Statement: stringlengths 7 24.3k |
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If the list "ls" is not empty, then the function "c_fold" applied to the list that has "x" as its first element and "ls" as the rest of its elements, is equal to the function "c_pair" applied to "x" and "c_fold" applied to "ls". |
The option 'None' is less than 'Some x'. |
The lemma named "combinable_and" assumes that both "A" and "B" are combinable with "Delta". It shows that the conjunction (And) of "A" and "B" is also combinable with "Delta". |
A pair 'p' is in the intersection of pair sets 'A' and 'B' if and only if 'p' is in pair set 'A' and 'p' is in pair set 'B'. |
The iteration of a function 'f' and a value 'x' is equal to the value 'x' followed by the stream map of the function 'f' and the iteration of the function 'f' and 'x'. |
If 'n' is greater than 1 and 'χ' is not the principal Dirichlet character of 'n' and is a Dirichlet character of 'n', or if 'χ' is the principal Dirichlet character of 'n' and 1 is not in 'A', then the Dirichlet L-function is continuous on 'A'. |
The lemma "find_base_vectors_i" is a transfer rule that states: if a matrix relation "R" holds, then the function "find_base_vectors_i ops" corresponds to the function "find_base_vectors" under the condition that all elements in the list satisfy the vector relation "R". |
For a fixed Gaussian integer 'p', if 'p' is a prime element, the real part of 'p' is greater than zero, the imaginary part of 'p' is greater than zero, and the imaginary part of 'p' is not equal to the real part of 'p', then the multiplicity of 'p' in the natural number 'c' is equal to the multiplicity of the norm of '... |
A field "f" in a method "n" is static if and only if the field "f" itself is static. |
If a state 's' transitions to a state 's'' through a series of thread actions 'ttas', and if a new thread 't' with state 'x' and method 'm'' is in the set of all thread actions in 'ttas', and if the wait set of state 's' is correctly associated with the thread state of 's', then the thread state of 's'' for thread 't' ... |
The integral from 0 to 4 of the function that takes a real number x and returns the square root of the quantity (4 - x) divided by x, is equal to 2 times pi. |
If 'j' is less than the length of 'M', then the set of black elements in 'M' is a subset of the set of black elements in 'M' where the 'j'-th element is black. |
The function "r" applied to "s" equals "Some t" if and only if "s" is not in the set "F" and "s" equals "t". |
Prepending the machine language value 0 to any machine language value sequence is equivalent to the original machine language value sequence. |
The lemma named "monomial_split_card" assumes that "P" is an element of the indexed power set "I" of the carrier "R". It also assumes that "m" is an element of the monomials of "R" and "P". It then shows that the cardinality of the monomials of "R" and the restricted polynomial to the monomial set "R", "P", and the dif... |
The image of the image of a set X under the dual function is equal to the set X itself. |
Given the assumptions that "y" is an element of the sequence set "ys", "xs" is a subsequence of "ys", and "xs" is not the least element, there exists an element "x" in the sequence set "xs" that is a subsequence of "y". |
The nth term of the Dirichlet series obtained by multiplying a given Dirichlet series 'f' by a constant 'c' is equal to the nth term of the original Dirichlet series 'f' multiplied by the constant 'c'. |
If the initial state of a Turing machine is represented by the tuple (s, CL@Bk^z1, Oc#rs) and the machine transitions to the state (s', l', r') after one step, and if za is less than z1, then there exists a tape configuration CL' and a number zb such that the left tape after the transition is represented by CL'@Bk^za@B... |
If process P is deadlock-free according to version 2 definition, then the divergence set of process P is empty. |
The variable 'vars1' from the process 'ps2_P' is equal to 'vars1'. |
:
1. The NGBA is a function that takes an input set of labels (L), an input set of states (S), a transition function from labels and states to a set of states, and a list of sets of states, and returns an NGBA scheme.
2. The alphabet function takes an NGBA scheme and returns a set of labels.
3. The initial function ... |
The set resulting from the operation "minusset" on a set A is equal to the set obtained by applying the inverse function to the intersection of set A and set G. |
If a non-zero integer w is multiplied by two integers z and z' respectively, and the results are equal, then z must be equal to z'. |
The lemma named "diff_hmem_imp_less" assumes that "x" is an element of "y". It shows that the representative of the difference between "y" and the set containing "x" is less than the representative of "y". |
For any elements "x" and "y" and lists "xs" and "ys", if "x" is not equal to "y", then the property "P" holds for the list formed by adding "x" at the beginning of "xs" and the list formed by adding "y" at the beginning of "ys". The second rule states that for any elements "x" and "y" and lists "xs" and "ys", if "x" is... |
If the function 'f' applied to 0 equals 0, then the application of 'f' to the raw partial deviation of 'xs' at 'i' is equal to the raw partial deviation of the mapped tuple '(i, f x)' in 'xs' at 'i'. |
For every 'v', there exists a 'u' such that 'v' equals the horizontal chop of 'v' and 'u'. |
The lemma named "bilateral_substitutes_lemma" assumes three conditions:
1. The image of x under Xd is not in the image of the set X under the function Ch h.
2. d is not in the image of the set X under the function Ch h.
