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If a point 'a' is in a set 'S', and 'S' is an open set that is a subset of 'T', then the neighborhood of 'a' within 'T' is the same as the neighborhood of 'a'.
: - The most significant bit (msb) of a numeral integer 'n' is False. - The most significant bit (msb) of a negative numeral integer '-n' is True.
The affine transformation of the pair (0, zero_pdevs) is equal to the set containing only the element 0.
If a tau-execution mover for a program P, thread t, and expression e' from a state with heap h, stack stk, local variables loc, program counter pc, and exception xcp to a state with stack stk', local variables loc', program counter pc', and exception xcp' exists, then a tau-execution mover for the same program P, threa...
The first element of the pair resulting from unpacking the tensor product of A and B, which is itself the first element of the pair resulting from unpacking the tensor product of A*B and C at index i, is equal to the first element of the pair resulting from unpacking the tensor product of A and B*C at index i.
The lemma named "nn_integral_conv_infsetsum" assumes that the function "f" is absolutely summable over the set "A" and that for any element "x" in set "A", the value of function "f" at "x" is greater than or equal to zero. Under these assumptions, it shows that the non-negative integral of function "f" over the count s...
The lemma named "infinite_sumset_aux" assumes that the intersection of sets A and G is infinite. It shows that the set resulting from the sumset operation on sets A and B is infinite if and only if the intersection of sets B and G is not empty.
If 'x' is an element of the space 'M', then the return of 'M' and 'x' is a subprobability space.
If "a" is an element of set "A" and there does not exist an element "m" in "A" such that all elements "x" in "A" are less than or equal to "m", then the "n"-th term of the ascending sequence of "A" starting from "a" is less than the "(n+1)"-th term of the same sequence.
The function "GS_rec_am_code'" applied to variables "v" and "d" is equal to a function that takes an input "s'" and performs the following operations: First, it initializes two variables "v'" and "d'" using a fold operation over the sorted list of all elements in the universe. For each element "s" in this list, it upd...
The lemma named "gen_binomial_real" is about a real number 'z'. It assumes that the absolute value of 'z' is less than 1. It shows that the sum of the series where each term is the generalized binomial coefficient of 'a' and 'n' multiplied by 'z' to the power of 'n', is equal to '1 + z' raised to the power of 'a'.
If a program P transitions from an expression "e" with heap "h", local variables "l", shared heap "sh", and block "b" to an expression "e'" with heap "h'", local variables "l'", shared heap "sh'", and block "b'", and if the initial configuration of the shared heap and the expression is valid, and if the block configura...
If a relation "A" is bi-unique, then the function "ffilter" preserves this relation. This is expressed in terms of a relation between two functions and two finite sets. Specifically, if two functions are related by the relation "A" and equality, and two finite sets are related by the relation "A", then the result of ap...
The lemma named "wellformed_union" assumes that both "G" and "H" are well-formed digraphs and that they are compatible. It shows that the union of "G" and "H" is also a well-formed digraph.
If x is not the bottom element and y is not the bottom element, then the product of x and y is not the bottom element.
There exists a strictly increasing function 'k' that maps a natural number to a natural number such that for any 'x' in 'S', there exists a limit 'l' to which the sequence 'f(k(n), x)' converges as 'n' tends to infinity.
The subdegree of a formal power series 'f' raised to the power of 'n' is equal to 'n' times the subdegree of 'f'. Here, 'f' is a formal power series over a semiring with no zero divisors.
If a function 'f' is defined as the recursive list of 'f1' and a lambda function that takes 'x', 'xs', and 'rec' as parameters and returns 'f2' applied to 'x', 'xs', and 'rec', then applying 'f' to an empty list will return 'f1'.
The lemma rng_to_rng_of_frac_is_ring_hom shows that the function rng_to_rng_of_frac is a ring homomorphism from the ring R to the ring rec_rng_of_frac.
If a graph 'g' is acyclic (i.e., it has no cycles) and inserting an edge (x,y) into 'g' makes it cyclic (i.e., it now has at least one cycle), then the edge (y,x) must have already been in the transitive closure of 'g'.
The lemma named "assign_indices_uint64_bij_betw" assumes that the size of M is less than 2^64. It shows that the function "uint64_of_nat" composed with "assign_indices" applied to the states of M is a bijective function between the states of M and the image of the states of M under the function "uint64_of_nat" composed...
The interior of the set B under the function phi is a two-element set.
The lemma "get_ancestors_never_empty" assumes that the heap "h" is well-formed and that when "h" is used to get the ancestors of "child", it results in "ancestors". It shows that "ancestors" is never an empty list.
