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If 'i' is less than the dimension of the row of the second quantum state, then the following two equations hold: 1. The sum of (1 minus the function of 'i' divided by 2) and the negative of the function of 'i' divided by 2, all multiplied by 1 divided by 2 to the power of 'n' plus 1, equals negative 1 to the power of ...
: 1. An empty list is a matrix with 0 rows and m columns. 2. If a list of lists 'xss' is a matrix with 'n' rows and 'm' columns, and the length of a list 'xs' is 'm', then adding 'xs' to the front of 'xss' results in a matrix with 'n+1' rows and 'm' columns.
The lemma named "has_subprob_density_distr_Fst" is defined with the following conditions: - It fixes t1, t2, and f. - It defines "N" as the stock measure of the product of t1 and t2. - It defines "N'" as the stock measure of t1. - It defines "fst'" as the operation semantics of Fst. - It defines "f'" as the function t...
If "A" is less than or equal to the length of "i", the set of registers enumerated by "a" is empty, the set of arithmetic expression inputs enumerated by "a" is not empty, and the maximum value in the set of arithmetic expression inputs enumerated by "a" is less than "A", then the arithmetic value of "a" when evaluated...
The lemma named "lincomb_list_as_mat_mult" assumes that for every vector 'w' in the set 'ws', the dimension of 'w' is 'n'. It shows that the linear combination of the list 'c' and 'ws' is equal to the multiplication of the matrix of columns 'n' and 'ws' with the vector of length 'ws' and 'c'. This is denoted as "?l ws ...
The lemma named "Reds_dom_lcl" assumes that "P" is a well-formed Jinja program. It shows that if "P" enables a transition from the expression "e" with heap "h" and local variables "l" to the expression "e'" with heap "h'" and local variables "l'", then the domain of "l'" is a subset of the union of the domain of "l" an...
The lemma named "formula_induct" is of the type "formula". It fixes a variable "x" and assumes the following: 1. For any set "xset", if "xset" is finite, and for any element "x" in the set "xset" the property "P" holds, then the property "P" also holds for the conjunction of "xset". 2. For any formula, if the propert...
The system classes in P are well-formed if and only if for every class C in the set consisting of Object, Throwable, and Thread, C is a class in P, and for every class C in the system exceptions, C is a subtype of Throwable in P.
If 'f' is a bounded complex bilinear function, then applying 'f' to any complex number 'x' results in a bounded complex linear function.
The lemma named "sfis_char" assumes that "M" is a measure space and "A" is a measurable set in "M". It shows that the measure of set "A" in measure space "M" is in the set of simple functions integrable over set "A" in measure space "M".
If 'p' is a prime number and 'p' divides the product of all elements in multiset 'M', then there exists an element 'n' in multiset 'M' such that 'p' divides 'n'.
If the list "xs" is not empty, then the result of the function "choose_u" applied to "xs" is an element of the set of elements in "xs".
The lemma named "box_conjE" assumes that if the state "s" necessarily satisfies the conditions "F", "G", and both "F" and "G" together, then "P" is implied. The lemma shows that "P" holds true under these assumptions.
In the context of Minkowski Spacetime, we have a lemma called "wlog_endpoints_distinct". This lemma assumes the following: 1. A is a path. 2. Q is a symmetric relation, meaning if Q holds between a and b, then Q also holds between b and a. 3. If Q holds between a and b within a subset I of A, then both a and b are ele...
If a pair (x, y) is in the set of Solutions but not in the set of Special_Solutions, and if the i-th element of x is greater than or equal to dij i j and the j-th element of y is greater than or equal to eij i j, where i is less than m and j is less than n, then this pair (x, y) is not in the set of Minimal_Solutions.
If twice the value of 'j' is less than 'l' and twice the value of 'x' equals 'l', then the 'j'th element of the list created by filtering even numbers from the list of numbers from 0 to 'l' (exclusive) is equal to twice 'j'.
The size of a multiset after removing an element 'x' from multiset 'M' is less than the size of the original multiset 'M' if and only if the element 'x' is in the multiset 'M'.
