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

Statement: stringlengths 7 24.3k
The neighborhood filter of a point $x$ in an ordered topological space is the intersection of the principal filters generated by the sets $\{y \in X : y < x\}$ and $\{y \in X : x < y\}$.
If $f$ converges to $L$ on the set $A \cap \{x \mid P(x)\}$ and $g$ converges to $L$ on the set $A \cap \{x \mid \lnot P(x)\}$, then the function $h$ defined by $h(x) = f(x)$ if $P(x)$ and $h(x) = g(x)$ if $\lnot P(x)$ converges to $L$ on $A$.
If $f$ converges to $L$ along the filter $G$ on the set $\{x \mid P(x)\}$ and $g$ converges to $L$ along the filter $G$ on the set $\{x \mid \lnot P(x)\}$, then the function $h$ defined by $h(x) = f(x)$ if $P(x)$ and $h(x) = g(x)$ if $\lnot P(x)$ converges to $L$ along the filter $G$ on the set $\{x\}$.
In a linearly ordered topological space, the set of points that are close to $x$ is the intersection of the sets of points that are less than $x$ and the sets of points that are greater than $x$.
If $y < x$, then the set of all open sets that contain $x$ is equal to the intersection of all open sets that contain $x$.
If $y < x$, then the predicate $P$ holds eventually at $x$ from the left if and only if there exists $b < x$ such that $P$ holds for all $y > b$ with $y < x$.
If $x < y$, then the set of all open sets containing $x$ is equal to the set of all open sets containing $x$ and not containing $y$.
If $x < y$, then the following are equivalent: $P(y)$ holds eventually as $y$ approaches $x$ from the right. There exists a $b > x$ such that $P(y)$ holds for all $y > x$ with $y < b$.
For any $x$, there exists a sequence of points $y_n$ such that $x < y_n$ and $y_n \to x$ from the right.
The limit at the right of the top element of a linearly ordered topological space is the empty set.
The limit at the left of the bottom element of a linearly ordered topological space is the bottom element.
The limit at the left of any real number is not trivial.
The limit at the right of any real number is not trivial.
In a linearly ordered topological space, the set of points that converge to $x$ is the union of the sets of points that converge to $x$ from the left and from the right.
In a linearly ordered topological space, the set of points at which a property $P$ holds is the intersection of the sets of points at which $P$ holds to the left of $x$ and to the right of $x$.
If $P$ holds for all $x$ in the open interval $(a,b)$, then $P$ holds eventually at $b$ from the left.
If $P$ holds for all $x$ in the open interval $(a,b)$, then $P$ holds eventually at $a$ from the right.
The filtercomap of the nhds filter is the filter of open sets.
The filtercomap of the filter at a point is the filter of the complement of the point.
For any real number $x$, the following are equivalent: $P(y)$ holds for all $y > x$ sufficiently close to $x$; there exists a real number $b > x$ such that $P(y)$ holds for all $y > x$ with $y < b$.
For any real number $x$, the following are equivalent: $P(y)$ holds for all $y$ sufficiently close to $x$ from the left. There exists a real number $b < x$ such that $P(y)$ holds for all $y > b$ with $y < x$.
If $f$ converges to $x$ in $F$, and $x = y$, then $f$ converges to $y$ in $F$.
A function $f$ tends to $l$ with respect to a filter $F$ if and only if for every open set $S$ containing $l$, the set $\{x \in X : f(x) \in S\}$ is in $F$.
If two functions $f$ and $g$ are eventually equal, then they converge to the same limit.
If $F$ is a filter on a set $X$ and $F'$ is a filter on $X$ such that $F \leq F'$, then if $f$ converges to $l$ with respect to $F'$, then $f$ converges to $l$ with respect to $F$.
The identity function tends to $a$ at $a$ within $s$.
The constant function $f(x) = k$ converges to $k$.
The filter $F$ converges to $b$ within $s$ if and only if $F$ eventually contains points $x$ such that $f(x) \in s$ and $f(x) \neq b$, and $F$ converges to $b$.
