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If a policy "p" is not a "MT" of "C", then applying "C" to the firewall policy list obtained by removing "MT" rules from "p" is equivalent to applying "C" to the firewall policy list of "p".
If k is greater than 0, then the generalized binomial coefficient of a and k is equal to the sum of the generalized binomial coefficient of (a - 1) and (k - 1) and the generalized binomial coefficient of (a - 1) and k, for a given field of characteristic 0.
For any two functions 'f' and 'g' that map to integers, given a finite set 'A', if for every element 'x' in 'A', 'g' of 'x' is not zero and 'g' of 'x' divides 'd', and 'd' is not zero, then there exists an integer 'c' such that the sum of 'f' of 'x' divided by 'g' of 'x' for all 'x' in 'A' equals the real number equiva...
Given the assumptions that 'mu' is an arrow in category C, 'nu' is also an arrow in category C, and the source of 'mu' in category C is the target of 'nu' in category C, it shows that 'Phi' of the pair ('mu', 'nu') is an arrow in category D. Furthermore, the source of 'Phi' of the pair ('mu', 'nu') in category D is the...
If a type variable 'n' is new to a substitution 'S' and a type 't', then 'n' is also new to the result of applying the substitution 'S' to the type 't'.
Zero is in the real interval of zero.
If for almost all 'x', 'P x' holds true, and if for any 'x', 'P x' implies 'Q x', then for almost all 'x', 'Q x' holds true.
If the function "find" applied to "f" and "x" equals "Some y", then "f" applied to "y" is true and "y" is an element of the set "x".
The closure of a set S in Euclidean space is equal to the closure of S.
The value of a prime number raised to the power of a natural number k is equal to the extended integer k.
The weight of the result of merging a sibling with a tree 't' is equal to the weight of the original tree 't'.
The lemma "exec_preserve_deadlocked" assumes that the Java Virtual Machine program P is well-formed according to a certain property Phi. It shows that the state of deadlock is preserved under the conditions of JVM finality, the execution of program P, the conversion of resource allocation, and the correctness of the JV...
For any relation "ate_ref_rel", the union of all majorities for a given round "r", vote function "v_f", and decision function "d_f", when transitioned by the "Ate_trans_step" with respect to the higher order functions "HOs", will result in a relation that is greater than or equal to the initial "ate_ref_rel".
If a class C sees methods Mm' in the compiled program cP, and cP is the compiled version of program P using function f, then there exists a set of methods Mm such that class C sees these methods in program P, and Mm' is the result of applying a certain mapping function to Mm. This mapping function takes a method and re...
The lemma named "transaction_fresh_subst_domain" is defined with a fixed protocol substitution, denoted as sigma. Assuming that the transaction fresh substitution of sigma, T, and A holds true, it shows that the substitution domain of sigma is equal to the set of transaction fresh T.
The lemma named "psiCasesFresh" is defined with two fixed elements: "x", which is a name, and "Cs", which is a list of tuples. Each tuple in the list consists of an element of type 'c (which is a fresh sequence name) and a psi element of types 'a, 'b, 'c. The lemma shows that "x" is fresh for the psi cases of "Cs" if a...
"not not P and not not (P implies Q) implies not not Q".
The signed cast of 2 to the power of any number 'x' in the context of 'a' length signed word is equal to 2 to the power of 'x' in the context of 'a' word.
The CP-rank of a tensor A is less than or equal to the product of the dimensions of tensor A.
If a configuration 'cfg' is in the set of valid configurations and another configuration 'cfg'' is in the set K_cfg of 'cfg', then the pair (state of 'cfg', state of 'cfg'') is in the set E.
The identity function, represented by "λx. x", is contour integrable on the line path from point a to point b.
The join of zero and any arity (denoted as 'x') is always zero.
The monitor function for a given formula and trace is equal to the M function applied to the trace.
If a pair (x, y) is in the transitive symmetric closure of a relation r, and if another pair (y, z) is also in the transitive symmetric closure of the same relation r, then the pair (x, z) must also be in the transitive symmetric closure of the relation r.
If "x" is not equal to "S" and "S" is in the set of states of the root of "A" in a Hierarchical Automaton, then the pair "(x, S)" is not in the transitive closure of "A".
The class of a NullPointer in the context E is defined as an Object with no fields and no methods.
For an executable Euclidean space, if 'n' is less than the dimension of the space, then the 'n'th element of the list representation of a point 'x' in the space is equal to the dot product of 'x' and the 'n'th element of the basis list.
