url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_20_iii' | [337, 1] | [352, 29] | rwa [pow_inj_iff_of_orderOf_eq_zero h] at hxy | case hi
G : Type u_1
inst✝ : Group G
g : G
h : orderOf g = 0
x y : ℕ
hxy : (fun i => g ^ i) x = (fun i => g ^ i) y
⊢ x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hi
G : Type u_1
inst✝ : Group G
g : G
h : orderOf g = 0
x y : ℕ
hxy : (fun i => g ^ i) x = (fun i => g ^ i) y
⊢ x = y
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | Int.gcd_val | [362, 1] | [363, 89] | rw [← natAbs_euclideanDomain_gcd, EuclideanDomain.gcd_val, natAbs_euclideanDomain_gcd] | a b : ℤ
⊢ gcd a b = gcd (b % a) a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℤ
⊢ gcd a b = gcd (b % a) a
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | coe_mul_inv_eq_one'' | [367, 1] | [377, 84] | cases n with
| zero =>
simp [isCoprime_zero_right] at h
rw [ZMod.mul_inv_eq_gcd, Nat.gcd_zero_right, Nat.cast_eq_one]
simp only [ZMod.val]
rw [Int.cast_id, Int.IsUnit.natAbs_eq h]
| succ n =>
haveI := n.succ_pos
rw [← Int.gcd_eq_one_iff_coprime, Int.gcd_comm, Int.gcd_val] at h
rw [ZMod.mul_inv_eq_gcd, ← Int.coe_nat_gcd, ZMod.val_int_cast, h, Nat.cast_one] | n : ℕ
x : ℤ
h : IsCoprime x ↑n
⊢ ↑x * (↑x)⁻¹ = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x : ℤ
h : IsCoprime x ↑n
⊢ ↑x * (↑x)⁻¹ = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | coe_mul_inv_eq_one'' | [367, 1] | [377, 84] | simp [isCoprime_zero_right] at h | case zero
x : ℤ
h : IsCoprime x ↑Nat.zero
⊢ ↑x * (↑x)⁻¹ = 1 | case zero
x : ℤ
h : IsUnit x
⊢ ↑x * (↑x)⁻¹ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
x : ℤ
h : IsCoprime x ↑Nat.zero
⊢ ↑x * (↑x)⁻¹ = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | coe_mul_inv_eq_one'' | [367, 1] | [377, 84] | rw [ZMod.mul_inv_eq_gcd, Nat.gcd_zero_right, Nat.cast_eq_one] | case zero
x : ℤ
h : IsUnit x
⊢ ↑x * (↑x)⁻¹ = 1 | case zero
x : ℤ
h : IsUnit x
⊢ ZMod.val ↑x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
x : ℤ
h : IsUnit x
⊢ ↑x * (↑x)⁻¹ = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | coe_mul_inv_eq_one'' | [367, 1] | [377, 84] | simp only [ZMod.val] | case zero
x : ℤ
h : IsUnit x
⊢ ZMod.val ↑x = 1 | case zero
x : ℤ
h : IsUnit x
⊢ Int.natAbs ↑x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
x : ℤ
h : IsUnit x
⊢ ZMod.val ↑x = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | coe_mul_inv_eq_one'' | [367, 1] | [377, 84] | rw [Int.cast_id, Int.IsUnit.natAbs_eq h] | case zero
x : ℤ
h : IsUnit x
⊢ Int.natAbs ↑x = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
x : ℤ
h : IsUnit x
⊢ Int.natAbs ↑x = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | coe_mul_inv_eq_one'' | [367, 1] | [377, 84] | haveI := n.succ_pos | case succ
x : ℤ
n : ℕ
h : IsCoprime x ↑(Nat.succ n)
⊢ ↑x * (↑x)⁻¹ = 1 | case succ
x : ℤ
n : ℕ
h : IsCoprime x ↑(Nat.succ n)
this : 0 < Nat.succ n
⊢ ↑x * (↑x)⁻¹ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
x : ℤ
n : ℕ
h : IsCoprime x ↑(Nat.succ n)
⊢ ↑x * (↑x)⁻¹ = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | coe_mul_inv_eq_one'' | [367, 1] | [377, 84] | rw [← Int.gcd_eq_one_iff_coprime, Int.gcd_comm, Int.gcd_val] at h | case succ
x : ℤ
n : ℕ
h : IsCoprime x ↑(Nat.succ n)
this : 0 < Nat.succ n
⊢ ↑x * (↑x)⁻¹ = 1 | case succ
x : ℤ
n : ℕ
h : Int.gcd (x % ↑(Nat.succ n)) ↑(Nat.succ n) = 1
this : 0 < Nat.succ n
⊢ ↑x * (↑x)⁻¹ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
x : ℤ
n : ℕ
h : IsCoprime x ↑(Nat.succ n)
this : 0 < Nat.succ n
⊢ ↑x * (↑x)⁻¹ = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | coe_mul_inv_eq_one'' | [367, 1] | [377, 84] | rw [ZMod.mul_inv_eq_gcd, ← Int.coe_nat_gcd, ZMod.val_int_cast, h, Nat.cast_one] | case succ
x : ℤ
n : ℕ
h : Int.gcd (x % ↑(Nat.succ n)) ↑(Nat.succ n) = 1
this : 0 < Nat.succ n
⊢ ↑x * (↑x)⁻¹ = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
x : ℤ
n : ℕ
h : Int.gcd (x % ↑(Nat.succ n)) ↑(Nat.succ n) = 1
this : 0 < Nat.succ n
⊢ ↑x * (↑x)⁻¹ = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ZMod.pow_totient' | [388, 1] | [391, 25] | cases n | n : ℕ
x : (ZMod n)ˣ
⊢ x ^ φ n = 1 | case zero
x : (ZMod Nat.zero)ˣ
⊢ x ^ φ Nat.zero = 1
case succ
n✝ : ℕ
x : (ZMod (Nat.succ n✝))ˣ
⊢ x ^ φ (Nat.succ n✝) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x : (ZMod n)ˣ
⊢ x ^ φ n = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ZMod.pow_totient' | [388, 1] | [391, 25] | apply ZMod.pow_totient | case succ
n✝ : ℕ
x : (ZMod (Nat.succ n✝))ˣ
⊢ x ^ φ (Nat.succ n✝) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n✝ : ℕ
x : (ZMod (Nat.succ n✝))ˣ
⊢ x ^ φ (Nat.succ n✝) = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ZMod.pow_totient' | [388, 1] | [391, 25] | simp | case zero
x : (ZMod Nat.zero)ˣ
⊢ x ^ φ Nat.zero = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
x : (ZMod Nat.zero)ˣ
⊢ x ^ φ Nat.zero = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_23 | [393, 1] | [395, 99] | rw [← CharP.intCast_eq_intCast (ZMod n), Int.cast_pow, Int.cast_one, ←
coe_unitOfIsCoprime' _ h, ← Units.val_pow_eq_pow_val, Units.val_eq_one, ← ZMod.pow_totient' _] | n : ℕ+
x : ℤ
h : IsCoprime x ↑↑n
⊢ x ^ φ ↑n ≡ 1 [ZMOD ↑↑n] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
x : ℤ
h : IsCoprime x ↑↑n
⊢ x ^ φ ↑n ≡ 1 [ZMOD ↑↑n]
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | n1_2_24 | [406, 1] | [425, 13] | refine' ⟨ℚ ⧸ AddSubgroup.zmultiples (1 : ℚ), inferInstance, fun g => _, _⟩ | ⊢ ∃ G x, (∀ (g : G), IsOfFinAddOrder g) ∧ AddMonoid.exponent G = 0 | case refine'_1
g : ℚ ⧸ AddSubgroup.zmultiples 1
⊢ IsOfFinAddOrder g
case refine'_2
⊢ AddMonoid.exponent (ℚ ⧸ AddSubgroup.zmultiples 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∃ G x, (∀ (g : G), IsOfFinAddOrder g) ∧ AddMonoid.exponent G = 0
