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If X is a qtable with parameters n, A, P, and Q, and x is a well-formed tuple with parameters n and A, and both P and Q hold for x, then x is an element of X.
The lemma "FOREACHc_filter_deforestation" assumes that the set "S" is finite. It shows that the computation of "FOREACHc" over the subset of "S" for which a predicate "P" holds, using a function "f" and a starting value "s", is equivalent to the computation of "FOREACHc" over the entire set "S", using a function that a...
The lemma named "nth_elem_Max" assumes that the cardinality (or size) of set S is greater than 0. It shows that the nth element of set S, where n is the cardinality of S minus 1, is equal to the maximum element of set S.
The lemma named "unit_cpx_vec_length" assumes that "i" is less than "n". It shows that the length of the unit vector with dimensions "n" at position "i" is equal to 1.
The lemma named "finite_lepoll_infinite" assumes that set A is infinite and set B is finite. It shows that set B is less than or equipollent to set A.
The lemma named "in_span_delete" assumes two conditions: first, that "a" is in the span of set "S", and second, that "a" is not in the span of the set "S" after removing "b". Under these assumptions, it shows that "b" is in the span of the set obtained by inserting "a" into the set "S" after removing "b".
Z guarantees the intersection of X and Y is equal to the intersection of Z guarantees X and Z guarantees Y.
The lemma named "minE" assumes that the following conditions are met: the invariant "s" holds, "x" is an element of the abstraction of "s", and the predicate "P" holds for "x". Under these assumptions, it is possible to obtain an "x'" such that the minimum element of "s" for which "P" holds is "x'", "x'" is an element ...
The product of the modulus of a complex number 'z' and the cosine of the argument of 'z' in the range of 2π is equal to the real part of 'z'. Similarly, the product of the modulus of 'z' and the sine of the argument of 'z' in the range of 2π is equal to the imaginary part of 'z'.
The lemma "wfG_member_subst" assumes that the tuple (x1, b1, c1) is in the set formed by the concatenation of Gamma' and Gamma, that the well-formedness condition wfG holds for Theta, B, and the concatenation of Gamma' and the list consisting of the tuple (x, b, c) followed by Gamma, and that x is not equal to x1. It s...
If an extended natural number "i" is less than or equal to another extended natural number "j", then the product of "i" and any extended natural number "k" is also less than or equal to the product of "j" and "k".
The lemma named "zero_polynom_imp_zero_coeffs" is defined for a function 'c' that maps natural numbers to a type 'a', where 'a' is a member of the classes 'ab_semigroup_mult' and 'real_normed_div_algebra'. The lemma assumes that for any 'w', the sum of 'c i * w^i' for all 'i' less than or equal to 'n' equals zero, and ...
A hypercomplex number, represented as "HComplex x y", is equal to a hyperreal number, represented as "hcomplex_of_hypreal xa", if and only if the real part "x" is equal to "xa" and the imaginary part "y" is equal to zero.
In the context of a digraph homomorphism, if beta is an integer and alpha is an element of beta, then the functor F is a tiny digraph homomorphism from digraph A to digraph B with respect to beta.
If a game-playing view (gpv) is well-typed under an interface (\<I>), and if the gpv is not colossless, then another gpv (gpv') is also well-typed under the same interface. This implies that the well-typedness of the combined game-playing view (try_gpv gpv gpv') under the same interface is guaranteed.
If epsilon is greater than zero, then the partial Gauss-Seidel iteration with epsilon and v as parameters is equal to the Gauss-Seidel iteration with epsilon and v as parameters.
For any object, if the pair (x, s) conforms to (G, L), and in the context of G and s, the object conforms to a valid type r, and s is less than or equal to the state s updated with the object at type r, then the pair (x, the state s updated with the object at type r) also conforms to (G, L).
The interpretation function "Ifm" applied to the parameters "bbs" and "bs" on the preprocessed version of "p" is equal to the interpretation function "Ifm" applied to the same parameters on "p" itself.
If a proposition "phi" is true, then it is necessarily true.
