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Statement: stringlengths 7 24.3k
The outputs generated by the 'generat' function are countable.
The data access at level 1 for system A32 is equal to system A41.
If for all elements 'x' in the set 'xs', a property 'P' holds for 'x' and 'x', then the property 'P' successively holding for the list 'xs' with adjacent duplicates removed is equivalent to the property 'P' successively holding for the original list 'xs'.
The lemma named "cindependent_butterfly" is defined with respect to two sets, basisA and basisB, which are of type 'a and 'b Hilbert spaces respectively. It assumes that both basisA and basisB are orthogonal sets. It also assumes that the norm of any element 'a' in basisA and any element 'b' in basisB is 1. The lemma t...
The lemma named "submachine_reachable_subset" assumes that "A" is a submachine of "B". It shows that the set of states reachable from "A" is a subset of the set of states reachable from "B".
The theorem named "wls_vsubstAbs_eq_swapAbs" assumes that "A" is a well-sorted abstraction with sort "s" and variables "us", and that "y1" equals "y2" or "y1" is a fresh variable for the abstraction "A" with respect to the variables "ys". It shows that the variable substitution of "y2" for "y1" in "A" with respect to "...
The theorem named "euler_maclaurin_strong_int" 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 "S" is equal to the function "F" applied to "b", plus "C", plus the function "T" applied to "b", minus the Euler-Maclaurin re...
The lifting of the identity relation to the zeroth variable context is equivalent to the unprimed and primed zeroth variable being equal in the unprimed state.
The identity function, when applied to the "l1_learn" function with parameter "m", refines or improves the relation "R01iar".
If there is a transition from state 1 to state 0 in the "drinks2" machine that is not labeled with "t", then the second element of the configuration "c" is the number 0.
The lemma named "strategy_attracts_invalid_path" assumes the following conditions: 1. P is a sequence that starts with v, followed by w, and then the rest of the sequence P'. 2. v is an element of set A but not in set W. 3. w is not an element of either set A or set W. Under these assumptions, the lemma shows that th...
Lemma: borel_measurable_piecewise_mono Let f be a function from real numbers to real numbers, and let C be a set of real number sets. If the following conditions hold: 1. C is countable. 2. For every set c in C, c is a Borel set. 3. For every set c in C, the function f is monotonic on c. 4. The union of all sets in C...
If the following conditions are met: 1. The list obtained by mapping the first element of each pair in list l0 is distinct, 2. The map m is the map of list l0, 3. The list l0 is sorted with respect to relation R, 4. The iterator iti is the result of folding list l0, then the iterator iti is a map iterator with respec...
In the context of a finite cyclic group, the group G is isomorphic to its character group.
The lemma "arg1_vsubset_VPow_Vfrom" assumes that "A" is a transitive set. It shows that "A" is a subset of the power set of "Vfrom A i".
The formal Laurent series base factor converted to a formal power series with a base of 1 is equal to 1.
The lemma named "timpl_closure_set_mono_timpl_closure_set" assumes that set N is a subset of the term implication closure set of M with respect to TI. It shows that the term implication closure set of N with respect to TI is also a subset of the term implication closure set of M with respect to TI.
For any instance 'i' of the protocol AODV, the following conditions hold: 1. If the label 'l' is in the set {PRreq-:16..PRreq-:18}, then the destination IP address 'dip' is in the set of known destinations 'kD' of the routing table 'rt'. 2. If the label 'l' is PRreq-:17, then the origin IP address 'oip' is in the set ...
The lemma "betaupright_obtain_support" assumes that "beta_upright" holds for variables "va" and "vb", and that the "zcount" of "va" at time "t" is greater than 0. Under these assumptions, it is possible to obtain a time "s" which is less than "t", and where the "zcount" of "va" at time "s" is less than 0 or the "zcount...
The function 'f' is continuous at the point 'x' within the set 'T' if 'x' is an element of the source carrier.
"x" is not in the set that results from removing all instances of "x" from the list "xs".
The coproduct of quasi-Borel spaces indexed by I and X is closed under the operation defined by qbs_closed3.
For a fixed function φ of type 'a to 'b, where 'a is a preorder, the evolution of φ under a condition G and a set U, followed by a relation R restricted to a condition Q, is less than or equal to the relation R restricted to a condition that for all t in U, if for all τ in the downward closure of U at t, G holds for φ ...
