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The lemma named "expand_until_propag" is defined with a fixed variable "n_ns" which is a pair consisting of an unspecified type and a set of nodes. The lemma assumes that the following conditions are met: the "until_propag__assm" function for parameters "mu", "eta", and "n_ns" returns true; the "until_propag__rslt" fun... |
The norm of the signed number one is equal to one. |
If r1 is a renaming and r2 is a renaming, then the composition of r1 and r2 (r1 composed with r2) is also a renaming. |
The lemma named "has_fields_mono_lem2" assumes the following:
1. "P" has fields "FDTs" in class "C" (denoted by "P \<turnstile> C has_fields FDTs").
2. The class "C" in "P" is defined as a tuple of "D", "fs", and "ms" (denoted by "class P C = Some(D,fs,ms)").
3. The mapping of "FDTs" to the tuple "(F,C)" results in a f... |
For a set 'S' in a Euclidean space, if 'S' is bounded and convex, then the Euclidean Neighbourhood Retract (ENR) of the relative frontier of 'S' holds true. |
If set A is finite and the maximum value in set A is less than the minimum value in set B, then sets A and B are disjoint, meaning they have no elements in common. |
The inverse of the public key of entity A in session B is not equal to the session key argument. |
The lemma named "T_Tinv_of_set" assumes that a quasi-measure preserving transformation is invertible and that a set A is in the sigma-algebra of M. It shows that the preimage of the intersection of the inverse image of A and the space M under transformation T, intersected with the space M, is equal to A. |
If set A is a subset of set B, and both sets A and B are measurable in measure space M, then the measure of set A is less than or equal to the measure of set B. |
If B receives a message containing a nonce M and an encrypted message (using B's shared key) that includes X, a nonce Nb, and a key Kab, and this event is part of a set of events in the Otway-Rees protocol, then X is part of the information known by the Spy in these events. |
If x and y are real numbers and the real part of x is greater than or equal to zero, then the power of x and y is equal to the real part of x raised to the power of the real part of y. |
There exists a constant C such that the iteration of the widening operation on the interval step function with the greatest element and the least element of c equals to some C. |
In the context of encoding with respect to barbs, we have a lemma named "indRelTEQ_impl_TRel_weakly_reflects_barbs". This lemma takes a fixed relation "TRel" which is a set of pairs of 'procT'. It assumes that the relation "indRelTEQ TRel" weakly reflects barbs with respect to the source and target calculi barbs "SWB" ... |
The function 'ego_other2' is continuous on the set 'T'. |
If the axiom AxPTB4 holds for a proposition 'p', and if 'M' is an equivalence relation, and if 'w' is an element of the set of all possible worlds in 'M', then in the model 'M', at the world 'w', the proposition 'p' is necessarily true. |
For all x and x', if x is less than or equal to x', then x raised to the power of y is less than or equal to x' raised to the power of y, where y is an ordinal number. |
The lemma named "has_field_derivative_transform_within_open" assumes the following conditions:
1. The function 'f' has a field derivative 'f'' at a point 'a'.
2. 'S' is an open set.
3. The point 'a' is in the set 'S'.
4. For all 'x' in the set 'S', the value of function 'f' at 'x' is equal to the value of function 'g'... |
The lemma named "condition2_part5" is defined as follows:
Let A be a matrix in a Bezout domain, k be a column index, bezout be a Bezout function, and i be defined as 0 if every row up to k in A is a zero row, otherwise it is the greatest row index for which the row is not a zero row up to k in A, plus 1. Let B be the ... |
The lemma named "comp1" assumes that the function "g" is a bounded linear function. It shows that the function which takes an input "x" and maps it to the operation of "g x" being multiplied by another value, is a bounded bilinear function. |
For any given packet type ('pkt_ext), interface (iface), and rule set (rs), the difference between the set determined by the final allow decision and the set determined by the final deny decision is equal to the set determined by the final allow decision. |
The lemma named "esqrt_ereal_ereal_sqrt" assumes that a real number 't' is greater than or equal to zero. It shows that the square root of 't' in the extended real number system is equal to the extended real number of the square root of 't'. |
If the transition set "TS" is a subset of the event trace of a statechart "ST", then the source of "TS" is a subset of the configuration of "ST". |
If a list 's' is not empty, then there exists an element 'a' and a list 't' such that 's' is the concatenation of 't' and a list containing 'a'. |
A set 's' is a tagged division of 'i' if the following conditions are met:
1. 's' is a finite set.
