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If for every element 'x' in set 'X', 'x' is finite, then the minimal product set of 'X' is also finite.
The lemma for the formal power series of sine in terms of the formal power series of the exponential function is: the formal power series of sine of a constant c is equal to the difference of the formal power series of the exponential of imaginary unit times c and the formal power series of the exponential of negative ...
If 'fs' and 'newFs' are distinct lists and the intersection of the sets 'fs' and 'newFs' is a subset of the set containing 'oldF', then the list resulting from replacing 'oldF' with 'newFs' in 'fs' is also distinct.
If the pair (x, y) is in the relation R, then the pair (x, y) is also in the partial equivalence relation union of R, a, and b.
The lemma named "bfun_prob_space_integrable" assumes that "S" is a probability space, "v" is a Borel measurable function, and "v" is a bounded function. Under these assumptions, it shows that "v" is integrable over the probability space "S". Here, 'a and 'b represent arbitrary types, with 'b belonging to the class of s...
The lemma Prim1_imp_R1 implies that if function 'f' belongs to the set Prim1, then function 'f' also belongs to the set R.
The limit of function 'f' as it approaches 'l' at point 'x' within the union of sets 's' and 't' is equivalent to the limit of function 'f' as it approaches 'l' at point 'x' within set 's' and the limit of function 'f' as it approaches 'l' at point 'x' within set 't'.
The lemma lm003 assumes that for every element x1 in set X, x1 is not an empty set and for every other element x2 in set X that is not x1, the intersection of x1 and x2 is an empty set. Based on these assumptions, it shows that set X is non-overlapping.
For a given multiset A and an element 'a', adding 'a' to the multiset A is a multiset extension of A under the relation P.
The set of all elements in the index list up to 'n' is equal to the set of all numbers less than 'n'.
Theorem T3: "It is possibly true that God necessarily exists."
The count of a particular element 'x' in the location 'loc' of the pointstamps of a configuration 'c' remains the same even after the pointstamps of the configuration 'c' are updated to 'M'.
The ASCII representation of the literal zero is an empty list.
For any expression 'e' and environment 'ρ', the restriction of the evaluation of 'e' in 'ρ' to the set 'C·r' is equal to the restriction of the evaluation of 'e' in the restricted environment 'ρ' to the set 'r', further restricted to the set 'C·r'.
The lemma "SNC_visPost" assumes that the visibility is not equal to "FriendV", the transition is valid, the user ID is strongly not connected with the transition, and the source of the transition is reachable. It shows that the visibility of the target of the transition is not equal to the visibility.
Taking the bit at position 'n' of the number 1 is equivalent to checking if 'n' is greater than 0, and converting the result to a boolean value.
If 'z' is a real number and the real part of 'z' is greater than zero, then the complex representation of 'z' belongs to the positive x-axis.
If R is an element of set R, and v is an element of R, and t is greater than or equal to 0, and the sum of v and t is an element of R', and R' is an element of set R, then R' is an element of the successor set of R in set R.
: if for all s1 and s2, the build of s1 in X is equal to the build of s2 in Y implies a contradiction, then X is different from Y.
The lemma named "omit_branch_mem_diff_opening" assumes that "i" only touches the "branch" at points "xs", that the pair "(ps, a)" is in the "branch", and that "i" is not equal to "a". Under these assumptions, it shows that the pair "(ps, a)" is in the "branch" from which "xs" has been omitted.
The lemma named "nemp" assumes that "u" is an element of "C". It shows that "u" is not equal to "epsilon".
If a triple [b', a, f] is an object in the comma category of b and functor F, and b is an object in category B, then it can be shown that a is an object in category A, b' is equal to 0, and f is a morphism from b to the object mapping of a under functor F in category B.
In the context of a functor, there is a lemma named 'cf_universal_arrow_of_if_is_iso_ntcf'. It assumes that 'r' is an object in category 'A', 'c' is an object in category 'B', and 'phi' is an isomorphism natural transformation from the hom-functor of 'r' in category 'A' to the composition of the hom-functor of 'c' in c...
The evaluation of a ternary AND operation between t1 and t2 results in TernaryFalse if and only if either t1 or t2 is TernaryFalse.
: 1. The star of a regular expression 'r' is equal to the mapping of a regular expression function 'f' over 'x' if and only if there exists a regular expression 'r'' such that 'x' is the star of 'r'' and the mapping of the regular expression function 'f' over 'r'' is 'r'. 2. The mapping of a regular expression functi...
The lemma mset_nths: Assuming that "l" is less than or equal to "r" and "r" is less than or equal to the length of list "xs", Assuming that for all "i", if "i" is less than "l", then the "i"th element of list "xs" is equal to the "i"th element of list "ys", Assuming that for all "i", if "i" is greater than or equal to ...
