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Statement: stringlengths 7 24.3k
If a filter F exists, then for all elements a, b, and c, the implication of a to b or the implication of b to c is equivalent to the top element.
The star-transform of the cosine function at zero equals one.
The lemma named "private_const_deduct" assumes that a function 'c' is not public and that 'c' can be deduced from a set 'M'. It shows that either the function 'c' is in the set 'M', or there exists a term 'm' in the subterms of 'M' such that 'm' can be deduced from 'M' and for all 'k' in the first part of the analysis ...
The lemma "tendsto_Lambert_W_2" assumes that a function 'f' tends towards 'L' under a filter 'F' and 'L' is greater than negative of the exponential of negative one. It shows that the function 'Lambert_W' applied to 'f' of 'x' tends towards 'Lambert_W' of 'L' under the same filter 'F'.
A set of edges 's' is acyclic if and only if the set of back edges of 's' is empty.
If the following conditions are met: 1. The type update of a type sequence 'ts' with an input type list 't_in' and an output type 't_out' equals to 'ts''. 2. The type sequence 'ts' is not undefined (Bot). 3. The type sequence 'ts'' is not undefined (Bot). 4. The types 'ts'' and 'tm' agree according to the function 'c_...
The hypernatural number of the successor of a natural number 'n' is equal to the hypernatural number of 'n' plus one.
A certain element 's' is in the path weight between location 1 and location 2 if and only if 's' is in the minimal antichain for the set of all 's' where the path weight property holds between location 1 and location 2.
The lemma "Lim_add" is about two functions 'f' and 'g' that map to a topological space that is both a T2 space and a topological monoid under addition. If there exists a limit 'y' such that 'f' converges to 'y' under filter 'F' and there exists a limit 'y' such that 'g' converges to 'y' under filter 'F', and filter 'F'...
The lemma named "val_of_power" assumes that "a" is an element of the non-zero p-adic numbers. It shows that the valuation of "a" raised to the power of a natural number "n" is equal to "n" times the valuation of "a".
Applying the transformation B to the line resulting from applying the transformation A to line l is equivalent to applying the composition of transformations A and B to line l.
The theorem named "path_contains_arc" is defined with a fixed function 'p' that maps real numbers to a complete space or real normed vector. Assuming that 'p' is a path, the start of the path 'p' is 'a', the end of the path 'p' is 'b', and 'a' is not equal to 'b', there exists a function 'q' such that 'q' is an arc, th...
A function 'f' is continuous on a set 'S' in the projective space if the function 'f' composed with the projection is continuous on the preimage of 'S' under the projection and 'S' is an open set. This is applicable for any function 'f' that maps from a projective space to any other type.
The lemma named "ac_ord_prop" makes several assumptions: 1. "a" is an element of the set of non-zero p-adic numbers. 2. "b" is also an element of the set of non-zero p-adic numbers. 3. The order of "a" is equal to the order of "b". 4. The order of "a" is equal to "n". 5. The absolute component of "a" at "m" is equal t...
If M has valid requirements, then the Intrusion Detection System (IDS) obtained from M also has valid requirements.
If a map 'f' is a submap of 'g', then the domain of 'f' is a subset of the domain of 'g'.
The lemma named 'absolutely_integrable_measurable_real' is defined for a function 'f' that maps from a Euclidean space to a real number. It assumes that 'S' is a set in the Lebesgue measure. It shows that 'f' is absolutely integrable on 'S' if and only if 'f' is Borel measurable on the Lebesgue measure restricted to 'S...
If the lengths of list a and list b are equal, then the function l2 applied to a and b is equal to the product of the function l2_phi applied to the d-th element of a and the d-th element of b for all d less than the length of a.
For every element 'k' in 'I', if 'k' is an abelian group in 'A', then the partial operation on the direct sum of abelian groups 'I' and 'A' is equal to the product of the partial operations on 'I' and 'A'.
