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Statement: stringlengths 7 24.3k
If for any relations R, sets A and B, given that R is a total preorder on a set of alternatives 'alts', and both A and B are non-empty subsets of 'alts', the preference of set A over set B according to the set extension of R is equivalent to the preference of set A over set B according to the set extension of the lifte...
For a fixed word 'a' of any length, if 'x' equals -1, then the bitwise AND operation between 'a' and 'x' will result in 'a'.
The division and modulus of two polynomials 'f' and 'g' can be computed as follows: First, we extract the coefficients of 'g' into 'cg'. If 'cg' is an empty list, then the division of 'f' by 'g' is zero and the modulus is 'f' itself. Otherwise, we extract the coefficients of 'f' into 'cf' and calculate the inverse of...
Given a finite set of real numbers 's', a tagged division 'p' of the interval from 'a' to 'b', a function 'γ' that refines 'p', and the condition that for every pair (x, K) in 'p' there exists real numbers 'c' and 'd' such that 'c' is less than 'd' and 'K' equals the interval from 'c' to 'd', and given that 'γ' is a ga...
The set of limit points of a set S in a metric space is closed.
The theorem of Hahn decomposition shows that there exist two sets, M1 and M2, such that they form a Hahn space decomposition.
Substituting a variable 'j' with a value 'w' in a sentence 'A' under the predicate 'SentP' is equivalent to applying the substitution first in 'A' and then applying the 'SentP' predicate.
If "eta" is a natural transformation, then making "eta" into a natural transformation (using the function "MakeNT") will indeed result in a natural transformation.
The lemma named "valid2_hd_Rcons" assumes that the transition sequence "tr" is valid according to the definition "valid2". It then shows that the head of the sequence resulting from appending the transition "tran" to "tr" is the same as the head of the original sequence "tr".
The lemma "lists_not_eq" assumes that list "xs" is not equal to list "ys". It then establishes three possible scenarios: 1. List "xs" is a strict prefix of list "ys". This means that "xs" is a subset of "ys" and appears at the beginning of "ys", but "xs" is not equal to "ys". 2. List "ys" is a strict prefix of list "...
The substitution of a composition of two functions, pi1 and pi2, into a formula Phi is equal to the substitution of pi1 into the result of the substitution of pi2 into Phi.
A function 'f' is a simple function on a measure space 'M' if and only if the image of 'f' on the space 'M' is finite and 'f' is Borel measurable on 'M'. Here, 'f' is a function from type 'a' to type 'x', where 'x' is a T2 space (also known as a Hausdorff space).
If a transition from state 's' with configuration 'c' to state 's'' with configuration 'c'' exists in the transitive closure of the non-terminal reduction pairs of the flowgraph (as assumed in 'A'), and if the flowgraph is valid with the multiset addition of state 's' and configuration 'c' (as stated in 'V'), then the ...
If "x" is in the domain of the function "Cp" applied to the firewall policy obtained from the list "a" concatenated with the list containing "y" followed by "z", then "x" is also in the domain of the function "Cp" applied to the firewall policy obtained from "a" concatenated with the firewall policy obtained from the l...
For any constant function on a closed box in a Euclidean space, the integral of the function over the box is equal to the product of the content (volume) of the box and the constant value.
The lemma "staysuntouched" assumes that the first element of S is distinct, and both x and y are in the set of the first element of S. It shows that if the set of qs is a subset of the set of the first element of S, and both x and y are not in the set of qs, then the order of x and y in the first element of the configu...
The preimage of a ring epimorphism, denoted by eta, from a ring R with operations addition (+), multiplication (·), zero element (0), and unit element (1), to its image under eta in another ring with operations addition (+'), multiplication (·'), zero element (0'), and unit element (1'), is indeed a ring epimorphism.
The first-order variables in the negation of a formula in numerical monadic second-order logic are the same as the first-order variables in the negation of a formula in first-order logic.
