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If a list 'xs' contains distinct elements and the index of an element 'x' in 'xs' is less than a certain value 'i', then 'x' is not in the set obtained by dropping the first 'i' elements from 'xs'.
The lemma named "subset_append_diff_cancel" assumes that the set of elements 'ys' is a subset of the set of elements 'xs'. It shows that the set of elements obtained by appending the elements of 'xs' not in 'ys' to 'ys' is equal to the set of elements 'xs'.
If a block "bl" is an element of a block design "B", then the size of the derived block design "B" is equal to the block size "b" minus one.
The lemma named "pnopE" makes the following assumptions: - The function "f" is a partial n-ary operation on the set "A". From these assumptions, it concludes the following: - The function "f" is a very simple function (vsv). - The number "n" is a natural number. - The domain of the function "f" is a subset of the Cart...
The pre-condition of the disjunction of 'x' and 'y' with respect to 'q' is equal to the conjunction of the pre-conditions of 'x' and 'y' with respect to 'q'.
The theorem of reference map soundness: Given two input/output automata A and B, with states 'sA' and 'sB' and actions 'a', and a function 'f' that maps from 'sB' to 'sA'. If 'f' is a reference map from B to A, and the external behavior of A is the same as the external behavior of B, then it can be shown that the set...
The product of the rational polynomials B and A equals 1.
The lemma named "iso_digraph_refl" assumes that "\<AA>" is a digraph in the context of a certain set "\<alpha>". It shows that "\<AA>" is isomorphic to itself in the context of the digraphs of set "\<alpha>".
The relative interior of an empty set is an empty set.
"q'" is in the successors of "q" after an epsilon step with "bs" if and only if an epsilon step from "q" to "q'" with "bs" is possible.
If a list 'xs' is a sublist of the concatenation of lists 'A' and 'B', and if an element 'x' is in the set of elements of 'xs', then 'x' must be in the set of elements of either 'A' or 'B'.
The lemma 'proj_stoch_adapted_if_adapted' assumes that 'M' is a filtration of 'F' and 'f' is an adapted stochastic process with respect to 'F' and 'N'. It shows that the projection of the stochastic process 'f' is also an adapted stochastic process with respect to 'F' in the stream space of 'N'.
In the context of UP_cring, the lemma named "lin_part_eq" is stated. It assumes that "f" is an element of the carrier "P". It shows that "f" is equal to the linear part of "f" plus the product of the shift of "f" by 2 and the monomial of "P" with degree 2.
There is no spoofing on the interface 'eth0' when the following conditions are met: 1. The interface 'eth0' is mapped to the IP address 192.168.0.0 with a subnet mask of 24. 2. There are two rules in place. The first rule drops any packets that both originate from the IP address 192.168.0.0/24 and are not from the ...
The result of applying the function 'cava_sys_agn' to 'cfg', 'sysi', and 'phi' can be one of three cases: 1. If the result is 'None', then the language of 'sys' is a subset of the language defined by 'phi'. 2. If the result is 'Some None', then the language of 'sys' is not a subset of the language defined by 'phi'. 3....
The lemma named "cone_iff_finite_cone" assumes that set A is a subset of the carrier vector of dimension n, and that set A is finite. It shows that the cone of set A is equal to the finite cone of set A.
For any two configurations 'c' and 'c'', if the transition relation 'S' can transit from 'c' to 'c'' with a sequence of events 'w', then the following two cases hold: 1. If 'w' is an empty list and 'c' is equal to 'c'', then some property 'P' holds. 2. For any configuration 'ch', event 'e', and sequence of events 'wh'...
The restriction of the negation of a proposition P over an alphabet 'a' is equivalent to the negation of the restriction of the proposition P over the same alphabet 'a'.
: Assuming that: - A satisfies the invariant P on the sequence S and the set U under the context Gamma, - Gamma is well-formed, - The transition relation of A is the sequential process sos of Gamma and i, - l' is a label in Gamma of q, - The pair (sigma, p) is operationally reachable in A from S' and U', - The transit...
If the length of the point set "ps" is greater than 0 and the pair "(δc, c)" is equal to the result of the function "find_closest_code" applied to the point "p", the distance "δ", and the point set "ps", then "δc" is equal to the distance from point "p" to point "c" according to the "dist_code" function.
An element 'a' is in the domain of a map that is otherwise empty except for a single key-value pair (k, v) if and only if 'a' is equal to 'k'.
In the field K, if the valuation v is a valuation on K, and x and y are elements in the carrier of K, and the valuation of x is less than the valuation of y, then the valuation of x is equal to the valuation of the sum of x and y.
