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For a given constant 'c' of a type that is an Archimedean field, a commutative monoid under addition, a linearly ordered topology, and a topological semigroup under multiplication, the infinite sum (infsum) of 'c' over a set 'A' is equal to the product of the cardinality (size) of 'A' and 'c'. This lemma suggests that ... |
The set of all natural transformations from functor F to functor G, where both functors map from semicategory A to semicategory B in the context of a theory morphism, is small. |
If the set S is finite and every element C in S is also finite, then there exists a set T such that the resolution of S equals to some T. |
If for all elements x, evaluating A with x implies evaluating B with x, and evaluating B with x implies evaluating A with x, then A is equal to B. |
The restriction of the partial order relation to a set A is transitive over the universal set. |
The lemma named "of_nat_diff_poly_mapping_nat" assumes that "m" is greater than or equal to "n". It shows that the natural number representation of the difference between "m" and "n" is equal to the difference between the natural number representations of "m" and "n", in the context of a monoid addition structure mappi... |
The lemma "frees_tm" consists of two parts:
1. The set of free variables in the application of function 'f' to 'x' is the union of the set of free variables in 'f' and the set of free variables in 'x'.
2. The set of free variables in the head 'h' of a term is empty if 'h' is a symbol, and it contains the name of 'h' ... |
The set of all vectors x such that the dot product of x and a is zero forms a subspace. |
The lemma named "semIntAbs_prSubstAbs" assumes that the semantic well-formedness condition "wlsSEM" holds for a given semantic model "SEM". It then shows that the function "prSubstAbs" preserves the relationship between the interpretation of the semantics "semInt" and the interpretation of the abstract semantics "semIn... |
The lemma named "lv_SCR" assumes that the function "gsrstep" with parameters "H" and "R" satisfies the Strong Church-Rosser property (SCR). It then shows that the function "srstep" with parameters "F" and "R" also satisfies the Strong Church-Rosser property (SCR). |
If a state variable 'sv' is in the set of atoms produced by encoding the initial state of a problem 'Prob', then 'sv' is a valid state variable. |
If 'x' is less than or equal to 'z' and 'y' is less than or equal to 'z', then the maximum value of 'x' and 'y' is also less than or equal to 'z'. |
If zero is less than y, then x is less than the product of y and the quotient of x divided by y, plus y. Here, y is an ordinal number. |
For a given predicate "Pred" which is a function that takes a pair of processes and returns a boolean, the statement "for all pairs (P, Q) in the relation indRelR, Pred(P, Q) holds" is equivalent to the statement "for all relations TRel, the statement 'for all pairs (TP, TQ) in TRel, Pred(TargetTerm TP, TargetTerm TQ) ... |
If the lengths of two lists 'ts' and 'ts'' are equal, and for all indices 'i' less than the length of 'ts'', the property 'P' of the 'i'th element of 'ts'' is equal to the property 'Q' of the 'i'th element of 'ts', then the union of the set of all elements 'x' in 'ts'' for which 'P' holds is equal to the union of the s... |
Subtracting the "Tip" from any tree "t" will result in the same tree "t". |
The theorem named "Ch_siso_Sbis" assumes that the channel "ch" is compatible and that "c1" is strongly bisimilar to "d1" and "c2" is strongly bisimilar to "d2". It shows that the channel "ch" with "c1" and "c2" is strongly bisimilar to the channel "ch" with "d1" and "d2". |
If for all 's', 'P' of 's' implies 'Q' of 's', then 1 is less than or equal to the refinement of 'P' and 'Q'. |
If a pair "p" is in the set created by zipping lists "as" and "bs" together, then the second element of the pair "p" is in the set "bs". |
