src/HOL/Analysis/normarith.ML

+(*  Title:      HOL/Analysis/normarith.ML
+    Author:     Amine Chaieb, University of Cambridge
+
+Simple decision procedure for linear problems in Euclidean space.
+*)
+
+signature NORM_ARITH =
+sig
+ val norm_arith : Proof.context -> conv
+ val norm_arith_tac : Proof.context -> int -> tactic
+end
+
+structure NormArith : NORM_ARITH =
+struct
+
+ open Conv;
+ val bool_eq = op = : bool *bool -> bool
+  fun dest_ratconst t = case Thm.term_of t of
+   Const(@{const_name divide}, _)$a$b => Rat.make(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
+ | Const(@{const_name inverse}, _)$a => Rat.make(1, HOLogic.dest_number a |> snd)
+ | _ => Rat.of_int (HOLogic.dest_number (Thm.term_of t) |> snd)
+ fun is_ratconst t = can dest_ratconst t
+ fun augment_norm b t acc = case Thm.term_of t of
+     Const(@{const_name norm}, _) $ _ => insert (eq_pair bool_eq (op aconvc)) (b,Thm.dest_arg t) acc
+   | _ => acc
+ fun find_normedterms t acc = case Thm.term_of t of
+    @{term "op + :: real => _"}$_$_ =>
+            find_normedterms (Thm.dest_arg1 t) (find_normedterms (Thm.dest_arg t) acc)
+      | @{term "op * :: real => _"}$_$_ =>
+            if not (is_ratconst (Thm.dest_arg1 t)) then acc else
+            augment_norm (dest_ratconst (Thm.dest_arg1 t) >= @0)
+                      (Thm.dest_arg t) acc
+      | _ => augment_norm true t acc
+
+ val cterm_lincomb_neg = FuncUtil.Ctermfunc.map (K ~)
+ fun cterm_lincomb_cmul c t =
+    if c = @0 then FuncUtil.Ctermfunc.empty else FuncUtil.Ctermfunc.map (fn _ => fn x => x * c) t
+ fun cterm_lincomb_add l r = FuncUtil.Ctermfunc.combine (curry op +) (fn x => x = @0) l r
+ fun cterm_lincomb_sub l r = cterm_lincomb_add l (cterm_lincomb_neg r)
+ fun cterm_lincomb_eq l r = FuncUtil.Ctermfunc.is_empty (cterm_lincomb_sub l r)
+
+(*
+ val int_lincomb_neg = FuncUtil.Intfunc.map (K ~)
+*)
+ fun int_lincomb_cmul c t =
+    if c = @0 then FuncUtil.Intfunc.empty else FuncUtil.Intfunc.map (fn _ => fn x => x * c) t
+ fun int_lincomb_add l r = FuncUtil.Intfunc.combine (curry op +) (fn x => x = @0) l r
+(*
+ fun int_lincomb_sub l r = int_lincomb_add l (int_lincomb_neg r)
+ fun int_lincomb_eq l r = FuncUtil.Intfunc.is_empty (int_lincomb_sub l r)
+*)
+
+fun vector_lincomb t = case Thm.term_of t of
+   Const(@{const_name plus}, _) $ _ $ _ =>
+    cterm_lincomb_add (vector_lincomb (Thm.dest_arg1 t)) (vector_lincomb (Thm.dest_arg t))
+ | Const(@{const_name minus}, _) $ _ $ _ =>
+    cterm_lincomb_sub (vector_lincomb (Thm.dest_arg1 t)) (vector_lincomb (Thm.dest_arg t))
+ | Const(@{const_name scaleR}, _)$_$_ =>
+    cterm_lincomb_cmul (dest_ratconst (Thm.dest_arg1 t)) (vector_lincomb (Thm.dest_arg t))
+ | Const(@{const_name uminus}, _)$_ =>
+     cterm_lincomb_neg (vector_lincomb (Thm.dest_arg t))
+(* FIXME: how should we handle numerals?
