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If function 'f' tends towards 'a' and function 'g' tends towards 'b' (both under filter 'F'), then the function that adds the results of 'f' and 'g' for any given 'x' will tend towards the sum of 'a' and 'b' under the same filter 'F'.
The inverse image of the set containing only element 'a' under relation 'r' is the set of all elements 'b' such that the ordered pair (a, b) is in the relation 'r'.
, given the assumptions that a heap context A for an element x and its implementation xi implies a combination of a frame F and a modified frame F', and that x is an element of a list l, it can be shown that a heap refinement exists. This heap refinement is between the abstract program (Pi xi) and the concrete program ...
The inverse of a string 'u' followed by the string 'u' itself equals the empty string.
The statement shows several lemmas: 1. If the application of a lambda function with a list of arguments is in the calls of x, then the set of arguments is a subset of the values of x. 2. If the application of a lambda function with a list of arguments is in the callsC of y, then the set of arguments is a subset of th...
If we have two categories, denoted as "AA" and "BB", and a functor "NN" from the product of "AA" and "BB" to another category "CC", and if "a" is an object in "AA", then the flipped binary natural transformation of "NN" with respect to "BB" and "AA" at "a" is equal to the binary natural transformation of "NN" with resp...
If a relation 'A' is single-valued and its inverse is also single-valued, then the pair (index, index) is in the relation from the list of 'A' to 'A' that results in a natural number relation.
The set of ground terms of the image of the function "lift_sig_RR2" applied to a set "F" is equal to the set of ground pairs "t" and "u" such that the pair "(t, u)" is in the restriction of the identity relation to the set of ground terms of "F".
The lemma "weakBisimBangPresAux" is defined with the fixed variables Psi, P, Q, and R, where Psi is of type 'b and P, Q, and R are of type psi. It assumes that Psi is in weak bisimulation with P and Q, and that both P and Q are guarded. It then shows that Psi is in weak bisimulation with R in parallel with the repl...
The lemma F_inj_composition_permutation: Assuming that phi is a permutation of the set {0..<n} and g is an injective function, it shows that the composition of g and phi is also an injective function.
The lemma named "has_sum_superconst_infinite_ereal" is defined with a function 'f' that maps any type 'a' to an extended real number. It assumes that for any 'x' in set 'S', the value of function 'f' at 'x' is greater than or equal to 'b'. It also assumes that 'b' is greater than 0 and that the set 'S' is infinite. Und...
For a given binary predicate 'Pred' on processes 'procS' and 'procT', the relation 'indRelR' respects the binary predicate if and only if the relation 'indRelL' respects the same binary predicate.
The lemma named "order_fun" is given a fixed set "S" of an arbitrary type "'a". It assumes that "S" is a subset of "T". It then provides a function "f" from "'a" to natural numbers such that for all elements "x" and "y" in "S", "f x" is less than "f y" if and only if "x" is less than "y" in some predefined order. Addit...
"ReadMem ad al v" is not an element of the set "start_heap_obs".
In the context of array relations, the value of an array is a subset of the product of the array's domain and codomain.
The lemma defines an inch as equivalent to 25.4 millimeters.
The lemma named "binary_expression_gteq0" assumes that for any natural number 'n', 'a n' is either 0 or 1. It shows that 0 is less than or equal to the sum of the real number conversion of 'a (n + k)' multiplied by half to the power of the successor of '(n + k)'.
The function 'to_dtree' preserves well-formedness for directed acyclic graphs.
The function "list_all2_cons1" with parameters x, sg', and sh is defined as follows: If the generator function h with parameter sh results in "Done", then "list_all2_cons1" returns False. If the generator function h with parameter sh results in "Skip", then "list_all2_cons1" is recursively called with parameters x,...
: assuming the restriction of function ρ to the complement of the domain of h equals the restriction of function ρ' to the same set, it follows that the heap semantics of h under ρ equals the heap semantics of h under ρ'.