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | n1_2_24 | [406, 1] | [425, 13] | induction g using QuotientAddGroup.induction_on' with | H g => ?_ | case refine'_1
g : ℚ ⧸ AddSubgroup.zmultiples 1
⊢ IsOfFinAddOrder g | case refine'_1.H
g : ℚ
⊢ IsOfFinAddOrder ↑g | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1
g : ℚ ⧸ AddSubgroup.zmultiples 1
⊢ IsOfFinAddOrder g
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | n1_2_24 | [406, 1] | [425, 13] | rw [isOfFinAddOrder_iff_nsmul_eq_zero] | case refine'_1.H
g : ℚ
⊢ IsOfFinAddOrder ↑g | case refine'_1.H
g : ℚ
⊢ ∃ n, 0 < n ∧ n • ↑g = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1.H
g : ℚ
⊢ IsOfFinAddOrder ↑g
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | n1_2_24 | [406, 1] | [425, 13] | refine' ⟨g.den, g.pos, _⟩ | case refine'_1.H
g : ℚ
⊢ ∃ n, 0 < n ∧ n • ↑g = 0 | case refine'_1.H
g : ℚ
⊢ g.den • ↑g = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1.H
g : ℚ
⊢ ∃ n, 0 < n ∧ n • ↑g = 0
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | n1_2_24 | [406, 1] | [425, 13] | rw [← QuotientAddGroup.mk_nsmul, nsmul_eq_mul, QuotientAddGroup.eq_zero_iff,
AddSubgroup.mem_zmultiples_iff] | case refine'_1.H
g : ℚ
⊢ g.den • ↑g = 0 | case refine'_1.H
g : ℚ
⊢ ∃ k, k • 1 = ↑g.den * g | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1.H
g : ℚ
⊢ g.den • ↑g = 0
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | n1_2_24 | [406, 1] | [425, 13] | use g.num | case refine'_1.H
g : ℚ
⊢ ∃ k, k • 1 = ↑g.den * g | case h
g : ℚ
⊢ g.num • 1 = ↑g.den * g | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1.H
g : ℚ
⊢ ∃ k, k • 1 = ↑g.den * g
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | n1_2_24 | [406, 1] | [425, 13] | rw [Int.smul_one_eq_coe, ← Rat.mul_den_eq_num, mul_comm] | case h
g : ℚ
⊢ g.num • 1 = ↑g.den * g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
g : ℚ
⊢ g.num • 1 = ↑g.den * g
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | n1_2_24 | [406, 1] | [425, 13] | rw [AddMonoid.exponent_eq_zero_iff, AddMonoid.ExponentExists] | case refine'_2
⊢ AddMonoid.exponent (ℚ ⧸ AddSubgroup.zmultiples 1) = 0 | case refine'_2
⊢ ¬∃ n, 0 < n ∧ ∀ (g : ℚ ⧸ AddSubgroup.zmultiples 1), n • g = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
⊢ AddMonoid.exponent (ℚ ⧸ AddSubgroup.zmultiples 1) = 0
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | n1_2_24 | [406, 1] | [425, 13] | push_neg | case refine'_2
⊢ ¬∃ n, 0 < n ∧ ∀ (g : ℚ ⧸ AddSubgroup.zmultiples 1), n • g = 0 | case refine'_2
⊢ ∀ (n : ℕ), 0 < n → ∃ g, n • g ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
⊢ ¬∃ n, 0 < n ∧ ∀ (g : ℚ ⧸ AddSubgroup.zmultiples 1), n • g = 0
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | n1_2_24 | [406, 1] | [425, 13] | refine' fun n hn => ⟨(1 / (2 * n) : ℚ), _⟩ | case refine'_2
⊢ ∀ (n : ℕ), 0 < n → ∃ g, n • g ≠ 0 | case refine'_2
n : ℕ
hn : 0 < n
⊢ n • ↑(1 / (2 * ↑n)) ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
⊢ ∀ (n : ℕ), 0 < n → ∃ g, n • g ≠ 0
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | n1_2_24 | [406, 1] | [425, 13] | rw [← QuotientAddGroup.mk_nsmul, nsmul_eq_mul, Ne.def, QuotientAddGroup.eq_zero_iff,
AddSubgroup.mem_zmultiples_iff, not_exists] | case refine'_2
n : ℕ
hn : 0 < n
⊢ n • ↑(1 / (2 * ↑n)) ≠ 0 | case refine'_2
n : ℕ
hn : 0 < n
⊢ ∀ (x : ℤ), ¬x • 1 = ↑n * (1 / (2 * ↑n)) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
n : ℕ
hn : 0 < n
⊢ n • ↑(1 / (2 * ↑n)) ≠ 0
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | n1_2_24 | [406, 1] | [425, 13] | refine' fun x => ne_of_apply_ne Rat.den _ | case refine'_2
n : ℕ
hn : 0 < n
⊢ ∀ (x : ℤ), ¬x • 1 = ↑n * (1 / (2 * ↑n)) | case refine'_2
n : ℕ
hn : 0 < n
x : ℤ
⊢ (x • 1).den ≠ (↑n * (1 / (2 * ↑n))).den | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
n : ℕ
hn : 0 < n
⊢ ∀ (x : ℤ), ¬x • 1 = ↑n * (1 / (2 * ↑n))
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | n1_2_24 | [406, 1] | [425, 13] | rw [Int.smul_one_eq_coe, Rat.coe_int_den, one_div, mul_inv_rev, ← mul_assoc,
mul_inv_cancel ((@Nat.cast_ne_zero ℚ _ _ _).mpr hn.ne'), one_mul, ← Int.cast_two,
Rat.inv_coe_int_den] | case refine'_2
n : ℕ
hn : 0 < n
x : ℤ
⊢ (x • 1).den ≠ (↑n * (1 / (2 * ↑n))).den | case refine'_2
n : ℕ
hn : 0 < n
x : ℤ
⊢ 1 ≠ if 2 = 0 then 1 else Int.natAbs 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
n : ℕ
hn : 0 < n
x : ℤ
⊢ (x • 1).den ≠ (↑n * (1 / (2 * ↑n))).den
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | n1_2_24 | [406, 1] | [425, 13] | norm_num | case refine'_2
n : ℕ
hn : 0 < n
x : ℤ
⊢ 1 ≠ if 2 = 0 then 1 else Int.natAbs 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
n : ℕ
hn : 0 < n
x : ℤ
⊢ 1 ≠ if 2 = 0 then 1 else Int.natAbs 2
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_25' | [433, 1] | [438, 21] | cases fintypeOrInfinite G | G : Type u_1
inst✝ : Group G
⊢ d1224 G ∣ Nat.card G | case inl
G : Type u_1
inst✝ : Group G
val✝ : Fintype G
⊢ d1224 G ∣ Nat.card G
case inr
G : Type u_1
inst✝ : Group G
val✝ : Infinite G
⊢ d1224 G ∣ Nat.card G | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
⊢ d1224 G ∣ Nat.card G
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_25' | [433, 1] | [438, 21] | rw [Nat.card_eq_fintype_card] | case inl
G : Type u_1
inst✝ : Group G
val✝ : Fintype G
⊢ d1224 G ∣ Nat.card G | case inl
G : Type u_1
inst✝ : Group G
val✝ : Fintype G
⊢ d1224 G ∣ Fintype.card G | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
G : Type u_1
inst✝ : Group G
val✝ : Fintype G
⊢ d1224 G ∣ Nat.card G
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_25' | [433, 1] | [438, 21] | exact l1_2_25 | case inl
G : Type u_1
inst✝ : Group G
val✝ : Fintype G
⊢ d1224 G ∣ Fintype.card G | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
G : Type u_1
inst✝ : Group G
val✝ : Fintype G
