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The lemma named "convex_less_ri_domain" is about a function 'f' that maps from a Euclidean space to an extended real number. The lemma assumes that the function 'f' is convex on the entire universe of discourse. It also assumes that there exists a point 'x' such that the value of function 'f' at 'x' is less than 'a'. U...
If "x" is an element in the set "xs", then the greatest common divisor of the list "xs" divides "x".
The lemma named "non_empty_cycle_root_msc" assumes that there is a non-empty cycle rooted at a point 'x' in relation 'r'. It shows that 'x' is strongly connected to many points.
A path in the heap 'h' from 'Null' following the sequence 'xs' to 'y' is equivalent to the condition where the sequence 'xs' is empty and 'y' is 'Null'.
If "St" is a final deterministic state, then the deterministic resolution prover step applied to "St" will result in "St" itself.
The theorem K_root2 assumes that the pattern is not undefined. It shows that the function K applied to the root2 of the pattern is true.
The enabling function of the supremum (or union) of two properties "phi" and "psi" is equal to the union of the enabling function of "phi" and the enabling function of "psi".
The second domain name chop of "i8--CoffeeMachine--Leaf" results in "Leaf".
The lemma about the components of the small category of categories (smc_CAT) shows the following: 1. The objects of smc_CAT are the set of all categories in a given universe of discourse, denoted by alpha. 2. The arrows of smc_CAT are all the category functors in the given universe of discourse, alpha. 3. The domain...
If a list "ys" comes after a list "xs", then there are no elements from "ys" that appear in "xs".
A relation R is single-valued in its inverse if and only if R is left-unique.
The product of 'r' and the inverse of 'u' concatenated with the inverse of 'w' and 'v', and 's' is equal to the product of 'r' and the concatenation of the inverse of 'u', 'w', 'v', and 's'.
An object is cast to a document if and only if the document is cast from the object.
If the set of vertices V is equal to the set of vertices V' and the set of blocks B is equal to the set of blocks B', then the transitive closure of the set of blocks B is equal to the system with multiple blocks of size 2.
The lemma named "dim_eq_full" shows that the dimension of a set S is equal to the universal dimension if and only if the span of set S is equal to the universal set.
The lemma "inserted_ids_last_ins" demonstrates that the list of inserted identifiers ("inserted_ids") for a list "xs" appended with a tuple (consisting of "oid", "Insert", "ref", and "val") is equivalent to the list of inserted identifiers for "xs" appended with "oid".
If there is a reflexive and transitive reduction from p to q using the "top_tail" ordering and the "sing_reg" signature in the set F, then there is also a reflexive and transitive reduction from p to q using the less than or equal to ordering and the "sing_reg" signature in the set F.
In the context of a program, the theorem named "virtual_simulates_direct_step" is given. It assumes a step from state (ts, m, S) to state (ts', m', S') in the direct semantics. It then shows that in the virtual semantics, a step from state (remove_rels ts, m, S) to state (remove_rels ts', m', S') also exists. Here, "re...
The set created from applying function "f" to "s" and "x" is finite.
The lemma "branch_instr_low_equal_sub1" assumes that two states, s1 and s2, are low equal. It also assumes that t1 is the second element of the first element of the set_annul function applied to True and the second element of the first element of the write_cpu function applied to the sum of the PC register value of s2 ...
If the following conditions are met: 1. The expression 'e' is of low type according to the typing context 'Γ' (denoted by "\<Gamma> \<turnstile>\<^sub>b e \<in> Low"). 2. The mode state 'mds' is consistent with the typing context 'Γ' (denoted by "mds_consistent mds \<Gamma>"). 3. The variable 'x' is in the set of varia...
If a sequence "as" is valid at a certain time "t", then taking the first "r t" elements of this sequence will also result in a valid sequence at time "t".
If time 't' is less than 'n', then the next instance 'n' in the interval that is strictly greater than 't' is equal to the next instance 'n' in the interval 'I'.
If A is a set in the Borel sigma-algebra, f is continuous on A, and g is continuous on the complement of A, then the function that maps x to f(x) if x is in A and g(x) if x is not in A is Borel measurable.
The roots of a polynomial that is a constant are an empty multiset.
If D is a derivative operator, then the strong negation satisfies McKinsey-Tarski axiom 3.
