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Statement: stringlengths 7 24.3k
The lemma named "out_arcs_in_subarcs" assumes that a node "r" with a set of subtrees "xs" is a subtree of "t", and that an edge "e" is in the set of outgoing arcs from the directed tree derived from "dt" and "dh" at root "r". It shows that the edge "e" is in the image of the second component of the finite set "xs".
The lemma "tmf_ivl_cs_correct" is about a fixed list of float intervals "A" and a float arithmetic "f". It assumes that "a" is entirely within "I" and that the function "tmf_ivl_cs" with parameters "prec", "ord", "I", "a", and "f" equals "Some cs". It shows that for any "i" less than "ord", the function "tmf_c" with pa...
If set A is finite, the maximum value of set A is less than the minimum value of set B, and if a is an element of set A and b is an element of set B, then a is less than b.
The lemma named "position_less_eq_induct" assumes three conditions. The first condition is that position 'p' is less than or equal to position 'q'. The second condition is that a property 'P' holds for any position 'p'. The third condition is that for any positions 'p', 'q', and 'r', if 'p' is less than or equal to 'q'...
The lemma named "partn_lst_greater_resource" assumes that there is a partition list "partn_lst" of a resource "r" into a set "B" with a certain ordinal number "\<alpha>" and a natural number "n". It also assumes that set "B" is a subset of set "C". Under these assumptions, it shows that there is a partition list "partn...
The topological concept of "at_top" is equivalent to the mapping of the inverse function over the filter "at_right" at zero for real numbers.
The element 'x' is in the multiset 'XS' when 'x' is added to 'XS'.
The lemma "ins_list_split_right_general" assumes that the result of splitting a tree 'ts' at a value 'x' equals a tuple consisting of a list 'ls' and a list that starts with a tuple '(sub,sep)' followed by a list 'rs'. It also assumes that the inorder traversal of the tree 'ts' is sorted in ascending order and that 'se...
If the image multiset of the negation normal form (nnf) of a set Γ implies Γ in the context of a normal form, then Γ implies itself in the context of a normal form.
The lemma named "update_square" assumes that "y" is a point. It shows that the result of updating "x" at "y" with the value of "x" at the index of "x" at "y" is less than or equal to the product of "x" with itself union "x".
If 'y' is in the set of 'older_seniors' of 'x' at time 'n', and the set of 'older_seniors' of 'x' at time 'n' is equal to the set of 'older_seniors' of 'x' at time 'n+1', then the set of 'older_seniors' of 'y' at time 'n' is equal to the set of 'older_seniors' of 'y' at time 'n+1'. This is to say, the left-hand side of...
The composition of a formal Puiseux series with power 0 and any real number r is equal to 0.
The unlabelled keyserver state transition system is a well-formed state transition system.
: assuming a function 'h' is injective on a set 'A' and the image of 'A' under 'h' is a subset of 'C', and for every element 'x' in 'A', the set 'B x' is less than or equipollent to the set 'D' of 'h x', it can be shown that the Cartesian product of 'A' and 'B' is less than or equipollent to the Cartesian product of 'C...
If the lengths of two lists, xs and ys, are not equal, then it is not possible for all corresponding elements in xs and ys to satisfy a certain relation R.
The map of the natural transformation from the right adjoint to the left adjoint in the categories \<CC> and \<DD> under the functors \<FF> and \<GG>, with natural isomorphisms \<Phi> and \<Psi>, and a new functor \<GG>', is a valid set-valued function.
It is not the case that for all 7-dimensional real vectors c, a, and b, the cross product of c and a, then cross product with b, equals the dot product of c and b scaled by a minus the dot product of c and a scaled by b.
: Assuming for all 'q' in the domain of 'Γ', and for all 'Z', if 'Γ' and 'Θ' can transition from a state 's' (where 's' equals 'Z' and 'Γ' does not lead to a 'Call q' in a 'Normal s' state that results in either a 'Stuck' state or a 'Fault' not in 'F', and 'Γ' can terminate 'Call q' in a 'Normal s' state, and the pair...
If "m" is less than or equal to "n", then the harmonic number of "m" is less than or equal to the harmonic number of "n", where the harmonic number is a real number.
