[sledge] Add Equality.solve_for_vars

Summary:
```
val solve_for_vars : Var.Set.t list -> t -> Subst.t
(** [solve_for_vars \[v₁;…\] r] is a solution substitution that is
    entailed by [r] and consists of oriented equalities [x ↦ u] such that
    [fv x ⊈ vᵢ ⊇ fv u] where [i] is minimal such that [vᵢ]
    distinguishes [fv x] and [fv u], if one exists. *)
```

To be used for existential witnessing and quantifier elimination.

Reviewed By: ngorogiannis

Differential Revision: D19282636

fbshipit-source-id: c5b006cea

sledge/src/import/import.ml

@@ -215,6 +215,9 @@ module List = struct
     in
     remove_ [] xs
 
+  let remove ?equal xs x =
+    try Some (remove_exn ?equal xs x) with Not_found -> None
+
   let rec rev_init n ~f =
     if n = 0 then []
     else

sledge/src/import/import.mli

@@ -177,6 +177,7 @@ module List : sig
       argument, or raise [Not_found] if no such element exists. [equal]
       defaults to physical equality. *)
 
+  val remove : ?equal:('a -> 'a -> bool) -> 'a list -> 'a -> 'a list option
   val rev_init : int -> f:(int -> 'a) -> 'a list
 
   val symmetric_diff :

sledge/src/llair/term.ml

@@ -1020,6 +1020,8 @@ let is_false = function Integer {data} -> Z.is_false data | _ -> false
 (** Solve *)
 
 let solve_zero_eq ?for_ e =
+  [%Trace.call fun {pf} -> pf "%a%a" pp e (Option.pp " for %a" pp) for_]
+  ;
   ( match e with
   | Add args ->
       let+ c, q =
@@ -1034,7 +1036,12 @@ let solve_zero_eq ?for_ e =
       let r = div n d in
       (c, r)
   | _ -> None )
-  |> check (fun soln ->
+  |>
-         match (for_, soln) with
+  [%Trace.retn fun {pf} s ->
+    pf "%a"
+      (Option.pp "%a" (fun fs (c, r) ->
+           Format.fprintf fs "%a ↦ %a" pp c pp r ))
+      s ;
+    match (for_, s) with
     | Some f, Some (c, _) -> assert (equal f c)
-         | _ -> () )
+    | _ -> ()]

sledge/src/symbheap/equality.ml

@@ -10,7 +10,7 @@
 (** Classification of Terms by Theory *)
 
 type kind = Interpreted | Simplified | Atomic | Uninterpreted
-[@@deriving compare]
+[@@deriving compare, equal]
 
 let classify e =
   match (e : Term.t) with
@@ -20,6 +20,17 @@ let classify e =
   | Ap1 _ | Ap2 _ | Ap3 _ | ApN _ -> Uninterpreted
   | RecN _ | Var _ | Integer _ | Float _ | Nondet _ | Label _ -> Atomic
 
+let rec fold_max_solvables e ~init ~f =
+  match classify e with
+  | Interpreted ->
+      Term.fold e ~init ~f:(fun d s -> fold_max_solvables ~f d ~init:s)
+  | _ -> f e init
+
+let rec iter_max_solvables e ~f =
+  match classify e with
+  | Interpreted -> Term.iter ~f:(iter_max_solvables ~f) e
+  | _ -> f e
+
 (** Solution Substitutions *)
 module Subst : sig
   type t [@@deriving compare, equal, sexp]
@@ -211,16 +222,27 @@ let pp_diff fs (r, s) =
   in
   Format.fprintf fs "@[{@[<hv>%t%t@]}@]" pp_sat pp_rep
 
+let ppx_cls x = List.pp "@ = " (Term.ppx x)
+let pp_cls = ppx_cls (fun _ -> None)
+let pp_diff_cls = List.pp_diff ~compare:Term.compare "@ = " Term.pp
+
+let ppx_clss x fs cs =
+  List.pp "@ @<2>∧ "
+    (fun fs (key, data) ->
+      Format.fprintf fs "@[%a@ = %a@]" (Term.ppx x) key (ppx_cls x)
+        (List.sort ~compare:Term.compare data) )
+    fs (Map.to_alist cs)
+
+let pp_clss fs cs = ppx_clss (fun _ -> None) fs cs
+
+let pp_diff_clss =
+  Map.pp_diff ~data_equal:(List.equal Term.equal) Term.pp pp_cls pp_diff_cls
+
 (** Invariant *)
 
