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Given the assumptions that M is equal to the norm-preserving extensions of E, p, F, and f, that c is an element of the chains of M, that the graph of H and h is equal to the union of c, and that x is an element of H, it can be shown that there exists H' and h' such that the graph of H' and h' is an element of c, and th...
, given a set D which is not empty and is a subset of the Basis, there exists a point 'a' in the relative interior of the convex hull of the set obtained by inserting 0 into D. This point 'a' belongs to a Euclidean space.
A linear function is invertible if and only if there exists another linear function such that the composition of the two functions in either order results in the identity function.
The Hamming stream is equivalent to an empty lazy list if and only if the condition is false.
If "j" is less than or equal to "j'", then the progress of "phi" at "j" is less than or equal to the progress of "phi" at "j'".
If the polynomial coefficient of the function 'f' at the index 'Suc n' in the set 'S' exists, then the polynomial expression of the extended coefficient function at the index 'n + j' for 'Suc n' is equal to the polynomial expression of the extended coefficient function at the same index 'n + j' for 'n'.
The function "td_ext" applied to the signed integer function "sint" (which converts a word of type 'a to an integer), the function "word_of_int" (which converts an integer to a word), the set of signed integers of length 'a, and the function "signed_take_bit" (which takes a signed bit at a certain position in a word) o...
If "k" is greater than 0, then the interleaved join of the function "f" mapped with "k" and the list of "n * k + (k - 1), mod k" is equal to the interleaved join of the function "f" divided by "k" and the list from "n" to the end.
The lemma named "bind_count_space_singleton" assumes that "f x" is a sub-probability space. It shows that binding the count space of the singleton set {x} with the function f is equal to the function f applied to x.
If a shadow root is deleted from a heap 'h' resulting in a heap 'h'', then any reference 'r' in the locations of the tag name of a pointer 'ptr'' is preserved between 'h' and 'h''.
The lemma named "csmall_to_cbig_prop" makes several assumptions: 1. For any program P, states σ and σ', and collection coll, if the pair (σ', coll) is in the csmall relation with P and σ, and if the relation R holds between P and coll, and if the relation Q holds between P and σ, then the relation R' holds between P, ...
If S is a subspace and x is an element of S, then the scaled version of x by a constant c is also an element of S.
If a positive integer 'n' is greater than 0 and an element 'a' is in the multiset 'A', then the element 'a' is also in the multiset resulting from repeating 'A' 'n' times.
If for any vertex 'v', 'v' belongs to the set 'VV' of player 'p' and 'v' is not a deadend, then 'v' leads to another vertex according to the strategy 'σ', it implies that 'σ' is a strategy of player 'p'.
The lemma named "get_deterministic" assumes that the state 's' of revision 'r' is a tuple consisting of sigma, tau, and a read operation on a location 'l'. It shows that a revision step from 's' to 's'' is equivalent to 'l' being in the domain of the composition of sigma and tau, and 's'' being the state where 'r' is m...
If one of the conditions A, B, C, D, or E is true, and if any of these conditions implies P, then P is true.
If a relation "r" is reflexive on a set "A", then if "r" relates "a" and "b", "b" must be an element of set "A".
The lemma named "mult_noteq_inf" assumes that neither z nor w are equal to infinity. It shows that the product of z and w is not equal to infinity.
The system conclusion is sound when the list is empty.
The function "lift_p P f" holds if for any heap, x, and heap', if P holds for the heap and running the state of f with the heap results in x and heap', then P also holds for heap'.
For any matrix A, the operation of A multiplied by the sum of matrices B and C is equal to the sum of the results of A multiplied by B and A multiplied by C. This is the distributive property of matrix multiplication over matrix addition.
If a function 'f' expands to a multiseries 'MS' with a leading term 'C' and exponent 'e' (where 'e' is less than zero) with respect to a well-formed basis 'basis', then the function 'exp(f(x))' expands to a multiseries generated by the power series expansion of the exponential function applied to the original multiseri...
If a method "mthd" is defined for a class "C" with parameters "M" and "Cs" in a program "P" according to "MethodDefs'", and if the "LeastMethodDef_unique" condition holds for the same parameters in the program, then the program "P" confirms that the class "C" has the least method "mthd" via "Cs".
The lemma named "valid_init_reach" assumes that the transition "tr" is valid and the source of the first transition is the initial state. It shows that the target of the last transition is reachable.
