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The lemma named "nu_star2" assumes that for any 'x', the star of 'x' multiplied by the star of 'x' is less than or equal to the star of 'x'. Under this assumption, it shows that the star of the nu of 'x' multiplied by the sum of 1 and the tau of 'x' is less than or equal to the star of 'x'.
, given the following assumptions: 1. A predicate P holds for the least element (bot), for any element x transformed by a function f, and for the supremum (SUP) of any sequence M indexed over natural numbers, where each element of the sequence satisfies P. 2. For any monotone function M, if P holds for each element o...
Mapping the second element of each pair in a list obtained by zipping lists 'xs' and 'ys' is equivalent to taking the minimum length of lists 'xs' and 'ys' from the list 'ys'.
The lemma named "mkCone_cone" assumes that "a" is a cone with apex "chi". It shows that making a cone with "chi" at the zero object of category "J" is equal to "chi".
A pair of bags, "Bag xs" and "Bag ys", is in the extension of the nonstandard multiplication relation if and only if the second element of the extended multiplication function applied to the associated lists of "xs" and "ys" is true. The nonstandard multiplication relation is defined by two sets: one where the second e...
The lemma named "observable_moves_equal_length" assumes that there is a transition from state (ms, s) to state (ms', s') under a system S and function f. It shows that the length of ms is equal to the length of s, and the length of ms' is equal to the length of s'.
The main implementation of the resultant function for G1 and G2 is equal to the resultant of G1 and G2.
Mapping a converter with the function "Inl" and functions "f", "g", "h" over a parallel composition of two converters "conv1" and "conv2" is equivalent to mapping a converter with the identity function and functions "f" composed with "Inl", "g", "h" over the converter "conv1". This is denoted by the equivalence of the ...
The lemma named "open_LCons'" assumes that "A" is open. It shows that the image of "A" under the function "LCons x" is also open.
Given the assumptions that a relation R is right-unique, its inverse is also right-unique, and x is in the domain of R, it shows that the image of the set {x} under the relation R, and then under the inverse of R, is equal to the set {x}.
If a program P transitions from state s to state s' with thread t and thread action ta according to the Java Virtual Machine dynamics, then the shared state of s is a heap extension of the shared state of s'.
The lemma named "lookup_in_rig_b" assumes that the pair (b2, c2) is the result of looking up x' in the context where x is mapped to c', and that the pair (b1, c1) is the result of looking up x' in the original context. It then shows that b1 is equal to b2.
If the length of a list 'xs' is less than or equal to 1, then the inversion number of the list 'xs' is 0.
The matrix ring of size n and base b forms an abelian monoid, where 'a' is any element of a semiring.
If the sum of "i" and "n" is less than or equal to the length of "w", then the set of elements obtained from the function "cyk2" with inputs "w", "i", and "n" is equal to the set of elements obtained from the function "cyk" with the input being the first "n" elements of "w" after dropping the first "i" elements.
An empty table is a valid table with 'n' columns and a set 'V' of variable columns.
The function 'p' with parameters 's' and 's'' is continuous on the set 'A'.
An element 'x' belongs to the set of arrows of the directed graph 'dg_Set' parameterized by 'α' if and only if 'x' is an arrow in the set 'arr_Set' parameterized by 'α'.
If z is a non-positive integer, then the Gamma function of z equals zero. Similarly, if z is a non-positive integer, then the reciprocal Gamma function of z also equals zero.
The lemma named "stutterd_valid" asserts that the "stutterd_axiom" is valid.
If set A is independent of set B, then set A is a subset of the complement of set B.
A character 'c' is a zero digit if and only if the bit at position 0 of the integer representation of the character 'c' is set.
The lemma named "prod_A_is_p_B_bound" assumes that the function B of n equals 1. It shows that p is equal to the product of the function A of i raised to the power of i plus 1, where i ranges from 0 to n-1.
The mirrored bounds of the union of two sets A and B with respect to left and right bounds l and r, is equal to the union of the mirrored bounds of set A and the mirrored bounds of set B with respect to the same left and right bounds.
The implication from phi to psi is equivalent to the implication test of the if-then-else expression of LTL (Linear Temporal Logic) for phi and psi.
If the type relation domain is of type 'a' and the type relation R is a set of pairs where the second element is of type 'a', then the type relation R holds.
The function which takes a set Y and returns the union of set X and the result of applying function L with parameters c and Y, is monotonic.
If 'n' is greater than 1, then the invariant 'ahm_invar' holds true for a new hashmap with 'n' elements, where 'bhc' is the bucket hash code.
