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The infimum (greatest lower bound) of two multihole contexts 'C' and 'D' is commutative. That is, the infimum of 'C' and 'D' is the same as the infimum of 'D' and 'C'.
Dropping the first 'n' elements from a list 'xs' is equivalent to selecting the elements from 'xs' at positions 'i' where 'i' is greater than or equal to 'n'.
: if for every element x in set A, if P(x) implies Q(x), and there exists an element x in set A such that P(x) is not true but Q(x) is true, then the filtered set of A by P is a proper subset of the filtered set of A by Q.
If a vertex 'v' is in the directed vertices of 't1', then there exists a vertex 'v2' in the directed vertices of the fully normalized 't1' such that 'v' is a sublist of 'v2'.
The lemma named "concat_inf_truncate_nonempty" assumes that the ith element of sequence z is not an empty list. It shows that the concatenation of the infinite truncation of sequence z at index i is not an empty list.
"alpha is not equal to infinity".
If a cryptographic key K encrypts some data X and is in a set H, and if K is a symmetric key, then K is in the set of keys for H.
The unsigned integer representation of the bitwise mask of a word of arbitrary length 'a is equal to the mask of the minimum value between the length of 'a and a given number 'n'.
If for all 'r' less than 'k', the 'r'th digit of 'a' in base 'b' is 0, then 'a' modulo 'b' to the power of 'k' is 0.
If "i" is an element of "I", and if the semicategory "CC" is the product of semicategories "AA" and "BB", and if the semicategory "DD" is equal to "AA", then the first projection from the product semicategory "AA" and "BB" is a semicategorical morphism from "CC" to "DD".
The lemma named "special_hyperplane_span" is about a set S in a Euclidean space of dimension 'n. It assumes that 'k' is an element of the Basis. It shows that the set of all 'x' such that the dot product of 'k' and 'x' equals zero, is equal to the span of the Basis excluding 'k'.
The product of the star of x and x is less than or equal to the star of x.
The extraction of bad elements from the set "IK" after inserting an element "X" into the set "CH" is equal to the union of the extraction of bad elements from the set "IK" with the element "X" and the extraction of bad elements from the set "IK" with the set "CH".
The lemma named "norm_columns_fst_QR_decomposition" is about a fixed matrix A of real numbers. The matrix A is of type 'n by 'm, where 'n and 'm are module types. The lemma assumes that the rank of matrix A is equal to the number of columns in A. It then shows that the norm (or length) of the i-th column of the first m...
The lemma named "eq_fin_end" assumes that "u1 is finitely equal to u2" and "the concatenation of u1 and v1 is finitely equal to the concatenation of u2 and v2". It shows that "v1 is finitely equal to v2".
The abscissa of convergence for any numeral in a Dirichlet series is negative infinity.
The lemma named "strict_dominate_trans2" assumes that if node1 dominates node2 and node2 strictly dominates node3, it shows that node1 strictly dominates node3.
The lemma "LS_from_LS_acyclic" assumes that the finite state machine "M" is acyclic. It shows that the language recognized by the machine "M" is equal to the acyclic language recognized by the machine "M" starting from its initial state.
For any formal power series 'a' of a certain type (specifically, a type that is a commutative monoid under addition, has a multiplicative zero, and is a monoid under multiplication), the multiplication of 'fps_X' and 'a' is commutative. That is, 'fps_X' multiplied by 'a' is equal to 'a' multiplied by 'fps_X'.
The lemma named "weakCongE" is defined with the following parameters: a fixed Psi of type 'b, P and Q of type psi (which is a tuple of 'a, 'b, 'c), and sigma of type list of tuples (name list and 'a list). The lemma assumes that P is weakly congruent to Q and that sigma is a well-formed substitution. It shows that ...
The lemma named "absolutely_integrable_sin_product" assumes that the function 'f' is absolutely integrable over the interval from negative pi to pi. It shows that the function 'f' multiplied by the sine of 'k' times 'x' is also absolutely integrable over the same interval.
If the antipode of z is not equal to the antipode of w, then the negative of the antipode of z is not equal to the negative of the antipode of w.
A vector "v" is in the fold of "lQuot" over a vector "w" and a set "L" if and only if the concatenation of "w" and "v" is in "L".
The application of a binary operation instruction, given the parameters bop, P, pc, mxs, T\<^sub>r, (T2#T1#ST,LT), is equivalent to the condition where the predicate holds when the binary operation is well-typed in the context of P, T1, bop, and T2, and this is bound to a predicate that always holds true.
