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The lemma named "subprob_density_distr_real_exp" assumes that "f" is a subprobability density function on the measure space "M" with respect to the Lebesgue measure "lborel". It shows that the distribution of the exponential function "exp" on "M" with respect to the Borel measure also has a subprobability density. This...
If 'd' is an ordinal according to set 'H', then 'H' also confirms that the function 'HFun_Sigma' applied to the expression 'Eats Zero (HPair d d')' is true.
The lemma named "valid_single" asserts that a flow graph "fg" is valid for a singleton multiset containing "s".
If 'n' is greater than 0, then the sum of the Bernoulli number of '2n' and the sum of the reciprocals of all prime numbers 'p' such that 'p-1' is a divisor of '2n', is an integer.
The function 'lsumr' is injective on the set 'A'.
A type T is an Array if and only if there exists a type U such that T is an array of U.
For a function 'f' mapping from a Euclidean space 'a' to another Euclidean space 'b', if 'S' is Borsukian, 'f' is continuous on 'S', the image of 'S' under 'f' is 'T', 'S' is compact, and for any subset 'U' of 'S' that is open in the topology induced by 'S', the image of 'U' under 'f' is open in the topology induced by...
An empty set is single-valued.
The monomials of a constant indexed by a ring R is equal to an empty set if the constant k is zero, otherwise it is a set containing an empty set.
If a, b, x, and y are elements of the carrier set R, then the product of the sum (or difference) of a and b with the sum (or difference) of x and y is equal to the sum (or difference) of the products of a and x, a and y, b and x, and b and y.
The lemma ideI: Assuming "f" is an arrow, the domain of "f" is equal to the codomain of "f", and for any "x" in the domain of "f", the function of "f" at "x" equals "x", it can be shown that "f" is an identity.
The sum of 't' and the pair '(s, i)' is equal to the pair '(t + s, i)'.
For any real number y, if x raised to the power of y is true and z is greater than or equal to zero, then the function of m, which is z times x of m, raised to the power of z times y, is also true.
If set I is not empty and n is not the smallest element in set I, then the smallest element of the set obtained by removing n from I is the same as the smallest element in set I.
The lemma "eval_mod_exp" states the following: 1. The modular exponentiation of any base 'b' to the power of 0 modulo any 'm' is equivalent to 1 modulo 'm'. 2. The modular exponentiation of any base 'b' to the power of 1 modulo any 'm' is equivalent to 'b' modulo 'm'. 3. The modular exponentiation of any base 'b' to t...
The renaming function 'h' applied to 'c' equals 'Spec r' if and only if 'c' equals 'Spec r'.
The value of the function n at x is less than or equal to the value of the function n at the product of n(x) and L.
The lemma named "matrix_matrix_mult_inner_mult" shows that the element at the ith row and jth column of the product of two matrices A and B is equal to the dot product of the ith row of matrix A and the jth column of matrix B.
If a set S is pairwise orthogonal (meaning every pair of elements in S is orthogonal) and does not contain the zero vector, then the set S is independent.
The inverse of "thetaParZOL1" is a subset of the retracement of the union of the inverse of "thetaParZOL1" and "ZObis".
Zero is less than infinity in the context of extended nonnegative real numbers.
In the context of a field, the lemma "exists_maximal_extension" shows that there exists a set M within the set S such that for all sets L within S, if M is a subset of or equal to L, then L is equal to M. This essentially means that M is the maximal extension within the set S.
If the length of a list 'xs' is 'n' and 'xs' is not an empty list, then converting 'xs' from a binary vector to a natural number and then back to a binary vector of length 'n' will result in the original list 'xs'.
The lemma named "ac_p_int_pow_factor" assumes that "a" is an element of the set of non-zero p-adic numbers. It shows that the angular component of the product of "a" and the p-adic integer raised to an integer power "n" is equal to the angular component of "a".
The density of the measure space M with respect to the function f, and then with respect to the function (g(x) / f(x)), is equal to the density of the measure space M with respect to the function g.
The count of equal pairs in a list is less than or equal to the length of the list.
If the lengths of two lists, xs and ys, are equal, then the result of zipping the paths of these two lists appended with elements x and y respectively, is equal to the result of zipping the paths of the original lists appended with the result of zipping the paths of lists containing x and y only.
The lemma "SN_on_weakening" assumes that the relation R1 is strongly normalizing on set A. It shows that the intersection of relations R1 and R2 is also strongly normalizing on set A.
