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If "k" is greater than 0, then "x" multiplied by "k" is an element of the interval "I" multiplied by "k" if and only if "x" is an element of the interval "I".
If the following conditions are assumed: A: The function "offs_pred" applied to the parameters "ns", "ub", "x # xs", "index", "key", "mi", "ma" is true. B: The function "index" applied to the parameters "key", "x", "length ns", "mi", "ma" equals "i". C: "i" is less than the length of "ns". D: The function "offs_num" a...
Zero right-implicated with any element 'a' equals one.
If the solution set of a system of equations (represented by A and b) is infinite, then the system of equations is consistent.
For every clause of type "Clause", if the clause is an element of "formula0", then it is also an element of "formula". The lemma then shows that if "formula" is satisfiable, then "formula0" is also satisfiable.
For any element x in the acquisition history of the list (e#w') and a given m, if m is in the first element of e and x is in the union of the first element of e, the second element of e, and the monitored list w', then a property P holds. Alternatively, if m is not in the first element of e and x is in the acquisition ...
: assuming a natural transformation "NN" maps a functor "FF" to a functor "GG" in the semicategory of functors from category "AA" to category "BB" with a fixed Grothendieck universe "alpha", then the domain of the natural transformation map of the vertical composition of a natural transformation "MM" and "NN" is the se...
The lemma belongs to a subgroup and is named "l_coset_eq_rcong". It assumes that "G" is a group and "a" is an element of the carrier "G". It shows that the left coset of "a" and "H" is equal to the right congruence of "H" and the set containing "a".
The result of the function "put" with parameters sigma and v is an element of the set S.
The largest value of a floating point type, represented as ('e, 'f)float, is equal to the product of two terms: the first term is 2 raised to the power of the sum of the length of 'f and 1, minus 1; the second term is 2 raised to the power of the real value of the difference between 2 raised to the power of the length ...
"nu of x equals x if and only if x times 0 equals 0".
The set of all possible outcomes (or the universe of discourse) for a Poisson probability mass function is the entire universe (or the set of all possible values).
If there is a path from 'p' to 'Ref a' in the function 'f' with 'a' mapped to 'q', then the path from 'p' to 'q' in the same function is equivalent to the path from 'p' to 'Ref a' followed by 'a'.
Given the assumptions that a pair (a, b) is in the set of arcs from i to j in xs, b is not equal to j, the property P is successive in the arcs from i to j in xs, P holds for the pair (a, b), and xs is not an empty list, it can be shown that there exists a c such that either xs is a list containing only b and c equals ...
The matchC_ZOC of theta, d, and c is defined as follows: For all s, t, d', and t', if s is equivalent to t and the pair (d,t) transitions to (d',t') under c, then either s is equivalent to t' and the pair (d',c) is in theta, or there exists a c' and s' such that (c,s) transitions to (c',s') under c and s' is equivale...
If you represent a Skew Binomial Heap 'q' and then abstract it back, you will get the original Skew Binomial Heap 'q'.
The set L is finite.
The lemma "reachable_constraints_preserves_labels'" assumes that for every transaction T in the set P, and for every action a in the strand of the transaction T, the action a either has a label N with a certain value l or has a label S. It also assumes that a certain set of constraints, denoted by \<A>, is reachable fr...
The minimum set of an empty set is None.
In the context of ring functions, there is a lemma called "function_uminus_eval". Given that "a" is an element of set "S" and "f" is an element of the carrier "F", it shows that the evaluation of the unary minus function on "f" at "a" is equal to the negation of the value of "f" at "a".
The composition of the union of 'e', the inverse of 'ov', the inverse of 's', the inverse of 'd', 'f', the inverse of 'm', and the inverse of 'f' with 'm' is a subset of the union of 'm', 'ov', the inverse of 'f', the inverse of 'd', 's', the inverse of 's', and 'e'.
The right fold operation of inserting elements from a list 'xs' into a set 'A' is equivalent to the union of set 'A' and the set of elements in the list 'xs'.
The lemma named "prec_spec_splitE" assumes that the precondition specification of h for the product of A and B is the same as the product of C and D. From this assumption, it obtains that the precondition specification of h for A is the same as for C, and the precondition specification of h for B is the same as for D.
If both x and y are greater than zero, then the sum of x and y is also greater than zero.
Mapping a function 'f' over an empty doubly-linked list results in an empty doubly-linked list.
1. For all x, there exists a y such that f(x, y) equals f(x, g(x)). 2. It is not the case that there does not exist an x for which P(x) holds if and only if it is not the case that for all x, P(x) does not hold. 3. For all u, there exists a v such that for all w, there exists an x where f(u, v, w, x) equals f(u, g(u), ...
