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The lemma named "stable_rank_bounded" assumes three conditions: 1. The rank of x is stable at j. 2. The successor of i is infinite. 3. q₀ is not in F. If these conditions are met, it shows that j is less than i.
The default nth element of a list, which is extended by appending a default element at the end, is the same as the default nth element of the original list.
The lemma named "out_diff1" assumes that "A is outside of B and C". It shows that "B is not equal to A".
The Bochner integral of a function that maps x to the complex number equivalent of f(x) is equal to the real number equivalent of the Bochner integral of f.
The lemma is about the formal power series integral. It fixes a formal power series 'a' in a semiring with an inverse. It shows two things: 1. The 0th term of the formal power series integral of 'a' with a constant 'a0' is 'a0'. 2. The (n+1)th term of the formal power series integral of 'a' with a constant 'a0' is the...
If "sim_move01" holds for the program P, thread t, thread action ta0, expressions e0 and e, heap h, local variables xs, thread action ta, expressions e' and e'', heap h', and local variables xs', then "sim_move01" also holds for various transformations of these expressions. These transformations include creating a new ...
The lemma named "ultra_filter" assumes that "F" is a proper filter. It shows that there exists a filter "G" such that "G" is an ultra filter and "F" is a subset of "G".
If "x" is greater than zero, then the base 10 logarithm of "x" is equal to the product of the base 10 logarithm of "e" (the base of natural logarithms) and the natural logarithm of "x".
The pre-condition of "q" applied to the top element is equal to 1.
The lemma named "maxR_domain" shows that if the set containing elements 'x' and 'y' is a subset of set 'A', then the maximum of 'x' and 'y' according to relation 'R' is an element of set 'A'.
If we have a disjoint family of sets A indexed by I, and each set A(i) for i in I is finite, and I itself is finite, then the cardinality of the union of all sets A(i) for i in I is equal to the sum of the cardinalities of each individual set A(i) for i in I.
The infimum (greatest lower bound) of the composition of a set M and a function f is equal to the infimum of the composition of each element in set M and the function f.
A function 'f' is a complex linear transformation if it satisfies the following two conditions: 1. The function of the sum of two complex numbers 'b1' and 'b2' is equal to the sum of the function of 'b1' and the function of 'b2'. 2. The function of a complex number 'b' multiplied by a scalar 'r' is equal to the funct...
The lemma named "valid_decomp_concat" assumes that for all 'ps', if 'ps' is in the set 'pss', then 'ps' is a valid decomposition of 'X'. It shows that the concatenation of 'pss' is a valid decomposition of 'X'.
If "n" is less than the length of the list "qs", then the sum of "t_BIT" of "n" and "Phi" of "n + 1" minus "Phi" of "n" is less than or equal to 7/4 times "t_A" of "n" minus 3/4.
The lemma "nxt_monotone" assumes that the next element in sequence 'b' is 'b''. It shows that the length of sequence 'b' is less than the length of sequence 'b''.
The lemma "insert_simps" provides three simplification rules for inserting an element into a binomial queue: 1. Inserting an element 'a' with value 'v' into an empty queue results in a queue with a single node containing 'a' and 'v', and no children. 2. Inserting an element 'a' with value 'v' into a queue where the f...
The lemma states two conditions: if 'b' is an element in the set 'bs', then 'b' is greater than 0. Additionally, if 'b' is an element in the set 'bs', then 'b' is less than 1.
For any real number x, if x is greater than -1, then the natural logarithm of (1 + x) is less than or equal to x.
For any word 'x' of length 'a', the bitwise AND operation between 'x' and the mask of length 'a' equals 'x'.
A function F is okay with the union of G and H if and only if F is okay with both G and H separately.
The lens formed by the sum of lenses "x", "y", and "more" is a bijective lens.
If the product of "n x" and "n z" equals the product of "n y" and "n z", and if the sum of "n x" and "n z" equals the sum of "n y" and "n z", then "n x" must be equal to "n y".
If x is equal to the function D with parameters m, i, j, and k, then x belongs to the set of lengths from i to j in m for all paths xs, where the set of xs is a subset of the set from 0 to the successor of k.
The well-formed action schema function with parameters STG, mp_constT, and s is equivalent to the well-formed action schema function with parameter s.
If the limit of a function g(x) as x approaches from the right is positive infinity, and if it is eventually true that the derivative of g(x) is not zero as x approaches from the right, and if it is eventually true that the derivative of f(x) exists as x approaches from the right, and if it is eventually true that the ...