The double substitution of a multiset of atomic formulas "AAs" with a composition of substitutions "τs" and "σs" is equal to the double substitution of "AAs" with "τs" followed by the double substitution with "σs".
: given a subring K of R, and two polynomials f and g that belong to the carrier of K[X] and are not equal to zero, the degree of the product of f and g is equal to the sum of the degrees of f and g.
If two lists 'xs' and 'ys' are distinct, then the selection function 'f' applied to these lists is equal to the selection function 'f' applied to the sets of 'xs' and 'ys'.
For any infinitesimal star number 'x' in a real normed field or Banach space, the product of the star-transformed sine of 'x' and the star-transformed cosine of 'x' is approximately equal to 'x'.
If "a" divides the greatest common divisor of "b" and "c", then "a" also divides "b".
: "If for every D in the set of ground terms of the first element of Cl, there exists an element El in N such that El is strictly less than Cl and D is in the set of ground terms of the first element of El, then Cl is in the set of fully reduced terms with respect to N."
The lemma named "mem_add_set" assumes that both "A" and "B" are cuts. It shows that the addition of set "A" and set "B" is also a cut.
There exist functions f and g such that h is the parallel composition of f and g.
The extra call-context bindings in a context Γ at a position i is equal to the join of the second element of i, the call-context bindings in Γ at position i, and the join over all pairs (x, e) in the map of Γ of the product of the free variables in e and the call-context neighbors of x in the second element of i.
The distance from a point 's' to itself in a directed graph is zero.
The norm of the projection of a complex Hilbert space (or a chilbert space) onto a closed convex subspace is less than or equal to 1.
For all sets A and B, if A is closed, B is closed, and A and B are disjoint, then A and B are separated.
Set A is a subset of set B if and only if for every element t, if t is in set A then t is also in set B.
For the function "bind_id" and "bind_pmf", if the first argument is a computational representation of identity probability of type A, and the second argument is a function from type A to a computational representation of identity probability of type B, then the result is a computational representation of identity proba...
The top element is equal to the healthiness condition applied to "false".
The function "funas_term" applied to the result of "adapt_vars" function on a term 't' is equal to the function "funas_term" applied directly to the term 't'.
The inverse of the strict set of pairs (x, y), where x is a constant and x is less than or equal to y, is a subset of the measure m_const.
The lemma named "rank_mat_mul_left" is about matrices. It takes two fixed matrices A and B. A is a matrix of type 'a, where 'a is a conjugatable ordered field. The lemma assumes that A is a matrix with n rows and d columns, and B is a matrix with d rows and nc columns. The lemma shows that the rank of the product of A ...
If the compiled code of a command 'c' is an empty list, then the execution of the command 'c' on a state 's' results in the same state 's'.
If the substitution of a term 't' with a substitution function 'σ' is equal to the substitution of the same term 't' with another substitution function 'τ', then for all variables 'x' in the term 't', the result of applying the substitution function 'σ' to 'x' is equal to the result of applying the substitution functio...
The function "poly_nate" applied to "bs" and "p" is equal to the function "polypoly'" applied to "bs" and the result of "polynate p".
Dropping a certain number of elements 'i' from the concatenation of two lists 'xs' and 'ys' is equivalent to dropping 'i' elements from 'xs' and dropping 'i' minus the length of 'xs' elements from 'ys', then concatenating the results.
If x and y are both in the Standard set, then the difference of x and y is also in the Standard set.
The theorem named "thm_2_3_6_indets_Dube" is defined for any given 'fs'. It defines 'X' as the union of the indeterminates in the set 'fs'. It shows that the set generated by the Macaulay list of the degree shifts of 'X' with respect to the Dube function (which takes the successor of the cardinality of 'X' and the maxi...
, given the assumptions that a constant "c" is not equal to zero and a polynomial "p" is not equal to zero, the degree of the polynomial resulting from the monomial multiplication of "c", "t", and "p" is equal to the degree of the polynomial "t" plus the degree of the polynomial "p". This is under the condition that "a...
Word "a" is equal to word "b" if and only if the unsigned integer representation of "a" is equal to the unsigned integer representation of "b".
For any given x, the function C applied to a list p is equal to the function C applied to the list obtained by removing duplicates from p.
The lemma P1_update_P1: assumes that function 'f' is in set 'P' shows that the function 'f' with 'x' replaced by 'z' is also in set 'P'.
For all natural numbers i and j, a property P holds for all elements in the list from i to j if and only if the property P holds for all elements in the natural number interval from i to j.
