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If a natural number 'm' is less than or equal to another natural number 'n', then the difference of 'l' and 'n' is less than or equal to the difference of 'l' and 'm'.
The range of variables in a substitution 's' is equal to the union of free variables 't' in the set of 't' such that there exists a variable 'v' where 's v' equals 't' and 't' is not equal to the variable 'v'.
The lemma named "bigtheta_mult_eq_set_mult" shows that the big theta of the function that multiplies f(x) and g(x) is equal to the set multiplication of the big theta of f and the big theta of g.
The maximum value of a list that is the result of appending two lists (xs and ys) is equal to the maximum value between the maximum value of list xs and the maximum value of list ys.
The set Sigma is not empty.
The domain of the sequence of P and Q is equal to the domain of the sequence of P and the closure of the domain of Q.
The product of the real number representations of x and y is equal to the real number representation of the product of x and y.
The lemma pos_oriented shows that either H is positively oriented or the opposite of H is positively oriented.
If neither of the lists 'xs' and 'ys' are empty, then the head of the merged list 'lmerge xs ys' is equal to the head of 'xs' if it is less than the head of 'ys', otherwise it is equal to the head of 'ys'.
The set of all natural transformations, denoted by "NN", is small. Here, each natural transformation "NN" exists between two functors "FF" and "GG", where "FF" and "GG" are mappings from category "AA" to category "BB" in the context of elementary category theory.
The lemma named "find_dep" assumes that "Insert m'" is in the union of "M" and the set containing "Insert m". It also assumes that "i" is in the dependencies of "Insert m'". Under these assumptions, it shows that the interpretation of "i" is in the set of external IDs of "s".
The height of a binary tree node, which is constructed with a subtree 'sub', an element 'x', and a list of other elements 'ls', is equal to the maximum of two values: the height of a node constructed with the list 'ls' and the height of the subtree 'sub' incremented by one.
The list representation of the result of the function "nrcons" applied to a tree "t" and a node "a" is equal to the concatenation of the list representation of the tree "t" and the list representation of the node "a".
For any integer 'x', the 'n'-th bit of 'x' is odd if and only if the 'n'-th bit is dropped from 'x'.
: "For all functions 'f' and types 'T', if 'Fun f T' is an element of 'M', then 'P f T' holds" is equivalent to "For all 'u' in 'M', if 'u' is a function 'Fun f T', then 'P f T' holds, otherwise it's true." "For all functions 'f' and types 'T', if 'Fun f T' is an element of 'M', then 'P f T' holds" is equivalent to "...
If a pair (a, b) is an element of the relation r and b is an element of the set A, then a is an element of the inverse image of A under the relation r.
Replicating any element or list zero times results in an empty list.
Shifting the word length number -1705 to the left by three places is equal to -13640.
The greatest common divisor of any integer 'm' and 1 is always 1.
The projection of a probability (PROB) onto a set of variables (vs1) and then onto another set of variables (vs2) is equal to the projection of the same probability onto the intersection of the two sets of variables.
For any list 'l' and state 's' where the value of 'x' in 'l' is the natural number equivalent of the value of 'x' in 's', if we continuously subtract 1 from 'x' until 'x' is no longer greater than 0, and for each iteration, we also continuously subtract 1 from 'y' (which is initially set to the current value of 'x') un...
The function 'l' applied to the result of 'update_rightmost' function with parameters 'f' and 't' is equal to the function 'l' applied to 't'.
The optimal binary search tree 2 for indices i and j is equal to the optimal binary search tree for indices i and j.
: - The pair (bf_True, x) is in the relation bf_ifex_rel if and only if x equals Trueif. - The pair (bf_False, x) is in the relation bf_ifex_rel if and only if x equals Falseif.
If "F or G" is enabled, then either "F" is enabled or "G" is enabled.
If set S is a subset of set S' and set T is a subset of set T', then the Cartesian product of set S and set T is a subset of the Cartesian product of set S' and set T'.
The lemma "node_silent_change_only" demonstrates that for any operation 'opaodv i' on node 'i' and for any queued message 'qmsg' in the set of received messages 'Ri', if a message arrives or any other event occurs (with no specific conditions on these events), then for any global state transition (sigma, a, sigma'), if...
The language resulting from the shuffle of two languages A and B is equal to the shuffle of the languages resulting from A and B individually.
For a given matrix A of type 'a::comm_ring_1 mat', if A is within the carrier matrix of dimensions n by m, and A is in Smith Normal Form, and given that j is less than the minimum of n and m, and i is less than or equal to j, then the element at position (i,i) in matrix A divides the element at position (j,j) in matrix...
