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If "j" is a non-negative integer and less than "n", then the auxiliary interference freedom holds between the function "Mul_Redirect_Edge" with parameters "j" and "n", and the function "Mul_Count" with parameter "n".
If a pair of states (s, s') is in a certain relation, the stack implementation 's' is not empty, 'vs' is in 'Vs' but not in 'R', and the head of the stack 's' is a pair (u, Vs), then the head of the stack 's'' is a pair (u, insert vs (Vs-R)). This means that 'vs' is inserted into the set difference of 'Vs' and 'R'.
If "E" is a unit group, "h" is a group chain of length "Suc n" (successor of n), "k" is not equal to "Suc n" but less than or equal to "Suc n", and "h k" is not isomorphic to "E", then the composition of "h" and the transposition of "k" and "Suc n" at "Suc n" is not isomorphic to "E".
The tau-divergence of the initial-final transition system for a given thread 't' and initial-final tau-move, given the pair ((status, x), m), is equivalent to the tau-divergence of 't' given the pair (x, m) and the status being 'Running'.
The value range of the object map of the Yoneda functor is a subset of the objects in the category of functors from the original category to the category of sets.
Given the assumptions that the execution of steps from state "s" with left tape "l" and right tape "r" to state "s'" with left tape "l'" and right tape "r'" in "stp" steps using Turing machine "A" starting from position 0, and that the final state "s'" is not 0, it shows that the execution of steps from state "s + off"...
If B is an element of the multiset F, and A is less than or equal to C under the multiset extension of a well-quasi-order P on F, and B is less than or equal to D under the same order, then the sum of A and B is less than or equal to the sum of C and D under the same order.
The lemma named "Ord_linear_le" assumes that "k" and "l" are ordinals. It then concludes that either "k" is less than or equal to "l", or "l" is less than or equal to "k".
The reversed string 'u' is in the set of reversed strings 'ws' if and only if the string 'u' is in the original set 'ws'.
: Given a set Y, for all pairs of elements xs and ys in Y, either xs is equal to ys or the intersection of the sets xs and ys is empty. Also, U and V are elements of Y, and V is not an empty set. If for all elements xs in Y, xs is a sublist of the list formed by concatenating as, U, bs, V, and cs in that order, then f...
The forward weak delay bisimulation divergence with parameters final2, r2, final1, r1, the flipped version of bisim function, the flipped version of bisim_wait function, tau move2, and tau move1 is equal to the forward weak delay bisimulation divergence with parameters final1, r1, final2, r2, bisim function, bisim_wait...
The set of propositions in a formula "phi" is a subset of the nested propositions in the same formula "phi".
The term bound at 0 for a term 't' is equivalent to the term bound0 for the same term 't'.
The size of a multiset obtained by repeating a multiset 'A' 'n' times is equal to 'n' times the size of 'A'.
, given the assumptions that a relation 'A' is bi-unique and right total, it shows that the semiring_0_ow function applied to the collection of the domain of 'A' is equivalent to the class.semiring_0, when both are applied to the same arguments. This is denoted by the placeholder "?PR". The functions are applied to two...
The iteration of "x" is less than or equal to 1.
The lemma "FOREACHoci_rule" in the Refine_Monadic refinement framework states the following: Assume that: 1. "S" is a finite set. 2. The predicate "I" holds for the set "S" and the initial state "σ0". 3. For any element "x" in a subset "it" of "S", and any state "σ" such that the condition "c" holds for "σ", "I" holds...
: Assuming that the steps from state (s, l, r) with machine (A, off) for a certain number of steps 'stp' results in state (s', l', r'), and that s' is not equal to 0, it shows that the steps from state (s, l, r) with machine (A appended with B, off) for the same number of steps 'stp' will also result in state (s', l',...
Lemma subtr2_rootL_in: Assuming that "tr2" is a subtree of "tr1" with respect to the set "ns", it can be shown that the root of "tr1" is an element of the set "ns".
