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The lemma "newShadowRoot_getCharacterData" assumes that a new shadow root "h" is equal to a pair consisting of a new shadow root pointer and "h'". It shows that getting the character data from a pointer in "h" is the same as getting the character data from the same pointer in "h'".
The first element is a composite lens.
The nth element of the stream f shrunk by k is equal to the last message of the stream f up to the nth element multiplied by k and then cut down by k.
The negation of the representation of a dense element 'x' is equivalent to the bottom element.
If A0 and A1 are elements of the set Gamma, which is a subset of deductions of A, and Disjs is a subset of or equal to A, then the conjunction of the negation of A0 and A1, along with Gamma, is an element of the deductions of A.
The extended nonnegative real number representation of x equals zero if and only if x is less than or equal to zero.
The lemma named "type_e_subst_fresh" is about variables 'x' and 'z'. It assumes that the atom 'z' is fresh for the pair (x,v) and the atom 'x' is fresh for 'e'. It shows that if we substitute 'x' with 'v' in the type expression "{ z : b | CE_val (V_var z) == e }", the result is the same as the original type expression ...
There exists an element 'y' and a list 'ys' such that the list 'xs' is equal to the list that starts with 'y' and is followed by the elements of 'ys'.
The infimum (greatest lower bound) of any generic model context 'C' and 'C' itself is equal to 'C'.
The characteristic of the semiring is equal to 0.
The intersection of the transition relation of r and the transition relation of r' is equal to the transition relation of the intersection of the function that takes x and y if x is an input of r and x is an input of r', and the union of r and r'.
The first element of a sequence obtained by appending a sequence 'σ' to a single element 's' is 's'.
If the idle deque is not empty and popping the idle deque results in a pair consisting of an element 'x' and a modified deque 'idle'', then the size of the modified deque 'idle'' incremented by one is equal to the size of the original idle deque.
If "j'" is less than "j", then the function "fwi" with parameters "m", "n", "k", "i'", "j", "i'", and "j'" is equal to the function "fwi" with parameters "m", "n", "k", "i'", "j'", "i'", and "j'".
The map that applies a function to its argument is a continuous map from the Euclidean topology to the strong operator topology.
The theorem of Validity: Assuming that there is a run "SHORun EIG_M rho HOs SHOs", and for all rounds "r", "EIG_commPerRd (HOs r) (SHOs r)" holds, and for all processes "q", the value of the root node at the initial round is "v", and a process "p" decides on some value "w" at round "r", then it can be shown that "v" i...
The function "pre_newman" is analytic on the set of complex numbers where the real part is greater than 1/2. This lemma is then used to assert that a subset is analytic and introduces analytic properties.
The list "p" becomes distinct after removing duplicates.
If the evaluation of expression e1 is equal to the evaluation of expression e1', and the evaluation of expression e2 is equal to the evaluation of expression e2', then the evaluation of the primitive function f applied to expressions e1 and e2 is equal to the evaluation of the primitive function f applied to expression...
The negation of the equality of the next states of F and G is equivalent to the next state of F not being equal to the next state of G.
The lemma named "add_leaf_awalk_orig_aux" is defined with fixed variables u, v, and a. It defines a new directed graph T' which includes all vertices and arcs of the original directed graph T, plus the vertex v and arc a. The tail of arc a in T' is u, and the head of arc a in T' is v. The lemma assumes that arc a is no...
A subpath in graph 'g' from vertex 'v1' to vertex 'v2' consisting of a single edge 'e' exists if and only if the source of edge 'e' is 'v1', edge 'e' is in the set of edges of graph 'g', and the target of edge 'e' is 'v2'.
The intersection of set d and the inverse of set m is an empty set.
The lemma named "numer_cf7_deriv" shows that the derivative of the function "numer_cf7" at a point "x" is "numer_cf7_deriv x".
The cardinality of set A is less than the cardinality of set C if and only if there exists a set B that is a subset of C such that the cardinality of A is equivalent to the cardinality of B and the cardinality of B is less than the cardinality of C.
For a given matrix A, if A is in reduced row echelon form and the first element of A is zero, then the first column of A is a zero column.
If for any two elements 'a' and 'b' in a set 'Y', either 'a' is related to 'b' or 'b' is related to 'a' under a relation 'r', then 'Y' is a chain under the relation 'r'.
If the set "NI" is a subset of "payload", then "NI" is clean of encryption keys.
"X is strongly equal to X" is always true.
The function "apfst" is parametric over two relations. Specifically, it maps a relation from "Ra" to "Rb" to a product relation between "Ra" and "Rc", resulting in a product relation between "Rb" and "Rc".
The lemma named "poly_bigtheta" assumes that "p" is not equal to zero and "F" is less than or equal to infinity. It shows that the polynomial "p" is in Big Theta of the function "x to the power of the degree of p" when "F" tends to infinity.
