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If "y" is in the spray of "x", then "y" is in the set of events "E".
The real part of the cosine of a real number 'x' is equal to the cosine of 'x'.
Taking the first 'n' elements from the shifted derivation 'D' (shifted by 'left' and 'right') is equivalent to shifting the first 'n' elements of the derivation 'D' by 'left' and 'right'.
If a stack is valid, then the first element of the pair resulting from the step function applied to the stack and rv is also valid.
Running the nondeterministic failure operation is equivalent to returning an empty set.
If 'a' is less than or equal to 'b' in the ordinal category 'A', then there exists a morphism from 'a' to 'b' in the same ordinal category 'A'.
If two finite sets are related by a relation A, then their union is also related by the same relation A. This is a rule for transferring properties between structures in Isabelle.
If 'n' is less than or equal to the length of 'a', then the mask of 'n' for 'a' (which is a word of length 'len') is equal to -1 right shifted by the difference of the length of 'a' and 'n'.
The union of set R is equal to V.
If a state 's' is not in the set 'S', then the function 'n' applied to 's' is undefined.
If the split of a floating point interval X results in two intervals A and B, then any real number x that belongs to interval X must also belong to either interval A or interval B.
If 'j' is less than or equal to 'i' and the 'j'-th element of list 't' is 'LNil', then the 'i'-th element of list 't' is also 'LNil'.
Shifting by zero in the exception table "xt" results in the same exception table "xt".
: Assuming the Kruskal spanning invariant holds for f, g, and h, and h is not the bottom element, it shows that: 1. If the chosen arc from h is not in the forest components of f, then the Kruskal spanning invariant still holds for the updated f, g, and h. Moreover, the cardinality of the set of regular elements that ...
If "m" is less than "n", then the minimum integer in the set of integers from "m" to "n" (including "m" but excluding "n") is "m".
The lemma Z_I_carrier shows that Z_I is an element of the carrier matrix of size 4 by 4.
The association of x and y in a map t is defined as follows: if mapping of t at x gives some value y', then the result is y', otherwise the result is y.
"0 is an element of the set of rational numbers."
The lemma named "eqExcPID_N2_sym" assumes that "s" is equivalent to "s1" with the exception of PID_N2. It shows that "s1" is also equivalent to "s" with the exception of PID_N2.
For any list "xs" appended with an element "x", when joined with a set "I" using the f-join operation, is equal to the list "xs" joined with the set "I" using the f-join operation, appended with the element "x" if the length of the list "xs" is in the set "I", otherwise appended with an empty list.
The lemma named "abe_ade_bcd_ace" assumes three conditions: 1. "a" is before "e" and "b" is between "a" and "e" (denoted as "[a;b;e]"), 2. "a" is before "e" and "d" is between "a" and "e" (denoted as "[a;d;e]"), 3. "b" is before "d" and "c" is between "b" and "d" (denoted as "[b;c;d]"). Under these assumptions, it...
The function "p" of 0 equals "s0".
If 's' is an open set, then the function 'f' is differentiable on 's' if and only if 'f' is differentiable at every point 'x' in 's'.
The set difference closure of a set 'a' with a function that maps any element to a singleton set containing 'x', is equivalent to applying 'a' to 'x'.
If "p" is a test, then "y" is less than or equal to the product of "x" and "p" if and only if the product of "y" and the converse of "p" is less than or equal to "x".
The lemma named "degree_lagrange_interpolation_poly" shows that the degree of the Lagrange interpolation polynomial is less than or equal to the length of the list of x and y coordinates minus one.
"x star z" is less than or equal to "(x join y) star z".
Appending an empty list to any list 'xs' results in the original list 'xs'.
If "f" is a disjunction, then the conversion of "f" from conjunctive normal form (CNF) to DIMACS format is equivalent to the conversion of "f" from a disjunction to DIMACS format.
The lemma "SeqAbstFormP_Ex" assumes that the atom 's' is distinct from the set of elements (k,s1,k1,x,x',v,i) and the atom 'k' is distinct from the set of elements (s1,k1,x,x',v,i). It shows that if the sequence abstraction form 'SeqAbstFormP' of 'v' at the successor of 'i' with respect to 'x' and 'x'', 's1', and 'k1' ...
