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The lemma "supp_is_subset" is defined for a set S of atoms and an element x of type 'a, which is a permutation type. It assumes that S supports x and that S is finite. It shows that the support of x is a subset of S.
If you represent a binomial heap 'q' and then abstract it back, you will get the original binomial heap 'q'.
If 'x' and 'y' are elements of the group 'G', then the right coset of 'H' with respect to the product of 'H' and 'x', when multiplied by 'y', is equal to the right coset of 'H' with respect to the product of 'x' and 'y'.
A stochastic vector 'v' is a non-negative vector.
If the Fr_2 property holds for a frame F, then for all A and B, if B is an operator, then the co-power of A and B negates C.
If the following conditions are met: 1. The composition of relations 's' and 'ns' is a subset of 's'. 2. The composition of relations 'ns' and 's' is a subset of 's'. 3. The relation 'ns' is reflexive. 4. The relation 'ns' is transitive. 5. The relation 's' is locally irreflexive for the multiset 'X' with an added ele...
The function PhiWhileP with parameters A, b, and Phi is equal to a lambda function that takes a pair of values (s,t) and checks if the tuple (b,A,Phi,s,t) is in the set var.
If a rule set is a terminal chain, then there is no matching 'Goto' for a given packet and decision function in that rule set.
The lemma "adjunction_unit_determines_counit" assumes that there are two adjunctions in a bicategory, both with the same source, target, unit, and morphisms, but potentially different counits. It then shows that the counits of these two adjunctions must actually be the same.
A list 'xs' is sorted if and only if for any two indices 'i' and 'j', where 'i' is less than 'j' and 'j' is less than the length of the list 'xs', the element at index 'i' is less than or equal to the element at index 'j'.
"infer0" is equal to the set of all "Wrapp q" such that for some function "f" and state "q", "q" is equal to the reachable states of "A" under function "f" with an empty list, and "q" is not equal to the empty set.
: Given the following conditions: 1. A set iterator abstraction 'it' with invariant 'invar' and abstraction function 'α' over a set 'S0'. 2. For any 'xc' that satisfies the invariant 'invar', the function 'f' applied to 'xc' is equal to the function 'f'' applied to the abstraction of 'xc'. 3. An invariant 'I' holds fo...
The lemma named "list_succ_rotate1" assumes that the list "xs" contains distinct elements. It shows that the successor of the list obtained by rotating "xs" by one position is the same as the successor of the list "xs" itself.
The formula for the Möbius function of a number n is equal to the Dirichlet inverse of a function that always returns 1, evaluated at 1 and n.
The lemma named "infinite_params_available" assumes that the set of all parameters not in any p in set S is infinite. It shows that there exists an x such that x is not in the set of all parameters in any p in the union of the set obtained by extending S with C, f, and n and the set containing f of n.
The imaginary unit (i) is not equal to zero.
The count of "l" at position 0 is 1, at position 1 is 1, at position 2 is 1, and at position 3 is 2.
If the following conditions are met: 1. The function "asi" is applied to "rank", "r", and "cost". 2. The rank of the reverse of "U" is less than the rank of the reverse of "V". 3. "U" is not an empty list. 4. "V" is not an empty list. 5. The list formed by concatenating "A", "U", "V", and "B" in that order contains di...
The scalar multiplication of "w1" and "w2" is equal to the scalar multiplication of "w2" and "w1".
If a formula "phi" is quantifier-free and for all "a" in the set of zero atoms of "phi", the divisor "a" divides "d", then the interpretation of the infimum of "phi" with respect to the list ((x - k*d)#xs) is equal to the interpretation of the infimum of "phi" with respect to the list (x#xs).
, given the assumption "relto_fun A B n as sel m (a,b)", the following can be shown: 1. The first element of the sequence 'as' is 'a'. 2. The 'm'th element of the sequence 'as' is 'b'. 3. For any 'i' less than 'm', if 'sel i' is true, then the pair formed by the 'i'th element and the next element of 'as' belongs to se...
If for every element 'a' in set A, 'a' is also in set B, and for every element 'b' in set B, 'b' is also in set A, then set A is equal to set B.
: "The intersection of 'x' multiplied by the bottom element and 'L' is equal to 'd' of 'x' multiplied by the bottom element times 'L'."
: Assuming that a set M is applicable to a set Omega, and that M is completely specified. Also, for every element t1 in set T1 and every element t2 in set T2, the r_dist_set of M, Omega, t1, and t2 holds true. Furthermore, T1 is a subset of the nodes in M. Under these conditions, the intersection of T1 and T2 is an em...
