1(* Title: Provers/Arith/cancel_numerals.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 i + #m*u + j ~~ i' + #m'*u + j' == #(m-m')*u + i + j ~~ i' + j' 8 9where ~~ is an appropriate balancing operation (e.g. =, <=, <, -). 10 11It works by (a) massaging both sides to bring the selected term to the front: 12 13 #m*u + (i + j) ~~ #m'*u + (i' + j') 14 15(b) then using bal_add1 or bal_add2 to reach 16 17 #(m-m')*u + i + j ~~ i' + j' (if m'<=m) 18 19or 20 21 i + j ~~ #(m'-m)*u + i' + j' (otherwise) 22*) 23 24signature CANCEL_NUMERALS_DATA = 25sig 26 (*abstract syntax*) 27 val mk_sum: typ -> term list -> term 28 val dest_sum: term -> term list 29 val mk_bal: term * term -> term 30 val dest_bal: term -> term * term 31 val mk_coeff: int * term -> term 32 val dest_coeff: term -> int * term 33 val find_first_coeff: term -> term list -> int * term list 34 (*rules*) 35 val bal_add1: thm 36 val bal_add2: thm 37 (*proof tools*) 38 val prove_conv: tactic list -> Proof.context -> thm list -> term * term -> thm option 39 val trans_tac: Proof.context -> thm option -> tactic (*applies the initial lemma*) 40 val norm_tac: Proof.context -> tactic (*proves the initial lemma*) 41 val numeral_simp_tac: Proof.context -> tactic (*proves the final theorem*) 42 val simplify_meta_eq: Proof.context -> thm -> thm (*simplifies the final theorem*) 43end; 44 45signature CANCEL_NUMERALS = 46sig 47 val proc: Proof.context -> cterm -> thm option 48end; 49 50functor CancelNumeralsFun(Data: CANCEL_NUMERALS_DATA): CANCEL_NUMERALS = 51struct 52 53(*For t = #n*u then put u in the table*) 54fun update_by_coeff t = 55 Termtab.update (#2 (Data.dest_coeff t), ()); 56 57(*a left-to-right scan of terms1, seeking a term of the form #n*u, where 58 #m*u is in terms2 for some m*) 59fun find_common (terms1,terms2) = 60 let val tab2 = fold update_by_coeff terms2 Termtab.empty 61 fun seek [] = raise TERM("find_common", []) 62 | seek (t::terms) = 63 let val (_,u) = Data.dest_coeff t 64 in if Termtab.defined tab2 u then u else seek terms end 65 in seek terms1 end; 66 67(*the simplification procedure*) 68fun proc ctxt ct = 69 let 70 val prems = Simplifier.prems_of ctxt 71 val t = Thm.term_of ct 72 val (t', ctxt') = yield_singleton (Variable.import_terms true) t ctxt 73 74 val (t1,t2) = Data.dest_bal t' 75 val terms1 = Data.dest_sum t1 76 and terms2 = Data.dest_sum t2 77 78 val u = find_common (terms1, terms2) 79 val (n1, terms1') = Data.find_first_coeff u terms1 80 and (n2, terms2') = Data.find_first_coeff u terms2 81 and T = Term.fastype_of u 82 83 fun newshape (i,terms) = Data.mk_sum T (Data.mk_coeff(i,u)::terms) 84 val reshape = (*Move i*u to the front and put j*u into standard form 85 i + #m + j + k == #m + i + (j + k) *) 86 if n1=0 orelse n2=0 then (*trivial, so do nothing*) 87 raise TERM("cancel_numerals", []) 88 else Data.prove_conv [Data.norm_tac ctxt'] ctxt' prems 89 (t', Data.mk_bal (newshape(n1,terms1'), newshape(n2,terms2'))) 90 in 91 (if n2 <= n1 then 92 Data.prove_conv 93 [Data.trans_tac ctxt' reshape, resolve_tac ctxt' [Data.bal_add1] 1, 94 Data.numeral_simp_tac ctxt'] ctxt' prems 95 (t', Data.mk_bal (newshape(n1-n2,terms1'), Data.mk_sum T terms2')) 96 else 97 Data.prove_conv 98 [Data.trans_tac ctxt' reshape, resolve_tac ctxt' [Data.bal_add2] 1, 99 Data.numeral_simp_tac ctxt'] ctxt' prems 100 (t', Data.mk_bal (Data.mk_sum T terms1', newshape(n2-n1,terms2')))) 101 |> Option.map 102 (Data.simplify_meta_eq ctxt' #> 103 singleton (Variable.export ctxt' ctxt)) 104 end 105 (* FIXME avoid handling of generic exceptions *) 106 handle TERM _ => NONE 107 | TYPE _ => NONE; (*Typically (if thy doesn't include Numeral) 108 Undeclared type constructor "Numeral.bin"*) 109end; 110