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The function "nhds" (which stands for a neighborhood in topology) preserves the relation "A" under the condition that "A" is bi-unique and bi-total, and the function "open" (which stands for an open set in topology) preserves the relation "A" when applied to sets.
The union of "ani(x)" and "ni(x)" equals L.
"Bool b" is a value.
The lemma "fsyms_satC_cong" assumes two structures, S1 and S2, which have the same weight function for function symbols (wtFsym), weight function for predicate symbols (wtPsym), arity function (arOf), result function (resOf), interpretation for types (intT), and interpretation for predicates (intP), but possibly differ...
If the following conditions are met: 1. A normal state transition from (xcp, h, stk) to (xcp', h', frs') under the program P and thread t is valid according to the JVM's deterministic execution model (denoted by -ta-jvmd→). 2. The thread actions (ta) are non-speculative with respect to the program P and the value sta...
1 is less than 4 in the context of 3-bit words.
For any two atoms 'a' and 'b', 'a' is equal to 'b' if and only if the sort of 'a' is equal to the sort of 'b' and the natural number representation of 'a' is equal to the natural number representation of 'b'.
If the "mopup_left_moving" function is applied to the tuple (2n + 5, l, xs) and the variables lm, n, and ires, then the list l is not empty.
The finite set of inputs of a machine M is equal to the inputs of machine M.
The lemma "wf_implies_termi_reach_step_case" assumes a hypothesis "hyp". According to this hypothesis, for any c' and s', if the transition from (c, Normal s) to (c', s') is valid under the context Γ, then c' terminates at s' under the context Γ. Based on this assumption, the lemma shows that c terminates at Normal s u...
The set of unfinished nodes in a graph S is finite.
For all natural numbers 'n', taking 'n' elements from a list 'xs' twice in succession is equivalent to just taking 'n' elements from 'xs' once.
The lemma from the pair automaton derivation: 1. The pair (None, None) is not in the derivation of the pair automaton A B from the term of the generalized term s. 2. If the pair (Some p, None) is in the derivation of the pair automaton A B from the term of the generalized term s, then there exists a term t such that ...
The tau instruction of the composition of a function f and a program P with heap h, stack stk, and instruction i is equivalent to the tau instruction of the program P with heap h, stack stk, and instruction i.
The lemma named "simple_rotations_bij" assumes that "r" is in "simple_rotations". It shows that "r" is a bijection.
For any two points 'a' and 'b' in an ordered Euclidean space, the set including all points from 'a' to 'b' is a closed set.
If a function 'f' is injective and the composition of 'f' and another function 'g' equals the identity function, then 'f' is equal to the inverse of 'g'.
There is no microstep transition from the process "{l}deliver(s_data).p" to the process "q" under the context "Gamma".
If the function "gSwapCls" is applicable to the model "MOD", then "gSwapCls" is also applicable to the model obtained from "MOD" using the function "fromMOD".
The inorder traversal of the leaves of a tree 'as' is equal to 'as' itself.
The lemma named "br_inj_run_conv" assumes that the function "alpha" is injective, and that "R" is equal to the function "br" applied to "alpha" and "I". It shows that a run in system "b" under the function "alpha" composed with "ra" is equivalent to a run in system "a" under "ra".
"P implies the supremum of S" is equivalent to "for all X in S, P implies X".
Given a valuation 'v' of a field 'K' and an element 'x' in the valuation ring of 'K' with respect to 'v', and a non-negative integer 'n', it is true that 'n' is less than or equal to the valuation of 'x' if and only if 'x' is in the nth power of the valuation ideal of 'K' with respect to 'v' in the valuation ring of 'K...
The lemma named "a_redu_OrR2_elim" assumes that if "OrR2 <a>.M b" implies "R", then there exists some "M'" such that "R" equals "OrR2 <a>.M' b" and "M" implies "M'".
: - The pair (x, Trueif) is in the relation "bf_ifex_rel" if and only if x is equal to "bf_True". - The pair (x, Falseif) is in the relation "bf_ifex_rel" if and only if x is equal to "bf_False".
The lemma "if_redT_read_knows_addr" assumes that there is a reduction from state 's' to state 's'' with thread 't' and thread action 'ta' according to the 'mthr.if.redT' rule. It also assumes that the normal action of reading memory at address 'ad' with lock 'al' and value 'v' is in the set of observed actions of 'ta'....
If the "inv_max_hi" property holds for a tree 't', then this property will still hold after deleting an element 'x' from the tree 't'.
For any complex number 'z' that is not a non-positive integer and 'z + 1/2' is also not a non-positive integer, the product of the Gamma function of 'z' and the Gamma function of 'z + 1/2' is equal to the exponential of the product of '1 - 2z' and the natural logarithm of 2, multiplied by the square root of pi, and the...
