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For a set A, the function that adds a real normed vector 't' to each element 'x' in A is an isometry. An isometry is a transformation that preserves distances, meaning the distance between any two points in A remains the same after the transformation.
For any lock number "ln", if for any lock "l", the lock expression and locks are okay when considering the lock status "ls" at lock "l", thread "t", lock number "ln" at lock "l", and the expression lock status "els" at lock "l", then the lock status and expression lock status are okay for lock status "ls", thread "t", ...
The homomorphism of the supremum (or join) of two elements x and y is equal to the supremum of the homomorphisms of x and y.
The natural logarithm of -1 is equal to the imaginary unit times pi.
Rounding down a floating point number with any precision to zero equals zero.
Applying the uncurry0 function to any function 'c' and any value 'x' will always result in the function 'c'.
The lemma named "not_empty_eff_in_as_proj" is defined with fixed variables as, a, and vs. It assumes that the domain of the function "fmrestrict_set" applied to vs and the second element of a is not an empty set. Under this assumption, it shows that the projection of the list "a # as" onto vs is equal to the list forme...
The function "at_map", which maps from an atomic boolean algebra to its atoms, is injective. This means that each element in the atomic boolean algebra is mapped to a unique atom.
The left product function is a bounded linear function.
If a string "u" is not empty, then the empty string is a proper suffix of "u".
If a sextuple (M, l, s, kb, l1, t) is in MStep and another sextuple (M, l, s, k, hh, v) is in Exec, then there exists a number 'n' such that the sextuple (M, l1, t, n, hh, v) is in Exec.
If the following conditions are met: 1. There is a relation 'rel_fundef' between function definitions 'fd1' and 'fd2' where the relation is defined by equality and a relation 'R'. 2. There is an instruction 'instr1' at a certain location 'l' and program counter 'pc' in the function definition 'fd1'. 3. The relation 'R...
Lemma: borel_measurable_cINF_real (measurable, raw): Fix a function F with type "_ => _ => real" Assume the following: 1. The set I is countable. 2. For all i in I, the function F i is borel measurable with respect to the measure space M. Then, the function (lambda x. the greatest lower bound of F i x for...
For any two similar matrices A and B, where A and B are square matrices over a field, there exist two matrices X and Y and a linear function f such that the matrix representation of f with respect to the bases X and X is A, and with respect to the bases Y and Y is B. Furthermore, the set of vectors of X and Y form a ba...
The lemma consists of two parts: 1. If Q is true at time t, and P is true for all times t' in the interval I that are less than t, and t is in interval I, then there exists a time t' in interval I such that P is true until time t, at which point Q becomes true. 2. If t is in interval I, Q is true at time t, and P is ...
"x" is an element of the set determined by "y" (where if "y" is the negation of a formula "phi", the set is {"phi"}, otherwise the set is empty) if and only if "y" is the negation of "x".
For an infinite mapping function 'f' applied to the concatenation of lists 'xs1' and 'xs' and another list 'ys', it is equivalent to the concatenation of the mapping function 'f' applied to 'xs1' and the first 'length xs1' elements of 'ys', and the infinite mapping function 'f' applied to 'xs' and the remaining element...
The bitwise AND operation of 'a' and 'b', multiplied by 2 to the power of 'c', is equal to the bitwise AND operation of 'a' multiplied by 2 to the power of 'c' and 'b' multiplied by 2 to the power of 'c'.
, under the assumption that for any element 'q' in set 'B', the leading term of 'q' does not add to the leading term of 'p', and that 'p' reduces to 'r' with respect to set 'B', it can be shown that 'r' is not equal to zero.
The convolution of delta and any function f is equal to the function f itself.
"there exists some 'x' such that if 'P' is true, then 'Q(x)' is true" is equivalent to "if 'P' is true, then there exists some 'x' for which 'Q(x)' is true".
The lemma named "nprv_exiE_var" assumes the following: 1. There is a proof in the system 'n' that from the set of formulas 'F', the existential quantification of 'x' in the formula 'phi' can be derived. 2. There is a proof in the system 'n' that from the set of formulas 'F' where 'phi' is substituted by the variable '...
, assuming "lhv M A B R XA XB", it shows that for almost every 'w' in 'M', for all 'a' in the spectrum of 'A', 'XA a w' is greater than or equal to 0 and less than or equal to 1, and the sum of 'XA a w' over all 'a' in the spectrum of 'A' equals 1.
