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The absolute value function for 32-bit unsigned integers, denoted as "Abs_uint32'", when applied to a variable 'x', is equivalent to converting 'x' into an unsigned 32-bit integer after 'x' has been converted into an integer from an unsigned integer.
For any integers 'j' less than or equal to 'n' and 'm', if the function 'f' applied to 'j' and the function 'g' applied to 'j' are both in the carrier 'A', then the sum of the joint function of 'n', 'f', 'm', and 'g' for the successor of the sum of 'n' and 'm' is equal to the sum of the function 'f' for 'n' added to th...
The lemma named "lad_onD" assumes that a function 'f' is a least-adjusted decision on a set 'A'. It also assumes that set 'C' is a subset of set 'B', and set 'B' is a subset of set 'A'. Under these assumptions, it shows that the cardinality (or size) of the set resulting from applying function 'f' to set 'C' is less th...
"x is less than the real part of extended real y" if and only if "the absolute value of y is not infinity implies that the extended real of x is less than y" and "the absolute value of y equals infinity implies that x is less than 0".
The lemma named "two_pers" assumes three conditions: 1) The word 'w' is a prefix of the concatenation of 'u' and 'w'. 2) The word 'w' is a prefix of the concatenation of 'v' and 'w'. 3) The combined length of 'u' and 'v' is less than or equal to the length of 'w'. If these conditions are met, it shows that the con...
The lemma "nullhomotopic_into_contractible_space" assumes that there is a continuous map 'f' from space 'X' to space 'Y', and that space 'Y' is a contractible space. It then concludes that there exists a constant 'c' such that 'f' is homotopic to the constant function at 'c' in space 'Y', for all points in space 'X'. T...
The lemma "InvariantUniqAfterApplyDecide" assumes the following conditions: 1. The variables in the elements of the state's model do not include all variables in Vbl. 2. The state's model maintains the unique invariant. 3. The state's formula, first watch, and second watch maintain the invariant that watches are eleme...
If the function 'f' is applied to the parameters 'self', 'X', 'invalid', and 'Z', the result will be 'invalid'.
The dual of a complete lattice is also a complete lattice.
The relation "R_a0m1a_rs" is preserved when transitioning from the initial state to the state after executing "m1a_step1 Ra A B".
The lemma named "det_wf_nfa" assumes that "A" is a well-formed non-deterministic finite automaton with "n" states. It shows that the deterministic finite automaton obtained by determinizing "A" is also well-formed with "n" states.
, under certain conditions, the new values of a process at a certain level in a system are uniform. Assuming that: 1. There is a run of the EIG_M algorithm with a given system state, set of heard-of sets, and set of sent messages. 2. The communication per round for the round corresponding to the length of the label i...
The floor function, when applied to real numbers, tends towards negative infinity as the input tends towards negative infinity.
The lemma named "match_final_backward" assumes that "s1" matches "s2" and that "s2" is final according to the function "Fubx_get" with "Ubx.IReturn". It shows that "s1" is also final according to the function "Finca_get" with "Inca.IReturn".
If a vertex 'v' is in the set of vertices of the decision tree converted to a list, and an element 'x' is in the set 'v', then 'v' is equal to the list containing only 'x' and 'x' is in the set of vertices of the tree 'T'.
If the conjunction of protocol 1 (p1) and protocol 2 (p2) equals None, then it is not the case that both protocol 1 and protocol 2 match the packet (pkt).
The composition of bounded linear functions is associative. That is, composing function F with the composition of functions G and H is the same as composing the composition of functions F and G with function H.
The lemma P_ring_ldistr: assumes that "a", "b", and "c" are elements of the coefficient carrier set. It shows that "a" multiplied by the sum of "b" and "c" is equal to "b" multiplied by "a" plus "c" multiplied by "a".
If a polynomial 'p' belongs to the carrier 'R', and 'c' is a polynomial coefficient in 'S', and 'p' is equal to the polynomial expression 'R' with variable 'X' and coefficient 'c', then the degree of 'p' in 'R', 'S', and 'X' is equal to the maximum coefficient 'c' in 'S'.
The support of the substitution of y by x in expression e is a subset of the union of the support of e minus the atom of y and the atom of x.
The lemma named "omega_sum_length_leD" assumes that the omega sum of the sequence "ms" is less than or equal to the omega sum of the sequence "ns" and that the sequence "ms" is normal. It shows that the length of the sequence "ms" is less than or equal to the length of the sequence "ns".
For a function 'f' mapping real numbers to a real normed field 'a', if 'f' is continuously differentiable on a set 'S', then the function that maps 'x' to 'f(x) / c' is also continuously differentiable on 'S'.
