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If the Herbrand addition of x and y equals z, then the Herbrand addition of x and the successor of y equals the successor of z.
The lemma "inf_concat_Suc" assumes the following conditions: 1. There is an infinite number of indices 'i' such that 'n i' is greater than 0. 2. For any index 'i', 'f i (n i)' equals 'f (Suc i) 0'. 3. 'inf_concat n k' equals '(i, j)'. 4. 'inf_concat n (Suc k)' equals '(i', j')'. Under these assumptions, the lemma sho...
The lemma named "C_case_cong" in Launchbury's C.thy assumes that for any 'r'', if 'r' equals 'C' applied to 'r'', then 'f' applied to 'r'' equals 'g' applied to 'r''. Under this assumption, it shows that 'C_case' applied to 'f' and 'r' equals 'C_case' applied to 'g' and 'r'.
The lemma named "proper1_valid_istate" assumes that the trace "tr" is valid and the source of the head of "tr" is the initial state. It also assumes that the user ID "UID'" is in the friend IDs of the target of the last trace for the user ID "UID". Under these assumptions, it shows that the trace "tr" is proper.
If node 'n' is in the set of phi definition nodes of graph 'g' for variable 'v', then 'n' is in the set of nodes of graph 'g'.
The lemma named "soundness" assumes that "check_eqv r s" is true. It then shows that the language of "r" is equal to the language of "s".
The pair of recursive list functions is included in the relation where the first function maps from a relation 'Ra' and the second function maps from a relation 'Rb' to a list relation of 'Rb' and then to 'Ra'. This pair of functions is then mapped to a list relation of 'Rb' and finally to 'Ra'.
"x" is not equal to "TypeError" if and only if there exists a "t" such that "x" equals "Normal t".
If n is greater than 0, then 0 is an nth power in any semiring.
A sequence "es" is a "Throws" sequence if and only if there exist some values "vs", an exception "a", and a sequence "es'", such that "es" is equal to the concatenation of the sequence of values "vs", the exception "a", and the sequence "es'".
If 'a' is an object in the category 'CC' and 'i' is an element of the index set 'I', then the 'i'-th component of the object map of the digraph homomorphism 'dghm_up' applied to 'a' is equal to the 'i'-th component of the object map of 'phi' applied to 'a'.
The composition of function 'f' and the top element, added to the composition of the adjoint of function 'f' and the bottom element, equals the top element.
Lifting an error transformer applied to the bind operation on 'm' and 'k' is equivalent to binding the error transformer lifted 'm' and the composition of lifting error transformer and 'k'.
The lemma "store_cap_correct_dom" shows that for a given offset "off", the keys in the mapping resulting from storing a capability "cap" in memory "m" at offset "off" with a size of 32, include "off" and the 31 subsequent integers (from "off + 1" to "off + 31").
, given the assumptions that a relation "A" is bi-unique and right total, it shows that a commutative monoid on the set of all elements that satisfy the domain predicate of "A" is equivalent to a commutative monoid on the set "A". This is done under the condition that the binary operation and the identity element of th...
Given a well-formed Java program "P", a well-formed start state with program "P", class "C", method "M", and variables "vs", and given that the program "P" can transition from the start state to some state "s" following a sequence of thread actions "ttas", if a thread "t" in state "s" is not final and is not deadlocked...
The composition of functions f and g is equal to the composition of the dagger of g and f.
The dimension of the universal set is equal to the cardinality of the basis.
The empty priority queue maintains the skew invariant, which is a property of the skew heap data structure.
The length of the list obtained by applying the substitution "eta" to the list "AAs" is equal to the length of the original list "AAs".
The lemma named "mult_cancelL" assumes that "s" is a transitive and irreflexive relation. It shows that the pair (Z + X, Z + Y) is in the multiset relation "mult s" if and only if the pair (X, Y) is in the multiset relation "mult s".
If the maximum value in a list is less than a certain number, then that number is not in the set of values in the list.
The lemma named "cong_4321" assumes that the segments AB and CD are congruent. It shows that if this is true, then the segments DC and BA are also congruent.
The lemma "term_ok'_bind_fv2" assumes that the term 't' is valid under the set 'Σ'. It shows that the term 't', when bound to a free variable 'v' of type 'T' at level 'lev', is also valid under the set 'Σ'.
The parallel operation of "x" and "y" multiplied by "U" is equal to the parallel operation of "x" multiplied by "U" and "y" multiplied by "U".
If a sequence of events 'evs' is in 'sr', then the key 'K' is in the analysis of the insertion of another key 'K'' into what the Spy knows from 'evs'. This is equivalent to either 'K' being equal to 'K'' or the key 'K' being in the analysis of what the Spy knows from 'evs'.
