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The lemma "Suc_div_unat_helper" assumes that "sz" is less than the length of 'a (where 'a is a type with a length) and "us" is less than or equal to "sz". It shows that 2 raised to the power of (sz - us) is equal to the successor of the unsigned natural number representation of ((2 raised to the power of sz minus 1) di...
If a list composed of the list "xs" concatenated with the element "xa" has the property of "singleCombinators", then the list "xs" also has the property of "singleCombinators".
The lemma named "subset_bp_card" assumes that "P" is a base point, "M" is a subset of "P", and for all "B" in "M", "B" intersects with "L" non-emptily. Under these assumptions, it shows that the cardinality of "M" is less than or equal to the cardinality of "L".
If the mapping of the sum of a list 'xs' using function 'F' equals to a list composed of 'ds' followed by 'd', 'e', then 'es', followed by 'f', 'g', and finally 'gs', then there exist lists 'as', 'bs', 'cs' and elements 'a', 'b' such that 'xs' equals to the concatenation of 'as', 'a', 'bs', 'b', and 'cs', and 'd' equal...
The free submonoid generated by a set G is equal to the intersection of all sets M such that G is a subset of M and M is equal to the free submonoid generated by M itself.
The set of Atoms, each of which is an Operator with a certain time 't' and an index derived from the operators of a given strips problem 'Pi', mapped from the filtered list of operators of 'Pi' that are members of a list 'vs', is equal to the set of Atoms, each of which is an Operator with the same time 't' and index, ...
For any list L, if the successor of i is less than the length of L, and if L is in the valid set of A, n, and B, and if the replacement of the successor of i in L with M is in the valid set of A, n, and B, then M is equal to the ith element of the successor of i in L.
For any two star elements x and y, which are of type real normed vector, the absolute value of the difference between the hyperreal norm of x and the hyperreal norm of y is less than or equal to the hyperreal norm of the difference between x and y.
If a valuation v of a field K exists, then the unit element (1) in the valuation ring of K with respect to v is equal to the unit element (1) in the field.
If a term 't' has a type 'T' under a list of types 'Ts', then after substituting types according to a substitution function 'ρ', the substituted term and type also hold the same relationship under the substituted list of types.
If the "at_begin_norm" function holds true for the tuple (as, am) and the tuple (q, aaa, []) with the result "ires", then it also holds true for the tuple (as, am) and the tuple (q, aaa, [Bk]) with the same result "ires".
The lemma named "card_subset_ex" assumes that "A" is a finite set and "n" is less than or equal to the cardinality of "A". It shows that there exists a subset "B" of "A" such that the cardinality of "B" is "n".
If for every index 'i' in 'I', 'M i' is a module over 'R', and 'm1' and 'm2' are elements of the carrier of the product module over 'R', 'I', and 'M', and for every index 'i' in 'I', 'm1 i' equals 'm2 i', then 'm1' equals 'm2'.
The lemma named "blocking_on_Field" assumes that there is a blocking pair on a set X with respect to a matching function h and a set X''. It then shows that the set of all pairs (d, X'') where d is an element of X, is a subset of the field of the preference relation for d.
If the operation "oclIsKindOf" is applied to an invalid object of any type, the result will be invalid.
The sum of the binomial coefficients "k choose m" for all k less than or equal to n is equal to the binomial coefficient "Suc n choose Suc m", where "Suc" represents the successor function (i.e., adding 1 to the argument).
The quotient of the set of all numbers less than the length of a list 'xs' by the kernel of 'xs' is equal to the induced partition of 'xs'.
The lemma SeqStTermP_Ind assumes that the atom s is distinct from the tuple (k,v,t,i) and the atom k is distinct from the tuple (v,t,i). It shows that if the variable v is a valid variable and t is an inductive set, then there exists an atom s and an atom k such that the sequence of structured terms from v, i, t, t to ...
The length of a Turing machine is less than or equal to the length of the Turing machine made composable.
Applying the dehomogenize function twice on a polynomial 'p' with respect to a variable 'x' is the same as applying the dehomogenize function once.
The vector derivative of the function "circlepath" with parameters z and r, at a point x within a set X, is equal to 2 times pi times the imaginary unit times r times the exponential of 2 times the real part of pi times the imaginary unit times x. This lemma is used for introducing derivatives.
If a function 'f', which maps from one metric space to another, is uniformly continuous on a set 'S' and is also continuous on the closure of 'S', then it is uniformly continuous on the closure of 'S'.
"if it is always the case that if F then eventually G, then if eventually F, then eventually G".
The supremum preserves the bounded functor.