The theorem named "br_while_algo" assumes that the transition function "delta" is finite. It shows that the breadth-first search algorithm "br_algo" can be applied in a while-loop algorithm.
The disjoint union of the set A indexed by I is equal to the union of the set of ordered pairs (i, x) for each i in I and each x in the set A indexed by i.
The first element (head) of the sequence produced by the function "nat_explode'" on any natural number 'n' is less than 10.
The recursive function "rec2" with parameters "s1" and "s2" applied to the constructor "ctor2" with parameter "x" is equal to the function "s2" applied to the function "F2map" with parameters "id" and a pair of "id" and "rec1 s1 s2", and another pair of "id" and "rec2 s1 s2", and "x".
For a system A, the resource generated by an oracle named "callee1" with state "s1" is equivalent to the resource generated by another oracle named "callee2" with state "s2" if and only if the system A considers the callee1 with state s1 to be equivalent to callee2 with state s2.
Theorem 5: Assuming that the set 'ds' is stable on 'X' and 'x' is an element of 'X', it can be shown that there exists an element 'y' in the cop of 'ds' such that the pair (x, y) is in the preference list of 'x' in the set 'Xd x'.
The function Lxy applied to a list 'xs' appended with an element 'x' and a set 'S' is equal to: if 'x' is in 'S', then the function Lxy applied to 'xs' and 'S' appended with 'x', otherwise the function Lxy applied to 'xs' and 'S'.
If a knowledge base K is the deductive closure of a set A, and a formula phi is not in the contraction of K by the conjunction of phi and psi under the total meet contraction function, then the contraction of K by the conjunction of phi and psi is a subset of the contraction of K by phi under the same function.
The lemma states two conditions: 1. If the list 'xs' is not undefined, then the length of 'xs' is not less than 0. 2. If the list 'xs' is not undefined, then the length of 'xs' is less than the length of 'xs' plus 1.
In any world 'v', the equivalence of the conjunction of 'phi' and 'psi' with the conjunction of 'chi' and 'psi' is equivalent to the implication that if 'psi' is true, then 'phi' is equivalent to 'chi'.
If the pair (x, y) is in the set of P, then P is true for x and y.
The m-th power of x, when bounded below by the least element, is less than or equal to the bound of x.
If "R1" and "R2" are both equivalence relations under the conditions of "Quotient3" with corresponding abstraction and representation functions "Abs1", "Rep1", "Abs2", and "Rep2", then the first element function "fst" respects the product of these relations. This means that if two pairs are related by the product of "R...
The polynomial of a natural number 'n' evaluated at any 'x' is equal to the natural number 'n' itself.
The real part of the division of complex numbers a and b is greater than 0 if and only if the real part of the product of a and the conjugate of b is greater than 0. Similarly, the imaginary part of the division of complex numbers a and b is greater than 0 if and only if the imaginary part of the product of a and the c...
If "ma" is less than 2 to the power of the sum of "n" and "m", then the high value of "ma" and "n" is less than 2 to the power of "m".
The function "addN'" is related to the function "addN" through a certain relation. This relation is defined as a function relation that preserves equality, and it operates on two other function relations that also preserve equality. These two inner function relations operate on mappings that preserve equality.
If "v" and "w" are elements of set "V", then "F" is always in the state where "v" is reachable or the number of messages between "v" and "w" is zero.
If "a" left derives "b" through rule "r" at position "i", then "a" derives "b" through rule "r" at position "i".
If "a" and "b" are both identities and diagrams, then the reduction of "a" and "b" is in the hom-set from the tensor product of "a" and "b" to the floor of the tensor product of "a" and "b".
If a polynomial 's'' belongs to the syzygy module of the set 'bs', and 's' is the atomization of the polynomial 's'' with respect to the index polynomial of 'bs', then 's' belongs to the syzygy module list of 'bs'.
The derivative of the polynomial 1 is 0.
If S is a subring of R, and if g and f are elements of the polynomial ring over S, then the product of f and g is also an element of the polynomial ring over S.
If a sequence 's' is Cauchy, then the range of 's' is bounded.
If the method "exec_meth" is executed with the parameters "ci", "P", "ins", "[]", "t", "h", "(stk, loc, pc, xcp)", "ta", "h'", and "s'", then the program counter "pc" is less than the length of the instructions "ins".
