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A complex bounded linear function 'A' mapping to a 1-dimensional complex space is equal to zero.
If A' is less than or equal to A, then the relation predicate of A is less than or equal to the relation predicate of A'.
The product of any two real numbers x and y is less than or equal to half of the square of x plus half of the square of y.
For any natural numbers 'n' and 'k', given that 'p' is a prime number, 'k' is greater than or equal to 1 and less than or equal to 'p' to the power of 'n' minus 1, and 'n' is greater than 0, then the binomial coefficient of 'p' to the power of 'n' and 'k', when divided by 'p', has a remainder of 0.
The lemma named "expands_to_power_0" assumes that for a function 'f' which approaches 0 as 'x' goes to infinity, a well-formed basis, and the length of the basis equals the expansion level of a type 'a' which is a multiseries. It also assumes that 'n' is defined as 'n'. Under these assumptions, it shows that the functi...
If you normalize the quotient of zero and any number 'x', the result will be (0, 1). Similarly, if you normalize the quotient of any number 'x' and zero, the result will also be (0, 1).
The lemma tame5 assumes that there is a sequence of transformations from the seed under the next_plane0 function to g, that g is a final state, and that g is tame. Under these assumptions, it shows that p is less than or equal to 3.
In the test, the following steps are performed: 1. A temporary variable, tmp0, is assigned the element with the ID 'test_slot_in_slot' from the slots document. 2. A test tree is created with tmp0 as the root node and assigned to variable n. 3. The 'test_slot_in_slot' property of n is assigned to tmp1. 4. Any text node...
The function "F" composed with function "f" is equal to the function "F" composed with function "g" if and only if function "f" is equal to function "g".
For any property P, P holds for all i of type 7 if and only if P holds for 1, 2, 3, 4, 5, 6, and 7.
There exists an integer x and y such that x is greater than 0, y is greater than or equal to 0, and the equation 3 times x minus 5 times y equals 1 holds true.
Lifting a natural number function, where the function multiplies a variable 'n' by 'm', is equivalent to a function that multiplies an integer 'm' by the lifted natural number function of 'k'.
For any process P of type CCS, P running in parallel with the null process is strongly bisimilar to P itself.
In a linearly ordered lattice, the infimum (greatest lower bound) of any two elements x and y is less than or equal to x.
The image of a multiset containing a single element under a function is a multiset containing the function applied to that single element.
The lemma shows two statements: 1. The absolute value of 'a' is not greater than or equal to the absolute value of 'b' if and only if the absolute value of 'a' is less than the absolute value of 'b'. 2. The absolute value of 'a' is not greater than the absolute value of 'b' if and only if the absolute value of 'a' is...
If a set S is finite and for every polynomial p in S, the degree of the function f applied to p is less than or equal to n, then the degree of the sum of the function f over all elements in S is also less than or equal to n.
The function F applied to the union of a set R is equal to the supremum of the function F applied to each element in the set R.
If the list 'xs' is not empty, then the first element of the polychain of 'p' and 'xs' is 'p'.
If a function "lift2 f" is closed over the error set of a set A, then the function "lift2 (sup f)" is also closed over the error set of the set of n-lists over A.
If a function 'f' belongs to the set of primitive recursive functions of one variable, then the function 'pr_conv_1_to_3 f' belongs to the set of primitive recursive functions of three variables.
The product of multisets A and B is empty if and only if either A is empty or B is empty.
If a certain execution move "exec_move" is valid for a given configuration consisting of the program "P", thread "t", expression "e", heap "h", and stack "s", resulting in a new heap "h'" and stack "s'", then the same execution move is also valid when the expression "e" is replaced with a "compareAndSwap" operation on ...
It is not the case that the claim of a certain car is followed by any state, which is then followed by the release of the same car.
The lemma named "quadratic_shape1a" is defined with fixed real numbers a, b, c, x, y. It assumes that a is greater than 0 and that x is less than y, and that both x and y are roots of the quadratic equation ax^2 + bx + c = 0. It shows that for any real number z, if z is greater than x and less than y, then z is not a r...
The address of an array, which is the first element of the pair resulting from allocating an array with elements 'x' in the heap 'h', is equal to the limit of the heap 'h'.
The product of a scalar 'a' and a vector 'x' belongs to the linear span of 'x'.
