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The weight of an empty graph is zero. |
If the evaluation of expressions e1, e2, and e3 are equal to the evaluation of expressions e1', e2', and e3' respectively, then the evaluation of the "if" expression with e1, e2, and e3 is equal to the evaluation of the "if" expression with e1', e2', and e3'. |
If H and N are submodules of a module M over a ring R, and H is a subset of N, then the quotient of the quotient of M by H by the quotient of N by H is isomorphic (in the ring R) to the quotient of M by N. |
If there is a reflexive transitive closure from r to s, then there is also a reflexive transitive closure from the result of eliminating zeros from r to the result of eliminating zeros from s. |
The lemma named "pickT_pred_pickT" assumes that "II" is an element of the union of partitions of "cl" and "P". It shows that the function "pickT_pred" applied to "cl", "P", "II", and the result of the function "pickT" applied to "cl" and "P" and "II" is true. |
If function f is parallel to function g, then function g is also parallel to function f. |
The reverse of the concatenation of sets A and B is equal to the concatenation of the reverse of set B and the reverse of set A. |
Given the assumptions that q\<^sub>c, q\<^sub>i, and q\<^sub>f are not elements of \<Q> \<A> and q\<^sub>c and q\<^sub>f are not elements of Q, and that q\<^sub>c, q\<^sub>i, and q\<^sub>f are all distinct, it shows that the language of the generalized tree automaton with the final state q\<^sub>f and the automaton gen... |
The lemma named "nominal_in_names" assumes that "A" is a Hintikka set and there exists a block in "H" such that "i" is in the nominals of that block. It shows that "i" is in the names of "H". |
If the program counter is greater than or equal to the length of the compiled expression, then there is no tau move in the second position for the given program, heap, stack, expression, program counter, and exception. |
: executing a while loop with the condition "FOREACH_cond c" and the body "FOREACH_body f" on a list "l" and a state "σ", then returning the state "σ", is less than or equal to applying the function "nfoldli" with the list "l", the condition "c", the function "f", and the state "σ". |
The negation of the polynomial constructed with coefficient 'a' and polynomial 'p' is equal to the polynomial constructed with the negation of 'a' and the negation of 'p'. |
The integral of a function (f) multiplied by a constant (c) over a quasi-Borel space (s) is equal to the constant (c) multiplied by the integral of the function (f) over the same quasi-Borel space (s). |
If x is a zero vector, then x is also a dense relation. |
If function 'f' preserves the measure from set M to N, and function 'g' preserves the measure from set N to P, then the composition of 'g' and 'f' (g o f) also preserves the measure from set M to P. |
If the execution move of a certain expression "e" in a given context "ci", program "P", thread "t", heap "h", and state "s" leads to a new heap "h'" and state "s'", then the same execution move can be applied when the expression "e" is part of a binary operation with another expression "e'". |
The permutation function "permutep" applied to the sum of "m1" and "m2" is equal to the sum of the permutation function applied to "m1" and the permutation function applied to "m2". |
The lemma named "null_space_eq_solution_set" demonstrates that the null space of a matrix A is equal to the solution set of the system of equations represented by matrix A with a zero vector. |
The standard simplex of dimension 'p' is connected in the power set topology of the universal set of real numbers. |
