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If the imaginary part of a complex number z is zero, and the absolute value of the real part of z is less than 1, then the value of 1 minus the square of z does not belong to the set of non-positive real numbers.
The second version of observability (observable2) with parameters ob and P is equivalent to the first version of observability (observable) with the same parameters ob and P.
If x is less than or equal to 1 in the sigma algebra and y is also less than or equal to 1 in the sigma algebra, then the product of x and y is equal to the parallel product of x and y.
The lemma named "filter_bool_list_inj_aux" assumes that the length of list bs1 is equal to the length of list xs, the length of list xs is equal to the length of list bs2, and all elements in list xs are distinct. It shows that if the result of filtering list xs by list bs1 is equal to the result of filtering list xs b...
The lemma named "one_time_pad_units" assumes two conditions: 1. The function 'f' is injective (or one-to-one) on the set of all numbers less than 'q' excluding zero. 2. The image of the set of all numbers less than 'q' excluding zero under the function 'f' is equal to the set of all numbers less than 'q' excluding ze...
If x is a transitive set and y is a transitive set, then the intersection of x and y is also a transitive set.
If W is an element of the set WWa (defined by D, g, and a), then a is an element of W.
An ordered map containing a single pair of elements (x, y) is equivalent to a list containing the same pair (x, y).
A match list, denoted by "ms", matches with a given action "a" and packet "p" under a certain condition "γ", if and only if there exists a match "m" within the set of "ms" that matches the condition "γ", action "a", and packet "p".
A map 'f' is continuous from topological space 'X' to 'Y' if and only if for all subsets 'S' of the topological space 'X', the image of the closure of 'S' under 'f' is a subset of the closure of the image of 'S' under 'f' in 'Y'.
The one-dimensional isomorphism of the sum of two elements (a and b) is equal to the sum of the one-dimensional isomorphisms of these two elements individually.
If two lists "xs" and "ys" are equal, and for every element "x" in the set of elements of "ys", the predicate "P" applied to "x" is equal to the predicate "Q" applied to "x", then the result of taking elements from the list "xs" while they satisfy the predicate "P" is equal to the result of taking elements from the lis...
The intersection of the multiset obtained by adding an element 'a' to multiset 'A' and the multiset obtained by adding the same element 'a' to multiset 'B' is equal to the multiset obtained by adding 'a' to the intersection of multiset 'A' and 'B'.
Prepending any list "xs" to an empty lazy list results in a lazy list of "xs".
If a unique factorization exists for the polynomial obtained by scalar multiplication of a polynomial 'f' by 'd' (denoted as 'smult d f'), and if the multiplication of 'di' and 'd' under a certain function 'M' equals 1, then a unique factorization also exists for the original polynomial 'f' with the factors being 'c * ...
For a given initial formula (F0), a set of decision variables, an initial state (state0), and a final state (state), under the assumptions that the variables of the initial formula are a subset of the decision variables, the initial state is the initial state of the formula, the final state is derived from the initial ...
The lemma named "rec_disjointify_is_disjoint" assumes that both A and B are in the recursive disjointify of m and f. If A is not equal to B, then the intersection of A and B is an empty set.
The lemma named "node_addressD" assumes that a pair, consisting of a state sigma and a node with index i, state zeta, and a set of rules R, is operationally reachable from a node with index ii, a set of transitions T, and a set of initial states R_i, under the conditions S and U. It then shows that the index i of the n...
In the context of a well-founded directed graph, if there is an Euler trail from a vertex 'u' back to itself and 'v' is a vertex in the graph 'G', then the in-degree of 'v' in 'G' is equal to the out-degree of 'v' in 'G'.
For any two environments E and F, and any two sequences t and s, the semantics of the composition of t and s at a point n in the combined environment of E and F is equal to the composition of the semantics of t and the function from m to the semantics of s at m, evaluated at n in the same combined environment.
