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"succ_impl" is an implementation of the successor function "succ" in the abstract graph "absG", according to the implementation specification "Impl_Succ" for the graph "isG". |
If a function 'f' is continuous at a point 'a', then the function that maps 'x' to the imaginary part of 'f(x)' is also continuous at 'a'. This is for 'f' which is a function from a real normed vector to a complex number. |
A multivariate polynomial 'p' is symmetric with respect to a set 'A' if and only if the negation of 'p' is also symmetric with respect to the same set 'A'. |
Given a finite set of indices called "inds", the finite map created from the indices "inds" and a function "f" is equal to the restriction of the function "f" to the set of indices "inds". |
Given the following assumptions:
1. "ns\<phi>'" is defined as the filter of "ns\<psi>" where "n" is not in the set of "ns\<phi>"
2. "ns\<psi>'" is defined as the filter of "ns\<phi>" where "n" is not in the set of "ns\<psi>"
3. "X\<phi>" is defined as the normalized projection values of "R\<phi>" on "ns\<phi>"
4. "X\<... |
If a list 'xs' is a prefix of another list 'ys', then the set of elements in 'xs' is a subset of the set of elements in 'ys'. |
The degree of the polynomial 'p' at a given point 's' is equal to 's'. |
If "a" is in the field of the equivalence relation "Qeq", then the pair "(a, a)" is in the transitive symmetric closure of "Qeq". |
The pair of pairs ((x1, y1), (x2, y2)) is in the real relation if and only if the sum of x1 and y2 is equal to the sum of x2 and y1. |
: if the encryption of the hash of the agent C and nonce NC1 using the private key of the certification authority i is part of what the spy knows in the event sequence, and if the event sequence is part of the set of cardholder registration, and if the certification authority i is not compromised, then there exists som... |
If the set of keys (domain) of a finite map N is a subset of a set A, then restricting the finite map N to the set A results in the original finite map N. |
Restricting the function 'f' first to the set 'Y' and then to the set 'X' is equivalent to restricting the function 'f' to the intersection of sets 'X' and 'Y'. |
The lemma named "power_eq_imp_eq_norm" is defined for a real normed division algebra. It takes a variable 'w' of any type 'a' in this algebra. It assumes that 'w' to the power of 'n' equals 'z' to the power of 'n' and that 'n' is greater than 0. Under these assumptions, it shows that the norm of 'w' equals the norm of ... |
If 'a' is less than or equal to 'b', both 'a' and 'b' are elements of set 'I', predicate 'P' is not true for 'b' but is true for 'a', then there exists an element 'n' in set 'I' such that 'a' is less than 'n', 'n' is less than or equal to 'b', predicate 'P' is not true for 'n', and predicate 'P' is true for the immedia... |
"kc8x7 is not an empty list". |
The lemma "h_method_via_h_framework_completeness_and_finiteness" is about two finite state machines (FSMs), M1 and M2. The FSMs are of a certain type, where 'a, 'b, and 'c are linearly ordered types. The lemma assumes that both M1 and M2 are observable and minimal, and that the size of M1 is less than or equal to m and... |
For any two hypernatural numbers 'm' and 'n', 'm' is less than 'n' if and only if the standard part of 'm' is less than the standard part of 'n' in any linearly ordered semidomain. |
"x is a vector if and only if x is a co-vector". |
The data access at level 0 for sA3 is equal to the set containing sA4. |
The lemma "wfP_silent_move_from_unroll" assumes that for any state 's'', if 's' transitions to 's'' silently, then the well-formed property (wfP) holds when the function 'silent_move_from' is applied to 's'' and its arguments are flipped. Under this assumption, it shows that the well-formed property (wfP) also holds wh... |
:
Given a fixed 'Psi' of type 'b', and three psi-calculi 'P', 'Q', and 'R' of types 'a', 'b', and 'c' respectively.
If the following conditions are met:
1. 'Psi' tensor 'Psi_R' directs 'P' to be bisimilar to 'Q'.
2. The frame extracted from 'R' equals to the pair of 'A_R' and 'Psi_R'.
3. 'A_R' is fresh for 'Psi'.
