internlm/Lean-Github huggingface dataset:
Theorem: exercise4_5
Proof state before:
case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ False
Tactic:
have h2 : 2 ^ 2 ≤ 2 ^ a := by gcongr; norm_num
Proof state after:
case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
⊢ False
Theorem: exercise4_5
Proof state before:
case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
⊢ False
Tactic:
have h3 : 1 ≤ 2 ^ a := by linarith
Proof state after:
case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
⊢ False
Theorem: exercise4_5
Proof state before:
case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1
⊢ False
Tactic:
have h6' : 2 ^ 0 ≤ 2 ^ (a * b) := by gcongr; norm_num; exact?
Proof state after:
case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1
h6' : 2 ^ 0 ≤ 2 ^ (a * b)
⊢ False
Theorem: exercise4_5
Proof state before:
case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : 2 ≤ 2 ^ (a * b) - 1 ∧ ∀ m < 2 ^ (a * b) - 1, m ∣ 2 ^ (a * b) - 1 → m = 1
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1
h6' : 2 ^ 0 ≤ 2 ^ (a * b)
h6 : 1 ≤ 2 ^ (a * b)
h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ False
Tactic:
apply h4
Proof state after:
case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : 2 ≤ 2 ^ (a * b) - 1 ∧ ∀ m < 2 ^ (a * b) - 1, m ∣ 2 ^ (a * b) - 1 → m = 1
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1
h6' : 2 ^ 0 ≤ 2 ^ (a * b)
h6 : 1 ≤ 2 ^ (a * b)
h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ 2 ^ a - 1 = 1
Theorem: exercise4_5
Proof state before:
case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : 2 ≤ 2 ^ (a * b) - 1 ∧ ∀ m < 2 ^ (a * b) - 1, m ∣ 2 ^ (a * b) - 1 → m = 1
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1
h6' : 2 ^ 0 ≤ 2 ^ (a * b)
h6 : 1 ≤ 2 ^ (a * b)
h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ 2 ^ a - 1 = 1
Tactic:
apply hn.2 _ h5
Proof state after:
case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : 2 ≤ 2 ^ (a * b) - 1 ∧ ∀ m < 2 ^ (a * b) - 1, m ∣ 2 ^ (a * b) - 1 → m = 1
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1
h6' : 2 ^ 0 ≤ 2 ^ (a * b)
h6 : 1 ≤ 2 ^ (a * b)
h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
Theorem: exercise4_5
Proof state before:
case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : 2 ≤ 2 ^ (a * b) - 1 ∧ ∀ m < 2 ^ (a * b) - 1, m ∣ 2 ^ (a * b) - 1 → m = 1
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1
h6' : 2 ^ 0 ≤ 2 ^ (a * b)
h6 : 1 ≤ 2 ^ (a * b)
h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
Tactic:
norm_cast at h
Proof state after:
no goals
Theorem: exercise4_5
Proof state before:
case inl
hn : Nat.Prime (2 ^ 0 - 1)
⊢ False
Tactic:
simp at hn
Proof state after:
no goals
Theorem: exercise4_5
Proof state before:
case inr.inl
hn : Nat.Prime (2 ^ 1 - 1)
⊢ False
Tactic:
norm_num at hn
Proof state after:
no goals
Theorem: exercise4_5
Proof state before:
case h
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
⊢ 2 ^ (a * b) - 1 ≡ 0 [ZMOD 2 ^ a - 1]
Tactic:
calc (2 : ℤ) ^ (a * b) - 1
    ≡ ((2 : ℤ) ^ a) ^ b - 1 [ZMOD (2 : ℤ) ^ a - 1] := by rw [pow_mul]
  _ ≡ (1 : ℤ) ^ b - 1 [ZMOD (2 : ℤ) ^ a - 1] := by gcongr; exact?
  _ ≡ 0 [ZMOD (2 : ℤ) ^ a - 1] := by simp
Proof state after:
no goals
Theorem: exercise4_5
Proof state before:
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
⊢ 2 ^ (a * b) - 1 ≡ (2 ^ a) ^ b - 1 [ZMOD 2 ^ a - 1]
Tactic:
rw [pow_mul]
Proof state after:
no goals
Theorem: exercise4_5
Proof state before:
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
⊢ (2 ^ a) ^ b - 1 ≡ 1 ^ b - 1 [ZMOD 2 ^ a - 1]
Tactic:
gcongr
Proof state after:
case h.h
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
⊢ 2 ^ a ≡ 1 [ZMOD 2 ^ a - 1]
Theorem: exercise4_5
Proof state before:
case h.h
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
⊢ 2 ^ a ≡ 1 [ZMOD 2 ^ a - 1]
Tactic:
exact?
