(* Title: HOLCF/Tr.ML
ID: $Id$
Author: Franz Regensburger
License: GPL (GNU GENERAL PUBLIC LICENSE)
Introduce infix if_then_else_fi and boolean connectives andalso, orelse
*)
(* ------------------------------------------------------------------------ *)
(* Exhaustion and Elimination for type one *)
(* ------------------------------------------------------------------------ *)
Goalw [FF_def,TT_def] "t=UU | t = TT | t = FF";
by (induct_tac "t" 1);
by (fast_tac HOL_cs 1);
by (fast_tac (HOL_cs addss simpset()) 1);
qed "Exh_tr";
val prems = Goal "[| p=UU ==> Q; p = TT ==>Q; p = FF ==>Q|] ==>Q";
by (rtac (Exh_tr RS disjE) 1);
by (eresolve_tac prems 1);
by (etac disjE 1);
by (eresolve_tac prems 1);
by (eresolve_tac prems 1);
qed "trE";
(* ------------------------------------------------------------------------ *)
(* tactic for tr-thms with case split *)
(* ------------------------------------------------------------------------ *)
bind_thms ("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),
(REPEAT(asm_simp_tac (simpset() addsimps
[o_def,flift1_def,flift2_def,inst_lift_po]@tr_defs) 1))
]);
(* ------------------------------------------------------------------------ *)
(* distinctness for type tr *)
(* ------------------------------------------------------------------------ *)
bind_thms ("dist_less_tr", map prover [
"~TT << UU",
"~FF << UU",
"~TT << FF",
"~FF << TT"
]);
val dist_eq_tr = map prover ["TT~=UU","FF~=UU","TT~=FF"];
bind_thms ("dist_eq_tr", dist_eq_tr @ (map (fn thm => (thm RS not_sym)) dist_eq_tr));
(* ------------------------------------------------------------------------ *)
(* lemmas about andalso, orelse, neg and if *)
(* ------------------------------------------------------------------------ *)
bind_thms ("andalso_thms", map prover [
"(TT andalso y) = y",
"(FF andalso y) = FF",
"(UU andalso y) = UU",
"(y andalso TT) = y",
"(y andalso y) = y"
]);
bind_thms ("orelse_thms", map prover [
"(TT orelse y) = TT",
"(FF orelse y) = y",
"(UU orelse y) = UU",
"(y orelse FF) = y",
"(y orelse y) = y"]);
bind_thms ("neg_thms", map prover [
"neg$TT = FF",
"neg$FF = TT",
"neg$UU = UU"
]);
bind_thms ("ifte_thms", map prover [
"If UU then e1 else e2 fi = UU",
"If FF then e1 else e2 fi = e2",
"If TT then e1 else e2 fi = e1"]);
Addsimps (dist_less_tr @ dist_eq_tr @ andalso_thms @
orelse_thms @ neg_thms @ ifte_thms);
(* ------------------------------------------------------------------- *)
(* split-tac for If via If2 because the constant has to be a constant *)
(* ------------------------------------------------------------------- *)
Goalw [If2_def]
"P (If2 Q x y ) = ((Q=UU --> P UU) & (Q=TT --> P x) & (Q=FF --> P y))";
by (res_inst_tac [("p","Q")] trE 1);
by (REPEAT (Asm_full_simp_tac 1));
qed"split_If2";
val split_If_tac =
simp_tac (HOL_basic_ss addsimps [symmetric If2_def]) THEN' (split_tac [split_If2]);
(* ----------------------------------------------------------------- *)
section"Rewriting of HOLCF operations to HOL functions";
(* ----------------------------------------------------------------- *)
Goal
"!!t.[|t~=UU|]==> ((t andalso s)=FF)=(t=FF | s=FF)";
by (rtac iffI 1);
by (res_inst_tac [("p","t")] trE 1);
by Auto_tac;
by (res_inst_tac [("p","t")] trE 1);
by Auto_tac;
qed"andalso_or";
Goal "[|t~=UU|]==> ((t andalso s)~=FF)=(t~=FF & s~=FF)";
by (rtac iffI 1);
by (res_inst_tac [("p","t")] trE 1);
by Auto_tac;
by (res_inst_tac [("p","t")] trE 1);
by Auto_tac;
qed"andalso_and";
Goal "(Def x ~=FF)= x";
by (simp_tac (simpset() addsimps [FF_def]) 1);
qed"Def_bool1";
Goal "(Def x = FF) = (~x)";
by (simp_tac (simpset() addsimps [FF_def]) 1);
qed"Def_bool2";
Goal "(Def x = TT) = x";
by (simp_tac (simpset() addsimps [TT_def]) 1);
qed"Def_bool3";
Goal "(Def x ~= TT) = (~x)";
by (simp_tac (simpset() addsimps [TT_def]) 1);
qed"Def_bool4";
Goal
"(If Def P then A else B fi)= (if P then A else B)";
by (res_inst_tac [("p","Def P")] trE 1);
by (Asm_full_simp_tac 1);
by (asm_full_simp_tac (simpset() addsimps tr_defs@[flift1_def,o_def]) 1);
by (asm_full_simp_tac (simpset() addsimps tr_defs@[flift1_def,o_def]) 1);
qed"If_and_if";
Addsimps [Def_bool1,Def_bool2,Def_bool3,Def_bool4];
(* ----------------------------------------------------------------- *)
section"admissibility";
(* ----------------------------------------------------------------- *)
(* 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 *)
Goal "(x~=FF) = (x=TT|x=UU)";
by (res_inst_tac [("p","x")] trE 1);
by (TRYALL (Asm_full_simp_tac));
qed"adm_trick_1";
Goal "(x~=TT) = (x=FF|x=UU)";
by (res_inst_tac [("p","x")] trE 1);
by (TRYALL (Asm_full_simp_tac));
qed"adm_trick_2";
bind_thms ("adm_tricks", [adm_trick_1,adm_trick_2]);
Goal "cont(f) ==> adm (%x. (f x)~=TT)";
by (simp_tac (HOL_basic_ss addsimps adm_tricks) 1);
by (REPEAT ((resolve_tac (adm_lemmas@cont_lemmas1) 1) ORELSE atac 1));
qed"adm_nTT";
Goal "cont(f) ==> adm (%x. (f x)~=FF)";
by (simp_tac (HOL_basic_ss addsimps adm_tricks) 1);
by (REPEAT ((resolve_tac (adm_lemmas@cont_lemmas1) 1) ORELSE atac 1));
qed"adm_nFF";
Addsimps [adm_nTT,adm_nFF];