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The function "mulextp" applied to two bags "xs" and "ys" with a function "f" is equivalent to the function "mul_ext" applied to the same function "f" and the single lists obtained from the implementation of "xs" and "ys" using the function "ass_list_to_single_list".
The lemma named "Units_invertible" is defined for a matrix A of type 'a in a semiring. It assumes that A is an element of the unit group of the ring of matrices of type 'a, size n, and base b. It then shows that the matrix A is invertible.
The weakest precondition of applying a function 'f' is sublinear.
If a set of formulas Γ implies a normal form, and for all formulas γ in the multiset of Γ, γ is in conjunctive normal form (CNF), then the union of the CNF of all formulas in the multiset of Γ entails a contradiction.
If a square relationship exists between relations R and S, and their reflexive transitive closures, then the reflexive transitive closures of R and S commute.
The synchronization of processes P and Q over a set A is equal to the union of two sets. The first set consists of pairs (s, R) such that there exist sequences t and u, and sets X and Y, where (t, X) is in the failures of P, (u, Y) is in the failures of Q, the sequence s interleaves with the pair (t, u) over the unio...
If the k+1-th power of u equals the k+1-th power of v, then u equals v.
The paths of the inserted element 'xs' in an empty set equals to the list containing 'xs' only.
The lemma named "short_ch_card_2" assumes that "f" is a chain ordered by "X". It shows that "X" is a short chain if and only if the cardinality of "X" is 2.
A constant function with value zero is equivalent to the zero element of the function ring.
Given a well-formed Java program "P", if a class "C" in "P" has a static method "M" with parameters "pns" and a body "body", and if the evaluation of "body" with initial heap "h", local variables "vs", and shared heap "sh" results in an exception "e'" with final heap "h'", local variables "l'", and shared heap "sh'", a...
: 1. For any state 'st', lower bound 'lb', and left value 'lv', the expression 'ex' evaluated with parameters 'ep', environment 'env', and code 'cd' in state 'st' is equal to the expression obtained by updating 'ex' evaluated with the same parameters 'ep', environment 'env', and code 'cd' in the same state 'st'. 2. For...
If the diamond of the product of "d t" and "x" is equal to zero, then "d p" is equal to the product of "d t" plus "d q" and the fbox of the product of "d t" and "x" applied to "p" if and only if "d p" is equal to the fbox of "while t do x od" applied to "q".
In the context of an infinite coin toss space, there is a lemma regarding natural filtration information. Given a function 'f' that maps a boolean stream to a type 'b' in a T0 space, if 'f' is borel measurable in the natural filtration 'n', then the composition of 'f' and the pseudo projection 'False' at 'n' is equal t...
If the deletion map of 'j' applied to two functions 'g' and 'g'' are equal, and 'i' is not equal to 'j', then the value of 'g' at 'i' is equal to the value of 'g'' at 'i'.
The lemma named "dioph_rhs_invariant" assumes that the absolute value of 'm' is less than or equal to the maximum of the absolute value of 'l' and the sum of the absolute values of each element in 'ks'. It shows that the absolute value of the result of the operation '(m - eval_dioph ks (map (lambda function that return...
If an integer n is greater than or equal to zero, then the p-adic order of p raised to the power of n is equal to n.
If in a graph G, vertex v is reachable from vertex w, and vertex u is not equal to either vertex v or vertex w, then vertex v is still reachable from vertex w even if vertex u is removed from the graph G.
The lemma named "in_set_permutationss" assumes that the list "xss" contains distinct elements. It shows that another list "yss" is a permutation of "xss" if and only if "yss" also contains distinct elements and the set of elements in "xss" is equal to the set of elements in "yss".
The greatest common divisor of the primitive parts of two polynomials 'f' and 'g' is equal to the normalized form of the primitive part of the greatest common divisor of 'f' and 'g'.
The arbitrary union of a set P relative to a set U is equal to the arbitrary union of the set P relative to U. This is denoted by "?lhs = ?rhs", where "?lhs" represents the left-hand side of the equation and "?rhs" represents the right-hand side of the equation.
: given the assumptions that a natural transformation arrow 'NN' is equal to 'NN'', the object map of a functor 'FF' at 'a' is equal to 'FFa', the object map of a functor 'GG' at 'b' is equal to 'GGb', 'NN' is a natural transformation from 'FF' to 'GG' in the category 'CC' with a morphism to the category of sets 'cat_S...
The function "jumpNestingOk" when applied to the parameters "jmps" and "In3 es" always returns true.
, assuming the determinant of a matrix M is not equal to zero, the congruence of M with the congruence of the inverse of M with any matrix H is equal to H.
The data sources at level 1 for sA31 are equal to the set containing sA22.
