Datasets:
UDACA
/
Split-Isa-AF-MMA

Dataset card Data Studio Files
xet
 Community
Statement:
stringlengths
7
24.3k
If 'n' is greater than 0, then the function 'vebt_buildupi' applied to 'n' will result in 'r', such that 'r' satisfies the 'vebt_assn' relation with 'n' and an empty set.
The lemma named "mult_rat_1_pos" is defined with a fixed rational number "r1". It assumes that "r1" is greater than 0 and that "y" satisfies the condition "invariant_1". It defines "z" as the result of the function "mult_rat_1_pos" applied to "r1" and "y". The lemma then shows that "z" satisfies the condition "invarian...
The lemma named "oinvariantI" is being defined with the variables T, TI, S, U, and P. It assumes the following: 1. For any state 's', if 's' is in the initial state of 'A', then property 'P' holds for 's'. 2. For any states 'g' and 'g'' and any label 'l', if the tuple (g, l) is reachable in 'A' using 'S' and 'U', and...
If the set I is infinite, then the previous element of the next element of n in set I is equal to n.
The lemma named "effect_ureturnE" assumes that the effect of returning a value 'x' in a heap 'h' results in a new heap 'h'' and a result 'r' with a time complexity 'n'. Under this assumption, it is obtained that the result 'r' is equal to 'x', the new heap 'h'' is equal to the original heap 'h', and the time complexity...
It is not always the case that a condition "phi" is not true is equivalent to saying that there is an event where "phi" is true.
If a function "f" is summable, then the function that maps each element "n" to the negation of "f(n)" is also summable.
If for all addresses, the expression locks of 'e' equals zero, then 'e' is in synchronization. And the lemma "expr_lockss_sync_oks" states that if for all addresses, the expression locks of 'es' equals zero, then 'es' is in synchronization.
For all 'y' in the set 'ys', if 'y' is not in the set 'xs', then mapping the function 'f' with 'ys' replaced by 'vs' over 'xs' is equal to mapping the function 'f' over 'xs'.
The lemma for the generalized Akiyama-Tanigawa algorithm when applied to any function 'f' with 0, results in the original function 'f'.
If node 'n' is in the set of predecessors of node 'n'' in graph 'g', and node 'n' is in the set represented by 'αn' of graph 'g', and the graph 'g' satisfies the invariant 'invar', then node 'n'' is in the set represented by 'αn' of graph 'g'.
The power of a relation "X" to the "i"th degree is equal to "X" raised to the power of "i".
If a set S is locally path-connected, then it is also locally connected.
If a set C is connected within a set X, and sets S and T are separated within X, and if C is a subset of the union of S and T, then C must be a subset of either S or T.
The lemma "depth_ctxt_nt_hole_inc" assumes that "C" is not equal to the empty context. It shows that the depth of the tree "t" is less than the depth of the context "C" with the tree "t" inserted.
The relation of probability mass functions (pmf) holds if and only if a certain condition P is true and the relation of pmf Q holds for x and y. This is true regardless of whether P and the relation Q are in conjunction in the order of "P and Q" or "Q and P".
If "x" is an element of the set of hypercomplex numbers "SComplex", then the hypercomplex modulus of "x" is an element of the set of real numbers.
If a set F satisfies the properties Fr_1a, Fr_2, and Fr_4, then for all sets A, the operation Op applied to the intersection of set A is true.
For any integer 'i', dividing 'i' by 2 is equivalent to dropping the least significant bit of 'i'.
If "z" plus the product of "y" and "x" is less than or equal to "y", then the product of "z" and "x" raised to the power of "n" is also less than or equal to "y".
The lemma "literalNotInEarlierLevelsThanItsLevel" assumes that a given level is less than the element level of 'e' in 'M'. It shows that 'e' is not in the set of elements in the prefix to the level of 'M'.
The lemma named 'comp_primroot_conv'' assumes that 'u' is not equal to the empty word and 'v' is not equal to the empty word. It shows that 'u' concatenated with 'v' is equal to 'v' concatenated with 'u' if and only if the primitive root of 'u' concatenated with the primitive root of 'v' is equal to the primitive root ...
A tree 't' is sorted if and only if the in-order traversal of the tree 't' is strictly sorted.
For a given function 'f' and a real number 'y', the function 'f' raised to the power 'y' will eventually be non-negative under the filter 'F'.
The lemma named "independent_length_le_dimension" assumes three conditions: 1) The dimension of the set E over the field K is n, 2) The set Us is independent over the field K, 3) The set Us is a subset of E. Under these assumptions, it shows that the length of the set Us is less than or equal to n.
