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If the indices 'i' and 'j' are less than the dimensions of the rows and columns of matrix 'A' respectively, then the element at position (i,j) in the matrix obtained by swapping rows 'k' and 'l' in 'A' is equal to the element at position (l,j) in 'A' if 'k' equals 'i', or the element at position (k,j) in 'A' if 'l' equ...
The function "sorted_wrt" is parameterized. It takes two arguments: a binary relation on elements of type A and a list of elements of type A. The function returns a boolean value indicating whether the list is sorted according to the given relation. The relation between the input and output of this function is a boolea...
The bit representation of a word 'a' is equal to the bit representation of the unsigned integer value of 'a'.
For any field with characteristic 0, the interpretation of a number 'x' is equal to the interpretation of a number 'y' if and only if the normalized form of 'x' is equal to the normalized form of 'y'. This is denoted by "?lhs" which stands for the left-hand side of the equivalence.
If the reduction of a single term "p" to "q" by a function "f" at a term "t" is assumed, then the lookup of the term "t" added to the leading term of "f" in "q" equals zero.
If the length of the bit list "blb" is greater than or equal to the length of the bit list "bla", then the result of the function "rbl_mult" applied to "bla" and the first "length of bla" elements of "blb" is equal to the result of "rbl_mult" applied to "bla" and "blb".
If A and B are complex subspaces, then the sum of A and B is also a complex subspace.
Twice the value of the function "two_phase" applied to "r", plus the value of the function "two_step" applied to "r", equals "r".
A complex expression is non-negative when variables are shifted twice in a set V and mapped to a function, if and only if the same complex expression is non-negative when variables are shifted once in the same set V. The function in the first case is a natural number case function with t1 and t2 as arguments and in the...
For a given relation R and a program P, if the program transitions from a state represented by the tuple (e, (h, l, sh), b) to a state represented by the tuple (e', (h', l', sh'), b') in zero or more steps, and if two conditions hold: 1. The relation R holds for any state represented by the tuple (e, h, l, sh, b) wit...
If x is less than zero, then the function f has a derivative of zero at x.
The quasi-Borel space of the product of a set of quasi-Borel spaces is equal to the product of the quasi-Borel spaces of each individual element in the set.
The lemma named "plus_monotone_strict_left" assumes that "s" is strictly less than "t". It shows that "u" added to "s" is strictly less than "u" added to "t".
If a function application "t $ u" has a type "R" under a type environment "Ts", and if the function "t" has a function type "U -> R" under the same type environment "Ts", then the argument "u" must have the type "U" under the type environment "Ts".
If the pair of well-formed program P and type safety property P leads to a correct state with no exception, a heap 'h', and a list of frame stacks 'frs' where 'a' is the first element, then the same pair also leads to a correct state with no exception, the same heap 'h', and the list of frame stacks 'frs' without the f...
If tree 't' is in the set 'bq', then the priority of the minimum tree obtained from 'bq' is less than or equal to the priority of 't'.
"r_prefixes_aux" is a recursive function with 3 arguments.
If an arrow 'f' is an isomorphism from 'a' to 'b' in the category, then 'f' is also a monomorphism from 'a' to 'b' in the same category.
If a rule set, represented by Γ, maps a chain to a list containing a rule with a match condition m and a call to the same chain, and if a packet p matches this condition under a certain match environment γ, then the rule set cannot decide the fate of the packet, which contradicts the assumption. Therefore, such a recur...
The lemma named "mult_sv_ex_id2" shows that there exists a complex number 'k' such that 'k' is not equal to zero and 'v' equals 'k' multiplied by 'v'.
If the height of tree 'm' is greater than 0, then the inorder traversal of a 4-node tree (with keys 'a', 'b', 'c', and subtrees 'l'', 'm', 'n', 'r') is equal to the concatenation of the inorder traversal of the left subtree 'l'', the key 'a', the inorder traversal of the tree 'm', the key 'b', the inorder traversal of ...
The lemma named "mertens_bound_real_strong" is about a fixed real number 'x'. Assuming 'x' is greater than or equal to 1, it shows that the difference between the Mertens function of 'x' and the natural logarithm of 'x' is within the range of greater than -1 minus 9 over pi squared minus the natural logarithm of 1 plus...
If "t" is in the set of ranked trees "A", then there exists a "q" in the set "Q" such that "q" is accessible from "t" under the transition function "delta".
The lemma named "regular_registerI" assumes that "F" is a register and "F" and "G" are complements. It also assumes that the operation of "F" followed by "G" on the tensor product of "selfbutterket default" and "a" equals "selfbutterket default". If these conditions are met, it shows that "F" is a regular register.
