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Statement: stringlengths 7 24.3k
If for all variables in the atoms of a formula, the variable is greater than zero, and if the formula is in conjunctive normal form (CNF), and if a model M satisfies the formula, then the DIMACS model of the model M, with the range from 1 to the maximum variable in the formula plus 2, is a valid model for the DIMACS re...
The lemma "initial_time_bounds" demonstrates two things: 1. If the existence interval of the initial value problem at time t0 and state x0 is bounded above, then t0 is less than the supremum (least upper bound) of this existence interval. 2. If the existence interval of the initial value problem at time t0 and state x...
If a sequence of commands "cs" starting from state "s" eventually leads to a sequence of commands "cs'" and state "s'", then appending another sequence of commands "cs2" to "cs" and "cs'" will still lead from state "s" to state "s'", regardless of the additional commands.
The lemma named "phi'_nonzero" assumes that "m" is greater than zero. It then shows that "phi' m" is also greater than zero.
The lemma named "venv_lookup" assumes that when looking up "x" in "\<Gamma>", we get "Some A". It then shows that when looking up "x" in "Venv \<Gamma>", we get "Some (V A)".
For any extended nonnegative real number 'x', if 'x' is divided by zero, the result is zero if 'x' is zero, otherwise the result is infinity.
The sum of two formal Laurent series constants, x and y, is equal to the formal Laurent series constant of the sum of x and y.
The c_assoc_value function is in the set of primitive recursive functions of two variables.
The sum of x and y is equal to the unique z such that for all v, v overlaps with z if and only if v overlaps with x or v overlaps with y.
: if a six-tuple (M, l, s, kb, l1, t) is an element of MStep, then for all hh and v, if there exists a k such that the six-tuple (M, l, s, k, hh, v) is an element of Exec, it implies that there exists an n such that the six-tuple (M, l1, t, n, hh, v) is also an element of Exec.
In the protocol AODV, for any instance 'i', if the label 'l' is in the range from PRrep-:4 to PRrep-:6, then the destination IP 'dip' is in the set of valid destination IPs 'vD' of the routing table 'rt'.
The lemma assumes that the pair (s, s') is in the restricted relation R. It then shows two things: 1. The pair formed by the stacks of s and s' is in the list relation, which is the identity relation cross the relation where U minus R equals U'. 2. The intersection of the visited set of s' and R is empty.
Executing the 'lookup' function on a heap 'h' with reference 'r' in the Heap Monad will return a pair consisting of the value at reference 'r' in the heap 'h' and the heap 'h' itself.
The composition of a parallel converter "conv1" with another parallel converter "conv2", followed by a swap operation, is equivalent to a swap operation followed by the composition of "conv2" with "conv1".
The lemma named "content_ff_normalize_content_ff_1" assumes that polynomial "p" is not equal to zero. It shows that the content of the field finite (ff) of the normalized content of the field finite (ff) of "p" is equal to 1.
"x", which is a pair of elements of any type 'a, belongs to the universe set.
The lemma R₀_in_hom shows that R₀ is in the hom-set from the identity map on the source of r.s₀ to the identity map on the source of r in the category of maps.
If "s" can be transformed into "s'" through zero or more beta-reductions, then the abstraction of "s" with respect to type "T" can also be transformed into the abstraction of "s'" with respect to the same type "T" through zero or more beta-reductions.
If the following conditions are met: 1. The list obtained by mapping the first element of each pair in list l0 is distinct (i.e., it contains no duplicates), 2. The map m is equivalent to the map of list l0, 3. The list obtained by mapping the first element of each pair in list l0 is sorted, 4. The iterator iti is ...
The nth power of the function T is measurable from the tail algebra of (m + n) to the tail algebra of m.
The lemma named "valid_abss_reps" assumes that the configuration 'cfg' is a valid configuration in the probabilistic timed automata. It shows that the abstraction of the representative state of the configuration 'cfg' is equal to the state of the configuration 'cfg'.
For any element 'x' of type 'a', the condition 'P' is infinitely often true if and only if 'P' is true and the universal set of type 'a' is infinite.
The set Z at indices k and n is a subset of the set X at index k.
If a program P with thread t transitions from state sigma to state sigma' with thread action ta, and a new thread t' with object x and memory m is in the set of thread actions of t, then the memory m is equal to the first element of the second part of the state sigma'.
If 'x' is a vector, then the negative of 'x' is also a vector.
The lemma named "contains_star" is defined with fixed variables s, r, and rs. It assumes that r is contained in s. It then shows that r is also contained in the result of the operation star applied to s and rs.
