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The lemma named "cl_in_sub_cong" assumes that "X" is a subset of "E prime" and "E prime" is a subset of "E". It shows that the closure of "X" in "E prime" according to "E" is equal to the closure of "X" in "E prime".
If x is not in the domain of the function C at b, then the value of the function C at the sum of a and b for x is equal to the value of the function C at a for x.
If a function 'f' is a linear transformation from a vector space 's1' to another vector space 's2' and is surjective (meaning that for every element in the codomain 's2', there is at least one element in the domain 's1' that maps to it), then there exists a function 'g' that is a linear transformation from 's2' to 's1'...
The multiplication of two functions 'f' and 'g' is equal to the inverse NTT polynomial of the product of the NTT polynomials of 'f' and 'g' with their coefficients multiplied in a ring.
The group closure of the set obtained by multiplying every element of set S by a constant k is equal to the set obtained by multiplying the constant k with every element of the group closure of set S.
If a list 'xs' contains distinct elements and 'x' is an element in the list 'xs', then the second element of the pair resulting from splitting the reversed list 'xs' at 'x' is equal to the reverse of the first element of the pair resulting from splitting the list 'xs' at 'x'.
The lemma "disjunct_set_mset_diff" assumes that the intersection of multisets M and N is an empty multiset. It shows that the set of elements in the multiset resulting from subtracting N from M is equal to the set of elements in the original multiset M.
The alpha function applied to an empty hashed tree automaton is equal to an empty tree automaton. The second part states that an empty hashed tree automaton is a hashed tree automaton.
If 'i' is less than 'k', then the 'i'-th element of the sequence 'ts' is in the set of 'ts'.
The lemma named "H_edge_succ_tail_eqD" assumes that the successor of edge 'a' in the half-edge map 'HM' is 'b'. It shows that the tail of 'a' in the graph 'H' is equal to the tail of 'b' in the same graph.
The theorem LT2 assumes that if proposition F holds, then proposition G also holds. It shows that if proposition F holds, then it is possible that proposition G will hold.
If a set A is an element of the Vset i, then the union of set A is also an element of the Vset i.
The lemma "step_redexes" shows that for any 'r' and 'r'', if the pair (r, s) transitions to the pair (r', s') under the context 'Γ' and 'r' is in the redexes of 'c', then there exists a 'c'' such that the pair (c, s) transitions to the pair (c'', s') under the context 'Γ' and 'r'' is in the redexes of 'c''.
For any three lazy lists of unbounded domain (denoted as xs, ys, and zs), the operation "plusU" is associative. This means that adding the lists xs and ys together first and then adding zs is the same as adding ys and zs together first and then adding xs.
If a message containing an encrypted block (which includes Agent C, a nonce N, two keys KC2 and cardSK, and an arbitrary value X) and another arbitrary value Y is received by Agent B in the event set, and if cardSK is not in the symmetric keys and the event set is in the cardholder registration set, then either the key...
The lemma seq_charSbC shows that sequence g f is equivalent to the assertion that f is an arrow, g is an arrow, and g f is a sequence in category C.
Any element multiplied by the identity element (1) on the right side results in the same element.
The minimum value "kmin" of the function "f" over the set "PR" is finite.
If the following conditions are met: - "ys" is an element of the set "Joints YS" - The length of "xs" is equal to the length of "XS" - The length of "YS" is equal to the length of "XS" - The set created by pairing elements from "xs" and "XS" is the same as the set created by pairing elements from "ys" and "YS" Then, ...
The precondition set "U" for the "Askip" action at location "l" and state "s" is equal to the postcondition set "U" at the same location "l" and state "s".
If a function 'f' is holomorphic on a set 'S', 'S' is open, a complex number 'ξ' is in 'S', and 'n' is a positive integer, and if the 'n'-th derivative of 'f' at 'ξ' is not zero, and for any integer 'i' between 0 and 'n', the 'i'-th derivative of 'f' at 'ξ' is zero, then there exist a function 'g' and a positive real n...
The lemma "select_object_any_defined_Set" assumes that the selection of any object from a set, represented by the function "select_object_any0_Set", is defined for a given state tau. It then shows that the set from which the object is selected is not empty.
The size of the combination of a list and a function is equal to the size of the function plus the size of the list.
The set generated from a base set 'a' and the union of two ordinals 'i' and 'j' is equal to the union of the set generated from the base set 'a' and ordinal 'i' and the set generated from the same base set 'a' and ordinal 'j'.
