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If $M_i$ is a chain of measures on a common measurable space, then the supremum of the $M_i$ is the measure that assigns to each measurable set $X$ the supremum of the $M_i(X)$. |
The space of a supremum of a family of topological spaces is the union of the spaces of the family. |
If $M$ is a nonempty set of $\sigma$-algebras on $X$, then the $\sigma$-algebra generated by the union of the $\sigma$-algebras in $M$ is equal to the $\sigma$-algebra generated by the $\sigma$-algebra generated by the union of the $\sigma$-algebras in $M$. |
If $m$ is a measure on $X$ and $M$ is a set of measures on $X$, then $A \in \mathcal{M}$ if and only if $A \in \mathcal{M}_m$ for all $m \in M$. |
If $k(A_i) = k(B_i)$ for all $i$, and $R(c(A_i))$ and $R(c(B_i))$ for all $i$ such that $k(B_i) = \sup_i k(B_i)$, and $R(s(A_i))$ and $R(s(B_i))$ for all $i$, then $R(\sup_i A_i)$ and $R(\sup_i B_i)$. |
If two families of measures agree on the sets they measure, then the supremum of the measures in the two families agree on the sets they measure. |
If $M$ is a nonempty set of $\sigma$-algebras on the same space $X$, then the $\sigma$-algebra generated by the union of the $\sigma$-algebras in $M$ is contained in the $\sigma$-algebra generated by $X$. |
If $f$ is measurable with respect to a measure $m$ and $m$ is a member of a set of measures $M$ such that all measures in $M$ have the same space, then $f$ is measurable with respect to the supremum of $M$. |
If $f$ is measurable with respect to each measure space in a nonempty set $M$ of measure spaces, and if all the measure spaces in $M$ have the same underlying set, then $f$ is measurable with respect to the supremum measure space. |
If $f$ is measurable with respect to each of the measures $M_i$, then $f$ is measurable with respect to the supremum of the measures $M_i$. |
If $M$ is a nonempty set of subsets of $\Omega$, then the $\sigma$-algebra generated by the union of $M$ is the same as the $\sigma$-algebra generated by the union of the $\sigma$-algebras generated by the elements of $M$. |
If $M$ is a nonempty set of subsets of $\Omega$, then $\sigma(\bigcup M) = \sigma(\bigcup_{m \in M} m)$. |
If $M$ is a nonempty set and $f$ is a function from $M$ to the power set of $\Omega$, then $\sigma(\bigcup_{m \in M} f(m)) = \sigma(\bigcup_{m \in M} \sigma(\Omega, f(m)))$. |
If $f$ is a function from $X$ to $Y$ and $M$ is a set of measures on $Y$, then the sets of the measure algebra of the supremum of $M$ under $f$ are the same as the sets of the supremum of the measure algebras of the measures in $M$ under $f$. |
The sets of a restricted space are the same as the sets of the vimage algebra of the intersection of the space and the set. |
If $A$ is a subset of the power set of $X$, then the $\sigma$-algebra generated by $X$ and $A$ is a subset of the $\sigma$-algebra $N$ if and only if $X$ is in $N$ and $A$ is a subset of $N$. |
A function $f$ is measurable if and only if it is a function from the space of $M$ to the space of $N$ and the preimage of every measurable set in $N$ is measurable in $M$. |
The set $X$ is in the $\sigma$-algebra generated by the image of $f$. |
If $N'$ is a sub-$\sigma$-algebra of $N$ and $M'$ is a super-$\sigma$-algebra of $M$, then any $N$-measurable $M$-measurable set is $N'$-measurable $M'$-measurable. |
If $f$ is a function from a measurable space $N$ to a set $A$, then $f$ is measurable with respect to the smallest $\sigma$-algebra on $A$ that makes $f$ measurable. |
If $a \subseteq M$ and $\Omega' \in M$, then $\sigma(a) \subseteq M$. |
If $X$ is a measurable set in the measure space $(Y, \mathcal{M}_i)$, then $X$ is a measurable set in the measure space $(Y, \mathcal{M})$, where $\mathcal{M}$ is the supremum of the $\sigma$-algebras $\mathcal{M}_i$. |
If $f$ is measurable with respect to each $M_i$, then $f$ is measurable with respect to $\sup_{i \in I} M_i$. |
If $X$ is a measurable set and $f$ is a measurable function, then the image of $X$ under $f$ is measurable. |
If $M$ is a sub-$\sigma$-algebra of $N$, then the image of $M$ under $f$ is a sub-$\sigma$-algebra of the image of $N$ under $f$. |
If $M$ is a sub-$\sigma$-algebra of $N$, then the restriction of $M$ to $X$ is a sub-$\sigma$-algebra of the restriction of $N$ to $X$. |
The sets of a measure space $M$ are empty if and only if $M$ is the trivial measure space. |
The only $\sigma$-algebra on a set $X$ that contains only the empty set is the trivial $\sigma$-algebra. |
If $M$ is a finite measure, then the set of points $x$ such that $M(\{x\}) \neq 0$ is countable. |
If $x$ is an integer, then $x^n$ is an $n$th power. |
If $x = y^n$, then $x$ is an $n$th power. |
If $x$ is an $n$th power, then there exists $y$ such that $x = y^n$. |
A number $x$ is a zeroth power if and only if $x = 1$. |
$1$ is the first power of any number. |
The first power of any number is itself. |
If $n$ is a positive integer, then $0$ is an $n$th power. |
A number is an $n$th power if and only if it is zero and $n > 0$. |
$1$ is an $n$th power. |
The number $1$ is an $n$th power. |