If $f$ converges to $L$ in the filter of neighbourhoods of $L$, then $f$ converges to $L$ from the left or from the right, depending on whether $f$ is eventually greater than $L$ or eventually less than $L$.
If $f$ converges to $c$ in the filter of neighborhoods of $c$, and $f$ eventually takes values in $A - \{c\}$, then $f$ converges to $c$ in the filter of neighborhoods of $c$ within $A$.
If $f$ converges to $c$ in the neighborhood filter of $c$, and $f$ is eventually not equal to $c$, then $f$ converges to $c$ in the filter at $c$.
If $f$ is a function from a filter $F$ to a topological space $X$ and $f$ converges to $l$ in the sense that for every open set $S$ containing $l$, the set $\{x \in X \mid f(x) \in S\}$ is eventually in $F$, then $f$ converges to $l$ in the sense of filters.
If $f$ converges to $l$ in a filter $F$, and $S$ is an open set containing $l$, then $f$ eventually takes values in $S$.
If $f$ tends to $a$, then $f$ tends to $-\infty$.
If $f$ converges to $l$ eventually, then $f$ converges to $l$.
If $f$ converges to $l$ at $x$ within $S$, then $f$ converges to $l$ at $x$ within $T$ for any $T \subseteq S$.
If $f$ tends to $L$ within $S$, then $f$ tends to $L$ within any subset of $S$.
A function $f$ tends to $x$ in the order topology if and only if for every $l < x$ and every $u > x$, there exists a neighborhood of $x$ such that $l < f(y) < u$ for all $y$ in that neighborhood.
If $f$ is a function from a totally ordered set $X$ to a totally ordered set $Y$, and if for every $y \in Y$ there exists $x \in X$ such that $f(x) < y$, and for every $y \in Y$ there exists $x \in X$ such that $y < f(x)$, then $f$ is continuous.
If $f$ converges to $y$ in the order topology, then $f$ is eventually greater than $a$ if $y$ is greater than $a$, and $f$ is eventually less than $a$ if $y$ is less than $a$.
If $X$ and $Y$ are nets that converge to $x$ and $y$, respectively, then the net $\max(X,Y)$ converges to $\max(x,y)$.
If $X$ and $Y$ are nets that converge to $x$ and $y$, respectively, then the net $\min(X,Y)$ converges to $\min(x,y)$.
In the order topology, the limit points of an interval are the endpoints.
If $a < x < b$, then the filter of neighbourhoods of $x$ within the interval $[a,b]$ is the same as the filter of neighbourhoods of $x$ in the real line.
If $f$ converges to two different limits $a$ and $b$, then $a = b$.
In a T2 space, a sequence converges to a constant if and only if all the terms of the sequence are equal to that constant.
If $F$ is not the empty filter, then there is at most one limit of $f$ with respect to $F$.
If $f$ is a function that converges to $l$ and $f$ eventually takes values in a closed set $S$, then $l \<in> S$.
If $A$ is an open set containing $x$, then there is a closed set $A'$ containing $x$ and contained in $A$ such that $A'$ is in the filter of neighbourhoods of $x$.
If $f$ is an increasing sequence of real numbers that is bounded above, then $f$ converges to its supremum.
If $f$ is a sequence of real numbers such that $f_n \geq l$ for all $n$ and $f_n < x$ for all $n$ whenever $x > l$, then $f$ converges to $l$.
If $f_n \leq g_n \leq h_n$ for all $n$ and $\lim_{n \to \infty} f_n = \lim_{n \to \infty} h_n = c$, then $\lim_{n \to \infty} g_n = c$.
If $f$ converges to $d$ on a set $F$ that is not empty and on which $f$ is frequently equal to $c$, then $d = c$.
If a function $f$ converges to $c$ in a topological space, then for any $c' \neq c$, there exists a neighborhood of $c$ such that $f$ does not take the value $c'$ in that neighborhood.
If $f$ and $g$ are real-valued functions defined on a topological space $X$, and if $f$ and $g$ converge to $x$ and $y$, respectively, then $y \leq x$.
If $f$ is a sequence of real numbers that converges to $x$, and if $a$ is a lower bound for the sequence, then $a \<le> x$.
If $f$ is a sequence of real numbers that converges to $x$, and if $a$ is an upper bound for $f$, then $a \geq x$.
If $f$ converges to $b$ within $s$, then $f$ eventually takes values in $s$ that are not equal to $b$.
