[sledge] Rework first-order solver to use a list of pending equations

Summary:
Currently when solving an equation requires solving one or more other
equations, they are handled recursively. This diff reworks the solver
to instead queue pending equations in a list rather than eagerly
solving them eagerly. The pending list will be needed when changing
the Context.t representation since some equations will need to be
delayed until representation invariants are re-established.

Additionally, the solution is accumulated as an association list
rather than being eagerly incorporated into the context
representation. This enables the uses in equality solving and
quantifier elimination to use different representations.

Reviewed By: jvillard

Differential Revision: D25883727

fbshipit-source-id: 2f6b5efa3

sledge/src/fol/context.ml

@@ -9,7 +9,7 @@
 
 open Exp
 
-(** Classification of Terms by Theory *)
+(* Classification of Terms ================================================*)
 
 type kind = Interpreted | Atomic | Uninterpreted
 [@@deriving compare, equal]
@@ -34,7 +34,8 @@ let rec max_solvables e =
 
 let fold_max_solvables e s ~f = Iter.fold ~f (max_solvables e) s
 
-(** Solution Substitutions *)
+(* Solution Substitutions =================================================*)
+
 module Subst : sig
   type t [@@deriving compare, equal, sexp]
 
@@ -58,7 +59,6 @@ module Subst : sig
   val extend : Trm.t -> t -> t option
   val map_entries : f:(Trm.t -> Trm.t) -> t -> t
   val to_iter : t -> (Trm.t * Trm.t) iter
-  val fv : t -> Var.Set.t
   val partition_valid : Var.Set.t -> t -> t * Var.Set.t * t
 
   (* direct representation manipulation *)
@@ -84,14 +84,6 @@ end = struct
   let for_alli = Trm.Map.for_alli
   let to_iter = Trm.Map.to_iter
 
-  let vars s =
-    s
-    |> to_iter
-    |> Iter.flat_map ~f:(fun (k, v) ->
-           Iter.append (Trm.vars k) (Trm.vars v) )
-
-  let fv s = Var.Set.of_iter (vars s)
-
   (** look up a term in a substitution *)
   let apply s a = Trm.Map.find a s |> Option.value ~default:a
 
@@ -126,7 +118,7 @@ end = struct
     | _ when Trm.equal key data -> r
     | _ ->
         assert (
-          (not (Trm.Map.mem key r))
+          Option.for_all ~f:(Trm.equal key) (Trm.Map.find key r)
           || fail "domains intersect: %a" Trm.pp key () ) ;
         let s = Trm.Map.singleton key data in
         let r' = Trm.Map.map_endo ~f:(norm s) r in
@@ -208,7 +200,7 @@ end = struct
   let remove = Trm.Map.remove
 end
 
-(** Theory Solver *)
+(* Theory Solver ==========================================================*)
 
 (** prefer representative terms that are minimal in the order s.t. Var <
     Sized < Extract < Concat < others, then using height of sequence
@@ -235,155 +227,163 @@ let orient e f =
   | Zero -> None
   | Pos -> Some (f, e)
 
-let norm (_, _, s) e = Subst.norm s e
+type solve_state =
-
+  { wrt: Var.Set.t
-let compose1 ~var ~rep (us, xs, s) =
+  ; no_fresh: bool
-  let s = Subst.compose1 ~key:var ~data:rep s in
+  ; fresh: Var.Set.t
-  Some (us, xs, s)
+  ; solved: (Trm.t * Trm.t) list option
-
+  ; pending: (Trm.t * Trm.t) list }
-let fresh name (wrt, xs, s) =
+
-  let x, wrt = Var.fresh name ~wrt in
+let pp_solve_state ppf = function
-  let xs = Var.Set.add x xs in
+  | {solved= None} -> Format.fprintf ppf "unsat"
-  (Trm.var x, (wrt, xs, s))
+  | {solved= Some solved; fresh; pending} ->
+      Format.fprintf ppf "%a%a : %a" Var.Set.pp_xs fresh
+        (List.pp ";@ " (fun ppf (var, rep) ->
+             Format.fprintf ppf "@[%a ↦ %a@]" Trm.pp var Trm.pp rep ))
+        solved
+        (List.pp ";@ " (fun ppf (a, b) ->
+             Format.fprintf ppf "@[%a = %a@]" Trm.pp a Trm.pp b ))
+        pending
+
+let add_solved ~var ~rep s =
+  match s with
+  | {solved= None} -> s
+  | {solved= Some solved} -> {s with solved= Some ((var, rep) :: solved)}
+
+let add_pending a b s = {s with pending= (a, b) :: s.pending}
+
+let fresh name s =
+  if s.no_fresh then None
+  else
+    let x, wrt = Var.fresh name ~wrt:s.wrt in
+    let fresh = Var.Set.add x s.fresh in
+    Some (Trm.var x, {s with wrt; fresh})
 