If the multiset of variables in term 't' is a superset of the multiset of variables in term 's', then the multiset of variables in the substitution of term 't' is also a superset of the multiset of variables in the substitution of term 's'.
If the position of "a" in the list "xs" is None, then "a" is not in the set of "xs".
The lemma named "Atm_transC_invert" assumes that there is a transition from the atomic action "atm" and state "s" to the configuration "cf". If this assumption holds, then the property "phi" is shown or proven.
The insertion sort is correct. If a list 'xs' is sorted and all elements in 'xs' are distinct, then the following conditions hold true: 1. The list obtained by inserting an element 'x' into 'xs' using insertion sort is distinct, meaning it has no duplicate elements. 2. The list obtained by inserting 'x' into 'xs' usin...
: Given two assumptions: 1. The semantic evaluation of expression e1 in the context of a heap where variable x is associated with expression e2 and the rest of the heap is denoted by Γ, under environment ρ, is less than or equal to the semantic evaluation of expression e2 in the same context. 2. The semantic evaluation...
The lemma named "Uact_uPaperC_step_eqButPID" assumes that "a" is equal to the user action of changing a paper with a certain conference id, user id, paper id, and content type. It also assumes that the step from state "s" with action "a" results in an output "ou" and a new state "s'". Under these assumptions, it shows ...
The lemma named "matroid_subset" assumes that a set E is a subset of the carrier. It shows that a matroid can be formed from the set E and the independent sets within E.
If the following conditions are met: 1. The Key Authentication Server (Kas) sends a message to Agent A, which includes an encrypted message (using A's shared key) containing the authentication key, the Ticket Granting Server (Tgs), and a timestamp (Ta), and this message is part of the event set; 2. An encrypted messag...
For all X, the dual of the projection of delta on X equals the dual of the existence of X such that the projection of delta on X holds.
The mask of a number equals zero if and only if the number itself is zero.
The data sources at level 0 for sA9 are equal to the set containing sA8.
If a function 'f' maps every 'j' to an element in the field 'K' and 'b' is an element in the field 'K' such that 'f' converges to 'b' with respect to a valuation 'v' on 'K', then the function that maps every 'j' to the additive inverse of 'f(j)' converges to the additive inverse of 'b' with respect to the same valuatio...
The lemma named "contour_integral_spike_finite_simple_path" assumes that "A" is finite, "g" is a simple path, "g" is equal to "g'", and for any "x" in the path image of "g" excluding "A", "f x" is equal to "f' x". Under these assumptions, it shows that the contour integral of "g" with respect to "f" is equal to the con...
The hypercomplex part of the conjugate of a quaternion q is equal to the negative of the hypercomplex part of q.
The lemma named "path_compression_invariant_simplify" assumes that "w" is a point, the transitive closure of the relation "p" composed with "w" is less than or equal to the negation of "w", and "w" is not equal to "y". Under these assumptions, it shows that the image of "w" under the relation "p" is not equal to "w".
The function "wti" for a given code "c", program counter "pc", and state "s" is defined as the result of merging the step of the state at the program counter with the code at the position of the program counter incremented by one.
The complex dimension of the complex number space is 1.
The lemma named "uniform_complement" assumes that "k" is less than or equal to "v" minus 1. It shows that the block design of "V" and the complement of "B" is "v" minus "k".
: Assuming the rank of set {A, B, P} is 3, the rank of set {A, A'} is 2, the rank of set {A, A', P} is 2, the rank of set {Q, A', a} is 2, the rank of set {A, B, C, A', B', C', P, Q} is at least 4, the rank of set {A, B, C, A', B', C', P} is 3, the rank of set {P, Q, R} is 2, the rank of set {P, R} is 2, the rank of se...
