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The set of free variables in the transformed expression (transform a e) is a subset of the free variables in the original expression (e).
For any set A, if P A is true, then A is finite and the cardinality of A is less than or equal to N. It also assumes that P B is true. The lemma then shows that there exists a set A which is finite and is maximal with respect to P.
If there is a proof of the existence of a variable 'x' in a formula 'phi' within a finite set 'F' of formulas, and there is a proof that the substitution of 'x' with another variable 'y' in 'phi' results in the equality of 'y' and 'x', given that 'F' is a subset of the set of all formulas, 'F' is finite, 'phi' is a for...
If the index 'i' is less than the length of the list 'xs' and the 'i'-th element of the list 'xs' is 'b', then 'b' is in the range of the list 'xs'.
If 'k' is less than or equal to 'm', then the property of being 'closed' for the successor of 'm' in relation to 'p' is equal to the property of being 'closed' for 'm' in relation to the substitution of 'p' with the application of 'c' to an empty list at 'k'.
If both "phi1" and "phi2" are true, and if "phi1" implies "chi", then "chi" is true.
If "c" is greater than or equal to the upper bound of "s" times "x" in the given direction, then a certain property, "P'", holds. The second assumption states that if "c" is not greater than or equal to the upper bound of "s" times "x" in the given direction, but is less than the lower bound of "s" times "x" in the g...
: Given the assumptions that "t" is an element of "SPR.jkbpC" and "spr_simAbs ec" is equal to "SPRdet.sim_equiv_class a t", it can be shown that the set of "spr_simAction a ec" is equal to the set of "jAction SPR.MC t a". This is denoted as "?lhs = ?rhs".
The lemma named "binop_plus" is about a fixed list "b" of pairs, where each pair consists of a natural number and an element from a Euclidean space. It shows that the sum of the product of a function "e" applied to the first element of each pair and the second element of the pair, over all pairs resulting from the bina...
The product of the identity function over the set F is congruent modulo p to the product of the identity function over the set E multiplied by negative one to the power of the cardinality of E.
The nth term of a constant formal power series is equal to the constant if n is zero, otherwise it is zero.
Assuming that 'f' is a continuous map from topological space T1 to topological space T2, the preimage of the topological space T2 under 'f' intersected with the topological space T1 is equal to the topological space T1.
: Assuming a user with identifier "uid" is not honest, and that this user is active at a certain time "n", and there exists a time "n'" less than "n" at which the user is also active, it can be shown that: There exists another user with identifier "nid'" who is active at the time when "uid" was last active, and eithe...
If the function 'g' is equivalent to a function that always returns 'False', or a function that returns its input (identity function), or a function that always returns 'True', or a function that returns the negation of its input, then the proposition 'P' holds.
The lemma named "tconcat'_code" has two parts: 1. The concatenation of 'b' and an empty list 'b'' results in the empty list 'b''. 2. The concatenation of 'b' and a list consisting of an element 'a' and a tail 'tr' is equivalent to appending 'a' to the concatenation of 'b' and 'tr'.
The lemma "sums_vec_nth" assumes that a sequence "f" sums to "a". It shows that the "i"-th component of the function "f" at "x" sums to the "i"-th component of "a".
The range of the function, which takes an input 'xa' and returns the absolute infinity quotient of the product of 's' (converted to an integer modulo ring) and the coefficient of 'x' (converted to a quotient ring) at 'xa', is finite.
If the result of adding map 'n' to map 'm' at key 'k' equals 'Some x', then either map 'n' at key 'k' equals 'Some x' or map 'n' at key 'k' equals 'None' and map 'm' at key 'k' equals 'Some x'.
If 'n' is less than or equal to 'm' and 'n' is a new type variable in 't', then 'm' is also a new type variable in 't'.
The evaluation of the logical expression "less" with variables "x" and "y" in the context of natural number environments E, F, and G is false.
