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An empty set is a Groebner basis.
If the rank invariant holds for a list consisting of an element 't' followed by a binomial queue 'bq', and if 'bq' is not an empty list, then the rank of 't' is less than the rank of the first element in 'bq'.
The pell valuation of the pair (a, 0) is equal to the integer value of a.
If "p" is a real linear form and "q" is a real linear form, then the conjunction of "p" and "q" is also a real linear form.
The lemma named "spectrum_root_char_poly" assumes that a matrix A, which belongs to a field, is an n x n matrix. It shows that the spectrum of matrix A is equal to the set of all values k such that the characteristic polynomial of matrix A evaluated at k equals zero.
The lemma named "formulaFalseAppend" is defined with respect to a formula, another formula (formula'), and a valuation. It assumes that the given formula is false under the given valuation. It then shows that the result of appending the second formula (formula') to the first formula is also false under the same valuati...
: given a sequence of events 'evs' that belongs to the trace 'p', and 'p' only has 'Says' actions, and 'p' preserves keys for 'n' and 'Ks', and 'p' is monotonic with respect to keys, and the initial state of the Spy is guarded by 'n' and 'Ks', then if 'Ks' is safe from being spied in 'evs', and if 'p' is fresh for 'R',...
For a fixed domain name department 'x', if 'x' is less than or equal to 'y' and 'y' is less than or equal to 'z', then 'x' is less than or equal to 'z'.
The function which multiplies A(t) and the derivative of f(t) is integrable over the interval from y to x. The "partial_summation_strong'" states that the sum of the product of a(n) and f(n) for all real numbers n in the open interval from y to x is equal to the product of A(x) and f(x) minus the product of A(y) and ...
If polynomial 'p' is not equal to zero, then the order of 'x' in polynomial 'p' is zero if and only if the value of polynomial 'p' at 'x' is not equal to zero.
A point "a + x" is an extreme point of the set obtained by translating all points in set "S" by "a" if and only if "x" is an extreme point of the set "S".
If M and N are modules over the ring R, P is a function from M to N, m is an element of the carrier of M, and n is an element of the carrier of N, then the tensor product of m and n with respect to R and P is an element of the carrier of the tensor product of M and N with respect to P and R.
If a binary number 'a' is in the set of unsigned integers of length 'l', and 'm' is greater than or equal to the sum of 'l' and 'n', then the binary number resulting from concatenating 'n' bits and 'b' to 'a' is in the set of unsigned integers of length 'm'.
The Digamma function of a natural number 'n' plus one-half, in the context of a real normed field or a Banach space, is equal to the sum (for all 'k' less than 'n') of two divided by the natural number of two times 'k' plus one, minus the Euler-Mascheroni constant, minus twice the natural logarithm of two.
The lemma named "lattice_of_finsum" assumes that the set of vectors 'fs' is a subset of the set of vectors of dimension 'n'. It shows that the lattice generated by 'fs' is equal to the range of the function that maps each integer 'c' to the finite sum of the product of 'c' and the 'i'-th vector in 'fs', where 'i' range...
If the atoms m, n, sm, sm', sn, and sn' are all distinct from the set of variables (v,u,x,y,s,k,sm,sm',sn,sn'), then the sequence substitution form of v, u, x, y, s, k implies that either the substitution atomic form of v, u, x, y holds, or there exist variables m, n, sm, sm', sn, sn' such that m and n are in k, the se...
The power operation is equivalent to applying the standard power operation to the nonstandard extension of a function. This function takes an input 'x' and raises it to the power of 'n'.
For any elements 'x' and 'y', the product of 'd' of 'x' and the sum of 'd' of 'x' and 'd' of 'y' is equal to 'd' of 'x'. This is a simplification rule.
The real part of the hypercomplex number 1 is 1.
The lemma named "vertices_cases" assumes that "v" is an element of "vertices". It then establishes two possible cases: 1. There exists a "c" such that "c" is in the set of conclusions and "v" is equal to the pair "(c, [])". 2. There exists a "c" and "is" such that "c" is in the set of conclusions, "is" is in the set o...