The lemma named "construct_NofI_prop" is defined with two fixed variables: "p", which is a real polynomial, and "I", which is a list of real polynomials. It assumes that "p" is not equal to zero. The lemma shows that the function "construct_NofI" with parameters "p" and "I" is equal to the rational number derived from ...
If a certain time 't' is vacant up to the communication records of the first state in a sequence 's', and if the next state in the sequence 's' is updated by sending, then 't' is also vacant up to the communication records of the next state in the sequence 's'.
If the following conditions hold: 1. The expression 'e' has a reference type 'statT' in the environment 'E' and datatype 'dt'. 2. The parameter sequence 'ps' has types 'pTs' in the environment 'E' and datatype 'dt'. 3. The maximum specification of the program 'prg E' with class 'cls E' and static type 'statT' has a met...
: Assuming that "q\<^sub>c", "q\<^sub>i", and "q\<^sub>f" are not elements of "\<Q> \<A>", and that "q\<^sub>c" is not an element of "Q", "q\<^sub>f" is not equal to "q\<^sub>c", "q\<^sub>f" is not equal to "q\<^sub>i", and "q\<^sub>i" is not equal to "q\<^sub>c", it can be shown that the language of the generalized t...
Lemma tendsto_proj: If the function (λx. x) converges to 'a' in the filter 'F', then the function (λx. x with index 'i' in the finite map) converges to 'a' with index 'i' in the finite map in the filter 'F'.
If a set C is an exponential set and an idempotent set, then if the relation R to the power of C on set C is antisymmetric, it implies that set C is a T0-separated set.
A function 'p' partitions a number 'n' if and only if the set of all 'x' for which 'p(x)' is greater than 0 is finite, and 'p' is a number partition of 'n' in the absolute multiset.
The lemma tauChainBangI: fixes a type 'b called Psi and a psi type P and another psi type P' assumes that Psi leads to P in parallel with P, transitioning through tau to P' and that P is guarded obtains a Q where Psi leads to the bang of P, transitioning through tau to Q and Psi leads to Q being similar...
The lemma named "rel_interior_interior" is about a set S in an n-dimensional Euclidean space. If the affine hull of set S equals the universal set, then the relative interior of set S is equal to the interior of set S.
The composition of the function "map_IInf" and "map_us" is equivalent to the identity function, which maps a type 'a to a type 'b, both of which are complete lattices with dual.
The lemma named "glbH_is_fixp" assumes that "H" is the set of all "x" in "A" such that the pair "(f x, x)" is in the relation "r". It shows that the greatest lower bound of "H" in the closure "cl" is in the set "P". This is a Tarski lemma for greatest lower bound.
: Assuming that "io" is in the language of the canonical separator of M, q1, and q2, and that M is observable, and q1 and q2 are states in M, and q1 is not equal to q2, it shows that: - If "io" is in the language of the state separator of M and q1 but not in the language of the state separator of M and q2, then the t...
The lemma named "vD_nsqn_sqn" assumes that an IP address is in the set of valid destination routes. It shows that the sequence number of the route for this IP address is equal to the sequence number of the IP address itself.
If 'k' is greater than 0 and 'n' is less than the length of the list 'xs', then the execution of the composition stream function 'f_Exec_Comp_Stream' with the transformation function 'trans_fun', the list 'xs' repeated 'k' times, and the constant 'c', truncated at the product of 'n' and 'k' and then dropped 'k' element...
For any fixed element 'a' of a modular ring with a prime cardinality, the element 'a' raised to the power of the cardinality of 'a' to the power of 'n' is equal to the element 'a' itself.
The lemma named "stirling_integral_bound" assumes that "n" is greater than 0. It obtains a constant "c" such that for any complex number "s" with a positive real part, the norm (or absolute value) of the Stirling integral of "n" and "s" is less than or equal to "c" divided by the real part of "s" raised to the power of...
The imaginary part of the line path between two real numbers "a" and "b" at a point "x" is equal to zero.
The lemma named "ln_upper_5_pos" assumes that "x" is greater than or equal to 1. It shows that the natural logarithm of "x" is less than or equal to the function "ln_upper_5" of "x".
If "b" is a lower bound of "x" and "y", then "b" is less than or equal to "y".
If you rename a formula A and then rename it again with the same formula A, you will get the original formula L.
The lemma named "nos_incD" assumes that "nos_inc" is a function that takes two arguments, "ξ" and "ξ'". It shows that the number "no" of "ξ" is less than or equal to the number "no" of "ξ'".
The infimum (or greatest lower bound) of sigma-1 and nc is sigma-1.
The function at a particular position in the mapped general term is equal to the mapped option of the function at the same position in the original general term.
