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If for any configuration 'c', the local states are equivalent, then the execution equivalence 'Equiv_Exec' holds for the given model 'm', state equivalence 'equiv_states', local state 'localState', input function 'input_fun', output function 'output_fun', transition function 'trans_fun', and configuration 'c' at any st...
The lemma named "expectation_finite_uniform_indicator" assumes that "M" is a uniform count measure of "S" and that "S" is finite. It shows that the expectation of the function that maps "x" to the indicator of "T(x)" in "y" is equal to the probability of the set of "x" in "S" for which the indicator of "T(x)" in "y" eq...
If the invariant "rqrm_invar" holds for "rqrm", then the lookup of "q'" in the "rqrm" after adding a pair "q, r" to "rqrm" is equal to: if "q" is equal to "q'", then "r" is inserted into the lookup of "q'" in "rqrm", otherwise, it is just the lookup of "q'" in "rqrm".
The variables of a linear polynomial (lp) are equal to the finite set of keys in the domain of the linear polynomial map of lp.
The converse of the relation of complex conjugate subspaces with respect to a relation R is equal to the relation of complex conjugate subspaces with respect to the converse of R.
The Mangoldt function of any number 'd' is always greater than or equal to zero.
The representation of x is less than or equal to n.
A function 'f' has a vector derivative 'x' at a point 'F' if and only if the real part of the function 'f' at point 'x' has a field derivative equal to the real part of 'x' at point 'F' and the imaginary part of the function 'f' at point 'x' has a field derivative equal to the imaginary part of 'x' at point 'F'.
The lemma named "discr" introduces the following statements: 1. If an atomic action 'atm' preserves atomicity, then 'atm' is discrete. 2. If command 'c1' is discrete and command 'c2' is discrete, then the sequence of 'c1' and 'c2' is discrete. 3. If command 'c1' is discrete and command 'c2' is discrete, then the condi...
, assuming "X" is a subset of "E", "X" is not independent in "E" if and only if there exists a circuit "C" in "E" that is a subset of "X".
If the binary trees 'l' and 'r' are both height-balanced, and the height of 'l' is less than or equal to the height of 'r', then the height of the binary tree resulting from joining 'l' and 'r' with 'x' on the left is equal to the height of 'r'.
The derivative of a function 'f' at a point 'x' equals 'l' if and only if there exists a function 'g' such that for all 'z', the difference between the function 'f' at 'z' and 'f' at 'x' equals the product of the function 'g' at 'z' and the difference between 'z' and 'x'. Additionally, the function 'g' is continuous at...
The weakest precondition of the union of X and Y with respect to Q is less than or equal to P if and only if the weakest precondition of X with respect to Q and the weakest precondition of Y with respect to Q are both less than or equal to P.
The lemma named "lin_indpt_dim_col_lt_dim" is about a matrix A of arbitrary type 'a. It assumes that A is a matrix with n rows and nc columns, that the columns of A are distinct, and that the set of columns of A is linearly independent. It then shows that the number of columns nc is less than or equal to the dimension.
If a variable 'x' is in the subterms of the set of terms 'A' in a stateful strand, then 'x' is also in the variables of the stateful strand 'A'.
The lemma eeqButPID_sym: Assuming "eeqButPID s s1", it shows that "eeqButPID s1 s". This means that if s is equivalent to s1 except for the PID, then s1 is also equivalent to s except for the PID.
A register is formed from a getter-setter pair if the getter-setter pair is valid.
The lemma named "bal_i_ins_Unbal" assumes that the result of inserting a node "n" with data "d" and key "x" into a tree "t" is an unbalanced tree "t'". It then shows that the tree "t'" is balanced in terms of its size and height.
The right-hand side of the command "c" is finite.
The value range of the object map of the parallel functor, constructed from functors F and F' and objects aa, bb, aa', and bb', is a subset of the objects of the category CC.
If "a" is an element of the set "Vset α" and "b" is also an element of the set "Vset α", then the set difference "a - b" is also an element of the set "Vset α".
The lemma named "blinfun_bij1" is given a function 'f' that maps from a type 'a' to a Euclidean space of the same type 'a'. It assumes that the composition of 'f' and 'g' equals the identity bounded linear function. It then shows that the application of the bounded linear function 'f' is a bijection.
If the following conditions are met: 1. There does not exist an 'x' such that both 'S(x)' and 'Q(x)' are true. 2. For all 'x', if 'P(x)' is true then either 'Q(x)' or 'R(x)' is true. 3. If there does not exist an 'x' such that 'P(x)' is true, then there exists an 'x' such that 'Q(x)' is true. 4. For all 'x', if either...
The function that maps U to the equivalence class of "freeright U" under the relation "msgrel" respects the relation "msgrel".
