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The lemma named "canon_ang_diff" shows that the canonical angle of the difference between two angles alpha and beta is equal to the difference of the canonical angles of alpha and beta.
The lemma named "closure_of_hull" assumes that set S is a subset of the topological space X. It shows that the closure of set S in topological space X is equal to the closure of set S in the closed subset of topological space X.
: Assuming there is a transition from "root" to "root'" represented by "xs", and "root" is an environment with attributes "att" and directory "dir", then there must exist attributes "att" and directory "dir" such that "root'" is also an environment with these attributes and directory.
If there is a tau-reduction from the state (e, xs) to the state (e', xs') in the context of the program P, thread t, and heap h with external actions extTA, then there is also a tau-reduction from the state (try e catch(C V) e2, xs) to the state (try e' catch(C V) e2, xs') under the same conditions.
The lemma named "is_tangle_diagram_weak_matrix_match" assumes that "w1*ws1" is a tangle diagram and the codomain block of "w2" is 0. Under these assumptions, it shows that the weak matrix match of the block matrix of "w1", the Kauffman matrix of "ws1", and the block matrix of "w2" holds true.
Filtering a list twice, first by a condition Q and then by a condition P, is equivalent to filtering the list once by a condition that checks if both P and Q are true for each element in the list.
The degree of the negation of a polynomial deviation 'x' is equal to the degree of 'x'.
: 1. There is no "While" loop in an atomic statement. 2. If there is no "While" loop in command 1 and command 2, then there is no "While" loop in the sequence of command 1 and command 2. 3. If there is no "While" loop in command 1 and command 2, then there is no "While" loop in the "If" statement that tests a condition...
If D is a monotonically increasing function, then the function 'dgr D' is also monotonically increasing.
If proposition phi holds for Br_1, then it also holds for Br_6.
If "alpha" is a bijective function, then "alpha" is a global renaming of "beta" in state "s" with respect to the global revision function "R_G" of "alpha", "beta", and "s".
The lemma named "strict_subtree_subset" assumes that "t" is a strict subtree of a node "r" with children "xs", and that "xs" is a subset of "ys". It shows that "t" is also a strict subtree of a node "r" with children "ys".
For any type 'a that is a linear order, the "less than" relation between 'x' and 'y' is equivalent to the "less than or equal to" relation between 'x' and 'y' and 'x' is not equal to 'y'.
The lemma named "affine_independent_insert" is about a fixed point 'a' in a Euclidean space. It shows that if 'S' is not affine dependent and 'a' is not in the affine hull of 'S', then the set formed by inserting 'a' into 'S' is also not affine dependent.
The lemma named "eqExcPID_N_Paper" assumes that "s" and "s1" are equivalent except for the PID_N. It also assumes that the paper in "s" with the id "pid" has the title, abstract, content, reviews, dis, and decs, and the paper in "s1" with the same id "pid" has the title1, abstract1, content1, reviews1, dis1, and decs1....
The function "psi_l" with parameter "l" belongs to the carrier vector space of dimension "N".
The lemma named "lleq_less" establishes that for any two fixed characters x and y, the condition of x being less than y is equivalent to x being less than or equal to y and y not being less than or equal to x.
The long sequence of elements (lnths) from a set (xs) with no elements ({}) is an empty long sequence (LNil).
For any two integers 'a' and 'b', if 'b' is greater than 0, then the greatest common divisor of 'a' and 'b' is less than or equal to 'b'.
: assuming a pair [a, b] is an object in the product of the opposite of category C and category C itself, then the object map of the Hom-functor of category C applied to the pair [a, b] is equal to the Hom-set from a to b in category C.
If x is equivalent to y, and y is less than or equal to z, and x, y, and z are all elements of set A, then x is less than or equal to z. This is a statement of transitivity in an equivalence order.
The lemma named "DC_refines" is defined with three fixed elements: a program 'a', and two unspecified elements 'b' and 'c'. It assumes that 'a' is a refinement of both 'b' and 'c'. The lemma then shows that 'a' is also a refinement of the greatest lower bound of 'b' and 'c'.