The lemma named "term_variants_pred_eq_case" is defined with two fixed terms, t and s, of a generic type ('a,'b) term. The lemma assumes that the predicate "term_variants_pred P" holds true for terms t and s, and that for all functions f in the set of functions of term t, the predicate P applied to f equals an empty li...
The lemma named "WbisT_coind" assumes that "theta" is symmetric and "theta" is a subset of "WretrT" of the union of "theta" and "WbisT". If these conditions are met, it shows that "theta" is a subset of "WbisT".
If A is an ideal of R, H is a subset of the carrier M, a is an element of A, and h is an element of H, then the scalar multiplication of a and h is an element of the linear span of R, M, A, and H.
All operator effects in a set of operations are consistent if and only if for every pair of operations in the set, the intersection of the set of added effects of the first operation and the set of deleted effects of the second operation is empty, and the intersection of the set of deleted effects of the first operatio...
The lemma named "iam_to_list_set_correct" assumes that the pair (a, m) is in the relation 'iam_rel'. It shows that the set obtained from mapping 'm' is equal to the set obtained from converting 'a' to a list using 'iam_to_list'.
: Assuming that the set of geometric entities containing the point 'p1' and the empty set is not equal to the set of geometric entities containing the point 'p2' and the empty set, and the line through points 'p1' and 'p2' does not contain the point 'p3', it can be shown that the set of geometric entities containing t...
The evaluation of term 't' after applying substitution 'σ' under environment 'E' and function 'F' is equal to the evaluation of term 't' under the same function 'F' but with an environment that is the result of applying the substitution 'σ' to the original environment 'E'.
The lemma named "imp_wf_dfa" assumes that both "M" and "N" are well-formed deterministic finite automata with "n" states. It shows that the deterministic finite automaton resulting from the implication operation on "M" and "N" is also a well-formed deterministic finite automaton with "n" states.
If "a" is in the initial state, then the parallel composition of two sequences, the first sequence being a fusion of a relation where "u" is in the initial state and "x" equals "y", and a predicate "p", and the second sequence being a predicate "p'" and a relation "r'", is equal to the fusion of a relation where "u" is...
The lemma named "compute_is_pot_ord" states the following: 1. The function "is_pot_ord_pprod" when applied to the lexicographic order (LEX) of a pair of natural numbers and a natural number, returns false. 2. The function "is_pot_ord_pprod" when applied to the degree reverse lexicographic order (DRLEX) of a pair of ...
The next element 'n' in the interval 'I' multiplied by 'k' is equal to: if 'n' modulo 'k' equals zero, then the next element of 'n' divided by 'k' in interval 'I' multiplied by 'k', otherwise it is 'n'.
"x" is less than or equal to the star of "y" multiplied by "x" and then multiplied by the star of "y".
The lemma named "cupcake_state_preserve" assumes that the evaluation of an expression 'e' in a certain environment 'env' and state 's' results in 'res', the environment 'env' is a cupcake environment, and the expression 'e' is a cupcake expression. It shows that the state 's' is equal to the first element of the result...
The lemma "FOREACHc_refines_FOREACHcd_aux" shows that "FOREACHc" with parameters s, c, f, and sigma is less than or equal to "FOREACHcd" with the same parameters s, c, f, and sigma.
The lemma named "FrameStatEqImpCompose" is given four fixed frames: F, G, H, and I. It assumes that frame F is statically equivalent to frame G, frame G is a subframe of frame H, and frame H is statically equivalent to frame I. Based on these assumptions, it shows that frame F is a subframe of frame I.
If 'y' is greater than or equal to negative one and less than or equal to one, then the arcsine of 'y' is greater than or equal to negative half pi and less than or equal to half pi, and the sine of the arcsine of 'y' is equal to 'y'.
The space of the limit stream for a given probability and element x is equal to the streams of the space M.
If "O!" applies to a property "x", then it necessarily follows that there does not exist a function "F" such that the pair "x, F" holds true in the universe "v".
The lemma named "sum_wt" assumes that "c" is proper. It shows that the sum of the weight "wt" of "c" in state "s" over all branches less than the branch "brn" of "c" equals 1.
The lemma is named "linorder_finite_Mapping_induct" and it consumes one assumption. It has two case names: "empty" and "update". It fixes a mapping 'm' of type 'a' and 'b', where 'a' is a linear order. It assumes that the keys of the mapping 'm' are finite. It also assumes that a property 'P' holds for an empty m...
The lemma named "eeqExcPID2_cong" assumes the following: 1. "paps" and "paps1" are equivalent except possibly at PID2. 2. If "pid" is equal to PID, then "uu" and "uu1" are equivalent except possibly at D2. 3. If "pid" is not equal to PID, then "uu" is equal to "uu1". Under these assumptions, it shows that "paps" with ...
The product of a multiplicative character "k" and the inverse of the same multiplicative character "k" equals the principal character of "G" at "k".
