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The square of the imaginary unit (denoted as iii) is equal to negative one.
The function 'option.mono_body' takes a lambda function with 'beta_norm' as an argument. This function operates as follows: - If 'beta_norm x' is None, it returns None. - If 'beta_norm x' is Some (Ct list typ), it applies the function 'Ct list typ' to the result of 'beta_norm u', if it exists. - If 'beta_norm x' is So...
For a given matrix A of a field type, and given indices i and j of a modular type, let's define A' as the result of multiplying the row i of the matrix A (after interchanging rows i and the least n such that the element at position (n, j) in A is not zero and i is less than or equal to n) by the reciprocal of the eleme...
If the set F is finite, then the reduced Groebner basis of F is a Groebner basis.
The lemma named "events_ex_path" assumes that the set of all paths, denoted by P, is not empty. It shows that for every event x in the set of all events, denoted by E, there exists a path Q in the set of all paths such that x is an element of Q.
The lemma "lt_mid2__lt13" assumes that "M" is the midpoint of "A" and "B", "M'" is the midpoint of "A'" and "B'", and the distance from "A" to "M" is less than the distance from "A'" to "M'". It shows that the distance from "A" to "B" is less than the distance from "A'" to "B'".
The alpha_l of any l belongs to the set of real numbers.
: Assuming that a chain subdivides a path 'γ' into a 'one_chain', and that this 'one_chain' is a boundary chain. Also, assuming that a line integral exists for the function 'F' and the basis 'basis' on the path 'γ', and that every pair of 'k' and 'γ' in the 'one_chain' forms a valid path. Lastly, assuming that the bas...
The determinant of a 3-element set {a, b, c} is the negative of the determinant of the set {a, c, b}.
If the degree is greater than or equal to 2, then the delete function on a tree node (which has a pair of minimum values and a summary) with respect to 'x' is less than or equal to 9.
If a function 'f' is in 'perms', then for most 'x', 'f' of 'x' equals 'x'.
For a function 'f' mapping from a complete lattice with dual to an ordered set with dual, the right adjoint of 'f' at 'x' is equal to the dual of the left adjoint of the dual of 'f' at the dual of 'x'.
The lemma "HEndPhase1_dblock_allInput" assumes that the action "HEndPhase1" transitions from state s to state s' for process p, and that the invariants "HInv1" and "Inv2a" hold in state s. It shows that the input of the disk block of process p in state s' is included in all inputs of state s'.
Normalizing the binding of an expression 'e' and a function 'c' is equivalent to binding the expression 'e' to a lambda function that normalizes the result of applying function 'c' to a variable 'v'.
Zero is not included in the set of non-zero elements of R.
If t1' is equivalent to t1 and t2' is equivalent to t2, then the append of t1' and t2' is equivalent to the append of t1 and t2.
The lemma named "open_rayE" assumes that "a" is less than "beta". It shows that "a" is in the open ray from "alpha" to "beta".
The last element of the list that results from appending list 'xs' to the list containing elements 'a', 'b', and 'c', is the same as the last element of the list that results from prepending 'c' to list 'xs'.
The function "T_chop" applied to an empty list and any value "d" will always return 1.
The "signed" function is injective.
For a given heap 'h', the function 'a_ptr_kinds_M' when applied to 'h' returns 'xa' if and only if 'xa' is equal to the sorted list of the set obtained by applying the function 'ptr_kinds' to 'h'.
The function "l" of "discharge a" is equal to "l a".
: Given a functor F, categories CC and DD, functors FF and GG, and natural transformations eta_m and eta, where FF is defined as the left adjoint of the right adjoint F and GG, and eta is defined as the tuple consisting of eta_m, the identity functor on CC, the composition of GG and FF, and CC twice. Assuming that G...
A function 'f' belongs to the set of quantale closed maps if and only if for all 'x' and 'y', the product of 'f(x)' and 'f(y)' is less than or equal to 'f' of the product of 'x' and 'y'.
: Given a point z inside the convex hull of points a, b, and c (forming a triangle), and assuming that the imaginary parts of the following expressions are positive: 1. (b-a) * conjugate(b-z) 2. (c-b) * conjugate(c-z) 3. (a-c) * conjugate(a-z) Then, the winding number of the closed curve formed by the line segments fr...