The alpha function of 2 and n equals n.
For a minimal principal ideal generated by an element 'a', if 'J' is an ideal of the ring 'R' and 'a' is an element of 'J', then the principal ideal generated by 'a' is a subset of 'J'.
The lemma named "Cong_sub_0_subst_right" assumes that "u" is approximately equal to "u'" under some equivalence relation denoted by "\<approx>\<^sub>0", and "t" is concatenated with "u". It shows that "t" can also be concatenated with "u'", and the difference between "t" and "u", subtracted from the difference between ...
If the product of y and x is less than or equal to the product of x and the reflexive-transitive closure of y, then the product of the reflexive-transitive closure of y and the reflexive-transitive closure of x is less than or equal to the product of the reflexive-transitive closure of x and the reflexive-transitive cl...
The principal part of the formal Laurent series obtained from the regular part of a given formal Laurent series 'f' is equal to 0.
For any two given regular expressions L and R, the final state of the start state of the combined regular expression (L or R) is equivalent to the final state of the start state of L or the final state of the start state of R.
If a term 't' is a subterm or equal to the substitution of term 's' by 'θ', and for any term 's'' that is a subterm or equal to 's', 't' is not equal to the substitution of 's'' by 'θ', then there exists a term 'w' in the free variables of 's' such that 't' is a proper subterm of the substitution of 'w' by 'θ'.
The lemma is about the size step of a K33 component. It assumes that the size invariant of a K33 component in graph G with index k and color blue is maintained, and that the vertex at index k in graph G is blue. It shows that the size invariant of a K33 component in graph G with index k+1 and color blue is also maintai...
"w" is less than or equal to "w" in the infimum order.
The function "class_ex" applied to the set of relations "rel" and a class "c" is equivalent to the function "execlass_ex" applied to the relations "rel" and the class "c".
A type "T" is a reference type if and only if there exists a type "T'" such that "T" is a reference to "T'".
The lemma named "semIntAll_prSubstAll" assumes that "SEM" is a well-sorted semantic domain. It shows that the function "prSubstAll" applied to the interpretation of semantics "semInt SEM", the interpretation of abstract semantics "semIntAbs SEM", and "SEM" itself, is true.
There exists functions f and g such that the morphism from the universal set to the universal set, with constructors ctor1 and ctor2 and sets s1 and s2, is valid for f and g.
If 'b' is accessible from set 'A' under predicate 'P', and 'a' is an element of 'A' such that predicate 'P' holds between 'a' and 'b', then 'a' is also accessible from set 'A' under predicate 'P'.
If the function 'derives' applied to delta and an empty list holds true, then there exists a 'D' such that the function 'LeftDerivation' applied to delta, 'D', and an empty list also holds true.
For any natural numbers a, b, and k, if a modulo m equals b modulo m, then k times a modulo m equals k times b modulo m.
If the negation of B is in H, then adding B to H logically entails A.
If a valuation v is applied to the field K, then there exists a unique element x0. This element x0 is in the image of v applied to the carrier of K excluding the zero element, and for every element w in the image of v applied to the carrier of K, there exists an integer a such that w equals a times x0. Additionally, x0...
The lemma in the context of the geometric random walk probability model shows that the geometric process is adapted to the natural filtration in the Borel sigma-algebra.
Mapping the function, which increments a natural number by one, over the list containing the positive number 0 and the negative number 1 results in a new list containing the positive number 1 and the negative number 2.
If x is not in the image of the set A under the function hcomplex_of_complex, then for all y in A, x is not equal to the hcomplex_of_complex of y.
The lemma named "less_multiset_doubletons" assumes that either "y" is less than "t" or "y" is less than "s", and that either "x" is less than "t" or "x" is less than "s". If these conditions are met, it shows that the multiset containing "y" and "x" is less than the multiset containing "t" and "s".
The representation of a finger tree after adding an element to the right end (rcons operation) is equivalent to adding the same element to the right end of the structure of the original finger tree.
Either 'f' is equal to zero (in the case named "zero"), or there exist integers 'm' and 'e' (in the case named "powr") such that the real number equivalent of 'f' is equal to 'm' times 2 raised to the power of 'e', 'm' is not divisible by 2, and 'f' is not equal to zero.