If a program P with thread t transitions from state s with expression e to state s' with expression e' through thread action ta, and if an address ad is in the allocated heap of state s' but not in the allocated heap of state s, then there exists a class type CTn such that a new heap element with address ad and class t...
The case sum of functions f1 and f2 and x is equivalent to the binding of the evaluation of x and the selection procedure of the case sum of functions f1 and f2 and x.
The line path from point a to point b is a path.
If the term 'p' can be reduced to 'q' through zero or more steps of signature reduction (as defined by the set 'F' and the relation '<'), then the leading term of the representation list of 'q' is equal to the leading term of the representation list of 'p'. This is assumed under the condition that the signature reducti...
If variable V is not in the low security level, then variable V is in the high security level.
If you push bits by a certain number 'n' from a number 'a' where 'n' bits have been dropped, and then add the result to the number 'a' where only 'n' bits are taken, the result will be equal to the original number 'a'.
The negation of an indexed infimum (greatest lower bound) of a predicate P(i) is equivalent to the indexed supremum (least upper bound) of the negation of the same predicate P(i).
If a system is not secure, given by the parameters Pc, Ic, and id, then there exists a partition R such that this partition is a view partition of Pc and id, is weakly future consistent with Pc, Ic, and id, is weakly step consistent with Pc and id, and locally respects Pc, Ic, and id.
The modular sum of x and y is equal to the remainder of the sum of x and y divided by m.
The union of the function "f" applied to each element in the list "xs" from the 0th index to the index less than the length of "xs", is equal to the union of the function "f" applied to each element in the set of "xs".
The lemma named "mu prime" assumes that "i" is less than "m" and "j" is less than or equal to "i". It shows that "mu prime" of "i" and "j" is equal to "d mu" of "i" and "j". If "j" is equal to "i", then "mu prime" of "i" and "j" is equal to "d fs" of "i" incremented by one.
The lemma named "concat_index_split_unique" assumes the following conditions: 1. The index 'i' is less than the length of the concatenated list 'xs'. 2. The lengths of the lists 'xs' and 'ys' are equal. 3. For all indices 'i' less than the length of 'xs', the length of the 'i'-th element of 'xs' is equal to the length...
The lemma for the separation logic refinement framework imports a parameter stating that the addition function for some type 'a maps from the identity relation to the identity relation.
If a term 't' is okay under a given signature 'Σ', then the term 't' with its bound variable incremented by 'inc' at level 'lev' is also okay under the same signature 'Σ'.
If a constraint model A is in the set of reachable constraints P, and if A is a well-typed constraint model according to I, and if every transaction T in the set P is admissible, and if the occurrence of t is in the intruder knowledge of A substituted by I, then the occurrence of t is in the intruder knowledge of A.
The lemma named "lim_stream_Suc" is measurable. Assuming "p" is a policy, it shows that the function that maps "a" to the limit stream of the successor policy of "p" at "a" and the transition kernel at "a", belongs to the measurable space "M" mapped to the probability algebra of the stream space of "M".
The nth derivative of the identity function at a point z is equal to z if n equals 0, 1 if n equals 1, and 0 for all other values of n.
The set of ground terms at a given term 't' is equal to the set of all possible ground functions at term 't' for every possible position 'p' in the ground positions of 't'.
"r_step_Mn" is a primitive recursive function with one argument.
If a sequence "w" satisfies the second Pi normal form of a formula "phi" with a set "N", and "N" is a subset of "N'", then "w" also satisfies the second Pi normal form of "phi" with the set "N'".
"Bignorabimus of alpha" is equal to the set of all M such that for all I, omega, and X, if omega is in the game semantics of I, alpha, and X, then omega is also in the game semantics of I, alpha, and the selection like X and omega of M.
The function "lrev_work2" applied to the bottom element returns the bottom element.
For any types "tau" and "sigma", "tau" is less than or equal to the supremum (least upper bound) of "tau" and "sigma".
If for any elements 'a' and 'a'' such that 'r a a'' holds, both 'a' and 'a'' are in set 'A', and if for any element 'a' in set 'A', 'r a a' holds, then 'r' is reflexive on set 'A'.
The series is summable where for each term, if the index k is equal to 0 or if the integer k divides n, then the term is 0, otherwise the term is 1 divided by q to the power of k.