If a formula 'phi' is mu-stable in a world 'w', and if another formula 'psi' is a subformula of 'phi', then 'psi' is also mu-stable in the same world 'w'.
It is false if for every float number 'x', 'x' equals zero.
The lemma named "zfact_iso_0" assumes that "n" is greater than 0. It shows that the function "zfact_iso" with parameters "n" and 0 equals to the zero element of the ring "ZFact" with parameter "n" converted to an integer.
The union of 1 and the product of x and the star of x is equal to the star of x.
The lemma named "prfx_Lam_iff" shows that "u" is a prefix of "Lam t" if and only if "u" is a lambda and the result of un-lambda-ing "u" is a prefix of "t".
If 'x' is up-closed, then the supremum is distributive over the complement of 'x'.
If set A is finite and is a subset of G, then the sum of the function d over set A is less than or equal to the cardinality (or size) of set A.
"CHAR equals 0 if and only if for all natural numbers greater than 0, the value of the natural number is not equal to 0 in the type 'a'".
The lemma named "preservation_let_val" assumes that either the tuple consisting of Theta, Phi, Delta, delta, the let expression with variable x and value v, and statement s is a valid typing judgement for tau, or the tuple consisting of Theta, Phi, Delta, delta, the let2 expression with variable x, type t, value v, and...
If a formula "phi" is quantifier-free, then the substitution of "phi" in "rcs" is also quantifier-free.
If list "l" is not empty, then applying the function "marl_butlast" to array "a" (which is an array list representation of "l" with a maximum size of "N") results in an array list that represents the "butlast" of list "l" (i.e., the list "l" without its last element). This is a rule for separation logic over heap memor...
For any list 'ys', and for all programs 'p', 'p'', and instruction sequences 'is', if the program state buffer 'p' and 'p'' with instruction sequence 'is' is not in the set 'xs', then the head program 'q' of the concatenation of 'xs' and 'ys' is equal to the head program 'q' of 'ys'.
, in the nonstandard analysis, the sequence defined by the function "r plus the inverse of the successor of n, where n is a natural number" tends to the limit "r" as n tends to infinity.
The lemma "decseq_emeasure" assumes that the range of B is a subset of the sets in M and that B is a decreasing sequence. It shows that the sequence of the measure of each set B[i] in M is also a decreasing sequence.
If H is a submodule of the Module M over the ring R, and if a is an element of the ring R and h is an element of the submodule H, then the scalar multiplication of a and h is also an element of the submodule H.
For every element X in the used events, the parts of X are a subset of the used events.
The function "omega_sum" is injective (or one-to-one) on the set of lists "l" where the length of "l" equals "n".
There exists a natural number such that the image of that number under I is equal to the image of the next natural number.
If 'i' is greater than or equal to zero, then the 'i'-th element of the concatenation of lists 'xs' and 'ys' is equal to the 'i'-th element of 'xs' if 'i' is less than the size of 'xs', otherwise it is the '(i - size xs)'-th element of 'ys'.
Skl1 implements a0i with the mediation of med0sk1iai.
The lemma "expands_to_MSLNil_imp_filterlim" assumes that the function "f" expands to the multiseries "MS MSLNil f'" on a given basis. It shows that "f" converges to 0 at the top.
The signed representation of a word (a fixed-size data type) created from a natural number 'n' is equal to the integer representation of the signed bit taken from the length of 'b' (a type with a length) minus one, converted to an integer from 'n'.
If a sequence 'xs' is a prefix of 'i', then the node delivery messages of 'xs' are distinct.
If X is an element of G and G is a Guard with parameters n and Ks, then X is an element of the guard with parameters n and Ks.
The lemma "observable_path_suffix" assumes the following: 1. The input/output sequence of path 'p' concatenated with 'io' is in the language of the system 'M' starting from state 'q'. 2. 'p' is a path in the system 'M' starting from state 'q'. 3. The system 'M' is observable. Under these assumptions, there exists a pa...
The lemma prv_imp_neg_encF_N_aux: Assuming that "\<phi>" is a formula and the free variables of "\<phi>" are empty, it can be proven that if the Peano Arithmetic representation of the negation of "\<phi>" is true, then the Peano Arithmetic representation of the encoding of "\<phi>" with a natural number "N" is also tru...