For any two polynomials 'p' and 'q' in a factorial semiring, a semiring with greatest common divisor, a ring with greatest common divisor, an integral domain with division, and a normalization semidomain with multiplication, the primitive part of the product of 'p' and 'q' is equal to the product of the primitive parts...
The lemma named "finsum_val_ultrametric_diff'" assumes the following: 1. The function 'g' maps from set 'A' to the carrier of the p-adic number field. 2. The set 'A' is finite. 3. The set 'A' is not empty. 4. For any two distinct elements 'a' and 'b' in set 'A', the value of 'g' at 'a' is not equal to the value of 'g'...
, given the assumptions that a transition from state "l" with clock zone "D" to state "l'" with clock zone "D'" exists in the automaton "A" under the normalized semantics, the global clock numbering of "A" is "v" and "n", "A" is a valid abstraction with respect to "X" and "k", "D" is a valid clock zone, and the clock z...
If portfolio 1 and portfolio 2 are both trading strategies, then replacing the quantity of a component in portfolio 1 with that in portfolio 2 also results in a trading strategy.
The lemma named "ord_gpv_coinduct" assumes two conditions. The first condition is that a relation "X" holds between two functions "f" and "g". The second condition is that for any such functions "f" and "g" for which "X" holds, the order of the sub-probability mass functions (ord_spmf) of the generative probabilistic v...
The lemma named "set_to_map_simp" assumes that the function "fst" is injective on the set "S". It shows that the function "set_to_map" applied to "S" and "k" equals "Some v" if and only if the pair "(k, v)" is an element of the set "S".
If the abstract syntax tree goal "v" is not equal to "None", then the value of the abstract syntax tree goal "v" is included in the set of values in the range map of "v".
If the product of two numbers 'm' and 'k' is less than or equal to a natural number 'n', then the modulus of the difference between 'n' and the product of 'm' and 'k' with 'm' is equal to the modulus of 'n' with 'm'.
The lemma named "eccentricity_obtains_inf" assumes that the set V is not equal to the set containing only v. It shows that the eccentricity of v is either infinity or there exists a u in the set V (excluding v) such that the shortest path from v to u is equal to the eccentricity of v.
If a system "\<R>" is a ground system and "t" is a ground term, and if the pair "(s, t)" is in the transitive closure of the single reduction step "\<F> \<R>", then "s" is a ground term.
The first element of the application of the substitution function "usubstappp" to the set U and the action alpha is a superset of U.
Given a linear polynomial "p" (as per the assumption "islin p"), if the polynomial "p" is neither negative infinity (as per the assumption "not Ifm vs (x#bs) (minusinf p)") nor positive infinity (as per the assumption "not Ifm vs (x#bs) (plusinf p)"), and there exists a value "x" such that the interpretation function "...
Taking the first 'n' elements from the list that results from prepending 'n' elements of 'b' to the concatenation of lists 'xs', 'ys', and 'zs', is equivalent to prepending 'n' elements of 'b' to an empty list.
The lemma named "lm04_22_multiple_register" is defined with the following parameters: a natural number 'c', a register 'l', a configuration 'ic', a program 'p', and two natural numbers 'q' and 'a'. It also defines 'b' as B of 'c', 'm' as the length of 'p', and 'n' as the length of the second element of 'ic'. The le...
If the dimension of vector 'v' is 'n', then 'v' belongs to the set of vectors with dimension 'n'.
If "x" is a subtree of "t", then "x" is a ranked decision tree with original values, given the rank, cost, comparison function, tree "T", and root.
"n of x" is less than or equal to "n of y" if and only if "n of x" multiplied by L is less than or equal to y.
If a sequence 's' is a constant sequence in the p-adic integers, then 's' is a Cauchy sequence in the p-adic integers. This is assumed under the condition that 's' is a constant sequence in the p-adic integers.
If a natural number 'x' is greater than 'y', 'y' is greater than 'z', and 'z' is greater than 'x', then this situation is impossible or false.
The lemma assumes that 't' is a valid tree map. It shows that the result of the function 'mapOf' applied to the result of 'mremove' function with parameters 'i' and 't', when further applied to 'i', is None.
"Agent X has a right to FWB according to Agent Y".
If 'a' is an element of set 'A' and 'ys' is a list containing only 'a', then 'ys' is an element of the listset of 'A'.
The lemma named "coprod_qbs_canonical1'" assumes that "I" is countable. Furthermore, for every "i" in "I", the function that maps "x" to "f(i,x)" is a quasi-Borel measurable function from "X i" to "Y". Under these assumptions, it shows that "f" is a quasi-Borel measurable function from the disjoint union of "X i" for a...
