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The lemma for the corecursive function on lazy lists in the code is defined as follows: The corecursive function on lazy lists, given the parameters IS_LNIL, LHD, endORmore, LTL_end, LTL_more, and b, is equal to a lazy list function that takes an arbitrary parameter and checks if IS_LNIL b is true. If it is, it retur...
The lemma "cnf_of_encode_interfering_operator_exclusion_is_iii" fixes a variable 'Pi' of type 'variable strips_problem'. It shows that the set of conjunctive normal form (CNF) of the encoding of interfering operator pair exclusion for 'Pi', 'k', 'op1', and 'op2', where 'op1' and 'op2' are filtered such that their indic...
The set of real numbers that are within the range of an integer and fall between 'a' and 'b' is finite.
The product of 18, 40, the square of any real number 't', and 2 raised to the power of the negative of any real number 'r' is less than or equal to 1.
If command "c" does not modify "x", then the combination of command "c" and annotations "annos" also does not modify "x".
"B" is trimonotonic if and only if for all orders in the function order of triorders, "B" is monotone with respect to that order.
If the function "tm_to_nat_list" applied to two lists "xs" and "ys" yields the same result, then the two lists "xs" and "ys" must be identical.
The cost of Karatsuba's algorithm is always greater than or equal to zero.
If sequence 'xs' is a prefix of sequence 'ys', then the sequence resulting from executing the computation stream function 'f_Exec_Comp_Stream' with transition function 'trans_fun' and initial state 'c' on 'xs' is also a prefix of the sequence resulting from executing the same function on 'ys' with the same initial stat...
The lemma named "interpret_floatariths_eq_take_max_VarI" assumes that taking the maximum variable from the floatariths of 'ea' in both 'ys' and 'zs' results in the same value. It then shows that interpreting the floatariths of 'ea' in 'ys' is equal to interpreting the floatariths of 'ea' in 'zs'.
Given a configuration "cfg" in the set of configurations on the abstraction of state "st", and given that "st" is in the set "S", and given that the configuration "cfg" is alternating, it shows that for almost every outcome "ω" in the Markov Decision Process's Markov Chain's transition function applied to the repeated ...
: Assuming 1) a sequence consisting of a constant 32-bit integer 'n' and a conditional branch 'i' leads to some sequence 'e', 2) under the context of a store type and a constant, the sequence of the constant 32-bit integer 'n' and the conditional branch 'i' has a type that transforms 'ts' into 'ts'', and 3) 'e' is eit...
The lemma named "eval_set" assumes that a formula "phi" belongs to the set of monotone formulas and to the set "A". It shows that the evaluation of "phi" under a certain condition "theta" is equal to the evaluation of the set of "phi" under the same condition "theta".
If "S" is a chain, then either "S i" is a subset of or equal to "S k", or "S k" is a subset of or equal to "S i".
In the context of a state space, the lemma demonstrates that if the initial state satisfies the condition that 'a' equals 'a' and 'b' equals 'b', then after the execution of the commands 'M' is assigned the value of 'a' and 'N' is assigned the value of 'b', the final state will satisfy the condition that 'M' equals 'a'...
If "A" says to "B" the message "X" in the set of events "evs", then "X" is in the set of things the "Spy" knows in the set of events "evs".
For all lists 'xs', the evaluation of 'VSEqualityBlocksLimited' applied to a formula 'phi' is equal to the evaluation of 'phi' applied to the same list 'xs'.
If a transformation 't' is equivalent to a transformation 'u', and 'u' is less than or equal to a transformation 'v', then 't' is also less than or equal to 'v'. This is a transitivity property in the context of transformations.
In the context of p-adic integers, there is a lemma for the decomposition of an integer. If we assume that an integer 'n' is equal to 'l' times 'p' raised to the power of 'k', and 'l' modulo 'p' is not equal to zero, then the order of 'n' in the ring of p-adic integers is equal to 'k'.
