diff --git a/src/cc/CC.ml b/src/cc/CC.ml
index a1bfa4c2..577519dd 100644
--- a/src/cc/CC.ml
+++ b/src/cc/CC.ml
@@ -25,7 +25,7 @@ let[@inline] find _ n = find_ n
 module Sig_tbl = CCHashtbl.Make (Signature)
 module T_tbl = Term.Tbl
 
-type propagation_reason = unit -> Lit.t list * Proof_term.step_id
+type propagation_reason = unit -> Lit.t list * Proof.Pterm.delayed
 
 module Handler_action = struct
   type t =
@@ -38,7 +38,7 @@ end
 
 module Result_action = struct
   type t = Act_propagate of { lit: Lit.t; reason: propagation_reason }
-  type conflict = Conflict of Lit.t list * Proof_term.step_id
+  type conflict = Conflict of Lit.t list * Sidekick_proof.Step.id
   type or_conflict = (t list, conflict) result
 end
 
@@ -50,7 +50,7 @@ type t = {
   view_as_cc: view_as_cc;
   tst: Term.store;
   stat: Stat.t;
-  proof: Proof_trace.t;
+  proof: Proof.Tracer.t;
   tbl: e_node T_tbl.t; (* internalization [term -> e_node] *)
   signatures_tbl: e_node Sig_tbl.t;
   (* map a signature to the corresponding e_node in some equivalence class.
@@ -108,7 +108,7 @@ let n_bool self b =
     n_false self
 
 let[@inline] term_store self = self.tst
-let[@inline] proof self = self.proof
+let[@inline] proof_tracer self = self.proof
 let[@inline] stat self = self.stat
 
 let allocate_bitfield self ~descr : bitfield =
@@ -216,7 +216,8 @@ let[@unroll 2] rec reroot_expl (self : t) (n : e_node) : unit =
 
 exception E_confl of Result_action.conflict
 
-let raise_conflict_ (cc : t) ~th (e : Lit.t list) (p : Proof_term.step_id) : _ =
+let raise_conflict_ (cc : t) ~th (e : Lit.t list) (p : Sidekick_proof.Step.id) :
+    _ =
   Profile.instant "cc.conflict";
   (* clear tasks queue *)
   Vec.clear cc.pending;
@@ -287,59 +288,72 @@ module Expl_state = struct
   type t = {
     mutable lits: Lit.t list;
     mutable th_lemmas:
-      (Lit.t * (Lit.t * Lit.t list) list * Proof_term.step_id) list;
+      (Lit.t * (Lit.t * Lit.t list) list * Proof.Pterm.delayed) list;
+    mutable local_gen: int;
   }
 
-  let create () : t = { lits = []; th_lemmas = [] }
+  let create () : t = { lits = []; th_lemmas = []; local_gen = -2 }
   let[@inline] copy self : t = { self with lits = self.lits }
   let[@inline] add_lit (self : t) lit = self.lits <- lit :: self.lits
 
+  let[@inline] new_local_id (self : t) : Proof.Pterm.local_ref =
+    let n = self.local_gen in
+    self.local_gen <- n - 1;
+    n
+
   let[@inline] add_th (self : t) lit hyps pr : unit =
     self.th_lemmas <- (lit, hyps, pr) :: self.th_lemmas
 
   let merge self other =
-    let { lits = o_lits; th_lemmas = o_lemmas } = other in
+    let { lits = o_lits; th_lemmas = o_lemmas; local_gen = o_l } = other in
     self.lits <- List.rev_append o_lits self.lits;
     self.th_lemmas <- List.rev_append o_lemmas self.th_lemmas;
+    self.local_gen <- min self.local_gen o_l;
     ()
 
