diff --git a/src/th-bool-dyn/Sidekick_th_bool_dyn.ml b/src/th-bool-dyn/Sidekick_th_bool_dyn.ml
index 5f90af14..0d985914 100644
--- a/src/th-bool-dyn/Sidekick_th_bool_dyn.ml
+++ b/src/th-bool-dyn/Sidekick_th_bool_dyn.ml
@@ -65,18 +65,18 @@ end = struct
   (* TODO: share this with th-bool-static by way of a library for
      boolean simplification? (also handle one-point rule and the likes) *)
   let simplify (self : state) (simp : Simplify.t) (t : T.t) :
-      (T.t * Proof_step.id Iter.t) option =
+      (T.t * Proof.Step.id Iter.t) option =
     let tst = self.tst in
 
     let proof = Simplify.proof simp in
     let steps = ref [] in
     let add_step_ s = steps := s :: !steps in
-    let mk_step_ r = Proof_trace.add_step proof r in
+    let mk_step_ r = Proof.Tracer.add_step proof r in
 
     let add_step_eq a b ~using ~c0 : unit =
       add_step_ @@ mk_step_
       @@ fun () ->
-      Proof_core.lemma_rw_clause c0 ~using
+      Proof.Core_rules.lemma_rw_clause c0 ~using
         ~res:[ Lit.atom tst (A.mk_bool tst (B_eq (a, b))) ]
     in
 
@@ -170,7 +170,7 @@ end = struct
   let preprocess_ (self : state) (_p : SMT.Preprocess.t) ~is_sub:_ ~recurse:_
       (module PA : SI.PREPROCESS_ACTS) (t : T.t) : T.t option =
     Log.debugf 50 (fun k -> k "(@[th-bool.dyn.preprocess@ %a@])" T.pp_debug t);
-    let[@inline] mk_step_ r = Proof_trace.add_step PA.proof r in
+    let[@inline] mk_step_ r = Proof.Tracer.add_step PA.proof_tracer r in
 
     match A.view_as_bool t with
     | B_ite (a, b, c) ->
@@ -201,8 +201,6 @@ end = struct
       SI.add_clause_permanent solver acts c pr
     in
 
-    let[@inline] mk_step_ r = Proof_trace.add_step (SI.proof solver) r in
-
     (* handle boolean equality *)
     let equiv_ ~is_xor a b : unit =
       (* [a xor b] is [(¬a) = b] *)
@@ -218,33 +216,37 @@ end = struct
       add_axiom
         [ Lit.neg lit; Lit.neg a; b ]
         (if is_xor then
-          mk_step_ @@ fun () -> Proof_rules.lemma_bool_c "xor-e+" [ t ]
+          fun () ->
+        Proof_rules.lemma_bool_c "xor-e+" [ t ]
         else
-          mk_step_ @@ fun () ->
-          Proof_rules.lemma_bool_c "eq-e" [ t; Lit.term a ]);
+          fun () ->
+        Proof_rules.lemma_bool_c "eq-e" [ t; Lit.term a ]);
 
       add_axiom
         [ Lit.neg lit; Lit.neg b; a ]
         (if is_xor then
-          mk_step_ @@ fun () -> Proof_rules.lemma_bool_c "xor-e-" [ t ]
+          fun () ->
+        Proof_rules.lemma_bool_c "xor-e-" [ t ]
         else
-          mk_step_ @@ fun () ->
-          Proof_rules.lemma_bool_c "eq-e" [ t; Lit.term b ]);
+          fun () ->
+        Proof_rules.lemma_bool_c "eq-e" [ t; Lit.term b ]);
 
       add_axiom [ lit; a; b ]
         (if is_xor then
-          mk_step_ @@ fun () ->
-          Proof_rules.lemma_bool_c "xor-i" [ t; Lit.term a ]
+          fun () ->
+        Proof_rules.lemma_bool_c "xor-i" [ t; Lit.term a ]
         else
-          mk_step_ @@ fun () -> Proof_rules.lemma_bool_c "eq-i+" [ t ]);
+          fun () ->
+        Proof_rules.lemma_bool_c "eq-i+" [ t ]);
 
       add_axiom
         [ lit; Lit.neg a; Lit.neg b ]
         (if is_xor then
-          mk_step_ @@ fun () ->
-          Proof_rules.lemma_bool_c "xor-i" [ t; Lit.term b ]
+          fun () ->
+        Proof_rules.lemma_bool_c "xor-i" [ t; Lit.term b ]
         else
-          mk_step_ @@ fun () -> Proof_rules.lemma_bool_c "eq-i-" [ t ])
+          fun () ->
+        Proof_rules.lemma_bool_c "eq-i-" [ t ])
     in
 
