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+import Lean
+import Std.Data.Int.Lemmas
+import Mathlib.Tactic.Linarith
+
+namespace Arith
+
+open Lean Elab Term Meta
+
+-- We can't define and use trace classes in the same file
+initialize registerTraceClass `Arith
+
+-- TODO: move?
+theorem ne_zero_is_lt_or_gt {x : Int} (hne : x ≠ 0) : x < 0 ∨ x > 0 := by
+  cases h: x <;> simp_all
+  . rename_i n;
+    cases n <;> simp_all
+  . apply Int.negSucc_lt_zero
+
+-- TODO: move?
+theorem ne_is_lt_or_gt {x y : Int} (hne : x ≠ y) : x < y ∨ x > y := by
+  have hne : x - y ≠ 0 := by
+    simp
+    intro h
+    have: x = y := by linarith
+    simp_all
+  have h := ne_zero_is_lt_or_gt hne
+  match h with
+  | .inl _ => left; linarith
+  | .inr _ => right; linarith
+
+-- TODO: move?
+theorem add_one_le_iff_le_ne (n m : Nat) (h1 : m ≤ n) (h2 : m ≠ n) : m + 1 ≤ n := by
+  -- Damn, those proofs on natural numbers are hard - I wish Omega was in mathlib4...
+  simp [Nat.add_one_le_iff]
+  simp [Nat.lt_iff_le_and_ne]
+  simp_all
+
+/- Induction over positive integers -/
+-- TODO: move
+theorem int_pos_ind (p : Int → Prop) :
+  (zero:p 0) → (pos:∀ i, 0 ≤ i → p i → p (i + 1)) → ∀ i, 0 ≤ i → p i := by
+  intro h0 hr i hpos
+--  have heq : Int.toNat i = i := by
+--    cases i <;> simp_all
+  have ⟨ n, heq ⟩  : {n:Nat // n = i } := ⟨ Int.toNat i, by cases i <;> simp_all ⟩
+  revert i
+  induction n
+  . intro i hpos heq
+    cases i <;> simp_all
+  . rename_i n hi
+    intro i hpos heq
+    cases i <;> simp_all
+    rename_i m
+    cases m <;> simp_all
+
+-- We sometimes need this to make sure no natural numbers appear in the goals
+-- TODO: there is probably something more general to do
+theorem nat_zero_eq_int_zero : (0 : Nat) = (0 : Int) := by simp
+
+end Arith