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The multiset of elements in a tree with a root node "r", an element "a", and a list of trees "ts" is equal to the multiset containing the element "a" plus the multiset of elements in the list of trees "ts".
The lemma named "powreal_surj" assumes that "a" is greater than 1 and "x" is greater than 0. It shows that there exists a "y" such that "a" to the power of "y" in real numbers equals "x".
The function new_pairs_naive satisfies the specification np_spec.
: Assuming that S is a convex set, k is a finite set, and f is a function that is continuous on S. Also, for any x in the interior of S but not in k, f is field differentiable at x. Furthermore, z is an element in the interior of S but not in k, γ is a valid path, the image of γ is a subset of S excluding z, and the e...
If the clauses C1 and C2 are satisfiable under the interpretations F and G, and if the resolution rule is applicable to C1 and C2 with literals L1 and L2 and substitution σ, then the result of the resolution of C1 and C2 with L1, L2, and σ is also satisfiable under the interpretations F and G.
The lemma named "transversal_segment_reverse" assumes that there is a transversal segment from point x to point y. It shows that there is also a transversal segment from point y to point x.
For all i less than or equal to the successor of n, a property P holds for i if and only if P holds for 0 and for all i less than or equal to n, P holds for the successor of i.
The product of 'a' and 'b' under the condition 'P' is equal to 'a' multiplied by 'b' under the same condition 'P'.
If the length of list "ys" is equal to the length of the enumerated list of type 'a, then there exists an element 'y' such that the function 'map_of' applied to the zipped list of the enumerated list and 'ys', when evaluated at 'x', gives 'Some y'.
Bisimulation is transitive. If there is a bisimulation R1 between systems A and B, and a bisimulation R2 between systems B and C, then there is a bisimulation (R1 O R2) between systems A and C.
The function "subtype" is a bijective function on the transitive closure of the sequence.
If there is a transition from state 's' to state 's'' in a system 'SES' and the initial state 's' is in the state set 'S' of the system 'SES', then the resulting state 's'' is also in the state set 'S' of the system 'SES'.
For any two dimension vectors "xs" and "ys" of type 'n and 'a respectively, their multiplication is defined as follows: - If the lengths of both "xs" and "ys" are equal to the cardinality of 'a, then the result is a new dimension vector where each element is the sum of the corresponding elements in "xs" and "ys". - If...
If the function "gen" is true for the variables x, Q, and G, then the function "eqs" for the variables z and G equals to an empty set.
The lemma "mayDiverge_Seq_R" assumes two conditions: first, that there is a transition from state "s" to state "s'" under command "c1", and second, that command "c2" may diverge in state "s'". Under these assumptions, it shows that the sequence of commands "c1" followed by "c2" may diverge in state "s".
The natural logarithm of 272/100 is greater than 1 in the real number system.
For a function 'f' from real numbers to a linearly ordered topology 'a', if 'a' is less than or equal to 'b' and for all 'x', 'a' is less than or equal to 'x' and 'x' is less than or equal to 'b' implies that 'f' is continuous at 'x', then there exists a 'M' such that for all 'x', 'a' is less than or equal to 'x' and '...
The lemma named "terminal_unity" in the Set Category shows that the unity is terminal.
The recursive swapping of a red-black tree comparator intersection is defined as follows: First, we define the black heights of two trees, t1 and t2, as bh1 and bh2 respectively. Then, we swap the order of the trees and their black heights if bh1 is less than bh2. Next, we define a new function, f', which is either t...
If a set 'I' is finite, non-empty, and not a singleton set (a set with only one element), then the immediately previous element 'n' in the set 'I' is not the maximum element in the set 'I'.
An IP address (addr) is in the IP set derived from a base address and a netmask if and only if the IP address is greater than or equal to the result of a bitwise AND operation between the base address and the netmask, and less than or equal to the result of a bitwise AND operation between the base address and the netma...
If we have a compact set K and an open set W, and the Cartesian product of the set containing only the element x0 and K is a subset of W, then there exists an open set X0 containing x0 such that the Cartesian product of X0 and K is a subset of W.
There exists a sequence 'xs' such that the first element of 'xs' is 'x', the last element of 'xs' is 'y', and 'xs' follows a sequence of steps, if 'x' can reach 'y' through zero or more steps.
For any hyperreal number 'x', the square root of 'x' raised to the power of 2 (which is a hypernatural number) is equal to the absolute value of 'x'.
The lemma in the state space shows that for any given state, if we increment N by 1 and then increment M by 1, and then set R to be the sum of the original values of N and M, then R will be equal to the new values of N and M minus 2.
