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If the function "compile_fundef" applied to F and fd1 equals to Some fd2, then the return value of fd1 is equal to the return value of fd2.
The lemma named "exp_pref_cancel" assumes that the m-th power of a term 't' concatenated with 'y' equals the k-th power of 't'. It shows that 'y' equals the (k - m)th power of 't'.
The If function in Higher Order Logic (HOL), which takes a condition P and two outcomes t and f, is equivalent to the case_bool function with parameters t, f, and P.
If 'x' is injective and a vector, then the intersection of the root of 'p' with respect to 'x' and the transpose of 'x' is less than or equal to the function 'fc' of 'p'.
The function "funas_term" applied to the result of the function "term_to_sig" with parameters \<F>, v, and t is a subset of \<F>.
If function 'f' is an elliptic isometry, then the inverse of function 'f' is also an elliptic isometry.
If set S is a subset of set T, then the left ideal generated by set S is also a subset of the left ideal generated by set T.
For a fixed function 'f' that maps any element 'a' to a real number, given a finite set 'X', if the product of 'f' over all elements in 'X' is greater than or equal to the product of another function 'g' over all elements in 'X', and for every element 'x' in 'X', 'f' of 'x' is greater than or equal to zero, 'g' of 'x' ...
The lemma named "head_precedes" assumes that "y" is an element in the set that consists of "x" followed by elements of "xs". It shows that "x" precedes "y" in the sequence that starts with "x" and is followed by elements of "xs".
The lemma named "divide_zero_left" assumes that "z" is not equal to zero in homogeneous coordinates. It shows that zero in homogeneous coordinates divided by "z" is equal to zero in homogeneous coordinates.
If the length of a list x is greater than or equal to 2, then the tail of list x is not empty.
The lemma named "glb_common" is defined with fixed variables s1, s2, s, rs1, and rs2. It assumes that s1 is equal to the result of s combined with rs1, and that s2 is equal to the result of s combined with rs2. It then shows that there exists a variable rs such that the greatest lower bound of s1 and s2 is equal to the...
: Given the assumptions that "mi" is less than or equal to the downward refinement of "m" with respect to "E'" and "R", and for all "ei" and "e", if "(ei,e)" is in "E'", then "hi ei" is less than or equal to the downward refinement of "h e" with respect to "E" and "R", it can be shown that "CATCH mi hi" is less than or...
, under the assumption that points A, B, and C are not collinear (do not lie on the same straight line), it follows that no permutation of these points (A, B, C; A, C, B; B, A, C; B, C, A; C, A, B; C, B, A) are collinear either.
If the ordered pair (x, y) is an ordinal, then it can be inserted into set H to derive A.
For any given point 'a' in an ordered Euclidean space, the set of all points greater than or equal to 'a' is a closed set.
The auxiliary purge function for a trace, when applied to a set with an inserted value and a list, is equivalent to the auxiliary purge function applied to the original set and the purge function applied to the value and the list.
The lemma named "const_mem_subst_cases" assumes that the function "c" with no arguments is in the set "M" after applying the substitution "θ". It shows that the function "c" with no arguments is either in the set "M" or in the image of the set of free variables of "M" under the substitution "θ".
The convolution of functions f and g over a set R and an element x is equal to the diamond modal operator applied to the function that maps each pair (x, y) to the product of the results of applying f to x and g to y, evaluated at x. The set over which the diamond modal operator is applied is the set of all triples (x,...
In a probability space, if S and T are sigma-finite measures, X is distributed with respect to M and S with probability function Px, and Y is distributed with respect to M and T with probability function Py, and the distribution of M with respect to S and X is independent of the distribution of M with respect to T and ...
The real part of the square root of a complex number is greater than or equal to zero.
The weight of a term 't' being equal to 0 is equivalent to false.
If function 'f' is asymptotically equivalent to function 'g' at filter 'F', then 'f' is a member of Big Omega of 'g' at filter 'F'. This means that 'f' grows at least as fast as 'g' for large inputs.
For any real numbers a, b, c, d and any real vector e, the inequality "a times e plus c is less than or equal to b times e plus d" is equivalent to the inequality "(a minus b) times e plus c is less than or equal to d".
The lemma named "cone_iff_commutative_square" shows that a cone with the domain of h and constructed with h and k is equivalent to a commutative square in category C with f0, f1, h, and k.
