natural-number-game

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lemma one_add_le_self (x : ℕ) : x ≤ 1 + x :=
begin
  rw le_iff_exists_add,
  use 1,
  rw add_comm,
end

lemma le_refl (x : ℕ) : x ≤ x :=
begin
  use 0,
  rw add_zero,
  refl,
end

theorem le_succ (a b : ℕ) : a ≤ b → a ≤ (succ b) :=
begin
  intro h,
  cases h with c hc,
  use c + 1,
  rw add_one_eq_succ,
  rw add_succ,
  rw hc,
  refl,
end

lemma zero_le (a : ℕ) : 0 ≤ a :=
begin
  use a,
  rw zero_add,
  refl,
end

theorem le_trans (a b c : ℕ) (hab : a ≤ b) (hbc : b ≤ c) : a ≤ c :=
begin
  cases hab with b' hb',
  cases hbc with c' hc',
  use b' + c',
  rw ← add_assoc,
  rw ← hb',
  exact hc',
end

theorem le_antisymm (a b : ℕ) (hab : a ≤ b) (hba : b ≤ a) : a = b :=
begin
  cases hab with c hc,
  cases hba with d hd,
  rw hd at hc,
  rw add_assoc at hc,
  symmetry at hc,
  have p := eq_zero_of_add_right_eq_self hc,
  have q := add_right_eq_zero p,
  rw hd,
  rw q,
  rw add_zero,
  refl,
end

lemma le_zero (a : ℕ) (h : a ≤ 0) : a = 0 := le_antisymm _ _ h (zero_le _)

lemma succ_le_succ (a b : ℕ) (h : a ≤ b) : succ a ≤ succ b :=
begin
  cases h with c hc,
  use c,
  rw succ_add,
  rw hc,
  refl,
end

theorem le_total (a b : ℕ) : a ≤ b ∨ b ≤ a :=
begin
  induction b with c,
  right,
  exact zero_le a,
  cases b_ih with hac hca,
  left,
  apply le_succ,
  exact hac,
  cases hca with d hd,
  cases d with e,
  rw add_zero at hd,
  left,
  rw hd,
  rw succ_eq_add_one,
  use 1,
  refl,
  right,
  use e,
  rw succ_add,
  rw ← add_succ,
  exact hd,
end

lemma le_succ_self (a : ℕ) : a ≤ succ a :=
begin
  use 1,
  exact succ_eq_add_one a,
end

theorem add_le_add_right {a b : ℕ} : a ≤ b → ∀ t, (a + t) ≤ (b + t) :=
begin
  intros h t,
  cases h with c hc,
  use c,
  rw add_assoc,
  rw ← add_comm c,
  rw ← add_assoc,
  rw hc,
  refl,
end

theorem le_of_succ_le_succ (a b : ℕ) : succ a ≤ succ b → a ≤ b :=
begin
  intro h,
  cases h with c hc,
  use c,
  rw succ_add at hc,
  apply succ_inj,
  exact hc,
end

theorem not_succ_le_self (a : ℕ) : ¬ (succ a ≤ a) :=
begin
  intro h,
  cases h with c hc,
  rw succ_add at hc,
  rw ← add_succ at hc,
  symmetry at hc,
  apply succ_ne_zero,
  apply eq_zero_of_add_right_eq_self,
  exact hc,
end

theorem add_le_add_left {a b : ℕ} (h : a ≤ b) (t : ℕ) : t + a ≤ t + b :=
begin
  repeat {rw add_comm t},
  exact add_le_add_right h t,
end

lemma lt_aux_one (a b : ℕ) : a ≤ b ∧ ¬ (b ≤ a) → succ a ≤ b :=
begin
  intro h,
  have p := le_total (succ a) b,
  cases p with hab hba,
  exact hab,
  cases hba with c hc,
  cases c with d,
  use 0,
  rw add_zero,
  rw add_zero at hc,
  symmetry at hc,
  exact hc,
  exfalso,
  apply h.2,
  use d,
  apply succ_inj,
  rw add_succ at hc,
  exact hc,
end

lemma lt_aux_two (a b : ℕ) : succ a ≤ b → a ≤ b ∧ ¬ (b ≤ a) :=
begin
  intro h,
  split,
  exact le_trans _ _ _ (le_succ_self a) h,
  intro p,
  apply not_succ_le_self b,
  exact le_trans _ _ _ (succ_le_succ _ _ p) h,
end

lemma lt_iff_succ_le (a b : ℕ) : a < b ↔ succ a ≤ b :=
begin
  split,
  intro h,
  apply lt_aux_one,
  exact h,
  intro h,
  apply lt_aux_two,
  exact h,
end