answer_id int64 22 5.14k | informal_proof stringclasses 7
values | formal_proof stringlengths 224 2.62k | informal_statement stringlengths 38 446 | task_id int64 2 2.15k | formal_statement stringlengths 16 393 | problem_id int64 1 3.32k | header stringclasses 1
value | passed bool 1
class | validation_detail stringclasses 9
values |
|---|---|---|---|---|---|---|---|---|---|
856 | import Mathlib
example { G : Type* } [ Group G ] { M N : Subgroup G } ( hM : M.Normal ) ( hN : N.Normal ) ( h : M ⊓ N = ⊥ ) { a : G } ( ha : a ∈ M ) { b : G } ( hb : b ∈ N ) : a * b = b * a := by
have : (h : b * a * b⁻¹ * a⁻¹ = 1) → a * b = b * a := by
intro h
calc
_ = 1 * a * b := by group
_ = ... | Let $M$ and $N$ be normal subgroups of the group $G$. If $M \cap N = 1$, then for any $a \in M$ and $b \in N$, we have $ab = ba$. | 64 | import Mathlib
| 1,841 | import Mathlib | true | {"env": 0}
| |
2,161 | import Mathlib
--1203
example {G : Type*} [Group G]: ∃! (le : G), ∃! (re : G), ∀ (g : G), le * g = g ∧ g * re = g := by
use 1
dsimp
constructor
· use 1
dsimp
constructor
· group
simp only [and_self, implies_true]
· intro y H
have : 1 * y = 1 := (H 1).2
calc
_ = 1 * y :... | Prove that a group have only one left identity element and only one right identity element. | 10 | import Mathlib
example {G : Type*} [Group G]: ∃! (le : G), ∃! (re : G), ∀ (g : G), le * g = g ∧ g * re = g := by sorry
| 18 | import Mathlib | true | {"env": 0}
| |
694 | import Mathlib
def Func (S : Type*) := S → ℝ
instance (S : Type*) : Add (Func S) where
add := λ f g => λ x => f x + g x
instance (S : Type*) : Neg (Func S) where
neg := λ f => λ x => - f x
instance (S : Type*) : Zero (Func S) where
zero := λ _ => 0
instance (S : Type*) : AddGroup (Func S) := by
apply AddG... | Consider a set $S$, let $G:=\{f:S\to\mathbb R\}$ to be all maps from $S$ to $\mathbb R$. Then $G$ is a group under function addition. | 15 | import Mathlib.Tactic
import Mathlib.Algebra.Group.MinimalAxioms
def Func (S : Type*) := S → ℝ
instance (S : Type*) : Add (Func S) where
add := sorry
instance (S : Type*) : Neg (Func S) where
neg := sorry
instance (S : Type*) : Zero (Func S) where
zero := sorry
noncomputable instance (S : Type*) : AddGroup (... | 1 | import Mathlib | true | {"messages":
[{"severity": "warning",
"pos": {"line": 25, "column": 10},
"endPos": {"line": 25, "column": 22},
"data": "`add_left_neg` has been deprecated, use `neg_add_cancel` instead"}],
"env": 0}
| |
83 | import Mathlib
-- 著者：秦宇轩
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic
-- Let `G` be a group.
-- Prove that `(a ^ {-1}) ^ {-1} = a` for any element `a ∈ G`.
example {G : Type*} [Group G] {a :G} : (a⁻¹)⁻¹ = a := by
group
| Let $G$ be a group. Prove that $\left(a^{-1}\right)^{-1}=a$ for any element $a\in G$. | 8 | import Mathlib
example {G : Type*}[Group G]{a :G}: (a⁻¹)⁻¹=a:=by sorry | 31 | import Mathlib | true | {"env": 0}
| |
2,508 | Proof
1. For the first theorem, we use the theorem that m is larger or equal than 2 if m isn't 1 and isn't 0.
2. And then, the conclusion becomes obvious, for it is exactly h0 and h1.
3. For the example, we use the theorem that that there is a prime which n could be divided by if n is not 1.
4. And then use n is larger... | import Mathlib
theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by
refine (Nat.two_le_iff m).mpr ?_
--For the first theorem, we use the theorem that m is larger or equal than 2 if m isn't 1 and isn't 0.