: Assuming that the well-formed program 'wfp' is represented as a pair of a program and its procedures, and another well-formed program 'wfp'' is represented as a pair of either the same program (if a condition 'b' is true) or a different program 'c2', and its procedures. Then, if 'wfp' has a path from a node 'n' in ...
The observation is consistent between "R0sk1iai" and "med0sk1iai" for "a0i" in the first level of the skeleton.
If for all Z, under the context Gamma, the Hoare triple (P Z) p (Q Z), (A Z) is valid with an empty set of faults, then for all Z, under the same context Gamma, the Hoare triple (P Z) p (Q Z), (A Z) is also valid with any set of faults F.
The lemma "Can_Nmlize_Can" assumes that "t" is a canonical morphism. It shows that the normal form of "t" is also a canonical morphism.
Mapping a function 'f' over two lists 'x' and 'y', each replicated 'n' times, is equivalent to replicating the result of applying the function 'f' to 'x' and 'y' 'n' times.
If 'x' is equivalent to a logical implication, then this leads to a contradiction or falsehood.
If 'r' is an order, and 'x' is less than or equal to 'y' according to the order 'r', and 'y' is also less than or equal to 'x' according to the same order 'r', then 'x' must be equal to 'y'. This is a statement of the antisymmetry property of an order relation.
If a function 'f' is a continuous map from a topological space 's' to another topological space 't'', and 't'' is finer than 't', then 'f' is also a continuous map from 's' to 't'.
The lemma named "tuple_pr" assumes that "a" and "b" are identities, and "h" is a function from the domain of "h" to the product of "a" and "b". It shows that the tuple of the composition of "pr1 a b" and "h", and the composition of "pr0 a b" and "h", is equal to "h".
If an element 'g' belongs to a set 'G' and for all elements 'x' in 'G', the binary operation of 'g' and 'x' equals 'x', then 'g' must be the identity element 'e'.
If a function 'f' tends towards a non-zero constant 'c' under a filter 'F' which is not the bottom element, then the sign of 'f' under 'F' is equal to the sign of 'c'.
The operation of updating the heap map "hm" with key "k" and value "v" is less than or equal to the specification of maintaining the heap map invariant "hmr_invar".
The lemma, in the context of Max_impl, named Max_spec2, demonstrates that for all states 'σ' and 'leq', it is provable in the logic system 'Γ' that if the state 'σ' intersects with the set of all states 'τ' such that 'r' is defined as the result of the procedure 'compare' applied to 'i' and 'j' and 'r' equals to the re...
The negation of 'a' is equivalent to 'a' or not 'a'.
The lemma named "proper_map_snd" assumes that "S" is compact, "K" is a subset of "T", and "K" is compact. It shows that the intersection of the Cartesian product of "S" and "T" and the preimage of "K" under the second projection function is compact.
Given the assumption that a natural transformation 'pi prime' maps from the index set I and object A in the category CC to a cocone U prime in the category I, there exists a unique morphism 'f prime' from U to U prime in the category CC such that for all elements j in the index set I, the natural transformation map of ...
If there are silent moves from state 's' to state 's'' in a multithreaded system 'mthr', and if the thread states of 's' are okay with respect to a program 'P' and shared state 'shr s', then the thread states of 's'' are also okay with respect to the same program 'P' and shared state 'shr s''.
An ordinal number 'x' is less than another ordinal number 'y' if and only if 'x' is less than or equal to 'y' and 'y' is not less than or equal to 'x'.
The ordinate (or y-coordinate) of the point negative A is equal to the negative of the ordinate of point A.
The actions are okay in state 's' for thread 't' with thread action 'ta' if and only if the following conditions are met: 1. The lock actions are okay with the current locks in state 's' for thread 't' with thread action 'ta'. 2. The thread actions are okay with the current threads in state 's' with thread action 'ta'...
If the intersection of the bound variables of a stateful strand S and the domain of a substitution θ is empty, then the set of operations on the stateful strand S under the substitution θ is equal to the set of operations on the stateful strand S, where each operation is transformed by applying the substitution θ to it...
'v' is greater than zero.
