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train · 499k rows

Statement: stringlengths 7 24.3k
A complete partial order (ccpo) exists with a least upper bound (lub), an order (ord), and a function to make less (mk_less) based on the order. 2. "mcont" which states that a monotone continuous function exists with supremum (Sup), less than or equal to relation (\<le>), least upper bound (lub), order (ord), and a fun...
The maximum border property for the reverse of strings u and w is equivalent to the maximum border property for strings u and w.
The lemma named "assoc'_naturality" assumes that "f0", "f1", and "f2" are arrows. It shows that the composition of the tensor product of "f0" and "f1" with "f2", followed by the inverse of the associator at the domains of "f0", "f1", and "f2", is equal to the inverse of the associator at the codomains of "f0", "f1", an...
If "R" is a ring and "I" is an ideal of "R", then the projection map "pj" from "R" to the quotient ring "qring R I" is a ring homomorphism.
Given variables A, B, M, a, a', as, and b, and given that as is a list of integers from 0 to n-1, A, B, and M are lists of length n with all elements initialized to 0 or False respectively, the following loop will execute: While the list as is not empty, the loop maintains the invariants that invAM A ({<n} - set as), ...
The lemma named "ma" assumes that "m1" is a matrix with "nr" rows and "n" columns, "m2" is a matrix with "n" rows and "nc" columns, "i" is less than "nr", and "j" is less than "nc". It shows that the element at the "i"th row and "j"th column of the matrix product of "m1" and "m2" is equal to the scalar product of the "...
If the size of the finite set created from list 'b' is greater than 1, and if 'x' is an element in the set 'b', then there exists another element 'x'' in the set 'b' that is not equal to 'x'.
The set of all streams (infinite sequences) over a set A is equal to the set of all things (the universe) if and only if A is equal to the set of all things (the universe).
The lemma named "gf_G_nonneg" establishes that for any real number 'z', if 'z' is greater than or equal to 0 and less than 1, then the function 'gf_G' with parameters 'x', 'y', and 'z' will always return a value that is greater than or equal to 0.
There exists a state such that for all instances of any type, each instance is of the type 'Any'.
If x is compatible with y, then y is also compatible with x.
The dual of the infimum (greatest lower bound) of x and y is equal to the supremum (least upper bound) of the dual of x and the dual of y.
If a binary numeral is greater than zero, then the binary split of this numeral and a word 'w' is equal to the binary split of the numeral decremented by one and the word 'w' divided by two. The result of this split is a pair consisting of 'w1' and the sum of the boolean value of 'w' being odd and twice 'w2'.
The integral of the difference between two Dirichlet series "a" and "b" and the difference between two functions "f" and "g" is equal to the difference between the integral of Dirichlet series "a" with function "f" and the integral of Dirichlet series "b" with function "g".
The set of free type variables in the substitution of a natural number variable "v" by a substitution "S" is a subset of the set obtained by inserting "v" into the codomain of "S".
If N is a module over a ring R and f is a module homomorphism from M to N, then the kernel of f with respect to the module image of f from M to N is equal to the kernel of f from M to N.
If a set of items "I" is well-formed, then the set of items resulting from the completion operation "Complete" at some level "k" on "I" is also well-formed.
The minimum elements of the set of all front_tickFree elements is the set containing only the empty list.
If "t" is an element in the set produced by the function "build_nfa_impl" with parameters "r", "q0", "qf", and "phis", and if "n" is an element in the formula set of "t", then "n" is less than the length of the collection of subformulas of "r" in "phis".
For any point 'a' in a Euclidean space, the affine hull of an open segment from 'a' to another point 'b' is equal to the affine hull of the set containing 'a' and 'b', unless 'a' and 'b' are the same point, in which case it is an empty set.
The lemma named "unique_ocircline_01inf" shows that there exists exactly one oriented circline, denoted as H, such that the points 0, 1, and infinity are in the set of the oriented circline H, and the point ii is not in the disc of H.
Given a matrix A with dimensions n by m, and a vector v with dimension m, the result of multiplying matrix A with the ket representation of vector v is equal to the ket representation of the result of multiplying matrix A with vector v.
If a list 'xs' is a sublist of 'v' and 'v' is an element of the directed vertices of 't', then there exists a 'v2' in the directed vertices of the merged 't' such that 'xs' is a sublist of 'v2'.
The lemma named "partial_pullback_carrier" assumes that the function 'f' maps from the carrier of the m-th power of the p-adic numbers to the carrier of the p-adic numbers. It shows that the partial pullback of 'f' at the l-th level on the carrier of the (l+1)-th power of the p-adic numbers equals the carrier of the (m...
The composition of the function "root_bot" and the function "term_to_bot_term" is equal to the function "root".
