1(* Title: Provers/Arith/cancel_numeral_factor.ML 2 Author: Lawrence C Paulson, Cambridge University Computer Laboratory 3 Copyright 2000 University of Cambridge 4 5Cancel common coefficients in balanced expressions: 6 7 u*#m ~~ u'*#m' == #n*u ~~ #n'*u' 8 9where ~~ is an appropriate balancing operation (e.g. =, <=, <, div, /) 10and d = gcd(m,m') and n=m/d and n'=m'/d. 11 12It works by (a) massaging both sides to bring gcd(m,m') to the front: 13 14 u*#m ~~ u'*#m' == #d*(#n*u) ~~ #d*(#n'*u') 15 16(b) then using the rule "cancel" to reach #n*u ~~ #n'*u'. 17*) 18 19signature CANCEL_NUMERAL_FACTOR_DATA = 20sig 21 (*abstract syntax*) 22 val mk_bal: term * term -> term 23 val dest_bal: term -> term * term 24 val mk_coeff: int * term -> term 25 val dest_coeff: term -> int * term 26 (*rules*) 27 val cancel: thm 28 val neg_exchanges: bool (*true if a negative coeff swaps the two operands, 29 as with < and <= but not = and div*) 30 (*proof tools*) 31 val prove_conv: tactic list -> Proof.context -> thm list -> term * term -> thm option 32 val trans_tac: Proof.context -> thm option -> tactic (*applies the initial lemma*) 33 val norm_tac: Proof.context -> tactic (*proves the initial lemma*) 34 val numeral_simp_tac: Proof.context -> tactic (*proves the final theorem*) 35 val simplify_meta_eq: Proof.context -> thm -> thm (*simplifies the final theorem*) 36end; 37 38 39functor CancelNumeralFactorFun(Data: CANCEL_NUMERAL_FACTOR_DATA): 40 sig val proc: Proof.context -> cterm -> thm option end = 41struct 42 43(*the simplification procedure*) 44fun proc ctxt ct = 45 let 46 val prems = Simplifier.prems_of ctxt; 47 val t = Thm.term_of ct; 48 val (t', ctxt') = yield_singleton (Variable.import_terms true) t ctxt 49 50 val (t1,t2) = Data.dest_bal t' 51 val (m1, t1') = Data.dest_coeff t1 52 and (m2, t2') = Data.dest_coeff t2 53 val d = (*if both are negative, also divide through by ~1*) 54 if (m1<0 andalso m2<=0) orelse 55 (m1<=0 andalso m2<0) then ~ (abs (Integer.gcd m1 m2)) else abs (Integer.gcd m1 m2) 56 val _ = if d=1 then (*trivial, so do nothing*) 57 raise TERM("cancel_numeral_factor", []) 58 else () 59 fun newshape (i,t) = Data.mk_coeff(d, Data.mk_coeff(i,t)) 60 val n1 = m1 div d and n2 = m2 div d 61 val rhs = if d<0 andalso Data.neg_exchanges 62 then Data.mk_bal (Data.mk_coeff(n2,t2'), Data.mk_coeff(n1,t1')) 63 else Data.mk_bal (Data.mk_coeff(n1,t1'), Data.mk_coeff(n2,t2')) 64 val reshape = (*Move d to the front and put the rest into standard form 65 i * #m * j == #d * (#n * (j * k)) *) 66 Data.prove_conv [Data.norm_tac ctxt'] ctxt' prems 67 (t', Data.mk_bal (newshape(n1,t1'), newshape(n2,t2'))) 68 in 69 Data.prove_conv 70 [Data.trans_tac ctxt' reshape, resolve_tac ctxt' [Data.cancel] 1, 71 Data.numeral_simp_tac ctxt'] ctxt' prems (t', rhs) 72 |> Option.map 73 (Data.simplify_meta_eq ctxt' #> 74 singleton (Variable.export ctxt' ctxt)) 75 end 76 (* FIXME avoid handling of generic exceptions *) 77 handle TERM _ => NONE 78 | TYPE _ => NONE; (*Typically (if thy doesn't include Numeral) 79 Undeclared type constructor "Numeral.bin"*) 80 81end; 82