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If function 'f' is a bounded linear function, then 'f' is continuous at any point 'a'.
The i-th element of a vector created from a list is the same as the i-th element of the original list.
The lemma named "xml_of_term_code" states the following: 1. The XML representation of a term that is a constant (Code_Evaluation.Const) with a given name 'x' and type 'ty' is a single XML tag with the tag name '0', the attribute 'x', and the XML representation of the type 'ty'. 2. The XML representation of a term tha...
The lemma named "pow_res_p_pow_factor" assumes that "n" is greater than 0 and "a" is an element of the carrier Qp. It shows that the power residue of "n" and "a" is equal to the power residue of "n" and the product of "p" raised to the power of "l" times "n" and "a".
If "xs" is a walk, then taking the first "n" elements from "xs" will also result in a walk.
If the atom 'i' is not in the set 'is' and the set of atoms in 'is' is fresh for 't', then substituting 'i' with 't' in the term obtained by transforming 't'' with 'vquot_dbtm' and 'trans_tm' using the set '{i}' is equal to the term obtained by transforming the term resulting from substituting 'i' with 't' in 't'' with...
The size of a product in the "Tagged_Prod_Sum" is equal to the sum of the function 'f' applied to the first element of the product 'p', the function 'g' applied to the second element of the product 'p', and the successor of 0 (which is 1).
The valuation of the derivative of the function f at point a in the ring of p-adic integers is greater than or equal to zero.
The supremum (or join) of the complement of x and the complement of y is equal to the supremum of the complement of x and the infimum (or meet) of the double complement of x and the complement of y.
In the context of a value-based relation, if the value range of relation 'r' is disjoint from the value range of relation 's', then it can be shown that relation 'r' is disjoint from relation 's'.
If "a" is an element of the set "A s", then the value of the function "L_a" at "a", "v", and "s" is less than or equal to the value of the function "L_a" at the argument that maximizes the function "(λa. L_a a v s)" over the set "A s", "v", and "s".
The lemma named "funpow_cycle_offset" is defined with a fixed function 'f' that maps an element 'a to another element 'a. Given the assumptions that the function 'f' raised to the power of 'k' and applied to 'x' equals the function 'f' raised to the power of 'i' and applied to 'x', 'i' is less than or equal to 'k', and...
If the function "bind_fv2" with parameters "v", "lev", and "tm" does not equal "tm", then the function "occs" with the parameter being the product case of "Fv" and "v" applied to "tm" holds true.
In the context of ring functions, there is a lemma or proposition named "function_ring_not_car". It assumes that "a" is not an element of set "S" and "f" is an element of the carrier "F". It then shows or proves that the function "f" at "a" is undefined.
The lemma named "sylvester_carrier_mat" establishes that for any given polynomials 'p' and 'q', if 'd' is defined as the sum of the degrees of 'p' and 'q', then the Sylvester matrix of 'p' and 'q' is an element of the set of 'd x d' matrices.
The set of first elements from the list resulting from the function "map_raw" applied to "f" and "xs", is a subset of the set of first elements from "xs" after applying the function "f".
Adding an element "a" to a multiset "M" is equivalent to adding an element "b" to a multiset "N" if and only if multiset "M" is equal to multiset "N" and element "a" is equal to element "b", or multiset "M" is equal to the multiset formed by adding element "b" to the multiset "N" after removing element "a", and multise...
The lemma named "weakBisimCoinductAux" with case names "weakBisim" and case conclusion "step" consumes 1 assumption. It assumes that the pair (P, Q) is in set X. It also assumes that for any P and Q, if the pair (P, Q) is in set X, then P weakly bisimulates Q with respect to the union of set X and the set of weak bisim...
The cosine of (2 times pi minus x) is equal to the cosine of x.
Assuming that the power of 'y' to the 'n' equals 'x', it shows that the power of the numeral 'x' to the ratio of 'm' and the numeral 'n' is equivalent to the power of the numeral 'y' to 'm'. This is in the context of real numbers.
