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Filtering an empty map with any condition "c" will result in an empty map.
The lemma named "coeff_xy_power" assumes that "k" is less than or equal to "n". It shows that the coefficient of the power "n" of the polynomial "x_y" in a commutative ring with unity is equal to the monomial of the product of the binomial coefficient "n choose (n - k)" and "(- 1) to the power k", with degree "(n - k)"...
If a valuation 'v' satisfies the atomic formula restricted to a set 'I' of the set of atomic formulas 'as', then if 'i' is in 'I', 'v' satisfies the atomic formula 'a'. This implies that 'v' satisfies the atomic formula restricted to 'I' of the set of the pair '(i, a)' and 'as'.
The lemma named "strand_fv_subst_subset_if_subst_elim'" assumes three conditions: 1) The substitution 'delta' eliminates the variable 'v', 2) The variable 'v' is in the set of free variables of the strand 'S', 3) The range of variables of the substitution 'delta' is a subset of the set of free variables of the stran...
The theorem named "exp_cf5_lower_bound_pos" assumes that "x" is less than or equal to zero. It shows that the function "exp_cf5" of "x" is less than or equal to the exponential function of "x".
The lemma named "nle__lt" assumes that "A B" is not less than or equal to "C D". It shows that "C D" is less than "A B".
If 'n' is not equal to infinity and 'n' is greater than or equal to zero, then the product of 'n' and 'a' is also greater than or equal to zero.
If the composition of the inverse of a relation 'r' with 'r' itself equals 'r', then 'r' is a feedback relation.
The function which multiplies a real number by its exponential is injective (or one-to-one) on the interval from -1 to infinity.
The lemma named "fold_sup" is defined with a fixed variable 'v' of type 'timestamp'. It demonstrates that the left fold operation with the supremum function and 'v' over a list 'vs' is equivalent to the right fold operation with the supremum function over 'vs' starting with 'v'.
The reachability of a transition system T is equivalent to the union of the set of all behaviors in T.
The function f of the product of a and b is equal to the product of the function f of a and the function f of b.
For a fixed 'a' which is a real normed vector, if 'c' is not equal to zero, then the image of the ball 'a' with radius 'r' under the scaling operation by 'c' is equal to the ball with center at 'c' times 'a' and radius 'c' times 'r'.
The carrier of the integer group is the universal set.
The Stirling row of any number 'n' is not empty.
The lemma named "hermitian_real_diag_decomp" is about a complex matrix A. It assumes that A is an n by n matrix, n is greater than 0, and A is Hermitian. It then obtains matrices B and U such that A can be decomposed into B and U where B is a real diagonal matrix.
If for every element 'i' in a set 'I', the function 'f i' has a derivative 'f' prime 'i' at 'F', then the function that maps 'x' to the sum of 'f i x' for all 'i' in 'I' has a derivative that maps 'x' to the sum of 'f' prime 'i x' for all 'i' in 'I' at 'F'.
For a given set U, the function "atU" applied to the addition of an element "c" to a multiset "c2" is equivalent to the logical disjunction (or operation) of the function "atU_s" applied to the set U and element "c" or the function "atU" applied to the set U and the multiset "c2".
"x" is equal to the real part of the abstraction of "x".
The raw pure assertion associated with 'b' is proper.
The lemma named "nu_roll" assumes that the function g is isotone (monotone in both directions), and that the composition of functions f and g, in both orders, has a greatest fixpoint. Under these assumptions, it shows that the greatest fixpoint of the composition of g and f is equal to the result of applying g to the g...
If x is an element of V, then the negation of x is also an element of V.
If list "l'" is not empty, then for a predicate "is_flatten_it" with parameters "l", "p", "l'", and "it", applying the function "flatten_it_next" to "it" results in a pair "(a, it')". This pair satisfies the predicate "is_flatten_it" with parameters "l", "p", the tail of "l'", and "it'", and the condition that "a" equa...
A unity root of 'k' and 'n' is equal to the exponential of the product of the imaginary unit 'i' and the real number obtained from dividing the product of '2*pi*n' by 'k'.
If "x" is greater than or equal to zero, then the square of the complex square root of "x" is equal to the complex version of "x".
The set of indices 'k' for which the 'k-th' element of the list 'xs' does not equal the default value 'dflt' is finite.
If a set "I" is finite and not empty, then the result of the operation "I plus negative maximum of I" equals the set containing only the element zero.
A grammar 'g' derives a string if and only if there exist strings 's1', 's2', a non-terminal symbol 'lhs', and a string 'rhs' such that the concatenation of 's1', the non-terminal symbol 'lhs', and 's2' equals the left string list 'lsl', and the concatenation of 's1', 'rhs', and 's2' equals the right string list 'rsl',...