, assuming a derivation is a chain with respect to the transition relation, and the empty multiset is in the closure of the limit inferior state, it follows that the grounding of the least head element of the states is not satisfiable.
If you convert a string 's' to a namespace using the function 'ns_of_s' and then convert it back to a string using the function 's_of_ns', you will get the original string 's'.
The inverse function of "lsumr" over the universal set is equal to "rsuml".
If F is a group and x is an element in the carrier of F, then the identity operation on x in F equals x.
The lemma named "natpermute_max_card" assumes that "n" is not equal to zero. It shows that the cardinality (or the number of elements) of the set of natural number permutations of "n" with "k + 1" elements, where "n" is in the set of "xs", is equal to "k + 1".
If a list 'ss' is a member of the set of all lists 'S', then the permutation of the induced signed left conjugate permutation of the inverse induced identity of the sum of the list 'ss' is equal to the signed list left conjugate action of 'ss'.
The lemma named "conjugate_unit_circle_set" shows that the set of conjugate points on the unit circle is equal to the set of points on the unit circle.
The composition of relations R and S, followed by the composition with T, is a subset of the composition of R with the composition of S and T.
The function "nat_to_sch" applied to the ordered pair (1, 0) results in "Base_suc".
There exists a lemma named propex_2 that states "there exists some 'r' such that 'Some r' equals 'd_2 F'".
The lemma named "regular_division_eqI" assumes that the set K is equal to the range from "a + (b-a)(real k / n)" to "a + (b-a)((1 + real k) / n)", where "a" is less than "b" and "k" is less than "n". It then shows that "K" is in the regular division of "a", "b", and "n".
The right coset relation in a group G is equivalent to the image of the left coset relation in G under the function that maps each element to its additive inverse.
The lemma named "sum_infsetsum" assumes that for every element 'x' in set 'A', the function 'f' of 'x' is absolutely summable on set 'B'. It then shows that the sum over 'x' in set 'A' of the sum over 'y' in set 'B' of the function 'f' of 'x' and 'y' is equal to the sum over 'y' in set 'B' of the sum over 'x' in set 'A...
For a vertex 'v' that belongs to the set of vertices, the output ports of the node of 'v' are empty if and only if the second element of 'v' is an empty list.
The function "int_encode" is injective on the set A.
The lemma named "unit\<^sub>0_simp" assumes that "a" is an object in category "C". It shows that the unit of the equivalence of bicategories for "a" is equal to the composition of the unit of the functor "F" applied to the source of the projection of "a" in category "C", and the unit of the equivalence of bicategories ...
For all "ys" in the set of natural number vectors of length "n", the function "bdd_all" applied to "P" and "make_bdd" of "f", "n", and "xs" is equivalent to "P" applied to "f" of the concatenation of "xs" and "ys".
If a number 'x' belongs to the set of hyperreal infinite numbers, then the inverse of 'x' belongs to the set of infinitesimal hyperreal numbers. This is applicable for 'x' which is a type of star real normed division algebra.
The function p, which maps from some type 'a to a boolean, is monotonic.
Appending any list "xs" to the bottom element results in the bottom element.
Evaluating a list in the algebra of combinatory logic with the functions e, f, and g on the ordered form of a list p is equivalent to evaluating the same list with the same functions without ordering the form of the list.
If a function f approaches a limit L as it approaches a, then the imaginary part of the function f, when evaluated at x, also approaches the imaginary part of the limit L as x approaches a. This is applicable for the function f which maps a real normed vector to a complex number.
A part of a circle path is a simple path if and only if the radius is not zero, the start and end of the path are not the same, and the absolute difference between the start and end of the path is less than or equal to 2*pi.
The function "lists_fc_succ" applied to any list "xss" results in a permutation.
: Assuming the angles (formed by points p11, p12, p13) and (formed by points p21, p22, p23) are defined, it shows that either the geometric set of the first angle added to the empty set is congruent to the geometric set of the second angle added to the empty set, or the geometric set of the first angle added to the em...