The lemma named "phi_in_hom" assumes that "y" is an identity in category D and "f" is a morphism from "F y" to "x" in category C. It shows that the morphism "phi y f" is from "y" to "G x" in category D.
If we have a reflexive relation "r" (as given by the assumption "REFL") and an element "a" that belongs to the field of "r" (as given by the assumption "IN"), then the set of elements "above" "a" in the relation "r" is equal to the union of the set of elements "strictly above" "a" in the relation "r" and the set contai...
The abstract commitment of the AND Sigma protocol is correct.
If a set A is not almost full with respect to a property P, then there exists elements x and y in A such that the property P does not hold for x and y, and there exists an element x in A such that A is not almost full with respect to the property that holds for y and z or x and y.
If a state 's' is in 'l2_inv7', then for any secret 'x' in that state, 'x' is not in the synthesized analysis of the generators of state 's'.
If the vector angle between two vectors 'u' and 'v' is zero, then there exists a scalar 'k' such that 'v' is equal to 'k' times 'u'.
Adding zero to any element 'a' from a commutative monoid under addition results in the element 'a' itself.
The lemma named "nth_non_equal_first_eq" assumes that "x" is not equal to "y". It shows that the "n"th element of the list that starts with "x" and continues with the elements of "xs" is equal to "y" if and only if the "(n - 1)"th element of "xs" is equal to "y" and "n" is greater than 0.
The O-notation of omega is equal to omega_O.
If the execution of command 'c' in heap 'h' results in an optional pair consisting of result 'r' and heap 'h'', then the effect of command 'c' on heap 'h' is heap 'h'' and result 'r'.
The monadic union of returning A and returning B is equivalent to returning the sum of A and B, for any given A and B.
If 'x' is a well-formed item, the beta part of 'x' is equal to the concatenation of delta and omega, the end of 'x' is 'k', and delta derives an empty list, then the increment of the dot by the length of delta in 'x' is in the set of items 'pi' at 'k' with no lookahead and 'x' as the only item in the set.
For any type 'tytok', the positive distribution condition for the relation 'rel_S' is equal for all its parameters.
If "b" is equal to "c", then the conditional statement "if b then P else Q" is equal to "if c then P else Q".
: if a property P holds for 0, and for any natural number n, if P holds for all natural numbers less than or equal to n, then P also holds for the successor of n (n+1), then we can conclude that P holds for any natural number n.
The lemma named "currentLevelPrefix" assumes that "a" is a prefix of "b". It then shows that the current level of "a" is less than or equal to the current level of "b".
The function 'lextdown' applied to an index 'i' and a lazy list 'LCons' composed of an element 'x' and a list 'xs', is equal to the following: if 'x' is less than or equal to 'i', then 'lextdown' is applied to 'x' and 'xs', otherwise, a new lazy list 'LCons' is created with 'i' and the result of the function 'lextup' a...
The lemma "sp_instrs_generalize0" assumes that the function "sp_instrs" with parameters F, ret, instrs, Σi, and Σo holds true. It also assumes that Σi' is a mapping of Σi where all elements are replaced with None, and Σo' is a mapping of Σo where all elements are replaced with None. Given these assumptions, it shows th...
The sequence defined by the function "r times (1 minus the inverse of the successor of n)" tends towards "r" in the nonstandard sense as "n" goes to infinity.
The lemma named "job_dist_lower_bound_makespan" assumes "lb T A j". It shows that the sum of "t x" for all "x" in the set from 1 to "j", divided by "m", is less than or equal to the makespan of "T".
Q3''_a is a monotonic function.
The balance tree function ensures that the tree 't' is weight-balanced.
The lemma named "uminus_2" assumes that "x" satisfies the property "invariant_2". It shows two things: first, the real number representation of the negation of "x" (under the operation "uminus_2") is equal to the negation of the real number representation of "x". Second, the negation of "x" (under the operation "uminus...
The lemma named "Src_Src" assumes that "t" is an arrow. It shows that the source of the source of "t" is equal to the source of "t".
The tail of the list obtained by mapping a function 'a' over a list 'b' is equal to the list obtained by mapping the function 'a' over the tail of the list 'b'.
The lemma SA_is_cring shows that the semialgebraic set SA of order n forms a commutative ring.
If 'x' is a singleton, then 'x' is injective. This is based on the assumption that 'x' is a singleton.
