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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


/- 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

end Arith