+ | Const(@ {const_name vec},_)$_ =>
+   let
+     val b = ((snd o HOLogic.dest_number o term_of o Thm.dest_arg) t = 0
+               handle TERM _=> false)
+   in if b then FuncUtil.Ctermfunc.onefunc (t,@1)
+      else FuncUtil.Ctermfunc.empty
+   end
+*)
+ | _ => FuncUtil.Ctermfunc.onefunc (t,@1)
+
+ fun vector_lincombs ts =
+  fold_rev
+   (fn t => fn fns => case AList.lookup (op aconvc) fns t of
+     NONE =>
+       let val f = vector_lincomb t
+       in case find_first (fn (_,f') => cterm_lincomb_eq f f') fns of
+           SOME (_,f') => (t,f') :: fns
+         | NONE => (t,f) :: fns
+       end
+   | SOME _ => fns) ts []
+
+fun replacenegnorms cv t = case Thm.term_of t of
+  @{term "op + :: real => _"}$_$_ => binop_conv (replacenegnorms cv) t
+| @{term "op * :: real => _"}$_$_ =>
+    if dest_ratconst (Thm.dest_arg1 t) < @0 then arg_conv cv t else Thm.reflexive t
+| _ => Thm.reflexive t
+(*
+fun flip v eq =
+  if FuncUtil.Ctermfunc.defined eq v
+  then FuncUtil.Ctermfunc.update (v, ~ (FuncUtil.Ctermfunc.apply eq v)) eq else eq
+*)
+fun allsubsets s = case s of
+  [] => [[]]
+|(a::t) => let val res = allsubsets t in
+               map (cons a) res @ res end
+fun evaluate env lin =
+ FuncUtil.Intfunc.fold (fn (x,c) => fn s => s + c * (FuncUtil.Intfunc.apply env x))
+   lin @0
+
+fun solve (vs,eqs) = case (vs,eqs) of
+  ([],[]) => SOME (FuncUtil.Intfunc.onefunc (0,@1))
+ |(_,eq::oeqs) =>
+   (case filter (member (op =) vs) (FuncUtil.Intfunc.dom eq) of (*FIXME use find_first here*)
+     [] => NONE
+    | v::_ =>
+       if FuncUtil.Intfunc.defined eq v
+       then
+        let
+         val c = FuncUtil.Intfunc.apply eq v
+         val vdef = int_lincomb_cmul (~ (Rat.inv c)) eq
+         fun eliminate eqn = if not (FuncUtil.Intfunc.defined eqn v) then eqn
+                             else int_lincomb_add (int_lincomb_cmul (FuncUtil.Intfunc.apply eqn v) vdef) eqn
+        in (case solve (remove (op =) v vs, map eliminate oeqs) of
+            NONE => NONE
+          | SOME soln => SOME (FuncUtil.Intfunc.update (v, evaluate soln (FuncUtil.Intfunc.delete_safe v vdef)) soln))
+        end
+       else NONE)
+
+fun combinations k l = if k = 0 then [[]] else
+ case l of
+  [] => []
+| h::t => map (cons h) (combinations (k - 1) t) @ combinations k t
+
+fun vertices vs eqs =
+ let
+  fun vertex cmb = case solve(vs,cmb) of
+    NONE => NONE
+   | SOME soln => SOME (map (fn v => FuncUtil.Intfunc.tryapplyd soln v @0) vs)
+  val rawvs = map_filter vertex (combinations (length vs) eqs)
+  val unset = filter (forall (fn c => c >= @0)) rawvs
+ in fold_rev (insert (eq_list op =)) unset []
+ end
+
+val subsumes = eq_list (fn (x, y) => Rat.abs x <= Rat.abs y)
+
+fun subsume todo dun = case todo of
+ [] => dun
+|v::ovs =>
+   let val dun' = if exists (fn w => subsumes (w, v)) dun then dun
+                  else v:: filter (fn w => not (subsumes (v, w))) dun
+   in subsume ovs dun'
+   end;
+
+fun match_mp PQ P = P RS PQ;
+
+fun cterm_of_rat x =
+let val (a, b) = Rat.dest x
+in
+ if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
+  else Thm.apply (Thm.apply @{cterm "op / :: real => _"}
+                   (Numeral.mk_cnumber @{ctyp "real"} a))
+        (Numeral.mk_cnumber @{ctyp "real"} b)
+end;
+
+fun norm_cmul_rule c th = Thm.instantiate' [] [SOME (cterm_of_rat c)] (th RS @{thm norm_cmul_rule_thm});
+
+fun norm_add_rule th1 th2 = [th1, th2] MRS @{thm norm_add_rule_thm};
+
+  (* I think here the static context should be sufficient!! *)
+fun inequality_canon_rule ctxt =
+ let
+  (* FIXME : Should be computed statically!! *)
+  val real_poly_conv =
