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Statement: stringlengths 7 24.3k
If D is an order, E is an order, f is an order-preserving injection from D to E, and D1 is a subset of the carrier of D, then the order induced by D on D1 is equivalent to the order induced by E on the image of D1 under f.
The lemma named "ident_rel_refl" assumes that "G" is a graph and "idt" is an identity relation on "G". It shows that the reflexivity rule for "idt" is maintained in "G".
The lemma named "uminus_id_iff" shows that the negation of a homogeneous coordinate 'z' equals 'z' if and only if 'z' equals zero or 'z' equals infinity in the context of homogeneous coordinates.
If a set S is convex, bounded, and its affine dimension is not equal to 1, then the relative frontier of S is path-connected.
If a chain is adjacent (represented by "x#xs@[y]"), then the cardinality of x is equal to the cardinality of y.
Extracting a list based on a predicate P from another list xs results in None if and only if separating the predicate P with an accumulator acc from the list xs results in a tuple of the accumulator and the list xs.
The function that takes a pair (x, y) and returns the first element x is equal to the function 'fst' (which also returns the first element of a pair).
The lemma named "filterlim_at_bot_at_right" is defined with respect to a function 'f' that maps from a linearly ordered topological space to a linear order. The lemma makes the following assumptions: 1. The function 'f' is monotonic, i.e., for any two elements 'x' and 'y' that satisfy condition 'Q', if 'x' is less tha...
: The result of the function 'infer1' applied to 'pq' and 'X' is the set of all pairs '(fst pq, r)', where 'q' and 'r' are such that '(q, r)' is in the epsilon closure of 'A' and 'q' is equal to the second element of 'pq'. This set is unioned with the set of all pairs '(r, snd pq)', where 'r' and 'p'' are such that '(r...
If a vertex list 'vol' is invalid for a graph 'g' and a function 'f', then adding a vertex 'v' to the start of 'vol' will also result in an invalid vertex list for 'g' and 'f'.
The domain of the product of domain x and domain y is equal to the product of domain x and domain y.
For any state 's1', if there is a transition from state 's' to 's1' under the evaluation of expression 'e' with a boolean result 'b' and a transition from 's1' to 's2' under the evaluation of either expression 'e1' or 'e2' depending on the boolean 'b' with a result 'v', then there is a transition from state 's' to 's2'...
The lemma named "current_bipartiteI" makes the following assumptions: 1. For any 'x', the outgoing degree of 'f' at 'x' is less than or equal to the weight of 'x' in graph 'Γ'. 2. For any 'x', the incoming degree of 'f' at 'x' is less than or equal to the weight of 'x' in graph 'Γ'. 3. For any edge 'e', if 'e' is not i...
The lemma named "vpairs_eqD" assumes that the set of all ordered pairs in relation 'r' is equal to 'r'. It shows that for any element 'x' in 'r', there exist elements 'a' and 'b' such that 'x' is the ordered pair of 'a' and 'b'.
The proposition "P" is true for either "a" or "b" if and only if it is not the case that "a" is true and "P" is not true for "True", or "b" is true and "P" is not true for "True", or both "a" and "b" are not true and "P" is not true for "False".
If an element 'e' is in the directed arcs of a directed tree 'to_dtree', then the directed head of 'to_dtree' with a default value 'def' for 'e' is equal to the head of tree 'T' for 'e'.
If the negation of a proposition A is in a clause C, then the proposition A is in the atoms of the clause C.
If a relation 'A' is right total, then the relation 'rel_gpv' derived from 'A' is also right total. This is a transfer rule, which means it is used to transfer properties or theorems between different representations of data.
It is admissible, in the context of a complete partial order (ccpo) with least upper bound (lSup) and order relation (\<sqsubseteq>), to apply a function that checks if a list (xs) is not lexicographically sorted.
If A is pure, then the list set order assumption of A is also pure.
"if P implies P, then it is true".