The lemma named "hyp2_in_middle" assumes that points p and q are in set S, point r is in the union of hyp2 and S, points p, q, and r are collinear, and points p and q are not the same. Under these assumptions, it shows that in the real plane, point r (when converted from projective to Cartesian coordinates) lies betwee...
: - If the difference between a real number 'a' and another real number 'b' is less than zero, then 'a' is less than 'b'. - If the difference between 'a' and 'b' is equal to zero, then 'a' is equal to 'b'. - If the difference between 'a' and 'b' is greater than zero, then 'a' is greater than 'b'.
The lemma named "subtype_valid_simple" assumes the following conditions: 1. Under contexts Theta, B, and Gamma, type t1 is a subtype of type t2. 2. The atom z is fresh in context Gamma. 3. Type t1 is defined as a record type with a single field z of type b and constraint c1. 4. Type t2 is defined as a record type with...
The lemma named "nth_minus_list" is about lists of elements that support the subtraction operation. It assumes that we have two lists, xs and ys, and an index i, such that i is less than the length of both lists. It shows that the i-th element of the list resulting from subtracting ys from xs is equal to the difference...
If the kinds of object pointers in heap 'h' are equal to the kinds of object pointers in heap 'h'', then the known pointers in heap 'h' are equal to the known pointers in heap 'h''.
The lemma "errMOD_igSubstIGOpSTR" assumes that the abstract interpretation "MOD" satisfies the following conditions: 1. "MOD" is well-sorted in the abstract interpretation of the bar operator. 2. "MOD" preserves the well-sortedness of variables in the abstract interpretation. 3. "MOD" preserves the well-sortedness of ...
The lemma named Ferison assumes that "A is entirely within C" and "B is partially within C". It shows that "A does not intersect with B".
The Free Group of a set S is a Group.
The lemma "acc_imp_SN_on" assumes that "x" belongs to the accessible points of relation "r". It shows that "x" is strongly normalizing on the set of all pairs "(y, z)" where "(z, y)" belongs to the relation "r".
For a list composed of 'xs' followed by 'x', the operation 'strip_while' with a predicate 'P' is equivalent to: if 'P' holds for 'x', then 'strip_while' with 'P' applied to 'xs'; otherwise, it is equivalent to the list 'xs' followed by 'x'.
If "L" is a ladder for "D", and if "index" is greater than or equal to 1 and less than the length of "L", then the "index - 1" position in the ladder "L" is less than the length of "D".
The accessible deterministic finite automaton (dfa) is indeed a deterministic finite automaton.
The lemma named "infsum_reindex_bij_betw" assumes that the function "g" is a bijective function between sets A and B. It shows that the infinite sum of the function "f" composed with "g" over the set A is equal to the infinite sum of the function "f" over the set B.
The complex conjugate of a number 'c' is a real number if and only if 'c' itself is a real number.
The lemma named "infinite_infinite_Fpow" assumes that set A is infinite. It then shows that the function power set of A (Fpow A) is also infinite.
The lemma named "end_smult_assoc" assumes that the range of T is a subset of V. It shows that the scalar multiplication of a and b with T is equal to the scalar multiplication of the product of a and b with T.
The size of term u1 is less than the size of the application of term u1 to term u2.
Given an assumption that an element "c" belongs to the union of two sets "A" and "B", it can be inferred that either "c" belongs to set "A", or "c" does not belong to set "A" but belongs to set "B".
Filtering a list with a function that always returns true will result in the original list.
The condition where h1 and h2 result in h0, and h3 and h4 result in h1, is equivalent to the existence of some h5 such that h3 and h5 result in h0, h2 and h4 result in h5, h1 and h2 result in h0, and h3 and h4 result in h1.
Joining function "f" with a sequence up to "n" is equivalent to applying function "f" to the successor of "n".
If a filter F is ultra, then the third condition of logical connectives preservation holds for implication and negation.
For any natural number 'm' that is greater than or equal to 1, 'm' is an element of its own divisor set.
Zero is not in some basis V.
If the set "qs" is a subset of the set containing "x" and "y", and "x" is not equal to "y", then the function "T" with parameters "p", "x", "y", "qs", and "OPT2 qs [x,y]" is greater than or equal to the function "T" with parameters "p_opt", "x", "y", and "qs".
The lemma named "dimension_sum_space" assumes that the dimensions of spaces E, F, and the intersection of E and F are n, m, and k respectively over the field K. It shows that the dimension of the sum of spaces E and F over the field K is (n + m - k).
The composition of the partial function on F, the feedback function on F, and the Kleisli operation is equal to the backdoor function on F.
The lemma named "restrict_index_in" assumes that "i" is less than the length of "X" and "i" is in "I". It shows that the "i"-th element of the restriction of "I" on "X" is equal to the "i"-th element of "X".
