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In the lemma within the Worder, if set X is a subset of the carrier D and there exists an upper bound b for X, then the supremum of set X in D is an element of the carrier D.
In an amicable pair where 'm' is greater than 'n', and 'm' is amicably related to 'n', it can be shown that 'n' is an abundant number.
Lemma of nonnegative integral monotone convergence for simple functions: If the sequence of functions 'f' is increasing, and for all indices 'i', the function 'f i' is a simple function on the measure space 'M', then the supremum of the integral of 'f i' with respect to 'M' over all indices 'i' is equal to the nonnegat...
If there is a path 'p' from source 's' to target 't', and if an edge 'e' is in the set of 'p', then there exist vertex sets 'vs1' and 'vs2' such that the vertices in the forward direction from 's' to 'p' are equal to 'vs1' followed by the first vertex of 'e', the second vertex of 'e', and then 'vs2'.
Given a polynomial coefficient 'c' from a set 'S' and a non-negative integer 'k' that is less than or equal to the first element of 'c', the polynomial expression of 'R', 'X', 'k', and 'c' equals zero if and only if the set of all 'j' such that 'j' is less than or equal to 'k' and the 'j'th element of the second elemen...
The lemma named "porder_linorder_tranclpE" assumes five conditions: 1. "r" is a partial order on set "A". 2. "s" is a linear order on set "B". 3. The orders "r" and "s" are consistent with each other. 4. Set "B" is a subset of set "A". 5. The pair "(x, y)" is in the transitive closure of the union of "r" and "s". Und...
If the route from rt1 to a specific IP is not fresher than the route from rt2 to the same IP, then rt1 is not fresher than rt2 with respect to that IP.
A generator forms an algebra in the space of the product measure "PiM" over the index set "I" and the sigma-algebra "M".
If process P is less than or equal to process Q, then the process of writing x to channel c followed by process P is less than or equal to the process of writing x to channel c followed by process Q.
In the lemma named "pdivides_zero" within the domain, assuming that "K" is a subring of "R" and "p" is an element of the carrier of "K[X]", it shows that "p" polynomially divides an empty list.
For any two floats 'x' and 'y', if the value of 'x' is less than or equal to the value of 'y', then the exponent of 'x' is less than or equal to the exponent of 'y'.
The function "unwind_cont" continues with the set that includes \<Delta>3, \<Delta>4, and \<Delta>e.
A lazy list "xs" is a prefix of another lazy list "ys" if and only if the following conditions are met: - If the list "xs" is empty, then it is a prefix of "ys" regardless of the contents of "ys". - If the list "xs" is not empty and contains an element "x" followed by the rest of the list "xs'", then it is a prefix of...
NullPointer, OutOfMemory, ClassCast, NoClassDefFoundError, IncompatibleClassChangeError, NoSuchFieldError, and NoSuchMethodError are all included in system exceptions.
The lemma "LIMSEQ_powrat_inverse_of_nat_ge_one" assumes that "a" is greater than or equal to 1. It shows that the sequence of "a" raised to the power of the inverse of a natural number "n" converges to 1 in the limit as "n" goes to infinity.
Getting the negation from the mapping of "NegPos_map" function on "X" and the positive mapping of "src" results in an empty list.
"crel_vs R a b" holds if "a" and "b" are related by the composition of "state_dp_consistency.crel_vs R" and "rel_state (=)".
If "i" strictly dominates "j", and "j" is a successor of "k", then "i" dominates "k".
AX0 is a subset of AX1.
It is false if the following three conditions are met: the relation "R" holds between "a" and "b" according to "crel_vs1", the execution of "b" with the given "heap" results in "None", and the predicate "P" holds for the "heap".
If a policy 'p' consists of single combinators and all networks in 'p' are distinct, then the policy 'p' after the removal of shadow rules will still consist of single combinators.
The lemma named "split_list_last_propE" assumes that there exists an element 'x' in the set 'xs' that satisfies the property 'P'. It then obtains lists 'ys' and 'zs', and an element 'x', such that 'xs' is the concatenation of 'ys', 'x', and 'zs', 'x' satisfies the property 'P', and no element in the set 'zs' satisfies ...
The factorial of any number "n" is greater than or equal to 1.
"x is less than or equal to x".
"x" is a surjective function if and only if the composition of the identity function (1) and "x" equals the identity function (1).