The infinite sum of the function 'f' over an empty set equals zero.
The lemma named "tsp__ncop2" assumes that points A, B, C, and TSP form a tetrahedron with P and Q. It shows that points A, B, C, and Q are not coplanar.
The lemma Gale_Shapley6_1 from the Gale_Shapley1.thy file in Gale_Shapley assumes that the ranking of b (Rb) is equal to the map ranking of Pb. It shows that given the variables A, B, M, a, a', ai, b, and r, and the initial conditions where ai and a are 0, A is a replication of 0 n times, the length of B is n, and M is...
If a function 'f' satisfies the following three conditions: 1. The function 'f' preserves addition, i.e., applying 'f' to the sum of two elements 'b1' and 'b2' is the same as the sum of the results of applying 'f' to 'b1' and 'b2' separately. 2. The function 'f' preserves scalar multiplication, i.e., applying 'f' to t...
For all packets 'p' of a certain type, if the input interface of 'p' is in the domain of the IP assignment, then the source IP of 'p' is in the union set of the IP assignment of the input interface of 'p'. This implies that there is no IP spoofing in the rule set 'rs'.
If R1 and R2 are right total relations, then the relation of resumption between R1 and R2 is also a right total relation.
The sum of the absolute values of the elements in the list "ks" is greater than or equal to zero. This is for integers.
The lemma named "common_factor_via_short" is about integer polynomials. It fixes three integer polynomials, namely f, g, and u. It defines n as the degree of f and k as the degree of g. It assumes that n is greater than 0 and k is greater than 0. It also assumes that u is monic (meaning its leading coefficient is 1) an...
For any given parameters 'vs' and 'bs', the interpretation function 'Ifm' applied to the simplified form of 'p' (denoted by 'simpfm p') is equivalent to the interpretation function 'Ifm' applied directly to 'p'.
Theorem aval_afold_N: Assuming that the approximation of "t" is "s", if the "afold" function applied to "a" and "t" equals "N n", then the "aval" function applied to "a" and "s" equals "n".
The degree of a polynomial that is scaled by a constant is equal to the degree of the original polynomial, unless the constant is zero, in which case the degree is zero. This is true in any semiring without zero divisors.
"x is less than or equal to y" if and only if "x is less than y" or "x is approximately equal to y".
In the context of a commutative ring, the set of units forms a commutative group.
If matrix A and B are unitarily diagonalizable by matrix U, then matrix A and B are unitarily equivalent by matrix U. This is under the assumption that matrix A and B are unitarily diagonalizable by matrix U.
For any elements a, b, c, and f of a pseudohoop N, if a is right residual to b and this result is right residual to b again, and if a times c is right residual to a times f, and b times c is right residual to b times f, then c is right residual to f. This is under the condition that the operations and relations involve...
The lemma named "closed_orbit_omega_limit_set" assumes that "x" is a closed orbit. It shows that the image of "x" under the flow for all time is equal to the omega limit set of "x".
For all 'i' less than or equal to 'n', if the matrix 'M' at position (i, i) equals the neutral element, and if the matrix 'M' is cycle-free up to 'n', then for all 'i' less than or equal to 'n', the Floyd-Warshall algorithm applied to 'M' up to 'n' at position (i, i) will also equal the neutral element.
The ordered difference of the sum of x and y, and the sum of x and z, is equal to the ordered difference of y and z.
The range of the function "af" when lifted and applied to an abstract syntax tree representation of a logical formula "phi", is a subset of the set of all abstract syntax tree representations of logical formulas "psi", such that the nested propositional atoms in "psi" are a subset of the nested propositional atoms in "...
If for all M, if M is an element of S then M is an element of Bignorabimus alpha, then the union of S is an element of Bignorabimus alpha.
If in the context of a set of hypotheses H, x is proven to be equal to x', then under the same set of hypotheses H, the predicate VarP applied to x is logically equivalent to the predicate VarP applied to x'.
