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The lemma named "block_sum_szero_bound" shows that for all 't', the sum of 'S0 p j' for a function 'k' that returns the 't'-th digit of the result of 's k' and 'e - z (l k)' is less than or equal to 1.
If "y" is greater than "x", then the length-lexicographic extension of "gt" for the list "xs @ y # xs'" is greater than the list "xs @ x # xs'".
The lemma named "seg_path" assumes that "x" is in the segment between points "a" and "b" (which implies that the segment is not empty). It then establishes that there exists a path "ab" from "a" to "b" such that the segment from "a" to "b" is a subset of this path "ab".
If a formula F is provable from a set of formulas Γ, then the negation of the negation of F is also provable.
If E is a well-ordered set and for every element 'a' in the carrier of D, there exists an element 'b' in the carrier of E such that the order equivalence between the segment of 'a' in D and the segment of 'b' in E is preserved, then the function 'Tw' from D to E is injective on the carrier of D.
If "q" is an element of "W" and the "brw_inner_invar" holds for the set of "r" in "delta" where "q" is in the set of "rhsq r", "q", and the tuple "(Q,W-{q},rcm)", with an empty set and "Sigma'", then the tuple "((Q,W,rcm),Sigma')" is an element of the "brw_step" of "delta".
The set of "ikkbz_sub r" where "r" is an element of the vertices of "G" is finite.
If "DenyAll" is an element in the set converted from the policy list "p", and all networks in the policy list "p" are distinct, and all elements in the policy list "p" are in list "l", then the function "C" applied to the firewall policy list obtained from normalizing the manual order "p" with list "l" is equal to the ...
The lemma "exp_times_self_strict_mono" assumes that a real number "x" is greater than or equal to -1 and less than another real number "y". It shows that the product of "x" and the exponential of "x" is less than the product of "y" and the exponential of "y".
If you map a function over a list, where the function takes an element and pairs it with a constant 'k', and then map the 'snd' function over the result (which extracts the second element of each pair), you will end up with a list that is just the constant 'k' repeated as many times as there are elements in the origina...
The lemma named "r_distinguishable_alt_def" assumes that "q1" and "q2" are states in a machine "M". It shows that "q1" and "q2" are r-distinguishable in "M" if and only if there exists a number "k" such that "q1" and "q2" are r-distinguishable at level "k" in "M".
If S and T are both elements of the set of the completion of M, then the exterior measure of the union of S and T in the completion of M is equal to the exterior measure of the union of the main parts of S and T in M.
The distance between the value function v and the Bellman operator applied to v is less than or equal to twice the distance between the value function v and the optimal Bellman value function.
If a tuple "fs" is a semialgebraic function tuple of length "m" and the length of "fs" is 1, then "fs" is a semialgebraic map tuple of length "m".
The lemma named "S_transform_Conj" assumes that the support of a set 'xset' is finite. It shows that the S transformation of the conjunction of 'xset' is equal to the conjunction of the S transformation applied to each element of 'xset'.
The stuttering step function "stutstep" applied to the sequence "t" prepended with its first element is equal to 0.
An element is a member of the intersection of a red-black tree set 't' and a list 'xs' if and only if the element is a member of 't' and the element is in the set of elements of 'xs'.
If an extended integer 'n' is less than another extended integer 'm', then there exists an integer 'k' such that 'n' is equal to the extended integer 'k'.
"an of x" is less than or equal to "an of y" if and only if the product of "an of x" and "an of y" equals "an of x".
The lemma "d_lt_logq" assumes that the natural number representation of "d" is less than the ceiling function of the logarithm base 2 of "q" (where the result is a real number). It shows that "d" is less than the logarithm base 2 of "q".
If a body of code "bdy" terminates normally with an initial state "init s", and for all states "t", if the body of code "bdy" with the initial state "init s" leads to a normal state "t", then the code "c s t" also terminates normally with the return state "return s t", then the block of code consisting of the initial s...
