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Statement: stringlengths 7 24.3k
A function 'f' is square integrable over a set 'S' if and only if the negative of the function 'f' is also square integrable over the same set 'S'.
If a pair (w, x) is in a relation r, and r is a well-founded and transitive relation, and both w and x are elements of a set A, and A is a small set, then the order map of w in relation to A and r is less than the order map of x in relation to A and r.
The lemma named "order_cycle_non_elem" assumes that "a" is not an element of the set "as". It shows that the order of "a" in the cycle of "as" is 1.
If set A is finite and the cardinality (or size) of set A is less than r, then the set of all n-element subsets of set A where n equals r is an empty set.
The space of a null measure of a measure space M is equal to the space of M itself.
If a function symbol 'f' is well-typed and the result of 'f' is not unprotected, then 'f' is well-typed in the ground expressions.
The lemma named "diamond_y_valid" demonstrates that the function "diamond_y" is piecewise continuously differentiable on the interval from negative half of d to half of d. This is represented by two equivalent statements, ?P and ?Q.
If a function 'f' is updated such that 'x' maps to 'y', then the value of the function 'f' at 'x' is 'Some y'.
The cosine of (x + pi) is equal to the negative cosine of x, indicating that the cosine function has a period of pi.
The determinant of a 3x3 matrix with 0, (a - x scaled by a), and b as its rows is 0 if and only if x equals 1 or the determinant of a 3x3 matrix with 0, a, and b as its rows is 0.
: if for all 'z' and 'n', every element in the union of sets 'A' and 'B' is 'nvalid' at 'n' implies that every element in the image of 'f z' over set 'Ms' is 'nvalid' at 'n', and if 'Cm' is an element of 'Ms', and every element in 'A' and 'B' is 'nvalid' at 'na', then 'f z Cm' is 'nvalid' at 'na'.
The Inverse Discrete Fourier Transform (IDFT) of a sequence "a" at index "i" with a length of "2m" is equal to the sum of the IDFT of the even-indexed elements of "a" at index "i" with a length of "m" and the product of the "i"th power of the reciprocal of the "2m"th root of unity and the IDFT of the odd-indexed elem...
The polynomial composition of the negation of a polynomial 'p' and another polynomial 'r' is equal to the negation of the polynomial composition of 'p' and 'r'. This is true for any polynomials 'p' and 'r' in a commutative ring.
The lemma named "SeqWRP_SUCC_lemma" assumes that the atom "y'" is distinct from the set of "s", "k", and "y". It shows that if "SeqWRP s (SUCC k) y" holds, then there exists a "y'" such that "SeqWRP s k (Var y')" and "y" is equal to the successor of "Var y'" hold.
The lemma named "Mid_perm" assumes that "A is the midpoint between B and C". It shows that "A is the midpoint between B and C" and "A is the midpoint between C and B".
If M and N are modules over the ring R, z is a module homomorphism from the module of integers modulo e over R to M, f is a module homomorphism from M to N, and z1 is a module homomorphism from N to the module of integers modulo e over R, and if the sequence of module homomorphisms and modules is exact at the fourth po...
If a problem is a valid SAS+ problem and there is a parallel solution for this problem, then for all operators in the solution, these operators are also in the STRIPS problem derived from the SAS+ problem.
The lemma named "list_all_list" assumes that "A" is a finite set. It shows that the function "list_all" applied to the function "phi" and the list of elements in "A" is equivalent to stating that for all elements "a" in "A", the function "phi" applies to "a".
The number of equivalence relations R on a set A, such that the number of equivalence classes in A/R is k, is equal to the number of partitions P on set A, such that the number of parts in P is k.
If 'a' is congruent to 'b' modulo 'm', and 'a' is coprime with 'm', then 'b' is also coprime with 'm'. This is applicable for any 'a' and 'b' that belong to a unique Euclidean semiring with a Euclidean semiring greatest common divisor.
If the term 't' does not have a loose bound variable at level 'lev'', and 'lev'' is less than or equal to 'lev', and 't' does not have a loose bound variable at the successor of 'lev', then 't' with 'lev'' decreased does not have a loose bound variable at level 'lev'.
The hom-set of the opposite semicategory of a given semicategory for objects a and b is equal to the hom-set of the original semicategory for objects b and a.
If a set 'U' is a part of the real borel sets and 'U' is not an empty set, then the standard borel of the restricted space of real borel to 'U' is shown.
