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Statement: stringlengths 7 24.3k
The set of all indices 'i' for which the raw truncated pseudo-deviation 'r' of 'x' at index 'i' is not zero, is a subset of the set of all indices 'i' for which 'x' at index 'i' is not zero.
, assuming 'r' is a transitive relation and 'r' is an irreflexive relation, and the pair (M, N) is in the multiplicative equality of 'r', it can be shown that there exist multisets I', J', and K' such that N equals the sum of I' and J', M equals the sum of I' and K', the intersection of J' and K' is an empty multiset, ...
An empty set is a 2-SAT (2-satisfiability problem). The second rule states that inserting a clause "cl" into a CNF (Conjunctive Normal Form) formula "cnf" is a 2-SAT if and only if the clause "cl" is a proper unordered pair and the CNF formula "cnf" is a 2-SAT.
Mapping a function over a singleton set in the option monad is equivalent to binding the function to the option. In other words, if you have an option value and a function that produces a singleton set, applying the function to the option value (if it exists) will give you the same result as if you had bound the functi...
If the function "lift_instrs" with parameters F, L, ret, N, Σi, and xs returns a value of Some (Σo, ys), then the list xs is not empty if and only if the list ys is not empty.
XCoP of eta is equivalent to the conjunction of ECQw of eta and TNDw of eta.
In a graph G, the tail of the successor of an edge M at a point a is equal to the tail of the edge at point a.
The lemma named "is_reducible_lit_remove1_Cons_iff" assumes that "L" is an element of the set "C'". It shows that the condition "L" is a reducible literal in the list formed by appending the list "C" and the list obtained by removing the first occurrence of "L" from the list "M # C'" is equivalent to the condition "L" ...
If there is a transition from state 'q' to a list of states 'qs' with label 'l' in the transition relation 'δ', and if 'q'' is an element of the set 'qs', and if 'x' is in the set of states that are finitely accessible from 'q'' under the transition relation 'δ', then 'x' is also in the set of states that are finitely ...
The lemma named "subdegree_div" is defined for any two formal power series 'p' and 'q' over a field. It assumes that 'q' is a divisor of 'p'. It then shows that the subdegree of the division of 'p' by 'q' is equal to the subdegree of 'p' minus the subdegree of 'q'.
In the context of a group, the lemma named "pow_nat_mod_ord" assumes that "a" is an element of the carrier group "G" and the order of "a" is not zero. It shows that the power of "a" raised to "n" (where "n" is a natural number) is equal to the power of "a" raised to the modulus of "n" and the order of "a".
The conjugate of a Gaussian integer 'z' divides 'z' if and only if the real part of 'z' is zero, or the imaginary part of 'z' is zero, or the absolute value of the real part of 'z' is equal to the absolute value of the imaginary part of 'z'.
The lemma named "composite_closed" assumes that "t" is an element of set "N", "u" is also an element of set "N", and "R" is a composite of "t", "u", and "v". It shows that "v" is also an element of set "N".
The precondition list of a sequence of two commands (C1 and C2) with respect to a list (l) and a state (s) is equal to the precondition list of the first command (C1) with respect to the same list and state, but with updates (upds) that have been mapped to include the precondition and function of the second command (C2...
The set of elements 'x' in the carrier of the power 'k' of the p-adic numbers, such that the valuation of 'g' at 'x' is less than the valuation of 'f' at 'x', is equal to the set of elements 'x' in the carrier of the power 'k' of the p-adic numbers, such that the valuation of 'g' at 'x' is less than or equal to the val...
The product of the absolute values of the elements in the list "xs" is equal to the absolute value of the product of the elements in the list "xs".
For all formulas 'A', if for all formulas 'B' with a size less than the size of 'A' a property 'P' holds, then the property 'P' also holds for 'A'. This implies that for all formulas 'A', if the size of 'A' equals 'n', then the property 'P' holds for 'A'.