If a real-valued function 'f' has an interval integral 'i' over the interval from 'a' to 'b', and if 'f' is non-negative when 'x' is between 'a' and 'b', and 'f' is non-positive when 'x' is between 'b' and 'a', then 'i' is non-negative.
The lemma named "hom_of_snd" assumes that "G" is a group. It shows that the function "f" composed with the second element of a pair is a homomorphism from the direct product of groups "G" and "H" to "K" if and only if "f" is a homomorphism from "H" to "K".
The operation "op_lso_bex" is equivalent to a function that takes two arguments, P and s, and applies the operation "op_lso_bex" to a function that takes one argument, x, applies P to x, and then applies the result to s. This is specified in the context of the sepref_monadify_arity.
The lemma named "filter_neq_takeWhile" assumes that filtering a list "l" to exclude an element "a" does not result in the same list as taking elements from "l" until an element "a" is encountered. If this assumption holds, then it shows that there exist indices "i" and "j" such that "i" is less than "j", "j" is less th...
If there is a run from the initial state "x0" through the states in the stream "xs", then for each state "x" in the stream "xs", there exists a path from the initial state "x0" to that state "x" in the graph represented by the relation "E".
The lemma named "arr_single" assumes that "x" is an element of "Arr". It shows that "arr" of "MkArr" with parameters "D x", "C x", and a list containing "x" is true.
"a" is in the interval between "a" and "b", and "b" is in the interval between "a" and "b".
The set of an empty list is equal to the set that contains only an empty list, for any type 'a.
If all coefficients in a polynomial are zero, then the value of the polynomial at any point x is also zero.
If n is less than the sum of the subdegrees of formal Laurent series f and g, then the nth coefficient of the product of f and g is equal to 0.
"If F leads to either G or H, and it is always the case that G does not hold, then F leads to H."
The lemma named "the_riesz_rep_unique" assumes that for all x, the product of f and x equals the inner product of t and x. It shows that t is equal to the Riesz representation of f.
If the XCoP property holds for a certain entity 'eta', then the rDNI property also holds for that same entity 'eta'.
Brzozowski's algorithm produces a minimal deterministic finite automaton.
The lemma "boxed_b_eq_eq" assumes that "boxed_b" holds for the parameters "\<Theta>, n1, b1, bv, b', n1'" and "\<Theta>, n2, b1, bv, b', n2'", and that "s" is equal to "SBool (n1 = n2)" and "\<Theta>" is well-formed. It also assumes that "s'" is equal to "SBool (n1' = n2')". It shows that "s" is equal to "s'".
The lemma named "primepow_power_iff" is defined for a factorial semiring multiplicative 'p'. Assuming that the unit factor of 'p' equals 1, it shows that 'p' raised to the power of 'n' is a prime power if and only if 'p' is a prime power and 'n' is greater than 0.
The function which takes an input 'x' and overrides 'm' with 'x' in the context of a specific domain 'S' is continuous.
: - The af of the conjunction of two propositions, phi and psi, with respect to a world w, is equivalent to the conjunction of the af of phi and the af of psi, both with respect to the same world w. - The af of the disjunction of two propositions, phi and psi, with respect to a world w, is equivalent to the disjunction...
Assuming the initial list "s0" is not empty, all elements in "s0" are distinct, and all elements in the request sequence "rs" are included in "s0", it shows that the time complexity of the Move-to-Front strategy on "s0" and "rs" is less than or equal to (2 - 1 divided by the size of "s0") times the optimal time complex...
Dropping the first 'p' elements from the list that results from applying the function 'f' to each element and its index in the list 'xs', is equivalent to applying the function 'f' to each element and its index in the list that results from dropping the first 'p' elements from 'xs'. Here, 'p' is an offset that is added...
There is an invariant such that the set of elements in the Tarjan stack of a state is a subset of the domain of the discovered elements in that state.
If you convert a list "l" to a tree using the "toTree" function and then convert it back to a list, you will get the original list "l".
If the final state of a Turing machine is reached after a certain number of steps starting from a state with a tape filled with blank symbols, then the final state will also be reached in the same number of steps starting from a state with an empty tape.
For any elements y and z, the union of the product of y and the omega power of y or the star power of y times z, and z itself, is equal to the union of the omega power of y and the star power of y times z.
Casting 'x' to an unsigned 64-bit word is less than or equal to casting 'y' to an unsigned 64-bit word if and only if 'x' is less than or equal to 'y' cast as an 8-bit word.
The square root of the sum of the squares of (a + c) and (b + d) is less than or equal to the sum of the square root of the sum of the squares of a and b, and the square root of the sum of the squares of c and d.
The lemma named "ab_specD2" assumes a specification "ab_spec ab". It shows that if the function "ab" is applied to the arguments "gs", "bs", an empty list, and "data", it will return "bs".
