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The lemma shows that there is a natural transformation from functor G to functor F over categories A and B using the given map.
For any polynomial 'x' in a modular ring, the power of the sum of 'x' and 'y' to the cardinality of 'a' is equal to the sum of 'x' to the power of the cardinality of 'a' and 'y' to the power of the cardinality of 'a'.
Updating an account "i" with the sum of two amounts "\<alpha>" and "\<beta>" is equivalent to the sum of updating the account "i" with amount "\<alpha>" and updating the same account "i" with amount "\<beta>".
The lemma "Hermite_Hermite_of_row_i_mx1" assumes that a matrix A is in the carrier matrix of dimensions m by 1 and that A is in echelon form. It shows that the Hermite normal form of the matrix A, when the range of the associated Euclidean function and the range of the residue class of integers are applied, is equal to...
A state "s" is concurrent with itself.
The determinant of the matrix obtained by updating the columns of matrix M with the function that multiplies the scalar c(i) with vector a(i) is equal to the product of the scalars c(i) for all i in set T, multiplied by the determinant of the matrix obtained by updating the columns of matrix M with vector a(i).
The lemma named "eval_e_restrict" is about the evaluation of an expression 'e' of type 'ce' under certain conditions. It fixes an expression 'e' and a finite set 'B' of bound variables. It assumes that the expression 'e' evaluates to 's' under the interpretation 'i'', and that 'i' is the restriction of 'i'' to 'd'. It ...
The data access at level 1 for system A93 is an empty set.
The return value of the function "foldli" with parameters list "l", condition "c", function "f", and state "s" is equal to the return value of the function "nfoldli" with the same parameters, where "nfoldli" applies the function "f" to each element "x" in the list "l" and the state "s", and then returns the result.
The set of an empty trie is an empty set.
If X is an element of the Vset of alpha and Y is also an element of the Vset of alpha, then the union over A in X and the union over B in Y of the Homomorphisms from the parallel digraph of alpha to A and B is also an element of the Vset of alpha.
If a space X is a Hausdorff space and S is a finite set, then the derived set of S in X is an empty set.
The composition of S and NS is S, and the composition of S, NS, and T is the composition of S and T.
The lemma named "map_merge_wrt" assumes two conditions. First, the image of the function 'f' applied to the sets 'xs' and 'ys' have no intersection. Second, for any elements 'x' from set 'xs' and 'y' from set 'ys', the relation 'r2' applied to 'f(x)' and 'f(y)' is equivalent to the relation 'r1' applied to 'x' and 'y'....
A list starting with "x" followed by list "xs" is a prefix of a list starting with "y" followed by list "ys" if and only if "x" is equal to "y" and list "xs" is a prefix of list "ys".
For any predicate P of type 'alpha upred, the conjunction of P and the disjunction of not P or Q is equivalent to the conjunction of P and Q.
The subgroup generated by the subgroup generated by a set S in group G is equal to the subgroup generated by set S in group G.
"r_univ" is equivalent to the following expression: first, let "u" be the composition of the function "r_dec" with the function "r_result" with arity 3, and let "t" be the composition of the function "r_not" with the function "r_result" with arity 3. Then, the composition of the function "u" with the minimal function "...
"the bottom element is equal to the infimum of the transpose of x and y if and only if the bottom element is equal to the infimum of x and the transpose of y".
The Borel sigma-algebra on the real numbers is equal to the sigma-algebra generated by the collection of all intervals of the form {x ..}, where x is a real number.
If a sequence X converges to L and if function f is strictly monotonically increasing, then the composition of X and f also converges to L.
Converting an empty doubly-linked list (Dlist.empty) to a regular list results in an empty list ([]).
If 'r' is greater than zero, then the composition of the formal Puiseux series power function with 1 and 'r' equals 1.
If 'n' is less than the length of the list 'xs', then the last element of the sublist of 'xs' that includes the first 'n+1' elements is the 'n'th element of 'xs'.
A class C has a method M in the compilation of program P into its second version if and only if the class C has the method M in the original program P.
If a match "m" is in normalized negation normal form (as per assumption "n"), and if the pair (disc, sel) is a well-formed discriminator-selector pair for a set "C" (as per assumption "wf_disc_sel"), and if the function "negate_f" ensures that for any match "m" that has a negated discriminator "disc" with a value of Fa...
If "T" satisfies the Zermelo-Fraenkel axioms and "[A,e,B]" is a morphism in the canonical category of "T", then the codomain of the morphism "[A,e,B]" in the canonical category of "T" is "B".
