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Statement: stringlengths 7 24.3k
If a list 'xs' is sorted and contains distinct elements, and there exists an element 'a' in the list 'xs' that is greater than or equal to 'n', then the element at the index equal to the length of the sublist of 'xs' containing elements less than 'n' is greater than or equal to 'n'.
When the function "dummies" is applied to the result of mapping the function "Dummy" over a list "l", the result is the set of elements in "l".
The interpretation of a term "T" under a mapping "U.int" with a variable "ξ" is equal to the interpretation of the same term "T" under a different mapping "M.int" with the same variable "ξ".
: Assuming "A and B are on the same side of line segment CD", "angle BCA is a right angle", "angle BDA is a right angle", "points C, D, and E are collinear", "line segment AE is perpendicular to line segment CD", and "angle CAB is congruent to angle DAB", it can be shown that "angle BAC is congruent to angle DAE", "ang...
The destination port of a packet, represented by the tuple (a,d,(x::ipv4),e), is equal to the second element of the tuple x.
The domain of the product of the domain of x and y is equal to the product of the domain of x and the domain of y.
, under certain conditions, a point 'p' belongs to a set 'S'. These conditions are as follows: 1. The point 'p' is related to points 'a', 'b', and 'c' through a linear combination. This is represented by the equation "x \<^sub>R p = u \<^sub>R a + v \<^sub>R b + w \<^sub>R c", where 'x', 'u', 'v', and 'w' are sca...
, in the context of G, n is true for the set obtained by inserting t into A if and only if n is true for t and n is true for A in the context of G.
The lemma named "subterms_union_ground" assumes that a set M is ground. It shows that the set of all subterms of M is also ground.
: - For any boolean `b` and variable `var`, a property `P1` holds for the variable `var`. - For any expression `exp` and variable `var`, if the property `P1` holds for the expression `exp`, then `P1` also holds for the application of `exp` to `var`. - For any heap `Γ` and expression `exp`, if the property `P2` holds f...
The lemma named "div_minus_1" assumes that an integer "x" is in the range from negative "b" (exclusive) to 0 (inclusive). It shows that the result of "x" divided by "b" is -1.
The lemma "init_fin_preserve_Status_no_wait_locks" assumes that the status of no wait locks is true for the thread state "s" (as stated by "Status_no_wait_locks (thr s)"). It also assumes that a multithreaded base transition from state "s" to state "s'" occurs, with the transition being defined by "init_fin_final", "in...
The lemma named "ccompatible_inv" shows that the "ccompatible" function is applicable to the inverse of A at location l within the set R.
The lemma named "continuous_eval_fps" is about a fixed element 'z' of a type that is both a real normed field and a Banach space. If the norm of 'z' is less than the convergence radius of a formal power series 'F', then the evaluation of 'F' at 'z' is continuous within a set 'A'.
The dual of x equals y if and only if x equals the inverse dual of y.
The length of the list obtained by filtering a list 'xs' at indices 'I' according to a predicate 'P' is less than or equal to the length of the list obtained by filtering the original list 'xs' according to the same predicate 'P'.
, assuming "x" is in the carrier of group "G" and "G" is a group, the inverse of "x" is in subgroup "H" if and only if "x" is in subgroup "H".
: Assuming a quasi-Borel space 's' has a probability space equal to 'X', and for any 'x' in the space of 'X', 'f' of 'x' equals 'g' of 'x', and 'f' is integrable over 's'. Then, it can be shown that 'g' is also integrable over 's'.
The successor of 0 is less than or equal to the start of "ly" as "as".
A type 'ty' is either a base type 'TBase', or there exist types 'σ' and 'τ' such that 'ty' is a function from 'σ' to 'τ', or there exists a type 'σ' such that 'ty' is a type constructor 'T' applied to 'σ'.
The omega power of the product of x and omega power of 1 is less than or equal to the product of x and omega power of 1.
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 merchant registration, then the analysis of the union of the image of the key inserted into KK and what the Spy knows from 'evs' is equal to the insertion of the key K into the analysis of the union of the...
If a set Φ is finite, then the disjunctive normal form (dnf) of the big OR operation over the set Φ is equal to the result of folding the set Φ with a function that takes an element φ from the set and unions it with the disjunctive normal form of φ, starting with an empty set.
The set resulting from the closure implementation function 'f' applied to a set 'S' is equal to the closure of the set 'S'.
The lemma named "c_futures_tr" assumes that the pair "(yps, Y)" is in the futures of the c_process with parameters step, out, and s0, and xps. It shows that "yps" is equal to the c_tr function with parameters step, out, and the result of folding the step function over s0 and the first element of each pair in xps, and t...
If the angle phi of player A is equal to pi/2 and the angle theta of player A is also equal to pi/2, then the unitary transformation of player A is equal to M.
