Datasets:
UDACA
/
Split-Isa-AF-MMA

Dataset card Data Studio Files
xet
 Community
Statement:
stringlengths
7
24.3k
For all 'x', if 'x' is in the direct sum of the carrier 'I' and module 'M', then 'x' is also in the product of the carrier 'I' and module 'M'.
The lemma named "multisetUnionLessMono2" demonstrates that if "r" is a transitive relation, and "b1" is related to "b2" in the multiset "r", then "a + b1" is also related to "a + b2" in the multiset "r".
The function of scaling by a factor of 'k' is twice field differentiable at any point 'x'.
The boolean value of the negation of a boolean expression 'b' in a state 's' is equivalent to the negation of the boolean value of 'b' in the same state 's'.
If a graph 'g' is symmetric, then there exists a spanning forest 'f' for this graph 'g'.
The lemma named "assumptions_impI" assumes that the assumptions of both P and A hold true. It then shows or proves that P is true.
The sum from M to N of the hyperreal series f is equal to the standard part of the function that takes m and n and returns the sum of f from m to n.
The index of an element 'x' in a list that is formed by appending 'x' to another list 'xs', is equal to the size of the list 'xs'.
The lemma named "rule_sub_remark_3_autosubst" assumes that for any variables 'v' and 'x', the property 'A!' of 'x' is equivalent to the negation of the possibility of the property 'E!' of 'x' in 'v'. Given this, if there exists an 'x' such that the property 'A!' of 'x' is in 'v', then it can be shown that there exists ...
The lemma "prv_scnj_mono_imp" assumes that "F1" is a subset of "fmla", "F2" is a subset of "fmla", both "F1" and "F2" are finite, and for every formula "phi2" in "F2", there exists a formula "phi1" in "F1" such that "phi1" implies "phi2". Under these assumptions, it shows that the conjunction of all formulas in "F1" im...
If B receives a message encrypted with the shared key of A and B, containing a nonce Nb, and if B is not the Spy and there is no illegal use of B's card, in the context of the Shoup-Rubin-Bella protocol, then there exists a nonce Na such that the card of B outputs to B a message containing the nonce Nb, the agent A, a ...
: Assuming a term 't' where the function symbols in 't' are a subset of a given finite set 'F', the variables in 't' are a subset of a given finite set 'Q', and 't' is not an empty set of variables. It shows that if 't' is a variable, then 'q_i' is in the transition system derived from the reflexive closure over a non-...
"x is less than or equal to y plus the negation of z if and only if the product of x and z is less than or equal to y".
The function "state_update_before" with parameter "sc" is monotonic.
The lemma D2_ALL shows that "for all d, property P holds for d" is equivalent to "for all d in the set consisting of D1 and D2, property P holds for d".
The lemma "keys_monomial_pmdl" assumes that "F" is a set of monomials, "p" is in the ideal generated by "F", and "t" is in the set of keys of "p". Under these assumptions, it is possible to obtain a non-zero monomial "f" from "F" such that the leading term of "f" adds to "t".
A topological space X is a T1 space if and only if for every element x in the top space of X, the set containing only x is closed in X.
The lemma named "lift" assumes that "d" is less than "dm" and "p" is in the sparse grid of "dm" and "lm". It also introduces a function "F" with parameters "l", "b", "p", and "α". This function "F" is defined such that if "b" is in the level grid starting at "dm" and excluding "d", with level "lm", and the sum of "l" a...
In the context of the Relabel-To-Front algorithm for maximum flow problems, the neighbor invariant holds for a given capacity function 'c', source 's', sink 't', flow 'f', labeling 'l', and node 'n'.
The function "get_disconnected_nodes_locs" is a new character data getter for disconnected nodes. This is an instance of the general concept "l_new_character_data_get_disconnected_nodes".
The components of a set 'u' are pairwise disjoint, meaning that the intersection of any two different components is an empty set.
The set resulting from merging two sets 'xs' and 'ys' with respect to a relation 'rel' is equal to the union of the set 'xs' and the set 'ys'.
