(* Title: FOL/simpdata
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Simplification data for FOL
*)
(*** Rewrite rules ***)
fun int_prove_fun s =
(writeln s;
prove_goal IFOL.thy s
(fn prems => [ (cut_facts_tac prems 1),
(IntPr.fast_tac 1) ]));
val conj_simps = map int_prove_fun
["P & True <-> P", "True & P <-> P",
"P & False <-> False", "False & P <-> False",
"P & P <-> P", "P & P & Q <-> P & Q",
"P & ~P <-> False", "~P & P <-> False",
"(P & Q) & R <-> P & (Q & R)"];
val disj_simps = map int_prove_fun
["P | True <-> True", "True | P <-> True",
"P | False <-> P", "False | P <-> P",
"P | P <-> P", "P | P | Q <-> P | Q",
"(P | Q) | R <-> P | (Q | R)"];
val not_simps = map int_prove_fun
["~(P|Q) <-> ~P & ~Q",
"~ False <-> True", "~ True <-> False"];
val imp_simps = map int_prove_fun
["(P --> False) <-> ~P", "(P --> True) <-> True",
"(False --> P) <-> True", "(True --> P) <-> P",
"(P --> P) <-> True", "(P --> ~P) <-> ~P"];
val iff_simps = map int_prove_fun
["(True <-> P) <-> P", "(P <-> True) <-> P",
"(P <-> P) <-> True",
"(False <-> P) <-> ~P", "(P <-> False) <-> ~P"];
val quant_simps = map int_prove_fun
["(ALL x.P) <-> P", "(EX x.P) <-> P"];
(*These are NOT supplied by default!*)
val distrib_simps = map int_prove_fun
["P & (Q | R) <-> P&Q | P&R",
"(Q | R) & P <-> Q&P | R&P",
"(P | Q --> R) <-> (P --> R) & (Q --> R)"];
(** Conversion into rewrite rules **)
fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th;
(*Make atomic rewrite rules*)
fun atomize r =
case concl_of r of
Const("Trueprop",_) $ p =>
(case p of
Const("op -->",_)$_$_ => atomize(r RS mp)
| Const("op &",_)$_$_ => atomize(r RS conjunct1) @
atomize(r RS conjunct2)
| Const("All",_)$_ => atomize(r RS spec)
| Const("True",_) => [] (*True is DELETED*)
| Const("False",_) => [] (*should False do something?*)
| _ => [r])
| _ => [r];
val P_iff_F = int_prove_fun "~P ==> (P <-> False)";
val iff_reflection_F = P_iff_F RS iff_reflection;
val P_iff_T = int_prove_fun "P ==> (P <-> True)";
val iff_reflection_T = P_iff_T RS iff_reflection;
(*Make meta-equalities. The operator below is Trueprop*)
fun mk_meta_eq th = case concl_of th of
Const("==",_)$_$_ => th
| _ $ (Const("op =",_)$_$_) => th RS eq_reflection
| _ $ (Const("op <->",_)$_$_) => th RS iff_reflection
| _ $ (Const("Not",_)$_) => th RS iff_reflection_F
| _ => th RS iff_reflection_T;
(*** Classical laws ***)
fun prove_fun s =
(writeln s;
prove_goal FOL.thy s
(fn prems => [ (cut_facts_tac prems 1),
(Cla.fast_tac FOL_cs 1) ]));
(*Avoids duplication of subgoals after expand_if, when the true and false
cases boil down to the same thing.*)
val cases_simp = prove_fun "(P --> Q) & (~P --> Q) <-> Q";
(*At present, miniscoping is for classical logic only. We do NOT include
distribution of ALL over &, or dually that of EX over |.*)
(*Miniscoping: pushing in existential quantifiers*)
val ex_simps = map prove_fun
["(EX x. x=t & P(x)) <-> P(t)",
"(EX x. t=x & P(x)) <-> P(t)",
"(EX x. P(x) & Q) <-> (EX x.P(x)) & Q",
"(EX x. P & Q(x)) <-> P & (EX x.Q(x))",
"(EX x. P(x) | Q) <-> (EX x.P(x)) | Q",
"(EX x. P | Q(x)) <-> P | (EX x.Q(x))",
"(EX x. P(x) --> Q) <-> (ALL x.P(x)) --> Q",
"(EX x. P --> Q(x)) <-> P --> (EX x.Q(x))"];