A pair (x, y) is in the multilevel lexicographic extension of a function 'f' and a relation 'r' if and only if the function 'f' applied to 'x' is less than the function 'f' applied to 'y', or the function 'f' applied to 'x' is equal to the function 'f' applied to 'y' and the pair (x, y) is in the relation 'r'.
The set of elements in the tree resulting from the "baliL" operation on a left subtree "l", a node "a", and a right subtree "r", is equal to the union of the set of elements in the left subtree, the set containing the node "a", and the set of elements in the right subtree.
For any complex vector 'x', scaling 'x' by 2 is equivalent to adding 'x' to itself.
: assuming "f" is equal to "semi-simplicial function \<ff>", it shows that the domain of "f" in the category of semi-simplicial objects is equal to the semi-simplicial object "o".
: 1. The final interpretation result of a model M and a proposition P is an empty set when the condition is true. 2. The final interpretation result of a model M is an empty set when both the proposition and the condition are empty.
The lemma named "quadratic_real_max_eq" is defined with fixed real numbers a, b, c, and x. It also defines "xm" as the negative of b divided by twice of a. It shows that if a is less than zero, then the value of the quadratic equation a*x² + b*x + c is always less than or equal to the value of the quadratic equation a*...
If the function 'f' is in the set of all products of 'b', 'u', and 'v' for some 'u' and 'v' in set 'A', and for all 'u' and 'v', the property 'P' holds for the product of 'b', 'u', and 'v', then the property 'P' holds for the function 'f' applied to 'y'.
An element 'a' is a member of the function map of 'f' and 'X' if and only if there exists an element 'b' such that 'b' is a member of 'X' and 'a' is equal to the function 'f' applied to 'b'.
If a state 's' is error-free, then the state remains error-free even after an abrupt update. This abrupt update is conditional, i.e., it occurs if a condition 'p' is met, and the update involves a potential jump 'j'.
The theorem 'wls_substAbs_vsubstAbs_compose1' assumes that 'A' is a well-sorted abstraction of sort '(us, s)', 'Y' is a well-sorted term of sort 'ys', and 'y' is not equal to 'y1'. It shows that substituting 'y1' for 'y' in 'A' and then substituting 'Y' for 'y' results in the same term as just substituting 'y1' for 'y'...
The mapping function 'f' applied to the iteration of 'x' is equal to the iteration of the function 'f' applied to 'x'.
If the set B represents the sequence s as 0, then the sequence s is an element of the syzygy module of B.
"x is less than or equal to the intersection of the double complement of x and x".
The lemma named "list_length_hint2a" assumes that the length of list "l" is equal to the successor of 0 (which means the length is 1). It then shows that the tail of list "l" is an empty list.
The lemma named "cross_ratio_swap_12" assumes that the cross ratios of points p, q, r, s and q, p, r, s are both correct. It shows that the cross ratio of points q, p, r, s is equal to the reciprocal of the cross ratio of points p, q, r, s.
The lemma named "permute_set_eq_image" shows that the permutation of a set X by p is equal to the image of the set X under the permutation p.
The lemma "Union_supports_set" shows that the union of the support of each element 'x' in set 'S' supports the set 'S'.
The lemma named "symmetric associative coherence" assumes that "a", "b", and "c" are identities. It shows that the composition of the associator of "b", "c", and "a", the braiding of "a" and the tensor product of "b" and "c", and the associator of "a", "b", and "c" is equal to the composition of the tensor product of "...
The relation "JF2rel R" holds between elements "a" and "b" if and only if the relation "F2rel R" holds between the relations "JF1rel R" and "JF2rel R" and the results of applying the function "dtor2" to "a" and "b", respectively.
The sum of an empty list under any function "f" is zero.
If A' is less than or equal to A and B is less than or equal to B', then the relation of A and B in deterministic data structures is less than or equal to the relation of A' and B'.
If x is less than or equal to y, then the supremum (or least upper bound) of x and z is less than or equal to the supremum of y and z.
The set that is closed downwards is a down-set for any given element 'x'.
A list is valid if and only if the list remains valid after prepending it with B.
The greatest common divisor (GCD) of a set containing only one element 'a' is equal to the normalized value of 'a'.
If a (of any arbitrary type) is equal to b, then a is less than or equal to b.
: Assuming that a finite state machine "M" is observable and minimal, and "q1" and "q2" are different states in "M", it shows that the function "get_distinguishing_sequence_from_ofsm_tables_with_provided_tables" distinguishes between "q1" and "q2" when it is applied to the output of the function "compute_ofsm_tables" ...