"the_Transaction_h" is a type definition within the universal set, and it is associated with "Transaction_h".
The lemma named "has_codomain_iff_arr" shows that the codomain of function 'f' is not empty if and only if 'f' is an arrow (or morphism in category theory).
In the context of an infinite coin toss space, the lemma named "pseudo_proj_True_Suc_False_proj" demonstrates that for any natural number 'n' and any outcome 'w', the pseudo projection of 'True' at the successor of 'n' on the pseudo projection of 'False' at 'n' on 'w' is equal to the pseudo projection of 'False' at 'n'...
The type of the value 'v' is 'Integer' if and only if there exists an integer 'i' such that 'v' is equal to 'i'.
The lemma "LocalSecretsComposition_neg_s" assumes the following conditions: 1. PQ is a subcomponent of P and Q. 2. PQ is a correct local composition. 3. PQ is a correct key-secret composition. 4. The specific key-secret of message m is not in the specific keys-secrets of P. 5. The specific key-secret of message m is n...
"if Q holds for some x, then Q holds for the unique element satisfying Q."
The system purge of a given system action 'sa' is equal to the system with the user of the action 'sa' and an empty password.
If the measure of 'phis' in graph 'g' at vertex 'v' after updating the mapping 'k' with an empty list is less than or equal to the original measure of 'phis', then the measure of 'phis' after updating the mapping 'k' with any list 'a' is also less than or equal to the original measure of 'phis'.
"a is less than or equal to a or b".
If the pair (x, y) is in all the edges between sets X and Y, then the set {x, y} is in E.
"in the set of components, any component 'i' in the 'chSet' is a subset of the abstraction level 'i'".
The set of indices for the cluster of the function "Some" composed with "f" applied to "X" is equal to "X".
The lemma named "inv_one_sub_Q" is defined with a fixed Q, which is a bounded linear function from a Banach space to itself. It assumes that the norm of the difference between the identity bounded linear function and Q is less than 1. It then shows two things: first, that the composition of Q and the infinite sum of th...
A well-formed formula atom is equivalent to a predicate atom that is also a well-formed formula.
If for all natural numbers 'n', the function 'f' applied to 'n' is equal to the function 'g' applied to 'n', then 'f' sums to 'c' if and only if 'g' sums to 'c'.
The proposition "Q" is true for the object at position "r" in the system heap "s" if and only if the following two conditions are met: 1. If the system heap at position "r" is empty (None), then "Q" is false. 2. For all objects, if the system heap at position "r" contains some object, then "Q" is true for the property...
The lemma cf_adjunction_unit_components shows the following: 1. The natural transformation map of the unit of the adjunction ηC Φ is equal to a function that, for each object x in the domain of the hom-set of the left adjoint of Φ, returns the arrow value of the pair (x, the object map of the left adjoint of Φ at x) u...
If "H" is a subring of "R", then the following conditions hold: 1. "H" is a subset of the carrier of "R". 2. The zero element of "R" is in "H". 3. The unity (or one) element of "R" is in "H". 4. "H" is not an empty set. 5. For any element "h" in "H", the additive inverse of "h" in "R" is also in "H". 6. For any two el...
"for an empty test, the function returns a result that is equivalent to the expected test result, and this is always true."
If strong fairness (SF) is satisfied for an action A with respect to a variable v, then weak fairness (WF) is also satisfied for the same action A with respect to the same variable v.
The lemma named "pure_state_bounded_clinear_right" assumes that F and G are compatible registers. It shows that the function that maps any state φ to the tensor product of the state ψ transformed by F and the state φ transformed by G is a bounded and continuous linear transformation.
The lemma "valid_plan_mems" assumes that a sequence "as" is in the set of valid plans "PROB" and an action "a" is an element of the sequence "as". It shows that the action "a" is an element of the set "PROB".
, assuming for any 't' and 'u', if 't' and 'u' are coinitial in 'R' then 't' is followed by 'u', it shows that if 't' is an arrow in 'R', 'U' is an arrow, and the sources of 't' in 'R' are the same as the sources of 'U', then 't' is transitively followed by 'U'.
The lemma named "bprv_prv'" assumes that the formula "\<phi>" is in the set of all formulas and that "\<phi>" is provable in the system "bprv". It then shows that "\<phi>" is also provable in the system "prv".