If a list created by mapping the function "Val" over the list "vs" and then appending the exception "throw ex" and list "es", is equal to another list created in the same way but with "vs'", "ex'", and "es'", then the exceptions "ex" and "ex'" must be equal.
If "ram1" and "ram2" are pre-existing points between a set of points "vs", then "ram1" and "ram2" will also be pre-existing points between the set of points obtained by rotating "vs" once.
The lemma named "affine_parallel_expl_aux" is defined for two sets S and T of real vectors. It assumes that for any vector x, x is in set S if and only if the vector obtained by adding a to x is in set T. Under this assumption, it shows that set T is equal to the image of set S under the function that adds a to each el...
A function that always returns a constant value 'c' is differentiable on any set 'S'.
If the following conditions are equal in importance: 1) the roots of a mutator, 2) the honorary roots of a mutator, 3) the fact that a reference is in the write list of any processor, 4) the system's free memory, 5) the union of the ranges of the object fields in the system heap for a given reference, 6) the mark...
The lemma named "e_take_init" assumes that a function "f" belongs to the set "R" and that "k" is less than the successor of "n". It shows that taking the successor of "k" elements from the sequence generated by applying "f" to "n" is equal to applying "f" to "k".
If a list 'xs' is sorted, and if 'i' is less than or equal to 'j', and 'j' is less than the length of the list 'xs', then the element at index 'i' in the list 'xs' is less than or equal to the element at index 'j' in the same list.
The lemma named "to_fun_Fun_mult" assumes that "x" and "y" are elements of the carrier "P". It shows that the function "to_fun" applied to the product of "x" and "y" under the operation in "P" is equal to the product of the function "to_fun" applied to "x" and the function "to_fun" applied to "y" under the operation in...
If 'n' is less than 'i', then the difference between 'i' and 'n' is equal to the successor of the difference between 'i' and the successor of 'n'.
The composition of the identity function on set A with itself is equal to the identity function on set A.
If the function sig_red_single is applied to the single regular signature sing_reg, the top tail top_tail, the polynomials p, q, and f, and the term t, then the term t plus the leading term of the representation list of f is included in the keys of the representation list of p.
The lemma AA23: assumes "If P and N are true, then either P will be true in the next state or Q will be true in the next state" and "If P, N and action A are true, then Q will be true in the next state" and "If P is true and there is no change in w, then P will be true in the next state" shows "If N is ...
If 'm' is less than infinity and 'n' is also less than infinity, then the product of 'l' and the difference of 'm' and 'n' is equal to the difference of the product of 'l' and 'm' and the product of 'l' and 'n'.
The star of the join of x and y, followed by z, is equal to the star of x followed by the star of the product of y and the star of x with 1, followed by z.
If the guess, the prize, and the clue in a game are all different from each other, and each of them is valid (as indicated by inv_prize, inv_guess, and inv_clue), then the set of these three elements {guess, prize, clue} is equal to the set {1, 2, 3}. This implies that the guess, the prize, and the clue must each be on...
If invoking the function "invoke" with arguments "xs", "ptr", and "args" in the heap "h" is successful, then the pointer "ptr" is in the set of object pointer kinds of the heap "h".
If negative q implies negative r given x, and negative r is less than or equal to negative s, then negative q implies negative s given x.
If the function "match_exception_table" with parameters G (a context), xcpt (an exception), pc (a program counter), and et (an exception table) returns some value pc' (another program counter), then there exists a class C such that C is in the set returned by the function "match_any" with parameters G, pc, and et, and ...
If a program G is well-formed according to the well-formedness condition "wf_mb", and for any class C, superclass S, field set fs, method set ms, and method m, if the tuple (C, S, fs, ms) is in the set G and m is in the set ms, then the well-formedness condition "wf_mb'" holds for G, C, and m, then the program G is wel...
If a tree 't' belongs to the set 'B h', then updating the tree 't' with 'x' and 'y' results in a tree that belongs to either the set 'B h' or the set 'B' of the successor of 'h'.
