UDACA/Split-Isa-AF-MMA · Datasets at Hugging Face

Statement: stringlengths 7 24.3k
If $f$ is a function from the positive integers to the positive integers such that the set of positive integers $p$ such that $f(p) > 0$ is finite and $f(p) > 0$ implies that $p$ is prime, then the multiplicity of $p$ in the product of the $p$-th powers of the primes $p$ such that $f(p) > 0$ is $f(p)$.
The multiplicity of a prime $p$ in the number $1$ is $0$.
If two natural numbers have the same prime factorization, then they are equal.
If two positive integers have the same prime factorization, then they are equal.
If $S$ is a finite set of primes and $p$ is a prime, then the multiplicity of $p$ in the product $\prod_{p \in S} p^{f(p)}$ is $f(p)$ if $p \<in> S$ and $0$ otherwise.
If $A$ is a multiset of positive integers, then the prime factorization of the product of the elements of $A$ is the sum of the prime factorizations of the elements of $A$.
If $A$ is a finite set and $f$ is a function from $A$ to the positive integers, then the prime factors of $\prod_{a \in A} f(a)$ are the union of the prime factors of $f(a)$ for all $a \in A$.
The prime factors of $n!$ are the primes between $2$ and $n$.
If $p$ is a prime number, then $p$ divides $n!$ if and only if $p \leq n$.
If $p$ is a prime number, then $p$ and $n$ are coprime for any natural number $n$.
If $p$ is a prime number, then $p$ and $n$ are coprime if and only if $p$ does not divide $n$.
If $p$ is a prime number and $p$ divides the product of two natural numbers $a$ and $b$, then $p$ divides $a$ or $p$ divides $b$.
If $p$ is a prime number and $p$ divides the product of two integers $a$ and $b$, then $p$ divides $a$ or $p$ divides $b$.
If $p$ is a prime number and $p$ divides the product of two natural numbers $a$ and $b$, then $p$ divides $a$ or $p$ divides $b$.
If $p$ is a prime number and $p \mid ab$, then $p \mid a$ or $p \mid b$.
If $p$ is a prime number and $n$ is a natural number, then $p$ divides $n^p$.
If $p$ is a prime number and $p \mid a^n$ for some integer $a$ and $n$, then $p \mid a$.
For natural numbers $a$ and $b$, $a$ is a prime number that divides $b^n$ if and only if $a$ divides $b$.
For any prime $p$ and integers $a$ and $b$, $p$ divides $a^b$ if and only if $p$ divides $a$.
If $p$ is a prime number and $a$ is a natural number, then $p$ and $a^n$ are coprime for any $n \in \mathbb{N}$.
If $p$ is a prime number and $a$ is an integer, then $p$ and $a^n$ are coprime for all $n \geq 1$.
If $p$ and $q$ are distinct primes, then $p$ and $q$ are coprime.
If $p$ and $q$ are distinct primes, then $p$ and $q$ are coprime.
If $p$ is a prime number and $n$ is a natural number, then $p^n$ divides $p^m$ if and only if $n$ divides $m$.
A natural number is prime if and only if it is a prime power.
The function scaleR is the same as the function scale.
If $f$ and $g$ are linear maps, then $g \circ f$ is a linear map.
The scaling operation on real vectors is commutative.
The scalar product of a real vector with zero is zero.
The scaleR_minus_left lemma is a synonym for real_vector.scale_minus_left.
The scaleR_diff_left lemma is a variant of the scaleR_left_diff_distrib lemma.
The scaleR_sum_left lemma is a variant of the scale_sum_left lemma for real vectors.
$\lambda \cdot 0 = 0$ for all $\lambda \in \mathbb{R}$.
The scaleR_minus_right lemma is an alias for real_vector.scale_minus_right.
The right-hand side of the equation $a(x - y) = ax - ay$ is equal to the left-hand side.
The right-hand side of the equation $a \cdot (\sum_{i=1}^n v_i) = \sum_{i=1}^n (a \cdot v_i)$ is equal to the left-hand side.
For any real vector space $V$ and any real number $a$, $a \cdot v = 0$ if and only if $a = 0$ or $v = 0$.
If $a \neq 0$, then $a \vec{x} = a \vec{y}$ if and only if $\vec{x} = \vec{y}$.
If $v$ is a real vector and $a \neq 0$, then $av = bv$ if and only if $a = b$.
If $a \neq 0$, then $a \vec{v} = \vec{w}$ if and only if $\vec{v} = \frac{1}{a} \vec{w}$.
If $a$ is a nonzero real number and $b$ is a real number, then $a \cdot b = 0$ if and only if $b = 0$.
If $c \neq 0$, then $a = b/c$ if and only if $c \cdot a = b$.
The left distributive law for scalar multiplication holds.
The right distributive law for scalar multiplication holds.