If 'f' is a polynomial function, then the function that maps 'x' to 'c' times 'f(x)' is also a polynomial function.
If 'f' and 'g' are both injective integral domain homomorphisms, then the composition of 'f' and 'g' (i.e., 'f' followed by 'g') is also an injective integral domain homomorphism.
For any real number 'x', given that 'x' is greater than or equal to 0 and less than or equal to 1/2, the value of '-x - 2x²' is less than or equal to the natural logarithm of '(1 - x)'.
The lemma named "Subst_Raise" shows that if "v" is not equal to the sharp symbol, "d" is less than or equal to "m", and "m" is less than or equal to the sum of "n" and "d", then the substitution of "m" with "v" in the raise of "d" to the successor of "n" in "t" equals the raise of "d" to "n" in "t".
In the context of a transformation morphism, the lemma shows two things: 1. The domain application of the functor diagram, when applied to the arrow of the transformation functor, equals the map of the functor F. 2. The codomain application of the functor diagram, when applied to the arrow of the transformation funct...
For a matrix "A" of type 'a (where 'a is a commutative semiring), the multiplication of "A" with a vector "x" is equal to the sum of the product of the i-th element of "x" and the i-th column of "A", where "i" ranges over all elements in the universal set of type 'n.
The lemma named "prod_prime_subset" makes the following assumptions: - Sets A and B are finite. - For any element x in set A, x is a prime number. - For any element x in set B, x is also a prime number. - The product of all elements in set A divides the product of all elements in set B. Under these assumptions, the le...
If the function "wadjust_erase2" is applied to the variables "m", "rs", and the pair "(c, [])", then two conditions hold. The first condition is that if "c" is an empty list, then the function "wadjust_on_left_moving" applied to "m", "rs", and the pair "([], [Bk])" is true. The second condition is that if "c" is not an...
For a given point 'a' in a Euclidean space, if the open ball centered at 'a' with radius 'r' is a subset of set 'T', and set 'T' is a subset of the closed ball centered at 'a' with radius 'r', then set 'T' is convex.
The lemma named "not_in_segment1" assumes that "y" is in the orbit of "x" under function "f". It shows that "x" is not in the segment from "x" to "y" under function "f".
The lemma named "setops\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst'" assumes that "a" is not a negative check. It shows that the set of operations on the stateful strand step "a" substituted by "θ" is equal to the set of operations on the stateful strand step "a" itself, which is then substituted by "θ".
An element 'x' belongs to the Cartesian product of 'I' and 'X' if and only if the bounded function applied to 'x' belongs to the Cartesian product of 'I' and 'X'.
If set A is finite, then the sigma finite measure of the count space of A is also finite.
In the context of a normal subgroup, the lemma named "inv_FactGroup" assumes that "X" is an element of the factor group "G Mod H". It shows that the inverse of "X" in the factor group "G Mod H" is equal to the set inverse of "X".
The theorem named "Hintikka_counter_model" assumes a set S is a Hintikka set. It shows that if a proposition 'p' is in set S, then 'p' is not modeled by S, and if the negation of 'p' is in set S, then 'p' is modeled by S.
The lemma named "tagged_division_finer" is defined with a fixed parameter 'p', which is a set of pairs, each consisting of a point in an Euclidean space and a set of such points. It assumes that 'p' is a tagged division of the closed box defined by points 'a' and 'b', and that 'd' is a gauge. It then concludes the exis...
If there is a single step of reduction in the thread "t" from the state "(e, xs)" to the state "(e', xs')" under the program "P" and the heap "h", then there is also a single step of reduction in the same thread from the state "(e.F{D} := e2, xs)" to the state "(e'.F{D} := e2, xs')". Here, "F{D}" represents a field "F"...
The abscissa of the point zero is zero.
Executing a serial plan in SAS+ after removing no-operations is equivalent to executing the original serial plan in SAS+.
The trace in a state space 'ss' at time 't' is equal to the least fixed point of a function 'F' at state 's'. This function 'F' is defined for a pair consisting of a time 't'' and a state 's'', and a possibly infinite sequence 'ω'. If 't' is less than 't'', then the state 's' is in the state space 'ss'. Otherwise, the ...
If the heap node context of "b_assn A P" for x and y is true or the heap node invalidity of A for x and y is true, then the heap node invalidity of A for x and y is true.
Swapping the elements of pairs produced by the infinite zipping function "i_zip" with functions "g" and "f" is equivalent to applying the infinite zipping function to "f" and "g" in that order.