For all variables 'n', 's', and 'p', if 's' is not a free variable in 't' and 'p' is not a free variable in 't', and 's' is not equal to 'p', then replacing 'n' with '[s,p]' and then replacing 'n' with '[Fvar s, Fvar p]' in 't' will result in 't' itself.
The opposite natural transformation of Z is an initial object in the category of functors from the identity category to the opposite category of C, parameterized by alpha.
For any type 'tytok', the negative distribution condition of the relation 'U' is equal to itself.
The lemma "ii_in_disc_o_x_axis" shows that the hyperbolic imaginary unit "ii_h" is not in the disc of the x-axis origin.
The lemma named "Ball_eqvt" shows that for a permutation 'p', if for all 'x' in a set 'S', a property 'P' holds for 'x', then for all 'x' in the permuted set 'p' applied to 'S', the permuted property 'p' applied to 'P' holds for 'x'.
The first element of the tuple resulting from the function "sum_to_tup" applied to "x" is equal to the function that takes "xa" and applies "x" to the discrete left injection of the undiscriminated "xa".
The lemma "no_ddep_same_state" assumes the following: 1. There is a path from node 'n' to node 'n'' represented by 'n -as\<rightarrow>* n''. 2. The variable 'V' is used in node 'n''. 3. The predicates of the kinds of 'as' hold for state 's'. 4. For all 'as', 'a', and 'as''', if 'as' equals 'as' appended with 'a' and '...
The function "ipurge_fail_aux_t_inv_1" with parameters I, D, U, and xs holds true when applied to a record with fields Pol equal to I, Map equal to D, Doms equal to U, List equal to xs, ListOp equal to None, and Set equal to X.
: 1. Lifting the substitution of term 'u' for variable 'm' in term 't' is equivalent to substituting the lifted term 'u' for variable 'm + 1' in the lifted term 't'. 2. Lifting the substitution of term 'u' for variable 'm' in a list of terms 'l' is equivalent to substituting the lifted term 'u' for variable 'm + 1' in ...
"there exists some x such that the property phi of x is in v" is equivalent to "there exists some y such that the property phi of y is in v".
The lemma named "distinct_sorted_list_lem1" makes the following assumptions: 1. The list 'xs' is sorted. 2. The list 'ys' is sorted. 3. The list 'xs' contains distinct elements. 4. The list 'ys' contains distinct elements. 5. For every element 'x' in the list 'xs', 'x' is less than 'e'. 6. For every element 'y' in the...
If "x" is an element of the carrier "G", then the order of "x" raised to the power "k" is equal to 1 if "k" equals 0; otherwise, it is equal to the order of "x" divided by the greatest common divisor of the order of "x" and "k".
The substitution of a type "T" with instances "insts" is equivalent to the type substitution of "T" with a function that, for a given identifier "idn" and type "S", defaults to the type variable "Tv" of "idn" and "S" if no match is found in the lookup of pairs equal to "(idn, S)" in "insts".
If an empty context proves that the triple (v, (Lam x vP'), vV) is of the function type from tau1 to tau2, then there exist v' and vV' such that v equals to the lambda abstraction of x and v', vV equals to the lambda abstraction of x and vV', and the empty context with x replaced by tau1 proves that the triple (v', vP'...
If a variable "x" is in the set of indeterminates of the substitution of a polynomial "t" by a function "f", then there exists a variable "y" such that "y" is in the set of keys of "t" and "x" is in the set of indeterminates of the function "f" applied to "y".
If U is equivalent to V, then the free left of U is equivalent to the free left of V.
If a set S is infinite and a function f is injective on S, then the image of S under f is also infinite.
If the separation logic implication from "sep_true" to "P" holds for a heap "h", then "P" also holds for the heap "h".
The theta function of two to the power of r is less than two to the power of r plus one times the natural logarithm of two.
In the context of a co-span of a covariant functor, there is a lemma that states: if a function 'f' is equal to 'ff' and an object 'b' is equal to 'oo', then 'f' maps from 'bb' to 'b' in the category 'CC'.
Two elements, x and y, are approximately equal in set A if and only if the result of applying the fold function with Deriv and x on set A is equal to the result of applying the fold function with Deriv and y on set A.
The set of keys for a mapping 'm' is empty if and only if the mapping 'm' is equivalent to an empty mapping.
The theorem named "matrix_equation_R" is defined with the following parameters: a real polynomial 'p', a list of real polynomials 'qs', a list of pairs of natural number lists 'subsets', and a list of lists of rational numbers 'signs'. The theorem assumes that the polynomial 'p' is not zero, the 'signs' are distinct, t...