If for all C, the strip of f(C) is equal to the strip of C, and the well-founded partial fixed point of function f and C is Some C', then the strip of C' is equal to the strip of C.
The domain of the identity morphism of the product category, for each i in I of the category A, is equal to the objects of the product category, for each i in I of the category A.
If the function 'f' is continuous at a point 'x' and 'f' at 'x' is not equal to zero, and the sign of 'g' to the left of 'x' is equal to the sign of 'g' to the right of 'x', then the jump of the function 'f' divided by 'g' at 'x' is equal to zero.
For all states 's', if 'P' of 's' is greater than or equal to the successor of 'Q' of 's' with 'a' substituted for 'x', then a Hoare triple is valid where 'P' is the precondition, 'x ::= a' is the command, and 'Q' is the postcondition.
, given the assumptions that "G" is right total, and that "G" preserves the equality of "αe", "αn", "invar", "inEdges'", "Entry", "defs", and "uses", it follows that "G" also preserves the equality of "defs'" and "CFG_Construct.defs' defs2".
The lemma named "bisimE" is defined with two fixed variables, P and Q, both of which are of type pi. The lemma assumes that P is bisimilar to Q. Under this assumption, the lemma shows that P leads to Q under bisimulation.
The counter assignment "v" is an object in the counter model "f".
If the concatenation of list 'a' and list 'b' belongs to the set of 'n'-lists over set 'A', then for any 'n1' and 'n2' such that 'n' equals the sum of 'n1' and 'n2', if list 'a' belongs to the set of 'n1'-lists over 'A' and list 'b' belongs to the set of 'n2'-lists over 'A', then proposition 'P' holds.
If we have a finite set 'I' and for each element 'i' in 'I', the function 'f i' is Borel measurable with respect to the measure space 'M', then the function that maps 'x' to the minimum value of the set of all 'f i x' for 'i' in 'I' is also Borel measurable with respect to the measure space 'M'. This holds for any type...
The lemma "f_of_newton_seq_closed" shows that the function "f" applied to the "m-th" element of the Newton sequence "ns" is in the carrier of the p-adic integers "Zp".
If 'x' is an element of the carrier of the fraction field of 'R', 'n' is greater than 0, and 'x' raised to the power of 'n' (where 'n' is a natural number) is an element of the non-zero elements of the fraction field of 'R', then 'x' is also an element of the non-zero elements of the fraction field of 'R'.
The "uconnected" relation for a set E is symmetric.
The evaluation of a list 'l' under the environment 'E' and the Herbrand function 'HFun' with respect to 'G' is equal to the evaluation of the list 'l' after applying the substitution function 'sub_of_denot' under the same environment 'E'. Furthermore, the list 'l' after applying the substitution function 'sub_of_denot'...
In the lemma (in the field), the power of a sum: if 'x' is an element of the carrier 'K' and 'x' is not zero, then the power of 'x' in 'K' to the sum of 'i' and 'j' is equal to the product of the power of 'x' in 'K' to 'i' and the power of 'x' in 'K' to 'j'.
The lemma named "residue_mult_comm" in the context of p-adic integers shows that the multiplication operation in the residue ring of Zp at level k is commutative. That is, for any two elements x and y in this ring, the result of multiplying x by y is the same as the result of multiplying y by x.
The lemma "rrb_simps" states the following: - The relational range bound (rrb) of a Boolean value is always true. - The relational range bound of a predicate is always true. - The relational range bound of an equality is always true. - The relational range bound of a negation is the same as the relational range bound ...
The lemma "finrank_imp_findim" assumes that "V" has a finite rank. It shows that "V" also has a finite dimension.
The function that maps each natural number to its order is injective when applied to the image set. In other words, no two different natural numbers will be mapped to the same order.
If the image of the closure of a set S under a function f is bounded, then the image of the set S under the same function f is also bounded.
The language of a restricted automaton is equal to the language of the original automaton.