The sum of the multiset obtained by applying a function "f" to each element of a multiset "A" and then wrapping each result in a singleton multiset, is equal to the multiset obtained by directly applying the function "f" to each element of the multiset "A". |
The copair quasi-Borel measure space of X and Y is a closed set in the binary coproduct of the quasi-Borel spaces of X and Y. |
If the set delta is finite, and if Sigma is in the condition "br_cond", the invariant "br_invar" of delta, and the step "br_step" of delta from Sigma to Sigma', then Sigma' is also in the invariant "br_invar" of delta. |
If F and G are groups, and f is a group homomorphism from F to G, and S is a right coset of the kernel of f in F, and s1 and s2 are elements of S, then the image of s1 under f is equal to the image of s2 under f. |
The lemma named "mult_mm_cancel_l" assumes that the determinant of matrix M is not equal to zero and that the result of matrix M multiplied by matrix A is equal to the result of matrix M multiplied by matrix B. Under these assumptions, it shows that matrix A is equal to matrix B. |
For any event in Otway, for all keys K and sets of keys KK, if KK is a subset of the complement of the range of shared keys, then the key K is in the analysis of the union of the image of KK under the Key function and the knowledge of the Spy in the event if and only if K is in KK or the key K is in the analysis of the... |
If for every variable 'v', the coefficient of 'v' in polynomial 'p' is equal to the coefficient of 'v' in polynomial 'q', then polynomial 'p' is equal to polynomial 'q'. |
If both 'j' and 'i' are greater than or equal to zero, then the matrix 'A' moved by 'j' and 'i' is greater than or equal to zero if and only if the matrix 'A' itself is greater than or equal to a zero matrix. This is applicable for any matrix 'A' of a type 'a' that has order and zero properties. |
The lemma named "residue_primroot_is_generator" assumes that "m" is greater than 1 and "g" is a primitive root modulo "m". It shows that there is a bijective function between the set of all "i" such that "g" to the power of "i" modulo "m", and the set of totatives of "m", where the range of "i" is less than the totient... |
The lemma named "shuffle_to_top_Total" assumes that "Y" is an allocation. It then shows that the operation "shuffle_to_top" with parameters "Y" and "d" is total. |
If 'y' is greater than or equal to -1 and less than or equal to 1, then the arcsine of 'y' is greater than or equal to -pi/2 and less than or equal to pi. Furthermore, the sine of the arcsine of 'y' is equal to 'y'. |
Given two complex numbers 'u' and 'v' that are within the unit disc and are real numbers, and neither of them is zero, the sum of the Poincare distance from 'u' to the origin and the Poincare distance from the origin to 'v' is equal to the Poincare distance from 'u' to 'v' if and only if the product of the real parts o... |
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Assuming that the set "us" is a subset of the uses of graph "g" at node "n", the lookup of node "n" in the mapping "u" is None, and the result of folding function "aux_1 g n" over the set "us" with initial value "(u,p)" is "(u',p')". Also assuming that the keys of mapping "p" form a finite set, for any node "n" and ... |
If 'n' is equal to the length of the list 'xs', then the list interval assertion of the set from 0 to 'n' for 'A', 'xs', and 'xsi' multiplied by 'F' is equal to the list assertion for 'A', 'xs', and 'xsi' multiplied by 'F'. |
The lemma named "injectionPowerset" assumes that the function 'f' is an injection on the set 'Y' and that the set 'X' is a subset of 'Y'. It shows that the image of 'f' is an injection on the power set of 'X'. |
:
Assuming the following conditions:
- For any initial state 's0', if 's0' is an initial state of event system 'E', then 'h s0' is an initial state of event system 'F'.