If a list "xs" is sorted according to the first element of its tuples, and for all "t" in the first element of the set "xs", "t" is less than or equal to "nt", then taking elements from the list "xs" while the difference between "nt" and "t" (converted to an extended natural number) is greater than "c" is equivalent to...
If "t" can be transformed into "t'" through zero or more beta-reduction steps, then the application of "s" to "t" can also be transformed into the application of "s" to "t'" through zero or more beta-reduction steps.
Given the assumptions that "x" is not equal to zero and "y" is not equal to zero, the prime factorization of "x" is a subset of the prime factorization of "y" if and only if "x" divides "y".
The lemma named "added_ids_Deliver_Create_diff_collapse" in the context of imap shows that if "e" is not equal to "e'", then the added ids from delivering a create operation with index "i" and element "e" for element "e'" is an empty list.
The lemma p is simply defined as the cardinality of p.
A list 'xs' followed by an element 'x' is equal to another list 'ys' if and only if 'ys' is not an empty list and the list 'ys' without its last element is equal to 'xs' and the last element of 'ys' is 'x'.
Mapping a function that takes two arguments, x and y, and returns true if x is true and y is false, or vice versa, over a list xs and a list of n false values, is equivalent to taking the first n elements of the list xs.
The extended nonnegative real number of the real part of "r" is equal to zero if "r" is infinite, otherwise it is equal to "r".
"RES X is less than or equal to RES Y" if and only if "X is a subset of Y".
The properties of psi are as follows: - Psi raised to the power of two times any integer equals one. - Psi raised to the power of any integer times two times any integer plus one equals negative one. - Psi is not equal to one.
The lemma named "aux_powertwo_digits" is defined with two fixed natural numbers, k and c. It assumes that k is less than c. The lemma then shows that the kth bit of 2 to the power of c is 0.
The second element of the result from the function "write_cpu" with inputs w, r, and s is False.
If "X" is a well-behaved prism, then the prism obtained from the lens derived from "X" is equal to "X".
If the complement of block 1 is equal to block 2, then the intersection of block 1 and block 2 is empty.
: Assuming "dm" is a directed graph map created from the input graph "iG" and the map of the input graph "iG" and "iM", and assuming "pred" is a condition where for any integer "i" less than the length of the edges of the input graph "iG", the predecessor of the successor of "iM" at "i" equals "i", we can obtain that t...
: Given a Hermitian matrix 'A' that belongs to the set of finite complex matrices, and assuming that the projective measurement of 'A' equals the pair '(p, M)', it shows that the set of real parts of elements in the spectrum of 'A' is equal to the set of measurement outcome values for all indices less than 'p'.
The lemma "rquot_dfs_invariant" assumes that "x" is a node in the deterministic finite automaton "A". It shows that the "rquot_invariant" of "A" holds for the result of the depth-first search "rquot_dfs" on "A" starting from node "x" with a maximum depth of "n".
The lemma named "picard_lindeloef_autonomous_affine" fixes a matrix A, which belongs to a Banach space, a real normed field, and satisfies the Heine-Borel property. It shows that the Picard-Lindelöf theorem holds for the function that maps any pair of real numbers (t, s) to the result of multiplying matrix A by vector ...
If the set of successors of a graph 'g' for a list 'ys' is a subset of the union of the set of 'xs' and the set of 'ys', then the set of successors of the graph 'g' for the depth-first search of 'g' with 'xs' and 'ys' is a subset of the set of the depth-first search of 'g' with 'xs' and 'ys'.
Given a group G and a unit group E, if a word comparison series w_cmpser for G at the successor of n (n+1) exists for a function f, and i is less than or equal to n, and the i-th element of the word comparison series for G and f is not isomorphic to E, then the i-th element of the function f is not equal to the (i+1)-t...
Running the "tell_writer" function with parameters "w" and "m" is equivalent to binding the result of running the "m" writer with a function that takes a pair "(a, ws)" and returns a new pair "(a, w # ws)".
