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If a simplex tableau "s" is in a valid state, and if "x_i" is in the left variables of "s", and "x_j" is in the right variables of the left variable "x_i" in "s", then the left variables of the tableau after a pivot operation on "x_i" and "x_j" in "s" is equal to the left variables of "s" minus "x_i" plus "x_j".
: Assuming: - A' is a matrix in the carrier space of dimensions m by n, - a is less than m, - n is greater than 0, - A is defined as A' appended by D times the identity matrix of size n, - The element at position (a,0) in matrix A is not 0, - m is greater than or equal to n, - i is an element in the set xs, - all elem...
If the graph 'g' maintains its invariance, then the function 'alpha edges aux' applied to 'g' is symmetric.
The product of the identity function on set A, multiplied by negative one to the power of the cardinality of set E, multiplied by a to the power of the cardinality of set A, multiplied by negative one to the power of the cardinality of set E, is equal to the product of the identity function on set A multiplied by a to ...
If the length of a list of numbers is equal to 2 raised to the power of l, and 2 raised to the power of l is less than or equal to n, then the Inverse Fast Number Theoretic Transform (IFNTT) of the list of numbers is equal to the Inverse Number Theoretic Transform (INTT) generalized to the length of the list of numbers...
If the maximum degree of a node 'r' in a set 'xs' is less than or equal to 1, and 'x' is an element of the finite set 'xs', then 'xs' is a singleton set containing only 'x'.
If function 'f' is negligible and 'c' is greater than zero, then 'f' is an element of the set of functions that grow slower than the inverse of 'x' raised to the power of 'c'.
The destination port of a packet, which is represented as a tuple (a,d,x,e) of type (DatatypePort, 'beta), is equal to the second element of the tuple x.
If 'F' is a functor from category 'J' to the composition of categories 'A' and 'B', and 'a' is an arrow in category 'A', then the functor 'F' applied at the domain of 'a', the functor 'F' applied at the codomain of 'a', and the functor 'F' applied at 'a' form a natural transformation from category 'J' to category 'B'.
The lemma "foreach_refine" assumes that the set "adjs" is equal to the set of pairs (v,d) such that (u,v) is in the set of edges of graph g and the weight of the edge {u,v} is d. It shows that the function "foreach" applied to u, adjs, and the pair (Q,π) is equal to the pair (Qinter Q π u, π' Q π u).
The kth power of r under the Kleene star operation is less than or equal to the product of r and the kth power of r under the Kleene star operation.
The function relation, when both its arguments are equality relations, is itself the equality relation.
The inverse of the subclass relation in E is well-founded.
The lemma 'cat_comma_Obj_def' defines the set of objects in the comma category of two functors, GG and HH. This set consists of triples [a, b, f], where a is an object in category AA, b is an object in category BB, and f is a morphism from the object mapped by GG from a to the object mapped by HH from b in category CC.
The integral with respect to the measure of a set A in the probability mass function is equal to the sum of the function f over the set A divided by the cardinality of the set A.
The lemma named "partial_equiv_iff" shows that a partial equivalence relation on a set A with relation R exists if and only if there exists a subset A' of A such that A' is an equivalence relation with R.
If a cryptogram containing the agents A and B, a service key, and a timestamp, encrypted with B's shared key, is part of what the spies can see, and if B is not the Ticket Granting Service (Tgs) and is not compromised (not in 'bad'), and the events are part of the Kerberos IV protocol, then there exists a timestamp and...
The lemma named "map_o_incl_eq" shows that mapping the inclusion function of A over t is equal to the function F applied to t.
The lemma named "latestAct_prop" assumes that there exists a number 'n'' less than 'n' such that 'nid' is parallel at time 'n'' in trace 't'. It shows that 'nid' is parallel at time 'latestAct nid t n' in trace 't' and that 'latestAct nid t n' is less than 'n'.
: 1. The union of the universal set and any set 'a' is the universal set. 2. The union of any set 'a' and the universal set is the universal set.
If S is a subgroup of G, N is a normal subgroup of G, and N is a subset of S, then N is a normal subgroup of the supergroup formed by restricting the carrier of G to S.
A property P holds successively for the reversed list of xs if and only if the property P holds in reverse order for the list xs.
The lemma named "vars_of_cl_lem" assumes that "L" is an element of "C". It shows that the variables of the literal "L" are a subset of the variables of the clause "C".