The function "igWlsAbs" applied to "termMOD" is equal to "wlsAbs".
If an atom 'i' is fresh in the permutation of 'p' and 'V', and 'V' is a subset of 'Vs', then the atom 'i' is fresh in the substitution of 'vquot_dbfm V A' with 'V' in 'F'.
If there is a coalgebra between B1 and B2 for states s1 and s2, then B1 is equivalent to the result of the lsbis1 operation on B1, B2, s1, and s2.
The lemma named "primroot_take_shortest" assumes that "u" is not equal to the empty word. It shows that the primitive root of "u" is equal to the shortest possible take of "u", where the length of the take is the least "d" such that there exists a word "r" where "u" is in the closure of "r" and the length of "r" is "d"...
If "x" reduces to "y", then "x" applied to "z" reduces to "y" applied to "z".
"the composition of x with the composition of y and z is congruent to the composition of the composition of x and y with z".
The lemma "eqExcPID_ex_isNthReview" assumes that "s" and "s1" are reachable states, and "s" and "s1" are equivalent except for the process ID (PID). It also assumes that in state "s", "uid" is the nth reviewer of the conference ID (cid) for the process ID (PID). The lemma shows that there exists a user ID "uid1" such t...
A new element's shadow root is a type well-formed location of getting a shadow root and its locations. This is an instance of the function "l_new_element_get_shadow_root".
For a function 'f' from a complete lattice 'a' to another complete lattice 'b', if 'f' preserves the supremum (denoted by "Sup_pres f"), then 'f' also preserves the supremum operation (denoted by "sup_pres f").
If "G" is a well-structured program and "C" is a class in "G", then for all "D", if the method with signature "sig" in the class "C" in "G" is defined and equals to the tuple (D, mh, code), it implies that "D" is a class in "G" and the method with signature "sig" in the class "D" in "G" is also defined and equals to th...
If a relation 'r' is antisymmetric, and 'r' relates 'a' to 'b' and 'b' to 'a', then 'a' must be equal to 'b'.
The collection of path components of X at a point x is a subset of the topological space X.
The lemma named "space_case_nat" assumes that omega is in the space of H i and s is in the space of Ms. It shows that the case of natural number s and the first element of the composition of omega at index i is in the space of Ms.
The dimension of the cotangent space at a point p in the set X is equal to the dimension of X, given that X is a subset of the cotangent space at point p.
If A and B are both order filters, then the union of A and B is also an order filter.
If P is a carrier coefficient, then the m-th element of P belongs to the carrier R. The assumption here is that P is a carrier coefficient.
If a function 'f' is continuous on a set 'S' and the image of 'f' is a subset of 'T', and if 'S' is contractible, then there exists a constant 'c' such that 'f' is homotopic to the constant function that maps every element of 'S' to 'c'. This is under the condition that the homotopy does not have to satisfy any additio...
If "\<FF>'" is a functor from the category of ordinal 3 to the category "\<AA>", and "\<TT>" is a functor from the category of ordinal 2 to the category "\<AA>", and "\<epsilon>'" is a natural transformation from the composition of "\<FF>'" and "\<KK>23" to "\<TT>" in the category of ordinal 2 to the category "\<AA>", ...
If p' is an element of the grid defined by p and ds, then the length of p' is equal to the length of p.
If ll, lr, and r are all greater than or equal to 1, then the logarithm base alpha of the sum of ll and lr plus beta times the logarithm base alpha of the sum of lr and r is less than or equal to beta times the logarithm base alpha of the sum of ll and lr plus the logarithm base alpha of the sum of ll, lr, and r.
If a list 'a' concatenated with another list 'c' is equal to the result of dropping 'n' elements from the concatenation of lists 'b' and 'c', then the list 'a' is equal to the result of dropping 'n' elements from the list 'b'.
If a is well-valued0, and a is not a Leaf, then the value of r of a is less than or equal to the length of as.
If variable x is independent of variable y, then the precondition of variable x is also independent of the precondition of variable y.
If a computation "c" refines to an abstract value "a" under the heap-relations "Γ" and "Γ'" and the result relation "R", and "c" can be converted to "c'", then "c'" also refines to "a" under the same heap-relations and result relation.
The cost function c_smj with parameters cf and f is symmetric.
"projecting C h F X' " applies to the universal set.
If n is less than p, then the encoding of n is less than the encoding of p.
