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If a pair (x, y) is an element of a finite set 'xs' of pairs, where each pair consists of a decision tree 'x' and a value 'y', then the size of the decision tree 'x' is less than the successor of the sum of the successors of the second element of the first element of each pair in the image of the finite set 'xs' under ...
If the nonstandard derivative of a function f at a point x is D, and h is an infinitesimal, then the difference between the hyperreal extension of the function evaluated at the hyperreal number (x + h) and the hyperreal extension of the function evaluated at x is approximately equal to the hyperreal extension of D time...
If a function F is not the least element (or bottom) and the absolute value of a constant c is not infinity, then the lower limit (or limit inferior) of the function F applied to the sum of a sequence g and the constant c is equal to the sum of the lower limit of the function F applied to the sequence g and the constan...
The polynomial of a two-variable constructed function (pCons a p) with variables x and y is equal to the polynomial of a with x plus y times the polynomial of the two-variable function p with variables x and y.
The inverse image of a set B under the right restriction of a relation r to a set A is equal to the intersection of set A and the image of set B under relation r.
In the context of a program G, if there is an exception at location 'a', then catching class C is equivalent to the address 'a' fitting into class C in the state 's' under the program G.
If "f" is a weak composition series of a group "G" with length "n", then "f" is also a weak composition series of the same group "G" with the same length "n".
If a set S is closed, then the preimage of S under the function that maps x to its i-th component is also closed.
If H and K are subfields of R, and if a set I is a subset of both H and K, then the field generated by I within K is a subset of the field generated by I within H.
The argument of a complex number 'z' in [0, 2π) is either 0 or π if and only if 'z' is a real number.
The low parts of t1 and t2 are equal if the low parts of s1 and s2 are equal and t1 is the second element of the first pair resulting from the multiplication instruction with s1, and t2 is the second element of the first pair resulting from the multiplication instruction with s2.
In the protocol AODV, for any instance 'i', the set of known destinations at any state 'ξ' is a subset of the set of known destinations at the next state 'ξ''. This means that the set of known destinations can only increase or stay the same, it cannot decrease.
The "locks" function, when applied to the pair consisting of "ls" and "tsmws", equals "ls".
The length of the result of the function "fo_nmlz_rec" with inputs i, m, AD, and xs is equal to the length of xs.
: given the assumptions that the Java Virtual Machine program P is well-formed with respect to a type safety policy Φ, and that Φ ensures the type safety of the thread state (xcp, h, f # frs), it can be shown that there exists a thread action ta and a subsequent state σ' such that P, with thread t, transitions from the...
Two streams 'u' and 'v' are related by a relation 'P' if and only if for all indices 'i', the 'i'-th elements of 'u' and 'v' are related by 'P'.
The lemma named "beta_numbound0" assumes "p" is an integer-coefficient quantifier-free formula (iszlfm p). It shows that for all "b" in the set generated by the beta function of "p", "b" is a number that does not depend on the variable (numbound0 b).
"it is necessarily the case that if agent A knows statement 'a', then it is known by the community 'c' that agent A knows statement 'a'".
The absolute value of "an(x)" being greater than 1 is equivalent to "an(x)" itself.
The lemma named "invar_step" assumes that "invar C" holds true and the cardinality of set C is less than k. It shows that "invar" also holds true for the union of set C and the set containing the element furthest from set C.
The boolean representation of the negation of a disjunctive normal form (dnf) is equivalent to the negation of the boolean representation of the dnf itself.
The lemma named "prefixE" assumes that "xs" is a prefix of "ys". It then concludes that there exists a list "zs" such that "ys" is the concatenation of "xs" and "zs".
If "S" is an element of the HAStates of "A", then the value of the computation function "CompFun" of "A" and "S" is a subset of the Statecharts Assemblies of "A".
The lemma "CLIQUE_solution_imp_ACC_cf_empty" assumes that for every graph G in the set of graphs G, G is in the clique if and only if the evaluation of the function theta_g of G and phi holds true. It also assumes that phi is in the set of total-formulae and phi is in the set A. Under these assumptions, it shows that t...
