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train · 499k rows

Statement: stringlengths 7 24.3k
The function h of i0, when applied to x and y, is equal to the sum of x, the application of function k of i0 to y, the application of function i of i0 to x and y, and the application of function j of i0 to x and y.
A set A is less than the union of sets B and B' if and only if set A is less than set B and set A is less than set B'.
For any element 'v', if 'v' is in set 'V', then 'v' is not in set 'VV p' if and only if 'v' is in the set 'VV p**'.
If "x" is an element of "G", then "x" is also an element of the free hull of "G".
In the context of an antisymmetric relation, the lemma named "well_ordered_iff_well_related" assumes that set X is a subset of set A. It shows that set X is a well-ordered set with respect to the relation (\<sqsubseteq>) if and only if set X is a well-related set with respect to the same relation (\<sqsubseteq>).
The left residual of the join (or union) of x and y with respect to 1 is equal to the join of the left residual of x with respect to 1 and the left residual of y with respect to 1.
If a natural transformation, denoted as NN, maps a semicategory functor FF to another semicategory functor GG from semicategory BB to semicategory CC, and if there is a semicategory functor HH from semicategory AA to semicategory BB, and if FF' is the composition of FF and HH, and if GG' is the composition of GG and HH...
The lemma "eval_Inv_Can" shows that if "t" is a canonical morphism, then the evaluation of the inverse of "t" is equal to the inverse of the evaluation of "t".
If the function 'ipv6_preferred_to_compressed' applied to an IPv6 address (represented by the eight components a, b, c, d, e, f, g, h) results in a list starting with 'None' followed by 'xs', then 'xs' is equal to the list [a, b, c, d, e, f, g, h] after removing all leading zeros.
For any real number 'p' greater than zero, the limit of the function 'x powr p' (x to the power of p) approaches infinity as x approaches infinity.
If you transpose a matrix A twice, you will get back the original matrix A.
"ldeep_T" is equal to "ldeep_T'".
If the binary height of the left subtree 'l' is equal to the binary height of the right subtree 'r', and both 'l' and 'r' maintain the invariant 'invh', then the function 'baliL' applied to 'l', 'a', and 'r' will also maintain the invariant 'invh'.
If there is a reflexive transitive closure path "r" from "x" to "y" through a list of distinct elements "xs", and the set containing "x" and the elements of "xs" has no intersection with the set "ys", then there is a reflexive transitive closure table "r" from "x" to "y" with respect to "ys".
The function "efficient_funpow" with parameters y, x, and n is defined as follows: - If n equals 0, then the function returns y. - If n equals 1, then the function returns the result of another function f with parameters x and y. - If n is an even number, then the function calls itself recursively with parameters y...
If Z is a member of beta and alpha is a subset of beta, then the functor maps from category A to category B in a tiny way with respect to beta.
If a function 'f' is continuous on a set 'A' and 'n' is greater than 1, and if 'χ' is not equal to the principal Dirichlet character of 'n' and 'χ' is a Dirichlet character of 'n', or if 'χ' is equal to the principal Dirichlet character of 'n' and for all 'x' in 'A', 'f' of 'x' is not equal to 1, then the function 'Dir...
If "phi" is a formula, then it is provable that the equivalence of the conjunction of "false" and "phi" is "false".
The unique factorization in modulus p of a polynomial f and a coefficient sequence cfs is equal to the unique factorization of the polynomial f and the coefficient sequence cfs.
"Set{}" is a constant.
The lemma named "moebius_dilatation_zero" shows that the dilatation of a Moebius transformation with a factor of 1 is equal to the identity Moebius transformation.
If a function 'f' is measurable with respect to the product measure of 'L' and 'M' and mapped to 'N', and if another function 'g' is measurable with respect to 'L' and the subprobability algebra of 'M', then the function that maps 'x' to the distribution of 'g' of 'x' under 'N' with 'f' of 'x' is measurable with respec...
The lemma named "extendRuleEmpty" shows that extending a rule with an empty set on both sides of the implication results in the original rule "r".
The lemma "invs_imp_msg_in_glob" is defined for a fixed configuration 'c' of type "('p::finite, 'a) configuration". It assumes the following conditions: 1. 'M' is an element of the set of messages in configuration 'c' from 'p' to 'q'. 2. 't' is a member of the multiset 'M' with non-zero count. 3. The global invariant ...
If "s" is a superterm of "t", then "t" is not a superterm of "s".
If p and q are prime elements and q divides p, then the normalization of q is equal to the normalization of p.
Applying the "nxt" function y times to the "holds P" function on a stream xs is equivalent to applying the predicate P to the y-th element of the stream xs.