Theorem C10: "There necessarily exists an entity x such that x is Godlike."
The lemma named "ground_term_der_ta_reach" assumes that "t" is a ground term and "q" is in the derivative of "t" with respect to automaton "A". It shows that "q" is in the reachability set of automaton "A".
If 'I' is an ideal of ring 'R', then the set of right cosets of 'I' in the additive group of 'R' is equal to the set of additive right cosets of 'I' in 'R'.
The set of the concatenation of lists "xs" is equal to the listset of the map of the set of "xs".
If a term 't' is well-formed according to a signature 'Σ', then the term 't' is also okay according to the same signature 'Σ'.
The lemma assumes that there is a chain of edges 'ces' from vertex 'v1' to vertex 'v2' with no additional vertices. It shows that this chain of edges 'ces' from 'v1' to 'v2' forms a graph.
If a list is not empty, then an element is either the head of the list or in the set of the tail of the list if and only if the element is in the set of the list.
If the disjunction (logical "or") of two formulas f1 and f2 is a disjunction, then both f1 and f2 are disjunctions.
The function "find_ce" is less than or equal to the function "find_ce_spec".
The cardinality of the set F2set1 of x is less than the ordinal sum of the cardinalities of bd_F1 and bd_F2.
For a function 'g' mapping real numbers to a Euclidean space 'a', the property of being a simple path remains unchanged if the path is translated by a vector 'a'. In other words, if we shift every point on the path by the same vector 'a', the path remains a simple path.
The adjoint of the Hadamard operator is equal to the Hadamard operator itself.
The synthesis of set G union the synthesis of set H is a subset of the synthesis of the union of set G and set H.
If for all values of x and y, the predicate P holds for x and y, then it is true that for all pairs of (x, y), the predicate P holds for x and y.
Function f is equal to function g if and only if for all values of n, the nth term of function f is equal to the nth term of function g.
The first epsilon value of the hyperdual extension of function f at beta of x is equal to the derivative of function f at x.
The natural extension of nu applied to the negation of the natural extension of nu applied to x unioned with the natural extension of nu applied to y is equal to the negation of the natural extension of nu applied to x unioned with the natural extension of nu applied to y.
The frontier of a set S is always closed.
Lemma: Convex Set Sum Assumption: For any element i in set A, the set B(i) is convex. Conclusion: The sum of the sets B(i) for all elements i in set A is also convex.
For any two sets A1 and A2 of any type 'a, and another set B of any type 'b, if A1 is a subset of A2 and B is not an empty set, then the cardinality of the set of optional functions from A1 to B is less than or equal to the cardinality of the set of optional functions from A2 to B.
The lemma named "infsum_prod_PiE_abs" is about a function 'f' that maps from 'a' to 'b' to 'c', where 'c' is a Banach space, a real normed division algebra, and a commutative semiring. It assumes that 'A' is a finite set and for every element 'x' in 'A', the function 'f' of 'x' is absolutely summable over 'B' of 'x'. I...
The product of the embedding of a proposition P in state s with itself is equal to the embedding of the proposition P in state s.
The reverse of the bit list representation of a signed cast of a word 'x' is equal to the result of filling the last bits of the reverse of the bit list representation of 'x' with false until it reaches the length of 'a'. Here, 'a' is a type with a length, and 'x' is a word.
If "R" is a subset of the states of the non-deterministic finite automaton "nfa'", then the run of "R" with input "bss" is equal to the run of "nfa'" with "R" and the same input "bss".
If for all pairs (x, v) in the set xs, the value v is greater than or equal to 0, then the result of the lookup function applied to 0, xs, and x (where 0, xs, and x are of a type that is an ordered commutative monoid with addition) is also greater than or equal to 0.
The range of a finite map after applying a function is equal to the function applied to the range of the original finite map.
The distance between the product of a constant 'c' and vector 'x', and the product of the same constant 'c' and another vector 'y', is equal to the absolute value of the constant 'c' multiplied by the distance between vector 'x' and vector 'y'.
If the length of the Turing machine 'tm' is greater than zero and even, then the length of the Turing machine 'tm' after being processed by the function 'mk_composable0' remains the same as its original length.
If the tree "t" satisfies the invariant property "inv_12", then the tree obtained by applying the "rbt_recolor" function to "t" also satisfies the invariant property "inv_12".
The disjunction of an empty set is equivalent to the Boolean value of False.
