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If for every natural number 'n', the set 'S n' is Lebesgue measurable, and if the intersection of any two such sets 'S m' and 'S n' is negligible (i.e., has measure zero), and if the union of all such sets 'S n' is bounded, then this union is also Lebesgue measurable. The second part states that under the same conditi...
The operation of restricting the right side of a graph, represented by "op_graph_restrict_right Vr E", is equivalent to the operation of creating a set of edges from the successors of a vertex "v", where each successor "x" is in the set of edges E that start from "v" and also in the set "Vr".
The statement is a lemma named "Euclid_induct" with case names "swap", "zero", and "add". It fixes a predicate P that takes two natural numbers. It assumes three conditions: 1. The predicate P is symmetric, i.e., P(a, b) is true if and only if P(b, a) is true. 2. The predicate P is true for any natural number a and ...
The lemma "esum_list_LCons" assumes that "x" is greater than or equal to 0 and for any "x" in the list "xs", "x" is also greater than or equal to 0. It shows that the sum of the list that starts with "x" and is followed by the list "xs" is equal to "x" plus the sum of the list "xs".
An empty list is not in "cptn".
If a pair (S,T) is in the ChiRel (Chidamber and Kemerer's relationship) of a Hierarchical Automaton (HA) ST, and if T is in the configuration of ST, then S is also in the configuration of ST.
The theorem named "associative_rat_poly_mat" assumes that m1, m2, and m3 are matrices with rational polynomial entries, where the number of columns in m1 is equal to the number of rows in m2, the number of columns in m2 is equal to the number of rows in m3, and m3 has a certain number of columns denoted by nc. Under th...
The function "raw_match" with parameters T, U, and subs is equal to the function "raw_match'" with the same parameters T, U, and subs.
If from the union of set A and the set containing the implication from p to q, we can infer p, then we can infer p from set A alone.
The lemma "list_updates_nth_notmem" assumes that the lengths of lists "xs" and "ys" are equal and that the element "i" is not in the set of "xs". It shows that the "i"-th element of the list resulting from updating the list "vs" with "xs" and "ys" is equal to the "i"-th element of the original list "vs".
The lemma named "integral_ineq_eq_0_then_AE" is about functions f and g that map to real numbers. It assumes that almost everywhere in the measure space M, the value of function f is less than or equal to the value of function g. It also assumes that both functions f and g are integrable over the measure space M, and t...
A diagonal matrix of size 'n' with all diagonal elements as '1' is equal to the identity matrix of size 'n'.
The lemma "Hoare_unhalt_impl_not_reaches_final" assumes that for a given program 'p', the initial state of the tape is empty on the left and contains some sequence 'ns' on the right. If this program 'p' does not halt, then it is shown that the program 'p' does not reach a final state with the sequence 'ns'.
If "a" is not in "B", then hiding "B" in the process of "a" leading to "P" is equivalent to "a" leading to the process of hiding "B" in "P".
The lemma "eval\<^sub>1_preserves_unmod" consists of two parts: 1. If a program P evaluates an expression 'e' in a state with heap 'h' and local variables 'ls' to a new state with expression 'e'', heap 'h'', and local variables 'ls'', and if the expression 'e' does not modify the variable at index 'i' and 'i' is withi...
If X is in the equivalence classes of Qeq and Y is also in the equivalence classes of Qeq, then X is equal to Y or the intersection of X and Y is empty. This means that the classes are disjoint, they either coincide or have no elements in common.
The lemma named "delta_ss_accs_precise" assumes that given "accs delta_ss t s" and "q is an element of s", it shows that "accs delta t q" holds true.
For any given state 's' in the system 'G', 's' can execute all operations to return to its original state 's'.
The lemma named "soundness_mat_leqb_eqc1" assumes that the number of rows in matrix A is equal to the dimension of vector b. It also assumes that the result of the simplex algorithm on the system of inequalities and equations represented by "mat_leqb_eqc A b c" is a satisfiable solution X. Under these assumptions, it s...
If a proposition 'phi' is in 'v' and if 'phi' implies 'psi' is in 'v', then 'psi' must also be in 'v'.