: Assuming the tree 't' is consistent and 'a' is not equal to 'b', - If 'a' is in the alphabet of 't' and 'b' is also in the alphabet of 't', then the weight of the tree obtained by swapping the leaves 'a' and 'b' with weights 'w_a' and 'w_b' respectively, plus the frequency of 'a' and 'b' in 't', is equal to the we...
The lemma named "con_char" shows that "t" is connected to "u" if and only if "t" is an arrow, "u" is an arrow, and "t" is connected to "u" in the residuated transition system "R".
The lemma fixes a function 'f' from 'a' to 'b', where 'b' is a semiring. It assumes that 'A' is finite. It shows two things: 1. The sum of the product of 'f x' and the indicator function of 'B x' evaluated at 'g x', for all 'x' in 'A', is equal to the sum of 'f x' for all 'x' in the set of 'x' in 'A' where 'g x' is in...
: Assuming for every set A in the collection F, A is a subset of X, and the cardinality (size) of X is greater than or equal to 1, and the cardinality of F is equal to the cardinality of X. It can be shown that there exists a set D such that D is a subset of X, the cardinality of D is less than the cardinality of X, an...
The program counter set of the stack lift operation with a parameter 'n' on 'xt' is equal to the program counter set of 'xt'.
: Given the assumptions that the simple ownership is distinct for the first element of the sequence 'c' at position 'n', and that the second element of the sequence 'c' at position 'n' is unowned and shared with the first element of 'c' at position 'n', it can be shown that for any 'l', if 'l' is less than or equal to...
The function "find_cheapest_distinguishing_trace" applied to a machine M, a distance function distFun, a list of input-output pairs ios, a partial test m1, a state q1, another partial test m2, and another state q2, is equal to the result of the following process: First, define a function f that takes a pair of a trace...
The lemma named "eval_at_point_closed" assumes that "a" is an element of the nth power of the ring R and "p" is an element of the polynomial ring in n variables over R. It shows that the evaluation of the polynomial "p" at the point "a" is an element of the ring R.
"i" is in "below k" if and only if "i" is less than "k".
The Kullback-Leibler divergence is greater than or equal to zero for any given sets a and e, even when the set is empty.
The lemma "init_fin_\<tau>diverge_RunningI" assumes that the system diverges at a given state (x, m) for a thread 't'. It shows that the system also diverges at the same state ((Running, x), m) when the thread 't' is in the 'Running' state, according to the transition system 'init_fin_\<tau>move' initialized with 'init...
Given a left derivation fix of a sequence of symbols (\<delta>@\<alpha>) at position i to a sequence of symbols \<gamma> at position j (denoted as "LeftDerivationFix (\<delta>@\<alpha>) i D j \<gamma>"), and given that the derivation D is greater than or equal to the length of \<delta> (denoted as "derivation_ge D (len...
A relation "r" is transitive if and only if the composition of the identity relation and "r", followed by "r", equals "r". Similarly, it is also transitive if the composition of "r" and the union of the identity relation and "r" equals "r".
"False and any boolean value P is always False".
If A and B are both in the set M, then the extended measure of the union of A and B in M is less than or equal to the sum of the extended measures of A and B in M.
"A is greater than or equal to 0".
The lemma named "den_simps" states the following: - The denominator of the root is 1. - The denominator of the left child is the sum of the numerator and the denominator. - The denominator of the right child is the same as the denominator.
For a given matrix A of real numbers, if A is an orthogonal matrix, then the columns of A are pairwise orthogonal.
The lemma "basis_in_augment" assumes that "B1" and "B2" are bases in a set "E", and "x" is an element of "B1" but not "B2". It shows that there exists an element "y" in "B2" but not in "B1", such that "y" along with "B1" minus "x" forms a basis in "E".
The bitwise NOT operation on an integer numeral m ANDed with an integer numeral n is equal to the result of the function 'and_not_num' applied to n and m. If 'and_not_num' returns None, the result is 0, otherwise, it's the numeral of the returned value.
The composition of R and H2 is equal to the composition of H2 and R.
The binary sign of the mask function with any input 'n' is always zero.
If set Y is a subset of the elements of x, then Y is small.
There is a path from vertex v to vertex v' through the combination of paths p1 and p2 if and only if there exists a vertex u such that there is a path from vertex v to vertex u through path p1 and a path from vertex u to vertex v' through path p2.