The quotient of a fraction converted to a fraction using the "to_fract" function is equal to the original number "x" over 1.
The associative property holds for the operation "times multiset". In other words, for any three multisets A, B, and C, the multiset of (A times B) times C is equal to the multiset of A times (B times C). This is achieved by mapping each element (a, b, c) in the multiset A times B times C to the element ((a, b), c).
The function "a" applied to "dblo.d x" is equal to "a" applied to "x".
The keys of a mapping function applied to 't' are equal to the set of keys of 't' in a Red-Black Tree.
If the program counter "pc" is less than the length of the compiled expression "compE2 e", then the execution move of the binary operation "bop" on expressions "e" and "e'" in the context of class instances "ci", program "P", thread "t", heap "h", stack "stk", local variables "loc", program counter "pc", and exception ...
The substitution of a multiset list "AAs" by "eta" results in an empty list if and only if "AAs" is an empty list.
If 'f' is an arrow (or morphism), then the domain of 'f' is an identity.
The dual of an empty list in a stateful strand is an empty list.
The lemma named "sorted_wrt_dist_last_insort_key_eq" assumes that the list "ps" is sorted with respect to the distance from a point "p", and that inserting a point "q" into the list "ps" based on the distance from "p" does not result in the list "ps" followed by "q". Under these assumptions, it shows that the last elem...
The function 'of_int' is a ring homomorphism.
The theorem "Thm (Abs_idx (0, 6)) (\<Psi> 5)" is true.
The lemma named "L_infinity_I'" assumes that the function 'f' is Borel measurable on the measure space 'M', and almost everywhere in 'M', the extended non-negative real number of the absolute value of 'f(x)' is less than or equal to 'C'. Under these assumptions, it shows that the essential supremum norm of 'f' on the m...
The partial derivative of a regular expression 'r' with respect to a symbol 'a' is finite.
If the context to statement function of E and c equals "If e p q", then c equals "If e p q" and E equals an empty list.
The basis of a finite map is countable.
Mapping the "Not" function over a Bernoulli probability mass function with probability "r" is equivalent to a Bernoulli probability mass function with probability "1 - r".
If 'A' is in the set of boolean axioms, then the evaluation of the formula 'A' under the environment 'e' holds true.
The lemma named "spectral_radius_max" assumes that "v" is an eigenvalue of matrix "A". It shows that the norm of "v" is less than or equal to the spectral radius of "A".
The lemma named "lindemann_weierstrass_integral_bound" is about a fixed complex number 'u'. It assumes that 'C' is greater than or equal to zero and that for any 't' in the closed segment from 0 to 'u', the norm of the polynomial 'f' at 't' is less than or equal to 'C'. Under these assumptions, it shows that the norm o...
The Möbius transformation preserves the cross ratio. Given that three points z1, z2, and z3 are distinct, the cross ratio of the image of a point z under a Möbius transformation M and the images of z1, z2, and z3 under the same transformation is equal to the cross ratio of z, z1, z2, and z3.
A function 'x' is injective if and only if the composition of the negation of the identity relation and 'x' is less than or equal to the negation of 'x'.
If a function 'f' tends towards a limit in a network 'net', and for almost all 'x' in 'net', 'b' is less than or equal to the sixth imaginary part of 'f(x)', and 'net' is not the least element, then 'b' is less than or equal to the sixth imaginary part of the limit.
The lemma named "urel_integer_plus" assumes that there is a relationship "urel_integer" between x and y, and between x' and y'. It shows that this relationship also holds for the sum of x and x' (using the function "plus_p_integer") and the sum of y and y' (using the function "plus_p").
There is no call to "x" in the tail of the path "p" if and only if either there is no call to "x" in the path "p" or there is exactly one call to "x" in the path "p" and "x" is the first element of the path "p".
The lemma named "lo_power" assumes that "x" is greater than 1. It then shows that the logarithm base "x" of "x" to the power of "r" is equal to "r".
If r and s are both in the set of computably enumerable relations, then the intersection of r and s is also in the set of computably enumerable relations.
The first projection of 'k' is positive.
If the remainder of the first component of a pair 'n' divided by 7 equals 3, then the function 'pr_gr' applied to the upper bound 'loc_upb' of 'n' and 'x' does not contain the key formed by pairing 'n' and 'x'.
The lemma is fixing variables delta1, delta2, w, q1, and q2. It defines rho as the run of the product of delta1 and delta2 with the pair (q1, q2) over w. It also defines rho1 as the run of delta1 with q1 over w, and rho2 as the run of delta2 with q2 over w. Assuming that the range of rho is finite, it shows that the ra...
