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The statement is a lemma that fixes a function 'f' which maps a natural number to a real normed field, banach, and second countable topology. It assumes that the function 'f' is summable for the norm of 'f' and that 'f' is a completely multiplicative function. The lemma shows three things: 1. The absolute convergent p...
The set of process names in a given state space and process is finite.
For any positive integer p, 2 is congruent to 0 modulo p if and only if p equals 1 or 2.
For any two extended non-negative real numbers 'a' and 'b', if 'b' is not equal to infinity, then the result of subtracting 'b' from the sum of 'a' and 'b' is equal to 'a'.
: Assuming that: 1. A satisfies the invariant property that if I holds, then Q holds. 2. For any state s, if s is in the initial state of A, then property P holds for s. 3. For any states s and s', and action a, if s is reachable from A under condition I, and property P holds for s, and Q holds for the transition from...
Concurrent operations commute. It assumes that 'xs' is a prefix of 'i'. It shows that the happen-before relation of concurrent operations commutes when node delivers messages 'xs'.
If set A is not empty, then the minimum element of set A is in set A.
If 's' is equal to the sum of 'm' and 'n', and 'x' is less than or equal to the mask of 'n', then the left shift of 'x' by 'm' is less than or equal to the mask of 's'. This is for 'x' which is a word of any length 'a'.
The intersection of multiset A with the intersection of multisets B and C is equal to the intersection of multiset B with the intersection of multisets A and C. This demonstrates the left-commutative property of intersection operations on multisets.
If the ordered pair (as, bs) is an element of the lexicographic product of S, and if S is a subset of R, then the ordered pair (as, bs) is also an element of the lexicographic product of R.
If the list "p" with "a" as its first element satisfies the condition of "singleCombinators", and if "p" does not contain any "DenyAll" elements, then the list "p" with "a" as its first element will not contain any "DenyAll1" elements.
If matrix A has the same diagonal as matrix B, and matrix B is an eigenvalue block, then matrix A is also an eigenvalue block.
If "x" is an element in the set generated by the dense list of piecewise-defined expressions "X", then "x" is also in the range of the function that applies these piecewise-defined expressions to "X".
If a set S is finite, then the set of type variables in S is also finite.
If S is less than or equal to S' and c is less than or equal to c', then the step' of S and c is less than or equal to the step' of S' and c'.
There is an infinite number of elements 'a' in the sequence 'w' for which there exists a natural number 'k' less than 'n' that satisfies the property 'P' if and only if there exists a natural number 'k' less than 'n' for which there is an infinite number of elements in the sequence 'w' that satisfy the property 'P'.
The size of a multiset M from which a single element x has been removed is less than or equal to the size of the original multiset M.
The complementary error function of a real number 'x' is greater than 1 if and only if 'x' is less than 0.
If a state 's' satisfies the condition 'Enabled F', and for any state 'u' such that 'first s' is compatible with 'u' and satisfies 'F', then 'Q' is true.
If the function "input_stored_in_reg" with parameters t2, t1, and a returns a pair (i, r), then the pair (r, V (I i)) is an element of the set produced by the function "Updates" with parameter t2.
If a program P allows class C to see a method M with types Ts returning type T, which is equal to the body in class D, then the tauMove2 of the compiled program P2 and the state (which includes the exception, heap, stack, local variables, class, method, and program counter) is equal to either the tauMove2 of the progra...
For a polynomial 'p' from a conjugatable ordered ring with no zero divisors, the square of the norm of 'p' is greater than zero if and only if 'p' is not equal to zero.
If two monoidal categories, denoted as A and B, are defined over the same type alpha, and if their monoidal category, monoidal functor, unit object, associator, left unitor, and right unitor are all equal, then the two monoidal categories A and B are equal.
The lemma named "cycle_root_non_empty" assumes that "x" is not equal to zero. It shows that "cycle_root r x" is equivalent to "non_empty_cycle_root r x".
The value of an if-then-else expression 'b' under an assignment that sets the variable 'var' to the value 'val' is equivalent to the value of the expression 'b' restricted to 'var' equals 'val' under the original assignment.
The number of variables in the sequence that results from prepending the element "x" to the sequence "A" is equal to the number of variables in the original sequence "A".
The lemma named "gctxt_of_gpos_gterm_gsubt_at_to_gterm" assumes that "p" is in the set of possible positions of "t". It shows that the context at position "p" in "t", when applied to the subtree at position "p" in "t", equals "t" itself.
A pair (x, y) is in the set "less_than_bool" if and only if x is False and y is True.
If x is equal to y, then y is equal to x, in the context of v.
