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Statement: stringlengths 7 24.3k
If the subdegree of a formal power series 'f' is greater than zero, then 'f' is not equal to zero.
The lemma states "It is not the case that a certain property Q, which applies to natural numbers, holds for 'The Q'."
The post condition is positive.
The Mersenne modulus of k and n is congruent to k modulo (2 to the power of n minus 1).
There is an equivalence in the bicategory for the composition of morphisms h and f, and g and k, with unit transformations eta and epsilon.
The theorem "Thm" is true for the absolute index "(0, 0)" in the context of the formula "Phi".
The set of coefficients of a polynomial 'p' is a subset of the set containing only the element '0' if and only if the polynomial 'p' is equal to '0'.
The lemma named "perms_inv" assumes that a function 'f' belongs to the set 'perms'. It shows that the inverse of function 'f' also belongs to the set 'perms'.
The lemma "tauChainResChainPres" is defined with the following parameters: a fixed 'Psi' of type 'b, 'P' and 'P'' of type psi (which is a tuple of 'a, 'b, 'c), and 'xvec' which is a list of names. The lemma assumes that 'Psi' directly leads to 'P' through a tau chain to 'P'', and that 'xvec' is fresh for 'Psi'. It sh...
For any two functions 'f' and 'g' from any set to a Banach space with a second countable topology, if both 'f' and 'g' are set integrable over a set 'A' with respect to a measure 'M', then the function that maps each element 'x' in the set to the sum of 'f(x)' and 'g(x)' is also set integrable over 'A' with respect to ...
For any two real polynomials 'p' and 'q', the jump of polynomial 'q' at a point 'x' with respect to polynomial 'p' is equal to the jump of the modulus of polynomial 'q' with respect to polynomial 'p' at the same point 'x'.
The lemma named "poly_compose_eval" 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 "R" with "n" variables. 4. "as" is an element of the carrier of the power of the ring "R" to "m". Under these assumptions, the...
The lemma BIT_c assumes that "x" is not equal to "y", "init" is in the set {[x,y],[y,x]}, and "v" is in the language of the sequence [Times (Atom y) (Atom x), Star (Times (Atom y) (Atom x)), Atom x]. It shows that the function Tp_on_rand' BIT (type0 init x y) v equals 0.75 times the length of v minus 0.5, and that the ...
There exists a decision 'd' such that 'd' is a decision.
The lemma named "chain_split" assumes that "C" is a chain in a complete partial order defined by the relation "ord" and that "x" is an element of "C". It shows that "C" can be split into two sets: one set contains elements "y" in "C" such that "x" is related to "y" by "ord", and the other set contains elements "y" in "...
If function 'f' maps from set 'A' to set 'B' and function 'g' maps from set 'B' to set 'C', then the composition of function 'g' and function 'f' maps from set 'A' to set 'C'.
The configuration of q0 at the step "Suc n" (which means the successor of n, or n+1) is equal to the set containing "Suc n" union with the set of all configurations of q' at step n, where q0 is equal to the transition function δ applied to q' and the nth element of w.
In the context of a tiny monoidal semicategory functor (is_tm_ntsmcf), there is a lemma (a proven statement) that introduces semicategory operations (smc_op_intros). This lemma assumes that the semicategory functor 'GG' is the opposite of 'GG'', the semicategory functor 'FF' is the opposite of 'FF'', the semicategory '...
The exponential function is continuous on the entire universe of discourse.
If A is an element of the sets of N, then the function that maps each element 'a' to the extended measure of A in the kernel 'K' at 'a', is Borel measurable on M.
The lemma named "small_insert_V" is defined for a set X of type V. It assumes that X is small. It then shows that the set resulting from inserting an element 'a' into X is also small.
The codomain function of the product of three directed graphs, denoted as \<AA>, \<BB>, and \<CC>, is a very special variable (vsv).