If vector 'v' is in the carrier space of dimension 'n', then the scalar product of the zero vector of dimension 'n' and 'v' is zero.
"23 is not less than 27 as a 2-bit word."
The lemma named "zmult_div_aux2" assumes that "b" is not equal to zero. It shows that the remainder of the division of "a" (an integer) minus the remainder of the division of "a" by "b", by "b" itself, is equal to zero.
The lemma named "ground_gctxt_of_ctxt_apply_gterm" assumes that the context "C" is a ground context. It shows that the term of a ground term "t" applied to the ground context of context "C" is equal to the context "C" applied to the term of the ground term "t".
If "p", "q", and "r" are tests, then the sum of "p" and "q", multiplied by the sum of the negation of "p" and "r", is equal to the product of the negation of "p" and "q" added to the product of "p" and "r".
If set A is a subset of set B, then the upper set of B is a subset of the upper set of A in a lattice L.
An abelian group with addition defined on a set S with operations plus (pls), zero (z), minus (mns), and unary minus (um) is equivalent to a commutative monoid with addition defined on the same set and operations, and the following conditions hold: for every element a in S, the result of applying the unary minus operat...
Lemma: Finite Set Sum Assumptions: 1. Set A is finite. 2. For every element i in set A, the set B(i) is finite. Conclusion: The sum of the sets B(i) for all elements i in set A is finite.
If an element 'x' is in the set 'xs', then the index of 'x' in 'xs' is less than the length of 'xs'.
For any three points A, B, and C in a two-dimensional real space, the content (area) of the convex hull (the smallest convex set that contains these points) is equal to the absolute value of the difference between the product of the difference in the first coordinates of C and A and the difference in the second coordin...
If the function "wcode_goon_right_moving_1" is true for the inputs "ires", "rs", and the pair "(aa, Bk # list)", and if "b" is equal to "aa" and the list starting with "Oc" is equal to "ba", then the function "wcode_backto_standard_pos" is true for the inputs "ires", "rs", and the pair "(aa, ba)".
Given the assumptions that 'C' is a locally small category, 'F' is a functor from the opposite of 'C' to the category of sets, 'X' is an object in 'C', and 'x' is an element of the set 'F' applied to 'X', it shows that applying the Yoneda map to the inverse of the Yoneda map applied to 'F' and 'x', equals 'x'.
: Assuming we have a profile P for a set of alternatives A, and x and y are alternatives in A. If the number of individuals who prefer x over y is greater than the number of individuals who prefer y over x, then we can find a new profile P' and an individual k such that: 1. P' is a valid profile for the set of altern...
The lemma named "prefixToElementAppend" shows that the prefix to an element 'e' in the concatenation of two lists 'M1' and 'M2' is equal to the prefix to 'e' in 'M1' if 'e' is in the set of elements of 'M1'. Otherwise, it is equal to 'M1' concatenated with the prefix to 'e' in 'M2'.
The operation of taking the modulus of two integers 'a' and 'b', and then converting the result to a real number, is equivalent to first converting 'a' and 'b' to real numbers and then taking the real modulus.
In the context of an infinite coin toss space, we have a lemma named 'nat_filtration_AE_eq'. This lemma is about two functions 'f' and 'g', both of which map a boolean stream to a real number. The lemma assumes that almost everywhere in the measure space 'M', 'f' and 'g' are equal. It also assumes that 'p' is a positiv...
The lemma named "cindex_combine" assumes that the set of x such that the jump of function f at x is not zero and x is between a and c is finite, and that a is less than b and b is less than c. It shows that the complex index of f from a to c is equal to the sum of the complex index of f from a to b, the jump of f at b,...
An infinite tau-divergence from a state 's' is not equal to a well-founded relation on the flipped silent move from the same state 's'.
The lemma "exitI2" assumes that the state "s" is reachable in a non-terminating way (denoted by "reachNT s"), and that some condition "K" holds for the state "s" (denoted by "K s"). It also assumes that "K" is an invariant in non-terminating states (denoted by "invarNT K") and that "K" does not hold any value "v" (deno...