2. For every pair (x, K) in 's', 'x' is an element of 'K'.
3. For every pair (x, K) in 's', 'K' is a subset of 'i'.
4. For every pair (x, K) in 's', 'K' is a closed box defined by two points 'a' and 'b'.
5. For any two d... |
The lemma named "first is less than or equal to second safe distance" assumes that the acceleration of the ego vehicle is less than the acceleration of the other vehicle. It shows that the first safe distance is less than or equal to the second safe distance. |
, assuming 't' is an arrow, the evaluation of the inverse of the right unit of 't' is equal to the composition of the inverse of the right unit of the codomain of 't' and the evaluation of 't'. |
The lemma "recurrence_time_flip_sign" assumes that "T" is in the existence interval of "x" and the flow of "x" at time "T" equals "x". It shows that "-T" is also in the existence interval of "x" and the flow of "x" at time "-T" equals "x". |
The natural number representation of a 16-bit unsigned integer "x" is equal to the natural number representation of the integer representation of the same 16-bit unsigned integer "x". |
For any three extended real numbers a, b, and c, if the absolute value of c is not infinity, then the product of c and a being less than or equal to the product of c and b is equivalent to the following two conditions: if c is greater than 0, then a is less than or equal to b; and if c is less than 0, then b is less th... |
:
Given a polynomial 'p' that belongs to the carrier 'P', and its degree is less than or equal to 'n', and a number 'a' that belongs to the carrier 'R', it can be shown that 'p' is equal to the direct sum over 'i' from 0 to 'n' of the Taylor series of 'p' at 'i' times the 'i'-th power of 'X' minus 'a' in the polynomial... |
The lemma named "structCongBisimSubst" is given two variables P and Q, both of which are of the type psi with three generic parameters 'a, 'b, and 'c. It assumes that P is structurally equivalent to Q. Based on this assumption, it shows that P is also bisimilar to Q under substitution. |
If for all points 'z' in the path image of 'g', the function 'f' evaluated at 'z' equals zero, then 'f' has a contour integral of zero along the path 'g'. |
The lemma named "MI_pred_MI" assumes that "MModel" is defined by "intT", "intF", and "intP". It then shows that "MI_pred" is true for "MI". |
If x is independent of y, then the sequential composition of the frame extension of x with respect to P and the frame extension of y with respect to Q is equal to the sequential composition of the frame extension of y with respect to Q and the frame extension of x with respect to P. |
A word of integer 'k' is less than a word of integer 'l' (both of type 'a' word with length 'len') if and only if the bit length of 'k' is less than the bit length of 'l'. |
If the list "xys" contains distinct elements, then there exists another list "xys'" such that "xys'" also contains distinct elements, the set of elements in "xys" is equal to the set of elements in "xys'", and the conjunction of disjoint conditions "Q" with "xys" is equal to the conjunction of conditions "Q" with "xys'... |
The set of insecure implementations "implInsecSet" under a symmetric key agreement protocol "G" is a subset of the auxiliary set of insecure implementations "implInsecSet_aux" under the same protocol "G". |
If "hom" is a digraph isomorphism and "H" is a subgraph of "G", then the application of the isomorphism "hom" to "H" results in a subgraph of the application of "hom" to "G". |
For all lists Ps, if we lift the function "Rev" to map to the body of the function "Rev_body", then for any call to "Rev" with a list of strings and a next string, it will return a list with the next string and the reverse of the original list of strings. |
The lemma named "find_fas_correct_aux" assumes a condition, denoted as "NC", where the state "s" does not meet a certain condition. Under this assumption, it shows that the state "s" is a feedback arc set (fas) of the graph "G" according to the definition of a feedback arc set in the state "fas_state". |
The top level summands of the flattened sum (using the PLUS operation) of the ACI normalized images of the top level summands of r and s, are equal to the union of the ACI normalized images of the top level summands of r and s. |
If the intersection of Omega and the space of M is in the sets of M, then A is in the sets of the restricted space of M to Omega if and only if A is a subset of Omega and A is in the sets of M. |
"it" and "it'" are proper iterators. Second, it assumes "A", which states that for any key-value pair "kv" and any "x", the function "f" applied to "kv" and "x" is flatly greater than or equal to the function "f'" applied to the same "kv" and "x".
