use lia instead of omega

theories/collections.v

@@ -1051,22 +1051,22 @@ Section seq_set.
     x ∈ seq_set (C:=C) start len ↔ start ≤ x < start + len.
   Proof.
     revert start. induction len as [|len IH]; intros start; simpl.
-    - rewrite elem_of_empty. omega.
-    - rewrite elem_of_union, elem_of_singleton, IH. omega.
+    - rewrite elem_of_empty. lia.
+    - rewrite elem_of_union, elem_of_singleton, IH. lia.
   Qed.
 
   Lemma seq_set_start_disjoint start len :
     {[ start ]} ## seq_set (C:=C) (S start) len.
-  Proof. intros x. rewrite elem_of_singleton, elem_of_seq_set. omega. Qed.
+  Proof. intros x. rewrite elem_of_singleton, elem_of_seq_set. lia. Qed.
 
   Lemma seq_set_S_disjoint start len :
     {[ start + len ]} ## seq_set (C:=C) start len.
-  Proof. intros x. rewrite elem_of_singleton, elem_of_seq_set. omega. Qed.
+  Proof. intros x. rewrite elem_of_singleton, elem_of_seq_set. lia. Qed.
 
   Lemma seq_set_S_union start len :
     seq_set start (S len) ≡@{C} {[ start + len ]} ∪ seq_set start len.
   Proof.
-    intros x. rewrite elem_of_union, elem_of_singleton, !elem_of_seq_set. omega.
+    intros x. rewrite elem_of_union, elem_of_singleton, !elem_of_seq_set. lia.
   Qed.
 
   Lemma seq_set_S_union_L `{!LeibnizEquiv C} start len :

theories/gmultiset.v

@@ -80,14 +80,14 @@ Lemma multiplicity_union X Y x :
   multiplicity x (X ∪ Y) = multiplicity x X + multiplicity x Y.
 Proof.
   destruct X as [X], Y as [Y]; unfold multiplicity; simpl.
-  rewrite lookup_union_with. destruct (X !! _), (Y !! _); simpl; omega.
+  rewrite lookup_union_with. destruct (X !! _), (Y !! _); simpl; lia.
 Qed.
 Lemma multiplicity_difference X Y x :
   multiplicity x (X ∖ Y) = multiplicity x X - multiplicity x Y.
 Proof.
   destruct X as [X], Y as [Y]; unfold multiplicity; simpl.
   rewrite lookup_difference_with.
-  destruct (X !! _), (Y !! _); simplify_option_eq; omega.
+  destruct (X !! _), (Y !! _); simplify_option_eq; lia.
 Qed.
 