4... |
The left-hand sides of the substitutions of all elements in set Y on the right-hand side yrhs, within the equivalence system ES, are equal to the left-hand sides of the equivalence system ES. |
If "h" is a homomorphism from group "G" to group "H", then the image of the carrier (or set of elements) of "G" under "h" is a subset of the carrier of "H". |
The function which maps x to the k-th power of x divided by the exponential of x, tends to 0 as x tends to infinity. |
If the atom 'i' is independent of the tuple (A, i, j, t), then the substitution of 'j' with 't' in the existential quantification of 'i' over 'A' is equal to the existential quantification of 'i' over the result of swapping 'i' and 'i' in 'A' and then substituting 'j' with 't'. |
The diamond operator applied to a false proposition and any proposition "Phi" results in a false proposition. |
The auxiliary insertion function applied to a constant zero polynomial with a coefficient 'c' equals 'c'. |
The lemma Osum_antisym:
Assuming that the intersection of the field of relation r and the field of relation r' is empty, and
that both relation r and relation r' are antisymmetric,
it shows that the ordered sum of relation r and relation r' is also antisymmetric. |
The lemma named "DERIV_real_root" assumes that "n" is greater than 0 and "x" is greater than 0. It shows that the derivative of the nth root of "x" is equal to the inverse of the product of "n" and the nth root of "x" raised to the power of "n - 1". |
If a search edge "se" extends from "prb" to "re" at a node "c" to form "prb'", and if "prb" is marked at the source of "re", then the fringe of "prb'" is equal to the fringe of "prb". |
A cocone in category theory, represented as "cf_Cocone" with parameters alpha, beta, and functor F, is defined as the composition of the hom-functor from the category of functors (cat_FUNCT) with parameters alpha, A, and B applied to the functor map of F and the delta functor with parameters alpha, A, and B. |
For any given element 'a' and any list 'xs' of elements from a commutative ring, if 'a' is an element of the set 'xs', then the product of all elements in 'xs' is equal to the product of all elements in the list 'xs' with one instance of 'a' removed, multiplied by 'a'. |
The lemma "missing_cd_or_loop" assumes that both "pi" and "pi prime" are paths, and that the first element of "pi" is the same as the first element of "pi prime". It also assumes that there does not exist any "k prime" such that the "k-th" control state of "pi" is the same as the "k prime-th" control state of "pi prime... |
The inverse of the single reduction step with respect to a set of rewrite rules and a relation is equal to the single reduction step with respect to the set of rewrite rules and the inverse of the relation. |
The group of integers is indeed a group. |
For a fixed value 'y' of a type 'a' that is a finite meet complete partial order (cpo), the function that takes 'x' and returns the meet (greatest lower bound) of 'x' and 'y' is monotonic. This means that if 'x1' is less than or equal to 'x2', then the meet of 'x1' and 'y' is less than or equal to the meet of 'x2' and ... |
The lemma named "zero_in_unit_disc" shows that the hyperbolic point zero is in the unit disc. |
If the function "g" is measurable from a sigma algebra "N" to a sub-probability algebra "N'", and the function "f" is measurable from a sigma algebra "M" to a sub-probability algebra "M'", and the function "h" (when applied to a pair of elements) is measurable from the product sigma algebra of "M" and "M'" to "N", then... |
The lemma named "insert_spec_distinct" is given. It fixes an object identifier 'oid' of any linearly ordered type. It assumes that the list 'xs' contains distinct elements, and that for any element 'x' in the set of 'xs', 'x' is less than 'oid'. It also assumes that if 'ref' is equal to 'Some r', then 'r' is less than ... |
The omega sum of the omega of Cantor function applied to alpha and m is equal to the Cantor sum of alpha and m. |
If a number 'n' is a Carmichael number, then 'n' is not a prime number. |
If a block 'bl' is an element of the set 'B', and the cardinality (or size) of 'bl' is equal to 'v', then 'bl' is equal to 'V'. |
The lemma for the IPv6 access matrix in a pretty format is equivalent to the code for the IPv6 access matrix in a pretty format. |
All necessary input channels for level 1 include data1, data10, and data11. |
If for all threads 't', 't' is not in the thread set 'ts' if and only if 't' is not in the thread set 'ts'', then the result of updating the thread 't' in the thread set 'ts' with the thread action 'ta' using the function 'redT_updT'' is None if and only if the result of updating the thread 't' in the thread set 'ts'' ... |
If a function 'f' is injective on a set 'S' and 'A' is a countable set, then the intersection of 'S' and the preimage of 'A' under 'f' is also countable. |
The lemma named "alternate_antimono" assumes that for all x, the function f(x) is less than or equal to x and the function g(x) is less than or equal to x. Under these assumptions, it shows that the function "alternate" of f and g is anti-monotonic. |
