src/HOL/Library/normarith.ML
author huffman
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permissions -rw-r--r--
instance ^ :: (metric_space, finite) metric_space
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(* Title:      Library/normarith.ML
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   Author:     Amine Chaieb, University of Cambridge
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   Description: A simple decision procedure for linear problems in euclidean space
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*)
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  (* Now the norm procedure for euclidean spaces *)
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signature NORM_ARITH = 
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sig
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 val norm_arith : Proof.context -> conv
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 val norm_arith_tac : Proof.context -> int -> tactic
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end
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structure NormArith : NORM_ARITH = 
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struct
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 open Conv Thm Conv2;
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 val bool_eq = op = : bool *bool -> bool
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  fun dest_ratconst t = case term_of t of
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   Const(@{const_name divide}, _)$a$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
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 | Const(@{const_name inverse}, _)$a => Rat.rat_of_quotient(1, HOLogic.dest_number a |> snd)
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 | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
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 fun is_ratconst t = can dest_ratconst t
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 fun augment_norm b t acc = case term_of t of 
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     Const(@{const_name norm}, _) $ _ => insert (eq_pair bool_eq (op aconvc)) (b,dest_arg t) acc
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   | _ => acc
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 fun find_normedterms t acc = case term_of t of
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    @{term "op + :: real => _"}$_$_ =>
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            find_normedterms (dest_arg1 t) (find_normedterms (dest_arg t) acc)
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      | @{term "op * :: real => _"}$_$n =>
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            if not (is_ratconst (dest_arg1 t)) then acc else
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            augment_norm (dest_ratconst (dest_arg1 t) >=/ Rat.zero) 
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                      (dest_arg t) acc
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      | _ => augment_norm true t acc 
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 val cterm_lincomb_neg = Ctermfunc.mapf Rat.neg
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 fun cterm_lincomb_cmul c t = 
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    if c =/ Rat.zero then Ctermfunc.undefined else Ctermfunc.mapf (fn x => x */ c) t
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 fun cterm_lincomb_add l r = Ctermfunc.combine (curry op +/) (fn x => x =/ Rat.zero) l r
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 fun cterm_lincomb_sub l r = cterm_lincomb_add l (cterm_lincomb_neg r)
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 fun cterm_lincomb_eq l r = Ctermfunc.is_undefined (cterm_lincomb_sub l r)
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 val int_lincomb_neg = Intfunc.mapf Rat.neg
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 fun int_lincomb_cmul c t = 
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    if c =/ Rat.zero then Intfunc.undefined else Intfunc.mapf (fn x => x */ c) t
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 fun int_lincomb_add l r = Intfunc.combine (curry op +/) (fn x => x =/ Rat.zero) l r
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 fun int_lincomb_sub l r = int_lincomb_add l (int_lincomb_neg r)
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 fun int_lincomb_eq l r = Intfunc.is_undefined (int_lincomb_sub l r)
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fun vector_lincomb t = case term_of t of 
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   Const(@{const_name plus},Type("fun",[Type("Finite_Cartesian_Product.^",_),_])) $ _ $ _ =>
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    cterm_lincomb_add (vector_lincomb (dest_arg1 t)) (vector_lincomb (dest_arg t))
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 | Const(@{const_name minus},Type("fun",[Type("Finite_Cartesian_Product.^",_),_])) $ _ $ _ =>
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    cterm_lincomb_sub (vector_lincomb (dest_arg1 t)) (vector_lincomb (dest_arg t))
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 | Const(@{const_name vector_scalar_mult},Type("fun",[Type("Finite_Cartesian_Product.^",_),_]))$_$_ =>
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    cterm_lincomb_cmul (dest_ratconst (dest_arg1 t)) (vector_lincomb (dest_arg t))
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 | Const(@{const_name uminus},Type("fun",[Type("Finite_Cartesian_Product.^",_),_]))$_ =>
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     cterm_lincomb_neg (vector_lincomb (dest_arg t))
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 | Const(@{const_name vec},_)$_ => 
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   let 
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     val b = ((snd o HOLogic.dest_number o term_of o dest_arg) t = 0 
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               handle TERM _=> false)
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   in if b then Ctermfunc.onefunc (t,Rat.one)
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      else Ctermfunc.undefined
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   end
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 | _ => Ctermfunc.onefunc (t,Rat.one)
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 fun vector_lincombs ts =
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  fold_rev 
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   (fn t => fn fns => case AList.lookup (op aconvc) fns t of
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     NONE => 
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       let val f = vector_lincomb t 
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       in case find_first (fn (_,f') => cterm_lincomb_eq f f') fns of
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           SOME (_,f') => (t,f') :: fns
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         | NONE => (t,f) :: fns 
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       end
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   | SOME _ => fns) ts []
