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Split-Isa-AF-MMA

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There exist four elements, C1, C2, C3, and C4, such that: 1. The set of parties is exactly the set containing C1, C2, C3, and C4. 2. C1, C2, C3, and C4 are all distinct. 3. The result of applying function r to the set containing the singleton sets {C1}, {C2}, the set {C1, C2}, the singleton sets {C3}, {C4}, and the se...
The matrix representation of the difference between two bounded linear functions (f and g) is equal to the difference between the matrix representations of these two functions individually.
The observation of a failure resource is equivalent to a constant resource fault, according to the lemma from the Observe Failure theory in Constructive Cryptography CM.
If 'l' is less than 'r' and 'r' is less than the length of the list 'xs', then the operation 'partition a l r' on the array 'a' that initially contains the list 'xs', results in an array that contains the second element of the pair resulting from the 'Quicksort.partition' operation on 'xs' with 'l' and 'r', and a retur...
Repeating an operation "mx" for "n + m" times is equivalent to first repeating the operation "mx" for "n" times and then repeating the operation "mx" for "m" times.
The lemma named "lex_irrfl" is defined with two fixed natural number lists, "ps1" and "ps2". It assumes that "ps1" is strictly less than "ps2" in lexicographical order. It then shows that "ps1" is not equal to "ps2".
The undiscriminated value of zero is zero.
The function, which takes a predicate 'p' and two arguments 'x1' and 'x2' and checks if there exist 'P' and 'Q' such that 'x1' equals 'P', 'x2' equals 'Q', 'P' weakly simulates 'Q' under the relation defined by 'p', and 'Q' weakly simulates 'P' under the same relation, is monotonic.
For a given global state 'G', updating the local state 's' with a variable 'vn' mapped to a value 'v' and checking if 'l' conforms to 'L' is equivalent to checking if 'l' conforms to 'L' in the original state 's'.
The lemma named "avl_height_upperbound" is defined with a constant "phi" equal to "(1 + square root of 5) / 2". Assuming "t" is an AVL tree, it shows that the height of "t" is less than or equal to "(1/log base 2 of phi) * log base 2 of (size1 of t)".
The lemma "language_initial_path_append_transition" assumes that the sequence "ios" followed by "io" is in the language of machine "M". It then concludes that there exists a path "p" and a transition "t" in machine "M" starting from its initial state, such that when "t" is appended to "p", the input-output sequence of ...
If the lengths of the lists 'rgs' and 'rcs' are equal to 'k', 'k' is greater than 0, and for all 'j' less than 'k', the recursive execution of the 'j'-th element of 'rcs' with 'xs' as arguments equals 0, then the Embranch function applied to the zipped lists of the recursive execution of 'rgs' and the map of a function...
If natural transformation eta1 is naturally isomorphic to eta2, then the domain of the composition of eta1 and eta2 is the same as the domain of eta1.
The lemma named "map_eq_append_replicate_conv" assumes that if we apply a function 'f' to every element of a list 'xs', the result is equivalent to a list that is a combination of 'n' repetitions of an element 'x' and another list 'ys'. Under this assumption, the lemma shows that if we apply the function 'f' to the fir...
If 1 is less than or equal to x, then y is less than or equal to the product of x and y.
A tree "t" is complete if and only if the height of the tree "t" is equal to the minimum height of the tree "t".
If there is a many-one reduction from set A to set B via function f, and a many-one reduction from set B to set C via function g, then there is a many-one reduction from set A to set C via the composition of functions g and f.
If a property P holds for x and z, and for y and z, and for all w, if property P holds for w and z then w overlaps with x or w overlaps with y, then the fusion of z such that z equals x or z equals y, is equal to z.
The lemma named "always_induct" assumes two conditions. First, it assumes that a property "P" holds for a state "σ". Second, it assumes that for all states, if "P" holds then "P" will hold in the next state. Given these assumptions, it shows that "P" always holds for state "σ".
If R is equal to the universal set, then the universal set is equal to R.