If a relation is valid with the properties flipped for P and Q and the commands switched, then the original relation is also valid with the original properties and commands.
If a Dickson grading 'd' exists, the set of components of terms in 'F' is finite, and 'F' is a subset of the set of polynomials of degree at most 'm' with respect to 'd', then the reduced Groebner basis of 'F' is indeed a reduced Groebner basis.
The minimum integer polynomial of a rational number 'r', when 'r' is considered as an element of a field with characteristic 0 and a field with a greatest common divisor, is equal to the polynomial of the rational number 'r'.
The hyperplane defined by the set of all points x such that the dot product of a and x equals b is a face of the half-space defined by the set of all points x such that the dot product of a and x is less than or equal to b.
If a match between a packet and a rule in the first list is equivalent to a match in the first list of rules, and a match between a packet and a rule in the second list is equivalent to a match in the second list of rules, then a match between a packet and a conjunction of rules from the first and second lists is equiv...
If the heap 's' is black, then the white heap is defined as the heap 's' with its system state modified such that 'fM' is the negation of the system's 'fM' and the memory store buffers of the garbage collector are set to 'ws'.
The theorem "TESL_interp_decreases_setinc" assumes that the set Φ is a subset of the set Φ'. It shows that the TESL interpretation of Φ is a superset of the TESL interpretation of Φ'.
If alpha prime is equal to alpha, then alpha prime is a partially ordered category.
The order of a point 'a' in a polynomial 'p' is equal to the successor of the order of 'a' in the derivative of 'p'. This is true if the derivative of 'p' is not zero and the order of 'a' in 'p' is not zero. This is applicable for 'p' which is a polynomial over a field with characteristic zero.
If you replace a subterm at a certain position with another subterm that has been replaced at a certain position, it is equivalent to replacing the subterm at the concatenated position.
If the finite set "l" is in the analysis of "H", and if the finite set of "l" is a subset of the analysis of "H", then a certain property "P" holds.
If from a list of propositions 'ps' and a proposition 'p', we can infer a proposition 'r', and if from the same list of propositions 'ps' and a proposition 'q', we can infer the proposition 'p', then from the list of propositions 'ps' and the proposition 'q', we can infer the proposition 'r'.
For a given polynomial 'p' of any field type 'a', if 'p' and 'q' are both non-zero polynomials of degree 'n0' and 'n1' respectively, then the product of 'p' and 'q' equals zero if and only if either 'p' or 'q' equals zero.
The maximum variable in the float arithmetic expression of (a / b) is the maximum of the maximum variable in float arithmetic expressions a and b. The maximum variable in the float arithmetic expression of (a * b) is the maximum of the maximum variable in float arithmetic expressions a and b. The maximum variable in th...
If "t1" is a subtree of "t", "v1" is in the directed vertices of "t1", "x" is an element of the set "v1", and there is a directed edge from "x" to "y" in the tree "T", then "y" is in the directed leaf vertices of "t1".
: For a channel of type "Channel" with capacity "cap", type sequence "ts", and queue "q", the channel is valid if and only if the following conditions are met: - The capacity is less than or equal to the maximum array size and is also greater than or equal to 0. - All elements in the type sequence are valid according ...
Given a well-formed non-deterministic finite automaton (NFA) "N" with "Suc n" states, a value "v" that is less than or equal to "n", a word "w" that is an alphabet of "n", and a node "q" that is a node of the NFA "N", the set of bit vectors resulting from the transition function of the quantified NFA with respect to "v...
The while loop with condition "b", command "c", and invariant "Inv" satisfying postcondition "P" is less than or equal to "w" if and only if there exists another while loop with condition "b", command "c'", invariant "Inv'", and postcondition "P'" such that "w" is equal to this new while loop and the command "c" is les...
If "-p" implies "-r" given "x", and "-q" implies "-r" given "x", then the disjunction of "-p" and "-q" (i.e., "-p" or "-q") also implies "-r" given "x".
If "r" and "s" are positive natural numbers and "i" is less than or equal to the product of "r" and "s" minus one, then the remainder of the function "rtos" of "r", "s", and "i" divided by "r" is equal to the integer division of "i" by "s".
The lemma named "binomial_antimono" assumes that "k" is less than or equal to "k'", "k" is greater than or equal to half of "n", and "k'" is less than or equal to "n". It shows that the binomial coefficient of "n" and "k" is greater than or equal to the binomial coefficient of "n" and "k'".
The lemma assumes that for a program 'P', there is a sequence of subclasses '(Cs@ C'#Cs')' for a class, where 'C'' is the head of this sequence according to the relation 'Subobjs\<^sub>R'. It shows that under these conditions, 'C'' and its sequence of subclasses '(C'#Cs')' also have a 'Subobjs' relation in the program ...