The lemma named "run_effectI" assumes that the command "c" runs from some state "h" to another state "h'" with a result "r". It shows that the command "c" has an effect from state "h" to state "h'" with the result "r".
The lemma "bddh_subsetbdd" assumes that all elements in the first component of A satisfy the "bddh l" condition, that "bddh l bdd'" is true, and that "q" is a node of the NFA A. Under these assumptions, it shows that "bddh l" is true for the subset of the first component of A determined by "q" and "bdd'".
The lemma "sortedP_right_of_append1" assumes that a list "zs" appended with an element "z" is sorted according to a certain predicate "P". It also assumes that an element "y" is in the set of elements of the list "zs". It then shows or proves that "y" is less than or equal to "z".
In the lemma of a partial order, the greatest lower bound (infimum) of a singleton set, where the element 'x' is in the carrier 'L', is equal to 'x' itself.
If "a" is less than or equal to "b" in the ordinal category "A", then it can be inferred that there exists a morphism from "a" to "b" in the ordinal category "A".
If a function 'f' is a continuous map from a topological space 'X' to the real numbers, and for all 'x' in the top space 'X', 'f' of 'x' is not equal to zero, then the function that maps 'x' to the inverse of 'f' of 'x' is also a continuous map from 'X' to the real numbers.
If for every element 'x' in the set 'p', the relation 'P' holds between the result of function 'f' applied to 'x' and the result of function 'g' applied to 'x', then the relation 'P' also holds between the result of binding 'p' with 'f' and the result of binding 'p' with 'g'.
The lemma from the Oriented_Circlines theory in Complex Geometry shows that the function "of_circline" applied to the "unit_circle" equals "ounit_circle".
If for every pair (x, y) in the set l, a property P holds for x and y, then for all x and y, if the mapping of l to x equals Some y, the property P also holds for x and y.
The lemma assumes that the program P is well-formed according to the metadata well-formedness condition. It shows that Object is a class in program P and Thread is also a class in program P.
The lemma named "a_rcos_equation" in the context of an abelian subgroup shows that if the sum of 'ha' and 'a' equals the sum of 'h' and 'b', and 'a', 'b', 'h', 'ha', and 'hb' are all elements of the carrier 'G' with 'h', 'ha', and 'hb' also being elements of 'H', then the sum of 'hb' and 'a' is an element of the union ...
The lemma named "same_range_equivalent" is defined with two fixed functions, F and G. F is a function that maps from a finite update of type 'a to a finite update of type 'c, and G is a function that maps from a finite update of type 'b to a finite update of type 'c. It is assumed that both F and G are registers, and t...
The lemma obligation2 assumes that the mapping of a probability mass function 'con' on 's' equals the sum of probability mass functions 'Da' and 'Db' weighted by 8/10. It also assumes that the sets of 'Da' and 'Db' are finite. It shows that the transition probability on a random 's' and 'qs' for the combination of an e...
The lemma named "exec_as_proj_valid_2" is defined for a fixed element 'a'. It assumes that 'a' belongs to the set 'PROB'. It shows that the action domain of the first element of 'a' and the second element of 'a' is a subset of the probability domain of 'PROB'.
For all lists 'ys', the length of the list resulting from appending 'ys' to 'xs' with function 'f' is equal to the minimum of the lengths of 'xs' and 'ys'.
The lemma named "add_get2_1" is defined with respect to a fixed environment 'e' of type 'a, a string 'x', an element 'A' of type 'a, another string 'y', and another element 'B' of type 'a. It assumes that the environment 'e' with 'x' mapped to 'A' and 'y' mapped to 'B' is okay. It then shows that retrieving the value o...
The lemma "rtrancl_is_path" assumes that there is a reflexive transitive closure between "u" and "v" in a set "E". It then obtains a list "l" where there is a path from "u" to "v" in set "E".
If a tree 't' is in the set 'T h', and the inorder traversal of 't' is sorted, then the inorder traversal of the tree resulting from deleting 'x' from 't' is equal to the result of deleting 'x' from the list obtained by the inorder traversal of 't'.