If we assume "x" is a constant function on set "A" with value "c", then we can obtain an element "a" from set "A" such that "x" is an ordered pair of "a" and "c".
The set resulting from applying the merge sort function to a list 'xs' is equal to the original set 'xs'.
If the product of two natural numbers 'a' and 'b' is congruent to zero modulo a prime number 'p', then either 'a' or 'b' must be congruent to zero modulo 'p'.
For any two integers p and q, given that p and q are coprime (they have no common factors other than 1), both are greater than 2, and both are odd, the product of the Jacobi symbol of p and q and the Jacobi symbol of q and p is equal to -1 raised to the power of the product of half the decrements of p and q (rounded do...
The set of "ikkbz_sub r" for each "r" in the vertices of "G" is not empty.
If a function 'f' is quasi-Borel integrable over the product of two quasi-Borel probability spaces (X, α, μ) and (Y, β, ν), then two conditions hold. First, 'f' is a quasi-Borel function from the product of X and Y to the set of real numbers. Second, 'f' composed with the product map of α and β is integrable with respe...
If a set C is not affine dependent and the integer value of the cardinality of C is equal to n + 1, then the n-simplex of the convex hull of C holds.
The lemma named "example1" is defined with the following conditions: - It fixes an expression 'e' and variables 'f', 'g', 'x', 'y', and 'z'. - It assumes that for any 'a', the A-expression of 'e' applied to 'a' equals the singleton set of 'x' applied to the up function of 'a' union the singleton set of 'y' applied to ...
The function f3, when applied to the arguments a, b, c, d, a, produces the same result as when it is applied to the arguments a, d, c, d, a.
If a machine "M" is observable and "q" is a state in "M", and if "f" is an equivalence relation on the states of "M", then the observable finite state machine (OFSM) table for "M", "f", "k", and "q" is equal to the set of all states "q'" in "M" such that for all input-output sequences "io" with a length less than or eq...
In the context of UP_cring, the lemma ltrm_smult_cring is stated as follows: Assuming that 'p' is an element of the carrier 'P', 'a' is an element of the carrier 'R', and the leading coefficient of 'p' multiplied by 'a' is not zero, it can be shown that the leading term of the scalar multiplication of 'a' and 'p' is e...
A proposition "p" is not equal to the conjunction of "p" and another proposition "q".
If a list "l" is not empty and is valid, and if a pair (m, l') is the result of removing the maximum element from "l", then the size of the multiset representation of "l" is greater than the size of the multiset representation of "l'".
The restriction of the predicate 'false' to any alphabet 'a' is 'false'.
The lemma named "value_plus_flow" assumes that there is a well-formed residual network and two flows, f and g, where g is a flow in the residual network of f. It shows that the value of the flow of the sum of f and g is equal to the sum of the value of the flow of f and the value of the flow of g.
The lemma "hfun_in_hexp" assumes that "F" is a function from set "B" to set "C". It shows that "F" is an element of the exponential object of "B" and "C".
The lemma "inv_upto_mono" assumes that for any indices i and j, if i is less than n and j is less than k, and if the predicate p holds for the matrix A at indices i and j, then the predicate q also holds for the matrix A at these indices. It shows that if the predicate p holds for all indices up to k in the matrix A, t...
The inverse character of a character is also a character in group G.
Normalizing the greatest common divisor in a factorial ring A does not change the value, it remains the same as the greatest common divisor in A.
For a fixed element 'a' of an integral domain, the count of polynomial roots of the polynomial [:-a,1:] in a set 's' is 1 if 'a' is in 's', otherwise it is 0.
If the successor of 0 (which is 1) is less than a certain number 'k', then there exists an interval 'I' in the set 'i_set0' such that the interval 'I' divided by 'k' is not in the set 'i_set0'.
If a function 'f' is of the same order of growth as 'h' (denoted as "f =o O(h)"), and for all 'x', 'g' of 'x' is greater than or equal to zero and 'g' of 'x' is less than or equal to 'f' of 'x', then 'g' is also of the same order of growth as 'h' (denoted as "g =o O(h)").