For any component 'c', given that 'k' is greater than 0, the transition function 'trans_fun' is deterministic with respect to the local state, and the local state is correctly defined with respect to the pointwise shrink function, then the accumulated local state after executing a stream of computations 'k' times with ...
For any extended real number 'x', the inverse of '-x' is equal to infinity if 'x' is zero, otherwise it is the negative of the inverse of 'x'.
Shifting a Dirichlet series that has been scaled by a factor of "c" is equivalent to scaling the shifted Dirichlet series by the same factor "c".
If a sequence "a" followed by "b" left-derives in one step at position "i" to sequence "c" using rule "r", and "j" is the leftmost position in sequence "a", then "i" must be equal to "j".
If an object "vo" is compatible with a heap "h", and heap "h" is less than or equal to heap "h'", then the object "vo" is also compatible with heap "h'".
The distribution of the "i-th" component of a random variable "f" in a measure space "M" over the entire countable set is equal to the measure of a Bernoulli distribution with probability "p".
The theorem "PfP_implies_SubstForm_PfP" assumes that if "PfP of y" is provable in theory H and "SubstFormP of x, t, y, and z" is provable in theory H, then it shows that "PfP of z" is also provable in theory H.
If a deque (double-ended queue) satisfies the invariant property, then the deque will still satisfy the invariant property after an element is enqueued at the left end.
For any state 'X', the operation 'oclIsModifiedOnly' applied to 'X' at a given state 'τ' is equivalent to the operation 'oclIsModifiedOnly' applied to the function that always returns 'X' at the same state 'τ'.
The lemma named "eq_monom_sum_var_list" assumes that "m" and "n" are inverse monomials. It shows that "m" is equal to "n" if and only if for all "x", the sum of the variable list of "m" evaluated at "x" is equal to the sum of the variable list of "n" evaluated at "x". This is denoted as "?l = ?r".
"p or (q and r)" is equivalent to "(p or q) and (p or r)".
If for all 'x', 's' equals 'Inl x' implies 'P', and 's' equals 'Inr x' implies 'P', then 'P' is true.
If r and s are both greater than zero, and if both f and g are composition series of the group G, then the operation of the first successor of f with the intersection of the initial elements of f and g equals the carrier of the group G.
The lemma shows the following: 1. There exists a non-negative constant c such that for all non-negative x, the difference between x times A(x) and T(x) is between 0 and c times x. 2. The function A(x) minus the natural logarithm of x is of the same order of magnitude as the constant function 1. 3. There exists a non...
, given a function 'f' mapping a set 'a to extended non-negative real numbers (ennreal) and a sequence 'A' of subsets of 'a, if 'f' is positive on a sigma-algebra 'M', 'f' is additive on 'M', 'f' is increasing on 'M', the range of 'A' is a subset of 'M', and 'A' is a disjoint family, then the sum of 'f' over all 'A' is...
If 'i' is less than the length of the list 'cs', then the length of the 'i'-th element of the second component of the function 'poly_pfd_simple' applied to 'p' and 'cs' is equal to the second component of the 'i'-th element of 'cs' plus one.
For any two hypermultisets 'm' and 'n', '1' is less than or equal to the product of 'm' and 'n' if and only if '1' is less than or equal to 'm' and '1' is less than or equal to 'n'.
The lemma named "guard_elims" has two parts: 1. If the guard function applied to a boolean 'b' equals 'Some x', then if 'b' implies a proposition 'P', 'P' is true. 2. If the guard function applied to a boolean 'b' equals 'None', then if the negation of 'b' implies a proposition 'P', 'P' is true.
If the term 't' is not dependent on any numeric variable (indicated by "numbound0 t"), then both the first part (indicated by "fst") and the second part (indicated by "snd") of the split integer of 't' are also not dependent on any numeric variable.
An empty list is in the concatenation of sets A and B if and only if the empty list is in both set A and set B.
For any three real numbers, the result of the multiplication is the same regardless of how they are grouped. In other words, multiplication of real numbers is associative.
If 'b' is not less than or equal to 'a', then the result of 'a' subtract 'b' is zero.
If a record containing the agents A and B, an email, a number AO, a clear text number, a nonce q, and S2TTP is received in the set of events, and these events are part of the certified mail, then S2TTP is in the analysis of what the spies know from these events.
The image of the set P under the function smap is equal to the PosetComplex of P.
A list of expressions 'e#es' is well-typed with types 'T#Ts' in a given program 'P', environment 'E', heap 'h', and static heap 'sh', if and only if the expression 'e' is well-typed with type 'T' and the rest of the expressions 'es' are well-typed with the rest of the types 'Ts' under the same conditions.