constructor
· exact h0
· exact h1
--And then, the conclusion becomes obvious, for it is exact... | Given an nature number $n$ which is not less than 2, prove either it is a prime or it can be divided by a prime nature number. | 78 | import Mathlib
theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := sorry
example {n : Nat} (h : 2 ≤ n) : ∃ p : Nat, p.Prime ∧ p ∣ n := sorry
| 1,854 | import Mathlib | true | {"env": 0}
|
382 | import Mathlib
example {G : Type*} [Group G]: ∀ (g : G), ∃! (y : G), g * y = 1 := by
intro g
use g⁻¹
constructor
· apply mul_right_inv
intro y h
rw [← mul_right_inv g] at h
apply mul_left_cancel at h
exact h | Prove that in a group, every element has exactly one inverse. | 9 | import Mathlib
example {G : Type*} [Group G]: ∀ (g : G), ∃! (y : G), g * y = 1 := by sorry | 19 | import Mathlib | true | {"messages":
[{"severity": "warning",
"pos": {"line": 7, "column": 10},
"endPos": {"line": 7, "column": 23},
"data": "`mul_right_inv` has been deprecated, use `mul_inv_cancel` instead"},
{"severity": "warning",
"pos": {"line": 9, "column": 8},
"endPos": {"line": 9, "column": 21},
"data":
"`mul_r... | |
3,061 | Proof that if every element in a group \(G\) satisfies \(x \cdot x = 1\) for all \(x \in G\), then \(G\) is a commutative grouTo show that G is a commutative group, apply the constructor for CommGroup
a * b = a * b * (a * b) * (a * b)⁻¹ = 1 * (a * b)⁻¹
= b⁻¹ * (b⁻¹⁻¹ * b) * a = b⁻¹ * (b⁻¹⁻¹ * b) * (a * a * a⁻¹)
= 1 * (... | import Mathlib.Tactic
example {G : Type*} [Group G] (h: ∀ (x : G), x * x = 1) : CommGroup G := by
-- To show that G is a commutative group, apply the constructor for CommGroup
apply CommGroup.mk
intro a b
-- a * b = a * b * (a * b) * (a * b)⁻¹ = 1 * (a * b)⁻¹
rw [← mul_inv_cancel_right (a * b) (a * b), h]
-- = b⁻... | Show that every group $G$ with identity $e$ and such that $x * x=e$ for all $x \in G$ is Abelian. | 17 | import Mathlib
example {G : Type*} [Group G] (h: ∀ (x : G), x * x = 1) : CommGroup G := by sorry | 26 | import Mathlib | true | {"env": 0}
|
22 | import Mathlib
example {G : Type*} [Group G] (a b : G) (h : a * b = b * a⁻¹) : b * a = a⁻¹ * b := by
have h₀ : a⁻¹ * (a * b) * a = a⁻¹ * (b * a⁻¹) * a := by rw [h]
calc
_ = 1 * (b * a) := by rw [one_mul]
_ = a⁻¹ * a * (b * a) := by rw [← mul_left_inv]
_ = a⁻¹ * (a * b) * a := by rw [mul_assoc a⁻¹, ← mul_asso... | Suppose that $G$ is a group and $a, b \in G$ satisfy $a * b=b * a^{-1}$. Prove that $b * a=a^{-1} * b$. | 2 | import Mathlib
example {G : Type*} [Group G] (a b : G) (h : a * b = b * a⁻¹) : b * a = a⁻¹ * b := sorry | 23 | import Mathlib | true | {"messages":
[{"severity": "warning",
"pos": {"line": 7, "column": 36},
"endPos": {"line": 7, "column": 48},
"data": "`mul_left_inv` has been deprecated, use `inv_mul_cancel` instead"},
{"severity": "warning",
"pos": {"line": 11, "column": 28},
"endPos": {"line": 11, "column": 40},
"data": "`mul_le... | |