, given the assumption that the length of a polynomial 'p' is greater than zero, the evaluation of the sum of the polynomial 'p' with respect to 'd' and 'f' at a point 'a' is equal to the sum of the polynomial 'p' with respect to 'd' and the evaluation of 'f' at 'x' at the same point 'a'.
The lemma named "perp2_comm" assumes that point P is perpendicular to both line segments AB and CD. It shows that point P is also perpendicular to both line segments BA and DC.
: Given the assumptions that - p is an element of the set P, - the character length of p is k, - X is an element of the set T, - T is a subset of the set X at index k, - q is defined as the concatenation of p and the list containing X, - the pair (N, the concatenation of alpha and beta) is an element of the set R, - ...
"for any real numbers x, y, and z, the sum of x and y being less than or equal to the sum of x and z implies that y is less than or equal to z."
If the sum of 'a' and 'b' is less than one, then the probabilistic conjunction of 'a' and 'b' is zero.
This is a lemma named "gen_Sigma" in the context of generic algorithms. It assumes the following: 1. A priority tag for a generic algorithm is present. 2. There are two side generic algorithms, one converting a set to a list with relation Rk and Rs1 for iterator it1, and another with relation Rl and Rs2 for iterator i...
Lemma: Fix i as an element of a type that has both second countable topology and real inner product properties. Then, the following sets are Borel measurable: 1. The set of all elements x such that a is less than the inner product of x and i. 2. The set of all elements x such that the inner product of x and i is ...
Setting the third bit of the binary number 0 to true results in the binary number 1000, in the context of a word of any length.
In the context of a flowgraph, the lemma named "ntrs_modify_context_s" is stated as follows: Assuming that: - A pair of states and contexts ((s,c),ee,(s',c')) is in the non-terminal relation set of the flowgraph, and - The intersection of the write monitor of the flowgraph and the context monitor of the flowgraph for ...
If X and Y are non stuttering invariant, then the function f of X and Y is also non stuttering invariant.
If a number 'n' is greater than zero, then the product of 'n' and infinity is infinity.
The property P holds for the subpath from 0 to 0 of the function g at point z. 2. The second case (subEq) states that if a value s is in the interval from 0 to 1 (excluding 1), and either s is 0 or the real part of g at s is equal to the real part of z, and for all t in the interval from s to 1 (excluding 1) the real ...
The lemma states two things about making a graph G symmetric: 1. The set of pointed vertices in the symmetric version of graph G is the same as the set of vertices in graph G. 2. The set of pointed arcs in the symmetric version of graph G is the union of two sets for each arc in graph G: one set contains the tail and...
The image of a single element 'a' under the function 'f' belongs to the core of the range of function 'f'.
Given variables a, b, and t, if A and B are both greater than 0 and t equals 0, then assign A to a and increment t by 1, assign B to b and increment t by 1. Then, while a is not equal to b, maintain the invariant that a and b are both greater than 0, the greatest common divisor (gcd) of A and B equals the gcd of a and ...
The lemma named "lmap_preserve" with the attribute "quot_preserve" assumes two conditions: 1. "R1", "abs1", and "rep1" form a Quotient3 relation, denoted by "a". 2. "R2", "abs2", and "rep2" form another Quotient3 relation, denoted by "b". It then shows two results: 1. The function "lmap" remains unchanged when it is ...
For any given functions m, k, and h, and variables x and y, the operation of binding y to the result of binding x to m and then applying k to x, and then applying h to y, is equivalent to the operation of binding x to m, and then binding y to the result of applying k to x, and then applying h to y. This is a statement ...
If an extended thread action (extTA), a program (P), and a thread (t) can transition from a state with expressions (es) and a state (s) to a new state with expressions (es') and a new state (s'), and if the program and the heap of the initial state are well-typed for the thread, then the program and the heap of the new...
The lemma named "nonneg_bdd_above_has_sum" is about a function 'f' that maps from any type 'a to a type 'b. The type 'b is a conditionally complete linear order, an ordered commutative monoid with addition, and a linear order topology. The lemma assumes two things: 1. For any 'x' in the set 'A', the value of functio...
The unsigned integer representation of an unsigned 16-bit integer "x" is equal to the absolute unsigned integer representation of the unsigned cast of the representative of the unsigned 16-bit integer "x".
It is not possible for the value of v in the next state to be equal to the value of v in the current state.
A finite function (finfun) is equal to the universal set (UNIV) of all functions from a finite set 'a to any set 'b.