Assuming that "sl" is a well-formed formula path, all elements in "sl" are states, and the list of user commands in "sl" (excluding the first element) is "ucl", there exists a list "sl'" such that the first element of "sl'" is "ss'", "sl'" is a well-formed formula path, "sl'" (excluding the first element) is related to...
If 'i' is less than the length of the array 'a', then the sub-array of 'a' at the position 'i' converted to an integer is equal to the sub-array of 'a' at position 'i'.
The lemma named "less_restrictive_saturated" assumes two conditions: firstly, that rule R1 is less restrictive than rule R2, and secondly, that rule R1 is a saturated binary rule with respect to set S. Given these assumptions, it shows or proves that rule R2 is also a saturated binary rule with respect to set S.
The lemma named "higher_deriv_mult" is about complex numbers. It assumes that two functions 'f' and 'g' are holomorphic on a set 'S', which is open, and a complex number 'z' is in 'S'. It shows that the 'n'-th derivative of the function that multiplies 'f' and 'g' at 'z' is equal to the sum from 'i' equals 0 to 'n' of ...
The lemma named "gcd_bezout_sum_nat" is defined for a natural number 'a'. It assumes that the sum of 'a' times 'x' and 'b' times 'y' equals 'd'. It then shows that the greatest common divisor of 'a' and 'b' divides 'd'.
: given the assumptions that the set of variables 'vs1' is a subset of the domain of the finite map 'fma', the set of variables 'vs2' is a subset of the domain of the finite map 'fmb', 'fma' and 'fmb' agree on their common domain, the restriction of 'fma' to 'vs1' is a submap of 's', and the restriction of 'fmb' to 'vs...
The function tau has a real derivative, denoted as tau prime at a specific time t, within a certain range u.
The lemma named "image_poly_iso_incl" assumes that a polynomial "q" is in the carrier of the polynomial ring "L" and that the degree of "q" is greater than 0. It shows that the identity function is a ring homomorphism from "L" to the image of the polynomial "q".
If x, y, and z are real numbers such that each of them is greater than or equal to 2 and less than or equal to 4, then the value of twice the sum of the products of x and z, x and y, and y and z, minus the sum of the squares of x, y, and z, is greater than or equal to 12.
The lemma "FOREACHi_refine" is about refining a "FOREACHi" operation. It takes as input two sets 'S' and 'S'' and two functions 'f' and 'f'', along with a relation 'R' and two initial states '\sigma>0' and '\sigma>0''. It also takes two predicates '\Phi>' and '\Phi>'' and a function '\alpha>' that maps elements of 'S' ...
The lemma "run_propag_on_create_graph" assumes that sigma is a valid path in the graph of the generalized Büchi automaton (gba). It shows that the k-th element of sigma is in the set of states (qs) and the name of the k-th element of sigma is in the incoming set of the (k+1)-th element of sigma.
The less than or equal to function of the natural term order "to" is equivalent to the function that takes two inputs "u" and "v", and applies the less than or equal to function of the key order of the inverse of the natural term order "to" to "v" and "u".
"z plus x times the omega power of x equals the omega power of x plus the star power of x times z".
For a given square matrix 'A' of type 'a' in a commutative ring, 'A' is invertible if and only if the determinant of 'A' divides 1.
The set of phi definitions in a graph 'g' at a node 'n' is finite.
The lemma named "arg_rcis_cis" assumes that "n" is greater than zero. It shows that the argument of the complex number in rectangular coordinates (rcis) with parameters "n" and "x" is equal to the argument of the complex number in polar coordinates (cis) with parameter "x".
The symmetric closure of the significant step relation under a set of rewrite rules is equal to the significant step relation under the symmetric closure of the set of rewrite rules.
The lemma l1 assumes that 'a' is not a quadratic residue modulo 'p' and 'x' is an element of set S1. It shows that 'x' is not equal to the function of 'x' under f_1.
: Given the assumptions that: 1. A pair (x, x') is in a relation R, 2. For any pair (x, x') in the relation R, the condition function 'b' applied to 'x' is equal to the condition function 'b'' applied to 'x'', and 3. For any pair (x, x') in the relation R where the condition functions 'b' and 'b'' applied to 'x' and '...
: Assuming a well-formed interpretation 'i' for a given context 'Γ' and theory 'Θ', and assuming that both expressions 'e1' and 'e2' evaluate to the same state 's' under the interpretation 'i'. Also, assuming that the interpretation 'i' satisfies the condition that the value of variable 'x' is equal to 'e1'. Under thes...
The lemma named "ZX_is_gate" demonstrates that "ZX" is a gate of size 1.
If x is an element of set A, then x is not congruent to 0 modulo p.
For any real number 'x', the difference between the exponential of 'i times x' and the exponential of 'negative i times x' equals '2 times i times the sine of x'.
