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The Cartesian product of the set containing 'x' and the set containing 'y' is equal to the set containing the pair '(x, y)'.
The lemma named "decl_evalw'" assumes two conditions: first, that the memory state 'mem' remains unchanged as 'mem'', and second, that the mode state 'mds'' is the result of updating the mode state 'mds' with 'upd'. Under these assumptions, it shows that the transition from the configuration with a Skip command followe...
If proposition P is true, then the generalized operation tag of P is also true.
The lemma named "bilinear_eq_stdbasis" is defined with a fixed function 'f' that maps from one Euclidean space to another. It assumes that both 'f' and 'g' are bilinear functions. If for all 'i' and 'j' in the basis, the function 'f' applied to 'i' and 'j' equals the function 'g' applied to 'i' and 'j', then it shows t...
For any fixed complex vector 'u', the inner product of 'u' with itself belongs to the set of non-negative real numbers.
For any two elements X and Y, if X and Y are in the carrier of the quotient ring of R by the additive kernel of the ring homomorphism from R to S, and if the image of X under the homomorphism is equal to the image of Y under the homomorphism, then X is equal to Y.
: Assuming there exists a 'y' such that property 'P' holds for 'x' and 'y', and 'f' is defined as some function 'P', it can be shown that property 'P' holds for 'x' and the function 'f' applied to 'x'.
There exists a value 'x' such that the assertion 'P x' or the assertion 'exists a value 'x' for which 'Q x' holds' is equivalent to 'there exists a value 'x' such that the assertion 'P x' or 'Q x' holds'.
If 'f'' is a morphism from 'd' to 'r' in category 'A' and 'u' is a morphism from the object mapped by functor 'F' at 'r' to 'c' in category 'B', then the value of the arrow in the functorial map 'umap_fo' of 'F' at 'c', 'r', 'u', and 'd' applied to 'f'' is equal to the composition in category 'B' of 'u' and the arrow m...
The view of a zipper model with an empty list and a tree 't' is equal to the tree 't'.
For any linearly ordered integral domain, if x is less than or equal to 0 or x is greater than or equal to 1, then the product of x and (x minus 1) is greater than or equal to 0.
Under the assumptions that "p0L" is not equal to "p1L", "p0R" is not equal to "p1R", and "ps" are distinct, if the pair "(c0, c1)" is equal to the result of the function "combine" applied to the pairs "(p0L, p1L)" and "(p0R, p1R)" with parameters "l" and "ps", then "c0" is not equal to "c1".
The lemma named "has_redex_iff_not_Ide_leftmost_strategy" shows that for any term 't', 't' is an arrow if and only if 't' has a redex (reducible expression) is equivalent to the statement that the leftmost strategy of the source of 't' is not an identity.
The opposite of semicategory morphism FF is an isomorphism from the opposite of semicategory AA to the opposite of semicategory BB.
The lemma named "weakPsiCongScopeExt" is defined with respect to a name 'x', a type 'b' named 'Psi', and two psi types 'P' and 'Q'. It assumes that 'x' is fresh for 'Psi' and 'P' is fresh for 'x'. It shows that under these conditions, 'Psi' in parallel with the restriction of 'x' on the composition of 'P' and 'Q' is we...
If a point 'z' is on a circline 'H', then 'z' is in the set of points that make up the circline 'H'. This is assumed to be a simple, or 'simp', lemma.
If "i" is an element of the set "is" and the lowest value of "is" is equal to "vr", and if "i" is ordered, then for any assignment "ass", the value of "fi" at "ass" is equal to the value of "i" at "ass". Therefore, the value of "restrict_top" function applied to "i", "vr", and "vl" at "ass" is equal to the value of "bf...
The number of relations in a tree 't' is equal to the length of the inorder traversal of the tree 't'.
If we have a category 'BB' and two functors 'GG' and 'SS' such that 'GG' maps from 'AA' to 'CC' and 'SS' maps from the opposite category of 'BB' cross product 'CC' to 'DD', and 'b' is an object in 'BB', then the right composition of the contravariant functor with 'SS' and 'GG' at the opposite category of 'BB' and 'AA' ...
