Datasets:
UDACA
/
Split-Isa-AF-MMA

Dataset card Data Studio Files
xet
 Community
Statement:
stringlengths
7
24.3k
The responses of a uniform interactive oracle (\<I>_uniform) with sets A and B, given an input x, is equal to set B if x is in set A, otherwise it is an empty set.
The range of a map 'm' updated at key 'k' with value 'v' is a subset of the union of the range of map 'm' and the set containing 'v'.
: Assuming that for any two distinct points y and z in the set Ps, they are not collinear with the origin (0), and that the points in Ps are sorted in a counterclockwise order around the origin. Also, assuming that all points in Ps are distinct. Under these assumptions, it can be shown that the points in Ps are sorted...
For all real numbers delta, pi equals for all X such that pi is a power set of delta and X.
If the function 'wprepare_loop_start_on_rightmost' with parameters 'm', 'lm', and a pair consisting of 'b' and a list starting with 'Oc', holds true, then the function 'wprepare_loop_start_on_rightmost' with parameters 'm', 'lm', and a pair consisting of a list starting with 'Oc' and 'b', also holds true.
If B says to A a set containing a nonce Nb and a certificate in a set of events 'evs' that is in 'sr', then the nonce Nb is in the analysis of what the Spy knows in 'evs'.
A zeroth-order lambda abstraction of a proposition is equal to the proposition itself in any possible world.
If "U" is a strongly connected component of the directed graph "E", then "U" is not an empty set.
If set A is a subset of or equal to set B, then the list of elements in set A is a subset of or equal to the list of elements in set B.
The carrier of the reduced homology group of order p for a topological space X is equal to the kernel of the homomorphism induced by the function that maps every element of X to the single point set in the discrete topology, between the homology group of order p for X and the homology group of order p for the discrete ...
The lemma named "delete_dom_inj_on" assumes two conditions: First, "i" is an element of the set that includes all integers from 0 to the successor of "n" (exclusive). Second, the function "f" is injective (or one-to-one) on this set. Under these assumptions, the lemma shows that the function "f" remains injective on...
The lemma "get_dom_component_get_child_nodes" assumes the following: 1. The heap 'h' is well-formed. 2. The heap 'h' has a well-defined type. 3. The heap 'h' has known pointers. 4. The component of the pointer 'ptr' in heap 'h' is 'c'. 5. The child nodes of the pointer 'ptr'' in heap 'h' are 'children'. 6. 'child' is o...
The lemma named "Gr_not_in" shows that if "x" is not in "F" or "f" of "x" is not equal to "y", then the pair "(x,y)" is not in the graph defined by "F" and "f" in the BNF (Bounded Natural Functors) Definition.
If we assume that "wf_darcs" (well-formed directed acyclic relations) is true for a node "r" with a finite set "xs", and if we have two distinct tuples "(t1,e1)" and "(t2,e2)" both in the finite set "xs", then "e1" and "e2" must be different.
The ordered Γ function on a set S is defined as the set of all inferences where the inference is from the clause CA to the clause DA, resulting in the clause E, such that the ordered resolution with renaming from S, CA, DA, AA, A, σ results in E.
If a function 'f' maps a set 'S' to a set 'T', then the image of 'S' under the function 'f' is a subset of 'T'.
Lemma: Borel Measurable Times (measurable in raw form) Fix a function f with the type 'a mapping to 'b, where 'b is a second countable topology and a real normed algebra. Assume that f is Borel measurable with respect to the measure space M. Assume that g is also Borel measurable with respect to the measure space M. Th...
All roots of the product of two functions 'p' and 'q' are real if and only if all roots of both 'p' and 'q' are real.
The lemma named "nall_tuples_rec_fo_nmlz_rec_complete" assumes that the normalization function "fo_nmlz_rec" applied to the identity map of "j", the set "AD", and the list "xs" results in the original list "xs". Under this assumption, it shows that the list "xs" is an element of the set of all tuples "nall_tuples_rec" ...
In the context of a normal function, if 'x' is less than the inverse of 'F' at 'a', then 'F' at the successor of 'x' is less than or equal to 'a'.