In the context of a partially directed graph base, if J is a subset of I, then J is also a partially directed graph base with respect to alpha and AA.
If "s" is equal to "t", then "s" is a superterm of "t".
The product of the function "a" applied twice to "x" and "y" is less than or equal to "y".
The reduction of a term 't' using an environment 'env' is reduced with respect to the keys of the environment 'env'.
The set of all natural transformations from a functor 'F' to a functor 'G' in the category of tiny digraph homomorphisms is small, given that 'F' and 'G' are tiny digraph morphisms from category 'A' to category 'B' in the context of an arbitrary universe 'alpha'.
For a real-valued function 'f', if the derivative of 'f' at a point 'x' is 'l' and 'l' is greater than zero, then there exists a positive 'd' such that for all positive 'h' less than 'd', the value of 'f' at 'x - h' is less than the value of 'f' at 'x'. This essentially means that if the derivative of a function at a p...
The lemma named "analz_Un_implInsecSet_aux_aux" assumes that the encryption keys are clean in the union of sets G and H. It shows that the analysis of the union of the implementation of insecure set auxiliary function G and set H is a subset of the union of the implementation of insecure set auxiliary function G, Tags,...
"ZObis" is symmetric.
The Secure Multi-Party Computation (S2_14) with parameters c and out is lossless.
If tau satisfies the definedness of x, then tau satisfying the implication from x to y is equivalent to the statement that if tau satisfies x, then tau satisfies y.
If a future invariant holds for a formula Phi, and a future test does not hold for the same formula, then a future specification must hold for that formula.
The lemma "append_set_prefix" assumes that a list "xs" is in the appended set of "T" and "X". It then shows that the list "xs" without its last element is in the set "T".
The current list resulting from pushing an element 'x' onto a 'small' list is equivalent to the list formed by prepending 'x' to the current 'small' list.
If "t" is embedded into "s", then the hierarchical size of "t" is greater than the hierarchical size of "s".
Ml_Rel is an isomorphism natural transformation. It assumes that 'a' is an object in the category of relations of type alpha. It shows that Ml_Rel of the category of relations of type alpha and 'a' maps from the product of 2 relations in the category of relations of type alpha (with the set containing 'a' and any other...
The logical 'and' operation between 'true' and 'false' equals 'false'.
A certain condition "ces" holds for variables v1 and v2 and a list made by appending es1 and es2, with a certain subset "subs", if and only if there exists a variable v such that the condition "ces" holds for v1 and es1 with the subset "subs", and also for v and es2 with the subset "subs".
For any two polynomials 'p' and 'q' in a commutative semiring, where 'm' is defined as the degree of 'p' and 'n' is defined as the degree of 'q', if 'i' is less than 'n', then the sum of the monomials of the Sylvester matrix of 'p' and 'q' at the position (i,j) for all 'j' less than 'm+n', where each monomial is raised...
If a vertex 'v' is in the set of vertices of 'to_list_dtree', then there exists an 'x' such that 'v' is equal to a list containing 'x' and 'x' is in the set of vertices of 'T'.
If a monad is defined with the return function and a bind function of type 'a and 'm, then a monad can also be defined with the same return function and a bind function of type 'a and a pair of 'r and 'm in the environment type.
The set of all 'y' such that 'x' is less than 'y' and 'y' is less than 'z' is empty if and only if 'x' is not less than 'z'.
"n of x" times "y" times "n of z" equals "y" times "n of z" if and only if "t of x" times "y" times "n of z" equals zero.
: If the following conditions are met: 1. l is greater than 0 and less than or equal to k. 2. The execution state of the local state and the transition function are equal. 3. The local state is idle with respect to the output function and the transition function. 4. For all c and m, if the local state is idle with res...
If the width is greater than zero, then there exists a value 'x' such that 'x' is equal to the width and 'x' is greater than zero.
If 'z' is an imaginary number, then 'z' lies on the y-axis of the circle line.
If the convergence abscissa of a Dirichlet series 'f' is not infinity, and if the limit of the function 'S x' projected onto 1 tends to infinity under a filter 'F', then the function that evaluates 'f' at 'S x' tends to the first term of 'f' under the filter 'F'.
The lemma named "pnopD" makes the following assumptions and shows the following results: Assumption: - The function 'f' is a partial n-ary operation on the set 'A'. Results: - The function 'f' is a value-single-valued function. - 'n' is a natural number. - The domain of the function 'f' is a subset of the Cartesian ...