The lemma named "mono_Follows_o" assumes that the function "h" is monotonic. It shows that the follows relation of "f" and "g" is a subset of the follows relation of "h composed with f" and "h composed with g".
The lemma named "heterogeneous_secrets_anti_mono" assumes that a heterogeneous list "t" with security level "Sec" is a subset of "Sec'". It shows that the heterogeneous list "t" with security level "Sec'" is also a subset of "Sec".
The lemma "nth_in_AffineI" assumes that a list "xs" is in the set of "Joints XS" and that an index "i" is less than the length of "XS". It shows that the "i"-th element of "xs" is in the Affine of the "i"-th element of "XS".
If three conditions are met: 1) the auxiliary function "interfree_aux" is true for the command of x, the postcondition of x, and the command of y, 2) the auxiliary function "interfree_aux" is true for the command of y, the postcondition of y, and the command of x, and 3) the function "interfree_swap" is true for the...
The lemma SA_oneE assumes that "a" is in the carrier of the nth power of the p-adic numbers. It shows that the unit element of the semialgebraic function ring of "n" evaluated at "a" is equal to the unit element.
The minimum set of components 'i' in the set 'chSet' is a subset of the abstraction level 'i'.
The lemma "funtyp_simple_check" is defined with fixed variables s, \<Delta>, \<tau>, and v. It assumes that the function type check for \<Theta>, \<Phi>, an empty set of bound variables, and the function type AF_fun_typ with parameters x, b, c, \<tau>, and s is valid. It also assumes that under the context of \<Theta>,...
If the partition factors main function applied to 's' and 'xs' equals the pair (a, b), then the multiset of 'xs' is equal to the sum of the multisets of 'a' and 'b'.
The pair (c, 0) is not in the function relation S.
If set X is finite, the cardinality (or size) of set Y is n, and set Y is a subset of set X, then the cardinality of set X is greater than or equal to n.
If an element 'x' is in the multiset 'darcs_mset' of a node 'r' with a finite set 'xs', and 'x' is not in the image of the second component of the finite set 'xs', then there exists a pair '(t1, e1)' in the finite set 'xs' such that 'x' is in the multiset 'darcs_mset' of 't1'.
If "f" is a nucleus, and "y" is a fixed point of "f", then the result of the operation "y left-residuated by x" is also a fixed point of "f".
The ceiling of a number 'x' is equal to 'a' if and only if 'x' is greater than 'a' minus one and less than or equal to 'a'.
For any fixed word 'a' of any length, the left shift operation over the bitwise AND operation of 'a' and 'b' is equal to the bitwise AND operation of the left shift of 'a' and the left shift of 'b'.
The intersection of the product of x and y with the complement of the product of x and z is equal to the product of x and the intersection of y with the complement of z, all intersected with the complement of the product of x and z.
The lemma L2_Boiler assumes that if there is a steam boiler from x to y in state s, and x is in a true state, then y is also in a true state.
For any two quadratic functions qf1 and qf2, defined by the complex coefficients A1, A2, b1, b2, B1, B2, c, and C, if A1 and A2 are not zero, and if there are two distinct points (x1, y1) and (x2, y2) that are roots of both functions, then any other point (x, y) that is a root of both functions must be either (x1, y1) ...
If the polynomials K, n, and P are distinct and for all j less than or equal to m, the function f(j) is in the carrier of the ring O with parameters K, P, and n, then the n-th sum of the function f in the ring O with parameters K, P, and n is equal to the n-th sum of the function f in the field K.
The lemma named "lens_scene_indep_compl" assumes that "x" and "y" are very well-behaved lenses. It shows that the scene of "x" is independent of the complement of the scene of "y" if and only if "x" is a sub-lens of "y".
If a program 'P' is well-formed according to the metadata 'wf_md' and 'P' is a class 'C', then there exists a set of field data types 'FDTs' such that 'P' has fields 'FDTs' in class 'C'.
For any two real numbers 'a' and 'b', the closed segment between 'a' and 'b' is an interval.
If the assignment of the value 'v' to the variable 'x' in the state 'ps' results in the state 'ps'' in a deterministic manner, and if the partial arithmetic evaluation of 'a' in the state 'ps' equals 'v', then the assignment of 'a' to 'x' in the state 'ps' also results in the state 'ps'' in a deterministic manner.
The extension function 'f' is irreflexive when applied to the 'extf' function.
In the context of a monoid, given that "x" and "y" are elements of the carrier "G", if the product of "x" and "y" equals the identity element, and the product of "y" and "x" also equals the identity element, then "y" is the inverse of "x". This lemma was contributed by Martin Baillon.
If "x" is less than or equal to "y" and "t_stop" is less than or equal to "x", then "p" of "y" is less than or equal to "p" of "x".
