Statement: stringlengths 7 24.3k |
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The lemma named "vlrestriction_Set_components" shows three things:
1. The value of the arrow in the left restriction of T to the set C is equal to the left restriction of the value of the arrow in T to the set C.
2. The domain of the arrow in the left restriction of T to the set C is equal to C.
3. The codomain of t... |
The lemma 3 assumes that r directly standardizes to r'. It shows that if s directly standardizes to s', then the standardization of r with s substituted for x also directly standardizes to the standardization of r' with s' substituted for x. |
If every element of set A is in set B, and every element of set B is in set A, then set A is equal to set B. |
"for all x, x is in the consumption set". |
If the initial state s0 satisfies state s, then there exists a state s' and two variables S and SK such that state s satisfies state s', the first element of SK is equal to some S, and the pair (Asset n, Crypt (Auth_ShaKey n) (PriKey S)) is not in state s. Furthermore, the set containing the pairs (Owner n, Crypt (SesK... |
The frame satisfies the properties Fr_1, Fr_2, and Fr_4. |
If for a given key 'k', there is no value 'v' that satisfies the property 'P', then looking up key 'k' in a mapping 'm' that has been filtered by property 'P' will result in 'None'. |
The function f2 is associative. |
, assuming "a" is not equal to zero, the point "z" under the Moebius transformation defined by the similarity transformation with parameters "a" and "b" is equal to the homogeneous multiplication of "z" with the complex number "a", plus the homogeneous addition of the complex number "b". |
The assertion set contains the element 'p'. |
If "A" is equal to a record with "nrm" equal to "B" and "brk" equal to a function that maps any "l" to the universal set, then in the environment "Env", "B" is definitely assigned after the execution of the "NewC C" command, resulting in "A". |
"epsilon is greater than or equal to zero". |
Casting a signed word 'a' to an unsigned word and then comparing it with another signed word 'b' casted in the same way, is equivalent to directly comparing 'a' and 'b'. Here, 'a' and 'b' are of the same length. |
The lemma "L3_Controller_Zero" assumes that:
1. The function "Controller_L" takes the parameters y, the finite or infinite append of [Zero] and l, and l and z.
2. The function l at time t equals Zero.
3. The head of the function y at time t+1 is not greater than 300.
Under these assumptions, it shows that the function... |
The supremum (greatest element) of two case expressions is equal to the case expression of the supremum of their corresponding elements. This is done for a variety of cases, including new, newArray, Cast, InstanceOf, Val, BinOp, Var, LAss, AAcc, AAss, ALen, FAcc, FAss, CompareAndSwap, Call, Block, Synchronized, InSynch... |
The lemma named "univ_asAbs_alphaAbsGood" assumes that the function "fAbs" respects the property "alphaAbsGood" and that the abstract "A" is good (as defined by "qGoodAbs"). Under these assumptions, it shows that applying the universal function "univ" to the result of applying the function "asAbs" to "A" is equal to ap... |
The lemma named "path_append_elim" assumes that there is a path "p" that follows the sequence of transitions "r" followed by "s". Under this assumption, it concludes that there is a path "p" that follows the sequence of transitions "r" and there is a path from the target of "r" that follows the sequence of transitions ... |
If all prerequisites of a match "m" are met, then the result of the function "OF_match_fields" with parameters "m" and "p" is equal to the result of the function "OF_match_fields_unsafe" with the same parameters "m" and "p". |
The incorrectness relation "ir_hoare" between two programs "c" and "c'" with precondition "P" and postcondition "Q" is equivalent to the conjunction of two conditions. The first condition is that for all possible initial states "t'", the Hoare triple consisting of the precondition "P" applied to the initial state "t'",... |
