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Statement: stringlengths 7 24.3k
"star_of x is less than star_of y" if and only if "x is less than y".
The lemma "llrg_monadic_rstep_pres_groundness" assumes that "llrg" is a relation "R", "monadic" is a function "F", and "(s, t)" is in the single reduction step of "F" and "R". It shows that "t" is a ground term.
The statement consists of two lemmas related to graph theory. 1. If the list 'p' is not empty, then the condition of 'v' being joinable with the list 'p' followed by 'q' is the same as 'p' being joinable with 'q'. 2. If the list 'q' is not empty, then the condition of 'p' being joinable with the list 'q' followed by...
If you apply a function 'f' to an updated list 'xs' at index 'k' with a new element 'y', it is equivalent to updating the list 'xs' after mapping it with the function 'f' at the same index 'k' with the new element 'y' processed through the function 'f'.
If the function "gen" with parameters x, Q, and G holds true, then the set G is finite.
If H is a submodule of M in ring R, then the map from M to the quotient of M by H is surjective.
The brick matrix of a cap has a dimension of 4 by 1.
For any given nodes 'u', 's'', and 'c'', if a transition relation exists from 'u' to 's'' and 'c'' in the flowgraph and the set of 's'' and 'c'' is at UV, then there exist sets 'Ml' and 'Me' such that 'u', 'Ml', and 'Me' are in the RUV constraint system of the flowgraph for U and V, and 'Ml' and 'Me' are subsets of the...
In the context of a finite directed graph, the set of all paths, where each path is a trail from one point 'u' to another point 'v', is finite.
A multiset M is less than or equal to the positive part of the multiset M.
For any set Y, the order string of Y is less than or equal to the updated character of Y.
If vertex 'v' is an element of set 'X', then the neighborhood of 'v' is equal to the subset of neighbors of 'v' in set 'Y'.
"For every number x, there exists a number y such that y is less than x."
If a group G acts on a set H, and if X and Y are both in the set of right cosets of G in H, and if the intersection of X and Y is not empty, then X is equal to Y.
The lemma named "mismatchId" is defined with fixed variables 'a' and 'b' of type 'name', and 'P' of type 'pi'. It assumes that 'a' is not equal to 'b'. It shows that the process 'P' under the condition that 'a' is not equal to 'b' is bisimilar to the process 'P' itself.
If function 'f' is equivalent to function 'g', and for all 'x', the result of function 'f' applied to 'x' and 's' is equal to the result of function 'f' applied to 'x' and 's'', then the result of function 'g' applied to 's' is equal to the result of function 'g' applied to 's''.
The lemma named "le_trivial" shows that "A is less than or equal to C D".
If the product of 'x' and 'an(z)' equals the product of 'y' and 'an(z)', and the product of 'x' and 'n(z)' equals the product of 'y' and 'n(z)', then 'x' must be equal to 'y'.
If the lengths of the lists 'pns' and 'Ts' are equal, the elements in 'pns' are distinct, and 'i' is less than the length of 'Ts', then mapping 'pns' to 'Ts' and looking up the 'i'-th element of 'pns' will yield the 'i'-th element of 'Ts'.
A condition Q holds for a logical condition (lcond) with parameters lp, lp', b, and s if and only if it is not the case that either the boolean expression b holds for s and Q does not hold for lp, or b does not hold for s and Q does not hold for lp'.
If the set 'pset' is equal to P and the 'order' is induced by P and r, then this structure belongs to the Complete Lattice.
If there is a τ-reduction from the state (e, xs) to the state (e', xs') in the context of the program P, thread t, heap h, and external actions extTA, then there is also a τ-reduction from the state (V := e, xs) to the state (V := e', xs') in the same context.
The product of (1 - l) and the distance between v and nu_b_opt is less than or equal to the distance between v and L_b of v.
: Assuming a function 'c' with no arguments is a subterm of the substitution of a set 'M' by 'σ', it shows that either the function 'c' with no arguments is a subterm of 'M', or there exists an 'x' in the free variables of 'M' such that 'x' is in the domain of the substitution 'σ' and the function 'c' with no arguments...
The lemma assumes that the product of y, x, x, y, and z equals the product of y, x, y, y, x, and v. It shows that the product of x and y equals the product of y and x.
If for any game 'x', the property 'P' holds for all games 'y' that are in the left product of 'option_of' with 'x', then the property 'P' also holds for the game 'a'.