The lemma named "subgraph_strongly_connected_iso" assumes that "H" is a subgraph of "G". It shows that "H" is strongly connected under the application of an isomorphism "hom" if and only if "H" is strongly connected.
If a polynomial 'p' is in the carrier 'P', and given that for any two polynomials 'p' and 'q' in the carrier 'P', if 'Q' holds for 'p' and 'q', then 'Q' also holds for the sum of 'p' and 'q' in the carrier 'P'. Additionally, if 'a' is in the carrier 'R', then 'Q' holds for the monomial of 'P' with coefficient 'a' and d...
The initial state of l3 is a subset of l3_inv1.
If scene A and scene B are both idempotent scenes, then the union of scene A and scene B is also an idempotent scene.
The lemma named "max_projection_propI" makes the following assumptions: 1. For any 'x', 'x' is greater than or equal to 'f x'. 2. For any 'x', property 'P' holds for 'f x'. 3. For any 'x' and 'p', if property 'P' holds for 'p' and 'x' is greater than or equal to 'p', then 'f x' is greater than or equal to 'p'. If thes...
The lemma lm101 assumes that an element 'x' is not in the set 'xs', the set of 'injections_alg xs Y' is equal to the injections of the set 'xs' into 'Y', and 'Y' is a finite set. It shows that the set of 'injections_alg' of the list consisting of 'x' and 'xs' into 'Y' is equal to the injections of the union of the set ...
"D is okay with G" if and only if "G preserves the output of distribution" and "D is allowed in G".
The product of the star of x and the bottom element is less than or equal to the omega of x.
If it is admissible to assert that "not P(x)" and it is admissible to assert that "not Q(x)", then it is also admissible to assert that "not both P(x) and Q(x)".
The identity relation is a subset of the weakest precondition of the differential equation "x' = f" with guard "G" on the set "U S" at time "t0", under the condition "G".
The lemma named "decomp_size_mset_less" assumes that the lengths of two lists, ss and ts, are equal. It shows that the size of the multiset formed by adding the multiset M to the multiset of the zipped lists ss and ts is less than the size of the multiset formed by adding the multiset M to the multiset containing the p...
"eq_nextl" is an equivalence relation over the universal set.
The multiset resulting from the substitution operation on the list of atoms 'As' with the substitution 'σ' is equal to the multiset of 'As' after the multiset substitution operation with 'σ'.
The lemma is_iso_semifunctorE': If "\<FF> is an isomorphism from semicategory \<AA> to semicategory \<BB> with respect to \<alpha>", then we can obtain the following: - "\<FF> is a semicategory morphism from \<AA> to \<BB> with respect to \<alpha>" - The object map of \<FF> is a bijection - The arrow ma...
If a proposition "phi" is valid, then the proposition "top hchop phi" is also valid. Here, "hchop" is a binary operator that represents the horizontal chopping of two propositions, and "top" represents the truth value true.
The lemma named "mersenne_mod_less'" assumes that "k" is less than or equal to "5 times 2 to the power of n". It shows that "the Mersenne modulus of k and n" is less than "2 to the power of n plus 5".
The lemma is about the semantics of propositional linear temporal logic (PLTL). It states the following: 1. A word 'w' satisfies the negation of a proposition 'phi' if and only if 'w' does not satisfy 'phi'. 2. A word 'w' always satisfies the proposition 'true'. 3. A word 'w' satisfies the disjunction of propositions ...
The substitution of a variable 'j' with a term 'w' in a sequence of complex terms 'SeqCTermP' with parameters 'vf', 's', 'k', and 't' is equal to the sequence of complex terms 'SeqCTermP' with parameters 'vf', the substitution of 'j' with 'w' in 's', the substitution of 'j' with 'w' in 'k', and the substitution of 'j' ...
If the lengths of two lists, xs and ys, are equal, then mapping the first element function (fst) over the zipped version of xs and ys will result in the original list xs.