The lemma named "cmp_in_hom" assumes that "\<mu>\<nu>" is an arrow in the category "B.VV". It shows that the composition "cmp \<mu>\<nu>" is a morphism from the domain of "\<mu>\<nu>" under the functor "H\<^sub>DoGF_GF.map" to the codomain of "\<mu>\<nu>" under the functor "GFoH\<^sub>B.map" in the category "D".
If the term "t" has a loose bound variable at the position "Suc k" (which means the successor of k or k+1), then the term "t" also has a loose bound variable at the position "k".
, given a real number 'c' and assuming that the function 'f' tends to 'c' at filter 'F' and 'c' is greater than zero, then the limit of the product of 'f' and 'g' as 'x' approaches 'F' is negative infinity if and only if the limit of 'g' as 'x' approaches 'F' is negative infinity.
The lemma named "dom_inv" assumes that "f" is an isomorphism. It shows that the domain of the inverse of "f" is equal to the codomain of "f".
The operation of truncating upwards at a certain precision 'p' the product of 'x' and 2 to the power of a real number derived from an integer 'k', is equivalent to the product of truncating upwards at precision 'p' 'x' and 2 to the power of 'k'.
If the following conditions are met: 1. The variable "states" satisfies the invariant "invar". 2. The variable "n" is less than the remaining steps in "states". 3. The function "size_ok'" applied to "states" and the difference between the remaining steps in "states" and "n" holds true. Then, the function "size_ok" app...
The function "s f g k" is periodic with period "k" in the context of arithmetic operations.
The lemma named "vset_neq_1" assumes that "b" is not an element of set "A" and "a" is an element of set "A". It shows that "b" is not equal to "a".
For a fixed function 'f' from type 'a' to type 'b' (where 'b' is an aggregation order), the sum over all 'k' of 'f(k)' is equal to the sum over all 'k' of 'f(k)' plus the lowest possible value (bot).
"If 'a' equals 'b' or 'c' equals 'd', and 'a' equals 'c' or 'b' equals 'd', then 'a' equals 'd' or 'b' equals 'c'."
The lemma named "is_pr_Hom_trans" assumes two conditions: first, that "f" is a preservation relation homomorphism from "A" to "B", and second, that "g" is a preservation relation homomorphism from "B" to "C". Given these assumptions, it shows that the composition of "g" and "f" is a preservation relation homomorphism f...
The extended natural number 0 is less than the length of P.
The evaluation of a polynomial at a point 'a' for a monomial composed of a coefficient 'c' and a term 't' is equal to the product of 'c' and the product of 'a' raised to the power of the lookup of 't' for each 'x' in the keys of 't'.
If a shadow root pointer "shadow_root_ptr" is updated with a setter function "setter" and a value "v" in a heap "h" to produce a new heap "h'", then the getter function "getter" for an element pointer "element_ptr" is preserved between the original heap "h" and the new heap "h'".
The lemma named "cop_osp__ex_cop2" assumes that points A, B, C, and P are coplanar and points A, B, C are on the same plane as points D, E. It shows that there exists a point Q such that points A, B, C, and Q are coplanar, points D, E, P, and Q are coplanar, and point P is not equal to point Q.
There exists a function 'f' such that 'f' is in the set of ground heads of 'zeta' and for all functions 'g' in the set of ground heads of 'zeta', the weight of 'g' plus the delta of 'h' times the arity of 'h' in 'g' is greater than or equal to the weight of 'f' plus the delta of 'h' times the arity of 'h' in 'f'.
The lemma named "region_not_empty" assumes that "X" is finite and "r" is a valid region with respect to "X", "k", and "I". It shows that there exists a "u" that belongs to the region defined by "X", "I", and "r".
The lemma states two things: 1. The degree of the radian value of 'w' is equal to 'w' itself. 2. The radian of the degree value of 'w' is also equal to 'w' itself.
If there exists a sequence of transitions from the expression "cast C on e" in state s to the expression e' in state s', and e' is a final expression, then there exists an expression e'' and a state s'' such that there is a sequence of transitions from the expression e in state s to the expression e'' in state s'', and...
If the duplication of a point 'p' at an index 'i' in a branch is in 'xs', then 'xs' only touches the index 'i' in the branch.
If 'x' is a vector, then 'x' is idempotent.
If a number 'n' is greater than or equal to the length of a certain type 'a', then 2 raised to the power of 'n' is equal to zero for the word length of type 'a'.
For any element 'r' of a multiplicative monoid, the hypernatural power of 'r' to 2 is equal to 'r' multiplied by 'r'.
For any two variables 'x' and 'y', 'x' is equal to 'y' if and only if their corresponding terms are equal.
