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train · 499k rows

Statement: stringlengths 7 24.3k
A sparse grid is defined as a set of points 'p' such that the length of 'p' equals 'dm', the level of 'p' is less than 'lm', and for all 'd' less than 'dm', the index of 'p' at 'd' is odd, greater than 0, and less than 2 to the power of the level of 'p' at 'd' plus 1.
The lemma named "samp_uni_plus_one_time_pad" shows that mapping a function, which takes a variable 'b' and returns the modulus of the sum of 'y' and 'b' with respect to 'q', over a uniformly distributed sample of 'q', is equal to the uniformly distributed sample of 'q' itself.
If the function "f" applied to "x" using the diamond operator is in the set "CF", then there exists some "x'" such that "x" is equal to "Opaque x'".
The lemma "rank_1_proj_carrier" shows that the rank 1 projection of a vector "v" is in the carrier matrix of the dimension of vector "v" by the dimension of vector "v".
The lemma named "BlueDFS_invar_eq" shows that the DFS (Depth-First Search) invariant for the graph G with blue DFS parameters and state s is equivalent to the BlueDFS invariant for the graph G with accepted state s.
If "rects" is a list of rectangles, then the intersection of these rectangles with an empty set at the 0th index is equivalent to these rectangles having an overlap.
The application of the function "app\<^sub>i" with the parameters "(AStore, P, pc, mxs, T\<^sub>r, (T1#T2#T3#ST,LT))" is equal to the condition where "T2" is an Integer and "T3" is either a Ty or NT, otherwise it is false.
The probability mass function (pmf) of the stochastic dominance strategy (sds) R8 for outcome 'd' is equal to 1 minus the pmf of the sds R8 for outcome 'a'. The lemma R8_c states that the pmf of the sds R8 for outcome 'c' is equal to 0. The lemma R26_a states that the pmf of the sds R26 for outcome 'a' is equal to 1...
If there is a path from l1 to l3 through the concatenation of lists xs and zs, and there is a path from l1 to l2 through list ys, and there is a path from l1 to l2 through list xs, then there is a path from l1 to l3 through the concatenation of lists ys and zs.
If two sets A and B are mutually distinct, and the union of the states of all elements in A does not intersect with the union of the states of all elements in B, then the union of sets A and B is also mutually distinct.
Mapping a function "f" over a range from 0 to the successor of "n" is equivalent to applying the function "f" to 0 and then prepending it to the list obtained by mapping the function "f" applied to the successor of each element in the range from 0 to "n".
The lemma "sep_combine1" states four equivalences: 1. The conjunction (or combination) of the assertion P and the assertion Q is equivalent to the assertion of P and Q. 2. The conjunction of the pure assertion P and the pure assertion Q is equivalent to the pure assertion of P and Q. 3. The conjunction of the assertio...
The lemma named "distinct_paths" assumes that Q is an element of P, R is also an element of P, d is not an element of Q, and d is an element of R. Under these assumptions, it shows that R is not equal to Q.
The function "ap_set" preserves the relation between two sets of functions, where the functions themselves preserve a relation between sets A and B. This is also true for the relation between two functions that preserve the relation between two sets A and B.
If the following conditions are met: 1. The function 'f' commutes with the augmented power-product set 'aug_pm_set' for a given 'z', 'i', and set 'A', 2. The function 'f' is associative with the augmented power-product set 'aug_pm_set' for a given 'z', 'i', and set 'A', 3. The first condition for an ideal power-prod...
The lemma named "iso_lunit" assumes that "f" is an identity. It shows that the left unit of "f" is an isomorphism.
If a state 's' is less than 'n' and is assigned the value 'i', then multiplying 's' by 'm' is equal to 's' multiplied by 'm'.
For any proposition P and any time point t', t' releases t over an interval I is false if and only if the interval I is empty.
If the negation of p and the negation of q implies that x is within the scope of the negation of q, then the negation of q implies that the star of the negation of p and x is within the scope of the double negation of p and the negation of q.
