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In the context of Kleene Algebra with Tests (KAT), if 'p' is a test and 'p' multiplied by the Kleene star of 'x' is less than or equal to the Kleene star of 'x' multiplied by 'p', then 'p' multiplied by 'x' is less than or equal to 'x' multiplied by 'p'.
: Assuming two conditions: 1. For any elements x and y, if x is in the set xs, y is in the set ys, and the first element of the pair (P x y) is true, then the first element of the pair (P' x y) is also true. 2. For any elements x and y, if x is in the set xs, y is in the set ys, and the second element of the pair (P x...
The substitution of a polynomial function 'f' into the sum of two polynomials 'p' and 'q' is equal to the sum of the substitution of 'f' into 'p' and the substitution of 'f' into 'q'.
The intersection of 'x' with 'nc' added to the intersection of 'y' with 'nc' is equal to the intersection of the sum of 'x' and 'y' with 'nc'.
If three conditions are met: 1) 'm' is less than or equal to 'b' raised to the power of 'n', 2) 'b' is greater than 1, and 3) 'm' is greater than 0, then the logarithm base 'b' of 'm' is less than or equal to 'n'.
The topology T is Hausdorff.
If the following conditions are met: 1. The right-hand side of the interference freedom auxiliary function for the tuple (r, com, ann) is free of interference under the contexts Gamma and Theta and the function F. 2. The right-hand side of the interference freedom auxiliary function for the tuple (Q, com, ann) is free...
: if Phi is finite, then A satisfies the And-sub-n of Phi if and only if for every element phi in Phi, A satisfies phi.
The lemma "secrets_are_homogeneous" assumes that for all 's' in 'Sec', if 'P s' is true, then for all 's'' in the subterms of 's', either '{}' entails 's'' or 's'' is in 'Sec'. Given that 's' is in 'Sec' and 'P s' is true, it shows that 's' is homogeneous in 'Sec' with respect to 'A'.
If we rearrange the alternatives in a preference profile and the anonymous profiles of two preference profiles are the same, then the social choice functions of these profiles will also be the same after applying the same rearrangement.
B is an independent set of vectors and the cardinality of B is equal to the cardinality of the set of natural numbers, if and only if B is a basis.
The lemma named "division_disjoint_union" assumes that "p1" is a division of "s1" and "p2" is a division of "s2", and that the interior of "s1" and "s2" do not intersect (i.e., they are disjoint). Under these assumptions, it shows that the union of "p1" and "p2" is a division of the union of "s1" and "s2".
The function "face_cycle_succ" is a permutation.
If 'v' and 'w' are elements of 'V', then there exists a unique path 'xs' that leads from 'v' to 'w'.
If "x" is an element of the set "f" acting on "X", and "v" is an element of "x", then the function "s" acting on "v" is equal to the function "g" acting on "v".
For the relation set "pcr_CoCalls", the union operation preserves the relation under the least upper bound (lub). This is a transfer rule, which means it is used to transfer properties between different representations of data.
The lemma named "is_head_reduction_resid'" shows that if "t" is a head reduction, "u" is an internal reduction, and the source of "t" is the same as the source of "u", then the head reduction of the residuation of "t" and "u" exists.
The function that maps each element 'w' in the set 'W' to 'w`\<rightarrow>C0' is injective on 'W'. An injective function is one where different inputs always produce different outputs.
The lemma named "substitution_in_allVars" assumes that "g" is redundant. It shows that the substitution of "g" is included in all variables of "g".
The lemma named "has_bochner_integral_nn_integral" assumes the following conditions: 1. The function 'f' is Borel measurable on the measure space 'M'. 2. Almost everywhere in 'M', the function 'f' is non-negative. 3. 'x' is non-negative. 4. The non-negative extended real-valued integral of 'f' over 'M' equals 'x'. If...
The lemma named "Int_interval_mixed_eq_empty" is about a fixed point 'a' in a Euclidean space. It assumes that the open box 'c d' is not empty. It shows that the intersection of the open box 'a b' and the closed box 'c d' is empty if and only if the intersection of the open box 'a b' and the open box 'c d' is empty.
The lemma named "card_orbit_eq_funpow_dist1" assumes that "x" is in the orbit of "x" under function "f". It shows that the cardinality of the orbit of "x" under function "f" is equal to the function power distance from "x" to "x" under function "f".
If set A is a subset of set C, and a proposition p holds true when we subtract set B from set A, then the same proposition p will also hold true when we subtract set B from set C.
The lemma "spr_simAbs_fst_snd_trc" assumes that "t" is in the set of common knowledge of the justified knowledge-based programs (SPR.jkbpC) and that the abstraction of "ec" under the spr_simAbs function is equivalent to the deterministic simulation equivalence class of "a" and "t". It shows that the set converted from ...
