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If an element 'a' is in the set 'b', then 'a' is included in 'b'.
: Assuming that R1 is a subset of upRules, R2 is a subset of nprovRules, and R3 is a subset of provRules. Also, R is the union of Ax, R1, R2, and R3, and r is an element of either R1, R2, or R3, and r is equal to the pair (ps,c). Given the induction hypothesis (IH) that for all m less than n, for all sets of formulas...
The lemma named "cmp_simps" from the theory "Pseudofunctor" in the context of Bicategory states the following: Assuming that "\<mu>\<nu>" is an arrow in the category "S.VV", it shows that "cmp \<mu>\<nu>" is an arrow in the category "B". It also shows that the source of "cmp \<mu>\<nu>" in category "B" is the same as ...
If 'x' and 'z' are in the range from 0 to 1 inclusive, then the function 'tl' applied twice in succession to 'x' and 'y', and then to 'z', is equal to the function 'tl' applied to 'x' and the result of 'tl' applied to 'y' and 'z'.
If k is greater than 0, then x is an element of the interval I divided by k if and only if the intersection of interval I and the interval from x times k to k minus 1 is not an empty set.
The lemma named "lcmof_leastUpper" is about a structure G. It assumes that both 'a' and 'b' are elements in the carrier of G. It shows that 'x' is an element in the carrier of G and 'x' is the least common multiple of 'a' and 'b'. This is equivalent to 'x' being the least element in the division relation of G that is a...
For a given unsigned cast of a word 'a' of any length 'a::len', the bit at position 'n' is the same as the bit at position 'n' in 'a' if and only if 'n' is less than the length of 'a::len'.
If "M" is a module over "R" and "T" is a subset of the carrier of "M", then "T" is a subset of the left scalar multiplication of "M" and "T" over "R".
If you substitute variable 'x' with variable 'z' in the abstract expression of value 'x', the result will be the abstract expression of value 'z'.
The lemma named "closed_map_snd" is defined with two fixed sets S and T, where S is a set in a Euclidean space 'a and T is a set in a Euclidean space 'b. The lemma assumes that S is compact and c is closed in the topological space of the Cartesian product of S and T. It shows that the image of c under the function 'snd...
If "Y" is a well-sorted term of sort "ys", and "inp" and "binp" are well-sorted inputs and binary inputs of sort "delta" respectively, then the substitution of "Y" for "y" in the operation "Op" with sort "delta", "inp", and "binp" is equal to the operation "Op" with sort "delta", the substitution of "Y" for "y" in "inp...
A function composition 'g' after 'f' is a continuous map on a topological space 'T' to a topological space 'X' if and only if 'g' is a continuous map from the topological space obtained by mapping 'f' onto 'T' to 'X'.
The range of the variable renaming function and its inverse are well-formed terms, for any natural number n.
The length of the nearest neighbors from a point in a k-dimensional tree is equal to the minimum of n and the size of the k-dimensional tree.
The function "check_all_list" with parameters P, l, msg, and msgf returns a right unit (Inr ()) if and only if for all elements x in the set l, the predicate P holds true for x.
The lemma named "useless_leq_useless" is defined as follows: Let "r" be defined as the cardinality of set A. If "x" is useless, "x" is less than or equal to "y", and the length of "x" equals "r", then "y" is also useless.
Lemma LIM_NSLIM: Assuming that the function f has a limit L as it approaches a, it is shown that f has a nonstandard limit L as it approaches a in the nonstandard analysis.
The lemma named "find_path_fields" implies that if the function "find_path_ty_f" applied to parameters P and ty returns some path, then there exists a field set such that the function "fields_in_path_f" applied to this path equals this field set.
The complex interval of integer 0 is equal to 0.
The lemma named "natural_transformation_α" shows that "α_SB" is a natural transformation in the composition of triple V and vertical composition from HoHV to HoVH.
The composition of the function "sfilter" applied to "p" and "sfilter" applied to "q" is equal to "sfilter" applied to a function that takes an input "x" and returns the logical conjunction of "q" applied to "x" and "p" applied to "x".
If a pair (s, t) is in the relation gctxtex_onp P R and t is in the set T_G F, then the pair (s, t) is also in the relation gctxtex_onp for a function C such that the function applied to gctxt C is a subset of F and P C holds.
If 'x' is not in the Gromov boundary, then there exists an 'a' such that 'x' is equal to the Gromov completion of 'a'.
