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The lemma named "check_mult_l_check_mult" assumes the following conditions: 1. The pair (A, B) is in the relation defined by fmap_polys_rel. 2. The pair (r, r') is in the composition of the relations sorted_poly_rel and mset_poly_rel. 3. The pair (q, q') is in the composition of the relations sorted_poly_rel and mset_...
, assuming "a" and "b" are elements of the union of "hyp2" and "S", it shows that the endpoint in "S" of "a" and "b" is an element of "S". Additionally, it shows that the real number "B" is the Cartesian product of "a", "b", and the endpoint in "S" of "a" and "b".
The lemma named "contractible_convex_tweak_boundary_points" is about a set S in a Euclidean space. It assumes that S is convex and that the relative interior of S is a subset of T, and T is a subset of the closure of S. Under these conditions, it shows that T is contractible.
The integer value of a rational number 'r' is equal to the following: First, 'r' is divided into a quotient consisting of 'a' and 'b'. If 'b' equals 1, then 'a' is the result. However, if 'b' does not equal 1, the code will abort and display the message "Failed to convert rat to int", and the function will attempt to c...
The lemma named "imp_compat_eqvL" assumes that the formulas phi1, phi2, and chi are in the set of all formulas. It also assumes that phi1 is provably equivalent to phi2. Under these assumptions, it shows that the implication of phi1 and chi is provably equivalent to the implication of phi2 and chi.
The lemma named "sign_r_pos_mult" is about real polynomials "p" and "q". It assumes that neither "p" nor "q" are zero. It shows that the sign of the real part of the product of "p" and "q" at a point "x" is positive if and only if the sign of the real part of "p" at "x" is the same as the sign of the real part of "q" a...
: Assuming that 1) there is a sequence of steps in program P that transforms expression e1 in state s0 with breakpoint b0 into an address a in state s1 with breakpoint b1, 2) there is another sequence of steps in program P that transforms expression e2 in state s1 with breakpoint b1 into a value v in a state charact...
If gamma1 holds and PsiInv holds in the current state, then PsiInv will hold in the next state.
The theorem named "rate_lower_bound" shows that the entropy of X is less than or equal to the code rate of encoding X.
The lemma "lastMarkedNotInPrefixBeforeLastMarked" assumes that the elements in M are unique and that the marked elements in M are not empty. It shows that the last marked element in M is not in the set of elements in the prefix before the last marked element in M.
The bit at position "m" of the result of dropping "n" bits from an integer "x" is the same as the bit at position "n + m" of the integer "x".
The lemma ConD shows the following: 1. If the composition of t and u is congruent to the composition of t' and u', then t is congruent to t' and u is congruent to u'. 2. If the lambda abstraction of v applied to u is congruent to the lambda abstraction of v' applied to u', then the lambda abstraction of v is congruent...
For any formal power series 'a' in a division ring with characteristic 0, the integral from 0 of the derivative of 'a' is equal to 'a' minus the constant term of 'a'.
If alpha prime is equal to alpha, J prime is the opposite category of J, C prime is the opposite category of C, and F prime is the opposite of the covariant functor F, then the opposite natural transformation of N is a cocone from F prime to c in the category J prime, which is a functor in the category C prime with res...
In the lemma for a group, if H is a subset of the carrier G and K is also a subset of the carrier G, and if a is an element of H and b is an element of K, then the product of a and b is an element of the set formed by the binary operation on H and K in the group G.
The keys of an empty map are an empty set.
The application of the instruction "Throw" with parameters P, pc, mxs, T\<^sub>r, and s is true if and only if there exists a type T, a stack type ST, and a local type LT such that s is equal to the pair (T#ST, LT) and T is a reference type.
The initial state of s0g is a subset of s0g_secrecy.
If a list starting with element 'x' followed by list 'xs1' is equal to list 'ys', and list 'xs' is equal to the concatenation of list 'xs1' and list 'zs', then the list starting with 'x' followed by list 'xs' is equal to the concatenation of list 'ys' and list 'zs'.
The probability mass function (pmf) of the transition function "K_st" for a given decision "d" and state "s" at a state "t" is equal to the integral over all actions "a" of the pmf of the transition function "K" for the state-action pair "(s, a)" at state "t", with respect to the decision "d" at state "s".
The function "read" applied to elements "el", "s", and "t" is equal to the function "map_option" applied to "pnet_\<alpha>" and the function "read'" applied to "el", "s", and "t".
The lemma named "coplanar_perm_3" assumes that points A, B, C, and D are coplanar. It then shows that points A, C, D, and B are also coplanar.