1,639 | import Mathlib
example {G : Type*} [Group G] (h: ∀ (x : G), x * x = 1) : CommGroup G := {
mul_comm := by
intro a b
have h1 : (a * b) * (a * b) = 1 := by rw [h (a * b)]
apply (mul_left_inj (a * b)⁻¹).mpr at h1
rw [mul_assoc, mul_right_inv (a * b), mul_one, one_mul] at h1
rw [h1]
apply (mul_lef... | Show that every group $G$ with identity $e$ and such that $x * x=e$ for all $x \in G$ is Abelian. | 17 | import Mathlib
example {G : Type*} [Group G] (h: ∀ (x : G), x * x = 1) : CommGroup G := by sorry | 26 | import Mathlib | true | {"messages":
[{"severity": "warning",
"pos": {"line": 8, "column": 19},
"endPos": {"line": 8, "column": 32},
"data":
"`mul_right_inv` has been deprecated, use `mul_inv_cancel` instead"}],
"env": 0}
| |
1,912 | import Mathlib
example {G : Type*} [Group G] {x y : G}
(h : (x * y) * (x * y) = (x * x) * (y * y)): x * y = y * x := by
rw[←mul_assoc, ←mul_assoc] at h
have h1 : (x * y * x) * y * y⁻¹ = (x * x * y) * y * y⁻¹ := by
rw[h]
simp only [mul_inv_cancel_right] at h1
have h2 : x⁻¹ * (x * y * x) = x⁻¹ * (x * x * y) ... | Suppose that $G$ is a group with operation $\circ$; suppose that $x, y \in G$. Show that if
$$
(x \circ y) \circ(x \circ y)=(x \circ x) \circ(y \circ y),
$$
then $x \circ y=y \circ x$. | 7 | import Mathlib
example {G : Type*} [Group G] {x y : G} (h : (x * y) * (x * y) = (x * x) * (y * y)): x * y = y * x := by sorry | 25 | import Mathlib | true | {"messages":
[{"severity": "warning",
"pos": {"line": 12, "column": 13},
"endPos": {"line": 12, "column": 25},
"data": "`mul_left_inv` has been deprecated, use `inv_mul_cancel` instead"}],
"env": 0}
| |
214 | import Mathlib
example {G : Type*} [Group G] (a b c : G) : ∃! x : G, a * x * b = c := by
use a⁻¹ * c * b⁻¹
constructor
. group
. intro x
intro h
have l : a⁻¹ * (a * x * b) * b⁻¹ = a⁻¹ * c * b⁻¹ := by
rw [h]
group at l
group
exact l | Suppose that $G$ is a group and $a, b, c\in G$. Prove that $a * x * b=c$ has a unique solution. | 6 | import Mathlib
example {G : Type*} [Group G] (a b c : G) : ∃! x : G, a * x * b = c := sorry | 22 | import Mathlib | true | {"env": 0}
| |
4,583 | import Mathlib
example (n : ℕ) (k j : Fin n) :
MulAction.stabilizer (Equiv.Perm (Fin n)) k ≃ MulAction.stabilizer (Equiv.Perm (Fin n)) j where
toFun := by
intro α
use (Equiv.swap k j) * α * (Equiv.swap k j)
sorry
invFun := sorry
left_inv := sorry
right_inv := sorry | 16. Let $n$ be positive integer, and $k, j$ fixed integers with $1 \leq k, j \leq n$. Define the function $\Phi: \operatorname{Stab}(k) \rightarrow \operatorname{Stab}(j)$ on $\alpha \in \operatorname{Stab}(k)$ by setting $\Phi(\alpha)=(k j) \alpha(k j)$ (where $(k j)$ is the 2-cycle switching $k$ and $j$ ). Prove that... | 900 | import Mathlib
| 1,005 | import Mathlib | true | {"sorries":
[{"proofState": 0,
"pos": {"line": 8, "column": 4},
"goal":
"case property\nn : ℕ\nk j : Fin n\nα : ↥(MulAction.stabilizer (Equiv.Perm (Fin n)) k)\n⊢ Equiv.swap k j * ↑α * Equiv.swap k j ∈ MulAction.stabilizer (Equiv.Perm (Fin n)) j",