 (* Collection *)
@@ -97,12 +97,12 @@ Proof. done. Qed.
 Global Instance gmultiset_simple_collection : SimpleCollection A (gmultiset A).
 Proof.
   split.
-  - intros x. rewrite elem_of_multiplicity, multiplicity_empty. omega.
+  - intros x. rewrite elem_of_multiplicity, multiplicity_empty. lia.
   - intros x y. destruct (decide (x = y)) as [->|].
     + rewrite elem_of_multiplicity, multiplicity_singleton. split; auto with lia.
     + rewrite elem_of_multiplicity, multiplicity_singleton_ne by done.
       by split; auto with lia.
-  - intros X Y x. rewrite !elem_of_multiplicity, multiplicity_union. omega.
+  - intros X Y x. rewrite !elem_of_multiplicity, multiplicity_union. lia.
 Qed.
 Global Instance gmultiset_elem_of_dec : RelDecision (∈@{gmultiset A}).
 Proof. refine (λ x X, cast_if (decide (0 < multiplicity x X))); done. Defined.
@@ -110,11 +110,11 @@ Proof. refine (λ x X, cast_if (decide (0 < multiplicity x X))); done. Defined.
 (* Algebraic laws *)
 Global Instance gmultiset_comm : Comm (=@{gmultiset A}) (∪).
 Proof.
-  intros X Y. apply gmultiset_eq; intros x. rewrite !multiplicity_union; omega.
+  intros X Y. apply gmultiset_eq; intros x. rewrite !multiplicity_union; lia.
 Qed.
 Global Instance gmultiset_assoc : Assoc (=@{gmultiset A}) (∪).
 Proof.
-  intros X Y Z. apply gmultiset_eq; intros x. rewrite !multiplicity_union; omega.
+  intros X Y Z. apply gmultiset_eq; intros x. rewrite !multiplicity_union; lia.
 Qed.
 Global Instance gmultiset_left_id : LeftId (=@{gmultiset A}) ∅ (∪).
 Proof.
@@ -127,7 +127,7 @@ Proof. intros X. by rewrite (comm_L (∪)), (left_id_L _ _). Qed.
 Global Instance gmultiset_union_inj_1 X : Inj (=) (=) (X ∪).
 Proof.
   intros Y1 Y2. rewrite !gmultiset_eq. intros HX x; generalize (HX x).
-  rewrite !multiplicity_union. omega.
+  rewrite !multiplicity_union. lia.
 Qed.
 Global Instance gmultiset_union_inj_2 X : Inj (=) (=) (∪ X).
 Proof. intros Y1 Y2. rewrite <-!(comm_L _ X). apply (inj _). Qed.
@@ -185,15 +185,15 @@ Proof.
   unfold elem_of at 2, gmultiset_elem_of, multiplicity; simpl.
   rewrite elem_of_list_bind. split.
   - intros [[??] [[<- ?]%elem_of_replicate ->%elem_of_map_to_list]]; lia.
-  - intros. destruct (X !! x) as [n|] eqn:Hx; [|omega].
+  - intros. destruct (X !! x) as [n|] eqn:Hx; [|lia].
     exists (x,n); split; [|by apply elem_of_map_to_list].
-    apply elem_of_replicate; auto with omega.
+    apply elem_of_replicate; auto with lia.
 Qed.
 Lemma gmultiset_elem_of_dom x X : x ∈ dom (gset A) X ↔ x ∈ X.
 Proof.
   unfold dom, gmultiset_dom, elem_of at 2, gmultiset_elem_of, multiplicity.
   destruct X as [X]; simpl; rewrite elem_of_dom, <-not_eq_None_Some.
-  destruct (X !! x); naive_solver omega.
+  destruct (X !! x); naive_solver lia.
 Qed.
 
 (* Properties of the size operation *)
@@ -250,7 +250,7 @@ Lemma gmultiset_subseteq_alt X Y :
   map_relation (≤) (λ _, False) (λ _, True) (gmultiset_car X) (gmultiset_car Y).
 Proof.
   apply forall_proper; intros x. unfold multiplicity.
-  destruct (gmultiset_car X !! x), (gmultiset_car Y !! x); naive_solver omega.
+  destruct (gmultiset_car X !! x), (gmultiset_car Y !! x); naive_solver lia.
 Qed.
 Global Instance gmultiset_subseteq_dec : RelDecision (⊆@{gmultiset A}).
 Proof.
@@ -264,12 +264,12 @@ Proof. apply strict_include. Qed.
 Hint Resolve gmultiset_subset_subseteq.
 
 Lemma gmultiset_empty_subseteq X : ∅ ⊆ X.
-Proof. intros x. rewrite multiplicity_empty. omega. Qed.
+Proof. intros x. rewrite multiplicity_empty. lia. Qed.
 
 Lemma gmultiset_union_subseteq_l X Y : X ⊆ X ∪ Y.
-Proof. intros x. rewrite multiplicity_union. omega. Qed.
+Proof. intros x. rewrite multiplicity_union. lia. Qed.
 Lemma gmultiset_union_subseteq_r X Y : Y ⊆ X ∪ Y.
-Proof. intros x. rewrite multiplicity_union. omega. Qed.
+Proof. intros x. rewrite multiplicity_union. lia. Qed.
 Lemma gmultiset_union_mono X1 X2 Y1 Y2 : X1 ⊆ X2 → Y1 ⊆ Y2 → X1 ∪ Y1 ⊆ X2 ∪ Y2.
 Proof. intros ?? x. rewrite !multiplicity_union. by apply Nat.add_le_mono. Qed.
 Lemma gmultiset_union_mono_l X Y1 Y2 : Y1 ⊆ Y2 → X ∪ Y1 ⊆ X ∪ Y2.
@@ -278,12 +278,12 @@ Lemma gmultiset_union_mono_r X1 X2 Y : X1 ⊆ X2 → X1 ∪ Y ⊆ X2 ∪ Y.
 Proof. intros. by apply gmultiset_union_mono. Qed.
 
