Statement: stringlengths 7 24.3k |
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If a set of known pointers "a_known_ptrs" exists in a heap "h", and a pointer "ptr" is in the set of object pointers "object_ptr_kinds" in the same heap "h", then the pointer "ptr" is a known pointer "known_ptr". |
If for every pair (h, U) in the set ps, h is an element of P[X] and U is a subset of X, and if for every pair (h, U) in the set (ps+) and every pair (h', U') in the same set, the polynomial degree of h equals the polynomial degree of h' implies that the pair (h, U) equals the pair (h', U'), then the exact decomposition... |
The equivalence of the pair (x, F) to the pair (y, G) is equivalent to the necessity of the equivalence of the pair (x, F) to the pair (y, G) in the context of v. |
The lemma named "fr_refinement_FrameTransmission" assumes the following conditions:
1. There is a cable with parameters n, nSend, and recv.
2. For all i less than n, there is a FlexRayController with parameters (nReturn i), recv, (nC i), (nStore i), (nSend i), and (nGet i).
3. The schedules for n and nC are disjoint.
... |
The support of the existential path quantifier of x is equal to the support of x. |
For any two fixed integers x and y, if x and y are odd, n is even, n is greater than 0, and the absolute value of x is not equal to the absolute value of y, then the multiplicity of 2 in the difference of x to the power of n and y to the power of n is equal to the sum of the multiplicity of 2 in the difference of x and... |
The theorem of tableau completeness:
Given a list G of formulas, each containing natural numbers,
Assuming that for all natural number functions e, and for all natural number terms f and g, if all elements in G evaluate to true under e, f, and g, then p also evaluates to true under e, f, and g,
It can be shown that the... |
The set of states "R" is a subset of the set obtained by inserting the bottom element into the closure of the set resulting from applying the function "psubt_lhs_bot" to "R". |
The range of a finite map 'm' when used with the function 'fmran'' is equal to the set of the range of the same finite map 'm' when used with the function 'fmran'. |
The lemma named "sauer_shelah_0" establishes that for any set F of sets, if the union of F is finite, then the cardinality (or size) of F is less than or equal to the cardinality of the set shattered by F. |
For a queue 'q' and a list 'xs', if we fold the list 'xs' by appending each element to the queue 'q', the linearized result will be equivalent to the concatenation of the linearized queue 'q' and the list 'xs'. |
The difference between the fraction with numerator 'a' and denominator 'numeral b', and the fraction with numerator 'c' and denominator 'numeral d', is equal to the fraction with numerator and denominator normalized by the rational number obtained from the difference of 'a' times 'numeral d' and 'c' times 'numeral b', ... |
The lemma named "cindexP_rectpath_edge_base" assumes that the real part of 'a' is less than the real part of 'b', the imaginary part of 'a' is less than the imaginary part of 'b', 'r' is not a rectangle vertex of 'a' and 'b', and 'r' is in the path image of the rectangle path from 'a' to 'b'. Under these assumptions, i... |
Assuming that "c" is greater than 0, the lemma demonstrates that the limit of the function that divides a natural number "x" by "c" approaches infinity as "x" approaches infinity. |
The multiset of a vector is equal to the multiset of the list representation of that vector. |
The lemma is about geodesic segments in a space. It assumes that there is a geodesic segment G between points x and y.
It then shows several things:
1. The geodesic segment parameter of G at point x and time 0 is x.
2. The geodesic segment parameter of G at point x and time equal to the distance between x and y is y... |
The final factorial body is equal to the factorial body prime. This is represented symbolically as "?lhs = ?rhs", where "?lhs" is the left-hand side (fac_body_final) and "?rhs" is the right-hand side (fac_body'). |
If the evaluation of a formula "alpha" in a certain environment "e" equals the negation of a value "x", then there exists a formula "A" such that "alpha" equals the negation of "A" and the evaluation of "A" in the same environment "e" equals "x". |
For a function 'f' mapping from one Euclidean space to another, if 'f' is a linear and injective function, then the image of the closure of a set 'S' under 'f' is equal to the closure of the image of 'S' under 'f'. |
The lemma "sum_e_below_sum_x_when_outgoing_same_component" assumes that:
- "g" is symmetric,
- "j" is a vector,
- "h" is a forest,
- "h" is regular,
- "x" is less than or equal to the negation of the forest components of "h" intersected with the complement of "g",
- the transpose of "x" multiplied by the top element is... |
The space of the tail algebra at a given point 'n' is equal to the product of the space 'M' over all elements of the universal set. |
If x is an element of set A, then the cardinality (or the number of elements) of the set obtained by removing x from A is equal to the cardinality of A minus one. |
The lemma named "adjoint_eqI" is defined with two fixed functions: G, which is a complex bounded linear function from a Hilbert space 'b to a complex inner space 'a, and F, which is a complex bounded linear function from 'a to 'b. The lemma assumes that for any two elements x and y, the complex inner product of the res... |
The lemma named "ccInf_mono" assumes that both sets A and B are countable. If for every element 'b' in set B there exists an element 'a' in set A such that 'a' is less than or equal to 'b', then the infimum (greatest lower bound) of set A is less than or equal to the infimum of set B. |
The lemma named "swap_permutes" assumes that the effect of swapping elements i and j in array a changes the state from h to h' with a resulting state rs. It shows that the multiset of the array a after the swap (h') is equal to the multiset of the array a before the swap (h). |
The signed division of zero by any integer "x" is zero. |
If "a" is less than or equal to "b" and "c" is less than or equal to "d", then the product of "a" and "c" is less than or equal to the product of "b" and "d". |
In the lemma of a commutative ring, the generated ideal is a principal ideal, assuming that 'i' is in the carrier of 'R'. |
The lemma named "caseCases" consumes 1 assumption and has a case name "cCase". It fixes Psi as type 'b, Cs as a list of tuples containing 'c and psi of types 'a, 'b, 'c, alpha as an 'a action, and P' as psi of types 'a, 'b, 'c.