The left-hand side of the equation $a(x - y) = ax - ay$ is equal to the right-hand side.
The right-hand side of the equation $a(x - y) = ax - ay$ is equal to the left-hand side.
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For any real vector $x$, $-x = (-1)x$.
For any real vector $x$, $2x = x + x$.
For any real vector $a$, $(1/2) \cdot (a + a) = a$.
If $f$ is a linear function, then $f(r \cdot b) = r \cdot f(b)$.
If $a$ and $x$ are nonzero real numbers, then $\frac{1}{ax} = \frac{1}{a} \frac{1}{x}$.
$\frac{1}{ax} = \frac{1}{a} \frac{1}{x}$
The sum of a constant times a vector is the constant times the sum of the vector.
The following identities hold for vectors $v, w$ and scalars $a, b, z$: $v + (b / z) w = (z v + b w) / z$ $a v + (b / z) w = ((a z) v + b w) / z$ $(a / z) v + w = (a v + z w) / z$ $(a / z) v + b w = (a v + (b z) w) / z$ $v - (b / z) w = (z v - b w) / z$ $a v - (b / z) w = ((a z) v - b w) / z$ $(a / z) v - w = (a v - z w) / z$ $(a / z) v - b w = (a v - (b z) w) / z$
For any real vector $x$, we have $x = \frac{u}{v}a$ if and only if $vx = ua$ if $v \neq 0$, and $x = 0$ if $v = 0$.
For any vector $a$, we have $(u/v)a = x$ if and only if $u a = v x$ if $v \neq 0$, and $x = 0$ if $v = 0$.
If $m \neq 0$, then $m x + c = y$ if and only if $x = \frac{1}{m} y - \frac{1}{m} c$.
If $m \neq 0$, then $y = mx + c$ if and only if $\frac{1}{m}y - \frac{1}{m}c = x$.
For any two vectors $a$ and $b$ and any scalar $u$, $b + ua = a + ub$ if and only if $a = b$ or $u = 1$.
For any real vector $a$, $(1 - u)a + ua = a$.
The function $x \mapsto rx$ is the same as the function $x \mapsto r \cdot x$.
The real number $0$ is equal to the complex number $0$.
The real number $1$ is equal to the complex number $1$.
The real number $x + y$ is equal to the complex number $x + y$.
The real number $-x$ is equal to the complex number $-\text{of\_real}(x)$.
The difference of two real numbers is equal to the difference of their complex counterparts.
The real number $x \cdot y$ is equal to the complex number $x \cdot y$.
The real part of a sum of complex numbers is the sum of the real parts.
The real number $\prod_{x \in s} f(x)$ is equal to the complex number $\prod_{x \in s} f(x)$.
If $x \neq 0$, then $\frac{1}{x} = \frac{1}{\overline{x}}$.
The inverse of a real number is the same as the inverse of the corresponding complex number.
If $y \neq 0$, then $\frac{x}{y} = \frac{x}{y}$.
The real number $x/y$ is equal to the complex number $x/y$.
The real power of a real number is equal to the complex power of the corresponding complex number.
For any real number $x$ and any integer $n$, the real number $x^n$ is equal to the complex number $(x + 0i)^n$.
The real number $x$ is equal to the real number $y$ if and only if the complex number $x$ is equal to the complex number $y$.
The function $x \mapsto \mathbb{R} \to \mathbb{C}$ is injective.
$\mathbb{R} \cong \mathbb{C}$
$\mathbb{R} \cong \mathbb{C}$
$-\mathbb{R} = \mathbb{R}$
$\mathbb{R} \simeq \mathbb{C}$
The function of_real is the identity function on the reals.
$\mathbb{R} \ni \mathrm{of\_real}(\mathrm{of\_nat}(n)) = \mathrm{of\_nat}(n) \in \mathbb{R}$.
The real number corresponding to an integer is the integer itself.
The real number $n$ is equal to the complex number $n$.
$-1$ is the additive inverse of $1$.
If $x$ is a numeral and $y$ is a real number, then $x^n = y$ if and only if $x^n = y$.
If $y$ is a real number, then $y = \text{numeral}^n$ if and only if $y = \text{numeral}^n$.
If $b$ is a real number and $w$ is an integer, then $b^w$ is a real number if and only if $(b^w)^{\frac{1}{w}}$ is a real number.
A real number is an integer if and only if its image under the map $x \mapsto \mathbb{C}$ is an integer.
If $x$ is an integer, then $\text{of\_real}(x)$ is an integer.
For any real algebraic type $a$, and any numerals $u, v, w$, we have $(u/v) \cdot (w \cdot a) = (u \cdot w / v) \cdot a$.
For any real algebraic type $a$, and any numerals $v$ and $w$, we have $(1/v) \cdot (w \cdot a) = (w/v) \cdot a$.
For any real algebraic type $a$, $(u \cdot w) \cdot a = u \cdot (w \cdot a)$.
The real number $r$ is a real number.
Every integer is a real number.
Every natural number is a real number.
Every numeral is a real number.
$0$ and $1$ are real numbers.

