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If "f" is a recursive function with "a" arguments, then "f" is extensionally equivalent to the wrapped identity function of "f" with "n" arguments.
For a fixed Euclidean space 'a', the relative frontier of the set obtained by adding 'a' to each element of set 'S' is equal to the set obtained by adding 'a' to each element of the relative frontier of 'S'.
The length of the power polynomials, when multiplied by 'u' and 'curr_p' for 'n' times, is equal to 'n'.
A list containing a natural number 'x' is equal to the 'Oc' function applied 'Suc x' times. 'Suc' is the successor function, which returns the next natural number.
The lemma named "goedel_rosser_first_theHardHalf" assumes that the system is consistent. It shows that the negation of the Rosser sentence cannot be proven.
If set A is finite and set B is finite, then the product of set A and set B is also finite.
The lemma named "nhop_update_changed" is defined with fixed variables rt, dip, dsn, dsk, flg, hops, and sip. It assumes that updating the variable rt with the values dip, dsn, dsk, flg, hops, and sip does not result in the original rt. It then shows that the next hop (nhop) of the updated rt for the destination dip is ...
The product of "a" and "b" (where "a" and "b" are extended natural numbers) equals infinity if and only if "a" equals infinity and "b" is not zero, or "b" equals infinity and "a" is not zero.
If the binary split of 'w' at position 'n' results in a pair '(u, v)', then 'w' can be reconstructed by concatenating 'u' and 'v' at position 'n'.
The function domain of the union of A and B is equal to the supremum of the function domains of A and B.
The lemma named "rel_mset_OO" assumes two conditions: "rel_mset R A B" and "rel_mset S B C". If these two conditions are met, it shows that "rel_mset (R OO S) A C" is also true. This means that if there is a multiset relation R between A and B, and a multiset relation S between B and C, then there is a composite multis...
The lemma named "integral_by_parts" is defined for four functions f, g, F, and G, each mapping real numbers to real numbers. It assumes that 'a' is less than or equal to 'b'. It also assumes that both f and g are continuous functions in the interval from 'a' to 'b'. Furthermore, it assumes that the derivative of F at a...
The indicator function of the intersection of sets A and B, evaluated at point x, is equal to the product of the indicator functions of sets A and B, evaluated at the same point x, in the context of a semiring_1 type class 'a.
The lemma named "fun_upd_same_oexp" assumes that the pair (f, g) belongs to the ordinal exponentiation relation "oexp", the function f at x equals the function g at x, x is in the field of the relation s, and y is in the field of the relation r. It shows that if you update the functions f and g at x to y, the resulting...
If the "wadjust_loop_check" function is called with parameters "m", "rs", and a pair consisting of "c" and an empty list, then "c" cannot be an empty list.
The lemma named "ims_hd_eq_comp" assumes that the string "u" is not empty and that the function "g" applied to "u" equals the function "h" applied to "u". It shows that the function "g" applied to the first element of "u" is equivalent to the function "h" applied to the first element of "u".
For all 'n', if the sequence 'X n' does not belong to the set 'M', then the hyperreal extension of the sequence 'X', denoted as 'star_n X', does not belong to the hyperreal extension of the set 'M', denoted as '*s* M'.
If for every element x in set A, B x is a transitive set, then the supremum of the image of set A under function B is also a transitive set.
The recursive function "r_s" has arity 1. It also asserts that for any input "x", the evaluation of the function "r_s" with "x" as an argument is defined. Furthermore, it states that for any "x", the function "s" of "x" is equal to the evaluation of the function "r_s" with "x" as an argument.
If the proposition "Q" holds at a state "sigma" at time "i", and the proposition "P" holds at all states "sigma" at times less than "i", then the "until" operator holds for propositions "P" and "Q" at state "sigma".
The distance from x to y plus the distance from x to z is greater than or equal to zero.
If a list "l" is in the set generated by the function "faxCD" with input "σ", then the function "GE.nv2L" applied to "l" equals the empty set.
If a simple distribution M is defined by X and P, and if a is an element in the space of M under X, then the probability of a is equal to the measure of M in the preimage of {a} intersected with the space of M.
The lemma "mayDiverge_transC" assumes that a command "c" may diverge from a state "s". It shows that there exists a command "c'" and a state "s'" such that command "c" transitions to command "c'" in state "s'" and command "c'" may diverge from state "s'".
If the following conditions are met: 1) A phase 1 or phase 2 read operation has transitioned the system from state 's' to state 's'' for process 'p', disk 'd', and request 'r', 2) The inner invariant 5 holds for the system state 's' and process 'q', 3) The invariant 2c holds for the system state 's', 4) The inner i...
The support of the function, which maps each element 'l' to the time of 'C', is an empty set.
"fib_final" is equal to "fib".
