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If "y" is an element of the set "r", then "y" is also an element of the set created by the node with left child "l", value "x", color "False", and right child "r".
The "strictly_subsumes" relation is a well-founded relation.
The lemma named "size_cnf_encode_exactly_one" assumes that the function "cnf_encode_exactly_one" applied to the variable "vars" equals the pair "(one, conds)". It shows that the size of the conjunctive normal form (CNF) of "one" plus the sum of the sizes of the CNF of each element in "conds" is less than or equal to 2 ...
The black height of a red-black tree 't' is equal to the basic height of the same tree 't'.
The lemma named "beta_boundedness_diag_lt" is defined with a fixed integer 'm'. Assuming that negative 'k y' is less than or equal to 'm', 'm' is less than or equal to 'k x', 'x' is in 'X', and 'y' is in 'X', it shows that there exists a subset 'U' of 'R' such that the union of 'U' equals the set of 'u' in 'V' where 'u...
For a given square matrix "mone" of a finite type 'a and a Kleene algebra type 'b, the restriction of the matrix to 'l' plus the restriction of a function 'f' to 'l' multiplied by the restriction of the star of 'f' to 'l' is equal to the restriction of the star of 'f' to 'l'.
In the context of a tm_functor, the identity natural transformation of a functor FF is a tm natural transformation from FF to FF in the double-arrowed category from A to B with a tm structure.
If "LeftDerivationIntro" holds for the parameters alpha, i, the second element of e, ix, E, j, and gamma, then the i-th element of alpha is a nonterminal.
The set of words obtained from a word interval, which is in turn derived from a prefix, is equal to the set of words obtained directly from the same prefix.
The lemma "reverse_arc_in_G_only" assumes that "G" is a graph and "e1" is an arc in "T". It shows that there exists an arc "e2" in "G" such that the head of "e2" in "G" is the tail of "e1" and the head of "e1" is the tail of "e2", and "e2" is not an arc in "T".
The lemma named "use_spec_rule" makes two assumptions: 1. The function 'm' is less than or equal to the specification of function 'Psi'. 2. The function 'm' is less than or equal to the specification of a function that, for any state 's', if 'Psi' of 's' is true, then 'Phi' of 's' is also true. Given these assumptions...
If a natural number 'a' is not equal to another natural number 'b', and an integer 'i' is not equal to another integer 'j', and 'a' is less than 2, and 'b' is less than 2, then the sum of 'a' and 'b' is less than or equal to the natural number equivalent of the maximum absolute value between 'i' and 'j'.
For any integer n, if n is less than the length of the list cl, then the partition of the set of all integers less than the branch (brn) of the nth element of cl, according to the predicate P(n), exists and the empty set is not in P(n). The second condition is that the predicate "pickT_pred" holds for the list cl, the ...
The computation pattern of "V X" is equal to "V\<^sub>M\<^sub>L X".
If the evaluation of a set of terms 'ts' under a substitution 'σ' is equal to the evaluation of the same set of terms 'ts' under a different substitution 'σ'', and if a natural number 'n' is in the set of free variables of the terms 'ts', then the substitution of 'n' under 'σ' is equal to the substitution of 'n' under ...
The function "return" applied to the "high" function of x and n is equal to the function "highi" applied to x and n.
If a tuple 'fs' is a function tuple over the p-adic numbers 'Qp' with a given integer 'm', and for any integer 'k' and set 'S' in the semialgebraic sets of the length of 'fs' plus 'k', the tuple partial pullback of 'm', 'fs', 'k', and 'S' is semialgebraic with 'm' plus 'k', then 'fs' is a semialgebraic map tuple with '...
If a list 'xs' is not empty and all elements in the list satisfy a certain condition 'pred', then there exists at least one element in the list that satisfies the condition 'pred'.
The lemma named "partition_on_mset_add_part" assumes three conditions: 1. There is a partition on the multiset A denoted by P. 2. The multiset X is not empty. 3. The sum of the multisets A and X equals A'. Under these assumptions, it shows that there is a partition on the multiset A' when X is added to the multise...
The reference union is closed under the second property P2.
The lemma, in the context of encoding, is about the symmetric closure of the indexed relation LST. It fixes two sets, SRel and TRel, which are sets of pairs of 'procS' and 'procT' respectively. It assumes that TRel is reflexive, and both SRel and TRel are symmetric. The lemma then shows that the indexed relation ST of ...
Grouping a list based on the identity function is equivalent to grouping the list without specifying any function.
"x is less than or equal to 'None' with respect to the order 'le' defined on the set 'r', if and only if x is equal to 'None'".
"TUM to Leaf" is not less than or equal to "TUM to i8 to Leaf".