Running the state of the return function with any input 'x' and state 's' will result in a pair consisting of 'x' and 's'.
The lemma "admits_tupling_respects_bij" assumes that the set "I" admits tupling and there is a bijective function "phi" between "I" and "I'". It shows that the set "I'" also admits tupling.
If a hash associative map "m" is associated with a pointer "p", then deleting a key "k" from the pointer "p" results in a hash associative map that is the restriction of "m" to the complement of the set containing "k".
For a function 'f' mapping elements of type 'a' to extended real numbers (ereal), and given a set 'Ω' that intersects with the space of a measure 'M' and belongs to the sets of 'M', the function 'f' is Borel measurable in the restricted space of 'M' and 'Ω' if and only if the function 'f' multiplied by the indicator fu...
The least common multiple of two integers 'm' and 'n' is equal to the integer value of the least common multiple of 'm' and 'n'.
If 'i' is the infimum of 'x' and 'y', and 'z' is less than or equal to both 'x' and 'y', then 'z' is also less than or equal to 'i'.
The extended interval between two points 'a' and 'b' is included in the Borel sets.
If a power series 'cs' converges to a function 'f' under the condition of 'convergent_powser'', then the power series 'cs' is convergent under the condition of 'convergent_powser'.
The set of all paths in a tree 't', after filtering out those paths that are not in the set 'S', is equal to the set of all paths in the tree obtained by restricting 't' to the complement of 'S'.
The nth vector of the result of multiplying a column in matrix A by a scalar a at position q is equivalent to the result of multiplying a row in matrix A by a scalar a at position q.
The lemma named "bindWT_unitWT_right_counterexample" is given. It fixes a variable "m" which is of type "'a\<cdot>('m::monad,'w::monoid) writerT". It assumes that "m" is equal to "WriterT\<cdot>(return\<cdot>\<bottom>)" and that "return\<cdot>\<bottom>" is not equal to "\<bottom>" for the type "('a\<cdot>'w writer)\<cd...
For any function that is constantly equal to 'c', the sign of 'c' is preserved under the filter 'F'.
The lemma named "diag_def_poly_eval_at_diagonal" assumes that "x" is an element of the diagonal "n" and "i" is less than "n". It shows that the evaluation at point "R" of "x" for the diagonal defining polynomial of "n" and "i" equals zero.
Adding the successor of a number 'n' to a number 'm' is equal to the successor of the sum of 'm' and 'n'.
If a regular expression "r" is well-formed with respect to the successor of a natural number "n", then "r" is also well-formed with respect to "n" when it is in the prefix position.
The lemma named "two_conjugs_imprim" assumes that the concatenation of "u" and "v" equals the concatenation of "r" and "s", and the concatenation of "v" and "u" equals the concatenation of "s" and "r". It also assumes that "u" is not equal to the empty string, "v" is not equal to the empty string, and "u" is not equal ...
The lemma "WhileAnnoFix'" assumes the following: 1. A consequence that the precondition P is a subset of the set of states s, where there exists a Z such that s is in the invariant I of Z and for all states t, if t is in the invariant I of Z and not in the condition b, then t is in the postcondition Q. 2. A body assu...
The lemma named "deg_Poly'" assumes that the polynomial represented by the list of coefficients "xs" is not equal to zero. It shows that the degree of this polynomial is less than or equal to the length of the list "xs" minus one.
The zero lens is independent of any lens X.
For a function 'f' from a complete lattice 'a' to another complete lattice 'b', if 'f' satisfies the property of being an Infimum dual, then 'f' also satisfies the property of being an infimum dual.
A route request (rreq) with parameters: number of hops, destination IP, destination sequence number, destination key, origin IP, origin sequence number, source IP, and whether it's handled, is equal to a Route Request (Rreq) with the same parameters.
A property P holds for a case resumption of a terminal, linear, or branch resumption if and only if the following conditions are met: - For any terminal resumption 'a', if 'r' equals this terminal resumption, then the property P holds for 't' of 'a'. - For any linear resumption 'b' and any other resumption 'r'', if 'r...
The function 'ctxt_of_gctxt' is injective on the set 'X'. This means that for any two different elements in 'X', their images under the function 'ctxt_of_gctxt' are also different.
For a function 'f' mapping from a first countable and T3 space to another first countable and T3 space, if for any sequence 'u' and any element 'b', the sequence 'u' converges to 'b' if and only if the sequence obtained by applying 'f' to each element of 'u' converges to 'f(b)', and if the closure of a set 'S' equals t...
If from a set of hypotheses H we can deduce the negation of A, and from the same set of hypotheses H we can deduce the negation of B, then from the set of hypotheses H we can also deduce the negation of the disjunction (A OR B).
