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The lemma Exe1E[meta_elim] assumes that the pair (F, x) is in v. It shows that there exist r and o₁ such that Some r equals the first derivative of F and Some o₁ equals the kappa derivative of x, and o₁ is in the existential quantifier of r in v.
If the abscissa of convergence of a Dirichlet series 'f' and another Dirichlet series 'g' are both less than or equal to 'd', then the abscissa of convergence of the difference between 'f' and 'g' is also less than or equal to 'd'.
The lemma named "prime_factors_prod" assumes that "A" is a finite set and "0" is not in the image of the function "f" over the set "A". It shows that the prime factors of the product of the function "f" over the set "A" are equal to the union of the prime factors of the function "f" over the set "A".
The translation length of function f is greater than or equal to zero.
The bound of a problem plan for "PROB" is equal to the supremum (or greatest element) of the set of all infimum (or least elements) of the plan length sequence "PLS" for each state-action pair "(s, as)", where "s" is a valid state in "PROB" and "as" is a valid plan in "PROB".
For a real non-negative matrix A with a Jordan normal form represented by n_as, if there exists a pair (m, lam) in the set n_as such that the absolute value of lam equals the spectral radius of A, then there exists a number M that is greater than or equal to m, such that the pair (M, spectral radius of A) is in the set...
The first 'n' elements of vector 'v' belong to the 'n'-dimensional vector space.
If a set M is equal to the sum of sets A and B, and A and B are additively independent, then M is equal to the direct sum of A and B.
If you convert a 64-bit word to an integer and then convert it back to a 64-bit word, you will get the original 64-bit word.
"some best" is included in "best".
"isChair_isPC" is an invariant under all possible updates.
The lemma "accept_GR_LTS" assumes that the range of w is a subset of Σ. It shows that the acceptance condition for a generalized Rabin automaton (GR) with transition function δ, initial state q0, and acceptance condition α on input w is equivalent to the acceptance condition for a generalized Rabin automaton with a tra...
The upper bound of the cosine of a floating point interval is equal to the second element of the cosine bounds, given a precision and the lower and upper bounds of the interval.
The theorem named "parallel_correct" assumes the following conditions: 1. The first elements of the tuples in the list "ts" are distinct. 2. For every tuple in the list "ts", the first element of the tuple, which is also a tuple, has its first element less than "n". 3. "n" is greater than 0, indicating there is at lea...
The function "find_fund_sol" for a given D equals to (0, 0) if D is a square, otherwise it equals to the fundamental solution of D in the context of Pell's equation.
If policy 'p' is not an empty list, then the function 'Cp' applied to the firewall policy obtained from the separated list 'p' and 'a' is equal to the function 'Cp' applied to the firewall policy obtained from the list 'p' and 'a'.
If "x" is less than or equal to "y", then the star operation applied to "x" and the result of the star operation applied to "y" and "z" is equal to the star operation applied to "y" and "z".
The lemma named "some_the_fst_netgmap" assumes that "i" is an element of the network IP addresses of "s". It shows that the first element of the network map of "sr" and "s" at "i", when it definitely exists, is equal to the first element of the network map of "sr" and "s" at "i".
A function, defined as raising x to the power of n, is field differentiable at any point x.
The function "poly_pfd_simple" applied to a variable "x" and a list "cs" is defined as follows: If the list "cs" is empty, then the result is a pair consisting of "x" and an empty list. Otherwise, a list "zs" is created by zipping the list "cs" with the result of decomposing "x" using a list created by mapping a fu...
The integer representation of the natural number 1 is 1.
The lemma "INFM_sort_of_eq" shows that there is an infinite number of elements 'a' such that the sort of 'a' is equal to 's'.
: Assuming the following: - For any natural number, predicate P1 holds for the variable of that natural number. - For any two expressions, if predicate P1 holds for both expressions, then predicate P1 also holds for the sum of these two expressions. - For any natural number and list, if predicate P2 holds for the list,...
If "indexlt k' u' k u" holds true, then the set "\<P> k' (Suc u')" is a subset of the set "\<P> k u".
The division of 10 by 3 equals 3 in the natural numbers.
If the following conditions are met: 1. The tuple (n, i, is) is equal to ni. 2. i is greater than or equal to 2. 3. n is greater than or equal to 2. 4. For any number j, if j is greater than or equal to 2 and less than i, then j does not divide n. 5. For any number j, if j is greater than or equal to i and less than j...
If both x and y are non-negative real numbers and their product equals 1, then the product of x, y, and the sum of x and y is less than or equal to the sum of the squares of x and y.