A set T is a face of another set S under translation by a vector 'a' if and only if T is a face of S.
If a set of assumptions A is consistent and maximal with respect to a set of propositions V, and if a proposition p can be derived from A, then p is a member of V.
If the clauses C1 and C2 are satisfiable under the evaluation function evalc with respect to the formula F and the assignment G, and if the resolution rule is applicable to C1 and C2 with literals L1 and L2 and substitution σ, then the result of the resolution of C1 and C2 with L1, L2, and σ is also satisfiable under t...
If a formula phi belongs to the set of all formulas, and if the set of free variables in phi is empty, and if phi is provable, then the provability predicate of the Gödel number of phi is also provable.
The lemma presents three simplification rules related to the abstraction function (\<Up>R): 1. The abstraction of a failure under any relation R is a failure. 2. If set X is a subset of the domain of relation R, then the abstraction of the restricted set X under relation R is the restricted set of the image of X under...
If T is a subset of the carrier D, and both a and b are elements of T, then the order relation (denoted by \<preceq>) in the semi-ideal order D on T between a and b is the same as the order relation in D between a and b.
If tau of x equals x and y is less than or equal to x, then tau of y equals y.
Applying the null measure function twice on a measure 'M' is equivalent to applying it once.
For any real normed field or Banach space 'x', the series that sums over all 'n' less than or equal to 'p', where if 'p' is even, the summand is ((-1) raised to the power of (p divided by 2) times (p choose n) divided by (factorial of p)) times (x raised to the power of n) times (y raised to the power of (p minus n)), ...
If a number 'n' is a bound of 'p' and 'q', then 'n' is also a bound of the disjunction of 'p' and 'q'.
The lemma named "sylvester_mat_const" is about a fixed element 'a' from a semiring. It shows that the Sylvester matrix of 'a' and 'q' is equal to 'a' times the identity matrix of the degree of 'q'. Similarly, the Sylvester matrix of 'p' and 'a' is equal to 'a' times the identity matrix of the degree of 'p'.
If the initial state of the Turing machine, after performing a step with input A, is not a final state (as per the assumption "unfinal"), then performing a step with the combined input of A and B is equivalent to performing a step with just input A.
If j is less than i, then replacing the i-th variable in the variable j with u results in the variable j.
The function "enn2ereal" applied to the maximum value results in positive infinity.
If 'r' and 's' are both in the standard set, then the hypercomplex number formed by 'r' and 's' is also in the standard set.
The function "lss_ins_dj" maintains the invariant "lss_invar" and correctly implements the "set_ins_dj" operation according to the abstraction function "lss_α".
The lemma named "same_one_same_zero" assumes that "R" is an element of "C". It shows that the identity element of the union ring of "C" is equal to the identity element of "R", and the zero element of the union ring of "C" is equal to the zero element of "R".
A multiset M is equal to the multiset M with an element x added to it if and only if this is false.
: Given the following conditions: 1. X securely executes command c1 with security level Q p K, equality relation EQ G q, equality relation EQ K q, and property Phi. 2. X securely executes command c2 with security level Q p H, equality relation EQ K q, equality relation EQ H q, and property Psi. 3. For all x, the secur...
There are two binary code morphisms, denoted as g_m and h_m.
The leading coefficient of a polynomial raised to the power of n is equal to the leading coefficient of the polynomial itself raised to the power of n. This is true for any polynomial p that belongs to a commutative semiring with unity and a semiring with no zero divisors.
A conclusion 'c' is in the set of nodes if and only if 'c' is in the set of conclusions.
If T is an isomorphism from A to B in the category of sets, then the following are true: 1. T is an arrow from A to B in the category of sets. 2. The value of the arrow T is a one-to-one function (or a bijection). 3. The domain of the function represented by the arrow T is A. 4. The codomain of the function represente...
The lemma "no_call_simps" states the following conditions: 1. There is no call in the program P with heap h for the creation of a new instance of class C. 2. There is no call in the program P with heap h for the creation of a new array of type T with elements e, if and only if there is no call in the program P with he...
If "m" is less than or equal to "t" and "t" is less than "n", then the next element in the set from "m" to "n" is the successor of "t".
The lemma named "Complex_scale2" demonstrates that the complex number created by multiplying the real part 'a' and the imaginary part 'b' by a constant 'c' is equal to the complex number formed by 'a' and 'b' multiplied by the complex conjugate of 'c'.
The lemma named "AR_Times" fixes two sets, S and T, where S is a set in a Euclidean space 'a and T is a set in a Euclidean space 'b. If S and T are both absolute retracts (AR), then the Cartesian product of S and T is also an absolute retract (AR).