The lemma fixes a variable 'n' as a type of 'zhmultiset'. It assumes that 'm' is greater than or equal to 0. It then shows two things:
1. 'n' is less than or equal to 'n + m' (le_add1_hmset)
2. 'n' is less than or equal to 'm + n' (le_add2_hmset) |
Zero is not equal to one. |
If function 'f' is a polynomial function and function 'g' is a real polynomial function, then the composition of 'g' and 'f' (g(f(x))) is also a real polynomial function. |
The lemma named "det_swaprows" assumes that "k" and "l" are less than "n", "k" is not equal to "l", and "A" is an n by n matrix. It shows that the determinant of the matrix "A" after swapping rows "k" and "l" is the negative of the determinant of the original matrix "A". |
For any integer x, the result of setting the nth bit of x to b and then dividing by 2 is equal to: if n is greater than 0, then setting the (n-1)th bit of the result of dividing x by 2 to b, otherwise, it's just the result of dividing x by 2. |
The lemma named "node_lift" assumes that "T satisfies the global property P with respect to the set S and the set U, both excluding the element ii, and the received message I". It also assumes that "for all pairs of elements xi and xi-prime, if they belong to the set S, then they also belong to the set U". Under these ... |
The lemma named "mean_square_density_bounded" assumes that there is a finite graph partition for the vertex set of graph G into k parts, denoted by P, and that the vertex set of graph G is finite. It shows that the mean square density of graph G with respect to partition P is less than or equal to 1. |
If the following conditions are assumed:
1. The binary decision diagram (BDD) 's' is sane,
2. The nodes 'ni1' and 'ni2' are valid in the BDD 's',
3. The result of the 'ifmi' function with inputs 'v', 'ni1', 'ni2', and 's' is '(ni, s')',
Then it can be shown that the node 'ni' is valid in the new BDD 's''. |
The function f91 of n equals n minus 10 if n is greater than 100, otherwise it equals 91. |
The lemma named "yun_monic_factorization_res" assumes that the result of the function "yun_monic_factorization" applied to "p" equals "bs". It shows that there exists a number "m" such that the set of "bs" equals the set of pairs "(A i, i)" where "i" is less than "m" and "A i" is not equal to 1, and "B m" equals 1, and... |
"A entails (p implies q)" is equivalent to "the set of all consequences of {q} is a subset of the set of all consequences of the union of A and {p}".
And the lemma Cn_imp_bis states that "A entails (p implies q)" is equivalent to "the set of all consequences of the union of A and {q} is a subset of the set of all cons... |
A is in the domain of relation r if and only if a is an element of set A. |
The function ASIF applied to the function ifopt with parameter AS of a is equivalent to ifopt being equal to Some a. |
The lemma named "small_det_csmall_det" assumes that for all sigma, the small P of sigma equals an empty set or there exists a sigma-prime such that the small P of sigma equals the set of sigma-prime. It shows that for all sigma, the csmall P of sigma equals an empty set or there exists an o-prime such that the csmall P... |
The lemma named "underS_under" assumes that "i" is in the field of relation "r". It shows that the strict under set of "r" with respect to "i" is equal to the under set of "r" with respect to "i", minus the set containing "i" itself. |
"e" is not operationally definable as a Partial Order Relation (POR). |
If a word 'w' is not empty, then the period of 'w' divides the length of 'w'. |
If the maximum speed is less than the initial distance, then there will be no collision for any time from zero onwards. |
The lemma named "uniform_limit_set_lebesgue_integral_at_top" is defined with the following conditions:
It fixes two functions, 'f' which maps from 'a' to real numbers and then to 'b' (where 'b' is a Banach space with a second countable topology), and 'g' which maps from real numbers to real numbers.