The norm of a vector 'x' is less than or equal to the sum of the norms of its components for all elements in the universe.
The lemma named "root_equals_end_points" assumes that "r" is the root of a backward terminating path of the transpose of "x", and "x" is not equal to zero. Under these assumptions, it shows that "r" is equal to the end points of "x".
The lemma named "bisimResComm" establishes that for any given names 'x' and 'y', and any given psi-calculus process 'P' under a context 'Psi', the process of restricting 'x' in the process of restricting 'y' in 'P' is bisimilar to the process of restricting 'y' in the process of restricting 'x' in 'P'. This essentially...
A point 'x' is in the relative interior of a set 'S' if and only if 'x' is in 'S' and there exists a positive number 'e' such that the intersection of the open ball centered at 'x' with radius 'e' and the affine hull of 'S' is a subset of 'S'.
The sum of the product of the Stirling number of the first kind of 'n' and 'k' and 'x' raised to the power of 'k', for all 'k' less than or equal to 'n', is equal to the Pochhammer symbol of 'x' and 'n'. This is valid in the context of a commutative semiring.
The lemma "meridian_reflect_K2_isometry" shows that the meridian reflection is a K2 isometry.
If "t" is an element of the frontier of the impact of "c" on "loc2", then there exists a "loc1" and a sequence "xs" such that the longest impact witness path from "c" from "loc1" to "loc2" through "xs" exists for "t".
The lemma named "accD_pos" assumes that the pair (x, y) is in the set "acc". It shows that there exists a positive integer n such that the probability p of transitioning from state x to state y in n steps is greater than zero.
If the list "lma" corresponds to the list "lmb" according to the function "abc_list_crsp", then the value of the function "abc_lm_v" applied to "lma" and "n" is equal to the value of the function "abc_lm_v" applied to "lmb" and "n".
The lemma named "mod_exE" assumes that there exists an 'x' such that 'P x' holds in the heap 'h'. It concludes that there is an 'x' for which 'P x' holds in the heap 'h'.
The lemma named "Image_outside_Domain" assumes that the intersection of the domain of relation R and set A is an empty set. It shows that the image of set A under relation R is also an empty set.
: Assuming that the machine M2 is observable and minimal, and the size of M2 is less than or equal to the size of the prime version of M1 plus some additional states represented as an integer. Also, the inputs and outputs of M2 are the same as those of M1. If M1 is already in its prime form, then M1 is observable and ...
For any two elements 'x' and 'y' in an ordered additive abelian semigroup, if 'x' is in the interval 'X' and 'y' is in the interval 'Y', then the sum of 'x' and 'y' is in the interval of the sum of 'X' and 'Y'.
The external IDs in the sequence "s" are distinct.
The lemma named "phis'_code_correct" assumes that the graph "g" is redundant according to the SSA (Static Single Assignment) form. It shows that the function "phis'" applied to the graph "g" is equal to the result of looking up the function "phis'_code" applied to the graph "g" and the function "chooseNext'" applied to...
If the length of list "xs" is 3, then there exist elements x, y, and z such that the list "xs" is composed of [x, y, z].
For any two sets S and T, where S and T are sets of elements from commutative monoids with addition, the Cartesian product of S and T is equal to the set of all sums of u and v, where u is an element of the image of S under the function that pairs each element with 0, and v is an element of the image of T under the fun...
In the context of an infinite coin toss space, there is a lemma regarding the natural filtration image being finite. Given a function 'f' that maps a boolean stream to a 'b' type in a T2 space, if 'f' is in the Borel measurable set of the natural filtration 'n', then the image of 'f' in the space 'M' is finite.
If function 'f' is a homomorphism from group G to group H, and function 'g' is a homomorphism from group H to group K, and the composition of 'g' and 'f' is a monomorphism from group G to group K, then function 'f' is a monomorphism from group G to group H.
