Datasets:
UDACA
/
Split-Isa-AF-MMA

Dataset card Data Studio Files
xet
 Community
Statement:
stringlengths
7
24.3k
If 'f' is an isometry, then the Busemann function at the Gromov extension of 'f' and 'xi' for 'f' of 'x' and 'f' of 'y' is equal to the Busemann function at 'xi' for 'x' and 'y'.
The lemma named "msubstneg_nb" assumes that 'a is a field with no zero characteristic, the linear polynomial 'p' is a line, and 't' is a term that is not bound to zero. Under these assumptions, it shows that the result of the negative substitution of 'p' with 'c' and 't' is not bound to zero.
If a well-formed tuple "v" of size "n" exists in set "B", and the intersection of sets "A" and "B" equals set "C", then a well-formed tuple of size "n" also exists in set "C" when set "A" is restricted to tuple "v".
The reach of m2 is a subset of m2_inv3_msg2 in the context of the m2_confid_chan.thy file in the Auth_simple of the Security_Protocol_Refinement.
The zeroth prime number is 2.
For all x, if the function f of x is not equal to zero, then the big O of the product of functions f and g is equal to the product of the big O of function f and the big O of function g. This is applicable for a function f that maps from any type 'a to any type 'b in a linearly ordered field.
If the initial state of a transition system T is a subset of a set I, and if the intersection of set I and the reachable states of T under the transition function of T is a subset of I, then the reachable states of T are a subset of I.
: If a property P holds for the expression (L x), and if the property P holds for the expression (V (I x)), and if the property P holds for the expression (V (R x)), and if for any two expressions x1a and x2a, if P holds for both x1a and x2a, then P also holds for the expression (Plus x1a x2a), and if for any two expr...
The lemma named "unitary_col_inner_prod" assumes that "A" is a square matrix of order "n", "n" is greater than 0, "A" is a unitary matrix, and "j" and "k" are less than "n". It shows that the inner product of the "j"-th column and the "k"-th column of matrix "A" is equal to the element at the "j"-th row and "k"-th colu...
The square root of the sum of the squares of x and y is less than or equal to the sum of the absolute values of x and y.
The lemma named "dis_loop2_refine" is a refinement, which assumes the following: - "XF" is a relation between "x" and "cf", and "f" in the context of "xf_rel". - "CLC" is a relation between "clc" and "l" in the context of "clc_rel". - "ami" is identical to "am", "Qi" is identical to "Q", and "ui" is identical to "u". ...
, under the assumptions that the length of 'b is a multiple of the length of 'a, and the length of the list 'w' is the quotient of the length of 'b and 'a, the operation of splitting a word 'b into a list of words 'a (word_rsplit), when applied to the result of concatenating a list of words 'a into a word 'b (word_rcat...
The function "psitau" applied to the result of "f" and "n" combined with the right triangle operator, when evaluated at "x", is equal to: if "x" is less than the successor of "n", then it returns the value of "f" at "x" (assuming it exists), otherwise it returns the value of the function "tau" evaluated at "x" with the...
For a formal power series 'f' multiplied by a constant 'c', the nth term of the resulting series is equal to the nth term of 'f' multiplied by 'c'. Here, 'c' is of a type that supports addition and multiplication with zero.
The lemma named "integral_subalgebra2" is defined for a function 'f' from 'a' to 'b', where 'b' is a Banach space with a second countable topology. The lemma assumes that 'F' is a subalgebra of 'M' and that 'f' is Borel measurable in 'F'. The lemma shows that the integral of 'f' over the restriction of 'M' to the subal...
"is_invar" is a property of a state "s" such that for every strongly connected component "scc" in the set of strongly connected components "sccs" of "s", "scc" is a strongly connected component of the graph "E".
The lemma named "ball_fun_contains_open_aux" is defined with a function 'x' that maps from 'a' to 'b' and a real number 'e'. Assuming that 'e' is greater than zero, it shows that there exists a set 'U' such that 'U' is open, 'x' is an element of 'U', and 'U' is a subset of the set of all 'y' such that the distance betw...
