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If a formula "a" is well-formed with respect to an index "idx" and "a" is a level 0 logical formula, then the well-formedness of the left derivative of "a" with respect to a variable "x" is guaranteed under the operation that increments the index by one.
If A, B, and C are elements of the set Vset α, then the composition of the function Mα_Rel_arrow_rl with the function Mα_Rel_arrow_lr, both applied to A, B, and C, in the category of sets of α, is equal to the identity morphism of the product of A and B with C in the category of sets of α.
There exists a 'c' in the interval from 'a' to 'b' such that the integral of the function 'g' times 'f' over the interval from 'a' to 'b' is equal to 'g' at 'a' times the integral of 'f' from 'a' to 'c' plus 'g' at 'b' times the integral of 'f' from 'c' to 'b'.
: Assuming that "M" is observable, and "q1" and "q2" are states in "M", and the observable finite state machine (ofsm) table of "M" at the (k+1)th step for "q1" is not equal to the ofsm table of "M" at the (k+1)th step for "q2", then there exist "x" and "y" such that the selection of diverging ofsm table input/outpu...
If command 'c' satisfies the Strong Coherence for Weak Bisimulation (SC_Wbis) property, then 'c' is weakly bisimilar to itself.
, assuming "xs" is not a finite list, the set of elements in the stream generated from the list "xs" is equal to the set of elements in the list "xs".
The product of "x implies y" and "y implies z" is less than or equal to "x implies z".
The lemma "cexpr_typing_dist_dens_cexpr" assumes that under context Γ, expression e1 is of type "dist_param_type dst" and expression e2 is of type "dist_result_type dst". It shows that under the same context Γ, the distribution density expression "dist_dens_cexpr dst e1 e2" is of type REAL.
: "For any list 'l' in an empty set, property 'P' holds for 'l'. If property 'P' holds for an element 'x' and for any list 'l' in the set 'xs', then property 'P' holds for any list 'l' in the set that consists of 'x' followed by 'xs'."
, assuming a run of Ute_M with parameters rho, HOs, and SHOs, and assuming that for all rounds r, the communication per round of Ute_M with the sets of heard-of and scheduled heard-of processes is as expected, and assuming that the global communication of Ute_M with the sets of heard-of and scheduled heard-of processes...
If a pair (p, q) is in the set 'ps', and 'h' is equal to the result of the 'trdsp' function applied to the first elements of the set 'bs' and the pair (p, q), and 'h' is not equal to zero, then 'h' is in the set resulting from the 'gb_red_aux' function applied to 'bs' and 'ps'.
The lemma named "signed_measure_summable" assumes the following conditions: 1. M is a signed measure with measure function µ. 2. The range of a function A is a subset of M. 3. A is a disjoint family. 4. The union of the range of A is in the set of M. 5. The absolute value of the measure of the union of the range of A ...
If two lists, xs and ys, are equal and for every element y in the set of ys, the function f of y is equal to the function g of y, then the sum of the list xs using function f is equal to the sum of the list ys using function g.
The lemma named "measurable_hld" is measurable. It assumes that "t" is an element of the sets in "M". It shows that the predicate "HLD t" is measurable in the stream space of "M".
If set A is finite, then the iterated sumset of A for any integer r is also finite.
For a given function 'f' and a limit 'l' in a real normed field, if the function 'f' applied to the reciprocal of 'x' tends to the limit 'l' as 'x' approaches zero, then the function 'f' also tends to the limit 'l' as 'x' approaches infinity.
The lemma named "set_to_list_empty" assumes that the cardinality (or size) of set S is zero. It then shows that converting this set S to a list results in an empty list.
The function "mapM" applied to a function "f" and a list "xs" equals "None" if and only if there exists an element "x" in the set of "xs" such that applying the function "f" to "x" equals "None".
If for every element 'x', 'f' is a continuous function at 'x', then the set of all sequences 'xs' for which the sequence 'f xs' is lexicographically sorted, is a closed set.