+    Semiring_Normalizer.semiring_normalize_wrapper ctxt
+     (the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"}))
+ in
+  fconv_rule (arg_conv ((rewr_conv @{thm ge_iff_diff_ge_0}) then_conv
+    arg_conv (Numeral_Simprocs.field_comp_conv ctxt then_conv real_poly_conv)))
+end;
+
+ val apply_pth1 = rewr_conv @{thm pth_1};
+ val apply_pth2 = rewr_conv @{thm pth_2};
+ val apply_pth3 = rewr_conv @{thm pth_3};
+ val apply_pth4 = rewrs_conv @{thms pth_4};
+ val apply_pth5 = rewr_conv @{thm pth_5};
+ val apply_pth6 = rewr_conv @{thm pth_6};
+ val apply_pth7 = rewrs_conv @{thms pth_7};
+ fun apply_pth8 ctxt =
+  rewr_conv @{thm pth_8} then_conv
+  arg1_conv (Numeral_Simprocs.field_comp_conv ctxt) then_conv
+  (try_conv (rewr_conv (mk_meta_eq @{thm scaleR_zero_left})));
+ fun apply_pth9 ctxt =
+  rewrs_conv @{thms pth_9} then_conv
+  arg1_conv (arg1_conv (Numeral_Simprocs.field_comp_conv ctxt));
+ val apply_ptha = rewr_conv @{thm pth_a};
+ val apply_pthb = rewrs_conv @{thms pth_b};
+ val apply_pthc = rewrs_conv @{thms pth_c};
+ val apply_pthd = try_conv (rewr_conv @{thm pth_d});
+
+fun headvector t = case t of
+  Const(@{const_name plus}, _)$
+   (Const(@{const_name scaleR}, _)$_$v)$_ => v
+ | Const(@{const_name scaleR}, _)$_$v => v
+ | _ => error "headvector: non-canonical term"
+
+fun vector_cmul_conv ctxt ct =
+   ((apply_pth5 then_conv arg1_conv (Numeral_Simprocs.field_comp_conv ctxt)) else_conv
+    (apply_pth6 then_conv binop_conv (vector_cmul_conv ctxt))) ct
+
+fun vector_add_conv ctxt ct = apply_pth7 ct
+ handle CTERM _ =>
+  (apply_pth8 ctxt ct
+   handle CTERM _ =>
+    (case Thm.term_of ct of
+     Const(@{const_name plus},_)$lt$rt =>
+      let
+       val l = headvector lt
+       val r = headvector rt
+      in (case Term_Ord.fast_term_ord (l,r) of
+         LESS => (apply_pthb then_conv arg_conv (vector_add_conv ctxt)
+                  then_conv apply_pthd) ct
+        | GREATER => (apply_pthc then_conv arg_conv (vector_add_conv ctxt)
+                     then_conv apply_pthd) ct
+        | EQUAL => (apply_pth9 ctxt then_conv
+                ((apply_ptha then_conv (vector_add_conv ctxt)) else_conv
+              arg_conv (vector_add_conv ctxt) then_conv apply_pthd)) ct)
+      end
+     | _ => Thm.reflexive ct))
+
+fun vector_canon_conv ctxt ct = case Thm.term_of ct of
+ Const(@{const_name plus},_)$_$_ =>
+  let
+   val ((p,l),r) = Thm.dest_comb ct |>> Thm.dest_comb
+   val lth = vector_canon_conv ctxt l
+   val rth = vector_canon_conv ctxt r
+   val th = Drule.binop_cong_rule p lth rth
+  in fconv_rule (arg_conv (vector_add_conv ctxt)) th end
+
+| Const(@{const_name scaleR}, _)$_$_ =>
+  let
+   val (p,r) = Thm.dest_comb ct
+   val rth = Drule.arg_cong_rule p (vector_canon_conv ctxt r)
+  in fconv_rule (arg_conv (apply_pth4 else_conv (vector_cmul_conv ctxt))) rth
+  end
+
+| Const(@{const_name minus},_)$_$_ => (apply_pth2 then_conv (vector_canon_conv ctxt)) ct
+
+| Const(@{const_name uminus},_)$_ => (apply_pth3 then_conv (vector_canon_conv ctxt)) ct
+
+(* FIXME
+| Const(@{const_name vec},_)$n =>
+  let val n = Thm.dest_arg ct
+  in if is_ratconst n andalso not (dest_ratconst n =/ @0)
+     then Thm.reflexive ct else apply_pth1 ct
+  end
+*)
+| _ => apply_pth1 ct
+
+fun norm_canon_conv ctxt ct = case Thm.term_of ct of
+  Const(@{const_name norm},_)$_ => arg_conv (vector_canon_conv ctxt) ct
+ | _ => raise CTERM ("norm_canon_conv", [ct])
+
+fun int_flip v eq =
+  if FuncUtil.Intfunc.defined eq v
+  then FuncUtil.Intfunc.update (v, ~ (FuncUtil.Intfunc.apply eq v)) eq else eq;
+
+local
+ val pth_zero = @{thm norm_zero}
+ val tv_n =
