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train · 499k rows

Statement: stringlengths 7 24.3k
If a set S is an invariant of a graph G, then the set of unknown vertices in S is equal to the total set of vertices in G minus the set of known vertices in S.
The lemma named "merkle_sum" assumes that there are two Merkle interfaces, one with parameters rha, boa, and ma, and the other with parameters rhb, bob, and mb. It shows that a new Merkle interface can be constructed using the hash sum of rha and rhb, the blinding of sum of boa and bob, and the merge sum of ma and mb.
, assuming "M" is a matching in a graph with vertices "V" and edges "E", the cardinality (or size) of the "i"-th matching "Mi" is equal to the cardinality of the set of edges in the "i"-th matching "Mi".
The lemma named "matrix_to_lpolies_valuate_scalarP" assumes that "i" is less than the number of rows in matrix A and the number of columns in matrix A is equal to the dimension of vector x. It shows that the value of the i-th polynomial in the list of linear polynomials derived from matrix A, when evaluated at vector x...
If a condition function always returns true, then the result of the 'condition' function with this condition, and any two actions A and B, is equivalent to action A.
If the function "rbtreeify_g" takes the inputs "n" and "kvs" and returns the tuple "(t, kvs')", and "n" is less than or equal to the successor of the length of "kvs", then the entries of the tree "t" are equal to the first "n - 1" elements of "kvs" and "kvs'" is equal to the remaining elements of "kvs" after removing t...
"op_iam_empty_sz$N" is equivalent to "UNPROTECT (op_iam_empty_sz N)".
If the ordered pair (r, A) is an element of the set D, then the ordered pair ((r, A), r restricted to A) is an element of the set D restricted to v.
If R is a member of congruences, then the congruence of the normal filter of R is equal to R.
The invariant "inv_count" is a precondition for the weakest precondition of the countdown function to be 1.
For any given label 'x', swapping 'z' and 'c' in 'x' is equivalent to swapping 'z'' and 'c' in 'x'.
The well-foundedness of the relation "listrel1 r" is equivalent to the well-foundedness of the relation "r".
Given the following conditions: 1. A well-formed program 'P' with respect to 'wf_md' (denoted by "wf_prog wf_md P"). 2. An external method call 'a.M(vs)' on thread 't' with heap 'h' transitions to a value 'va' and a possibly updated heap 'h' (denoted by "P,t \<turnstile> \<langle>a\<bullet>M(vs), h\<rangle> -ta\<right...
The observation function in level 2 is the identity function.
If the arithmetic compiler for an arithmetic expression 'a' transitions from the initial state (0, s, stk) to the final state (size of the arithmetic compiler for 'a', s', stk') in 'n' steps, then the final state 's' is equal to the initial state 's' and the final stack 'stk' is equal to the value of the arithmetic exp...
If you linearize the result of prepending an element 'a' to a queue 'q', it is equivalent to adding the element 'a' to the front of the linearized queue 'q'.
The lemma named "MEM_SPLIT" is defined for any object 'x' and list 'l'. It assumes that 'x' is not a member of the list 'l'. It then shows that for all lists 'l1' and 'l2', the list 'l' is not equal to the concatenation of 'l1', 'x', and 'l2'.
If a configuration is valid and the probability of the state of the configuration is zero, then the value of the configuration is also zero.
The boundary of the rotated diamond cube is equal to the set that includes the following elements: 1. A pair consisting of the integer 1 and a function that takes a real number x and returns a pair. The first element of this pair is the result of the operation (2 * x - 1) multiplied by the diamond_y of the diamond_x ...
If "t" is an element of "ik_uinfo", then "t" is also an element of "ik_hfs".
If a number 'n' is in the RSpan of a list consisting of 'm' followed by 'ms', then there exists a number 'r' and a list of numbers 'rs' such that the set of 'r' and 'rs' is a subset of R, the length of 'rs' is equal to the length of 'ms', and 'n' is equal to 'r' multiplied by 'm' plus the dot product of 'rs' and 'ms'.
If the relation 'r' is well-founded for the element 'x' and the relation 'step1 r' is well-founded for the list 'xs', then the relation 'step1 r' is also well-founded for the list formed by prepending 'x' to 'xs'.
The lemma named "bin_rep_index_0" is defined with two fixed natural numbers, n and m. It assumes that m is less than 2 to the power of n and that k is greater than n. It shows that the first element (at index 0) of the binary representation of k and m is 0.
The knight's path on an 8 by 9 upper left board is not empty.
The lemma named "induced_arrowI'" assumes that "b" is an identity of category "C" and for any arrow "j" in category "J", "F j" is an arrow from "b" to "D j". It shows that for any arrow "j" in category "J", the composition of "j" and the induced arrow from "b" to "F" equals "F j".
