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Statement: stringlengths 7 24.3k
If a complex number 'z' is not equal to zero, then the function 'cis' applied to the argument of 'z' is equal to the sign of 'z'.
If "a" of "ps" is less than or equal to "d", and for any "h'" and "U'" in the set "ps", if "i" is less than or equal to the cardinality of "U'", then the polynomial degree of "h'" is less than "d", then "b" of "ps" at index "i" is less than or equal to "d".
If a type tau is equal to an OrderedSet of type rho, and rho is less than sigma, then tau is less than an OrderedSet of type sigma.
If 'p' is a prime element and 'n' is greater than zero, then 'p' divides 'x' raised to the power of 'n' if and only if 'p' divides 'x'.
If a pair (u,x) is in the set created by zipping the list 'us' with the second elements of the list 'txs', and if a pair (t,u) is in the set created by zipping the first elements of the list 'txs' with the list 'us', and if the second elements of the list 'txs' are distinct, and if the list 'us' is distinct, and if the...
The lemma named "diff_fresh_vsubst" assumes that "X" is good and "y" is not equal to "y1". It shows that "y" is fresh in the variable substitution of "y1" for "y" in "X" under the context of "ys".
: Assuming a well-formed theory "Theta", a finite set "Gamma", a term "t" that is valid in the theory "Theta", a term "u" that is the eta-normal form of "t", and that every term in "Gamma" is valid in the theory "Theta" and has the type "propT", it can be shown that the theory "Theta" and the set "Gamma" can prove the ...
: assuming the domain of a function 'f' does not intersect with the range of 'f' and 'f' is an injective function on its domain, it shows that the function formed by the union of the inverse map of 'f' and 'f' itself is also an injective function on the union of the domain and range of 'f'.
The tree resulting from the function "add_distinguishing_sequence_and_complete_if_required" applied to the distinguishing function "dist_fun", the constant "c", the machine "M", and the pair of sequences "alpha" and "beta" with their respective states after the initial state of "M", is a finite tree.
If n1 is less than n2, then the primitive recursive index enumerator of n1 and m is less than the primitive recursive index enumerator of n2 and m.
If the inverse image of a set T under the projection function is open, then the set T itself is open.
If H is a subgroup of G, H1 is also a subgroup of G, and the quotient group of G and H is a normal subgroup of H1, then for any element h in H and any element h1 in H1, the product of h, h1, and the inverse of h is an element of H1.
For any numeral n, the variable x does not affect it (x is unrestrictive to it).
The lemma named "padic_mult_comm0" assumes that "p" is a prime number. It shows that the multiplication of "p-adic" numbers "x" and "y" is commutative, meaning that the order of multiplication does not change the result. In other words, multiplying "x" by "y" is the same as multiplying "y" by "x".
If a program P and a heap h are in static heap conformance with a shared heap sh, and if the heap h at address a contains an object of class C with fields fs, then the program P and the updated heap h (where the object at address a now has fields fs') are still in static heap conformance with the shared heap sh.
The lemma named "restrict_nonempty" is defined for a fixed square function 'f' of type 'a' and 'b', where 'b' is a bounded semilattice with a supremum and a bottom element. It shows that the function 'f' applied to the concatenation of 'k' and 'ks' and 'l' and 'ls' is equal to the sum of the function 'f' applied to 'k'...
The theorem named "ikkbz_query_graph_if_simple_h" is defined with two parameters: "cost" and "h'". "Cost" is defined as the IKKBZ cost function with parameters h, cf, and match_sel. "h'" is defined as a function of x, where it applies the function h to x and the result of applying cf' to x. The theorem shows that the I...
If for every pair of elements x and y, where x is in set S and y is in set S', the relation x--y is in set G, then the Cartesian product of set S and set S' is a subset of set G.
'f' is surjective in the context of a matrix_bounded_idempotent_semiring if and only if for every element 'j', there exists an element 'i' such that the matrix element at position (i, j) is the maximum possible value (top).
The set of lower bounds "as" is equal to the set of all integers "i" such that the statement "i+1 is less than 0" is in the set "as".
The universal quantification (or infimum) over a function 'f' applied to each element in a list 'xs' from index 0 to the length of the list is equivalent to folding the list with the logical disjunction (or) operator after mapping the function 'f' to each element in the list, starting with the value 'false'.
The length of the modified Turing program is twice the length of the original Turing program.
If 'a' and 'b' are elements of the carrier Qp, then the evaluation of the Qp_val_poly at [a, b] is equal to the fourth power of 'b' plus the product of the third power of 'p' and the fourth power of 'a' in the field Qp.
Applying the function "blinfun_apply" to the composition of "a" and "b" is equivalent to applying "a" to "b" in the context of bounded linear functions.