Two requests (represented by "reqm A r n I B" and "reqm A' r' n' I' B'") are equal if and only if their corresponding components are equal. That is, A is equal to A', r is equal to r', n is equal to n', I is equal to I', and B is equal to B'.
The transitive implication closure of a set 'c' is a subset of the transitive implication closure of the reflexive transitive closure of 'c'.
The lemma shows two things: 1. The measure of the distribution function 'f' from measure space 'Mf' to 'Nf' is measurable in 'Mf' if and only if 'Nf' is measurable in 'Mf'. 2. The measure of 'Mg' in the distribution function 'g' from 'Mg' to 'Ng' is measurable if and only if 'Mg' is measurable in 'Ng'.
If two functions are related by the relation "rel_fun" (which means they are functionally equivalent), and the first function is a bounded sesquilinear function and the second function is a bounded antilinear function, then these two functions are related by the relation "pcr_cblinfun" (which means they are equivalent ...
Any number 'x' is less than or equal to the result of rounding down 'x' to a certain precision 'prec' plus 2 raised to the power of negative 'prec'.
The lemma "event_can_go_back_if_no_sender_trace" shows that if the following conditions are met: 1. There is a trace from configuration 'c' to configuration 'c'' through a series of events 't'. 2. All events in 't' do not occur on the same process as event 'ev''. 3. Event 'ev'' can occur in configuration 'c''. 4. Even...
The lemma SN_SSN: Assuming "a" that the strong normalization of term "t" applied to "k" is true, it shows that the strong sequential normalization of "k" is also true.
The lemma named "post_del_adjR" assumes that the invariant holds for the triple (l, (a,lv), r), the pre-adjustment condition holds for the triple (l, (a,lv), r'), and the post-deletion condition holds between r and r'. Under these assumptions, it shows that the post-deletion condition holds between the triple (l, (a,lv...
If a set 's' is restricted to a set 'k' resulting in a set 's'', and an element 'a' is in 'k', then the restriction of the set 's' combined with the ordered pair of 'a' and 'b' to the set 'k' results in the set 's'' combined with the ordered pair of 'a' and 'b'.
Given a measurable predicate P and assuming that for all time points t, it is almost everywhere in T t that the intersection of P and the intersection of the hold set X and the next hold set X until P does not hold, then the expected measure of the set of outcomes in the space of T s where the hold set X holds until P ...
The domain of the identity morphism of the product category of three categories A, B, and C is equal to the objects of the product category of A, B, and C.
The lemma named "effect_run" assumes that the effect of command "c" changes the state from "h" to "h'" and returns "r". Under this assumption, it shows that running the command "c" changes the state from "Some h" to "Some h'" and returns "r".
If h1 and h2 imply h0, and h1 and h3 also imply h0, then h1 and h2 imply h0 and h2 is equal to h3.
If for all 'x', the function 'f' with 'x' and 'y' as arguments is monotone continuous when using the least upper bound 'lub'' and order 'ord'' for 'y', and the supremum and less than or equal to order for 'x'; and if for all 'y', the function 'f' with 'x' and 'y' as arguments is monotone continuous when using the least...
For any two fixed points 'u' and 'v' in a one-dimensional Euclidean space, the image of the closed interval between 'u' and 'v' under the function that maps each point to its first coordinate is equal to the closed interval between the first coordinates of 'u' and 'v'.
The coefficient of the term in the product of 'f' and 'g', whose degree is the sum of the degrees of 'f' and 'g', is equal to the product of the coefficients of the terms in 'f' and 'g' of their respective degrees.
"a" is greater than zero.
If all chains in a rule set "rs" do not call a chain "c", then the semantics of the rule set "rs" under the environment "Γ" updated with a mapping from "c" to "x" and the packet "p" leading to the target "t" is equivalent to the semantics of the rule set "rs" under the original environment "Γ" and the packet "p" leadin...
