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The theorem shows that unwinding implies non-interference unrelated.
If the function 'f' is continuous at 'F', then the function that applies the arctan operation to 'f(x)' is also continuous at 'F'.
For a function 'f' that maps from a complete Boolean algebra with a dual to another complete Boolean algebra with a dual, if 'f' composed with the dual operation equals 'g', then 'g' composed with the dual operation equals 'f'.
In the context of a probability space, the lemma "exponential_distributed_memoryless" assumes that a random variable X follows an exponential distribution with rate parameter l, and that a, l, and t are all non-negative, with l being strictly positive. It shows that the probability of X exceeding a value of a + t given...
A subpath from 0 to 1 of a path 'g' is equivalent to the path 'g' itself.
If a monadic computation "m" is less than or equal to a specification function "f" for all "x", then the bind operation of "m" and "f" is also less than or equal to the specification function "Phi".
: given the assumptions that a complex matrix U is unitary, and both U and B are elements of finite complex matrices, it shows that the measurement outcome projection of the jth measurement outcome, created by B and U, is also an element of finite complex matrices.
The mask of the successor of 0 is equal to 1, for any length word 'a.
: Assuming that for all locations 'l' and for all configurations 'cfg' in the intersection of the set of configurations on 'x' and 'X', where 'x' is in the preimage of 'u' under 'f', the measure of the action configuration in the preimage of 'u' under 'f' is less than or equal to 'r'. Also, assuming that 'r' is less t...
If for all elements 'x' in a sequence 'xs', a property 'P' holds, and if for any 'x', 'P x' implies that 'S x' is equivalent to 'R x', then the statement "always eventually 'S' holds in 'xs'" is equivalent to the statement "always eventually 'R' holds in 'xs'".
If there is a shortest path 'p' from point 's' to point 't' and the pair (u,v) is in the set 'p', and if there is another path 'p'' from 's' to 't' where the pair (v,u) is in the set 'p'', then the length of the path 'p'' is greater than or equal to the length of the path 'p' plus 2.
A shadow root pointer cannot be cast to a node pointer kind.
: Assuming there is a transition 'x' from the initial state 'root' to the final state 'root'', and the initial state 'root' is an environment with attributes 'att' and directory 'dir', then there must exist attributes 'att' and directory 'dir' such that the final state 'root'' is also an environment with these attribut...
If a natural number 'a' is less than or equal to another natural number 'b', and the remainder of 'a' divided by 'm' is greater than the remainder of 'b' divided by 'm', then the quotient of 'a' divided by 'm' is less than the quotient of 'b' divided by 'm'.
The set of suffixes of a single element set {xs} is equal to the set of suffixes of the element xs.
The lemma named "cop4_perp_in2__col" assumes that points X, Y, A, A' are coplanar, points X, Y, A, B' are coplanar, points X, Y, B, A' are coplanar, points X, Y, B, B' are coplanar, point P is perpendicular to line AB at points X and Y, and point P is also perpendicular to line A'B' at points X and Y. Given these assum...
If "k" is greater than 0, then taking the first "n * k" elements from the stream "f" and then shrinking it by "k" is equivalent to first shrinking the stream "f" by "k" and then taking the first "n" elements.
The evaluation of the expression "e" with type "t1" in a given environment "env" and state "st" is equal to the evaluation of the expression "e" in the same environment and state.
If "a" is greater than or equal to 1, then the sequence of "a" raised to the power of the increasing sequence of "x" is a bounded sequence.
In the context of Jozsa's algorithm, assuming that the function is balanced, the probability of the first 'n' qubits in Jozsa's algorithm being 0 is shown to be 0.
: Given a key 'k' of type 'a', if 'k' is less than all keys in trees 'a' and 'b', and 'k' is less than 'x', then 'k' is also less than all keys in the tree resulting from balancing the left side of the tree with 'a', 'x', 't', and 'b'.