For all elements 'g' in the set 'G', the value of 'g' in the state 's' is not false if and only if for all elements 'g' in the set 'G', the value of 'g' in the state 's' is true or there exists an element 'g' in the set 'G' where the value of 'g' in the state 's' is invalid.
The data space of the representation of an update U on a data point DP is equal to the data space of the data point DP itself. This is a simplification rule in Isabelle.
: 1. Updating any element of an empty list with a value results in an empty list. 2. Updating the first element (at index 0) of a list that starts with 'x' and continues with 'xs' with a value 'y' results in a list that starts with 'y' and continues with 'xs'. 3. Updating the element at the position after the first on...
If 'p' is a policy, and 'x' is an element in the space of the probability algebra 'Ms', then the Ionescu-Tulcea theorem applies to the function 'K' prime of 'p' and 'x', with 'M' as a constant function.
If prefix 1 and prefix 2 are both valid, and they are not the same, and they have the same length, and both match the semantics of a given word, then this situation is impossible (False).
The lemma named "stability" assumes that the time at index j is less than or equal to t1. It defines "incrs" as a function of h, t, and x, which is the sum of the increment of h, t, and x and the function s of h, t, and x. It then shows that the distance between the grid function of the discrete evolution of "incrs" wi...
: 1. If a list 'l' is not empty, then taking the successor of 0 (which is 1) elements from the list 'l' will result in a new list containing only the first element of 'l'. 2. If the length of the list 'l' is greater than 0, then taking the successor of 0 (which is 1) elements from the list 'l' will result in a new lis...
The lemma named "verts3_slim_in_verts3" assumes that "v" is an element of "verts3 slim". It shows that "v" is also an element of "verts3 G".
The lemma named "dual_wellformed_transaction_ident_cases'" assumes that the transaction "T" is well-formed. It shows that the dual of the transaction checks of "T" is equal to the transaction checks of "T" itself, and the dual of the transaction updates of "T" is equal to the transaction updates of "T" itself.
If you extend a function 'f' over a set 'A' and a set 'S' for 'n' times, the resulting set will be a subset of the set resulting from extending the same function 'f' over the same sets 'A' and 'S' for 'n+1' times.
Partial and total correct recursion are equal if we can provide a termination proof. This is under the conditions that 'body' is trimonotonic, 'V' is well-founded, 'pre a x' is true, and for any 'f', 'arb', and 'x', if for any 'arb'' and 'x''', 'pre arb'' x''' and '(x'', x) is in V' imply 'f x'' is less than or equal t...
The lemma named "atoms_pltl_AND" assumes that the set "Phi" of PLTL (Propositional Linear Temporal Logic) expressions is finite. It shows that the set of atomic propositions in the conjunction of all expressions in "Phi" is equal to the union of the sets of atomic propositions in each expression in "Phi".
For a given set S of real numbers, if S is not empty, S is bounded above, and S is closed, then the supremum (the least upper bound) of S is an element of S.
The lemma named "eventually_not_in_closed" makes the following assumptions: 1. P is a closed set. 2. The function f evaluated at time t is not in P. 3. The time t is in the set T. 4. The function f is continuous on the set T. Under these assumptions, it shows that for all future times t in the set T, the function f ev...
The lemma named "lookup_split" is defined with a fixed variable Gamma of type Gamma. It assumes that the pair (b,c) is the result of a lookup operation on Gamma with x. The lemma then shows that there exists two variables G and G', such that Gamma is equal to the concatenation of G' and the list containing the tuple (x...
The lemma named "lin_indpt_lin_comb_list" makes the following assumptions: 1. The list "ss" contains distinct elements. 2. The set of elements in "ss" is linearly independent. 3. The set of elements in "ss" is a subset of the carrier vector of dimension "n". 4. The linear combination of the list "ss" with coefficients...
If the tail of the path P is not empty, then there is no dead end in the path P from the initial vertex v0 in the game G according to the strategy σ.
The lemma named "select_memberE" assumes that "y" is in the set obtained by applying function "f" to set "S". It then concludes that there must exist an element "x" in set "S" such that when function "f" is applied to "x", it results in "y".
The lemma named "graph_formula" assumes that "g" is a function from the nth power of the p-adic number field to the p-adic number field. It shows that the graph of "g" over the integer "n" is equal to the set of elements "as" in the (n+1)th power of the p-adic number field such that "g" applied to the first "n" element...
The restriction of function 'f' to the complement of a singleton set containing 'x' is equal to the function 'f' where 'x' is mapped to 'None'.
If predicate P is less than or equal to predicate Q, then the feedback function of F applied to P is less than or equal to the feedback function of F applied to Q.
The transformation of the unit element from the polynomial ring indexed by i to the univariate polynomial ring over R is equal to the unit element of the univariate polynomial ring over R.