It assumes the fo... |
The session key K is not in the image of the shared key function on the set AA. |
If P type-checks the type inference function "ty\<^sub>i'" with the stack type ST, the environment E appended with type T, and the address type A, then this is less than or equal to the type inference function "ty\<^sub>i'" with the stack type ST, the environment E, and the address type A. |
Enumerating a list that is the concatenation of two lists, "xs" and "ys", starting from a certain number "n", is equivalent to enumerating the list "xs" starting from "n", and then appending the result of enumerating the list "ys" starting from "n" plus the length of "xs". |
The elements of the order type of a set A with respect to a relation r are a subset of the image of the set A under the order map with respect to the same relation r. |
If 'l' is greater than zero, then the probability space of the exponential density 'l' with respect to the Lebesgue measure is a valid probability space. |
If the function "f" applied to the predecessor of "n" is continuous at "x", and if the function "g" is also continuous at "x", then the function that applies "g" to "x" in the case where "n" is infinity, and applies "f" to "n" and "x" otherwise, is also continuous at "x". |
The lemma "rep_props" makes several assertions given the assumption that "r" is an identity:
1. The representative of "r" is a morphism from the tensor product of "tab\<^sub>1 r" and the dual of "tab\<^sub>0 r" to "r".
2. The representative of "r" is an isomorphism.
3. "r" is isomorphic to the isomorphism class repres... |
If a formula "phi" is safe, then the progress of the "convert_multiway" function applied to "phi" with parameters "sigma", "P", and "j" is equal to the progress of "phi" with the same parameters "sigma", "P", and "j". |
If point 'p' is not in set 'V', then deleting point 'p' from set 'V' will result in set 'V' itself. |
If 'i' is less than the length of 'ts' and for all store operations 'sop' in 'is', 'sop' is valid, then the store operations remain valid even after updating the 'i'-th element of 'ts' with the tuple (p,is,xs,sb,\<D>,\<O>,\<R>). |
If a natural number 'm' is greater than 1 and is not a prime number, then there exist natural numbers 'n' and 'k' such that 'n' equals the product of 'm' and 'k', 'm' is greater than 1 and less than 'n', and 'k' is greater than 1 and less than 'n'. |
The lemma named "Psi_simp" assumes that "y" is an identity of category D and "x" is an identity of category C. It shows that the function "Psi" applied to the pair (y, x) is an arrow in category S. Furthermore, "Psi" of (y, x) is equal to the function "S.mkArr" applied to the set "HomD.set" of (y, Gx), the set "HomC.se... |
The fraction of the negation of a floating point number 'f' is equal to the fraction of 'f'. |
If "t" is an arrow and the source of "t" is "a", then the composition of "a" and "t" is equal to "t". |
The lemma "freshFreeTransition" is defined with respect to a process 'P', a free resource 'α', another process 'P'', and a name 'c'. It assumes that process 'P' transitions to process 'P'' with free resource 'α', and that name 'c' is fresh for both process 'P' and free resource 'α'. Under these assumptions, it shows th... |
In the context of a ring, the lemma named "eval_monom" assumes that "b" and "a" are elements of the carrier set "R". It shows that the evaluation of the monomial "b" with degree "n" at "a" is equal to "b" multiplied by "a" raised to the power of "n". |
The lemma named "finite_sum_nth_digit_commute" is fixed for a natural number M. It shows that for all t and for all k less than or equal to M, if the nth digit of the function k at time t in base b is less than 2 to the power of c, and for all t, the sum from i equals 0 to M of the nth digit of the function i at time t... |
If the network map of subnet s1 equals the network mask of the IP addresses in network tree n1, and the network map of subnet s2 equals the network mask of the IP addresses in network tree n2, and s is a subnet of s1 and s2, then the network map of subnet s equals the network mask of the IP addresses in the parallel ne... |
The relation of T with the inverse of L1, the inverse of Co1, the inverse of Co2, the inverse of Contra1, and the inverse of Contra2 is equal to the inverse of the relation of T with L1, Co1, Co2, Contra1, and Contra2. |
The function "app" is monotonic with respect to the partial order defined by "sup_state_opt G". This function "app" is applied to each element of the list "bs" at position "pc", with additional parameters "G", "maxs", "rT", "pc", and "et". The length of the list "bs" and the optional states "opt_states G maxs maxr" are... |
