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If the sum of 'i' and 'j' is less than or equal to the length of 'w', and 'k' is greater than 0 but less than 'j', then the subword of 'w' from 'i' to 'j' is equal to the concatenation of the subword of 'w' from 'i' to 'k' and the subword of 'w' from 'i + k' to 'j - k'.
If a Dirichlet series 'f' converges at a complex number 's', then evaluating the Dirichlet series 'f' that has been transformed into real numbers at the real part of 's' is equivalent to transforming the evaluation of the Dirichlet series 'f' at 's' into real numbers. This is true in the context of a field that is both...
The tangent of a value x is equal to the sine of twice that value divided by the sum of the cosine of twice that value and one. This is valid for x which is of a type that is a real normed field, a Banach space, and a field.
The observable reach of "a0i" is a subset of "a0n_agreement".
If the state handler 'sh' for class 'C' is in an error state, and program 'P' can transition from the initial state of throwing a 'NoClassDefFoundError' for class 'C' to some other state, then program 'P' can also transition from the initial state of initializing a new class 'C'' to the same other state.
The quality of a term X remains unchanged after a quasi-swap operation between x and y within the set zs.
The lemma named "inv_hcomp\<^sub>R\<^sub>W\<^sub>C" assumes that "\<mu>" and "\<nu>" are isomorphisms and the intersection of the sources of "\<nu>" and the targets of "\<mu>" is not empty. It shows that the inverse of the horizontal composition of "\<nu>" and "\<mu>" is equal to the horizontal composition of the inver...
The lemma named "proots_ball_plane_eq" is defined with "q1" as the complex number "i - 1" and "q2" as the complex number "i + 1". It assumes that "p" is not equal to zero. It shows that the count of the roots of polynomial "p" within the unit ball centered at the origin is equal to the count of the roots of the polynom...
If the following conditions are met: 1. "Measure x1 x2a x3a" is a well-formed command, 2. "x4" is an element of the carrier matrix of dimension "d" by "d", 3. "x4" is a partial density operator, 4. For any "x3aa" and "ρ" such that "x3aa" is in the set "x3a", "x3aa" is a well-formed command, "ρ" is an element of the ca...
If we assume 'inv_imp', which asserts that for all states 's', if 's' is in the invariant of the deterministic while algorithm 'WA' and 's' is not in the condition of 'WA', then 'Q' holds for 's'. Under this assumption, the lemma shows that 'Q' holds for the loop.
The lemma "valid_refs_invD" consists of several statements: 1. If 'x' is in the mutable roots of 'm' in state 's', 'x' reaches 'y' in state 's', and the state 's' has valid references, then 'y' is a valid reference in state 's'. 2. Under the same conditions as above, there exists an object such that 'y' is associated ...
The set S is connected if and only if for every pair of elements x and y in S, the connected component of x in S is equal to the connected component of y in S.
The conversion of an empty bit-vector to a natural number results in zero.
: If the axiom EQU001_0 is equal and the axiom BOO002_0 is equal to the inverse of the multiplicative identity, additive identity, multiplication, product, addition, and sum, and if the axiom BOO002_0 is equal to the inverse of multiplication, addition, product, sum, and equality, then the negation of the sum of any el...
In the context of a fully faithful semicategorical functor, given the assumptions that the semicategory 'A' is the opposite of 'A' and the semicategory 'B' is the opposite of 'B', it can be shown that the opposite of the semicategorical functor 'F' maps 'A' to 'B' in a fully faithful manner.
For any integer 'k', taking the 'n' bits of the negation of 'k' after taking 'n' bits of 'k' is equal to taking 'n' bits of the negation of 'k'.
If a key K is not used in a set of events 'evs', and K is in the set of symmetric keys, and 'evs' is in the set of purchase events, then it implies that if key K is not used in 'evs' and K is in the set of symmetric keys, then K is not in the set of keys for the union of the parts of key K and what the Spy knows in 'ev...