If a natural number 's' is less than 't', then the sum of the function 'sf' applied to 's' and 's' itself is less than the function 'sf' applied to 't'.
If an element 'x' is in the set 'xs', then the element at the index of 'x' in the list 'xs' is 'x'.
For any word "n", the bitwise AND operation between "n" and 1 is equal to 1 if the first bit of "n" is 1, otherwise it is 0.
The union of ((the negation of z or the negation of y) and (the negation of x or the double negation of y)) and the double negation of y is equal to the union of the negation of x and the double negation of y.
The lemma named "card_low_digits" assumes that "m" is greater than 0 and for any "d" in the set "ds", "m" is less than or equal to "d". It shows that the cardinality of the set of "i" such that "i" is less than the product of the list "ds" and for all "i0" in the set resulting from encoding the digits of "ds" in "i", "...
For any fixed element 'x' of type 'a' (which is a permutation type), the permutation of 'x' under 'p' after it has been restricted by a set 'as' is equal to the restriction of the permutation of 'x' under 'p' by the set obtained by permuting 'as' under 'p'.
The interpretation of a TESL specification where a master event is time-delayed by delta-tau on a measuring event implies a slave event, is equal to the intersection of all sets Y, where Y is the interpretation of the same TESL specification but with a restriction that the events must occur after a certain natural numb...
The maximum value between the distance from x to y and the absolute difference between the distance from x to z and the distance from z to y is equal to the distance from x to y.
If X is an element of the synthetic analysis of H, then for all G, if H is a subset of G, the presence of the key K in the analysis of the insertion of X into G is equivalent to the presence of the key K in the analysis of G.
The relation "between" is commutative. This means that if point A is between point B and some other point, then point B is also between point A and that other point.
If a matrix A belongs to the set of finite complex matrices, and A is Hermitian, and i is less than the size of the projective measurement of matrix A, then the measurement outcome of the projective measurement outcomes of matrix A at index i is a projector.
"s" is equivalent to itself excluding the User ID.
For any psi-calculus expression, the parallel composition operation is associative. Specifically, for any psi expression and any three psi processes P, Q, and R, the parallel composition of P and Q, followed by a parallel composition with R, is bisimilar to the parallel composition of P with the result of the parallel ...
If a formula "phi" is safe, then the satisfaction of the converted multiway formula is equivalent to the satisfaction of the original formula "phi". This is evaluated in the context of a given structure "sigma", variable assignment function "V", variable "v", and index "i".
The first-order extension of a decision procedure "dec_I1" with a parameter "FO" at position 0, applied to a structure "\<AA>" and a set "P", and then applied to a list consisting of "x" and "bs1", is equal to the decision procedure "dec_I1" applied to "\<AA>" and "bs1", where "x" is mapped to the minimum element of th...
The set resulting from applying function V to a trace "tr" is a subset of the set of all function "f" applied to a trace "trn" where "trn" is an element of "tr" and satisfies the property "φ".
For a function 'f' mapping natural numbers to a linearly ordered integral domain, the square of the sum of the absolute values of 'f' from 0 to 'n' is equal to the sum of the squares of 'f' from 0 to 'n' plus the sum of the products of the absolute values of 'f' for all distinct pairs of 'i' and 'j' from 0 to 'n'.
If "x" and "y" are vectors, then the infimum (greatest lower bound) of "x" and "y" is also a vector.
The lemma named "enat_unfold_unique" assumes a function "h" such that for any "x", "h" of "x" equals zero if "stop" of "x" is true, otherwise it equals the successor of "h" of "next" of "x". It then shows that "h" of "x" is equal to the unfolding of "enat" with "stop" and "next" at "x".
If the binary search trees "l" and "r" both satisfy the black depth invariant (bd_inv), and if the black depth of "l" is equal to the black depth of "r", then the node created with "l" and "r" as its left and right children, "c" as its color, and "k" and "v" as its key and value respectively, also satisfies the black d...
For any regular expression 'r', the character 'c' is not in the set of final characters of the modified regular expression 'empty_mrexp3 r'.
