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Statement: stringlengths 7 24.3k
The identity element of the reduced homology group of X at index p is equal to the identity element of the homology group of X at index p.
If "DenyAll" is an element in the set converted from the policy list "p", then the result of the function "normalize_manual_orderPrQ" with inputs "p" and "l" cannot be an empty list.
The lemma assumes "B" is true. It shows that "B" is true and either "True" or "False" is true.
The limit of the function 'f' as it approaches 'a' from the left in the filter 'F' is equivalent to the limit of the function 'f' with its argument negated as it approaches the negation of 'a' from the right in the same filter 'F'. This is applicable for 'a' which is a real number.
If a list "xs" is in the set of Joints "XS", and for all elements "Y" in the list "XS", the intersection of the pdevs_domain of the second element of "X" and the pdevs_domain of the second element of "Y" is empty, and if "x" is in the Affine set of "X", then the list consisting of "x" followed by "xs" is in the set of ...
If "x" is a test, then the converse of "x" is less than or equal to 1.
For any complex number 'z', the sine of 'z' equals -1 if and only if there exists an integer 'n' such that 'z' equals the complex number derived from the real number (2 times 'n' times pi) plus 3/2 times pi.
If a pair (l, r) is in the set R, and if s is equal to the context C applied to the term l substituted with sigma, and if t is equal to the context C applied to the term r substituted with sigma, then the pair (s, t) is in the reduction step of R.
The lemma assumes that if "a" is true, then "b" is true, if "a" is not true, then "c" is true, and "d" is true. It shows that if "a" is true then "b" is true, otherwise "c" is true, and in both cases "d" is true.
The lemma named "swap_continuous" assumes that the function 'f' is continuous on the closed interval from (a,c) to (b,d). It then shows that the function 'f' is also continuous on the closed interval from (c,a) to (d,b) when the inputs 'x' and 'y' are swapped.
If the following conditions are met: 1. The encrypted message containing the number T3 with the server key is part of the events known to the spies. 2. The encrypted message containing the server key, agent B, the number Ts, and the server ticket with the authentication key is part of the events known to the spies. 3....
If the result of the function ins3 with inputs n, d, x, t is Unbal2 with outputs t', n', h', and the balance condition bal_i n (height t + d) holds, then the result of the function T_ins3 with inputs n, d, x, t is less than or equal to two times the size of t plus one plus the height of t.
If a function 'f' is surjective on set 'A' to set 'B', and set 'A' is a subset of set 'A'', then the function 'f' is also surjective on set 'A'' to set 'B'.
If a list 'xs' contains distinct elements and the set of elements in 'xs' is equal to 'X', then the index of 'xs' in 'X' is equal to the set of all numbers from 0 to the size of 'xs' minus one.
The function 'msort_monoms_impl' composed with 'return', when compared to 'sort_coeff', is in a Kleene-star relation from 'monom_assn' to 'monom_assn'.
The result of applying the merge sort algorithm on a list 'xs' is a sorted list.
Two lists are equal when mapped with functions f and g over the same list xs if and only if for every element x in the set of xs, applying function f to x equals applying function g to x.
A flow in the network Delta is zero.
The lemma named "wt_ge_0" assumes that "c" is proper and "i" is less than the branching factor of "c". It shows that the weight of "c" at state "s" and index "i" is greater than or equal to 0.
The lemma "SN_trancl_imp_SN" assumes that if a relation "R" is strongly normalizing in its transitive closure, then "R" is strongly normalizing.
"n" is less than or equal to the next element in the interval "I".
The lemma named "stake_preimage_measurable" is given a variable "x" of type "bool stream". It assumes that the successor of 0 is less than or equal to "n" and that "M" is equal to a Bernoulli stream with probability "p". It shows that the set of all "w" in the space of "M" where the stake of "w" at "n" is equal to the ...
: given a category 'CC' and an index set 'I', if 'a' is an arrow in 'CC' and 'i' is an element of 'I', then the 'i'-th component of the arrow map of the digraph morphism 'dghm_up' applied to 'a' is equal to the arrow map of the morphism 'phi' applied to 'i' and then to 'a'.
The function "obtain_sorted_carrier'''" is refined by the function "obtain_sorted_carrier''" in the context of the non-deterministic result monad, with the size relation being added.
The lemma named "bi_unique_union_l" assumes the following: 1. "T" is a bi-unique relation. 2. There is a relation "T" between set "a" and set "a'". 3. There is a relation "T" between set "b" and set "b'". 4. There is a relation "T" between set "xl" and the union of sets "a'" and "b'". Under these assumptions, it shows...
