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The lifting of 'a' to the power of 'n' is equal to the lift of 'a' itself to the power of 'n'.
The lemma "slice_kind_Call_in_slice_Formal_in_not" assumes that the source node 'a' is in the HRB slice of 'S', and the kind of 'a' is a call to a function 'p' with return site 'r' and actual parameters 'fs'. It also assumes that for all 'x' less than the length of 'fs', the formal input at the target node 'a' and posi...
The order of a single-probability mass function (spmf) with the return value of 'x' is equal to 'p' if and only if 'p' is equal to the return of the single-probability mass function of 'x'.
If "x" is reachable from "y" while avoiding "E", then "x" is reachable from "y".
The lemma shows that the parallel strategy is a reduction strategy.
If "S" is in the HAStates of "A" and "S" is not in the States of the HARoot of "A", then there exists a unique "T" such that the pair "(T, S)" is in the ChiRel of "A".
If a pair (r, (Some p, Some p')) is an element of the epsilon closure of the pair automaton A B, then the first element of r is not None and the second element of r is not None. Furthermore, either the first element of r is equal to Some p or the second element of r is equal to Some p'. Additionally, if the first eleme...
The lemma "snd_eucl_of_list_prod" shows that the second element of the pair resulting from applying the function "eucl_of_list" to a list "xs" of type "'b::executable_euclidean_space \<times> 'c::executable_euclidean_space" is equal to the result of applying "eucl_of_list" to the list obtained by dropping the first "D...
The sum of the Stirling numbers of the first kind for all 'k' less than or equal to 'n' is equal to the factorial of 'n'.
The statement asserts that if a set X is stable on a set of preferences ds, then there exists a pair XD_XH such that this pair is stable on ds and X is the match of this pair.
If "m" is greater than zero, then the interval from "r" to "mod m" is infinite.
If two integers have the same relationship under the modular ring relation, then they are equal. This equality is preserved under the modular ring relation.
The word "w" is not in the language "Zero".
The lemma named "arctan_series'" assumes that the absolute value of x is less than or equal to 1. It shows that the series, where each term is (-1) raised to the power of k times (1 divided by the real number equivalent of (k times 2 plus 1) times x raised to the power of (k times 2 plus 1)), sums up to the arctangent ...
The lemma named "graph_mem_closed" assumes that a function 'f' maps from the carrier of the m-th power of the p-adic numbers to the carrier of the p-adic numbers. It also assumes that 'x' is an element of the graph of 'f' with respect to 'm'. It shows that 'x' is an element of the carrier of the (m+1)-th power of the p...
The function "IntVal" is injective.
The negation of the relation F with parameters x, y, and z is equivalent to the relation F with parameters x, y, and z in the context of v.
The relation "mod_ring_rel64" is right total.
The binary operation on set A forms a group.
If "ufa_invar l" holds true (meaning that the list 'l' satisfies the invariant of the union-find algorithm), and if 'i' is less than the length of the list 'l', and if the 'i'-th element of the list 'l' is equal to 'i', then the representative of 'i' in the list 'l' is 'i'.
The Kauffman matrix of the composition of the basic sequence of a cup and a vertical line followed by the basic sequence of a vertical line and a cap is equal to the Kauffman matrix of the composition of two basic sequences of vertical lines.
The satisfaction of a formula "at" in the generic encoding "GE" is equivalent to the satisfaction of the same formula "at" in the Ikeda encoding "Ik".
The lemma named "fresh_empty_fset" shows that "a" is fresh for the empty finite set.
The exclusive OR operation (denoted as \<oplus>) is associative. That is, for any three elements A, B, and C, the result of applying the exclusive OR operation to A and B first, and then to the result and C, is the same as the result of applying the exclusive OR operation to B and C first, and then to the result and A.
The composition of CSP1 and CSP2 is commutative, meaning that applying CSP1 to the result of CSP2 is the same as applying CSP2 to the result of CSP1.
The lemma "sp_instrs_deterministic" assumes that given the same input state "Σi" and instruction "instr", the function "sp_instrs" with parameters "F" and "ret" produces two output states "Σo" and "Σo'". It shows that these two output states must be the same, thus demonstrating that "sp_instrs" is deterministic.