The lemma named "subseq_Higman_antichains" assumes that for all elements 'x' and 'y' in a set 'A', where 'x' and 'y' are lists of a finite type 'a', if 'x' is not equal to 'y', then 'x' is not a subsequence of 'y' and 'y' is not a subsequence of 'x'. Under this assumption, it shows that the set 'A' is finite.
For any integer 'a' not equal to zero, 'a' divided by 'b' equals negative 'a' if and only if 'b' equals negative one.
The ability to lock (may_lock) the lock after it has been locked (lock_lock l t) by a thread (t) is equivalent to the ability to lock the lock before it has been locked (may_lock l t).
The natural term comparison of the degree of a term order is equal to the degree comparison of the natural term comparison of the term order.
If a problem is a valid SAS+ problem and there is a parallel solution for this problem in STRIPS semantics, then all operators in the inverse transformation of the problem and its solution are indeed operators of the original SAS+ problem.
: Assuming that for every natural number 'n', 'X n' is integrable with respect to measure 'M', and 'F' is a filtration of 'M'. Also, for every natural number 'm', 'F m' is a sigma-finite subalgebra of 'M' and 'X m' is Borel measurable with respect to 'F m'. Furthermore, for every natural number 'n', almost everywhere ...
The lemma named "prv_cnj_imp_monoR5" assumes that four expressions, namely phi1, phi2, chi1, and chi2, are all formulas. It shows that if it is provable that if phi1 implies chi1, then it is also provable that if phi2 implies chi2, then phi1 implies that if phi2, then chi1 and chi2 are both true.
The length of a list with adjacent duplicates removed is less than or equal to the original length of the list.
If 'x' and 'y' are invertible and both belong to the set 'M', then the product of 'x' and 'y' is also invertible.
The lemma shows that the set of partial derivatives of a language A with respect to a regular expression r is finite.
The observable reach of "a0n" is a subset of "a0n_inv1_niagree".
If a certain event 'evs' is in 'p2' and 'A' is not in 'bad', then if the encryption of the hash of 'X' with 'A's private key is in the parts that spies can see in 'evs', it implies that there exists some 'B' and 'Y' such that 'A' says to 'B' something 'Y' in the set of events 'evs' and the signature of 'A' on 'X' is in...
Lemma 5: Assuming that 'r' reduces to 'r' prime in a leftmost reduction (as indicated by 'lred: "r \<rightarrow>\<^sub>l r'"), it can be shown that 'r' is weakly normalizing (as indicated by "WN r") and 'r' prime is in normal form (as indicated by "NF r'").
The lemma named "card_mostly_constant_maps" assumes that "y" is an element of set "B". It shows that the cardinality (or size) of the set of all functions "f" whose range is a subset of "B" and for which "f(x)" equals "y" for all "x" greater than or equal to "n", is equal to the cardinality of set "B" raised to the pow...
The lemma named "gen_isEmpty" is assumed to be a general algorithm priority tag. It also assumes that the generic operation "ball" corresponds to the operation "Ball" with a relation that maps from a key relation "Rk" to an identity relation, and this whole mapping is itself an identity relation. The lemma shows that t...
The lemma "postdominate_implies_path" assumes that "n'" postdominates "n". It then obtains a sequence "as" where there is a path from "n" to "n'" following the sequence "as".
The lemma named "fun_bit_count_est" assumes two conditions. First, function 'f' is extensional over the set 'xs'. Second, for any 'x' in the set 'xs', the bit count of 'e' applied to 'f' of 'x' is less than or equal to 'a'. Given these assumptions, it shows that the bit count of 'e' applied to 'f' over the set 'xs' is ...
If for all k less than or equal to the successor of n, J k is an ideal of R, then the set of all I such that there exists a k less than or equal to the successor of n for which I equals J k, is equal to the set of all J k such that k is an element of the set of all j less than or equal to the successor of n.
A list "xs" is in the set of all suffixes of "xs" if and only if "xs" is not an undefined or bottom value.