: Assuming for all m less than k plus 3, the request at time t plus m is not equal to 'send', and m is less than or equal to k, it shows that the request at time t plus m plus 2 is not equal to 'send'.
If P is an equivalence relation under a function phi, X is a quotient set of P under phi, x is an element of X, and phi relates x and y, then y is also an element of X.
The lemma named "finite_bindI" assumes two conditions: first, that the set 'm' is finite, and second, that for any element 'x' in the set 'm', the function 'f' applied to 'x' results in a finite set. Given these assumptions, it shows that the set resulting from applying the function 'f' to each element 'x' in the set '...
The spanning forest is equivalent to the basis of 's'.
A list "xs" is sorted as an empty list if and only if "xs" is an empty list.
The Bell-LaPadula model specification is a valid requirement.
The interaction is bounded by the lift of a single probability mass function (spmf), given a certain consideration and a number 'n'.
There exists a function 'f' that maps from a finite type 'a' to a natural number, such that 'f' is injective (i.e., no two different elements in 'a' map to the same natural number).
Walking through an indexed array with the function 'f' applied to an array that starts with 'a' followed by some elements 'xs', starting from the element at position 'x', with a list 'l', and a step count of 'p+1', is equivalent to walking through the same array without the first element 'a', starting from the same pos...
If set X is finite, then the cardinality (or size) of the Cartesian product of X with itself is equal to the square of the cardinality of X.
The integer "i" adheres to the show law as defined by the function "showsp_int".
The function "recognises_execution" with parameters e, s, r, t is equivalent to the function "recognises_prim" with the same parameters e, s, r, t. This is a code lemma, meaning it is used for code generation.
: assuming the rate of return at time 0 is 0, the rate of return at any time is less than 1, and the interest rate is non-negative, then the balance of a ledger with a certain capital is equal to the balance of the updated ledger with the capital multiplied by (1 plus the interest rate). This is represented symbolicall...
If "a" is less than or equal to "b", then the infimum (greatest lower bound) of "a" and "c" is less than or equal to the infimum of "b" and "c".
If the execution of writer transformer 'a' is less than or equal to the execution of writer transformer 'b', then 'a' is less than or equal to 'b'.
In a parallel computation, for any index 'i' ranging from 0 to 'n', if we start with a true precondition and set the 'i'-th element of array 'a' to 0, the postcondition that the 'i'-th element of array 'a' is 0 will hold. At the end of the computation, it is guaranteed that all elements of array 'a' with indices less t...
Lemma: 1. The natural number 0 is equal to 0. 2. The natural number 1 is equal to 1. 3. The natural number 0 is less than 1. 4. The natural number 0 is less than or equal to 1. 5. The natural number 123,456,789 is less than 2,345,678,901.
The symmetric closure of the union of two sets A and B is equal to the union of the symmetric closures of A and B.
If 't' is a proper subterm of 'u' and 'u' is a subterm or equal to 'v', then 't' cannot be equal to 'v'.
If there exists some 'y' and 'c' such that for some 'x', the conjunction of 'P x', 'R x', 'Q y', and the assertion that both 'b' and 'c' are true holds, then it necessarily follows that there exists some 'x' for which the assertion 'b' is true and the conjunction of 'P x', 'R x', 'Q y', and the assertion 'c' is true.
The lemma "transT_not_mustT" assumes that from state "s", "c" leads to state "s'" in a transition "t". It shows that "c" must transition from state "s".
For any category C, if a signature S supports a morphism from A to B with variable x and expression e, and also supports a morphism from B to C with variable x and expression d, then the signature S also supports a morphism from A to C with variable x and the expression obtained by substituting e in d.
If a set A has the property of commutation, then for any two elements x and y in set A, the product of x and y is equal to the product of y and x.
If a node "r" with a finite set "xs" is well-formed according to the "wf_darcs" function, and if "t1" is an element in the set of first elements of "xs", then "t1" is also well-formed according to the "wf_darcs" function.
For any set Y of type V, if Y is small, then the multiplication of x with the supremum of Y is equal to the supremum of the set obtained by multiplying x with each element of Y.
If a well-formed tuple "xs" has a length of "b + n" and belongs to set "A", and set "B" is defined as the set of all elements "x" from set "A" that are greater than or equal to "b" with "b" subtracted from them, then dropping the first "b" elements from the tuple "xs" results in a well-formed tuple of length "n" that b...
The hyper-union of A and B, then hyper-union with C, is equal to the hyper-union of A with the hyper-union of B and C. This is the associativity property of hyper-union.