Lemma: winding_number_pos_lt_lemma Assumptions: 1. γ is a valid path. 2. z is not in the path image of γ. 3. e is a positive number. 4. For any x such that 0 < x < 1, e is less than or equal to the imaginary part of the vector_derivative of γ at x divided by (γ(x) - z). Conclusion: The real part of the winding number ...
If the encryption of X with key K is in the analyzed set S, and if the set S is clean of encryption keys, then the key K is either in the range of long-term keys or in the payload.
If "r" is a vertex in graph "G", then the "ikkbz_sub" of "r" is distinct.
If function f is partially equivalent to function g, and if x and y are in the domain, and x is partially equivalent to y, then the result of function f applied to x is partially equivalent to the result of function g applied to y.
If a generative probabilistic value (gpv) bound to a function f is well-typed under a context Γ, then the gpv itself is also well-typed under the same context Γ.
The numeral function applied to any number 'n' is equivalent to the pure function applied to the numeral function of 'n'.
If 'a' is in 'c', 'a' is not in 'd', and 'c' is equal to 'd', then this leads to a contradiction.
The lemma named "step_simulation" assumes that "cs" is related to "cm" under the relation "rel_S₀", and that "cm" is related to "cm'" under the "step" relation. It shows that there exists a "cs'" such that "cs" is related to "cs'" under the "st.step" relation raised to the power of "(10 + 5 * Max (range (mt_pos cm)))",...
The lemma named "bin_long_mismatch" assumes that the length of alpha is less than the length of the function f applied to w. It shows that the concatenation of alpha and the code of the head of w is a prefix of the function f applied to w.
If the product of relation R and an empty set is a subset of the product of relation S and an empty set, then the product of the product of relation R and relation T and an empty set is a subset of the product of the product of relation S and relation T and an empty set.
For any state "omega" and any variable "x", replacing the value of variable "x" in state "omega" with its own value in the same state "omega" will result in the original state "omega".
If automaton A is a subset of automaton B, then the set of states reachable by A is also a subset of the states reachable by B.
The lemma named "poly_roots_factorization" is about a polynomial 'p' and constants 'c' and 'A'. It assumes that 'c' is not equal to zero. The polynomial 'p' is defined as the scalar multiplication of 'c' and the product of a multiset, where the multiset is the image of a function that maps 'x' to a polynomial with root...
If the standard size is equal to the size of 'c' minus the size of 'b', and if splitting the word 'c' results in a pair '(a, b)', then 'a' is equal to the conversion of the binary list obtained by taking the first 'std' elements from the binary list of 'c', and 'b' is equal to the conversion of the binary list obtained...
The lemma named "vec_cnj_vec_cnj" shows that the complex conjugate of the complex conjugate of a vector v equals the original vector v.
If a list formed by appending list A and list B is a sublist of list C, then both list A and list B are sublists of list C.
Lifting a context E twice, first by n and then by 0, is equivalent to lifting the context E first by 0 and then by n+1.
If A, B, and Y form a polynomial ring, h is a ring homomorphism from S to B, and (n, f) is a polynomial coefficient in S, then the coefficient function of h applied to (n, f) equals the pair (n, the composition of h and f).
: Given the conditions that k is greater than 0, m is not equal to NoMsg, the event occurs at time t if and only if t modulo k equals 0, t0 equals t times k, and s equals the composition of the output function and the execution of the computation stream with the transformation function, the input scaled by k, and the ...
If the following conditions are met: 1. Entity A says to Entity B a certain Certificate, which is part of a set of events, 2. For all p and q, the Certificate is not equal to the set containing p and q, 3. Entity A is not a Spy, 4. The set of events is in srb (Shoup-Rubin-Bella protocol), Then there exists a Key ...
For any set 'S' in a Euclidean space, the vector is differentiable on 'S'.
The product of L and L equals L.
The lemma "is_red_alt" shows that "a" is reducible by the set "F" if and only if there exists a "b" such that "a" can be reduced to "b" by the set "F".
If the sum of x and z equals the sum of y and z, and the product of x and z equals the product of y and z, then x must be equal to y.