If we have two generic operations "eqa" and "eqb" that are equivalent to the equality operation (=) in the context of relations Ra and Rb respectively, then the product equality of "eqa" and "eqb" is equivalent to the equality operation (=) in the context of the product relation of Ra and Rb.
If 'a' is greater than zero, then the function that maps 'x' (a real number) to the natural logarithm of 'a*x' raised to the power of 'd' is of the same order of growth (Big Theta) as the function that maps 'x' to the natural logarithm of 'x' raised to the power of 'd'.
The lemma ecl_ord_trans demonstrates that the ecl_ord relation is transitive.
If the next element in the list xs from the element x is y, then one of the following cases must be true: 1. The element x is not in the set xs and y is equal to d. 2. The list xs is not empty, x is the last element in xs, y is equal to d, and x is not in the set of all elements in xs except for the last one. 3. There...
The interior of set S with respect to topology X is empty if and only if for all sets T, if T is open in X and T is not empty, then the difference of T and S is not empty.
Given a digraph isomorphism "hom", and vertices "u" and "v" that are in the image of the isomorphism applied to the vertices of graph G, then "u" is reachable from "v" in the graph obtained by applying the isomorphism to G if and only if the inverse isomorphism applied to "u" is reachable from the inverse isomorphism a...
If the identity of a CEQ of type 'a is equal to some eq, then eq is equal.
The lemma "setToListFresh" establishes the following: Given a list of names 'xvec', an entity 'C' of type 'fs_name', another list of names 'yvec', a set of names 'Xs', and a name 'x', it shows that: 1. 'xvec' is fresh for the set of 'yvec' if and only if 'xvec' is fresh for 'yvec'. 2. 'Xs' is fresh for the set of 'yv...
If "g" is an element of the carrier "P", "f" is also an element of the carrier "P", the degree of "f" is "n", and the degree of "g" is greater than 0, then the leading term of the composition of "f" and "g" is equal to the leading coefficient of "f" scaled by the nth power of the leading term of "g" in the carrier "P".
For the given types and functions - type_wf, set_mode, set_mode_locs, get_disconnected_nodes, and get_disconnected_nodes_locs - the function l_set_mode_get_disconnected_nodes is a valid instance.
If there exists an x such that property φ holds for x, then there exists a z such that for all y, y overlaps with z if and only if there exists an x such that property φ holds for x and y overlaps with x.
The function "f" is integrable over the measure "M".
The second pass of a function on the result of the first pass of a function on a heap structure is equal to merging pairs in the heap structure.
For all 'a', if 'g' of 'a' equals 'a', then there exist natural numbers 'b1' through 'b11' such that 'b1' is less than 'b11' and 'f5' of 'g' and 'x' equals 'f5' of the function that returns 'a' if 'b1' is less than 'b11', otherwise it returns 'h' of 'b2', evaluated at 'x'.
The sum of the absolute values of two real numbers, represented as equivalence classes under the real relation, is equal to the absolute value of the real number represented by the equivalence class of the sum of the corresponding elements.
The minimum null of tree 't' is less than or equal to 1.
If "f", "g", and "h" are identities and the source of "f" is the target of "g" and the source of "g" is the target of "h", then the inverse of the associator of "f", "g", and "h" is in the hom-set from the source of "h" to the target of "f". Furthermore, the inverse of the associator of "f", "g", and "h" is a morphism ...
Matrix A is disjoint with the zero matrix.
An InterruptedException is not an Object.
"w" is less than or equal to the floor logarithm of "x" with base "b", if "b" to the power of "w minus 1" is less than or equal to "x" and "b" is greater than 1.
The powerset of the identity function is equal to the identity function itself.
The lemma from the Isometries.thy in Gromov_Hyperbolicity states the following: 1. There is a geodesic segment between point x and itself within the set containing only x. 2. The set containing only x forms a geodesic segment. 3. There is a geodesic segment between point x and itself within set G if and only if set G ...
The natural number representation of an integer is equal to its identity, under the condition of ZN relation. This is a transfer rule used for transferring properties between types.
If hypothesis set H proves A, then hypothesis set H proves "A OR B".
If a heapmap "hm" satisfies the invariant "hmr_invar" and the application of the abstraction function "heapmap_α" to "hm" and "k" is not equal to None, then the function "hm_valid" applied to "hm" and the result of "hm_index" applied to "hm" and "k" is true.
: Assuming that 'r' is stable with respect to 'step', 'phi', and 'pc', and that 'pc' is less than the length of 'phi', it can be shown that the well-typedness condition 'wtc' for 'c', 'pc', and the 'pc'-th element of 'phi' is not equal to 'top'.
If a collection S is a constant predicate (cp), and for all elements x, P x is also a constant predicate, then the constant predicate of a function that applies to all elements X of S, such that for all elements x in the bag, P x X holds, is also a constant predicate.