If $n$ is a positive integer, then $x$ is an $n$th power if and only if $n$ divides the multiplicity of $p$ in $x$ for every prime $p$. |
If $n$ is a positive integer, then $x$ is an $n$th power if and only if $n$ divides the multiplicity of $p$ in $x$ for every prime $p$. |
If $a$ and $b$ are $n$th powers, then $a \cdot b$ is an $n$th power. |
If $a$ and $b$ are coprime and $a b$ is an $n$th power, then $a$ and $b$ are $n$th powers. |
If $a$ and $b$ are coprime, then $a$ is an $n$th power if and only if $b$ is an $n$th power. |
If $p$ is a prime number, then $p^k$ is an $n$th power if and only if $n$ divides $k$. |
If $n$ divides $n'$, then $m^{n'}$ is an $n$th power. |
The function is_nth_power_nat is defined by the following code: |
The $n$th root of $0$ is $0$. |
If $k > 0$, then the set of all $m \in \mathbb{N}$ such that $m^k \leq n$ is a subset of the set $\{0, 1, \ldots, n\}$. |
If $k > 0$, then the set of all $m \in \mathbb{N}$ such that $m^k \leq n$ is finite and non-empty. |
If $k > 0$ and $n \geq 1$, then $k$-th root of $n$ is the largest integer $x$ such that $x^k \leq n$. |
If $k > 0$ and $x^k > n$, then the $k$th root of $n$ is less than $x$. |
If $m^k \leq n < (m+1)^k$, then $n^{1/k} = m$. |
The $k$th root of $0$ is $0$. |
If $k > 0$, then the $k$th root of $1$ is $1$. |
If $k > 0$, then the $k$th root of $1$ is $1$. |
The first root of $n$ is $n$. |
The first root of a natural number $n$ is $n$. |
The $k$th root of $n$ is the largest integer $m$ such that $m^k \leq n$. |
If $k^m \leq n$, then $n^{1/m} \leq n$. |
If $m > 0$, then $(\text{nth\_root\_nat\_aux}(m, k, k^m, n) + 1)^m > n$. |
If $k^m \leq n$ and $m > 0$, then the function $nth\_root\_nat\_aux$ returns the same value as $nth\_root\_nat$. |
The naive code equation for the nth root of a natural number. |
If $k$ is a positive integer, then the $k$th root of $n^k$ is $n$. |
If $k$ is a positive integer that divides $m$, then the $k$th root of $n^m$ is $n^{m/k}$. |
If $m \leq n$, then $\sqrt[k]{m} \leq \sqrt[k]{n}$. |
The list $0$ followed by the empty list is the empty list. |
The list constructor $x \# y$ is the same as the list constructor $x # y$. |
The concatenation of a list with a singleton list is the same as the concatenation of the list with the singleton list. |
If $x \neq 0$, then $x ## xs = x # xs$. |
If $x$ is not $0$, then the result of stripping $x$ from the list $x :: xs$ is $x$ followed by the result of stripping $x$ from $xs$. |
The tail of a list with a single element is the empty list. |
Two polynomials are equal if and only if they have the same coefficients. |
If two polynomials have the same coefficients, then they are equal. |
All but finitely many coefficients of a polynomial are zero. |
If the degree of a polynomial $p$ is less than $n$, then the coefficient of $x^n$ in $p$ is zero. |
If the coefficient of $x^n$ in a polynomial $p$ is nonzero, then $n$ is less than or equal to the degree of $p$. |
If the coefficients of a polynomial $p$ are zero for all indices greater than $n$, then the degree of $p$ is at most $n$. |
If $n$ is less than the degree of a polynomial $p$, then there exists an index $i > n$ such that the coefficient of $x^i$ in $p$ is nonzero. |
The coefficient of $x^n$ in the polynomial $0$ is $0$. |
The degree of the zero polynomial is zero. |
If a polynomial is nonzero, then its leading coefficient is nonzero. |
The leading coefficient of a polynomial is zero if and only if the polynomial is zero. |
If the coefficient of the $n$th power of a polynomial $p$ is zero, then either $p$ is the zero polynomial or the degree of $p$ is less than $n$. |
If the coefficient of $x^d$ in a polynomial $p$ is zero, and $d$ is at least the degree of $p$, then the degree of $p$ is at most $d-1$. |
The coefficient of a polynomial is equal to the coefficient of its representation. |
The coefficient of the constant term of a polynomial is the constant term. |
The coefficient of $x^n$ in $a + x p$ is the coefficient of $x^{n-1}$ in $p$. |
The degree of a polynomial with a leading coefficient is at most one more than the degree of the polynomial without the leading coefficient. |
If $p$ is a nonzero polynomial, then the degree of $p$ is one more than the degree of $p$ with a leading coefficient of $a$. |
The degree of a polynomial of the form $a + 0x$ is $0$. |
The degree of a polynomial with a nonzero constant term is one more than the degree of the polynomial without the constant term. |
The polynomial $0x^0$ is equal to $0$. |
Two polynomials are equal if and only if their leading coefficients are equal and their tails are equal. |
A polynomial is zero if and only if its leading coefficient is zero and its tail is zero. |
Every polynomial can be written as $a + bx + cx^2 + \cdots$, where $a, b, c, \ldots$ are coefficients. |
If $P(0)$ and $P(p)$ implies $P(a + xp)$ for all $a$ and $p$, then $P(p)$ for all $p$. |
If the degree of a polynomial is zero, then it is a constant polynomial. |
The polynomial whose coefficients are all zero is the zero polynomial. |
A polynomial is zero if and only if all of its coefficients are zero. |
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