The lemma named "distance_unique" assumes that "u" and "v" are in the unit disc. It shows that there exists a unique "d" such that "d" is the Poincare distance between "u" and "v".
The Nonce X is atomic.
For a given formal Laurent series 'f' with elements from a semiring with 1, inverse, and uminus operations, if the inverse of 1 is 1, then dividing 'f' by the inverse of the formal Laurent series 'fls_X' raised to the power of 'n' is equal to shifting the formal Laurent series 'f' by -n.
The lemma named "subcocycle_avg_add" assumes that "u" and "v" are subcocycles and that the subcocycle averages of "u" and "v" are greater than negative infinity. It shows that the subcocycle average of the function that adds "u" and "v" for any given "n" and "x" is also greater than negative infinity. Furthermore, the ...
If "y" is an element of the finite set "Y" and "x" is an element of the multiset "f y", then "x" is an element of the multiset sum of "f y" for each "y" in the finite set "Y".
If H is a subgroup of G, a is an element of the carrier of G, and x is an element of the right coset of H by a, then x is an element of the set of all elements z such that there exists an element v in the right coset of H by a and an element h in H such that the product of h and the right translation of v equals z.
The function which adds a fixed extended natural number 'x' to another extended natural number 'y' is monotone. This means that if 'y1' is less than or equal to 'y2', then 'x + y1' is also less than or equal to 'x + y2'.
The lemma named "pre_to_univ_poly_inv_is_hom" assumes that "i" is less than the successor of "n". It shows that the function "pre_to_univ_poly_inv" with parameters "Suc n" and "i" is a ring homomorphism from the polynomial ring "Pring (R[X_n]) {i}" to the polynomial ring "R[X_Suc n]".
If the following conditions are met: 1) R and pi are observationally consistent with Sa and Sc, 2) bc is in the behavior of Sc, 3) ba is in the behavior of Sa, 4) the pair (ba, bc) is in the sequence lift of R, then the mapping of pi to the mapping of the observation of Sc on bc is equal to the mapping of the obse...
An atom 'a' is equal to an atom 'b' if and only if 'a' is equal to 'b'.
The Dirichlet L-function at 0 for a given natural number 'n' and a character function 'χ' is equal to 0 if the character function of 'n - 1' equals 1. Otherwise, it is equal to the negative of the sum from 'k = 1' to 'n - 1' of the product of the natural number 'k' and the character function of 'k', all divided by the ...
If the following conditions are met: a command 'c' and state 's' result in state 't', the stripped version of a set 'C' equals to command 'c', the state 's' is in set 'S', and the step function applied to set 'S' and set 'C' is less than or equal to 'C', then the state 't' is in the postcondition of 'C'.
If relation "r" is a subset of relation "s", then the domain of relation "r" is a subset of the domain of relation "s".
If a semantic domain SEM is well-sorted, then the interpretation of SEM preserves well-sortedness.
The composition of the function "F2set" and the function "F2map" applied to a function "f" is equal to the composition of the function "image" applied to the function "f" and the function "F2set".
If 's' is an element of 'S' and 't' is also an element of 'S', then the pair relator of 's' and 't' in the context of a free word is an element of 'Q'.
If the following conditions are met: 1. The encrypted message containing the timestamp 'Ta' is part of what the spies have. 2. The encrypted message containing the timestamp 'Tk', the agent 'B', the key 'K', and some other data 'X' is part of what the spies have. 3. The key 'K' is not part of what the spies have analy...
The transformation of an action formula in the S system is equivalent to the action formula in the S system with the transformed action and formula.
A match occurs between the parameters "a" and "p" in the "andfold_MatchExp" function applied to "ms" if and only if a match occurs between the parameters "a" and "p" for every element "m" in the set "ms".
If "v" is a clock numbering, then the abstraction of "cc" and "M" with respect to "v", "c1", and "c2" is equal to "M" applied to "v c1" and "v c2".
For all elements 'a' and 'b', if the pair (a, b) is in set S if and only if the pair (a, b) is in set T, then set S is equal to set T.
The sum, over all outcomes, of the probability mass function of an outcome times its utility, is equal to the sum, over all outcomes, of the utility of an outcome times its probability mass function.
If there is a proper path from 'as' to 'tr', then there exists a 'tr'' in the subset of 'tr' such that either 'phi' of 'stl as' and 'tr'' holds true or there is a proper path from 'stl as' to 'tr''.
The lemma "stern_brocot_simps" states three things: 1. The root of the recursive Stern-Brocot tree is the pair (1, 1). 2. The left child of any node in the recursive Stern-Brocot tree is obtained by taking the reciprocal of the node's value, then incrementing that reciprocal, and then taking the reciprocal of the resu...
