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The base of a null capability is zero.
A match between a command and a memory state, represented by "matchC_M theta d c", is defined as follows: For all states 's' and 't', and all memory states 'd' and 'd'', if 's' is equivalent to 't' and there is a command transition from (d,t) to (d',t'), then one of the following must be true: 1. There exists a comma...
: Assuming the following conditions: - D' is less than or equal to D, - D' is an element of N and the set of S D' is empty, or the maximum of the atoms of D' is less than the maximum of the atoms of D, - The interpretation of D does not satisfy D', - D is less than C, - The interpretation of C is a subset of the union...
The set Q is a subset of the kernel of the function "lift_f" in the free group over the set S.
The composition of the functions "dm" and "eta" is equal to the downset function, which maps elements of a preorder to sets of elements.
The discriminator function applied to a Nonce with any value N equals 0.
The lemma named "linord_of_list_sorted_on_subseqs" assumes that "ys" is an element in the set of subsequences of "xs" and that "xs" contains distinct elements. It shows that "ys" is sorted according to the linear order of the list obtained by reversing "xs".
The set of mu-subformulas in the list for a given formula phi is equal to the mu-subformulas of phi.
"there exists some 'r' such that 'Some r' equals 'd_1 F'".
The lemma named "all_nstri_imp_mul_nstri" assumes that for all indices less than the length of the list 'ys', the second element of the function 'f' applied to the 'i'th elements of lists 'xs' and 'ys' is true. It also assumes that the lengths of lists 'xs' and 'ys' are equal. Under these assumptions, it shows that the...
The lemma named "c_list_prepend_f_disjunct" assumes that the lists "ys" and "xs" combined are distinct. Under this assumption, it shows that the cost of the list resulting from the function "ldeep_s" applied to the combined list "ys@xs" with parameters "cf", "h", and "r", when evaluated over "xs", is equal to the cost ...
The function that takes a function 'g' and returns a function 'h' which in turn returns a list function (constructed by applying 'g' and 'h' to 'c' and 'n' respectively), is continuous. Additionally, the concatenation of the results of applying 'g' and 'h' to the 'build' function is also part of this continuous functio...
The set of elements in the concatenation of list x and list y is equal to the set of elements in the concatenation of list y and list x.
The lemma Comm2: fixes a process P, names a and b, a process P', a process Q, a name x, a process Q'', and a process Q'. It assumes that process P can make a transition with action a[b] to process P', and process Q can make a transition with action b in Q'' leading to a<x> to process Q'. It shows that if proc...
The keys of the range of the vectorized polynomial p are equal to the power product of the term's keys of p.
, given the assumption named "context_sharing_Cut" which says that for any formula A and any contexts Gamma and Delta, if Gamma implies A and Delta, and A and Gamma imply Delta, then Gamma implies Delta. Under this assumption, it shows "context_free_Cut" which says that for any formula A and any contexts Gamma, Delta, ...
The lemma "get_owner_document_is_component_unsafe" assumes the following conditions: 1. The heap 'h' is well-formed. 2. The heap 'h' has a valid type. 3. The heap 'h' has known pointers. 4. The owner document of the pointer 'ptr' in heap 'h' is 'owner_document'. 5. The root node of the pointer 'ptr' in heap 'h' is not...
For all x in the set from 1 to m, the value of function T at min_arg T m is less than or equal to the value of function T at x.
, assuming 's' added to 's' equals zero, if 'w' is an element of the set generated by 's', then 'w' is either zero or 's'.
The C_merge_adj of a list xss is less than or equal to the length of the concatenated list xss.
The lemma "toggle_friends12_True" assumes the following conditions: 1. The system is in a reachable state 's'. 2. The identifiers for users UID1 and UID2 are valid in state 's'. 3. Users UID1 and UID2 are not friends in state 's'. Under these assumptions, it obtains an action list 'al' and an output list 'oul' such t...
