author  haftmann 
Wed, 03 Dec 2008 15:58:44 +0100  
changeset 28952  15a4b2cf8c34 
parent 28262  src/HOL/int_arith1.ML@aa7ca36d67fd 
child 29269  5c25a2012975 
permissions  rwrr 
23164  1 
(* Title: HOL/int_arith1.ML 
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ID: $Id$ 

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Authors: Larry Paulson and Tobias Nipkow 

4 

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Simprocs and decision procedure for linear arithmetic. 

6 
*) 

7 

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structure Int_Numeral_Base_Simprocs = 

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struct 

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fun prove_conv tacs ctxt (_: thm list) (t, u) = 

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if t aconv u then NONE 

12 
else 

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let val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (t, u)) 

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in SOME (Goal.prove ctxt [] [] eq (K (EVERY tacs))) end 

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fun prove_conv_nohyps tacs sg = prove_conv tacs sg []; 

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fun prep_simproc (name, pats, proc) = 

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Simplifier.simproc (the_context()) name pats proc; 

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fun is_numeral (Const(@{const_name Int.number_of}, _) $ w) = true 
23164  22 
 is_numeral _ = false 
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fun simplify_meta_eq f_number_of_eq f_eq = 

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mk_meta_eq ([f_eq, f_number_of_eq] MRS trans) 

26 

27 
(*reorientation simprules using ==, for the following simproc*) 

23881  28 
val meta_zero_reorient = @{thm zero_reorient} RS eq_reflection 
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val meta_one_reorient = @{thm one_reorient} RS eq_reflection 

25481  30 
val meta_number_of_reorient = @{thm number_of_reorient} RS eq_reflection 
23164  31 

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(*reorientation simplification procedure: reorients (polymorphic) 

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0 = x, 1 = x, nnn = x provided x isn't 0, 1 or a Int.*) 
23164  34 
fun reorient_proc sg _ (_ $ t $ u) = 
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case u of 

25481  36 
Const(@{const_name HOL.zero}, _) => NONE 
23164  37 
 Const(@{const_name HOL.one}, _) => NONE 
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 Const(@{const_name Int.number_of}, _) $ _ => NONE 
23164  39 
 _ => SOME (case t of 
25481  40 
Const(@{const_name HOL.zero}, _) => meta_zero_reorient 
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 Const(@{const_name HOL.one}, _) => meta_one_reorient 

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 Const(@{const_name Int.number_of}, _) $ _ => meta_number_of_reorient) 
23164  43 

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val reorient_simproc = 

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prep_simproc ("reorient_simproc", ["0=x", "1=x", "number_of w = x"], reorient_proc) 

46 

47 
end; 

48 

49 

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Addsimprocs [Int_Numeral_Base_Simprocs.reorient_simproc]; 

51 

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structure Int_Numeral_Simprocs = 

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struct 

55 

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(*Maps 0 to Numeral0 and 1 to Numeral1 so that arithmetic in Int_Numeral_Base_Simprocs 

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isn't complicated by the abstract 0 and 1.*) 

25481  58 
val numeral_syms = [@{thm numeral_0_eq_0} RS sym, @{thm numeral_1_eq_1} RS sym]; 
23164  59 

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(** New term ordering so that ACrewriting brings numerals to the front **) 

61 

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(*Order integers by absolute value and then by sign. The standard integer 

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ordering is not wellfounded.*) 

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fun num_ord (i,j) = 

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(case int_ord (abs i, abs j) of 
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EQUAL => int_ord (Int.sign i, Int.sign j) 
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 ord => ord); 
23164  68 

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(*This resembles Term.term_ord, but it puts binary numerals before other 

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nonatomic terms.*) 

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local open Term 

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in 

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fun numterm_ord (Abs (_, T, t), Abs(_, U, u)) = 

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(case numterm_ord (t, u) of EQUAL => typ_ord (T, U)  ord => ord) 

