src/FOL/simpdata.ML
 author paulson Tue, 15 Jan 2002 15:07:41 +0100 changeset 12765 fb3f9887d0b7 parent 12725 7ede865e1fe5 child 12825 f1f7964ed05c permissions -rw-r--r--
new theorem
```
(*  Title:      FOL/simpdata.ML
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright   1994  University of Cambridge

Simplification data for FOL.
*)

(* Elimination of True from asumptions: *)

bind_thm ("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) ]));

bind_thms ("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)"]);

bind_thms ("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)"]);

bind_thms ("not_simps", map int_prove_fun
["~(P|Q)  <-> ~P & ~Q",
"~ False <-> True",   "~ True <-> False"]);

bind_thms ("imp_simps", map int_prove_fun
["(P --> False) <-> ~P",       "(P --> True) <-> True",
"(False --> P) <-> True",     "(True --> P) <-> P",
"(P --> P) <-> True",         "(P --> ~P) <-> ~P"]);

bind_thms ("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*)
bind_thms ("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!*)
bind_thms ("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 **)

bind_thm ("P_iff_F", int_prove_fun "~P ==> (P <-> False)");
bind_thm ("iff_reflection_F", P_iff_F RS iff_reflection);

bind_thm ("P_iff_T", int_prove_fun "P ==> (P <-> True)");
bind_thm ("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;

(*Replace premises x=y, X<->Y by X==Y*)
val mk_meta_prems =
rule_by_tactic
(REPEAT_FIRST (resolve_tac [meta_eq_to_obj_eq, def_imp_iff]));

(*Congruence rules for = or <-> (instead of ==)*)
fun mk_meta_cong rl =
standard(mk_meta_eq (mk_meta_prems rl))
handle THM _ =>
error("Premises and conclusion of congruence rules must use =-equality or <->");

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 (the_context ()) 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.*)
bind_thm ("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*)
bind_thms ("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*)
bind_thms ("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))"]);

bind_thms ("ex_simps", int_ex_simps @ cla_ex_simps);

(*universal miniscoping*)
bind_thms ("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*)
bind_thms ("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))"]);

bind_thms ("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 (the_context ()) thm (fn _ => [Blast_tac 1]);

int_prove "conj_commute" "P&Q <-> Q&P";
int_prove "conj_left_commute" "P&(Q&R) <-> Q&(P&R)";
bind_thms ("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)";
bind_thms ("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_imp" "~(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)))";

local
val uncurry = prove_goal (the_context()) "P --> Q --> R ==> P & Q --> R"
(fn prems => [cut_facts_tac prems 1, Blast_tac 1]);

val iff_allI = allI RS
prove_goal (the_context()) "ALL x. P(x) <-> Q(x) ==> (ALL x. P(x)) <-> (ALL x. Q(x))"
(fn prems => [cut_facts_tac prems 1, Blast_tac 1])
val iff_exI = allI RS
prove_goal (the_context()) "ALL x. P(x) <-> Q(x) ==> (EX x. P(x)) <-> (EX x. Q(x))"
(fn prems => [cut_facts_tac prems 1, Blast_tac 1])

val all_comm = prove_goal (the_context()) "(ALL x y. P(x,y)) <-> (ALL y x. P(x,y))"
(fn _ => [Blast_tac 1])
val ex_comm = prove_goal (the_context()) "(EX x y. P(x,y)) <-> (EX y x. P(x,y))"
(fn _ => [Blast_tac 1])
in

(** 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;
fun dest_imp((c as Const("op -->",_)) \$ s \$ t) = Some(c,s,t)
| dest_imp _ = None;
val conj = FOLogic.conj
val imp  = FOLogic.imp
(*rules*)
val iff_reflection = iff_reflection
val iffI = iffI
val iff_trans = iff_trans
val conjI= conjI
val conjE= conjE
val impI = impI
val mp   = mp
val uncurry = uncurry
val exI  = exI
val exE  = exE
val iff_allI = iff_allI
val iff_exI = iff_exI
val all_comm = all_comm
val ex_comm = ex_comm
end);

end;

local

val ex_pattern =
read_cterm (Theory.sign_of (the_context ())) ("EX x. P(x)", FOLogic.oT)

val all_pattern =
read_cterm (Theory.sign_of (the_context ())) ("ALL x. P(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 ***)

bind_thm ("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 delsplits     = Splitter.delsplits;
val Delsplits        = Splitter.Delsplits;

(*** Standard simpsets ***)

structure Induction = InductionFun(struct val spec=IFOL.spec end);

open Induction;

bind_thms ("meta_simps",
[triv_forall_equality,   (* prunes params *)
True_implies_equals]);  (* prune asms `True' *)

bind_thms ("IFOL_simps",
[refl RS P_iff_T] @ conj_simps @ disj_simps @ not_simps @
imp_simps @ iff_simps @ quant_simps);

bind_thm ("notFalseI", int_prove_fun "~False");
bind_thms ("triv_rls", [TrueI,refl,reflexive_thm,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
setSSolver (mk_solver "FOL safe" safe_solver)
setSolver (mk_solver "FOL unsafe" unsafe_solver)
setmksimps (mksimps mksimps_pairs)
setmkcong mk_meta_cong;

(*intuitionistic simprules only*)
val IFOL_ss = FOL_basic_ss
addsimps (meta_simps @ IFOL_simps @ int_ex_simps @ int_all_simps)

bind_thms ("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);

val simpsetup = [fn thy => (simpset_ref_of thy := FOL_ss; thy)];

(*** integration of simplifier with classical reasoner ***)

structure Clasimp = ClasimpFun
(structure Simplifier = Simplifier and Splitter = Splitter
and Classical  = Cla and Blast = Blast
val iffD1 = iffD1 val iffD2 = iffD2 val notE = notE
val cla_make_elim = cla_make_elim);
open Clasimp;

val FOL_css = (FOL_cs, FOL_ss);
```