diff --git a/heap_lang/derived.v b/heap_lang/derived.v
index a8fd8ea2ea70476435d7874b6e0809544da8572d..6ce1fad8eb0c1aa80266ac853771a209f5bfb78b 100644
--- a/heap_lang/derived.v
+++ b/heap_lang/derived.v
@@ -15,18 +15,18 @@ Implicit Types P : iProp heap_lang Σ.
 Implicit Types Q : val → iProp heap_lang Σ.
 
 (** Proof rules for the sugar *)
-Lemma wp_lam E x ef e v Q :
+Lemma wp_lam' E x ef e v Q :
   to_val e = Some v → ▷ wp E (subst ef x v) Q ⊑ wp E (App (Lam x ef) e) Q.
-Proof. intros. by rewrite -wp_rec ?subst_empty. Qed.
+Proof. intros. by rewrite -wp_rec' ?subst_empty. Qed.
 
-Lemma wp_let E x e1 e2 v Q :
+Lemma wp_let' E x e1 e2 v Q :
   to_val e1 = Some v → ▷ wp E (subst e2 x v) Q ⊑ wp E (Let x e1 e2) Q.
-Proof. apply wp_lam. Qed.
+Proof. apply wp_lam'. Qed.
 
 Lemma wp_seq E e1 e2 Q : wp E e1 (λ _, ▷ wp E e2 Q) ⊑ wp E (Seq e1 e2) Q.
 Proof.
   rewrite -(wp_bind [LetCtx "" e2]). apply wp_mono=>v.
-  by rewrite -wp_let //= ?to_of_val ?subst_empty.
+  by rewrite -wp_let' //= ?to_of_val ?subst_empty.
 Qed.
 
 Lemma wp_le E (n1 n2 : Z) P Q :
diff --git a/heap_lang/lifting.v b/heap_lang/lifting.v
index 2904ef540e372e7cf542096b14b6b270781152f1..bcea9d2e95110b8ab5e503cb97a67f354836dab5 100644
--- a/heap_lang/lifting.v
+++ b/heap_lang/lifting.v
@@ -84,7 +84,9 @@ Proof.
   by rewrite -(wp_value' _ _ (Lit _)) //; apply const_intro.
 Qed.
 
-Lemma wp_rec E f x e1 e2 v Q :
+(* For the lemmas involving substitution, we only derive a preliminary version.
+   The final version is defined in substitution.v. *)
+Lemma wp_rec' E f x e1 e2 v Q :
   to_val e2 = Some v →
   ▷ wp E (subst (subst e1 f (RecV f x e1)) x v) Q ⊑ wp E (App (Rec f x e1) e2) Q.
 Proof.
@@ -137,7 +139,7 @@ Proof.
     ?right_id -?wp_value' //; intros; inv_step; eauto.
 Qed.
 
-Lemma wp_case_inl E e0 v0 x1 e1 x2 e2 Q :
+Lemma wp_case_inl' E e0 v0 x1 e1 x2 e2 Q :
   to_val e0 = Some v0 →
   ▷ wp E (subst e1 x1 v0) Q ⊑ wp E (Case (InjL e0) x1 e1 x2 e2) Q.
 Proof.
@@ -145,7 +147,7 @@ Proof.
     (subst e1 x1 v0) None) ?right_id //; intros; inv_step; eauto.
 Qed.
 
-Lemma wp_case_inr E e0 v0 x1 e1 x2 e2 Q :
+Lemma wp_case_inr' E e0 v0 x1 e1 x2 e2 Q :
   to_val e0 = Some v0 →
   ▷ wp E (subst e2 x2 v0) Q ⊑ wp E (Case (InjR e0) x1 e1 x2 e2) Q.
 Proof.
diff --git a/heap_lang/substitution.v b/heap_lang/substitution.v
index 27d57f967b1d7d4aab07894394400ba1a4949ae0..74176c30ba507b28319d4ce16d09139740865031 100644
--- a/heap_lang/substitution.v
+++ b/heap_lang/substitution.v
@@ -120,32 +120,32 @@ Implicit Types P : iProp heap_lang Σ.
 Implicit Types Q : val → iProp heap_lang Σ.
 Hint Resolve to_of_val.
 