The lemma consists of two parts: 1. If the result of zipping two lists, xs and ys, using the option monad is Some zs, then if the length of ys is equal to the length of xs, the length of zs is equal to the length of xs, and zs is the result of zipping xs and ys, then a property P holds. 2. If the result of zipping tw...
Applying a function 'f' to the result of an 'if-then-else' operation on three elements 'x', 'y', and 'z' is the same as performing the 'if-then-else' operation on the results of applying the function 'f' to each of the three elements individually.
If 'a' is greater than 1 and 'a' is less than or equal to 'x' which is less than or equal to 'b', then the minimum of 'a' to the power of 'p' and 'b' to the power of 'p' multiplied by the minimum of the natural logarithm of 'a' to the power of 'p'' and the natural logarithm of 'b' to the power of 'p'' is less than or e...
Applying a function 'f' numeral 'k' times is equivalent to applying the function 'f' once and then applying the function 'f' numeral 'k-1' times.
For all polynomials that satisfy the condition "isnpoly p", the interpretation function "Ifm" applied to the variables "vs", the boolean values "bs", and the "neq" function of "p" is equal to the interpretation function "Ifm" applied to the same variables "vs", boolean values "bs", and the "NEq" function of "p".
Checking the precondition with parameters of ego vehicle's state, velocity, and acceleration (s\<^sub>e, v\<^sub>e, a\<^sub>e) and other vehicle's state, velocity, and acceleration (s\<^sub>o, v\<^sub>o, a\<^sub>o) is equivalent to maintaining a safe distance between the ego vehicle and the other vehicle.
The binary number 00010000 is equal to the hexadecimal number 10.
"A" is less than or equal to "None".
The lemma "nested_subalg_is_sigma_finite" assumes that G is a subalgebra of M and F is a subalgebra of G. Under these assumptions, it shows that G is a sigma-finite subalgebra of M.
The row of an empty matrix at any index i is also empty.
"x is an element of the explosion of X" if and only if "x is an element of X".
If A, U, and B are n x n matrices, and A is the conjugate of the product of U and B, and the adjoint of U times U equals the n x n identity matrix, then the product of A and A equals the conjugate of the product of U and B times B.
If state 's' leads to an infinite computation, then the computation step 's' is not finished.
Either "s" is in degree reverse lexicographic order with "t" or "t" is in degree reverse lexicographic order with "s".
A number 'z' is less than the floor of 'x' if and only if the integer value of 'z' plus one is less than or equal to 'x'.
The lemma named "compatible_tensor_id_update_right" is defined with two fixed functions, F and G. F is a function that updates a type 'a to a type 'c, and G is a function that updates a type 'b to a type 'c. The lemma assumes that F and G are compatible. It then shows that the function that applies F to an input 'a and...
The type configuration of a program P with respect to a heap h is equal to the type configuration of the program P.
The lemma named "fst_prs" with the attribute "quot_preserve" assumes two quotients: "Quotient3 R1 Abs1 Rep1" and "Quotient3 R2 Abs2 Rep2". It shows that mapping the product of "Rep1" and "Rep2" to "Abs1" using the function "fst" is equal to the function "fst" itself.
The bound variables of the extended strand term of the list consisting of "x" followed by "X" is equal to the union of the bound variables of the extended strand term of the list containing "x" and the bound variables of the extended strand term of "X".
The lemma named "cosh_dist_general" assumes that "p" and "q" are endpoints in "S" of "a" and "b". It shows that the hyperbolic cosine of the distance between "a" and "b" is equal to the sum of the square root of the cross ratio of "p", "q", "a", "b" and the reciprocal of the square root of the cross ratio of "p", "q", ...
The union of the partition generated by P is equal to the union of P.
The lemma named "forward_if_ldeep_rev_no_cross" assumes that "r" is a vertex in graph "G", there are no cross products in the reverse list created by "create_ldeep_rev" function applied to "xs", the first element of the reversed list "xs" is "r", and all elements in "xs" are distinct. Under these assumptions, it shows ...
The lemma named "ptautology" assumes that "p" is a propositional tautology. It shows that for any model "M" and world "w", "p" is true in the pointed model.
The topological space of a Euclidean space of dimension m is a subset of the topological space of a Euclidean space of dimension n if and only if m is less than or equal to n.
The lemma "spy_method_via_spy_framework_completeness_and_finiteness" is about two finite state machines (FSMs), M1 and M2. These FSMs are defined over a linear order of types 'a, 'b, and 'c for M1, and types 'd, 'b, and 'c for M2. The lemma assumes that both M1 and M2 are observable and minimal. It also assumes that ...