The lemma named "matrix_equation_helper_step" is defined with the following conditions: - It fixes a real polynomial 'p', a list of real polynomials 'qs', a list of natural numbers 'I', and a list of lists of rational numbers 'signs'. - It assumes that the polynomial 'p' is not zero. - It assumes that all elements in ...
Given the assumption that for any natural number "i", the set of elements "x" in the space "M" for which the property "P" holds for "i" and "x" is in "M", it shows that the preimage of a set "A" under the function that maps each "x" to the least natural number "j" for which "P" holds, intersected with the space "M", is...
The lemma "RDirProd_list_is_ring" assumes that for any index 'i', if 'i' is less than the length of the list 'Rs', then the 'i'-th element of 'Rs' is a ring. Under this assumption, it shows that the direct product of the list 'Rs' is also a ring.
The lemma "samequot_soundness" assumes that the function "while_option" with parameters "fst", "rderiv_and_add as", and "(True, [r])" returns "Some (b, rs)". It shows that the language "n" of the sum of "rs" is equal to the union of the language "n" of the set of derivatives of "r" replicated "m" times for any "m".
The application of the instruction "Swap" with parameters P, pc, mxs, T\<^sub>r, and s is equivalent to the existence of T1, T2, ST, and LT such that s is equal to the pair (T1#T2#ST,LT).
If a sublist of a list 'xs' mapped by a function 'h' and ordered by a relation 'R' is sorted from index 'hi'+1 to 'hi', and if 'lo' is less than or equal to 'lo', and 'hi' is less than the length of 'xs', and for all 'j', if 'hi' is less than 'j' and 'j' is less than or equal to 'hi', then the 'j'th element of 'xs' is ...
The real value is measurable in the context of Borel sets and real numbers.
The stateful strand database of the concatenation of two lists A and B, with a given set I and database D, is equal to the stateful strand database of list B with the same set I and the stateful strand database of list A with the same set I and database D.
If the strong topological negative definiteness holds for a function eta, then the weak topological negative definiteness also holds for the function eta.
A single polynomial mapping of key 'k' and value 'v' is equivalent to a sparse zero map with a pair (k, v).
The lemma "ptrm_fvs_finite" shows that the set of free variables in the preterm X is finite.
: Assuming that for every element x in set A, x is between a and b, and for every element x in set B, x is between c and d. Also, set A and set B are subsets of set Q, and set Q is a member of a set P. Furthermore, the cardinality of set A is greater than 1 or set A is infinite, and the cardinality of set B is greater...
"less_aexp_aux" of x and y is equal to the negation of "less_aexp_aux" of y and x, and x is not equal to y.
If there is no instance of a label "l" in a list "A", then the set of terms in the projection of "A" with respect to label "l" is a subset of the set of terms in the projection of "A" with respect to any other label "l'".
The lemma named "segments_of_aform_closed" assumes that the list of segments of aform X is not empty. It shows that the first element of the first segment is equal to the second element of the last segment.
The set S' is irreflexive.
The lemma named "division_of_subset" assumes that "p" is a division of the union of "p", and "q" is a subset of "p". It shows that "q" is a division of the union of "q".
The lemma named "prv_exi_cong" assumes that both "phi" and "chi" are formulas and "x" is a variable. If it is proven that "phi" implies "chi", then it can be shown that there exists a "x" such that "phi" implies there exists a "x" such that "chi".
The lemma named "comp_bounded_clinear" is about two functions, A and B. A is a function that maps from a complex normed vector space 'b to another complex normed vector space 'c. B is a function that maps from a complex normed vector space 'a to 'b. The lemma assumes that both A and B are bounded and linear over the co...
The hereditarily finite set of the ordinal of a number n is equal to the image of the ordinal of the set of all numbers from 0 to n (exclusive).
The lemma "drop_nonempty_simps" consists of three statements: 1. Dropping a nonempty value 'v' from the successor of 'n' in a list that starts with 'x' and continues with 'xs' is equivalent to dropping the nonempty value 'x' from 'n' in the list 'xs'. 2. Dropping a nonempty value 'v' from 0 in a list that starts with...
The lemma named "rel_interior_convex_nonempty_aux" is about a set S in a Euclidean space of any dimension 'n. It assumes that the set S is convex and contains the point 0. Under these assumptions, it shows that the relative interior of S is not empty.
The value type of x is BOOLEAN if and only if x belongs to the universal set of Boolean values.
