diff --git a/include/cvc5/cvc5_proof_rule.h b/include/cvc5/cvc5_proof_rule.h
index 301620d1e96..26fe15ef3e0 100644
--- a/include/cvc5/cvc5_proof_rule.h
+++ b/include/cvc5/cvc5_proof_rule.h
@@ -1917,9 +1917,13 @@ enum ENUM(ProofRule)
    *            {(x_1 \diamond x_2) = (y_1 \diamond y_2)}
    *
    * where :math:`\diamond \in \{<, \leq, =, \geq, >\}` for arithmetic and
-   * :math:`\diamond \in \{=\}` for bitvectors. :math:`c_x` and :math:c_y` are
+   * :math:`\diamond \in \{=\}` for bitvectors. :math:`c_x` and :math:`c_y` are
    * scaling factors. For :math:`<, \leq, \geq, >`, the scaling factors have the
    * same sign. For bitvectors, they are set to :math:`1`.
+   *
+   * If :math:`c_x` has type :math:`Real` and :math:`x_1, x_2` are of type
+   * :math:`Int`, then :math:`(x_1 - x_2)` is wrapped in an application of
+   * `to_real`, similarly for :math:`(y_1 - y_2)`.
    * \endverbatim
    */
   EVALUE(ARITH_POLY_NORM_REL),
diff --git a/proofs/eo/cpc/Cpc.eo b/proofs/eo/cpc/Cpc.eo
index 4567a1ac56b..bd6e9dc58ee 100644
--- a/proofs/eo/cpc/Cpc.eo
+++ b/proofs/eo/cpc/Cpc.eo
@@ -37,8 +37,8 @@
 ; is prefixed by MACRO_ are not formalized in this signature and will not
 ; appear in final proofs.
 ;
-; The cases of ProofRule::DSL_REWRITE are by default provided on demand in
-; the preamble of the proof that is output by cvc5.
+; The cases of ProofRule::DSL_REWRITE are each formalized as individual
+; proof rules in ./rules/Rewrites.eo.
 ;
 ; The rules ProofRule::TRUST and ProofRule::TRUST_THEORY_REWRITE are output in
 ; proofs using a catch-all rule trust defined at the end of this file. This
diff --git a/proofs/eo/cpc/programs/PolyNorm.eo b/proofs/eo/cpc/programs/PolyNorm.eo
index a4c0e693ace..216a7943056 100644
--- a/proofs/eo/cpc/programs/PolyNorm.eo
+++ b/proofs/eo/cpc/programs/PolyNorm.eo
@@ -239,6 +239,18 @@
   )
 )
 
+; program: $remove_to_real
+; args:
+; - t U: The term to process.
+; return: The result of removing an application of to_real from t, if applicable.
+(program $remove_to_real ((U Type) (x U) (T Type))
+  (U) T
+  (
+  (($remove_to_real (to_real x)) x)
+  (($remove_to_real x)           x)
+  )
+)
+
 ; program: $mk_poly_norm_rel
 ; args:
 ; - t U: The left hand side of the premise of ProofRule::ARITH_POLY_NORM_REL.
@@ -247,14 +259,18 @@
 ; return: >
 ;   The conclusion of ProofRule::ARITH_POLY_NORM_REL. This program does not
 ;   evaluate if the input to this proof rule is invalid.
