Datasets:
UDACA
/
Split-Isa-AF-MMA

Dataset card Data Studio Files
xet
 Community
Statement:
stringlengths
7
24.3k
The lemma named "compatible_iff_seq" shows that the intersection of the domains of g and the codomains of f is not empty if and only if g is sequentially followed by f.
The lemma assumes that for any 's', if 's' is generic then 's' is an instance. It shows that if 'A' and 'B' are jointly present in 's', and 'generic' and 'Y' are also jointly present in 's', then 'X' and 'instance' must be jointly present in 's'.
For the transfer rule, the function "Option.bind" on the generative probabilistic value (gpv) with respect to the computational relation (cr) of option gpv, is equivalent to "bind_gpv" when the functions are equal and the computational relation of option gpv is maintained.
The cardinality (or the number of elements) of the set of isomorphic arcs of a graph G under a homomorphism is equal to the cardinality of the set of arcs of the graph G.
A pair (x, y) is an Inductive pair if and only if x equals the 6-tuple and y is an ordinal.
The lemma named "plus_p_mod_def" assumes that "x" is an element in the set from 0 to "p" (exclusive) and "y" is also an element in the set from 0 to "p" (exclusive). It shows that the function "plus_p" with parameters "p", "x", and "y" is equal to the modulus of the sum of "x" and "y" with respect to "p".
For any two lists of type "tvar" (type variables) called "xs" and "ys", the support (supp) of the difference between these two lists (xs - ys) is equal to the difference between the support of xs and the support of ys.
If W is equal to zero, then V is also equal to zero.
If the following conditions are met: - g' is an element in the set produced by the function "generatePolygon" with parameters n, v, f, and g, - g satisfies the properties of a minimal graph, - f is an element in the set of non-final elements of g, - v is a vertex of f, - n is greater than or equal to 3, then there ex...
If p is a subset of q, and d is fine with respect to q, then d is also fine with respect to p.
The lemma named "retract_of_locally_path_connected" assumes that the set T is locally path-connected and that the set S is a retract of T. It shows that under these assumptions, the set S is also locally path-connected.
Applying the function "rprodl" to the result of applying the function "lprodr" to any input "x" will return the original input "x".
The plain non-deterministic result relation when parameterized with a relation R is equivalent to the composition of the bounded relation of the RETURN function and any function that always returns true, with the non-deterministic result relation parameterized with the same relation R.
Interpreting a floating point arithmetic operation with Nr as the operation and U as the operand, given a vector of values vs, is equivalent to converting U to a floating point number and then converting it to a real number.
In the lemma for a variable finite sequence, if k is an element of the set of natural numbers, then the restriction of the sequence xs to k is also a variable finite sequence.
The lemma named "right_iff_right_inv" assumes that "mu" is an isomorphism. It shows that "sigma" being right to "mu" is equivalent to "sigma" being right to the inverse of "mu".
The function which takes three arguments (x, y, z) and returns the third one (z) is in the set of primary recursive functions of one argument.
The theorem named "Dugundji" is defined as follows: Let's consider a function 'f' from a metric space with a second countable topology to a real inner product space. Assume that 'C' is a convex set and it is not empty. Also, 'S' is a closed subset within the topological space 'U'. The function 'f' is continuous on 'S'...
The lemma named "minus_one_cong_solve" is defined with two fixed natural numbers, n and x. It assumes that n is greater than 1, and that x is congruent to n minus 1 modulo n, and that the product of x and y is congruent to 1 modulo n. It also assumes that x and n are coprime, as well as y and n. Under these conditions,...
If r and s are positive natural numbers, and l is less than the product of s and r, then the integer division of l by s is less than or equal to r minus 1.
A stochastic process monad function (spmf) 'p' is lossless if and only if the probability mass function (pmf) of 'p' being 'None' is zero.
The lemma "srsteps_with_root_step_to_grsteps" assumes that the pair (s, t) is in the relation "srsteps_with_root_step" under a set of rewrite rules \<R> and a set of function symbols \<F>. It shows that the pair (s \<cdot> \<sigma>\<^sub>c, t \<cdot> \<sigma>\<^sub>d) is in the reflexive transitive closure of the relat...
One is not less than or equal to zero in the context of extended real numbers.
The "keys" function is monotonous with respect to the "ns" set.
If S is less than or equal to S', then the abstract value of e under S is less than or equal to the abstract value of e under S'.
If a relator R is equal to the identity relation, then the identity relation is equal to R. The second proposition states that a relator t is equal to itself.