A relation 'x' is asymmetric if and only if the infimum (greatest lower bound) of the square of 'x' is irreflexive.
The mapping of a probability mass function (pmf) that has been bound with another function 'f' is equal to the auxiliary bind pmf function applied to 'p', 'f', and the set of the pmf 'p'.
Zero is less than one plus the absolute value of x, where x is a real number.
"a" is greater than or equal to 1 divided by "c minus 1".
The lemma max_flow states two things: 1. The maximum flow is a network flow from source 's' to sink 't' with capacity 'c' (referred to as ?thesis1). 2. There does not exist a path 'p' such that the maximum flow can be increased by augmenting along this path (referred to as ?thesis2).
The lemma named "size_pratt_fpc" assumes that "a" is less than or equal to "p", "r" is less than or equal to "p", "a" is greater than 0, "r" is greater than 0, "p" is greater than 0, and the product of the list "qs" equals "r". It shows that for all "x" in the set that is built by the function "build_fpc" with paramete...
If the flowsto condition holds for the set X0, time T, set CX, and set Y, and if the pair (x, d) is an element of X0, and if T is a subset of the set of all positive real numbers, then for all future times t (approaching from the right of 0), the pair consisting of the flow at time t starting from x and the derivative ...
If the absolute value of x is less than or equal to 1 and the absolute value of y is less than or equal to 1, then the arcsine of x is less than the arcsine of y if and only if x is less than y.
If two states s1 and s2 are equal in memory at a given physical address 'pa' (as stated by the assumption a1: "mem_equal s1 s2 pa"), then the alternative memory values at 8-bit chunks of the physical address (and the next three addresses) remain the same in both states even after updating the 10th and 11th memory locat...
If 'x' satisfies the condition of 'co_test', then 'x' also satisfies the condition of 'sup_dist_contact'.
A match occurs in the context of a logical "not" applied to a logical "or" between two matches (m1 or m2) for a given packet and action, if and only if a match occurs for both the logical "not" of m1 and the logical "not" of m2 for the same packet and action.
If 's' is an element of 'S', then the function 'F' applied to the absolute value of the free letter 's' in the Free Group 'S' is equal to the function 'f' applied to 's'.
The lemma "LLL_dµ_bound" assumes that "i" is less than "m" and "j" is less than "i". It shows that the absolute value of "dµ" of "fs", "i", and "j" is less than or equal to "N" raised to the power of twice "m - 1", times 2 raised to the power of "m - 1", times "m".
The lemma named "finite_ample_reachable" assumes that "prog" is a type of program. It shows that the set of all states reachable from the initial state in the program "prog", when transitions are defined by the function "impl4_succ_impl" and visibility is determined by "is_vis_var", is finite. This set is referred to a...
There is a single-pass instruction from state Sigma to state Sigma prime. It shows that there exists a state Sigma double prime and an instruction prime such that the optimized instruction from state Sigma equals to some value of instruction prime and state Sigma double prime, and the length of state Sigma prime equals...
If from the assumption that a statement phi is in v, it follows that another statement psi is in v, then it can be concluded that "phi implies psi" is in v.
For a given polynomial 'p' of type 'v' (where 'v' is a linear order) and 'a' (where 'a' is a polynomial carrier), if the function "check_poly_weak_mono_smart" for the discrete variable 'v' holds true for the polynomial 'p', then the polynomial 'p' is weakly monotonically increasing with respect to the variable 'v'.
If the function "mute" is applied to the parameters q, k, l, r, and A, resulting in r' and A', and if i is less than the number of rows in A, j is less than the number of columns in A, l is less than the number of rows in A, k is less than the number of rows in A, and i is not equal to k, then the element at the i-th r...
There exists exactly one 'x' such that 'x' satisfies properties 'P' and 'Q' and 'x' equals 'a'. This is equivalent to saying that 'a' satisfies properties 'P' and 'Q'.
An integer "a" is less than a real number "b" if and only if "a" is less than the floor of "b" or "a" is equal to the floor of "b" and the real number equivalent of the floor of "b" is less than "b".
