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The integer part of a number 'x' is equal to 'x' if and only if there exists an integer 'n' such that 'x' is equal to the integer 'n'.
For any two extended real numbers 'a' and 'b', if 'b' is greater than zero and less than infinity, then the product of 'b' and the quotient of 'a' divided by 'b' equals 'a'.
The heap node value of the unit relation for any x and y is empty.
"0" is not in the Braun indices of the tree "t".
If a set of propositions, denoted as "\<G>", satisfies the condition "Only_G", and if "\<G>" satisfies a property or condition denoted as "\<phi>", then "\<G>" also satisfies the property or condition resulting from the application of the function "af_G_letter" to "\<phi>" and "\<nu>".
If the function f applied to the bottom element equals the bottom element, then the function f applied to the result of the bind operation on g and h equals the bind operation on g and a function that applies f to the result of applying h to x.
If the list "l'" is not empty, then for a given "it" that is a flatten iterator over a list "l", a list "p", and the list "l'", when the function "flatten_it_next" is applied to "it", it results in a pair (a, it'), where "it'" is a flatten iterator over the list "l", the list "p", and the tail of the list "l'", and "a"...
If x is less than or equal to y, then the extended nonnegative real number representation of x is less than or equal to the extended nonnegative real number representation of y.
The lemma named "weight_minus_web" assumes that "f" is a current of a web "\<Gamma>". It shows that the weight of "x" in the web "\<Gamma>" subtracted by "f" is equal to the weight of "x" in "\<Gamma>" minus the outgoing degree of "f" at "x" if "x" is in the set "A" of "\<Gamma>", otherwise it is equal to the weight of...
'S prime' is regular.
The product of the function A over the set containing the element i is equal to the product of the function A over the set containing the element j.
The lemma named "set_disconnected_nodes_get_disconnected_nodes_wf_is_l_set_disconnected_nodes_get_disconnected_nodes_wf" is an instance of the following conditions: "l_set_disconnected_nodes_get_disconnected_nodes_wf", "type_wf", "known_ptr", "heap_is_wellformed", "parent_child_rel", "get_child_nodes", "get_disconnecte...
For any regular expressions L and R, the starting state of the 'or' operation between L and R, paired with some state q, is not in the step function of the 'or' operation between L and R with respect to some input a.
For a given set 's' in a Euclidean space, 's' is not affine dependent if and only if 's' is finite and the affine dimension of 's' is equal to the integer value of the cardinality of 's' minus one.
If a path 'p' conforms with a strategy 'σ' on a path 'P', then the path 'p' also conforms with the same strategy 'σ' on the path 'P' after dropping 'n' elements from the beginning.
Theta is in the hom-set from the source of u to the source of r, and theta is also in the hom-set from the tensor product of f and w to u.
If "r" is a cardinal number, then the product of "r" and the cardinal number one is order-isomorphic to "r".
If "x" is a well-behaved (moderately well-behaved) lens, then "x" is also a weak lens.
The universal infimum (greatest lower bound) of the function g applied to the set of all elements i such that i is in the image of the function f over the set A, is equal to the infimum over all elements i in A of g applied to f of i.
If the function "f" in the substitution "σ" maps to some value "r" in the set of state functions "SFuncs", then the function "f" in the adjoint of "σ" with respect to an interpretation "I" and a state "ω" maps to the semantic interpretation of the term "r" after replacing "I" with "dotid" in the context of a differenti...
If a function "cr_spmf_gpv" is applied to the "bind_spmf" function and the "bind_gpv" function, under the condition that the "bind_gpv" function is equal to the "cr_spmf_gpv" function, then the result will be a "cr_spmf_gpv" function.
1 is an infimum distribution kernel.
For a floating point number represented as "Float m e", the operation "float_down p" is defined as follows: If the sum of p and e is less than zero, then the result is a new floating point number where the mantissa is "m" divided by two to the power of the absolute value of "p + e", and the exponent is "-p". Otherwise,...
The lemma named "ex_min_finite_set" establishes that for a given set 'S' of any type 'a' and a function 'f' that maps 'a' to a natural number, if 'S' is finite and not empty, then there exists an element 'x' in 'S' such that for all elements 'y' in 'S', the value of 'f' at 'x' is less than or equal to the value of 'f' ...
The lemma named "make_pm_decomp" in the context of complex square matrices shows that the function "make_pm" applied to a matrix A is equal to a tuple. This tuple consists of the size of the projective measurement of "make_pm" applied to A and the outcomes of the projective measurement of "make_pm" applied to A.
The lemma named "iso_\<i>" assumes that "a" is an object in category "C". It shows that the identity on "a" is an isomorphism in category "D".
