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If the initial state of the Turing machine "fourtimes_tm" after a certain number of steps "stp" is a specific configuration (with a certain tape content and head position), then the initial state of a more complex Turing machine (constructed by appending "fourtimes_tm" to other Turing machines and shifting their positi...
Setting the bit at the position "Suc n" (which is the successor of n, or n+1) in a number "a" is equal to the remainder of "a" divided by 2 plus 2 times the result of setting the bit at position "n" in the quotient of "a" divided by 2.
The set defined by the iMODb2_set function is equal to the set defined by the iMODb2_UN_set function.
"v1" is a vector of length "n", and "wf2" which states that "v2" is also a vector of length "n". It then shows that "v1" is equal to "v2" if and only if for all indices "i" less than "n", the "i"-th element of "v1" is equal to the "i"-th element of "v2". This is denoted as "?l = ?r".
In the context of bounded linear functions, if a function 'g' is uniformly convergent on a set 'A', then the function that applies 'f' to the result of 'g' (denoted as 'f (g x y)') is also uniformly convergent on the set 'A'.
If a directed group chain of length 'n+1' exists for a function 'f' in group 'G', then a directed group chain of length 'n' also exists for the same function 'f' in the same group 'G'.
The tiny natural transformation functor is an element of the set Vset alpha.
If "x" is less than or equal to "y", then the join (or union) of "x" and "y" is "y".
"m3" refines "m2". This is under the condition that the intersection of "R23" and the intersection of "m2_inv3a_sesK_compr" and "m2_inv3b_sesK_compr_non", when paired with the intersection of "m3_inv3_sesK_compr", "m3_inv2_ticket", and "m3_inv1_lkeysec", refines the medium "med32" from model "m2" to model "m3".
: given the assumptions that K is a subfield of R, and both p and q are elements of the carrier of K[X], with q not being an empty set, it shows that long division of p by q results in the pair (p divided by q, remainder of p divided by q).
The lemma named "tainting_iff_blp_trusted" defines a function called "project". This function takes two arguments, "a" and "ts", and returns a record with two fields: "security_level" and "trusted". The "security_level" field is set to 1 if "a" is in the set difference of "taints ts" and "untaints ts", otherwise it i...
The cardinality of the function "sucs" applied to the result of merging "t1" is less than or equal to 1.
The lemma named "less_fun_strongI" assumes that for any positive number 'c', there exists a number 'n' such that for all numbers 'm' greater than 'n', the value of 'c' times the function 'f' at 'm' is less than the function 'g' at 'm'. If this assumption holds, it shows that the function 'f' is of smaller order of grow...
The delay bisimulation lift function, when applied to the parameters final2, r2, final1, r1, a lambda function that flips the bisimulation at time t, tau move2, and tau move1, holds true.
The lemma named "circline_type_zero_card_eq1_0" assumes that the type of a given circline H is 0 and that the zero homogenous point is in the set of points on the circline H. It shows that the set of points on the circline H is equal to the set containing only the zero homogenous point.
For all i less than n, if each tree in the enumeration of trees of size i is distinct, then each tree in the enumeration of trees of size n, where each tree is composed of a left subtree from the enumeration of trees of size i, an empty node, and a right subtree, is also distinct.
The nth term of the difference between two formal power series f and g is equal to the difference between the nth terms of f and g.
The lemma "shifted_transitions_observable_against_distinguishing_transitions_left" assumes that transition 1 is in the set of shifted transitions for a machine M between states q1 and q2, transition 2 is in the set of distinguishing transitions on the left for the same machine and states, and both states q1 and q2 are ...
The stable segment of a list "xs" sorted using a comparison function "cmp" and an element "x" is equal to the stable segment of the original list "xs" with the same comparison function "cmp" and element "x".
If q is in the finite acceptance condition of a transition function delta and a natural number n, then the acceptance condition of delta, n, and q holds true.
If "i" strictly dominates "j", and "j" is in the set of vertices of graph "G", then "i" is not equal to "j".
If you apply the function "case_prod" to the result of the co-recursive product of functions "f" and "g" with input "s", it is equivalent to applying function "h" to the results of applying "f" and "g" to "s" separately.
, given an empty set, the abstraction of a term, represented by "AbstTermP", of a variable "i", an ordinal number "n", and a quoted database term "u", is equal to the quoted database term of the abstraction of the database term "i", "n", and "u".
A polynomial 'p' is equivalent to an empty polynomial if and only if all coefficients 'c' in the set of 'p' are equal to zero.