 let solve_poly p q s =
   [%trace]
     ~call:(fun {pf} -> pf "@ %a = %a" Trm.pp p Trm.pp q)
-    ~retn:(fun {pf} -> function
+    ~retn:(fun {pf} -> pf "%a" pp_solve_state)
-      | Some (_, xs, s) -> pf "%a%a" Var.Set.pp_xs xs Subst.pp s
-      | None -> pf "false" )
   @@ fun () ->
   match Trm.sub p q with
-  | Z z -> if Z.equal Z.zero z then Some s else None
+  | Z z -> if Z.equal Z.zero z then s else {s with solved= None}
-  | Var _ as var -> compose1 ~var ~rep:Trm.zero s
+  | Var _ as var -> add_solved ~var ~rep:Trm.zero s
   | p_q -> (
     match Trm.Arith.solve_zero_eq p_q with
     | Some (var, rep) ->
-        compose1 ~var:(Trm.arith var) ~rep:(Trm.arith rep) s
+        add_solved ~var:(Trm.arith var) ~rep:(Trm.arith rep) s
-    | None -> compose1 ~var:p_q ~rep:Trm.zero s )
+    | None -> add_solved ~var:p_q ~rep:Trm.zero s )
 
 (* α[o,l) = β ==> l = |β| ∧ α = (⟨n,c⟩[0,o) ^ β ^ ⟨n,c⟩[o+l,n-o-l)) where n
    = |α| and c fresh *)
-let rec solve_extract ?(no_fresh = false) a o l b s =
+let solve_extract a o l b s =
   [%trace]
     ~call:(fun {pf} ->
-      pf "@ %a = %a@ %a%a" Trm.pp
+      pf "@ %a = %a" Trm.pp (Trm.extract ~seq:a ~off:o ~len:l) Trm.pp b )
-        (Trm.extract ~seq:a ~off:o ~len:l)
+    ~retn:(fun {pf} -> pf "%a" pp_solve_state)
-        Trm.pp b Var.Set.pp_xs (snd3 s) Subst.pp (trd3 s) )
-    ~retn:(fun {pf} -> function
-      | Some (_, xs, s) -> pf "%a%a" Var.Set.pp_xs xs Subst.pp s
-      | None -> pf "false" )
   @@ fun () ->
-  if no_fresh then Some s
+  match fresh "c" s with
-  else
+  | None -> s
-    let n = Trm.seq_size_exn a in
+  | Some (c, s) ->
-    let c, s = fresh "c" s in
+      let n = Trm.seq_size_exn a in
-    let n_c = Trm.sized ~siz:n ~seq:c in
+      let n_c = Trm.sized ~siz:n ~seq:c in
-    let o_l = Trm.add o l in
+      let o_l = Trm.add o l in
-    let n_o_l = Trm.sub n o_l in
+      let n_o_l = Trm.sub n o_l in
-    let c0 = Trm.extract ~seq:n_c ~off:Trm.zero ~len:o in
+      let c0 = Trm.extract ~seq:n_c ~off:Trm.zero ~len:o in
-    let c1 = Trm.extract ~seq:n_c ~off:o_l ~len:n_o_l in
+      let c1 = Trm.extract ~seq:n_c ~off:o_l ~len:n_o_l in
-    let b, s =
+      let b, s =
-      match Trm.seq_size b with
+        match Trm.seq_size b with
-      | None -> (Trm.sized ~siz:l ~seq:b, Some s)
+        | None -> (Trm.sized ~siz:l ~seq:b, s)
-      | Some m -> (b, solve_ l m s)
+        | Some m -> (b, add_pending l m s)
-    in
+      in
-    s >>= solve_ a (Trm.concat [|c0; b; c1|])
+      add_pending a (Trm.concat [|c0; b; c1|]) s
 