If a recursive function 'f' has two arguments, then evaluating the function 'r_swap f' with the arguments 'x' and 'y' is equivalent to evaluating the function 'f' with the arguments 'y' and 'x'. In other words, 'r_swap' swaps the order of the arguments of the function 'f'.
The m-th iteration of function L with parameter d applied to v is equal to the m-th term of the sequence defined by the function νb_fin applied to the stationary version of d, plus the m-th power of the product of l and the transition probability of d, all applied to v.
The lemma named "map_nth_append2" assumes that the length of the list "ys" is "d". It shows that if you map the function that gets the nth element of the concatenated list of "xs" and "ys" over the range from the length of "xs" to the length of "xs" plus "d", you will get the list "ys".
If "a" is less than or equal to "c", then the interval from "r" to "c" with a step size of "m" at the position "a" is equal to "r" plus "m" times "a".
Updating the identity map at the index "i" with the value "i" for the right part of the disjoint sum, results in the identity map at the successor of "i".
If "ta" is an empty list, then there exist "t1" and "u1" such that if "s" concatenated with "t" interleaves with the pair "(ta, u)" over the set "A", then "t1" is a subsequence of "ta" and "u1" is a subsequence of "u", and "s" interleaves with the pair "(t1, u1)" over the set "A".
For a certain function 'return_id' and 'return_pmf', if 'A' is related to 'cr_id_prob A', then 'return_id' is related to 'return_pmf' in the same way.
If there exists an alpha such that both phi of alpha and psi of alpha are true, then it implies that there exists an alpha for which phi of alpha is true and there exists an alpha for which psi of alpha is true, in the context of v.
If Gamma and the relation between h1 and h2 implies h0, and the relation between h2 and h1 also implies h0, then this implies Delta. Therefore, if Gamma and the relation between h1 and h2 implies h0, then this also implies Delta.
The weakest precondition of an action with an empty set is the negation of the domain of the action.
The flopped half chamber system of a chamber system C is equal to the half chamber system of C under a group action g.
The finite map created by mapping a set of elements 'xs' to a value 'v' unioned with the finite map created by mapping a set of elements 'ys' to the same value 'v' is equal to the finite map created by mapping the union of the sets 'xs' and 'ys' to the value 'v'.
The lemma named "Disjoint_E" assumes that under the hypothesis set H, t and u are disjoint sets, and that a is an element of both t and u. Under these assumptions, it shows that the hypothesis set H implies A.
If the following conditions are met: - A phase 1 or 2 read operation has transitioned the system from state 's' to state 's' for process 'p', disk 'd', and request 'r' (as per the assumption "act"). - The invariant "HInv4c" holds for the system state 's' and process 'p' (as per the assumption "inv4c"). - The inner inv...
The set of states on the right-hand side of a tree automaton is a subset of the set of states of the tree automaton.
The lemma named "is_auto_reducedD" assumes that "B" is auto-reduced and "b" is an element of "B". It shows that "b" is not reducible by the set "B" excluding "b".
If the depth-first search (dfs) values of two sets 'xs' and 'ys' are distinct, and the intersection of these two sets is empty, then the depth-first search values of the meld of 'xs' and 'ys' are also distinct.
If x is an element of set A, then set A is not empty.
If the free variables of a stateful strand S are a subset of the domain of a substitution sigma, then the free variables of the stateful strand S after applying the substitution sigma are a subset of the range variables of sigma.
The length of the list 'ts' is equal to 2 if and only if there exist elements 'a' and 'b' such that 'ts' is a list consisting of 'a' and 'b'.
The value of the function "psi_odd_2" at any given number "n" is less than or equal to the value of the function "psi_even_2" at the same number "n".
The lemma Z_I_hermitian demonstrates that Z_I is Hermitian.
The set of all pairs (x,e), where if e is in the output arcs of tree T at root r then x is equal to the auxiliary to_dtree function applied to the head of tree T at e, is finite.
In the context of finite measure-preserving transformations (fmpt), we have a lemma called "fmpt_factor_projection". This lemma takes a function 'f' that maps from type 'a' to type 'b', where 'b' is a second countable topology. It assumes that 'f' is Borel measurable in 'M'. The lemma shows that the measure-preserving ...
For a given binary relation 'R' and elements 'x', 'y', and 'z', if 'R' is right-unique (meaning for any 'x' and 'y', there is at most one 'z' such that 'R x z' and 'R y z' are both true), then if there is a transitive closure of 'R' from 'x' to 'y' and there is no element 'w' such that 'R y w' is true, and if there is ...
A specific key 'n' is not in the guard key set 'Ks' for the same key 'n'.
The lemma named "sqrt_lower_bound_id" assumes that "x" is greater than or equal to 0 and less than or equal to 1. It shows that "x" is less than or equal to the square root of "x".
