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If point p is projectively equivalent to point p', then point p' is also projectively equivalent to point p.
The lemma named "onode_sos_dest_is_net_state'" assumes that a triple, consisting of a pair (\<sigma>, NodeS i p R) and a state s', belongs to the set onode_sos S. It shows that there exist some \<sigma>', \<zeta>', and R' such that the state s' is equal to the pair (\<sigma>', NodeS i \<zeta>' R').
The initial state of level 1 is a subset of the invariant 5 of level 1.
A list containing a pair of a variable and a term is equivalent to an empty list if and only if the term is the same as the variable.
The next word after the 8-bit word 255 is 255.
If 'e' is less than zero, then the closed ball centered at 'x' with radius 'e' is an empty set.
The lemma Con_consI, which can be used as an introduction rule and simplification rule, shows two things: 1. If T is not an empty list, U is not an empty list, [t] is concatenated with U, and T is concatenated with U and then with [t], then t followed by T is concatenated with U. 2. If T is not an empty list, U is no...
The hereditary multiset "zhmset_of M" is less than the hereditary multiset "zhmset_of N" if and only if "M" is less than "N".
The induced function is a group homomorphism from the quotient group G/L (where G is a group and L is a normal subgroup of G) with operation [\<cdot>] and identity element (contained.Class \<one>), to the group G' with operation (\<cdot>') and identity element \<one>'.
If an action "act" is single-valued, then the set of weakly enabling conditions for a set B under a program constructed with initial conditions "init", the action "act", and allowed actions "allowed", is a subset of the set formed by inserting the weakly enabling condition for a single action "act" on set B into the ra...
The lemma named "effect_insertion_sort" assumes that the effect of applying the insertion sort algorithm on array 'a' changes the initial heap 'h' to the final heap 'h' and results in an output 'u'. It shows that the multiset of elements in the array 'a' from the initial heap 'h' is equal to the multiset of elements in...
The negation of the disjunction (or operation) between two propositions (phi and psi) is equivalent to the conjunction (and operation) of the negations of each proposition.
The function "extend_distance" is well-ordered and admissible according to the well-ordering principle.
If sets S and T are separated in set X, then sets S and T are disjoint.
: If the size of a multiset M is less than or equal to 1, then the following holds: If M is an empty multiset, then a property P holds. Furthermore, for any element m, if M is a multiset containing only m, then the property P also holds. Therefore, in both cases, the property P holds.
The set of elements obtained from the list "rs" after removing duplicates in reverse order is equal to the set of elements in the original list "rs".
The row of the matrix created from row 'y' at index 0 is 'y' itself.
If a path 'g' is valid and the line integral of a function 'F' along this path equals 'c', then the line integral of the same function along the reverse of this path equals '-c'. This is true for all 'x' in the interval from 0 to 1.
The lemma "sqrt_approx" assumes that epsilon is greater than zero. It shows that the absolute difference between the square of the square root approximation of epsilon and the absolute value of x, and epsilon is less than epsilon.
Given a countable set D, a positive number e, and assuming that for every element d in D, d is Lebesgue measurable, and for any subset D' of D that is finite, the Lebesgue measure of the union of D' is less than or equal to B, then there exists a subset D' of D that is finite and the Lebesgue measure of the union of D ...
The inverse of function 'f' composed with 'ov' is a subset of 'ov'.
If a regular expression 'r' is in the set of modified regular expressions 'mrexps3' of a string 's', then the result of the function 'move3' applied to 'x', 'r', and 'a' is also in the set of modified regular expressions 'mrexps3' of the same string 's'.
If 'n' is a multiple of 'k' (i.e., 'n mod k' equals zero), then the 'i_aggregate' of the 'n'-th element of the function 'f' with parameters 'k' and 'ag' is equal to the 'i_aggregate' of the function 'f' with parameters 'k' and 'ag' shifted up by 'n' divided by 'k'.
The lemma named "expand_forest_modulo_equivalence" assumes that "H" is a forest modulo equivalence with respect to "d". It shows that the transitive closure of the product of the transpose of "d" and "H", multiplied by the transitive closure of the product of "H" and "d", is equal to the union of the transitive closure...
For any 's'', if there is a transition relation from '(c,s)' to 's'' and 's'' is similar to 't'', then the thesis holds. The second case, named 'MatchO', states that for any 'c'' and 's'', if there is a transition relation from '(c,s)' to '(c'',s'')', 's'' is similar to 't'', and 'c'' is discrete, then the thesis hold...
The lemma l13_1_aux assumes that points A, B, and C are not collinear, point P is the midpoint of B and C, point Q is the midpoint of A and C, and point R is the midpoint of A and B. It shows that there exist points X and Y such that R is perpendicular to line XY at points A and B, line XY is perpendicular to line PQ, ...