     match v with
@@ -254,7 +256,7 @@ end = struct
     | B_bool false ->
       SI.add_clause_permanent solver acts
         [ Lit.neg lit ]
-        (mk_step_ @@ fun () -> Proof_core.lemma_true (Lit.term lit))
+        (fun () -> Proof.Core_rules.lemma_true (Lit.term lit))
     | _ when expanded self lit -> () (* already done *)
     | B_and l ->
       let subs = List.map (Lit.atom self.tst) l in
@@ -265,14 +267,12 @@ end = struct
           (fun sub ->
             add_axiom
               [ Lit.neg lit; sub ]
-              ( mk_step_ @@ fun () ->
-                Proof_rules.lemma_bool_c "and-e" [ t; Lit.term sub ] ))
+              (fun () -> Proof_rules.lemma_bool_c "and-e" [ t; Lit.term sub ]))
           subs
       else (
         (* axiom [¬(and …t_i)=> \/_i (¬ t_i)], only in final-check *)
         let c = Lit.neg lit :: List.map Lit.neg subs in
-        add_axiom c
-          (mk_step_ @@ fun () -> Proof_rules.lemma_bool_c "and-i" [ t ])
+        add_axiom c (fun () -> Proof_rules.lemma_bool_c "and-i" [ t ])
       )
     | B_or l ->
       let subs = List.map (Lit.atom self.tst) l in
@@ -283,13 +283,12 @@ end = struct
           (fun sub ->
             add_axiom
               [ Lit.neg lit; Lit.neg sub ]
-              ( mk_step_ @@ fun () ->
-                Proof_rules.lemma_bool_c "or-i" [ t; Lit.term sub ] ))
+              (fun () -> Proof_rules.lemma_bool_c "or-i" [ t; Lit.term sub ]))
           subs
       else (
         (* axiom [lit => \/_i subs_i] *)
         let c = Lit.neg lit :: subs in
-        add_axiom c (mk_step_ @@ fun () -> Proof_rules.lemma_bool_c "or-e" [ t ])
+        add_axiom c (fun () -> Proof_rules.lemma_bool_c "or-e" [ t ])
       )
     | B_imply (a, b) ->
       let a = Lit.atom self.tst a in
@@ -297,19 +296,16 @@ end = struct
       if Lit.sign lit then (
         (* axiom [lit => a => b] *)
         let c = [ Lit.neg lit; Lit.neg a; b ] in
-        add_axiom c
-          (mk_step_ @@ fun () -> Proof_rules.lemma_bool_c "imp-e" [ t ])
+        add_axiom c (fun () -> Proof_rules.lemma_bool_c "imp-e" [ t ])
       ) else if not @@ Lit.sign lit then (
         (* propagate [¬ lit => ¬b] and [¬lit => a] *)
         add_axiom
           [ a; Lit.neg lit ]
-          ( mk_step_ @@ fun () ->
-            Proof_rules.lemma_bool_c "imp-i" [ t; Lit.term a ] );
+          (fun () -> Proof_rules.lemma_bool_c "imp-i" [ t; Lit.term a ]);
 
         add_axiom
           [ Lit.neg b; Lit.neg lit ]
-          ( mk_step_ @@ fun () ->
-            Proof_rules.lemma_bool_c "imp-i" [ t; Lit.term b ] )
+          (fun () -> Proof_rules.lemma_bool_c "imp-i" [ t; Lit.term b ])
       )
     | B_ite (a, b, c) ->
       assert (T.is_bool b);
@@ -319,10 +315,10 @@ end = struct
       let lit_a = Lit.atom self.tst a in
       add_axiom
         [ Lit.neg lit_a; Lit.make_eq self.tst t b ]
-        (mk_step_ @@ fun () -> Proof_rules.lemma_ite_true ~ite:t);
+        (fun () -> Proof_rules.lemma_ite_true ~ite:t);
       add_axiom
         [ Lit.neg lit; lit_a; Lit.make_eq self.tst t c ]
-        (mk_step_ @@ fun () -> Proof_rules.lemma_ite_false ~ite:t)
+        (fun () -> Proof_rules.lemma_ite_false ~ite:t)
     | B_equiv (a, b) ->
       let a = Lit.atom self.tst a in
       let b = Lit.atom self.tst b in
diff --git a/src/th-bool-dyn/intf.ml b/src/th-bool-dyn/intf.ml
index 8552ced8..18f6d8f7 100644
--- a/src/th-bool-dyn/intf.ml
+++ b/src/th-bool-dyn/intf.ml
@@ -1,4 +1,5 @@
 open Sidekick_core
+module Proof = Sidekick_proof
 module SMT = Sidekick_smt_solver
 module Simplify = Sidekick_simplify
 
diff --git a/src/th-bool-dyn/proof_rules.ml b/src/th-bool-dyn/proof_rules.ml
index 82288385..f57986a3 100644
--- a/src/th-bool-dyn/proof_rules.ml
+++ b/src/th-bool-dyn/proof_rules.ml
@@ -1,19 +1,20 @@
 open Sidekick_core
+module Proof = Sidekick_proof
 
 type term = Term.t
 type lit = Lit.t
 
-let lemma_bool_tauto lits : Proof_term.t =
-  Proof_term.apply_rule "bool.tauto" ~lits
+let lemma_bool_tauto lits : Proof.Pterm.t =
+  Proof.Pterm.apply_rule "bool.tauto" ~lits
 
-let lemma_bool_c name terms : Proof_term.t =
-  Proof_term.apply_rule ("bool.c." ^ name) ~terms
+let lemma_bool_c name terms : Proof.Pterm.t =
+  Proof.Pterm.apply_rule ("bool.c." ^ name) ~terms
 
