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If the second elements of the pairs in the list 'xs' are all distinct, and if the pair (x, y) is in the set 'xs', then the inverse permutation of the list 'xs' at the position 'y' is 'x'. |
If a conclusion B can be derived from a set of assumptions H, where the assumptions A1, A2, A3, A4, and A5 are added in the order A5, A1, A2, A3, A4, then the same conclusion B can also be derived from the same set of assumptions H, but with the assumptions added in the order A1, A2, A3, A4, A5. |
The lemma named "Std_imp_Arr" shows that if "T" is standard and "T" is not an empty list, then "T" is an arrow. |
In the lemma named "cring_monom_degree" within the context of UP_cring, we assume the following:
- "a" is an element of the carrier "R"
- "p" is defined as the monomial of "P" with coefficient "a" and degree "m"
- The nth power of "a" is not equal to zero
Under these assumptions, the lemma shows that the degree of the... |
Combining the top element with any relation R will result in the relation R itself. |
The Frechet derivative of the square root of a function 'f' at a point 'x' is equal to the function 'v' multiplied by the Frechet derivative of 'f' at 'x', scaled by 1 or -1 (depending on whether 'f' at 'x' is greater than 0) divided by twice the square root of 'f' at 'x'. This is under the condition that 'f' is differ... |
The lemma named "anonymity_prefs_from_table" assumes that two preference tables, xs and ys, are well-formed with respect to the same set of agents and alternatives. It also assumes that the multiset of preferences in xs is the same as the multiset of preferences in ys. Under these assumptions, it shows that the social ... |
The second projection map is a quotient map from the product topology of X and Y to Y if and only if the topological space of X is empty implies that the topological space of Y is empty. |
, given two faithful digraph homomorphisms, one from a digraph B to a digraph C, denoted as "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>a\<^sub>i\<^sub>t\<^sub>h\<^sub>f\<^sub>u\<^sub>l\<^bsub>\<alpha>\<^esub> \<CC>", and another from a digraph A to B, denoted as "\<FF> : \<AA> \<mapsto>\<ma... |
The excess of 'f prime' at 'v' is equal to the excess of 'f' at 'v' plus 'Delta'. |
Evaluating the statement, which assigns to the variable 'v1' the result of the external call to the method 'm1' of the address 'extaddr' with arguments 3 and 4 and gas limit 0, in the context where 'myaddr' has a balance of 100 and 'extaddr' has a method 'm1' that adds its two integer arguments, and 'v1' is initially 0... |
If the program counter pc1 is at location b, then action N1 is enabled for the variables x, y, sem, pc1, and pc2. |
The keys of the next OFSM (Observable Finite State Machine) table, given a machine M and a previous table, are equal to the states of the machine M. |
The difference of the integer powers of the formal Laurent series X (fls_X_intpow) with indices i and j is equal to the product of the formal Laurent series X with integer power i and the formal Laurent series X with integer power -j, where 'a is an instance of the semiring_1 class. |
The reduction relation "τred0r" for the external thread action "extTA", program "P", thread "t", and heap "h" from the state "(Val v, xs)" to state "s'" if and only if state "s'" is equal to "(Val v, xs)". |
The lemma named "j_gt_m_pigeonhole" assumes that "T" is a lower bound of "A" at "j" and "j" is greater than "m". It shows that there exists an "x" in the set from 1 to "j", a "y" in the same set, and a "z" in the set from 1 to "m", such that "x" is not equal to "y" and both "x" and "y" are in the set "A" at "z". |
If a functor 'F' maps from category 'A' to category 'B' under a certain condition 'alpha', and both 'A' and 'B' are tiny categories under the same condition 'alpha', then 'F' is a tiny functor from 'A' to 'B' under the condition 'alpha'. |
The evaluation of a polynomial 'p' under the mapping 'phi' from ring 'R' to semiring 'S' belongs to the set of functions that are extensional on the carrier of polynomial ring 'P'. |
The function 'to_nat' is strictly monotonically increasing. |
If the set containing elements "a" and "b" is equal to the set containing elements "c" and "d", and "a" is not equal to "c", then "a" must be equal to "d". |
The theorem named "wls_substEnv_psubstEnv_compose" assumes that "rho" is a well-sorted environment, "Y" is well-sorted of sort "ys", and "rho'" is also a well-sorted environment. It shows that the composition of the substitution of "rho'" into "rho" and then the substitution of "Y" for "y" of sort "ys" is equal to the ... |
If the axiom T holds for a proposition 'p', and if 'M' is a reflexive model, and if 'w' is an element of the set of possible worlds in 'M', then 'p' is true in the model 'M' at the world 'w'. |
"x precedes x in xs" is equivalent to "x is in the set xs". |
The set of free variable indices in a term 't' with base 'b' is finite. |
If 'f' is a closure operator on a set of type 'a', then the application of 'f' to the bottom element of the set equals the bottom element of the set. |
The transition function "delta" applied to the final state "qf" and any string of input symbols "bs" results in an empty set. |
A Hoare triple, which is a tool used in program correctness proofs, for an if-then-else statement is equivalent to the conjunction of two Hoare triples: one for the then-branch under the condition that both the precondition and the boolean condition hold, and one for the else-branch under the condition that the precond... |