If x3 equals the absolute value of x2 minus x1, x4 equals the absolute value of x3 minus x2, x5 equals the absolute value of x4 minus x3, x6 equals the absolute value of x5 minus x4, x7 equals the absolute value of x6 minus x5, x8 equals the absolute value of x7 minus x6, x9 equals the absolute value of x8 minus x7, x1...
The lemma P23_invariant shows that the property P23 is an invariant under the operation of composition.
The security invariant with offending flows, specifically the "sinvar" function, applies to all edges in a normal form that is not reflective, given the allowed sink flow.
Given a list "xs" of natural numbers with a length equal to the dimension of an executable Euclidean space, and assuming that all elements in "xs" are distinct, the Euclidean representation of the list obtained by mapping each element "xa" in "xs" to 1 if "xa" equals "x" and 0 otherwise, is equal to the element at the ...
"ws" is zero if and only if for all "i", "i" is not found in "ws".
The lemma "weak_diamond_imp_CR" assumes that "r" has the weak diamond property. It shows that "r" is confluent.
The lemma named "store_sub2_low_equal" assumes that s1 and s2 are low equal, and t1 is the second element of the first pair resulting from the function "store_sub2" applied to the parameters instr, curr_win, rd, 10, addr, and s1, and similarly, t2 is the second element of the first pair resulting from the function "sto...
The relational composition of X and the Kleene star of Y is equal to the union over all natural numbers n of the relational composition of X and the nth power of Y in the context of a relational dioid.
The lemma named "Bang" is defined with two fixed variables: P of type pi and Rs of type residual. It demonstrates two conditions: 1. If P is in parallel with the replication of P, and P leads to a new name 'x' with a process P', then the replication of P also leads to a new name 'x' with the same process P'. 2. If...
The lemma "steps_preserves_types" assumes that a well-formed C++ program "P" and a sequence of steps "P,E \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>]* \<langle>es',s'\<rangle>". It shows that for any type "Ts", if "P,E" confirms the state "s" and "P,E,hp s" assigns the expressions "es" to the type "Ts", then th...
: Assuming the following conditions: 1. The program P evaluates the expression e in the state s0 to an address a in the state s1. 2. The program P evaluates the expressions ps in the state s1 to a sequence of events evs in the state (h2, l2). 3. The sequence of events evs maps to the values vs. 4. The heap h2 at addre...
If "D y z" and "D x (y times z)" are true, then "D x y" and "D (x times y) z" must also be true.
The lemma "rank_invar_rank_skew" assumes that the rank invariant holds for a skew binomial heap "q". It shows that the rank skew invariant also holds for the same skew binomial heap "q".
The function "heap_array_empty" with no arguments, when paired with the function "RETURN op_list_empty" also with no arguments, is included in the unit assertion in the Kleene star form, which implies an "is_array" assertion.
If the heap 'h' at address 'a' is some object consisting of class 'C' and fields 'fs', then the type of 'x' in the context of the program 'P' and the heap 'h' updated at address 'a' with the object consisting of class 'C' and updated fields 'fs'' is less than or equal to 'T' if and only if the type of 'x' in the contex...
The property of being locally Lipschitz is preserved under a relation that respects the structure of sets A, B, and a binary operation on these sets. This is true if the following conditions are met: the sets A and B are bi-total and bi-unique, and the distance function on sets A, B, and C is preserved under the same r...
If the number of ground holes in context C is equal to 1, then the ground meta-context of the ground context derived from the ground meta-context of C is equal to C itself.
If you substitute a base type 'b' for a base variable 'bv' in the constraint 'c_of' of a type 'τ' and a variable 'z', it is equal to the constraint 'c_of' of the type 'τ' where 'b' has been substituted for 'bv', and the variable 'z'.