If a list 'ss' is a subset of 'S', then the inverse induced identity of the sum of the list 'ss' is equal to the quotient group of the absolute free list of 'ss' over the free group of 'S'.
The lemma named "gsubt_at_gpair_poss" assumes that "p" is in the possible values of "s" and "p" is also in the possible values of "t". It shows that the subtree at position "p" in the pair of "s" and "t" is equal to the pair of the subtree at position "p" in "s" and the subtree at position "p" in "t".
There exists a function g such that: 1. g is a continuous map from X to Y. 2. For any element i in I and any element x in the intersection of the topological space of X and T i, the function g applied to x equals the function f applied to i and x.
If a graph 'g' is extended by an edge 'e' to form a new graph 'g'', and there is a subpath in 'g'' from a vertex 'v' to the target of 'e' (denoted as 'tgt e') that is a subset of 'subs', and 'v' is not equal to the target of 'e', then there exists a sequence of edges 'es'' such that 'es' is equal to 'es'' followed by '...
If S is conjunctive, then S is also sequentially conjunctive.
If the concatenation of two lists "u" and "v" equals a list "w", and if "w" is a list within the set "A", then "v" must also be a list within the set "A".
"All" is equivalent to a function 'P' that maps from any type 'A' to a boolean. This function 'P' is defined as equal to another function that maps any 'x' of type 'A' to 'True'.
The function "apsnd" is parametric over two relations. Specifically, it states that given a relation from Rb to Rc, "apsnd" maps a pair of elements (where the pair is in the product relation of Ra and Rb) to another pair (which is in the product relation of Ra and Rc).
The knowledge of A after an event sequence 'evs', combined with the private encryption key of B, the private signature key of B, the shared key of B, and any notes from B in the event sequence, minus the same set of keys and notes, is equal to the knowledge of A after the event sequence 'evs'.
The lemma of reflexivity shows that for any points A and B, A is less than or equal to B.
The lemma named "convex_polychain_segments_of_aform" assumes that the length of the half segments of a form X is greater than 1. If this condition is met, it shows that the segments of the form X form a convex polychain.
The set of all real numbers less than or equal to a certain real number 'a' is path-connected.
If the following conditions are met: 1) r is less than or equal to r', 2) the remainder of r' divided by m is equal to the remainder of r divided by m, and 3) the remainder of m' divided by m is 0, then the interval from r' to the remainder of m' is a subset of the interval from r to the remainder of m.
The lemma named "element_carrier_P" is defined as follows: Let P be a function that takes two arguments, x and n, and checks if x raised to the power of (the cardinality of 'a' raised to the power of n) equals x, in the ring R. Assuming that the monomial with coefficient 1 and degree 1 is in the carrier of the ring R...
The generic size function is a valid operation under the given invariant and alpha mapping in the context of set data structures.
The theorem "wls_obtain_fresh_paramS" assumes that "P" is a well-sorted parameter. It shows that there exists a "z" such that for all "xs", "z" is not in the variables of "P" with respect to "xs". Furthermore, for all "s" and for all "X" in the terms of "P" with respect to "s", "z" is fresh with respect to "zs" and "X"...
A condition "phi" holds until a condition "psi" on a constant stream "x" if and only if the condition "psi" holds on the constant stream "x".
There exists a function "f" such that the inner product of the affine form value of "e" and the affine form of the interval [a, b] with "c" is equal to the affine form value of "f" and the affine form of the interval [a.c, b.c]. Furthermore, "f" is a function mapping any value to a number between -1 and 1.
For any set of ownership records, the set of reads acquired from a store buffer is a subset of the union of the set of true acquisitions from the store buffer and the set of all shared elements in the store buffer.
If X and Y are both very well-behaved lenses and X is a sublens of Y, then X and Y are compatible lenses.
If the following conditions are met: 1. The method 'm' is the method of the dynamic type 'dynT' with the method name 'mn' and parameter types 'pTs'. 2. The dynamic type 'dynT' is the first element of the heap 'h' at address 'a'. 3. 'dynT' is a class in the Java program 'G'. 4. The Java program 'G' is well-formed. 5. T...
The length of the list produced by the negative group orbit function, when applied to the parameters gs, T, and as, is equal to the length of the list gs.
The lemma named "timpl_apply_inv'" assumes that "s" is an element in the set that results from applying the function "a" to "b" on the function "f" with the set "T". It then shows that there exists a function "g" and a set "S" such that "s" is equal to the function "g" applied to the set "S".