If the action "act" is single-valued, then the union of the function "wens_single_finite" applied to the action "act", a set "B", and a value "k", and the weakest precondition of the action "act" on the function "wens_single_finite" applied to the action "act", the set "B", and the value "k", is within the range of the...
The lemma named "r_result1_diverg_phi" assumes that the function phi, when applied to i and x, is undefined. Under this assumption, it shows that the evaluation of the function r_result1, when applied to the list [t, i, x], is defined and equals 0.
"S is less than or equal to the binding of ASSUME function on Phi and a function that ignores its input and returns S prime" if and only if "S is not the least element (or bottom) implies Phi" and "S is less than or equal to S prime".
Both Fax and Wax satisfy the property GE.satPB according to the model \<xi>.
If "i" is not in the set "xs", "j" is not in the set "xs", the elements of "xs" are distinct, and "i" is not equal to "j", then if the pair "(a,b)" is in the set of arcs from "i" to "j" through "xs", "a" is not equal to "b".
For the state interpolation function, the valuation agrees with the initial state over the set S, and the valuation agrees with the final state over the complement of the set S.
If a single step of unobservable reduction (τred1gt) under the given conditions (uf P t h) transforms the expression-state pair (e, xs) into (e', xs'), then the same unobservable reduction also transforms the synchronized expression-state pair (sync(e) e2, xs) into (sync(e') e2, xs'). Here, "sync" denotes synchronizati...
The lemma named "drop_rk" assumes that the list "xs" contains distinct elements. It shows that the set obtained by dropping the first "i" elements from the list "xs" is equal to the set of all elements "q" for which there exists a number "j" greater than or equal to "i" such that the rank of "q" in the list "xs" is "j"...
"edges_basic" is an invariant.
Given the assumptions that "is_bounded_hashcode Id (=) bhc" and "def_size > 1", it can be shown that the pair "(ahm_empty def_size, Map.empty)" is in the relation "ahm_map_rel' bhc". In simpler terms, if we have a bounded hashcode and the defined size is greater than 1, then an empty associative hash map of that defi...
The lemma "WHILE_refine'" in the context of refinement is stated as follows: Assume the following conditions: 1. If the invariant "I'" holds for "x'", then the pair "(x, x')" is in the relation "R". 2. For any pair "(x, x')" in the relation "R" where the invariant "I'" holds for "x'", the boolean function "b" applied ...
The lemma named "Gleason9_34_contra" assumes that "A" is a cut. It shows that if for every "x" in "A", "x + u" is also in "A", and "u" is greater than 0, and "y" is greater than 0, but "y" is not in "A", then this leads to a contradiction.
The implementation of the probability mass function of a multiset (pmf_of_multiset_impl) applied to a multiset of elements (mset xs) is defined as follows: - If the list of elements (xs) is empty, then the code aborts with the message "pmf_of_multiset of empty multiset". In this case, the function mapping_of_pmf is ap...
If "k" is greater than zero, then the cardinality (or size) of the interval "I" multiplied by "k" is equal to the cardinality of the interval "I".
Given two functions f and g, where both are functions from some type 'a to natural numbers, and assuming that f is a simple function on a measure space M, then the function that maps each element x of type 'a to the real number corresponding to f(x) is also a simple_function on the measure space M.
If specification s is true, then SafeRecordsMono is always true for s.
If a functor maps a category to another category, if there is a morphism from one object to another in the second category, and if there are universal arrows from the first object and the second object to the right adjoint of the functor applied to the object map of the unit at the first object and the second object re...
For a given element 'x' in a T1 space, 'x' is isolated in the set obtained by inserting an element 'a' into set 'S' if and only if 'x' is isolated in set 'S' or 'x' is equal to 'a' and 'x' is not a limit point of set 'S'.
The non-negative conditional expectation is the conditional expectation. It assumes that for every element in the space of the measure, the function X is non-negative. It also assumes that X is integrable with respect to the measure, that Y is a discrete function, and that Y is point measurable with respect to the spac...