 (* α₀^…^αᵢ^αⱼ^…^αᵥ = β ==> |α₀^…^αᵥ| = |β| ∧ … ∧ αⱼ = β[n₀+…+nᵢ,nⱼ) ∧ …
    where nₓ ≡ |αₓ| and m = |β| *)
-and solve_concat ?no_fresh a0V b m s =
+let solve_concat a0V b m s =
   [%trace]
-    ~call:(fun {pf} ->
+    ~call:(fun {pf} -> pf "@ %a = %a" Trm.pp (Trm.concat a0V) Trm.pp b)
-      pf "@ %a = %a@ %a%a" Trm.pp (Trm.concat a0V) Trm.pp b Var.Set.pp_xs
+    ~retn:(fun {pf} -> pf "%a" pp_solve_state)
-        (snd3 s) Subst.pp (trd3 s) )
-    ~retn:(fun {pf} -> function
-      | Some (_, xs, s) -> pf "%a%a" Var.Set.pp_xs xs Subst.pp s
-      | None -> pf "false" )
   @@ fun () ->
-  Iter.fold_until (Array.to_iter a0V) (s, Trm.zero)
+  let s, n0V =
-    ~f:(fun aJ (s, oI) ->
+    Iter.fold (Array.to_iter a0V) (s, Trm.zero) ~f:(fun aJ (s, oI) ->
-      let nJ = Trm.seq_size_exn aJ in
+        let nJ = Trm.seq_size_exn aJ in
-      let oJ = Trm.add oI nJ in
+        let oJ = Trm.add oI nJ in
-      match solve_ ?no_fresh aJ (Trm.extract ~seq:b ~off:oI ~len:nJ) s with
+        let s = add_pending aJ (Trm.extract ~seq:b ~off:oI ~len:nJ) s in
-      | Some s -> `Continue (s, oJ)
+        (s, oJ) )
-      | None -> `Stop None )
+  in
-    ~finish:(fun (s, n0V) -> solve_ ?no_fresh n0V m s)
+  add_pending n0V m s
-
+
-and solve_ ?no_fresh d e s =
+let solve_ d e s =
-  [%Trace.call fun {pf} ->
+  [%trace]
-    pf "@ %a@[%a@ %a@ %a@]" Var.Set.pp_xs (snd3 s) Trm.pp d Trm.pp e
+    ~call:(fun {pf} -> pf "@ %a = %a" Trm.pp d Trm.pp e)
-      Subst.pp (trd3 s)]
+    ~retn:(fun {pf} -> pf "%a" pp_solve_state)
-  ;
+  @@ fun () ->
-  ( match orient (norm s d) (norm s e) with
+  match orient d e with
   (* e' = f' ==> true when e' ≡ f' *)
-  | None -> Some s
+  | None -> s
   (* i = j ==> false when i ≠ j *)
-  | Some (Z _, Z _) | Some (Q _, Q _) -> None
+  | Some (Z _, Z _) | Some (Q _, Q _) -> {s with solved= None}
   (*
    * Concat
    *)
   (* ⟨0,a⟩ = β ==> a = β = ⟨⟩ *)
   | Some (Sized {siz= n; seq= a}, b) when n == Trm.zero ->
       s
-      |> solve_ ?no_fresh a (Trm.concat [||])
+      |> add_pending a (Trm.concat [||])
-      >>= solve_ ?no_fresh b (Trm.concat [||])
+      |> add_pending b (Trm.concat [||])
   | Some (b, Sized {siz= n; seq= a}) when n == Trm.zero ->
       s
-      |> solve_ ?no_fresh a (Trm.concat [||])
+      |> add_pending a (Trm.concat [||])
-      >>= solve_ ?no_fresh b (Trm.concat [||])
+      |> add_pending b (Trm.concat [||])
   (* ⟨n,0⟩ = α₀^…^αᵥ ==> … ∧ αⱼ = ⟨n,0⟩[n₀+…+nᵢ,nⱼ) ∧ … *)
   | Some ((Sized {siz= n; seq} as b), Concat a0V) when seq == Trm.zero ->
-      solve_concat ?no_fresh a0V b n s
+      solve_concat a0V b n s
   (* ⟨n,e^⟩ = α₀^…^αᵥ ==> … ∧ αⱼ = ⟨n,e^⟩[n₀+…+nᵢ,nⱼ) ∧ … *)
   | Some ((Sized {siz= n; seq= Splat _} as b), Concat a0V) ->
-      solve_concat ?no_fresh a0V b n s
+      solve_concat a0V b n s
   | Some ((Var _ as v), (Concat a0V as c)) ->