The lemma "heap_copies_known_addrs_ReadMem" assumes that a heap copy operation from address 'a' to address 'a'' with a list of addresses 'als' transforms heap 'h' to heap 'h'' with a set of observations 'obs'. If a read memory operation with address 'ad', address list 'al', and value 'v' is in the set of observations, ...
A set of formulas Γ entails a contradiction if and only if Γ is not satisfiable.
The lemma named "mem_affine_3" assumes that "S" is an affine set, and "x", "y", and "z" are elements in "S". It also assumes that the sum of "u", "v", and "w" equals 1. Under these assumptions, it shows that the result of "u" times "x" plus "v" times "y" plus "w" times "z" is also an element in "S".
For any positive integer n, the nth root of the absolute value of x is equal to the absolute value of the nth root of x.
If a function 'f' converges to zero for a filter 'F' and if it is eventually true that 'f' is greater than zero for 'F', then the limit of the inverse of 'f' as 'x' approaches 'F' is positive infinity.
The lemma named "bounded_depth_finite" assumes the following conditions: 1. The set F is finite. 2. The union of the image of the function "funas_term" applied to each element in set S is a subset of F. 3. For every element t in set S, the depth of t is less than or equal to n. 4. For every element t in set S, t is a ...
If there are homeomorphic maps 'f' and 'g' between two topological spaces 'X' and 'Y', and the image of 'f' mapped from 'S' is a subset of 'T', and the image of 'g' mapped from 'T' is a subset of 'S', then there exist group isomorphisms between the relative homology groups of 'X' and 'S', and 'Y' and 'T'. These group i...
: Assuming that: - for every element l in set C, the absolute difference between function f at l and function g at l is less than or equal to x, - for every pair of elements l and m in set C, the absolute difference between function f at l and function f at m is less than or equal to y, - for every pair of elements l ...
The inverse image of the intersection of two sets A and B under a function f is equal to the intersection of the inverse images of the sets A and B under the same function.
A list containing only one element "x" is considered sorted with respect to any property "P".
The signed cast of a signed cast of a word 'w' is equal to the word 'w' itself, given the condition that the signed cast is an upward cast.
The normalization of a first-order formula with respect to a list of variables "vs" is equal to the normalization of the same first-order formula with respect to a different list of variables "vs'" if and only if the list of variables "vs" agrees with the list of variables "vs'" according to the agreement relation defi...
The lemma named "insertAssertionExtractFrameFreshImp" is defined with respect to a list of names "xvec", a type 'b "\Psi", and a psi type P. It assumes that "xvec" is fresh for "\Psi". It shows that the insertion of an assertion into the extraction of a frame from a restricted "xvec" in P is less than or equal to the r...
If the set V is empty, then for any element x in the unit_table of size n, x is a well-formed tuple of size n over the set V.
If the successor of 0 (which is 1) is not less than or equal to x, then x equals 0.
For any natural numbers 'a' and 'b' where 'a' is less than or equal to 'b', the modulus of the difference of 'b' and 'a' with respect to 'm' is equal to the modulus of the expression '(m plus the modulus of 'b' with respect to 'm' minus the modulus of 'a' with respect to 'm')' with respect to 'm'.
The lemma named "permutation_funpow" assumes that "p" is a permutation. It shows that the nth power of "p" is also a permutation.
If a heap 'h' is known to the Document Class, then it is also known to the Shadow Root Class.
The lemma named "pos_assign_poly" assumes a positive assignment "alpha" and a polynomial "p" that is greater than or equal to zero. It shows that the evaluation of the polynomial "p" with the assignment "alpha" is also greater than or equal to zero.
"x is less than or equal to the difference of z and y" if and only if the following two conditions are met: 1) If y is infinity, then z must not be infinity and x must be negative infinity. 2) If the absolute value of y is not infinity, then the sum of x and y must be less than or equal to z.
The lemma named "connectedin_nonseparated_union" assumes that both S and T are connected within the set X and that S and T are not separated within X. It shows that the union of S and T is also connected within X.
The block identifier of NULL is 0.
If a polynomial "p" is in the carrier set "P" and the degree of "p" is less than a certain value "k", then the coefficient of the term of "p" with degree "k" is zero.
A sequence (a@b) is a prefix of D if and only if sequence a is a prefix of D and sequence b is a prefix of the sequence obtained by dropping the first (length of a) elements from D.
"null" is not equal to "true", which is always "false".
The lemma "longest_prefix_gr_0" assumes "case_two j". It shows that the longest prefix of j is greater than 0.
The lemma "r_0_distinguishable_from_not_completely_specified_initial" assumes that the product of the states q1 and q2 from the finite state machine M is not a completely specified state. It also assumes that both q1 and q2 are states in the finite state machine M. Under these assumptions, it shows that the states q1 a...