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The interval "I" shifted by "k" is finite if and only if the interval "I" is finite.
For any three points A, B, and C in a real inner product space, if A-C is orthogonal to B-C, then the square of the distance between B and C plus the square of the distance between C and A equals the square of the distance between A and B. This is a formal statement of the Pythagorean theorem.
If a machine "M" is observable and minimal, and "q1" and "q2" are different states in this machine, then the distinguishing sequence from the observable finite state machine tables for "M" from "q1" to "q2" is the same as the distinguishing sequence from "q2" to "q1".
If 2 raised to the power of the sum of m and n is not equal to zero, then 2 raised to the power of m is not equal to zero and 2 raised to the power of n is not equal to zero.
"l equals the root node if and only if for all l', l is not in the children of l'".
A method declaration in the Java programming language is well-formed with respect to a program P and a class C, and the method M with parameters types Ts, return type T, parameter names pns, and body of the method, if and only if the following conditions are met: 1. The number of parameter types is equal to the number...
The adjoint of the product of a scalar 'a' and a complex matrix 'A' is equal to the product of the scalar 'a' and the adjoint of the complex matrix 'A'.
The lemma "Lambert_W'_gtI" assumes that the product of "y" and the exponential of "y" is greater than "x", "y" is less than or equal to -1, and "x" is greater than or equal to the negative exponential of -1. If these conditions are met, it shows that the derivative of the Lambert W function of "x" is greater than or eq...
The lemma, marked for quotient respect, demonstrates three things: 1. The 'permute' function preserves equality and transforms 'alpha_abs_set' in a way that respects the 'alpha_abs_set' relation. 2. The 'permute' function preserves equality and transforms 'alpha_abs_res' in a way that respects the 'alpha_abs_res' rela...
The mapping of the natural logarithm function (ln) at positive infinity (at_top) is equal to positive infinity (at_top) for real numbers.
For any multisets M, N, and K, the union of M and K is equal to the union of N and K if and only if M is equal to N.
The quantity conversion of the composition of two conversions (c1 and c2) applied to a quantity x is equal to the quantity conversion of c1 applied to the quantity conversion of c2 applied to x.
If the set V is a subset of Vs, the support of alpha and beta are subsets of the set of atoms of Vs, and if the premises are that H proves the PfP of the substitution of the floor of alpha in V and F, and the quote of all p in V is a subset of H, then it shows that H proves the PfP of the substitution of the floor of a...
The lemma named "insertion_irrelevant_vars" is about a fixed polynomial 'p' of type 'a', where 'a' is a commutative ring. The lemma assumes that for any variable 'v' in the variables of 'p', the function 'f' at 'v' equals the function 'g' at 'v'. Under this assumption, the lemma shows that the insertion of 'f' into 'p'...
If from the negation of F and the negation of G, we can derive a contradiction under the assumptions in Gamma, then from the negation of (G or F), we can also derive a contradiction under the same assumptions in Gamma.
If the function body "bdy" is associated with the function "p" in the context "Γ", and if executing "bdy" with the initial state "init s" results in a "Stuck" state after "n" steps, then executing the function call "call init p return c" with the normal state "s" will result in a "Stuck" state after "n+1" steps.
Given the assumptions that the program P is well-formed (as stated by "wwf_prog P"), the program P under environment E evaluates the expression e with heap h and local variables l to the expression e' with heap h' and local variables l' (as stated by "P,E \<turnstile> \<langle>e,(h,l)\<rangle> \<Rightarrow> \<langle>e'...
The conditional probability mass function of the first element after mapping a function 'f' to the first element of a stochastic process 'p' and then conditioning on 'f x', is equal to the conditional probability mass function of the first element of 'p' conditioned on 'x'. This holds true if 'f' is an injective functi...
If a function 'g' is continuous on a set 's' and for every element 'xa' in 's', 'g xa' is in the existence interval of 't0' and 'x0', then the flow of 't0', 'x0' and 'g xa' is continuous on 's'.
Zero multiplied by any quantity 'x' equals zero.
The lemma "dgrad_p_set_closed_uminus" assumes that the polynomial "p" belongs to the set of polynomials of degree "d" with monomial order "m". It shows that the negation of the polynomial "p" also belongs to the set of polynomials of degree "d" with monomial order "m".
The lemma "fv_disj_Fun_subterm_param_cases" assumes that the intersection of the free variables of term 't' and set 'X' is empty, and that the function 'f' with parameters 'T' is in the subterms of 't'. It shows that either 'T' is an empty list or there exists a parameter 's' in the set 'T' that is not in the image of ...
The star of y times the star of x is less than or equal to the star of x times the star of y if and only if the star of the sum of x and y is equal to the star of x times the star of y.
There exists a value 'v' and a 'heap'' such that when function 'f' is executed with 'heap', it returns 'Some (v, heap')' and 'P heap'' holds true.