       if not (Var.Set.mem (Var.of_ v) (Trm.fv c)) then
         (* v = α₀^…^αᵥ ==> v ↦ α₀^…^αᵥ when v ∉ fv(α₀^…^αᵥ) *)
-        compose1 ~var:v ~rep:c s
+        add_solved ~var:v ~rep:c s
       else
         (* v = α₀^…^αᵥ ==> ⟨|α₀^…^αᵥ|,v⟩ = α₀^…^αᵥ when v ∈ fv(α₀^…^αᵥ) *)
         let m = Trm.seq_size_exn c in
-        solve_concat ?no_fresh a0V (Trm.sized ~siz:m ~seq:v) m s
+        solve_concat a0V (Trm.sized ~siz:m ~seq:v) m s
   (* α₀^…^αᵥ = β₀^…^βᵤ ==> … ∧ αⱼ = (β₀^…^βᵤ)[n₀+…+nᵢ,nⱼ) ∧ … *)
   | Some (Concat a0V, (Concat _ as c)) ->
-      solve_concat ?no_fresh a0V c (Trm.seq_size_exn c) s
+      solve_concat a0V c (Trm.seq_size_exn c) s
   (* α[o,l) = α₀^…^αᵥ ==> … ∧ αⱼ = α[o,l)[n₀+…+nᵢ,nⱼ) ∧ … *)
-  | Some ((Extract {len= l} as e), Concat a0V) ->
+  | Some ((Extract {len= l} as e), Concat a0V) -> solve_concat a0V e l s
-      solve_concat ?no_fresh a0V e l s
   (*
    * Extract
    *)
   | Some ((Var _ as v), (Extract {len= l} as e)) ->
       if not (Var.Set.mem (Var.of_ v) (Trm.fv e)) then
         (* v = α[o,l) ==> v ↦ α[o,l) when v ∉ fv(α[o,l)) *)
-        compose1 ~var:v ~rep:e s
+        add_solved ~var:v ~rep:e s
       else
         (* v = α[o,l) ==> α[o,l) ↦ ⟨l,v⟩ when v ∈ fv(α[o,l)) *)
-        compose1 ~var:e ~rep:(Trm.sized ~siz:l ~seq:v) s
+        add_solved ~var:e ~rep:(Trm.sized ~siz:l ~seq:v) s
   (* α[o,l) = β ==> … ∧ α = _^β^_ *)
-  | Some (Extract {seq= a; off= o; len= l}, e) ->
+  | Some (Extract {seq= a; off= o; len= l}, e) -> solve_extract a o l e s
-      solve_extract ?no_fresh a o l e s
   (*
    * Sized
    *)
   (* v = ⟨n,a⟩ ==> v = a *)
-  | Some ((Var _ as v), Sized {seq= a}) -> s |> solve_ ?no_fresh v a
+  | Some ((Var _ as v), Sized {seq= a}) -> s |> add_pending v a
   (* ⟨n,a⟩ = ⟨m,b⟩ ==> n = m ∧ a = β *)
   | Some (Sized {siz= n; seq= a}, Sized {siz= m; seq= b}) ->
-      s |> solve_ ?no_fresh n m >>= solve_ ?no_fresh a b
+      s |> add_pending n m |> add_pending a b
   (* ⟨n,a⟩ = β ==> n = |β| ∧ a = β *)
   | Some (Sized {siz= n; seq= a}, b) ->
-      ( match Trm.seq_size b with
+      s
-      | None -> Some s
+      |> Option.fold ~f:(add_pending n) (Trm.seq_size b)
-      | Some m -> solve_ ?no_fresh n m s )
+      |> add_pending a b
-      >>= solve_ ?no_fresh a b
   (*
    * Splat
    *)
   (* a^ = b^ ==> a = b *)
-  | Some (Splat a, Splat b) -> s |> solve_ ?no_fresh a b
+  | Some (Splat a, Splat b) -> s |> add_pending a b
   (*
    * Arithmetic
    *)
@@ -398,27 +398,16 @@ and solve_ ?no_fresh d e s =
   | Some (rep, var) ->
       assert (not (is_interpreted var)) ;
       assert (not (is_interpreted rep)) ;
-      compose1 ~var ~rep s )
+      add_solved ~var ~rep s
-  |>
-  [%Trace.retn fun {pf} ->
-    function
-    | Some (_, xs, s) -> pf "%a%a" Var.Set.pp_xs xs Subst.pp s
-    | None -> pf "false"]
 