If a term 't' is locally closed, and a variable 'x' is not in the free variable set of 's' and 'p', then substituting 'x' with 't' in the term 't' and then opening 't' with 's' and 'p' at index 'n' is the same as first opening 't' with 's' and 'p' at index 'n' and then substituting 'x' with 't' in 't'.
If the following conditions are met: 1. N is a module over a ring R, 2. H is a free generator of a module M over the ring R, 3. f is a function from H to the carrier of N, 4. C is a subset of the set of free sequences of partial sums (fsps) of R, M, N, f, and H, 5. For every pair of elements a and b in C, either the f...
The sequence of operation c followed by the omega repetition of operations d and c is equivalent to the omega repetition of operations c and d followed by operation c.
The lemma named "of_alist_transfer" is a transfer rule. Assuming that "R1" is bi-unique, it shows that if "list_all2" is related to the product of "R1" and "R2", then "map_of" is related to "of_alist" through the partial correctness relation of "R1" and "R2" in the context of mapping.
If a value "v" is in the set resulting from looking up a list "xs" in the tries structure created from a list "vs" using function "f", then "v" must also be in the set of "vs".
The bit list representation of the result of applying the "slice1" operation on a word "w" at position "n" is equivalent to reversing the list obtained by filling with "False" up to the length of the word type "'a", and then reversing the list obtained by filling with "False" up to position "n" in the bit list represen...
The abstraction of the initial state of a target observation, "tObsC_abs (tInit s)", is equal to the set of all pairs of states "(s', s')" such that "s'" is in the initial environment set "envInit" and the environment observation of "s'" is equal to the environment observation of the initial state "s".
The evaluation of the recursive function "rec_fact_aux" with inputs "x" and "y" is equal to the factorial of "x".
Appending a list "xs" with a list that starts with element "y" and is followed by list "ys" is equivalent to appending list "xs" with a list containing only element "y", and then appending this result with list "ys".
"S2P r x" is true if and only if "x" is an element of "r".
: "if a word 'w' is in the set S3, then the length of the list of elements 'x' in 'w' that are equal to 'a' is the same as the length of the list of elements 'x' in 'w' that are equal to 'b'."
The lemma named "ordertype_VWF_finite_nat" assumes that set A is finite. It shows that the order type of set A under the well-founded relation VWF is equal to the order of the cardinality of set A.
A natural transformation "N" is a tiny natural transformation from functor "F" to functor "G" in the tiny category from "A" to "B" if and only if the following conditions are met: 1. "N" is a natural transformation from functor "F" to functor "G" in the category from "A" to "B". 2. Functor "F" is a tiny functor in the...
If the relation R is locally antisymmetric over the union of sets A and B, then the relation R is also locally antisymmetric over set B.
If the pair (a, b) is an element of the multiset product of A and B, then b is an element of the multiset B of a.
The polynomial with an empty set of power products is equal to the set containing only the zero polynomial.
The nonces (random numbers used in cryptographic communications) of an encrypted message (Crypt K X) are equal to the nonces of the original message (X).
The codomain function of the product of two directed graphs, denoted as (\<AA> \<times>\<^sub>D\<^sub>G \<BB>), is a very special variable (vsv).
The set of all input-output pairs, where the pair is in the language of machine M and the length of the pair is less than or equal to k, is finite.
The lemma named "factorize_rat_poly" assumes a result "res" where the function "factorize_rat_poly_generic" applied to a polynomial "f" equals a pair "(c,fs)". It shows two things: 1. The polynomial "f" has a square-free factorization represented by the pair "(c,fs)". 2. If a pair "(fi,i)" is an element of the set "f...
If the natural transformation 'N' maps functor 'F' to functor 'G' in the context of categories 'A' and 'B' under a certain condition 'alpha', and both functors 'F' and 'G' are tiny morphisms from category 'A' to category 'B' under the same condition 'alpha', then the natural transformation 'N' is a tiny morphism that m...
For a given integer p, topological space X, and subset S, the singular relative boundary with respect to the negative of a chain c is equivalent to the singular relative boundary with respect to the chain c itself.
The function "map_values" applied to a function "f" and a Red-Black Tree (RBT) mapping "t" is equivalent to folding over the RBT with a function that takes an element "x", a value "v", and a mapping "m". If the function "f" applied to "x" and "v" returns None, the mapping "m" is returned. If it returns Some value "v'",...
If the following conditions are met: 1. The sharing is consistent with the thread states (denoted by "sharing_consis \<S> ts"). 2. The index "i" is less than the length of the thread states (denoted by "i < length ts"). 3. The "i-th" thread state is a 7-tuple consisting of program counter, instruction sequence, regist...
The intersection of 'e' and the inverse of 'ov' is an empty set.
: Assuming that "x" is greater than "mi" and less than or equal to "ma", "mi" is not equal to "ma", the degree is greater than or equal to 2, the high value of "x" divided by 2 is "h", the low value of "x" divided by 2 is "l", "newnode" is the result of deleting "l" from the tree list at index "h", "newlist" is the tr...