⊢ d1224 G ∣ Fintype.card G
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_25' | [433, 1] | [438, 21] | rw [Nat.card_eq_zero_of_infinite] | case inr
G : Type u_1
inst✝ : Group G
val✝ : Infinite G
⊢ d1224 G ∣ Nat.card G | case inr
G : Type u_1
inst✝ : Group G
val✝ : Infinite G
⊢ d1224 G ∣ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
G : Type u_1
inst✝ : Group G
val✝ : Infinite G
⊢ d1224 G ∣ Nat.card G
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_25' | [433, 1] | [438, 21] | exact dvd_zero _ | case inr
G : Type u_1
inst✝ : Group G
val✝ : Infinite G
⊢ d1224 G ∣ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
G : Type u_1
inst✝ : Group G
val✝ : Infinite G
⊢ d1224 G ∣ 0
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | n1_2_26_ii | [443, 1] | [443, 83] | group | G : Type u_1
inst✝ : Group G
g h : G
⊢ ⁅g, h⁆ = ⁅h, g⁆⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
g h : G
⊢ ⁅g, h⁆ = ⁅h, g⁆⁻¹
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | n1_2_26_ii' | [445, 1] | [446, 84] | rw [commutatorElement_def, mul_inv_eq_iff_eq_mul, mul_inv_eq_iff_eq_mul, one_mul] | G : Type u_1
inst✝ : Group G
g h : G
⊢ ⁅g, h⁆ = 1 ↔ g * h = h * g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
g h : G
⊢ ⁅g, h⁆ = 1 ↔ g * h = h * g
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_26_iii' | [451, 1] | [472, 31] | simp only [d1226Iii] | G : Type u_1
inst✝ : Group G
⊢ ↑(d1226Iii G) = {1} ↔ ∀ (g h : G), g * h = h * g | G : Type u_1
inst✝ : Group G
⊢ ↑(Subgroup.closure {x | ∃ g h, ⁅g, h⁆ = x}) = {1} ↔ ∀ (g h : G), g * h = h * g | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
⊢ ↑(d1226Iii G) = {1} ↔ ∀ (g h : G), g * h = h * g
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_26_iii' | [451, 1] | [472, 31] | rw [Set.ext_iff] | G : Type u_1
inst✝ : Group G
⊢ ↑(Subgroup.closure {x | ∃ g h, ⁅g, h⁆ = x}) = {1} ↔ ∀ (g h : G), g * h = h * g | G : Type u_1
inst✝ : Group G
⊢ (∀ (x : G), x ∈ ↑(Subgroup.closure {x | ∃ g h, ⁅g, h⁆ = x}) ↔ x ∈ {1}) ↔ ∀ (g h : G), g * h = h * g | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
⊢ ↑(Subgroup.closure {x | ∃ g h, ⁅g, h⁆ = x}) = {1} ↔ ∀ (g h : G), g * h = h * g
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_26_iii' | [451, 1] | [472, 31] | simp_rw [← n1_2_26_ii'] | G : Type u_1
inst✝ : Group G
⊢ (∀ (x : G), x ∈ ↑(Subgroup.closure {x | ∃ g h, ⁅g, h⁆ = x}) ↔ x ∈ {1}) ↔ ∀ (g h : G), g * h = h * g | G : Type u_1
inst✝ : Group G
⊢ (∀ (x : G), x ∈ ↑(Subgroup.closure {x | ∃ g h, ⁅g, h⁆ = x}) ↔ x ∈ {1}) ↔ ∀ (g h : G), ⁅g, h⁆ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
⊢ (∀ (x : G), x ∈ ↑(Subgroup.closure {x | ∃ g h, ⁅g, h⁆ = x}) ↔ x ∈ {1}) ↔ ∀ (g h : G), g * h = h * g
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_26_iii' | [451, 1] | [472, 31] | simp only [SetLike.mem_coe, Set.mem_singleton_iff, Set.setOf_exists, Subgroup.closure_iUnion,
Set.setOf_eq_eq_singleton'] | G : Type u_1
inst✝ : Group G
⊢ (∀ (x : G), x ∈ ↑(Subgroup.closure {x | ∃ g h, ⁅g, h⁆ = x}) ↔ x ∈ {1}) ↔ ∀ (g h : G), ⁅g, h⁆ = 1 | G : Type u_1
inst✝ : Group G
⊢ (∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1) ↔ ∀ (g h : G), ⁅g, h⁆ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
⊢ (∀ (x : G), x ∈ ↑(Subgroup.closure {x | ∃ g h, ⁅g, h⁆ = x}) ↔ x ∈ {1}) ↔ ∀ (g h : G), ⁅g, h⁆ = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_26_iii' | [451, 1] | [472, 31] | constructor | G : Type u_1
inst✝ : Group G
⊢ (∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1) ↔ ∀ (g h : G), ⁅g, h⁆ = 1 | case mp
G : Type u_1
inst✝ : Group G
⊢ (∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1) → ∀ (g h : G), ⁅g, h⁆ = 1
case mpr
G : Type u_1
inst✝ : Group G
⊢ (∀ (g h : G), ⁅g, h⁆ = 1) → ∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
⊢ (∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1) ↔ ∀ (g h : G), ⁅g, h⁆ = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_26_iii' | [451, 1] | [472, 31] | intro h g g' | case mp
G : Type u_1
inst✝ : Group G
⊢ (∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1) → ∀ (g h : G), ⁅g, h⁆ = 1 | case mp
G : Type u_1
inst✝ : Group G
h : ∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1
g g' : G
⊢ ⁅g, g'⁆ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_1
inst✝ : Group G
⊢ (∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1) → ∀ (g h : G), ⁅g, h⁆ = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_26_iii' | [451, 1] | [472, 31] | rw [← h] | case mp
G : Type u_1
inst✝ : Group G
h : ∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1
g g' : G
⊢ ⁅g, g'⁆ = 1 | case mp
G : Type u_1
inst✝ : Group G
h : ∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1
g g' : G
⊢ ⁅g, g'⁆ ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_1
inst✝ : Group G
h : ∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1
g g' : G
⊢ ⁅g, g'⁆ = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_26_iii' | [451, 1] | [472, 31] | apply Subgroup.mem_iSup_of_mem g | case mp
G : Type u_1
inst✝ : Group G
h : ∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1
g g' : G
⊢ ⁅g, g'⁆ ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} | case mp
G : Type u_1
inst✝ : Group G
h : ∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1
g g' : G
⊢ ⁅g, g'⁆ ∈ ⨆ i, Subgroup.closure {⁅g, i⁆} | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_1
inst✝ : Group G
h : ∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1
g g' : G
⊢ ⁅g, g'⁆ ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆}
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_26_iii' | [451, 1] | [472, 31] | apply Subgroup.mem_iSup_of_mem g' | case mp
G : Type u_1
inst✝ : Group G
h : ∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1
g g' : G