The lemma l8_16_1: Assuming that points A, B, and X are collinear, and points A, B, and U are collinear, and line AB is perpendicular to line CX, it can be shown that points A, B, and C are not collinear and line CX is perpendicular to line U.
The output function "h_out" for a finite state machine "M" is defined as follows: for any state and input pair "qx", it maps to a set of outputs "yqs" if such a set exists in the transition function of "M" when viewed as a map. If no such set exists, it maps to an empty set.
The lemma named "vcpower_vsubset_mono" assumes that set A is a subset of set B. It shows that the nth Cartesian power of set A is a subset of the nth Cartesian power of set B.
The supremum of a set F composed with a function g is equal to the supremum of the set of all functions in F composed with g.
If a policy "p" has single combinators, is distinct, has all distinct networks, is a well-formed policy according to definitions 1 and 2, then it is also a well-formed policy according to definition 3.
The division function for integers (ZN) corresponds to the division function for natural numbers (ZN).
Assuming that "f" is a marked binary morphism (a function that transforms one binary string into another), there exists a binary string "c" such that the first element of the binary string resulting from applying "f" to "c" is "a".
In the context of a ring, the lemma named "finsum_ring_hom" is stated as follows: Assuming that "S" is a ring, "h" is a ring homomorphism from "R" to "S", "F" is a function from "I" to the carrier of "R", and "I" is finite, it can be shown that the homomorphism of the finite sum over "I" of the function "F" is equal t...
The result of the function "cyk_ix" with parameters (i, j) is equal to the result of the function "cyk_ix_impl" with parameters i and j when applied to an empty heap.
If 'a' is less than or equal to 'c' and 'b' is less than or equal to 'd', then the logical 'or' of 'a' and 'b' is less than or equal to the logical 'or' of 'c' and 'd'.
If "x" is a well-behaved (moderately) lens, then the input variable "x" is also a well-behaved (moderately) lens.
If a list "xs" is in the valid lists of "S CoCalls", then the list "xs" without its last element is also in the valid lists of "S CoCalls".
The lemma ide_w shows that "w" is an identity in the bicategory "B".
For all terms 't' in the set 'ts' and for all indices 'i' in the free variables of term 't' that are less than 'n', if 'Some f' equals the match of 'ts' and 'xs', then mapping the evaluation of term 't' (where 't' is tabulated by the function 'f' from 0 to 'n') over the set 'ts' equals 'xs'.
: given a sequence of events 'evs' in 'ya' and an agent 'A' not in 'bad', if the cipher 'A' containing the agent 'B', key 'K', and nonces 'NA' and 'NB' is in the parts that spies can see in 'evs', then the event 'ya3' with parameters 'A', 'B', 'NA', 'NB', and 'K' is in the set of events 'evs'.
The lemma named "rev_iterateo_rule_P" with case names "minv", "inv0", "inv_pres", and "i_complete" is assumed. It assumes the following: 1. "MINV": The map "m" is invariant. 2. "I0": The initial state "σ0" is in the domain of the abstraction of "m". 3. "IP": For any key "k", value "v", iterator "it", and state "σ", if...
The application of the right complex bounded linear function to a vector psi is equal to the ordered pair (0, psi).
The last message hold of an empty list is an empty list.
The lemma named "ns_mul_ext_union_compat" assumes that the pair (A, B) is in the set defined by the function "ns_mul_ext" with parameters "ns" and "s", and that the function "locally_refl" with parameters "ns" and "C" holds true. Under these assumptions, it shows that the pair (A + C, B + C) is also in the set defined ...
The lemma named "NextRecvCapD" assumes that "NextRecvCap'" is true for the parameters c0, c1, p, and t. If this assumption holds, it shows three things: 1. The function "next_recvcap'" with parameters as the exchange configuration of c0, the exchange configuration of c1, p, and t in the cri (presumably a context or a ...
If "Y" is a chain, then the synchronization of "S" and the supremum of "Y" over "i" with respect to "A" is equal to the supremum over "i" of the synchronization of "S" and "Y i" with respect to "A".
If "x" is a unit, then the inverse of the inverse of "x" is equal to "x".
If 'x' is regular and 's' is a selection of 'x', then the negation of 's' is also a selection of 'x'.
The lemma named "path_inv_choose_point_decrease" assumes that a path is constructed inversely from point 'q' to point 'x' and 'y' in a domain 'D' and a set 'W', and 'q' is not equal to 'x'. It shows that if 'W' is not equal to zero, then the chosen point in 'D*q' is less than or equal to the negative of the transpose o...
"Set X is a subset of Set Y if and only if the downward closure of Set X is a subset of the downward closure of Set Y."