A list 'xs' is a prefix of the concatenation of 'xs' and 'ys' if and only if 'ys' is an empty list.
For any formula "phi", the function "create_gba" applied to "phi" is less than or equal to the specification of a function "A" such that "nds_invars" applied to "g_V of A" holds true.
Given a matrix A of size n by n, and assuming that A is an upper triangular matrix in a real normed field, and that for any pair (n, a) in the set resulting from converting A to a Jordan Normal Form vector, the norm of a is less than or equal to 1, and if the norm of a is equal to 1, then n is less than or equal to N. ...
The lemma is about cubic reduction. It fixes a complex number 'a'. It assumes that 'a' is not equal to zero, 'x' is equal to 'y' minus 'b' divided by three times 'a', 'p' is equal to three times 'a' times 'c' minus 'b' squared, all divided by nine times 'a' squared, and 'q' is equal to nine times 'a' times 'b' times 'c...
If the representation of permutation p1 is equal to the representation of permutation p2, then permutation p1 is equal to permutation p2.
The lemma "Putstatic_nInit_correct" is about a specific state in the Java Virtual Machine (JVM). It assumes the following: 1. The program P is well-formed according to the typing rules wt. 2. The program P can see a method M with a certain signature and body in class C. 3. The instruction at the program counter pc is ...
If the function "lift_opts" applied to the mapping of a function (which finds the type of a variable in a given context) over a list of variable declarations results in some type list, then the length of the list of variable declarations is equal to the length of the type list.
If the second element of the tuple resulting from applying the function "usubstappp" to the parameters sigma, U, and Loop alpha is not undefined, then both the second element of the tuple resulting from applying "usubstappp" to sigma, U, and alpha, and the second element of the tuple resulting from applying "usubstappp...
If the size "sz" is less than the length of a certain type 'a, then the unsigned natural number representation of the bitwise AND operation between a pointer "ptr" of type 'a and the bitwise NOT operation on the mask of size "sz", added to the unsigned natural number representation of the bitwise AND operation between ...
If the exception configuration "conf_xcp" is conformant with the program "P", the heap "h", the exception "xcp", and the type "i", and if heap "h" is a sub-heap of heap "h'", then the exception configuration "conf_xcp" is also conformant with the program "P", the heap "h'", the exception "xcp", and the type "i".
If n is greater than 0, then the following concurrent program is valid: The program starts with a scheme that runs for each i from 0 to n (exclusive). In each iteration, it increments the variable y by 1 and sets the i-th element of the array C to 1. The pre-condition of the program is that y equals the sum of all e...
If a relation R is single-valued and its inverse is also single-valued, then the subset relation is in the set relation of R, mapping to a boolean relation.
The maximum degree of a node in a decision tree is equal to the maximum value between the cardinality of the node's children and the maximum degree of the first element in the set of the node's children.
Lemma Almost Everywhere Implies Nonnegative Integral: Given a set of elements {x} in the space M such that the property P holds for x, and this set is in the sets M, then the statement "almost everywhere in M, the property P holds for x" is equivalent to the nonnegative integral of the indicator function for the set of...
In the context of complex square matrices, if A is a finite complex matrix and is unitary, and if j and k are less than the dimension of the matrix and are not equal to each other, then the product of the rank 1 projection of the jth column of matrix A and the rank 1 projection of the kth column of matrix A is equal to...
The product algebra 'I M' is intersection-stable.
The lemma named "cartesian_power_cons" assumes that "as" is in the carrier of the nth power of R and "a" is in the carrier of R. It shows that the list consisting of "a" followed by "as" is in the carrier of the (n+1)th power of R.
The Catalan number for a given integer 'n' is equal to the auxiliary Catalan number with parameters 'n', 0, and 1.
If F is a monic arrow from A to B in the category of sets with a fixed cardinality alpha, then the map of variable finite sequences F from A to B is also a monic arrow in the same category.
The lemma named "rebase_eq_0" shows that for any 'x' of type 'a mod_ring', 'x' is equal to zero if and only if 'x' is equal to zero when cast to type 'b mod_ring'.
If a node pointer, when used to get a parent in a heap, returns None (indicating that the node has no parent), then getting the root node in shadow-including mode with the cast of this node pointer will return the cast of the node pointer itself.
Interpreting the float arithmetic of 1 with any list of values (xs) will always result in 1.
The front of the list containing only the element "a" is free of ticks.
The downset of the supremum of a set X is equal to the downset of the supremum of the union of the downsets of the elements of the set X, where X is a set in a quantale.