⊢ ⁅g, g'⁆ ∈ ⨆ i, Subgroup.closure {⁅g, i⁆} | case mp
G : Type u_1
inst✝ : Group G
h : ∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1
g g' : G
⊢ ⁅g, g'⁆ ∈ Subgroup.closure {⁅g, g'⁆} | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_1
inst✝ : Group G
h : ∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1
g g' : G
⊢ ⁅g, g'⁆ ∈ ⨆ i, Subgroup.closure {⁅g, i⁆}
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_26_iii' | [451, 1] | [472, 31] | rw [Subgroup.mem_closure_singleton] | case mp
G : Type u_1
inst✝ : Group G
h : ∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1
g g' : G
⊢ ⁅g, g'⁆ ∈ Subgroup.closure {⁅g, g'⁆} | case mp
G : Type u_1
inst✝ : Group G
h : ∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1
g g' : G
⊢ ∃ n, ⁅g, g'⁆ ^ n = ⁅g, g'⁆ | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_1
inst✝ : Group G
h : ∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1
g g' : G
⊢ ⁅g, g'⁆ ∈ Subgroup.closure {⁅g, g'⁆}
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_26_iii' | [451, 1] | [472, 31] | use 1 | case mp
G : Type u_1
inst✝ : Group G
h : ∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1
g g' : G
⊢ ∃ n, ⁅g, g'⁆ ^ n = ⁅g, g'⁆ | case h
G : Type u_1
inst✝ : Group G
h : ∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1
g g' : G
⊢ ⁅g, g'⁆ ^ 1 = ⁅g, g'⁆ | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_1
inst✝ : Group G
h : ∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1
g g' : G
⊢ ∃ n, ⁅g, g'⁆ ^ n = ⁅g, g'⁆
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_26_iii' | [451, 1] | [472, 31] | rw [zpow_one] | case h
G : Type u_1
inst✝ : Group G
h : ∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1
g g' : G
⊢ ⁅g, g'⁆ ^ 1 = ⁅g, g'⁆ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_1
inst✝ : Group G
h : ∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1
g g' : G
⊢ ⁅g, g'⁆ ^ 1 = ⁅g, g'⁆
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_26_iii' | [451, 1] | [472, 31] | intro h g | case mpr
G : Type u_1
inst✝ : Group G
⊢ (∀ (g h : G), ⁅g, h⁆ = 1) → ∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1 | case mpr
G : Type u_1
inst✝ : Group G
h : ∀ (g h : G), ⁅g, h⁆ = 1
g : G
⊢ g ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ g = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_1
inst✝ : Group G
⊢ (∀ (g h : G), ⁅g, h⁆ = 1) → ∀ (x : G), x ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ x = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_26_iii' | [451, 1] | [472, 31] | constructor | case mpr
G : Type u_1
inst✝ : Group G
h : ∀ (g h : G), ⁅g, h⁆ = 1
g : G
⊢ g ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ g = 1 | case mpr.mp
G : Type u_1
inst✝ : Group G
h : ∀ (g h : G), ⁅g, h⁆ = 1
g : G
⊢ g ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} → g = 1
case mpr.mpr
G : Type u_1
inst✝ : Group G
h : ∀ (g h : G), ⁅g, h⁆ = 1
g : G
⊢ g = 1 → g ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_1
inst✝ : Group G
h : ∀ (g h : G), ⁅g, h⁆ = 1
g : G
⊢ g ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} ↔ g = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_26_iii' | [451, 1] | [472, 31] | intro h' | case mpr.mp
G : Type u_1
inst✝ : Group G
h : ∀ (g h : G), ⁅g, h⁆ = 1
g : G
⊢ g ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} → g = 1 | case mpr.mp
G : Type u_1
inst✝ : Group G
h : ∀ (g h : G), ⁅g, h⁆ = 1
g : G
h' : g ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆}
⊢ g = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.mp
G : Type u_1
inst✝ : Group G
h : ∀ (g h : G), ⁅g, h⁆ = 1
g : G
⊢ g ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} → g = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_26_iii' | [451, 1] | [472, 31] | symm | case mpr.mp
G : Type u_1
inst✝ : Group G
h : ∀ (g h : G), ⁅g, h⁆ = 1
g : G
h' : g ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆}
⊢ g = 1 | case mpr.mp
G : Type u_1
inst✝ : Group G
h : ∀ (g h : G), ⁅g, h⁆ = 1
g : G
h' : g ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆}
⊢ 1 = g | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.mp
G : Type u_1
inst✝ : Group G
h : ∀ (g h : G), ⁅g, h⁆ = 1
g : G
h' : g ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆}
⊢ g = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_26_iii' | [451, 1] | [472, 31] | simpa only [h, ciSup_const, Subgroup.mem_closure_singleton, one_zpow, exists_const] using h' | case mpr.mp
G : Type u_1
inst✝ : Group G
h : ∀ (g h : G), ⁅g, h⁆ = 1
g : G
h' : g ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆}
⊢ 1 = g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.mp
G : Type u_1
inst✝ : Group G
h : ∀ (g h : G), ⁅g, h⁆ = 1
g : G
h' : g ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆}
⊢ 1 = g
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_26_iii' | [451, 1] | [472, 31] | rintro rfl | case mpr.mpr
G : Type u_1
inst✝ : Group G
h : ∀ (g h : G), ⁅g, h⁆ = 1
g : G
⊢ g = 1 → g ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} | case mpr.mpr
G : Type u_1
inst✝ : Group G
h : ∀ (g h : G), ⁅g, h⁆ = 1
⊢ 1 ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.mpr
G : Type u_1
inst✝ : Group G
h : ∀ (g h : G), ⁅g, h⁆ = 1
g : G
⊢ g = 1 → g ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆}
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_26_iii' | [451, 1] | [472, 31] | exact Subgroup.one_mem _ | case mpr.mpr
G : Type u_1
inst✝ : Group G
h : ∀ (g h : G), ⁅g, h⁆ = 1
⊢ 1 ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.mpr
G : Type u_1
inst✝ : Group G
h : ∀ (g h : G), ⁅g, h⁆ = 1
⊢ 1 ∈ ⨆ i, ⨆ i_1, Subgroup.closure {⁅i, i_1⁆}
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_8 | [480, 1] | [485, 13] | refine Commute.of_orderOf_dvd_two (fun g => ?_) a b | G : Type u_1
inst✝ : Group G
