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If the continuation of a command "c1" with a state "s" and an index "i" is finished, then the continuation of the sequence of commands "c1" and "c2" with the same state "s" and index "i" equals to the command "c2".
The empty set is a member of the power set of any set B.
The function Phi of tree t is always greater than or equal to zero.
If phi is equivalent to psi, then phi is also language-equivalent to psi.
The operation of taking the opposite of the opposite of a natural transformation results in the original natural transformation. This is a property of the category theory operation "op_ntcf", which reverses the direction of morphisms.
If 'Bot' is less than 'x', then there exists a 'z' such that 'x' is equal to the value of 'z'.
The homomorphism of a scalar multiplication of a matrix is equal to the scalar multiplication of the homomorphism of the matrix.
A set U is open in the final topology generated by a function f from a set X to a family of topological spaces Y if and only if U is a subset of X and for every index i, the preimage of U under the function f i, intersected with the topological space Y i, is open in Y i.
: Assuming that for every element 't' in the first component of the finite set 'xs', the function 'g' applied to 'f' of 't' is less than or equal to 'g' of 't'. And there exists an element 't' in the first component of the finite set 'xs' such that 'g' applied to 'f' of 't' is strictly less than 'g' of 't'. Also, for ...
If K is a subset of the carrier R and x is an element of the carrier R, then the simple extension of K by x is also a subset of the carrier R.
The set of elements in the list resulting from appending list 'ys' to list 'xs' is a subset of the union of the set of elements in list 'xs' and the set of elements in list 'ys'.
A string "str" is in the image of the function "confIDAsStr" applied to the set of "confIDs" if and only if the configuration ID of the string "str" is in the set of "confIDs".
The lemma "spyh_method_via_h_framework_completeness_and_finiteness" is about two finite state machines (FSMs), M1 and M2. These FSMs are fixed and have types 'a, 'b, and 'c for M1 and 'e, 'b, and 'c for M2, where 'a and 'e are linearly ordered. The lemma assumes that both M1 and M2 are observable and minimal. It also...
The top element is in the set of measurable functions from a measure space M to the countable universal set.
The box of the omega power of x unioned with the star power of x applied to n of y is equal to n of y times the box of n of y times x applied to the omega power of x unioned with the star power of x applied to n of y.
An element 'x' is in the set 'I' cut at a value less than or equal to 't' if and only if 'x' is in the set 'I' and 'x' is less than or equal to 't'.
Trying to return a single probability mass function "x" and if that fails, then doing "q", is equivalent to just returning the single probability mass function "x".
The lemma named "lookup_zip_idx_eq" makes the following assumptions: 1. The length of the list "params" is equal to the length of the list "args". 2. The index "i" is less than the length of the list "args". 3. The elements in the list "params" are distinct. 4. The variable "k" is equal to the "i"-th element of the li...
The lemma named "imp_cnj_compat_eqvR" assumes that phi, chi1, and chi2 are formulas. It shows that if chi1 is equivalent to chi2, then the conjunction of phi and chi1 is equivalent to the conjunction of phi and chi2. This is proven.
If a list 'ds' is in the evaluation of a query 'Q' with respect to an interpretation 'I', then for any state 'σ', if the length of 'ds' is equal to the cardinality of the free variables in 'Q' and 'Q' is satisfied by 'I' under the state 'σ' where the sorted list of the free variables in 'Q' is replaced by 'ds', then 'R...
The post-measurement of the Bell state "beta_10" at the second position with measurement outcome 0 is equal to the zeroth unit vector in a four-dimensional space.
The lemma named "transfer_monoidl" assumes that "f" and "g" are isomorphic according to "Derive.iso". It shows that if a left monoid law holds for the operations "NEUTRAL" and "MULT", then a left monoid law also holds for the operations "g NEUTRAL" and "g applied to MULT of f x and f y".
A function that always returns a constant value 'c' is monotone with respect to any order 'ord' and the less than or equal to order.
The value of function d at x is less than or equal to 1.
Any number "a" right-shifted by 0 is equal to "a" itself.
If there exists exactly one x such that property P holds for x, and if property P holds for both x and y, then x must be equal to y.