If set T is finite and set S is a subset of T, and for every element i in the difference of T and S, the function g applied to i equals the identity element, then the function F applied to g and S equals the function F applied to g and T.
The lemma named "empty_prob_dom_finite" is defined for a problem "PROB" of any type 'a. It assumes that the problem domain of "PROB" is empty. Under this assumption, it shows that "PROB" is finite.
For a matrix A with a certain number of columns and rows, the length of the first row (or column) when the matrix is converted to an integer array is equal to the cardinality (or total number) of the columns.
If the following conditions are met: 1. The graph 'g' is valid. 2. The starting node 'v0' is in the set of nodes of the graph 'g'. 3. For any vertices 'v' and 'v'', and weight 'w', if the edge (v,w,v') is in the set of edges of the graph 'g', then the weight 'w' is non-negative. Then, the following outcomes are true:...
If "v" equals "c" implies "u", then "u" equals the owner of "v" implies "v".
The third element from the right in the tuple resulting from the left fold operation on the function "echelon_form_of_column_k_det_iarrays" with initial tuple (n, A, 0, bezout) and list [0..<k] is equal to "bezout".
A variable 'j' is fresh in the substitution of a variable 'i' with an expression 'x' in a context 'A' if and only if either 'i' is fresh in 'A' and 'j' is fresh in 'A', or 'j' is fresh in 'x' and ('j' is fresh in 'A' or 'j' is equal to 'i').
The lemma named "arctan_diff_small" assumes that the absolute value of the product of two real numbers x and y is less than 1. It shows that the difference between the arctan of x and the arctan of y equals the arctan of the fraction (x - y) divided by (1 + x * y).
If function "f" considers "iA" at "n", then function "f" contains "iA" at "n".
The star of x multiplied by the omega of x under the n operation, further multiplied by the top element, is equal to the star of x multiplied by the bottom element, joined with the omega of x under the n operation multiplied by the top element.
The lemma named "enum_v_repeat" assumes that "x" is an element of the set "V", and "x" is not equal to the source or the sink of "Delta". It shows that there exists an "i'" greater than "i" such that "enum_v" of "i'" equals "x".
The function "domainNameDeptChopOne" applied to the string "i8--CoffeeMachine--Leaf" results in the string "i8--Leaf".
The lemma named "prm_apply_unequal" is defined with the fixed variables "pi" of type "'a prm" and "x" and "y" of type 'a. It assumes that "x" is not equal to "y". It then shows that applying "pi" to "x" is not equal to applying "pi" to "y".
If removing an element L from a multiset x1 results in a multiset M, then the following two conditions hold: 1. If L is an element of x1, then x1 is equal to the multiset M with the element L added. 2. If L is not an element of x1, then x1 is equal to M. In either case, the proposition P holds.
: Assuming that M1 and M2 are observable, and for any u and v, if u and v are members of x and u is in the language of M1 and M2, then M1 and M2 converge on u and v. Also, for any x2, u, and v, if x2 is in the set list of 'don', and u and v are members of x2, and u is in the language of M1 and M2, then M1 and M2 con...
Inserting an element 'a' into the union of sets 'A' and 'B' is equivalent to the union of set 'A' and the set obtained by inserting the element 'a' into 'B'.
The theorem named "leftmost_reduction_theorem" demonstrates that the leftmost strategy is a normalizing strategy in Lambda Calculus.
The mirror of a lazy list of elements 'xs' is equal to the lazy list of elements 'xs' appended with the reverse of 'xs'.
The set resulting from the list difference of two lists, xs and ys, is equal to the set of xs minus the set of ys.
1 is a co-test.
If the set V consists of only one element 'v', then the diameter is equal to the eccentricity of 'v'.
A null value, when considered as an integer in the context of an arbitrary system state, is not equal to any integer 'n'. This is always false.
The lemma "boundOutputFreshSet" establishes the following: 1. For a given set of names 'Xs', a list of names 'xvec', and a bound output 'B', 'Xs' is fresh for the bound output with respect to 'xvec' if and only if for every name 'x' in 'Xs', 'x' is either in 'xvec' or 'x' is fresh for 'B'. 2. For a given list of name...
The set resulting from the extension upwards operation on an element 'a' and a set 'xs', is a subset of the union of the set containing 'a' and the set 'xs'. Similarly, the set resulting from the extension downwards operation on an element 'a' and a set 'xs', is also a subset of the union of the set containing 'a' and ...
For a given program P and execution environment E, the synchronization order (sync_order) is a total order on the set of all synchronization actions (sactions) in that program.
"a when P when Q" is equivalent to "a when P and Q".
The language of the pair automaton Q with respect to a set of rules G is equal to the language of the abstract syntax tree transducer obtained from the pair automaton Q with respect to the same set of rules G.