: Assuming that the expression "e", "eP", and "Rec x eV" are of the function type "Fun τ1 τ2" under the context "Γ", and that the atom "x" is fresh in the context "Γ". Then, there exist "y", "e'", "eP'", and "eV'" such that "e" is equivalent to "Rec x (Lam y e')", "eP" is equivalent to "Rec x (Lam y eP')", "eV" is e...
"a" is less than or equal to the product of the encoding of "a" and "b".
The lemma named Mu_Sim: Assuming that for any X and Y, if X is less than or equal to Y in the simulation order, then the function P applied to X is less than or equal to the function Q applied to Y. Also assuming that P and Q are monotonic functions. It can be shown that the least fixed point of P is less than or equal...
Given fixed values for dsn, dsk, dsk', flag, hops, nhip, and nhip', the tuple (dsn, dsk, flag, hops, nhip) is less than or equal to the tuple (dsn, dsk', flag, hops, nhip').
The lemma "fail_reachable" assumes that M1 is not a submachine of M2, both M1 and M2 are well-formed, and the product of M2 and M1 with a failure state is PM. Under these assumptions, it shows that the failure state is reachable from the initial state in PM.
The lemma named "to_fun_monom'" assumes that "c" is an element of the ring "R", "c" is not zero, and "x" is also an element of the ring "R". It shows that the function "to_fun" applied to the scalar multiplication of "c" and the power "n" of "X" in the polynomial ring "P" evaluated at "x", is equal to the product of "c...
Infinity plus the additive inverse of any number z equals infinity.
The truth value remains unchanged when a partial function 'a' is applied to it.
The lemma named "prveqlPT_prv_instInp_instInpP" assumes the following conditions: 1. The formula "phi" belongs to the set of all formulas. 2. The free variables of "phi" are a subset of the set containing only "inp". 3. The term "tau" belongs to the set of all 0-arity pseudo-terms. 4. The term "t" belongs to the set o...
The lemma named "angle_partition" assumes that point A is not equal to point B and point B is not equal to point C. It shows that the angle ABC is either acute, right (perpendicular), or obtuse.
If regular expression 'r' satisfies the property 'ok2', then shifting 'r' by two positions with a character 'c' and a number 'm' will also satisfy the property 'ok2'.
If a heap map 'hm' is related to a heap 'h' by 'hmr_rel', and the same heap map 'hm' is related to a map 'm' by 'heapmap_rel', and the map 'm' at key 'k' equals to some value 'v'', then the operation of decreasing the key 'k' with value 'v' in the heap map 'hm' is refined by the operation of decreasing the key at the i...
If the following conditions are met: 1. There is a silent move from state (ms, s) to state (ms', s') under system S and function f. 2. The tail of ms equals the target nodes of rs. 3. The length of rs equals the length of cs. 4. For all i less than the length of cs, the call of return node of the i-th element of the t...
The function "pcs" applied to an empty list returns an empty set.
The function "merge_coeffs" applied to a parameter "p" and then returned, is equivalent to a tail-recursive function "f" applied to the same parameter "p". This function behaves as follows: - If "p" is an empty list, it returns an empty list. - If "p" is a list with only one element, it returns "p" as it is. - If "p" ...
The lemma named "poly_asymp_equiv" assumes that the polynomial "p" is not equal to zero and "F" is less than or equal to infinity. It shows that the polynomial "p" is asymptotically equivalent to the function that multiplies the leading coefficient of "p" by "x" raised to the degree of "p" when "x" approaches infinity.
In the context of encoding with respect to barbs, there is a lemma stating that if an indexed relation with respect to a transition relation weakly reflects barbs, then the transition relation itself weakly reflects barbs. This is fixed for a set of process transition relations. The assumption is that the indexed relat...
For any floating point arithmetic expression 'fa' and any substitution function 's', folding constants in the expression after applying the substitution is equivalent to applying the substitution after folding constants in the expression.