The lemma tauLaw3PsiCong: It establishes a condition for a psi calculus. It fixes a psi denoted by \<Psi> and a psi term denoted by P. It shows that for these fixed psi and psi term, the application of an action alpha to the tau prefix of P is congruent to the application of the same action alpha to P itself.
If a set P has a lowest element in the domain, and if an element x belongs to P such that the function of x equals the lowest element of the image of P in the codomain, then x must be the lowest element of P in the domain.
If the length of a list 'xs' is divisible by 'k' (i.e., the remainder when the length of 'xs' is divided by 'k' is 0), then the result of the function 'list_slice2' applied to 'xs' and 'k' is equal to the result of the function 'list_slice' applied to 'xs' and 'k'.
If there exists exactly one 'x' such that property 'P' holds for 'x', and if property 'P' holds for both 'x' and 'y', then 'x' must be equal to 'y'.
If 'v' and 'v'' are valuations on the field 'K', 'x' is an element of the carrier of 'K', the value of 'v'' at 'x' is greater than 0, and the carrier of the valuation ring of 'K' with respect to 'v' is a subset of the carrier of the valuation ring of 'K' with respect to 'v'', then the value of 'v' at 'x' is greater tha...
The lemma "wsX_if1" assumes that "xcs" is well-scoped under context "G". It shows that the set of atoms from the first elements of "xcs" is a subset of the atom domain of "G", and that the list of first elements of "xcs" is distinct.
A retraction map from the product topology of X and Y to Y using the second projection function exists if and only if when the topological space of X is empty, the topological space of Y is also empty.
The lemma named "semigroup_transfer" is a transfer rule. It assumes that a relation "A" is bi-unique and right-total. It shows that a semigroup in the domain of "A" is equivalent to a semigroup in the original structure, under the condition that the binary operation of the semigroup respects the relation "A".
If set A is finite, then the list representation of set A contains distinct elements and the set of elements in the list is equal to set A.
The Aleph of any set x is a cardinal.
If the order of generative probabilistic value 'f' is less than or equal to the order of generative probabilistic value 'g', then the order of the sub-probability mass function of 'f' and 'g', with respect to the relation of the generator and the function, is less than or equal to the order of the sub-probability mass ...
"cf" is a residual graph with respect to capacity "c", source "s", and target "t".
The lemma named "composite_of_arr_normal" shows that if "t" is an arrow, "u" is in the set of natural numbers, and "t" is a composite of "u" and "t'", then "t'" is approximately equal to "t".
If the derivative of a polynomial p is not zero, and p is equal to the product of q and d, and the derivative of p is equal to the product of e and d, and d is equal to the product of r and p plus the product of s and the derivative of p, then for all a, the order of a in q is equal to zero if the order of a in p is ze...
: Assuming that: 1. The relation Q is symmetric, meaning if Q holds between a and b in interval I, then Q also holds between b and a in the same interval. 2. The relation P is symmetric under the condition that there exist points a and b such that Q holds between them in interval I. 3. If Q holds between points a and ...
The theorem is named "real_cond_exp_jensens_inequality". It fixes a function 'q' that maps real numbers to real numbers. It assumes that a function 'X' is integrable over a measure space 'M' and almost everywhere in 'M', 'X' falls within a set 'I'. The set 'I' is either an open interval between 'a' and 'b', a half-open...
The lemma named "flow_summation_aux" demonstrates that the sum of the function 'f' applied to each element 'v' in the set of all elements related to 'u' in relation 'E', subtracted by the sum of the function 'f' applied to each element 'v' in the same set but with 'v' and 'u' reversed, is equal to the sum of the functi...
The lemma consists of two statements: 1. If 'i' is greater than or equal to 'u', then the imperative for-loop from 'i' to 'u' with condition 'c', function 'f', and state 's' is equivalent to simply returning the state 's'. 2. If 'i' is less than 'u', then the imperative for-loop from 'i' to 'u' with condition 'c', fu...
The lemma named "lemma_st_part_major" is defined with three variables x, r, and t, all of which are hyperreal numbers. It assumes that x is a finite hyperreal number, r is a real number and is greater than 0, and t is the least upper bound of the set of all real numbers s such that s is less than x. The lemma then show...
For any given point 'a' of type 'real_normed_vector', the closed segment between point 'a' and another point 'b' is a closed set.
If the set of free variables in a term 't' is empty, then 't' is a ground term.
The lemma "mem_read_rip_write" demonstrates that the memory read operation with the updated instruction pointer set to a new value 'a' is equivalent to the memory read operation with the original updates, for any given state 'σ' and size 'si'.
If set A is in the sets of measure M, then the outer measure of M for set A is equal to the extended measure of M for set A.