Proof state after:
no goals
Theorem: exercise4_5
Proof state before:
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
⊢ 1 ^ b - 1 ≡ 0 [ZMOD 2 ^ a - 1]
Tactic:
simp
Proof state after:
no goals
Theorem: exercise4_5
Proof state before:
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ 2 ^ 2 ≤ 2 ^ a
Tactic:
gcongr
Proof state after:
case ha
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ 1 ≤ 2
Theorem: exercise4_5
Proof state before:
case ha
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ 1 ≤ 2
Tactic:
norm_num
Proof state after:
no goals
Theorem: exercise4_5
Proof state before:
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
⊢ 1 ≤ 2 ^ a
Tactic:
linarith
Proof state after:
no goals
Theorem: exercise4_5
Proof state before:
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
⊢ 2 ^ a < 2 ^ (a * b)
Tactic:
gcongr
Proof state after:
case h
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
⊢ 1 < 2

case h2
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
⊢ a < a * b
Theorem: exercise4_5
Proof state before:
case h2
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
⊢ a < a * b
Tactic:
refine lt_mul_right ?h2.hn hb
Proof state after:
case h2.hn
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
⊢ 0 < a
Theorem: exercise4_5
Proof state before:
case h2.hn
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
⊢ 0 < a
Tactic:
linarith
Proof state after:
no goals
Theorem: exercise4_5
Proof state before:
case h
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
⊢ 1 < 2
Tactic:
norm_num
Proof state after:
no goals
Theorem: exercise4_5
Proof state before:
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1
⊢ 2 ^ 0 ≤ 2 ^ (a * b)
Tactic:
gcongr
Proof state after:
case ha
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1
⊢ 1 ≤ 2

case h
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1
⊢ 0 ≤ a * b
Theorem: exercise4_5
Proof state before:
case ha
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1
⊢ 1 ≤ 2

case h
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1
⊢ 0 ≤ a * b
Tactic:
norm_num
Proof state after:
case h
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1
⊢ 0 ≤ a * b
Theorem: exercise4_5
Proof state before:
case h
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1
⊢ 0 ≤ a * b
Tactic:
exact?
Proof state after:
no goals
Theorem: exercise4_6
Proof state before:
a b : ℕ
ha : 0 < a
hb : 0 < b
⊢ ¬IsSquare (a ^ 2 + b) ∨ ¬IsSquare (b ^ 2 + a)
Tactic:
have h1 : ∀ {a b : ℕ}, 0 < a → a ≤ b → ¬ IsSquare (b ^ 2 + a)
Proof state after:
case h1
a b : ℕ
ha : 0 < a
hb : 0 < b
⊢ ∀ {a b : ℕ}, 0 < a → a ≤ b → ¬IsSquare (b ^ 2 + a)

a b : ℕ
ha : 0 < a
hb : 0 < b
h1 : ∀ {a b : ℕ}, 0 < a → a ≤ b → ¬IsSquare (b ^ 2 + a)
⊢ ¬IsSquare (a ^ 2 + b) ∨ ¬IsSquare (b ^ 2 + a)
Theorem: exercise4_6
Proof state before:
a b : ℕ
ha : 0 < a
hb : 0 < b
h1 : ∀ {a b : ℕ}, 0 < a → a ≤ b → ¬IsSquare (b ^ 2 + a)
⊢ ¬IsSquare (a ^ 2 + b) ∨ ¬IsSquare (b ^ 2 + a)
Tactic:
obtain h|h := le_total a b
Proof state after:
case inl
a b : ℕ
ha : 0 < a
hb : 0 < b
h1 : ∀ {a b : ℕ}, 0 < a → a ≤ b → ¬IsSquare (b ^ 2 + a)
h : a ≤ b
⊢ ¬IsSquare (a ^ 2 + b) ∨ ¬IsSquare (b ^ 2 + a)

case inr
a b : ℕ
ha : 0 < a
hb : 0 < b
h1 : ∀ {a b : ℕ}, 0 < a → a ≤ b → ¬IsSquare (b ^ 2 + a)
h : b ≤ a
⊢ ¬IsSquare (a ^ 2 + b) ∨ ¬IsSquare (b ^ 2 + a)
Theorem: exercise4_6
Proof state before:
case h1
a b : ℕ
ha : 0 < a
hb : 0 < b
⊢ ∀ {a b : ℕ}, 0 < a → a ≤ b → ¬IsSquare (b ^ 2 + a)
Tactic:
intro a b ha hab ⟨c, hc⟩
Proof state after:
case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b ^ 2 + a = c * c
⊢ False
Theorem: exercise4_6
Proof state before:
case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b ^ 2 + a = c * c
⊢ False
Tactic:
rw [pow_two] at hc
Proof state after:
case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
⊢ False
Theorem: exercise4_6
Proof state before:
case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
⊢ False
Tactic:
have : b * b < c * c := by linarith
Proof state after:
case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this : b * b < c * c
⊢ False
Theorem: exercise4_6
Proof state before:
case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this : b * b < c * c
⊢ False
Tactic:
have : b < c := by exact?