Theorem 2.5: If segment AB is congruent to segment CD, then segment AB is also congruent to segment DC.
If the concatenation of words 'w' and 's' equals 'v', then the left quotient of 'v' by 's' equals 'w'.
If the following conditions are met: 1) In the context of program P and environment E, expression e is of class D, 2) C is a class in program P, 3) Either the path from D to C in program P is unique or there does not exist a well-typed dynamic cast from D to C in program P, then, in the context of program P and env...
Lemma: Pole at Infinity Assuming that the function 'f' is holomorphic on the universal set and the limit of the inverse of 'f' as it approaches infinity is 'l', we can obtain constants 'a' and 'n' such that for any complex number 'z', 'f(z)' can be expressed as the sum of 'a(i) * z^i' for all 'i' less than or equal to ...
The lemma (as stated in the paper) asserts that if all prerequisites are satisfied by 'p', and if 'L4Src x y' is an element of 'p', then the intersection of 'p' and the set containing 'TCP', 'UDP', and 'L4_Protocol.SCTP' under 'IPv4Proto' is not empty.
Lemma: Convex Function on a Convex Set Assume that the set S is convex. Then, a function f is convex on the set S if and only if for all natural numbers k, and for all functions u and x, the following condition holds: if for all i in the range from 1 to k, u(i) is non-negative and x(i) is in the set S, and the sum of ...
If 'i' is less than or equal to the length of list 'ys', then the condition of list 'xs' being a sublist at index 'i' of list 'ys' is equivalent to the condition of list 'xs' being a sublist at index 'i' of list 'ys' according to the second definition of sublist.
If a polynomial 'p' is homogeneous and 't' is in the keys of 'p', then 't' excluding 'x' is in the keys of the dehomogenized version of 'p' with respect to 'x'.
The function 'get_host' applied to 'element_ptr' is pure with respect to the heap 'h'. In other words, the function 'get_host' does not have any side effects and its result is solely determined by its input 'element_ptr'.
For any word 'w' of a certain length 'a', the unsigned integer representation of 'w' modulo 2 raised to the power of the length of 'a' is equal to the unsigned integer representation of 'w'.
If "x" and "y" are weak lenses and "x" is orthogonal to "y", then the quotient of "x" and the sum of "x" and "y" is equal to the first lens.
If set N is a subset of set N', then the result of function Ri on set N is also a subset of the result of function Ri on set N'.
A list "xs" is sorted if and only if for every index "i", where the successor of "i" is less than the length of the list "xs", the element at index "i" is less than or equal to the element at the successor of index "i".
If "k" is less than the difference of "j" and "i", then the "k"-th element of the subsequence of "w" from "i" to "j" is equal to the element of "w" at the position "i + k".
The contour integral of a function 'f' along a partial circular path from point 'a' to point 'b' is equal to the negative of the contour integral of the same function along the same partial circular path but from point 'b' to point 'a'.
An empty tuple is in relation to a unit relation.
If set A is equipollent to set B, and if element 'a' is not in set A and element 'b' is not in set B, then the set resulting from inserting 'a' into set A is equipollent to the set resulting from inserting 'b' into set B.
The lemma named "bisimWeakBisimulation" is defined with two fixed variables, P and Q, both of which are of type CCS. The lemma assumes that P is bisimilar to Q. If this assumption holds, the lemma shows that P is weakly bisimilar to Q.
Theorem C10: "There exists at least one entity that is Godlike."
"there exists a complex number 'x' such that the polynomial 'p' evaluated at 'x' is not zero if and only if the polynomial 'p' is not the zero polynomial."
If a plane has different sides with respect to line l1 and points p1 and p2, then the plane also has different sides with respect to line l1 and points p2 and p1. This assumes that the plane has different sides for line l1 and points p1 and p2.
The exception effect function, when applied to the parameters i, P, pc, and type state τ with an empty list, will return an empty list.
The lemma "LS_unique_same" assumes that the concatenation of string x and y is not equal to the concatenation of string y and x. It also assumes that j and k are both greater than or equal to 1, and that the concatenation of string x repeated j times and string y repeated k times is not primitive. Similarly, it assumes...
The lemma "InvariantCllCharacterizationAfterApplyLearn" assumes the following conditions for a given state: 1. The state maintains a unique invariant (InvariantUniq) for the current assignment (getM). 2. The state maintains a characterization invariant (InvariantClCharacterization) for the current clause (getCl), the ...
For all sets of natural numbers I and all natural numbers d, if the cardinality of I divided by d plus 0 (if the cardinality of I modulo d equals 0) or 1 (otherwise) is less than or equal to the function f of the cardinality of I and d, and this function is less than or equal to the cardinality of the set I divided by ...
