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The lemma named "bij_coord" is defined for a fixed X, which is a square matrix of a field. If X is assumed to be a basis of the set of vectors, then it is shown that the coordinate function of X is a bijection.
A pair (s,t) is in the reflexive transitive closure of a relation R if and only if there exists a list such that the last element of the list starting with s is t, and for all indices i less than the length of the list, the pair formed by the i-th element and the successor of the i-th element of the list starting with ...
The lemma named "sameDom_lift2_1" assumes that the domain of "inp'" is the same as the domain of "inp". It shows that the domain of "inp'" is the same as the domain of the function "lift2 f inp' inp", and also that the domain of "inp" is the same as the domain of the function "lift2 f inp' inp".
The inverse function of "map_dual" when applied to the identity function equals the identity function.
The lemma named "bij_Var_subst_compose_Var" assumes that "g" is a bijective function. It shows that if you first apply the inverse of "g" to a variable and then apply "g" to the result, you get back the original variable.
If a certain condition "P" has a time bound "TBOUND" of "c" and "b", and "b" is less than or equal to "b'", then the condition "P" has a time bound "TBOUND" of "c" and "b'".
The range of a function "f" permuted by a permutation "π" is equal to the range of the function "f" itself after being permuted by the same permutation "π".
For a given real vector 'x' and a real matrix 'A', if 'A' is invertible and the product of 'x' and 'A' equals zero, then 'x' must be equal to zero.
Given the following conditions: 1. The context "\<Theta>; B" well forms the context "\<Gamma>' appended with the tuple (x, b0, c0) and the context "\<Gamma>", 2. The context "\<Theta>; B" well forms the context "\<Gamma>' appended with the tuple (x, b0, c0') and the context "\<Gamma>", 3. The result of looking up "y" ...
The lemma named "infsum_le_finite_sums" is about a fixed 'b' which is a type that includes commutative monoid addition, topological space, and linear order topology. It assumes that 'f' is summable on 'A'. It also assumes that for any finite 'F' that is a subset of 'A', the sum of 'f' over 'F' is less than or equal to ...
If the following conditions are met in state 's': 1. 'x' is black, 2. 'x' points to 'y', 3. 'y' is a valid reference, 4. the strong tricolour invariant holds, then 'y' must be marked in state 's'.
The lemma named "Cong_subst" assumes that "t is equivalent to t prime", "u is equivalent to u prime", "t is concatenated with u", and "the sources of t prime in R are equal to the sources of u prime in R". It shows that "t prime is concatenated with u prime" and "the difference of t and u is equivalent to the differenc...
The lemma named "coprime_linear_comp" is defined with the assumption that it needs to be generalized. It fixes two real numbers, b and c, and defines r0 as the polynomial with coefficients b and c. It assumes that two polynomials p and q are coprime and that c is not equal to zero. It shows that the composition of p wi...
If for all elements x in list xs, y in list ys, and z in list zs, a predicate P holds true when applied to the functions f(x), g(y), and h(z), then the predicate P also holds true for all elements in the lists obtained by mapping the functions f, g, and h to the lists xs, ys, and zs respectively.
Any string 's' belongs to the set of all ground terms.
The first element of the result of applying the function "lift Q" to the last element of the non-empty list "x#xs" is not None.
The lemma OPT2_C2 assumes that "x" is not equal to "y" and "qs" is in the language generated by the sequence of an atom "x", an atom "y", an atom "x", a star of times of an atom "y" and an atom "x", and an atom "x". It shows that the transition function Tp with parameters [x,y], "qs", and OPT2 "qs" [x,y] equals to the ...
The lemma named "Map_cod" assumes that "\<mu>" is an arrow. It shows that the map of the codomain of "\<mu>" is equal to the codomain of the partial magma of the Hom-set from the source of "\<mu>" to the target of "\<mu>", applied to the map of "\<mu>".
A function, which takes two natural numbers 'k' and 'l' and returns a list of natural numbers from 'k' to 'l+1', is related to the 'atLeastAtMost' function in the context of natural number relations. This is expressed in the list set relation of natural number relations.
If the transfer relation R holds between x and y, then the relation match R also holds between x and y.
The lemma named "Ide_resid_Join" shows that if t and u are coinitial, then the identity of u subtracted from the join of t and u is true.
The cardinality of the set of all numbers up to a certain extended natural number 'k' is equal to the cardinality of the set of all numbers up to the natural number 'k'.
