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If there is a subprobability mass function (spmf) relationship between p and q such that for every element x in p and y in q, applying function f to x equals applying function g to y, then the result of mapping function f over spmf p is equal to the result of mapping function g over spmf q.
A function 'f' is a proper map from a topological space 'X' to a topological space 'Y' if and only if 'f' is a closed map from 'X' to 'Y' and for every set 'K' that is compact in 'Y', the preimage of 'K' under 'f' (i.e., the set of all points 'x' in the topological space 'X' such that 'f(x)' is in 'K') is also compact ...
In a test, the following steps are performed: 1. A document element with the ID 'test_slot_not_in_shadow_2' is retrieved and stored in tmp0. 2. A test tree is created with tmp0 and stored in n. 3. The 'test_slot_not_in_shadow_2' property of n is retrieved and stored in tmp1. 4. All white space only text nodes are remo...
The lemma "fin_make_untimed_inf_truncate_Nonempty" assumes that the kth element of sequence z is not empty and that k is less than or equal to i. Under these conditions, it shows that the result of applying the function "fin_make_untimed" to the result of "inf_truncate" of z at i is not an empty list.
If the set A strictly supersedes the set B (as denoted by "A \<sqsupset>p B") and if the polynomial p can be reduced by the set B (as denoted by "is_red B p"), then the polynomial p can also be reduced by the set A (as denoted by "is_red A p"). The lemma assumes the conditions "A \<sqsupset>p B" and "is_red B p" to sho...
The lemma named "cancel" assumes that the curve defined by coefficients a and b is nonsingular, and points p1, p2, and p3 are on this curve. If the addition of point p1 and point p2 equals the addition of point p1 and point p3 (under the elliptic curve group law), then it shows that point p2 is equal to point p3.
For a function 'f' mapping from any type 'a' to a real normed vector type 'b', if 'f' and 'g' are both continuous maps in a Euclidean space 'X', then the function that maps 'x' to the difference between 'f(x)' and 'g(x)' is also a continuous map in the same Euclidean space 'X'.
If a command 'c' is well-typed in a context 'G' with a type 'A', then for all 'n', 'c' is semantically sound in the context 'G' with respect to 'n' and type 'A'.
If x is less than or equal to y, then the weakly connected component of x is less than or equal to the weakly connected component of y.
The lemma named "resid_Subst" assumes that "t" is compatible with "t'" and "u" is compatible with "u'". It shows that the substitution of "n" in "t" and "u", subtracted by the substitution of "n" in "t'" and "u'", equals the substitution of "n" in the subtraction of "t" and "t'", and the subtraction of "u" and "u'".
The lemma named "lhv_integrable" is within the context of complex square matrices. It fixes a real number 'c'. It assumes that 'lhv' holds for 'M', 'A', 'B', 'R', 'XA', and 'XB'. It also assumes that 'a' is in the spectrum of 'A' and 'b' is in the spectrum of 'B'. It shows that the function 'w' mapping to the product o...
The lemma prj_in_homs shows the following: 1. The projection of the 0th leg of mu and the 1st leg of the composition of H, nu, pi, and rho is a morphism from the chine of the composition of H, mu, and the composition of H, nu, pi, and rho to the chine of the composition of H, nu, pi, and rho. 2. The projection of the...
Handling an exception 'b' on the lifted expression 'a' is equivalent to just the lifted expression 'a'.
Appending an empty transaction list (LNil) to any transaction list (tr) will result in the original transaction list (tr).
The function "ldrop' n" is continuous at the point "l".
"A" can reach by processing "t".
The univariate power of a variable monomial is equal to one if the variable x is equal to y, otherwise it is undefined (None).
If a system is well-formed and there exists a process number such that a process 'p' is a subterm of the system at that process number, and if a transition from state 'p' to state 'p'' under action 'a' exists in the sequential semantics of the system, then there exists a process number such that the process 'p'' is a s...
The set of "hs_noop_locs" is a subset of the set of "hs_pending_loaded_locs".
If a set 's' of real normed vectors is compact, then the set obtained by subtracting a fixed vector 'a' from each element of 's' is also compact.
, under certain conditions, the language of two machines M and I are equal. These conditions are as follows: 1. Both M and I are observable. 2. M is minimal. 3. The size of I is less than or equal to m. 4. m is greater than or equal to the size of the range of M. 5. The inputs of I are the same as the inputs of M. 6....
An element 's' is not in the set 'E i' if and only if 's' is in the set 'H i' or 's' is in the set 'T i'.
"if F is initially true, then F is eventually true".
If for all j less than or equal to n, f(j) is in the carrier of the Abelian group, and both g and h are functions from the set of all j less than or equal to n to itself, then the n-fold sum over the Abelian group of the composition of f and h, then g, is equal to the n-fold sum over the Abelian group of the compositio...
