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If 'b' is an object in the category, then the domain of the object map of the contravariant functor Hom from 'b' is equal to the objects of the opposite category.
If a list of elements 'xs' is a subtype of 'E', if the Java program (prg E) is well-formed, if the statement 'st' is well-typed in 'E', and if the Java program (prg E) can transition from 'xs' to 'xs' through the statement 'st', then the list of elements 'xs' after the transition is also a subtype of 'E'.
If two multisets are related by the "MM_Rel" relation, then the relation between their corresponding matrix modulus is equality. This is a transfer rule used for reasoning about multisets in the context of matrix modulus.
The function "gen_refine.mem\<^sub>A_of" applied to any element "x" equals "x" itself.
"For all x, x is necessary."
The lemma named "sublist_not_mid" assumes that "U" is a sublist of the list formed by concatenating "A", "V", and "B" in that order, that the intersection of the sets of "U" and "V" is empty, and that "V" is not an empty list. It shows that "U" is a sublist of either "A" or "B".
The lemma named "echelon_form_Hermite_of_row_nontriv_mod_ring" is about a fixed integer matrix A. It assumes that A is in the carrier matrix of the cardinality of a nontrivial modular ring 'm and the cardinality of another nontrivial modular ring 'n. It also assumes that A is in Jordan Normal Form (JNF) echelon form. I...
If a command 'c' is type-checked under context 'Γ' and results in context 'Γ'', and if this command 'c' is 'Stop', then context 'Γ' is a subcontext of context 'Γ''.
The lemma named "flat_ang" assumes that points A, B, and C are collinear, and points A', B', and C' are also collinear. It also assumes that the angle ABC is equal to angle a1, and the angle A'B'C' is equal to angle a2. Given these assumptions, it shows that angle a1 is equal to angle a2.
The lemma named "loc_reach_mem_diff_eq" assumes that for all x, if x is in the set "mds AsmNoWrite", then the value of x in mem' is equal to the value of x in mem. Under this assumption, it shows that the local reachability of the system state defined by the command c, the memory domain separation (mds), and the memory...
The lemma named "full_componentE" assumes that the second element of the representation of a natural term 'x' is 'i'. It then concludes that there exists a 'y' such that the representation of 'y' is a pair consisting of 't' and 'i'.
The lemma named "plossless_converter_coinduct" assumes that there is a converter "X conv". It also assumes that for any converter and any output "a" in the output set of an interface, if "X conv" is true and "a" is in the output set of the interface, then the probability of lossless generalized process variable (gpv) o...
The lemma named "added_files_Deliver_Append_diff_collapse" in the context of imap shows that if event 'e' is not equal to event 'e'', then the added files from the delivery of an append operation with index 'i' and event 'e' to event 'e'' is an empty list.
The interpretation of the inverse of a floatarith function 'f' with respect to 'xs' is equal to the inverse of the interpretation of the floatarith function 'f' with respect to 'xs'.
The infimum (or greatest lower bound) of the product of 'x' and 'bot' and the product of 'y' and 'bot' is equal to the product of the infimum of the product of 'x' and '1' and 'y' and 'bot', and 'bot'. Here, '*' represents a binary operation, 'bot' represents the least element in a bounded lattice, and '1' represents t...
The predicates of the adjoint of a substitution sigma, an interpretation I, and a state omega, when applied to a predicate p, is equal to the following: if the substituted predicates of sigma applied to p is None, then it is the predicates of the interpretation I applied to p; otherwise, if it is some r, then it is a f...
The address of the initialization of a contract 'ct' with an index 'i' and an event 'e' is equal to the address of the event 'e'. Similarly, the sender of the initialization of the contract 'ct' with an index 'i' and an event 'e' is equal to the sender of the event 'e'.
If a composition series of group G with length n and function f exists, then the weighted composition series of group G with length n and function f also exists.
The size of the linear temporal logic formula in normal form, obtained from the conversion of a given linear temporal logic formula, is less than or equal to twice the size of the original formula.
If an almost ordered binary tree "sub" with a maximum number of children "k" exists, and for all subtrees "s" in the set of subtrees of the list formed by concatenating "ls" and "rs", the order of "s" is "k", and the order of tree "t" is "k". Additionally, if the first element of "rs" is a pair consisting of a subtree ...