If 'gf' is a morphism from the ordered pair [a, b] to the ordered pair [c, d] in the opposite category of the product of the category with itself, then the hom-functor of 'gf' in the category maps the hom-set from 'a' to 'b' in the category to the hom-set from 'c' to 'd' in the category of sets.
The lemma named "maintain_invar1" assumes two conditions: 1. "prim_invar1 A" which means that A satisfies the first invariant of Prim's algorithm. 2. "light_edge (S A) u v" which means that the edge from u to v is the lightest edge crossing the cut (S A). The lemma then shows that: 1. The set A, when unioned with the...
If a program P and environment E type-check an expression e to a type T under the single-threaded semantics, then the maximum stack size for the expression e is at least 1.
If 'I' is an index set and 'i' is not in 'I', and for all 'x', the evaluation of 'r_phi' at '[i, x]' is equal to the evaluation of 'r_phi' at '[j, x]', then 'j' is not in 'I'.
Shifting a formal power series 'f' multiplied by 'X' to the power of 'n' by 'n' places will result in the original formal power series 'f'.
The lemma named "normal_is_Cong_closed" assumes that "t" is a member of the set of normal elements, denoted as "N", and "t" is congruent to "t'". It shows that "t'" is also a member of the set of normal elements.
The lemma named "element_ptr_kinds_M_eq" assumes that the set of node pointers in memory model 'h' is equal to the set of node pointers in memory model 'h''. It then shows that under this assumption, the set of element pointers in memory model 'h' is also equal to the set of element pointers in memory model 'h''.
The lemma "setops\<^sub>s\<^sub>s\<^sub>t_pair_image_subst_cons" consists of several statements: 1. The image of the pair function applied to the set operations of the stateful strand term (x#S) substituted by θ is equal to the union of the image of the pair function applied to the set operations of the stateful stran...
For any given Java-like programming environment G and heap hp, replicating a value v n times is approximately located to replicating an error Err n times.
If the sum of "n" and "i" equals "Entry", then "n" must be equal to "Entry".
If "x" is an element of the kth power of "m1" under a certain alphabet "AA", then the maximum element of the kth power of "m1" under "AA" is less than the length of "AA".
If "M" is in the set of valid elements, then the intersection of "R23s" and the universal set, along with the intersection of "l3_inv5", "l3_inv6", "l3_inv7", "l3_inv8", and "l3_inv9", unioned with the set of "M'" such that "M'" is in the "l2_dy_fake_chan" of "M" and "l3_dy" of "M", is a strict superset of "R23s".
If a natural number 'x' is less than or equal to another natural number 'y', then 'x' is also less than or equal to 'y' minus the modulus of 'y' minus 'x' with respect to 'm'.
If the removal lookup of "a" in "h" equals to some array with type "T", size "si", function "f", and exception "e", then "h" is a sub-collection of the removal update of "a" with a new array with the same type "T", size "si", new function "f'", and exception "e'".
For any list 'x' and proposition 'P', if 'P' holds when 'x' is an empty list, or 'x' is a list with one element 'y', or 'x' is a list with two elements 'y' and 'z', or 'x' is a list with three or more elements where 'w', 'y', and 'z' are the first three elements and 'zs' represents the rest of the elements, then 'P' mu...
The lemma named "pref_iffD2" assumes that "F is equivalent to G". If this is true, it shows that "if G is true, then F is also true".
For a formal power series 'f', which belongs to a set 'a' that has the properties of an additive abelian group, an inverse, and can multiply by zero, the subdegree of the inverse of 'f' is always equal to zero.
The lemma named "ord_power_aux" is defined with fixed natural numbers m, x, k, and a. It also defines "l" as the order of a with respect to m. The lemma then shows that the product of the order of a to the power of k with respect to m and the greatest common divisor of k and l equals l.