The lemma named "coprime_generic" assumes that "bs" is a subset of "as" and for every pair (a, i) in "bs", the function "f" applied to "i" is greater than 0. It shows that the product of all pairs (a, i) in "bs" is coprime with the sum of the product of all pairs (b, j) in "bs" excluding the current pair (a, i), multip...
: The inference of a pair 'pq' and a set 'X' is equal to the union of three sets: 1. The set of pairs '(p, snd pq)', where 'p' and 'q' are such that '(p, q)' is in 'X' and '(q, fst pq)' is in the set 'Delta_epsilon B A'. 2. The set of pairs '(fst pq, v)', where 'q', 'r', and 'v' are such that '(snd pq, r)' is in the...
The lemma named "poly_at_top_or_bot_at_bot" is about a polynomial 'p' with real coefficients. It assumes that the degree of 'p' is greater than or equal to 1 and that the coefficient of 'p' at its degree is greater than 0. It shows that the limit of the polynomial 'p' as 'x' approaches negative infinity is either posit...
The lemma named "finite_pred_imp_se_updatable" assumes that the set of variables in the conjunction of the predicate of command 'c' is finite (denoted as "finite ?V"). Under this assumption, it shows that the command 'c' is updatable.
The value of "x" in V is equal to the value of "x" in Vidx at the 0th position.
The lemma "sint_word_ariths" establishes several properties for words of a certain length 'a. 1. The signed integer of the sum of two words 'a' and 'b' is equal to the signed take bit of the sum of their signed integers, where the number of bits taken is one less than the length of 'a'. 2. The signed integer of the...
The number of blocks is less than or equal to the sum of the block representations for all elements in the set V.
The lemma "has_sum_Sigma" is defined with respect to a set 'A', a function 'B' that maps elements of 'A' to sets of 'b', and a function 'f' that maps pairs of 'a' and 'b' to 'c', where 'c' is a commutative monoid with addition and a uniform space. The lemma assumes that the function that adds any two elements of 'c' ...
, given an object 'a' in the bicategory 'B', the following conditions hold: 1. 'd a' and 'e a' are identities in 'B'. 2. The morphisms 'η a' and 'ε a' are isomorphisms in 'B'. 3. The source of 'e a' is 'a' and its target is 'P₀ a'. Similarly, the source of 'd a' is 'P₀ a' and its target is 'a'. 4. The domain of 'e a' ...
'a of class C is of a certain type class.
The lemma named "sum_1" assumes the following conditions: 1. "I" is a finite set. 2. The sum of the function "f" over all elements in "I" equals 1. 3. "i" is an element in the set "I" but not in the subset "I1". 4. "I1" is a subset of "I". 5. For any element "i" in "I", the function "f" of "i" is greater than or equal...
If x is less than or equal to y, then the residuum of x and z is less than or equal to the residuum of y and z.
The lemma named "worker_wrapper_fix" establishes a relationship between two functions, "wrap" and "unwrap", and a computation. It assumes that the "wrap" function, when composed with the "unwrap" function and a body of code, is a fixed point of the body of code. It also assumes that the computation is a fixed point of ...
: Assuming that both M and N are sigma-finite measures, the space of M is not empty, and for every x in the space of M, the function f(x) has a density g(x) with respect to N. Also, the product of the case and g is Borel measurable in the product sigma algebra of M and N, and f is measurable from M to the subprobabilit...
If a valuation "v'" satisfies the non-standard form of a set of constraints "cs" when flattened, then the valuation "v'" transformed from the non-standard form using the flattened list of the non-standard form of "cs", also satisfies the standard form of the set of constraints "cs" when flattened.
If a variable "states" satisfies the invariant "invar", then the size of "new_big" after a step operation on "states" is equal to the size of "new_big" in the original "states".
If 'v' is an element of the carrier 'M' and 'f' is a function from 'A' to the carrier 'R', then the finite sum of the function 'f' over the set 'A', scaled by 'v', is equal to the finite sum over the set 'A' of the function 'f' scaled by 'v'.
If the range of the substitution function theta is well-formed in terms of terms, then the result of applying the substitution function theta to any term x is also well-formed.
The coercion of a functor 'f from type 'a to type 'b is equivalent to the functor map of the coercion.
The measure as a quasi-Borel space is an element of the monad of quasi-Borel spaces over the set of real numbers.
Mapping the composition of the functions "projr" and "Inr" over the list "vl" results in the same list "vl".
The multiplication of a positive rational number 'a' with the sum of two positive rational numbers 'b' and 'c' is equal to the sum of the multiplication of 'a' with 'b' and the multiplication of 'a' with 'c'. This is a property of distributivity in multiplication over addition for positive rational numbers.