The derivative of the identity function at any point 'a' is the identity function itself.
The lemma named "msetext_dersh_wf" assumes that the well-founded relation "gt" is such that for any pair of elements (x, y), "gt y x" holds. Under this assumption, it shows that for any pair of sets (xs, ys), the well-founded relation "msetext_dersh gt" is such that "msetext_dersh gt ys xs" holds.
If in a given system P, the class C is less than or equal to type T, then there exists a class D such that type T is equal to class D.
The lemma named "termination_variant_invar" is defined with a fixed natural number "bound" and a function "I" that takes a state 's' and returns a boolean. It assumes the following: 1. For any state 's', if the guard function returns true for 's' and the function 'I' also returns true for 's', then the function 'f' ap...
If a program P can transition from a state represented by the expressions 'es', the state 's', and the boolean 'b' to a new state represented by the expressions 'es'', the state 's'', and the boolean 'b'' through zero or more steps, then the program P can also transition from a state where the value 'v' is added to the...
If X is an element of the parts of the parts of H, then X is an element of the parts of H.
The reduced calculus, which is defined by the bottom element and the infimum, implies the interaction between two reductions and the reduction function.
"the always operator applied to a relation 'r' composed with the always operator applied to a relation 'r'' is less than or equal to the always operator applied to the composition of the relations 'r' and 'r''."
If "q" is an element in the derivative of "gta" with respect to "t" in the automaton "A", then there exists a "p" and a sequence "qs" such that "A" has a run on the generalized function "GFun" with parameters "p" and "q" and sequence "qs" on "t".
If "A" is not a Spy and "evs" is an element of "srb", then the knowledge of the Spy about the inputs of "A", "C", and "X" in the event "evs" is equal to the Spy's knowledge about the event "evs" itself.
: Given the conditions that the function "find_endpoint_path_inv" is valid for the graphs G and H, the length, and the vertices u, v, w, and x, the graph G is a valid instance of an IGraph, the graph created from G is loop-free, the function "select_nodes_prop" is valid for the graphs G and H, the length is greater tha...
For a fixed constant 'c' of a type that satisfies the properties of Heine-Borel theorem and Banach space, the weakest precondition of the differential equation 'x' equals 'c' and a condition 'G', with respect to a postcondition 'Q', is equivalent to the condition that for all non-negative time 't', if 'G' holds for all...
An application of a term 's' to a term 't' is greater than the term 't' itself, in terms of term order.
If a type variable "t" is new for a natural number "n", then "t" is also new for the successor of "n".
The set of all entries "e" such that there exists a key-value pair "k maps to v" in the map "m" and "False" is always an empty set.
The interpretation of the term subtraction 'tmsub' of 'a' and 'b' with respect to variables 'vs' and values 'bs' is equal to the interpretation of the subtraction 'Sub' of 'a' and 'b' with respect to the same variables 'vs' and values 'bs'.
The lemma named "card_of_Sigma_mono" assumes the following conditions: 1. The function 'f' is injective on the set 'I' (denoted by "inj_on f I"). 2. The image of the set 'I' under the function 'f' is a subset of the set 'J' (denoted by "f ` I \<le> J"). 3. For every element 'j' in the set 'J', the cardinality of the s...
The lemma named "restrict_in_SA_car" assumes that "f" is a semialgebraic function of "n". It shows that if we restrict the function "f" to the carrier of the p-adic number field raised to the power of "n", then this restricted function is in the carrier of the semialgebraic set of "n".
If 'x' is an element of 'assertion', then the product of 'x' and 'x' is equal to 'x'.
: If for any predicate P, which is a function from a topological space to a boolean, it holds that if P is continuous on a set S, then there exists a constant c such that for all elements s in S, P of s equals c, then the set S is connected.
The lemma named "factorize_polys_map_square_free" assumes that the factorization of a set of polynomials 'ps' results in a pair '(fs, data)', and that 'fss' is the mapping of 'fs' to real numbers. It shows that for any polynomial 'f' in the set 'fss', 'f' is square-free (i.e., 'f' does not have any repeated roots).