Proof state after:
case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this✝ : b * b < c * c
this : b < c
⊢ False
Theorem: exercise4_6
Proof state before:
case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this✝ : b * b < c * c
this : b < c
⊢ False
Tactic:
have : c * c < (b + 1) * (b + 1) := by linarith
Proof state after:
case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this✝¹ : b * b < c * c
this✝ : b < c
this : c * c < (b + 1) * (b + 1)
⊢ False
Theorem: exercise4_6
Proof state before:
case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this✝¹ : b * b < c * c
this✝ : b < c
this : c * c < (b + 1) * (b + 1)
⊢ False
Tactic:
have : c < b + 1 := by exact?
Proof state after:
case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this✝² : b * b < c * c
this✝¹ : b < c
this✝ : c * c < (b + 1) * (b + 1)
this : c < b + 1
⊢ False
Theorem: exercise4_6
Proof state before:
case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this✝² : b * b < c * c
this✝¹ : b < c
this✝ : c * c < (b + 1) * (b + 1)
this : c < b + 1
⊢ False
Tactic:
exact?
Proof state after:
no goals
Theorem: exercise4_6
Proof state before:
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
⊢ b * b < c * c
Tactic:
linarith
Proof state after:
no goals
Theorem: exercise4_6
Proof state before:
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this : b * b < c * c
⊢ b < c
Tactic:
exact?
Proof state after:
no goals
Theorem: exercise4_6
Proof state before:
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this✝ : b * b < c * c
this : b < c
⊢ c * c < (b + 1) * (b + 1)
Tactic:
linarith
Proof state after:
no goals
Theorem: exercise4_6
Proof state before:
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this✝¹ : b * b < c * c
this✝ : b < c
this : c * c < (b + 1) * (b + 1)
⊢ c < b + 1
Tactic:
exact?
Proof state after:
no goals
Theorem: exercise4_6
Proof state before:
case inl
a b : ℕ
ha : 0 < a
hb : 0 < b
h1 : ∀ {a b : ℕ}, 0 < a → a ≤ b → ¬IsSquare (b ^ 2 + a)
h : a ≤ b
⊢ ¬IsSquare (a ^ 2 + b) ∨ ¬IsSquare (b ^ 2 + a)
Tactic:
exact .inr (h1 ha h)
Proof state after:
no goals
Theorem: exercise4_6
Proof state before:
case inr
a b : ℕ
ha : 0 < a
hb : 0 < b
h1 : ∀ {a b : ℕ}, 0 < a → a ≤ b → ¬IsSquare (b ^ 2 + a)
h : b ≤ a
⊢ ¬IsSquare (a ^ 2 + b) ∨ ¬IsSquare (b ^ 2 + a)
Tactic:
exact .inl (h1 hb h)
Proof state after:
no goals
Theorem: exercise5_3
Proof state before:
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_1
inst✝¹ : Field K
inst✝ : CharP K p
x y : K
⊢ (x + y) ^ p = x ^ p + y ^ p
Tactic:
rw [add_pow]
Proof state after:
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_1
inst✝¹ : Field K
inst✝ : CharP K p
x y : K
⊢ ∑ m in range (p + 1), x ^ m * y ^ (p - m) * ↑(Nat.choose p m) = x ^ p + y ^ p
Theorem: exercise5_3
Proof state before:
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_1
inst✝¹ : Field K
inst✝ : CharP K p
x y : K
⊢ ∑ m in range (p + 1), x ^ m * y ^ (p - m) * ↑(Nat.choose p m) = x ^ p + y ^ p
Tactic:
have h2 : p.Prime := hp.out
Proof state after:
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_1
inst✝¹ : Field K
inst✝ : CharP K p
x y : K
h2 : Nat.Prime p
⊢ ∑ m in range (p + 1), x ^ m * y ^ (p - m) * ↑(Nat.choose p m) = x ^ p + y ^ p
Theorem: exercise5_3
Proof state before:
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_1
