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Viewing the right end of an empty finger tree results in None.
For any natural number 'n', the value of 'n' is less than 2 raised to the power of 'n'. This applies in the context of a linearly ordered integral domain.
If a pair of values (v1, v2) is well-formed under an empty context and an empty typing environment, then the type of the first value v1 is a subtype of the type of the pair (v1, v2).
The composition of zero with any other element is zero.
The number of changes in the list "xs" is equal to the number of changes in the list obtained by applying the sign function to each element of "xs".
A sublist of the list 'xs' from the list 'ys' is empty if and only if the list 'ys' is empty.
The interrupts of the result of the function "activate_cond_actions1" with inputs s1, s2, and cts are equal to the interrupts of s1. This is a simplification rule in Isabelle.
If "Y" is not an empty set and "Y" is a subset of "X", then the image of "X" under the function "subsetFn" applied to "X" and "Y" equals "Y".
The numeral of any variable 'x' is equivalent to unprotecting the numeral of the same variable 'x'.
: Assuming that the policy does not interfere with the partition of the task identifier 'tid' and user 'u', and that the atomic step is invariant in state 's', and that the precondition for the atomic step in state 's' with task identifier 'tid' is the signal event finish for the partner, and that the interrupt type is...
If a variable "x" is in the set of indeterminates of the polynomial obtained by substituting a function "f" into a polynomial "p", then there exists a variable "y" such that "y" is in the set of indeterminates of "p" and "x" is in the set of indeterminates of the function "f" applied to "y".
The interpretation of the concatenation of symbolic runs Γ1, Γ2, and Γ3 is associative. This means that concatenating Γ1 and Γ2 first and then concatenating the result with Γ3 is the same as concatenating Γ2 and Γ3 first and then concatenating the result with Γ1.
The square of the formal power series cosine of a number 'c' plus the square of the formal power series sine of the same number 'c' equals 1.
The alpha function applied to the transitive closure of a red-black tree, given a relation 'r' and a set 'as', is equal to the set of all 'b' such that there exists an 'a' in the set 'as' where the pair (a, b) is in the transitive closure of the set 'r'.
In the context of p-adic integers, the lemma named "residue_mult_assoc" shows that the multiplication operation in the residue ring of Zp at level k is associative. That is, for any elements x, y, and z in this ring, multiplying x and y together first and then multiplying the result by z gives the same result as multip...
For any natural number power, if 'x' is a non-zero element of the ring 'R', then 'x' raised to the power of 'n' is also a non-zero element of the ring 'R'.
For a given polynomial 'p' and a set 'S', the degree-graded polynomial set 'd' is less than or equal to the set containing only 'p', excluding the set 'S'.
If a pair (s, X) is in the set F P, and for all elements c in set Y, the pair (s, X union {c}) is not in the set F P, then for all elements c in set Y, the pair (s concatenated with [c], empty set) is in the set F P.
"d of x times U equals x" if and only if "x times the identity element is parallel to U equals x".
The generative probabilistic value of failure is finite.
The interaction with a generative probabilistic value (gpv) is bounded by the count of bad events. This is under the following conditions: 1. The interaction with the gpv is bounded by a certain number 'n'. 2. A state 's' satisfies a certain condition 'I'. 3. The state 's' is not bad. 4. For any state 's', input 'x', ...
The lemma "summable_on_comm_additive_general" strengthens the theorem "summable_on_comm_additive_general". It fixes two functions, g and f. The function g maps from any type 'a to a type 'b that is a commutative monoid with addition and a topological space. The function f maps from type 'b to a type 'c that is also a c...
The lemma named "fps_div_fps_X_power_nth'" is defined for a formal power series 'f' of type 'a', where 'a' is a semiring with additional properties of having an inverse and a unary minus operation. The lemma assumes that the inverse of 1 for type 'a' is 1. It shows that the k-th term of the series obtained by dividing ...
The lemma named "triangle" assumes that "a" and "b" are identities. It shows that the tensor product of "a" and the left identity of "b", when composed with the associator of "a", the unit object, and "b", is equal to the right identity of "a" tensor "b".
The lemma "nAct_not_active" is defined with the fixed natural numbers 'n' and 'n'', a trace 't' of configuration lists, and an identifier 'c'. It assumes that the extended natural number 'i' is less than the length of the trace 't' and that 'c' is not active in the 'i'-th element of the trace 't'. Under these assumptio...
