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A discrete target category morphism is a target category morphism functor. This maps from I to the category C, in the context of an alpha. |
The lemma named "lemma_termdiff2" is defined as follows:
Let 'h' be a variable of any field type. Assuming 'h' is not equal to zero, it shows that the difference between the nth power of (z + h) and the nth power of z, divided by h, minus the product of the natural number equivalent of n and the (n - 1)th power of z, ... |
Given the assumptions that a context "\<C>" validates that an expression "e" has a type transition from "ts" to "ts'", and that the expression "e" is equal to "Br_table is i", it can be shown that there exist type sequences "ts_c" and "ts''" such that for all indices in the list "is@[i]", each index is less than the le... |
In the context of a weak lattice, the inverse of a given lattice order also forms a weak lattice. |
The lemma named "perp_right_comm" assumes that line segment AB is perpendicular to line segment CD. It shows that line segment AB is also perpendicular to line segment DC. |
If a function g converges to 0, and there exists a function f such that for infinitely many values of n, f(n) is non-negative and less than or equal to g(n), then f also converges to 0. |
If the function 'f' is in the carrier 'P', and 'a' is in the carrier 'R', and 'g' is defined as the root division of 'f' by 'a', and the function 'f' applied to 'a' equals zero, then 'f' is equal to the product of 'X' minus 'a' and 'g' in the context of 'P'. |
The lemma named "cos_sin_csqrt" assumes that the cosine of the real part of a complex number z is greater than 0, or the cosine of the real part of z is equal to 0 and the product of the imaginary part of z and the sine of the real part of z is less than or equal to 0. Under these assumptions, it shows that the cosine ... |
The set of all downsets is equal to the set of all upsets in the dual order. |
The lemma named "vcg_bb" states three conditions:
1. Updating the state index of 's' where 'x' is replaced by 'w', with 'x', 'i', and 'v' is equivalent to 's' where 'x' is replaced by the updated index of 'w' with 'i' and 'v'.
2. If 'x' is not equal to 'y', then updating the state index of 's' where 'x' is replaced b... |
In the context of a public non-secret event sequence, the knowledge of entity A combined with the private encryption key of B, the private signing key of B, and the shared key of B, is equivalent to the knowledge of entity B combined with the private encryption key of A, the private signing key of A, and the shared key... |
If a proposition Φ implies a proposition Ψ applied to an empty tuple, then assuming Φ is less than or equal to (in the context of refinement) the specification of Ψ. |
If a list "a" and a list "b" both satisfy the property "list_constr" with respect to a certain element "n", then the concatenated list "a@b" also satisfies the "list_constr" property with respect to the same element "n". |
If the ordered associative list "ys" is valid (as per the condition "oalist_inv ys") and the raw form of the ordered associative lists "xs" and "ys" are equal (as per the condition "oalist_eq_raw xs ys"), then the lookup function applied to the raw form of "xs" is equal to the lookup function applied to the raw form of... |
If an element 'x' is in the multiset sum of the function 'f' applied to all elements in the finite set 'Y', then there exists an element 'y' in the finite set 'Y' such that 'x' is in the multiset of the function 'f' applied to 'y'. |
The lemma pe_cond1:
Assumes that the time event of q at i+1 is less than or equal to the time event of p at i+1 (ie: "te q (i+1) <= te p (i+1)"),
and that p, q, and l are correct at the time event of p at i+1 (corr_p: "correct p (te p (i+1))", corr_q: "correct q (te p (i + 1))", and corr_l: "correct l (te p (i+1))").