inst✝¹ : Field K
inst✝ : CharP K p
x y : K
h2 : Nat.Prime p
⊢ ∑ m in range (p + 1), x ^ m * y ^ (p - m) * ↑(Nat.choose p m) = x ^ p + y ^ p
Tactic:
have h3 : 0 < p := h2.pos
Proof state after:
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_1
inst✝¹ : Field K
inst✝ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
⊢ ∑ m in range (p + 1), x ^ m * y ^ (p - m) * ↑(Nat.choose p m) = x ^ p + y ^ p
Theorem: exercise5_3
Proof state before:
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_1
inst✝¹ : Field K
inst✝ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
⊢ ∑ m in range (p + 1), x ^ m * y ^ (p - m) * ↑(Nat.choose p m) = x ^ p + y ^ p
Tactic:
have h4 : range p = insert 0 (Ioo 0 p)
Proof state after:
case h4
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_1
inst✝¹ : Field K
inst✝ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
⊢ range p = insert 0 (Ioo 0 p)

p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_1
inst✝¹ : Field K
inst✝ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
⊢ ∑ m in range (p + 1), x ^ m * y ^ (p - m) * ↑(Nat.choose p m) = x ^ p + y ^ p
Theorem: exercise5_3
Proof state before:
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_1
inst✝¹ : Field K
inst✝ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
⊢ ∑ m in range (p + 1), x ^ m * y ^ (p - m) * ↑(Nat.choose p m) = x ^ p + y ^ p
Tactic:
have h5 : ∀ i ∈ Ioo 0 p, p ∣ Nat.choose p i := by sorry
Proof state after:
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_1
inst✝¹ : Field K
inst✝ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
h5 : ∀ i ∈ Ioo 0 p, p ∣ Nat.choose p i
⊢ ∑ m in range (p + 1), x ^ m * y ^ (p - m) * ↑(Nat.choose p m) = x ^ p + y ^ p
Theorem: exercise5_3
Proof state before:
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_1
inst✝¹ : Field K
inst✝ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
h5 : ∀ i ∈ Ioo 0 p, p ∣ Nat.choose p i
⊢ ∑ m in range (p + 1), x ^ m * y ^ (p - m) * ↑(Nat.choose p m) = x ^ p + y ^ p
Tactic:
have h6 : ∑ i in Ioo 0 p, x ^ i * y ^ (p - i) * Nat.choose p i = 0 :=
calc
  _ =  ∑ i in Ioo 0 p, x ^ i * y ^ (p - i) * 0 := by sorry
  _ = 0 := by sorry
Proof state after:
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_1
inst✝¹ : Field K
inst✝ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
h5 : ∀ i ∈ Ioo 0 p, p ∣ Nat.choose p i
h6 : ∑ i in Ioo 0 p, x ^ i * y ^ (p - i) * ↑(Nat.choose p i) = 0
⊢ ∑ m in range (p + 1), x ^ m * y ^ (p - m) * ↑(Nat.choose p m) = x ^ p + y ^ p
Theorem: exercise5_3
Proof state before:
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_1
inst✝¹ : Field K
inst✝ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
h5 : ∀ i ∈ Ioo 0 p, p ∣ Nat.choose p i
h6 : ∑ i in Ioo 0 p, x ^ i * y ^ (p - i) * ↑(Nat.choose p i) = 0
⊢ ∑ m in range (p + 1), x ^ m * y ^ (p - m) * ↑(Nat.choose p m) = x ^ p + y ^ p
Tactic:
sorry
Proof state after:
no goals
Theorem: exercise5_3
Proof state before:
case h4
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_1
inst✝¹ : Field K
inst✝ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
⊢ range p = insert 0 (Ioo 0 p)
Tactic:
ext (_|_) <;> simp [h3]
Proof state after:
no goals
Theorem: exercise5_3
Proof state before:
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_1
inst✝¹ : Field K
inst✝ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