If we assume that K is a subfield of R, p is an element of the carrier of K[X], and p is a prime in K, then it shows that p is not an empty set, p is not an element of the Units of K[X], and for any q and r, if q and r are elements of the carrier of K[X] and p divides the product of q and r, then p divides either q or ...
If a function 'f', which maps from a real normed vector to a linear function from a real vector of size 'n' to another real vector of size 'n', has a derivative 'f'' at a point 'x' within a set 'S', then the function that applies 'f' to 'x' and then converts the result to a matrix also has a derivative at 'x' within 'S...
If the product of x with itself equals x, then the star of x equals 1 plus x.
The lemma named "cat_ordinal_is_arrE" assumes that "f" is a morphism from "a" to "b" in the ordinal category "A". It concludes that "a" is an element of "A", "b" is an element of "A", "a" is less than or equal to "b", and "f" is the ordered pair of "a" and "b".
The lemma named "real_transfer_operator_sum" is about a function 'f' that maps from 'b' to 'a' to a real number. It assumes that for any 'i', 'f i' is integrable over the measure space 'M'. It shows that for almost every 'x' in 'M', the real transfer operator of the sum of 'f i x' over 'i' in 'I' is equal to the sum of...
If the function "holds" with parameters "pre", "l", and "None" is true, then one of the following conditions must be true: 1. The list "l" is empty. 2. There exists an element "x" such that the list "l" consists only of "x" and the function "P" with parameters "pre", "x", and "None" is true. 3. There exist elements "x...
The lemma for the factorial of the successor of a number n is equal to the successor of n times the factorial of n.
The multiplication of a monomial with a polynomial, where the monomial is the sum of two terms (c and d), is equal to the sum of the multiplication of each term (c and d) with the polynomial. This is a distribution property over multiplication for monomials and polynomials.
If M transitions to M' under the relation "a*", then "NotL <a>.M x" also transitions to "NotL <a>.M' x" under the same relation "a*".
If integer 'a' is less than integer 'b' and 'b' is less than integer 'c', then there exists a set 'A' such that 'a' is not an element of 'A', 'b' is an element of 'A', and 'c' is not an element of 'A'. This is an example 4.
The length of the sequence obtained by marking out the 'n' elements in 'bs' is equal to the length of 'bs'.
The equality of word intervals r1 and r2 is commutative, meaning that r1 is equal to r2 if and only if r2 is equal to r1.
If the substitution of an abstraction preserves the well-formedness of the abstraction string in a given module, then the substitution of a binary input also preserves the well-formedness of the binary input string in the same module.
If the permutation of variables "vs" is less than "n", then for every "i" in the set of variables "vs", "i" is less than "n".
The theorem named "discr_Seq" assumes that "c1" and "c2" are discrete. It shows that the sequence of "c1" and "c2" is also discrete.
The lemma states several properties of the function "biQuot" with parameters a and b: 1. If the function is applied to an empty set, it returns an empty set. 2. If the function is applied to a set containing only an empty list, it returns an empty set. 3. If the function is applied to a set containing only a single ch...
If "m" is less than "n", then the "m"-th element of the stream "s" truncated at "n" is equal to the "m"-th element of the original stream "s".
The omega power of the product of x and y, multiplied by x, is less than or equal to x multiplied by the omega power of the product of y and x.
The agent A is included in the payload.
"p" is greater than 2 and "p" is odd.
If a well-formed formula property (wffp) holds for the concatenation of two lists (sl1 and sl2), and the first list (sl1) is not empty, then the well-formed formula property also holds for the first list (sl1).
If the exclusive OR (xor) of two sets 'xs' and 'ys' is an empty set, then 'xs' is equal to 'ys'.
"The product of the k-th bit of 'a' and the k-th bit of 'b' is less than or equal to 1."
The lemma named "atoms_closed" assumes that "Xs" is finite, "A" is an element of the atoms of the generated Boolean algebra of "S" and "Xs", and "S" is the union of "Xs". It shows that "A" is an element of the generated Boolean algebra of "S" and "Xs".
If a list "xs" is not empty, then the last element of the list "xs" is equal to the element at the position "length of xs - 1" in the list.
If a set V is a subset of the span of set B in module m1, then the image of set V under function f is a subset of the span of the image of set B under function f in module m2.
The lemma named "rank_matrix_dim_range" assumes that "f" is a linear function. It shows that the rank of the matrix "f" is equal to the dimension of the range of "f". Here, "f" is a function in a field, and the dimensions of the matrix are defined by a module type.