'x' is either a 'Check' or an 'Assignment' if and only if 'x' is 'Check_or_Assignment'.
If a set A is equal to the internal direct product of a group G and a set S over an index set I, every element in the index set I is a subgroup of G, and S and I form a complementary family, then A is an internal direct product of S and I.
If x is independent of y, then deleting x from the substitution of y by v in sigma is the same as first deleting x from sigma and then substituting y by v.
The lemma named "set_UNIV_bin_lcp_pref" assumes that the set of all elements "w" is equal to the universal set. It shows that the product of alpha and the binary code of the head of "w" is less than or equal to the prefix of the function "f" applied to "w".
The complement of the bottom element is the top element.
If the invariant "S" holds, then the map iterator function "iteratei" applied to "S" is equivalent to the abstraction function "\<alpha>" applied to "S".
The lemma "mult_poly_by_monom_simps" states three things: 1. Multiplying a polynomial by a monomial with an empty set is equal to an empty set. 2. Multiplying a polynomial by a monomial with the sum of two sets (ps and qs) is equal to the sum of the polynomial multiplied by the monomial with each set separately. 3. Mu...
If the system is in a state where it is reachable by B (given the conditions c, n, and ts), and if the time t is less than N, and if the ticket lock status of t is tk, then the ticket tk must be equal to the condition c.
There exists a command 'c''', such that 'c' is equal to 'While b c''' and 'c''' is a subset of 'c'', if 'c' is a subset of 'While b c''.
The top level summands of the sum of "r" and "s" under the ACI_nPlus operation are equal to the top level summands of the direct sum of "r" and "s".
The universal map of the opposite co-functor is equal to the opposite of the universal map of the functor.
If a function 'f' has an integral 'k' over a set 'S', then the function that multiplies 'f' by a constant 'c' has an integral that is 'c' times 'k' over the same set 'S'.
The finite power series numerator of a linear homogeneous recurrence sequence, given the difference between the length of the sequence and the length of the coefficients plus one, and the coefficients and function, is equal to the finite power series numerator of the same sequence, given the same parameters, where the ...
The net asset value of alpha is equal to the sum of pi of alpha for all i up to the shortest period of alpha.
The lemma named "triangle_mid_par" assumes that points A, B, and C are not collinear, point P is the midpoint of line segment BC, and point Q is the midpoint of line segment AC. Under these assumptions, it shows that line AB is strictly parallel to line QP.
If the minimum variable not in bounds of a state 's' is some 'x', then 'x' is in the set of left-hand side variables of the tableau of 's'.
If a node is left-aligned with a list "ts" appended with a tuple of "sub" and "sep", and "t" and "u" are aligned with "sep" and "lt" and "a" respectively, and "a" and "rt" are aligned with "u", then the node is left-aligned with the list "ts" appended with the tuples "(sub,sep)" and "(lt,a)", and "rt" and "u".
If set S is a subset of set A and element x is in set S, then the greatest lower bound of set S in the closure cl is related to element x in relation r.
The lemma named "con_prfx_composite_of" assumes that "w" is a composite of "t" and "u". It shows that if "t" is concatenated with "w" and "w" is concatenated with "v", then "t" can be concatenated with "v".
An order relation "R" holds for an option "x" and "Some b" if and only if for all "a", if "x" equals "Some a", then the relation "R" holds between "a" and "b".
The lemma named "span_mem" assumes that "E" is a subset of the carrier "V" and "u" is an element of "E". It shows that "u" is an element of the span of "E".
The lemma named "insert_spec_none" shows that the set resulting from the operation "insert_spec" on a set "xs" with a pair "(oid, None)" is equal to the union of the set "xs" and the set containing "oid".
If 'u' is an element of the Complex Basis, then the norm of 'u' is equal to 1.
The lemma 3.6 assumes that 'r' is a transitive relation, 'τ' is not an empty list, and 'D' is a decreasing diagram with parameters 'r', 'τ', 'σ', 'σ'', and 'τ''. It shows that the pair consisting of the sum of the relation 'r' restricted to 'σ'' and the relation 'r' restricted to 'upsilon', and the sum of the relation ...
The lemma named "sources_are_Cong" assumes that "a" is in the sources of "t" in "R" and "a'" is also in the sources of "t" in "R". It shows that "a" is approximately equal to "a'".
The scalar multiplication of the fragment by zero equals zero.
