diff --git a/CHANGES.md b/CHANGES.md
index b3ee0895a..9260309c6 100644
--- a/CHANGES.md
+++ b/CHANGES.md
@@ -1,3 +1,11 @@
+# UNRELEASED
+
+Language:
+  - Change CHR syntax now accepts any term in eigen variables position `E` in
+    sequent `E : _ ?- _`. Meaningful terms are lists or unification variables
+  - Change CHR eigen variables are a list of names (used to be an integer)
+  - Fix CHR relocation/scope-checking for new goals
+
 # v1.18.2 (January 2024)
 
 Requires Menhir 20211230 and OCaml 4.08 or above.
diff --git a/ELPI.md b/ELPI.md
index 1f6a386bf..bbd8783dd 100644
--- a/ELPI.md
+++ b/ELPI.md
@@ -620,9 +620,13 @@ As soon as a new constraint C is declared:
    If the name context for the new goal is given, then the new goal
    is checked to live in such context before resuming.  If the name
    context is not specified the resumed goals lives in the disjoint union
-   of all name contexts.
+   of all name contexts (*).
    
-The application of CHR rules follows the [refined operation semantics](https://en.wikipedia.org/wiki/Constraint_Handling_Rules).
+The application of CHR rules follows the [refined operational semantics](https://en.wikipedia.org/wiki/Constraint_Handling_Rules).
+
+(*) The name context `C` is the first component of the sequent, eg
+`C : G ?- P`. `C` is unified with the list of eigen variables, that is
+`[c0, c1, .. cn]`. 
 
 ## Quotations
 
diff --git a/src/parser/error_messages.txt b/src/parser/error_messages.txt
index 18f2df143..8f376624d 100644
--- a/src/parser/error_messages.txt
+++ b/src/parser/error_messages.txt
@@ -108,7 +108,6 @@ program: CONSTANT RPAREN
 
 Unexpected keyword.
 
-goal: AFTER USE_SIG
 program: AFTER FAMILY_QMARK USE_SIG
 program: AFTER VDASH FLOAT USE_SIG
 program: AFTER FAMILY_TIMES FLOAT USE_SIG
@@ -183,6 +182,7 @@ goal: AFTER VDASH FLOAT USE_SIG
 goal: AFTER FAMILY_TIMES FLOAT USE_SIG
 goal: AFTER FAMILY_SHARP FLOAT USE_SIG
 goal: LPAREN AFTER RPAREN USE_SIG
+goal: AFTER USE_SIG
 goal: AFTER AFTER USE_SIG
 goal: AFTER FAMILY_TICK FLOAT USE_SIG
 goal: AFTER SLASH FLOAT USE_SIG
@@ -272,6 +272,8 @@ Malformed macro declaration. Example:
 macro @foo X Y :- p X => q Y.
 
 program: COLON VDASH
+program: COLON REPLACE VDASH
+program: COLON INDEX LPAREN FRESHUV RPAREN VDASH
 
 Attribute expected. Examples:
 :name "some name"
@@ -317,7 +319,6 @@ type lam (tm -> tm) -> tm.
 type (++) list A -> list A -> list A.
 
 program: RULE LPAREN USE_SIG
-program: RULE LPAREN AFTER USE_SIG
 program: RULE VDASH
 program: RULE IFF AFTER VDASH
 program: RULE LPAREN AFTER COLON VDASH
diff --git a/src/parser/grammar.mly b/src/parser/grammar.mly
index 001eca261..3ce282967 100644
--- a/src/parser/grammar.mly
+++ b/src/parser/grammar.mly
@@ -265,9 +265,9 @@ sequent:
     let context, conclusion = decode_sequent t in
     { Chr.eigen = underscore (); context; conclusion }
   }
-| LPAREN; c = constant; COLON; t = term; RPAREN {
+| LPAREN; c = closed_term; COLON; t = term; RPAREN {
     let context, conclusion = decode_sequent t in
-    { Chr.eigen = Const c; context; conclusion }
+    { Chr.eigen = c; context; conclusion }
   }
 