If for all pairs of lists in two list of lists (xss and yss), the lengths of each pair of lists are equal, and if the concatenation of all lists in xss is equal to the concatenation of all lists in yss, then xss is equal to yss.
The lemma named "blinding_of_T_mono" assumes that "bo" is less than or equal to "bo'". It shows that the function "blinding_of_T" with parameters "rh" and "bo" is less than or equal to the function "blinding_of_T" with parameters "rh" and "bo'".
The exception of a single polynomial mapping of 'u' and 'c' with respect to 'S' is equal to the single polynomial mapping of 'u' and 'c' when 'u' is not an element of 'S'.
The lemma named "dim_col_take_rows" shows that the dimension of the columns of the matrix obtained by taking certain rows (specified by "inds") from matrix A is equal to the dimension of the columns of the original matrix A.
"It is necessarily the case that God exists".
The unordered pair consisting of p and q is equal to the unordered pair consisting of q and p.
If a match 'm' does not have a negated discriminator 'disc' and 'neg', and for all 'a', 'disc' does not apply to the output interface 'a', and if 'm' is a normalized negation normal form match, then if the function 'compress_normalize_output_interfaces' applied to 'm' equals 'm'', then 'm'' is a normalized negation nor...
The set S is strongly normalizing with respect to an empty relation.
The lemma named "nhop_update_changed" is established under the assumption that updating the routing table "rt" with destination IP "dip", destination sequence number "dsn", destination sequence key "dsk", flag "flg", hop count "hops", source IP "sip", and an empty set does not result in the original routing table "rt"....
Lemma fls_pow_subdegree_eqI: If the nth power of the coefficient of the formal Laurent series f at its subdegree is not equal to 0, then the subdegree of the nth power of f is equal to n times the subdegree of f.
If M has a finite rank in the ring R, then there exists a number n such that for all sets ms, if ms is a subset of M and ms is linearly independent in R, then the length of ms is less than or equal to n.
The lemma named "nop_ontoI" makes the following assumptions: 1. "f" is a very simple function (vsv). 2. "n" is a natural number. 3. The domain of "f" is the n-fold Cartesian product of set "A". 4. The range of "f" is set "A". If these conditions are met, it shows that "f" is a natural number onto function from "A" to ...
If "n" is greater than 0, then the primorial of "n" is equal to the primorial of the (n-1)th prime number.
If "f" and "g" are inverse arrows, and if the identity of the composition of "g" and "f", as well as the identity of the composition of "f" and "g", implies "T", then "T" is shown or proven.
Applying the function "ap_sum" with the function "f" to the result of applying "ap_sum" with the function "f" to the result of applying "ap_sum" with the function "f" to the left injection of the composition function "o", and then applying this to "h", "g", and "x" in sequence, is equivalent to applying "ap_sum" with t...
The lemma fp0_correct: Assuming that the condition 's' is not met, it shows that in the case of 'ppath s' being None, there does not exist any 'v0' in 'V0' such that there exists a 'v' where '(v0,v)' is in the reflexive transitive closure of 'E' and 'P v' holds. Alternatively, if 'ppath s' is 'Some (p,v)', there exists...
The first set of uses from the phi functions in graph g is a subset of the set of nodes in graph g.
The lemma named "sumZero" establishes that for any process P in the pi calculus, the summation of P and zero is structurally congruent to P.
If two lists 'xs' and 'ys' are sorted in ascending order, then the merged list of 'xs' and 'ys' will also be sorted in ascending order.
The lemma named "cinner_divide_left" establishes that for any complex number 'a' that belongs to the set of complex inner products and complex division algebras, the complex inner product of 'a' divided by a complex number 'm' and another complex number 'b' is equal to the complex inner product of 'a' and 'b' divided b...
The lemma "nxt_nontotal_hyp" assumes that the next element of b is undefined and that the function s of b is defined and equals i. It shows that there exists an x such that the function phi of i and x is undefined.
The function "deleteFriend" is symmetric with respect to user IDs. This means that deleting a friend with user ID 'uid' from the friend list of a user with user ID 'uid'' in a given state 's' and privacy setting 'p' is the same as deleting a friend with user ID 'uid'' from the friend list of a user with user ID 'uid' i...
The lemma establishes that for a fixed boolean B, the permutation of p acting on the implication (A implies B equals C) holds.
The lemma named "mut_get_roots_mark_object_invL" shows that if the following conditions are met: the mutual exclusion lock invariant for a given mutator 'm' holds, the handshake invariant for the same mutator holds, the mark object invariant for the mutator in the 'mut_get_roots' phase holds, and the system is in a sta...