Given a certain number of steps 'stp' in the 'tm_dither' Turing machine, starting from the first state with 'm' blank symbols followed by an 'Oc' symbol and 'n' blank symbols, the machine will either remain in the same state and configuration, or it will move to the second state, with an 'Oc' symbol followed by 'm' bla...
The function 'is_type' applied to the composition of 'f' and 'P' is equal to 'is_type' applied to 'P'. This means that the type of the composition of 'f' and 'P' is the same as the type of 'P'.
The lemma named "os_ts__sams" assumes that "B P TS A C" and "A B OS P C". If these conditions are met, it shows that "SAMS A B P P B C".
Performing the "unset_break_status" operation twice in sequence before performing operation "M" is equivalent to performing "unset_break_status" once before operation "M". This is an idempotent property of the "unset_break_status" operation.
The decoding of a value 'M' with a parameter 'e' is equal to the sum of 'base' raised to the power of decoding '0' and 'm' for each 'm' in the multiset 'M', all divided by 'base' raised to the power of 'e'.
The function "get_owner_document" is an instance of "l_get_owner_document".
The real part of the complex number z divided by a real number r is equal to the real part of z divided by r.
The product of "n" applied to "top" and "L" equals "L".
When applying the function "takeWhile" with a predicate P to the concatenation of two lists (xs and ys), the result is as follows: if all elements in the list xs satisfy the predicate P, then the result is the concatenation of xs and the result of applying "takeWhile" with predicate P to ys. Otherwise, the result is th...
The lemma for the greatest common divisor (gcd) of two integers k and l is defined as follows: if l equals zero, then the gcd is the absolute value of k. Otherwise, the gcd is the absolute value of the gcd of l and the remainder of the absolute value of k divided by the absolute value of l.
The relation of the inverse of C with respect to equality is the same as the inverse of the relation of C with respect to equality.
The non-negative integral of a function 'f' with respect to the measure 'INTEG' is equal to the non-negative integral of the function 'f' applied to 'IntVal x' with respect to the count_space over the universal set.
When the function "make_full_observation" is applied with parameters e, None, r, p, and i, and then the function "statename" is sequentially mapped over the result, it is equivalent to a constant sequence of None.
The lemma named "hd_pow" assumes that "n" is not equal to zero. It shows that the head of the sequence "u" raised to the power of "n" is equal to the head of the sequence "u".
The lemma named "initial_d_sb" assumes that the initial state of a thread system "ts" with shared memory "\<S>" is valid under the "initial\<^sub>d" condition. It shows that the initial state of the same thread system "ts" with shared memory "\<S>", after being mapped by a function that removes the store buffer "sb" fr...
The lemma named "cartesian_power_tail" assumes that "as" is an element of the carrier of the ring R raised to the power of (n+1). It shows that the tail of "as" is an element of the carrier of the ring R raised to the power of n.
If there is a reduction sequence from polynomial 'p' to polynomial 'q' under a set 'F', and for all terms 'u' and 'v', if 'u' is in the keys of 'p' and 'v' is in the keys of 'r', then 'u' is less than 'v' in term order, then there is a reduction sequence from the sum of 'p' and 'r' to the sum of 'q' and 'r' under the s...
The lemma named "diag_block_mat_cong_comp" assumes the following conditions: 1. The lengths of lists Al and Bl are equal. 2. For all indices i less than the length of Al, the number of rows in the i-th element of Al is equal to the number of rows in the i-th element of Bl. 3. For all indices i less than the length of ...
The implementation of "find_ce" is refined by or equal to "gfind_ce" under the identity relation.
The semantics of an extended strand space theory with a given initial multiset M\<^sub>0, an interpretation \<I>, and a strand space \<A> is equal to the semantics of a configuration with the same initial multiset M\<^sub>0, the same interpretation \<I>, and the strand space \<A> transformed to a stateful strand.
If for every element 'y' in set 'A', the function 'g' is monotone with respect to 'x' under the order 'ord' and the less than or equal to relation, then the function that maps 'x' to the supremum of 'g x y' for all 'y' in 'A' is also monotone under the same order and relation.
The least common multiple of 'a' and 'b' is zero if and only if 'a' is zero or 'b' is zero.
The lemma named "resChainAppend" is defined with three fixed elements: "xvec" which is a list of names, "yvec" which is also a list of names, and "P" which is a psi of type 'a, 'b, 'c where 'a, 'b, 'c are names in the file system. The lemma shows that the restriction of the concatenation of "xvec" and "yvec" on "P" is ...
