Datasets:
UDACA
/
Split-Isa-AF-MMA

Dataset card Data Studio Files
xet
 Community
Statement:
stringlengths
7
24.3k
If a process R is followed by a relation of A, where R is defined as a process that for some value 'a', maintains the trace of A, does not wait, and increases the 'more' of A by 'a', and the relation of A is defined as a process that maintains the trace of A, does not wait, and decreases the 'more' of A, then this sequ...
The lemma "Subspace_eqdim_imp_equal" assumes that "U" is a subspace and the dimension of "U" is equal to the dimension of "V". It shows that "U" is equal to "V".
If the function 'f' is injective on set 'A', then a linear order on 'A' can be established using the linear order derived from the keys of 'A' through function 'f'.
For a relation R and a monad M, if a function transforms R into M and another function transforms R into an option type based on M, then these transformations preserve the relation between the "ask_option" operations in the context of the option type based on M.
If the absolute value of a real number 'z' is less than another real number 'a', then the series formed by the terms '((1/2) gchoose n) * a powr (1/2 - real_of_nat n) * z ^ n' converges to the square root of the sum of 'a' and 'z'. Here, 'n' is a natural number, 'gchoose' is a generalized binomial coefficient, and 'pow...
The limit superior (limsup) of a sequence "u" shifted by a constant "k" is equal to the limit superior of the original sequence "u".
If a polynomial 'p' is irreducible, then it is also square-free.
"four times x is equal to x union x union x union x".
The lemma named "smooth5" asserts that the number 5 is smooth.
The lemma named "subterm_wpo_ns_arg" assumes that "i" is an element in the set of the function "σ" applied to "f" and the length of "ss". It shows that the function "f" applied to "ss" is greater than or equal to the "i-th" element of "ss".
The intersection over the set containing elements i and j of the function f is equal to the intersection of the function f at i and the function f at j.
If the set of vertices 'uverts' of a graph 'G' is finite and 'G'' is a subgraph of 'G', then the set of vertices 'uverts' of the subgraph 'G'' is also finite.
The interpretation function "Ifm" applied to the boolean list "bs" and the disjunction "disj" of propositions "p" and "q" is equal to the interpretation function "Ifm" applied to the boolean list "bs" and the logical "Or" of propositions "p" and "q".
If 'c' is a constant and 't' is a term, then substituting the term 't' for the variable 'w' in the constant 'c' results in the constant 'c'.
The lemma named "ultra_filter_prime" assumes that "F" is an ultra filter. It shows that "F" is also a prime filter.
The lemma named "distincts_orbits_list" assumes that a list "as" contains distinct elements and "f" is a permutation. It shows that the list of orbits of "f" on "as" also contains distinct elements.
For a prime number 'p', the Legendre symbol of 'a' and 'p' equals zero if and only if 'a' and 'p' are not coprime.
If the ordermap of 'A' and 'r' with respect to 'y' is an element of the ordermap of 'A' and 'r' with respect to 'x', and 'r' is a well-founded relation, and 'r' is total on 'A', and 'x' and 'y' are elements of 'A', and 'A' is small, then the pair '(y, x)' is in the relation 'r'.
The lemma "InvariantUniqAfterApplyBackjump" assumes the following conditions: 1. The state maintains consistency (InvariantConsistent) in its assignment of literals (getM state). 2. The state ensures uniqueness (InvariantUniq) in its assignment of literals (getM state). 3. The state ensures that the watches (getWatch1...
: 1. The relation of interface property holds for the interface of relation S and S itself. 2. The relation of interface property holds for the interface of relation R and R itself.
The function C applied to the combination of a, b, and c in the order (a combined with b) then combined with c is equal to the function C applied to the combination of a, b, and c in the order a combined with (b combined with c).
If in a given context G, the state s1 transitions to state s1' through t1 in n1 steps with the result w1, and the state s2 transitions to state s2' through t2 in n2 steps with the result w2, then in the same context G, the state s1 can transition to state s1' through t1 in maximum of n1 and n2 steps with the same resul...
A match against the negation of the negation of a condition 'm' for a given packet 'p' and abstract firewall 'a' under a certain matching scheme 'γ' is equivalent to a match against the condition 'm' itself for the same packet 'p' and abstract firewall 'a' under the same matching scheme 'γ'.