The function 'f' restricted to a set 'A' is injective on 'A' if and only if the function 'f' itself is injective on 'A'.
Either "a" is less than "b", or "a" is equal to "b", or "b" is less than "a" according to the "less_than_char" relation.
For a natural number 'n', 'a' raised to the power 'd' is congruent to 1 modulo 'n' if and only if the order of 'a' modulo 'n' divides 'd'.
The power of a polynomial 'a' modulo 'm', raised to the power 'n', is equal to the following: If 'n' is zero, then it is 1 modulo 'm'. Otherwise, let 'd' and 'r' be the quotient and remainder when 'n' is divided by 2. Then, recursively calculate the power of the polynomial 'a' squared modulo 'm', raised to the power 'd...
The lemma named "step_Stuck" assumes a step from (c, s) to (c', s') in a given context Γ. It shows that for any function f, if the initial state s is Stuck, then the final state s' is also Stuck.
If p is the negation of p, q is the negation of q, and r is the negation of r, then the total sum of p, x, q, and r is equal to the negation of the total sum of p, x, q, and r.
The lemma named "pmin_is" assumes that "a" is greater than or equal to "b". It then shows that the minimum of "a" and "b" is "b".
The correct composition of different levels is applicable to sA23.
: Assuming the following conditions: - GlobVacantUpto (shd s) q t holds true, - InvGlobVacantUptoImpliesNrec (shd s) holds true, - InvGlobalRecordCount (shd s) holds true, - InvGlobalIncomingInfoUpright (shd s) holds true, - InvGlobalInfoAtBetaUpright (shd s) holds true, - and the next state is a result of a receive up...
The conjugate of the inverse of any octonion 'x' is equal to the inverse of the conjugate of 'x'.
There exists a pair of elements x and y, from a linearly ordered field, such that x is not equal to y, ten times x is not equal to nine times y, and if ten times x is less than y, then x is less than y.
If the dot product of 's' and 1 is greater than the convergence abscissa of 'f', then the evaluation of the Dirichlet series 'f' is continuous at 's' within the set 'A'.
If two generators are equivalent (as defined by the "pcr_generator" predicate), then applying the "next" function from the list theory to the second element of each generator will yield equivalent results. This is a rule for transferring properties between equivalent generators.
The lemma named "sylvester_index_mat2" is defined for any two polynomials 'p' and 'q' over a commutative semiring. It defines 'm' as the degree of polynomial 'p' and 'n' as the degree of polynomial 'q'. The lemma assumes that 'i' is less than the sum of 'm' and 'n', and 'j' is also less than the sum of 'm' and 'n'. It ...
If "v" is an element of the function "agent_ii_abs" for a given agent "a" and time "t", then the environmental observation of agent "a" at "v" is equal to the environmental observation of agent "a" at the last time of "t".
If edge 'e' is in the set 'E', and 'e' is a subset of 'V'', and 'V'' is a subset of 'V', then edge 'e' is in the set of all edges of 'V''.
The lemma "remove_prime_factor" assumes that the result of the function "remove_prime_factor" with inputs i, n, and ps is equal to (m, qs). It also assumes that i is greater than 1 and n is not equal to 0. It shows that there exists a list rs such that qs is equal to rs appended to ps, and n is equal to m times the pro...
: Assuming a graph 'g' has a path from node 'n' to node 'm' through nodes 'ns', and a variable 'v' is used in node 'm' but not defined in any node in 'ns', and the number of predecessors of node 'n' in graph 'g' is not equal to 1, then node 'n' is in the set of nodes where 'v' is defined by a phi function in graph 'g'.
The lemma "last_leq_paths" assumes that the pair (Cs, Ds) is in the transitive closure of the path1 relation on P and C, and that the program P is well-formed with respect to wf_md. It shows that the pair formed by the last elements of Cs and Ds is in the transitive closure of the subcls1 relation on P.
