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Statement: stringlengths 7 24.3k
For any type 'a that is a finite meet complete partial order (cpo), the meet of 'r' and the meet of 'r' and 'x' is equal to the meet of 'r' and 'x'.
If "v" is a pointer in the heap "h", then "v" is either null or there exists an address "a" such that "v" equals to this address and there exists a "Cfs" for which the heap at address "a" equals to "Cfs".
For all elements 'y' and all lists 'ys', 'x2' is not equal to the list formed by prepending 'y' to 'ys' if and only if 'x2' is an empty list.
The most significant bit of a word 'a' is true if and only if 'a' is greater than or equal to 2 raised to the power of (the length of 'a' minus 1). Here, 'a' is of any word length.
The empty set is semialgebraic in n dimensions.
The lemma "InvariantNoDecisionsWhenConflictEnsuresCurrentLevelCl" assumes the following conditions: 1. The invariant "InvariantNoDecisionsWhenConflict" holds for the formula F, model M, and the current level of M. 2. The clause is an element of the formula F. 3. The clause is false under the model M. 4. The elements o...
The lemma named "commutativity2_lemma" assumes the following: 1. For any action 'act' and states 's' and 't', if 'act' is in the set of actions of 'F', 's' is in 'T', and the pair '(s, t)' is in the reflexive and transitive closure of 'L', then either 's' is in 'B', 't' is in 'B', or the pair resulting from applying '...
The operation "or" is commutative, meaning that the order of the operands does not change the result. In other words, "X or Y" is equal to "Y or X".
If the evaluation of a list, under the condition that the evaluation of an environment, a state, and an expression results in a result and that if the environment is a cupcake environment then the expression is a cupcake expression which leads to a single cupcake evaluation, then if the environment is a cupcake environ...
If there is a beta transition from state "l, Z" to state "l', Z'" in system "A", and if "Z" is a subset of "V", then "Z'" is also a subset of "V".
The function "of_int" applied to the result of the function "int_of_integer" on any integer 'k', is equal to 'k' itself.
"coalg" holds for the parameters "carT1", "carT2", "strT1", and "strT2".
Binding a function "f" with "ureturn" is equivalent to the function "f" itself.
For any integers 'a', 'b', and 'd', if 'd' is greater than or equal to zero and 'd' divides both 'a' and 'b', and for all 'e' such that 'e' divides both 'a' and 'b' it also divides 'd', then 'd' is equal to the greatest common divisor of 'a' and 'b'.
The function "last_H" is measurable with respect to the sigma-algebra S and the counting measure on the universal set.
If a sequence 'f' forms a descending chain of subgroups of 'G' of length 'Suc n' (successor of n), then the composition series of 'G' at 0 with the constant map that maps 0 to the successor of n in 'f' is well-defined.
The extension of a set function 'h' over the union of a set 'A' is equal to the union of the extension of the set function 'h' over each element 'X' in the set 'A'.
A monoid is equivalent to a monoid over the universal set.
For any table 'tab' and any list 'qs', if the length of list 'ps' is equal to the length of list 'qs', then updating the table 'tab' with the list 'ps' mapped to the list 'qs' appended with 'zs' is equal to updating the table 'tab' with the list 'ps' mapped to the list 'qs'.
The lemma named "not_in_interior_convex_hull" is about a set S in a Euclidean space. It assumes that S is finite, the cardinality of S is less than or equal to the successor of the dimension of the Euclidean space, and x is an element of S. It shows that x is not an element of the interior of the convex hull of S.
If the heap 'h' at 'obj_id' is equal to 'Some obj', then updating the slots of the heap 'h' at 'obj_id' with 'obj' will result in 'obj'.
If 'a' is a real number, then 'a' raised to the power of 'n' is also a real number. This is applicable for 'a' which is of type 'real_algebra_1'.
If S and T are arrows in a set with alpha elements, and if x is an element in the domain of the arrow value of T, and the arrow value of T at x is an element in the domain of the arrow value of S, then the arrow value at x of the composition of S and T is equal to the arrow value of S at the arrow value of T at x.
The lemma named "det_nfa_accepts" assumes that all elements in the list 'w' are alphabets of 'n'. It shows that the deterministic finite automaton (DFA) accepting 'w' when determined by non-deterministic finite automaton (NFA) 'A' is equivalent to the NFA 'A' accepting 'w'.
The lemma named "errMOD_igFreshIGOpSTR" is fixed for a model "MOD" that consists of various parameters such as 'index', 'bindex', 'varSort', 'sort', 'opSym', 'var', 'gTerm', and 'gAbs'. Assuming that the model "MOD" preserves the well-sortedness of operation symbols in the interpretation of ground terms and that the mo...
