23274

1 
structure LinZTac =


2 
struct


3 


4 
val trace = ref false;


5 
fun trace_msg s = if !trace then tracing s else ();


6 


7 
val cooper_ss = @{simpset};


8 


9 
val nT = HOLogic.natT;


10 
val binarith = map thm


11 
["Pls_0_eq", "Min_1_eq"];


12 
val intarithrel =


13 
(map thm ["int_eq_number_of_eq","int_neg_number_of_BIT",


14 
"int_le_number_of_eq","int_iszero_number_of_0",


15 
"int_less_number_of_eq_neg"]) @


16 
(map (fn s => thm s RS thm "lift_bool")


17 
["int_iszero_number_of_Pls","int_iszero_number_of_1",


18 
"int_neg_number_of_Min"])@


19 
(map (fn s => thm s RS thm "nlift_bool")


20 
["int_nonzero_number_of_Min","int_not_neg_number_of_Pls"]);


21 


22 
val intarith = map thm ["int_number_of_add_sym", "int_number_of_minus_sym",


23 
"int_number_of_diff_sym", "int_number_of_mult_sym"];


24 
val natarith = map thm ["add_nat_number_of", "diff_nat_number_of",


25 
"mult_nat_number_of", "eq_nat_number_of",


26 
"less_nat_number_of"]


27 
val powerarith =


28 
(map thm ["nat_number_of", "zpower_number_of_even",


29 
"zpower_Pls", "zpower_Min"]) @


30 
[simplify (HOL_basic_ss addsimps [thm "zero_eq_Numeral0_nring",


31 
thm "one_eq_Numeral1_nring"])


32 
(thm "zpower_number_of_odd")]


33 


34 
val comp_arith = binarith @ intarith @ intarithrel @ natarith


35 
@ powerarith @[thm"not_false_eq_true", thm "not_true_eq_false"];


36 


37 


38 
val zdvd_int = thm "zdvd_int";


39 
val zdiff_int_split = thm "zdiff_int_split";


40 
val all_nat = thm "all_nat";


41 
val ex_nat = thm "ex_nat";


42 
val number_of1 = thm "number_of1";


43 
val number_of2 = thm "number_of2";


44 
val split_zdiv = thm "split_zdiv";


45 
val split_zmod = thm "split_zmod";


46 
val mod_div_equality' = thm "mod_div_equality'";


47 
val split_div' = thm "split_div'";


48 
val Suc_plus1 = thm "Suc_plus1";


49 
val imp_le_cong = thm "imp_le_cong";


50 
val conj_le_cong = thm "conj_le_cong";


51 
val nat_mod_add_eq = mod_add1_eq RS sym;


52 
val nat_mod_add_left_eq = mod_add_left_eq RS sym;


53 
val nat_mod_add_right_eq = mod_add_right_eq RS sym;


54 
val int_mod_add_eq = @{thm "zmod_zadd1_eq"} RS sym;


55 
val int_mod_add_left_eq = @{thm "zmod_zadd_left_eq"} RS sym;


56 
val int_mod_add_right_eq = @{thm "zmod_zadd_right_eq"} RS sym;


57 
val nat_div_add_eq = @{thm "div_add1_eq"} RS sym;


58 
val int_div_add_eq = @{thm "zdiv_zadd1_eq"} RS sym;


59 
val ZDIVISION_BY_ZERO_MOD = @{thm "DIVISION_BY_ZERO"} RS conjunct2;


60 
val ZDIVISION_BY_ZERO_DIV = @{thm "DIVISION_BY_ZERO"} RS conjunct1;


61 


62 
(*


63 
val fn_rews = List.concat (map thms ["allpairs.simps","iupt.simps","decr.simps", "decrnum.simps","disjuncts.simps","simpnum.simps", "simpfm.simps","numadd.simps","nummul.simps","numneg_def","numsub","simp_num_pair_def","not.simps","prep.simps","qelim.simps","minusinf.simps","plusinf.simps","rsplit0.simps","rlfm.simps","\\<Upsilon>.simps","\\<upsilon>.simps","linrqe_def", "ferrack_def", "Let_def", "numsub_def", "numneg_def","DJ_def", "imp_def", "evaldjf_def", "djf_def", "split_def", "eq_def", "disj_def", "simp_num_pair_def", "conj_def", "lt_def", "neq_def","gt_def"]);


64 
*)


65 
fun prepare_for_linz q fm =


66 
let


67 
val ps = Logic.strip_params fm


68 
val hs = map HOLogic.dest_Trueprop (Logic.strip_assums_hyp fm)


69 
val c = HOLogic.dest_Trueprop (Logic.strip_assums_concl fm)


70 
fun mk_all ((s, T), (P,n)) =


71 
if 0 mem loose_bnos P then


72 
(HOLogic.all_const T $ Abs (s, T, P), n)


73 
else (incr_boundvars ~1 P, n1)


74 
fun mk_all2 (v, t) = HOLogic.all_const (fastype_of v) $ lambda v t;


75 
val rhs = hs


76 
(* val (rhs,irhs) = List.partition (relevant (rev ps)) hs *)


