(* 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
fun mk_meta_eq r = r RS @{thm eq_reflection};
fun safe_mk_meta_eq r = mk_meta_eq r handle Thm.THM _ => r;
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 @{thm Eq_FalseI}
| _ => th RS @{thm Eq_TrueI}
fun mk_eq_True r =
SOME (r RS @{thm meta_eq_to_obj_eq} RS @{thm 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
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 @{thm 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 @{thms simp_implies_def},
REPEAT (ares_tac (@{thm meta_eq_to_obj_eq} ::
map (rewrite_rule @{thms 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);
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;
fun unsafe_solver_tac prems =
(fn i => REPEAT_DETERM (match_tac @{thms simp_impliesI} i)) THEN'
FIRST' [resolve_tac (reflexive_thm :: @{thm TrueI} :: @{thm refl} :: prems), atac,
etac @{thm 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 @{thms simp_impliesI} i)) THEN'
FIRST' [match_tac (reflexive_thm :: @{thm TrueI} :: @{thm refl} :: prems),
eq_assume_tac, ematch_tac @{thms FalseE}];
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);
val split_tac = Splitter.split_tac;
val split_inside_tac = Splitter.split_inside_tac;
val op addsplits = Splitter.addsplits;
val op delsplits = Splitter.delsplits;
val Addsplits = Splitter.Addsplits;
val Delsplits = Splitter.Delsplits;
(* 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});
open Clasimp;
val _ = ML_Context.value_antiq "clasimpset"
(Scan.succeed ("clasimpset", "Clasimp.local_clasimpset_of (ML_Context.the_local_context ())"));
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 HOL_basic_ss =
Simplifier.theory_context @{theory} 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 hol_simplify rews = Simplifier.full_simplify (HOL_basic_ss addsimps rews);
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;
(** simprocs **)
(* 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 @{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 @{theory} "neq_simproc" ["x = y"] neq_prover;
end;
(* simproc for Let *)
val use_let_simproc = ref true;
local
val (f_Let_unfold, x_Let_unfold) =
let val [(_$(f$x)$_)] = prems_of @{thm Let_unfold}
in (cterm_of @{theory} f, cterm_of @{theory} x) end
val (f_Let_folded, x_Let_folded) =
let val [(_$(f$x)$_)] = prems_of @{thm Let_folded}
in (cterm_of @{theory} f, cterm_of @{theory} x) end;
val g_Let_folded =
let val [(_$_$(g$_))] = prems_of @{thm Let_folded} in cterm_of @{theory} g end;
in
val let_simproc =
Simplifier.simproc @{theory} "let_simp" ["Let x f"]
(fn thy => fn ss => fn t =>
let val ctxt = Simplifier.the_context ss;
val ([t'], ctxt') = Variable.import_terms false [t] ctxt;
in Option.map (hd o Variable.export ctxt' ctxt o single)
(case t' of (Const ("Let",_)$x$f) => (* x and f are already in normal form *)
if not (!use_let_simproc) then NONE
else if is_Free x orelse is_Bound x orelse is_Const x
then SOME @{thm Let_def}
else
let
val n = case f of (Abs (x,_,_)) => x | _ => "x";
val cx = cterm_of thy x;
val {T=xT,...} = rep_cterm cx;
val cf = cterm_of thy f;
val fx_g = Simplifier.rewrite ss (Thm.capply cf cx);
val (_$_$g) = prop_of fx_g;
val g' = abstract_over (x,g);
in (if (g aconv g')
then
let
val rl =
cterm_instantiate [(f_Let_unfold,cf),(x_Let_unfold,cx)] @{thm Let_unfold};
in SOME (rl OF [fx_g]) end
else if Term.betapply (f,x) aconv g then NONE (*avoid identity conversion*)
else let
val abs_g'= Abs (n,xT,g');
val g'x = abs_g'$x;
val g_g'x = symmetric (beta_conversion false (cterm_of thy g'x));
val rl = cterm_instantiate
[(f_Let_folded,cterm_of thy f),(x_Let_folded,cx),
(g_Let_folded,cterm_of thy abs_g')]
@{thm 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 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 [@{thm conjE}, @{thm exE}] 1 ORELSE
filter_prems_tac test 1 ORELSE
etac @{thm disjE} 1) THEN
((etac @{thm notE} 1 THEN eq_assume_tac 1) ORELSE
ref_tac 1);
in EVERY'[TRY o filter_prems_tac test,
REPEAT_DETERM o etac @{thm rev_mp}, prep_tac, rtac @{thm ccontr}, prem_nnf_tac,
SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]
end;
end;
val defALL_regroup =
Simplifier.simproc @{theory}
"defined ALL" ["ALL x. P x"] Quantifier1.rearrange_all;
val defEX_regroup =
Simplifier.simproc @{theory}
"defined EX" ["EX x. P x"] Quantifier1.rearrange_ex;
val simpset_simprocs = HOL_basic_ss
addsimprocs [defALL_regroup, defEX_regroup, neq_simproc, let_simproc]
end;
structure Splitter = Simpdata.Splitter;
structure Clasimp = Simpdata.Clasimp;