If the proposition "P x" is true when condition "c" is true, and the proposition "P y" is true when condition "c" is not true, then the proposition "P" is true for "x" if "c" is true and for "y" if "c" is not true.
For any square matrix A of type 'a (where 'a is any commutative ring), the degree of the characteristic polynomial of A is equal to the cardinality of 'n (the dimension of the matrix), and the characteristic polynomial of A is monic (its leading coefficient is 1).
The lemma named "sum2Chain" is defined with three fixed elements: P, Q, and Q', all of which are of type pi. It assumes two conditions: first, that Q implies Q' through a tau transition, and second, that Q is not equal to Q'. If these conditions are met, it shows that the sum of P and Q implies Q' through a tau trans...
: "If set A is finite and the cardinality of set A is the successor of the successor of n, then there exists an element x that belongs to set A."
The interpretation of the composition of an interpretation function 'int' with a substitution function 'rho' applied to a term 'T' is equal to the interpretation function 'int' applied to the substitution of 'rho' in 'T'.
If the following conditions are met: 1. I is deterministic, 2. the round robin step of n0, sigma, s, and t' equals the floor of (t, the floor of (ta, x', m'), sigma'), 3. the round robin invariant of sigma and the domain of the thread s is true, 4. s is in I, then the round robin invariant of sigma' and the union of ...
The count of roots for a polynomial 'p' of real numbers is defined as follows: If 'p' equals zero, then the count of roots is zero. Otherwise, the count of roots is the difference between the number of sign changes at negative infinity and the number of sign changes at infinity for the Sturm sequence of 'p'.
Lemma: Tendency to Finite Map Let f be a function from natural numbers to finite maps from 'i' to 'a' (where 'a' is a metric space). Assume the following: 1. For all natural numbers n, the domain of f(n) is equal to the domain of g. 2. For all i in the domain of g, the sequence of f(n) applied to i converges to g(i)....
If a heap node value 'y' corresponds to a value 'x' under the relation of a pair of natural numbers mapped to 'A', then this implies that the same heap node value 'y' corresponds to the same value 'x' under the relation of a pair of natural numbers mapped to the pure form of 'A'.
If the function 'nu' is less than 'mu' and is a differential invariant under the conditions 'f', 'U', 'S', 't0', and 'G', and if the function 'nu' is greater than 'mu' and is also a differential invariant under the same conditions, then it can be shown that 'nu' is not equal to 'mu' and is a differential invariant unde...
The lemma "min_key_val_in" assumes that the first element of the ordered association list "xs" is not empty. It shows that the minimum key-value pair in the ordered association list "xs" is in the set of the first element of the list of the ordered association list "xs".
If the function "inv_after_left_moving" holds true for the given parameters (as, lm) and (s, l, []) with the result "ires", then the function "inv_stop" will also hold true for the parameters (as, lm) and (s', Bk # l, []) with the same result "ires".
For a given Van Emde Boas tree 't' and its corresponding implementation 'ti', the insertion operation 'vebt_inserti' on 'ti' with an element 'x' is equivalent to the insertion operation 'vebt_insert' on 't' with the same element 'x'. The time complexity of this operation is proportional to '13 + 13 * height of t'.
If we apply the 'mergesort_remdups' function to the list of tuples [(1, 2), (8,0), (8,1), (2,2), (2,4), (1,2), (2,2)], where the first element of the tuple is an IPv4 address and the second element is a natural number, the result will be a sorted and duplicate-removed list [(1, 2), (2, 2), (2, 4), (8, 0), (8, 1)].
The lemma bit_b assumes that "x" is not equal to "y", and "init" is either in the set {[x,y],[y,x]}. It also assumes that "qs" is in the language of the sequence [Plus (Atom x) One, Atom y, Atom x, Star(Times (Atom y) (Atom x)), Atom y, Atom y]. Under these assumptions, it shows that the random transition probability o...
Given two matrices A and B, where A has "nr" rows and "nc1" columns, and B has "nr" rows and "nc2" columns, the following two conditions hold: 1. If "i" is less than the number of columns in A ("nc1"), then the "i"-th column of the matrix formed by appending B to A column-wise is the same as the "i"-th column of A. 2...