If a program P transitions from an expression state e with heap h, local variables l, shared heap sh, and bytecode b to a new state with expression e', heap h', local variables l', shared heap sh', and bytecode b', then for any type T and environment E, if the program P, environment E, heap h, and shared heap sh type-c...
In the context of a state space, the lemma asserts that if the value of N is initially 5, then after the operation of doubling N is performed, the resulting value of N will be 10.
If the well-formedness of a formula "phi" is indexed by "idx", then the well-formedness of the restricted version of the formula "phi" is also indexed by "idx".
If a branch is a subset of Gamma function 'f', and 'm' is less than 'n', then if the terminal is a subset of function 'f' at 'm', it implies that the terminal is also a subset of function 'f' at 'n'.
The lemma named "multpw_cancel" assumes that "ns" is reflexive and transitive. It shows that a pair (X + Z, Y + Z) is in the multiset power of "ns" if and only if the pair (X, Y) is in the multiset power of "ns".
If J and K are disjoint subsets of I, and if g is a morphism from b to c in the product semicategory over J, g' is a morphism from b' to c' in the product semicategory over K, f is a morphism from a to b in the product semicategory over J, and f' is a morphism from a' to b' in the product semicategory over K, then the ...
The net asset value of the difference between alpha and beta is equal to the difference between the net asset value of alpha and the net asset value of beta.
The lemma named "mult_left_bounds" is fixing two functions, f and g, that map real numbers to real numbers. It shows two things: 1. For all real numbers x, if the value of g at x is within the range from l(x) to u(x), and if the value of f at x is greater than or equal to 0, then the product of f(x) and g(x) is within...
The first step in model 1 (m1_step1) refines the first step in model 1a (m1a_step1) with respect to the refinement relation R1a1 for the parameters Ra, A, B, and Na.
The lemma named "valid_rule_disj" assumes two conditions: first, that "-p" is valid for "x" in relation to "-r", and second, that "-q" is valid for "x" in relation to "-s". Given these assumptions, it shows that the disjunction of "-p" and "-q" is valid for "x" in relation to the disjunction of "-r" and "-s".
The lemma named "vreal_mult_transfer" shows that the multiplication operation in the complex-real number space, denoted by "(*\<^sub>\<real>)", corresponds to the standard multiplication operation "*", under the condition that the operands are complex-real numbers. This is a rule for transferring properties between dif...
The Boolean function derived from the simplicial complex "sc_threshold_2_3" with 4 variables, where all variables are set to False, will return False.
The lemma named "register_empty" assumes that "F" is a pre-register. It shows that applying "F" to an empty map results in an empty map.
The Floyd-Warshall algorithm applied to a matrix M with parameters n and k results in the original matrix M, if the matrix M is a canonical substitution over the range from 0 to k and k is less than or equal to n.
: - The application of a swap operation between 'a' and 'b' on 'c' equals 'a' if and only if 'c' equals 'b'. - The application of a swap operation between 'a' and 'b' on 'c' equals 'b' if and only if 'c' equals 'a'.
The set difference between PR and C is finite.
If a function F leads from set A to the union of set A' with itself, then function F leads from set A to set A'.
"false or maybe x equals x".
The lemma named "times_in_interval" is defined for any two elements x and y of a type 'a, which is a linearly ordered ring. The lemma assumes that x is in the interval X and y is in the interval Y. It then shows that the product of x and y is in the interval formed by the product of intervals X and Y.
The transition from state m2 to state m3 refines the relation R23. This is under the condition that R23 intersects with the universal set and the intersection of m3_inv2_badkeys and m3_inv1_lkeysec. The result is a subset of R23.
If a process "c" is discrete and process "c" is bisimilar to process "d" under strong bisimulation, then process "d" is also discrete.
The lemma named "pairwise_coprime_dvd" assumes that "A" is a finite set, the elements of "A" are pairwise coprime (i.e., any two different elements in "A" are coprime), "n" is a natural number equal to the product of all elements in "A", and every element in "A" divides "j". Under these assumptions, it shows that "n" d...
The variables of a constant are an empty set.
If query Q1 is equivalent to query Q1' and query Q2 is equivalent to query Q2', then the disjunction (logical OR) of Q1 and Q2 is equivalent to the disjunction of Q1' and Q2'.
The lemma named "equivalenceI" is defined with respect to two functions, P and E. P is a binary relation on elements of type 'a, and E is a set of elements of type 'a. The lemma assumes three properties: 1. Reflexivity: For any element x in set E, the relation P holds between x and itself. 2. Symmetry: For any two ele...