If 'p' is a real polynomial function, then the function that maps 'x' to 'p(x) / c' is also a real polynomial function, where 'c' is a constant.
If the variables (x, y, sem, pc1, pc2) remain unchanged and the PsiInv property holds in the current state, then the PsiInv property will also hold in the next state.
If a binary function respects a relation between type A and type B, and if a list respects the relation between all pairs of type A, then the fold function also respects the relation between type B.
The set of variables in any arithmetic expression (a::aexp) is finite.
The product of the "an" function of "x" multiplied by "L" and "x" is equal to the product of the "an" function of "x" multiplied by "L", "x", and "bot".
If a pair (x, z) is in relation R and another pair (y, z) is also in relation R, then the pair (x, y) is in the relation "joinable R".
The union of the set of terms of pairs of transitions in the set of pairs of transitions of F and D is a subset of the union of the pair of the set F and the pair of the set D.
The multiplication of two matrices, m1 and m2, using the "matrix_mult" function from the "rat_poly" module is equivalent to the multiplication of the same matrices using the "mat_mult" function with the row length of m1 as an additional parameter.
The star-transform of the insertion of an element 'x' into a set 'A' is equal to the insertion of the star-transform of 'x' into the star-transform of set 'A'.
The lemma shows that the distance on the Riemann sphere between any two points M1 and M2 is always greater than or equal to zero.
If 'A' and 'B' are finite sets and the maximum value in 'A' is less than the minimum value in 'B', then the join operation on the union of 'A' and 'B' with respect to a function 'f' is equal to the concatenation of the join operation on 'A' with respect to 'f' and the join operation on 'B' with respect to 'f'.
If the following three conditions are met: 1. NI is a subset of payload, 2. K is a subset of the range of LtK, 3. I is a subset of valid, then the analysis of the union of the extraction of Bad and NI from the abstraction of I, K, and Tags is a subset of the union of the analysis of the extraction of Bad and NI fr...
The set "R" is a subset of the set "S" if and only if the function of "R" is less than or equal to the function of "S".
The variable of the function 'g' after substituting 'v' in the next position is equal to the variable of the function 'g' with 'v'.
If a list 'xs' is not empty, 'r' is a vertex in graph 'G', and 'xs' is a forward direction in the directed tree rooted at 'r', then there are no cross products in the left-deep tree created from 'xs'.
If the first elements of a list of pairs, when mapped, are distinct and reverse sorted, then the set resulting from updating the list with a new pair (n, x) is equal to the set formed by inserting the pair (n, x) into the original set, after removing any pair whose first element is n.
The lemma named "map_iteratei_I" assumes that for any map "m", if "m" is invariant, then the map iterator "iti m" is a valid iterator over the abstraction of "m". Given this assumption, it shows that "map_iteratei" is a valid iterator over the abstraction of any invariant map, using the iterator "iti".
The distance between two points x and y is not less than zero.
If "c" is an instance of "I", then the identity relation holds between "c" and "c" under "I".
The lemma named "closure_difference" is defined for a language 'A' of a finite type 'a'. It assumes that both 'A' and 'B' are regular languages. It shows that the difference of 'A' and 'B' (i.e., the set of elements in 'A' but not in 'B') is also a regular language.
The lemma is named "has_bochner_integral_discrete_difference". It is about a function 'f' that maps from 'a' to 'b', where 'b' is a Banach space with a second countable topology. The lemma assumes that 'X' is countable, the measure of each element 'x' in 'X' is zero, each singleton set containing an element 'x' from 'X...
If the following conditions hold: 1. k minus the integer division of n by 2 equals the successor of k' (k - n div 2 = Suc k') 2. n is less than or equal to k (n ≤ k) 3. The remainder of n divided by 2 is 1 (n mod 2 = Suc 0) Then, k' minus the integer division of n by 2 equals k minus n (k' - n div 2 = k - n).
