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If the degree lexicographic power monomial of 's' is less than or equal to 't' and the degree lexicographic power monomial of 't' is less than or equal to 's', then 's' must be equal to 't'.
If the following conditions are met: 1. There is a refinement relation between 's' and 's''. 2. For any 's' and 's'', if they are in a refinement relation and 'I s' holds, then 'I' s'' also holds. 3. For any 's' and 's'', if they are in a refinement relation, 'I s' and 'I' s'' hold, then the result of 'b' s'' is less ...
The constant of the negation of a number 'n' is equal to the negation of the constant of the number 'n'.
If a formula 'F' is not in the propositional calculus proof sequence 'pcp_seq' for a given 'C' and 'S' at the successor of the natural number representation of 'F', then 'F' will not be in the 'pcp_seq' at the successor of the natural number representation of 'F' plus any natural number 'n'.
The lemma "cycle_needs_back_edge" assumes that the set of back edges of a structure 's' is empty. It shows that the edges of the structure 's' are acyclic, meaning there are no cycles in the structure 's'.
The proposition "P" is true for "E".
If there exists an element in the set 'xs' for which the property 'P' does not hold, then the prefix length of 'P' in 'xs' is less than the length of 'xs'.
: Assuming a well-formed Java program "P", and given that "P" can transition from the initial state "s0" and boolean "b0" to a state where the expressions "es" are evaluated to values "vs", and the state is "(h2, l2, sh2)" with boolean "b2". Also, given that "P" sees a static method "M" in class "C" with parameter typ...
If the set of keys in a mapping 'm' is not finite, then the ordered entries of 'm' are an empty list.
If the co-power operation holds for a certain value eta, then the third co-power property also holds for eta.
The condition "value_gt a b equals true" is equivalent to the existence of numbers x and y such that a equals the option of the number x and b equals the option of the number y, and x is greater than y.
The image of a set A under the transposition of elements a and b is equal to the set A itself, given the condition that element a is in set A if and only if element b is in set A.
If a class declaration "C,c" is well-formed in a given program "G" and "I" is in the set of super interfaces of "c", then "I" is an accessible interface in "G" for the package identifier of "C" and for all signatures "s" and for all interface methods in "G" for "I" and "s", there exists a class method in "G" for "C" an...
The set of bad pairs in sets S and T with a constant c is equal to the set obtained by filtering the Cartesian product of set S with itself, where the filter condition is that a pair is considered bad if the first and second elements of the pair form a bad pair in set T with constant c.
, given a function 'f' that is twice field differentiable at a point 'x', and a sequence of coefficients 'coef' indexed by natural numbers, the function defined as the sum from 'i' to 'n' of the product of the 'i'-th coefficient and the 'i'-th power of 'f' at 'x', is also twice field differentiable at 'x'. This is unde...
If the initial state of T is a subset of I, and the intersection of I and J after the transition of T is a subset of I, and the reachable state of T is a subset of J, then the reachable state of T is a subset of I.
The function "slp_index_lookup" applied to the reverse of a list "xs" and an index "i" is equal to the function "slp_index_lookup" applied to the original list "xs" and the same index "i".
If the optimal most recently used guard function "opt_mru_guard" is true for the set of votes "vs", the quorum "Q", and the vote "v", then "Q" belongs to the set of quorums "Quorum".
If the function "min_range_mono" with parameters p, f, f', i, g, h returns a value of Some ((a, b), ba), then both a and ba are elements of the set of floating point numbers.
The lemma named "cod_char" shows that the codomain of a function 'f' is equal to the identity on the codomain of 'f' if 'f' is an arrow (i.e., a morphism in a category), otherwise it is null.
"the equivalence of p and q is equivalent to r if and only if p is equivalent to the equivalence of q and r".
The lemma named "init_strict" has two parts: 1. The initialization of the bottom element is the bottom element itself. 2. The initialization of a sequence where the first element is the bottom element results in the bottom element.
The lemma named "cubic" assumes that "a" is not equal to zero. It shows that: Firstly, we define several variables: - "p" is defined as the result of (3 times "a" times "c" minus "b" squared) divided by (9 times "a" squared). - "q" is defined as the result of (9 times "a" times "b" times "c" minus 2 times "b" cubed mi...
The lemma "sxalloc_no_jump" assumes that from state s0 to state s1, there is no abrupt jump due to "sxalloc" (a function or operation in the system). If there is no abrupt jump from state s0, then there will not be an abrupt jump in state s1.
Lemma AE_upper_bound_inf: Let F and G be functions from type 'a to real numbers. Assume that for any positive number e, almost everywhere in the measure space M, F(x) is less than or equal to G(x) + e. Then, almost everywhere in the measure space M, F(x) is less than or equal to G(x).