: 1. The product of set A and the union of sets B and C is equal to the union of the product of set A and set B and the product of set A and set C. 2. The product of the union of sets A and B and set C is equal to the union of the product of set A and set C and the product of set B and set C.
The natural number equivalent of an integer 'k' is defined as follows: If 'k' is less than or equal to zero, then the natural number equivalent is zero. Otherwise, we divide 'k' by 2 and get a quotient 'l' and a remainder 'j'. We then convert 'l' to a natural number 'l''. We double 'l'' to get 'l'''. If 'j' is zero, th...
The set of variables in the linear program obtained by substituting variable xj in lp with lp' is a superset of the set obtained by removing xj and the variables in lp' from the variables in lp.
If regular expression r is approximately equal to regular expression s, then r is equivalent to s.
The "ni" function of the product of "x" and "n" of "y" is equal to the "ni" function of "x".
The lemma named "ramanujan_sum_dvd_eq_totient" assumes that "k" is a divisor of "n". It shows that the Ramanujan sum of "k" and "n" is equal to the totient of "k".
If 'c' and 'd' are polynomial coefficients in a ring 'S', then there exists a polynomial coefficient 'e' in the same ring such that the first element of 'e' is the sum of the first elements of 'c' and 'd', and the value of the second element of 'e' at the first element of 'e' is the product of the values of the second ...
"mzero" is equal to "Nothing".
If two conditions 'p' applied to 'x' and 'y', and 'p' applied to 'u' and 'v' are true, then it is equivalent to saying that if 'p' is true then both 'x' and 'u' are true, otherwise both 'y' and 'v' are true.
"the product of (a implies b) and (b implies c) is less than or equal to (a implies c)".
A monomial with degree 0 and coefficient equal to a natural number 'n' is equivalent to the natural number 'n' itself.
For all 'x' in the range from (1.41, 2.25) to (1.42, 2.25), the Van der Pol oscillator (vdp) returns to the range from (1, 2.25) to (2, 2.25) at 'x', and the Poincare map of the Van der Pol oscillator at 'x' in the range from (1, 2.25) to (2, 2.25) is in the range from (1.41, 2.25) to (1.42, 2.25). The theorem notes re...
For a product case, applying a function that returns the result of another function 'f' applied to elements 'a' and 'b', to a product 'p', is equivalent to returning the result of applying the function 'f' to the product 'p'.
The lemma named "common_refinement_coarsest" assumes the following conditions: 1. For any set P in the collection P, P is a partition on set A. 2. R is a partition on set A. 3. For any set P in the collection P, R refines P on set A. 4. The collection P is not empty. Under these assumptions, the lemma shows that R re...
If the type of the heap 'h' is well-formed and the element pointer is in the set of element pointer kinds of 'h', then the operation 'get_shadow_root' on the element pointer is successful in the heap 'h'.
The lemma named "convex_improper" is about a function 'f' that maps from a Euclidean space to an extended real number. It assumes that the function 'f' is convex on the entire universe and that 'f' is not a proper function. It then shows that for all 'x' in the relative interior of the domain of 'f', the value of 'f' a...
"x is less than or equal to the supremum (least upper bound) of y and z if and only if x is less than or equal to y or x is less than or equal to z".
The lemma named "strong_barbed_bisimulation_and_closures" is about a fixed relation "Rel" of type set of pairs of processes and a fixed "CWB" of type calculus with barbs. It assumes that "Rel" is a strong barbed bisimulation in "CWB". It then shows that the reflexive closure of "Rel", the symmetric closure of "Rel", th...
A0i refines a0n under the refinement mapping R0n0i and mediator med0n0i.
: If D is greater than 0 and there exists an integer z such that for all integers x, if x is less than z then P(x) equals P1(x), and for all integers x, if there does not exist an integer j in the range from 1 to D and an integer b in the set B such that P(b+j) is true, then if P(x) is true, P(x-D) is also true, and ...
If "s" can be transformed into "s'" through zero or more beta-eta reductions, then "s" applied to "t" can also be transformed into "s'" applied to "t" through zero or more beta-eta reductions.
Given two complex matrices A and B, and two vectors u and v, where u is in the carrier vector of dimension d2, v is in the carrier vector of dimension d3, A is in the carrier matrix of dimensions d1 and d2, and B is in the carrier matrix of dimensions d3 and d4, then the product of A, the outer product of u and v, and ...