If "I" and "J" are ideals of the ring "R" and the sum of "I" and "J" equals the carrier of "R", then the quotient ring of "R" by the intersection of "I" and "J" is isomorphic to the direct product of the quotient rings of "R" by "I" and "J" respectively.
For any natural number 'n' and integer 'w', the function 'take_bit' applied to 'n' and 'w' will always be greater than or equal to zero.
The lemma named "pre_zeta_aux_eq_pre_zeta" assumes that the real part of s is greater than negative two times the real number n, and a is greater than zero. Under these assumptions, it shows that the auxiliary pre-zeta function of n, a, and s equals the pre-zeta function of a and s.
The lemma named "endpoints_in_S_cross_ratio_correct" assumes that "a" is not equal to "b", and both "a" and "b" are in the hyperbolic plane "hyp2". Under these assumptions, it shows that the cross ratio is correct for the endpoints "a" and "b" in the set "S", where the endpoints are determined by the function "endpoint...
If 'b' is a reduced ordered binary decision tree (ro_ifex b), then for all assignments (ass), if the value of 'b' under these assignments is true (val_ifex b ass), 'b' must be equivalent to 'Trueif'.
The lemma M_fin\<^sub>C_af\<^sub>\<UU>_correct: Assuming that "t" is in the reach of the transition function "delta\<^sub>C" of the Rabin automaton constructed from the LTL formula "phi" with the abstract function "af\<^sub>\<UU>" and the unfolder "Unf\<^sub>G" over the alphabet "Sigma", and the initial state is determ...
"invar'" holds for the expression "unitET" applied to any "x".
The lemma named "val_ring_add_pow" assumes that "a" is an element of the carrier Qp and the valuation of "a" is greater than or equal to 0. It shows that the valuation of the product of "a" and a natural number "n" is also greater than or equal to 0.
If the list of vertices 'f' is not empty, then the minimum vertex of 'f' is an element of the vertex set of 'f'.
"an empty imperative array map of size 'n' is equivalent to an empty immutable map".
There exists a function 'd' such that for all integers 'p', topological spaces 'X', and subsets 'S' of 'X', 'd' applied to 'p', 'X', and 'S' is an element of the homomorphism from the relative homology group of 'X' with respect to 'S' at dimension 'p' to the homology group of the subtopology of 'X' induced by 'S' at di...
The lemma named "weight_le_stable" assumes that the weight of 's' is less than or equal to the weight of 't'. It shows that this relationship remains stable under a transformation represented by 'σ', i.e., the weight of 's' transformed by 'σ' is still less than or equal to the weight of 't' transformed by 'σ'.
The invariant l2_inv1 is preserved under the transition relation l2.
If a formula G is in the set of subformulas of F, then for all formulas H in the set resulting from the transformation tseytin_tran1 applied to the result of tseytin_toatom applied to F and G, if a model A satisfies H, then the model A satisfies G if and only if the model A satisfies the result of tseytin_toatom applie...
The lemma named "poly_mod_dvd_drop_smult" assumes that "u" is monic, "p" is prime, "c" is not equal to zero and the absolute value of "c" is less than "p" raised to the power of "l". It also assumes that "u" divides "c" times "f" under the modulus "p" raised to the power of "l" in the polynomial ring. Under these assum...
The property of a function being injective on a set A respects the permutation of the domain and range between sets A and B.
If the lengths of list t and list s are equal, then the result of applying a function f to the tail of list t and the tail of list s (using the map2 function) is equal to the tail of the result of applying the function f to list t and list s (using the map2 function).
If the pair (x, fx) is in the graph of the function f over the set A, then x is an element of the set A.
If a set is an interior of a two-element set, then this set is equivalent to the filter of a set.
The function "cofactorM" is equivalent to the composition of the functions "to_vec", "Square_Matrix.cofactor", and "from_vec".
If the index 'i' is greater than or equal to the maximum variable in the floating point arithmetic 'fa', and if the derivative of 'fa' with respect to 'j' exists at 'xs', then the derivative of 'fa' with respect to 'i' also exists at 'xs'.