If the set of elements 'xs' is a subset of 'V', and the set of edges in the walk of 'xs' is a subset of 'E', and 'xs' is not an empty set, then 'xs' is a walk.
The combination of two Red-Black Tree (RBT) Mappings 't' and 'u' using a function 'f' is equal to the following: If the identity comparison of 'a' is None, then the program will abort with the message "combine RBT_Mapping: ccompare = None", otherwise, if there is some comparison, it will return a new RBT Mapping that j...
A binomial heap is formed by a single node with rank 0, a value x, and an empty list of children.
In the context of a group, the lemma named "pow_int_mod_ord" assumes that "a" is an element of the carrier group "G" and the order of "a" is not zero. It shows that the power of "a" raised to an integer "n" is equal to the power of "a" raised to the modulus of "n" and the order of "a".
The lemma named "sparse_mult_eq_sum_lookup" is defined with a fixed list "xs" of pairs, where each pair consists of a natural number and an element of a commutative semiring. It assumes that for all pairs in the list, the natural number is less than a certain value "M", and that the first elements of the pairs in the l...
If "f" is a semialgebraic function of "n" and "g" is an element of the carrier of "SA n", then the sum of "f" and "g" in the context of "SA n" is also an element of the carrier of "SA n".
If action G is enabled, then either action F or action G is enabled.
If the type of a value 'v' is 'Some T', then the type of 'v' in the context of 'h' is also 'Some T'.
The lemma "lift_instrs_all_not_jump_not_return" assumes that the function "lift_instrs" with parameters F, L, ret, N, Σi, and xs returns a pair (Σo, ys). It shows that all elements in the list xs are neither Inca jumps nor Inca returns if and only if all elements in the list ys are neither Ubx jumps nor Ubx returns.
The predicate "tpOfV_pred" holds for "tpOfV".
Given the assumptions that 'a' and 'b' are elements of the carrier 'R', 'p1', 'p2', and 'p3' are on the curve defined by 'a' and 'b', 'p1' and 'p2' are generic points, 'p2' and 'p3' are tangent points, the point resulting from the addition of 'p1' and 'p2' is generic with 'p3', and 'p1' is tangent to the point resultin...
: Assuming 1) "G" is a feedback graph, 2) there is a relation between "Gi" and "G" in the extended implementation relation of "Rm" and "Id", 3) the cycle checker code for "Gi" returns "x", it can be shown that "x" is equivalent to the statement "G" is not acyclic when considering the intersection of the edge set of "G...
: Assuming that if the gas in the state is less than or equal to the cost of transferring an address with an expression, then the gas in the state is thrown, otherwise, the gas in the state is reduced by the cost of the transfer. This is equal to the normal state. And assuming that the expression of the address, expr...
The lemma named "eval_fds_inverse" is defined with a fixed 's' of type 'a. 'a' is a natural power, real normed field, Banach space, and second countable topology. The lemma assumes that the absolute value of the formal Dirichlet series 'f' converges at 's', the inverse of 'f' also converges at 's', and the first term o...
Applying the zero function from the algebraic entity zero function set to any element x results in zero.
The right fold operation with the truth-or operator and false as the base case is equivalent to the 'the_or' operation.
"if a real number x is greater than or equal to zero, then 1 minus x is less than or equal to 1 divided by the sum of 1, x, and the square of x."
The lemma "construct_lhs_vector_drop_consistent_R" is defined with the following parameters: a real polynomial "p", a list of real polynomials "qs", a pair of lists of natural numbers "I", and a list of lists of rational numbers "signs". It assumes that the polynomial "p" is not zero, the "signs" are distinct, and th...
If the assumption that "a" does not fail implies that "a" is refined by "c" under the heap node refinement relation "R" with preconditions "Γ" and postconditions "Γ'", then "a" is refined by "c" under the heap node refinement relation "R" with preconditions "Γ" and postconditions "Γ'".
The lemma named "Gamma_times_zeta_has_integral" assumes that the real part of z is greater than 1. It shows that the function which maps x to x raised to the power of (z - 1) divided by (the exponential of x minus 1) has an integral from 0 to infinity that equals the product of the Gamma function of z and the Zeta func...
For any functions f, g, and h, the suprema of f and g combined with h is equal to the suprema of f and the combination of g and h.
If p equals p', then the pair consisting of the fp0_state_impl with ppath equal to p and the fp0_state with ppath equal to p' is in the fp0_erel relation.
The lemma named "mono_instantiate" assumes three conditions: 1. The function "applies_eq A" is monotonic. 2. The element "x" is in the set that "applies_eq A" maps from the pair of "Delta" and "s". 3. "Delta" is smaller than or equal to "Delta'" according to the "smaller_interp" relation. If these conditions are met,...