For a natural number 'm', the sum of the square matrix diagonals of a function 'g' for all 'n' less than 'm' is equal to the diagonal of the sum of the function 'g' applied to 'n' for all 'n' less than 'm'.
The theorem "wls_subst_preserves_fresh" assumes that "X" is a well-sorted term of sort "s" and "Y" is a well-sorted term of sort "ys". It also assumes that "z" is a fresh variable for both "X" and "Y" with respect to the set of variables "zs". It shows that "z" remains a fresh variable for the term resulting from subst...
If for all j less than or equal to n, f(j) belongs to the set of extended integers, and if j is less than or equal to n and for all l in the set of h such that h is less than or equal to n excluding j, f(l) equals zero, then the sum from 1 to n of f equals f(j).
If x is left-sided, then the product of y and x is less than or equal to x.
The theorem named "Helly_selection" is defined with respect to a function 'f' that maps a natural number to a real number. It assumes three conditions: 1. The function 'f' is right-continuous for all natural numbers 'n' and real numbers 'x'. 2. The function 'f' is monotonically increasing for all natural numbers 'n'...
If all chains do not call to "c" in the rule set "rs" under the context "Γ", and the context "Γ" at "c" equals to some rule set "rs", then the semantics of the rule set "rs" under the context "Γ", packet "p", and decision "t" is equivalent to the semantics of the rule "Call c" that matches any packet under the same con...
: If an element 'q' is in the set 'xs' and 'xs' contains distinct elements, then: - If the index of 'q' in 'xs' is less than 'i' and 'i' is less than the length of 'xs', then the index of the 'i'-th element of 'xs' in the list obtained by moving 'q' 'n' places towards the front of 'xs' is equal to the index of the 'i...
The lemma named "infsum_cinner_left" assumes that the function "phi" is summable over the set "I". It shows that the inner product of "psi" and the infinite sum of "phi" over "I" is equal to the infinite sum of the inner product of "psi" and "phi" over "I".
The lemma named "set1_FGcontra_bound" fixes a variable "x" of type "FGcontra" with various type parameters. It shows that the cardinality of the set "set1_FGcontra" of "x" is less than the cardinality of the boundary "bd_FGcontra" of the type "FGcontra" with the same type parameters, in the order of the relation "rel".
The nth element of the formal Dirichlet series (fds) of a function f is equal to 0 if n is 0, otherwise it is equal to the value of the function f at n.
If a property P holds for all time points t in set A or for all time points t in set B, then the property P also holds for all time points t in the intersection of sets A and B.
There is no collision over the entire range starting from 0 in the context of delayed safe distance if and only if there is no collision in the reaction context starting from delta.
The lemma named "map_simp_ide" assumes that "a" is an identity of category "A". It shows that the mapping of "a" is equal to the composition of "τ a" and "σ a" in category "B".
A new packet is not equal to a Route Request (Rreq) with the given parameters: number of hops, request ID, destination IP, destination sequence number, destination key, origin IP, origin sequence number, source IP, and whether it has been handled or not.
If "x" and "y" are tests, then the infimum (greatest lower bound) of "x" and "y" is equal to the infimum of "y" and "x". In other words, the operation of finding the infimum is commutative for tests "x" and "y".
If a real number 'x' is greater than or equal to zero, 'a' is less than or equal to zero, and 'b' is greater than or equal to zero, then the real number 'x' belongs to the complex cone defined by 'a' and 'b'.
"None" is an element of the set "A-bottom".
For a deterministic fold map operation on a set X, if the operation negates each element in X, applies a predicate P to each element, starts with a false value, and checks if there exists an element in the map that satisfies the predicate P, then this operation is equivalent to the operation of checking if there exists...
The lemma named "GFP_idem" assumes that the function 'f' maps elements from the carrier set 'L' to itself, 'f' is monotone, and 'f' is idempotent. It shows that the greatest fixed point of 'f' is equal to the result of applying 'f' to the greatest element in the set.
The lemma named "r01_eq_iff" assumes that both "r1" and "r2" are greater than 0 and less than 1. It shows that "r1" is equal to "r2" if and only if the binary expansion of "r1" in the range of 0 to 1 is equal to the binary expansion of "r2" in the same range.
The lemma named "ev2_aux" is defined as follows: Let "f" be a function of "x", such that "f" equals to "(1 + 1/ln (bx) ln (1 + hb / b * ln x to the power of (-1-e))) to the power of (-e/2)". Then, there exists a function "h" such that, eventually for large values of "x", "f(x)" is greater than or equal to "1 + h(x)"...
The set M1 is finite.