For a given element 'x' of any field, if 'x' is a root of a polynomial 'p' (i.e., the polynomial evaluated at 'x' equals zero) and the degree of 'p' is greater than 1, then 'p' is not irreducible.
The restriction of a transfer tree to a set S' is equal to the transfer tree of the intersection of S and S' in a call graph G.
This lemma, named "partial_term_of_tllist_code", is about a function "partial_term_of" applied to a type token "tytok" of a lazy ternary list (tllist). The lemma shows three cases: 1. When "partial_term_of" is applied to a narrowing variable, it is equivalent to a free variable of the type of the lazy ternary list. 2...
If a type τ is less than an enumeration type Enum E, then if τ is equal to OclVoid, the proposition P holds.
If "n" is less than the number of rows in matrix A, and the natural number equivalent of "a" is greater than "n", then the first element of the pair resulting from the row_add_iterate_PA function applied to the pair (P,A) with parameters n, i, and j, when accessed at indices a and b, is equal to the element of matrix P...
For any hypercomplex number 'a', the 'n' power of 'hcis a' is equal to 'hcis' of the product of the hyperreal representation of 'n' and 'a'.
"The union of F and G is in the intersection of the set where NF of s equals k and the set where BB of s holds, leads to the set where k is less than NF of s."
If 'r' and 'r'' are both reflexive relations, then the ordinal sum of 'r' and 'r'' is also a reflexive relation.
If 'g' is an inverse of 'f' in the category 'CC', then there exist objects 'a' and 'b' such that 'g' is a morphism from 'b' to 'a', 'f' is a morphism from 'a' to 'b', the composition of 'g' and 'f' is the identity morphism at 'a', and the composition of 'f' and 'g' is the identity morphism at 'b'.
If an environment 'e' is okay and a variable 'x' is not in the domain of the environment 'e', then the environment 'e' with 'x' mapped to 'X' is also okay.
The element at the position (i, j) in the decision matrix P_dec for decision function d is equal to the probability mass function of the transition kernel K at the state-action pair (i, d i) for state j.
A relation 'r' is order-isomorphic to the empty ordinal (ozero) if and only if 'r' is an empty set.
The representative list of the S-polynomial of two polynomials p and q is equal to the S-polynomial of the representative lists of p and q.
The function "rec_noteq" is primitive recursive at the point "Suc (Suc 0)", which is equivalent to the number 2.
The execution of the instruction "IfFalse i" in a Java Virtual Machine, given a program P, a heap h, a stack with v at the top followed by other stack elements stk, a local variable array loc, a class C₀, a method M₀, a program counter pc, and a list of frame records frs, is defined as follows: First, a new program c...
The size of a tree that has had a function applied to every element is equal to the size of the original tree.
The Exchange Lemma: Assuming that T is a finite set, S is an independent set, and S is a subset of the span of T, it can be shown that there exists a set t' such that the cardinality of t' is equal to the cardinality of T, t' is a finite set, S is a subset of t', t' is a subset of the union of S and T, and S is a subse...
If the function "raw_match'" applied to T, U, and subs results in Some subs', then subs is less than or equal to subs' in the map ordering.
If N is a normal subgroup of G, and a, b, a1, b1 are elements of the carrier of G, and the product of a and the inverse of a1, as well as the product of b and the inverse of b1, are in N, then the product of a and b and the inverse of the product of a1 and b1 is also in N.
The zero lens is a well-behaved lens.
The evaluation of "r_cons_else" with the arguments [e, es] is defined and equals the successor of the product encoding of the length of "es" and the product encoding of "e" and the second predecessor of "es" minus one.
The set of "fs" is a subset of the set that results from the second element of the "sym_preproc_aux" function when applied to "gs", "ks", and the pair "vs, fs".
The omega power of y is equal to the supremum of 1 and the product of y and the omega power of y.
Accessing any property of an invalid object results in an invalid value.
An empty set is in the set of all single-valued partial maps over sets A and B.
The numbers 0 and 1 are algebraic integers. It also states that any numeral, any natural number 'k', and any integer 'm' are also algebraic integers.
If "r" is an equivalence relation on a set "A" and "C" is an equivalence class in the set of all equivalence classes of "A" under "r", then for every pair of elements "x" and "y" in "C", the pair "(x, y)" is in the relation "r".
If a function 'f' has a contour integral 'i' along a circular path centered at 'z' with radius 'r', and 'k' is a finite set, 'B' is a non-negative number, and 'r' is a positive number. Furthermore, for any 'x' such that the norm of 'x - z' equals 'r' and 'x' is not in 'k', the norm of 'f(x)' is less than or equal to 'B...
There exists a 'ws'' such that the function 'redT_updWs' updates the thread 't' with the work-stealing deque 'ws' and the list of stolen work 'was' to produce 'ws''.