If the function "remdups_fwd_acc" applied to the accumulator "Acc" and the list "xs" equals the concatenation of lists "ys" and "zs", then there exist lists "us" and "vs" such that "xs" equals the concatenation of "us" and "vs", the function "remdups_fwd_acc" applied to the accumulator "Acc" and "us" equals "ys", and t...
The count space of a singleton set {x} is equal to the return of the count space of the same singleton set {x} when applied to x.
The length of the list 'xs' being greater than 0 is equivalent to the list 'xs' not being empty.
The lemma named "solve_ratfps" assumes that the solution of the rational function "f" is "sol". It then shows that the absolute value of the function "interp_ratfps_solution" applied to "sol" is equal to the function "fps_of_ratfps" applied to "f".
The lemma named "theta_conv_MM_integral" assumes that "x" is greater than or equal to 2. It shows that the Mellin transform "MM" has an integral from 2 to "x" that equals the product of "MM" and "x" minus the theta function of "x".
The lemma named "moebius_dilatation_uminus" assumes that "a" is greater than zero. It shows that the negative of the Moebius dilatation of "a" is equal to the Moebius dilatation of "1/a".
"x is less than or equal to y if and only if the converse of x is less than or equal to the converse of y".
If a variable 'ox' is an optional element of a set 'S', and for all elements 'x' in 'S', if 'ox' equals 'Some x' then the function 'f' applied to 'x' is in 'S', it follows that the function 'f' mapped to the optional 'ox' is an optional element of the set 'S'.
The lemma named "pderiv_coeff" assumes that "K" is a subring of "R" and "f" is an element of the carrier of "K[X]". It shows that the coefficient of the polynomial derivative of "f" at "k" is equal to the integer embedding of "R" at "k+1" times the coefficient of "f" at "k+1".
The lemma named "mk_linear_orders_total_on_decisive_on" assumes three conditions: 1. The relation 'r' is in the set of linear orders 'C' on 'A'. 2. The set 'A' is a subset of the set 'r'. 3. The function 'f' satisfies the independence of irrelevant alternatives (iia) condition on 'A'. If these conditions are met, the...
If the equivalence relation "eq\<rho>" holds for the arity environments "ae", "\<rho>1", and "\<rho>2", and if "ae" at point "x" equals the least upper bound of "a", then "a" is equivalent to "\<rho>1" at point "x" and "\<rho>2" at point "x".
The binary to bit-list conversion of the concatenation of two binary numbers 'w' and 'v', with lengths 'nw' and 'nv' respectively, is equal to the binary to bit-list conversion of 'v' with 'nv' bits, appended to the binary to bit-list conversion of 'w' with 'nw' bits.
The function "r01_to_r01_r01" is measurable with respect to the product of the real Borel sigma-algebra and the real Borel sigma-algebra, given that its domain is the real Borel sigma-algebra.
The lemma named "well_founded_extend" is about a fixed binary relation "less" denoted by "<". It assumes three things: 1. The relation "<" is well-founded on set A. 2. The relation "<" is well-founded on set B. 3. For all elements 'a' in set A and 'b' in set B, 'b' is not less than 'a'. Given these assumptions, it sh...
If under the program P and environment E, the expression e with state s can reduce to expression e' with state s', then under the same program P and environment E, the expression e accessing field F with a sequence of classes Cs and state s can reduce to the expression e' accessing the same field F with the same sequen...
For a complete tree with a degree greater than or equal to 1, if a value 'x' is a member of both the minimum and maximum options of the tree, then 'x' is either a member of both the minimum and maximum options of the child tree at the index calculated by the high function of 'x' and half the degree, or 'x' is equal to ...
For a graph G, if G is a graph and G is connected, then the number of arcs in G is greater than or equal to twice the number of vertices in G minus one.
The function "isolate_variable_sparse" applied to a monomial (represented by "MPoly_Type.monom m a") with respect to a variable "v" and an integer "i" is equal to the following: if the coefficient "a" is zero or the lookup of "v" in the monomial "m" is not equal to "i", then the result is zero; otherwise, the result is...
