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import Lean
import Mathlib.Tactic.Linarith -- Introduces a lot of useful lemmas
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 omega
simp_all
have h := ne_zero_is_lt_or_gt hne
match h with
| .inl _ => left; omega
| .inr _ => right; omega
-- 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
-- This is mostly used in termination proofs
theorem to_int_to_nat_lt (x y : ℤ) (h0 : 0 ≤ x) (h1 : x < y) :
↑(x.toNat) < y := by
simp [*]
-- This is mostly used in termination proofs
theorem to_int_sub_to_nat_lt (x y : ℤ) (x' : ℕ)
(h0 : ↑x' ≤ x) (h1 : x - ↑x' < y) :
↑(x.toNat - x') < y := by
have : 0 ≤ x := by omega
simp [Int.toNat_sub_of_le, *]
end Arith
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