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If 'q' is a prefix of the sequence 'p' without its last element, then 'q' is also a prefix of the sequence 'p'.
The lemma named "swap_diff" assumes that "A" is not equal to "B". It then shows that "B" is not equal to "A".
The lemma "line_integral_exists_smooth" assumes that the function "gamma" is continuously differentiable on the interval from 0 to 1. It also assumes that for every element "b" in the basis set of a Euclidean space, the function "F" is continuous on the image of the path "gamma". Additionally, it assumes that the basis...
The lemma named "girth_obtains" assumes that the girth is not the maximum possible value. It then obtains a cycle 'c' where 'c' is in the set of all cycles and the walk length of 'c' is equal to the girth.
In the context of a well-founded sub-relation, we have a lemma named "sp_in_extends_not_using_sub". This lemma assumes three conditions: 1. The graph 'g' extends from 'subs' to '(v,v')' in 'subs''. 2. There is a subpath in the graph 'g' from 'v1' to 'v2' in 'subs''. 3. The subpath from 'v1' to 'v2' does not use the s...
The lemma named "non_empty_multiset_induct" is about induction on non-empty multisets. It assumes three things: 1. The multiset M is not empty. 2. For any element x, a property P holds for the singleton multiset containing only x. 3. For any element x and any multiset M, if the property P holds for M, then it also hol...
If a set S is connected, and there are elements a and b in S, and a relation R is symmetric and transitive for all elements in S, and for every element a in S there exists an open set T in the topology of S such that a is in T and for all elements x in T, R relates a and x, then R relates a and b.
The lemma fixes a structure R. Assuming that R is a ring and that both a and b are elements in the carrier of R, it shows that subtracting (a subtract b) from a equals b.
If "q" is a reference in the heap "h" for the reference set "rs", then "q" is an element of the set "rs".
If the Van Emde Boas tree 't' maintains the invariant 'n', then the minimum element of the tree 't' is equal to 'None' if the set of elements in the tree 't' is empty; otherwise, it is 'Some' with the minimum element of the set of elements in the tree 't'.
The lemma "LLL_iso_main" assumes the following: 1. The state implementation invariant holds for parameters p, mfs, dmu, and state. This is denoted by "state_impl_inv p mfs dmu state". 2. The LLL invariant under weak modular conditions holds for parameters fs, mfs, dmu, p, first, and b. This is denoted by "LLL_invaria...
The lemma named "interp_ops_distinct" assumes that "ops" is a list of operations. It shows that the first element of the interpreted operations is distinct.
The program F is compatible with the SKIP operation.
The lemma named "interval_list_union_correct" assumes that the list "S" is not empty and that "i" is less than the length of "S". It shows that the set of elements at the "i"-th position in "S" is a subset of the set of elements in the union of the intervals in the list "S".
The set resulting from updating a list "al" with a value "v" at key "k" using function "f" is equal to the set of "al" minus the set containing all pairs of "k" and any value, union the set containing the pair of "k" and the result of function "f" applied to the value at key "k" in "al". If there is no value at key "k"...
If an ordered pair (r, s) is in the converse of set A, then r is an element of set A and s is the inverse of r.
The lemma named "project_at_indices_consecutive_ind_length" assumes that for any natural number 'i' which is less than 'j', and 'j' is less than or equal to 'n', and the length of 'a' is 'n'. It shows that the length of the projection at indices from 'i' to 'j' of 'a' is equal to 'j' minus 'i'.
The set defined by the function "givenBy v" is equal to the set of all sets A, where there exists a property P such that A is the set of all states s for which P holds true when applied to "v s".
"x or possibly y equals false" if and only if "x equals false and y equals false".
The projection of the union of a set A under a function h is equal to the union of the projection of each element in set A under the same function h.
The lemma named "total_preorder_of_weak_ranking" assumes that the union of the set of elements in xs is equal to A and that xs is a weak ranking. Under these assumptions, it shows that a total preorder on A can be derived from the weak ranking xs.
If a polynomial "p" is a normalized polynomial with respect to "n0" and "p" is a polynomial with no free variables, then the polynomial obtained by decreasing the degree of "p" is a normalized polynomial with respect to "n0 - 1".
The permutation of the non-recursive analysis of a co-call arity expression, a variable, and an expression is equal to the non-recursive analysis of the permuted co-call arity expression, the permuted variable, and the permuted expression.
The minimum height of a tree that has been transformed by a function 'f' is equal to the minimum height of the original tree 't'.
Adding an empty sparse matrix to any sparse matrix "as" will result in the original sparse matrix "as".
If a function 'f' is injective, then the size of the finite set obtained by applying function 'g' to the image of 'f' is equal to the size of the finite set obtained by applying 'g' after 'f'.