If the list "zs" is not empty, then the firewall policy resulting from the concatenation of "x" and the list consisting of "y" followed by "z" (when converted to a firewall policy), when applied to "p", is equal to the firewall policy resulting from the concatenation of "x" and the firewall policy converted from the li...
If the intersection of the network IP addresses of two subnets s1 and s2 is empty, and for every IP address in the subnet of s1 and s2, the network map of the subnet of s1 and s2 at that IP address equals to a function sigma at that IP address, then for every IP address in the subnet s1, the network map of s1 at that I...
If a tree "t" is complete, then the size of the tree "t" is equal to 2 raised to the power of the height of the tree "t" minus 1.
The lemma named "replicate_eq_map" assumes that the length of the list "xs" is "n" and that applying the function "f" to any element "y" in the list "xs" results in "x". It then shows that creating a list of "n" repetitions of "x" is equivalent to applying the function "f" to every element in the list "xs".
, given the assumption that "ps" is true or the type 'a is a scalar in the context of natural term comparison, the natural term order equality between the power-of-two function applied to "to1" and "to2" with parameters "dg" and "ps" is equivalent to the natural term order equality between "to1" and "to2" with the same...
The lemma named "set_glob_var_glob_bit_lift" assumes that setting a global variable in a certain state with a given value results in a successful operation, producing a new state. It then shows that the capability associated with this operation has global permissions.
The segments of an affine form "X" are not equal to a singleton list containing "x".
The lemma "isMinimalBackjumpLevelEnsuresIsNotUnitBeforePrefix" establishes the following: Given a literal trail 'M', a conflict flag, a clause 'c', and a literal 'l', under the assumptions that the elements of 'M' are consistent and unique, the clause 'c' is false under the elements of 'M', the backjump level is minim...
For all elements 'x' in the set 'xs', if a property 'P' holds true for 'x', then filtering 'xs' with 'P' will result in the original list 'xs'.
The lemma named "c_IKKBZ_eq_c_list_nlj" is defined with fixed variables f, cf, and t. It defines "xs" as the reverse order of t. It assumes that the relations in t are distinct, the cards cf and f in t are reasonable, and t is left deep. It shows that the cost function c_IKKBZ with identity function, cf, f, and t is eq...
For a given process 'P' of type 'Proc' and a given 'barb' 'a', the barb 'a' in the process 'P' in the context of the STCalWB calculus is the same as the barb 'a' in the process 'P'.
The lemma named "fresh_star_singleton" establishes that for any fixed atom 'a', 'a' being fresh in the set containing only 'a' is equivalent to 'a' being fresh in 'a'.
If tree "t" is less than or equal to tree "t'", then the paths of tree "t" are less than or equal to the paths of tree "t'".
If the mode use is sound for a given pair of command sequences and memory (cms, mem), and if a variable x is in the guarantee of no read or write of the i-th command sequence, and if i is less than the length of the command sequences, then the i-th command sequence does not read or modify the variable x.
The set of all elements 'a' such that there exists an element 'b' in set 'A' where the ordered pair (a, b) is in relation 'r' is small.
The function "a" of "x" equals 1 if and only if "x" is less than or equal to "Z".
The points of the function "pmap_\<Gamma>" applied to "f" and "\<pi>" are equal to the points of "\<pi>".
If the following conditions are met: 1. evs0 is an event in the Transport Layer Security (TLS) protocol. 2. Agent A notes the session ID (SID), Agent A, Agent B, and a nonce (M) in the event set evs0. 3. Agent B also notes the same session ID (SID), Agent A, Agent B, and nonce (M) in the event set evs0. 4. For all eve...
If the length of the list "y_cl_var_var'_v_list" is equal to the length of the list "y_ty_list", then mapping the first element of the zipped lists "y_cl_var_var'_v_list" and "y_ty_list" is equivalent to mapping the fourth element of the list "y_cl_var_var'_v_list".
If the pair (l1, l2) is in the implication graph of a conjunctive normal form (cnf), then the pair (negate l2, negate l1) is also in the implication graph of the same conjunctive normal form.
If a natural transformation "N" forms a cone from a functor "F" mapping category "A" to category "B" over an object "c", and if there is a functor "G" mapping category "B" to category "C", then the composition of "G" and "N" forms a cone from the composition of "G" and "F" mapping category "A" to category "C" over the ...
Given two sets S and S' in Euclidean spaces, if S is an Absolute Neighborhood Retract (ANR) and S is homeomorphic to S' (i.e., there exists a continuous, bijective map between them with a continuous inverse), and if S' is closed in the topological space U, then there exist two sets V and W such that V is open in U, W i...
If "l" is a heap and "r" is a heap, then the merge of "l" and "r" is also a heap.
If the function "iso_tuple_update_accessor_cong_assist" holds for the update function "upd" and the accessor function "ac", and if the record "r" is equal to the record "r'", and if applying the accessor function "ac" to the record "r'" results in the value "v'", and if for all values "v" that are equal to "v'", applyi...