: given an assumption that for any element 'i' in the set 'I', the function 'f' of 'i' is a subset of 'C', it can be shown that the union of the function 'f' for all elements 'i' in the set 'I' is also a subset of 'C'.
The square root of a number n is less than or equal to n.
The lemma named "et" makes the following assumptions: 1. "v" is a vector. 2. "e" is less than or equal to the product of "v" and the transpose of the negation of "v". 3. "t" is less than or equal to the product of "v" and the transpose of "v". Under these assumptions, the lemma shows two things: 1. The product of "e" ...
The lemma "has_sum_Im" assumes that the function "f" has a sum "M" at "a". It shows that the function that gives the imaginary part of "f(x)" also has a sum "M" at the imaginary part of "a".
The index of a constant at the successor of a given number 'u' is equal to the Cantor pairing of 4 and the Cantor pairing of 2 and the index of the constant at 'u'.
The outputs of an interface \<I> are equal to A.
: 1. If the information of a real algebraic number 'x' is equal to the pair (p,n), then the polynomial 'p' represents the real part of 'x', the cardinality of the set of 'y' such that 'y' is less than or equal to the real part of 'x' and the polynomial 'p' evaluated at 'y' equals zero is 'n', and 'p' is irreducible. ...
Lemma fls_lr_inverse_of_int: Let x be an element of a ring with unity and zero multiplication (denoted as 'a::{ring_1,mult_zero}'). Then, the left inverse of the formal Laurent series of an integer n with respect to x is equal to the formal Laurent series with constant x. Similarly, the right inverse of the forma...
The semigroup operation of the boldface number 1 and the function F applied to a function g and a set A is equal to the function F applied to the function g and the set A. This essentially means that the boldface number 1 acts as an identity for the semigroup operation with the function F.
There exists an element p that belongs to both set S and set T.
, under the assumption that there is no match between 'x' (of any type belonging to the field class) and 'b', and that 'b' (of any type belonging to the euclidean semiring cancel class) is not equal to zero, it can be shown that the division of the sum of 'a' and 'b' by 'b' is equal to the sum of the division of 'a' by...
There is no one-step reduction in the uf, P, t system from the state consisting of the value v and s to any state consisting of e' and s', with any thread action ta.
The pair (h1, h2) right-triangle 0 is equal to the condition where the pair (0, 0) right-triangle 0 and both h1 and h2 are equal to 0.
The set of nodes in the deterministic co-Büchi automaton "AA_sub_1" for a given formula "phi" and a list "xs" is finite.
The lemma named "compact_Inter" is about a set of sets, denoted as F, where each set is of a type that satisfies the Heine-Borel theorem. The lemma assumes two things: first, that every set S in F is compact, and second, that F is not an empty set. Given these assumptions, the lemma shows that the intersection of all s...
If a function 'f' is negligible when 'c' is not equal to zero, then the function 'f' multiplied by 'c' is also negligible.
The lemma named "pow_res_classes_mem_eq" assumes three conditions: 1. m is greater than 0, 2. a is an element of the power residue classes of m, 3. x is an element of a. If these conditions are met, it shows that a is equal to the power residue of m and x.
For every list 'xs' in the set of lists 'xss' that is finite, the filter 'P' applied to the concatenation of 'xss' is equal to the concatenation of the map of the filter 'P' applied to 'xss'.
A function 'f' is continuous at a point 'x' if and only if the function 'f' applied to '(x + h)' tends to 'f x' as 'h' tends to zero. This is applicable for a function 'f' that maps from a real normed vector to a topological space.
The lemma named "ine_nonempty_exprChannelSet" assumes that the expression channel set "E" of process "P" is not in channel set "ChSet" and that "ChSet" is not empty. If these conditions are met, it shows that "E" is not in process "P".
There is a transition from state s0 to state s1 under the operation sxalloc in the context of G, and "error_free_s0" which states that the state s0 is error-free. Under these assumptions, it shows or proves that the state s1 is also error-free.
