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The lemma named "transaction_negchecks_comp_notin_fv_empty" assumes that "x" is not in the free variables of "C". It shows that the transaction negative checks component of "C" with respect to "x" is an empty set.
If the implementation of list 'xs' is equal to the implementation of list 'ys', then list 'xs' is equal to list 'ys'.
The omega power of "x" multiplied by "y" under the "ni" operation is equal to the omega power of "x" under the "ni" operation.
The lemma shows that the weight of a sub-probability mass function (SPMF) is a monotone continuous function. This is demonstrated by showing that the weight function is monotone continuous under the least upper bound of the SPMF, the order of the SPMF (which is equality), the supremum operation, and the less than or eq...
The lemma named "index_tensor_gate_unitary2" assumes that "G1" is a gate of size "m" and "G2" is a gate of size "n". It also assumes that "i" is less than the number of rows in a unit matrix of size equal to the product of the number of columns in "G1" and "G2", and "j" is less than the number of columns in the same un...
The multiplication of the imaginary unit 'i' with itself equals negative one.
If the following three conditions are met: 1. Property P holds for any number x and 0, 2. Property P holds for 0 and the successor of any number y, 3. If Property P holds for any two numbers x and y, then it also holds for their successors, then we can conclude that Property P holds for any two numbers m and n.
There exists a delta such that delta is logically equivalent to alpha with i replaced by delta, and the support of delta is the support of alpha minus the atom i.
The function pair (return composed with Weight.val, RETURN composed with Weight.val) is included in the set of function pairs that transform a non-infinity value into a value in the abstract data type A, under the assumption that the input is not infinity. This is expressed in the assertion language of the separation l...
The lemma named "axis_eq_0_iff" shows that the axis m of x equals 0 if and only if x equals 0.
The product of the assertions A and B on the pairs (a1, b1) and (a2, b2) respectively, is equal to the product of the assertion A on a1 and a2 and the assertion B on b1 and b2.
The bottom edges of the diamond are not equal to the top edges of the diamond.
The lemma named "almost_full_on_Int" assumes that two properties, P1 and P2, are almost full on two sets, A1 and A2, respectively. It shows that the property of both P1 and P2 being true is almost full on the intersection of A1 and A2.
The data access at level 1 for system A11 is equal to the set consisting of systems A21, A22, and A23.
The lemma named "rel_frontier_of_polyhedron_alt" is about a set S in a Euclidean space. If S is a polyhedron, then the relative frontier of S is equal to the union of all sets F such that F is a face of S and F is not equal to S.
The lemma named "dflt_inter_refine" shows that the return of the default intersection is less than or equal to the intersection specification of S and G.
If an interpretation "I" satisfies the constraints of a strand "S" with a substitution "theta", then the interpretation "I" also satisfies the strand "S" with the substitution "theta".
The result of folding over A using "res_fold_A" is equal to the result of folding over A using "fold_A".
If the ordered pair [a, b] is an element of the relation r, then b is an element of the function field of r. This is assumed in the context of the Isabelle proof assistant.
Substituting a term 't' in the function 'sterm_to_pterm' with an environment 'env' that has been mapped with the function 'sterm_to_pterm' is equivalent to applying the function 'sterm_to_pterm' to the result of substituting the term 't' with the environment 'env'.
The polynomial of a scalar multiplied by a polynomial 'p' with respect to 'x' and 'y' is equal to the polynomial of the scalar with respect to 'x' multiplied by the polynomial 'p' with respect to 'x' and 'y'.
If a function f is summable, then the sequence of partial sums of f converges to the infinite sum of f.
The lemma named "omit_branch_oob" assumes that the length of the branch is less than or equal to v. It shows that omitting a branch from the union of the set containing the pair (v, v') and xs is the same as omitting the branch from xs.
The lemma named "dvd_smult" assumes that polynomial "p" divides polynomial "q". It shows that polynomial "p" also divides the polynomial obtained by scalar multiplication of "q" by a constant "a".
The lemma named "freshAbs_swapAbs_id" assumes that "A" is a good abstraction, "x1" is a fresh abstraction of "A" with respect to "xs", and "x2" is also a fresh abstraction of "A" with respect to "xs". It shows that swapping "x1" and "x2" in "A" with respect to "xs" results in "A" itself.
If H is a submodule of the module M over the ring R, then H is equal to the direct sum of the zero element and H in the module M.
If the multiset obtained by applying a function 'f' to each element of a multiset 'P' equals a multiset containing a single element 'y', then for any element 'x' such that 'P' is a multiset containing only 'x' and 'f' applied to 'x' equals 'y', we can conclude 'Q'. This lemma consumes one assumption and the case is nam...