h : ∀ (g : G), g = 1 ∨ orderOf g = 2
a b : G
⊢ a * b = b * a | G : Type u_1
inst✝ : Group G
h : ∀ (g : G), g = 1 ∨ orderOf g = 2
a b g : G
⊢ orderOf g ∣ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
h : ∀ (g : G), g = 1 ∨ orderOf g = 2
a b : G
⊢ a * b = b * a
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_8 | [480, 1] | [485, 13] | obtain rfl | h := h g | G : Type u_1
inst✝ : Group G
h : ∀ (g : G), g = 1 ∨ orderOf g = 2
a b g : G
⊢ orderOf g ∣ 2 | case inl
G : Type u_1
inst✝ : Group G
h : ∀ (g : G), g = 1 ∨ orderOf g = 2
a b : G
⊢ orderOf 1 ∣ 2
case inr
G : Type u_1
inst✝ : Group G
h✝ : ∀ (g : G), g = 1 ∨ orderOf g = 2
a b g : G
h : orderOf g = 2
⊢ orderOf g ∣ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
h : ∀ (g : G), g = 1 ∨ orderOf g = 2
a b g : G
⊢ orderOf g ∣ 2
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_8 | [480, 1] | [485, 13] | rw [orderOf_one] | case inl
G : Type u_1
inst✝ : Group G
h : ∀ (g : G), g = 1 ∨ orderOf g = 2
a b : G
⊢ orderOf 1 ∣ 2 | case inl
G : Type u_1
inst✝ : Group G
h : ∀ (g : G), g = 1 ∨ orderOf g = 2
a b : G
⊢ 1 ∣ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
G : Type u_1
inst✝ : Group G
h : ∀ (g : G), g = 1 ∨ orderOf g = 2
a b : G
⊢ orderOf 1 ∣ 2
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_8 | [480, 1] | [485, 13] | exact one_dvd _ | case inl
G : Type u_1
inst✝ : Group G
h : ∀ (g : G), g = 1 ∨ orderOf g = 2
a b : G
⊢ 1 ∣ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
G : Type u_1
inst✝ : Group G
h : ∀ (g : G), g = 1 ∨ orderOf g = 2
a b : G
⊢ 1 ∣ 2
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_8 | [480, 1] | [485, 13] | rw [← h] | case inr
G : Type u_1
inst✝ : Group G
h✝ : ∀ (g : G), g = 1 ∨ orderOf g = 2
a b g : G
h : orderOf g = 2
⊢ orderOf g ∣ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
G : Type u_1
inst✝ : Group G
h✝ : ∀ (g : G), g = 1 ∨ orderOf g = 2
a b g : G
h : orderOf g = 2
⊢ orderOf g ∣ 2
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_9 | [487, 1] | [497, 24] | refine' Set.eq_singleton_iff_unique_mem.mpr ⟨_, fun x hx => _⟩ | G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
⊢ ↑A ∩ ↑B = {1} | case refine'_1
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
⊢ 1 ∈ ↑A ∩ ↑B
case refine'_2
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
x : G
hx : x ∈ ↑A ∩ ↑B
⊢ x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
⊢ ↑A ∩ ↑B = {1}
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_9 | [487, 1] | [497, 24] | rw [Set.mem_inter_iff, SetLike.mem_coe, SetLike.mem_coe] at hx | case refine'_2
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
x : G
hx : x ∈ ↑A ∩ ↑B
⊢ x = 1 | case refine'_2
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
x : G
hx : x ∈ A ∧ x ∈ B
⊢ x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
x : G
hx : x ∈ ↑A ∩ ↑B
⊢ x = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_9 | [487, 1] | [497, 24] | rw [← pow_one x, ← h] | case refine'_2
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
x : G
hx : x ∈ A ∧ x ∈ B
⊢ x = 1 | case refine'_2
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
x : G
hx : x ∈ A ∧ x ∈ B
⊢ x ^ gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
x : G
hx : x ∈ A ∧ x ∈ B
⊢ x = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_9 | [487, 1] | [497, 24] | apply pow_gcd_eq_one | case refine'_2
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
x : G
hx : x ∈ A ∧ x ∈ B
⊢ x ^ gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1 | case refine'_2.hm
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
x : G
hx : x ∈ A ∧ x ∈ B
⊢ x ^ Monoid.exponent ↥A = 1
case refine'_2.hn
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
x : G
hx : x ∈ A ∧ x ∈ B
⊢ x ^ Monoid.exponent ↥B = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
x : G
hx : x ∈ A ∧ x ∈ B
⊢ x ^ gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_9 | [487, 1] | [497, 24] | simp only [SetLike.mem_coe] | case refine'_1.hb
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
⊢ 1 ∈ ↑B | case refine'_1.hb
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
⊢ 1 ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1.hb
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
⊢ 1 ∈ ↑B
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_9 | [487, 1] | [497, 24] | exact Subgroup.one_mem _ | case refine'_1.hb
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
⊢ 1 ∈ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1.hb
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
⊢ 1 ∈ B
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_9 | [487, 1] | [497, 24] | rw [← Subgroup.coe_mk A x hx.1, ← Subgroup.coe_pow, Monoid.pow_exponent_eq_one,
Subgroup.coe_one] | case refine'_2.hm
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
x : G
hx : x ∈ A ∧ x ∈ B
⊢ x ^ Monoid.exponent ↥A = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2.hm
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
x : G
hx : x ∈ A ∧ x ∈ B
⊢ x ^ Monoid.exponent ↥A = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_9 | [487, 1] | [497, 24] | rw [← Subgroup.coe_mk B x hx.2, ← Subgroup.coe_pow, Monoid.pow_exponent_eq_one,
Subgroup.coe_one] | case refine'_2.hn
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
x : G
hx : x ∈ A ∧ x ∈ B
⊢ x ^ Monoid.exponent ↥B = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2.hn
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
x : G
hx : x ∈ A ∧ x ∈ B
⊢ x ^ Monoid.exponent ↥B = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_10_i | [499, 1] | [500, 69] | rw [Nat.card_eq_fintype_card, Fintype.card_perm, Fintype.card_fin] | n : ℕ
⊢ Nat.card (Equiv.Perm (Fin n)) = n ! | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ Nat.card (Equiv.Perm (Fin n)) = n !
