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The lemma named "on_circline_unit_circle" shows that a point is on the unit circle if and only if the complex modulus of that point is equal to 1.
Marking guards with a function 'f' in a 'whileAnno' loop with condition 'b', invariant 'I', variant 'V', and command 'c', is equivalent to a 'whileAnno' loop with the same condition, invariant, and variant, but with the command 'c' having its guards marked by the function 'f'.
The recursive disjunction function is a primitive recursive function when its argument is the successor of the successor of zero (which is essentially 2).
In the context of a probability space, the lemma named "integrable_iexp" assumes that a function 'f' is Borel measurable and the imaginary part of 'f' for any 'x' is zero. Under these assumptions, it shows that the function 'f' is integrable with respect to the measure 'M' and the function 'exp(i * f(x))'.
If a class P C is defined as having some base classes Bs, fields fs, and methods ms, then the set of base classes Bs is finite.
If a traffic scenario 'ts' transitions to another traffic scenario 'ts' through a non-collision event 'd' that is not equal to 'c', then the restriction of the traffic scenario 'ts' to 'c' is equal to the restriction of the traffic scenario 'ts' to 'c'.
"eventually always F is equivalent to always F".
The lemma named "total_linorder_matrix_1" is defined with a fixed square matrix 'f'. This matrix 'f' is of a type that is finite and expands a linear order stone relation algebra. Assuming that the matrix 'f' satisfies the property of being a total variable in the matrix stone relation algebra, it is shown that there e...
The set of all numbers greater than or equal to the successor of a number 'n' is equal to the set of all numbers strictly greater than 'n'.
If 'f' is a ring homomorphism from ring R to ring S, then the function that maps 'a' to the list containing 'f a' is a ring homomorphism from ring R to the direct product of the list containing ring S.
For any B, if the signature S is applied to the expression 'e' with variable 'x' of type 'A' resulting in type 'B', then both 'A' and 'B' must be in the base types of the signature 'S'.
The lemma named "fds_abs_converges_comparison_test" is defined for a variable 's' of type 'dirichlet_series'. It assumes that for sufficiently large 'n', the norm of the nth term of the function 'f' is less than or equal to the nth term of the function 'g'. It also assumes that the function 'g' converges at the point '...
The lemma "slice_kind_only_one_True_edge" assumes that the source node of edge 'a' is equal to the source node of edge 'a'', but the target node of edge 'a' is not equal to the target node of edge 'a''. It also assumes that both edges 'a' and 'a'' are valid, and the slice kind of 'S' on edge 'a' is always true. Under t...
"not ECQ and Frame 1 and Frame 2 and Frame 4 and not XCoP".
The lemma "alg_path_equiv_implies_valid" shows that if the context Γ proves that s is equivalent to t of type T, then Γ is valid. And if the context Γ proves that s is bi-directionally equivalent to t of type T, then Γ is also valid.
The number 3844 raised to the power of 256 is equivalent to 1 in the finite field of 7681 elements under modular arithmetic.
The predicate "before_C B" is measurable in the probability space S.
The lemma named "diamond_imp_CR'" assumes that "s" is a diamond property, "r" is a subset of "s", and "s" is a subset of the reflexive transitive closure of "r". It shows that "r" is confluent.
The outer measure function 'emeasure' for a measure space 'M' is additive over the sets in 'M'.
The existence of a value 'x' such that both properties 'P' and 'Q' hold is equivalent to the existence of a value 'x' for which property 'P' holds and property 'Q' holds.
A synchronous heard-of run (SHORun) with parameters A, rho, HOs, and SHOs is equivalent to the initial configuration of the heard-of model (HOinitConfig) with parameters A and rho at time 0, and for all rounds r, the next configuration of the synchronous heard-of model (SHOnextConfig) with parameters A, r, rho at round...
Configuration 'cf' can transition to itself via zero or more confidential transitions.
A function is measurable in the domain D if and only if it is measurable in the stream space of the product measure of all countable sets, where the product measure is taken over all elements in the universal set.