The lemma named "ntcf_terminal_components" demonstrates the following: 1. The natural transformation map of the terminal object in the category "CC" with respect to "z" is equal to the function "f" such that "f" maps an object "c" in the category "CC" to "z". 2. The domain of the natural transformation of the termina...
"FG phi" for a given world "w" is equal to the set of all "psi" in the subformulas of "phi" for which "w" satisfies the temporal logic condition "Fn (Gn psi)".
The lemma named "formSubst" shows that if "y" is fresh for "x" and "y" is fresh for "A", then the substitution of "x" in "A" under the scope of "F" is equal to the substitution of "y" in "A" under the scope of "F" after swapping "y" and "x" in "A".
The lemma named "extends_closed" assumes two conditions: first, that T is closed in Gamma, and second, that Delta extends Gamma. If these two conditions are met, it shows that T is also closed in Delta.
The pair (\<rho>,\<eta>) is well-formed according to \<Gamma>, and "ln" which states that looking up the nth element in \<Gamma> yields some \<tau>. Given these assumptions, it shows that there exists a value v such that the nth element of \<rho> is v and v belongs to the set T of \<tau> under \<eta>.
The lemma named "orthogonal_fst_QR_decomposition" demonstrates that the columns of the first element of the QR decomposition of a matrix A are pairwise orthogonal.
The lower bound (LB) of the function "f" for the set "xs" is finite.
The lemma named "out_arcs_single_per_vert" assumes that "v" and "u" are vertices in a graph "G". It also assumes that "e1" and "e2" are outgoing arcs from vertex "v" in graph "G", and that the head of both "e1" and "e2" is vertex "u". Given these assumptions, it shows that "e1" and "e2" must be the same arc.
If the ordinal number of the function f at 0 is less than the limit ordinal of the function f, then the limit ordinal of the function f at the successor of n is equal to the limit ordinal of the function f.
The set of a list 'xs' updated at index 'i' with value 'k' is equal to: if 'i' is less than the length of 'xs', then it is the union of the set of the first 'i' elements of 'xs' and the set of the elements of 'xs' after the 'i+1'th element, with 'k' inserted; otherwise, it is just the set of 'xs'.
The lemma named "lin_indpt_take_cols_id" shows that the set of columns taken from the identity matrix of size n, according to the indices specified by "inds", is linearly independent.
The length of the list 'xs' is equal to the integer 'n' plus one if and only if there exist elements 'u' and 'us' such that 'xs' is the list formed by 'u' followed by 'us' and the length of 'us' is equal to the integer 'n'.
The maximum value is the negation of the minimum value.
The list of keys obtained from applying the function "Poly_Mapping.map_key PP" to a polynomial 'p' is equal to the list obtained by mapping the function "mapping_of" over the list of keys of 'p'. This is within the context of polynomial power-products 'pp_pm' and 'punit'.
The count of leading zeros in a word "w" is less than or equal to the size of the word "w", where "w" is any word of arbitrary length.
The satisfaction of a pseudo-Boolean expression "satPB" after a substitution "substPB" with a substitution rule "rho" is equal to the satisfaction of the pseudo-Boolean expression where the substitution rule is applied to the integer representation of the expression.
If there is no matching comparison in the reverse of a list "vs" according to a certain rule "nm", then there is also no matching rule "nm" in the list "vs" when each element of "vs" is mapped to a certain function "dtermML".
For any index 'i', given that 'ord' is a transitive relation, 'ord' is the order of the list 'xs', 'j' is less than the length of 'xs', and 'i' is less than 'j', it follows that the element at index 'i' in 'xs' is related to the element at index 'j' in 'xs' by the order 'ord'.
If 'f' is a finite set, and for every 's' in 'f' there exist 'a' and 'b' such that 's' equals the closed box defined by 'a' and 'b', and the interior of 's' is a subset of 't', and 't' is a closed set, then the interior of the union of 'f' is a subset of 't'.
The union of two sets, iterated using the foldli function on sets 'as' and 'bs', is equivalent to the foldli function applied to the concatenation of sets 'as' and 'bs'.
If the binary composition of 'x' satisfies the condition 'cfs' and the second element of the pair 'x' is greater than or equal to zero, then the binary predicate of 'x' holds true for the first element of 'cfs' and the last element of 'cfs'.
If the set D(P) is not empty, then the events of P are equal to the universal set.
The lemma named "alpha_rp_eqE" assumes that "r" is equal to "r'". Under this assumption, it is obtained that the "ok" of "r" is equal to the "ok" of "r'", the "tr" of "r" is equal to the "tr" of "r'", the "wait" of "r" is equal to the "wait" of "r'", the "ref" of "r" is equal to the "ref" of "r'", and the "more" of "r"...