The lemma named "mult_reals" assumes that "a" and "b" are real numbers. It shows that the product of "a" and "b" is also a real number.
The lemma named "empty_mulex_on" assumes that a multiset M is not empty and that M is a member of the set of all multisets of A. Under these assumptions, it shows that the multiset extension of a well-quasi-order P on A is less than or equal to M.
The reverse of a list 'xs' appended with an element 'x' is equal to the reverse of a list where 'x' is the first element followed by the list 'xs'.
If the angles A B C and A' B' C' are both obtuse and the sum of the angles A B C and A B C equals the sum of the angles A' B' C' and A' B' C', then the angle A B C is congruent to the angle A' B' C'.
The basis of a group G is a subset of the group G itself.
: Given the assumptions that "H1" and "H2" are hashed tree automata (TA), "H1" has an index for states (hta_has_idx_s) and "H2" has an index for state and final state (hta_has_idx_sf), and "(Q,W,δd)" is in the invariant of "H1" and "H2" (pa_invar H1 H2), and the condition "pa_cond (Q,W,δd)" holds, It shows that: 1. ...
The lemma named "set_integrable_cong" assumes that "M equals M'", "A equals A'", and "for all x in A, f of x equals f' of x". It shows that "the set integrability of f over A under measure M is equal to the set integrability of f' over A' under measure M'".
: Assuming that two states 's' and 's'' are low-equivalent, and that 'High' is not in the backward slice of 'S', and 'Low' is in 'S'. Also, assuming that there is a path from 'Entry' to 'Exit' denoted by 'as', and the predicates of the kinds of 'as' hold in state 's'. Similarly, there is a path from 'Entry' to 'Exit' d...
If all networks in a set 'p' are distinct, and 'x' and 'y' are elements of this set, with 'a' being the source network of 'x', 'b' being the destination network of 'x', 'c' being the source network of 'y', and 'd' being the destination network of 'y', and either 'a' is not equal to 'c' or 'b' is not equal to 'd', and '...
: Assuming the following conditions: 1. The first elements of the tuples in the list 'cs' are distinct. 2. 'q1' and 'q2' are states in the machine 'M'. 3. For any state 'qq' and input 'x' in the set 'cs', 'qq' is a state in the canonical separator of 'M', 'q1', and 'q2', and 'x' is an input in 'M'. 4. The tuple '(q1, ...
A semigroup is defined by the function that, given any two elements x and y, always returns x.
The lemma named "sc_eq_trivial" assumes that "M" is a strongly connected Markov decision process and "s" is a state in "M" with no actions. It shows that "M" is a trivial Markov decision process with "s" as its only state.
The lemma "det_nfa_reach'" is defined with the fixed variables "bd" of type "nat option bdd" and "ls" of type "bool list list". It assumes that the subset depth-first search of automaton A starting from its start node equals the pair (bd, ls). It also assumes that there exists a list "bs" such that automaton A can reac...
The lemma "reach_AID_used_valid" assumes that the system state 's' is reachable. If the API ID 'aid' is in the server API IDs of 's', or in the client API IDs of 's', or in the pending server API requests of 's', or in the pending client API requests of 's', then it shows that the admin of 's' is in the user IDs of 's'...
The ordinal number omega raised to the power of omega is less than or equal to the order type of the set WW under the length-lexicographic order relation.
If the size of the list 'xs' is 'n', then the last index of 'x' in 'xs' being 'n' is equivalent to 'x' not being an element of the set 'xs'.
The initial state of machine M is accessible.
If 'n' is greater than 0, then the function 'f' of 'n' is equal to the product of the function 'f_prime_power' of 'p' and the multiplicity of 'p' in 'n', where 'p' is in the set of prime factors of 'n'.
The lemma named "Der_conc" shows that the derivative of a concatenation of two sets A and B with respect to a character c is equal to the concatenation of the derivative of set A with respect to c and set B, union with the derivative of set B with respect to c if the empty list is in set A, otherwise it's union with an...
The lemma named "comp_arr_inv" assumes that "f" and "g" are inverse arrows. It shows that the composition of "f" and "g" equals the domain of "g".
Applying the function "lift_RealVal" to the function "f" and the real value "x" is equal to applying the function "f" to the real value "x" and then wrapping the result in "RealVal".
The lemma named "interior_subset" assumes that the set U is a subset of the carrier of Qp. It shows that the interior of U is a subset of U.
The negation normal form (nnf) of any formula F is indeed in negation normal form (is_nnf).
In the Mutex system, any state 's' where property 'p' is not true and 'm' equals 2, inevitably leads to a state where 'm' equals 3.