(*Miniscoping: pushing in universal quantifiers*)
val all_simps = map prove_fun
["(ALL x. x=t --> P(x)) <-> P(t)",
"(ALL x. t=x --> P(x)) <-> P(t)",
"(ALL x. P(x) & Q) <-> (ALL x.P(x)) & Q",
"(ALL x. P & Q(x)) <-> P & (ALL x.Q(x))",
"(ALL x. P(x) | Q) <-> (ALL x.P(x)) | Q",
"(ALL x. P | Q(x)) <-> P | (ALL x.Q(x))",
"(ALL x. P(x) --> Q) <-> (EX x.P(x)) --> Q",
"(ALL x. P --> Q(x)) <-> P --> (ALL x.Q(x))"];
fun int_prove nm thm = qed_goal nm IFOL.thy thm
(fn prems => [ (cut_facts_tac prems 1),
(IntPr.fast_tac 1) ]);
fun prove nm thm = qed_goal nm FOL.thy thm (fn _ => [fast_tac FOL_cs 1]);
int_prove "conj_commute" "P&Q <-> Q&P";
int_prove "conj_left_commute" "P&(Q&R) <-> Q&(P&R)";
val conj_comms = [conj_commute, conj_left_commute];
int_prove "disj_commute" "P|Q <-> Q|P";
int_prove "disj_left_commute" "P|(Q|R) <-> Q|(P|R)";
val disj_comms = [disj_commute, disj_left_commute];
int_prove "conj_disj_distribL" "P&(Q|R) <-> (P&Q | P&R)";
int_prove "conj_disj_distribR" "(P|Q)&R <-> (P&R | Q&R)";
int_prove "disj_conj_distribL" "P|(Q&R) <-> (P|Q) & (P|R)";
int_prove "disj_conj_distribR" "(P&Q)|R <-> (P|R) & (Q|R)";
int_prove "imp_conj_distrib" "(P --> (Q&R)) <-> (P-->Q) & (P-->R)";
int_prove "imp_conj" "((P&Q)-->R) <-> (P --> (Q --> R))";
int_prove "imp_disj" "(P|Q --> R) <-> (P-->R) & (Q-->R)";
int_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" "(~ (ALL x.P(x))) <-> (EX x.~P(x))";
prove "imp_all" "((ALL x.P(x)) --> Q) <-> (EX x.P(x) --> Q)";
int_prove "not_ex" "(~ (EX x.P(x))) <-> (ALL x.~P(x))";
int_prove "imp_ex" "((EX x. P(x)) --> Q) <-> (ALL x. P(x) --> Q)";
int_prove "ex_disj_distrib"
"(EX x. P(x) | Q(x)) <-> ((EX x. P(x)) | (EX x. Q(x)))";
int_prove "all_conj_distrib"
"(ALL x. P(x) & Q(x)) <-> ((ALL x. P(x)) & (ALL x. Q(x)))";
(*Used in ZF, perhaps elsewhere?*)
val meta_eq_to_obj_eq = prove_goal IFOL.thy "x==y ==> x=y"
(fn [prem] => [rewtac prem, rtac refl 1]);
(*** case splitting ***)
qed_goal "meta_iffD" IFOL.thy "[| P==Q; Q |] ==> P"
(fn [prem1,prem2] => [rewtac prem1, rtac prem2 1]);
local val mktac = mk_case_split_tac meta_iffD
in
fun split_tac splits = mktac (map mk_meta_eq splits)
end;
local val mktac = mk_case_split_inside_tac meta_iffD
in
fun split_inside_tac splits = mktac (map mk_meta_eq splits)
end;
(*** Standard simpsets ***)
structure Induction = InductionFun(struct val spec=IFOL.spec end);
open Simplifier Induction;
(*Add congruence rules for = or <-> (instead of ==) *)
infix 4 addcongs delcongs;
fun ss addcongs congs =
ss addeqcongs (congs RL [eq_reflection,iff_reflection]);
fun ss delcongs congs =
ss deleqcongs (congs RL [eq_reflection,iff_reflection]);
fun Addcongs congs = (simpset := !simpset addcongs congs);
fun Delcongs congs = (simpset := !simpset delcongs congs);
val IFOL_simps =
[refl RS P_iff_T] @ conj_simps @ disj_simps @ not_simps @
imp_simps @ iff_simps @ quant_simps;
val notFalseI = int_prove_fun "~False";
val triv_rls = [TrueI,refl,iff_refl,notFalseI];
fun unsafe_solver prems = FIRST'[resolve_tac (triv_rls@prems),
atac, etac FalseE];
(*No premature instantiation of variables during simplification*)
fun safe_solver prems = FIRST'[match_tac (triv_rls@prems),
eq_assume_tac, ematch_tac [FalseE]];
val FOL_basic_ss = empty_ss setsubgoaler asm_simp_tac