If "x" is in the set "Standard", then the inverse of "x" is also in the set "Standard".
If 'p' is a prime element and 'p' to the power of 'n' divides the product of 'a' and 'b', where 'n' is greater than 0 and 'p' does not divide 'a', then 'p' to the power of 'n' divides 'b'.
"if a real number x is in the range from 0 to 1, then the square of x is less than or equal to x."
The lemma assumes that if 'a' is in the read-only section of 'S', then 'a' is also in the read-only section of 'S prime'. It also assumes that 'a' is in the read-only section of the combination of 'S' and 'R' under 'W'. Under these assumptions, it shows that 'a' is in the read-only section of the combination of 'S prim...
The lemma named "span_substd_basis" assumes that "d" is a subset of the Basis. It shows that the span of "d" is equal to the set of all "x" such that for every "i" in the Basis, if "i" is not in "d", then the inner product of "x" and "i" is zero.
"BisT" is a subset of "RetrT BisT".
If a function 'f' is continuous on a set 's' and for all 'x' in 's', the value of 'f' is between -1 and 1 inclusive, then the function that applies the arccosine to 'f' is also continuous on 's'.
In the context of location-based values semantics (lbvs), we have a lemma named 'wtl_suc_pc'. It assumes two conditions: firstly, the well-typedness of the list of instructions 'ins' with the counter 'c' starting from 0 and the initial state 's0' is not undefined. Secondly, the program counter 'pc' incremented by 1 is ...
The lemma "merge1_dom_wedge" assumes that a node "r" with a set of children "xs" is a subtree of the result of the function "merge1" applied to a tree "t", and that the cardinality of the set "xs" is greater than 1. Under these assumptions, it shows that the domain of the children of a node "r" with a set of children r...
In the context of a vector space, the lemma named "dim_subadditive" assumes that W1 and W2 are subspaces of V over a field K, and that both W1 and W2 have finite dimensions. It then shows that the dimension of the subspace sum of W1 and W2 is less than or equal to the sum of the dimensions of W1 and W2.
A point 'x' is in the closed line segment between two distinct points 'a' and 'b' if and only if the dot product of the vector from 'a' to 'x' and the rotation of the vector from 'a' to 'b' is zero, and the dot product of 'x' and the vector from 'a' to 'b' is within the range of the dot products of 'a' and 'b' with the...
If the frame conditions Fr_1, Fr_2, Fr_3, and Fr_4 hold for a frame \<F>, then the denial of the double negation interpretation (DNI) implies the first local contraposition property (lCoP1) for the implication operator and negation.
The function Mp when applied to 1 equals 1.
Mapping the identity function over all elements of a set S does not change the set, i.e., it remains the same.
The inverse of the empty word prepended to any word 'u' is equal to the word 'u' itself.
, in a given context P, a type T is less than or equal to a class C if and only if either T is a null type, or there exists a class D such that T is equal to class D and D is a subtype of C in context P, or C is an object and there exists an array A such that T is equal to array A.
The execution of a sequence "s" followed by two sequences "rs" and "rs'" is equivalent to the execution of "s" followed by the concatenation of "rs" and "rs'".
For a fixed integer 'i', if 'K' is a subring of 'R' and 'k' is an element of 'K', then the addition power of 'R', 'i', and 'k' is equal to the addition power of 'R' with its carrier set to 'K', 'i', and 'k'. This is expressed as the left-hand side being equal to the right-hand side.
If state 's' is updated such that 'A1' is assigned the value 'a', the value of 'A100' in the updated state is the same as the value of 'A100' in the original state 's'.
Applying the function "fBex" to the finite set obtained from a list "l" and a function "f" is equivalent to applying the function "list_ex" to the list "l" and the function "f".
If the result of deleting the left element from a tree 't' is a pair (x, t'), and if 't' is a Braun tree and 't' is not a leaf, then 't'' is also a Braun tree and the size of 't' is one more than the size of 't''.
