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If a set 'I' is finite and not empty, then the mirror element of the maximum value in set 'I' is equal to the minimum value in set 'I'.
If the following conditions are met: 1. The length of mfs is m. 2. i is less than or equal to m. 3. The length of fs is m. 4. The lattice of fs is L. 5. The list fs is linearly independent over the rational numbers. 6. The list fs is weakly reduced at index i. 7. The list mfs is obtained by applying the function "map_...
The lemma "freshInputTransition" is defined with fixed variables P, a, u, P', and c, where P and P' are processes in the pi calculus, and a, u, and c are names. The lemma assumes that: 1. There is a transition from process P to process P' with input action a<u>. 2. The name c is fresh for process P, meaning it does n...
"x" is an element of the limit inferior of "A" if and only if for all "n" in a sequence, "x" is an element of "A" at "n".
If the set obtained by cutting the set 'I' at a point 't' is not empty, then the minimum element of this cut set is equal to the minimum element of the original set 'I'.
: given the assumptions that for any element 'x' in set 'A', 'x' is a subset of 'B', and that set 'A' is not empty, it can be shown that the intersection of set 'A' is a subset of 'B'.
: Given two conditions, "leq" and "geq". "leq" is a condition where for a list A and an index i, all elements at indices less than i are less than or equal to a pivot value. "geq" is a condition where for a list A and an index i, all elements at indices greater than i and less than the length of A are greater than or ...
A probabilistic function (pmf) mapping 'f' is lossless if and only if for every element 'x' in the set of the probabilistic function 'p', the function 'f' applied to 'x' does not equal to None.
If the number of columns in the transpose of matrix A equals the dimension of vector x, and the number of rows in the transpose of matrix A equals the dimension of vector c, and for all i less than the dimension of vector c, the i-th element of the product of the transpose of matrix A and vector x equals the i-th eleme...
If the following conditions are met: 1. The pair (out, callee', rpv') is in the set of outcomes produced by the inline1 function applied to the callee, gpv, and s. 2. The pair (x, s') is in the set of results produced by the callee' function applied to the input. 3. The input is in the set of responses produced by the...
If the "icheck" function for a program P and class C on expression e' holds true, and the expression e is equal to e', then the "curr_init" function for the program P on expression e equals the class C.
The image of a set 'S' under the function 'h' is a polyhedron if and only if 'S' itself is a polyhedron.
If a system P is secure with respect to an interference function I and a domain function D, then for any pair of sequences (xs, ys) that belong to the relation defined by the ipurge function with respect to P, I, D, and the domain of x, if the set containing x is in the refusals of P with respect to xs, then the set co...
The transpose of a list 'xs' is empty if and only if every element 'x' in the set of 'xs' is an empty list.
An element 'k' is in the successor of 'k'.
If a system is deterministic, and a thread in the system's state is not waiting for any locks, and the thread transitions from one state to another with some action trace, and these actions are acceptable in the system's state, and the system's state is in a set of states I, then the unique tuple of action trace, new s...
The lemma named "dynmethd_access_prop" assumes the following conditions: 1. The method "statM" is defined in the class "statC" with the signature "sig" in the program "G". 2. The method "statM" of the class "statC" with the signature "sig" is accessible from the class "accC" in the program "G". 3. The dynamic method "...
If the initial time "t₀" is in the set "U(s)", and "U(s)" is a subset of "T(s)", and "X" is a solution to the differential equation system "f" with time set "T", state space "S", initial time "t₀", and initial state "s", then "X" is also a solution to the same differential equation system but with time set "U" instead ...
The lemma named "has_integral_restrict_closed_subinterval" is about a function 'f' that maps from a Euclidean space to a Banach space. It assumes that the function 'f' has an integral 'i' over the closed interval from 'c' to 'd'. It also assumes that the closed interval from 'c' to 'd' is a subset of the closed interva...
