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"x equals tau_steps of x and n if and only if n equals zero".
The lemma named "cmod_eq_Re" assumes that the absolute value of x is equal to the real part of x. It then shows that the real part of x is indeed equal to x.
For a given polynomial 'f' in a field with characteristic 0, factorial ring gcd, and semiring gcd mult normalize, if 'f' is square-free, then 'f' is separable.
If x is not an element of E, then the SPRAY of x is an empty set.
If a function F leads from set A to the union of sets A' and B, and also leads from the difference of sets B and A' to B', then the function F leads from set A to the union of sets A' and B'.
Binding any function "f" to "ABORT" results in "ABORT".
The lemma named "sincos_total_pi_half" assumes that both "x" and "y" are greater than or equal to zero and the sum of their squares equals one. It shows that there exists a "t" such that "t" is greater than or equal to zero and less than or equal to pi/2, and "x" equals the cosine of "t" and "y" equals the sine of "t".
The lemma named "next_eventually" assumes a stuttering invariant "F". It shows that if "F" eventually holds, and "F" does not currently hold, then "F" will hold at the next state or at some point in the future.
The function that composes x with y is a complex linear function when x is varied.
The composition of the relation "ov" and function "f" is a subset of the union of the relations "s", "ov", and "d".
The sequence extraction of a list (xs @ ys) with respect to a set A is equal to the concatenation of the sequence extraction of xs with respect to A and the sequence extraction of ys with respect to the set of all j such that j plus the length of xs is in A.
If set Q is a subset of the states of machine M, then set Q is finite.
The operation "Op" with parameters delta, inp, and binp is not a variable in the term s.
For all arguments, the domain of the optimal binary search tree weighted path length 2 is defined.
If a function 'f' is injective on a set 'A', and 'X' and 'Y' are subsets of 'A', and the image of 'X' under 'f' is a subset of the image of 'Y' under 'f', then 'X' is a subset of 'Y'.
Given a set of tensors 'As' where each tensor 'A' in 'As' has dimensions 'ds', and an index 'i' that is less than the length of 'As', the subtensor at index 'i' of the combined subtensor of 'As' with dimensions 'ds' is equal to the 'i'-th element of 'As'.
The theorem "distinct_nearest_neighbors" assumes that the k-d tree is invariant. It shows that the nearest neighbors in the k-d tree are distinct for any given point 'p' and number 'n'.
If a predicate "P" with variables "x" and "y" implies another predicate "Q" with the same variables "x" and "y", then substituting "x" and "y" with some value "v" in both "P" and "Q" will preserve this implication relationship. That is, if "P" implies "Q", then the substituted "P" will also imply the substituted "Q".
If 'x' is not in set 'A', then the constant map of 'v' over set 'A' at 'x' equals 'None'.
If "a" is equivalent to "a'" and both "a" and "a'" as well as "b" are elements of the carrier "G", then the product of "a" and "b" is equivalent to the product of "a'" and "b".
The action "Deq inp q1 mid" is enabled with respect to the variables if and only if the queue "q1" is not empty.
The lemma named "am_to_adj_nodes_refine" assumes that "am" is an adjacency map. It shows that the pair consisting of "am u" and "adjacent_nodes u" is in the relation of the list set of natural numbers.
The median of a list 'xs' is indeed a median of that list 'xs'.
A space is path-connected in the subtopology of the Euclidean space S if and only if it is path-connected in S.
If the set of radicals of an expression 'e' is not empty, then there exists a radical 'r' in the set of radicals of 'e'. There also exist expressions 'a' and 'b' such that the value of 'e' is equal to the value of the addition of 'a' and the multiplication of 'b' and the square root of 'r', and the value of 'r' is grea...
The concatenation of list 'a' and 'b' is equal to the concatenation of list 'a' and 'c' if and only if the length of list 'a' is infinite or list 'b' is equal to list 'c'.
The lemma named "dg_SemiCAT_components" shows the following: 1. The object component of the digraph of semi-categories for a given cardinal number alpha is the set of all semi-categories for that alpha. 2. The arrow component of the digraph of semi-categories for a given cardinal number alpha is all small category fu...
The lemma named "gmctxtex_onp_arg_monoI" assumes that "P" is a ground meta-hole. It shows that "R" is a subset of the ground meta-context extension on "P" of "R".
"Red_I N" is a subset of "Inf".
