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The lemma named "distinguishing_formula_supp_aux" assumes that P is not equivalent to Q. It shows that the support of the absolute value of the set, which contains the permutation of some x (where the support of x is a subset of the support of the permutation of P and x distinguishes the permutation of P from the permu...
The lemma named "card_geq_3_iff_contains_3_elems" shows that the cardinality of set A is greater than or equal to 3 if and only if set A is finite and there exist three elements x, y, and z such that x is not equal to y, x is not equal to z, y is not equal to z, and x, y, and z are all in set A.
The lemma named "updown_eq_iff" is defined with two fixed variables, x and y. x is of a type that is parameterized by 'd1 (a dimension type) and 's (a unit system), and y is of a type that is parameterized by 'd2 (another dimension type) and the same 's (unit system). The lemma assumes that the quantity dimension of 'd...
If a transition from state "s" with command "c" to state "s'" with command "c'" exists in the transition relation "trss" of a flow graph "fg", then the set of monitors for event "e" in the flow graph "fg" and the set of monitors for command "c" in the flow graph "fg" have no elements in common.
If for all 'x', 'P x' is a proper assertion, then the assertion that there exists an 'x' such that 'P x' holds for a given heap 'h' is also proper.
The theorem "Thm" holds for the value 15 of "Psi".
: if for all 'n', the body 'A' is true in the context of 'A' union 'G', then if 'G' is true for 'n', the call 'A' is also true for 'n'.
The lemma named "traces_dom_events" assumes that "u" is in the range of "D". It shows that "xs" is in the traces of "P" if and only if the union of the next domain events of "P", "D", "u", and "xs" and the reference domain events of "P", "D", "u", and "xs" is not an empty set.
The lemma for the long-term logic of stakeWhile is simple: the long-term logic of stakeWhile applied to a predicate P and a stream xs is equal to, if the predicate P holds for the head of the stream xs, then stakeWhile applied to P and the tail of xs, otherwise it is an empty lazy list.
The lemma named "mkVarChannelProc_cl_inv" assumes that the clause invariant holds for a pair (g,p). It then shows that the clause invariant also holds for the process "mkVarChannelProc v g p".
If 'A' is a finite set, then the cardinality (or size) of the set of all multisets 'M' such that the set of elements in 'M' is a subset of 'A', the size of 'M' is 'k', and for all elements 'x', the count of 'x' in 'M' is less than or equal to 1, is equal to the number of ways to choose 'k' elements from the set 'A'.
"pi" permuted with "f" equals "f" if and only if for all "x", "pi" permuted with "f x" equals "f" of "pi" permuted with "x". This is denoted as the equivalence of "?LHS" (left-hand side) and "?RHS" (right-hand side).
If A is a prefix of B, then the unlabelled version of A is also a prefix of the unlabelled version of B.
The set resulting from the function "f4_sel_aux" applied to "t" and "ps" is a subset of the set "ps".
The square of the sum of two real numbers (f and g) is equal to the square of the first number plus twice the product of the two numbers plus the square of the second number.
If a function 'f' is residuated, and 'x' is less than or equal to 'y', then the residual of 'f' at 'f x' is less than or equal to the residual of 'f' at 'f y'.
The lemma named "delete_cnj" assumes that the order of "x" in the product of the sequence "xis", where each element of the sequence is raised to the power of its successor, is greater than or equal to "si", and "si" is not equal to zero. It shows that the product of the sequence "xis", where each element of the sequenc...
: given the assumptions that "x" is an element of the group "G", "y" is an element of the subgroup "H", and "G" is a group, it shows that "x" multiplied by "y" is an element of "H" if and only if "x" is an element of "H".
The lemma named "insert_dom_inj_on" assumes the following conditions: 1. The function 'f' is injective on the set of all numbers from 0 to 'n' (exclusive). 2. 'i' is less than 'n+1'. 3. 'j' is less than 'n+1'. 4. The image of the set of all numbers from 0 to 'n' (exclusive) under the function 'f' is equal to the set o...
If command 'c' does not read from 'x', then the sequence of command 'c' followed by any other command 'c'' also does not read from 'x'.
The identity relation and the action "m1a_leak Rs" refines the relation "R_a0m1a_rs".
The size of the conjunction of A and B is equal to the size of A plus the size of B plus 4.
If X is not in the synthetic set H, then the condition "Sign X K is in the synthetic set H" is equivalent to "Sign X K is in the set H".
If for all expressions 'e'', if 'e'' is a subexpression of 'e' and 'e'' does not satisfy the property 'sub_RI', then 'e'' is in an initial configuration with shape 'sh'. Then, if 'e' does not satisfy the property 'sub_RI', 'e' is also in an initial configuration with shape 'sh'.