The lemma named "span_same" assumes that the set S is a subset of the matrix kernel of A. It shows that the span of S is equal to the nullspace-complement span of S.
The lemma named "functor_implies_equals_dom" assumes that tau is a functor from category A to category B. It shows that functor F is equal to tau.
If the proposition "P" is not true for a state "σ", then the result of evaluating the symbolic execution "if P then B1 else B2" in the state "σ" is the same as the result of evaluating "B2" in the state "σ".
If the determinant of matrix B is not zero and the product of matrices A and B equals matrix C, then matrix A is equal to the product of matrix C and the inverse of matrix B.
If "x" and "y" are both very well-behaved lenses and "y" is equivalent to "x" in terms of lenses, then the lens quotient of "x" and "y" is a bijective lens.
If the following conditions are met: the well-formedness of the code invocation "ci", the code invocation "ci" is inserted into the program "P" at the heap "h", stack "stk", location "loc", class "C", method "M", program counter "pc", and frame stack "frs", and the instruction "ins" is okay to jump to the program count...
The lemma named "NTT_gen_NTT_full_length" assumes that the length of the list "numbers" is equal to "n". It shows that the general Number Theoretic Transform (NTT_gen) of "n" and "numbers" is equal to the Number Theoretic Transform (NTT) of "numbers".
Vertex 'v' is connected to vertex 'v'' if and only if there exists a distance 'd' such that 'v' is at distance 'd' from 'v''.
The expected measure of a probability mass function M is equal to 1.
The lemma named "dom_eqI'" assumes that "a" is in the domain of "f". It shows that "a" equals the domain of "f".
Given the assumptions that there is a path from node 'n' to node 'n''', node 'n'' is postdominated by node 'n', and node 'n' is not postdominated by node 'n', we can obtain an edge 'a' and two paths 'as'' and 'as''', such that the original path 'as' is the concatenation of 'as'', 'a', and 'as''', the edge 'a' is valid,...
If x is greater than or equal to 1, then the natural logarithm of the Gamma function of x is greater than or equal to the natural logarithm of (2*pi/x) divided by 2 plus x times the natural logarithm of x minus x. Additionally, the natural logarithm of the Gamma function of x is less than or equal to the natural logari...
If the function kappa_mu_nu applies to f, then f has an approximate least fixed point.
The set of free variables in a strand step, denoted by "fv\<^sub>s\<^sub>t\<^sub>p x", is finite.
The lemma named "dist_midpoint" establishes four facts about the distance between points in a real normed vector space. It fixes two points, a and b. 1. The distance from point a to the midpoint of a and b is half the distance from a to b. 2. The distance from point b to the midpoint of a and b is also half the dista...
If the following conditions are met: 1. Predicate P holds for α, an empty list, and another empty list. 2. For any elements ad and ae, Predicate P holds for α, an empty list, and a list containing ad followed by ae. 3. For any elements z and aa, Predicate P holds for α, a list containing z followed by aa, and an empty...
If 'a' is greater than 1 and 'x' is greater than 0, then 'a' is less than 'x' if and only if the logarithm of 'x' to the base 'a' is greater than 1.
There is no microstep transition from the process that is receiving a message at location "l" to another process "q" under the context "Γ".
The lemma named "always_imp_trans" assumes two conditions: first, that P implies Q is true for a given state sigma, and second, that Q implies R is also true for the same state sigma. Under these assumptions, it shows that P implies R is also true for the state sigma.
: assuming for every element 'x' in set 'A', there exists an element 'y' in set 'B' such that 'x' is a subset of 'y', it can be shown that the union of set 'A' is a subset of the union of set 'B'.
For variables tl, p, q, pp, qq, given that a list tl from p to Ps and a list tl from q to Qs exist, and the intersection of the sets Ps and Qs is empty, and the size of Qs is less than or equal to the size of Ps, then the following operations can be performed: - pp is assigned the value of p - While q is not Null, the...
If a tau-execution move is valid for a given configuration in the context of a certain program, thread, expression, heap, and two states, then a sequence of tau-execution moves is also valid for the same configuration but with the expression being part of a list of expressions.
For a given tree "t", the set of all paths in "t" is a subset of the set that includes the empty path and the union of all sets of paths in the children of "t", where each path in a child is prefixed with the index of that child in the list of children of "t".
In the context of a finite measure, the measure of the difference between two sets A and B is equal to the measure of set A minus the measure of the intersection of sets A and B, given that both sets A and B are in the measure space M.
: If there exists a pair (i, l) in the set of inputs (get_Ins) of the guards (Guards) of transition t2, such that the filter of guards that constrain the variable (V) of input i equals [(In (I i) l)], and there exists another pair (i', l') in the set of inputs of the guards of transition t1, where i equals i' and the ...