The lemma named "sd_path_id" assumes that the pair (x,y) and its reverse (y,x) are not elements of the set p. Under these assumptions, it shows that the semi-directed path from x to y in the weighted graph w with path p is equal to p itself.
"x is a limit point of set S if and only if for all positive numbers e, the intersection of set S and the closed ball around x with radius e is infinite."
The lemma "distinguishing_transitions_right_observable_against_distinguishing_transitions_right" assumes that both "t1" and "t2" are in the set of distinguishing transitions to the right of states "q1" and "q2" in machine "M". Furthermore, it assumes that the source of "t1" is the same as the source of "t2", the input ...
The lemma named "convex_affine_rel_frontier_Int" is about sets S and T in a Euclidean space of dimension n. It assumes that S is a convex set, T is an affine set, and the intersection of the interior of S and T is not empty. Under these assumptions, it shows that the relative frontier of the intersection of S and T is ...
If two lists 'xs' and 'ys' are well-formed lists of vertices, and for every pair of vertices 'v1' from 'xs' and 'v2' from 'ys', the intersection of the sets of 'v1' and 'v2' is empty, then the result of merging 'xs' and 'ys' using a comparison function 'cmp' is also a well-formed list of vertices.
The function "insert1" in combination with the "RETURN" function and the "insert" function is included in the identity function mapped to the "gap_rel" function, which in turn is mapped to the "nres_rel" function within the context of "gap_rel".
If the prices of a market asset form a Borel-adapted stochastic process with respect to a filtration F, and if a portfolio follows a trading strategy that is supported and adapted to the market, then the remaining quantity of the asset in the portfolio forms a Borel-predictable stochastic process with respect to the sa...
There exists a primitive root, which is a modulo ring of a prime number, such that the primitive root raised to the power of (p-1) equals 1, the primitive root is not equal to 1, and the residue of the primitive root with respect to p (converted to an integer from the modulo ring) is a primitive root.
The lemma named "poincare_line_inversion" assumes that "z" is not equal to "w". It shows that the inversion of "z" is on the Poincare line of "z" and "w", and the inversion of "w" is also on the Poincare line of "z" and "w".
The lemma "mem_read_mem_write_separate" assumes that "a" and "si" are separate from "a'" and "si'". It shows that the memory read operation on the updated state, where the update includes a memory write operation at address "a" with size "si" and value "v", is the same as the memory read operation on the state with oth...
The infimum (Inf) of the upward closure (\<up>) of a set is equal to the identity function (id).
The content and primitive part of the polynomial (4 * X * Y^2 * Z^3 + 6 * X^2 * Y^4 + 8 * X^2 * Y^5) are 2 and (2 * X * Y^2 * Z ^ 3 + 3 * X^2 * Y ^ 4 + 4 * X^2 * Y ^ 5) respectively.
The lift is always moving down.
If the list representation of array 'xs' is equal to the list representation of array 'ys', then array 'xs' is equal to array 'ys'.
The second element of the function atom derived from a Herbrand atom is a ground term.
If the security invariant "sinvar" holds for a given network policy "nP" on a graph "G", then the set of offending flows in the network policy "nP" on the graph "G" is empty.
The lemma "converged_cs_single" assumes that both "pi" and "pi prime" are paths, and that "l" is less than "m". It also assumes that the "m"th element of the sequence "cs" along the path "pi" is equal to the "m prime"th element of the sequence "cs" along the path "pi prime". Furthermore, it assumes that the 0th element...
The diamond operator applied to "x" and the negation of the interior of "y" is equal to the negation of the interior of the product of "x" and "y".
The Legendre symbol of 'a' and 'p' is congruent to 'a' raised to the power of '(p - 1) divided by 2', modulo 'p'.
The pre-processed term of the difference between variables v and t is equal to the difference of the pre-processed term of v and t.
The application of a bounded bilinear operator is itself a bounded bilinear operation.
The set of all reachable states from the initial state in the transition system defined by "delta" and "Sigma", for a given logical proposition "phi", is finite.
If a set H is a subset of the carrier M, and if there are three functions s, f, and g such that s maps any natural number less than or equal to n to the carrier R, and both f and g map any natural number less than or equal to n to the set H, and for all natural numbers less than or equal to n, f and g map to the same e...
For all 'a', if 'a' is not a discriminator of 'Extra a', and if there is no negated discriminator in 'm', then there will be no negated discriminator in the abstraction of the primitive 'm' with the selection function 'sel_f'.
If set A is a subset of set B, then the set resulting from subtracting set C from set A is a subset of the set resulting from subtracting set C from set B.
If a face 'f' is an element of the face set 'g' and 'f' is not final, then the cardinality (or size) of the set of final faces after making 'f' final is equal to the cardinality of the set of final faces in 'g' plus one.