The lemma named "helper_masked_ucast_equal_generic" is defined with a fixed 16-bit word 'b'. It assumes that 'n' is less than or equal to 112 (which is 128 minus 16). It shows that if you cast 'b' to a 128-bit word, shift it left by 'n' bits, perform a bitwise AND operation with the result of shifting a 16-bit mask lef...
The lemma named "dpWeak" assumes the following conditions: First, the pair consisting of the set Gamma implying E with multiplicity n is derivable from the reflexive transitive closure of R. Second, the set R' is a subset of the upward rules. Third, R is the union of the axiom rules and R'. Under these assumptions...
The composition of the inverse of d and f is a subset of the union of the inverses of ov, s, and d.
The initial state of the power set 2 (ps2) of the union of the set containing vertex v and set S in graph G is equal to the initial state of ps2 of set S in graph G minus the set containing vertex v, plus the function mapping the set containing vertex v to the bottom element.
The "list_eq" relation is symmetric.
In the context of a co-functor adjunction, the lemma named "op_cf_adj_components" demonstrates the following: 1. The left adjoint of the opposite co-functor adjunction is the opposite co-functor of G. 2. The right adjoint of the opposite co-functor adjunction is the opposite co-functor of F. 3. The natural transformat...
The lemma named "extend_map_lemma" assumes the following conditions: 1. Set F is finite. 2. Set G is a subset of F. 3. T is a convex set. 4. T is a bounded set. 5. For any set X in F, X is a polytope. 6. For any set X in F that is not in G, the affine dimension of X is less than the affine dimension of T. 7. For any s...
The lemma assumes that "P" is a well-formed Java program. It shows two things: 1. If "P" can transition from state "s" with expression "e" to state "s'" with expression "e'" in one big step, then "P" can also transition from state "s" with expression "e" to state "s'" with expression "e'" in zero or more small steps. ...
The lemma named "before_in_index1" is defined for a list 'l'. It assumes that the set of elements in 'l' is {x, y}, the length of 'l' is 2, and 'x' is not equal to 'y'. It shows that if 'x' is before 'y' in 'l', then the index of 'x' in 'l' is 0, otherwise, the index is 1.
"x is not an element of an empty set".
"m is OK if and only if there exists an x such that m equals the return of x".
If "I" is a subset of "valid", then the intersection of "parts I" and the range of "LtK" is an empty set. This means that there are no common elements between "parts I" and the range of "LtK".
The quantifier elimination interior of a formula is quantifier-free.
The function "fwi" with inputs m, n, k, n, n, i, i is greater than or equal to 0 if the following conditions are met: the function "cyc_free_subs" with inputs n, I, m is true, m at position k, k is greater than or equal to 0, k is less than or equal to n, k is an element of I, and i is less than or equal to n.
The square root of 4 in the extended non-negative real numbers is 2.
In the context of the network flow invariant, there is a lemma for the invariant successor step. Assuming that 'v' is an element of 'C' and 'dst' is not an element of the image of 'E' under 'v' minus the domain of 'PRED', it shows that the network flow invariant holds for 'c', 'src', 'dst', the map updated by 'PRED' an...
If a heap 'h' is not a leaf, then the minimum element obtained from 'h' is in the set of elements of the tree 'h'.
If a function is closed under a relation R and the restriction of the identity relation to the set of all ground terms of the function is a subset of R, then all contexts of the function are closed under R.
The lemma named "sub3_det" assumes that a matrix A is a square matrix of dimension d, the function "sub3" applied to the parameters d, l, and the pair (r,A) results in the pair (r'',A''), r is not equal to zero, and l is less than or equal to d. Under these assumptions, it shows that the product of r and the determinan...
If a quotient set is formed by an equivalence relation R with abstraction function Abs and representation function Rep over a set T, then a quotient set can also be formed by the relation "rel_bset R" with the map "map_bset Abs" and "map_bset Rep" over the relation "rel_bset T".