The sum of two lists, where the first list starts with 'y' followed by list 'ys' and the second list starts with 'x' followed by list 'xs', is equal to a new list. This new list starts with the sum of 'y' and 'x', followed by the sum of the lists 'ys' and 'xs'.
The nth power of a word 'a' is equal to the word of an integer which is the nth power of the unsigned integer representation of 'a'. This is applicable for any word 'a' of length 'len'.
The lemma named "swap_cols_rows_block_similar" assumes that "A" is a square matrix of order "n", and "j" is less than "n" and "i" is less than or equal to "j". It shows that the matrix "A" is similar to the matrix obtained by swapping the "i"-th and "j"-th columns and rows of "A".
For every object 'w', 'w' overlaps with the sum of 'x' and 'y' if and only if 'w' overlaps with 'x' or 'w' overlaps with 'y'.
The present value of a life insurance policy on a life aged s for n years, with interest rate i and payments made at the beginning of each year (denoted as $(I^{l}s)^{m}_n), is equal to (1+i) raised to the power of n times the present value of a life annuity on a life aged s for n years, with payments made at the begin...
The lemma "SN_imp_minimal" assumes "SN A". It shows that for all sets Q and elements x, if x is in Q, then there exists an element z in Q such that for all elements y, if the pair (z, y) is in A, then y is not in Q.
If 'x' and 'y' are both units in the ring 'R', and the set containing 'x' and its inverse is not equal to the set containing 'y' and its inverse, then the intersection of these two sets is empty.
If "x" is less than or equal to the join of "z" and the product of "y" and "x", then "x" is less than or equal to the iteration of "y" over "z".
The lemma named "perm_inputs_diff" assumes that "V''" is a permutation of "V" with respect to "M", and "vs1" and "vs2" are elements of "V''". It also assumes that "vs1" is not equal to "vs2". Under these assumptions, it shows that the mapping of "vs1" is not equal to the mapping of "vs2" when the function "fst" is appl...
If the size of the list "xs" is equal to "n", then the condition that the index at which predicate "P" is found in the list "xs" is equal to "n" is equivalent to the condition that for all elements "x" in the list "xs", the predicate "P" is not true.
The lemma named "TameEnum_comp" assumes that there is a sequence starting from the seed of a given plane 'p' that reaches 'g' through repeated applications of the function 'next_plane' on 'p', and that 'g' is a final and tame element. It shows that there is also a sequence starting from the seed of 'p' that reaches 'g'...
The lemma named "q65_upto_def" defines "q65" as a function that, for any integer 'k' less than 65, returns the remainder of the square of 'k' divided by 65.
The lemma named "column_gr_k_Gram_Schmidt_upt" is defined with a fixed matrix A of real numbers. It assumes that "i" is greater than "k" and "i" is less than the number of columns in matrix A. It shows that the "i"th column of the matrix obtained by applying the Gram-Schmidt process up to "k" on matrix A is equal to th...
The lemma states several properties about the start and end points of partial circular paths: 1. The start point of a partial circular path centered at z0, with radius r, starting at 3*pi/2 and ending at tt, is z0 minus a complex number with real part 0 and imaginary part r. 2. The start point of a partial circular p...
If "u" is a prefix of the product of "z" and "w", and "u" is also a prefix of "w", then "u" is a prefix of the product of "z" and "u".
The product of a set A and a singleton set containing only x is equal to the image of the set A under the function that inserts x. This is true regardless of whether the singleton set is on the left or right side of the product.
The lemma named "mem_box_componentwiseI" is about a set "S" in a Euclidean space. It assumes that "S" is an interval. It also assumes that for any element "i" in the basis, the dot product of "x" and "i" is in the image of the function that maps each element of "S" to its dot product with "i". If these assumptions hold...
The hyper-imaginary part of an Octonion, represented as "Octo w x y z u v q g", is a vector consisting of the elements x, y, z, u, v, q, and g.
The box of the product of "x" and the antidomain of "p", followed by "q", is equal to the box of "x" followed by the sum of the domain of "p" and the domain of "q".
The cardinality of set S is less than the cardinality of the universal set of variable type, or set S is finite, if and only if the cardinality of set S is less than the cardinality of the universal set of variable type.
The function that maps 'a' to the tensor product of 'a' and 'b' is a complex linear map.
If x is an element of B in the context of Γ, then there is no edge in Γ from x to y.
If a word "u" is a prefix of another word "v", then applying a function "f" to the result of removing the prefix "u" from "v" is equivalent to removing the image of "u" under "f" from the image of "v" under "f".
The lemma named "min_per_primitive" assumes that the word "w" is not equal to the empty word. It shows that the period of the word "w" is primitive.