If a function 'f' is smooth on a set 'S', then the Frechet derivative of 'f' at any point 'x' in 'S', when applied to 'v', is also smooth on 'S'.
If you convert a formula 'p' to a natural number using the function 'nat_of_fm' and then convert it back to a formula using the function 'fm_of_nat', you will get the original formula 'p'.
The measure of a probability mass function p is not equal to zero.
If the function "stored_reused" applied to two terms "t'" and "t" equals "Some (p, r)", then "p" and "r" are a generalized output of "t'" and "t".
The ribbon diagram associated with the assignment "P" is equal to the ribbon diagram interpretation of "P".
If for any function 'f', where 'f' at 0 equals to a tuple (cs, css, s) and for any integer 'i', the transition from 'f' at 'i' to 'f' at the successor of 'i' is valid under a context 'Γ', leads to a contradiction, then it is not the case that 'inf' under the context 'Γ' for 'cs', 'css', and 's' is true.
Given an object 'a' in the category of sets, the inverse of the natural transformation of the pointed inverse of 'a' is equal to the natural transformation of the pointed 'a'.
If x and y are less than z, and x, y, and z are all greater than or equal to zero, then the modulus of the sum of x and y with respect to z is equal to the sum of x and y if the sum of x and y is less than z, otherwise it is equal to the sum of x and y minus z. This is applicable for x, y, and z as integers.
The theorem named "standard_cone_decomp_snd_split" is defined with a fixed set F. It also defines G as the reduced Gröbner basis of F, ss as the split of 0, X, and the leading power product of G, and d as the successor of the maximum of the degree of the polynomial in the set of the second element of ss. It assumes tha...
The lemma named "integrable_ln_Gamma_aux_real" assumes that "s" is greater than 0. It shows that the function which maps "x" to the Bernoulli polynomial of "x" divided by "(x + s) to the power of n" is integrable on the interval from 0 to the real number "N".
: 1. The initial state of a finite state machine (FSM) with precomputations from a list where 't' is the first element and 'ts' are the remaining elements, is the source of 't'. 2. The states of this FSM are the union of the sources and targets of all elements in the list (including 't' and all elements in 'ts'). 3. T...
If a red-black tree (RBT) "t" is a valid RBT, then inserting a new key-value pair (k, v) into the tree using the function "f" will also result in a valid RBT.
If x is a monotonic write-protected lens, then the right-hand side user variable of x is also a monotonic write-protected lens.
Applying the "upds" function from the "TWu" module to "upd_uinfo", "uinfo", and "hfs" results in "uinfo".
The length of the list obtained by applying the transform function 't' to each element 'ae' in the list 'xs' is equal to the length of the original list 'xs'.
For a given vertex 'v' in the directed graph of a node 'r' with a finite set 'xs', 'v' is either equal to 'r' or 'v' is in the union of the directed vertices of 't' for each pair '(t,e)' in the finite set 'xs'.
The lemma named "inst_ok_code1" assumes that the theory 'Theta' is well-formed according to 'exe_wf_theory'. It shows that for a list of instances 'insts', checking if each type in the list is okay according to 'exetyp_ok' and 'Theta' is equivalent to checking if each type is okay according to 'typ_ok' and the translat...
The lemma named "is_rewrite_ord_canon_rewriterD1" assumes that "rword" is a rewrite order, "A" is a canonical rewriter for "u" with respect to "p" and "v" with respect to "q", the leading term of "p" adds to "v", and the leading term of "q" adds to "u". Under these assumptions, it shows that the leading term of "p" is ...
The instructions in a list 'xs' are equivalent to the concatenation of the instructions in the sublist of 'xs' taken while a certain condition 'P' is met and the instructions in the sublist of 'xs' after dropping elements while the same condition 'P' is met.
The sequence generated by the function 'incseq_of' applied to 'x' is an increasing sequence.