-let solve ~wrt ~xs d e =
+let solve ~wrt ~xs d e pending =
-  [%Trace.call fun {pf} -> pf "@ %a@ %a" Trm.pp d Trm.pp e]
+  [%trace]
-  ;
+    ~call:(fun {pf} -> pf "@ %a@ %a" Trm.pp d Trm.pp e)
-  ( solve_ d e (wrt, xs, Subst.empty)
+    ~retn:(fun {pf} -> pf "%a" pp_solve_state)
-  |>= fun (_, xs, s) ->
+  @@ fun () ->
-  let xs = Var.Set.inter xs (Subst.fv s) in
+  solve_ d e {wrt; no_fresh= false; fresh= xs; solved= Some []; pending}
-  (xs, s) )
-  |>
-  [%Trace.retn fun {pf} ->
-    function
-    | Some (xs, s) -> pf "%a%a" Var.Set.pp_xs xs Subst.pp s
-    | None -> pf "false"]
 
-(** Equality classes *)
+(* Equality classes =======================================================*)
 
 module Cls : sig
   type t [@@deriving equal]
@@ -460,7 +449,7 @@ end = struct
   let pp_diff = List.pp_diff ~cmp:Trm.compare "@ = " Trm.pp
 end
 
-(** Equality Relations *)
+(* Conjunctions of atomic formula assumptions =============================*)
 
 (** see also [invariant] *)
 type t =
@@ -470,7 +459,10 @@ type t =
   ; rep: Subst.t
         (** functional set of oriented equations: map [a] to [a'],
             indicating that [a = a'] holds, and that [a'] is the
-            'rep(resentative)' of [a] *) }
+            'rep(resentative)' of [a] *)
+  ; pnd: (Trm.t * Trm.t) list
+        (** pending equations to add (once invariants are reestablished) *)
+  }
 [@@deriving compare, equal, sexp]
 
 let classes r =
@@ -484,9 +476,12 @@ let cls_of r e =
   let e' = Subst.apply r.rep e in
   Trm.Map.find e' (classes r) |> Option.value ~default:(Cls.of_ e')
 
-(** Pretty-printing *)
+(* Pretty-printing ========================================================*)
+
+let pp_eq fs (e, f) = Format.fprintf fs "@[%a = %a@]" Trm.pp e Trm.pp f
+let pp_pnd = List.pp ";@ " pp_eq
 
-let pp_raw fs {sat; rep} =
+let pp_raw fs {sat; rep; pnd} =
   let pp_alist pp_k pp_v fs alist =
     let pp_assoc fs (k, v) =
       Format.fprintf fs "[@[%a@ @<2>↦ %a@]]" pp_k k pp_v (k, v)
@@ -494,9 +489,14 @@ let pp_raw fs {sat; rep} =
     Format.fprintf fs "[@[<hv>%a@]]" (List.pp ";@ " pp_assoc) alist
   in
   let pp_term_v fs (k, v) = if not (Trm.equal k v) then Trm.pp fs v in
-  Format.fprintf fs "@[{@[<hv>sat= %b;@ rep= %a@]}@]" sat
+  let pp_pnd fs pnd =
+    if not (List.is_empty pnd) then
+      Format.fprintf fs ";@ pnd= @[%a@]" pp_pnd pnd
+  in
+  Format.fprintf fs "@[{@[<hv>sat= %b;@ rep= %a%a@]}@]" sat
     (pp_alist Trm.pp pp_term_v)
     (Iter.to_list (Subst.to_iter rep))
+    pp_pnd pnd
 
