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If the evaluation of expression "e" is equal to the evaluation of expression "e'", then the evaluation of the lambda expression "ELam x e" is equal to the evaluation of the lambda expression "ELam x e'".
If the following conditions are met: 1. The precondition "wf_matchF_aux" holds for the given parameters, where "wf_matchF_aux" is a well-formedness condition for the auxiliary function in the matchF algorithm. 2. The well-formedness condition "wf_matchF_invar" holds for the given parameters, where "wf_matchF_invar" is...
The lemma named "itrm_brackets_Cons_SomeE" assumes that the function "itrm_brackets" applied to a list consisting of "v" followed by "vs" and "x" equals "Some x'". Under this assumption, it is possible to obtain a "y'" such that the function "itrm_brackets" applied to "vs" and "x" equals "Some y'" and the function "itr...
, assuming "pre_adjust" is applied to a node with left child 'l', value 'a', level 'lv', and right child 'r', then the level 'lv' is equal to the level of the adjusted node or the level of the adjusted node plus one.
The Gaussian function is measurable in the product of real numbers and is a sub-probability on the set of real numbers.
There is a relationship between the implementation of an NGBA (ngbai) and the specification of an NGBA (ngba). This relationship is defined over a set of lists of type L, a set of lists of type S, a function from L to S that returns a set of lists of type S, and a list of functions from S to a boolean relation. 2. The...
The negative of a vector 'v' is in the vector space of dimension 'n' if and only if the vector 'v' itself is in the vector space of dimension 'n'.
The set of deadends in a parity game p is a subset of the set V.
The lookup of a key 'k' in the ordered association list 'xs' with the exception of the minimum key 'ko' is equal to 0 if for all keys 'k'' in the first element of the set of the first element of the list of the ordered association list 'xs', 'ko' is less than or equal to 'k' and 'k'. Otherwise, it is equal to the looku...
The lemma named "alignUp_le_greater_al" is defined for a fixed 'x' which is a word of arbitrary length 'a'. It assumes three conditions: 'a' is less than or equal to 'x', 'n' is less than the length of 'a', and 'x' is aligned to 'n'. Under these assumptions, it shows that the result of aligning 'a' up to 'n' is less th...
In the context of quasi-Borel spaces, the lemma defines 'qbs_integrable' as follows: a function 'f' is integrable in the quasi-Borel probability space (X, α, μ) if and only if 'f' is a quasi-Borel measurable function from 'X' to the set of real numbers and the composition of 'f' and 'α' is integrable with respect to th...
The set of standard real numbers is a subset of the set of hyperfinite numbers.
If under the context of Gamma and Theta, the program P with annotation "anno" ensures the postcondition Q with the assertion A, and if command c is equivalent to the annotation "anno", then under the same context, the program P with command c also ensures the postcondition Q with the assertion A.
If 'c' is a polynomial coefficient in set 'S', then the pair consisting of the maximum 'c' in set 'S' and the second element of 'c' is also a polynomial coefficient in set 'S'.
The lemma named "a_star_redu_ImpL_elim" assumes that there is an implication relation "ImpL" between "<a>.M", "(y).N", and "z" that reduces to "R" under the "a-star" reduction. It shows that there exist "M'" and "N'" such that "R" is equal to the implication relation "ImpL" between "<a>.M'", "(y).N'", and "z", and "M" ...
The integer value of the word obtained by dropping 'n' bits from the word 'w' is equal to the result of dropping 'n' bits from the integer value of the word 'w'.
If 'm' is greater than 'n', then shifting the formal power series 'f' multiplied by 'X' to the power of 'm' by 'n' places is equal to 'f' multiplied by 'X' to the power of 'm - n'.
: Assuming that for any elements x2, u, and v, if x2 is in the set of elements of list xs, and both u and v are in x2, and u is in the languages of both M1 and M2, then u and v converge in both M1 and M2. Furthermore, if x2 is in the set of elements of the second element of the pair that is the second element of the...
If a well-formed program "S" with well-formed method bodies "wf_mb" exists, then the execution of "S" preserves the type. This is determined by the function "pres_type" which checks the preservation of types during execution. The parameters for the execution are "S", the maximum stack size "maxs", the return type "rT",...
If a new constant 'c' is in the conjunction of a list 'G', then 'c' is new for all elements in the list 'G'.
For any given number 'n', the following conditions hold: 1. 'n' is a proper multiplier. 2. The set of roots is a subset of the set of black elements in 'M'. 3. The set of old black elements is a subset of the set of black elements in 'Ma'. 4. The set of black elements in 'Ma' is a subset of the set of black elements i...
