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The type function "Tf" of any type "t" is a subtype of the top type.
The theorem named "gen_ramanujan_sum_dirichlet_expr" is defined as follows: Let's consider two functions 'f' and 'h' that map natural numbers to complex numbers, and two natural numbers 'n' and 'k'. We define 'g' as a function that multiplies the Moebius function of 'k' with the value of function 'h' at 'k'. We define...
The lemma named "qo_on_predecessor_subset_conv'" assumes that "P" is a quasi-order on "A", and "B" and "C" are subsets of "A". It shows that the set of all elements "x" in "A" for which there exists an element "y" in "B" such that "P x y" holds, is a subset of the set of all elements "x" in "A" for which there exists a...
Lemma T_get_min_bound: Assuming that "ts" satisfies the invariant property, and "ts" is not an empty list, it shows that the T_get_min of "ts" is less than or equal to the logarithm base 2 of (the size of the multiset of trees in "ts" plus 1).
If the set of alternatives is not empty, then the Random Serial Dictatorship Pareto equivalence class of an empty set, given the set of alternatives, a relation R, and again the set of alternatives, holds true.
The lemma "chainTransitionAppend" establishes two conditions for the pi calculus. 1. If process P can transition to process P'' via a silent step, and process P'' can transition to process P' via an alpha step, then process P can transition to process P' via an alpha step. 2. If process P can transition to process ...
The lemma "insert_index_inj_on" shows that the function "insert_index i" is injective on the set S.
If the elements of matrix M are a subset of the set of integers from 0 to m-1, then a column j in the modified matrix (mat_mod M) is non-empty if and only if the same column j in the original matrix M is non-empty.
If set A and set B are countable, then the supremum of the union of set A and set B is equal to the supremum of the supremum of set A and the supremum of set B.
If two lists 'xs' and 'ys' have the same length, and if the result of applying a function 'f' to all elements of 'xs' is the same as applying a function 'g' to all elements of 'ys', then for any index 'i' less than the length of 'xs', applying 'f' to the 'i'-th element of 'xs' will yield the same result as applying 'g'...
The infimum (greatest lower bound) of a set 'x' and its double complement is the bottom element.
If you take a weighted state 'St' and extract its 'wN', 'wP', 'wQ', and 'n' components, and then reassemble them in the same order, you will get back the original weighted state 'St'.
The action of the permutation 'p' on the result of the lookup function with arguments 'xs', 'n', and 'x' is equal to the result of the lookup function with arguments being the action of the permutation 'p' on 'xs', 'n', and 'x' respectively.
The timestamps of a multiset, where each element is a pair consisting of a function 'f' applied to 'x' and 't', and where each pair '(x, t)' belongs to a multiset 'M', are equal to the timestamps of the multiset 'M'.
The analysis of the insertion of a hashed value X into a set H is equal to the insertion of the hashed value X into the analysis of the set H.
The lemma "has_subprob_densityI'" assumes that the function 'f' is Borel measurable on the measure space 'N', the space 'N' is not empty, 'M' is the density of 'f' on 'N', and the integral of 'f' over 'N' is less than or equal to 1. Under these assumptions, it shows that 'M' has a subprobability density on 'N' with res...
If room r is not equal to room r', then the safety of room r in a timeline where guest g enters room r' with card c is the same as the safety of room r in the original timeline.
The "initial_root_problem" is equivalent to both "initial_root_problem_complex" and "initial_root_problem_real".
The lemma named "rotation_minus_pi_2_positive_y_axis" shows that the image of the positive y-axis under a Möbius transformation represented by a rotation of negative pi over 2 equals the positive x-axis.
The lemma named "span_not_univ_orthogonal" is about a set S in a Euclidean space. It assumes that the span of S is not the entire universe. It then shows that there exists a non-zero vector 'a' such that for every vector 'x' in the span of S, the dot product of 'a' and 'x' is zero. This means 'a' is orthogonal to every...
If 'x' is the least element of set 'A' according to order 'L' and 'y' is also the least element of the same set 'A' according to the same order 'L', then 'x' is equivalent to 'y'.
There is a Quotient3 relationship with respect to R1, abs1, and rep1. 2. "b" states that there is a Quotient3 relationship with respect to R2, abs2, and rep2. It shows that if we have a function that maps abs1 to a function that maps abs2 to rep1, and this function is mapped to rep1, and then to a function that maps a...
