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The lemma named "straight_path_differentiable_y" is defined with fixed real numbers 'b', 'y1', and 'y2'. It assumes a function 'gamma' defined as 'gamma = (y2 + y1 * x , b)'. The lemma shows that for all 'x', 'gamma' is differentiable at 'x'. |
The lemma named "ssuf_induct" assumes that for any string 'u', if property 'P' holds for all strings 'v' that are strict suffixes of 'u', then property 'P' also holds for 'u'. It concludes that property 'P' holds for 'u'. |
Given a finite set S that is not empty and is a subset of the carrier of the union ring C, there exists a ring R in C such that S is a subset of the carrier of R. |
The lemma named "sorted_wrt_quicksort_by_rel" is about a relation R, which is a function that takes two arguments of type 'x and returns a boolean. It assumes two conditions:
1. For any two elements x and y, either R x y is true or R y x is true. This is stated in the assumption "lin".
2. If R x y and R y z are true, ... |
The lemma "equivalence_maps_compose" assumes that "f" and "f'" are equivalence maps and that the source of "f'" is the target of "f". It shows that the composition of "f'" and "f" is also an equivalence map. |
The exception of a polynomial mapping function applied to a set of variables 'xs' and a set 'S' is equal to the polynomial mapping function applied to the filtered set of variables 'xs', where the filter condition is that the variable 'k' is not in the set 'S'. |
The function that always returns "Some 0" is in the set of R01 predicates. |
The sum of an alternating list with length 2n, alternating between s and t, is equal to the sum of s and t raised to the power of n. |
: given an element 'x' from the carrier 'R' of the field, and assuming that 'x' is not equal to zero, it can be shown that there exists another element 'y' in the carrier 'R' (excluding zero), such that the product of 'y' and 'x' equals one. |
Multiplying 1 with any element 'x' in an associative algebraic structure will result in 'x'. |
The step symbol set of {qf} is an empty set. |
The lemma named "g_set_to_listr_correct" makes the following assertions:
1. Assuming that the set 's' is invariant, the set of elements obtained from converting the list representation of the set 's' to a list using the function 'g_set_to_listr' is equal to the set 's' itself.
2. The list obtained from converting the... |
For an integer k, the operation "take_bit" applied to n and k is less than or equal to k if and only if k is greater than or equal to 0. |
Subtracting a set I from a natural number k is equivalent to mirroring the set I that is less than or equal to k, then adding k, and then subtracting the sum of the minimum value of the set I that is less than or equal to k and the maximum value of the set I that is less than or equal to k. |
A command "c" satisfies the Hoare triple "{P}c{Q}" if and only if for all states "s", the weakest precondition for command "c" and postcondition "Q" at state "s" is less than or equal to the precondition "P" at state "s". |
If the encryption keys are clean in a set H, and the intersection of the set Ag and the broken parts of the set H in the range of LtK is empty, then the analysis of the union of the implementation of the confidential set Ag with G and H is a subset of the synthesis of the analysis of H union with the complement of the ... |
If x is not in the domain of sigma, then the memory mapping of x with respect to sigma is equal to the memory of x. |
The lemma named "fundef_ex1_eqvt_at" is defined with the following conditions:
1. It fixes 'x' as an instance of a type 'a' which is a permutation type.
2. It assumes a function 'f' is defined as a lambda function that takes 'x' of type 'a' and returns the unique value 'y' that satisfies the function 'G' applied to 'x... |
If a polynomial "p" is bound to zero, then "p" is quantifier-free. |
The lemma named "integral_on_memE" assumes that the function 'f' is an integral on 'm' over the set 'B'. It shows two things:
1. The function 'f' is in the carrier of the ultraproduct of the semialgebraic set 'm'.
2. For any 'x' in the set 'B', the transformation of the semialgebraic polynomial 'f' to a p-adic poly... |
If a word 'w' and a prefix 'p' both belong to the set of words generated by 'C', and 'p' is a prefix of 'w', then the decomposition of 'p' with respect to 'C' is also a prefix of the decomposition of 'w' with respect to 'C'. |
The lemma named "invertible_bezout_matrix_JNF_n1" is defined with the following conditions:
- It fixes a matrix 'A' of a type that belongs to the 'bezout_ring_div' class.