Rotating the 4-bit binary number 0110 two places to the right results in the 4-bit binary number 1001.
If a sequence "s" has a next element, then the termination of the sequence "s" within the context of the functions "has_next" and "next" is equivalent to mapping an option that increments by 1 to the termination of the sequence that starts from the next element of "s" within the same context.
The set abstraction of x with respect to the atom a is equal to the set abstraction of y with respect to the atom b if and only if for all atoms c, if atom c is fresh for z and fresh for the tuple (a, b, x, y), then the permutation of x by swapping a and c is equal to the permutation of y by swapping b and c. 2. The s...
If the product of a rational number "c" and some value "a" is less than "b", and "c" is less than zero, then "a" is greater than the result of dividing "b" by "c" and multiplying by 1.
The theorem 'wls_substEnv_vsubstEnv_compose1' assumes that 'rho' is a well-sorted environment, 'Y' is well-sorted of sort 'ys', and 'y' is not equal to 'y1'. It shows that substituting 'y1' for 'y' in 'ys' in 'rho' and then substituting 'Y' for 'y' in 'ys' results in the same as just substituting 'y1' for 'y' in 'ys' i...
The lemma named "access_unalive" assumes that the reference "l" is not alive in the state "s". It shows that accessing the state "s" at location "l" is equal to initializing the type of location "l".
The lemma "eigenvalue_hom" assumes that a matrix A is an n x n matrix and that "ev" is an eigenvalue of A. It then shows that the homomorphism of "ev" is an eigenvalue of the homomorphism of matrix A.
If 'v' is a valuation on field 'K', and 'x' and 'y' are elements in the field 'K', then the minimum of the valuation of 'x' and the valuation of 'y' is less than or equal to the valuation of the sum of 'x' and 'y'.
The lemma "pref_overlap_imp_same" assumes that both "u" and "v" are elements of a set "C", and that the product of "p" and "u" is less than or equal to the product of "q" and "v". Additionally, it assumes that the absolute value of "q" is less than the sum of the absolute values of "p" and "u". Under these conditions, ...
The power of a set A raised to the successor of a number n is equal to the power of the set A raised to the number n multiplied by the set A.
The incoming flow "h" at a node "x" is less than or equal to the value of the flow "h" in the network flow model "Delta". This is represented as "?lhs <= ?rhs".
If the binary height of the left subtree 'l' is equal to the binary height of the right subtree 'r', then the binary height of the balanced tree resulting from the right rotation operation 'baliR' on these subtrees with 'a' as the root, is equal to the successor of the binary height of the left subtree 'l'.
The set of states in the reversed version of a non-deterministic finite automaton (NFA) is equal to the set of states in the deterministic finite automaton (DFA).
There is an order isomorphism between the function "us" and the upset of any element in an ordered set with a dual.
The condition where interval 'p' meets 's' and there exists an interval 't' such that 'p' meets 't' and 't' meets 's', and there exists an interval 't' such that 'r' meets 't' and 't' meets 'q', is false.
An arithmetic progression with a common difference of 0 is a set containing only the initial term 'a'.
The action order in a given environment E is reflexive on the actions in that environment E.
The set of all elements 'z' such that property 'P' holds for the dot product of 'z' and the pair (0,i) is equal to the set of all pairs (x,y) such that property 'P' holds for the dot product of 'y' and 'i'.
If you restrict the domain of a function to its own domain, the function remains unchanged.
If relation 'r' is a subset of relation 's', then the symmetric factor of relation 'r' is also a subset of the symmetric factor of relation 's'.
The lemma for the partial derivatives of the union of two languages shows that the partial derivatives of the union of language A and language B with respect to a regular expression r is equal to the union of the partial derivatives of language A and language B with respect to the same regular expression r.
The value of the Riemann Zeta function at 3 is not a rational number.
In the context of an elliptic field, the lemma named "pdouble_closed" assumes that "a" and "b" are elements of the carrier "R" and that the point "p" is on the elliptic curve defined by "a" and "b". It shows that the point resulting from doubling "p" (denoted as "pdouble a p") is also on the same elliptic curve.
The countable intersection of a set P relative to a set U is equal to the countable intersection of the set P relative to U, relative to U again. This is denoted by the equation "?lhs = ?rhs", where "?lhs" represents the left-hand side of the equation and "?rhs" represents the right-hand side.