: Assuming that the sum of the square of a prime number 'p' and the product of a positive integer 'N' greater than 1 and the square of an integer 'q' is a prime number, and the cube of this sum equals the sum of the square of an integer 'a' and the product of 'N' and the square of an integer 'b', and 'a' and 'b' are co...
If "invar'" holds for "m" and "invar'" holds for "k" applied to any "x", then "invar'" also holds for the expression "bindET" applied to "m" and "k".
The mutual information between two variables X and Y is equal to the entropy of X minus the conditional entropy of X given Y. This is under the assumption that both X and Y are simple functions in the measure space M.
The variable "x" does not affect the literal value "v".
If an element 'x' is in the tail of a list 'xs' and the list 'xs' contains distinct elements, then 'x' is not equal to the head of the list 'xs'.
For any set 'A' and element 'x', the order map of 'A' and 'x' with respect to the well-founded relation 'VWF' is an ordinal.
The lemma "Semantics_optimize_primitive_univ_common_matcher" assumes that the matcher agrees on exact matches for a common matcher. It shows that the semantics matches for the optimized primitive universal matcher and a packet are equal to the semantics matches for the matcher and the packet.
If we have a polynomial ring A B Y, a ring homomorphism h from S to B, and three polynomial coefficients c, d, and e such that the first element of e is the sum of the first elements of c and d, and the product of the polynomial expressions of c and d (with respect to ring R and variable X) is equal to the polynomial e...
Parallel reduction implies beta reduction. If A can be parallel reduced to B (denoted as "A >> B"), then A can also be beta reduced to B (denoted as "A ->β* B").
The lemma named "list_combE" assumes a proposition "P" for a term "t" and a combination of a name and arguments. It then concludes that there exists a list of arguments "args'" such that "t" equals the combination of the name and "args'", and the proposition "P" holds true for all corresponding pairs in "args'" and "ar...
The intersection of "m" and "ov" is an empty set.
The methods of the body of an interface "i" are equal to the methods of the interface "i" itself.
The lemma assumes that "x" is an element of the carrier "L". It then shows four things: 1. The XOR operation of "x" and the bottom element equals "x". 2. The XOR operation of the bottom element and "x" equals "x". 3. The XOR operation of "x" and the top element equals the negation of "x". 4. The XOR operation of the t...
The limit of a function 'f' is a subset of the image of 'f' over the set of all natural numbers starting from 'n'.
The lemma named "arctan_upper_14" assumes that "x" is greater than 0. It shows that the arctangent of "x" is less than the upper bound of arctangent for "x", denoted as "arctan_upper_14 x".
The strongly normalizing property on the transitive closure of a relation R over a set T is equivalent to the strongly normalizing property on the relation R over the set T.
The function 'ipv4_cidr_toString' when applied to the IPv4 address in dot-decimal notation (192,168,0,1) with a subnet mask of 22, will result in the string '192.168.0.1/22'.
If a relation "A" is bi-unique and right total, then the transitive closure on the set of elements for which "A" is defined (i.e., the domain of "A") is related to the transitive closure of "A" itself. This is expressed in terms of the relation between sets of ordered pairs (i.e., "rel_set (rel_prod A A)").
The lemma named "globalize_local_consts" assumes that "is'" is an element of the paths in the globalized tree (obtained by applying the function "globalize" to "is", "f", and "t"). It shows that the substitution of local constants in the substitution at the tree (obtained by applying "globalize" to "is", "f", and "t") ...
The lemma named "elementary_Int" is about two sets 's' and 't' in a Euclidean space. It assumes that 'p1' is a division of 's' and 'p2' is a division of 't'. It shows that there exists a 'p' such that 'p' is a division of the intersection of 's' and 't'.
The set of all elements x that are subtrees of a tree t is equal to the set obtained by inserting the tree t into the union of sets of all elements x that are subtrees of t1, where t1 is in the first component of the finite set of successors of t.
The lemma named "validt_augment_Faults" assumes two conditions. The first condition is that under the program context Γ, the triple (P, c, Q, A) is totally correct with respect to the fault set F. The second condition is that the fault set F is a subset of F'. Under these assumptions, the lemma shows that under the sam...
The conjunction of the mapping of a function, which negates the atom of an operator at a certain time and index or negates the atom of a state at the next time and index, over the delete effects of an operator, is in conjunctive normal form.
If the following conditions are met: 1. The set of nodes in NBA A is finite. 2. The set of nodes in NBA B is finite. 3. The alphabet of NBA A is a subset of the alphabet of NBA B. 4. The pair (Ai, A) is in the relation defined by the identity function and the natural number relation, under the NBA to NBAi relation. 5....