The lemma "hfref_weaken_pre_nofail" assumes that the pair (f,g) is in the relation from the assertion [P] to the relation R and the set S. It shows that the pair (f,g) is also in the relation from the assertion that if there is no failure in g(x), then P(x) holds, to the relation R and the set S.
The lemma named "has_antiderivative_sequence" is defined for a function 'f' that maps a natural number to a real normed vector and then to a Banach space. It assumes that a set 'S' is convex and for every natural number 'n' and every element 'x' in 'S', the function 'f' at 'n' has a derivative 'f'' at 'x' within 'S'. I...
The lemma named "lifting_q" assumes that "q" is an element of "Q". It shows that the function "labeled_tiebreaker_lifting" with parameters "Bot_F", "Inf_F", "Bot_G", "entails_q q", "Inf_G_q q", "Red_I_q q", "Red_F_q q", "\<G>_F_q q", "\<G>_I_q q", a lambda function that always returns false, and "Inf_FL", is true.
If s1 and s2 are low equal, then the user register value of s1 is equal to the user register value of s2 for any given window and user register.
If 'x' is a preorder, then the transpose of 'x' is also a preorder.
The product of a polynomial 'p' and the negation of another polynomial 'q' is equal to the negation of the product of 'p' and 'q'.
The tail of a list that starts with 'x' and is followed by list 'S' is equal to 'S'.
The lemma "get_all_matching_src_ips" assumes that a match "m" is in normalized negation normal form. It shows that the set of all source IP addresses that match "m" on a given interface "iface" is a subset of the set of all IP addresses for which every packet "p" (of any type and length) that matches "m" on the same in...
If a composition series exists for a subgroup H of group G, and if the quotient group G/H is a simple group with H being a normal subgroup of G, then a composition series exists for group G, which is the extension of the composition series of subgroup H with the carrier of group G.
The operation "splus" is associative. That is, for any elements a, b, and c, adding a and b together first and then adding c is the same as adding a to the sum of b and c.
The lemma named "join_of_un_upto_cong" assumes that "v" and "v'" are both join of "t" and "u". It shows that "v" is similar to "v'".
The lemma named "qbs_strength_law3" assumes that "x" is an element of the quasi-Borel space of the product of "X" and "Y". It shows that the quasi-Borel return of "x" in the product of "X" and "Y" is equal to the composition of the quasi-Borel strength of "X" and "Y" and the mapping of the product of the identity and t...
The lemma named "fstar_Inf_pres" establishes that for a function 'f' mapping from a complete lattice to the same complete lattice, if 'f' preserves infimum (greatest lower bound), then the Kleene star of 'f' (denoted as 'fstar f') also preserves infimum.
If "C" is the maximum simplex in the simplicial complex "Y", then "C" belongs to the opposite half apartment of the chamber system.
The lemma named "relH_in_rangeI" assumes that there is a relation "relH" between "as", "h", and "h'". It shows that "as" is in the range of both "h" and "h'".
A function 'f' is continuous with respect to the least upper bounds 'luba' and 'lubb', and the order relations 'orda' and 'ordb', if and only if for all 'x', the function 'f' applied to 'y' and 'x' is continuous with respect to the least upper bounds 'luba' and 'lubb', and the order relations 'orda' and 'ordb'.
"the product of c and a is less than or equal to the product of c and b if and only if a is less than or equal to b."
For a given natural number 'n' and a word 'k' of arbitrary length 'a', the operation of taking the signed bit from 'k' after taking the bit of length 'a' from 'k' is equivalent to the operation of directly taking the signed bit from 'k'. This equivalence holds under the condition that the two operations are performed o...
For any integer n, dropping n bits from -1 results in -1.
For any list 'xs', if the result of dropping 'n' elements from 'xs' is a list starting with 'y' and followed by the list 'ys', then the result of dropping 'n+1' elements from 'xs' will be the list 'ys'.
If the unsigned natural number representation of 'y' is less than 2 raised to the length of 'b', and the length of 'b' is less than the length of 'a', and 'x' is less than or equal to 'y', then the unsigned cast of 'x' is less than or equal to the unsigned cast of 'y'. This is for any 'x' and 'y' of type 'a' word.
The k-smooth compatibility of x and y is equivalent to the k-smooth compatibility of y and x.
The lemma named "type_less_eq_x_Integer_intro" introduces that if a type tau is equal to UnlimitedNatural, then tau is less than or equal to Integer.
The product of the intersection of 1 and the product of x and the transpose of x, with x, is equal to x.
