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An integer 'm' is a divisor of another integer 'n' in the context of Gaussian integers if and only if 'm' is a divisor of 'n' in the context of regular integers.
The lemma named "vec_l_eq" assumes that "i" is less than "n". It shows that the "i"-th element of a vector of length "Suc n" (successor of n, which is n+1) created by function "f" is equal to the "i"-th element of a vector of length "n" created by the same function "f".
The lemma named "tuple_partial_pullback_memE'" assumes the following: 1. "as" is an element of the carrier of the nth power of the p-adic numbers. 2. "bs" is an element of the carrier of the kth power of the p-adic numbers. 3. "x" is equal to the concatenation of "as" and "bs". 4. "x" is an element of the tuple partial...
The transition from "a0n" to "a0i" refines the relation "R0n0i". This is true if and only if the set "R0n0i" is a subset of the transition from "a0n" to "a0i".
The reflexive transitive closure of a relation R applied twice to a set s is the same as the reflexive transitive closure of the relation R applied once to the set s.
For a given point 'a' of any real normed vector, if 'S' is a convex set and 'a' and 'b' are elements of 'S' where 'a' is not equal to 'b', then 'S' is uncountable.
If the cardinality of set A is the successor of 0 (which means the set contains exactly one element), then there exists an element 'a' such that set A is a singleton set containing only 'a'.
If a function 'f' is continuous and 'C' is a chain, then applying the function 'f' to the natural union of 'C' is equal to the natural union of the function 'f' composed with 'C'.
If u is greater than 0, then the function c_hd of u is equal to: if the length of u is 1, then the second component of (u minus 1), otherwise, the first component of the second component of (u minus 1).
If a global invariant, Person_label_globalinv, holds at a state tau, and if the set of all instances of Person includes an object X at that state, then there exists a recursive function, REC, such that at state tau, REC of X is equivalent to the conjunction of the invariant Person_label_inv of X and the condition that ...
The antidomain of x combined with the antidomain of y, multiplied by the antidomain of z, is equal to the combination of the antidomain of x and the antidomain of y, multiplied by the combination of the antidomain of x and the antidomain of z.
The lemma named "inj_finite_single'" assumes that the function 'f' is injective. It shows that the set of all 'z' such that 'f(z)' equals 'x' is finite.
The function "find_index" applied to the list "xs" is injective on the set of "xs". This means that for any two different elements in the set of "xs", their corresponding outputs from the "find_index" function will also be different.
If R1 and R2 are both left unique, then the relation of resumption between R1 and R2 is also left unique.
For a given function 'f', a starting value 'a', and a number 'n', the function for_rec2' with these parameters repeated three times is equivalent to the function for_rec applied twice with the function 'f' and the starting value 'a' and the number 'n' as parameters.
The field of all pairs (x, y) where y is less than or equal to x, for any type x in a preorder, is equal to the universal set.
The universal set of any type 'a is not finite.
If a tuple ((s,c),e,(s',c')) is in the transition relation "trss" of a flow graph "fg", then there exists a counter set "csp" such that "c'" is equal to "csp+c" and "csp" is either an empty multiset or there exists a process "p" such that "e" equals "LSpawn p" and "csp" equals the multiset containing the entry of "p" i...
If a function 'f' is injective on a set 'S', and for all 't' in set 'T', 't' is a subset of 'S', then the function 'f' is also injective on the image of set 'T'.
For any formula phi, the assertion that phi is equivalent to itself holds in the universe v.
, given the conditions that the source is not equal to the sink, there is no edge leading out of the sink, and there is no direct edge from the source to the sink (otherwise, the sets A and B of the web would not be disjoint), it can be shown that the flow of current is orthogonal to the cut of the separator.
The lemma named "transfer_val1" assumes that the transfer function, when applied to the parameters ads, addr, val, and acc, results in Some acc'. It then shows that the read operation on the integer value 'val' is greater than or equal to zero.
For the abstract machine code "asm" and any arbitrary program "ap", zero steps of the abstract machine code will result in the original abstract machine code.
The polynomial is strongly normalizing with respect to the polynomial greater than relation.
The post-measurement of the Bell state "beta_10" at the 2nd position with measurement result 0 is equal to the unit vector in 4-dimensional space at position 0.
Given two finite sets A and B, the multiset of set A is a subset of the multiset of set B if and only if set A is a subset of set B.
The lemma named "strict_extends_if_strict_extends_reflc" assumes that "r_ext" strictly extends the reflexive closure of "r". It shows that "r_ext" also strictly extends "r".