If a pair of state and naive collection set is in the big-step semantics of the JVM naive collection semantics and also in the n-step small-step semantics of the JVM naive collection semantics, and the same pair of state and a smart collection set is in the n'-step small-step semantics of the JVM smart collection seman...
The lemma "subst_b_valid" is about the substitution of a variable in a context and a constraint. It takes as input a state 's' and a variable 'b'. It assumes that 'b' is well-formed under an empty context, that 'B' is the set containing 'bv', and that a constraint 'c' is valid under the context containing 'bv'. It then...
The lemma Aux assumes that for all elements 's' in the set 'ss', 's' is an empty list. It shows that the concatenation of all elements in 'ss' results in an empty list.
The composition of 'x' with the infimum of 'y' and the composition of 'z' with 'top' is equal to the composition of the infimum of 'x' and the composition of 'top' with the transpose of 'z', all multiplied by 'y'.
Binding the assumption "\<Phi>" to a function that returns "S", is less than or equal to "S'", if and only if when "\<Phi>" is true, "S" is less than or equal to "S'".
A sort exists in the set 'S' with classes 'cs' if and only if for all 'c' in 'S', a class exists in 'cs' for 'c'.
: assuming a function 'f' is a measurable function from measure space 'M' to 'N', and the null sets of 'N' are a subset of the null sets of the distribution of 'M' over 'N' under 'f', and a set 'A' is in the completion of 'N'. Then, the preimage of the null part of 'N' under 'A' intersected with the space 'M' is in the...
In the context of a tiny functor, there is a lemma that introduces operations on categories. Assuming that the category 'A' is the opposite of 'A', the category 'B' is the opposite of 'B', and 'alpha' remains the same, it shows that the opposite of the functor 'F' maps from the category 'A' to the category 'B' in the c...
The probability mass function (pmf) of the stochastic dominance strategy (sds) R24 at point a is 0, the pmf of sds R24 at point b is also 0, and the pmf of sds R24 at point d is equal to 1 minus the pmf of sds R24 at point c.
If R is equal to the identity relation, then the set relation of R is also the identity relation.
: given the assumptions that A is an object in the category of parallel morphisms, B is also an object in the category of parallel morphisms, AB is the Cartesian product of A and B, B' is equal to B, and the category of parallel morphisms is denoted as CC', it can be shown that the second projection arrow from A to B i...
If 'a' is less than or equal to zero and 'b' is less than or equal to zero, then the sum of 'a' and 'b' is also less than or equal to zero.
The infimum (greatest lower bound) is a closed set for the range of the left-sided natural extension of nu.
The environment of the mapping of "LOC" over "w" is an empty list.
If "D" and "E" are orders, "f" is an order isomorphism from "D" to "E", and "D1" is a subset of the carrier of "D", then the order equivalence between the strict induced order of "D" on "D1" and the strict induced order of "E" on the image of "D1" under "f" holds.
The lemma named "ipurge_tr_rev_ipurge_tr_sources_1" assumes that the domain of y is not in the sources of the list consisting of y and zs with respect to the interference relation I, the domain function D, and the process u. Under this assumption, it shows that the reverse intransitive purge of the list obtained by app...
The homomorphism of x divides the homomorphism of y if and only if x divides y.
The lemma "SA_is_ring" demonstrates that "SA n" forms a ring.
The lemma named "fset_map_snd_id" assumes that for any elements 'a', 'b', and 'e', if the pair '(a, b)' is in the finite set 'cs', then the function 'f' applied to 'a' and 'b' equals 'b'. Under this assumption, it shows that the set of all pairs '(a, f a b)' for each '(a, b)' in 'cs' is equal to 'cs'.
For any vertex 'v', memory state 'M', and set 'P' that belong to the constraint system of the flowgraph at a given level 'k', there exists a path 'p', a new memory state 'c'', and a sequence 'w' such that the following conditions are met: 1. There is a transition from the entry of the flowgraph along the path 'p' to t...
The lemma "friendsOfUID_cong" assumes that the received outer friend IDs of state 's' are equal to the received outer friend IDs of state 's''. It then shows that the friends of user ID in state 's' are equal to the friends of user ID in state 's''.