If for every element 'i' in set 'A', the function 'f' is continuous at a point 'a', then the function that maps 'x' to the sum of 'f i x' for each 'i' in 'A' is also continuous at 'a'. This is for a function 'f' that maps an element of type 'a' and an element of type 'b' from a Hausdorff space to an element of type 'c'...
The lemma "list_all_iff_butlast_last" assumes that the list "xs" is not empty. It shows that a property "P" holds for all elements in the list "xs" if and only if the same property "P" holds for all elements in the list obtained by removing the last element from "xs" and the property "P" holds for the last element of "...
The lemma named "Amicable_pair_equiv_def" assumes that "m" and "n" are an amicable pair and both "m" and "n" are greater than or equal to 1. It shows that the sum of the proper divisors of "m" is equal to the sum of the proper divisors of "n", and this sum is also equal to the sum of "m" and "n".
If a polynomial 'p' is signature-reduced by the set containing only the polynomial 'f' with respect to a term ordering 'prec_t' and the top_tail function, then the leading term of 'f' is less than the leading term of 'p' with respect to the term ordering 'prec_t'.
If an element 'a' is not in the set 'A', then the set 'A' minus the singleton set containing 'a' is equal to the set 'A'.
If y is less than x, then the left neighborhood of x is equal to the infimum of the principal set where a is less than x, for all a in the set of elements less than x.
The binomial coefficient of n and the successor of n is equal to the difference between the binomial coefficient of the successor of n and itself, and the binomial coefficient of n and itself.
The component of a graph 'g' containing a node 'r' is connected.
The lemma for formal power series derivative is linear. This means that the derivative of the sum of two formal power series, each multiplied by a constant, is equal to the sum of the derivatives of the individual series, each multiplied by the corresponding constant.
The lemma "Diagonal_to_Smith_row_i_PQ'" assumes that the tuple (P',B',Q') is equal to the result of the function "Diagonal_to_Smith_row_i_PQ" with parameters i, bezout, and (P,B,Q). It also assumes that B is a diagonal matrix and that i is less than the minimum of the number of rows of A and the number of columns of A....
"ll" is less than or equal to "l".
If 'x' is of the shadow root kind, then the document type of the cast of 'x' with its shadow root mode replaced by 'arg' is equal to the document type of 'x'.
A word, when converted to an integer and then back to a word, remains the same word.
The lemma named "finer_nchotomy" assumes that P is a partition of I0, Q is a partition of I0, and P is finer than Q. It also assumes that I is an element of P and II is an element of Q. It shows that either I is a subset of II or the intersection of I and II is empty.
If the triple "uf, P, t" can transition from the state "e, s" to "e', s'" under the action "ta", and the first call in "e" is "aMvs", then "aMvs" is a synthesized call in the program "P" with heap state "hp s". Similarly, the lemma "reds1_calls_synthesized" states that if the triple "uf, P, t" can transition from the...
In the context of separated temporary variables, there is a lemma named 'tmps_le''. This lemma assumes that 'i' is less than the length of 'ts' and that the 'i'th element of 'ts' is a tuple consisting of '(s,t)', 'is', 'θ', 'sb', 'D', and 'O'. The lemma then shows that for all 'i' in the domain of 'θ', 'i' is less than...
The lemma RDirProd_list_zero' shows that the zero element of the direct product list of a ring R and other rings Rs is the list of the zero element of R and the zero element of the direct product list of the other rings Rs.
A literal is an element of a formula if and only if there exists a clause such that this clause is an element of the formula and the literal is an element of this clause.
: Given a function B that maps a function of type 'a to a program with 'heap, exception, and 'result, if B is a monotonically increasing function in the domain of the program (as stated by the assumption "mf"), and for any value 'y', the function that maps 'f' to 'C y f' is also a monotonically increasing function in ...
: Assuming the following conditions: 1. For all 'x', 'G x' is less than or equal to 'H x' or 'p' is less than or equal to 'H x'. 2. There does not exist an 'x' such that 'p' is less than or equal to 'H x' and 'H x' is less than or equal to 'q'. 3. 'X' enforces 'c1' under the security context 'Q p H', 'G q' equals 'H q'...