The lemma named "poly_root_induct" is about a polynomial "p" over an integral domain. It assumes three conditions: 1. The property "Q" holds for the zero polynomial. 2. For any polynomial "p", if for all "a" such that property "P" holds for "a", "a" is not a root of "p", then property "Q" holds for "p". 3. For any "a"...
The product of a single element set r and delta2 under the function delta_prod_sng1 is equal to the product of the set {r} and delta2 under the function delta_prod. The second part states that the product of delta1 and a single element set r under the function delta_prod_sng2 is equal to the product of delta1 and the s...
The strip of the abstract command obtained by mapping a function 'f' over an abstract command 'c' is equal to the strip of the original abstract command 'c'.
: Assuming that "X" is finite and "ps" is a valid decomposition, and "ps" is a cone decomposition of a set "T" of type "(('x maps to nat) maps to 'a)" where "'a" is a commutative ring with no zero divisors, and "ps" is a standard decomposition of "k", and "k" is less than or equal to "d", then there exists a "qs" such ...
If 'f' is an element of the carrier 'P', 'a' is an element of the carrier 'R', and 'm' is greater than or equal to 'n', then the product of the monomial 'P' with 'a' at the 'n' position and 'f' at the 'm' position is equal to 'a' times the value of 'f' at the position 'm - n'.
The edges of a graph 'g' are symmetrical.
The lemma named "is_minimal_basis_trd_distinct" assumes that "x # xs" forms a minimal basis (as per the assumption "min") and that "x # xs" are distinct elements (as per the assumption "dist"). Under these assumptions, it shows that the elements of the list formed by the term rewrite descent of "x" with respect to "xs"...
The lemma named "strict0'" simplifies that if the function 'f' is applied to invalid arguments 'X' and 'Y', the result will be invalid.
The lemma named "dependent_with_transfer" is a transfer rule. It includes lifting syntax and assumes that "A" is right total and bi-unique. It shows that the function "dependent.with" is preserved under a relation that is defined by equality, the relation "A", a binary operation on "A", a binary relation involving equa...
The lemma named "word_mult_le_iff" is defined with the following conditions: - It fixes 'i' as a word of any length 'a'. - It assumes that 'k' is greater than 0. - It assumes that the product of the unsigned natural number representations of 'i' and 'k' is less than 2 raised to the length of the type 'a'. - It also as...
If 'n' is less than the cardinality of the set 'I' cut at 't', then the 'n'th element of the set 'I' cut at 't' is equal to the 'n'th element of the set 'I'.
If the key (shrK A) is part of what the Spy knows in the events (evs) and these events are part of the recursive protocol, then A is considered bad.
In the context of an integral domain with characteristic 0, for a non-zero polynomial 'p', the polynomial [-a, 1] raised to the power of the order of 'a' in 'p' divides 'p'. However, the polynomial [-a, 1] raised to the power of the successor of the order of 'a' in 'p' does not divide 'p'.
The lemma named "symmetry3" assumes that the relation "R" is symmetric. It shows that the transitive closure of "R" is also symmetric.
If an unordered resolution of clauses C, D, and E is true, and if clause C is satisfied by interpretation I, and if clause D is also satisfied by interpretation I, then clause E must also be satisfied by interpretation I.
The lemma named "nat_div" assumes that a natural number "b" is greater than 0 and "a" is greater than "b". It shows that the division of "a" by "b" is equal to the division of the difference of "a" and "b" by "b", plus 1.
The lemma named "cylinder_prob" assumes that "A" is a subset of "S_edges", "B" is a subset of "S_edges", and "A" and "B" have no intersection. It shows that the probability of the cylinder function applied to "S_edges", "A", and "B" equals "p" raised to the power of the cardinality of "A" times (1 - p) raised to the po...
If the nth element is less than the prefix length of a polynomial P with respect to a list xs, then the polynomial P holds true for the nth element of the list xs.
If the dual check for equivalence between n, phi, and psi holds, then the language of phi in M2L at n is equal to the language of psi in M2L at n.
Applying the function 'f' to the infinite matrix 'A' at the positions 'j' and 'i' is equivalent to applying the function 'f' to the element at position 'j' and 'i' in the matrix 'A'.
If a rule "r" is a member of the semantic path rules for a given configuration "q\<^sub>c", initial state "q\<^sub>i", and final state "q\<^sub>f", then there exists a number "n" and an index "i" such that the root of the rule "r" paired with "n" is a member of the set "F", "i" is less than "n", the right-hand side of ...
The lemma named "close_open_invariant" assumes that action I, under condition U, leads to P in system A. It shows that in system A, condition I directly leads to P.
If the second elements of the tuples in the list 'xys' are distinct, then the function 'map_of xys' is injective on its domain.