Both closed boxes (cbox) and open boxes (box) with corners at points a and b are measurable with respect to the Lebesgue measure (lborel).
: Assuming that "q1" and "q2" are states in the FSM "M", and that the fixed OFSM table for "M" with respect to "q1" at position 0 is not equal to the fixed OFSM table for "M" with respect to "q2" at position 0, it shows that the first distinct OFSM table for "M" between "q1" and "q2" is equal to the minimum value "k" ...
If a list 'xs' is a sublist of 'v', and 'v' is an element of the directed vertices of 't', then 'xs' is a sublist of the denormalized 'ikkbz_sub' of 't'.
A relation "x" is one strongly connected if and only if the transposed relation of "x" composed with the identity relation and "x" is equal to the transitive closure of "x".
If the invariant "Inv2b" holds in state "s" and the "HEndPhase2" transition occurs from state "s" to state "s'" for process "p", then the invariant "Inv2b" also holds in the new state "s'".
Dropping a finite set S from a finite map m, after dropping an element b from the same map, is equivalent to dropping a finite set, which is the insertion of the element b into the set S, from the map m.
The lemma named "product_transition_split_ob" assumes that "t" is in the transitions of the product of A and B. It then obtains "t1" and "t2" such that "t1" is in the transitions of A and the source, input, output, and target of "t1" are the same as the first element of the source and target, and the input and output o...
The lemma "fresh_Cons" shows that "a" is fresh for the list consisting of "x" followed by "xs" if and only if "a" is fresh for "x" and "a" is fresh for "xs".
The lemma "embed_measure_prod" assumes that functions 'f' and 'g' are injective, and that measures 'M' and 'N' are sigma-finite. It shows that the product of the embedded measures of 'M' and 'N' using 'f' and 'g' respectively, is equal to the embedded measure of the product of 'M' and 'N' using a function that applies ...
If a regular expression "r" is safe in the past and strict, and if "s" is in the right-past derivative of "r" with respect to a test and an index "i", then the set of free variables in "r" is a subset of the set of free variables in "s".
The composition of a relation "s" with itself is a subset of "s".
If the function "wprepare_add_one2" with parameters m, lm, and a pair consisting of b and an empty list is true, then the function "wprepare_add_one2" with parameters m, lm, and a pair consisting of b and a list containing "Oc" is also true.
: Given two lambda expressions "Lam [x]. e" and "Lam [x']. e'" are equal, and for any permutation "pi", if the support of the inverse of "pi" is disjoint from the set of free variables in "Lam [x]. e", and the support of "pi" is disjoint from "Lam [x]. e", then applying the function "F" to the permuted expression, perm...
If you have a record with xpos equal to m, ypos equal to n, and other fields equal to p, and you set xpos to 0, the resulting record will have xpos equal to 0, ypos still equal to n, and the other fields still equal to p.
For any natural number 'n' and any function 'c' from natural numbers to a domain that is an integral domain and a real normed division algebra, the following is true: for all 'x', the sum from 'i' to 'n' of 'c i' times 'x' to the power of 'i' equals the sum from 'i' to 'n' of 'd i' times 'x' to the power of 'i' if and ...
: Given a list "nl" that is equal to a list of zeros of length "xs" concatenated with the result of the recursive execution of a function "f" with parameters "gs" on "xs", followed by a list of zeros of length equal to the maximum of the successor of the length of "xs" and the maximum of the set obtained by inserting ...
: Assuming that for any function f_i and any integer i, if (f_i, i) is in the set fis, then either the number of roots of the polynomial f_i is not equal to the degree of f_i, or the polynomial f_i evaluated at 1 is zero, or the polynomial f_i evaluated at -1 is zero, then the successor of i is less than or equal to d...
If 'p' is a prime element and 'p' divides the product of a list 'A', then there exists an element 'a' in the set 'A' such that 'p' divides 'a'.
The lemma named "finite_Psegments_less_eq1" assumes that the set of P-segments from point 'a' to point 'c' is finite and that 'b' is less than or equal to 'c'. Under these assumptions, it shows that the set of P-segments from point 'a' to point 'b' is also finite.
The function "n" of the product of "x" and "top" is equal to the function "n" of the product of "x" and "L".
The lemma named "conv_radius_shift" is defined with a fixed function 'f' that maps a natural number to a type 'a', where 'a' is a Banach space and a real normed division algebra. This lemma shows that the convergence radius of the function 'f' shifted by 'm' is equal to the convergence radius of the function 'f'.
If "x" is an element of the set "evnodd A b", then "x" is also an element of the set "A".
The lemma named "union_list_nodes_finite" assumes that for all elements in the list AA, the set of nodes in the deterministic cellular automaton (DCA) is finite. It shows that the set of nodes in the DCA of the union of all elements in the list AA is also finite.