If a function 'f' is integrable over a measure 'M', and another function 'g' is measurable from 'N' to 'M', then the function that applies 'f' to the output of 'g' for any input 'x', is Borel measurable over 'N'.
If a sequence 'e' is single-input single-output (siso) and sequence 'c2' is weakly bisimilar to sequence 'd2', then the sequence composed of 'e' followed by 'c2' is weakly bisimilar to the sequence composed of 'e' followed by 'd2'.
A function 'f' is 'n' times differentiable on a set 'S' if for every element 'x' in 'S', there exists a subset 'T' such that 'x' is in 'T', 'T' is open, and the function 'f' is 'n' times differentiable on 'T'.
The operation "oplus" is commutative over the interval I and the elements a and b. That is, adding a to I and then b is the same as adding b to I and then a.
If time 't' is within the existence interval of 't0' and 'x0', then 't0' is in set 'T' and 'x0' is in set 'X'.
The set of differences between each element of a set S and a specific element a (denoted as x - a) forms a subspace, given that S is an affine set and a is an element of S.
For a function 'f' mapping from type 'a' to a real normed vector type 'b', and given a set 'Ω' that intersects with the space of a measurable space 'M' and belongs to the sets of 'M', the function 'f' is Borel measurable in the restricted space of 'M' and 'Ω' if and only if the function that maps 'x' to the product of ...
"bindU" is equal to "bind".
The composition of theta and i under substitution is equal to the function substitution to index substitution of the composition of the index substitution to function substitution of theta and the index substitution to function substitution of i.
The lemma "AE_embed_measure" assumes that the function "f" is injective. It shows that for almost every "x" in the measure space obtained by embedding "M" with "f", the property "P" holds for "x" if and only if for almost every "x" in "M", the property "P" holds for "f(x)".
If a binary search tree 't' is a valid binary search tree, then the binary search tree resulting from deleting element 'x' from 't' is also a valid binary search tree.
The cardinality (or the number of elements) of the intersection of incoming arcs and outgoing arcs of a graph G at a vertex w is even. This is denoted as "even (card ?S)" where ?S represents the intersection of incoming and outgoing arcs.
If 'a' is a member of 'b', and 'x' is not in the domain of the function 'C' applied to 'b', then 'x' is not in the domain of the function 'C' applied to 'a'.
The set of keys of the polynomial obtained by decreasing the grading of polynomial 'p' by 'd' and 'n', is a subset of the set obtained by applying the decreasing grading term function to 'd', 'n' and the keys of polynomial 'p'.
For any extended real numbers 'a', 'b', and 'c', if 'a' is less than or equal to 'b', then 'a' plus 'c' is also less than or equal to 'b' plus 'c'.
If x and y are nodes in the product of automata A and B, then y is a member of the product of the depth-first search on A and B starting from x if and only if the pair (x, y) is in the reflexive transitive closure of the successor relation on the product of the successors of A and B.
The size of set A plus the size of set B is equal to the size of the union of set A and B plus the size of the intersection of set A and B.
The function "board" with parameters n and m is equal to the function "board_exec" with the same parameters n and m.
"the number X is not equal to the hash of X'".
The lemma named "has_ticket_simps" states the following conditions: 1. There is no ticket in the initial state for any key 'k'. 2. A key 'k' has a ticket in the waiting state if and only if 'k' is equal to 'k''. 3. A key 'k' has a ticket in the hold state if and only if 'k' is equal to 'k''. 4. A key 'k' has a ticket ...
The lemma named "sub_code" states the following: 1. Subtracting one from one results in zero. 2. Subtracting one from a binary number ending in zero results in the natural number equivalent of a binary number ending in one less than the original number. 3. Subtracting one from a binary number ending in one results in ...
If a term 't' is ground (i.e., it contains no variables), then the set of variables in the term 't' is empty.