Updating the polynomial mapping at index 'i' with value 'k' for the difference of 'm' and the monomial of 'i' and 'k', is equal to updating the polynomial mapping at index 'i' with value 'k' for 'm'.
Extracting from the result of binding an identifier 'x' with a function 'f' is equivalent to extracting from the function 'f' applied to the extraction of 'x'.
Applying the function af_F to a logic formula phi and the abstraction of another logic formula psi, with respect to a word w, is equivalent to the abstraction of the result of applying the function af_F to phi, psi, and w.
If for every element 'x', if 'x' is a member of list A implies 'x' is also a member of list B, then list A is a sublist of list B.
The converse of a set containing the ordered pairs (a, b) and (c, d) is the set containing the ordered pairs (b, a) and (d, c).
If a string 'u' belongs to the set of all strings composed solely of the element 'a', and 'u' is not an empty string, then the set of unique elements in 'u' is exactly the set containing 'a'.
The lemma named "wf_plan_act_is_sound'" assumes that the plan action, represented by "\<pi>", is well-formed (as indicated by "wf_plan_action \<pi>"). It shows that the operation of resolving and instantiating "\<pi>" is sound (as indicated by "sound_opr (resolve_instantiate \<pi>)") when it is converted from a PDDL op...
The lemma "add_edge_preserve_subgraph" assumes that H is a subgraph of G and that the edge (a, w, b) is in E. It shows that adding the edge (a, w, b) to H still results in a subgraph of G.
A map 'g' is continuously mapping from the Euclidean space to the Euclidean space if and only if 'g' is continuous on the universal set.
The lemma named "norm_FW_valid_preservation" assumes that the DBM (Difference Bound Matrix) M is valid. It shows that the DBM resulting from normalizing the Floyd-Warshall algorithm applied to M, with respect to a function k composed with v' and n, is also valid.
The set of IP addresses resulting from the conversion of the IP address range interval (from ip1 to ip2) is equal to the set of all IP addresses from ip1 to ip2.
The norm of a one-dimensional vector x is equal to the norm of its first component.
If function f is convex on set S and set S is convex, then the epigraph of function f on set S is also convex.
The head element of the sequence 'e' at position 0 is 0.
For any predicate P, if it is not the case that P holds for all x, then it is not the case that P holds for some x for which P does not hold.
If 'r' is a well-order and '(i, j)' is in 'r', then the set of elements under 'i' in 'r' is a subset of the set of elements under 'j' in 'r'.
If a set 'xs' has pairwise scene-distinct elements and 'a' is a very well-behaved lens, then the supremum of the map lens composition of 'xs' and 'a' is equal to the supremum of the scene composition of 'xs' and 'a'.
"x is equal to the domain of x if and only if x is less than or equal to 1".
The lemma named "GcdI" assumes the following: 1. For any element 'a' in set 'A', 'b' divides 'a'. 2. For any 'c', if 'c' divides every element 'a' in set 'A', then 'c' also divides 'b'. 3. 'b' is equal to its normalized form. Given these assumptions, it shows that 'b' is equal to the greatest common divisor of set 'A'...
The lemma named "transcendental_cos" assumes that "z" is an algebraic number and "z" is not equal to zero. It shows that the cosine of "z" (interpreted as a complex number) is not an algebraic number.
The sequence of operations "c" followed by the star of "d" and "c" is equal to the star of "c" and "d" followed by "c".
The lemma named "Markov's Inequality" assumes that for any element 'a', 'X a' is greater than or equal to zero, 'X' is integrable over the measure space 'M', and 't' is greater than zero. It shows that the probability of the set of all elements 'a' in the measure space 'M' for which 't' is less than or equal to 'X a' i...
The complex conjugate of x equals the complex conjugate of y if and only if x equals y.
If "r" is not equal to "s", then getting "r" from the hash "h" after setting "s" to "x" in "h" is the same as getting "r" from "h" before the setting operation.
Given two sets S and S' in Euclidean spaces, if S is an Euclidean Neighbourhood Retract (ENR) and S is homeomorphic to S', and S' is a subset of U, then there exists a set V such that V is an open set in the topology induced by U and S' is a retract of V.
The lemma named "path_join_eq" is about two paths, g1 and g2, in a metric space. It assumes that both g1 and g2 are paths. It shows that the path formed by joining g1 and g2 is a valid path if and only if the end of path g1 is the same as the start of path g2.
"y" is not in the segment from "x" to "y" in function "f".