The lemma "emeasure_exponential_Ioi_cutoff" assumes that "l" is greater than 0. It shows that the expected measure of an exponential distribution with rate "l" for all values less than a certain value "x" is equal to the exponential of the negative product of "l" and the maximum of 0 and "x".
Concatenating the flattened 64-bit values at the indices 0 through 7 of a list "vs" is equal to the original list "vs".
The relation "rel_U" with all its arguments being the equality relation is equivalent to the equality relation itself.
If a reference "r" is present in the heap "h" and another reference "r'" is not present in the same heap "h", then "r" is not equal to "r'".
For all elements 'x' in the set of 'f k v' where 'k' maps to 'v' in 'm', property 'P' holds for 'x' if and only if for all 'k' and 'v', if 'm' at 'k' equals 'Some v', then property 'P' holds for 'f k v'.
If the class of the program "tprg" is equal to some value "z", then the class "C" must be either "Object", "Base", "Ext", "Xcpt NP", "Xcpt ClassCast", or "Xcpt OutOfMemory".
The lemma named "properties_for_sort" assumes that the multiset of "ys" is equal to the multiset of "xs" and that "ys" is sorted. It then shows that the sorted version of "xs" is equal to "ys".
The set of undirected edges in the graph G, excluding the vertex u, is a subset of the set of undirected edges in the graph G.
For a semicategory mapping from B to C denoted as G, and another semicategory mapping from A to B denoted as F, and an arrow f in the semicategory A, the arrow mapping of the composition of G and F at f is equal to the arrow mapping of G at the arrow mapping of F at f.
The lemma named "finite_ipoly_roots" assumes that "p" is not equal to zero. It shows that the set of real numbers "x" for which the integer polynomial "p" evaluated at "x" equals zero is finite.
If tree "t" is in the set "bq", then the priority of the minimum tree obtained from "bq" is less than or equal to the priority of tree "t".
The stabilizer of B is a subset of the stabilizer of the sumset of A and B.
A source term S can transition to a process P' in the source-target calculus (STCal) if and only if there exists a source term S' such that S' is in the set of processes derived from P' and S can transition to S' in the source calculus. The second condition states that a target term T can transition to a process P' in...
The difference between the bounded linear function of matrix x and the bounded linear function of matrix y is equal to the bounded linear function of the difference between matrix x and matrix y.
If x is less than y, then the division of x by y equals zero, in the context of ordinal numbers.
The first element of the binary list representation of a word 'w' is equal to whether the signed integer representation of 'w' has a binary sign of -1.
The Beukers integral 3 is equal to the real number conversion of Beukers non-negative integral 3.
Given three assumptions: 1) a function 'g' maps from set 'I' to set 'J', 2) a polynomial 'P' is in the carrier of the polynomial ring over 'R' with indeterminates 'I', and 3) a scalar 'a' is in the carrier of the ring 'R', it can be shown that the relabeling of variables in the scalar multiplication of 'a' and 'P' ...
If "h" is a bijective function, then the image of the set "Always A" under the function "rename h" is equal to the set "Always (h ` A)".
If we have two morphisms of presheaves of rings, the first one from a presheaf of rings denoted by \<FF> to another presheaf of rings denoted by \<FF>', and the second one from \<FF>' to a third presheaf of rings denoted by \<FF>'', then the composition of these two morphisms is also a morphism of presheaves of rings f...
If "ks" is a subsequence of "ls", then the result of the function "from_cnf" applied to "ks" is less than or equal to the result of the function "from_cnf" applied to "ls".
If U is a unit register, then U is a register.
If 'a' and 'b' are both singular chains of the same dimension 'p' in a topological space 'X', then the difference between 'a' and 'b' is also a singular chain of dimension 'p' in the same topological space 'X'.
For any two integers 'x' and 'y', the comparison of 'x' and 'y' results in an undefined value if and only if either 'x' or 'y' is undefined.