The lemma named "powr_closureE" assumes that a function "g" is in the power closure of a function "f". It then obtains a value "p" such that "g" is equal to the function "f" raised to the power of "p".
The lemma named "pole_theorem_analytic_open_superset" assumes the following conditions: 1. The function g is analytic on set S. 2. Set S is a subset of set T. 3. Set T is open. 4. For any element z in the set T excluding a, the function g at point z equals (z - a) times the function f at point z. Under these assumpti...
If "l" satisfies the invariant "l_invar" and "u" is an element of the set "V", then the function "relabel_effect" applied to "f", "l", and "u" also satisfies the invariant "l_invar".
The lemma named "Polygamma_real_strict_antimono" assumes that a real number 'x' is greater than 0 and less than another real number 'y', and that 'n' is an odd number. It shows that the Polygamma function of 'n' and 'x' is greater than the Polygamma function of 'n' and 'y'.
There exists a point C such that A, B, and C are collinear (in that order) and B is not equal to C.
Applying a filtermap operation with a predicate and a function to a list is equivalent to first filtering the list with the predicate and then mapping the function over the filtered list.
The length of the interaction of two lists, xs and ys, is equal to the sum of the lengths of each list in xs plus the sum of the lengths of each list in ys. This statement is a lemma in the context of an additive monoid.
The lemma named "splitFace_face_face_op" assumes that "g" has minimum graph properties (mgp), "g" is ready to split face with vertices "u", "v", "f", and "vs" (pre), and the result of splitting the face of "g" with "u", "v", "f", and "vs" is "(f1, f2, g')" (fdg). It shows that the face-face operation is applicable to t...
The lemma named "f_recurrence" assumes that "k" is greater than 0 and less than or equal to "n". It shows that the function "f" of "k" is equal to the negative of the natural number "k" multiplied by negative one raised to the power of "k" and multiplied by "e'" of "k", minus the sum from "i" equals 1 to less than "k" ...
The lemma named "vrat_identity_law_addition" assumes that "x" is an element of the set of rational numbers. It shows that adding zero to "x" results in "x".
The composition of a function "f" and a sequence "σ", when restricted to a suffix starting at index "i", is equal to the restriction to the same suffix of the composition of "f" and "σ".
It is not the case that for any time point 't'', 't'' is strictly before 't' in interval 'I' where the proposition 'False' holds.
The lemma named "closed_arc_image" is fixing a function 'g' that maps real numbers to a 't2_space' (a topological space with certain properties). It shows that if 'g' is an arc (a continuous function whose start and end points are distinct), then the image of the path under 'g' is closed (contains all its limit points)...
If points p and q are distinct and both incident to line l, and the images of p and q under a collineation C are incident to line m, then the image of line l under collineation C is equal to line m.
The function that multiplies a given number 'b' by a fixed number 'a' is a bounded linear function.
If a function 'f' is additive over a measure space 'M', and if 'x' and 'y' are disjoint sets within 'M', then the value of 'f' applied to the union of 'x' and 'y' is equal to the sum of 'f' applied to 'x' and 'f' applied to 'y'.
The lemma named "flowsto_union" assumes two conditions: 1. There is a flow from set X0 to set Y within time T and crossing through set CX. 2. There is a flow from set Z to set W within time S and crossing through set CZ. Given these assumptions, it shows that there is a flow from the union of sets X0 and Z to the uni...
The sum of function 'f' applied to the concatenation of list 'al' and 'bl' is equal to the sum of function 'f' applied to list 'al' plus the sum of function 'f' applied to list 'bl'.
If a sequence of events, a server, an agent, and a user are part of a protocol, then if the user is not a spy, the password of the user cannot be analyzed or deciphered from the union of the agent's knowledge and the information the spies have gathered from the sequence of events.
The property P holds for an element x of type X.
Sorting a list "xs" is equivalent to folding the list "xs" with the "insort" function starting from an empty list.