The legal minimum assignment is a legal integer assignment.
If the sequences "u" and "v" converge to "xi" and "eta" in the Gromov completion of a Gromov hyperbolic space respectively, then the limit superior of the Gromov product at "e" of "u" and "v" is less than or equal to the extended Gromov product at "e" of "xi" and "eta" plus two times the Gromov hyperbolicity constant o...
The lemma "bet2_suma__out" assumes that points A, B, and C are collinear in that order, and points D, E, and F are also collinear in that order. It also assumes that the sum of the angles ABC and DEF is equal to the angle GHI. Under these assumptions, the lemma shows that point H is outside the line segment GI.
If the list "a # t1" is less than or equal to "t2", then "t1" is also less than or equal to "t2".
The lemma named "col_ABC_ABD_2" assumes that point A is not equal to point B, and that points A, B, and C are collinear, as well as points A, B, and D. Under these assumptions, it shows that points A, C, and D are also collinear.
The function "ennexp" is a bijection.
A function that always returns "None" is equivalent to an operation that results in an empty map.
If for any pair of numbers (n', x') such that ((n',x'), (n,x)) is in the lexicographic order, the function "c_assoc_have_key" applied to the function "pr_gr" of the location upper bound of n' and x', and the pair (n', x') equals zero, then the function "c_assoc_have_key" applied to the function "pr_gr" of the location ...
The evaluation of a term "eval_tm" with parameters "\<AA>" and "is" is less than or equal to the product of "(2 to the power of the length of \<AA> minus 1)" and the sum of the list of "i" where "i" is drawn from "is" and "i" is greater than 0.
: Assuming that points A' and B are not the same, and points D' and E are not the same. Also, assuming that point B lies between points A and A', and point E lies between points D and D'. Furthermore, assuming that the sum of angles ABC and DEF is equal to the angle GHI. Under these assumptions, it can be shown that th...
The execution of a stream with access to local state and initialized at the first step (Suc 0) using a local state, a transition function (trans_fun), an input, and a computation (c) is equivalent to the local state composed with the execution of a stream initialized with the transition function, the input, and the com...
If "x" is not equal to "y", and if the maximum simplicial chain is from "x" through "xs" to "y", and for any "ys", if the maximum simplicial chain is from "x" through "ys" to "y" then the length of "xs" is less than or equal to the length of "ys", then the minimum maximum simplicial chain is from "x" through "xs" to "y...
The "ahm_iteratei" function is a proper iterator in the context of an Array Hash Map.
If an infinite subset "A" is in "w", then the intersection of the sequence set "w" and "A" is not empty.
If a function 'p' belongs to the set of primitive recursive functions of two variables, then the function that takes two variables 'n' and 'm' and returns the sum of the function 'p' applied to 'x' and 'm' over all 'x' less than 'n' also belongs to the set of primitive recursive functions of two variables.
If a key "K" is not in the set of keys for the analysis of "H", then the analysis of the insertion of key "K" into "H" is equal to the insertion of key "K" into the analysis of "H".
The inverse of 'b' composed with the union of 'e', 's', 'd', 'ov', the inverse of 'ov', the inverse of 's', 'f', the inverse of 'f', and the inverse of 'd' is a subset of the union of the inverse of 'b', the inverse of 'm', the inverse of 'ov', 'f', and 'd'.
The lemma named "map2_length" assumes that the lengths of list Al and list Bl are equal. It then shows that the length of the result of applying a function 'f' to each pair of corresponding elements in lists Al and Bl (using the map2 function) is equal to the length of list Al.
If X is an element of G, G is a GuardK with parameters n and Ks, and Y is an element of the kparts of the set containing X, then Y is an element of guardK with parameters n and Ks.
If 'a' is a real integer and 'b' is a real integer, then the sum of 'a' and 'b' is also a real integer.
The lemma "entailment_subset" assumes that "S2" is a subset of "S1". It shows that "S1" entails "S2".
The lemma named "calc_ideal_point1_mix" assumes that the pair (z1, z2) is equal to the result of the function "calc_ideal_point1_cvec" with inputs A and B, that A is a real number, and that the square of the complex modulus of B is greater than the square of the complex modulus of A. Under these assumptions, it shows t...
"it is not necessarily the case that a proposition φ holds if and only if it is possibly the case that φ does not hold" in the context of a given world v.
The lemma named "ik_rcv_map" assumes that a strand "ts" is an element of the set "L". It shows that the set of "ts" is a subset of the intruder knowledge after receiving the map of "L".