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Assuming "do_opapp vs" equals "Some (env', exp')", we can obtain the following two cases:
1. Closure case: There exist an environment "env", a number "n", and a value "v0" such that "vs" equals "[Closure env n exp', v0]" and "env'" equals the environment "env" with its "sem_env.v" field updated to bind "n" to "v0" i... |
The second element of the pair resulting from the function "grab" applied to N and n is a subset of N. |
If a group G acts on a subgroup H and G normally acts on a subgroup N, then G acts normally on the set resulting from the group operation of N and H within G. |
: if 'a' is not equal to zero, then the equation 'a' times 'x' squared plus 'b' times 'x' plus 'c' equals zero if and only if the square of (2 times 'a' times 'x' plus 'b') equals the discriminant of 'a', 'b', and 'c'. |
The lemma named "seq_pr_tuple" assumes that "f" and "g" form a span. It shows that the sequence of the projection at the codomain of "f" and the codomain of "g" with the pair "f, g" is valid. |
The full checker implementation, when applied to a specification and a set A, is in the relation with the PAC checker specification for the same inputs. This relation is defined over the cartesian product of the full polynomial assignment, the full polynomial input assignment, and the fully PAC assignment. The result o... |
The greatest common divisor (gcd) of two polynomials 'f' and 'g' is either 1 or 'f' if 'f' is a monic polynomial and the degree of 'g' is 0. This is applicable for any 'f' and 'g' that are polynomials over a field with a gcd operation. |
For any two natural numbers 'm' and 'n', their conversion to a type 'zhmultiset' is equal if and only if 'm' and 'n' are equal. |
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Assuming for all 'a', the co-call restriction 'S' on the co-call expression 'e' with 'x' substituted by 'y' and applied to 'a' is equal to the co-call restriction 'S' on the co-call expression 'e' applied to 'a'. Under this assumption, it shows that the co-call restriction 'S' on the function update 'fup' applied to... |
The lemma named "ru_t_Some_tau" assumes that the function "reaches_on" with parameters "ru_t", "l_t0", and "ts" reaches the state "Some (e, t)". It shows that "t" is equal to the function "tau" applied to "sigma" and the length of "ts". |
Lifting the failure state in the extended nondeterministic result monad results in a failure state. |
The Markov Decision Process Control (MDPC) is closed for a loop at state 's'. |
Given the following conditions:
1. The result of the function "merge1" applied to tree "t" is tree "t2".
2. The node "Node r' xs'" is a subtree of tree "t".
3. The cardinality of the set "xs'" is greater than 1.
4. For all trees "t" in the first element of the set "xs'", the maximum degree of "t" is less than or equal... |
If the free variables of a term 't' are empty, then there exists a function 'f' and a set 'X' such that 't' is the function 'f' applied to the set 'X'. |
If the proposition "Q of x" is not true implies that the proposition "P of x" is true, then either "P of x" or "Q of x" must be true. |
If a triple of morphisms (f, f', f'') belongs to the set of arrows of the product of three digraphs (A, B, C), then the codomain of this triple in the product digraph is equal to the triple formed by the codomains of each individual morphism in their respective digraphs. |
The lemma named "wfI_wfC_eval_c" is about a fixed command 'c'. It assumes that the command 'c' is well-formed under the environment 'P', the type 'B', and the context 'G'. It also assumes that under the environment 'P' and the context 'G', the instruction 'i' is valid. Under these assumptions, it shows that there exist... |
If a pair (x, y) is in the generalized prefix relation r, then for any z, if the pair (y, z) is in the generalized prefix relation s, it follows that the pair (x, z) is in the generalized prefix of the composition of relations r and s. |
If "t1" is a subtree of "t2" and "t2" is in the first element of the finite set "xs", then "t1" cannot be equal to the node "r" with children "xs". |