The function "return o hd_tl" in the imperative setting corresponds to the function "RETURN o hd_tl" in the functional setting. This correspondence is valid under the assumption that the input list is not empty. The input list is associated with the assertion "list_assn A", and the output is a pair consisting of an ele...
If 'e' is an element of the directed arcs of tree 't', then the directed head of 'e' in tree 't' is an element of the set of directed vertices of 't', excluding the root of 't'.
If a property P holds for a time point t and for all time points in a time interval I, then the property P also holds for all time points in the time interval that results from inserting the time point t into I. 2. The second part is a conversion rule that states that the property P holds for all time points in the ti...
Filtering a multiset difference (M - N) with a predicate P is equivalent to the difference of the filtered multisets of M and N with the same predicate P.
If the ASID of 'ahi' is equal to the ASID of 'ahi', and if 'downif' is an ASIF of the DownIF of 'ahi' and 'upif' is an ASIF of the UpIF of 'ahi', and similarly 'downif' is an ASIF of the DownIF of 'ahi' and 'upif' is an ASIF of the UpIF of 'ahi', then 'ahi' is equal to 'ahi'.
If a well-formed Java program P transitions from state s to state s' through thread t and trace action ta, and if the lock status and synchronization events of state s are acceptable, then the lock status of state s' is also acceptable.
If the lengths of two lists, xs and ys, are equal, then the reverse bit list order of the function F applied to the concatenation of xs with a list containing x and ys with a list containing y is equal to the logical disjunction of the following two conditions: y and not x, or the logical conjunction of y or not x and ...
The nth elements of the list obtained by applying a function 'f' to each element of a list 'xs' is equal to the result of applying the function 'f' to the nth elements of the list 'xs'. Here, 'I' represents the indices of the elements.
The transition from state m2 to state m3 refines the relation R23. This is under the condition that R23 intersects with the combination of the m2_inv3a_sesK_compr state and the intersection of the m3_inv3_sesK_compr, m3_inv2_ticket, and m3_inv1_lkeysec states.
The "dockermynet4policy" is a well-formed list graph.
For any given numbers 'n' and 'i', and any polynomial map 'm', the operation of computing the higher polynomial of 'm' with respect to 'n' and 'i' is equivalent to the operation of applying the polynomial map function to the higher polynomial of 'm' with respect to 'n' and 'i'.
The lemma named "inv_partD" assumes that "inv_part p A m_i m_j", "i" is less than "n", and "j" is less than "n". It shows that if "j" is less than "m_j", then "p A i j" holds. Additionally, if "j" equals "m_j" and "i" is greater than or equal to "m_i", then "p A i j" also holds. Finally, if either "j" is less than "m_j...
The lemma named "merge_interact" assumes that the function "merge" is applied to the lists "as", "bs", "us", and "vs", and that the concatenation of the lists "as" and "bs" are strictly sorted. It also assumes that "bs" belongs to the set of all lists excluding the empty list. Under these assumptions, it shows that the...
The antecedent of 0 is equal to 0, in the context of antecedents.
If a variable 'v' is in the set of free variables of the term 't' substituted by 's', and 'v' is not in the range of variables of 's', then 'v' must be in the set of free variables of the term 't'.
If 'I' is a finite set and 'a' and 'b' are elements in 'I', then 'b' mirrored in 'I' is less than or equal to 'a' mirrored in 'I' if and only if 'a' is less than or equal to 'b'.
The lemma named "spmf_run_obsf_oracle_obsf_distinguisher" is defined with two terms "eg1" and "eg2", both of which are defined as "exec_gpv". This lemma shows that the probability mass function (spmf) of the first element of the pair resulting from executing "eg1" with the "obsf_oracle" oracle, the "obsf_distinguisher"...
The lemma "ivl_integral_nonneg" is about a function "f" that maps real numbers to real numbers. It assumes that "f" is integrable on the closed interval from "a" to "b". It also assumes that for any "x", if "x" is greater than or equal to "a" and less than or equal to "b", then "f(x)" is greater than or equal to 0. Mor...
