[sledge] Do not solve polynomials for Mul or Div terms

Summary:
This diff adds enough interpretation of Mul and Div terms to be able
to exclude them from the domain of solution substitutions. While
non-linear arithmetic is still treated very incompletely, this change
increases the propagation power of the equality constraints that are
deduced. Mainly, this appears to be enough to avoid operations that
are semantically equivalence-preserving such as solve_for_vars from
producing equalities that are unprovable from their inputs.

Reviewed By: jvillard

Differential Revision: D20863528

fbshipit-source-id: fca74cba3

sledge/lib/equality.ml

@@ -14,8 +14,10 @@ type kind = Interpreted | Simplified | Atomic | Uninterpreted
 
 let classify e =
   match (e : Term.t) with
-  | Add _ | Mul _ -> Interpreted
+  | Add _ | Mul _
-  | Ap2 (Memory, _, _) | Ap3 (Extract, _, _, _) | ApN (Concat, _) ->
+   |Ap2 ((Div | Memory), _, _)
+   |Ap3 (Extract, _, _, _)
+   |ApN (Concat, _) ->
       Interpreted
   | Ap2 ((Eq | Dq), _, _) -> Simplified
   | Ap1 _ | Ap2 _ | Ap3 _ | ApN _ -> Uninterpreted
@@ -308,6 +310,8 @@ and solve_ ?f d e s =
       ( ((Add _ | Mul _ | Integer _ | Rational _) as p), q
       | q, ((Add _ | Mul _ | Integer _ | Rational _) as p) ) ->
       solve_poly ?f p q s
+  (* e = n / d ==> e × d = n *)
+  | Some (rep, Ap2 (Div, num, den)) -> solve_ ?f (Term.mul rep den) num s
   | Some (rep, var) ->
       assert (non_interpreted var) ;
       assert (non_interpreted rep) ;

sledge/lib/import/map.ml

@@ -61,6 +61,9 @@ end) : S with type key = Key.t = struct
 
   let pop m = choose m |> Option.map ~f:(fun (k, v) -> (k, v, remove m k))
 
+  let pop_min_elt m =
+    min_elt m |> Option.map ~f:(fun (k, v) -> (k, v, remove m k))
+
   let find_and_remove m k =
     let found = ref None in
     let m =

sledge/lib/import/map_intf.ml

@@ -47,8 +47,17 @@ module type S = sig
     -> 'r
 
   val choose : 'a t -> (key * 'a) option
+  (** Find an unspecified element. [O(1)]. *)
+
   val pop : 'a t -> (key * 'a * 'a t) option
+  (** Find and remove an unspecified element. [O(1)]. *)
+
+  val pop_min_elt : 'a t -> (key * 'a * 'a t) option
+  (** Find and remove minimum element. [O(log n)]. *)
+
   val find_and_remove : 'a t -> key -> ('a * 'a t) option
+  (** Find and remove an element. *)
+
   val pp : key pp -> 'a pp -> 'a t pp
 
   val pp_diff :

sledge/lib/import/qset.ml

@@ -55,6 +55,7 @@ struct
     M.change m x ~f:(function Some j -> if_nz Q.(i + j) | None -> if_nz i)
 
   let remove m x = M.remove m x
+  let find_and_remove = M.find_and_remove
 
   let union m n =
     M.merge m n ~f:(fun ~key:_ -> function
@@ -77,8 +78,8 @@ struct
   let count m x = match M.find m x with Some q -> q | None -> Q.zero
   let choose = M.choose
   let pop = M.pop
-  let min_elt_exn = M.min_elt_exn
   let min_elt = M.min_elt
+  let pop_min_elt = M.pop_min_elt
   let to_list m = M.to_alist m
   let iter m ~f = M.iteri ~f:(fun ~key ~data -> f key data) m
   let exists m ~f = M.existsi ~f:(fun ~key ~data -> f key data) m

sledge/lib/import/qset_intf.ml

@@ -54,13 +54,19 @@ module type S = sig
   (** Multiplicity of an element. [O(log n)]. *)
 
   val choose : t -> (elt * Q.t) option
-  val pop : t -> (elt * Q.t * t) option
+  (** Find an unspecified element. [O(1)]. *)
 