The lemma named "consts_tm" consists of two parts: 1. The set of constants in the application of a function 'f' to an argument 'x' is the union of the set of constants in 'f' and the set of constants in 'x'. 2. The set of constants in the head 'h' of a term is empty if 'h' is a variable, and it is the set containing ...
If set S is convex, then point x is an extreme point of set S if and only if x is an element of set S and the set S without point x remains convex.
If process P is less than or equal to process Q in the failures model, then process P is also less than or equal to process Q in the traces model.
The extended Gromov product at a point 'e' and 'y' for the Gromov completion of 'x' is equal to the infimum (greatest lower bound) of the set of all limit inferior (liminf) of the sequence of the Gromov product at 'e', 'v n' and 'x', where 'v' is a function such that the sequence of the Gromov completion of 'v n' conve...
The lemma named "llrg_UNF" assumes the following conditions: 1. The relation of function signatures, denoted as "funas_rel \<R>", is a subset of a set denoted as "\<F>". 2. The pair (c, 0) is not in the set "\<F>". 3. The property "llrg" holds for the relation "\<R>". 4. The property "UNF" holds for the "gsrstep" of t...
The theorem prop3: Assuming that the property "bar" holds for a list "xs", it shows that for any list "zs", if "xs" is not an empty list and relation "R" holds between an element "a", the list "xs", and the list "zs", then the property "bar" also holds for the list "zs".
: Assuming the following conditions: 1. "it" is a reverse linear order set iterator over the set "S0". 2. The predicate "I" holds for the set "S0" and the state "\sigma>0". 3. For any set "S", state "\sigma>", key "k", and value "v", if the condition "c" holds for the state "\sigma>", the pair "(k, v)" is in the set "...
The infinity of the product of a set 'xs' is equivalent to the condition that every element 'x' in the set 'xs' is infinite.
The target of the union of s and t is equal to the union of the target of s and the target of t.
The function 'insert_before' maintains the well-formedness of the heap in the Shadow DOM when a new node is inserted before an existing one. This is under the conditions that the type of the ShadowRootClass is well-formed, the known pointer of the ShadowRootClass is well-defined, and the known pointers of the ShadowRoo...
If "a" is a positive extra, then the conjunction of "a" and the list "as" is equal to the match conjunction of the match of the extra of the positive "a" and the conjunction of the list "as".
The set of elements that are greater than or equal to another element is a subset of the nonstrict Weighted Path Order.
If for every element 'a' in set 'A', the function 'f' applied to 'a' equals zero, then the L2 norm (or Euclidean norm) of the function 'f' over the set 'A' equals zero.
If a note from agent A containing information X is in the set of events, and if agent A is in the set of compromised agents, then the information X is in the set of information known by the spy in these events.
Lemma fls_subdegree_ge0I: If for all k less than 0, the k-th coefficient of the formal Laurent series f is 0, then the subdegree of f is greater than or equal to 0.
If scene A is independent of scene B, then the quotient of scene A by X is also independent of the quotient of scene B by X.
The lemma named "split_float_interval_bounds" demonstrates four things: 1. The lower bound of the first part of the split float interval is equal to the lower bound of the original float interval. 2. The lower bound of the second part of the split float interval is equal to the midpoint of the original float interval....
The checker for the variables of ego vehicle (sₑ, vₑ, aₑ) and other vehicle (sₒ, vₒ, aₒ) is equivalent to the condition where the preconditions for these variables are checked and there is no collision between the ego vehicle and other vehicle for the given range of time starting from 0.
The lemma Eq_1I[meta_intro]: "If F is equal to G, then F is strongly equal to G in v."
If a set, which includes 'x' and other elements 'xs', is a minimal basis and 'x' is not an element of 'xs', then the set that includes the remainder of 'x' when divided by 'xs' and 'xs' itself is also a minimal basis.
If a set A is a subset of the lower bound set of the antecedent of z and A is not an empty set, then there exists a unique element m in A such that m is less than or equal to every element x in A.
Filtering a set of transitions 'e' based on three conditions: the origin state equals 's', the label of the transition equals 'l', and the transition can be taken given input 'i' and register 'r', is equivalent to first filtering the set of transitions based on the origin state equals 's', and then filtering the result...
The norm of x equals a if and only if a is greater than or equal to zero and the complex inner product of x with itself equals the complex number derived from the square of a.