⊢ ∀ i ∈ Ioo 0 p, p ∣ Nat.choose p i
Tactic:
sorry
Proof state after:
no goals
Theorem: exercise5_3
Proof state before:
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_1
inst✝¹ : Field K
inst✝ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
h5 : ∀ i ∈ Ioo 0 p, p ∣ Nat.choose p i
⊢ ∑ i in Ioo 0 p, x ^ i * y ^ (p - i) * ↑(Nat.choose p i) = ∑ i in Ioo 0 p, x ^ i * y ^ (p - i) * 0
Tactic:
sorry
Proof state after:
no goals
Theorem: exercise5_3
Proof state before:
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_1
inst✝¹ : Field K
inst✝ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
h5 : ∀ i ∈ Ioo 0 p, p ∣ Nat.choose p i
⊢ ∑ i in Ioo 0 p, x ^ i * y ^ (p - i) * 0 = 0
Tactic:
sorry
Proof state after:
no goals
Theorem: exercise5_4
Proof state before:
R : Type u_1
M : Type u_2
M' : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Nontrivial M
inst✝¹ : NoZeroSMulDivisors R M
inst✝ : Module R (M →ₗ[R] M)
h : ∀ (r : R) (f : M →ₗ[R] M) (x : M), ↑(r • f) x = r • ↑f x
r s : R
⊢ r * s = s * r
Tactic:
sorry
Proof state after:
no goals
Theorem: C05S03.primes_infinite
Proof state before:
n : ℕ
⊢ ∃ p > n, Nat.Prime p
Tactic:
have : 2 ≤ Nat.factorial (n + 1) + 1 := by
  apply Nat.succ_le_succ
  exact Nat.succ_le_of_lt (Nat.factorial_pos _)
Proof state after:
n : ℕ
this : 2 ≤ Nat.factorial (n + 1) + 1
⊢ ∃ p > n, Nat.Prime p
Theorem: C05S03.primes_infinite
Proof state before:
case intro.intro
n : ℕ
this : 2 ≤ Nat.factorial (n + 1) + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ Nat.factorial (n + 1) + 1
ple : p ≤ n
⊢ False
Tactic:
have : p ∣ Nat.factorial (n + 1) := by
  apply Nat.dvd_factorial
  apply pp.pos
  linarith
Proof state after:
case intro.intro
n : ℕ
this✝ : 2 ≤ Nat.factorial (n + 1) + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ Nat.factorial (n + 1) + 1
ple : p ≤ n
this : p ∣ Nat.factorial (n + 1)
⊢ False
Theorem: C05S03.primes_infinite
Proof state before:
case intro.intro
n : ℕ
this✝ : 2 ≤ Nat.factorial (n + 1) + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ Nat.factorial (n + 1) + 1
ple : p ≤ n
this : p ∣ Nat.factorial (n + 1)
⊢ False
Tactic:
have : p ∣ 1 := by
  convert Nat.dvd_sub' pdvd this
  simp
Proof state after:
case intro.intro
n : ℕ
this✝¹ : 2 ≤ Nat.factorial (n + 1) + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ Nat.factorial (n + 1) + 1
ple : p ≤ n
this✝ : p ∣ Nat.factorial (n + 1)
this : p ∣ 1
⊢ False
Theorem: C05S03.primes_infinite
Proof state before:
case intro.intro
n : ℕ
this✝¹ : 2 ≤ Nat.factorial (n + 1) + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ Nat.factorial (n + 1) + 1
ple : p ≤ n
this✝ : p ∣ Nat.factorial (n + 1)
this : p ∣ 1
⊢ False
Tactic:
have := Nat.le_of_dvd zero_lt_one this
Proof state after:
case intro.intro
n : ℕ
this✝² : 2 ≤ Nat.factorial (n + 1) + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ Nat.factorial (n + 1) + 1
ple : p ≤ n
this✝¹ : p ∣ Nat.factorial (n + 1)
this✝ : p ∣ 1
this : p ≤ 1
⊢ False
Theorem: C05S03.primes_infinite
Proof state before:
case intro.intro
n : ℕ
this✝² : 2 ≤ Nat.factorial (n + 1) + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ Nat.factorial (n + 1) + 1
ple : p ≤ n
this✝¹ : p ∣ Nat.factorial (n + 1)
this✝ : p ∣ 1
this : p ≤ 1
⊢ False
Tactic:
linarith [pp.two_le]
Proof state after:
no goals
