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Split-Isa-AF-MMA

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If for any function 'f', all values 'i' of 'f' are in set 'A' implies that 'f' is good with respect to property 'P', then 'P' is almost full on set 'A'.
Equality in the context of higher-order logic is defined as a function that takes two arguments, x and y, and checks if the pair (x, y) belongs to the identity relation over natural numbers.
The left-hand side variables (lhs) in the tableau resulting from preprocessing the constraints (cs) and the start state are distinct when mapped. This means that there are no duplicate left-hand side variables in the tableau after the preprocessing step.
The lemma named "positions_of_an_atomic_term" assumes that "t" is either a variable or a constant. It shows that the position of "t" is equal to the set that contains only the empty list.
If the length of 'b is less than or equal to the length of 'a, then the untyped cast of 'a to 'b of x shifted left by n is equal to the untyped cast of x shifted left by n.
The statement consists of three lemmas related to a Markov Decision Process: 1. If a configuration (cfg) is valid, then the state of that configuration is in the set S. 2. If a configuration is valid, and a state (s) is in the action of that configuration, then that state is in the set S. 3. If a configuration is vali...
The lemma named "unitary_between_apply" is defined with respect to two sets E and F, both of which are of type 'a in a Hilbert space. The lemma assumes that E and F are orthonormal bases (is_onb E and is_onb F) and that a certain element 'e' belongs to set E. The lemma then shows that the result of applying the unitary...
If the encryption keys are clean in a set H, and the intersection of the set Ag and the broken parts of set H in the range of LtK is empty, then the analysis of the union of the implementation of a secure set Ag with a set G and set H is a subset of the synthesis of the analysis of set H union with the set that does no...
If you create a 'foo' object with parameters 'a' and 'b', and then set 'field_1' of 'foo' to 'y', it is equivalent to creating a 'foo' object with 'y' as the first parameter and 'b' as the second parameter.
If there is a retraction from set S to set T using function f, and a retraction from set T to set U using function g, then there is a retraction from set S to set U using the composition of functions g and f.
If 'x' is an element of the carrier group 'G', then the homomorphism of 'x' raised to the power of an integer 'n' is equal to the homomorphism of 'x' raised to the power of 'n' in the group 'H'.
, given the assumptions that both 'g' and 'f' are equal to '\<bb>', it can be shown that the composition of 'g' and 'f' in the category '\<Up>\<^sub>C \<aa> \<bb> F' is equal to 'g' and also equal to 'f'.
The function "assign_indices" creates a bijective (one-to-one and onto) mapping between the states of a finite state machine M and the image of the states under the function "assign_indices".
The substitution of any set of atoms with an empty set, under any substitution sigma, results in an empty set.
The lemma named "mkVarChannelProc_gState_progress_rel" assumes that the program "prog" maintains the global state "g" invariant. It shows that the global state "g" and the first element of the process created by "mkVarChannelProc" with parameters "v", "g", and "p" are in the progress relation of the global state for th...
The function "ikkbz_sub" applied to any "t" results in well-formed directed labeled vertices.
The identity element for the extended field lens is the zero lens.
In the context of an encoding, the lemma named "indRelLTPO_refl" is defined. It fixes a relation "TRel" which is a set of pairs of 'procT'. Assuming that "TRel" is reflexive, it shows that the induced relation "indRelLTPO TRel" is also reflexive.
If K is equal to the deductive closure of A, then the intersection of the contraction of K by phi and the contraction of K by psi is a subset of the contraction of K by the conjunction of phi and psi.
The partial term of "x" with respect to "nt" is equivalent to "t".
For a given strand S: 1. The union of the set of free variables in the send actions of the strand is a subset of the well-formed restricted variables of the strand. 2. The union of the set of free variables in the receive actions of the strand is a subset of the well-formed restricted variables of the strand. 3. The u...
If there is a descending group chain G of length n represented by function f, and if i and j are less than or equal to n with i being less than j, then the value of function f at j is a subset of the value of function f at i.