If F is in the set where A ensures B, then F is also in the set where the intersection of the reachable set of F and A ensures B. |
The lemma named "raw_match_extend_map_preserve'" assumes that the function "raw_match_term" applied to terms t and u with substitutions subs results in some substitutions subs', and that the map subs' is less than or equal to the map subs''. It then shows that the term substitution of t with the converted substitutions... |
If the length of the list "ns" is equal to "n" and "i" is less than the length of the list "ls", then the evaluation of the "i"th element of the list "ls" (after each element of "ls" has been processed by the function "push_param" with "n" as an argument) with the list "ns" appended to "a" is equal to the evaluation of... |
The lemma named "linear_simps" assumes that "f" is a bounded linear function. It then shows the following properties:
1. The function of the sum of "a" and "b" is equal to the sum of the function of "a" and the function of "b".
2. The function of the difference between "a" and "b" is equal to the difference between th... |
The resultant of a multivariate polynomial and a polynomial, given by "resultant_mpoly_poly x p q", is equivalent to the basic implementation of the resultant, given by "resultant_impl_basic", applied to the multivariate polynomial converted to a multivariate polynomial of polynomials, "mpoly_to_mpoly_poly x p", and th... |
For a function 'f' mapping real numbers to a Banach space 'a', if 'f' is integrable on the interval from 'a' to 'b', 'x' is an element of the set from 'a' to 'b' excluding 'S', 'S' is a finite set, and 'f' is continuous at 'x' within the set from 'a' to 'b' excluding 'S', then the function that maps 'u' to the integral... |
If 'x' is an integer, then 'x + y' is an integer if and only if 'y' is an integer. |
If "K" is a subring of "R" and "K" is finite, then the set of all polynomials "f" that belong to the carrier of "K[X]" and have a degree of "n" is finite (denoted as "finite ?A"). Additionally, the set of all polynomials "f" that belong to the carrier of "K[X]" and have a degree less than or equal to "n" is also finite... |
If x is adjacent to y and x is not equal to y, then the intersection of x and y is a proper face of y. |
For any vector 'y' in a group with addition, if 'y', 'x', and 'z' are all vectors in the same carrier space of dimension 'n', then the sum of 'y' and the difference of 'x' and 'z' is equal to the difference of the sum of 'y' and 'x' and 'z'. |
The associative value of the construct "c_cons" with the pair of "x" and "y" and "z" as its arguments, when evaluated with "x", equals "y". |
The lemma named "norm_pn_in_cf" assumes that "x" is an element of "CF". It shows that the norm of "pn" function applied to "x" is also an element of "CF". |
The lemma named "enc_length" assumes that the sequence "w" is not empty and that the encoding of the pair (w, I) is equal to the encoding of the pair (w', I'). Under these assumptions, it shows that the length of sequence I is equal to the length of sequence I'. |
For a given complex matrix A of size n by n, if k is in the spectrum of A, then k is an eigenvalue of A. |
The length of an undefined or bottom lazy list is also undefined or bottom. |
If a program G successfully transitions from state s to state s' through term t, resulting in output Y', and if the type of term t is okay in state s under program G, and if program G is a well-formed program, then there exists a natural number n such that the program G can transition from state s to state s' through t... |
The lemma named "conv_abscissa_mono" assumes that for any 's', if the Dirichlet series 'g' converges at 's', then the Dirichlet series 'f' also converges at 's'. Under this assumption, it shows that the abscissa of convergence of 'f' is less than or equal to the abscissa of convergence of 'g'. |
"Pred x" is true and either "Pred [x]" is true or "Pred [x]" is not true. |
Extending the state of the combined oracle (obtained by adding oracle1 and oracle2) is equivalent to the combination (addition) of the extended states of oracle1 and oracle2 individually. |
The lemma "FalseD" assumes that "False" is in the set of probabilities of the primality test P for a number n. It shows that the number n is not prime. |
The list "lift_start_obs" for a given thread "t" and observations "obs" is distinct if and only if the observations "obs" are distinct. |
If from a set of assumptions A, we can deduce that the set of propositions ps implies the proposition q, and if the set of propositions ps is a subset of another set of propositions ps', then from the same set of assumptions A, we can deduce that the set of propositions ps' also implies the proposition q. |
, under the assumptions that the function "f" applied to the successor of any natural number "n" is equivalent to the function "h" applied to "n", and that the function "f" applied to 0 is equivalent to "g", it can be shown that the function "f" applied to any natural number "n" is equivalent to "g" if "n" equals 0, ot... |
All edges of a set A are a subset of the power set of A. |
If the multiset "m" is a subset of the set "s", and if the count of element "x" in multiset "m" is less than "n", then "x" is an element of the set "s". |
The lemma named "globIA" is defined with the following parameters: a constant 'c' of any type 'id', two functions 't' and 't'' that map a natural number to a configuration and a component respectively, and a natural number 'n'.