The lemma named "ord_resolve_sound" assumes the following: 1. "ord_resolve CAs DA AAs As E" holds, where "ord_resolve" is a function or relation involving the variables CAs, DA, AAs, As, and E. 2. The multiset "CAs" is true under interpretation "I", denoted by "I \<Turnstile>m mset CAs". 3. The assertion "DA" is true u...
The identity function applied to "Some x" equals "Some x".
If process P is weakly bisimilar to process Q, then process P is weakly equivalent to process Q.
"path" is indeed a path according to the definition of "is_path" in the context of a given state "σ".
The text obtained from inserting 'x' at position 'b' is equal to the text taken from the start to the position 'b' in the original text, followed by 'x', and then followed by the text from position 'b' to the end in the original text.
The lemma named "insert_dom_image" assumes the following conditions: 1. "i" is less than the successor of "n", 2. "j" is less than the successor of "n", 3. The image of the function "f" over the set of all numbers from 0 to "n" (exclusive) equals the set of all numbers from 0 to the successor of "n" (exclusive) excl...
If a polynomial 'p' is irreducible, then it is also square-free. This is assumed for any 'p' in the integral domain.
The length of the Braun array, created with a key 'k' and a list 'xs', is equal to the minimum of the length of the list 'xs' and 2 raised to the power of 'k'.
If "m" is less than or equal to the specification of "P", and for any "r1" and "r2" such that "P" holds for the pair "(r1, r2)", "g" of "(r1, r2)" is less than or equal to the specification of "P", then "m" is less than or equal to the specification of a function that maps any "r'" to "g r'" being less than or equal to...
"x" is less than or equal to "x" circled.
In the context of a finite group, the order of group G is greater than 0.
The lemma named "exp_2dist_positive" assumes that "a" and "b" are elements in the hyperbolic plane "hyp2". It shows that the exponential of twice the distance between "a" and "b" is greater than zero.
If Rm and Rf are both single-valued relations, then the relation generated by Rm and Rf through the function "gen_gbg_impl_rel_eext" is also single-valued.
The lemma named "Nml_Cod" shows that if "t" is normal, then the codomain of "t" is also normal.
The set of 5-tuples from (0, -100, -500, -1000, -1000) to (125, -100, 0, 1000, 1000) in the real numbers is equal to the set from the Euclidean representation of the list [0, -100, -500, -1000, -1000] to the Euclidean representation of the list [125, -100, 0, 1000, 1000].
If "t" is a red-black tree, then the lookup function for the red-black tree "t" is equivalent to the red-black tree lookup of "t".
The lemma named "finite_INF_in" is about a function 'f' that maps from some type 'a to extended real numbers. It assumes that 'S' is a finite set and that 'S' is not an empty set. It shows that the infimum (greatest lower bound) of the function 'f' over the set 'S' is an element of the image of 'S' under the function '...
If a list "xs" is represented by "i" and "ba", and "i" is less than the length of the list "xs", then the successor of "i" represents the list "xs" when "ba" is incremented by "i".
The carrier of the chain group at a certain level 'p' in a topological space 'X' is equal to the set of singular chains at that same level 'p' in 'X'.
The lemma named "run_new_ref" in the context of "run_elims" is assuming that the function "run" with parameters "ref x", "\<sigma>", "\<sigma>'", and "r" is true and that "\<sigma>" is not an exception. If these conditions are met, it can be concluded that "\<sigma>'" is equal to the second element of the allocation of...
Applying the "ValD_case" function to the values "ValF", "ValTT", "ValFF", and "ValN" results in the identity function "ID".
, under the assumptions that: 1. Event 'ev' is a transition from state 'c' to state 'd', 2. Event 'ev'' is a transition from state 'd' to state 'e', 3. Both 'ev' and 'ev'' are receiving events, 4. Event 'ev'' is also a transition from state 'c' to state 'd'', 5. Event 'ev' is also a transition from state 'd'' to state...
If a natural transformation arrow "N" belongs to the set of tiny morphisms of natural transformation arrows between two categories "A" and "B", then there exist a tiny morphism "M" and two functors "F" and "G" such that "N" is the arrow of "M" and "M" is a tiny morphism from "F" to "G" in the category of tiny morphisms...
The lemma named "indets_map_indets" assumes that the function "f" is injective (or one-to-one) on the set of individual terms (indets) of the polynomial "p". It shows that the set of individual terms of the polynomial obtained by mapping "f" onto "p" is equal to the image of the set of individual terms of "p" under "f"...
Entity A knowing the secret key set of message m is equivalent to entity A knowing the single encryption of message m.
If X is in filters, then X is filtered.