The imaginary part function is a bounded linear operation.
The relations in a tree "t" are distinct if and only if for all elements "x" in the multiset of relations of "t", the count of "x" in the multiset is equal to 1.
The lemma named "b_e_type_set_local" assumes that a context "C" validates that an expression "e" has a type transition from "ts" to "ts'", and that "e" equals "Set_local i". It then shows that there exists a type "t" such that "ts" equals the concatenation of "ts'" and "t", and that the "i"-th element of the local cont...
Filtering a finite map "m" first by predicate "P" and then by predicate "Q" is equivalent to filtering "m" first by predicate "Q" and then by predicate "P".
If a' is not Null, a' is a subclass of Class C, dynT is the first element of the pair returned by the function h applied to the address of a', dynT is a class in G, and G is a well-structured program, then dynT is a subclass of Class C in G.
The transition from state 's' to state 'u' through commands 'c' and 'd' is equivalent to the existence of an intermediate state 't' such that 's' transitions to 't' through command 'c' and 't' transitions to 'u' through command 'd'.
If there is a polynomial relationship, then the leading coefficient of the polynomial in the "ops" structure is related to the leading coefficient of the polynomial by some relation R. This is a transfer rule, which means it is used to transfer properties between different representations of the same mathematical conce...
If epsilon is a coequalizer in the category E with respect to the set containing the parallel morphisms gPL and fPL, where each morphism in this set is mapped to f if it equals fPL and to g otherwise, then epsilon is also a coequalizer in the category E with respect to the morphisms g and f. This is under the assumptio...
The lemma named "BisT_coind" assumes that "theta" is symmetric and "theta" is a subset of "RetrT" of the union of "theta" and "BisT". If these conditions are met, it shows that "theta" is a subset of "BisT".
If a pair (ainf, l) belongs to the set St and l' is a suffix of l, then the pfragment of ainf and l' also belongs to the set St.
: if the set of all pairs x in the product space of M1 and M2, such that a property P holds for the first and second elements of x, belongs to the set of measurable sets in the product space of M1 and M2, then the statement "for almost every x in M1, for almost every y in M2, the property P holds for x and y" is equiva...
The lemma named "cols_upt_k_insert" is defined for a matrix 'A' of arbitrary type. It assumes that the successor of 'k' is less than the number of columns in 'A'. It shows that the set of columns up to the successor of 'k' in 'A' is equal to the set obtained by inserting the column at the position of the successor of '...
If the function sets M and N are disjoint, then the function sets N and M are also disjoint.
If variable 'x' does not affect 'u', then 'x' does not affect the sign of 'u'.
For a fixed Dirichlet series 'f' of a certain type (specifically, a type that is both a natural number powered normed field and a Banach space), the Dirichlet series 'f' converges in norm at a point 's' if and only if the absolute value of the Dirichlet series 'f' converges at the real part of 's'.
The parallel morphism of F with respect to "aa" is not equal to the parallel morphism of F with respect to "bb".
The function that takes a tuple of a left tree, a node, and a right tree and constructs a new tree from them, is a measurable function from the product sigma algebra of the tree sigma algebra, the given sigma algebra M, and the tree sigma algebra again, to the tree sigma algebra.
Converting an empty list using the function "from_list" results in an empty output.
"For all x, the property P holds for x."
The uniqueness of a record type substitution (where the kth field is substituted with type U) is equivalent to the uniqueness of the original field sequence.
: given a protocol with events, state, agent, and user, if the internal mapping of the state with a certain agent, nonce, and run is None, then the external mapping of the state with the same agent, nonce, and run is also None.
If "x" and "y" form a direct product, then "y" is total.
The identity function on set A is a single-valued function.
The lemma named "map_of_rev_zip_upto_length_eq_nth" assumes that "i" is less than the length of list "B" and "d" equals the length of list "B". It shows that mapping over the reversed zipped list of a range from 0 to "d" (exclusive) and list "B", at index "i", is equal to the "i"-th element of list "B".
The multiplication of the matrix representation of a bounded linear function 'f' and a vector 'v' is equal to the function 'f' applied to the bounded function of 'v' indexed by 'i'.
The lemma "det_swap_initial_rows" assumes that a matrix A is an m by m matrix, and that the sum of k and n is less than or equal to m. It shows that the determinant of A is equal to (-1) raised to the power of the product of k and n, times the determinant of a new m by m matrix. This new matrix is defined such that for...
The lemma "append_subset" assumes that the set of elements in list "xs" is equal to the set of elements in the combined list of "ys" and "zs". It then shows that the set of elements in list "ys" is a subset of the set of elements in list "xs", and the set of elements in list "zs" is also a subset of the set of elements...