The lemma named "sshiftr_clamp" assumes that the length of 'a is less than or equal to n. It shows that the logical right shift of 'x' (of type 'a word) by 'n' is equal to the logical right shift of 'x' by the length of 'a'.
The inverse of the relational binding of a function phi is equal to the set of all pairs (x, y) such that y is in the composition of phi and eta applied to x.
The lemma named "Ord_mem_succ_cases" assumes that "k" is an ordinal number and "l" is an element of "k". It shows that either the successor of "l" is equal to "k" or the successor of "l" is an element of "k".
If a term 't' evaluates to a value 'v' under a set of rules 'rs' and a value environment 'Γ', and if the term 't' and the value environment 'Γ' are well-formed, then the value 'v' is also well-formed.
A singleton set, represented by {p}, is deterministic.
If 'null' is the condition in an 'if-then-else' statement, the result will be 'invalid', regardless of the outcomes 'B1' or 'B2'.
For an empty heap, creating an array of size 'n' with elements generated by function 'f' results in an array 'r' that maps to the array formed by applying function 'f' to each element in the range from 0 to 'n' (exclusive).
An element 'z' is in the hereditary restriction of 'r' to 'A' if and only if 'z' is in 'r' and there exist elements 'x' and 'y' such that 'z' is the ordered pair of 'x' and 'y', and 'x' is in 'A'.
If a function 'f' is injective on the set formed by inserting an element 'x' into the set of elements 'xs', then mapping 'f' over the list 'xs' with 'x' removed is equal to removing the image of 'x' under 'f' from the list obtained by mapping 'f' over 'xs'.
The lemma named "ideal_resource_alt_def" defines "ideal_resource" as follows: First, a simulator 'sim' is defined as the conversion of 'sim_callee' with no input. Then, an initial state 's_init' is defined as a tuple containing 'einit', 'key.PState_Store', an empty set, and the initial state of 'ideal_rest_alt'. ...
The lemma named "prod_le_prodinf" is defined with respect to a function 'f' that maps natural numbers to a type 'a', which is a linearly ordered integral domain with a linear order topology. The lemma assumes that 'f' has a product 'a', all values of 'f' are greater than or equal to zero, and for all 'i' greater than o...
The function "frees'" applied to the combination of a list "f" and "xs" is equal to the concatenation of the reverse of the mapped "frees'" over "xs" and the "frees'" of "f".
"ok" is enabled if and only if "ok" is a constant sequence.
The lemma named "invariance_of_dimension_closedin_Euclidean_space" assumes that "S" is a closed subset in the n-dimensional Euclidean space. It shows that the subtopology of the n-dimensional Euclidean space restricted to "S" is homeomorphic (topologically equivalent) to the n-dimensional Euclidean space if and only if...
The sequence X is Cauchy if and only if for every property P that eventually holds in the uniformity, there exists a natural number N such that for all natural numbers n and m greater than or equal to N, the property P holds for the pair (X n, X m).
The lemma named "stereographic_sphere_circle_set" shows that the image of the sphere circle set H under the stereographic projection is equal to the set of points on the stereographic circline of H.
If a pointer 'ptr' is not equal to the cast of a new character data pointer, and if 'h' leads to new character data resulting in a new character data pointer, and if 'h' leads to new character data resulting in 'h', then 'r' is in the locations of the get child nodes of 'ptr' and 'r' is a relation between 'h' and 'h'.
The lemma "eqExcPID2_step_\<phi>" assumes that "s" and "s1" are equivalent except for PID2, and that the state "s" transitions to state "s'" with output "ou" and action "a", and similarly, the state "s1" transitions to state "s1'" with output "ou1" and the same action "a". It shows that the property "\<phi>" holds for ...
Two raised to the power of ten equals 1024, in the context of integer numbers.
The lemma "Redirect_Edge" asserts that the "Redirect_Edge" is a valid transition from the precondition to the "Color_Target".
If the condition is True, then raising an exception with parameters x and y does not result in None.