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 numterm_ord 

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(Const(@{const_name Int.number_of}, _) $ v, Const(@{const_name Int.number_of}, _) $ w) = 
23164  77 
num_ord (HOLogic.dest_numeral v, HOLogic.dest_numeral w) 
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 numterm_ord (Const(@{const_name Int.number_of}, _) $ _, _) = LESS 
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 numterm_ord (_, Const(@{const_name Int.number_of}, _) $ _) = GREATER 
23164  80 
 numterm_ord (t, u) = 
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(case int_ord (size_of_term t, size_of_term u) of 

82 
EQUAL => 

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let val (f, ts) = strip_comb t and (g, us) = strip_comb u in 

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(case hd_ord (f, g) of EQUAL => numterms_ord (ts, us)  ord => ord) 

85 
end 

86 
 ord => ord) 

87 
and numterms_ord (ts, us) = list_ord numterm_ord (ts, us) 

88 
end; 

89 

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fun numtermless tu = (numterm_ord tu = LESS); 

91 

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(*Defined in this file, but perhaps needed only for Int_Numeral_Base_Simprocs of type nat.*) 

93 
val num_ss = HOL_ss settermless numtermless; 

94 

95 

96 
(** Utilities **) 

97 

98 
fun mk_number T n = HOLogic.number_of_const T $ HOLogic.mk_numeral n; 

99 

100 
fun find_first_numeral past (t::terms) = 

101 
((snd (HOLogic.dest_number t), rev past @ terms) 

102 
handle TERM _ => find_first_numeral (t::past) terms) 

103 
 find_first_numeral past [] = raise TERM("find_first_numeral", []); 

104 

105 
val mk_plus = HOLogic.mk_binop @{const_name HOL.plus}; 

106 

107 
fun mk_minus t = 

108 
let val T = Term.fastype_of t 

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in Const (@{const_name HOL.uminus}, T > T) $ t end; 
23164  110 

111 
(*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*) 

112 
fun mk_sum T [] = mk_number T 0 

113 
 mk_sum T [t,u] = mk_plus (t, u) 

114 
 mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts); 

115 

116 
(*this version ALWAYS includes a trailing zero*) 

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fun long_mk_sum T [] = mk_number T 0 

118 
 long_mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts); 

119 

120 
val dest_plus = HOLogic.dest_bin @{const_name HOL.plus} Term.dummyT; 

121 

122 
(*decompose additions AND subtractions as a sum*) 

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fun dest_summing (pos, Const (@{const_name HOL.plus}, _) $ t $ u, ts) = 

124 
dest_summing (pos, t, dest_summing (pos, u, ts)) 

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 dest_summing (pos, Const (@{const_name HOL.minus}, _) $ t $ u, ts) = 

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dest_summing (pos, t, dest_summing (not pos, u, ts)) 

127 
 dest_summing (pos, t, ts) = 

128 
if pos then t::ts else mk_minus t :: ts; 

129 

130 
fun dest_sum t = dest_summing (true, t, []); 

131 

132 
val mk_diff = HOLogic.mk_binop @{const_name HOL.minus}; 

133 
val dest_diff = HOLogic.dest_bin @{const_name HOL.minus} Term.dummyT; 

134 

135 
val mk_times = HOLogic.mk_binop @{const_name HOL.times}; 

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fun one_of T = Const(@{const_name HOL.one},T); 
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(* build product with trailing 1 rather than Numeral 1 in order to avoid the 
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unnecessary restriction to type class number_ring 
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which is not required for cancellation of common factors in divisions. 
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*) 
23164  143 
fun mk_prod T = 
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let val one = one_of T 
23164  145 
fun mk [] = one 
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 mk [t] = t 

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 mk (t :: ts) = if t = one then mk ts else mk_times (t, mk ts) 

148 
in mk end; 

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(*This version ALWAYS includes a trailing one*) 

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fun long_mk_prod T [] = one_of T 
23164  152 
 long_mk_prod T (t :: ts) = mk_times (t, mk_prod T ts); 
153 

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val dest_times = HOLogic.dest_bin @{const_name HOL.times} Term.dummyT; 

