(* Title: HOL/simpdata.ML
ID: $Id$
Author: Tobias Nipkow
Copyright 1991 University of Cambridge
Instantiation of the generic simplifier for HOL.
*)
(** tools setup **)
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 = HOLogic.conj
val imp = HOLogic.imp
(*rules*)
val iff_reflection = thm "eq_reflection"
val iffI = thm "iffI"
val iff_trans = thm "trans"
val conjI= thm "conjI"
val conjE= thm "conjE"
val impI = thm "impI"
val mp = thm "mp"
val uncurry = thm "uncurry"
val exI = thm "exI"
val exE = thm "exE"
val iff_allI = thm "iff_allI"
val iff_exI = thm "iff_exI"
val all_comm = thm "all_comm"
val ex_comm = thm "ex_comm"
end);
structure Simpdata =
struct
local
val eq_reflection = thm "eq_reflection"
in fun mk_meta_eq r = r RS eq_reflection end;
fun safe_mk_meta_eq r = mk_meta_eq r handle Thm.THM _ => r;
local
val Eq_FalseI = thm "Eq_FalseI"
val Eq_TrueI = thm "Eq_TrueI"
in fun mk_eq th = case concl_of th
(*expects Trueprop if not == *)
of Const ("==",_) $ _ $ _ => th
| _ $ (Const ("op =", _) $ _ $ _) => mk_meta_eq th
| _ $ (Const ("Not", _) $ _) => th RS Eq_FalseI
| _ => th RS Eq_TrueI
end;
local
val meta_eq_to_obj_eq = thm "meta_eq_to_obj_eq"
val Eq_TrueI = thm "Eq_TrueI"
in fun mk_eq_True r =
SOME (r RS meta_eq_to_obj_eq RS Eq_TrueI) handle Thm.THM _ => NONE;
end;
(* Produce theorems of the form
(P1 =simp=> ... =simp=> Pn => x == y) ==> (P1 =simp=> ... =simp=> Pn => x = y)
*)
local
val meta_eq_to_obj_eq = thm "meta_eq_to_obj_eq"
val simp_implies_def = thm "simp_implies_def"
in fun lift_meta_eq_to_obj_eq i st =
let
fun count_imp (Const ("HOL.simp_implies", _) $ P $ Q) = 1 + count_imp Q
| count_imp _ = 0;
val j = count_imp (Logic.strip_assums_concl (List.nth (prems_of st, i - 1)))
in if j = 0 then meta_eq_to_obj_eq
else
let
val Ps = map (fn k => Free ("P" ^ string_of_int k, propT)) (1 upto j);
fun mk_simp_implies Q = foldr (fn (R, S) =>
Const ("HOL.simp_implies", propT --> propT --> propT) $ R $ S) Q Ps
val aT = TFree ("'a", HOLogic.typeS);
val x = Free ("x", aT);
val y = Free ("y", aT)
in Goal.prove_global (Thm.theory_of_thm st) []
[mk_simp_implies (Logic.mk_equals (x, y))]
(mk_simp_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (x, y))))
(fn prems => EVERY
[rewrite_goals_tac [simp_implies_def],
REPEAT (ares_tac (meta_eq_to_obj_eq :: map (rewrite_rule [simp_implies_def]) prems) 1)])
end
end;
end;
(*Congruence rules for = (instead of ==)*)
fun mk_meta_cong rl = zero_var_indexes
(let val rl' = Seq.hd (TRYALL (fn i => fn st =>
rtac (lift_meta_eq_to_obj_eq i st) i st) rl)
in mk_meta_eq rl' handle THM _ =>
if can Logic.dest_equals (concl_of rl') then rl'
else error "Conclusion of congruence rules must be =-equality"
end);
fun mk_atomize pairs =
let
fun atoms thm =
let
fun res th = map (fn rl => th RS rl); (*exception THM*)
fun res_fixed rls =
if Thm.maxidx_of (Thm.adjust_maxidx_thm ~1 thm) = ~1 then res thm rls
else Variable.trade (K (fn [thm'] => res thm' rls)) (Variable.thm_context thm) [thm];
in
case concl_of thm
of Const ("Trueprop", _) $ p => (case head_of p
of Const (a, _) => (case AList.lookup (op =) pairs a
of SOME rls => (maps atoms (res_fixed rls) handle THM _ => [thm])
| NONE => [thm])
| _ => [thm])
| _ => [thm]
