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If you first curry a function 'f' and then uncurry it, you will get back the original function 'f'.
An empty list is concurrently set with any list 'x'.
The conjugate of the hypercomplex number "Hv v" is equal to the negative of "Hv v".
The lemma named "tllist_all2_preserve" with the attribute "quot_preserve" assumes two quotient relations, "R1" with associated abstraction function "Abs1" and representation function "Rep1", and "R2" with associated abstraction function "Abs2" and representation function "Rep2". It shows that if we map the representati...
For any IPv6 address, if you convert it to a natural number and then convert it back to an IPv6 address, you will get the original IPv6 address.
The opposite of the strict monoidal functor of a functor is equal to the strict monoidal functor of the opposite of the functor.
The lemma named "comp_of_ords" assumes that "le" and "lt" form a linear order according to the class "linorder". It shows that a comparator can be formed from "le" and "lt" using the function "comp_of_ords".
The lemma named "nn_transfer_operator_cmult" assumes that the function "g" is Borel measurable on the measure space "M". It shows that for almost every "x" in "M", the non-negative transfer operator of the function that multiplies "g" by a constant "c" at "x", is equal to "c" times the non-negative transfer operator of...
The lemma named "adjust_not_mem" assumes the following conditions: 1. A vector 'w' is an element of the vector carrier of dimension 'n'. 2. The set of vectors 'us' is a subset of the vector carrier of dimension 'n'. 3. The vectors in 'us' are distinct. 4. The vector 'w' is not in the span of the set of vectors 'us'...
The "nodes_it" function is a proper iterator.
The lemma named "similar_mat_wit_refl" assumes that matrix A is an n x n matrix. It shows that matrix A is similar to itself with the identity matrix of size n as the transformation matrix.
If a key "K" is in the set of symmetric keys, then "K" cannot be the public key of "A" for any "b".
If the operation 'isEmpty' on a set 'X' is defined at a state 'τ', then 'X' is valid at that state 'τ'.
The auxiliary function of Sturm for p and q equals an empty list if and only if the condition is false.
If the set of type variables in term 't' is empty, then the type substitution of term 't' with any substitution function 'ρ' will be unchanged, meaning it will remain as 't'.
A numeral n is less than or equal to an extended nonnegative real number m if and only if the numeral n is less than or equal to m.
Taking bits from 'l' to 'h' of the result of taking bits from 'l'' to 'h'' of 'w' is equal to taking bits from 'l+l'' to 'h'' of 'w' if the minimum of the length of 'a' and 'h' is greater than or equal to 'h'' minus 'l''. Otherwise, it is equal to taking bits from 'l+l'' to 'l'' plus the minimum of the length of 'a' an...
The lemma "Units_nonzero_Qp" assumes that "u" is an element of the unit group of p-adic numbers. It shows that "u" is also an element of the non-zero p-adic numbers.
The quotient of a ring R and an ideal I is also a ring.
The lemma named "ex_anga" assumes that angle ABC is acute. It shows that there exists an angle 'a' such that 'a' is congruent and acute to angle ABC.
Given the assumptions that "c" is a non-standard polynomial, "d" is a non-standard polynomial, and "Lt (CNP 0 a r)" is a linear polynomial, the interpretation function "Ifm" applied to the variables "vs", the list "x#bs", and the result of the function "msubstlt" applied to "c", "t", "d", "s", "a", and "r" is equivalen...
For any fixed numbers 'n' and 'n'', the action of 'c' at time 't' in the context of 'n' is less than or equal to 'n'.
: Assuming there does not exist a 'k' such that 'M', 'q1', and 'q2' are 'k'-distinguishable, and 'q1' and 'q2' are states in 'M', then for all sequences 'xs', if the set of 'xs' is a subset of the inputs of 'M', there exists a path 'p' such that 'p' is a path in the product of the FSMs derived from 'M' starting at 'q1...
For any lists L and R, the pair (True#p, q) is in the step of the concatenation of L and R with respect to a, if and only if either there exists a list r such that q equals True#r and the pair (p, r) is in the step of L with respect to a, or the final state of L is p and a is None and q equals False#start of R.
The valuation of a linear polynomial monomial, when applied to a variable 'v', is equal to the coefficient 'c' multiplied by the value of 'v' at 'x'.
"it is not the case that the social choice function for R5 is strictly preferred, according to the Fishburn preference relation on the set A1, to the social choice function for R21."
If set A is equivalent to the universal set, then the lens defined by the type definition is a bijective lens.
The subdegree of a formal Laurent series, represented by the absolute value function Abs_fls, with coefficients such that the coefficient is 'a' when the index n is 0 and 0 otherwise, is equal to 0.