The lemma lm079 assumes that 'x' is in the domain of function 'f' and 'f' is a function with a unique range. It shows that the Cartesian product of the set containing 'x' and the image of 'x' under function 'f' is equal to the set containing the ordered pair of 'x' and the value of function 'f' at 'x'.
"thetaIfWT" is a subset of "WbisT".
If a polynomial "p" is linear, then the function "plusinf" applied to "p" is bounded from below.
In the context of a directed graph, if 'a' is an element of the set of arrows of the graph, then 'a' is also an element of the set Vset alpha.
If a program command 'c' starting in a state satisfying precondition 'P' ensures postcondition 'Q', and for all states 's', if 'Q' holds then 'Q'' also holds, then the program command 'c' starting in a state satisfying precondition 'P' ensures postcondition 'Q''.
For all uu1 and uu2, if s and s1 are equal excluding PID_N2 and uu1 equals uu2, then the following hold: - s with confIDs replaced by uu1 and s1 with confIDs replaced by uu2 are equal excluding PID_N2. - s with conf replaced by uu1 and s1 with conf replaced by uu2 are equal excluding PID_N2. - s with userIDs replaced ...
"if X is obligatory with respect to Y, then the intersection of X and Y is ideal."
If a variable 't' is within the set {1 .. 1} and a pair (s, x) is within the set {(0, 1) .. (0, 1)}, then 't' is within the existence interval of the initial value problem for the exponential function with initial values (s, x), and the flow of the initial value problem at time 't' is within the set {(0.99, 2.718281) ....
The lemma states the law of inverses for multiplication. It assumes that 'x' is a rational number and 'x' is not equal to zero. It shows that the product of 'x' and the inverse of 'x' is equal to one.
The lemma named "gmctxtex_onp_pred_dist" assumes that for any set C, the predicate P holds for C if and only if either the predicate Q holds for C or the predicate R holds for C. Under this assumption, it shows that the set of all elements for which the predicate P holds in the context of relation R is equal to the uni...
If a sequence 's' interleaves with a pair of sequences '(t1, u)' over a set 'C', and the intersection of the set of elements in sequence 't2' with set 'C' is empty, then the sequence formed by appending 't2' to 's' interleaves with the pair of sequences formed by appending 't2' to 't1' and 'u', over the set 'C'.
The lemma N_d assumes that 'i' is less than or equal to 'm'. It shows that the Gramian determinant of 'fs' and 'i' is less than or equal to 'N' raised to the power of 'i'.
If Q follows f, then the weakest precondition of applying f is Q.
If a term "w" is in the keys of the product of two polynomials "p" and "q", then there exist terms "u" and "v" such that "u" is in the keys of "p", "v" is in the keys of "q", the component of the term "u" is equal to the component of the term "v", and "w" is equal to the term of the pair formed by the sum of the pair o...
The function "funas_gmctxt" applied to the intersection of C and D is a subset of the function "funas_gmctxt" applied to D.
The lemma shows three things: 1. The space of the bottom element is an empty set. 2. The sets of the bottom element is a set containing an empty set. 3. The extended measure of the bottom element for any set X is zero.
The lemma named "addBalance_val" assumes that adding a value to the balance of an account results in a new account state. It shows that the sum of the initial balance of the account and the value added is less than 2 to the power of 256.
If "n" is greater than 0, then the function "bin_to_bl_aux" with parameters "n", "-1", and "bl" is equal to the function "bin_to_bl_aux" with parameters "n - 1", "-1", and a list consisting of "True" followed by "bl".
If the polynomial of the head of the sequence 'ps' evaluated at 'x₀' is not zero, then it is eventually true that the number of sign changes in the sequence 'ps' at any 'x' is equal to the number of sign changes in 'ps' at 'x₀', in the vicinity of 'x₀'.
The set of paths in the "Either" tree structure is equal to the set that includes the empty path and the union of the sets of paths in all trees in the "ts" set.
The semantics of a loop in a game, represented by "Loop \<alpha>", is equivalent to the least fixed point of the function that unites the set X with the semantics of the game \<alpha> applied to Z.
If the concatenation of two lists 'u' and 'v' belongs to the set of all lists 'A', then the list 'v' also belongs to the set of all lists 'A'.
A numeral m is less than or equal to the extended real representation of a natural number n if and only if the numeral m is less than or equal to n.
The lemma named "sum_mset_eq_sum_on_supersets" assumes that "A" is a finite set and the set of elements in the multiset "M" is a subset of "A". It shows that the sum of each element in the multiset "M" multiplied by its count is equal to the sum of each element in the set "A" multiplied by its count in the multiset "M"...
A function 'h' is a ring homomorphism from ring 'A' to ring 'B' if and only if 'h' is a ring homomorphism from the law-restricted ring 'A' to the law-restricted ring 'B'.