The lemma assumes three conditions:
1. There exists a natural number 'i' greater than or... |
If X is an element of kparts of H, then X is also an element of parts of H. |
The lemma assumes that the application "app" of instruction "i" with program "P", program counter "pc", type "T", maximum stack size "mxl", and exception table "xt" is compatible with the type "\<tau>". It then shows that the effect "eff" of instruction "i" on the compiled program "compP f P" with program counter "pc",... |
Setting a bit at position 'n' for the integer representation of 'k' is equal to the integer representation of setting a bit at position 'n' for 'k'. |
If matrix M is unitary, then the conjugate transpose of M (denoted as M^dagger) is the inverse of M. |
The formal power series exponential of any element 'c' from a field with characteristic 0 is not equal to 0. |
The set of vertices in the graph after removing a vertex "x" is a subset of the original set of vertices. |
If 'n' is less than or equal to 'd', then the interval from 0 to 'n' is a subset of the interval from 0 to 'd'. |
If the condition Fr_1b holds for a set F, then the set CoPw holds for the intuitionistic negation. |
The substitution of a polynomial 'p' raised to the power 'n' with a function 'f' is equal to the substitution of the polynomial 'p' with the function 'f' raised to the power 'n'. |
The relation "R23" refines the "m2_leak Rs" into "m3_leak Rs". |
The lemma named "find_set_path_compression_path_compression_semantics" assumes a precondition called "find_set_precondition" for a pair 'p' and 'x'. Under this assumption, it shows two things:
1. The function 'fc' applied to the result of 'path_compression' function with inputs 'p', 'x', and 'find_set' of 'p' and 'x' ... |
For any natural numbers i and j, and for any elements u, m, and n of a type 'a that is both a cancellative commutative monoid under addition and an ordered additive semigroup where addition implies less than or equal to, if i is less than or equal to j, then the sum of the i-fold addition of u and m being less than the... |
If "rpr A r" is true and "x" and "y" are elements of set "A", then either "x" is less than or equal to "y" according to relation "r", or "y" is less than or equal to "x" according to relation "r". |
The lemma named "invimage_vsingleton_nin" assumes that the element 'b' is not in the set 'A'. It shows that the inverse image of the set 'A' under the set consisting of the ordered pair (a, b) is the empty set. |
The lemma named "connected_iff6" assumes that "x" is an injective function and a partial function. It shows that "x" is connected if and only if the transitive closure of "x" followed by 1 and then followed by the transitive closure of the transpose of "x" is less than or equal to the reflexive transitive closure of "x... |
The set of states in the minimal Markov Decision Process is equal to the union of two sets. The first set is the set of all left injections of some trivial element in R. The second set is simply the set containing the right injections of True and False. |
The relation of a set with a sum relation is equivalent to the relation of the set with the left relation and the right relation. Specifically, the relation of set A1 to set A2 under the sum relation (composed of relations chi and phi) is equivalent to the relation of the preimage of A1 under the left injection to the ... |
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Assuming a proposition "phi" is a formula, and both "phi" and the negation of "phi" are provable, then falsehood is provable. |
The "set_filter" function preserves the "lss_invar" and "lss_α" properties when applied to the "lss_filter". |
If "e" is not an element of the set "E", then the value of "g" at "e" equals zero. |
: "The description of an object x, such that the relation A! holds between x and its property, and for all properties F, the relation between x and F is equivalent to the condition phi F, is equivalent to the condition phi G, in the domain of discourse dw." |
If a sparse vector is sorted (represented by "a#b#t"), then the first element of 'a' is less than the first element of 'b'. |
The supremum (or greatest element) of the set obtained by applying a function 'f' to each element 'x' in set 'Y' and then taking the supremum of the resulting set, is equal to the supremum of the set obtained by applying the function 'f' to each element 'x' in set 'Y' such that 'f' is applied to 'x' itself. |
The set of modified regular expressions of the third kind derived from a regular expression 'r' is finite. |