The function "vsplit" applied to the arguments "c" and a pair "\<langle>a,b\<rangle>" is equal to the function "c" applied to the arguments "a" and "b".
The lemma named "convex_hull_convex_sum" assumes that set A is a subset of the vector carrier of dimension n. It also assumes that vectors x and y are in the convex hull of set A, and that a is a real number between 0 and 1 inclusive. Under these assumptions, the lemma shows that the vector resulting from the operation...
The lemma named "rm_constant_equations_dioph" is defined with fixed variables b, c, d, e, f, q. It defines "DR" as a constant relation with these variables. It then shows that this relation "DR" is a Diophantine relation.
If X is small, then the set of elements of the supremum of X is equal to the union of the set of elements of X.
A vertex 'v' is in the set 'Vd' of vertices at distance 'd' if and only if 'v' is connected to the source 'src' and the shortest path distance from 'src' to 'v' is 'd'.
The lemma named "radical_mult_distrib" is about formal power series in a field with characteristic 0. It takes a formal power series 'a' and assumes the following: 'k' is greater than 0, the k-th root of the 0-th coefficient of 'a' raised to the power of 'k' equals the 0-th coefficient of 'a', the k-th root of the 0-th...
If the transition from state (e, xs) to state (e', xs') is a silent (tau) reduction in a given thread t, heap h, and program P with an external action extTA, then the transition from state (V := e, xs) to state (V := e', xs') is also a silent (tau) reduction under the same conditions.
: 1. The essential norm of the function 'f' in the L0 space over a measure 'M' is 0 if 'f' is Borel measurable in 'M', otherwise it is infinity. 2. The defect of the L0 space over a measure 'M' is 1.
"a" is in the set of normal forms "NF" of "A" if and only if for all "b", the pair "(a, b)" is not in "A".
The lemma named "rule_sub_remark_8" assumes that for any variable "v", the assertion of a statement "phi" is equivalent to "phi" in "v". It also assumes that the necessity of the assertion of "phi" is in "v". Under these assumptions, it shows that the necessity of "phi" is in "v".
If for any subset A of the carrier L, there exists a least upper bound and a greatest lower bound, then L is a weak complete lattice.
The lemma named "inversion_id_iff" demonstrates that "a" is equal to its inversion if and only if "a" is not equal to infinity and the product of "a" (converted to a complex number) and the conjugate of "a" (also converted to a complex number) is equal to 1.
If a function 'f' maps from the image of 'h' over a set 'A' to a set 'M', and 'h' is an injective function on 'A', then the finite composition of 'f' with the image of 'h' over 'A' is equal to the finite composition of 'f' with 'h' over 'A'.
If an obligation from X to Y exists and X is a subset of Z, then there is an obligation from Z to the union of the intersection of Z and the complement of X, and Y.
If memory states s1 and s2 are equal at a particular physical address pa, then even after modifying the CPU register of s1 to new_cpu1 and s2 to cpu_reg2, the memory states remain equal at the same physical address pa.
The sum relation of the identity relation with itself is equal to the identity relation.
If the multiseries "xs" is not equal to the empty multiseries, then the leading exponent of the inverse of the multiseries "xs" is the negation of the leading exponent of the multiseries "xs".
The box star of "x" applied to "n(y)" is equal to "n(y)" multiplied by the box of "n(y)" multiplied by "x" applied to the box star of "x" applied to "n(y)".
If 'x' is an argument that minimizes the function 'f' over a set 'S', and if there is a 'y' in 'S' such that the function value of 'y' is less than or equal to the function value of 'x', then 'y' is also an argument that minimizes the function 'f' over the set 'S'.
The lemma part12_pred assumes that the set V, of any type 'a, is infinite. It shows that there exists a pair V1_V2 such that the predicate part12_pred holds for V and V1_V2.
If the well-formed program "P" with well-formed method bodies "wf_mb" is true, then the semi-lattice property holds for the JVM semi-type with parameters "P", "mxs", and "mxl".
The function "add_swap" with inputs m and n equals the sum of n and m.
The lemma "prv_sdsj_mono_imp" assumes that "F1" is a subset of "formula" and "F2" is also a subset of "formula", both "F1" and "F2" are finite. It also assumes that for every formula "phi1" in "F1", there exists a formula "phi2" in "F2" such that "phi1" implies "phi2" is provable. Under these assumptions, it shows that...
The lemma named "monom_valuate" assumes that "m" is a monomial. It shows that the valuation of "m" is equal to the coefficient of the monomial "m" multiplied by the value of the variable of the monomial "m" in the valuation "v".