If a scene "S" is idempotent, then "x" is equal to "y" if and only if the pair (x, y) is in both the region "S" and the coregion "S".
If the double negation elimination property holds for a proposition, then the double negation or property also holds for that proposition.
The IP address is in the IP set that is derived from the CIDR block with the IPv4 address of 0.0.0.0 and a prefix length of 0.
If from a set of hypotheses "H" with an additional proposition "p", a proposition "q" can be deduced, then from the original set of hypotheses "H" alone, it can be deduced that "p" implies "q".
The lemma named "list_order_trans" assumes that "op_list" is a list of insertion operations, and that there is a list order from "x" to "y" and from "y" to "z" in the "op_list". It shows that there is also a list order from "x" to "z" in the "op_list". This is a demonstration of the transitive property in the context o...
If from the assumption set H with A added, B can be inferred, and from the assumption set H with the negation of A added, B can also be inferred, then B can be inferred from the assumption set H alone.
For a given square matrix A of real numbers, if A is invertible, then the transpose of A is also invertible.
The restriction of a relation 'r' to a set 'A' is a subset of the relation 'r'.
If x comes before y in the list xs, then it is true that y does not come before x in the list xs.
If the function "read" applied to the parameters "es", "s", and "t" equals to a specific structure with fields "pn_c", "pn_V", "pn_succ", "pn_pred", "pn_adjmap", "pn_s_node", and "pn_t_node", then the following conditions hold: 1. The pair ("es", "c") is in the relation "ln_rel". 2. The set of vertices of the graph "c...
The coefficient of the polynomial that only contains the constant "a" at index "i" is "a" if "i" is zero, otherwise it is zero.
The lexicographic order fusion of g1 and g2 for s1 and s2 is equivalent to the following condition: If list s1 has a next element under g1, then check if list s2 also has a next element under g2. If it does, assign the next element of s1 under g1 to x and the next element of s2 under g2 to y. Then check if x is less ...
If a card A is in the set of cloned cards, then the spy knows the card key of card A, the shared key of A, the pin of A, and for every B, the spy knows the paired key of B and A in any given sequence of events.
If regular expression "r" is equal to regular expression "s", then "eq_mrexp2" of "r" and "s" is true. This is a simplified version of the lemma.
Given a set I and a function M such that for every element i in I, M(i) is a probability space. Also given a set J that is a subset of I and is finite, and a function A such that for every element i in J, A(i) is in the set of measurable sets of M(i). Then, the extended measure of the product space of I and M, over the...
"a" is fresh for the ordinal not equal predicate on "x" and "y" if and only if "a" is fresh for both "x" and "y".
If "ufa_invar l" holds true (meaning that the list 'l' satisfies the invariant of the union-find algorithm), and 'i' is less than the length of the list 'l', then the representative of the element at index 'l ! i' in the list 'l' is equal to the representative of the element at index 'i' in the same list 'l'.
If word 'w' is a prefix of word 'u' and word 'v' is also a prefix of word 'u', then word 'w' is a prefix of word 'v'.
For all edges between two sets X and Y in a graph (V, E), where E is the union of all edge sets E i for i in I, is equivalent to the union of all edges between X and Y in each individual graph (V, E i) for i in I.
In the context of a program P, class C0, and main method, there is a reachable state where the main class of the program P, the main method of the program P, the floor function of 0, and the return command are executed.
The lemma named "not_zeroD" assumes that "x" is not zero in the region "[u]", "u" is an element of set "V", and "x" is an element of set "X". Under these assumptions, it shows that the value of "u" at "x" is greater than zero.
If the type 't' is equal to the union of the image of the function 'to_total' applied to all variables in the arithmetic expression 'e' under the context 'Γ', then under the same context 'Γ', the arithmetic expression 'e' is of type 't'.
The set of distances from C to S is not empty.
The lemma named "index_states_integer_observable" assumes that the machine "M" is observable. It shows that the machine "M" remains observable even when its states are indexed by integers.
The lemma named "phi_in_O" assumes that "V" is an open set in "Y" and "q" is an element of the structure "O" on "V". It shows that the function "phi_f" applied to "V" and "q" is an element of the structure "O" on the preimage of "V" under "f" in "X".