In the context of a probability space, the lemma named "char_distr_add" is defined. It fixes two functions X1 and X2 that map from an arbitrary type to real numbers, and a real number t. It assumes that the variables X1 and X2 are independent with respect to the Borel sigma-algebra. The lemma shows that the characteris...
Applying the function "foldri" after the function "tsl" to a proper iterator results in a proper iterator.
The lemma named "dr_Apply" is defined with a fixed function 'phi' that maps from type 'c' to type 'b'. It assumes that the composition of function 'f' and 'phi' is equal to the composition of 'phi' and function 'g'. Under this assumption, it shows that 'phi' refines 'G' from the application of function 'f' to the appli...
The lemma named "proots_count_of_root_set" assumes that the polynomial P is not zero, the set R is a subset of S, and for any element x in R, the polynomial P evaluated at x equals zero. Under these assumptions, it shows that the count of polynomial roots in set S is greater than or equal to the cardinality of set R.
If "n" is less than "m" and the "n-th" bit of integer "k" is true, then the "take_bit" operation on "m" and "k" will yield a result greater than zero. This is for "k" as an integer.
The lemma named "is_monom_vars_monom_var" assumes that "l" is a monomial. It shows that the set of variables in "l" is equal to the set containing only the variable of the monomial "l".
If the content (measure) of the closed box from point 'a' to point 'b' is zero, and 's' is a subset of this closed box, then the content of 's' is also zero.
For any given condition P, if for any element z, x is equal to the absolute ordinal of the equivalence class of z under the relation ord0rel, then the condition P holds.
: Given two real numbers A and B, and two integers p and n, if A is less than or equal to B, n is greater than 0, and p is greater than 0, then the result of dividing A by n and then multiplying by p is less than or equal to the result of dividing B by n and then multiplying by p.
If the mapping of the value "Val" over a list "vs" is equal to the mapping of the value "Val" over another list "vs'", then the two lists "vs" and "vs'" are equal.
The lemma named "Fun_tuple" assumes that "f" and "g" form a span. It shows that the function of the tuple of "f" and "g" is equal to the restriction of a function that maps "x" to the ordered pair of the element of the function of "f" at "x" and the element of the function of "g" at "x", to the domain of "f".
If "j" is less than the dimension of the column of matrix A, then the j-th element of the product of vector v and matrix A is equal to the dot product of vector v and the j-th column of matrix A.
The lemma named "m2_inv6_ikk_init_derived" assumes that the pair (s, t) is in the relation R12, s is in the invariant m1_inv2i_serv, and t is in the invariant m2_inv5_ikk_sv. Under these assumptions, it shows that t is in the invariant m2_inv6_ikk_init.
The lemma named "urel64_uminus" assumes a relation "urel64" between x and y. It shows that this relation "urel64" also holds between the unary minus of x and y under the conditions p64 and p respectively.
If 't' is a generalisation of 't'' with respect to 'i' and 'r', then for all guard expressions 'g' in the set of guards of 't'', 'g' does not constrain the variable 'i'.
If a map 'm' maps 'x' to 'Some y', 'm' is injective on its domain, and 'z' is not in the range of 'm', then the range of the map 'm' updated at 'x' with 'Some z' is equal to the range of 'm' minus the set containing 'y', union the set containing 'z'.
The pair (d, 0) is not in the functional relation of the inverse of R.
The sequence [True, False, True] is in the language of palindromes.
If a match test on a regular expression "r" from index "i" to index "j" is true, then "i" is less than or equal to "j".
The infimum (greatest lower bound) of any indexed set where each element is 'P', is 'P' itself.
If 'a' is a non-positive real number and 'b' is a non-negative real number, then the product of 'a' and 'b' is a non-positive real number.
The lemma named "suppa_distincts" assumes that "A B C SuppA D E F". It shows that "A is not equal to B", "B is not equal to C", "D is not equal to E", and "E is not equal to F".
If the intersection of sets X and Y is empty, and if e is an element of all bi-edges of X and Y, then e is not a subset of Y.
The deadlock-free refinement of the union of the range of left and the range of right is less than or equal to the COPY in the failure-divergence model.
The inverse image of the composition of two functions f and g is equal to the composition of the inverse images of g and f.
The set of all elements 'i' that are less than the successor of 'n' is equal to the set that includes 'n' and all elements 'i' that are less than 'n'.
If 'n' is less than the product of 'k' and 'm', then the integer division of 'n' by 'm' is less than 'k'. Here, 'k', 'm', and 'n' are natural numbers.
If F is a member of the set of filters, then 1 is a member of F.