-  val min_elt_exn : t -> elt * Q.t
+  val pop : t -> (elt * Q.t * t) option
-  (** Minimum element. *)
+  (** Find and remove an unspecified element. [O(1)]. *)
 
   val min_elt : t -> (elt * Q.t) option
-  (** Minimum element. *)
+  (** Minimum element. [O(log n)]. *)
+
+  val pop_min_elt : t -> (elt * Q.t * t) option
+  (** Find and remove minimum element. [O(log n)]. *)
+
+  val find_and_remove : t -> elt -> (Q.t * t) option
+  (** Find and remove an element. *)
 
   val to_list : t -> (elt * Q.t) list
   (** Convert to a list of elements in ascending order. *)

sledge/lib/term.ml

@@ -502,8 +502,16 @@ module Prod = struct
     assert (match term with Integer _ | Rational _ -> false | _ -> true) ;
     Qset.add prod term Q.one
 
-  let singleton term = add term empty
+  let of_ term = add term empty
   let union = Qset.union
+
+  let to_term prod =
+    match Qset.pop prod with
+    | None -> one
+    | Some (factor, power, prod')
+      when Qset.is_empty prod' && Q.equal Q.one power ->
+        factor
+    | _ -> Mul prod
 end
 
 let rec simp_add_ es poly =
@@ -547,21 +555,21 @@ and simp_mul2 e f =
   | Rational {data= c}, x | x, Rational {data= c} ->
       Sum.to_term (Sum.of_ ~coeff:c x)
   (* (∏ᵤ₌₀ⁱ xᵤ) × (∏ᵥ₌ᵢ₊₁ⁿ xᵥ) ==> ∏ⱼ₌₀ⁿ xⱼ *)
-  | Mul xs1, Mul xs2 -> Mul (Prod.union xs1 xs2)
+  | Mul xs1, Mul xs2 -> Prod.to_term (Prod.union xs1 xs2)
   (* (∏ᵢ xᵢ) × (∑ᵤ cᵤ × ∏ⱼ yᵤⱼ) ==> ∑ᵤ cᵤ × ∏ᵢ xᵢ × ∏ⱼ yᵤⱼ *)
   | (Mul prod as m), Add sum | Add sum, (Mul prod as m) ->
       Sum.to_term
         (Sum.map sum ~f:(function
-          | Mul args -> Mul (Prod.union prod args)
+          | Mul args -> Prod.to_term (Prod.union prod args)
           | (Integer _ | Rational _) as c -> simp_mul2 c m
-          | mono -> Mul (Prod.add mono prod) ))
+          | mono -> Prod.to_term (Prod.add mono prod) ))
   (* x₀ × (∏ᵢ₌₁ⁿ xᵢ) ==> ∏ᵢ₌₀ⁿ xᵢ *)
-  | Mul xs1, x | x, Mul xs1 -> Mul (Prod.add x xs1)
+  | Mul xs1, x | x, Mul xs1 -> Prod.to_term (Prod.add x xs1)
   (* e × (∑ᵤ cᵤ × ∏ⱼ yᵤⱼ) ==> ∑ᵤ e × cᵤ × ∏ⱼ yᵤⱼ *)
   | Add args, e | e, Add args ->
       simp_add_ (Sum.map ~f:(fun m -> simp_mul2 e m) args) zero
   (* x₁ × x₂ ==> ∏ᵢ₌₁² xᵢ *)
-  | _ -> Mul (Prod.add e (Prod.singleton f))
+  | _ -> Prod.to_term (Prod.add e (Prod.of_ f))
 
 let rec simp_div x y =
   match (x, y) with
@@ -1222,6 +1230,11 @@ let fold_terms e ~init ~f =
 let iter_vars e ~f =
   iter_terms e ~f:(function Var _ as v -> f (v :> Var.t) | _ -> ())
 
+let exists_vars e ~f =
+  with_return (fun {return} ->
+      iter_vars e ~f:(fun v -> if f v then return true) ;
+      false )
+
 let fold_vars e ~init ~f =
   fold_terms e ~init ~f:(fun s -> function
     | Var _ as v -> f s (v :> Var.t) | _ -> s )
@@ -1248,42 +1261,69 @@ let height e =
 
 (** Solve *)
 
