1(* Author: Various *) 2 3section \<open>Combination and Cancellation Simprocs for Numeral Expressions\<close> 4 5theory Numeral_Simprocs 6imports Divides 7begin 8 9ML_file "~~/src/Provers/Arith/assoc_fold.ML" 10ML_file "~~/src/Provers/Arith/cancel_numerals.ML" 11ML_file "~~/src/Provers/Arith/combine_numerals.ML" 12ML_file "~~/src/Provers/Arith/cancel_numeral_factor.ML" 13ML_file "~~/src/Provers/Arith/extract_common_term.ML" 14 15lemmas semiring_norm = 16 Let_def arith_simps diff_nat_numeral rel_simps 17 if_False if_True 18 add_0 add_Suc add_numeral_left 19 add_neg_numeral_left mult_numeral_left 20 numeral_One [symmetric] uminus_numeral_One [symmetric] Suc_eq_plus1 21 eq_numeral_iff_iszero not_iszero_Numeral1 22 23declare split_div [of _ _ "numeral k", arith_split] for k 24declare split_mod [of _ _ "numeral k", arith_split] for k 25 26text \<open>For \<open>combine_numerals\<close>\<close> 27 28lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)" 29by (simp add: add_mult_distrib) 30 31text \<open>For \<open>cancel_numerals\<close>\<close> 32 33lemma nat_diff_add_eq1: 34 "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)" 35by (simp split: nat_diff_split add: add_mult_distrib) 36 37lemma nat_diff_add_eq2: 38 "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))" 39by (simp split: nat_diff_split add: add_mult_distrib) 40 41lemma nat_eq_add_iff1: 42 "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)" 43by (auto split: nat_diff_split simp add: add_mult_distrib) 44 45lemma nat_eq_add_iff2: 46 "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)" 47by (auto split: nat_diff_split simp add: add_mult_distrib) 48 49lemma nat_less_add_iff1: 50 "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)" 51by (auto split: nat_diff_split simp add: add_mult_distrib) 52 53lemma nat_less_add_iff2: 54 "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)" 55by (auto split: nat_diff_split simp add: add_mult_distrib) 56 57lemma nat_le_add_iff1: 58 "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)" 59by (auto split: nat_diff_split simp add: add_mult_distrib) 60 61lemma nat_le_add_iff2: 62 "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)" 63by (auto split: nat_diff_split simp add: add_mult_distrib) 64 65text \<open>For \<open>cancel_numeral_factors\<close>\<close> 66 67lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)" 68by auto 69 70lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)" 71by auto 72 73lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)" 74by auto 75 76lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)" 77by auto 78 79lemma nat_mult_dvd_cancel_disj[simp]: 80 "(k*m) dvd (k*n) = (k=0 \<or> m dvd (n::nat))" 81by (auto simp: dvd_eq_mod_eq_0 mod_mult_mult1) 82 83lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)" 84by(auto) 85 86text \<open>For \<open>cancel_factor\<close>\<close> 87 88lemmas nat_mult_le_cancel_disj = mult_le_cancel1 89 90lemmas nat_mult_less_cancel_disj = mult_less_cancel1 91 92lemma nat_mult_eq_cancel_disj: 93 fixes k m n :: nat 94 shows "k * m = k * n \<longleftrightarrow> k = 0 \<or> m = n" 95 by auto 96 97lemma nat_mult_div_cancel_disj [simp]: 98 fixes k m n :: nat 99 shows "(k * m) div (k * n) = (if k = 0 then 0 else m div n)" 100 by (fact div_mult_mult1_if) 101 102lemma numeral_times_minus_swap: 103 fixes x:: "'a::comm_ring_1" shows "numeral w * -x = x * - numeral w" 104 by (simp add: mult.commute) 105 106ML_file "Tools/numeral_simprocs.ML" 107 108simproc_setup semiring_assoc_fold 109 ("(a::'a::comm_semiring_1_cancel) * b") = 110 \<open>fn phi => Numeral_Simprocs.assoc_fold\<close> 111 112(* TODO: see whether the type class can be generalized further *) 113simproc_setup int_combine_numerals 114 ("(i::'a::comm_ring_1) + j" | "(i::'a::comm_ring_1) - j") = 