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For any two functions that transform elements from set A to set B, the application of these functions to any element from set A will also result in an element in set B. This is a transfer rule, meaning it is used to transfer properties between different representations of data.
If "s" is an element of "S", then "s" is an endomorphism of the chamber complex "X" under permutation.
The finite set of transitions in a finite state machine M is equal to the set of transitions in M.
If E is in the grounding of class D, and D is in the grounding of class C, then E is equal to D.
Applying the function "H" to the fixed point of "R" and then finding the fixed point of the result is equivalent to simply finding the fixed point of "R".
The lemma states two things: 1. The smallest prime number greater than 'n' is indeed a prime number. 2. The smallest prime number greater than 'n' is always greater than or equal to 'n'.
In the context of a probability space, the lemma named "integrable_squareD" is contributed by Johannes Hölzl. It fixes X as a function mapping to real numbers. The lemma assumes that the square of X is integrable over the measure space M and that X is Borel measurable over M. If these conditions are met, it shows that ...
If the following conditions are met: 1. The index 'i' is less than the length of the list 'Fss'. 2. The index 'i' is an element of the set 'is'. 3. The set 'is' contains distinct elements. 4. The normalized faces of the 'i'-th element of 'Fss' are distinct. 5. The normalized faces of 'newFs' are distinct. 6. The inter...
The set of edges E is the union of the set of edges in E that are loops and the set of edges in E that are simple edges.
The lemma named "fundfoldpair_unique_half_chamber_system_f" assumes that the pairs (f,g) and (f',g') are both fundamental folding pairs. If the induced automorphism of the pair (f',g') is equal to the induced automorphism of the pair (f,g), then it shows that the half-chamber system of f' is equal to the half-chamber s...
If expression "e" is not a sub-RI, then "e" is not initialized in class "C".
The lemma named "inf_continuous_sup" is about two functions P and Q, which are both defined on a countable complete lattice and map to a countable complete distributive lattice. The lemma assumes that both P and Q are inf-continuous. It then shows that the function that maps any element x to the supremum of the images ...
The well-formed signature is a tuple consisting of an empty map, another empty map, and an empty ordered signature.
For any two matrices A and B, the determinant of the product of A and B is equal to the sum over all subsets I of the set of column indices of A that have the same cardinality as the number of rows in A, of the product of the determinants of the submatrices of A and B obtained by restricting to the rows and columns in ...
A message pair "X, Y" is in the synthetic analysis of a set "H" if and only if both "X" and "Y" are in the synthetic analysis of "H".
If a is not equal to the successor of 0 (i.e., a is not 1) and the set of multiples of a is a subset of the set of multiples of b, then the remainder of a divided by b is 0.
The lemma named "combinable_imp" assumes that "A" is pure and "B" is combinable with "Delta". Under these assumptions, it shows that the implication of "A" and "B" is also combinable with "Delta".
If 'a' is an object in the category, then the domain of the object map of the contravariant functor from 'a' to another object is equal to the set of objects in the category.
For any element 'a' in the carrier set 'R', the sum of 'a' and the 'n'-th power of the polynomial 'X' is also in the carrier set of the univariate polynomial ring. Here, 'n' is a natural number.
If A implies B, then A combined with C also implies B combined with C.
The reach of "a0i" is a subset of "a0n_inv1_niagree".
Pi is in the range of the Taylor model for the precision "prec".
The set of nested propositional atoms of the second element of the function "af_letter\<^sub>\<nu>" applied to X, p, and \<nu>, is a subset of the union of the set of nested propositional atoms of the first element of p with respect to X and the set of nested propositional atoms of the second element of p.
If for all 'x', 'A x' and 'B x' are in the space of the subprobability algebra 'N', and for all 'i' in the set of probability mass function 'x', 'A i' equals 'B i', then the bind of the measure of probability mass function 'x' with 'A' equals the bind of the measure of probability mass function 'x' with 'B'.
The run of a set A is equivalent to the condition where for every element z in set A, the run of set A continues.
: 1. Sorting an empty list using any key function 'f' will result in an empty list. 2. Sorting a list that has 'x' as its first element and 'xs' as the rest of the elements, using a key function 'f', is equivalent to inserting 'x' into the list that results from sorting 'xs' with the same key function 'f'. This inserti...
For all elements 'x' in the set that results from grouping 'xs' by 'f', 'x' is not an empty list.
"u" is an element of "V".
For any integers 'a' and 'p', if 'p' is a prime number, then 'a' is congruent to 0 modulo 'p' if and only if 'a' and 'p' are not coprime.
In the context of a group, if H is a subgroup of G, then the product of H and the identity element is H.
The 8-bit word representation of the number 42 is less than or equal to negative 1705.