If A is a bounded sesquilinear function, then the function that takes x and y and returns the negative of A applied to x and y is also a bounded sesquilinear function.
The lemma named "semIntAll_prSwapAll" assumes that "SEM" is a well-sorted semantic domain. It shows that the function "prSwapAll" applies to the interpretation of semantics "SEM", the interpretation of abstract semantics "semIntAbs SEM", and "SEM" itself.
If variable 'x' does not affect 'u' and 'v', then 'x' will not affect the product of 'u' and 'v'.
The lemma named "consistent_extend'" is defined with respect to a set S of blocks, where a block is a pair of elements of types 'a and 'b. It assumes that the set of all elements of type 'b is infinite, the set A is finite, the set A is consistent with the set S, and the union of the nominals of the blocks in set S is ...
The second element of the infimum (greatest lower bound) of a set A is equal to the infimum of the second elements of all elements in set A.
If the function f of y equals to a list where x is the first and second element followed by a list xs, then x must be equal to zero (in the context of natural numbers) and the list xs must be empty.
In the context of a complete lattice framework (CLF), the least upper bound (lub) of a set P and a complete lattice cl is equal to the least upper bound of the set of all elements x such that x is related to its function image f(x) under a relation r and x is an element of set A, with the same complete lattice cl.
: Assuming - "\<A> is within the reachable constraints of P" - "\<I> is a well-typed constraint model of \<A>" - "Every transaction T in the set P is admissible" - "f is in the union of the function terms of the image of \<I> over the free variables of \<A>" It can be shown that: - "There does not exist a number n su...
The lemma named "extend_measure_caratheodory" is defined with respect to a function G that maps an index set to a set of elements. It assumes that M is the extension of a measure on a set Ω, indexed by I, with the function G and measure µ. It also assumes that the index is a member of I and that the image of G over I f...
Given the assumptions that "Derives1" applies to delta, i, r, and the concatenation of a and b, and that "splits_at" applies to delta, i, alpha, N, and beta, it can be shown that either beta is a suffix of b or b is a suffix of beta.
The restriction of the join (or union) of two trees 't' and 't'' over a set 'S' is equal to the join of the restrictions of the two trees 't' and 't'' over the same set 'S'.
The lemma named "plus_or_minus_sqrt" is defined for any two real numbers x and y. It assumes that y is greater than or equal to zero. It shows that x squared equals y if and only if x equals the square root of y or x equals the negative square root of y.
If 'aa' is an element of the set 'Vset alpha' and 'ff' is also an element of the set 'Vset alpha', then the category of 'alpha' and 'cat_1 aa ff' is a category.
The lemma named "fo_nmlz_eqI" assumes that the list "vs" is agreeable to the list "vs'" under the condition "AD". It then shows that the normalization of the list "vs" under the condition "AD" is equal to the normalization of the list "vs'" under the same condition "AD".
: Given a natural number store, if 'x' is greater than or equal to 0, then the following sequence of operations is valid: 1. Set 'q' to 0. 2. Set 'r' to 'x'. 3. While 'y' is less than or equal to 'r', maintain the invariant that 'x' equals 'q' times 'y' plus 'r', and 'r' is greater than or equal to 0. In this loop, d...
Lemma rbt_sorted_entries_right_unique: "Assuming that (k, v) is an element of the set of entries of the tree t, (k, v') is also an element of the set of entries of the tree t, and the tree t is sorted according to the red-black tree properties, then v must be equal to v'."
If the invariant "Inv2c" holds for a state 's', and an action "HPhase1or2Write" transitions the system from state 's' to state 's'' with parameters 'p' and 'd', and the invariant "Inv2a" holds for state 's', then the invariant "Inv2c" will also hold for the new state 's''.
The lemma named "alice_state" assumes that "\<phi>" is a state of 1. It then shows that "alice \<phi>" is a state of 3.