: 1. The locks of the abstract state representation of 's' are equal to the locks of 's'. 2. The thread of the abstract state representation of 's' is equal to the abstract thread representation of the thread of 's'. 3. The shared resources of the abstract state representation of 's' are equal to the shared resources ...
The lemma "get_owner_document_disconnected_nodes" assumes that the heap "h" is well-formed, the document pointer "document_ptr" can successfully retrieve the disconnected nodes "disc_nodes" from the heap "h", and the node pointer "node_ptr" is in the set of disconnected nodes. It also assumes that the heap "h" contains...
The function 'p' has a vector derivative 'p' prime at the point 't' within the set 's'.
The lemma named "obtain_partition" assumes two conditions: 1. Set A is finite. 2. The number of partitions of the cardinality of set A equals N. If these conditions are met, it shows that there exists a partition P on set A such that the multiset of the cardinalities of the sets in partition P equals N.
Given three finite sets U, V, and W, all subsets of a set G, the cardinality (or size) of set U multiplied by the cardinality of the difference set between V and W is less than or equal to the cardinality of the difference set between U and V multiplied by the cardinality of the difference set between U and W.
Partial and total correct recursion are equal if total recursion does not fail. This is under the assumption (denoted by "NT") that the total correct recursion of the body and x is not equal to the top. If this condition is met, then the total correct recursion of the body and x is equal to the partial correct recursio...
If there is a transition from the psi process P, permuted by the pair (x, y), under the context Psi, to the psi process P' via the output action M with the new names xvec and yvec and the object N, and if y is in the support of N, x is fresh for N and P', x is not equal to y, y is fresh for xvec
If the initial set is invalid, then the result of the iterate operation on this set, regardless of the accumulator 'A' and the body 'P a x' of the iteration, will also be invalid.
There exists a number 'y' such that 'y' is greater than negative pi and less than or equal to pi, and the sine of 'y' is equal to the sine of 'x' and the cosine of 'y' is equal to the cosine of 'x'.
Mapping a function "f" over a probabilistic function "p" is equal to mapping a function "g" over a probabilistic function "q" if and only if there exists a relation between "p" and "q" such that for every pair of elements "x" from "p" and "y" from "q", the result of applying "f" to "x" is equal to the result of applyin...
The "nolb" of the conjunction (And) of two propositions (phi1 and phi2), with respect to a list of variables (xs), a list (l), and a variable (x), is equivalent to the conjunction of the "nolb" of each proposition with respect to the same list of variables, list, and variable.
If the count of an element 'x' in a filtered multiset 'M' (filtered by a predicate 'P') is greater than zero, then the count of 'x' in the original multiset 'M' is also greater than zero.
The pair of operations (hm_lookup_op, RETURN composed with op_map_lookup) is included in the relation that maps an identity to a heapmap relation, which in turn maps to a nested result relation that may optionally contain an identity.
The lemma named "gap_relabel_aux_refine" assumes that "C" is equal to the cardinality of "V" and "l" maintains an invariant. It shows that the function "gap_relabel_aux" with parameters "C", "f", "l", and "u" is less than or equal to the function "gap_relabel" with parameters "f", "l", and "u".
If "A" and "B" are objects in the category and "f" is a morphism from "A" to "B", then the domain of "f" is "A" and the codomain of "f" is "B".
If for every element 'x' in the set 'm1', the function 'f1' applied to the first and second elements of 'x' equals the function 'f2' applied to the first and second elements of 'x', and if 'm1' equals 'm2', then the mapping of the range of 'f1' over 'm1' equals the mapping of the range of 'f2' over 'm2'.
If 'a' is an object in category 'C', then the following conditions hold: 1. There exists an isomorphism in category 'D' between the tensor product of the transformation 'τ₀' prime of 'a' and 'τ₀' of 'a', and the functor 'F₀' of 'a'. 2. There exists an isomorphism in category 'D' between the tensor product of the trans...
The lemma named "match_rel" assumes that the predicate "P" holds for terms "t" and "u". It then shows that the relation "rel_option P_env" holds between the results of matching pattern "p" with terms "t" and "u".