The set of variables in the variable "Var N" is equal to the set containing only "N".
The lemma named "card_extensional_funcset_not_inj_on" assumes that both sets S and T are finite and the cardinality of set S is less than or equal to the cardinality of set T. It shows that the cardinality of the set of all functions from set S to set T that are not injective is equal to the cardinality of set T raised...
If the nonstop function of a Turing machine 'm' with input 'bl2wc (<lm>)' and state 'stpa' equals zero, then the configuration of the Turing machine 'm' with the same input 'bl2wc (<lm>)' at state 'stpa' is equal to the configuration of the Turing machine 'm' with the same input 'bl2wc (<lm>)' at state 'stpa + n'.
Looking up a value 'v' in the environment 'ρ' restricted by 'r' is the same as looking up 'v' in 'ρ' and then restricting by 'r'.
There exists a substitution, denoted as theta, of type ('f,'v) such that the domain of this substitution is equal to X and the range of this substitution is grounded.
If "Y" is an idempotent scene and "X" is a subscene of "Y", then the operation of adding scene "s2" to scene "s1" on "X" and then adding scene "s3" on "Y" is equivalent to just adding scene "s3" to scene "s1" on "Y".
The lemma "LeftDerivation_take_derive" assumes that "a" is left-derived to "b" through a sequence "D". It shows that if you take the first "n" elements of "D", "a" is left-derived to the result of deriving "a" with the first "n" elements of "D".
In the context of a univariate polynomial ring (UP_cring), the lemma named "to_fun_taylor_term" is stated. It assumes that "p" is an element of the carrier of P, "a" is an element of the carrier of R, and "c" is also an element of the carrier of R. It shows that the function "to_fun" applied to the Taylor term of "p" a...
If "t1" and "t2" are binary search trees, then the union of "t1" and "t2" is also a binary search tree.
If the following conditions are met: 1. 'e' is greater than 0, 2. the multiseries 'MS (MSLCons (C, e) xs) g' is trimmed positive, 3. the basis is well-formed, 4. and 'f' expands to the multiseries 'MS (MSLCons (C, e) xs) g' with respect to the given basis, then the function 'arctan (f x)' expands to the multiseri...
If the atom 'a' is not present in 'A', then substituting 'a' with 'x' in 'A' will not change 'A'.
A polynomial (pCons a p) is positive if and only if either the polynomial p is positive or the polynomial p equals zero and a is greater than zero.
The sum of the fourth powers of all integers from 0 to a given integer m, multiplied by 30, is equal to the product of m, m minus 1, 2m minus 1, and 3m squared minus 3m minus 1.
For any real number 'a', if the sum of 'a', 'b', 'c', and 'd' is less than or equal to the sum of 'i', 'j', 'k', and 'l', and if 'a' is less than or equal to 'b', 'b' is less than or equal to 'c', 'c' is less than or equal to 'd', 'i' is less than or equal to 'j', 'j' is less than or equal to 'k', and 'k' is less than ...
There exists a sequence 'xs' such that the set of 'xs' is a subset of the set from 0 to 'k', and the Floyd-Warshall algorithm applied to 'm', 'n', 'k', 'i', 'j' equals the length of the path from 'i' to 'j' in 'm' using the sequence 'xs'. This is true if the matrix 'm' is cycle-free for the subset from 0 to 'k', and 'i...
If lens X is bijective, then the inverse of lens X is also bijective.
If the function 'f' applied to 'x' equals bottom, and 'f' is less than or equal to 'g', then 'f' is less than or equal to the environment after deleting 'x' from 'g'.
The left associative operation on interfaces I1, I2, and I3 is valid in the constructive cryptography context.
The lemma named "remove_complete_blocks_set_pbd" assumes the following conditions: 1. A certain value 'x' is less than 'Λ'. 2. The size of set 'A' is equal to 'x'. 3. Set 'A' is a proper subset of set 'Β'. 4. For any element 'a' in set 'A', 'a' is equal to 'V'. Given these assumptions, it shows that the function 'pai...
The set of all complex numbers 'x' for which the imaginary part is greater than 'b' is infinite.
If a string 's' is a certified associated string of 'M' and another string 't' is also a certified associated string of 'M', then 's' and 't' must be the same string.
If a message "m" is okay in a state "σ" and "σ" transitions to another state "σ'" where the set of nodes increases except for node "i", then the message "m" is still okay in the new state "σ'".
If the following conditions are met: 1. The function g has finite real segments at point z. 2. The function g is a valid path. 3. The point z is not in the image of the path g. 4. The end of the path g is the same as the start of the path g. 5. The negative of the real part of the contour index of the path g at point ...
"c is a subset of itself in terms of guards".
The lemma named "distinct_set_notin" assumes that the list starting with "x" and followed by "xs" is distinct. It shows that "x" is not in the set of "xs".