 (** test membership in carrier *)
 let in_car r e = Subst.mem r.rep e
 
-let rec iter_max_solvables e ~f =
-  match classify e with
-  | Interpreted -> Term.iter ~f:(iter_max_solvables ~f) e
-  | _ -> f e
-
 let invariant r =
   Invariant.invariant [%here] r [%sexp_of: t]
   @@ fun () ->
@@ -411,17 +433,8 @@ let fold_vars r ~init ~f =
   fold_terms r ~init ~f:(fun init -> Term.fold_vars ~f ~init)
 
 let fv e = fold_vars e ~f:Set.add ~init:Var.Set.empty
-
+let pp_classes fs r = pp_clss fs (classes r)
-let ppx_classes x fs r =
+let ppx_classes x fs r = ppx_clss x fs (classes r)
-  List.pp "@ @<2>∧ "
-    (fun fs (key, data) ->
-      Format.fprintf fs "@[%a@ = %a@]" (Term.ppx x) key
-        (List.pp "@ = " (Term.ppx x))
-        (List.sort ~compare:Term.compare data) )
-    fs
-    (Map.to_alist (classes r))
-
-let pp_classes fs r = ppx_classes (fun _ -> None) fs r
 
 let ppx_classes_diff x fs (r, s) =
   let clss = classes s in
@@ -439,3 +452,195 @@ let ppx_classes_diff x fs (r, s) =
         (List.pp "@ = " (Term.ppx x))
         (List.sort ~compare:Term.compare cls) )
     fs (Map.to_alist clss)
+
+(** Existential Witnessing and Elimination *)
+
+type 'a zom = Zero | One of 'a | Many
+
+(* try to find a [term] in [cls] such that [fv (poly - term) ⊆ us ∪ xs] and
+   [poly - term] has at most one maximal solvable subterm, [kill], where [fv
+   kill ⊈ us]; solve [poly = term] for [kill]; extend subst mapping [kill]
+   to the solution *)
+let solve_interp_eq us us_xs poly (cls, subst) =
+  [%Trace.call fun {pf} ->
+    pf "poly: @[%a@]@ cls: @[%a@]@ subst: @[%a@]" Term.pp poly pp_cls cls
+      Subst.pp subst]
+  ;
+  ( if not (Set.is_subset (Term.fv poly) ~of_:us_xs) then None
+  else
+    List.find_map cls ~f:(fun trm ->
+        if not (Set.is_subset (Term.fv trm) ~of_:us_xs) then None
+        else
+          let diff = Subst.norm subst (Term.sub poly trm) in
+          let max_solvables_not_ito_us =
+            fold_max_solvables diff ~init:Zero ~f:(fun solvable_subterm ->
+              function
+              | Many -> Many
+              | zom when Set.is_subset (Term.fv solvable_subterm) ~of_:us ->
+                  zom
+              | One _ -> Many
+              | Zero -> One solvable_subterm )
+          in
+          match max_solvables_not_ito_us with
+          | One kill ->
+              let+ kill, keep = Term.solve_zero_eq diff ~for_:kill in
+              Subst.compose1 ~key:kill ~data:keep subst
+          | Many | Zero -> None ) )
+  |>
+  [%Trace.retn fun {pf} subst' ->
+    pf "@[%a@]" Subst.pp_diff (subst, Option.value subst' ~default:subst) ;
+    Option.iter subst' ~f:(fun subst' ->
+        Subst.iteri subst' ~f:(fun ~key ~data ->
+            assert (Set.is_subset (Term.fv key) ~of_:us_xs) ;
+            assert (
+              Subst.mem subst key
+              || not (Set.is_subset (Term.fv key) ~of_:us) ) ;
+            assert (Set.is_subset (Term.fv data) ~of_:us) ) )]
+
+(* move equations from [cls] to [subst] which are between [Interpreted]
+   terms and can be expressed, after normalizing with [subst], as [x ↦ u]
+   where [us ∪ xs ⊇ fv x ⊈ us ⊇ fv u] *)
+let rec solve_interp_eqs us us_xs (cls, subst) =
