x = TT"
by (induct x) simp_all

subsection ‹Case analysis›

default_sort pcpo

definition tr_case :: "'a → 'a → tr → 'a" where
"tr_case = (Λ t e (Def b). if b then t else e)"

abbreviation
cifte_syn :: "[tr, 'c, 'c] ⇒ 'c"  ("(If (_)/ then (_)/ else (_))" [0, 0, 60] 60)
where
"If b then e1 else e2 == tr_case⋅e1⋅e2⋅b"

translations
"Λ (XCONST TT). t" == "CONST tr_case⋅t⋅⊥"
"Λ (XCONST FF). t" == "CONST tr_case⋅⊥⋅t"

lemma ifte_thms [simp]:
"If ⊥ then e1 else e2 = ⊥"
"If FF then e1 else e2 = e2"
"If TT then e1 else e2 = e1"
by (simp_all add: tr_case_def TT_def FF_def)

subsection ‹Boolean connectives›

definition
trand :: "tr → tr → tr" where
andalso_def: "trand = (Λ x y. If x then y else FF)"
abbreviation
andalso_syn :: "tr ⇒ tr ⇒ tr"  ("_ andalso _" [36,35] 35)  where
"x andalso y == trand⋅x⋅y"

definition
tror :: "tr → tr → tr" where
orelse_def: "tror = (Λ x y. If x then TT else y)"
abbreviation
orelse_syn :: "tr ⇒ tr ⇒ tr"  ("_ orelse _"  [31,30] 30)  where
"x orelse y == tror⋅x⋅y"

definition
neg :: "tr → tr" where
"neg = flift2 Not"

definition
If2 :: "[tr, 'c, 'c] ⇒ 'c" where
"If2 Q x y = (If Q then x else y)"

text ‹tactic for tr-thms with case split›

lemmas tr_defs = andalso_def orelse_def neg_def tr_case_def TT_def FF_def

text ‹lemmas about andalso, orelse, neg and if›

lemma andalso_thms [simp]:
"(TT andalso y) = y"
"(FF andalso y) = FF"
"(⊥ andalso y) = ⊥"
"(y andalso TT) = y"
"(y andalso y) = y"
apply (unfold andalso_def, simp_all)
apply (cases y, simp_all)
apply (cases y, simp_all)
done

lemma orelse_thms [simp]:
"(TT orelse y) = TT"
"(FF orelse y) = y"
"(⊥ orelse y) = ⊥"
"(y orelse FF) = y"
"(y orelse y) = y"
apply (unfold orelse_def, simp_all)
apply (cases y, simp_all)
apply (cases y, simp_all)
done

lemma neg_thms [simp]:
"neg⋅TT = FF"
"neg⋅FF = TT"
"neg⋅⊥ = ⊥"
by (simp_all add: neg_def TT_def FF_def)

text ‹split-tac for If via If2 because the constant has to be a constant›

lemma split_If2:
"P (If2 Q x y) = ((Q = ⊥ ⟶ P ⊥) ∧ (Q = TT ⟶ P x) ∧ (Q = FF ⟶ P y))"
apply (unfold If2_def)
apply (cases Q)
apply (simp_all)
done

(* FIXME unused!? *)
ML ‹
fun split_If_tac ctxt =
simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm If2_def} RS sym])
THEN' (split_tac ctxt [@{thm split_If2}])
›

subsection "Rewriting of HOLCF operations to HOL functions"

lemma andalso_or:
"t ≠ ⊥ ⟹ ((t andalso s) = FF) = (t = FF ∨ s = FF)"
apply (cases t)
apply simp_all
done

lemma andalso_and:
"t ≠ ⊥ ⟹ ((t andalso s) ≠ FF) = (t ≠ FF ∧ s ≠ FF)"
apply (cases t)
apply simp_all
done

lemma Def_bool1 [simp]: "(Def x ≠ FF) = x"
by (simp add: FF_def)

lemma Def_bool2 [simp]: "(Def x = FF) = (¬ x)"
by (simp add: FF_def)

lemma Def_bool3 [simp]: "(Def x = TT) = x"
by (simp add: TT_def)

lemma Def_bool4 [simp]: "(Def x ≠ TT) = (¬ x)"
by (simp add: TT_def)

lemma If_and_if:
"(If Def P then A else B) = (if P then A else B)"
apply (cases "Def P")
apply (auto simp add: TT_def[symmetric] FF_def[symmetric])
done

subsection ‹Compactness›

lemma compact_TT: "compact TT"
by (rule compact_chfin)

lemma compact_FF: "compact FF"
by (rule compact_chfin)

end
```