- It assumes that 'A' is in the carrier matrix of dimensions 'm' by 'n'.
- It assumes that 'bezout' is a Bezout extension.
- It assumes that 'a' is ... |
If a function 'f' maps from set 'A' to set 'B', and 'A1' and 'A2' are subsets of 'A', then the image of the union of 'A1' and 'A2' under the function 'f' is equal to the union of the image of 'A1' under 'f' and the image of 'A2' under 'f'. |
For all 'a', 'n', and 's', if the interpretation of 'a' with respect to the list 'x#bs' is equal to the interpretation of the complex number 'CN 0 n s' with respect to the same list, and 's' is numerically bounded by 0, then the interpretation of the greater than function 'gt n s' with respect to the list 'x#bs' is equ... |
, given an assumption that for any 'x' and 'y', if 'y' is an element associated with 'x' in the finite map 'm', then the property 'P' does not hold for 'x'. Under this assumption, if we apply the filter 'P' to the map 'm', the result will be an empty finite map. |
If 'w' is greater than zero, then 'w' times 'x' being less than 'w' times 'y' is equivalent to 'x' being less than 'y'. Here, 'w', 'x', and 'y' are ordinals. |
: if for any transition 't' and conditions 'c1' and 'c2', obtaining 's1', 'c1', 'e1', 0 not equal to 't' implies obtaining 's2', 'c2', 'e2', 0 not equal to 't' and this implies 'f' of 'c2', and if for any condition 'c', 'f' of 'c' implies that 't1' subsumes 't2', then 'e1' directly subsumes 'e2' from 's1' to 's2' for t... |
The function 'both' is continuous with respect to its first argument 'x', while 'y' is kept constant. |
The lemma named "halting_condition_04_27" is defined with the following parameters: a natural number 'c', a register 'l', a configuration 'ic', a program 'p', a natural number 'q', and another natural number 'a'.
It also defines 'b' as the result of function 'B' applied to 'c', and 'm' as the length of the program 'p... |
The lemma "count_image_mset_lt_imp_lt" assumes that the count of element 'b' in the multiset obtained by applying function 'f' to multiset 'M' is less than the count of 'b' in the multiset obtained by applying 'f' to multiset 'N'. Under this assumption, it shows that there exists an element 'x' such that 'f' applied to... |
If a precondition P guarantees that M will result in a postcondition Q, and if Q is a precondition for M', then P is also a precondition for the sequence of M followed by M'. |
The lemma P_G proves that if the Gödel sentence (phi G) is provable, then falsehood is provable. |
If the evaluation of the disjunction (logical "or") of two conditions, gamma and gamma prime, is true at a certain time point in a certain context, then either the evaluation of gamma or the evaluation of gamma prime must be true at that same time point in that same context. |
The lemma named "Maclaurin_minus" is defined with respect to a natural number 'n' and a real number 'h'. It assumes that 'h' is less than 0, 'n' is greater than 0, and the derivative at 0 is 'f'. It also assumes that for all 'm' and 't', if 'm' is less than 'n' and 'h' is less than or equal to 't' and 't' is less than ... |
For a fixed function 'f' from 'a' to 'b' (where 'b' is an aggregation order), the sum of the function 'f' over all 'k' plus the lowest possible value (bot) is equal to the sum of the function 'f' over all 'k'. |
The lemma named "comp_in_normal_iff" shows that the composition of "t" and "u" is in the set of normal elements if and only if "t" is in the set of normal elements, "u" is in the set of normal elements, and "t" and "u" are sequentially related. |
The lemma is from the context of a complete partial order (ccpo). It fixes a bottom element (bot), a least upper bound (lub), and an order relation (ord). It assumes that the ccpo is a class of ccpo with the given lub, ord, and a less than relation (lt). It also assumes that the function f is monotonic, meaning that if... |
The radical of the radical of a set F is equal to the radical of F. |
The lemma named "cf_cat_prod_12_of_3_ObjMap_vrange" assumes that there are three categories, denoted as "A", "B", and "C". It shows that the range of the object map of the product of the first two categories of the three-category product is a subset of the objects of the product of category "A" and the product of categ... |
If x is less than or equal to y, then the value of x truncated upwards at precision p is less than or equal to the value of y truncated upwards at precision p. |
The extended pre-distance between points x and z is less than or equal to the square root of 2 times the maximum of the extended pre-distance between points x and y and the extended pre-distance between points y and z. |
The sources of "sS5" at "level2" are "sS2" and "sS4". |
:
- Given the assumption "MINV" that "s" is invariant,
- And the assumption "I0" that the function "I" applied to the abstraction of "s" and "sigma0" is true,
- And the assumption "IP" that for any "k", "it", and "sigma", if "k" is an element of "it", all elements "j" in "it" are greater than or equal to "k", all elem... |
The lemma tauLaw2CongSimRight:
fixes a type 'b element Psi
and a psi type element P
assumes condition C1: "For any Psi, P, and Q, if Psi bisimulates P and Q, then the tuple (Psi, P, Q) is in the relation Rel"
shows "Psi leads to tau of P in relation Rel, given P composed with tau of P" |
:
Assuming the following:
- There is a function refinement from 'f' to 'g' with precondition 'P' and relations 'R1' and 'R2'.