The lemma named "Resid_Arr_Src" shows that if "T" is an arrow and "a" is in the source of "T", then the residual of "T" with respect to "a" is equal to "T".
The lemma named "path_sum_a" assumes that the length of "r" is equal to the length of "w" and that "a" is a path in automaton "A" that corresponds to the parallel composition of "w" and "r" starting from point "p". Under these assumptions, it shows that "c" is a path in the sum of automata "A" and "B" that corresponds ...
If we have two functions 'f' and 'g' that map to real numbers, and both are continuous at a point 'a' within a set 's', and the value of function 'f' at point 'a' is not zero, then the function that maps 'x' to the power of 'f(x)' raised to 'g(x)' is also continuous at the point 'a' within the set 's'.
The least common multiple of the set obtained by multiplying each element of set A by b is equal to the normalization of the product of b and the least common multiple of set A, given that set A is not empty.
If A is a transitive set, then the set generated from A at any stage i is also a transitive set.
The lemma named "degree_aform_err_powr_aform_err" makes the following assumptions: 1. The power of an affine form with error 'X' to 'Z' equals 'Y' (represented by "powr_aform_err p X Z = Some Y"). 2. The degree of the affine form with error 'X' is less than or equal to 'n' (represented by "degree_aform_err X \<le> n")....
The lemma named "add_row_to_multiple_index_change" assumes that "i" is less than the number of rows in matrix A, "j" is less than the number of columns in matrix A, "i" is in the set "ks", the elements in "ks" are distinct, "l" is not in the set "ks", and "l" is less than the number of rows in matrix A. It shows that t...
: "U_b of (Successor of m) v equals v if and only if v equals v_b_opt".
The multiset of the list range from "a" between "l" and "h" is equal to the multiset of the range of "a" from "l" to less than "h".
If a category "CC" is small and complete in a certain universe "alpha", and if a functor "FF" maps a small category "JJ" to "CC" in a tiny manner, then there exist objects "u" and "r" such that "u" is a limit cone of "FF" from "JJ" to "CC" in the universe "alpha".
If a computation tree "cpt" with parameters P, env, and πl exists, then a computation tree with parameters P, env, and the mapping of the function "sdrop i" onto πl also exists.
The lemma named "card4_eq" assumes that the cardinality of set X is 4. It shows that there exist four distinct elements a, b, c, and d such that set X is equal to the set of these four elements.
The lemma "non_null_if_pivot" assumes that "x" is a solution for the matrix "A" with dimensions "m" by "n", and that "q" is less than "m". It shows that there exists a "p" less than "m" such that the element at position (p, q) in matrix "A" is not equal to zero.
If a variable set 'vs' is bounded by 'VS' and 'isFDERIV_approx' holds true for the given precision 'prec', number 'n', list 'xs', and function 'fas' with respect to 'VS', then 'isFDERIV' also holds true for the same number 'n', list 'xs', and function 'fas' with respect to 'vs'.
If angle ABC is less than or equal to angle DEF, and points A', C', D', and F' are on the same side of lines BA, BC, ED, and EF respectively, then angle A'BC' is less than or equal to angle D'EF'.
: Assuming that the theory 'Theta' is well-formed, the set 'Gamma' is finite, the term 't' is valid in the context of 'Theta' and 'Gamma', the term 't' after beta normalization equals 'u', and for all 'A' in 'Gamma', 'A' is a valid term in 'Theta' and the type of 'A' is 'propT'. Then, it can be shown that the term '...
The lemma named "basis_reduction_iso_main" assumes that the "LLL_invariant_mod_weak" holds for the variables fs, mfs, dmu, p, first, and b. It also assumes that the result of the function "basis_reduction_iso_main" with parameters p, first, mfs, dmu, g_idx, and j equals (p', mfs', dmu'). It then shows that there exist ...
The floor of a number 'x' is less than a numeral 'v' if and only if 'x' itself is less than the numeral 'v'.