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_10_ii_aux_1 | [502, 1] | [514, 18] | rw [orderOf_eq_iff two_pos] | x : Equiv.Perm (Fin 4)
⊢ orderOf x = 2 ↔ x ≠ 1 ∧ x ^ 2 = 1 | x : Equiv.Perm (Fin 4)
⊢ (x ^ 2 = 1 ∧ ∀ m < 2, 0 < m → x ^ m ≠ 1) ↔ x ≠ 1 ∧ x ^ 2 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x : Equiv.Perm (Fin 4)
⊢ orderOf x = 2 ↔ x ≠ 1 ∧ x ^ 2 = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_10_ii_aux_1 | [502, 1] | [514, 18] | constructor | x : Equiv.Perm (Fin 4)
⊢ (x ^ 2 = 1 ∧ ∀ m < 2, 0 < m → x ^ m ≠ 1) ↔ x ≠ 1 ∧ x ^ 2 = 1 | case mp
x : Equiv.Perm (Fin 4)
⊢ (x ^ 2 = 1 ∧ ∀ m < 2, 0 < m → x ^ m ≠ 1) → x ≠ 1 ∧ x ^ 2 = 1
case mpr
x : Equiv.Perm (Fin 4)
⊢ x ≠ 1 ∧ x ^ 2 = 1 → x ^ 2 = 1 ∧ ∀ m < 2, 0 < m → x ^ m ≠ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x : Equiv.Perm (Fin 4)
⊢ (x ^ 2 = 1 ∧ ∀ m < 2, 0 < m → x ^ m ≠ 1) ↔ x ≠ 1 ∧ x ^ 2 = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_10_ii_aux_1 | [502, 1] | [514, 18] | rintro ⟨h1, h2⟩ | case mp
x : Equiv.Perm (Fin 4)
⊢ (x ^ 2 = 1 ∧ ∀ m < 2, 0 < m → x ^ m ≠ 1) → x ≠ 1 ∧ x ^ 2 = 1 | case mp.intro
x : Equiv.Perm (Fin 4)
h1 : x ^ 2 = 1
h2 : ∀ m < 2, 0 < m → x ^ m ≠ 1
⊢ x ≠ 1 ∧ x ^ 2 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
x : Equiv.Perm (Fin 4)
⊢ (x ^ 2 = 1 ∧ ∀ m < 2, 0 < m → x ^ m ≠ 1) → x ≠ 1 ∧ x ^ 2 = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_10_ii_aux_1 | [502, 1] | [514, 18] | refine ⟨?_, h1⟩ | case mp.intro
x : Equiv.Perm (Fin 4)
h1 : x ^ 2 = 1
h2 : ∀ m < 2, 0 < m → x ^ m ≠ 1
⊢ x ≠ 1 ∧ x ^ 2 = 1 | case mp.intro
x : Equiv.Perm (Fin 4)
h1 : x ^ 2 = 1
h2 : ∀ m < 2, 0 < m → x ^ m ≠ 1
⊢ x ≠ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro
x : Equiv.Perm (Fin 4)
h1 : x ^ 2 = 1
h2 : ∀ m < 2, 0 < m → x ^ m ≠ 1
⊢ x ≠ 1 ∧ x ^ 2 = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_10_ii_aux_1 | [502, 1] | [514, 18] | have := h2 1 one_lt_two one_pos | case mp.intro
x : Equiv.Perm (Fin 4)
h1 : x ^ 2 = 1
h2 : ∀ m < 2, 0 < m → x ^ m ≠ 1
⊢ x ≠ 1 | case mp.intro
x : Equiv.Perm (Fin 4)
h1 : x ^ 2 = 1
h2 : ∀ m < 2, 0 < m → x ^ m ≠ 1
this : x ^ 1 ≠ 1
⊢ x ≠ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro
x : Equiv.Perm (Fin 4)
h1 : x ^ 2 = 1
h2 : ∀ m < 2, 0 < m → x ^ m ≠ 1
⊢ x ≠ 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_10_ii_aux_1 | [502, 1] | [514, 18] | rwa [pow_one] at this | case mp.intro
x : Equiv.Perm (Fin 4)
h1 : x ^ 2 = 1
h2 : ∀ m < 2, 0 < m → x ^ m ≠ 1
this : x ^ 1 ≠ 1
⊢ x ≠ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro
x : Equiv.Perm (Fin 4)
h1 : x ^ 2 = 1
h2 : ∀ m < 2, 0 < m → x ^ m ≠ 1
this : x ^ 1 ≠ 1
⊢ x ≠ 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_10_ii_aux_1 | [502, 1] | [514, 18] | rintro ⟨h1, h2⟩ | case mpr
x : Equiv.Perm (Fin 4)
⊢ x ≠ 1 ∧ x ^ 2 = 1 → x ^ 2 = 1 ∧ ∀ m < 2, 0 < m → x ^ m ≠ 1 | case mpr.intro
x : Equiv.Perm (Fin 4)
h1 : x ≠ 1
h2 : x ^ 2 = 1
⊢ x ^ 2 = 1 ∧ ∀ m < 2, 0 < m → x ^ m ≠ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
x : Equiv.Perm (Fin 4)
⊢ x ≠ 1 ∧ x ^ 2 = 1 → x ^ 2 = 1 ∧ ∀ m < 2, 0 < m → x ^ m ≠ 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_10_ii_aux_1 | [502, 1] | [514, 18] | refine ⟨h2, ?_⟩ | case mpr.intro
x : Equiv.Perm (Fin 4)
h1 : x ≠ 1
h2 : x ^ 2 = 1
⊢ x ^ 2 = 1 ∧ ∀ m < 2, 0 < m → x ^ m ≠ 1 | case mpr.intro
x : Equiv.Perm (Fin 4)
h1 : x ≠ 1
h2 : x ^ 2 = 1
⊢ ∀ m < 2, 0 < m → x ^ m ≠ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
x : Equiv.Perm (Fin 4)
h1 : x ≠ 1
h2 : x ^ 2 = 1
⊢ x ^ 2 = 1 ∧ ∀ m < 2, 0 < m → x ^ m ≠ 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_10_ii_aux_1 | [502, 1] | [514, 18] | intros m hm1 hm2 | case mpr.intro
x : Equiv.Perm (Fin 4)
h1 : x ≠ 1
h2 : x ^ 2 = 1
⊢ ∀ m < 2, 0 < m → x ^ m ≠ 1 | case mpr.intro
x : Equiv.Perm (Fin 4)
h1 : x ≠ 1
h2 : x ^ 2 = 1
m : ℕ
hm1 : m < 2
hm2 : 0 < m
⊢ x ^ m ≠ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
x : Equiv.Perm (Fin 4)
h1 : x ≠ 1
h2 : x ^ 2 = 1
⊢ ∀ m < 2, 0 < m → x ^ m ≠ 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_10_ii_aux_1 | [502, 1] | [514, 18] | obtain rfl : m = 1 := by linarith | case mpr.intro
x : Equiv.Perm (Fin 4)
h1 : x ≠ 1
h2 : x ^ 2 = 1
m : ℕ
hm1 : m < 2
hm2 : 0 < m
⊢ x ^ m ≠ 1 | case mpr.intro
x : Equiv.Perm (Fin 4)
h1 : x ≠ 1
h2 : x ^ 2 = 1
hm1 : 1 < 2
hm2 : 0 < 1
⊢ x ^ 1 ≠ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
x : Equiv.Perm (Fin 4)
h1 : x ≠ 1
h2 : x ^ 2 = 1
m : ℕ
hm1 : m < 2
hm2 : 0 < m
⊢ x ^ m ≠ 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_10_ii_aux_1 | [502, 1] | [514, 18] | rwa [pow_one] | case mpr.intro
x : Equiv.Perm (Fin 4)
h1 : x ≠ 1
h2 : x ^ 2 = 1
