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For a given matrix "m" and elements "i" and "j", one of two cases is obtained: either the pair "(i,j)" is in the set of non-zero elements of the matrix "m", or the element at position "(i,j)" in the matrix "m" is zero.
If the exception class with name xn is a direct or indirect subclass of another exception class with the same name xn in the program tprg, then the result is R.
If there is a path from node 'n' to node 'n'' in the program 'prog', and 'n' is not the entry node, and the path 'as' is not empty, then there is a corresponding path in the sequence of 'c1' followed by 'prog', where the nodes and the path are offset by the length of 'c1'.
For a direct product of quotient rings, if for all 'i' less than or equal to 'n', 'I i' is an ideal of 'R', then the direct product of the list of these quotient rings is a ring.
"it is possible that it is possible for a condition P to hold in the context of a function f" is equivalent to "it is possible for a condition P to hold in the context of a function f".
If polynomial 'p' belongs to the set of polynomials 'P[X]' and polynomial 'q' also belongs to the set of polynomials 'P[X]', then the product of 'p' and 'q' also belongs to the set of polynomials 'P[X]'.
Two lists, when appended with the same list 'xs', are equal if and only if the two lists themselves are equal.
Generating a valid topology "M" from a graph "G" is equivalent to deleting edges from "G" that are part of the union of offending flows obtained from "M" and "G".
If a multiset E is unifiable, then E is either an empty multiset or there exists another multiset E' such that the pair (E, E') is in the set of unifiable pairs.
The lemma named "image_map_of" assumes that the list of first elements from the pairs in the list "xs" contains distinct elements. Under this assumption, it shows that the mapping of the list "xs" over the set of first elements from the pairs in the list "xs" is equal to the set of second elements from the pairs in the...
The evaluation of the unary function of R and P at a point a is equal to the application of R to the evaluation of P at the same point a.
If 'x' and 'y' are elements of the carrier of Zp, and the p-adic valuation of 'x' is not equal to the p-adic valuation of 'y', then 'x' is not equal to 'y', the difference of 'x' and 'y' is not zero, and the sum of 'x' and 'y' is not zero.
If alpha prime is equal to alpha and I prime is equal to I, then the finite product category of alpha prime, I prime, and AA is shown.
Transitivity holds under certain conditions. It fixes three elements P, Q, and R of type pi, and three relations Rel, Rel', and Rel'' of type set of pairs of pi. It assumes the following: 1. Q is weakly late bisimilar to R with respect to Rel'. 2. The relation Rel is equivariant. 3. The relation Rel'' is also equivar...
If we have a subset A of the set of complex numbers z such that the real part of f(z) is greater than or equal to zero or the imaginary part of f(z) is not zero, and for any z in A, if f(z) equals zero then the real part of g(z) is greater than zero, and if f and g are continuous on A, then the function that maps z to ...
For any natural numbers n, i, j, and m, the sum of n and the maximum of i and j is less than or equal to m if and only if both the sum of n and i and the sum of n and j are less than or equal to m.
: Assuming for any polynomial 'p' of type 'a', if the degree of 'p' is greater than 1, the leading coefficient of 'p' is 1, the coefficient of 'p' at the degree of 'p' minus 1 is 0, and the coefficient of 'p' at 0 is not 0, then 'p' is not irreducible. Given these assumptions, it shows that 'a' is a field with charact...
For the "drinks" machine, it is always the case that if the value of register 2 is not greater than or equal to 100, then it implies that the next state is not equal to 2.
If we assume that for all characters 'c' in the set 'w', 'c' equals 'a', then we can obtain a value 'k' such that 'w' equals 'a' repeated 'k' times.
If set A is a subset of the carrier of the nth power of a ring R, and set B is a subset of the carrier of the mth power of the same ring R, and if the cardinality of set S is n and the cardinality of set S' is m, and if the union of sets S and S' equals the set of all numbers less than n + m, then the twisted cartesian...
The lemma named "frechet_derivative_eq_vector_derivative" assumes that the function 'f' is differentiable at a point 'x'. It shows that the Frechet derivative of the function 'f' at the point 'x' is equal to a function that multiplies a real number 'r' with the vector derivative of the function 'f' at the point 'x'.
The lemma named "content_smult" is defined for a fixed 'c' which belongs to the type class 'normalization_semidom_multiplicative' and 'semiring_gcd'. It shows that the content of the scalar multiplication of 'c' and 'p' is equal to the product of the normalization of 'c' and the content of 'p'.
The derivative of the function f at point a belongs to the set of non-zero elements in the ring of p-adic integers Zp.