"ZObis" is a subset of "ZOretr ZObis".
The natural number representation of 0 is equal to the algebraic zero.
If the degree of a form 'X' is less than or equal to 'd', and for all forms 'X' in the set 'XS', their degree is also less than or equal to 'd', and if the list consisting of 'x' followed by 'xs' is in the Joints of the list consisting of 'X' followed by 'XS', and 'y' is in the Affine of 'Y', then the list consisting o...
The lemma named "poly_eval_cartesian_prod" assumes that "a" is in the carrier of the nth power of R, "b" is in the carrier of the mth power of R, and "p" is in the carrier of the polynomial ring in n variables over R. It shows that the evaluation of the polynomial "p" at the point "a" in the ring R is equal to the eval...
A new character data is a type of new character data that is well-formed.
If a monad state alternative with return, bind, get, put, and altc functions is defined for a pair of a value and a list of 'w' elements in the 'm' context, then a monad state alternative with the same functions can also be defined for a value in the context of a writerT monad, which includes 'w', 'a', and 'm'.
For a given relation 'Rel' and a calculus with barbs 'CWB', the relation respects barbs if and only if for all sets of barbs, the relation respects the barb set.
If 'v' is an element of 'V', then 'v' is less than 'N'.
If "k" is less than or equal to "l", then taking "k" elements from the sequence of "l" repetitions of "r" will result in a sequence of "k" repetitions of "r".
The lemma "incseq_emeasure" assumes that the range of B is a subset of the sets in M and that B is an increasing sequence. It shows that the function that maps each index i to the extended measure of the set B[i] in M is also an increasing sequence.
In the table of classes, there is no entry for the class "HasFoo".
The signed integer representation of a word 'w' is less than 2 raised to the power of (the length of 'a' minus 1). Here, 'w' is any word of type 'a', where 'a' is a type with a defined length.
The lemma NEST_nonempty assumes that S is a nest. It shows that S is not an empty set.
The lemma named "conorm_componentwise" assumes that the function 'f' is a bounded and complex linear function. It shows that the operator norm of 'f' is less than or equal to the sum of the norms of 'f' over the complex basis.
The set of Recursive Pattern Databases (RPDs) for a given pattern 'r' is the least fixed point of the function 'addRPD' applied to 'r'.
The predicate "visit I L" is measurable in the probability space S.
If property P holds for the Control Register (CR), Context Table Pointer (CTP), Context Number Register (CNR), Fault Status Register (FTSR), and Fault Address Register (FAR), then property P holds for any register x.
The definitions of a partial function are complete when using the "greater than or equal to" relation and the greatest lower bound (Infimum).
If sets A and B are finite, then there is a bijective relationship between the set of functions from A to B and the set of number partitions of the cardinality of A, where the size of the partition equals the cardinality of B. This bijective relationship is modulo the permutation of the domain and range of the function...
The length of the normalized version of a polynomial 'p' is less than or equal to the length of the original polynomial 'p'.
, under the assumptions that the set of the concatenation of lists xs and ys forms a minimal basis, the concatenation of xs and ys contains distinct elements, and p is an element in the set resulting from the function "comp_red_basis_aux" applied to xs and ys, it can be shown that p is not equal to zero.
If there is a derivation from 'a' to 'b' using 'D', then 'a' derives 'b'.
If two lists 'xs' and 'ys' are both sorted and contain distinct elements, then the set of elements in 'xs' is strictly less than the set of elements in 'ys' if and only if the lexicographic order of 'xs' is greater than 'ys'.
If a string 'u' is not empty, then the function 'f' applied to 'u' is equal to the function 'f' applied to 'u' without its last element, concatenated with the function 'f' applied to the last element of 'u'.
If the ordinal category of 'm' is equal to the ordinal category of 'n', then 'm' must be equal to 'n'. This is assumed under the condition "cat_ordinal m = cat_ordinal n".
The function "terminates_on" applied to a function that always returns "Done" is equal to the universal set.
The list of propositions "p" is distinct.
For any term 't' and any substitution 'σ', the head of the term 't' is less than or equal to the head of the term resulting from applying the substitution 'σ' to 't'.