If a function F constrains A to the union of A' and B, and F also constrains B to B', then F constrains A to the union of A' and B'.
The product of t of x and t of y is equal to the product of t of y and t of x.
The lemma named "cat_SemiCAT_obj_isoE" assumes that object "\<AA>" is isomorphic to object "\<BB>" in the category "cat_SemiCAT" with a parameter "\<alpha>". If this assumption holds, it obtains that "\<AA>" is semantically equivalent to "\<BB>" under the same parameter "\<alpha>".
For any element pointer, the reading of the object model from the heap 'h' to the heap 'h' is preserved when no changes are made to the object.
If the following conditions are met: 1. The list of deltas is not empty. 2. For any two value lists, if they satisfy condition B, then the initial state of the first value list in the head of the deltas list is the same as the initial state of the second value list. 3. For any two consecutive deltas in the deltas list...
The lemma "lemLineContainsEndpoint" demonstrates that the point 'q' is on the line that joins points 'p' and 'q'.
The lemma named "distinct_delete" assumes that the list of first elements from the pair list "al" is distinct. It shows that the list of first elements from the pair list "al" remains distinct even after deleting the element "k".
The predicate "p" holds for the "index n".
The lemma named "cyclic_on_image" assumes two conditions. The first condition is that the function "f" is cyclic on the set "S". The second condition is that for any element "x" in the set "S", applying function "g" to the result of function "h" applied to "x" is equal to applying function "h" to the result of function...
If a list (xs) strictly descends according to a preorder, then the elements in the list are distinct.
The bra of the ket vector "v" is equal to the bra vector "v".
If a pair (S, U) is in the ChiRel of A, and another pair (U, T) is also in the ChiRel of A, and if S is in the states of SA, and SA is in the SAs of A, then T is not in the states of SA.
The lemma named "iteratei_postfixed_correct" assumes that the invariant "invar_trie" holds for a trie "t" of type "('key, 'val) trie". It shows that the set iterator "iteratei_postfixed ks0 t" of type "('key list \<times> 'val, '\<sigma>) set_iterator" is equivalent to the set formed by applying the function "(\<lambda...
Converting an empty bit-vector to an integer results in zero.
In the system context, there is a lemma I stating that the system preserves the invariant I.
For any instance 'i' of the protocol 'paodv', it holds that for any state and action pair '(ξ, l)' in the AODV model, if the action 'l' is the first step of the PRerr process, then for every IP address in the domain of the destinations of 'ξ', the following conditions must be met: the IP address is in the set of valid ...
The lemma "eOp_invert" assumes that the operation "eOp" with the modulus "delta", the input "einp", and the binary input "ebinp" equals "OK X". It shows that there exist an input "inp" and a binary input "binp" such that "einp" equals "OKI inp" and "ebinp" equals "OKI binp", and "X" equals the operation "igOp" with the...
The lemma named "weakStatImpMonotonic" is defined with the following parameters: a fixed Psi of type 'b, P of type psi ('a, 'b, 'c), A of type set containing tuples of 'b and psi ('a, 'b, 'c), Q of type psi ('a, 'b, 'c), and B of type set containing tuples of 'b and psi ('a, 'b, 'c). The lemma assumes that Psi entails...
The lemma "Abs_lst_fcb" is defined with the following conditions: - It fixes 'xs' and 'ys' as variables of type 'a' which is a finite set, and 'S' and 'T' as variables of type 'b' which is also a finite set. - It assumes that the abstraction of 'T' with respect to the list 'ba xs' is equal to the abstraction of 'S' wi...
"x is a Just value and the value inside x is equal to y" is the same as "x is equal to Just y".
In the context of a recursive event sequence, a key (shrK A) is part of what the spies can see if and only if A is in the set of compromised or "bad" agents.
"L" is greater than or equal to zero if "f" is Lipschitz on "X" with Lipschitz constant "L".