The sum of the function 'f' for all 'x' less than 'n' is equal to the sum of the function 'f' for all 'x' in the list from 0 to 'n' excluding 'n'.
The lemma named "monad_state_nondetT" is being proven. This lemma is related to the concept of monads in the context of state and non-determinism. Although it's not typically logical to assume a commutative state monad, the proof is being carried out regardless. The lemma takes two fixed elements, "get" and "put". It a...
: Given the assumptions that: - The set "G" is finite, - Only "G" is in "G", - For any "chi" and "j", if "chi" is in "G", then the suffix of "w" starting at position "i" satisfies "chi" if and only if the suffix of "w" starting at position "i + j" also satisfies "chi", - "G psi" is in the intersection of "G" and the s...
If "a" is less than or equal to "b" under the K-order, then the function of "s" and "a" is less than or equal to the function of "s" and "b" under the S-order.
If s1 and s2 are low equal, then the state variable of s1 is equal to the state variable of s2.
If sigma is a valid strategy for player p and there is a move from v0 to w0, then the strategy where sigma is updated at v0 to w0 is also a valid strategy for player p.
The nu of the meet (or intersection) of x and y is equal to the meet (or intersection) of the nu of x and the nu of y.
The star of omega of any element 'x' is equal to 1 plus the omega of 'x'.
If a six-tuple (M, l, s, k, ll, t) is in the multistep relation MStep and k is greater than or equal to 1, then there exist integers n, m and r, and a list l1 such that the five-tuple (M, l, s, n, l1, r) is in the step relation Step, the five-tuple (M, l1, r, m, ll, t) is in the multistep relation MStep, and the sum of...
The inner product of the negation of vector xs and vector ys is equal to the negation of the inner product of vector xs and vector ys.
The greatest common divisor (gcd) of the rational polynomial of an integer polynomial 'f' and the rational polynomial of an integer polynomial 'g' is equal to the scalar multiple (smult) of the inverse of the leading coefficient of the gcd of 'f' and 'g' and the integer polynomial of the gcd of 'f' and 'g'.
The lemma states a test scenario. First, it gets the body of the document. Then, it sets a test ID as 'test7'. It creates a new 'div' element and assigns the test ID to this element. The element is then appended to the body of the document. It retrieves the element by its ID and asserts that the retrieved element is th...
For a given less than or equal to function "cless_eq" and a less than function "cless", both of type 'a to 'a to bool, a linear order is established.
The lemma named "store_size" assumes that if a value 'v' is stored at an offset 'off' in a memory 'm' at index 'n' with a length 'l', resulting in a new memory 'm'', then the size of the original memory 'm' is equal to the size of the new memory 'm''.
If "h" is in the existence interval of "x0" and "i" is in the existence interval of the flow of "x0" at "h", then the derivative of the flow at "x0" at the time "h + i" is equal to the composition of the derivative of the flow at the flow of "x0" at "h" at time "i" and the derivative of the flow at "x0" at time "h".
: - The globally operator applied to a proposition P on a state sigma is equivalent to the conjunction of the proposition P and the next state of the globally operator applied to P on the same state sigma. - The eventually operator applied to a proposition P on a state sigma is equivalent to the disjunction of the pro...
If 'f' and 'g' are both bijective functions, then the sum of the absolute permutations of 'f' and 'g' is equal to the absolute permutation of the composition of 'f' and 'g'.
For any three ordinals x, y, and z, the sum of x and y, when further added to z, is equal to the sum of x and the sum of y and z. This is the associative property of addition for ordinals.
If the sequence 'sb' is appended with a volatile write operation 'Write\<^sub>s\<^sub>b' with address 'a', data function '(D,f)', value 'f \<theta>', and attributes 'A', 'L', 'R', 'W' to form 'sb'', and if the ghost state 'ghst' is '(\D, \O, \R, \S)' and the new ghost state 'ghst'' is '(True, \O, \R, \S)', then a trans...
If the following conditions are met: 1. The tag of a capability 'c' is true. 2. The load permission of 'c' is true. 3. The offset of 'c' plus the size of a type 't' is less than or equal to the base of 'c' plus the length of 'c'. 4. The offset of 'c' is greater than or equal to the base of 'c'. 5. The offset of 'c' is...
If x is equivalent to y, then y is equivalent to x.
If a configuration (c, Stuck) is not final, then there exists some configuration (c', s') such that the system transitions from (c, Stuck) to (c', s') according to the transition relation Γ.
The lemma named "qp_close_after_exexH_k" is about a fixed complex matrix P. It assumes that P is a quantum predicate. It shows that if k is less than n, then the result of multiplying the adjoint of "exexH_k k" with P and "exexH_k k" is also a quantum predicate.
For any fixed function 'f' and polynomials 'p' and 'q', the insertion function applied to the sum of 'p' and 'q' is equal to the sum of the insertion function applied to 'p' and the insertion function applied to 'q'.