+  [%Trace.call fun {pf} ->
+    pf "cls: @[%a@]@ subst: @[%a@]" pp_cls cls Subst.pp subst]
+  ;
+  let rec solve_interp_eqs_ cls' (cls, subst) =
+    match cls with
+    | [] -> (cls', subst)
+    | trm :: cls -> (
+        let trm' = Subst.norm subst trm in
+        match classify trm' with
+        | Interpreted -> (
+          match solve_interp_eq us us_xs trm' (cls', subst) with
+          | None -> (
+            match solve_interp_eq us us_xs trm' (cls, subst) with
+            | None -> solve_interp_eqs_ (trm' :: cls') (cls, subst)
+            | Some subst -> solve_interp_eqs_ cls' (cls, subst) )
+          | Some subst -> solve_interp_eqs_ cls' (cls, subst) )
+        | _ -> solve_interp_eqs_ (trm' :: cls') (cls, subst) )
+  in
+  let cls', subst' = solve_interp_eqs_ [] (cls, subst) in
+  ( if subst' != subst then solve_interp_eqs us us_xs (cls', subst')
+  else (cls', subst') )
+  |>
+  [%Trace.retn fun {pf} (cls', subst') ->
+    pf "cls: @[%a@]@ subst: @[%a@]" pp_diff_cls (cls, cls') Subst.pp_diff
+      (subst, subst')]
+
+(* move equations from [cls] (which is assumed to be normalized by [subst])
+   to [subst] which are between non-[Interpreted] terms and can be expressed
+   as [x ↦ u] where [us ∪ xs ⊇ fv x ⊈ us ⊇ fv u] *)
+let solve_uninterp_eqs us us_xs (cls, subst) =
+  [%Trace.call fun {pf} ->
+    pf "cls: @[%a@]@ subst: @[%a@]" pp_cls cls Subst.pp subst]
+  ;
+  let rep_ito_us, cls_not_ito_us, cls_delay =
+    List.fold cls ~init:(None, [], [])
+      ~f:(fun (rep_ito_us, cls_not_ito_us, cls_delay) trm ->
+        if not (equal_kind (classify trm) Interpreted) then
+          let fv_trm = Term.fv trm in
+          if Set.is_subset fv_trm ~of_:us then
+            match rep_ito_us with
+            | Some rep when Term.compare rep trm <= 0 ->
+                (rep_ito_us, cls_not_ito_us, trm :: cls_delay)
+            | Some rep -> (Some trm, cls_not_ito_us, rep :: cls_delay)
+            | None -> (Some trm, cls_not_ito_us, cls_delay)
+          else if Set.is_subset fv_trm ~of_:us_xs then
+            (rep_ito_us, trm :: cls_not_ito_us, cls_delay)
+          else (rep_ito_us, cls_not_ito_us, trm :: cls_delay)
+        else (rep_ito_us, cls_not_ito_us, trm :: cls_delay) )
+  in
+  ( match rep_ito_us with
+  | None -> (cls, subst)
+  | Some rep_ito_us ->
+      let cls =
+        if List.is_empty cls_delay then [] else rep_ito_us :: cls_delay
+      in
+      let subst =
+        List.fold cls_not_ito_us ~init:subst ~f:(fun subst trm_not_ito_us ->
+            Subst.compose1 ~key:trm_not_ito_us ~data:rep_ito_us subst )
+      in
+      (cls, subst) )
+  |>
+  [%Trace.retn fun {pf} (cls', subst') ->
+    pf "cls: @[%a@]@ subst: @[%a@]" pp_diff_cls (cls, cls') Subst.pp_diff
+      (subst, subst') ;
+    Subst.iteri subst' ~f:(fun ~key ~data ->
+        assert (Set.is_subset (Term.fv key) ~of_:us_xs) ;
+        assert (
+          Subst.mem subst key || not (Set.is_subset (Term.fv key) ~of_:us)
+        ) ;
+        assert (Set.is_subset (Term.fv data) ~of_:us) )]
+
+(* move equations between terms in [rep]'s class [cls] from [classes] to
+   [subst] which can be expressed, after normalizing with [subst], as [x ↦