The lemma "shortest_path_not_inf" assumes that there is a connection between vertex u and vertex v. It shows that the shortest path between vertex u and vertex v is not infinite.
If set A is countable, then the supremum (the least upper bound) of the set resulting from inserting an element 'a' into set A is equal to the supremum of 'a' and the supremum of set A.
For any list of pairs 'xs', the length of the result of applying the function 'ci' (which takes a layout of 'xs' appended with the length of 'x' as its first argument) to each pair in 'xs', is equal to the length of the result of applying the function 'ci' (which takes a layout of 'xs' as its first argument) to each pa...
If the projection of list 'l' on set 'X' is an empty list, then the projection of list 'l' on the intersection of sets 'X' and 'Y' is also an empty list.
If you drop 3 bits from the signed 16-bit integer -1705, the result will be -214.
The lemma named "routing_ipassmt_wi_disjoint" assumes that "tbl" is a valid set of prefixes, "a1" is not equal to "a2", and both "(a1, b1)" and "(a2, b2)" are elements in the set produced by the function "routing_ipassmt_wi" applied to "tbl". Under these assumptions, it shows that the intersection of the sets produced ...
If a map "m" maintains an invariant and the selection function "sel'" applied to "m" and a predicate "P" returns a pair (u, v), then for any "u" and "v" such that "u" maps to "v" in the abstract version of "m" and the pair (u, v) satisfies the predicate "P", a certain unspecified statement (thesis) holds. If these cond...
, given the assumption that the pair (q, CInr p) is an element of the transitive closure of the epsilon function applied to the Inf_automata A with parameter Q, we can obtain a q' such that q is equal to CInr q'.
If a location "al" at address "a" in a heap "h" under a program "P" is of type "T", then the type of the address location "al" at "a" in "h" under "P" is "T".
The lemma named "update_by_fun_raw_eq_update_by_raw" assumes that the ordered association list "xs" is valid. It shows that updating the list "xs" by a function "f" at a key "k" is equivalent to updating the list "xs" at the key-value pair "(k, f (lookup_raw xs k))".
If the product of Lambda and (v minus 1) equals the product of k and (k minus 1), then the balanced incomplete block design with parameters V, B, k, and Lambda is symmetric.
The lemma named "ab_semigroup_add_on_with_transfer" includes lifting syntax and assumes that "A" is bi-unique. It shows that the relation of an additive abelian semigroup over a set "A" is preserved under a certain transformation, specifically, a transformation that relates sets and functions on "A". This is expressed ...
The count of elements satisfying a predicate P in a stream ω is infinite if and only if it is always the case that there is some event where P is satisfied in the stream ω.
A matrix A with the element at position i, j removed (denoted as "mat_erase A i j") belongs to the set of matrices with "nr" rows and "nc" columns (denoted as "carrier_mat nr nc") if and only if the original matrix A also belongs to the set of matrices with "nr" rows and "nc" columns.
If a function 'f' is injective on a set 'S' and 'A' is a countable set, then the set of all 'x' in 'S' such that 'f(x)' is in 'A' is also countable.
The lemma named "Zero_int_eqvt" shows that the action of any permutation "pi" on the integer zero results in zero.
If 'x' is a part of the set containing 'e1' and 'e2', and 'x' is not a part of the set containing only 'e1', then 'x' must be a part of the set containing only 'e2'.
The logical disjunction (or) of "a" and "c" being equal to the logical disjunction of "b" and "c" is equivalent to "a" being equal to "b" or "c" being true.
The lemma ssaBS_I: assumes that there is a path from the entry of graph g to node n through nodes ns then there exists a state s such that the nodes ns are in the semi-pruned SSA form in graph g.
For a given function 'f' that maps real numbers to real numbers, if 'f' is continuous on the interval from 0 to 1, and 'e' is a positive real number, then there exists a natural number 'N' such that for all natural numbers 'n' and all real numbers 'x' in the interval from 0 to 1, if 'n' is greater than or equal to 'N',...
If the sets of A are a subset of the sets of B, then the space of A is a subset of the space of B.
If a function F follows another function g, then F leads to a state where the prefix of g is less than or equal to the prefix of f.
The absolute value of a Dirichlet series with a constant term converges for any complex number 's'.
The lemma Bpow_ordLeq_Func_Field assumes that 'r' is a cardinal order, the field of 'r' is not empty, and 'A' is not finite. It shows that the cardinality of the B-power of 'r' and 'A' is less than or equal to the cardinality of the function from the field of 'r' to 'A'.