The sum of a list that replicates a certain element 'x' 'n' times is equal to 'n' times 'x'.
In a successful transition of a graph with nodes "vertex_1" and "vertex_2", and an edge from "vertex_2" to "vertex_1", the successor of "vertex_2" is "vertex_1".
The head omega function of a natural number n is equal to 0 if n is 0, otherwise it is equal to 1.
If there is a reflexive transitive closure path 'r' from 'x' to 'y' through a list 'xs', and another reflexive transitive closure path 'r' from 'y' to 'z' through a list 'ys', then there is a reflexive transitive closure path 'r' from 'x' to 'z' through the concatenation of lists 'xs' and 'ys'.
The zeroth element of the list obtained by dropping the first 'n' elements from the list 'f' is equal to the 'n'th element of the original list 'f'.
The lemma named "closed_consistent" assumes that the free variables of expression 'e' equals an empty set of variables. It shows that the consistency holds for the given configuration (bottom element, 0, empty list) and the triple (empty list, expression 'e', empty list).
The lemma about the components of a hom-functor in a category states the following: 1. The value of an arrow in the hom-functor of a category with a given function is equal to the composition of the second component of the function with an arrow from the hom-set of the codomain of the first component of the function a...
If "x" is an element of the set "xs", then the element at the position in "xs" given by the index of "x" is "x".
"P is equal to the union of P and the Cartesian product of the set containing x and the image of the set containing x under P."
The sum of "d of y" and the box of "ad of y" times "x", all inside the box of "x star", equals to the box of "x star" of "y".
The lemma named "stoch_proc_filt_adapt" assumes that for all natural numbers 'n', 'X n' is a measurable function from the sigma-algebra 'M' to the sigma-algebra 'N'. Under this assumption, it shows that the stochastic process 'X' is adapted to the filtration generated by the stochastic process 'X' and the sigma-algebra...
The sphere with center at 0 and radius 'r' consists of only these two points 'x' and 'y', and the distance between 'x' and 'y' is twice the radius 'r'.
The leading term of the negation of a polynomial "p" is the same as the leading term of the polynomial "p" itself.
The expression "eWls MOD s ERR" is always false.
If the following conditions are met: 1. The invariant I is equal to the invariant I'. 2. The set of possible outputs for the interface \<I> is equal to the set of possible outputs for the interface \<I>'. 3. For any state 's' and output 'x' where 's' satisfies the invariant I' and 'x' is in the set of possible outputs...
For a prime number 'p' in a factorial semiring multiplicative, 'p' raised to the power of 'n' is a prime power if and only if 'n' is greater than zero.
"to_poly" is a ring homomorphism from ring R to ring P.
The special equivalence of the list obtained by mapping a function sigma over a list of elements ns is the same as the special equivalence of the list obtained by mapping a function tau over the same list of elements ns if and only if the special equivalence of sigma and tau is the same as the set of elements ns.
The lemma named "current_Sup" is about a structure called "\<Gamma>". It assumes that "Y" is a non-empty set and forms a chain in a complete partial order with respect to the less than or equal to relation. It also assumes that for every function "f" in "Y", "f" is a current in "\<Gamma>", and that the support flow of ...
If the term of a gterm 's' is equal to 'C' applied to 'l', then applying the gterm of 'l' to the gctxt of 'C' results in 's'.
The negation of the top element is equal to the bottom element.
If K is a simplicial complex, then the boolean function derived from the simplicial complex K is a monotone boolean function. This is under the assumption that K is a set of sets of natural numbers.
There is no reduction sequence in the extended Thread Action semantics of a program P at time t, where the initial state is a list of mapped values followed by a thrown exception and other expressions, and the final state is a different list of expressions and a different state. This is always false.
If for every element 'x' in set 'X', the function 'f' applied to 'x' results in an element in set 'A', then the product of the function 'f' applied to all elements in set 'X' is also an element in set 'A'.
If the most general unifier of two terms 's' and 't' is some substitution 'σ', then the domain of this substitution 'σ' is finite.
If there is a reflexive transitive closure of a reduction from p to q over a set F, then there is also a reflexive transitive closure of a reduction from the monomial multiplication of c, t, and p to the monomial multiplication of c, t, and q over the same set F.