If a problem 'prob' is a well-formed AST problem, and for every operator 'op' in the set of operators of the problem 'prob', the operator 'op' is consistent and is a standard operator, and if a state 's' is in the valid states of the problem 'prob', then the path from state 's' to state 's'' in the problem 'prob' with ...
The left restriction of a relation 'r' to a set 'A' is also a variable set value.
If relation R is strongly normalizing (SN), then relation R is acyclic.
The lemma named "fresh_fin_union" is defined with respect to sets X and Y of type 'a, and an element a of type 'x. It assumes that 'a is a permutable type with 'x, 'x is an atom type, and 'a is a finitely supported type over 'x. It also assumes that both X and Y are finite sets. The lemma then shows that a is fresh for...
The lemma named "matchs" assumes that the function "matchs" applied to the patterns "pats" and the list "ts" returns some environment "env", and that all elements in the list "ts" satisfy a certain property "P". Under these assumptions, it shows that the function "fmpred" applied to a function that maps any input to th...
If the vertices of a face 'f' are distinct, and the pair (a, b) is in the edges of the face 'f', then 'b' is the next element after 'a' in the vertices of the face 'f'.
If a set S is a subset of the carrier G, then S is also a subset of the carrier of the subgroup generated by G and S.
If a well-formed graph G exists and there is a node v in the nodes of G such that the value of nP at v is less than the value of the dependability_fix_nP function at v (with G as the graph and a function that returns 0 for all inputs), then the security invariant function sinvar with inputs G and nP does not hold true.
The remainder of an integer 'i' when divided by 42 is less than or equal to 41.
The lemma assumes that "A" is a square matrix of order "n", "A" is a non-negative matrix, for any "x", if "x" is a root of the characteristic polynomial of "A", then "x" is less than or equal to 1, for any complex number "x", if the norm of "x" is 1, then the order of "x" in the characteristic polynomial of "A" (with r...
The lemma "if_red_known_addrs_new_thread" assumes that there is a transition from state (x, m) to state (x', m') for thread t with thread action ta, and that a new thread t' with state (x'', m'') is in the set of thread actions of ta. It shows that the known addresses of the new thread t' are a subset of the known addr...
If the relation RR implies the relation RR', then the assertion that the relation RR leads to state S under precondition P is a subset of the assertion that the relation RR' leads to state S under the same precondition P.
"for all a and b, the property P holds for a and b" if and only if "for all z, the property P holds for the first element and the second element of z".
A command 'c' is valid with respect to pre-condition 'P', post-condition 'Q', and time complexity 'n' if and only if the command 'c' satisfies the Hoare triple with pre-condition 'P' and post-condition 'Q', where 'P' and 'Q' are embedded into a three-prime and two-prime system respectively, and the time complexity is '...
If the transition from state (e, xs) to state (e', xs') is a silent reduction in the context of program P, thread t, heap h, and external actions extTA, then the transition from state (e instanceof T, xs) to state (e' instanceof T, xs') is also a silent reduction under the same conditions.
The concatenation of b and a is an element of the interleave of a and b.
If event e1 simulates event e2 and event e2 simulates event e1, then event e1 and event e2 are trace equivalent.
The reverse of list xs is less than or equal to the reverse of list ys if and only if list xs is less than or equal to list ys.
The lemma named "Cauchy_Schwarz_eq_iff" shows that the square of the inner product of vectors x and y is equal to the product of the inner product of x with itself and the inner product of y with itself if and only if there exists a scalar k such that x is equal to k times vector y, or vector y is zero.
If the sequence (x#y#xs) is forward and z is an element in the set of this sequence, then the head of the sequence (x#y#xs) can reach z in the transition system T.
Given the assumptions that there is a lowlink path from 's' to 'v' via 'p' to 'w', and that the pair '(u,v)' is in the transitive closure of the tree edges of 's', and that 'u' is either in the domain of the finished 's' or 'u' is the head of the stack 's' when the stack 's' is not empty, it can be shown that there exi...
The lemma named "in_angle_two_sides" assumes that point P is not collinear with points B and A, and also not collinear with points B and C. It also assumes that point P is within the angle ABC. Given these assumptions, it shows that point P is on the same side of line BC as point A.
: Assuming "b1" is equal to the substitution of "b" for "bv" in "b2" and "b2" is not equal to the variable "bv", it shows that: - If "b1" is an integer, then "b2" is also an integer. - If "b1" is a boolean, then "b2" is also a boolean. - If "b1" is a unit, then "b2" is also a unit. - If "b1" is a bit vector, then "b2...
