Datasets:
UDACA
/
Split-Isa-AF-MMA

Dataset card Data Studio Files
xet
 Community
Statement:
stringlengths
7
24.3k
If a function "f" applied to the successor of "n" tends to a limit "l" as "n" goes to infinity, then the function "f" itself also tends to the same limit "l" as "n" goes to infinity.
If a function 'f' is anti-monotonic and a set 'A' is bounded above, then the image of 'A' under 'f' is bounded below.
: Given the assumptions that a pair (xs, ys) is in the list relation of Ra, a pair (c, d) is in the relation Rs mapping to the boolean relation, for any pair (x, y) in Ra, if x is in the set xs and y is in the set ys, then the pair (f x, g y) is in the relation Rs mapping to Rs, and a pair (a, b) is in Rs, it can be sh...
For any two elements 'a' and 'b' of the type 'floatarith', the difference of 'a' and 'b' is equal to 'a' added to the negation of 'b'.
"fa2" is included in its own subterms.
There is a trail from node 'u' to node 'w' through the edge 'e' followed by a sequence of edges 'es' in graph 'G' if and only if the following conditions are met: 1. The edge 'e' is an arc in the graph 'G'. 2. The node 'u' is the tail of the edge 'e' in the graph 'G'. 3. The edge 'e' is not in the set of edges 'es'....
Encoding and transition relation imply that the induced relation weakly respects success. It fixes 'success' as a type of barbs and 'TRel' as a set of processT pairs. It assumes that encoding weakly respects the barb set of success, the relation weakly preserves the barb set of success, and the relation weakly reflects...
If "f" is a nucleus, then the fixed point set of "f" is an infimum-closed set.
If Fr_1b holds for a set F, then the set of all boundaries A (denoted as Br A) is closed under infimum.
There exists a function 'f' such that 'f' establishes an ordering 'ord' for the set consisting of 'a' and 'b'.
If c is an element of Wax, then the mcalc of sigma and c in the TE theory holds true.
The function C applied to the product of x and y is equal to the product of the function C applied to x and y.
For any fixed bound variables bv1 and c, the permutation of bv1 and c applied to the function type AF_fun_typ with parameters x1, b1, c1, τ1, and s1 is equal to the function type AF_fun_typ with parameters x1, and the permutation of bv1 and c applied to b1, c1, τ1, and s1 respectively.
If the successor of a number 'n' is greater than or equal to the size of a list 'xs', then swapping the 'n' element in the list 'xs' will result in the same list 'xs'.
If list L and list ls are distinct, then the list resulting from the transitive closure function applied to L and ls is also distinct.
The "strong until" operator (represented by "suntil") for two propositions phi and psi is equivalent to the least fixed point (represented by "lfp") of a function P on a stream s. This function P returns true if either psi holds for the stream s, or if phi holds for the stream s and P holds for the tail of the stream (...
: Assuming "Q u" equals "enat d" (as stated in the assumption "UNS") and for all "v", "enat d" is less than or equal to "Q v" (as stated in the assumption "MIN"), we can obtain a "v" such that the shortest path from "u" to "v" exists and the edge from "v" to "u" is a light edge in the graph "S" with the set of vertices...
If "\<FF>" is a semicategory morphism from "\<AA>" to "\<BB>" in the context of an arbitrary universe "\<alpha>", then "\<FF>" is also a morphism from "\<AA>" to "\<BB>" in the digraph of semicategories in the same universe "\<alpha>".
If we assume "R" to be a refinement from a heap node "P" to a computation "c" with postcondition "Q" and return value "R" for a method "m", and if we also assume that "Q" implies "Q'", then "R" is also a refinement from "P" to "c" with postcondition "Q'" and return value "R" for the method "m".
"H function applied to parameters p, Ref function applied to parameters p and q, and q".
The lemma states the identity law of multiplication for real numbers. It assumes that 'x' is a real number. It shows that the product of 'x' and 1 is 'x'.