The lemma named "bet_cop__tsp" assumes that points A, B, C, and X are not coplanar, point P is not equal to point Y, points A, B, C, and P are coplanar, and point X lies between points P and Y. Under these assumptions, it shows that points A, B, and C form a tetrahedron space partition (TSP) with points X and Y.
If from a set of assumptions A, it can be inferred that if q then r, then it can also be inferred from A that if p implies q, then p also implies r.
The set of matrices of a certain size (with number of rows 'nr' and number of columns 'nc') over a type 'a' that forms a commutative ring, forms an abelian group under matrix addition.
If 'a', 'b', and 'c' are elements in the carrier 'G', then the greatest common divisor of the greatest common divisor of 'a' and 'b' with 'c' is equivalent to the greatest common divisor of 'a' with the greatest common divisor of 'b' and 'c'.
If "x" is an upper bound for "S" in relation "R", then "S" is less than or equal to "x".
The set of items less than or equal to "Suc k" in "J" is equal to the union of the set of items less than "k" in "J" and the set of items equal to "Suc k" in "J". Here, "Suc k" represents the successor of "k", or "k+1".
If a command "c" transforms a state "s" into a state "s'", and state "s" is equivalent to state "t" on a set "L c X", then there exists a state "t'" such that the command "c" with the set "X" buried transforms the state "t" into the state "t'", and the state "s'" is equivalent to the state "t'" on the set "X".
For a well-formed heap, type, known pointer, and known pointers, the function 'get_owner_document' works well with the function 'get_root_node'. This is an instance of 'l_get_owner_document_wf_get_root_node_wf'.
Appending list 'ys' to list 'xs' is a prefix of the concatenation of list 'xs' and function 'f' if and only if list 'ys' is a prefix of function 'f'.
If the invariant holds for a binomial queue with a tree 't' at the front, and there is another tree 't'' with a rank less than the rank of 't' and this tree 't'' also maintains the tree invariant, then the invariant will also hold for a binomial queue where 't'' is added to the front of the queue.
In the context of a ring, if for any non-empty set C, where C is a chain of ideals of R, the union of C is in C, then R is a Noetherian ring.
If eta is a negative NOR operation, and eta is also a negative digital NOR operation, then eta is a real double negative operation.
The lemma named "REP_CLS" assumes that "f" is a left adjoint in the bicategory "B". It shows that the representation of "f" is isomorphic to "f" in the bicategory "B".
If the union of v1 and v2 equals some v12, then v2 is less than or equal to v12.
For any two frames 'F' and 'G', and any permutation 'p' of names, applying the permutation to the statement "F implies G" (denoted as "F \<hookrightarrow>\<^sub>F G") is equivalent to the statement "the permutation of F implies the permutation of G" (denoted as "(p \<bullet> F) \<hookrightarrow>\<^sub>F (p \<bullet> G)...
If 'p' is a list in heap 'h', then 'Null' is not in the set of elements of the list 'p' in heap 'h'.
The lemma named "hom_frag_diff" assumes that "h" is a homomorphism from the free Abelian group over set "S" to the free Abelian group over set "T", and that the keys of the poly mappings "a" and "b" are subsets of "S". It shows that the homomorphism of the difference between "a" and "b" is equal to the difference betwe...
The lemma "alive_reach_alive" assumes that "x" is alive in state "s" and that "l" is reachable from "x" in state "s". It shows that the reference of "l" is alive in state "s".
A field 'f' is equal to the tuple of its field name and field class.
The set of acquired elements when there is no pending write (False) in the sequence xs from set A is a subset of the set of acquired elements when there is a pending write (True) in the same sequence xs from set B.
The image of a set under the monic function preserves the property of being a monic set.
If a process P is secure with respect to an interference function I and a domain function D, and if a pair (xs, X) is in the failures of P, then the result of applying the auxiliary function "ipurge_fail_aux_t" with parameters I, D, U, xs, and X is also in the failures of P.
The lemma named "hermitean_congruence" assumes that H is a Hermitian matrix. It shows that the congruence transformation of H with any matrix M results in another Hermitian matrix.
If there is a list relation predicate such that for every pair of elements x and y, both R and S hold true, then there is a list relation predicate where S holds true for every pair of elements x and y.