If we have a function 'g' mapping from a set 'A' to the carrier of the p-adic numbers, and 'A' is a finite set, and for every element 'i' in 'A', the valuation of 'g(i)' is greater than a constant 'c', and 'c' is less than infinity, then the valuation of the finite sum of 'g' over 'A' is also greater than 'c'.
If digraph homomorphism 'GG' maps from digraph 'BB' to 'CC' and digraph homomorphism 'FF' maps from digraph 'AA' to 'BB', then the domain of the arrow map of the composition of 'GG' and 'FF' is equal to the set of arrows of 'AA'.
The lemma named "finite_ta_Regions'" is defined with a fixed timed automaton 'A'. Assuming that 'A' is a finite timed automaton, it is possible to obtain variables 'v', 'n', and 'x' such that they satisfy the condition 'Regions' with the clock set of 'A', 'v', 'n', and 'x'.
If a heap 'h' is well-formed, has a valid type, and has known pointers, and if a node 'node_ptr' is adopted by a document 'document_ptr' resulting in a new heap 'h'', then if a pointer 'ptr' is not in the set of the DOM component of the document, the node, or the owner document of the node in the original heap 'h', the...
For the function "int of_nat" (which converts a natural number to an integer), if two natural numbers are equal, then their corresponding elements in the residue ring are also equal. This is a transfer rule for the residue ring type.
The lemma named "ordering_sym_loc" assumes the following: 1. There is an order symmetry, denoted as "ord_sym", such that for any three elements a, b, and c, if a is ordered with respect to b and c, then c is also ordered with respect to b and a. 2. The set X is finite. 3. There is a local ordering function f that es...
The lemma named "line_integral_exists_smooth_one_base" assumes that the function "gamma" is continuously differentiable on the interval from 0 to 1. It also assumes that the function F, when applied to the image of the path "gamma" and then projected onto the direction "b", is continuous. Under these assumptions, the l...
If for all subsets S of the topological space X, the image of the closure of S under the function f is a subset of the closure of the image of S under f in the topological space Y, then the function f is a continuous map from X to Y.
The function "e2lenas" applied to the encoding of a configuration consisting of a list with elements (f, xs, ls) followed by ss, and rv, is equal to the length of xs.
The restriction of a relation 'r' to a set 'A' is a subset of the set of ordered pairs in 'r'.
The lemma named "prappend_signed_list_assoc_conv_snoc2Cons" assumes that both "xs@[y]" and "y#ys" are proper signed lists. It then shows that the operation "prappend_signed_list" applied to the list "xs@[y]" and the list "ys" is equal to the operation "prappend_signed_list" applied to the list "xs" and the list "y#ys".
Taking the bit at the position specified by the numeral "l" from the numeral represented by "Num.Bit1 k" is equal to taking the bit at the position specified by the predecessor of the numeral "l" from the numeral "k", multiplying that by 2, and then adding 1.
If an expression 'e' is of type 't' under context 'Γ' and 'e' is random-free, then the expression 'e' converted from random-free to a C-expression is also of type 't' under the same context 'Γ'.
The lemma L7_Controller_One: assumes that there is a controller L with parameters y, a finite or infinite list appended with zero, l, and z, and that the t-th element of list l is one, and that the first element of y is 500 (a natural number), it shows that the last element of the finite, untimed version of the...
If the invariant holds for both the left and right elements, then the invariant will also hold for the result of joining the two elements.
, given the assumption of a weak universal property for a type 'x, a relation R, two labeled graphs G1 and G2, and two functions f1 and f2, it shows that for any functions h1 and h2 and a labeled graph G of type ('a, 'x), if h1 is a graph homomorphism of the second component of R and G, and if h2 is a graph homomorphis...
The lemma named "extended_ord_lp" assumes that "None" is not in the indices of polynomial "p". It shows that the leading power product of the polynomial "p" with restricted indices is equal to the leading power product of the polynomial with restricted indices.
For any context 'cn', if a transition from state 's' with context 'c' to state 's' with context 'c'' through event 'e' exists in the transition relation of the flowgraph, and the intersection of the set of monitors of event 'e' and the set of monitors of context 'cn' is empty, then there exists a context 'csp' such tha...
The lemma named "graph_domain_funct" assumes that for any x, y, and z, if the pair (x, y) is in g and the pair (x, z) is also in g, then z must be equal to y. This assumption implies uniqueness. Under this assumption, the lemma shows that the graph of the function g, when restricted to its domain, is equal to g itself.
