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The lemma named "pd_antisym" assumes two conditions: "x pd→ y" and "y pd→ x". If these two conditions are met, it shows that "x = y".
If 'b' is in the finite set resulting from the function 'child2' applied to 'a', 'ys', and 'xs', and 'b' is not in the finite set 'ys', then there exists 'rs' and 'e' such that the pair (Node 'a' 'rs', 'e') is in the finite set 'xs' and 'b' is in the finite set 'rs'.
The bisimilarity relation is reflexive.
The lemma "sorted_wrt_dist_last_upd_nbors_mono" assumes that the list "ps" is sorted with respect to the distance from a point "p", the number "n" is less than or equal to the length of "ps", and "n" is greater than 0. It shows that the distance from "p" to the last element of the updated neighbors list (obtained by up...
For a function 'f' which maps from a complete lattice with dual to another complete lattice with dual, if there exists a function 'g' such that 'f' is left adjoint to 'g', then 'f' preserves the supremum.
If for all 'n' in the free variables of formula 'phi', the function 'sigma' at 'n' equals the function 'sigma prime' at 'n', then the satisfaction of 'phi' under interpretation 'I' and function 'sigma' is equivalent to the satisfaction of 'phi' under the same interpretation 'I' and function 'sigma prime'.
The optional value "The_opt p" is equal to "Some x" if and only if "p x" is true and for all "y", if "p y" is true then "x" is equal to "y". This is denoted as "?lhs = ?rhs".
The lemma belongs to the context of a terminal category cocone. It introduces category theory and small category theory. It assumes that alpha prime is equal to alpha, c prime is equal to c, J prime is equal to J, C prime is equal to C, and F prime is equal to F. It shows that N is a terminal morphism cocone from F pri...
If function 'f' is a bijective function between set 'A' and set 'B', then the multiset image of 'f' applied to the multiset of set 'A' is equal to the multiset of set 'B'.
The flip operation permutes the set containing 0 and 1.
The lemma "STCal_divergent" establishes two things for any given processes 'S' and 'T'. First, it shows that if the source term 'S' diverges in the source-target calculus (STCal), then 'S' also diverges in the source calculus. Second, it shows that if the target term 'T' diverges in the source-target calculus (STCa...
The transpose of the complement of x is equal to the complement of the transpose of x.
The function 'cast from shadow root pointer to document pointer' is injective. In other words, if the function applied to 'x' equals the function applied to 'y', then 'x' must be equal to 'y'.
If a function 'f' belongs to the induced space of a group 'G', a finite-dimensional Hilbert space 'FH', and a vector space 'V', then 'f' is also a member of the intersection of the set of additive functions over 'FH' and 'V', and the set of scalar multiplication functions over 'G' and 'FH'.
The set generated by the function "pivot_positions_main_gen" with parameters ze, A, nr, nc, i, j is a subset of the Cartesian product of the set of all integers from 0 (inclusive) to nr (exclusive) and the set of all integers from 0 (inclusive) to nc (exclusive).
The lemma named "AbGroup_inner_dirsum_el_decomp_ex1_double" assumes that "G" and "H" are Abelian groups and that they are additively independent. If "x" is an element of the direct sum of "G" and "H", it shows that there exists a unique pair "gh" such that the first element of the pair is in "G", the second element is ...
"if it is necessary that a proposition phi is actual, then it is necessary that phi is actual in the world v."
"if F is always true, then F is true initially."
Given two assumptions: 1) there exists a reflexive transitive closure between x and y, and 2) there is no reflexive transitive closure between z and y, it can be shown that there exists a reflexive transitive closure between x and y in the restricted set of R, where the restriction is the set difference of R and the...
If 'f' is a bounded linear function, then the operator norm of 'f' is less than or equal to the sum of the norms of 'f' applied to each element in the basis.
The orbit of a function 'f' at a point 'a' is equal to the set of all images of 'a' under the function 'f' raised to the power of 'n', where 'n' is less than the order of 'f' at 'a'. This is denoted as "?A".