-let lemma_bool_equiv t u : Proof_term.t =
-  Proof_term.apply_rule "bool.equiv" ~terms:[ t; u ]
+let lemma_bool_equiv t u : Proof.Pterm.t =
+  Proof.Pterm.apply_rule "bool.equiv" ~terms:[ t; u ]
 
-let lemma_ite_true ~ite : Proof_term.t =
-  Proof_term.apply_rule "bool.ite.true" ~terms:[ ite ]
+let lemma_ite_true ~ite : Proof.Pterm.t =
+  Proof.Pterm.apply_rule "bool.ite.true" ~terms:[ ite ]
 
-let lemma_ite_false ~ite : Proof_term.t =
-  Proof_term.apply_rule "bool.ite.false" ~terms:[ ite ]
+let lemma_ite_false ~ite : Proof.Pterm.t =
+  Proof.Pterm.apply_rule "bool.ite.false" ~terms:[ ite ]
diff --git a/src/th-bool-dyn/proof_rules.mli b/src/th-bool-dyn/proof_rules.mli
index 0379b4c5..38a72872 100644
--- a/src/th-bool-dyn/proof_rules.mli
+++ b/src/th-bool-dyn/proof_rules.mli
@@ -1,20 +1,21 @@
 open Sidekick_core
+module Proof = Sidekick_proof
 
 type term = Term.t
 type lit = Lit.t
 
-val lemma_bool_tauto : lit list -> Proof_term.t
+val lemma_bool_tauto : lit list -> Proof.Pterm.t
 (** Boolean tautology lemma (clause) *)
 
-val lemma_bool_c : string -> term list -> Proof_term.t
+val lemma_bool_c : string -> term list -> Proof.Pterm.t
 (** Basic boolean logic lemma for a clause [|- c].
-      [proof_bool_c b name cs] is the Proof_term.t designated by [name]. *)
+      [proof_bool_c b name cs] is the Proof.Pterm.t designated by [name]. *)
 
-val lemma_bool_equiv : term -> term -> Proof_term.t
+val lemma_bool_equiv : term -> term -> Proof.Pterm.t
 (** Boolean tautology lemma (equivalence) *)
 
-val lemma_ite_true : ite:term -> Proof_term.t
+val lemma_ite_true : ite:term -> Proof.Pterm.t
 (** lemma [a ==> ite a b c = b] *)
 
-val lemma_ite_false : ite:term -> Proof_term.t
+val lemma_ite_false : ite:term -> Proof.Pterm.t
 (** lemma [¬a ==> ite a b c = c] *)
diff --git a/src/th-bool-static/Sidekick_th_bool_static.ml b/src/th-bool-static/Sidekick_th_bool_static.ml
index a82937ae..7cc60b91 100644
--- a/src/th-bool-static/Sidekick_th_bool_static.ml
+++ b/src/th-bool-static/Sidekick_th_bool_static.ml
@@ -55,18 +55,18 @@ end = struct
     aux T.Set.empty t
 
   let simplify (self : state) (simp : Simplify.t) (t : T.t) :
-      (T.t * Proof_step.id Iter.t) option =
+      (T.t * Proof.step_id Iter.t) option =
     let tst = self.tst in
 
     let proof = Simplify.proof simp in
     let steps = ref [] in
     let add_step_ s = steps := s :: !steps in
-    let mk_step_ r = Proof_trace.add_step proof r in
+    let mk_step_ r = Proof.Tracer.add_step proof r in
 
     let add_step_eq a b ~using ~c0 : unit =
       add_step_ @@ mk_step_
       @@ fun () ->
-      Proof_core.lemma_rw_clause c0 ~using
+      Proof.Core_rules.lemma_rw_clause c0 ~using
         ~res:[ Lit.atom tst (A.mk_bool tst (B_eq (a, b))) ]
     in
 
@@ -160,7 +160,7 @@ end = struct
   let cnf (self : state) (_preproc : SMT.Preprocess.t) ~is_sub:_ ~recurse
       (module PA : SI.PREPROCESS_ACTS) (t : T.t) : T.t option =
     Log.debugf 50 (fun k -> k "(@[th-bool.cnf@ %a@])" T.pp t);
-    let[@inline] mk_step_ r = Proof_trace.add_step PA.proof r in
+    let[@inline] mk_step_ r = Proof.Tracer.add_step PA.proof_tracer r in
 
     (* handle boolean equality *)
     let equiv_ (self : state) ~is_xor ~t ~box_t t_a t_b : unit =
diff --git a/src/th-bool-static/intf.ml b/src/th-bool-static/intf.ml
index 8552ced8..18f6d8f7 100644
--- a/src/th-bool-static/intf.ml
+++ b/src/th-bool-static/intf.ml
@@ -1,4 +1,5 @@
 open Sidekick_core
+module Proof = Sidekick_proof
 module SMT = Sidekick_smt_solver
 module Simplify = Sidekick_simplify
 
diff --git a/src/th-bool-static/proof_rules.ml b/src/th-bool-static/proof_rules.ml
index 82288385..f57986a3 100644
--- a/src/th-bool-static/proof_rules.ml
+++ b/src/th-bool-static/proof_rules.ml
@@ -1,19 +1,20 @@
 open Sidekick_core
+module Proof = Sidekick_proof
 
 type term = Term.t
 type lit = Lit.t
 
-let lemma_bool_tauto lits : Proof_term.t =
-  Proof_term.apply_rule "bool.tauto" ~lits
+let lemma_bool_tauto lits : Proof.Pterm.t =
+  Proof.Pterm.apply_rule "bool.tauto" ~lits
 