 Lemma gmultiset_subset X Y : X ⊆ Y → size X < size Y → X ⊂ Y.
-Proof. intros. apply strict_spec_alt; split; naive_solver auto with omega. Qed.
+Proof. intros. apply strict_spec_alt; split; naive_solver auto with lia. Qed.
 Lemma gmultiset_union_subset_l X Y : Y ≠ ∅ → X ⊂ X ∪ Y.
 Proof.
   intros HY%gmultiset_size_non_empty_iff.
   apply gmultiset_subset; auto using gmultiset_union_subseteq_l.
-  rewrite gmultiset_size_union; omega.
+  rewrite gmultiset_size_union; lia.
 Qed.
 Lemma gmultiset_union_subset_r X Y : X ≠ ∅ → Y ⊂ X ∪ Y.
 Proof. rewrite (comm_L (∪)). apply gmultiset_union_subset_l. Qed.
@@ -292,9 +292,9 @@ Lemma gmultiset_elem_of_singleton_subseteq x X : x ∈ X ↔ {[ x ]} ⊆ X.
 Proof.
   rewrite elem_of_multiplicity. split.
   - intros Hx y; destruct (decide (x = y)) as [->|].
-    + rewrite multiplicity_singleton; omega.
-    + rewrite multiplicity_singleton_ne by done; omega.
-  - intros Hx. generalize (Hx x). rewrite multiplicity_singleton. omega.
+    + rewrite multiplicity_singleton; lia.
+    + rewrite multiplicity_singleton_ne by done; lia.
+  - intros Hx. generalize (Hx x). rewrite multiplicity_singleton. lia.
 Qed.
 
 Lemma gmultiset_elem_of_subseteq X1 X2 x : x ∈ X1 → X1 ⊆ X2 → x ∈ X2.
@@ -303,7 +303,7 @@ Proof. rewrite !gmultiset_elem_of_singleton_subseteq. by intros ->. Qed.
 Lemma gmultiset_union_difference X Y : X ⊆ Y → Y = X ∪ Y ∖ X.
 Proof.
   intros HXY. apply gmultiset_eq; intros x; specialize (HXY x).
-  rewrite multiplicity_union, multiplicity_difference; omega.
+  rewrite multiplicity_union, multiplicity_difference; lia.
 Qed.
 Lemma gmultiset_union_difference' x Y : x ∈ Y → Y = {[ x ]} ∪ Y ∖ {[ x ]}.
 Proof.
@@ -314,14 +314,14 @@ Qed.
 Lemma gmultiset_size_difference X Y : Y ⊆ X → size (X ∖ Y) = size X - size Y.
 Proof.
   intros HX%gmultiset_union_difference.
-  rewrite HX at 2; rewrite gmultiset_size_union. omega.
+  rewrite HX at 2; rewrite gmultiset_size_union. lia.
 Qed.
 
 Lemma gmultiset_non_empty_difference X Y : X ⊂ Y → Y ∖ X ≠ ∅.
 Proof.
   intros [_ HXY2] Hdiff; destruct HXY2; intros x.
   generalize (f_equal (multiplicity x) Hdiff).
-  rewrite multiplicity_difference, multiplicity_empty; omega.
+  rewrite multiplicity_difference, multiplicity_empty; lia.
 Qed.
 
 Lemma gmultiset_difference_subset X Y : X ≠ ∅ → X ⊆ Y → Y ∖ X ⊂ Y.

theories/infinite.v

@@ -52,15 +52,15 @@ Section Fresh.
     revert n.
     induction s as [s IH] using (well_founded_ind collection_wf); intros n.
     setoid_rewrite fresh_generic_fixpoint_unfold; unfold fresh_generic_body.
-    destruct decide as [Hcase|Hcase]; [|by eauto with omega].
+    destruct decide as [Hcase|Hcase]; [|by eauto with lia].
     destruct (IH _ (subset_difference_elem_of Hcase) (S n))
       as (m & Hmbound & Heqfix & Hnotin & Hinbelow).
-    exists m; repeat split; auto with omega.
+    exists m; repeat split; auto with lia.
     - rewrite not_elem_of_difference, elem_of_singleton in Hnotin.
-      destruct Hnotin as [?|?%inject_injective]; auto with omega.
+      destruct Hnotin as [?|?%inject_injective]; auto with lia.
     - intros i Hibound.
       destruct (decide (i = n)) as [<-|Hneq]; [by auto|].
-      assert (inject i ∈ s ∖ {[inject n]}) by auto with omega.
+      assert (inject i ∈ s ∖ {[inject n]}) by auto with lia.
       set_solver.
   Qed.
 
