Datasets:
UDACA
/
Split-Isa-AF-MMA

Dataset card Data Studio Files
xet
 Community
Statement:
stringlengths
7
24.3k
The parts of "ik_oracle" are equal to "ik_oracle" itself.
If for any property P, if P is unitary then the transformation of P (denoted as t P) is also unitary, then the greatest fixed point expression of the transformation (denoted as gfp_exp t) is also unitary.
Multiplying any quantity "x" by the unit quantity on the left side results in a quantity equivalent to "x".
If "m" is less than or equal to "m'", then the VCG frame of "m" is also less than or equal to "m'".
If the interval from 'a' to 'b' equals some 'i', then the lower limit of 'i' is 'a' and the upper limit of 'i' is 'b'.
If a match 'm' does not have a negated discriminator 'disc'' and for all strings, 'disc'' does not apply to the extra string, then the abstract primitive of the match 'm' with respect to the discriminator 'disc'' also does not have a negated discriminator 'disc''.
If the norm of a bounded function 'f' is less than or equal to 'B', then the sum over all 'x' of 'l' to the power of 'x' times the application of the bounded function 'f' to 'x' is less than or equal to the sum over all 'x' of 'l' to the power of 'x' times 'B'.
The lemma named "concat_len_one" assumes that the length of the list "us" is 1. It shows that the concatenation of the list "us" is equal to the head of the list "us".
The lemma named "mono2mono_sup" assumes two functions 'f' and 'g'. Both 'f' and 'g' are monotone with respect to the order 'ord' and the less than or equal to relation. The lemma shows that a function that takes an input 'x' and returns the supremum (or least upper bound) of 'f x' and 'g x' is also monotone with respec...
If for every element 'i' in a set 'I', the function 'F' applied to 'i' is in the set 'X' applied to 'i' and guarantees the set 'Y' applied to 'i', and if 'F' is OK for 'I', then the join of 'I' and 'F' is in the intersection of the image of 'X' over 'I' and guarantees the intersection of the image of 'Y' over 'I'.
If the assumption "is_sig_red sing_reg top_tail F p" holds true, then it shows that the "ordered_powerprod_lin.is_le_rel top_tail" is also true.
The lemma named "finite_long_chain_with_alt2" assumes that: 1. Q is a finite set, 2. there is a local long chain ordered by function f on set Q, 3. the function f maps 0 to x, 4. the function f maps the cardinality of Q minus 1 to z, 5. x, y, and z are elements such that y is in set Q. Under these assumptions, it show...
The difference between the polynomial of variables with removed duplicates from the multiset "xs" and the polynomial of variables "xs" belongs to the ideal of the boolean polynomial in the module "More_Modules".
If an integer 'b' is greater than 2, then the alpha function of 'b' applied to 'n+1' is less than or equal to 'b' raised to the power of 'n'.
A document pointer reference of any kind is not a shadow root pointer kind.
If we apply the function 'decompV' to the result of mapping the function 'LVal source' over a list 'vl', with 'source' as an argument, we will get back the original list 'vl'.
The codomain of the identity substitution is an empty set.
The "icrit_spec" function is specified for a lambda function that always returns false, regardless of its inputs. This lambda function is of type "icritT" with generic types 't, 'b, 'c, 'd, where 'b is a field.
The lemma named "vcg_of_SPEC_np" assumes that a function "f" is defined as the specification of a postcondition. It shows two things: 1. If the specification of the postcondition is less than or equal to "m", then "f" is also less than or equal to "m". 2. If the specification of the postcondition is less than or eq...
The lemma named "semantics_in_vertices" assumes that "A" is a graph and "(a,b)" is an edge in "A". It shows that both "a" and "b" are vertices in the graph "A".
If set V is a subset of the set of all integers from 0 to a (not inclusive), then the cardinality (or size) of set V is equal to a if and only if set V is equal to the set of all integers from 0 to a (not inclusive).
The set resulting from folding the list "xs" with the append operation, starting from the initial list "i", is equal to the union of the set of elements in "i" and the union of the sets of elements in each list in "xs".
If a problem "Pi" is a valid STRIPS problem and a model "A" satisfies the formula "Phi" of problem "Pi" at time "t", then the mapping of the decoded state of problem "Pi" and model "A" over the range from 0 to the successor of the length of the inverse of the formula "Phi" of problem "Pi" and model "A" at time "t" is e...
An element "x" is in the combination of "s" and "t" at position "k" if and only if "x" is in "t" at position "k" or "t" at position "k" is empty and "x" is in "s" at position "k".
Converting the intersection of two lists, xs and ys, to a list is equivalent to finding the intersection of the lists obtained by converting xs and ys to lists individually.