The lemma, in the context of a Cox-Ross-Rubinstein (CRR) market viability, shows that the discounted value of a put option at time T for a given world state w is equal to the deterministic discount at time T multiplied by the maximum value between the strike price K minus the logarithmic geometric process of the initia...
If the absolute value of an extended real number 'x0' is not infinity, then the function 'real_of_ereal' is continuous at 'x0'.
If 'f' is a bounded linear operator, then the function that maps 'x' to the negation of 'f(x)' is also a bounded linear operator.
For every element Q in the set Qs union with the set containing Q, if Q is cpropagated and the free variable of Q equals A, then the free variable of the cpropagated version of the disjunction of Qs and Q also equals A.
The lemma named "ins_inorder" assumes the following conditions: 1. The value of k is greater than 0. 2. The values l, t, and u are aligned. 3. The leaves of t are sorted in ascending order. 4. The root order of k and t is maintained. 5. The value of l is less than x and x is less than or equal to u. Under these assum...
The intersection of the restriction of v to the resolution of ts and the restriction of v to the claim of ts is an empty set.
: Assuming the following conditions: - The tag of capability 'c' is true. - The permission to store in capability 'c' is true. - For any capability value 'cv', if 'v' is equal to the value of 'cv' and the tag of 'cv' is true, then the permission to store in capability 'c' is true and either the permission to store loc...
If a set A is a subset of the carrier of the m-th power of the p-adic numbers, and if b is an element of the carrier of the n-th power of the p-adic numbers, and if the cartesian product of A and the set containing only b is semialgebraic in the (m+n)-th power of the p-adic numbers, then the set A is semialgebraic in t...
: if there exists an alpha such that the property phi holds for alpha in v, and for any alpha, if the property phi holds for alpha in v then the property psi holds in v, then we can conclude that the property psi holds in v.
If the minimum value is split from tree 't' resulting in 'x' and a new tree 't'', and if the height of tree 't' is greater than 0 and tree 't' is complete, then the height difference of the new tree 't'' is equal to the height of the original tree 't'.
Applying the function "R2" twice to any input "x" is equivalent to applying it once.
If "next_recvupd'" holds for c0, c1, p, and q, then the following conditions are true: 1. The records of c1 are equal to the records of c0. 2. The incoming information at c1, 0, p', and q' is equal to the incoming information at c0, 0, p', and q' minus the first message from c0 to p and q if p' equals p and q' equals ...
The lemma named "in_set_R" is defined with the following conditions: - It fixes a polynomial 'p' of real numbers and assumes that 'p' is not equal to zero. - It fixes a list 'qs' of polynomials of real numbers. - It fixes a list 'sign' of rational numbers. - It fixes a real number 'x'. - It assumes that 'x' is a root ...
The diamond operation of "x" and "d" of "y" is equal to "d" of the product of "x" and "y".
The function "no_ofail" with parameters P and f is a triple judgement with parameters P, f, and a lambda function that takes s and f and returns true if the function f applied to s is not None.
If the intersection of l2_inv3 and l2_inv2 holds before the transition l2, then l2_inv3 also holds after the transition.
The lemma "weakSimE" is defined with the following parameters: a fixed 'b' type parameter 'F', a psi type parameter 'P', a set 'Rel' of tuples containing 'b' and two psi type parameters, and a psi type parameter 'Q'. The lemma assumes that 'P' weakly simulates 'Q' under the relation 'Rel' in the context of 'Psi'. I...
Lemma rbt_union_rec: If tree t1 and tree t2 both satisfy the invariant property inv_12, then the result of the function rbt_union_rec with input f, t1, and t2 also satisfies the invariant property inv_12.
: Given the assumption "a" that "t" applied to "s" reduces to "r" in one step, it shows that either there exist "t'" and "s'" such that "r" is equal to "t'" applied to "s'", and both "t" reduces to "t'" and "s" reduces to "s'" in one step, or there exist "x", "p", "p'", and "s'" such that "r" is equal to "p'" with "x"...