If the set I is not empty, then the intersection over the set obtained by inserting element j into I, under the function f, is equal to the intersection of the function of j and the intersection over the set I under the function f.
The lemma tauLaw2SimRight: fixes a type 'b element Psi and a psi type element P assumes condition 1: For any Psi, P, and Q, if Psi bisimulates P and Q, then the tuple (Psi, P, Q) is in the relation Rel. shows that Psi simulates the tau action on P leading to P composed with the tau action on P under the relat...
The lemma "basis_reduction_step" assumes the following: 1. "LLL_impl_inv state i fs" which means that the LLL implementation invariant holds for the given state, index i, and the basis vectors fs. 2. "LLL_invariant upw i fs" which means that the LLL invariant holds for the given upper bound upw, index i, and the basi...
If the atom 'a' is fresh for 'A', then substituting 'a' in 'A' with 'x' will result in 'A' itself.
: given the assumption that "a" is equal to "b", it can be shown that "a" is an object in the double lift category of "a", "b", "g", and "f".
The lemma named "derivative_component_fun_component" is about a function 'f' in a Euclidean space. It assumes that 'f' is differentiable at a point 'x'. It then shows that the dot product of the vector derivative of 'f' at 'x' and 'i' is equal to the vector derivative at 'x' of the function that maps 'x' to the dot pro...
The lemma "infer_v_form_consp" assumes that under contexts Theta, B, and Gamma, the variant constructor expression "V_consp" with sort "s", data constructor "dc", base type "b", and value "v" infers the type "tau". It shows that the base type of "tau" is the application of sort "s" to base type "b".
For any permutation 'p' and any elements 'x' and 'y', applying the permutation 'p' to the function 'f' with arguments 'x' and 'y' is equivalent to applying the function 'f' to the results of applying the permutation 'p' to 'x' and 'y' respectively. The second and third conditions, named "fresh1" and "fresh2", state tha...
: given the assumptions that the lambda term "LAM [x].t" has type "T" under the context "Γ", and that "x" is fresh for "Γ", it can be shown that there exist types "T₁" and "T₂" such that "T" is a function type from "T₁" to "T₂", and under the context where "x" has type "T₁" and is added to "Γ", "t" has type "T₂".
The lemma named "neg_diag_empty" assumes that for all 'k' less than or equal to 'n', if 'k' is greater than 0, then there exists a 'c' such that 'v' of 'c' equals 'k'. It also assumes that 'i' is less than or equal to 'n' and 'M' of 'i' and 'i' is less than one. Under these assumptions, it shows that the matrix 'M' wit...
Delta hedging replicates a portfolio. This is under the assumptions that the derivative is Borel measurable with respect to the maturity, and that N is a Bernoulli stream with the probability of success being the ratio of (1 + interest rate - down factor) to (up factor - down factor). The lemma then shows that the port...
The lemma named "vector_space_row_space" assumes that a matrix "A" is in the carrier matrix of "nr" rows and "n" columns. It shows that the row space of "A" forms a vector space over the class ring.
The lemma named "env_add_dom" is about an environment 'e' of any type 'a' and a string 'x'. It assumes that the environment 'e' is okay and that 'x' is not in the domain of the environment 'e'. It shows that the domain of the environment when 'x' is added to 'e' is equal to the union of the domain of 'e' and the set co...
The lemma of logical pseudo reflexivity: Assuming that under the context Gamma', term t is equivalent to term s over Gamma, it can be shown that under the same context Gamma', term s is equivalent to itself over Gamma.
The lemma sim_move10_SyncLocks consists of three parts: 1. If the thread actions ta1 and ta are both locking a, then the simulation move 10 for program P and thread t from the synchronized expression with respect to V (sync\<^bsub>V\<^esub>(addr a) e0) to e1' is equivalent to the move from the synchronized expression ...
"1 is less than the floor of x if and only if 2 is less than or equal to x".