The union of the image of the segment function applied to all numbers less than "k" is equal to an empty set if "k" equals zero, otherwise it is equal to the set of all real numbers from zero to "k" divided by "n".
If for all initial states 's0', a function 'f' is valid under normal form (NF) with a precondition 'P' that depends on 's0' and 's', and a postcondition 'Q' that depends on 's0', the return value 'rv', and the state 's', then it is also valid under normal form with a precondition that there exists an initial state 's0'...
If alpha is in the set of equality axioms, then alpha is a provable formula.
If a function 'f' from real numbers to real numbers is an element of little-o of 'g', then the lambda function that takes a natural number 'x' and maps it to 'f' of the real value of 'x', is an element of little-o of the lambda function that takes 'x' and maps it to 'g' of the real value of 'x'.
The lemma named "closure_atLeastLessThan" is about a fixed pair of elements 'a' and 'b' from a set that has both a linear order topology and a dense order. Given the assumption that 'a' is less than 'b', it shows that the closure of the set of all elements from 'a' to 'b' (including 'a' but excluding 'b') is equal to t...
Removing duplicates from the difference of two lists 'a' and 'b' is the same as finding the difference between the two lists after duplicates have been removed from each.
If 'n' belongs to the set of hypernatural infinite numbers, then the product of the hyperreal sine of the inverse of 'n' and the hyperreal representation of 'n' is approximately equal to 1.
In a locally path-connected space X, the path component of a set X containing a point x is equal to the connected component of the same set X containing the same point x. This is under the assumption that the space X is locally path-connected.
If "s" is a last sequence at "k" for "y", and "l" is an element of "k", then "s" is a last sequence at "l" for the application of "s" at "l".
The number of rows in the matrix resulting from the application of the 'diagonal_to_Smith_i' function with parameters 'xs', 'A', 'i', and 'bezout' is equal to the number of rows in matrix 'A'.
If 'r' is a cardinal order, then the successor of 'r' in cardinality is a well order.
If 'n' is less than 'l', then the probability mass function mapped over 'y' at index 'n' for a Bernoulli variable of length 'l' is equal to a Bernoulli probability mass function with a probability of 0.5.
The set of ground terms of a term 't' is equal to the set of function symbols in the term obtained from the ground term 't'.
The lemma "typeII_cube_explicit_spec" assumes that "twoC" is a type II two-cube. It shows that there exist numbers "a" and "b", and functions "g1" and "g2" such that "a" is less than "b", and for all "x" in the interval from "a" to "b", "g2" of "x" is less than or equal to "g1" of "x". The image of the cube "twoC" is...
The lemma Union_regular_division: Assuming "a" is less than or equal to "b", it shows that the union of the regular division of "a" and "b" over "n" equals to an empty set if "n" equals to zero, otherwise it equals to the set from "a" to "b".
If a function 'f' is differentiable at 'x', and another function 'g' is differentiable at the value of 'f' at 'x', then the derivative of the composition of 'g' and 'f' at 'x' is equal to the product of the derivative of 'g' at the value of 'f' at 'x' and the derivative of 'f' at 'x'.
A topological space X is equivalent to a discrete topology containing a single element 'a' if and only if the topological space of X is a set containing the single element 'a'.
The function "int_decode" is injective on the set A.
The union of 'a' and 'b' minus 'c' is equal to the union of 'a' minus 'c' and 'b' minus 'c'.
The interpretation of "false" at any state is the double floor of "False".
The reference union is closed under the sequential composition of P1 and Q.
The cardinality (or size) of the set of senior states for 'p' at 'n' is less than or equal to the cardinality of the set of senior states for 'q' at 'n' if and only if the set of senior states for 'p' at 'n' is a subset of the set of senior states for 'q' at 'n'.
If a natural number 'p' is prime and even, then 'p' must be equal to 2.
If for any element 'x', the aggregate function 'ag' applied to a list containing only 'x' equals 'x', then applying the function 'f_aggregate' to a list 'xs' with the successor of 0 (which is 1) and the aggregate function 'ag' equals the list 'xs' itself.
"S" does not hold for the variable "x" and "t".
If 'y' is not an empty word property and 'x' equals the concatenation of 'x' and 'y' plus 'z', then 'x' equals 'z' concatenated with the Kleene star of 'y'.
The generic function "g_isEmpty" correctly implements the abstract function "set_isEmpty" under the assumption of invariant "invar".
The lemma named "card_of_Fpow_mono" assumes that the cardinality of set A is less than or equal to the cardinality of set B. It shows that the cardinality of the function power set of A is also less than or equal to the cardinality of the function power set of B.