I... |
The norm of x is less than the norm of y if and only if the inner product of x with itself is less than the inner product of y with itself. |
If m is equal to stat_inner_c, then m. |
The first element of the trace of a parallel plan in SAS+ is the initial state. |
In the context of a base partial semicategory, there is a lemma that can be introduced into the semicategory product case. This lemma assumes that alpha-prime is equal to alpha and I-prime is equal to I. It shows that the base partial semicategory is defined by alpha-prime, I-prime, and the category A. |
:
Assuming that the heap 'h' is well-formed, contains known pointers, and has a valid type. If 'parent' is an object in the heap and 'node' is a node in the heap, and 'parent' is not a character data pointer kind, and the cast of 'node' is not in the set of ancestors of 'parent', and the parent of 'ref' is 'parent'. I... |
The lemma named "rstate_down_up" shows that applying the down operation followed by the up operation to a state "s" results in the original state "s". |
If the atom j is not a part of term t, and the atom i is not a part of term u, and i is not equal to j, then substituting j with u in the term resulting from substituting i with t in term trm is the same as substituting i with t in the term resulting from substituting j with u in term trm. This lemma is marked as a sim... |
, given the axiom of equality EQU001_0_ax, if 'a' is equal to 'b' or 'c' is equal to 'd', and if 'a' is equal to 'c' or 'b' is equal to 'd', but 'a' is not equal to 'd' and 'b' is not equal to 'c', then this leads to a contradiction. |
The tag of X is not equal to the long-term key of X'. |
The lemma named "clauseFalseAppendValuation" is defined with three fixed elements: a clause of type Clause, and two valuations of type Valuation. It assumes that the clause is false under the given valuation. It then shows that the clause remains false when the valuation is appended with another valuation. |
The lemma named "bot_test" tests the bottom element. |
The bit length of the product of a positive number 'b' and 2 raised to the power of 'c' is equal to the sum of the bit length of 'b' and 'c'. This is under the condition that 'b' is greater than 0. |
For any three lists of names A, B, and C, A is fresh for the concatenation of B and C if and only if A is fresh for both B and C. |
If "braun" is a node with left child "l", value "a", and right child "r", then the set of elements in the tree resulting from the "sift_down" operation on "l", "a", and "r" is equal to the set containing "a" union with the set of elements in the left subtree "l" and the set of elements in the right subtree "r". |
The lemma demonstrates that the domain of the substitution heap, when an element y is substituted with z in the heap Gamma, is equal to the domain of the original heap Gamma. This is always true. |
The lemma named "floor_floor_div_eq" is defined with two fixed variables: x, which is a real number, and d, which is a natural number. It assumes that x is greater than or equal to 0. It shows that the floor of the natural number of the floor of x divided by the real number d is equal to the floor of x divided by the r... |
The outgoing capacity from the Source is equal to 1. |
The type of a value 'v' in a heap 'h' is 'Class C' if and only if there exists an address 'a' and a field 'fs' such that 'v' is equal to the address 'a' and 'h' at address 'a' is equal to the pair (Class C, fs). |
, assuming a mapping 'beta' exists from the equivalence class of 'alpha' in set S to set T, and that the composition of 'beta' and the class equals 'alpha', it can be shown that 'beta' is equal to the induced function. |
, given a polynomial 'p' that belongs to the set of polynomials 'P' with variable 'X', the focus of 'X' in 'p' is equal to the sum of the monomials, where each monomial is formed by looking up the coefficient of 't' in 'p' and setting it as the coefficient of a monomial with degree 't'. This sum is taken over all 't' i... |
The function "nat_to_sch" applied to the successor of the successor of a natural number x is equal to the following: Let u be the remainder when the first component of the successor of the successor of x is divided by 7, let v be the second component of the successor of the successor of x, let v1 be the first component... |
:
- For the "zero" case, the property P holds for 0.
- For the "swap" case, given any two elements a and b, if they are of the same sort and are not equal to each other, then the property P holds for the permutation of a and b.
- For the "plus" case, given any two elements p1 and p2, if the property P holds for both p... |
The vector of an ell2 space with dimension 1 is equivalent to the vector of a list containing only the number 1. |
The measure of set B under the conditional probability mass function of p given A is equal to the measure of the intersection of sets A and B under the probability mass function p, divided by the measure of set A under the probability mass function p. This is true if the intersection of the set defined by the probabili... |
In the context of trace topology, the lemma named "liveness_dense_iff" assumes that P is a subset of the infinite power set of A. It shows that the liveness of A and P is equivalent to P being dense. |
The relation of a deterministic Rabin automaton with explicit transitions, when both components are the identity relation, is equal to the identity relation. |
If you abstract a view result and then represent it again, you will get the original view result. This is a simple and code abstract type. |
The lemma assumes that the function invocation result for pairs (B, B) and (A, A) is F with respect to C and D. It shows that the function invocation query for the product of A, B, and C is F with respect to the product of D, B, and A. |