If "g" is an element of the carrier of the semialgebraic set "SA k" and "f" is also an element of the carrier of the semialgebraic set "SA k", then the set of all "x" in the carrier of the p-adic field "Q_p" to the power of "k" for which the valuation of "g x" is less than the valuation of "f x" is semialgebraic.
: Assuming the following conditions: 1. The sequence of basic expressions "b_es" is well-typed from type sequence "ts" to "ts'" under the context "\<C>". 2. The sequence of commands "cs" is well-typed from an empty type sequence to "ts" under the store "\<S>" and context "\<C>". 3. There is no level-zero label hole "l...
The lemma named "horner_eval_Suc'" is defined for a variable 'x' of a type that is both a commutative monoid under addition and supports multiplication. It shows that the evaluation of the Horner scheme at the successor of a natural number 'n' is equal to the value of the function 'f' at 0 plus 'x' times the evaluation...
If the first component of a natural number 'n' modulo 7 equals 4, then the function 'nat_to_pr' applied to 'n' and 'x' equals the function 'nat_to_pr' applied to the first component of the second component of 'n' and the function 'nat_to_pr' applied to the second component of the second component of 'n' and 'x'.
The lemma named "ent_wandI2" assumes that if proposition P implies (in the sense of entailment) the wand of Q and R, then it shows that the combination of Q and P implies R.
"If there is an obligation of B given A, then there is an obligation of B given both A and B."
Encoding a term, which is a product of a constant natural number 1 and a factor of a constant natural number 2, results in a specific sequence of boolean values: [False, False, True, False, True, True, True, False, True, True, False, False, True, True, False, True, True].
The lemma named "reflects_isomorphic" assumes that "f" and "f'" are identities in category "A", and that the functor "F" maps "f" and "f'" to isomorphic objects in category "B". It shows that "f" and "f'" are isomorphic in category "A".
The lemma named "pr1_in_hom" assumes that "a" and "b" are identities. It shows that the first projection from the product of "a" and "b" maps to "a".
The lemma named "fmap_neq" is defined with two fixed variables, s and s', both of which are finite maps from some type 'a to boolean. It assumes that the domain of s is equal to the domain of s'. It then shows that s is not equal to s' if and only if there exists a value v in the domain of s such that the value of v in...
"v" is equal to the sum of the product of "g" and "groups_of_size g" for each "g" in "group_sizes".
If a natural number 'i' is less than the difference of 'j' and 'm', and 'n' is less than or equal to 'm', then 'i' is less than the difference of 'j' and 'n'.
The lemma named "bit_numeral_simps" states three things: 1. The bit at position "numeral n" of the number 1 is not set. 2. The bit at position "numeral n" of a number represented by "numeral (Num.Bit0 w)" is the same as the bit at position "pred_numeral n" of the number represented by "numeral w". 3. The bit at positi...
The lemma named "not_cong_is_anga1" assumes that angle ABC is not congruent to angle A'B'C' and that angle ABC is acute. It shows that angle A'B'C' is not acute.
Applying the "K_statefun" function to any state 'x' with a constant 'c' will always result in the constant 'c'.
If the proposition "A equals A" is false, then the result is R.
The set of accessible components at level 1 for component "sA6" is the set consisting of components "sA71", "sA72", "sA81", "sA82", "sA91", "sA92", and "sA93".
For a fixed element 'a' of a type that is both a real normed field and a second countable topology, the Lebesgue integral over a set 'A' of 'a' multiplied by a function 'f' at 't' is equal to 'a' multiplied by the Lebesgue integral over the set 'A' of the function 'f' at 't'.
The function 'set_val' is an instance of 'l_set_val' given the conditions 'type_wf', 'set_val', and 'set_val_locs'.
The lemma named "correct" assumes two conditions. First, it assumes that the pair of pairs ((x0,x1), (w0,w1)) belongs to the relation Rel_AND. Second, it assumes that "e" belongs to the challenge space. Given these assumptions, it shows that the completeness game of Sigma_AND with inputs (x0,x1), (w0,w1), and e returns...
