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The omega exponentiation of the sum of two numbers 'n' and 'm' is equal to the product of the omega exponentiation of 'n' and the omega exponentiation of 'm'.
The function f is in the polynomial set P[X].
: Assuming a system 'R' is linear, the pair (c, 0) is not in the functional relation of 'R', and the pair (c, 0) is not in the functional term 's'. If (s, t multiplied by the constant substitution of c) is in the reflexive transitive closure of the reduction step of 'R', then it shows that the pair (s, t) is also in th...
The lemma named "padic_mult_in_padic_set" assumes that "f" and "g" are elements of the p-adic set for a prime number "p". It shows that the p-adic multiplication of "f" and "g" is also an element of the p-adic set for the same prime number "p".
If the product of a rational number "c" and a variable "a" is less than or equal to "b", and "c" is less than 0, then "a" is greater than or equal to the product of the reciprocal of "c" and "b".
The function "insert" is parametric over a relation R. This means that if two elements are related by R, then their insertions into two sets that are related by the set relation of R will result in two new sets that are also related by the set relation of R.
Applying the sequence operator "..;" to a variable function "vf" and a predicate "P", then applying the result to a variable "Y" and a state "s", is equivalent to applying the predicate "P" to the variable resulting from applying the variable function "vf" to the state "s", and the second element of the pair resulting ...
If phi is a local formula, then the restriction of phi is also a local formula.
The function "the_default" is an autoref rule, where "the_default" is related to itself under the relation R, and this relation is mapped from R to an optional relation of R.
"x is less than or equal to the join of y and Z" if and only if "x is less than or equal to the product of the relative domain of x and y joined with Z".
The lemma named "span_subset_spanI" assumes that set A is a subset of the span of set B. It shows that the span of set A is also a subset of the span of set B.
The lemma named "index_ordered_less" assumes that "i" is less than "j". It shows that the index of "i" is less than the index of "j".
If "C" is a class in the program "P", then "C" is a subclass of itself in the program "P".
The limit stream of K given x is a probability space.
If a function 'f' is Borel measurable in the natural filtration at a given point 'n', then 'f' is integrable with respect to measure 'M'.
The FORK function with parameter i is equal to the normalized FORK function with parameters i and 0.
Theorem (smn theorem): Assuming that "n" is greater than 0, it can be shown that there exists a function "s" which is a primitive recursive function of "m+1" such that for all "p", "cs", and "xs", if the length of "cs" is "m" and the length of "xs" is "n", then the evaluation of the universal function of "m+n" with inp...
If the image of a function 'f' restricted to a finite set 'xs' is not equal to 'xs', then there exists an element 'x' in the finite set 'xs' such that 'f' of 'x' is not equal to 'x'.
If F is an isomorphism class and f and f' are elements of F, then f is isomorphic to f'.
If a word sequence 'ws' belongs to the list of elements in group 'G', then the set of references in group 'G' for 'ws' is equal to the union of the set of decompositions of 'G' applied to the set of 'ws'.
If set L is a subset of set R, then the function "valid_mmuaux" applied to the initial arguments (I, n, L, R, b) and 0, and the function "init_mmuaux" applied to the initial arguments (I, n, L, R, b) with an empty list, is valid.
The distance from vertex 'v' to vertex 'v'' is zero if and only if vertex 'v'' is equal to vertex 'v'.
In the context of Minkowski Spacetime, we can establish the following lemma without loss of generality for distinct endpoints: Assume that A is a part of the set of all paths. Also, assume that for any a, b, and I, if Q holds for I, a, and b, then Q also holds for I, b, and a. Moreover, if I is a subset of A and Q hol...
If 'x' is up-closed, then the complement of 'x' is inf-distributive.
For any given 'x' of type 'a' (where 'a' is pure), 'a' is fresh for 'x'.
The function "T_up3" with parameters x, sib, twist, and u is defined as follows: - If u is "Same2", then the result is 1. - If u is "Bal2 t", then the result is 1. - If u is "Unbal2 t n1 h1", then we define n2 as the size of sib, t' as the node with parameters twist, t, x, and sib, h' as h1 plus 1, and n' as the sum o...
The function "expand" takes as input a product of a node and a set of nodes, and returns a non-deterministic result which is a product of a natural number and a set of nodes.
If point A is on different sides of line l1 from points B and C, and the geometric points of B and C are not equal, then points B and C are on the same side of line l1.
If x and y are compatible, then the join of x with the join of x and y is equal to the join of x and y.