155 

156 
fun dest_prod t = 

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let val (t,u) = dest_times t 

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in dest_prod t @ dest_prod u end 
23164  159 
handle TERM _ => [t]; 
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(*DON'T do the obvious simplifications; that would create special cases*) 

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fun mk_coeff (k, t) = mk_times (mk_number (Term.fastype_of t) k, t); 

163 

164 
(*Express t as a product of (possibly) a numeral with other sorted terms*) 

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fun dest_coeff sign (Const (@{const_name HOL.uminus}, _) $ t) = dest_coeff (~sign) t 

166 
 dest_coeff sign t = 

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let val ts = sort Term.term_ord (dest_prod t) 

168 
val (n, ts') = find_first_numeral [] ts 

169 
handle TERM _ => (1, ts) 

170 
in (sign*n, mk_prod (Term.fastype_of t) ts') end; 

171 

172 
(*Find first coefficientterm THAT MATCHES u*) 

173 
fun find_first_coeff past u [] = raise TERM("find_first_coeff", []) 

174 
 find_first_coeff past u (t::terms) = 

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let val (n,u') = dest_coeff 1 t 

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in if u aconv u' then (n, rev past @ terms) 
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else find_first_coeff (t::past) u terms 
23164  178 
end 
179 
handle TERM _ => find_first_coeff (t::past) u terms; 

180 

181 
(*Fractions as pairs of ints. Can't use Rat.rat because the representation 

182 
needs to preserve negative values in the denominator.*) 

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fun mk_frac (p, q) = if q = 0 then raise Div else (p, q); 
23164  184 

185 
(*Don't reduce fractions; sums must be proved by rule add_frac_eq. 

186 
Fractions are reduced later by the cancel_numeral_factor simproc.*) 

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fun add_frac ((p1, q1), (p2, q2)) = (p1 * q2 + p2 * q1, q1 * q2); 
23164  188 

189 
val mk_divide = HOLogic.mk_binop @{const_name HOL.divide}; 

190 

191 
(*Build term (p / q) * t*) 

192 
fun mk_fcoeff ((p, q), t) = 

193 
let val T = Term.fastype_of t 

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in mk_times (mk_divide (mk_number T p, mk_number T q), t) end; 
23164  195 

196 
(*Express t as a product of a fraction with other sorted terms*) 

197 
fun dest_fcoeff sign (Const (@{const_name HOL.uminus}, _) $ t) = dest_fcoeff (~sign) t 

198 
 dest_fcoeff sign (Const (@{const_name HOL.divide}, _) $ t $ u) = 

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let val (p, t') = dest_coeff sign t 

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val (q, u') = dest_coeff 1 u 

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in (mk_frac (p, q), mk_divide (t', u')) end 
23164  202 
 dest_fcoeff sign t = 
203 
let val (p, t') = dest_coeff sign t 

204 
val T = Term.fastype_of t 

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in (mk_frac (p, 1), mk_divide (t', one_of T)) end; 
23164  206 

207 

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(*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1, 1*x, x*1, x/1 *) 
23164  209 
val add_0s = thms "add_0s"; 
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val mult_1s = thms "mult_1s" @ [thm"mult_1_left", thm"mult_1_right", thm"divide_1"]; 
23164  211 

212 
(*Simplify inverse Numeral1, a/Numeral1*) 

213 
val inverse_1s = [@{thm inverse_numeral_1}]; 

214 
val divide_1s = [@{thm divide_numeral_1}]; 

215 

216 
(*To perform binary arithmetic. The "left" rewriting handles patterns 

217 
created by the Int_Numeral_Base_Simprocs, such as 3 * (5 * x). *) 

25481  218 
val simps = [@{thm numeral_0_eq_0} RS sym, @{thm numeral_1_eq_1} RS sym, 
219 
@{thm add_number_of_left}, @{thm mult_number_of_left}] @ 

220 
@{thms arith_simps} @ @{thms rel_simps}; 