The cardinality of the set I cut at the nth element is equal to the successor of n if n is less than the cardinality of I or if I is infinite, otherwise it is equal to the cardinality of I.
If class C is initialized in the global state of s, and s is less than or equal to s', then class C is also initialized in the global state of s'.
The lemma named "meet_top" assumes that "P" is a healthiness condition. It shows that the meet (or intersection) of "P" and top (the maximum or universal set) is equal to "P". This is a simplification rule in Isabelle.
If the following conditions are met: 1. Predicate P holds for the 0th element of list s. 2. For any natural number n, if the successor of n is less than the length of list s and predicate P holds for the nth element of list s, then predicate P also holds for the successor of n. 3. The index i is less than the length o...
If 'p' is a prime element and 'p' does not divide 'q', then the multiplicity of 'p' in the product of 'q' and 'x' is equal to the multiplicity of 'p' in 'x'. Here, multiplicity refers to the number of times 'p' divides 'x' or 'q*x'.
If for every element 'x' in set 'A', the function 'f' of 'x' equals the function 'g' of 'x', then the sum of the function 'f' for all elements in set 'A' equals the sum of the function 'g' for all elements in set 'A'.
The right associative operation followed by the left associative operation is equivalent to the identity operation in constructive cryptography.
The method "make_list_ins" with the test name "test_name", which takes no arguments and returns void, is well-typed in the environment "E". The method has a maximum stack size of 3 and a maximum number of local variables of 2. The method's bytecode verifier state is represented by "[] \<phi>\<^sub>m".
The function "from_cnf" applied to the concatenation of two lists "ks" and "ls" is equal to the sum of the function "from_cnf" applied to "ks" and "ls" separately.
The lemma named "extend_eq_linorder" assumes that "cmp" is an equivalence linear order on the set "D". It shows that the extension of "cmp" to "D" is also an equivalence linear order.
The function 'fwi'' with parameters m, n, k, i, j is less than or equal to the specification function that returns a result r, where r is equal to the uncurried version of the function 'fwi' applied to the curried version of m, with parameters n, k, i, j.
A state 'st1' in the finite state machine 'fsm1' can reach a state 'st'' through a sequence of transitions 'xs' if and only if there exists a state 'st1'' such that 'st'' is equivalent to 'st1'' and 'st1' can reach 'st1'' in 'fsm1' through the same sequence of transitions 'xs'. This is in the context of the combined fi...
A method declaration in the J1 language is well-formed with respect to a program P and a class C if and only if the following conditions are met: 1. There exists a type T' such that the body of the method is well-typed under the context of the class C and the types Ts, and T' is a subtype of T according to the program...
The lemma named "cf_blcomp_is_functor" assumes that a functor "SS" maps a product category "CC x CC" to a category "CC" in a context of a fixed universe "alpha". It shows that the bifunctor left composition of "SS" maps a triple product category "CC x CC x CC" to a category "CC" in the same context of the universe "alp...
The square root of 196 is 14.
The lemma "atc_io_reduction_on_sets_alt_def" shows that the adaptive test case input-output reduction on sets for M1, T, and Omega with respect to M2 is equivalent to the condition where the input language of M1 and T is a subset of the input language of M2 and T, and the union over the input language of M1 and T of th...
If for all elements x and y in the set xs, x is equal to y, and for any element x in the set xs, x is equal to y, then the right fold of the addition operation over the set xs with the initial value of 0 is equal to the product of the length of xs and y.
The lemma with equivariance property shows that the permutation of the free variables in the product of A and B (x, y) is equal to the product of the permutation of A, the permutation of B, and the permutation of (x, y).
The composition of the permutation function with input 'x' and the permutation function with input 'y' is equal to the permutation function with the input being the sum of 'x' and 'y'.
The lemma assumes that the mapping of 't' at index 'i' is 'None'. It shows that looking up the first element 'fst' at index 'i' in 't' also results in 'None'.
: - A basic value "b" under the action of a context "B" is not similar to a continuous function "g" under the action of a context "CFn" at any level "n". - A function "f" under the action of a context "Fn" is not similar to a basic value "b" under the action of a context "CB" at any level "n".
If we prefer the identification of Rv and Rw, then the function that takes a state and a vertex and returns the list of successors of the vertex in the state, is related to the function 'succ_to_list' through the relation that maps pairs of Rv and Rw to a list of pairs of Rw and Rv, in the context of nondeterministic r...
If 'x' is in the space of the stock measure of real numbers, then the function 'lift_RealIntVal' applied to 'f', 'g', and 'x' is equal to the function 'lift_RealVal' applied to 'g' and 'x'.
