src/Provers/Arith/cancel_numerals.ML
author wenzelm
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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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Cancel common coefficients in balanced expressions:
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     i + #m*u + j ~~ i' + #m'*u + j'  ==  #(m-m')*u + i + j ~~ i' + j'
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where ~~ is an appropriate balancing operation (e.g. =, <=, <, -).
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It works by (a) massaging both sides to bring the selected term to the front:
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     #m*u + (i + j) ~~ #m'*u + (i' + j') 
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(b) then using bal_add1 or bal_add2 to reach
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     #(m-m')*u + i + j ~~ i' + j'       (if m'<=m)
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or
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     i + j ~~ #(m'-m)*u + i' + j'       (otherwise)
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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: 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 -> Sign.sg -> 
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                  thm list -> term * term -> thm option
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  val trans_tac: thm option -> tactic (*applies the initial lemma*)
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  val norm_tac: tactic                (*proves the initial lemma*)
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  val numeral_simp_tac: tactic        (*proves the final theorem*)
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  val simplify_meta_eq: thm -> thm    (*simplifies the final theorem*)
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end;
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functor CancelNumeralsFun(Data: CANCEL_NUMERALS_DATA):
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  sig
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  val proc: Sign.sg -> thm list -> term -> thm option
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  end 
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=
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struct
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(*For t = #n*u then put u in the table*)
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fun update_by_coeff (tab, t) =
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  Termtab.update ((#2 (Data.dest_coeff t), ()), tab);
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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 = foldl update_by_coeff (Termtab.empty, terms2)
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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 is_some (Termtab.lookup (tab2, u)) then u
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		  else seek terms
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	      end
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  in  seek terms1 end;
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(*the simplification procedure*)
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fun proc sg hyps t =
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  let (*first freeze any Vars in the term to prevent flex-flex problems*)
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      val rand_s = gensym"_"
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      fun mk_inst (var as Var((a,i),T))  = 
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	    (var,  Free((a ^ rand_s ^ string_of_int i), T))
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      val t' = subst_atomic (map mk_inst (term_vars t)) t
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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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      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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      fun newshape (i,terms) = Data.mk_sum (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] sg hyps
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			(t', 
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			 Data.mk_bal (newshape(n1,terms1'), 
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				      newshape(n2,terms2')))
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  in
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      apsome Data.simplify_meta_eq
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       (if n2<=n1 then 
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	    Data.prove_conv 
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	       [Data.trans_tac reshape, rtac Data.bal_add1 1,
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		Data.numeral_simp_tac] sg hyps
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	       (t', Data.mk_bal (newshape(n1-n2,terms1'), 
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				 Data.mk_sum terms2'))
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	else
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	    Data.prove_conv 
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	       [Data.trans_tac reshape, rtac Data.bal_add2 1,
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		Data.numeral_simp_tac] sg hyps
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	       (t', Data.mk_bal (Data.mk_sum terms1', 
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				 newshape(n2-n1,terms2'))))
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  end
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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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end;