The function "paths" in the context of non-deterministic Büchi automata (nba) is a parametricity relation. Specifically, it states that a pair of "paths" functions (from two possibly different non-deterministic Büchi automata) is included in the relation from the non-deterministic Büchi automata relation over the label...
The lemma "types_exist_lfilled" assumes that "lfilled" is a well-formed sequence of instructions nested within "k" layers of label nodes, and that "lfilled" is type correct under the context "\<S>\<bullet>\<C>", transforming a stack with types "ts" to a stack with types "ts'". It shows that there exist some types "t1s"...
"If G is a well-formed graph and for all vertices v in the nodes of G, the property nP of v is greater than or equal to the cardinality of the nodes of G, then the security invariant sinvar holds for the graph G and the property nP."
The support of the negation of a formula is equal to the support of the formula itself.
The function "faceListAt_len" holds true for the seed "p".
A pair of equality relations is in the function space from K to K to boolean relation if and only if K is single-valued and the inverse of K is also single-valued.
If there is not a cycle-free path from m to n, then there exists an i less than or equal to n such that the Floyd-Warshall algorithm applied to m, n, n, n, n, i, i is less than one.
If the following conditions are met: 1. 'l' and 'u' form a Galois connection, 2. 'g' and 'h' are isotone functions, 3. 'g' has a least prefixpoint, 4. 'h' has a least fixpoint, 5. the result of applying 'l' to the result of applying 'g' to the result of applying 'u' to the least fixpoint of 'h' is less than or equal t...
The function "fetch" with parameters p, n, and b is defined as follows: First, we define a variable "len" as the length of p. Then, if n equals 0, the function returns a pair (Nop, 0). Otherwise, if b equals Bk, then if the length of p is less than or equal to 2n - 2, the function returns (Nop,0). If not, it returns...
An element 'x' is in the set difference of 'A' and 'B' if and only if 'x' is in 'A' and 'x' is not in 'B'.
If a plain float arithmetic operation is performed on 'N' elements of 'fa', and 'N' is less than or equal to the length of 'XS', and for all 'i' less than 'N', the 'i'th element of 'XS' is not 'None', then the approximation of 'fa' with precision 'p' and 'XS' as the input is not 'None'.
The lemma named "fmrel_to_rtrancl" assumes two conditions: 1. "r" is a reflexive relation (as stated by "reflp r"). 2. There exists a transitive closure of the relation "r" between the finite maps "xm" and "ym" (as stated by "fmrel r\<^sup>*\<^sup>* xm ym"). If these two conditions are met, it shows that there exists...
The lemma named "add_block_index_in" assumes that "ps" is a subset of "b". It shows that the index of "ps" in the added block "b" is equal to the index of "ps" in the block set plus one.
A function that takes any state 's' and always returns 'P' is equivalent to 'P' itself.
If a function, which takes two arguments 'a' and 'b' and returns either 'Q a b' if 'P a b' is true or 'R a b' if 'P a b' is false, is commutative, and 'P x z' is not true, then applying the fold function to this function, with initial value 'z' and a singleton set containing 'x', will result in 'R x z'.
Applying the closure operator twice is the same as applying it once.
In a program, if there is a call edge from node 'n' to node 'n'' with procedure 'p', edge summary 'es', and return edges 'rets', and there is an intra-procedural edge from node 'n''' to node 'n'', then this situation is impossible.
The lemma named "subseteq_Upto_Star" shows that the lexical value of "Upto r n" of string "s" is a subset of the lexical value of "Star r" of string "s".
This lemma, named "update_simps", is fixing several variables: r, s, nrt, nr, nr', ns, rt, and ip. It also defines four terms: "s" is defined as the route of rt and ip, "nr" is defined as the addition of prefix r to the seventh element of s, "nr'" is defined as a tuple consisting of the second element of s and the thir...
