diff --git a/Lattice/FiniteValueMap.agda b/Lattice/FiniteValueMap.agda
index db6b45f..d046bbc 100644
--- a/Lattice/FiniteValueMap.agda
+++ b/Lattice/FiniteValueMap.agda
@@ -185,7 +185,7 @@ module IterProdIsomorphism where
                 narrow₂ {fm₁} {fm₂ = (_ ∷ fm'₂ , push k≢ks _) , kvs≡ks@refl} kvs₁⊆kvs₂ k' v' k',v'∈fm'₁
                     with kvs₁⊆kvs₂ k' v' k',v'∈fm'₁
                 ...   | (v'' , (v'≈v'' , here refl)) rewrite sym (proj₂ fm₁) =
-                        ⊥-elim (All¬-¬Any k≢ks (forget {m = proj₁ fm₁} k',v'∈fm'₁))
+                        ⊥-elim (All¬-¬Any k≢ks (forget k',v'∈fm'₁))
                 ...   | (v'' , (v'≈v'' , there k',v'∈fm'₂)) =
                         (v'' , (v'≈v'' , k',v'∈fm'₂))
 
@@ -196,7 +196,7 @@ module IterProdIsomorphism where
         k,v∈pop⇒k,v∈ : ∀ {k : A} {ks : List A} {k' : A} {v : B} (fm : FiniteMap (k ∷ ks)) →
                        (k' , v) ∈ᵐ pop fm → (¬ k ≡ k' × ((k' , v) ∈ᵐ fm))
         k,v∈pop⇒k,v∈ {k} {ks} {k'} {v} (m@((k , _) ∷ fm' , push k≢ks uks') , refl) k',v∈fm =
-            ( (λ { refl → All¬-¬Any k≢ks (forget {m = (fm' , uks')} k',v∈fm) })
+            ( (λ { refl → All¬-¬Any k≢ks (forget k',v∈fm) })
             , there k',v∈fm
             )
 
@@ -215,11 +215,11 @@ module IterProdIsomorphism where
         Provenance-union fm₁@(m₁ , ks₁≡ks) fm₂@(m₂ , ks₂≡ks) {k} {v} k,v∈fm₁fm₂
             with Expr-Provenance-≡ ((` m₁) ∪ (` m₂)) k,v∈fm₁fm₂
         ...   | in₁ (single k,v∈m₁) k∉km₂
-                with k∈km₁ ← (forget {m = m₁} k,v∈m₁)
+                with k∈km₁ ← (forget k,v∈m₁)
                 rewrite trans ks₁≡ks (sym ks₂≡ks) =
                 ⊥-elim (k∉km₂ k∈km₁)
         ...   | in₂ k∉km₁ (single k,v∈m₂)
-                with k∈km₂ ← (forget {m = m₂} k,v∈m₂)
+                with k∈km₂ ← (forget k,v∈m₂)
                 rewrite trans ks₁≡ks (sym ks₂≡ks) =
                 ⊥-elim (k∉km₁ k∈km₂)
         ...   | bothᵘ {v₁} {v₂} (single k,v₁∈m₁) (single k,v₂∈m₂) =
@@ -324,11 +324,9 @@ module IterProdIsomorphism where
           | from-first-value (fm₁ ⊔ᵐ fm₂)
           | from-first-value fm₁ | from-first-value fm₂
     ...   | (v , k,v∈fm₁fm₂) | (v₁ , k,v₁∈fm₁) | (v₂ , k,v₂∈fm₂) | refl | refl | refl
-            with Expr-Provenance k ((` m₁) ∪ (` m₂)) (forget {m = proj₁ (fm₁ ⊔ᵐ fm₂)} k,v∈fm₁fm₂)
-    ...       | (_ , (in₁ _ k∉km₂ , _)) = ⊥-elim (k∉km₂ (forget {m = m₂}
-                                                                k,v₂∈fm₂))
-    ...       | (_ , (in₂ k∉km₁ _ , _)) = ⊥-elim (k∉km₁ (forget {m = m₁}
-                                                                k,v₁∈fm₁))
+            with Expr-Provenance k ((` m₁) ∪ (` m₂)) (forget k,v∈fm₁fm₂)
+    ...       | (_ , (in₁ _ k∉km₂ , _)) = ⊥-elim (k∉km₂ (forget k,v₂∈fm₂))
+    ...       | (_ , (in₂ k∉km₁ _ , _)) = ⊥-elim (k∉km₁ (forget k,v₁∈fm₁))
     ...       | (v₁⊔v₂ , (bothᵘ {v₁'} {v₂'} (single k,v₁'∈m₁) (single k,v₂'∈m₂) , k,v₁⊔v₂∈m₁m₂))
                 rewrite Map-functional {m = m₁} k,v₁∈fm₁ k,v₁'∈m₁
                 rewrite Map-functional {m = m₂} k,v₂∈fm₂ k,v₂'∈m₂
@@ -387,10 +385,10 @@ module IterProdIsomorphism where
                     (v , (IsLattice.≈-refl lB , here refl))
             ...   | here refl | there k',v₂∈fm₂' =
                     ⊥-elim (All¬-¬Any k≢ks' (subst (k' ∈ˡ_) (proj₂ fm₂')
-                                                   (forget {m = proj₁ fm₂'} k',v₂∈fm₂')))
+                                                   (forget k',v₂∈fm₂')))
             ...   | there k',v₁∈fm₁' | here refl =
                     ⊥-elim (All¬-¬Any k≢ks' (subst (k' ∈ˡ_) (proj₂ fm₁')
-                                                   (forget {m = proj₁ fm₁'} k',v₁∈fm₁')))
+                                                   (forget k',v₁∈fm₁')))
             ...   | there k',v₁∈fm₁' | there k',v₂∈fm₂' =
                         let
                             k',v₁v₂∈fm₁'fm₂' =
diff --git a/Lattice/Map.agda b/Lattice/Map.agda
index 42d91e7..f5c8c33 100644
--- a/Lattice/Map.agda
+++ b/Lattice/Map.agda
@@ -488,7 +488,9 @@ _∈k_ k m = MemProp._∈_ k (keys m)
 locate : ∀ {k : A} {m : Map} → k ∈k m → Σ B (λ v → (k , v) ∈ m)
 locate k∈km = locate-impl k∈km
 