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fun replacenegnorms cv t = case term_of t of 
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  @{term "op + :: real => _"}$_$_ => binop_conv (replacenegnorms cv) t
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| @{term "op * :: real => _"}$_$_ => 
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    if dest_ratconst (dest_arg1 t) </ Rat.zero then arg_conv cv t else reflexive t
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| _ => reflexive t
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fun flip v eq = 
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  if Ctermfunc.defined eq v 
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  then Ctermfunc.update (v, Rat.neg (Ctermfunc.apply eq v)) eq else eq
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fun allsubsets s = case s of 
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  [] => [[]]
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|(a::t) => let val res = allsubsets t in
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               map (cons a) res @ res end
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fun evaluate env lin =
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 Intfunc.fold (fn (x,c) => fn s => s +/ c */ (Intfunc.apply env x)) 
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   lin Rat.zero
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fun solve (vs,eqs) = case (vs,eqs) of
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  ([],[]) => SOME (Intfunc.onefunc (0,Rat.one))
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 |(_,eq::oeqs) => 
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   (case filter (member (op =) vs) (Intfunc.dom eq) of (*FIXME use find_first here*)
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     [] => NONE
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    | v::_ => 
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       if Intfunc.defined eq v 
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       then 
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        let 
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         val c = Intfunc.apply eq v
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         val vdef = int_lincomb_cmul (Rat.neg (Rat.inv c)) eq
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         fun eliminate eqn = if not (Intfunc.defined eqn v) then eqn 
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                             else int_lincomb_add (int_lincomb_cmul (Intfunc.apply eqn v) vdef) eqn
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        in (case solve (vs \ v,map eliminate oeqs) of
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            NONE => NONE
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          | SOME soln => SOME (Intfunc.update (v, evaluate soln (Intfunc.undefine v vdef)) soln))
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        end
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       else NONE)
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fun combinations k l = if k = 0 then [[]] else
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 case l of 
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  [] => []
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| h::t => map (cons h) (combinations (k - 1) t) @ combinations k t
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fun forall2 p l1 l2 = case (l1,l2) of 
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   ([],[]) => true
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 | (h1::t1,h2::t2) => p h1 h2 andalso forall2 p t1 t2
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 | _ => false;
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fun vertices vs eqs =
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 let 
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  fun vertex cmb = case solve(vs,cmb) of
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    NONE => NONE
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   | SOME soln => SOME (map (fn v => Intfunc.tryapplyd soln v Rat.zero) vs)
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  val rawvs = map_filter vertex (combinations (length vs) eqs)
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  val unset = filter (forall (fn c => c >=/ Rat.zero)) rawvs 
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 in fold_rev (insert (uncurry (forall2 (curry op =/)))) unset [] 
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 end 
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fun subsumes l m = forall2 (fn x => fn y => Rat.abs x <=/ Rat.abs y) l m 
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fun subsume todo dun = case todo of
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 [] => dun
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|v::ovs => 
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   let val dun' = if exists (fn w => subsumes w v) dun then dun
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                  else v::(filter (fn w => not(subsumes v w)) dun) 
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   in subsume ovs dun' 
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   end;
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fun match_mp PQ P = P RS PQ;
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fun cterm_of_rat x = 
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let val (a, b) = Rat.quotient_of_rat x
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in 
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 if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
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  else Thm.capply (Thm.capply @{cterm "op / :: real => _"} 
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                   (Numeral.mk_cnumber @{ctyp "real"} a))
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        (Numeral.mk_cnumber @{ctyp "real"} b)
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end;
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fun norm_cmul_rule c th = instantiate' [] [SOME (cterm_of_rat c)] (th RS @{thm norm_cmul_rule_thm});
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fun norm_add_rule th1 th2 = [th1, th2] MRS @{thm norm_add_rule_thm};
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  (* I think here the static context should be sufficient!! *)
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fun inequality_canon_rule ctxt = 
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 let 
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  (* FIXME : Should be computed statically!! *)
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  val real_poly_conv = 
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    Normalizer.semiring_normalize_wrapper ctxt
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     (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
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 in fconv_rule (arg_conv ((rewr_conv @{thm ge_iff_diff_ge_0}) then_conv arg_conv (field_comp_conv then_conv real_poly_conv)))
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end;