23164  221 

222 
(*Binary arithmetic BUT NOT ADDITION since it may collapse adjacent terms 

223 
during rearrangement*) 

224 
val non_add_simps = 

25481  225 
subtract Thm.eq_thm [@{thm add_number_of_left}, @{thm number_of_add} RS sym] simps; 
23164  226 

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(*To evaluate binary negations of coefficients*) 

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val minus_simps = [@{thm numeral_m1_eq_minus_1} RS sym, @{thm number_of_minus} RS sym] @ 
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@{thms minus_bin_simps} @ @{thms pred_bin_simps}; 
23164  230 

231 
(*To let us treat subtraction as addition*) 

232 
val diff_simps = [@{thm diff_minus}, @{thm minus_add_distrib}, @{thm minus_minus}]; 

233 

234 
(*To let us treat division as multiplication*) 

235 
val divide_simps = [@{thm divide_inverse}, @{thm inverse_mult_distrib}, @{thm inverse_inverse_eq}]; 

236 

237 
(*push the unary minus down:  x * y = x *  y *) 

238 
val minus_mult_eq_1_to_2 = 

239 
[@{thm minus_mult_left} RS sym, @{thm minus_mult_right}] MRS trans > standard; 

240 

241 
(*to extract again any uncancelled minuses*) 

242 
val minus_from_mult_simps = 

243 
[@{thm minus_minus}, @{thm minus_mult_left} RS sym, @{thm minus_mult_right} RS sym]; 

244 

245 
(*combine unary minus with numeric literals, however nested within a product*) 

246 
val mult_minus_simps = 

247 
[@{thm mult_assoc}, @{thm minus_mult_left}, minus_mult_eq_1_to_2]; 

248 

249 
(*Apply the given rewrite (if present) just once*) 

250 
fun trans_tac NONE = all_tac 

251 
 trans_tac (SOME th) = ALLGOALS (rtac (th RS trans)); 

252 

253 
fun simplify_meta_eq rules = 

254 
let val ss0 = HOL_basic_ss addeqcongs [eq_cong2] addsimps rules 

255 
in fn ss => simplify (Simplifier.inherit_context ss ss0) o mk_meta_eq end 

256 

257 
structure CancelNumeralsCommon = 

258 
struct 

259 
val mk_sum = mk_sum 

260 
val dest_sum = dest_sum 

261 
val mk_coeff = mk_coeff 

262 
val dest_coeff = dest_coeff 1 

263 
val find_first_coeff = find_first_coeff [] 

264 
val trans_tac = fn _ => trans_tac 

265 

266 
val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @ 

23881  267 
diff_simps @ minus_simps @ @{thms add_ac} 
23164  268 
val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps 
23881  269 
val norm_ss3 = num_ss addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms mult_ac} 
23164  270 
fun norm_tac ss = 
271 
ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1)) 

272 
THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2)) 

273 
THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3)) 

274 

275 
val numeral_simp_ss = HOL_ss addsimps add_0s @ simps 

276 
fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss)) 

277 
val simplify_meta_eq = simplify_meta_eq (add_0s @ mult_1s) 

278 
end; 

279 

280 

281 
structure EqCancelNumerals = CancelNumeralsFun 

282 
(open CancelNumeralsCommon 

283 
val prove_conv = Int_Numeral_Base_Simprocs.prove_conv 

284 
val mk_bal = HOLogic.mk_eq 

285 
val dest_bal = HOLogic.dest_bin "op =" Term.dummyT 

25481  286 
val bal_add1 = @{thm eq_add_iff1} RS trans 
287 
val bal_add2 = @{thm eq_add_iff2} RS trans 

23164  288 
); 
289 

290 
structure LessCancelNumerals = CancelNumeralsFun 

291 
(open CancelNumeralsCommon 

292 
val prove_conv = Int_Numeral_Base_Simprocs.prove_conv 

23881  293 
val mk_bal = HOLogic.mk_binrel @{const_name HOL.less} 
294 
val dest_bal = HOLogic.dest_bin @{const_name HOL.less} Term.dummyT 