If a binary decision diagram (bdd) is defined by the triple (i,t,e) and equals to p, then looking up the triple (i, t, e) in the decision diagram's implementation (dcli bddi) will yield a result (pi) that equals to p. This is true under the precondition that the binary decision diagram implementation (bddi) is a valid ...
For all x of type 'a lambda, the property P holds for x.
If the following conditions are met: 1) A is a unary normal form (UNC), 2) a is equal to b implies P, 3) a is not in the normal form of A implies P, 4) b is not in the normal form of A implies P, 5) the pair (a, b) is not in the reflexive transitive closure of A implies P, then P is true.
The second element of a pair is a bounded linear function.
Zero in the context of real numbers is not equal to null, which is always false.
If a function 'f' tends towards a limit 'l' as 'F' approaches infinity, then the function that applies the extended nonnegative real number function 'e2ennreal' to 'f' also tends towards the extended nonnegative real number of 'l' as 'F' approaches infinity. This is under the assumption that 'f' indeed tends towards 'l...
The lemma named "R_divergent_divergent_aux" is given a stream of values 'xs'. It assumes that 'xs' is a stream of transitions and all elements of 'xs' belong to the set 'V'. It also assumes that the 'i'th element of 'xs' for 'c1' is 0 and there exists an index 'k' greater than 'i' and less than or equal to 'j' such tha...
The function "H2" applied to the result of the function "CSP1" with input "P" is equal to the function "CSP1" applied to the result of the function "H2" with input "P".
If the intersection of "m2_inv3_extrKey" and "m2_inv3a_sesK_compr" holds before the transition "m2", then "m2_inv3_extrKey" also holds after the transition.
The lemma named "restrict_lub_spmf" assumes that "Y" is a chain with respect to the order "ord_spmf (=)". It shows that the restriction of the least upper bound of "Y" to the set "A" is equal to the least upper bound of the set obtained by restricting each element of "Y" to "A".
"In the next 10 time units, the value of time 't' plus 10 is greater than 10."
If 'x' is less than or equal to the successor of 'n', then 'x' minus the successor of 0 is in the set of 'i' such that 'i' is less than or equal to 'n'.
The lemma "transversal_segment_exists" assumes that "x" is an element of "X" and "f x" is not equal to zero. It then obtains "a" and "b" such that "x" is an element of the open interval from "a" to "b" and this interval is a transversal segment.
For any matrices A, B, and C, where A and B are of the same size and C has the same number of rows as A and B have columns, the product of the difference of A and B with C is equal to the difference of the product of A and C and the product of B and C. This is under the assumption that the elements of the matrices are ...
If the absolute value of x and y are less than or equal to 1, then the arccosine of x is less than the arccosine of y if and only if y is less than x.
The validity of the hierarchy position for the department "TUM" with no sub-departments is false when "TUM" is connected to "facilityManagement" at the leaf level.
The lemma named "rep_tensor" assumes that "f" and "g" are arrows. It shows that the representation of the tensor product of "f" and "g" is equal to the tensor product of the representations of "f" and "g".
If a variable 'x' does not affect a predicate 'P', then the composition of 'x' and any other variable 'y' also does not affect the predicate 'P'.
If an extended finite state machine 'e' is not deterministic, then there exist states 's', labels 'l', inputs 'i', and outputs 'r', and there exist transitions 'd1', 'd2', 't1', and 't2', such that either 'd1' is not equal to 'd2' or 't1' is not equal to 't2', and the transitions from 's' to 'd1' and 't1' are in 'e', a...
If 'a' and 'b' are elements of a set 'U', and the sum of 'a' and 'b' equals zero, then the additive inverse of 'a' is equal to 'b'.
If the list "a#l" is an interleaving of lists "l1" and "l2", then there exists a list "lh" such that either "l1" is the list "a#lh" and "l" is an interleaving of "lh" and "l2", or "l2" is the list "a#lh" and "l" is an interleaving of "l1" and "lh".