For a given function 'f' and a value 'x', the lower semicontinuous hull of 'f' at 'x' is less than or equal to 'f' at 'x'.
The homomorph components of a polynomial 'p' are equal to an empty set if and only if the polynomial 'p' is equal to zero.
"The value of function H at point x is zero if x is not within the ball of radius 2 centered at 0."
The signed modulus of negative k and l is equal to the negative of the signed modulus of k and l, for any integers k and l.
The lemma named "update_acyclic_4" assumes the following conditions: 1. The set "p" minus 1 is acyclic. 2. "y" is a point. 3. "w" is a point. 4. The intersection of "y" and the star of "p" multiplied by "w" is the bottom element. Under these assumptions, it shows that the set "p" with "w" mapped to "y" minus 1 is also...
The unfreezing function applied to the step function of a formula and a letter is equal to the unfreezing function applied to the abstracted formula and the letter.
The lemma named "psi_altdef" assumes that "x" is greater than 0. It shows that the psi function of "x" is equal to the sum up to "log 2 x" of the function that takes "m" and returns the prime sum up to the natural logarithm of the m-th root of "x". This right-hand side is denoted as "?rhs".
The well-founded relation (wf) holds for the set "VWO" minus the identity relation (Id).
If a pair (A, B) is in a relation R, then the pair (A, B) is in the relation R.
The lemma "rewrite_MultiportPorts_unchanged_if_not_has_disc" assumes that a match "m" is in normalized NNF (Negation Normal Form). It also assumes that the match "m" does not have a discrepancy with "is_MultiportPorts". Under these assumptions, it shows that rewriting "MultiportPorts" with match "m" will result in a li...
The composition of the "strip" function and the "strip2" function applied to the set "mrexps2 r" results in a set containing only the element "r".
For a given complex number 'z', if a function 'g' is field differentiable at 'z' within a set 'S' and 'g' at 'z' is not equal to zero, then the function 'g' raised to the power of an integer 'n' is also field differentiable at 'z' within the set 'S'.
The nonstrict multiplicative extension of the identity relation and a relation R is equal to the reflexive transitive closure of the strict multiplicative extension of the identity relation and R.
The composition of the function "JF1set" with the function "JF1map" applied to a variable "u" is equal to the composition of the image of "u" with the function "JF1set".
The lemma named "ln_upper_13_neg" assumes that "x" is greater than 0 and "x" is less than or equal to 1. It shows that the natural logarithm of "x" is less than or equal to the function "ln_upper_13" of "x".
The lemma named "lspace_iff" assumes that infinity is greater than a positive number 'p'. It shows that the Lebesgue space on a set 'S' with exponent 'p' is equal to the set of functions 'f' that are Borel measurable on the Lebesgue measure on 'S' and are integrable on the Lebesgue measure on 'S' with respect to the fu...
The lemma named "empty_pow_add" assumes that an empty list is an element of set A and a string 's' is an element of the nth power of set A. It shows that the string 's' is also an element of the sum of nth and mth power of set A.
For any elements 'b' and 'c' of the type 'a' in a commutative ring, the formal power series derivative of 'a' with respect to the difference of 'b' and 'c' is equal to the difference of the formal power series derivative of 'a' with respect to 'b' and the formal power series derivative of 'a' with respect to 'c'.
For any two integers 'a' and 'b', the division of 'a' by 'b' equals negative 'a' if and only if 'b' equals negative one. This is under the condition that 'a' is less than zero.
, assuming "f" belongs to the set of value ring polynomials of degree "d", it follows that "f" belongs to the polynomial residue class of "n" and "d" for "f".
If a state 's' is in OV_inv2, then the set of votes (vote_set) observed in that state 's' for a quorum 'Q' is finite.
If 'y' is an element of 'Y' and 'y' is adjacent to 'x', then 'x' must be an element of 'X'.
The application of the function "ABinds" to the restriction of the function "ae" to its own domain in the context "Gamma" is equal to the application of the function "ABinds" to the context "Gamma" and then to "ae".
For a given function 'f' that maps from a Euclidean space 'a' to a Banach space 'b' with a second countable topology, the Lebesgue integral of 'f' over an empty set is equal to 0.
Given the conditions that "d" is a Dickson grading, the set of keys "F" has finite components of terms, and "F" is a subset of the set of polynomials of degree at most "m" with respect to "d", there exists a unique set "G" such that "G" is a subset of the set of polynomials of degree at most "m" with respect to "d", "G...