Lemma fls_lr_inverse_mult_ring1: Let f and g be formal Laurent series over a ring with unity. Assume that x times the coefficient of f at its subdegree equals 1, and the coefficient of f at its subdegree times x equals 1. Also assume that y times the coefficient of g at its subdegree equals 1, and the coefficient...
If there is a fold graph with function 'f', base case 'z', and set 'B' for 'x', and another fold graph with the same function 'f', base case 'z', and set 'B' for 'y', and if set 'B' is a subset of set 'A', and 'z' is an element of 'A', then 'y' is equal to 'x'.
The lemma "rank_None_Suc" assumes that "x" is less than or equal to "n" and the rank of "x" at "n" is "None". It shows that the rank of "x" at the successor of "n" is also "None".
For any two elements 'x' and 'y' of a linearly ordered set, one of the following three conditions must be true: 'x' is equal to 'y', 'x' is less than 'y', or 'y' is less than 'x'.
The lemma named "firstPosLast" assumes that the list "l" is not empty and that "l" contains unique elements. It shows that the first position of "x" in the list "l" being equal to the length of the list "l" minus one is equivalent to "x" being the last element in the list "l".
The lemma named "isometry_on_UNIV_iterates" is about a function 'f' that operates in a metric space. If 'f' is an isometry on the universal set, then the lemma shows that the 'n' iterations of 'f' is also an isometry on the universal set.
If 'x' and 'y' are elements of the ring 'R' and either 'x' or 'y' is zero, then the ring multiplication of 'x' and 'y' is zero.
The validity of the heap map representation alpha of a heap map is equivalent to the validity of the heap map itself.
The leading element of the Hamming sequence is 1.
The value of "a'_n" is equal to "(1 - v to the power of n) divided by delta" under the condition that "i" is not equal to zero and "n" is greater than or equal to zero. This is applicable for any real number "n".
The norm bound of a matrix A is the maximum of the norms of the elements of A, where each element is identified by its row index i and column index j, and i is less than the number of rows in A and j is less than the number of columns in A.
The pell power of a number 'z' to the power of 'm + n' is equal to the multiplication of the pell power of 'z' to the power of 'm' and the pell power of 'z' to the power of 'n'.
A sequence satisfies the next step of the negation of a property if and only if it satisfies the negation of the next step of the property.
The sum of the time complexity of merging two trees (t1 and t2), the potential of the merged tree, and the negation of the potentials of the original trees, is less than or equal to the sum of the rightmost leaf rank of the merged tree, the rightmost leaf rank of the first tree, the rightmost leaf rank of the second tr...
: Assuming that there is no edge leading out of the sink, no edge leading into the source, and every edge in the edge set has a positive capacity, it can be shown that there exists a flow 'f' and a cut 'S' such that 'f' is a flow in the network and 'S' is a cut in the network, and 'f' and 'S' are orthogonal to each oth...
If the following conditions are met: 1. I is a finite set. 2. For every element i in I, F leads from A(i) to A'(i). 3. For every element i in I, F stabilizes A'(i). Then, F leads from the intersection of all A(i) for i in I to the intersection of all A'(i) for i in I.
The domain of the map of system C3PO is a subset of the set of nodes in the policy.
The lemma "obtain_qFresh_card_of" assumes that the cardinality of set V is less than the cardinality of the universal set of 'var' type, and the cardinality of set XS is also less than the cardinality of the universal set of 'var' type. It also assumes that for all X in XS, X is 'qGood'. The lemma shows that there exis...
If for all k less than or equal to 1, the ideal R is Jk, Bk equals the quotient ring R of Jk, and Sk equals the projection of R on Jk, then the product of the quotient rings R of J0 and J1 mapped from A to the product of the additive group R where j is less than or equal to 1 and Sk, is equal to the mapping from A to t...
If a valid edge 'a' exists in the well-formed program 'wfp' with a kind 'a' equal to 'Q:r↪pfs', and 'a'' is in the set of return edges of 'a' in 'wfp', then there exists 'Q'' and 'f'' such that the kind of 'a'' is equal to 'Q'↩pf''. In other words, only return edges are in the set of return edges for a given edge in a ...