The lemma named "rel_FGl_neg_distr_imp" is given certain fixed elements: two pairs of functions 'Co1' and 'Co1'', 'Co2' and 'Co2'', two pairs of functions 'Contra1' and 'Contra1'', 'Contra2' and 'Contra2'', and three types 'tytok_F', 'tytok_G', and 'tytok_FGl'. It assumes that the negative distribution condition hold...
The nth term of the Bernoulli formal power series is equal to the nth Bernoulli number divided by the factorial of n.
The norm of the octonion units e1, e2, e3, e4, e5, e6, and e7 is equal to 1.
The inverse of one ("ni(1)") is equal to the bottom element ("bot").
The lemma named "cube_restrict" assumes that "j" is less than "n" and "y" is in the "n"-dimensional cube with side length "t". It shows that the function "g", which is less than "j", when applied to "y", is in the "j"-dimensional cube with side length "t".
The lemma P_D2_H2_h_substitutes shows that the function P_D2_H2_h for a given h has the property of substitutes.
Mapping the composition of the mirror point function with respect to a point x0 and the second element function over a polychain of y and xs is equal to mapping the second element function over a polychain of the mirror point of y with respect to x0 and the negation of xs.
For any given element 'x' and two functions 'f' and 'g' that map 'x' to a natural number, if for every 'x' in multiset 'A', 'f' of 'x' is greater than or equal to 'g' of 'x', then the sum of the differences of 'f' and 'g' for each 'x' in 'A' is equal to the difference of the sum of 'f' of each 'x' in 'A' and the sum of...
If the union of 'w' and 'z' equals 1, and if the product of 'w' and 'x' equals the product of 'w' and 'y', and if the product of 'z' and 'x' equals the product of 'z' and 'y', then 'x' must equal 'y'.
: given a Hermitian matrix 'A' that belongs to the set of finite complex matrices, and a number 'a' that is less than the cardinality of the spectrum of 'A', it shows that the measurement outcome projection of the projection measurement outcomes of 'A' with 'a' belongs to the set of finite complex matrices.
The flipped version of a relation R is equal to the inverse of R.
If queue 'q' is not empty, then the list obtained from removing the minimum element from 'q' is equal to the tail of the list representation of 'q'.
An element "x" is in the set of "xs" if and only if there exist lists "ys" and "zs" such that "xs" is equal to the concatenation of "ys", "x", and "zs", and "x" is not in the set of "ys".
The lemma named "aL_pre_below_t_seq2" assumes that "p" is a test expression and "q" is a pre-expression. It shows that the expression "(p star x left-arrow q) times aL" is less than or equal to the sequence "tseq2" of "(-p times q)" and "p", "x", "(p star x left-arrow q)", and "(-p times q union p times (x left-arrow (...
For any security level "e", "e" is less than or equal to itself.
The function "f", which is defined as the identity function with "x" mapped to "x'" and "x'" mapped to "x", is a bijection.
The conjunctive normal form (cnf) of the encoded interfering operator pair exclusion for a given problem, step, and two operators is equal to the set containing the inverses of the operators at the given step and their indices in the list of operators of the problem.
The lemma named "disjoint_dlverts_img" assumes that "xs" is a set of disjoint dlverts and for every pair (t,e) in the finite set "xs", the dlverts of "f t" is a subset of the dlverts of "t". It shows that the dlverts remain disjoint when a function "f" is applied to "t" in each pair (t,e) in the set "xs". This is refer...
Lemma 1: rbt_del_from_left_rbt_less: If lt is less than v, rt is less than v, and k is less than v, then the result of rbt_del_from_left with parameters x, lt, k, y, and rt is also less than v. Lemma 2: rbt_del_from_right_rbt_less: If lt is less than v, rt is less than v, and k is less than v, then the result of rbt_d...
The singleton environment "esing x" multiplied by "a" is less than or equal to "ae" if and only if "a" is less than or equal to "ae" at "x".
The lemma assumes that time 't' is within the existence interval of 't0' and 'x0'. It shows three things: 1. The closed segment between 't0' and 't' is a subset of the existence interval of 't0' and 'x0'. 2. The interval from 't0' to 't' is a subset of the existence interval of 't0' and 'x0'. 3. The interval from 't' ...
If "x" is greater than or equal to zero, then the result of the function "power_down_fl" with parameters "p", "x", and "n" is less than or equal to "x" raised to the power of "n".
Taking the first 'n' elements from the normalized version of a first-order formula 'xs' in an algebraic data type 'AD' is equivalent to normalizing the first 'n' elements of the first-order formula 'xs' in the same algebraic data type 'AD'.
There exists a function 'h' which is also holomorphic on the same ball, and for any complex number 'z' with a norm less than 'r', the value of 'f' at 'z' is equal to 'z' times the value of 'h' at 'z'. Furthermore, the derivative of 'f' at 0 is equal to the value of 'h' at 0.
