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"thetaParZOL2" is a subset of the "ZOretr" of the union of "thetaParZOL2" and "ZObis".
The support of the function 'f' multiplied by 'r' is a subset of 'R'.
If the lengths of two lists 'xs' and 'ys' are equal, and a variable 'x' is not equal to another variable 'y' and 'x' is in the free variables of 'Q', then for any two lists 'xs' and 'ys' with equal lengths, the generalized substitution of 'x' with 'xs' and 'ys' in 'Q' implies the generalized substitution of 'x' with 'x...
If we assume that trace "w1" is equivalent to trace "w2", then we can obtain that "w1" is less than or equal to "w2" and "w2" is less than or equal to "w1".
Property 'P' holds for the empty set. The 'insert' case states that for any element 'a' and set 'F', if 'F' is finite, 'a' is an element of 'A', 'F' is a subset of 'A', 'a' is not an element of 'F', and property 'P' holds for 'F', then property 'P' also holds for the set that results from inserting 'a' into 'F'. Th...
The lemma named "prob_space_step" assumes that "p" is a policy and "x" is an element in the space "M". It shows that the limit stream of the successor policy "p" at "x" and "K" at "x" is a probability space.
A variable 'a' is fresh for the sequence form predicate with parameters 's', 'k', and 't' if and only if 'a' is fresh for 's', 'k', and 't' individually. 2. The second lemma (eval_fm_SeqFormP) states that the evaluation of the sequence form predicate with parameters 's', 'k', and 't' under a certain environment 'e' is...
If x1 is less than or equal to the finite value y1, and x2 is less than or equal to the finite value y2, then the sum of x1 and x2 is less than or equal to the finite sum of y1 and y2. This is under the condition that the type 'a is an ordered additive abelian group.
The lemma fixes three integers a, b, and c. It assumes that the function f of the sum of (a - a) and (a - a), plus the function f of the sum of 0 and c, equals the sum of the function f of 0 and the function f of c. It shows that the function f of the sum of (a - a) and (a - a), plus the function f of the sum of (a - a...
The first element (head) of the concatenation of two lists (xs and ys) is equal to the first element of the second list (ys) if the first list (xs) is empty, otherwise it is equal to the first element of the first list (xs).
The lemma named "pfrag_extr_auth" assumes that "hf" is an element of the set "p" and that the pair "(ainfo, p)" is in the "auth_seg2" of "uinfo". If these conditions are met, it shows that the "pfragment" of "ainfo" in the extraction of "HVF hf" is in "auth_seg0".
A well-formed ruleset, under a certain interpretation and a certain predicate, is the concatenation of two rulesets if and only if each of the two rulesets is well-formed under the same interpretation and predicate.
The lemma named "sub_mat_index" assumes that "m" is a matrix with "nr" rows and "nc" columns, "sr" is less than or equal to "nr", "sc" is less than or equal to "nc", "j" is less than "sr", and "i" is less than "sc". It shows that the element at the "i"th row and "j"th column of the sub-matrix of "m" with "sr" rows and ...
The lemma A34_ub assumes that "t" is a binary search tree. It shows that the amortized cost of the function S34.A with input "a" and "t" is less than or equal to 1.5 times the base 2 logarithm of the size of the tree increased by 1.
A is disjoint from the union of t and u if and only if a is disjoint from both t and u.
The lemma named "comp_preserves_can" assumes that both "f" and "g" are canonical, and the domain of "f" is equal to the codomain of "g". It shows that the composition of "f" and "g" is also canonical.
In the context of a ring, let's consider a lemma called generate_ring_in_carrier. Given the assumption that a set H is a subset of the carrier R, the lemma shows that if an element h belongs to the generated ring of R and H, then h must also belong to the carrier R.
The substitution of a set containing a single atom A with a substitution function sigma is equal to the set containing the atom A substituted with the same substitution function sigma.
The inverse of the generalized model context extension on a predicate P and a relation R is equal to the generalized model context extension on the same predicate P and the inverse of the relation R.
The function 'minus_ms_aux' with 'MSLNil' and 'ys' as arguments is equal to the function 'uminus_ms_aux' with 'ys' as an argument.
If 'x' is an element of the infinite set 'A' and it is false that 'x' is an element of the universal set, then 'x' is an element of the omega set of 'A'.
For ordinal numbers, 'x' is less than or equal to zero if and only if 'x' is equal to zero.
If the length of "pss" is greater than zero, then the compression of "pss" at layer 4 ports does not match everything.
If a value 'x' is in the inverse image of 'D' and 'D' is a subset of the Cartesian product of 'R' and 'X', then there exist values 'r' and 'A' such that 'x' is equal to the ordered pair of 'r' and the preimage of 'A' under 'r', and 'r' is in 'R' and 'A' is in 'X'.
