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The lemma for the conditional probability mass function after mapping with a function 'f' is equivalent to mapping with function 'f' the conditional probability mass function with the preimage of 'A' under 'f'.
The count of "results_in t s" in the "after_summary" of the "zmset_frontier" of "c_imp c l1" and the "summary l1 l2" is less than or equal to the count of "results_in t s" in the "union_frontiers c l2".
The lemma named "key_map_is_coherent" assumes that set V is a subset of set U, both U and V are Zariski open sets, a point p is in set V, and s is an element of the structure sheaf of U. Under these assumptions, it shows that the composition of the key map of V and the sheaf morphisms from U to V applied to s is equal ...
If 't' is a pattern and for all 'i' in the free variables of 't', the function 'σ' at 'i' equals the function 'σ'' at 'i', then the substitution of 'σ' in the compiled pattern of 't' equals the substitution of 'σ'' in the compiled pattern of 't'.
The inverse of 'd' composed with the inverse of 'ov' is a subset of the union of the inverse of 'ov', the inverse of 's', and the inverse of 'd'.
If an object "obj" is well-formed in a heap "h" according to a program "P", and if heap "h" is a sub-heap of another heap "h'", then the object "obj" is also well-formed in the heap "h'" according to the same program "P".
: Assuming that for every state transition X or not in F, appended to S, is in the state set S, and that the new state set S' is the update of S with the state transition X or not in F, and that the new action set A' is the action set A appended with the step of the state transition X or not in F, it shows that: 1. T...
: if a function 'p' is a primitive recursive function of three variables, and for all 'x' and 'y', the pair '(x, y)' is in relation 'r' if and only if there exists a 'u' such that 'p' of 'x', 'y', and 'u' equals zero, then the relation 'r' is in the set of computably enumerable relations.
1 is an nth power for any integer n.
The lemma "incl_reflects_con" assumes that there is a transition from state "t" to state "u". It shows that "t" is connected to "u".
For a given set 'S' in Euclidean space, if 'S' is an Absolute Neighborhood Retract (ANR), then 'S' minus a single element 'a' is also an Absolute Neighborhood Retract (ANR).
If 'x' belongs to the set of integers, then 'x' is algebraic.
: Assuming a configuration 'c' is active at a time 'i' in a trace 't', and for all time 'n' greater than 'n', the configuration 'c' is not active in the trace 't' at time 'n', it can be shown that the configuration 'c' is active at the time when 'c' and 't' are combined in the trace 't'.
The lemma suppI: Assuming that for any 'a', 'b', and 's', if 'a' and 'b' are in the delta of 's', then there exists an 'x' such that 'a' is greater than or equal to 'x', 'b' is greater than or equal to 'x', and 'x' is in the delta of 's'. This shows that the support of delta exists.
The program dependence graph (pdt) is transitive.
The height of a binary tree after an insertion operation is equal to the height of the original tree.
If the function "top_on_opt" holds for options o1 and o2 over a set X, then it also holds for the supremum (or join) of o1 and o2 over the same set X. Here, o1 and o2 are options of some state type.
If "x implies y", then "x implies the superset of y and z".
The domain of the constant map with value 'v' over the set 'S' is equal to 'S'.
The lemma named "rel_spmf_inf" assumes that "p" is less than or equal to "q", "q" is less than or equal to "p", "ord" is reflexive, and "ord" is transitive. Under these assumptions, it shows that the relation "rel_spmf" holds for the infimum of "ord" and the inverse of "ord" with respect to "p" and "q".
The lemma named "nn_transfer_operator_intg" assumes that both functions "f" and "g" are Borel measurable on a measure space "M". It shows that the positive integral of the product of function "f" and the non-negative transfer operator of function "g", with respect to measure "M", is equal to the positive integral of th...
The arctangent of a real number x is equal to twice the arctangent of the quotient of x and the sum of 1 and the square root of 1 plus the square of x.
If the count of iterations of a program "b" under a state "s" with a command "c" and a path "\<pi>" is "n", and if the count of iterations of the same program "b" under the same state "s" with the same command "c" and the same path "\<pi>" is "n'", then "n'" must be equal to "n".