If a relation "ir_valid" holds for propositions P, Q, and commands c, c', then it also holds for a new proposition which is the conjunction of Q and the existence of states s, s' such that P holds for s, s' and the commands c, c' transition to states t, t' respectively.
The signed cast of a numeral 'w' as a word of length 'a' is equal to the word representation of the integer obtained by taking the signed bit of the numeral 'w' at the position 'LENGTH('a) - 1'.
The probability mass function of a Bernoulli distribution with probability 1 is equal to the probability mass function that always returns true.
A formula "Since \<phi> I \<psi>" is satisfied at time 0 in a given system state, variable valuation, and variable, if and only if 0 is an element of the interval I and the formula \<psi> is satisfied at time 0 under the same conditions.
The product of two monomials, one with coefficient 'c' and term 'u', and the other with coefficient 'd' and term 'v', is equal to a new monomial. The coefficient of this new monomial is the product of 'c' and 'd', and its term is the sum of the terms 'u' and 'v', represented as a pair with the sum of the polynomial pow...
The lemma named "setdist_eq_0_compact_closed" assumes that "S" is a compact set and "T" is a closed set. It shows that the set distance between "S" and "T" equals zero if and only if either "S" is an empty set, "T" is an empty set, or the intersection of "S" and "T" is not an empty set.
For any boolean algebra, the operation "xor" (exclusive or) is associative. This means that for any elements x, y, and z in this algebra, the result of xor-ing x with y, and then xor-ing the result with z, is the same as the result of xor-ing y with z, and then xor-ing x with the result.
The lemma "rank_of_indep_in_le" assumes that "X" is an independent set in "E". It shows that the cardinality of "X" is less than or equal to the rank of "E".
The set of edges in a walk that includes the sequences xs, followed by v, then w, and then xs', is equivalent to the union of the set of edges in a walk that includes the sequence xs followed by v, the edge from v to w, and the set of edges in a walk that starts with w and is followed by xs'.
For a given interval [r, mod m] cut at a point less than or equal to t, if t is less than r, then the result is an empty set. Otherwise, the result is a new interval [r, mod m, (t - r) divided by m].
The equality of values 'a' and 'b' being false is equivalent to the existence of some 'x' and 'y' such that 'a' equals 'Some x' and 'b' equals 'Some y', and 'x' is not equal to 'y'.
The opposite of a co-spanning functor from a category is equal to a co-spanning functor in the opposite category.
The memory is correctly initialized as empty. This is done using two lambda functions. The first lambda function retrieves the current state of the memory and returns the value associated with a given key. The second lambda function retrieves the current state of the memory and updates it by associating a given key wit...
The lemma named "wfLearnLessState" shows that the "learnLessState" is well-founded.
If action A is true, then action A is enabled.
The lemma named "CAST_to_list" under the category of "autoref_rules_raw" assumes the priority tag general algorithm and the side general algorithm for the conversion of a set to a list. It shows that the transformation operation "tol" is in relation to the "CAST" operation within the context of a set and list set relat...
The conjugate of a complex number, derived from a real number, is equal to the original real number.
Under the assumptions that "i" is the encoding of function "f", "f" is a recursive function of the length of "x", and the first element of the iteration of "t" steps of the function "step" starting from the tuple "[(f, list_decode x, [])], None" is not an empty list, it can be shown that the evaluation of "r_result" wi...
If a set 'a' is an element of set 'y' and the union of set 'x' and set 'y' is set 'z', then 'a' is an element of set 'z'.
The function "poly_set_iterateoi" is a correct implementation of iteration over a sorted list set, given the abstraction function "lss_α" and the invariant "lss_invar" for the sorted list set "lss_iterateoi".
For any set A, the union of the monadic return of each element in A (where each element is wrapped in a list) is equal to the monadic return of A itself.
If an empty list is not an element of the set 'xs' and 'y' is not an empty list, then an empty list is not an element of the set resulting from the function 'combine_lists_P' applied to 'P', 'y', and 'xs'.
The hyperbolic cosine of any real number is greater than or equal to 1.
Given three assumptions: 1. There exists a transition from the initial state (empty environment, expression 'e', empty stack) to a state (environment 'Γ', expression 'e'', stack with dummy values mapped from 'r') under the Sestoft's garbage collection semantics. 2. The expression 'e'' is a value. 3. The free variabl...
The lemma named "inter_ov" assumes that the pair (i,j) is in the intersection of three sets. The first set is the union of b, m, ov, the inverse of f, and the inverse of d. The second set is the union of e, the inverse of b, the inverse of m, the inverse of ov, ov, the inverse of s, s, the inverse of f, f, the inverse ...