If the module MOD preserves the well-formedness of all structures under all substitutions of interpretations, then the module MOD also preserves the well-formedness of all under all substitutions of interpretations.
The function "ltake" is monotone with respect to the product relation of less than or equal to and the prefix relation, when applied to a pair of arguments.
If two lists, where all elements satisfy a certain condition A, are appended using the "lappend" function, the resulting list will also satisfy the condition A. This rule is applicable for transfer reasoning.
: given an iterator "iterate" over a set "s" that generates elements in an order defined by "ordR" (as per assumption A), and given that for any element "x" and state "σ", the result of applying function "fi" is less than or equal to the result of applying function "f" (as per assumption R), then the result of applying...
The function "llast_worker" is equivalent to the composition of the functions "unwrap", "llast_body", and "wrap" in that order.
The representation of zero with respect to any basis is a function that assigns zero to every basis element.
Given a state, if the following conditions are met: 1. The state is consistent (InvariantConsistent). 2. The state is unique (InvariantUniq). 3. The watches in the state are elements of the formula (InvariantWatchesEl) and they are different (InvariantWatchesDiffer). 4. The watches in the state characterize the formul...
: given a protocol event sequence "evs" involving a server "S", an agent "A", and a user set "U", if the user set "U" contains a session key "sesK K", an agent "User m", and two numbers "n" and "run", then the session key "sesK K" is also in the user set "U".
The number 127 is equivalent to -1 in a 6-bit word system.
There are two possible cases for the function "list_comb" applied to a function "f" and a list "xs". The first case, labeled "app", asserts that the result of "list_comb f xs" is an application. The second case, labeled "empty", asserts that the result of "list_comb f xs" is equal to "f" and the list "xs" is empty.
For any field 'a, the vector space over the ring of class 'a is the module of vectors of type 'a with dimension equal to the number of rows in matrix A. The carrier of this module is the span of the set of columns of matrix A in the vector space over the ring of class 'a.
The hereditary multiset of the sum of A and B is equal to the sum of the hereditary multiset of A and the hereditary multiset of B.
The theorem named "finite_developments" demonstrates that "t" has finite developments.
, given the assumptions that matrix A is upper triangular, there is a modular type connection between matrix A and another matrix A' of integer type, and there is a modular type connection between index i and another index i', it can be shown that there is a modular type connection between the Hermite form of row i in ...
If the following conditions are met: 1. The problem 'prob' is well-formed according to the ast_problem definition. 2. Every element in the set of actions of the problem 'prob' is a standard operator. 3. The sequence of actions 'pis' is a valid plan for the problem 'prob'. 4. The length of the sequence of actions 'pis'...
If in a given context H, term t is equivalent to term t', and term u is equivalent to term u', then in the same context H, the term resulting from the operation 'tms' on t and u is equivalent to the term resulting from the operation 'tms' on t' and u'.
The lemma "recvcap_preserves_InvInfoJustifiedWithII" assumes that the initial condition "InvInfoJustifiedWithII" is true for "c0". If the function "next_recvcap'" transitions "c0" to "c1" with parameters "p" and "t", then the condition "InvInfoJustifiedWithII" is also true for "c1".
The lemma named "is_image_is_image_spec" assumes that A is not equal to B. It shows that the property P' of P being the reflection of A over B is equivalent to the property P' of P being the reflection of A over the line B.
The lemma named "connected_below_halfspace" makes the following assumptions: 1. A point 'x' is in the below halfspace of a section 'sctn'. 2. The same point 'x' is also in a set 'S' which is connected. 3. The intersection of set 'S' and the plane of the section 'sctn' is empty. Under these assumptions, it shows that t...
For any elements x and y, the product of x, the bottom element (which can be considered as zero in some structures), and y is equal to the product of x and the bottom element. This essentially means that multiplication with the bottom element (or zero) is associative.
The value range of the arrow map of a functor from 'a' through 'g' to 'o' and from 'o' through 'f' to 'b' in a category 'C' is a subset of the set of arrows in the category 'C'.
If a property P holds for an element z, and for all elements x in the set obtained by removing z from the set xs, then if x is in the set xs, the property P holds for x.
If a formula "phi" is normal according to the Z set, then the interpretation of the result of applying the Cooper's quantifier elimination to "phi" with respect to a set "xs" is equal to the interpretation of "phi" with respect to the same set "xs".
The type function of t1 intersected with the type function of t2 is a subtype of the type function of the concatenation of t1 and t2.
If two lists 'xs' and 'ys' are sorted and each has distinct elements, then the union of these two lists, when sorted, will also have distinct elements.