The lemma "shift_instr_privilege" assumes that the state 's' is the second element of the first pair resulting from the shift instruction on the SPARC state 's', and that the 'S' bit of the PSR register in state 's' is 0. It then shows that the 'S' bit of the PSR register in the new state 's' is also 0.
The lemma named "resid_ide" assumes that "a" is an identity and "a" and "t" are co-initial. It shows that if "t" is restricted by "a", it remains "t", and if "a" is restricted by "t", it becomes the target of "t".
A function f is analytic on the union of sets S and T if and only if the function f is analytic on set S and the function f is analytic on set T.
If digraph homomorphism 'GG' maps from digraph 'BB' to 'CC' and digraph homomorphism 'FF' maps from digraph 'AA' to 'BB', then the domain of the object map of the composition of 'GG' and 'FF' is equal to the objects of digraph 'AA'.
If a query Q is cpropagated, and x is in the free variable of Q, then Q has no copy propagation.
If "r" is an accepting run, then the composition of "f" and "r" is also an accepting run in the graph "G'".
The lemma named "rtr_nomac_e3" is defined with a fixed variable "pi". It assumes that a simple Linux router without layer 12, given the routing table "rt" and firewall "fw" and packet "pi", results in some packet "po". It also assumes that the interface packet check with interface list "ifl" and packet "pi" results in ...
The proposition "True equals False" is False.
If "i" is less than "j", "k" is greater than 0, and "k" is not infinity, then the product of "i" and "k" is less than or equal to the product of "j" and "k".
The theorem named "exec_correct" in the context of "mojmir_to_rabin" assumes that the set Sigma is equal to the set Sigma prime. It shows that a word is accepted if and only if it is accepted by the Rabin Less Than or equal to System (R_LTS) when the Mojmir to Rabin execution function is applied to Sigma prime, delta, ...
The lemma named "merkle_tree" assumes that a Merkle interface exists with parameters h, bo, and m. It then shows that a Merkle interface also exists with parameters as the hash tree of h, the blinding of tree h with bo, and the merge tree of h with m.
If 'b' is an element of the carrier of Zp, then the k-th component of the difference between 'a' and 'b' is equivalent to the difference between the k-th components of 'a' and 'b', modulo p to the power of k.
The lemma rel_R₂_S₀: assumes that "rel_R₂ cs0 w 0" and the set of w is a subset of Σ. It shows that there exists a cs1 such that (cs0, cs1) is in the st.dstep and rel_S₀ cs1 is the initial configuration of w.
A singleton multiset containing the element 'x' is in the set of all multisets of 'A' if and only if 'x' is in 'A'.
The set of all lists 'l' that are empty and valid is equal to the set containing only the empty list.
The lemma named "obtain_function_on_ext_funcset" assumes that "A" is a finite set and the size of "M" is equal to the cardinality of "A". It shows that there exists a function "f" from "A" to the multiset "M" such that the image of the multiset of "A" under "f" is equal to "M".
The orthogonal complement of the set containing only the zero vector is the entire universe of vectors.
The lemma named "compatible_heaps_sum" assumes that heap 'a' is compatible with heap 'b' and heap 'a' is also compatible with heap 'c'. It shows that heap 'a' is compatible with the sum of heap 'b' and heap 'c'.
If a function 'g' is absolutely summable over a set 'A', and for every 'x' in 'A', the norm of 'f(x)' is less than or equal to 'g(x)', then the function 'f' is also absolutely summable over the set 'A'.
, given an empty set, the function "MakeFormP" applied to the negation of "x" ("Q_Neg x") and "x" itself results in "y".
If the 'steps0' function is applied to the tuple (1, [], []) with 'tm_semi_id_gt0' and 3 as arguments, the result will be the tuple (1, [], [Bk]).
The union of the set of all comparables of S and M is a subset of the union of S and M.
: Given a matrix A of a certain type, where A is a carrier matrix with dimensions nr by nc. Also given are vectors b and c, where b is a carrier vector of length nr and c is a carrier vector of length nc. Assuming that there exists a vector x, which is a carrier vector of length nc, such that when matrix A is multip...
If 'r' is an ordered relation and 'r' is accessible, then the list of 'n' elements ordered by 'r' is also accessible.
The converter 'conv' is collision-less at the bottom level according to the interface 'I'.
There exist sets A, B, and C such that the multiset M can be decomposed into the multiset of A and C, and the multiset N can be decomposed into the multiset of B and C.
The lemma named "coplanar_6" assumes that the rank of the set {A, B, C} is 3, the rank of the set {B, C, α} is 2, the rank of the set {A, B, γ} is 2, and the rank of the set {A, C, β} is 2. Under these assumptions, it shows that the rank of the set {A, B, C, α, β, γ} is 3.