For any two formal Laurent series 'f' and 'g' over a division ring 'a', the division of 'f' by the shifted version of 'g' by 'm' is equal to the shifted version of the division of 'f' by 'g' by '-m'.
The complementary error function of a real number 'x' is less than 1 if and only if 'x' is greater than 0.
If 'x' and 'y' are elements of the carrier of the ring, then the product of 'x' and the n-th power sum of the product of the n-th power of 'x' and the i-th power of 'y', scaled by the binomial coefficient 'n choose i', is equal to the n-th power sum of the product of the (n+1)-th power of 'x' and the (i+1)-th power of ...
For any set I and state s: 1. The lower bound of the set_unsat function applied to I and s is equal to the lower bound of s. 2. The upper bound of the set_unsat function applied to I and s is equal to the upper bound of s. 3. The tableau of the set_unsat function applied to I and s is equal to the tableau of s. 4. The...
The function 'f' belongs to the small omega of 'F' at 1 if and only if the limit of the function 'f' approaches infinity at 'F'.
A relation is one-strongly-connected if and only if its transpose is also one-strongly-connected.
For any three trilean values a, b, and c, the operation of "or" between a and b, followed by "and" with c, is equal to the operation of "and" between a and c, "or" with the operation of "and" between b and c. This is a statement of distributivity property in trilean logic.
"sswSts" is a full chain with respect to the "leads to" relation. Additionally, "wSts" is embedded in "sswSts". Furthermore, the leftmost element of "sswSts" is the same as the leftmost element of "wSts", and the last element of "sswSts" is the same as the last element of "wSts".
, assuming for all natural numbers 'n', 'a n' is either 0 or 1, the limit of the sum from 'i=0' to 'n' of the real number of 'a i' times one-half to the power of the successor of 'i', is equal to the sum over all natural numbers 'n' of the real number of 'a n' times one-half to the power of the successor of 'n'.
If a system A satisfies the invariants P and Q under condition I, and if for any states s and s' and action a, if P holds for s and s', Q holds for the transition from s to s' under action a, and I holds for action a, then R holds for the transition from s to s' under action a, then the system A satisfies the invariant...
The fold operation on a Mapping object, which is constructed from a list "xs" and an initial value "a", is equivalent to performing a List.fold operation on the sorted and cleared list "xs" with the same function "f" and initial value "a". The function "f" is applied to each pair in the list, where "case_prod f" is a f...
The predecessor of the identity of zero (in natural numbers) is the identity of zero itself. Furthermore, for all natural numbers 'n', the predecessor of the successor of 'n' is 'n' itself.
The lemma named "jumpF_polyR_coprime'" assumes that the polynomial p evaluated at x is not equal to zero or the polynomial q evaluated at x is not equal to zero. It shows that the jump of the polynomial q over the polynomial p at x is equal to: if p is not equal to zero and q is not equal to zero and the polynomial p e...
The lemma named "Z_I_ZmX_eq" shows that the product of "Z_I" and "I_ZmX" equals "Z_ZmX".
If we have an if-then-else operation with "x" as the condition, "-q" as the then clause, and "y" as the else clause, and this entire operation is again used as the condition in another if-then-else operation with "-p" as the then clause and "y" as the else clause, then this is equivalent to having an if-then-else opera...
The observation function of skl1 is the identity function.
, given a function 'f' and a condition 'G' on a set 'U' at a specific time 't0', the derivative of 'x' equals 'f' and 'G' holds true for 's'.
The lemma named "check_s_v_unit" assumes that under the contexts Theta, B, and Gamma, the type constraint set {z : B_unit | TRUE} is a subtype of tau. It also assumes that Delta is a well-formed context under Theta, B, and Gamma, and that Phi is a well-formed context under Theta. Given these assumptions, it shows that ...