If a set A is countable, and for every element 'i' in set A, the function 'f' of 'i' is less than or equal to 'u', then the supremum (the least upper bound) of the function 'f' for all elements 'i' in set A is also less than or equal to 'u'. |
If 'w' is an element of the quasi-Borel space 'W', and 'sx' is the quasi-Borel probability space of the tuple (X, α, μ), then the quasi-Borel probability of the tuple (W, X) with the function λr. (w, αr) and μ is valid. Furthermore, the strength of the quasi-Borel spaces 'W' and 'X' with the tuple (w, sx) is equal to t... |
If a formula "phi" belongs to the set of all formulas and the free variables of "phi" are empty, then it is provable in bounded Peano arithmetic that "phi" is equivalent to the existence of a variable "yy" such that the formula "phi" is a property of "yy". |
The lemma named "iterative_HSem" assumes that "x" is not in the domain of "Γ". It shows that the heap semantics of the pair (x,e) appended to "Γ" under the environment "ρ" is equal to the least fixed point of "ρ'", where "ρ'" is the update of "ρ" with the heap semantics of "Γ" under "ρ'" restricted to the domain of "Γ"... |
The function "cmp_lex" applied to the comparison function "cmp" and lists "l" and "m" is equal to the function "cmp_lex'" applied to the same comparison function "cmp" and lists "l" and "m". |
The lemma named "mrbst_push_down_correct'" assumes that "A" is a finite set of linearly ordered elements and "x" is an element in "A". It shows that if you perform the following operations: first, create a random binary search tree "l" from all elements "y" in "A" that are less than "x"; second, create a random binary ... |
The ACC (Approximate Counter Circuit) of the union of two sets X and Y is equal to the union of the ACC of set X and the ACC of set Y. |
If "cl" and "dl" are proper and "cl" is strongly bisimilar to "dl", then the parallel composition of "cl" is strongly bisimilar to the parallel composition of "dl". |
The lemma named "sum_clauses" demonstrates two things:
1. The sum of function 'f' over an empty set is zero.
2. If set 'S' is finite, then the sum of function 'f' over the set obtained by inserting element 'x' into 'S' is equal to the sum of function 'f' over 'S' if 'x' is already in 'S'. Otherwise, it is equal to the... |
"tdev of x" is less than or equal to "tdev' of p and x". |
The set of nominals in a branch is finite. |
For a fixed square matrix 'f' of type 'a' (where 'a' is finite) and 'b' (where 'b' is a non-trivial linear order stone relation algebra expansion), the negation of the matrix 'f' multiplied by the maximum top element equals the minimum bottom element if and only if for every 'i', there exists a 'j' such that the elemen... |
If two temporal linked lists "xs" and "ys" satisfy the conditions P and Q, then the finiteness of "xs" is equivalent to the finiteness of "ys". |
The signed cast of a signed cast of a word 'w' is equal to the signed cast of the word 'w' itself, given the condition that the signed cast is an upward cast. |
For any context G, the domain of atoms in the context that includes the triple (x, b, c) is the same as the domain of atoms in the context that includes the triple (x, b, c'), regardless of the values of c and c'. |
If 'm' is less than or equal to 'n', then taking 'm' elements from the list obtained by filling 'n' places with 'fill' in list 'w' is equivalent to filling 'm' places with 'fill' in list 'w'. |
The lemma establishes a Depth-First Search (DFS) invariant without considering the parameterization. It assumes the following:
1. For any element 'e', 'I' holds for the empty state of 'e'.
2. For any states 's' and 's'', and any vertex 'v0' and element 'e', if 'I' holds for 's', 's' satisfies the condition 'cond', 's'... |
The lemma named "tuple_decode_nonzero" assumes that "n" is greater than 0. It shows that the decoding of a tuple "n" and "i" is equal to the first part of the decoding of "i" followed by the decoding of the tuple of "n - 1" and the second part of the decoding of "i". |
Running the "alt_option" function with two parameters m1 and m2 is equivalent to running the "alt" function with the results of running the "run_option" function on m1 and m2 respectively. |
"sA7 is not an elementary component in the data dependency." |
If "a" and "b" are elements of the carrier of the p-adic integers, then the image of the difference of "a" and "b" under the map "iota" is equal to the difference of the images of "a" and "b" under the same map. |
If "Notes A X" is in the set of events "evs", then "X" is in the used set of "evs". |
"for all x, P of x and Q of x holds true" is equivalent to "for all x, P of x holds true" and "for all x, Q of x holds true". |
The theorem "eval_fps_deriv" assumes that the norm of z is less than the convergence radius of the formal power series f. It shows that the evaluation of the derivative of the formal power series f at z is equal to the derivative of the evaluation of the formal power series f at z. |
The concatenation of a bottom element results in a bottom element. |
The lemma named "cycle_leave_ex_enter" assumes that "p" is a cycle in graph "G" and that the ordered pair "(x, y)" is an element of the set of cycle edges of "p". It shows that there exists a "z" such that the ordered pair "(z, x)" is an element of the set of cycle edges of "p". |
