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

Statement: stringlengths 7 24.3k
If $r$ and $s$ are strictly monotone functions, then so is $r \circ s$.
If $X$ is an increasing sequence, then it is a monotone sequence.
If a sequence is decreasing, then it is monotone.
A sequence is decreasing if and only if the sequence of negated terms is increasing.
If $F$ is a decreasing sequence of sets, then $\bigcap_{i=1}^\infty F_i = \bigcap_{i=n}^\infty F_i$ for any $n$.
A sequence converges to a constant if and only if the constant is the limit.
If $X$ is an increasing sequence of real numbers, then $\lim_{n \to \infty} X_n = \sup_{n \in \mathbb{N}} X_n$.
If $X$ is a decreasing sequence of real numbers, then $\lim_{n \to \infty} X_n = \inf_{n \in \mathbb{N}} X_n$.
If $f$ converges to $a$, then the sequence $f(n + k)$ also converges to $a$.
If the sequence $f(n + k)$ converges to $a$, then the sequence $f(n)$ converges to $a$.
If $f$ converges to $l$, then $f(n+1)$ converges to $l$.
If the sequence $(f(n+1))_{n \in \mathbb{N}}$ converges to $l$, then the sequence $(f(n))_{n \in \mathbb{N}}$ converges to $l$.
$\lim_{n \to \infty} f(\{1, 2, \ldots, n\}) = x$ if and only if $\lim_{n \to \infty} f(\{1, 2, \ldots, n-1\}) = x$.
If $X$ is a sequence in a T2 space, then there is at most one limit point of $X$.
If a sequence converges to two different limits, then the two limits are equal.
If $X_n$ converges to $x$ and there exists $N$ such that $a \leq X_n$ for all $n \geq N$, then $a \leq x$.
If $X_n$ and $Y_n$ are sequences of real numbers such that $X_n \leq Y_n$ for all sufficiently large $n$, and if $X_n$ converges to $x$ and $Y_n$ converges to $y$, then $x \leq y$.
If $X_n$ converges to $x$ and there exists $N$ such that for all $n \geq N$, $X_n \leq a$, then $x \leq a$.
If $f$ converges to $l$ and is bounded above by $C$ for all $n \geq M$, then $l \leq C$.
If $f$ is a sequence of real numbers that converges to $l$, and if $f$ is eventually greater than or equal to $C$, then $l$ is greater than or equal to $C$.
If $X_n \leq Y_n$ for all $n \geq N$ and $\lim_{n \to \infty} X_n = x$ and $\lim_{n \to \infty} Y_n = y$, then $x \leq y$.
If $b_n$ is a sequence of elements of a set $s$ that converges to $a$, then $a$ is less than or equal to the supremum of $s$.
If $b_n$ is a sequence in $s$ that converges to $a$, then $a$ is greater than or equal to the infimum of $s$.
If $X$ is an increasing sequence of real numbers that converges to $l$, then $\sup_n X_n = l$.
If $X$ is a decreasing sequence of real numbers that converges to $l$, then $\inf_{n \in \mathbb{N}} X_n = l$.
If a sequence converges, then it converges to some limit.
If a sequence converges, then it is convergent.
A sequence $X$ converges if and only if $X$ converges to its limit.
The constant sequence $c, c, c, \ldots$ converges.
If a sequence is monotone and converges to a limit, then it is either eventually less than or equal to the limit, or eventually greater than or equal to the limit.
If $X_n$ converges to $L$, and $f$ is a strictly increasing function, then $X_{f(n)}$ converges to $L$.
If $X$ is a convergent sequence and $f$ is a strictly increasing function, then $X \circ f$ is a convergent sequence.
If $X$ converges to $L$, then $\lim X = L$.
If $f$ is a convergent sequence and $f(n) \leq x$ for all $n$, then $\lim_{n \to \infty} f(n) \leq x$.
The limit of a constant sequence is the constant.
If $X$ is an increasing sequence of real numbers that converges to $L$, then $X_n \leq L$ for all $n$.
If $X$ is a decreasing sequence that converges to $L$, then $L \leq X_n$ for all $n$.
If $X$ is a first-countable topological space, then for every $x \in X$, there exists a sequence of open sets $A_i$ such that $x \in A_i$ for all $i$ and $A_i \subseteq S$ for all $i$ whenever $S$ is an open set containing $x$.