Given these assumptions, the lemma shows that the iterator "it'" appli... |
The lemma named "backward_terminating_path_root_2" assumes that there is a backward terminating path root "r" at "x". It shows that "x" is backward terminating. |
The lemma named "vderiv_cosI" in the category of "poly_derivatives" assumes that the derivative of a real-valued function 'f' equals 'f'' on a set 'T', and 'g' is defined as a function that for each 't' returns the negative product of 'f'' at 't' and the sine of 'f' at 't'. It shows that the derivative of the function ... |
The locations in the sweep loop are a subset of the locations where fA is equal. |
The lemma "steps_preserves_terminations" assumes a sequence of steps from the state "(cs, css, s)" to the state "(cs', css', t)" in the context of a program Γ. It shows that if the program Γ terminates at the state "(cs, css, s)", then it also terminates at the state "(cs', css', t)". |
If the priority annotation invariant holds for a set 's' and the abstract version of the set 's' is not empty, and if the annotation is valid for the set 's', then the unwrapped priority of the annotation of 's' is an element of the abstract version of the set 's'. |
If a Java program P is well-formed (as assumed by "wwf_J_prog P"), and if a reduction from the state (e, es) to the state (e', es') under program P and thread t is possible (as denoted by "P,t \<turnstile>0 \<langle>e / es, h\<rangle> -ta\<rightarrow> \<langle>e' / es', h'\<rangle>"), and if the state (e, es) is a well... |
A pair of identical lists (xs, xs) is in the reflexive transitive closure of the list relation r. |
The negation of a property P after a state transition sigma-t is equivalent to the state transition sigma-t not having the property P. |
The lemma "lunit_commutes_with_L" assumes that "f" is an identity of "S". It shows that the left unit of "L f" is equal to "L" of the left unit of "f". |
The lemma named "backward_terminating_path_start_points_3" assumes that "x" is a backward terminating path. It shows that the end points of "x" are less than or equal to the transitive closure of "x" followed by the start points of "x". |
It is not possible for the "dickson_less_v" function with parameters d, m, and v to be less than v itself. |
"possible anything x'" is always true, regardless of the value of x'. |
"In (I v) l" is an element of the set G if and only if the pair (v, l) is an element of the set obtained by the function "get_Ins" applied to G. |
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Assuming that the probability of the quasi-Borel space (qbs) of a monad 'X' with respect to a measure 'μ' and a function 'β' is a valid qbs probability, and 'ssx' is the probability space of the same qbs with the same measure and function. Also, assuming that 'α' is a member of the set of all qbs measures on 'X', an... |
If the ASID of "hf" equals "nF" and the concatenation of list "zs", "hf", and list "ys" equals "rightpath", then this leads to a contradiction. This implies that these conditions cannot be true at the same time. |
Given two assumptions:
1) For any x, B x is a measurable set in M, and
2) For any distinct x and y, the measure of the intersection of B x and B y is zero,
it can be shown that the sum of the measures of B x over all x equals the measure of the union of B x over all x. |
The strong normalization on the relation to R and S on set T is equivalent to the strong normalization on the relation of the reflexive transitive closure of S composed with R on set T. |
The length of the list resulting from applying a function 'f' to each pair of elements from lists 'xs' and 'ys' (using the 'zip_with' function) is equal to the minimum length of the two lists 'xs' and 'ys'. |
