(* Title: FOL/simpdata
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Simplification data for FOL
*)
(* Elimination of True from asumptions: *)
val True_implies_equals = prove_goal IFOL.thy
"(True ==> PROP P) == PROP P"
(K [rtac equal_intr_rule 1, atac 2,
METAHYPS (fn prems => resolve_tac prems 1) 1,
rtac TrueI 1]);
(*** 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"];
(*The x=t versions are needed for the simplification procedures*)
val quant_simps = map int_prove_fun
["(ALL x. P) <-> P",
"(ALL x. x=t --> P(x)) <-> P(t)",
"(ALL x. t=x --> P(x)) <-> P(t)",
"(EX x. P) <-> P",
"(EX x. x=t & P(x)) <-> P(t)",
"(EX x. t=x & P(x)) <-> P(t)"];
(*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;
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("op =",_)$_$_) => th RS eq_reflection
| _ $ (Const("op <->",_)$_$_) => th RS iff_reflection
| _ =>
error("conclusion must be a =-equality or <->");;
fun mk_eq th = case concl_of th of
Const("==",_)$_$_ => th
| _ $ (Const("op =",_)$_$_) => mk_meta_eq th
| _ $ (Const("op <->",_)$_$_) => mk_meta_eq th
| _ $ (Const("Not",_)$_) => th RS iff_reflection_F
| _ => th RS iff_reflection_T;
fun mk_meta_cong rl = standard (mk_meta_eq rl);
(*FIXME: how about the premises?*)
val mksimps_pairs =
[("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
("All", [spec]), ("True", []), ("False", [])];
(* ###FIXME: move to Provers/simplifier.ML
val mk_atomize: (string * thm list) list -> thm -> thm list
*)
(* ###FIXME: move to Provers/simplifier.ML *)
fun mk_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 mksimps pairs = (map mk_eq o mk_atomize pairs o gen_all);
(*** 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";
(*** Miniscoping: pushing quantifiers in
We do NOT distribute of ALL over &, or dually that of EX over |
Baaz and Leitsch, On Skolemization and Proof Complexity (1994)
show that this step can increase proof length!
***)
(*existential miniscoping*)
val int_ex_simps = map int_prove_fun
["(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))"];
(*classical rules*)
val cla_ex_simps = map prove_fun
["(EX x. P(x) --> Q) <-> (ALL x. P(x)) --> Q",
"(EX x. P --> Q(x)) <-> P --> (EX x. Q(x))"];
val ex_simps = int_ex_simps @ cla_ex_simps;
(*universal miniscoping*)
val int_all_simps = map int_prove_fun
["(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))"];
(*classical rules*)
val cla_all_simps = map prove_fun
["(ALL x. P(x) | Q) <-> (ALL x. P(x)) | Q",
"(ALL x. P | Q(x)) <-> P | (ALL x. Q(x))"];
val all_simps = int_all_simps @ cla_all_simps;
(*** Named rewrite rules proved for IFOL ***)
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 _ => [Blast_tac 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)";
prove "imp_disj1" "(P-->Q) | R <-> (P-->Q | R)";
prove "imp_disj2" "Q | (P-->R) <-> (P-->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]);
(** make simplification procedures for quantifier elimination **)
structure Quantifier1 = Quantifier1Fun(
struct
(*abstract syntax*)
fun dest_eq((c as Const("op =",_)) $ s $ t) = Some(c,s,t)
| dest_eq _ = None;
fun dest_conj((c as Const("op &",_)) $ s $ t) = Some(c,s,t)
| dest_conj _ = None;
val conj = FOLogic.conj
val imp = FOLogic.imp
(*rules*)
val iff_reflection = iff_reflection
val iffI = iffI
val sym = sym
val conjI= conjI
val conjE= conjE
val impI = impI
val impE = impE
val mp = mp
val exI = exI
val exE = exE
val allI = allI
val allE = allE
end);
local
val ex_pattern =
read_cterm (sign_of FOL.thy) ("EX x. P(x) & Q(x)", FOLogic.oT)
val all_pattern =
read_cterm (sign_of FOL.thy) ("ALL x. P(x) & P'(x) --> Q(x)", FOLogic.oT)
in
val defEX_regroup =
mk_simproc "defined EX" [ex_pattern] Quantifier1.rearrange_ex;
val defALL_regroup =
mk_simproc "defined ALL" [all_pattern] Quantifier1.rearrange_all;
end;
(*** Case splitting ***)
val meta_eq_to_iff = prove_goal IFOL.thy "x==y ==> x<->y"
(fn [prem] => [rewtac prem, rtac iffI 1, atac 1, atac 1]);
structure SplitterData =
struct
structure Simplifier = Simplifier
val mk_eq = mk_eq
val meta_eq_to_iff = meta_eq_to_iff
val iffD = iffD2
val disjE = disjE
val conjE = conjE
val exE = exE
val contrapos = contrapos
val contrapos2 = contrapos2
val notnotD = notnotD
end;
structure Splitter = SplitterFun(SplitterData);
val split_tac = Splitter.split_tac;
val split_inside_tac = Splitter.split_inside_tac;
val split_asm_tac = Splitter.split_asm_tac;
val op addsplits = Splitter.addsplits;
val op delsplits = Splitter.delsplits;
val Addsplits = Splitter.Addsplits;
val Delsplits = Splitter.Delsplits;
(*** Standard simpsets ***)
structure Induction = InductionFun(struct val spec=IFOL.spec end);
open Induction;
(* Add congruence rules for = or <-> (instead of ==) *)
(* ###FIXME: Move to simplifier,
taking mk_meta_cong as input, eliminating addeqcongs and deleqcongs *)
infix 4 addcongs delcongs;
fun ss addcongs congs = ss addeqcongs (map mk_meta_cong congs);
fun ss delcongs congs = ss deleqcongs (map mk_meta_cong congs);
fun Addcongs congs = (simpset_ref() := simpset() addcongs congs);
fun Delcongs congs = (simpset_ref() := simpset() delcongs congs);
val meta_simps =
[triv_forall_equality, (* prunes params *)
True_implies_equals]; (* prune asms `True' *)
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]];
(*No simprules, but basic infastructure for simplification*)
val FOL_basic_ss = empty_ss setsubgoaler asm_simp_tac
addsimprocs [defALL_regroup,defEX_regroup]
setSSolver safe_solver
setSolver unsafe_solver
setmksimps (mksimps mksimps_pairs);
(*intuitionistic simprules only*)
val IFOL_ss =
FOL_basic_ss addsimps (meta_simps @ IFOL_simps @
int_ex_simps @ int_all_simps)
addcongs [imp_cong];
val cla_simps =
[de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2,
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)"];
(*classical simprules too*)
val FOL_ss = IFOL_ss addsimps (cla_simps @ cla_ex_simps @ cla_all_simps);
simpset_ref() := FOL_ss;
(*** integration of simplifier with classical reasoner ***)
structure Clasimp = ClasimpFun
(structure Simplifier = Simplifier
and Classical = Cla
and Blast = Blast);
open Clasimp;
val FOL_css = (FOL_cs, FOL_ss);