The lemma named "dj_perm_forget" is defined with a fixed permutation 'pi' of type 'y' and a variable 'x' of type 'x'. It assumes that the types 'x' and 'y' are disjoint. The lemma then shows that applying the permutation 'pi' to 'x' results in 'x' itself.
If m is less than or equal to n, then taking the first n elements of a function f and then taking the first m elements of the result is equivalent to just taking the first m elements of the function f.
The upper closure of a set A is up-closed.
If x is less than or equal to y, then the residuum of y with respect to z is less than or equal to the residuum of x with respect to z.
For all pairs of elements x and y in the lists xs and ys, the property "P x y and Q x y" holds if and only if the property P holds for all pairs x and y in xs and ys and the property Q holds for all pairs x and y in xs and ys.
The last element of the sequence generated by the function "prefixes" with parameters "j" and "Suc t" (which represents the successor of "t") is equal to 1.
If the number of ground holes in a context C is zero, then the term of the ground term of the context C is equal to the term of the context C.
The function "F" totalized is in a stable state "A" if and only if the function "F" itself is in the stable state "A".
The lemma named "queue_invar_simps" states three things: 1. An empty queue maintains the queue invariant. 2. A queue with an element 't' at the front followed by queue 'q' maintains the queue invariant if and only if the tree 't' maintains the tree invariant and the queue 'q' maintains the queue invariant. 3. The conc...
The lemma named "transaction_renaming_subst_is_wf_trm" is fixed for a protocol substitution denoted as "α". It assumes that the transaction renaming substitution "α" is applicable to protocol "P" and agent "A". It shows that the well-formed term (wf_trm) is a function of the protocol substitution "α" and variable "v".
The conditional distribution of a conjunction (P and Q) given a condition b is equivalent to the conjunction of the conditional distributions of P and Q given the same condition b, where P, Q, R, and S are propositions.
The ball with center "a" and radius "r" is Lebesgue measurable.
If under the assumptions of a given context (Theta, B, Gamma), the variable x is inferred to have the type tau, then there exist a variable z, a type b, and a constraint c such that the pair (b,c) is the result of looking up x in the context Gamma, and tau is the type that contains z of type b with the constraint that ...
The sum of an infinite list generated by a function 'f' is equal to zero.
A function 'K' is measurable from a measure space 'M' to the subprobability space over 'M' if and only if two conditions are met. First, for every element 'x' in the space of 'M', 'K x' is a subprobability space and the sets of 'K x' are equal to the sets of 'M'. Second, for every set 'A' in the sets of 'M', the functi...
The lemma from Complex_Roots_Of_Unity.thy in Gauss_Sums states two things: 1. The sum of the roots of unity for 0 and any number 'n' is 0. This is represented by the statement "unity_root_sum 0 n = 0". 2. If 'k' is greater than 0, then the sum of the roots of unity for 'k' and 0 is 'k'. This is represented by the sta...
The list obtained from updating an ordered association list (OAlist) using a function 'f' and a key 'k', when converted to a simple list, is equivalent to updating the corresponding simple list using the same function and key.
: 1. An empty set is equivalent to the operation of creating an empty set. 2. The uncurried function of returning whether an element is in a set is equivalent to the uncurried function of checking set membership. 3. The operation of checking if a set is empty in the interface set is equivalent to the operation of chec...
The lemma named "uniform_limit_cong" is defined with respect to two functions 'f' and 'g' that map from type 'a' to a function from type 'b' to a metric space of type 'c', and two functions 'h' and 'i' that map from type 'b' to type 'c'. It assumes that eventually, for all 'x' in set 'X', the function 'f' applied to 'y...
If a red-black tree "t" is sorted, then the tree remains sorted even after inserting elements from another tree "t'" into "t" using a fold operation.
The language of the trimmed regular relation "R" is equal to the language of the regular relation "R".
The lemma named "pcancel1_cancel1_confluent" assumes that "xs" is partially cancellable to "ys" and "zs" is fully cancellable to "ys". It shows that there exists a "us" such that "us" is fully cancellable to "xs" and partially cancellable to "zs".