If "k" is greater than 0, then the join operation on the function "f" mapped with "k" and the list [n * k + (k - 1), mod k, d] is equal to the join operation on the function "f" divided by "k" and the list [n through to the end, d].
If the integer relation "urel_integer" holds between x and z, and also between y and z, then x must be equal to y.
An empty set overridden by any set 'p' is equal to 'p'.
The function 'trunc_ell2 M' is a bounded linear operator.
The rational number derived from an integer (which itself is derived from an arbitrary integer 'x') is equal to the rational number directly derived from the arbitrary integer 'x'.
The delay bisimulation final base of two transition systems (trsys2 and trsys1) with the flip of a bisimulation, the second tau-move, the first tau-move, the second final state, and the first final state is equal to the delay bisimulation final base of the two transition systems (trsys1 and trsys2) with the original bi...
If 't' is a Braun tree, then the list of 't' in ascending order is less than or equal to twice the binary logarithm of the size of 't' plus one, all multiplied by the size of 't'.
The set of constants in the polynomial term obtained from the normal term 't' is equal to the set of constants in the normal term 't' itself.
When the function "strip_all_vars" is applied to the result of the function "mk_all" with inputs "s", "ty", and "t", it is equivalent to the list that has "ty" as its first element and the result of "strip_all_vars" applied to "t" as the rest of the list.
The function map_acom f applied to c equals to an if-then-else statement with c1' and c2' and a set S', if and only if there exist a set S and two conditions c1 and c2 such that c equals to an if-then-else statement with c1 and c2 and a set S, and the function map_acom f applied to c1 equals to c1', and the function ma...
The lemma named "unif_rtrancl_empty_imp_unifiable" assumes that the pair (E, empty multiset) is in the reflexive transitive closure of the unification relation. It shows that the multiset E is unifiable.
If the signed cast operation (scast) is an upward cast from type 'a word to type 'b word, and the unsigned cast operation (ucast) is a downward cast from type 'b word to type 'c word, then casting 'a word to 'c word through 'b word using ucast after scast is equivalent to directly applying scast on 'a word.
If 'x' is equal to the 'n'th element of the list 'xs' and 'n' is less than the length of the list 'xs', then 'x' is in the function range of 'xs'.
If the successor of x equals the successor of y in a set H, then x equals y.
If 'd' is greater than 0, then the composition of the power series 'fls_X_intpow n' with 'd' is equal to the power series 'fls_X_intpow' with the integer 'd' multiplied by 'n'.
The polynomial of 'p' modulus 'q' evaluated at 'x' is equal to the polynomial 'p' evaluated at 'x' if the polynomial 'q' evaluated at 'x' equals zero.
The lemma named "inverts_mat_symm" is about two matrices A and B, both of which are of type 'a::field mat'. The lemma assumes that both A and B are n x n matrices, and that matrix A is the inverse of matrix B. Given these assumptions, the lemma shows that matrix B is also the inverse of matrix A.
The invariant of the relation "rel" is always true, regardless of its input.
The value of the logical AND operation on 'a' and 'b' in a given state 's' is equal to the value of the logical AND operation on 'b' and 'a' in the same state 's'. This demonstrates the commutative property of the logical AND operation.
: Assuming that the set P is a subset of the set of states s such that there exists a set Z where the initial state s is in the set P' of Z and for all states t in the set Q' of Z, the return state s to t is in the set R of s and t, and for all states t in the set A' of Z, the return state s to t is in the set A. And...
For a given relation on options of type A and a monad M, the function "return_option" preserves this relation when lifted to the option monad transformer of M.
The partial Hoare triple "HT_partial" for a program counter "c", precondition "P", postcondition "Q", and program "π" is equal to the modified partial Hoare triple "HT_partial_mods" for the same program counter, precondition, postcondition, and program, with the addition of an analysis of the left-hand side variable of...
The lemma named "hoare_pre" makes two assumptions. The first assumption is that function "f" takes the system from a state satisfying condition "R" to a state satisfying condition "Q". The second assumption is that for any system state "s", if condition "P" holds, then condition "R" also holds. Given these assumptions,...
The range of the domain function on the ordinal category A is a subset of the objects of the ordinal category A.
There exists a result such that the evaluation of declarations with a clocked value of true, given a module name, environment, state, and declarations, produces this result.
The domain of the function "set_to_map" applied to a set S is equal to the set of first elements of the pairs in S.
If a Dickson grading 'd' exists, the set of keys 'F' has finite components of terms, and 'F' is a subset of the set of polynomials of degree 'm' with respect to 'd', then the reduced Groebner basis of 'F' is finite.