If a heap 'h' is preallocated and 'h' is a subheap of another heap 'h'', then the heap 'h'' is also preallocated.
For any given parameters p, not, and no, if a directed acyclic graph (Dag) exists with root p and another Dag exists with root no, and no is in the set of t, then not is less than or equal to t.
The norm (or magnitude) of the difference between a complex number 'z' and an integer 'm' is always greater than or equal to the norm of the difference between 'z' and the integer part of the real part of 'z' rounded to the nearest integer.
If "t1" and "t2" are binary search trees, then the difference of "t1" and "t2" is also a binary search tree.
The lemma named "combine_map3" assumes that the lengths of lists ys, ys', and ys'' are equal to the lengths of lists xs, xs', and xs'' respectively. It also assumes that the elements in lists xs, xs', and xs'' are distinct and that these lists do not share any common elements. Under these assumptions, it shows that the...
The first element of the tail of a sequence is equal to the second element of the sequence.
In the context of an order, if a set X is a subset of the carrier D, and if b is an element of the carrier D such that every element x in X is less than or equal to b, then b is an upper bound of X.
The lemma named "poly_shift_monom'" assumes that "a" is an element of the ring "R". It shows that the shift of the polynomial "a" multiplied by "X" to the power of "n+1" is equal to "a" multiplied by "X" to the power of "n".
The imaginary part of the cosine of a complex number z is equal to the sine of the real part of z times the difference of the exponential of the negative imaginary part of z and the exponential of the imaginary part of z, all divided by 2.
If sets A and B are finite, then the cardinality (or size) of the union of A and B is equal to the sum of the cardinalities of A and B minus the cardinality of their intersection.
The lemma named "strip_iter_widen" is given a function 'f' that maps from a type 'a' (which has the properties of order and widen) to another 'a'. It assumes that for all 'C', the result of applying the function 'f' to 'C' and then stripping it is the same as just stripping 'C'. If the iterative widening of 'f' on 'C' ...
The function "ennreal_of_enat" is continuous at any point x.
There exists a constant C such that for all n greater than or equal to 1, the norm of R(n) is less than or equal to C divided by the real number n to the power of (2*N+2).
If for any two geodesic segments G and H, and any two points x and y, the condition that a geodesic segment exists between x and y in both G and H, and the intersection of G and H is the set containing only x and y, implies that x equals y. Given that a geodesic segment exists between x and y in both G and H, it can be...
If a function 'f' is in the small omega of 'g' with respect to a filter 'F', then it is eventually true that the norm of 'f(x)' is greater than or equal to 'c' times the norm of 'g(x)' for the same filter 'F'.
The lemma "enabled_qsend_indis" assumes that "cf" is enabled in the trace "ω", the quantity of "qsend" in the combined trace of "cf" and "ω" is less than or equal to "n", and the quantity of "qsend" in the combined trace of "cf" and "ω" is less than or equal to "m". It shows that the second element of the "n-th" item i...
The lemma named "eqI" assumes that there is a natural transformation from the composition of the opposite category to the functor Y applied to a, and that the transformation at a is equal to the transformation of tau at a. It then shows that the transformation tau prime is equal to the transformation tau.
Deleting an element 'x' from the join (or union) of two environments 'f' and 'g' is equivalent to the join of deleting 'x' from 'f' and deleting 'x' from 'g'. Here, 'f' and 'g' are functions from some type 'a' to some type 'b', where 'b' is a finite join complete partial order (cpo) and a pointed cpo.
If x is greater than or equal to zero, then x is less than or equal to the square root of y if and only if the square of x is less than or equal to y.
, given the assumption that the pair (X1 + X2, Y) is in the multiset power (multpw) of ns, it is possible to obtain two sets Y1 and Y2 such that Y equals the sum of Y1 and Y2, and both pairs (X1, Y1) and (X2, Y2) are in the multiset power of ns.
If set A is a subset of set B, and if there is a chain on predicate P and set C on set A, then there is also a chain on predicate P and set C on set B.
If 'i' is less than the length of the list 'ls', then the evaluation of the 'i'-th element of the list 'ls' (after each element of 'ls' has been transformed by the function 'push_param' with parameter 1) with the function 'peval' and argument 'push a n' is equal to the evaluation of the 'i'-th element of the list 'ls' ...
The intersection of back edges and transitive closure of tree edges in a depth-first search (DFS) is an empty set. This means that back edges and tree edges in a DFS do not share any common elements.
If "y" is not in the set of "xs", then the rank of "y" in the list formed by appending "y" and list "zs" to "xs" is equal to the length of "xs".
If for every number 'n' less than the length of 'ias', the interrupt action is okay when 'redT_updIs' is applied to 'is' and the first 'n' elements of 'ias', and the 'n'th element of 'ias' is considered, then the interrupt actions are okay for 'is' and 'ias'.