Theorem: C05S03.primes_infinite
Proof state before:
n : ℕ
⊢ 2 ≤ Nat.factorial (n + 1) + 1
Tactic:
apply Nat.succ_le_succ
Proof state after:
case a
n : ℕ
⊢ 1 ≤ Nat.factorial (n + 1)
Theorem: C05S03.primes_infinite
Proof state before:
case a
n : ℕ
⊢ 1 ≤ Nat.factorial (n + 1)
Tactic:
exact Nat.succ_le_of_lt (Nat.factorial_pos _)
Proof state after:
no goals
Theorem: C05S03.primes_infinite
Proof state before:
n : ℕ
this : 2 ≤ Nat.factorial (n + 1) + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ Nat.factorial (n + 1) + 1
ple : p ≤ n
⊢ p ∣ Nat.factorial (n + 1)
Tactic:
apply Nat.dvd_factorial
Proof state after:
case a
n : ℕ
this : 2 ≤ Nat.factorial (n + 1) + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ Nat.factorial (n + 1) + 1
ple : p ≤ n
⊢ 0 < p

case a
n : ℕ
this : 2 ≤ Nat.factorial (n + 1) + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ Nat.factorial (n + 1) + 1
ple : p ≤ n
⊢ p ≤ n + 1
Theorem: C05S03.primes_infinite
Proof state before:
case a
n : ℕ
this : 2 ≤ Nat.factorial (n + 1) + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ Nat.factorial (n + 1) + 1
ple : p ≤ n
⊢ 0 < p

case a
n : ℕ
this : 2 ≤ Nat.factorial (n + 1) + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ Nat.factorial (n + 1) + 1
ple : p ≤ n
⊢ p ≤ n + 1
Tactic:
apply pp.pos
Proof state after:
case a
n : ℕ
this : 2 ≤ Nat.factorial (n + 1) + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ Nat.factorial (n + 1) + 1
ple : p ≤ n
⊢ p ≤ n + 1
Theorem: C05S03.primes_infinite
Proof state before:
case a
n : ℕ
this : 2 ≤ Nat.factorial (n + 1) + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ Nat.factorial (n + 1) + 1
ple : p ≤ n
⊢ p ≤ n + 1
Tactic:
linarith
Proof state after:
no goals
Theorem: C05S03.primes_infinite
Proof state before:
n : ℕ
this✝ : 2 ≤ Nat.factorial (n + 1) + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ Nat.factorial (n + 1) + 1
ple : p ≤ n
this : p ∣ Nat.factorial (n + 1)
⊢ p ∣ 1
Tactic:
convert Nat.dvd_sub' pdvd this
Proof state after:
case h.e'_4
n : ℕ
this✝ : 2 ≤ Nat.factorial (n + 1) + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ Nat.factorial (n + 1) + 1
ple : p ≤ n
this : p ∣ Nat.factorial (n + 1)
⊢ 1 = Nat.factorial (n + 1) + 1 - Nat.factorial (n + 1)
Theorem: C05S03.primes_infinite
Proof state before:
case h.e'_4
n : ℕ
this✝ : 2 ≤ Nat.factorial (n + 1) + 1
p : ℕ
pp : Nat.Prime p
pdvd : p ∣ Nat.factorial (n + 1) + 1
ple : p ≤ n
this : p ∣ Nat.factorial (n + 1)
⊢ 1 = Nat.factorial (n + 1) + 1 - Nat.factorial (n + 1)
Tactic:
simp
Proof state after:
no goals
Theorem: Nat.Prime.eq_of_dvd_of_prime
Proof state before:
p q : ℕ
prime_p : Nat.Prime p
prime_q : Nat.Prime q
h : p ∣ q
⊢ p = q
Tactic:
cases prime_q.eq_one_or_self_of_dvd _ h
Proof state after:
case inl
p q : ℕ
prime_p : Nat.Prime p
prime_q : Nat.Prime q
h : p ∣ q
h✝ : p = 1
⊢ p = q

case inr
p q : ℕ
prime_p : Nat.Prime p
prime_q : Nat.Prime q
h : p ∣ q
h✝ : p = q
⊢ p = q
Theorem: Nat.Prime.eq_of_dvd_of_prime
Proof state before:
case inr
p q : ℕ
prime_p : Nat.Prime p
prime_q : Nat.Prime q
h : p ∣ q
h✝ : p = q
⊢ p = q
Tactic:
assumption
Proof state after:
no goals
Theorem: Nat.Prime.eq_of_dvd_of_prime
Proof state before:
case inl
p q : ℕ
prime_p : Nat.Prime p
prime_q : Nat.Prime q
h : p ∣ q
h✝ : p = 1
⊢ p = q
Tactic:
linarith [prime_p.two_le]
Proof state after:
no goals