If 'x' is in the set of While programs, and 'p', 'q', and 'r' are in the set of pre-expressions, then the sequential composition of 'p', 'x', 'q', 'r', and 'm' is also in the set of pre-expressions.
If the function s1_1 applied to x1 and y1 equals the function s1_1 applied to x2 and y2, then y1 must be equal to y2.
The function "en" applied to the trivial function of "s" equals to an empty set.
If the following conditions are met: 1. NRO is equal to the encryption of a set containing a number f_nro, an agent B, a nonce L, and an encrypted message M using A's private key. 2. NRO is a part of what the spies can see in the event list. 3. There is an event in the event list where A says to B a set containing a n...
For a given set 'I', 'I' has products if and only if 'I' is not equal to the universal set and 'I' admits tupling.
If 'f' is a morphism from 'r' to 's' in the category, and if we define several terms such as 'beta' equal to 'alpha', 'A' as the Hom-functor applied to 'r', 'B' as the Hom-functor applied to 's', 'C' as the opposite category of the original category, and 'D' as the category of sets with a fixed cardinality 'alpha', the...
If A, B, and C are objects in the category of sets of a certain size α, and if A' is the Cartesian product of A and B, and then the Cartesian product of the result with C, and if B' is the Cartesian product of A with the Cartesian product of B and C, and if the category is the category of sets of size α, then the arrow...
If the pair (k, a) is in the set produced by the function "annotate_rlen" applied to the list "l", then "k" is less than the length of the list "l".
The semi-functor "FF" is an element of the set "Vset" of the ordinal number "alpha" plus seven.
The lemma named "monad_catch_envT" assumes that a monad catch operation is defined with the functions "return", "bind", "fail", and "catch". If this assumption holds, it shows that a monad catch operation is also defined with the environment versions of these functions, namely "return_env", "bind_env", "fail_env", and ...
The substitution of a variable 'x' in 's' with a value 'v' is equal to the substitution of 'x' in 's' with the same value 'v'.
Applying the function "case_sum_bot" with parameters f, g, h on the result of "sumbot" function with parameter p is equivalent to applying the function "case_sum" with parameters g, h on p.
The probability "p" is equal to the probability of the event "sKs p".
The inverse of the feedback operator applied to the identity function equals the identity relation.
"I" is reflexive, and "S" which states that the process is secure under the conditions "step", "out", "s0", "I", and "D". It shows that if a pair consisting of a list "yps" appended with "yp" and a set "Y" is in the futures of the process defined by "step", "out", and "s0" with inputs "xps", then a pair consisting of t...
The lemma "rev_pointwise" assumes that the effect of reversing the array 'a' from index 'i' to index 'j' changes the state from 'h' to 'h'' and returns 'r'. It shows that the element at index 'k' in the array 'a' after the state change is equal to: the element at index 'k' before the state change if 'k' is less than 'i...
The lemma L_10_5_\<chi>'_arrow_components shows the following: 1. The value of the arrow function L_10_5_\<chi>'_arrow for parameters \<alpha>, \<beta>, \<TT>, \<KK>, c, and a is equal to a function that, for each \<tau> in the object map of the cone functor applied to the composition of \<TT> and c with the infimum o...
The lemma named "fps_right_inverse_idempotent_ring1" is about a formal power series (fps) in a ring. It takes a formal power series 'f' and two elements 'x' and 'y' from the ring. The lemma assumes that the product of the zeroth term of 'f' and 'x' equals 1, and the product of 'x' and 'y' equals 1. Although these assum...
If a transformer "t" is healthy, then "t" is less than or equal to the transformation that takes a predicate "P" and a state "s" and returns the bound of "P".
The traces of the concurrent composition of processes P and Q with respect to the processes p and q, is equal to the domain of the concurrent composition of failures of P and Q with respect to p and q.
If a list 'xs' contains distinct elements, then the size of the binary search tree created from this list is equal to the length of the list 'xs'.
For every vertex 'v' in the successors of 'n', the pair (n, v) is an element of the edge set 'g_E' of the graph 'G'.
Applying the function "zipwith0" with function "f", an empty list, and list "ys" is equivalent to mapping the function "f" with 0 over the list "ys".
The weight of a choice operation "Ch" with choices "c1" and "c2" on a channel "ch" in a state "s" at the (n+1)th step is equal to 1 minus the value of the channel "ch" in the state "s".
The star of x plus the star of y is equal to the plus of x plus the star of y.
If the previous element of 'a' in set 'I' is equal to the previous element of 'b' in set 'I', and the minimum element of set 'I' is not equal to 'a' or 'b', then 'a' must be equal to 'b'.
The lemma named "stopping_time_less_const" assumes that "T" is a stopping time with respect to filtration "F". It shows that the predicate "T < t" is measurable with respect to the sigma-algebra "F(t)".