If a command 'c' takes a state from satisfying precondition 'P'' to satisfying postcondition 'Q' under a context 'G', and for all states 's' and all variables 'Z', if 'P' holds then 'P'' also holds, then it can be inferred that the command 'c' takes a state from satisfying precondition 'P' to satisfying postcondition '...
The function map of the infimum (greatest lower bound) of two elements x and y is equal to the infimum of the function maps of x and y individually.
If for all processes in the set of processes, the union of the set G and the set of all pairs (P p, Q p) such that P p is the precondition and Q p is the postcondition of the BODY p, implies the set of all pairs (P p, Q p) such that P p is the precondition and Q p is the postcondition of the body of p, and if pn is a p...
For a given set S of complex normed vectors, the closure of the set obtained by scaling each element of S by a complex number a is equal to the set obtained by scaling each element of the closure of S by the same complex number a.
A word "w" satisfies the weak until operator of "phi" and "psi" if and only if "w" satisfies the disjunction of "phi" and the globally operator of "psi" with the greatest fixed point of "psi" in "w", and the next operator of "psi".
: If a pair of states (\<langle>c, mds', mem\<rangle>, \<langle>c', mds', mem'\<rangle>) is in the relation "rel_inv", then the following conditions hold: 1. For all variables x, and any pair of values (a, b) such that A x equals to the pair (a, b), if the memory at x equals to a and the memory at x in the new state ...
The indices found by the function "find_indices" for a given element "x" in a list "xs" are distinct.
"for almost every x in a measure space M, for all countable i, a property P holds for i and x" if and only if "for all countable i, for almost every x in a measure space M, the property P holds for i and x".
If the negative of 'r' times 'x' is directed towards '0', then 'r' directed towards 'x' equals '1'.
The lemma named "bit_b''" assumes the following conditions: 1. x is not equal to y. 2. The set {x, y} is equal to the set {x0, y0}. 3. The BIT_inv function applied to s, x, and the list [x0, y0] is true. 4. The set of elements in qs is a subset of the set {x, y}. 5. qs is in the language generated by the sequence [Plu...
If for all states 's', 'P' holds whenever 'P'' holds, and there is no failure of 'f' under precondition 'P', then there is no failure of 'f' under precondition 'P''.
Let x be an element of a ring with unity and inverse. If the inverse of 1 is 1, then the integral of the formal Laurent series with constant term x is equal to the formal Laurent series with constant term x multiplied by the formal Laurent series with variable X.
: Assuming the following conditions: 1. For all states 's' and 'ss', if 's' and 'ss' are equivalent under the function 'G' at point 'p', then the evaluation of boolean expression 'b' at 's' is equal to the evaluation of 'b' at 'ss'. 2. For all 'x', either 'G(x)' is less than or equal to 'H(x)' or 'p' is less than or e...
The lemma is in the context of a raw converter invariant. It defines 'inline1' as 'inline1' and assumes the following: 1. The game process value (gpv) is well-typed under the interface (\<I>). 2. The gpv is p-finite under the interface (\<I>). 3. For any state 's' that satisfies the invariant 'I', the function 'f' app...
"x is less than or equal to the star of x".
If a natural number 'n' is less than 'N', then the reciprocal of 'N' is less than or equal to 1.
The lemma "InvariantUniqQAfterApplyUnitPropagate" assumes the following conditions: 1. The "InvariantWatchesEl" condition holds for the formula (getF state), and the first and second watches (getWatch1 state and getWatch2 state). 2. The "InvariantWatchListsContainOnlyClausesFromF" condition holds for the watch list (g...
The lemma named "DERIV_compose_FDERIV" assumes two conditions. The first condition is that the derivative of the function 'f' at the point 'g x' exists and is equal to 'f''. The second condition is that the function 'g' has a derivative 'g'' at the point 'x' within the set 's'. If these two conditions are met, it shows...
If "I" is a subset of "valid", then the long-term key "ltk" is not part of the parts of the extractable set "K" from "I".
The lemma named "mojmir_accept_token_set_def1" assumes "accept". It shows that there exists an 'i' less than the maximum rank such that the 'fail' set is finite, the 'merge i' set is finite, the 'succeed i' set is infinite, and for all 'j' less than 'i', the 'succeed j' set is finite.