The parallel product of x and y is equal to the sum of the parallel product of x and the nu of y, the parallel product of the nu of x and y, and the tau of the parallel product of x and y.
The relation "rel_T" is equal to the equality relation when all its arguments are the equality relation.
If x is a vector, then the intersection of x and y is less than or equal to the transpose of x multiplied by y.
The function composition of 'return' and 'E2' in the Isabelle language is in correspondence with the function composition of 'RETURN' and 'E2' in the assertion language, given that it is mapped from the 'd'-th power of 'A' to the enumeration assertion of 'A'.
The lemma, in the context of TC1, is named "compT_LT_prefix". It consists of two parts: 1. For any E, A, and ST₀, if the pair (ST, LT) is in the set produced by the function compT with parameters E, A, ST₀, and e, and if the function B applied to e and the size of E holds, then P implies that the pair (ST, LT) is less...
If the row index 'i' is less than the number of rows in matrix 'A', the column index 'j' is less than the number of columns in matrix 'A', 'i' is less than 'low', and 'i' is greater than or equal to 'up', then adding a row to multiple rows indexed from 'low' to 'up' in matrix 'A' does not change the element at position...
If the function "pre_splitFace" holds true for the graph 'g', vertex 'v', vertex 'a', face 'f', and vertices 'vs', and if the graph 'g' possesses the properties of a minimal graph, then the graph resulting from the "splitFace" operation on graph 'g', vertex 'v', vertex 'a', face 'f', and vertices 'vs' will also possess...
If the first element of pair P is not None, then the result of the function OptionOverride on P is equal to the value of the first element of P.
The lemma named "omega_simps" demonstrates the following: - "omega" is an arrow. - The source of "omega" is "a". - The target of "omega" is "a". - The domain of "omega" is the horizontal composition of "w" with itself. - The codomain of "omega" is "w".
If the length of the function sequence 'fs' is greater than zero, and 'fs' is a function tuple of 'Qp' with 'n' elements, and 'x' is an element of the carrier of the power of 'Qp' with 'n+l' elements, and 'i' is less than 'l', then the 'i'th element of the tuple partial image of 'n' and 'fs' and 'x' is equal to the 'i'...
The lemma named "subst_preserves_good" assumes that both X and Y are good. It shows that the substitution of X for x in Y, within the context of xs, is also good.
The lemma named "filterlim_tendsto_add_at_top_iff" assumes that function f tends to c under filter F. It shows that the limit of the sum of functions f and g as x approaches F towards positive infinity is equivalent to the limit of function g as x approaches F towards positive infinity. Here, both f and g are real-valu...
If x is not equal to y and the atomic formula with x is independent of u, then substituting x with t in formula A and then substituting y with u in the result is equal to first substituting y with u in formula A and then substituting x with the result of substituting y with u in t.
The lemma named "tensor_char" assumes that "f" and "g" are arrows. It shows that the tensor product of "f" and "g" is equal to the result of making an arrow from the tensor product of the representations of "f" and "g" in the free monoidal category.
If "n" is less than or equal to the length of "xs" and the balance of "n" and "xs" equals the pair "(t,zs)", then the minimum height of "t" is equal to the natural number of the floor of the logarithm base 2 of "(n + 1)".
The key parts of an empty set is an empty set.
If "c" is not equal to zero, then the polynomial "p" multiplied by the constant "c" evaluated at "k" is equal to zero if and only if the polynomial "p" evaluated at "k" is equal to zero. This is valid in an integral domain.
: 1. The matrix created from a list of rows "vs" with "n" columns is in the carrier matrix with the number of rows equal to the length of "vs" and "n" columns. 2. The row dimension of the matrix created from a list of rows "vs" with "n" columns is equal to the length of "vs". 3. The column dimension of the matrix creat...
For any hyperdual number 'a' (where 'a' belongs to a set that has the properties of an inverse, a commutative ring with unity, and a division ring), the inverse of the negation of 'a' is equal to the negation of the inverse of 'a'.