Applying the lifted af_letter function to X, nu, and a pair of abstract values (Abs x, Abs y) is equivalent to a pair of abstract values where the first element is the abstraction of the first element of the result of applying the af_letter function to X, the pair (x, y), and nu, and the second element is the abstracti...
The transpose of the product of two square matrices A and B (where A and B belong to a commutative semiring) is equal to the product of the transpose of B and the transpose of A.
If the result of removing duplicates from a list "dl" is a list starting with "h" followed by list "t", then "h" is not an element of the set of elements in "t".
The lower bound of the negation of a polynomial 'p' with respect to a variable 'v' is equal to the negation of the lower bound of the polynomial 'p' with respect to the same variable 'v'.
If "b implies a" is left-implied by "a", then it right-implies "a", and this is equivalent to "b implies a".
"y times x, left-residual of x equals y" is equivalent to "there exists a z such that y equals z left-residual of x".
The lemma assumes that a lasso from state 's' equals 'Some (pr,pl)'. It shows four things: 1. The loop 'pl' is not empty. 2. The head of the loop 'pl' is accepted. 3. The loop 'pl' is a path in the graph 'E' that starts and ends at the head of 'pl'. 4. There exists a vertex 'v0' in the initial set of vertices 'V0' suc...
Given a polynomial 'p' with a square-free factorization represented by (c, bs), where 'c' is a constant and 'bs' is a list of pairs (b, i) representing the irreducible factors 'b' raised to the power 'i'. If for every pair (b, i) in 'bs', the function 'f' applied to 'b' results in a pair (d, fs), where 'd' is a constan...
If 'h' is an element of 'H', 'H' is a subset of the carrier 'G', and 'x' is an element of the carrier 'G', then the sum of 'h' and 'x' is an element of the right coset of 'H' in 'G' with respect to 'x'.
The lemma named "cross_alt_coprime_0" assumes that the polynomials p and q are coprime and that either p or q is equal to zero. Under these assumptions, it shows that the cross-alternation of p and q between the points a and b is equal to zero.
The lemma "cancels_to_preserves_generators" assumes that "l" cancels to "l'" and "l" is in the list of pairs that includes all possible elements and generators. It shows that "l'" is also in the list of pairs that includes all possible elements and generators.
The lemma named "profile_swapAltsP" assumes that there is a profile P for a set A, and that the set containing elements a and b is a subset of A. It shows that swapping elements a and b in profile P still results in a valid profile for set A.
N is greater than zero.
The uncurrying of a function, where the function is defined as a lambda function that takes a pair of arguments (ps, cf) and applies the function f to them, is equivalent to the uncurrying of the function f when it is treated as a function of two arguments.
In the program "tprg", the class declaration "Base" is well-formed with respect to the "Object" class.
The function "f" tends towards negative infinity if and only if for every real number "B", there exists a natural number "N" such that for all natural numbers "n" greater than or equal to "N", the extended real representation of "B" is greater than or equal to the function "f" at "n".
The lemma "SubsetGuards" assumes that command 'c' is a subset of command 'c'' with respect to guards. It also assumes that under the context of program state 'Γ' and function specifications 'Θ', command 'c'' ensures that if precondition 'P' holds before execution, then either postcondition 'Q' holds after execution or ...
The cardinality of 'a' is greater than or equal to 2. This implies that if there's only one action, the whole situation is irrelevant. The second assumption, labeled "ms", states that there exists a function 'ff' such that the matcher 'γ' accepts the packet 'p' for any match condition. This implies that if our matcher ...
If for every element 'x' in set 'u', the function 'f' applied to 'x' results in an element in set 'U', then the function 'Lambda' applied to set 'u' and function 'f' is an element of the function space from 'u' to 'U'.
An expression "P" is equivalent to the same expression "P" with an unbound existential quantifier over "x".