If a net has a non-trivial limit, then the limit of the net is equal to the limit of the function.
If the limit of a function $f$ at $x$ within $s$ is not trivial, then the limit of $f$ at $x$ within $s$ is $x$.
If $P$ holds for all points in a neighborhood of $x$ except possibly $x$ itself, then $P$ holds for all points in a neighborhood of $x$.
If $f$ is a monotone function and $g$ is a bijection such that $f \circ g$ is the identity function, then $f$ is continuous at $a$ if and only if $g$ is continuous at $a$.
If $f$ is a monotone function and $g$ is a bijection such that $f \circ g$ is the identity function, then $f$ is a filter limit at infinity.
If $f$ converges to $L$ from the left and from the right at $x$, then $f$ converges to $L$ at $x$.
The limit of a function $f$ at a point $x$ is the same as the limit of $f$ from the left and from the right at $x$.
If $b$ is less than the top element of a linearly ordered topological space, then the following are equivalent: $P$ holds eventually in a neighborhood of the top element. There exists $b < top$ such that $P$ holds for all $z > b$.
The limit of a function $g$ at a point $l$ within a set $S$ is the same as the limit of $g$ at $l$ in the neighborhood of $l$ that is contained in $S$.
The limit of a sequence $X$ is the unique number $L$ such that $X$ converges to $L$.
A sequence $f$ converges to $f_0$ if and only if for every open set $S$ containing $f_0$, there exists an $N$ such that for all $n \geq N$, $f_n \in S$.
A sequence $X$ is increasing if and only if for all $m$ and $n \geq m$, we have $X_n \geq X_m$.
A sequence $X$ is decreasing if and only if for all $m$ and $n \geq m$, we have $X_n \leq X_m$.
If $r$ is a strict monotone function and $m \leq n$, then $r(m) \leq r(n)$.
The identity function is strictly monotonic.
If $X_n \leq X_{n+1}$ for all $n$, then $X_n$ is an increasing sequence.
If $f$ is an increasing sequence, then $f_i \leq f_j$ whenever $i \leq j$.
If $A$ is an increasing sequence, then $A_i \leq A_{i+1}$.
A sequence $f$ is increasing if and only if $f(n) \leq f(n+1)$ for all $n$.
The sequence $(k)_{n \in \mathbb{N}}$ is increasing.
If $X_{n+1} \leq X_n$ for all $n$, then $X_n$ is a decreasing sequence.
If $f$ is a decreasing sequence, then $f(j) \leq f(i)$ for all $i \leq j$.
If $A$ is a decreasing sequence, then $A_{i+1} \leq A_i$.
A sequence $f$ is decreasing if and only if $f(n+1) \leq f(n)$ for all $n$.
The constant sequence $k, k, k, \ldots$ is decreasing.
A sequence is monotonic if and only if it is either increasing or decreasing.
A sequence is monotone if and only if it is either increasing or decreasing.
If $X_m \leq X_n$ for all $m \leq n$, then $X$ is monotone.
If $X_n \leq X_m$ for all $n \geq m$, then $X_n$ is monotone.
If $X_n \leq X_{n+1}$ for all $n$, then $X_n$ is monotone.
If $X(n+1) \leq X(n)$ for all $n$, then $X$ is monotone decreasing.
If $a$ is a monotone sequence, then $-a$ is a monotone sequence.
A function $f$ is strictly monotone if and only if $f(n) < f(n+1)$ for all $n$.
The function $n \mapsto n + k$ is strictly monotone.
Every sequence of real numbers has a monotone subsequence.
If $f$ is a strictly increasing function from the natural numbers to the natural numbers, then $n \leq f(n)$ for all $n$.
If $r$ is a strictly increasing sequence of natural numbers, and $P$ is an eventually true property of the natural numbers, then $P$ is also an eventually true property of $r$.
If a property $P$ does not hold eventually, then there exists a strictly increasing sequence of natural numbers $r$ such that $P$ does not hold for any $r(n)$.
If a property $P$ holds for all sufficiently large $i$, then it also holds for all sufficiently large $i + k$.
If $f$ converges to $l$ as $i \to \infty$, then $f(i - k)$ converges to $l$ as $i \to \infty$.
If $f$ is a strictly monotone function, then $f$ is a filter limit of the sequence $f(n)$.
If $r$ and $s$ are strictly monotone functions, then $r \circ s$ is strictly monotone.

