src/HOLCF/Tr.thy
 author huffman Sat, 04 Jun 2005 00:22:08 +0200 changeset 16221 879400e029bf parent 16121 a80aa66d2271 child 16228 9b5b0c92230a permissions -rw-r--r--
New theory with lemmas for the fixrec package
```
(*  Title:      HOLCF/Tr.thy
ID:         \$Id\$
Author:     Franz Regensburger

Introduce infix if_then_else_fi and boolean connectives andalso, orelse.
*)

header {* The type of lifted booleans *}

theory Tr
imports Lift Fix
begin

types
tr = "bool lift"

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

consts
TT              :: "tr"
FF              :: "tr"
Icifte          :: "tr -> 'c -> 'c -> 'c"
trand           :: "tr -> tr -> tr"
tror            :: "tr -> tr -> tr"
neg             :: "tr -> tr"
If2             :: "tr=>'c=>'c=>'c"

syntax  "@cifte"        :: "tr=>'c=>'c=>'c" ("(3If _/ (then _/ else _) fi)" 60)
"@andalso"      :: "tr => tr => tr" ("_ andalso _" [36,35] 35)
"@orelse"       :: "tr => tr => tr" ("_ orelse _"  [31,30] 30)

translations
"x andalso y" == "trand\$x\$y"
"x orelse y"  == "tror\$x\$y"
"If b then e1 else e2 fi" == "Icifte\$b\$e1\$e2"
defs
TT_def:      "TT==Def True"
FF_def:      "FF==Def False"
neg_def:     "neg == flift2 Not"
ifte_def:    "Icifte == (LAM b t e. flift1(%b. if b then t else e)\$b)"
andalso_def: "trand == (LAM x y. If x then y else FF fi)"
orelse_def:  "tror == (LAM x y. If x then TT else y fi)"
If2_def:     "If2 Q x y == If Q then x else y fi"

text {* Exhaustion and Elimination for type @{typ tr} *}

lemma Exh_tr: "t=UU | t = TT | t = FF"
apply (unfold FF_def TT_def)
apply (induct_tac "t")
apply fast
apply fast
done

lemma trE: "[| p=UU ==> Q; p = TT ==>Q; p = FF ==>Q|] ==>Q"
apply (rule Exh_tr [THEN disjE])
apply fast
apply (erule disjE)
apply fast
apply fast
done

text {* tactic for tr-thms with case split *}

lemmas tr_defs = andalso_def orelse_def neg_def ifte_def TT_def FF_def
(*
fun prover t =  prove_goal thy t
(fn prems =>
[
(res_inst_tac [("p","y")] trE 1),
[o_def,flift1_def,flift2_def,inst_lift_po]@tr_defs) 1))
])
*)
text {* distinctness for type @{typ tr} *}

lemma dist_less_tr [simp]: "~TT << UU" "~FF << UU" "~TT << FF" "~FF << TT"

lemma dist_eq_tr [simp]: "TT~=UU" "FF~=UU" "TT~=FF" "UU~=TT" "UU~=FF" "FF~=TT"

text {* lemmas about andalso, orelse, neg and if *}

lemma ifte_simp:
"If x then e1 else e2 fi =
flift1 (%b. if b then e1 else e2)\$x"
apply (unfold ifte_def TT_def FF_def flift1_def)
done

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

lemma andalso_thms [simp]:
"(TT andalso y) = y"
"(FF andalso y) = FF"
"(UU andalso y) = UU"
"(y andalso TT) = y"
"(y andalso y) = y"
apply (unfold andalso_def, simp_all)
apply (rule_tac p=y in trE, simp_all)
apply (rule_tac p=y in trE, simp_all)
done

lemma orelse_thms [simp]:
"(TT orelse y) = TT"
"(FF orelse y) = y"
"(UU orelse y) = UU"
"(y orelse FF) = y"
"(y orelse y) = y"
apply (unfold orelse_def, simp_all)
apply (rule_tac p=y in trE, simp_all)
apply (rule_tac p=y in trE, simp_all)
done

lemma neg_thms [simp]:
"neg\$TT = FF"
"neg\$FF = TT"
"neg\$UU = UU"
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=UU --> P UU) & (Q=TT --> P x) & (Q=FF --> P y))"
apply (unfold If2_def)
apply (rule_tac p = "Q" in trE)
apply (simp_all)
done

ML {*
val split_If_tac =
simp_tac (HOL_basic_ss addsimps [symmetric (thm "If2_def")])
THEN' (split_tac [thm "split_If2"])
*}

subsection "Rewriting of HOLCF operations to HOL functions"

lemma andalso_or:
"!!t.[|t~=UU|]==> ((t andalso s)=FF)=(t=FF | s=FF)"
apply (rule_tac p = "t" in trE)
apply simp_all
done

lemma andalso_and: "[|t~=UU|]==> ((t andalso s)~=FF)=(t~=FF & s~=FF)"
apply (rule_tac p = "t" in trE)
apply simp_all
done

lemma Def_bool1 [simp]: "(Def x ~= FF) = x"

lemma Def_bool2 [simp]: "(Def x = FF) = (~x)"

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

lemma Def_bool4 [simp]: "(Def x ~= TT) = (~x)"

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

text {*
The following rewrite rules for admissibility should in the future be
replaced by a more general admissibility test that also checks
chain-finiteness, of which these lemmata are specific examples
*}

apply (rule_tac p = "x" in trE)
apply (simp_all)
done

apply (rule_tac p = "x" in trE)
apply (simp_all)
done