If a word 'a' of any length 'l' is less than or equal to another word 'b', then the word from 'a' to 'b' is equivalent to the map of the natural number representation of the range from 'a' to the successor of 'b'.
The universal set is equal to the set containing elements a1 and a2.
If the decomposition "qs" is valid with respect to "X", is a standard decomposition with respect to "k", is an exact decomposition with respect to "m", and is a homogeneous decomposition, then the auxiliary function "exact_aux" with parameters "k", "m", and "qs" will also result in a homogeneous decomposition.
If a message containing a session key, a receiver B, a timestamp, and a ticket, encrypted with key K, is said by S to A and is in the set of events, then the ticket is in the parts that spies can see in the set of events.
The lemma named "Domainp_fun" assumes that "A" is left unique. It shows that the domain of the function related to "A" and "B" is equal to the function "f" such that the image of "f" over the collection of the domain of "A" is a subset of the collection of the domain of "B".
The hyperbolic sine of a real number x is greater than or equal to zero if and only if x is greater than or equal to zero.
The greatest lower bound of the reflexive transitive closure of a relation P at a point B and x is equivalent to the greatest lower bound of the relation P at the same point B and x.
The lemma "PosOrd_RightE" assumes that if "Right v1" is a strict subvalue of "Right v2" and "v1" has the same flat value as "v2", then "v1" is a strict subvalue of "v2".
If "x" is an element of set "A", then the supremum (or least upper bound) of "x" and "top" is "top". This is a simplification and elimination rule.
An element 'x' is in the domain of 'Gamma' if and only if the atom 'x' is not disjoint from 'Gamma'.
The lemma named "heap_copy_loc_deterministic" assumes that the heap operations are deterministic, as stated by "deterministic_heap_ops". It also assumes that the heap copy location function, "heap_copy_loc", is applied twice with the same parameters "a", "a'", "al", "h", but with different operation sequences "ops" and...
The validation rules for C_info are equivalent to the abstract syntax validation rules.
The length of the list 'ps' is equal to the length of the mirrored version of the list 'ps'.
If a substitution sigma extends a substitution tau, and substitution tau extends a substitution rho, then substitution sigma extends substitution rho. This is a statement of transitivity in the context of substitution extension.
The lemma named "union" demonstrates that the linear choice function of the union of two sets X and Y is equal to the following conditions: if the linear choice function of X is an empty set, then it is equal to the linear choice function of Y. Otherwise, if the linear choice function of Y is an empty set, then it is e...
The successor of 'a_neigh' in the face cycle of graph 'G' is 'a'.
The lemma named "actionCases" has three case names: cInput, cOutput, and cTau. It fixes an action 'alpha' of type 'a', where 'a' is a finite set of names. The lemma assumes three conditions: 1. If 'alpha' equals 'M' inputting 'N', then a property 'Prop' holds. 2. If 'alpha' equals 'M' restricting a vector 'xvec' and...
The function "get_owner_document" is an instance of "l_get_owner_document".
If a message "M" is in the payload, then the intersection of "R23s" and the universal set with "l3_inv5" refines the fake message "l2_dy_fake_msg M" into "l3_dy M" under the relation "R23s".
"top" is less than or equal to "x" if and only if "x" equals "top".
If 'p' is a prime number and an odd number, and 'a' is a number in the range from 2 to one less than 'p', then 'a' is a Solovay-Strassen witness for 'p'.
If "x" is less than or equal to the join of "y" and "z", then the meet of "x" and the negation of "z" is less than or equal to "y".
: Assuming that for all 'i' less than or equal to 'n', the enumeration of trees 'i' are distinct, and that the list 'xs' is distinct. Also, for all 'i' in the set of 'xs', 'i' is less than 'n', and 'xs' is sorted in ascending order. Then it shows that the list obtained by mapping each 'i' in 'xs' to a list of trees ...
A structure R is an abelian monoid if it satisfies the following conditions: 1. For any two elements x and y in the carrier of R, their sum is also in the carrier of R. 2. The zero element is in the carrier of R. 3. For any three elements x, y, and z in the carrier of R, the sum of x and y, then added to z, is equal t...