The list obtained from the function "iam_to_list" applied to any input "a" is distinct, meaning it contains no duplicate elements.
If 'S' is an Integral Domain and 'P' is an ideal of 'S', and if 'R'' is a Polynomial Ring over the quotient ring 'S / P' with variable 'Y', and '(n, c')' is a polynomial coefficient in the quotient ring 'S / P', then the homomorphism from the ring 'R' to the ring 'R'' induced by the projection map 'pj' from 'S' to 'S /...
The lemma "NewHeapElem_start_heap_obsD" assumes that the system classes "P" are well-formed and that a new heap element "a" of type "hT" is in the set of starting heap observations. It shows that the type of address "a" in the starting heap is "hT".
In the context where 'is_ntcf' is true, the lemma 'ntcf_arrow_NTCod_in_Vset' assumes that 'beta' is a member of the class of all sets, and 'alpha' is an element of 'beta'. It shows that the codomain of the natural transformation arrow 'NN' is an element of the set 'Vset beta'.
The lemma named "sublens_iff_subscene" assumes that "X" and "Y" are very well-behaved lenses. It shows that "X" is a sublens of "Y" if and only if the scene of "X" is less than or equal to the scene of "Y".
If two charts c1 and c2 are k-smooth compatible, then the composition of chart c1 and the inverse of chart c2 is k-smooth on the image of the intersection of the domains of c1 and c2 under chart c2.
If for all pairs of X and Y, the pair is not in H; for all xs, L xs is not in H; for all xs, FS xs is not in H; and for all X and K, Crypt K X is not in H, then the analysis of H is equal to H.
If a triple ((ξ, {l}send(s_msg). p), a, (ξ', q)) is in the sequential process semantics (seqp_sos) under a context Γ, and if the action a is the sending of the message s_msg evaluated at ξ, ξ' is equal to ξ, and q is equal to p, then the proposition P holds.
If the following conditions are met: 1. The concatenation of lists 'vs' and 'xs' is in the set 'C' with parameters 'M2', 'M1', 'Ω', 'V', 'm', and 'i'. 2. The most common prefix of the concatenation of lists 'vs', 'xs', and a list containing 'x', with respect to 'V', is 'vs'. 3. 'x' is an input of 'M2'. 4. 'i' is great...
If the ordered pair "MPair X Y" is equal to the ordered pair "MPair X' Y'", then "Y" is equal to "Y'".
The lemma named "sfilter_eq" assumes that there exists an event where the predicate P holds for the stream s. It shows that the stream filter of P on s equals to the stream starting with x and followed by s' if and only if the predicate P holds for x and it is not the case that the predicate P holds until the stream s ...
The lemma named "compact_infdist_le" is about a set A of type 'a, where 'a is a Heine-Borel set. It assumes that set A is not empty, that A is compact, and that a real number e is greater than 0. It shows that the set of all elements x, for which the infimum distance from x to set A is less than or equal to e, is also ...
If list 'xs' is a prefix of list 'ys' and list 'xs' is not empty, then list 'ys' cannot be empty.
The lemma "lift_exists" consists of two parts: 1. If it is false that a property P holds for all elements, and sk is some element for which P does not hold, then P does not hold for sk. 2. If it is true that there exists an element for which property P holds, and sk is some element for which P holds, then P holds for...
The lemma named "filterlim_cong'" assumes two conditions. The first condition is that the function 'f' has a limit in the filter 'F' that is within the filter 'G'. The second condition is that eventually, for some 'x', 'f' of 'x' equals 'g' of 'x' within the filter 'G'. If these two conditions are met, it shows that th...
The sum of the series with all terms equal to zero is convergent.
A function of three variables (x, y, z) that always returns a constant natural number "n" is in the set of primitive recursive functions of three arguments.
: Assuming "q" is a state in the machine "M", and "f" is an equivalence relation on the states of "M", it shows that the output finite state machine table for "M", "f", "k", and "q" is equal to the set of all states "q'" in "M" such that for all input-output pairs "io" with a length less than or equal to "k", the follo...