The omega power of the supremum of the product of x and L and y is equal to the product of the omega power of y and L, the supremum of the star power of y, and the product of the star power of y, x, and L.
For all 'x' greater than 'a', the polynomial 'p' evaluated at 'x' is greater than 0 if and only if the polynomial 'p' is not equal to 0, the polynomial 'p' at infinity is 1, and the count of roots of the polynomial 'p' above 'a' is 0.
For any numeral word 'w' and any number 'n', the first 'n' bits of 'w' (obtained using the 'slice1' function) are equal to the result of converting the first 'n' bits of 'w' (represented as a binary list) back into a word. If 'w' has less than 'n' bits, the 'takefill' function pads the binary list with 'False' values (...
If a state sigma satisfies the condition where a value x is obtained from an assertion of a predicate P and then used in a monadic computation M, then the predicate P holds for the state sigma and the state sigma satisfies the monadic computation M with the value True.
For any four sets C, D, E, and F of elements that can be added together, if set C is a subset of set D and set E is a subset of set F, then the set formed by adding each element of C with each element of E is a subset of the set formed by adding each element of D with each element of F.
The lemma named "stutter_free_reduced" assumes a condition, represented by "sigma", that is stutter-free. It shows that the natural transformation of "sigma" is equal to "sigma" itself.
If a state "q" is in the set "Q" and the same state "q" is in the derivative of automaton "A" with respect to the term of a generalized term "t", then "t" is in the language of the generalized tree automaton "A" with respect to the set "Q".
A number list "nl" is in the set of number lists "nls" if and only if the characteristic function "chi_fun" applied to "nls" and "nl" returns "Some 1".
If there is a transition from state "l" with clock valuation set "Z" to state "l'" with clock valuation set "Z'", and if "Z" is a subset of "V", then "Z'" is also a subset of "V".
: The function f with inputs a, b, c, and d equals the function f with inputs a', b', c', and d' if and only if a equals a', b equals b', c equals c', and d equals d'.
The lemma "permChainAppend" is defined with five fixed variables: xvec1, yvec1, xvec2, yvec2, and p, where xvec1, yvec1, xvec2, yvec2 are lists of names and p is a point in a space with a name type. The lemma assumes that the length of xvec1 is equal to the length of yvec1. It then shows two things: 1. The permuta...
If 'x' is an element of the unit group 'G', then the inverse of the inverse of 'x' is equal to 'x'.
The lemma is named "cov_invertible_real". It fixes two functions, f and g, which map from a real vector space of dimension 'n to a real number and a real vector space of dimension 'n, respectively. The lemma assumes the following: 1. For every x in set S, g has a derivative at x within S. 2. For every y in set T, h h...
If the compiled version of expression 'e' at position 'pc' is a 'Goto' instruction with offset 'i', and 'pc' is less than the length of the compiled expression, then the natural number equivalent of 'pc' plus 'i' cannot be equal to 'pc'. The second proposition, named "compEs2_Goto_not_same", states that if the compile...
The lemma assumes that for any G and phi, the function that maps x to the pair (G, the unique y such that phi y x) is proper in X. It shows that for any H, there exists an F such that for all x, the pair (F, the power of x) is necessarily equivalent to the pair (the power of x, H) in v.
The interval starting from "n'" is not a subset of the interval defined by "r", "mod m", and "c".
If a key 'k' is greater than 0 and the root order of 'k' in a binary tree 't' is maintained, then the root order of 'k' will still be maintained in the binary tree 't' after inserting 'k' and 'x' into 't'.
The transition equivalence relation is reflexive for any transition 't', meaning any transition 't' is equivalent to itself.
The lemma named "ac_in_res_ring" assumes that "x" is an element of the set of non-zero p-adic numbers. It shows that the angular component of "x" at level "n" is an element of the carrier of the residue ring at level "n".
If policy 'p' is not an empty list, then the Cp function applied to the firewall policy derived from list 'p' is equal to the Cp function applied to the firewall policy derived from the list 'p' after removing all duplicates.
The remainder of the division of k by l plus the result of the division of k by l multiplied by l equals k.
For any table "tab", the domain of the table after updating it with a new mapping from "x" to "y" is equal to the union of the original domain of the table and the set containing "x".
The lemma for the floor value of the square root of a natural number 'x' is equivalent to the floor value of the square root of the positive integer representation of 'x'.