The weakest precondition of SetPC with function f and probability measure p, given a postcondition P, is a function that, for a given state s, sums over all elements a in the support of the probability measure at state s, the product of the probability of a at state s and the weakest precondition of applying function f...
The infinite sequence in the context Γ, starting with the sequence of commands c1 followed by c2 and then the rest of the commands cs, and the configuration css starting in the normal state s, is equivalent to the infinite sequence in the same context Γ, starting with the command c1, followed by the command c2 and then...
The lemma named "finite_sset_sfilter_decomp" assumes that the set of elements (sset) from the stream (xs) filtered by a predicate P is finite, and that the predicate P always (alw) eventually (ev) holds true for the stream (xs). Under these assumptions, it obtains a stream element x, and streams ws, ys, zs such that th...
If set S is a subset of set T, then the closure of set S in topological space X is also a subset of the closure of set T in the same topological space X.
Noninterference in a system with a single initial state is equivalent to noninterference in a system.
The set of annotations resulting from annotating a command 'c' with a function that always returns 'a', is a set containing only 'a'.
If the atom 'j' is fresh for 'A', then substituting 'j' with 'u' in the result of substituting 'i' with 't' in 'A' is the same as substituting 'i' with the result of substituting 'j' with 'u' in 't' in 'A'.
The lemma named "composite_of_unq_upto_cong" assumes that there are two composites, one of "u", "t", and "v", and another of "u", "t", and "v'". It shows that "v" is equivalent to "v'".
The function "return composed with op_list_last" is refined by the function "RETURN composed with op_list_last". This refinement relation is valid under the assumption that the input list is not empty, and it holds within the abstract state relation "list_assn R" in the domain and the relation "R" in the range.
For the function f(x) = ((x + 4) to the power of 3/2 + e to the power of x - 9) / x, as x approaches 0, the value of the function approaches 4. Here, 'real' refers to the real number system, 'exp' refers to the exponential function, and 'powr' refers to the power function.
The lemma "left_adjoints_compose" assumes that "f" is a left adjoint and "f'" is also a left adjoint, and that the source of "f'" is the target of "f". It shows that the composition of "f'" and "f" is also a left adjoint.
The lemma named "map_upds_overwrite" assumes that "x" is an element of the set "xs" and that the length of "xs" is equal to the length of "ys". It shows that mapping "xs" to "ys" in the map "m" for the element "x" is the same as directly mapping "x" in the map from "xs" to "ys".
The real number 'r' belongs to the standard set when it is converted using the 'of_real' function.
The signed right shift by one of a numeral word 'w' is equal to the word representation of the integer obtained by taking the signed bit of the numeral 'w', divided by 2, where the length of the bit is one less than the length of 'a'. Here, 'a' is a word of arbitrary length.
If a list 'xs' is in the list 'n A', and for all pairs '(p,x)' in the set 'ps', 'p' is less than the size of 'xs' and 'x' is in 'A', then 'xs' is less than or equal to the result of the function 'merges' applied to 'f', 'ps', and 'xs' in the order relation 'r'.
If 'v' is an element of 'V', then applying the polymap function with parameters '[-c, 1]' to 'v' is equivalent to subtracting 'c' times 'v' from 'T' times 'v'.
The lemma "eeqExcPID_N2_sym" assumes that "s" and "s1" are equivalent except for PID_N2. It shows that this equivalence is symmetric, meaning if "s" is equivalent to "s1" except for PID_N2, then "s1" is also equivalent to "s" except for PID_N2.
The lemma named "absolutely_integrable_bounded_measurable_product" is defined for a function 'h' that maps from a Euclidean space 'a' to a Euclidean space 'b' and then to a Euclidean space 'c'. The lemma assumes that 'h' is bilinear, 'f' is Borel measurable on the Lebesgue measure restricted to a set 'S', 'S' is a set ...
The theorem named "sum_of_two_squares_prime" assumes that "p" is a prime number. It shows that "p" is the sum of two square natural numbers if and only if "p" is not equal to 3 when considered modulo 4.
The value of the function f at the point (x, y) is less than or equal to the out-degree of x in the function f.
Running the function "run_accept_eps" with parameters R and an empty list on the state "cs" is equivalent to accepting the epsilon transition of R on the state "cs".
If the following conditions are met: 1. A cryptographic key K encrypting a pair of nonces NB is part of what the spies can see in the events. 2. A cryptographic key, shared by B, encrypting a key K and an agent A is part of what the spies can see in the events. 3. For all nonces NA and NB, a note from the Spy containi...
The lemma named "has_vderiv_on_domain" assumes that "s" is an element of "S". It shows that the derivative of the function "phi" with respect to "t" at the point "s" is equal to the function "f" applied to "phi" at the point "t" on the domain "T".
These two conditions are equivalent for any term "t".