The successor of the edge 'a' in graph 'G' as defined by the mapping 'M' is 'a' itself.
If "x" is univalent, then the function "fc" applied to "x" forms an equivalence relation.
The symmetric factor of the reflexive closure of a relation 'r' is equal to the reflexive closure of the symmetric factor of the relation 'r'.
The lemma named "denom_dvd_ints" is about an integer 'i'. It assumes that the quotient of 'r' equals the pair (z, n) and that the product of 'i' and 'r' is an integer. Under these assumptions, it shows that 'n' divides 'i'.
If a function symbol 'f' is well-typed, then the term 'rOfFax f' is also well-typed.
If a set S is not empty and is a subset of another set T, then the set distance from T to a point u is less than or equal to the set distance from S to the same point u.
The inverse of the first stereographic projection of "xa" is not equal to the top sphere.
For any two elements 'c' and 'd' of the type 'classes1', 'c' is less than or equal to the supremum of 'c' and 'd'.
If a natural number 'm' is less than or equal to another natural number 'n', then the maximum value of the absolute natural integer set from 'm' to 'n' is 'n'.
The lemma named "cong3_symmetry" assumes that three points A, B, and C are congruent to three other points A', B', and C' respectively. It shows that if this is true, then the three points A', B', and C' are also congruent to the three points A, B, and C respectively. This is a statement of symmetry in congruence.
If a sequence 'xs' evaluates to a sequence 'xs' prime under the execution of exception statements with successful values, and if the function 'gx' applied to 'xs' prime is None, then the function 'gx' applied to 'xs' is also None.
If the function 'f' applied to 0,0 equals 0, then the combination of two singleton matrices using the function 'f' is equal to a singleton matrix with elements resulting from the function 'f' applied to the elements of the original singleton matrices.
The lemma "elimder_gcast" assumes the following: 1. The process 'p' is defined as a groupcast from 'fips' with message 'fmsg' to process 'qq' with label 'l'. 2. The label 'l'' is in the set of labels of process 'q' in the system 'Γ'. 3. The transition from state 'ξ' with process 'p' to state 'ξ'' with process 'q' is in...
In the context of an encoding, the lemma "indRelRTPO_relates_source_target" is defined. It fixes a relation "TRel" which is a set of pairs of 'procT', and two processes 'S' and 'T' of types 'procS' and 'procT' respectively. The lemma assumes that the source term 'S' is related to the target term 'T' under the relation ...
The theorem named Atm_Sbis assumes that the atomic action 'atm' is compatible. It shows that the atomic action 'atm' is strongly bisimilar to itself.
The lemma named "subprob_imp_subprob_cexpr" assumes that there is a density context for variables V and V' and context Γ, where the function maps a state σ to the extraction of a real number from the semantics of the complex expression δ under state σ. Under this assumption, it shows that δ is a subprobability complex ...
If 'j' is an element of the set of elements less than 'i' (denoted as underS i), and 'i' is an element of set 'A', and 'A' is a subset of the field of relation 'r', and the set of elements greater than all elements of 'A' (denoted as Above A) is not empty, then 'j' is an element of the set of elements less than the sup...
The open ball with center 'x' and radius 'e' is an open set.
If "n" is equal to the sum of "m" and "k", then filling up to "m" elements in the list "l" with the element "x" after filling up to "n" elements in the same list "l" with the element "x" is equivalent to just filling up to "m" elements in the list "l" with the element "x".
Lemma: fls_right_inverse' Let f be a formal Laurent series over a ring_1 type 'a. Assume that the product of x and the coefficient of f at its subdegree is 1, and the product of the coefficient of f at its subdegree and y is also 1. Note that these assumptions imply x equals y, but we do not need to assume that. The le...
The initial conversion of transition system "ta" is equal to the initial conversion of transition system "ta'" if and only if "ta" is equal to "ta'".
The dehomogenization of polynomial 'p' with respect to 'x' equals 'p' if and only if 'x' is not in the set of indeterminates of 'p'.
The lemma named "Bloch_lemma" assumes the following: 1. The function 'f' is holomorphic (complex differentiable) on the closed ball centered at 'a' with radius 'r', and 'r' is greater than 0. 2. For any complex number 'z' in the open ball centered at 'a' with radius 'r', the norm (magnitude) of the derivative of 'f' at...
The product of x and y is less than or equal to the star of the sum of x and y.