3. d is not equal to the image of x under Xd.
Given these assumptions, it shows that d is not in t... |
If "i" and "j" are non-negative integers, both less than "n", and "i" is not equal to "j", then the function "interfree_aux" applied to the pair of "Mul_Redirect_Edge i n" and "Mul_Redirect_Edge j n" is true. |
For any three octonions x, y, and z, the product of x and y, when multiplied by x, is equal to the product of x and the product of y and x. |
The composition of the function IF2map with f and the constructor function ctor2 is equal to the composition of ctor2 and the function F2map with parameters f, IF1map with f, and IF2map with f. |
The lemma named "wt_Wit" assumes that the weight of the function "Wit" with type sigma and applied to "Tl" is true according to the theory "TE". It shows that "Tl" is an empty list. |
The number 1 is not a Carmichael number. |
If A is an object in the comma category of the functor F and b, and b is an object in the category B, then there exist an object a and a morphism f such that A is equal to the triple [a, 0, f], a is an object in the category A, and f is a morphism from the object mapped by the functor F at a to b in the category B. |
The product of 'x' and 'n(y)' and 'L' is less than or equal to the least upper bound of the product of 'x' and 'bot' and the product of 'n(x * y)' and 'L'. |
If "a" is an arc, then the transitive closure of the union of "a" and "x" is equal to the union of the transitive closure of "x" and the reflexive transitive closure of "x" multiplied by "a" and the reflexive transitive closure of "x". |
The lemma named "map_nth_append_upt" assumes that a number 'a' is greater than or equal to the length of a list 'xs'. It shows that mapping the function '(!)' (which retrieves the element at a given index) over the range from 'a' to 'b' on the appended list of 'xs' and 'ys' is equal to mapping the same function over th... |
If a pair (A, B) is in the relation "polys_rel", and if x12 is in the domain of A, then the value that corresponds to x12 in A is in B. |
The nth vector of the result of interchanging rows a and b in matrix A is equal to the result of the function "interchange_rows_row" applied to the same matrix A and rows a and b. |
If N is a module over a ring R, and f is a module homomorphism from M to N that is also a bijection, then the inverse function of f is a bijection from N to M. |
If an ordered sequence of probability mass functions "p" and "q" are equal, then the measure of any set "A" under the probability mass function "p" is less than or equal to the measure of the same set "A" under the probability mass function "q". |
If "m0", "cond", and "step" are invariants of "is_rwof_invar" and "rwof" includes "m0", "cond", and "step" with state "s", then state "s" is an invariant "I". |
If time "t" is less than "h", and if an operator "op" is in the set of operators of the problem with no operation (noop) abstraction problem, and if a state variable "sv" is in the atoms of the conjunction of the map (which maps each variable "v" to the disjunction of the negation of the atom of the operator at time "t... |
The theorem named "MPair_MPair_eq" shows that the pair (X, Y) is equal to the pair (X', Y') if and only if X is equal to X' and Y is equal to Y'. |
The lemma named "worklist_tree_aux_state_inv" assumes the following conditions:
1. The function "worklist_tree_aux" with parameters "succs", "f", and a pair "(ws,s)" returns a pair "(ws',s')".
2. The predicate "I" holds for the state "s".
3. For any "x" and "s", if the predicate "I" holds for "s", then "I" also holds ... |
If the following conditions are met:
1. Set A is a subset of the reflexive transitive closure of set B under relation Rb,
2. Set B is a subset of the reflexive transitive closure of set A under relation Ra,
3. For all pairs (x, y) in the set difference of Ra and Rb, y is in the reflexive transitive closure of set B ... |
If the function F is continuous, then the function that takes a single argument x and applies F to it is equivalent to the function that takes a triple of arguments (x, y, z) and applies F to the triple. |
If a morphism 'f' is equal to 'aa', then the application of the ArrMap function to 'f' in the double-up-arrow functor with parameters 'CC', 'aa', 'bb', 'gg', 'ff', 'aa''', 'bb''', 'gg''', and 'ff''' results in the identity morphism of 'aa''' in the category 'CC'. |
The result of the Karatsuba multiplication algorithm on two polynomials f and g, represented by their coefficients lists f and g and their degrees n and m respectively, is equal to the product of the two polynomials. |
If it is true that whenever "S'" does not fail, then "S" does not fail and for all "x", if "x" is in the result of "S", then "x" is also in the result of "S'", then "S" is less than or equal to "S'". |
Lemma: Borel Measurable Limit Sequence for Real Numbers
Let u be a function that takes a natural number and an element from a measurable space M and returns a real number. Assume that for every element x in the space M, the sequence of real numbers generated by applying u to each natural number i (i.e., u(i, x)) conve... |
If memory 1 is equivalent to memory 2 with respect to the low security level under a certain mode state (mds), then the direct memory access (dma) of memory 1 is equal to the dma of memory 2. |
Taking the signed 3-bit representation of the integer 1705 results in -7. |
The lemma named "square_free_separable_GFp" is about a polynomial 'f' in a modular ring with a prime cardinality 'a'. It assumes that the cardinality of 'a' is greater than the degree of the polynomial 'f' and that 'f' is square-free. Under these conditions, it shows that the polynomial 'f' is separable. |
The cardinality of the set of number partitions N of a number n, where the size of N is k, is equal to the partition of n into k parts. |
The logical equivalence in the Floyd Logic (FL) is a bisimulation in the L language. |
If a type τ is less than a type σ[1], and for any type ρ, if τ equals ρ[1] and ρ is less than σ, then a proposition P holds. |
For any word of length 'a, the sum of negative one, 'b', and 'c' is equal to the sum of 'b' and the difference of 'c' and one. |
The lemma "replace_in_g_subtyped_cons" assumes that "replace_in_g_subtyped" is true for the context \<Theta>, the set \<B>, the environment G, the list of replacements xcs, and the environment G'. It also assumes that the well-formedness of the environment G with the addition of the tuple (x,b,c) is true under the cont... |
For any two objects x and y, the strict reference equality of x and y at a given state tau is the same as the strict reference equality of the constant functions of x and y at the same state tau. |
There exists a list 'xs' of a finite type 'a' such that the elements of 'xs' are distinct and the set of elements in 'xs' is equal to 'Y'. |
For a function 'g' mapping real numbers to a Euclidean space 'b', if 'g' is a polynomial function, then 'g' is also a path. |
If qc is not equal to qf and qc is not equal to qi, and qc is in the transition system derived from the reflexive closure over a non-hole context transition system with parameters Q, F, qc, qi, and qf for a term list u, then the variables in the term list u are all qc, repeated for the length of the variables in the te... |
The method declaration "foo_Ext" in the class "Ext" that extends the base class "Base" with the method "foo" is well-formed according to the rules of the Java programming language, as defined by the type program "tprg" and the well-formedness predicate "wf_java_mdecl". |
: if T is a finite set, S is a subset of T, for every element i in the difference of T and S, the function g applied to i equals the identity element, and for every element x in S, the function g applied to x equals the function h applied to x, then the function F applied to g and T equals the function F applied to h a... |
, given four fixed sets: Delta_A and Delta_B, which are both sets of tree automaton rules parameterized by a linearly ordered type 'q and 'f; and Delta_A_epsilon and Delta_B_epsilon, which are both sets of pairs of the type 'q, the function Delta_trancl_impl applied to these four sets is not equal to None. |
The relation-to-predicate function applied to a multiset relation parameterized by A is equal to the multiset relation of the relation-to-predicate function applied to A. |
The lemma named "valid_decompD" assumes that "ps" is a valid decomposition of "X" and that the pair "(h, U)" is an element of the set "ps". Under these assumptions, it shows that "h" is an element of the polynomial ring "P" over "X", "h" is not equal to zero, and "U" is a subset of "X". |
If 'x' is greater than 0 and 'i' is less than 1, then the product of 'i' and 'x' is less than 'x'. This is applicable for real numbers 'i' and 'x'. |
A set is finite if and only if it belongs to the range of the set function. |
The lemma named "arr_char" shows that a function "f" is an arrow if and only if "f" equals Zero, or "f" equals One, or "f" is in the image of the function "fromArr" applied to the set "Arr". |
If the diamond of p is less than or equal to the weakest precondition of x and q, then x is less than or equal to the relation of p and q. |