The modulus function, which takes two arguments 'a' and 'b' and returns the remainder of 'a' divided by 'b', is a primitive recursive function of two variables.
: given a local hidden variable model M with observables A and B, and response functions Va and Vb, if the set of real parts of the spectrum of A is not empty and is a subset of {-1, 1}, and if A is a Hermitian matrix that belongs to the set of finite complex matrices, then for almost every w in M, the absolute value o...
The absolute convergence abscissa of the nth derivative of a function f is equal to the absolute convergence abscissa of the function f itself, where f is a Dirichlet series.
If the size of the current object is greater than 0, the current object is invariant, and popping the current object results in a pair (x, current'), then the size of current' is equal to the size of the current object minus 1.
The lemma named "mismatch_pref_comm" assumes that "w1" and "w2" are in the set containing "u" and "v". If the concatenation of "u", "w1", and "v" is a prefix of the concatenation of "v", "w2", and "u", then it shows that the concatenation of "u" and "v" is equal to the concatenation of "v" and "u".
The lemma named "valid_conj" assumes two conditions. The first condition is that a certain property, represented by the function "valid ds", holds true when three conditions - R, P, and Q - are all true for any given ds and fp. The second condition is similar to the first, but it specifically asserts the validity of Q ...
If the function "enum_pos" applied to 'x' equals "enum_pos" applied to 'y', then 'x' must be equal to 'y'. This implies that the function "enum_pos" is injective.
If there is a chain according to the LGC relation in the set Ns, then there is also a chain according to the L relation in the set obtained by applying the function "snd" to each element of Ns.
The lemma named "permute_set_eq" shows that the permutation of a set X by p is equal to the set of all elements x such that the inverse permutation of x by p is in X.
The wait set of the result of activating conditional actions between two states (s1 and s2) with a given set of conditional thread actions (cts) is equal to the wait set of the first state (s1).
If the first elements of each pair in a list 'xs' are distinct, then the list obtained by applying a function 'f' to the first element of each pair in 'xs' is also distinct.
The lemma named "nat_mod" assumes that a natural number "m" is greater than another natural number "n". It shows that the modulus of "m" and "n" is equal to the modulus of the difference of "m" and "n" with "n".
Given the assumption that "f" is a morphism from the tensor product of the identity object and "a" to "a", and that the tensor product of the identity object and "f" is equal to the tensor product of the unit object and "a" composed with the inverse of the associator of the identity object, the identity object, and "a"...
The lemma named "context_unique" assumes that "Gamma" is valid, and that the pair (x, T) and the pair (x, U) are both elements of the set "Gamma". It then shows that T must be equal to U.
A lazy list starting with element 'x' followed by the list 'xs' is a prefix of another lazy list starting with element 'y' followed by the list 'ys' if and only if 'x' is equal to 'y' and 'xs' is a prefix of 'ys'.
If the following conditions are met: the successor of 0 is less than b, x equals b raised to the power of y times q plus r, q is greater than 0, q is less than b, and r is less than b raised to the power of y, then the ordinal logarithm of x with base b equals y.
If 'n' is less than the length of lists 'xs' and 'bs', and the 'n'th element of 'bs' is true, then the 'n'th element of 'xs' is in the set obtained by filtering 'xs' with 'bs'.
The lemma named "epred_Max" assumes that "A" is a finite set and "A" is not an empty set. It shows that the predecessor of the maximum element in "A" is equal to the maximum element in the set of predecessors of "A".
If the size of a multiset A is the successor of a number n, then for any element 'a' and any multiset 'B', if A is equal to the multiset formed by adding 'a' to 'B' and the size of 'B' is 'n', then a property 'P' holds.
The residue of a natural number 'n' is equal to the integer 'n' modulo 'm'.
The third step in protocol m2 refines the third step in protocol m1, given that the relation R12 intersects with the universal set and the set of all authenticated messages in protocol m2.
If the output of a communication from a smartcard C to a spy X is in the set of events, and the message is secure, then X is in the set of entities known by the spy in the set of events.
The lemma named "gamma_Step_subcomm" is defined with respect to two fixed commands C1 and C2 in the context of a semilattice. It assumes two conditions: 1. For all x, e, and S, the application of function f1 to x, e, and the gamma function of S is a subset of the gamma function of the application of function f2 to x,...
The lemma "token_stays_in_sink" assumes that "q" is a sink and that the token run "x" at position "n" equals "q". It shows that the token run "x" at position "n + m" also equals "q".