The lemma named "effect_freezeE" is an effect elimination rule. It assumes that the effect of freezing array 'a' transitions the heap 'h' to 'h'' and results in 'r'. From this assumption, it can be obtained that 'h'' is equal to 'h' and 'r' is equal to the value obtained from heap 'h' at array 'a'.
"alpha is an element of the sum of alpha and omega".
The lemma named "max_flow_min_cut'" assumes the following conditions: 1. There is no edge in the graph that leads to the source. 2. There is no edge in the graph that originates from the sink. 3. For any edge in the graph from node x to node y, there exists another node z such that there is either an edge from y to z ...
If "n" is greater than 0, then the bitwise AND operation between a word "w" and a mask "n" is equal to the division of the product of "w" and 2 raised to the power of the difference between the size of "w" and "n" by 2 raised to the power of the same difference. This is applicable for any word "w" of any length 'a.
The lemma named "tuple_disj" shows that the set of all elements 'y' such that 'y' is equal to 'x' or 'y' is equal to 'z' is the same as the set containing 'x' and 'z'.
For any function 'f' mapping from a metric space to an extended real number, for every 'x', the lower semicontinuous hull of 'f' at 'x' is equal to the minimum of the value of 'f' at 'x' and the limit inferior of 'f' at 'x'.
If the pair (N, α) is in the set R, then N is a nonterminal.
The Euclidean representation of a list containing a single element "x" is equal to "x" itself, where "x" is a real number.
For an octonion created from a real number x, the real part is x and the imaginary parts (Im1 through Im7) are all zero.
The lemma named "word_add_offset_less" is defined with respect to a fixed 'x' which is a word of some length 'a'. It assumes the following conditions: 'y' is less than 2 raised to the power of 'n', 'x' is less than 2 raised to the power of 'm', 'sz' is less than the length of 'a', 'x' is less than 2 raised to the power...
If a set S is finite and every subset T of S is closed, then the union of S is also closed.
The natural logarithm of 'x' is greater than or equal to zero if and only if 'x' is greater than or equal to one.
If the product of a rational number 'c' and another number 'a' is greater than 'b', and 'c' is greater than 0, then 'a' is greater than 'b' divided by 'c'.
Updating a single-element list with an optional value in an environment with a base and a list of elements is equal to creating a new environment with the same base and the list of elements updated with the optional value at the position of the single-element list.
Deleting an arc from a finite directed graph results in a finite directed graph.
The lemma "HT_mods_adjust_scope" assumes that "HT_mods" is a function of parameters pi, vs, P, c, and Q. It shows that "HT_mods" is also a function of parameters pi, the set of vs filtered by the global scope, the adjustment of the pre-scope of P, the scope of c, and the adjustment of the post-scope of Q.
For any list "xs" of elements from a linearly ordered integral domain, the function "changes" applied to the list "xs" appended with the elements "b" and "a" is equal to: - 1 plus the function "changes" applied to the list "xs" appended with "b", if the product of "a" and "b" is less than 0, - the function "changes"...
The lemma named "transp_sequence" assumes that "r" is a transitive relation and for any integer "i", the relation "r" holds between the "i+1"th element and the "i"th element in the sequence "seq". If "i" is less than "j", it shows that the relation "r" holds between the "j"th element and the "i"th element in the sequen...
The alphabet of the result of splitting a leaf in a tree 't' with weights 'wₐ' and 'wₑ' and symbols 'a' and 'b' is equal to the alphabet of 't'. If 'a' is in the alphabet of 't', then 'b' is added to the alphabet of 't'. If 'a' is not in the alphabet of 't', then the alphabet remains the same.
If 'p' is a prime number, then the value of function 'f' at 'p' is equal to the value of function 'f_prime' at 'p'.
The lemma named "s_mul_ext_bottom" assumes that multiset A is not empty. It shows that the pair (A, empty multiset) is in the relation defined by s_mul_ext with parameters ns and s.
If 'k' is less than the length of 'ap', then the 'k'th element of the Turing machine states of 'ap' is equal to the configuration instruction of the layout of 'ap' at the start of the layout of 'ap' for 'k', applied to the 'k'th element of 'ap'.
If n is greater than 0 and for all x in set A, the degree of function f(x) is less than n, then the degree of the sum of function f(x) for all x in set A is also less than n.
: if x equals 0 or x equals -1 or n equals 0, then the result is x. Otherwise, the result is the division of x by 2, with the power of n decreased by 1.
The carrier of the ccTTree function, when applied to S and G, is equal to S.
If "s" applied to "t" is a pattern, then there exist some "nm" and "ss" such that "s" is equal to "C nm" applied to "ss".