The lemma named "preserves_cones" is defined with a fixed component 'J'. It assumes that 'a' is a cone from 'A' to 'D' over 'J'. It then shows that 'F a' is also a cone from 'B' to 'F o D' over 'J' with 'F o \<chi>' as the cone's morphism.
If the encryption keys are clean for the union of sets G and H, then the analysis of the union of the implementation of the authorization set G and set H is a subset of the synthesis of the analysis of the union of sets G and H, union the complement of the payload.
The lemma named "ln_upper_7_neg" assumes that "x" is greater than 0 and "x" is less than or equal to 1. It shows that the natural logarithm of "x" is less than or equal to the function "ln_upper_7" of "x".
If a filter F is ultra, then weak truth negation duality holds for negation.
If "x" is in "tfin", then there exists a "j" such that "τ" of "s" and "j" is not less than or equal to "τ" of "s" and "i" plus "x".
If for every pattern 'p' in the set of patterns 'ps' and every value 'v' in the set of values 'vs', when all elements in the environment 'env' are cupcake values, the function 'if_match' applied to the function 'cupcake_pmatch' with parameters 'cenv', 'p', 'v', and 'env' preserves the property of being a cupcake value,...
If there exists an element 'w' such that 'w' is a part of 'x', 'y', and 'z', then for all elements 'w', 'w' is a part of the product of 'x' and 'y', and then 'z', if and only if 'w' is a part of 'x', 'y', and 'z'.
In an abelian group G, if x and y are elements of the group G and the difference of x and y equals the identity element of the group, then x equals y.
A sequence is free of ticks (represented as "tickFree(s@t)") if and only if both the individual sequences 's' and 't' are free of ticks (represented as "tickFree s" and "tickFree t").
The lemma "lub_upper" assumes that "S" is a subset of "A" and "x" is an element of "S". It shows that the pair "(x, lub S cl)" is an element of "r".
The formal power series of the Lambert W function is equal to the formal power series of X times the composition of the formal power series of the exponential function at -1 and the formal power series of the Lambert W function.
If "k" is greater than 0, then the equality of the "i"-th expansion of "f'" by "k" to "f" is equivalent to the condition that for all "i", "f" at "i" equals "f'" at "i" divided by "k" if "i" modulo "k" equals 0, otherwise it equals "NoMsg".
The result of the function "T_deq" applied to the pair (fs,rs), plus the function "Phi" applied to the result of the function "deq" applied to the pair (fs,rs), minus the function "Phi" applied to the pair (fs,rs), is less than or equal to 3.
The supremum (or greatest element) of the range of a function 'p' (which maps from type 'W' to a type 'a' that forms a complete distributive lattice) when asserted is equal to the supremum of the range of the function 'p' composed with the assert function.
If a transition system "TS" is in the set of Hierarchical Protocol Transition systems of a statechart "ST", and if a transition "t" is in "TS", and the source of "t" is in the states of a statechart "A", and "A" is in the set of Statechart Assemblies of the Hierarchical Automaton of "ST", then "t" is in the transition ...
The z-th omega syntactic ordinal is equal to the z-th hereditary multiset of omega.
If alpha prime is equal to alpha, G prime is equal to G, K prime is equal to K, T prime is equal to T, H prime is equal to H, B prime is equal to B, A prime is equal to A, C prime is equal to C, and D prime is equal to D, then epsilon maps from the composition of G prime and K prime to T prime in the category of right ...
The lemma "gauss_int_norm_mod_less" assumes that "b" is not equal to zero. It shows that the Gaussian integer norm of the modulus of "a" and "b" is less than the Gaussian integer norm of "b".
The product topology of a set T and an empty set is equal to the discrete topology of a set that contains only a function that is undefined for all inputs x.
If the following conditions are met: 1. The product of 'x' and the Kleene plus of 'z' and 'y' equals the bottom element, 2. The product of 'y' and the Kleene plus of 'z' and 'y' equals the bottom element, 3. The product of 'z' and 'x' equals the bottom element, then the Kleene star of the join of the Kleene star of 'x...