If a tree "t" is ordered and balanced, then the bound of the height of the tree "t" with respect to "k" is less than or equal to the product of the number of nodes in the tree "t" and "k".
"S is a face of S if and only if S is convex".
If an element "x" is in the set "S", then the set of terms of the step "x" is a subset of the set of terms of the state "S".
For a given matrix A of type 'a in a commutative ring, if A is an n x n matrix and there exists another n x n matrix B such that the product of A and B equals the identity matrix of order n, then the product of B and A also equals the identity matrix of order n.
If a condition P is false, then the result of the conditional operation between P and Q will be Q.
The simplicial complex induced by the monotone boolean function of 'n' with the boolean function 'and' applied to 'f' and 'g', is equal to the intersection of the simplicial complex induced by the monotone boolean function of 'n' with 'f' and the simplicial complex induced by the monotone boolean function of 'n' with '...
If the function 'f' is equal to the ordered pair [0, 2], then the application of the ArrMap function of the RK23 functor to 'f' is equal to the application of the ArrMap function of the functor 'F' to the ordered pair [0, 1].
The lemma 'blinding_of_view_induct' assumes that 'x' is a blinding of 'y' view. It also assumes that if 'x' and 'y' are blindings of a product of a hash of view content and a hash list of views, then a property 'P' holds for the 'View' of 'x' and 'y'. Under these assumptions, it shows that the property 'P' holds for 'x...
: In parallel, given the precondition that 'i' is not equal to 'j', the following two processes occur: 1. The variable 'out' is set to True. Then, while True (in an infinite loop) with the invariant that 'i' is not equal to 'j', the process waits until 'out' is True, then sets 'out' to False and 'who' to 'i'. If 'out...
The complex number representation of the real number representation of a rational number 'q' is equal to the rational number 'q' itself.
If a state variable 'sv' is in the atoms of the encoded operators of a problem with no operation at time 't', and 't' is less than the successor of 'h', then 'sv' is a valid state variable.
The lemma "preserves_adjunction" assumes that there is an adjunction in the bicategory defined by parameters V\<^sub>C, H\<^sub>C, \<a>\<^sub>C, \<i>\<^sub>C, src\<^sub>C, trg\<^sub>C, f, g, \<eta>, and \<epsilon>. It then shows that there is also an adjunction in another bicategory defined by parameters V\<^sub>D, H\<...
If "x" is a vector, then the product of "x" with itself is equal to "x".
If the multiplication of elements x and y is commutative (i.e., x multiplied by y equals y multiplied by x) and both x and y are elements of the group G, then the order of the product of x and y divides the product of the order of x and the order of y.
The factorial of the successor of 0 is equal to 1.
If 'c' is a divisor of 'b', then the symmetric modulo of 'a' and 'b', further symmetric modulo 'c', is equal to the symmetric modulo of 'a' and 'c'.
If the truth value of p is less than or equal to the truth value of p', and the truth value of q' is less than or equal to the truth value of q, then if the hybrid system H holds for p', x, and q', it also holds for p, x, and q.
The lemma named "chop_subst_emb0" assumes two conditions. First, "s" is an application. Second, the result of applying the substitution "rho" to "s" and then chopping is not equal to the result of chopping "s" first and then applying the substitution "rho". Under these assumptions, it shows that the embedding step at t...
If a vector 'vs' is orthogonal, and 'i' and 'j' are less than the length of the vector 'vs', then the dot product of the 'i'-th and 'j'-th elements of the vector 'vs' is zero if and only if 'i' is not equal to 'j'.
, given the assumptions that the function "rb_aux_inv" is applied to the tuple (bs, ss, p # ps), the signature criterion "sig_crit" is not satisfied for the parameters bs, the new syzygy signatures "new_syz_sigs" for ss, bs, and p, and that b is an element in the set bs, it can be shown that the leading term "lt" of b ...
The distance between the scaled versions of two vectors 'x' and 'y' by a scalar 'a' is equal to the absolute difference between 'x' and 'y' multiplied by the norm of 'a'. This is applicable for 'a' which is of the type 'real_normed_vector'.
If from the set H with B added, C can be inferred, then from the set H with (A OR B) added, (A OR C) can be inferred.
If the lists 'as' and 'bs' are distinct and their intersection is empty, and if the list [x,y] is a sublist of the concatenation of 'as' and 'bs', then the list [x,y] is either a sublist of 'as', a sublist of 'bs', or 'x' is the last element of 'as' and 'y' is the first element of 'bs'.