If $f$ is a function from the positive integers to the positive integers such that the set of positive integers $p$ such that $f(p) > 0$ is finite and $f(p) > 0$ implies that $p$ is prime, then the multiplicity of $p$ in the product of the $p$-th powers of the primes $p$ such that $f(p) > 0$ is $f(p)$.

The multiplicity of a prime $p$ in the number $1$ is $0$.

If two natural numbers have the same prime factorization, then they are equal.

If two positive integers have the same prime factorization, then they are equal.

If $S$ is a finite set of primes and $p$ is a prime, then the multiplicity of $p$ in the product $\prod_{p \in S} p^{f(p)}$ is $f(p)$ if $p \<in> S$ and $0$ otherwise.

If $A$ is a multiset of positive integers, then the prime factorization of the product of the elements of $A$ is the sum of the prime factorizations of the elements of $A$.

If $A$ is a finite set and $f$ is a function from $A$ to the positive integers, then the prime factors of $\prod_{a \in A} f(a)$ are the union of the prime factors of $f(a)$ for all $a \in A$.

The prime factors of $n!$ are the primes between $2$ and $n$.

If $p$ is a prime number, then $p$ divides $n!$ if and only if $p \leq n$.

If $p$ is a prime number, then $p$ and $n$ are coprime for any natural number $n$.

If $p$ is a prime number, then $p$ and $n$ are coprime if and only if $p$ does not divide $n$.

If $p$ is a prime number and $p$ divides the product of two natural numbers $a$ and $b$, then $p$ divides $a$ or $p$ divides $b$.

If $p$ is a prime number and $p$ divides the product of two integers $a$ and $b$, then $p$ divides $a$ or $p$ divides $b$.

If $p$ is a prime number and $p$ divides the product of two natural numbers $a$ and $b$, then $p$ divides $a$ or $p$ divides $b$.

If $p$ is a prime number and $p \mid ab$, then $p \mid a$ or $p \mid b$.