The lemma "enfoldli_rule" is stated as follows: Assume the following conditions: 1. "I [] l0 \<sigma>0" holds. 2. For any elements x, l1, l2, and \<sigma>, if l0 equals the concatenation of l1, x, and l2, and "I l1 (x#l2) \<sigma>" and "c \<sigma>" both hold, then "f x \<sigma>" is less than or equal to "ESPEC E (I (l...
The set of terms in a stateful strand S is equal to the set of terms in the list representation of the stateful strand S.
If a semicategory exists for a given alpha and category, then the zero semicategory morphism from the zero semicategory to the given category is a semifunctor.
The star transformation of the scaled vector "r x" is equal to the scaling of the star transformation of the vector "x" by "r".
: Given the following conditions: - The program G is well-formed with respect to the type-checking relation wt. - The method corresponding to the signature sig in class C of program G is defined and has the properties (C,rT,maxs,maxl,ins,et). - The instruction at the program counter pc is Ifcmpeq with a certain branch...
: "In any world v, the statement 'phi or (psi and chi)' is logically equivalent to the statement '(phi or psi) and (phi or chi)'."
If the extraction of a bad element from the set IK on a channel is a subset of T, then the dynamic fake message from the bad element on the set IK on the channel is a subset of the synthetic analysis of T.
For any process 'P' and any name 'a', 'a' is fresh for the restricted process 'P' under the restriction of 'a'.
: if a certain condition "atU U" holds for the union of "c1" and "c2", and if assuming "atU U" holds for "c1" leads to a proposition "P", and also assuming "atU U" holds for "c2" leads to the same proposition "P", then we can conclude that the proposition "P" is true.
The function "SCAST" is injective on the set A if "scast" is a downward cast from 'b word to 'a word.
The lemma "phiUses_exI" assumes that "m" is in the set of predecessors of "n" in graph "g", that the function "phis" applied to "n,v" in graph "g" equals "vs", and that "n" is in the set of nodes in graph "g". It then obtains a value "v'" such that "v'" is in the set of phi uses of "m" in graph "g" and "v'" is in the s...
The lemma "succ_tableau_valuated" assumes that "s" is succeeded by "s'" and "s" is valuated. It shows that "s'" is also valuated.
If "i" is less than "j", and there exists a derivation from "a" to "b" at position "j" using production "p", and another derivation from "b" to "c" at position "i" using production "q", then there exists some "b'" such that there is a derivation from "a" to "b'" at position "i" using production "q", and a derivation fr...
The abstract state of an array hash map, after an update operation with key 'k' and value 'v' on a hash map 'hm', is equal to the abstract state of the original hash map 'hm' with the key 'k' mapped to the value 'v'.
The lemma named "circline_set_moebius_circline" shows that the set of points on a circline transformed by a Möbius transformation M is equal to the image of the original circline's set of points under the same Möbius transformation M.
For any pointer 'p', if 'p' is not Null, then 'p' is not in the list of pointers 'ps' in the heap 'h' where the heap at location 'p' has been updated to 'p'.
The cosine coefficient of the successor of a number 'n' is equal to the negative of the sine coefficient of 'n' divided by the real value of the successor of 'n'.
If a list "xs" is not finitely long (i.e., it is infinitely long), then the last element of the list "xs" is undefined.
The join (supremum) of an empty set is equal to the meet (infimum) of the universal set.
The lemma named "strong_int_gr_induct_helper" assumes that an integer "k" is less than another integer "i". It also assumes that for any integer "i" greater than "k", if a property "P" holds for all integers "j" that are greater than "k" and less than "i", then the property "P" also holds for "i". Given these assumptio...
There is a delay bisimulation base between transition system 1 and transition system 2, given the bisimulation relation "bisim", the tau-moves "tau_move1" and "tau_move2", and the final states "final1" and "final2".
If the "prim_spanning_invariant" holds for a tree 't', a vertex 'v', a graph 'g', and a root 'r', and if the intersection of the graph 'g' and the transpose of the complement of the vertex 'v' is not the bottom element, then the "prim_spanning_invariant" also holds for the union of the tree 't' and the minimum arc of t...
The zhmset of the product of A and B is equal to the product of the zhmset of A and the zhmset of B.
For all elements 'q' in the set produced by the function 'conjuncts' applied to 'p', the interpretation function 'Ifm' applied to 'bs' and 'q' holds true if and only if the interpretation function 'Ifm' applied to 'bs' and 'p' holds true.
If 'r' is an order, then 'x' is not less than itself according to the order 'r'.
If the set of all states reachable from the initial state q₀ in a transition system with transition function δ and alphabet Σ is finite, then the set of all states reachable from the initial state q₀ in a system with the same transition function δ and alphabet Σ is also finite.
The lemma named "tau_converg'" assumes that the deterministic time function 'i' applied to 'x' converges to 't'. It shows that the tau function applied to 'i' and 'x' also converges to the 'x'-th element of the block function applied to 'i' and 't'.