A nonce "X" is not equal to a number "X'".
The lemma named "arctan_upper_13" assumes that "x" is greater than or equal to 0. It shows that the arctangent of "x" is less than or equal to the upper bound of arctangent function defined as "arctan_upper_13" at "x".
The first level protocol step 2 refines the initial zero level protocol for running A and B with the exponent of gnx and the nonce of Rb$ny. This is expressed in the relation R01iai.
If two systems, M1 and M2, are observable and minimal, and both systems converge at points u and v, then the preservation of divergence for M1 and M2 when u is added to set X is the same as the preservation of divergence for M1 and M2 when v is added to set X.
If the most general unifier of two terms t1 and t2 is some term u, and if a new type variable n is not present in both t1 and t2, then this new type variable n is also not present in the term u.
: Given the following conditions: - k is greater than 0, - The local state is idle after executing a computation stream with a given local state, transition function, input, and time, - The message m is not a null message, - The time t0 is equal to the product of t and k, - The computation stream s is executed with a ...
: 1. For any "nhip" and "pre", the function "update_arg_wf" is well-formed when applied to the tuple (0, unk, val, Suc 0, nhip, pre). 2. For any "n", "hops", "nhip", and "pre", the function "update_arg_wf" is well-formed when applied to the tuple (Suc n, kno, val, hops, nhip, pre).
: "For all 'x', there exists a unique 'y' such that a certain condition 'Q' holds between 'x' and 'y'. This implies that there exists a unique function 'f' such that for all 'x', the condition 'Q' holds between 'x' and the value of function 'f' at 'x'."
The lemma named "on_closure_eqI" is about two functions 'f' and 'g' that map from a topological space 'a' to a T2 space 'b'. It assumes that for any element 'x' in set 'S', 'f' of 'x' equals 'g' of 'x'. It also assumes that 'x' is in the closure of set 'S', and both 'f' and 'g' are continuous on the universal set. Unde...
The undiscr function, when applied to 1, returns 1.
The lemma named "alt_group_inv_equality" assumes that "p" is an element of the alternating group of order "n". It shows that the inverse of "p" in the alternating group of order "n" is equal to the inverse of "p" under the operation "inv'".
The theorem named "closure_sound" assumes that the result of the closure function on the pair (r,s) with an empty list as the second argument is Some([],ps). It also assumes that the union of the atoms of r and s is a subset of the set as. Under these assumptions, it shows that the language of the set as with respect t...
The theorem "positional_strategy_exists" assumes that "v0" is an element of "V". It shows that there exists a "p" such that "v0" is in the winning region of "p".
The negation of the supremum (or greatest element) of a set A, where each element is transformed by a function B, is equal to the infimum (or least element) of the same set A, where each element is transformed by the negation of function B. This is applicable for any function B that maps from a type 'a to a type 'b, wh...
The set of all lists 'xs' that are lexicographically sorted is closed.
The domain of A in Gamma is a subset of the free variables in Gamma.
A logical literal is positive if and only if the negation of that literal is not positive.
The multiplication sheaf specification of U, a, and b is in the extensional of U.
In the context of complex square matrices, if A and B are both finite-dimensional complex matrices, then the product of A and B is also a finite-dimensional complex matrix.
If a list 'xs' contains distinct elements and another list is formed by appending an element 'x' to the concatenation of lists 'ys' and 'zs' such that it equals 'xs', then the function that multiplies the result of applying function 'ldeep_s' with parameters 'f', 'xs', and 'a' by 'cf a' at 'x' is equal to the function ...
The diagonal sequence at position 'n' is equal to the fold reduction at position 'n' with initial value 0 and list length 'n'.
The lemma named "ncoplanar_perm_5" assumes that points A, B, C, and D are not coplanar. It shows that points A, D, C, and B are also not coplanar.
The normalization of the negation of any element 'x' from a commutative ring with a normalization semidomain is equal to the normalization of 'x' itself.
If a tuple (cl_k, var_k, ty_k) is in the set cl_var_ty_list, and if the list of variables obtained by mapping each tuple (y, cl, var, u) in y_cl_var_var'_v_list to its variable component is the same as the list of variables obtained by mapping each tuple (cl, var, ty) in cl_var_ty_list to its variable component, then t...
The lemma named "in_set_CollectI" assumes that a property "P" holds for an element "x" and that the set of all elements for which "P" holds is small. It then shows that "x" is an element of the set of all elements for which "P" holds.
Let S be a set of complex numbers. Suppose that S is convex, and for every natural number n and every complex number x in S, the function f(n) has a field derivative f'(n) at x within S. Also, suppose that for any positive real number e, there exists a natural number N such that for all natural numbers n and complex nu...

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