The function 'f' is equal to the 'case_prod' function.
The lemma named "consume_top_leq" assumes the following conditions: 1. The function "consume" when applied to "TopType ts" and "ts'" equals "c_t". 2. The length of "ts" is less than or equal to the length of "ts'". 3. "c_t" is not equal to "Bot". If these conditions are met, it shows that "c_t" equals "TopType []".
If the set "bad" is a subset of "bad'", then the extraction from "bad" in the context of "IK" and "H" is also a subset of the extraction from "bad'" in the same context.
Given a set of conditions, there exists a decomposition D of the intruder knowledge and assignment right-hand side of a stateful protocol A under an interpretation I, such that for all terms t in the set ts, the intruder knowledge of the composition of A and D under the interpretation I entails the term t under the int...
If n1 is less than or equal to n2, then the branching left function of cl with n1 is less than or equal to the branching left function of cl with n2.
The function "p2r" applied to the negation of "P" is equal to the function "rel_n" applied to "p2r" of "P".
The lemma named "filt_equiv_sets" assumes that F is a filtration equivalent between M and N. It then shows that the set of events in M is equal to the set of events in N.
The lemma named "unlift_b_e" assumes that a sequence of basic expressions "b_es" is well-typed under the typing context "\<C>" and store type "\<S>". It shows that "b_es" is also well-typed under the typing context "\<C>" alone, without considering the store type "\<S>".
If an element 'a' is in the domain of a relation 'r', then there exists an element 'b' such that the ordered pair (a, b) is in the relation 'r'.
The language of G' is equal to the language of G.
The function "theNFTBoth" is symmetric with respect to its arguments "cl" and "dl". This means that swapping the order of "cl" and "dl" does not change the result of the function.
If a natural number 'm' is less than or equal to another natural number 'n', then the set of natural numbers from 'm' to 'n' is not empty.
The lemma demonstrates two things: 1. The function that maps each element x to itself has a vector derivative of 1 on the set T. This is known as the identity function. 2. The function that maps each element x to a constant c has a vector derivative of 0 on the set T. This is known as a constant function.
The sequence of the infimum (greatest lower bound) of a set A for a function P, followed by Q, is equal to the infimum of the set A for the sequence of the function P followed by Q.
The lemma named "composable_tm_from_abacus" assumes that "tp" is equal to "tm_of ap". It shows that the Turing machine "tp" concatenated with the shifted "mopup_n_tm n" (shifted by the length of "tp" divided by 2) is a composable Turing machine.
The lemma named "writes_union_minus_split" assumes that there are writes from set S excluding set S2 and from set S' excluding set S2, both resulting in the same final state. It shows that there are writes from the union of set S and set S', excluding set S2, also resulting in the same final state.
The lemma "vwalkI_vwalk_join_l" assumes that the list "p" is not empty and that there is a vertex walk in graph "G" that follows the sequence of vertices in the combined list "p" and "q" (denoted as "p \<oplus> q"). Under these assumptions, it shows that there is a vertex walk in graph "G" that follows the sequence of ...
If the square of a natural number 'n' equals the product of a prime number 'p' and the square of another number 'k', and if 'p' is a prime number, then 'p' divides 'n'.
The lemma named "finite_lessThan_enatI" assumes that "m" is not equal to infinity. It shows that the set of all elements less than "m" in the extended natural numbers is finite.
If "x" is not equal to the sink of Delta, then the inward degree of "x" in "g" is not equal to the top.
The set M is regular.
Given the following assumptions: 1. A state measure of a set V' in a context Γ (denoted as ρ) exists in the space of that state measure. 2. A measurable function M exists from the state measure of the union of sets V and V' in context Γ to the subprobability algebra of the stock measure t. 3. A Borel measurable functi...
The difference between the result of the "alignUp" operation on a word 'q' of any length 'a' and 'q' itself is less than or equal to the mask of 'sz'.