+  (dest_TVar o Thm.typ_of_cterm o Thm.dest_arg o Thm.dest_arg1 o Thm.dest_arg o Thm.cprop_of)
+    pth_zero
+ val concl = Thm.dest_arg o Thm.cprop_of
+ fun real_vector_combo_prover ctxt translator (nubs,ges,gts) =
+  let
+   (* FIXME: Should be computed statically!!*)
+   val real_poly_conv =
+      Semiring_Normalizer.semiring_normalize_wrapper ctxt
+       (the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"}))
+   val sources = map (Thm.dest_arg o Thm.dest_arg1 o concl) nubs
+   val rawdests = fold_rev (find_normedterms o Thm.dest_arg o concl) (ges @ gts) []
+   val _ = if not (forall fst rawdests) then error "real_vector_combo_prover: Sanity check"
+           else ()
+   val dests = distinct (op aconvc) (map snd rawdests)
+   val srcfuns = map vector_lincomb sources
+   val destfuns = map vector_lincomb dests
+   val vvs = fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom) (srcfuns @ destfuns) []
+   val n = length srcfuns
+   val nvs = 1 upto n
+   val srccombs = srcfuns ~~ nvs
+   fun consider d =
+    let
+     fun coefficients x =
+      let
+       val inp =
+        if FuncUtil.Ctermfunc.defined d x
+        then FuncUtil.Intfunc.onefunc (0, ~ (FuncUtil.Ctermfunc.apply d x))
+        else FuncUtil.Intfunc.empty
+      in fold_rev (fn (f,v) => fn g => if FuncUtil.Ctermfunc.defined f x then FuncUtil.Intfunc.update (v, FuncUtil.Ctermfunc.apply f x) g else g) srccombs inp
+      end
+     val equations = map coefficients vvs
+     val inequalities = map (fn n => FuncUtil.Intfunc.onefunc (n,@1)) nvs
+     fun plausiblevertices f =
+      let
+       val flippedequations = map (fold_rev int_flip f) equations
+       val constraints = flippedequations @ inequalities
+       val rawverts = vertices nvs constraints
+       fun check_solution v =
+        let
+          val f = fold_rev FuncUtil.Intfunc.update (nvs ~~ v) (FuncUtil.Intfunc.onefunc (0, @1))
+        in forall (fn e => evaluate f e = @0) flippedequations
+        end
+       val goodverts = filter check_solution rawverts
+       val signfixups = map (fn n => if member (op =) f n then ~1 else 1) nvs
+      in map (map2 (fn s => fn c => Rat.of_int s * c) signfixups) goodverts
+      end
+     val allverts = fold_rev append (map plausiblevertices (allsubsets nvs)) []
+    in subsume allverts []
+    end
+   fun compute_ineq v =
+    let
+     val ths = map_filter (fn (v,t) => if v = @0 then NONE
+                                     else SOME(norm_cmul_rule v t))
+                            (v ~~ nubs)
+     fun end_itlist f xs = split_last xs |> uncurry (fold_rev f)
+    in inequality_canon_rule ctxt (end_itlist norm_add_rule ths)
+    end
+   val ges' = map_filter (try compute_ineq) (fold_rev (append o consider) destfuns []) @
+                 map (inequality_canon_rule ctxt) nubs @ ges
+   val zerodests = filter
+        (fn t => null (FuncUtil.Ctermfunc.dom (vector_lincomb t))) (map snd rawdests)
+
+  in fst (RealArith.real_linear_prover translator
+        (map (fn t => Drule.instantiate_normalize ([(tv_n, Thm.ctyp_of_cterm t)],[]) pth_zero)
+            zerodests,
+        map (fconv_rule (try_conv (Conv.top_sweep_conv (K (norm_canon_conv ctxt)) ctxt) then_conv
+                       arg_conv (arg_conv real_poly_conv))) ges',
+        map (fconv_rule (try_conv (Conv.top_sweep_conv (K (norm_canon_conv ctxt)) ctxt) then_conv
+                       arg_conv (arg_conv real_poly_conv))) gts))
+  end
+in val real_vector_combo_prover = real_vector_combo_prover
+end;
+
+local
+ val pth = @{thm norm_imp_pos_and_ge}
+ val norm_mp = match_mp pth
+ val concl = Thm.dest_arg o Thm.cprop_of