A connected component of X for x is equivalent to y if and only if x is in the top space of X and y is in the top space of X, and the connected component of X for x is equal to the connected component of X for y. |
If a reparameterization exists between two paths, γ1 and γ2, and if γ2 is piecewise continuously differentiable on the interval from 0 to 1, and if for every element in the basis, the function F is continuous on the path image of γ2, and if the basis is finite, and if for every element in the basis, the line integral o... |
The infimum distance of the function's output from the image of a set under the function is equal to the infimum distance of the input from the set. This holds true if the function is an isometry on a set X, the set A is a subset of X, and the input is an element of X. |
The lemma named "mul_add_right" assumes that the pair (M, N) is in the multiplication relation of r. It shows that the pair (M, N+P) is also in the multiplication relation of r. |
The theorem "implies_sub" assumes that "sub" is a proper subset of "r" and "sub" is in the TESL interpretation of "c1 implies c2". It shows that "r" is also in the TESL interpretation of "c1 implies c2". |
The arguments of 's' are empty if and only if 's' is a head. |
If "n" is less than or equal to "m", then the predecessor of "n" is less than or equal to the predecessor of "m". |
The lemma named "conjE" is defined with a fixed variable "n". It assumes that if both conditions "γ" and "γ'" hold at time "t" and position "n", and if from this it follows that condition "γ''" holds at the same time and position, then it can be shown that condition "γ''" indeed holds at time "t" and position "n". |
If a pair (c1, c2) is in the delta step rule, then the pair (c1, c2) is also in the step. |
The sum of the function "f" over all elements "i" in the set "I" is equal to a function that, for any given "x", sums the result of applying "f" to each "i" in the set "I". |
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Assuming "iD" is a derivative in the n-th dimension of a function "fas" with respect to a variable "x" and a direction "d", and the maximum variable in the function "fas" is less than or equal to twice the dimension of the Euclidean space 'a. Then, for almost all 'x' in the neighborhood of 'x', "iD" is also a derivat... |
If there is a shortest path 'p' from 'm' to 'n' in graph 'G', then there exists a distance from 'm' to 'n' in graph 'G'. |
The lemma named "Gamma_seriesI" assumes that the sequence formed by the function g(n) divided by the Gamma series of z at n tends to 1 as n tends to infinity. Under this assumption, it shows that the function g tends to the Gamma function of z as n tends to infinity. |
If for any node with root r1 and subtree (t1,e1), being a subtree of t2 implies that the rank of the reversed r1 is less than or equal to the rank of the reversed root of t1, then the list obtained from t2 by the function "dtree_to_list" is sorted according to the comparison function cmp''. |
The lemma named "exp_conj_1_right" assumes that "A" is non-negative. It shows that the conjunction of "A" and a function that always returns 1 is equal to "A". |
If there is a transition from state S1 with context θ1 to state S2 with context θ2, and if S1 satisfies the LI_preproc_prop property, then S2 also satisfies the LI_preproc_prop property. |
The zero element of the function ring over the p-adic numbers with n arguments is the function that maps every element in the n-fold Cartesian product of the p-adic numbers to the zero element of the p-adic numbers. |
The lemma named "agtt_langE" assumes that the pair (s, t) is in the language of the abstract game tree \<G>. It then concludes that there exists a state q such that q is in the derivative of the first element of \<G> with respect to s and q is also in the derivative of the second element of \<G> with respect to t. |
The negative of the result of function r applied to the negative of function l applied to x equals x. |
The union of "OK x" and "OK y" under the function "lift2 f" is equal to the union of "x" and "y" under the function "f". |
The lemma named "zero_simps" states the following properties for a floating point number of any type ('e, 'f):