In the context of a set Z, there is a lemma for Z's indiscrete topological space. Assuming that A is an element of the set V indexed by alpha, it shows that the topological space of Z indexed by alpha is the indiscrete topological space of A.
There is a local flow for the vector field scaled by K, defined for all points and all times, where the flow at time t from point s is given by the function ((t * K) squared divided by 2 plus (t * K) plus 1) scaled by the vector s.
The leading coefficient of the product of a list of polynomials is equal to the product of the leading coefficients of each polynomial in the list.
The lemma named "cong_sum_mono_neutral_right" assumes that "T" is a finite set, "S" is a subset of "T", and for all elements "i" in the difference of "T" and "S", the function "g" applied to "i" is congruent to zero modulo "n". Under these assumptions, it shows that the sum of the function "g" applied to all elements i...
If a pair (s, t) is in the transitive closure of the single rewrite step relation (srstep) with respect to a set of function symbols (F) and a rewrite system (R), then the following conditions hold: 1. The pair (s, t) is in the transitive closure of the rewrite step relation (rstep) with respect to the rewrite system...
The cardinality (or the number of elements) of the set 'bit' is 2.
If angle DEF is less than or equal to angle D'E'F', and if A, B, C, D', E', F' form a strictly antipodal angle sum, and if the sum of angles ABC and DEF equals angle GHI, and the sum of angles ABC and D'E'F' equals angle G'H'I', then angle GHI is less than or equal to angle G'H'I'.
If set S is a subset of set T, then the connected component set of S with respect to x is also a subset of the connected component set of T with respect to x.
'x' is not a rational number.
The lemma states two things about inserting a node into an undirected graph. First, the set of nodes in the graph after inserting a node 'v' is the same as the set of nodes in the original graph with 'v' added. Second, the set of edges in the graph after inserting a node 'v' remains the same as the set of edges in the ...
"it is not the case that the reduction of an empty set can be applied to any two elements p and q".
The lemma named "bounded_coeffs_imp_fds_abs_converges'" is defined with two fixed elements: 's' which is a type of Dirichlet series and 'f' which is a type of formal Dirichlet series. It assumes two conditions: first, the sequence of the nth coefficient of 'f' multiplied by the nth power of 's0' is bounded, and second,...
The lemma named "totient_less" assumes that "n" is greater than 1. It shows that the totient of "n" is less than "n".
If a list "xs" permutes a set "A", and if a pair (x,y) is in the set "xs", then the inverse permutation of the list "xs" at the position "y" is "x".
If a function 'f' is monotonic on a set 'A' with respect to orders 'orda' and 'ordb', and 'orda' is transitive, then 'f' is also monotonic on the set of all multisets 'M' whose elements are all in 'A', with respect to the multiset extensions of 'orda' and 'ordb', when 'f' is applied to each element of the multiset.
If there are homotopic paths on a space 's' between the joined subpaths from 'u' to 'v' and from 'v' to 'w' on a path 'g', and the subpath from 'u' to 'w' on the same path 'g', then there are also homotopic paths on the same space 's' between the joined subpaths from 'w' to 'v' and from 'v' to 'u' on the path 'g', and ...
If "h" is an element of set "H", then the inverse of "h" under operation "T" is an element of set "G".
The lemma named "extremal" assumes that "P" is an extreme of "b" under the condition "Extreme". It then shows that "b" is also an extreme of "F P".
The lemma named "csol" makes the following assumptions and shows the following results: Assumption: The time 't' is within the interval of existence for the initial value problem at time 't0' and initial condition 'x0'. It then shows the following: 1. The time 't' is within the set 'T'. 2. The closed interval from '...
A predicate P holds for all corresponding pairs in the list xs and the list obtained by applying a function g to all elements of the list ys if and only if the predicate P holds for all corresponding pairs in the list xs and the list ys, where the second element of each pair in ys is transformed by the function g.
If "a" derives to "b" and "b" derives to "c", then "a" derives to "c".