The lemma named "before_if_dom_children_wf_conform" assumes the following conditions: 1. The domain of children of the node "r" with the set containing the pair "(t1,e1)" is "T". 2. For all vertices "v" in the directed vertices of the node "r" with the set containing the pair "(t1,e1)", "v" conforms to a sequence. 3. ...
If a string "u" is not empty and belongs to the submonoid generated by "G", then the first element of the sequence obtained by decomposing "u" with respect to "G" is in "G".
A one-matrix of size n is upper triangular.
For a list created by replicating a value 'y' 'n' times, all elements in the list are equal to 'y'.
The reference ipurge (an operation that removes certain elements from a list based on a security policy) applied to a list that is the concatenation of list 'xs' and element 'x', with respect to a security policy 'I', a domain assignment function 'D', a user 'u', and a set 'X', is equal to: - the reference ipurge appl...
The initial learning state "s0_learn m" is refined to "l1_learn m" under the relation "R01s".
If 'x' is greater than 0 and less than pi/2, then the tangent of 'x' is greater than 0.
For any given value 'x' of type 'u', the sum of 'x' and the bottom element is the bottom element, and the sum of the bottom element and 'x' is also the bottom element.
The lemma named "paperIDs_mono" assumes that a step from state 's' to state 's'' with action 'a' results in output 'ou'. If a paper ID 'pid' is in the set of paper IDs for conference 'cid' in state 's', then the lemma shows that 'pid' is also in the set of paper IDs for conference 'cid' in the new state 's''.
The lemma named "vcg_disj_lift" assumes two conditions: first, that the command "c" transforms a state satisfying precondition "P" into a state satisfying postcondition "Q", and second, that the same command "c" transforms a state satisfying precondition "P'" into a state satisfying postcondition "Q'". Given these assu...
If the Dickson's condition "dickson_le" holds true for the parameters d, m, s, and t, then the value of d applied to t is less than or equal to m.
If there is a valid edge 'a' in the control flow graph of the main function in the program P and class C0, and the predicate of the kind of 'a' holds for a state 's', and for all variables 'V' in the use set of the source node of 'a' in program P, the state value of 'V' in state 's' is equal to the state value of 'V' i...
If the ordered pair (x, y) is an element of f, then x is an element of the hereditary domain of f.
The function field of a constant function on an empty set is also an empty set.
The lemma P2I [intro] assumes that "r" is a recursive function of two variables and that the evaluation of "r" with inputs "x" and "y" equals the function "f" of "x" and "y". It shows that the function "f" belongs to the set of partial functions of two variables.
If "x" is less than or equal to "y", then "nabla of x" is less than or equal to "nabla of y".
The unsigned natural number representation of 1 as a 10-bit word is equal to 1.
The lemma named "Trg_dom" assumes that "mu" is an arrow. It shows that the target of the domain of "mu" is equal to the target of "mu".
: Assuming that 'A' is a preference profile, and for all 'z' in set 'Z', the count of 'z' in 'A' is greater than or equal to 2. Also, 'X' is an element of 'A' and there exists a 'Y' such that 'Y' is an element of 'A' excluding 'X', 'Y' is a subset of 'X' and 'Y' is not a subset of 'Z'. Under these conditions, it is not...
If a function "f" is contour integrable on "g", then the function that multiplies "f" by a constant "c" at each point "x" is also contour integrable on "g".
The lower bound of the sum of two polynomials 'p' and 'q' at a variable 'v' is equal to the sum of the lower bounds of 'p' and 'q' at the same variable 'v'.
The column vector of the product of a matrix A and a vector v is equal to the matrix A multiplied by the column vector of v.
Theorem 14, Second Existence Theorem 3: Assuming that Q and R are paths, and Q is not equal to R. Also, assuming that x is an event on both paths Q and R, a is an event on path R but not equal to x, and b is not an event on path Q. Furthermore, assuming that there exists a path P from b to an event q on path Q. Then, i...
The factor group of A and H is a group.
If 'a' and 'b' are elements of set 'A', then the join of 'a' and 'b' is less than or equal to the join of 'b' and 'a'.