The lemma named "in_targets_respects_Cong" assumes that "t" is equivalent to "t'", "b" is in the targets of "t", and "b'" is in the targets of "t'". It shows that "b" is equivalent to "b'".
In the context of a co-span in a category functor, there is a lemma that states: if a morphism 'f' is equal to the co-span morphism 'g', then the arrow map of the co-span functor applied to 'f' is equal to 'g'.
If H is a subgroup of G and y is an element of the carrier of G, and the right coset of H with respect to x equals the right coset of H with respect to y, then y is an element of the right coset of H with respect to x.
The lemma named "valid_eval_e_eq" is defined with respect to two expressions, e1 and e2. It assumes that for all environments (Γ) and integers (i), if the environment is well-formed under a given set of type definitions (Θ), then there exists a state (s) such that both e1 and e2 evaluate to s. It also assumes that both...
If a maximum simplex chain is formed by appending 'x' to 'xs', and 'y' is a maximum simplex, and 'x' is related to 'y', then a maximum simplex chain can be formed by appending 'x' and 'y' to 'xs'.
The size of a decision tree node "r" with successors "t1" is less than the size of a decision tree node "v" with a set containing "t1" and "e1".
The lemma named "has_field_derivative_powser" is about a function in a Banach space and a real normed field. It assumes that the norm of a complex number 'z' is less than the convergence radius of a function 'f'. It shows that the derivative of the power series of 'f' at 'z' within a set 'A' is equal to the sum of the ...
For all values of 'k', if 'i' is less than or equal to 'k' and 'k' is less than 'j', and the function 'interfree_aux' with parameters 'Γ', 'Θ', 'F', and a tuple consisting of 'c k', 'Q k', 'x', and 'a' holds true, then the function 'inter_aux_Par' with parameters 'Γ', 'Θ', 'F', and a tuple consisting of the map of 'c' ...
If the product of n(y) and x is less than or equal to the product of x and n(y), then the product of n(y) and the omega power of x equals the product of n(y) and the omega power of the product of n(y) and x.
The lemma named "append_lq" assumes that the concatenation of strings "u" and "v" is a prefix of string "w". It shows that the right quotient of string "w" by the concatenation of "u" and "v" is equal to the right quotient of the right quotient of "w" by "u" by "v".
The lemma named "S_ground_total" assumes that the function as term "s" is a subset of "F", "s" is a ground term, the function as term "t" is also a subset of "F", and "t" is a ground term. It shows that either "s" equals "t", "S" of "s" is less than "t", or "S" of "t" is less than "s".
The lemma named "cos_mono_le" assumes that the absolute value of x is less than or equal to pi and the absolute value of y is less than or equal to pi. It shows that the cosine of x is less than or equal to the cosine of y if and only if the absolute value of y is less than or equal to the absolute value of x.
"xq" is contained in "Q" if and only if the predicate of the sequence "xq" is less than or equal to "Q".
The bitwise OR operation on two natural numbers m and n is equal to the following: if m is zero, then it's n; if n is zero, then it's m; otherwise, it's the maximum of the remainder of m divided by 2 and the remainder of n divided by 2, plus twice the bitwise OR operation on m divided by 2 and n divided by 2.
The lemma holds that the correct decomposition of the variable system is true.
If 'f' is a co-closure operator on a set of type 'a', then applying 'f' to the top element of the set results in the top element of the set.
The lemma named "convert_substInfinitesimalLinear" makes several assumptions: 1. The conversion of an atomic formula 'a' with respect to a variable 'var' and a list of variables 'xs' appended with 'x' and 'xs' again, results in 'a''. 2. The insertion of 'b' into the list of variables 'xs' appended with 'x' and 'xs' a...
Applying the inverse of a function 'f' to 'a' is equivalent to applying the function 'f' raised to the power of the order of 'f' with respect to 'a' minus one, to 'a'.
If F is less than or equal to G, then the projection of F on C through h is less than or equal to the projection of G on C through h.
If G is a graph and G is connected, and if v is a vertex in G, then there exists a path p in G from the source to v.
The lemma named "oalist_inv_raw_sort_oalist_aux" assumes that the ordered association list "xs" is valid. It shows that after sorting the ordered association list "xs" using the auxiliary function "sort_oalist_aux" with key order "ko", the resulting list is also valid according to the raw invariant of ordered associati...