The lemma named "to_dtree_aux_self" assumes that a node "r" with a finite set "xs" is equal to the result of the function "to_dtree_aux" applied to "r". It also assumes that the pair "(y,e)" is an element of the finite set "xs". Under these assumptions, it shows that "y" is equal to the result of the function "to_dtree...
The Dickson lexicographic order of a power product "s" is reflexive with itself.
If the function "sig_red_single" is applied to the single regular sequence "sing_reg", the top tail "top_tail", the polynomials "p", "q", the polynomial "f", and the term "t", then the linearly ordered power product "top_tail" is less than or equal to its relation.
The lemma introduces SbisL. It assumes that the lengths of list cl and list dl are equal. Furthermore, for any integer n, if n is less than the length of both lists, then the nth element of list cl is approximately equal to the nth element of list dl. If these conditions are met, it shows that cl and dl are SbisL.
"r_inconsist" is a recursive function of two variables.
The lemma "I_lift_nnf_qe_normal" assumes the following conditions: 1. For any formula 'phi', if 'phi' is quantifier-free, then the quantifier elimination of 'phi' is also quantifier-free. 2. For any formula 'phi', if 'phi' is quantifier-free and normal, then the quantifier elimination of 'phi' is also normal. 3. For a...
If a property P holds for the first element of the initial state, and if for any state s, query q, and action a, the property P holding for state s implies that P holds for the state resulting from the step function applied to s, q, and a, then the property P holds for the first element of the configuration resulting f...
If 'k' is an element of set 'I', then the minimum integer of set 'I' is also an element of 'I' and is less than or equal to 'k'.
If the frame stack "frs" is conformant with the program "P", heap "h", bytecode verifier state "Φ", method "M", natural number "n", and return type "T\<^sub>r", and heap "h" is a sub-heap of heap "h'", then the frame stack "frs" is also conformant with the program "P", heap "h'", bytecode verifier state "Φ", method "M"...
The lemma "Gets_imp_Says" implies that if "B" gets "X" from the set of events "evs" and "evs" is in "otway", then there exists an "A" such that "A" says to "B" about "X" in the set of events "evs".
The composition of a formal Puiseux series 'f' with the power of 1 is equal to 'f' itself.
The lemma named "unique_periodic_arithmetic_extension" assumes the following conditions: 1. k is greater than 0. 2. For all j less than k, the function g at j equals the function h at j. 3. Both g and h are periodic arithmetic functions with period k. Under these assumptions, it shows that the function g at any point ...
If "A" is logically equivalent to "A'" under the hypothesis "H", and for all "C" in "H", the atom "i" is independent of "C", then "H" entails that the existence of "i" in "A" is logically equivalent to the existence of "i" in "A'".
The bitwise exclusive OR (XOR) operation between negative 1705 and negative 42, both interpreted as integers, results in 1665.
Any numeral "i" is in the set of floating point numbers.
The lemma named "invariant_next_subseqs" assumes two conditions: 1. The "invariant" condition holds for the variables "base", "elements", "n", and "state". 2. The function "next_subseqs" applied to "state" results in a pair "(out, state')". Under these assumptions, it shows that the "invariant" condition still holds ...
If a program P satisfies a configuration sequence cfs, then P also satisfies the configuration sequence obtained by dropping the first n elements from cfs.
A sequence 'u' followed by 'v' equals to an element 'a' followed by a sequence 'w' if and only if either 'u' is an empty sequence and 'v' equals to 'a' followed by 'w', or there exists a sequence 'u'' such that 'u' equals to 'a' followed by 'u'' and 'u'' followed by 'v' equals to 'w'.
The layout of the concatenation of the list 'ap' and the list containing 'p' is equal to the concatenation of the layout of 'ap' and the list containing the length of 'p'.
The time bound of the "swap" operation in an array, represented as "Array_Time.swap i x a", is 2.
Taking the first 'n' elements from the list 'xs' equals the entire list 'xs' if and only if the length of 'xs' is less than or equal to 'n'.
The upward closure of the function 'Unf' applied to the abstraction of 'phi' is equal to the abstraction of the function 'remove_and_or' applied to the simplified 'Unf' of 'phi'.
If "s" is an element in the set of arguments of "t", then "t" embeds "s".