If 'u' is an element of 'V' and 's' is an element in the set from 'u' to 't', then 'u' is in the left tree of 's' and 't'. |
The function "ValN" is monotonically increasing with respect to the "below" relation in the context of continuous functions. |
The lemma "Constrains_UN" assumes that for any element "i" in set "I", "F" is constrained from "A i" to "A' i". It shows that "F" is constrained from the union of "A i" over all "i" in set "I" to the union of "A' i" over all "i" in set "I". |
If a variable 'v' is in all variables of graph 'g', then there exists a node 'n' in the set of nodes of graph 'g' such that 'v' is in all definitions of graph 'g' at node 'n'. |
The successor implication graph of a conjunctive normal form (cnf) and a literal (l) is equal to the finite set obtained by folding the function "succs_of_clause l" over the empty set and the conjunctive normal form. |
Given the assumption that the set Ws is a subset of the vector carrier of dimension n, if the set Vs is also a subset of the vector carrier of dimension n, then the linear combination of the list f with the concatenation of Vs and Ws is equal to the sum of the linear combination of the list f with Vs and the linear com... |
The lemma named "pvar_pow" assumes that "i" is an element of "I". It shows that the power of "pvar R i" in the polynomial ring "Pring R I" with a natural number exponent "n" is equal to the image under the function "mset_to_IP R" of the multiset representation of the natural number "n" with "i" as the base. |
If a quaternion 'q' belongs to the set of real numbers, then the norm of 'q' is equal to the absolute value of the real part of 'q'. |
In the context of a probability space, the probability of event A is less than or equal to 1. |
The degree of any numeral 'n' is equal to zero. |
For any complex number 'v', the product of 'v' and its conjugate belongs to the set of real numbers. |
If a pair (cl, f) is an element of CLF, and CLF is a subset of the set of all functions from the power set of any element cl' of CL to the power set of cl', then f is a function from the power set of cl to the power set of cl. |
If "h" is a bijective function, then the renaming of "F" by "h" being okay with the renaming of "G" by "h" is equivalent to "F" being okay with "G". |
For any set of owned locations 'O' and any pending write operation, the set of locations read from the pending write operation in the store buffer 'sb' that are also in 'O', is a subset of the union of 'O' and all locations acquired by the store buffer 'sb'. |
If there is no await command in a program 'c', and if the program 'c' with memory state 'mds' and memory 'mem' transitions to a new state with program 'c'', memory state 'mds'', and memory 'mem'', then there is no await command in the new program 'c''. |
Zero is included in the span of set S. |
A preference profile 'A' is valid if and only if the set of 'A' is a subset of the power set of parties excluding the empty set, and the size of 'A' is equal to the number of voters. |
If for all 'a', the rational number of integer 'f a' equals the polynomial 'p' of the rational number of integer 'a', then 'f' is a polynomial function. |
The lemma named "drop_apply_apply" assumes three conditions:
1. Function 'f' is in the carrier of the semialgebraic function ring 'SA k'.
2. 'a' is in the carrier of the p-adic number field 'Q_p' raised to the power 'n'.
3. 'b' is in the carrier of the p-adic number field 'Q_p' raised to the power 'k'.
Under these as... |
The set resulting from mapping a function that applies a logical OR operation to the results of functions A and B on each element v in a list vs, is equal to the set of all elements where a logical OR operation is applied to the results of functions A and B on each element v that is in the set of elements in the list v... |
If the result of the function "nat_to_bv_length" with inputs x and y is not an empty list, then converting this result back to a natural number using the function "bv_to_nat" will yield the original input x. |
Applying a substitution sigma to a ternary operation (trop) with functions f, u, v, and w is equivalent to applying the ternary operation to the results of applying the substitution sigma to u, v, and w respectively. |
If 'x' is less than or equal to 'y', and the supremum (least upper bound) of 'x' and 'z' is equal to the supremum of 'x' and 'w', then the supremum of 'y' and 'z' is equal to the supremum of 'y' and 'w'. |
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