1(* Title: ZF/arith_data.ML 2 Author: Lawrence C Paulson, Cambridge University Computer Laboratory 3 4Arithmetic simplification: cancellation of common terms 5*) 6 7signature ARITH_DATA = 8sig 9 (*the main outcome*) 10 val nat_cancel: simproc list 11 (*tools for use in similar applications*) 12 val gen_trans_tac: Proof.context -> thm -> thm option -> tactic 13 val prove_conv: string -> tactic list -> Proof.context -> thm list -> term * term -> thm option 14 val simplify_meta_eq: thm list -> Proof.context -> thm -> thm 15 (*debugging*) 16 structure EqCancelNumeralsData : CANCEL_NUMERALS_DATA 17 structure LessCancelNumeralsData : CANCEL_NUMERALS_DATA 18 structure DiffCancelNumeralsData : CANCEL_NUMERALS_DATA 19end; 20 21 22structure ArithData: ARITH_DATA = 23struct 24 25val iT = Ind_Syntax.iT; 26 27val zero = Const(\<^const_name>\<open>zero\<close>, iT); 28val succ = Const(\<^const_name>\<open>succ\<close>, iT --> iT); 29fun mk_succ t = succ $ t; 30val one = mk_succ zero; 31 32val mk_plus = FOLogic.mk_binop \<^const_name>\<open>Arith.add\<close>; 33 34(*Thus mk_sum[t] yields t+#0; longer sums don't have a trailing zero*) 35fun mk_sum [] = zero 36 | mk_sum [t,u] = mk_plus (t, u) 37 | mk_sum (t :: ts) = mk_plus (t, mk_sum ts); 38 39(*this version ALWAYS includes a trailing zero*) 40fun long_mk_sum [] = zero 41 | long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts); 42 43val dest_plus = FOLogic.dest_bin \<^const_name>\<open>Arith.add\<close> iT; 44 45(* dest_sum *) 46 47fun dest_sum (Const(\<^const_name>\<open>zero\<close>,_)) = [] 48 | dest_sum (Const(\<^const_name>\<open>succ\<close>,_) $ t) = one :: dest_sum t 49 | dest_sum (Const(\<^const_name>\<open>Arith.add\<close>,_) $ t $ u) = dest_sum t @ dest_sum u 50 | dest_sum tm = [tm]; 51 52(*Apply the given rewrite (if present) just once*) 53fun gen_trans_tac _ _ NONE = all_tac 54 | gen_trans_tac ctxt th2 (SOME th) = ALLGOALS (resolve_tac ctxt [th RS th2]); 55 56(*Use <-> or = depending on the type of t*) 57fun mk_eq_iff(t,u) = 58 if fastype_of t = iT then FOLogic.mk_eq(t,u) 59 else FOLogic.mk_iff(t,u); 60 61(*We remove equality assumptions because they confuse the simplifier and 62 because only type-checking assumptions are necessary.*) 63fun is_eq_thm th = 64 can FOLogic.dest_eq (FOLogic.dest_Trueprop (Thm.prop_of th)); 65 66fun add_chyps chyps ct = Drule.list_implies (map Thm.cprop_of chyps, ct); 67 68fun prove_conv name tacs ctxt prems (t,u) = 69 if t aconv u then NONE 70 else 71 let val prems' = filter_out is_eq_thm prems 72 val goal = Logic.list_implies (map Thm.prop_of prems', 73 FOLogic.mk_Trueprop (mk_eq_iff (t, u))); 74 in SOME (prems' MRS Goal.prove ctxt [] [] goal (K (EVERY tacs))) 75 handle ERROR msg => 76 (warning (msg ^ "\nCancellation failed: no typing information? (" ^ name ^ ")"); NONE) 77 end; 78 79 80(*** Use CancelNumerals simproc without binary numerals, 81 just for cancellation ***) 82 83val mk_times = FOLogic.mk_binop \<^const_name>\<open>Arith.mult\<close>; 84 85fun mk_prod [] = one 86 | mk_prod [t] = t 87 | mk_prod (t :: ts) = if t = one then mk_prod ts 88 else mk_times (t, mk_prod ts); 89 90val dest_times = FOLogic.dest_bin \<^const_name>\<open>Arith.mult\<close> iT; 91 92fun dest_prod t = 93 let val (t,u) = dest_times t 94 in dest_prod t @ dest_prod u end 95 handle TERM _ => [t]; 96 97(*Dummy version: the only arguments are 0 and 1*) 98fun mk_coeff (0, t) = zero 99 | mk_coeff (1, t) = t 100 | mk_coeff _ = raise TERM("mk_coeff", []); 101 102(*Dummy version: the "coefficient" is always 1. 