The lemma named "par_LHS_RHS" demonstrates that the left-hand side (LHS) and the right-hand side (RHS) are parallel in the context of a Span.
The lemma named "max_Var_floatariths_replicateI" assumes that the maximum variable in a floatarith "v" is less than or equal to "m". It shows that the maximum variable in a list of "n" replications of "v" is also less than or equal to "m".
If Justified Minimal Logic (JML) holds, then the third law of contraposition (lCoP3) for the implication operator and negation also holds.
: Assuming 1) D is an element of the set P_of_state at the i-th position of the sequence Sts, 2) The sequence Sts is fair, and 3) The natural number i is less than the length of the sequence Sts, it can be shown that there exists a natural number l such that 1) D is an element of the set P_of_state at the l-th positi...
The lemma named "succ_rel_functional" assumes that the pair (a, b1) and the pair (a, b2) are both in the successor relation of a list xs, and that the list xs contains distinct elements. Under these assumptions, it shows that b1 must be equal to b2.
The length of the product of the polynomial represented by the list [-a,1] and another polynomial q is equal to the successor of the length of polynomial q.
The function "real_of_int" is equivalent to the composition of the functions "Code_Real_Approx_By_Float.real_of_integer" and "integer_of_int".
The lemma is named qTerm_templateInduct with case names Var, Op, and Abs. It fixes a quasi-term X and a quasi-abstraction A, and assumes the existence of functions phi, phiAbs, and a relation rel. The lemma assumes the following: 1. REL: For any quasi-terms X and Y, if the pair (X, Y) is in the relation rel, then th...
, assuming 'q' is in the carrier of Qp_x, then converting 'q' from Qp_x to Qp_x will result in 'q'.
The lemma named "rule_statesE" is an elimination rule. It assumes that a state "q" is in the set of rule states "Delta". It then obtains a function "f", a list of states "ps", and a state "p" such that the tree automata rule "TA_rule f ps p" is in "Delta", and "q" is either in the finite set of the list "ps" or "q" equ...
If f and g form a conjugation pair, then f and the function that takes x to the negation of g of the negation of x form a residuated pair.
The composition of the identity relation (1) and the transitive closure of a relation (x) is equal to the composition of the identity relation and the relation itself.
For any real numbers 's' and 'a', given that 'r' is greater than 0 and 'a' is also greater than 0, the function 'g' defined as 'g(x) = x to the power of s times the exponential of negative a times x divided by (1 minus the exponential of negative x)' is set integrable over the set of all real numbers greater than or eq...
If the transition function of an operational automaton (OA) is equal to the sequential process system (oseqp_sos) with a given context (Γ) and index (i), then the operational automaton is subreachable with respect to any other index except for the given index (i).
If 'i' is less than 2 and 'j' is less than 2, and 'i' is not equal to 'j', then the successor of 0 minus 'i' equals 'j'.
The lemma "locally_faithful_iff_faithful" shows that being locally faithful is equivalent to being a faithful functor from V sub C to V sub D under F.
If there is a weak reduction simulation between the induced relation of a transition relation (TRel) and the source-target calculus (STCal), then there is also a weak reduction simulation in the transition relation's transitive closure with respect to the target.
The lemma named "continuous_transform_within" is about two functions 'f' and 'g' that map from a metric space 'a' to a topological space 'b'. It assumes that the function 'f' is continuous at a point 'x' within a set 's', that 'd' is a positive number, and that 'x' is an element of 's'. It also assumes that for any 'x'...
If 'a' is an element of 'A' and 'A' is small, then the order map of 'a' in 'A' with respect to relation 'r' is an element of the set of elements of the order type of 'A' with respect to 'r'.
If a new character data is derived from 'h' to 'h'' and a new character data pointer is derived from 'h' to 'new_character_data_ptr', and the pointer is not equal to the new character data pointer, then the character data getter preserves the character data from 'h' to 'h''.
For all sets of states that intersect with the universal set and the invariant l2_inv7 and the universal set and the invariant l2_inv8, the transition from the state "l1_learn m" to the state "l2_dy_fake_msg m" refines the relation R12s.
If the boolean value of 'b' is not true in the state 's', then the execution of the while-loop 'b' with the command 'c' in the state 's' will result in the state 's'.
If n is greater than 0, then the function "bin_to_bl_aux" with parameters n, the numeral of (Num.Bit1 w), and bl is equal to the function "bin_to_bl_aux" with parameters (n - 1), the numeral of w, and a list starting with True followed by bl.
The weak equality of p and q is less than or equal to the downward boolean relation of the weak equality of p' and q'. This is true if p and p' are in the sorted polynomial relation composed with the multiset polynomial relation, and q and q' are also in the sorted polynomial relation composed with the multiset polynom...