For a given set S, which is a subspace of a Euclidean space, it is homeomorphic (topologically equivalent) to the Euclidean space of its own dimension.
The lemma named "lex_two_order_pair" assumes two order pairs, "s1 ns1" and "s2 ns2". It shows that the lexicographic order pair of "s1 ns1 s2" is the same as the lexicographic order pair of "s1 ns1 ns2".
The lemma named "partition_by_nth_nth" makes two assertions, given three assumptions. The assumptions are: 1. The sum of the list "ys" is equal to the length of the list "xs". 2. The index "i" is less than the length of the list "ys". 3. The index "j" is less than the element at index "i" in the list "ys". Under...
If the "StartBallot" function transitions from state 's' to state 's'' for a given 't', and if the "HInv3_inner" function holds true for state 's' and parameters 'p', 'q', and 'd', and 't' is not equal to 'p' or 'q', then the "HInv3_inner" function will also hold true for the new state 's'' and the same parameters 'p',...
The lemma "int_on_path" assumes that "l" is in "L", "r" is in "R", and "Q" is in "P". It also assumes that "L" is a subset of "Q", "L" is not empty, "R" is a subset of "Q", "R" is not empty, and the union of "L" and "R" equals "Q". Given these assumptions, it shows that the interval between "l" and "r" is a subset of "...
For any real normed vector 'a', the open ball centered at (c - a) with radius r is equal to the image of the open ball centered at c with radius r under the function that subtracts 'a' from 'x'.
The longest common prefix of a word with itself is the word itself.
The restriction of the function "m" to the universal set is equal to "m" itself.
The lemma "extended_Gromov_product_tendsto_PInf_a_b" assumes that the extended Gromov product at point 'a' for sequences 'u' and 'v' tends to infinity as 'n' goes to infinity under filter 'F'. Under this assumption, it shows that the extended Gromov product at point 'b' for the same sequences 'u' and 'v' also tends to ...
For a function 'f' that operates on a complete lattice, if 'f' preserves the infimum (greatest lower bound), then the composition of 'f' and the Kleene star of 'f' is equal to the composition of the Kleene star of 'f' and 'f'.
If "Lfilled n lholed es LI" is assumed, then there exists "lholed'", "ves", and "es_c" such that "ves" is a constant list and "Lfilled_exact n lholed' (ves@es@es_c) LI" is true.
The lemma named "while_simulate_right" assumes that the product of z and x is less than or equal to the product of y and z union w. It shows that the product of z and the star of x and v is less than or equal to the star of y and the union of the product of z and v and the product of w and the star of x and v.
The function "st" applied to the global and local states of "s" equals "s" itself. This is a simplification rule.
If "r" is less than or equal to 1, then the pair of the probability mass function "p" and the scaled probability mass function "q" scaled by "r" is equal to the scaled pair of the probability mass function "p" and "q" by "r". This lemma needs to be generalized for any arbitrary "r".
The lemma named "id_is_local_ring_morphism" assumes that "A" is a local ring with operations "add" and "mult", and elements "zero" and "one". It shows that the identity function on "A" is a local ring morphism from "A" to itself, with the same operations and elements.
The lemma named "H_cosets_aux" assumes that "x" is an Euler witness for a prime number "p". It shows that the intersection of the coset of "x" in group "G" with "H" and "H" itself is an empty set.
The weakest precondition of a conditional statement where 'p' is the condition, 'x' is the action if 'p' is true, and 'y' is the action if 'p' is false, when 'p' is true, is equivalent to the weakest precondition of the action 'y' when 'p' is true.
The lemma named "child2_commute_aux" assumes a function "f" that takes a pair (t, _) and a variable "b". In the case where "t" is a node with root "r" and children "ys", if "r" equals "a", then the union of "ys" and "b" is returned, otherwise "b" is returned. The lemma shows that the composition of function "f" applied...
For any value 'z' of type 'a', where 'a' is a Banach space, a real normed field, and a field with characteristic zero, the evaluation of the formal power series (fps) of cosine at 'c' and 'z' is equal to the cosine of the product of 'c' and 'z'.
The number 6 is equal to the successor of the successor of the successor of the successor of the successor of the successor of 0.
The lemma "lexer_correct_Some" demonstrates that a string "s" is in the language of a regular expression "r" if and only if there exists a value "v" such that when the lexer function is applied to "r" and "s", it returns "v" and "s" is in the image of "r" under the function that maps to "v".
If 'a' is in the set 'L_P', then there exists a 'b' such that 'a' concatenated with 'b' left-derives from the start symbol 'S'.
If no MT rules are applied to a combination of a firewall configuration C and a packet p, then no MT rules will be applied to the same firewall configuration C and the packet p after removing any duplicate elements.