If "i" is a node in deterministic finite automaton "A" and "j" is a node in deterministic finite automaton "B", then the first element of the product of depth-first search on "A" and "B" starting from nodes "i" and "j" respectively, at position "i" and "j", is equal to "Some 0".
If 'Top' is less than or equal to 'x', then 'x' equals 'Top'.
The lemma "zippers_view_blinding_of" assumes that "z" is an element in the set produced by the function "zippers_view" when applied to the pair "(p, t)". It shows that the function "blinding_of_view" when applied to the view of the zipper "z" is equal to the view of the zipper when the function "blind_view_path" is app...
If the union of two relations 'r' and 's' is a valid binary relation, then both 'r' and 's' are valid binary relations.
The lemma "performop_preserves_InvJustifiedII" assumes that "InvJustifiedII" holds for a given state "c0", the operation "next_performop'" transitions the state from "c0" to "c1" with parameters "p", "Δneg", "Δmint_msg", and "Δmint_self", and "InvCapsNonneg" holds for the state "c0". It shows that under these assumptio...
The value of the function T_vebt_buildupi with input n is greater than 0.
If a program P is well-formed, and if a certain type relationship (denoted by "sty_one") holds between two types "ty_x_d" and "ty_x_s", and if a field type "f" of type "ty_f" exists in the type "ty_x_s" in the program P, then the same field type "f" of type "ty_f" also exists in the type "ty_x_d" in the program P.
The lemma named "m_diff_image_weak" is about a function 'f' that maps from a real vector space to another real vector space. It assumes that a set 'S' is Lebesgue measurable, and that for every element 'x' in 'S', the function 'f' has a derivative 'f' prime at 'x' within 'S'. It also assumes that the absolute value of ...
For any value 'x' of type 'V', the multiplication of 'x' with the sum of 'y' and 'z' is equal to the sum of the product of 'x' and 'y' and the product of 'x' and 'z'.
The least common multiple of 'a' and 'b' multiplied by the greatest common divisor of 'a' and 'b' is equal to the normalized value of 'a' multiplied by the normalized value of 'b'.
There exists a value 'v' such that 'v' is the result of the function 'input2state' applied to 'i' and 'x', if and only if 'x' is less than the length of 'i'.
The lemma named "linear_lim_0" assumes that the function "f" is a bounded linear function. It shows that the limit of the function "f" approaches 0 at the point 0.
If the function "auth_restrict" is true for the parameters "ainfo", "uinfo", and the list consisting of "x", "y", and "hfs", then it is also true for the parameters "ainfo", the updated "uinfo" with "y", and the list consisting of "y" and "hfs".
The number of agents is 4 and the number of alternatives is also 4.
Theorem 4: Assuming that the function "cop_F_rejected_inv" holds for the set "ds" and the function "fp", and that "x" is an element of the set "RH" of "fp", it can be shown that "x" is also an element of the set "RH" of the function "cop_F" applied to "ds" and "fp".
If the list "ws" is not empty, then the result of applying the function "step_invariant" to the tuple (ws, ps, N) and then mapping the second element of each pair in the resulting list "ws'" and the resulting "N'" is equal to the result of applying the function "step" to the tuple formed by mapping the second element o...
The affine algebraic set in the ring R with n variables, where the set only contains the zero element of the polynomial ring in n variables over R, is equal to the carrier of the nth power of the ring R.
If the function "add_spmat" applied to the pair (a,b) followed by the array "arr" and the array "brr" equals the pair (ab, bb) followed by a list, and if "aa" is less than "a", and if the sparse vector consisting of the pair (aa, ba) followed by "brr" is sorted, then "aa" is less than "ab".
The lemma named "ValD_copy_chain" asserts that "ValD_copy_i" forms a chain. This is an introductory and simplifying lemma.
The lemma named "R_component_ob" assumes that "pR'" is an element of "R" with parameters "M", "q", "q'", "pP", and "p". It then obtains "pR" where "pR'" is equal to the concatenation of "pP" and "pR".