A list "xs" is a maximum if and only if the following two conditions are met: 1. For every index "i" less than the length of "xs", the element at index "i" in "xs" is not an empty list. 2. For every index "i" less than the length of "xs" minus one, one of the following two conditions is met: a. The list at index "...
Function f is not equal to function g if and only if there exists an x such that the value of function f at x is not equal to the value of function g at x.
: Assuming the representative list of polynomial 'p' is not zero and the leading term of the representative list of 'p' plus the leading term of 'q' is less than the leading term of the representative list of 'q' plus the leading term of 'p', it shows that the leading term of the S-pair of 'p' and 'q' equals the least ...
The weakest precondition of binding function g and function f is equivalent to the weakest precondition of applying function f to the result of applying function g to state s, for any given postcondition P and state s.
If G is a well-formed graph, then the security invariant of G with respect to a security policy nP holds if and only if for every node v in the nodes of G, for every successor node v' reachable from v in G, the security level of v is less than or equal to the security level of v'.
If "m" is an invertible pair, then the union of the mappings of the first and second elements of "m" is a submap of the mapping of the pair "m".
The set of all trees is equal to the universal set.
"hindered" in the context of Γ is equivalent to the existence of an epsilon greater than zero for which Γ is "hindered_by" epsilon.
If x is less than the limit of the function ord0_Lim f, then there exists a number n such that x is less than or equal to f(n).
If the function "sp_instrs" with parameters F, ret, an empty list, and states Sigma_i and Sigma_o holds, then Sigma_i is equal to Sigma_o.
The product of a constant "c" and the evaluation of a monomial (given by the constant "c'" and the exponent vector "es") at a point "x" in a certain basis, is equal to the evaluation of a monomial (given by the product of the constants "c" and "c'" and the exponent vector "es") at the same point "x" in the same basis.
The weakest precondition for the player winning when they switch is 2/3.
If a function 'f' maps elements from the carrier set 'D' to itself, and there exists an element 'a' in the carrier set 'D' such that for all elements 'x' in the carrier set 'D', 'x' is less than or equal to 'f(x)', and if 'W1' and 'W2' are elements of the set 'WWa D f a', and the order equivalence relation holds betwee...
The principal ideal generated by the set of all multiples of a given integer 'a' is a subset of the set of all integers.
The lemma named "zip_path_eq_left" assumes that the lengths of four lists (xs1, xs2, ys1, ys2) are all equal and that the result of zipping together xs1 and xs2 is the same as the result of zipping together ys1 and ys2. Under these assumptions, it shows that the list xs1 is equal to the list ys1.
There is a transfer rule between the cardinality function on variable finite sequences and the length function, where the cardinality function is represented by "cr_cr_vfsequence" and the length function is represented by "cr_omega".
The function "lift i F" is in a stable state with respect to the set "lift_set i A" if and only if the function "F" is in a stable state with respect to the set "A".
The allowed allocation in the Alloc theory is equal to the preservation of allocGiv.
The lemma named "return_app_return_cong" assumes that the function f applied to x equals the function g applied to y. It shows that the application of f to x is equal to the application of g to y.
The lemma named "pareto_efficient''" assumes that "R" is a preference profile, "i" is an agent, for all agents "i", "y" is at least as preferred as "x" according to "R i", and "y" is not less preferred than "x" according to "R i". It shows that "x" is not in the social choice function "R".
If for every 'i' in 'I', 'M i' is a module over 'R', and 'a' is an element of the carrier of 'R', and 'm' is an element of the cartesian product of 'I' and 'M', then the product of 'a' and 'm' under the scalar multiplication operation in the ring 'R' is an element of the cartesian product of 'I' and 'M'.
For all 'k', if 'i' is less than or equal to 'k' and 'k' is less than 'j', then the command 'c' at 'k' with postcondition 'Q' at 'k' is valid. This implies that the validity holds for the mapping of each 'k' in the range from 'i' to 'j' (exclusive) to the pair of 'Some' of the command 'c' at 'k' and the postcondition '...
If for any location, the work of a configuration 'c' at that location is an empty multiset, and if the configuration 'c' is inv_safe, then the configuration 'c' is safe.
If a set F is a frontier set, then the closure of the set is equivalent to the interior of the set's complement.
If vertex 'v' is in the directed vertices of the normalized form of 't', then 'v' conforms to the sequence.
The lemma named "trg_char'" shows that the target of a transition system is the set of all B such that the transition system is an arrow and B is an identity of P, and the targets of the congruence class representative of the transition system are the sources of B.