hm1 : 1 < 2
hm2 : 0 < 1
⊢ x ^ 1 ≠ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
x : Equiv.Perm (Fin 4)
h1 : x ≠ 1
h2 : x ^ 2 = 1
hm1 : 1 < 2
hm2 : 0 < 1
⊢ x ^ 1 ≠ 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_10_ii_aux_1 | [502, 1] | [514, 18] | linarith | x : Equiv.Perm (Fin 4)
h1 : x ≠ 1
h2 : x ^ 2 = 1
m : ℕ
hm1 : m < 2
hm2 : 0 < m
⊢ m = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : Equiv.Perm (Fin 4)
h1 : x ≠ 1
h2 : x ^ 2 = 1
m : ℕ
hm1 : m < 2
hm2 : 0 < m
⊢ m = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_10_ii | [522, 1] | [525, 9] | rw [Nat.card_eq_fintype_card] | ⊢ Nat.card { x // orderOf x = 2 } = 9 | ⊢ Fintype.card { x // orderOf x = 2 } = 9 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Nat.card { x // orderOf x = 2 } = 9
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_10_ii | [522, 1] | [525, 9] | simp_rw [ex_1_2_10_ii_aux_1] | ⊢ Fintype.card { x // orderOf x = 2 } = 9 | ⊢ Fintype.card { x // x ≠ 1 ∧ x ^ 2 = 1 } = 9 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Fintype.card { x // orderOf x = 2 } = 9
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_10_ii | [522, 1] | [525, 9] | decide | ⊢ Fintype.card { x // x ≠ 1 ∧ x ^ 2 = 1 } = 9 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Fintype.card { x // x ≠ 1 ∧ x ^ 2 = 1 } = 9
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_10_iii_5 | [527, 1] | [529, 16] | native_decide | ⊢ Nat.find (_ : Monoid.ExponentExists (Equiv.Perm (Fin 5))) = 60 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Nat.find (_ : Monoid.ExponentExists (Equiv.Perm (Fin 5))) = 60
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_10_iii_6 | [531, 1] | [533, 16] | native_decide | ⊢ Nat.find (_ : Monoid.ExponentExists (Equiv.Perm (Fin 6))) = 60 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Nat.find (_ : Monoid.ExponentExists (Equiv.Perm (Fin 6))) = 60
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | Nat.card_subtype | [538, 1] | [541, 29] | rw [← Fintype.card_subtype] | α : Type u_1
inst : Fintype α
p : α → Prop
inst✝ : DecidablePred p
⊢ Nat.card { x // p x } = (Finset.filter p Finset.univ).card | α : Type u_1
inst : Fintype α
p : α → Prop
inst✝ : DecidablePred p
⊢ Nat.card { x // p x } = Fintype.card { x // p x } | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst : Fintype α
p : α → Prop
inst✝ : DecidablePred p
⊢ Nat.card { x // p x } = (Finset.filter p Finset.univ).card
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | Nat.card_subtype | [538, 1] | [541, 29] | rw [@card_eq_fintype_card] | α : Type u_1
inst : Fintype α
p : α → Prop
inst✝ : DecidablePred p
⊢ Nat.card { x // p x } = Fintype.card { x // p x } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst : Fintype α
p : α → Prop
inst✝ : DecidablePred p
⊢ Nat.card { x // p x } = Fintype.card { x // p x }
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_11_ii | [552, 1] | [557, 16] | rintro ⟨h⟩ | n : ℕ
hn : 3 ≤ n
⊢ ¬Std.Commutative fun x x_1 => x * x_1 | case mk
n : ℕ
hn : 3 ≤ n
h : ∀ (a b : DihedralGroup n), a * b = b * a
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
hn : 3 ≤ n
⊢ ¬Std.Commutative fun x x_1 => x * x_1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_11_ii | [552, 1] | [557, 16] | have := h (r 1) (sr 0) | case mk
n : ℕ
hn : 3 ≤ n
h : ∀ (a b : DihedralGroup n), a * b = b * a
⊢ False | case mk
n : ℕ
hn : 3 ≤ n
h : ∀ (a b : DihedralGroup n), a * b = b * a
this : r 1 * sr 0 = sr 0 * r 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
n : ℕ
hn : 3 ≤ n
h : ∀ (a b : DihedralGroup n), a * b = b * a
⊢ False
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_11_ii | [552, 1] | [557, 16] | simp only [r_mul_sr, zero_sub, sr_mul_r, zero_add, sr.injEq] at this | case mk
n : ℕ
hn : 3 ≤ n
h : ∀ (a b : DihedralGroup n), a * b = b * a
this : r 1 * sr 0 = sr 0 * r 1
⊢ False | case mk
n : ℕ
hn : 3 ≤ n
h : ∀ (a b : DihedralGroup n), a * b = b * a
this : -1 = 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
n : ℕ
hn : 3 ≤ n
h : ∀ (a b : DihedralGroup n), a * b = b * a
this : r 1 * sr 0 = sr 0 * r 1
⊢ False
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_11_ii | [552, 1] | [557, 16] | rw [neg_eq_iff_add_eq_zero, one_add_one_eq_two, ← ZMod.val_eq_zero, ZMod.val_ofNat_of_lt hn] at this | case mk
n : ℕ
hn : 3 ≤ n
h : ∀ (a b : DihedralGroup n), a * b = b * a
this : -1 = 1
⊢ False | case mk
n : ℕ
hn : 3 ≤ n
h : ∀ (a b : DihedralGroup n), a * b = b * a
this : 2 = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
n : ℕ
hn : 3 ≤ n
h : ∀ (a b : DihedralGroup n), a * b = b * a
this : -1 = 1