The lemma RD_on_mono demonstrates that the function RD_on, when applied to the variable ds, is monotonic.
The condition for negative distribution over the relation set A and A' is true.
For a function 'f' mapping from type 'b' to a complete boolean algebra 'c', the diamond modality of 'f' at 'x' is equal to the negation of the box modality of the negation of 'f' at 'x', where 'R' is a relation.
The lemma named "streamValue2" assumes that for all natural numbers 'j' less than or equal to 'D' plus some natural number 'z', the value of the stream at time 't + j' equals 'x'. It then shows that for all natural numbers 'j' less than or equal to 'D', the value of the stream at time 't + j + z' also equals 'x'.
The lemma named "viewR_correct_nonEmpty" assumes that the finger tree "t" satisfies the invariant "ft_invar" and "t" is not empty. It then obtains an element "a" and a sequence "s" where the right view of "t" equals the pair of "a" and "s", and "s" satisfies the invariant "ft_invar" and the list representation of "t" e...
: 1. The set of operations in the unlabeled version of B is a subset of the set of operations in the unlabeled version of the concatenation of A and B. 2. The set of operations in the projection of the unlabeled version of B at position n is a subset of the set of operations in the projection of the unlabeled version...
For any natural numbers a, b, and d, the difference of a and the modulus of a and b is less than or equal to the difference of d and b, given that a is less than d and b divides d.
Lemma Chernoff_inequality_nn_integral_ge: Assume s is greater than 0 and A is in the sets of M. Also assume that the function (f(x) * indicator A x) is borel measurable in M. Then, the emeasure of M for the set of x in A such that f(x) is greater than or equal to a is less than or equal to the product of ennrea...
: Assuming that the function 'asi' takes 'rank', 'root', and 'cost' as parameters, and for all pairs of elements 'xs' and 'ys' in set 'Y', either 'xs' equals 'ys' or the intersection of the sets 'xs' and 'ys' is empty. Also, for every element 'xs' in 'Y', the function 'forward' applied to 'xs' holds true. The empty se...
If there is a descending group chain 'f' of length 'n' in group 'G' and the 'n'th element of 'f' is not the same as the first element of 'f', then the smallest 'j' such that 'f j' is in the set of 'f i' for 'i' less than or equal to 'n' and not equal to the first element of 'f', is less than or equal to 'n' and 'f' of ...
The domain of a finite map, after a set A has been dropped from it, is equal to the original domain of the finite map minus the set A.
, given the assumptions that a polynomial 'p' is in the positive hull of the set 'ps' and that 'p' is not equal to zero, there exists a polynomial 'g' such that 'g' is in the set generated by the Macaulay list of 'ps', 'g' is not equal to zero, and the leading term of 'p' is equal to the leading term of 'g'.
The lemma named "h_minus_aux_cycle" is defined with a fixed variable "j" and "C" is defined as the enumeration of cycle "j". Assuming that the set of cycles "Delta" is not empty, it shows that there exists an edge "e" in the set of cycle edges "C" such that the function "h_minus_aux" applied to the successor of "j" and...
The lemma named "ginitial_impl_refine" assumes that "v0" is not in "goD s0" and "go_is_no_brk s0" is true. It also assumes that "s0i" is related to "s0" in "goGS_rel" and "v0i" is identical to "v0". Under these assumptions, it shows that the initial implementation of "g" with "v0i" and "s0i" is related to the initial "...
The pair consisting of the set 'intY1' and the order induced by 'intY1' and 'r' belongs to the Complete Lattice.
The keys of the map obtained by filtering the map 'm' using the predicate 'P' are a subset of the keys of the original map 'm'.
The lemma named "close_sym" assumes that "g" has minimum graph properties (mgp), "u" is a vertex in graph "g" (ug), and "v" is close to "u" in graph "g" (cl). It shows that "u" is also close to "v" in graph "g". This lemma demonstrates the symmetry of closeness in the graph.
The proposition "R" is implied by both "P" and "Q" is equivalent to the proposition "R" being implied by "P" and then "Q".