The lemma named "norm_eq" assumes that "l" is in the carrier and that the norm of "P1" is equal to the norm of "P2". Under these assumptions, it shows that the Ipolex of "l" and "P1" is equal to the Ipolex of "l" and "P2".
The lemma named "convex_with_UNIV_domain" is about a function 'f' that maps from a Euclidean space to an extended real number. It assumes that the function 'f' is convex on the universal set and that the domain of 'f' is the universal set. It shows that for all 'x', the function 'f' at 'x' is greater than negative infi...
In the context of encoding, the lemma named "indRelEQ_symm" demonstrates that "indRelEQ" is symmetric.
The maximum value of a finite set created from a list (with elements 'a' followed by 'as') is equivalent to folding the list with the 'max' function starting from 'a'.
If a multiset A is the image of another multiset B under a function f, then the filtered multiset of A under a predicate P is equal to the image of the filtered multiset of B under the same function f, where the predicate P is applied to the function f of x.
The probability mass function (pmf) of the symmetric digital system (sds) R29 for 'a' is 1/2 and the pmf of the sds R29 for 'd' is also 1/2.
The lemma named "soundness_graphical_helper" assumes that there is no variable interference, meaning that for any command 'c', the set of variables written by 'c' is empty. Under this assumption, the lemma shows three things: 1. If a diagram 'G' proves a relation between propositions 'P' and 'Q', then 'P' is the top p...
The lemma named "dg_GRPH_is_arrE" assumes that "\<FF>" is a mapping from "\<AA>" to "\<BB>" in the context of a directed graph "dg_GRPH" parameterized by "\<alpha>". If this assumption holds, it obtains that "\<FF>" is a double-arrow mapping from "\<AA>" to "\<BB>" in the context of a directed graph parameterized by "\...
A method declaration is well-formed in a program G and a class C with a signature and method pair (sig, m) if the following conditions are met: 1. The method header is well-formed in the program G, the package id of the class C, the signature sig, and the method header m. 2. The local variables in the method body m ar...
If property P holds for x, then x is preferred according to property P.
The lemma "m2_inv34_M2b_authorized" assumes that a secure message containing a key K, an agent A, and a number T is in the channel state s, the state s is in the invariant m2_inv3_extrKey, the state s is also in the invariant m2_inv4_M2b, and the key K is not in the leaked information of state s. Under these assumption...
If a function "phi" is monotonic, then for all sets A, B, and C, if A is a subset of B, then the value of "phi" applied to the implication from C to A is a subset of the value of "phi" applied to the implication from C to B.
If a list "xs" consists of single combinators, then any sublist of "xs" that satisfies a certain property "P" will also consist of single combinators.
There exists an 'x' such that 'x' is in the subnet of the pair (a, b), where 'a' is an integer and 'b' is a port.
If a policy "p" consists of single combinators and all networks in "p" are distinct, then all networks will remain distinct even after the application of the function "removeShadowRules2" to the policy "p".
If function B is monotonically non-decreasing according to the flatf_mono_ge property and is also a monotone function, then function B has the trimono property.
For all 'a' and 'b', and for all 'x' in set 'A' and for all 'y', if 'a' equals the function 't' of 'x' and 'y', and the function 'h' of 'x' and 'y' holds true, then either the function 'g' of 'x' and 'y' holds true or 'b' is not equal to the function 'u' of 'x' and 'y'. This is equivalent to saying that for all 'x' in ...
If a function 'f' is injective (or one-to-one) on the set 'Vs', and 'bl' is a subset of the set 'Vs', then the incidence vector of 'Vs' and 'bl' is equal to the incidence vector of the mapped function 'f' on 'Vs' and the image of 'bl' under the function 'f'.
For any elements 'x' and 'y', the product of 'n x' and the sum of 'n x' and 'n y' is equal to 'n x'. This is a simplification rule in the logic system.