If a node "r" with a single edge "e1" to a subtree "t1" is a subtree of "t", and the maximum degree of this node is greater than 1, and "v" is a vertex in the subtree "t1" but not the root of "t1", then for every element "y" in the set "v", there exists an element "x" in the set of the root of "t1" such that "x" can re...
: Assuming that the function "find_endpoint_path_inv" is valid for the graphs G and H, the length 'len', and the vertices u, v, w, and x. Also assuming that the graph G is a valid instance of an IGraph, the graph G is loop-free, and the function "select_nodes_prop" is valid for the graphs G and H. Additionally, assumi...
For every point 'q' in the set 'ps', the distance between point 'p' and the closest point to 'p' in 'ps' found by brute force (find_closest_bf p ps) is less than or equal to the distance between 'p' and 'q'.
If a function H is Functional Conjunctive, then it is also Monotonic.
If I is an ideal of ring R, and if x and y are elements of I, then the sum of x and y (denoted as x \<plusminus> y) is also an element of I. This property is known as the closure of an ideal under addition.
The lemma named "unop_v11E" assumes that the unary operation "unop_v11" is valid for a set A and a function f. From this assumption, it can be obtained that the nullary operation "nop_v11" is also valid for the set A, the natural number 1, and the function f.
The sum of Matrix A and Matrix B is equal to a new Matrix, where each element at position (i, j) is the sum of the corresponding elements at position (i, j) in Matrix A and Matrix B.
If for all pairs of states c1 and c2, the equivalence of the local state of c1 to the local state of c2 is the same as the equivalence of the local state of c2 to the local state of c1, then the execution equivalence of a model with respect to these states, input functions, output functions, transition functions, and c...
The lemma named "uminus_rel_frac" assumes that the pair (r, s) belongs to the Cartesian product of sets R and S. It then shows that the additive inverse relation of the fraction r/s is equal to the fraction formed by the additive inverse of r divided by s.
If "x" is an element of the carrier Zp, and the successor of "n" applied to "x" is not equal to zero, then "n" is greater than or equal to the order of "x" in Zp.
If a value 't' is an element of set 'U', then for all values 'x', 'l', and 'u', if for all 't', 't' is greater than 'l' and less than 'u' and 't' is not an element of 'U', and 'x' is greater than 'l' and less than 'u' and 'x' equals 't', then for all 'y', if 'y' is greater than 'l' and less than 'u', 'y' must equal 't'...
If x is less than or equal to the maximum possible value (top), then x is less than or equal to the result of function d applied to x.
If 'x' and 'y' are both less than 2 raised to the power of the length of 'a', then 'x' being equal to 'y' when both are converted to natural numbers is equivalent to 'x' being equal to 'y'. Here, 'a' is a word of some length.
The application of the function "Omega_fun" with parameters f, g, h, and p is equal to the intersection of the application of function g with parameter p and the application of function f with parameter h applied to p.
: 1. The semantics of a term, when shifted by 0 and lifted, is equal to the semantics of the original term. 2. The semantics of a list, when shifted by 0 and lifted, is equal to the semantics of the original list.
The lemma named "del_point_order" assumes that "p" is an element of "V". It shows that the cardinality (or the number of elements) of the set obtained by deleting the point "p" is equal to "V" minus 1.
The lemma named "direct_decompE" assumes that "A" is a direct decomposition of "ss" and "a" is an element of "A". It then concludes that there exist "qs" such that "qs" is in the listset of "ss" and "a" is equal to the sum of the list "qs".
If the type of the heap 'h' is well-formed and the element pointer is in the set of element pointer kinds of 'h', then the operation of setting the tag name of the element pointer to 'tag' is a valid operation in the heap 'h'.
The positive part of a hereditary multiset 'a' plus the positive part of a hereditary multiset 'b' plus the difference between the negative part of 'a' and the positive part of 'b', added to the difference between the negative part of 'b' and the positive part of 'a', is equal to the negative part of 'a' plus the negat...
If the function "sig_red_single" is applied to the single regular sequence "sing_reg", the top tail "top_tail", and the polynomials "p", "q", "f", and "t", then the leading term of the representative list of "q" is less than or equal to the leading term of the representative list of "p" in the polynomial unit order.
The join of a set X is equal to the union of the set of all elements in X.
If 'x' is an element of the set of extended integers and 'x' is not equal to infinity, then there exists an integer 'z' such that 'x' is the antipode of 'z'.
The lemma named "runit_char_eqn" assumes that "a" is an identity. It shows that the product of the right unit of "a" and the identity element is equal to the product of "a" and the unit object, composed with the associativity of "a" and two identity elements.
If the following conditions are met: 1. The limit inferior (Liminf) of the sequence of local classes of states (lclss_of_state) in the state sequence (Sts) is saturated according to the First-Order Logic (FL) saturation definition, and 2. The set of passive elements in the limit inferior of the sequence of local class...