If a string "s" is derived from a string "δ" at position "i" using rule "r", and if "δ" splits at position "i" into "α", "N", and "β", then "α" is equal to the first "i" characters of "δ" and "s", and the length of "α" is "i".
"if for all 'x' there exists a 'y' such that property 'P' holds for 'x' and 'y', then there exists a function 'f' such that for all 'x', property 'P' holds for 'x' and 'f(x)'".
An empty list paired with any list 'xs' does not belong to the relation 'listrel1 r'.
If a set F is a subset of a set G, then the restriction of the identity relation to the transformation of F under G is a subset of the restriction of the identity relation to the transformation of G under G.
: Given a function 'f' from any type 'b' to a well-ordered type 'a', and a set 'A' that is not empty, if a property 'P' holds for the least value 'y' such that there exists an 'x' in 'A' for which 'y' equals 'f(x)', then there exists an 'x' in 'A' for which the property 'P' holds for 'f(x)'.
If two conditions, L and R, are both valid in the normal form (NF) with respect to a postcondition Q, then a third condition, which is dependent on a condition C and chooses between L and R based on the state s, is also valid in the normal form with respect to the same postcondition Q. This is achieved through the oper...
If a sequence of events 'evs' is in 'srb', then the key 'pairK(A,B)' is in the analysis of what the Spy knows from 'evs' if and only if the card 'B' is cloned.
If a knowledge base K is equal to the deductive closure of a set A, and the intersection of the truth-related selection function of K applied to the conjunction of two propositions phi and psi, and the bottom element of K and phi is not empty, then the truth-related selection function of K applied to phi is a subset of...
The polynomial obtained by mapping a homomorphism on a polynomial 'p' is equal to the evaluation of the polynomial 'p' with the same homomorphism.
: if a configuration 'cfg' is a valid configuration, then the value of 'cfg' is 0 if and only if for almost every 'x' in the transition function of 'cfg', it is not the case that the state 'cfg' holds until it transitions from state S1 to state S2.
The intersection of the connected component set S of a and the connected component set S of b is empty if and only if a is not in the connected component set S of b.
If "G" is a group and "w_cmpser G (Suc n) f" holds, then "Gchain n (Qw_cmpser G f)" also holds.
If the event "ev_at" occurs at the hold "HLD" of the single set {x} at the nth position in the stream "omega", then the nth element of the stream "omega" is x.
The lemma mayDiverge_Seq_L: Assuming "c1 may diverge from state s", it shows that "the sequence of c1 and c2 may diverge from state s".
The iteration of a function 'c' is equal to the minimum of 'nil' and the composition of 'c' with its own iteration.
A zero vector of size zero is equivalent to an empty vector.
If 'r' is a member of complex rotations, then the following two conditions hold: 1. For all 'v', if 'r' applied to 'v' does not equal 'v', then if 'r' applied to 'v' equals 'w', 'r' applied to 'w' equals 'v'. 2. If 'r' applied to 'v' equals 'v' and 'r' applied to 'w' equals 'w' and 'v' is not equal to 'w', then 'r' is...
The lemma named "fps_deriv_lr_inverse_comm" is about formal power series (fps). It fixes a variable 'x' of a commutative ring. It assumes that the product of 'x' and the zeroth term of the formal power series 'f' equals 1. It then shows two things: 1. The derivative of the left inverse of 'f' with respect to 'x' equa...
If n is greater than 0, if there is a decreasing group chain of length n for some function f, and if k is less than or equal to (n - 1), then the successor of k in the function f is a subset of f at k.
The lemma named "degree_sum_smaller" assumes that "n" is greater than 0 and "A" is a finite set. It shows that if for every "x" in "A", the degree of "f(x)" is less than "n", then the degree of the sum of "f(x)" for all "x" in "A" is also less than "n".
The lemma is named "leibniz_rule_field_derivative". It fixes a function 'f' from a real normed field and Banach space to a Euclidean space. It assumes the following: 1. For any 'x' and 't' where 'x' is in 'U' and 't' is in the closed interval from 'a' to 'b', the function 'f' with respect to 'x' and 't' has a field de...
Running the "writer" function on the alternative of two monads 'm' and 'm'' is equivalent to the alternative of running the "writer" function on each monad 'm' and 'm'' separately.
"If set A is a subset of set B, then every element in set A is also an element in set B."
The lemma named "split_max_height" assumes that the result of the function "split_max" with parameters "k" and "t" equals "(sub,sep)", the last tree "t" is not empty, and "t" is not a leaf. It shows that the height of the subtree "sub" is equal to the height of the tree "t".