If the trie "t" satisfies the invariant property of a trie, then the domain of the lookup function applied to trie "t" is empty if and only if the trie "t" is empty.
The theorem named "wls_freshEnv_psubst_ident" assumes that "X" is a well-sorted term of sort "s", and "rho" is a well-sorted environment. It also assumes that for any set "zs" and element "z", either "z" is fresh for "rho" with respect to "zs", or "z" is fresh for "X" with respect to "zs". Under these assumptions, the ...
If a message encrypted with the private key of RCA, containing the agent C, a session key SKi, and only a signature, is part of what the Spy knows from the events, and these events are in the set of purchase protocol, then the session key SKi must be the public session key of agent C.
The lemma "prec_ev_signal_dom_independent" assumes that the current state 's' is not equal to 'd', the atomic step is invariant in state 's', and the atomic step precondition is met for state 's', 'd', and 'a'. Under these assumptions, it shows that the atomic step precondition is also met for the atomic step event sig...
The lemma named "distinguishing_formula_distinguishes" assumes that P is not equivalent to Q. It shows that the distinguishing formula for P and Q distinguishes P from Q.
For a given type scheme "sch", if the set of free type variables in "sch" is equal to the set of free type variables in another type scheme "sch'", and if "n" is a new type variable in "sch", then "n" is also a new type variable in "sch'".
If the natural number n1 is equal to the natural number n2, then n1 is indeed equal to n2.
For any psi-calculus expressions P, Q, and R, and any environment Psi, the parallel composition of P and Q, followed by a parallel composition with R, is bisimilar under substitution to the parallel composition of P with the result of the parallel composition of Q and R.
If assumption A2 holds for variables x and a, and postcondition Q with a pair (ps, n), then assumption A4 also holds for the same variables, postcondition, and pair.
: given a subfield K of R, and two polynomials p and q in the carrier of K[X], the rupture_surj function applied to K, p, and q equals zero in the rupture of K and p if and only if p divides q.
Filtering the list "xs" with the function "ipurge_tr_aux" with parameters "I", "D", and "U", and then applying predicate "P" to each element "x" in the list, is equivalent to applying the function "ipurge_tr_aux_foldr" with parameters "I", "D", "P", and "U" to the list "xs".
For a given set 'S' of real normed vectors, if 'x' is an extreme point of 'S' and 'S' is not a set containing only 'x', then 'x' is not in the relative interior of 'S'.
The lemma named "sum_content_null" assumes that the content of the closed box from point 'a' to point 'b' is zero and 'p' is a tagged division of the closed box from 'a' to 'b'. It shows that the sum of the product of the content 'K' and the real-numbered function 'f' at point 'x', for all (x, K) in 'p', is equal to ze...
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 initial signals of states s and t are equal, the response signals of states s and t are equal, the channel of s...
If for all pairs of elements 'a' and 'b', the relation 'phi1' implies that the relation 'F1rel' holds between the destructors 'dtor1 a' and 'dtor1 b' under the equality relation, and similarly, the relation 'phi2' implies that the relation 'F2rel' holds between the destructors 'dtor2 a' and 'dtor2 b' under the equality...
The theorem assumes three premises: 1. There exists a being in the understanding that is greater than any conceivable being. 2. If the being that is greater than any conceivable being does not exist in reality, then there exists a being in the understanding that is greater than the being that is greater than any conce...
If 'f' is an isomorphism, then the composition of 'f' and the inverse of 'f' is equal to the identity of the domain of 'f'.
The output list 'O' of a trace 'tr' is empty if and only if there does not exist an element in the trace 'tr' that satisfies the property 'γ'.
The lemma named "subobj_aux_rev" assumes that "C''" is a sub-object of "C" in the reverse order of "Cs" appended with "C'" under a certain property "P". It shows that "C'" is also a sub-object of "C" in the reverse order of "Cs" under the same property "P".
The lemma named "merge_bot_args_bless_eq_merge" assumes that the pair (s, t) is in the set defined by mergeP. It shows that s is less than or equal to the result of the operation s raised to t, in the context of bot (denoted by the subscript b).