If the function "mopup_aft_erase_a" is true for the input of a pair consisting of the successor of the successor of twice a number 'n', a variable 'l', and a list starting with 'Oc' followed by 'xs', and the variables 'lm' and 'ires', then the function "mopup_aft_erase_b" is also true for the input of a pair consisting...
The confluent quotient function, when applied to the double function and the dlist_eq relation from the Dlist module, is equal to the dlist_eq relation for all six instances. Furthermore, this lemma involves the mapping of the first and second elements of a pair, as well as the list_all2 function and the set function.
The lemma "subadditive_converges_bounded" assumes that the function "u" is subadditive and the set of "u n/n" for all "n" greater than 0 is bounded below. It shows that the sequence "u n/n" as "n" goes to infinity converges to the infimum (greatest lower bound) of the set of "u n/n" for all "n" greater than 0.
The lemma is_bintree_list_triv states two conditions: 1. A list of binary trees 'ts' is considered to be of level 0 if and only if 'ts' is an empty list. 2. A list is considered to be of level 'l' if and only if 'l' is 0 when the list is empty.
The lemma named "disjoint_effects_no_effects" is defined for a fixed state 's'. It assumes that for all actions 'a' in the list 'as', the function 'fmdom'' applied to the restriction of the function 'snd a' to the set 'vs' results in an empty set. Under this assumption, it shows that the restriction of the function 'ex...
If the function "wcode_backto_standard_pos" is applied to the variables "ires", "rs", and the pair "(b, Oc # list)", and if the tail of "b" equals "aa", and the head of "b" followed by "Oc" and "list" equals "ba", then the function "wcode_backto_standard_pos" applied to the variables "ires", "rs", and the pair "(aa, ba...
Set A is a subset of the sum of sets G and A.
A given atomic proposition "a" satisfies the Hybrid Automata (H) if and only if the set "S" is an element of the configuration of the initial status of "a".
The lemma "get_scdom_component_ptrs_same_scope_component" assumes that the heap "h" is well-formed, the type of "h" is well-formed, and "h" knows its pointers. It also assumes that the scope component of the pointer "ptr" in heap "h" is "sc", and that the pointer "ptr'" is in the set "sc". Under these assumptions, it s...
If the sum of 1, x, and the product of y and y is less than or equal to y, then the star of x is less than or equal to y.
The lemma named "length_drop_mask" is defined for a word 'w' of any length. It shows that the length of the list obtained by dropping elements from the beginning of the binary representation of the word 'w' ANDed with a mask 'n', until a true (or 1) is found, is less than or equal to 'n'.
If two temporal linked lists "xs" and "ys" satisfy the properties "P" and "Q" respectively, and the extended natural number "n" is less than the temporal length of "ys", then the property "P" holds for the "n"-th elements of "xs" and "ys".
If H is a subgroup of G, then the product of the cardinality (or size) of the right cosets of H and the cardinality of H equals the order of G.
The Jacobi symbol of a number 'a' and the negative of a number 'n' is equal to the Jacobi symbol of 'a' and 'n'.
If "mxs" is less than or equal to "mxs'", then the set of states in G with "mxs" and "mxr" is a subset of the set of states in G with "mxs'" and "mxr".
For all indices 'j' less than 'l', if the 'j'-th element of string 'w' is equal to the element at the position of the length of 'w' minus 'l' plus 'j' in 'w', then the substring of 'w' up to length 'l' is a border of 'w'.
The set of all paths in a tree 't' is a subset of the set of all paths in the tree resulting from many calls to 'x', excluding 'x' from 't'.
The lemma "subtree_to_list_dtree_dom" assumes that "to_list_dtree" is a subtree of the node "r" with a finite set "xs", and that "t" is in the first element of the image of the finite set "xs". It shows that the head of "r" is directly connected to the head of the root of "t" in a transition system "T".