If the inorder traversal of a binary tree node (composed of a left subtree 'l', a value 'x', and a right subtree 'r') contains distinct elements, then the cost of this node is equal to the sum of the function 'p' applied to the set of elements in the left subtree, plus the cost of the left subtree, plus the function 'p...
The function phi does not consistently learn the set V01 in the scenario s.
If set A is uncountable, then set A is infinite.
The representation of the product of a monomial with coefficient 'c' and degree 0, and another element 'r', is equal to 'c' times the representation of 'r'.
For all sigma, the function "subst_list1" applied to sigma and a list l is equal to the function "map" applied to the function "substs" with sigma and the list l.
If 'y' is a partial function, then the composition of 'x' and 'z' followed by the converse of 'y', and then followed by 'y' itself, is equal to the composition of 'x' and 'y' followed by 'z'.
If 'x' is greater than or equal to 0 and less than or equal to pi, then the sine of 'x' is greater than or equal to 0.
The negation of the supremum (or least upper bound) of a set X is equal to the infimum (or greatest lower bound) of the negation of each element in set X.
The lemma named "sum_unique_nonzero" is defined with respect to a fixed variable 'i' of any finite type 'sv' and a function 'f' that maps from 'sv' to real numbers. It assumes that for any 'j' in the universal set of 'sv', if 'j' is not equal to 'i', then the function 'f' applied to 'j' equals zero. Under these conditi...
If a program P concatenated with a command c and another program P' can transition from the state (size P + i, s) to the state (j, s'), given that i is greater than or equal to 0 and less than the size of the command c, then the command c can transition from the state (i, s) to the state (j - size P, s').
: given the assumption that a function F is a 3-tuple consisting of h, g, and f, and this 3-tuple belongs to the set of arrows of the product of three categories A, B, and C, it can be shown that the arrow map of the 12th categorical product of A, B, and C applied to F equals a 2-tuple consisting of h and a 2-tuple of ...
: Assuming that "as" is a path on the same level as "cs" and the length of "cs" is equal to the length of "cfsx", we can obtain "cfx" and "cfsx'" such that the transfer of "as" kinds to the concatenation of "cfsx", "cf", and "cfs" equals the concatenation of "cfsx'", "cfx", and "cfs". Similarly, the transfer of "as" ki...
The set S is empty if and only if the affine dimension of S is -1.
The disjunction (logical "or") operation is commutative for predicates P and Q. That is, "P or Q" is equivalent to "Q or P".
The lemma named "pi_approx'" establishes a condition for an approximation of pi. It fixes a natural number 'n' and assumes that 'n' is even and greater than 0. It also assumes that 'k' is less than or equal to 13 times 'n' minus 2. Under these conditions, it shows that the absolute difference between pi and the approxi...
If the length of a list 'xs' is 'n' and all elements of 'xs' are contained within set 'A', then 'xs' is a list of length 'n' with all elements from set 'A'.
If a list S is not empty, sorted, and contains distinct elements, then the cardinality (or size) of the set of pairs (x, y) where x and y are elements of S and x is less than y, is equal to the product of the length of S and one less than the length of S, divided by 2.
The set of the union of two lists, xs and ys, is equal to the union of the set of xs and the set of ys.
Lemma fls_left_inverse_delta: Let b be an element of a type 'a, which is an abelian group with addition and multiplication by zero. Assume that b is not equal to 0. Then, the left inverse of the formal Laurent series with coefficients given by the function (\<lambda>n. if n=a then b else 0) at point x is equal to...
"lss_iteratei" is a proper iterator according to the "proper_it'" property.
The conjunction (logical AND) of two conditional expressions, where each expression is either P1 or Q1 depending on the boolean value b, and P2 or Q2 depending on the same boolean value b, is equivalent to a new conditional expression where the condition is still b, but the true case is the conjunction of P1 and P2, an...
The function 't' applied to the product of 'n x' and 'n y' is equal to the product of 'n x' and 'n y'. This implies that the function 't' does not change the product of 'n x' and 'n y'.
For any state 'tau', if 'X' at state 'tau' is not equal to 'true' at state 'tau', then the logical OR operation between 'bot' and 'X' at state 'tau' is equal to 'bot' at state 'tau'.
If "pre_between" is true for vertices "vs" and "ram1" and "ram2", then "pre_between" will also be true when the vertices "vs" are rotated by "n" positions, with "ram1" and "ram2" remaining the same.
The index 'i' of the vector implementation 'v' is equal to the implementation of the vector index of 'v' at 'i'.
Taking the first 'n' elements from the concatenation of stream 'u' and 's' is equivalent to taking the minimum of 'n' and the length of 'u' from 'u', and then appending it with the result of taking 'n' minus the length of 'u' from 's'.