"there exists at most one derivative D such that the derivative of function f at point x is greater than D".
The right restriction of a relation 'r' to a set 'A' is equal to the set of ordered pairs '(a, b)' where 'b' is an element of 'A' and the ordered pair '(a, b)' is an element of the relation 'r'.
: Given the fixed elements Psi, Psi_Q, P, M, N, P', A_P, Psi_P, Q, K, xvec, Q', and A_Q, if the following conditions are met: 1. Psi tensor Psi_Q directs P to transition to P' with the message M carrying N, 2. The frame extracted from P is a pair consisting of A_P and Psi_P, 3. Psi tensor Psi_P directs Q to transit...
The lemma named "red_mthr_deterministic" assumes that the heap operations are deterministic, as stated by "deterministic_heap_ops". It shows that the method thread reduction, represented by "red_mthr", is deterministic for all possible states, represented by "P UNIV", in the program.
If the modulus of 'b' and 'N' equals the modulus of 'a' and 'N', and 'a' is less than or equal to 'b', and 'b' is less than 'a' plus 'N', then 'b' equals 'a'. Here, 'a', 'b', and 'N' are natural numbers.
"Sign X Y equals Sign X' Y'" if and only if "X equals X' and Y equals Y'".
The lemma named "gpair_assoc" defines two functions, "f" and "g". Function "f" takes a pair (f, gh) and returns a triple (f, gh bound to the first element, gh bound to the second element). Function "g" takes a pair (fg, h) and returns a triple (fg bound to the first element, fg bound to the second element, h). The lemm...
The lemma named "symm_strong_reduction_simulation_is_bisimulation" establishes a relationship between two concepts: strong reduction simulation and strong reduction bisimulation in the context of a process calculus. It takes two parameters: a relation 'Rel' which is a set of pairs of processes, and a process calculus '...
The lemma named "set_renaming_perm" assumes that "bs" is a finite set. It shows that there exists a permutation "q" such that for every element "b" in "bs", the action of "q" on "b" is equal to the action of "p" on "b", and the support of "q" is a subset of the union of "bs" and the action of "p" on "bs".
"a" is an element of the Vset of the successor of the rank of "a".
If 'i' is less than the length of the list 'xs', then a sorted sublist from 'i' to 'i' in the list 'xs' is reflected.
The lemma named "inv_partI" assumes that for any indices i and j, if i and j are both less than n, and if j is either less than m_j or equal to m_j and i is greater than or equal to m_i, then the predicate p holds for the matrix A at indices i and j. Under these assumptions, it shows that the inverse partition of the p...
A pair of states (q1, q2) is in the set of reachable states of the product of two finite state machines A and B if and only if there exist paths p1 and p2 such that p1 is a path in A starting from its initial state, p2 is a path in B starting from its initial state, the target of p1 in A is q1, the target of p2 in B is...
If there is a relationship between two probability mass functions A and B, and a function that maps A to another relationship with B, then there is a relationship between B and the bind of the probability mass function. This is a transfer rule, which means it is used to transfer properties between different representat...
If an event that is not holding at a certain state 'y' is occurring in the sequence starting with 'y' followed by 'xs', then there is an event where the first element of the sequence is not equal to the second element of the sequence in the sequence starting with 'y' followed by 'xs'.
The lemma for correct composition at different levels is true for system S4.
If "x" is a non-zero p-adic integer, then "x" can be expressed as the product of "p" raised to the power of the order of "x" in Zp and the associated component of "x" in Zp.
For any set B, if x is an element of the roofed_gen function with inputs G, B, and S, and S is a subset of S', then x is also an element of the roofed_gen function with inputs G, B, and S'.
A list of values 'vs' converted into a list of expressions is a constant list.
If "I" is an element of "i_set0", then "I" plus the negative of "k" is also an element of "i_set0".
The universal set of "ex_lbl" is equal to the set containing "Label1" and "Label2".
If a list representation "list_repr" exists for the variables i, ba, and xs, then the function "of_list_repr" applied to ba equals xs.
If 'r' is a cardinal order, then 'r' is less than or equal to the order type of the list of elements in the field of 'r'.
If G is a group, then the automorphism group of G is also a group.
If variable x is independent of variable y, then the output of variable x is also independent of the output of variable y.
If "Gen z Q" holds and the lengths of lists xs and ys are equal, then "Gen (subst_var xs ys z) (cp (Q[xs \<^bold>\<rightarrow>\<^sup>* ys]))" also holds. Here, "Gen" is a generic function, "subst_var" is a function that substitutes variables, "cp" is a function (possibly a composition), and "[xs \<^bold>\<rightarrow>\<...