: Assuming the following conditions: 1. The program state 'prog' and process 'p' satisfy the invariant 'pState_inv'. 2. The pair (g,p) satisfies the invariant 'cl_inv'. 3. For any 'k', 'v'', and 'cs', if the function 'toVariable' applied to 'g', 'p', and 'v' equals the tuple (k,v',cs), then the pair resulting from upd...
The lemma fr_nGet2a assumes two conditions: 1. The FlexRayController is defined with parameters nReturn k, recv, nC k, nStore k, nSend k, and nGet k. 2. The time t modulo the cycle length of nC k is not a member of the schedule of nC k. If these two conditions are met, it shows that the value of nGet k at time t is...
The lemma "dim_kernel_non_zero_jordan_block_pow" assumes that "a" is not equal to zero. It shows that the dimension of the kernel of the power of a Jordan block of size "n" and eigenvalue "a" is zero.
For a given predicate P, a stream composed of an element x followed by a stream xs, and another stream ys, it is true if and only if there exists an element z and a stream zs such that ys is composed of z followed by zs, and the predicate P holds for x and z, and also holds for the streams xs and zs.
If a command "c" starting in a state satisfying precondition "P" always ends in a state satisfying postcondition "Q", and for all states "s", "Q" of "s" is greater than or equal to "Q'" of "s", then it can be inferred that the command "c" starting in a state satisfying precondition "P" always ends in a state satisfying...
The lemma named "at_set_avoiding2_atom" assumes that the support of c and x are finite, and that a is fresh for x. It shows that there exists a permutation p such that the permuted a is fresh for c and the support of x is fresh for p.
If the security level of an expression is Low or the security level of a variable is High, then the assignment of the variable to the expression is compatible in the Example Programming Language.
If functor G maps category A to category C, functor H maps category B to category C, and functor F is a tiny functor from category J to the comma category of G and H, then the composition of the pullback functor of G and H with F is a tiny functor from category J to category A.
The lemma named "line_integral_sum_gen" assumes the following: 1. The basis is finite. 2. The line integral exists for the function F with respect to basis1 and basis2 along the path γ. 3. The union of basis1 and basis2 equals the basis, and the intersection of basis1 and basis2 is empty. Under these assumptions, it ...
The lemma named "path_append_elim" assumes that there is a path in machine M from state q following the sequence of transitions (p1 followed by p2). Under this assumption, it concludes two things: first, there is a path in machine M from state q following the sequence of transitions p1, and second, there is a path in m...
, in a loop where the condition "q" is checked and the operation "R" is performed on the combination of "t p" and "t q" with "p", the result is less than or equal to the operation "R" performed on "p" and the combination of "t p" and "n q".
The power of a literal value 'x' raised to 'n' is equal to the literal value of 'x' raised to 'n'.
The function "to_qr", which maps a polynomial over a modular ring to a quotient ring, is injective (or one-to-one) on the set of all polynomials whose degree is less than the degree of the quotient ring.
The lemma named "Arcsin_sin" assumes that the absolute value of the real part of a complex number 'z' is less than pi/2, or that the absolute value of the real part of 'z' is equal to pi/2 and the imaginary part of 'z' is zero. Under these assumptions, it shows that the arcsine of the sine of 'z' is equal to 'z'.
For any given option 'x', applying a function 'f' to the value 'v' if 'x' is 'Some v', or returning a default value 'a' if 'x' is 'None', and then wrapping the result in 'RETURN', is equivalent to first wrapping the default value 'a' or the result of applying 'f' to 'v' in 'RETURN', and then applying 'case_option' to '...
The lemma consists of several statements: 1. If the function 'f' is not a continuous map from space 'X' to space 'Y', or the image of the intersection of the topological space 'X' and set 'S' under 'f' is not a subset of 'T', or 'c' is not an element of the carrier of the relative homology group of 'p', 'X', and 'S', ...
The lemma named "non_interference_induct_case" assumes the following: 1. For all 'i' and 't', if 'i' is less than or equal to 'n' and the sequence 'SEQ' at index 'i' for 's1' equals 't', then 't' is a good context, the delayed pool of 't' is empty, and the trap set of 't' is empty. The same conditions apply for 's2'. ...
If a list "l" is finitely long, then the neighborhood of "l" is equal to the principal set of "l".
For any two formulas phi and chi, if phi and chi are in the set of formulas and they have no free variables, then it is provable that if the provability predicate of the implication of phi and chi holds, then the provability predicate of phi implies the provability predicate of chi.