The expression "Exp X Y" is not equal to the expression "Enc X' Y'". |
The lemma "execute_preserves_valid" assumes that "s" is in the set of valid states and that the action "name" is enabled in state "s". It shows that executing the action "name" in state "s" results in a state that is also in the set of valid states. |
If a variable 'v' is not in the domain of 'Γ' and 'v' is not in the first element of 'S', then adding the relation 'Rel v AsmNoReadOrWrite' to 'Γ' does not change 'Γ'. This is an idempotent operation. |
For a given set A of complex normed vectors, if A is a complex subspace, then the closure of A is also a complex subspace. |
For two functors 'G' and 'F', where 'G' maps category 'B' to category 'C' and 'F' maps category 'A' to category 'B', and for an object 'a' in category 'A', the object mapping of the composition of 'G' and 'F' applied to 'a' is equal to the object mapping of 'G' applied to the object mapping of 'F' applied to 'a'. |
The degree of the function "of_qr" applied to any element "x" of type 'a in the quotient ring is less than the degree of the quotient ring of type 'a. |
The lemma named "canon_ang_pi_3pi" assumes that "alpha" is greater than "pi" and less than or equal to "3 times pi". It shows that the canonical angle of "alpha" is equal to "alpha" minus "2 times pi". |
: if set A is finite and the cardinality (or size) of set A is the successor of n (n+1), then there exists a function f that maps from the set of all i such that i is less than or equal to n, to set A, and this function f is surjective (onto) from the set of all i such that i is less than or equal to n, to set A. |
The absolute value of any extended real number 'x' is always greater than or equal to zero. |
If a matrix M is cycle-free for a given number n, and for all numbers k up to and including n, the kth diagonal element of M is less than or equal to zero, then for all numbers i up to and including n, the ith diagonal element of M is equal to zero. |
The natural number representation of a 64-bit unsigned integer "x" is equal to the natural number representation of the integer representation of the same 64-bit unsigned integer "x". |
"for all alpha, the necessity of alpha applied to tau is equivalent to the necessity of alpha applied to tau." |
The set "P_set n" is a subset of the carrier "Q_p". |
For a map where each element 'x' is mapped to a function 'f' of 'x', if a condition 'P' holds for 'x', then the result is 'Some' of function 'f' applied to 'x'. Otherwise, the result is 'None'. |
In the context of a group, the lemma named "generate_pow" assumes that "a" is an element of the carrier "G". It then shows that the generation of "G" by the set containing "a" is equal to the set of "a" raised to the power of "k" (where "k" is an integer), for all possible values of "k". |
If the cast of a shadow root pointer is not equal to another pointer, and if deleting the shadow root from a heap results in a new heap, then if a relation exists in the locations of the disconnected nodes of the other pointer, this relation holds between the original heap and the new heap. |
The lemma "next_thread_eq_SomeD" assumes that the next thread function, given a seed and an active thread, returns a tuple of a thread and a new seed. It also assumes that the active thread is in a valid state according to the "thr'_invar" invariant and that the seed is in a valid state according to the "random_schedul... |
For any function from worlds to entities (denoted as 'τ'), the necessity of the property 'φ' holding for 'τ' is equivalent to the necessity of 'φ' holding for 'τ'. |
If 'K' is a subset of the carrier 'R' and 'I' is an ideal of 'R', then 'K' is a subalgebra of 'I' in 'R'. |
The lemma named "qfp_as_extreme_bound" assumes that for all elements 'x' in set 'A', 'x' is less than or equal to 'f(x)' or for all elements 'y' in set 'A', if 'y' is less than or equal to 'x' then 'f(y)' is less than or equal to 'f(x)'. Under these assumptions, it shows that 'f(c)' is approximately equal to 'c'. |
The lemma named "const_list_cons_last" assumes that the list "es" appended with element "e" is a constant list. It then shows that the list "es" is a constant list and the element "e" is a constant. |
"a" is not lexicographically less than "b" if and only if "b" is lexicographically less than "a" and "b" is not equal to "a". |
If D and E are orders, f is an order isomorphism from D to E, and a is in the carrier of D, then there is an order isomorphism from the strict initial segment of D at a to the strict initial segment of E at f(a). This order isomorphism is defined by the function f on the carrier of the strict initial segment of D at a. |
The value of function f1' is greater than zero. |
The lemma named "proj2_incident_Col" assumes that points p, q, and r are all incident on line l. It shows that points p, q, and r are collinear. |
The lemma named "row_mat1" assumes that "i" is less than "nr". It shows that the row of the matrix "mat1I" with zero elements on "nr", indexed by "i", is equal to the vector "vec1I" with zero elements on "nr" at index "i". |
The reach of m3 is a subset of m3_inv7b_sesK_compr_non. |
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