The lemma named "tauLaw3StatImpRight" is defined with fixed variables: Psi of type 'b, and P and Q of type psi ('a, 'b, 'c). It assumes a condition C1: for any Psi', the tuple (Psi tensor Psi', alpha times P, alpha times tau of Q) is in the relation Rel. The lemma then shows that under this assumption, Psi turnstile...
If a node "Node r0 {(t0,e0)}" is a subtree of the decision tree derived from the list "v1 ys", then there exists an event "e", and two lists "as" and "bs", such that the concatenation of "as", the pair "(r0,e)", the pair "(root t0, e0)", and "bs" equals the list that starts with the pair "(v1,e1)" and is followed by "y...
For almost all 'n', the coefficient of the polynomial 'p' at 'n' is equal to 0.
The lemma named "compress_mod" assumes that "x" is in the set from "q" minus the ceiling of "q" divided by "2" raised to the power of "d" to "q" minus 1, and that the natural number "d" is less than the ceiling of the logarithm base 2 of "q". If these conditions are met, it shows that the result of the function "compre...
The theorem "DivergenceOfHarmonicSeries" demonstrates that the harmonic series (the sum of the reciprocals of the natural numbers) is not summable.
The validity of "null" is true.
: Assuming that the relation R is a subset of the Cartesian product of the set F with itself under the transformation T_G, and that the predicates P and Q are compatible. Also, assuming that for any context C, if P holds for C, then the function "funas_gmctxt" applied to C is a subset of F. Furthermore, for any two co...
If there is a relation 'p' between two functions 'f' and 'g' in an environment, then for any 'x', if 'p' holds for 'f(x)' and 'g(x)', it implies the thesis. If this condition is met, then the thesis is true.
If 'x' is a limit ordinal, then 'x' cannot be the successor of any ordinal 'p'.
For a matrix A, the multiplication of A with the sum of two vectors x and y is equal to the sum of the multiplication of A with x and the multiplication of A with y.
A set S is connected in Euclidean space if and only if S is connected.
The static property of the method header "mhd" of method "m" is equivalent to the "is_static" property of method "m".
The generalized implementation relation for implicit generalized Büchi automata (IGBA) with extended edges, denoted as "\<langle>Rm,Rl\<rangle>gen_igba_impl_rel_eext", is the set of all pairs of IGBA implementations and IGBA models (Li, mi) and (L, m) such that (Li, L) is in the relation Rl and (mi, m) is in the relati...
The Robbins algebra is a Huntington algebra.
If a set S has unique keys and a pair (k, v) is in S, then the mapping of the set S minus the pair (k, v) is equal to the deletion of the key k from the mapping of the set S.
For the function "create_graph" applied to any proposition "phi", it is guaranteed that for all nodes "q" in the set of nodes "nds", there exists a unique node "q'" in "nds" such that the name of "q" is equal to the name of "q'".
The lemma named "left_connected" assumes that the sequence of "nu" and "nu prime" exists. It shows that "nu" combined with "mu" is not null if and only if "nu prime" combined with "mu" is not null.
If there exists a decider 'K1D1' such that for all 'nl', if 'nl' is in 'K1' then 'K1D1' yields a numerical result of 1, and if 'nl' is not in 'K1' then 'K1D1' yields a numerical result of 0, then this leads to a contradiction.
The lemma named "depthSubst" is defined with fixed variables P of type pi, and a and b of type name. It demonstrates that the depth of P, where a is substituted by b, is equal to the depth of P.
"delta" is equal to the function "F" applied to the pair consisting of "dual delta" and "delta".
If a vertex is isolated, then its degree is zero.
The product of "n(x)" and the join (or union) of "an(x)" and "an(y)" is equal to the product of "n(x)" and "an(y)".
: 1. The root of a tree represented by a numeral 'n' is the numeral 'n' itself. 2. The left child of a tree represented by a numeral 'n' is the numeral 'n' itself. 3. The right child of a tree represented by a numeral 'n' is the numeral 'n' itself.
The lemma is from the stateful_protocol_model in the Automated_Stateful_Protocol_Verification. It is named timpl_closure_Ana_keys and it fixes s and t as protocol terms. The lemma assumes that t is in the term implication closure of s and TI, and that the analysis of s and t results in (Ks, Rs) and (Kt, Rt) respectivel...
The lemma named "greater_Card" assumes "\<kappa>" is a cardinal number. It shows that "\<kappa>" is less than the cardinality of the power set of "\<kappa>".
In the context of program P, environment E, and heap h, the sequential composition of expressions e1 and e2 has the type T2 if and only if there exists a type T1 such that e1 has the type T1 and e2 has the type T2 in the same context.
The lemma named "absolutely_integrable_absolutely_integrable_ubound" is about functions 'f' and 'g' that map from a Euclidean space 'm' to real numbers. It assumes that function 'f' is integrable on set 'S' and function 'g' is absolutely integrable on set 'S'. Furthermore, for any element 'x' in set 'S', the value of f...