The lemma named "Gr_insert" shows that the graph of the function "f" over the set obtained by inserting an element "x" into a set "F" is equal to the set obtained by inserting the pair "(x, f x)" into the graph of the function "f" over the set "F".
Given the assumptions that "H" is a Poincare line, the ideal points of "H" are "i1" and "i2", and the complex number representation of "i1" is a constant "k" times the complex number representation of "i2", it can be shown that the zero hyperbolic point is in the set of points on the circline "H".
The lemma named "cindex_pathE_linepath_on" assumes that "z" is in the closed segment between "a" and "b". It shows that the complex index of the line path from "a" to "b" at "z" equals zero.
If we have elements 'e' in the arcs of graph G and 'w' not in the vertices of graph G, and a subset 'V' of the vertices of graph G, and a path 'p' from 'u' to 'v' in the graph obtained by subdividing graph G at 'e' with 'w', then two things hold: 1. There is a path from 'u' to 'v' in the graph obtained by replacing 'e...
The lemma "recv_marker_means_snapshotted_1" assumes that an event "ev" is the receipt of a marker from process "i" by process "p" from process "q", and that this event transitions the system configuration from "c" to "c'". It shows that after this event, process "p" has taken a snapshot in the new system configuration ...
An empty sequence in the context of variable-free sequences corresponds to an empty list.
The valuation of the function f at point a is greater than the valuation of the derivative of the function f at point a in the ring of p-adic integers.
Mapping a homomorphism onto a polynomial after applying a polynomial square substitution is equivalent to applying a polynomial square substitution onto the polynomial after mapping the homomorphism.
The function 'f' is differentiable up to order 'l' between the chart sets 'charts1' and 'charts2', if 'l' is less than or equal to 'k'.
If a is equal to b, then the equality EQ of a and b holds true.
: given a polynomial 'p' that belongs to the carrier of the polynomial ring 'R' and has a degree of 1, 'x' is a root of 'p' if and only if 'x' is equal to the negation of the product of the inverse of the leading coefficient of 'p' and the constant term of 'p'.
The lemma named "match_matches" assumes that the match function, when applied to terms 't' and 'p', results in an optional substitution 'σ'. It then shows that applying this substitution 'σ' to 'p' results in the term 't'.
If the term "t" does not contain an absolute value, then the interpretation function "Ifm" applied to the boolean list "bs" and the result of the function "gt" applied to the result of "rsplit0" on "t" is equal to the interpretation function "Ifm" applied to "bs" and "Gt t". Additionally, the result of the function "gt...
, for a given state sigma, the validity of the sequence where we first perform the monadic bind operation with an empty list and a function f, and then perform the unit operation with a predicate P applied to the state s, is equivalent to the predicate P applied to an empty list.
If "t" reduces to "t'" (denoted by "t \<rightarrow>* t'"), then "s" applied to "t" (denoted by "s \<bullet> t") also reduces to "s" applied to "t'" (denoted by "s \<bullet> t'").
The composition of the exponential of a formal power series "d" (where "d" is from a field with characteristic 0) with the product of a constant "c" and the formal power series "X", is equal to the exponential of the product of "c" and "d".
If 'f' is an open map from topological space 'X1' to 'X2', and 'S' is a subset of the topological space 'X1', then the image of 'S' under 'f' is a subset of the topological space 'X2'.
For a function 'f' mapping from any type 'a' to extended real numbers, the product of 'f' over all elements in a set 'A' equals zero if and only if 'A' is a finite set and there exists at least one element in 'A' for which the function 'f' equals zero.
The property P holds for the heap' if executing function f on the heap results in an optional value (v, heap').
If a function f has a sub-probability density with respect to measures M and N, then it also has a probability density with respect to measures M and N.
If p is less than zero, then the relative homology group of p, X, and S is a trivial group.
For a given set 'S' of type 'x' and an element 'x' of type 'a', assuming that 'a' is a permutation type of 'x', 'x' is an atom type, 'S' supports 'x', 'S' is finite, and for all sets 'S'' such that 'S'' supports 'x', 'S' is a subset of 'S'', it can be shown that 'S' is equal to the support of 'x'.