: "If x equals y then the result is a, otherwise the result is b. This is equivalent to saying that if for all i in the union of the keys of x and y, the lookup of i in x equals the lookup of i in y, then the result is a, otherwise the result is b."
The lemma named "ac_p_nat_pow_factor_right" assumes that "m" is greater than 0 and "a" is an element of the carrier of the p-adic numbers. It shows that the angular component of "m" times the product of "a" and the p-adic prime raised to the power of a natural number "n" is equal to the angular component of "m" times "...
If the "mopup_left_moving" function is true for the parameters (2n + 5, l, Oc # xs) lm n ires, then the "mopup_jump_over2" function will also be true for the parameters (2n + 6, tail of l, head of l # Oc # xs) lm n ires.
The lookup of the least common subsequence (lcs) of two sequences s and t is equal to the least common subsequence of the lookup of s and the lookup of t, where t is a function from 'a to 'b.
If the head of the tail of a list 'xs' is not in 'second_vertices', then 'xs' is not in 'paths'.
The left fold function 'foldl' applied to a function 'f', an initial accumulator state 'σ', and a finger tree 't' is equivalent to applying the left fold function 'foldl' from the List library to the same function 'f', the same initial accumulator state 'σ', and the list representation of the finger tree 't'.
The lemma "cd_impl_icd_cd" assumes that "i" directly influences "l" under a policy "π", and "i" indirectly influences "k" under the same policy "π", but "i" does not indirectly influence "l" under the policy "π". Under these assumptions, it shows that "k" directly influences "l" under the policy "π".
If the following conditions are met: 1. The state 's' completes a happens-before relation 'E' (hb_completion s E). 2. The state 's' transitions to state 's' through a series of thread actions 'ttas' (s -\<triangleright>ttas\<rightarrow>* s'). 3. The thread actions 'ttas' are non-speculative with respect to the program...
The root of rcut is equal to the root of tr0.
The lemma named "always_weakly_fair" shows that "weakly_fair enabled taken" is always true.
The lemma fg_invariant2b: Assuming that "fg_run A rho HOs SHOs coords" holds true, and that a message "m" is received by process "p" from process "q" at step "n" (denoted as "?rcvd n"), and that process "p" is in the set of strongly heard of processes by process "q" at round "?rd n". It shows that the message "m" is eq...
The lemma named "eval_simps" makes the following assumptions and shows the following results: Assumptions: 1. "b" is an identity. 2. "c" is an identity. Results: 1. The evaluation of "b" and "c" is an arrow (a morphism in a category). 2. The domain of the evaluation of "b" and "c" is the product of the exponentiation...
For any natural number 'n' and any integer 'm', the power of the prime number 'p' raised to the product of 'n' and 'm' is equal to the power of 'p' raised to 'n' and then raised to 'm'.
The lemma named "some_src_composable" assumes that "tau" is an arrow. It shows that "tau" composed with "mu" is not null if and only if "some source of tau" composed with "mu" is not null.
The overflow value of an unsigned integer in word interval variable semantics is greater than zero.
For any list of expressions 'xs' and any list of values 'vs', the interpretation of the list of expressions 'xs' after applying the function 'fold_const_fa' to each expression in 'xs' is equal to the interpretation of the original list of expressions 'xs' with the values 'vs'. This is true if the length of the list 'xs...
For a given set 'S' of real normed vectors, if 'S' is closed, 'g' is a path, the start of path 'g' is in 'S', and the end of path 'g' is not in 'S', then there exists a path 'h' such that 'h' is a path, the start of path 'h' is the same as the start of path 'g', the image of path 'h' is a subset of the intersection of ...
The lemma named "papp_multiset_3_aux" assumes the following conditions: 1. The size of set A is 3. 2. x is an element in set A. 3. y is an element in set A. 4. The set of elements in set A is a subset of the set {x, y}. 5. x is not equal to y. 6. The count of x in set A is greater than or equal to the count of y in se...
The right fold of an empty list with any function and initial value is the initial value.
The lemma named "base_case_matrix_eq" is defined with two fixed real polynomials, p and q. It assumes that p is not equal to zero. The lemma shows that the multiplication of a matrix (created from a list of rows with 3 elements each: [1,1,1], [0,1,0], [1,0,-1]) and a vector (constructed from the left-hand side vector R...
If a special equivalence list is formed by appending a value 'v' after list 'xs' and before list 'ys', and another one by appending a value 'w' after list 'xs' and before list 'ys', then if 'v' is in the union of the sets of 'xs' and 'ys' or 'w' is in the union of the sets of 'xs' and 'ys', then 'v' must be equal to 'w...
The subdegree of 0 is 0.