If 'b' is an object in category 'B', and 'BCG' is a triple consisting of 'B', 'C', and a pair of morphisms 'h'' and 'g'', and 'ABF' is a triple consisting of 'A', 'B', and a pair of morphisms 'h' and 'g', and 'BCG' is a morphism from 'B' to 'C' in the comma category of 'b' and functor 'F', and 'ABF' is a morphism from ...
The lemma named "fix_index" is defined with respect to a matrix M of type 'a DBMEntry, where 'a is a time. The lemma assumes that the sum of the elements at positions (0,i) and (i,0) in the matrix M is greater than one, that the matrix M is cycle-free up to the nth element, that the matrix M is canonical up to the nth ...
"if it is possible that there exists some property X, then it is possible that there exists some property P."
In the context of the Congruence Rule, there is a lemma or proposition known as the Segment Between relation. This proposition assumes a point 'p3' is between a segment from 'p1' to 'p2'. It then shows that the geometric set of the segment from 'p1' to 'p2' added to an empty set is not equal to the geometric set of the...
Applying the CSP1 operation to the result of the R1 operation on process P is equivalent to applying the R1 operation to the result of the CSP1 operation on process P.
The intersection of the application of function d to x and y, when multiplied by U, is equal to the intersection of the product of the application of function d to x and U, and the product of the application of function d to y and U.
If a quantified proposition (qp) exists for some variable x and proposition Q, then if the quantified proposition Q is true and x is a free variable in Q, it implies a result R. Therefore, the result R is true.
The lemma "next_append_generator" consists of two parts: 1. The next element in the sequence generated by appending generator g1 and g2, starting from the state (s1, s2) in the left branch (Inl), is determined as follows: If there is a next element in the sequence generated by g1 starting from s1, then the next elemen...
If the function rb_aux_inv2 is applied to the tuple (bs, ss, ps) and if j is less than the length of fs and the right injection of j is not in the set ps, then the jth element of fs is in the ideal of the set obtained by applying the function rep_list to the set obtained by filtering bs with the condition that the comp...
For any system state 's', if 'r' is in the worklist of any processor 'p' and 'r' is marked in the system heap, then 'r' is black. This is equivalent to the conjunction of the system's free memory and the marking of 'r' in the system heap.
The fractional part of a number tends to 0 as we approach 0 from the right. This is applicable for any number of a type 'a, where 'a has the properties of floor-ceiling and order topology.
The order of a list, where an element 'x' is appended at the end of another list 'xs', is equivalent to either the list 'xs' being empty or the last element of 'xs' being in order with 'x' and the order of the list 'xs' being maintained.
The lemma named "store_typing_imp_mem_length_eq" assumes that the store "s" is of a certain type "\<S>". It shows that the length of the memory of the store "s" is equal to the length of the memory of the type "\<S>".
The condition of the transpose of matrix 'Q' multiplied by vector 'v' being equal to vector 'v' is equivalent to the condition of the result of subtracting the identity matrix (of the same dimension as the number of rows in 'Q') from the transpose of 'Q', and then multiplying the result by vector 'v', being equal to th...
The head term of the sequence "s" concatenated with "ts" is equal to the head term of "s".
The composition of a polynomial constructed from a constant 'a' and another polynomial 'p' with a polynomial 'q' is equal to the sum of the polynomial constructed from 'a' and the product of 'q' and the composition of 'p' and 'q'.
The lemma named "gen_model_jAction_eq" assumes the following: 1. The generated model "M" at world "w" is equal to the generated model "M'" at the same world "w". 2. The world "w'" is included in the worlds of the generated model "M'" at world "w". 3. "M" is a Kripke model. 4. "M'" is also a Kripke model. Under these a...
The lemma named "in_Chains_finite" is a translation from the theorem "in_chain_finite". It assumes that "M" is in the chains of "leq", "M" is not an empty set, and "M" is a finite set. It shows that the least upper bound of "M" is an element of "M".
The "stream_raw" function will always terminate.
The Smith normal form is valid up to the 0th index for any matrix A.
The lemma named "pflat_len_outside" assumes that "p" is not in the position "v1". It shows that the flat length of "v1" at position "p" is -1.
The dual of the concatenation of two strands A and B is equal to the concatenation of the dual of strand A and the dual of strand B.
The function "enum" is injective on the set of all numbers from 0 to n.