Restricting the call-context (cc) by removing a specific element 'x' from it in a graph 'G' is equivalent to deleting 'x' from the call-context in the same graph 'G'.
The lemma named "iterateoi_rule_P" with case names "minv", "inv0", "inv_pres", "i_complete", and "i_inter" makes the following assumptions: 1. "MINV": The map "m" is invariant. 2. "I0": The initial state "σ0" is in the domain of the abstraction of "m". 3. "IP": For any key "k", value "v", iterator "it", and state "σ",...
The inverse of a formal power series (fps) of a natural number 'n' in a semiring with additional operations (times, uminus, inverse) is equal to the formal power series constant of the inverse of the natural number 'n'.
: given two elements 'x' and 'y' from the carrier of the Ring 'R', the product of 'y' and the n-th power sum of the product of the n-th power of 'x' subtracted by 'i' and the i-th power of 'y', all scaled by the binomial coefficient 'n choose i', is equal to the n-th power sum of the product of the n-th power of 'x' su...
If a composite of 't' and the difference of 'u' and 't' equals 'v', and a composite of 'u' and the difference of 't' and 'u' equals 'v', then 'v' is the join of 't' and 'u'.
If the pair (fi, i), (ft, t), and (fe, e) are in the relation bf_ifex_rel, then the pair consisting of the boolean function "bf_ite" applied to fi, ft, and fe, and "ifex_ite" applied to i, t, and e, is also in the relation bf_ifex_rel.
If "A" is an object in the category of relations of a certain size "alpha", and "b" is also an object in the same category, then the first component of the arrow from "A" to the set containing "b" is an isomorphism in the category of relations of size "alpha" from the product of "A" and the set containing "b" to "A".
If a transition system "TS" is in the high priority transitions of a state "ST", and "U" is in the resolution of racing conditions of "TS", then the tuple consisting of the hierarchical automaton of "ST", the step configuration of the hierarchical automaton of "ST" and the configuration of "ST" with "TS", the step acti...
The lemma named "arctan_aform_err" assumes that "x" is in the error form "aform_err" of "e" and "X", and that the error form of the arctangent function "arctan_aform_err" with parameters "p" and "X" equals "Some Y". It also assumes that "e" is a function mapping the entire universe to the interval between -1 and 1. Und...
The length of the sequence extracted from 'xs' at positions 'A' is equal to the cardinality of the intersection of 'A' and the domain of 'xs'.
In the context of an infinite coin toss space, the lemma named "set_filt_contain" assumes that a set A belongs to the sets of the natural filtration at time n, and an element w belongs to set A. It shows that the preimage of the set containing the pseudo-projection of w at time n under the pseudo-projection at time n i...
The function f2 with five arguments all equal to zero is equal to the function f2 where the first argument is 1 minus 1 and the rest of the arguments are zero.
The lemma named "repeatable_both_both2" assumes that "t" is repeatable. It shows that if "t" is repeated twice after "t'", the result is the same as if "t" was just repeated once after "t'".
If A and B are elements in the set created by the function "build_ip_partition" with parameters c and rs, and A is not equal to B, then for all IP addresses ip1 in the set created by the function "wordinterval_to_set" with parameter A, and for all IP addresses ip2 in the set created by the function "wordinterval_to_set...
The lemma named "is_cofiniteE" assumes that the set F is a subset of the carrier of the nth power of the p-adic numbers. It also assumes that F is cofinite in the nth power of the p-adic numbers. Under these assumptions, it shows that the set difference between the carrier of the nth power of the p-adic numbers and F i...
The operation "hm_change_key_op" is related to the operation "mop_map_update_ex" in the context of identity functions and the heapmap relation. This relation is also applicable to the result of these operations under the heapmap relation.
The step symbol set of the union of R and S is equal to the union of the step symbol set of R and the step symbol set of S.
The lemma named "positive_antisym" assumes that a matrix A is positive and its negation (-A) is also positive. Under these assumptions, it shows that matrix A is a zero matrix with dimensions equal to the number of columns in A.
If it is eventually the case that both the negative of function f(x) and the negative of function g(x) are non-negative under a filter F, then it is also eventually the case that the ratio of function f(x) to function g(x) is non-negative under the same filter F. This is assumed to be true.
The function 'of_complex' is injective. This means that if the 'of_complex' function applied to two complex numbers 'x' and 'y' gives the same result, then 'x' and 'y' must be the same complex number.
If two lists 'xs' and 'ys' are distinct and sorted, then the set resulting from the intersection of 'xs' and 'ys' is equal to the intersection of the sets 'xs' and 'ys'.
The representation of the difference between two elements 'r' and 's' is equal to the difference between the representations of 'r' and 's'.