The lemma named "cone_decomp_shift" assumes that "qs" is a valid decomposition of "X", a standard decomposition of "k", and an exact decomposition of "Suc m". It also assumes that "qs" is a cone decomposition of "T". Under these assumptions, it shows that "shift qs" is also a cone decomposition of "T".
If 'a', 'x', and 'b' are elements of the group and 'x' plus 'a' equals 'b', then 'x' is equal to 'b' plus the additive inverse of 'a'.
The U-variation of two functions 'ν' and 'ν'' over an empty set is equal to the statement that 'ν' is equal to 'ν''.
The blinding of a transaction in the mediator transaction tree is equivalent to embedding the transaction tree.
The function 'g' has a derivative 'g'' at a point 'x', if 'g' has a derivative 'g'' at 'x' within the interval up to 'x' and 'g' also has a derivative 'g'' at 'x' within the interval starting from 'x'. This is for any real number 'x'.
For any vectors 'x' and 'y', the dot product of vector 'a' and the rotation of the difference between 'x' and 'y' is equal to the negative of the dot product of vector 'a' and the rotation of the difference between 'y' and 'x'.
The type of the object, which is a combination of class C and state S, is equal to class C.
Given a valid network 'm', where the dimensions of the inputs match the input sizes of the network 'm', and 'j' is less than the output size of the network 'm', the evaluation of the network 'm' with the inputs at index 'j' is equal to the sum of the product of the inputs at index 'k' and the lookup of the tensors from...
A run is accepted with epsilon transitions in a non-deterministic finite automaton (NFA) R on a list of input symbols (bs # bss) and a list of states (cs) if and only if a run is accepted with epsilon transitions in the NFA resulting from the transition function delta applied to R and bs, on the remaining list of input...
The lemma "Lcm_factorial_least" assumes that for any element 'y' in set 'A', 'y' divides 'x'. It then shows that the least common multiple (Lcm) of the factorial of set 'A' also divides 'x'.
The lemma named "of_complex_ii_iff" shows that a complex number 'x' is equal to the homogeneous coordinate 'ii' if and only if 'x' is equal to the imaginary unit 'i'.
The lemma named "mono2mono_ldropn" shows that the function "ldropn" is monotone with respect to the "lprefix" relation. This means that if one list is a prefix of another, then dropping a fixed number of elements from the start of each list preserves this prefix relationship.
The lemma named "list_of_invariant" assumes two conditions. The first condition is that a list "xs" is represented by a node "r" in a heap "h" of a certain type 'a. The second condition is that for all references, if the references of node "r" in heap "h" are included in these references, then for every reference in th...
The lemma named "expands_to_MSLCons_0_asymp_equiv_hd" assumes that a function "f" expands to a multiseries "MS" with a leading term "C" and exponent "0", and the rest of the series "xs", and a function "g". It also assumes that the multiseries "MS" with the leading term "C" and exponent "0", and the rest of the series ...
The lemma named "filterlim_at_bot_linear_iff" is about a function 'f' that maps from a linearly ordered field to any type 'b'. Given that 'c' is not equal to zero, it shows that the limit of the function 'f' as 'x' approaches negative infinity (where 'x' is multiplied by 'c' and then added to 'b') converging to 'F2' is...
If "x" is less than or equal to "s" and "y" is less than or equal to "s", and for any "z", if "x" is less than or equal to "z" and "y" is less than or equal to "z" then "s" is less than or equal to "z", then the join of "x" and "y" is equal to "s".
The lemma named "callee_invariant_restrict_relp" includes lifting syntax. It assumes that the relation between callee1 and callee2 is a sub-probability monad function (rel_spmf) of the product of relation R and S. It also assumes that callee1 and callee2 are invariant under conditions I1 and I2 respectively. The lemma ...
If A is an object in the digraph of the set alpha, B is also an object in the digraph of the set alpha, A is a subset of B, A' is equal to A, B' is equal to B, and the digraph of the set alpha is equal to CC', then the inclusion of A in B is a morphism from A' to B' in the context of CC'.
Two papers, "pap" and "pap1", are considered equivalent excluding decision if and only if their title, abstract, content, reviews, and discussions are the same.
If 'x' is a unit, then 'x' is an 'n'th power up to a unit for any 'n'.