 goal:
diff --git a/src/runtime.ml b/src/runtime.ml
index 6e7ac47c3..74f8d175e 100644
--- a/src/runtime.ml
+++ b/src/runtime.ml
@@ -688,7 +688,7 @@ module HO : sig
 
   val shift_bound_vars : depth:int -> to_:int -> term -> term
 
-  val subtract_to_consts : amount:int -> depth:int -> term -> term
+  val map_free_vars : map:constant C.Map.t -> depth:int -> to_:int -> term -> term
 
   val delay_hard_unif_problems : bool Fork.local_ref
     
@@ -2029,12 +2029,16 @@ let shift_bound_vars ~depth ~to_ t =
   in
     if depth = to_ then t else shift depth t
 
-let subtract_to_consts ~amount ~depth t =
+let map_free_vars ~map ~depth ~to_ t =
+  let shift_bound = depth - to_ in
   let shift_db d n =
     if n < 0 then n
-    else if n < amount then
-      error "The term cannot be put in the desired context"
-    else n - amount in
+    else if n < depth then try C.Map.find n map with Not_found ->
+      error (Printf.sprintf "The term cannot be put in the desired context since it contains name: %s" @@ C.show n)
+    else if n >= depth && n - depth <= d then n - shift_bound
+    else
+      error (Printf.sprintf "The term cannot be put in the desired context since it contains name: %s" @@ C.show n)
+    in
   let rec shift d = function
   | Const n as x -> let m = shift_db d n in if m = n then x else mkConst m
   | Lam r as orig ->
@@ -2059,16 +2063,16 @@ let subtract_to_consts ~amount ~depth t =
     if args == args' then orig
     else AppArg (i, args')
   | (UVar (r,_,_) | AppUVar(r,_,_)) when !!r != C.dummy ->
-      anomaly "subtract_to_consts: non-derefd term"
-  | (UVar _ | AppUVar _) ->
-      error "The term cannot be put in the desired context (leftover uvar)"
-      | Builtin(c,xs) as orig ->
-        let xs' = smart_map (shift d) xs in
-        if xs == xs' then orig
-        else Builtin(c,xs')
+      anomaly "map_free_vars: non-derefd term"
+  | UVar(r,lvl,ano) -> UVar(r,min lvl to_,ano)
+  | AppUVar (r,lvl,args) -> AppUVar (r,min lvl to_,smart_map (shift d) args)
+  | Builtin(c,xs) as orig ->
+    let xs' = smart_map (shift d) xs in
+    if xs == xs' then orig
+    else Builtin(c,xs')
   | CData _ as x -> x
   in
-    if amount = 0 then t else shift 0 t
+    if depth = to_ && C.Map.is_empty map then t else shift 0 t
 
 let eta_contract_flex ~depth t =
   eta_contract_flex depth 0 ~argsdepth:0 empty_env t
@@ -3084,7 +3088,7 @@ module Ice : sig
       freezer -> freezer * term
 
   val defrost :
-    maxd:int -> term -> env -> to_:int -> freezer -> term
+    from:int -> term -> env -> to_:int -> map:constant C.Map.t -> freezer -> term
 
   val assignments : freezer -> assignment list
 
@@ -3162,17 +3166,17 @@ end = struct (* {{{ *)
     [%spy "dev:freeze:out" ~rid (uppterm maxground [] ~argsdepth:0 empty_env) t];
     !f, t
 