The lemma named "simply_connected_eq_homotopic_circlemaps2b" establishes a condition for a set to be path-connected. It fixes a set 'S' of real normed vectors. It assumes that for any two functions 'f' and 'g' from complex numbers to 'a', if both functions are continuous on the unit sphere and their images under these ...
There exists a number 'n' in the set {0, 2, 4, 6, 8} such that 'n' is not even.
The lemma named "rank_take_cols_geq" assumes the following conditions: 1. Matrix R is an n by nc matrix. 2. The matrix obtained by taking certain columns (ls) from matrix R is an n by r matrix. 3. Matrix B is an r by nc matrix. 4. Matrix R is equal to the product of the matrix obtained by taking certain columns (ls) f...
For any given point 'y' in a topological space of type 'a' (which is a T1 space), if 'I' is a finite set, then eventually for all 'x' in the neighborhood of 'y' within 'Y', 'x' will not belong to 'I'.
Reading a natural number after showing it results in the original natural number.
Converting a disjunctive normal form (dnf) to a boolean using the function "dnf_to_bool" is equivalent to the existence of a disjunct in the set of the disjunctive normal form that can be converted to a boolean using the function "cnf_to_bool".
The lemma named "grab_Un_eq" assumes that "N" is infinite. It shows that the union of the first element (fst) and the second element (snd) of the function "grab" applied to "N" and "n" equals "N".
The lemma "before_in_mtf" assumes that "z" is an element in the set "xs". It shows that "x" is less than "y" in the move-to-front operation on "z" in "xs" if and only if "y" is not equal to "z" and if "x" equals "z" then "y" is an element in the set "xs", otherwise "x" is less than "y" in "xs".
"for all 'x', if there exists a 'y' such that 'x' equals 'y', then the statement is true, otherwise it is false."
The advantage of A is greater than or equal to zero.
The union of F and G is in the transient state A if and only if either F is in the transient state A or G is in the transient state A.
The conversion of a physical quantity 'x' scaled by a factor 'd' using a conversion factor 'c' is equivalent to scaling the converted quantity 'x' by the same factor 'd'.
The lemma "isArrLenLoc_simps" states the following conditions: - The function "isArrLenLoc" returns false when applied to an object location "objLoc cf a". - The function "isArrLenLoc" returns false when applied to a static location "staticLoc f". - The function "isArrLenLoc" returns true when applied to an array leng...
For all pairs of elements (x, y) in the set xys, if the function f applied to x and y equals the function g applied to x and y, then the sum of the results of applying function f to each pair in xys equals the sum of the results of applying function g to each pair in xys.
The greatest common divisor of two natural numbers 'x' and 'y' is equal to the greatest common divisor of 'y' and the remainder of 'x' divided by 'y'.
If 'v' is directed towards the root, then the parent of 'v' is the root.
The supremum (or greatest element) of 'OK x' and 'OK y' in the context of a function 'f' is equal to 'OK' of the supremum of 'x' and 'y' under the same function 'f'.
The bitwise AND operation between a word of integer 'i' and a mask 'n' is equal to the word of integer obtained by taking the 'n' least significant bits of 'i'.
If S is a cring, and if there exists a ring homomorphism from R to S, and if r is an element of the carrier of R, then the polynomial lift homomorphism from R to S applied to the monomial in the univariate polynomial ring over R with coefficient r and degree n, is equal to the monomial in the univariate polynomial ring...
The number 5 is equal to the successor of the successor of the successor of the successor of the successor of 0.
: If for any class C, method M, method body b, list of types Ts, type T, and method m, the following conditions hold: - The method declaration is well-formed according to the well-formedness condition wf1 in program P for class C and method (M, b, Ts, T, m), and - The program P sees that class C has method M with body...
If the implication of phi and psi holds for a path pi, and phi holds for the same path, then psi must also hold for that path.
In the context of group homomorphisms, the lemma named "generate_hom" assumes that a set A is a subset of the carrier G. It shows that the image of the generate function applied to G and A under the homomorphism h is equal to the generate function applied to H and the image of A under the homomorphism h.
Given the assumption that the image of the function 'f' over the set 'S' is a subset of 'T', the function 'f' is continuous on 'S' if and only if for all 'U', if 'U' is closed in the topology of 'T', then the intersection of 'S' and the preimage of 'U' under 'f' is closed in the topology of 'S'.