"gcp xs ys" is a prefix of "ys".
If set A is the union of set A1 and set A2, then the image of set A under function f is the union of the image of set A1 under function f and the image of set A2 under function f.
If 'r' is in the Standard set and 's' is also in the Standard set, then the hyperreal complex number 'hrcis' of 'r' and 's' is also in the Standard set.
If a hierarchical tree automaton H has an index function, then a tree t is a member of H if and only if t is in the language of the automaton H.
If the pair (t1,e1) is in the successor set of t, then the set of descendant vertices of the node (which is the root of t with the pair (t1,e1)) excluding the root of t, is equal to the set of descendant vertices of t1.
If the following conditions are met: 1. The Java Virtual Machine program P is well-formed according to the type safety policy Φ (denoted by "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"). 2. The heap configuration is typeable for the program P (denoted by "heap_read_typeable hconf P"). 3. The start state of the program P with...
The lemma shows the following: 1. The dehomogenization of zero with respect to any variable x is zero. 2. The dehomogenization of one with respect to any variable x is one. 3. The dehomogenization of a monomial with respect to a variable x is the monomial with the variable x removed. 4. The dehomogenization of the sum...
, given the assumption that the application of term t1 to term t2 in context Γ has type T (represented as "Γ ⊢ App t1 t2 : T"), it can be shown that there exists a type T' such that term t1 in context Γ has type T' → T and term t2 in context Γ has type T'.
The minimal integer polynomial of 1, in any field with characteristic 0 and a defined greatest common divisor, is equal to the polynomial -1 + x.
The lemma "qFreshInp_preserves_alphaInp_aux" assumes that either "qinp" or "qinp'" is a good input (as stated by "qGoodInp qinp" or "qGoodInp qinp'") and that "qinp" is alpha-equivalent to "qinp'" (as stated by "qinp %= qinp'"). It also assumes that "x" is a fresh variable in the input "qinp" with respect to "xs" (as s...
The lemma named "ap_map_list_cast_Dyn_replicate" assumes that applying the function "cast_Dyn" to each element of the list "xs" using "ap_map_list" results in the list "ys". It then shows that mapping the function "typeof" to each element of "xs" results in a list of "None" values, with the length of this list being th...
A number X is not equal to a pair of X' and Y'.
If a memory restriction "R" applies to a list "xs", then the same memory restriction "R" also applies to the list "xs" after updating the "i"-th element to "None".
In the context of an equivalence relation, the lemma named "classes_eq" assumes that set A is a subset of the carrier set S. It then shows that the equivalence class of set A is equal to the closure of set A.
If the arithmetic compiler "acomp" for an arithmetic expression "a" is correct with respect to the configuration sequence "cfs", then the arithmetic predicate "apred" holds true for the first and the last element of the configuration sequence.
The lemma named "error_abs_euclE" is defined with a fixed error "err" of a type that belongs to the ordered Euclidean space. It assumes that the absolute value of the error is less than or equal to "k". It then obtains a function "e" that maps from the type of "err" to real numbers, where "err" is equal to the sum of t...
If "n" is less than or equal to "i" and "i" is less than "m", then the sequence obtained by updating the "i"-th element of the sequence "xs" to "v" and then taking the subsequence from "n" to "m" is the same as the sequence obtained by first taking the subsequence from "n" to "m" of "xs" and then updating the "(i-n)"-t...
The lemma "enabled_iff_sconst" assumes that the set of probability mass function (pmf) of K at point x is equal to {x}. It shows that x is enabled in the sequence ω if and only if ω is a constant sequence of x.
The function "get_distinguishing_sequence_from_ofsm_tables_refined" for a given machine M and states q1 and q2 is defined as follows: First, compute the OFSM tables for the machine M up to the size of M minus 1 and assign it to "tables". Then, define a variable "k". If both q1 and q2 are states in the machine M and ...
The lemma "oghoare_step" shows that if a configuration "cf" transitions to a configuration "cf'" under a program context "Γ", and if "cf" is a normal state "s" under command "c", and "cf'" is a normal state "t" under command "c'", and if "Γ" and "Θ" prove that command "c" satisfies the assertions "Q" and "A" under the ...
If we have a cocone under a functor (denoted by NN) from a category AA to a category BB, and we have a functor (denoted by GG) from BB to another category CC, and the composition of GG and FF (denoted by GG ∘ FF) is a functor from AA to CC, then the composition of GG and the natural transformation of NN (denoted by GG ...