If a state 's' is reachable, and there is a sequence of steps leading from state 's' to another state 's'', then the state 's'' is also reachable.
The tail of the list converted to a natural number using the function "c_tl" is equal to the natural number obtained by first taking the tail of the list and then converting it to a natural number using the function "list_to_nat".
A list is considered a finite weak ranking if and only if the following conditions are met: the first element of the list is not an empty set, the first element is finite, the rest of the list is a finite weak ranking, and the intersection of the first element and the union of the rest of the list is an empty set.
The cardinality of a cone is less than or equal to two.
The span of the set containing only the zero vector is equal to the set containing only the zero vector.
For all 'i' less than the length of 'subsets', the 'i'-th row of the alternate matrix 'A' constructed from 'signs' and 'subsets' is equal to a vector of the same length as 'signs'. This vector is defined by a function that takes an index 'j' and returns the value of the function 'z' applied to the 'i'-th subset and the...
The function 'mkClockAuto' implements the 'Clock'. This is applicable for the given parameters: agents, jkbp, envInit, envAction, envTrans, envVal, and envObs.
The lemma named "gen_image" is assumed to have a priority tag for generic algorithms. It also assumes that there is a side effect free generic algorithm that converts a set to a list, denoted by "is_set_to_list Rk Rs1 it1". It further assumes that there is a generic operation "ins2" that corresponds to the insertion op...
The lemma named "pell_valuation_neg_neg" assumes that "x" is less than 0, "y" is less than 0, and "(x, y)" is a solution. It shows that the pell valuation of "(x, y)" is less than -1.
The real number representation of the division of two algebraic numbers x and y is equal to the division of the real number representations of x and y.
For any two natural numbers 'n' and 'x', if 'n' is greater than 1 and 'x' and 'n' are coprime (i.e., their greatest common divisor is 1), then the modulus of 'x' and 'n' is greater than 0.
The lemma named "diameter_le" assumes that the set S is not empty or the value d is greater than or equal to zero. It also assumes that for any two elements x and y in the set S, the norm of the difference between x and y is less than or equal to d. Under these assumptions, it shows that the diameter of the set S is le...
The lemma named "preserves_join" assumes that "v" is the join of "t" and "u" in system "A". It shows that the function "F" applied to "v" is equal to the join of the function "F" applied to "t" and "u" in system "B".
If 'h' is less than the length of 'treeList' and 'xaa' is equal to the result of deleting the 'h'-th element of 'treeList', then the following holds: The raw Van Emde Boas tree association of 'xaa' and 'x', multiplied by the assertion that 'xb' is equal to the minimum element in the Van Emde Boas tree 'xaa', multiplie...
The conjunction (logical "and") of two pure assertions "a" and "b" is equivalent to the star operation applied to these two pure assertions.
: Assuming that the abscissa (x-coordinate) and ordinate (y-coordinate) of points A, B, C, D, E, and F are all in the set of numbers that can be expressed as the square root of a rational number (radical_sqrt), and that the distance from point A to point X is equal to the distance between points B and C, and the dista...
The lemma "subseq_imp_sorted" assumes that "s" is a subsequence of "t" and "t" is sorted with respect to "p". It shows that "s" is also sorted with respect to "p".
For a given set of polynomials 'fs', and 'X' defined as the union of the indeterminates of the polynomials in 'fs', the number 1 is in the ideal generated by 'fs' if and only if the number 1 is in the set produced by the Macaulay list operation on the degree shifts of 'X' and the Dube operation on the successor of the ...
The function "cmp_prod" is parametric over two relations. Specifically, it states that given two relations, one from type 'a' to 'b' and another from type 'c' to 'd', the "cmp_prod" function preserves these relations when applied to pairs of elements from 'a' and 'c' to 'b' and 'd' respectively.
The lemma maxR_absorb1 shows that if the pair (y, x) is in relation r, then the maximum relation of x and y is x.
: if for all xA, the memory of xA is equal to the memory of the variable C of xA, then the memory of A is equal to the memory of A of memory C.