A string 'w' is accepted by the star of automaton 'A' if and only if there exists a list of strings 'us' such that every string 'u' in 'us' is accepted by 'A' and 'w' is the concatenation of all strings in 'us'.
The maximum arity (the number of arguments or operands that the function can take) of the transformed inference rule set is equal to the maximum arity of the original inference rule set minus one.
Taking 'a' elements from the sequence 's' and then taking 'b' elements from the result is the same as taking 'b' elements from the sequence 's' and then taking 'a' elements from the result.
The set of elements in list x is equal to the set of elements in list y if and only if the multiset of the list x with duplicates removed is equal to the multiset of the list y with duplicates removed.
: Assuming that the theory 'Theta' is well-formed, and that 'Theta' and 'Gamma' prove the equality of two abstract terms 's' and 't' (both of which are closed), where each term is bound to a free variable 'x' with type 'tau-prime'. Also assuming that the pair (x, tau-prime) is not a free variable in 'Gamma' and that...
Two reaction functions, React f and React f', are equal if and only if the functions f and f' themselves are equal.
The lemma named "map_of_inject_set" assumes that the first elements of the pairs in the lists "xs" and "ys" are distinct. It shows that the mapping of the list "xs" is equal to the mapping of the list "ys" if and only if the set of pairs in "xs" is equal to the set of pairs in "ys".
The previous element of a singleton set containing 'a' with respect to 'n' is 'n'.
If a set S is connected and there is a continuous function P on S, then there exists a constant c such that for all elements s in S, the value of function P at s equals to c. This is for a function P that maps from a topological space to a boolean value.
A singular relative cycle 'c' of dimension 'p' in a topological space 'X' with a subspace 'S' is equivalent to 'c' being a singular chain of dimension 'p' in 'X' and the boundary of 'c' being a singular chain of dimension 'p-1' in the subtopology of 'X' restricted to 'S'.
The specification of a function that takes a pair consisting of a key-value pair and a map, is equivalent to the operation of picking and removing from a map. Specifically, the function should satisfy the condition that the map contains the key-value pair, and the new map is the original map restricted to the complemen...
If "e" is final, then the first component of "e" compiled with respect to variables "Vs" is equal to the first component of "e" being final.
If for any list 'zs', if 'zs' is a valid ordered association list, then 'f zs' is also a valid ordered association list, and if the set of first elements of 'f zs' is a subset of the set of first elements of 'zs', then 'f' is compatible with the function 'map2_val'.
: Assuming that for any finite set G, if G is a subset of M, then for almost all H in F, H is a superset of G (big assumption). And also assuming that for almost all H in F, H is a subset of M (small assumption). Then it shows that the function that applies trunc_ell2 to S and ψ tends to apply trunc_ell2 to M and ψ ...
The lemma "InvariantConsistentAfterApplyDecide" assumes the following conditions: 1. The set of variables in the elements of the state's model does not contain all variables in Vbl. 2. The state's model is consistent. 3. The state's formula, first watch, and second watch satisfy the InvariantWatchesEl condition. 4. Th...
For any natural number 'm', the Cartesian product of a set 'A' and the set of all natural numbers up to 'm' is equal to the union of two sets. The first set is the set of all pairs where the first element is from 'A' and the second element is a natural number up to 'm' minus the first element. The second set is the set...
If the image of a value 'v' under a relation 'R' is empty, then the image of 'v' under the reflexive transitive closure of 'R' is a set containing only 'v'.
If the bag X is empty in a given state tau, then X is valid in that state tau.
For any index 'i', if 'P j' holds true and 'P' does not hold true for the element at index 'i' in the list, and 'i' is less than the length of the list, then the length of the list obtained by filtering the list with 'P' is less than the length of the list obtained by filtering the list where the element at index 'i' i...
The lemma named "convex_rel_open_sum" is about sets S and T in a Euclidean space. The lemma assumes that both S and T are convex and relatively open. Under these assumptions, the lemma shows that the sum of S and T is also convex and relatively open.