If "C" is a class in the program "G" and "G" is a well-structured program, then the fields of the class "C" in the program "G" are unique. |
For a function 'f' mapping from a Euclidean space 'a' to another Euclidean space 'b', if 'S' is in the set of Lebesgue measurable sets and 'f' is integrable on 'S', then 'f' is Borel measurable on the Lebesgue measure restricted to 'S'. |
The limit of a function at a point 'a' within an empty set is the lowest element or bottom element. |
For a given alternative sample option, if we have a relation that maps from type A to a function (which itself maps from type P to type M), then this relation also holds for the alternative sample option. This is under the condition that the relation is a transfer rule, meaning it preserves the structure of the operati... |
The lemma named "variant_names_fresh" assumes that "S" is a finite set. It shows that for all "s" in the set of the first element of the variant variables of "n", "s", and "S", "s" is not an element of "S". |
If the least upper bound (lub) of U and V is T according to a certain property P, then according to the same property P, U is less than or equal to T and V is also less than or equal to T. |
If the function "ipt_iprange_to_interval" applied to an IP address "ip" equals the tuple (s,e), then the set of all elements from "s" to "e" is equal to the set obtained by applying the function "ipt_iprange_to_set" to the IP address "ip". |
The Little Picard Theorem:
Assuming that the function 'f' is holomorphic over the entire complex plane, and that 'f' does not map to either 'a' or 'b' (which are distinct), we can conclude that 'f' must be a constant function. |
The zeta function of the complex conjugate of a number z is equal to the complex conjugate of the zeta function of the number z. |
If 'n' is an odd number, then the square part of 'x' raised to the power of 'n' is equal to the normalization of 'x' raised to the power of 'n' divided by 2, multiplied by the square part of 'x'. |
If a set Q is finite, and for every element Q' in Q, the free variable of Q' equals X, and for every element Q in Q, the safe range of Q holds, then the safe range of the disjunction over the set Q also holds. |
Zero multiplied by the additive inverse of any number z equals zero. |
The lemma named "norm_proj_fmla_esat_sat" is given a formula 'phi' of first-order logic, where 'phi' is of type 'a' (which is infinite) and 'b'. It assumes that 'phi' is well-formed with respect to the interpretation 'I'. It then shows that the normalization of the active domain of 'phi' under interpretation 'I', when ... |
: the multiset of the function 's' mapped over the range from 0 to 'n' and the multiset of the function 't' mapped over the range from 0 to 'm' belongs to the extension of the multiset ordering 's_mul_ext' with respect to 'NS' and 'S' if and only if there exists a value 'v' such that 'v' evaluates all elements in the f... |
The lemma named "real_inprod_cvec" shows that the inner product of a complex vector with itself is a real number. |
The transitive closure of "c" followed by "d" is equal to the greatest fixed point of the function that takes "x" and returns the infimum of the composition of "c" and "x" and "d". |
The set of all total morphisms of digraphs, denoted by "\<FF>", that exist between any two digraphs "\<AA>" and "\<BB>", is small. |
If 'a' is greater than 0, 'x' is greater than or equal to 0, and 'x' is less than 'y', then 'x' raised to the power of 'a' is less than 'y' raised to the power of 'a'. This is for 'x' being a real number. |
The lemma "add_nonneg_eq_0_iff" assumes that both x and y are greater than or equal to 0. It shows that the sum of x and y equals 0 if and only if both x and y are 0. |
Binding the function "f" to the uplifted value "a" is equivalent to applying the function "f" directly to "a". |
Lemma 12:
Assuming that the reduced Buchberger algorithm invariant holds for the tuple (bs, ss, ps), and that the set bs is a reduced Gröbner basis up to the degree-graded term ordering rword for the term u, and that the degree of the power-product of term u is less than or equal to the maximum degree of the degree-gra... |
The lemma "remove_cycles_neg_cycles_aux" is defined with respect to a fixed 'i', 'xs', and 'ys', and a new list 'xs'' is defined as 'i' followed by 'ys'. Assuming that 'i' is not an element of 'ys', 'i' is an element of 'xs', and 'xs' is equal to the concatenation of 'as', the map of 'i' prepended to each list in 'xss'... |
If A and B are elements in the set produced by the function "groupWIs" with parameters c and rs, and A is not equal to B, then for every ip1 in the set produced by the function "wordinterval_list_to_set" with parameter A, and for every ip2 in the same set but with parameter B, the function "same_fw_behaviour_one" with ... |
If a list of formulas "phis" is sorted, distinct, all formulas in the list satisfy the property "nf_ACI", and none of the formulas in the list is of type "FAnd", then the property "nf_ACI" holds for the formula obtained by applying the function "nFAND" to the list "phis". |
, assuming "A" is a subset of the universal set "S.Univ", the image of the function "arr_of" applied to the image of the function "elem_of" applied to "A" is equal to "A". |
If the length of a set or list 'AA' is zero, then the result of the function 'nullable0' applied to 'a' is equivalent to 'AA' satisfying 'a' under the zeroth interpretation. |
The lemma named "empty_interior_affine_hull" is about a set S in a Euclidean space. If S is finite and the cardinality of S is less than or equal to the dimension of the Euclidean space, then the interior of the affine hull of S is an empty set. |