If from the set containing A and B, we can infer C, and if by inserting C into a set H, we can infer D, and if from set H we can infer B, then by inserting A into set H, we can infer D.
The composition in the semicategory of graphs is a function that takes a pair of composable arrows in the semicategory of graphs and returns the composition of the first arrow with the second one.
The lemma named "span_list_as_span" assumes that the set of vectors 'vs' is a subset of the carrier vector of dimension 'n'. It shows that the span of the list 'vs' is equal to the span of the set 'vs'.
If a well-formed graph is obtained from a list graph "G" using the function "list_graph_to_graph", then a well-formed graph will also be obtained when a node "v" is added to the list graph "G" using the function "add_node", and this new list graph is converted to a graph using the same "list_graph_to_graph" function.
The set of subterms of a term 't' after substitution by 'σ' is a subset of the subterms of the term 't' after it has been substituted by 'σ'.
If G is a well-structured program, then the supremum state option of G is ordered.
The lemma proves_eq_combination_rule_better: Assuming that the theory 'Theta' is well-formed, and 'Theta' and 'Gamma' prove the equality of 'f' and 'g', and the equality of 'x' and 'y', and the type of 'f' is some function from 'tau' to 'tau prime', and the type of 'x' is some 'tau', and 'Gamma' is finite, every term i...
If a property P holds for all elements in the set that results from inserting an element y into set A, then it can be concluded that property P holds for element y and for all elements in set A.
If the following conditions are met: 1. TS is an element of the high-level Petri net of ST. 2. S is an element of the states of the hierarchical automaton of ST. 3. SA is an element of the composition function of the hierarchical automaton of ST applied to S. 4. T is an element of the states of SA. 5. T is an element ...
If H is a subgroup of G, then the following conditions hold: 1. H is a subset of the carrier of G. 2. H is not an empty set. 3. For any element 'a' in H, the inverse of 'a' is also in H. 4. For any two elements 'a' and 'b' in H, the product of 'a' and 'b' is also in H.
Zero steps of the semi-identity Turing machine, when started in state 1 with empty input and output tapes, results in the machine remaining in state 1 with empty input and output tapes.
The lemma named "map_upt_zip" assumes that the lengths of two lists, xs and ys, are equal. It shows that mapping a function f on the elements of xs and ys at each index i from 0 to the length of ys is equivalent to mapping the same function f on the pairs of elements (x, y) from the zipped lists xs and ys.
If a hypothesis set "H" entails a statement "A", and for all statements "B" in "H", the atom "i" is fresh in "B", then "H" entails the statement "A" where "i" is substituted by "x".
The lemma named "Ball_scf" assumes that "R" is a preference profile. It shows that for every element "x" in the social choice function of "R", property "P" holds for "x" if and only if either "a" is not in the social choice function of "R" or property "P" holds for "a", and either "b" is not in the social choice functi...
If 'x' and 'y' are finite, then the next maximum value of the union of 'x' and 'y' is equal to the maximum of the next maximum value of 'x' and the next maximum value of 'y'.
If X is a decreasing sequence and X converges to L, then L is less than or equal to X at any given point n. This is for L in the context of a linear order topology.
The lemma is named "linear_system_homogenous_3_2". It fixes variables a11, a12, a13, a21, a22, a23, x1, y1, z1, x2, y2, z2 which belong to a field. It assumes two functions f1 and f2, where f1 is defined as a11 times x plus a12 times y plus a13 times z, and f2 is defined as a21 times x plus a22 times y plus a23 times z...
The lemma named "adjust_orthogonal" assumes the following conditions: 1. The set of vectors 'us' is a subset of the carrier vector of dimension 'n'. 2. The vectors 'us' are corthogonal, meaning they are mutually orthogonal and each vector has a norm of 1. 3. The vector 'w' is in the carrier vector of dimension 'n'. 4....
In the context of node histories, the lemma named "node_total_order_antisym" assumes that event 1 is strictly less than event 2 and event 2 is strictly less than event 1. This leads to a contradiction, hence it shows "False". This means that it's impossible for both conditions to be true at the same time, reflecting th...