The greatest common divisor of 'a' and 'b' is zero if and only if both 'a' and 'b' are zero.
For any two variables x and y, the recursive execution of the function "rec_less" with x and y as arguments, is equal to 1 if x is less than y, otherwise it is 0.
If the pair (a, b) is in the relation R, then the pair ([a, c], [b, c]) is in the lexicographic product of the relation R.
The lemma named "has_chain_integral_chain_integral4" assumes that the function 'f' has a contour integral 'i' over the chain of line paths from point 'a' to 'b', 'b' to 'c', 'c' to 'd', and 'd' to 'e'. It shows that the sum of the contour integrals of function 'f' over each individual line path (from 'a' to 'b', 'b' to...
The lemma named "current_support_flow" is about a fixed structure, denoted as "Gamma". It assumes that "Gamma" is a current flow "f". It shows that the support flow of "f" is a subset of the set of all edges in "Gamma".
The lemma SpecAnno: Assumes a consequence: "P is a subset of the set of states s such that there exists a Z where s is in P' of Z and Q' of Z is a subset of Q and A' of Z is a subset of A." Assumes a specification: "For all Z, the total correctness assertion holds that under the context of program \<Gamma> and the set ...
If a program P, environment E, heap h, and static heap sh type-check an expression e to a type T, and if static heap sh is a subheap of static heap sh', and if e is not a sub-RI, then the program P, environment E, heap h, and static heap sh' also type-check the expression e to the type T. The lemma WTrts_shext_mono st...
The height of a binary tree node with a list of elements 'ls' appended with an element 'a' is equal to the height of a binary tree node with the element 'a' prepended to the list of elements 'ls'. Both nodes have the same tree 't'.
If a natural number 'n' is greater than 1 and the real part of a complex number 's' is not equal to zero, then the 'n' raised to the power of 's' will not be equal to 1 in the complex number system.
The complement of x is equal to the complement of the union of the complement of the union of the complement of x and the complement of y, and the complement of the union of the complement of x and the double complement of y.
"isRevNth_geq_revPH" is an invariant in the context of "cIsInvar".
If 'I' is an ideal of ring 'R' and 'a' is an element in the carrier of 'R', then the coset representative function of 'R' and 'I' applied to the sum of 'a' and 'I' and then added to 'I' is equal to the sum of 'a' and 'I'.
The lemma named "info_2_card" assumes that "x" satisfies the condition "invariant_2". It shows two things: 1. If "info_2 x" equals "Inr (p,n)", then the following conditions hold: "p" satisfies the "poly_cond" condition, the polynomial "p" evaluated at "real_of_2 x" equals zero, the degree of "p" is greater than or eq...
If "c" is an element of the relative homology group of "p" with respect to "X" and "S", then applying the identity function "id" to "c" through the homomorphism induced by "p", "X", "S", "X", and "S" results in "c" itself.
The interval up to "n'" is a subset of the interval up to "n" if and only if "n'" is less than or equal to "n".
The lemma named "subspace_image_invertible_mat" is defined with a fixed P, which is a matrix of elements from a field. It assumes that P is invertible and W is a vector subspace. It shows that the image of W under the linear transformation defined by matrix multiplication with P is also a vector subspace.
The lemma named "round_num_denom_cost" demonstrates two things. First, it shows that the result of the function "round_num_denom_cost" with parameters n and d is equal to the result of the function "round_num_denom" with the same parameters. Second, it shows that the cost of the function "round_num_denom_cost" with par...
: assuming the intersection of the alphabets of a set of deterministic complete automata (DCA) equals the union of the alphabets of the same set, it shows that the language of the union of the list of these automata equals the union of the languages of the set of these automata.
The lemma named "subst_at_ctxt_at_eq_termI" assumes that "p" is a possible position in both "s" and "t", the subterms of "s" and "t" at position "p" are equal, and the context at position "p" in "s" and "t" are also equal. Under these assumptions, it shows that "s" is equal to "t".
The lemma "awalk_verts_append_cas" assumes that there is a walk from node 'u' to node 'v' following the path 'p' appended with 'q'. It shows that the vertices visited in this walk are the same as the vertices visited in the walk following path 'p' from 'u', appended with the vertices visited in the walk following path ...
"b" is not an internal node of a leaf.
If 's' is an element of S_r, then 's' is not an element of S2.
The lemma named "stirling_Suc_n_2" assumes that "n" is greater than or equal to 1. It shows that the Stirling number of the second kind for "n+1" is equal to the sum of the factorial of "n" divided by "k", where "k" ranges from 1 to "n".