The neighborhood filter of a point $x$ in an ordered topological space is the intersection of the principal filters generated by the sets $\{y \in X : y < x\}$ and $\{y \in X : x < y\}$.

If $f$ converges to $L$ on the set $A \cap \{x \mid P(x)\}$ and $g$ converges to $L$ on the set $A \cap \{x \mid \lnot P(x)\}$, then the function $h$ defined by $h(x) = f(x)$ if $P(x)$ and $h(x) = g(x)$ if $\lnot P(x)$ converges to $L$ on $A$.

If $f$ converges to $L$ along the filter $G$ on the set $\{x \mid P(x)\}$ and $g$ converges to $L$ along the filter $G$ on the set $\{x \mid \lnot P(x)\}$, then the function $h$ defined by $h(x) = f(x)$ if $P(x)$ and $h(x) = g(x)$ if $\lnot P(x)$ converges to $L$ along the filter $G$ on the set $\{x\}$.

In a linearly ordered topological space, the set of points that are close to $x$ is the intersection of the sets of points that are less than $x$ and the sets of points that are greater than $x$.

If $y < x$, then the set of all open sets that contain $x$ is equal to the intersection of all open sets that contain $x$.

If $y < x$, then the predicate $P$ holds eventually at $x$ from the left if and only if there exists $b < x$ such that $P$ holds for all $y > b$ with $y < x$.

If $x < y$, then the set of all open sets containing $x$ is equal to the set of all open sets containing $x$ and not containing $y$.

If $x < y$, then the following are equivalent: $P(y)$ holds eventually as $y$ approaches $x$ from the right. There exists a $b > x$ such that $P(y)$ holds for all $y > x$ with $y < b$.

For any $x$, there exists a sequence of points $y_n$ such that $x < y_n$ and $y_n \to x$ from the right.

The limit at the right of the top element of a linearly ordered topological space is the empty set.

The limit at the left of the bottom element of a linearly ordered topological space is the bottom element.

The limit at the left of any real number is not trivial.

The limit at the right of any real number is not trivial.

In a linearly ordered topological space, the set of points that converge to $x$ is the union of the sets of points that converge to $x$ from the left and from the right.

In a linearly ordered topological space, the set of points at which a property $P$ holds is the intersection of the sets of points at which $P$ holds to the left of $x$ and to the right of $x$.

If $P$ holds for all $x$ in the open interval $(a,b)$, then $P$ holds eventually at $b$ from the left.

If $P$ holds for all $x$ in the open interval $(a,b)$, then $P$ holds eventually at $a$ from the right.

The filtercomap of the nhds filter is the filter of open sets.

The filtercomap of the filter at a point is the filter of the complement of the point.

For any real number $x$, the following are equivalent: $P(y)$ holds for all $y > x$ sufficiently close to $x$; there exists a real number $b > x$ such that $P(y)$ holds for all $y > x$ with $y < b$.

For any real number $x$, the following are equivalent: $P(y)$ holds for all $y$ sufficiently close to $x$ from the left. There exists a real number $b < x$ such that $P(y)$ holds for all $y > b$ with $y < x$.

If $f$ converges to $x$ in $F$, and $x = y$, then $f$ converges to $y$ in $F$.

A function $f$ tends to $l$ with respect to a filter $F$ if and only if for every open set $S$ containing $l$, the set $\{x \in X : f(x) \in S\}$ is in $F$.

If two functions $f$ and $g$ are eventually equal, then they converge to the same limit.

If $F$ is a filter on a set $X$ and $F'$ is a filter on $X$ such that $F \leq F'$, then if $f$ converges to $l$ with respect to $F'$, then $f$ converges to $l$ with respect to $F$.

The identity function tends to $a$ at $a$ within $s$.

The constant function $f(x) = k$ converges to $k$.

The filter $F$ converges to $b$ within $s$ if and only if $F$ eventually contains points $x$ such that $f(x) \in s$ and $f(x) \neq b$, and $F$ converges to $b$.

If $f$ converges to $L$ in the filter of neighbourhoods of $L$, then $f$ converges to $L$ from the left or from the right, depending on whether $f$ is eventually greater than $L$ or eventually less than $L$.