The lookup function for polynomial power-products represented as ordered associative lists (PP_oalist) is equivalent to the lookup function for ordered associative lists (OAlist_tc_lookup) with the same input.
Applying the identity operator (represented by 1) to any element x results in x itself.
The lemma named "presAtm_compatAtm" assumes that the atomic action "atm" is preserved. It shows that the atomic action "atm" is compatible.
Restricting a finite map twice, first to a set S and then to a set T, is equivalent to restricting the map to the intersection of sets S and T.
For every element 'y' in the set produced by the function 'enumBase' with 'nmax' as an argument, every element 'z' in the set 'y' is less than or equal to 'nmax'.
If there is a message from agent A to agent B containing a ticket and a cryptic message with agent A and a number Ta in the set of events, and agent A is not in the set of bad agents, and the events are in the banker's gets, then there exists a number Tk such that agent A gets a cryptic message containing the number Tk...
The lemma named "convergent_subseq_res'" assumes that "s" is in the set of closed sequences in Zp and "n" is greater than "l". It shows that the convergent subsequence of "s" with parameters "n" and "l" is equal to the convergent subsequence of "s" with parameters "l" and "l".
If a function 'f' from any type 'a' to any type 'b' (where 'b' is a Banach space with a second countable topology) is Borel measurable in the count space of the universal set, then the integral of 'f' with respect to the probability mass function measure 'M' is equal to the integral of 'f' with respect to the restricte...
The lemma shows that the right unitor is a natural isomorphism in the composition of a bicategory.
The carrier relation of a probability mass function's cset is equal to the group defined by the universal set and the cset of the probability mass function.
The lemma named "convergent_prod_iff_shift" shows that the product series converges for the function "f" shifted by "n" if and only if the product series converges for the function "f".
The lemma named "sameDom_substBinp_gSubstBinp" assumes that "binp'" is a well-sorted binary predicate under "delta", and "binp" is a well-sorted binary predicate under the model "MOD" and "delta". It shows that the domain of the substitution of "binp'" with "ys", "Y'", and "y" is the same as the domain of the substitut...
The bitwise AND operation of 'a' and 'b' equals -1 if and only if both 'a' and 'b' are -1.
If the nonstandard derivative of a function f at a point x is D, and a hyperreal number u is infinitesimally close to the real number x, then the hyperreal extension of the function that maps z to the difference between f(z) and f(x), evaluated at u, is infinitesimally close to the product of the hyperreal number D and...
: given a functor 'SS' from the product category of 'CC' with itself to 'CC', and a triple 'A' consisting of objects 'a', 'b', and 'c' in 'CC', the object mapping of the bifunctor left composition of 'SS' applied to 'A' equals the tensor product of 'a' and 'b' under 'SS', tensor product with 'c' under 'SS'.
Given a subring K of a ring R, the function h is injective on K if and only if the kernel of the function h, when the carrier of R is restricted to K and mapped to S, is trivial (i.e., it only contains the zero element).
The lemma named "negligibleD_real" assumes that "f" is negligible. It shows that "f" is an element of the set of functions that grow slower than the inverse of "x" raised to the power of "c".
A floating point number with a negative numeral 'k' and exponent '0' is equal to the negative of the numeral 'k'.
The weakest precondition of Abort for any postcondition P is a function that returns zero for any state s.
The statement consists of two parts: 1. If for every element 'x' in set 'X', 'x' is less than or equal to 'b' (denoted as 'x \<sqsubseteq> b'), then 'b' is a bound of set 'X' with respect to the relation '\<sqsubseteq>'. 2. If 'b' is a bound of set 'X' with respect to the relation '\<sqsubseteq>', then for any statem...
If 'p' and 'q' are tests, then the product of 'p' and 'q' is equal to the product of 'q' and 'p'. This implies that the multiplication operation is commutative for tests 'p' and 'q'.
The lemma Xd1'_Xd2_stable shows that the set {Xd1', Xd2} is stable according to the BSI stability definition.
If for all x in the domain of Γ, the well-formed object P H (L x) (Γ x) holds, and for all x in the set of y_ty_list, the function (λ(y, ty). sty_option P (Γ y) (Some ty)) x holds, then there exists a vs such that the function lift_opts applied to the map of the function (λ(y, ty). L y) over y_ty_list equals Some vs.
If a Van Emde Boas tree "t" of size "n" maintains its invariants, then the predecessor operation "T_pred" on this tree with an element "x" is less than or equal to 29 times the height of the tree plus 1.