Given a subgroup I of group G, the function h is injective on I if and only if the kernel of the function h, when the carrier of group G is set to I and the group is H, is equal to the set containing only the identity element.
The list 'xs' is equal to the empty list 'TNil' with an element 'x'.
The guard of the intersection of S and T is equal to the union of the guard of S and the guard of T.
If "p" is a prefix of "w", then the inverse of the left quotient of "w" by the inverse of the right quotient of "p" by "w" is equal to "p".
The implementation of Kruskal's algorithm, denoted as "Kruskal_Impl", with parameters E, V, vertices, joins, forest, connected, weight, alpha, endpoints, getEdges, getEdges_impl, superE, and endpoints_impl, holds true.
If set B is not empty, then the prefixes of the concatenation of sets A and B is equal to the union of the prefixes of set A and the concatenation of set A with the prefixes of set B.
If a relation "r" is well-founded for a natural number "n", then the relation "r" in the star closure is also well-founded for the same natural number "n".
The lemma named "canI_associator_3" assumes that "a" is an object. It shows that the associator of "a", "a", and "a" is equal to the canonical isomorphism of the tensor product of "a", "a", and "a", where the tensor product is taken in two different ways: first "a" tensor "a" tensor "a", and then ("a" tensor "a") tenso...
If polynomial p1 is greater than or equal to polynomial p2, and polynomial p2 is greater than or equal to polynomial p3, then it logically follows that polynomial p1 is greater than or equal to polynomial p3. This is a transitive property of polynomials.
: Assuming that the triple [[a, b, f], [a', b', f'], [g, h]] is an arrow in the comma category of the functors G and H, where G is a functor from category A to category C, and H is a functor from category B to category C. Then, it can be shown that: 1. g is an arrow from object a to object a' in category A. 2. h is...
The lemma named "fps_unit_factor_mult'" is defined with two fixed formal power series 'f' and 'g'. These series are of a type that supports both commutative addition and multiplication by zero. The lemma assumes that the product of the coefficients of 'f' and 'g' at their respective subdegrees is not zero. Under this a...
For any constant 'c', context 'G', value 'v', and variable 'xa', given that the context 'G' is well-formed and 'v' is of type 'tau' under the assumptions 'Theta', 'B', and 'Gamma', and 'G' is equal to the concatenation of 'Gamma' and a list containing a tuple of 'x', 'b', and 'c' where 'z' is replaced by 'x', and 'xa' ...
The lemma "get_elements_by_tag_name_is_strongly_dom_component_safe_step" assumes the following conditions: 1. The heap 'h' is well-formed. 2. The heap 'h' has a valid type. 3. The heap 'h' has known pointers. 4. The function 'get_component' when applied to the pointer 'ptr' in the heap 'h' returns the component 'c'. 5...
The Map Operations are a deterministic simulation abstraction of the SPRdet.jkbpSEC according to the Map Operations.
The lemma "measurable_on_Pair" assumes that functions 'f' and 'g' are measurable on a set 'S'. It shows that a function that maps 'x' to a pair consisting of 'f(x)' and 'g(x)' is also measurable on 'S'.
The supremum (or least upper bound) of two sequences, f and g, is equal to a new sequence defined by a function that takes an element u and returns the following: 1. If the result of applying the function f to the empty tuple is Empty, then return the result of applying the function g to the empty tuple. 2. If the res...
The sum of the function 'g' applied to the function 'f' over a finite set 'S' is equal to the sum of the product of the size of the preimage of 'm' under 'f' intersected with 'S' and 'g' of 'm', for each 'm' from 0 to 'bd' - 1.
The set of terms in the state transition system after applying the Lazy Intruder preprocessing is equal to the set of terms in the original state transition system.
There exists a function 'g' such that: - 'g' is continuous on 'S'. - The image of 'S' under 'g' is a subset of 'U'. - For any 'x' in 'C', 'g' at 'x' equals 'f' at 'x'.