If the number of lanes is one, and the lane is occupied (not free), then there exists a car either in the right lane or in the left lane.
The nth power of x under the operations of the cring class is equal to the nth power of x.
The lemma named "dhops_update_another" is established under the assumption that "ip" is not equal to "dip". It shows that the number of hops in the updated routing table for "ip" is the same as the number of hops in the original routing table for "ip", regardless of the updates made to "dip". Here, "rt" represents the ...
If 'x' and 'y' are elements of set 'A', then the interval 'r' between 'x' and 'y' is equal to the interval of the order of the dual closure 'cl' between 'y' and 'x'.
The exponential quasi-Borel space of X and Y is a closed set in the first argument.
The lemma named "quietFrame" is defined with fixed variables: Psi of type 'b and P of type psi, which is a tuple of 'a, 'b, 'c. The lemma assumes that P is quiet. It then shows that the insertion of an assertion (extracted from the frame of P) into Psi is structurally equivalent to the frame of an empty process wit...
If a value 'v' is a member restricted to the intersection of sets 'A' and 'B', then 'v' is a member restricted to both set 'A' and set 'B'.
Subtracting any number 'n' from infinity results in infinity, in the context of extended natural numbers.
The result of applying the function "set_bits" to a function "f" and then dividing by 2 (for integer numbers) is equal to the result of applying the function "set_bits" to the composition of function "f" and the successor function.
If we have a standard signature 'Σ', a term 't' that is okay according to 'Σ', a list of instances 'insts' where each instance is unique (no two instances have the same first element), and all types in 'insts' are okay according to 'Σ', then the term 't' with substituted types from 'insts' is also okay according to 'Σ'...
The set of constants in the combination of a list and a function is equal to the set of constants in the function unioned with the set of constants in the list, where the list has been mapped to its set of constants.
In the PAODV protocol at any instance 'i', the sequence number 'sn' of any state 'ξ' is always less than or equal to the sequence number of the next state 'ξ''.
The computation of the "case_option_compute" function with "None" as an argument is equivalent to a function that takes two arguments "a" and "f", and returns "a".
The lemma named "mat_of_cblinfun_Proj_ccspan" is defined with a fixed list "S" of type "'a::onb_enum". It shows that the matrix of the complex bounded linear function of the projection of the closed convex span of the set "S" is equal to the result of creating an orthogonal projector with dimension "d" (where "d" is th...
The norm of the adjoint of a function A is equal to the norm of A itself. Here, A is a function from a Hilbert space to a complex inner product space.
If an interface I is accessible in a program P with a given environment G, then I is an interface in the environment G.
Running the "return_state" function with any input 'x' and state 's' is equivalent to returning a pair consisting of 'x' and 's'.
If the domain of a relation R intersects with the range of the same relation R resulting in an empty set, then the composition of the relation R with itself is also an empty set.
An oriented simplex "p" with labeling "l" is equal to an oriented simplex "p" with labeling "l'" if and only if for all "i" such that "i" is less than or equal to "p", the labeling "l" at "i" is equal to the labeling "l'" at "i".
If a list 'xs' is updated according to a set 'S' and a function 'f' to produce a new list 'xs', and if 'i' is less than the length of the new list 'xs', then the 'i'-th element of the new list 'xs' is equal to the result of the function 'f' applied to 'i' if 'i' is in the set 'S', otherwise it is equal to the 'i'-th el...
The lemma named "distinct_edgesI" assumes that the path "p" is distinct. It shows that the edges of the undirected walk "p" are also distinct.
If an element is in the carrier set 'l', then the interpretation of the product of polynomials 'P' and 'Q' is equal to the product of the interpretations of 'P' and 'Q' individually.
The coefficient of X at the index i is 1 if i equals 1, otherwise it is 0.
: Assuming that for every element 't' in set 'M', there exists a constant 'c' such that 't' equals the function 'c' with no arguments, and that for every element 't' in set 'M', the analysis of 't' yields an empty pair of lists, and that the set 'N' is a term-for-rewriting set, it can be shown that the union of sets '...
The lemma consists of two parts: 1. If the program P, with initial class C0 and main method Main, transitions from state (C', M', pc', nt') to state (C, M, pc, nt) via an exception or a kill, and neither C nor C' are the main class of P, then the following holds: - If pc is None, then the statement is true. - ...
The lemma named "uncountable_ball" is about a fixed point 'a' in a Euclidean space. It assumes that 'r' is greater than 0. It shows that the ball with center 'a' and radius 'r' is uncountable.