The lemma named "comparator_composition" assumes that "cmp" is a comparator and "f" is an injective function. It shows that a function that compares "f x" and "f y" is also a comparator.
The successor of any number 'k' is not equal to zero.
If the relations in the join of t and r' are distinct and t is left deep, then the number of left deep nodes in the reverse order of t is equal to the cardinality of the function f applied to t.
The data sources at level 1 for sA92 are equal to the set containing only sA81.
If a term 't' is equal to a variable 'v', then the domain of the substitution function, where 'v' is replaced by 't', is equal to the domain of the original substitution function excluding the variable 'v'.
If a token 't' is in the set 'Z' with parameters 'k' and 'u', then 't' is a well-formed token.
If alpha is within the range from a to b, and r is greater than or equal to 0, then the result of the function rcis with parameters r and alpha is within the complex cone defined by a and b.
"if x is less than or equal to y, then it implies x".
The hereditary union of an empty set is equal to zero.
The positive part of zero is zero.
The Ute_M protocol satisfies the SHOcommPerRd property for the set of higher order servers (HOs) and the set of second higher order servers (SHOs) at the next round (Suc r). The second condition, named "threshold", states that the number of servers q for which the message sent by q to p at the next round (Suc r) accord...
A function which is the quotient of two functions 'a' and 'b' (i.e., a(x) / b(x)) is differentiable with respect to the manifold 'k' and the chart 'charts'. This is true if both 'a' and 'b' are differentiable with respect to the same manifold and chart, and if 'b' is not equal to zero for any 'x' in the carrier set. Th...
The lemma "path_has_prefixes" makes the following assumptions: 1. There is a path 'pth' from vertex 'v' to vertex 'v''. 2. The second element of the vertex 'v'' is an empty list. 3. The list formed by appending 'i' to 'is' is a prefix of the second element of the vertex 'v'. Given these assumptions, the lemma shows t...
If a program P with environment E can transition from a state s0 to a state s1 by evaluating a boolean expression b to true, and if the same program P with environment E can transition from state s1 to state s2 by evaluating a command c to a thrown exception r, then the program P with environment E can transition from ...
Starting from the initial state (1, [], []) and taking zero steps in the semi-identical Turing machine with a positive initial state, the system remains in the initial state (1, [], []).
: "Set B is a subset of the image of set A under relation r if and only if for every element b in set B, there exists an element a in set A such that relation r holds between a and b."
The lemma named "Complex_Re_express_cnj" shows that the complex number with real part "Re z" and imaginary part 0 is equal to the average of the complex number "z" and its conjugate.
If the polynomial "p" is equal to "Fext" and the list of parameters "al" of "p" does not satisfy the condition "intT" for all elements, then the interpretation "I_intP" of "p" with "al" is not true.
The degree of the zero polynomial power product is zero.
If 'as' is a sublist of 'bs', then 'as' is also a sublist of the list formed by appending any lists 'xs' and 'ys' to 'bs'.
The lemma named "homeomorphic_locally" is about two sets, S and T, where S is a set in a metric space of type 'a and T is a set in a metric space of type 'b. The lemma assumes that S is homeomorphic to T, meaning there is a continuous, bijective function from S to T with a continuous inverse. It also assumes that for a...
The lemma "parenthesis_impl_tree_path_not_finished" assumes that: - "v" is in the domain of the "discovered" state, - "w" is in the domain of the "finished" state, - the delta of state "s" at "v" is less than the delta of state "s" at "w", - and "v" is not in the domain of the "finished" state. It shows that the pair ...
The RR1 specification of the relabeled regular automaton A for a tree T is equivalent to the RR1 specification of the original automaton A for the same tree T.
If for all elements 't', the multiplicity count in multiset 'M' is non-negative, then the multiplicity count of 't' in the multiset obtained by applying function 'f' to all elements of 'M' is also non-negative.
If a polynomial 'p' belongs to the set of univariate polynomials over a ring 'R', then the function that maps each coefficient 'i' of 'p' to its additive inverse also belongs to the set of univariate polynomials over 'R'.
The lemma named "pp_init_xcf2_refine" assumes that "am" is an adjacency map, as stated by "is_adj_map am". It shows that the function "pp_init_xcf2" applied to "am" is less than or equal to the function "pp_init_f" in the context of "xf_rel", when the latter is returned. This is denoted by "pp_init_xcf2 am \<le>\<Down>...
The function "c_in" is in the set of primitive recursive functions of two variables.
The image of 'v' under relation 'E' is equivalent to the successors of 'v' in the directed graph 'E'.
If a formula specification is valid for a set of formulas \<F>, a set of relations Rs, a set of variables vs, a register A, and a formula fm, then the same formula specification is also valid for the same sets of formulas, relations, and variables, but with the register A relabeled.