The lemma E_wfD assumes that the pair (v, v') is in the fixed flows of a network security policy T. It shows that both v and v' are in the hosts of the network security policy T.
If a pair of terms 'a' and 'b' are related by the term_pp_rel relation, and both 'a' and 'b' are canonical term_pp_pairs, then 'a' and 'b' must be equal.
The lemma named "from_index_geq0_vector_simplex2" assumes that a vector x satisfies the constraints set by the vector v starting from index i. It also assumes that i is less than or equal to j and that j is less than the sum of the dimension of vector v and i. Under these assumptions, the lemma shows that the jth eleme...
If "dickson_grading d" is assumed and "red F p q" is true, then the Dickson grading of the set containing "q" is less than or equal to the Dickson grading of the set obtained by inserting "p" into "F".
For any extended natural number "n", "n" is less than or equal to the sum of "m" and "n".
: Given a function 'f' from a linear continuum topology to a linorder topology, if the value of 'f' at 'b' is less than or equal to 'y', and 'y' is less than or equal to the value of 'f' at 'a', and 'a' is less than or equal to 'b', and for all 'x' such that 'x' is between 'a' and 'b' (inclusive), 'f' is continuous at ...
The general depth-first search step is equal to the step.
The lemma named "separating_tensor" is defined with two fixed variables, A and B, which are sets of updates for any finite type 'a and 'b respectively. It assumes that both A and B are separating for a certain finite type 'c. The lemma then shows that the set of all tensor products of elements from A and B is also sepa...
Shifting the formal Laurent series 'f' by 'm' is equal to shifting the formal Laurent series 'g' by 'm' if and only if 'f' is equal to 'g'.
For a certain function "return_pmf" and "return_nondet", if they satisfy the relation "A" then they also satisfy the relation "cr_prob_ndi A". This is a part of the transfer rules for the monomorphic probability monad in non-deterministic computations.
If a single element vector 'r' is appended to a vector created by a function 'f' of length 'n', then the 'n-th' element of the resulting vector will be 'r'.
If "xa" is not in the domain of "e", "e" is okay, and the value of "xa" in "e" is "T" after "U" is added to "xa", then "U" must be equal to "T".
If H is a subgroup and is normal, and if g is an element of G and h is an element of H, then the left conjugate of h by g is an element of H.
The square of the norm (or magnitude) of the difference between a complex number 'z' and the Gaussian integer to which 'z' is rounded, is always less than or equal to one-half.
The entire universe is an order filter of the natural numbers under the less than or equal to relation.
If proposition "p" is quantifier-free and proposition "q" is quantifier-free, then the disjunction of "p" and "q" (represented as "disj p q") is also quantifier-free.
If the elements of matrix N are a subset of the set of integers from 0 to m (exclusive), then the matrix point index of the modulus of matrix N and the matrix point index of matrix N itself are equal for any index I.
If either proposition phi or proposition psi is true in a given context v, then either proposition phi is true in context v or proposition psi is true in context v.
The lemma named "blinding_of_rt_Fm_hash" assumes that "boa" is less than or equal to the inverse image of the function "ha" under the equality relation, and "bob" is less than or equal to the inverse image of the function "hb" under the equality relation. It shows that the blinding of "rt_Fm" with respect to "ha", "boa...
The lemma named "subtype_refl'" assumes that under context Γ, term t has type T. It shows that under the same context Γ, type T is a subtype of itself.
A complete lattice is a ccpo (complete chain partial order) with the supremum operation "Sup", the less than or equal to relation "(\<le>)", and the less than relation is derived from the less than or equal to relation.
The sum of x and y is less than or equal to the extended integer n if and only if there exist extended integers x' and y' such that x equals x', y equals y', and the sum of x' and y' is less than or equal to n.
The hiding operation in process algebra on a process P with respect to a set A is equal to the divergence hiding of process P with respect to set A.
If there is a path from node 'u' to node 'v' in the graph 'g' restricted to the edges 'E', then there is also a path from 'u' to 'v' in the original graph 'g'.
The least significant bit of a numeral binary word of any length 'a is true if and only if the numeral binary as an integer is odd.
The lemma named "concurrent_set_ConsE" assumes that a set "a" is concurrent with the set that has "x" as its first element and "xs" as the rest of its elements. It also assumes that if the set "a" is concurrent with the set "xs" and "x" is concurrent with "a", then "G" holds. The lemma shows that "G" holds.
If the closed interval from 'a' to 'b' is not empty, then the upper bound of this interval is 'b' and the lower bound of this interval is 'a'.

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