Lemma Pi'_eq_empty [simplified]: Assuming "A" is a finite set, the product set "Pi' A B" is empty if and only if there exists an element "x" in "A" such that the set "B x" is empty.
If x is strictly less than y, then x is less than or equal to y.
If command 'c' executed in state 's' results in state 's'', then executing the sequence of command 'c' followed by command 'c' in state 's' also results in state 's''.
The tree created from a list is a finite tree.
The identity function is an isometry on any set A. This is expressed in two equivalent ways: once using the lambda notation (\<lambda>x. x>), and once using the predefined identity function (id). An isometry is a transformation that preserves distances, so this lemma is saying that if you don't move any points (i.e., y...
If "aa" is an element of the carrier set "K", then the expression "1 plus or minus the negation of ((1 plus or minus the negation of (aa to the power of N))^ to the power of N)" is also an element of the carrier set "K".
The lemma named "u_fresh_t" establishes that for any given universe 'u' and type 't', the atom 'u' is fresh for 't'. This means that 'u' does not occur in 't'.
In a Hausdorff space X, the derived set of S is closed in X.
If a strong reduction simulation exists between the inverse of the induced relation of a transition relation (TRel) and the source-target calculus (STCal), then a strong reduction simulation also exists between the inverse of the reflexive transitive closure of the transition relation (TRel) and the target. This is ass...
The predecessor of a given element 'c' is equal to the intersection of the set of all predecessors of 'c'.
The union of incident loops for all vertices in set V is equal to the set of all edges in set E that are loops.
The count of "z" in the "lower_result" of "C" is equal to the count of "z" in "C" minus 1 if "z" is equal to "x", otherwise it remains the same.
The lemma "count_image_mset_le_imp_lt" assumes that the count of the image of a function 'f' applied to a multiset 'M' at 'f a' is less than or equal to the count of the image of the function 'f' applied to a multiset 'N' at 'f a', and that the count of 'a' in multiset 'M' is greater than the count of 'a' in multiset '...
If "s" eta-reduces to "t", then the eta-reduction of the substitution of "s" at index "i" with "u" is the same as the substitution of "t" at the same index with "u".
If a node is left-aligned with a list "ts" appended with two tuples "(sub',sep')" and "(sub,sep)" and a tree "t" at a value "u", and if "sep'" is aligned with "lt" at "a", and "a" is aligned with "rt" at "u", then the node is left-aligned with the list "ts" appended with the tuples "(sub',sep')" and "(lt,a)" and the tr...
If the state 's' prime is equal to the second element of the first element of the result of the function "write_cpu" with parameters 'w', 'nPC', and 's', and if the 'S' bit of the 'PSR' CPU register in state 's' is zero, then the 'S' bit of the 'PSR' CPU register in state 's' prime is also zero.
If the second invariant holds true for two numbers 'x' and 'y', and if the second information of 'x' is equal to the second information of 'y', then the real representation of 'x' is equal to the real representation of 'y'.
The lemma "normalized_add" assumes that the list "xs" is normalized. It shows that adding an element "x" to the list "xs" will result in a list that is also normalized.
If the selfbutter function applied to x equals the selfbutter function applied to y, then there exists a complex number c such that the absolute value (or modulus) of c equals 1 and x equals c times y.
"Test__Lazy__Case_i" is equal to "i".
If a function 'f' has a non-essential singularity at a point 'z' and 'f' has an isolated singularity at the same point 'z', then the function 'g(w) = 1 / f(w)' also has a non-essential singularity at the point 'z'.
If a method "m" with a signature "sig" is found in the table of methods of a class "c", and class "C" is indeed class "c" in the context of a program "G", then it can be inferred that the method declaration "(sig, m)" is declared in class "C" in the context of the program "G".
The lemma named "inner_prod_with_itself_nonneg_reals_non0" assumes that the vector "u" is not the zero vector in the vector space of its own dimension. It shows that the inner product of the vector "u" with itself is not zero.
Evaluating a polynomial at zero is equivalent to looking up the coefficient of the zeroth degree term in the polynomial.
If "A" and "R" are rings and "f" is a ring homomorphism from "A" to "R", then the image of the multiplicative identity of "A" under "f" is the multiplicative identity of "R".
A directed graph is defined as a four-element structure consisting of objects, arrows, domains, and codomains.
If X and Y are tables with the same number of columns 'n' and the same set of variables 'V', then the union of X and Y is also a table with 'n' columns and the set of variables 'V'.