25481  295 
val bal_add1 = @{thm less_add_iff1} RS trans 
296 
val bal_add2 = @{thm less_add_iff2} RS trans 

23164  297 
); 
298 

299 
structure LeCancelNumerals = CancelNumeralsFun 

300 
(open CancelNumeralsCommon 

301 
val prove_conv = Int_Numeral_Base_Simprocs.prove_conv 

23881  302 
val mk_bal = HOLogic.mk_binrel @{const_name HOL.less_eq} 
303 
val dest_bal = HOLogic.dest_bin @{const_name HOL.less_eq} Term.dummyT 

25481  304 
val bal_add1 = @{thm le_add_iff1} RS trans 
305 
val bal_add2 = @{thm le_add_iff2} RS trans 

23164  306 
); 
307 

308 
val cancel_numerals = 

309 
map Int_Numeral_Base_Simprocs.prep_simproc 

310 
[("inteq_cancel_numerals", 

311 
["(l::'a::number_ring) + m = n", 

312 
"(l::'a::number_ring) = m + n", 

313 
"(l::'a::number_ring)  m = n", 

314 
"(l::'a::number_ring) = m  n", 

315 
"(l::'a::number_ring) * m = n", 

316 
"(l::'a::number_ring) = m * n"], 

317 
K EqCancelNumerals.proc), 

318 
("intless_cancel_numerals", 

319 
["(l::'a::{ordered_idom,number_ring}) + m < n", 

320 
"(l::'a::{ordered_idom,number_ring}) < m + n", 

321 
"(l::'a::{ordered_idom,number_ring})  m < n", 

322 
"(l::'a::{ordered_idom,number_ring}) < m  n", 

323 
"(l::'a::{ordered_idom,number_ring}) * m < n", 

324 
"(l::'a::{ordered_idom,number_ring}) < m * n"], 

325 
K LessCancelNumerals.proc), 

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("intle_cancel_numerals", 

327 
["(l::'a::{ordered_idom,number_ring}) + m <= n", 

328 
"(l::'a::{ordered_idom,number_ring}) <= m + n", 

329 
"(l::'a::{ordered_idom,number_ring})  m <= n", 

330 
"(l::'a::{ordered_idom,number_ring}) <= m  n", 

331 
"(l::'a::{ordered_idom,number_ring}) * m <= n", 

332 
"(l::'a::{ordered_idom,number_ring}) <= m * n"], 

333 
K LeCancelNumerals.proc)]; 

334 

335 

336 
structure CombineNumeralsData = 

337 
struct 

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type coeff = int 
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val iszero = (fn x => x = 0) 
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val add = op + 
23164  341 
val mk_sum = long_mk_sum (*to work for e.g. 2*x + 3*x *) 
342 
val dest_sum = dest_sum 

343 
val mk_coeff = mk_coeff 

344 
val dest_coeff = dest_coeff 1 

25481  345 
val left_distrib = @{thm combine_common_factor} RS trans 
23164  346 
val prove_conv = Int_Numeral_Base_Simprocs.prove_conv_nohyps 
347 
val trans_tac = fn _ => trans_tac 

348 

349 
val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @ 

23881  350 
diff_simps @ minus_simps @ @{thms add_ac} 
23164  351 
val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps 
23881  352 
val norm_ss3 = num_ss addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms mult_ac} 
23164  353 
fun norm_tac ss = 
354 
ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1)) 

355 
THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2)) 

356 
THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3)) 

357 

358 
val numeral_simp_ss = HOL_ss addsimps add_0s @ simps 

359 
fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss)) 

360 
val simplify_meta_eq = simplify_meta_eq (add_0s @ mult_1s) 

361 
end; 

362 

363 
structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData); 

364 

365 
(*Version for fields, where coefficients can be fractions*) 

366 
structure FieldCombineNumeralsData = 

367 
struct 

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368 
type coeff = int * int 
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val iszero = (fn (p, q) => p = 0) 
23164  370 
val add = add_frac 
371 
val mk_sum = long_mk_sum 

