paulson@8760: (*  Title:      Provers/Arith/cancel_numerals.ML
paulson@8736:     ID:         $Id$
paulson@8736:     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
paulson@8736:     Copyright   2000  University of Cambridge
paulson@8736: 
paulson@8760: Cancel common coefficients in balanced expressions:
paulson@8736: 
paulson@8760:      i + #m*u + j ~~ i' + #m'*u + j'  ==  #(m-m')*u + i + j ~~ i' + j'
paulson@8736: 
paulson@8736: where ~~ is an appropriate balancing operation (e.g. =, <=, <, -).
paulson@8760: 
paulson@8760: It works by (a) massaging both sides to bring the selected term to the front:
paulson@8760: 
wenzelm@17223:      #m*u + (i + j) ~~ #m'*u + (i' + j')
paulson@8760: 
paulson@8760: (b) then using bal_add1 or bal_add2 to reach
paulson@8760: 
paulson@8760:      #(m-m')*u + i + j ~~ i' + j'       (if m'<=m)
paulson@8760: 
paulson@8760: or
paulson@8760: 
paulson@8760:      i + j ~~ #(m'-m)*u + i' + j'       (otherwise)
paulson@8736: *)
paulson@8736: 
paulson@8736: signature CANCEL_NUMERALS_DATA =
paulson@8736: sig
paulson@8736:   (*abstract syntax*)
paulson@14387:   val mk_sum: typ -> term list -> term
paulson@8736:   val dest_sum: term -> term list
paulson@8736:   val mk_bal: term * term -> term
paulson@8736:   val dest_bal: term -> term * term
paulson@15965:   val mk_coeff: IntInf.int * term -> term
paulson@15965:   val dest_coeff: term -> IntInf.int * term
paulson@15965:   val find_first_coeff: term -> term list -> IntInf.int * term list
paulson@8760:   (*rules*)
paulson@8760:   val bal_add1: thm
paulson@8760:   val bal_add2: thm
paulson@8736:   (*proof tools*)
wenzelm@20114:   val prove_conv: tactic list -> Proof.context -> thm list -> term * term -> thm option
wenzelm@16973:   val trans_tac: simpset -> thm option -> tactic (*applies the initial lemma*)
wenzelm@16973:   val norm_tac: simpset -> tactic                (*proves the initial lemma*)
wenzelm@16973:   val numeral_simp_tac: simpset -> tactic        (*proves the final theorem*)
wenzelm@16973:   val simplify_meta_eq: simpset -> thm -> thm    (*simplifies the final theorem*)
paulson@8736: end;
paulson@8736: 
paulson@8736: 
paulson@8760: functor CancelNumeralsFun(Data: CANCEL_NUMERALS_DATA):
paulson@8760:   sig
wenzelm@20044:   val proc: simpset -> term -> thm option
wenzelm@17223:   end
paulson@8760: =
paulson@8736: struct
paulson@8736: 
paulson@8779: (*For t = #n*u then put u in the table*)
wenzelm@17223: fun update_by_coeff t =
wenzelm@17412:   Termtab.update (#2 (Data.dest_coeff t), ());
paulson@8760: 
paulson@8760: (*a left-to-right scan of terms1, seeking a term of the form #n*u, where
paulson@8760:   #m*u is in terms2 for some m*)
paulson@8760: fun find_common (terms1,terms2) =
wenzelm@17223:   let val tab2 = fold update_by_coeff terms2 Termtab.empty
wenzelm@17223:       fun seek [] = raise TERM("find_common", [])
wenzelm@17223:         | seek (t::terms) =
wenzelm@17223:               let val (_,u) = Data.dest_coeff t
wenzelm@17223:               in if Termtab.defined tab2 u then u else seek terms end
paulson@8760:   in  seek terms1 end;
paulson@8736: 
paulson@8736: (*the simplification procedure*)
wenzelm@20044: fun proc ss t =
wenzelm@15027:   let
wenzelm@20114:     val ctxt = Simplifier.the_context ss
wenzelm@20114:     val prems = prems_of_ss ss
wenzelm@20114:     val ([t'], ctxt') = Variable.import_terms true [t] ctxt
wenzelm@20114:     val export = singleton (Variable.export ctxt' ctxt)
wenzelm@20114: 
wenzelm@20114:     val (t1,t2) = Data.dest_bal t'
wenzelm@20114:     val terms1 = Data.dest_sum t1
wenzelm@20114:     and terms2 = Data.dest_sum t2
wenzelm@20114: 
wenzelm@20114:     val u = find_common (terms1, terms2)
wenzelm@20114:     val (n1, terms1') = Data.find_first_coeff u terms1
wenzelm@20114:     and (n2, terms2') = Data.find_first_coeff u terms2
wenzelm@20114:     and T = Term.fastype_of u
wenzelm@20114: 
wenzelm@20114:     fun newshape (i,terms) = Data.mk_sum T (Data.mk_coeff(i,u)::terms)
wenzelm@20114:     val reshape =  (*Move i*u to the front and put j*u into standard form
wenzelm@17223:                        i + #m + j + k == #m + i + (j + k) *)
wenzelm@20114:         if n1=0 orelse n2=0 then   (*trivial, so do nothing*)
wenzelm@20114:           raise TERM("cancel_numerals", [])
wenzelm@20114:         else Data.prove_conv [Data.norm_tac ss] ctxt prems
wenzelm@20114:           (t', Data.mk_bal (newshape(n1,terms1'), newshape(n2,terms2')))
paulson@8736:   in
wenzelm@20114:     Option.map (export o Data.simplify_meta_eq ss)
wenzelm@20114:       (if n2 <= n1 then
wenzelm@20114:          Data.prove_conv
wenzelm@20114:            [Data.trans_tac ss reshape, rtac Data.bal_add1 1,
wenzelm@20114:             Data.numeral_simp_tac ss] ctxt prems
wenzelm@20114:            (t', Data.mk_bal (newshape(n1-n2,terms1'), Data.mk_sum T terms2'))
wenzelm@20114:        else
wenzelm@20114:          Data.prove_conv
wenzelm@20114:            [Data.trans_tac ss reshape, rtac Data.bal_add2 1,
wenzelm@20114:             Data.numeral_simp_tac ss] ctxt prems
wenzelm@20114:            (t', Data.mk_bal (Data.mk_sum T terms1', newshape(n2-n1,terms2'))))
paulson@8736:   end
wenzelm@20114:   (* FIXME avoid handling of generic exceptions *)
skalberg@15531:   handle TERM _ => NONE
skalberg@15531:        | TYPE _ => NONE;   (*Typically (if thy doesn't include Numeral)
wenzelm@17223:                              Undeclared type constructor "Numeral.bin"*)
paulson@8736: 
paulson@8736: end;