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: Assuming "v" is less than or equal to "u", "Bb" of "u" is greater than or equal to zero, and "u" is improving with respect to "d'", it can be shown that "Lb" of "v" is less than or equal to "W" of "d'" at the successor of "m" applied to "u".
The lemma named "subset_cls_val_process" assumes that "A" is a finite set and the support set of "p" is a subset of "A". It shows that for all "n" and "w", the closing value process of a market "Mkt" and a portfolio "p" at time "Suc n" (the successor of "n") and state "w" is equal to the sum of the product of the price...
There exists a matrix B such that when it is multiplied from the right by matrix A, the result is the identity matrix if and only if the span of the columns of matrix A is the entire universe. This is denoted as "?lhs = ?rhs", where "?lhs" represents the left-hand side of the equivalence and "?rhs" represents the right...
The lemma named "subst_cv_var_flip" is established with a fixed constant 'c'. It assumes that the atom 'y' is independent from 'c'. It then shows that swapping 'y' and 'x' in 'c' is equivalent to substituting 'x' with 'y' in 'c'.
The lemma named "odd_distinct" assumes that for any integer 'i', if 'p i' is not equal to zero, then 'i' is odd. Under this assumption, it shows that the function 'odd_of_distinct' applied to the result of the function 'distinct_of_odd' applied to 'p' is equal to 'p'.
If 'i' is an ordinal and 'j' is an element of 'i', then 'j' is a transitive set.
If the card B outputs a set containing a nonce Nb, an agent A, a key K, and two certificates (Cert1 and Cert2) in the event set evs, and if this event set is in the Shoup-Rubin-Bella protocol, then there exists a nonce Na such that B inputs a set containing the agent A and the nonce Na in the event set evs.
The scalar multiplication of any polynomial 'p' and zero equals zero.
For a function 'f' that maps a natural number to a real number, the sum of the product of 'f k' and 'g j' for all 'k' and 'j' in the range from 1 to 'n' is equal to the sum of the average of the product of 'f k' and 'g j' and the product of 'f j' and 'g k' for all 'k' and 'j' in the same range.
If the invariant m2_inv9_M5 holds before the execution of step 4 in protocol m2, involving roles Ra, A, B, Na, and Kab, then the invariant m2_inv9_M5 will still hold after the execution of this step.
The lemma named "kronecker_inverse_index" assumes that "r" is less than the number of rows in matrix A, "s" is less than the number of columns in matrix A, "v" is less than the number of rows in matrix B, and "w" is less than the number of columns in matrix B. It shows that the element at the position (dim_row B*r+v, d...
: Given the assumptions that the pair (f u, v') is in the reflexive transitive closure of the edge relation of graph G' and that u is an element of the set V, it can be shown that there exists an element v such that v' is equal to f v, v is an element of the set V, and the pair (u, v) is in the reflexive transitive clo...
The lemma "LLL_d_bound" assumes that "i" is less than or equal to "m". It shows that the absolute value of "d fs i" is less than or equal to "N" raised to the power of "i" and the absolute value of "d fs i" is also less than or equal to "N" raised to the power of "m".
If in the context of a program P, the type of a value v, when evaluated in a heap h, is some null type (NT), then the value v must be null.
The projection of trace 't' on configuration 'c' is an empty list if and only if for all 'k' in the lset of 't', 'c' is not parallel to 'k'.
The lemma named "constraint_model_Value_in_constr_prefix_fresh_action'" is given a list of protocol transactions 'P'. It assumes that 'A' is within the reachable constraints of 'P'. It also assumes that for every transaction 'T' in 'P', the transaction terms are admissible, the transaction declaration is empty, and the...
: Assuming the support of X is finite and the action coherence (AC) of the tuple (f, F, P) leads to the L-transition pair (\<alpha>\<^sub>L, P\<^sub>L'), it can be shown that there exists an action \<alpha> and a state P' such that P leads to the pair (\<alpha>, P') and the L-transition pair (\<alpha>\<^sub>L, P\<^sub>...
The arithmetic progression with a common difference of 1 (Successor of 0) and starting from any number 'a', encompasses the entire universe of discourse (all possible values).
The lemma named "alphaBinp_imp_asBinp_equal" assumes that "qbinp" is a good binary predicate and that "qbinp" is alpha-equivalent to "qbinp'". Under these assumptions, it shows that the standard form of "qbinp" is equal to the standard form of "qbinp'".
The box of the product of x and d(y), applied to z, is equal to the box of the product of x and d(y), applied to the product of d(y) and z.
The nth element of a column vector is equal to the row of the column vector v.