The operation "\<Sqinter>" (infimum or greatest lower bound) is associative. That is, for any three elements a, b, and c, the infimum of a with the infimum of b and c is equal to the infimum of the infimum of a and b with c.
For all rules 'r', if 'r' is in the set 'R', then 'r' is persistent.
If "k" is less than or equal to "m", then "k" minus "n" is also less than or equal to "m". This is applicable for "k", "m", and "n" as extended natural numbers.
The lemma of reachable states assumes that state 'q' is reachable from state 'p' and there is a path from state 'r' to state 'q'. It shows that the set of states from 'r' to 'q' is a subset of the states reachable from 'p'.
If a function F leads from set A to the union of sets B and A', and also leads from set B to set B', then it follows that function F leads from set A to the union of sets B' and A'.
If "L" is less than or equal to "x", and the product of "x" and "y" union "z" is less than or equal to "x", then the product of "z" and the transitive closure of "y" is less than or equal to "x".
In the context of a subgroup, if the carrier of group G is finite, then the cardinality of subgroup H is greater than zero.
A well-formed policy, in a strong sense, is maintained when a deny rule is inserted after removing shadow rules from policy p.
If M is a sigma-finite measure and A is a set in M, then the sigma-finite measure of the restricted space of M to A is also a sigma-finite measure.
The imaginary part of the signum of a complex number z is equal to the imaginary part of z divided by the modulus of z.
The restriction of a relation 'r' to a set 'A' is a subset of the relation 'r'.
If the pair (ra, rb) is in the run relation R, then the composition of a.L and ra is equal to the composition of b.L and rb.
The private key of x is not in the image of the shared key function over the set AA.
The function "replace_groups" applied to a list "xs" and an element "e" is equivalent to applying a fold function. This fold function takes a function as its first argument, which itself takes a pair (id, t) and an accumulator as arguments and applies the "replace_transition" function to the accumulator, id, and t. Thi...
If 'p' is an element of the carrier 'P', then the product of 'p' and 'X' raised to the power 'n' (where 'n' is a natural number) evaluated at 'n + k' equals 'p' evaluated at 'k'.
The lemma "depth_poss_split" assumes that the successor of the sum of the depth of the term of a general term 't' and a number 'n' is less than the depth of the term of a general term 'u'. It shows that there exist two sequences 'p' and 'q' such that 'p' followed by 'q' is in the general positions of 'u', 'n' is less t...
If for every element 'j' in set 'I', 'aGroup (A j)' holds true, and 'X' is an element of the direct sum of the additive group 'A' over 'I', then the product of the multiplicative operation 'mOp' over 'I' and 'A' and 'X' is also an element of the direct sum of the additive group 'A' over 'I'.
If x is greater than or equal to -1, then the Lambert W function of the product of the exponential of x and x itself equals x.
The lexical value of a regular expression 'r' repeated 'n' times in a string 's' is a subset of the concatenation of the lexical value of the star of 'r' in 's' and the union of the lexical values of 'r' repeated 'i' times for all 'i' less than or equal to 'n'.
A property P holds for the combination of a function f and a list of elements xs if and only if the property P holds for the function f and for all elements in the list xs.
If the context "G" is valid under the "n" semantics and the call "A" is also valid under the "n" semantics, then the union of "A" and "G" is also valid under the "n" semantics.
The lemma named "finite_reach_t" assumes that the reachability set of the initial state q0 under the transition function δ and the set of states Σ is finite. It also assumes that the set of states Σ is finite. Under these assumptions, it shows that the reachability set of the initial state q0 under the transition funct...
If for every index 'i' less than the length of list 'T', the 'i'-th element of list 'S' is in the set resulting from applying the function 'b' to the 'i'-th element of list 'T', and if the lengths of lists 'T' and 'S' are equal, then the function 'b' applied to list 'S' is in the set resulting from applying the functio...
The inverse of permutation 'p' at 'y' equals 'x' if and only if permutation 'p' at 'x' equals 'y'.
The lemma named "ground_sys_nf_eq_lift" is defined with a fixed term relation 'R'. It assumes two ground systems 'R' and 'S', and that the normal form of the ground system rewrite step of 'R' is equal to the normal form of the ground system rewrite step of 'S'. It then shows that the normal form of the simple rewrite s...
The supremum of the product of complex conjugate subspaces A and B and the product of complex conjugate subspaces C and D is equal to the product of the supremum of A and C and the supremum of B and D.