If x is greater than 0, then the canonical number form (cnf) of x is equal to the ordinal logarithm of x with base omega, followed by the cnf of the difference between x and omega raised to the power of the ordinal logarithm of x with base omega.
If a tree "t" is a red-black tree, then the tree that results from inserting a value "v" at position "x" in "t" is also a red-black tree.
For a multiset, adding an element to a multiset B and then unioning it with a multiset A is the same as unioning A and B first and then adding the element to the resulting multiset.
The maximum non-zero element "x" implied by "a" is less than or equal to the maximum value in the set of a list of natural numbers "a".
The lemma named "subst_prm" shows that applying a permutation to the substitution of a term N for a variable z in a term M is equivalent to substituting the permuted term for the permuted variable in the permuted term M.
The lemma G_lower assumes that x is greater than 0 and n is greater than or equal to 1. It shows that the Gamma series of x and n is less than or equal to G of x.
If the liveness property holds for a set A and a property P, and if for any infinite sequence t, t being in the infinite closure of A and the concatenation of s and t being in P implies R, and if s is not in the finite closure of A implies R, then R holds.
"PR_CONST of x" is equivalent to "PR_CONST of x".
The transitive closure of a relation 'x' composed with the identity relation is equal to the relation 'x' composed with the identity relation.
If "d" is not equal to "d'", then the "d'"th element of the child of "p" in direction "dir" at depth "d" is equal to the "d'"th element of "p".
The theorem named "n_sequences_card" assumes that "A" is a finite set. It shows that the cardinality (or the number of elements) of the set of all "n" sequences from "A" is equal to the cardinality of "A" raised to the power of "n".
Extracting from the merge of an identity function on a set A and another function f is equivalent to the union over A of the extraction of the function f.
The lemma "Mr_Rel_components" shows that: 1. The natural transformation map of "Mr_Rel" in the category "CC" with respect to "b" is equal to the first component of the arrow "A" in the set of objects of "CC" to the set containing "b". 2. The domain of the natural transformation of "Mr_Rel" in the category "CC" with r...
For a given set S in a Euclidean space, if S is a polytope, then S is also a polyhedron.
The lemma "term_instance_of_reach_state" assumes that "q" is an element in the transition action derivation of the normal form of a transition action "R" with respect to a set of variables "F" adapted to a term "t", and that "t" is a ground term. It shows that "q" is less than or equal to the bottom element of "t".
"Fls" is a Sigma formula.
If you apply the context 'C' to the term 's' at the hole position of 'C', you will get 's'.
If the law of non-contradiction holds for a certain condition, then the existential closure of the quasi-metric space also holds for that condition.
A function 'f' is admissible under the conditions of a complete partial order (ccpo) with least upper bound (lub) and order (ord), if and only if the function 'f' is monotonically continuous (mcont) with respect to the least upper bound (lub), order (ord), supremum (Sup), and less than or equal to relation (\<le>).
The lemma "ltlr_con_param" states the following rules for the autoref tool: 1. The "True_ltlr" function is related to itself under the "ltlr_rel" relation parameterized by "R". 2. The "False_ltlr" function is related to itself under the "ltlr_rel" relation parameterized by "R". 3. The "Prop_ltlr" function is related t...
A constant function, which always returns a certain value 'c', belongs to the set 'Prim1'.
The well-formedness of a Java method declaration is a well-formed program in the context of the program "tprg".
The lemma P is a simple and automatically introduced statement that P belongs to the formula. And the free variables in P are equal to the set containing xx.
The lemma named "mask_out_add_aligned" assumes that "p" is aligned with "n". It shows that adding "p" to the result of a bitwise AND operation between "q" and the bitwise NOT of the mask of "n" is equal to the result of a bitwise AND operation between the sum of "p" and "q" and the bitwise NOT of the mask of "n".
A trace between states 's' and 's'' exists in the concatenation of traces 'tr1' and 'tr2' if and only if there exists an intermediate state 'sh' such that there is a trace between 's' and 'sh' in 'tr1' and a trace between 'sh' and 's'' in 'tr2'.
The inner product of vector x with the difference of vectors y and z is equal to the difference of the inner product of vector x with vector y and the inner product of vector x with vector z.
The lemma named "less_is_drop_replicate" is defined for a fixed 'x' which is a word of any length 'a'. It assumes that 'x' is less than 2 raised to the power of 'n'. It shows that the binary list representation of 'x' is equal to a list that starts with (length of 'a' minus 'n') number of 'False' values, followed by th...