If two things are equal, then the relationship between their states is the same as the relationship between the option of their equality. This is in the context of looking up the first element in memory.
The lemma named "ntsmcf_comp_interchange_law" refers to a law of composition interchange in the context of semicategories. This is referenced in Chapter II-5 of "Categories for the Working Mathematician" by Mac Lane (2010). The assumptions are as follows: - There is a semicategory morphism from \<GG> to \<HH> denoted ...
The lemma named "convex_fx_plus_fy_ineq" is about a function 'f' that maps real numbers to real numbers. It assumes that the function 'f' is convex on the set of all non-negative real numbers, that 'x' and 'y' are non-negative real numbers, and that the function 'f' at 0 equals 0. It shows that the sum of the function ...
The lemma named "sym_preproc_aux_term_wf" assumes that "d" is a Dickson grading. It shows that the function "sym_preproc_aux_term" with "d" as an argument is well-founded.
The lemma named "euler_maclaurin_strong_int_aux" assumes that "a" is less than or equal to "b". It defines "S" as the sum of the function "f" applied to each integer from "a" to "b". It shows that the difference between "S", "F" applied to "b", and "T" applied to "b" equals the Euler-Maclaurin remainder of "N", the fun...
: Assuming that for any two processes P and Q, if P is weakly bisimilar to Q, then P satisfies a property x if and only if Q also satisfies x. Also, given that P is weakly bisimilar to Q, and P satisfies the conjunction of the predicate φ and the property x under a silent transition, then Q also satisfies the conjunct...
If the assumption "invar" holds for a tuple consisting of "l", a pair of "a" and "lv", and "r", then "invar" also holds for "l".
The lemma shows that the permutation of a function, which takes an input 'x' of type 'a' (where 'a' is a permutable type) and returns a logical implication between 'A' and the equality of function 'B' applied to 'x' and 'C', is equal to 'foo'.
If "a" is an element of "s" and "n" is less than or equal to "j", then the "j"-th element of "a" is equal to "p".
The lemma named "mcp_ex" assumes that an empty list is an element of W and that W is finite. It then obtains a value p where the function "mcp" with parameters z, W, and p holds true.
If 'a' is not equal to 'b', and the set {a, b} is equal to the set {x, y}, and 'z' is an element of the set {x, y}, then the function TS_inv with parameters 'c', 'z', and the list [a, b] is equal to the function TS_inv with parameters 'c', 'z', and the list [x, y].
The lemma named "eq_mat_on_vecI" is about two matrices M and N, which are elements of a field. The lemma assumes that for any vector v in the carrier vector of nA, the product of M and v is equal to the product of N and v. It also assumes that M and N are in the carrier matrix of nB and nA. The lemma then shows that M ...
"if it is possible that there exists some property X, then it is possible that there exists some property X that is necessarily true."
If we have a Program Dependence Graph (PDG) with certain properties (defined by the variables sourcenode, targetnode, kind, valid_edge, Entry, Def, Use, state_val, Exit, and the weak control dependence from the StrongPostdomination theory), and if we have Strong Postdomination with certain properties (defined by the va...
The lemma named "cod_simp" assumes that "f" is an arrow. It shows that the codomain of "f" is equal to the pair consisting of the codomain of the first element of "f" in category C1 and the codomain of the second element of "f" in category C2.
Lemma: winding_number_nearby_loops_eq Assumptions: 1. g is a path. 2. h is a path. 3. The path finish of g is equal to the path start of g. 4. The path finish of h is equal to the path start of h. 5. For all t in the range [0, 1], the norm of (h(t) - g(t)) is less than the norm of (g(t) - z). Conclusion: Under these a...
If G is a group, r and s are positive integers, and f and g are composition series of G, then the operation of the function f at the index "Suc ((r * s - Suc 0) div s)" with the intersection of the function f at the index "(r * s - Suc 0) div s" and the function g at the index "(r * s - Suc 0) mod s" is equal to the in...