A path 'g' is an arc if and only if 'g' is a simple path and the finish of the path 'g' is not equal to the start of the path 'g'.
If the function "stop" applied to an argument "a" holds true, then applying the function "enat_unfold" with the arguments "stop", "next", and "a" will result in zero.
If the function 'g' has a limit 'G' at 'x' within the set 'S', and 'G' is less than or equal to 'F', and 'd' is greater than 0, and for all 'x'' in 'S' the distance from 'x'' to 'x' is greater than 0 and less than 'd' implies that 'f' of 'x'' equals 'g' of 'x'', then the function 'f' has a limit 'F' at 'x' within the s...
The lemma named "siso_transC" assumes that "c" is a single-input single-output system (siso) and that there is a transition from the state "s" under the system "c" to the state "s'" under the system "c'". It shows that "c'" is also a single-input single-output system.
The length of the result of the function "mopup_n_tm" with input "k", when divided by 2, has a remainder of 0. In other words, the length of the result is an even number.
The third bit of the 16-bit word representation of -1705 is not set.
: Assuming T and S are sigma-finite measures, X is distributed according to Px on measure S, and the pair (X, Y) is distributed according to Pxy on the product measure of S and T. Then, it can be shown that for almost every x in S, the probability of x according to Px is equal to the integral over all y of the probabi...
If "a" is an element of set "s", then the base of "j" is less than or equal to "a" of "j".
The primitive part of a polynomial p is equal to zero if and only if the polynomial p is equal to zero.
The bitwise AND operation between 0 and 5 results in 0, in the context of 8-bit words.
"Timer" leads to a state where the count is zero for all possible states.
If the following conditions are met: 1. Alpha is a member of the set of integers. 2. FF is a sequence of variable functions. 3. AA and BB are semicategories under alpha. 4. The cardinality of FF is 4. 5. The domain of homomorphisms in FF is AA. 6. The codomain of homomorphisms in FF is BB. 7. The object map of FF is a...
Two "Pause" instances with outputs "out" and "out'" and continuations "c" and "c'" are equal if and only if both their outputs and continuations are equal.
"ExcessTable g vs" is a table for the function "ExcessAt g" over the set "vs".
The lemma shows that the mapping of unit and counit induces an adjunction between categories C and D through functors F and G.
The lemma named "discounted_measurable" assumes that "X" is in the Borel measurable space "N". It shows that the discounted value of "X" at time "m", with a discount rate "r", is also in the Borel measurable space "N".
If the inorder traversal of a tree 't' is sorted according to the function 'sorted1', then the inorder traversal of the result of deleting element 'x' from the tree 't' is equal to the result of deleting 'x' from the inorder traversal of the tree 't'.
The derivative of the epsilon-free transition function applied to a term derived from a generalized term is equal to the derivative of the transition function applied to the same term. This is denoted as "?Ls = ?Rs".
If an element 'c' belongs to the set 'A' at index 'k', then 'c' also belongs to the union of all sets 'B' that are mapped from 'A' at the same index 'k'.
If 'x' and 'y' are elements of a certain set (the carrier), then the utility of 'x' is greater than or equal to the utility of 'y' if and only if 'x' is preferred to or is indifferent to 'y' according to a certain preference relation.
If the invariant "invar" holds for the state "Left" with parameters "big" and "small", then the list on the left of the state "Left" is the reverse of the list on the left of the state "Right" with the same parameters "big" and "small".
The difference of two formal Puiseux series 'f' and 'g', when shifted by a factor 'r', is equal to the difference of the individual shifts of 'f' and 'g' by the same factor 'r'.
The lemma named "sp_equiv_list_subset" assumes that the set of "ms" is a subset of the set of "ns" and that the list obtained by mapping a function "σ" over "ns" is equivalent to the list obtained by mapping a function "σ'" over "ns". Under these assumptions, it shows that the list obtained by mapping the function "σ" ...
A pair (i, j) is in board b if and only if the pair (i + k1, j + k2) is in the transformed board with transformation parameters (k1, k2) applied to board b.
If the height of a tree 't' is greater than 0, then there exist left and right subtrees such that 't' is the join of these subtrees.
The lemma named "strict_upper_solution" is defined with respect to two functions, w and w', both of which map real numbers to real numbers. It assumes the following: 1. y is a solution to the ordinary differential equation defined by function f over the interval T in the set X. 2. The derivative of w with respect to ...
A matrix A is less than or equal to a matrix B if and only if the transpose of matrix A is less than or equal to the transpose of matrix B.