-let lemma_bool_c name terms : Proof_term.t =
-  Proof_term.apply_rule ("bool.c." ^ name) ~terms
+let lemma_bool_c name terms : Proof.Pterm.t =
+  Proof.Pterm.apply_rule ("bool.c." ^ name) ~terms
 
-let lemma_bool_equiv t u : Proof_term.t =
-  Proof_term.apply_rule "bool.equiv" ~terms:[ t; u ]
+let lemma_bool_equiv t u : Proof.Pterm.t =
+  Proof.Pterm.apply_rule "bool.equiv" ~terms:[ t; u ]
 
-let lemma_ite_true ~ite : Proof_term.t =
-  Proof_term.apply_rule "bool.ite.true" ~terms:[ ite ]
+let lemma_ite_true ~ite : Proof.Pterm.t =
+  Proof.Pterm.apply_rule "bool.ite.true" ~terms:[ ite ]
 
-let lemma_ite_false ~ite : Proof_term.t =
-  Proof_term.apply_rule "bool.ite.false" ~terms:[ ite ]
+let lemma_ite_false ~ite : Proof.Pterm.t =
+  Proof.Pterm.apply_rule "bool.ite.false" ~terms:[ ite ]
diff --git a/src/th-bool-static/proof_rules.mli b/src/th-bool-static/proof_rules.mli
index 0379b4c5..5f98bfd2 100644
--- a/src/th-bool-static/proof_rules.mli
+++ b/src/th-bool-static/proof_rules.mli
@@ -1,20 +1,21 @@
 open Sidekick_core
+module Proof = Sidekick_proof
 
 type term = Term.t
 type lit = Lit.t
 
-val lemma_bool_tauto : lit list -> Proof_term.t
+val lemma_bool_tauto : lit list -> Proof.Pterm.t
 (** Boolean tautology lemma (clause) *)
 
-val lemma_bool_c : string -> term list -> Proof_term.t
+val lemma_bool_c : string -> term list -> Proof.Pterm.t
 (** Basic boolean logic lemma for a clause [|- c].
       [proof_bool_c b name cs] is the Proof_term.t designated by [name]. *)
 
-val lemma_bool_equiv : term -> term -> Proof_term.t
+val lemma_bool_equiv : term -> term -> Proof.Pterm.t
 (** Boolean tautology lemma (equivalence) *)
 
-val lemma_ite_true : ite:term -> Proof_term.t
+val lemma_ite_true : ite:term -> Proof.Pterm.t
 (** lemma [a ==> ite a b c = b] *)
 
-val lemma_ite_false : ite:term -> Proof_term.t
+val lemma_ite_false : ite:term -> Proof.Pterm.t
 (** lemma [¬a ==> ite a b c = c] *)
diff --git a/src/th-cstor/Sidekick_th_cstor.ml b/src/th-cstor/Sidekick_th_cstor.ml
index 6708ab35..f9663f38 100644
--- a/src/th-cstor/Sidekick_th_cstor.ml
+++ b/src/th-cstor/Sidekick_th_cstor.ml
@@ -1,4 +1,5 @@
 open Sidekick_core
+module Proof = Sidekick_proof
 module SMT = Sidekick_smt_solver
 module SI = SMT.Solver_internal
 module T = Term
@@ -9,7 +10,7 @@ let name = "th-cstor"
 
 module type ARG = sig
   val view_as_cstor : Term.t -> (Const.t, Term.t) cstor_view
-  val lemma_cstor : Lit.t list -> Proof_term.t
+  val lemma_cstor : Lit.t list -> Proof.Pterm.t
 end
 
 module Make (A : ARG) : sig
diff --git a/src/th-cstor/Sidekick_th_cstor.mli b/src/th-cstor/Sidekick_th_cstor.mli
index b292ba6b..b3eb22cb 100644
--- a/src/th-cstor/Sidekick_th_cstor.mli
+++ b/src/th-cstor/Sidekick_th_cstor.mli
@@ -7,7 +7,7 @@ type ('c, 't) cstor_view = T_cstor of 'c * 't array | T_other of 't
 
 module type ARG = sig
   val view_as_cstor : Term.t -> (Const.t, Term.t) cstor_view
-  val lemma_cstor : Lit.t list -> Proof_term.t
+  val lemma_cstor : Lit.t list -> Sidekick_proof.Pterm.t
 end
 
 val make : (module ARG) -> SMT.theory
diff --git a/src/th-data/Sidekick_th_data.ml b/src/th-data/Sidekick_th_data.ml
index c86bda62..fb548bc1 100644
--- a/src/th-data/Sidekick_th_data.ml
+++ b/src/th-data/Sidekick_th_data.ml
@@ -2,6 +2,7 @@
 
 open Sidekick_core
 open Sidekick_cc
+module Proof = Sidekick_proof
 include Th_intf
 module SI = SMT.Solver_internal
 module Model_builder = SMT.Model_builder
@@ -174,7 +175,7 @@ end = struct
         Some { c_n = n; c_cstor = cstor; c_args = args }, []
       | _ -> None, []
 