For any two integers 'a' and 'b', the division of 'a' by 'b' equals 'a' if and only if 'b' equals 1. This is true given the condition that 'a' is greater than 0.
The lemma named "adds_termE" assumes that "u" adds to "v" in terms. If this is true, then it obtains a term "t" where "v" is equal to "t" added to "u".
If there is a vertex walk 'p' in a graph 'G', and 'a' is a vertex in 'G', and there is an arc from 'a' to the first vertex in 'p' in 'G', then there is a vertex walk starting from 'a' and then following 'p' in 'G'.
For a saturated natural number "Sat n" of type 'a with a length constraint, the function "nat_of" applied to "Sat n" is equal to the minimum of the length of type 'a and the natural number n.
The "set_isEmpty" function is implemented for the "lsnd" (list set nondeterministic) data structure, given the abstraction function "lsnd_α" and the invariant "lsnd_invar".
The lemma named "ang_vec_plus_pi1" assumes that the angle between vectors z1 and z2 is greater than 0. It shows that the absolute value of the angle between z1 and z2 plus pi is equal to the angle between z1 and z2 minus pi.
'p' is equal to 'q' if and only if the word representation of 'p' is equal to the word representation of 'q'.
The composition of the inverse of the dual operation and the dual operation itself equals the identity operation.
For a given binomial heap represented by the list of trees "ts" and assuming that the heap satisfies the invariant property "invar ts", the length of the list of trees "ts" is less than or equal to the logarithm base 2 of the size of the multiset of trees "mset_trees ts" plus 1.
The lemma named "c1_on_open_euclidean_anchor" is true.
The generalized context extension on predicate P of the relational composition of R and L is a subset of the relational composition of the generalized context extension on predicate P of R and the generalized context extension on predicate P of L.
Lemma borel_comp [measurable]: If set A is in the Borel sets, then the complement of set A is also in the Borel sets.
The lemma named "BuildSeq_insf" assumes the following conditions: 1. A sequence "s" is built using functions "B" and "C" with parameters "k" and "z" (denoted by "BuildSeq B C s k z"). 2. "m" is an element of the successor of "k" (denoted by "m \<^bold>\<in> succ k"). 3. "n" is also an element of the successor of "k" (...
In the context of a tiny model target digraph homomorphism, given the assumptions that alpha prime equals alpha, A prime equals A, B prime equals B, F prime equals F, and G prime equals G, it can be shown that N is a tiny model digraph homomorphism from F prime to G prime in the tiny model digraph from A prime to B pri...
The binary number represented by a list with a leading 'True' followed by 'n' number of 'False' is equal to the binary number represented by a list with 'n' number of 'True' plus one.
If a key 'k' is in the set of keys resulting from a fold operation that updates a set 'm' with a function that maps to 't' for all elements in the set 'Z t X', then 'k' is either in the set of keys of 'm' or there exists a pair '(t, X)' in the set 'xs' such that 'k' is in the set 'Z t X'.
The lemma in the network context, named "node_deliver_messages_append", demonstrates that the delivery of messages to a node from the combination of two lists (xs and ys) is equivalent to the concatenation of the delivery of messages from each list individually (xs and ys).
The lemma assumes that both 'a' and 'b' are real numbers. It shows two things: 1. 'a' is not greater than or equal to 'b' if and only if 'a' is less than 'b'. 2. 'a' is not greater than 'b' if and only if 'a' is less than or equal to 'b'.
: Given the following conditions: 1. k is greater than 1. 2. The local state is idle after executing the k-th computation stream. 3. The message m is not a null message. 4. The time t0 is equal to the product of t and k. 5. The computation stream s is the result of executing the computation stream with the input scale...
The function "list_to_set" correctly implements the abstract set data type, given the functions "lsnd_α" and "lsnd_invar" and the list "list_to_lsnd".
: Assuming that: - "s" is an element of "V" - "p" is the shortest path from "s" to "t" - "edges" is a subset of the set "p" - The set of edges in the graph "c'" is a superset of the set difference of "E" and "edges" union the product of the swap of "edges", and is a subset of the union of "E" and the product of the sw...