If two paper applications paps and paps1 are equivalent except for their PID2, then the title, abstract, content, reviews, and discussions of the paper in paps and paps1 are the same, and the first decision on the paper in paps is the same as the first decision on the paper in paps.
The probability mass function probability of "p" is equal to the measure of probability mass function of "p".
If the natural number 'r' is greater than zero, 's' is greater than zero, and 'i' is less than or equal to the product of 'r' and 's' minus one, then the quotient of the function 'rtos' with inputs 'r', 's', and 'i' divided by 'r' is equal to the remainder of 'i' divided by 's'.
If the first component of 'n' modulo 7 equals 6 and the first component of 'x' equals 0, then the upper bound location of 'n' and 'x' equals the pair of 'n' and 'x' with the upper bound location of the first component of the second component of 'n' and the second component of 'x', all plus 1.
If a relation 'aRel' is transitive, then the interpreted value 'IV' is true.
If we have a transitive relation 'r', and a pair (M,N) that belongs to the multiplicative equality of 'r', and the multiset 'N' is a subset of the decreasing sequence of 'r' and 'S', then the multiset 'M' is also a subset of the decreasing sequence of 'r' and 'S'.
The lemma named "lookup_pair_tl'" assumes that the list "xs" satisfies the invariant of an ordered association list. It shows that the lookup of a key "k" in the tail of the list "xs" is equal to zero if "k" is equal to the first element of the head of "xs", otherwise it is equal to the lookup of the key "k" in the lis...
If the following conditions are met: 1. The set I is not empty. 2. For every element j in I, A j is an abelian group. 3. S is an abelian group. 4. For every element j in I, g j is a group homomorphism from S to A j. 5. ff is a function from the carrier of the direct product of A over I to B. 6. ff is a bijection from ...
In the context of a semiring of sets, there is a lemma for the generated ring of disjoint union. If a set A is finite, then if A is a subset of the generated ring and A is disjoint, it follows that the union of A is in the generated ring.
The lemma "lists_succ_not_in" assumes that "x" is not in the union of the sets of "xss". It shows that the list of successors of "xss" for "x" is equal to "x".
If a type tau is less than OclVoid, then proposition P holds.
If R behaves like a set, then the transitive closure of R under the extensionality axiom of Zermelo-Fraenkel set theory is equal to the extensionality hull of R and s.
The lemma named "at_different" assumes that "at" is a type 'x'. It shows that there exists a 'b' of type 'x' such that 'a' is not equal to 'b'.
The lemma is named "stronger interior version of the fundamental theorem of calculus". It is defined for a function 'f' that maps real numbers to elements of a Banach space. The lemma assumes that 'S' is a finite set, 'a' is less than or equal to 'b', and for any 'x' in the open interval from 'a' to 'b' excluding 'S', ...
The lemma named "complete_imp2" assumes that there is some value for the function "exhaustive" from the "exhaustive_class", when applied to the function "f" and the natural number "n" converted to a natural number type. If this assumption holds, then there exists a value "v" of a type belonging to the "complete" class ...
The binary length to natural number function, when applied to the y-th power of Oc and 0, equals 2 to the power of y minus 1.
The lemma named "foldl_Gauss_condition_9" is defined with the following conditions: - It fixes a matrix 'A' of any field type, and a natural number 'k'. - It assumes that 'k' is less than the number of columns in 'A'. - It assumes that 'A' is in reduced row echelon form up to 'k'. - It assumes that the row 'm' in 'A' ...
For every real number x, there exists a real number y such that the value of 4 times x plus 3 times y is less than or equal to zero and greater than or equal to negative one.
A predicate P holds for a stream w if and only if P holds for every element at index i in the stream w.
The third element of the ZF-Triple (a, b, c) is c.
: 1. If we have two weakened heap node refinement postconditions, P and P' under a context Γ, and Q and Q' under a context Γ', then we can create a new weakened heap node refinement postcondition (P*Q) and (P'*Q') under the combined context (Γ*Γ'). 2. The weakened heap node refinement postcondition of a heap node con...
If a set A is in the sigma-algebra of measurable sets of a measure space M, then the extended measure of A in M is equal to the measure of A in M.
The paths are equal pointwise for a given 'k'.
The parallel composition of "x" intersected with the identity element and "y" intersected with the identity element is equal to the product of "x" intersected with the identity element and "y" intersected with the identity element.
The lemma named "bisimParPresSym" is defined with fixed variables Psi, P, Q, and R, all of which are of type psi. The lemma assumes that Psi is in bisimulation with P and Q. It then shows that Psi is also in bisimulation with R in parallel with P and R in parallel with Q.
If 'f' and 'g' are both bounded and linear transformations, then the function that maps 'x' to the sum of 'f(x)' and 'g(x)' is also a bounded and linear transformation.