If there is an equivalence in a bicategory defined by parameters V, H, \<a>, \<i>, src, trg, f, g, \<eta>, \<epsilon>, and there is an isomorphism from f to f' denoted by \<phi>, then there is also an equivalence in the bicategory for f', g, with the morphisms (g \<star> \<phi>) \<cdot> \<eta> and \<epsilon> \<cdot> (i...
The matrix Q related to decision d is invertible.
If for every edge in the directed arcs of a tree, the head of the directed tree does not equal a vertex, and if the path is not empty, and if there is a directed path from a vertex 'u' to a vertex 'v' in the directed tree, then the set of vertices in the path is not a subset of the directed arcs of the tree.
Dropping the signed bit at position 2 from the binary number 10011 results in the 8-bit binary number 100.
The interaction bound of a function 'f' applied to a lifted stochastic probability monad 'p' is equal to the supremum (or greatest) interaction bound of 'f' applied to each element in the set of 'p'.
The identity function, which maps each element to itself, is a quasi-Borel morphism from set X to itself.
The lemma named "mpoly_base_conv" establishes three facts about polynomials in the context of a commutative ring with unity, denoted by "'a::comm_ring_1". 1. The evaluation of the zero polynomial at any point 'x' is zero. 2. The evaluation of a constant polynomial 'c' at any point 'x' is equal to the constant 'c' its...
If "t1" is a subtree of "t2", and for any "r1", "t1", and "e1", if a node with root "r1" and a set containing the pair "(t1,e1)" is a subtree of "t2", then the rank of the reverse of "r1" is less than or equal to the rank of the reverse of the root of "t1". Under these conditions, the list obtained from the dtree (deci...
The γ-representation of the infimum representation of two points p1 and p2 is equal to the intersection of the γ-representations of p1 and p2.
Given a quasi-Borel space 's' which is equal to 'X', a function 'f' that is integrable over 's', and a positive constant 'c', the measure of the set of elements 'x' in the space 'X' for which 'c' is less than or equal to the absolute value of 'f(x)', is less than or equal to the measure of the product of '1/c' and the ...
If you substitute a variable 'i' with a value 'u' in a restricted predicate 'RestrictedP' with parameters 'f', 'k', and 'g', it is equivalent to creating a new restricted predicate where 'f', 'k', and 'g' have all had 'i' substituted with 'u'.
The sequence of the conjunction of P and the lifted predicate q, followed by R, is equal to the sequence of P followed by the conjunction of the lifted predicate q and R.
For any two prime numbers 'p' and 'q' in a factorial semiring, if 'm' is greater than 0 and 'p' raised to the power of 'm' equals 'q' raised to the power of 'n', then 'p' must be equal to 'q'.
The lemma "tm_semi_id_eq0_loops'" shows that for the Turing machine "tm_semi_id_eq0", there exists a tape configuration where the tape is empty on the left and has two occurrences of "Oc" followed by a sequence of "Bk" of length "l" on the right.
A matchT_T of d and c is defined as for all states s, t, and t', if s is equivalent to t and there is a transition from the pair (d,t) to t', then there exists a state s' such that there is a transition from the pair (c,s) to s' and s' is equivalent to t'.
The lemma named "preserves_src" assumes that "\<mu>" is an arrow in category "C". It shows that the source of "F \<mu>" in category "D" is the image under "map\<^sub>0" of the source of "\<mu>" in category "C".
The lemma "acyclic_d_in_forest_modulo_equivalence" assumes that "x" is a forest modulo equivalence with respect to "d". It shows that "d" is acyclic.
The lemma named "senior_states_cases_subseteq" has two case names: "le" and "ge". It assumes two conditions: 1. If the senior states of 'p' at 'n' are a subset of the senior states of 'q' at 'n', then 'P' holds. 2. If the senior states of 'p' at 'n' are a superset of the senior states of 'q' at 'n', then 'P' also hold...
If a hotel state "s" exists, and a pair of keys (k, k') belongs to the cards of guest "g" in state "s", then key "k'" must be in the set of keys issued in state "s".
The lemma named "even_sum_list" assumes that for any element 'x' in the set 'xs', the parity (evenness) of the function 'f' applied to 'x' is the same as the parity of the function 'g' applied to 'x'. Under this assumption, it shows that the parity of the sum of the list obtained by applying function 'f' to each elemen...
"For all x, if P of x is true, then P of x is true."
If 'p' is not equal to zero, 'p' does not divide 1, and for any 'a' and 'b' such that 'p' equals the product of 'a' and 'b', either 'a' divides 1 or 'b' divides 1, then 'p' is irreducible.
If a function 'f' is an injection on a set 'S' of natural numbers, and 'f' maps elements of 'S' to 'S', and 'S' is finite, and 'f' maps any element not in 'S' to itself, then 'f' permutes 'S'. This means that 'f' rearranges the elements of 'S' without any duplication or omission.