The lemma named "compact_lemma_cart" is about a function 'f' that maps natural numbers to a vector in a Heine-Borel space. The lemma assumes that the range of 'f' is bounded. It then shows that there exist a limit 'l' and a strictly monotonically increasing function 'r' such that for any positive number 'e', eventually...
The multiplication of matrix A and vector v is equal to the matrix vector multiplication in NTT of matrix A and vector v.
Traversing a tree according to a path extended by 'ext' is the same as first traversing the tree according to the original path, and then traversing the resulting subtree according to 'ext'.
The lemma named "pre_oldest_configuration_obtain" assumes that "x" is in the set of pre-oldest tokens of "q" at time "n", excluding the successor of "n". It then concludes that there exists a "q'" such that the oldest token of "q'" at time "n" is "x", and "q" is the transition function of "q'" at time "n".
If the condition c is not satisfied by the state sigma, then the postfixed iteration of the function f over the trie t with keys ks and condition c results in the state sigma.
The relation "rel_envir" between functions "f" and "g" with respect to a predicate "p" is equivalent to the statement that for all "x", the predicate "p" holds for the pair of values "(f x)" and "(g x)".
The theorem named "wls_psubst_Var_simp1" assumes that "rho" is a well-sorted environment and that the value of "rho" at "xs" and "x" is None. It shows that the parallel substitution of "rho" into the variable "Var xs x" results in the original variable "Var xs x".
If 'n' is a unit, then the square-free part of 'n' equals 'n' itself.
If "x" is a border of "w", then "w" has a period of the length of "w" minus the length of "x".
The lemma named "meet_below2" is defined for any two elements x and y of a type 'a that is a finite meet complete partial order (cpo). It assumes that y is less than or equal to z. It shows that the meet (greatest lower bound) of x and y is less than or equal to z.
The lemma "all_ipd_imp_ret" assumes that "pi" is a path and for all "i", if "pi" of "i" is not equal to return, then there exists a "j" greater than "i" such that "pi" of "j" is equal to the immediate post dominator (ipd) of "pi" of "i". It shows that there exists a "j" such that "pi" of "j" is equal to return.
If 'p' is an element of 'PP', then 'p' is admissible.
If set A is finite, not empty, and is a subset of the set of all numbers up to n, then the maximum value in set A is less than or equal to n.
For a tropical semiring, the operation "trop" on elements f, u, v, w and a, where a is added to the result of the operation, is equivalent to the operation "trop" where a is added to each of u, v, and w before the operation.
There is a binary mismatch prefix between 'x' and 'y', where 'y' is repeated.
The postcondition of a command "C" is included in the set of annotations of the same command "C".
In the context of a probability space, the lemma named "distributed_RN_deriv" assumes that X is distributed with respect to measures M and S, and probability density function Px. It shows that almost everywhere in S, the Radon-Nikodym derivative of the density of S with respect to Px is equal to Px at x.
A set A is a proper subset of the universal set if and only if set A is not equal to the universal set.
If the following conditions are met: 1. The term 't' evaluates to the value 'v' under the rule set 'rs' and the variable environment 'Γ', 2. The term 't' is closed except for the free variables in the domain of 'Γ', 3. The variable environment 'Γ' is closed, 4. The term 't' is well-formed, 5. The variable environment ...
The function "ports_toString" with the parameters "spt: " and the tuple (1024,2048) will return the string "spt: 1024:2048".
The lemma named "ofilter_subset_iso" assumes that "r" is a well-order, and "A" and "B" are order filters of "r". It shows that "A" is equal to "B" if and only if there is an isomorphism between the restrictions of "r" to "A" and "B" that is the identity function.
If "u" is a prefix of "v" and the length of "u" is equal to the length of "v", then "u" is equal to "v".
For any given list 'is' appended with 'i', the function 'edge_to' with parameters 'c' and this list is equal to a tuple. This tuple consists of another tuple with 'c' and the list 'is' prepended with 0, and the result of the function 'in_port_at' with the same tuple as its parameter and 'i' as its index.
The lemma "trancl_map_both_Restr" assumes that the function "f" is injective on the set "X". It shows that the transitive closure of the image of the restriction of relation "R" to "X" under the function "f" applied to both elements of each pair, is equal to the image under the function "f" applied to both elements of ...
"ClassMain P" cannot be a class in the program "P". If "ClassMain P" is a class in the program "P", then it leads to a contradiction or false.
Let f be a function from a Euclidean space 'a to another Euclidean space 'b. If S is a set in the Lebesgue measure, then the function that maps x to f(x) if x is in S and 0 otherwise is Borel measurable with respect to the Lebesgue measure if and only if f is Borel measurable with respect to the Lebesgue measure restri...