The lemma named "card_bijections_range_permutation_eq_0" assumes that the cardinality of set A is not equal to the cardinality of set B. It shows that the cardinality of the set of all bijective functions from A to B, when divided by the range permutation of A and B, is equal to zero.
The set of predicates for the initialization of an object (represented by "init_obj_pred P C") is equal to the collection of the initialization of an object (represented by "init_obj' P C").
The lemma named "pickE_alphaEnv_iff_equal" assumes that "rho" and "rho'" are both good environments. It shows that the function "pickE" applied to "rho" is equal to "pickE" applied to "rho'" if and only if "rho" is equal to "rho'".
The identifier of a message "m" is not in the set of identifiers of the inserted messages "M".
For any 'k' and 'ns', applying the function 'from_WW' to the result of 'into_WW' applied to 'k' and 'ns' will yield 'ns'. This suggests that 'from_WW' and 'into_WW' are inverse functions with respect to 'k' and 'ns'.
The edge density between two sets X and Y in a graph G is equal to the edge density between Y and X in the same graph G. This means that the edge density function is commutative.
The lemma "strong_postdominate_refl" assumes that "n" is a valid node and "n" is not equivalent to "Exit". It shows that "n" strongly postdominates itself.
The pre-condition of "p" with input 1 is equivalent to the negation of "p".
If F is an isomorphic register, then F is a bijective function.
Inserting an element 'y' into a set 'A' is equivalent to creating a set of elements 'x' such that 'x' is either equal to 'y' or is an element of 'A'.
If 'n' is less than the length of the list 'xs', then the length of 'xs' is equal to the length of the list obtained by taking the first 'n' elements of 'xs', appending 'y', and then appending the elements of 'xs' after the 'n+1'th position.
, given certain conditions, two matrices H1 and H2 are equal. The conditions are as follows: 1. H1 and H2 are both Hermite normal form matrices associated with a result 'res'. 2. H1 and H2 are both n x n matrices. 3. There exist matrices P1 and P2, which are also n x n matrices and are invertible, such that the matr...
The lemma named "fps_shift_mult_right" assumes that "n" is less than or equal to the subdegree of "g", where "g" is a formal power series in a commutative semiring. It shows that shifting "n" in the product of "g" and "h" is equal to the product of "h" and the shift of "n" in "g".
If a set A is finite, then the sorted list of this set A is the unique list 'xs' such that 'xs' is sorted, 'xs' is distinct (has no duplicate elements), and the set of 'xs' is equal to A.
The lemma named "f_open_OVal" assumes that a step from state 's' to state 's'' with action 'a' results in output 'ou'. It also assumes that the 'open' status of 's' is not equal to the 'open' status of 's'' and that action 'a' is not a COMact action. Furthermore, it assumes that there does not exist a user action 'ua' ...
The absolute value of the product of 'an(x)' and 'y' being greater than 'z' is equivalent to 'an(x)' times the absolute value of 'y' being greater than 'z'.
The monadic bind operation "mbind" applied to a state "S" and a function "f" on a state "\sigma" is not equal to "None".
The theorem named "IK_card" shows that the cardinality (or size) of the set of all functions 'f' that satisfy the condition 'IK' is less than or equal to 7.
If the domain of children of a node "r0" with a set containing a pair "(t1,e1)" is "T", the set of vertices "t1" is well-formed, and the maximum degree of "t1" is less than or equal to 1, then the domain of children of the node "r0" with a set containing a pair "(normalize1 t1,e1)" is also "T".
The Gaussian integer is not equal to zero, one, any natural number 'n', any integer 'n', or any numeral 'm'. This is true regardless of whether the Gaussian integer is on the left or right side of the inequality.
In the context of a vector space, if x is an element of V, then x is also an element of the linear span of x.
The lemma named "fun_sum_commute" assumes that the function f of 0 equals 0 and for any x and y, the function f of the sum of x and y equals the sum of the function f of x and the function f of y. It shows that the function f of the sum of function g over a set A equals the sum of the function f of function g of each e...