The lemma named "eqExcNth_trans" assumes that the sequence "xs" is equal to the sequence "xs1" except at the nth position and that the sequence "xs1" is equal to the sequence "xs2" except at the nth position. It shows that the sequence "xs" is equal to the sequence "xs2" except at the nth position.
"None" is in the results of the generative probabilistic value (gpv) after a catch operation if and only if the gpv is not colossless.
The extension of a set through a function h, where the set is defined by a property P of its elements, is equal to the set defined by the same property P applied to the function f of its elements.
The function which maps a real number x to the natural logarithm of x divided by x, tends to 0 as x tends to infinity.
If an extension order "gt" holds for the pair of lists "ys" and "xs", then the same extension order "gt" also holds for the pair of lists obtained by appending another list "zs" to the left of both "ys" and "xs".
The lemma "RModuleHom_restrict0_submodule" assumes that "N" is a submodule. It shows that the restriction of the module homomorphism to "N" is also a module homomorphism. Here, "R" is the ring, "smult" is the scalar multiplication in the original module, "smult'" is the scalar multiplication in the submodule, and "T" i...
If "x" is a very well-behaved lens, then the sequential program of "p" with "x" assigned to "e" is equivalent to there existing a value "v" such that "p" with "v" substituted for "x" and "x" is unconditionally equal to "e" with "v" substituted for "x".
The lemma named "vsubst_compose1" assumes that "X" is good. It shows that substituting "y1" for "y" in "X" and then substituting "y2" for "y" in the result is equal to substituting "y1" with "y" replaced by "y2" in "X".
If a method "im" is in the accessible methods of a package "pack" for an interface "I" with a signature "sig" in a given context "G", then this method "im" is in the methods of the interface "I" with the same signature "sig" in the same context "G", and the interface "I" is accessible in the package "pack" according to...
The imaginary unit 'i' raised to the power of 'n' times 2 is equal to negative one raised to the power of 'n'.
The lemma assumes a bisimulation between E and E' with respect to a set of variables VS and XS, and that E is in its final state. It shows two things: 1. If there is a bisimulation between e and e' with respect to a set of variables Vs and xs, and if the function call of e equals aMvs, and if the free variables of e a...
If a certain atom 'A' is in the atoms of clause 'C', and if clause 'C' is a ground clause, then atom 'A' is a ground atom.
If a variable 'v' is in the domain of 'Γ' or 'v' is in 'C', then adding the annotation 'Acq v AsmNoWrite' to 'Γ' does not change 'Γ'. This is an idempotent operation.
The set of edges in a walk "ys" is a subset of the set of edges in a walk that is the concatenation of "xs" and "ys".
The lemma named "intruder_deduct_GSMP_induct" assumes the following conditions: 1. A term 't' is derivable from a set of messages 'M' and a set of terms 'T' under the GSMP deduction system. 2. For any term 't' in 'M', a property 'P' holds for 'M' and 't'. 3. For any function 'f' and a list 'S' such that the length of ...
, assuming "z" is not a non-negative real number, the function "Arg2pi" is continuous at "z".
If Q is in the flat disjunction of Q', and if Q' satisfies the restricted range bounds condition, then Q also satisfies the restricted range bounds condition.
If for all k less than or equal to (Suc n), the following conditions hold: 1) J k is an ideal of the ring R, 2) B k is the quotient ring of R and J k, 3) S k is the projection of R onto J k, 4) for all i and j less than or equal to (Suc n), if i is not equal to j, then the ideals J i and J j of R are coprime, then...
The lemma states two things about tensor products. Firstly, the vector of the tensor product of A and B is equal to the concatenation of the map of the scalar multiplication of a and the vector of B over the vector of A. Secondly, the dimensions of the tensor product of A and B are equal to the concatenation of the...
Lemma: Simple Function Property Assumptions: Let g be a simple function on a measure space M. Conclusion: Then, the following properties hold: 1. The image of g on the space M is finite. 2. The preimage of any set X under g, intersected with the space M, belongs to the sets of M.
A table with no elements and 'n' attributes, denoted as "table n {}", is equivalent to 'X' if and only if 'X' is an empty table or 'X' is a unit table with 'n' attributes.
If a list 'xs' is a permutation of a set 'A', then the length of the list 'xs' is equal to the cardinality (or size) of the set 'A'.
The relation "MV_Rel" is right unique.
If the pair (x, y) is a solution to the linear Diophantine equations a and b, and x is not zero, and there does not exist a pair (u, v) that is also a solution to the equations a and b where u is not zero and the vector (u, v) is less than the vector (x, y), then the pair (x, y) is a part of the minimal solutions to th...
The subgroup generated by a set S in group G is also a group.
If a function 'f' is integrable over a set 's' and for all 'x' in 's', 'f' at 'x' equals 'g' at 'x', then 'g' is also integrable over 's'.