If A and B are objects in the category of partial maps of a given ordinal number α, and if AB is the Cartesian product of A and B, and A' is equal to A, and the category C' is the category of partial maps of α, then the first component function from A to B is a morphism from AB to A' in the category C'.
The interleaving of sequences "w" and "w'" with respect to a specific alphabet "α" is a subset of the interleaving of sequences "w" and "w'".
The lemma "sources determine composability" assumes that "a" is in the sources of "tau". It shows that "tau" composed with "mu" is not null if and only if "a" composed with "mu" is not null.
The lemma named "timpl_closure'_inv''" assumes that a pair of functions, "Fun f S" and "Fun g T", is in the transitive closure of the term implication relation "TI". Under this assumption, it shows three things: 1. The lengths of the lists "S" and "T" are equal. 2. For any index "i" that is less than the length of "T"...
A function f is regular on a derivation tree tr if and only if for all nodes ns and all sub-trees tr' of tr, if tr' is a sub-tree of tr rooted at ns, then tr' is equal to the function f applied to the root of tr'.
The lemma named "composable_imp_seq" assumes that "t" and "u" are composable. It shows that a sequence exists between "t" and "u".
If the abstract machine evaluation of the concrete global configuration 'gc\<^sub>C' after 'n' steps results in 'gc\<^sub>C'', and the length of the first element of 'gc\<^sub>C' is equal to the length of 'cms', and the modes respect privacy for the second elements of the first element of 'gc\<^sub>C', then the modes r...
If "t'" is a generalization of "t" at position "i" with respect to relation "r", then "i" is less than the arity of "t".
The lemma named "reduction_to_dis_log" shows that the relative advantage of an adversary in the Chaum Pedersen Sigma Commitment scheme is equal to the advantage of the same adversary in the discrete logarithm problem, when using the third alternative adversary model.
If "x" and "y" both maintain non stuttering invariant (NSTUTINV), then the sum of "x" and "y" also maintains non stuttering invariant (NSTUTINV).
A function F is totally defined if and only if the totalized version of function F is equal to function F itself.
The security invariant "sinvar" with default node properties does not allow for receiver violation.
The degree of the product of a multiset 'fs' is less than or equal to the sum of the degrees of the multiset 'fs'.
: Assuming the following conditions: 1. The initial state of the iterator is valid (I S0 \<sigma>0). 2. For any set S, state \<sigma>, and element x, if the following conditions are met: - The state \<sigma> is valid (c \<sigma>). - The element x is valid (invar x). - The abstract representation of x is in th...
The supremum of two elements x and y is unique. If s is the supremum of x and y, and s' is also the supremum of x and y, then s and s' must be the same.
The lemma named "finite_perm_lemma" demonstrates that the set of all atoms 'a' such that the permutation 'p' applied to 'a' does not equal 'a' is finite.
The lemma named "mu_bound_rowI" assumes that for any integer 'j' less than or equal to 'i', the square of 'mu' of 'fs' at 'i' and 'j' is less than or equal to 'bnd'. If this assumption holds, it shows that 'mu' is bounded in the row of 'fs' by 'bnd' at 'i'.
Given the assumption that "H" is a Poincare line, it shows that "H" intersects the positive x-axis if and only if the calculation of the x-axis intersection of "H" is within the unit disc and the calculation of the x-axis intersection of "H" is in the intersection of the circline set of "H" and the positive x-axis.
If set A and set B are finite, then the set that consists of the product of the function f applied to each element a in set A and the function g applied to each element b in set B is also finite.
The evaluation domain of the evaluated heap is a subset of the domain of the original heap.
"ya" is regular in the Yahalom protocol.
: 1. A map is equivalent to an empty map if and only if the map is empty. 2. An empty map is equivalent to a map if and only if the map is empty.
If function 'f' is a bijection, then permuting the polynomial mapping that maps 'a' to 'b' using function 'f' is equivalent to a polynomial mapping that maps the inverse of 'a' with respect to function 'f' in the universal set to 'b'.
The combination of two Red-Black Tree (RBT) Mappings 't' and 'u' using a function 'f' is equal to the following: If there is no comparison function for type 'a' (CCOMPARE('a) is None), then the program will abort with the message "combine RBT_Mapping: ccompare = None" and the original combination operation will be retu...
The lemma named "cSup_add" is about a fixed real number 'c'. It assumes that the set 'S' is not empty and that for any element 'x' in 'S', 'x' is less than or equal to 'b'. It shows that the supremum (least upper bound) of 'S' plus 'c' is equal to the supremum of the set of all 'x' plus 'c' where 'x' is an element in '...