-    let merge cc state n1 c1 n2 c2 e_n1_n2 : _ result =
+    let merge _cc state n1 c1 n2 c2 e_n1_n2 : _ result =
       Log.debugf 5 (fun k ->
           k "(@[%s.merge@ (@[:c1 %a@ %a@])@ (@[:c2 %a@ %a@])@])" name E_node.pp
             n1 pp c1 E_node.pp n2 pp c2);
@@ -189,14 +190,12 @@ end = struct
           pr
       in
 
-      let proof = CC.proof cc in
       if A.Cstor.equal c1.c_cstor c2.c_cstor then (
         (* same function: injectivity *)
         let expl_merge i =
           let t1 = E_node.term c1.c_n in
           let t2 = E_node.term c2.c_n in
-          mk_expl t1 t2 @@ Proof_trace.add_step proof
-          @@ fun () -> Proof_rules.lemma_cstor_inj t1 t2 i
+          mk_expl t1 t2 @@ fun () -> Proof_rules.lemma_cstor_inj t1 t2 i
         in
 
         assert (List.length c1.c_args = List.length c2.c_args);
@@ -212,8 +211,7 @@ end = struct
         (* different function: disjointness *)
         let expl =
           let t1 = E_node.term c1.c_n and t2 = E_node.term c2.c_n in
-          mk_expl t1 t2 @@ Proof_trace.add_step proof
-          @@ fun () -> Proof_rules.lemma_cstor_distinct t1 t2
+          mk_expl t1 t2 @@ fun () -> Proof_rules.lemma_cstor_distinct t1 t2
         in
 
         Stat.incr state.n_conflict;
@@ -297,7 +295,7 @@ end = struct
 
   type t = {
     tst: Term.store;
-    proof: Proof_trace.t;
+    proof: Proof.Tracer.t;
     cstors: ST_cstors.t; (* repr -> cstor for the class *)
     parents: ST_parents.t; (* repr -> parents for the class *)
     cards: Card.t; (* remember finiteness *)
@@ -353,15 +351,15 @@ end = struct
            with exhaustiveness: [|- is-c(t)] *)
         let proof =
           let pr_isa =
-            Proof_trace.add_step self.proof @@ fun () ->
+            Proof.Tracer.add_step self.proof @@ fun () ->
             Proof_rules.lemma_isa_split t
               [ Lit.atom self.tst (A.mk_is_a self.tst cstor t) ]
           and pr_eq_sel =
-            Proof_trace.add_step self.proof @@ fun () ->
+            Proof.Tracer.add_step self.proof @@ fun () ->
             Proof_rules.lemma_select_cstor ~cstor_t:u t
           in
-          Proof_trace.add_step self.proof @@ fun () ->
-          Proof_core.proof_r1 pr_isa pr_eq_sel
+          Proof.Tracer.add_step self.proof @@ fun () ->
+          Proof.Core_rules.proof_r1 pr_isa pr_eq_sel
         in
 