   (* proof of [\/_i ¬lits[i]] *)
-  let proof_of_th_lemmas (self : t) (proof : Proof_trace.t) : Proof_term.step_id
-      =
+  let proof_of_th_lemmas (self : t) : Proof.Pterm.delayed =
+    Proof.Pterm.delay @@ fun () ->
+    let bs = ref [] in
+    let bind (t : Proof.Pterm.t) : Proof.Pterm.local_ref =
+      let n = new_local_id self in
+      bs := (n, t) :: !bs;
+      n
+    in
+
     let p_lits1 = List.rev_map Lit.neg self.lits in
     let p_lits2 =
       self.th_lemmas |> List.rev_map (fun (lit_t_u, _, _) -> Lit.neg lit_t_u)
     in
-    let p_cc =
-      Proof_trace.add_step proof @@ fun () ->
-      Proof_core.lemma_cc (List.rev_append p_lits1 p_lits2)
-    in
+    let p_cc = Proof.Core_rules.lemma_cc (List.rev_append p_lits1 p_lits2) in
     let resolve_with_th_proof pr (lit_t_u, sub_proofs, pr_th) =
+      let pr_th = pr_th () in
       (* pr_th: [sub_proofs |- t=u].
-            now resolve away [sub_proofs] to get literals that were
-            asserted in the congruence closure *)
+          now resolve away [sub_proofs] to get literals that were
+          asserted in the congruence closure *)
       let pr_th =
         List.fold_left
           (fun pr_th (lit_i, hyps_i) ->
             (* [hyps_i |- lit_i] *)
             let lemma_i =
-              Proof_trace.add_step proof @@ fun () ->
-              Proof_core.lemma_cc (lit_i :: List.rev_map Lit.neg hyps_i)
+              bind
+              @@ Proof.Core_rules.lemma_cc (lit_i :: List.rev_map Lit.neg hyps_i)
             in
             (* resolve [lit_i] away. *)
-            Proof_trace.add_step proof @@ fun () ->
-            Proof_core.proof_res ~pivot:(Lit.term lit_i) lemma_i pr_th)
+            Proof.Core_rules.proof_res ~pivot:(Lit.term lit_i) lemma_i
+              (bind pr_th))
           pr_th sub_proofs
       in
-      Proof_trace.add_step proof @@ fun () ->
-      Proof_core.proof_res ~pivot:(Lit.term lit_t_u) pr_th pr
+      Proof.Core_rules.proof_res ~pivot:(Lit.term lit_t_u) (bind pr_th)
+        (bind pr)
     in
     (* resolve with theory proofs responsible for some merges, if any. *)
-    List.fold_left resolve_with_th_proof p_cc self.th_lemmas
+    let body = List.fold_left resolve_with_th_proof p_cc self.th_lemmas in
+    Proof.Pterm.let_ (List.rev !bs) body
 
   let to_resolved_expl (self : t) : Resolved_expl.t =
     (* FIXME: package the th lemmas too *)
-    let { lits; th_lemmas = _ } = self in
+    let { lits; th_lemmas = _; local_gen = _ } = self in
     let s2 = copy self in
     let pr proof = proof_of_th_lemmas s2 proof in
     { Resolved_expl.lits; pr }
@@ -507,10 +521,10 @@ let n_is_bool_value (self : t) n : bool =
    asserted literals needed in the explanation (which is useful for
    the SAT solver), and [pr] is a proof, including sub-proofs for theory
    merges. *)
-let lits_and_proof_of_expl (self : t) (st : Expl_state.t) :
-    Lit.t list * Proof_term.step_id =
-  let { Expl_state.lits; th_lemmas = _ } = st in
-  let pr = Expl_state.proof_of_th_lemmas st self.proof in
+let lits_and_proof_of_expl (_self : t) (st : Expl_state.t) :
+    Lit.t list * Proof.Pterm.delayed =
+  let { Expl_state.lits; th_lemmas = _; local_gen = _ } = st in
+  let pr = Expl_state.proof_of_th_lemmas st in
   lits, pr
 
 (* main CC algo: add terms from [pending] to the signature table,
@@ -602,6 +616,7 @@ and task_merge_ self a b e_ab : unit =
 
       (* regular conflict *)
       let lits, pr = lits_and_proof_of_expl self expl_st in
+      let pr = Proof.Tracer.add_step self.proof pr in
       raise_conflict_ self ~th:!th (List.rev_map Lit.neg lits) pr
     );
     (* We will merge [r_from] into [r_into].
@@ -759,6 +774,7 @@ and raise_conflict_from_expl self (expl : Expl.t) : 'a =
   let lits, pr = lits_and_proof_of_expl self st in
   let c = List.rev_map Lit.neg lits in
   let th = st.th_lemmas <> [] in
+  let pr = Proof.Tracer.add_step self.proof pr in
   raise_conflict_ self ~th c pr
 
 let add_iter self it : unit = it (fun t -> ignore @@ add_term_rec_ self t)
@@ -840,7 +856,8 @@ let[@inline] on_propagate self = Event.of_emitter self.on_propagate
 let[@inline] on_is_subterm self = Event.of_emitter self.on_is_subterm
 
 let create_ ?(stat = Stat.global) ?(size = `Big) (tst : Term.store)
-    (proof : Proof_trace.t) ~view_as_cc : t =
+    (proof : #Proof.Tracer.t) ~view_as_cc : t =
+  let proof = (proof : #Proof.Tracer.t :> Proof.Tracer.t) in
   let size =
     match size with
     | `Small -> 128
@@ -938,7 +955,11 @@ end
 
 module type BUILD = sig
   val create :
-    ?stat:Stat.t -> ?size:[ `Small | `Big ] -> Term.store -> Proof_trace.t -> t
+    ?stat:Stat.t ->
+    ?size:[ `Small | `Big ] ->
+    Term.store ->
+    #Proof.Tracer.t ->
+    t
   (** Create a new congruence closure.
 