The cardinality or the number of elements in the set 'sdim' is 7.
: Assuming that 'u' is a unit in the monoid 'G', 'a' is not a unit in 'G', 'as' are factors of 'a' in 'G', and the set of 'as' is a subset of the carrier of 'G', it can be shown that the factors of 'a' multiplied by 'u' in 'G' are the factors 'as' with the first element replaced by the first element of 'as' multiplied...
If two variables x and x' are equal, and for all pairs of variables a and b, the relation R holds between the permutation of x' by swapping a and b and x' if and only if the same holds for the relation R', then the support of the relation R on x is equal to the support of the relation R' on x'.
If a document pointer "document_ptr" is updated with a setter function "setter" and a value "v" in a heap "h" resulting in a new heap "h'", then the getter function "getter" for character data from a character data pointer "character_data_ptr" is preserved between the original heap "h" and the new heap "h'".
The lemma named "exp_integer_2pi_plus1" assumes that "n" is an integer. It shows that the exponential of the complex number obtained by multiplying "n" by 2πi and then adding πi is equal to -1.
If "f" and "g" are Zermelo-Fraenkel functions and the cardinality of the domain of "g" is equal to the cardinality of the codomain of "f", then the composition of "f" and "g" is also a Zermelo-Fraenkel function.
There is a bijective correspondence (bijection) between the set of natural numbers less than 'k' and the one-dimensional cube of size 'k'. This is achieved through a function that maps each element 'x' in the set of natural numbers less than 'k' to an element 'y' in the set of natural numbers less than 1.
An empty list is less than a list that has at least one element.
The lemma "omit_redundant_points_2" assumes that "p" is a point. It shows that the intersection of "p" and the reflexive transitive closure of "x" is equal to the union of the intersection of "p" and the identity relation, and the product of the intersection of "p", "x", and the complement of the transpose of "p", and ...
If C is an Object, then in the context of P, an Array A is less than or equal to Class C.
Given the following assumptions: 1. "wf_prog G" which means G is a well-formed program, 2. "class G C = Some c" which means that in program G, class C is defined and equals to some class c, 3. "C \<noteq> Object" which means that class C is not the Object class, 4. "methd G (super c) sig = Some old" which means that t...
The lemma named "dist_lemmas_aux" makes several assumptions and shows several results: Assumptions: 1. The set C is finite. 2. The set C is not empty. Results: 1. The set of distances from a point 's' to each point in set C is finite. 2. If the set of distances from a point 's' to each point in set C is finite, then ...
If function 'f' is positive on a set 'M', then function 'f' is also positive on the lambda system generated by set 'M' and function 'f' in the space 'Omega'.
The lemma named "final_notin_to_finaln" assumes that a given command 'c' and state 's' under context 'Γ' does not lead to a set 'T'. It shows that the same command 'c' and state 's' under context 'Γ' does not lead to the set 'T' after 'n' steps.
In the context of Yoneda, the lemma named "sandwich_type" assumes that A and B are objects, and "a" is an element of the object function of A. It then shows that the sigma function of A and "a" applied to B is an element of the hom-set from the set of morphisms from A to B to the object function of B.
If 'a' and 'b' are elements in the domain of 'r', and the value of 'r' at 'a' is equal to the value of 'r' at 'b', then 'a' must be equal to 'b'. This implies that the function 'r' is injective, meaning each element in the domain of 'r' maps to a unique element in the range of 'r'.
The lemma named "corthogonal_inv_result" assumes that a matrix A, which belongs to a field that can be conjugated, is corthogonal. It shows that the inverse of this corthogonal matrix A is its own inverting matrix.
If the length of the list "xs" is not zero, then the first element (head) of the list "xs" is included in the set of elements of the list "xs".
The lemma named "integral_integrable_2dim" is about a function 'f' that maps from a pair of Euclidean spaces 'a' and 'b' to a Banach space 'c'. The lemma assumes that the function 'f' is continuous on the closed box defined by the points (a,c) and (b,d). It then shows that the function 'x' that maps to the integral ove...