If "m" is an element of "M", then the RSpan of the single-element list containing "m" is a subset of "M".
The lemma named "maxR_idem" shows that the maximum relation of a value 'x' with itself is equal to 'x'.
If natural number 'i' is less than natural number 'j', then the 'j'th element of the sequence obtained by shifting the sequence 'e' at position 'i' to true ('T') is equal to the '(j - 1)'th element of the original sequence 'e'.
The lemma named "bind_game_eq_dis_log" demonstrates that the binding game in Pedersen commitment is equivalent to the discrete logarithm game when the discrete logarithm adversary is the same as the adversary in the Pedersen commitment.
The lemma "CondRuleTC" assumes the following: 1. The set 'p' is a subset of the set of all 's' such that if 's' is in 'b', then 's' is in 'w', and if 's' is not in 'b', then 's' is in 'w''. 2. The triple (w, c1, a1) is valid with respect to the condition 'q' under the total correctness assertion 'ValidTC'. 3. The tripl...
Setting the program counter to a specific value 'a' for a program 'p' is equivalent to binding the program 'p' to a function that sets the program counter to 'a' regardless of its input.
The lemma named "reduce_basis_external'" assumes that the result of the function "reduce_basis_external'" when applied to HNF and fs_init equals fs. It shows that fs is reduced with respect to m and that the LLL invariant holds true for m and fs.
"r_take" is a primitive recursive function with two arguments.
If X and Y are both good, then the equality of the result of the function "pick" applied to X and Y implies the equality of X and Y themselves.
The lemma named "inf_neg_E_set" assumes that the infimum of the negative part of set E is greater than 0. If this condition is met, it shows that set E belongs to the sets in M.
A pair of finite dimensional vector spaces s1 and s2 with bases b1 and b2 respectively, is equivalent to a pair of finite dimensional vector spaces defined on the universal set with the standard operations of addition, subtraction, negation, and zero.
The lemma "has_sum_0" assumes that for any element x in set M, the function f of x equals 0. It shows that the function f has a sum over the set M equal to 0.
If 'a' is greater than 1 and 'x' is greater than 0, then the logarithm of 'x' with base 'a' is less than 0 if and only if 'x' is less than 1.
The predicate of a set created from a list "xs" is equal to the result of folding the "sup" function over the list obtained by mapping the "Predicate.single" function on the list "xs", starting with the bottom element "bot".
If 'i' is an element of 'V', then the function that maps 'x' to 'x i' is measurable with respect to the state measure of 'V' under 'Γ' and 'Γ i'. This is under the context of raw measurability.
The lemma named "prv_imp_LLq_neg_Num_suc" assumes that "t" is an arithmetic term. It shows that if "t" is less than or equal to the successor of a number "n", then if "t" is not equal to the successor of "n", it implies that "t" is less than or equal to "n".
In the context of a set Z, the lemma named "tiny_category_cat_GRPH" assumes that Z is a member of beta and alpha is a subset of beta. It shows that the graph category of alpha is a tiny category in beta.
A program counter value (pc') is a target in a step of a program if and only if there exists a program counter (pc) in the range from 0 to the length of the program (phi) such that pc' is not equal to pc+1 and pc' is a member of the list obtained by mapping the first element of the step function applied to pc and the p...
If the following two conditions are true: 1. Under the condition that both 'b' and 'p' are true, 'S' ensures 'q' holds. 2. Under the condition that 'b' is not true and 'p' is true, 'T' ensures 'q' holds. Then, under the condition 'p', if 'b' is true then 'S' ensures 'q' holds, otherwise 'T' ensures 'q' holds. This is...
The lemma named "deg_leqI" makes the following assumptions and shows a result. It assumes that "p" is an element of the carrier "P". It also assumes that for any integer "n" that is greater than "k", the value of "p" at "n" is zero. Given these assumptions, it shows that the degree of "p" is less than or equal to "k".
If 'f' is an injective function and 'g' is a Borel measurable function under the embedded measure of 'M' with 'f', then the non-negative integral of the embedded measure of 'M' with 'f' for 'g' is equal to the non-negative integral of 'M' for the function 'g' composed with 'f'.
The lemma "lowner_le_trace" assumes that A and B are both n x n matrices. It shows that A is less than or equal to B in the Lowner order if and only if for all density operators ρ that are n x n matrices, the trace of the product of A and ρ is less than or equal to the trace of the product of B and ρ.
Running the "pause" function with parameters "out" and "c" in the context of an option monad is equivalent to pausing with "out" and a function that takes "input" and runs the option monad on the result of calling "c" with "input".
If "x" is in the set of numbers from 0 (inclusive) to 1 (exclusive), then the fractional part of "x" is equal to "x" itself.