If a state 's' belongs to the set 'wset_Suspend_ok I', then 's' also belongs to the set 'I'.
If the function "wcode_erase2" is true for the inputs "ires", "rs", and a pair consisting of "b" and a list starting with "Bk", then the function "wcode_on_right_moving_2" is also true for the inputs "ires", "rs", and a pair consisting of a list starting with "Bk" and "b".
The initial state satisfies the condition t.
The polynomial set of the intersection of X and Y is equal to the intersection of the polynomial set of X and the polynomial set of Y.
The degree of a polynomial set is less than or equal to the negative of a polynomial, given that the polynomial set only contains that polynomial.
Dropping 'n' elements from list 'xs' results in an empty list if and only if the length of list 'xs' is less than or equal to 'n'.
The interpretation of the negation of a formula phi with respect to a list of variables xs is equal to the interpretation of the negation of phi with respect to the same list of variables xs.
, under the precondition of an empty state, the function 'uf_init' with parameter 'n' results in a state that is a union-find data structure of size 'n', as defined by the relation 'uf_init_rel' with parameter 'n'. This is expressed within the framework of Hoare logic, which is used to reason about the correctness of c...
If a function 'f' has the greatest postfix point and 'f' is isotone, then the greatest fixpoint of the function 'f' is the postfix point of 'f'.
For any two new thread actions "nt1" and "nt2", the bisimulation relation "nta_bisim" with "init_fin_bisim" as its argument and applied to the conversion of these new thread actions with "PreStart" as a pair, is equal to the bisimulation relation "nta_bisim" applied directly to "nt1" and "nt2" with "bisim" as its argum...
Given two projective lines 'l' and 'm', if the sum of 'i' times the representative of line 'l' and 'j' times the representative of line 'm' equals zero, and either 'i' or 'j' is not zero, then line 'l' must be equal to line 'm'.
For a foldr function, which takes a function 'f' and a list, if the list is empty, the result is the initial value 'b'. If the list is not empty and has a head 'h' and a tail 't', the result is the function 'f' applied to the head 'h' and the result of foldr applied to the tail 't' and the initial value 'b'.
A function (specifically an indicator function of a set A) is integrable over a measure space M if and only if the intersection of set A and the space M is in the set of measurable subsets of M, and the extended measure of the intersection of set A and space M is less than infinity.
The theorem named "atom_ref_back_struth" assumes the following conditions: 1. s is less than or equal to the product of s and q. 2. a is less than or equal to the product of q and a. 3. The product of q and b equals zero. 4. The product of r and b is less than or equal to the product of b and r. 5. The product of r an...
Given a non-zero constant "c" from a commutative monoid with addition, and a polynomial "p" whose leading term is less than "v", the leading coefficient of the sum of the monomial "c*v" and the polynomial "p" is equal to "c".
A group G is cyclic if and only if there exists an element x in the carrier of G such that the carrier of G is equal to the range of the function that maps each integer n to the nth power of x.
If a command 'c' takes a state satisfying property 'P' to a state satisfying property 'Q', and for all states 's', if 'P'' holds then 'P' also holds, then the command 'c' can also take a state satisfying property 'P'' to a state satisfying property 'Q'. This is a rule of transitivity in the context of Hoare logic.
The lemma named "ghost_honorary_grey_mo_co_mark" consists of three parts: 1. If the ghost honorary grey set of the state of process p is empty, then the state of process p is black if and only if it was black before and it is not equal to r. 2. If the ghost honorary grey set of the state of process p is empty, then t...
The integral from negative pi to pi of the function cos(nx), where n is a real number, is equal to 2*pi if n is zero, otherwise it is zero.
If the lookup of 'as' in 'ts' equals 'Some t', and for all 'X', if the pair '(t, X)' is in the set 'ds', then 'as' is not in 'X', then the lookup of 'as' in the mapping obtained by folding the function (which takes a pair '(t, X)' and a mapping 'ts', and returns a new mapping by filtering 'ts' according to the conditio...
Mapping a function over the variables of the sum of a set of multivariate polynomials is equivalent to the sum of mapping the function over the variables of each multivariate polynomial in the set.