The lemma named "leq_add_right" shows that "x" is less than or equal to "x plus y".
The lemma named "theta_conv_pi_integral" assumes that "x" is greater than or equal to 2. It shows that the integral from 2 to "x" of the function "pi(t) / t" is equal to "pi(x) * ln(x) - theta(x)".
The lemma named "completion_upper" assumes that A is an element of the sets in the completion of M. It shows that there exists a set A' in the sets of M such that A is a subset of A' and the exterior measure of A in the completion of M is equal to the exterior measure of A' in M.
The function 'set_to_sorted_list' preserves the invariance property under the mapping 'g_to_sorted_list'.
The pair "(gen, gen)" is in the relation that maps "A" to the predicate relation "B", and this is related to the function that maps a list of "A" to the predicate relation "B".
The lemma named "m_rcancel" assumes that "a" is not equal to zero and that "a", "b", and "c" are elements in the ring "R". It shows that if the product of "b" and "a" equals the product of "c" and "a", then "b" must be equal to "c".
In the context of a category with a right Kan extension, we have a lemma about the existence of category limits. This is based on the elements of Chapter X-5 in Mac Lane's "Categories for the Working Mathematician" (2010). Assuming that the functor KK maps category BB to category CC, and the functor TT maps category ...
If a monomial 'm' is greater than the leading monomial of a polynomial 'p', then the coefficient of 'm' in 'p' is zero. This is under the assumption that 'm' is greater than the leading monomial of 'p'.
If a map h₀ is disjoint from another map h₁ and a certain element x is in the domain of h₀, then the mapping of x in h₁ is None.
The result of applying function RR_a to the result of applying function R1'_a to the result of applying function R1_a to the SetMarkInv of the final set is less than or equal to ClassicFinal.
Lemma subtrOf_root [simplified]: Assuming tr is a well-formed tree and tr' is a subtree contained in tr, the subtree of tr with the root of tr' is equal to tr'.
The Bellman-Ford algorithm is correct. It establishes a relationship between the computational result and the actual result. If there is a negative cycle, then the computational result is 'None'. Otherwise, it is 'Some' value which is the shortest path for all vertices from 0 to n. This is verified using the Bellman-Fo...
If Q is equal to R, then the separation conjunction of Q and P is equal to the separation conjunction of R and P.
If 'e' is greater than zero, then the determinant of the 3x3 matrix with 0, 'e' scaled by 'xr', and 'P' as its rows is non-negative if and only if the determinant of the 3x3 matrix with 0, 'xr', and 'P' as its rows is non-negative.
The lemma "rule_statesD" states the following: 1. If rule 'r' is in the set of rules for automaton A, then the right-hand side of rule 'r' is in the set of states for automaton A. 2. If rule 'r' is in the set of rules for automaton B, then the right-hand side of rule 'r' is in the set of states for automaton B. 3. If ...
The data space consists of the universal set partitioned into two parts, one for "Inl" and one for "Inr".
If two relations 'A' and 'B' are bi-unique, then the relation 'rel_gpv' that is generated from 'A' and 'B' is also bi-unique.
The reverse of the mapping function 'f' applied to the word 'w' raised to the power of 'n' is equal to the reverse of the function 'f' applied to the reverse of 'w', all raised to the power of 'n'.
The combination of a partial equivalence relation (PER) P with the HOL equality is equal to the intersection of the HOL equality and a function that applies P to x and y.
A set containing a single element 'a' is not empty.
If 'a', 'b', and 'c' are elements of the carrier 'D', and 'a' is less than or equal to 'b', and 'b' is less than 'c', then 'a' is less than 'c'. This is a statement of the transitivity of the less than relation.
If a floating point arithmetic expression "ode_e" is fresh with respect to an index "i", and if a list of such expressions "ds" is also fresh with respect to the same index, and if the length of this list equals the dimension of an executable Euclidean space 'a, then the floating point derivative of the expression "ode...
The reverse of a list obtained by filtering list 'xs' using predicate 'P' is equal to the result of filtering the reverse of list 'xs' using the same predicate 'P'.
The lemma "converged_cd_diverge_cs" assumes that both "π" and "π'" are paths, and that the control states of "π" at point "j" and "π'" at point "j'" are equal. It also assumes that "j" is less than "l", and there does not exist an "l'" such that the control state of "π" at point "l" is equal to the control state of "π'...
The lemma named "count_in_diffI" assumes that for any number 'n', the count of 'x' in multiset 'N' is not equal to 'n' plus the count of 'x' in multiset 'M'. If this assumption holds, it shows that 'x' is in the difference of multiset 'M' and 'N'.
The vector of an empty list is equivalent to an empty vector.