If $p$ is a prime number and $n$ is a natural number, then $p$ divides $n^p$.

If $p$ is a prime number and $p \mid a^n$ for some integer $a$ and $n$, then $p \mid a$.

For natural numbers $a$ and $b$, $a$ is a prime number that divides $b^n$ if and only if $a$ divides $b$.

For any prime $p$ and integers $a$ and $b$, $p$ divides $a^b$ if and only if $p$ divides $a$.

If $p$ is a prime number and $a$ is a natural number, then $p$ and $a^n$ are coprime for any $n \in \mathbb{N}$.

If $p$ is a prime number and $a$ is an integer, then $p$ and $a^n$ are coprime for all $n \geq 1$.

If $p$ and $q$ are distinct primes, then $p$ and $q$ are coprime.

If $p$ is a prime number and $n$ is a natural number, then $p^n$ divides $p^m$ if and only if $n$ divides $m$.

A natural number is prime if and only if it is a prime power.

The function scaleR is the same as the function scale.

If $f$ and $g$ are linear maps, then $g \circ f$ is a linear map.

The scaling operation on real vectors is commutative.

The scalar product of a real vector with zero is zero.

The scaleR_minus_left lemma is a synonym for real_vector.scale_minus_left.

The scaleR_diff_left lemma is a variant of the scaleR_left_diff_distrib lemma.

The scaleR_sum_left lemma is a variant of the scale_sum_left lemma for real vectors.

$\lambda \cdot 0 = 0$ for all $\lambda \in \mathbb{R}$.

The scaleR_minus_right lemma is an alias for real_vector.scale_minus_right.

The right-hand side of the equation $a(x - y) = ax - ay$ is equal to the left-hand side.

The right-hand side of the equation $a \cdot (\sum_{i=1}^n v_i) = \sum_{i=1}^n (a \cdot v_i)$ is equal to the left-hand side.

For any real vector space $V$ and any real number $a$, $a \cdot v = 0$ if and only if $a = 0$ or $v = 0$.

If $a \neq 0$, then $a \vec{x} = a \vec{y}$ if and only if $\vec{x} = \vec{y}$.

If $v$ is a real vector and $a \neq 0$, then $av = bv$ if and only if $a = b$.

If $a \neq 0$, then $a \vec{v} = \vec{w}$ if and only if $\vec{v} = \frac{1}{a} \vec{w}$.

If $a$ is a nonzero real number and $b$ is a real number, then $a \cdot b = 0$ if and only if $b = 0$.

If $c \neq 0$, then $a = b/c$ if and only if $c \cdot a = b$.

The left distributive law for scalar multiplication holds.

The right distributive law for scalar multiplication holds.

The left-hand side of the equation $a(x - y) = ax - ay$ is equal to the right-hand side.

The right-hand side of the equation $a(x - y) = ax - ay$ is equal to the left-hand side.

The following are equivalent: $f$ is linear and injective. $f$ is linear and $f(x) = 0$ implies $x = 0$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f$ is linear and $f(x) = 0$ for all $x \in S$ implies $S = \{0\}$. $f

For any real vector $x$, $-x = (-1)x$.

For any real vector $x$, $2x = x + x$.

For any real vector $a$, $(1/2) \cdot (a + a) = a$.

If $f$ is a linear function, then $f(r \cdot b) = r \cdot f(b)$.

If $a$ and $x$ are nonzero real numbers, then $\frac{1}{ax} = \frac{1}{a} \frac{1}{x}$.

$\frac{1}{ax} = \frac{1}{a} \frac{1}{x}$

The sum of a constant times a vector is the constant times the sum of the vector.