If X is a nonstandard bounded sequence and N is an element of infinitely large hypernatural numbers, then the hyperreal extension of X at N is a finite hyperreal number.
The intersection of the range from "a" to "b" and the range from "c" to "d" is empty if and only if for all "x", it is not the case that "x" is greater than or equal to "a" and less than or equal to "b" and also greater than or equal to "c" and less than or equal to "d".
If 't' is a red-black tree, then looking up in the red-black tree obtained by folding the function (which inserts key-value pairs into the tree) over a list 'xs' and the tree 't', is equivalent to looking up in the original tree 't' and appending the reverse of the map of the list 'xs'.
"y" multiplied by "a(z)" is less than or equal to "a(x)" multiplied by "y" if and only if "d(x)" multiplied by "y" and "a(z)" equals the bottom element.
The complex conjugate of the tangent of a complex number z is equal to the tangent of the complex conjugate of z.
The set generated from a set A and the rank of an element x is equal to the set generated from A and the element x itself.
The constant "#c" preserves non stuttering invariance.
If a property P does not hold for an element x, then removing x from the list that results from filtering list xs by property P, will result in the same list as filtering xs by property P.
The composition of the set union of 'ov', 's', 'm', 'the inverse of f', 'the inverse of d', 'e', and 'the inverse of s' with the set union of 'the inverse of ov', 'the inverse of s', 'the inverse of m', 'f', 'd', 'e', and 's' is a subset of the set union of 's', 'the inverse of s', 'f', 'the inverse of f', 'd', 'the in...
The polynomial created by mapping each element of polynomial 'p' to be multiplied by 'a' is the same as the polynomial 'p' scaled by 'a'.
If 'k' is an ordinal and property 'P' holds for 'k', then there exists an ordinal 'n' such that property 'P' holds for 'n' and for all ordinals 'm' where property 'P' holds, 'n' is less than or equal to 'm'. This essentially means that 'n' is the smallest ordinal for which property 'P' holds.
The lemma named "qbs_morphism_comp" assumes that there is a quasi-Borel morphism from set X to set Y denoted by "f" and another quasi-Borel morphism from set Y to set Z denoted by "g". It shows that the composition of "g" and "f" is a quasi-Borel morphism from set X to set Z.
The opposite natural transformation epsilon maps the opposite functor TT to the composition of the opposite functors GG and KK in the category of left Kan extensions indexed by alpha. This mapping occurs from the opposite category BB to the opposite category CC, and is mapped to the opposite functor HH from the opposit...
The set of elements in the incidence vector of Vs with respect to bl is a subset of the set {0, 1}.
: If the following conditions are met: - The axiom EQU001_0_ax of equality is satisfied, - The axioms GRP003_0_ax and GRP003_0_eq of group theory, which involve the operations of multiplication, inversion, identity, and product, are satisfied, - For all pairs of elements A and B, if A is equal to B, then the function ...
The lemma named "matchNilLeft" is defined with fixed variables "a" and "b" of type "name", and "P" of type "pi". It assumes that "a" is not equal to "b". It shows that the null process leads to the process "P" under the relation "Rel", where "P" is prefixed by the match of "a" and "b".
If the product of "an of y" and "x" is less than or equal to the product of "x" and "an of y", then the product of "an of y" and the omega power of "x" is equal to the product of "an of y" and the omega power of the product of "an of y" and "x".
If X is an element of the set of items H, then inserting X into H does not change the set of items in H.
There exists some FSMs (Finite State Machines) M2 and M1, and a natural number m, for which the adaptive state counting fault domain holds true.
Any complex number 'a' is equal to the sum of its real part and its imaginary part multiplied by the imaginary unit 'i'.
For a given permutation "p" of type "name prm" and a list "xvec" of type "'a::pt_name list", the length of the list obtained by applying the permutation "p" to "xvec" is equal to the length of "xvec".
For any natural number 'n', the sequence defined by 'a''^{m}_n' as 'm' goes to infinity, converges to 'a'_n.