An empty set entails MGT CALL.
If the first component of a pair "x" is equal to "x" itself, then "x" must be equal to zero.
The lemma named "indspace_fspanning_set" is defined with the following conditions: It fixes two lists, "vs" of type 'v and "hs" of type 'g. It also defines "hfvss" as the negative horizontal orbit list of "hs" induced vector "vs". It assumes that the set of "vs" is a subset of "V" and "V" is the finite span of "vs". ...
For a given function 'f' applied to 't', the function is monotone continuous. This is determined using the least upper bound 'lub', the order 'ord', and the function order 'fun_ord' and function least upper bound 'fun_lub'. This lemma is used for continuity introduction and simplification.
If 'i' is less than or equal to the degree of polynomial 'p', then the 'i'-th element of the vector representation of 'p' is equal to the coefficient of 'p' at the position (degree of 'p' - 'i').
If a term 's' is in the term implication closure set of the union of all substitution sets 'M' applied to a substitution 'delta' where 'P' holds for 'delta', then there exists a substitution 'delta' such that 'P' holds for 'delta' and the term 's' is in the term implication closure set of the substitution set 'M' appli...
If the low-level security condition "low_mds_eq" holds between memory states "mem" and "mem'", and if the pair of configurations (\<langle>c, mds, mem\<rangle>, \<langle>c, mds, mem'\<rangle>) is in the relation "rel_inv", then the pair of configurations (\<langle>c, mds, mem'\<rangle>, \<langle>c, mds, mem\<rangle>) i...
If there is a path 'p' from node 'u' to node 'v', and edge 'e' is in the set of edges that make up this path, then edge 'e' must be in the edge set 'E'.
The lemma named "ex_consume_NO" assumes that the list "vl" is not empty and that the function "f" applied to the transition "trn" equals the first element of the list "vl". Under these assumptions, it obtains a list "vl'" such that the function "consume" applied to the transition "trn" and the list "vl" results in the ...
The lemma named "fold_update" assumes that the keys of a map 'm' are finite. It also assumes that a key 'k' is not in the keys of 'm' and that for any key 'k'' in the keys of 'm', 'k'' is less than or equal to 'k'. Under these assumptions, it shows that the fold function applied to the updated map (with key 'k' and val...
The function "allow_pfun" applied to the composition of the identity function and the "Some" function is equivalent to the assertion that for all "Ax", "x" is defined.
If machine M weighs type A at a certain point, then type A is weighed at that point.
The implementation of "goinitial", "ginitial", "gpath_is_empty", "gselect_edge", "gis_on_stack", "gcollapse", "last_is_acc", "ce", "gis_done", "gpush", "gpop", "goBrk", "gto_outer", "go_is_done", "is_done_o", and "go_is_no_brk" are equivalent to their respective operation patterns.
The multiplication of the negative of a polynomial 'p' and another polynomial 'q' is equal to the multiplication of the polynomial 'p' and the negative of the polynomial 'q'. This demonstrates the commutative property of multiplication with respect to scalar multiplication in the context of polynomials.
When finding the indices of a list where a predicate P holds, if the predicate P holds for the first element 'a' of the list 'xs', then the result is 0 followed by the indices in the rest of the list 'xs' incremented by 1. If the predicate P does not hold for the first element 'a', then the result is just the indices i...
If h0 is disjoint from h1, then the restriction of the union of h0 and h0' to the domain of h1 is equal to the restriction of h0' to the domain of h1.
If R is an element of the set of beta regions, then R is a subset of V.
: Assuming that: - A is a subset of the set of n-dimensional vectors, - B is also a subset of the set of n-dimensional vectors, - C is the sum of A and B, - There exists a bound for C such that C is a subset of the set of vectors bounded by this bound, Then it shows that: - If A is not an empty set, then there exists...
If alpha derives beta, and if the sequence formed by concatenating u, alpha, and v is a derivation, then the sequence formed by concatenating u, beta, and v is also a derivation.
The lemma named "LS_unique_distinct" assumes the following conditions: 1. The concatenation of x and y is not equal to the concatenation of y and x. 2. j is greater than or equal to 2. 3. The concatenation of j copies of x with y is not primitive. 4. k is greater than or equal to 2. 5. The concatenation of x with k co...
If a function 'f' is 'n' times differentiable on an open set 'S', then the functions 'sin(f(x))' and 'cos(f(x))' are also 'n' times differentiable on the same set 'S'.
If a natural chop operation is performed on elements i, j, and k, then j is a subset of i.
The lemma named "hermitian_decomp_unitary" assumes that A and B are decomposed into a Hermitian form U. It then shows that U is unitary.