115 \<open>fn phi => Numeral_Simprocs.combine_numerals\<close> 116 117simproc_setup field_combine_numerals 118 ("(i::'a::{field,ring_char_0}) + j" 119 |"(i::'a::{field,ring_char_0}) - j") = 120 \<open>fn phi => Numeral_Simprocs.field_combine_numerals\<close> 121 122simproc_setup inteq_cancel_numerals 123 ("(l::'a::comm_ring_1) + m = n" 124 |"(l::'a::comm_ring_1) = m + n" 125 |"(l::'a::comm_ring_1) - m = n" 126 |"(l::'a::comm_ring_1) = m - n" 127 |"(l::'a::comm_ring_1) * m = n" 128 |"(l::'a::comm_ring_1) = m * n" 129 |"- (l::'a::comm_ring_1) = m" 130 |"(l::'a::comm_ring_1) = - m") = 131 \<open>fn phi => Numeral_Simprocs.eq_cancel_numerals\<close> 132 133simproc_setup intless_cancel_numerals 134 ("(l::'a::linordered_idom) + m < n" 135 |"(l::'a::linordered_idom) < m + n" 136 |"(l::'a::linordered_idom) - m < n" 137 |"(l::'a::linordered_idom) < m - n" 138 |"(l::'a::linordered_idom) * m < n" 139 |"(l::'a::linordered_idom) < m * n" 140 |"- (l::'a::linordered_idom) < m" 141 |"(l::'a::linordered_idom) < - m") = 142 \<open>fn phi => Numeral_Simprocs.less_cancel_numerals\<close> 143 144simproc_setup intle_cancel_numerals 145 ("(l::'a::linordered_idom) + m \<le> n" 146 |"(l::'a::linordered_idom) \<le> m + n" 147 |"(l::'a::linordered_idom) - m \<le> n" 148 |"(l::'a::linordered_idom) \<le> m - n" 149 |"(l::'a::linordered_idom) * m \<le> n" 150 |"(l::'a::linordered_idom) \<le> m * n" 151 |"- (l::'a::linordered_idom) \<le> m" 152 |"(l::'a::linordered_idom) \<le> - m") = 153 \<open>fn phi => Numeral_Simprocs.le_cancel_numerals\<close> 154 155simproc_setup ring_eq_cancel_numeral_factor 156 ("(l::'a::{idom,ring_char_0}) * m = n" 157 |"(l::'a::{idom,ring_char_0}) = m * n") = 158 \<open>fn phi => Numeral_Simprocs.eq_cancel_numeral_factor\<close> 159 160simproc_setup ring_less_cancel_numeral_factor 161 ("(l::'a::linordered_idom) * m < n" 162 |"(l::'a::linordered_idom) < m * n") = 163 \<open>fn phi => Numeral_Simprocs.less_cancel_numeral_factor\<close> 164 165simproc_setup ring_le_cancel_numeral_factor 166 ("(l::'a::linordered_idom) * m <= n" 167 |"(l::'a::linordered_idom) <= m * n") = 168 \<open>fn phi => Numeral_Simprocs.le_cancel_numeral_factor\<close> 169 170(* TODO: remove comm_ring_1 constraint if possible *) 171simproc_setup int_div_cancel_numeral_factors 172 ("((l::'a::{euclidean_semiring_cancel,comm_ring_1,ring_char_0}) * m) div n" 173 |"(l::'a::{euclidean_semiring_cancel,comm_ring_1,ring_char_0}) div (m * n)") = 174 \<open>fn phi => Numeral_Simprocs.div_cancel_numeral_factor\<close> 175 176simproc_setup divide_cancel_numeral_factor 177 ("((l::'a::{field,ring_char_0}) * m) / n" 178 |"(l::'a::{field,ring_char_0}) / (m * n)" 179 |"((numeral v)::'a::{field,ring_char_0}) / (numeral w)") = 180 \<open>fn phi => Numeral_Simprocs.divide_cancel_numeral_factor\<close> 181 182simproc_setup ring_eq_cancel_factor 183 ("(l::'a::idom) * m = n" | "(l::'a::idom) = m * n") = 184 \<open>fn phi => Numeral_Simprocs.eq_cancel_factor\<close> 185 186simproc_setup linordered_ring_le_cancel_factor 187 ("(l::'a::linordered_idom) * m <= n" 188 |"(l::'a::linordered_idom) <= m * n") = 189 \<open>fn phi => Numeral_Simprocs.le_cancel_factor\<close> 190 191simproc_setup linordered_ring_less_cancel_factor 192 ("(l::'a::linordered_idom) * m < n" 193 |"(l::'a::linordered_idom) < m * n") = 194 \<open>fn phi => Numeral_Simprocs.less_cancel_factor\<close> 195 196simproc_setup int_div_cancel_factor 197 ("((l::'a::euclidean_semiring_cancel) * m) div n" 198 |"(l::'a::euclidean_semiring_cancel) div (m * n)") = 199 \<open>fn phi => Numeral_Simprocs.div_cancel_factor\<close> 200 201simproc_setup int_mod_cancel_factor 202 ("((l::'a::euclidean_semiring_cancel) * m) mod n" 203 |"(l::'a::euclidean_semiring_cancel) mod (m * n)") = 204 \<open>fn phi => Numeral_Simprocs.mod_cancel_factor\<close> 