-forget : ∀ {k : A} {v : B} {m : Map} → (k , v) ∈ m → k ∈k m
+-- defined this way because ∈ for maps uses projection, so the full map can't be guessed.
+-- On the other hand, list can be guessed.
+forget : ∀ {k : A} {v : B} {l : List (A × B)} → (k , v) ∈ˡ l → k ∈ˡ (ImplKeys.keys l)
 forget = ∈-cong proj₁
 
 Map-functional : ∀ {k : A} {v v' : B} {m : Map} → (k , v) ∈ m → (k , v') ∈ m → v ≡ v'
@@ -594,7 +596,7 @@ Expr-Provenance k (e₁ ∩ e₂) k∈ke₁e₂
 
 Expr-Provenance-≡ : ∀ {k : A} {v : B} (e : Expr) → (k , v) ∈ ⟦ e ⟧ → Provenance k v e
 Expr-Provenance-≡ {k} {v} e k,v∈e
-    with (v' , (p , k,v'∈e)) ← Expr-Provenance k e (forget {m = ⟦ e ⟧} k,v∈e)
+    with (v' , (p , k,v'∈e)) ← Expr-Provenance k e (forget k,v∈e)
     rewrite Map-functional {m = ⟦ e ⟧} k,v∈e k,v'∈e = p
 
 module _ (≈₂-dec : IsDecidable _≈₂_) where
@@ -609,7 +611,7 @@ module _ (≈₂-dec : IsDecidable _≈₂_) where
             let (v , k,v∈m₁) = locate-impl k∈km₁
             in no (λ m₁⊆m₂ →
                    let (v' , (_ , k,v'∈m₂)) = m₁⊆m₂ k v k,v∈m₁
-                   in k∉km₂ (∈-cong proj₁ k,v'∈m₂))
+                   in k∉km₂ (forget k,v'∈m₂))
         SubsetInfo-to-dec {m₁} {m₂} (mismatch k v₁ v₂ k,v₁∈m₁ k,v₂∈m₂ v₁̷≈v₂) =
             no (λ m₁⊆m₂ →
                 let (v' , (v₁≈v' , k,v'∈m₂)) = m₁⊆m₂ k v₁ k,v₁∈m₁
@@ -979,7 +981,7 @@ updating-via-k∈ks-forward m {ks} f k∈ks k∈km rewrite transform-keys-≡ (p
 updating-via-k∈ks-≡ : ∀ (m : Map) {ks : List A} (f : A → B) {k : A} {v : B} →
                     k ∈ˡ ks → (k , v) ∈ (m updating ks via f)→ v ≡ f k
 updating-via-k∈ks-≡ m {ks} f k∈ks k,v∈um
-    with updating-via-k∈ks m f k∈ks (forget {m = (m updating ks via f)} k,v∈um)
+    with updating-via-k∈ks m f k∈ks (forget k,v∈um)
 ...   | k,fk∈um = Map-functional {m = (m updating ks via f)} k,v∈um k,fk∈um
 