If a property P holds for an element x, and if the heap state after execution (h') is the same as the heap state before execution (h), and if the result of the execution (r) is the same as the element x, then the effect of asserting the property P on the element x has occurred in the heap from state h to state h' with ...
In the context of a finite incidence system, the set of replication numbers is finite.
If "n1" is less than "n2" and "n2" is less than or equal to the length of list "al", then the sublist of "al" taken from the start to "n1" is less than the sublist of "al" taken from the start to "n2".
If a word 'x' of type 'b' is less than another word 'y', and 'y' is less than 2 raised to the length of type 'a', then the unsigned cast of 'x' is less than the unsigned cast of 'y' of type 'a'.
A strict order 'x' is equivalent to 'x' being both transitive and acyclic.
Lemma: above_Above_trans Assumptions: 1. The relation "r" is transitive. 2. "a" is above "b" in relation "r". 3. "b" is in the set "Above" of elements greater than all elements in set "C" with respect to relation "r". Conclusion: "a" is in the set "Above" of elements greater than all elements in set "C" with respect to...
If point x is in the convex hull of set S and point y is also in the convex hull of set S, then the closed segment between point x and point y is a subset of the convex hull of set S.
The accessible left language of a state 'q' is equal to the left language of the state 'q'.
The double negation of the product of x and y is equal to the negation of the product of x and y.
Adding the maximum possible value (top) to any value x will always result in the maximum possible value (top).
If the intersection of set N and the set of subformulas of phi is a subset of set S, then the second projection of phi over the intersection of sets N and S is equal to the second projection of phi over set N.
The lemma named "lists_succ_set_cyc_permutationss" assumes that "xss" is a list of distinct elements. It shows that the image of the set of cyclic permutations of "xss" under the function "lists_succ" is equal to the set of all functions "f" that permute the union of the sorted set of "xss" and for all "c" in the sorte...
If a set F is in the set of offending flows G nP and F is an empty set, then this is a contradiction, or in other words, this situation cannot occur.
If for every element 'x' in the nth power of the ring 'R', the function 'p' applied to 'f' of 'x' equals 'Q' of 'x', then the inverse of 'f' with respect to 'n' applied to the set of all elements that satisfy 'p' equals the intersection of the set of all elements that satisfy 'Q' and the nth power of the ring 'R'.
If a list "xs" is a well-formed probability mass function (pmf), and the set of first elements from the pairs in the list "xs" is a subset of a set "A", then the probability mass function of the list "xs" belongs to the set of lotteries on "A".
The lemma named "similar_mat_wit_smult" is about matrices A and B of a type that is a commutative ring. It assumes that A and B are similar matrices with witnesses P and Q. It shows that if A and B are similar, then the matrices obtained by multiplying A and B by a scalar k are also similar with the same witnesses P an...
If a rule set "rs" when processed with parameters "Γ", "γ", and "p" transitions from an "Undecided" state to a state "t", then processing the return of the rule set "rs" with the same parameters will also transition from an "Undecided" state to the same state "t".
If a function 'f' has a least fixpoint and a greatest fixpoint, and if 'f' is isotone (i.e., preserves the order) in both the standard and approximate sense, and if the approximate meet is below the least fixpoint of 'f', then the kappa-mu-nu property holds for the function 'f'.
If we assume a partial derivative exists such that "\<Gamma>,\<Theta> proves under the frame F that program c, when started in a state satisfying precondition P, either diverges or terminates in a state satisfying postcondition Q or exceptional postcondition A", and we also assume "termination" such that "for all state...
The lemma, named "code_unfold", demonstrates that the inverse of a matrix A, represented as an integer array, is equal to the following: First, let A' be the result of applying the Gauss-Jordan elimination method to the matrix A. Also, let "nrows" be the length of A. If the length of the list, which is created by filte...
If for any value 'x', 's' is equal to 'x' implies 'P', then 'P' is true.
If a sequence 'xs' is a prefix of 'j' and 'i' is in the set of added IDs of 'xs' for an element 'e', then the delivery of the pair (i, Add i e) is in the set 'xs'.
A pair of numbers (a, b) belongs to the function "square2" with parameters n, i, and j if and only if the following conditions are met: i is less than or equal to a, a is less than 2 to the power of n plus i, j is less than or equal to b, and b is less than 2 to the power of n plus j.
If the representative of a state s is in the set S, then the state s is in the set S.
The sum of the function 'g' over the set 'I', where 'g' is applied to each element 'i' in 'I', is equal to the sum of the function 'g' over the set 'I' without any restrictions.
Adding a matrix 'f' to itself results in the matrix 'f' itself.
The tau function of the meet (or intersection) of x and y is equal to the meet of the tau function of x and the tau function of y.