The logical 'and' operation between 'false' and 'invalid' results in 'false'.
If H, H1, K, and K1 are subgroups of G, and if H1 is a normal subgroup of the group formed by G and H, and K1 is a normal subgroup of the group formed by G and K, then the group formed by G and the set operation of H1 with the intersection of H and K is a normal subgroup of the group formed by G and the set operation o...
If under the context of Γ and Θ, with a fault set F, R implies c2 with postconditions Q and A, and P implies c1 with postconditions Q and R, then under the same context and fault set, P implies the catch of c1 and c2 with postconditions Q and A.
A word "w" bitwise AND with a mask of "n" is equal to "w" if and only if the unsigned integer representation of "w" is less than 2 to the power of "n".
The lemma named "mask_eqs" states the following: 1. The bitwise AND of 'a' with a mask of 'n' bits, added to 'b', and then bitwise AND with the same mask equals the sum of 'a' and 'b' bitwise AND with the same mask. 2. The sum of 'a' and the bitwise AND of 'b' with a mask of 'n' bits, then bitwise AND with the same ma...
The lemma named "sconcat_scons" shows that the concatenation of a sequence that starts with an element 'x' followed by a sequence 'xs' is equal to the sequence that starts with 'x' followed by the concatenation of the sequence 'xs'.
The formal power series of the sum of a function f over a set A is equal to the sum of the formal power series of the function f for each element in the set A.
The negation of p followed by the negation of q is equal to the negation of p multiplied by the greatest lower bound of the negation of q and the double negation of q multiplied by the top element.
The cardinality or size of set B is equal to the cardinality or size of set A.
The lemma named "partition_set" assumes that partitioning a list "xs" using a predicate "P" results in two lists: "yes" and "no". It shows that the union of the sets of "yes" and "no" is equal to the set of "xs".
If function f1 is related to function f1' by relation R and equality, and function f2 is related to function f2' by relation R and equality, then the function that takes an input x and returns the result of applying function g to the results of applying functions f1 and f2 to x, is related to the function that takes an...
The lemma named "initialiI" is an introduction rule. It assumes two conditions: 1. For any state-transition pair (\<sigma>, p) in the set OI, the pair (\<sigma> i, p) is in the set CI. 2. For any state-transition pair (\<xi>, p) in the set CI, there exists a state-transition function \<sigma> such that \<xi> equals...
The identity function is a regular register.
If the future reach of a formula φ is not infinite, then there exists a number j such that i is less than or equal to the progress of the sequence σ with respect to the formula φ at j.
Given the precondition that variable 'x' equals 'a' and variable 'y' equals 'b', after the following operations: assigning the value of 'x' to 'z', then assigning the value of 'y' to 'x', and finally assigning the value of 'z' to 'y', the postcondition will be that variable 'x' equals 'b' and variable 'y' equals 'a'. T...
If the function "inv_on_right_moving" holds true for the input parameters "(as, lm)" and "(x, l, Oc # r) ires", then it also holds true for the input parameters "(as, lm)" and "(y, Oc # l, r) ires".
If the intersection of sets A and B is empty, then the sum of the indicator function of set A and the indicator function of set B, for any element x in a monoid with addition, is equal to the indicator function of the union of sets A and B for the same element x.
If 'h' is a root, then the size of 'h' in the heap is equal to the size of the homomorphism of 'h'.
If a list 'l' is in the carrier and the monomial polynomial list 'mps' is an interpretation of 'l', then the interpretation of the list 'l' with the polynomial substitution 'psubstl1' applied to polynomial 'P1', 'mps', and integer 'n' is equal to the interpretation of the list 'l' with polynomial 'P1'.
The second element of the image of a map's graph is equal to the range of the map.
The lemma named "multiset_filter_mono'" assumes that for any element 'x' in multiset 'A', if 'P x' holds true, then 'Q x' also holds true. Under this assumption, it shows that the multiset obtained by filtering 'A' with predicate 'P' is a subset of the multiset obtained by filtering 'A' with predicate 'Q'.