If a natural number 'm' is less than or equal to 'n' and 'n' plus one is less than or equal to 'l', then the union of the set of natural numbers from 'm' to 'n' and the set of natural numbers from 'n+1' to 'l' is equal to the set of natural numbers from 'm' to 'l'.
If set X and set Y are both in InternalSets, then the union of set X and set Y is also in InternalSets.
"False or a proposition P is equal to the proposition P".
The truncation of an atom "a" at the successor of any number "k" is equal to the atom "a" itself.
The slice of a word 'w' at position 'n' is equal to the unsigned cast of the word 'w' right shifted by 'n' positions.
The function "lift" applied to a function "f" and an input "inp" maintains the same domain as the input "inp".
: Assuming that "d" is a homogeneous grading, "B" is a subset of the set of polynomial degrees "d" with maximum degree "m", and "p" is a polynomial in both the set of polynomial degrees "d" with maximum degree "m" and the ideal generated by "B". Then, there exists a polynomial "r" such that the keys of "r" are a subset...
: 1. If the intersection of the states of automaton A and automaton B is empty, and if there is a rule in automaton A that transitions from a set of states to a state p, then there is no epsilon transition from state p to any state q in automaton B. 2. Similarly, if the intersection of the states of automaton A and a...
: Assuming that set A is a subset of U2, set B is a subset of U3, for every element x in U2, the function f(x) is in U, and for every element x in U3, the function g(x) is in U. Furthermore, for every element n in set A, there exists an element m in set B such that f(n) is less than or equal to g(m). Under these condit...
If the state "tau" satisfies the definedness of "X", then the double ceiling of the representation of the base bag "X" in the state "tau" at the bottom element is equal to zero.
The property "PhiWhileP" of a loop with condition "A" and body "b", evaluated at states "s" and "t", is equivalent to the 5-tuple "(b, A, Phi, s, t)" being in the set "var".
The Pochhammer symbol of 'a' and 'n' equals 1 if 'n' equals 0. Otherwise, it equals 'a' multiplied by the Pochhammer symbol of 'a+1' and 'n-1'.
Points A, A, and B are collinear.
Converting the result of the left-cons operation on an element 'a' and a tree 't' to a list is equivalent to prepending 'a' to the list obtained by converting 't' to a list.
If 'k' is less than the length of list 'D', then 'D' can be split into three parts: the first part is the list of the first 'k' elements, the second part is a list containing only the 'k'th element, and the third part is the list of elements after the 'k'th element.
If a word 'w' is bordered, then there exists a word 'x' such that 'x' is a border of 'w' and 'x' is not bordered.
If the lengths of two lists, xs and ys, are equal, then the interleaved zip of the concatenation of xs with a function f and ys with a function g is equal to the concatenation of the zip of xs and ys with the interleaved zip of f and g.
The lemma named "pos_signed_to_meas_space" assumes that "M1" is a positive measure set. It also assumes that "m1" is a function that maps any set "A" to the measure of the intersection of "A" and "M1". Under these assumptions, it shows that there exists a measure space with the same space and sets as "M", but with "m1"...
The conversion of a nondeterministic generalized Büchi automaton (NGBA) A to an implicit generalized Büchi game (IGBG) results in an IGBG with the following properties: 1. The set of vertices (g_V) is the universal set (UNIV). 2. The set of edges (g_E) is defined by the function E_of_succ, which takes a state p and re...
The lemma named "FL_bisimilar_eqvt" assumes that P is bisimilar to Q under F. It shows that if P is permuted by p, then it remains bisimilar to Q permuted by p, under F permuted by p.
The "holdsIstate" holds for "geq_noPH_confIDs".
If the set V is finite, then the set resulting from applying the function "quote_all" with parameters p and V is also finite.
For a local function 'f', command 'c', and function 'g', applied to a state 's', it is equivalent to applying command 'c' to the result of function 'f' applied to state 's'.
The lemma "roots4_generic" assumes that the degree of polynomial p is 4. It also assumes a solver such that for any polynomials p, q, r and any number y, y is in the set produced by the depressed_solver for p, q, r if and only if the equation y^4 + p * y^2 + q * y + r equals 0. The lemma shows that the set produced by ...
The lemma named "depth_prm" shows that the depth of the permutation of A is equal to the depth of A.
The function "remove_Union" applied to sets A, X, and B is equal to the following: if set X is finite, then apply the function "set_fold_cfi" to the function "remove_Union_cfi B", set A, and set X. Otherwise, subtract the union of all elements x in set X applied to function B from set A.