A ring element 'r' is irreducible if the following conditions are met: 1. 'r' is a non-zero element of the ring 'R'. 2. 'r' is not a unit in 'R'. 3. For any two elements 'a' and 'b' in 'R', if 'r' is the product of 'a' and 'b', then either 'a' or 'b' must be a unit in 'R'.
If 'e' is a value, then 'e' is valid and the set of free variables in 'e' is empty.
: 1. There exists exactly one value 'v' such that the function U_b of 'm+1' applied to 'v' equals 'v'. 2. The sequence generated by repeatedly applying the function U_b of 'm+1' to 'v' converges to the unique value 'v' for which U_b of 'm+1' applied to 'v' equals 'v'.
The property "P is H healthy" is defined as "H applied to P equals P".
If a policy "p" is well-formed according to the first and third definitions of well-formed policies, all networks in "p" are distinct, and "p" only contains single combinators, then if the rule "C" applied in reverse to "p" results in denying all from "c" to "d", and if allowing a port from "a" to "b" with port "po" is...
The lemma named "cake_times_correct" assumes two conditions. First, the evaluation of 'u' in the environment 'env' and state 's' results in a new state 's'' and a return value of an integer literal 'y'. Second, the evaluation of 't' in the environment 'env' and state 's'' results in a new state 's''' and a return value...
If the length of array 'a' in heap 'h' is equal to the length of array 'a' in heap 'h'', then the subarray from index 'i' to 'j' of array 'a' in heap 'h' is equal to the subarray from index 'i' to 'j' of array 'a' in heap 'h'' if and only if for all 'i'' such that 'i'' is greater than or equal to 'i' and less than 'j',...
The set of backflows from the difference between set X and the backflows of set Y is equal to the difference between the backflows of set X and set Y.
If 'a' and 'b' are single combinators, the set of 'a' is equal to the set of 'b', and all networks in 'b' are distinct, then it logically follows that all networks in 'a' are also distinct.
For all elements in the universe, the universal quantification is equivalent to the assertion that a property holds for all elements.
If "C" is a category, then the identity functor of "C" is a functor.
The lemma named "nat_k_induct" is about a natural number 'k'. It assumes that a property 'P' holds for all natural numbers 'i' less than 'k'. It also assumes that for any given natural number 'n₀', if the property 'P' holds for all 'i' less than 'k' added to 'n₀', then the property 'P' also holds for 'k' added to 'n₀'....
Set A is a subset of set B if and only if for all elements a1 and a2, if they are in set A then they are also in set B.
The product of matrix "ZmX" with itself equals the 2x2 identity matrix.
If "k" is greater than 0, then the degree of polynomial "p" with respect to "x" is greater than or equal to "k" if and only if there exists a monomial "m" in the monomials of "p" such that the lookup of "m" with respect to "x" is greater than or equal to "k".
Given an acyclic relation R, the following process is defined: 1. Set W to 0. 2. Set q to the point chosen by the function "choose_point" which takes the minimum value of R and 1 as arguments. 3. Set v to q. 4. While v is not equal to 1, maintain the invariant that the topological sort of q, v, R, and W is valid. 5. D...
The set of vertices of a given face (f) is equal to the union of the set of all first elements (a) of each pair (a, b) in the set of edges of the face (f).
Flipping the order of arguments x and y in function f is equivalent to applying function f to y and x.
If two functions 'f' and 'g' are continuous at a point 'F', then the cross product of these two functions 'f' and 'g' is also continuous at the same point 'F'.
For any two extended non-negative real numbers 'a' and 'b', if 'b' is not equal to zero and 'b' is not equal to infinity, then the result of multiplying 'a' and 'b' and then dividing by 'b' is equal to 'a'.
For any streams 'sg', the result of unstreaming the transformation of 'g' and 'h' streams with only 'g' stream present (Left_only sg) is equivalent to the result of unstreaming 'g' with 'sg'.
If "Y" is a chain, then the function that maps each index "i" to the greatest lower bound of the set of all "z" in "A" mapped to "Y i z" is also a chain.
The function T_delete2, when applied to an element x and a tuple (t,n,dl), is equal to the sum of the function T_del applied to x and t, and a case expression. This case expression checks the result of the function del applied to x and t. If the result is None, it returns 1. If the result is Some t', it defines n' as n...
The lemma for the upper bound of the sum of two floating point intervals is equal to the sum of the upper bounds of the two intervals, rounded up to the nearest floating point number.
For a function 'f' mapping from a complete lattice to itself, if 'f' is a closure operator, then the set of fixed points of 'f' is supremum-closed.
The lemma consists of two parts: 1. The function "map_generat" applied to "generat" equals "IO out c" if and only if there exist "out'" and "c'" such that "generat" equals "IO out' c'" and "out" equals the application of "g" to "out'" and "c" equals the application of "h" to "c'". 2. "IO out c" equals the function "m...