It assumes a transition from Psi to the Cases of Cs resulting in Rs, and a rule "rCase" th... |
The lemma tft_3 assumes that the dimension of vector x is 3, and the elements of x at positions 0, 1, and 2 are True, False, and True respectively. It shows that the zeros of the boolean input x are at position 1. |
If the list 'ys' is empty, then for all lists 'xs', if 'xs' is empty, it equals to a list consisting of the element 'y' followed by the list 'ys'. |
Given a well-formed theory "Theta", a term "B" that is valid in "Theta", and a type of "B" that equals "propT", and for all pairs of identifier and type in the set "l", if the type is valid in "Theta", then the term "mk_all_list l B" is valid in "Theta" and the type of "mk_all_list l B" equals "propT". |
The lemma named "as_qbs_measure_retract" makes the following assumptions:
1. 'a' is a measurable function from a probability space 'P' to the real Borel space.
2. 'b' is a measurable function from the real Borel space to the probability space 'P'.
3. For any 'x' in the space 'P', the composition of 'b' and 'a' applied... |
3.26 times a light year is less than a parsec. |
The negation of a polynomial 'p' is equal to the multiplication of the monomial with coefficient -1 and degree 0 by the polynomial 'p', in the context of a commutative ring with unity. |
The function "heap_copies" with the parameters "heap_read_typed" (which is a function that reads the type of an address in a heap), "heap_write", "a", "a'", "als", "h", "obs", and "h'" is equivalent to the function "heap_copies" with the parameters "heap_read", "heap_write", "a", "a'", "als", "h", "obs", and "h'" and t... |
If the basic rule applies to the pair (lhs, rhs) and the term "rhs" is well-formed, then the rule applies to the pair (lhs, rhs). |
For a circular singly-linked list "l" with pointer "p", if we apply the "cs_rotate" operation on "p", it will result in a circular singly-linked list that is equivalent to rotating the list "l" once. |
If 'a' and 'b' are elements of 'A' and the interval 'r' between 'a' and 'b' is not empty, then the interval 'r' between 'a' and 'b' is a sublattice of 'cl'. |
The lemma named "lookup_inside2" assumes that the result of looking up "y" in the concatenation of the list "Γ'" and the list that contains the tuple "(x,b0,c0)" followed by the list "Γ" is "Some (b1,c1)", and that "x" is not equal to "y". It then shows that the result of looking up "y" in the concatenation of the list... |
If c is not equal to infinity, and both a and b are less than or equal to c, then if the difference between c and a is equal to the difference between c and b, it implies that a is equal to b. |
The measure of the stochastic probability mass function (SPMF) after applying a function 'f' to 'p' is equal to the distribution of the measure of the SPMF 'p' over the countable universal set, under the function 'f'. |
The lemma named "supp_fun_app_eqvt3" assumes that the function "f" is equivariant. It shows that the support of the function "f" applied to "x", "y", and "z" is a subset of the union of the supports of "x", "y", and "z". |
If the function "next_propagate'" is applied to the parameters c0, c1, loc, and t, and the summary from location loc to location loc' is not an empty set, then the work at location loc' in context c1 is equal to the work at location loc' in context c0 plus the result of applying the function "after_summary" to the fron... |
A specific nonce "N" is not in the synthetic analysis of the set of all messages "M" that exist such that "M" is the hash of some nonce. |
The transition function "\<T>" for a policy "p" and state "s" is equal to the measure of the probability mass function (pmf) for the policy "p" and state "s", bound to a function that distributes the transition function "T" for the successor policy and the reward function "K" for the state-action pair "(s,a)", over the... |
If you create three 'div' elements (a, b, c) in a document using the 'Node_insertBefore_document' function, then trying to insert 'b' before 'c' in 'a' will throw a 'NotFoundError'. This is tested on the 'Node_insertBefore_heap'. |
Secure Alice is in the image of the secure party of key party function applied to a set P if and only if key Alice is in set P. |
If 'a' and 'b' are objects in the category, then the inverse of the natural transformation Hom map from 'a' to 'b' is a one-to-one function. |
The lemma named "list_split_two_elems" assumes that the list "xs" contains distinct elements, and that "x" and "y" are elements in the set of "xs". It also assumes that "x" is not equal to "y". It shows that there exist lists "pre", "mid", and "suf" such that "xs" is equal to the concatenation of "pre", "x", "mid", "y"... |
The set "BrnFT cl dl" is finite. |
The lemma named "isNewArr_simps" states two conditions:
1. The function "isNewArr" returns false when a new instance of class C is created.
2. The function "isNewArr" returns true when a new array of type T with length l is created. |
A formal power series 'f' (from a field) divides 1 if and only if the zeroth coefficient of 'f' is not zero. |
Two words 'x' and 'y' are equal if and only if for all 'n' less than the length of 'a', the 'n'th bit of 'x' is equal to the 'n'th bit of 'y'. This is applicable for 'x' and 'y' which are words of any length 'a'. |
In the context of a ring, the set of units forms a group under the operation defined by the units group of the ring R. |
The limit of the function (x^2 + 3x - 10) / (7x^2 - 5x + 4) as x approaches 1/7 from the right is positive infinity. |
The lemma named "CX5h_singular" shows that the function "X5d" is injective (or one-to-one) on the set "CX5h h X". |
If the program state invariant holds for a program 'prog' and process 'p', and the channel list invariant holds for a pair (g,p), then the program state invariant will also hold for the program 'prog' and the second element of the pair resulting from the function 'mkChannels' applied to 'g', 'p', and 'cs'. |
For any hypercomplex number 'z', multiplying 'z' by negative one is equal to the negative of 'z'. |
The set of actions of a total program created with initial state "init", actions "acts", and allowed states "allowed" is equal to the set of "acts" after being transformed by the "totalize_act" function, with the identity function "Id" inserted into this set. |
If "i" is less than or equal to "j", and if taking "j" elements from lists "a" and "b" results in the same list, then taking "i" elements from lists "a" and "b" will also result in the same list. |
The theorem named "complement_language" assumes that the set of nodes in A is finite. It shows that the language of the complement of A is equal to the set of all possible streams over the alphabet of A, excluding the language of A. |
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Given the assumptions that 'C' is a locally small category, 'X' is an object in 'C', and 'f' is a morphism that is right-cancellable with 'g' in 'C', it shows that the composition of the hom-functor applied to 'X' and 'f', and the hom-functor applied to 'X' and 'g', is equal to the hom-functor applied to 'X' and the... |
The lemma named "proj2_incident_abs" assumes that neither "v" nor "w" are zero. It shows that a point "proj2_abs v" is incident on a line "proj2_line_abs w" if and only if the dot product of "v" and "w" is zero. |
If list l is not empty, then there exists an element x such that list l is equal to x followed by the tail of list l. |
If a flow "F" is in the set of offending flows "G" with respect to a network policy "nP", then "F" cannot be an empty set. |
Applying the function "pdevs_apply" to the polynomial deviation "Pdevs" of a variable "x" at a point "i" is equivalent to applying the function "slist_apply" to the same variable "x" at the same point "i". This is a piece of code. |
"True and P" is equivalent to "P". |
If "i" is less than "m" and "i" is not equal to zero, then the function "fim1_state" applied to the state equals the "i-1" element of the sequence "fs". |
If node 'u' is connected to node 'v', and node 'v' is connected to node 'w' through an edge in the graph 'E', then node 'u' is connected to node 'w'. |
The lemma named "relChain_under" assumes that "r" is a well-order. It shows that the relation chain of "r" is under "r" for any given index "i". |
A list is null after being filtered by a predicate P if and only if for all elements in the original list, the predicate P is not true. |
If "k" is greater than 0, then the "f" expansion of the concatenation of list "xs" and element "x" is equal to the "f" expansion of list "xs" concatenated with element "x" and the replication of "NoMsg" for "k - 1" times. |
The function 'unstables' for a given relation 'r', step 'step', and state 'ss', is equivalent to a fold operation. This operation iterates over a list from 0 to the size of 'ss'. For each element 'p' in this list, it checks if 'p' is not stable with respect to 'r', 'step', and 'ss'. If 'p' is not stable, it is inserted... |
The odd mirror function applied to the bird function is equal to the recursive function of Stern-Brocot. |
A pair consisting of 'x' and 'T' belongs to 'U' if and only if 'T' is equal to 'U'. |
Attaching a stateless callee to an oracle is equivalent to extending the state oracle with a function that takes a state and a query, and executes the oracle with the callee and the given state. |
The natural logarithm of x is equal to the logarithm of x with base e (where e is the exponential constant). |