The lemma is named "parAssocLeft". It fixes a psi-calculus with elements Psi, P, Q, and R, and a relation Rel. It assumes that the relation Rel is equivariant (eqvt). It also assumes four conditions (C1 to C4): C1: For any Psi', S, T, and U, the tuple (Psi', (S parallel T) parallel U, S parallel (T parallel U)) is i...
The lemma named "mat_col_first_last_append" assumes that the number of columns in matrix A equals m plus n. It shows that if you append the last n columns of matrix A to the first m columns of matrix A, you will get matrix A itself.
If there is no match in the comparison of "nm" and "vs", then there will also be no match in the comparison of "nm" and the list formed by prepending "v" to "vs".
The lemma named "abc_only_cba" assumes an order of "a, b, c". It shows that the orders "b, a, c", "a, c, b", "b, c, a", and "c, a, b" are not true.
The lemma named "subgame_ParityGame" demonstrates that the subgame "V'" is a Parity Game.
For any process P in the pi calculus, if tau is applied to P, then it weakly simulates tau leading to P.
The lemma named "awalk_ends" assumes two arc walks: one from point u to point v following path p, and another from point u' to point v' following the same path p. It shows that either the path p is not empty and the starting points (u and u') as well as the ending points (v and v') of both walks are the same, or the pa...
If x is less than or equal to y, then the star operation applied to x and z is less than or equal to the star operation applied to y and z.
The lemma named "compact_bot" assumes that "x" is the supremum of an empty set. It shows that "x" is compact in the complete partial order defined by the supremum operation and the less than or equal to relation.
A tuple (x', b', c') is in the set created from the list that is the concatenation of list Γ' and the list containing the single tuple (x, b, c) followed by list Γ if and only if the tuple (x', b', c') is in the union of the set created from list Γ', the set containing the single tuple (x, b, c), and the set created fr...
If X is a part of G, and G is a subset of the parts of H, then X is a part of H.
Five times the arctangent of one-seventh plus two times the arctangent of three-seventy-ninths equals one-fourth of pi.
The Kauffman matrix of the right-over braid is equal to the Kauffman matrix of the left-over braid.
The representation of the partial order relation "r" is in the set of admissible sequences.
The lemma named "formulaTrueAppend" is defined with two fixed formulas, "formula1" and "formula2", and a valuation. It shows that the truth of the combined formula (formula1 appended with formula2) under a given valuation is equivalent to the truth of formula1 and the truth of formula2 under the same valuation.
The lemma neqs1 shows that: 1. The function that maps x to the pair (diamond_x x, diamond_y of diamond_x x) is not equal to the function that maps x to the pair (diamond_x x, negative of diamond_y of diamond_x x). 2. The function that maps y to the pair (negative of diamond_y of diamond_x y, diamond_x y) is not equal...
The lemma named "sshiftr_clamp_neg" assumes that the length of 'a is less than or equal to n and the signed integer of x is less than 0. It shows that the logical right shift of x by n equals -1, where x is a word of length 'a.
"0 is congruent to 1 modulo n" if and only if "n equals 1 or n equals -1", for any integer n.
The lemma named "bij_betw_image_Fpow" assumes that there is a bijective function 'f' between sets 'A' and 'B'. It shows that there is also a bijective function between the image of 'f' and the function power sets of 'A' and 'B'.
If the complex derivative of function f at point x is df, and the complex derivative of function g at point x is dg, then the complex derivative of the function that subtracts g from f at point x is equal to df subtracted by dg.
If set S is a subset of set T, then the right ideal generated by set S is also a subset of the right ideal generated by set T.
The lemma named "nonneg_sqrt_nat_or_irrat" assumes that the square of "x" equals the real number "a" and that "x" is greater than or equal to zero. It shows that "x" is either a natural number or not a rational number.
The lemma named "state_ext" assumes that for any state 's', running state 'm' with 's' is equal to running state 'n' with 's'. It then shows that state 'm' is equal to state 'n'.
If 'x' is the least upper bound of 'S' in relation 'R', then 'x' is less than or equal to all upper bounds of 'S' in relation 'R'.
The lemma "on_circline_cmat_cvec_unit" shows that a point (z1, z2) is on the unit circle defined by the circline matrix if and only if the product of z1 and its complex conjugate equals the product of z2 and its complex conjugate.
: Assuming that two models M1 and M2 are observable, and they both converge at points u' and v'. Also, for any x2, u, and v, if x2 is in the set of list g, u is in x2, v is in x2, u is in the language of M1, and u is in the language of M2, then M1 and M2 converge at u and v. Furthermore, if x2 is in the set of the l...
If "phi is in v" and "not psi implies not phi is in v", then if "phi is in v", it implies that "psi is in v".