The set of free variables in a function 'f' is a subset of the set of free variables in the differential of the function 'f'. |
If the hyperreal representation of a real number x is not equal to zero, then this hyperreal representation of x belongs to the set of finite hyperreal numbers and does not belong to the set of infinitesimal hyperreal numbers. |
The implication "A implies B" is not in the set of subformulas of the negation normal form of F. |
If X is an independent set in E, then the intersection of X and E' is an independent set in E'. |
If set A is a subset of set C, then the intersection of set A and set B is also a subset of set C. |
The lemma named "not_prim_expE" assumes that "x" is not primitive and "x" is not equal to the empty string. It then obtains "r" and "k" such that "r" is primitive, "k" is greater than or equal to 2, and "r" raised to the power of "k" equals "x". |
A mapping 'x' is equal to itself if and only if it is true. |
If a set P has a bottom element in the domain and x is pseudominimal in set P of the domain, then the image of x under function f is pseudominimal in the image of set P under function f in the codomain. |
, from an empty set of assumptions, either a predicate "p" is true or the negation of the predicate "p" is true. This is also known as the law of excluded middle in propositional logic. |
If the sum of v and w is equal to the sum of x and y, and x overlaps with both v and w, then the sum of the product of x and v and the product of x and w is equal to x. |
If the function 'u' (which maps from a certain type 'a' to a process type 'b') is continuous, and the function 'v' is monotonically increasing, then it is admissible that for any 'x', 'u' of 'x' is less than or equal to 'v' of 'x' in the domain-theoretic ordering. |
If "x" is a maximal simplex, then the image of "x" under a function "f" is a maximal simplex in the simplicial complex resulting from applying "f" to "X". |
If process P is less than or equal to process Q in the domain of termination, then hiding process P with respect to a set A is also less than or equal to hiding process Q with respect to the same set A in the domain of termination. |
"For all x, P of x is true, or it is not the case that P is true for all x." |
If "I" is a finite set, "F" is a function from "I" to the carrier of the univariate polynomial over the semialgebraic set "SA m", "f" is the direct sum over "I" of "F i", and "x" is an element of the carrier of the p-adic number field of "Suc m", then the function "SA_poly_to_SA_fun" applied to "m", "f", and "x" is equ... |
For a given index 'i', an element 'x', a list 'xs', and a function 'g', the function of the list with 'x' as the head and 'xs' as the tail is equal to the function of the list 'xs' with the index 'i' incremented by 1 and the function 'g' updated with the value 'x' at index 'i'. |
The lemma "alllsts_mono" assumes that set A is a subset of set B. It shows that the infinite repetition of set A is also a subset of the infinite repetition of set B. |
The lemma named "affine_hull_UNIV" is about a set S in a Euclidean space of dimension 'n. If the affine dimension of S is equal to the integer value of the dimension of 'n, then the affine hull of S is equal to the entire Euclidean space of dimension 'n. |
The lemma named "gen_sets" assumes that set G is a subset of set S and set S is a subset of the generated set of G. It shows that the generated set of S is equal to the generated set of G. |
The lemma named "n_in_obs_intra" assumes that "n" is a valid node and "n" is an element of "S". It shows that the intra-observable behavior of node "n" in set "S" is equal to the set containing only "n". |
The lemma named "integrable_by_parts_interior_strong" is defined with a fixed product function of type 'b in a Banach space. It assumes that the product function is a bounded bilinear operation. It also assumes that 's' is finite and 'a' is less than or equal to 'b'. It assumes that the functions 'f' and 'g' are contin... |
If under program P, expression 'e' evaluates to 'false' from initial state 's0' and initial heap 'b0' to state 's1' and heap 'b1', and if under the same program P, expression 'e2' evaluates from state 's1' and heap 'b1' to some expression 'e'' and state 's2' and heap 'b2', then under program P, the conditional expressi... |
Given a sequence A, where the range of A is a subset of the sets in measure space M and A is a decreasing sequence, and assuming that the measure of each element in A is not infinite, it can be shown that the measure of each element in A converges to the measure of the intersection of all elements in A. |
If the invariant holds for v0, D0, and PDPE, and if the invariant for v0, D0, and PDPE implies the partial invariant for (B, PDPE), then the full invariant holds for v0, D0, (B, PDPE). |
The lemma named "inangle1123" assumes that point A is not the same as point B, and point C is not the same as point B. Under these assumptions, it shows that point A is in the angle formed by points A, B, and C. |
The diamond of the sum of x and y applied to z is equal to the sum of the diamond of x applied to z and the diamond of y applied to z. |
The lemma named 'help2'' assumes that the extended natural number 'p' plus the inverse of 't' is less than or equal to the inverse of 's', 'p' is greater than 0, and the inverse of 's' equals the extended natural number 'n'. It shows that the inverse of 't' is less than the inverse of 's'. |
The lemma named "image_set_plus" assumes that "U" is a linear operation. It shows that the image of the union of sets "A" and "B" under "U" is equal to the sum of the image of set "A" under "U" and the image of set "B" under "U". |
A function from R to S is equivalent to a refinement function where the condition is always true for R and S. |
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