If a logical relation "R" exists, and if a function "f" belongs to this relation "R", and if the set of arguments "args" is a subset of the relation "R", then the application of function "f" to the vector of arguments "args" also belongs to the relation "R".
If the ordered pair "X, Y" is equivalent to the ordered pair "X', Y'", then "Y" is equivalent to "Y'".
There exists a proposition P such that the option of applying function f in a while loop until f A is not equal to A, starting with initial value a, results in Some P.
The normalization of a match, which is the conjunction of a match expression mapped over an empty list with each element negated by a layer 4 port, is equivalent to matching any.
If x is not equal to -1, then the product of x and the exponential of x is greater than the negative exponential of -1, where x is a real number.
The language of the reversed non-deterministic finite automaton (NFA) is equal to the reverse of the language of the deterministic finite automaton (DFA).
If a step from state 's' to state 's prime' exists, then the step 's' is not finished.
If function 'f' and function 'g' are both bounded functions, then the function that maps 'x' to the sum of 'f(x)' and 'g(x)' is also a bounded function. This is applicable for any functions 'f' and 'g' that map from any type 'a' to any type 'b'.
If a variable "v" is in the keys of the second element of the symmetric preprocessing of "gs" and "fs", and if "g" is in the set "gs", and if the leading term of "g" adds to "v", then the monomial multiplication of 1 and the difference between the packed pair of the term "v" and the leading pair of "g" and "g" is in th...
If a matrix A is within the carrier matrix of dimensions nr by nc, and a vector v is in the kernel of matrix A, then it shows two things: firstly, the vector v is within the carrier vector of dimension nc, and secondly, the multiplication of matrix A and vector v equals to the zero vector of dimension nr.
The lemma named "irreflexiveDecisionLess" shows that the pair "(x, x)" is not in the relation "decisionLess".
The lemma named "pvalid_admissible" assumes that "p" is valid for "x" (expressed as "pvalid p x"). It shows that "p" is admissible.
If the derivative of function f at point x is d, and the derivative of function g at point x is e, and the value of function g at point x is not zero, then the derivative of the function that is the quotient of f and g at point x is equal to the difference of the product of d and the value of g at point x and the produ...
The domain of the piecewise-defined expression 'x' is finite.
If a function 'f' is monotonic and there exists a natural number 'n' such that 'f(n)' is greater than zero, then the function 'f(n) + k' is asymptotically equivalent to the function 'f'.
If the inorder traversal of a tree 't' is sorted, then the inorder traversal of the tree after deleting an element 'x' is equal to the inorder traversal of 't' with 'x' removed from the list.
If the following conditions are met: - For any object pointer, the preservation property holds between the states h and h' for the operation of getting the object with no additional properties. - For any node pointer, the preservation property holds between the states h and h' for the operation of getting the node wit...
If a function is balanced, then the sum of all terms from 0 to 2^n (where each term is (-1) raised to the power of the function's value at that term plus 1) equals 0.
The quotient of a fraction 'x' is included in the set of normalized fractions.
For any natural numbers 'a', 'b', and 'c', the modulus of 'a' with the product of 'b' and 'c', divided by 'c', is equal to the modulus of the quotient of 'a' and 'c' with 'b'.
The lemma "degen_lasso_sound" assumes that "degen.is_lasso T m L" is true. It then shows that "is_lasso (map_lasso fst L)" is also true.
For any element 'a' of an additive group, if the left coset of 'a' with respect to set A is equal to the left coset of 'a' with respect to set B, then set A is equal to set B.
: given a natural transformation "psi" from functor "F" to functor "G" in the category "B" to the category "C" under a fixed Grothendieck universe "alpha", and an object "a" in the opposite category of "C", the domain of the ArrVal (Arrow Value) of the right composition of the Hom-component of "psi" with "a" and "b" is...
If sequence w1 is less than or equal to sequence w2 in the finite and infinite prefix order, and sequence w2 is a prefix of sequence w3 in the infinite prefix order, then sequence w1 is a prefix of sequence w3 in the finite and infinite prefix order. This is assumed to be transitive.
"p" is odd.
For natural numbers 'x', the unsigned cast of 'x' as a 32-bit word to a signed 32-bit word is equal to 'x'. Similarly, the unsigned cast of 'x' as a 32-bit word to a 16-bit signed word or an 8-bit signed word is also equal to 'x'. The same holds true when 'x' is cast as a 16-bit word to a 16-bit signed word or an 8-bit...