If the logarithm base 2 of u is less than x, then the function c_in evaluated at x and u equals 0.
The lemma named "DirProds_iso2" assumes that the function 'f' is injective on the set 'A' and that the direct product of the group 'g' and the image of 'A' under 'f' forms a group. Under these assumptions, it shows that the direct product of the group 'g' composed with 'f' on 'A' is isomorphic to the direct product of ...
, assuming "learn_fin \<psi> U s" holds, it shows two things: 1. The environment is defined by "\<psi> U s". 2. For any function "f" that belongs to "U", there exist an "i" and a "n₀" such that "\<psi> i" equals "f" and for all "n" less than "n₀", "s" applied to "f" after "n" steps equals 0, and for all "n" greater t...
The lemma named "xeq1_trans" is a transitive property. It assumes two conditions: first, "s" is equivalent to "t" with respect to "x", and second, "t" is equivalent to "r" with respect to "x". If these two conditions are met, it shows that "s" is equivalent to "r" with respect to "x".
For any memory ownership (\<O>), shared memory (\<S>), and pending write operation, if the pending write operation is non-volatile and either owned or read-only, then the outstanding references that are non-volatile reads from the store buffer (sb) are a subset of the memory ownership, all acquired store buffers, or re...
The infimum (greatest lower bound) of p3 and q4 is not the bottom element.
The lemma "tr_NegChecks_constr_iff" consists of two parts: Part A: For all G in the set L, G is an inequality model of I, X, and the concatenation of F and G if and only if the derivation of I is from M and the map of G such that for all X, the disjunction is not equal to the concatenation of F and G in the stateful t...
If there exists an 'x' such that 'F x' is true and there exists a 'y' such that 'G y' is true, then the statement 'for all 'x', 'F x' implies 'H x' and for all 'y', 'G y' implies 'J y' is equivalent to the statement 'for all 'x' and 'y', 'F x' and 'G y' implies 'H x' and 'J y'.
The dimensions of the tensor Aw are equal to the replication of the last element in rs, repeated 2 * N_half times.
The lemma named "after_futures" assumes that a sequence "xs" is in the traces of a process "P". It shows that the futures of the process "P" after the sequence "xs" is removed, when applied to a sequence "ys", is equal to the futures of the process "P" when applied to the concatenation of the sequences "xs" and "ys".
The product of 'x' and the star of 'x' and 'y' is equal to the star of 'x' and the product of 'x' and 'y'.
If C is equal to the class of cf, then the object location of cf and a is reachable from the object value of C and a in the state s.
If a clock numbering 'v' for 'n' exists and the set derived from 'M' using 'v' and 'n' is not empty, then this set is equal to the set derived from a function that, for each pair of indices 'i' and 'j', returns 'one' if 'i' equals 'j' and otherwise returns the value of 'M' at 'i' and 'j', using the same 'v' and 'n'.
The lemma named "approximation3" is defined with a fixed natural number 's'. It assumes that the value of "(k - 1) to the power of m divided by 3" is less than "s times (L squared times (k - 1) to the power of m) divided by 2 to the power of (p - 1)". If this assumption holds, it shows that "((m - l) divided by k) to t...
The correct composition of different levels is applicable to system A2.
If 'n' is a prime power, then the prime omega function of 'n' equals 1.
The set of all elements between 'a' and 'b', including 'a' but excluding 'b', is connected. This is applicable for 'a' and 'b' in any linear continuum topology.
If 'r' is an order on 'A', and 'xs' is in the union of 'nlists' 'n' 'A' for 'n' less than or equal to 'mxs', then 'xs' is less than or equal to itself according to the list 'n' order 'r'.
The first element of the list produced by the "dom_kildall" function with "start" as an argument is an empty list.
If the pair (a, x) is in the function group defined by function f and set s, and the pair (a, y) is also in the same function group, then x must be equal to y.
The projection of a vector "psi" onto a set "S" is equal to "psi" itself, given that "psi" is in the space represented by the set "S".
An empty set is incomparable.
The function that maps each function to its image on set A respects the domain permutation between set A and set B.
The exponential of 'x' to the power of 'y' equals zero if and only if 'x' equals zero and 'y' is greater than zero.
The "atom" function is injective.
The lemma named "poly_squarefree_decomp" is about a polynomial 'p' in a field with characteristic 0. It assumes that the derivative of 'p' is not zero, 'p' is equal to 'q' times 'd', the derivative of 'p' is equal to 'e' times 'd', and 'd' is equal to 'r' times 'p' plus 's' times the derivative of 'p'. It shows that 'q...