If a room "r" is owned by a guest "g" in a state "s", then there exist states "s1" and "s2" and a card "c" such that "s" is the result of appending "s2", the action of checking in guest "g" into room "r" with card "c", and "s1" in that order, and there is no check-in action for room "r" in state "s2". |
Lemma borel_measurable_minus_ennreal [measurable (raw)]:
Given two functions f and g of type "'a → ennreal",
if f is borel measurable and g is borel measurable,
then the function that maps x to (f x - g x) is also borel measurable. |
If "F" is enabled, then either "G" or "F" is enabled. |
In the context of lbvc, the lemma "wti_less" is stated. It assumes that "r" is stable with respect to "step", "phi", and "pc". It also assumes that the successor of "pc" is less than the length of "phi". Under these assumptions, it shows that the weight of "c" and "pc" at the position "pc" in "phi" is less than or equa... |
If an element 'x' is not in the set of second elements of the pairs in the list 'xs', then applying the function 'inverse_permutation_of_list' to 'xs' and 'x' will return 'x'. |
If the composition of functions h and f is equal to the composition of functions j and g, then the composition of the function "iso_tuple_fst_update" with isomorphism "isom" and function h, followed by the composition of the function "iso_tuple_fst_update" with isomorphism "isom" and function j, is equal to the composi... |
The action of a permutation 'p' on the action 'Act' with parameters 'α' and 'x' is equal to the action 'Act' with parameters 'p' acting on 'α' and 'p' acting on 'x'. This lemma is marked as equivariant (eqvt) and a simplification rule (simp). |
If two states 's' and 's1' are equal except for their process IDs (PID), and a given process ID 'pid' is not equal to the PID, then the post of 'pid' in state 's' is equal to the post of 'pid' in state 's1'. |
If the following conditions are met:
1. The function 'lower_on' is applied to the substitution functions 'sigma' and 'eta' on the variables of the clause 'C',
2. The ordered pair consisting of the result of applying the substitution function 'sigma' to the variable 'x' and the result of applying the substitution funct... |
The negation of the negation of three times a variable x combined with x is equal to the negation of the combination of the negation of the combination of the negation of three times x and the negation of the combination of the negation of three times x and five times x. |
The height of the mirrored version of a tree 't' is equal to the height of the original tree 't'. |
The lemma named "qbs_closed2I" assumes that for any element 'x' in set 'X', the function that maps any input to 'x' is in 'Mx'. Under this assumption, it shows that the set 'X' is quasi-Borel closed with respect to 'Mx'. |
The lemma "WN_UNF_imp_CR" assumes that "r" is weakly normalizing (WN) and uniformly normalizing (UNF). It shows that "r" is confluent (CR). |
The count of phi functions in a graph 'g' is equal to the cardinality (or size) of the domain of the phi functions in 'g'. |
The set of elements in a vector obtained by mapping a function 'f' over a vector 'v' is equal to the image of the set of elements in 'v' under the function 'f'. |
If the types are well-formed according to the context Γ, and if the boolean expression 'e' is of type 't' under the context Γ, then the type 't' is well-formed. |
The lemma named "nsubst_fml" is established with the following conditions:
1. An interpretation 'I' is fixed, which is a function mapping from the set of state functions ('sf'), state constants ('sc'), and state variables ('sz').