The lemma named "mult_eq_is_mul_eq" assumes that "r" is transitive. It shows that "mult_eq r" equals "mul_eq r".
For a given relation "TRel" of type set of pairs of 'procT', if the induced relation "indRelTEQ TRel" forms a weak barbed bisimulation with the source and target weak barb sets "STCalWB SWB TWB", then the transitive closure of "TRel" also forms a weak barbed bisimulation with the target weak barb set "TWB".
The first sequence is sequential.
The lemma named "valid_path_aux_split" assumes that a valid auxiliary path exists for a sequence 'cs' followed by the concatenation of sequences 'as' and 'as''. It then shows that a valid auxiliary path exists for the sequence 'cs' followed by 'as', and a valid auxiliary path exists for the sequence updated by 'as' in ...
The product of negative p and p equals zero.
The lemma L10_Controller: assumes that there is a controller L on a stream s, a finite or infinite list l, and a list z, such that the finite or infinite list is appended with zero. It also assumes that the last element of the untimed version of the truncated list z at index i is not zero. It assumes that the las...
: assuming for any 'm', if the lookup of 'm' in 'Poly_Mapping' is greater than 'n', then the coefficient of 'p' at 'm' in 'MPoly_Type' is zero. Under this assumption, it shows that the degree of 'p' at 'i' in 'MPoly_Type' is less than or equal to 'n'.
If X is not in the synthetic analysis of H, then the presence of the hash of the pair {X,Y} in the synthetic analysis of H is equivalent to the presence of the hash of the pair {X,Y} in the analysis of H.
If 'f' is an equivalence map, then the following conditions hold: 1. The quasi-inverse of 'f' is also an equivalence map. 2. The quasi-inverse of 'f' is an identity. 3. The source of the quasi-inverse of 'f' is the target of 'f'. 4. The target of the quasi-inverse of 'f' is the source of 'f'. 5. The domain of the quasi...
If 'n' is less than or equal to the length of the arrays 'xs' and 'xs'', then the first 'n' elements of the array resulting from copying 'xs' into 'xs'' are the same as the first 'n' elements of 'xs'.
The lemma "types_preserved_get_local" assumes that: 1. Under the context of "\<S>\<bullet>\<C>", the instruction "$Get_local i" transforms the type sequence "ts" into "ts'". 2. The length of the sequence "vi" is equal to "i". 3. The local context of "\<C>" is the type of the concatenation of the sequences "vi", "[v]", ...
The lemma named "bell_11_index" shows that the value of the Bell state "beta_11" at index (0,0) is 0, at index (1,0) is 1 divided by the square root of 2, at index (2,0) is negative 1 divided by the square root of 2, and at index (3,0) is 0.
"False" is not in the language of abSSa.
The inner product of vector x and the difference of vectors y and z is equal to the difference of the inner product of x and y and the inner product of x and z.
A key 'k' is not in the keys of function 'f' if and only if looking up 'k' in function 'f' equals zero.
If vertex 'v' is not an element of the vertices of graph 'G', then the out-degree of vertex 'v' in graph 'G' is 0.
The difference operation "-" on matrices is preserved under the matrix correspondence "HMA_M". This is a transfer rule, which means it is used to transfer properties between abstract and concrete representations.
The equality of predicates x and y is symmetric, meaning if x is equal to y then y is equal to x.
If a vertex-port pair (v,p) is valid and another vertex v' is in the set of vertices but not in the scope of (v,p), then the intersection of the set of local variables of the node of v (after being freshened by the vertex index of v) and the set of substituted local constants of the instance of v' is empty.
The norm (or length) of the i-th component of a vector x is less than or equal to the norm of the vector x itself, given that i is within the basis.
If a pair (S, U) is in the set "ChiPlus A", and another pair (U, T) is in the set "ChiRel A", and if S is in the set of states of SA, and SA is in the set of SAs of A, then T is not in the set of states of SA.