"endPos": {"line": 8, "column": 9}},
{"proofState": 1,
... | |
279 | import Mathlib
example {G : Type*} [Group G] {x y : G} (h : (x * y) * (x * y) = (x * x) * (y * y)): x * y = y * x := by
have lemma1 : x * y * x = x * x * y := by
calc
_ = (x * y) * (x * y) * y⁻¹ := by group
_ = (x * x) * (y * y) * y⁻¹ := by rw [h]
_ = _ := by group
calc
_ = x⁻¹ * (x * x * y) :=... | Suppose that $G$ is a group with operation $\circ$; suppose that $x, y \in G$. Show that if
$$
(x \circ y) \circ(x \circ y)=(x \circ x) \circ(y \circ y),
$$
then $x \circ y=y \circ x$. | 7 | import Mathlib
example {G : Type*} [Group G] {x y : G} (h : (x * y) * (x * y) = (x * x) * (y * y)): x * y = y * x := by sorry | 25 | import Mathlib | true | {"env": 0}
| |
1,460 | 1、证明x的存在性
2、证明x的唯一性 | import Mathlib
section Basic_Calculations
example {G : Type*} [Group G] (a b c : G) : ∃! x : G, a * x * b = c := by
·use a⁻¹ * c * b⁻¹
group
rw[true_and]
intro y h
calc
_ = a⁻¹ *(a*y*b)* b⁻¹ := by group
_ = _ := by simp[h]
| Suppose that $G$ is a group and $a, b,a_1,a_2 \in G$. Prove that $(bab^{-1})^{-1}=ba^{-1}b^{-1}$ and $(ba_1b^{-1})(ba_2b^{-1})=ba_1a_2b^{-1}$. | 5 | import Mathlib
example {G : Type*} [Group G] (a b a₁ a₂ : G) : (b * a * b⁻¹)⁻¹ = b * a⁻¹ * b⁻¹ ∧ (b * a₁ * b⁻¹) * (b * a₂ * b⁻¹) = b * a₁ * a₂ * b⁻¹ := sorry | 24 | import Mathlib | true | {"env": 0}
|
5,140 | We just enumerate all the conditions and use ‘decide’ to calculate the results.
| import Mathlib.Tactic
example : ∀ x : ZMod 7, 3 * x = 2 ↔ x = 3 := by
intro x
constructor
-- We start by simplifying the negation of the existential quantifier.
· by_contra h0
push_neg at h0
have h1 : 3 * x = 2 := h0.1
have h2 : x ≠ 3 := h0.2
absurd h1
-- We use pattern matching to check the e... | Solve the equation $3 x=2$ in the field $\mathbb{Z}/7\mathbb{Z}$ and in the field $\mathbb{Z}/23\mathbb{Z}$. | 2,149 | import Mathlib
| 3,324 | import Mathlib | true | {"messages":
[{"severity": "warning",
"pos": {"line": 33, "column": 14},
"endPos": {"line": 33, "column": 15},
"data":
"unused variable `a`\nnote: this linter can be disabled with `set_option linter.unusedVariables false`"}],
"env": 0}
|
1,677 | import Mathlib
-- 著者：秦宇轩
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic
import Init.Prelude
example {G : Type*} [Group G] (a : G) (k : ℕ) (h : Odd k) (pow_one : a ^ k = 1):
Odd (orderOf a) := by
have : (orderOf a) ∣ k := by
exact orderOf_dvd_of_pow_eq_one pow_one
exact Odd.of_dvd_nat h this... | Let $a$ be an element of a group $G$. If $a^{k}=e$ where $k$ is odd, then the order of $a$ is odd. | 28 | import Mathlib
example {G : Type*} [Group G] (a : G) (k : ℕ) (h : Odd k) (pow_one : a ^ k = 1): Odd (orderOf a) := by sorry | 1,838 | import Mathlib | true | {"env": 0}
| |
2,151 | 1. We define a structure two_div_sum with two integer $x,y$ components and a proof $h$ that $2 ∣ x + y$.