The complex conjugate span of set A union the complex conjugate span of set B is equal to the complex conjugate span of the union of set A and set B.
If the length of the list 'xs' is 'n' and 'ft' is the maximum of 'n+3' and the maximum of 'fft' and 'gft', then for any list 'nl' that can be formed by appending 'xs', a single 0, 'ft - n' number of 0s, and any other elements, applying the function 'mv_box' with parameters 'n' and 'ft' to 'nl' will result in a list tha...
The lemma named "sandwitch_real" is defined with two fixed functions 'l' and 'r' that map from natural numbers to real numbers. It assumes that both 'l' and 'r' converge to 'a' in the limit. It also assumes that for all 'i', 'l' of 'i' is less than or equal to 'm' of 'i', and 'm' of 'i' is less than or equal to 'r' of ...
The lemma named "lift_hom" in the context of a binary operation set group, fixes a function 'T' that maps from type 'a' to type 'b' in an additive group. It assumes that for all elements 'g' and 'h' in the group 'G', the function 'T' applied to the binary operation of 'g' and 'h' equals the sum of the function 'T' appl...
The lemma named "confluentpE" is given two assumptions: that "r" is confluent and that "r" can reach both "y1" and "y2" from "x". Under these assumptions, it is guaranteed that there exists a "z" such that "r" can reach "z" from both "y1" and "y2".
If "s" is an operator on "X", then the well-sorted input of the first and second elements of the operation "castOp" applied to "s" and "X" holds true.
For any hereditarily finite set 'x', the difference between zero and 'x' is always zero.
Lemma: Measure Equality for Countable Sets Assumptions: 1. The sets M and N have the same power set of A. 2. The set A is countable. 3. For every element 'a' in set A, the measure of the singleton set {a} in M is equal to the measure of the singleton set {a} in N. Conclusion: Under these assumptions, the measure M is...
If 'p' is a single combinator, then the removal of duplicates from 'p' is also a single combinator.
"if it is necessarily the case that either proposition phi or proposition psi is true, then it is necessarily the case that phi is true or it is possibly the case that psi is true, in any possible world v."
The lemma named "dom_MkArr" assumes that "MkArr A B F" is an arrow. It shows that the domain of "MkArr A B F" is "MkIde A".
The binary decision tree (bdt) of a tree 't' and a variable 'var' equals 'Some Zero' if and only if there exists a path 'p' such that 't' equals a node with 'Tip' as both its left and right children and the variable of the path 'p' equals zero.
If the function "sum_list" is injective on the set of all lists "ss", and the image of the set of all lists "ss" under the function "sum_list" equals to "A", then "A" is a direct decomposition of "ss".
: Assuming that for every element 'a' in set 'A', the value of 'm' at 'a' is equal to the value of 'm'' at 'a', and that the outstanding references of volatile reads in 'sb' are a subset of 'A'. We could be even more restrictive here, only the non-volatile reads that are not buffered in 'sb' have to be the same. If 'm...
If a tree "t" is an AVL tree and is ordered, then the tree remains ordered after deleting an element "x" from it.
For every element 'x' in the set 'xs', if 'x' is less than the length of 'as', then the function 'f' mapped onto the reverse of the map of 'as' at each 'x' in 'xs' is equal to the reverse of the map of the function 'f' mapped onto 'as' at each 'x' in 'xs'.
If a proposition phi holds for the derivative operator Der_4, then it also holds for the derivative operator Der_4e.
If there is an infinite number of elements 'x' for which the property 'P' holds, and for most elements 'x' the property 'Q' holds, then there is an infinite number of elements 'x' for which both properties 'P' and 'Q' hold.
The lemma named "p_pos" assumes that the pair (s, t) is in the reflexive transitive closure of the set formed by the union of the set_pmf D for all D in K s, where s is in S1. It also assumes that t is in S2. Under these assumptions, it shows that the probability of s is greater than 0.