The lemma named "one_outcome_card_lotteries_1" assumes that the cardinality (or the number of elements) of "outcomes" is 1. It shows that the cardinality of the set of all lotteries on "outcomes" is also 1.
If there exist polynomial coefficients 'f' and 'g' for 'n' and 'm' in the set 'S' respectively, then there exists a polynomial coefficient 'e' for the sum of 'n' and 'm' in the set 'S' such that 'e' of the sum of 'n' and 'm' equals the product of 'f' of 'n' and 'g' of 'm' in the ring 'S'. Furthermore, the polynomial ex...
The lemma named "linear_lower" is about a real number 'x'. It assumes three conditions: 1. For any 'x' in the range from 'a' to 'b', the function 'f' has a field derivative 'f' prime at 'x' within the same range. 2. For any 'x' in the range from 'a' to 'b', 'f' prime of 'x' is less than or equal to 'u'. 3. 'x' is...
The less than or equal to operation is equivalent to a function called "subclass1_fun" applied to two variables x and y, or x is equal to y.
For any integer 'x', the bitwise AND operation between 1 and 'x' is equal to the remainder of 'x' divided by 2.
A discrete topology U is equivalent to a topological space X if and only if the topological space X is equal to U and the derived set of U in X is empty.
If a tree "t" satisfies the tree maximum invariant and "t" is not a tip, then the tree that results from deleting the minimum element from "t" also satisfies the tree maximum invariant.
The set difference between the set of all numbers from 0 to the length of dims0 (exclusive) and the set vars2' is equal to the intersection of the set of all numbers from 0 to the length of dims0 (exclusive) and the set vars1'.
Zero, when considered as a real number, is an integer.
The length of an array remains the same even after setting a value at a specific index in the array.
The lemma named "union_nodes_finite" assumes that the sets of nodes in deterministic co-Büchi automata A and B are finite. It shows that the set of nodes in the union of automata A and B is also finite.
If 'z' is less than the least 'b' such that 'f' applied to 'x' and 'b' is true, then 'f' applied to 'x' and 'z' equals zero.
If for any elements 'a', 'b', and 'c' such that 'a' is in set 'A' and both pairs '(a, b)' and '(a, c)' are in relation 'r', then the pair '(b, c)' is in the join of relation 'r', it implies that the Weak Church-Rosser property holds on relation 'r' for set 'A'.
If "ram1" comes before "ram2" in the list "vs" and the list "vs" contains distinct elements, then the first element of the pair resulting from splitting the list "vs" at "ram2" and then splitting the resulting first part at "ram1" is equal to the first element of the pair resulting from splitting the list "vs" at "ram1...
An integer x is equal to an integer y if and only if for all natural numbers n, the nth bit of x is equal to the nth bit of y.
In the context of a vector space, the vector space V is a subspace of itself.
The lemma named "order_prod" makes two assumptions: 1. For any element 'x' in set 'A', the function 'f' applied to 'x' is not equal to zero. 2. For any two distinct elements 'x' and 'y' in set 'A', the function 'f' applied to 'x' and 'y' are coprime (i.e., their greatest common divisor is 1). Given these assumptions, ...
The lemma named "independent_explicit" shows that a set B is independent if and only if B is finite and for all scalars c, the sum of the product of c and vector v for all v in B equals zero implies that for all v in B, c equals zero.
A node is a leaf in the Modified Prefix Tree (MPT) if and only if the mapping 'm' is empty.
If the "dec_left_move" function is applied to the tuples "(as, am)" and "(s, l, r)" with "ires" as an additional parameter, then the list "l" cannot be empty.
The lemma named "semIntAbs_prSwapAbs" assumes that "SEM" is a well-sorted semantic domain. It shows that the function "prSwapAbs" applied to the interpretation of an abstraction in the semantic domain "SEM" is indeed in "SEM".
Lifting a natural number function, where the function multiplies an input 'n' by 'm', is equivalent to lifting the natural number function first and then multiplying the result by the integer 'm'.
The lemma named "Natural Transformation Composition Definition" assumes that natural transformation eta1 is naturally isomorphic to eta2 and that X is an object in the domain category of eta1. It shows that the application of eta1 to X is naturally isomorphic in the codomain category of eta1 to the application of eta2 ...
Setting the 0th bit of a number 'a' is equivalent to adding 1 to twice the integer division of 'a' by 2.
"level1 is not a data source for sA12".
The composition of the bottom element and any set S is equal to the bottom element.

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