src/Provers/Arith/cancel_numerals.ML

merged

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