-  let defrost ~maxd t env ~to_ f =
+  let defrost ~from t env ~to_ ~map f =
   [%spy "dev:defrost:in" ~rid (fun fmt () ->
-    Fmt.fprintf fmt "maxd:%d to:%d %a" maxd to_
-      (uppterm maxd [] ~argsdepth:maxd env) t) ()];
-    let t = full_deref ~adepth:maxd env ~depth:maxd t in
+    Fmt.fprintf fmt "from:%d to:%d %a" from to_
+      (uppterm from [] ~argsdepth:from env) t) ()];
+    let t = full_deref ~adepth:from env ~depth:from t in
   [%spy "dev:defrost:fully-derefd" ~rid (fun fmt ()->
-    Fmt.fprintf fmt "maxd:%d to:%d %a" maxd to_
-      (uppterm maxd [] ~argsdepth:0 empty_env) t) ()];
-    let t = subtract_to_consts ~amount:(maxd - to_) ~depth:maxd t in
+    Fmt.fprintf fmt "from:%d to:%d %a" from to_
+      (uppterm from [] ~argsdepth:0 empty_env) t) ()];
+    let t = map_free_vars ~map ~depth:from ~to_ t in
   [%spy "dev:defrost:shifted" ~rid (fun fmt () ->
-    Fmt.fprintf fmt "maxd:%d to:%d %a" maxd to_
+    Fmt.fprintf fmt "from:%d to:%d %a" from to_
       (uppterm to_ [] ~argsdepth:0 empty_env) t) ()];
     let rec daux d = function
       | Const c when C.Map.mem c f.c2uv ->
@@ -3446,7 +3450,7 @@ let try_fire_rule (gid[@trace]) rule (constraints as orig_constraints) =
     (pats_to_match @ pats_to_remove) ([],[],[]) in
   
   let match_eigen i m (dto,d,eigen) pat =
-    match_goal (gid[@trace]) i max_depth env m (dto,d,Data.C.of_int eigen) pat in
+    match_goal (gid[@trace]) i max_depth env m (dto,d,list_to_lp_list @@ C.mkinterval 0 eigen 0) pat in
   let match_conclusion i m g pat =
     match_goal (gid[@trace]) i max_depth env m g pat in
   let match_context i m ctx pctx =
@@ -3466,6 +3470,7 @@ let try_fire_rule (gid[@trace]) rule (constraints as orig_constraints) =
     (* no meta meta level *)
     chr = CHR.empty;
     initial_goal =
+      (* TODO: maybe we don't need to do this if we don't reach the guard *)
       move ~argsdepth:max_depth ~from:max_depth ~to_:max_depth env
         (shift_bound_vars ~depth:0 ~to_:max_depth guard);
     assignments = StrMap.empty;
@@ -3494,12 +3499,12 @@ let try_fire_rule (gid[@trace]) rule (constraints as orig_constraints) =
            Format.fprintf f "(lives-at:%d, to-be-lifted-to:%d) %a"
              dt dto (uppterm dt [] ~argsdepth:0 empty_env) t) ";") constraints_goals];
 
-      let m = fold_left2i match_eigen m
-        constraints_depts patterns_eigens in
       let m = fold_left2i match_conclusion m
         constraints_goals patterns_goals in
       let m = fold_left2i match_context m
         constraints_contexts patterns_contexts in
+      let m = fold_left2i match_eigen m
+        constraints_depts patterns_eigens in
 