If the result of merging a node "r" with a set containing a single tuple "(t1,e1)" equals a node "r" with a set containing a single tuple "(t2,e2)", then the root of "t1" must be equal to the root of "t2".
If a formula φ is provable to be false, then it is provable that if any formula χ is true, then it is not the case that if χ is true then φ is true. This is under the assumption that both φ and χ are well-formed formulas.
For any distributive lattice type 'a, the infimum (greatest lower bound) of 'x' and the supremum (least upper bound) of 'y' and 'z' is equal to the supremum of the infimum of 'x' and 'y' and the infimum of 'x' and 'z'.
The carrier of the intersection of tree 't' and tree 't'' is a subset of the intersection of the carrier of tree 't' and the carrier of tree 't''.
This lemma, named "bangInduct", is about the behavior of a parallel composition of a process and its replication in the pi calculus. It takes a function F, a process P, a residual Rs, and a constant C as inputs. The lemma assumes that the process P can transition to the residual Rs. It then defines several cases for ...
For any real normed vector 'a' and 'd', the filter map of 'x' subtracted by 'd' at 'a' is equal to 'a' subtracted by 'd' at 'x'.
The lemma named "chine_composite" shows that the chine of the composite of two morphisms mu and nu is equal to the whiskered product of the leg0 of mu and the leg1 of nu.
The lemma named "partial_vector_derivative_works" assumes that the function F is partially vector differentiable at a point 'a' in the direction of a base vector. It shows that F has a partial vector derivative at the point 'a' in the direction of the base vector, and this derivative is equal to the partial vector deri...
"k is not equal to 0".
If the selectivity function 'f' is reasonable, then the auxiliary list selectivity of 'f' with respect to 'x' and 'y' is less than or equal to 1 and greater than 0.
If you convert a word to an integer and then convert that integer back to a word, you will get the original word.
The implementation of the function "hm_isEmpty" is an empty imperative hashmap, as defined by the function "imp_map_is_empty is_hashmap".
Given the assumptions that the composition of the relations "ns" and "s" is a subset of "s", "ns" is reflexive, and "s" is transitive, then the pair (X, Y) is in the multiset relation "mult2_s" of "ns" and "s" if and only if there exist multisets X1, X2, Y1, Y2 such that X is the union of X1 and X2, Y is the union of Y...
If a command 'c' takes state 's' to state 's'' and 'c' is well-typed at security level 0, and if state 's' is equivalent to state 't' at security level 'l', then there exists a state 't'' such that the command 'c' takes state 't' to state 't'' and state 's'' is equivalent to state 't'' at security level 'l'.
The function which maps any two elements x and y to the condition "x equals c" is right total.
If a relation "r" is reflexive on a set "A", then "A" is equal to the domain of the relation "r".
The lemma named "sig_red_zero_monom_mult" assumes that "r" significantly reduces to zero under the single regular sequence "F". It shows that the significant reduction to zero under the single regular sequence "F" also applies when "r" is multiplied by the monomial "c" and "t".
If a function 'f' is in the order of 'g' plus the order of 'h', then the function that maps 'x' to the sum (for 'y' in 'A' of 'x') of 'l' of 'x' and 'y' times 'f' of 'k' of 'x' and 'y', is in the order of the function that maps 'x' to the sum (for 'y' in 'A' of 'x') of 'l' of 'x' and 'y' times 'g' of 'k' of 'x' and 'y'...
The multiplicity of any prime number 'p' in the successor of 0 (which is 1) is 0. This means that no prime number 'p' divides 1 more than once.
If 'x' is a hyperreal number and 'x' is an infinitesimal, then 'x' must be equal to zero.
The lemma named "runit_naturality" assumes that "a" is an identity. It shows that the right unit of the codomain of "a" composed with "a" tensor the identity equals "a" composed with the right unit of the domain of "a".
"for all properties X of individuals, if X is a positive property, then it is possible that there exists an individual that has property X."
If "i" is less than the dimension of vector "c", then "i" is in the mapping of column to block of "c" if and only if the "i"-th element of "c" equals 1.
The nth power of a negative numeral as a hyperreal number is equal to the hyperreal number derived from the nth power of the same negative numeral.
If the reference get function applied to h and q equals Empty, then the function list_of' applied to h, q, and xs is equivalent to xs being an empty list.
If 'l' is greater than zero and 'X' is exponentially distributed with rate 'l', then the variance of 'X' is equal to 1 divided by the square of 'l'.