If you drop 3 bits from the 16-bit word representation of the number 1, the result will be 0.
The lemma named "costBIT_4x" assumes that "x" is not equal to "y", "x" is an element of the set {x0, y0}, and "y" is also an element of the set {x0, y0}. It shows that the expected value of the function "type4" applied to the list [x0, y0] and the elements "x" and "y", which is then bound to a function "BIT_step" appli...
The coercion is equal to the structured product map applied to two coercions.
The results of the generative probabilistic value (gpv) when it fails is an empty set.
The weakest precondition of the differential equation "x' = f & G on U S at time t0" with respect to the predicate "I" is less than or equal to the ceiling of "I" if and only if "I" is a differential invariant of the differential equation "x' = f & G on U S at time t0".
If 'x' and 'y' are elements of HFinite, then the product of 'x' and 'y' is also an element of HFinite. This is applicable for 'x' and 'y' as any elements of the star extension of a real normed algebra.
If the relation "rel" is reflexive and the list "ys" contains distinct elements, then the result of applying the function "filter_min_aux" to the lists "xs" and "ys" will also contain distinct elements.
The sum of the function "f" applied to each element "x" in the list "xs", evaluated at "y", is equal to the sum of the function "f" applied to each element "x" in the list "xs" and then evaluated at "y".
Converting a real matrix to an interactive array and then displaying it is equivalent to first converting the matrix to an interactive array and then displaying the real matrix in interactive arrays.
The carrier of the p-adic numbers is a universal semialgebraic set.
For the lemma named "model_to_dimacs_model_complete_max_var": If for all variables in the atoms of a formula, the variable is greater than 0, and if the formula is in conjunctive normal form (CNF), and if the DIMACS model of the DIMACS representation of the formula is true, then the abstract DIMACS model with a defaul...
The order of applying function 'f' to the result of applying function 'f' to 'a' is equal to the order of applying function 'f' to 'a'.
A list "xs" is not empty if and only if there exists an element "x" and a list "xs'" such that "xs" is the concatenation of "xs'" and a list containing "x".
The lemma named "swap_neighbors" shows that if "i+1" is less than the length of list "l", then swapping the elements at positions "i" and "i+1" in the list "l" results in a new list that is the concatenation of the first "i" elements of "l", followed by the element at position "i+1", then the element at position "i", a...
The sum of lengths in a set is zero if and only if the number of elements in the set is zero. If the number of elements is not zero, then the set is empty.
If m is not equal to n, the maximum of m and n is less than p, the length of lm is greater than p, and the p-th element of lm is 0, then the function "addition_inv" applied to the tuple (0, lm) with parameters m, n, p, and lm holds true.
The lemma named "has_derivative_add" assumes two functions, f and g, each with their own derivatives, f' and g', respectively, within a filter F. It shows that the derivative of the sum of functions f and g within the filter F is equal to the sum of the derivatives of f and g.
If "rho" is a renaming, then applying "rho" as a renaming to the atomic multiset list "AAs", and then applying the inverse renaming of "rho", results in the original atomic multiset list "AAs".
: Assuming that: - A' is a matrix in the carrier of size m by n, - a is less than m, - j is less than n, - A is defined as A' appended by D times the identity matrix of size n, - The element at the position (a,0) in matrix A is not zero, - m is greater than or equal to n, - i is not in the set xs, - xs contains distin...
If "A" does not contain "x", then evaluating the negation of the predicate "contains_pred A x" will be true.
In the context of a ring, there is a lemma stating that if set J is a subset of set I, then the Pring set of R with respect to J is a subset of the Pring set of R with respect to I.
The logical 'and' operation between 'false' and 'true' results in 'false'.
If you try a generative probabilistic value (gpv) and if it fails, then the result is still the gpv.
, assuming a run "fg_run Ate_M rho HOs SHOs" with an undefined function for "r q", and assuming that "SHOcommPerRd Ate_M (HOs r) (SHOs r)" holds for all "r", and assuming "SHOcommGlobal Ate_M Ate_M HOs SHOs", it can be shown that "weak_consensus" holds for the initial state function "?inits", the decision function "dec...
The application of the chart 'c' is continuous at a point 'x' within the set 'X', given the condition that 'x' is in the domain of the chart 'c'.
The function "f(x)" multiplied by a constant "c" is in Big O of the function "f(x)".
If 'a' is well-defined and 'b' is well-defined, then the sequence of 'a' followed by 'b' is also well-defined.
If the address of an array 'r' is not in 'as' and 'as' is in the range of 'h', then the relation 'relH' holds for 'as', 'h', and the array 'r' set with 'x' in 'h'.
It is not not true that the equivalence of P and Q is the same as the conjunction of the disjunction of Q and not P, and the disjunction of not Q and P.

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