The lemma named "locked_preserved" assumes the following: 1. The state 's' is in the set 'Vinv1'. 2. The round 'r' is the next round after state 's'. 3. The votes in state 's'' are equal to the votes in state 's' with the round 'r' replaced by 'v_f'. Under these assumptions, it shows that the set of locked states in '...
The lemma named "mcs_conjunction_mult" assumes that a set A is consistent and maximal with respect to a set V. It shows that if a set S (which is a list of formulas) is a subset of V and is finite, then the conjunction of S is in V.
For all Z, if a total Hoare triple is valid under a certain program context and fault set, then the same total Hoare triple is valid under the same program context and fault set.
If for all elements x and y in the carrier set L, either x is less than or equal to y or y is less than or equal to x, then L is a total order.
The validity of the function, which maps any value to the base type of the absolute void with no value, is true.
If a relation "r" is transitive, then the multiset relation "multp r" is also transitive.
A multiset replicated 'm' times with element 'x' is less than or equal to a multiset replicated 'n' times with the same element 'x' if and only if 'm' is less than or equal to 'n'.
The lemma named "suffs_cond1I" assumes the following conditions: 1. For every element 'y' in the set 'aa', 'y' is less than or equal to the maximum non-zero value implemented in 'aaa' and 'b'. 2. The length of 'aa' is equal to the length of 'a'. 3. The dot product of 'a' and 'aa' is equal to the dot product of 'b' and...
The lemma named "substn_crename_comm" assumes that "c" is not equal to "a" and "c" is not equal to "b". It shows that substituting "x" with "<c>.P" in "M" and then renaming "a" to "c" and "c" to "b" in the result is equal to renaming "a" to "c" and "c" to "b" in "M" first and then substituting "x" with "<c>.(P[a\<turns...
The lemma named "combine_append" assumes that the length of lists Ks and Us are equal, and the sets of Ks, Us, Ks', and Vs are all subsets of the carrier R. It shows that the combination of Ks and Us, when added to the combination of Ks' and Vs, is equal to the combination of the concatenation of Ks and Ks', and the co...
The interpretation of the time relation between K1 and K2 being in R in TESL is equal to the intersection of all sets Y, where Y is the interpretation of the time relation between K1 and K2 being in R in TESL for all natural numbers n or greater.
Binding function 'f' to the lifted function 'g' is equivalent to binding function 'f' to a function that returns the result of applying function 'g' to 'x'.
If two states 's' and 's1' are equivalent except for PID_N2, then the following conditions hold: - The conference IDs of 's' and 's1' are the same. - The configurations of 's' and 's1' are the same. - The user IDs of 's' and 's1' are the same. - The passwords of 's' and 's1' are the same. - The users of 's' and 's1' a...
The lemma named "statelist'_TAKE_i" assumes that the successor of a number 'm' is less than or equal to the length of a list that consists of an element 'a' and a list 'as'. It shows that 'm' is less than or equal to the length of the list 'as'.
The positive multiset of the negation of a multiset A is equal to the negative multiset of A.
The lemma named "in_ivp_sols_ivl" assumes that "t" is in "T" and "s" is in "S". It shows that the function of "t", which is "phi of t and s", is in the solutions of the function of "t", which is "f", with the interval from 0 to "t" for "s", in the set "S", starting at 0, with "s".
The lemma named "min_list_param_in" assumes that the list "xs" is not empty. It shows that the minimum element in the list "xs" according to a certain relation "rel" is indeed an element of the list "xs".
If P is true, and if Q is true, and if R is true, then P and (Q and R) are true.
If 'p' is an element of the carrier 'R', then there exists a 'c' such that the coefficient solution 'cf_sol' for the rings 'R' and 'S', and the set 'X', and the element 'p' and 'c' holds true.
If 'i' is less than or equal to 'j' and 'i' is less than or equal to 'j-k', then the list of integers from 'i' to 'j' (excluding 'j') is equal to the concatenation of the list of integers from 'i' to 'j-k' (excluding 'j-k') and the list of integers from 'j-k' to 'j' (excluding 'j').