The lemma named "max_ideal_UNIV_radical" assumes that the set "F" generates the maximum ideal in the universe. It shows that the radical of the ideal generated by "F" is equal to the ideal generated by "F" itself.
Lemma less_eq_multiset_HO: "M is less than or equal to N if and only if for all y, if the count of y in N is less than the count of y in M, then there exists an x such that y is less than x and the count of x in M is less than the count of x in N."
The concatenation of sequences "u", "v", and "w" is equal to the concatenation of "u" and "v", followed by "w". This is represented symbolically as "u \<frown> v \<frown> w = (u @ v) \<frown> w", where "?lhs" refers to the left-hand side of the equation and "?rhs" refers to the right-hand side of the equation.
If the intersection of the states of SA and the HAStates of A is empty, and if S is in the HAStates of A, and if T is not in the states of SA and T is not equal to S, then the Chi function of the addition of A and the pair (S,SA) with respect to T is equal to the Chi function of A with respect to T.
The function "Abs_bit" is equal to the function "of_bool". This is a code abbreviation in Isabelle.
: Assuming that the length of the list "fresh_idns" is equal to the length of the list "insts", and that both "fresh_idns" and the first elements of the tuples in "insts" are distinct. Also, assuming that for every identifier in "fresh_idns", it does not appear in the first elements of the tuples in the type variables...
The domain of the empty set is the empty set.
The lemma "some_src_in_sources" assumes that "\<mu>" is an arrow. It shows that "some source of \<mu>" is in the set of sources of "\<mu>".
Given the assumptions that M is equal to the norm-preserving extensions E, p, F, and f, that c is an element in the chains of M, that the graph of H and h is equal to the union of c, and that x is an element in H, it can be shown that there exists H' and h' such that x is an element in H', the graph of H' and h' is a s...
The lemma "h_obs_state" assumes that for a finite state machine M, the function h_obs applied to states q, x, and y yields some state q'. It then shows that this resulting state q' is included in the set of states of the finite state machine M.
Theorem T3: "It is possible that God exists."
If the pair (S,T) is in the set ChiPlus A, then if (S,T) is not in the set ChiRel A, there exists a U such that the pair (S,U) is in the set ChiPlus A and the pair (U,T) is in the set ChiRel A.
If a set 's' is bounded, then the set of first elements (obtained by applying the 'fst' function) of 's' is also bounded.
The set of ordered pairs from the union of sets A and B is equal to the union of the set of ordered pairs from set A and the set of ordered pairs from set B.
, according to a given typing relation P, the Object is a subtype of C if and only if C is equal to Object.
If 'x' and 'y' are elements of 'A', then 'x' is less than or equal to 'y' if and only if the join of 'y' and 'x' equals 'y'.
The lemma named "lateEarlyOutput" is defined with fixed variables P and P' of type pi, and a and b of type name. It assumes that P transitions to P' with late output of b on a. The lemma then shows that P can also transition to P' with early output of b on a.
If there is a well-founded relation on the transitive closure of a predicate P over a set A, then there is a well-founded relation on the predicate P over the same set A.
The lemma named "funpow_dist1_least" assumes that "n" is greater than 0 and less than the distance from x to y under repeated application of function f. It shows that applying function f "n" times to x does not result in y.
The system composed of the renamed system of clients (obtained by applying the client_map function to each client in the set of clients with indices less than Nclients) and the union of the Network and the renamed system of Alloc, is equal to the System.
The lemma named "sum_image_inj_card_helper" is defined with a fixed natural number "l". It assumes two conditions: first, for any "i" that is less than or equal to "l", the set "I i" is finite; second, for any "i" and "j" that are both less than or equal to "l" and not equal to each other, the intersection of the sets ...
The lemma named "point_loop_not_acyclic" assumes that "p" is a point and "p" is less than or equal to the successor of "n" times "x" followed by "p". It shows that "x" is not acyclic.
For all substitutions "σ" in the set of "renamings_apart Cs", for every "x", "σ x" is a variable.
The lemma named "eqvt" demonstrates that the bisimulation relation is equivariant.
The integer representation of a 32-bit unsigned integer 'n' is defined as follows: If the 31st bit of 'n' is true (or 1), then the integer is the result of a bitwise OR operation between the integer representation of the signed 32-bit integer obtained by performing a bitwise AND operation on 'n' and the hexadecimal num...
If 'x' is not in the set 'xs', then 'x' is not in the set that lies between 'a' and 'b' in the list 'xs'.