The lemma named "rel_FGco_neg_distr_imp" is about certain conditions related to negative distribution in the context of relations. It takes several arguments, including multiple relations (Co1, Co1', Co2, Co2', Co5, Co5') and type tokens (tytok_F, tytok_G, tytok_FGco). The lemma assumes two conditions: 1. The negati...
The image of the function "f" over the interval "I" added to a number "n" is equal to the image of the function that adds "n" to each element of "f" over the interval "I".
The cardinality or size of the edge set E is equal to the sum of the degrees of all vertices in set Y.
The lemma named "blinding_of_prod_hash" assumes that "boa" is less than or equal to the inverse image of the relation "rha" under itself and equality, and "bob" is less than or equal to the inverse image of the relation "rhb" under itself and equality. It shows that the blinding of the product of "boa" and "bob" is les...
For a given matrix A, if the rank of A is equal to the number of columns in A, then the columns of A are linearly independent. Here, A is a matrix of any field type and the dimensions of the matrix are modular types.
, assuming a context Γ in which the type abstraction of s over S has type T (denoted as "\<Gamma> \<turnstile> (\<lambda><:S. s) : T"), if Γ also proves that T is a subtype of a universal type over U and U' (denoted as "\<Gamma> \<turnstile> T <: (\<forall><:U. U')"), then for any type S', if Γ proves that U is a subty...
The lemma named "arctan_0_1_bounds_round" assumes that a real number 'x' is greater than or equal to 0 and less than or equal to 1. It shows that the arctangent of 'x' is within the range defined by the lower bound and upper bound. The lower bound is the product of 'x' and the lower bound arctan horner for 'p1', the ev...
The lemma "join_ti_hint1" assumes that the join time index of a stream 's' and a value 'x' at the successor of index 'i' is an empty list. It shows that the join time index of the same stream 's' and value 'x' at index 'i' is also an empty list.
The lemma "remove_preserves_known_ptrs" assumes that the heap "h" is well-formed, the type of "h" is well-formed, and "h" contains known pointers. It also assumes that removing a child from "h" results in a new heap "h'". The lemma then shows that "h'" also contains known pointers.
The length of the weak integer composition enumeration of 0 with the successor of any integer n is 1.
If process P is deadlock-free and process Q is deadlock-free, then the non-deterministic choice between process P and process Q is also deadlock-free.
: Given the assumptions that a pair (P, Q) is in set X, set X is equivariant (i.e., invariant under permutations), for any P and Q in set X, P weakly simulates Q with respect to the bisimulation relation composed with set X and the bisimulation relation, and for any P and Q in set X, if (P, Q) is in set X then (Q, P) ...
If the intersection of set A and the field of relation R is an empty set, then the restriction of relation R to set A is equal to R itself.
If the intersection of "m3_inv3_sesK_compr", "m3_inv2_ticket", and "m3_inv1_lkeysec" holds before the transition "m3", then "m3_inv3_sesK_compr" holds after the transition.
If the schedule "as" takes the global configuration from "gc" to "gc'", and the schedule "bs" takes the global configuration from "gc'" to "gc''", then the combined schedule "as@bs" (which is the concatenation of "as" and "bs") takes the global configuration from "gc" to "gc''".
If "x" is an element in the set of the stack "s", then "x" is not in the domain of the finished "s".
: Assuming that "u" and "v" are distinct points in the unit disc, and that for any Moebius transformation "M" that fixes the unit disc, if the property "P" holds for the images of "u" and "v" under "M", then "P" also holds for "u" and "v" themselves. Furthermore, assuming that for any real number "x" with positive rea...
The universal set is equal to the range of the function that takes a pair (x, pos) and returns a literal with x and pos.
If "x" is greater than negative half pi and less than half pi, then the arctangent of the tangent of "x" equals "x".
The function S applied to a term t is equal to the function EFSM.S applied to the term tm of t.
There exists a polynomial 'f' in the modulo ring of 'p' such that 'u' is equal to the product of the coefficients of 'v' and 'w' plus the product of 'q' (converted to a natural number) and the coefficient of 'f'.
If M is an equivalence, then the restriction of M (denoted as M[r!]) is also an equivalence.
The probability mass function "n" is defined over the set consisting of the elements "C1", "C2", and "C3".