+ fun conjunct1 th = th RS @{thm conjunct1}
+ fun conjunct2 th = th RS @{thm conjunct2}
+fun real_vector_ineq_prover ctxt translator (ges,gts) =
+ let
+(*   val _ = error "real_vector_ineq_prover: pause" *)
+  val ntms = fold_rev find_normedterms (map (Thm.dest_arg o concl) (ges @ gts)) []
+  val lctab = vector_lincombs (map snd (filter (not o fst) ntms))
+  val (fxns, ctxt') = Variable.variant_fixes (replicate (length lctab) "x") ctxt
+  fun instantiate_cterm' ty tms = Drule.cterm_rule (Thm.instantiate' ty tms)
+  fun mk_norm t =
+    let val T = Thm.typ_of_cterm t
+    in Thm.apply (Thm.cterm_of ctxt' (Const (@{const_name norm}, T --> @{typ real}))) t end
+  fun mk_equals l r =
+    let
+      val T = Thm.typ_of_cterm l
+      val eq = Thm.cterm_of ctxt (Const (@{const_name Pure.eq}, T --> T --> propT))
+    in Thm.apply (Thm.apply eq l) r end
+  val asl = map2 (fn (t,_) => fn n => Thm.assume (mk_equals (mk_norm t) (Thm.cterm_of ctxt' (Free(n,@{typ real}))))) lctab fxns
+  val replace_conv = try_conv (rewrs_conv asl)
+  val replace_rule = fconv_rule (funpow 2 arg_conv (replacenegnorms replace_conv))
+  val ges' =
+       fold_rev (fn th => fn ths => conjunct1(norm_mp th)::ths)
+              asl (map replace_rule ges)
+  val gts' = map replace_rule gts
+  val nubs = map (conjunct2 o norm_mp) asl
+  val th1 = real_vector_combo_prover ctxt' translator (nubs,ges',gts')
+  val shs = filter (member (fn (t,th) => t aconvc Thm.cprop_of th) asl) (Thm.chyps_of th1)
+  val th11 = hd (Variable.export ctxt' ctxt [fold Thm.implies_intr shs th1])
+  val cps = map (swap o Thm.dest_equals) (cprems_of th11)
+  val th12 = Drule.instantiate_normalize ([], map (apfst (dest_Var o Thm.term_of)) cps) th11
+  val th13 = fold Thm.elim_implies (map (Thm.reflexive o snd) cps) th12;
+ in hd (Variable.export ctxt' ctxt [th13])
+ end
+in val real_vector_ineq_prover = real_vector_ineq_prover
+end;
+
+local
+ val rawrule = fconv_rule (arg_conv (rewr_conv @{thm real_eq_0_iff_le_ge_0}))
+ fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
+ fun simple_cterm_ord t u = Term_Ord.term_ord (Thm.term_of t, Thm.term_of u) = LESS;
+  (* FIXME: Lookup in the context every time!!! Fix this !!!*)
+ fun splitequation ctxt th acc =
+  let
+   val real_poly_neg_conv = #neg
+       (Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
+        (the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"})) simple_cterm_ord)
+   val (th1,th2) = conj_pair(rawrule th)
+  in th1::fconv_rule (arg_conv (arg_conv (real_poly_neg_conv ctxt))) th2::acc
+  end
+in fun real_vector_prover ctxt _ translator (eqs,ges,gts) =
+     (real_vector_ineq_prover ctxt translator
+         (fold_rev (splitequation ctxt) eqs ges,gts), RealArith.Trivial)
+end;
+
+  fun init_conv ctxt =
+   Simplifier.rewrite (put_simpset HOL_basic_ss ctxt
+    addsimps ([(*@{thm vec_0}, @{thm vec_1},*) @{thm dist_norm}, @{thm right_minus},
+      @{thm diff_self}, @{thm norm_zero}] @ @{thms arithmetic_simps} @ @{thms norm_pths}))
+   then_conv Numeral_Simprocs.field_comp_conv ctxt
+   then_conv nnf_conv ctxt
+
+ fun pure ctxt = fst o RealArith.gen_prover_real_arith ctxt (real_vector_prover ctxt);
+ fun norm_arith ctxt ct =
+  let
+   val ctxt' = Variable.declare_term (Thm.term_of ct) ctxt
+   val th = init_conv ctxt' ct
+  in Thm.equal_elim (Drule.arg_cong_rule @{cterm Trueprop} (Thm.symmetric th))
+                (pure ctxt' (Thm.rhs_of th))
+ end
+
+ fun norm_arith_tac ctxt =
+   clarify_tac (put_claset HOL_cs ctxt) THEN'
+   Object_Logic.full_atomize_tac ctxt THEN'
+   CSUBGOAL ( fn (p,i) => resolve_tac ctxt [norm_arith ctxt (Thm.dest_arg p )] i);
+
+end;