If the "holds" function is true for the list consisting of "x" followed by "xs" and the next element "nxt", then there exists an element "y" such that the predicate "P" is true for "pre", "x", and "y".
There exists a 'h prime' such that when the insertion sort algorithm is applied to array 'a' with initial state 'h' and final state 'h prime', the following conditions hold: 1. The operation has an effect from 'h' to 'h prime' on 'u'. 2. The multiset of elements in the array 'a' before sorting (in state 'h') is equal...
The next state of the product of two finite state machines (fsm1 and fsm2), given an input x and a pair of states (st1, st2), is equal to a new pair of states (st1', st2') if and only if st1' is the next state of fsm1 given st1 and x, and st2' is the next state of fsm2 given st2 and x.
The generalized parallel reduction step function with respect to a relation R is equal to the generalized parallel reduction step function with respect to the universe set and the generalized substitution closure of the relation R. This is denoted as "?Ls = ?Rs".
The lemma named "set_annul_low_equal" assumes that s1 and s2 are low equal, and t1 is the second element of the first pair resulting from setting annul to true in s1, and t2 is the second element of the first pair resulting from setting annul to true in s2. Under these assumptions, it shows that t1 and t2 are low equal...
The lemma assumes that for all integers x in the set from 0 to infinity and for all integers y in the set from 0 to infinity, the result of multiplying x and y and then dividing by y equals x. It then shows that this is also true for all natural numbers x and y.
The lemma "inf_notin_unit_disc" shows that infinity does not belong to the unit disc.
The statement is a lemma about induction on lists that satisfy a certain property. It assumes that there is a property P that holds for all pairs of elements in two lists xs and ys. It also assumes that a property R holds for two empty lists. Furthermore, it assumes that if a pair of elements x and y satisfies property...
"P allows Integer to be less than or equal to T" is equivalent to "T equals Integer".
In the context of an ORSet, the lemma named "add_id_valid" assumes that the sequence 'xs' is a prefix of 'j' and that the delivery of the pair (i1, Add i2 e) is in the set 'xs'. It then shows that 'i1' is equal to 'i2'.
The function "t" of 0 equals 0.
If semicategory 'A' is isomorphic to semicategory 'B' under a certain condition 'alpha', and semicategory 'B' is also isomorphic to semicategory 'C' under the same condition 'alpha', then semicategory 'A' is isomorphic to semicategory 'C' under the condition 'alpha'. This is a statement of the transitivity property in ...
Concatenating any list 'xs' with an empty list results in the original list 'xs'.
The lemma named "int_pow_ord" assumes that "a" is a non-zero element of the p-adic numbers. It shows that the order of "a" raised to the power of an integer "i" is equal to "i" times the order of "a".
The lemma named "plain_nres_rel_def" defines that a pair (c, a) is in the relation "plain_nres_rel" parameterized by R if and only if the return of c is less than or equal to the downward closure of R applied to a.
If a function 'f' is homotopic to the identity function on a topological space 'X' (meaning that 'f' can be continuously deformed into the identity function without leaving 'X'), and the image of 'f' is a subset of 'S', then the homology group of order 'p' of the subtopology of 'X' restricted to 'S' is isomorphic to th...
For any relation defined by "ct_ref_rel", the union of the function "majorities.opt_mru_step2" over all rounds "r" and decision functions "dec_f", is refined by the transition step "CT_trans_step" with parameters "HOs", "crds", "next2", "send2", and "2" under the relation "ct_ref_rel".
The function "iterate_to_set" with parameters "emp", "ins_dj", and a set iterator "it" of type 'x,'x_set is equal to the function "it" with parameters being a lambda function that always returns true, a lambda function that applies "ins_dj" to x and σ, and the result of calling "emp" with no arguments.
If the initial state does not affect the condition 'b', then the sequential composition of a process 'P' with the conditional choice between processes 'Q' and 'R' based on 'b', is equivalent to the conditional choice between the sequential composition of 'P' with 'Q' and the sequential composition of 'P' with 'R', base...
If a function 'f' mapping real numbers to real numbers is continuous on a set 'A', and for all 'x' in 'A', 'f(x)' is greater than or equal to 1, then the function that maps 'x' to the inverse hyperbolic cosine of 'f(x)' is also continuous on 'A'.
If "i" is less than the length of the list "xss" and the "i"th element of "xss" is "ys", and if "j" is less than the length of "ys" and the "j"th element of "ys" is "z", then there exists a "k" less than the length of the concatenated list "xss" such that: 1. Taking the first "k" elements of the concatenated list "xss...