The supremum is closed for the fixed point of the set of relations R in the Alexandrov topology.
Mapping a function that takes a pair (x, y) and returns the second element y over a list is equivalent to mapping the function "snd" (which returns the second element of a pair) over the same list.
The lemma provides simplification rules for the join operation on lazy lists: 1. The join of an undefined lazy list of lazy lists is undefined. 2. The join of an empty lazy list is an empty lazy list. 3. The join of a lazy list that has a head 'xs' and a tail 'xss' is the monadic plus of 'xs' and the join of 'xss'.
If a non-negative real number 'c' is less than infinity and a function 'f' is supremum continuous, then the function that multiplies 'f(x)' by 'c' is also supremum continuous. This is in the context of extended real numbers.
For any two polynomials P and Q, if for every value of x, the evaluation of polynomial P at x equals the evaluation of polynomial Q at x, then the polynomials P and Q are equal. This is under the assumption that the coefficients of the polynomials belong to a commutative ring with no zero divisors and of characteristic...
The lemma named "chine_hcomp_ide_ide" assumes that "g" and "f" are identities and the source of "g" is the target of "f". It shows that the horizontal composition of "g" and "f" is equal to the double down arrow between the Leg0 of the domain of "g" and the Leg1 of the domain of "f".
For a given binary relation "less" on a set A, the relation is asymmetric if and only if it is both irreflexive and antisymmetric.
The lemma named "union_wit_subset" shows that the union of the set "E" with any other set is a subset of "E".
If 'g' and 'h' are elements of 'G', then the difference between the permutation of 'g' and the permutation of 'h' is equal to the permutation of the binary operation of 'g' and the inverse of 'h'.
The lemma named "sum_rel_def" defines the relation "sum_rel" parameterized by two relations "Rl" and "Rr". This relation consists of two sets. The first set contains pairs of left injections ("Inl") of elements "a" and "a'" such that "a" and "a'" are related by the relation "Rl". The second set contains pairs of right ...
It is not the case that the program Γ transitions from the state (c, Fault x) to an infinite state.
The identity function is defined as a function that, for any input of type 'A, returns the same input.
The lemma named "eval_lowerFm_helper" assumes that "i" is a free variable in "F" and the length of "init" equals "i". It shows that the evaluation of the function "lowerFm" with parameters "i", "1", and "F", when applied to the concatenation of "init" and "xs", is equal to the evaluation of "F" when applied to the conc...
The lemma "hn_case_prod'" is a rule for preparing and combining separation logic refinements. It assumes two conditions: 1. The frame rule (FR): Given a context Γ, there exists a heap node context (hn_ctxt) for a product of two assumptions P1 and P2, denoted as p' and p, and another context Γ1 such that Γ implies the ...
If a product sequence 'f' is convergent, then the product of the inverses of each term in the sequence 'f' is equal to the inverse of the product of the sequence 'f'.
For any pair of natural numbers 'k', if 'k' is greater than the pair (0, 0) in the context of the 'less_int_raw' function, then there exists a natural number 'n' greater than 0 such that 'k' is equivalent to the integer representation of 'n' under the 'int_of_nat_raw' function.
The set of all complex numbers z, such that the argument of z in [0, 2π) is less than or equal to a given number t, is a closed set.
The lemma named "d_states_size" assumes that "q" is in the states of "M". It shows that the length of the "d_states" of "M" and "q" is less than or equal to the size of "M" minus one.
The lemma named "substEnv_preserves_good" assumes that "X" is good and "rho" is a good environment. It shows that the environment "rho" remains good even after the substitution of "X" for "x" in "xs".
If a function phi is monotonic and satisfies the infinite multiplicative property, then the fixed point set of phi is closed under infimum.
: 1. If the final state of 'i' is less than or equal to the first element of 'x' and 'x' is an interval, then the size of the interval derived from the left of 'x' is less than the size of the interval 'x'. 2. If the second element of 'x' is less than or equal to the final state of 'i' and 'x' is an interval, then the ...