The count of tree "t" is greater than or equal to 0.
If 's' is in the space 'Ms' and 'a' is in the space 'Ma', then the pair (s,a) is in the space 'M'.
The homomorphism of the difference between two elements 'x' and 'y' is equal to the difference between the homomorphism of 'x' and the homomorphism of 'y'.
The product case matches with the "match_brz" function for the "M 128" case.
Mapping the variables of the negative of a multivariate polynomial p using a function f is the same as the negative of the mapping of the variables of the polynomial p using the same function f.
The lemma "elem_list_splits_elem" assumes that the tuple (xs, y, ys) is an element of the set produced by splitting the list zs. It shows that y is an element of the set zs.
The validity of the heap map after exchanging elements at positions i and j is the same as the validity of the original heap map.
The integer "z" belongs to the "Standard" set when it is converted using the function "of_int".
For any expression 'e' of any type 'a', the evaluation of 'e' is defined when the context is None. Similarly, for any list of expressions 'es' of any type 'a', the evaluation of 'es' is defined when the context is None.
If "s" is an element of "S", then the sum of "w" and the singleton set of "S" excluding "s" is not in the singleton map of the sum of "w" and the singleton set of "s".
The lemma named "PREC_case_aux" assumes two conditions: 1. For any list 'l', the sum of function 'f' applied to 'l' and the sum of the elements in 'l' is less than the Ackermann function applied to 'kf' and the sum of the elements in 'l'. 2. For any list 'l', the sum of function 'g' applied to 'l' and the sum of the e...
For all flow sets F that are part of the offending flows obtained from the Access Control System (ACS) of a model M and converted to a network graph of a stateful policy T, F is a subset of the backflows of the state flows of T.
The lemma "exe_complete_tcsigs_imp_complete_tcsigs" assumes that the conditions for "exe_ars" are met for "arss" and that "exe_complete_tcsigs" is true for "cs" and "arss". If these conditions are met, it shows that "complete_tcsigs" is true for the set of "cs" and the translation of "arss".
The nth bit of a word (of any length 'a) converted from a bit list (bl) is equal to the nth element of the reversed bit list, given that 'n' is less than the length of 'a' and the length of the bit list.
The star transformation of zero equals zero.
The lemma named "not_weak" assumes that "x" is an element of set "A" and "y" is also an element of set "A". It shows that "x" is not weakly less than or equal to "y" if and only if "y" is strictly less than "x".
The function which takes three arguments (k, v, f) and updates the function f at the point k with the value v, is related to the function that updates a map, when uncurried. This relationship is within the context of a function space where the input is a pair of a pair and a function (where the pair consists of two ide...
If a function 'f' applied to a value 'v' equals 'v', then 'v' is a trivial code for the mapping of function 'f' over a set of values 'vs' if and only if there exists a value 'v'' in the set 'vs' such that applying function 'f' to 'v'' does not equal 'v' and the set of all values obtained by applying function 'f' to the...
The lemma named "after_reachable" assumes that the machine "M" is observable, the input-output pair "io" is in the labeled states of "M" at state "q", and "q" is in the reachable states of "M". It shows that the state of "M" after "q" with input-output pair "io" is also in the reachable states of "M".
If v equals the difference of u and w, then the cardinality (or size) of the restriction of function f to the set w with condition c is less than or equal to the cardinality of the restriction of function f to the set v with the same condition c.
The lemma "netmap_none_not_in_net_ips" assumes that the network map of a certain network state 'ns' at an index 'i' is 'None'. It shows that 'i' is not in the network IPs of the network state 'ns'.
: Assuming that the Ute model satisfies the safety property of the Heard-Of model for each round, and that the next state of process p in round r, given the current state and message vector, is the state of process p in round r+1, and that the message vector for process p is in the set of possible message vectors for ...
If there is an equivalence implication between the union of all worklists for a given state and the system's free memory, as well as the mapping of an object's mark in the system's heap, then this implies a black heap.
If there is no loose bound variable in term 't' at level 'lev', then the substitution of bound variables in term 't' at level 'lev' with 'us' will result in term 't' itself.
The integer one is not equal to the bottom element in the domain of integers.
The sum of the differences of a series in an integral is monotonically sequenced.