"SN" (Strong Normalization) holds for the set of all pairs of natural numbers (x, y) where x is less than y and y is less than or equal to a certain bound b.
The assertion function applied to a predicate 'p' of type 'a', where 'a' is a distributive lattice, belongs to the set of disjunctive functions in the Apply module.
The lemma named "freshTop" establishes three things: 1. The name 'x' is fresh for the top element. 2. The list of names 'xvec' is collectively fresh for the top element. 3. The set of names 'Xs' is collectively fresh for the top element. Here, "fresh" means that the names are not used or do not appear in the top ele...
If the predicate P holds for 1, then the value of the function "fds_subseries" for P and 1 is 1.
The lemma named "distinct_mset_mono" assumes that the list "ys" contains distinct elements and the multiset of "xs" is a subset of the multiset of "ys". Under these assumptions, it shows that the list "xs" also contains distinct elements.
"lex_comp" is equal to a comparator of type 'a.
The lemma named "length_gt_1_Cons_snoc" assumes that the length of the list "ys" is greater than 1. It then obtains elements "x1", "xs", and "x2" such that the list "ys" can be expressed as the element "x1" followed by the list "xs" and ending with the element "x2".
If the image of set A under the function "of_complex" is equal to the image of set B under the same function, then set A is equal to set B. This is assumed under the condition that "of_complex ` A = of_complex ` B".
The lemma named "tl_x_not_x" shows that if "x" is not an empty list, then the tail of "x" is not equal to "x".
If 's' is greater than 1 (where 's' is a real number), then the absolute value of the Dirichlet series for the Möbius function converges at 's'.
The supremum preserves the operation on K.
The lemma named "diag_mat_map" assumes that the matrix M is an n by n matrix. It shows that the diagonal of the matrix obtained by applying a function f to all elements of M is equal to the result of applying the function f to the diagonal of the matrix M.
If "rects" is a list of rectangles, "ops" is the list of all operations on "rects", and "k" is a number less than the length of "ops", then the operation at the k-th index of "ops" is an interval.
The compilation of program P using function f does not change the subtype relation between type τ and type τ'. In other words, if type τ is a subtype of type τ' in program P, then it remains a subtype in the compiled program.
If a finger tree 'm' is at level 'n+1' and is measured, and if a digit 'pr' is at level 'n' and is measured, and if every element 'x' in the set 'sf' is a measured node at level 'n' and the length of 'sf' is less than or equal to 4, then the finger tree resulting from the operation 'deepR' with 'pr', 'm', and 'sf' as a...
If "p" is less than or equal to the diamond of "y" and "d i", and "d i" is less than or equal to the diamond of "x" and "i", and "d i" is less than or equal to "d q", then "p" is less than or equal to the diamond of "y" and the loop of "x" with inverse "i" and "q".
The box of the star of x applied to n of y is equal to the box of the star of the product of n of y and x applied to n of y.
The lemma named "max_pos_imp_Interp" assumes the following conditions: 1. C is an element of N. 2. C is not an empty multiset. 3. The maximum element of the multiset C is the positive A. 4. The result of applying function S to C is an empty multiset. Under these assumptions, it shows that the interpretation of C sati...
If for any element 'q', 'q' being reducible by set 'B' implies 'q' is reducible by set 'A', and if 'p' is reducible by set 'A' but not reducible by set 'B', then set 'A' strictly reduces 'p' more than set 'B'.
If a run reaches from state 's' to state 's'' through a sequence of states 'vs', and if it also reaches from state 's'' to another state 's''' through a sequence of states 'vs'', then it reaches from state 's' to state 's''' through the sequence of states 'vs' followed by 'vs''.
In the context of a category with finite object product, there is a lemma for the operation of a natural transformation. Assuming that the category 'CC' prime is the opposite of the category 'CC', it shows that the opposite of the natural transformation 'pi' maps from 'A' to the finite coproduct of 'P' over 'I' in the ...
The lemma named "una_Top" assumes that "P" is a profile and "S" is not an empty set. It shows that the function "f" applied to the top of "S" composed with "P" is an element of "S".