Two Euclidean vectors, x and y, are equal if their corresponding lists of Euclidean coordinates are equal.
If a pair "p" and "q" is a regular S-pair, then the leading term of the S-pair of "p" and "q" is equal to the maximum of two terms. The first term is the sum of the leading term of "p" and the difference between the least common multiple of the leading terms of "p" and "q" and the leading term of "p". The second term i...
If a type τ is less than OclSuper, then assuming τ is not equal to OclSuper implies P. Therefore, P is true.
If the end of phase 0 in state 's' transitions to state 's' prime for process 'p', then for all blocks read in state 's' by process 'p', the message ballot number of the block is less than the message ballot number of the disk block in state 's' prime for process 'p'.
For a generic preflow push, given the assumption A that the initial preflow and label, along with the final preflow and label, are in the transitive closure of the preflow push algorithm's labeled transition system, it shows two things: 1. The sum of the cost estimates of all elements in the preflow is less than or eq...
The lemma named "box_conjE" assumes that the statements "sigma satisfies always F", "sigma satisfies always G", and if "sigma satisfies always both F and G" then a proposition R holds. It shows that the proposition R holds.
Given a sequence of events "evs" in the Otway-Rees protocol and a key "KAB" that is not in the range of shared keys, the key "K" is in the analysis of the knowledge of the Spy after inserting the key "KAB" if and only if "K" is equal to "KAB" or the key "K" is in the analysis of the knowledge of the Spy in the sequence...
The lemma named "is_algebraicI" assumes the following: 1. The set "ps" is finite. 2. The set "ps" is a subset of the carrier of the polynomial ring R in variable X with subscript n. 3. The set "S" is equal to the affine algebraic set over the ring R in n variables defined by the set "ps". Under these assumptions, it s...
If set G is a subset of set H, then the set of fake messages created by a malicious entity in the context of set G and some set C is also a subset of the set of fake messages created by the same malicious entity in the context of set H and set C.
For any two complex numbers 'a' and 'b', if 'S' is a compact set, 'T' is a closed set, and the intersection of 'S' and 'T' is connected, and if 'a' and 'b' are in the same connected component of the complement of 'S' and also in the same connected component of the complement of 'T', then 'a' and 'b' are in the same con...
If "f", "g", and "h" are identities and the source of "f" is the target of "g" and the source of "g" is the target of "h", then the associator of "f", "g", and "h" is in the hom-set from the source of "h" to the target of "f". Furthermore, the associator of "f", "g", and "h" is a morphism from the tensor product of the...
The adjoint operation is a bounded antilinear function.
The permutation of the fixed point of the co-call arity analysis on an expression and a context is equal to the fixed point of the co-call arity analysis on the permuted expression and context.
If 'm' is less than the length of list 'xs' divided by 'k', and 'n' is less than 'k', then the 'n'th element of the 'm'th slice of list 'xs' when divided into 'k' slices is equal to the element at the position 'm * k + n' in the original list 'xs'.
The initialization of a total program, which is constructed using an initialization, actions, and allowed conditions, is equal to the initial state.
If E is a subset of the Cartesian product of V and V, then the stutter extension of edges V and E is also a subset of the Cartesian product of V and V.
For a given matrix A, which is of a certain field and of a certain modular type, there exists a matrix P such that P is invertible and the Gauss-Jordan elimination of matrix A is equal to the product of P and A.
The lemma "ran_map_add" assumes that the domain of map 1 and the domain of map 2 have no elements in common. It shows that the range of the union of map 1 and map 2 is equal to the union of the range of map 1 and the range of map 2.
The identity morphism of the product category of two categories, denoted as (\<AA> \<times>\<^sub>C \<BB>), is a very special variable (vsv).
A function 'f' is a simple function in the count space of a set 'A' if and only if the image of 'f' applied to 'A' is finite.
For any words 'a', 'b', and 'c' of any length, if 'c' is greater than 0 and 'a' is less than the product of 'b' and 'c', then the quotient of 'a' divided by 'c' is less than 'b'.
If the relations "Re" and "R" are single-valued, then the node relation constructed from "Re" and "R" is also single-valued.

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