If a vertex 'v'' is an element of the vertex set 'V' of a face 'f', and the vertices of 'f' are distinct, then if 'v' is the next element after the last element in the sequence of vertices starting from 'v'' in 'f', then 'v' must be equal to 'v''.
If a relation "S" holds between two probability mass functions "p" and "q", and if for every input "ia" in "IA", the result of applying the adversary function "rfunc_adv" to "rest1" and "rest2" and then mapping the first element of the result is the same for "p" and "q", and if for every state "s1" and "s2" in the supp...
"x is less than or equal to y" if and only if "x is less than or equal to the join of y and L" and "x is less than or equal to the join of y and the product of n(y) and top".
The product of any value 'a' and zero is zero.
If a relation "r" is accessible, then the list relation "Listn.le r" is also accessible.
"it is always the case that the value of 'v' changes over time."
The lexicographic order is asymmetric. It assumes that the relation R is asymmetric and that the pair (a, b) is in the lexicographic order of R. It then shows that the pair (b, a) is not in the lexicographic order of R.
: given two functors, "GG" from category "AA" to category "CC" and "HH" from category "BB" to category "CC", the range of the object flip operation on the comma category of "GG" and "HH" is equal to the object set of the comma category of the opposite of "HH" and the opposite of "GG".
The length of a list remains the same after sorting it by a key function.
If 'f' and 'g' are continuous maps from an n-sphere to itself, then the Brouwer degree of the composition of 'g' and 'f' is equal to the product of the Brouwer degree of 'g' and the Brouwer degree of 'f'.
If two converters "conv1" and "conv2" are related by the relation "rel_converter" with parameters A, B, C, and R, then the function "run_converter" applied to "conv1" and "conv2" is related by the function relation "rel_fun" with parameters A and the relation "rel_gpv''" applied to the product relation "rel_prod" of B ...
If x is greater than 0, then the natural logarithm of x is less than or equal to the function ln_upper_3 of x.
If the coefficient of a polynomial 'p' at any index 'i' belongs to set 'A' and 'x' also belongs to set 'A', then the coefficient of the polynomial resulting from the synthetic division of 'p' by 'x' at index 'i' will also belong to set 'A'.
The lemma named "is_projection_on_eqI" is defined with respect to a set M of a complex inner product space. It assumes four conditions: 1. M is a closed complex subspace. 2. The difference between h and x belongs to the orthogonal complement of M. 3. x is an element of M. 4. Pi is a projection on M. Given these assu...
If "C" is a class in the program "G", the accessible field "en" in class "C" under the scope "S" is some field "f" with declaration "fd", and "G" is a well-structured program, then "fd" is a class in the program "G".
A type T is equivalent to a function from Ts to B if and only if Ts is equivalent to the function's destination type T.
The lemma named "set_diff" is about two sets, xs and ys, of a certain type 'a that has an equality operation. It assumes that both xs and ys are distinct, meaning they don't have any duplicate elements. It also assumes that for every element 'a' in set ys and every element 'x' in set xs, the equality operation between ...
The lemma named "rot_circle_div_circle" assumes that a real number "d" is greater than 0. It shows that the general division of the image of the circle cube is the image of the set containing the rotated circle cube.
If a trie "t" maintains the invariant "invar_trie", then applying a predicate "P" to all elements in the trie "t" is equivalent to applying the predicate "P" to all elements in the range of the function "lookup_trie" applied to the trie "t".
"m" is less than or equal to the result of the function "triangle" applied to "n + m".
The theorem that the differential constant axiom is valid.
A paper 'p' is equivalent to a paper with a specific title, abstract, content, reviews, dis, and decs if and only if the title of 'p' is the same as the specified title, the abstract of 'p' is the same as the specified abstract, the content of 'p' is the same as the specified content, the reviews of 'p' are the same as...
The lemma named "H_values_right_bottom" is defined with two fixed natural numbers i and j. It assumes that both i and j are equal to 1. It then shows that the value of the matrix H at position (i,j) is equal to negative one divided by the square root of 2.
If "i" is greater than or equal to 0, then the binary scalar function "bin_sc" with inputs "n", "b", and "i" is also greater than or equal to 0.
The rank of the hereditarily finite set obtained by inserting element 'a' into set 'b' is equal to the union of the rank of set 'b' and the successor of the rank of element 'a'.