For any two quantities 'x' and 'y' of a certain dimension type 'd' and unit system 's', 'x' is equivalent to 'y' if and only if 'x' equals 'y'.
, given the assumptions that a pair (s, s') is in a relation "rel", a generic condition "gen_cond" holds for s, and the stack implementation "s" is not empty, it shows that the operation "get_pending_impl" on s is less than or equal to the supremum of two sets. The first set is the downward closure of the relation co...
The sources at level 1 for sA6 are an empty set.
The nth coefficient of a formal Laurent series 'f' at its subdegree is zero if and only if the formal Laurent series 'f' is equal to zero.
For a function 'f' mapping from a Euclidean space to a 1-dimensional real space, 'f' is integrable over a set 'S' if and only if the function that maps 'x' to the first component of 'f x' is also integrable over 'S'.
Given fixed values for dsn, dsk, dsk', flag, hops, nhip, nhip', pre, and pre', the tuple (dsn, dsk, flag, hops, nhip, pre) is less than or equal to the tuple (dsn, dsk', flag, hops, nhip', pre').
"S t t" is not true, or in other words, the relation "S" is irreflexive for any term "t".
If H1 is a maximum subgraph satisfying property P, and H2 is also a maximum subgraph satisfying property P, and if H1 is a subgraph of H2, then H1 must be equal to H2.
For any state 'tau', if 'X' evaluated at 'tau' is not equal to 'true' evaluated at 'tau' and 'X' evaluated at 'tau' is not equal to 'bot' evaluated at 'tau', then the logical OR operation between 'X' and 'null' evaluated at 'tau' is equal to 'null' evaluated at 'tau'.
If the following conditions are met: 1. Set A is finite, 2. Set A is not empty, 3. The maximum value in Set A is less than the minimum value in Set B, 4. The nth element of Set A is not the maximum value of Set A, then the nth element of the union of Set A and Set B is equal to the nth element of Set A.
The function "check_all_index" applied to a function "f" and a list "xs" is OK if and only if for all indices "i" less than the length of the list "xs", the function "f" applied to the "i"-th element of "xs" and "i" itself is true.
"r_delay" is a recursive function of two variables.
If 'i' is not less than the length of 'x', then the 'i'-th element of 'x' is equal to the 'i - length of x'-th element of the empty list.
If the sort of 'a' is equal to the sort of 'b', and the sort of 'b' is equal to the sort of 'c', and swapping 'a' and 'c' in 'x' leaves 'x' unchanged, and swapping 'b' and 'c' in 'x' also leaves 'x' unchanged, then swapping 'a' and 'b' in 'x' will also leave 'x' unchanged.
If there is a surjective function 'f' from set A to set B, and another surjective function 'g' from set B to set C, then the composition of 'g' and 'f' (denoted as 'cmp g f') is also a surjective function from set A to set C.
If the image of set A under function I is a subset of the sets in N, and if an element i is in set I, and if function f is measurable from M to N, then the predicate function that maps x to whether f(x) is in A(i) is measurable from M.
If for all elements 'h', 'h' substitutes on set 'A' for the choice function 'Ch h', then 'A' is Pareto separable.
The lemma "isNotDSource_EmptyAcc" assumes that "i" is not a dependency source in "S". It then shows that the accumulation of "i" in "S" is an empty set.
The set of TLS messages is a type-consistent set.
For any integers n, m, and c, and any binary sequences bs and cs, the function "bin_rsplit_aux" applied to n, m, c, and the concatenation of bs and cs is equal to the concatenation of the function "bin_rsplit_aux" applied to n, m, c, and bs with cs.
If event e1 is less than event e2 in the total order of a node, and event e2 is less than event e3 in the same total order, then event e1 is less than event e3 in the total order of the node.
, under certain conditions, an arc can be formed in the context of Boruvka's algorithm for finding minimum spanning trees. These conditions are: 1. F is an equivalence relation. 2. v is a forest. 3. e is an arc. 4. F is regular. 5. F is a subset of the forest components of the intersection of F and v. 6. v is regular....
A point 'l' is a limit point of a set 'S' in a T1 space if and only if for every open set 'U' containing 'l', the intersection of 'U' and 'S' is infinite.
In the context of an algebra, there is a lemma called sigma_property_disjoint_lemma. This lemma assumes that M is a subset of C and that C is a closed disjoint interval over the set Omega. The lemma shows that the sigma sets over the set Omega and M are a subset of C.
The circular omega of x is equal to 1 combined with the omega of x.