If T is a minimum weight basis with no additional elements, then T is a minimum basis.
If the following conditions are met: 1) 'a' is greater than or equal to 1, 2) 'r' is an increasing sequence, 3) for all 'n', the rational number 'r n' is less than or equal to 'x' (where 'x' is an element of an archimedean field), then the sequence defined by 'a' to the power of 'r n' is a bounded sequence.
The extraction of a list based on a predicate P from a list that starts with an element x followed by a list xs is defined as follows: If the predicate P holds true for the element x, then the result is a tuple containing an empty list, the element x, and the list xs. Otherwise, the result depends on the extraction of ...
: given a natural transformation "psi" from functor "F" to functor "G", both mapping from category "B" to category "C" in a universe "alpha", and an object "a" in the opposite category of "C", the domain of the arrow in the right composition of the Hom-component of "psi" with "a" and "b" is equal to the Hom-set in cate...
: Assuming that: 1. Function 'x' uniquely solves the ordinary differential equation (ODE) 'f' from time 'tx' over interval 'T' in set 'X'. 2. For any time 't' in the intersection of the closures of sets 'S' and 'T', function 'y' uniquely solves the ODE 'g' from time 't' over interval 'S' in set 'X'. 3. The intersectio...
The lemma "deriv_cong_ev" assumes that for values close to 'x', the functions 'f' and 'g' produce the same result, and that 'x' is equal to 'y'. Under these assumptions, it shows that the derivative of function 'f' at point 'x' is equal to the derivative of function 'g' at point 'y'.
: 1. "max_v" is an eigenvector of the matrix "A" with eigenvalue "max_ev". 2. The norm of "max_ev" is equal to "sr". 3. The 1-norm of "max_v" is equal to 1.
If the function "stored_reused_aux" applied to the parameters t', t, mr, and mp results in Some (p, r), then p and r are a generalized output of t' and t.
If a function 'f' is continuous on a closed segment from 'a' to 'b', and 'k' is a real number between 0 and 1 inclusive, and 'c' is a point on the line segment from 'a' to 'b' such that the vector from 'a' to 'c' is 'k' times the vector from 'a' to 'b', then the function 'f' is also continuous on the closed segment fro...
If M is less than or equal to N, then the result of the function In0 applied to M is less than or equal to the result of the function In0 applied to N.
If almost everywhere in a measure space M, function f is less than or equal to function g, and both functions f and g are integrable over M, then almost everywhere in M, the real conditional expectation of f with respect to M and a sigma-algebra F is less than or equal to the real conditional expectation of g with resp...
The theorem named "unfold_contents" assumes that "E" is a constant predicate and "self" is defined. It shows that the evaluation of "E" applied to the contents of "self" is equivalent to the evaluation of "E" applied to either the set containing only the salary of "self" (if the boss of "self" is null), or the contents...
The lemma named "c_ballE" assumes that "x" is an element of the closed ball with center "c" and radius "n". It shows two things: first, "x" is an element of the carrier of the p-adic numbers. Second, the valuation of the difference between "x" and "c" is greater than or equal to "n".
The lemma named "mlex_snd_decrI" is defined with fixed natural numbers a, a', b, b', and N. It assumes that a is equal to a' and b is less than b'. Under these assumptions, it shows that the expression "a times N plus b" is less than "a' times N plus b'".
The lemma "init_fin_silent_move_RunningI" assumes that there is a silent move from state (x, m) to state (x', m') for thread t. It shows that there is also a silent move in the transition system "tau trsys" from state ((Running, x), m) to state ((Running, x'), m') when initializing and finalizing thread t, where the si...
If the first element of function 'f' at program counter 'pc' is 'Some' with a tuple of 'T' at the head of the stack 'ST' and local table 'LT', and 'T' is a subtype of 'rT' in the context of 'G', and 'pc' is less than 'max_pc', and the type check for the translation of 'G' with maximum stack 'mxs' and maximum registers ...
The lemma named "word_enum_decomp_elem" assumes that a list consisting of the word 'x' is equal to the concatenation of list 'as', element 'a', and list 'bs', where 'y' is a word of any length 'a'. It shows that 'x' is less than or equal to 'a' and 'a' is less than or equal to 'y'.