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 fun absc cv ct = case term_of ct of 
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 Abs (v,_, _) => 
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  let val (x,t) = Thm.dest_abs (SOME v) ct
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  in Thm.abstract_rule ((fst o dest_Free o term_of) x) x (cv t)
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  end
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 | _ => all_conv ct;
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fun sub_conv cv ct = (comb_conv cv else_conv absc cv) ct;
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fun botc1 conv ct = 
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  ((sub_conv (botc1 conv)) then_conv (conv else_conv all_conv)) ct;
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 fun rewrs_conv eqs ct = first_conv (map rewr_conv eqs) ct;
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 val apply_pth1 = rewr_conv @{thm pth_1};
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 val apply_pth2 = rewr_conv @{thm pth_2};
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 val apply_pth3 = rewr_conv @{thm pth_3};
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 val apply_pth4 = rewrs_conv @{thms pth_4};
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 val apply_pth5 = rewr_conv @{thm pth_5};
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 val apply_pth6 = rewr_conv @{thm pth_6};
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 val apply_pth7 = rewrs_conv @{thms pth_7};
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 val apply_pth8 = rewr_conv @{thm pth_8} then_conv arg1_conv field_comp_conv then_conv (try_conv (rewr_conv (mk_meta_eq @{thm vector_smult_lzero})));
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 val apply_pth9 = rewrs_conv @{thms pth_9} then_conv arg1_conv (arg1_conv field_comp_conv);
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 val apply_ptha = rewr_conv @{thm pth_a};
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 val apply_pthb = rewrs_conv @{thms pth_b};
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 val apply_pthc = rewrs_conv @{thms pth_c};
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 val apply_pthd = try_conv (rewr_conv @{thm pth_d});
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fun headvector t = case t of 
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  Const(@{const_name plus}, Type("fun",[Type("Finite_Cartesian_Product.^",_),_]))$
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   (Const(@{const_name vector_scalar_mult}, _)$l$v)$r => v
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 | Const(@{const_name vector_scalar_mult}, _)$l$v => v
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 | _ => error "headvector: non-canonical term"
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fun vector_cmul_conv ct =
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   ((apply_pth5 then_conv arg1_conv field_comp_conv) else_conv
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    (apply_pth6 then_conv binop_conv vector_cmul_conv)) ct
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fun vector_add_conv ct = apply_pth7 ct 
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 handle CTERM _ => 
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  (apply_pth8 ct 
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   handle CTERM _ => 
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    (case term_of ct of 
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     Const(@{const_name plus},_)$lt$rt =>
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      let 
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       val l = headvector lt 
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       val r = headvector rt
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      in (case TermOrd.fast_term_ord (l,r) of
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         LESS => (apply_pthb then_conv arg_conv vector_add_conv 
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                  then_conv apply_pthd) ct
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        | GREATER => (apply_pthc then_conv arg_conv vector_add_conv 
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                     then_conv apply_pthd) ct 
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        | EQUAL => (apply_pth9 then_conv 
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                ((apply_ptha then_conv vector_add_conv) else_conv 
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              arg_conv vector_add_conv then_conv apply_pthd)) ct)
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      end
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     | _ => reflexive ct))
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fun vector_canon_conv ct = case term_of ct of
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 Const(@{const_name plus},_)$_$_ =>
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  let 
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   val ((p,l),r) = Thm.dest_comb ct |>> Thm.dest_comb
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   val lth = vector_canon_conv l 
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   val rth = vector_canon_conv r
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   val th = Drule.binop_cong_rule p lth rth 
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  in fconv_rule (arg_conv vector_add_conv) th end
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| Const(@{const_name vector_scalar_mult}, _)$_$_ =>
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  let 
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   val (p,r) = Thm.dest_comb ct
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   val rth = Drule.arg_cong_rule p (vector_canon_conv r) 
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  in fconv_rule (arg_conv (apply_pth4 else_conv vector_cmul_conv)) rth
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  end
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| Const(@{const_name minus},_)$_$_ => (apply_pth2 then_conv vector_canon_conv) ct
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| Const(@{const_name uminus},_)$_ => (apply_pth3 then_conv vector_canon_conv) ct
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| Const(@{const_name vec},_)$n => 
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  let val n = Thm.dest_arg ct
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  in if is_ratconst n andalso not (dest_ratconst n =/ Rat.zero) 
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     then reflexive ct else apply_pth1 ct
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  end
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| _ => apply_pth1 ct
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fun norm_canon_conv ct = case term_of ct of
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  Const(@{const_name norm},_)$_ => arg_conv vector_canon_conv ct
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 | _ => raise CTERM ("norm_canon_conv", [ct])
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fun fold_rev2 f [] [] z = z