The prefix match of the range union of the word intervals (0x40000000 to 0x5FEFBBCC) and (0x5FEEBB1C to 0x7FFFFFFF) is equal to the prefix match of 0x40000000 with a prefix length of 2 in a 32-bit word.
If the function 'f' is differentiable at the point 'g(x)' within the range of 'g' and if the function 'g' is differentiable at the point 'x' within the set 's', then the composition of the functions 'f' and 'g', denoted as 'f(g(x))', is also differentiable at the point 'x' within the set 's'.
The Dirichlet series for function 'f' converges at 's' if and only if the series formed by the nth term of the Dirichlet series for 'f' divided by the nth natural number raised to the power 's' sums to the evaluation of the Dirichlet series for 'f' at 's'.
The set of outputs represented as a list in machine M is equal to the outputs of machine M.
The prime factors of a number raised to the power of n are a subset of the prime factors of the original number.
The lemma "lst_of_sp_is_vert" assumes that there is a subpath in graph 'g' from vertex 'v1' through edges 'es' to vertex 'v2'. It shows that vertex 'v2' is in the vertices of graph 'g'.
Binding the result of binding a function "f" to a function "g" to another function "k" is equivalent to binding the function "f" to a lambda function that binds the result of function "g" applied to an input "x" to the function "k". This is a property of the bind operation in the context of a Heap monad.
If class C is a direct subclass of class D in a well-structured program G, then class C cannot be the same as class D.
If a key K is not used in the event sequence 'evs', and K is in the set of symmetric keys, and 'evs' is in the set of purchase events, then K is not in the set of keys for the analysis of the union of the set of keys 'KK' and the knowledge of the Spy in the event sequence 'evs'.
If a condition "x" is maintained at a state "τ", then at the same state "τ", the precondition "x" at "H1" and the postcondition "x" at "H2" are both valid.
The step function is bounded by the size of the instruction sequence.
The element X is in the carrier set P.
For all 'k', if 'i' is less than or equal to 'k' and 'k' is less than 'j', then given the context of 'Γ' and 'Θ', the triple '(P k) (c k) (Q k), (A k)' is derivable under the set 'F'. If this is true, then given the same context, the list of triples '(P k, Q k, A k)' for all 'k' in the range from 'i' to 'j' (excluding ...
If the intersection of the states of SA and the HA states of A is empty, then the set of pairs 'p' where the first element of 'p' is not the initial state of SA and the second element of 'p' is not the initial state of SA, and 'p' is in the set formed by inserting the initial state of SA into the HA initial states of A...
If an event 'ev' transitions the system from configuration 'c' to 'c'', and the second element of the channel status 'cs' of channel 'cid' in configuration 'c' is not 'NotStarted', and the channel 'cid' is associated with a pair of processes (p, q), then the second element of the channel status 'cs' of channel 'cid' in...
: Given a structure specification "sel ap ab compl", a complete connection "compl", a Dickson grading "d", and sets "gs", "bs", "ps", and "D" with certain properties, and a unique index for the concatenation of "gs" and "bs" in the second element of "data". If for any sets "B", "a", and "b" such that the union of "g...
If we have two natural transformations, phi from functor F to functor G in the category from A to C, and psi from functor F' to functor G' in the category from B to C, then the Hom functor applied to phi and psi is a natural transformation from the Hom functor applied to G and F' to the Hom functor applied to F and G' ...
The lemma named "tag_Times_injI" assumes that the function "tag_Times" applied to sets A and B and elements x and y results in the same output. It also assumes that the concatenation of x and some element z is in the product of sets A and B. Under these assumptions, it shows that the concatenation of y and the same ele...
The degrees of the filtered piecewise-deviation list "I" from "x" is less than or equal to the degrees of "x".
Given the assumptions that Y is simply distributed in measure M with probability Py, and the pair (X, Y) is also simply distributed in measure M with probability Pxy, it shows that the conditional entropy of X given Y is equal to the negative sum over all (x, y) in the space M of the product of Pxy (x, y) and the logar...
The lemma named "semialg_val_ineq_set_plus" assumes that "N" is greater than 0, "c" is in the carrier of "SA N", and "a" is in the carrier of "SA N". It shows that the set of "x" in the carrier of "Qp^N" such that the valuation of "c x" is less than or equal to the valuation of "a x" plus an extended integer "n" is sem...
"for all n, if the function f of n is less than or equal to x, then the limit of the function f is also less than or equal to x".
The well-founded relation "wcode_double_case_le" is well-founded.
There exist a heap 'h', a heap 'h'', a host, and a new shadow root pointer such that the heap 'h' is well-formed, the type of heap 'h' is well-formed, and the heap 'h' is known. Furthermore, the heap 'h' can attach a shadow root to the host and return a new shadow root pointer, resulting in the heap 'h''. However, it i...