The first element of a pair is equal to the second element of the flipped pair and the second element of the pair is equal to the first element of the flipped pair.
If x and y are elements of negative infinity integers, then the maximum of x and y is also an element of negative infinity integers.
If a packet p matches a rule set rs under the context Γ and the packet property γ, resulting in a decision t, and if the packet p matches a match condition m under the packet property γ, and if all chains in the rule set rs under the context Γ do not call to a chain c, then under the context Γ where chain c is mapped t...
If a list containing a single element "x" is less than or equal to another list "xs", then the result of subtracting the list containing "x" from "xs" is equal to the tail of "xs".
If the function "bind_fv2" applied to a variable "v", a level "lev", and a term "tm" does not equal to the term "tm", then the variable "v" is in the set of free variables of the term "tm".
The lemma named "dim_col_of_iter_tensor" is defined with fixed variables A and n. It assumes that n is greater than or equal to 1. It shows that the column dimension of the n-times iterated tensor product of A is equal to the nth power of the column dimension of A.
A constant map with any value 'v' mapped to an empty set is equal to an empty map.
If the following conditions are met: 1. 'n' is less than or equal to 'n' prime, 2. 'n' prime is less than or equal to the successor of 'n' plus 'd', and 3. 'n' plus 'd' is less than or equal to 'n' prime plus 'd' prime, then the union of the interval from 'n' to 'd' and the interval from 'n' prime to 'd' prime is ...
The lemma named "retract_of_locally_connected" assumes that a topological space T is locally connected and that another topological space S is a retract of T. Under these assumptions, it shows that the topological space S is also locally connected.
The first element of a pair (denoted by "fst") is equal to some element of type 'A, such that there exists an element of type 'B where the pair is equal to the combination of 'A and 'B.
The sub-probability mass function of a bound probability mass function is equal to the bound probability mass function of p and the function that maps x to the sub-probability mass function of the function f at x.
The lemma "AE_not_suntil_coinduct_strong" assumes the following: 1. A property P holds for a state s. 2. For any state s, if property P holds, then state s does not belong to the set psi. 3. For any states s and t, if property P holds for state s, and state s belongs to the set phi, and state t belongs to the transiti...
The freshness at a given tree 't' and index 'is' is defined as follows: If the index 'is' is an empty list, then the result is an empty set. Otherwise, the result is the image of the function 'freshenLC' applied to the annotation of the tree at the index 'butlast is', on the set of fresh annotations of the in-ports of ...
If "rs" is an element of set "R", then the function "freeliftid" applied to the "relator_freeword" of "rs" equals zero.
The set of keys in a mapping 'm', filtered by the condition that 'p + n' is less than or equal to 'x', is equal to the set of keys in the mapping resulting from storing the capability 'c' at position 'p' in 'm' for 'n' length, filtered by the same condition.
If 'a' and 'b' are both elements of the set of non-zero p-adic numbers, and the valuation of the fraction 'a' divided by 'b' is zero, then the valuation of 'a' is equal to the valuation of 'b'.
The function that divides a natural number 'x' by another natural number 'k' is injective on the interval from 'r' to the modulus of 'k' times 'm' and 'c'.
Mapping a function 'f' over a co-recursive list generated by the functions 'IS_LNIL', 'LHD', 'endORmore', 'TTL_end', 'TTL_more', and 'b' is equivalent to generating a co-recursive list with the same functions, but with 'f' composed with 'LHD' and 'lmap f' composed with 'TTL_end'.
If a function F constrains a set A to a set A', and a function G constrains a set B to a set B', then the join of F and G constrains the intersection of A and B to the union of A' and B'.
"b divided by c is less than or equal to a" if and only if "b is less than or equal to c times a", given that "c is greater than 0". This is under the context of real vector spaces.
If "select_first" function with inputs x and p results in y, then the value of "select_first" function with inputs x and p is y.