In the context of an infinite continuous-time filtration, the lemma named "proj_stoch_range_singleton" is established. It fixes X as a function from natural numbers to a stream of boolean values, mapped to a second topological space ('b). The lemma assumes that X is a Borel-adapted stochastic process with respect to fi...
The function "x por y" is continuous with respect to x.
There exists an element 'x' in the set 'l' for which the property 'P' holds if and only if the recursive function 'list_ex_rec' applied to 'P' and 'l' is true.
In the context of the Congruence Rule, there is a lemma called "Angle Greater than transitive Greater than Greater than". This assumes that the angles formed by points p11, p12, p13; p21, p22, p23; and p31, p32, p33 are all defined. It also assumes that the geometric set of the first angle added to an empty set is grea...
The function "Wrapp" is injective.
The lemma named "not_vset_not_disj_eff_prod_dom_diff" is defined with fixed variables PROB, a, and vs. It assumes that "a" is an element of "PROB" and that the action "a" is not a variable set action with respect to "vs". Under these assumptions, it shows that the intersection of the domain of the second element of "a"...
"TUM" department with no sub-departments is a valid leaf in the positive hierarchy.
If a list 'l' is not empty, then the minimum value of the list 'l' is equal to the minimum value of the set of 'l'.
A hypernatural number derived from a natural number 'n' is less than or equal to zero if and only if 'n' is zero.
, within a given set A, at a certain point 'a', the function that maps 'x' to 'x' will eventually be non-zero.
: - If tree t1 is in the set of balanced plus trees of height h and tree t2 is in the set of trees of height h, then the tree resulting from the operation n2 on t1, a, and t2 is in the set of balanced plus trees of height h+1. - If tree t1 is in the set of trees of height h and tree t2 is in the set of balanced plus tr...
If "m" is less than or equal to "n", then the odd psi function of "m" is less than or equal to the odd psi function of "n".
For a fixed element 'x' of type 'a' in a complete partial order (cpo) with a join operation, if 'x' is an element of a set 'S', then 'x' is less than or equal to the least upper bound (lub) of 'S'.
If a pointer is of the type "element pointer", then it is also of the kind "element pointer kind".
If set A and set B are both finite, then the fold of function f over the union of set A and set B with a starting value x is equal to the fold of function f over set B, starting with the fold of function f over set A with a starting value x.
The element at the position (i, j) in the matrix A, when converted to an integer array using the function "matrix_to_iarray", is the same as the element at the position (i, j) in the original matrix A.
The lemma named "sum_setv_subset_eq" assumes that "A" is a finite set and the set of vector "v" is a subset of "A". It shows that the sum of the count of vector "v" for each element in the set of vector "v" is equal to the sum of the count of vector "v" for each element in set "A".
If "d" and "e" are greater than or equal to zero, and if there is a transition from state "l" with clock valuation "u" to state "l'" with clock valuation "u'" after time "d" in automaton "A", and if there is a transition from state "l'" with clock valuation "u'" to state "l''" with clock valuation "u''" after time "e" ...
The lemma named "con_char_CC'" shows that a transition system T is connected to a transition system U if and only if both T and U are arrays and for all transition systems T and U, if T is in the set of T and U is in the set of U, then T is connected to U.
The arrival message for any index "i" with the truth value "TT" is equal to "TT".
The lemma named "flop_adj" assumes that "C" and "D" are chambers and "C" is adjacent to "D". It shows that the "flop" of "C" is adjacent to the "flop" of "D".
The lemma named "fds_converges_cong" assumes that for sufficiently large values of n, the nth term of the formal Dirichlet series f is equal to the nth term of the formal Dirichlet series g, and that s is equal to s'. Under these assumptions, it shows that the formal Dirichlet series f converges at s if and only if the...
The "mon_pl" of any "w" is equal to the union of the first and second elements of "e", for all "e" that belong to the set "w".
An element 'x' is in the set of arrows of the parallel digraph 'dg_Par' parameterized by 'α' if and only if 'x' is an arrow in the parallel arrangement 'arr_Par' parameterized by 'α'.
If the function 'fine' with parameters 'delta', 'a', and 'b' holds true for 'D', then the sum of the differences 'f v - f u' for each tuple '(u, x, v)' in 'D' is equal to the difference 'f b - f a'.
The lemma named "CondInv'" makes several assumptions and provides a conclusion. The assumptions are as follows: 1. The set P is a subset of I. 2. The set I is a subset of the set of all states s such that if s is in b, then s is in P1, and if s is not in b, then s is in P2. 3. The set I is a subset of Q. 4. Under the ...
