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If $p$ is a polynomial over a Euclidean domain $R$, then $p$ has a unique factorization into irreducible polynomials. |
If $p$ is a prime number, then the polynomial $x^p - x$ is irreducible over $\mathbb{Z}_p$. |
The Euclidean algorithm can be used to compute the quotient and remainder of a division. |
If $x$ and $y$ are polynomials and $x = qy + r$ for some polynomials $q$ and $r$, then $x \div y = q$. |
If $x$ and $y$ are polynomials and $(q, r)$ is the euclidean division of $x$ by $y$, then $x \bmod y = r$. |
If $p$ and $q$ are polynomials, then $p(x) = a + xq(x) + r(x)$ where $r$ is the remainder of $p$ divided by $q$. |
The remainder of a polynomial $p$ divided by a polynomial $q$ is equal to $p$ minus the product of the leading coefficient of $q$ and the quotient of $p$ divided by $q$. |
The quotient and remainder of two polynomials $p$ and $q$ can be computed by a folding operation on the coefficients of $p$. |
If $y \neq 0$, then either $x \equiv 0 \pmod y$ or $\deg(x \bmod y) < \deg(y)$. |
If $b \neq 0$ and $a \bmod b \neq 0$, then the degree of $a \bmod b$ is less than the degree of $b$. |
If the degree of $x$ is less than the degree of $y$, then $x$ divided by $y$ is zero. |
If the degree of $x$ is less than the degree of $y$, then $x \bmod y = x$. |
If $x$ and $y$ are polynomials such that $x = qy + r$ for some polynomials $q$ and $r$, then $ax = aqy + ar$. |
For any polynomial $x$ and $y$, and any scalar $a$, we have $(ax) \div y = a(x \div y)$. |
If $a$ is a scalar and $x$ and $y$ are vectors, then $(ax) \bmod y = a(x \bmod y)$. |
For any polynomials $x$ and $y$, $-x \div y = -(x \div y)$. |
For any polynomials $x$ and $y$, $-x \mod y = -(x \mod y)$. |
If $x$ and $x'$ are both related to $y$ by the Euclidean relation, then $x + x'$ is related to $y$ by the Euclidean relation. |
For any polynomials $x$, $y$, and $z$, $(x + y) \div z = x \div z + y \div z$. |
For polynomials $x$, $y$, and $z$, $(x + y) \mod z = (x \mod z) + (y \mod z)$. |
For any polynomials $x$, $y$, and $z$, we have $(x - y) \div z = x \div z - y \div z$. |
For polynomials $x, y, z$, we have $(x - y) \mod z = x \mod z - y \mod z$. |
If $a \neq 0$, then the Euclidean relation between $x$ and $y$ is the same as the Euclidean relation between $x$ and $a y$. |
If $a \neq 0$, then $\frac{x}{ay} = \frac{1}{a} \frac{x}{y}$ for polynomials $x$ and $y$. |
If $a \neq 0$, then $x \bmod ay = x \bmod y$. |
For any polynomial $x$ and any nonzero polynomial $y$, $x \div (-y) = -(x \div y)$. |
The remainder of a polynomial $x$ divided by $-y$ is the same as the remainder of $x$ divided by $y$. |
If $x = qy + r$ and $q = q'z + r'$, then $x = q'yz + yr' + r$. |
For polynomials $x, y, z$, we have $x/(yz) = (x/y)/z$. |
For polynomials $x$, $y$, and $z$, $x \mod (y \cdot z) = y \cdot (x \div y \mod z) + x \mod y$. |
If $y$ is a nonzero polynomial, then the remainder of $x$ divided by $y$ is $x \bmod y = x - b y$, where $b$ is the coefficient of the leading term of $x$ divided by the coefficient of the leading term of $y$. |
Subtracting a list of zeros from a list does nothing. |
The length of the list of coefficients of the difference of two polynomials is the same as the length of the list of coefficients of the first polynomial. |
If $a = 0$, then $x$ is equal to $x$ minus the list $y$ multiplied by $a$. |
The polynomial corresponding to the concatenation of two lists is the sum of the polynomials corresponding to the two lists, with the second multiplied by $x^{n}$, where $n$ is the length of the first list. |
If $p$ is a polynomial of degree at least $n$, then $p - x^n q$ is equal to the polynomial whose coefficients are the coefficients of $p$ minus the coefficients of $q$ (with zeros added to the end of $q$ if necessary). |
If $f$ is a polynomial, then $a \cdot (bx^n \cdot f) = (abx^n) \cdot f$. |
If $d$ is a non-empty list of coefficients of a polynomial $p$ and $r$ is a list of coefficients of a polynomial $q$ such that $p$ divides $q$, then the remainder of $q$ divided by $p$ has a zero leading coefficient. |
If $p$ is a polynomial, then $a \mapsto ap$ is a polynomial. |
The last coefficient of a polynomial is the first element of the reversed list of coefficients. |
If $d$ is a non-empty list of coefficients of a polynomial $D$ such that $D$ has a non-zero leading coefficient $l$, and $q$ is a list of coefficients of a polynomial $Q$, and $r$ is a list of coefficients of a polynomial $R$, and $n$ is a natural number, and $q'$ and $r'$ are lists of coefficients of polynomials $Q'$ ... |
The pseudo-division of two polynomials is the same as the pseudo-division of their coefficients. |
The second component of the result of pseudo_divmod_main_list is equal to the result of pseudo_mod_main_list. |
The pseudo remainder of two polynomials is the polynomial whose coefficients are the pseudo remainder of the coefficients of the two polynomials. |
The Euclidean relation between two polynomials $p$ and $q$ is equivalent to the equality of the quotient and remainder of $p$ divided by $q$. |