- For any states 's' and 's'' and any action 'a', if 's' transitions to 's'' in event system 'E' via action 'a', and 's' satisfies invariant 'I' and '... |
The lemma named "d_reaches_unique_obtain" assumes that there is a deterministic path in machine M from its initial state to state q following a sequence of inputs xs. It shows that the set of all states p that can be reached from the initial state of M following the sequence xs is exactly the set containing only state ... |
The lemma l8_14_3 assumes that point X is perpendicular at line segment AB to line segment CD and point Y is also perpendicular at line segment AB to line segment CD. It shows that point X is equal to point Y. |
The lemma "convert_list2" assumes that the function "convert_atom_list" with parameters "var", "L", and a list consisting of "xs'", "x", and "xs" equals "Some L'". It also assumes that "l'" is an element of the set "L'". Given these assumptions, it shows that there exists an element "l" in the set "L" such that the fun... |
If the domain block of the list consisting of the element x alone is greater than zero, or if the domain block of the list y is greater than zero, then the domain block of the list formed by appending y to the list consisting of the element x alone is also greater than zero. |
The derivative of the formal power series 'a' at the integer 'b' is equal to zero. |
In the context of polynomial rings, the degree of the polynomial X with respect to the ring R and the set S is 1. |
The lemma named "hom_componentsE" assumes that "q" is an element of the homomorphous components of "p". It then concludes that there exists a "t" such that "t" is an element of the keys of "p" and "q" is the homomorphous component of "p" with the degree of the polynomial map "t". |
If d is an element of D and u is less than or equal to v, then the function R applied to d and u is less than or equal to the function R applied to d and v. |
The composition of 'x' and 'y' is less than or equal to the composition of 'x' and the supremum of 'y' and 'z'. |
The lemma named "countableE_infinite" assumes that a set "S" is countable and infinite. It then obtains a function "e" from any type "'a" to natural numbers, where "e" is a bijective function between the set "S" and the set of all natural numbers. |
The function "worklist_aux" with parameters "succs", "f", and a pair "(ws, s)" is defined as follows:
If "ws" is an empty list, then the function returns a pair consisting of an empty list and "s".
If "ws" is a list with "x" as the first element and "ws'" as the rest of the list, then the function calls itself recu... |
If a term 't' is pure and for all 'i', the substitution of 'i' in sigma prime is also pure, and if sigma double prime is defined as the substitution of sigma in sigma prime of 'i', then the substitution of sigma in the substitution of sigma prime in 't' is equal to the substitution of sigma double prime in 't'. |
The function "real_of_int" tends to infinity at the top filter. |
Lemma: holomorphic_on_union_open
Assumptions:
1. For every n in I, the function f is holomorphic on A n.
2. For every n in I, the set A n is open.
Conclusion:
The function f is holomorphic on the union of A n for all n in I. |
If a new constant "c" is not in "z", and "d" is not in the image of function "f" applied to the union of parameters of elements in the set "z", then "d" is new in the list obtained by mapping the function "psubst" (which substitutes "c" with "d" in "f") over "z". |
The lemma named "simulation_language" makes the following assumptions:
1. The alphabet of automaton A is a subset of the alphabet of automaton B.
2. For any state 'p' in the initial states of automaton A, there exists a state 'q' in the initial states of automaton B such that the pair (p, q) is in the relation R.
3. F... |
"A" is weakly contingent in the universe "v". |
The lemma named "fresh_fun_simp_NotL" assumes that "x'" is fresh for "P" and "M". It shows that the freshness function applied to the function "x'" mapped to "Cut <c>.P (x').NotL <a>.M x'" is equal to "Cut <c>.P (x').NotL <a>.M x'". |
The hash of X is not equal to the ephemeral key of X'. |
If a position "p @ q" is in "t", then the position "p" is also in "t". |
If a well-formed substitution weight (wlsFSw) is a module (MOD), then the term formed by the substitution weight morphism (termFSwMorph) of the recursion (rec) and the recursion of the abstraction (recAbs) is also a module. |
If an arc 'a' is not in the set of arcs 'G', then applying the function 'face_cycle_pred' to 'a' 'n' times will result in 'a'. |