If "t1" and "t2" are binary search trees, then the intersection of "t1" and "t2" is also a binary search tree.
If "a times b" implies "c", then "a" implies that "b" implies "c".
The lemma "se_star_mono_for_states_eq" assumes that the states of c1 are equal to the states of c2. It also assumes that there is a symbolic execution from c1 to c1' over a list of labels ls, and a similar symbolic execution from c2 to c2' over the same list of labels. Under these assumptions, it shows that the states ...
If we assume that a set A is almost full on a property P, and for all natural numbers i, the function f(i) belongs to A, then there exists a function φ from natural numbers to natural numbers such that for all natural numbers i and j, if i is less than j, then φ(i) is less than φ(j) and the property P holds for f(φ(i))...
The closest bound of Qb and Q is equivalent to the existence of a function f such that the closest bound of Qb and Q is f.
The lemma named "syzygy_module_list_subset" assumes that the elements of the list "bs" are distinct. It shows that the syzygy module of the list "bs" is a subset of the polynomial module of the set that is initialized by the syzygy list of "bs".
If a snapshot is reachable in state 's' and 'roots'' is a subset of the roots of the mutator 'm' in state 's', then the snapshot remains reachable even if the roots of the mutator 'm' are replaced with 'roots''.
If the function F is injective on the set P, then the inverse of F applied to the image of P under F equals P. This means that if each element in P maps to a unique element in the function F, then applying the inverse function to the set of all images of P under F will give us back the original set P.
If relation R holds between a, vs, and ws, and if list L contains as and vs, then list L also contains a and as in ws.
If the inverse of 'aform' prime of 'p' and 'x' equals to the tuple ((a, bb), baa), then 'a' belongs to the set of floats.
If the first elements of both lists xs and ys are sorted, then the value of the key k in the map of the sorted union with function f of xs and ys is equal to the following: - If the key k is not found in the map of xs, then the value of the key k in the map of ys. - If the key k is found in the map of xs with value v,...
The lemma named "jordan_nf_block_size_order_bound" assumes that "A" is in Jordan Normal Form with "n_as". It also assumes that the pair "(n,a)" is an element of the set "n_as". Under these assumptions, it shows that "n" is less than or equal to the order of "a" in the characteristic polynomial of "A".
The length of the block matrix is equal to 2 raised to the power of the natural number form of the codomain block.
The "min" function is a relation in natural numbers that maps from one natural number to another.
The lemma shows that "L" is an equivalence functor with respect to the composition operation "S.comp" in the source and target categories.
: given a function mapping "f" to "f'" denoted as "theta", if for all "x" and "x'" such that "eta" maps "x" to "x'" implies "eta" maps "f" applied to "x" to "f'" applied to "x'", then the property "P" holds for "f" and "f'".
The lemma I_hd_coeff1_mult_a assumes that "m" is greater than 0. It shows that if the highest coefficient of "a" divides "m" or the highest coefficient of "a" equals 0, then the interpretation of the highest coefficient of "m" times "a" in the context of "m*x#xs" is equal to the interpretation of "a" in the context of ...
The lemma named "iso_charSbC" shows that "f" is an isomorphism if and only if "f" is an isomorphism in category "C", "f" is an arrow, and the inverse of "f" in category "C" is also an arrow.
If for every element 'a' in set 'A', the product of function 'f' at 'a' and function 'g' at 'a' equals function 'h' at 'a', and if functions 'f', 'g', and 'h' all map from set 'A' to the carrier 'G', then the finite product of function 'h' over set 'A' equals the product of the finite product of function 'f' over set '...
The set of terms in the dual of a strand S is equal to the set of terms in the strand S itself.
If the zeroth term of a formal power series 'f' is not zero in a field, then 'f' divides 'g'.
For any given word 'x' of any length 'a', a property 'P' holds for 'x' if and only if for any 'x', the same property 'P' holds for the word representation of an integer 'x'.