Mapping the function 'Dummy' over list 'l' is equal to mapping 'Dummy' over list 'l'' if and only if list 'l' is equal to list 'l''. This implies that the function 'Dummy' is injective, meaning it assigns a unique value to each element of the list.
The "iteratei" function of "ahs" is a proper iterator.
If set B is a subset of set C, then the intersection of sets A and B is also a subset of set C.
: If two commands 'c' and 'd' are approximately equal in terms of termination, and it is necessary that 'c' terminates in state 's' and 'd' terminates in state 't', and states 's' and 't' are equivalent, and command 'd' can transition from state 't' to state 't'' through zero or more steps, then there exists a state '...
The i-th element of the zero vector is zero.
The lemma named "lActive_equality" assumes that "c" is parallel at an index "i" in a trace "t", and for any "x", if "c" is parallel at "x" in the trace "t", then "x" is less than or equal to "i". Under these assumptions, it shows that the index of the first occurrence of "c" in the trace "t" is equal to "i".
For a list 'xs' of elements from a preorder set, appending 'ys' to 'xs' is less than appending 'zs' to 'xs' if and only if 'ys' is less than 'zs'.
The lemma named "subprob_density_distr_real_inverse_aux" assumes that "f" is a subprobability density function on the measure space "M" with respect to the Lebesgue measure "lborel". It shows that the function "g", defined as the product of "f" evaluated at the negative inverse of "x" and the non-negative real number "...
The lemma named "dim_eq_card" assumes that the span of set B is equal to the span of set V and that set B is independent. It shows that the dimension of set V is equal to the cardinality of set B.
The lemma named "eval_ring_hom" assumes that "a" is an element of the carrier set "R". It shows that the evaluation of "a" in the ring "R", using the identity function, results in a ring homomorphism from "P" to "R".
: Assuming that the product of 'p' and 's' is less than or equal to the product of 's', 'p', and 's', and the product of 'p' and 'w' is less than or equal to the product of 'w', 'p', and 'w', and the Hoare triple with precondition 'p' times 's', command 'x', and postcondition 'p' holds true, then it shows that the Hoa...
The lemma named "algebraic_int_inverse" assumes that the polynomial p evaluated at x equals zero, every coefficient of the polynomial p is an integer, and the coefficient of the zeroth term of the polynomial p is 1. Under these assumptions, it shows that the inverse of x is an algebraic integer.
If a pair of worlds (w', w'') is in the transitive closure of the union of relations of a general model M at world w for all actions in as, then the world w'' is in the set of worlds of the general model M at world w.
The lemma named "right_angle_to_right_angle" assumes that there is a right angle at point 'a' between points 'p' and 'q', and another right angle at point 'b' between points 'r' and 's'. It shows that there exists a transformation 'J' that is a K2 isometry (a distance-preserving transformation) such that when 'J' is ap...
The field of the transitive symmetric closure of a relation 'r' is equal to the field of the relation 'r' itself.
If a function 'f' is analytic on a set 'S', then the function that maps 'z' to the negation of 'f(z)' is also analytic on the set 'S'.
For the lemma named iter_aux: If for all states 's' and 's'', if the semantics of 'c' transition 's' to 's'' implies that 's' is in 'I' and 's' is in 'b', then 's'' is in 'I'. Then, for any states 's' and 's'', if 's' is in 'I' and 's' transitions to 's'' through 'n' iterations of 'b' using the semantics of 'c', then ...
The theorem "construct_tree_total" declares variables e, v, and D. It starts with the precondition that a tree is constructed from x to y in relation R. It initializes D to 0 and v to x. Then, it enters a loop that continues as long as y is not less than or equal to v. The loop maintains the invariant that the tree con...
The lemma "nsqn\<^sub>r_invalidate" assumes that "dip" is an element of "kD(rt)" and "dip" is also in the domain of "dests". It shows that the "nsqn\<^sub>r" of the invalidation of "rt", "dests", and "dip" is equal to the value of "dests dip" minus one.
The lemma named "dg_10_components" demonstrates the following: - The object component of the 10-digraph of "aa" is equal to the set containing "aa". - The arrow component of the 10-digraph of "aa" is equal to zero. - The domain component of the 10-digraph of "aa" is equal to zero. - The codomain component of the 10-di...
If "t" is approximately equal to "t'" and "t'" is approximately equal to "t''", then "t" is approximately equal to "t''". This lemma assumes transitivity of the approximate equality relation.
The lemma named "Bag_inv_lemma'" assumes that "X" is defined at state "tau" and that the element "e" is a member of the base representation of "X" at state "tau" with a count of at least 1. It shows that "e" is valid at state "tau".
The difference between any list 'xs' and an empty list is the list 'xs' itself.