The theorem named "list_order_consistent" assumes that "some_ops" and "all_ops" are both list operations, and that the set of "some_ops" is a subset of the set of "all_ops". If the list order of "some_ops" is such that x comes before y, then the theorem shows that the list order of "all_ops" will also have x come befor...
The lemma named BOLZANO: It fixes a predicate P that takes two real numbers. It assumes three things: 1. The real number 'a' is less than or equal to the real number 'b'. 2. For any real numbers 'a', 'b', and 'c', if P holds for 'a' and 'b', and 'b' and 'c', and 'a' is less than or equal to 'b', and 'b' is less than or...
If the following conditions are met: 1) The requirements M are valid, 2) The graph with nodes V and edges E is well-formed, and 3) The set of edges E' is a subset of the set of edges E, then the union of the offending flows obtained from the graph with nodes V and edges E' is a subset of the union of the offending ...
A new type variable "n" is in the type variable "TVar".
If the following conditions are met: 1. A phase 1 or phase 2 write operation transitions the system from state 's' to state 's'' for process 'p' and disk 'd' (as per the assumption "act: HPhase1or2Write s s' p d"). 2. The inner invariant 5 holds for the system state 's' and process 'q' (as per the assumption "inv: HIn...
If the authentication key "authK" is not used in the event list "evs" and "evs" is in the Kerberos IV gets, then the function "AKcryptSK" with parameters "authK", "servK", and "evs" cannot be true.
If the following conditions are met: 1. The process 'p' satisfies the assertion that for any state 'σ', the arrival of a message 'I' leads to a property 'P'. 2. The pair of states '(σ, s)' is reachable in the process 'p' with the other states satisfying the condition 'S' and the incoming messages being 'I'. 3. The tra...
: given a function 'f' that is a very special variable (vsv) and the domain of 'f' is 'I', if for every 'i' in 'I', the value of 'f' at 'i' is in the set of arrows of the digraph 'AA' at 'i', then 'f' is in the product of the set of arrows of the digraphs 'AA' at each 'i' in 'I'.
The lemma named "mk_linear_orders_range" assumes that "r" is in the set of linear orders made from "C" and "B". It also assumes that "B" is a subset of "A". Under these assumptions, it shows that the set "r" is a subset of "B".
The bitwise AND operation between the word-length number 1705 and 1 equals 1.
For any two extended non-negative real numbers 'x' and 'y', the function 'e2ennreal' applied to the difference of 'x' and 'y' (when 'x' and 'y' are converted to extended real numbers using the 'enn2ereal' function) is equal to the difference of 'x' and 'y'.
The closure of a set C is subset-closed.
The open interval between two real numbers 'a' and 'b' is path-connected.
For a probability mass function, if we map the return function to a set A and then join the resulting probability mass function, we will get back the original set A.
If sets H and J are finite and are subsets of the carrier of the group, and the cardinality of the set multiplication of H and J equals the product of the cardinalities of H and J, then for any two distinct elements a and b from H, the intersection of the set obtained by multiplying each element of J by a and the set o...
The type of the function "Fn" with the tag "Tag" and the type list "Tl" is equal to "sigma".
The lemma named "po_on_subset" assumes that set A is a subset of set B and that there is a partial order on set B with respect to predicate P. It shows that there is also a partial order on set A with respect to the same predicate P.
The signed integer of negative one equals negative one.
If x is in the range from pi/2 to 3*pi/2, then the cosine of x is less than or equal to zero.
The lemma named "term_cases" has three cases: 1. If the term is a constant (denoted by "const name"), then the result is "if_const name". 2. If the term is free (denoted by "free name"), then the result is "if_free name". 3. If the term is an application (denoted by "app t u"), then the result is "if_app t u". In ot...
If the inverse of the subclass relation in a graph G is well-founded, then the field of the composition of G and a class C is equal to the field of G and the class C.
The lemma named "image_mset_respects_domain_permutation" shows that the function which applies "image_mset" to a function "f" and a multiset created from set "A", respects the domain permutation between set "A" and set "B".
The resource of an oracle operating in parallel (with states s1 and s2) is equivalent to the parallel resource of the resources of two separate oracles (oracle1 with state s1 and oracle2 with state s2).
Lemma summable_sum: "If for all i in I, the function f(i) is summable, then the function that maps n to the sum of f(i) for all i in I is also summable."
If process P is less than or equal to process Q, then the domain of Q is a subset of the domain of P.