In any reachable state 's', if a post ID 'PID' is not in the set of post IDs of 's', then the post corresponding to 'PID' in 's' is an empty post.
The lemma named "expands_to_hd" assumes that a function 'f' expands to a multiseries 'MS' with a leading series 'MSLCons' composed of a multiseries 'MS ys h', an exponent 'e', and a series 'xs', and a function 'g', in the context of a basis that includes 'b'. If this assumption holds, it shows that the function 'h' exp...
If the union of the directed arcs of tree t1 and element e1 has no intersection with the union of the directed arcs of tree t2 and element e2, then the set of second elements from the list converted from the directed tree with a node r and a set containing a pair of tree t1 and element e1 also has no intersection with ...
If an integer 'm' is greater than or equal to zero, then the sum of 'm' and the integer representation of 'n' is equal to the integer representation of the sum of the natural number representation of 'm' and 'n'.
If a match "m" does not have a primitive, then the equivalence of "m" matching any is the negation of "m" not matching any.
The Hoare triple with precondition p, program 1 (which represents the skip command), and postcondition p is valid.
If x is an element of set S, then the negative of x is also an element of set S.
For any given destination IP 'dip', the next hop in the invalidated routing table for the destinations in the system state 'xi' is the same as the next hop in the original routing table for the system state 'xi'.
The lemma "undo_factorize_correct" assumes that the factorization of a monic rational polynomial "p" equals "(c,fs)". It also assumes that for any pair "(f,p)" in the set "fs", "f" is in the set "ftrs". It shows that the result of the function "undo_factorize" applied to "(c,map (\<lambda>(f,pow). (index_of ftrs f, pow...
If the post-state of variable 'x' does not affect the merge operation 'M', then the post-state of 'x' will also not affect the parallel composition of processes 'P' and 'Q' with respect to 'M'.
The range of the intersection of a set F is equal to the power set of F.
Function g is in Big Omega of function f if and only if function f is in Big O of function g.
The lemma named "jvm_Invokestatic" assumes the following conditions: 1. The program P is equivalent to the second compilation of the program P₁. 2. The program P, class C, method M, and program counter pc are executing the static method invocation of method M' in class C' with the length of Ts. 3. The state of the sys...
If "x" is in the set of assertions, then "x" is also in the set of conjunctives.
If there exists an element 'x' in the set that consists of 'y' and the elements of list 'xs', such that 'x' is a sublist of 'as', then there exists an element 'x' in the set of combined lists of 'P', 'y', and 'xs', such that 'x' is a sublist of 'as'.
Vertices v1 and v2 are adjacent if and only if v1 is incident with the set {v1, v2}, v2 is incident with the set {v1, v2}, and the set {v1, v2} is an element of the edge set E.
: Assuming for any elements 'a' and 'b' in set 'A', the relation 'r' between 'a' and 'b' is equivalent to the relation 'r'' between 'a' and 'b', it can be shown that the total pseudo-ordered set 'A' with relation 'r' is equivalent to the total pseudo-ordered set 'A' with relation 'r''.
The negation of a monomial with coefficient 'a' and degree 'n' is equal to a monomial with coefficient '-a' and the same degree 'n'.
If a state sigma satisfies the symbolic execution of an if-then-else statement (if P then B1 else B2) followed by M, then if P holds in sigma and sigma satisfies the symbolic execution of B1 followed by M implies Q, and if P does not hold in sigma and sigma satisfies the symbolic execution of B2 followed by M implies Q...
If a list 'us' is not empty and is a part of the lists generated from a set 'G', then refining 'us' with respect to 'G' is equivalent to decomposing the first element of 'us' with respect to 'G' and then refining the tail of 'us' with respect to 'G'.
If the input-output pair "IO out c" is an element of the set of sub-probability mass functions (set_spmf) of the generative probabilistic value (the_gpv gpv), then the interaction bound of the continuation function "c" with input "input" is less than or equal to the interaction bound of gpv. If the output "out" is cons...
The matrix "mat_Ph" is equal to twice the matrix "pre" minus the N-dimensional identity matrix.

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