The lemma named "pinv_double_half" assumes that "p" is a positive rational number. It shows that the product of "half" and the multiplicative inverse of "p" is equal to the multiplicative inverse of the sum of "p" and "p".
: if 'n' is a valid node, and for any edge 'e', if 'e' is a valid edge and 'n' is the source node of 'e', it implies the thesis, and for any edge 'e', if 'e' is a valid edge and 'n' is the target node of 'e', it also implies the thesis, then the thesis holds.
If from the insertion of proposition A into a set H, proposition B can be inferred, then from the insertion of the negation of proposition B into set H, the negation of proposition A can be inferred.
If 'ls' is a subfunction of 'univ_for_pr', then 'g_step ls key' is also a subfunction of 'univ_for_pr'.
If a program P sees that class C has a method M with parameters of types Ts, returning type T, and the method's body is m in class D, then the program P also sees that class D has the same method M with the same parameters of types Ts, returning the same type T, and the method's body is m in class D.
The holding condition for a set "x" on the stream map of a function "f" and a stream "s" is equivalent to the holding condition for the preimage of "f" under "x" on the stream "s".
The lemma named "election" takes two parameters, "agents" and "alts".
The simplicial subdivision of any point 'p' at 0 is equal to 0.
Applying the identity combinator to any element x is equivalent to the element x itself.
The lemma named "countable_image" assumes that set A is countable. It then shows that the image of set A under function f is also countable.
The lemma "Solutions_snd_not_0" assumes that if a pair (x, y) is in the set of Solutions and x is not zero, then y is also not zero.
The lemma named "example" is defined with fixed real numbers x and t, and a finite variable i. It assumes that t is greater than 0. It shows that x is equal to the flow on an open set, which is a function of time and state, evaluated at time t and state x, and then projected onto the i-th coordinate.
The lemma for the function-based relation of the insertion of the ordered pair (a, b) into the relation r is equivalent to the function-based relation of r.
If the function 'f' applied to 'x' is less than or equal to the function 'f'' applied to 'x'', then the result of binding 'x' to 'f' is less than or equal to the result of binding 'x'' to 'f''.
The type substitution of a term 't' with a substitution function 'ρ' is equal to the substitution of type' with a mapping function that maps each pair (x, y) to ((x, y), ρ x y) for some list 'l' that is distinct and the type variables of 't' are a subset of the set of elements in 'l'.
If a well-sorted term 'X' exists for a sort 's' and 'y1' is fresh for 'X' with respect to 'ys', then substituting 'y1' for 'y' in 'X' and then substituting 'y2' for 'y1' results in the same term as directly substituting 'y2' for 'y' in 'X'.
The Ford-Fulkerson method is less than or equal to the specification function such that the function is a maximum flow.
: Assuming that a command 'c' is typeable from context 'Γ' to 'Γ'' (as per the assumption 'typeable'), the mode state 'mds' is consistent with the context 'Γ' (as per the assumption 'mds_consistent'), and for all variables 'x', if the total function of 'Γ' and 'x' is 'Low', then the memory state 'mem₁' of 'x' is equal ...
"x" is less than or equal to the function "R" with inputs 0 and 1.
If 'm' is less than 'n' and 'S' is an infinite set, then the enumeration of 'S' at 'm' is less than the enumeration of 'S' at 'n'.
The lemma named "monadic_WCR" assumes the following conditions: 1. The function-as-relation 'funas_rel' of 'R' is a subset of 'F'. 2. 'F' is monadic. 3. 'R' is a left-linear right-ground rewrite system (llrg). 4. The weak Church-Rosser property (WCR) holds for the generalized single-step reduction 'gsrstep' of 'F' and ...
If a parallel composition "A" is secure according to the security model "Sec", and "A'" is an element in the set resulting from the transition relation "tr_pc" applied to "A" and an empty list, then the parallel composition "A'" is also secure according to the security model "Sec".
The unit interval in a Euclidean space, represented as a closed box from 0 to 1, is equal to the convex hull of the set of points x such that for every i in the basis of the space, the inner product of x and i is either 0 or 1.
If there exists an element 'w' such that 'w' is related to 'x', 'y', and 'z', then for all 'w', 'w' is related to the product of 'x' and the product of 'y' and 'z' if and only if 'w' is related to 'x', 'y', and 'z'.
The lemma named "wfT_wfG" is fixing variables x, z of type x, a context Gamma of type Gamma, and variables c of type c and b of type b. It assumes that under the context of Theta, B, and Gamma, the dependent pair (z : b | c) is well-formed, and that the atom x is fresh for Gamma. It then shows that under the context of...