The lemma named "infinite_convex_domain_iff" is about a function 'f' that maps from a Euclidean space to an extended real number. It assumes that for all 'x', 'f' of 'x' equals positive infinity or negative infinity. It shows that 'f' is convex on the entire universe if and only if the domain of 'f' is convex.
If the following conditions are met: 1. "prim_EP w v e" is regular, 2. "r" is a vector, 3. "e" is less than or equal to the product of "v" and the transpose of "-v", 4. "v" is a vector, 5. The transpose of "v" is equal to the product of the transpose of "r" and the star of "t", 6. "t" is less than or equal to "w", 7. "...
For any three extended real numbers x, y, and z, if x is greater than 0 and x is not infinity, then y divided by x is less than z if and only if y is less than x times z.
A list starting with 'x' followed by 'xs' is sorted according to a comparison function 'cmp', if 'xs' is sorted according to 'cmp' and for any 'y' and 'ys' such that 'xs' is a list starting with 'y' followed by 'ys', the comparison of 'x' and 'y' using 'cmp' is not 'Greater'.
If the initial list is distinct, the index 'i' is less than the length of the list 'qs', and both 'a' and 'b' are in the set of the initial list. Also, if the set of 'qs' is a subset of the set of the initial list, and the 'i'-th element of 'qs' is not equal to 'a' or 'b', then if 'a' is less than 'b' in the sequence '...
The composition of the identity function on a set A with itself is equal to the identity function on A.
The product of the absolute values of two real numbers, represented as equivalence classes under the real relation, is equal to the absolute value of the equivalence class under the real relation of the pair formed by the sum of the products of the first elements and the second elements of the original pairs.
If a function "g" permutes a set "B" and another function "f" is cyclic on a set "C", and if the intersection of sets "A" and "B" is empty and "C" is a subset of "A", then the function "f" composed with "g" is cyclic on set "C".
The lemma named "Nom'" assumes the following: 1. Proposition 'p' is at position 'b' in the branch that starts with the pair '(ps, a)'. 2. The nominal 'a' is at position 'b' in the branch that starts with the pair '(ps, a)'. 3. For all 'i', if 'p' is equal to the nominal 'i' or 'p' is equal to the diamond operator appl...
The cardinality of the set of relations in a tree "t" is less than or equal to the length of the inorder traversal of the tree "t".
If "x" is greater than 0 and "b" is greater than 1, then the floor of the logarithm base "b" of "x" equals "k" if and only if "b" to the power of "k" is less than or equal to "x" and "x" is less than "b" to the power of "k + 1".
The lemma named "prefix_transfer" is a transfer rule. Assuming that the relation "A" is bi-unique, it shows that if two lists are related by "A", then their prefix relationship is preserved.
If we have a finite set 'I', and for every subset 'J' of 'I', 'f' converges uniformly to 'g' on 'J' with respect to a filter 'F', then 'f' also converges uniformly to 'g' on the union of 'I' with respect to the same filter 'F'.
If alpha is an element of group G and x is not an element of set S, then the transformation of x by alpha is undefined.
If a set A is a subset of the carrier G, then the generation of G by the union of the generation of G by each element in A is equal to the generation of G by A.
The lemma named "tick_count_sub" assumes that "f" is a dilating function from "sub" to "r". It shows that the tick count in "sub" at a point "c" and time "n" is equal to the tick count in "r" at the same point "c" and time "f n".
The error function is holomorphic on the set A.
The lemma min_mult_left assumes that a real number 'x' is greater than 0. It then shows that the product of 'x' and the minimum of 'y' and 'z' is equal to the minimum of the product of 'x' and 'y', and the product of 'x' and 'z'.
, under the assumption that the observation within the same procedure (obs_intra) of the parent node of n in the HRB slice of S is equal to the observation within the same procedure of the parent node of n' in the HRB slice of S (obs_eq), and that x is in the reverse reachable set (rv) of S from n, it follows that x is...
If term "t" is a zero-bound term, then the equation of term "t" is also zero-bound.
The dagger lift of the eta function is equivalent to the identity function on a set of type 'a.
Lifting a "Pause" resumption, which consists of an output and a continuation function "c", results in a "Pause" with the same output and a lifted continuation function.
The lemma named "basic_semialg_set_memE" assumes that "q" is an element of the basic semialgebraic set with parameters "m", "n", and "P". It shows two things: 1. "q" is an element of the carrier of the p-adic number field raised to the power "m". 2. There exists a "y" in the carrier of the p-adic number field such ...
If for all pairs of lists in "rss" and "mss", the length of each pair of lists is equal, then the linear combination of the concatenation of "rss" and "mss" is equal to the sum of the linear combinations of each pair of lists in "rss" and "mss".
If a function 'f' is injective on a set 'Z', then the image of 'Z' under 'f' is infinite if and only if 'Z' itself is infinite.