The lemma named "BuildSeq_trunc" assumes that "BuildSeq B C s k y" is true and "l" is an element of "k". It shows that "BuildSeq B C s l (app s l)" is also true.
The lemma l7_3 assumes that M is the midpoint of point A to point A. It shows that M is equal to A.
The lemma named "CH_irc_idem" shows that applying the function "CH" twice to a set "A" is equivalent to applying it once.
The length of the alternating list of xs multiplied by 2 is either equal to 1 plus the length of xs or equal to the length of xs.
If a trace is valid, then proposition P holds.
The multiset created from the set of a given multiset M is a subset of the multiset M.
The lemma named "fetch_bef_erase_b_b" assumes that "n" is less than the length of "lm", "s" is greater than 0, "s" is less than or equal to twice "n", and "s" is an even number. Under these conditions, it shows that the fetch function, when applied to the concatenation of "mopup_a n" and "shift mopup_b (2 * n)" with "s...
"DSources for a given component C and interface i is a subset of Sources for the same component C and interface i".
A proposition is free of the "next" operator if and only if both of its constituent propositions are free of the "next" operator.
If the copy property holds for a person P, then the copy property also holds for the function that takes a person X and applies the OclAsType operation to P, casting it to OclAny type.
For the reactive program "Rprg", it universally leads to a fixed point.
The lemma named "dist_inf_mono" assumes that the function F distributes over infimum (as stated by "dist_over_inf F"). Under this assumption, it shows or proves that the function F is monotone.
It is admissible, under the least upper bound operator "lSup" and the order relation "sqsubseteq", to substitute any list "x" with the property that it is not distinctly listed.
"y" is less than or equal to the infimum of "y" and "x".
The constant formal Laurent series of 0 is equal to 0.
If a room "r" is owned by some guest "g" in a state "s", then there exist states "s1" and "s2", and a card "c", such that the state "s" can be represented as the concatenation of "s2", the check-in of guest "g" to room "r" with card "c", and "s1". Furthermore, there is no check-in for room "r" in the state "s2".
The lemma "MAC_synth" assumes that the following conditions are met: 1. "hf_valid ainfo uinfo m z" - the function hf_valid with parameters ainfo, uinfo, m, and z is true. 2. "HVF m \<in> synth ik" - the function HVF with parameter m is an element of the set produced by the function synth with parameter ik. 3. "mac_for...
The lemma named "is_le_relE" assumes that "rel" is a less than or equal to relation. It concludes that "rel" is either an equality relation, a less than or equal to relation, or a less than relation.
: Assuming the absolute value of the difference between z and z0 is not equal to r, r is greater than 0, st is greater than or equal to 0, st is less than tt, and tt is less than or equal to 2*pi, then the complex index of the path E (part of a circle path from z with radius r from st to tt) at z0 is equal to: - If t...
The composition of bisimulations is associative. That is, composing bisimulation 12 with bisimulation 23 and then composing the result with bisimulation 34 is the same as composing bisimulation 12 with the result of composing bisimulation 23 with bisimulation 34.
The product of a number 'c' and the least common multiple of two numbers 'a' and 'b' is equal to the product of the unit factor of 'c' and the least common multiple of the products of 'c' and 'a', and 'c' and 'b'.
The lemma named "b_e_check_single_type_type" assumes that the function "check_single" applied to the parameters "\<C>", "e", and "xs" equals "(Type tm)". If this is true, it shows that there exists a "tn" such that "xs" equals "(Type tn)".
If "A" is a good abstraction, then applying the identity environment to "A" leaves "A" unchanged.
If "a" equals "b", "x" equals "y", "S" equals "T", and for any "x" that is not equal to "b" and is an element of "T", "f x" equals "g x", then the limit of "f" as it approaches "x" at "a" within "S" is equivalent to the limit of "g" as it approaches "y" at "b" within "T".
Given the assumptions that the set of elements in multiset N is a subset of set A, the set of elements in multiset M is also a subset of set A, and for all elements x, y, and z in set A, if relation R holds between x and y, and between y and z, then it also holds between x and z. Under these conditions, the relation "l...
If "next_propagate" holds true for the pair of configurations "c" and "c'", then the "implied_frontier_alt" for the location "loc" is the same for both configurations "c" and "c'".
If a graph 'g' has minimum graph properties, and 'f' is an element of the face set of 'g', and 'v' is an element of the vertex set of 'f', then 'f' is an element of the set of faces at vertex 'v' in the graph 'g'.
The value of a generalized expression, when folded right with a generalized AND operation and a base case of True, is equivalent to the value of the same generalized expression when folded (without specifying direction) with the same generalized AND operation and base case of True, for any state 's'.