372 
val dest_sum = dest_sum 

373 
val mk_coeff = mk_fcoeff 

374 
val dest_coeff = dest_fcoeff 1 

25481  375 
val left_distrib = @{thm combine_common_factor} RS trans 
23164  376 
val prove_conv = Int_Numeral_Base_Simprocs.prove_conv_nohyps 
377 
val trans_tac = fn _ => trans_tac 

378 

379 
val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @ 

23881  380 
inverse_1s @ divide_simps @ diff_simps @ minus_simps @ @{thms add_ac} 
23164  381 
val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps 
23881  382 
val norm_ss3 = num_ss addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms mult_ac} 
23164  383 
fun norm_tac ss = 
384 
ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1)) 

385 
THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2)) 

386 
THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3)) 

387 

388 
val numeral_simp_ss = HOL_ss addsimps add_0s @ simps @ [@{thm add_frac_eq}] 

389 
fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss)) 

390 
val simplify_meta_eq = simplify_meta_eq (add_0s @ mult_1s @ divide_1s) 

391 
end; 

392 

393 
structure FieldCombineNumerals = CombineNumeralsFun(FieldCombineNumeralsData); 

394 

395 
val combine_numerals = 

396 
Int_Numeral_Base_Simprocs.prep_simproc 

397 
("int_combine_numerals", 

398 
["(i::'a::number_ring) + j", "(i::'a::number_ring)  j"], 

399 
K CombineNumerals.proc); 

400 

401 
val field_combine_numerals = 

402 
Int_Numeral_Base_Simprocs.prep_simproc 

403 
("field_combine_numerals", 

404 
["(i::'a::{number_ring,field,division_by_zero}) + j", 

405 
"(i::'a::{number_ring,field,division_by_zero})  j"], 

406 
K FieldCombineNumerals.proc); 

407 

408 
end; 

409 

410 
Addsimprocs Int_Numeral_Simprocs.cancel_numerals; 

411 
Addsimprocs [Int_Numeral_Simprocs.combine_numerals]; 

412 
Addsimprocs [Int_Numeral_Simprocs.field_combine_numerals]; 

413 

414 
(*examples: 

415 
print_depth 22; 

416 
set timing; 

417 
set trace_simp; 

418 
fun test s = (Goal s, by (Simp_tac 1)); 

419 

420 
test "l + 2 + 2 + 2 + (l + 2) + (oo + 2) = (uu::int)"; 

421 

422 
test "2*u = (u::int)"; 

423 
test "(i + j + 12 + (k::int))  15 = y"; 

424 
test "(i + j + 12 + (k::int))  5 = y"; 

425 

426 
test "y  b < (b::int)"; 

427 
test "y  (3*b + c) < (b::int)  2*c"; 

428 

429 
test "(2*x  (u*v) + y)  v*3*u = (w::int)"; 

430 
test "(2*x*u*v + (u*v)*4 + y)  v*u*4 = (w::int)"; 

431 
test "(2*x*u*v + (u*v)*4 + y)  v*u = (w::int)"; 

432 
test "u*v  (x*u*v + (u*v)*4 + y) = (w::int)"; 

433 

434 
test "(i + j + 12 + (k::int)) = u + 15 + y"; 

435 
test "(i + j*2 + 12 + (k::int)) = j + 5 + y"; 

436 

437 
test "2*y + 3*z + 6*w + 2*y + 3*z + 2*u = 2*y' + 3*z' + 6*w' + 2*y' + 3*z' + u + (vv::int)"; 

438 

439 
test "a + (b+c) + b = (d::int)"; 

440 
test "a + (b+c)  b = (d::int)"; 

441 

442 
(*negative numerals*) 

443 
test "(i + j + 2 + (k::int))  (u + 5 + y) = zz"; 

444 
test "(i + j + 3 + (k::int)) < u + 5 + y"; 

445 
test "(i + j + 3 + (k::int)) < u + 6 + y"; 

446 
test "(i + j + 12 + (k::int))  15 = y"; 