The absolute infinity of a polynomial x is equal to zero if and only if x is equal to zero.
An empty list is related to the evaluation of a "Let" expression, which binds variable "x" to a lambda function that takes an argument "y" and returns "y", in the context of variable "x". This evaluation is done in the context of an empty list and results in a list containing a single pair, where "x" is paired with the...
The lemma named "ucast_sub_ucast" is defined with a fixed variable 'x' of type 'a word, where 'a is a length. It assumes that 'y' is less than or equal to 'x' and that the length of 'a' is less than or equal to the length of 'b'. It shows that the unsigned cast of the difference between 'x' and 'y' is equal to the diff...
For any natural number 'n', regular automaton 'A', and tree 'T', the specification of 'n' in the relabeled version of 'A' is equivalent to the specification of 'n' in 'A' itself.
If "s" can be transformed into "s'" through zero or more beta-reductions, then "s" applied to "t" can also be transformed into "s'" applied to "t" through zero or more beta-reductions.
If the union of G and the function that maps each process to its body satisfies the Hoare triple for all processes, and the set of processes is finite, then G satisfies the Hoare triple for the function that maps each process to its body.
The elements of set 'a' are less than or equal to the elements of the sum of sets 'a' and 'b'.
The lemma named "eqButUIDl_trans" assumes two conditions: first, that "uidl1" is equivalent to "uidl" except for the user ID "uid", and second, that "uidl2" is equivalent to "uidl1" except for the user ID "uid". Given these assumptions, it shows that "uidl2" is equivalent to "uidl" except for the user ID "uid".
If C is an element of the carrier of the function fTo D, and m is maximal in the induced order of the function fTo D for all sets S in the carrier of fTo D where C is a subset of S, then m is also maximal in the function fTo D.
The lemma named "card_deg_sect" assumes that the set X is finite and not empty. It shows that the cardinality of the degree section of X with respect to d is equal to the combination of (d plus the cardinality of X minus 1) and (the cardinality of X minus 1).
The lemma "same_state_implies_same_result_state" assumes that if two systems c and d are in the same state at point p, and if system c transitions to state c' after event ev, and system d transitions to state d' after the same event ev, then the states of systems c' and d' at point p will be the same.
The lemma named "list_nth_append8" assumes that "i" is not less than the successor of the sum of the lengths of list "x" and list "y". It also assumes that "i" is less than the successor of the successor of the sum of the lengths of list "x" and list "y". Under these assumptions, it shows that "i" equals the successor ...
Dropping a set A from an empty finite map results in an empty finite map.
The relational interpretation of R is a dual negation operator.
If "Notes A X" is in the set of events, and A is bad, then X is in the set of things the Spy knows in the events.
If alpha prime is equal to alpha, then alpha prime is an array relation with respect to T. This is an introduction rule for both digraphs and relation digraphs.
A set 'S' is locally compact if and only if for any subsets 'K' and 'T' of 'S', where 'K' is compact and 'T' is open and 'K' is a subset of 'T', there exist subsets 'U' and 'V' such that 'K' is a subset of 'U', 'U' is a subset of 'V', 'U' is a subset of 'T', 'V' is a subset of 'S', 'U' is an open set in the topology of...
For any formal power series 'a' in a commutative ring with unity, the k-th power of the formal power series derivative operator applied to 'a' is equal to the formal power series whose n-th coefficient is the product of the n-th power of the natural number 'n' and the n-th coefficient of 'a'.
The conjugate of a natural number 'n' is equal to the natural number 'n' itself.
If "n" is less than or equal to the length of the list "xs", then the first element of the result of the function "heapify" applied to "n" and "xs" satisfies the Braun condition.
If "I" is a finite set, then if for every polynomial "q" in "I", "p" and "q" are coprime, it implies that "p" is coprime with the product of all polynomials in "I".
The function C is Borel measurable in the Borel sigma-algebra.
The termMOD is an instance of the igSubstCls class.
The deletion of an undirected edge from vertex v to vertex v' in graph g is the same as the deletion of an undirected edge from vertex v' to vertex v in the same graph g.
If the following conditions are met: 1. Set A is a subset of the sumset of set {x} and the stabilizer of set B, 2. x is an element of set A, 3. the sumset of set {x} and the stabilizer of set B is not a subset of set C, 4. set A is a subset of set C, and 5. set C is a subset of set G, then the stabilizer of set ...
If a set S is Borsukian and S is homeomorphic to T, then T is also Borsukian.
The least common multiple of an integer x and the negative of an integer y is equal to the least common multiple of x and y. This is true for any integers x and y.