If the function "find_path_rec_f" with parameters P, ctx, cl, and path' returns some path, then for all fully qualified names (fqn) and context class declarations (ctxcld), if the class (cl) is equal to the fully qualified name (fqn) and the function "find_cld_f" with parameters P, ctx, and fqn returns some context cla...
The lemma "NF_no_trancl_step" assumes that "a" is in the normal form of "A". It shows that for all "b", there is no transition from "a" to "b" in the transitive closure of "A".
The lemma "decode_plan_step_element_then" is defined for a fixed strips problem 'Pi'. It assumes that 'k' is less than 't' and that 'op' is an element in the set at the 'k' index of the inverse of 'Phi' applied to 'Pi' and 'A' at time 't'. If these conditions are met, it shows that 'op' is an element in the set of oper...
If a command "c" transforms state "s" into state "t", and command "c" is well-typed under typing context "l", then the states "s" and "t" are identical for all variables with names less than "l".
The post-measurement of the Bell state "beta01" at the second position with measurement outcome 1 is equal to the unit vector in a 4-dimensional space at position 1.
The first element of the pair resulting from the function "strip_comb" applied to the application of term "t" to term "u" is equal to the first element of the pair resulting from the function "strip_comb" applied to term "t".
The equality on a property P for an environment that has been updated with a process 'p' and a path 'π', is equivalent to the equality on the property P excluding the process 'p' for the original environments.
The function 'f', which maps from 16-bit unsigned integers to a unit, is equal to the function 'g'.
Two intervals A and B are disjoint if and only if the start of A is greater than the end of B or the start of B is greater than the end of A or the start of A is greater than its own end or the start of B is greater than its own end.
The function "f_bnd" when applied to "outside" is less than or equal to the function "f_bnd" when applied to "False".
The lemma alpha_test3 demonstrates that letting a variable 'x' be equal to another variable 'y', and 'y' be equal to 'x', in the context of 'x', is the same as letting 'y' be equal to 'x', and 'x' be equal to 'y', in the context of 'y'.
The join operation of any arity 'x' with zero results in zero.
The pair of integers (0, 0) is an element of the set Ainteg.
If a property φ holds, then there exists a context c where φ holds.
The lemma "onet_nhop_quality_increases" shows that for a given network tree 'p', if the quality increases for any other network tree in the set of all network tree IPs of 'p', then for all nodes 'i' in the set of all network tree IPs of 'p' and for all destination IPs 'dip', if 'dip' is in the intersection of the set o...
The lemma H_on_ket_one shows that the product of H and ket one equals psi_01.
The lemma "PO_refines_implies_R_image_invariant" assumes that "PO_refines R Ta Tc". It shows that the reach of Tc is a subset of the image of the reach of Ta under R.
The lemma named "fds_exp_ln_strong" is about a fixed Dirichlet series 'f'. It assumes that the first non-zero term of the series 'f' is not zero. It shows that the exponential of the logarithm of the series 'f' scaled by a factor 'l' is equal to the series 'f' scaled by the constant term 'exp l' divided by the first no...
For any hyperreal numbers 'u' and 'x', if 'x' is less than zero and 'x' is not an infinitesimal, and 'u' is in the monad of 'x', then 'u' is less than zero.
The items of zero are equal to an empty list.
Looking up a key in an associative list after updating the list with a new key-value pair is equivalent to looking up the key in the original list, but with the key mapped to the new value.
If a function B is monotonically increasing when applied to an option, then a function that maps another function g onto the result of B is also monotonically increasing when applied to an option.
The negative of a number 'a' is equal to the bitwise NOT operation on 'a' plus 1.
If the intersection of set S and set T is an empty set, then the intersection of all edges in set S and all edges in set T is also an empty set.
Two intervals A and B are not disjoint if and only if, for A defined by a start point 's' and an end point 'e', and B defined by a start point 's'' and an end point 'e'', the start point of A is less than or equal to the end point of B, the start point of B is less than or equal to the end point of A, the start point o...
: Assuming that for all 'v', if 'ifp' is reflexive and transitive on 'v' and 'u', then 'vpeq' holds between 'v', 's', and 't'. Also, assuming that the current state of 's' is the same as the current state of 't'. It shows that if the precondition of the next state of 's' and 't' after executing 'execs' is 'a', and if...
If a system A satisfies the invariant P for states S and updates U, and if a state s is in the initial states of system A, then the property P holds for the state s.

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