This is a lemma for step invariant induction in the context of a call arity. It assumes the following: 1. There is a transition from configuration 'c' to 'c''. 2. The configuration 'c'' is not a boring step. 3. The invariant 'I' holds for the transition relation. 4. The invariant 'I' holds for the configuration 'c'. 5...
Converting a formal Laurent series with a constant term "c" to a formal power series results in a formal power series with the same constant term "c".
If 'h' is an inverse of 'f' in category 'CC' and 'g' is also an inverse of 'f' in the same category 'CC', then 'h' must be equal to 'g'. This is referenced from Chapter I-5 in "Categories for the Working Mathematician" by Mac Lane, 2010.
There exist values 'u' and 'v' such that the function 'tm_to_nat_list' applied to a list starting with 'x' and followed by 'xs' is equal to a list starting with 'u', followed by 'v', and then followed by the result of 'tm_to_nat_list' applied to 'xs'.
Running a resource that is a constant resource 'y' is equivalent to a function that, for any input, returns a sub-probability monad of the pair consisting of 'y' and the constant resource 'y'.
If a set of lists M is finite, then there exists a number n such that for all lists s in M, the size of s is less than n.
Adding the zero game to any game G on the left side results in the game G itself.
If two linearly infinite sequences "u" and "v" are given, and "s" is a stutter selection of "u" and "t" is a stutter selection of "v", and if the linear selection of "s" from "u" is equal to the linear selection of "t" from "v", then the nth element of "u" is equivalent to the nth element of "v".
"there exists a value 'v' such that the equality of the value of 'I i' and 'L v' is in the set 'G' if and only if there exists a 'g' in the set 'G' that is a literal equality 'g' with 'i'."
The lemma "formulaEntailsClauseRemoveEntailedLiteralOpposites" assumes that a formula F entails a clause and a valuation. It shows that the formula F entails a clause that is the difference between the original clause and the opposite of the literal list of the valuation.
The image of the set of all integers less than 'n' under the function 'zfact_iso n' is equal to the carrier of the ring of integers modulo 'n'.
The lemma named "chamber_image_complement_subset" assumes that "D" is an element of the set difference between "C" and the image of "C" under "f". It defines "C" as the image of "D" under "f", and "closerToD" as the set of all elements "B" in "C" such that the chamber distance from "B" to "D" is less than the chamber d...
The product of "n + 2" and the Catalan number of "n + 1" is equal to twice the product of "2n + 1" and the Catalan number of "n".
Applying the function "fix_clock" to the result of "fun_evaluate" with parameters "s1", "env", and "e" is equivalent to just applying "fun_evaluate" with the same parameters "s1", "env", and "e". In other words, the "fix_clock" function does not change the result of "fun_evaluate".
The lemma named "out2_bet_out" assumes three conditions: 1. B is outside of line segment AC. 2. B is outside of line segment XP. 3. Point X is between points A and C. If these conditions are met, it shows two results: 1. B is outside of line segment AP. 2. B is outside of line segment CP.
"B2 of a and B1 of b and c is not equal to a" and "B2 of a and B1 of b and c is not equal to c".
The lemma named "abc_comp_first_steps_halt_eq" assumes that the function "abc_steps_l" with inputs (0, lm) and A, after n steps, results in (length A, lm'). It then shows that there exists a number n' such that the function "abc_steps_l" with inputs (0, lm) and the concatenation of A and B, after n' steps, also results...
The lemma named "perpendicular_to_x_axis_y_axis" assumes that a Poincare line through the origin and a complex number 'z' is perpendicular to the x-axis and that 'z' is not equal to zero. It shows that 'z' is an imaginary number.
The optimal policy "opt_policy_gs'" for a given decision problem "d" and value "v" belongs to the set of deterministic decisions "D\<^sub>D".
For any two extended natural numbers 'a' and 'b', if 'a' is not infinity, then the result of adding 'b' to 'a' and then subtracting 'a' will be 'b'.
