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A number x is an integer rational number if and only if x belongs to the set of integers.
If a proposition "q" can be inferred from a set of propositions "ps", and if the set "ps" is a subset of another set "ps'", then the proposition "q" can also be inferred from the set "ps'".
The term domain of a substitution (where X is replaced by T) in a context Gamma is equal to the term domain of the original context Gamma.
The lemma named "fin_inf_cases" has two case names: "finite" and "infinite". The statement asserts that if 'r' belongs to the set of all finite sequences (denoted by UNIV\<^sup>\<star>) then 'P' holds, and if 'r' belongs to the set of all infinite sequences (denoted by UNIV\<^sup>\<omega>) then 'P' also holds. Therefor...
The lemma named "deg_pm_minus" assumes that "s" adds to "t", where "t" and "s" are elements of a commutative monoid with addition. It shows that the degree of the polynomial "t - s" is equal to the degree of the polynomial "t" minus the degree of the polynomial "s".
If there is a left derivation from the concatenation of strings u and v to string w, and the derivation D is greater than or equal to the length of string u, then there is a left derivation from string v to the string obtained by dropping the first 'length of u' characters from w, using the derivation obtained by shift...
The representation "p" is empty if and only if the gamma representation of "p" is an empty set.
The lemma named "lowest_vertex" is defined with respect to two affine forms, X and Y, each of which is a pair of real numbers. It assumes the following: 1. The function 'e' maps every element in the universal set to a value between -1 and 1 inclusive. 2. For every index 'i', the second element of the pair resulting fr...
If "p" is a division of "S", then there exist points "a" and "b" such that "S" is a subset of the closed box defined by "a" and "b" in a Euclidean space.
The lemma named "line_is_dim1_subspace_t_ge_1" assumes that "n" is greater than 0, "t" is greater than 1, and "L" is a line with parameters "n" and "t". It shows that the restriction of the function "L" to the cube of dimension 1 and side length "t", when applied to the first element of any input, is a subspace of dime...
: Assuming that the function "sub1" applied to the element at position (l,l) in matrix A, with parameters (dim_row A - Suc l), l, and (r,A), results in (r',A'), and that the matrix A is triangular up to index l, and that the number of rows in matrix A is greater than 0, and that l is less than the number of rows and co...
If a function 'f' maps from a set of natural numbers 'j' less than or equal to 'n' to the set of all integers, then the maximum integer value 'f' can take on this domain plus one is not in the image of 'f' over the set of 'j' less than or equal to 'n'.
If an entity B gets a message X in a set of events 'evs' that are part of the 'bankerb_gets' process, then the message X is part of the knowledge known to the Spy in the same set of events 'evs'.
The lemma named "integrable_from_subalg" is about a function 'f' that maps from 'a' to 'b', where 'b' is a Banach space with a second countable topology. The lemma assumes that 'F' is a subalgebra of 'M' and that the function 'f' is integrable when restricted to this subalgebra. Under these assumptions, the lemma shows...
The lemma named "FrameStatImpClosed" is defined with three fixed elements: F and G, which are both frames of type 'b, and p, which is a name permutation. The lemma assumes that frame F implies frame G. Given this assumption, the lemma shows that the permutation of F implies the permutation of G.
The lemma "extend_eval" assumes that the program P is well-formed (wf: "wwf_prog P") and that there is a reduction sequence from expression e with state s to expression e'' with state s'' (reds: "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e'',s''\<rangle>"). It also assumes that expression e'' with ...
If a list of values 'vs' mapped to 'Val' and appended with a 'throw ex' followed by another list 'es' is equal to another list of values 'ws' mapped to 'Val' and appended with a 'throw ex'' followed by another list 'es'', then the original lists of values 'vs' and 'ws' are equal, the exceptions 'ex' and 'ex'' are equal...
The lemma named "orthogonal_extension_aux" is about a set S in a Euclidean space. Given that T is finite, S is finite, and all pairs of elements in S are orthogonal, it shows that there exists a set U such that all pairs of elements in the union of S and U are orthogonal, and the span of the union of S and U equals the...
If the polynomial coefficient 'c' belongs to the set 'S', then for all 'j' less than or equal to the first element of 'c', the second element of the extended coefficient function of 'S', '0', and 'c' at 'j' is equal to the second element of 'c' at 'j'.
The theorem named "affine_unop" assumes the following: 1. "x" is an element of the affine form with error "e" and set "X". 2. The absolute difference between the function "f" at "x" and the linear combination of "a" and "x" plus "b" is less than or equal to "d". 3. The error "e" is a function that maps any element to a...
The lemma "SA_nonzero_set_is_semialgebraic" assumes that the function 'f' is in the carrier of the semialgebraic set 'SA' of dimension 'n'. It shows that the set of non-zero elements of 'f' in the semialgebraic set 'SA' of dimension 'n' is also semialgebraic.
