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Statement: stringlengths 7 24.3k
The continued fraction of a positive polynomial "f" equals zero if and only if the polynomial "f" itself equals zero.
The function f, when applied to negative one, gives the same result as when it is applied to the sum of 65,535 (represented in hexadecimal as 0xFFFF) and one, in the context of words of length 'a.
The set of all numbers from 0 to the successor of i (not including the successor of i) is equal to the set of all numbers from 0 to i (not including i) with i inserted into it.
In the context of the relation R01iai, the identity function and the function l1_step1 with parameters Ra, A, and B refine or improve upon the relation R01iai.
If "I" is reflexive and the pair "(D x, u)" does not belong to "I", then the concurrent ipurge of "I", "D", "u", and the list "xs" appended with "x" is equal to the concurrent ipurge of "I", "D", "u", and "xs".
If a function 'f' tends to a limit 'l' under a filter 'F', and if 'e' is a positive number, then eventually the distance between the function 'f' at a point 'x' and the limit 'l' will be less than 'e' under the filter 'F'.
The projection on the left component of the sum type of an element "x" inserted on the left is equal to "x" itself.
The open interval from 'a' to 'b' is connected, for any 'a' and 'b' in a linear continuum topology.
If a pair (hm, h) is in the relation hmr_rel, then the operation hm_pop_min_op' on hm is less than or equal to the operation h.pop_min_op on h, when these operations are related by the relation (kvi_rel hm 1 times hmr_rel).
The lemma named "inprod_cvec_ge_zero" shows that the real part of the inner product of a complex vector with itself is greater than or equal to zero.
The lemma named "bound_of_least" assumes that P is bounded by b (as stated by "bounded_by b P"). It shows that the bound of P is less than or equal to b.
The keys of the polynomial expression (X squared times Z cubed plus 2 times Y cubed times Z squared) are equal to the set containing the sum of the monomial of degree 2 with variable 0 and the monomial of degree 3 with variable 2, and the sum of the monomial of degree 3 with variable 1 and the monomial of degree 2 with...
The lemma named "path_prefix" assumes that there is a path in machine M from state q that follows the sequence of states (p1 followed by p2). It shows that there is also a path in machine M from state q that follows the sequence of states p1.
If a word 'w' is not empty, then 'w' is a prefix of its own infinite repetition.
The lemma named "ideal_induct_Polys" assumes four conditions and shows a result. The conditions are: 1. A set F is a subset of the set of polynomials P[X]. 2. A polynomial p is an element of the set of polynomials P[X]. 3. The polynomial p is an element of the ideal generated by F. 4. The polynomial 0 satisfies the pr...
If a signature 'Σ' is standard and a type 'T' is okay in the signature 'Σ', then for any pair (x, S) in the set of type variables of 'T', 'S' is a well-formed sort in the subclass of the object signature 'Σ'.
The head (first element) of a list, which is chosen based on a condition p, is equal to the head of the list chosen directly by the same condition p. If the condition p is true, the head of the list xs is chosen, otherwise the head of the list ys is chosen.
If a market process 'pf1' is an arbitrage process, and another market process 'pf2' is self-financing and a trading strategy, and for all instances 'n' and 'w', the closing value process of 'pf1' equals the closing value process of 'pf2', then it can be shown that 'pf2' is also an arbitrage process.
If "x" is an element in the set "xs", then the set "xs" is congruent to the set obtained by rotating "xs" to "x".
If a relation A is right total, then the relation of predicate A is right unique.
The real part of the product of the inner product of x and y and the conjugate of the inner product of x and y is greater than or equal to zero.
: assuming K is a subfield of R, F is a subfield of R, the dimension of F over K is n, and the dimension of E over F is 1, it can be shown that the dimension of E over K is also n.
The lemma "is_element_ptr_kind_obtains" assumes that "node_ptr" is of the kind "is_element_ptr_kind". It then obtains an "element_ptr" where "node_ptr" is equal to the cast of "element_ptr" to a node pointer.
The head of a list filtered by a predicate P is equal to the head of a list from which elements not satisfying the predicate P have been dropped.
The complex inner product of 1 (where 1 is an instance of a complex inner product and a complex normed algebra) and a complex number 'x' converted from its complex form, is equal to 'x' itself. This is a simplification rule in Isabelle.
If a sequence sigma is updated at positions specified by a list xs with the values obtained by mapping sigma over xs, the result is the same as the original sequence sigma.
For any program P and heap h, an empty map is less than or equal to E in the context of P and h.
If "k" is less than the difference of "j" and "i", and "j" is less than or equal to the length of the list "xs", then the "k"-th element of the sublist of "xs" from index "i" to "j" is equal to the element at the index "i + k" in the list "xs".