The symmetric part of the restriction of a relation 'r' to a set 'A' is equivalent to the restriction of the symmetric part of the relation 'r' to the set 'A'.
Four times the sum of the cubes of all natural numbers from 0 to n is equal to the square of the product of n and (n + 1).
The space of the interval measure F is equal to the universal set.
If set B and set A' are finite, then the set that consists of all elements produced by the function f for each pair of elements a and b, where a is in both sets A and A', and b is in set B, is also finite.
Inserting a key "K" into a set "H" does not change the public parts of the set "H".
In the context of an imap, the lemma named "concurrent_append_store_independent_technical" assumes the following conditions: 1. The variable 'i' is equal to 'mo'. 2. The sequence 'xs' is a prefix of 'j'. 3. The pair '(i, Append i e)' is an element in the set of messages delivered by the node 'xs'. 4. The pair '(r, Sto...
The lemma named "is_fmapD" assumes that "M" is a finite map, and that the pairs "(a, b1)" and "(a, b2)" are both in "M". It then shows that "b1" must be equal to "b2".
The lemma shows three things: 1. The number of comparison sequences in a list 'xs' is less than or equal to the length of 'xs' minus one. 2. The number of ascending comparisons between an element 'x' and a list 'ys' is less than or equal to the length of 'ys'. 3. The number of descending comparisons between an element...
The lemma named "cartesian_product_car_complement_right" assumes that set A is a subset of the carrier of the m-th power of ring R. It shows that the difference between the carrier of the sum of the n-th and m-th powers of ring R and the Cartesian product of the carrier of the n-th power of ring R and set A is equal to...
If k is equal to the sum of m and n, then converting m bits of the binary number obtained after dropping n bits from c is equivalent to taking the first m bits from the bit list obtained by converting the binary number c to a bit list of length k.
The rotation of a diamond cube is a valid two-cube.
If a list 'xs' is sorted according to a relation 'Q', and another list 'ys' is also sorted according to the same relation 'Q', and for any elements 'a', 'x', and 'y', if 'Q' holds between 'x' and 'y', then 'P' holds between 'f(x, a)' and 'f(y, a)', and for any elements 'x', 'y', 'u', and 'v', if 'x' and 'y' are in the ...
If we have a category 'BB' and a functor 'GG' from 'AA' to 'CC', and if 'b' is an object in the opposite category of 'BB' and 'a' is an object in 'AA', then the object map of the right composition of the contravariant functor 'SS' and 'GG' applied to the pair 'b, a' is equal to the object map of 'SS' applied to the pai...
The lemma named "power_neg" assumes that "a" is less than zero. It shows that if "n" is odd, then "a" to the power of "n" is less than zero, and if "n" is even, then "a" to the power of "n" is greater than zero.
The relational structure "rell_F" for three relations L1, L2, and L3 is equivalent to the composition of two group actions. The first group action is the inverse of the group formed by the set of all elements x such that the first set of x is a subset of pairs (x, y) where L1 relates x and y, the second set of x is a s...
The application of the function "ValK_copy" to the bottom element results in the bottom element.
The validity of the function that maps any value to the base void type with no value is false.
Lemma disjoint_sets_insert: If x is not an element of set N, and sets M and N are disjoint, then the sets formed by inserting x into M and N remain disjoint.
The set of edges in the policy created by applying the invariants, excluding the edges already present in the original policy, is equal to the union of the following sets: 1. The set of all pairs of identical nodes in the original policy. 2. The set containing the pairs ("Encryption_A", "Sensors_A"), ("Client_A_out", ...
The second tau move function, when applied to a program P, a heap h, a stack stk, a block of code (blocks1 n Ts body) and exception pointers pc' and xcp', is equivalent to the same function applied to the same program, heap, stack, and exception pointers, but with the body of the code block instead of the entire block.
If X is in the guard of nonce n and keys Ks, then nonce n is not in the key parts of the set containing X.