The lemma named "ex_Loop_stream_enc" assumes that for every element 'x' in the set 'I', if 'x' is of the form 'Inr P', then 'P' is finite, otherwise it's true. Under this assumption, it shows that there exists a number 'n' such that if you drop the first 'n' elements from the stream encoded by '(w, I)', you get a const...
The theorem of checker soundness assumes that the effect of the checker function with parameters n, p, and i on h results in h' and cs. It shows that cs is inconsistent.
For a given matrix relation (MM_Rel), if the indices i and j are less than the number of rows and columns of the matrix M respectively, then the element at the position (i, j) in the matrix M is returned. If the indices i and j are not less than the number of rows and columns of the matrix M respectively, then 0 is ret...
The lemma named "src_eqI" assumes that "a" is an identity and "a" is concatenated with "t". It then shows that the source of "t" is equal to "a".
The supremum (greatest value) up to a certain point in a lazy list "Xs" at the successor of a given point "j" is equal to: - The supremum up to the lazy list "Xs" at the successor of a natural number "k" if "j" is equal to the extended natural number "k". - The supremum of the lazy list "Xs" if "j" is infinity.
If "A" entails "B", then "A" entails either "B" or "C".
The lemma named "converge_trans_2" assumes that a machine "M" is observable and minimal, and that "M" converges for inputs "u" and "v". It shows that the convergence of "M" for the combined inputs "u" and "w1", and "u" and "w2", is equal to the convergence of "M" for the combined inputs "v" and "w1", and "v" and "w2". ...
The factorial of an integer 'n' remains the same even when it is converted to another type.
For any quasi-term 'X', any quasi-abstraction 'A', any variables 'x', 'y', 'x'', 'y''', and any variable sorts 'zs', 'zs'', the operation of swapping 'x' and 'y' in 'X' or 'A' under the sort 'zs', and then swapping 'x'' and 'y'' under the sort 'zs'', is equivalent to first swapping 'x'' and 'y'' under the sort 'zs'', a...
If command 'c' does not read from variable 'x', then the command 'c' combined with annotations does not read from variable 'x' either.
Given an Observable Finite State Machine (OFSM) M, if the successor of the product of the number of nodes in M and a certain number m is less than or equal to the length of a list xs, and a list formed by appending xs to another list vs is in the language of M, then there exists a node q in M such that the cardinality ...
: Assuming that for any elements x2, u, and v, if x2 is an element of the set of lists xss, and both u and v are elements of x2, and u is an element of the languages M1 and M2, then u and v converge in both M1 and M2. Furthermore, if x2 is an element of the set of the second element of the pair resulting from the fun...
The lemma named 'seq'' assumes that a rule set 'rs' is the concatenation of two other rule sets 'rs1' and 'rs2'. It also assumes that under the context 'Γ', packet 'p', and state 's', 'rs1' leads to state 't', and under the same context and packet, 'rs2' leads from state 't' to state 't''. Given these assumptions, it s...
The lemma "terminates_splice_raw" assumes that both "g" and "h" terminate. It shows that the raw splice of "g" and "h" also terminates.
In the context of map_size_abort, there is a lemma for size_abort_autoref that is part of the autoref_rules. If the preference is given to the identifier Rk, then the pair (size_abort, op_map_size_abort) is in the relation from the identifier to the pair (Rk, Rv), and this is also an identifier.
If a natural transformation "MM" maps a semicategory "GG" to a semicategory "HH" from "AA" to "BB" in a context "alpha", and another natural transformation "NN" maps a semicategory "FF" to "GG" from "AA" to "BB" in the same context "alpha", then the range of the vertical composition of "MM" and "NN" is a subset of the ...
If the following conditions are met: 1. The function "pres_type" with parameters "step", "n", and "A" holds true (as per the assumption "pres"). 2. The element at the position "pc+1" in the list "c" is in the set "A" (as per the assumption "cert"). 3. The element "s" is in the set "A" and "pc" is less than "n" (as per...