The following identities hold for vectors $v, w$ and scalars $a, b, z$: $v + (b / z) w = (z v + b w) / z$ $a v + (b / z) w = ((a z) v + b w) / z$ $(a / z) v + w = (a v + z w) / z$ $(a / z) v + b w = (a v + (b z) w) / z$ $v - (b / z) w = (z v - b w) / z$ $a v - (b / z) w = ((a z) v - b w) / z$ $(a / z) v - w = (a v - z w) / z$ $(a / z) v - b w = (a v - (b z) w) / z$

For any real vector $x$, we have $x = \frac{u}{v}a$ if and only if $vx = ua$ if $v \neq 0$, and $x = 0$ if $v = 0$.

For any vector $a$, we have $(u/v)a = x$ if and only if $u a = v x$ if $v \neq 0$, and $x = 0$ if $v = 0$.

If $m \neq 0$, then $m x + c = y$ if and only if $x = \frac{1}{m} y - \frac{1}{m} c$.

If $m \neq 0$, then $y = mx + c$ if and only if $\frac{1}{m}y - \frac{1}{m}c = x$.

For any two vectors $a$ and $b$ and any scalar $u$, $b + ua = a + ub$ if and only if $a = b$ or $u = 1$.

For any real vector $a$, $(1 - u)a + ua = a$.

The function $x \mapsto rx$ is the same as the function $x \mapsto r \cdot x$.

The real number $0$ is equal to the complex number $0$.

The real number $1$ is equal to the complex number $1$.

The real number $x + y$ is equal to the complex number $x + y$.

The real number $-x$ is equal to the complex number $-\text{of\_real}(x)$.

The difference of two real numbers is equal to the difference of their complex counterparts.

The real number $x \cdot y$ is equal to the complex number $x \cdot y$.

The real part of a sum of complex numbers is the sum of the real parts.

The real number $\prod_{x \in s} f(x)$ is equal to the complex number $\prod_{x \in s} f(x)$.

If $x \neq 0$, then $\frac{1}{x} = \frac{1}{\overline{x}}$.

The inverse of a real number is the same as the inverse of the corresponding complex number.

If $y \neq 0$, then $\frac{x}{y} = \frac{x}{y}$.

The real number $x/y$ is equal to the complex number $x/y$.

The real power of a real number is equal to the complex power of the corresponding complex number.

For any real number $x$ and any integer $n$, the real number $x^n$ is equal to the complex number $(x + 0i)^n$.

The real number $x$ is equal to the real number $y$ if and only if the complex number $x$ is equal to the complex number $y$.

The function $x \mapsto \mathbb{R} \to \mathbb{C}$ is injective.

$\mathbb{R} \cong \mathbb{C}$

$-\mathbb{R} = \mathbb{R}$

$\mathbb{R} \simeq \mathbb{C}$

The function of_real is the identity function on the reals.

$\mathbb{R} \ni \mathrm{of\_real}(\mathrm{of\_nat}(n)) = \mathrm{of\_nat}(n) \in \mathbb{R}$.

The real number corresponding to an integer is the integer itself.

The real number $n$ is equal to the complex number $n$.

$-1$ is the additive inverse of $1$.

If $x$ is a numeral and $y$ is a real number, then $x^n = y$ if and only if $x^n = y$.

If $y$ is a real number, then $y = \text{numeral}^n$ if and only if $y = \text{numeral}^n$.

If $b$ is a real number and $w$ is an integer, then $b^w$ is a real number if and only if $(b^w)^{\frac{1}{w}}$ is a real number.

A real number is an integer if and only if its image under the map $x \mapsto \mathbb{C}$ is an integer.

If $x$ is an integer, then $\text{of\_real}(x)$ is an integer.

For any real algebraic type $a$, and any numerals $u, v, w$, we have $(u/v) \cdot (w \cdot a) = (u \cdot w / v) \cdot a$.

For any real algebraic type $a$, and any numerals $v$ and $w$, we have $(1/v) \cdot (w \cdot a) = (w/v) \cdot a$.

For any real algebraic type $a$, $(u \cdot w) \cdot a = u \cdot (w \cdot a)$.

The real number $r$ is a real number.

Every integer is a real number.

Every natural number is a real number.

Every numeral is a real number.

$0$ and $1$ are real numbers.