If the set of all regions (denoted by "\<R>") is equal to the set of all regions defined by the parameters X, I, and r that form a valid region, and if a certain element 'u' belongs to a region 'R', and this region 'R' is in the set of all regions, then the equivalence class of 'u' under the relation defined by the set...
The set of external outputs from the generative probabilistic value when lifted from a subprobability monad is an empty set.
: 1. If 't' is well-formed, then the expression 'mk_exp S t' is a valid cupcake expression. 2. If 'cs' is a set of well-formed clauses, then the clauses 'mk_clauses S cs' are valid cupcake clauses and also valid cake linear clauses. 3. If 't' is well-formed, then the expression 'mk_con S t' is a valid cupcake expressio...
The inverse of zero in the context of formal power series (abbreviated as fps) for any type 'a that is a commutative monoid under addition, supports multiplication by zero, unary minus, and has an inverse, is equal to the formal power series constant of the inverse of zero.
If the following conditions are met: 1. The function f when applied to 0 equals the pair (c # cs, s), 2. For all i, the function f when applied to i leads to the function f when applied to the successor of i, 3. The first element of the pair resulting from applying the function f to k equals cs, 4. For all i less than...
The lemma named "inter_plus_addition_in" assumes that the key "k" is not present in the first map "m1" and the key "k" is associated with the value "v'" in the second map "m2". It shows that if we add the key-value pair "k-v" to the first map and then take the "plus" intersection of the two maps, it is the same as taki...
"1 combined with the circular (transitive closure) of x multiplied by x is less than or equal to the circular (transitive closure) of x".
For all natural numbers a, x, and y, if a is less than or equal to x, then a is also less than or equal to the maximum of x and y.
The lemma "all_derived_clauses_are_wellconstrained" shows that if a clause C is derivable from a set S, and all elements in S are well-constrained, then the clause C is also well-constrained.
The lemma named "ide_eval_Ide" shows that if "t" is an identity object, then the evaluation of "t" is also an identity.
The lemma "targets_are_isomorphic" assumes that "b" is in the targets of "mu" and "b'" is also in the targets of "mu". It shows that "b" is isomorphic to "b'".
A property P always holds for the stream obtained by applying a function f to each element of a stream s if and only if it is always the case that the composition of P and f holds for the head of the stream s.
If sets A and B are subsets of the carrier R, and for all j less than or equal to n, the function f(j) is in the set multiplication of R, A, and B, then if i is less than or equal to n, the function f(i) is in the carrier R.
The intersection of multiset A with the union of multisets B and C is empty if and only if the intersection of multiset A with B is empty and the intersection of multiset A with C is also empty.
The function "eq_newpkt" is not equal to the Route Request (Rreq) with parameters hops, destination IP (dip), destination sequence number (dsn), destination key (dsk), origin IP (oip), origin sequence number (osn), and source IP (sip).
: for all x, if the function f when applied to 0 and x equals x, and for all i, if i is less than n then the sequence s at index i equals 0, then the fold sequence of function f and sequence s over n equals the sequence s at index n.
: for all 'b', if the multiplicity of 'b' in multiset 'A' is non-negative, and the multiplicity of 'b' in multiset 'B' is also non-negative, and if 'A' is a subset of 'B' (in the sense of multisets), and for every 'b' in 'B' there exists an 'a' in 'A' such that 'a' is less than or equal to 'b', then the frontier of 'A'...
Given a matrix A of type 'a, which is a strictly ordered commutative semiring, and assuming that A is a carrier matrix with dimensions nr by nc, and vectors b, c, and x are carrier vectors with dimensions nr and nc respectively, and the product of matrix A and vector x is less than or equal to vector b, and vector y is...
If a function 'f' maps from a set 'A' to the carrier of the ring 'R', then the homomorphism of the finite product of 'f' over all elements in 'A' is equal to the finite product in the ring 'S' of the composition of 'h' and 'f' over all elements in 'A'.
The language of the product of tree automata A and B, with the state set being the product of the state sets of A and B, is equal to the intersection of the languages of tree automata A and B with their respective state sets.
The set of a multiset, where an element 'a' is added to multiset 'A', is equal to the set where the element 'a' is inserted into the set of multiset 'A'.
The lemma named "has_derivative_zero_unique" is defined with a function 'f' that maps from a real normed vector 'a' to another real normed vector 'b'. It assumes that 's' is a convex set. It also assumes that for any 'x' in 's', the derivative of 'f' at 'x' within 's' is zero. Given that 'x' and 'y' are in 's', it show...
: assuming for all 'k', if 'k' is less than or equal to the length of 'p' minus one and 'p' at index 'k' is a subsequence and 'l' equals the destination of 'p' at index 'k', then 'f' at index 'k' equals 'g' at index 'k'. Under these conditions, the sum over the set 'S' minus 'p' at index 'l' for function 'f' is equal t...