If a list 'xs' consists of single combinators and there do not exist any elements 'a' and 'b' such that 'x' is the sum of 'a' and 'b', then the list consisting of 'x' followed by 'xs' also consists of single combinators.
The execution of the least upper bound function "exec_lub" with parameters "r", "f", and "T" twice is equal to "T".
The multiset resulting from the function "lower_result" applied to "C" is a subset of the multiset formed by inserting "x" into the multiset "C" after removing "y".
The lemma named "count_image_mset_lt_imp_lt_raw" assumes the following conditions: 1. A is a finite set. 2. A is the union of the multisets M and N. 3. The count of the element b in the multiset obtained by applying function f to M is less than the count of b in the multiset obtained by applying function f to N. Unde...
The lemma named "power_in_totatives" assumes that "m" is greater than 1 and "m" and "g" are coprime (i.e., their greatest common divisor is 1). It shows that the remainder of "g" raised to the power of "i" divided by "m" is in the set of totatives of "m". (Note: A totative of a number "m" is a positive integer less tha...
If the "dec_after_write" function is true for the parameters (as, am) and (s, l, Bk # r) along with "ires", then it is also true for the parameters (as, am) and (s', l, Oc # r) along with "ires".
The lemma named "minimum_spanning_forest_impl_tree2" assumes that "F" is a minimum spanning forest of a graph "G" and that the graph "G" is connected. It shows that under these assumptions, "F" is a minimum spanning tree of "G".
If from a set of hypotheses H, we can deduce that for all i, A holds, and if by adding the assumption that A holds for a specific value x to the set H, we can deduce B, then we can deduce B from the original set of hypotheses H alone.
For a given ideal of a field B, it either equals to the set containing only the zero element, or it equals to the universal set.
This is a lemma stating that "Theorem (of type boolean) 0 is less than or equal to Phi0".
The lemma named "normf_upper" assumes that the function 'f' is continuous on the set 'S' and 'x' is an element of 'S'. It shows that the absolute value of 'f' at 'x' is less than or equal to the norm of 'f'.
The lemma named "pow_mono" assumes two conditions: "a" is greater than or equal to "b", and "b" is greater than or equal to 0. Under these assumptions, it shows that "a" to the power of "n" is greater than or equal to "b" to the power of "n", and "b" to the power of "n" is greater than or equal to 0.
: Assuming for all i and j less than or equal to n, if the matrix M at position i, j is not infinity, then the constant value at that position is an integer. Also, a, b, c, i, and j are all less than or equal to n. If the value at position i, j in the matrix resulting from the Floyd-Warshall algorithm applied to M with...
The lemma named "can_divide" assumes that the product of p and a equals the product of p and b. It shows that a equals b.
The lemma named "erf_remainder_integral_aux_bound" assumes that "x" is greater than 0. It shows two things: 1. The absolute value (or norm) of the integral from "x" to "x + m" (where "m" is a real number) of the function "e to the power of negative t squared divided by t to the power of 2n" is less than or equal to "e...
The negation of a value 'a' under a condition 'P' is equal to the negation of 'a' under the same condition 'P'.
If the length of list x is equal to the length of list y, and appending a constant sequence z to both x and y results in the same sequence, then x and y must be equal.
The set of all vectors x, where the k-th component of x is zero, forms a subspace.
If 'n' is less than or equal to the infinite prefix length of lists 'xs' and 'ys', then taking 'n' elements from list 'xs' is equal to taking 'n' elements from list 'ys'.
If "v" is a closed term in the metalogic at level 0, then the substitution of "v" with a function "V" applied to a function "sigma" of "n" is equal to the substitution of "v" with "sigma" in the metalogic.
The lemma named "dilated_ticks" assumes that a function 'f' is dilating 'sub' into 'r'. It shows that the set of all 'i' such that 'f m' is less than or equal to 'i', 'i' is less than or equal to 'f n', and 'hamlet' applied to the 'i'-th element of the representation of 'r' at 'c' is true, is equal to the image of 'f' ...
If function f is differentiable on set S and function g is also differentiable on set S, then the function that is the difference of f and g (i.e., for each z in S, the value is f(z) minus g(z)) is also differentiable on set S.
If for every element 'x', predicate 'P' of 'x' is equal to predicate 'Q' of 'x', then the statement "for every 'x', 'P' of 'x' holds" is equivalent to the statement "for every 'x', 'Q' of 'x' holds".
The exponential function is twice differentiable at any point x.