The lemma named "LIST_FOREACH'_param" shows that the function "LIST_FOREACH'" is related to itself under certain conditions. These conditions are defined by a relation that involves a nested list relation of "Rv", a non-deterministic result relation, a boolean relation on "R" and "sigma", a relation between "Rv" and "R...
The map of a finite function 'xs' belongs to the set of finite functions.
The lemma named "generating_card_n_basis" assumes three conditions: 1. The set X is a subset of the carrier vector of dimension n. 2. The carrier vector of dimension n is a subset of the span of X. 3. The cardinality (or size) of set X is n. If these conditions are met, it shows that X is a basis.
The execution preserves the type, given that the step function is applied to the size of the instruction sequence and A.
: if for all 'k' less than or equal to 'n', 'k' is greater than 0 implies there exists a 'c' such that 'v' of 'c' equals 'k', and for all 'c', 'v' of 'c' is less than or equal to 'n' and 'v' of 'c' is greater than 0, then the Floyd-Warshall algorithm applied to matrix 'M' and 'n' under 'v' and 'n' is empty if and only ...
The lemma named "eval_e_uniqueness" is about a fixed expression 'e' of type 'ce'. It assumes that the evaluation of expression 'e' in the context 'i' results in the state 's' and also in the state 's''. It then shows that 's' must be equal to 's''.
For a function 'f' mapping from the product of sets X and Y to the set of real numbers, the integral of 'f' with respect to the product measure of the quasi-Borel probability spaces (X, α, μ) and (Y, β, ν) is equal to the integral of the function 'f' composed with the product map of α and β, with respect to the product...
If "x" is an element of the prediction of "Y" at position "k", then the end of the item "x" is "k" or "x" is an element of "Y".
If "m" is less than or equal to "n", then the greatest common divisor of the "m-th" Fibonacci number and the Fibonacci number at the position "n - m" is equal to the greatest common divisor of the "m-th" Fibonacci number and the "n-th" Fibonacci number.
Comparing a double value "x" with itself will always result in 0.
The successor of a specific extended natural number, represented as "numeral k", is equal to the numeral of "k + 1".
If set A is a subset of the identity relation, then the pair (f, f) is in the function refinement relation of A to the identity relation, given the precondition P.
If b is approximately equal to c, then the sum of d and b is approximately equal to the sum of d and c.
The floor function of "x" in base 3 is equal to the floor of the real number representation of "x" in base 3.
If the preferences from a table are well-formed for a given set of agents and alternatives, then the preference profile generated from this table is also well-formed for the same set of agents and alternatives.
If a pair (a, b) is in the reflexive transitive closure of a relation r, and both a and b are elements of a set F, and it is consistent that for any pair (a, b) where a is in F and (a, b) is in the relation r, b is also in F, then the pair (a, b) is in the reflexive transitive closure of the relation r on the set F.
The lemma named "val_ring_res" assumes that "a" and "b" are elements of the ring of p-adic integers. It shows that the difference of "a" and "b" in the p-adic integers, when mapped to the ring of p-adic integers modulo N, is equal to the difference of the images of "a" and "b" under the same mapping.
The lemma named "uminus_key_order_sameD" assumes that the negation of a key order 'ko' is equal to 'ko' itself. Under this assumption, it shows that 'x' is equal to 'y'. Both 'x' and 'y' are of the same type 'a'.
The lemma named "inj_on_impl_disjoint_family_on_singleton" assumes that the function "f" is injective on the set "A". It shows that this leads to a disjoint family on the set "A" when each element "x" in "A" is mapped to a singleton set containing "f(x)".
The set of first order variables in the normalized form of a formula is equal to the set of first order variables in the original formula.
The domain of the function 'clearjunk0' applied to 'xs' is equal to the domain of the function 'xs' excluding the set of 'x' for which the lookup of 'x' in 'xs' equals 'Some 0'.