Functions f and g are conjugates if and only if both f of bottom equals bottom and g of bottom equals bottom, and for all x and y, the intersection of f of x and y is less than or equal to f of the intersection of x and g of y, and the intersection of g of y and x is less than or equal to g of the intersection of y and...
If a unary operation is onto a set A with function f, then a no-parameter operation is also onto the same set A with the same function f and a natural number 1.
If r' is less than or equal to the identity relation, then the ordinal product of r and r' minus the identity relation is less than or equal to the set of pairs ((x1,y), (x2,y)) where (x1, x2) is in the set difference of r and the identity relation and y is in the field of r'.
The lemma named "transitive" is defined with three fixed variables P, Q, and R, all of which are of type pi. It assumes that P is strongly late bisimilar to Q and that Q is strongly late bisimilar to R. Under these assumptions, it shows that P is strongly late bisimilar to R.
If there exists a relation P that holds for all pairs of elements in lists xs and ys, and for any pair of elements x and y where P holds, the condition Q1 holds for x if and only if Q2 holds for y, then the relation P also holds for all pairs of elements in the filtered lists of xs and ys, where the lists are filtered ...
The lemma shows that the functor G has a right adjoint between categories C and D.
If X is an element of the analysis of G, and G is a subset of the analysis of H, then X is an element of the analysis of H.
The lemma "isRev_getRevRole2" assumes that "isRev s confID uID papID". It shows that there exists a number 'n' such that the function 'getRevRole' applied to 's', 'confID', 'uID', and 'papID' equals 'Some (Rev papID n)'.
The top space of the topology generated by a set S is equal to the union of the set S.
If x is an equivalence relation, then the intersection of the composition of x and y with the composition of x and z is equal to the composition of x with the intersection of y and the composition of x and z.
Mapping a function 'f' and another function 'g' over 'x' is equivalent to composing 'g' with the mapping of function 'f' over the identity function applied to 'x'.
If "i" is less than "j", then the result of the function "subst_bv2" with parameters "Bv j", "i", and "u" is equal to "Bv" of "j - 1".
: Assuming the following conditions: - A' is an m by n matrix - a is less than m - n is greater than 0 - A is defined as A' appended with D times the identity matrix of size n - The element at the a-th row and 0-th column of A is not 0 - m is greater than or equal to n - The elements in the list xs are distinct - Ever...
If K' is a subfield of K, then the set of elements (carrier) of K is a subset of the set of elements (carrier) of K'.
The lemma about the components of the opposite category states: - The set of objects in the opposite category is the same as the set of objects in the original category. - The set of arrows in the opposite category is the same as the set of arrows in the original category. - The domain of an arrow in the opposite categ...
If "x" is an element of set "A", then "x" is less than or equal to the maximum element of set "A".
Pushing a bit 'n' times into the negative of 'a' is equal to the negative of pushing a bit 'n' times into 'a'.
The lemma named "compact_differences'" is defined for a set 'S' of elements of type 'real_normed_vector'. It assumes that both 'S' and 'T' are compact sets. It demonstrates that the set of all differences between elements of 'S' and 'T' is also a compact set.
The lemma "adopt_node_is_l_adopt_node" is an instance that states: for the functions "type_wf", "known_ptr", "known_ptrs", "get_parent", "adopt_node", "adopt_node_locs", "get_child_nodes", and "get_owner_document", the "adopt_node" function is a "l_adopt_node" function.
The lemma named "has_densityD" assumes that "f" has a density "N" with respect to a measure "M". It shows that "f" is Borel measurable with respect to "N", "M" is equal to the density of "N" with respect to "f", and the space of "N" is not empty.
If the difference between set A and a single element set {x} is an empty set, then set A must either be an empty set or a set containing only the element x.
The head of tree T is equal to the head of graph G.
If the evaluation of expression 'e' in memory 'mem' is not true, then the evaluation of the 'If' statement with expressions 'e', 'c1', and 'c2', along with annotations 'annos', mode states 'mds', and memory 'mem', will lead to the evaluation of 'c2' with updated mode states 'mds' combined with 'annos', and the same mem...
The lemma named "diamond_eq_characterisations" shows that the function "diamond_cube" applied to the pair (x, y) is equal to the function "diamond_cube_gen" applied to the same pair (x, y).