"X" is an element of the tangent space at point "p" if and only if "X" is a derivation at point "p" and "X" is an extensional on the differential function space.
If the following conditions are met: 1) there is a correspondence between list "lma" and list "lmb", 2) "m" is less than the length of list "lma", and 3) "m" is not less than the length of list "lmb", then there is a correspondence between the list "lma" with the "m"th element replaced by "n" and the list "lmb" app...
"If it is obligatory that 'B' given 'A', and it is possible that 'A' and 'B' can both be true, and it is possible that 'A' is true and 'B' is not true, then it is ideally obligatory that if 'A' is true then 'B' must be true."
If the function "f" is summable, then the limit of the sequence "f" approaches 0.
The lemma named "degree_greater_zero_in_keys" assumes that the degree of a function 'f' is greater than zero. It shows that the result of subtracting one from the degree of 'f' is included in the keys of 'f'.
: Given the assumptions that: 1. There is a specification that a program P, when calling a procedure with initialization 'init', parameter 'p', return function 'return'', and continuation 'c', satisfies the postcondition Q and the abort condition A under the context of global and local variables (\<Gamma>,\<Theta>) an...
The lemma named "bound_max_state_set" assumes that "R" is finite. It shows that the maximum bound of "t" in the subset of "R" (where the left-hand side is not the bottom element) is included in the states of "R".
If the weak topological negation distributivity property holds for a certain function, then the strong topological negation distributivity property also holds for that function.
If a list is not empty, then there exists a list such that prepending an element 'a' to the given list is equivalent to appending the last element of the given list to 'ls'.
The lemma named "lfp_the_while_option_lattice" is defined with a fixed function 'f' that maps from a complete lattice 'a' to 'a'. Assuming that 'f' is a monotone function and the universal set of 'a' is finite, it shows that the least fixed point of 'f' is equal to the value of the 'while_option' function applied to a ...
The function FSwSbImorph preserves the identity when applied to the identity function and the MOD function.
The product of "d x" and "y" is less than or equal to "y".
The theorem named "safe" assumes that "s" is within reach. It shows that if "s" is safe for "r", and "g" is in "s" for "r", then "r" owns "g" in state "s".
The list of first elements (obtained by mapping the function 'fst') from the associative list of an array 'a' is distinct.
The set containing the elements uv and vu is a subset of the arcs of graph G.
The graph G is a bidirected digraph with respect to the reverse arc function 'arev'.
If for all variables 'v', 'phi' implies 'psi', then it is necessarily the case that 'phi' implies 'psi' in 'v'.
If an element 'x' belongs to the function 'fconst_on' with parameters 'A' and 'c', which is left-restricted to 'B', then 'x' also belongs to the function 'fconst_on' with parameters 'B' and 'c', which is left-restricted to 'A'. This is under the assumption that 'x' is a member of the set 'fconst_on A c' left-restricted...
The relation of x is healthy according to the CSP1 criterion.
The lemma named "Cond" makes the following assumptions and shows a result: Assumptions: 1. The set P is a subset of the set of all states 's' such that if 's' is in 'b', then 's' is in P1, and if 's' is not in 'b', then 's' is in P2. This is denoted by the 'wp' assumption. 2. There is a total correctness derivation fr...
The product of two lists, xs and ys, is equal to the concatenation of the mapping of each element x in xs to the mapping of each element y in ys into a pair (x, y).
If for all permutations 'p' of a finite set 'UNIV', the sum of the degrees of the function 'to_fun A' applied to each element 'i' and its image under 'p' is less than or equal to 'x', and there exists a permutation 'p' such that this sum equals 'x', then the maximum degree of a permutation of 'A' is 'x'.
The lemma 'almost_additive_converges' is defined with a fixed function 'u' that maps natural numbers to real numbers. It assumes that for any two natural numbers 'm' and 'n', the absolute value of the difference between 'u' applied to the sum of 'm' and 'n', and the sum of 'u' applied to 'm' and 'u' applied to 'n', is ...
"well-founded partial order on P and A" is equivalent to "inductive on P and A".
If the mode is insecure, then whatever the spy knows in the event sequence where A inputs X to C is equal to X added to what the spy knows in the original event sequence.
If "i" is less than or equal to "n", then "x" to the power of "i" is less than or equal to "x" at the base "0" to the power of "n".