end;
in atoms end;
fun mksimps pairs =
map_filter (try mk_eq) o mk_atomize pairs o gen_all;
local
val simp_impliesI = thm "simp_impliesI"
val TrueI = thm "TrueI"
val FalseE = thm "FalseE"
val refl = thm "refl"
in fun unsafe_solver_tac prems =
(fn i => REPEAT_DETERM (match_tac [simp_impliesI] i)) THEN'
FIRST'[resolve_tac(reflexive_thm :: TrueI :: refl :: prems), atac, etac FalseE];
end;
val unsafe_solver = mk_solver "HOL unsafe" unsafe_solver_tac;
(*No premature instantiation of variables during simplification*)
local
val simp_impliesI = thm "simp_impliesI"
val TrueI = thm "TrueI"
val FalseE = thm "FalseE"
val refl = thm "refl"
in fun safe_solver_tac prems =
(fn i => REPEAT_DETERM (match_tac [simp_impliesI] i)) THEN'
FIRST'[match_tac(reflexive_thm :: TrueI :: refl :: prems),
eq_assume_tac, ematch_tac [FalseE]];
end;
val safe_solver = mk_solver "HOL safe" safe_solver_tac;
structure SplitterData =
struct
structure Simplifier = Simplifier
val mk_eq = mk_eq
val meta_eq_to_iff = thm "meta_eq_to_obj_eq"
val iffD = thm "iffD2"
val disjE = thm "disjE"
val conjE = thm "conjE"
val exE = thm "exE"
val contrapos = thm "contrapos_nn"
val contrapos2 = thm "contrapos_pp"
val notnotD = thm "notnotD"
end;
structure Splitter = SplitterFun(SplitterData);
(* integration of simplifier with classical reasoner *)
structure Clasimp = ClasimpFun
(structure Simplifier = Simplifier and Splitter = Splitter
and Classical = Classical and Blast = Blast
val iffD1 = thm "iffD1" val iffD2 = thm "iffD2" val notE = thm "notE");
val mksimps_pairs =
[("op -->", [thm "mp"]), ("op &", [thm "conjunct1", thm "conjunct2"]),
("All", [thm "spec"]), ("True", []), ("False", []),
("HOL.If", [thm "if_bool_eq_conj" RS thm "iffD1"])];
val simpset_basic =
Simplifier.theory_context (the_context ()) empty_ss
setsubgoaler asm_simp_tac
setSSolver safe_solver
setSolver unsafe_solver
setmksimps (mksimps mksimps_pairs)
setmkeqTrue mk_eq_True
setmkcong mk_meta_cong;
fun simplify rews = Simplifier.full_simplify (simpset_basic addsimps rews);
fun unfold_tac ths =
let val ss0 = Simplifier.clear_ss simpset_basic addsimps ths
in fn ss => ALLGOALS (full_simp_tac (Simplifier.inherit_context ss ss0)) end;
(** simprocs **)
(* simproc for proving "(y = x) == False" from premise "~(x = y)" *)
val use_neq_simproc = ref true;
local
val thy = the_context ();
val neq_to_EQ_False = thm "not_sym" RS thm "Eq_FalseI";
fun neq_prover sg ss (eq $ lhs $ rhs) =
let
fun test thm = (case #prop (rep_thm thm) of
_ $ (Not $ (eq' $ l' $ r')) =>
Not = HOLogic.Not andalso eq' = eq andalso
r' aconv lhs andalso l' aconv rhs
| _ => false)
in if !use_neq_simproc then case find_first test (prems_of_ss ss)
of NONE => NONE
| SOME thm => SOME (thm RS neq_to_EQ_False)
else NONE
end
in
val neq_simproc = Simplifier.simproc thy "neq_simproc" ["x = y"] neq_prover;
end;
(* simproc for Let *)
val use_let_simproc = ref true;
local
val thy = the_context ();
val Let_folded = thm "Let_folded";
val Let_unfold = thm "Let_unfold";
val Let_def = thm "Let_def";
val (f_Let_unfold, x_Let_unfold) =
let val [(_$(f$x)$_)] = prems_of Let_unfold
in (cterm_of thy f, cterm_of thy x) end
val (f_Let_folded, x_Let_folded) =
let val [(_$(f$x)$_)] = prems_of Let_folded