"P" is true when applied to "B" which is applied to "C F". |
The theorem named "wls_vsubstEnv_updEnv_fresh" assumes that the term "X" is well-sorted under the sort "s" and that the variable "y" is fresh for the set of variables "ys" in the term "X". It shows that if we first update the environment "rho" by substituting "X" for "x" under the sort "xs" and then perform a variable ... |
The substitution of a word in a list, where the list is composed of an element 'x' followed by a list 'xs', is equivalent to the function 'f' applied to 'x' concatenated with the substitution of the word in the list 'xs' using the same function 'f'. |
The lemma named "drop_at_shift" assumes that "n" is less than or equal to "index" and "index" is less than the length of "D". It shows that the element at the position "(index - n)" in the list obtained by dropping the first "n" elements from "D" is the same as the element at the position "index" in the original list "... |
The nth bit of the result of flipping the mth bit of a is true if and only if m is equal to n and the nth bit of a is not true, and it is possible for the type 'a to have an nth bit. |
In the context of a composition series, the lemma named "composition_series_snd_simple" assumes that the length of the series is greater than or equal to 2. It shows that the second group in the series is a simple group. |
:
Given the conditions that a positive integer k is greater than 1, an event occurs at time t if and only if the remainder of t divided by k equals k minus 1, t0 equals t times k, and s equals the composition of the output function and the execution of a computation stream with a transformation function, an input stre... |
If 'l' is less than or equal to 'n', then the function of 'l' is less than or equal to the maximum function of 'n' and 'f'. |
The theorem of call-by-value type preservation:
Assuming that
a: term 't' reduces to term 't'' in a call-by-value manner,
and
b: under context 'Γ', term 't' has type 'T',
it can be shown that under the same context 'Γ', term 't'' also has type 'T'. |
"the star of x times y is less than or equal to y" if and only if "x times y is less than or equal to y". |
The hereditary addition operation is associative. That is, for any three elements x, y, and z, adding x and y together first and then adding z is the same as adding y and z together first and then adding x. |
The lemma named "dim_gen_eigenspace_max_jordan_block" assumes a Jordan Normal Form (jnf) of a matrix A with a list of pairs n_as. It shows that the dimension of the generalized eigenspace of A with eigenvalue l and dimension d is equal to the order of l in the characteristic polynomial of A if and only if for all n, if... |
The word of an integer obtained from the unsigned integer representation of a word 'w' is equal to the word 'w' itself. |
Applying a composite wiring function (fg composed with fg') to a result (res) is equivalent to applying the wiring function fg to the result of applying the wiring function fg' to the same result (res). |
The lemma named "Disjoint_I" assumes the following:
1. The insertion of the variable 'i' in both 't' and 'u' into the set 'H' leads to a contradiction (falsehood).
2. The variable 'i' is independent of (or does not appear in) the pair (t, u).
3. For all elements 'B' in the set 'H', the variable 'i' is independent of 'B... |
The lemma named "exponential_initial_condition" makes several assumptions:
1. The time 't' is within the existence interval of 't0' and 'y0'.
2. The time 't' is also within the existence interval of 't0' and 'z0'.
3. The set 'Y' is a subset of 'X'.
4. For any 's' in the closed segment from 't0' to 't', the flow from '... |
For any two elements x and y in the carrier set K, the result of the field multiplication operation on x and y is the same as the result of the field multiplication operation on y and x. This is the commutative property of multiplication in a field. |
If block 'b' is a subset of the set 'V', then adding block 'b' results in an incidence system on 'V'. |
, given the assumptions that "C" is a part of the set of all possible communicating classes and for any "x" and "y", if "x" is in "C" and "x" can reach "y", then "y" is also in "C", it can be shown that "C" is an essential class. |
The single probability mass function of fast_uniform at a point x is equal to 1 divided by n if x is less than n, otherwise it is 0. |
The lemma named "length_prefix_ex'" assumes that "n" is less than the length of the list "xs". It shows that there exist lists "ys" and "zs" such that "xs" is equal to the concatenation of "ys", the nth element of "xs", and "zs", and the length of "ys" is equal to "n". |
If there is a path from node 'n' to node 'm' through a list of nodes 'ns' in graph 'g', then there is also a path from the head of the list 'ns' to node 'm' in the same graph 'g'. |
, assuming "p" is a linear polynomial (as indicated by "islin p"), the existence of a value "x" such that the interpretation function "Ifm" applied to the variable state "vs", the list "x#bs", and the polynomial "p" is true, is equivalent to either the interpretation function "Ifm" applied to "vs", "x#bs", and the poly... |
The guards of the Van der Pol oscillator are invariant in a four-dimensional real space. This is demonstrated through two cases. The first case involves the fourth derivative of y with respect to time at (1,t2) in the interval (-2, 0) and (-10, 10), and the second case involves the fourth integral of y with respect to ... |
For any type, the set of all update1 functions for any pair of x and y is separating. |
The lemma defines an alternative definition for watt, stating that a watt is equivalent to a joule per second. |
The successor of an empty set with any vertex 'v' is an empty set. |
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