There exists some 'y', 'a', 'b', and list 'zs' such that 'x' is a list beginning with 'y', followed by a pair 'a' and 'b', and then the list 'zs'.
If the reduction of polynomial p to polynomial q by a set of polynomials F is valid (denoted as "red F p q") and the set of keys of polynomial p and polynomial r have no common elements (denoted as "keys p \<inter> keys r = {}"), then the reduction of the sum of polynomials p and r to the sum of polynomials q and r by ...
If the term "t" is not bound to 0 and the inequality "Lt (CNP 0 a r)" is linear, then the result of the function "msubstltpos" with parameters "c", "t", "a", and "r" is also not bound to 0.
An ordered pair (a, b) is not equal to its second element b.
The lemma "Proc_CFG_edge_rhs_pred_eq" assumes the following conditions: 1. There is a program control flow graph (CFG) edge from the label 'l' to 'n' in the program 'prog' with the edge type 'et'. 2. The edge type 'et' is a predecessor of the state 's'. 3. For all variables 'V' in the right-hand side (rhs) of the labe...
If "mprev_next I xs ts" equals "(ys, xs', ts')" and all pairs in the lists "xs" and "ts" from the index "Suc i" to "j'" and from "i" to "j" respectively satisfy the predicate "P" and the function "\<tau> \<sigma>", and if "Suc i" is less than or equal to "j'" and "i" is less than "j", then all pairs in the lists "ys", ...
The results of the game playing value (gpv) when intercepting the universal probabilistic function (upf) with parameters k, b, LD, and x, under the full interaction (both \<I>_full), are a subset of the responses under the full interaction (both \<I>_full) with x and any element from the universal set.
The lemma named "sum_vimage" is defined with a function 'g' that maps a natural number to a real number. It assumes that for any 'w' in set 'S', the value of function 'f' at 'w' is less than a certain bound 'bd', and that 'bd' is greater than 0. It also assumes that the set 'S' is finite. The lemma then shows that the ...
The lemma "borel_measurable_ereal_iff" shows that the function that maps x to the extended real number (ereal) of f(x) is Borel measurable with respect to the measure space M if and only if the function f itself is Borel measurable with respect to the measure space M.
Under the assumption that "f" is not an abstraction, the set of all "s" such that "s" is transformable to a predicate "A" through the function "f" applied to "T", is a subset of the image of the function "f" applied to all "S" such that the length of "T" equals the length of "S" and for all "s" in the set "S", there ex...
For any function 'f', the uncurried version of 'f' and the uncurried version of the function that returns the sum of two inputs are in the Kleene star of the identity assertion, which is a relation between two identity assertions. 2. The second rule states that for any function 'f', the uncurried version of 'f' and th...
Lemma fls_left_inverse_eq_fls_right_inverse_comm: Let f be a formal Laurent series over a commutative ring with unity. If the product of x and the coefficient of the term with the subdegree of f is 1, then the left inverse of f with respect to x is equal to the right inverse of f with respect to x.
If the sum of an index 'i' and the length of list 'xs' is less than the length of list 'ys', and if list 'xs' is a sublist of list 'ys' starting at index 'i', and if the element at the index 'i + length xs' in list 'ys' is 'x', then the list 'xs' appended with 'x' is a sublist of list 'ys' starting at index 'i'.
If the sum of 'n' and 'd' is less than or equal to the sum of 'l' and 'r', then the mirrored bounds of the interval from 'n' to 'd' is equal to the interval from 'l + r - n - d' to 'd'.
The domain of the product of x and y is less than or equal to the domain of x.
The nth element of the list representation of a Euclidean vector x is equal to the nth basis of x if n is less than the dimension of the Euclidean space. Otherwise, it is equal to the nth element of an empty list subtracted by the dimension of the Euclidean space.
The action of a permutation 'p' on the action 'Act' with parameters 'f', 'α', and 'x' is equal to the action 'Act' with parameters 'f', 'α', and 'x' each permuted by 'p'.
If "x" is a vector, then "x" is transitive.
The lemma named "conjugate_circline_set'" shows that the set of conjugates of the circline set H is a subset of the circline set of the conjugate of H.