If a is greater than b (where a and b are elements of a linearly ordered topology), then eventually for values to the immediate right of b, x will fall within the open interval between b and a.
The lemma named "prv_expl" assumes two conditions. First, the formula phi is in the set of all formulas. Second, the formula "fls" is provable. Given these assumptions, it shows that the formula phi is provable.
The lemma "inf_step_into_τinf_step" assumes that the state "s" transitions infinitely often via some thread-local steps "tls". It shows that there exists a set "A" such that the state "s" makes infinite silent transitions according to the thread-local steps "tls" in "A".
If a label instruction with certain properties leads to a sequence of values and expressions, then the type of the sequence of values and expressions is preserved. Specifically, the assumptions are: 1. A label instruction with a certain number of values, a sequence of expressions, and a label instruction sequence lead...
The lemma named "ν_le_ν_opt" assumes that "p" is a policy. It shows that the value of policy "p" at state "s" is less than or equal to the optimal value at state "s".
If the following conditions are met: 1. The variable 'result' is equal to the execution of the function 'trans_fun' on the 'input' and 'c' in the context of a computation stream. 2. For all 'x_n' and 'c_n', if 'P' holds for 'c_n', then 'Q' holds for 'x_n' and the result of applying 'trans_fun' to 'x_n' and 'c_n'. 3. F...
The closure of a set S in a topological space X is equal to the intersection of the topological space X and the union of the set S and the derived set of S in X.
The lemma named "weakBisimWeakCoinductAux" takes one assumption and has three fixed elements: P, Q, and X, which are all of type "ccs" or "ccs set". The assumption is that the pair (P, Q) is in set X and for any P and Q in set X, P weakly leads to Q under X and the pair (Q, P) is also in set X. If these conditions are ...
The ring multiplication of n and m in the ring of integers is equal to the standard multiplication of n and m.
For all lists "ss" within the set "S", and for all lists "ts", if "ss" is a flipped alternate sublist adjacent to "ts", then the sum of the elements in list "ts" is equal to the sum of the elements in list "ss".
If 'b' is equal to the function 'f' applied to 'x', and 'c' is equal to the function 'g' applied to 'x', and 'x' is an element of set 'A', then the pair '(b, c)' is an element of the image of the set 'A' under the functions 'f' and 'g'.
If 'f' is an element of the morphisms of the category exponential of 'A' and 'B', then the domain of 'f' in the category exponential of 'A' and 'B' is equal to the natural transformation domain of 'f'.
The leading term of the product order (with respect to the degree reverse lexicographic order) of the sum of the vector with index 1 and the polynomial X squared times Z, and the vector with index 0 and the polynomial 2 times Y cubed times Z squared, is equal to the sparse vector with index 0 and degree 2, and index 2 ...
If 'x' is greater than 0 and less than pi/2, then the cotangent of 'x' is greater than 0.
In the context of a weak quantale, the product of 'x' and the bottom element is equal to the bottom element.
The lemma is proving that the positive semi-linear program (slp) with no variables, a precision of 20, using the Caprasse method with a maximum of 10000 iterations, and a set of affine forms of intervals (each replicated 4 times with a value of 1/2), is true.
The lemma named "mono_D22" assumes that "P" is less than or equal to "Q" in some order. It shows that for all "s", if "s" is not in the domain of the sequential composition of "S" and "P", then the range of the sequential composition of "S" and "P" for "s" is equal to the range of the sequential composition of "S" and ...
If the function 'f' applied to 'x' equals 'Some y', and 'z' is not in the domain of the function 'f', then 'x' is not equal to 'z'.
The lemma named "subst2" is defined with the following fixed variables: a list of names "xvec", a list "Tvec" of type 'a, a name "x", a psi "P" of type ('a, 'b, 'c), an input "I" of type ('a, 'b, 'c), and a psiCase "C" of type ('a, 'b, 'c). The lemma assumes that the name "x" is fresh for the list "Tvec" and the list...