The size of the multiset that results from filtering multiset M using function f is less than or equal to the size of the original multiset M.
There exist a heap 'h', an element pointer, an optional shadow root pointer, and a heap 'h'', such that the heap 'h' is well-formed, has a valid type, and contains known pointers. Moreover, the operation of safely getting the shadow root of the element pointer in heap 'h' results in the optional shadow root pointer and...
If the pair (a, b) is in the relation "bf_ifex_rel", then "b" is a read-only if-expression.
The lemma named "boundOutputPar2Dest'" is defined with the following parameters: a list of names "xvec", an element "M" of type 'a which is a name with a finite support, a psi calculus "P" of types 'a, 'b, and 'c which are names with finite support, a list of names "yvec", an element "N" of type 'a, a psi calculus "Q" ...
If "n" is greater than 0, then the set of positions in a row (where the row is defined as "x" concatenated with a constant sequence of "any" and "n" number of "False") starting from position 0, is finite.
If the first call in the expression 'e' is defined and equals to (a, M, vs), then the tau-move of 'e' is equivalent to the condition that if a call is synthesized for (a, M, vs), then the tau-external of 'a' and 'M' holds. The second condition, named 'tau moves1 not calls1', states that if the first call in the expres...
The lemma named "pi_lsi" asserts that "foldli" is a proper iterator, according to the definition of "proper_it'".
Casting from a ShadowRoot to an Object is injective. In other words, if the result of casting ShadowRoot 'x' to an Object is equal to the result of casting ShadowRoot 'y' to an Object, then 'x' must be equal to 'y'.
The conjugate of a number 'x' raised to the power 'n' is equal to the conjugate of 'x' raised to the power 'n'.
The vector derivative of a function "g" at a point "t", subtracted by a constant "z", with respect to a filter "net", is equal to the vector derivative of the function "g" at the same point "t" with respect to the same filter "net".
If A and B are ideals of the ring R, and if a is an element of A and b is an element of B such that the sum of a and b equals the multiplicative identity in R, then the projection of b onto A in R equals the multiplicative identity in the quotient ring of R and A, and the projection of a onto B in R equals the multipli...
For any list 'xs' and function 'f', taking the first 'n' elements of the concatenation of 'xs' and 'f' is equivalent to appending the first 'n' elements of 'xs' and the first 'n - length of xs' elements of 'f'.
If a function 'f' applied to 'x' equals 'x', then applying the function 'f' 'n' times to 'x' will also equal 'x'.
The composition of relation R with the union of relations S and T is equal to the union of the composition of relation R with S and the composition of relation R with T.
If list 'xs' is less than list 'ys' and both lists 'xs' and 'ys' are sorted, then the set of list 'xs' is less than the set of list 'ys'.
The flattening of zero is equal to zero.
If "iota" is an element of "Inf_F" and the grounded interpretation of "iota" is not equal to "None", then the grounded interpretation of "iota" is a subset of "Inf_G".
For any set X, inserting the element "Inr i" into the union of set X and the image of the set of all numbers less than i under the function Inr, is equal to the union of set X and the image of the set of all numbers less than or equal to i under the function Inr.
The function type of the generalized function definition "fd" is equal to the function type of the original function definition "fd".
If 'f' and 'g' are elements of the set 'P', the degree of 'g' is less than 'n', and the degree of 'f' equals 'n', then the degree of the sum of 'f' and 'g' (denoted as 'f \<oplus>\<^bsub>P\<^esub> g') is also 'n'.
: Given the assumptions that: 1. List 'xs' is equal to list 'xs''. 2. If list 'xs'' is empty, then 'n' is equal to 'n''. 3. For any elements 'y' and 'ys', if list 'xs'' is equal to the list formed by prepending 'y' to 'ys', and both 'y' and 'ys' are not undefined, then the function 'c' applied to 'y' and 'ys' is equal...
The graph relation of the identity relation paired with itself is equal to the identity relation.
For any function 'f' that maps natural numbers to complex numbers, the sum of 'f' over the range from 0 to less than 2 is equal to the value of 'f' at 0 plus the value of 'f' at 1.
If 'x' is an ordinal and the successor of 'y' is in the successor of 'x', then 'y' is in 'x'.
If the function 'wcode_goon_right_moving_1' is true for the inputs 'ires', 'rs', and the pair '(aa, [])', and if 'b' is equal to 'aa' and '[Oc]' is equal to 'ba', then the function 'wcode_backto_standard_pos' is true for the inputs 'ires', 'rs', and the pair '(aa, ba)'.
The Kernel is a right submodule.
If a set 'I' is not empty and finite, and a value 't' is less than the maximum value in 'I', then the set of elements in 'I' that are greater than 't' is not empty.