In a first-countable topological space, every point has a countable neighborhood basis.
If $X$ is a first-countable topological space, then there exists a countable basis for $X$.
If every sequence in $s$ that converges to $a$ eventually satisfies $P$, then $P$ holds in every neighborhood of $a$ that is contained in $s$.
In a first-countable topological space, a property holds eventually in the intersection of a neighborhood of a point and a set if and only if it holds for all sequences converging to the point in the set.
In a first-countable topological space, a property holds in a neighborhood of a point if and only if it holds along every sequence converging to that point.
If $A$ is a nonempty set of real numbers, then there exists a sequence of elements of $A$ that converges to the infimum of $A$.
A function $f$ tends to $a$ at $x$ within $s$ if and only if for every sequence $X$ that converges to $x$ and is contained in $s - \{x\}$, the sequence $f \circ X$ converges to $a$.
If $x < y$, then there exists a sequence $u$ such that $u_n > x$ for all $n$ and $u_n \to x$.
If $x$ is greater than $y$ in a dense linear order, then there exists a sequence $u$ such that $u_n < x$ for all $n$ and $u$ converges to $x$.
If $f$ tends to $l$ at $a$ within $S$, then $f$ converges to $l$ at $a$ in $S$.
If $f$ tends to $l$ at $a$ within $S$, then $f$ tends to $l$ at $a$.
If $k$ is a constant function and $k \neq L$, then $k$ does not converge to $L$.
If a sequence converges to a nonzero constant, then it does not converge to zero.
If a constant function converges to a limit, then the limit is the constant.
If a function converges to two different limits at a point, then the two limits are equal.
If $f$ is a function from a perfect space $X$ to a T2 space $Y$, then $f$ has at most one limit at $a \in X$.
If two functions $f$ and $g$ agree everywhere except at a point $a$, then they have the same limit at $a$.
If $f$ and $g$ are functions that agree on all points except $b$, and $f(b) = g(b) = l$, then $f$ converges to $l$ at $a$ if and only if $g$ converges to $l$ at $b$.
If $f$ converges to $l$ in $F$, and $k = l$, then $f$ converges to $k$ in $F$.
$g$ tends to $g(l)$ at $l$ if and only if $g$ tends to $g(l)$ in a neighborhood of $l$.
If $g$ is continuous at $l$ and $f$ tends to $l$ in $F$, then $g \circ f$ tends to $g(l)$ in $F$.
If $g$ converges to $m$ and $f$ converges to $l$ except possibly at a set of points in $F$, then $g \circ f$ converges to $m$ in $F$.
If $f$ converges to $b$ and $g$ converges to $c$, and $f$ is eventually different from $b$, then $g \circ f$ converges to $c$.
If $f$ is a function from a topological space $X$ to a topological space $Y$, and $g$ is a function from $Y$ to a topological space $Z$, then the composition $g \circ f$ is a function from $X$ to $Z$.
If $f$ tends to $y$ and $g$ tends to $z$ at $y$, then $g \circ f$ tends to $z$ at $y$.
If every sequence in $s$ that converges to $a$ eventually satisfies $P$, then $P$ holds eventually at $a$ within $s$.
If every sequence converging to $a$ eventually satisfies $P$, then $P$ holds in a neighborhood of $a$.
If $f$ converges to $l$ at $a$, then for any sequence $S$ that converges to $a$ and does not converge to $a$, the sequence $f(S)$ converges to $l$.
If $f$ is a function from a first-countable topological space to a topological space, and if for every sequence $S$ that converges to $a$, the sequence $f(S)$ converges to $l$, then $f$ converges to $l$ at $a$.
A sequence $S$ converges to $a$ if and only if the sequence $X(S)$ converges to $L$.
If $b < a$ and for every sequence $f$ such that $b < f(n) < a$ for all $n$ and $f$ is increasing and converges to $a$, then $P(f(n))$ holds for all but finitely many $n$, then $P(x)$ holds for all but finitely many $x < a$.