The lemma named "fixing_ide_gives_natural_transformation_2" assumes that "a2" is an identity in the category A2. It shows that there is a natural transformation from category A1 to B, where the transformation is defined by the functions F and G applied to the pair (f1, a2), and the natural transformation is given by th...
The monitoring of a list of events 'w' with an event 'e' at the beginning, under a flow graph 'fg', is equal to the union of the monitoring of the event 'e' under the same flow graph and the monitoring of the list of events 'w' under the same flow graph.
If a point 'p' is in the set produced by the function 'ExcessTable' applied to a graph 'g' and the vertices of 'g', then 'p' is a final vertex of the graph 'g'. The function 'fst' is used to extract the first element of the point 'p'.
If D is an inductive set, X is a chain in D, and there exists a predecessor of the supremum of D and X in the interval of D and the set including the supremum of D and X, then the supremum of D and X is an element of X.
In the context of a ring, the lemma named "canonical_proj_vimage_mem_iff" is assumed to be an ideal I of R and a subset J of the carrier of the quotient ring R/I. It shows that for any element 'a' in the carrier of R, 'a' is in the union of J if and only if the coset of 'a' with respect to I is in J.
Looking up a value in a nested environment (Env b es) with a key (x # xs) is equal to the following: 1. If there is no value associated with the key x in the environment es, then the result is None. 2. If there is a value (Some e) associated with the key x in the environment es, then the result is the lookup of the re...
In the context of a well-founded sub-relation, the lemma named "subsumed_by_two_imp" is stated. It assumes that 'v' is related to 'v1' and 'v' is also related to 'v2' in the sub-relation 'subs'. It then shows or proves that 'v1' must be equal to 'v2'.
The lemma named "fps_left_inverse'" is defined with a fixed function 'f' of type "'a::ring_1 fps". It assumes that the product of 'x' and the 0th element of 'f' equals 1, and that the product of the 0th element of 'f' and 'y' equals 1. These assumptions imply that 'x' equals 'y', but it is not necessary to assume that....
Updating the array 'a' at index 'i' with value 'x' using the 'Array_Time.upd' function is equivalent to updating 'a' at the index converted from a natural number 'i' with value 'x' using the 'upd'' function, and then returning 'a'.
: Assuming that: 1. "\<Psi>" is a valid SAS+ problem, 2. the domain of "s" is a subset of the set of variables in "\<Psi>", 3. for every variable "v" in the domain of "s", the value of "s" at "v" is in the domain of "\<Psi>" at "v", 4. the set of operators "ops'" is a subset of the set of operators in the translation ...
If the degree of a function 'f' is less than or equal to 'n', then the value of the partial deviation of the mixed sum of 'f' and 'g' at 'e' is equal to the sum of the partial deviation of 'f' at 'e' and the partial deviation of 'g' at the function 'i' shifted by 'n'.
If the function "fgl_outer_invar" holds for the iteration "it" and the pair "(None, D)", then the function "outer_invar" also holds for the iteration "it" and the data "D".
The lemma named "wf_subcls_nCls'" assumes that the program P is well-formed according to the well-formedness condition wf_md, and that C0 is not a class in program P. It shows that for any class descriptor cd and any class D' in the set of classes in program P, the first element of the class descriptor cd is not a supe...
A list 'xss' is finitely long after concatenation if and only if the list 'xss' is finitely long after filtering out all null elements and all elements in the set of 'xss' are finite.
The star operation applied to "x" and the star operation applied to "y" and "1", when further applied to "z", is equal to the star operation applied to "x" and the star operation applied to "y" and "1", when further applied to the star operation applied to "x" and the star operation applied to "y" and "z".
If set S is a subset of set S', and the intersection of set A with set S' is equal to the intersection of set B with set S', then the intersection of set A with set S is equal to the intersection of set B with set S.
If a well-formed program P and a program state P are valid for a given state (None, h', frs'), then converting this state to a JVM state using the function "state_to_jvm_state" and updating the stack and local variables with "update_stk" and "update_loc" respectively, will result in the original state (None, h, frs').
Zero plus infinity equals infinity.
The lemma "ZFact_prime_is_domain" assumes that "p" is a prime number in the Factorial Ring. It shows that the integer factorization of "p" is a domain.
If "D" and "E" are orders, "f" is an order isomorphism from "D" to "E", and "D1" is a subset of the carrier of "D", then there is an order isomorphism from the induced order on "D1" to the induced order on the image of "D1" under "f". This isomorphism is given by the restriction of "f" to "D1".
If 'x' is a transitive set and 'a' is less than or equal to 'x', then the transitive closure of 'a' is also less than or equal to 'x'.
The theorem 'laplace_exists_laplace_integrandI' assumes that the Laplace integrand of 's' is integrable on the interval from 0 to infinity. Under this assumption, it is possible to obtain a function 'F' such that 'f' has a Laplace transform 'F' at 's'.