If the successor of the successor of the length of the list "xs" is equal to 2 raised to the power of "l", then the length of the list formed by prepending "x" to the list obtained by calling the function "evens_odds" with arguments "True" and "xs" is equal to 2 raised to the power of "l - 1".
If for all natural numbers 'n', 'f' of 'n' equals 'g' of 'n', then 'f' has a product 'c' if and only if 'g' has a product 'c'.
The function "blank'" for a given program P and class C is equal to the pair consisting of class C and the set of predicates obtained by initializing the object predicate for program P and class C.
If a variable "v" is in all uses of a graph "g" and module "m", and if this module "m" is in the set of nodes of the graph "g", then the definition node of the variable "v" is also in the set of nodes of the graph "g".
The lemma named "backward_finite_path_root_acyclic_terminating_iff" assumes that "x" is the root of a backward finite path "r". It shows that "x" is acyclic if and only if the product of "x" and "r" equals zero.
If function 'f' is a lifting from 'basis' to 'basis'', then the 'map_ms' function applied to 'f' and 'C' is trimmed if and only if 'C' is trimmed.
The K axiom is valid.
If the following conditions are met: 1. 'r' is a vector, 2. 'r' is injective, 3. the transpose of 'v' is equal to the product of the transpose of 'r' and the star of 't', 4. 'w' is a forest, 5. 't' is less than or equal to 'w', 6. 'w' is less than or equal to the product of the transpose of the product of the transpos...
If the handshake invariant and phase invariant hold before garbage collection, and the phase relation invariant is a part of the system's long-term state properties, then the phase relation invariant will still be a part of the system's long-term state properties after garbage collection.
The lemma named "measurable_envelopeD2" assumes that E is a measurable envelope of set A with respect to measure M. It shows that the measure of E with respect to M is equal to the outer measure of set A with respect to M.
The function 'compress_normalize_primitive_monad' preserves the properties 'normalized_nnf_match' and 'P'. Assuming that for any 'm', 'm'' and 'f', if 'f' is in the set 'fs' and 'm' is a normalized NNF match, then if 'f' applied to 'm' equals 'm'', 'm'' is also a normalized NNF match. And also assuming that for any...
The lemma named "lim_Ln_over_power" is defined for a complex number 's'. It assumes that the real part of 's' is greater than 0. It shows that the limit of the sequence defined by the natural logarithm of 'n' divided by 'n' raised to the power of 's', as 'n' goes to infinity, is 0.
The strong deletion of a block 'b' is equal to the multiset where 'b' is subtracted from each block 'bl' in the multiset obtained by removing all instances of 'b' from the multiset 'B'.
: Assuming a game-playing value (gpv) is lossless over the full interface, and the interaction is bounded by a certain quantity 'q' for the left side of the interface and 'q'' for the right side of the interface, the advantage of an adversary in an indistinguishable chosen ciphertext attack (ind_cca) is less than or eq...
For any real matrix A of dimensions 'n by 'm, the multiplication of A with a vector (which is the product of a real number c and a vector x) is equal to the product of the real number c and the result of the multiplication of A with the vector x.
The chamber complex of the image of a function 'f' applied to 'X' is equal to the intersection of the codomain's chamber complex and the image of the function 'f' applied to 'X'.
If the absolute values of x and y are less than or equal to 1, then the arccosine of x equals the arccosine of y if and only if x equals y.
The lemma named "subgraph_clique" assumes that a graph G is well-formed, C is a subgraph of G, and C is a complete graph with vertices of C. It shows that the set of edges in G that are subsets of the vertices of C is equal to the set of edges in C.
The outstanding references of non-volatile memory in the store buffer are equal to the union of the outstanding references of non-volatile write operations and the outstanding references of non-volatile read operations in the store buffer.
If n is greater than 0, then taking the n-th bit of the result of taking the (n-1)-th signed bit of an integer w is equal to taking the n-th bit of w. This is a property of the functions "take_bit" and "signed_take_bit" in the context of binary truncation.
If a unary boolean operation 'op' has the same type 'τ' and results in two types 'σ1' and 'σ2', then 'σ1' and 'σ2' must be equal.