If $X$ converges to $L$ along any sequence $S$ that converges to $a$ from the left, then $X$ converges to $L$ at $a$ from the left.
If $f$ is a decreasing sequence of real numbers that converges to $a$, then $f$ eventually takes on any value less than $a$.
If $a < b$ and $X$ is a function such that for every sequence $S$ with $a < S_n < b$ and $S_n \to a$, the sequence $X(S_n)$ converges to $L$, then $X$ converges to $L$ at $a$ from the right.
If two sets are equal and two functions are equal on the sets, then the functions are continuous on the sets if and only if the other function is continuous on the other set.
If $s = t$ and $f(x) = g(x)$ for all $x \in t$, then $f$ is continuous on $s$ if and only if $g$ is continuous on $t$.
A function $f$ is continuous on a set $S$ if and only if for every $x \in S$ and every open set $B$, if $f(x) \in B$, then there exists an open set $A$ such that $x \in A$ and $f(y) \in B$ for all $y \in A \cap S$.
A function $f$ is continuous on a set $s$ if and only if for every open set $B$, there exists an open set $A$ such that $A \cap s = f^{-1}(B) \cap s$.
A function $f$ is continuous on an open set $S$ if and only if the preimage of every open set is open.
If $f$ is continuous on $s$, $s$ is open, and $B$ is open, then $f^{-1}(B)$ is open.
If $f$ is a continuous function and $s$ is an open set, then $f^{-1}(s)$ is open.
A function $f$ is continuous on a set $S$ if and only if for every closed set $B$, the preimage $f^{-1}(B)$ is closed in $S$.
A function $f$ is continuous on a closed set $S$ if and only if the preimage of every closed set is closed.
If $f$ is a continuous function from a closed set $t$ to a topological space $X$, and $s$ is a closed subset of $X$, then $f^{-1}(s) \cap t$ is closed in $t$.
If $s$ is a closed set and $f$ is a continuous function, then $f^{-1}(s)$ is a closed set.
The empty set is a closed set.
A function defined on a singleton set is continuous.
If $f$ is continuous on each open set $s \in S$, then $f$ is continuous on $\bigcup S$.
If $f$ is continuous on each open set $A_s$ for $s \in S$, then $f$ is continuous on $\bigcup_{s \in S} A_s$.
If $f$ is continuous on two open sets $s$ and $t$, then $f$ is continuous on the union of $s$ and $t$.
If $f$ is continuous on two closed sets $s$ and $t$, then $f$ is continuous on $s \cup t$.
If $f$ is continuous on each closed set $U_i$ in a finite collection of closed sets, then $f$ is continuous on the union of the $U_i$.
If $f$ and $g$ are continuous functions on closed sets $S$ and $T$, respectively, and if $f$ and $g$ agree on $S \cap T$, then the function $h$ defined by $h(x) = f(x)$ if $x \in S$ and $h(x) = g(x)$ if $x \in T$ is continuous on $S \cup T$.
If $f$ and $g$ are continuous functions on closed sets $S$ and $T$, respectively, and if $f$ and $g$ agree on the intersection of $S$ and $T$, then the function $h$ defined by $h(x) = f(x)$ if $x \in S$ and $h(x) = g(x)$ if $x \in T$ is continuous on $S \cup T$.
The identity function is continuous.
The identity function is continuous on any set.
The constant function $f(x) = c$ is continuous on any set $S$.
If $f$ is continuous on $s$, then $f$ is continuous on any subset $t$ of $s$.
If $f$ is continuous on $s$ and $g$ is continuous on $f(s)$, then $g \circ f$ is continuous on $s$.
If $g$ is continuous on $t$ and $f$ is continuous on $s$, and $f(s) \subseteq t$, then $g \circ f$ is continuous on $s$.
If $f$ is a function from a set $A$ to a topological space $X$ such that for every open set $B$ in $X$, there exists an open set $C$ in $A$ such that $C \cap A = f^{-1}(B) \cap A$, then $f$ is continuous on $A$.
If $f$ is a monotone function defined on a set $A$ and $f(A)$ is open, then $f$ is continuous on $A$.