, given the assumptions that "A" is right total, and that "A" preserves the equality of "αe", "αn", "invar", and "inEdges'", it can be shown that "A" also preserves the equality of "predecessors" in the context of the base path of a graph with respect to "inEdges2".
If 'i' is less than the cardinality of set 'A', then the 'i'-th element of 'aA' belongs to set 'G'.
If set B is finite and set A is a subset of B, then the multiset of set A is a subset of the multiset of set B.
The lemma "succss_closed" assumes that the image of the successor function "succss" applied to a set X under the abstraction function "nodeAbs" is a subset of the image of X under "nodeAbs". It also assumes that X is a subset of the set of all elements x that are nodes. Under these assumptions, it shows that the image ...
If 'x' is strongly connected in one direction, then the relation of 'x' interacting with 1 and then 'x' again is equivalent to the transitive closure of 'x'.
This lemma, named proj_Cons, consists of six parts: 1. The projection of a list that starts with a pair consisting of the label of n and a, onto n, is a list that starts with the same pair followed by the projection of the rest of the list onto n. 2. The projection of a list that starts with a pair consisting of a st...
For any natural numbers 'm', 'n', and 'k', if 'm' is less than or equal to 'k' and 'n' is less than or equal to 'k', then 'k' minus 'm' being equal to 'k' minus 'n' is equivalent to 'm' being equal to 'n'.
The lemma "Resid_along_normal_preserves_Cong₀" assumes that "t" is approximately equal to "t'" (under some equivalence relation), "u" is in the set of normal elements, and the sources of "t" in relation "R" are the same as the sources of "u". Under these assumptions, it shows that the residuated product of "t" and "u" ...
The thread action bisimulation function, when applied to the flip of the bisimulation function for a given thread, is equal to the flip of the thread action bisimulation function applied to the bisimulation function.
Applying substitution σ1 to literal l and then applying substitution σ2 to the result is equivalent to applying the composition of substitutions σ1 and σ2 to literal l.
: Given the assumptions that 1. Functor KK maps category BB to category CC, 2. Functor TT maps category BB to category AA, 3. There is a morphism g from object c to object c' in category CC, 4. The limit object of c with respect to functor UArr is a limit of the composition of functor TT and object c under the infimu...
If the function rb_aux_inv is applied to the tuple (bs, ss, ps), then the list ps is sorted with respect to the order defined by pair_ord.
If 'w' is a chain and for all 'k', 'stake k v' is less than or equal to 'w' at 'l k' in the finite order, then 'v' is less than or equal to the limit of 'w' in the infinite order.
If a block 'b' is a subset of a set 'V', then the cardinality (or size) of the complement of 'b' is equal to the cardinality of 'V' minus the cardinality of 'b'.
The lemma "InvariantQCharacterizationAfterAssertLiteral" assumes the following conditions: 1. The state of the system is consistent after appending the pair (literal, decision) to the current state. 2. The watch list of the state only contains clauses from the formula of the state. 3. The watch list of the state is un...
The lemma "uniqueness_of_ctxt" is about a fixed environment "Gamma". It assumes that "Gamma" is okay, and that both "TVarB X T" and "TVarB X S" are in the set "Gamma". It then shows that "T" equals "S".
If a pair (x, y) is in the reflexive transitive closure of a relation r, then there exists a number n such that the pair (x, y) is in the nth power of the relation r.
The lemma named "poly_degE" assumes that the polynomial "p" is not equal to zero. It then obtains a term "t" where "t" is in the keys of "p" and the degree of the polynomial "p" is equal to the degree of the power monomial "t".
The lemma named "effect_tapI" assumes that "h'" is equal to "h" and "r" is equal to the function "f" applied to "h". Under these assumptions, it shows that the effect of tapping function "f" on "h" results in "h'", "r", and a time complexity of 1.