 updating-via-k∉ks-forward : ∀ (m : Map) {ks : List A} (f : A → B) {k : A} {v : B} →
@@ -1017,11 +1019,11 @@ module _ {l} {L : Set l}
                     with Expr-Provenance-≡ ((` (f' l₁)) ∪ (` (f' l₂))) k,v∈f'l₁f'l₂
                 ...   | in₁ (single k,v∈f'l₁) k∉kf'l₂ =
                         let
-                            k∈kfl₁ = updating-via-∈k-backward (f l₁) ks (updater l₁) (forget {m = f' l₁} k,v∈f'l₁)
+                            k∈kfl₁ = updating-via-∈k-backward (f l₁) ks (updater l₁) (forget k,v∈f'l₁)
                             k∈kfl₁fl₂ = union-preserves-∈k₁ {l₁ = proj₁ (f l₁)} {l₂ = proj₁ (f l₂)} k∈kfl₁
                             (v' , k,v'∈fl₁l₂) = locate {m = (f l₁ ⊔ f l₂)} k∈kfl₁fl₂
                             (v'' , (v'≈v'' , k,v''∈fl₂)) = fl₁fl₂⊆fl₂ k v' k,v'∈fl₁l₂
-                            k∈kf'l₂ = updating-via-∈k-forward (f l₂) ks (updater l₂) (forget {m = f l₂} k,v''∈fl₂)
+                            k∈kf'l₂ = updating-via-∈k-forward (f l₂) ks (updater l₂) (forget k,v''∈fl₂)
                         in
                             ⊥-elim (k∉kf'l₂ k∈kf'l₂)
                 ...   | in₂ k∉kf'l₁ (single k,v'∈f'l₂) =
@@ -1044,7 +1046,7 @@ module _ {l} {L : Set l}
 
                 f'l₂⊆f'l₁f'l₂ : f' l₂ ⊆ ((f' l₁) ⊔ (f' l₂))
                 f'l₂⊆f'l₁f'l₂ k v k,v∈f'l₂
-                    with k∈kfl₂ ← updating-via-∈k-backward (f l₂) ks (updater l₂) (forget {m = f' l₂} k,v∈f'l₂)
+                    with k∈kfl₂ ← updating-via-∈k-backward (f l₂) ks (updater l₂) (forget k,v∈f'l₂)
                     with (v' , k,v'∈fl₂) ← locate {m = f l₂} k∈kfl₂
                     with (v'' , (v'≈v'' , k,v''∈fl₁fl₂)) ← fl₂⊆fl₁fl₂ k v' k,v'∈fl₂
                     with Expr-Provenance-≡ ((` (f l₁)) ∪ (` (f l₂))) k,v''∈fl₁fl₂
@@ -1058,8 +1060,8 @@ module _ {l} {L : Set l}
                         with k∈-dec k ks
                 ...       | yes k∈ks with refl ← updating-via-k∈ks-≡ (f l₂) (updater l₂) k∈ks k,v∈f'l₂ =
                             let
-                                k,uv₁∈f'l₁ = updating-via-k∈ks-forward (f l₁) (updater l₁) k∈ks (forget {m = f l₁} k,v₁∈fl₁)
-                                k,uv₂∈f'l₂ = updating-via-k∈ks-forward (f l₂) (updater l₂) k∈ks (forget {m = f l₂} k,v₂∈fl₂)
+                                k,uv₁∈f'l₁ = updating-via-k∈ks-forward (f l₁) (updater l₁) k∈ks (forget k,v₁∈fl₁)
+                                k,uv₂∈f'l₂ = updating-via-k∈ks-forward (f l₂) (updater l₂) k∈ks (forget k,v₂∈fl₂)
                                 k,uv₁uv₂∈f'l₁f'l₂ = ⊔-combines {m₁ = f' l₁} {m₂ = f' l₂} k,uv₁∈f'l₁ k,uv₂∈f'l₂
                             in
                                 (updater l₁ k ⊔₂ updater l₂ k , (IsLattice.≈-sym lB (g-Monotonicʳ k l₁≼l₂) , k,uv₁uv₂∈f'l₁f'l₂))