The lemma named "grid_shift_base" assumes three conditions: 1. The intersection of ds and ds' is an empty set, 2. b is an element of the sparse grid with dimension dm, 3. p is an element of the grid with base as the union of ds' and ds and the union of ds' and ds as the dimension set. Under these assumptions, it s...
If the size of the k-d tree is equal to 2 raised to the power of the minimum height of the k-d tree, then the k-d tree is complete.
If the term-order of the term "tt p" is less than "v", then the term-order of "lower p v" is equal to the term-order of "tt p". Here, "tt" refers to the highest term of a polynomial "p", "lower" refers to a function that lowers the degree of a polynomial, and "\<prec>\<^sub>t" denotes a term-order relation.
The Paxos protocol refines the Three-Step Most Recently Used (MRU) protocol with respect to the relation "paxos_ref_rel". This is done under the conditions defined by the timestamp "majorities.ts_mru_TS" and the Paxos timestamp scheme "Paxos_TS" with given higher order functions "HOs" and coordinators "crds".
If 'f' and 'g' are both residuated functions, then the conjugate of the composition of 'f' and 'g' at a point 'x' is equal to the conjugate of 'g' at the point which is the conjugate of 'f' at 'x'.
The weight of a scaled probability mass function (scale_spmf r p) is equal to the minimum of 1 and the product of the maximum of 0 and r, and the weight of the original probability mass function p.
The lemma named "cblinfun_blinfun_transfer_plus" in the context of Complex Bounded Linear Functions and Complex Bounded Operators, includes lifting syntax. It shows that the equality of complex bounded linear functions is preserved under the addition operation. This means that if two pairs of complex bounded linear fun...
The function which maps each element 'x' to the probability mass function of 'M x' is measurable with respect to the countable universal set and the subprobability algebra of the countable universal set.
The lemma 7 assumes that r standardizes to r'. It shows that if r' is in normal form, then r locally reduces to r'.
If 'p' implies 'q', and 'q' implies 'r', then 'p' implies 'r'.
If the list 'l' satisfies the 'ufa_invar' condition and 'i' is less than the length of 'l', then the representative of 'i' in 'l' is 'i' if the 'i'-th element of 'l' is 'i'. Otherwise, the representative of 'i' is the representative of the 'i'-th element of 'l'.
The modulus of any natural number 'i' with zero is 'i' itself.
If a nonce "n" is in the analyzed set "G", and "n" is guarded by a set of keys "Ks" in "G", and "G" is finite, then there exists a key "K" such that "K" is in "Ks" and the key "K" is in the analyzed set "G".
For a given system A, the observed failure oracle of callee1 with a fault is equivalent to the observed failure oracle of callee2 with a fault.
The lemma named "gen_lossless_bind_gpvI" assumes that a generative probabilistic value (gpv) is lossless with respect to a certain context (co) and interface (\<I>). It also assumes that for any result (x) within the results of the gpv, the function (f) applied to this result is also a lossless gpv under the same conte...
The lemma named "pcr_SomeE" is a case predicate. It assumes that "pcr_Some R x y" is true. If this is the case, it obtains a value "z" such that "y" is equal to "Some z" and "R x z" is true.
The lemma named "targets_respects_Cong" assumes that transition system T is congruent to transition system T'. It shows that the set of target states in transition system T is equal to the set of target states in transition system T'.
Looking up a key set 'ks' in a trie 't' after updating 't' with a value 'v' for 'ks', is the same as looking up 'ks' in 't' and then mapping 'ks' to 'v'.
The transitive closure of the supremum (or union) of x and y is less than or equal to the transitive closure of the product of the transitive closures of x and y.
The lemma HVZK1 assumes that the pair (x1, w1) is in the relation Rel1. It shows that for all c in the challenge space, the relation R of Sigma OR with parameters (x0, x1), (Inr w1), and c is equal to the relation S of Sigma OR with parameters (x0, x1), and c.
In the context of the generative probabilistic value (gpv) monad, the 'Fail' operation behaves as a failure in a monad, with 'Done' as the return operation and 'bind_gpv' as the binding operation.