The call-context restriction of an empty set on any graph G is equivalent to the bottom element.
If the height of the tree 'r' is greater than 0, then the inorder traversal of the tree resulting from the function 'treeD' applied to the node 'node21' with parameters 'l', 'a', and 'r', is equal to the concatenation of the inorder traversal of the tree resulting from the function 'treeD' applied to 'l', the element '...
The lemma named "finite_inf_linorder_ne_ex" is about a function 'f' that maps from any type to a type that forms a complete lattice and a linear order. Given the assumptions that 'S' is a finite set and 'S' is not an empty set, it shows that there exists an element 'x' in 'S' such that the infimum (greatest lower bound...
The lemma named "poly_strict_mono_E" assumes the following conditions: 1. The polynomial "p" is strictly monotonically increasing with respect to the variable "v" (denoted by "poly_strict_mono p v"). 2. For any variable "w" that is not equal to "v", the function "f" at "w" is equal to the function "g" at "w" (denoted ...
The lemma named "test_s_prod_aux2" assumes that the pair (a, A) is in the relation R and the pair (a, {a}) is in the relation P. It shows that the pair (a, A) is in the composition of the relations P and R.
The encryption function of the cipher for a given parameter eta is a perfectly lossless converter. This converter operates between two interfaces: the first interface is a uniform interface over all lists of length eta, and the second interface is a combination of a uniform interface over all lists of length eta and a ...
The lemma named "exists_asSort_wlsAbs" assumes that the pair (us, asSort xs) is in the set defined by the function "isInBar". It shows that there exists an A such that A is a well-sorted abstraction of the pair (us, asSort xs).
The greatest common divisor of any number 'a' and zero is equal to the normalized value of 'a'.
The set of positions in a term 's' is a subset of the positions in the term resulting from the substitution of 's' with 'σ'.
: Given two unique roots "x" and "y" of two polynomials "p1" and "p2" respectively, with "x" defined as the unique root of (p1,l1,r1) and "y" as the unique root of (p2,l2,r2), and assuming that both polynomials satisfy certain conditions (poly_cond2), and that for any given bounds (l1,r1,sr1) and (l2,r2,sr2), the func...
There exists a vertex path 'r' in graph 'G' that starts at vertex 'u' and ends at vertex 'w'.
The lemma named "crel_vs1_execute_Some" assumes that "crel_vs1 R a b" and "P heap". It then concludes that there exist some 'x' and 'heap'' such that the execution of 'b' with 'heap' results in 'Some (x, heap')' and 'P heap'' is true.
If the program counter "pc1" is equal to "a", and the semaphore "sem" is greater than 0, then "alpha1" is enabled.
If "ms" is an element of "RPD mr", then the set of free variables in the regular expression obtained from "ms" and "φs" is a subset of the set of free variables in the regular expression obtained from "mr" and "φs".
The reflexive transitive closure of the join of 'x' and 'y', when multiplied by 'z', is equal to the reflexive transitive closure of 'x' multiplied by the join of 'y' multiplied by the reflexive transitive closure of the join of 'x' and 'y' multiplied by 'z', and 'z'.
The interpretation function "Irint" of the integer constant "IC 0" with respect to any variable state "vs" equals 0.
The lemma named "akra_bazzi_term_floor_subtract" in the Akra_Bazzi theory introduces the Akra-Bazzi term. It assumes that: 1. A real number 'b' is greater than 0. 2. 'b' is less than 1. 3. The real value of 'x₀' is less than or equal to 'b' times the real value of 'x₁' minus 'c'. 4. The sum of 'c' and the product of (1...
If we have a well-formed graph with nodes V and edges that include a new edge e in addition to existing edges E, and if the set of offending flows for the graph with nodes V and edges E under policy nP is empty, but the set of offending flows for the graph with nodes V and edges that include the new edge e is not empty...
The lemma "proj_proj_set_subset" states four conditions: 1. The set of the projection of the projection of A at indices n and m is a subset of the set of the projection of A at index n. 2. The set of the projection of the projection of A at indices n and m is a subset of the set of the projection of A at index m. 3. T...
If a set A is a subset of the lower bound set of the antecedent of z and A is not an empty set, then the minimum element of A is in A and for all elements x in A, the minimum element of A is less than or equal to x.
: if the evaluation of 'b' under the context 'Γ' equals 'false', then the evaluation of the conditional expression '(P when b otherwise Q)' under the same context 'Γ' equals the evaluation of 'Q' under the same context 'Γ'.
The lemma named "less_subset_eq_sum_list" assumes that "i" is less than the length of "CAi". It shows that the "i"-th element of "CAi" is a subset of the sum of the list "CAi".
The set of all polynomial powers is equal to the universal set.