The lemma Loss_L5: Assuming that for all t1 less than or equal to t, a t1 equals an empty list, and there is a loss at a certain point i in the sequence a, denoted as "Loss lose a i2 y i", and t2 is less than or equal to t, and there is a timestamp loss, denoted as "ts lose", it can be shown that y at t2 equals an emp...
If 'v' and 'w' are elements of 'V', then 'F' is in the stable state where the number of messages equals zero between 'v' and 'w', or 'v' is reachable.
The function "return composed with op_list_is_empty" is in correspondence with the function "RETURN composed with op_list_is_empty", under the condition that the input is a list of elements of type A. The output of this function is a boolean value. This is expressed in the context of separation logic, which is used for...
If the condition is false, then the "If" function will return the second argument.
If "ps" is an element of set "P", then the free lift id of the absolute free word of "ps" equals zero.
Given two distinct points "u" and "v" in the unit disc, there exists a Moebius transformation "M" that fixes the unit disc and maps "u" to the origin and "v" to a point on the positive x-axis.
If the infliveness of an empty set and a property P holds, then it leads to a contradiction or falsehood.
The lemma named "cong_exp_mod" states two equivalent conditions in modular arithmetic: 1. The remainder of 'a' divided by 'n', raised to the power of 'b', is congruent to 'c' modulo 'n' if and only if 'a' raised to the power of 'b' is congruent to 'c' modulo 'n'. 2. 'c' is congruent to the remainder of 'a' divided by...
Given the assumptions that a system "\<R>" is linear, the pair "(c, 0)" is not in the functional relation of "\<R>", the pair "(c, 0)" is not in the functional term "s", and the pair "(s, t)" with "t" substituted by a constant "c" is in the reflexive transitive closure of the reduction step of "\<R>", it can be shown t...
The theorem named "abstract_correct" demonstrates that Pedersen's commitment scheme is correct.
If 'x' is a unit in group 'G', and 'y' and 'z' are elements in the carrier of 'G', then 'x' multiplied by 'y' equals 'x' multiplied by 'z' if and only if 'y' equals 'z'.
There is no infinite computation from the configuration (c, s) under the context Γ. It shows that under the context Γ, the command c terminates on the state s.
If "D" is an order and "T" is a subset of the carrier of "D", then the order of the subset "SIod D T" is also an order.
If the free variables of term 't' are a subset of 'Q', and 'q' is an element of the derivative of the restriction of automaton 'A' to 'Q' with respect to term 't', then 'q' is an element of 'Q'.
The successor of zero (which is one) modulo the cardinality of a nontrivial type 'a is equal to one.
The lemma named "bhv2cnf_cnf2bhv" assumes that "n" is greater than or equal to the configuration "c" and time "t". It shows that applying the function "c" to the time "t" on the result of applying the function "t" to the configuration "c" on "n" equals "n". This is expressed as "?lhs = ?rhs".
If the local state is not idle and becomes idle after executing a transition function with no message for 'n' times, then there exists a number 'i' less than 'n' such that for all numbers 'j' less than or equal to 'i', the local state is not idle after executing the transition function with no message for 'i' times, an...
For a fixed matrix 'A' of a type 'bezout_ring', and for two indices 'a' and 'b' of a modular type, if the natural number representation of 'a' is in the set from 'i' to the minimum of the number of rows of 'A' and the number of columns of 'A', and if the natural number representation of 'a' is equal to the natural numb...
If 'x' is an element of 'assertion', then the negation of 'x' is also an element of 'assertion'.
If a group G acts on a set H and G also acts on a set N, then the group operation on the set resulting from the group operation on H and N in G is a group.
The lemma named "scalar_prod_smult_distrib" assumes that vectors v1 and v2 are in the carrier vector space of dimension n. It shows that the scalar product of vector v1 and the scalar multiple of a and vector v2 is equal to the scalar a times the scalar product of vectors v1 and v2. This is true for any a in the commut...
The lemma "nth_visible_eff" assumes that the nth visible element in sequence s is i. It shows that the extended set of i is a subset of the set of sequence s mapped by function I.
If 'e' is an element of 'E', then the cardinality of 'e' is either 1 or 2.
If 'G implies p' is a tautology, then 'the conjunction of G implies p' is also a tautology.
The identity function, which maps each element to itself, is measurable with respect to the sigma algebra M.
If an edge 'e' is not in the directed arcs of a tree 't' and a vertex 'v' is not in the directed vertices of the tree 't', then the function 'insert_between' with parameters 'v', 'e', 'x', 'y', and 't' will result in well-formed directed arcs.
: if for any given 'y', the functions p1, p2, p3, and p4 are monotonically non-decreasing with respect to the partial order defined by sum_bot, then the function xml3to4elements, which takes the parameters t, p1, p2, p3, p4, f, and x, is also monotonically non-decreasing with respect to the same partial order.