 let pp_diff fs (r, s) =
   let pp_sat fs =
@@ -505,9 +505,14 @@ let pp_diff fs (r, s) =
   in
   let pp_rep fs =
     if not (Subst.is_empty r.rep) then
-      Format.fprintf fs "rep= %a" Subst.pp_diff (r.rep, s.rep)
+      Format.fprintf fs "rep= %a;@ " Subst.pp_diff (r.rep, s.rep)
   in
-  Format.fprintf fs "@[{@[<hv>%t%t@]}@]" pp_sat pp_rep
+  let pp_pnd fs =
+    Format.fprintf fs "pnd= @[%a@]"
+      (List.pp_diff ~cmp:[%compare: Trm.t * Trm.t] ";@ " pp_eq)
+      (r.pnd, s.pnd)
+  in
+  Format.fprintf fs "@[{@[<hv>%t%t%t@]}@]" pp_sat pp_rep pp_pnd
 
 let ppx_classes x fs clss =
   List.pp "@ @<2>∧ "
@@ -548,7 +553,7 @@ let ppx var_strength fs clss noneqs =
     "@ @<2>∧ " (Fml.ppx var_strength) fs noneqs ~suf:"@]" ;
   first && List.is_empty noneqs
 
-(** Basic queries *)
+(* Basic queries ==========================================================*)
 
 (** test membership in carrier *)
 let in_car r e = Subst.mem e r.rep
@@ -560,7 +565,7 @@ let semi_congruent r a' b = Trm.equal a' (Trm.map ~f:(Subst.norm r.rep) b)
 (** terms are congruent if equal after normalizing subterms *)
 let congruent r a b = semi_congruent r (Trm.map ~f:(Subst.norm r.rep) a) b
 
-(** Invariant *)
+(* Invariant ==============================================================*)
 
 let pre_invariant r =
   let@ () = Invariant.invariant [%here] r [%sexp_of: t] in
@@ -586,6 +591,7 @@ let pre_invariant r =
 let invariant r =
   let@ () = Invariant.invariant [%here] r [%sexp_of: t] in
   pre_invariant r ;
+  assert (List.is_empty r.pnd) ;
   assert (
     (not r.sat)
     || Subst.for_alli r.rep ~f:(fun ~key:a ~data:a' ->
@@ -596,13 +602,44 @@ let invariant r =
                || fail "not congruent %a@ %a@ in@ %a" Trm.pp a Trm.pp b pp r
                     () ) ) )
 
-(** Core operations *)
+(* Representation helpers =================================================*)
+
+let add_to_pnd a a' x =
+  if Trm.equal a a' then x else {x with pnd= (a, a') :: x.pnd}
+
+(* Propagation ============================================================*)
+
+let propagate1 (trm, rep) x =
+  [%trace]
+    ~call:(fun {pf} ->
+      pf "@ @[%a ↦ %a@]@ %a" Trm.pp trm Trm.pp rep pp_raw x )
+    ~retn:(fun {pf} -> pf "%a" pp_raw)
+  @@ fun () ->
+  let rep = Subst.compose1 ~key:trm ~data:rep x.rep in
+  {x with rep}
+
+let rec propagate ~wrt x =
+  [%trace]
+    ~call:(fun {pf} -> pf "@ %a" pp_raw x)
+    ~retn:(fun {pf} -> pf "%a" pp_raw)
+  @@ fun () ->
+  match x.pnd with
+  | (a, b) :: pnd -> (
+      let a' = Subst.norm x.rep a in
+      let b' = Subst.norm x.rep b in
+      match solve ~wrt ~xs:x.xs a' b' pnd with
+      | {solved= Some solved; wrt; fresh; pending} ->
+          let xs = Var.Set.union x.xs fresh in
+          let x = {x with xs; pnd= pending} in
+          propagate ~wrt (List.fold ~f:propagate1 solved x)
+      | {solved= None} -> {x with sat= false; pnd= []} )
+  | [] -> x
+
+(* Core operations ========================================================*)
 