If the filter is not an empty list, then the function C applied to the policy list derived from the filter and p is equal to the function C applied to the firewall policy derived from the reversed filter and p.
The lemma named "transitive" is defined with respect to a process P, a relation Rel, a process Q, a relation Rel', a process R, and a relation Rel''. It assumes that process P is related to process Q via relation Rel, and process Q is related to process R via relation Rel'. It also assumes that the composition of rel...
If an element 'a' is in set 'C' and set 'B' is a subset of set 'D', then the set resulting from the operation 'a +o B' is a subset of the set resulting from the operation 'C + D'.
The conditional probability mass function of the first element in the attempt to combine two probability mass functions "p" and "q" with respect to "x" is equal to "None" if "p" is lossless. Otherwise, it is equal to the conditional probability mass function of the first element in "q" with respect to "x". This is unde...
The set of options resulting from zipping two options x and y is equal to the Cartesian product of the set of options of x and the set of options of y.
If the function of x, which is the product of the prime-counting function of x and the natural logarithm of the prime-counting function of x, is asymptotically equivalent to the function of x at positive infinity, then it implies that the function of the nth prime number is asymptotically equivalent to the function of ...
For a given matrix 'A' of a certain field and modular type, the rank of 'A' is equal to the vector dimension of the range of the function that multiplies 'A' with a vector 'x'.
The cosine of pi, when pi is considered as a real number, is equal to -1.
Any natural number power of the identity element equals the identity element.
There is a valid positive hierarchy within the "TUM" department, which includes the "i8" and "i20" departments, with "TUM" being a leaf node.
The lemma named "tree_recurse_fusion" assumes that the composition of functions h and l equals the composition of functions l' and h, and that the composition of functions h and r equals the composition of functions r' and h. Under these assumptions, it shows that mapping function h over the tree recursion of functions...
If for every element P in set S, P is less than or equal to Q, and Q is non-negative, then the supremum of the exponential set S is less than or equal to Q.
The multiset of the merge sorted list is equal to the multiset of the original list.
The set of all elements of the ring R is an ideal of R.
If a pair of lists (xs, ys) is in the relation "listrel1 r" and lists us and vs are identical, or if lists xs and ys are identical and a pair of lists (us, vs) is in the relation "listrel1 r", then the concatenation of lists xs and us, and ys and vs, is also in the relation "listrel1 r".
For any given values of 'd' and 'b', the parents of 'b' in 'd' are only 'b' itself.
For any values of x and y, the function M with inputs x, x, and y equals x, the function M with inputs x, y, and x equals x, and the function M with inputs y, x, and x also equals x.
The lemma named "ray_path" assumes that "x" is in the ray from "a" to "b" and "x" is not equal to "b". It then obtains a path "ab" from "a" to "b" such that the ray from "a" to "b" is a subset of this path "ab".
The lemma is about the induction on a structure called 'sterm'. It fixes an 'sterm' t, a predicate P1 on 'sterm', a function f from Label to 'sterm', and a predicate P3 on such functions. It assumes the following: - P1 holds for 'Bvar b' for any b. - P1 holds for 'Fvar x' for any x. - If P3 holds for a function f, the...
, under the assumptions that the LLL invariant mod weak holds for the given parameters fs, mfs, dmu, p, first, and b, and that c equals the rounded number over the denominator of the element at position (i,j) in dmu, and that i is less than m, and that j is less than i, it can be shown that c equals the rounded value o...
The lemma named "RG_proj_mult_rightdelta" is defined with a fixed variable 'r' of any semiring. Assuming 'g' is an element of 'G', it shows that the projection of the product of 'x' and the double delta of 'r' and 'g' is equal to the product of the projection of 'x' and the double delta of 'r' and 'g'.
The lemma named "poly_subst_cong" assumes that polynomial "p" is equal to polynomial "q" and for all "x" in the set of indeterminates of polynomial "q", the function "f" at "x" is equal to the function "g" at "x". Under these assumptions, it shows that the polynomial substitution of function "f" into polynomial "p" is ...