If a pair (s, t) is in the relation R23_msgs and the same pair (s, t) is in the relation R23_ink, and t is in the m3_inv1_lkeysec, then the abstract message of the synthesized analysis of the intruder knowledge of t is a subset of the fake initial knowledge domain of the runs of s and the channel of s. |
The transitive closure of the product of 'y' and the transitive closure of 'x' is equal to the transitive closure of 'y' union the product of the transitive closure of 'y', 'y', 'x', the transitive closure of 'x', and the transitive closure of the product of 'y' and the transitive closure of 'x'. |
If there is a sequence from 't' to 'u' (as assumed), and if 't' is an arrow, 'u' is an arrow, and the target of 't' is the source of 'u', then 'T' is implied. The lemma concludes that 'T' is true. |
If the sum of the positive divisors of a number 'm' (denoted by Esigma m) is less than twice the number 'm', and 'm' is greater than or equal to 1, then 'm' is a deficient number. |
The union over the set I of the function f(i) unioned with the union over the set I of the function g(i) is equal to the union over the set I of the union of f(i) and g(i). |
The lemma named "s_antimono" assumes that "x" is less than or equal to "y" and "t_stop" is less than or equal to "x". It shows that "s of y" is less than or equal to "s of x". |
In the context of a valid simple undirected graph, if there is an adjacent path from a vertex 'v' through a list of vertices 'ps', then there is also an adjacent path from the same vertex 'v' through the list of vertices 'ps' excluding the last vertex. |
The entries of the mapping red-black tree "t1" are equal to the entries of "t2" if and only if the lookup of "t1" is equal to the lookup of "t2". |
If an expression 'e' is low and the states 's' and 't' are equivalent, then updating the first part of 's' and 't' with the evaluation of 'e' in their respective first parts, while keeping their second parts the same, will result in equivalent states. |
For all lists 'xs', evaluating 'VSLuckyBlocksLimited' with 'xs' is equal to evaluating 'phi' with 'xs'. |
The lemma named "equivalence_functor_G" demonstrates that "G" is an equivalence functor between the composition of hom' and the composition of hom. |
A property "P" holds for a list containing a single element "x". The "snoc" case states that for any element "x" and any non-empty list "xs", if the property "P" holds for the list "xs", then it also holds for the list "xs" appended with the element "x". The lemma shows that the property "P" holds for the list "xs". |
The lemma named "factorize_polys_square_free_prod_list" assumes that the factorization of a list of polynomials "ps" results in a pair "(fs, data)". It then shows that the product of the list "fs" is square-free. |
The lemma named "convert_substInfinitesimalQuadratic_less" makes several assumptions and shows a result. The assumptions are:
1. The conversion of a polynomial 'p' with respect to a variable 'var' and a list of elements 'xs' appended with 'x' and 'xs' gives 'p''.
2. The insertion of 'a' into the list 'xs' appended wit... |
"for all x, the codomain of the domain of x is isomorphic to the domain of x." |
If function 'f' is in the set of scalar multiplication functions 'smultfunset' for a group 'G' and a set 'FH', and if 'g' is an element of 'G' and 'x' is an element of 'FH', then the function 'f' applied to the scalar multiplication of 'a' and 'g' with 'x' is equal to the scalar multiplication of 'a' and 'g' with the f... |
If a graph mapping "GG" exists from "BB" to "CC" and another graph mapping "FF" exists from "AA" to "BB" in the semicategory of graphs "smc_GRPH" parameterized by "alpha", then the composition of "GG" and "FF" in the semicategory "smc_GRPH" is equal to the composition of "GG" and "FF" in the digraph homomorphism. |
The lemma named "ty_cases" is defined with a fixed type "T". It shows that "T" can be one of the following cases: "T" equals "PR s" for some "s", "T" equals "NOT T'" for some "T'", "T" equals "S OR U" for some "S" and "U", "T" equals "S AND U" for some "S" and "U", or "T" equals "S IMP U" for some "S" and "U". |
The lemma named "semIntAll_prCons" assumes that "SEM" is a well-sorted semantic domain. It shows that "prCons" (probably a constructor for a certain type of objects) applied to the interpretation of "SEM" in the semantic domain, the interpretation of "SEM" in the abstract semantic domain, and "SEM" itself, holds true. |
If the pair (u,v) is in the edge set of graph G', then the pair (fi u, fi v) is in the edge set E. |
For any two extended real numbers 'a' and 'b', negative infinity is equal to the sum of 'a' and 'b' if and only if either 'a' is negative infinity and 'b' is not positive infinity, or 'b' is negative infinity and 'a' is not positive infinity. |