"north" is not in the set that includes "east" and "west".
If C is a part of the connected components of X, then C is a subset of the topological space of X.
If 'd' is less than the length of 'p', then the index 'ix' of the child 'p' in direction 'dir' at depth 'd' is odd.
The lemma named "expectation_bernoulli_pmf" is about a function 'f' that maps a boolean to a type 'a' which belongs to the classes 'banach' and 'second_countable_topology'. Given the assumption that 'p' is a real number between 0 and 1 inclusive, it shows that the expected value of the function 'f' under the Bernoulli ...
For any two hypercomplex numbers 'w' and 'z', the conjugate of the product of 'w' and 'z' is equal to the product of the conjugates of 'w' and 'z'.
If U is a subset of the carrier, m is open, x is an element of m, and m is a subset of U, then U is in the neighborhood of x.
If 's' and 't' both add to 'l', and for any 'u' if 's' and 't' both add to 'u' then 'l' also adds to 'u', then 'l' is equal to the least common sum of 's' and 't'.
If a τ-execution move is possible from a certain state (stk, loc, pc, xcp) to another state (stk', loc', pc', xcp') for the expression e' under the context ci, program P, and thread t with heap h, then a τ-execution move is also possible from the state ((stk @ [v]), loc, (length (compE2 e) + pc), xcp) to the state ((st...
If "x" and "y" are tests, then the union of "x" and the intersection of "y" and "x" is equal to "x".
The initial value of q at time 0 is s0.
There exist a set C and a function 'g' such that: 1. Set C is finite. 2. Set C and set U are disjoint. 3. The cardinality of set C is less than or equal to the cardinality of set F. 4. Function 'g' is continuous on the union of all sets
If for all elements 'a' and 'b' in set 'A', the relation 'r' between 'a' and 'b' is equivalent to the relation 'r'' between 'a' and 'b', then the reflexivity of the relation 'r' on set 'A' is equivalent to the reflexivity of the relation 'r'' on set 'A'. Here, a relation is reflexive if every element is related to itse...
A component 'i' is in the intersection of the set of states 's' where the pair (i,j) is in 's' equals 'b', and the complement of the highest priority 'i', co-related to the set of states 's' where the pair (i,j) is in 's' equals 'b'.
If M is a subterm of N, then the set of free variables in M is a subset of the set of free variables in N.
The lemma named "scopeFresh" is defined with a fixed process 'P' of type 'pi' and a fixed name 'a'. It assumes that 'a' is fresh for 'P'. It shows that the process 'P' under the restriction of 'a' is structurally congruent to 'P'.
The lemma named "poly_at_bot_at_top" is about a polynomial "p" of real numbers. It assumes that the degree of "p" is greater than or equal to 1 and that the coefficient of "p" at its degree is less than 0. Under these assumptions, it shows that the limit of the polynomial "p" as "x" approaches positive infinity is nega...
In the context of a probability space, if we assume that there are k-wise independent variables in a subset M' of X indexed by I, and J is a subset of I that is finite and its cardinality is less or equal to k, then the variables in M' of X indexed by J are independent.
The set {C1, C2, C3} is aperiodic in A.
The lemma named "orthogonal_App_cong" assumes that "T" and "U" are arrows. It shows that the sequence obtained by mapping each element of "T" with a function that composes it with the source of the first element of "U", and then appending the result of mapping each element of "U" with a function that applies the target...
The lemma named "Greatest_not_less" is defined with a fixed natural number 'k'. It assumes that 'k' is greater than the greatest 'k' for which a property 'P' holds. It also assumes that for any 'k', if property 'P' holds for 'k', then 'k' is less than or equal to 'l'. Under these assumptions, it shows that property 'P'...
If 'x' belongs to the set difference of 'HFinite' and 'Infinitesimal', then the hypercomplex modulus of 'x' also belongs to the set difference of 'HFinite' and 'Infinitesimal'.
For a given state 's', the sum of the natural number 'n+1' and the embedding of the equality between 's' at 'x' and the integer 'n', is equal to the assignment of 'y' to the natural number 0, followed by the weighted sum of the function that always returns 0.
The lemma named "greater_sum_both" assumes that "a" is greater than or equal to the sum of "b" and "c". It shows that there exist "a1" and "a2" such that "a" is equal to the sum of "a1" and "a2", and "a1" is greater than or equal to "b", and "a2" is greater than or equal to "c".
The lemma named "coeff_mult" assumes that "K" is a subring of "R", and that "f" and "g" are elements of the polynomial ring "K[X]". It shows that the coefficient of the product of "f" and "g" at index "i" is equal to the sum (over all "k" from 0 to "i") of the product of the coefficient of "f" at index "k" and the coef...