 let empty =
   let rep = Subst.empty in
-  (* let rep = Option.get_exn (Subst.extend Trm.true_ rep) in
+  {xs= Var.Set.empty; sat= true; rep; pnd= []} |> check invariant
-   * let rep = Option.get_exn (Subst.extend Trm.false_ rep) in *)
-  {xs= Var.Set.empty; sat= true; rep} |> check invariant
 
 let unsat = {empty with sat= false}
 
@@ -656,17 +693,13 @@ let extend a r =
   let rep = extend_ a r.rep in
   if rep == r.rep then r else {r with rep} |> check pre_invariant
 
-let merge ~wrt a b r =
+let merge ~wrt a b x =
-  [%Trace.call fun {pf} -> pf "@ %a@ %a@ %a" Trm.pp a Trm.pp b pp r]
+  [%trace]
-  ;
+    ~call:(fun {pf} -> pf "@ %a@ %a@ %a" Trm.pp a Trm.pp b pp x)
-  ( match solve ~wrt ~xs:r.xs a b with
+    ~retn:(fun {pf} x' ->
-  | Some (xs, s) ->
+      pf "%a" pp_diff (x, x') ;
-      {r with xs= Var.Set.union r.xs xs; rep= Subst.compose r.rep s}
+      pre_invariant x' )
-  | None -> {r with sat= false} )
+  @@ fun () -> propagate ~wrt (add_to_pnd a b x)
-  |>
-  [%Trace.retn fun {pf} r' ->
-    pf "%a" pp_diff (r, r') ;
-    pre_invariant r']
 
 (** find an unproved equation between congruent terms *)
 let find_missing r =
@@ -684,12 +717,12 @@ let find_missing r =
           in
           Option.return_if need_a'_eq_b' (a', b') ) )
 
-let rec close ~wrt r =
+let rec close ~wrt x =
-  if not r.sat then r
+  if not x.sat then x
   else
-    match find_missing r with
+    match find_missing x with
-    | Some (a', b') -> close ~wrt (merge ~wrt a' b' r)
+    | Some (a', b') -> close ~wrt (merge ~wrt a' b' x)
-    | None -> r
+    | None -> x
 
 let close ~wrt r =
   [%Trace.call fun {pf} -> pf "@ %a" pp r]
@@ -718,7 +751,7 @@ let and_eq_ ~wrt a b r =
 
 let extract_xs r = (r.xs, {r with xs= Var.Set.empty})
 
-(** Exposed interface *)
+(* Exposed interface ======================================================*)
 
 let is_empty {sat; rep} =
   sat && Subst.for_alli rep ~f:(fun ~key:a ~data:a' -> Trm.equal a a')
@@ -880,7 +913,7 @@ let trms r =
 let vars r = Iter.flat_map ~f:Trm.vars (trms r)
 let fv r = Var.Set.of_iter (vars r)
 
-(** Existential Witnessing and Elimination *)
+(* Existential Witnessing and Elimination =================================*)
 
 let subst_invariant us s0 s =
   assert (s0 == s || not (Subst.equal s0 s)) ;
@@ -922,6 +955,19 @@ let solve_poly_eq us p' q' subst =
   [%Trace.retn fun {pf} subst' ->
     pf "@[%a@]" Subst.pp_diff (subst, Option.value subst' ~default:subst)]
 