The lemma named "asEnv_pickE" assumes that "rho" is a good environment. It shows that the function "asEnv" applied to the result of the function "pickE" with "rho" as an argument, and then applied to "xs" and "x", is equal to the function "rho" applied to "xs" and "x".
If the tuple (y, cl, var_k, var', v) is an element of the set y_cl_var_var'_v_list, then the tuple (x_var var_k, x_var var') is an element of the set obtained by mapping the function (which takes a tuple (y, cl, var, var', v) and returns a tuple (x_var var, x_var var')) over the set y_cl_var_var'_v_list.
: Assuming that: - A system P is secure with respect to an interference policy I and a domain map D (assumption A), - A value y is in the range of either function p or function q (assumption B), - The set of all values ys is a subset of the union of the ranges of functions p and q (assumption C), - A set Y is a subset...
If 'u' is a primitive root of 'r', then the primitive root of 'u' is 'r'. This is under the assumption that 'u' is a primitive root of 'r'.
The lemma named "untame_negFin" assumes three conditions: "inv g", "final g", and "tame g". If these conditions are met, it shows that "g" is tame. In other words, if "g" is an inverse function, if "g" is final, and if "g" is tame, then it can be shown that "g" is indeed tame.
If the function "input_stored_in_reg_aux" applied to the parameters t', t, mr, and mi returns Some (i, r), then t' is a generalisation of t with respect to i and r.
A generative probabilistic value (gpv) is well-typed according to an interface (\<I>) if and only if the set of outputs from the gpv according to the interface is a subset of the set of outputs defined by the interface.
If set S is a retract of set T, then set S is a subset of set T.
The lemma "Lim_bounded2" is defined with a function 'f' that maps natural numbers to a type 'a' in a linear order topology. It assumes that the function 'f' converges to 'l' and for all natural numbers greater than or equal to 'N', the function 'f' at 'n' is greater than or equal to 'C'. It shows that 'l' is greater th...
If 'h' is a bijective function, then the image of the rename function 'h' applied to the constrains relation between sets 'A' and 'B' is equal to the constrains relation between the image of 'A' under 'h' and the image of 'B' under 'h'.
A set T is in the comparables of S and M if and only if T is finite and either T is an initial segment of S or S is an initial segment of T, and T is a subset of the union of S and M.
'P' is less than or equivalent to 'Q' under the 'LT' relation with respect to 'TRel'. The lemma then shows four things: 1. For all 'SP' and 'SQ', if 'SP' is in the source of 'P' and 'SQ' is in the source of 'Q', then 'SP' is equal to 'SQ'. 2. For all 'SP' and 'TQ', if 'SP' is in the source of 'P' and 'TQ' is in the ...
If a set S is homotopy equivalent to a set T, and there are two functions f and g that are continuous on a set U and map U into T, then if for any two functions that are continuous on U and map U into S are homotopic, then the functions f and g are also homotopic. Here, the sets S, T, and U are sets of real normed vect...
The lemma named "progress_L0" assumes that a transition from a state "s" with values "vs" and expressions "es" to a state "s'" with values "vs'" and expressions "es'" exists under the context "i". It also assumes that "cs" is a list of constants. It shows that a transition from a state "s" with values "vs" and expressi...
The lemma named "unrest_var_res_diff" under the category "unrest" assumes that "x" is independent of "y". It shows that "y" does not affect the restriction of "P" to "x".
The difference of the polynomial maps of 'f' and 'g' with respect to 'xs' is equal to the polynomial map of 'f' with respect to 'xs' minus the polynomial map of 'g' with respect to 'xs'.
The semantic environment is a complete cupcake environment.
The theorem named "msgrel_imp_eq_freediscrim" assumes that U is equivalent to V. It shows that the freediscrim of U equals the freediscrim of V.
The lemma named "sources_composite_of" assumes that "v" is a composite of "u" and "t". It shows that the sources of "v" are the same as the sources of "u".
Mapping a function 'f' over a set derived from a list 'xs' is equivalent to creating a set from the result of mapping and filtering the function 'f' over the list 'xs'.
The group G is equal to the subgroup generated by the carrier of G.
"QR_disc is greater than or equal to 0".
Binding any value "x" with "Ok" results in the original value "x".
A list starting with 'x' followed by list 'xs' is less than list 'ys' if and only if there exists an element 'y' and a list 'ys'' such that 'ys' is a list starting with 'y' followed by 'ys'', and either 'x' is less than 'y' or 'x' is equal to 'y' and 'xs' is less than 'ys''.
The domain of the left restriction of a relation 'r' to a set 'A' is equal to the intersection of the domain of 'r' and 'A'.
The function "swap a b" is injective (or one-to-one) on the set of keys of the map "m".
There exists a number 'n' such that 'n' is greater than zero and the absolute difference between 'x' and the natural logarithm of 'n' plus the square root of 'n' squared minus one, all divided by pi, is less than one-half.