The lemma named "awalk_cyc_decomp_has_prop" assumes that there is an arc walk from node u to node v following path p, and that the vertices visited in this arc walk (awalk_verts u p) are not all distinct. Under these assumptions, it shows that the cyclic decomposition of the path p (awalk_cyc_decomp p) is indeed a vali...
, given the assumptions that "c" is equal to the tuple "(((Pr n f g, xs, ls) # fs), rv)", "ls" is not an empty list, the length of "ls" is not equal to the successor of the head of "xs", "rv" is None, and "recfn" of the length of "xs" and "(Pr n f g)" is true, it shows that the function "estep_Pr" applied to the encodi...
The first selection function is a determinant selection function.
If the following conditions are met: 1. The evaluation of expression 'e' in the normal state 's0' under the context 'G' results in a value 'a'' and a state 's1'. 2. The evaluation of the parameters 'ps' in the state 's1' under the context 'G' results in parameter values 'pvs' and a state 's2'. 3. 'D' is the declaratio...
In the context of a category with a coequalizer_2, the application of the natural transformation map epsilon to PL2 of the category B is equal to the composition in the category C of the application of the natural transformation map epsilon to PL2 of the category A and g. Similarly, the application of the natural trans...
The cardinality of the set containing elements 'a' and 'b' equals 2 if and only if 'a' is not equal to 'b'.
If the sum of two natural numbers x1 and y1 is less than the sum of two other natural numbers x2 and y2, then the Cantor pairing of x1 and y1 is less than the Cantor pairing of x2 and y2.
If the precondition is that 'N' equals 'n', then after the procedure 'Fac' is called on 'N', the postcondition is that 'R' equals the factorial of 'n'.
In the context of a cancellative commutative monoid, we have a lemma named properfactorI3. This lemma assumes that a variable 'p' is equal to the product of 'a' and 'b', that 'b' is not a unit in the group 'G', and that both 'a' and 'b' are elements of the carrier of 'G'. Under these assumptions, the lemma shows that '...
The composition of the power of a complex bounded linear function A with exponent n and the same function with exponent m is equal to the power of the function A with the exponent being the sum of n and m.
, under the assumptions that "x" is in the set of variables, the second elements of the tuples in the set "txs" are all in the set of variables, the first elements of the tuples in the set "txs" are all in the set of terms, and "x" is not among the second elements of the tuples in the set "txs", it shows that the term ...
If a trie 't' maintains its invariant, then the invariant is also maintained after deleting keys 'ks' from the trie 't'.
For any list "LT", the condition that "P" allows the replication of "Err" "n" times to be less than or equal to "LT" in the lattice order, is equivalent to "LT" being equal to the replication of "Err" "n" times.
The relation R under the context Gamma prime strongly bisimulates the low-level memory management.
The restriction of a value 'v' to the result of the function 'res' applied to 'ts' and 'c' is less than or equal to the result of 'res' applied to 'ts' and 'c'.
: Assuming that: - In program P, class C sees method M, which is static, has no arguments, returns void, and has body m, in class D. - Method M is not the class initializer. - There is no class named "Start" in program P. - The type of method "start_m" in class "Start" is "start_φm". - The type of the class initialize...
The lemma named "pickE_asEnv" assumes that "qrho" is a good environment in the quasi-terms. It shows that the result of applying the "pickE" function to the environment "qrho" converted to an actual environment (using the "asEnv" function) is equivalent to the original "qrho".
The lemma named "card_of_Times_cong1" assumes that the cardinality of set A is order-isomorphic to the cardinality of set B. It shows that the cardinality of the Cartesian product of set A and set C is order-isomorphic to the cardinality of the Cartesian product of set B and set C.
The lemma "lgrid_sum" is about a function F that maps grid points to real numbers. It assumes that 'd' is less than the length of 'b' and the level of 'b' is less than 'lm'. It shows that the sum of the function F over all points 'p' in the lgrid of 'b' with parameters 'd' and 'lm' is equal to the sum of F over all poi...
If a Möbius transformation M preserves the unit disc, and u, v, and w are points in the unit disc, then the Poincaré betweenness relation for the images of u, v, and w under M is equivalent to the Poincaré betweenness relation for u, v, and w themselves.