The lemma named "sqrt_int_pos" assumes a result "res" where the square root of an integer "x" equals a pair consisting of "s" and "ms". It shows that "s" is greater than or equal to 0.
The lemma named "multiplicity_gt_zero_iff" shows that the multiplicity of 'p' in 'x' is greater than zero if and only if 'p' divides 'x'.
The lemma named "mat_det_12_mm_r" shows that the determinant of the matrix product of A and M, and B and M (in that order) is equal to the product of the determinant of M and the determinant of the matrix product of A and B.
The minimal integer polynomial of the real number representation of 'x' is equal to the minimal integer polynomial of 'x'.
The application of a sequence of terms "ss" to a head term "zeta" is equal to the application of a sequence of terms "ts" to a head term "xi" if and only if "zeta" is equal to "xi" and "ss" is equal to "ts".
If a command 'c' is totally correct with respect to precondition 'P' and postcondition 'Q', and for all states 's', if 'Q' holds then 'Q'' also holds, then the command 'c' is totally correct with respect to precondition 'P' and postcondition 'Q''.
Expanding a "NoMsg" stream by a factor of "k" is equivalent to replicating the "NoMsg" stream "n" times and then multiplying it by "k".
: 1. If the substitution of a stateful strand step 'a' with a substitution 'σ' equals sending a term sequence 'ts', then there exists a term sequence 'ts'' such that 'ts' equals the substitution of 'ts'' with 'σ' and 'a' equals sending 'ts''. 2. If the substitution of a stateful strand step 'a' with a substitution 'σ...
The lemma named "discr0_raw_coind" assumes three conditions: 1. The condition "phi c" holds for some command "c". 2. For any commands "c" and "c'", and states "s" and "s'", if "c" must terminate in state "s", "phi c" holds, and there is a command transition from "(c,s)" to "(c',s')", then "s" is equivalent to "s'" and...
: given the assumptions that "lmax" is less than or equal to the length of list "l" and "i" is less than or equal to "lmax", it shows that the operation "first_memb" on "lmax", "a", and "i" in the context of "a" being mapped to list "l", results in a value "k" such that "a" is still mapped to list "l" and the intersect...
If "T" is a homomorphism between two vector spaces "V" and "W" with scalar multiplication operations "fsmult" and "fsmult'", and if the image of "V" under "T" is a subset of "W", then "T" is an element of the set of homomorphisms from "V" to "W" with respect to "fsmult" and "fsmult'".
The variables in a function call "FnCall" with function "F" and arguments "X" followed by list of arguments "Xs" is the union of the variables in "X" and the variables in the list "Xs".
If two functions 'f' and 'g', which map from a topological space 'a' to a linearly ordered topological space 'b', are continuous on a set 'A', then the function that maps each element 'x' in 'A' to the minimum of 'f(x)' and 'g(x)' is also continuous on 'A'. This lemma is used for proving continuity of functions.
The lemma named "hyps_free_path_length" assumes two conditions: first, there is a path from vertex 'v' to vertex 'v'' named 'pth', and second, the path 'pth' is hypothesis-free. Given these assumptions, it shows that the sum of the length of the path 'pth' and the length of the second element of vertex 'v'' is equal to...
"list_all function f applied to list l is equivalent to applying function all to list l with function f".
: 1. If the set of first elements from the pairs in set A is a singleton set containing only 'x', then set A is equal to the Cartesian product of the set of first elements and the set of second elements from the pairs in set A. 2. If the set of second elements from the pairs in set A is a singleton set containing only ...
Unstreaming an infinite list generated by a function 'f' at a position 'n' is equivalent to dropping 'n' elements from the infinite list generated by the same function 'f'.
The difference between the concatenation of lists "l" and "m", and the concatenation of lists "l" and "t", is equal to the difference between lists "m" and "t".
If "x" and "y" are elements of Hyperfinite numbers, then the standard part of the product of "x" and "y" is equal to the product of the standard parts of "x" and "y".
If a digraph with projection 'G' is loop-free, then 'G' is a pair loop-free digraph.
The leading coefficient of the product of two polynomials "p" and "q" is equal to the product of the leading coefficient of polynomial "p" and the leading coefficient of polynomial "q". Here, "p" is a scalar and "q" is a polynomial over a semiring without zero divisors.
"a" is an element of the document formed by appending "u" and "v" if and only if either "a" is an element of the document "u", or "a" is not an element of the set "u" and "a" is an independent element of the set "u" and "a" is an element of the document "v".
If 0 is not in the set of free variable indices of a formula φ at index b, and if a well-formed tuple exists for the successor of index b for the same formula φ and a vector v, then a well-formed tuple also exists for the successor of index n for the formula φ at index b, with None prepended to the vector v.