For any two elements x and y in a factorial semiring, if x and y are coprime and the product of x and y is an nth power up to a unit multiple, then x is also an nth power up to a unit multiple.
If "t" is a binary search tree, then the set of elements in the first part of the binary search tree split by "x" is equal to the set of elements in the tree "t" that are less than "x".
The lemma named GTRS_SCR assumes that a system is a ground system, denoted as "ground_sys \<R>", and that the system satisfies the Simplification Church-Rosser property (SCR) for a ground simplification rewrite step, denoted as "SCR (gsrstep \<F> \<R>)". It then shows that the system also satisfies the Simplification C...
, given the assumption H: "\<Gamma> is a well-formed type T", it shows that if the mapping of is_TVarB over \<Gamma>' is equal to the mapping of is_TVarB over \<Gamma>, then \<Gamma>' is also a well-formed type T.
If the length of the list "xs" is greater than zero, then the second element of the last pair in the polychain of "p" and "xs" is equal to "p" plus the sum of the list "xs".
If for all 'm', 'F' is in the co-relation of 'm' and the maximum of 'f' and 'g' at 'm', and 'n' is in 'common', then 'F' is in the stable state at most 'n'.
In the context of an isomorphism in a digraph homomorphism, the domain of the object map of the inverse digraph homomorphism is equal to the object of the digraph B.
If you zip together two lists, each created by replicating a value 'x' 'i' times and a value 'y' 'j' times respectively, the result will be a list that replicates the pair (x, y) for the minimum number of times between 'i' and 'j'.
The lemma "rquot_accepting" assumes that "q" is a node in the right quotient of automaton "A" by "n". It shows that "q" is an accepting state in the right quotient of "A" by "n" if and only if there exists a number "m" such that the state reached by taking "m" steps of zeros in "A" from "q" is an accepting state.
For all 'a' and 'b', 'a' strictly implies 'b' is less than or equal to 'a' implies 'b'.
If we have a shift function 'f' with a shift amount 'k' and a deletion 'del' that maps to another function 'g', and if an element 'a' is in 'g', then there exist elements 'x', 'x'', and 'y' such that 'a' is equal to the ordered pair of 'x'' and 'y', the sum of 'del' and 'x' equals 'x'', the ordered pair of 'x' and 'y' ...
A singleton multiset containing 'x' is equal to the union of multiset 'M' and a singleton multiset containing 'y' if and only if multiset 'M' is empty and 'x' is equal to 'y'.
If "a" is either "Accept" or "Drop", and "alpha" is packet-independent, then "alpha" not matching any unknown is equivalent to "alpha" matching all unknowns.
If a module is fresh with respect to generic variables, then the instantiation of the generic fresh variable from that module is also fresh.
The lemma named "help_lemma" assumes the following conditions: 1. The count of zero in the property configuration of c and p at location 0 is greater than 0 at time t0. 2. The pair (location 0, time 0) is less than or equal to the pair (location 1, time 1) in the product order. 3. The summary of the path from location...
Lemma fls_integral_X_inv_power: Assuming "n" is greater than or equal to 2, the integral of the formal Laurent series of the inverse of X raised to the power of "n" (fls_X_inv ^ n) for a given type 'a (which is a ring with an inverse) is equal to the formal Laurent series of the constant (inverse of (1 - the intege...
The lemma named "gState_progress_rel_channels_update" assumes that the global state "g" satisfies the invariant "gState_inv" for a given program "prog", and that the channel "c" satisfies the invariant "channel_inv". If the index "i" is less than the length of the channels in the global state "g", then the lemma shows ...
In the context of finite groups, the group G is a group.
If a program 'prog' and procedures 'procs' are well-formed according to the 'wf' (well-formedness) property, then the procedures 'procs' are well-formed.
For any word 'x' of length 'a', the unsigned natural number representation of 'x' is less than 2 raised to the power of the length of 'a'.
If the condition "b" implies that "t" refines to "t'", and the negation of "b" implies that "e" refines to "e'", then "If b t e" refines to "If b t' e'". This is a rule for refining conditional statements.
"BSP is valid for SIA parameterized by rho".
The prefix of a set L is equal to the reverse of the suffix of the reverse of set L.
The product of the semicategory diagrams of semicategories A, B, and C is equal to the semicategory diagram of the product of semicategories A, B, and C.
If a key K is not used in the event sequence evs, then K is not in the range of the card key function crdK.