If alpha is an element of set M and x is an element of set S, then the transformation of x by alpha is also an element of set S.
The lemma named "clock_submodel_aux" assumes that "t" is an element of Clock.jkbpC and "s" is an element of the worlds in the clock_simMCt of "t". It shows that the general model of Clock.MCS at state "s" is equal to the general model of the clock_simMCt of "t" at state "s".
The lemma named "subset_mult2_alt" assumes three conditions: 1. Multiset X is a subset of multiset Y. 2. The pair (Y, Z) belongs to the multiset relation "mult2_alt" with parameters b, ns, and s. 3. If b is true, then b' is also true. Under these assumptions, it shows that the pair (X, Z) also belongs to the multiset...
The lemma named "Span_append_eq_set_add" assumes that the set of elements "Us" and the set of elements "Vs" are both subsets of the carrier "R". It shows that the span of the union of "Us" and "Vs" in a field "K" is equal to the sum of the span of "Us" in "K" and the span of "Vs" in "K" within the ring "R".
If the difference between relation R and the projection relation 1 is equal to the difference between relation S and the projection relation 1, then the difference between the composition of relations T and R and the projection relation 1 is equal to the difference between the composition of relations T and S and the p...
The set of all elements of a certain type 'a that are less than or equal to a certain value is equal to the image of the set of all values less than the length of 'a under the function Abs_atMost.
Prepending a bit vector with a successor of a natural number 'n' and a bit 'b' is equal to prepending the bit 'b' followed by prepending the bit vector with the natural number 'n' and the bit 'b'.
The roots of a polynomial along a line are calculated as follows: If the polynomial evaluated at the start and end points of the line is not zero, then if the start and end points are not the same, we compose the polynomial with a linear polynomial defined by the start and end points. We then map the real and imaginar...
If "x" is an element of "assertion", then the dual of "x" is an element of "disjunctive".
If a relation "rel_R" holds between two states, where in the first state variable "z" equals "a" and variable "x" equals "b", and in the second state variable "x" equals "b" and variable "y" equals "a", then this implies that the relation "rel_R" also holds after the assignment of the value of variable "z" to variable ...
The number of rows in matrix Fs is equal to m, the number of columns in matrix Fs is equal to n, and matrix Fs is a member of the set of all m by n matrices.
If "u" is a unit in the p-adic numbers, and "a" and "b" are elements in the p-adic numbers, and the valuation of the product of "u" and "a" is less than or equal to the valuation of the product of "u" and "b" plus some real number "α", then the valuation of "a" is less than or equal to the valuation of "b" plus "α".
The uncurried function "mtx_pointwise_binop_fold_impl1 N M get_impl set_impl fi" is in the same assertion type as the uncurried function "RETURN oo PR_CONST (mtx_pointwise_binop f)". This is under the condition of the assertion being in the domain of the function and the assertion being in the codomain of the function.
A sequence 'w' satisfies the condition 'phi Until psi' if and only if either 'w' satisfies 'psi' or 'w' satisfies 'phi' and the next state of 'w' satisfies 'phi Until psi'.
The lemma named "gen_boolean_algebra_elem_uni_of_atoms" assumes that "Xs" is a finite set, "S" is the union of "Xs", and "X" is in the generated Boolean algebra of "S" and "Xs". It shows that "X" is equal to the union of the set of all "a" in the atoms of "Xs" such that "a" is a subset of "X".
If set A is a subset of the set formed by inserting element x into set B, then either set A is a subset of set B and element x is not in set A, or there exists a set B' such that set A is equal to the set formed by inserting element x into set B' and set B' is a subset of set B.
If for all states 's', 'Q' of 's' implies 'Q' of 's' with 'v' replaced by 'e s', then the relation 'rel_R' of 'P' and 'Q'' followed by the assignment of 'e' to 'v' is a subset of the relation 'rel_R' of 'P' and 'Q'.
It is not the case that for any time point t0, there exists a time point t1 such that the property P holds at t0 until t1, and the property Q holds at t1, where the interval between t0 and t1 is empty.
The function "aial_length" is related to the function "mop_list_length" in the context of "ial_rel1 maxsize", mapping to a natural number relation in the result space. This is a statement about the relationship between two functions within a specific context.
If from a set of hypotheses H, we can deduce that p implies falsehood, then from the same set of hypotheses H, we can deduce that p implies any proposition q.
If set A is finite, then the preimage of the image of set A under a function phi is also finite.
The function "additive_principal" applied to the successor of 0 (which is 1 in ordinal numbers) is true.

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