The lemma named "lift_preserves_sameDom" assumes that the inputs 'inp' and 'inp'' have the same domain. It shows that after applying the functions 'f' and 'g' to 'inp' and 'inp'' respectively, the resulting outputs still have the same domain.
The lemma "LLL_add_rows_loop" assumes that the state implementation invariant holds for parameters p, mfs, dmu, and state. It also assumes that the LLL invariant modulo holds for fs, mfs, dmu, p, b, first, and i. It further assumes that the basis reduction modulo add rows loop for parameters p, mfs, dmu, i, and j equal...
For a function 'g', the product of each prime number 'p' less than or equal to a natural number 'n', raised to the power of the value of 'g' at 'p', belongs to the set of functions from the set of prime numbers less than or equal to 'n' to the universal set. This is mapped to the set of natural numbers 'm' greater than...
, assuming a context Γ where P is a subtype of Q, it shows that both P and Q are well-formed in the context Γ.
The mapping of a column to a block is equal to the set of all indices 'i' such that 'i' is less than the dimension of the vector 'c' and the 'i'-th element of 'c' is equal to 1.
If the type of address 'a' in heap 'h' is the class type 'D', and 'D' is not a subtype of 'C' in the program 'P', then the simulation relation 'sim_move10' holds between the 'try-catch' statement 'try (Throw a) catch(C V') e0' and 'try (Throw a) catch(C V) e1' in the program 'P' with thread 't', heap 'h', and exception...
Filtering a property "p" on the concatenation of a list "xs" is equivalent to concatenating the result of mapping the filter of property "p" over the list "xs".
If an ordered pair (a, b) is an element of the relation r, then a is an element of the field of the relation r. This is assumed in the context of CZH_Sets_BRelations.thy in CZH_Foundations/czh_sets.
If the normalization of x is not an empty list, and if the function C applied to the firewall policy list derived from the normalization of x equals to the function C applied to x, then the policy list derived from the function C applied to the reversed normalization of x equals to the function C applied to x.
The lemma "apath_imp_progressing" assumes that there is an alternate path "p" from point "u" to point "v". It shows that the path "p" is progressing.
The lemma named "inconsistent_imply" assumes that the set A is not consistent with the union of the negation of p and the set G. Based on this assumption, it shows that within the system A, G logically implies p.
The lemma named "proots_card_sphere_axis_eq" is defined with "q1" as the complex number with imaginary part 1 and real part -1, and "q2" as the complex number with imaginary part 1 and real part 1. It assumes that "p" is not equal to 0. It shows that the cardinality (or the number of elements) of the roots of "p" withi...
The meet (or intersection) of x and y is a subset of or equal to y.
If the application of polynomial partial deviation "pdevs_apply" to "x" and "j" is not equal to zero, then the degree of "x" is greater than "j".
Lemma: Non-zero Neighbour Alternative Assumptions: 1. The function 'f' is holomorphic on the set 'S'. 2. The set 'S' is open. 3. The set 'S' is connected. 4. The point 'z' is in the set 'S'. 5. The point 'β' is in the set 'S' and the value of the function 'f' at 'β' is not equal to 0. Conclusion: For almost all point...
Under the context of Gamma and Theta, the program 'c' totally terminates from precondition 'P' to postcondition 'Q' and 'A' in the universal set. The second assumption, 'part', states that under the same context, the program 'c' partially terminates from precondition 'P' to postcondition 'Q' and 'A' in the set 'F'. Giv...
A map 'f' from a discrete topology 'U' to a topological space 'X' is continuous if and only if the image of 'U' under 'f' is a subset of the topological space 'X'.
The lemma "Coinitial_Raise_Raise" shows that if t and u are coinitial, then raising t and u by the same degree d and number n will also result in coinitial values.
The composition of the negation of function p with functions q and r is equal to the negation of the composition of function p with functions q and r.
If the left tree (lt) and the right tree (rt) both satisfy the cl_inv condition, then if the color (cl) of the left tree is Black (B) and the color of the right tree is also Black, the combined tree of the left and right trees satisfies the cl_inv condition. Furthermore, the combined tree of the left and right trees sa...
The negation of 'X or Y' is equivalent to 'not X' and 'not Y'.
, under certain conditions, the cost of a list is greater than zero. These conditions are: 1. The selection function 'sf' is reasonable. 2. For every element 'x' in the set 'xs', the cost function 'cf' of 'x' is greater than zero. 3. For every element 'x' in the set 'ys', the cost function 'cf' of 'x' is greater than ...