       @param term_store used to be able to create new terms. All terms
diff --git a/src/cc/CC.mli b/src/cc/CC.mli
index e64de856..9d1ad0aa 100644
--- a/src/cc/CC.mli
+++ b/src/cc/CC.mli
@@ -1,6 +1,20 @@
-(** Main congruence closure type. *)
+(** Main congruence closure signature.
 
-open Sidekick_core
+    The congruence closure handles the theory QF_UF (uninterpreted
+    function symbols).
+    It is also responsible for {i theory combination}, and provides
+    a general framework for equality reasoning that other
+    theories piggyback on.
+
+    For example, the theory of datatypes relies on the congruence closure
+    to do most of the work, and "only" adds injectivity/disjointness/acyclicity
+    lemmas when needed.
+
+    Similarly, a theory of arrays would hook into the congruence closure and
+    assert (dis)equalities as needed.
+*)
+
+open Types_
 
 type e_node = E_node.t
 (** A node of the congruence closure *)
@@ -20,22 +34,6 @@ type bitfield = Bits.field
     All fields are initially 0, are backtracked automatically,
     and are merged automatically when classes are merged. *)
 
-(** Main congruence closure signature.
-
-    The congruence closure handles the theory QF_UF (uninterpreted
-    function symbols).
-    It is also responsible for {i theory combination}, and provides
-    a general framework for equality reasoning that other
-    theories piggyback on.
-
-    For example, the theory of datatypes relies on the congruence closure
-    to do most of the work, and "only" adds injectivity/disjointness/acyclicity
-    lemmas when needed.
-
-    Similarly, a theory of arrays would hook into the congruence closure and
-    assert (dis)equalities as needed.
-*)
-
 type t
 (** The congruence closure object.
       It contains a fair amount of state and is mutable
@@ -44,7 +42,7 @@ type t
 (** {3 Accessors} *)
 
 val term_store : t -> Term.store
-val proof : t -> Proof_trace.t
+val proof_tracer : t -> Proof.Tracer.t
 val stat : t -> Stat.t
 
 val find : t -> e_node -> repr
@@ -78,7 +76,7 @@ val set_bitfield : t -> bitfield -> bool -> E_node.t -> unit
 (** Set the bitfield for the e_node. This will be backtracked.
       See {!E_node.bitfield}. *)
 
-type propagation_reason = unit -> Lit.t list * Proof_term.step_id
+type propagation_reason = unit -> Lit.t list * Proof.Pterm.delayed
 
 (** Handler Actions
 
@@ -120,7 +118,7 @@ module Result_action : sig
           to not propagate and only trigger conflicts. *)
 
   type conflict =
-    | Conflict of Lit.t list * Proof_term.step_id
+    | Conflict of Lit.t list * Sidekick_proof.Step.id
         (** [raise_conflict (c,pr)] declares that [c] is a tautology of
           the theory of congruence.
           @param pr the proof of [c] being a tautology *)
@@ -168,10 +166,7 @@ val on_conflict : t -> (ev_on_conflict, unit) Event.t
       closure triggers a conflict by asserting the tautology [c]. *)
 
 val on_propagate :
-  t ->
-  ( t * Lit.t * (unit -> Lit.t list * Proof_term.step_id),
-    Handler_action.t list )
-  Event.t
+  t -> (t * Lit.t * propagation_reason, Handler_action.t list) Event.t
 (** [ev_on_propagate Lit.t reason] is emitted whenever [reason() => Lit.t]
       is a propagated lemma. See {!CC_ACTIONS.propagate}. *)
 
@@ -266,7 +261,11 @@ end
 
 module type BUILD = sig
   val create :
-    ?stat:Stat.t -> ?size:[ `Small | `Big ] -> Term.store -> Proof_trace.t -> t
+    ?stat:Stat.t ->
+    ?size:[ `Small | `Big ] ->
+    Term.store ->
+    #Proof.Tracer.t ->
+    t
   (** Create a new congruence closure.
 