The function M is asymptotically equivalent to the function (lambda x. natural logarithm of x plus two times Euler-Mascheroni constant minus one) plus a function that is big O of (lambda x. one divided by the square root of x).
The lemma shows the following: - A terminal is not a linear. - A linear is not a terminal. - A terminal is not a branch. - A branch is not a terminal. - A linear is not a branch. - A branch is not a linear. - A terminal equals another terminal if and only if their respective elements are equal. - A linear equals anothe...
The right injection function is a well-behaved prism.
If "v" is an element of "V", then the transformation "T" applied to the product of "a" and "v" (where "a" acts on "v") is equal to the product of "a" and the transformation "T" applied to "v" (where "a" acts on "T v").
If a pair (a1, b1) is in the set produced by the function "routing_ipassmt_wi" applied to a table "tbl", and another pair (a1, b2) is in the set produced by the function "routing_port_ranges" applied to the same table "tbl" and the universal word interval, then the set produced by the function "wordinterval_to_set" app...
In the context of a given system G and a set s, the set of classes initialized by s in G is a subset of the set of all classes in G.
"sch is less than or equal to FVar n" if and only if "sch is equal to FVar n".
The integer embedding of 0 in ring R is equal to the additive identity (zero) of the ring R.
The lemma named "gt_embedding_property" assumes that "t" is an embedding of "s" and "t" is not equal to "s". It shows that "t" is greater than "s".
The identity function (i.e., the function that maps each element to itself) is holomorphic (i.e., complex differentiable) on any given set 's'.
The first element of a pair (x, y) is x.
If we have two functors, "GG" from category "AA" to category "CC" and "HH" from category "BB" to category "CC", and if we have two morphisms, "BCG" from object "B" to object "C" in the comma category of "GG" and "HH", and "ABF" from object "A" to object "B" in the same comma category, then the composition of "BCG" and ...
The lemma establishes that for any two elements 'a' and 'b' of the type 'name2', swapping 'a' and 'b' in the pair '(a, b)' results in the pair '(b, a)'.
If a list "xs" contains distinct elements, then the cardinality (or size) of the inverse of "xs" under the operation "swaps sws xs" is less than or equal to the length of "sws".
If for any states sigma and sigma', and any actions a, if (sigma, s) is reachable in system A with state transition function S and update function U, and there is a transition from (sigma, s) to (sigma', s') in system A with action a, and the state transition function S holds for sigma, sigma', and a, then property P h...
The lemma, in the context of SecurityInvariant_preliminaries, is named "sinvar_valid_remove_flattened_offending_flows". It assumes that the graph with nodes as nodesG and edges as edgesG is a well-formed graph. The lemma shows that the security invariant holds for the graph with nodes as nodesG and edges as the set dif...
If "sz'" is less than or equal to "sz", then the mask of "sz'" is less than or equal to the mask of "sz" for any word length 'a.
"weakly bisimilar" is a weak bisimulation.
If the intersection of set A and the set from the probability mass function p is empty, then set A belongs to the null sets of the measure from the probability mass function p.
If "x" is an object in the category "C", "f" is a morphism from the object mapped by functor "F" at "x" to "a" in the category "D", and "h" is a morphism from "x'" to "x" in the category "C", then the composition of the arrow value of the natural transformation component of the adjunction at "(x, a)" with "f" and "h" i...
If the mode state 'mds' is equal to 'mds'' and the memory state 'mem'' is equal to the memory state 'mem' with 'x' updated to the evaluation of 'e', then the transition from the configuration of assigning 'e' to 'x' in the mode state 'mds' and memory state 'mem' leads to the configuration of stopping in the mode state ...
The lemma named "finite_enum_ext" assumes the following conditions: for any index 'i', if 'i' is less than the cardinality (size) of set X, then the 'i'th element of set X is equal to the 'i'th element of set Y. Additionally, both sets X and Y are finite and have the same cardinality. Under these assumptions, the lemma...
The lemma named "base_case_satisfy_equation_alt_R" is defined with two fixed variables: "p" which is a real polynomial, and "qs" which is a list of real polynomials. It assumes two conditions: first, the length of "qs" is 1, and second, "p" is not equal to 0. Given these conditions, it shows that the equation "satisfy_...
If a relation 'r' is totally ordered on a set 'A', then the relation 'r' restricted to the intersection of sets 'A' and 'B' is also totally ordered on this intersection.
If "t" is a Braun tree, and "m" equals the size of "t", then the size of the tree resulting from updating the tree "t" at position "Suc m" (the successor of m, i.e., m+1) with the element "x" is equal to the size of "t" plus one.