The lemma named "L_transform_Conj" assumes that the support of a set 'xset' is finite. It shows that the L_transform of the conjunction of 'xset' is equal to the conjunction of the map_bset of the L_transform of 'xset'.
If for any list 'xs', the function 'f' applied to 'xs' equals the result of the function 'list_rec' applied to 'xs', 'c', and 'h', then the following two conditions hold: the function 'f' applied to an empty list equals 'c', and the function 'f' applied to a list consisting of an element 'x' followed by a list 'xs' equ...
The assigned nodes are instances of the local assigned nodes.
In the lemma named "factorization_property" within the context of a Noetherian domain, given the assumptions that "a" is an element of the carrier set of ring R and is not equal to zero, and "a" is not a unit in ring R, it can be shown that there exists a set "fs" such that "fs" is a subset of the carrier of the multip...
The leading term of the polynomial X squared times Z plus 2 times Y cubed times Z squared, under the degree reverse lexicographic order (DRLEX), is the sparse polynomial with the term list [(1, 3), (2, 2)].
There is a relationship between "efficient_nat_div2" and "h.parent" in the identity function from the identity set to the identity set.
The function composition operator "(o)" is a parametric relation. This means that if we have a relation Ra that maps to Rb, and another relation Rc that maps to Ra, then we can compose these two relations to get a new relation that maps Rc to Rb.
The lemma named "eqgfE" makes two assumptions. The first assumption is that "t" is equivalent to the pair "(D, ss)" under the ground function. If this is true, then it shows two things. First, "t" is equal to the result of filling ground holes in "D" with "ss". Second, the number of ground holes in "D" is equal to the ...
The set of vertex indices in the complete planar graph with 5 vertices, denoted as "c_K5", is the set of integers from 0 to 4.
The lemma named "set_attribute_get_disconnected_nodes_is_l_set_attribute_get_disconnected_nodes" is an instance of "l_set_attribute_get_disconnected_nodes" with respect to the type well-formedness ("type_wf"), the "set_attribute" function, the locations of "set_attribute" ("set_attribute_locs"), the "get_disconnected_n...
The set of all elements 'v' that are less than or equal to 'w' is equal to the set of all elements obtained by taking the first 'k' elements of 'w', where 'k' is less than or equal to the length of 'w'.
The identity function permutes any set S.
The set of distinguishable state pairs with separators in a given machine M is equal to the union of the image of a function. This function takes a pair of states (q1, q2) and a separator A, and returns two pairs: ((q1, q2), A) and ((q2, q1), A). This function is applied to the set of all pairs (qq, A) such that A is n...
If the execution of expression "e1" in a certain thread "t" with a certain heap "h" and a certain state "s" results in a new state "s'", then the execution of the binary operation "bop" between expressions "e1" and "e2" in the same thread "t" with the same heap "h" and the same initial state "s" will also result in the...
The function "merge" applied to a term "t" produces a well-formed doubly-linked vertex set.
The index of any element 'x' of an enumerated type 'a' is less than the cardinality (or total number of elements) of that type 'a'.
There exists a non-negative constant K such that for all real vectors a and b, the norm of the product of a and b is less than or equal to the product of the norm of a, the norm of b, and K.
The lemma "rank_1_proj_square_mat" shows that the rank 1 projection of a vector 'v' results in a square matrix.
For a fixed matrix 'A' of any size, the statement "for all 'm' greater than 'i', the element at the 'm'th row and 'k'th column of 'A' is zero" is equivalent to the statement "the vector obtained from the 'k'th column of 'A' is all zero from the index 'i+1' onwards". Here, 'to_nat' is a function that converts a value to...
The lemma named "card_keys_homogenize" assumes that "x" is not in the set of individual terms of polynomial "p". It shows that the cardinality (or the number of elements) of the keys of the homogenized version of polynomial "p" with respect to "x" is equal to the cardinality of the keys of the original polynomial "p".
For any three configurations (cfg1, cfg2, cfg3) and a set Q of type 'p, if cfg1 can reach cfg2 and cfg2 can reach cfg3, then it is possible for cfg1 to reach cfg3.
The lemma named "unitary_col_norm_square" assumes that "U" is a unitary matrix, "U" is an n by n matrix, and "i" is less than "n". It shows that the square of the norm of the "i"th column of "U" is equal to 1.
The strict filtering of a bottom element with any predicate results in a bottom element.
The length sum set of the successor of a number 'r' and a number 'n' is equivalent to the Cartesian product of the set of all numbers up to 'n' and the function that maps each number 'i' to the length sum set of 'r' and the difference between 'n' and 'i'. This is denoted as "?L is equivalent to ?R".