The lemma named "reciprocal_mult" establishes a property about polynomials in a field with characteristic 0. It takes two polynomials P and Q as inputs. The lemma assumes that the degree of the product of P and Q is less than or equal to a certain value p, and that the degrees of both P and Q are also less than or equa...
If 'i' is less than the successor of the successor of 0, then 'i' equals 0 or 'i' equals 1.
The lemma "some_trg_in_targets" assumes that "mu" is an arrow. It shows that "some target of mu" is in the set of targets of "mu".
If a set 'gs' is a complete family of generators for the group and is a subset of the carrier of the group, and if 'A' is a subset of 'gs', then 'A' is also a complete family of generators for the group.
The results of "ABORT" are an empty set.
For a fixed tree 't' where 'u' is defined as '2 to the power of n', if 't' maintains the invariant 'invar_vebt' at 'n', then the time complexity of the successor function 'T_succ' on tree 't' with input 'x' is less than or equal to '54 plus 27 times the binary logarithm of the binary logarithm of u'.
For any morphism 'h' and any object 'C', if we have a morphism 'g' such that it is the interpretation of a term 'e' of type 'B' under a variable 'Vx' of type 'A', and a morphism 'h' is the interpretation of a term 'd' of type 'C' under a variable 'Vx' of type 'B', then the interpretation of the substitution of 'e' in '...
If a message "m" is part of what the agent "C" knows from the events "evs", and the message "m" is a static secret of agent "A", and agent "A" is not the same as agent "C", and the events "evs" are part of the public namespace, then the message "m" is part of what agent "D" knows from the events "evs".
The lemma named "exFresh_s_p_cof" is defined with a fixed set "L" of type "fVariable". Assuming that "L" is finite, it shows that there exist elements "s" and "p" such that "s" is not in "L", "p" is not in "L", and "s" is not equal to "p".
For any property P, if P holds for an empty multiset, and if P holds for any multiset M when an element x is added to M, then P holds for all multisets M.
A list is lexicographically sorted if and only if the corresponding lazy list is also lexicographically sorted.
The lemma named "integrable_combine" is about a function 'f' that maps real numbers to a Euclidean space. It assumes that the function 'f' is integrable over the interval from 'a' to 'c' and also over the interval from 'c' to 'b', with 'a' being less than or equal to 'c' and 'c' being less than or equal to 'b'. The lem...
The Horner sum of a function 'f' and a number 'a' over a list that is the concatenation of two lists 'xs' and 'ys', is equal to the sum of the Horner sum of 'f' and 'a' over 'xs' and the product of 'a' raised to the power of the length of 'xs' and the Horner sum of 'f' and 'a' over 'ys'.
The lemma named "eqButUID_remove1_UID_recvOuterFriends" assumes that "s" and "s1" are equal except for the User ID (UID). It shows that if you remove the pair (AID, UID) from the received outer friend IDs of "s" for a certain user ID "uid", the result will be the same as if you remove the pair (AID, UID) from the recei...
If a map 'm' is a submap of another map 'm'', and if 'm' maps 'x' to 'y', then 'm'' also maps 'x' to 'y'.
If a class is a preorder with respect to an order and a corresponding "less than" relation, then the order is transitive.
If "x" is a maximum subgraph satisfying property "Q", and all elements satisfying property "P" also satisfy property "Q", and "x" satisfies property "P", then "x" is also a maximum subgraph satisfying property "P".
The substitution of an empty set with any substitution function results in an empty set.
If the set 'b' is finite, then the incidence system resulting from adding the block 'b' to the set of vertices 'V' is also finite.
If the length of 'a is less than the length of 'b, then casting 'x (which is a word of length 'a) to a word of length 'b and then casting it back to a word of length 'a will result in the original 'x.
If a relation 'r' is single-valued and a pair (x, y) is in 'r', then the function of 'r' at 'x' equals 'y'.