If the statement "phi or psi is in v", "if phi then chi is in v", and "if psi then Theta is in v" are all true, then it logically follows that "chi or Theta is in v".
The relation of a regular expression "r" with respect to a list of variables "vs" is equal to the relation of "r" with respect to the function that retrieves the "i"-th element from "vs".
If the transition function 'delta' for regular expressions 'R' and 'S' with input 'bs' are equal, then the acceptance of a run with input list 'bs # bss' and output 'cs' for 'R' is equivalent to the acceptance of a run with the same input list and output for 'S'.
If the subdegree of a formal Laurent series 'f' is 0, then converting the base factor of 'f' to a formal power series results in the regular part of 'f'.
Given a well-formed heap 'h' with well-typed elements and known pointers, if 'h' can be transformed into a tree order 'nodes' with a pointer 'ptr', and 'h' can get child nodes 'children' from a parent node 'parent', and a child node 'child' (which is not equal to 'ptr' when cast) is in the set of 'children' and the set...
For all subsets 'ps' of 'V', if the cardinality of 'ps' is 't', then the index of 'ps' in 'B' is greater than or equal to 'Lambda_t'.
, given the assumptions that a relation "A" is bi-unique and right-total, it can be shown that the function "comm_monoid_mult_ow" over the collection of domain elements of "A" is equivalent to the class of commutative monoids under multiplication, when both are considered as functions that take two arguments from "A" a...
The relation "gen_sa_impl_rel_eext" over two relations Rm and Rl is defined as the set of all pairs of states (Li, L) and (mi, m) such that (Li, L) is in relation Rl and (mi, m) is in relation Rm.
The number of symbols in a term 't' is always greater than or equal to the number of variables in the same term 't'.
For any process P, the replication of P is bisimilar to the parallel composition of P and its replication.
The set of states in a loop of state 's' is equal to the set containing only 's'.
If there is a directed graph isomorphism defined by 'hom', then the application of this isomorphism to a graph 'G' results in a directed graph map with the map of the isomorphism 'hom'.
The lemma named "HcompNml_Trg_Nml" assumes that "t" is normal. It shows that the target of "t" when horizontally composed with "t" is equal to "t".
For any graph 'E'', if 'E'' is a subgraph of 'E', then 'E'' is a valid graph.
If the vertices from function 'f' starting from vertex 'v' are distinct, and if the list of vertices from function 'f' starting from vertex 'v' that are either equal to 'a' or belong to the set 'vs' is equal to the list 'a' followed by 'vs', then the list of vertices from function 'f' starting from vertex 'v' that belo...
If a tau-execution move is valid for a given configuration consisting of the program P, thread t, expression e, heap h, and states s and s', then a tau-execution move is also valid for the same configuration where the expression is accessing a field F of an object of type D.
If the sum of the diamond of z and the box of x applied to the nabla of the sum of x and y equals the nabla of the sum of x and y, then the nabla of the sum of x and y equals the nabla of x plus the box of the star of x applied to z.
In the context of an equivalence relation, the lemma named "disjoint_sum" assumes that the carrier of set S is finite. It then shows that the sum of function f over each element x in each class is equal to the sum of function f over each element x in the carrier of set S.
If a list 'xs' is in the set produced by the weak integer composition enumeration of 'i' and 'l', then the sum of the elements in the list 'xs' is equal to 'i'.
If from a set of assumptions A, we can infer that if "a" implies "b" implies "c", then we can also infer from the same set of assumptions A that if "a" and "b" are both true, then "c" is true.
If the time of operation c on heap h is less than or equal to t1, and the time of operation d on the result of c on heap h is less than or equal to t2, then the time of operation c bound to d on heap h is less than or equal to the sum of t1 and t2.
The lemma named "finite_pcategory_finite_psemicategoryI" assumes that for a given index set I, if there is a finite pre-semi-category with index i and category "cat_smc (\<AA> i)" and for every index i in I, there is a category (\<AA> i), then there is a finite pre-category with the same index set I and category \<AA>.