+   u] where [us ∪ xs ⊇ fv x ⊈ us ⊇ fv u] *)
+let solve_class us us_xs ~key:rep ~data:cls (classes, subst) =
+  let classes0 = classes in
+  [%Trace.call fun {pf} ->
+    pf "rep: @[%a@]@ cls: @[%a@]@ subst: @[%a@]" Term.pp rep pp_cls cls
+      Subst.pp subst]
+  ;
+  let cls, subst = solve_interp_eqs us us_xs (rep :: cls, subst) in
+  let cls, subst = solve_uninterp_eqs us us_xs (cls, subst) in
+  let cls =
+    List.remove ~equal:Term.equal cls (Subst.norm subst rep)
+    |> Option.value ~default:cls
+  in
+  let classes =
+    if List.is_empty cls then Map.remove classes rep
+    else Map.set classes ~key:rep ~data:cls
+  in
+  (classes, subst)
+  |>
+  [%Trace.retn fun {pf} (classes', subst') ->
+    pf "subst: @[%a@]@ classes: @[%a@]" Subst.pp_diff (subst, subst')
+      pp_diff_clss (classes0, classes')]
+
+(* move equations from [classes] to [subst] which can be expressed, after
+   normalizing with [subst], as [x ↦ u] where [us ∪ xs ⊇ fv x ⊈ us ⊇ fv u] *)
+let solve_classes (classes, subst, us) xs =
+  [%Trace.call fun {pf} -> pf "xs: {@[%a@]}" Var.Set.pp xs]
+  ;
+  let rec solve_classes_ (classes0, subst0, us_xs) =
+    let classes, subst =
+      Map.fold ~f:(solve_class us us_xs) classes0 ~init:(classes0, subst0)
+    in
+    if subst != subst0 then solve_classes_ (classes, subst, us_xs)
+    else (classes, subst, us_xs)
+  in
+  solve_classes_ (classes, subst, Set.union us xs)
+  |>
+  [%Trace.retn fun {pf} (classes', subst', _) ->
+    pf "subst: @[%a@]@ classes: @[%a@]" Subst.pp_diff (subst, subst')
+      pp_diff_clss (classes, classes')]
+
+let pp_vss fs vss =
+  Format.fprintf fs "[@[%a@]]"
+    (List.pp ";@ " (fun fs vs -> Format.fprintf fs "{@[%a@]}" Var.Set.pp vs))
+    vss
+
+(* enumerate variable contexts vᵢ in [v₁;…] and accumulate a solution subst
+   with entries [x ↦ u] where [r] entails [x = u] and [⋃ⱼ₌₁ⁱ vⱼ ⊇ fv x ⊈
+   ⋃ⱼ₌₁ⁱ⁻¹ vⱼ ⊇ fv u] *)
+let solve_for_vars vss r =
+  [%Trace.call fun {pf} -> pf "%a@ @[%a@]" pp_vss vss pp_classes r]
+  ;
+  List.fold ~f:solve_classes
+    ~init:(classes r, Subst.empty, Var.Set.empty)
+    vss
+  |> snd3
+  |>
+  [%Trace.retn fun {pf} subst ->
+    pf "%a" Subst.pp subst ;
+    Subst.iteri subst ~f:(fun ~key ~data ->
+        assert (entails_eq r key data) ;
+        assert (
+          List.exists vss ~f:(fun vs ->
+              match
+                ( Set.is_subset (Term.fv key) ~of_:vs
+                , Set.is_subset (Term.fv data) ~of_:vs )
+              with
+              | false, true -> true
+              | true, false -> assert false
+              | _ -> false ) ) )]

sledge/src/symbheap/equality.mli

@@ -62,3 +62,14 @@ val difference : t -> Term.t -> Term.t -> Z.t option
     offset. *)
 
 val fold_terms : t -> init:'a -> f:('a -> Term.t -> 'a) -> 'a
+
+(** Solution Substitutions *)
+module Subst : sig
+  type t [@@deriving compare, equal, sexp]
+end
+
+val solve_for_vars : Var.Set.t list -> t -> Subst.t
+(** [solve_for_vars \[v₁;…\] r] is a solution substitution that is
+    entailed by [r] and consists of oriented equalities [x ↦ u] such that
+    [fv x ⊈ vᵢ ⊇ fv u] where [i] is minimal such that [vᵢ]
+    distinguishes [fv x] and [fv u], if one exists. *)