-let find_for ?for_ args =
+let exists_fv_in vs qset =
-  let exists_var args ~f =
+  Qset.exists qset ~f:(fun e _ -> exists_vars e ~f:(Var.Set.mem vs))
-    with_return (fun {return} ->
+
-        Qset.iter args ~f:(fun arg _ ->
+let exists_fv_in4 vs w x y z =
-            iter_vars arg ~f:(fun v -> if f v then return true) ) ;
+  exists_fv_in vs w || exists_fv_in vs x || exists_fv_in vs y
-        false )
+  || exists_fv_in vs z
-  in
+
-  let remove_if_non_occuring rejected args c q =
+(* solve [0 = rejected_sum + (coeff × prod) + sum] *)
-    let args = Qset.remove args c in
+let solve_for_factor rejected_sum coeff prod sum =
-    let fv_c = fv c in
+  let rec find_factor rejected_prod prod =
-    if exists_var ~f:(Var.Set.mem fv_c) args then None
+    let* factor, power, prod = Qset.pop_min_elt prod in
-    else Some (c, q, Qset.union rejected args)
+    if
-  in
+      (not (Q.equal Q.one power))
-  let rec find_for_ rejected args =
+      || exists_fv_in4 (fv factor) rejected_sum rejected_prod prod sum
-    let* c, q = Qset.min_elt args in
+    then find_factor (Qset.add rejected_prod factor power) prod
-    remove_if_non_occuring rejected args c q
+    else Some (factor, Qset.union rejected_prod prod)
-    |> Option.or_else ~f:(fun () ->
-           find_for_ (Qset.add rejected c q) (Qset.remove args c) )
   in
-  match for_ with
+  let+ factor, prod = find_factor Qset.empty prod in
-  | Some c ->
+  (* solve [0 = rejected_sum + (coeff × factor × prod) + sum] yielding
-      let q = Qset.count args c in
+     [factor = (rejected_sum + sum) / (-coeff × prod)] *)
-      if Q.equal Q.zero q then None
+  ( factor
-      else remove_if_non_occuring Qset.empty args c q
+  , div
-  | None -> find_for_ Qset.empty args
+      (Sum.to_term (Qset.union rejected_sum sum))
-
+      (mul (rational (Q.neg coeff)) (Prod.to_term prod)) )
+
+(* solve [0 = rejected_sum + (coeff × mono) + sum] *)
+let solve_for_mono rejected_sum coeff mono sum =
+  match mono with
+  | Mul prod -> solve_for_factor rejected_sum coeff prod sum
+  | _ ->
+      if exists_fv_in (fv mono) sum then None
+      else
+        Some
+          ( mono
+          , Sum.to_term
+              (Sum.mul_const
+                 (Q.inv (Q.neg coeff))
+                 (Qset.union rejected_sum sum)) )
+
+(* solve [0 = rejected + sum] *)
+let rec solve_sum rejected_sum sum =
+  let* mono, coeff, sum = Qset.pop_min_elt sum in
+  solve_for_mono rejected_sum coeff mono sum
+  |> Option.or_else ~f:(fun () ->
+         solve_sum (Qset.add rejected_sum mono coeff) sum )
+
+let rec solve_div = function
+  (* [n / d = t] ==> [n = d × t] *)
+  | Some (Ap2 (Div, num, den), trm) -> solve_div (Some (num, mul den trm))
+  | o -> o
+
+(* solve [0 = e] *)
 let solve_zero_eq ?for_ e =
-  [%Trace.call fun {pf} -> pf "%a%a" pp e (Option.pp " for %a" pp) for_]
+  [%Trace.call fun {pf} -> pf "0 = %a%a" pp e (Option.pp " for %a" pp) for_]
   ;
   ( match e with
-  | Add args ->
+  | Add sum ->
-      let+ c, q, args = find_for ?for_ args in
+      ( match for_ with
-      let n = Sum.to_term (Qset.remove args c) in
+      | None -> solve_sum Qset.empty sum
-      let d = rational (Q.neg q) in
+      | Some mono ->
-      let r = div n d in
+          let* coeff, sum = Qset.find_and_remove sum mono in
-      (c, r)
+          solve_for_mono Qset.empty coeff mono sum )
+      |> solve_div
   | _ -> None )
   |>
   [%Trace.retn fun {pf} s ->
@@ -1292,5 +1332,7 @@ let solve_zero_eq ?for_ e =
            Format.fprintf fs "%a ↦ %a" pp c pp r ))
       s ;
     match (for_, s) with
+    | Some (Mul prod), Some (var, _) ->
+        assert (not (Q.equal Q.zero (Qset.count prod var)))
     | Some f, Some (c, _) -> assert (equal f c)
     | _ -> ()]