The lemma named "rel_interior_scaleR" is about a set S in a Euclidean space of any dimension 'n'. It assumes that a scalar 'c' is not equal to zero. It shows that the relative interior of the set S scaled by 'c' is equal to the relative interior of the set S itself scaled by 'c'.
The lemma named "unitarily_equiv_eq" assumes that matrices A and B are unitarily equivalent through matrix U. It then shows that matrix A is equal to the product of matrix U, matrix B, and the adjoint of matrix U.
The lemma named "find_remove_2_removeAll" assumes that the function "find_remove_2" with parameters P, xs, and ys returns a value of Some (x, y, xs'), and that the list xs contains distinct elements. It shows that the list xs', is equal to the list xs with all occurrences of x removed.
If "t1" is a term and "t2" is also a term, and if "yy" is not in the free variables of "t1", then the free variables of the function "NN" applied to "t1" and "t2" is the union of the free variables of "t1" and the free variables of "t2".
The intersection of the difference of multiset A and multiset B with the difference of multiset B and multiset A is an empty multiset.
If for all elements 'x' in set 'A', function 'f' of 'x' equals function 'g' of 'x', and if set 'A' is equal to set 'B', then the absolute summability of function 'f' on set 'A' is equivalent to the absolute summability of function 'g' on set 'B'.
If "ceq" is equal to "Some eq", then "eq" is an equality.
For any two hyperdual numbers 'a' and 'b' of a type 'a' that is an inverse, commutative ring, and division ring, if the base of 'a' is not zero and the base of 'b' is not zero, then the sum of the inverses of 'a' and 'b' is equal to the product of the inverse of 'a', the sum of 'a' and 'b', and the inverse of 'b'.
If the number 'n' is greater than or equal to 1, and the sum of the length of list 'xs' and 'n' is greater than or equal to 0, then this situation is impossible or false.
If the property Fr_2 holds for a set F and the property Fr_3 also holds for the same set F, then for all sets A, if the property Br holds for A, then the double negation elimination property holds for the complement of A.
"H is a submonoid of G". This is a contribution by Martin Baillon.
The lemma "boundOutputBindObject" is defined with the following parameters: a fixed 'b type 'Psi', a psi type 'P', an 'a type 'M', a name list 'yvec', an 'a type 'N', another psi type 'P'', and a name 'y'. It assumes that 'Psi' is in a relation with 'P' that leads to 'P'', the bound names of 'alpha' are fresh for the...
The relation with the target implies that the transitive closure of the target relation is a strong reduction simulation. It fixes two sets, TRel, a set of pairs of 'procT', and Rel, a set of pairs of processes 'procS' and 'procT'. It assumes three conditions: 1. Rel is a strong reduction simulation in the STCal of S...
The lemma named "free_global_cap" assumes that "c" is not NULL, the tag of "c" is true, and the global permission of "c" is true. Under these assumptions, it shows that freeing "c" from the heap "h" results in an error, specifically a logic error indicating an unhandled case with the code 0.
If "a" is in the inverse image of "A" under the function "f" at the nth position, then the function "f" applied to "a" is in "A".
The lemma named "continuous_on_inv" is about a function 'f' that maps from a topological space 'a' to a T2 space 'b'. It assumes that 'f' is continuous on a set 's', that 's' is compact, and that for all 'x' in 's', applying function 'g' to 'f(x)' gives 'x'. Under these assumptions, it shows that 'g' is continuous on t...
If for all x, the value of function A at x implies the value of function B at x, then function A is less than or equal to function B.
The lemma named "docking_station_arith" assumes that a real number "d" is greater than "x" and "v" is greater than 0. It shows that "v" equals "v" squared times "t" divided by twice "d" minus twice "x" if and only if "v" times "t" minus "v" squared times "t" squared divided by four times "d" minus four times "x" plus "...