If an element 'x' is not in set 'A', then the cardinality (or size) of the set obtained by inserting 'x' into 'A' is equal to the successor (or one more than) the cardinality of 'A'.
If the function 'x' bound to 'f' is not done, then resuming 'x' bound to 'f' is equivalent to: if 'x' is done, then resume 'f' with the result of 'x', otherwise, for any input 'inp', resume 'x' with 'inp' and then bind it to 'f'.
The list that starts with element 'a' followed by list 'xs' is equal to the list that starts with element 'b' followed by the same list 'xs'.
The function "an" of "x" equals 1 if and only if the function "an" of the star of "x" equals 1.
If function 'f' is monotone and 'f' maps from the carrier 'L' to the carrier 'L', then the greatest fixed point of 'f' is equal to the function 'f' applied to the greatest fixed point of 'f'.
The validity of the null value for an integer in the type system 'A' is true.
If the following conditions are met: 1. The state maintains the uniqueness invariant (InvariantUniq) for its current assignment (getM). 2. The state maintains the element invariant (InvariantWatchesEl) for its current formula (getF) and its two watch variables (getWatch1 and getWatch2). 3. The state's watch list (getW...
An empty set cannot be a strongly connected component (SCC) in a directed graph E.
The lemma assumes that M is a measure space and for any two distinct natural numbers j1 and j2 in a set R, the intersection of B j1 and B j2 is empty. It also assumes that for every natural number j in R, B j is in the measurable sets of M, and that R is a finite set. Given these assumptions, it shows that the measure ...
The Gamma function is asymptotically equivalent to the function sqrt(2*pi/x) * (x / e)^x for real numbers, as x approaches infinity.
The base type of a type expression "tau" remains the same even after a substitution of a variable "x" with a value "v" is performed on "tau".
For any streams 'g', 'h', 'sh', and 'sh'', the result of unstreaming the appended transformation of 'g' and 'h' on 'sh' with 'Inr sh'' is equal to the result of unstreaming 'h' on 'sh''.
The representation of the Gromov completion of a value 'y' when converted back to Gromov completion is equal to the original value 'y' for all 'n'.
The lemma named "flowD_outside" assumes "g" is a flow in the network "\<Delta>". It shows that if "e" is not an edge in the network, then the flow "g" through "e" is zero.
The lemma named "cross_orthogonal_transformation" assumes that "f" is an orthogonal transformation. It shows that the cross product of "f x" and "f y" is equal to the determinant of the matrix of "f" multiplied by the result of "f" applied to the cross product of "x" and "y".
: Assuming that the time event of process q at instance i+1 is less than or equal to the time event of process p at instance i+1, and process p, q, and l are correct at the time event of process p at instance i+1, it shows that the absolute difference between the difference of the program counters of process l at the t...
The lemma l8_4: Assuming "A, B, C" forms a right angle at B and "B" is the midpoint of line segment "C C'", it can be shown that "A, B, C'" also forms a right angle at B.
The product of the projection "proj_O" and the vector "alpha" equals the zero vector of dimension N.
If the relation set of A and B is equal, then A is equal to B.
If a function 'f' is injective (or one-to-one) on the union of the sets of elements 'xs' and 'ys', then the equality of the lists obtained by mapping 'f' over 'xs' and 'ys' implies the equality of 'xs' and 'ys' themselves.
If you store a 32-bit unsigned integer 'x' at a certain offset 'off' in an object 'obj', and then retrieve the bytes from the same offset, when these bytes are concatenated back into a 32-bit unsigned integer, you will get the original integer 'x'.
The lemma named "qbs_space_eq_Mx" assumes that the quasi-Borel space of X is equal to the quasi-Borel space of Y. Under this assumption, it shows that the quasi-Borel space of X is equal to the quasi-Borel space of Y.
The lemma "w_method_via_spy_framework_completeness_and_finiteness" is about two finite state machines, M1 and M2. These machines are of a certain type, where 'a, 'b, and 'c are linearly ordered. The lemma assumes that both M1 and M2 are observable and minimal, and that the size of M1 and M2 is less than or equal to a c...
The lemma processedD3: Assuming that the pair (a, b) is processed with respect to xs and ps, it shows that the pair (a, b) is not an element of the set ps.
The set of infinitesimal numbers is a subset of the set of finite hyperreal numbers.
The lemma X_ZmX_carrier shows that X_ZmX is in the carrier matrix of size 4 by 4.