The sorted list of sequences in a mapping prefix tree (MPT) is equal to the following: If the keys of the mapping are empty, then the result is a list containing an empty list. Otherwise, the result is an empty list concatenated with the result of mapping a function over the sorted list of keys of the mapping. This ...
For a given polynomial 'p' and degree 'd', the degree-graded polynomial set of the tail of 'p' is less than or equal to the degree-graded polynomial set of 'p' itself.
The equalizer of functions 'g' and 'f' with respect to an object 'a' is a subset of the object 'a'.
If sets A, B, and C are finite, and the intersection of sets A and B is not empty, and set C is the difference between sets A and B, then the cardinality (or size) of set C is less than the cardinality of set A.
An empty list is progressing.
The composition of the butterfly function with parameters psi1 and psi2 and the butterfly function with parameters psi3 and psi4 is equal to the inner product of psi2 and psi3 multiplied by the butterfly function with parameters psi1 and psi4.
The left nesting of the combination of a list and a function is equal to the sum of the left nesting of the function and the length of the list.
If a memory location "m" is reachable from a reference "r" in a system state "s" where the heap has been modified such that the reference "r'" points to None, then "m" is also reachable from "r" in the original system state "s".
If "a", "b", and "c" are identities, then the associator of "a", "b", and "c" is a morphism from the tensor product of "a" and "b" tensor "c" to the tensor product of "a", "b", and "c".
The lemma named "valid_Act_fresh" assumes that the bound names of action alpha are fresh for process P. It shows that process P satisfies the action alpha of x if and only if there exists a process P' such that P transitions to P' with action alpha and P' satisfies x.
If a sequence 'ss' is in the list of 'S', then the empty set is not in the set of wall crossings that is mapped from the sum of 'ss' to 'C0'.
For any set 'r' of pairs of vertices, 'r' is acyclic if and only if 'r' is well-founded.
The lemma "foldr_cong_plus" is a more complex version of "foldr_cong". If we use f'=id, g'=id and s = set lst, we would get "foldr_cong" exactly. The main assumption is that for any a, b, and c, if b is an element of set s and f' of a equals f of b applied to f' of c, then g' of a equals g of b applied to g' of c. ...
The option relation of the intersection of two relations R1 and R2 is equal to the intersection of the option relations of R1 and R2.
The span of a set X is equivalent to the span with addition, zero, and scaling applied to the set X.
The lemma wtE is true for any element in the set F according to the function wtE.
The lemma named "compatible_awalk_verts" assumes that the graphs G and H are compatible. It shows that the set of vertices visited by an arbitrary walk in graph G is equal to the set of vertices visited by an arbitrary walk in graph H.
If the functor FF' is the digraph morphism of the functor FF, the functor GG' is the digraph morphism of the functor GG, the category AA' is the digraph of the category AA, and the category BB' is the digraph of the category BB, then the transformation digraph morphism of the natural transformation NN maps from FF' to ...
If "n" is greater than zero, then the Jacobi symbol of "a" and "2 to the power of n" is equal to the remainder of "a" divided by 2.
In the context of a ring, for any integer n, if x and y are elements of the ring's carrier set, then the nth power of the product of x and y is equal to the product of the nth power of x and the nth power of y.
If a set 'I' is finite and non-empty, and the cardinality of 'I' is less than or equal to the successor of 'n', then the 'n'-th previous element of 'I' is the minimum element of 'I'.
"(4 times the real number m plus 1) divided by (2 times 2 to the power of n)" is not equal to "(4 times the real number m' plus 3) divided by (2 times 2 to the power of n')".
The function 'f' is injective on the elements of set 'A' if and only if the function 'f' applied to each element 'a' in the set 'A' is a one-to-one function.
If 'x' is in the set of solutions that maximize the linear program defined by matrices A, b, and c, and 'y' is in the set of solutions that minimize the same linear program, then the dot product of 'x' and 'c' is less than or equal to the dot product of 'y' and 'b'.
If H and N are submodules of a module M over a ring R, and if H is a subset of N, and if X and Y are elements in the carrier of the quotient module M/H, then the result of applying the map mQmp (from M/H to M/N) to the coset sum of X and Y in M/H is equal to the coset sum in M/N of the images of X and Y under the map m...
If a pair (s, s') is in a relation R and a property Φ holds for s', then the lifted assertion of R and Φ holds for s.
