(* Title: HOL/simpdata.ML
ID: $Id$
Author: Tobias Nipkow
Copyright 1991 University of Cambridge
Instantiation of the generic simplifier for HOL.
*)
(* legacy ML bindings - FIXME get rid of this *)
val Eq_FalseI = thm "Eq_FalseI";
val Eq_TrueI = thm "Eq_TrueI";
val de_Morgan_conj = thm "de_Morgan_conj";
val de_Morgan_disj = thm "de_Morgan_disj";
val iff_conv_conj_imp = thm "iff_conv_conj_imp";
val imp_cong = thm "imp_cong";
val imp_conv_disj = thm "imp_conv_disj";
val imp_disj1 = thm "imp_disj1";
val imp_disj2 = thm "imp_disj2";
val imp_disjL = thm "imp_disjL";
val simp_impliesI = thm "simp_impliesI";
val simp_implies_cong = thm "simp_implies_cong";
val simp_implies_def = thm "simp_implies_def";
local
val uncurry = thm "uncurry"
val iff_allI = thm "iff_allI"
val iff_exI = thm "iff_exI"
val all_comm = thm "all_comm"
val ex_comm = thm "ex_comm"
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 = HOLogic.conj
val imp = HOLogic.imp
(*rules*)
val iff_reflection = HOL.eq_reflection
val iffI = HOL.iffI
val iff_trans = HOL.trans
val conjI= HOL.conjI
val conjE= HOL.conjE
val impI = HOL.impI
val mp = HOL.mp
val uncurry = uncurry
val exI = HOL.exI
val exE = HOL.exE
val iff_allI = iff_allI
val iff_exI = iff_exI
val all_comm = all_comm
val ex_comm = ex_comm
end);
end;
val defEX_regroup =
Simplifier.simproc (the_context ())
"defined EX" ["EX x. P x"] Quantifier1.rearrange_ex;
val defALL_regroup =
Simplifier.simproc (the_context ())
"defined ALL" ["ALL x. P x"] Quantifier1.rearrange_all;
(* simproc for proving "(y = x) == False" from premise "~(x = y)" *)
val use_neq_simproc = ref true;
local
val neq_to_EQ_False = thm "not_sym" RS 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 (the_context ())
"neq_simproc" ["x = y"] neq_prover;
end;
(* Simproc for Let *)
val use_let_simproc = ref true;
local
val Let_folded = thm "Let_folded";
val Let_unfold = thm "Let_unfold";
val (f_Let_unfold,x_Let_unfold) =
let val [(_$(f$x)$_)] = prems_of Let_unfold
in (cterm_of (the_context ()) f,cterm_of (the_context ()) x) end
val (f_Let_folded,x_Let_folded) =
let val [(_$(f$x)$_)] = prems_of Let_folded
in (cterm_of (the_context ()) f, cterm_of (the_context ()) x) end;
val g_Let_folded =
let val [(_$_$(g$_))] = prems_of Let_folded in cterm_of (the_context ()) g end;
in
val let_simproc =
Simplifier.simproc (the_context ()) "let_simp" ["Let x f"]
(fn sg => fn ss => fn t =>
let val ctxt = Simplifier.the_context ss;
val ([t'],ctxt') = Variable.import_terms false [t] ctxt;
in Option.map (hd o Variable.export ctxt' ctxt o single)
(case t' of (Const ("Let",_)$x$f) => (* x and f are already in normal form *)
if not (!use_let_simproc) then NONE
else if is_Free x orelse is_Bound x orelse is_Const x
then SOME (thm "Let_def")
else
let
val n = case f of (Abs (x,_,_)) => x | _ => "x";
val cx = cterm_of sg x;
val {T=xT,...} = rep_cterm cx;
val cf = cterm_of sg f;
val fx_g = Simplifier.rewrite ss (Thm.capply cf cx);
val (_$_$g) = prop_of fx_g;
val g' = abstract_over (x,g);
in (if (g aconv g')
then
let
val rl = cterm_instantiate [(f_Let_unfold,cf),(x_Let_unfold,cx)] Let_unfold;
in SOME (rl OF [fx_g]) end
else if Term.betapply (f,x) aconv g then NONE (*avoid identity conversion*)
else let
val abs_g'= Abs (n,xT,g');
val g'x = abs_g'$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
(*** Case splitting ***)
(*Make meta-equalities. The operator below is Trueprop*)
fun mk_meta_eq r = r RS HOL.eq_reflection;
fun safe_mk_meta_eq r = mk_meta_eq r handle Thm.THM _ => r;
fun mk_eq th = case concl_of th of
Const("==",_)$_$_ => th
| _$(Const("op =",_)$_$_) => mk_meta_eq th
| _$(Const("Not",_)$_) => th RS Eq_FalseI
| _ => th RS Eq_TrueI;
(* Expects Trueprop(.) if not == *)
fun mk_eq_True r =
SOME (r RS HOL.meta_eq_to_obj_eq RS Eq_TrueI) handle Thm.THM _ => NONE;
(* Produce theorems of the form
(P1 =simp=> ... =simp=> Pn => x == y) ==> (P1 =simp=> ... =simp=> Pn => x = y)
*)
fun lift_meta_eq_to_obj_eq i st =
let
val {sign, ...} = rep_thm st;
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 HOL.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 (HOL.meta_eq_to_obj_eq :: map (rewrite_rule [simp_implies_def]) prems) 1)])