If a certain condition "<P> do { bb \<leftarrow> b; if bb then t else e } <Q>" holds, then another condition "<P> mIf b t e <Q>" also holds. Here, "b" is a boolean condition, "t" and "e" are the actions taken if "b" is true or false respectively, and "P" and "Q" are preconditions and postconditions for the actions.
The lemma named "return_pd" assumes that "x" is an element of "nodes". It shows that "return pd" points to "x".
The parallel composition of the processes resulting from prefixing process P with action a and prefixing process Q with action b is equivalent to the internal choice between the process resulting from prefixing process P with action a and then parallel composing it with the process resulting from prefixing process Q wi...
: "For all x, P of x is true, or for all x, Q is true" is equivalent to "For all x, P of x is true or Q is true".
If the following conditions are met: 1) There are no forward arcs from 'y' to any element in the list 'xs', 2) There are forward arcs from 'y' to any element in the list formed by appending 'ys' to 'xs', 3) 'x' is an element in the set 'xs', then there is no directed edge from 'x' to 'y' in the graph 'T'.
The lemma named "trapped_sol" assumes that "K" is a compact set and is a subset of "X". It also assumes that "x" is an element of "X" and "x" is trapped in "K". Under these assumptions, it shows that the existence interval of "x" is equal to the universal set.
The absolute value of "unit" is equal to "unit".
The lemma named "diff_streams_if" assumes that there exists a number 'n' such that the 'n'th element of sequence 'w' is not equal to the 'n'th element of sequence 'x'. It then shows that sequence 'w' is not equal to sequence 'x'.
The set of unifiers for the insertion of the pair (s, t) into set E is equal to the set of unifiers for the insertion of the pair (t, s) into set E.
If set B is finite and function f is a mapping from set A to set B, then function f is a bijective function between set A and set B if and only if function f is an injective function on set A and the cardinality of set A is equal to the cardinality of set B.
If the pair of pairs ((x1, x2), (y1, y2)) is in the set 'acc', then the pair (x1, y1) is in the set 'K1.acc' and the pair (x2, y2) is in the set 'K2.acc'.
The lemma named "inv_num_wtn_shift_summable" assumes that "E" is an element of the set "M" and the absolute value of "mu" of "E" is less than infinity. Under these assumptions, it shows that the series formed by the function "(1/(num_wtn E n - 1))" is summable.
The lemma named "more_than_one_mset_mset_diff" assumes that an element "a" is in the multiset "M" after removing one occurrence of "a". It shows that the set of elements in the multiset "M" after removing one occurrence of "a" is equal to the set of elements in the original multiset "M".
If an item 'x' is well-formed and the item 'β' of 'x' is equal to a list consisting of 't' followed by 'δ', then the next symbol of 'x' is 't'.
If we assume that K is a subring of R, and p is an element of the polynomial ring K[X], and p is irreducible over K, then it shows that p is not an empty set, and p is not an element of the unit group of the polynomial ring K[X]. Furthermore, for any polynomials q and r in the polynomial ring K[X], if p is equal to the...
For any two elements 'x' and 'y' of a real normed algebra in the nonstandard analysis framework, if 'x' is an infinitesimal and 'y' is a finite hyperreal number, then the product of 'x' and 'y' is also an infinitesimal.
Route 1 is less than or equal to route 2 with respect to a given index 'i' if and only if the sequence number of route 1 at index 'i' is less than the sequence number of route 2 at the same index, or the sequence numbers are equal and the fifth element of route 2 at index 'i' is less than or equal to the fifth element ...
If a message encrypted with B's shared key, containing the agents A and B, a service key, and a timestamp, is part of what the spies can see, and if B is not the Ticket Granting Service (Tgs) and B is not compromised, and the events are in the Kerberos V protocol, then there exists an authentication key and a timestamp...
If a function 'h' is continuous on the product of sets 'T' and 'X', and if 't' is an element of 'T', then the function 'h' composed with the pairing of 't' is continuous on 'X'.
"unwind_cont Δ1" is applicable to the set that includes Δ1 and Δ11.
A subtopology X of a set S is a Lindelof space if and only if for all collections of sets U, where each set U is open in X and the intersection of the topological space X and set S is a subset of the union of the collection U, there exists a countable set V that is a subset of U and the intersection of the topological ...
The truncation of 'x' down to a certain precision 'prec' equals zero if and only if 'x' itself equals zero.
If "select_first x p y" holds true, then the infimum (greatest lower bound) of the set of all "p x" for "x" in "I x" is equal to "p y".

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