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

If $X$ is an increasing sequence, then it is a monotone sequence.

If a sequence is decreasing, then it is monotone.

A sequence is decreasing if and only if the sequence of negated terms is increasing.

If $F$ is a decreasing sequence of sets, then $\bigcap_{i=1}^\infty F_i = \bigcap_{i=n}^\infty F_i$ for any $n$.

A sequence converges to a constant if and only if the constant is the limit.

If $X$ is an increasing sequence of real numbers, then $\lim_{n \to \infty} X_n = \sup_{n \in \mathbb{N}} X_n$.

If $X$ is a decreasing sequence of real numbers, then $\lim_{n \to \infty} X_n = \inf_{n \in \mathbb{N}} X_n$.

If $f$ converges to $a$, then the sequence $f(n + k)$ also converges to $a$.

If the sequence $f(n + k)$ converges to $a$, then the sequence $f(n)$ converges to $a$.

If $f$ converges to $l$, then $f(n+1)$ converges to $l$.

If the sequence $(f(n+1))_{n \in \mathbb{N}}$ converges to $l$, then the sequence $(f(n))_{n \in \mathbb{N}}$ converges to $l$.

$\lim_{n \to \infty} f(\{1, 2, \ldots, n\}) = x$ if and only if $\lim_{n \to \infty} f(\{1, 2, \ldots, n-1\}) = x$.

If $X$ is a sequence in a T2 space, then there is at most one limit point of $X$.

If a sequence converges to two different limits, then the two limits are equal.

If $X_n$ converges to $x$ and there exists $N$ such that $a \leq X_n$ for all $n \geq N$, then $a \leq x$.

If $X_n$ and $Y_n$ are sequences of real numbers such that $X_n \leq Y_n$ for all sufficiently large $n$, and if $X_n$ converges to $x$ and $Y_n$ converges to $y$, then $x \leq y$.

If $X_n$ converges to $x$ and there exists $N$ such that for all $n \geq N$, $X_n \leq a$, then $x \leq a$.

If $f$ converges to $l$ and is bounded above by $C$ for all $n \geq M$, then $l \leq C$.

If $f$ is a sequence of real numbers that converges to $l$, and if $f$ is eventually greater than or equal to $C$, then $l$ is greater than or equal to $C$.

If $X_n \leq Y_n$ for all $n \geq N$ and $\lim_{n \to \infty} X_n = x$ and $\lim_{n \to \infty} Y_n = y$, then $x \leq y$.

If $b_n$ is a sequence of elements of a set $s$ that converges to $a$, then $a$ is less than or equal to the supremum of $s$.

If $b_n$ is a sequence in $s$ that converges to $a$, then $a$ is greater than or equal to the infimum of $s$.

If $X$ is an increasing sequence of real numbers that converges to $l$, then $\sup_n X_n = l$.

If $X$ is a decreasing sequence of real numbers that converges to $l$, then $\inf_{n \in \mathbb{N}} X_n = l$.

If a sequence converges, then it converges to some limit.

If a sequence converges, then it is convergent.

A sequence $X$ converges if and only if $X$ converges to its limit.

The constant sequence $c, c, c, \ldots$ converges.

If a sequence is monotone and converges to a limit, then it is either eventually less than or equal to the limit, or eventually greater than or equal to the limit.

If $X_n$ converges to $L$, and $f$ is a strictly increasing function, then $X_{f(n)}$ converges to $L$.

If $X$ is a convergent sequence and $f$ is a strictly increasing function, then $X \circ f$ is a convergent sequence.

If $X$ converges to $L$, then $\lim X = L$.

If $f$ is a convergent sequence and $f(n) \leq x$ for all $n$, then $\lim_{n \to \infty} f(n) \leq x$.

The limit of a constant sequence is the constant.

If $X$ is an increasing sequence of real numbers that converges to $L$, then $X_n \leq L$ for all $n$.

If $X$ is a decreasing sequence that converges to $L$, then $L \leq X_n$ for all $n$.