The application of the 'Store' function with parameters 'idx', 'G', 'pc', 'maxs', 'rT', and a pair '(ST,LT)', is equivalent to the existence of a type 'T' and a type stack 'ST'' such that 'ST' is the stack with 'T' at the top and 'ST'' as the rest, and 'idx' is less than the length of 'LT'.
The lemma named "card_lucas_lehmer_Units" assumes that "m" is greater than 1. It shows that the cardinality (or the number of elements) of the Units in the Lucas-Lehmer ring modulo "m" is less than "m" squared.
If a relation "p" holds between two functions "f" and "g" in a certain environment, and for any input "x", if "p" holds for "f(x)" and "g(x)" then "q" holds for "f'(x)" and "g'(x)", then it can be inferred that the relation "q" holds between the functions "f'" and "g'" in the same environment.
The comparison of two doubles "x" and "y" is less than or equal to zero if and only if "x" is not a number or the floating point comparison of "x" and "y" is equal to or less than zero.
If a matrix A is continuous on a set T, then the function that multiplies the matrix A by a vector s at any point t in the set T is also continuous on T.
The lemma named "max_list_mono" assumes that for any element 'x' in the set 'xs' but not in the set 'ys', there exists an element 'y' in the set 'ys' such that 'x' is less than or equal to 'y'. Under this assumption, it shows that the maximum element in the list 'xs' is less than or equal to the maximum element in the ...
If a set S is less than or equal to x, and if y is an element of set S, then y is less than or equal to x.
The lemma named "Sup_int" is about an integer 'x' and a set of integers 'X'. It assumes that the set 'X' is not empty and that it is bounded above. It shows that the supremum (the least upper bound) of 'X' is an element of 'X' and that all elements 'y' in 'X' are less than or equal to the supremum of 'X'.
For all 'i', if the second element of the function 'af\<^sub>\<nu>' applied to 'X', 'p', and the first 'i' elements of 'w' is not equivalent to 'false\<^sub>n', then the second element of the function 'af\<^sub>\<nu>' applied to 'X', 'p', and 'w' is equal to the function 'af' applied to the second element of 'p' and 'w...
: Assuming the following conditions: A: The sequence of programs P followed by P' of x followed by P'' satisfies the condition cfs. B: The integer k is less than the length of cfs. C: The first element of the kth element of cfs is equal to the size of P. D: The length of the second element of the first element of cfs ...
The omega power of x under the operation C is less than or equal to the star power of x multiplied by the bottom element, unioned with the n operation on the omega power of x multiplied by the top element.
If 'a' is an element of set 'A', then the function 'aezfun' applied to the projection of 'x' onto the set spanned by 'A' at 'a' is equal to the function 'aezfun' applied to 'x' at 'a'.
If a set H, when augmented with elements A1 through A11 in the order A11, A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, can prove B, then the same set H, when augmented with elements A1 through A11 in the order A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, can also prove B.
If a state 's' is reachable in the composition of two components 'C1' and 'C2', then the first component 'C1' is reachable in the first part of the state 's'.
For any type 't1', the set of free type variables in 't1' after a substitution operation, where each type variable 'x' is replaced by 'TVar' of 'x' if 'x' is in the free type variables of 'A', otherwise 'n + x', and then subtracting the set of free type variables in 'A', is a subset of the set of all 'x' such that ther...
The empty set is included in the computably enumerable sets.
If there is an arrow 'h'' from 'b' to 'c' and a zero arrow 'h' from 'a' to 'b', then the composition of 'h'' and 'h' is a zero arrow from 'a' to 'c'.
The function "splitset_fun" applied to any input "xs" does not result in an empty set.
If a set containing elements A15, A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14 and some set H can infer B, then a set containing elements A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, A15 and the same set H can also infer B. This essentially means that the order of elements in the set does...
The function 'uniform_real_density', when applied to a pair of arguments (as indicated by 'case_prod'), is measurable with respect to the product sigma-algebra of the Borel sigma-algebra with itself.