If the method "exec_meth" is executed with parameters "ci", "P", the compilation of expression "e", the compilation of expression "e" with starting index 0 and depth "d", "t", "h", and the tuple "(stk, loc, pc, xcp)", and this execution results in "ta", "h'", and "s'", then the program counter "pc" is less than the len...
If "accs" is a function of delta, t, and q, then q is in the set of b_accessible of delta.
For any number 'a' that is greater than zero and not equal to one, the hyperbolic logarithm of 'a' with base 'a' is equal to one.
The star-transform of sequence X is approximately equal to the star-transform of sequence Y if and only if for every natural number m, it is eventually the case that the norm of the difference between the nth term of sequence X and the nth term of sequence Y is less than the inverse of the real number successor of m, u...
The function pair "(Array.make N, PR_CONST init_ps)" is an element of the function space from "pure Id" to "is_am", where the function space is augmented with a Kleene star.
If the function type of "Fun f X" under a context Γ is equal to the type composition "TComp g Y", then the function identifiers "f" and "g" must be equal.
The square of the square root of 2 divided by 2, all multiplied by 2, equals 1.
If a function f is integrable over a measure M and a scalar c is not equal to zero, then the function f scaled by c (i.e., each value of f multiplied by c) is also integrable over the measure M.
The lemma named "consume_type_type" assumes that consuming the list "xs" with the function "cons" results in the type "t_int". It then shows that there exists a type "tn" such that the list "xs" is of type "tn".
The angle ABC is congruent to the angle A'B'C' if and only if the angle A'B'C' is congruent to the angle ABC.
If a function "phi" preserves the supremum, then there exists a relation "R" such that the bounded relation of "R" is equal to "phi".
The parallel beta reduction relation is a diamond property.
If the oldest token of a state 'q' at a certain point 'n' is 'x', then the token run of 'x' at point 'n' is 'q'.
For all 'i', if sigma of 'i' equals sigma prime of 'i', then the substitution of sigma in 'v' equals the substitution of sigma prime in 'v'.
For a sequence X of natural numbers mapped to a type 'a, where 'a is a conditionally complete linear order and a linear order topology, if the range of X is bounded below and X is a decreasing sequence, then X converges to the infimum (greatest lower bound) of the set of all X i.
If "s" is a hereditary function, then the function "insf" with parameters "s", "k", and "y" is also a hereditary function.
The composition of the functions "frame_of_frame'" and "frame'_of_frame" with respect to a given parameter P is equivalent to the identity function.
The product of x and 0 is less than or equal to 1.
The lemma named "cpx_vec_length_square" shows that the square of the length of a complex vector "v" is equal to the sum of the squares of the absolute values of its components, where the sum is taken over all indices from 0 to the dimension of the vector minus one.
If a list of field patterns "fps" matches a list of field types "fTs" resulting in a context "Δ", and for every pair (l, U) in the set of field types there exists a value "u" such that the field "l" in the record "fs" is equal to "u", then if the list of field patterns "fps" matches the record "fs" resulting in a list ...
: 1. The function "mapping_impl_choose2" when applied to any two arguments "x" and "y", returns "mapping_Choose". 2. The function "mapping_impl_choose2" when applied to two "mapping_Mapping" arguments, returns "mapping_Mapping". 3. The function "mapping_impl_choose2" when applied to two "mapping_Assoc_List" arguments,...
If an element 'xa' is not in the domain set of a finite map 'N', then restricting the finite map 'N' to the set obtained by inserting 'xa' into another set 'l1' is the same as restricting 'N' to the set 'l1' itself.
If the CoP2 condition holds for a function η, then the DS1 property also holds for the function η with respect to the operation represented by the bold arrow symbol.
The signed modulus of zero and any integer 'x' is zero.
The lemma named "SetToIndex" is defined with two fixed elements: 'a' and a list named 'liste' of type 'a'. It assumes that 'a' is an element in the set 'liste'. It then shows that there exists an index less than the length of 'liste' such that 'a' is equal to the element at that index in 'liste'.
The lemma named "prod_relE" assumes that the pair (p, p') is in the product relation of R1 and R2. It then concludes that there exist elements a, b, a', and b' such that p equals the pair (a, b) and p' equals the pair (a', b'), and that (a, a') is in R1 and (b, b') is in R2.
The lemma states two conditions: 1. If the equality of 'n' and '0' is true, then the 'n'th element of the list composed of 'x' followed by 'xs' is 'x'. 2. If the equality of 'n' and '0' is false, then the 'n'th element of the list composed of 'x' followed by 'xs' is the '(n - 1)'th element of 'xs'.
