--- a/src/HOL/simpdata.ML Mon Aug 19 13:03:17 1996 +0200
+++ b/src/HOL/simpdata.ML Mon Aug 19 13:06:30 1996 +0200
@@ -8,59 +8,71 @@
open Simplifier;
+(*** Integration of simplifier with classical reasoner ***)
+
+(*Add a simpset to a classical set!*)
+infix 4 addss;
+fun cs addss ss = cs addbefore asm_full_simp_tac ss 1;
+
+fun Addss ss = (claset := !claset addbefore asm_full_simp_tac ss 1);
+
+(*Maybe swap the safe_tac and simp_tac lines?**)
+fun auto_tac (cs,ss) =
+ TRY (safe_tac cs) THEN
+ ALLGOALS (asm_full_simp_tac ss) THEN
+ REPEAT (FIRSTGOAL (best_tac (cs addss ss)));
+
+fun Auto_tac() = auto_tac (!claset, !simpset);
+
+fun auto() = by (Auto_tac());
+
+
local
-fun prover s = prove_goal HOL.thy s (fn _ => [fast_tac HOL_cs 1]);
+ fun prover s = prove_goal HOL.thy s (fn _ => [fast_tac HOL_cs 1]);
-val P_imp_P_iff_True = prover "P --> (P = True)" RS mp;
-val P_imp_P_eq_True = P_imp_P_iff_True RS eq_reflection;
+ val P_imp_P_iff_True = prover "P --> (P = True)" RS mp;
+ val P_imp_P_eq_True = P_imp_P_iff_True RS eq_reflection;
-val not_P_imp_P_iff_F = prover "~P --> (P = False)" RS mp;
-val not_P_imp_P_eq_False = not_P_imp_P_iff_F RS eq_reflection;
+ val not_P_imp_P_iff_F = prover "~P --> (P = False)" RS mp;
+ val not_P_imp_P_eq_False = not_P_imp_P_iff_F RS eq_reflection;
-fun atomize pairs =
- let fun atoms th =
- (case concl_of th of
- Const("Trueprop",_) $ p =>
- (case head_of p of
- Const(a,_) =>
- (case assoc(pairs,a) of
- Some(rls) => flat (map atoms ([th] RL rls))
- | None => [th])
- | _ => [th])
- | _ => [th])
- in atoms end;
+ fun atomize pairs =
+ let fun atoms th =
+ (case concl_of th of
+ Const("Trueprop",_) $ p =>
+ (case head_of p of
+ Const(a,_) =>
+ (case assoc(pairs,a) of
+ Some(rls) => flat (map atoms ([th] RL rls))
+ | None => [th])
+ | _ => [th])
+ | _ => [th])
+ in atoms end;
-fun mk_meta_eq r = case concl_of r of
- Const("==",_)$_$_ => r
- | _$(Const("op =",_)$_$_) => r RS eq_reflection
- | _$(Const("not",_)$_) => r RS not_P_imp_P_eq_False
- | _ => r RS P_imp_P_eq_True;
-(* last 2 lines requires all formulae to be of the from Trueprop(.) *)
+ fun mk_meta_eq r = case concl_of r of
+ Const("==",_)$_$_ => r
+ | _$(Const("op =",_)$_$_) => r RS eq_reflection
+ | _$(Const("not",_)$_) => r RS not_P_imp_P_eq_False
+ | _ => r RS P_imp_P_eq_True;
+ (* last 2 lines requires all formulae to be of the from Trueprop(.) *)
-fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th;
-
-val imp_cong = impI RSN
- (2, prove_goal HOL.thy "(P=P')--> (P'--> (Q=Q'))--> ((P-->Q) = (P'-->Q'))"
- (fn _=> [fast_tac HOL_cs 1]) RS mp RS mp);
+ fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th;
-val o_apply = prove_goalw HOL.thy [o_def] "(f o g)(x) = f(g(x))"
- (fn _ => [rtac refl 1]);
-
-val simp_thms = map prover
- [ "(x=x) = True",
- "(~True) = False", "(~False) = True", "(~ ~ P) = P",
- "(~P) ~= P", "P ~= (~P)", "(P ~= Q) = (P = (~Q))",
- "(True=P) = P", "(P=True) = P",
- "(True --> P) = P", "(False --> P) = True",
- "(P --> True) = True", "(P --> P) = True",
- "(P --> False) = (~P)", "(P --> ~P) = (~P)",
- "(P & True) = P", "(True & P) = P",
- "(P & False) = False", "(False & P) = False", "(P & P) = P",
- "(P | True) = True", "(True | P) = True",
- "(P | False) = P", "(False | P) = P", "(P | P) = P",
- "(!x.P) = P", "(? x.P) = P", "? x. x=t", "(? x. x=t & P(x)) = P(t)",
- "(P|Q --> R) = ((P-->R)&(Q-->R))" ];
+ val simp_thms = map prover
+ [ "(x=x) = True",
+ "(~True) = False", "(~False) = True", "(~ ~ P) = P",
+ "(~P) ~= P", "P ~= (~P)", "(P ~= Q) = (P = (~Q))",
+ "(True=P) = P", "(P=True) = P",