The transition function "delta" applied to the epsilon-closure of a set R and a string bs is equal to the transition function "delta" applied to the set R and the same string bs.
If the absolute value of 'z' is not infinity and 'z' is not equal to zero, then the extended real number of the inverse of the real number of 'z' is equal to the inverse of 'z'.
If a property P holds for a natural number x, then the property P also holds for the unique element that satisfies property P.
If "n" is less than or equal to "n prime" and "n prime" plus "d prime" is less than or equal to "n" plus "d", then the interval from "n prime" to "d prime" is a subset of the interval from "n" to "d".
If a binary tree 't' with height 'h' is a sorted tree, then the tree that results from removing element 'e' from 't' at height 'h' is also a sorted tree.
An operator with a certain name cannot be found in the given problem if and only if the same operator cannot be found in the problem after removing implicitly preserved operators.
The lemma named "cfs_monom_mult_r" assumes that "p" is an element of the carrier set "P" and "a" is an element of the carrier set "R". It shows that the result of multiplying the monomial of "P" with "a" and "n", and then multiplying it with "p", evaluated at "k + n", is equal to the result of multiplying "a" with "p" ...
Zero adds to any polynomial vector 'v'.
In the lemma for a ZF-sequence, the domain of the ZF-sequence is in the Vset.
If points A, B, C are supplementary to points A', B', C' and points D, E, F are supplementary to points D', E', F', and if the sum of angles A, B, C and D, E, F equals to the sum of angles G, H, I, then the sum of angles A', B', C' and D', E', F' also equals to the sum of angles G, H, I.
For a list consisting of an element 'x' followed by a list 'xs', and given two natural numbers 'i' and 'j', the function 'nths'' with parameters 'i', 'j' and the list, is equal to the following: if 'i' is 0, then the result is a list starting with 'x' followed by the result of 'nths'' with parameters 0, 'j', and 'xs'; ...
The lemma named "combine_substs" makes the following assumptions: 1. "V1" is a finite set. 2. The intersection of "V1" and "V2" is an empty set, meaning they have no elements in common. 3. The function "η1" is ground on "V1", meaning all variables in "η1" are from the set "V1". Given these assumptions, the lemma shows...
The result of the function "pop" applied to a list that starts with "j" followed by the list "js" and a value "v" is equal to: if the "j"-th element of the list "xs" is greater than or equal to "v", then the result of the function "pop" applied to the list "js" and the value "v", otherwise the list that starts with "j"...
Under the context Γ and Θ, given the precondition 'precond', if we concurrently begin a scheme where 'm' ranges from 0 to less than 2 and call the function 'sumarr' with 'm', then the local postcondition 'local_postcond m' holds. The scheme ends with the postcondition 'postcond'. The false braces indicate that there ar...
If the comparison of 'x' and 'y' using the 'cmp'' function results in 'Greater', then the comparison of 'y' and 'x' using the 'cmp''' function cannot result in 'Greater'.
The complete graph with vertex set V is well-formed.
The supremum of a finite set A is equal to the result of folding the set A with the supremum operation.
The set of all r-element subsets of a singleton set {a} is equal to: an empty set containing an empty set if r equals 0, a set containing a set with element a if r equals 1, or an empty set in all other cases.
"a", "b", and "c" form a triangle. It shows that there do not exist any three points "d", "e", and "f" from the set of points {a, b, c} such that "d", "e", and "f" are in betweenness relation.
The extraction of a variable 'v' from the sum of two polynomials 'p' and 'p'' is equal to the sum of the extraction of the variable 'v' from each polynomial 'p' and 'p'' individually.
The function "sameDom" when applied to the input "inp" and the result of the function "lift" applied to "f" and "inp", will yield the same domain.
For a finite function 'f' updated at 'a' with value 'b', the function holds for all values except those in set 'A' if and only if 'a' is in set 'A' or 'b' is true, and the function holds for all values except 'a' and those in set 'A'.
If for all x, if phi of x implies necessarily phi of x, then if there exists a unique x such that phi of x, it implies that there exists a unique x such that necessarily phi of x. This statement is in the context of a variable v.
If "f" is an arrow in the category (as assumed by "Arr f"), then "f" is an arrow (as shown by "arr f").
If 'x' is an order, then the transpose of 'x' is also an order.
The lemma named "aux1_digit_sum_repr" assumes that "n" is less than 2 to the power of "c" and "r" is less than "c". It shows that the sum of the product of "n" exclamdown "k" and 2 to the power of "k" for all "k" less than "c", exclamdown "r", is equal to "n" exclamdown "r".