For any two elements x and y of a set that supports multiplication and zero, the product of y and either x or 0 (depending on whether a condition b is true or false) is equal to either the product of x and y (if b is true) or 0 (if b is false).
The Cauchy index of a path 'g' at a point 'z' is equal to the negative of the Cauchy index of the reverse of that path at the same point 'z'.
In the context of a certain program, if the initial state has the variable 'N' equal to 'n', then after the execution of the sequence of commands where 'N' is first assigned the value 10, then 'M' is executed, and finally 'N' is incremented by 2, the final state will still have the variable 'N' equal to 'n'.
The bitwise AND operation on 'x' and 'y' is equal to the bitwise OR operation on 'x' and 'y', where 'y' is a 32-bit unsigned integer.
The lemma "emeasure_lub_spmf" assumes that the set "Y" is not empty. It shows that the exterior measure of the set "A" under the measure of the least upper bound of the subprobability mass function is equal to the supremum of the exterior measures of the set "A" under the measure of each element in the set "Y".
The Euclidean size is equal to the natural number representation.
The lemma "kth_obs_stable" assumes that "is_kth_obs" is true for the parameters pi, l, and j, and that k is less than l. It shows that there exists an i such that "is_kth_obs" is true for the parameters pi, k, and i.
For any element 'x', 'x' is prefix greater than or equal to 'x' itself.
The lemma "converged_ipd_same_icd" assumes that both "pi" and "pi prime" are paths, and that "l" is less than "m". It also assumes that the control state of "pi" at "m" is equal to the control state of "pi prime" at "m prime", and that the control state of "pi" at "k" is equal to the control state of "pi prime" at "k p...
If a system is well-formed, the control is within the system for the initial state of a process A, and the transition of process A is sequential, then the union of sequential terms of the system reachable from process A is a subset of the concurrent terms of the system.
If the function "throw_of" applied to "e" equals "Some e'", then "e" must be equal to "throw e'".
A process is Markovian if and only if for all 'n', 'p n' is a decision process.
If a multiset M is an element of the multiset encoded by e and n, then there exists an e' greater than or equal to e such that M is equal to the encoding of 0 and e'.
The set of all sublists of the union of a single element 'w' and a set 'A' is equal to the union of the set of all sublists of 'w' and the set of all sublists of 'A'.
If set A is independent, and set B is a subset of set A, then set B is also independent.
For any given sets S and T, and random variables X and Y, if we define P as the distribution of X over S and Y over T, and Q as the distribution of the pair (X, Y) over the product set of S and T, then X and Y are independent variables if and only if X is a random variable over S, Y is a random variable over T, P is ab...
If $a$ is not in the topological space $X$, then the set of neighborhoods of $a$ in $X$ is empty.
The predicate $P$ holds eventually in the neighborhood filter of $a$ in $X$ if and only if $a$ is not in the topological space $X$ or there exists an open set $S$ in $X$ such that $a$ is in $S$ and $P$ holds for all $x$ in $S$.
A predicate $P$ holds eventually at $a$ in a topological space $X$ if and only if either $a$ is not in the topological space $X$ or there exists an open set $U$ containing $a$ such that $P$ holds for all points in $U$ except $a$.
The limit of a function $f$ at a point $l$ in a filter $F$ is the same as the limit of $f$ at $l$ in the Euclidean topology.
If $f$ converges to $l$ in $X$, then $l$ is in the topological space $X$.
A constant sequence converges to a point $l$ if and only if $l$ is in the topological space $X$.
The constant function $f(x) = l$ is a limit of the sequence $F$ in the Euclidean topology.
If $l$ is a limit point of $X$ and $f$ is eventually equal to $l$ on $F$, then $l$ is a limit point of $f$ on $F$.
If $f$ is a function defined on a topological space $X$ and $r$ is a strictly increasing sequence of natural numbers, then the subsequence $f \circ r$ converges to $l$ if and only if $f$ converges to $l$.
A sequence $f$ converges to a limit $l$ in the subspace topology on $S$ if and only if $l \in S$ and $f$ converges to $l$ in the topology on $X$ and $f$ eventually takes values in $S$.
If $S$ is an open set, then the limit of $f$ in $S$ is the same as the limit of $f$ in the topology of $S$ and $l \in S$.