For any rational polynomial 'p', raising 'p' to the power of 'm' and then raising the result to the power of 'n' is equivalent to raising 'p' directly to the power of 'm' times 'n'. Here, 'm' and 'n' are natural numbers.
The lemma named "subst_mono_neq" assumes that "t" is a proper subterm of "u". It shows that if you substitute "s" into "t", the result is a proper subterm of the result of substituting "s" into "u".
If "xs" is a terminal list, then the result of applying the function "tmap" with functions "f" and "g" to "xs" is a terminal list with "g" applied to the terminal element of "xs".
If 'n' is a number, 'φ' is a formula with no free variables, and 'n' does not prove 'φ', then it can be formally proven that the negation of the substitution of 'Pf' with 'n' and 'φ' is true.
For all lists 'ys', if a list 'xs' is equal to 'ys', then the length of 'xs' is equal to the length of 'ys', and for all indices 'i' less than the length of 'xs', the 'i'-th element of 'xs' is equal to the 'i'-th element of 'ys'.
"isViewOn" is valid for a view that is a union of V and N, with N being an empty set, and C being a subset of V, on the event set of SES.
A set P is dependent if and only if there exists an element 'a' in P such that 'a' is in the span of the set P excluding 'a'.
Observing the execution of the "drinks" machine starting from state 0 with the sequence of actions "select coke", "insert 50 cents", "insert another 50 cents", and "vend", results in the sequence of states: initial state, state with 50 cents, state with 100 cents, and state with a coke.
The lemma named "assign_vacuous_skip" assumes that "x" is a very well-behaved lens. It shows that assigning "x" to itself is equivalent to doing nothing (skip).
For any matrix A, where A is of a certain field type, the transformation of the basis of the null space of A into an iarray (integer array) is equal to the basis of the null space of the iarray form of matrix A.
If an element 'x' is in the set of all but the last elements of list 'xs' or in the set of all but the last elements of list 'ys', then 'x' is in the set of all but the last elements of the list formed by appending 'xs' and 'ys'.
Multiplying the constant zero with any polynomial P results in the constant zero.
If the pair (a, b) is in the monom_rel relation and the pair (a', b') is also in the monom_rel relation, then the pair resulting from merging monoms a and a', and b and b' is also in the monom_rel relation.
, assuming "w" is not an empty list, the property "P" holds infinitely often in the cycle of "w" if and only if there exists an element in the set of "w" for which property "P" holds.
The third element in the sequence of prime numbers is 5.
The lemma named "closed_shiftpath" assumes that the end of path 'g' is the same as the start of path 'g', and 'a' is a value between 0 and 1 inclusive. It shows that the end of the shifted path (shiftpath) 'a g' is the same as the start of the shifted path 'a g'.
For a fixed matrix A, which is of any field type, the null space of A is a vector subspace.
The substitution of a term 's' at position 'n' in a term 't' is equivalent to the term 't' where 's' is lifted by 'n' levels and then substituted at position 'n'.
If a function "x" is monotonically convergent to "y" (both are real-valued functions) and a function "a" is monotonically convergent to "b" (both are also real-valued functions), then the function that maps "m" and "w" to the sum of "x m w" and "a m w" is monotonically convergent to the function that maps "w" to the su...
A property Q holds for a conditional expression (lcond) with conditions lp, lp', and b evaluated at state s if and only if: when b evaluated at state s is true, property Q holds for lp, and when b evaluated at state s is false, property Q holds for lp'.
The domain of the finite map 'm' when using the function 'fmdom'' is equal to the finite set of the domain of the same map 'm' when using the function 'fmdom'.
The lemma named "path_change" assumes that "u" is in set "S", "v" is not in set "S", there is a path "p" from "u" to "v" in graph "g", and "p" is simple. It then concludes that there exist vertices "x" and "y", and paths "p1" and "p2", such that the edge "(x,y)" is in the set of edges of path "p", "x" is in set "S", "y...
The lemma named "cInf_abs_ge" is defined for a set 'S' of elements of a type that is both a linearly ordered integral domain and a conditionally complete linear order. It assumes that 'S' is not an empty set and that for any element 'x' in 'S', the absolute value of 'x' is less than or equal to 'a'. It shows that the a...
Given two real-valued functions Px and Py, and a real-valued function Pxy of a pair of values, under certain conditions, the mutual information between X and Y is equal to the entropy of X minus the conditional entropy of X given Y. The conditions are as follows: - S and T are sigma-finite measures. - Px and Py are ...
The number of parameters of the polynomial part of the normalized polynomial of a Taylor model "t" is less than or equal to "X", given that the number of parameters of the polynomial part of the Taylor model "t" is less than or equal to "X".