Negative x is in the set of non-positive integers if and only if x is in the set of natural numbers.
The lemma named "conga_ex_cong3" assumes that triangle ABC is congruent to triangle A'B'C'. It shows that there exist points AA and CC such that if B is an exterior point to both AA and CC, then the triangle formed by AA, B, and CC is congruent to triangle A'B'C'.
If a function 'f' tends to a limit 'l' under a filter 'F', and 'l' is not equal to zero, then the sign function of 'f' tends to the sign of 'l' under the same filter 'F'. This is for 'l' which is a type of real normed vector.
If a match "m" is in normalized negation normal form (as per assumption "n"), and another match "m'" is in the set resulting from the normalization of ports in "m" (as per assumption "inset"), and there is no negated discriminator in the rewritten match "m" (as per assumption "noNeg"), and for any matches "m" and "m'",...
The lemma named "prop_equiv" in PLM shows that if for all 'x', the property 'F' of 'x' is equivalent to the property 'G' of 'x', then 'F' is equal to 'G' in 'v'.
The composition pattern of "C nm" is equal to the untyped version of "C" with "nm" and an empty list.
If a program P indicates that class C has fields FDTs, then class C is indeed a class in program P.
"it is not the case that the function 'f' is executed infinitely often in the state 's'".
The denial function applied to the composition of the identity function and a function that floors a value x, is equivalent to the assertion that for all Dx, x is floored.
, assuming the execution of a command 'c' in a state 's' under a stripped semantic environment 'F' in 'n' steps results in a state 't' (denoted as "strip F \<Gamma>\<turnstile>\<langle>c,s\<rangle> =n\<Rightarrow> t"), it can be shown that there exists a state 't'' such that the execution of the same command 'c' in the...
The lemma assumes that the image of x under the function Xd is not in the image of the set Ch h X under the same function. It also assumes that x is an element of the set X. Under these assumptions, it shows two things: 1. x is not an element of the set Ch h of the union of sets X and Y. This is referred to as "no re...
The lemma named "lasttoshow" is defined with the following conditions: It fixes a function 'd' mapping natural numbers to real numbers, two functions 'a' and 'b' mapping natural numbers to integers, an integer 'q', and a natural number 's'. It assumes that for all natural numbers 'n', 'd' of 'n' is greater than 1, '...
The tau function applied to the parallel composition of x and y is equal to the parallel composition of the tau function applied to x and the tau function applied to y.
The product of "x" and the omega power of the product of "y" and "x" is less than or equal to the omega power of the product of "x" and "y".
If a statement 's' is well-typed under the contexts 'Theta', 'Phi', 'B', 'Gamma', 'Delta' with the type 'tau', then 's' is well-formed with respect to the base type of 'tau'. 2. The second part of the lemma states that if a branch 'cs' is checked under the contexts 'Theta', 'Phi', 'B', 'Gamma', 'Delta' with the type i...
The lemma named "sorted_list_of_set_eq_pick" assumes that "i" is less than the length of the sorted list of the set "I". It shows that the "i"-th element of the sorted list of the set "I" is equal to the "i"-th pick from the set "I".
For any extended integers n and m, n plus 1 is less than or equal to m plus 1 if and only if n is less than or equal to m.
: Assuming for all l and l', if l is not equal to l' then GSMP_disjoint (the projection of terms from a list A onto a state transition l) and (the projection of terms from a list A onto a state transition l') are secure. And, if a term t is in the GSMP (the set of terms that can be generated by a state transition) of t...
The length of the first element of the function "typeof_opubx" applied to "opubx" is equal to the arity of the function "deinl" applied to the function "deubx" applied to "opubx".
The lemma "Greater_unbounded" assumes that "X" is finite, "r" is a valid region with respect to "X", "k", and "I", "c" is an element of "X", and "I c" is greater than "k c". Under these assumptions, it shows that "c" is unbounded in the region defined by "X", "I", and "r".
For a given hyperreal number 'x', if 'x' is in the set of hyperfinite numbers, then the star function of the exponential of 'x' is also in the set of hyperfinite numbers.
If a function 'f' is differentiable at a point 'x' within a set 's', and if set 't' is a subset of 's', then the function 'f' is also differentiable at the point 'x' within the set 't'.
If a property or condition (phi) holds for all elements x, then it is universally true that the property or condition (phi) applies to all x.
If "i" is an element of "Vs", then the permutation of "i" by "p" is not an element of "Vs".
The sine of pi/2 is equal to 1, given the sort constraint that 'a is a real field, a Banach space, and a real normed algebra 1.
Running the "put_env" function with parameters "s" and "m" in the environment "r" is equivalent to putting "s" in the result of running the environment "m" in "r".

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