in (cterm_of thy f, cterm_of thy x) end;
val g_Let_folded =
let val [(_$_$(g$_))] = prems_of Let_folded in cterm_of thy g end;
in
val let_simproc =
Simplifier.simproc thy "let_simp" ["Let x f"]
(fn sg => fn ss => fn t =>
let val ctxt = Simplifier.the_context ss;
val ([t'], ctxt') = Variable.import_terms false [t] ctxt;
in Option.map (hd o Variable.export ctxt' ctxt o single)
(case t' of (Const ("Let",_)$x$f) => (* x and f are already in normal form *)
if not (!use_let_simproc) then NONE
else if is_Free x orelse is_Bound x orelse is_Const x
then SOME Let_def
else
let
val n = case f of (Abs (x,_,_)) => x | _ => "x";
val cx = cterm_of sg x;
val {T=xT,...} = rep_cterm cx;
val cf = cterm_of sg f;
val fx_g = Simplifier.rewrite ss (Thm.capply cf cx);
val (_$_$g) = prop_of fx_g;
val g' = abstract_over (x,g);
in (if (g aconv g')
then
let
val rl = cterm_instantiate [(f_Let_unfold,cf),(x_Let_unfold,cx)] Let_unfold;
in SOME (rl OF [fx_g]) end
else if Term.betapply (f,x) aconv g then NONE (*avoid identity conversion*)
else let
val abs_g'= Abs (n,xT,g');
val g'x = abs_g'$x;
val g_g'x = symmetric (beta_conversion false (cterm_of sg g'x));
val rl = cterm_instantiate
[(f_Let_folded,cterm_of sg f),(x_Let_folded,cx),
(g_Let_folded,cterm_of sg abs_g')]
Let_folded;
in SOME (rl OF [transitive fx_g g_g'x])
end)
end
| _ => NONE)
end)
end;
(* generic refutation procedure *)
(* parameters:
test: term -> bool
tests if a term is at all relevant to the refutation proof;
if not, then it can be discarded. Can improve performance,
esp. if disjunctions can be discarded (no case distinction needed!).
prep_tac: int -> tactic
A preparation tactic to be applied to the goal once all relevant premises
have been moved to the conclusion.
ref_tac: int -> tactic
the actual refutation tactic. Should be able to deal with goals
[| A1; ...; An |] ==> False
where the Ai are atomic, i.e. no top-level &, | or EX
*)
local
val conjE = thm "conjE"
val exE = thm "exE"
val disjE = thm "disjE"
val notE = thm "notE"
val rev_mp = thm "rev_mp"
val ccontr = thm "ccontr"
val nnf_simpset =
empty_ss setmkeqTrue mk_eq_True
setmksimps (mksimps mksimps_pairs)
addsimps [thm "imp_conv_disj", thm "iff_conv_conj_imp", thm "de_Morgan_disj", thm "de_Morgan_conj",
thm "not_all", thm "not_ex", thm "not_not"];
fun prem_nnf_tac i st =
full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset) i st;
in
fun refute_tac test prep_tac ref_tac =
let val refute_prems_tac =
REPEAT_DETERM
(eresolve_tac [conjE, exE] 1 ORELSE
filter_prems_tac test 1 ORELSE
etac disjE 1) THEN
((etac notE 1 THEN eq_assume_tac 1) ORELSE
ref_tac 1);
in EVERY'[TRY o filter_prems_tac test,
REPEAT_DETERM o etac rev_mp, prep_tac, rtac ccontr, prem_nnf_tac,
SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]
end;
end;
val defALL_regroup =
Simplifier.simproc (the_context ())
"defined ALL" ["ALL x. P x"] Quantifier1.rearrange_all;
val defEX_regroup =
Simplifier.simproc (the_context ())
"defined EX" ["EX x. P x"] Quantifier1.rearrange_ex;
val simpset_simprocs = simpset_basic
addsimprocs [defALL_regroup, defEX_regroup, neq_simproc, let_simproc]
end;
structure Splitter = Simpdata.Splitter;
structure Clasimp = Simpdata.Clasimp;