The sign of the normalized form of a floating point number x is 0 if x is greater than 0, otherwise it is 1.
Sequencing any property "P" with "false" results in "false".
The operation of unsetting the bit at position "Suc n" (which is the successor of n, or n+1) in a number 'a' is equal to the remainder of 'a' divided by 2 plus 2 times the operation of unsetting the bit at position 'n' in the quotient of 'a' divided by 2.
The lemma named "sum_div_1" is defined for a function 'f' that maps from any type 'b' to any field type 'c'. It assumes that the sum of the function 'f' over all elements in set 'A' is not equal to zero. Under this assumption, it shows that the sum of the function 'f' for each element in set 'A' divided by the sum of t...
If variable 'x' does not affect the outcome of predicates 'P' and 'Q', then 'x' does not affect the outcome of either 'P' or 'Q'.
If the canonical normal form (cnf) of a variable 'x' is equal to a list with 'a' as its first element, then 'a' is equal to the base-ω logarithm of 'x'.
In the context of a group, the lemma named "mono_generate" assumes that "K" is a subset of "H". It then shows that the generated group from "K" is a subset of the generated group from "H".
The lemma named "nat_diff_mod_right" is defined for three natural numbers a, b, and c. It assumes that b is less than a. It then shows that the modulus of the difference of a and b with respect to c is equal to the modulus of the difference of a and the modulus of b with respect to c, with respect to c again.
, assuming "A" is a residuated transition system (rts), there exists a residuated transition system with composites "C" and a simulation "I" from "A" to "C". Furthermore, for any extensional residuated transition system with composites "B" and any simulation "J" from "A" to "B", there exists a unique simulation "J'" fr...
: The infinite append of two lists 'xs' and 'f' is equal to the infinite append of two lists 'ys' and 'g' if and only if: - If the length of 'xs' is less than or equal to the length of 'ys', then 'xs' is equal to the prefix of 'ys' of length 'xs' and 'f' is equal to the infinite append of the suffix of 'ys' of length...
The lemma "normalize1_new_verts_not_reach1" assumes that vertex v1 is in the set of directed vertices of the normalized version of tree t and not in the set of directed vertices of tree t itself. The same conditions apply to vertex v2. Additionally, v1 and v2 are not the same vertex. Under these assumptions, the lemma ...
The lemma named "simple_bochner_integral_add" assumes that both functions 'f' and 'g' are simple Bochner integrable over the measure space 'M'. It shows that the simple Bochner integral over 'M' of the function that adds 'f' and 'g' at each point 'x', is equal to the sum of the simple Bochner integral over 'M' of 'f' a...
The join-irreducibles of the infimum of a set X is equal to the intersection of the join-irreducibles and the intersection of the join-irreducibles of each element in set X.
: Assuming that points p and q are not the same, both p and q are in set S, and a is in the hyperbolic plane. Also, assuming that points p, q, and a are all incident to line l. It can be shown that there exists a real number k such that k is greater than 0 and less than 1. Furthermore, the Cartesian coordinates of p...
The lemma named "injective_codomain" assumes that "x" is injective. It then shows that the product of "x" and the infimum (greatest lower bound) of "x" and 1 is equal to the infimum of "x" and 1.
If a list 'xs' is a permutation of a multiset 'A', then the length of the list 'xs' is equal to the size of the multiset 'A'.
The matrix M1 is Hermitian.
The lemma assumes that the set S is finite. It shows that the set of 4-tuples (a, b, c, d), where the pair of lists ([a, b], [c, d]) is in the set S, is also finite.
The lemma "linord_of_listP_empty" shows that for all elements 'x' and 'y', 'x' and 'y' are not in the linear order of the list 'xs' if and only if the list 'xs' is empty.
The lemma named "sum_add" assumes that both A and B are finite sets. It shows that the sum of the function f over set A plus the sum of the function g over set B equals the sum of the function f over the set difference of A and B, plus the sum of the function g over the set difference of B and A, plus the sum of the fu...