If 'a' is irreducible under a homomorphism, then 'a' is irreducible. This is assumed under the condition that 'a' is irreducible under the homomorphism.
The lemma in the context of the prisoner's dilemma shows that the Hermitian conjugate of J is equal to the complex conjugate of J.
The scalar multiplication of a sparse vector with a scalar 'y' and a list consisting of an element 'a' and an array 'arr' is equal to a list where the first element is the first element of 'a' and the second element is 'y' multiplied by the second element of 'a', followed by the scalar multiplication of 'y' and the arr...
: Assuming the following conditions: 1. For any initial state 's0' of event system 'E', there exists an initial state 't0' of event system 'F' such that 'R' relates 's0' and 't0'. 2. For any states 's', 't', 's'' and any action 'a', if 'R' relates 's' and 't', 'I' holds for 's', 'J' holds for 't', and 's' transitions ...
If the pair (l, R) is an element of S, then R is a subset of the set of invariants of automaton A at location l.
If an infinite set of any type 'a exists, then for most elements 'x' of type 'a', if property 'P' holds for 'x', then there are infinitely many elements 'x' of type 'a' for which property 'P' holds.
For all z, if F leads from the state where z is less than or equal to the function of s, to the state where z is less than or equal to the function of g of s, then F also leads from the state where z is prefix greater than or equal to the function of s, to the state where z is prefix greater than or equal to the functi...
Taking the first 'n' elements of a zipped list of 'xs' and 'ys' is equivalent to zipping the first 'n' elements of list 'xs' and the first 'n' elements of list 'ys'.
The image of a singleton set under a function f is a singleton set containing the image of the element under the function f.
The inverse of 'b' intersected with the inverse of 'm' results in an empty set.
The elements of set A are countable if and only if the cardinality of set A is less than or equal to Aleph null (the smallest infinite cardinal number).
: Given two assumptions: 1) For any element 'a' in set 'S', the function 'f' applied to 'a' results in an element in the carrier 'R'. 2) For any element 'a' not in set 'S', the function 'f' applied to 'a' is undefined. Under these conditions, it can be shown that the function 'f' is an element of the carrier 'F'.
The unary minus function is in the relation of integer to integer.
The sum of the modular ring representations of two integers x and y is equal to the modular ring representation of the sum of x and y, where 'a is a prime cardinality.
The lemma epm_4: Assuming that "e" is less than or equal to "w", "w" is injective, the product of "w" and "v" is less than or equal to "v", and "e" is less than or equal to the product of "v" and the transpose of the negation of "v", it can be shown that the product of the top element, "e", and the transpose of the pos...
The function A with subscript f is less than or equal to p in the context of an algebraic structure A.
The mapping of the merged lists "xs" and "ys" is equivalent to the mapping of list "xs" combined with the mapping of list "ys".
The lemma named "comp_gmctxt_sym" assumes that the pair (C, D) is in the set "comp_gmctxt". It then shows that the pair (D, C) is also in the set "comp_gmctxt". This suggests that the set "comp_gmctxt" is symmetric, meaning if (C, D) is in the set, then (D, C) is also in the set.
The strict but last element of an undefined or bottom list is also undefined or bottom.
The lemma named "inter_aform_plane_ortho_overappr" assumes that the intersection of an affine form with a plane orthogonal to a given vector results in some affine form X. It then shows that the set of all vectors x, such that for all basis vectors i, the dot product of x and i is in the set of all pairs (g, y) where (...
If the pair (p, r) is in the reflexive transitive closure of the epsilon transition function of A, then the pair (True appended to p, True appended to r) is in the reflexive transitive closure of the epsilon transition function of the Kleene star of A.
: the complement of x is equal to the union of the intersection of the complement of x and the complement of y, and the intersection of the complement of x and the double complement of y.