The counterclockwise orientation of points a, a+b, and a+c is equivalent to the counterclockwise orientation of points 0, b, and c.
Given the assumption A, where list 'xs' interleaves with the set containing lists 'ys', 'zs', and predicate 'P', if for all 'n' less than the length of list 'ws', predicate 'P' holds true for the 'n'th element of 'ws' and the concatenation of the list obtained by dropping the first 'n+1' elements of 'ws' and 'xs', then...
The configuration on state 's' is not empty.
There exists sets S and T such that both S and T are finitely satisfiable, and it is not the case that if S is a subset of T, then the extension of S is a subset of the extension of T, where T is a set of countable formulas.
The lemma named "absolutely_integrable_on_image_real" is defined with two functions, "f" and "g". "f" is a function from a real vector space of dimension 'n to real numbers, and "g" is a function from a real vector space of dimension 'n to another real vector space of dimension 'n. The lemma assumes two conditions. F...
The union of the image of a function 'f' over a set 'M' is equal to the bind of the set 'M' and the function 'f'.
If a program P with thread t transitions from calling method M with arguments vs on object a in heap h, producing thread action ta and resulting in value va and heap h', and if a new thread t' with exception ex and heap h'' is in the set of thread actions ta, then heap h'' must be equal to heap h'.
For any heap 'h' of type 'heap_sep_algebra', if 'p7' implies 'p4' and 'p6' implies 'p1', then if 'p4' implies 'p1' implies 'sep_empty' and 'p1' and 'p4', this implies that 'p1' implies 'p4' implies 'p4' and 'sep_empty' and 'p1', and 'p7' implies 'p6' implies 'p1' and 'p4'.
The inverse of the cancel1 function is semi-confluent.
If a list "tss" is a flip alternate sublist chain, then the list obtained by mapping the function that adds the element "t" to the beginning of each sublist in "tss" is also a flip alternate sublist chain.
If "x" is in the field "r", then "zero" is in the set of elements that are less than "x".
The lemma named "density_collapse_carrier" assumes that the number of rows in matrix R is greater than 0, and both matrices P and R are n by n matrices. It shows that the result of the function "density_collapse" applied to matrices R and P is also an n by n matrix.
If 'r' is a well-order and 'f' is an injective function on the field of 'r', then the order type of the image of 'r' under the function 'map_prod f f' is the same as the order type of 'r'.
The lemma "ms_of_greek_elem" assumes that the pair (x,xs) is an element of the multiset derived from the list "ys". It shows that "x" is an element of the list "ys".
The evaluation of the function F and G with the sets {Nb,Na}, {Pax}, {Pa}, and {Na,Pb,Naa} under the context of resolution in first-order logic is not true.
If you take the first 'n' elements from a list 'f' and append it with the remaining elements of the list 'f' after dropping the first 'n' elements, you will get the original list 'f'.
The lemma named "nf_rules_unique" assumes that the function "f" applied to "qs" leads to "q" and is in the set of normal form rules for "R" and "F", and that the same function "f" applied to "qs" leads to "q'" and is also in the same set of normal form rules. The lemma then shows that "q" must be equal to "q'".
The lemma named "diff_col_ex" shows that there exists a point C such that C is not equal to A and C is not equal to B, and points A, B, and C are collinear.
The space of the quasi-Borel space derived from the quasi-Borel probability space of a set 's' is equal to the space of the quasi-Borel probability measure of the set 's'.
If F is equal to the parallel composition of F1 and F2, then the composition of N and F followed by function f is equal to the parallel composition of the composition of N and F1 followed by function f, and the composition of N and F2 followed by function f.
If matrix A is diagonal, then the matrix resulting from applying the diagonal_to_Smith function to A with bezout as a parameter, is also diagonal. This is assumed under the condition that A is diagonal.
The theorem named "preimpsplit" assumes two conditions: 1. If I and N are true, then Q is true. 2. If I is true and the variable v remains unchanged, then Q is true. Based on these assumptions, it shows that if I is true and N is true in the context of v, then Q is true.
If the Basic Security Domain (BSD) is applicable for a modified view 'V' and a set of traces 'Tr' in an event system 'ES', and if this modified view 'V' is defined as the union of the original view 'V' and a set 'N' in 'V', with 'N' being an empty set and 'C' being a subset of 'V', then the Strong Security Domain (SD) ...
If 'a' is an element of the ring 'R', then the product of 'a' and the 'n'-th power of the polynomial 'X_poly R' is an element of the univariate polynomial ring over 'R'.
