(* Title: HOL/Decision_Procs/mir_tac.ML
Author: Amine Chaieb, TU Muenchen
*)
signature MIR_TAC =
sig
val mir_tac: Proof.context -> bool -> int -> tactic
end
structure Mir_Tac: MIR_TAC =
struct
val mir_ss =
simpset_of (@{context} delsimps [@{thm "of_int_eq_iff"}, @{thm "of_int_less_iff"}, @{thm "of_int_le_iff"}]
addsimps @{thms "iff_real_of_int"});
val nT = HOLogic.natT;
val nat_arith = [@{thm diff_nat_numeral}];
val comp_arith = [@{thm "Let_def"}, @{thm "if_False"}, @{thm "if_True"}, @{thm "add_0"},
@{thm "add_Suc"}, @{thm add_numeral_left}, @{thm mult_numeral_left(1)}] @
(map (fn th => th RS sym) [@{thm "numeral_One"}])
@ @{thms arith_simps} @ nat_arith @ @{thms rel_simps}
val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
@{thm of_nat_numeral},
@{thm "of_nat_Suc"}, @{thm "of_nat_1"},
@{thm "of_int_0"}, @{thm "of_nat_0"},
@{thm "div_by_0"},
@{thm "divide_divide_eq_left"}, @{thm "times_divide_eq_right"},
@{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"},
@{thm uminus_add_conv_diff [symmetric]}, @{thm "minus_divide_left"}]
val comp_ths = distinct Thm.eq_thm (ths @ comp_arith @ @{thms simp_thms});
fun prepare_for_mir 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 Term.is_dependent 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)
(Misc_Legacy.term_frees fm' @ Misc_Legacy.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 =
Object_Logic.atomize_prems_tac ctxt
THEN' simp_tac (put_simpset HOL_basic_ss ctxt
addsimps [@{thm "abs_ge_zero"}] addsimps @{thms simp_thms})
THEN' (REPEAT_DETERM o split_tac ctxt [@{thm "split_min"}, @{thm "split_max"}, @{thm "abs_split"}])
THEN' SUBGOAL (fn (g, i) =>
let
(* Transform the term*)
val (t,np,nh) = prepare_for_mir q g
(* Some simpsets for dealing with mod div abs and nat*)
val mod_div_simpset = put_simpset HOL_basic_ss ctxt
addsimps [refl, @{thm mod_add_eq},
@{thm mod_self},
@{thm div_0}, @{thm mod_0},
@{thm div_by_1}, @{thm mod_by_1}, @{thm div_by_Suc_0}, @{thm mod_by_Suc_0},
@{thm "Suc_eq_plus1"}]
addsimps @{thms add.assoc add.commute add.left_commute}
addsimprocs [@{simproc cancel_div_mod_nat}, @{simproc cancel_div_mod_int}]
val simpset0 = put_simpset HOL_basic_ss ctxt
addsimps @{thms minus_div_mult_eq_mod [symmetric] Suc_eq_plus1}
addsimps comp_ths
|> fold Splitter.add_split
[@{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 = put_simpset HOL_basic_ss ctxt
addsimps [@{thm "zdvd_int"}, @{thm "of_nat_add"}, @{thm "of_nat_mult"}] @
map (fn r => r RS sym) [@{thm "int_int_eq"}, @{thm "zle_int"}, @{thm "of_nat_less_iff"}, @{thm nat_numeral}]
|> Splitter.add_split @{thm "zdiff_int_split"}
(*simp rules for elimination of int n*)
val simpset2 = put_simpset HOL_basic_ss ctxt
addsimps [@{thm "nat_0_le"}, @{thm "all_nat"}, @{thm "ex_nat"}, @{thm zero_le_numeral},
@{thm "of_nat_0"}, @{thm "of_nat_1"}]
|> fold Simplifier.add_cong [@{thm "conj_le_cong"}, @{thm "imp_le_cong"}]
(* simp rules for elimination of abs *)
val ct = Thm.cterm_of ctxt (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 (put_simpset mir_ss ctxt) 1)]
(Thm.trivial ct))
fun assm_tac i = REPEAT_DETERM_N nh (assume_tac ctxt i)
(* The result of the quantifier elimination *)
val (th, tac) =
case Thm.prop_of pre_thm of
Const (@{const_name Pure.imp}, _) $ (Const (@{const_name Trueprop}, _) $ t1) $ _ =>
let
val pth = mirfr_oracle (ctxt, Envir.eta_long [] t1)
in
((pth RS iffD2) RS pre_thm,
assm_tac (i + 1) THEN (if q then I else TRY) (resolve_tac ctxt [TrueI] i))
end
| _ => (pre_thm, assm_tac i)
in resolve_tac ctxt [((mp_step nh) o (spec_step np)) th] i THEN tac end);
end