The hyperdual of the sine of x plus the hyperdual of the cosine of x is equal to the hyperdual of the function that adds the sine of x to the cosine of x, evaluated at x.
If "u" is an element of the Units, then "u" is invertible and "u" is an element of "M".
If there is a path in the heap 'h' from 'b' to the reference 'a' through a list of pointers 'Ps' and 'a' is not in the set of 'Ps', then there is a list from 'b' to 'a' in the heap 'h' where 'a' is replaced with 'Null', and this list includes 'Ps' followed by 'a'.
A calculus that is reducedly dynamically complete with respect to the bottom element and the infimum operation implies the reduction from Red_I to Red_F. This is equivalent to a calculus that is reducedly statically complete with respect to the bottom element and the infimum operation implying the reduction from Red_I ...
The length of the array at the reference 'm_ref2' in the heap is equal to 'size' if the weak pair invariant holds for the heap.
The lemma T_down_closed assumes that v is an element of T with parameters tau and eta, and that v' is a sub-element or equal to v. It shows that v' is also an element of T with parameters tau and eta.
If there is a message from Tgs to A, which includes an encrypted message with a server key, an agent B, and a timestamp Ts, and a server ticket, within a set of events that are part of the Kerberos V protocol, then there exists a timestamp Ta such that there is a message from Kas to A. This message includes two encrypt...
In the context of an additive principal, the statement asserts that there exists a number 'x' such that 'a' is equal to omega raised to the power of 'x'.
In the context of a given set of triples A and a program state G, the empty list is equivalent to the normal state where the predicate P is equal to the set of values in the empty list.
The complex conjugate function is a bounded linear operation.
: Assuming that the function "Fun f T" is equal to "t" modulo "TI" (a set of term implications), we can show that: 1. "t" is a function. 2. The length of "T" is equal to the length of the arguments of "t". 3. For any index "i" less than the length of "T", the "i"-th element of "T" is equal to the "i"-th argument of "...
The lemma named "ofilter_suc_Field" assumes that "A" is an order filter and "A" is not equal to the field of relation "r". It shows that the union of "A" and the successor of "A" is also an order filter.
If "rsx" is greater than 0, then there exists a "y" such that replacing the element at the position "Suc (length xs)" in the list "xs @ x # rsx # anything" with "rsx - Suc 0" results in the list "xs @ x # y # anything", and "y" is less than or equal to "rsx". Here, "Suc" is the successor function, which returns the nex...
The sum over all words in the set of k-words, of the quantity 1 divided by b to the power of the length of the concatenated codeword of the word, is equal to the sum from m equals 0 to the successor of the product of k and the maximum length, of the product of the cardinality of the intersection of the set of k-words a...
For a property P and a formula φ, if the following conditions are met: 1. P holds for any predicate with arguments x and θ. 2. P holds for any inequality between θ and η. 3. If P holds for a formula φ, then P also holds for the negation of φ. 4. If P holds for formulas φ and ψ, then P also holds for the conjunction of...
The lemma named "p_intpow" is defined for an integer 'n'. It assumes that 'n' is less than zero. It shows that the 'n'-th power of 'p' is equal to the fraction one over the '(-n)'-th power of 'p' in the field of p-adic integers.
The lemma named "nth_find_first" assumes that "a" is an element in the set "xs". It shows that the element at the position in "xs" where "a" is first found is indeed "a".
Appending a lazy tllist 'xs' to another list 'ys' is equivalent to creating a new lazy tllist where, for each element in 'xs', if the element is a pair of a value 'x' and a remaining list 'xs'', then the new element is a pair of 'x' and the result of appending 'xs'' to 'ys'. If the element is a value 'b', then the new ...
The lemma named "map_of_sorted_list_of_map" assumes that the domain of the map "m" is finite (as stated by "FIN: finite (dom m)"). It shows that the map obtained from the sorted list of the map "m" is equal to the original map "m".
If an operator "op" is well-formed, then for all pairs (v, a) in the precondition of the abstract syntax tree operator "op", there exists a "y" such that "v" maps to "y" in the abstract range map, and "a" is an element of the value that "v" maps to in the abstract range map.