If 'i' and 'j' are consecutive and 'n' is not an element of the union of the representations of 'i' and 'j', then 'n' is not an element of the union of 'i' and 'j'.
If there is a dynamic program dependence graph (DynPDG) path from node 'n' to node 'n'', then one of the following conditions must be true: 1. The path 'as' is empty and the nodes 'n' and 'n'' are the same. 2. There exists a node 'n''' and two paths 'asx' and 'asx'' such that 'n' is connected to 'n''' through a contro...
The weakest liberal precondition of a demonic choice between two programs 'a' and 'b' with respect to a predicate 'P' is a function that, for a given state 's', returns the minimum of the weakest liberal preconditions of 'a' and 'b' with respect to 'P' in state 's'.
If the height of the tree 'r' is greater than 0, then the inorder traversal of a 3-node tree (with 'l' as the left subtree, 'a' as the left value, 'm'' as the middle subtree, 'b' as the right value, and 'r' as the right subtree) is equal to the inorder traversal of 'l', followed by 'a', followed by the inorder traversa...
If two functions 'f' and 'g' are not equal and both belong to the set of finite functions (FINFUNC), then the maximum difference between 'f' and 'g' over the field 's' is the maximum value where 'f' and 'g' are not equal.
The space X is connected if and only if there do not exist two sets E1 and E2 such that both E1 and E2 are open in X, the union of E1 and E2 equals the entire space X, the intersection of E1 and E2 is empty, and neither E1 nor E2 are empty sets.
The lemma named "multiplicity_radical_prime" assumes that "p" is a prime number and "x" is not equal to zero. It shows that the multiplicity of "p" in the radical of "x" is equal to 1 if "p" divides "x", otherwise it is equal to 0.
The silent moves of the system, under the transition relation "mred1g uf P t" and the silence relation "tau MOVE1 P", from the state "((Val v, xs), []), h" to state "s" if and only if state "s" is equal to "((Val v, xs), []), h".
The lemma named "degree_eq_zeroE" is defined with a fixed polynomial 'p' of type 'a', where 'a' is a zero type. It assumes that the degree of 'p' is zero. It then concludes that there exists a value 'a' such that 'p' is equal to the polynomial constructed with 'a' as the coefficient and 0 as the degree.
The normalization of an unsigned natural number converted to a bit vector is equal to the original natural number converted to a bit vector.
If a function phi is a first, second, and fourth derivative, then for all sets A, the derivative of phi applied to A is less than or equal to phi applied to the derivative of phi applied to A.
If a hyperreal number 'r' is greater than or equal to zero and less than or equal to one, and if 'N' belongs to the set of infinitely large hypernatural numbers, then for all natural numbers 'n', the 'N'th power of 'r' is less than or equal to the 'n'th power of 'r'.
If the following conditions are met: 1. The instruction sequence of the method body in class C and signature S in the global state G is equal to a list consisting of a prefix 'pre', an instruction 'i', and a suffix 'post', and the program counter 'pc0' is equal to the length of the prefix 'pre'. 2. The execution of th...
The error function (erf), when applied to real numbers, is injective on the set A. This means that for any two different elements in set A, their images under the error function are also different.
The theorem, from the Set_Linorder.thy file in Containers, assumes that the universal set of type 'a is finite (denoted by "finite (UNIV :: 'a set)"). It also assumes that two lists, xs and ys, are sorted and contain distinct elements (denoted by "sorted xs", "distinct xs", "sorted ys", "distinct ys"). The theorem th...
The lemma "to_tree_order_ptrs_in_heap" assumes that the heap "h" is well-formed, has a well-defined type, and contains known pointers. It also assumes that the pointer "ptr" is transformed into a tree order "to" in the heap "h". If another pointer "ptr'" is in the set "to", then it shows that the pointer "ptr'" is in t...
The lemma l12_22_aux: Assuming that point P is not equal to point A, and point A is not equal to point C, and point P is between point A and point C, and point P and point A are on the same side of line segment BD, and angle BAP is congruent to angle DCP, it can be shown that line AB is parallel to line CD.