The lemma named "cong_append" assumes that there is a sequence from T to U, T is residually equivalent to T', and U is residually equivalent to U'. It shows that the concatenation of T and U is residually equivalent to the concatenation of T' and U'.
The lemma "InvariantVarsFAfterApplyLearn" assumes the following conditions: 1. The conflict flag of the state is invariant with respect to the assignment of variables (getM state) and the clause (getC state). 2. The conflict flag of the state is true. 3. The set of clauses (getF state) is invariant with respect to the...
The hyperreal part of a hypercomplex number derived from a complex number is equal to the hyperreal number derived from the real part of the complex number.
If 'f', 'g', and 'h' are identities in the category 'D', and the source of 'f' is the target of 'g', and the source of 'g' is the target of 'h', then the composition of the functor 'F' applied to the composition of the associator of 'D' with 'f', 'g', and 'h', the functor 'Φ' applied to the pair 'f' tensor 'g' and 'h',...
Given the assumptions that 'a' is in the carrier 'R', 'b' is in the carrier 'R', 'p1' is on the curve defined by 'a' and 'b', 'p2' is also on the curve defined by 'a' and 'b', the addition of 'p1' and 'p2' is equal to the addition of 'p1' and the opposite of 'p2', and 'p1' is not equal to the opposite of 'p1', it can b...
The interval from r' to 0 with a step size of mod 0 is a subset of the interval from r to m with a step size of mod m if and only if the remainder of r' divided by m is equal to the remainder of r divided by m and r is less than or equal to r'.
If "f" is an element of the carrier of the semialgebraic set "SA m", and "C" is an element of the residue classes of "Qp" at "n", then the inverse of "C" with respect to "m" in "f" is semialgebraic at "m".
, assuming for any positive integer n, g(n) is equal to the sum of f(d) for all d that divide n, and n is a positive integer, it can be shown that f(n) is equal to the Dirichlet product of the Moebius function and g(n).
A polynomial with only one coefficient, which is 1, is equal to 1.
The superset of "x implies y" and "x implies z" is less than or equal to "x implies the superset of y and z".
Lemma: Let f be a function from natural numbers to real numbers. Assume the following: 1. f is summable. 2. g is an injective function. 3. For all x, f(x) is greater than or equal to 0. Then, we can show the following: a. The composition of f and g (f ∘ g) is summable. b. The sum to infinity of (f ∘ g) is less than or...
If the difference between a natural number 'k' and a set 'I' is not an empty set, then the 'n'th element of the difference between 'k' and 'I' is equal to 'k' minus the 'n'th element of the set 'I' that is less than or equal to 'k'.
If the tuple (M,l,s,0,ll,r) is an element of MStep, then r is equal to s and ll is equal to l.
If "I" is an interpretation, then for all "x" and for all "i", the derivative of the function "i" at point "x" in the interpretation "I" exists and is equal to the Frechet derivative of the function "i" at point "x" in the interpretation "I".
In a given context G, if a state s transitions to a state s' through an intermediate term t, then there exists a value function vf such that the intermediate term is equal to vf and the state s transitions to state s' through vf.
The lemma "rem_effectless_works" is defined with fixed variables s, A, and as. It assumes the following: 1. The execution plan of 'as' starting from state 's' is the same as the execution plan of 'rem_effectless_act as' starting from the same state 's'. 2. If the preconditions for 'as' are satisfied starting from stat...
If "Phi" is finite, then the logical formula of "NFAND Phi" is equivalent to the logical formula of every element in "Phi".
If the set S2 is a subset of the set S, then the restriction of the environment rho1 to the set S, overridden on the set S2 by the environment rho2, is equal to the environment rho1 overridden on the set S2 by the environment rho2.
If a function 'f' tends to a limit 'l' at infinity, then the function 'f' when applied to a sequence 'n' also tends to the limit 'l'.
The lemma "HStartBallot_blocksOf_q" assumes that the action "HStartBallot" transitions from state s to state s' for a process p. It also assumes that process p is not equal to process q. Under these assumptions, it shows that the set of blocks for process q in state s' is a subset of the set of blocks for process q in ...
If the list "xs" is not empty, then the logical OR operation applied to the concatenation of lists "xs" and "ys" is equal to the logical OR operation applied to the list formed by prepending the result of the logical OR operation on "xs" to the list "ys".