The lemma states three things: 1. An empty multiset added to any multiset 'x' equals 'x'. Similarly, any multiset 'x' added to an empty multiset equals 'x'. This means that the empty multiset acts as a neutral element in addition. 2. Adding a non-empty multiset '{# x #}' to any multiset 'rs' does not result in an emp...
If "y" is an element of "V" and "x" is an element of "V", and "x" is not equal to "y", then the size of the set of all "bl" in "B" such that the set containing "x" and "y" is a subset of "bl", is equal to "Lambda".
If F is a member of filters, then the left congruence of F with a and b is equivalent to the existence of x and y such that x and y are members of F and the product of a and x equals the product of b and y.
If the zeroth homology group of space X is isomorphic to the zeroth homology group of space Y, then space X is path-connected if and only if space Y is path-connected.
The lemma named "sym_mpoly_insert" assumes that "X" is a finite set and "x" is not an element of "X". It shows that the symmetric multivariate polynomial of the set obtained by inserting "x" into "X" and the successor of "k" is equal to the product of the monomial of the set containing "x" and the symmetric multivariat...
The lemma named "nodes_root_height_lower_bound_simp" assumes that the root order of a binary tree 't' is 'k', the tree 't' is balanced, and 'k' is greater than 0. It shows that the expression "(2*((k+1)^(height of the tree 't' - 1) - 1)) divided by 'k' plus the boolean value of whether the tree 't' is not a leaf, is le...
The lemma "reachable_constraints_no_bvars" assumes that "\<A>" is in the reachable constraints of "P". It also assumes that for all "T" in the set "P", the bound variables of the transaction strand of "T" are empty. Under these assumptions, it shows that the bound variables of "\<A>" are also empty.
If "other P {i} σ σ'" holds, then for all "j" such that "j" is not equal to "i", "P (σ j) (σ' j)" also holds.
If psi is in the FG fragment of phi with respect to a word w, then w satisfies the next-step semantics of the next-step globally operator applied to psi.
For a substitution sigma, if we substitute the variable x with a value v in the precondition P, it is equivalent to first substituting v for x in P and then applying the substitution sigma.
The first component of a pair "u" is less than or equal to "u".
If "canAccessThis" is not equal to "vertex_1", then the successor transition of a graph with nodes "vertex_1" and "canAccessThis" and an edge from "vertex_1" to "canAccessThis", when applied to "canAccessThis", results in an empty set.
The evaluation of the formula "HF_axP x" under the environment "e" is true if and only if the interpretation of "x" under the environment "e" belongs to the set "HF_ax".
If a transition function "trans_fun" is deterministic for a given local state, then the accumulated transition function "Acc_Trans_Fun_Step" with parameters k, trans_fun, and pointwise_shrink is also deterministic for the same local state.
If "P" is a type of "T" and "U" is a reference type, then an "Instanceof T" instruction is well-typed, transitioning from a type state with "U" on top of the stack type "ST", to a type state with "Boolean" on top of the stack type "ST", with the same exception and operand stack types.
Given a finite set A, if A is independent and the sum of the scaled elements of A is zero, then for all elements in A, the scaling factor is zero.
The lemma shows that the unit circle is a circle.
If the height of tree t1 or the height of tree t2 is greater than 0, then the sibling of a node 'a' in a new tree (Node w t1 t2) is equal to the sibling of 'a' in tree t1 if 'a' is in the alphabet of t1, otherwise it's equal to the sibling of 'a' in tree t2.
The lemma named "lim_h_minus_OUT_IN" assumes that the set of cycles in the system, denoted by "cycles \<Delta>", is not empty. Under this assumption, it shows that the outgoing flow from a node 'x' in the limit, denoted by "d_OUT lim_h_minus x", is equal to the incoming flow to the same node in the limit, denoted by "d...