If the set of "cts" is a subset of the insertion of "Yield" into the range of "Join", then the conditional action "cond_action_oks'" is okay for "s" and "t".
If the invariant "bq" holds, then the size of the queue converted to a multiset "bq" is greater than or equal to 2 raised to the power of the length of "bq" minus 1, subtracted by 1.
If "x" is less than or equal to "y" and "w" is less than or equal to "z", then the union of the omega power of "x" intersected with "L" and the star power of "x" multiplied by "w" is less than or equal to the union of the omega power of "y" intersected with "L" and the star power of "y" multiplied by "z".
The lemma named "src_comp" assumes that the transition "t" is composable with transition "u". It shows that the source of the composition of "t" and "u" is the same as the source of "t".
"The interval I shifted negatively by k equals an empty set if and only if the interval I is an empty set or the interval I is finite and its maximum value is less than k."
If the substitution range of a variable 's' is ground (i.e., it contains no free variables), then the range of variables for 's' is an empty set.
The inverse of a given partial order on a set L is also a partial order.
For a given real number c and a function f that is measurable with respect to the Lebesgue measure, if c is not equal to 0, then the non-negative integral of f with respect to the Lebesgue measure is equal to the absolute value of c times the non-negative integral of f(t + c * x) with respect to the Lebesgue measure.
: 1. If "P" and "R" are true, and if "R" implies "Q", then if "L" implies "P", "Q" is also true. 2. If "P" is true, then if "L" is true, "P" remains true.
A pair of multisets (M, N) is in the multiset extension of the less-than relation {(x, y). x < y} if and only if M is less than N in the multiset ordering with respect to the higher-order function.
The norm of x is equal to the complex modulus of the one-dimensional isomorphism of x.
Decrypting a message "U" with a key "K" using the "Decrypt" function is equivalent to applying the "DECRYPT" function with the same key "K" to the message "U", and then applying the "Abs_Msg" function to the result. The "msgrel``{...}" notation is used to denote the set of all messages related to the message inside the...
The lemma named "butlast_take_length_helper" is defined with respect to a list of characters 'x'. It assumes three conditions: 1. The length of list 'i2' is less than or equal to the length of list 'i1'. 2. The first 'length of i2 - 1' elements of list 'i1' are equal to the list 'i2' with its last element removed. ...
The reverse of string u concatenated with the reverse of string v is equal to the reverse of string v concatenated with the reverse of string u if and only if string u concatenated with string v is equal to string v concatenated with string u.
The absolute value of the sine of any real number x is less than or equal to one.
Lemma borel_measurableI_ge: Let f be a function from type 'a to type 'b, where 'b' has a linear order topology and a second countable topology. If for all y, the set of all x in the space M such that y is less than or equal to f(x) belongs to the sets M, then f is borel measurable in M.
The domain of the body is finite.
The norm of x is less than the norm of y if and only if the inner product of x with itself is less than the inner product of y with itself.
The set of events in the restricted measure space is equal to the set of events in the inverse image algebra, where the inverse image algebra is defined on the intersection of the set Omega and the space of the measure M, with the identity function applied to M.
The lemma named "per_cong_mid" assumes that points B and C are not the same, point B is between points A and C, the distance from A to H is congruent to the distance from C to H, and the angle HBC is a right angle. Under these assumptions, it shows that point B is the midpoint of points A and C.
The lemma named "inverse_assoc_assoc'" assumes that "f", "g", and "h" are identities, and that the source of "f" is the target of "g", and the source of "g" is the target of "h". It shows that the arrows "assoc_SB f g h" and "assoc'_SB f g h" are inverses of each other.
If a new document is successfully created and assigned to "new_document_ptr" in a heap "h", and if this creation results in a new heap "h'", then any location "r" in the set of locations returned by "get_tag_name_locs" for any pointer "ptr'" must be a relation between the original heap "h" and the new heap "h'".
If X is an element of the synthetic analysis of H, then the parts of X are a subset of the union of the synthetic analysis of H and the parts of H.
If 'n' is less than 'm' and 'm' is a new type variable in the substitution 'S', then 'm' is also a new type variable in the substitution of 'n' by 'S'.
The lemma named "GB_subset" assumes that "G" is a Groebner basis, "G" is a subset of "G'", and the polynomial ideal generated by "G'" is equal to the polynomial ideal generated by "G". Under these assumptions, it shows that "G'" is also a Groebner basis.
There is a bijective function between the set of all complex numbers 'x' with an imaginary part equal to zero and the set of all points on a sphere with radius 1 centered at the origin, excluding the point -1. This function is defined as (i - x) / (i + x), where 'i' is the imaginary unit.
If a world model "M" is well-formed, then "M" is a basic world model.