If $f$ converges to $c$ in the filter of neighborhoods of $c$, and $f$ eventually takes values in $A - \{c\}$, then $f$ converges to $c$ in the filter of neighborhoods of $c$ within $A$.

If $f$ converges to $c$ in the neighborhood filter of $c$, and $f$ is eventually not equal to $c$, then $f$ converges to $c$ in the filter at $c$.

If $f$ is a function from a filter $F$ to a topological space $X$ and $f$ converges to $l$ in the sense that for every open set $S$ containing $l$, the set $\{x \in X \mid f(x) \in S\}$ is eventually in $F$, then $f$ converges to $l$ in the sense of filters.

If $f$ converges to $l$ in a filter $F$, and $S$ is an open set containing $l$, then $f$ eventually takes values in $S$.

If $f$ tends to $a$, then $f$ tends to $-\infty$.

If $f$ converges to $l$ eventually, then $f$ converges to $l$.

If $f$ converges to $l$ at $x$ within $S$, then $f$ converges to $l$ at $x$ within $T$ for any $T \subseteq S$.

If $f$ tends to $L$ within $S$, then $f$ tends to $L$ within any subset of $S$.

A function $f$ tends to $x$ in the order topology if and only if for every $l < x$ and every $u > x$, there exists a neighborhood of $x$ such that $l < f(y) < u$ for all $y$ in that neighborhood.

If $f$ is a function from a totally ordered set $X$ to a totally ordered set $Y$, and if for every $y \in Y$ there exists $x \in X$ such that $f(x) < y$, and for every $y \in Y$ there exists $x \in X$ such that $y < f(x)$, then $f$ is continuous.

If $f$ converges to $y$ in the order topology, then $f$ is eventually greater than $a$ if $y$ is greater than $a$, and $f$ is eventually less than $a$ if $y$ is less than $a$.

If $X$ and $Y$ are nets that converge to $x$ and $y$, respectively, then the net $\max(X,Y)$ converges to $\max(x,y)$.

If $X$ and $Y$ are nets that converge to $x$ and $y$, respectively, then the net $\min(X,Y)$ converges to $\min(x,y)$.

In the order topology, the limit points of an interval are the endpoints.

If $a < x < b$, then the filter of neighbourhoods of $x$ within the interval $[a,b]$ is the same as the filter of neighbourhoods of $x$ in the real line.

If $f$ converges to two different limits $a$ and $b$, then $a = b$.

In a T2 space, a sequence converges to a constant if and only if all the terms of the sequence are equal to that constant.

If $F$ is not the empty filter, then there is at most one limit of $f$ with respect to $F$.

If $f$ is a function that converges to $l$ and $f$ eventually takes values in a closed set $S$, then $l \<in> S$.

If $A$ is an open set containing $x$, then there is a closed set $A'$ containing $x$ and contained in $A$ such that $A'$ is in the filter of neighbourhoods of $x$.

If $f$ is an increasing sequence of real numbers that is bounded above, then $f$ converges to its supremum.

If $f$ is a sequence of real numbers such that $f_n \geq l$ for all $n$ and $f_n < x$ for all $n$ whenever $x > l$, then $f$ converges to $l$.

If $f_n \leq g_n \leq h_n$ for all $n$ and $\lim_{n \to \infty} f_n = \lim_{n \to \infty} h_n = c$, then $\lim_{n \to \infty} g_n = c$.

If $f$ converges to $d$ on a set $F$ that is not empty and on which $f$ is frequently equal to $c$, then $d = c$.

If a function $f$ converges to $c$ in a topological space, then for any $c' \neq c$, there exists a neighborhood of $c$ such that $f$ does not take the value $c'$ in that neighborhood.

If $f$ and $g$ are real-valued functions defined on a topological space $X$, and if $f$ and $g$ converge to $x$ and $y$, respectively, then $y \leq x$.

If $f$ is a sequence of real numbers that converges to $x$, and if $a$ is a lower bound for the sequence, then $a \<le> x$.

If $f$ is a sequence of real numbers that converges to $x$, and if $a$ is an upper bound for $f$, then $a \geq x$.

If $f$ converges to $b$ within $s$, then $f$ eventually takes values in $s$ that are not equal to $b$.

If a net has a non-trivial limit, then the limit of the net is equal to the limit of the function.