If the range of a function 'r' is an empty set, then the function 'r' itself is an empty set.
Running the "put" state with parameters "s" and "m" and then running the state "s'" is equivalent to running the state "m" with the parameter "s".
The fixed point of the dual closure of a proposition "phi" applied to a set "X" is equivalent to the fixed point of "phi" applied to the complement of "X" being approximately equal to the bottom element.
The lemma named "all_imp_alw" assumes that for all streams "xs", the property "phi" holds. It shows that "phi" is always true for the stream "xs".
The function relation "f_rel" is well-founded.
If the sort "s1" is less than or equal to the sort "s2" under a certain class structure "cs", then there exists a sort "s1'" such that for every class "c1" in "s1'", there exists a class "c2" in "s2" where "c1" is less than or equal to "c2" under the class structure "cs", and "s1'" is less than or equal to "s2" under t...
The negation of a hypercomplex number, consisting of x and y, is equal to the hypercomplex number of the negations of x and y.
If the following conditions are met: 1. A satisfies the global property P with respect to the set S and the function U, where S and U are defined for all processes except process i, and the messages received by the processes are determined by the function R. 2. The function U is an idempotent function, meaning that a...
Given two elements "g" and "f" that belong to the set of arrows of the direct product of digraphs indexed by "I", if for every index "i" in "I", the "i"-th component of "g" is equal to the "i"-th component of "f", then "g" is equal to "f".
If 'p' is a prime number, then the normalization of 'p' is equal to 'p' itself.
The lemma "iso_cancel_right" assumes that "g" is an isomorphism, the composition of "f" and "g" is equal to the composition of "f'" and "g", there is a sequence from "f" to "g", and "g" is an isomorphism. It shows that "f" is equal to "f'".
For a fixed 'a' in a Euclidean space, the set of all 'x' such that the dot product of 'a' and 'x' is less than or equal to 'b' forms a polyhedron.
The lemma named "homeomorphic_simple_path_images" is defined with two fixed variables, "gamma1" and "gamma2", which are functions from real numbers to two different types of topological spaces 'a and 'b respectively. The lemma assumes that both "gamma1" and "gamma2" are simple paths, and that the end of each path is th...
The reflexive transitive closure of the product of a variable and its reflexive transitive closure is equal to the reflexive transitive closure of the variable itself.
The carrier of the nth power of the p-adic numbers is in the set of semi-algebraic sets of dimension n.
The lemma named "evaldjf_qf" assumes that for all elements 'x' in the set 'xs', the function 'f' applied to 'x' is quantifier-free. It shows that the function 'evaldjf' applied to 'f' and 'xs' is also quantifier-free.
If the first property of a frame holds and the second property of a frame also holds, then the causal disjointness property is true.
The integer is a subtype of real. Assuming that X is defined for a state tau, it shows that if X is first cast to a real type and then cast back to an integer type, it is equivalent to the original X.
The visited edges are only between touched nodes.
For any transition system T, the function "mkM" produces a Kripke structure.
The theorem named "getEnv_psubstEnv_Some" assumes that the function "rho" applied to "xs" and "x" equals "Some X". It shows that if you apply the function "rho" to "xs" and "x" after it has been combined with "rho'", it will equal "Some (X #[rho'])".
The lemma named "pvar_trans_eval" assumes that "a" is an element of the carrier set "R", "b" is an element of the nth power of the carrier set "R", and "i" is less than "n". It shows that the evaluation at point "R" of "b" for the polynomial variable transformation of "n", "i", and "a" is equal to the ith element of "b...
The theorem named "correct" assumes that the first elements of the tuples in the list "ts" are distinct. It also assumes that for every tuple in the set "ts", the first element of the tuple is less than "n". Given these assumptions, it shows that the result of the function "alg" applied to "ts" and "x" is equal to the ...
The lemma named "auto_weaken_pre_uncurry_start" assumes that "P" is equivalent to "P'". It also assumes that if "P'" is true, then "Q" is true. Under these assumptions, it shows that if "P" is true, then "Q" is also true.