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 | fold_rev2 f (x::xs) (y::ys) z = f x y (fold_rev2 f xs ys z)
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 | fold_rev2 f _ _ _ = raise UnequalLengths;
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fun int_flip v eq = 
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  if Intfunc.defined eq v 
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  then Intfunc.update (v, Rat.neg (Intfunc.apply eq v)) eq else eq;
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local
29843
4bb780545478 Fixed theorem reference
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 val pth_zero = @{thm "norm_0"}
29841
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 val tv_n = (hd o tl o dest_ctyp o ctyp_of_term o dest_arg o dest_arg1 o dest_arg o cprop_of)
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             pth_zero
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 val concl = dest_arg o cprop_of 
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 fun real_vector_combo_prover ctxt translator (nubs,ges,gts) = 
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  let 
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   (* FIXME: Should be computed statically!!*)
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   val real_poly_conv = 
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      Normalizer.semiring_normalize_wrapper ctxt
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       (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
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   val sources = map (dest_arg o dest_arg1 o concl) nubs
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   val rawdests = fold_rev (find_normedterms o dest_arg o concl) (ges @ gts) [] 
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   val _ = if not (forall fst rawdests) then error "real_vector_combo_prover: Sanity check" 
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           else ()
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   val dests = distinct (op aconvc) (map snd rawdests)
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   val srcfuns = map vector_lincomb sources
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   val destfuns = map vector_lincomb dests 
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   val vvs = fold_rev (curry (gen_union op aconvc) o Ctermfunc.dom) (srcfuns @ destfuns) []
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   val n = length srcfuns
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   val nvs = 1 upto n
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   val srccombs = srcfuns ~~ nvs
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   fun consider d =
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    let 
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     fun coefficients x =
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      let 
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       val inp = if Ctermfunc.defined d x then Intfunc.onefunc (0, Rat.neg(Ctermfunc.apply d x))
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                      else Intfunc.undefined 
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      in fold_rev (fn (f,v) => fn g => if Ctermfunc.defined f x then Intfunc.update (v, Ctermfunc.apply f x) g else g) srccombs inp 
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      end
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     val equations = map coefficients vvs
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     val inequalities = map (fn n => Intfunc.onefunc (n,Rat.one)) nvs
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     fun plausiblevertices f =
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      let 
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       val flippedequations = map (fold_rev int_flip f) equations
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       val constraints = flippedequations @ inequalities
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       val rawverts = vertices nvs constraints
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       fun check_solution v =
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        let 
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          val f = fold_rev2 (curry Intfunc.update) nvs v (Intfunc.onefunc (0, Rat.one))
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        in forall (fn e => evaluate f e =/ Rat.zero) flippedequations
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        end
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       val goodverts = filter check_solution rawverts
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       val signfixups = map (fn n => if n mem_int  f then ~1 else 1) nvs 
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      in map (map2 (fn s => fn c => Rat.rat_of_int s */ c) signfixups) goodverts
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      end
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     val allverts = fold_rev append (map plausiblevertices (allsubsets nvs)) [] 
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    in subsume allverts []
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    end
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   fun compute_ineq v =
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    let 
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     val ths = map_filter (fn (v,t) => if v =/ Rat.zero then NONE 
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                                     else SOME(norm_cmul_rule v t))
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                            (v ~~ nubs) 
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    in inequality_canon_rule ctxt (end_itlist norm_add_rule ths)
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    end
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   val ges' = map_filter (try compute_ineq) (fold_rev (append o consider) destfuns []) @
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                 map (inequality_canon_rule ctxt) nubs @ ges
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   val zerodests = filter
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        (fn t => null (Ctermfunc.dom (vector_lincomb t))) (map snd rawdests)
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  in RealArith.real_linear_prover translator
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        (map (fn t => instantiate ([(tv_n,(hd o tl o dest_ctyp o ctyp_of_term) t)],[]) pth_zero)
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            zerodests,
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        map (fconv_rule (once_depth_conv (norm_canon_conv) then_conv
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                       arg_conv (arg_conv real_poly_conv))) ges',
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        map (fconv_rule (once_depth_conv (norm_canon_conv) then_conv 
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                       arg_conv (arg_conv real_poly_conv))) gts)