For any element 'x' of a type 'a', which is a field and a normalization semidomain, the unit factor of 'x' is 'x' itself.
If set A is finite, then the set probability mass function of the random binary search tree of A is a subset of the trees of A.
The lemma, in the context of encoding, is about combining conditions on sets of relations. It fixes two sets of relations, Rel and RelA, and two conditions, Cond and CondA, on pairs of processes. It assumes that for every pair of processes in RelA, the condition CondA holds. It also assumes that the condition Cond hold...
The lemma named "sym_mod_sym_div" assumes that "y" is greater than zero. It shows that the symmetric modulo of "x" and "y" is equal to "x" minus the product of the symmetric division of "x" and "y", and "y".
The lemma named "delete_point_index_eq" assumes that the set "ps" is a subset of the set obtained by deleting the point "p". It shows that the index of the set "ps" in the set of blocks obtained by deleting the point "p" is equal to the index of the set "ps" in the original set of blocks.
The lemma "limit_subseteq_reach" assumes that the range of "w" is a subset of "Σ". It shows that the limit of the run of "δ" and "q₀" over "w" is a subset of the reach of "Σ", "δ", and "q₀". It also shows that the limit of the run of "δ" and "q₀" over "w" with respect to time is a subset of the reach of "Σ", "δ", and "...
If a tree "t" is a Braun tree and the size of the tree "t" is "n", then the list obtained by adding elements from list "xs" to the tree "t" at position "n" is equal to the concatenation of the list representation of the tree "t" and the list "xs".
The infimum (greatest lower bound) of 'x' and the negation of the infimum of 'x' and 'y' is equal to the infimum of 'x' and the negation of 'y'.
: - Given any i, t, and e, if the lowest tops of the list [i, t, e] is None and i equals Trueif, then P holds for i, t, and e. - Given any i, t, and e, if the lowest tops of the list [i, t, e] is None and i equals Falseif, then P holds for i, t, and e. - Given any i, t, e, and a, if for any value, P holds for the rest...
The labels of l, c, and v are finite.
The lemma named "mult_sv_mv" shows that the multiplication of a scalar "k" with the result of a matrix "A" multiplied by a vector "v" is equal to the result of the matrix "A" multiplied by the product of the scalar "k" and the vector "v".
The lemma "nth_root_nat_aux_gt" assumes that "m" is greater than 0. It shows that the "m"th power of the sum of the function "nth_root_nat_aux" with parameters "m", "k", "k" to the power of "m", and "n", plus 1, is greater than "n".
The lemma named "crit_pair_cbelow_distinct_component" assumes that the component of the leading term of polynomial p is not equal to the component of the leading term of polynomial q. Under this assumption, it shows that the critical pair criterion is below a certain level on a given order d, module m, set of polynomia...
If A is a monomial set, then if p is an element of A, p is a monomial.
If an auxiliary interference freedom condition holds for a given context, program, fault set, and triple of any annotation, postcondition, and abrupt completion, then the same auxiliary interference freedom condition also holds when the program is a while loop with a given condition, body, and annotated while loop with...
There is a constraint between function f and relation R.
The function "harm n - (ln n + euler_mascheroni + 1 / (2 * n))" is a member of the Big O notation of the function "1 / n ^ 2". This means that the growth rate of the function "harm n - (ln n + euler_mascheroni + 1 / (2 * n))" is at most as fast as the growth rate of the function "1 / n ^ 2".
The reverse of the binary list representation of the product of two words 'x' and 'y' is equal to the result of the multiplication operation 'rbl_mul' on the reverse of the binary list representations of 'x' and 'y'. This is valid for any word 'x' of any length.
If the order of the sub-probability mass functions "p" and "q" are equal, then the order of the restricted sub-probability mass functions "p" and "q" to a set "A" are also equal.
The lemma named "eq_\<I>_gpv_coinduct_bind" is about two generalized probabilistic values (gpv and gpv') of types "('a, 'out, 'in) gpv" and "('a', 'out, 'in) gpv" respectively. It assumes that a relation X holds between gpv and gpv'. Furthermore, it assumes that for any two generalized probabilistic values for which th...
, assuming "phi" is a proper function in the variables X, Y, and Z, then the application of the lambda function, which takes three arguments x, y, and z and returns the result of applying "phi" to these arguments, to the arguments x, y, and z, is equivalent to the result of applying "phi" to x, y, and z, in the context...
If the multithreaded reduction relation tau-mredT takes state s to state s', and thread t does not exist in state s, then the multithreaded reduction relation tau-mredT also takes the state with added thread t from state s to the state with added thread t in state s'. The state is represented as a tuple of locks, threa...