1. The sign of 0 is 0.
2. The sign of -0 is 1.
3. The exponent of 0 is 0.
4. The exponent of -0 is also 0.
5. The fraction of 0 is 0.
6. The fraction of -0 is also 0. |
The upset of a set X is equal to the image of the downset of the image of set X under the boundary function. |
The floor of x plus the floor of y equals the floor of the sum of x and y, where x and y are real numbers. |
The moment of a normal distribution is integrable with respect to the Lebesgue measure. Specifically, the function that represents the density of a normal distribution with mean "mu" and standard deviation "sigma", multiplied by the k-th power of the difference between "x" and the mean "mu", is integrable. |
: given a trace from initial to final state, and given that 'i' is less than or equal to 'j' and 'j' is less than or equal to the length of the trace, if for all 'i', 'P' holds for 'i' and 'i', and if for any 'i' and 'j' where 'i' is less than 'j' and 'j' is less than or equal to the length of the trace, the state tran... |
For a function 'f' mapping from a first countable and T3 space to another first countable and T3 space, if the following conditions are met:
1. For any sequence 'u' and any element 'b' in set 'T', if all elements of 'u' are in set 'S' and 'u' converges to 'b', then the sequence obtained by applying 'f' to each element... |
If time 't' is in the existence interval of the initial value problem at time 't0' and initial condition 'x0', and if a sequence 'ts' tends to 't' in a filter 'F', then the flow at time 't0' and initial condition 'x0' for the sequence 'ts' also tends to the flow at time 't0' and initial condition 'x0' for 't' in the sa... |
The strip of a command 'c' equals to the command 'SKIP' if and only if there exists a condition 'P' such that 'c' equals to 'SKIP' with the condition 'P'. |
If a pair (xs, Y) is in the sequential composition failures of P and Q, then if X is a subset of Y, the pair (xs, X) is also in the sequential composition failures of P and Q. |
The lemma named "assigns_if_const_not_b_simp" assumes that the constant value of 'e' equals 'Some (Bool b)', symbolized as "?Const b e". It shows that if 'b' is not true, then 'e' is assigned to every element of the set, symbolized as "?Ass b e". |
The lemma named "supports_perm" demonstrates that the set of all elements 'a' such that the permutation of 'a' by 'p' is not equal to 'a' supports 'p'. |
The lemma named "objI_src" assumes that "a" is an arrow and the source of "a" is "a" itself. Under these assumptions, it shows that "a" is an object. |
The lemma named "Chn_RHS_eq" shows that the chain on the right hand side is equal to the composition of the chain of "TfHgh_HfHgh" and the chain of "TfTgh_TfHgh", and then composed with "tuple_ABC'". |
The lemma named "hd_drop" assumes that a number 'n' is less than the length of a list 'xs'. It shows that the head of the list obtained by dropping the first 'n' elements from 'xs' is equal to the 'n'th element of 'xs'. |
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Assuming that:
1. For a given state 'σ', if the condition 'cond' and the property 'P' hold for 'σ', then either 'M' preserves 'P' or 'M' leads to an error.
2. For all states 'σ', if 'cond' and 'P' hold for 'σ', then either 'M' of 'σ' is undefined or the measure of the result of 'M' applied to 'σ' is less than the me... |
If we have three matrices A, B, and C such that A is a matrix with 'nr' rows and 'nc' columns, B and C are square matrices with 'nc' rows and 'nc' columns, and B multiplied by C equals the identity matrix of size 'nc'. Also, if 'gen' is a finite set and it forms a basis for the kernel of the matrix product A and B, the... |
The lemma named "card_of_min_alg2" is given two fixed functions: "s1" which maps from a type F1 with three type variables 'a, 'b, 'c to 'b, and "s2" which maps from a type F2 with the same three type variables to 'c. The lemma shows that the cardinality of the minimum algorithm 2 applied to s1 and s2 is less than or eq... |
The lemma named "preprm_ext_uncompose" assumes that permutation pi is equivalent to permutation sigma, and that the composition of permutation pi with P is equivalent to the composition of permutation sigma with S. Under these assumptions, it shows that P is equivalent to S. |
The fold operation, using the supremum operation on scene spaces, applied to the map of the operation of scene space composition with 'a' on the variable 'Vars', and 'b', is equivalent to the scene space similarity relation of 'a' supremum with 'b'. |
The evaluation of a ternary AND operation between t1 and t2 is TernaryTrue if and only if both t1 and t2 are TernaryTrue. |