The lemma named "finite_input" is defined with fixed variables t, n, and nid. It assumes that nid is parallel to t n. It also defines "dep nid'" as the output of the function sigma applied to nid' and t n. The lemma then shows that the input of the composition of nid and t n is finite.
If a variable 'x' is not in the boolean expression 'e', then the 'While' loop with expression 'e' and command 'c', along with its annotations, does not read or modify 'x'.
The lemma named "in_set_upd_eq_aux" assumes that "i" is less than the length of list "l". It shows that "x" is in the set where list "l" at index "i" is replaced with "y" if and only if "x" equals "y" or for all "y", "x" is in the set where list "l" at index "i" is replaced with "y".
For all variables 'v' in the set 'vs', if 'v' is less than 'i', then the distribution of variables in 'vs' and 's' with 't' substituted for 'i' is equal to the distribution of variables in 'vs' and the substitution of 't' for 'i' in 's'.
The length of the iteration of function 'f' from 'm' to 'n' on 'x' is equal to 'n' minus 'm'.
For a formal Laurent series 'f' raised to the power of 'n', the coefficient of the term with the exponent 'n * fls_subdegree f' is equal to the coefficient of the term with the exponent 'fls_subdegree f' in the original series 'f', raised to the power of 'n'.
The lemma named "reachable_transitions\<^sub>C_eq" assumes that the set of all states, denoted as "\<Sigma>", is equal to the set "\<Sigma>'". Under this assumption, it shows that the set of reachable transitions "\<Sigma>'" under a formula "\<phi>" is equal to the set of reachable transitions "\<Sigma>" under the tran...
The multivariate polynomial 'p' is symmetric with respect to an empty set.
The lemma named "infer_v_wf" is about a fixed variable "v". It assumes that under the contexts of type variable set "Theta", base type set "B", and environment "G", the variable "v" is inferred to have the type "tau". Under this assumption, it shows that the variable "v" is well-formed with the base type of "tau" under...
, given a quotient set defined by the equivalence relation R and the abstraction and representation functions Abs and Rep respectively, the function LCons (which constructs a list) preserves this quotient structure. This is shown by demonstrating that applying Rep to the elements of the list, then applying LCons, and f...
If a function 'f' is injective (meaning it never maps distinct elements to the same element), then mapping 'f' over a list 'xs' with all occurrences of an element 'x' removed is the same as removing all occurrences of 'f(x)' from the list obtained by mapping 'f' over 'xs'.
If n is greater than 0, then the left recursive function of u, v, lc, rc, and the expression ((4 * real m + 3) / 2 ^ (n+1)) is equal to the right recursive function of the left recursive function and the right recursive function of the same parameters and the expression ((2 * real m + 1) / 2^n), and the midpoint of the...
"b times the right residual of a and c" is equal to "the right residual of a and the right residual of b and c".
The condition where a pair (h1, h2) is related to h0 and it is not the case that the operation (A ** B) on h0 holds, is equivalent to either of the following two conditions: 1. The pair (h1, h2) is related to h0 and it is not the case that either the operation A on h1 holds or the operation (A ** B) on h0 holds. 2. T...
The lemma named "merge_length" assumes that the length of list "a" is "n" and the length of list "b" is also "n". It then shows that the length of the result of merging lists "a" and "b" is "n".
The lemma named "uminus_of_complex" shows that the homogeneous coordinate of the negation of a complex number z is equal to the negation of the homogeneous coordinate of z.
If there are two messages in the set of events 'evs' which are part of the Kerberos V protocol, where both messages are sent by the Key Authentication Server (Kas) to an agent A and A' respectively, and both messages contain a timestamp T encrypted with the shared key of the Ticket Granting Server (Tgs), then the agent...
If a list 'xs' is sorted and contains distinct elements, and if 'xs' is the concatenation of list 'as', an element 'b', and list 'cs', then every element 'x' in list 'cs' is greater than 'b'.
If the inorder traversal of a tree 't' is sorted, then the inorder traversal of the tree after deleting an element 'x' is equal to the result of deleting 'x' from the inorder traversal of the original tree 't'.