Filtering a finite set "xs" twice, first by a condition "Q" and then by a condition "P", is equivalent to filtering the set once with a condition that requires both "Q" and "P" to be true for each element "x" in the set.
If "x" is an element in the set produced by the function "collect_subfmlas" applied to "r" and an empty list, then the size of "x" is less than the size of "r".
The lemma named "construct_cycle_pre" assumes that "R" is not acyclic, "y" is chosen from the reflexive transitive closure of "R" and "x" is chosen from the reflexive transitive closure of "R" with respect to "y". Under these assumptions, it shows that the invariant of the cycle construction holds for "x", "x", "y", 0,...
Either polynomial s is lexicographically greater than polynomial t according to the degree lexicographic order, or polynomial t is lexicographically greater than polynomial s according to the same order.
If 'a' and 'b' are singular chains of dimension 'p' in a topological space 'X', then the sum of 'a' and 'b' is also a singular chain of dimension 'p' in the same topological space 'X'.
The lemma named "bridge_proper_dia" assumes that there does not exist an 'a' such that 'p' equals 'Nom a'. It shows that the function 'bridge' with parameters 'k', 'j', 'xs', 'v', 'v'', and 'diamond p' equals 'diamond p'.
The universal set is in the collection of computably enumerable sets.
If the unsigned natural number representation of a word 'a' is less than or equal to the unsigned natural number representation of another word 'b' minus 1, then 'a' cannot be equal to -1.
The negation of a number 'a' is finite if and only if the number 'a' itself is finite.
The pair (l, u) is in the set S if l is in the set L, u is in the set V, and u satisfies the invariant of A at location l.
The lemma named "Lp_zero_space" assumes that a real number "p" is greater than zero. It shows that the zero space of Lp space with parameter p over a measure space M is the set of all functions that are Borel measurable on M and almost everywhere in M, the function equals zero.
The lemma named "conga_left_comm" assumes that the angle ABC is congruent to the angle DEF. It shows that if this is true, then the angle CBA is also congruent to the angle DEF.
There exists a state such that it is not the case that all instances of any type at the previous state are of the type 'Any'.
The lemma named "null_eqI" assumes that for any 't', the operation 'OP' with 'n' and 't' equals 'n', and the operation 'OP' with 't' and 'n' also equals 'n'. If these conditions are met, it shows that 'n' equals 'null'.
Function f is equal to function g if and only if for all r, the nth term of the Puiseux series of function f at r is equal to the nth term of the Puiseux series of function g at r.
The concatenation of the kth power of r and u is partially less than or equal to the concatenation of r, the kth power of r, and v if and only if u is partially less than or equal to the concatenation of r and v.
Normalizing a polynomial 'p' appended with 'n' number of zeros is equivalent to just normalizing the polynomial 'p'.
If under program P and environment E, the evaluation of expression 'e' from initial state 's0' results in 'null' and a new state 's1', then under the same program P and environment E, the operation 'throw e' from the same initial state 's0' will result in a 'NullPointer' exception and the same new state 's1'.
A pair of functions "f" and "g" is equivalent to the absolute finite function of a lambda function that takes an input "x" and returns a pair of the results of applying "f" and "g" to "x".
If a natural number 'n' is less than or equal to the natural number representation of the maximum IPv6 address, then the natural number representation of the IPv6 address obtained from 'n' is equal to 'n'.
Shifting the numeral value of 'w' to the left by one bit is equivalent to the numeral value of 'w' represented in binary form.
If 'l' is less than the length of the list 'xs', then picking an element from the list 'xs' (where each element is paired with 1) at the position 'l' (converted from a natural number to a natural number) is equal to the 'l'-th element of the list 'xs'.
If the binary representation of a word 'p' is equal to the concatenation of a list 'addr' and 'n' number of false values, then the word 'p' is aligned to 'n'.
The lemma "cart2_append1_between_right_strict" assumes that points p, q, and r are not at the origin (i.e., "z_non_zero p", "z_non_zero q", and "z_non_zero r"). It also assumes that in the real number plane, point q is between points p and r ("B\<^sub>\<real> (cart2_pt p) (cart2_pt q) (cart2_pt r)") and that points q a...