If the cast of an element pointer is not equal to a node pointer, and if applying the setter function to the element pointer in heap 'h' results in a new heap 'h', then the getter function applied to the node pointer is preserved between heap 'h' and heap 'h'.
For any element 'x' of a boolean algebra, the operation of 'x' exclusive OR 'x' exclusive OR 'y' will always result in 'y'.
Given a term 'tm' of any type 'a' that is finite, and assuming that the pair (dc, {x : b | c}) is in the set 'dclist', and that the algebraic data type 'AF_typedef_poly' with type identifier 'tyid', bound variable 'bv', and data constructor list 'dclist' is in the set 'P', and that 'P' is well-formed, then there exists...
The independent variables M' and X over the index set I are defined as follows: for all i in the index set I, i is a random variable of M' and X. Furthermore, the independent sets are defined as the sigma sets of the space M for all sets A in M' i, where A is the preimage of X i intersected with the space M.
In the context of a linear form, if V is a vector space, then the function f applied to the zero vector equals zero.
If "u" is a suffix of "v", then there exists a string "z" such that concatenating "z" and "u" results in "v".
If the following conditions are met: 1) E is a unit group, 2) h is a group chain of length (n+1), 3) k is not equal to (n+1), 4) k is less than or equal to (n+1), 5) the (n+1)th element of h is not isomorphic to E, 6) the kth element of h is not isomorphic to E, then the set of all i such that i is less than or...
The function "to_nat" applied to 1 equals 1.
The bit length of a number 'a' is defined as follows: if 'a' is greater than 0, then the bit length is the floor of the logarithm base 2 of 'a' plus 1, otherwise, if 'a' is not greater than 0, the bit length is 0.
If a set 'x' is orientable, then 'x' is irreflexively infinite.
If the function 'f' has a vector derivative 'X' at 'F', and 'X' is equal to 'Y', then the function 'f' also has a vector derivative 'Y' at 'F'.
The function "CompFun" for a Hierarchical Automaton (HA) is a composite function of the Statechart Automaton's (SAs) states of the same Hierarchical Automaton (HA).
If beta is an element of the derivative of alpha with respect to a, and if beta is in the context-free language generated by w, then alpha is in the context-free language generated by the list consisting of a followed by w.
"P unless Q and eventually Q" is equivalent to "P until Q".
The set of Standard numbers is a subset of the set of Hyperfinite numbers.
"it is necessarily true that not all x are actual objects" in the context of v.
If the starting heap is okay and C is in the system exceptions, then the address of the system exception C is in the set of starting addresses.
The falling factorial of 'n' and 'k' is equal to the factorial of 'k' multiplied by the binomial coefficient of 'n' and 'k'.
The entries of a mapping created from a list "xs" are equivalent to the set created by clearing the junk from the list "xs".
For any formal power series 'a' in a division ring, dividing 'a' by the formal power series of 'X' is equivalent to shifting 'a' by 1.
A pair "p" is in the set "A" if and only if either "p" is in "A" or the swapped version of "p" is in "A".
The Pochhammer symbol for a number 'a' and the successor of a number 'n' is equal to the Pochhammer symbol for 'a' and 'n' multiplied by the sum of 'a' and the natural number representation of 'n'.
If a body of code, when executed in a normal state initialized with 's', leads to a 'Stuck' state after 'n' steps, then executing a block of code, which includes the same body of code, an initialization, a return, and a continuation 'c', in a normal state 's', will also lead to a 'Stuck' state after 'n' steps.
If a single step of reduction with visible side effects occurs in a program P from state (e, xs) to state (e', xs') under thread t and heap h, then the same reduction can occur when the expression e is part of a field assignment to a field F of an object of type D with value v.
: Assuming that "t" is a transition in the product of two finite state machines (FSMs) derived from an FSM "M" with the first and second states of the source of "t", and that the first and second states of the source of "t" are states in "M". It shows that the path in the FSM derived from the product of two FSMs (wh...
The lemma named "design_iso_block_intersect_num_eq" assumes that "b1" is an element in the multiset "B" and "b2" is also an element in the multiset "B". It shows that the intersection of "b1" and "b2" is equal to the intersection of the image of "b1" under the function "pi" and the image of "b2" under the function "pi"...
If x is approximately equal to y, then the hyperreal part of x is approximately equal to the hyperreal part of y.