If "S" and "w" form an orthonormal system, then the L2 norm of "w i" in "S" is equal to 1.
If neither list 'xs' nor 'ys' is empty and the head of both lists are equal, then the predecessor of the longest common prefix of 'xs' and 'ys' is equal to the longest common prefix of the tail of 'xs' and the tail of 'ys'.
The selection sort of a list 'xs' is empty if and only if the list 'xs' itself is empty.
The lemma named "add_casew" assumes four conditions and four rules: Conditions: 1. "a" is in the carrier "R". 2. "b" is in the carrier "R". 3. "p" is on the curve defined by "a" and "b". 4. "q" is on the curve defined by "a" and "b". Rules: 1. For any point "p", property "P" holds for the point at infinity, "p", and ...
The lemma named "refl_gfp" assumes two conditions: 1. The function "Retr" is monotonic. 2. For any "theta", if "theta" is reflexive, then "Retr theta" is also reflexive. Under these assumptions, the lemma shows that the greatest fixed point of "Retr" is reflexive.
The conversion of a preferred IPv6 address to a compressed one (represented by the function "ipv6_preferred_to_compressed" applied to an IPv6 address consisting of eight parts: a, b, c, d, e, f, g, h) is defined as follows: If the longest sequence of zeros in the address (calculated by the function "max_zero_streak" a...
The lemma named "measure_subadditive" assumes that both A and B are measurable sets in the measure space M, and that the measures of A and B are not infinite. It then shows that the measure of the union of A and B is less than or equal to the sum of the measures of A and B.
The lemma named "pmf_Pn_pair" assumes that "sa" is in the set of probability mass functions (pmf) of "Pn" with parameters "p", "S0", and "n". It shows that the pmf of "Pn" with parameters "p", "S0", and "n" for "sa" is equal to the product of the pmf of "Y_cond_X" with parameters "p", "S0", "n", and the first element o...
If sequence w1 is a finite prefix of sequence w2, and sequence w2 is a finite or infinite prefix of sequence w3, then sequence w1 is a finite or infinite prefix of sequence w3. This is a transitive property.
The lemma named "inputSwapFrameSubject" is defined with the following parameters: a fixed 'Psi' of type 'b, 'P' of type psi ('a, 'b, 'c), 'M' and 'N' of type 'a, 'P'' of type psi ('a, 'b, 'c), and 'x' and 'y' of type name. The lemma assumes that 'Psi' is in a certain relation with 'P' that results in 'M' receiving 'N'...
For any two hypernatural numbers 'm' and 'n', 'm' is equal to 'n' if and only if the star-transform of 'm' is equal to the star-transform of 'n' in any linearly ordered semidomain 'a'.
The lemma "slice_kind_Pred_empty_obs_nearer_not_SOME" assumes the following conditions: 1. The source node of 'a' is not in the HRB slice of 'S' in the control flow graph (CFG). 2. The kind of 'a' is 'Q'. 3. The intra-procedural observable behaviors at the source node of 'a' in the HRB slice of 'S' in the CFG is empty...
: Assuming the following conditions: 1. The tableau "t" is in a valid state. 2. The property "P" holds for the empty set and the initial state of "t". 3. For any sets "as" and "bs" and any tableau "t", if "as" is a subset of "bs" and the property "P" holds for "as" and "t", then the property "P" also holds for "bs" an...
If a linear function 'f' is invertible with respect to scales B and C, then 'f' is bijective.
The lemma "summable_zeta" assumes that the real part of "s" is greater than 1. It shows that the series formed by the function "(n+1) to the power of -s" is summable. Here, "n" is a natural number.
The lemma "istate_sSys" assumes that the transition from an initial state "istate" through an action "a" results in an output "ou" and a new state "s'". It then concludes that either the action "a" is a system action "sSys" performed by a user "uid" with a parameter "p", or the new state "s'" is the same as the initial...
If "x" is an element in the set "G", then applying the function "apply_guards" to "G" and "s" is equivalent to applying "apply_guards" to "x" followed by "G" and "s".