2. The $Add$ on two_div_sum is defined as $(a+b).x=a.x+b.x, (a+b).y=a.y+b.y$ and $2 ∣ (a+b).x + (a+b).y$.
3. The $Neg$ on two_div_sum is defined as $(-a).x=-a.x, (-a).y=-a.y$ and $2 ∣ (-a).x + (-a).y$.
4. The $Zero$... | import Mathlib
-- Define a structure 'two_div_sum' with two integer $x,y$ components and a proof $h$ that $2 ∣ x + y$
@[ext]
structure two_div_sum where
x : ℤ
y : ℤ
h : 2 ∣ x + y
-- Define an 'Add' instance for 'two_div_sum' using the properties of integers
instance : Add two_div_sum where
add := by
-- In... | Prove that the elements $\{(x,y):x,y\in\mathbb Z, 2\mid x+y\}$ under elementwise addition is a group. | 55 | import Mathlib.Tactic
import Mathlib.Algebra.Group.MinimalAxioms
structure two_div_sum where
x : ℤ
y : ℤ
h : 2 ∣ x + y
instance : Add two_div_sum where
add := sorry
instance : Neg two_div_sum where
neg := sorry
instance : Zero two_div_sum where
zero := sorry
instance : AddGroup two_div_sum := by
appl... | 8 | import Mathlib | true | {"env": 0}
|
2,905 | 1. first we show that order of g * a * g⁻¹ is 2
2. by calculation, (g * a * g⁻¹) ^ 2 = g * (a * a) * g⁻¹ = 1
3. so the order of g * a * g⁻¹ can only be 1 or 2, we show it is not 1. if so a = 1, which contradicts the assumption that order of a is 2
4. since a is the unique element with order 2, we have g * a * g⁻¹ = a, ... | import Mathlib.Tactic
example {G : Type*} [Group G] {a : G} (h₁: orderOf a = 2) (h₂: ∃! (x : G), orderOf x = 2) : ∀ x : G, a * x = x * a := by
intro g
rw [ExistsUnique] at h₂
rcases h₂ with ⟨x, ⟨ _ , hx2⟩⟩
have haa: a * a = 1 := by
calc
a * a = a ^ 2 := by exact Eq.symm (pow_two a)
_ = 1 := ... | Let $G$ be a group and suppose $a \in G$ generates a cyclic subgroup of order 2 and is the unique such element. Show that $a x=x a$ for all $x \in G$. [Hint: Consider $\left(x a x^{-1}\right)^{2}$.] | 33 | import Mathlib
example {G : Type*} [Group G] {a : G} (h₁: orderOf a = 2) (h₂: ∃! (x : G), orderOf x = 2) : ∀ x : G, a * x = x * a := by sorry | 102 | import Mathlib | true | {"env": 0}
|
923 | import Mathlib
instance {G : Type*} [Group G] (h : IsCyclic G) : CommGroup G := by
apply CommGroup.mk
intro a b
rcases h with ⟨t, ht⟩
rcases ht a with ⟨na, hna⟩
simp at hna
rcases ht b with ⟨nb, hnb⟩
simp at hnb
rw [← hna, ← hnb]
group | Every cyclic group is a abelian group. | 35 | import Mathlib
instance : sorry := sorry | 1,840 | import Mathlib | true | {"env": 0}
| |
714 | import Mathlib
example {G : Type*} [Group G] (a b : G) (h : a * b = b * a⁻¹) : b * a = a⁻¹ * b := by
have h₁ : a * b * a = b := by
rw [h, mul_assoc, mul_left_inv, mul_one]
calc
_ = a⁻¹ * a * b * a := by
rw [mul_assoc, mul_left_inv, one_mul]
_ = _ := by
rw [mul_assoc,mul_assoc]
nth_rw ... | Suppose that $G$ is a group and $a, b \in G$ satisfy $a * b=b * a^{-1}$. Prove that $b * a=a^{-1} * b$. | 2 | import Mathlib
example {G : Type*} [Group G] (a b : G) (h : a * b = b * a⁻¹) : b * a = a⁻¹ * b := sorry | 23 | import Mathlib | true | {"messages":
[{"severity": "warning",
"pos": {"line": 5, "column": 22},
"endPos": {"line": 5, "column": 34},
"data": "`mul_left_inv` has been deprecated, use `inv_mul_cancel` instead"},
{"severity": "warning",
"pos": {"line": 8, "column": 21},
"endPos": {"line": 8, "column": 33},
"data": "`mul_left... |
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