The lemma named "coord_ring_to_UP_SA_constant" assumes that "c" is an element of the carrier Qp. It shows that the function "coord_ring_to_UP_SA" applied to "n" and "Qp.indexed_const c" equals "to_polynomial" applied to "SA n" and "\<cc>\<^bsub>n\<^esub> c".
If you apply the identity function to both the state and the instruction of a Minsky machine, the machine remains unchanged.
If space X is homotopy equivalent to space Y, then space X is contractible if and only if space Y is contractible.
"F leads to B from the union of a set S" is equivalent to "for all subsets A of S, F leads to B from A".
The initial state of level 3 is a subset of the invariant 10 of level 3.
If a natural number 'k' is greater than 0 and less than 2 raised to the power of the length of 'a (where 'a is a type with length), then the word representation of 'k' is not equal to the word representation of 0.
: if the sets of integers from 0 to n and from 0 to m are not empty, and if m and n are both greater than 1, and if a matrix A of type 'a (which is a bezout ring division) is in the carrier matrix of m by n, and if bezout is a bezout extension, and if a is less than b, and if b is less than m, and if j is less than n, ...
If a polynomial 'f' is separable in modulo 'p', then 'f' is square-free. Additionally, if the natural number 'p' is greater than the degree of 'f' and the separable bound of 'f', and 'f' is square-free, then 'f' is separable in modulo 'p'.
The negative interval translation of the union of two sets A and B by k is equal to the union of the negative interval translations of the sets A and B by k.
If the block map of 'b' equals the scan of 'r', and the fetch of 'tp', 'st', and 'b' equals the pair '(nact, ns)', and 'st' is less than or equal to the length of 'tp' divided by 2, then the new state of the code of 'tp', 'st', and 'r' equals 'ns'.
The lemma named "rel_E_E" assumes that there is a relation "rel_E" between the variables cs, cm, cm1, rule, and the successor of p'. It shows that there exists a variable cs1 such that cs1 is related to cs through four iterations of the deterministic step relation "st.dstep", and cs1 is also related to cm, cm1, rule, a...
The set resulting from the function "add_tvars'" applied to a term "t" and an accumulator "acc" is equal to the union of the set of the accumulator "acc" and the set of type variables in the term "t".
The tail of a list being equal to the list itself is equivalent to the list being empty.
If E is an Order, f is an order isomorphism from D to E, and D1 is a subset of the carrier of D, then there is an order isomorphism from the induced order on D restricted to D1 to the induced order on E restricted to the image of D1 under f, where the isomorphism is given by the function f restricted to D1.
For a given set 'S' in a Euclidean space, if 'S' is homeomorphic to a closed box between points 'a' and 'b', and the dimension of the Euclidean space is greater than or equal to 2, then the complement of 'S' is path-connected.
If "m" is an element of the intra-procedural observable set of "n" in the HRB slice of "S" in the control flow graph, then the intra-procedural observable set of "n" in the HRB slice of "S" in the control flow graph is equal to the set containing only "m".
The lemma named "orthogonal_sumlist" assumes that for any vector 'x' in the set 'S', the dot product of 'v' and 'x' is zero. It also assumes that the set 'S' is a subset of the vector space of dimension 'n', and that 'v' is a vector in the same vector space. Under these assumptions, it shows that the dot product of 'v'...
The lemma named "min_coin_dec" assumes that the function g applied to e equals the function h applied to f. It then obtains a sequence ps such that the concatenation of the first elements of ps equals e, the concatenation of the second elements of ps equals f, and for any pair p in the set of ps, the function g applied...
The range of a set A is empty if and only if the set A itself is empty.
The product of "n of x" and "n of x" is less than or equal to "n of x".
The lemma "returned_missing_cd_or_loop" assumes that both "pi" and "pi prime" are paths, and that the first element of "pi" is the same as the first element of "pi prime". It also assumes that there is no index "k prime" such that the "k-th" control state of "pi" is the same as the "k prime-th" control state of "pi pri...
"if the length of a word 'w' is less than or equal to 2, and 'w' is in the list of 'x', 'y', 'z', then property 'P' holds for 'w'."
The lemma named "mat_det_12_mm_l" shows that the determinant of the product of matrix M and matrices A and B is equal to the product of the determinant of matrix M and the determinant of matrices A and B.
If for every pair (x, y) in set A, property P holds for x and y, then if property P implies property Q, and if the pair (x, y) is not in set A this also implies property Q, then property Q must hold.