: Assuming for all variables 'v', if 'v' is free in expression 'e' and 'k' is less than 'v', then the expression obtained by applying the substitution function 'Γ' to '(v - Suc k)' is closed. Also assuming that 'X' is a closed expression. Then, the expression obtained by substituting 'X' at position 'k' in the express...
The cardinality of the set F2set2 of x is less than the ordinal sum of the cardinalities of bd_F1 and bd_F2.
The lemma named "mk_minsky_comp3_3" assumes that "f", "g", "h", and "k" are valid Minsky3 functions. It shows that a new Minsky3 function can be created by composing "f", "g", "h", and "k" such that the new function takes three inputs "x", "y", and "z", and applies "g", "h", and "k" to these inputs, and then applies "f...
The value of a boolean expression, when converted from a boolean expression 'b' and evaluated with a state 's', is equal to the value of the boolean expression 'b' evaluated with the same state 's'.
The lemma named "strict_subset_subsumption_redundant_clause" assumes that 1) The set D multiplied by the substitution sigma is a strict subset of C, and 2) The substitution sigma is a ground substitution. Under these assumptions, it shows that C is an element of the redundancy criterion function (Rf) applied to the...
If the interior of a set is a subset of the set itself, then the set is equivalent to the filter of the set.
If you have some information 'info', a sequence 'xs', and a next element 'nxt', then the function 'holds' applied to 'info' and the result of 'takeW' function applied to 'info', 'xs', and 'nxt' is equivalent to the 'extract' function applied to 'info', 'xs', and 'nxt'.
The degree of the quotient ring for a given type 'a is equal to n.
For a fixed matrix A, which belongs to a field, if A is invertible, then the linear function that maps each vector x to the product of A and x is also invertible.
If type tau is less than or equal to type sigma, and type sigma is less than or equal to type rho, then type tau is less than or equal to type rho. This is for any types tau, sigma, and rho of a certain type 'a.
The lemma "lookup0_fmap_of_list_simps" consists of two parts: 1. The value of the function "lookup0" applied to a function map created from a list that starts with a pair (x, y) and continues with a list xs, at the index i, is equal to y if x equals i, otherwise it is equal to the value of "lookup0" applied to a funct...
The bitwise AND operation between the word-length number 1705 and the negation of 42 equals 1664.
Multiplying any quantity 'x' by the unit quantity on the right side results in a quantity equivalent to 'x'.
If a chart 'c' is k-smooth compatible with another chart 'c'', then the chart 'c' restricted to a set 'S' is also k-smooth compatible with 'c''.
The absolute convergence of the Dirichlet series of a real function is equivalent to the absolute convergence of the function itself. This is true for any real number 's' in a Banach space or a natural power normed field.
The Cartesian product of a set G and the union of a set S' is equal to the supremum over all elements X in S', of the Cartesian product of S and X.
If the following conditions are met: 1. The function 'uniq' applied to parameters 'p', 'inff', and 'secret' is unique. 2. The event 'evs' is in the trace 'p'. 3. 'R' and 'R'' are in 'p'. 4. 'n' is a new nonce in 'R' and 'n'' is a new nonce in 'R''. 5. The nonce 's n' equals the nonce 's'' n''. 6. The nonce 's n' is no...
: Given the assumptions that - E is a set in the measure space M, - the absolute value of the measure of E is less than infinity, - C is a set in the measure space M, - C is a subset of the positive part of E, - and for any integer n, the weight of E at n is greater than 0, it can be shown that the measure of C is gre...
The composition of the formal power series 'X' with the exponential formal power series of a variable 'a' (where 'a' is a field) is equal to the exponential formal power series of 'a' minus 1.
The lemma "inductive_on_imp_accessible_on" assumes that a predicate P is inductive on a set A. It shows that for every element x in set A, x is accessible on predicate P and set A.
If 's' reduces to 't' in zero or more steps, then the lambda abstraction of 'T' in 's' also reduces to the lambda abstraction of 'T' in 't' in zero or more steps.