The valuation of a linear polynomial "lp" with a value "val" is equal to the sum of the list that is obtained by mapping a function over the list of variables in "lp". This function takes a variable "x", looks up its corresponding value in the linear polynomial map of "lp", and multiplies it with the value of "x".
For all sets A, the closure of A is equivalent to the fuzzy set of A being approximately equal to the crisp set of A.
, given the assumptions that "f'" is a morphism from the target of "e0" to the target of "e1" and "f'" is an identity, it can be shown that "psi of f'" is equal to the composition of several other operations. These operations include the right unitor of "f'", the composition of the left unitor of "f'" and the target of...
Route 1 is strictly fresher than route 2 with respect to node i if and only if route 1 is as fresh as or fresher than route 2 with respect to node i and it is not the case that route 2 is as fresh as or fresher than route 1 with respect to node i.
The lemma "no_change_if_ge_length_t" assumes that there is a trace from an initial state to a final state, and that the indices 'i' and 'j' are greater than or equal to the length of the trace. Under these assumptions, it shows that the state of the trace at index 'i' is equal to the state of the trace at index 'j'.
The lemma "awalk_empty_ends" assumes that there is an arc walk from node 'u' to node 'v' with no intermediate nodes. It then shows that node 'u' must be the same as node 'v'.
Given a non-zero value "a", the point at infinity is in the set of an oriented circline "H" if and only if the point at infinity is in the set of the oriented circline obtained by applying a Moebius similarity transformation (with parameters "a" and "b") to "H".
The lemma named "freshInputTransition" is defined with fixed variables P, a, b, P', and c, where P and P' are processes in the pi calculus, and a, b, and c are names. The lemma assumes that: 1. There is a weak late input transition from process P to process P' with input channel a and bound output name b. 2. The name...
For two functions 'f' and 'g' that map real numbers to real numbers, given a positive real number 'd', if 'g' is the inverse of 'f' within the range of 'd' from 'x' and if 'f' is continuous within the same range, then 'g' is continuous at the point 'f(x)'.
If "braun" is a node with left child "l", value "a", and right child "r", then the multiset of the tree resulting from the "sift_down" operation on "l", "a", and "r" is equal to the multiset containing "a" plus the multisets of the trees "l" and "r".
The product of 'x' and the infinity power of 'x' is less than or equal to the infinity power of 'x'.
The dimension of zero is equal to one.
The lemma named "Ex_surj_conv'" from the theory "Reverse_Symmetry" in the field of "Combinatorics_Words" assumes that the function "f" is surjective. It shows that there exists an element for which the property "P" holds after applying the function "f" if and only if there exists an element for which the property "P" h...
If 'c' is a polynomial coefficient in set 'S' and 'n' is less than or equal to the first element of 'c', then the pair consisting of 'n' and the second element of 'c' is also a polynomial coefficient in set 'S'.
The lemma named "permutes_memb" is defined with a fixed function 'p' that maps from a set 'b' to another set 'b'. It assumes that 'p' permutes 'B' and 'a' is an element of 'A'. It also defines 'ip' as the inverse of 'p' according to Hilbert's Choice. The lemma then shows that 'a' is an element of 'A', the mapping of 'a...
If alpha is an ordinal number, and if 'a' is an element of the set Vset of alpha, and 'b' is an element of the set Vset of the successor of alpha, then the insertion of 'a' into 'b' is an element of the set Vset of the successor of alpha.
The lemma "Rf_model" assumes that an interpretation "I" satisfies the set difference of "N" and "Rf N". It shows that the same interpretation "I" also satisfies "N".
The difference of the cosine coefficient is equal to the negative sine coefficient for any given number 'n'.
A Reduced Ordered Binary Decision Diagram (ROBDD) does not depend on a variable "v" when it is a leaf node with a function "f".
: the concatenation of 'r' and 'q', followed by the concatenation of 'r' and 'q' raised to the power of 'k', and then followed by 'v', is equal to the concatenation of 'r' and 'q' raised to the power of 'k', followed by 'u', if and only if the concatenation of 'r', 'q', and 'v' is equal to 'u'.
The function "maximal_repetition_sets_from_separators" applied to a machine M is equal to the set of results from the function "maximal_repetition_sets_from_separators_list_naive" applied to the same machine M.
In the context of a functor, the domain of the object map of the comma category projection of the functor is equal to the objects of the comma category of the functor and an object 'b'.
If "k" is greater than 0, "D" is greater than 0, and "D" is less than "k", then the result of the function "mydiv" with "D" and "k" as arguments is the successor of 0 (which is 1 in natural numbers).