- There is a function refinement from 'g' to 'h' with precondition 'Q' and relations 'S1' and 'S2'.
- For any 'x', if 'P''x' holds, then 'Qx' also holds.
- For any 'x' and 'y', if 'P''x' and '(... |
The lemma named "mu_roll" assumes that the function "g" is isotone (meaning it preserves a given order), and that both the composition of functions "f" after "g" and "g" after "f" have least fixpoints. Under these assumptions, it shows that the least fixpoint of the composition of "g" after "f" is equal to the result o... |
The lemma named "sqr_of_cmod_of_prod" shows that the square of the absolute value (or modulus) of the product of two complex numbers (z1 and z2) is equal to the product of the squares of the absolute values of these two complex numbers. |
The lemma "eval_at_point_ring_hom" assumes that "a" is an element in the nth power of the carrier set of ring "R". It shows that the evaluation at point "a" is a ring homomorphism from the coordinate ring of "R" over "I" to "R". |
The lemma named "vassn_dest" has two parts:
1. If the tag of the variable assertion is the product of two assertions (Gamma1 and Gamma2), then both Gamma1 and Gamma2 must have variable assertion tags.
2. If the tag of the variable assertion is the heap node context of R, a, and b, then a must be in the relational dom... |
The lemma named "cupcake_build_conv_preserve" is about a fixed variable 'v'. It assumes that all elements in the list 'vs' are cupcake values and that the function 'build_conv' with parameters 'static_cenv', 'Some (Short i)', and 'vs' returns 'Some v'. If these conditions are met, it shows that 'v' is a cupcake value. |
If term "t" can be reduced to term "t'" through zero or more beta reduction steps, and if term "t" is locally closed, then term "t'" is also locally closed. |
The dimension of the tangent space at a point 'p' in the source is equal to the dimension of the tangent space at the image of 'p' under a function 'f' in the destination, given that 'p' is in the carrier of the source and the dimension of the tangent space at the image of 'p' under 'f' in the destination is greater th... |
If a morphism 'f' is equal to 'ff' and an object 'b' is equal to 'bb', then 'f' is a morphism from the object 'oo' to 'b' in the category 'CC'. |
The lemma l7_13 assumes that A is the midpoint of both P' and P, and Q' and Q. It shows that the distance between P and Q is congruent to the distance between P' and Q'. |
, given the assumption that "j" is less than "n", it can be shown that the integer 1 is equal to the sum of the following terms:
1. The product of (-1) raised to the power of (n - j + 1) and the generalized binomial coefficient of (d - 1) and (n - j + 1),
2. The product of (-1) raised to the power of (n - j) and the g... |
The lemma named "carrierE" assumes that "x" is in the carrier. It then obtains a "c" where "c" is in the charts and "x" is in the domain of "c". |
The top-level summands of the derivative of the product of r and s with respect to xs is a subset of the set that includes the product of the derivative of r with respect to xs and s, and the set of all r' such that there exist ys and zs where r' is in the top-level summands of the derivative of s with respect to ys, y... |
If a memory capability 'c' is truncated and then extended with a tag 'b', it is equivalent to directly assigning the tag 'b' to the memory capability 'c'. |
The lemma named "sqrt_lower_bound_2_small" assumes that "x" is greater than or equal to 0. It shows that the value of the expression "32*x*(16*x + 1)*(256*x^2 + 96*x + 1) / (65536*x^4 + 114688*x^3 + 17920*x^2 + 448*x + 1)" is less than or equal to the square root of "x". |
SA41 is an elementary component in data dependencies. |
The lemma "lift_instrs_all_norm" assumes that the function "lift_instrs" with parameters F, L, ret, N, Σ1, and instrs1 equals Some (Σ2, instrs2). It shows that for all elements in the lists instrs1 and instrs2, the function "norm_eq" holds true. |
If "B" is trimonotonic, then the recursive operator "RECT" applied to "B" is equal to "B" applied to the recursive operator "RECT" applied to "B". |
The sum of zero and an integer 'x' is equal to 'x' if 'x' is defined and not null, otherwise it is invalid. |
The lemma LV_simps demonstrates the following:
- The lexical value of Zero for any string s is an empty set.