The lemma named "analz_Un_eq_Crypt" assumes that there does not exist a key K such that K is in the analysis of G, and there does not exist a key K and a message X such that the encryption of X with K is in the analysis of G. Under these assumptions, it shows that the analysis of the union of H and G is equal to the un...
The lemma "save_restore_instr_sub1_low_equal" assumes that s1 and s2 are low equal, and t1 is the second element of the first pair returned by the function "save_retore_sub1" with parameters result, new_cwp, rd, and s1, and t2 is the second element of the first pair returned by the same function but with s2 instead of ...
1 is less than the rational number "r" converted to a linearly ordered field type 'a if and only if 1 is less than the rational number "r".
If the index 'i' is less than the length of the list 'Basis_list' (which is a list of elements from an executable Euclidean space), then the 'i'-th element of the 'Basis_list' is not zero.
For any natural number 'n' and any integer 'w', the operation 'take_bit' applied to 'n' and 'w' results in an integer that is less than 2 raised to the power of 'n'.
For a given relation "MV_Rel", if we have a function that takes an element 'x' and a set 'Y' and checks if there exists an element 'y' in 'Y' such that the modulus of 'x' is equal to the modulus of 'y', then this function is equivalent to the membership function. This is a rule for transferring properties between diffe...
The event sequence of the spliced system state, composed of a system state 's' and another system state 's'', with an action 'a', is equal to the event sequence of the original system state 's'.
The lemma named "gen_qe_eval''" is assuming that all elements in list L are of degree 2 and the length of list xs' is equal to the variable 'var'. It shows that the existence of an 'x' such that the evaluation of the conjunction of all atoms in list L (when appended with 'x' and list 'Gamma' to list xs') is equivalent ...
If "a" is a unit, then the greatest common divisor of "b" and the quotient of "c" divided by "a" is equal to the greatest common divisor of "b" and "c".
The lemma named "not_tendsto_frequently_metricE" assumes that the function f does not tend to the limit l at F. It then obtains a positive number e such that there exists frequently an x in F where e is less than or equal to the distance between f(x) and l.
If the function 'f' at point 'x' equals zero, then 'x' is not in the support of the function 'f'.
If the sum of the list "ys" is equal to the length of the list "xs", then for every "zs" in the set created by partitioning "xs" by "ys", and for every "x" in the set "zs", the union of the function "f" applied to "x" is equal to the union of the function "f" applied to every "x" in the set "xs".
The lemma "forward_alt_aux1" assumes that "i" is within the range from 1 to the length of the list "xs" minus 1, "j" is less than "i", and the "j"th element of "xs" is related to the "i"th element of "xs" through a relation "T". It shows that there exist lists "as" and "bs" such that "as" concatenated with "bs" equals ...
If you update the identity function at a point 'x' with the same value 'x', the function remains the same as the identity function.
Infinity is not equal to 1 in the context of extended natural numbers.
There exists an element in the list "ps" that, when evaluated using the functions "e", "f", and "g", returns false if and only if there exists an element in the list obtained by mapping the complement function "compl'" over "ps" that, when evaluated using the same functions, returns true.
For a function 'g' mapping real numbers to a space 'a' (where 'a' is a Hausdorff space), if 'g' is a simple path, then the image of the path under 'g' is closed.
The lemma named "acc_eq_limit" assumes that the range of r is finite (denoted by "FIN: finite (range r)"). It shows that r is accepting (denoted by "is_acc r") if and only if for all A in F, the intersection of the limit of r and A is not empty (denoted by "\<forall>A\<in>F. limit r \<inter> A \<noteq> {}").
The lemma named "sigma_eqI" assumes that both M and N are subsets of the power set of Ω, and that the sigma sets of Ω with respect to M and N are equal. Under these assumptions, it shows that the sigma algebras of Ω with respect to M and N are also equal.
If 'g' is an identity and 'g' tensor 'g' is not null, then 'g' is an endofunctor with respect to the left action of 'g' on the horizontal composition 'H'.
If a command 'c' transitions a state 's' to a state 's'', then for all command sequences 'cs', the command 'c' followed by the command sequence 'cs' starting from state 's' will eventually reach the state 's'' with the command sequence 'cs'.
If the set of elements 'xs' is equal to the vertices of graph 'G', the elements in 'xs' are distinct, and there are no cross products in the deep creation of 'xs', then the cost on the right side with the head of 'xs' and the reverse of 'xs' is equal to the cost on the left side of 'xs'.
If H1, H2, and K are subgroups of G, and H1 is a subset of K, then the intersection of the group operation on H1 and H2 with K is equal to the group operation on H1 and the intersection of H2 and K.