: given the assumptions that f, g, and h are identities in bicategory B, and the source of f is the target of g, and the source of g is the target of h, it shows that the composition of the associator of f, g, and h with the composition morphism of the tensor product of f and g with h and the tensor product of the comp...
The argument that minimizes the function "f" over the list "xs" is determined as follows: If the list "xs" is empty, the result is undefined. If the list "xs" starts with an element "x" followed by the rest of the list, then we apply a fold operation on the rest of the list, starting with the pair of "x" and "f(x)". In...
The lemma named "antichain_obtain_foundation" assumes that "t" is an element of "M". It then obtains an "s" such that "s" is an element of "M", "s" is less than or equal to "t", and for all "u", if "u" is an element of "M", then "u" is not less than "s".
The sublist of the list obtained by applying a function 'f' to each element of list 'xs' from index 'i' to 'j' is equal to the list obtained by applying the function 'f' to each element of the sublist of 'xs' from index 'i' to 'j'.
If 'x' is less than 'y', both 'x' and 'y' are elements of set 'I', and for any 'k' such that 'x' is less than 'k' and 'k' is less than 'y', 'k' is not an element of 'I', then the immediate next element of 'x' in set 'I' is 'y'.
For a unique orthogonal subspace decomposition, given a set S in a Euclidean space, if x + y equals x' + y', where x and x' are in the span of S and y and y' are in the span of T, and for any a in S and b in T, a and b are orthogonal, then x equals x' and y equals y'.
If the function "gSwapGAbs" is applicable to the model "MOD", then "igSwapIGAbs" is applicable to the model obtained from "fromMOD MOD".
If 'p' is a prime number, then for any positive 'k', the square of 'm' cannot be equal to 'p' times the square of 'k'.
The alternative semantics of a formula F under a model A is equivalent to the model A satisfying the formula F.
In the context of Polynomial Rings, there is a lemma stating that if the polynomial coefficient 'c' belongs to the set 'S', 'n' is less than 'm', and 'm' is less than or equal to the first element of 'c', then the finite sum from 'n+1' to 'm' of the product of the 'j-th' element of the second element of 'c' and 'X' rai...
If a greedy conforming path from "v0" under strategies "σ" and "σ'" equals to the list constructed by "v" and "P", then "P" is either an empty list, or it is a greedy conforming path from "σ(v0)" under strategies "σ" and "σ'", or it is a greedy conforming path from "σ'(v0)" under strategies "σ" and "σ'".
The least common multiple of any integer 'n' in set 'N' is equal to the integer value of the least common multiple of the set 'N'.
The lemma named "wffp_Idle_State" assumes that a well-formed formula (wffp) is composed of a sequence of states (sl1), followed by an idle state (st1), followed by another state (st2) with associated actions (u2, c2), and then another sequence of states (sl2). It shows that either the second state (st2) is equal to the...
If the following conditions are met: 1. The program P is well-formed with respect to the type-checking relation wt. 2. The class C in program P sees the method M with the type signature Ts returning type T, which is defined by the method body (mxs, mxl0, ins, xt). 3. The instruction at the program counter pc is Push v...
This statement is a lemma or a proven proposition that states "There exists an arbitrary but fixed element that satisfies the property P". The "Eps P" is a function in Isabelle that returns some x such that P(x) holds, if such an x exists.
: given a subset J of I, a non-empty space of the product measure over I and M, and a set X that belongs to the sets of the product measure over J and M, if the product embedding of I, M, J, and X equals the empty set, then X must be the empty set.
If a value 'x' is in the result of either 'Q' or 'R', depending on a condition 'P', and if 'S' follows from 'P' and 'x' being in the result of 'Q', and 'S' also follows from 'not P' and 'x' being in the result of 'R', then 'S' must be true.
If the second element of x is equal to the second element of y, and the first element of y is equal to the first element of x, then x is equal to y.

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