The coefficient of the multivariate polynomial, obtained from a univariate polynomial 'p' at index 'i', at the single term mapping of index 'i' and degree 'n', is equal to the coefficient of the univariate polynomial 'p' at degree 'n'.
: "The image of the function 'nth l' over the set of elements 'n' from the list '0' to the size of 'l', where each element 'x' of 'l' is in 'X', is a subset of the intersection of 'X' and the set of elements in 'l'."
The lemma named "unit_register_domains_isomorphic" is given two functions, F and G. F is a function that maps from a finite update of type 'a to a finite update of type 'c. Similarly, G is a function that maps from a finite update of type 'b to a finite update of type 'd. The lemma assumes that both F and G are unit re...
The set of authenticated keys in a sequence of events, where the sequence starts with the event of Kas (the authentication server) sending a message to agent A. This message is encrypted with A's shared key and contains a new key K, the agent Peer, a number Ta, and another encrypted message. This inner message is encry...
The lemma named "integer_word8_bounded_eq" assumes that an integer 'i' is greater than or equal to 0 and less than or equal to 2 raised to the power of the length of 8 minus 1. Under these assumptions, it shows that the integer of a word8 (8-bit word) derived from the integer 'i' is equal to 'i' itself.
If a heap 'h' is well-formed and the disconnected nodes of a document pointer 'document_ptr' in heap 'h' are 'disc_nodes', then any node in the set of 'disc_nodes' is also in the set of node pointers in heap 'h'.
The spectrum is Zariski open.
For any two formulas 'F' and 'F'', any literal trail 'M', and any clause 'C', if the 'InvariantReasonClauses' holds for 'F' and 'M', and if 'F' entails the clause 'C', and if 'F'' is the result of appending 'C' to 'F', then the 'InvariantReasonClauses' also holds for 'F'' and 'M'.
The lemma named "prefs_from_table_agent_permutation" assumes two conditions. The first condition is that the preference tables for agents and alternatives, represented by "xs" and "ys", are well-formed. The second condition is that the multiset of the second elements of "xs" is equal to the multiset of the second eleme...
If there is a generalized internal arc path (gen_iapath) in a simplified graph (slim) from vertex u to vertex v via path p, then there is also a generalized internal arc path from vertex u to vertex v via path p in the original graph G.
The ith bit of a mask of length n (for any word length 'a) is true if and only if i is less than n and i is less than the size of the mask of length n (for any word length 'a).
If matrix A is in the carrier matrix of dimensions nr by nc, and matrix B is in the carrier matrix of dimensions nr by nc2, then the Gauss-Jordan elimination of the homogenized versions of matrices A and B is equal to the homogenized version of the Gauss-Jordan elimination of matrices A and B.
For any set 'A' of elements from a commutative monoid with addition, the sum of the set containing only the additive identity (0) and 'A' is equal to 'A' itself.
"q" is not equal to the logical conjunction of "p" and "q".
The lemma named "Iff_reduce'" assumes that 'p' is static. It shows that 'p' is equivalent in the context of 'A' and 'B' under the restriction 'r', to the reduction of 'r' and 'p'.
The lemma named "acyclic_plus_loop" assumes that the set "x" minus 1 is acyclic. It shows that the transitive closure of "x" intersected with 1 is equal to "x" intersected with 1.
In the SE_LR_RBAC_Policy, appending an entry with Alice as the user, Clerical as the role, patient1 as the patient, ei as the event identifier, and d as the data, to the initial state sigma0, results in a denial.
The degree of the Newton polynomial is less than or equal to n.
The representation of the quasi-borel function "monadP_qbs" applied to X is equal to the pair of functions "monadP_qbs_Px" applied to X and "monadP_qbs_MPx" applied to X.
The lemma named "empty_eff_exec_eq" is defined with fixed variables s and a. It assumes that the domain of the second element of a is empty. Under this assumption, it shows that the state successor of s and a is equal to s.