, given the assumption that "a" is an object in category "C", it shows two things. First, the counit at "a", denoted as "ε a", is a morphism in category "D" from "G₀ a" to "G₀ a". Second, "ε a" is also a morphism in category "D" from the tensor product of "τ₀ a" and "τ₀' a" to "G₀ a".
The lemma "create_character_data_is_weakly_dom_component_safe_step" assumes the following: 1. The heap 'h' is well-formed. 2. The heap 'h' is type correct. 3. The heap 'h' contains known pointers. 4. The heap 'h' can create character data for a given document pointer and text, resulting in a new heap 'h''. 5. The poin...
"X" is an element of the sets of the inverse image algebra of "X" under function "f" in measure space "M".
The lemma named "sumset_stabilizer_eq_self" assumes that set A is a subset of set G. It shows that the sumset of the stabilizer of set A and set A is equal to set A itself.
The lemma named "range_inter_code" shows that the code for the intersection of the range of functions g and h is in the set B.
The lemma named "forward_terminating_iff3" assumes that "x" is a path. It shows that "x" is forward terminating if and only if "x" is less than or equal to the reflexive transitive closure of "x" composed with the complement of "x" composed with 1.
The lemma named "succE" assumes that "x" is an element of the successor of "y". It then concludes that either "x" equals "y" or "x" is an element of "y".
If a list 'xs' is not empty, then flattening a stream that starts with 'xs' and continues with 'ws' is equivalent to appending the list 'xs' to the flattened version of the stream 'ws'.
If a set S in a Euclidean space is homeomorphic to a set T in another Euclidean space, and the affine dimensions of S and T are equal, then the set difference of S and its relative interior is homeomorphic to the set difference of T and its relative interior.
: Assuming "c" is a single-input single-output (siso) system, "s" is equivalent to "t", and "cfT" is enabled at (c, t), it can be shown that: 1. The system remains a single-input single-output (siso) system at the continuation of (c, s) at "cfT" at step "n". 2. The continuation of (c, s) at "cfT" at step "n" is equal t...
If 'r' is an order and 'A' is accessible under 'r', then the union over all 'n' of the list 'n' 'A' is accessible under the list order 'Listn.le r'.
Filtering a polynomial map "Pm_fmap" with a polynomial "P" is equivalent to applying "fmfilter" with the same polynomial "P" on the map "m".
The lemma named "update_total" assumes that "x" is total, "y" is a vector, "y" is regular, and "z" is surjective. Under these assumptions, it shows that the total of the function that results from updating "x" at "y" with "z" is also total.
Given the assumption that there exists a Turing machine 'tm' which transforms a tape initially containing only a list of natural numbers 'ns' into a tape containing a natural number 'n' surrounded by a certain number of blank spaces 'Bk', then there exists a natural number 'n' such that the function 'fun_of_tm' applied...
For any matrix 'A' of a certain field, the basis of the left null space of 'A' when converted to an iarray (integer array) is equal to the basis of the null space of the transpose of 'A' when 'A' is converted to an iarray.
The lemma named "empty_dom" is about a fixed variable 'g'. It assumes that the domain of 'g' is an empty set. Under this assumption, it shows that 'g' is equal to an empty map.
If "k" and "l" are both greater than zero, and "a" is an element of the carrier of the p-adic field with respect to "n", and "b" is an element of the carrier of the p-adic field with respect to "m", then the cartesian product of the basic semialgebraic set with respect to "n", "k", and "a" and the basic semialgebraic s...
If the following conditions are met: 1. There is a relationship between list "li" and list "l" in the set "S". 2. There is a relationship between state "si" and state "s" in the set "R". 3. There is a relationship between condition "ci" and condition "c" in the set "R" that returns a boolean value. 4. For any elements...
The set of all natural numbers is not finite.
If for all 'i' in the union of the network IPs of 's1' and 's2', the value of 'i' in the combined network map of 's1' and 's2' equals to 'σ i', and if the intersection of the network IPs of 's1' and 's2' is empty (meaning 's1' and 's2' have no common network IPs), then for all 'i' in the network IPs of 's1', the value ...