       @param term_store used to be able to create new terms. All terms
@@ -282,7 +281,7 @@ val create :
   ?stat:Stat.t ->
   ?size:[ `Small | `Big ] ->
   Term.store ->
-  Proof_trace.t ->
+  #Proof.Tracer.t ->
   t
 (** Create a new congruence closure.
 
@@ -292,7 +291,7 @@ val create :
   *)
 
 val create_default :
-  ?stat:Stat.t -> ?size:[ `Small | `Big ] -> Term.store -> Proof_trace.t -> t
+  ?stat:Stat.t -> ?size:[ `Small | `Big ] -> Term.store -> Proof.Tracer.t -> t
 (** Same as {!create} but with the default CC view *)
 
 (**/**)
diff --git a/src/cc/dune b/src/cc/dune
index cd929144..ffa28dd9 100644
--- a/src/cc/dune
+++ b/src/cc/dune
@@ -3,5 +3,6 @@
  (public_name sidekick.cc)
  (synopsis "main congruence closure implementation")
  (private_modules signature)
- (libraries containers iter sidekick.sigs sidekick.core sidekick.util)
+ (libraries containers iter sidekick.sigs sidekick.core sidekick.util
+   sidekick.proof)
  (flags :standard -open Sidekick_util))
diff --git a/src/cc/expl.mli b/src/cc/expl.mli
index efa26063..cfb5ac2f 100644
--- a/src/cc/expl.mli
+++ b/src/cc/expl.mli
@@ -26,7 +26,11 @@ val mk_list : t list -> t
 val mk_congruence : E_node.t -> E_node.t -> t
 
 val mk_theory :
-  Term.t -> Term.t -> (Term.t * Term.t * t list) list -> Proof_term.step_id -> t
+  Term.t ->
+  Term.t ->
+  (Term.t * Term.t * t list) list ->
+  Proof.Pterm.delayed ->
+  t
 (** [mk_theory t u expl_sets pr] builds a theory explanation for
     why [|- t=u]. It depends on sub-explanations [expl_sets] which
     are tuples [ (t_i, u_i, expls_i) ] where [expls_i] are
diff --git a/src/cc/resolved_expl.ml b/src/cc/resolved_expl.ml
index c16c1edd..78d4fb73 100644
--- a/src/cc/resolved_expl.ml
+++ b/src/cc/resolved_expl.ml
@@ -1,6 +1,6 @@
 open Types_
 
-type t = { lits: Lit.t list; pr: Proof_trace.t -> Proof_term.step_id }
+type t = { lits: Lit.t list; pr: Proof.Pterm.delayed }
 
 let pp out (self : t) =
   Fmt.fprintf out "(@[resolved-expl@ %a@])" (Util.pp_list Lit.pp) self.lits
diff --git a/src/cc/resolved_expl.mli b/src/cc/resolved_expl.mli
index 537a11be..b136970e 100644
--- a/src/cc/resolved_expl.mli
+++ b/src/cc/resolved_expl.mli
@@ -1,17 +1,17 @@
 (** Resolved explanations.
 
-      The congruence closure keeps explanations for why terms are in the same
-      class. However these are represented in a compact, cheap form.
-      To use these explanations we need to {b resolve} them into a
-      resolved explanation, typically a list of
-      literals that are true in the current trail and are responsible for
-      merges.
+    The congruence closure keeps explanations for why terms are in the same
+    class. However these are represented in a compact, cheap form.
+    To use these explanations we need to {b resolve} them into a
+    resolved explanation, typically a list of
+    literals that are true in the current trail and are responsible for
+    merges.
 
-      However, we can also have merged classes because they have the same value
-      in the current model. *)
+    However, we can also have merged classes because they have the same value
+    in the current model. *)
 
 open Types_
 
-type t = { lits: Lit.t list; pr: Proof_trace.t -> Proof_term.step_id }
+type t = { lits: Lit.t list; pr: Proof.Pterm.delayed }
 
 include Sidekick_sigs.PRINT with type t := t
diff --git a/src/cc/types_.ml b/src/cc/types_.ml
index 86fba51b..e82bd817 100644
--- a/src/cc/types_.ml
+++ b/src/cc/types_.ml
@@ -1,4 +1,5 @@
 include Sidekick_core
+module Proof = Sidekick_proof
 
 type e_node = {
   n_term: Term.t;
@@ -36,4 +37,4 @@ and explanation =
       Term.t
       * Term.t
       * (Term.t * Term.t * explanation list) list
-      * Proof_term.step_id
+      * Proof.Pterm.delayed