For any complex vector 'x', if 'm' is not equal to zero, then 'y' equals 'm' times 'x' plus 'c' if and only if 'x' equals the inverse of 'm' times 'y' minus the product of the inverse of 'm' and 'c'.
The lemma named "bisimWeakCoinductAux" consumes 1 assumption and fixes a variable 'Psi' of type 'b', and two variables 'P' and 'Q' of type psi, and a set 'X' of tuples containing 'b' and two psi types. It assumes that the tuple of 'Psi', 'P', and 'Q' is in set 'X'. And for any 'Psi', 'P', and 'Q' such that the tuple o...
The lemma "join_of_symmetric" assumes that "t" joins with "u" to form "v". It then shows that "u" joins with "t" to also form "v". This demonstrates the symmetry of the join operation.
If the strict reference equality of an object 'x' and an object 'y' (of type 'a', which can be null or an object) holds in a state 'tau', then 'x' and 'y' are both valid in the state 'tau'.
The function "force_enabled" with input "x" is measurable with respect to the sigma-algebra "S" and the range of the function "rT" with input "x".
The set of free variable indices in a formula (phi) is finite, given a bound (b).
The lemma named "uniform_limit_add" is about functions f and g that map from two arbitrary types 'a and 'b to a real normed vector 'c. It assumes that both f and g have uniform limits l and m respectively over a set X under a filter F. It shows that the uniform limit of the sum of f and g over the set X is the sum of t...
If the "inv_after_clear" function holds true for the given parameters (as, lm) and (x, l, Bk # r) with the result ires, then the "inv_on_right_moving" function will also hold true for the parameters (as, lm) and (y, Bk # l, r) with the same result ires.
The lemma named "HMA_spectral_radius" is a transfer rule that states: if a matrix HMA_M is transformed according to the function Spectral_Radius.spectral_radius, the result is equivalent to applying the function spectral_radius.
There is a path from node 'u' to node 'v' through the concatenation of list 'ls1' and 'ls2' if and only if there exists a node 'w' such that there is a path from 'u' to 'w' through 'ls1' and a path from 'w' to 'v' through 'ls2'.
If the product of a rational number 'c' and another number 'a' is less than 'b', and 'c' is greater than 0, then 'a' is less than the product of the reciprocal of 'c' and 'b'.
If Br_1 is a subset of B, then for all A, A is a border of B.
If R is not an empty set, S is not an empty set, and for all elements 'a', the pair (a, empty set) is not in the union of R and S, then the parallel product of R and S is a subset of the sequential product of R and S.
The initial state of s0 is a subset of s0_secrecy.
The function "iso_tuple_update_accessor_cong_assist" holds true when the identity function is used as both of its arguments.
The supremum operation in the context of JVM SemiType for given parameters S, m, and n is equivalent to lifting a binary operation. This binary operation is the supremum of an optional type and a product type. The product type is the supremum of a list type and a function that maps two inputs to the "OK" state of a lis...
If x is a symmetric lens, then x is also a pseudo-symmetric lens.
Zero and one are algebraic numbers in any field with characteristic zero.
For any element 'x' in the carrier 'L', the image of the carrier 'L' under the xor operation with 'x' is equal to the carrier 'L' itself. This assumes that 'x' is an element of the carrier 'L'.
If the function "wf_matchF_aux" is well-formed for a given sequence "sigma", a set "V", a natural number "n", a restriction "R", an interval "I", a regular expression "r", and a list "aux" with elements consisting of a time "t", a set of relations "rels", and a relation "rel", and a natural number "ne" and "i", then th...
The empty set is an element of the set given by v.
The filtered set of elements 'xs' based on the property 'P' is empty if and only if all elements 'x' in 'xs' do not satisfy the property 'P'.
If "s" is a subset of "u" and "t" is a subset of "u", then "s" and "t" are overlapping.

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