The lemma named "similar'_FnE" in the context of Launchbury's ValueSimilarity theory makes the following assumptions and shows a proposition: It assumes that the function 'f' is similar to the continuous function 'g' at the successor of 'n'. It also assumes that for any 'x' and 'y', if 'x' is similar to the continuo...
The lemma named "strict_mono_continuous_invD" is defined with a function 'f' that maps real numbers to real numbers. It assumes the following conditions: 1. The function 'f' is strictly monotonically increasing on the interval from 'a' to infinity. 2. The function 'f' is continuous on the interval from 'a' to infinity...
The function "next_tame" preserves the property "inv" for any given point "p". This is an invariant property.
If the pair (f, g) belongs to the identity relation and the pair (x, y) also belongs to the identity relation, then the pair (f x, g y) belongs to the identity relation.
For all lists 'xs' in the set of increasing lists, 'xs' is not an empty list.
The alphabet of the deterministic Rabin automaton (\<AA>' \<phi> xs ys) is the universal set.
The length of the derivation shift of D with respect to left and right is equal to the length of D.
The nth element of the sum of two vectors (a + b) is equal to the sum of the nth elements of the two individual vectors (a and b).
"hs_ins" is an imperative set insertion implementation for a hash set.
A natural number 'n' is greater than zero if and only if the extended real representation of 'n' is also greater than zero.
The lemma named "aux3lose" assumes two conditions: 1. For all 'j' less than or equal to 'k + d + 3', the function 'lose' applied to '(t + j)' equals '[False]'. 2. 'j' is less than or equal to 'k'. Under these assumptions, it shows that the function 'lose' applied to 'the successor of the successor of (t + j)' equals ...
The lemma named "bisimulation_upto_bisimulation" assumes that a function "f" is compatible and there is a bisimulation up to a relation "R" with respect to function "f". Under these assumptions, it is possible to obtain a bisimulation "S" such that "R" is less than or equal to "S".
If H is a submodule of M in ring R, then the quotient module of M and H forms an abelian group.
For any two elements x and y from a linearly ordered integral domain, if the product of x and y is greater than 0, then the number of sign changes in the list that starts with x, followed by y, and then the rest of the list (xs), is equal to the number of sign changes in the list that starts with y followed by the rest...
In the context of a variable finite sequence, if an element 'y' is not in the range of the sequence 'xs' and 'y' is not equal to 'x', then 'y' is not in the range of the sequence obtained by appending 'x' to 'xs'.
The lemma "HPhase1or2ReadThen_HInv4d_p" assumes that "HPhase1or2ReadThen s s' p d q" is an action and "HInv4d s p" is an invariant in state s for process p. It shows that the invariant "HInv4d s' p" holds in the state s' for the same process p. This means that if the action "HPhase1or2ReadThen" occurs transitioning fro...
The lemma named "list_all2_app_l" assumes two conditions: First, that "R" is a reflexive relation (as stated by "reflp R"). Second, that the relation "R" holds for all corresponding pairs of elements in the lists "l" and "r" (as stated by "list_all2 R l r"). Given these assumptions, it shows that the relation "R" a...
The lemma named "sumset_iterated_2" shows that the iterated sumset of set A for 2 times is equal to the sumset of set A with itself.
"eps test i r" is equivalent to "match test r i i".
The function "tm_to_nat" is injective.
The negation of "p" times "x" exported by the negation of "q" is equal to the double negation of "p" join (or union) "x" exported by the negation of "q".
If two lists, xs and ys, have the same length and for all indices in a set of indices, the elements at those indices in both lists are equal, then the elements at those indices in both lists are the same.
The lemma named "map_involution" assumes that applying function "f" twice is equivalent to the identity function. It shows that applying function "f" twice to each element in a list is equivalent to the identity function.
If a set F is a subset of the set of formulas and F is finite, then the conjunction of the set F is also a formula.
For a given matrix 'A' of a certain type 'bezout_domain', if 'bezout' is a Bezout extension, 'n' is less than the number of rows in 'A', the natural number equivalent of 'i' is less than or equal to 'n', and the element at the 'i'th row and 'j'th column of 'A' is not zero, then there exists a matrix 'P' such that 'P' i...
Any ordinal number 'x' modulo 0 is equal to 'x' itself.
The lemma named "mu_bound_Gramian_determinant" assumes that "l" is less than "k" and "k" is less than "m". It shows that the square of "mu" at indices "k" and "l" is less than or equal to the product of the Gramian determinant of "fs" at index "l" and the square of the norm of "fs" at index "k".