: Assuming the following conditions: 1. The cardinality of set I is w. 2. w is greater than 0. 3. For any number n in set K', n is greater than or equal to 2. 4. For any set B in the multiset B, the K_GDD (generalized difference digraph) of the Cartesian product of B and I, and the function f of B, is K' and the set o...
The result of adding a variable 'x' to itself, then subtracting 'x' twice, 'y', and 'z', is equal to zero minus 'y' minus 'z' for any integer.
The lemma "lunit_tensor'" assumes that "a" and "b" are identities. It shows that the left unit of the tensor product of "a" and "b", when composed with the associator of the unit object, "a", and "b", is equal to the tensor product of the left unit of "a" and "b".
Theorem TBtheorem3a: Assuming that neither P nor Q are in E, that the subcomponents of PQ are P and Q, and that PQ is correctly composed, it can be shown that PQ is not in E.
If the intersection of t and u is not null, then t is concatenated with u.
If the following conditions are met: 1. A cryptographic message containing the agent A and a number T3, encrypted with a server key (servK), is part of what the spies have seen (evs). 2. A message from the ticket granting server (Tgs) to agent A, containing the server key (servK), agent B, a number Ts, and a server ti...
Inserting the identity function into the set of actions of a system F does not change the set of actions of F.
Shifting the frequency domain series "a" by 1 results in 1.
The size of the lifted profile of a set A is equal to l times the size of set A.
For any permutation 'pi', term 't', variable 'X', and type 'T', applying the permutation 'pi' to the term 't' where 'X' is substituted by 'T', is equivalent to substituting the permuted 'X' by the permuted 'T' in the permuted term 't'.
The lemma named "trace_eq_intercept" is defined with two sub-definitions. The first sub-definition, "outs_adv", is defined as the union of six universal sets. The second sub-definition, "outs_usr", is defined as the union of four universal sets. The lemma assumes that the function "sample_triple" is lossless, meaning...
The lemma named "transcendental_Arccos" assumes that "z" is algebraic and "z" is not equal to 1. It shows that the arccosine of "z" is not algebraic.
For any block number and index i, given the following conditions: 1. The next free block invariant holds for the file cfile1. 2. The file cfile1, when its data is written to the CArray at the index obtained by dividing the byte index by the block size, and then the CArray at this index is read and written to at the in...
If the double negation introduction rule applies to a statement, and the double negation elimination rule also applies to the same statement, then the minimal logic rule applies to that statement.
If a, b, and c are elements in the ring R, and both b and c are not zero, then the result of dividing a by b and then dividing that result by c is the same as dividing a by the product of b and c.
Theorem TBtheorem1a: Assuming that "PQ" is not an element of "E", and the subcomponents of "PQ" are "P" and "Q", and "PQ" is correctly composed, it can be shown that either "P" is not an element of "E" or "Q" is not an element of "E".
The binary product of two quasi-Borel spaces, X and Y, is a closed set in the 3-dimensional quasi-Borel space.
If we restrict the function 'h' to the carrier of group 'G', it remains a homomorphism from group 'G' to group 'H'.
If the index of 'P' is equal to 'i' and 'i' is greater than 0, then 'P' of 'i - 1' is not true. The second property states that if the index of 'P' is equal to 'i' and 'j' is greater than or equal to 'i', then 'P' of 'j' is true.
If "s" is equivalent to "t" under some relation, then "s" is in the set of trees generated by "F". Similarly, if "s" is equivalent to "t" under the same relation, then "t" is also in the set of trees generated by "F".
If for any natural number 'n', the value of function 'X' at 'n' is less than or equal to the value of function 'X' at 'n+1', then 'X' is an increasing sequence.
Updating the mapping P2 at position 'p' with value 'y', and then taking the maximum mapping with P1 and updating at position 'p' with value 'x', is equivalent to taking the maximum mapping of P1 and P2 and then updating at position 'p' with value 'x'.
The lemma named "loop_root" assumes that "x" is injective (meaning that it has a unique output for every unique input) and that "x" is equal to the image of "x" under the function "p". Under these assumptions, it shows that "x" is equal to the root of "p" with respect to "x".
If a pair (\<sigma>, p) is reachable in OA under the initial condition I, and the initial condition i is the same for both OA and A, and if the transition in OA is equivalent to the sequential process sos in \<Gamma> and i, and the transition in A is equivalent to the sequential process sos in \<Gamma>, then there exis...
In the lemma for the semilattice with infimum, if 'a' is less than or equal to the infimum of 'b' and 'c', then 'a' is less than or equal to 'b'.