If 'n' is less than or equal to 'a', then the 'n'-th element of the 'e'-tuple with parameters 'a' and 'i' is equal to the 'n'-th element of the tuple decoded from 'a' and 'i'.
The pair (s, s) is not in the relation "ChiRel a".
: 1. If the function "BigStep.evaluate_list" with parameters "False", "v", "s", "es", and "bv'" evaluates to true, then the function "evaluate_list" with the same parameters "v", "s", "es", and "bv'" also evaluates to true. 2. If the function "BigStep.evaluate" with parameters "False", "v", "s", "e", and "bv" evaluates...
: assuming for all 'j' less than or equal to '2d' plus '4' plus 'k', the function 'lose' applied to 't' plus 'j' equals '[False]', it can be shown that for all 'j' less than or equal to '2d', the function 'lose' applied to 't' plus '3' plus 'k' plus 'j' equals '[False]'.
The lemma "mod_finite_set" assumes the following conditions: 1. "fs" is a linearly independent set (lin_indep fs). 2. The length of "fs" is "m" (length fs = m). 3. The set "I" is a subset of the set of pairs (i, j) where i is less than m and j is less than i (I \<subseteq> {(i,j). i < m \<and> j < i}). 4. The lattice ...
The set of measurable sets in the product measure space of two quasi-Borel spaces X and Y, when each is converted to a measure space, is a subset of the set of measurable sets in the quasi-Borel space formed by the product of X and Y.
For any formal Laurent series 'f' with elements from a division ring, dividing 'f' by the inverse of the formal Laurent series 'fls_X_inv' raised to the power of 'n' is equal to shifting the formal Laurent series 'f' by the negative integer value of 'n'.
The least common multiple of two natural numbers m and n is equal to n if and only if m divides n.
The unsigned integer representation of the 10-bit word number 234567 is 71.
If "R" is a uniquely injective relation, then the sequential composition of the conjunction of "P" and "Q" with "R" is equal to the conjunction of the sequential composition of "P" with "R" and "Q" with "R".
If a pointer 'ptr' is not equal to the cast of a new document pointer, and if a heap 'h' turns a new document into a new document pointer, and if the same heap 'h' turns a new document into a new heap 'h'', then a relation 'r' is in the locations of the get child nodes of the pointer 'ptr' and this relation 'r' holds b...
The lemma named "strict_prefix_diff_minus" assumes that "xs" is a prefix of "ys" and "xs" is not equal to "ys". It shows that the difference between "ys" and "xs" is not an empty list.
If an empty list, when evaluated with a given state 's' and a packet 'p' under a semantics 'γ', results in a state 't', then the initial state 's' must be equal to the final state 't'.
If "a" is less than or equal to "n", then the backward interval from "a" to "n" is equal to "n" minus "a".
If there is an epsilon transition from state 'q' to state 'q'' under the input 'bs', then the state 'q' must be an element of the set 'SQ'.
For a fixed function 'f' that maps any type to a Euclidean space 'a', if 'f' is a polynomial function, then 'f' is differentiable at a point 'a' within a set 'S'.
The least fixed point of games is equal to the fixed point of games at the least fixed point.
The natural order relation on 'm' is less than or equal to the natural order relation on 'n' if and only if 'm' is less than or equal to 'n'.
The lemma named "assoc_naturality" assumes that "f0", "f1", and "f2" are arrows. It shows that the associator of the codomains of "f0", "f1", and "f2", when composed with the tensor product of "f0", "f1", and "f2" (in that order), is equal to the tensor product of "f0", "f1", and "f2" (in that order) composed with the ...
For a fixed function 'f' mapping from any finite set 'b' to another set 'a', the supremum (or greatest element) of the transpose of 'f' over all 'k' is equal to the transpose of the supremum of 'f' over all 'k'.
The magnitude of the product of two physical quantities x and y is equal to the product of the magnitude of x and the magnitude of y.
If a list 'xs' is sorted according to an order 'orda' and a function 'f' is monotone from 'orda' to 'ordb', then the list resulting from mapping 'f' over 'xs' is sorted according to 'ordb'.