The lemma named "coplanar_perm_16" assumes that points A, B, C, and D are coplanar. It shows that if this is the case, then points C, D, A, and B are also coplanar.
The lemma named "replay_code" assumes that the theory 'thy' is well-formed according to 'exe_wf_theory'. It shows that the result of replaying 'P' in the translated theory 'thy' is equal to the result of executing replay on 'P' in the original theory 'thy'.
The lemma "LEX_chain_on_eq_upto_imp_ith_chain_on" assumes that there is a chain (a sequence of elements where each element is related to the next one) on the lexicographic order of P, restricted to the sequence A and equal up to the function f at index i. If this is the case, it shows that there is also a chain on P at...
The lemma "Getfield_correct" is about a specific state of the Java Virtual Machine (JVM), denoted as σ'. It assumes the following: 1. The program P is well-formed according to the typing rules wt. 2. The program P can see a method M with a certain signature (from types Ts to type T) and body (maximum stack size mxs, m...
If 'sd' is less than or equal to 'n', then the matrix ordered semiring, denoted as 'mat_ordered_semiring n sd (\<succ>) b', forms an ordered semiring scheme. This is represented as ('a mat,'b) ordered_semiring_scheme in the Isabelle language. The lemma is also stating that this ordered semiring is denoted as ?R.
If the well-sortedness conditions 'wls s X' and 'wlsEnv rho' are met, and if 'rho ys y' equals 'None' and 'fresh zs z (X #[rho])' is true, then 'fresh ys y X' is true or 'ys' is not equal to 'zs' or 'y' is not equal to 'z'.
The lemma named "translation_inverse_eq" assumes that "a" is an element of group "G". It shows that the inverse of "a" under the transformation subgroup is equal to the inverse of "a" in the group "G".
The lemma named "pt_list_inst" assumes that there is a permutation type between type 'a and type 'x. It then shows that there is also a permutation type between the list of type 'a and type 'x.
Focusing on any variable X with a power of 1 equals 1.
If a certain relation holds between the parameters of two randomized function adversaries, then the same relation holds between the outcomes of these adversaries. This is expressed in terms of a relation between the rest of the parameters (S, E, IA, IU, OA, OU, M), the initial state (S), and the initial adversary (IA)....
The lemma named "wf_hull_rel_relcomp" assumes that the relation "R" is well-founded and equivariant. It shows that the composition of the hull relation and "R" is also well-founded.
The lemma "strategy_attracts_via_no_deadends" assumes that a vertex 'v' belongs to a set 'V' and 'v' is in the difference of sets 'A' and 'W'. It also assumes that a strategy 'σ' attracts 'v' from 'A' to 'W' through a path 'p'. Under these assumptions, it shows that 'v' is not a deadend.
If the conjunction of A and B holds in the separation algebra, and the conjunction of A and Y holds in the state, then the conjunction of A and X also holds in the state.
The real number representation of the integer floor of a floating point number 'x' is less than or equal to the real number representation of 'x' itself.
The language of the transitive closure of a ground tree tuple G is equal to the transitive closure of the language of G.
The function 'gba_to_idx_ext' is correct. Assuming that for any set 'A' in 'F', 'A' is finite, it shows that 'gba_to_idx_ext' applied to 'ecnv' and 'G' is less than or equal to a specification of 'G''. This specification states that the acceptance condition of 'G'' is equal to 'accept', the set of vertices of 'G'' is e...
In the context of a ring, if M and N are modules over R, and P is a module homomorphism from M to N, then the set defined by the transformation rule TR3 from M to N under P is a subset of the carrier of P.
The minimum initial state of the expression φex is a record with the following fields: - The initial index is 0. - The monitor state is a conjunction of a predicate "A" with a single variable 0 and False, and an until operator that is always true, has a relation with no elements, an interval from 1 to 2, and an exist...