+let rec solve_pending s soln =
+  match s.pending with
+  | (a, b) :: pending -> (
+      let a' = Subst.norm soln a in
+      let b' = Subst.norm soln b in
+      match solve_ a' b' {s with pending} with
+      | {solved= Some solved} as s ->
+          solve_pending {s with solved= Some []}
+            (List.fold solved soln ~f:(fun (trm, rep) soln ->
+                 Subst.compose1 ~key:trm ~data:rep soln ))
+      | {solved= None} -> None )
+  | [] -> Some soln
+
 let solve_seq_eq ~wrt us e' f' subst =
   [%Trace.call fun {pf} -> pf "@ %a = %a" Trm.pp e' Trm.pp f']
   ;
@@ -935,11 +981,14 @@ let solve_seq_eq ~wrt us e' f' subst =
       | Some n -> (a, n)
       | None -> (Trm.sized ~siz:n ~seq:a, n)
     in
-    let+ _, xs, s =
+    solve_pending
-      solve_concat ~no_fresh:true ms a n (wrt, Var.Set.empty, subst)
+      (solve_concat ms a n
-    in
+         { wrt
-    assert (Var.Set.is_empty xs) ;
+         ; no_fresh= true
-    s
+         ; fresh= Var.Set.empty
+         ; solved= Some []
+         ; pending= [] })
+      subst
   in
   ( match ((e' : Trm.t), (f' : Trm.t)) with
   | (Concat ms as c), a when x_ito_us c a ->
@@ -1309,9 +1358,7 @@ let apply_and_elim ~wrt xs s r =
       let r = trim ks r in
       (zs, r, ks)
 
-(*
+(* Replay debugging =======================================================*)
- * Replay debugging
- *)
 
 type call =
   | Add of Var.Set.t * Formula.t * t

sledge/src/test/context_test.ml

@@ -91,23 +91,24 @@ let%test_module _ =
     let%expect_test _ =
       replay
         {|(Solve_for_vars
-            (((Var (id 0) (name 2)) (Var (id 0) (name 6)) (Var (id 0) (name 8)))
+           (((Var (id 0) (name 2)) (Var (id 0) (name 6)) (Var (id 0) (name 8)))
-              ((Var (id 5) (name a0)) (Var (id 6) (name b)) (Var (id 7) (name m))
+            ((Var (id 5) (name a0)) (Var (id 6) (name b)) (Var (id 7) (name m))
-                (Var (id 8) (name a)) (Var (id 9) (name a0))))
+             (Var (id 8) (name a)) (Var (id 9) (name a0))))
-            ((xs ()) (sat true)
+           ((xs ()) (sat true)
-              (rep
+            (rep
-                (((Var (id 9) (name a0)) (Var (id 5) (name a0)))
+             (((Var (id 9) (name a0)) (Var (id 5) (name a0)))
-                  ((Var (id 8) (name a))
+              ((Var (id 8) (name a))
-                    (Concat
+               (Concat
-                      ((Sized (seq (Var (id 5) (name a0))) (siz (Z 4)))
+                ((Sized (seq (Var (id 5) (name a0))) (siz (Z 4)))
-                        (Sized (seq (Z 0)) (siz (Z 4))))))
+                 (Sized (seq (Z 0)) (siz (Z 4))))))
-                  ((Var (id 7) (name m)) (Z 8))
+              ((Var (id 7) (name m)) (Z 8))
-                  ((Var (id 6) (name b)) (Var (id 0) (name 2)))
+              ((Var (id 6) (name b)) (Var (id 0) (name 2)))
-                  ((Var (id 5) (name a0)) (Var (id 5) (name a0)))
+              ((Var (id 5) (name a0)) (Var (id 5) (name a0)))
-                  ((Var (id 0) (name 6))
+              ((Var (id 0) (name 6))
-                    (Concat
+               (Concat
-                      ((Sized (seq (Var (id 5) (name a0))) (siz (Z 4)))
+                ((Sized (seq (Var (id 5) (name a0))) (siz (Z 4)))
-                        (Sized (seq (Z 0)) (siz (Z 4))))))
+                 (Sized (seq (Z 0)) (siz (Z 4))))))
-                  ((Var (id 0) (name 2)) (Var (id 0) (name 2)))))))|} ;
+              ((Var (id 0) (name 2)) (Var (id 0) (name 2)))))
+            (pnd ())))|} ;
       [%expect {| |}]
   end )

sledge/src/test/sh_test.ml

@@ -78,11 +78,11 @@ let%test_module _ =
       pp (star p q) ;
       [%expect
         {|
-          ∃ %x_7 .   emp
+        ∃ %x_7 .   emp
 
-            0 = %x_7 ∧ emp
+          0 = %x_7 ∧ emp
 
-            0 = %x_7 ∧ emp |}]
+          0 = %x_7 ∧ emp |}]
 
     let%expect_test _ =
       let q =