103 In the result, the factors are sorted terms*) 104fun dest_coeff t = (1, mk_prod (sort Term_Ord.term_ord (dest_prod t))); 105 106(*Find first coefficient-term THAT MATCHES u*) 107fun find_first_coeff past u [] = raise TERM("find_first_coeff", []) 108 | find_first_coeff past u (t::terms) = 109 let val (n,u') = dest_coeff t 110 in if u aconv u' then (n, rev past @ terms) 111 else find_first_coeff (t::past) u terms 112 end 113 handle TERM _ => find_first_coeff (t::past) u terms; 114 115 116(*Simplify #1*n and n*#1 to n*) 117val add_0s = [@{thm add_0_natify}, @{thm add_0_right_natify}]; 118val add_succs = [@{thm add_succ}, @{thm add_succ_right}]; 119val mult_1s = [@{thm mult_1_natify}, @{thm mult_1_right_natify}]; 120val tc_rules = [@{thm natify_in_nat}, @{thm add_type}, @{thm diff_type}, @{thm mult_type}]; 121val natifys = [@{thm natify_0}, @{thm natify_ident}, @{thm add_natify1}, @{thm add_natify2}, 122 @{thm diff_natify1}, @{thm diff_natify2}]; 123 124(*Final simplification: cancel + and **) 125fun simplify_meta_eq rules ctxt = 126 let val ctxt' = 127 put_simpset FOL_ss ctxt 128 delsimps @{thms iff_simps} (*these could erase the whole rule!*) 129 addsimps rules 130 |> fold Simplifier.add_eqcong [@{thm eq_cong2}, @{thm iff_cong2}] 131 in mk_meta_eq o simplify ctxt' end; 132 133val final_rules = add_0s @ mult_1s @ [@{thm mult_0}, @{thm mult_0_right}]; 134 135structure CancelNumeralsCommon = 136 struct 137 val mk_sum = (fn T:typ => mk_sum) 138 val dest_sum = dest_sum 139 val mk_coeff = mk_coeff 140 val dest_coeff = dest_coeff 141 val find_first_coeff = find_first_coeff [] 142 143 val norm_ss1 = 144 simpset_of (put_simpset ZF_ss \<^context> addsimps add_0s @ add_succs @ mult_1s @ @{thms add_ac}) 145 val norm_ss2 = 146 simpset_of (put_simpset ZF_ss \<^context> addsimps add_0s @ mult_1s @ @{thms add_ac} @ 147 @{thms mult_ac} @ tc_rules @ natifys) 148 fun norm_tac ctxt = 149 ALLGOALS (asm_simp_tac (put_simpset norm_ss1 ctxt)) 150 THEN ALLGOALS (asm_simp_tac (put_simpset norm_ss2 ctxt)) 151 val numeral_simp_ss = 152 simpset_of (put_simpset ZF_ss \<^context> addsimps add_0s @ tc_rules @ natifys) 153 fun numeral_simp_tac ctxt = 154 ALLGOALS (asm_simp_tac (put_simpset numeral_simp_ss ctxt)) 155 val simplify_meta_eq = simplify_meta_eq final_rules 156 end; 157 158(** The functor argumnets are declared as separate structures 159 so that they can be exported to ease debugging. **) 160 161structure EqCancelNumeralsData = 162 struct 163 open CancelNumeralsCommon 164 val prove_conv = prove_conv "nateq_cancel_numerals" 165 val mk_bal = FOLogic.mk_eq 166 val dest_bal = FOLogic.dest_eq 167 val bal_add1 = @{thm eq_add_iff [THEN iff_trans]} 168 val bal_add2 = @{thm eq_add_iff [THEN iff_trans]} 169 fun trans_tac ctxt = gen_trans_tac ctxt @{thm iff_trans} 170 end; 171 172structure EqCancelNumerals = CancelNumeralsFun(EqCancelNumeralsData); 173 174structure LessCancelNumeralsData = 175 struct 176 open CancelNumeralsCommon 177 val prove_conv = prove_conv "natless_cancel_numerals" 178 val mk_bal = FOLogic.mk_binrel \<^const_name>\<open>Ordinal.lt\<close> 179 val dest_bal = FOLogic.dest_bin \<^const_name>\<open>Ordinal.lt\<close> iT 180 val bal_add1 = @{thm less_add_iff [THEN iff_trans]} 181 val bal_add2 = @{thm less_add_iff [THEN iff_trans]} 182 fun trans_tac ctxt = gen_trans_tac ctxt @{thm iff_trans} 183 end; 184 185structure LessCancelNumerals = CancelNumeralsFun(LessCancelNumeralsData); 186 