The lemma named "filter_code" is defined for a mapping 't' of type 'a' (which has a ccompare) and 'b'. It shows three things: 1. The filter operation on a Mapping 'm' is equivalent to creating a new Mapping where for each key 'k', if 'm' at 'k' is None, the result is None, otherwise if the predicate 'P' holds for 'k' ...
A model 'ls' satisfies a DIMACS format 'cs' if and only if all clauses 'c' in 'cs' have at least one literal 'l' in 'ls' that is true, and all literals in 'ls' mapped to their variables are distinct.
The set of constraints generated from a matrix-vector pair A and b is finite.
The lemma named "EVEN_union_ODD" shows that the union of the set of even numbers and the set of odd numbers equals the universal set.
If beta is in the trace of the induced event structure of SES, and Adm of V, rho, the trace of the induced event structure of SES, beta, and e holds, then there exists a state s in the state set of SES such that s0 of SES, beta implies s in SES and En rho s e holds.
If a polynomial "p" is linear, then the lower bound of "p" when it approaches negative infinity is zero.
If a matrix A, over a field, has a set of columns that form a basis, and the cardinality (or size) of this set of columns is equal to the number of columns in A, then the rank of matrix A is equal to the number of its columns.
The opposite of the small category functor maps from the opposite of the small category A to the opposite of the small category B in a tiny way.
If a list "l" is unique and a pair (k, x) is in the set of "l", then the mapping of "l" with "k" will result in "Some x".
The relation with the target implies that the transitive closure of the target relation is a strong reduction bisimulation. It fixes two sets, TRel, a set of pairs of 'procT', and Rel, a set of pairs of processes 'procS' and 'procT'. It assumes that Rel is a strong reduction bisimulation in the source-target calculus, ...
The middle sums of a list consisting of an element 'x' followed by a list 'xs' is equal to the union of the front sums of the same list and the middle sums of the list 'xs'.
If a message M is in the dynamic fake message set of bad actors, keys, and channels, and if the extraction of bad actors, keys, and channels is a subset of the synthesized analysis of X, then the synthesized analysis of the insertion of message M and keys into the set is a subset of the synthesized analysis of X.
The partial pullback of the union of two sets (S1 and S2) under a function (f) and a natural number (m) is equal to the union of the partial pullbacks of the two sets under the same function and natural number.
If the i-th element of the list xs is not "NoMsg" and i is less than the length of xs, then the last message in xs is not "NoMsg".
If the data space of P is equal to the data space of Q, and if i is less than the number of parts in the data space of Q, then the i-th data value of the option override of P and Q is equal to the option override of the i-th data value of P and the i-th data value of Q.
The intruder knowledge function applied to the substitution of a stateful strand A with a substitution delta is equal to the intruder knowledge function applied to A, then substituted with delta.
The lemma "del_min_mset" is about a fixed heap, which is a function mapping integers to integers. It assumes the following conditions: 1. The integer 'l' is less than the integer 'r'. 2. The heap is a binary heap from 'l' to 'r'. 3. The modified heap is the same as the original heap, except at the position 'l', where ...
The lemma named "mat_inter_num_conv" assumes that "j1" and "j2" are less than the number of columns in matrix "N". It shows that the intersection of the j1-th and j2-th elements of the list "B" is equal to the result of the function "mat_inter_num" with inputs "N", "j1", and "j2".
If an integer m is greater than or equal to zero and less than p raised to the power of k, then m is an element of the carrier of the residue ring of Zp at k.
If the content (measure) of the closed box from 'a' to 'b' is zero, then the integral over this box for any function 'f' is also zero.
The lemma "Diagonalize_Tensor_Unity_Arr" assumes "u" is an arrow. It shows that the floor of the tensor product of the unit object and "u" is equal to the floor of "u".
The lemma named "map_iterator_dom_filter_correct" assumes that "it" is a valid map iterator over the map "m". It shows that if we apply a domain filter "P" to the map iterator "it", the resulting set iterator is valid over the set of keys "k" for which there exists a value "v" such that the map "m" maps "k" to "v" and ...
A formula "nFAND φs" is satisfied in a bounded model "A" if and only if every formula "φ" in the set of formulas "φs" is satisfied in the bounded model "A".
If the following three conditions hold: 1) If not P and not Q then R, 2) If P then R, 3) If not P and Q then R, then R must be true.
If for every element 'a' in set 'A', there is a corresponding element 'a_to_b a' in set 'B', and for every element 'b' in set 'B', there is a corresponding element 'b_to_a b' in set 'A'. Furthermore, if for every element 'a' in set 'A', applying 'b_to_a' to 'a_to_b a' returns 'a', and for every element 'b' in set 'B', ...
If the inverse of shift2 function with parameters d and qs holds, then the inverse of shift2 function with parameters c and the result of shift2 function with parameters c, d, and qs also holds.