The theorem 'wls_psubst_subst_compose' assumes that 'X' is a well-sorted term of sort 's', 'Y' is a well-sorted term of sort 'ys', and 'rho' is a well-sorted environment. It shows that the parallel substitution of 'rho' in the result of the substitution of 'Y' for 'y' in 'X' is equal to the substitution in 'X' of the r...
The lemma named "poly_inc_as_poly_lift_hom" assumes that "f" is an element of the carrier of the ring of p-adic integers. It shows that the polynomial increase of "f" is equal to the polynomial lift homomorphism from the ring of p-adic integers to the field of p-adic numbers, applied to "f".
A relation "r" is Church-Rosser if and only if for all elements "x", "y", and "z", if the pair "(x, y)" is in the relation "r" and the pair "(x, z)" is in the reflexive transitive closure of "r", then the pair "(y, z)" is in the join of "r".
The lemma named "after_reachable_initial" assumes that the machine "M" is observable and that the input-output pair "io" is in the language of "M". It shows that the state of "M" after processing "io" from the initial state is in the set of states reachable from the initial state of "M".
The function "map_FGcontra" preserves the relation between two sets of functions. These sets of functions are defined by the relations L1 to L1', Co1 to Co5', and Contra1 to Contra5'. The function "map_FGcontra" is a higher-order function that takes these sets of functions as arguments and returns a new function. The l...
The lemma named "monad_prob_envT" assumes that a monad probability exists for the functions "return", "bind", and "sample". If this assumption holds true, it shows that a monad probability also exists for the environment versions of these functions, namely "return_env", "bind_env", and "sample_env". This lemma is a loc...
If the following conditions are met: 1. Agent B' sends a message to Agent A containing a nonce NA and a cryptographic key K, encrypted with A's shared key. This message is part of the event set 'evs'. 2. Agent A sends a message to Agent B containing a nonce NA, identifiers for Agents A and B, and a cryptographic messa...
The lemma named "min_set_obtains" assumes that "A" is an element of "B". It then obtains a "C" such that "C" is a subset of "A" and "C" is an element of the minimal set of "B".
If "f" is a nucleus, then the composition of "f" with the product of "x" and the composition of "f" with the product of "y" and "z" is equal to the composition of "f" with the product of the composition of "f" with the product of "x" and "y" and "z".
Shuffling a singleton list (a list with only one element) will return the same list with a probability mass function.
If S5n holds for a Kripke model M, then the relation for an agent a in the model M is an equivalence relation over the worlds in M.
In the context of a finite measure, the theorem named "second moment method" is stated. It assumes that the function 'f' is measurable, and that the integral of the square of 'f' exists. It defines 'mu' as the Lebesgue integral of 'f'. It also assumes that 'a' is greater than zero. The theorem then shows that the measu...
: 1. If a term 't' has type 'T' in a context where a type variable 'Q' is bound and possibly other bindings are present (represented by '\<Delta>'), and if 'P' is a subtype of 'Q' in the context '\<Gamma>', then 't' still has type 'T' in a similar context where 'Q' is replaced by 'P'. 2. Similarly, if a list of terms...
If 'x' is an element of the set 'xs' and the size of 'xs' is less than or equal to the size of 'ys', then if the map 'm1' is a submap of the map 'm2' with 'xs' mapped to 'ys', it implies that the map 'm1' with 'x' mapped to 'y' is a submap of the map 'm2' with 'xs' mapped to 'ys' where the last index of 'xs' that conta...
If "tp" is equal to the concatenation of the elements taken from the "tms_of ap" up to the "as" index, and "as" is less than or equal to the length of "ap", then the start of the layout of "ap" at "as" is equal to the successor of the quotient of the length of "tp" divided by 2.
If S is a subring of R, and if g is an element of the univariate polynomial over the ring R with carrier S, and if a is an element of S, then the Taylor expansion of g at a is also an element of the univariate polynomial over the ring R with carrier S.