The star transformation of any element 'x' belongs to the set of hyperfinite numbers.
If F is in the co-relation of A with A, then F is in the stable set of A.
The lemma Conjunction Elimination: Assuming that from A and B, we can infer the conjunction of p and q, it shows that from A and B, we can infer p and from A and B, we can infer q.
Lemma: isUCont_isNSUCont Let f be a function from a real normed vector space 'a to a real normed vector space 'b. If f is uniformly continuous (isUCont), then f is also nonstandard uniformly continuous (isNSUCont).
The parallel composition of the relation R with the identity relation, when composed with R, is equal to R.
The lemma named "containID" assumes that: 1. The atomic proposition 'i' is in both sets Gamma and Delta. 2. The set of axioms 'Ax' is a subset of 'R'. Under these assumptions, it shows that the sequent (Gamma implies Delta, with rank 0) is derivable from 'R'.
If every element 'x' in the set 'xs' is less than or equal to zero, then the sum of the list 'xs' is also less than or equal to zero.
For two functions 'f' and 'g' mapping from a topological space to a real normed field, if both 'f' and 'g' are continuous at a point 'a' within a set 's' and 'g' at point 'a' is not equal to zero, then the function that maps 'x' to the quotient of 'f(x)' and 'g(x)' is also continuous at point 'a' within the set 's'.
If two generative probabilistic values (gpv) are related by a certain relation R, and this relation is preserved under the actions A and outcomes C, then the operation "try_gpv" (which presumably tries or tests these gpv) will also preserve this relation. This is stated for two pairs of gpv, hence the operation "try_gp...
For any alpha and beta of type 'a::id_act, alpha not being equal to beta is equivalent to the assertion that alpha is not equal to beta in the universe 'v'.
: Assuming the following: 1. A valid call exists in the context of Gamma and Theta, where the call is defined by the precondition P, the call procedure, the initialization function init, the postcondition Q, the abort condition A, and the return function return'. 2. For all states sigma, a valid modification exists in...
If the sets p and q are subsets of the carrier R, the lengths of p and q are equal, and a is an element of the carrier R, then evaluating the sum of the polynomials p and q at a is equal to the sum of evaluating p at a and evaluating q at a.
In the context of polynomial modulo prime, if the modulo of the polynomial obtained by multiplying a scalar 'p' with a polynomial 'f' by 'p' raised to the power 'k' times 'p' is zero, then the modulo of the polynomial 'f' by 'p' raised to the power 'k' is also zero.
For any two elements 'a' and 'b' from a semiring, the composition function 'of_comp' applied to the product of 'a' and 'b' is equal to the product of the composition function applied to 'a' and 'b' separately.
A list beginning with element 'c' followed by list 'w' is in the set of lists composed entirely of element 'a' if and only if 'c' is equal to 'a' and list 'w' is also in the set of lists composed entirely of 'a'.
If X is in the set of ideals, then the bottom element is in X.
If a string 'p' is a prefix of the concatenation of strings 'u' and 'p', and the inverse conjugate of 'p' with respect to the concatenation of 'u' and 'p' is an empty string, then 'u' must be an empty string.
The lemma named "set_integrable_restrict_space" is defined with a function 'f' that maps from 'a' to 'b', where 'b' is a Banach space with a second countable topology. Given the assumptions that 'f' is set integrable over a measure space 'M' and a set 'S', and 'T' is in the set of measurable subsets of the restricted m...
If a pair (c, aft) is in the fragments of another pair (c', aft') and c is not a response, then this pair (c, aft) is also in the vcg_fragments' of the pair (c', aft').
The power series of the cosine series stream is convergent. This is denoted as "convergent_powser ?cs".
The relationship between a host and its shadow root in a given host 'h' is finite.
The lemma named "sources_single_Trg" assumes that "t" is an arrow in the lambda calculus. It shows that the sources of the target of "t" are equal to the targets of "t".
A polynomial "p" is positive if and only if the coefficients of "p" are not empty and the last coefficient is greater than zero.
The Lebesgue measure of a singleton set (a set containing only one element x) is equal to zero.
The lemma named "maximalidealI" is about a fixed element R. It assumes that I is an ideal of R, and that the carrier of R is not equal to I. Furthermore, it assumes that for any ideal J of R, if I is a subset of J and J is a subset of the carrier of R, then J must be either equal to I or equal to the carrier of R. Unde...