The lemma named "WtNFT_WtFT_1" assumes that the length of "cl" is greater than 0 and that the weight of fault-tolerant "cl" is not equal to 1. Under these assumptions, it shows that the ratio of the weight of non-fault-tolerant "cl" to the difference between 1 and the weight of fault-tolerant "cl" is equal to 1.
The lemma named "equivalence_functor_L" shows that "L" is an equivalence functor with respect to vertical composition "vcomp".
If we have three functions: the first one transforms a state and an environment into a sub-probability monad of the state, the second one transforms a state and an input into a sub-probability monad of a pair consisting of an output and a state, and the third one transforms a state and an input into a sub-probability m...
The lemma named "H_tensor_Id_1_is_gate" shows that the tensor product of H and the identity of 1 is a gate of 2.
"qbs_bernoulli" is a morphism from the set of rational numbers to the monad of quasi-Borel spaces of rational booleans.
A value 'x' is in the range of a constant map of 'v' over a set 'S' if and only if the set 'S' is not empty and 'x' is equal to 'v'.
, given the assumptions that "r1" and "r2" are in the set of atomic terms, the second elements of the tuples in the set "txs" are all in the set of variables, the first elements of the tuples in the set "txs" are all in the set of atomic terms, and the second elements of the tuples in the list "txs" are all distinct, i...
The pair of functions "(is_empty, RETURN o op_list_is_empty)" is included in the Kleisli extension of "is_list" function, which maps to a pure Boolean relation. This means that the function "is_empty" in the abstract program corresponds to the function "RETURN o op_list_is_empty" in the concrete program, under the cond...
If under the program P and environment E, the expression 'e' with state 's' reduces to throwing an exception 'r' with state 's'', then under the same program P and environment E, the expression 'e' accessing field 'F' of classes 'Cs' and assigning it the value of expression 'e2' with state 's' also reduces to throwing ...
If the input 'inp' is well-sorted under the sort 'delta', and if the function 'h' preserves well-sortedness under the module 'MOD', then the function 'gWlsInp' under the module 'MOD' and sort 'delta' preserves the well-sortedness of the lifted version of 'h' applied to 'inp'.
If there is a path in the tree T that starts with the node 'a' followed by the nodes in the list 'ds', then there exists a left subtree 'l' and a right subtree 'r' such that T is a branching of 'l' and 'r'.
: assuming a Dickson grading "d", a set "F" that is a subset of the set of power products of degree less than or equal to "m" with respect to "d", and two elements "p" and "q" from "F" that are not zero, and the greatest common divisor of the leading terms of "p" and "q" is zero, it shows that the critical pair of "p" ...
The lemma named "decr" assumes the following conditions: 1. The length of the list 'bs' is greater than the value of 'p' as determined by the function 'boundslt'. 2. The value 'p' is a bound for 'm' as determined by the function 'bound'. 3. 'm' is less than the length of the list 'bs'. Under these assumptions, the le...
The lemma "splice_last" demonstrates that: 1. If the size of list 'ys' is less than or equal to the size of list 'xs', then splicing list 'xs' with an element 'x' added at the end and list 'ys' is equivalent to splicing list 'xs' and 'ys' first and then adding the element 'x' at the end. 2. If the size of list 'ys' i...
If the function "sd_chain" with input "n" equals "Some q", then "q" is an element of the set of lotteries on alternatives.
For any given index 'j' and any list 'x', taking the first 'k' elements of 'x' and applying the function 'Ej' to it is a subset of applying 'Ej' to the entire list 'x'.
If X is a T1 space and 'a' is an element of the topological space X, then the set containing only 'a' is a closed set in X.
If the initialization of an object P of class C with a tuple (Cs, fs) is successful, then the list Cs is not empty.
The dynamic logic box of the intersection of P and Q is equivalent to the conjunction of the dynamic logic box of P and the dynamic logic box of Q.
If polynomial p is strictly less than polynomial q, then it is not possible for polynomial q to be strictly less than polynomial p.
: Assuming "\<Gamma> reduces to \<Delta> with output z" and "the free variables of (\<Gamma>, e) are a subset of the set L union the domain of \<Gamma>", it can be shown that "the free variables of (\<Delta>, z) are a subset of the set L union the domain of \<Delta>".
If the pair of pairs ([[b, c], [a, b]]) is in the set of ordinal composable elements of A, then a, b, and c are elements of A, and a is less than or equal to b, and b is less than or equal to c.
The lemma named "partial_equiv_subst" assumes that a clause C, when substituted with a substitution τ, is falsified by a given set of ground clauses G. It then shows that the original clause C is also falsified by the set of ground clauses G.