447 
test "(i + j + 12 + (k::int))  15 = y"; 

448 
test "(i + j + 12 + (k::int))  15 = y"; 

449 
*) 

450 

451 

452 
(** Constant folding for multiplication in semirings **) 

453 

454 
(*We do not need folding for addition: combine_numerals does the same thing*) 

455 

456 
structure Semiring_Times_Assoc_Data : ASSOC_FOLD_DATA = 

457 
struct 

23881  458 
val assoc_ss = HOL_ss addsimps @{thms mult_ac} 
23164  459 
val eq_reflection = eq_reflection 
460 
end; 

461 

462 
structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data); 

463 

464 
val assoc_fold_simproc = 

465 
Int_Numeral_Base_Simprocs.prep_simproc 

466 
("semiring_assoc_fold", ["(a::'a::comm_semiring_1_cancel) * b"], 

467 
K Semiring_Times_Assoc.proc); 

468 

469 
Addsimprocs [assoc_fold_simproc]; 

470 

471 

472 

473 

474 
(*** decision procedure for linear arithmetic ***) 

475 

476 
(**) 

477 
(* Linear arithmetic *) 

478 
(**) 

479 

480 
(* 

481 
Instantiation of the generic linear arithmetic package for int. 

482 
*) 

483 

484 
(* Update parameters of arithmetic prover *) 

485 
local 

486 

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487 
(* reduce contradictory =/</<= to False *) 
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488 

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489 
(* Evaluation of terms of the form "m R n" where R is one of "=", "<=" or "<", 
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490 
and m and n are ground terms over rings (roughly speaking). 
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491 
That is, m and n consist only of 1s combined with "+", "" and "*". 
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492 
*) 
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493 
local 
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494 
val zeroth = (symmetric o mk_meta_eq) @{thm of_int_0}; 
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495 
val lhss0 = [@{cpat "0::?'a::ring"}]; 
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496 
fun proc0 phi ss ct = 
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497 
let val T = ctyp_of_term ct 
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498 
in if typ_of T = @{typ int} then NONE else 
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499 
SOME (instantiate' [SOME T] [] zeroth) 
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500 
end; 
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501 
val zero_to_of_int_zero_simproc = 
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502 
make_simproc {lhss = lhss0, name = "zero_to_of_int_zero_simproc", 
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503 
proc = proc0, identifier = []}; 
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504 

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505 
val oneth = (symmetric o mk_meta_eq) @{thm of_int_1}; 
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506 
val lhss1 = [@{cpat "1::?'a::ring_1"}]; 
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507 
fun proc1 phi ss ct = 
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508 
let val T = ctyp_of_term ct 
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509 
in if typ_of T = @{typ int} then NONE else 
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510 
SOME (instantiate' [SOME T] [] oneth) 
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511 
end; 
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512 
val one_to_of_int_one_simproc = 
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513 
make_simproc {lhss = lhss1, name = "one_to_of_int_one_simproc", 
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514 
proc = proc1, identifier = []}; 
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515 

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516 
val allowed_consts = 
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517 
[@{const_name "op ="}, @{const_name "HOL.times"}, @{const_name "HOL.uminus"}, 
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518 
@{const_name "HOL.minus"}, @{const_name "HOL.plus"}, 
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519 
@{const_name "HOL.zero"}, @{const_name "HOL.one"}, @{const_name "HOL.less"}, 
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520 
@{const_name "HOL.less_eq"}]; 
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521 

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522 
fun check t = case t of 
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523 
Const(s,t) => if s = @{const_name "HOL.one"} then not (t = @{typ int}) 
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524 
else s mem_string allowed_consts 
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525 
 a$b => check a andalso check b 
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526 
 _ => false; 
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527 