The lemma "sdescending_sset_sappend" assumes that the sequence formed by appending 'v' to 'u' is strictly descending. It also assumes that 'a' is an element of the set 'u' and 'b' is an element of the stream set 'v'. Under these assumptions, it shows that 'a' is greater than or equal to 'b'.
The first round of the protocol R1, with message M and state sigma, is lossless with respect to the probability measure defined by the sub-probability monad.
If the domain of finite map N is equal to the domain of finite map N', and for every element x in the domain of N, the lookup of x in N is equal to the lookup of x in N', then N is equal to N'.
If an object 'z' is initial in the category 'CC', then the zero natural transformation from the zero functor to 'z' in the zero category maps to 'CC' in the category 'CC'.
If the ordered resolution renaming of CAs, DA, AAs, As, and sigma as E is true, then the length of CAs is less than or equal to the size of DA.
The lemma named "split_block_hermitian_1" assumes that matrix A is Hermitian and that n is less than or equal to the number of rows in matrix A. It also assumes that the matrices A1, A2, A3, and A4 are the result of splitting matrix A at the nth row and nth column. Under these assumptions, the lemma shows that matrix A...
If the analysis of set X is a subset of the synthesis of the analysis of set X', and the analysis of set Y is a subset of the synthesis of the analysis of set Y', then the analysis of the union of sets X and Y is a subset of the synthesis of the analysis of the union of sets X' and Y'.
The lemma named "nearly_healthy_wlp_Bind" is defined with a fixed function 'f' that maps from 's' to 'a'. It assumes that for all 's', the function 'wlp' (weakest liberal precondition) applied to 'p' of 'f' of 's' is nearly healthy. The lemma then shows that this implies that the 'wlp' of the bind of 'f' and 'p' is als...
The value of a prime number raised to the power of an integer k is k.
The box operation applied to 1 and n of y equals n of y.
The "gena_rule" is locally sound.
If set A is a subset of or equal to set B, and set B is a subset of or equal to set C, then it logically follows that set A is a subset of or equal to set C.
The lemma eeqExcPID_imp implies two things: 1. If paps and paps1 are equivalent except for PID, then the exception of PID in paps and paps1 are also equivalent. 2. If paps and paps1 are equivalent except for PID, and pid is not equal to PID, then paps pid is equal to paps1 pid.
The theorem named "f0_exact_space_usage" assumes the following: 1. Epsilon is a number greater than 0 and less than 1. 2. Delta is a number greater than 0 and less than 1. 3. The set 'as' is a subset of the set of all numbers less than 'n'. It defines Omega as the result of folding the function 'f0_update a' over the ...
The exponential function of an extended real number "x" is equal to the extended real number of the exponential of "x".
The lemma named "map_nth_append1" assumes that the length of the list "xs" is "d". It shows that if you map the function, which gets the element at a certain index from the appended list of "xs" and "ys", over the list of numbers from 0 to "d" (exclusive), you will get the original list "xs".
For a given capacity function 'c', source 's', sink 't', flow 'f', and labeling 'l', there exists a height-bounded labeling.
If the execution of command 'c' in heap 'h' results in a value 'v' and a potentially modified heap 'h'', then the result of command 'c' in heap 'h' is 'v'.
If C strictly generalizes D, then D cannot strictly generalize C.
If there exists a state 's1'' such that, if a silent move from 's2'' to 's2' exists in the second transition system, then a corresponding silent move or silent move transition exists from 's1' to 's1'' in the first transition system, and 's1'' is bisimilar to 's2'', then 's1' is bisimilar to 's2'' and a reflexive trans...
If "j" is less than the length of "Bs" and "i" is less than the length of "Vs", then the element at position "j" in the "i"-th row of the incidence matrix of "Vs" and "Bs" is equal to the "j"-th element of the vector obtained by mapping each block "bl" in "Bs" to 1 if "Vs" at index "i" is in "bl", and 0 otherwise.