The lemma is about tree induction. It assumes that 'x' is an element in the set of tree 't'. It also assumes three conditions: 1. For any tree 't', property 'P' holds for the root of 't'. 2. For any element 'x' and tree 't', if 'x' is in the set of the left subtree of 't' and property 'P' holds for 'x' in the left sub...
For a given matrix A, an index i, and a scalar q, the operation of multiplying the i-th row of A by q (denoted as "multrow i q A") is equivalent to the operation "mult_row" under the conditions specified by the Mod_Type_Connect.HMA_M and Mod_Type_Connect.HMA_I functions. This equivalence holds in the context of a commu...
The lemma named "multiplicity_prime" assumes that "i" is a prime number and "j" is not equal to "i". It shows that the multiplicity of "j" in "i" is zero.
If the following conditions are met: 1. The axiom EQU001_0 is equal, 2. The axiom HEN002_0 establishes the identity of Zero Divide as equal to mless_equal, 3. The equation HEN002_0 equates mless_equal to Divide as equal, then, if it is not the case that a is less than or equal to itself, this leads to a contradiction o...
Rotating the 4-bit binary number 1110 two places to the left results in the 4-bit binary number 1011.
If a pair (cl, f) is an element of CLF, and CLF is defined as the set of all functions f from the power set of cl to the power set of cl for each cl in CL, then f is a function from the power set of cl to the power set of cl.
If a list 'ps' is sorted with respect to the distance from a point 'p', then the list remains sorted with respect to the distance from 'p' even after inserting a new point 'q' into the list 'ps' based on the distance from 'p' to 'q'.
In the context of the Incidence Rule, there is a unique line for each point. This is based on the assumptions that the set of geometrical objects on line 1 is not equal to the set of geometrical objects on line 2, point 1 and point 2 are on line 1, and point 1 and point 2 are also on line 2. This leads to the conclusio...
: Assuming that: 1. "S" is a finite set, 2. "I" is a predicate that holds for the set "S" and some initial state "\sigma>0", 3. For any element "x" in a subset "it" of "S", if "I" holds for "it" and some state "\sigma>", and "c" holds for "\sigma>", then applying function "f" to "x" and "\sigma>" is less than or equal...
The validity of the function f applied to x is equal to the definedness of the function f applied to x.
If a term "s" is strictly smaller than another term "t" in the Knuth-Bendix order, then a function "f" applied to a list of terms that includes "s" is also strictly smaller than "t". The list of terms is represented as a concatenation of two lists "bef" and "aft", with "s" in between.
The last element of a stream "s" when no initial element is provided (None) is equal to the last element of the unstreamed version of the same stream "s" when function "g" is applied.
If a category is a tiny category and a partial order category under a certain condition alpha, then it is also a tiny partial order category under the same condition alpha.
If the message authentication response (MsgAuthRes) is composed of a set containing the merchant's local identifier (LID_M), transaction identifier (XID), and purchase amount (PurchAmt), and the hash of the authentication code (authCode), and if the encryption of the hash of MsgAuthRes using the private key of the paym...
The supremum dual operation is applicable to the upward set in a complete lattice.
If a set A is a subset of the upper bound set of the element z, and set A is not empty, then the maximum element of set A is an element of set A.
If for all x in the domain of Delta, the function ae of x equals the function ae' of x, then the application of ABinds to Delta and ae equals the application of ABinds to Delta and ae'.
For any sequence 'w', 'w' satisfies the strong release of the disjunction of 'phi1' and 'phi2' and 'psi' if and only if 'w' satisfies the disjunction of the strong release of 'phi1' and 'psi' and the strong release of 'phi2' and 'psi'.
Taking the bit of the successor of the successor of the successor of zero from the negative one of a 16-bit word equals seven.
In the context of a group, if H is a subgroup of G and h is an element of H, then h is also an element of the carrier set of G.
If x is adjacent to y, then the cardinality of x is equal to the cardinality of y.
The mapping of the value 'vs' results in a final state.
If function 'f' is injective, then the function that applies 'f' after any function 'g' (represented as 'f' composed with 'g') is also injective.
If the set F is finite, then the reduced Groebner basis of F is also finite.
The signed division of any word "a" by any word "b", both of a certain length 'a, is always greater than or equal to the negative of 2 raised to the power of the size of "a" minus 1.
The lemma l2_11_b: assumes that points A, B, and C are collinear (in that order), and points A', B', and C' are also collinear (in that order), and the distance between A and B is equal to the distance between A' and B', and the distance between B and C is equal to the distance between B' and C'. It the...
If a value 'v' is closed under the ML language for a given 'k', and for all 'i' less than 'k', the substitution function 'σ' applied to 'i' equals the ML language value of 'i', then applying the substitution function 'σ' to 'v' equals 'v'.
