diff --git a/theories/algebra/big_op.v b/theories/algebra/big_op.v index e1207f7822f5bc88567181a438966faefb080204..18c6eaa286fe5bb5b5d3ea4b79b27aa36558eeaf 100644 --- a/theories/algebra/big_op.v +++ b/theories/algebra/big_op.v @@ -321,7 +321,9 @@ Section gmap. Qed. Lemma big_opM_unit m : ([^o map] k↦y ∈ m, monoid_unit) ≡ (monoid_unit : M). - Proof. by induction m using map_ind; rewrite /= ?big_opM_insert ?left_id // big_opM_eq. Qed. + Proof. + by induction m using map_ind; rewrite /= ?big_opM_insert ?left_id // big_opM_eq. + Qed. Lemma big_opM_fmap {B} (h : A → B) (f : K → B → M) m : ([^o map] k↦y ∈ h <$> m, f k y) ≡ ([^o map] k↦y ∈ m, f k (h y)). @@ -331,7 +333,8 @@ Section gmap. Qed. Lemma big_opM_omap {B} (h : A → option B) (f : K → B → M) m : - ([^o map] k↦y ∈ omap h m, f k y) ≡ [^o map] k↦y ∈ m, from_option (f k) monoid_unit (h y). + ([^o map] k↦y ∈ omap h m, f k y) + ≡ [^o map] k↦y ∈ m, from_option (f k) monoid_unit (h y). Proof. revert f. induction m as [|i x m Hmi IH] using map_ind=> f. { by rewrite omap_empty !big_opM_empty. } @@ -372,7 +375,8 @@ Section gmap. Lemma big_opM_union f m1 m2 : m1 ##ₘ m2 → - ([^o map] k↦y ∈ m1 ∪ m2, f k y) ≡ ([^o map] k↦y ∈ m1, f k y) `o` ([^o map] k↦y ∈ m2, f k y). + ([^o map] k↦y ∈ m1 ∪ m2, f k y) + ≡ ([^o map] k↦y ∈ m1, f k y) `o` ([^o map] k↦y ∈ m2, f k y). Proof. intros. induction m1 as [|i x m ? IH] using map_ind. { by rewrite big_opM_empty !left_id. } @@ -385,7 +389,9 @@ Section gmap. Lemma big_opM_op f g m : ([^o map] k↦x ∈ m, f k x `o` g k x) ≡ ([^o map] k↦x ∈ m, f k x) `o` ([^o map] k↦x ∈ m, g k x). - Proof. rewrite big_opM_eq /big_opM_def -big_opL_op. by apply big_opL_proper=> ? [??]. Qed. + Proof. + rewrite big_opM_eq /big_opM_def -big_opL_op. by apply big_opL_proper=> ? [??]. + Qed. End gmap. @@ -479,7 +485,8 @@ End gset. Lemma big_opM_dom `{Countable K} {A} (f : K → M) (m : gmap K A) : ([^o map] k↦_ ∈ m, f k) ≡ ([^o set] k ∈ dom _ m, f k). Proof. - induction m as [|i x ?? IH] using map_ind; [by rewrite big_opM_eq big_opS_eq dom_empty_L|]. + induction m as [|i x ?? IH] using map_ind. + { by rewrite big_opM_eq big_opS_eq dom_empty_L. } by rewrite dom_insert_L big_opM_insert // IH big_opS_insert ?not_elem_of_dom. Qed.