setSSolver safe_solver
setSolver unsafe_solver
setmksimps (map mk_meta_eq o atomize o gen_all);
val IFOL_ss = FOL_basic_ss addsimps IFOL_simps
addcongs [imp_cong];
val cla_simps =
[de_Morgan_conj, de_Morgan_disj, not_all, not_ex, cases_simp] @
map prove_fun
["~(P&Q) <-> ~P | ~Q",
"P | ~P", "~P | P",
"~ ~ P <-> P", "(~P --> P) <-> P",
"(~P <-> ~Q) <-> (P<->Q)"];
val FOL_ss = IFOL_ss addsimps (cla_simps @ ex_simps @ all_simps);
(*** Install simpsets and datatypes in theory structure ***)
simpset := FOL_ss;
exception SS_DATA of simpset;
let fun merge [] = SS_DATA empty_ss
| merge ss = let val ss = map (fn SS_DATA x => x) ss;
in SS_DATA (foldl merge_ss (hd ss, tl ss)) end;
fun put (SS_DATA ss) = simpset := ss;
fun get () = SS_DATA (!simpset);
in add_thydata "FOL"
("simpset", ThyMethods {merge = merge, put = put, get = get})
end;
add_thy_reader_file "thy_data.ML";
(*** Integration of simplifier with classical reasoner ***)
(* rot_eq_tac rotates the first equality premise of subgoal i to the front,
fails if there is no equaliy or if an equality is already at the front *)
fun rot_eq_tac i = let
fun is_eq (Const ("Trueprop", _) $ (Const("op =" ,_) $ _ $ _)) = true
| is_eq (Const ("Trueprop", _) $ (Const("op <->",_) $ _ $ _)) = true
| is_eq _ = false;
fun find_eq n [] = None
| find_eq n (t :: ts) = if (is_eq t) then Some n else find_eq (n + 1) ts;
fun rot_eq state = let val (_, _, Bi, _) = dest_state (state, i) in
(case find_eq 0 (Logic.strip_assums_hyp Bi) of
None => no_tac
| Some 0 => no_tac
| Some n => rotate_tac n i) end;
in STATE rot_eq end;
fun safe_asm_more_full_simp_tac ss = TRY o rot_eq_tac THEN'
safe_asm_full_simp_tac ss;
(*an unsatisfactory fix for the incomplete asm_full_simp_tac!
better: asm_really_full_simp_tac, a yet to be implemented version of
asm_full_simp_tac that applies all equalities in the
premises to all the premises *)
(*Add a simpset to a classical set!*)
infix 4 addSss addss;
fun cs addSss ss = cs addSaltern (CHANGED o (safe_asm_more_full_simp_tac ss));
fun cs addss ss = cs addbefore asm_full_simp_tac ss;
fun Addss ss = (claset := !claset addss ss);
(*Designed to be idempotent, except if best_tac instantiates variables
in some of the subgoals*)
type clasimpset = (claset * simpset);
val FOL_css = (FOL_cs, FOL_ss);
fun pair_upd1 f ((a,b),x) = (f(a,x), b);
fun pair_upd2 f ((a,b),x) = (a, f(b,x));
infix 4 addSIs2 addSEs2 addSDs2 addIs2 addEs2 addDs2
addsimps2 delsimps2 addcongs2 delcongs2;
fun op addSIs2 arg = pair_upd1 (op addSIs) arg;
fun op addSEs2 arg = pair_upd1 (op addSEs) arg;
fun op addSDs2 arg = pair_upd1 (op addSDs) arg;
fun op addIs2 arg = pair_upd1 (op addIs ) arg;
fun op addEs2 arg = pair_upd1 (op addEs ) arg;
fun op addDs2 arg = pair_upd1 (op addDs ) arg;
fun op addsimps2 arg = pair_upd2 (op addsimps) arg;
fun op delsimps2 arg = pair_upd2 (op delsimps) arg;
fun op addcongs2 arg = pair_upd2 (op addcongs) arg;
fun op delcongs2 arg = pair_upd2 (op delcongs) arg;
fun auto_tac (cs,ss) =
let val cs' = cs addss ss
in EVERY [TRY (safe_tac cs'),
REPEAT (FIRSTGOAL (fast_tac cs')),
TRY (safe_tac (cs addSss ss)),
prune_params_tac]
end;
fun Auto_tac () = auto_tac (!claset, !simpset);
fun auto () = by (Auto_tac ());