The lemma for the formal power series logarithm at 0, where the constant is 'c', is equal to 0.
If a state sequence "sigma" satisfies the condition "always F", and a proposition "W" holds, then the proposition "W" still holds.
The lemma named "runit_agreement" assumes that "a" is an identity. It shows that the right unit of "a" is equal to the right unit of "a" in the category "C".
If the set M is a subset of the set N, then the operation "setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t" applied to M is also a subset of the operation "setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t" applied to N.
The lemma named "sumZeroLeft" is defined with a fixed process 'P' of type 'pi' and a relation 'Rel' of type set of pairs of 'pi'. It assumes that the identity relation is a subset of 'Rel'. The lemma shows that the sum of process 'P' and the null process leads to process 'P' under the relation 'Rel'.
The lemma named "mismatchId" is defined with fixed variables 'a' and 'b' of type 'name', and 'P' of type 'pi'. It assumes that 'a' is not equal to 'b'. It shows that the process 'P' with a mismatch condition of 'a' not equal to 'b' is bisimilar to the process 'P' itself.
The lemma named "eventually_less_ceiling" is defined with a fixed function 'f' that maps from 'a' to 'b', where 'b' is an ordered topology with floor and ceiling properties. It assumes that 'f' tends towards 'l' in filter 'F' and 'l' is not an integer. It shows that, eventually for all 'x' in 'F', the value of function...
If the free variables of formula A are a subset of set B, then formula A is a ground formula with respect to the set of atoms generated from set B.
The arrow of a natural transformation is a valid finite sequence.
A variable "v" in a set "A" is equivalent to the state update before a function that takes a pair of states "(s, s')" and checks if there exists an "a" such that the second state "s'" is the result of increasing the variable "v" by "a" in the first state "s". This is equivalent to letting "v" be a part of set "A".
If we assume "exact_decomp m ps" and "(h, U) is an element of the set ps", it shows that "h is an element of P[X]" and "U is a subset of X". Furthermore, if "(h', U') is an element of the set ps" and the polynomial degree of h is equal to the polynomial degree of h', and m is less than the cardinality of U and m is les...
If a key K is not in a set H, then the condition of the encrypted message (with key K and message X) being in the synthetically generated set H is equivalent to the condition of the encrypted message being in the original set H.
The function "map_of" applied to the result of the function "update_list" with parameters n, x, and ys is equal to the function "map_of" applied to ys, with the nth element replaced by Some x.
Infinity is not equal to zero in the context of extended natural numbers.
If an element 'a' is in the field of a relation 'r', and if 'a' is in the union of the domain and range of 'r', then a certain proposition 'P' holds.
, given a matrix A of dimensions m by n, and assuming that n is less than 2, and for all i in the set from 1 to m (exclusive), the element at the i-th row and 0-th column of A is 0, it can be shown that the matrix A is in Jordan Normal Form (JNF) echelon form.
The lemma named "dgrad_p_set_closed_snd_crit_pair" assumes that "d" is a Dickson grading, and "p" and "q" are in the set of polynomials of degree "m" with respect to "d". It shows that the second element of the critical pair of "p" and "q" is also in the set of polynomials of degree "m" with respect to "d".
If the set p is a subset of the set q, then the globally always true property of p is also a subset of the globally always true property of q.
The lemma named "ad_agr_list_fo_nmlzd" assumes that the list "vs" agrees with the list "vs'" according to the agreement domain "AD", and both "vs" and "vs'" are normalized according to the agreement domain "AD". Under these assumptions, it shows that the list "vs" is equal to the list "vs'".
The lemma named "dom_simp" demonstrates that the domain of Zero is Zero, the domain of One is One, the domain of j0 is Zero, and the domain of j1 is Zero.
The lemma named "eval_evaln" assumes that under the context G, the state s0 evaluates to the state s1 with the value v through the term t. It shows that there exists a natural number n such that under the context G, the state s0 evaluates to the state s1 with the value v through the term t in n steps.