-(program $mk_poly_norm_rel ((U Type) (r (-> U U Bool)) (x1 U) (x2 U) (y1 U) (y2 U) (cx U) (cy U) 
+(program $mk_poly_norm_rel ((U Type) (r (-> U U Bool)) (x U) (y U) (cx U) (cy U)
                             (n Int) (c (BitVec n)) (nil (BitVec n) :list) 
                             (xb1 (BitVec n)) (xb2 (BitVec n)) (yb1 (BitVec n)) (yb2 (BitVec n)))
   (U U (-> U U Bool)) Bool
   (
-    (($mk_poly_norm_rel (* cx (- x1 x2)) (* cy (- y1 y2)) r) 
+    (($mk_poly_norm_rel (* cx x) (* cy y) r)
         (eo::requires ($is_poly_norm_rel_consts (r cx cy)) true
-          (= (r x1 x2) (r y1 y2))))
+        (eo::match ((x1 U) (x2 U) (y1 U) (y2 U))
+          (@pair ($remove_to_real x) ($remove_to_real y))
+          (
+          ((@pair (- x1 x2) (- y1 y2)) (= (r x1 x2) (r y1 y2)))))
+          ))
     (($mk_poly_norm_rel (bvmul c (bvsub xb1 xb2) nil) (bvmul c (bvsub yb1 yb2) nil) =)
         (eo::requires (eo::to_z c) 1
         (eo::requires (eo::to_z nil) 1
diff --git a/src/theory/arith/arith_poly_norm.cpp b/src/theory/arith/arith_poly_norm.cpp
index 72f5a62d169..08ca6816f7d 100644
--- a/src/theory/arith/arith_poly_norm.cpp
+++ b/src/theory/arith/arith_poly_norm.cpp
@@ -562,6 +562,15 @@ Node PolyNorm::getArithPolyNormRelPremise(TNode a,
     {
       cx = nm->mkConstReal(rx);
       cy = nm->mkConstReal(ry);
+      // add TO_REAL to avoid mixed arithmetic
+      if (x.getType().isInteger())
+      {
+        x = nm->mkNode(Kind::TO_REAL, x);
+      }
+      if (y.getType().isInteger())
+      {
+        y = nm->mkNode(Kind::TO_REAL, y);
+      }
     }
     lhs = nm->mkNode(Kind::MULT, cx, x);
     rhs = nm->mkNode(Kind::MULT, cy, y);
diff --git a/src/theory/arith/proof_checker.cpp b/src/theory/arith/proof_checker.cpp
index d8c01af6211..be882862706 100644
--- a/src/theory/arith/proof_checker.cpp
+++ b/src/theory/arith/proof_checker.cpp
@@ -432,7 +432,9 @@ Node ArithProofRuleChecker::checkInternal(ProofRule id,
         return Node::null();
       }
       Node lr = l[1];
+      lr = lr.getKind() == Kind::TO_REAL ? lr[0] : lr;
       Node rr = r[1];
+      rr = rr.getKind() == Kind::TO_REAL ? rr[0] : rr;
       if ((lr.getKind() != Kind::SUB && lr.getKind() != Kind::BITVECTOR_SUB)
           || (rr.getKind() != Kind::SUB && rr.getKind() != Kind::BITVECTOR_SUB))
       {
diff --git a/test/regress/cli/CMakeLists.txt b/test/regress/cli/CMakeLists.txt
index efa06409e53..e96497642b6 100644
--- a/test/regress/cli/CMakeLists.txt
+++ b/test/regress/cli/CMakeLists.txt
@@ -1308,6 +1308,7 @@ set(regress_0_tests
   regress0/proj-issue307-get-value-re.smt2
   regress0/proj-issue645-abs-value-subs.smt2
   regress0/proofs/alethe-res-need-or-step.smt2
+  regress0/proofs/arith-poly-norm-rel-mixed.smt2
   regress0/proofs/cyclic-ucp.smt2
   regress0/proofs/bvrewrite-concat-merge.smt2
   regress0/proofs/bvrewrite-extract.smt2
diff --git a/test/regress/cli/regress0/proofs/arith-poly-norm-rel-mixed.smt2 b/test/regress/cli/regress0/proofs/arith-poly-norm-rel-mixed.smt2
new file mode 100644
index 00000000000..e64b6b0a203
--- /dev/null
+++ b/test/regress/cli/regress0/proofs/arith-poly-norm-rel-mixed.smt2
@@ -0,0 +1,7 @@
+; EXPECT: unsat
+(set-option :proof-elim-subtypes true)
+(set-logic NIRA)
+(declare-const x Int)
+(declare-const y Int)
+(assert (distinct (<= x y) (<= (to_real x) (to_real y))))
+(check-sat)