The sign of literal 1 is not equal to the sign of literal 2 and the atom of literal 1 is equal to the atom of literal 2 if and only if literal 2 is the complement of literal 1.
If the injection of P into the right (Inr P) is in the set resulting from the interpretation of a first-order logic formula (dec_interp n FO w), and if a certain element p is in P, then p is less than the length of w.
If a term 't' is a subterm of the set of terms in a stateful strand 'S', and the free variables of 't' and 't' substituted by 'θ' do not intersect with the bound variables of 'S', then the free variables of 't' substituted by 'θ' are a subset of the free variables in the stateful strand 'S' substituted by 'θ'.
If a word "w" is in the submonoid generated by "G", then there exists a list of words "ws" in "G" such that the concatenation of "ws" equals "w".
The extended pre-distance of x and y is equal to the extended pre-distance of y and x.
The recursive lazy sequence function of 'f' and 'xq' is equal to the function 'f' applied to the list of lazy sequence 'xq'.
For any real number x that is greater than or equal to 2, the prime-counting function of x is less than three times the sum of Euler's number raised to the power of negative one and the natural logarithm of 2, all multiplied by x and divided by the natural logarithm of x.
The lemma named BlockSpec makes the following assumptions and shows a result: Assumptions: 1. There is a subset relationship between P and a set of states s such that there exists a Z where the initial state s is in P' Z, and for all states t, if t is in Q' Z then the return state s t is in R s t, and if t is in A' Z ...
The set obtained by removing isomorphic duplicates from the set "Us" is a subset of the original set "Us".
The lemma "snd_echelon_form_of_upt_k_det_eq" demonstrates that the second element of the pair resulting from the function "echelon_form_of_upt_k_det bezout" applied to the pair "(n, A)" and "k" is equal to the result of the function "echelon_form_of_upt_k A k bezout".
'valid' prime of 'l' is equal to 'valid' of 'l'.
The lemma named "non_empty_cycle_root_tc_start_points" assumes that "x" is a non-empty root of a cycle "r". It shows that the transitive closure of "x" followed by "r" is equal to "x" followed by the identity relation.
The lemma named "theSubjectFreshChain" is defined with two fixed elements: an action labeled as "alpha" and a list of names labeled as "xvec". It assumes that the action "alpha" is not equal to "tau". The lemma shows that the freshness of the list of names "xvec" for the subject of the action "alpha" is equal to the fr...
A product topology on a set X indexed by I is a Hausdorff space if and only if the product of the top spaces of each X indexed by I is empty, or every X indexed by I is a Hausdorff space.
The lemma named "match_ex_table_SomeD2" assumes the following conditions: 1. The function "match_ex_table" with parameters P, D, pc, and the exception table of P, C, M returns a pair (pc', d'). 2. The triple (P, C, M) is related to the pair (xt/I, d) in some way. 3. For all x in the set xt, x does not match the exce...
If "a" is not less than "b", then "a" is greater than "b".
If 'x' is less than or equal to 'y', then there exists a natural number 'n' such that 'y' is equal to 'x' plus 'n'.
If Z is a part of the set obtained by inserting X into H, and X is a synthetic element derived from the analysis of H, then Z is either a synthetic element derived from the analysis of H or a part of H.
"d" is a strong bisimulation under the relation "approximately equal to" in the context of "d".
If a set S is an absolute retract (AR) and is homeomorphic to another set S', and if S' is closed within the topological space U, then S' is a retract of U. This is under the condition that S and S' are subsets of Euclidean spaces.
For a transitive relation R, R is asymmetric if and only if R is irreflexive.
The lemma named "complex_increasing_LIMSEQ" is about a function 'f' that maps natural numbers to complex numbers. It assumes three conditions: 1. The function 'f' is increasing, i.e., the value of 'f' at any natural number 'n' is less than or equal to the value of 'f' at the next natural number 'n+1'. 2. The functi...
The lemma named "store_instr_privilege" assumes that the state 's' is the second element of the first pair resulting from the operation "store_instr" applied to an instruction 'instr' and a SPARC state 's' of any length 'a'. It also assumes that the Supervisor Mode bit (get_S) of the Processor State Register (PSR) in s...
If "t" is an element of the vertex set "V" of the tree "T", then the bag "t" is a subset of the vertex set "V".
If the value of the function "gval" applied to the fold of "gAnd" over a set "G" with a base case of "True" and a state "s" is "false", then there exists an element "g" in the set "G" such that the value of the function "gval" applied to "g" and "s" is "false".
If a type tau is less than Integer, then if tau equals OclVoid or tau equals UnlimitedNatural, the proposition P holds true.