If "a" is greater than or equal to 0, "a" is in "tfin", "a" is less than or equal to "c", "c" is in "tfin", "b" is greater than or equal to 0, and "b" is not in "tfin", then "c" is less than the sum of "a" and "b".
The union of the function symbols in the terms of the unlabelled version of the dual of a stateful strand S is equal to the union of the function symbols in the terms of the unlabelled version of S itself.
The theorem named "bertrand" is defined with a fixed natural number "n". Assuming "n" is greater than 1, it shows that there exists a prime number "p" in the range between "n" and "2n".
The lemma named "integral_distr" is about a function 'f' that maps from 'a' to 'b', where 'b' is a Banach space with a second countable topology. Given the assumptions that 'g' is measurable in the measure spaces M and N, and 'f' is Borel measurable in N, it shows that the Lebesgue integral of 'f' over the distribution...
If a property P holds for an empty finite map, and if for any element x, value y, and finite map m, the property P holds for m and the lookup of x in m is None, then the property P also holds for the updated finite map where x is associated with y. Then, it can be concluded that the property P holds for any finite map ...
In the context of a program P and environment E, the variable V is well-typed with type T if and only if the environment E maps the variable V to the type T.
The lemma "is_ground_cls_is_ground_on_var" assumes that 1) the clause C, after being substituted by sigma, is a ground clause (a clause with no variables), and 2) the variable v is in the set of variables of the clause C. Under these assumptions, it shows that the atom resulting from applying the substitution sigma ...
The lemma named "sumlist_as_summset" assumes that the set of elements 'xs' is a subset of the carrier 'G'. It shows that the sum of the list 'xs' is equal to the sum of the multiset 'xs'.
The lemma named "primerew" establishes that for a fixed natural number 'p', if 'm' is a divisor of 'p', and 'm' is not equal to 1 or 'p', then 'p' is not a prime number.
If a list of instructions "is" and exception table "xt" are well-typed with respect to a list of types "τs", then they are also well-typed with respect to the list of types "τs" extended with any additional list of types "τs'".
A closed segment from point a to point b is a subset of an open segment from point c to d if and only if both point a and point b are in the open segment from point c to d.
P1 is equal to p2, which is equal to p3, which is equal to p4, which is equal to p5, and this pattern repeats itself.
The lemma Abs_res_fcb: fixes sets xs and ys of type 'a (which is a base type of atoms) and objects S and T of type 'b (which is a type with finite support) assumes that the abstraction of T over the atoms in xs is equal to the abstraction of S over the atoms in ys and for any atom x in xs, if x is in the su...
The lemma named "at_in_mapi_branch" assumes that proposition "p" is at position "a" in a branch and that proposition "p" is not equal to the nominal "a". It shows that there exist values "v" and "v'" such that the function "f" applied to "v", "v'", and "p" is at position "a" in the mapped branch created by applying fun...
The union of an empty set is equal to zero.
If the precondition for union of sets "p" with elements "x" and "y" holds, then there exists a postcondition "p'" such that the postcondition for union of sets with elements "x", "y", and "p" holds.
If the following conditions are met: 1. For every pair (m_impl, m_spec) in the set M, the security invariant complies with the formal definition. 2. G is a well-formed list graph. 3. The set of edges Es is a subset of the set of edges in G. 4. The set accu is a subset of the set of edges in G. 5. The list formed by co...
The lemma named "uf_graph_invar_preserve" assumes that the graph invariant "uf_graph_invar" holds for the union-find structure "uf" and the set "T", that "a" and "b" are elements of the vertex set "V", that "e" is an edge that connects "a" and "b", that "e" is an element of the edge set "E", and that "T" is a subset of...
The coefficient of the composition of a polynomial 'p' and a linear polynomial with coefficients '0' and 'a' at a given index 'i' is equal to 'a' raised to the power of 'i' times the coefficient of the polynomial 'p' at the same index 'i'. This is under the condition that 'a' belongs to a commutative semiring.
The incoming degree of a vertex 'x' in a split function 'f' is equal to the incoming degree of 'x' in the function 'f'. This is denoted as "?lhs = ?rhs", where "?lhs" represents the left-hand side of the equation and "?rhs" represents the right-hand side.
The transition function "a0i" refines into the transition function "m1" under the relation "R01".
: if for any alpha, the statement phi of alpha holds in v, then it follows that for all alpha, the statement phi of alpha holds in v.
The lemma "qty_single_updated" shows that the closing value process of a market for a single quantity of an asset at a given time (n+1) in a certain state (w) is equal to the product of the price of the asset in the market at that time and state, and the quantity at that time and state.