The complex roots of a complex constant "c" is an empty list.
There exists a positive number 'x' in a linearly ordered field such that for all 'y', there exists a 'z' where 13 times the absolute value of 'z' is not equal to the absolute value of 12 times 'y' minus 'x', and 5 times 'x' minus 3 times the absolute value of 'y' is less than or equal to 7 times 'z'.
If the epsilon closure set of a step in R with respect to bs is equal to the union of R and S, then the epsilon closure set of a step in S with respect to bs is a subset of the union of R and S. This is assumed to be true under the given conditions.
The set of invariant sets is a subset of the tail events of the function that maps each index to the set of pre-images under the function f at that index, in the space of the probability measure P.
There exists a root 'r' for a path 'r' that starts at 'x' if and only if 'x' is a path.
The interior of the negation of set S is equal to the negation of the interior of set S.
The lemma consists of two parts: 1. For any states 's' and 't', if 's' is weakly bisimilar to 't' in terms of traces, then 't' matches 's' in the context of weak bisimulation and multilevel security policy. 2. For any states 's' and 't', if 's' is weakly bisimilar to 't' in terms of traces, then 't' matches 's' in th...
The nth vector of the product of matrix A and vector x is equal to the sum of the product of the ith row and jth column of matrix A and the jth element of vector x, where j ranges over all possible values.
The lemma named "closed_orbit_flow0" assumes that "x" is a closed orbit. It shows that the flow of "x" at time "t" is also a closed orbit.
The lemma named "Pring_universal_prop" makes the following assertions: Assuming that "S" is a commutative ring (as per "cring S"), "g" is a function from the set "I" to the carrier of "S" (as per "g \<in> I \<rightarrow> carrier S"), "\<phi>" is a ring homomorphism from ring "R" to ring "S" (as per "ring_hom_ring R S ...
The set of regular past-due debts (RPDs) of "MStar r" is a subset of the union of the set containing "MStar r" and the set resulting from the left multiplication of "MStar r" with the RPDs of "r".
If it is assumed that "not phi" is in v and if "not psi" being in v implies that "phi" is in v, then it can be concluded that if "not phi" is in v, then "psi" is in v.
For a fixed list of real numbers "xs", the product of all elements in the list that are equal to "m" is equal to "m" raised to the power of the length of the list of elements that are equal to "m".
"m3" refines "m2" under the relation "R23" intersected with the universal set cross product "m3_inv1_keys", with "med32" as the mediator.
The lemma STL4_3: assumes that if F holds then G holds (h1), and G is stuttering invariant (h2), it shows that if F always holds then G always holds.
, given the assumptions that "xe" is in the set generated by the function "compxE2" with parameters "e", "pc", and "d", and that "pc'" is either less than "pc" or greater than or equal to the sum of "pc" and the size of the result of the function "compE2" with parameter "e", it follows that the function "matches_ex_ent...
The lemma, named "inline1_fixp_induct", with case names "adm", "bottom", and "step", assumes three conditions: 1. It assumes that a function 'inline1' is admissible in the complete partial order (ccpo) defined by the least upper bound (lub) of a function on the space of sub-probability measures (spmfs) and the order r...
The set of all functions from a finite set 'k to the direction set 'dir' is finite.
The lemma named "represents_mult" assumes that polynomial "p" represents number "x", polynomial "q" represents number "y", and polynomial "q" is not equal to zero when evaluated at zero. It shows that the multiplication of polynomials "p" and "q" represents the multiplication of numbers "x" and "y".
If all corresponding elements in two lists, xs and ys, are equivalent and two elements b and c are equivalent, then the result of folding the lists xs and ys with the disjunction operation, starting with b and c respectively, are also equivalent.
The lemma named "mean_rv_var" assumes that "i" is less than "s2". It shows that the variance of the mean random variable "omega" at "i" is less than or equal to the square of the real number obtained from the product of "delta" and "F 2 as", divided by 3.
If a heap 'h' is well-formed, has a valid type, and contains known pointers, and if 'h' can successfully retrieve the child nodes of a pointer 'ptr' resulting in 'children', then if 'h' can successfully remove all 'children', resulting in a new heap 'h'', then 'h'' can successfully retrieve the child nodes of 'ptr', wh...
The domain of the function "HMA_I", which maps natural numbers to a modular type 'n, is the set of all natural numbers 'i' that are less than the cardinality of the modular type 'n.
In the lemma named "alg_mult_gt_zero_iff_is_root" within the domain, given the assumption that "p" belongs to the carrier of the polynomial ring "R", it shows that the algebraic multiplicity of "p" at "x" is greater than zero if and only if "x" is a root of "p".