The cardinality of the set of roots x of a polynomial p, where x is greater than a and less than or equal to b, is equal to the count of roots of the polynomial p between a and b.
If K' is a field, v is a valuation on K, v' is a valuation on K', and K is a completion of K' with respect to v and v', then the set of elements (carrier) of K is a subset of the set of elements (carrier) of K'.
: Assuming that "c" is equal to 2 raised to the power of "i", and that "mu" is less than or equal to the total number of carrots "c" implies that "c" is less than or equal to "lambda". Also, assuming that the sequence of the total number of carrots "c" is not equal to the sequence of the total number of carrots "c" pl...
If P implies the weakest liberal precondition of action a and postcondition R, and Q implies the weakest liberal precondition of action b and postcondition R, and p is a probability (i.e., a real number between 0 and 1 inclusive), then the function that maps each state s to the weighted average of P(s) and Q(s) with we...
The norm of the result of applying a complex bounded linear function 'f' to 'x' is less than or equal to the product of the norm of 'f' and the norm of 'x'.
If 'a' is in the reflexive transitive closure of the relation 'r' applied to set 'A', and the ordered pair (a, b) is in the reflexive transitive closure of 'r', then 'b' is also in the reflexive transitive closure of the relation 'r' applied to set 'A'.
The composition of the function "f" with a sub-block of a block transformed by the function "g" is equal to a sub-block of the block transformed by the composition of the functions "f" and "g".
The lemma named "transaction_renaming_subst_range_notin_vars" is given a protocol substitution, denoted as "α". It assumes that "α" is a transaction renaming substitution for a protocol "P" and a stateful protocol "A". The lemma then shows that there exists a variable "y" such that "α" applied to "x" equals "y", and "y...
If P' is a subset of P, then P entails P'.
, given the assumption that "b" is an identity in category "C" and "f" is a morphism from "a" to "a'", it can be shown that "Y f b" is a morphism from the hom-set of the pair "(b, a)" to the hom-set of the pair "(b, a')". Furthermore, "Y f b" is equal to the composition of the functor "phi" applied to the pair "(b, a')...
The lemma named "open_closed_invariant" is defined with a fixed variable J. It assumes that A enforces that I implies P, that A is subreachable with U and J, and that for all states sigma and sigma prime, and all s, if for all j in J, sigma prime of j equals sigma of j and P of sigma prime and s holds, then P of sigma ...
The sources at level 2 for "sS4" are equal to the set containing "sS2".
The lemma fixes a set S in the Euclidean space. It assumes that the union of set S and set T is not affine dependent. Under this assumption, it shows two things: 1. The intersection of the convex hull of set S and the convex hull of set T is a subset of the convex hull of the intersection of set S and set T. 2. The i...
The lemma named "while_completeness_1" assumes that the negation of p times (x shifted to the left by the negation of q) is less than or equal to the negation of q. Under this assumption, it shows that the negation of p star times (x shifted to the left by the negation of q) is less than or equal to the negation of q u...
The paths up to a certain length or condition with a witness in a given machine M, a property P, a length k, and a state q, is equal to the paths up to a certain length or condition with a witness' from the machine M, the property P, an empty list, the length k, and the state q, if the state q is in the states of the m...
If "k" is less than the number of columns in matrix "A" and "i" is less than or equal to the number of rows in matrix "A", then the first element of the second element of the tuple resulting from applying the function "echelon_form_of_column_k_det" with parameters "bezout", "n", "A", and "i" to "k", when converted to a...
The lemma named "modaliseEmpty" shows that the product of "a" and "Empty" is "Empty".
If set X is a subset of set S, then the set consisting of the empty set, X, the difference of S and X, and S itself forms an algebra on S.
The length of the list "xs" is less than or equal to the length of the encoded version of "xs".