The quotient and remainder of the division of two polynomials $f$ and $g$ can be computed by dividing $f$ by $g$ using the pseudo-division algorithm. |
The pseudo-division algorithm for polynomials is equivalent to the division algorithm for polynomials. |
The pseudo-division algorithm for polynomials with coefficients in $\mathbb{Z}$ is equivalent to the division algorithm for polynomials with coefficients in $\mathbb{Z}/n\mathbb{Z}$. |
If $r$ is the remainder of the division of a polynomial $p$ by a polynomial $q$, then the remainder of the division of $p$ by $q^2$ is the remainder of the division of $r$ by $q$. |
The function pdivmod_via_divmod_list is defined as the function pdivmod_via_pseudo_divmod_list, where the latter is unfolded using the definition of pseudo_divmod_main_list_1. |
The second component of the result of divmod_poly_one_main_list is equal to the result of mod_poly_one_main_list. |
The function mod_poly computes the remainder of the polynomial division of $f$ by $g$. |
The division of two polynomials is implemented by the function div_field_poly_impl. |
If $d$ is a non-empty list of coefficients of a polynomial $p$ and $lc$ is the last coefficient of $p$, then the list of coefficients of $p$ divided by $q$ is equal to the list of coefficients of $p$ divided by $q$ computed by the function divide_poly_main_list. |
The function $f \mapsto f/g$ is equal to the function $f \mapsto \text{divide\_poly\_list}(f,g)$. |
The product of a multiset of polynomials is equal to the polynomial whose coefficient is the product of the coefficients of the polynomials in the multiset. |
A constant polynomial is irreducible if and only if the constant is irreducible. |
If $c$ is a prime element, then $[:c:]$ is a prime element. |
A constant polynomial is a prime element if and only if its constant is a prime element. |
The content of a polynomial is the greatest common divisor of its coefficients. |
The content of the empty set is zero. |
The content of the unit interval is 1. |
The content of a constant polynomial is the normalization of the constant. |
A constant polynomial $c$ divides a polynomial $p$ if and only if $c$ divides the content of $p$. |
The content of a polynomial divides the polynomial. |
The content of a polynomial divides its coefficient. |
The content of a polynomial divides every coefficient of the polynomial. |
The content of a polynomial is always a unit. |
The content of a polynomial is a unit if and only if it is equal to 1. |
The content of a polynomial multiplied by a constant is the constant times the content of the polynomial. |
The content of a polynomial is zero if and only if the polynomial is zero. |
The primitive part of $0$ is $0$. |
The product of the content of a polynomial and its primitive part is the polynomial itself. |
The primitive part of a polynomial is zero if and only if the polynomial is zero. |
If $p$ is a nonzero polynomial, then the content of its primitive part is $1$. |
Every polynomial $p$ can be written as $p = c \cdot p'$, where $c$ is the content of $p$ and $p'$ has content $1$. |
If $p$ divides $q$, then the content of $p$ divides the content of $q$. |
The primitive part of a constant polynomial is the constant polynomial with the same leading coefficient. |
If the content of a polynomial is 1, then the primitive part of the polynomial is the polynomial itself. |
The degree of the primitive part of a polynomial is the same as the degree of the polynomial. |
The product of the content of a polynomial and the normalized primitive part of the polynomial is the normalized polynomial. |
$\text{to\_fract}(0) = 0$. |
The fractional part of $1$ is $1$. |
The fractional part of the sum of two real numbers is the sum of the fractional parts of the two real numbers. |
The difference of two fractions is equal to the difference of their numerators divided by the product of their denominators. |
The negation of a fraction is the negation of the numerator divided by the denominator. |
The fractional part of a product is the product of the fractional parts. |
The function to_fract is injective. |
The fractional part of a real number is zero if and only if the real number is zero. |
The denominator of a fraction is never zero. |
The fractional part of a fraction is the fraction itself. |
If the denominator of the rational number $x$ is $1$, then the numerator of $x$ is $x$ itself. |
If $y$ divides $x$, then the denominator of the fraction $x/y$ is $1$. |
The fraction $a/b$ is equal to the fraction $to\_fract(a)/to\_fract(b)$. |
The quotient of a fraction is equal to the fraction itself. |
The first component of the quotient of a fraction is zero if and only if the fraction is zero. |
The denominator of the fractional representation of a rational number is 1. |
The numerator and denominator of a fraction are coprime. |
The unit factor of the denominator of a fraction is 1. |
If the unit factor of $x$ is 1, then $x$ is already normalized. |
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