If A and B are finite sets, then there is a bijective correspondence between the set of functions from A to B and the set of subsets of B with the same cardinality as A, modulo the equivalence relation of domain permutation on A and B. This means that each function from A to B corresponds uniquely to a subset of B with...
If the filter of list "xs" based on a predicate "P" is not empty, then the first element of the filtered list is an element of the original list "xs".
If k and d are positive numbers and the remainder of k divided by d is also positive, then there exists an interval I, which is a multiple of k, such that the interval I divided by d is not an empty interval.
The lemma named "basis_image_linear" is defined with a fixed function 'f' that maps from a field 'b' to another field 'b'. It assumes two conditions: the function 'f' is invertible in the linear function space, and 'X' is a basis of a vector set. The lemma shows that the image of the vector set 'X' under the function '...
A path 'p' can be joined with the combination of paths 'q' and 'r' if and only if 'p' can be joined with 'q' or 'q' is an empty path and 'p' can be joined with the tail of 'r'.
If a constant "c" is not equal to zero and the function of "c" times "f" is in Big O of "h", then "f" is also in Big O of "h". This is applicable for "c" which is of a type that is a linearly ordered field.
: "If the FCD property holds for a system view represented by Γ1 and V1 in a set of traces Tr of an event structure ES, and if the intersection of V2 and the set of all events in Γ2 is a subset of the intersection of V1 and the set of all events in Γ1, and if the intersection of N2 and the set of all traces in Γ2 is a...
For a given comparison function 'cmp' and heap 'heap', if the left and right boundaries are the same (both 'r'), then it is a partial heap.
Theorem Radon: Assuming that the set "c" is affine dependent, there exist subsets "m" and "p" such that "m" is a subset of "c", "p" is a subset of "c", "m" and "p" are disjoint, and the intersection of the convex hulls of "m" and "p" is non-empty.
The representation of a signed 64-bit integer is equal to the following: if the integer is less than negative 2^63 or greater than or equal to 2^63, then the representation of an undefined 64-bit integer is used. Otherwise, the word (binary representation) of the integer is used.
The norm of the probability of transition function P_X at time t, with the decision rule r_dec_b applied to p at time t, is less than or equal to the maximum reward r_M.
The lemma named "poly_compose_indexed_pmult" assumes the following conditions: 1. "m fs" is a polynomial tuple. 2. The length of "fs" is "n". 3. "f" is an element of the carrier of the polynomial ring with "n" variables. 4. "i" is less than "n". Under these assumptions, it shows that the composition of the polynomial...
Applying the "List_case" function to the arguments "c", "h", and the list constructed by "CONS M N", is equal to applying the function "h" to the arguments "M" and "N".
The lemma "address_in_enclosed_region" assumes that a region is enclosed by "a'" and "si'" around "a" and "si". If "x" is in the region addresses of "a'" and "si'", it shows that the unsigned integer of "x - a" is greater than or equal to the unsigned integer of "a' - a", and the sum of the unsigned integer of "a' - a"...
If 'inf' is less than or equal to 'x' and 'inf' is less than or equal to 'y', and for any 'z' that is less than or equal to both 'x' and 'y', 'z' is also less than or equal to 'inf', then 'inf' is the infimum of 'x' and 'y'.
The bounded variation function of the composition of alpha and beta is a subset of the union of the bounded variation function of alpha and the bounded variation function of beta.
For the variables h, p, x, and rank, given the condition is always true, for each value of x, using h, under the invariant that the difference between p and h equals 1 minus h, and the difference between rank and h equals the transpose of Z minus h, the following operations are performed: p is assigned the result of th...
If a set D is a subset of the domain of a function m, and m is an injective function on D, then the composition of the function "the" and m is also an injective function on D.
If the accessible method "accmethd" in a given class "C" with a specific signature "sig" in a system "G" from a source "S" is some method "m", then the method "m" with the same signature "sig" in the same class "C" in the system "G" exists and the method "m" of class "C" is accessible from the source "S" in the system ...