2. A state 'v' is fixed, which belongs to the set of state variables 'sz'.
3. It is ass... |
"x is greater than y" is equivalent to "the negation of the parallel of x right-multiplied by the product of y and L". |
: if for all variables V, W, X, Y, Z and a state transition function τ, the function f applied to these variables and τ is equal to the function f applied to the values of these variables at state τ, then if N, P, Q, R, and S are all constant-preserving (cp), the function f applied to the values of N, P, Q, R, and S at... |
For the function that maps x to 0 if x equals 0 and to 1 otherwise, this function preserves the relation "mod_ring_rel64". |
The degree of the function "funpow k shift1 p" is equal to "k" plus the degree of "p". |
D is greater than 0. |
The lemma named "transaction_check_pre_NotInSetE" assumes two conditions. First, it assumes that a transaction check precondition "FPT T" is in delta. Second, it assumes that "u" is equal to the variable "x" not in the function of the set "s" and that "u" is in the set of unlabelled transaction checks "T". Given these ... |
The orbit of a function 'f' applied to the inverse of 'f' acting on 'a' is equal to the orbit of 'f' acting on 'a'. |
The lemma "in_orbit_imp_in_segment" assumes that "y" is in the orbit of "f x", "x" is not equal to "y", and "f" is a bijection. It shows that "y" is in the segment from "f x" to "f y". |
If a program G is well-formed according to the well-formed method body (wf_mb), then G is a well-structured program. |
The function "ltakeWhile" applied to a predicate P and a list xs equals the list xs if and only if the set of elements in xs is a subset of the set of elements for which the predicate P holds true. |
If a relation preserves barbs with respect to the induced relation of a transition relation (TRel), then the transition relation (TRel) itself preserves barbs. This is fixed for a set of process transitions 'procT'. The assumption is that the relation preserves barbs for the induced relation of the transition relation ... |
Given a valid problem in STRIPS representation, denoted as "\<Pi>", and a model "\<A>" that satisfies the formula "\<Phi>" of "\<Pi>" at time "t", if "k" is less than "t", and an operation "op" is in the set of operations at time "t" decoded from "\<Pi>" and "\<A>", and a variable "v" is in the set of preconditions of ... |
If a match "m" is in normalized negation normal form (as per assumption "n"), and if the pair (disc, sel) is a well-formed discriminator-selector pair for a set "C" (as per assumption "wf_disc_sel"), and if for every set of points "pts", the function "negate_f" does not have a negated discriminator "disc2" when "disc" ... |
Replicating the integer 1 with any element x results in a list containing only the element x. |
If Qinf is equivalent to Qinf' and the free variable of Qinf is equal to the free variable of Qinf', then the evaluation of Q with Qfin and Qinf is equal to the evaluation of Q with Qfin and Qinf'. |
If the set of elements 'xs' is equal to the set of all 'x' such that 'x' is greater than or equal to 'a' and less than or equal to 'b' (where 'a' and 'b' are words of some length), and if all elements in 'xs' are distinct, then the length of 'xs' is equal to the successor of the unsigned natural number representation o... |
If a function 'f' is measurable with respect to a measure 'M' and the count space of the universal set, then the predicate that checks if a constant 'c' is equal to the function 'f' applied to 'x' is also measurable with respect to 'M'. |
If the atom 's' is distinct from the set of elements (k, s1, s2, k1, k2, α1, α2, β) and the atom 'k' is distinct from the set of elements (s1, s2, k1, k2, α1, α2, β), then given the set of conditions {PrfP s1 k1 α1, PrfP s2 k2 α2, ModPonP α1 α2 β OR ExistsP α1 β OR SubstP α1 β}, there exists a 'k' and 's' such that Prf... |
The lemma named "expl_cond_exp_add" assumes that both X and Z are integrable over the measure space M. It shows that for all elements w in the space of M, the expected conditional expectation of M, Y, and the function that adds X and Z is equal to the sum of the expected conditional expectation of M and Y with respect ... |
The lemma named "lcoset_el_reduce" assumes that "a" is an element of the group "G". It shows that adding "a" to the group "G" (in the sense of left coset operation) results in the group "G" itself. |
If "x" is an element of the carrier Zp, and the nth term of "x" is zero, and "x" is not equal to zero, then the order of "x" in Zp is greater than or equal to "n". |
If "not actually phi" is true in a possible world v, then "actually not phi" is also true in that possible world v. |
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