If alpha is less than beta and gamma is less than delta, and if alpha, beta, gamma, and delta are all ordinal numbers, then the sum of alpha and gamma is less than the sum of beta and delta.
The lemma named "subprob_measurableD" assumes that "N" is a measurable function from a measure space "M" to a subprobability space "S" and "x" is an element of the space "M". It shows that the space of the function "N" at "x" is equal to the space "S", the sets of the function "N" at "x" are equal to the sets "S", the ...
The lemma named "ab_semigroup_mult_sep_conj" belongs to the class "ab_semigroup_mult".
A name 'x' is fresh for a function that takes a pattern 'p' and returns a term 't' if and only if 'x' is in the support of 'p' or 'x' is fresh for 't'.
The lemma named "le_fin_ind''" assumes that "a" is less than or equal to "w" in the finite order, "b" is less than or equal to "w" in the finite order, and "a" is not equal to "b". It shows that "a" is independent of "b".
For any two formal Puiseux series 'f' and 'g', shifting the series 'f + g' by 'r' is equivalent to shifting the series 'f' by 'r' and adding it to the series 'g' shifted by 'r'.
If 'x' is a limit ordinal, then 'oSuc w' is less than 'x' if and only if 'w' is less than 'x'.
If 'a' is not equal to 'b', then applying the function 'chg_map' with parameters 'f', 'a', 'm', and 'b' is equivalent to applying 'm' to 'b'.
If "A" is a very well-behaved lens, then the parallel substitution of an empty set on the left side of "A" in a state "rho" under a substitution "sigma" is equal to the identity substitution of an empty set on the left side of "A" in the state "rho".
If a relation R holds between two interfaces under a condition C, and the condition C holds, then the relation R also holds between the responses of these two interfaces. This is a transfer rule in Isabelle.
The lemma named "is_aligned_replicate" is about a word 'w' of any length 'a'. It assumes that the word 'w' is aligned at 'n' and that 'n' is less than or equal to the length of 'a'. It shows that the binary list representation of 'w' is equal to the concatenation of the first (length of 'a' minus 'n') elements of the b...
The lemma "Lim_norm_ubound" is about a function 'f' that maps from a set 'a' to a set 'b', where 'b' is a real normed vector. It assumes that the limit of the net is not trivial, the function 'f' converges to 'l' over the net, and eventually, the norm of 'f' at any point 'x' is less than or equal to 'e' over the net. I...
If the function "incrIndexList" is applied to the list "ls", the integer "m", and the maximum integer "nmax", and if the successor of "n" is less than the length of the list "ls", and if the "n"th element of the list "ls" is not equal to the successor of the "n"th element of the list "ls", then the "n"th element of the...
If a valuation M satisfies a formula φ, then the closed world interpretation of M also satisfies φ.
The action of alpha on the supremum (or join) of x and y is equal to the supremum of the action of alpha on x and the action of alpha on y.
If a formula "FOr \<phi> \<psi>" is well-formed for a given "n", then the language of "n" for the encoding of "n" and the first-order variables of "FOr \<phi> \<psi>" is a subset of the intersection of the language of "n" for the encoding of "n" and the first-order variables of "\<phi>", and the language of "n" for the...
The lemma named "det_identical_rows" is about a fixed matrix A of type 'a, where 'a is a commutative ring. It assumes that i is not equal to j (denoted by "i \<noteq> j") and that the i-th row of matrix A is equal to the j-th row of matrix A (denoted by "row i A = row j A"). The lemma then shows that the determinant of...
The lemma "Ide_Diagonalize_rep_ide" assumes that "a" is an identity. It shows that the representation of "a" is an identity in the diagonalized form.
The nth derivative of the sum of a formal power series function over a set S is equal to the sum of the nth derivatives of the formal power series function applied to each element in the set S, where the formal power series function is over a ring.
If "v" is trivial in graph "g", then there exists exactly one "v'" such that "v'" is a trivial phi-node in graph "g" for "v".

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