The composition of two function relations, the first from R1 to R2 and the second from R3 to R4, is a subset of the function relation from the composition of R1 and R3 to the composition of R2 and R4.
In the context of an order, the empty set is an order on the set D.
The Turing machine halting condition 'TSTD' for a given state 'st', left tape 'l', and right tape 'r', is equal to the condition where the state 'st' is zero and the standard tape configuration 'std_tap' for the left tape 'l' and right tape 'r' is true.
The lemma named "develops_at_withinD" assumes that "e" develops at "a" within "I". It shows that "a" is a subset of all elements in "I", and for any "i" that is less than the length of "I", the "i"-th element of "e" belongs to the range of the difference between the "i"-th element of "I" and the "i"-th element of "a".
In the context of an imap, the lemma "store_delete_ids_imply_messages_same" assumes that "i" is an element of "is", "xs" is a prefix of "j", "(i, Store e1 mo i)" is in the set of "node_deliver_messages xs", and "(ir, Delete is e2)" is also in the set of "node_deliver_messages xs". Under these assumptions, it shows that...
If a program P can transition from a state represented by the tuple (e, s, b) to another state represented by the tuple (e', s', b') through zero or more steps, and if the expression e is final, then the expressions e and e' are the same, the states s and s' are the same, and the conditions b and b' are the same.
Theta is not equal to zero.
The most significant bit (msb) of the result of taking the signed n-bit representation of a word 'w' is equivalent to the bit of 'w' at the position 'min n (word_length TYPE('a) - 1)'.
The lemma named "f_1_lemma_1" assumes that "x" is an element of "S1". It shows that the property "P" holds for "x" and the function "f_1" applied to "x".
A tree with depth "n" is balanced at depth "0".
If the list "a # as" is a sublist of "vs", then "as" is also a sublist of "vs".
If a heap 'h' is free of at least 'n+1' elements, then the heap 'h' with the mapping from 'a' to 'b' added is free of at least 'n' elements.
: 1. If a logical statement phi implies another logical statement psi (denoted as "\<phi> \<longrightarrow>\<^sub>P \<psi>"), then the evaluation of phi under a set S (denoted as "eval\<^sub>G S \<phi>") also implies the evaluation of psi under the same set S (denoted as "eval\<^sub>G S \<psi>"). 2. If a logical stat...
"d p" is less than or equal to the choice between "x" and "y" after "q", if and only if "d p" is less than or equal to "x" after "q" and "d p" is less than or equal to "y" after "q".
The left conjugation by "x" of the right conjugation by "x" of "y" is equal to "y".
For all 'n', if 'va' is not equal to 'LVar n', then 'G', 'L', and 's' allow 'va' to successfully transition to 'vf' in a manner that is less than or equal to 'Inl T'. This is equivalent to 'G' and 's' allowing the first element of 'vf' to be less than or equal to 'T' and 's' being less than or equal to the second eleme...
: Assuming 1) the difference between the original position and the expected position is greater than the safe distance for two reactions, 2) the change in position of the other object is less than or equal to the maximum speed, and 3) it is not the case that the original acceleration is greater than the expected acce...
For any given psi-calculus system (represented by Psi), any two processes (represented by P and P'), and any permutation of names (represented by p), the tau-chain equivalence of the system and the processes under the permutation is the same as the tau-chain equivalence of the system and the processes after applying th...
The function is continuous. This function takes an input 'x' of any type 'a' that is a partial order pointed chain complete (pcpo). If 'x' is less than or equal to 'd', then the function returns the bottom element, otherwise it returns 'c'.
The application of the WHILEIT function with invariant I, condition function b, function f, and state s is equivalent to binding the evaluation of state s to a function that applies the monadic version of WHILEIT with the same invariant I, condition function b (wrapped in an evaluation), function f, and state s.
The Fibonacci number of the greatest common divisor 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.
If an object sort signature "oss" is well-formed according to the execution model, then the translation of this object sort signature is also well-formed.
If two lists are related by the binary relation A, then rotating each list by one element will result in two new lists that are also related by the same binary relation A. This is a transfer rule, which means it is used to transfer properties between different representations of data.
The lemma named test_assoc3 assumes that the relation P is a subset of the singleton set. It then shows that the composition of relations R and S, followed by P, is equal to the composition of R with the composition of S and P.