If a formula "phi" with a substitution "t" for a variable "x" is provable in a finite set "F" of formulas, and if "phi" is a formula, "t" is a term, and "x" is a variable, then there exists a variable "x" in the formula "phi" that is provable in the set "F".
If the queue discipline invariant holds for a certain flow function 'f' and queue 'Q' with respect to a node 'u', and the excess flow at node 'u' is not zero, then the queue invariant holds for the flow function 'f' and the queue 'Q' appended with the node 'u'.
The process P being equal to the composition of processes R, R1, H1, and H2 applied to P is equivalent to the process P being equal to the composition of processes R, CSP1, and H2 applied to P.
The inverse of a rational polynomial A is equal to negative A.
The converter of a callee with an identity oracle and state 's' is equivalent to an identity converter.
The group G is a lemma.
: Assuming that 'a' is greater than 0, 'τ' is greater than or equal to 0, 'Tmin' is greater than 0, and 'Tmax' is less than 'L', It is shown that the reference of the interval 'I Tmin Tmax' is greater than or equal to the loop of the following control and dynamics: In the control, if the 4th state is 0 and the 3rd ...
If the negation of a formula (denoted as "Formula.Neg \<phi>") is safe, then either the free variable of the formula (denoted as "fv \<phi>") is empty, or there exists a variable 'x' such that the formula is equal to the equation of the variable 'x' with itself (denoted as "Formula.Eq (Formula.Var x) (Formula.Var x)").
Given two monads m1 and m2 of type "'e + 'a", and two functions f1 and f2 of type "'a to 'e + 'b", if m1 is equal to m2 and for any value y, if m2 is equal to the right injection of y then f1 of y is equal to f2 of y, then the result of binding m1 to f1 is equal to the result of binding m2 to f2.
"1 is less than or equal to the backwards implication of x by x".
The lemma named "emp_ref" assumes that "us" is in the list of positive elements of "G" and that the reflection of "G" over "us" equals the empty string. It shows that "us" must be the empty string.
If a pair (t, t') is in the reflexive transitive closure of the reduction relation, and if another pair (u, u') is also in the reflexive transitive closure of the reduction relation, then the pair formed by the application of t to u and the application of t' to u' is also in the reflexive transitive closure of the redu...
If a list 'xs' is in the set Lxx of 'x' and 'y', then the length of the list 'xs' is greater than or equal to 2.
If "j" is less than "n" and the "j"-th element of "x" is zero, then the reduced value of "n" and "x" is not equal to "j".
The norm of the negative of a numeral in a real normed algebra is equal to the numeral itself.
: For any states 's' and 's'', any commands 'c1', 'c2', and 'c'', and any sequence of actions 'w', if the pair of state and combined commands ((s, c1+c2), w, (s', c')) is in the transitive closure of the normalized transition relation of the flowgraph, and the flowgraph is valid for the multiset of the state 's' and t...
If "u" is less than or equal to "v" with respect to term ordering, then the lower part of the higher part of polynomial "p" at "v", evaluated at "u", is zero.
The vector set of the concatenation of two vectors 'v' and 'w' is equal to the union of the vector set of 'v' and the vector set of 'w'.
If a system is consistent, then it cannot prove Rosser's proposition and it cannot prove the negation of Rosser's proposition, and Rosser's proposition is true.
If "b1" is an element of set "B" and "b2" is an element of the set "B" excluding "b1", then the intersection of "b1" and "b2" is in the set of intersection numbers.
There exist a value 'b' and a function 'g' such that 'f' is equal to the absolute value of the function obtained by mapping the default value 'b' to 'g', the domain of 'g' is finite, and 'b' is not in the range of 'g'.
The function "to_rat_3" applied to a variable "x" is defined as follows: If the real number representation of "x" belongs to the set of rational numbers, then it returns the unique rational number "q" such that the real number representation of "x" equals the rational number representation of "q". If the real number re...