If state s1 transitions to state t1 through command c in n1 steps, and state s2 transitions to state t2 through expression e with value v in n2 steps, then state s1 can transition to state t1 through command c in maximum of n1 and n2 steps, and state s2 can transition to state t2 through expression e with value v in ma... |
The match of an exception table P, C, pc with a shifted value n of xt equals None if and only if pc is less than n or the match of the exception table P, C with the difference of pc and n in xt equals None. |
If a new character data is derived from 'h' to 'h'' and a new character data pointer is derived from 'h' to 'new_character_data_ptr', then the 'get_M_Element' function with parameters 'ptr' and 'getter' is preserved from 'h' to 'h''. |
If "w" belongs to the set of good values of the function "VFun f", then there exist such "x", "e", and "ρ" that "w" equals to the base closure of "x", "e", and "ρ", and the function "f" with parameters "x", "e", and "ρ" is a good function. |
The lemma named "dom_char" shows that the domain of a function 'f' is equal to the codomain of 'f' in category 'C'. |
The lemma named "getFreshPaperID_getAllPaperIDs" assumes that a conference ID is in the set of conference IDs of a system state. It shows that a fresh paper ID obtained from all paper IDs of the system state is not in the set of paper IDs of the system state for the given conference ID. |
The converter "conv" is well-typed with respect to the full interface "\<I>_full" on both the input and output side. |
The lemma named "symmetric" is defined with two fixed variables, P and Q, both of which are of type pi. It assumes that P is weakly late bisimilar to Q. It then shows that Q is weakly late bisimilar to P. |
"thetaParW" is a subset of "Wbis". |
The lemma named "uniform_limit_bcontfunE" is defined with the following conditions:
It fixes a function 'f' that maps from an arbitrary type 'i' to a bounded continuous function from a topological space 'a' to a metric space 'b', and a function 'l' that maps from the topological space 'a' to the metric space 'b'.
It... |
If a string "u" is a factorial prefix of another string "w", then "u" is also a factorial prefix of the string obtained by concatenating any string "p" with "w". |
If the pair (a, b) is in set s, then the pair of multisets containing a and b respectively is in the multiset extension of ns and s. |
The function which sums the real value of the divisor count of each number up to 'x' is asymptotically equivalent to the function 'x * ln x + (2 * euler_mascheroni - 1) * x', plus a term that is big-O of the square root of 'x'. |
The lemma named "up_ray_borel" assumes that "I" is an upward ray in a linearly ordered topology. It shows that "I" belongs to the Borel sigma-algebra. |
The lemma states "Try to prove that the greatest common divisor program correctly computes the greatest common divisor of 6 and 21 when the identity relation is used." |
The lemma named "folds_plus_simp" is defined with a fixed set "rs" of regular expressions. Assuming that the set "rs" is finite, it shows that the language of the union over the set "rs" is equal to the union of the languages of the elements in the set "rs". |
The lemma named "outputEmpty" assumes that under the condition of Psi, M outputs N to process P, which leads to a process transition to P' with a restriction on some vector xvec and an output of N'. It then shows that this vector xvec must be empty. |
The lemma named "segment_altdef" assumes that "b" is in the orbit of "f" at "a". It shows that the segment of "f" from "a" to "b" is equal to the set of all "n"th powers of "f" at "a", where "n" ranges from 1 to the distance from "f" at "a" to "b" minus 1. This is expressed as "?L = ?R". |
The set of the first 'n' elements taken from a list 'l' is equal to the set of elements in 'l' at index 'i', where 'i' is less than 'n' and 'i' is less than the length of the list 'l'. |
The set "ys" is a subset of the set that results from applying the function "comp_red_basis_aux" to "xs" and "ys". |
If a relation R is right total, then the matrix relation of R is also right_total. |
The lemma "mem_read_mem_write_enclosed_plus2" assumes that the unsigned natural number representation of the offset plus si' is less than or equal to si, and that si is less than 2 to the power of 64. It shows that the memory read operation, with the state sigma updated with the memory write operation at address a with... |
"there are infinitely many natural numbers 'i' for which the property 'P' holds" if and only if "there are infinitely many numbers 'i' such that the property 'P' holds for the sum of 'i' and 'j'". |
Given the following conditions:
1. "bezout" is a Bezout extension (as assumed by "is_bezout_ext bezout").