If a "pfragment" of "ainf" exists in "xs" and "St", and for any "zs1" and "zs2" such that the pair of "ainf" and the concatenation of "zs1", "xs", and "zs2" is in "St", then "P" holds. This implies that "P" is true.
S is an absolute retract (AR) if and only if S is an absolute neighborhood retract (ANR), S is contractible, and S is not an empty set.
The lemma named "weakenBisimCoinduct" is defined with the following conditions: 1. It fixes a function 'F' of type 'b', and two processes 'P' and 'Q' of type psi, and a set 'X' of tuples containing an element of type 'b' and two processes of type psi. 2. It assumes that the tuple containing the process environment 'P...
If a sequence of commands "cs1" followed by "cs2" transitions from state "s" to a new state "s'" and a new sequence of commands "cs1'" followed by "cs2", and if "cs1" is not an empty sequence, then "cs1" alone transitions from state "s" to the new state "s'" and the new sequence of commands "cs1'".
The result of imploding the string "select" is equal to the string "select".
The value of the function "find_closest_tm" with parameters p, delta, and ps is equal to the result of the function "find_closest" with the same parameters p, delta, and ps.
If alpha is less than or equal to beta, gamma is strictly less than delta, and alpha, beta, gamma, and delta are all ordinal numbers, then the sum of alpha and gamma is strictly less than the sum of beta and delta.
If a sequence of events 'evs' is in a protocol 'p', a nonce 'n' is not used in 'evs', the protocol 'p' gets correct, and 'p' is one step, then the guard 'n' with keys 'Ks' knows the maximum information that a friend 'C' knows in 'evs'.
If the next configuration of the system is such that event K1 weakly precedes event K2, then this configuration is included in the set of all configurations where the number of occurrences of K2 up to time n is less than or equal to the number of occurrences of K1 up to time n.
For any element "w" of an additive group, the left coset of the empty set generated by "w" is the set containing only "w".
The function 'check_no_overlap' applied to 'γ' and 'ft' is equivalent to the statement that for all 'a' and 'b' in the set 'ft', if 'a' is not equal to 'b' and the priority of 'a' is equal to the priority of 'b', then there does not exist a 'p' such that 'γ' applied to the fields of 'a' and 'p' and 'γ' applied to the f...
If G is a well-structured program, then the collection of all elements that satisfy the property "is_methd G" (i.e., being a method in G) is finite.
The implementation of set deletion is a hashset deletion.
The function "tap" with parameter "f" successfully executes in the heap "h".
The language of the regular closure of the non-hole multi-context under a set of functions and a language is equal to the set of all strings that fill the ground holes in a context, where the context has a number of ground holes equal to the length of the string, the number of ground holes is greater than zero, the con...
The XOR operation of any integer 'w' and 0 results in 'w'. The second rule states that the XOR operation of any integer 'w' and -1 results in the negation of 'w'. These rules apply for any integer 'w'.
The norm of the factorial of any number 'n' in a real normed algebra 1 is equal to the factorial of 'n'.
For a given x of a type that is a complete linear order and a dense linear order, the infimum of the function f over the set A is less than or equal to x if and only if for all y greater than x, there exists an i in A such that f(i) is less than or equal to y.
The union of the symmetric factor of a relation 'r' and the asymmetric factor of the same relation 'r' equals to the relation 'r' itself.
The lemma named "normalize_summation" assumes that a real number B is not equal to zero. It shows that the sum from i equals 0 to n+1 of the function f(i) times B to the power of (n - i) equals zero if and only if the sum from i equals 0 to n+1 of the function f(i) divided by B to the power of i equals zero. Here, n is...
The lemma named "meet_3_col_3" assumes that the point P is the intersection of three lines: l, m, and the line through points A and B. It shows that the points A, B, and P are collinear.
Given a total order 'D' and two elements 'a' and 'b' that are in the carrier of 'D', 'a' is less than or equal to 'b' in 'D' if and only if 'b' is not in the strict lower segment of 'D' with respect to 'a'.