The lemma named "int_of_nat" demonstrates that the function "of_nat" applied to any natural number 'm' is equal to the function "int_of_nat" applied to the same natural number 'm'.
The lemma named "triangle_strict_reverse_inequality" assumes the following conditions: 1. Point A is outside the line segment BD. 2. The distance from A to C is congruent to the distance from A to D. 3. Point A is not outside the line segment BC. Under these assumptions, it shows that the distance from B to D is less...
If the tuple (nm,ts,t') is an element of the set R, and x is an element of the free variables of the composition of the open term t', then there exists a term t in the set ts such that x is an element of the free variables of the composition of the pattern t.
The lemma "alphaAbs_weakestEx_imp_strongestEx" assumes that "X" is a good quasi-term ("qGood X") and that there exists an alpha abstraction that is either equal or fresh for the sets "xs", "x", "X", "xs'", "x'", and "X'". If these conditions are met, then it shows that there exists a distinct alpha abstraction that is ...
If the intersection of "m2_inv2b_corrKey_leaked" and "m2_inv1_keys" holds before the transition "m2", then "m2_inv2b_corrKey_leaked" also holds after the transition.
The function "abc_list_crsp" corresponds to the list "lm" appended with "m" number of zeros.
If "k" is greater than 0, then the join operation on the function "f" with parameters "k" and a list consisting of "n" multiplied by "k", the modulus of "k", and "d", is equal to the join operation on the function "f" with parameters a list consisting of "n" to any number and "d".
Padding m to the linear combination of n and a with W is equal to the linear combination of n+m and the composition of a and the unpadding of m, with the image of W under padding m. The lemma shows that this goal function holds true for the right padding and unpadding of U, and for the left padding and unpadding of U.
A null natural transformation from the zero functor to an object 'z' in the category is equivalent to the object 'z' being an initial object in the category.
The theorem named "wlsFSw_recAll_isInjAll" assumes that the well-sortedness of the free substitution with respect to a given module "MOD" and the abstraction renaming with respect to the same module are both true. It also assumes that the constructor injection with respect to the module is true. Under these assumptions...
The lemma named "extend" is about the extensibility into a new graph. It assumes that a graph homomorphism exists between the first element of R and the labeled graph LG E from 0 to n-1, with f as the function, and that R is a graph rule. It defines g as the extension of n, R, and f, and G' as the labeled graph LG with...
The lemma named "exist_dioph" in the context of Diophantine equations, fixes a relation D. It assumes that D is a Diophantine relation. It then shows that the existence of D is also a Diophantine relation.
The maximum bound of a term 't' in the left-hand side of the partial substitution 'R' (where 'R' is stripped of its bottom elements) is an element of the finite set of states of 'R'.
The tail of an unfolded lazy list, given a certain condition, a head function, and a tail function, is equal to either an empty lazy list if the condition is met, or another unfolded lazy list with the same parameters, but applied to the tail function.
For a fixed real normed vector 'a', the function that translates each element of a set 'S' by 'a' is a homeomorphism. This means it is a continuous, bijective function with a continuous inverse, which in this case is the function that translates each element of 'S' by '-a'.
The lemma named "thetaZOCh_Sretr" assumes that a channel "ch" is compatible, and that "c1" is probabilistically equivalent to "d1" and "c2" is probabilistically equivalent to "d2". It shows that the set of all pairs of states in the thetaZOCh relation for "ch", "c1", "c2", "d1", and "d2" is a subset of the set of all p...
The function which maps any real normed algebra element 's' to the dot product of 's' and 1 is surjective.
The function "nat_to_pr" when applied to 0 is equal to a function that returns 0 for any input x.
The lemma named "pmf_of_set_impl_code_alt" assumes that a set "A" is not empty and is finite. It shows that the implementation of the probability mass function of the set "A" is equal to the following: let "p" be the reciprocal of the cardinality of set "A" (i.e., 1 divided by the number of elements in set "A"). Then, ...
