structure LinrTac =
struct
val trace = ref false;
fun trace_msg s = if !trace then tracing s else ();
val ferrack_ss = let val ths = map thm ["real_of_int_inject", "real_of_int_less_iff",
"real_of_int_le_iff"]
in @{simpset} delsimps ths addsimps (map (fn th => th RS sym) ths)
end;
val nT = HOLogic.natT;
val binarith = map thm
["Pls_0_eq", "Min_1_eq",
"pred_Pls","pred_Min","pred_1","pred_0",
"succ_Pls", "succ_Min", "succ_1", "succ_0",
"add_Pls", "add_Min", "add_BIT_0", "add_BIT_10",
"add_BIT_11", "minus_Pls", "minus_Min", "minus_1",
"minus_0", "mult_Pls", "mult_Min", "mult_num1", "mult_num0",
"add_Pls_right", "add_Min_right"];
val comp_arith = binarith @ simp_thms
val zdvd_int = thm "zdvd_int";
val zdiff_int_split = thm "zdiff_int_split";
val all_nat = thm "all_nat";
val ex_nat = thm "ex_nat";
val number_of1 = thm "number_of1";
val number_of2 = thm "number_of2";
val split_zdiv = thm "split_zdiv";
val split_zmod = thm "split_zmod";
val mod_div_equality' = thm "mod_div_equality'";
val split_div' = thm "split_div'";
val Suc_plus1 = thm "Suc_plus1";
val imp_le_cong = thm "imp_le_cong";
val conj_le_cong = thm "conj_le_cong";
val nat_mod_add_eq = @{thm mod_add1_eq} RS sym;
val nat_mod_add_left_eq = @{thm mod_add_left_eq} RS sym;
val nat_mod_add_right_eq = @{thm mod_add_right_eq} RS sym;
val int_mod_add_eq = @{thm "zmod_zadd1_eq"} RS sym;
val int_mod_add_left_eq = @{thm "zmod_zadd_left_eq"} RS sym;
val int_mod_add_right_eq = @{thm "zmod_zadd_right_eq"} RS sym;
val nat_div_add_eq = @{thm "div_add1_eq"} RS sym;
val int_div_add_eq = @{thm "zdiv_zadd1_eq"} RS sym;
val ZDIVISION_BY_ZERO_MOD = @{thm "DIVISION_BY_ZERO"} RS conjunct2;
val ZDIVISION_BY_ZERO_DIV = @{thm "DIVISION_BY_ZERO"} RS conjunct1;
fun prepare_for_linr sg q fm =
let
val ps = Logic.strip_params fm
val hs = map HOLogic.dest_Trueprop (Logic.strip_assums_hyp fm)
val c = HOLogic.dest_Trueprop (Logic.strip_assums_concl fm)
fun mk_all ((s, T), (P,n)) =
if 0 mem loose_bnos P then
(HOLogic.all_const T $ Abs (s, T, P), n)
else (incr_boundvars ~1 P, n-1)
fun mk_all2 (v, t) = HOLogic.all_const (fastype_of v) $ lambda v t;
val rhs = hs
(* val (rhs,irhs) = List.partition (relevant (rev ps)) hs *)
val np = length ps
val (fm',np) = foldr (fn ((x, T), (fm,n)) => mk_all ((x, T), (fm,n)))
(foldr HOLogic.mk_imp c rhs, np) ps
val (vs, _) = List.partition (fn t => q orelse (type_of t) = nT)
(term_frees fm' @ term_vars fm');
val fm2 = foldr mk_all2 fm' vs
in (fm2, np + length vs, length rhs) end;
(*Object quantifier to meta --*)
fun spec_step n th = if (n=0) then th else (spec_step (n-1) th) RS spec ;
(* object implication to meta---*)
fun mp_step n th = if (n=0) then th else (mp_step (n-1) th) RS mp;
fun linr_tac ctxt q i =
(ObjectLogic.atomize_prems_tac i)
THEN (REPEAT_DETERM (split_tac [@{thm "split_min"}, @{thm "split_max"}, @{thm "abs_split"}] i))
THEN (fn st =>
let
val g = List.nth (prems_of st, i - 1)
val thy = ProofContext.theory_of ctxt
(* Transform the term*)
val (t,np,nh) = prepare_for_linr thy q g
(* Some simpsets for dealing with mod div abs and nat*)
val simpset0 = Simplifier.theory_context thy HOL_basic_ss addsimps comp_arith
val ct = cterm_of thy (HOLogic.mk_Trueprop t)
(* Theorem for the nat --> int transformation *)
val pre_thm = Seq.hd (EVERY
[simp_tac simpset0 1,
TRY (simp_tac (Simplifier.theory_context thy ferrack_ss) 1)]
(trivial ct))
fun assm_tac i = REPEAT_DETERM_N nh (assume_tac i)
(* The result of the quantifier elimination *)
val (th, tac) = case (prop_of pre_thm) of
Const ("==>", _) $ (Const ("Trueprop", _) $ t1) $ _ =>
let val pth = linr_oracle thy (Pattern.eta_long [] t1)
in
(trace_msg ("calling procedure with term:\n" ^
Sign.string_of_term thy t1);
((pth RS iffD2) RS pre_thm,
assm_tac (i + 1) THEN (if q then I else TRY) (rtac TrueI i)))
end
| _ => (pre_thm, assm_tac i)
in (rtac (((mp_step nh) o (spec_step np)) th) i
THEN tac) st
end handle Subscript => no_tac st | ReflectedFerrack.LINR => no_tac st);
fun linr_meth src =
Method.syntax (Args.mode "no_quantify") src
#> (fn (q, ctxt) => Method.SIMPLE_METHOD' (linr_tac ctxt (not q)));
val setup =
Method.add_method ("rferrack", linr_meth,
"decision procedure for linear real arithmetic");
end