: Assuming that the adaptive test case 'A' passes the machine 'M' with a failure set 'FS' and a maximum length 'k', and that 'A' is an adaptive test case, and 'M' is observable, and the inputs of 'A' are a subset of the inputs of 'M', and the sequence 'io' followed by 'ioA' is in the language of 'A', and the sequence '...
If a term is differentially free, then it is also differentially safe.
If the value of a database entry for a given user "u" and two optional parameters "c" and "d" is "a", and if the value of a database entry for the same user "u" with optional parameter "d" and no other parameter is "b", then the value of a database entry for the same user "u" with optional parameter "c" and no other pa...
Evaluating the formal power series "f" at 0 is equal to the nth term of the formal power series "f" at 0.
The method table of the combination of two functions, f1 and f2, applied to a state type, is equal to the concatenation of the method table of f1 applied to the state type and the method table of f2 applied to the state type of f1 applied to the state type.
'f' is continuous on a subset 's' if and only if for all 'x' and for all 'a' in 's', if for all natural numbers 'n', 'x(n)' is in 's' and 'x' converges to 'a' sequentially, then the composition of 'f' and 'x' also converges to 'f(a)' sequentially.
A route request (rreq) with parameters for number of hops, destination IP, destination sequence number, destination key, origin IP, origin sequence number, and source IP is equal to a Route Request (Rreq) with the same parameters.
The lemma shows two things related to the evaluation of expressions in a programming language: 1. If an expression 'e' with state 's' evaluates to 'e'' with state 's'' under the programming language 'P', and the initial configuration of 'e' is valid, then either 'e' is the initialization of a class 'C'' with some clas...
If a list 'xs' is in the list 'n A', and for all pairs '(p,x)' in the set 'ps', 'p' is less than the size of 'xs' and 'x' is in 'A', then 'xs' is less than or equal to 'merges f ps xs' in the relation 'r'.
In the program 'prog', if there is an edge labeled 'et' leading from node 'n' to the entry node, then this is a contradiction or false. This means that there should be no edge leading to the entry node in the control flow graph of the program.
The pair (g, id) is in the relation 'rel' implies 'id' on the range of 'rel'.
If "t" is an embedding of "s", then the set of variables in "s" is a subset of the set of variables in "t".
The lemma named "Lyndon_pref_suf_less" assumes that "w" is a Lyndon word, "p" is a prefix of "w", "s" is a non-empty suffix of "w", and "s" is not equal to "w". It shows that "p" is lexicographically less than "s".
If the following conditions are met: 1. Agent B receives a message containing two encrypted parts: one encrypted with B's shared key (shrK B) containing the identities of agents A and B, a key K, and a nonce NB; and the other part encrypted with key K containing the nonce NB. This message is part of a set of events (e...
: given an assumption that "A" is an element of the set "Vset" indexed by "alpha", it shows that the identity object of the category "cat_Set" indexed by "alpha" at "A" is equal to the identity of the set "A".
The semantics of a foldr operation with Implication on a list 'l' and a proposition 'p' is equivalent to the condition that for all propositions 'q' in the set of 'l', the semantics of 'q' implies the semantics of 'p'.
The lemma shows that the row dimension of a matrix A after row echelon form transformation is equal to the original row dimension of matrix A, and the column dimension of a matrix A after row echelon form transformation is equal to the original column dimension of matrix A.
If the set T2 is a subset of the set T1, then the set resulting from the operation "after" applied to T2 and xs is also a subset of the set resulting from the operation "after" applied to T1 and xs.
The lemma "find_remove_2_set" assumes that the function "find_remove_2" with parameters P, xs, and ys returns a tuple (x, y, xs'). Under this assumption, the lemma shows the following: 1. The predicate P holds for x and y. 2. x is an element of the set xs. 3. y is an element of the set ys. 4. If the elements of xs are...
The case generation of a map generation with parameters pure, io, a, b, d, and r is equivalent to the case generation with parameters being the composition of pure and a, and the function that takes an output and returns the composition of io with the output of b and d, with r as the final parameter.