⊢ False
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | ex_1_2_11_ii | [552, 1] | [557, 16] | contradiction | case mk
n : ℕ
hn : 3 ≤ n
h : ∀ (a b : DihedralGroup n), a * b = b * a
this : 2 = 0
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
n : ℕ
hn : 3 ≤ n
h : ∀ (a b : DihedralGroup n), a * b = b * a
this : 2 = 0
⊢ False
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.pos_iff_ex | [207, 1] | [230, 53] | have mul_ident := Impl.multiplication.mul_identity (ℕ := ℕ) | x : ℤ
⊢ Positive x ↔ ∃ n, Positive n ∧ x ≃ coe n | x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
⊢ Positive x ↔ ∃ n, Positive n ∧ x ≃ coe n | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
⊢ Positive x ↔ ∃ n, Positive n ∧ x ≃ coe n
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.pos_iff_ex | [207, 1] | [230, 53] | apply Iff.intro | x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
⊢ Positive x ↔ ∃ n, Positive n ∧ x ≃ coe n | case mp
x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
⊢ Positive x → ∃ n, Positive n ∧ x ≃ coe n
case mpr
x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
⊢ (∃ n, Positive n ∧ x ≃ coe n) → Positive x | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
⊢ Positive x ↔ ∃ n, Positive n ∧ x ≃ coe n
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.pos_iff_ex | [207, 1] | [230, 53] | case mp =>
intro (_ : Positive x)
show ∃ (n : ℕ), Positive n ∧ x ≃ coe n
have (Integer.NonzeroWithSign.intro
(n : ℕ) (_ : Positive n) (_ : x ≃ 1 * coe n)) :=
Impl.sign_props.positive_iff_sign_pos1.mp ‹Positive x›
exists n
apply And.intro ‹Positive n›
show x ≃ coe n
exact Rel.trans ‹x ≃ 1 * coe n› AA.identL | x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
⊢ Positive x → ∃ n, Positive n ∧ x ≃ coe n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
⊢ Positive x → ∃ n, Positive n ∧ x ≃ coe n
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.pos_iff_ex | [207, 1] | [230, 53] | case mpr =>
intro (Exists.intro (n : ℕ) (And.intro (_ : Positive n) (_ : x ≃ coe n)))
show Positive x
apply Impl.sign_props.positive_iff_sign_pos1.mpr
show Integer.NonzeroWithSign x 1
apply Integer.NonzeroWithSign.intro n ‹Positive n›
show x ≃ 1 * coe n
exact Rel.trans ‹x ≃ coe n› (Rel.symm AA.identL) | x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
⊢ (∃ n, Positive n ∧ x ≃ coe n) → Positive x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
⊢ (∃ n, Positive n ∧ x ≃ coe n) → Positive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.pos_iff_ex | [207, 1] | [230, 53] | intro (_ : Positive x) | x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
⊢ Positive x → ∃ n, Positive n ∧ x ≃ coe n | x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
x✝ : Positive x
⊢ ∃ n, Positive n ∧ x ≃ coe n | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
⊢ Positive x → ∃ n, Positive n ∧ x ≃ coe n
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.pos_iff_ex | [207, 1] | [230, 53] | have (Integer.NonzeroWithSign.intro
(n : ℕ) (_ : Positive n) (_ : x ≃ 1 * coe n)) :=
Impl.sign_props.positive_iff_sign_pos1.mp ‹Positive x› | x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
x✝ : Positive x
⊢ ∃ n, Positive n ∧ x ≃ coe n | x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
x✝ : Positive x
n : ℕ
pos✝ : Positive n
eqv✝ : x ≃ 1 * coe n
⊢ ∃ n, Positive n ∧ x ≃ coe n | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
x✝ : Positive x
⊢ ∃ n, Positive n ∧ x ≃ coe n
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.pos_iff_ex | [207, 1] | [230, 53] | exists n | x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
x✝ : Positive x
n : ℕ
pos✝ : Positive n
eqv✝ : x ≃ 1 * coe n
⊢ ∃ n, Positive n ∧ x ≃ coe n | x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
x✝ : Positive x
n : ℕ
pos✝ : Positive n
eqv✝ : x ≃ 1 * coe n
⊢ Positive n ∧ x ≃ coe n | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
x✝ : Positive x
n : ℕ
pos✝ : Positive n
eqv✝ : x ≃ 1 * coe n
⊢ ∃ n, Positive n ∧ x ≃ coe n
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.pos_iff_ex | [207, 1] | [230, 53] | apply And.intro ‹Positive n› | x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
x✝ : Positive x
n : ℕ
pos✝ : Positive n
eqv✝ : x ≃ 1 * coe n
⊢ Positive n ∧ x ≃ coe n | x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
x✝ : Positive x
n : ℕ
pos✝ : Positive n
eqv✝ : x ≃ 1 * coe n
⊢ x ≃ coe n | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
x✝ : Positive x
n : ℕ
pos✝ : Positive n
eqv✝ : x ≃ 1 * coe n
⊢ Positive n ∧ x ≃ coe n
TACTIC:
|
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