The sum of M1 and M0 equals the 1-sub-matrix K.
, assuming a well-formed C++ program P, if P and environment E can reduce expression e with state s to expression e' with state s', and if P, E, and s can type-check e to type T, then P, E, and the heap of state s' can type-check e' to type T under non-throwing execution.
: - The "Let" expression with parameters e1, x, and e2 is not a value. - The "App" expression with parameters v and v' is not a value. - The "Case" expression with parameters v, v1, and v2 is not a value. - The "Prj1" expression with parameter v is not a value. - The "Prj2" expression with parameter v is not a value. ...
For any ordinal number 'j', the successor of 'i' is less than or equal to 'j' if and only if 'i' is less than 'j'.
If a chamber "F" is part of the chamber complex "A", then the image of "F" under the function "res_fold_A" is equal to the image of "F" under the function "fold_A".
The mediant of F is equal to (1, 1) and the mediant of I is also equal to (1, 1).
The lemma named "pt_set_diff_eqvt" is defined with the fixed variables X and Y, which are sets of type 'a, and pi, which is of type 'x prm. It assumes that 'pt' is a function of types 'a' and 'x', and 'at' is a function of type 'x'. The lemma shows that applying the permutation 'pi' to the difference of the sets X and ...
The set of probability mass functions for a binomial distribution with parameters n and 1 is equal to the set containing n.
"x" if not "p" else ("x" if not "q" else "y") is equal to "x" if not "p" or not "q" else "y".
The property P holds for the abrupt_if function with parameters c, x', and x if and only if the following two conditions are met: 1) If c is true and x is None, then P holds for x'. 2) If it is not the case that both c is true and x is None, then P holds for x.
Applying a function 'f' to an immutable array 'as' is equivalent to applying 'f' to the list representation of 'as'.
The identity mapping on M is a bijective function from M to M.
The function v applied to the successor of 0 and x is equal to the function u applied to 1 and x.
The lemma demonstrates that the function 'lconcat' is monotone with respect to the partial order 'sqsubseteq'.
The one-step reduction of a global state in a multithreaded Java-like language with uncaught exceptions, represented by "tau_red1g uf P t h (e, xs) (e', xs')", is equivalent to the condition that there is a silent move from the expression "e" with heap "h" and local variables "xs" to the expression "e'" with the same h...
The lemma named "class_of_to_rel" shows that the equivalence class of the pair (r, s) with respect to the relation "rel" is equal to the relation "rel" applied to r and s.
If 'u' is a finite list, and 'u' is less than or equal to 'v', then the list of 'u' is less than or equal to the stream of the infinite list 'v'.
The value of a prime number to the power of negative integer k is equal to the negative of the extended integer k.
The lemma "optim_instrs_all_fun_call_in_range" assumes that the function "optim_instrs" with parameters F, ret, pc, Σi, and xs returns a pair consisting of Σo and ys. It shows that the condition of all elements in the list xs satisfying the function "fun_call_in_range" with parameter F is equivalent to all elements in ...
If the function f(x) is greater than or equal to zero, then the non-negative integral of f(x) with respect to the probability mass function of x is equal to f(x).
If an object 'Obj f T' can be reduced to 'z' through beta reduction, and for all 'lz' where the domain of 'lz' is equal to the domain of 'f' and 'z' is equal to the object 'Obj lz T', then 'Q' holds. This lemma consumes one assumption and is used in object case analysis.
The composition of relations "s" and "d" is a subset of relation "d".
If the action "HEndPhase0" transitions the system from state "s" to state "s'" for process "p", and if the inner invariant "HInv5" holds for the system state "s" and process "q", and if the invariant "Inv1" holds for the system state "s", and if process "p" is not equal to process "q", then the inner invariant "HInv5" ...
For any elements x and y in a unital quantale, the nu of the infimum of nu of x and nu of y is equal to the infimum of nu of x and nu of y.
The set of prefix pairs of a list "xs" is equal to the set of all pairs of lists "(zs, ys)" such that there exist lists "xs1" and "xs2" where "zs" followed by "xs1" equals "ys", "ys" followed by "xs2" equals "xs", and "xs1" is not an empty list.
The OCL integer 9 is not equal to null, which is always false.
The relation "oGS_rel" is single-valued.
If the inorder traversal of a tree 't' is sorted, then the inorder traversal of the tree resulting from inserting an element 'x' into 't' is equal to the result of inserting 'x' into the list obtained from the inorder traversal of 't'.
If C is an element of the carrier of the function fTo D, then there exists an element M in the carrier of the function fTo D such that C is a subset of M and M is maximal in the function fTo D.
If a list "xs" is not empty, then rotating the list by one position is equivalent to appending the head of the list to the tail of the list.

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