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  end
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in val real_vector_combo_prover = real_vector_combo_prover
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end;
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local
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 val pth = @{thm norm_imp_pos_and_ge}
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 val norm_mp = match_mp pth
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 val concl = dest_arg o cprop_of
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 fun conjunct1 th = th RS @{thm conjunct1}
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 fun conjunct2 th = th RS @{thm conjunct2}
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 fun C f x y = f y x
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fun real_vector_ineq_prover ctxt translator (ges,gts) = 
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 let 
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(*   val _ = error "real_vector_ineq_prover: pause" *)
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  val ntms = fold_rev find_normedterms (map (dest_arg o concl) (ges @ gts)) []
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  val lctab = vector_lincombs (map snd (filter (not o fst) ntms))
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  val (fxns, ctxt') = Variable.variant_fixes (replicate (length lctab) "x") ctxt
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  fun mk_norm t = capply (instantiate_cterm' [SOME (ctyp_of_term t)] [] @{cpat "norm :: (?'a :: norm) => real"}) t
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  fun mk_equals l r = capply (capply (instantiate_cterm' [SOME (ctyp_of_term l)] [] @{cpat "op == :: ?'a =>_"}) l) r
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  val asl = map2 (fn (t,_) => fn n => assume (mk_equals (mk_norm t) (cterm_of (ProofContext.theory_of ctxt') (Free(n,@{typ real}))))) lctab fxns
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  val replace_conv = try_conv (rewrs_conv asl)
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  val replace_rule = fconv_rule (funpow 2 arg_conv (replacenegnorms replace_conv))
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  val ges' =
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       fold_rev (fn th => fn ths => conjunct1(norm_mp th)::ths)
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              asl (map replace_rule ges)
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  val gts' = map replace_rule gts
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  val nubs = map (conjunct2 o norm_mp) asl
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  val th1 = real_vector_combo_prover ctxt' translator (nubs,ges',gts')
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  val shs = filter (member (fn (t,th) => t aconvc cprop_of th) asl) (#hyps (crep_thm th1)) 
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  val th11 = hd (Variable.export ctxt' ctxt [fold implies_intr shs th1])
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  val cps = map (swap o dest_equals) (cprems_of th11)
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  val th12 = instantiate ([], cps) th11
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  val th13 = fold (C implies_elim) (map (reflexive o snd) cps) th12;
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 in hd (Variable.export ctxt' ctxt [th13])
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 end 
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in val real_vector_ineq_prover = real_vector_ineq_prover
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end;
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local
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 val rawrule = fconv_rule (arg_conv (rewr_conv @{thm real_eq_0_iff_le_ge_0}))
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 fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
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 fun simple_cterm_ord t u = TermOrd.term_ord (term_of t, term_of u) = LESS;
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  (* FIXME: Lookup in the context every time!!! Fix this !!!*)
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 fun splitequation ctxt th acc =
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  let 
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   val real_poly_neg_conv = #neg
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       (Normalizer.semiring_normalizers_ord_wrapper ctxt
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        (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"})) simple_cterm_ord)
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   val (th1,th2) = conj_pair(rawrule th)
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  in th1::fconv_rule (arg_conv (arg_conv real_poly_neg_conv)) th2::acc
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  end
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in fun real_vector_prover ctxt translator (eqs,ges,gts) =
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     real_vector_ineq_prover ctxt translator
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         (fold_rev (splitequation ctxt) eqs ges,gts)
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end;
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  fun init_conv ctxt = 
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   Simplifier.rewrite (Simplifier.context ctxt 
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fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
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     (HOL_basic_ss addsimps ([@{thm vec_0}, @{thm vec_1}, @{thm vector_dist_norm}, @{thm diff_0_right}, @{thm right_minus}, @{thm diff_self}, @{thm norm_0}] @ @{thms arithmetic_simps} @ @{thms norm_pths})))
29841
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   then_conv field_comp_conv 
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   then_conv nnf_conv
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 fun pure ctxt = RealArith.gen_prover_real_arith ctxt (real_vector_prover ctxt);
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 fun norm_arith ctxt ct = 
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  let 
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   val ctxt' = Variable.declare_term (term_of ct) ctxt
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   val th = init_conv ctxt' ct
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  in equal_elim (Drule.arg_cong_rule @{cterm Trueprop} (symmetric th)) 
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                (pure ctxt' (rhs_of th))
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 end
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 fun norm_arith_tac ctxt = 
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   clarify_tac HOL_cs THEN'
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   ObjectLogic.full_atomize_tac THEN'
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   CSUBGOAL ( fn (p,i) => rtac (norm_arith ctxt (Thm.dest_arg p )) i);
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end;