2. The element at the position (ma, Suc k) in matrix A is not zero.
3. k is less than the number of columns in matrix A.
Then, there exists a row m such that it is not a zero row up to the index (Suc (Suc k)) in ... |
The set of all ket vectors forms an orthogonal set. |
One of the following must be true:
1. A user U1 sends a friend request to UID2, the output is OK, the function f applied to the transition results in a friend request value from U1, the IDs of U1 and UID2 are valid, they are not friends in either state, the friend request from U1 is pending in state s' but not in stat... |
The untyped cast of a word 'w' is equal to the conversion of the binary list of 'w' back to a word. |
The lemma named "subst_cv_v_flip" is about a constant 'c'. It assumes that the atom 'x' is independent from 'c'. It shows that if we swap 'x' and 'z' in 'c' and then substitute 'x' with 'v' in 'c', it is equal to substituting 'z' with 'v' in 'c'. |
The lemma named "least_contains" is about a set A, which is of a type that has a well-defined order. The lemma assumes that an element 'k' is in set A. It then shows that the smallest or least element in set A is indeed a member of set A. |
The lemma named "decompose_Max" assumes that set A is a subset of the carrier. It defines M as the maximum value with respect to the "le" relation among set A. It then shows that the restriction of the relation "le" to set A is equivalent to a function that, for any elements x and y, returns true if x is in set A and y... |
For a non-zero polynomial 'p', there exist real numbers 'l' and 'u' such that 'l' is less than or equal to 'u', the polynomial 'p' evaluated at 'l' and 'u' is not zero, and the set of 'x' such that 'x' is greater than 'l' and less than or equal to 'u' and the polynomial 'p' evaluated at 'x' is zero, is equal to the set... |
:
If the first axiom of PLA001_0, which states that putting down on pickup results in a difference between holding, table, clear, and EMPTY, holds true, and
if the first axiom of PLA001_1, which states that EMPTY is clear at the initial state on the table with different elements d, c, b, a, holds true, and
if for all s... |
Function f is less than or equal to function g if and only if the range of function f is a subset of the range of function g. |
The mirror of a lazy list that starts with an element 'x' and is followed by a lazy list 'xs', is equal to a lazy list that starts with the same element 'x' and is followed by the append of the mirror of 'xs' and a lazy list that only contains 'x'. |
The lemma named "index_of_nth" assumes that the list "xs" contains distinct elements and that "i" is less than the length of the list "xs". It shows that the index of the "i"-th element in the list "xs" is "i". |
The restriction of the disjoint union of two maps, h0 and h1, to the set difference of the domains of h1 and h0. |
: "For all 'x', there exists a 'y' in set 'E' such that 'x' is less than 'y'. If this is true, then the function 'injf_max' that takes 'n' as an argument and returns the maximum 'injf' value in set 'E' is injective." |
The lemma named "language" assumes that there is a path in automaton A with word w starting from the initial state of A, and that the test of the condition of A with word w, the states of A with word w starting from the initial state of A, and the initial state of A is true. If these conditions are met, it shows that t... |
Given the following assumptions:
1. All elements in the list 'xs' are lexicographically greater than 0.
2. No element in the list 'xs' is equal to 0.
3. The function 'e' maps any element from the universal set to the interval between -1 and 1.
Then, there exists a function 'e'' that maps any element from the universa... |
The bottom element of a closure operator is equal to the top element of its dual. |
The "find" operation maintains the invariants of a skew heap, which is represented by "skew_invar", and the abstract state of the skew heap is represented by "skew_α". |
The sum of integer i, integer j, 12, and integer k, plus 15 (since subtracting negative 15 is equivalent to adding 15), is equal to y. |
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