In the context of UP_cring, the lemma named "poly_lift_hom_X_var" is stated. It assumes that "S" is a cring and "\<phi>" is a ring homomorphism from "R" to "S". It shows that the polynomial lift homomorphism from "R" to "S" applied to the monomial of "R" with coefficient 1 and degree 1 equals the monomial of "S" with c...
In the context of the Reachability Problem, we have a lemma named "n_eq_0_closed". This lemma assumes that "s" is an element of "S'", "S'" is a subset of "S", "S'" and "S2" have no common elements, and for any "s" in "S'", there exists a "D" in "K s" that is a subset of "S'". Under these assumptions, the lemma shows th...
An empty list is in the prefixes of A if and only if A is not an empty set.
The lemma named "coneE" assumes that "p" is an element of the cone of "(h, U)". It then concludes that there exists an "a" such that "a" is an element of the polynomial ring "P[U]" and "p" is equal to the product of "a" and "h".
The lemma "signs_at_insert" assumes that "i" is less than or equal to the length of "qs". It shows that inserting at position "i" the result of squashing the polynomial "p" evaluated at "x" into the signs at "qs" evaluated at "x", is equal to the signs at the result of inserting at position "i" the polynomial "p" into ...
The lemma named "star_simulation_left" assumes that the product of x and z is less than or equal to the product of z and y. It shows that the star of x multiplied by z is less than or equal to z multiplied by the star of y.
A Pythagorean triple (x, y, z) is primitive if and only if there exist two numbers u and v such that v is positive and less than u, u and v are coprime (i.e., their greatest common divisor is 1), not both u and v are odd, and either x equals 2 times u times v and y equals u squared minus v squared, or x equals u square...
The lemma lm032 assumes that A is a finite set. It shows that the sum of the function f over the set A is equal to the sum of the function f over the intersection of sets A and B plus the sum of the function f over the difference of sets A and B.
The set of elements from the list obtained by taking elements while a condition P holds, is a subset of the intersection of the set of elements from the original list and the set of elements for which condition P holds.
If the following conditions are met: 1. The value 'n' is less than or equal to the norm of the application of a complex bounded linear function 'f' to 'x' divided by the norm of 'x'. 2. For any 'x', the norm of the application of the complex bounded linear function 'f' to 'x' is less than or equal to 'n' times the norm...
: 1. The extension upwards of a list 'xs' with an element 'a' is a prefix of the extension upwards of the concatenation of 'xs' and 'ys' with the same element 'a'. 2. The extension downwards of a list 'xs' with an element 'a' is a prefix of the extension downwards of the concatenation of 'xs' and 'ys' with the same ele...
The leading term of the lower polynomial of "p" with respect to variable "v" is less than or equal to the leading term of the polynomial "p".
If k is greater than n, then the Pochhammer symbol of the negative of n (where n is a natural number) and k equals zero, in the context of an integral domain.
: Assuming that for every set U in a collection C, U is open in a topological space X, and the topological space X is a subset of the union of the collection C, and there exists a singular relative cycle c in the topological space X with a subset S and a dimension p. Then, there exists another singular relative cycl...
The lemma named "eqButPID_paper" assumes that "s" and "s1" are equal except for their PID (Paper ID). It shows that the paper "s" is equal to the paper "s1" with the PID replaced by a new paper. This new paper has the same title, abstract, reviews, discussions, and decisions as the paper "s1", but the content is taken ...
If 'a' and 'b' are elements in the carrier 'R' and 'p' is an element in the carrier 'P', then the scalar multiplication of 'a' and 'b', then 'p' is equal to the scalar multiplication of 'a' and the scalar multiplication of 'b' and 'p'.
For any list of values 'vs' and expressions 'es', and any list 'xs', mapping the function 'Val' over 'vs' and appending 'es' to the result is equivalent to just applying 'els_loc_ok' to 'es' and 'xs'.
If the function 'f' is associative over the augmented partially ordered monoid set 'aug_pm_set' with parameters 'z', 'i', and 'A', and if 'x' and 'y' are elements in the addition set of 'f' over the same augmented partially ordered monoid set, then the addition of 'x' and 'y' under the function 'f' is also an element o...