The lemma AGTT_union_sound2 demonstrates that the language of AGTT1 is a subset of the language of the union of AGTT1 and AGTT2. Similarly, the language of AGTT2 is also a subset of the language of the union of AGTT1 and AGTT2.
If function 'f' is antilinear and function 'g' is also antilinear, then the composition of 'g' and 'f' (g o f) is clinear.
: Assuming that a list of natural numbers 'nl' is not empty, and that 'n' equals 1, and that 'n' is less than or equal to the length of 'nl', and that the last element of the first 'n' elements of 'nl' is 0, it can be shown that there exist lists 'CL' and 'CR' such that: - The list 'CL' appended to a list containing ...
The lemma "dp2_dispatch_int_also_works_for_honest" assumes that a packet fragment "pfragment" is associated with "ainfo" and "fut" in the authenticated segment "auth_seg2" for user information "uinfo". It also assumes that the past of message "m" is empty, the authentication information of "m" is "ainfo", the user info...
If 'y' is less than 'x', then the list of 'a' type words from 'x' to 'y' is an empty list.
The lemma named "list_split" assumes that "n" is less than the length of the list "xs". It then obtains two lists, "xs1" and "xs2", where "xs" is the concatenation of "xs1" and "xs2", "n" is equal to the length of "xs2", and the length of "xs1" is greater than zero.
The set of units in ring R is equal to the set of totatives of n.
The intersection of the union of the complement of x and the complement of y with the complement of x is equal to the complement of x.
A sequence 'u' is primitive if and only if the reverse of 'u' is primitive.
If X is an interval, then the image of X under the unary minus function (which negates each element of X) is also an interval.
A word 'w' is in the language generated by a grammar 'G' starting from a symbol 'S' and the length of 'w' is greater than 1 if and only if there exist symbols 'A' and 'B' such that the pair '(S, Branch A B)' is in the set 'G' and there exist words 'l' and 'r' such that 'w' is the concatenation of 'l' and 'r' and 'l' is...
The step symbol set of an empty set is also an empty set.
The set Sigma is finite.
The lemma named "pi_approx" is defined with a fixed natural number 'n'. It assumes that 'n' is an even number and 'n' is greater than zero. It shows that the absolute difference between pi and the approximation of pi with 'n' is less than or equal to the inverse of 2 raised to the power of (13 times 'n' minus 2).
If two atomic actions 'a1' and 'a2' are interference-free with any other action and a specific condition 'q', then the sequence of these two atomic actions (represented as 'AnnSeq a1 a2') is also interference-free with any other action and the same condition 'q'.
The lemma named "gcd_impl_primitive" assumes that the primitive part of G1 is equal to G1 and the primitive part of G2 is equal to G2. It shows that the implementation of the greatest common divisor for primitive numbers G1 and G2 is equal to the greatest common divisor of G1 and G2.
If a program G is well-formed, if the method "methd" in class C with signature "sig" in program G is some method m, and if the class C in program G is some class y, then the result type of method m is a type in program G.
The set of natural numbers from 1 (Successor of 0) to the successor of the successor of a natural number n is equal to the union of the set of natural numbers from 1 to the successor of n and the set containing the successor of the successor of n.
The lemma named "principalidealI" is about a structure R. It assumes that I is an ideal of R and there exists an element i in the carrier of R such that I is the ideal generated by the set containing only i. Under these assumptions, it shows that I is a principal ideal of R.
For any function F, when it is totalized, it becomes all total.
The product of "x" and the star of "x" is less than or equal to the star of "x".
If a function 'f' is monotone with respect to orders 'orda' and 'ordb', and is continuous with respect to least upper bounds 'luba' and 'lubb' and orders 'orda' and 'ordb', then 'f' is monotonically continuous with respect to 'luba', 'orda', 'lubb', 'ordb'.
The boundary of a set in the filter base of a formula phi is nonempty.
The ordered positive resolvent is complete.
The theorem named "Green's theorem type II" is defined as follows: Given a function F mapping pairs of real numbers to pairs of real numbers, four functions gamma1, gamma2, gamma3, gamma4 each mapping a real number to a pair of real numbers, two real numbers a and b, and two functions g1 and g2 each mapping a real num...