If "r" is a vertex in graph "G", there are no cross products in the creation of "ldeep" with "xs", the head of "xs" is "r", and "xs" are distinct, then the directed tree moves forward in the direction of the tree "r" with "xs".
For any terminal 'b', converting a tllist to an llist and then back to a tllist will result in the original tllist 'b'. This is essentially asserting the reversibility of the conversion between tllist and llist for a given terminal 'b'.
The lemma named "right_unit_mult_rng_of_frac" assumes that the pair (r, s) is in the carrier of the relation "rel". It shows that the product of the fraction (r | rel s) and the multiplicative identity (one) in the ring of fractions "rec_monoid_rng_of_frac" is equal to the fraction (r | rel s).
The lemma named "list_succ_unique" assumes that "s" is in the orbit of "f" at "s", and for any "x" that is not in the orbit of "f" at "s", "f" at "x" equals "x". Under these assumptions, it shows that there exists a unique list "xs" such that "f" equals the successor of the list "xs", the elements of "xs" are distinct,...
If 'c' is greater than or equal to zero, then the supremum expected value at state 's' for a function that always returns 'c' is 'c'.
The implementation of list pop operation in circular singly linked list (cs_list) is equivalent to the operation cs_pop.
For any integer 'k', if 'k' is divided by 2 after being shifted left by 'n' bits (represented by the function 'push_bit n x'), it is equivalent to shifting 'k' left by 'n-1' bits if 'n' is greater than 0. Otherwise, it is equivalent to 'k' divided by 2.
If a predicate P holds for all corresponding pairs of elements in lists xs and ys, then the same predicate P holds for all corresponding pairs of elements in the mirrored versions of these lists.
The list representation of a vector 'xs' is equal to the mapping of the function '($)' on 'xs' over the range from 0 to the dimension of the vector 'xs' minus one.
If a natural transformation arrow is equal to 'NN', then 'NN' is an element of the target morphism natural transformation arrows from category 'AA' to category 'BB' in a given ordinal 'alpha'.
If a state 's' is a hotel, then the current key 'currk' of room 'r' in state 's' is included in the set of keys issued in state 's'.
The image of the set H under the function invT is equal to the set G.
The first element of the minimum coin set of g and h is equal to the second element of the minimum coin set of h and g.
: 1. The size of the strand sequence (S followed by S') is less than the size of the strand sequence (S followed by sending ts and then S'). 2. The size of the strand sequence (S followed by S') is less than the size of the strand sequence (S followed by receiving ts and then S').
A matrix with dimensions m by 2n can be tiled with dominoes.
The lemma named "unat_ucast_10_64" is defined for a fixed variable 'x' of type "10 word". It shows that the unsigned natural number representation of the unsigned cast of 'x' to a 64-bit word is equal to the unsigned natural number representation of 'x'.
The lemma defines 'p' as the unique value such that the sum of the product of the 'i'th element of list 'as' and the 'i'th element of list 'bs' raised to the power of 'p', for all 'i' less than 'k', equals 1.
If the algorithm 'A' is lossless and the order of the group 'G' is greater than zero, then the lcdh (which stands for "Lossy Computational Diffie-Hellman") of algorithm 'A' is also lossless in the sense of a single-probability mass function (spmf).
"rword" is a rewrite order.
"w1 has a derivative (vderiv) on the interval from 0 to 1/a, where the derivative is represented by w1'".
For any list of real vectors 'xs', the scalar multiplication of a real number 'a' and the sum of the list 'xs' is equal to the sum of the list obtained by multiplying each element of 'xs' by 'a'.
The lemma named "restrict_IF_id" assumes two conditions: firstly, that the if-expression (IF v t e) is ordered, and secondly, that the value 'v' is greater than 'v''. If these conditions are met, it shows that restricting the if-expression (IF v t e) to 'v' with a certain value is equal to the original if-expression (I...
The domain of the lookup function applied to any map 'm' is finite.
Either there exists a string 's' and a variable 'X' such that 'eX' equals 'OK X' and 'igWls MOD s X' holds true, or 'eFresh MOD ys y eX' holds true.
The 32-bit memory value at address 'a' in address space 'asi' of state 's' remains the same even if the cpu register, user register, data write, state variable, traps, and undefined values of the state 's' are modified to new values.
The lemma "welltyped_leakage_free_occurssec_constI" is defined with "s" being equivalent to the function "OccursSec" with no arguments. It assumes the following conditions for all transactions "T" in the set "P": 1. Each transaction "T" is well-formed. 2. For each variable "x" in the union of the variables of the tran...