If $X$ is a first-countable topological space, then for every $x \in X$, there exists a sequence of open sets $A_i$ such that $x \in A_i$ for all $i$ and $A_i \subseteq S$ for all $i$ whenever $S$ is an open set containing $x$.

In a first-countable topological space, every point has a countable neighborhood basis.

If $X$ is a first-countable topological space, then there exists a countable basis for $X$.

If every sequence in $s$ that converges to $a$ eventually satisfies $P$, then $P$ holds in every neighborhood of $a$ that is contained in $s$.

In a first-countable topological space, a property holds eventually in the intersection of a neighborhood of a point and a set if and only if it holds for all sequences converging to the point in the set.

In a first-countable topological space, a property holds in a neighborhood of a point if and only if it holds along every sequence converging to that point.

If $A$ is a nonempty set of real numbers, then there exists a sequence of elements of $A$ that converges to the infimum of $A$.

A function $f$ tends to $a$ at $x$ within $s$ if and only if for every sequence $X$ that converges to $x$ and is contained in $s - \{x\}$, the sequence $f \circ X$ converges to $a$.

If $x < y$, then there exists a sequence $u$ such that $u_n > x$ for all $n$ and $u_n \to x$.

If $x$ is greater than $y$ in a dense linear order, then there exists a sequence $u$ such that $u_n < x$ for all $n$ and $u$ converges to $x$.

If $f$ tends to $l$ at $a$ within $S$, then $f$ converges to $l$ at $a$ in $S$.

If $f$ tends to $l$ at $a$ within $S$, then $f$ tends to $l$ at $a$.

If $k$ is a constant function and $k \neq L$, then $k$ does not converge to $L$.

If a sequence converges to a nonzero constant, then it does not converge to zero.

If a constant function converges to a limit, then the limit is the constant.

If a function converges to two different limits at a point, then the two limits are equal.

If $f$ is a function from a perfect space $X$ to a T2 space $Y$, then $f$ has at most one limit at $a \in X$.

If two functions $f$ and $g$ agree everywhere except at a point $a$, then they have the same limit at $a$.

If $f$ and $g$ are functions that agree on all points except $b$, and $f(b) = g(b) = l$, then $f$ converges to $l$ at $a$ if and only if $g$ converges to $l$ at $b$.

If $f$ converges to $l$ in $F$, and $k = l$, then $f$ converges to $k$ in $F$.

$g$ tends to $g(l)$ at $l$ if and only if $g$ tends to $g(l)$ in a neighborhood of $l$.

If $g$ is continuous at $l$ and $f$ tends to $l$ in $F$, then $g \circ f$ tends to $g(l)$ in $F$.

If $g$ converges to $m$ and $f$ converges to $l$ except possibly at a set of points in $F$, then $g \circ f$ converges to $m$ in $F$.

If $f$ converges to $b$ and $g$ converges to $c$, and $f$ is eventually different from $b$, then $g \circ f$ converges to $c$.

If $f$ is a function from a topological space $X$ to a topological space $Y$, and $g$ is a function from $Y$ to a topological space $Z$, then the composition $g \circ f$ is a function from $X$ to $Z$.

If $f$ tends to $y$ and $g$ tends to $z$ at $y$, then $g \circ f$ tends to $z$ at $y$.

If every sequence in $s$ that converges to $a$ eventually satisfies $P$, then $P$ holds eventually at $a$ within $s$.

If every sequence converging to $a$ eventually satisfies $P$, then $P$ holds in a neighborhood of $a$.

If $f$ converges to $l$ at $a$, then for any sequence $S$ that converges to $a$ and does not converge to $a$, the sequence $f(S)$ converges to $l$.

If $f$ is a function from a first-countable topological space to a topological space, and if for every sequence $S$ that converges to $a$, the sequence $f(S)$ converges to $l$, then $f$ converges to $l$ at $a$.

A sequence $S$ converges to $a$ if and only if the sequence $X(S)$ converges to $L$.

If $b < a$ and for every sequence $f$ such that $b < f(n) < a$ for all $n$ and $f$ is increasing and converges to $a$, then $P(f(n))$ holds for all but finitely many $n$, then $P(x)$ holds for all but finitely many $x < a$.