If K is a subset of the carrier R, then K is a subalgebra of the carrier R in R.
Lemma 3.7.i: If F is an element of filters, then F is equal to the set of all elements 'a' such that 'a' is congruent to 1 modulo F.
There exist a heap 'h', a node pointer 'node_ptr', a slot option 'slot_opt', and a heap 'h'', such that the heap 'h' is well-formed, the type of 'h' is well-formed, and 'h' contains known pointers. Furthermore, the assigned slot of 'node_ptr' in 'h' results in 'slot_opt' and 'h''. However, it is not the case that the w...
For any two processes P and Q in the pi calculus, the sum of P and Q is early bisimilar to the sum of Q and P.
The lemma named "iteration_step_ex" assumes the following: 1. "ES" is invariant. 2. The pair "(X, xrhs)" is an element of "ES". 3. The condition "Cond ES" holds. Under these assumptions, it shows that there exists an "xrhs'" such that the pair "(X, xrhs')" is an element of the iteration of "X" over "ES".
For any natural number x, if a is greater than 0 and less than or equal to b, then x raised to the power of a is less than or equal to x raised to the power of b.
Applying the function "unitK" to any value "v" and then applying the result to any continuation "c" is equivalent to applying the continuation "c" directly to the value "v".
The set of dyadic numbers is equal to the union over all sets K, where for each number k less than K, it takes the union over all numbers m, and includes the number obtained by dividing the natural number m by 2 raised to the power of k.
For any function 'f' and any list composed of an element 'x' followed by a list 'xs', the union over all pairs of lists '(ys, zs)' resulting from splitting the list '(x # xs)' and applying the function 'f' to these pairs, is equal to the union of the function 'f' applied to the pair of an empty list and the list '(x # ...
The edge set E is finite.
The tuple (a, b, ps) is related to the tuple (c, d, p followed by ps) in the well-founded relation "factorize_rat_poly_main_wf_rel".
If a substitution 'r' is idempotent, and 's' is a unifier for 't' and 'u' under the substitution 'r', then for any substitution 'q' that also unifies 't' and 'u' under 'r', the composition of 's' and 'q' is equal to 'q'. Under these conditions, the composition of 'r' and 's' is also idempotent.
The lemma named "powa_less_mono" assumes that "r" is less than "s" and "a" is greater than 1. It shows that "a" raised to the power of "r" is less than "a" raised to the power of "s".
If the transformation of a property P (denoted as t P) is less than or equal to P, and if P is sound, then the least fixed point of the transformation (denoted as lfp_exp t) is less than or equal to P.
The lemma "eqExcPID2_step_out" assumes the following conditions: 1. "eqExcPID2 s s1" - states s and s1 are equivalent except for PID2. 2. "step s a = (ou,s')" and "step s1 a = (ou1,s1')" - the state s transitions to state s' with output ou when action a is taken, and similarly for state s1. 3. "reachNT s" and "reach s...
The function "lunstream'" applied to the product of the "unfold_llist_prod" function with parameters "stop", "head", and "tail" is equal to the "unfold_llist" function with the same parameters "stop", "head", "tail" and an additional parameter "s".
The star of x multiplied by the omega of x multiplied by the bottom element unioned with 1 is equal to the omega of x multiplied by the bottom element unioned with the star of x.
The lemma "setops_sstp_subst" assumes that the intersection of the set of bound variables of a stateful strand protocol action 'a' and the domain of a substitution 'θ' is empty. Under this assumption, it shows that the set of operations of the substituted stateful strand protocol action 'a' by 'θ' is equal to the set o...
The function "typeof_h" applied to heap "h" is equal to the function "typeof_h" itself. This lemma is heap independent and can be used interchangeably.
If 'a' and 'b' are less than infinity, then the inverse of the product of 'a' and 'b' is equal to the product of the inverse of 'a' and the inverse of 'b'. This is in the context of extended nonnegative real numbers.