The polynomial monoid, when constructed with any unique factorization domain as its elements, is a factorial monoid.
If the extended measure of set B is not infinite, and both sets A and B are in the sigma-algebra of M, and B is a subset of A, then the extended measure of the difference of A and B is equal to the difference of the extended measures of A and B.
The most significant bit of the negation of a word 'w' is true if and only if the most significant bit of 'w' is false.
If the supremum of "h" and the composition of "g" and "f" is less than or equal to "g", then the composition of "h" and the "i"-th power of "f" in the Kleisli category is also less than or equal to "g".
The lemma named "errMOD_iwlsFSbSwTR" fixes a model named "MOD" with the types 'index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'gTerm, 'gAbs. Assuming that "MOD" satisfies the property "iwlsFSb", it shows that "MOD" also satisfies the property "iwlsFSbSwTR" when the "errMOD" function is applied to it.
If a path 'p' can be joined with a path 'q', and 'q' is not an empty path, then the last element of the joined path is the same as the last element of path 'q'.
An element 'b' is in the range of a relation 'r' if and only if there exists an element 'a' in the domain of 'r' such that the ordered pair (a, b) is in 'r'.
The image of the set {U} under the message relation is an element of the set of messages.
If 'x' and 'y' are elements of 'assumption', then the supremum (or join) of 'x' and 'y' is equal to the product of 'x' and 'y'.
The integer representation of the natural number 1 is 1.
The set of probability mass function (pmf) of "wt_pmf cf" is equal to a conditional statement. If the first element of "cf" is proper, then the set is composed of all "i" such that "i" is less than the branching factor of the first element of "cf" and the weight of the first element of "cf" at the second element of "cf...
The lemma named "tendsto_powr_complex'" is about complex functions f and g. It assumes that a is not a non-positive real number or that a is zero and the real part of b is greater than zero. It also assumes that f tends to a and g tends to b. Under these assumptions, it shows that the function that maps z to the power ...
The Goodstein sequence for a constant 'c' and a number 'n' is equal to the Goodstein sequence for the same constant 'c' and the ordinal representation of 'n'.
If the product of x and y is less than or equal to z', and z' is less than or equal to z, then the product of x and y is less than or equal to z.
The relations in a join of two sets 'l' and 'r' are distinct if and only if the relations in 'l' and 'r' are distinct and the intersection of the relations in 'l' and 'r' is an empty set.
The "just_cash" function, when applied to zero, returns zero.
If an expression "e" evaluates to a call to a method "l" on an object "f" of type "U" with a body "b" and this results in a type "T", then there exists a finite set "F" such that for all strings "s" and "p" that are not in "F" and are not equal to each other, the expression "e" with "s" of type "U" and "p" of the param...
The tail of a stream 'xs' is undefined if and only if 'xs' itself is undefined or 'xs' is an empty list.
"for all points a and b, if point a is between point b and itself on the hyperbolic plane, then point a is equal to point b."
The nth step of the process Pn with initial state S0 and parameter p is equal to the nth step of the process Xn with the same initial state and parameter, bound to the function that takes a state 's' and binds it to the conditional expectation of Y given X with parameters p, S0, n, and state 's', which is then bound to...
The negation of 'p until not p' is equivalent to 'always p'.
: Assuming that the machine M2 is observable and minimal, and the size of M2 is less than or equal to the size of the prime version of M1 plus some additional states represented as an integer. Also, the inputs and outputs of M2 are the same as those of M1. If M1 is already in its prime form, then M1 is observable and ...
The lemma named "fsymsPB_sappend" assumes two conditions: first, that the set Phi1 is not empty, and second, that the set Phi2 is not empty. Under these assumptions, it shows that the function fsymsPB applied to the concatenation of Phi1 and Phi2 is equal to the union of the function fsymsPB applied to Phi1 and the fun...
The lemma "diagonal_to_Smith_aux_PQ_JNF_dim" assumes that matrices P', A', and Q' are all square matrices of dimensions m by m, m by n, and n by n respectively. It also assumes that the function "diagonal_to_Smith_aux_PQ_JNF" with inputs xs, bezout, and the tuple (P',A',Q') returns the tuple (P,A,Q). Given these assump...
If vector v is equal to the parallel composition of vectors u and w, then the restriction of function f to vector u at point c is equal to the restriction of function f to vector w at point c.
If a sequence 'xs' is a subsequence of another sequence 'ys', then the multiset of 'xs' is a subset of the multiset of 'ys'.