If the limit of a function $f$ at $x$ within $s$ is not trivial, then the limit of $f$ at $x$ within $s$ is $x$.

If $P$ holds for all points in a neighborhood of $x$ except possibly $x$ itself, then $P$ holds for all points in a neighborhood of $x$.

If $f$ is a monotone function and $g$ is a bijection such that $f \circ g$ is the identity function, then $f$ is continuous at $a$ if and only if $g$ is continuous at $a$.

If $f$ is a monotone function and $g$ is a bijection such that $f \circ g$ is the identity function, then $f$ is a filter limit at infinity.

If $f$ converges to $L$ from the left and from the right at $x$, then $f$ converges to $L$ at $x$.

The limit of a function $f$ at a point $x$ is the same as the limit of $f$ from the left and from the right at $x$.

If $b$ is less than the top element of a linearly ordered topological space, then the following are equivalent: $P$ holds eventually in a neighborhood of the top element. There exists $b < top$ such that $P$ holds for all $z > b$.

The limit of a function $g$ at a point $l$ within a set $S$ is the same as the limit of $g$ at $l$ in the neighborhood of $l$ that is contained in $S$.

The limit of a sequence $X$ is the unique number $L$ such that $X$ converges to $L$.

A sequence $f$ converges to $f_0$ if and only if for every open set $S$ containing $f_0$, there exists an $N$ such that for all $n \geq N$, $f_n \in S$.

A sequence $X$ is increasing if and only if for all $m$ and $n \geq m$, we have $X_n \geq X_m$.

A sequence $X$ is decreasing if and only if for all $m$ and $n \geq m$, we have $X_n \leq X_m$.

If $r$ is a strict monotone function and $m \leq n$, then $r(m) \leq r(n)$.

The identity function is strictly monotonic.

If $X_n \leq X_{n+1}$ for all $n$, then $X_n$ is an increasing sequence.

If $f$ is an increasing sequence, then $f_i \leq f_j$ whenever $i \leq j$.

If $A$ is an increasing sequence, then $A_i \leq A_{i+1}$.

A sequence $f$ is increasing if and only if $f(n) \leq f(n+1)$ for all $n$.

The sequence $(k)_{n \in \mathbb{N}}$ is increasing.

If $X_{n+1} \leq X_n$ for all $n$, then $X_n$ is a decreasing sequence.

If $f$ is a decreasing sequence, then $f(j) \leq f(i)$ for all $i \leq j$.

If $A$ is a decreasing sequence, then $A_{i+1} \leq A_i$.

A sequence $f$ is decreasing if and only if $f(n+1) \leq f(n)$ for all $n$.

The constant sequence $k, k, k, \ldots$ is decreasing.

A sequence is monotonic if and only if it is either increasing or decreasing.

A sequence is monotone if and only if it is either increasing or decreasing.

If $X_m \leq X_n$ for all $m \leq n$, then $X$ is monotone.

If $X_n \leq X_m$ for all $n \geq m$, then $X_n$ is monotone.

If $X_n \leq X_{n+1}$ for all $n$, then $X_n$ is monotone.

If $X(n+1) \leq X(n)$ for all $n$, then $X$ is monotone decreasing.

If $a$ is a monotone sequence, then $-a$ is a monotone sequence.

A function $f$ is strictly monotone if and only if $f(n) < f(n+1)$ for all $n$.

The function $n \mapsto n + k$ is strictly monotone.

Every sequence of real numbers has a monotone subsequence.

If $f$ is a strictly increasing function from the natural numbers to the natural numbers, then $n \leq f(n)$ for all $n$.

If $r$ is a strictly increasing sequence of natural numbers, and $P$ is an eventually true property of the natural numbers, then $P$ is also an eventually true property of $r$.

If a property $P$ does not hold eventually, then there exists a strictly increasing sequence of natural numbers $r$ such that $P$ does not hold for any $r(n)$.

If a property $P$ holds for all sufficiently large $i$, then it also holds for all sufficiently large $i + k$.

If $f$ converges to $l$ as $i \to \infty$, then $f(i - k)$ converges to $l$ as $i \to \infty$.

If $f$ is a strictly monotone function, then $f$ is a filter limit of the sequence $f(n)$.

If $r$ and $s$ are strictly monotone functions, then $r \circ s$ is strictly monotone.