The lemma named "image_mset_thm" assumes that the multiset E is the image of another multiset E' under a function f. It also assumes that an element x is in the multiset E. It then shows that there exists an element y in the multiset E' such that x is equal to the image of y under the function f.
If "is_pr_Hom" holds for the function phi and sets A and B, then "is_pr_Hom" also holds for the function phi and the reverse images of sets A and B.
The join of "w" and "x" plus the join of "y" and "z" is less than or equal to the join of "w" and "y" with the join of "x" and "z".
The composition of substitutions "tau" and "sigma" applied to a set of atoms "AAA" is equivalent to first applying the substitution "sigma" to "AAA", and then applying the substitution "tau" to the result.
The lemma named "mult_mm_non_zero_l" assumes that the determinant of matrix A is not zero and matrix B is not a zero matrix. It shows that the product of matrix A and matrix B is not a zero matrix.
The set of type variables in the binding of a free variable "v" with type "T" in a term "t", union with the set of type variables in "T", is equal to the set of type variables in "t" union with the set of type variables in "T".
The complex conjugate of the sum of a function 'f' over a set 's' is equal to the sum of the complex conjugates of the function 'f' for each element 'x' in the set 's'.
The lemma named "arr_char" shows that the property "arr" of a transition "t" is equivalent to the property "Arr" of the same transition "t".
For any Y and x, if for every x the set Y x is countable, then the set F Y x is also countable. The second condition, named "cont", states that the function F is supremum-continuous. Given these assumptions, the lemma shows that the least fixed point of the function F at x, denoted as lfp F x, is countable.
The lemma named "fake_beta": Assuming that for any G and phi, the function (lambda x. {G, the unique y such that phi y x}) is proper in X, it shows that the equivalence of {the backdoor function applied to (lambda x. psi x), x to the power of P} and the universal quantification of psi applied to (x to the power of P) h...
The parallel composition of the product of (R minus the identity relation) and the identity relation, and the product of (S minus the identity relation) and the identity relation, with the non-conflicting relation, is equal to the product of the parallel composition of (R minus the identity relation) and the identity r...
If 'a' is an element of the carrier 'R' and 'a' is not equal to zero, then the inverse of the inverse of 'a' is equal to 'a'.
If "a" is less than or equal to "b" in a non-failing context, "b" does not fail, and "b" is less than or equal to "c" in a non-failing context, then "a" is less than or equal to "c" in a non-failing context.
If a function 'f' maps from a set of natural numbers less than or equal to 'n' to itself, and 'f' is an injective function on this set, and 'b' is less than or equal to 'n', then applying the function 'f' to the inverse of 'b' under 'f' and 'n' gives 'b'.
For any two real matrices A and B, the rank of the product of A and B is less than or equal to the rank of A.
If the minimum cost path from node u to node v, denoted as "mu f u v", is negative infinity, then there exists a path p from node u to node v such that there also exists a node w and a cycle c starting and ending at w, where w is in the set of vertices visited by path p, and the cost of cycle c is less than 0.
The minimum product operation is associative. That is, for any three elements A, B, and C, the minimum product of A and B, then with C, is equal to the minimum product of A with the minimum product of B and C.
The lemma named "invertible_Hermite_reduce_above" assumes that "n" is less than or equal to the natural number equivalent of "i". It shows that there exists a matrix "P" such that "P" is invertible and the result of the Hermite reduction of matrix "A" above a certain row "n" at positions "i" and "j" with a certain resu...
The lemma named "suminf_ennreal_2dimen" is about a function 'f' that takes a pair of natural numbers and returns an extended non-negative real number. It assumes that for any natural number 'm', the function 'g' at 'm' is equal to the sum of 'f' over all 'n' for a fixed 'm'. The lemma then shows that the sum of 'f' ove...
If 'x' is acyclic, and 'p' and 'q' are points such that 'p' is less than or equal to 'x;q', then 'p' is less than or equal to the negation of 'q' and 'p' is not equal to 'q'.
There exists a point x in the closed box from a to b such that for any positive number e, there exist points c and d such that x is in the closed box from c to d, this closed box is a subset of the ball with radius e centered at x, this closed box is also a subset of the closed box from a to b, and P does not hold for ...