If specification 7 holds true for a property p, then specification 6 also holds true for the same property p.
If an integer 'i' is not equal to 'j', a natural number 'a' is not equal to 'b', 'a' is less than 2, and 'b' is less than 2, then the sum of 'a' and 'b' is less than or equal to the natural number equivalent of the maximum absolute value between 'i' and 'j'.
For a fixed formula A, the symbol 's' is fresh for the quotation of A.
If property P holds for a binary value 'a' and also holds for '1-a' (which represents the binary complement of 'a'), then property P holds for any binary value 'c'.
"pap" is equivalent to "pap" with some exceptions in "D".
If the af-nu function of X applied to the pair (\<xi>, \<zeta>) and w equals the pair (af function of \<xi> applied to w, nu function of X applied to af function of \<xi> applied to w), then the af-nu function of X applied to the pair (\<xi>, \<zeta>) and the concatenation of w and w' equals the af-nu function of X app...
If a subset system V B is a subdesign of another subset system V' B', then the image of V under a function pi and the image of B under the block image function with pi, form a subdesign of another subset system V'' B''.
An empty list is equivalent to the trace "O tr" if and only if "gamma" never occurs in the trace "tr".
The Lebesgue measure of a ball with center c and radius r is finite.
If a set with an inserted element 'x' into 'xs' has disjoint directed arcs, then the original set 'xs' also has disjoint directed arcs.
If the function "find_adds" applied to the set "fs" and the element "u" returns "None", and "f" is an element of the set "fs", then the following holds: if "f" equals zero, then the thesis is true; if "f" does not equal zero and the leading term of "f" does not add to "u", then the thesis is also true.
The unit factor of the finite greatest common divisor of A is equal to the boolean value of whether the finite greatest common divisor of A is not equal to zero.
The lemma named "vint_uminus_transfer" which includes lifting syntax, demonstrates that the function "vint_uminus_app" when applied to the relation "cr_vint" results in the same output as the unary minus function "uminus" when applied to the same relation. This is a transfer rule.
The operations "init_nsc", "assert_nsc", "check_nsc", "solution_nsc", "checkpoint_nsc", "backtrack_nsc", "weak_invariant_nsc", "invariant_nsc", and "checked_nsc" are all part of the "Incremental_NS_Constraint_Ops" in the Simplex Incremental namespace.
If a triple (c,w,c') is in the transitive closure of the non-terminal relation of the flowgraph, and the intersection of the write-write memory conflict of the flowgraph w and the memory conflict of the flowgraph cn is empty, then the triple (c+cn,w,c'+cn) is also in the transitive closure of the non-terminal relation ...
If the valuation functions of omega and omega prime agree on the free variable function of phi, then omega is in the semantic formula of phi if and only if omega prime is also in the semantic formula of phi.
The lemma named "mat_rep_num_sum_alt" is defined with a fixed matrix 'M' of a type that is a ring. It shows that if the elements of matrix 'M' are a subset of the set {0, 1} and 'i' is less than the number of rows in matrix 'M', then the natural number representation of matrix 'M' at index 'i' is equal to the sum of th...
Zero is included in the set of non-positive integers.
The previous element of a function 'f' at the position 'Suc n' (successor of n) in the initialized list is equal to the function 'f' at position 'n'.
The product of the successor function applied to the set of numbers from 0 to n (exclusive) is equal to the factorial of n.
If the product of x and y equals the identity element, and both x and y are elements of the group of units G, then the product of y and x also equals the identity element.
The lemma named "summable_on_scaleR_left" is about a fixed element 'c' which is a real normed vector. It assumes that if 'c' is not equal to zero, then the function 'f' is summable on 'A'. It shows that the function 'f' multiplied by 'c' is also summable on 'A'.
The lemma for the initialization of the second invariant in the Prim's algorithm is valid for the initial queue and the initial predecessor function.
If "g" is a function from the nth power of the p-adic number field to the p-adic number field, and "n" is less than "k", then the partial pullback of "k" applied to the composition of "g" and the "take n" function at "m" and "S" equals the split Cartesian product of "n + m", "k - n", "n", the partial pullback of "n" ap...
If A is a set in the measure space M, then the nonnegative integral of the function (c times the indicator function of A) with respect to M is equal to c times the measure of A in M.
A match does not occur for a given packet and abstract firewall when the match is not for multiple ports, if and only if a match does not occur for the source ports and a match does not occur for the destination ports of the packet and the abstract firewall.
The states of the Audio Player Control are equal to the states of the Audio Player Control.
"eq_o_i" is equal to a single predicate.

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