If a path P is not finite, the set of P is a subset of S, P is a valid path, and the path P conforms with the well-ordered strategy p, then there exists a number n such that for all numbers m greater than or equal to n, the path strategy of P at n is equal to the path strategy of P at m. This implies that the path stra...
For any two polynomials 'f' and 'h' over a modular ring, if 'h' raised to the power of the cardinality of 'a' is congruent to 'h' modulo 'f', and if 'f' is monic and square-free, then 'f' is equal to the product of the greatest common divisor of 'f' and 'h' minus 'c' for all 'c' in the set of all elements of the modula...
"n is not an element of I or the minimum element of I is n" if and only if "the immediate predecessor of n in I is n".
The content of the box defined by points a and b is equal to the product of the differences between the coordinates of b and a for each basis, if for all bases, the coordinate of a is less than or equal to the coordinate of b. Otherwise, the content of the box is zero.
If the thread states are okay according to a certain condition Q for some index i, then there exists an index set I such that the thread states are invariant under the condition Q.
If the return of x is less than or equal to the refinement of M' under R', and for all x' such that (x, x') is in R', the return of the function f applied to x is less than or equal to the refinement of the function f' applied to x' under R, then the return of the Let function applied to x and f is less than or equal t...
Given two functors, "GG" from category "AA" to category "CC" and "HH" from category "BB" to category "CC", and two morphisms "G" and "F" in the comma category of "GG" and "HH", where "G" is from "B" to "C" and "F" is from "A" to "B", the composition of "G" and "F" in the comma category is a morphism from "A" to "C" wit...
The lemma named "pos_neg_rec_call_mono" assumes that "pos_neg_rec_call" is true for some boolean value "b" and some set "A". It shows that if "b" is true, then the function "applies_eq" with input "A" is monotonic. And if "b" is false, then the function "applies_eq" with input "A" is non-increasing.
If a function 'f' is residuated, then 'f' of 'x' is less than or equal to 'y' if and only if 'x' is less than or equal to the residual of 'f' at 'y'.
If a real number 'a' is greater than zero, then 'a' raised to any real power 'x' is also greater than zero.
For a Euclidean space of dimension 2 or greater, the space remains path-connected even after removing a single point.
The lemma named "beukers_nn_integral1_different" assumes that "r" is greater than "s". It shows that the Beukers non-negative integral of "r" and "s" is equal to the non-negative real number obtained by dividing the sum of the reciprocals of the squares of all numbers from "s" to "r" by the difference between "r" and "...
: Given a fixed identifier 'c', a function 't' mapping natural numbers to configurations, another function 't'' mapping natural numbers to components, and two natural numbers 'n' and 'n'', under the assumptions that there does not exist a natural number 'i' greater than or equal to 'n' such that 'c' is parallel in the...
If two states s1 and s2 are multi-threaded bisimilar (mbisim), and the set of conditional actions collected from ta1 is empty, and ta1 and ta2 are bisimilar with respect to bisim, then the actions performed in state s1 by thread t according to ta1 are valid if and only if the actions performed in state s2 by thread t a...
If S is an Integral Domain, A, B, Y are Polynomial Rings, h is a ring homomorphism from S to B, and s is an element of the carrier set S, then the evaluation ring homomorphism of R, S, X, A, B, Y, h, s is equal to the ring homomorphism of s.
"x is equal to the result of the function f4 applied to x and n".
The equality of a comparator (of an enumerable type 'a) to itself is always true.
The lemma named "one_sheaf_spec_in_sheaf_spec" assumes that "U" is a Zariski open set. It shows that the one-sheaf specification of "U" is an element of the structure sheaf of "U".
The lemma named "language_coinduct_upto_Plus" assumes a relation R between two languages L and K. It also assumes a hypothesis that for any two languages L and K, if the relation R holds between them, then the empty word of L is equal to the empty word of K and the derivative of L is related to the derivative of K thro...
The agent X is atomic.