If a program P indicates that class C has fields FDTs, and the tuple (FD, b, T) is in the set of FDTs, and if the program P is well-formed with respect to method declarations, then T is a type in program P.
The lemma "Rep_Label_root_node" simplifies to the statement that the representation of the root node label is an empty list.
The lemma "prv_lcnj_eqv" assumes that the set of formulas "phi1s" is a subset of "fmla" and that the set of formulas "phi2s" is equal to the set of formulas "phi1s". It shows that the proof of the equivalence between the logical conjunction of "phi1s" and the logical conjunction of "phi2s" is valid.
The set of non-positive integers is a subset of the set of non-positive real numbers.
A routing table has a default route if and only if there exists a route in the set of routes such that the prefix length of the routing match is zero.
: "If 'DenyAllFromTo a b' is in the set p, all networks in the list consisting of 'AllowPortFromTo c d po' and p are distinct, and either a is not equal to c or b is not equal to d, then the two networks a b and c d are distinct."
If the following conditions are met: 1. The code instruction stack offers "ci". 2. For any instruction, program, heap, stack, location, class, method, program counter, and frame stack, if "ci" applies to them, then the check instruction' applies to them as well. 3. For any instruction, program, heap, stack, location, ...
If 'f' has a real derivative 'l' at a point 'x' within a set 'S', and 'l' is less than zero, then there exists a positive 'd' such that for all positive 'h', if 'x + h' is in 'S' and 'h' is less than 'd', then 'f' at 'x' is greater than 'f' at 'x + h'. This essentially means that if the derivative of the function at a ...
The lemma named "fps_one_over_one_minus_fps_X_squared'" assumes that the inverse of 1 in a ring with an inverse operation is 1. It shows that the inverse of the square of (1 - fps_X) in the formal power series of 'a is equal to the absolute value of the formal power series of the function that maps each natural number ...
The set X is finite if X is a subset of the set of all integers n such that n is less than or equal to x. This is applicable for any X and x.
If the global vacancy is up to a certain point (shd s) in a queue (q) at a certain time (t), and the next operation is a perform operation (s), then the global vacancy will still be up to that point (shd (stl s)) in the queue (q) at that time (t).
The lemma named "gmctxt_closing_sup_args_ghole_or_gterm" assumes that "C" is a context closing over "D". It shows that for all "E" in the set of super ground meta-context arguments of "C" and "D", "E" is either a ground meta-hole or the number of ground meta-holes in "E" is zero.
The lemma named "perp_col1" assumes that point C is not equal to point X, line AB is perpendicular to line CD, and point X lies on line CD. Under these assumptions, it shows that line AB is perpendicular to line CX.
If a function 'f' maps from a set of natural numbers less than or equal to 'n' to a set 'B', and 'f' is injective on this set, and a certain element 'b' is not in the image of 'f' on this set, then the function 'jointfun' that combines 'f' and a constant function that maps any natural number to 'b', is also injective o...
: 1. The polynomial evaluation of the polynomial expression "PCnst 0" with any variable values is equal to zero. 2. The polynomial evaluation of the polynomial expression "PCnst 1" with any variable values is equal to one. 3. The polynomial evaluation of the polynomial expression "PCnst" with any numeral n is equal to...
The lemma named "assert_parametric" which includes lifting syntax shows that for two functions, monad_fail_base.assert and monad_fail_base.assert, if the first function transforms an input of type A into an output of type M, and the second function takes an input of type M and a function from A to M and returns an outp...
The set of subterms in the range of the composition of two substitutions, θ1 and θ2, excluding the range of the variable function, is a subset of the union of the set of subterms in the range of θ2, excluding the range of the variable function, and the set of subterms in the range of θ1, excluding the range of the vari...
There exists a real number 't' such that the absolute value of 't' is less than or equal to the absolute value of 'x', and the cosine of 'x' is equal to the sum (for 'm' less than 'n') of the cosine coefficient of 'm' times 'x' to the power of 'm', plus the cosine of 't' plus one-half times the real number 'n' times pi...
"Root_CTRL" is not equal to "TunerMode_CTRL".
If n is greater than 0, then the htuple of n is equal to the pair of 0 and the htuple of (n - 1).
The function "at_right" at a certain point "c" is equivalent to the mapping of the function "filtermap" which adds "c" to the inverse of "x", where "x" is a real number, at the top.
If set Y is finite, then there exists a set X such that the intersection of Y and X is empty, X is finite, and the cardinality of X is greater than or equal to n.
The application of the "Pop" operation in the context of G, maxs, rT, pc, et, and some state s, is equivalent to the existence of ts, ST, and LT such that s is equal to the pair (ts#ST, LT).