An empty set is a dual chain.
The sequence of "not p" then "not q", followed by "not r", is less than or equal to "not p" then the sequence of "not q" and "not r".
: 1. For any goal state and start state, it is not the case that both 'fails' and 'follows' conditions hold for the goal state and the start state. 2. For any goal state, intermediate state, and start state, if 'fails' condition holds for the goal state and the start state, then either 'fails' condition holds for the...
The universe of all elements with a bottom element is equal to the universe of all elements.
If the successor of the rank of 'a' is less than or equal to 'k' and 'k' is an ordinal number, then 'a' is an element of the set Vset 'k'.
The lemma named "qe_infinitesimals_helper" is defined with a fixed real number "k". It assumes the following: 1. For every triplet (d, e, f) in the set "a", there exists a number "y" greater than "k" such that for all "x" in the range from "k" to "y", the equation "d * x² + e * x + f = 0" holds true. 2. For every tri...
For a function 'f' mapping from type 'a' to type 'b', the map of a list 'xs' where each key-value pair is transformed such that the key is the result of applying 'f' to the original key, and the value remains the same, when applied to 'x', is equal to the map of the original list 'xs' applied to 'g' of 'x'. This holds ...
If a class is a preorder under a certain order and if the least upper bound exists for a singleton set, then the function 'f' is monotonically continuous with respect to the supremum, equality, the least upper bound, and the order.
The maximum coefficient 't' is greater than or equal to zero.
The lemma assumes three conditions: 1. There is a chain of elements in Y ordered by the relation (\<le>). 2. The set Y is not empty. 3. The support flow of the supremum of Y is countable. Under these assumptions, the lemma shows three things: 1. The outgoing degree of the supremum of Y at any point x is equal to th...
The lemma named "arr_Rel_eqI" assumes that "S" and "T" are arrow relations in a certain set "alpha". If the value, domain, and codomain of "S" are equal to the value, domain, and codomain of "T" respectively, then "S" is equal to "T".
"alpha_Tree" is an equivalence relation.
For any hypernatural number 'n', 'n' is less than or equal to zero if and only if 'n' is equal to zero.
The lemma named "RR1_difference" assumes that both A and T satisfy the RR1 specification, and B and U also satisfy the RR1 specification. It shows that the difference of A and B in terms of regular registration also satisfies the RR1 specification, as does the difference of T and U.
If the module MOD satisfies the property "igAbsRenSTR", then it also satisfies the property "igAbsCongUSTR".
For all indices 'i', registers 'r', and labels 'l', for every transition ((a, b), t) that is in the filtered set where the origin equals 's' from the set obtained by inserting the transition ((s, s'), t') into 'e', if the label of transition 't' equals 'l' and transition 't' can be taken given index 'i' and register 'r...
The lemma named "ivp_solsD" makes the following assumptions and shows the following results: Assumption: X is an element of the solutions of the initial value problem defined by the function f, the set U, the set S, the initial time t₀, and the initial state s. Results: 1. The derivative of X equals the function f ev...
For any point in time t1, if t1 is since t2 in the interval I and either condition Q1 or Q2 holds at t2, then it is equivalent to say that either t1 is since t2 in the interval I and condition Q1 holds at t2, or t1 is since t2 in the interval I and condition Q2 holds at t2.
There exists a prime number 'q' such that 'q' is a prime element of the Gaussian conjugate, the real part of 'q' is greater than zero, the imaginary part of 'q' is greater than zero, 'p' is equal to the product of 'q' and its Gaussian conjugate, and the Gaussian integer norm of 'q' is equal to 'p'.
: - The minimum of 'm' and 'n' is 'm' if and only if 'm' is less than or equal to 'n'. - The minimum of 'm' and 'n' is 'n' if and only if 'n' is less than or equal to 'm'.
The limit of the function "c * f(x)" as "x" approaches "F1" is greater than "l". This is true if the limit of the function "f" as it approaches "p" from the right is "F1", "c" is greater than zero, and "l" equals "c * p". This is for any "c" that is a member of a linearly ordered field with a linear order topology.
If there is a path from node 'u' to node 'v' in the graph 'g' restricted to the set of nodes 'E', then there is also a path from 'u' to 'v' in the original graph 'g'.
: Assuming that for any two transformations 't' and 'u', if 't' is less than or equal to 'u' in the unitary transformation order, and for any property 'P', if 'P' is unitary then both 't(P)' and 'u(P)' are unitary, then 'T(t)' is less than or equal to 'T(u)' in the unitary transformation order. And also assuming that...