Given three assumptions: 1) a function 'e' that maps any value to a range between -1 and 1, 2) a value 'x' that belongs to the error form 'aform_err' of 'e' and 'X', and 3) the range reduction of the error form 'aform_err' of 'X' with respect to 'p' equals 'Y', then there exists an integer 'n' such that 'x' plus 'n...
If M can be alpha-converted to M' and N can be alpha-converted to N', then the Cut of M with respect to a and N with respect to x can be alpha-converted to the Cut of M' with respect to a and N' with respect to x.
The order of Γ in the ground resolution is a subset of the ground sound Γ.
The "lifted_less_eq" of None and any value 'y' is equivalent to 'y' being None.
A map 'h' is continuous from the topological space of set 'S' to the subtopology of the Euclidean space 'T' if and only if 'h' is continuous on 'S' and the image of 'S' under 'h' is a subset of 'T'.
Substituting "x" with "F x" in the result of substituting "y" with "F y" in "t" is equal to substituting "y" with "F y" in the result of substituting "x" with "F x" in "t".
The equality of two matrix implementations, "mat_impl (m1)" and "mat_impl m2", in the HOL (Higher Order Logic) framework is equivalent to the equality of "m1" and "m2" as determined by the "mat_equal_impl" function.
For all 'n', if 'f' of 'n' is less than or equal to 'g' of 'n', then the limit of 'f' at 0 is less than or equal to the limit of 'g' at 0.
The lemma named "hausdorff_distance_zero" assumes that sets A and B are both non-empty and bounded. It shows that the Hausdorff distance between set A and set B is zero if and only if the closure of set A is equal to the closure of set B.
If 'i' is less than 'n', then 'x' is not in the differing variables lists of 'mem1' and 'mem2' when these are zipped together with 'n' repetitions of 'mem1' and 'mem2' at the 'i'-th position.
The lemma named "poincare_line_type" assumes that "z" is not equal to "w". It shows that the type of the Poincare line through points "z" and "w" is "-1".
If a graph 'g' has minimum graph properties, and 'v' is a vertex in graph 'g', and 'f' is in the set of faces at vertex 'v' in graph 'g', then 'f' is in the set of faces of graph 'g'.
The lemma named "sym_in_hom" assumes that "a" and "b" are identities. It shows that the symmetry of "a" and "b" is a morphism from the tensor product of "a" and "b" to the tensor product of "b" and "a".
The lemma "merge_status_eq_iff" states the following conditions: 1. The merge status of 'a' and 'SUCCESS' is 'SUCCESS' if and only if 'a' is 'SUCCESS'. 2. The merge status of 'a' and 'SUCCESS' is 'FOUND' if and only if 'a' is 'FOUND'. 3. The merge status of 'SUCCESS' and 'a' is 'SUCCESS' if and only if 'a' is 'SUCCESS...
The variables of a polynomial 'p' are equal to the union of the keys of 'm', where 'm' is in the set of all 'm' such that the coefficient of 'p' at 'm' is not zero.
If x is an element of the set derived from the natural number u, then 2 raised to the power of x is less than or equal to u.
The lemma "sequentially_consistent_most_recent_write_for" assumes that "E" is an element of "E" and "non_speculative P" is a function that maps any value to an empty set, applied to the last element of "E". It shows that "P" and "E" are sequentially consistent, where the function maps any "r" to the most recent write "...
The set resulting from the function "status" applied to the pair (f, n) is a subset of the set of all integers from 0 (inclusive) to n (exclusive).
If the following conditions are met: 1. The subset 'P' is inherited from 'subs', 2. 'fans' is a subset of 'subs', 3. 'P' is not a property of the tree derived from 'subs' and 'gamma', 4. The terminal of the path derived from 'subs', 'gamma', 'P', and 'n' is not a subset of 'subs', then the path derived from 'subs', '...
For two given generators g1 and g2, and states s1 and s2, and an optional value ao, the following conditions hold: - If there is a next element in both lists generated by g1 and g2 from states s1 and s2 respectively, then: - Let x be the next element from the list generated by g1 and s1' be the new state, and y be t...
If the most significant bit of 'x' is not true, then the signed integer representation of 'x' is equal to the unsigned integer representation of 'x'.
The opposite of a co-span in a category is equal to the co-span in the opposite category.
If the modulus of 'a' and 'm' equals the modulus of 'b' and 'm' (as per the assumption "modeq"), then the ZMod of 'm' and 'a' is equal to the ZMod of 'm' and 'b'.
If an orthogonal current 'f' exists on a set 'S' in a context 'Γ', and if 'x' is an element of 'S', then 'x' is also an element of 'SAT' in the context 'Γ' with respect to 'f'.