, under the assumption that the abstraction of the stateful symbolic trace database 'A' with respect to 'I' at position 'n' is not equal to the abstraction of the stateful symbolic trace database 'A' appended with 'B' with respect to 'I' at the same position 'n', it can be shown that there exists a term 'u' and a state...
The set of minimal coincidences for functions g and h is a subset of the set of coincidences for functions g and h.
If the "Client" is in the state of increasing "giv", it guarantees that always, for any state "s", the value of "rel" at state "s" is less than or equal to the value of "giv" at state "s".
, given a function 'f' mapping from 'i' to a topological space 'a' and a Banach space 'b' with a second countable topology, if for every 'i' in the set of probability mass function 'M', 'f' is continuous on 'A', and for every 'a' in 'A' and 'i' in the set of probability mass function 'M', the norm of 'f(i, a)' is less ...
The lemma "hvzk_z2_r2_tuple_rewrite" assumes that "xb" is less than the order of "\<G>". It shows that the tuple consisting of "\<bold>g" raised to the power of the sum of "x'" and "xa" times "x1", all modulo the order of "\<G>", plus the order of "\<G>" times "xa" times "x1", minus "xa" times "x1", all modulo the orde...
The lemma "wt_jvm_prog_impl_wt_start" assumes that the Java Virtual Machine program P is well-formed according to the specification Phi, and that the program P sees a method M with parameters of types Ts, returning type T, and consisting of maximum stack size mxs, local variable array size mxl0, instructions ins, and e...
The lemma in the Deutsch theory shows that the evaluation of the Deutsch algorithm is equal to the sum of the function evaluated at 0 and 1, all under modulo 2.
If 'a' is an element of the carrier 'R' and 'a' is not equal to zero, then 'a' raised to the power of any natural number 'n' will not be equal to zero.
The lemma named "map_insort_key_fst" assumes the following conditions: 1. The list obtained by mapping the first element of each pair in the list "ps" is distinct. 2. The key "k" is not in the set obtained by mapping the first element of each pair in the list "ps". 3. The function "f" is injective, meaning that it alw...
If 'n' is not in the variables of term 't', then updating the environment 'E' at 'n' with 'x' and evaluating term 't' is the same as evaluating term 't' with the original environment 'E'.
The lemma named "lift_instrs_complete" is defined with a fixed variable "Sigma" which is a list of optional types. It assumes the following conditions: 1. All elements in the list "instrs" are within the range of local variables defined by "N" in the Sinca model. 2. All elements in the list "instrs" are within the ran...
A complex vector of size 'n' forms a module over the complex number ring.
The lemma named "var_rename_fv_set_disjoint" assumes that "M" is a finite set and "s" is an element of "M". It shows that the intersection of the free variables of "s" and the free variables of "t" after renaming the variables based on the maximum variable in the free variable set of "M" is an empty set. This means tha...
If set v is a subset of set v', then the singleton map of k to v is a subset of the singleton map of k to v'.
The set C is finite.
The lemma named "rootE" assumes that "u" belongs to the closure of "r". It then concludes that there exists a number "k" such that "r" raised to the power of "k" equals "u".
If H is defined as the set of all x such that x is related to f(x) by relation r and x is an element of set A, then the least upper bound (lub) of H in the complete lattice (cl) is a fixed point of function f in set A.
The index of any point set 'ps' in an empty multiset is zero.
Swapping an element twice will result in the original element.
The lemma named "body_lc" is defined with three fixed elements: t, s, and p. It assumes that t is a body, and both s and p are locally closed (lc). It then shows that the term t, when substituted with s and p, is also locally closed.
The lemma named "poincare_collinear_doubleton" assumes that "u" and "v" are in the unit disc. It shows that "u" and "v" are Poincare collinear.
The unit factor of the finite least common multiple (Lcm) of a set A is equal to the boolean value of whether the finite least common multiple of A is not equal to zero.
Adding zero to any matrix A is equal to the matrix A itself. This is applicable for any matrix A of type 'a, where 'a is a monoid under addition, and 'm and 'n are the dimensions of the matrix.