The i-th digit of a number x in a certain base is equal to the remainder when x divided by the base raised to the power of i is further divided by the base.
The lemma named "bx_aux" demonstrates that if "b" is greater than 1, then the "t'"th digit of "b" to the power of "x" in base "b" is 1 if "x" equals "t'", otherwise it is 0.
For any two functions 'f' and 'g' that map from a set 'X' to a real normed vector space 'b', if both 'f' and 'g' are continuous maps with respect to the Euclidean topology, then the function that maps each element 'x' in 'X' to the sum of 'f(x)' and 'g(x)' is also a continuous map with respect to the Euclidean topology...
The lemma named "irreducible\<^sub>d_reflect_poly" is defined for a polynomial 'f' of a type 'a' that is an integral domain and a semiring with greatest common divisor. It assumes that the coefficient of 'f' at 0 is not equal to 0. The lemma shows that the irreducibility of the reflection of the polynomial 'f' is equal...
There exists an element 'x' in the union of 'y' and set 'A' such that property 'P' holds for 'x', if and only if either property 'P' holds for 'x' when 'x' is replaced by 'y', or there exists an element 'x' in set 'A' for which property 'P' holds.
The omega power of the star of x multiplied by y is less than or equal to the omega power of the join of x and y.
Filtering a list "xs" after removing duplicates is the same as removing duplicates after filtering the list "xs" based on a predicate "P".
If the states of SA do not intersect with the HA states of A, and S is in the initial configuration of A, then the initial configuration of A added to the pair (S, SA) is equal to the insertion of the initial state of SA into the initial configuration of A.
: Assuming a pair (\<sigma>, NodeS i \<zeta> R) is in the set of states reachable by the operational semantics of a network with node ii running process T and initial states R\<^sub>i, with the arrival of messages I being handled according to the function S, and all other nodes behaving according to the function U. Fu...
The lemma named "chine_char" shows that "chine" is congruent to the composition of "pi.chine" and "mu.chine".
For any real number 'm' that is greater than zero, the inequality 'm times x plus c is less than or equal to y' is equivalent to 'x is less than or equal to the inverse of m times y minus c divided by m'. This is valid for 'm' which is a member of any linearly ordered field.
If 3 is less than or equal to i, and i is less than or equal to k plus 1, and j is less than n of i minus 1, then the scalar multiplication of the product of alpha l to the power of n of l minus 1 minus j (where l ranges from 3 to i) and the subresultant of j, F1, and F2, is equal to the scalar multiplication of the pr...
The lemma named "h_framework_static_with_simple_graph_completeness_and_finiteness" is defined with two fixed finite state machines (FSMs), M1 and M2. M1 is of type ('a::linorder,'b::linorder,'c::linorder) fsm and M2 is of type ('e,'b,'c) fsm. The lemma assumes that both M1 and M2 are observable and minimal. The size ...
The absolute value of nu at state s is less than or equal to the sum over i of l to the power of i times r sub M.
For a given graph 'g', node 'n', and any generic 'ga', the successors of 'n' in the graph 'g' when ignoring 'gen_αn' and 'gen_inEdges'' are the same as the successors of 'n' in the graph 'g' when considering 'gen_αn' and 'gen_inEdges''.
The lemma "mat_mult_append" assumes that A is a matrix with "nr1" rows and "nc" columns, B is a matrix with "nr2" rows and "nc" columns, and v is a vector with "nc" components. It shows that the vector resulting from multiplying the matrix formed by appending B to A row-wise with v is the same as the vector formed by a...
: Given a process 'p' and a configuration 'c', under the assumptions that the configuration 'c' is non-uniform, there is pseudo termination for any configuration 'cc' and any set 'Q', and for any configuration 'cfg' reachable from 'c', there is agreement in 'cfg'. It can be shown that there exists a configuration 'c''...
The lemma named "subgraph_impl_connected" assumes that "H" is a connected graph and "H" is a subgraph of "G". It shows that if these conditions are met, then "G" is also a connected graph.
The intersection of sets m and s is an empty set.

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