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528 
val conv = 
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529 
Simplifier.rewrite 
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530 
(HOL_basic_ss addsimps 
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531 
((map (fn th => th RS sym) [@{thm of_int_add}, @{thm of_int_mult}, 
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532 
@{thm of_int_diff}, @{thm of_int_minus}])@ 
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533 
[@{thm of_int_less_iff}, @{thm of_int_le_iff}, @{thm of_int_eq_iff}]) 
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534 
addsimprocs [zero_to_of_int_zero_simproc,one_to_of_int_one_simproc]); 
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535 

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536 
fun sproc phi ss ct = if check (term_of ct) then SOME (conv ct) else NONE 
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537 
val lhss' = 
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538 
[@{cpat "(?x::?'a::ring_char_0) = (?y::?'a)"}, 
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539 
@{cpat "(?x::?'a::ordered_idom) < (?y::?'a)"}, 
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540 
@{cpat "(?x::?'a::ordered_idom) <= (?y::?'a)"}] 
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541 
in 
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542 
val zero_one_idom_simproc = 
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543 
make_simproc {lhss = lhss' , name = "zero_one_idom_simproc", 
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544 
proc = sproc, identifier = []} 
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545 
end; 
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546 

23164  547 
val add_rules = 
25481  548 
simp_thms @ @{thms arith_simps} @ @{thms rel_simps} @ @{thms arith_special} @ 
23164  549 
[@{thm neg_le_iff_le}, @{thm numeral_0_eq_0}, @{thm numeral_1_eq_1}, 
550 
@{thm minus_zero}, @{thm diff_minus}, @{thm left_minus}, @{thm right_minus}, 

26086
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New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
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551 
@{thm mult_zero_left}, @{thm mult_zero_right}, @{thm mult_Bit1}, @{thm mult_1_right}, 
23164  552 
@{thm minus_mult_left} RS sym, @{thm minus_mult_right} RS sym, 
553 
@{thm minus_add_distrib}, @{thm minus_minus}, @{thm mult_assoc}, 

23365  554 
@{thm of_nat_0}, @{thm of_nat_1}, @{thm of_nat_Suc}, @{thm of_nat_add}, 
555 
@{thm of_nat_mult}, @{thm of_int_0}, @{thm of_int_1}, @{thm of_int_add}, 

556 
@{thm of_int_mult}] 

23164  557 

23365  558 
val nat_inj_thms = [@{thm zle_int} RS iffD2, @{thm int_int_eq} RS iffD2] 
23164  559 

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560 
val Int_Numeral_Base_Simprocs = assoc_fold_simproc :: zero_one_idom_simproc 
23164  561 
:: Int_Numeral_Simprocs.combine_numerals 
562 
:: Int_Numeral_Simprocs.cancel_numerals; 

563 

564 
in 

565 

566 
val int_arith_setup = 

24093  567 
LinArith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} => 
23164  568 
{add_mono_thms = add_mono_thms, 
569 
mult_mono_thms = @{thm mult_strict_left_mono} :: @{thm mult_left_mono} :: mult_mono_thms, 

570 
inj_thms = nat_inj_thms @ inj_thms, 

25481  571 
lessD = lessD @ [@{thm zless_imp_add1_zle}], 
23164  572 
neqE = neqE, 
573 
simpset = simpset addsimps add_rules 

574 
addsimprocs Int_Numeral_Base_Simprocs 

575 
addcongs [if_weak_cong]}) #> 

24196  576 
arith_inj_const (@{const_name of_nat}, HOLogic.natT > HOLogic.intT) #> 
25919
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577 
arith_discrete @{type_name Int.int} 
23164  578 

579 
end; 

580 

581 
val fast_int_arith_simproc = 

28262
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back to dynamic the_context(), because static @{theory} is invalidated if ML environment changes within the same code block;
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582 
Simplifier.simproc (the_context ()) 
23164  583 
"fast_int_arith" 
584 
["(m::'a::{ordered_idom,number_ring}) < n", 

585 
"(m::'a::{ordered_idom,number_ring}) <= n", 

24093  586 
"(m::'a::{ordered_idom,number_ring}) = n"] (K LinArith.lin_arith_simproc); 
23164  587 

588 
Addsimprocs [fast_int_arith_simproc]; 