If a function 'f' is monotonic and set 'A' is not empty, then applying the function 'f' to the maximum element of set 'A' is equal to the maximum element of the set obtained by applying the function 'f' to each element of set 'A'.
, given a fixed final state, if process P at time t transitions from state "a.M(vs), h" to state "va, h'" with thread action "ta", then there exists a state "s" such that the thread actions are acceptable in the final state for time t and thread action "ta".
If the inner product of a basis 'a' with itself equals zero, and if 'B' is finite and 'a' is an element of 'S', then 'a' equals zero.
: Given variables v, s, cs, x, c, b, and two contexts Γ1 and Γ2, if the following conditions are met: 1. Under context Γ1, variable v is checked against the type { z : b \| c }. 2. Atom z is fresh for the pair (x, v). 3. Under context Γ, statement s is checked against type τ'. 4. Context Γ is equal to the concatenatio...
The lemma "fresh_star_insert" shows that "inserting a into as is fresh for all x" is equivalent to "a is fresh for x and as is fresh for all x".
: 1. The divergence of process P is empty if and only if the chaos of the entire universe is less than or equal to the divergence of process P in the context of skip. 2. The divergence of process P is empty if and only if the chaos of the entire universe is less than or equal to the divergence of process P. 3. The div...
If a grid 'g' is in the sparse grid with dimensions 'dm' and level multipliers 'lm', then the value at the index 'pm g' of the array 'a' is equal to the value of the grid 'g' at the vertex 'v'. This is under the assumption that the grid 'g' is represented by the array 'a' and the vertex 'v'.
The function "lunstream'" applied to the product of iterations of a function "f" and a variable "x" is equal to the iterations of the function "f" applied to "x".
If a state 's' is reachable in the parameterized network 'pnet' with parameters 'np' and 'p', then the network IP addresses of the state 's' are equal to the network tree IP addresses of 'p'.
The function "rm_iterateoi" from the module "RBT_add" is a proper iterator when applied to the tree "t", both when it is used as the initial iterator and the modified iterator.
The lemma "r_dist_dist" assumes that a transition "t" is applicable to a machine "M", the machine "M" is completely specified, there is a "r_dist" relation between the machine "M", transition "t", and states "q1" and "q2", and "q1" is a node in the machine "M". It shows that the states "q1" and "q2" are not the same.
The lemma named "res_seq_subseq''" assumes that "s" is in the set of closed sequences of Zp. It shows that the residue sequence of "s" at "n" is a subsequence of the residue sequence of "s" at "n + k".
The interior of a set S under the discrete topology U is equal to the intersection of U and S.
, given the assumption that node "n" is in the set of nodes of graph "g" and that graph "g" maintains its invariants, there exists a sequence of nodes "ns" such that there is a path from the entry of graph "g" to node "n" through "ns", and "ns" is an entry path in graph "g".
The flattening of the "whileAnno" construct with conditions "b", invariant "I", variant "V", and command "c" is equal to a list containing the "whileAnno" construct with the same parameters.
The minimum weight basis is empty for an empty set and a carrier.
For all x, if function f of x is not equal to zero, then the big O of the product of functions f and g is less than or equal to the product of function f and the big O of function g. Here, f is a function from type 'a to type 'b, where 'b is a linearly ordered field.
If points p, q, and r are not at the origin (i.e., they are non-zero), and point q is strictly between points p and r on the real number line, and points q, p, and r are distinct, then there exists a positive real number k, less than 1, such that the Cartesian coordinates of point q (augmented by 1) can be expressed as...
For a subring K of R, a unitary monomial of degree n is equal to the variable X raised to the power of n in the polynomial ring K[X].
, given a real number 'd' that is greater than 0 and less than or equal to 1, there exists a function 'delta' from natural numbers to real numbers such that for all 'l', 'delta' of 'l' is greater than 0, and 'delta' tends to 0 as 'l' tends to infinity. Furthermore, the measure of the set of all 'x' in the space 'M' for...
The lemma "prv_imp_neg_neg" assumes that "phi" is a formula. It shows that "phi" implies the negation of the negation of "phi" is provable.
The property I holds before the execution of the garbage collector.
Applying the function "apply_bfun" to the function "Bfun" of "f" (where "f" is a function from a finite type 'c to any type) is equal to "f".
If a Java program 'P' is well-formed, and if a class 'C' with superclass 'D', fields 'fds', and methods 'mths' is in the set of 'P', and if a method declaration 'jmdcl' is in the set of methods 'mths', then the method declaration 'jmdcl' in class 'C' in program 'P' is well-formed.
The component of the term resulting from the subtraction operation between variables v and t is equal to the component of the term v.
The function 'of_bool' preserves the relation 'pcr_natural' under logical equivalence.

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