, given a list 'p' with elements 'Ps' and a pointer 'next', we initialize 'q' to null. Then, while 'p' is not null, we maintain the invariant that there exist lists 'Ps'' and 'Qs'' such that 'p' points to 'Ps'', 'q' points to 'Qs'', the sets of 'Ps'' and 'Qs'' have no common elements, and the reverse of 'Ps'' concatena...
If the enumeration assertion "enum_assn A" for elements x and y equals z, then the heap node context of the enumeration assertion "enum_assn A" for x and y is also equal to z.
The lemma named "norm_set_diag" assumes that a matrix M is canonical with respect to n and that the region of M with respect to v and n is not empty. It then obtains another matrix M' such that the region of M with respect to v and n is equal to the region of M' with respect to v and n, the region of the normalized M w...
Given two Red-Black Trees "t1" and "t2", if both trees are sorted (as indicated by the assumptions "rbt_sorted t1" and "rbt_sorted t2"), then the entries of "t1" are equal to the entries of "t2" if and only if the lookup function for "t1" is equal to the lookup function for "t2".
The lemma named "merge_ffold_supset" assumes that the set "xs" is a subset of the set "ys" and that "ys" forms a list of decision trees with root "r". It shows that the result of applying the function "merge_f" with parameters "r" and "ys" to the fold function "ffold" with accumulator "acc" and set "xs" is equal to the...
The lemma named "measurable_on_restrict" assumes that a function 'f' is measurable on the universal set and that 'S' is a member of the Lebesgue measure sets. It shows that a lambda function, which returns 'f x' if 'x' is in 'S' and '0' otherwise, is also measurable on the universal set.
If the pair (l1, u1) is equal to the bounds of a floating point power with precision "prec", exponent "n", and lower and upper bounds "l" and "u", and if a real number "x" is within the range from "l" to "u", then the "n"th power of "x" is within the range from "l1" to "u1".
The lemma named "surj_imp_imT_carrier" assumes that the image of the carrier set V under transformation T is equal to the carrier set W. It shows that the image of transformation T is equal to the carrier set W.
The set generated by the function "n_sequence_enum" with parameters "xs" and "n" is equal to the set of "n" sequences from the set "xs".
If a digraph homomorphism "N" maps a functor "F" to a functor "G" from a digraph "B" to a digraph "C", and a digraph "H" maps from a digraph "A" to "B", then the composition of "N" and "H" in the context of a total digraph homomorphism and a digraph homomorphism maps the composition of "F" and "H" to the composition of...
If "x" and "y" are both in the set of Hyperreal Finite numbers, and "x" is approximately equal to "y", then the standard part of "x" is equal to the standard part of "y".
The identity is less than or equal to the natural join of nu with itself.
If we assume that a real number "c" is not equal to zero, and if we assume that the limit of the function "f(x) / g(x)" as "x" approaches "F" is equal to "c", then we can show that "f" is in the same big theta class as "g" at "F".
If for any class C, method M, types Ts, type T, and method body m, given that the program P sees the method M with parameters of types Ts returning type T as method body m in class C, and the method declaration (M,Ts,T,m) is well-formed under the well-formedness condition wf1 in the program P and class C, then the meth...
The length of the integer 0 is 0.
There exists a positive number K such that for all x, the norm of the function f(x) is less than or equal to the norm of x times K.
The negation of true is false.
When taking the 3rd bit of the negative integer 1705, the result is 7.
If a set S is Lebesgue measurable and the symmetric difference of S and T (i.e., the union of the difference of S and T and the difference of T and S) is negligible, then the Lebesgue measure of T is equal to the Lebesgue measure of S.
: 1. An element 'x' belongs to the set 'alts' if and only if 'x' belongs to the set that includes 'a', 'b', 'c', and 'd'. 2. For all elements 'x' in the set 'alts', the property 'P' holds for 'x' if and only if 'P' holds for 'a', 'b', 'c', and 'd'. 3. There exists an element 'x' in the set 'alts' for which the prope...
The lemma named "YFtor'Obj" assumes that "X" is an object in the domain of the functor "YFtor'" of the category "C", and that "C" is a locally small category. It shows that the application of the functor "YFtor'" of the category "C" to "X" equals the set of morphisms in the category "C" from any object to "X".
If "m" is an invertible pair, then the combination of the map of the first element of "m" and the map of the second element of "m" is equal to the map of the pair "m".
The lemma named "plus_adds_2" assumes that "t" contributes to "u" and "s" contributes to the difference of "u" and "t". It shows that the sum of "s" and "t" also contributes to "u".
The totient of the product of two numbers times the totient of their greatest common divisor is equal to the product of the totient of the first number, the totient of the second number, and their greatest common divisor.