+ "(True --> P) = P", "(False --> P) = True",
+ "(P --> True) = True", "(P --> P) = True",
+ "(P --> False) = (~P)", "(P --> ~P) = (~P)",
+ "(P & True) = P", "(True & P) = P",
+ "(P & False) = False", "(False & P) = False", "(P & P) = P",
+ "(P | True) = True", "(True | P) = True",
+ "(P | False) = P", "(False | P) = P", "(P | P) = P",
+ "(!x.P) = P", "(? x.P) = P", "? x. x=t",
+ "(? x. x=t & P(x)) = P(t)", "(! x. x=t --> P(x)) = P(t)" ];
in
@@ -71,6 +83,11 @@
val conj_assoc = prover "((P&Q)&R) = (P&(Q&R))";
+val disj_assoc = prover "((P|Q)|R) = (P|(Q|R))";
+
+val imp_disj = prover "(P|Q --> R) = ((P-->R)&(Q-->R))";
+
+
val if_True = prove_goalw HOL.thy [if_def] "(if True then x else y) = x"
(fn _=>[fast_tac (HOL_cs addIs [select_equality]) 1]);
@@ -100,12 +117,6 @@
fun Addcongs congs = (simpset := !simpset addcongs congs);
-(*Add a simpset to a classical set!*)
-infix 4 addss;
-fun cs addss ss = cs addbefore asm_full_simp_tac ss 1;
-
-fun Addss ss = (claset := !claset addbefore asm_full_simp_tac ss 1);
-
val mksimps_pairs =
[("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
("All", [spec]), ("True", []), ("False", []),
@@ -113,14 +124,30 @@
fun mksimps pairs = map mk_meta_eq o atomize pairs o gen_all;
+val imp_cong = impI RSN
+ (2, prove_goal HOL.thy "(P=P')--> (P'--> (Q=Q'))--> ((P-->Q) = (P'-->Q'))"
+ (fn _=> [fast_tac HOL_cs 1]) RS mp RS mp);
+
+val o_apply = prove_goalw HOL.thy [o_def] "(f o g)(x) = f(g(x))"
+ (fn _ => [rtac refl 1]);
+
val HOL_ss = empty_ss
setmksimps (mksimps mksimps_pairs)
setsolver (fn prems => resolve_tac (TrueI::refl::prems) ORELSE' atac
ORELSE' etac FalseE)
setsubgoaler asm_simp_tac
- addsimps ([if_True, if_False, o_apply, conj_assoc] @ simp_thms)
+ addsimps ([if_True, if_False, o_apply, imp_disj, conj_assoc, disj_assoc]
+ @ simp_thms)
addcongs [imp_cong];
+
+(*In general it seems wrong to add distributive laws by default: they
+ might cause exponential blow-up. This one has been added for a while
+ and cannot be removed without affecting existing proofs. Moreover,
+ rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
+ grounds that it allows simplification of R in the two cases.*)
+
+
local val mktac = mk_case_split_tac (meta_eq_to_obj_eq RS iffD2)
in
fun split_tac splits = mktac (map mk_meta_eq splits)
@@ -182,6 +209,10 @@
prove "conj_left_commute" "(P&(Q&R)) = (Q&(P&R))";
val conj_comms = [conj_commute, conj_left_commute];
+prove "disj_commute" "(P|Q) = (Q|P)";
+prove "disj_left_commute" "(P|(Q|R)) = (Q|(P|R))";
+val disj_comms = [disj_commute, disj_left_commute];
+
prove "conj_disj_distribL" "(P&(Q|R)) = (P&Q | P&R)";
prove "conj_disj_distribR" "((P|Q)&R) = (P&R | Q&R)";
@@ -189,18 +220,20 @@
prove "disj_conj_distribR" "((P&Q)|R) = ((P|R) & (Q|R))";
prove "imp_conj_distrib" "(P --> (Q&R)) = ((P-->Q) & (P-->R))";
-prove "imp_conj_assoc" "((P&Q)-->R) = (P --> (Q --> R))";
+prove "imp_conj" "((P&Q)-->R) = (P --> (Q --> R))";
prove "de_Morgan_disj" "(~(P | Q)) = (~P & ~Q)";
prove "de_Morgan_conj" "(~(P & Q)) = (~P | ~Q)";
+prove "not_iff" "(P~=Q) = (P = (~Q))";
prove "not_all" "(~ (! x.P(x))) = (? x.~P(x))";
+prove "imp_all" "((! x. P x) --> Q) = (? x. P x --> Q)";
prove "not_ex" "(~ (? x.P(x))) = (! x.~P(x))";
+prove "imp_ex" "((? x. P x) --> Q) = (! x. P x --> Q)";
prove "ex_disj_distrib" "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))";
prove "all_conj_distrib" "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";
-prove "ex_imp" "((? x. P x) --> Q) = (!x. P x --> Q)";
qed_goal "if_cancel" HOL.thy "(if c then x else x) = x"
(fn _ => [simp_tac (HOL_ss setloop (split_tac [expand_if])) 1]);