In the context of an Ordering Set Isomorphism, if P is a simplex-like element in the domain, then the image of P under the function f is a simplex-like element in the codomain.
In the context of a semiring with no zero divisors, if a polynomial is normalized (pnormal), then the length of the polynomial is greater than zero.
If two binary decision diagrams (b1 and b2) are equivalent, then their semantic interpretations (robdd_α b1 and robdd_α b2) are also equivalent.
If the set V is finite, then the set of all edges in V is also finite.
The lemma assumes that 'u' is greater than or equal to 0 and less than or equal to 1. It shows two things: 1. The absolute value of the sine of the product of 'u' and a complex number 'z' is less than or equal to the exponential of the absolute value of the imaginary part of 'z'. 2. The absolute value of the cosine...
The finite set created from a list "l", minus the singleton set containing "x", is equal to the finite set created from the list "l" after filtering out the element "x".
A list 'x' is not less than an empty list.
If the following conditions are met: 1. In the context of program P, expression e reduces to the address a in the state transition from s to s' with heap b changing to b'. 2. In the heap of state s', the address a points to an object of class C with fields fs. 3. The field F of class D in the fields fs has the value v...
If x is greater than 0, b is greater than the successor of 0, and x is less than b raised to the power y, then the ordinal logarithm of x with base b is less than y.
If 'a' is an element of the topological space 'X', then the reduced homology group of 'X' at index 'p' is isomorphic to the relative homology group of 'X' at index 'p' with respect to the singleton set containing 'a'.
The supremum of x and y is the maximum of x and y.
: Assuming "x" is an element of the carrier "L", and for any "u" such that "u" is an element of the carrier "L" and "f(u)" is less than or equal to "u", "x" is less than or equal to "u". It then shows that "x" is less than or equal to the least fixed point of "f".
The epsilon transition function of a tree automaton is a subset of the Cartesian product of the set of states of the automaton with itself.
For any two elements x and y of a tree structure, where the elements belong to a unique Euclidean semiring, the modulus of the sum of x and y with respect to y is equal to the modulus of x with respect to y.
A stream cylinder "S" with elements "xs" is a subset of the stream space "S".
If F is an element of the chamber complex of A, then the function fold_A fixes F pointwise when applied to the image of F under fold_A.
The set of all streams (infinite sequences) over the intersection of a set S is equal to the intersection of the set of all streams over each element in S.
The intersection of set 'd' and the inverse of set 's' is an empty set.
The lemma named "memory_read_privilege" assumes that the state 's' is the second element of the memory read operation at a specific address 'addr' and 'asi', and that the 'S' field of the 'PSR' CPU register in state 's' is zero. Under these assumptions, it shows that the 'S' field of the 'PSR' CPU register in the new s...
The lemma named "extract_comm_real_diag_decomp" is about a set of complex matrices, denoted as CjA. It assumes that for any subset of complex matrices, denoted as Af, and any natural number n, if Af is finite, the cardinality of Af is less than or equal to i, Af is not an empty set, every matrix in Af is an n by n matr...
The lemma "parabolic_imp_not_elliptic_loxodromic" assumes that "f" is a parabolic isometry. It shows that "f" is neither an elliptic isometry nor a loxodromic isometry.
A formal power series 'a' is equal to the product of (1 - X) and the absolute value of a formal power series, where the latter is defined as the sum of the series 'a' from 0 to 'n' for each 'n'. Here, 'a' is a formal power series in a ring with unity.
If all elements of list 'p' are in list 'l', then all elements of the list resulting from the application of the function 'removeShadowRules2' on 'p' are also in list 'l'.
The third step in protocol m3 refines the third step in protocol m2. This is under the condition that the relation R23 intersects with the combination of the invariant m2_inv3a_sesK_compr from protocol m2 and the intersection of the invariants m3_inv3_sesK_compr and m3_inv1_lkeysec from protocol m3. The result of this ...
If function 'f' and function 'g' are both bounded functions, then the function that maps 'x' to the difference of 'f(x)' and 'g(x)' is also a bounded function. This is applicable for any functions 'f' and 'g' that map from any type 'a' to any type 'b'.
If the sets F satisfy the properties Fr_1a, Fr_2, Fr_3, and Fr_4, then for all elements a, b, and c, the implication that 'a implies c' implies that 'a implies (c or b)' is equivalent to the top element (usually representing the universal set or truth in a Boolean algebra).