The lemma named "fresh_finite_union" is defined with respect to two sets S and T, both of which contain elements of a type 'a that satisfies the finite support (fs) property. The lemma assumes that both S and T are finite sets. It then shows that an element 'a' is fresh (i.e., not present) in the union of S and T if an...
The lemma named "subtermeq_Var_const" assumes that "s" is a subterm of "t". It shows two things: 1. If "t" is equal to the variable "v", then "s" is also equal to the variable "v". 2. If "t" is a function "f" with no arguments, then "s" is also a function "f" with no arguments.
The lemma named "closed_subsets_ideal_iff" assumes that both "aa" and "bb" are ideals in the ring R with addition (+), multiplication (·), zero element (0), and one element (1). It shows that the set of all points where the ideal generated by the product of "aa" and "bb" vanishes is equal to the union of the set of all...
If 'a' is not equal to 1, then there exists a number 'n' such that 'a' to the power of 'n' is less than or equal to 'b'.
The lemma named "refines_obtains_subset" assumes that "A" refines "P" to "Q" and "q" is an element of "Q". It shows that a partition on "q" is obtained from the set of all "p" in "P" such that "p" is a subset of "q".
The lemma named "nontrivialE" assumes that the function "g" is not trivial for the variable "p", the function "phi" of "g" and "p" is not equal to "None", and "p" is in the set of all variables of "g". Under these assumptions, it obtains two variables "r" and "s" such that "r" and "s" are arguments of the function "phi...
The lemma named "arrE" assumes that "f" is an arrow. If the first element of "f" is an arrow in category C1 and the second element of "f" is an arrow in category C2, then it concludes that "T" is true.
If a real number 'a' is greater than zero and less than one, then the inverse of 'a' is greater than one.
If 'a' is an element of the carrier 'R' and 'a' is also an element of the ideal 'I' of 'R', then the sum of 'a' and 'I' in 'R' is equal to 'I'.
The lemma named "con_char" shows that the conjunction of "t" and "u" is equivalent to the conjunction of the first element of "t" and the first element of "u", and the conjunction of the second element of "t" and the second element of "u" in the context of two residuated transition systems R1 and R2.
The lemma named "vifunion_vunion_index" assumes that the set B is not empty. It shows that the intersection over all sets C in B of the union of A and C is equal to the union of A and the intersection over all sets in B.
The function "remdups_fwd" applied to a list "xs" is equal to the function "rev" applied to the function "remdups" applied to the reversed list "xs".
For any given function 'f' and 'g', the product of the star function of 'f' at 'z' and the star function of 'g' at 'z' is equal to the star function of the product of 'f' and 'g' at 'z'.
The function "pickInp" is injective (or one-to-one) on the set of all elements that satisfy the property "goodInp".
A complex number "z" is not equal to zero if and only if the square of the real part of "z" plus the square of the imaginary part of "z" is greater than zero.
Given the real numbers 'a' and 'b' are greater than or equal to zero, and 'x' is not equal to zero, then 'a' raised to the power of 'x' is equal to 'b' raised to the power of 'x' if and only if 'a' is equal to 'b'.
The lemma "bifunctor_proj_fst_is_functor" assumes that a bifunctor maps from the product category of categories A and B to category D, and that an object b is in the set of objects of category B. It shows that the bifunctor, when applied to any object from category A and the specific object b from category B, is a func...
If points A and B are not the same, and points A and C are not the same, and line AB is not parallel to line AC, then points A, B, and C are not collinear.
The function "of_bl" applied to a bit sequence "bs" is equal to the sum of the "horner" method applied to the boolean value of "2" and the reverse of the bit sequence "bs".
The lemma "seq_char" shows that a sequence T U exists if and only if T is an arrow, U is an arrow, and the targets of T are the same as the sources of U.
In the context of an encoding, let's consider a lemma about the reflexivity of the induced relation between source and target processes. Let's fix two relations, SRel which is a set of pairs of source processes, and TRel which is a set of pairs of target processes. If TRel is reflexive, then the lemma shows that the in...