The lemma named "map2set_insert" under the assumption of "PRIO_TAG_GEN_ALGO" and "GEN_OP i op_map_update (Rk -> Id -> <Rk,Id>R -> <Rk,Id>R)" shows that the pair "(map2set_insert i, Set.insert)" is in the relation "Rk -> <Rk>map2set_rel R -> <Rk>map2set_rel R".
If N is a normal subgroup of G and x is an element of the carrier of G, then the projection of x in G with respect to N is equal to the coset of N containing x.
"F is equivalent to G in v" if and only if "F equals G".
The lemma named "part_permutes" assumes that the effect of the function "part1" with parameters "a", "l", "r", and "p" changes the state from "h" to "h'" and results in "rs". It shows that the multiset of the array "a" after the state change "h'" is equal to the multiset of the array "a" before the state change "h".
: Assuming that: 1. If the invariant I' holds for state s', then the pair (s, s') is in the relation R'. 2. For any pair (s, s') in the relation R', if the invariant I' holds for state s', then the invariant I holds for state s. 3. For any pair (s, s') in the relation R', if the invariants I' and I hold for states s' ...
If an element 'b' is in the image of a set 'A' under a relation 'r', then there exists an element 'a' such that 'a' is related to 'b' under 'r' and 'a' is an element of 'A'.
The lemma named "behead_isnpolyh" assumes that "p" is a normalized polynomial with respect to "n". It shows that the result of the operation "behead" on "p" is also a normalized polynomial with respect to "n".
The lemma named "fvi_Suc_bound" assumes that for all 'i' in the function 'fvi' applied to the successor of 'b' and 'phi', 'i' is less than 'n'. It shows that for all 'i' in the function 'fvi' applied to 'b' and 'phi', 'i' is less than the successor of 'n'.
The composition of C with the infinite power of C equals the infinite power of C.
If the length of a word 'w' is less than or equal to 'n', then dropping 'n' bits from 'w' will result in zero. This is applicable for any word 'w' of any length 'a'.
The big Theta of a function that raises a real number x to the power of e is equal to the big Theta of the evaluation of the primary function (LnChain 0, real e).
If the intersection of P and g is a subset of P', and if the intersection of P and the negation of g is not empty implies that f is in F, and if under the context of Γ and Θ, P' a c leads to Q,A with respect to F, then under the same context, P with the annotated recursion of r and a, and the guard of f, g, and c, also...
The lemma named "inc_to_Zp" assumes that "b" is an element of the carrier set of the p-adic integers. It shows that the function "to_Zp" applied to the image of "b" under the function "iota" equals "b".
The lemma named "weak_reduction_correspondence_simulation_impl_coupled_simulation" is about a fixed relation 'Rel' of pairs of processes and a fixed process calculus 'Cal'. It assumes that the relation 'Rel' and the process calculus 'Cal' satisfy the property of weak reduction correspondence simulation. Under this assu...
For all time points 't' in a time interval 'I' multiplied by a constant 'k', a property 'P' holds for 't', if and only if for all time points 't' in 'I', the property 'P' holds for 't' multiplied by 'k'.
The lemma named "select_append'" assumes that for any elements 'y' and 'z', if 'y' is in the set 'ys' and 'z' is in the set 'zs', then 'y' is less than or equal to 'z'. It also assumes that 'k' is less than the sum of the lengths of 'ys' and 'zs'. Under these assumptions, it shows that selecting the 'k'-th element from...
The lemma named "pred_alt_def" has three parts: 1. The predicate "pred" applied to a value constructor "Vconstr" with a name and a list of values "vs" is true if and only if the predicate "Q" applied to the name is true and the predicate "pred" is true for all values in the list "vs". 2. The predicate "pred" applied ...
If 'a' is an element of the carrier of the p-adic numbers, and the valuation of 'a' is not infinity, then 'a' is an element of the non-zero p-adic numbers.
The theorem assumes that if it is not the case that a certain property Phi holds for some x, then there exists a y such that y has the property G with respect to x and y has property C. Given this assumption, it shows that g exists.
For any given state 's', 's' is less than or equal to itself.

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