The lemma named "fds_truncate_ln" is about a fixed Dirichlet series 'f' in a real normed field. It shows that truncating the natural logarithm of the Dirichlet series 'f' at 'm' (represented as "fds_truncate m (fds_ln l (fds_truncate m f))") is equal to truncating the natural logarithm of 'f' at 'm' (represented as "fd...
: 1. If 'i' is less than 'n', then the 'n' element of the sequence 'Γ''', when applied to the pair ('f', 'i'), equals to some command of Example2'n 'i'. 2. If 'i' is less than 'n', then the 'n' element of the sequence 'Θ''', when applied to the pair ('f', 'i'), equals to some annotation of Example2'n 'i'. 3. The first...
The image of the ideal F under the focus function X is equal to the intersection of the ideal of the image of F under the focus function X and the range of the focus function X.
If the function phi of g and v equals some value vs, and if g, v, and v' form a trivial Phi, then the function "the_trivial" of v and vs equals some value v'.
A model M satisfies a formula phi under a substitution g if and only if M satisfies the formula obtained by mapping the substitution g over phi.
If a sequence 's' is strictly sorted, and 'i' is less than or equal to 'j', then the 'i'-th element of the sequence 's' is less than or equal to the 'j'-th element of the sequence 's'.
The lemma "DBM_val_bounded_len'" is defined with a fixed variable 'v'. It also defines a function 'vo' which returns 'None' if 'k' equals 0, otherwise it returns 'Some' value 'c' such that 'v c' equals 'k'. The lemma assumes that 'v' is a DBM (Difference Bound Matrix) value bounded by 'u', 'm', and 'n', and that the co...
The value of a parametric device, when evaluated with a function that maps all inputs to a constant 'c', is equal to 'c' times the sum of the list obtained by mapping the second element of each pair in the list of parametric devices 'X'.
The lemma named "entailsD" assumes that if "P" logically implies "Q" and "P" holds in heap "h", then it shows that "Q" also holds in heap "h".
The lemma named "card_vimage_inter" assumes that the function "f" is injective on set "A" and that set "X" is a subset of the image of set "A" under function "f". It shows that the cardinality of the intersection of the preimage of set "X" under function "f" and set "A" is equal to the cardinality of set "X".
The evaluation of the multiplication of two ordinal numbers 'n' and 'm' in base 'b' is equal to the product of the evaluations of 'n' and 'm' in the same base 'b'.
If 'x' belongs to the set of hyperfinite numbers, then there exists an 'e' in the set of infinitesimals such that 'x' is equal to the standard part of 'x' plus 'e'.
For a given multiset P of polynomials over a field, if the product of the elements in P equals the product of the elements in another multiset Q, and if both P and Q are subsets of the set of irreducible and monic polynomials, then P and Q must be equal.
The initialization of l1 is a subset of the relation R01iai applied to the initialization of a0i.
, given the assumptions that a formula "phi" is in the set of formulas, the second element of each pair in the set "txs" is a subset of the set of variables, the first element of each pair in the set "txs" is a subset of the set of terms, the second elements of the pairs in "txs" are distinct, and for any two indices "...
The language of the generalized tree automaton, which only contains the state "q" and the tree automaton that only contains the rule with function "f", no arguments, and state "q", is equal to the set containing only the generalized function with function "f" and no arguments.
The trilean addition of 'a', 'b', and 'c' is the same whether 'b' and 'c' are added first and then 'a' is added, or 'a' and 'b' are added first and then 'c' is added.
: 1. If all elements in the unlabelled list 'S' are Assignments, then the dual of 'S' is equal to 'S'. 2. If all elements in the unlabelled list 'S' are Checks, then the dual of 'S' is equal to 'S'. 3. If all elements in the unlabelled list 'S' are Updates, then the dual of 'S' is equal to 'S'.
If "x" is a test and "x" and "y" are complementary, then "y" is also a test.
If a state sigma satisfies the monadic computation where a value x is drawn from the assumption of a predicate P and then used in a computation M, then there exists a state sigma such that the predicate P holds for sigma and sigma satisfies the computation M with no arguments.