The outstanding references that are volatile in a store buffer (sb) are equivalent to the union of outstanding references that are volatile writes in the store buffer and outstanding references that are volatile reads in the store buffer.
If a set 'I' is finite, then the set of elements in 'I' that are less than the maximum element of 'I' is equal to the set 'I' with the maximum element removed.
: if for all elements 'x' in the set 'xs', 'x' is less than 2 to the power of 'n', and 'n' is greater than 0, then it implies that the time bound of the function 'test' with parameters 'n', 'xs', and 'ys' is less than or equal to 10 times 2 to the power of 'n' plus the sum of the length of the list 'xs' with 0 prepende...
The lemma named "valid_gc_join_mmsaux" assumes that "valid_mmsaux args cur aux auxlist" and "table (args_n args) (args_L args) rel1". It shows that "valid_mmsaux args cur (gc_join_mmsaux args rel1 aux) (map (\<lambda>(t, rel). (t, join rel (args_pos args) rel1)) auxlist)" is true. In other words, if a mmsaux is valid w...
Mapping the function "Some" over a list "xs" results in a normalized list.
If "P" is a process, and the sequence "s" with the set containing "x" is not in the failures of "P", and the sequence "s" with the empty set is in the failures of "P", then the sequence "s" followed by "x" is in the traces of "P".
The tensor product of the predicate "pre" and the projection on the 0th component is a quantum predicate.
If 'a' is less than or equal to 'c' and 'b' is less than or equal to 'c', then the supremum (or least upper bound) of 'a' and 'b' is also less than or equal to 'c'.
For any integer n, the support of the function I, when the value at position x is replaced with n, is a subset of the support of the function I.
The opposite of the semifunctor FF maps from the opposite of the semicategory AA to the opposite of the semicategory BB in a faithful manner.
For any element 'x', the single probability mass function of 'p' at 'x' is less than or equal to 'f' at 'x'. The second condition, "sum", states that the integral over all elements 'x' of 'f' at 'x' with respect to the count space of the universal set is equal to the weight of the single probability mass function 'p'. ...
The lemma named "consistent_prv_LLq_Num_leq" assumes that a system is consistent and that a proof exists for the less than or equal to relation between two numbers, "m" and "n". It shows that "m" is less than or equal to "n".
For a formal Laurent series 'f' with elements from an abelian group with addition, inverse, and multiplication by zero, the inverse of the base factor of 'f' is equal to the base factor of the inverse of 'f'.
: Assuming that two models M1 and M2 are observable, and M1 is minimal. Also, assuming that there are two sequences u' and v' that belong to the language of M1. Furthermore, assuming that both sequences u and v converge to u' and v' respectively in both models M1 and M2. Moreover, assuming that there is a sequence γ...
For any ring R and any natural number n, the identity function tuple of R and n is indeed a function tuple of R and n.
A set U is countably compact if and only if it is compact, in the context of a second countable topology.
If s and s1 are equivalent except for PID, then the title of the paper with PID in s is the same as the title of the paper with PID in s1, the abstract of the paper with PID in s is the same as the abstract of the paper with PID in s1, the content of the paper with PID in s is the same as the content of the paper with ...
The lemma named "reflp_fun1" includes lifting syntax and shows that if "A" is an equality and "B" is a reflexive relation, then the relation "(A ===> B)" is also reflexive.
The lemma "tank_arith" makes the following assumptions: - τ is a real number that is greater than or equal to 0. - c₀ is a positive real number. - c₀ is less than cᵢ. Given these assumptions, the lemma shows the following: - For all τ in the range from 0 to τ, if τ is less than or equal to the negative of the fraction...
The lemma named "a_AuthT_value_inv" assumes that "v", "vP", and "vV" are of type "AuthT τ" and that "v", "vP", and "vV" are values. It then obtains a value "vP'" such that "vP" is equal to the hashed value of "vP'", "vV" is equal to the hash of "vP'", and "vP'" is a value.
If set R is a subset of set S, then the extension of set R with element y is a subset of the extension of set S with the same element y.
The star function applied to a function 'f' and the star number of 'X' is equal to the star number of a lambda function that applies 'f' to 'n' and 'X' of 'n'.
If a function F is supremum-continuous and M is a monotonically increasing function, then the function F applied to the supremum of the sequence M is equal to the supremum of the sequence obtained by applying F to each element of M.
If a list formed by appending list 'ys' to list 'xs' is strictly sorted (i.e., the elements are in strictly increasing order), and if 'x' is an element of list 'xs', then 'x' is less than every element 'y' in list 'ys'.
The lemma "adds_pp_sminus" assumes that "t" adds to "v" in polynomial terms. It shows that "t" plus the difference of "v" and "t" equals "v".