, given the assumption "l" is an element of the set resulting from the addition of "x" and "y", it must be the case that either "l" is an element of the set "x", or there exists a "z" such that "z" is an element of the set "y" and "l" is equal to the addition of "x" and "z".
The lemma named "wt_int" assumes "wtE" which is the weight of Epsilon and "wt" which is the weight of T. It shows that the integer type of the type of T and the integer of Epsilon and T are true.
For a given function 'f' from any type 'a' to any type 'b' that is an ordered additive abelian group, if 'A' is a finite set and for all elements 'x' in 'A', 'f(x)' is less than or equal to zero, then the sum of 'f' over all elements in 'A' is equal to zero if and only if for all elements 'x' in 'A', 'f(x)' is equal to...
The lemma named "wrapS_unwrapS_id'" assumes that the function "f", which maps a natural number to some type 'a', when applied to the bottom element, results in the bottom element. Under this assumption, it shows that the nth element of the sequence obtained by applying the function "unwrapS" to "f" is equal to the resu...
If a function 'f' has an argument maximum over a set 'X', then the argument that maximizes the function 'f' over the set 'X' is indeed the argument maximum.
If "x" is less than or equal to -1, then the Lambert W' function of the exponential of "x" times "x" equals "x".
The function 'al_splits' maintains the abstract state and invariance of the finite tree, as represented by 'ft_α' and 'ft_invar' respectively, when applied to 'ft_splits'.
The lemma named "of_nat_mono" assumes that "n" is greater than or equal to "m". It shows that the natural number representation of "n" in an ordered semiring is also greater than or equal to the natural number representation of "m".
The transition from model 2 to model 3 refines the relation R23, given that the intersection of R23 and the universal set times the invariant "m3_inv1_keys" is a subset of the transition from model 2 to model 3. This is under the assumption that the transition from model 2 to model 3 preserves the relation R23.
If a singular chain 'c' of dimension 'p' exists in a topological space 'X', and if there is a continuous map 'g' from 'X' to another topological space 'X', then there exists a singular chain of the same dimension 'p' in the topological space 'X' that is the result of applying the chain map 'g' to 'c'.
The lemma named "produce_weaken_type" assumes that the function "produce" when applied to a type "tn" and a constant "cons" results in a type "tm". It then shows that the same function "produce" when applied to a type that is a concatenation of "ts" and "tn" with the same constant "cons" will result in a type that is a...
The tuple partial pullback of the intersection of two sets S1 and S2 is equal to the intersection of the tuple partial pullbacks of the sets S1 and S2.
If two polynomials 'p' and 'q' are in the carrier of the polynomial ring 'R' and are associated in the polynomial ring 'R', then 'x' is a root of 'p' if and only if 'x' is a root of 'q'.
If "xd" is not in the domain of "P", then the set "P" minus the set "{y} cross (successor of y minus ita)" minus the pair "(y, xd)" minus the set "{xd} cross the universal set" is equal to the set "P" minus the insertion of the pair "(y, xd)" into the set "{y} cross (successor of y minus ita)".
"x is not a proper part of itself."
The substitution of a universal quantification over a set of predicates, where each predicate is a function of an index, is equal to the universal quantification over the set of substituted predicates.
If "q" is not an integer or "1" is not in set "A", then the periodic zeta function of "q" is holomorphic on set "A".
The norm of the butterfly function applied to psi and phi is equal to the product of the norms of psi and phi.
The lemma named "cupcake_sem_env_preserve" assumes that a single evaluation of the cupcake function, with the semantic environment "sem_env" and the constructed term "mk_con S t", results in the return value "ml_v". It also assumes that the term "t" is well-formed. If these conditions are met, it shows that "ml_v" is a...
If a proof exists for a proposition ψ, given a set of assumptions that includes propositions φ1, φ2, and φ3, then a proof also exists for the same proposition ψ, given the same set of assumptions but with the order of φ1, φ2, and φ3 rearranged.