If the generalized multi-context extension on a predicate P and a relation R is a subset of R, and the identity relation on the second component of R is also a subset of R, then the generalized multi-context extension on P and the transitive closure of R is a subset of the transitive closure of R.
The lemma named "unitarily_equiv_rank_1_proj_col_carrier" assumes that matrix A is an n by n matrix, A is unitarily equivalent to B with U being the unitary matrix, and i is less than n. It shows that the rank 1 projection of the i-th column of matrix U is also an n by n matrix.
The lemma named "psi_mapsto" assumes that "x" is an identity of category "C". It shows that the function "psi" applied to "x" is in the hom-set from "y" to "G of x" in category "D", which implies a morphism from "F of y" to "x" in category "C".
The lemma named "unitary_gen_real" assumes that "M" is a general unitary matrix. It shows that there exists a real number "k" greater than zero such that the adjoint of matrix "M" multiplied by "M" equals "k" times the identity matrix.
If F and G are registers and they are compatible (meaning the composition of F and G is equal to the composition of G and F for all a and b), then the pair of F and G is also a register.
, given a static reference "rt" in a state "s", if the state "s" is not abrupt, then it will evaluate to "Null". Otherwise, the result is undefined.
If zero is less than 2 to the power of n for a word of any length 'a, then the unsigned integer representation of 2 to the power of n for a word of any length 'a is equal to 2 to the power of n.
If an element 'x' belongs to a set 'A', and this set 'A' is in the set 'xs', then 'x' belongs to the union of all sets in 'xs'.
If the well-formedness conditions 'n r' and 'n s' hold, then the language 'n' of the intersection of 'r' and 's' is equal to the language 'n' of the product of 'r' and 's'.
For a given matrix A, the columns up to the kth index, where k is the number of columns in A minus 1, are equal to all the columns of A.
The lemma named "invalidate_other" is defined with fixed variables rt, dests, and dip. It assumes that dip is not in the domain of dests. It shows that the function "invalidate" with parameters rt, dests, and dip is equal to rt dip.
If a pair (h', a) is in the set of allocations from h to hT according to the Java Memory Model, then for all hT' that are not equal to hT, the value of h' at hT' is equal to the value of h at hT'.
The lemma named "subtensor_combine_tensor_prod" assumes that for all A in the set As, the dimensions of A are equal to ds. It shows that the combination of subtensors of ds and As tensor product with B is equal to the combination of subtensors of ds appended with the dimensions of B and the map of each A in As tensor p...
"x" is in the interval starting from "n" if and only if "n" is less than or equal to "x".
If X is independent of Y, then the scene of X is also independent of the scene of Y.
If the product of y and x is less than or equal to the product of x and y, then the star operation applied to the join of x and y and z is equal to the star operation applied to x and the star operation applied to y and z.
A set 's' is a hereditarily finite function with a sigma order if and only if 's' is an ordinal domain and 's' is a hereditarily finite function with a sigma.
The lemma named "not_plus_prv_neg_eql_pls" assumes that the sum of "m" and "n" is not equal to "k". It shows that it is provable that the equality of the sum of the numbers "m" and "n" and the number "k" is false.
If G acts on H, G acts on N, H is a subset of N, and a is an element of the carrier G, then the quotient map of G, H, N and the coset of H and a is equal to the coset of N and a.
For any two functions 'f' and 'g' from a metric space to the extended real numbers, 'g' is equal to the lower semicontinuous hull of 'f' if and only if 'g' is lower semicontinuous and for all 'x', 'g(x)' is less than or equal to 'f(x)', and for any function 'h' that is lower semicontinuous and for all 'x', 'h(x)' is le...
The function "takeW" with parameters "pre", "l", and "nxt" is equal to "l" if and only if the function "holds" with parameters "pre", "l", and "nxt" is true.
"OK I F" is true if and only if for every "i" in "I", for every "j" in "I" excluding "i", "(F i) ok (F j)" is true.
If 'x' is an element of 'V', then applying the function 'phi' to 'x' and then applying the function 'phi'' to the result equals 'x'.
The generalized binomial coefficient (n gchoose k) is equal to the regular binomial coefficient (n choose k).
If 'n' is not equal to infinity, then the function 'enat' applied to 'the_enat' of 'n' is equal to 'n'.
The lemma named "cnv_refl" assumes that "a" is an identity. It then shows that the converse of "a" is "a".
If "f" is an element of the carrier of the power series ring over the p-adic numbers with "n" indeterminates, and "g" is also an element of the same carrier, then the set of values where "f" is not equal to "g" is semialgebraic.