If x is not equal to epsilon, then the primary root of x is not equal to epsilon.
The lemma "oelimder_assign" assumes the following conditions: 1. The process 'p' is equivalent to the process 'qq' with a single action 'fa' at label 'l'. 2. The label 'l'' is in the set of labels of the process 'q' in the context 'Γ'. 3. The triple '((σ, p), a, (σ', q))' is in the operational semantics of the sequenc...
The lemma named "Sup_pres_ladj_aux" establishes that for a function 'f' mapping from one complete lattice with dual to another, if 'f' preserves the supremum, then 'f' is left adjoint to the right adjoint of 'f'.
The lemma named "rel_interior_injective_on_span_linear_image" is about a function 'f' that maps from one Euclidean space to another and a set 'S' in the first Euclidean space. The lemma assumes that 'f' is a bounded linear function and that 'f' is injective on the span of 'S'. Under these assumptions, the lemma shows t...
If a program 'P' is well-formed according to the well-formedness condition 'wf_mb', then the subtype relation of 'P' is accessible. This lemma is both an introduction rule and a simplification rule.
The extension of a set S with a chain C and a function f is a chain.
The lemma named "outputSwapSubject" is defined with the fixed elements: Psi of type 'b, P of type psi ('a, 'b, 'c), M of type 'a, B of type boundOutput ('a, 'b, 'c), and x and y of type name. The lemma assumes the following conditions: 1. Psi is in a certain relation with P, which leads to M with a new vector xvec, ou...
If a natural number 'n' is not equal to zero, then the result of subtracting one from this natural number is less than the original natural number 'n'.
If a set A is not empty and for every element x in A, x is closed in U with respect to the set B of x, then the intersection of all B x for x in A is also closed in U.
The lemma named "span_add" assumes that "U" is a subset of the carrier "V", "v" is in the span of "U", and "w" is in the carrier "V". It shows that "w" is in the span of "U" if and only if the direct sum of "v" and "w" is in the span of "U".
The maximum number for an IPv4 address is 4294967295.
An empty array list is equivalent to an empty list in the array list representation.
If f and g are quasi-inverses of each other, then g and f are also quasi-inverses of each other. This is assumed under the condition that f and g are quasi-inverses.
If we assume that "As" is a finite set, "As" is a subset of the carrier of the semialgebraic function ring "SA m", "B" is a semialgebraic set in "m", and "S" is an element of the minimum set partition of "m", "As", and "B", then "S" is a semialgebraic set in "m".
The pair "(mpath', mpath')" is in the relation that maps an optional pair consisting of a list of triples (each triple consisting of an element from set A, an element from set B, and another element from set A) and an element from set B to an optional list of such triples.
The lemma named "test_sup_neg_2" assumes that "x" and "y" are tests. It shows that the supremum (or union) of "x" and "y", intersected with the infimum (or intersection) of the negations of "x" and "y", equals the bottom element.
If the length of the path from node i to node j after starting to remove cycles with node k is greater than the length of the path from node i to node j before removing cycles, then there exists a path ys such that ys is a subset of xs and the length of the cycle at node k in path ys is less than 0.
The term is related to the list set relation, the single-layer graph relation, and again the list set relation, all parameterized by "Rv".
The lemma named "vreal_inverse_transfer" shows that the function "vreal_inverse_app" corresponds to the function "inverse" under the relation "cr_vreal". This is a transfer rule in the context of lifting syntax.
The set of events in a stream space M is equal to the set of events in the supremum over all elements in the universe, of the inverse image algebra on the streams of the space M, where the function maps a stream to its i-th element.
The language of the regular expression "Plus r One" over a given alphabet set and length is equal to the language of the regular expression "Plus s One" over the same alphabet set and length if and only if the language of the regular expression r over the same alphabet set and length, excluding the empty string, is equ...
If the following conditions are met: the intersection of the initial knowledge of states s and t is equal to the payload, the secrets of states s and t are equal, the progress of states s and t are equal, the signals of states s and t are equal, the channel of state s is the absolute value of the initial knowledge of s...
The lemma named "eval_fds_ln'" is defined with a fixed variable "s0" of type "ereal". It assumes that the real part of "s" is greater than "s0". It also assumes that "s0" is greater than or equal to the absolute convergence abscissa of "f" and the absolute convergence abscissa of the derivative of "f" divided by "f". F...
The cardinality (or the number of elements) of an empty set is zero.
A particular event "e" (of any type 'a) is included in the set that is formed by inserting the "tick" event into the range of all possible events "ev".