77 
val np = length ps


78 
val (fm',np) = foldr (fn ((x, T), (fm,n)) => mk_all ((x, T), (fm,n)))


79 
(foldr HOLogic.mk_imp c rhs, np) ps


80 
val (vs, _) = List.partition (fn t => q orelse (type_of t) = nT)


81 
(term_frees fm' @ term_vars fm');


82 
val fm2 = foldr mk_all2 fm' vs


83 
in (fm2, np + length vs, length rhs) end;


84 


85 
(*Object quantifier to meta *)


86 
fun spec_step n th = if (n=0) then th else (spec_step (n1) th) RS spec ;


87 


88 
(* object implication to meta*)


89 
fun mp_step n th = if (n=0) then th else (mp_step (n1) th) RS mp;


90 


91 


92 
fun linz_tac ctxt q i = ObjectLogic.atomize_tac i THEN (fn st =>


93 
let


94 
val g = List.nth (prems_of st, i  1)


95 
val thy = ProofContext.theory_of ctxt


96 
(* Transform the term*)


97 
val (t,np,nh) = prepare_for_linz q g


98 
(* Some simpsets for dealing with mod div abs and nat*)


99 
val mod_div_simpset = HOL_basic_ss


100 
addsimps [refl,nat_mod_add_eq, nat_mod_add_left_eq,


101 
nat_mod_add_right_eq, int_mod_add_eq,


102 
int_mod_add_right_eq, int_mod_add_left_eq,


103 
nat_div_add_eq, int_div_add_eq,


104 
mod_self, @{thm "zmod_self"},


105 
DIVISION_BY_ZERO_MOD,DIVISION_BY_ZERO_DIV,


106 
ZDIVISION_BY_ZERO_MOD,ZDIVISION_BY_ZERO_DIV,


107 
@{thm "zdiv_zero"}, @{thm "zmod_zero"}, @{thm "div_0"}, @{thm "mod_0"},


108 
@{thm "zdiv_1"}, @{thm "zmod_1"}, @{thm "div_1"}, @{thm "mod_1"},


109 
Suc_plus1]


110 
addsimps add_ac


111 
addsimprocs [cancel_div_mod_proc]


112 
val simpset0 = HOL_basic_ss


113 
addsimps [mod_div_equality', Suc_plus1]


114 
addsimps comp_arith


115 
addsplits [split_zdiv, split_zmod, split_div', @{thm "split_min"}, @{thm "split_max"}]


116 
(* Simp rules for changing (n::int) to int n *)


117 
val simpset1 = HOL_basic_ss


118 
addsimps [nat_number_of_def, zdvd_int] @ map (fn r => r RS sym)


119 
[int_int_eq, zle_int, zless_int, zadd_int, zmult_int]


120 
addsplits [zdiff_int_split]


121 
(*simp rules for elimination of int n*)


122 


123 
val simpset2 = HOL_basic_ss


124 
addsimps [nat_0_le, all_nat, ex_nat, number_of1, number_of2, int_0, int_1]


125 
addcongs [conj_le_cong, imp_le_cong]


126 
(* simp rules for elimination of abs *)


127 
val simpset3 = HOL_basic_ss addsplits [abs_split]


128 
val ct = cterm_of thy (HOLogic.mk_Trueprop t)


129 
(* Theorem for the nat > int transformation *)


130 
val pre_thm = Seq.hd (EVERY


131 
[simp_tac mod_div_simpset 1, simp_tac simpset0 1,


132 
TRY (simp_tac simpset1 1), TRY (simp_tac simpset2 1),


133 
TRY (simp_tac simpset3 1), TRY (simp_tac cooper_ss 1)]


134 
(trivial ct))


135 
fun assm_tac i = REPEAT_DETERM_N nh (assume_tac i)


136 
(* The result of the quantifier elimination *)


137 
val (th, tac) = case (prop_of pre_thm) of


138 
Const ("==>", _) $ (Const ("Trueprop", _) $ t1) $ _ =>


139 
let val pth = linzqe_oracle thy (Pattern.eta_long [] t1)


140 
in


141 
((pth RS iffD2) RS pre_thm,


142 
assm_tac (i + 1) THEN (if q then I else TRY) (rtac TrueI i))


143 
end


144 
 _ => (pre_thm, assm_tac i)


145 
in (rtac (((mp_step nh) o (spec_step np)) th) i


146 
THEN tac) st


147 
end handle Subscript => no_tac st);


148 


149 
fun linz_args meth =


150 
let val parse_flag =


151 
Args.$$$ "no_quantify" >> (K (K false));


152 
in


153 
Method.simple_args


154 
(Scan.optional (Args.$$$ "("  Scan.repeat1 parse_flag  Args.$$$ ")") [] >>


155 
curry (Library.foldl op >) true)


156 
(fn q => fn ctxt => meth ctxt q 1)


157 
end;


158 


159 
fun linz_method ctxt q i = Method.METHOD (fn facts =>


160 
Method.insert_tac facts 1 THEN linz_tac ctxt q i);


161 


162 
val setup =


163 
Method.add_method ("cooper",


164 
linz_args linz_method,


165 
"decision procedure for linear integer arithmetic");


166 


167 
end 