The lemma named "Trans" assumes that if "H" proves "x" is equal to "y" and "H" proves "y" is equal to "z", then it shows that "H" proves "x" is equal to "z". This is a statement of the transitive property of equality.
The infinite product of 'f' is greater than 1 if and only if there exists a natural number 'i' such that 'f(i)' is greater than 1.
If the flow attainability axioms are assumed for Delta, then flow attainability is shown for Delta double prime.
The lemma named "cIsInvar" with case names "cUser", "cConf", "cPC", "cChair", "cPaper", "cAuthor", "cConflict", and "cReview" assumes the following: 1. For any state 's', user ID 'uID', password 'p', name, information 'info', and email, if the state 's' is reachable, the property 'phi' holds in state 's', and a user i...
If the sequence at function 'f' and value 'x' is infinite and the sequence 'f' mapped with 'Inr' is equal to 'g', then there exists a value 'x'' in the set of sequence 'g' such that 'x' is in the set of 'x''.
If for all states 's' in 'P', the Hoare triple "{s} intersect P, c, Q, A" is valid under the context of programs 'Γ' and 'Θ' with a fault set 'F', then the Hoare triple "P, c, Q, A" is also valid under the same context.
The weakest precondition for the "SKIP" command with respect to a postcondition "Q" is equivalent to the postcondition "Q" combined with a time cost of 1.
If 'a' is less than or equal to the extended real number 'x', and 'x' is less than or equal to 'y', then 'a' is less than or equal to the extended real number 'y'.
The lemma "InvariantUniqAfterDecide" is defined with respect to a literal trail 'M', a literal 'literal', and another literal trail 'M''. It assumes that 'M' has the property of 'InvariantUniq', meaning that it does not contain any duplicate literals. It also assumes that the variable of 'literal' is not in the variabl...
If the supervisor bit (get_S) of the processor status register (PSR) in the state 's' is 0 (indicating a non-privileged mode), and if the state 's' is transformed to state 's' prime by flushing the instruction cache, then the supervisor bit of the processor status register in the state 's' prime will also be 0 (indicat...
If p is a cycle, then p is a general path.
If x is an element of V, then the product of a and b, each multiplied by x, is equal to the product of a and b, which is then multiplied by x.
If an object "f" of type "U" is valid in the environment "e" and has type "A", then "A" must be equal to "U".
If the function "ci_app" is applied to the parameters "ci", "ins", "P", "h", "stk", "loc", "C", "M", "pc", and "frs", then it is also applicable when the "stk" parameter is concatenated with "stk'".
If the property Fr_6 holds for a filter F, then for all sets A and B, if A is closed, then the complement of the second power of the coproduct of A and B is a fixed point of the negation operator.
For octonions, the product of 'x' and the square of 'y' is equal to the product of 'x' and 'y', then multiplied by 'y' again.
The uncurried implementation of "cnt_incr_impl" is in correspondence with the uncurried "PR_CONST cnt_incr" function. This correspondence is defined over the cartesian product of the "cnt_assn" in the domain and "nat_assn" in the codomain, both mapped to "cnt_assn". This is a rule for the separation logic framework.
The lemma named "COND_terms_hf" assumes that "hf" is a valid hash function according to "ainfo" and "uinfo", that the terms of "hf" are a subset of the analyzed "ik", and that there is no oracle according to "ainfo" and "uinfo". If these conditions are met, it shows that there exists a set of hash functions "hfs" such ...
If the category 'A' is equal to the double lift of the categories 'a', 'b', the functors 'g', 'f', and the category 'C' is equal to 'C', then the double arrow of the double lift of the cofunctor from category 'C' to 'a', 'b', 'g', 'f', 'a'', 'b'', 'g'', 'f'' is a tiny monad functor from 'A' to 'C' in the context of a f...
The length of the merged list of 'as' and 'ls' is less than or equal to the length of the concatenated list of 'as' and 'ls'.
A pair (sigma-prime, coll) is in the set "cbig P sigma" if and only if there exists a natural number "n" such that the pair (sigma-prime, coll) is in the set "csmall_nstep P sigma n" and sigma-prime is in the "endset".
: given the assumptions that "x" is an element of set "V" and "x" is not equal to zero, and that the product of "a" and "x" equals zero, it shows or concludes that "a" must be equal to zero.
If the number of ground holes in a structure D is equal to 1, then the function 'funas_gctxt' applied to the ground context of the ground model context D is equal to the function 'funas_gmctxt' applied to D.