If "b" is less than or equal to the successor of "x", then "b" is also less than or equal to the result of dividing the sum of "2 times x" and the length of "mopup_n_tm n" by 2. |
The lemma named "residue_hom_p" assumes that for any natural number 'n' that is greater than or equal to 'm', and for any prime integer 'p', the residue of 'p' raised to the power of 'm' is in the ring homomorphism from the residue ring of 'p' raised to the power of 'n' to the residue ring of 'p' raised to the power of... |
The lemma named "fun_upd2_absorb" demonstrates that updating a function twice with the same arguments 'a' and 'b' but different values 'c' and 'c'', results in the same function as if it was updated only once with 'a', 'b', and 'c''. |
The epsilon of an atom "a" is an empty set. |
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Assuming "R w" holds, and for any "w", if "R w" holds, then there exist "u", "v", "x", and "y" such that "w" equals the concatenation of "u", "x", "y", and "v", and the steps of the concatenation of "u" and "x" hold, and "E x y" holds, and "R" of the concatenation of "y" and "v" holds.
If these conditions are met,... |
In the context of an infinite coin toss space, the lemma named "nat_filtration_null_set" assumes that a set A belongs to the sets of the natural filtration at a certain point n, the probability p is greater than 0 and less than 1, and the expected measure of set A in measure space M is 0. Under these assumptions, it sh... |
The set of free variables in the term 't' after substitution 's' is a subset of the union of the set of free variables in 't' and the range variables in 's'. |
A jump is valid in the appended sequence of 'xs' and 'xs'' at positions 'n' and 'n'' if and only if a jump is valid in sequence 'xs' at positions 'n' and 'n' plus the length of 'xs'', and a jump is valid in sequence 'xs'' at positions 'n' plus the length of 'xs' and 'n''. |
If "a" is an arrow in the semicategory "CC", and for any "i" in "I", the morphism "phi i" is a semicategory morphism from "CC" to "AA i", then the range of the arrow map of the semicategory morphism "smcf_up I AA CC phi" applied to "a" is a subset of the union of the arrows of "AA i" for all "i" in "I". |
The set of face cycle sets of graph H is equal to the set of face cycle sets of graph G minus the face cycle sets of vertices a and a' in graph G, union the image of the face cycle set of graph H when applied to the union of the face cycle sets of vertices a and a' in graph G minus the set containing vertices a and a'. |
The lemma "abs_pdevs_val_less_tdev" assumes that "e" is a function mapping from any value to a number between -1 and 1, and the degree of "x" is greater than 0. Under these assumptions, it shows that the absolute value of the polynomial deviation value of "e" and "x" is less than the total deviation of "x". |
The lemma named "microstep_call_or_choice" assumes that process p leads to process q under a certain context Γ. It shows that either process p is a call to some process pn, or process p is a choice between two processes p1 and p2. |
If 's' and 't' are closed sets, and if a function 'f' is continuous on both 's' and 't', then 'f' is also continuous on the union of 's' and 't'. |
A polynomial equals zero if and only if there exists a number 'n' such that the polynomial is equivalent to 'n' repetitions of zero. |
If the logical formula "FOr" applied to the formulas "phi" and "psi" is a well-formed formula, then the logical formula "nFOr" applied to the same formulas "phi" and "psi" is also a well-formed formula. |
If relation Q is equivalent to relation Q', then the cartesian product of relation Q is equivalent to the cartesian product of relation Q'. |
The set of keys for an empty set is also an empty set. |
A relation "R" is a subset of "NC" if and only if the intersection of "R" and "NC" is equal to "R". |
The lemma named "fpxs_nth_below_val" assumes that "n" is less than the value of "f" in the function "fpxs_val". It shows that the nth term of the function "fpxs_nth" for "f" is equal to zero. |
The lemma named "the_cat_sspan_Arr_\<oo>I" in the category of small signatures introduces the following statement: If "a" is equal to the initial object in the category of small signatures, then "a" is an arrow in the category of contravariant functors. |
The inverse of the negation of a formal Laurent series 'f' is equal to the negation of the inverse of the formal Laurent series 'f' itself, where 'f' belongs to a division ring. |
The length of the result from applying the function "extract_IPSets_generic0" with parameters "sel" and "rs" is equal to the length of "rs". |
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