For any program P and heap h, the reverse of sequence s is conformant to sequence t if and only if sequence s is conformant to the reverse of sequence t.
The lemma "dghm_const_is_tm_dghm" assumes that "\<CC>" is a tiny digraph, "\<DD>" is a digraph, and "f" is a function from "a" to "a" in "\<DD>". It shows that the constant digraph homomorphism from "\<CC>" to "\<DD>" at "a" with function "f" is a tiny digraph homomorphism from "\<CC>" to "\<DD>".
D is well-composed.
The lemma named "lincomb_scalar_prod_right" assumes that "W" is a subset of the vector space of dimension "n" and "v" is a vector in the same vector space. It shows that the dot product of "v" and the linear combination of "W" with coefficients "a" is equal to the sum of the product of "a w" and the dot product of "v" ...
The function "frpar" applied to a parameter "p" is quantifier-free, and the interpretation function "Ifm" applied to the variables "vs", the boolean values "bs", and the function "frpar" applied to "p" is logically equivalent to the interpretation function "Ifm" applied to the variables "vs", the boolean values "bs", a...
Given a list of configurations 'exec', a list of messages 'trace', a message 'msg', and a process 'p', under the assumptions that the message 'msg' is enabled in the last configuration of 'exec', the process 'p' is the receiver of the message 'msg', 'exec' and 'trace' form an execution starting from 'start' with transi...
If a polynomial 'P' belongs to the indexed polynomial set 'I' over the carrier 'R', and if 'ms' is a finite set of monomials, then the polynomial obtained by restricting 'P' to the monomial set 'ms' also belongs to the indexed polynomial set 'I' over the carrier 'R'.
The signed integer of an 8-bit unsigned integer 'n' is equal to the following: If the 7th bit of 'n' is true, then the result is undefined for the integer of the 8-bit unsigned integer 'n'. Otherwise, the result is the integer of the unsigned integer represented by 'n'.
The lemma named "vec_of_poly_rev_shifted_scalar_prod" is defined with fixed variables p and v. It also defines q as the polynomial of vector v. It assumes that the degree of p is less than or equal to m, the dimension of vector v is n, and j is less than the sum of m and n. The lemma shows that the scalar product of th...
If the domain of x exists or is defined (as assumed), then the auxiliary recursive function domain of x also exists or is defined.
If a set S is infinite, then the nth element enumerated in set S belongs to set S.
The convergence radius of a formal power series (fps) remains the same even after shifting the series by a certain number 'n'.
The lemma named "inv_inv" assumes that "f" is an isomorphism. It shows that the inverse of the inverse of "f" is equal to "f".
The probability mass function of the result of the function 'return_spmf' applied to 'x', evaluated at 'y', is equal to the indicator function applied to the set containing 'y' and the value 'Some x'.
The lemma named "not_col_permutation_3" assumes that points A, B, and C are not collinear. It shows that points C, B, and A are also not collinear.
The property Q holds for the orbit of a function gamma under the transformation phi if and only if for every state s in the set S, for all time t in the set T, the property Q holds for the transformation of the state s at time t.
The set T is a subset of the set that is the first element of the result of the function "distribute_extension" when applied to the parameters M, T, G, cg_lookup, cg_insert, u, w, b, and heuristic.
For all states 's', if the while-loop with condition 'b', body 'i', and continuation 'c' transitions to a parallel composition of threads 'Ts' in state 't' under the semantics of 'Pn', and if all threads in 'Ts' are none, then there exists a natural number 'k' such that the fixed-point while-loop with condition 'b', bo...
If "r" and "s" are well-formed duals with respect to "n", then the pnCoPlus of "b", "r", and "s" is also a well-formed dual with respect to "n".
The square of the norm of the 7-dimensional cross product of x and y is equal to the difference between the product of the squares of the norms of x and y and the square of the dot product of x and y.
The natural number form of the nth Catalan number is equal to the natural number form of the combination of 2n choose n divided by the natural number form of the successor of n, in any field with characteristic 0.
If X is an element of the known parts of G, and G is a subset of H, then X is an element of the analyzed parts of H.
The domain of the function "HMA_V", which maps a matrix vector of type 'a to a mod type of 'a to the power of 'n, is the set of all vectors 'v' that are in the carrier vector of the cardinality of 'n.
The lemma named "ordIso_finite_Field" assumes that the order of 'r' is isomorphic to the order of 's'. It shows that the field of 'r' is finite if and only if the field of 's' is finite.
If angle ABC is less than angle A'B'C', and angle DEF is less than or equal to angle D'E'F', and A'B'C' and D'E'F' are strictly supplementary angles, and the sum of angles ABC and DEF is equal to angle GHI, and the sum of angles A'B'C' and D'E'F' is equal to angle G'H'I', then angle GHI is less than angle G'H'I'.