The lemma "composite_with_leg0_is_map" shows that the composite of "s0" and "w" is a left adjoint.
: Given a machine M that is observable, and two input-output sequences io1 and io2 such that the state of the machine M after processing io1 from state q is the same as the state after processing io2 from state q, and both io1 and io2 are in the language of the machine M starting from state q, then for any additional i...
The lemma named "is_orthogonal_cspan" assumes that for any element 's' in set 'S', 'a' is orthogonal to 's'. It also assumes that 'x' is in the complex span of set 'S'. Under these assumptions, it shows that 'a' is orthogonal to 'x'.
: 1. The harmonic number of 0 is 0. 2. The harmonic number of 1 (successor of 0) is 1. 3. The harmonic number of any numeral 'n' is equal to the harmonic number of the predecessor of 'n' plus the inverse of 'n'.
The lemma named "rsplit_rcat_u32_eq" assumes that a 32-bit word "w" is equal to a list of 8-bit words "[a1, a2, b1, b2]". It shows that if you concatenate this list of 8-bit words into a 32-bit word using the function "word_rcat" and then split this 32-bit word back into a list of 8-bit words using the function "word_r...
Mapping a function 'f' over the concatenated list 'xss' is equivalent to concatenating the list obtained by mapping 'f' over each sublist of 'xss'. Here, 'xss' is a list of lists.
The lemma named "pair_automaton" assumes that "A" satisfies the RR1_spec with respect to "T" and "B" satisfies the RR1_spec with respect to "U". It shows that the pair automaton constructed from "A" and "B" satisfies the RR2_spec with respect to the Cartesian product of "T" and "U".
The set of all elements 'k' that are less than the successor of 'n' is equal to the union of the set of all elements 'k' that are less than 'n' and the set containing 'n' itself.
If d is an element of D and v is greater than or equal to 0, then the inverse of Q of d applied to v is also greater than or equal to 0.
The lemma named "IdNatTransMapR" assumes that "f" is a natural transformation. It shows that the composition of "f" and the identity natural transformation of the codomain of "f" is equal to "f".
The lemma "represented_at_post_objects_nonnull" assumes that for a given state tau, an object x is included in all instances of a certain type H. It then shows that in this state tau, the object x is not equal to null.
The function that takes a length 'k' and returns the first 'k' elements of a list 'w' is injective (or one-to-one) on the set of all lengths up to the length of 'w'.
Normalizing a dynamic call with initial state "init", procedure "p", return state "return", and continuation "c", is equivalent to a dynamic call with the same "init", "p", and "return", but with the continuation being the normalization of "c" applied to states "s" and "t".
The nabla of the star of x is equal to the nabla of x.
If 'f' is a linear function and 'g' is a simple Bochner integrable function over a measure space 'M', then the simple Bochner integral of the function 'f(g(x))' over 'M' is equal to 'f' applied to the simple Bochner integral of 'g' over 'M'.
The lemma "cs_length_g_one" assumes that the length of the sequence "cs" at index "i" under a policy "π" is not equal to 1. If this is the case, it obtains a value "k" such that the sequence "cs" at index "i" under policy "π" is equal to the sequence "cs" at index "k" under policy "π" appended with the policy "π" at in...
The lemma named "changes_drop_dup" assumes that the lists "xs" and "ys" are not empty, and if "ys" is not empty, then the last element of "xs" is the same as the first element of "ys". Under these assumptions, it shows that the number of changes in the combined list of "xs" and "ys" is equal to the number of changes in...
If a natural number 'x' is less than another natural number 'y', and 'y' is less than a third natural number 'z', then 'x' is less than 'z'.
If a frame satisfies the conditions Fr_2 and Fr_3, then for all A and B, if B is a regular open set, then the complement of A with respect to B is a fixed point of the negation operator.
Lemma: Isolated Singularity at Inverse Assumptions: 1. f has an isolated singularity at z (f_iso). 2. f is not essential at z (f_ness). Conclusion: The function g(w) = 1 / f(w) has an isolated singularity at z.
If there exists an IP assignment such that the IP assignment of the interface of a packet equals some IP set and the source of the packet is in the union set of the IP CIDR of the IP set, then it implies that if the IP assignment of the interface of the packet equals some IP set, the source of the packet is in the unio...
If set X is not empty and is a chain, then X is co-directed.
The function "check_plan" applied to a problem P and a plan πs returns no errors if and only if the problem P is well-formed according to the abstract syntax tree problem definition and the plan πs is valid for the problem P.
The predecessor of 0 is 0.
There exist functions 'u', 'e', and 'r' such that 'f' at (x, u x) equals 0, 'u' at 'x' equals 'y', for any 's' in the closed ball around 'x' with radius 'e', 'f' at (s, u s) equals 0, 'u' is continuous on the closed ball around '