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);
structure SplitterData =
struct
structure Simplifier = Simplifier
val mk_eq = mk_eq
val meta_eq_to_iff = HOL.meta_eq_to_obj_eq
val iffD = HOL.iffD2
val disjE = HOL.disjE
val conjE = HOL.conjE
val exE = HOL.exE
val contrapos = HOL.contrapos_nn
val contrapos2 = HOL.contrapos_pp
val notnotD = HOL.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;
val mksimps_pairs =
[("op -->", [HOL.mp]), ("op &", [thm "conjunct1", thm "conjunct2"]),
("All", [HOL.spec]), ("True", []), ("False", []),
("HOL.If", [thm "if_bool_eq_conj" RS HOL.iffD1])];
(*
val mk_atomize: (string * thm list) list -> thm -> thm list
looks too specific to move it somewhere else
*)
fun mk_atomize pairs =
let fun atoms th =
(case concl_of th of
Const("Trueprop",_) $ p =>
(case head_of p of
Const(a,_) =>
(case AList.lookup (op =) pairs a of
SOME(rls) => List.concat (map atoms ([th] RL rls))
| NONE => [th])
| _ => [th])
| _ => [th])
in atoms end;
fun mksimps pairs =
(List.mapPartial (try mk_eq) o mk_atomize pairs o gen_all);
fun unsafe_solver_tac prems =
(fn i => REPEAT_DETERM (match_tac [simp_impliesI] i)) THEN'
FIRST'[resolve_tac(reflexive_thm :: HOL.TrueI :: HOL.refl :: prems), atac, etac HOL.FalseE];
val unsafe_solver = mk_solver "HOL unsafe" unsafe_solver_tac;
(*No premature instantiation of variables during simplification*)
fun safe_solver_tac prems =
(fn i => REPEAT_DETERM (match_tac [simp_impliesI] i)) THEN'
FIRST'[match_tac(reflexive_thm :: HOL.TrueI :: HOL.refl :: prems),
eq_assume_tac, ematch_tac [HOL.FalseE]];
val safe_solver = mk_solver "HOL safe" safe_solver_tac;
val HOL_basic_ss =
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 unfold_tac ths =
let val ss0 = Simplifier.clear_ss HOL_basic_ss addsimps ths
in fn ss => ALLGOALS (full_simp_tac (Simplifier.inherit_context ss ss0)) end;
(*In general it seems wrong to add distributive laws by default: they
might cause exponential blow-up. But imp_disjL has been in 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 ex_simps = thms "ex_simps";
val all_simps = thms "all_simps";
val simp_thms = thms "simp_thms";
val cases_simp = thm "cases_simp";
val conj_assoc = thm "conj_assoc";
val if_False = thm "if_False";
val if_True = thm "if_True";
val disj_assoc = thm "disj_assoc";
val disj_not1 = thm "disj_not1";
val if_cancel = thm "if_cancel";
val if_eq_cancel = thm "if_eq_cancel";
val True_implies_equals = thm "True_implies_equals";
in
val HOL_ss =
HOL_basic_ss addsimps
([triv_forall_equality, (* prunes params *)
True_implies_equals, (* prune asms `True' *)
if_True, if_False, if_cancel, if_eq_cancel,
imp_disjL, conj_assoc, disj_assoc,
de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2, thm "not_imp",
disj_not1, thm "not_all", thm "not_ex", cases_simp,
thm "the_eq_trivial", HOL.the_sym_eq_trivial]
@ ex_simps @ all_simps @ simp_thms)
addsimprocs [defALL_regroup,defEX_regroup,neq_simproc,let_simproc]
addcongs [imp_cong, simp_implies_cong]
addsplits [thm "split_if"];
end;
fun hol_simplify rews = Simplifier.full_simplify (HOL_basic_ss addsimps rews);
(* default simpset *)
val simpsetup =
(fn thy => (change_simpset_of thy (fn _ => HOL_ss); thy));
(*** integration of simplifier with classical reasoner ***)
structure Clasimp = ClasimpFun
(structure Simplifier = Simplifier and Splitter = Splitter
and Classical = Classical and Blast = Blast
val iffD1 = HOL.iffD1 val iffD2 = HOL.iffD2 val notE = HOL.notE);
open Clasimp;
val HOL_css = (HOL_cs, HOL_ss);
(*** A general 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 nnf_simpset =
empty_ss setmkeqTrue mk_eq_True
setmksimps (mksimps mksimps_pairs)
addsimps [imp_conv_disj,iff_conv_conj_imp,de_Morgan_disj,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 [HOL.conjE, HOL.exE] 1 ORELSE
filter_prems_tac test 1 ORELSE
etac HOL.disjE 1) THEN
((etac HOL.notE 1 THEN eq_assume_tac 1) ORELSE
ref_tac 1);
in EVERY'[TRY o filter_prems_tac test,
REPEAT_DETERM o etac HOL.rev_mp, prep_tac, rtac HOL.ccontr, prem_nnf_tac,
SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]
end;
end;