The lemma named "find_fas_correct" assumes that "G" is a graph and that the set of all edges in "G" starting from the initial vertices "g_V0 G" is finite. It shows that the function "find_fas G" is less than or equal to the specification function "is_fas G" for the graph "G". This means that the function "find_fas G" c...
The composition of the negation of a complex bounded linear function 'a' and another complex bounded linear function 'b' is equal to the negation of the composition of 'a' and 'b'.
The set of nominals (or names) in the expression "bridge' k j p" combined with the set containing "k" and "j", is equal to the set of nominals in "p" combined with the set containing "k" and "j".
Appending a list "ms" to a list "ns" with an element "a" at the front is equivalent to appending "ms" to the list "ns" that has "a" appended at the end.
If the dequeue operation on the variables ic, q, oc is enabled, then the queue q is not empty.
: If a diagram D is monotonic, and for all u, i, j, the weakest precondition of X u i with respect to the diagram D from i to j is less than or equal to the supremum of the least fixed point of X paired with u and i, j. And if the precondition P is less than or equal to the supremum of the range of X, and the supremum...
The product of sigma-finite measures over an interval, where the measure of an interval is defined as its length, is sigma-finite.
The lemma "unfold_concatMapStep" is defined with two fixed functions: "ha" which is a function from 's to a step of 'a and 's, and "f" which is a function from 'a to a stream of 'b and 't. It also defines "h" as the concatenation of map steps of "f" and "ha", and "f'" as the composition of "unstream" and "f". The lem...
If a pair (s, t) is in the transitive closure of the relation "rstep" parameterized by R, and if the relation "funas_rel" parameterized by R is a subset of F, and if the function "funas_term" applied to t is a subset of F, then the pair (the function "term_to_sig" applied to F, v, and s, t) is in the transitive closure...
The lemma named "in_closure_connected_component" is about a set S of real normed vectors. It assumes that a vector x is an element of S and that S is an open set. It shows that x is an element of the closure of the connected component set of S and y if and only if x is an element of the connected component set of S and...
The lemma "fair_implies_Liminf_saturated" assumes that there is a chain "Ns" under the relation "\<rhd>", and that "Ns" is fair. Under these assumptions, it shows that the limit inferior (Liminf) of the list "Ns" is saturated.
An integer 'k' is less than an integer 'l' if and only if the integer representation of 'k' is less than the integer representation of 'l'.
The longest common prefix of the concatenation of strings "z" and "u", and "z" and "v", is equal to the concatenation of string "z" and the longest common prefix of strings "u" and "v".
The program counters (pcs) of the concatenation of two exception tables (xt₁ and xt₂) is equal to the union of the program counters of xt₁ and xt₂.
A list of expressions 'e#es' is well-typed with respect to a list of types 'Ts' in a given program 'P' and environment 'E', if and only if there exist types 'U' and 'Us' such that 'Ts' is a list with 'U' as its first element and 'Us' as the rest, and the expression 'e' is well-typed as 'U' and the list of expressions '...
: Given a set 'S' of pairs of states, and assuming three conditions: 1. The function 'is_bounded_hashcode' applied to 'S', 'seq', and 'bhc' is a side-effect-free generic algorithm, 2. The function 'is_valid_def_hm_size' applied to the type of 'statei' and 'hms' is a side-effect-free generic algorithm, 3. The operatio...
If the set of natural numbers "M" is infinite, then the least common multiple (Lcm) of "M" is equal to zero.
Let f be a formal Laurent series over a ring. If the coefficient of f at its subdegree times y equals 1, then the product of f and the right inverse of the base factor of f with respect to y is equal to the formal Laurent series X raised to the power of the subdegree of f.
If the star-transform of function f at point x is approximately equal to l, and the star-transform of function g at point x is approximately equal to m, and if l and m are both in the set of hyperfinite numbers, then the star-transform of the function that multiplies f and g at point x is approximately equal to the pro...
For any destination IP 'dip', the sequence number field 'sqnf' of the invalidated routing table 'rt' for all destinations 'dests' in the network state 'xi' is equal to the sequence number field 'sqnf' of the original routing table 'rt' in the network state 'xi'.
Converting a world model "M" to a state with a pair consisting of "p" and "as" is equivalent to the condition where the atom, which is the predicate atom of "p" and "as", is an element of the world model "M".
There exists a value 'v' such that the type of 'v' equals 't'.
The function "g_inj_image_filter" preserves the invariance of sets s1 and s2 under the transformation defined by their respective alpha functions.
Flipping the 0th bit of a number 'a' is equivalent to the sum of the boolean value of 'a' being even and twice the integer division of 'a' by 2.

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