If there exists a number 'j' such that 'j' is greater than 3 and less than or equal to 5, and this number is equal to 'x', then 'x' must be either 4 or 5.
The Knuth-Bendix order (kbo) of two terms 's' and 't' is defined as follows: First, we calculate the weights of 't' and 's'. If the multiset of variables in the subterm coefficient function (SCF) of 't' is a subset of the multiset of variables in the SCF of 's' and the weight of 't' is less than or equal to the weig...
Encoding a node with a value of 3 (of natural number type), which has two child nodes with values 1 and 2 respectively (both of which have no further child nodes), results in a list of boolean values: [True, True, True, True, True, True, False, False, True, True, True, False, False].
, assuming the well-foundedness of a set of tuples (t, s) where a predicate P holds for s, a condition b holds for s, and some t equals the result of a function c on s, it can be shown that the set of tuples (t, s) where a predicate P holds for an option of s, a condition b holds for an option of s, and t equals the re...
Getting the CharacterData pointer from a given object after putting the object in a heap is equal to the following condition: if the pointer is equal to the cast of the CharacterData pointer, then it is equal to the cast of the object, otherwise, it is equal to getting the CharacterData pointer from the heap.
If L is less than or equal to M, then the atomic part of L is less than or equal to the atomic part of M.
The boolean value of 'and' operation between boolean 1 and boolean 2 in a state 's' is equivalent to the 'and' operation between the boolean value of boolean 1 in state 's' and the boolean value of boolean 2 in state 's'.
The submatrix of the product of two matrices A and B, indexed by I and J, is equal to the product of the submatrix of A indexed by I and the universal set, and the submatrix of B indexed by the universal set and J.
The previous interval "iprev" at time "t" for a given interval [r, mod m, c] is equal to "t - m" if "t" is within the interval [r, mod m, c] and "r" is less than "t". Otherwise, it is equal to "t".
The units of the fraction field are equal to the carrier of the fraction field minus the zero element of the fraction field.
The lemma "awalk_verts_append" assumes that there is an arc walk from node 'u' to node 'v' following the path 'p' appended with 'q'. It shows that the vertices involved in this arc walk, when the paths 'p' and 'q' are appended, are equal to the vertices involved in the arc walk following path 'p' from 'u', appended wit...
The lemma named "Sup_lexord1" assumes that: 1. Set A is not empty. 2. For any element 'a' in set A, the function 'k' applied to 'a' equals the union of the function 'k' applied to all elements in set A. 3. Predicate 'P' holds for the function 'c' applied to set A. Under these assumptions, it shows that the predicate '...
The acceptance of a word "wacc" with parameters rho and q at the position (i, j) is equivalent to the acceptance of arguments "args" when running q on the substring of rho from position i to j.
If the set of rules in a tree automaton "A" is empty, then the set of reachable states in the same tree automaton "A" is also empty.
Applying a function 'f' to the attributes of a file using 'map_attributes' is the same as directly applying the function 'f' to the attributes of the file.
The probability mass function (pmf) of the stochastic dominance strategy (sds) R27 for 'a' is 1/2, for 'b' is 0, for 'c' is 0, and for 'd' is 1/2.
The relation "rel_blinfun A B" is bi-total. This is under the conditions that both A and B are bi-unique and bi-total, the addition and scalar multiplication operations on B and A are preserved under the relation, and the norm function on B and A is preserved under the relation. The comments in the code ("really?") sug...
If the density of a subprobability space M with respect to a function f is a subprobability space, then the space M is not empty.
The lemma named "min_product_set_empty" states three conditions: 1. The minimum product of an empty set is a set containing an empty sequence. 2. The minimum product of a set containing an empty set is an empty set. 3. The minimum product of a set containing a set with an empty sequence is a set containing an empty se...
The lemma named "topological_sort_post" assumes that "v" is not different from 1 and that the topological sort invariant holds for "q", "v", "R", and "W". Under these assumptions, it shows that "R" is less than or equal to the transitive closure of "W", that "W" is a terminating path, and that the sum of "W" and its tr...