The lemma named "older_seniors_recursive_card" is defined with a fixed 'x' and 'n'. It also defines "os" as the older seniors of 'x' and 'n'. Assuming that "os" is not an empty set, it shows that the cardinality of "os" is equal to the successor of the cardinality of the older seniors of the maximum value in "os" and '...
If set A is a subset of set B, then the restriction of set A by set C with respect to x is also a subset of the restriction of set B by set C with respect to x.
The set P has a least element.
, given a finite set F and an element 'a' not in F, and given a function 'f' that maps elements of F to a group G and 'f a' is in the group G, then the finite product of the function 'f' over the set obtained by inserting 'a' into F is equal to the product of 'f a' and the finite product of the function 'f' over F.
If the following conditions are met: 1. The operation "op_vis_insert" is related to the operation "insert" in the relation Rv, mapping from Rv to the visibility relation vis_rel. 2. The operation "op_vis_memb" is related to the membership operation "(∈)" in the relation Rv, mapping from Rv to the visibility relation v...
If "f" is a monotone function and "g" is a monotone function, then the function that is the infimum (greatest lower bound) of the composition of "f" with the floor function applied to "x" and the composition of "g" with the floor function applied to "y" is also a monotone function.
The mapping of a concatenation of two lists "as" and "bs" using function "NegPos_map" with a constant "C" is equal to the concatenation of the mapping of list "as" and the mapping of list "bs" using the same function "NegPos_map" with the same constant "C".
If the following conditions are met: 1. There is a bytecode-method type correspondence between bc', f', sttp0, cG, rT, mxr, and l1. 2. bc' is equal to the concatenation of bc1, inst, and bc3. 3. f' is the combination of f1, f2, and f3. 4. l1 is equal to the length of bc1. 5. The length of bc1 is equal to the length of...
If for every element 'j' in the set from 'n' to 'n + m', the function 'f' applied to 'j' equals zero, then the sum of the function 'f' over the set 'A' from 'n' to 'n + m' equals zero.
The lemma named "wfT_wf_cons3" assumes that under the contexts Theta, B, and Gamma, the term "{ z : b | c }" is well-formed, and the atom y is fresh for the pair (c, Gamma). It then shows that under the contexts Theta and B, the term "(y, b, c[z::=V_var y]\<^sub>c\<^sub>v) #\<^sub>\<Gamma> Gamma" is also well-formed.
If the shared key between A and B is in the intruder knowledge and the keys are bad in the intruder knowledge for the bad set, then either A or B is in the bad set.
If the global object "r" in the state "s" is None, then updating the global object "r" with a new value "v" at name "n" does not change the global objects in the state "s".
If there is a morphism from (B1, B2, s1, s2) to (B1', B2', s1', s2') with functions f and g, then the image of B2 under function g is a subset of B2'.
The norm of the axis 'i' times 'k' is equal to the norm of 'k'.
If a directed acyclic graph 't' is well-formed, then the successor nodes of 't' are disjoint.
The lemma D2_UNION shows that the union over all elements 'd' for a property 'P' is equal to the union over the elements 'd' in the set {D1, D2} for the same property 'P'.
If a word 'w' is compatible with a word 'r', then the parallel composition of 'w' and another word 'w'' is also compatible with 'r'.
The Euler-Mascheroni constant is greater than zero.
If a pair (w, x) is in a relation r or w equals x, and r is a well-founded and transitive relation, and both w and x are elements of a set A which is small, then the order map of w in relation to A and r is less than or equal to the order map of x in relation to A and r.
The lemma named "tensor_vec_add1" is defined with three fixed vectors v1, v2, and v3, which belong to a commutative ring. It assumes that v1 and v2 are vectors in a carrier space of dimension d1, and v3 is a vector in a carrier space of dimension d2. The lemma then shows that the tensor product of the sum of v1 and v2 ...
The lemma named "pure_smaller" assumes that "a" is pure and "a" is greater than or equal to "b". It shows that "b" is also pure.
The lemma named "affine_hull_2" establishes that for any two real vectors 'a' and 'b', the affine hull of the set containing 'a' and 'b' is equal to the set of all linear combinations of 'a' and 'b' where the coefficients 'u' and 'v' sum to 1.
The lemma "index_states_minimal" assumes that a finite state machine "M" is minimal. It shows that the finite state machine obtained by indexing the states of "M" is also minimal.
The lemma named "sync_\<phi>1_\<phi>2" assumes the following conditions: 1. "trn1" is a valid transition (v1). 2. The source of "trn1" is reachable (r1). 3. "trn2" is also a valid transition (v2). 4. The source of "trn2" is reachable as well (s2). 5. "trn1" is a communication transition of the first type (c1). 6. "trn...