If a set Y forms a chain under the order defined by the equivalence relation on the spmf (stochastic/probabilistic monadic function), then the least upper bound of the mapped set Y under the function f is equal to the mapping of the least upper bound of the set Y under the function f.
The lemma named "bounded_imp_convergent_prod" is defined with a fixed function 'a' that maps natural numbers to real numbers. It assumes two conditions: first, for any natural number 'n', the function 'a' of 'n' is greater than or equal to 1; second, the product of 'a' for all 'i' less than or equal to 'n' is less than...
The theorem named "wls_freshEnv_swapEnv_id" assumes that the environment "rho" is well-sorted and that "x1" and "x2" are fresh in the environment "rho" with respect to the set "xs". It shows that swapping "x1" and "x2" in the environment "rho" with respect to the set "xs" results in the original environment "rho".
If almost everywhere in a measure space M, two functions f and g are equal, and both f and g are Borel measurable, then the real conditional expectation of f and g are also equal almost everywhere in M.
If S is a subring of R, and if g and f are elements of the univariate polynomial over the ring R with carrier S, then the addition of g and f in the univariate polynomial over R is equal to the addition of g and f in the univariate polynomial over the ring R with carrier S.
If the sum of a list of numbers (xs) is equal to 1, then there exists an index (i) such that the element at this index in the list is 1, and all other elements in the list are 0.
If both "f" and "g" are monotone functions, then the function "Omega_fun" applied to "f" and "g" is also a monotone function.
If action "a" is enabled at state "p" and state "q" is reachable by executing action "a" at state "p", then state "q" is also reachable from state "p".
: Assuming "i" is an element of "j", and "A" equals "A", and "B" equals "B", it can be shown that: 1. "Vfrom A i" equals "Vfrom A" at the rank of "i", 2. The rank of "i" is an ordinal number, 3. "Vfrom B j" equals "Vfrom B" at the rank of "j", 4. The rank of "j" is an ordinal number, 5. The rank of "i" is an element of...
The projection of an empty trace on any set X is an empty trace.
The lemma named "eval_v_i_upd" is about a fixed variable "v". It assumes that the atom "x" is independent of "v" and that "i" evaluates "v" to "s'". It shows that the evaluation of "v" under the substitution of "x" in "i" is "s'".
If pair x is less than or equal to pair y, then the second element of pair x is less than or equal to the second element of pair y.
If there exists a 'K1D0' such that for all 'nl', if 'nl' is in 'K1' then 'K1D0' yields a numerical result of 0, and if 'nl' is not in 'K1' then 'K1D0' yields a numerical result of 1, then this leads to a contradiction.
If for all "h", the function "P" applied to "h" equals the function "Rep_assn" applied to "Pr" and "h", then the function "Abs_assn" applied to "P" equals "Pr". Conversely, if for all "h", the function "P" applied to "h" equals the function "Rep_assn" applied to "Pr" and "h", then "Pr" equals the function "Abs_assn" ap...
The theorem shows that unwinding implies non-interference from indirect sources.
The proposition P holds for a binary tree x of any type 'a.
The auxiliary iteration function "ahm_iteratei_aux" for an array hash map, given the parameters "a", "c", "f", and "\<sigma>", is equivalent to the index iteration function "idx_iteratei" applied to "a" and "c", with a lambda function that applies "foldli" to "x", "c", and "f", and the state "\<sigma>".
The lemma named "z_zero" assumes that the third element of the projection 2 representation of point p is zero. It shows that point p is incident to the zero point in the z direction.
The lemma named "distinct_enqueue_new_threads" is defined with fixed thread information 'ts' and a list of new thread actions 'ntas'. It assumes that the thread actions are valid for the given thread information, the set of the queue is equal to the domain of the thread information, and the queue is distinct. Under the...
The set of atoms of the norm of a regular expression 'r' is a subset of the set of atoms of 'r'.
The lemma named "drop_is_semialg_map" demonstrates that the function "drop k" is a semialgebraic map for the sum of "k" and "n", and "n".
As x approaches positive infinity (at_top), the function f(x) = e^(e^(e^x)) / e^(e^(e^x - e^(-e^(e^x)))) tends towards 1. Here, e is the base of the natural logarithm, approximately equal to 2.71828.
The lemma named "sigma_over_set_partition_count" assumes that "A" is a finite set, "P" is a partition on "A", and "x" is an element in the multiset sum of the multiset set of the image of "P". It shows that the count of "x" in the multiset sum of the multiset set of the image of "P" is 1.