       [%spy "dev:CHR:matching-assignments" ~rid
         (pplist (uppterm max_depth [] ~argsdepth:0 empty_env) ~boxed:false ",") (Array.to_list env)];
@@ -3522,15 +3527,19 @@ let try_fire_rule (gid[@trace]) rule (constraints as orig_constraints) =
       | Some { CHR.eigen; context; conclusion } ->
       let eigen =
         match full_deref ~adepth:max_depth env ~depth:max_depth eigen with
-        | CData x when Data.C.is_int x -> Data.C.to_int x
-        | Discard -> max_depth
-        | _ -> error "eigen not resolving to an integer" in
+        | (Nil | Cons _) as x -> lp_list_to_list ~depth:max_depth x
+        | Discard -> C.mkinterval 0 max_depth 0
+        | _ -> error "eigen not resolving to a list of names" 
+      in
+      let max_eigen,map = List.fold_left (fun (i,m) c ->
+          i+1,C.Map.add (Data.destConst c) i m)
+        (0,C.Map.empty) eigen in
       let conclusion =
-        Ice.defrost ~maxd:max_depth ~to_:eigen
+        Ice.defrost ~from:max_depth ~to_:max_eigen ~map
           (App(Global_symbols.implc,context,[conclusion])) env m in
       (* TODO: check things make sense in heigen *)
       let prog = initial_program in
-      Some (make_constraint_def ~rid ~gid:((make_subgoal_id gid (eigen,conclusion))[@trace]) eigen prog [] conclusion) in
+      Some (make_constraint_def ~rid ~gid:((make_subgoal_id gid (max_eigen,conclusion))[@trace]) max_eigen prog [] conclusion) in
     [%spy "dev:CHR:try-rule:success" ~rid];
     Some(rule_name, constraints_to_remove, new_goals, Ice.assignments m)
   with NoMatch ->
diff --git a/tests/sources/chr-scope-change-failure.elpi b/tests/sources/chr-scope-change-failure.elpi
new file mode 100644
index 000000000..e3ccf5435
--- /dev/null
+++ b/tests/sources/chr-scope-change-failure.elpi
@@ -0,0 +1,12 @@
+constraint foo good bad {
+    rule \ ([C1,C2] : D ?- foo G B) <=> ([] : D ?- foo G B).
+}
+
+pred good i:int.
+pred bad i:int.
+pred foo i:int, i:int.
+foo A B :- print A B, good A, bad B.
+
+main :-
+  pi x y\
+    good x => bad y => declare_constraint (foo x y) []. 
\ No newline at end of file
diff --git a/tests/sources/chr-scope-change.elpi b/tests/sources/chr-scope-change.elpi
new file mode 100644
index 000000000..8dbc52517
--- /dev/null
+++ b/tests/sources/chr-scope-change.elpi
@@ -0,0 +1,12 @@
+constraint foo good bad {
+    rule \ (L : D ?- foo G B) | (std.rev L L1) <=> (L1 : D ?- foo G B).
+}
+
+pred good i:int.
+pred bad i:int.
+pred foo i:int, i:int.
+foo A B :- print A B, good A, bad B.
+
+main :-
+  pi x y\
+    good x => bad y => declare_constraint (foo x y) []. 
\ No newline at end of file
diff --git a/tests/sources/chr-scope.elpi b/tests/sources/chr-scope.elpi
new file mode 100644
index 000000000..73c3cef99
--- /dev/null
+++ b/tests/sources/chr-scope.elpi
@@ -0,0 +1,18 @@
+kind tm type.
+type f tm -> tm -> tm.
+
+pred move i:tm, i:tm, i:tm -> tm, o:tm -> tm.
+move A B (x\f B x) (x\f A x).
+
+pred c i:tm, i:tm -> tm.
+
+constraint c {
+    rule (E : G ?- c (uvar X [A]) T1) \ (F : _ ?- c (uvar X [B]) T2)
+      | (move A B T2 T2') <=> (E : G ?- T1 = T2').
+}
+
+main :-
+  pi x\
+    declare_constraint (c (X x) (y\f x y)) [X x],
+    pi z\ 
+    declare_constraint (c (X x) (y\f x y)) [X x].
\ No newline at end of file
diff --git a/tests/suite/correctness_HO.ml b/tests/suite/correctness_HO.ml
index b825d0b7c..4f1d1df1b 100644
--- a/tests/suite/correctness_HO.ml
+++ b/tests/suite/correctness_HO.ml
@@ -320,3 +320,24 @@ let () = declare "bug_226"
   ~typecheck:true
   ~expectation:Success
   ()
+
+let () = declare "chr-scope"
+  ~source_elpi:"chr-scope.elpi"
+  ~description:"chr-relocation"
+  ~typecheck:true
+  ~expectation:Success
+  ()
+
+let () = declare "chr-scope-change"
+  ~source_elpi:"chr-scope-change.elpi"
+  ~description:"chr-relocation"
+  ~typecheck:true
+  ~expectation:Success
+  ()
+
+let () = declare "chr-scope-change-err"
+  ~source_elpi:"chr-scope-change-failure.elpi"
+  ~description:"chr-relocation"
+  ~typecheck:true
+  ~expectation:(FailureOutput (Str.regexp "cannot be put in the desired context"))
+  ()