The lemma lm047: Assuming that the range of P and the range of Q have no intersection, and that the inverses of P and Q are both right-unique (each value in the domain maps to a unique value in the range), it can be shown that the inverse of the union of P and Q is also right-unique.
The function "gfun_at" applied to the "glabel" of a function "f" and a tree "t" at a position "p" is equal to: if "p" is in the possible positions of "t", then it returns "Some" with the function "f" applied to "p", otherwise it returns "None".
If the function "inv_loop5_loop" is true for the input "x" and the pair "(b, Oc # list)", where "b" is not an empty list and the head of "b" is "Bk", then the function "inv_loop5_exit" is true for the input "x" and the pair "(tl b, Bk # Oc # list)".
The lemma "isREVNth_imp_isRevNth" assumes that the state 's' is reachable, the paper ID 'pid' is in the set of paper IDs for the conference ID 'cid' in state 's', and the user ID 'uid' is the 'n'th reviewer of the paper ID 'pid' in state 's'. It shows that the user ID 'uid' is also the 'n'th reviewer of the paper ID 'p...
The free variable list of pairs for the concatenation of two lists F and G is equal to the concatenation of the free variable list of pairs for F and the free variable list of pairs for G.
If term 't' is a subterm of 's', then there exists a function 'f' and a set 'T' such that the function 'f' applied to 'T' is a subterm or equal to 's' and 't' is an element of the set 'T'.
The lemma named "extreme_point_of_convex_hull_convex_independent" is about a set S in a Euclidean space. It assumes that S is compact and that for any element a in S, a is not in the convex hull of the set obtained by removing a from S. The lemma shows that an element x is an extreme point of the convex hull of S if an...
The lemma named "inf_WhileTrue" assumes that the state "s" is in "b". It shows that the infiniteness of the program execution sequence in the context "Γ" starting with a "While" loop with condition "b" and body "c", followed by commands "cs", and initial state "Normal s", is equal to the infiniteness of the program exe...
In the context of order topology, given that 'a' is less than 'b', it can be shown that 'at a within the closed interval from a to b' is equivalent to 'at right of a'. Similarly, 'at b within the closed interval from a to b' is equivalent to 'at left of b'.
The cube image generated by the diamond cube generator is equal to the cube image of the rotated diamond cube.
The lemma "isin_sorted_split" assumes that the inorder traversal of a node (consisting of a tree 't' and a list of trees 'ts') is sorted in ascending order, and that splitting the list of trees 'ts' at a certain element 'x' results in two lists 'ls' and 'rs'. It then shows that 'x' is in the set of elements obtained fr...
The lemma named "unitary_diag_trace_mult_sum" is about complex matrices. It takes as input a complex matrix A and makes several assumptions. It assumes that A is an n by n matrix, that there is another n by n matrix C, that A is Hermitian, and that A has a unitary diagonalization with B and U. It also assumes that n is...
If 'x' is greater than or equal to zero, then the square root of 'x' (denoted as (f sqrt) x) belongs to the set of Infinitesimals if and only if 'x' itself belongs to the set of Infinitesimals.
A vector of size 'm' is equal to a vector of size 'n' if and only if 'm' is equal to 'n'.
If the intersection of x and y equals the intersection of x and z, and the intersection of the complement of x with y equals the intersection of the complement of x with z, then y must be equal to z.
The ideal generated by the integer "k" is a subset of the ideal generated by the integer "l" if and only if "k" is in the ideal generated by "l".
The lemma named "success_effectE" assumes that the command "c" is successful in the heap "h". If this is true, it implies that there exist some result "r" and some heap "h'" such that the command "c" has an effect on the heap "h" that results in the heap "h'" and the result "r".
If a word 'x' bitwise AND with the mask of 'n + m' equals 'x' itself, and 'm' is less than the length of 'a', then the right shift of 'x' by 'n' is less than 2 to the power of 'm'. This is for 'x' which is a word of length 'a'.
If A implies A', B implies B', and the disjunction of A' and B' implies C, then the disjunction of A and B implies C.

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