If there are two messages in the set of events "evs" that are part of the Kerberos IV protocol, where the Key Distribution Center (Kas) tells agent A and agent A' respectively, encrypting the message with keys Ka and Ka' and including the same authentication key (authK), the Ticket Granting Server (Tgs), timestamps Ta ...
The lemma named "prv_cnj_assoc_imp1" assumes that phi, chi, and psi are formulas. It shows that if a conjunction of phi and a conjunction of chi and psi is provable, then a conjunction of a conjunction of phi and chi and psi is also provable.
The universal set is equal to the set containing TCP_SYN, TCP_ACK, TCP_FIN, TCP_RST, TCP_URG, and TCP_PSH.
If integer i is defined and integer j is defined, then the sum of i and j (where both i and j are integers) is also defined.
The strict polynomial order is transitive. If polynomial 'a' is strictly less than polynomial 'b', and polynomial 'b' is strictly less than polynomial 'c', then it follows that polynomial 'a' is strictly less than polynomial 'c'.
If for all C, if P C implies that the function symbols in the generalized context C are a subset of F and if P C implies that the function symbols in the generalized context C are a subset of G, and if the relation R is a subset of the product of the term sets of F and G, then the generalized context extension on P and...
If "X" is less than or equal to "X'", then the loop of "X" with invariant "I" is less than or equal to the loop of "X'" with the same invariant "I".
If T is a monomorphism from A to B in the category of sets, then the restriction of T to the range of the arrow value of T is an isomorphism from A to the range of the arrow value of T in the category of sets.
Applying the function "domainNameChop" to a domain name "dn" and a number "n" is equivalent to applying the function "domainNameDeptChopOne" to the domain name "dn" "n" times.
, assuming "FOr \<phi>\<^sub>1 \<phi>\<^sub>2" is a well-formed formula with respect to n, the language of "FOr \<phi>\<^sub>1 \<phi>\<^sub>2" under M2L with respect to n is a subset of the union of the languages of \<phi>\<^sub>1 and \<phi>\<^sub>2 under M2L with respect to n, intersected with the set of encodings (w,...
The existence interval of the initial value problem for the exponential function, denoted as "exp_ivp.existence_ivl0 p", is equal to the universal set.
If 'C' is a locally small category, and if for any two morphisms 'x' and 'y' in 'C', 'x' equals 'y' when their mappings to objects are the same, and if for any two objects 'X' and 'Y' in 'C', the collection of morphisms from 'X' to 'Y' in 'C' is in the range of the explode function, then 'R' holds. This implies that 'R...
The number of "heads" in a list "xs" is less than or equal to the length of the list "xs".
If a hash table 'l' satisfies the hash property, its length is greater than 1, and a key 'k' is in the domain of the map formed by the list at the index calculated by the bounded hashcode function on 'k' and the length of 'l', then 'k' is also in the domain of the map formed by concatenating the list 'l'.
If s and s1 are equivalent except for PID2, then the title, abstract, content, reviews, and discussions of the paper with PID in s are equal to those in s1.
The uncurried implementation of "ps_get_imp" is refined by the uncurried operation "ps_get_op". This is under the assumption that the abstract machine state and the identity function are in a Kleene star relation. The result is an assertion on a list that is in relation with the identity function.
The greatest fixed point of the function "x maps to y times x join z" under the operation C is less than or equal to the least fixed point of the same function join the result of applying the operation n to the greatest fixed point of the function, all times the top element.
The set of individual terms (indets) in the sum of two polynomials (p + q) is a subset of the union of the set of individual terms in polynomial p and the set of individual terms in polynomial q.
If a predicate P holds for all elements in two lazy lists xs and ys, then the predicate P also holds for all elements in the two lazy lists obtained by dropping the first n elements from xs and ys.
Applying the function "bliftM" to the function "f" applied to the boxed value of "x" is equal to the boxed value of the function "f" applied to "x".
If a state 's' transitions to a state 's'' through a thread 't' and a thread action 'ta', and if the initial state 's' satisfies the 'lock_thread_ok' condition with respect to its locks and threads, then the final state 's'' also satisfies the 'lock_thread_ok' condition with respect to its locks and threads.
The second element of the image of a set 'x' under a function that maps each element 'e' to a pair consisting of 'to_dtree_aux' applied to 'head T e' and 'e', is equal to 'x' itself.
In the context of a subcategory, the lemma named "subcategory_axioms''" can be introduced with category theory intros. Assuming that "alpha prime" is equal to "alpha" and "C prime" is equal to "C", it can be shown that "B" is a subcategory of "C prime" in the context of "alpha prime".
For any transition 't', if 't' is in the set of transitions of a finite state machine 'M', then the source of transition 't' is in the set of states of 'M'.