Given the assumptions that a function call "Callcl cl" is well-typed under a context "\<S>\<bullet>\<C>" with type "(t1s' _> t2s')" and that the function "cl" is a native function with type function "tf", local types "ts", and body "es", it can be shown that there exist types "t1s", "t2s", and "ts_c" such that "t1s'" i...
The domain of the function "redT_updT" applied to "ts" and "ta" is finite if and only if the domain of "ts" is finite.
If 'g' is an element of the carrier 'G' and 'x' is an element of the right coset of 'g' in 'M1', then 'x' is also an element of the carrier 'G'.
If 'x' is transitive, then 'x' is asymmetric if and only if 'x' is irreflexive.
The matrix extension of a d1-dimensional identity matrix equals a d-dimensional identity matrix.
For any real numbers 'x' and 'y', 'x' is less than or equal to 'y' if and only if 'x' is less than 'y' or 'x' is equal to 'y'.
The lemma named "polyadd_commute" assumes that 'a is a field with characteristic 0. Given two polynomials p and q, where p is a normalized polynomial of degree n0 and q is a normalized polynomial of degree n1, it shows that the addition of p and q is commutative, i.e., adding p to q is the same as adding q to p.
The lemma named "acute_per__sams" assumes that points A, B, and C are distinct, with B being the vertex of a right angle ABC, and that angle DEF is acute. Under these assumptions, it shows that the sum of the angles ABC and DEF is equal to a straight angle (SAMS stands for "Sum of Angles Makes Straight").
The lemma named "well_formed_path_io_containment" assumes that "M" is well-formed and that there is a path "p" to "q" in "M". It shows that the set of the first elements of "p" is a subset of the Cartesian product of the inputs and outputs of "M".
The function "OK" applied to "I" and the extension of function "h" with "F i" is equivalent to the function "OK" applied to "I" and "F".
If a relation R relates to S, and S is less than or equal to S' with respect to R, then S is less than or equal to S' with respect to R.
The lemma named "multeqp_code_iff_reflcl_mult" assumes that "R" is an irreflexive and transitive relation, and that for any "x" and "y", the predicate "P" holds if and only if the pair "(x, y)" is in the relation "R". Under these assumptions, it shows that the multiset equality predicate "multeqp_code" with respect to ...
The lemma named "polyfun_evaluate_net_plus_a" assumes three conditions: 1. The mapping of the dimension vector of inputs equals the input sizes of the model 'm'. 2. The model 'm' is valid. 3. The variable 'j' is less than the output size of the model 'm'. If these conditions are met, it shows that the polynomial func...
"p" is projectively equivalent to itself.
A natural number "n" is an element of the absolute natural interval that only contains "n".
For the lemma bisim1_Val_length_compE2D: If the bisimulation relation holds between the evaluation of expression 'e' to value 'v' in heap 'h' under program 'P' and the state of the stack 'stk', local variables 'loc', the length of the compiled expression 'e', and the exception 'xcp', then the stack equals to the list c...
The formal power series of a rational function with a cutoff at 'n' is equal to the formal power series of the rational function 'x' with a cutoff at 'n'.
If a list "xs" is a suffix of list "l" and "xs" is not equal to "l", then there exists an element "x" such that adding "x" to the beginning of "xs" still results in a suffix of "l".
If a tree instance 'ti' is equal to another tree instance 'ti'', and if two functions 'f1' and 'f2' refine their corresponding functions 'f1'' and 'f2'' for all values of their parameters, then the function that refines the case distinction on 'ti' (where 'ti' is either a leaf with parameters 'a' and 'b', in which case...
Assuming that for any 'x', the pre-post condition of 'x' is 'x', there exists a state 'τ' such that it is not the case that for all instances of 'OclAny' in the pre-post condition, every element 'X' in the set is of type 'OclAny'.