The lemma named "not_reset_unbounded" assumes the following conditions: 1. For all 'j' greater than or equal to 'i', the function 'zero' applied to 'c' and the 'j'th element of 'xs' under the reset operation is not true. 2. 'j' is greater than or equal to 'i'. 3. 'c' is an element of the set 'X'. 4. 'xs' is a stream t...
The lemma named "bangPres" is defined with fixed variables P and Q of type pi, and Rel of type set of pairs of pi. The assumptions are as follows: - P is weakly late step bisimilar to Q with respect to Rel' (denoted as "P \<leadsto><Rel'> Q"). - The pair (P, Q) is in the set Rel. - For any pair (P, Q) in the set Rel, ...
This is a lemma named "list_cases_Cons_snoc" with three cases: Nil, Single, and Cons_snoc. It assumes the following: 1. Nil case: If the list "xs" is empty, then proposition "P" holds. 2. Single case: For any element "x", if the list "xs" consists only of "x", then proposition "P" holds. 3. Cons_snoc case: For any ele...
The lemma named "poly_mapping_eqI" assumes that for any key 'k', the lookup of 'k' in function 'f' is equal to the lookup of 'k' in function 'g'. Under this assumption, it shows that function 'f' is equal to function 'g'.
If 'x' is greater than or equal to 'a' and less than or equal to 'b', then if 'x' is in the interval from 'a' to 'b', this implies 'P'. Therefore, 'P' is true.
The lemma named "per_col_unitary" assumes that "A" is a square matrix of order "n", "A" is a unitary matrix, and "f" is a bijective function between two sets of integers from 0 to "n-1". It shows that the matrix obtained by permuting the columns of "A" according to the function "f" is also a unitary matrix.
The lemma named "aff_dim_halfspace_lt" is about a fixed Euclidean space 'a'. It shows that the affine dimension of the set of all points 'x' such that the dot product of 'a' and 'x' is less than 'r' is equal to the dimension of 'a', unless 'a' equals zero and 'r' is less than or equal to zero, in which case it is -1.
A number 'k' is in the totatives of 'n' if and only if 'k' is greater than 0, 'k' is less than or equal to 'n', and 'k' and 'n' are coprime.
Given an open set 's' that contains the number 0, and a function 'f' that is holomorphic on 's', the residue of the function 'f' divided by 'w' at 0 is equal to the value of the function 'f' at 0.
The lemma named "msubst_nb" assumes that "p" is a linear polynomial, and both "t" and "s" are term-bound to 0. Under these assumptions, it shows that the result of the "msubst" function applied to "p" and the pair of pairs ((c,t),(d,s)) is also bound to 0.
If an encrypted message "Crypt K X" is in the analyzed knowledge of the Spy in a sequence of events "evs", and the key "K" is in the set of symmetric keys, and the sequence of events "evs" is in the set of purchase protocol events, then the key "K" is in the set of used keys in the sequence of events "evs".
: Assuming the function "dominate_False" applied to A_pos, L_pos, A_neg, and L_neg results in a specific structure, and the lengths of A_pos and L_pos are equal, as well as the lengths of A_neg and L_neg are equal. Then it can be shown that: - A_pos is equal to the concatenation of A_zs, A_x, and A_xs. - A_neg is equa...
, given the assumptions that a certain context (represented by Theta, Phi, and Delta) type-checks a statement involving the first or second element of a pair (v1, v2), and that the function "fst_snd" is either the first element function with v equal to v1 or the second element function with v equal to v2, it can be sho...
If a system transitions from state 's' to state 's'' through a series of thread actions 'ttas', and if the invariant 'ts_inv' holds for the initial state 's', then the same invariant 'ts_inv' will hold for the final state 's'', even after updating the invariants with the thread actions 'ttas'.
If 's' is an element of 'S', then 's' fixes pointwise the intersection of 'C0' and the image of 'C0' under the action of 's'.