205 206simproc_setup dvd_cancel_factor 207 ("((l::'a::idom) * m) dvd n" 208 |"(l::'a::idom) dvd (m * n)") = 209 \<open>fn phi => Numeral_Simprocs.dvd_cancel_factor\<close> 210 211simproc_setup divide_cancel_factor 212 ("((l::'a::field) * m) / n" 213 |"(l::'a::field) / (m * n)") = 214 \<open>fn phi => Numeral_Simprocs.divide_cancel_factor\<close> 215 216ML_file "Tools/nat_numeral_simprocs.ML" 217 218simproc_setup nat_combine_numerals 219 ("(i::nat) + j" | "Suc (i + j)") = 220 \<open>fn phi => Nat_Numeral_Simprocs.combine_numerals\<close> 221 222simproc_setup nateq_cancel_numerals 223 ("(l::nat) + m = n" | "(l::nat) = m + n" | 224 "(l::nat) * m = n" | "(l::nat) = m * n" | 225 "Suc m = n" | "m = Suc n") = 226 \<open>fn phi => Nat_Numeral_Simprocs.eq_cancel_numerals\<close> 227 228simproc_setup natless_cancel_numerals 229 ("(l::nat) + m < n" | "(l::nat) < m + n" | 230 "(l::nat) * m < n" | "(l::nat) < m * n" | 231 "Suc m < n" | "m < Suc n") = 232 \<open>fn phi => Nat_Numeral_Simprocs.less_cancel_numerals\<close> 233 234simproc_setup natle_cancel_numerals 235 ("(l::nat) + m \<le> n" | "(l::nat) \<le> m + n" | 236 "(l::nat) * m \<le> n" | "(l::nat) \<le> m * n" | 237 "Suc m \<le> n" | "m \<le> Suc n") = 238 \<open>fn phi => Nat_Numeral_Simprocs.le_cancel_numerals\<close> 239 240simproc_setup natdiff_cancel_numerals 241 ("((l::nat) + m) - n" | "(l::nat) - (m + n)" | 242 "(l::nat) * m - n" | "(l::nat) - m * n" | 243 "Suc m - n" | "m - Suc n") = 244 \<open>fn phi => Nat_Numeral_Simprocs.diff_cancel_numerals\<close> 245 246simproc_setup nat_eq_cancel_numeral_factor 247 ("(l::nat) * m = n" | "(l::nat) = m * n") = 248 \<open>fn phi => Nat_Numeral_Simprocs.eq_cancel_numeral_factor\<close> 249 250simproc_setup nat_less_cancel_numeral_factor 251 ("(l::nat) * m < n" | "(l::nat) < m * n") = 252 \<open>fn phi => Nat_Numeral_Simprocs.less_cancel_numeral_factor\<close> 253 254simproc_setup nat_le_cancel_numeral_factor 255 ("(l::nat) * m <= n" | "(l::nat) <= m * n") = 256 \<open>fn phi => Nat_Numeral_Simprocs.le_cancel_numeral_factor\<close> 257 258simproc_setup nat_div_cancel_numeral_factor 259 ("((l::nat) * m) div n" | "(l::nat) div (m * n)") = 260 \<open>fn phi => Nat_Numeral_Simprocs.div_cancel_numeral_factor\<close> 261 262simproc_setup nat_dvd_cancel_numeral_factor 263 ("((l::nat) * m) dvd n" | "(l::nat) dvd (m * n)") = 264 \<open>fn phi => Nat_Numeral_Simprocs.dvd_cancel_numeral_factor\<close> 265 266simproc_setup nat_eq_cancel_factor 267 ("(l::nat) * m = n" | "(l::nat) = m * n") = 268 \<open>fn phi => Nat_Numeral_Simprocs.eq_cancel_factor\<close> 269 270simproc_setup nat_less_cancel_factor 271 ("(l::nat) * m < n" | "(l::nat) < m * n") = 272 \<open>fn phi => Nat_Numeral_Simprocs.less_cancel_factor\<close> 273 274simproc_setup nat_le_cancel_factor 275 ("(l::nat) * m <= n" | "(l::nat) <= m * n") = 276 \<open>fn phi => Nat_Numeral_Simprocs.le_cancel_factor\<close> 277 278simproc_setup nat_div_cancel_factor 279 ("((l::nat) * m) div n" | "(l::nat) div (m * n)") = 280 \<open>fn phi => Nat_Numeral_Simprocs.div_cancel_factor\<close> 281 282simproc_setup nat_dvd_cancel_factor 283 ("((l::nat) * m) dvd n" | "(l::nat) dvd (m * n)") = 284 \<open>fn phi => Nat_Numeral_Simprocs.dvd_cancel_factor\<close> 285 286declaration \<open> 287 K (Lin_Arith.add_simprocs 288 [@{simproc semiring_assoc_fold}, 289 @{simproc int_combine_numerals}, 290 @{simproc inteq_cancel_numerals}, 291 @{simproc intless_cancel_numerals}, 292 @{simproc intle_cancel_numerals}, 293 @{simproc field_combine_numerals}] 294 #> Lin_Arith.add_simprocs 295 [@{simproc nat_combine_numerals}, 296 @{simproc nateq_cancel_numerals}, 297 @{simproc natless_cancel_numerals}, 298 @{simproc natle_cancel_numerals}, 299 @{simproc natdiff_cancel_numerals}]) 300\<close> 301 302end 303