If the end of phase 1 in states 's' and 's'' for process 'p' and the left-hand side of invariant 3 holds in state 's'' for processes 'p', 'q', and disk 'd', then the right-hand side of invariant 3 also holds in state 's'' for processes 'p', 'q', and disk 'd'.
If a multiset "F" is a multiset of factors of "p" (as per the assumption "mset_factors F p"), and if for all "f" in "F", "f" is irreducible, and "F" is not an empty multiset, and the product of all elements in "F" equals "p", then the thesis (or the main proposition being argued) holds true.
The sort function is equivalent to the quicksort function.
If a program P evaluates an expression e in a state s0 to null in a state s1, and P evaluates a sequence of expressions ps in state s1 to a sequence of values evs in state s2, and the values of evs map to vs, then P evaluates the method call e.M(ps) in state s0 to throw a NullPointer exception in state s2.
If 'a' and 'b' are elements in the carrier set 'D', then 'a' is less than 'b' if and only if the segment of 'D' containing 'a' is a proper subset of the segment of 'D' containing 'b'.
"x" is the least prefix point of a function "f" if and only if the function "f" has a least prefix point and "x" is equal to the least prefix point of function "f".
The lemma named "fundchamber_vertex_stabilizer1" is defined with a fixed variable "t". It also defines "v" as the fundamental antivertex of "t". It assumes that "t" is an element of "S", "w" is an element of "W", and "w" maps to "v" equals "v". The lemma shows that "w" is an element of the generated set "S" excluding "...
There are no cycles in the statecharts of the hybrid automaton "HA" when composed using the composition function "CompFun".
The lemma "isubexpr_reduct" assumes that "d1" is a subexpression of "e1". It shows that for any "d2", if "d1" reduces to "d2" under the class table "CT", then there exists an "e2" such that "e1" reduces to "e2" under the same class table "CT".
In the context of a security invariant with offending flows, there is an instance of weak reflexivity for the first elements of the set. It fixes the default node properties, represented as "\<bottom>". It assumes the following conditions: 1. The security invariant for all edges is in normal form and not reflexiv...
If for all elements 'x' in the set 'xs', the function 'P' applied to 'x' is pure in the heap 'h', then the function 'forall_M' applied to 'P' and 'xs' is also pure in the heap 'h'.
If a probability mass function 'x' is related to 'y' under a relation 'P', and 'P' is less than or equal to 'Q', then 'x' is also related to 'y' under the relation 'Q'.
If "X" is in the tangent space at point "p" and "p" is in the carrier, then the value of "X" at the restriction of the constant function "c" to the carrier is zero.
For every element 'k' in 'I', if 'k' is an aGroup (an associative group) in 'A k', then the zero element of the product aGroup 'prodag I A' is equal to the product zero 'prod_zero I A'.
For a permutation "π" and a function "f" of type 'a to 'a, where 'a is a permutable type, the permutation of the function "f" raised to the power "n" is equal to the permutation of the function "f" raised to the power of the permutation of "n".
If "g" is an element of the group's carrier "G", and "H1" and "H2" are subsets of the carrier "G", and the conjugation of "g" and the inverse of "g" with "H1" is equal to the conjugation of "g" and the inverse of "g" with "H2", then "H1" is a subset of "H2".
The tail of the list obtained by dropping the first 'n' elements from the list 'xs' is equal to the list obtained by dropping the first 'n+1' elements from the list 'xs'.
The set B is a subset of the power set of U.
The left and right Kleene star of the product of a language 'r' and the sum of the language 'One' and another language 's' is equal to the left and right Kleene star of the language 'r' and 's'.
If there is a relation "S" between states "s1" and "s2", and if "gpv1" and "gpv2" are equivalent under the interface "\<I>" and relation "A", then the inline transformations of "gpv1" and "gpv2" with "callee1" and "callee2" respectively, are also equivalent under the interface "\<I>'" and the product relation of "A" an...
The norm (or magnitude) of a complex number formed by cosine of t as the real part and sine of t as the imaginary part is equal to 1.
The function "lrev_body1" is equal to the composition of the functions "unwrapH", "lrev_body", and "wrapH".
The tensor product of 'a' with the sum of 'b' and 'c' is equal to the sum of the tensor product of 'a' with 'b' and the tensor product of 'a' with 'c'.
Taking the first 'n' elements from a sequence 'xs' will always result in a subsequence of 'xs'.
The order of a sub-probability mass function (spmf) R for two cases - when x is None and when x is Some y - is the same for both a and a', and for b y and b' y. This holds true if the order of spmf R for a and a' is the same, and for all y, the order of spmf R for b y and b' y is the same.
If the restriction of function ρ to the complement of the domain of h is less than or equal to the restriction of function ρ' to the same set, then the heap semantics of h with respect to ρ is less than or equal to the heap semantics of h with respect to ρ'.