- The lexical value of One for any string s is the set containing Void if s is an empty list, otherwise it's an empty set.
- The lexical value of an Atom c for any string s is the set containing Atm c if s is a... |
The lemma named "soundness_mat_x_leq" assumes that the number of rows in matrix A is equal to the dimension of vector b. It also assumes that the result of the simplex algorithm on the system of inequalities "A*x ≤ b" is a satisfiable assignment X. Under these assumptions, it shows that there exists a vector x such tha... |
If the following conditions are met:
1. The state is consistent (InvariantConsistent).
2. The state is unique (InvariantUniq).
3. The state's watches are elements of the state's formula (InvariantWatchesEl).
4. The state's watch list contains only clauses from the state's formula (InvariantWatchListsContainOnlyClauses... |
If x is less than y, then the square root of x is less than the square root of y. |
The lemma named "reach_PairI" assumes that a state "s" is reachable and that taking a step from state "s" with action "a" results in an output "ou" and a new state "s'". Under these assumptions, it shows that the new state "s'" is also reachable. |
The lemma named "subdegree_div'" is defined with respect to two formal power series 'p' and 'q' over a division ring. It assumes that there exists a scalar 'k' such that 'p' is equal to 'k' times 'q'. Under this assumption, it shows that the subdegree of the division of 'p' by 'q' is equal to the subdegree of 'p' minus... |
The lemma for the S-transform of a tree with a "Not" operator is equal to the "Not" operator of the S-transform of the tree. |
There is a delay bisimulation with divergence and finality between two multithreaded systems, represented by r1 and r2. This bisimulation is characterized by the transition relations r1.redT and r2.redT, the bisimulation relation mbisim, the bisimulation relation for thread actions mta_bisim, the silent move relations ... |
If the precondition P is a subset of the intersection of g and R, and if under the context of Γ and Θ, the command c from state R leads to state Q with exception A under the fault set F, then under the same context and fault set, the Guard command with fault f, guard g and command c from state P also leads to state Q w... |
The intersection of two elements "r" and "s" is equal to the following: First, we define "X" as the top-level intersections of the intersection of "r" and "s", excluding the "Full" set. Then, we flatten the intersection of "X". If "Zero" is in "X", then we only consider the set containing "Zero", otherwise we consider ... |
The carrier of the union ring of a set C is equal to the union of the carriers of the set C. |
The lemma named "supp_DNil" shows that the support of an empty list (DNil) is an empty set ({}). |
The lemma named "finfun_left_compose" assumes that "y" is a finite function. It shows that the composition of "g" and "y" is also a finite function. |
The Secure Multi-Party Computation (SPMF) of the XOR operation between A and B in Round 1 (R1) is lossless. |
The function 'well_base_h' applied to the encoding of 'e' and 'n' is well-defined. |
The set of variables in the polynomial obtained by extracting a variable 'v' from a polynomial 'p' is a subset of the set of variables in the original polynomial 'p'. |
The lemma named "hom_char" shows that function 'f' is in the hom-set from 'b' to 'a' if and only if in category 'C', function 'f' is in the hom-set from 'a' to 'b'. |
The function "lucas_lehmer_mult" with inputs m, x, and y is commutative with respect to x and y. This means that swapping the positions of x and y does not change the result of the function. |
The function "r01_to_r01_r01" applied to three-fourths (3/4) equals the pair (1/2, 1/2). |
The lemma named "graph_more_rel_def" defines the relation "graph_more_rel" over three relations Rm, Rv, and Rw. This relation holds between two graphs g and g' if and only if the following three conditions are met:
1. The set of nodes in g is related to the set of nodes in g' by the relation Rv.
2. The set of edges in... |
The pair consisting of the session key 'K' and 'A' is not in the 'keySetup'. |
If a list 'xs' is lexicographically sorted, then taking any number 'n' of elements from the start of this list will also result in a lexicographically sorted list. |
The lemma named "scalar_prod_block_size_lift_01" assumes that "i" is less than "b". It shows that the scalar product of the column "i" of the matrix "lift_01_mat N" with itself is equal to the cardinality of the "i-th" element of the set "B", where this cardinality is of natural number type and the set "B" is of a type... |
A pair of buffers (BUF, BUF) is included in the relation that maps a list of elements of type A to another list of elements of type A, which in turn maps to a text buffer of elements of type A. |
If you update the value of 'y' in 'x' to be the transposition of 'x' squared, the result will be less than or equal to the transitive closure of 'x'. |
The lemma named "list_eq2_refl" shows that the list "xs" is approximately equal to itself at the second level. |
The lemma named "lemma_2_reachability_s_i" is defined with a function 'f' that maps from type 'a' to type 'b', and two sets 's' and 't'. It assumes that 't' is a finite set and the cardinality of 't' is less than the cardinality of 's'. Under these assumptions, it shows that the function 'f' is not injective from 's' t... |
The lemma named "sw_preserves_mship" assumes that the set containing elements y1 and y2 is a subset of the variable "ys". It shows that swapping x with xs (where y1 and y2 are in ys) in the variable "xs" is equivalent to x being in the variable "xs". |
The lemma named "scope_find" assumes three conditions:
1. "v" is in the scope of "ps",
2. "v'" is a terminal vertex,
3. there is a path from "v" to "v'" named "pth".
If these conditions are met, it shows that "ps" is in the second element of the set "pth". |
The lemma named "finite_partGen" assumes that the set P is finite. It shows that the partition generated by P (partGen P) is also finite. |
The lemma named "children_subtrees_equal" assumes that "c" is a child of "l", "c'" is also a child of "l", and "c'" is a subtree of "c". Under these assumptions, it shows that "c'" must be equal to "c". |
This is a lemma named "two_lists_cases_Cons_Cons" with three cases: "Nil1", "Nil2", and "ConsCons". It assumes the following:
1. In the "Nil1" case, if list "as" is empty and list "bs" is equal to any list "ys", then proposition "P" holds.
2. In the "Nil2" case, if list "bs" is empty and list "as" is equal to any list... |
If the lengths of two lists of matrices, Al and Bl, are equal, and for every index i less than the length of Al, the number of rows of the i-th matrix in Al equals the number of columns of the i-th matrix in Al, and the number of rows of the i-th matrix in Al equals the number of rows of the i-th matrix in Bl, and the ... |
The lemma named "least_insert" assumes that "S" is a finite set. It shows that the least "k" elements of the set obtained by inserting "x" into the least "k" elements of "S" is equal to the least "k" elements of the set obtained by inserting "x" into "S". This is denoted as "?lhs = ?rhs". |
The lemma named "fin_ad_agr_list_iff" is defined with a fixed set "AD" of an infinite type 'a. It assumes that "AD" is finite, the length of any element "vs" in "Z" is "n", and "Z" is the set of all "ts" such that there exists a "ts'" in "X" where "ts" and "ts'" agree on "AD" when "ts" is mapped to "Inl ts". The lemma ... |
If "k" is greater than or equal to "p", and "t" is a loose bound variable for "k", then "t" is also a loose bound variable for "p". |
If "A" is a transitive relation, then the restriction of "A" to a set "S" is also a transitive relation. |
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