Mapping a logical connective substitution function 'p' over the result of another substitution function 's' applied to a formula 'f' is equivalent to applying a substitution function where 'p' is mapped over 's' to a formula where 'p' is also mapped over 'f'.
If a pair "d, d'" is in the set "r" and "d" is not in the domain of "r", then "A" is true.
The function F applied to the set A with the element a inserted is equal to the element a multiplied by the function F applied to the set A with the element a removed.
If a set 's' is finite, then it is negligible.
There exists a function f that is linear, the image of S under f is a subset of T, and for any element x in S, the norm of f(x) is equal to the norm of x.
If mapping a list of pairs (F, y) to ((F, D), y) using the function "map_of" results in the pair (F, D) being associated with some value T, then the original list of pairs fs, when mapped using "map_of", will associate F with the same value T.
If "A" satisfies the constraint of being pure, then the function pair (return composed with the identity function, RETURN composed with the function "the") is included in the set of functions from the default option assignment "dflt" with "A" to "A", under the condition that the input is not "None". This is a rule for ...
For any three lists (xs, ys, and zs), if list xs is less than or equal to list ys, and list ys is less than or equal to list zs, then it follows that list xs is less than or equal to list zs. This is a demonstration of the transitive property in the context of lists.
For a fixed matrix A, which is a field, the first element of the Gauss Jordan column k determinant P is equal to the determinant of the first element of the Gauss Jordan column k PA.
If process P is less than or equal to process Q, then the sequential composition of process S with P is also less than or equal to the sequential composition of process S with Q.
The lemma "distinct_lists_children" assumes that the lists in heap 'h' are distinct, the pointer 'ptr' is known, and that getting the child nodes of 'ptr' in heap 'h' results in 'children'. It shows that the 'children' are distinct.
: Given a transition from state (p, s) to state (p', s'), for any program p2, if p equals to the catch of Skip and p2, then the final state s' is equal to the initial state s and the final program p' is Skip.
The lemma "rroots2" assumes that the degree of polynomial p is 2. It shows that the set of x such that the polynomial p evaluated at x equals 0, is equal to the set of real roots of the polynomial p.
The trace set includes the empty set, a set with None, a set with some element 'a', and a set with some element 'a' followed by None.
The lemma named "insert_into_member_partition1" establishes that for any given element 'elem' of any type 'a', any set 'P' of sets of type 'a', and any set 'set' of type 'a', the union of the set resulting from inserting 'elem' into 'P' at 'set' is equal to the union of the set resulting from inserting the union of 'se...
The theorem named "CD_eq" demonstrates that if you encrypt a message 'X' with a key 'K' and then decrypt it with the same key 'K', you will get the original message 'X' back.
The lemma named "prv_zer_dsj_suc" assumes that "t" is an arithmetic term and "x" is a variable that does not appear in the free variables of "t". It shows that it is provable that "t" is equal to zero or there exists a variable "x" such that "t" is equal to the successor of "x".
There exists a 'y' such that the distance between 'x' and 'y' is zero.
If a path P is locally finite and the last element of P is a deadend, then P is a maximal path.
If for every pair (a, b) in the set AxB, the property P holds for a and b, then it is true that for all pairs (a, b) in the set AxB, the property P holds for a and b.
For a function 'f' and lists 'xs' and 'ys' where 'ys' starts with element 'y', the mapping of 'f' over 'xs' and 'ys' is equal to an empty list if 'xs' is empty. Otherwise, it is equal to the list that starts with the result of applying 'f' to the first element of 'xs' and 'y', followed by the mapping of 'f' over the re...
: Given a well-formed heap 'h' that has well-defined types and known pointers, if 'h' can successfully retrieve a list of child nodes from a pointer 'ptr' that starts with 'node_ptr' followed by other 'children', and if 'h' can successfully remove 'node_ptr', resulting in a new heap 'h'', then 'h'' can successfully ret...
Iterating over an array using the "array_get" function and "array_length" is equivalent to iterating over a list using the "nth" function and "length".