-Lemma wp_rec' E e1 f x erec e2 v Q :
+Lemma wp_rec E e1 f x erec e2 v Q :
   e1 = Rec f x erec →
   to_val e2 = Some v →
   ▷ wp E (gsubst (gsubst erec f e1) x e2) Q ⊑ wp E (App e1 e2) Q.
 Proof.
   intros -> <-%of_to_val.
   rewrite (gsubst_correct _ _ (RecV _ _ _)) gsubst_correct.
-  by apply wp_rec.
+  by apply wp_rec'.
 Qed.
 
-Lemma wp_lam' E x ef e v Q :
+Lemma wp_lam E x ef e v Q :
   to_val e = Some v → ▷ wp E (gsubst ef x e) Q ⊑ wp E (App (Lam x ef) e) Q.
-Proof. intros <-%of_to_val; rewrite gsubst_correct. by apply wp_lam. Qed.
+Proof. intros <-%of_to_val; rewrite gsubst_correct. by apply wp_lam'. Qed.
 
-Lemma wp_let' E x e1 e2 v Q :
+Lemma wp_let E x e1 e2 v Q :
   to_val e1 = Some v → ▷ wp E (gsubst e2 x e1) Q ⊑ wp E (Let x e1 e2) Q.
-Proof. apply wp_lam'. Qed.
+Proof. apply wp_lam. Qed.
 
-Lemma wp_case_inl' E e0 v0 x1 e1 x2 e2 Q :
+Lemma wp_case_inl E e0 v0 x1 e1 x2 e2 Q :
   to_val e0 = Some v0 →
   ▷ wp E (gsubst e1 x1 e0) Q ⊑ wp E (Case (InjL e0) x1 e1 x2 e2) Q.
-Proof. intros <-%of_to_val; rewrite gsubst_correct. by apply wp_case_inl. Qed.
+Proof. intros <-%of_to_val; rewrite gsubst_correct. by apply wp_case_inl'. Qed.
 
-Lemma wp_case_inr' E e0 v0 x1 e1 x2 e2 Q :
+Lemma wp_case_inr E e0 v0 x1 e1 x2 e2 Q :
   to_val e0 = Some v0 →
   ▷ wp E (gsubst e2 x2 e0) Q ⊑ wp E (Case (InjR e0) x1 e1 x2 e2) Q.
-Proof. intros <-%of_to_val; rewrite gsubst_correct. by apply wp_case_inr. Qed.
+Proof. intros <-%of_to_val; rewrite gsubst_correct. by apply wp_case_inr'. Qed.
 End wp.
 
diff --git a/heap_lang/tests.v b/heap_lang/tests.v
index 76bc32a4dcf1085b469aac21af4c50deadad8fd3..a459db55cafcab3d6305db015a5dfc8ed06d9f70 100644
--- a/heap_lang/tests.v
+++ b/heap_lang/tests.v
@@ -69,12 +69,12 @@ Module LiftingTests.
     rewrite (comm uPred_and (â–  _)%I) assoc; apply const_elim_r=>?.
     (* first need to do the rec to get a later *)
     rewrite -(wp_bindi (AppLCtx _)) /=.
-    rewrite -wp_rec' // =>-/=; rewrite -wp_value' //=.
+    rewrite -wp_rec // =>-/=; rewrite -wp_value' //=.
     (* FIXME: ssr rewrite fails with "Error: _pattern_value_ is used in conclusion." *)
     rewrite ->(later_intro (Q _)).
     rewrite -!later_and; apply later_mono.
     (* Go on *)
-    rewrite -wp_let' //= -later_intro.
+    rewrite -wp_let //= -later_intro.
     rewrite -(wp_bindi (LetCtx _ _)) -wp_bin_op //= -wp_let' //= -!later_intro.
     rewrite -(wp_bindi (IfCtx _ _)) /=.
     apply wp_lt=> ?.
@@ -88,7 +88,7 @@ Module LiftingTests.
 
   Lemma Pred_spec n E Q : ▷ Q (LitV (n - 1)) ⊑ wp E (Pred 'n)%L Q.
   Proof.
-    rewrite -wp_lam' //=.
+    rewrite -wp_lam //=.
     rewrite -(wp_bindi (IfCtx _ _)) /=.
     apply later_mono, wp_le=> Hn.
     - rewrite -wp_if_true.