The lemma named "the_wall_betw_nempty" assumes that the wall between chamber C and chamber D is not empty. It shows that this wall between chamber C and chamber D is included in the set of all walls, and that this wall separates chamber C and chamber D.
If we have a successful operation of updating the mode of a shadow root pointer in a heap 'h' to a new heap 'h'', then the well-formedness of types in the original heap 'h' is preserved in the new heap 'h''.
The coefficient of the inverse of f at the subdegree of the inverse of f is equal to the inverse of the coefficient of f at its subdegree.
If F is compatible with G, and the union of F and G is compatible with H, then F is compatible with the union of G and H.
If "x" is an element of "A" and the second element of each pair in the set "ss" is a subset of "A", and if the result of the function "merge" with inputs "c", "pc", "ss", and "x" is not equal to the top element, then the result of the function "merge" with inputs "c", "pc", "ss", and "x" is equal to the least upper bou...
If function 'f' is a ring homomorphism from ring 'A' to ring 'R', then 'f' is also an additive group homomorphism from 'A' to 'R'.
If "v" is a valuation on the field "K", "aa" is an element of the field "K", "aa" is not zero, and the valuation of "aa" is greater than or equal to zero, then the product of an integer "N" and the valuation of "aa" is less than or equal to the valuation of the expression "1 plus or minus the negative of (1 plus or min...
The lemma named "errMOD_igSwapIGOpSTR" assumes that the following conditions are true for a given module "MOD": 1. "igWlsAbsIsInBar MOD" - The well-sortedness of abstractions is preserved in the module. 2. "igOpIPresIGWls MOD" - The operation preserves the well-sortedness in the module. 3. "igSwapIPresIGWls MOD" - Th...
For a function 'r' defined on a domain, the function that maps each element 'x' in the domain of 'r' to its corresponding value in 'r' is equal to 'r' itself.
The lemma named "properties_mod_lambda" assumes that "mu" is less than or equal to "i". It shows that the "i"-th element of the sequence "seq" is equal to the element of the sequence "seq" at the position "mu" plus the remainder of the division of "(i - mu)" by "lambda".
: Assuming the sum of the scalar multiplication of 'j' and vector 'v', and 'k' and vector 'w' is not zero, it shows that the sum of the scalar multiplication of 'j' and the product of vector 'v' and the representation of collineation 'C', and 'k' and the product of vector 'w' and the representation of collineation 'C'...
The lemma named "infinite_eqpoll_lists" assumes that "A" is infinite. It shows that the set of all lists of "A" is equivalent to "A".
The counterclockwise orientation of three mirrored points with respect to a point 'x' is the same as the counterclockwise orientation of the original three points 'y', 'z', and 'w'.
If the keys are bad in the context of a set of compromised keys (Ik) and a set of bad agents (Bad), and if an element (x) is in the payload, then the keys remain bad even when this element is inserted into the set of compromised keys.
If there exists a function 'g' such that 'g' and 'f' form a conjugation pair, then 'f' is residuated.
The lemma named "triple_val_ineq_set_semialg'''" shows that the set of 3-tuples in the 3-dimensional p-adic number field, where the valuation of the second element is greater than or equal to the valuation of the third element, is semialgebraic.
The function domain scene of A is always compatible with the function domain scene of B.
Shifting the Fourier-Direct series 'f' by 0 results in the original series 'f'.
The symmetric logical relation is monotonic.
For a function 'f' from 'a' to 'b' (where 'b' is a Banach space with a second countable topology), if 'T' is a measure preserving transformation from measure space 'M' to 'N', and the function 'f' composed with 'T' is integrable over 'M' and 'f' is Borel measurable over 'N', then 'f' is integrable over 'N' and the inte...
If "m" is less than or equal to "t" and "t" is less than "n", then the immediate predecessor of the successor of "t" in the set from "m" to "n" is "t".
The sum of the list "C", where each element "x" in "C" is processed through the function "lec_rel (f x)", is equal to "Leq_Rel" if and only if for every element "x" in the set of "C", "lec_rel (f x)" is equal to "Leq_Rel".
If two lists 'xs' and 'ys' are distinct, then the selectivity of a function 'f' on these lists is equal to the selectivity of the same function on the sets derived from these lists.
If 1 is less than or equal to x, then the product of Z and x equals Z.
The parallel composition of two processes, where the first process is preconditioned by "p" and refined by "r", and the second process is preconditioned by "p'" and refined by "r'", is equal to a single process that is preconditioned by the conjunction of "p" and "p'" and refined by the conjunction of "r" and "r'".
The lemma named "fold_once_set_equiv" assumes that the sets "xs" and "ys" are equivalent. It shows that applying the function "fold_once" to both "xs" and "ys" will yield the same result.