For a given square matrix A of order n, if the order of a complex number 'lam' in the characteristic polynomial of A is 1, and both 'v' and 'w' are eigenvectors of A corresponding to 'lam', then there exists a complex number 'a' such that 'v' is a scalar multiple of 'w'.
Swapping elements i and j in list xs results in an empty list if and only if the original list xs is empty.
The lemma named "targets_eqI" assumes that the intersection of the targets of t and the targets of t' is not empty. It shows that the targets of t are equal to the targets of t'.
"acyclic_5a of x" is equivalent to "acyclic_5b of x".
The head of "ikkbz" is in the vertices of G.
Inserting an element 'x' into the difference of two sets 'A' and 'B' is equivalent to subtracting the set 'B' (excluding 'x') from the set 'A' with 'x' inserted.
For any hypernatural number 'n', 'n' is less than 1 if and only if 'n' is equal to 0.
The theorem named "equiv" assumes that "g" is a valid control flow graph for both "ns" and "ss'", and "v" is in the set of variables used in the last node of "ns". It shows that the state of the variable "v" in "g" under state "s" is equal to the state of "v" under state "s'".
If the set Q is finite, then the set of free variables in the disjunction of the closure of the image of Q is an empty set.
The lemma named "face_of_convex_hull_insert2" is about a fixed point 'a' in a Euclidean space. It assumes that 'S' is a finite set and 'a' is not in the affine hull of 'S'. It also assumes that 'F' is a face of the convex hull of 'S'. The lemma then shows that the convex hull of the set obtained by inserting 'a' into '...
The set of elements in any list 'xs' is small.
If the image of a function 'f' applied to a set 'S' is infinite and the function 'f' is injective on set 'S', then the set 'S' is also infinite.
In the lemma within the Z set, given that A, B, and C are objects in the category of sets with a cardinality of alpha, it is shown that the composition of the function Malpha_Rel_arrow_lr with Malpha_Rel_arrow_rl in the category of sets with a cardinality of alpha is equal to the identity function on the Cartesian prod...
The lemma named "dual_transaction_strand_subst" assumes that the transaction "T" is well-formed. It shows that the dual of the transaction strand of "T" after substitution "δ" is equal to the substitution "δ" applied to the concatenation of the dual of the transaction receive of "T", the transaction checks of "T", the ...
The set resulting from updating the list "xs" with the key-value pair (k, v) is a subset of the union of the set containing the pair (k, v) and the set derived from the original list "xs".
For any sigma, if sigma satisfies the condition that either "F" always holds or "G" always holds, then sigma satisfies the condition that "F or G" always holds.
"the equality of 'x' and 'true' is 'x'".
The lemma named "gds_new_root_refine" assumes a precondition "pre_new_root v0 s". It shows that the function "gds_new_root" with parameters "gds", "v0", and "s" is less than or equal to the specification function "SPEC" with a lambda function as its parameter. This lambda function returns a state 's' such that 's' is a...
The state interpolation of v and w over the set S agrees with the state interpolation of v and w over the set S with the element z inserted, excluding the set containing z.
If "mu" and "nu" are arrows and the source of "mu" is the target of "nu", then the source of the horizontal composition of "mu" and "nu" is the source of "nu".
If the edges of graph G1 are a subset of the edges of graph G2, and the vertices of graph G1 are a subset of the vertices of graph G2, then the union of graph G1 and graph G2 is equal to graph G2.
The GS recursion step for a given state 's' and decision 'd' with value 'v' is equal to the reward for the state-action pair (s, d s) plus a constant 'l' times the sum of two terms: the first term is the sum over all states 's'' less than 's' of the probability mass function of the transition kernel 'K' for the state-a...
A pair consisting of a conclusion 'c' and a list 'i#is' belongs to the set of vertices if and only if the conclusion 'c' is in the set of conclusions, the integer 'i' equals zero, and the list 'is' is in the set of paths of the inverted tree for the conclusion 'c'.
The lemma named "Ide_consE" makes the following assumptions: 1. "Ide (t # U)" - The function "Ide" is applied to the list that has "t" as its first element and "U" as the rest of the list. 2. A function that takes two arguments: "R.ide t" and two implications "U is not an empty list implies Ide U" and "U is not an emp...
For polynomial operations, the integral domain division operations hold true with respect to the polynomial relation.
If a command 'c' transforms a state 's' into a state 's'', and 's' is equivalent to 't' on the set 'L c X', then there exists a state 't'' such that the command 'c' transforms 't' into 't'' and 's'' is equivalent to 't'' on the set 'X'.
The set of constants "csts" in a term "t" is finite.