If a derivation D is greater than or equal to the successor of i, and a is the leftmost element at position i, and there is a derivation from a to b using D, then b is also the leftmost element at position i.
The product of the cosine of 'w' and the cosine of 'z' is equal to the sum of the cosine of the difference of 'w' and 'z' and the cosine of the sum of 'w' and 'z', all divided by 2.
The 256th power of 62 is equivalent to negative one in the modulo ring of 7681 for finite numbers.
The lemma named "eoutKnowCorrect_L1k" assumes that the function "eoutKnowCorrect" with parameters C and "kKS m" is true, and that the function "eout" with parameters C and "kE m" is also true. It then shows that "m" is an element of the function "specKeys" with parameter C, or the function "know" with parameters C and ...
The valuation of a linear polynomial monomial with coefficient 1 and variable x, under a valuation v, is equal to the valuation of x.
The lemma named "closed_transdb_inv_aux" assumes two conditions. The first condition is that for all values 'v', if 'v' is a free variable in the expression 'e', then 'v' is less than 'k'. The second condition is that for all values 'v', if 'v' is a free variable in the expression 'e', then the value of 'v' in the cont...
Lemma on action transfer: Assuming that "s" is in the set of states reachable by the closed transition system of the parameterized network "np" with network tree "n" under the total time model "TT", and the network state mapping of "s" under the state relation "sr" equals the network mask of the network tree IPs of "n"...
The Fibonacci number of the greatest common divisor (gcd) of two numbers 'm' and 'n' is equal to the greatest common divisor of the Fibonacci number of 'm' and the Fibonacci number of 'n'. This is referred to as Law 6.111.
The polynomial lift function "poly_lift2" applied to a polynomial "p" equals zero if and only if the polynomial "p" itself equals zero.
The pair of operations (hm_set_op, RETURN ooo op_map_update) is in the relation of the identity function to the identity function to the heapmap relation to the nres relation of the heapmap relation.
If x is equivalent to y, then x is in the domain.
If a list "xs" is in the permutations of a multiset "A", then the multiset of the list "xs" is equal to "A".
The lemma bet__le1213: assumes "Point B is between point A and point C" shows "The distance from point A to point B is less than or equal to the distance from point A to point C"
The lemma named "new_CharacterData_getElement" assumes that creating a new character data results in a new character data pointer and a modified heap. It shows that getting an element from the original heap is the same as getting the same element from the modified heap.
If 'n' is less than the sum of 'm' and the length of the binary list 'bl', then the 'n'-th element of the binary list obtained by converting the word 'w' to a binary list with auxiliary 'm' and 'bl' is equal to the 'm - 1 - n'-th bit of 'w' if 'n' is less than 'm', otherwise it is the '(n - m)'-th element of 'bl'.
The lemma "fpxs_val_add_ge" assumes that the sum of "f" and "g" is not equal to zero. It shows that the value of the formal Puiseux series of the sum of "f" and "g" is greater than or equal to the minimum of the values of the formal Puiseux series of "f" and "g".
If 'a' and 'b' are complex numbers and 'b' is not zero, then the result of 'a' divided by 'b' is also a complex number.
The composition of relation 'm' and relation 'b' is a subset of relation 'b'.
"the negation of q is less than or equal to the pre-test of negation of p and negation of q".
: Assuming - a sequence 'w' is in the tensor product of 'wa' and 'wb' with respect to the flowgraph 'fg', - there exists a transition from 'ca' to 'ca'' via 'wa' in the transitive closure of the non-terminal relation of 'fg', - there exists a transition from 'cb' to 'cb'' via 'wb' in the transitive closure of the non...
: 1. The core of a heap, when applied to a formula, is equal to the heap applied to the same formula. 2. The core of a mask, when applied to a formula, results in an empty mask.
The k-th iteration of the function "bw_diff" applied to the identity function is equal to the identity function if k equals 0, the constant function 1 if k equals 1, and the constant function 0 for all other values of k.
If A is in the set given by v, then there exists a property P such that A is the set of all states s where P(v s) holds true.
For all elements x in the range of the range of A, property P holds for x if and only if for all functions f in the range of A, property P holds for all x in the domain of f.
If a cryptographic message containing a service key, an agent B, a timestamp, and a service ticket, encrypted with an authentication key, is part of what the spies can see, and the authentication key is not part of what the spies can analyze, and the sequence of events is part of the Kerberos IV protocol, then the serv...