         Term.Tbl.add self.single_cstor_preproc_done t ();
@@ -416,8 +414,7 @@ end = struct
               "(@[%s.on-new-term.is-a.reduce@ :t %a@ :to %B@ :n %a@ :sub-cstor \
                %a@])"
               name Term.pp_debug t is_true E_node.pp n Monoid_cstor.pp cstor);
-        let pr =
-          Proof_trace.add_step self.proof @@ fun () ->
+        let pr () =
           Proof_rules.lemma_isa_cstor ~cstor_t:(E_node.term cstor.c_n) t
         in
         let n_bool = CC.n_bool cc is_true in
@@ -443,8 +440,7 @@ end = struct
               E_node.pp n i A.Cstor.pp c_t);
         assert (i < List.length cstor.c_args);
         let u_i = List.nth cstor.c_args i in
-        let pr =
-          Proof_trace.add_step self.proof @@ fun () ->
+        let pr () =
           Proof_rules.lemma_select_cstor ~cstor_t:(E_node.term cstor.c_n) t
         in
         let expl =
@@ -480,8 +476,7 @@ end = struct
              :sub-cstor %a@])"
             name Monoid_parents.pp_is_a is_a2 is_true E_node.pp n1 E_node.pp n2
             Monoid_cstor.pp c1);
-      let pr =
-        Proof_trace.add_step self.proof @@ fun () ->
+      let pr () =
         Proof_rules.lemma_isa_cstor ~cstor_t:(E_node.term c1.c_n)
           (E_node.term is_a2.is_a_n)
       in
@@ -509,8 +504,7 @@ end = struct
             k "(@[%s.on-merge.select.reduce@ :n2 %a@ :sel get[%d]-%a@])" name
               E_node.pp n2 sel2.sel_idx Monoid_cstor.pp c1);
         assert (sel2.sel_idx < List.length c1.c_args);
-        let pr =
-          Proof_trace.add_step self.proof @@ fun () ->
+        let pr () =
           Proof_rules.lemma_select_cstor ~cstor_t:(E_node.term c1.c_n)
             (E_node.term sel2.sel_n)
         in
@@ -618,8 +612,7 @@ end = struct
         | { flag = Open; cstor_n; _ } as node ->
           (* conflict: the [path] forms a cycle *)
           let path = (n, node) :: path in
-          let pr =
-            Proof_trace.add_step self.proof @@ fun () ->
+          let pr () =
             let path =
               List.rev_map
                 (fun (a, b) -> E_node.term a, E_node.term b.repr)
@@ -677,10 +670,7 @@ end = struct
         Log.debugf 50 (fun k ->
             k "(@[%s.assign-is-a@ :lhs %a@ :rhs %a@ :lit %a@])" name
               Term.pp_debug u Term.pp_debug rhs Lit.pp lit);
-        let pr =
-          Proof_trace.add_step self.proof @@ fun () ->
-          Proof_rules.lemma_isa_sel t
-        in
+        let pr () = Proof_rules.lemma_isa_sel t in
         (* merge [u] and [rhs] *)
         CC.merge_t (SI.cc solver) u rhs
           (Expl.mk_theory u rhs
@@ -706,14 +696,10 @@ end = struct
                (* TODO: set default polarity, depending on n° of args? *)
                lit)
       in
-      SI.add_clause_permanent solver acts c
-        ( Proof_trace.add_step self.proof @@ fun () ->
-          Proof_rules.lemma_isa_split t c );
+      SI.add_clause_permanent solver acts c (fun () ->
+          Proof_rules.lemma_isa_split t c);
       Iter.diagonal_l c (fun (l1, l2) ->
-          let pr =
-            Proof_trace.add_step self.proof @@ fun () ->
-            Proof_rules.lemma_isa_disj (Lit.neg l1) (Lit.neg l2)
-          in
+          let pr () = Proof_rules.lemma_isa_disj (Lit.neg l1) (Lit.neg l2) in
           SI.add_clause_permanent solver acts [ Lit.neg l1; Lit.neg l2 ] pr)
     )
 
@@ -789,10 +775,11 @@ end = struct
      early *)
 
   let create_and_setup ~id:_ (solver : SI.t) : t =
+    let proof = (SI.tracer solver :> Proof.Tracer.t) in
     let self =
       {
         tst = SI.tst solver;
-        proof = SI.proof solver;
+        proof;
         cstors = ST_cstors.create_and_setup ~size:32 (SI.cc solver);
         parents = ST_parents.create_and_setup ~size:32 (SI.cc solver);
         to_decide = N_tbl.create ~size:16 ();
diff --git a/src/th-data/proof_rules.ml b/src/th-data/proof_rules.ml
index 5e65ef3c..1496d902 100644
--- a/src/th-data/proof_rules.ml
+++ b/src/th-data/proof_rules.ml
@@ -1,27 +1,28 @@
 open Sidekick_core
+module Proof = Sidekick_proof
 
-let lemma_isa_cstor ~cstor_t t : Proof_term.t =
-  Proof_term.apply_rule ~terms:[ cstor_t; t ] "data.isa-cstor"
+let lemma_isa_cstor ~cstor_t t : Proof.Pterm.t =
+  Proof.Pterm.apply_rule ~terms:[ cstor_t; t ] "data.isa-cstor"
 
-let lemma_select_cstor ~cstor_t t : Proof_term.t =
-  Proof_term.apply_rule ~terms:[ cstor_t; t ] "data.select-cstor"
+let lemma_select_cstor ~cstor_t t : Proof.Pterm.t =
+  Proof.Pterm.apply_rule ~terms:[ cstor_t; t ] "data.select-cstor"
 
-let lemma_isa_split t lits : Proof_term.t =
-  Proof_term.apply_rule ~terms:[ t ] ~lits "data.isa-split"
+let lemma_isa_split t lits : Proof.Pterm.t =
+  Proof.Pterm.apply_rule ~terms:[ t ] ~lits "data.isa-split"
 
-let lemma_isa_sel t : Proof_term.t =
-  Proof_term.apply_rule ~terms:[ t ] "data.isa-sel"
+let lemma_isa_sel t : Proof.Pterm.t =
+  Proof.Pterm.apply_rule ~terms:[ t ] "data.isa-sel"
 
-let lemma_isa_disj l1 l2 : Proof_term.t =
-  Proof_term.apply_rule ~lits:[ l1; l2 ] "data.isa-disj"
+let lemma_isa_disj l1 l2 : Proof.Pterm.t =
+  Proof.Pterm.apply_rule ~lits:[ l1; l2 ] "data.isa-disj"
 