For any function f that is continuous on set U and maps set U into set S, there exists a constant c such that f is homotopic to the constant function at c on set S. This is equivalent to saying that for any function f that is continuous on set U and maps set U into set T, there exists a constant c such that f is homoto...
The set of elements in the list obtained by appending the list 'ys' and the element 'y' to the list 'xs' is equal to the set of elements in the list obtained by appending the list 'ys' to the list 'xs' with 'y' at the beginning.
The correspondence relation "cr_vreal" is bi-unique. This is a transfer rule.
The lemma named "block_mat_cond_rev" assumes that "j" is less than the length of the list "B". It shows that the set of "i" such that "i" is less than the length of the list "V" and the element at the position (i, j) in the matrix "N" equals 1, is equal to the image of the j-th element of the list "B" under the functio...
The lemma named "ball_rad'" assumes that "B" is a ball, "B" is equal to the ball with radius "n" centered at "c", and "B" is also equal to the ball with radius "m" centered at "d". It also assumes that "c" and "d" are elements of the carrier of the p-adic numbers. Under these assumptions, it shows that "n" must be equa...
If "phi" is an element of "fmla" (formula), then the encoding of "phi" is an element of "trm" (term).
: Given two assumptions: 1. The possibility of 'x' in 't' is equivalent to the possibility of 'x' in 't''. 2. If 'x' is possible in both 't' and 't'', then 'xs' is in the paths of the next 't' and 'x' if and only if 'xs' is in the paths of the next 't'' and 'x'. Then it can be shown that: The list consisting of 'x' fol...
The lemma named "vreal_transitive_law" assumes that x, y, and z are elements of the set of real numbers. If x is less than y and y is less than z, then it shows that x is less than z. This is a demonstration of the transitive property in the set of real numbers.
The lemma "cons_cluster_has_cons_cluster_slice" assumes that "C" is a consensus cluster and "p" is an element of "C". It then obtains a slice "Sl" such that "Sl" is in the slices of "p" and "Sl" is a subset of "C".
The lemma L4_8': Assuming that - "\<sigma>" is an infinite path in the graph of the generalized Büchi automaton (gba), - "\<sigma>" is accepted by the gba, - for all "i", the label of the state "\<sigma> i" in the gba accepts the symbol "\<xi> i", - and "\<psi>" is in the set of old states at the start of th...
If "A" is not equal to "Friend C", then the key (shrK A) is not part of the initial state of "Friend C".
The lemma named "mu'_via_sigma" assumes that "j" is less than or equal to "i" and "i" is less than "m". It shows that "mu' i j" equals to "fs ! i dot fs ! j" if "j" equals to zero. Otherwise, it equals to "mu' (j - 1) (j - 1) times (fs ! i dot fs ! j) minus sigma j i j".
The lemma "LLL_swap_row" assumes that the state implementation invariant holds for parameters p, mfs, dmu, and state, as indicated by "impl: state_impl_inv p mfs dmu state". It also assumes that the LLL invariant under weak conditions holds for parameters fs, mfs, dmu, p, first, and b, as indicated by "Linv: LLL_invari...
If a function 'f' is integrable over a closed interval from 'a' to 'b', and a set of functions 'F' is equi-integrable over the same interval, then the set of functions that includes 'f' and all functions in 'F' is also equi-integrable over the interval from 'a' to 'b'.
If a predicate P holds for a finite map m, then the same predicate P also holds for the finite map obtained by dropping the element with key a from m.
If "Class C" is a type in the context G, then C is a class in the context G.
If a generative probabilistic value 'f' is ordered with respect to another generative probabilistic value 'g', and 'g' is also ordered with respect to 'f', then 'f' and 'g' must be equal.
The conjugate of the inverse of x is equal to the inverse of the conjugate of x.
The lemma named "not_less_if_bot" assumes that "b" is less than or equal to "p". It shows that "p c" is not less than "p b".
If the multiplicity of a block is 1, then the residual of the block is a design.
"false" is not equal to "null", and this is always false.
If the function "c_in" of "x" and "u" equals 1, then "2 to the power of x" is less than or equal to "u".
If a set 'm' is open and 'm' is an element of 'T', then 'R' is implied.

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