The natural union of a function 'f' with respect to 'i' and 'j' is equal to the natural union of the same function 'f' with respect to 'j' and 'i'. This implies that the order of 'i' and 'j' does not affect the result of the natural union operation.
The lemma "root_in_start_points" assumes that "r" is connected to the root "x", "r" is a vector, "x" is not equal to zero, and the product of "x" and "r" is zero. It shows that "r" is less than or equal to the start points of "x".
For any set E, the fundamental circuit "x" in "B" is equal to the fundamental circuit in set E, "x" and "B".
The lemma for the Mahler measure of a polynomial, as stated in the Mahler_Measure theory in Berlekamp_Zassenhaus, is that the Mahler measure of a polynomial 'p' is equal to the absolute value of the leading coefficient of 'p' multiplied by the Mahler measure of 'p' when it is monic.
If the size of a word 'x' is equal to the length of 'a' times 'n', and 'n' is not equal to zero, then the right split of the word 'x' is equal to the map of the function that casts 'x' right shifted by 'i' times the length of 'a' to the word 'a', over the reverse of the range from 0 to 'n' (exclusive).
If a tree "t" is weight-balanced, then the imbalance of the tree "t" is zero.
The lemma named "mat_arc_pos_mult" assumes that: - n is greater than 0, - A is an n by n matrix, - B is an n by n matrix, - A is a matrix with all entries greater than 0, - B is a matrix with all entries greater than 0. It then shows that the product of A and B is also a matrix with all entries greater than 0.
If "t" can be transformed into "t'" through a beta step, then "t" is beta reducible.
If it is eventually the case that the function "f" is non-negative when negated (i.e., "-f(x)" is non-negative) under a filter "F", and it is also eventually the case that the function "g" is non-negative when negated (i.e., "-g(x)" is non-negative) under the same filter "F", then it can be shown that it is eventually ...
The length of the list obtained by dropping elements from the start of list 'xs' while condition 'P' is true, is equal to the length of 'xs' minus the length of the list obtained by taking elements from the start of 'xs' while 'P' is true, if there exists an element 'x' in 'xs' for which 'P' is not true. Otherwise, the...
If a is less than or equal to zero and x is greater than or equal to zero, then the product of a and x (where the multiplication is a complex scalar multiplication) is less than or equal to zero. This is for x which is an ordered complex vector.
For any process 'P' in the Pi calculus and any permutation 'p' of names, the validity of the process 'P' after applying the permutation 'p' is equal to the validity of the original process 'P'.
"thetaSD" is a subset of "Sretr thetaSD".
If you group a list 'l' by a function 'f' and then flatten it using the foldr function with list concatenation, you will get the original list 'l'.
The count of an element 'a' in the multiset 'xs' is equal to the count of 'a' in the vector form of the list 'xs'.
The lemma named "vwalkE" makes the following assumptions: 1. There is a vertex walk 'p' in the graph 'G'. 2. If the set of vertices in 'p' is a subset of the vertices in 'G' and the set of arcs in 'p' is a subset of the arc ends in 'G', and 'p' is not an empty set, then a certain condition 'P' holds. The lemma then sh...
The lemma is named "Refl_above_Above" and has the following assumptions and conclusion: Assumptions: 1. "Refl r" means that the relation "r" is reflexive. 2. "A \<noteq> {}" means that the set "A" is not empty. Conclusion: "Above r A = (\<Inter>a \<in> A. above r a)" means that the set of elements above all elements ...
The third level step 4 refines the second level step 4, given the set of relations R23s, and the parameters Rb, A, B, and gnx.
If a sequence 's' is not an element of the domain of process 'P', and the sequence 's' followed by an element 'c' is an element of the domain of process 'P', then the sequence 's' followed by 'c' is an element of the minimum elements of the domain of process 'P'.
The lemma named "simplex_extremal_le" is about a set S of real numbers. It assumes that S is finite and not empty. It shows that there exist two elements, u and v, in S such that for any two points, x and y, in the convex hull of S, the norm (or length) of the vector from x to y is less than or equal to the norm of the...
If all elements in the list "vl" are valid according to the "nValidV" condition, then all elements in the list resulting from applying the "invTranslateVal" function to each element of "vl" are valid according to the "compValidV" condition.
The lemma named "set_integral_sum" is about a function 'f' that maps from any two types to a type that is both a Banach space and a second countable topology. Given that 'B' is a finite set and for any 'x' in 'B', 'f' of 'x' is set integrable over a measure space 'M' and a set 'A', it shows that the set Lebesgue integr...
If the "Filter" is not an empty list, then the function "C" applied to the reverse of "Filter" converted to a policyR list is equal to the function "C" applied to the "Filter" converted to a FWpolicy list.