The embedding of integers into the ring R is a ring homomorphism from the ring of integers to R.
If 'k' is less than the length of the list 'xs', then augmenting the list 'xs' at position 'k' with the element 'x' is equivalent to updating the list 'xs' at position 'k' with the element 'x'.
The lemma "adopt_node_children_subset" assumes the following conditions: 1. A node is adopted by an owner document, transitioning the heap from 'h' to 'h''. 2. The child nodes of a pointer 'ptr' in heap 'h' are 'children'. 3. The child nodes of the same pointer 'ptr' in the new heap 'h'' are 'children''. 4. All pointe...
The space of essentially bounded measurable functions is the set of all functions 'f' that are Borel measurable on 'M', for which there exists a constant 'C' such that for almost every 'x' in 'M', the absolute value of 'f(x)' is less than or equal to 'C'.
If machine M is minimal, then the transformation of the finite set of states of M to a list of states will also result in a minimal machine.
The lemma named "tuple_partial_pullback_is_semialg_map_tuple" assumes that "fs" is a tuple of semialgebraic function on "m". It shows that "fs" is also a tuple of semialgebraic map on "m".
: given an object 'a' in category 'A' and an arrow 'g' in category 'B', the arrow mapping of the bifunctor applied to 'a' and 'g' is equal to the composition of the arrow mapping of the functor 'S' applied to the identity on 'a' in category 'A' and 'g'.
The lemma named "TBOUND_vebt_succi" defines a function "foo" for any tree "t" and value "x", which is equivalent to 7 times the sum of 1 and the height of the tree "t". It then shows that the time bound of the function "vebt_succi'" with parameters "t", "ti", and "x" is less than or equal to the value of the function "...
If condition "re(c)" holds true, then for all "d", "re(c)" also holds true at "d".
If three conditions are met: 1) x is greater than 0, 2) the function inv_loop5 applied to x and the pair consisting of b and a list starting with Bk holds true, 3) b is not an empty list, then the function inv_loop6 applied to x and the pair consisting of the tail of b and a list starting with the head of b, follow...
For any real number 't' that lies in the open interval between negative pi and pi, the cosine of 't' is greater than negative one.
If 'n' is less than the length of the list 'xs', then the length of the list 'xs' after deleting the element at index 'n' is equal to the original length of the list 'xs' minus one.
The function "basis_reduction_swap" applied to the index "i" and a tuple consisting of a pair of lists "f1" and "f2", a list of lists "dmus", and a list "ds", is equal to a tuple. This tuple is constructed as follows: First, several variables are defined: - "di" is the "i"-th element of "ds", - "dsi" is the "i+1"-th e...
If a function 'f' has an integral 'k' over a set 'S', then the function that is the negation of 'f' has an integral of '-k' over the same set 'S'.
If a function 'f' is an element of the Cartesian product of 'A' over 'i' in 'n', then 'f' is also an element of the Cartesian power of 'A' to 'n'.
The non-deterministic choice between P and Q is equal to the non-deterministic choice between Q and P.
The reflexive property holds for a comparison function where the comparison does not result in "Less".
The set resulting from updating a pair "kv" in a list "xs" is a subset of the set formed by inserting the pair "kv" into the set of "xs".
The signed division of two 32-bit unsigned integers x and y is equal to the representation of x divided by the representation of y, unless y is zero. If y is zero, then the result is undefined for the division operation on x and zero.
If there is a path from node u to node v that ends with edge e, then either the path is empty, or if the path is not empty, the second element of the last pair in the path is equal to the first element of edge e.
The lemma "learn_limE" assumes that the function "psi" learns the limit of the set "U" with respect to the strategy "s". Under this assumption, it shows two things: 1. The function "psi" is an environment for the set "U" with respect to the strategy "s". 2. For any function "f" in the set "U", there exists an index "...
The size of the list "t" is equal to the size of "t".
The lemma named "well_base_h_mono_base" assumes three conditions: 1. "well_base_h" is true for a given base and a multiset M, 2. the base is greater than or equal to 2, 3. the base is less than or equal to base'. If these conditions are met, it shows that "well_base_h" is also true for base' and the multiset M.
For the successor of a number 'k', the function 'alternate' applied to 'f' and 'g' is equal to the composition of either 'f' or 'g' (depending on whether 'k' is even) and the 'alternate' function applied to 'f', 'g', and 'k'.
If 'x' is in the transitive closure of 'R' or 'x' is in the transitive closure of 'S', then 'x' is in the transitive closure of the union of 'R' and 'S'.
If a set A is a subset of the carrier of the group and the relations of A are equal to the set containing only the function that maps every element in A to 0, then for every function f in the set of all functions from the set of all generated subgroups of single elements of A to the identity function, if the finite pro...