-let lemma_cstor_inj t1 t2 i : Proof_term.t =
-  Proof_term.apply_rule ~terms:[ t1; t2 ] ~indices:[ i ] "data.cstor-inj"
+let lemma_cstor_inj t1 t2 i : Proof.Pterm.t =
+  Proof.Pterm.apply_rule ~terms:[ t1; t2 ] ~indices:[ i ] "data.cstor-inj"
 
-let lemma_cstor_distinct t1 t2 : Proof_term.t =
-  Proof_term.apply_rule ~terms:[ t1; t2 ] "data.cstor-distinct"
+let lemma_cstor_distinct t1 t2 : Proof.Pterm.t =
+  Proof.Pterm.apply_rule ~terms:[ t1; t2 ] "data.cstor-distinct"
 
-let lemma_acyclicity ts : Proof_term.t =
-  Proof_term.apply_rule
+let lemma_acyclicity ts : Proof.Pterm.t =
+  Proof.Pterm.apply_rule
     ~terms:(CCList.flat_map (fun (t1, t2) -> [ t1; t2 ]) ts)
     "data.acyclicity"
diff --git a/src/th-data/proof_rules.mli b/src/th-data/proof_rules.mli
index a2010781..30b444a4 100644
--- a/src/th-data/proof_rules.mli
+++ b/src/th-data/proof_rules.mli
@@ -1,33 +1,34 @@
 open Sidekick_core
+module Proof = Sidekick_proof
 
-val lemma_isa_cstor : cstor_t:Term.t -> Term.t -> Proof_term.t
+val lemma_isa_cstor : cstor_t:Term.t -> Term.t -> Proof.Pterm.t
 (** [lemma_isa_cstor (d …) (is-c t)] returns the clause
       [(c …) = t |- is-c t] or [(d …) = t |- ¬ (is-c t)] *)
 
-val lemma_select_cstor : cstor_t:Term.t -> Term.t -> Proof_term.t
+val lemma_select_cstor : cstor_t:Term.t -> Term.t -> Proof.Pterm.t
 (** [lemma_select_cstor (c t1…tn) (sel-c-i t)]
       returns a proof of [t = c t1…tn |- (sel-c-i t) = ti] *)
 
-val lemma_isa_split : Term.t -> Lit.t list -> Proof_term.t
+val lemma_isa_split : Term.t -> Lit.t list -> Proof.Pterm.t
 (** [lemma_isa_split t lits] is the proof of
       [is-c1 t \/ is-c2 t \/ … \/ is-c_n t] *)
 
-val lemma_isa_sel : Term.t -> Proof_term.t
+val lemma_isa_sel : Term.t -> Proof.Pterm.t
 (** [lemma_isa_sel (is-c t)] is the proof of
       [is-c t |- t = c (sel-c-1 t)…(sel-c-n t)] *)
 
-val lemma_isa_disj : Lit.t -> Lit.t -> Proof_term.t
+val lemma_isa_disj : Lit.t -> Lit.t -> Proof.Pterm.t
 (** [lemma_isa_disj (is-c t) (is-d t)] is the proof
       of [¬ (is-c t) \/ ¬ (is-c t)] *)
 
-val lemma_cstor_inj : Term.t -> Term.t -> int -> Proof_term.t
+val lemma_cstor_inj : Term.t -> Term.t -> int -> Proof.Pterm.t
 (** [lemma_cstor_inj (c t1…tn) (c u1…un) i] is the proof of
       [c t1…tn = c u1…un |- ti = ui] *)
 
-val lemma_cstor_distinct : Term.t -> Term.t -> Proof_term.t
+val lemma_cstor_distinct : Term.t -> Term.t -> Proof.Pterm.t
 (** [lemma_isa_distinct (c …) (d …)] is the proof
       of the unit clause [|- (c …) ≠ (d …)] *)
 
-val lemma_acyclicity : (Term.t * Term.t) list -> Proof_term.t
+val lemma_acyclicity : (Term.t * Term.t) list -> Proof.Pterm.t
 (** [lemma_acyclicity pairs] is a proof of [t1=u1, …, tn=un |- false]
       by acyclicity. *)
diff --git a/src/th-lra/intf.ml b/src/th-lra/intf.ml
index 5e3f5672..8ed7ad90 100644
--- a/src/th-lra/intf.ml
+++ b/src/th-lra/intf.ml
@@ -1,5 +1,6 @@
 open Sidekick_core
 module SMT = Sidekick_smt_solver
+module Proof = Sidekick_proof
 module Predicate = Sidekick_simplex.Predicate
 module Linear_expr = Sidekick_simplex.Linear_expr
 module Linear_expr_intf = Sidekick_simplex.Linear_expr_intf
diff --git a/src/th-lra/proof_rules.ml b/src/th-lra/proof_rules.ml
index 72087162..81dddc96 100644
--- a/src/th-lra/proof_rules.ml
+++ b/src/th-lra/proof_rules.ml
@@ -1,3 +1,4 @@
 open Sidekick_core
+module Proof = Sidekick_proof
 