If it is proven under hypothesis H that set z is disjoint from set x, and if inserting the condition "y is an element of z" into H leads to a contradiction (false), then under hypothesis H, it can be proven that set z is disjoint from the set formed by adding element y to set x.
If we have two categories, denoted as B and C, in a universe α, and if every object in B is also an object in C, and if for every pair of objects a and b, every morphism from a to b in B is also a morphism from a to b in C, and if for every pair of morphisms f and g in B, the composition of g and f in B is the same as ...
For any real normed field or Banach space 'a', given a fixed 'K' and 'x' of type 'a', if for any 'y', the series of 'c n * y^n' is summable, then the function defined as the sum from 'n' of 'c n * x^n' has a field derivative at 'x' which is equal to the sum from 'n' of 'diffs c n * x^n'.
If there is a τ-reduction from expression "e" with state "xs" to expression "e'" with state "xs'" in a given thread "t" and heap "h" with respect to a program "P" and an external action "extTA", then there is also a τ-reduction from the conditional expression "if e then e1 else e2" with state "xs" to the conditional ex...
The lemma named "replace_in_g_dist" assumes that "x'" is not equal to "x". It shows that replacing "x'" with "c''" in the context "G" that starts with the tuple "(x, b, c)" is the same as starting with the tuple "(x, b, c)" and then replacing "x'" with "c''" in the context "G".
The lemma named "bounded_by_update" assumes that a list "xs" is bounded by "vs". It also assumes, under the condition named "bnd", that the "i"-th element of "xs" is in the interval "ivl". It shows that "xs" is still bounded by "vs" even after the "i"-th element of "vs" is replaced with the interval "ivl".
The lower limit of the bottom element under function f is the top element.
The hyperreal sign function of 1 equals 1.
The incorrectness relation hoare triple P, c, c', Q is equivalent to the conjunction of the incorrectness relation hoare triple P, c, SKIP, decomposed by P, c, c', Q and the incorrectness relation hoare triple decomposed by P, c, c', Q, SKIP, c', Q.
: 1. The evaluation of the first-order relation "r1" under the interpretation function "eval_rr1_rel" and the set of relations "Rs" in the structure "F" is a subset of the set of terms in the structure "F". 2. The evaluation of the first-order relation "r2" under the interpretation function "eval_rr2_rel" and the set o...
If the evaluation of a certain condition "exists bp. gamma p" is true for a given context identifier "cid", time "t", new time "t'", and natural number "n", then there exists a "p" such that the evaluation of "gamma p" is also true for the same "cid", "t", "t'", and "n".
If n is greater than 0 and less than N, and x is greater than 0 and less than 1, then the nth root of x is less than the Nth root of x.
The product of "n of x", "n of y", and "n of x" is equal to the product of "n of y" and "n of x".
The function "is_static" applied to a member scheme "m" of any type 'a is equal to the function "static" applied to "m".
The lemma named "closed_compactin" assumes that the set K is compact in the topological space X, the set C is a subset of K, and the set C is closed in the topological space X. Under these assumptions, it shows that the set C is also compact in the topological space X.
The final point of the path resulting from the composition of functions f and p is equal to the function f applied to the final point of path p.
The lemma named "eqvtI" is defined with the following parameters: P and Q are processes in the pi calculus, Rel is a set of pairs of pi calculus processes, and perm is a permutation of names. The lemma assumes the following: 1. There is a strong early simulation from process P to process Q with respect to the relati...
If "Iface I" is a type in the context G, then I is an interface in the context G.
The lemma named "midpoint_distinct_1" assumes that points A and B are not the same, and that I is the midpoint between A and B. It shows that I is not the same point as A and I is not the same point as B.
The composition of the function "mapl_F" with parameters l1, l2, l3 and the function "mapl_F" with parameters l1', l2', l3' is equal to the function "mapl_F" with parameters being the composition of l1 and l1', the composition of l2 and l2', and the composition of l3 and l3'.