The function 'set_child_nodes' is an instance of 'l_set_child_nodes' with respect to the type well-formedness 'type_wf' and the locations 'set_child_nodes_locs'.
The determinant of the matrix obtained by updating column i of matrix M with the sum of vectors a and b is equal to the sum of the determinant of the matrix obtained by updating column i of matrix M with vector a and the determinant of the matrix obtained by updating column i of matrix M with vector b.
If the set Ms is not additively independent, then the direct sum of all elements in Ms equals zero.
Given an integer k, if k is less than 0, then the function returns the negative of word_of_integer of the absolute value of k. If k is equal to 0, then the function simply returns 0. Otherwise, it divides k by 2 and gets the quotient (q) and the remainder (r). If the remainder is 0, then the function returns twice the ...
The lemma "scopeExtParLeft" is defined with fixed elements P and Q of type pi, a of type name, lst of type "name list", and Rel of type "(pi x pi) set". It assumes that "x" is fresh for "P", the identity relation is a subset of "Rel", "Rel" is equivariant, for any "R", "S", and "y", if "y" is fresh for "R", then the ...
The multiset extension function "mulex1" with any predicate "P" and any multiset "M" applied to an empty multiset is equivalent to "False".
The theorem named "pure_universal_release_simp" assumes that "nu" is purely universal. It shows that a sequence "w" satisfies the release of "phi" by "nu" if and only if "w" satisfies "nu".
The lemma "stable_config_after_final_ge_2'" assumes that after a certain number of steps "n1", the Turing machine reaches a stable configuration "(0, l', r')" from an initial configuration "(s, l, r)" under a given program "p". It also assumes that "n1" is less than or equal to "n2". It then shows that even after "n2" ...
If 'x' is not in set 'A', then the probability mass function of 'p' restricted to 'A' at 'x' is equal to zero.
The lemma "PhiDef_defNode_eq" assumes that a node "n" is in the set of nodes of a graph "g", that "n" is in the set of nodes where a phi function is defined for a variable "v" in the graph "g", and that "v" is a variable in the graph "g". It shows that the node where the variable "v" is defined by a phi function in the...
The lemma named "uniform_limit_bounded" is defined with respect to a function 'f' that maps from an arbitrary type 'i' to a topological space 'a' and then to a metric space 'b'. It assumes three conditions: 1. 'l' is a uniform limit of the function 'f' over a set 'S' with respect to a filter 'F'. 2. For almost all '...
If for any integer 'i', the pair consisting of 'r i' and 'r (Suc i)' belongs to 'E', then 'r' is an infinite path in 'E'.
The Cartesian product of the empty set with any set B is the empty set.
If "M" is a Kripke model, "w'" is in the set of worlds generated by the model "M" and "w", and there is a relation between "w'" and "w''" in the model "M" for an action "a", then there is also a relation between "w'" and "w''" in the set of relations generated by the model "M" and "w" for the same action "a".
The lemma named "cycle_rotate" is defined with a fixed matrix M, which is of type 'a. Here, 'a is a linearly ordered additive abelian monoid. The lemma assumes that the length of the list xs is greater than 1 and that the pair (i, j) is in the set of arcs' of xs. It then shows that there exist lists ys and zs such that...
If "x" is less than "y", then the function "enn2ereal" applied to "x" is less than "enn2ereal" applied to "y". This is under the assumption that "x" is less than "y".
If a tau-execution mover in a certain context, program, thread, and environment moves from state s to state s', then the same tau-execution mover can move from the same state s to the same state s' in a try-catch block in the same context, program, and thread.
Given the assumptions that a new shadow root is created in the heap 'h' resulting in a new heap 'h'' and a new shadow root pointer 'new_shadow_root_ptr', it shows that the list of disconnected nodes in the document pointed to by the new shadow root in the new heap 'h'' is empty.
A function 'f' has an integral 'y' over an interval 'i' if and only if either of the following conditions is true: 1. There exist numbers 'a' and 'b' such that 'i' is the closed interval from 'a' to 'b'. In this case, 'f' has an integral 'y' over 'i'. 2. For every positive number 'e', there exists a positive number '...
A finger tree "t" is not empty if and only if the view from the left of the tree "t" is not None.