The lemma named "list_R1_rel_trans" shows that if a relation preserves equality, then it also preserves the equality of lists under the condition of "rel_list_R1" and "list_all2". This lemma is a transfer rule in the context of lifting syntax.
: if the function "destrmi'" applied to "n" and "bdd" equals "Some r", then for any "bddi" that is an implementation of "bdd", the function "destrci" applied to "n" and "bddi" will result in a value "r'" such that "r'" equals "r", and "bddi" remains an implementation of "bdd".
The probabilistic encryption of a plaintext message with a given key is lossless if and only if the order of the group G is greater than zero.
A pair (u, v) is in the transitive closure of E if and only if v is reachable from u in E.
The cut greater than a certain value 't' in a set 'I' is equal to the following: if 't' is an element of 'I', then it is the set of elements in 'I' that are greater than or equal to 't', excluding 't' itself; otherwise, it is simply the set of elements in 'I' that are greater than or equal to 't'.
If a term 's' belongs to the term algebra of the finite set 'F' under a given theory 'G', then a state 'q' is in the transition system derived from the reflexive term algebra of 'F' and 'q' for the term corresponding to the generalized term 's'.
The Chaum Pedersen Sigma commitment scheme provides perfect hiding against chosen plaintext attacks for any given adversary 'A' and security parameter 'n'.
The lemma named "tau reduction zero t fv subset" assumes that the Java program P is well-formed. It shows that if tau reduction zero t from the state (e, xs) to the state (e', xs') occurs in the context of P, thread t, and heap h with an external action extTA, then the set of free variables in e' is a subset of the set...
The offset number of a list element 'n' in a list that starts with 'x' and continues with 'xs', given an index function, a key, a minimum value 'mi', a maximum value 'ma', and an index 'i', is equal to the following: If the index of the key 'x' in the list 'n' with minimum 'mi' and maximum 'ma' equals 'i', then the suc...
The function "set_disconnected_nodes_get_host" is an instance of "l_set_disconnected_nodes_get_host" with respect to the locations "set_disconnected_nodes_locs" and "get_host_locs".
If the measure of the entire space of a measure space M is equal to 1, then M is a probability space.
: For any state 's', if the vertices of the input graph 'iG' are distinct, and 'iM' is a digraph map of the graph created from 'iG', and 'c' is the cardinality of the strongly connected components of the graph created from 'iG', then the operation 'euler_genus iG iM c' is valid in normal form. The result 'r' of this...
If "phi" is necessarily true in a world "v", then "phi" is true in any world "v".
Given a non-empty set A which is a subset of alternatives, and given that the social decision scheme is not efficient with respect to a preference relation R and the probability mass function of set A, if R is a preference profile, then there exists an element in set A such that the probability mass function of the soc...
: Assuming that points A and B are not the same, points A' and B are not the same, points D and E are not the same, points D' and E are not the same, point B is between points A and A', point E is between points D and D', and the angle ABC is less than or equal to the angle DEF, it can be shown that the angle DEF is le...
The lemma named "ofilter_UnderS" assumes that a set A is a subset of or equal to the field of a relation r. It shows that the set of all elements that are strictly less than A forms an order filter.
If tau is a subtype of sigma, then it is not possible for sigma to be a subtype of tau.
If the successor of 0 (which is 1) is less than a certain number 'a', then the successor of 'a' minus the successor of the successor of 0 (which is 2) is less than 'a'.
The sum of the mapped and filtered elements of a list, where the filter condition is P, added to the sum of the mapped and filtered elements of the same list, where the filter condition is not P, is equal to the sum of all the mapped elements of the list.
If "t" is a well-formed directed acyclic graph and "x" is an element in the successor set of "t", then the graph "Node r {|x|}" is also a well-formed directed acyclic graph.
The lemma named "path_hd" assumes that the path "p" is not empty and there is a path "E" from vertex "v" to vertex "w". It shows that the first vertex in the path "p" is "v".
The lemma named "uniqLengthEqCardSet" assumes that the list "l" is unique. It shows that the length of the list "l" is equal to the cardinality of the set of "l".
For the function "create_graph" applied to a formula "phi", it is less than or equal to the specification that for all nodes "n" and "n'" in the set of nodes "nds", if "mu Rr eta" is in the old state of "n" and the name of "n" is in the incoming state of "n'", then either the set containing "mu" and "eta" is a subset o...
If the following conditions are met: - The program G is well-formed according to the method well-formedness criterion wf_mb (as stated by the assumption "wf_prog wf_mb G") - C is a class in the program G (as stated by the assumption "is_class G C") - The set of parameter types pTs is a subset of the types in the progr...
If a new shadow root is created in a heap 'h', resulting in a new heap 'h'', then there are no disconnected nodes in the new shadow root in the new heap 'h''.