The lemma "OclExcluding_excludes" assumes that "x" is excluded from the set "X" in the state "τ". It shows that excluding "x" from the set "X" in the state "τ" is equivalent to the set "X" itself.
If n1 is less than or equal to n2 and m1 is less than or equal to m2, then the board with dimensions n1 by m1 is a subset of the board with dimensions n2 by m2.
: Assuming two conditions, firstly, for any stream 'omega', if 'P omega' implies 'Q1 omega' and 'Q2 omega', and secondly, if 'P omega' implies 'R1 omega' and 'R2 omega'. If it is also assumed that 'Q1 suntil R1' holds for 'omega' and 'P omega' always holds, then it can be shown that 'Q2 suntil R2' also holds for 'omega...
There exists an 'x' such that if 'x' is equal to 'y', then for all 'y', 'y' is either equal to 'x' or not equal to 'x'. Otherwise, for all 'y', 'y' is either equal to the pair '(x, x)' or not equal to the pair '(x, x)'.
If X is an element of G, then the parts of the set formed by inserting X into H are a subset of the union of the parts of G and the parts of H.
The configuration of a virtual machine's heap "h" and a variable state "vs" with an inserted value "v" at the address "ad" and location "al" is equivalent to the configuration of the heap "h" and the original variable state "vs", if and only if there exists a type "T" such that the address "ad" at location "al" is of t...
The lemma named "dist_dens_integral_space" assumes that "v" is in the type universe of the distribution parameter type "dst". It shows that the positive integral of the distribution density "dst" for "v" and "u" with respect to the stock measure of the distribution result type "dst" is less than or equal to 1.
If "k" is not less than the length of list "l", then the imperative array map of list "l" at index "k" is None.
If the function "incrIndexList" is applied to the list "ls", the integer "m", and the maximum integer "nmax", then the length of the list "ls" is equal to "m".
: 1. If the pair (p, q) is an element of the set Delta_E_A, then p is an element of the set Q_A. 2. If the pair (p, q) is an element of the set Delta_E_A, then q is an element of the set Q_A. 3. If the pair (p, q) is an element of the transitive closure of the set Delta_E_A, then p is an element of the set Q_A. 4. If ...
The lemma named "primroot_len_mult" assumes that "u" is not equal to the empty word and "u" is in the closure of "q". It then obtains a "k" such that the length of "q" is equal to "k" times the length of the primitive root of "u".
Two 3-element tuples are equal if and only if their corresponding elements are equal. Specifically, the first elements of the tuples are equal, the second elements of the tuples are equal, and the third elements of the tuples are equal.
If 'n' is greater than 0 and for all 'j' less than or equal to 'n', 'f j' is in the carrier 'A', then the sum over 'A' of the composition of 'f' and the transposition of 'n - 1' and 'n' is equal to the sum over 'A' of 'f n'.
The lemma named "injectivity_subexpr2" assumes the following conditions: 1. "a" is not an element of set "A". 2. Inserting the set containing "a" into set "P" is equal to inserting the set containing "a" into set "P'". 3. Both "P" and "P'" are partitions on "A" and the cardinality (or size) of both "P" and "P'" is "k'"...
For a given set S of real numbers, if S is not empty and is bounded above, then the supremum of S is included in the closure of S.
The greatest common divisor of the absolute value of an integer 'x' and another integer 'y' is equal to the greatest common divisor of 'x' and 'y'. This is applicable for any integers 'x' and 'y'.
If "I" is a finite set, then the join of "f" over the insertion of "n" into "I" is equal to the join of "f" over "I" restricted to elements less than "n", followed by "f" of "n", followed by the join of "f" over "I" restricted to elements greater than "n".
The lemma named "has_integral_restrict" is about a function 'f' that maps from a Euclidean space to a Banach space. It assumes that 'S' is a subset of 'T'. It shows that the function 'f' has an integral 'i' over the set 'T' if and only if 'f' has the same integral 'i' over the subset 'S'. This is true when the function...
The lemma "smcf_proj_snd_is_semifunctor" assumes that "i" is an element of "I". It shows that the second projection of the semicategory product of "A" and "B" is a semifunctor from the semicategory product of "A" and "B" to "B".