-let lemma_lra lits : Proof_term.t = Proof_term.apply_rule "lra.lemma" ~lits
+let lemma_lra lits : Proof.Pterm.t = Proof.Pterm.apply_rule "lra.lemma" ~lits
diff --git a/src/th-lra/proof_rules.mli b/src/th-lra/proof_rules.mli
index 26a2688a..e6706f1e 100644
--- a/src/th-lra/proof_rules.mli
+++ b/src/th-lra/proof_rules.mli
@@ -1,5 +1,5 @@
 open Sidekick_core
+module Proof = Sidekick_proof
 
-val lemma_lra : Lit.t list -> Proof_term.t
+val lemma_lra : Lit.t list -> Proof.Pterm.t
 (** List of literals [l1…ln] where [¬l1 /\ … /\ ¬ln] is LRA-unsat *)
-
diff --git a/src/th-lra/sidekick_th_lra.ml b/src/th-lra/sidekick_th_lra.ml
index a5051141..e76c4115 100644
--- a/src/th-lra/sidekick_th_lra.ml
+++ b/src/th-lra/sidekick_th_lra.ml
@@ -125,7 +125,7 @@ module Make (A : ARG) = (* : S with module A = A *) struct
 
   type state = {
     tst: Term.store;
-    proof: Proof_trace.t;
+    proof: Proof.Tracer.t;
     gensym: Gensym.t;
     in_model: unit Term.Tbl.t; (* terms to add to model *)
     encoded_eqs: unit Term.Tbl.t;
@@ -145,7 +145,7 @@ module Make (A : ARG) = (* : S with module A = A *) struct
 
   let create (si : SI.t) : state =
     let stat = SI.stats si in
-    let proof = SI.proof si in
+    let proof = (SI.tracer si :> Proof.Tracer.t) in
     let tst = SI.tst si in
     {
       tst;
@@ -255,14 +255,15 @@ module Make (A : ARG) = (* : S with module A = A *) struct
 
   let add_clause_lra_ ?using (module PA : SI.PREPROCESS_ACTS) lits =
     let pr =
-      Proof_trace.add_step PA.proof @@ fun () -> Proof_rules.lemma_lra lits
+      Proof.Tracer.add_step PA.proof_tracer @@ fun () ->
+      Proof_rules.lemma_lra lits
     in
     let pr =
       match using with
       | None -> pr
       | Some using ->
-        Proof_trace.add_step PA.proof @@ fun () ->
-        Proof_core.lemma_rw_clause pr ~res:lits ~using
+        Proof.Tracer.add_step PA.proof_tracer @@ fun () ->
+        Proof.Core_rules.lemma_rw_clause pr ~res:lits ~using
     in
     PA.add_clause lits pr
 
@@ -405,14 +406,15 @@ module Make (A : ARG) = (* : S with module A = A *) struct
     )
 
   let simplify (self : state) (_recurse : _) (t : Term.t) :
-      (Term.t * Proof_step.id Iter.t) option =
+      (Term.t * Proof.Step.id Iter.t) option =
     let proof_eq t u =
-      Proof_trace.add_step self.proof @@ fun () ->
+      Proof.Tracer.add_step self.proof @@ fun () ->
       Proof_rules.lemma_lra [ Lit.atom self.tst (Term.eq self.tst t u) ]
     in
     let proof_bool t ~sign:b =
       let lit = Lit.atom ~sign:b self.tst t in
-      Proof_trace.add_step self.proof @@ fun () -> Proof_rules.lemma_lra [ lit ]
+      Proof.Tracer.add_step self.proof @@ fun () ->
+      Proof_rules.lemma_lra [ lit ]
     in
 
     match A.view_as_lra t with
@@ -477,10 +479,7 @@ module Make (A : ARG) = (* : S with module A = A *) struct
       |> CCList.flat_map (Tag.to_lits si)
       |> List.rev_map Lit.neg
     in
-    let pr =
-      Proof_trace.add_step (SI.proof si) @@ fun () ->
-      Proof_rules.lemma_lra confl
-    in
+    let pr () = Proof_rules.lemma_lra confl in
     Stat.incr self.n_conflict;
     SI.raise_conflict si acts confl pr
 
@@ -491,10 +490,7 @@ module Make (A : ARG) = (* : S with module A = A *) struct
       Stat.incr self.n_propagate;
       SI.propagate si acts lit ~reason:(fun () ->
           let lits = CCList.flat_map (Tag.to_lits si) reason in
-          let pr =
-            Proof_trace.add_step (SI.proof si) @@ fun () ->
-            Proof_rules.lemma_lra (lit :: lits)
-          in
+          let pr () = Proof_rules.lemma_lra (lit :: lits) in
           CCList.flat_map (Tag.to_lits si) reason, pr)
     | _ -> ()
 
@@ -538,10 +534,7 @@ module Make (A : ARG) = (* : S with module A = A *) struct
       if A.Q.(le_const <> zero) then (
         (* [c=0] when [c] is not 0 *)
         let lit = Lit.atom ~sign:false self.tst @@ Term.eq self.tst t1 t2 in
-        let pr =
-          Proof_trace.add_step self.proof @@ fun () ->
-          Proof_rules.lemma_lra [ lit ]
-        in
+        let pr () = Proof_rules.lemma_lra [ lit ] in
         SI.add_clause_permanent si acts [ lit ] pr
       )
     ) else (