187structure DiffCancelNumeralsData = 188 struct 189 open CancelNumeralsCommon 190 val prove_conv = prove_conv "natdiff_cancel_numerals" 191 val mk_bal = FOLogic.mk_binop \<^const_name>\<open>Arith.diff\<close> 192 val dest_bal = FOLogic.dest_bin \<^const_name>\<open>Arith.diff\<close> iT 193 val bal_add1 = @{thm diff_add_eq [THEN trans]} 194 val bal_add2 = @{thm diff_add_eq [THEN trans]} 195 fun trans_tac ctxt = gen_trans_tac ctxt @{thm trans} 196 end; 197 198structure DiffCancelNumerals = CancelNumeralsFun(DiffCancelNumeralsData); 199 200 201val nat_cancel = 202 [Simplifier.make_simproc \<^context> "nateq_cancel_numerals" 203 {lhss = 204 [\<^term>\<open>l #+ m = n\<close>, \<^term>\<open>l = m #+ n\<close>, 205 \<^term>\<open>l #* m = n\<close>, \<^term>\<open>l = m #* n\<close>, 206 \<^term>\<open>succ(m) = n\<close>, \<^term>\<open>m = succ(n)\<close>], 207 proc = K EqCancelNumerals.proc}, 208 Simplifier.make_simproc \<^context> "natless_cancel_numerals" 209 {lhss = 210 [\<^term>\<open>l #+ m < n\<close>, \<^term>\<open>l < m #+ n\<close>, 211 \<^term>\<open>l #* m < n\<close>, \<^term>\<open>l < m #* n\<close>, 212 \<^term>\<open>succ(m) < n\<close>, \<^term>\<open>m < succ(n)\<close>], 213 proc = K LessCancelNumerals.proc}, 214 Simplifier.make_simproc \<^context> "natdiff_cancel_numerals" 215 {lhss = 216 [\<^term>\<open>(l #+ m) #- n\<close>, \<^term>\<open>l #- (m #+ n)\<close>, 217 \<^term>\<open>(l #* m) #- n\<close>, \<^term>\<open>l #- (m #* n)\<close>, 218 \<^term>\<open>succ(m) #- n\<close>, \<^term>\<open>m #- succ(n)\<close>], 219 proc = K DiffCancelNumerals.proc}]; 220 221end; 222 223val _ = 224 Theory.setup (Simplifier.map_theory_simpset (fn ctxt => 225 ctxt addsimprocs ArithData.nat_cancel)); 226 227 228(*examples: 229print_depth 22; 230set timing; 231set simp_trace; 232fun test s = (Goal s; by (Asm_simp_tac 1)); 233 234test "x #+ y = x #+ z"; 235test "y #+ x = x #+ z"; 236test "x #+ y #+ z = x #+ z"; 237test "y #+ (z #+ x) = z #+ x"; 238test "x #+ y #+ z = (z #+ y) #+ (x #+ w)"; 239test "x#*y #+ z = (z #+ y) #+ (y#*x #+ w)"; 240 241test "x #+ succ(y) = x #+ z"; 242test "x #+ succ(y) = succ(z #+ x)"; 243test "succ(x) #+ succ(y) #+ z = succ(z #+ y) #+ succ(x #+ w)"; 244 245test "(x #+ y) #- (x #+ z) = w"; 246test "(y #+ x) #- (x #+ z) = dd"; 247test "(x #+ y #+ z) #- (x #+ z) = dd"; 248test "(y #+ (z #+ x)) #- (z #+ x) = dd"; 249test "(x #+ y #+ z) #- ((z #+ y) #+ (x #+ w)) = dd"; 250test "(x#*y #+ z) #- ((z #+ y) #+ (y#*x #+ w)) = dd"; 251 252(*BAD occurrence of natify*) 253test "(x #+ succ(y)) #- (x #+ z) = dd"; 254 255test "x #* y2 #+ y #* x2 = y #* x2 #+ x #* y2"; 256 257test "(x #+ succ(y)) #- (succ(z #+ x)) = dd"; 258test "(succ(x) #+ succ(y) #+ z) #- (succ(z #+ y) #+ succ(x #+ w)) = dd"; 259 260(*use of typing information*) 261test "x : nat ==> x #+ y = x"; 262test "x : nat --> x #+ y = x"; 263test "x : nat ==> x #+ y < x"; 264test "x : nat ==> x < y#+x"; 265test "x : nat ==> x le succ(x)"; 266 267(*fails: no typing information isn't visible*) 268test "x #+ y = x"; 269 270test "x #+ y < x #+ z"; 271test "y #+ x < x #+ z"; 272test "x #+ y #+ z < x #+ z"; 273test "y #+ z #+ x < x #+ z"; 274test "y #+ (z #+ x) < z #+ x"; 275test "x #+ y #+ z < (z #+ y) #+ (x #+ w)"; 276test "x#*y #+ z < (z #+ y) #+ (y#*x #+ w)"; 277 278test "x #+ succ(y) < x #+ z"; 279test "x #+ succ(y) < succ(z #+ x)"; 280test "succ(x) #+ succ(y) #+ z < succ(z #+ y) #+ succ(x #+ w)"; 281 282test "x #+ succ(y) le succ(z #+ x)"; 283*) 284