(* Title: HOL/Decision_Procs/mir_tac.ML
Author: Amine Chaieb, TU Muenchen
*)
signature MIR_TAC =
sig
val trace: bool Unsynchronized.ref
val mir_tac: Proof.context -> bool -> int -> tactic
val setup: theory -> theory
end
structure Mir_Tac =
struct
val trace = Unsynchronized.ref false;
fun trace_msg s = if !trace then tracing s else ();
val mir_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 nat_arith = map thm ["add_nat_number_of", "diff_nat_number_of",
"mult_nat_number_of", "eq_nat_number_of", "less_nat_number_of"];
val comp_arith = (map thm ["Let_def", "if_False", "if_True", "add_0",
"add_Suc", "add_number_of_left", "mult_number_of_left",
"Suc_eq_plus1"])@
(map (fn s => thm s RS sym) ["numeral_1_eq_1", "numeral_0_eq_0"])
@ @{thms arith_simps} @ nat_arith @ @{thms rel_simps}
val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
@{thm "real_of_nat_number_of"},
@{thm "real_of_nat_Suc"}, @{thm "real_of_nat_one"}, @{thm "real_of_one"},
@{thm "real_of_int_zero"}, @{thm "real_of_nat_zero"},
@{thm "Fields.divide_zero"},
@{thm "divide_divide_eq_left"}, @{thm "times_divide_eq_right"},
@{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"},
@{thm "diff_def"}, @{thm "minus_divide_left"}]
val comp_ths = ths @ comp_arith @ 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 imp_le_cong = @{thm "imp_le_cong"};
val conj_le_cong = @{thm "conj_le_cong"};
val mod_add_eq = @{thm "mod_add_eq"} RS sym;
val mod_add_left_eq = @{thm "mod_add_left_eq"} RS sym;
val mod_add_right_eq = @{thm "mod_add_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_mir thy 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) = List.foldr (fn ((x, T), (fm,n)) => mk_all ((x, T), (fm,n)))
(List.foldr HOLogic.mk_imp c rhs, np) ps
val (vs, _) = List.partition (fn t => q orelse (type_of t) = nT)
(OldTerm.term_frees fm' @ OldTerm.term_vars fm');
val fm2 = List.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 mir_tac ctxt q i =
Object_Logic.atomize_prems_tac i
THEN simp_tac (HOL_basic_ss addsimps [@{thm "abs_ge_zero"}] addsimps simp_thms) 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_mir thy q g
(* Some simpsets for dealing with mod div abs and nat*)
val mod_div_simpset = HOL_basic_ss
addsimps [refl, mod_add_eq,
@{thm "mod_self"}, @{thm "zmod_self"},
@{thm "zdiv_zero"},@{thm "zmod_zero"},@{thm "div_0"}, @{thm "mod_0"},
@{thm "div_by_1"}, @{thm "mod_by_1"}, @{thm "div_1"}, @{thm "mod_1"},
@{thm "Suc_eq_plus1"}]
addsimps @{thms add_ac}
addsimprocs [cancel_div_mod_nat_proc, cancel_div_mod_int_proc]
val simpset0 = HOL_basic_ss
addsimps [mod_div_equality', @{thm Suc_eq_plus1}]
addsimps comp_ths
addsplits [@{thm "split_zdiv"}, @{thm "split_zmod"}, @{thm "split_div'"}, @{thm "split_min"}, @{thm "split_max"}]
(* Simp rules for changing (n::int) to int n *)
val simpset1 = HOL_basic_ss
addsimps [@{thm "nat_number_of_def"}, @{thm "zdvd_int"}] @ map (fn r => r RS sym)
[@{thm "int_int_eq"}, @{thm "zle_int"}, @{thm "zless_int"}, @{thm "zadd_int"},
@{thm "zmult_int"}]
addsplits [@{thm "zdiff_int_split"}]
(*simp rules for elimination of int n*)
val simpset2 = HOL_basic_ss
addsimps [@{thm "nat_0_le"}, @{thm "all_nat"}, @{thm "ex_nat"}, @{thm "number_of1"},
@{thm "number_of2"}, @{thm "int_0"}, @{thm "int_1"}]
addcongs [@{thm "conj_le_cong"}, @{thm "imp_le_cong"}]
(* simp rules for elimination of abs *)
val ct = cterm_of thy (HOLogic.mk_Trueprop t)
(* Theorem for the nat --> int transformation *)
val pre_thm = Seq.hd (EVERY
[simp_tac mod_div_simpset 1, simp_tac simpset0 1,
TRY (simp_tac simpset1 1), TRY (simp_tac simpset2 1), TRY (simp_tac mir_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 =
(* If quick_and_dirty then run without proof generation as oracle*)
if !quick_and_dirty
then mirfr_oracle (false, cterm_of thy (Pattern.eta_long [] t1))
else mirfr_oracle (true, cterm_of thy (Pattern.eta_long [] t1))
in
(trace_msg ("calling procedure with term:\n" ^
Syntax.string_of_term ctxt 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);
val setup =
Method.setup @{binding mir}
let
val parse_flag = Args.$$$ "no_quantify" >> K (K false)
in
Scan.lift (Scan.optional (Args.$$$ "(" |-- Scan.repeat1 parse_flag --| Args.$$$ ")") [] >>
curry (Library.foldl op |>) true) >>
(fn q => fn ctxt => SIMPLE_METHOD' (mir_tac ctxt q))
end
"decision procedure for MIR arithmetic";
end