The lemma "append_vec_le" assumes that "v" is a vector in the n-dimensional vector space and "w" is also a vector in the same n-dimensional vector space. It shows that the vector formed by appending "v'" to "v" is less than or equal to the vector formed by appending "w'" to "w" if and only if "v" is less than or equal ...
If a proposition "p" can be inferred from a set of propositions "G", then "p" can also be inferred from the set of propositions obtained by adding a new proposition "a" to "G".
The connection of D with the observed resource of the ideal resource is equal to the connection of D with the observed resource of the fused resource. This fused resource is the result of fusing the lazy core with the DH1 sample and the lazy rest, and it is initialized with the basic core and the basic rest.
The lemma named "arctan_upper_34" assumes that "x" is greater than 0. It shows that the arctangent of "x" is less than the upper 34th arctangent of "x".
The lemma named "frameEqSupp" is given certain fixed elements: F, which is a frame of type 'a (where 'a is a name from a finite set), G, which is a frame of type 'a, and x and y, which are both names. The lemma assumes that the frame F restricted to the name x is equal to the frame G restricted to the name y, and that...
The sum of the function "d" applied to "y" and the fbox function applied to "x" and "y" is equal to the fbox function applied to the antidomain of "y" multiplied by "x" and "y".
The inverse location of 'b' is not empty when the function 'inv_locate_b' is applied to the parameters '(as, lm)', '(n, [], [])', and 'ires', and the result is False.
If there exists a 'y' such that the property 'phi' holds for 'y' in 'v', then it implies that there exists an 'x' such that the property 'phi' holds for 'x' in 'v'.
The lemma named "lens_override_irr" assumes that "X" is disjoint from "Y". It shows that the override of "S1" with the override of "S2" and "S3" on "Y", all on "X", is equal to the override of "S1" and "S2" on "X".
The lemma 2.6.5.a assumes that 'r' is a transitive relation, the downward closure of 'r' in 'S' is a subset of 'S', and the pair '(M,N)' is in the multiplicative equality of 'r'. It shows that the pair '(M -s S, N -s S)', where '-s' denotes set difference, is also in the multiplicative equality of 'r'.
The lemma named "discriminant_zero" is about real numbers a, b, c, and x. It assumes that a is not equal to zero and that the discriminant of a, b, and c is zero. It shows that the equation "a times x squared plus b times x plus c equals zero" is equivalent to "x equals negative b divided by two times a".
If a quantum program S is well-composed, and if a matrix ρ is a d by d matrix that is a partial density operator, then the trace of the result of applying the program S to ρ is less than or equal to the trace of ρ.
If 'S' is a finite set and for every element 'u' in 'S', 'f u' is a gauge, then the function 'f' is a gauge for all 'x' in the intersection of the set of 'f u x' where 'u' is in 'S'.
If a ruleset "rs" is well-formed under a certain condition "p" and a certain model "γ", then the ruleset remains well-formed under the same condition and model after it has been normalized using the "normalize_rules_dnf" function.
The lemma named "orthogonal_proj_set" assumes that "y" is an element of set "C", "C" is a finite set, and "C" is pairwise orthogonal. It shows that "a" minus the projection of "a" onto "C" is orthogonal to "y".
If the Secure Information Asymmetry (SIA) holds for a certain view V1 under a security policy rho1 and trace Tr in an Event Structure (ES), and if the security policy rho2 for a view V2 includes rho1 for V1, then the Secure Information Asymmetry (SIA) also holds for the view V2 under the security policy rho2 and trace ...
There exist lists 'xs' and 'ys' such that when a predicate 'P' is applied to filter 'xs' and 'ys', the results are 'xs' and 'ys' respectively, and 'as' is the concatenation of 'xs' and 'ys'. This is under the condition that when the predicate 'P' is applied to filter 'as', the result is the concatenation of 'xs' and 'y...
The sequence map of function f on sequence xs zipped with the sequence map of function g on sequence ys is equal to the sequence map of the product of function f and g on the zip of sequence xs and ys.
'S' is locally path-connected if and only if 'C' is locally path-connected.
The lemma "sort_eqv_imp_normalize_sort_eq" assumes that there exist sorts "s1" and "s2" in the context "cs", and that these sorts are equivalent in this context. It shows that the normalization of sort "s1" in the context "cs" is equal to the normalization of sort "s2" in the same context.
If the product of y and x is less than or equal to the product of x and the star of x joined with y, then the star of the union of x and y with z is equal to the star of x with the star of y and z.
The transitive closure of the union of a set X and the restriction of the identity relation to a set Y is equal to the union of the transitive closure of X and the restriction of the identity relation to Y.