The lemma "Inl_dec_interp_length" assumes that for every element 'i' in the set 'FO', there exists a unique 'p' such that 'p' is less than the length of 'w' and the second element of the 'p'-th element of 'w' contains 'i'. It shows that if 'Inl p' is an element of the set resulting from the interpretation of 'FO' in 'w...
If X is a member of filters, then X is also a member of upsets.
The lemma "snd_row_add_iterate_PA_eq_row_add_iterate" shows that the second element of the pair resulting from the function "row_add_iterate_PA" applied to the pair (P,A), and the parameters n, i, and j, is equal to the result of the function "row_add_iterate" applied to A with the same parameters n, i, and j.
The determinant of an identity matrix of any order 'n' is 1.
The lemma "canonical_separator_t_source_isl" assumes that transition 't' is in the set of transitions of the canonical separator of machine 'M' for states 'q1' and 'q2'. It also assumes that 'q1' and 'q2' are states in machine 'M'. It shows that the source of transition 't' is a left summand.
The lemma named "subst_cls_eq_grounding_of_cls_subset_eq" assumes that a set D substituted by σ equals to set C. It shows that the grounding of set C is a subset of the grounding of set D.
The SK function with parameters c, p, b, the successor of t, and k is equal to the product of the S2 function with parameters c, p, and the successor of t, and b raised to the power of the successor of t, plus the SK function with parameters c, p, b, t, and k.
A function 'f' is summable if and only if 'f' is nonstandard summable.
The lemma named "equiv_up_closed_subset" assumes that "A" is an upward closed set and "B" is equivalent to "C". It shows that "B" is a subset of "A" if and only if "C" is a subset of "A".
The polynomial "p" evaluated at the point "ys" is equal to the polynomial "p" inserted at the point "xs" if and only if for all "i" less than the number of parameters of "p", the "i"-th element of "xs" is equal to the "i"-th element of "ys".
, in an empty context, the sum of the variable 'xx' and zero is equal to the variable 'xx'.
The changes in a polynomial at point 'a' is zero when the polynomial is either empty or contains only one element.
The lemma named "sum_radd_block_bound" shows that the nth digit of the sum of the function R+ applied to parameters p and l, where the function is applied to each k in the sequence s, is less than or equal to 1 when evaluated at t and b.
The lemma named "lift_to_metric_completion" is defined as follows: Let's consider a function 'f' that maps from a metric space 'a' to a complete space 'b'. If it is assumed that 'f' is uniformly continuous on the universal set, then there exists a function 'g' such that 'g' is uniformly continuous on the universal set...
D-indistinguishability is symmetric. If expression e is d-indistinguishable from expression e', then expression e' is also d-indistinguishable from expression e.
Given a set of equations (eq) and inequalities (les), if one of the following conditions is met: 1. There exists a triplet (a', b', c') in the set of equations such that a' equals 0 and b' does not equal 0, and for all triplets (d, e, f) in the set of equations, the equation d * (- c' / b')² + e * (- c' / b') + f equa...
If y is less than or equal to y', then the sum of x and y is less than or equal to the sum of x and y', where y' is an ordinal number.
The possible allocations relation prime of A and B is equal to the possible allocations relation of A and B.
The set of nested propositional atoms in a formula "phi" is a subset of the subformulas in "phi".
If x and y are both integers, then the sum of x and y is also an integer.
The complete function applied to the parameters k, \<pi> k T I equals \<pi> k T I.
Mapping the identity function over the five arguments of the function "map_U" results in the identity function.
: 1. For any given values of hops, destination IP (dip), destination sequence number (dsn), destination key (dsk), origin IP (oip), origin sequence number (osn), and source IP (sip), the function "rreq_rrep_sn" applied to a Route Request (Rreq) returns true if and only if the origin sequence number (osn) is greater th...
For any matrix A of a certain ring, given that A, B, and C are matrices with appropriate dimensions, the product of A and the difference of B and C is equal to the difference of the product of A and B and the product of A and C.
If a pointer 'ptr' is not equal to the cast of a new shadow root pointer, and if 'h' leads to a new shadow root pointer under the new shadow root model, and if 'h' also leads to 'h' under the new shadow root model, then 'r' is in the location of the child nodes of 'ptr' in both 'h' and 'h''.
The permission to load from a null capability is false.
If a complex number 'z' is in the unit circle of an oriented circline, then the conjugate of 'z' is also in the unit circle of the same oriented circline.
The lemma named "ac_cong_set_is_univ_semialg" assumes that "n" is greater than zero and "k" is an element of the unit group of the residue ring of "n". Under these assumptions, it shows that the set defined by the arithmetic congruence of "n" and "k" is a universal semi-algebraic set.
The last message held in the concatenation of two lists (xs and ys) is equal to the last message held in the first list (xs) concatenated with the tail of the last message held in the list formed by the last message of the first list (xs) and the second list (ys).