The signed integer representation of a word, which is created from an integer 'x' and is of a certain length 'a', is equal to the result of the following operation: add 'x' to 2 raised to the power of ('a' minus 1), then take the modulus of this sum with 2 raised to the power of 'a', and finally subtract 2 raised to th... |
The theorem "wlsFSwSb_recAll_unique_presCons" assumes that the module "MOD" is well-sorted for free variables and substitutions ("wlsFSwSb MOD") and that the functions "h" and "hA" preserve the consistency of the module "MOD" ("presCons h hA MOD"). It shows that if a well-sorted term "X" exists for a sort "s" ("wls s X... |
The definedness of the function 'f' applied to 'x' and 'y' is equivalent to the conjunction of the definedness of 'x' and the definedness of 'y'. |
If a list 'ks' is sorted in ascending order, and a value 'x' is greater than 'l' and less than or equal to 'u', then if a leaf node of a B+ tree is aligned between 'l' and 'u', it will remain aligned even after 'x' is inserted into the list 'ks'. |
The lemma "actions_ok1_ex_actions_ok2" assumes that the action "ta1" is acceptable in state "s1" under thread "t" according to rule "r1", and that action "ta1" is similar to action "ta2" under some relation "m". Under these assumptions, it asserts that there exists a state "s2" in which action "ta2" is acceptable under... |
"P and (P implies Q)" is equivalent to "P and Q". |
The output variable "x" is independent of the input variable "y". |
The lemma ereal_less_eq states the following conditions:
1. Any extended real number 'x' is less than or equal to positive infinity.
2. Negative infinity is less than or equal to any extended real number 'x'.
3. An extended real number 'r' is less than or equal to another extended real number 'p' if and only if 'r' is... |
The lemma named "dickson_le_trans" assumes that if the Dickson's lemma holds true for the ordered pairs (d, m, s, t) and (d, m, t, u), then it also holds true for the ordered pair (d, m, s, u). This is a statement of transitivity in the context of Dickson's lemma. |
The lemma named "hmem_induct" with the case name "step" assumes that for any element 'x', if a property 'P' holds for all elements 'y' that are in 'x', then the property 'P' also holds for 'x'. It concludes that the property 'P' holds for 'x'. |
The lemma named "gfp_transfer" assumes that alpha, f, and g are all inf-continuous functions. It also assumes that the top element under alpha is the top element, and for any element x, applying alpha to f of x equals applying g to alpha of x. Under these assumptions, the lemma shows that applying alpha to the greatest... |
A match is non-strictly stable if and only if the match is either {(M1, W2), (M2, W1), (M3, W3)} or {(M1, W3), (M2, W2), (M3, W1)}. |
The opposite natural transformation of 'u' is a terminal morphism limit of the opposite covariant functor of 'F' from the opposite category of 'J' to the opposite category of 'C' in a category with objects of type 'alpha'. |
The set of elements greater than a certain value 'x' is not empty. |
The real interval of the product of two sets A and B is equal to the product of the real intervals of A and B. |
The dual of the relation algebra Ra with respect to a certain domain D is equal to the complement of the relation algebra Ra with respect to the inverse of the domain D. |
If "xs" is in the set of paths, then the second element of "xs" is in the set "V". |
Applying the function "nu" twice to any element "x" is equivalent to applying the function "nu" once to the same element "x". |
The lemma named "union" demonstrates that the optimal maximum rating (MaxR_opt) of the union of two sets A and B is equal to the following: If the optimal maximum rating of set A is None, then it is the optimal maximum rating of set B. Otherwise, if there is some maximum rating mA for set A, then it is equal to some ma... |
If a relation 'SRel' between source processes and a relation 'TRel' between target processes satisfy certain conditions, then 'SRel' is a weak reduction bisimulation in the source. The conditions are as follows: 'SRel' is fully abstract with respect to 'TRel', 'TRel' is operationally complete, 'TRel' is operationally s... |
If the copy property holds for 'self', then the copy property also holds for the precondition of 'self'. |
The lemma named "ideal_generated_pair_exists" assumes that the ideal generated by the set {a, b} is equal to the ideal generated by the set {d}. Under this assumption, it shows that there exist numbers p and q such that the equation p*a + q*b equals d. |
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