If $X$ converges to $L$ along any sequence $S$ that converges to $a$ from the left, then $X$ converges to $L$ at $a$ from the left.

If $f$ is a decreasing sequence of real numbers that converges to $a$, then $f$ eventually takes on any value less than $a$.

If $a < b$ and $X$ is a function such that for every sequence $S$ with $a < S_n < b$ and $S_n \to a$, the sequence $X(S_n)$ converges to $L$, then $X$ converges to $L$ at $a$ from the right.

If two sets are equal and two functions are equal on the sets, then the functions are continuous on the sets if and only if the other function is continuous on the other set.

If $s = t$ and $f(x) = g(x)$ for all $x \in t$, then $f$ is continuous on $s$ if and only if $g$ is continuous on $t$.

A function $f$ is continuous on a set $S$ if and only if for every $x \in S$ and every open set $B$, if $f(x) \in B$, then there exists an open set $A$ such that $x \in A$ and $f(y) \in B$ for all $y \in A \cap S$.

A function $f$ is continuous on a set $s$ if and only if for every open set $B$, there exists an open set $A$ such that $A \cap s = f^{-1}(B) \cap s$.

A function $f$ is continuous on an open set $S$ if and only if the preimage of every open set is open.

If $f$ is continuous on $s$, $s$ is open, and $B$ is open, then $f^{-1}(B)$ is open.

If $f$ is a continuous function and $s$ is an open set, then $f^{-1}(s)$ is open.

A function $f$ is continuous on a set $S$ if and only if for every closed set $B$, the preimage $f^{-1}(B)$ is closed in $S$.

A function $f$ is continuous on a closed set $S$ if and only if the preimage of every closed set is closed.

If $f$ is a continuous function from a closed set $t$ to a topological space $X$, and $s$ is a closed subset of $X$, then $f^{-1}(s) \cap t$ is closed in $t$.

If $s$ is a closed set and $f$ is a continuous function, then $f^{-1}(s)$ is a closed set.

The empty set is a closed set.

A function defined on a singleton set is continuous.

If $f$ is continuous on each open set $s \in S$, then $f$ is continuous on $\bigcup S$.

If $f$ is continuous on each open set $A_s$ for $s \in S$, then $f$ is continuous on $\bigcup_{s \in S} A_s$.

If $f$ is continuous on two open sets $s$ and $t$, then $f$ is continuous on the union of $s$ and $t$.

If $f$ is continuous on two closed sets $s$ and $t$, then $f$ is continuous on $s \cup t$.

If $f$ is continuous on each closed set $U_i$ in a finite collection of closed sets, then $f$ is continuous on the union of the $U_i$.

If $f$ and $g$ are continuous functions on closed sets $S$ and $T$, respectively, and if $f$ and $g$ agree on $S \cap T$, then the function $h$ defined by $h(x) = f(x)$ if $x \in S$ and $h(x) = g(x)$ if $x \in T$ is continuous on $S \cup T$.

If $f$ and $g$ are continuous functions on closed sets $S$ and $T$, respectively, and if $f$ and $g$ agree on the intersection of $S$ and $T$, then the function $h$ defined by $h(x) = f(x)$ if $x \in S$ and $h(x) = g(x)$ if $x \in T$ is continuous on $S \cup T$.

The identity function is continuous.

The identity function is continuous on any set.

The constant function $f(x) = c$ is continuous on any set $S$.

If $f$ is continuous on $s$, then $f$ is continuous on any subset $t$ of $s$.

If $f$ is continuous on $s$ and $g$ is continuous on $f(s)$, then $g \circ f$ is continuous on $s$.

If $g$ is continuous on $t$ and $f$ is continuous on $s$, and $f(s) \subseteq t$, then $g \circ f$ is continuous on $s$.

If $f$ is a function from a set $A$ to a topological space $X$ such that for every open set $B$ in $X$, there exists an open set $C$ in $A$ such that $C \cap A = f^{-1}(B) \cap A$, then $f$ is continuous on $A$.

If $f$ is a monotone function defined on a set $A$ and $f(A)$ is open, then $f$ is continuous on $A$.