If an atom 'x' is fresh in terms 't' and 's', then the atom 'x' is also fresh in the term resulting from substituting 's' for 'y' in 't'. |
The composition of "ov" and "d" is a subset of the union of "s", "ov", and "d". |
The lemma "proj_same_not_active" assumes that "n" is less than or equal to "n'", the extended natural number of "n'-1" is less than the length of "t", and the projection of the first "n'" elements of "t" is equal to the projection of the first "n" elements of "t". It shows that there does not exist a "k" such that "k" ... |
If 'l' is less than 'm', then the jump function applied to 'l' and the concatenation of the comparison of 'm' and 'e' with 'cs' is equal to the jump function applied to 'l' and 'cs'. |
The lemma named "add_row_to_multiple_det" assumes that "l" is not in the set "ks", "l" is less than "n", and "A" is an n by n matrix. It shows that the determinant of the matrix resulting from adding a multiple of row "l" to each row in the set "ks" in matrix "A" is equal to the determinant of matrix "A". |
There are no irrelevant channels at level 2 for the data sets "data1" and "data12". |
The m-th power of the shifted formal Puiseux series of x by n is equal to the shifted formal Puiseux series of x to the m-th power by the product of m and n. |
If a sort "s" exists in the context "cs", then a normalized sort of "s" in the context "cs" also exists. |
The set "W of P and k" is a subset of the set "X of k". |
The lemma named "compare_expansions_LT_I" assumes that the difference between "e1" and "e2" is less than zero. It then shows that comparing the expansions of two multiseries, where the first multiseries is constructed with the coefficient "C1" and exponent "e1" and the rest of the series "xs", and the function "f", and... |
If the first element of the pair 'i' belongs to the set that includes the load store types LDSB, LDUB, LDUBA, LDUH, LD, LDA, and LDD, then the second element of the pair resulting from the function 'load_instr' applied to 'i' and 's' is False. |
The lemma "fv_delete_heap" assumes that the mapping of x in the context Γ is equal to some expression e. It shows that the set of free variables in the pair (delete x Γ, e) union with the set containing x is a subset of the set of free variables in the pair (Γ, Var x), where the latter set is a set of variables. |
The lemma "adapt_aexp_is_subst" assumes that the symbol substitution sigma is consistent with the store s. It shows that adapting the arithmetic expression e to the store s and then applying the symbol substitution sigma is equivalent to applying sigma directly to e. |
If "x" is an element of the carrier of the first power of the ring "R", then "x" is equal to the head of "x". |
If the function "g" applied to "None" returns a probability mass function of "None", then binding a probability mass function to the result of binding a sub-probability mass function "p" with a function "f" and "g" is equal to binding the sub-probability mass function "p" with a lambda function that binds the result of... |
Lifting any value 'x' by 0 results in 0. |
If an effect is observed when a value 'x' is assigned the result of expression 't' and then function 'f' is applied to 'x', with the initial heap state being 'h' and the final heap state being 'h'', and the result of the effect being 'r', then it must also be true that the same effect is observed when function 'f' is d... |
The states of P with maximum stack size (mxs) and maximum number of local variables (mxl) is equal to the first element of the sequence list (sl) of P with mxs and mxl. |
The lemma named "prod_fun_nat_0" establishes that for any two functions 'f' and 'g' that map from a function returning natural numbers to a semiring 'b', the product function of 'f' and 'g' at 0 is equal to the product of 'f' at 0 and 'g' at 0. |
If a triple consisting of b', a, and f is mapped to another triple consisting of b'', a', and f' in the comma category of b and the functor, and if b is an object in the category B, then the following conditions hold:
1. b' equals 0.
2. b'' equals 0.
3. h equals 0.
4. g is a morphism from a to a' in the category A.
5.... |
If the intersection of sets I and J is empty, then the result of merging sets I and J with the pair (x, y) is the same as merging sets J and I with the pair (y, x). |
The lemma named p_LIMSEQ:
Assuming x is greater than 0,
it shows that the sequence p(n, x) converges to P(x) in the limit as n approaches infinity. |
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