The evaluation of the predicate "widen1p_i_i_o" with parameters P and T is equivalent to the function that takes U as an argument and checks if, according to P, T is less than or equal to U with a one-step widening.
If "x" is less than "y" and "t_stop" is less than or equal to "x", then the function "p" of "y" is less than the function "p" of "x".
The frontier of a set S in a topological space X is the same as the frontier of the intersection of the topological space X and the set S.
The implementation of getting offending flows from the security invariants in the policy results in an empty list.
The lemma named "uniform_complement_block_size" assumes that a block "bl" is an element of the complement of a block set "B". It shows that the cardinality or size of the block "bl" is equal to the difference between the total number of points "v" and the block size "k".
Given the assumptions that 'd' is a divisor of 'n' and 'd' is not equal to 1, and defining 'χ₁' as the principal Dirichlet character of 'n', the induced modulus of 'd' is equivalent to the existence of a Dirichlet character 'Φ' of 'd' such that for all 'k', the Dirichlet character 'χ' of 'k' is equal to the product of ...
The multiplication of a quaternion and its conjugate is commutative, meaning the order in which they are multiplied does not change the result.
If a certain element "x" is in the domain of a finite map "m", then there exists an element "y" such that "x" is mapped to "y" in the map "m".
"ESPEC Φ Ψ is less than or equal to ESPEC Φ' Ψ'" if and only if "for all e, if Φ e is true then Φ' e is also true" and "for all x, if Ψ x is true then Ψ' x is also true".
If the left subtree (l) is a sorted red-black tree, and the right subtree (r) is a sorted red-black tree, and all keys in the left subtree are less than the key (a), and the key (a) is less than all keys in the right subtree, then looking up a key (k) in the balanced red-black tree created by the function "rbt_baliR" w...
The lemma "enc_wf_interp" assumes that a well-formed formula is of the same length as the interpretation I, and that the interpretation for the formula (w, I) is well-formed. It shows that the interpretation for the formula (dec_word (enc (w, I)), dec_interp (length I) (FOV \<phi>) (enc (w, I))) is also well-formed. Th...
If H is a submonoid of G, and for every element 'a' in H, the inverse of 'a' is also in H, then H is a subgroup of G.
The lemma tt_2 assumes that the dimension of vector x is 2 and that the first and second elements of vector x are both true. It shows that the zeros of the boolean input x is an empty set.
: Assuming that "g" is a canonical rewriter for "u" with respect to "G" and "rword", and that "g" is not a signature-reduced with respect to the term order "\<prec>\<^sub>t", the equality "(=)", the set "G", and the monomial obtained by multiplying "g" by 1 and subtracting the leading power-product of "g" from the powe...
If a ring 'a of type comm_ring_1 is a Hermite ring, then it is also a Bezout ring. This is assumed under the condition H.
The sum of the successor of n, the successor of m, and the successor of k, where the addition is defined by the function add2, is equal to the third successor of the sum of n, m, and k.
The coercion of the universal domain to a type 'a is equal to the projection of type 'a.
The function "nfoldli", which takes a list of type 'a, a function from type 'b to boolean, another function from type 'a and 'b to 'b nres, and a type 'b, and returns a 'b nres, is equal to the function "nfoldli_alt".
The lemma "resActLeft" is defined with fixed variables "x" of type "name", "\<alpha>" of type "act", and "P" of type "ccs". It assumes that "x" is fresh for "\<alpha>", and that the identity relation is a subset of "Rel". It shows that the restriction of "x" on the action prefix of "\<alpha>" on "P" strongly simulates ...
"s1 is equivalent to s2 with respect to 0 in the lens algebra if and only if s1 is equal to s2".
If Y is an idempotent scene, and if X is a subscene of Y, and if Y is a subscene of Z, then X is a subscene of Z. This is a statement of transitivity in the context of scenes and subscenes.
If 'f' is a nucleus, then the value of 'f' at the product of 'x' and 'y' is equal to the value of 'f' at the product of 'f(x)' and 'y'.
The operation "ias_new" on an empty state results in an isolated array set with no elements.
If x is less than or equal to y, then the result of the operation "left residual" on x and z is less than or equal to the result of the same operation on y and z.
"A\<^sub>L'" is strongly independent.
If 'u' is locally closed, then substituting 'xT' into 'u' in the open term 'open i t v' is equivalent to opening 'i' in the substitution of 'xT' into 'u' in 't' and the substitution of 'xT' into 'u' in 'v'.