@@ -76,8 +76,8 @@ Section Fresh.
       destruct (fresh_generic_fixpoint_spec Y 0)
         as (mY & _ & -> & HnotinY & HbelowinY).
       destruct (Nat.lt_trichotomy mX mY) as [case|[->|case]]; auto.
-      + contradict HnotinX. rewrite HeqXY. apply HbelowinY; omega.
-      + contradict HnotinY. rewrite <-HeqXY. apply HbelowinX; omega.
+      + contradict HnotinX. rewrite HeqXY. apply HbelowinY; lia.
+      + contradict HnotinY. rewrite <-HeqXY. apply HbelowinX; lia.
     - intros X. unfold fresh, fresh_generic.
       destruct (fresh_generic_fixpoint_spec X 0)
         as (m & _ & -> & HnotinX & HbelowinX); auto.

theories/list.v

@@ -1484,7 +1484,7 @@ Proof.
     eapply Permutation_cons, (IH _ g).
     + rewrite length_delete by (rewrite <-Hl; eauto); simpl in *; lia.
     + unfold g. intros i j Hg.
-      repeat destruct (lt_eq_lt_dec _ _) as [[?|?]|?]; simplify_eq/=; try omega.
+      repeat destruct (lt_eq_lt_dec _ _) as [[?|?]|?]; simplify_eq/=; try lia.
       apply (inj S), (inj f); lia.
     + intros i. unfold g. destruct (lt_eq_lt_dec _ _) as [[?|?]|?].
       * by rewrite lookup_delete_lt, <-Hl.

theories/nat_cancel.v

@@ -78,25 +78,25 @@ Module nat_cancel.
   Instance nat_cancel_leaf_plus m m' m'' n1 n2 n1' n2' n1'n2' :
     NatCancelR m n1 m' n1' → NatCancelR m' n2 m'' n2' →
     MakeNatPlus n1' n2' n1'n2' → NatCancelR m (n1 + n2) m'' n1'n2' | 2.
-  Proof. unfold NatCancelR, NatCancelL, MakeNatPlus. omega. Qed.
+  Proof. unfold NatCancelR, NatCancelL, MakeNatPlus. lia. Qed.
   Instance nat_cancel_leaf_S_here m n m' n' :
     NatCancelR m n m' n' → NatCancelR (S m) (S n) m' n' | 3.
-  Proof. unfold NatCancelR, NatCancelL. omega. Qed.
+  Proof. unfold NatCancelR, NatCancelL. lia. Qed.
   Instance nat_cancel_leaf_S_else m n m' n' :
     NatCancelR m n m' n' → NatCancelR m (S n) m' (S n') | 4.
-  Proof. unfold NatCancelR, NatCancelL. omega. Qed.
+  Proof. unfold NatCancelR, NatCancelL. lia. Qed.
 
   (** The instance [nat_cancel_S_both] is redundant, but may reduce proof search
   quite a bit, e.g. when canceling constants in constants. *)
   Instance nat_cancel_S_both m n m' n' :
     NatCancelL m n m' n' → NatCancelL (S m) (S n) m' n' | 1.
-  Proof. unfold NatCancelL. omega. Qed.
+  Proof. unfold NatCancelL. lia. Qed.
   Instance nat_cancel_plus m1 m2 m1' m2' m1'm2' n n' n'' :
     NatCancelL m1 n m1' n' → NatCancelL m2 n' m2' n'' →
     MakeNatPlus m1' m2' m1'm2' → NatCancelL (m1 + m2) n m1'm2' n'' | 2.
-  Proof. unfold NatCancelL, MakeNatPlus. omega. Qed.
+  Proof. unfold NatCancelL, MakeNatPlus. lia. Qed.
   Instance nat_cancel_S m m' m'' Sm' n n' n'' :
     NatCancelL m n m' n' → NatCancelR 1 n' m'' n'' →
     MakeNatS m'' m' Sm' → NatCancelL (S m) n Sm' n'' | 3.
-  Proof. unfold NatCancelR, NatCancelL, MakeNatS. omega. Qed.
+  Proof. unfold NatCancelR, NatCancelL, MakeNatS. lia. Qed.
 End nat_cancel.