If you have three scenes, S1, S2, and S3, and you override S1 with S2 on a set X, and then override the result with S3 on the same set X, it is equivalent to just overriding S1 with S3 on set X.
The lengths of the elements in the list obtained by applying a function 'f' to each element in list 'xs' and then selecting elements from a set 'A', is equal to the list obtained by first selecting elements from set 'A' in list 'xs' and then applying the function 'f' to each element.
For any element 'v' in set 'V', there exists a region 'R' in the set of all regions 'R' such that 'v' is an element of 'R'.
The zero element of the image ring of a function f from a ring R is equal to the function f applied to the zero element of the ring R.
If P is a free module over the ring R with respect to the modules M and N, then P is a generator of the direct product of M and N over the ring R.
Running the "return" function with a certain value "x" in the context of an option monad, is equivalent to returning an option with that value "x" present. This is applicable for any value "x" of any type 'a.
The lemma named "range_mset_map" assumes that the multiset Ds is a subset of the range of a function f. It shows that Ds is in the range of a function that takes a variable Cl and returns a multiset of the map of function f over Cl.
The join of the irreducibles of the meet of 'x' and 'y' is equal to the intersection of the join of the irreducibles of 'x' and 'y'.
Theorem Hyp2 Axiom 9: "For all points p, q, a, b, c, if p is not equal to q and the distance from a to p is congruent to the distance from a to q, and the distance from b to p is congruent to the distance from b to q, and the distance from c to p is congruent to the distance from c to q, then either a, b, c are colline...
The division of x by y, after being extended by a, is equal to the extension of x by a divided by the extension of y by a.
The measure of a list created by replicating 'False' 'n' times and then appending 'True' at the end, when evaluated by the function 'harry_meas', is equal to '2n + 1'.
The lemma named "finite_config_BIT" assumes that the "init" is distinct. It shows that the set of probability mass functions (pmf) of the random configuration (which is initialized by BIT_init and stepped through by BIT_step) with respect to the initial values and queries is finite. This is denoted as "finite ?D".
The cardinality (or the number of elements) of the set obtained by mapping a function to a set is equal to the cardinality of the domain of the function.
If it is necessarily the case that either proposition phi or proposition psi is true, then it is necessarily the case that either phi or psi is true in all possible worlds.
The lemma ex1_pole shows that there exists exactly one z such that z is a pole of M.
The lemma named "Upsilon_cong_aux" assumes that for every pair (t,n) in the set U, the number t is not dependent on any variable (numbound0 t) and n is greater than 0. It shows that the set of all pairs ((t,n),(s,m)) in U, where each pair is transformed into ((m*t + n*s) / (2*n*m)), is equal to the set of all pairs ((t...
If the function 'g' is isotone (meaning it preserves the order), and if the composition of functions 'f' and 'g' has a greatest fixpoint, and if the composition of 'f' and 'g' is equal to the composition of 'g' and 'f', then 'g' is a fixpoint of the least fixpoint of the composition of 'f' and 'g'.
The set of elements obtained from an in-order traversal of a tree 't' is equal to the set of relations in 't'.
If a list "xss" is a prefix of another list "yss", then the concatenation of "xss" is also a prefix of the concatenation of "yss".
The recursive evaluation of the function "rec_max1" with inputs "x" and "y" is equal to the bounded maximum recursive function of "u" such that the recursive evaluation of function "f" with inputs "u" and "y" equals zero, where "u" is bounded by "x".
The lemma named "fresh_fact" is defined with a fixed variable "x" of type "name". It assumes that "x" is fresh for "v" and "x" is fresh for "t". It then shows that "x" is fresh for the term "t" where "y" is substituted by "v".
A state 's' is in the set of states resulting from the 'poke' function applied to state 's' and event 'x' if and only if one of the following conditions is met: 1. There exists a kernel state 's_kernel' and a set of parties such that the initial state 's' is a pair consisting of 's_kernel' and the set of parties, the ...
If node 'u' is directly reachable from node 'v', then 'u' is also reachable from 'v' through a path of zero or more edges.
If set A is equal to set B, then set A is a subset of set B.
If a natural transformation "N" is a tiny isomorphism from functor "F" to functor "G" in the context of tiny categories "A" and "B" with a fixed universe "alpha", then we can conclude three things: 1. The natural transformation "N" is an isomorphism from functor "F" to functor "G" in the context of categories "A" and ...
If the weakest precondition of command "c" and postcondition "Q" holds in state "s", then the weakest liberal precondition of command "c" and postcondition "Q" also holds in state "s".
The lemma named "degree_0_id" assumes that the degree of a polynomial 'p' is 0. It shows that the coefficient of 'p' at 0 is equal to 'p' itself.