If you substitute a term 't' for a variable 'i' in a sequence 'SeqWRP' with parameters 's', 'k', and 'y', the result is the same as creating a new sequence 'SeqWRP' where 's', 'k', and 'y' have each had 'i' replaced with 't'.
Flipping the nth bit of the integer representation of k is equivalent to the integer representation of the flipped nth bit of k.
The lemma named "delay_unbounded_monotone" is defined for a natural number 'n'. It assumes that 'f' belongs to the set of recursive functions, and for all natural numbers 'n', the function 'phi' applied to 'i' and 'f' delayed by 'n' is defined. It shows that there exists a natural number 'm0' such that for all 'm' grea...
If a function 'f' cycles on a set 'S', then the set 'S' is finite. This is based on the assumption that 'f' is cyclic on 'S'.
: 1. For all X, the product of the identity and X is X. 2. For all X, the product of the inverse of X and X is the identity. 3. For all X and Y, the product of X and Y is the multiplication of X and Y. 4. For all X, Y, Z, and W, if the product of X and Y is Z and the product of X and Y is W, then Z is equal to W. 5. F...
The lemma states the law of trichotomy for real numbers. It assumes that both 'x' and 'y' are real numbers. It then shows that one and only one of the following three conditions must be true: 1. 'x' is less than 'y', 'x' is not equal to 'y', and 'y' is not less than 'x'. 2. 'x' is not less than 'y', 'x' is equal to '...
If a set of rewrite rules "rs" can transform a term "t" into a term "t'", and if the term "t" is closed (meaning it has no free variables), then the term "t'" is also closed.
The positive iteration of a predicate "P" is equal to the infimum over all "i" of the predicate "P" raised to the power of the successor of "i".
If a function F is equal to a program created with initial state "init", a set containing action "act", and allowed actions "allowed", and if the action "act" has a single valued property, then the set of all weakly enabling states of function F for a set B is equal to the insertion of the single weakly enabling state ...
: Assuming for any state 's' in set 'S', '0' is in 'U s', 'U s' is an interval, and 'U s' is a subset of 'T', it shows that the solution 'x' of the differential equation 'f' with guard 'G' on 'U S' at '0' is less than or equal to the reference of the predicate 'P' if and only if for any state 's' in 'S', for all 't' i...
: for all i and j less than or equal to n, if the matrix element m[i][j] is not infinity, then the constant part of m[i][j] is an integer. Given that k, i, j, i', and j' are all less than or equal to n, if the updated matrix element (after applying the Floyd-Warshall update operation) is not infinity, then the constant...
If the following conditions are met, then Q is true: 1. The lock status 'ls' and thread status 'ts' satisfy the 'lock_oks1' condition. 2. For all threads 't', if 'ts' at 't' is None, then for all locks 'l', the number of locks 't' has on 'l' is zero. 3. For all threads 't', expressions 'e', states 'x', exception stack...
A set 'c' is in the components of 's' if and only if the following conditions are met: 1. 'c' is not an empty set. 2. 'c' is a subset of 's'. 3. 'c' is connected. 4. For any set 'd', if 'd' is not an empty set, 'c' is a subset of 'd', 'd' is a subset of 's', and 'd' is connected, then 'd' must be equal to 'c'.
The vector space defined over the set UVS with the operations of addition (+), zero (0), subtraction (-), and unary minus (uminus) is equivalent to the vector space on the set UVS.
The lemma 'theI'' assumes that a proposition 'P' holds for an element 'a' and that for any element 'x', if the proposition 'P' holds for 'x', then 'x' is equal to 'a'. It then shows that the proposition 'P' holds for the unique element 'x' for which 'P' holds, and that for any element 'y', if the proposition 'P' holds ...
The size of the word 'w' is equal to the length of 'a'. This is applicable for 'w' which is a word of type 'a' with a defined length.

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