The lemma, named "prob_long_aux", demonstrates two things: 1. The value of "2 divided by pi times the quantity of (l divided by d minus the square root of (l divided by d squared minus 1) plus the arccosine of (d divided by l))" is greater than or equal to zero. 2. The Buffon's needle problem, defined for a set of pa...
If A and B are both groups and f is a homomorphism from A to B, then the composition of A and f with the identity element of A is equal to f.
The function that takes a triple (P, t, Q) and maps it to the set {P} followed by t and then {Q} is injective.
The roots of a polynomial, when the polynomial is normalized and converted from an integer polynomial, are the same as the roots of the original integer polynomial. This is true for any type 'a that is an integral domain and has a ring characteristic of zero.
The division of x by y equals to zero if y equals zero, otherwise it equals the 32-bit unsigned integer division of x by y.
: Assuming a function 'f' maps from the carrier of the m-th power of the p-adic numbers to the carrier of the p-adic numbers, and a set 'S' is a subset of the carrier of the (n+1+l)-th power of the p-adic numbers. If 'Y' is defined as the partial pullback of 'f' with respect to m, (n + l), and the image under the funct...
: Given the assumptions that: - "I" holds for an empty list, a list "l0", and a state "σ0", - For any elements "x", lists "l1" and "l2", and state "σ", if "l0" is the concatenation of "l1" and the list consisting of "x" followed by "l2", "I" holds for "l1", the list consisting of "x" followed by "l2", and "σ", and "c"...
If a function 'f' maps from set 'X' to the space of sigma-algebra 'M', then the set of all sets in the inverse image algebra of 'X', 'f', and 'M' is equal to the set of all sets that are the intersection of the inverse image of 'A' under 'f' and 'X', where 'A' is a set in the sigma-algebra 'M'.
The lemma "LLL_invariant_mod_to_weak_m_to_i" assumes that the "LLL_invariant_mod" holds for the parameters fs, mfs, dmu, p, first, b, and m. It also assumes that "i" is less than or equal to "m". Under these assumptions, it shows that the "LLL_invariant_mod" also holds for the parameters fs, mfs, dmu, p, first, b, and ...
If points 'a' and 'b' are in the hyperbolic plane, and 'b' is incident to line 'l' but 'a' is not, then the perpendicular from 'a' to line 'l' is in the set 'S'. Furthermore, the points formed by the Cartesian coordinates of the foot of the perpendicular from 'a' to 'l', 'a' itself, and the point where the perpendicula...
For any precondition P, asserting P in the context of a state-exception monad will always result in a postcondition that is true, regardless of the state and result.
The lemma named "cod_char" shows that the codomain of a function 'f' is equal to 'One' if 'f' is equal to 'j0' or 'j1'. Otherwise, if 'f' is an arrow (i.e., a morphism in a category), then the codomain of 'f' is 'f' itself. If neither of these conditions are met, then the codomain of 'f' is 'null'.
The lemma "Jcc_pieces_SFAss" assumes that the function "Jcc_pieces" with parameters P, E, C, M, h0, vs, ls0, pc, ics, frs, sh0, I, h', ls', sh', v, xa, and (C' .sF{D}:=e) equals to (True, frs0, (xp',h',(v#vs',ls',C0,M',pc',ics')#frs',sh'), err). If this is true, then there exists an error such that the function "Jcc_pi...
The lemma named "before_first_time" assumes that the first occurrence of event K in the run ρ is at time τ and index n, and that m is less than n. It shows that the time of the m-th occurrence of event K in the run ρ is less than τ.
If 'q' is a prefix of an empty list, then 'q' must be an empty list itself.
For a given relation "Rel" of a set of ordered pairs of 'a', if "Rel" is a preorder, then the reflexive transitive closure of "Rel" is equal to "Rel" itself.
For any two formal power series 'f' and 'g' of a type 'a' that supports multiplication with zero and commutative addition, the nth term of the product of 'f' and 'g' is equal to the sum of the product of the ith term of 'f' and the (n-i)th term of 'g', where 'i' ranges from the subdegree of 'f' to 'n'.
, given the assumptions that "T" is a morphism from "A" to "B" in the category of sets (denoted as "smc_Set"), the function "T" is a one-to-one function (denoted as "v11"), and the domain of the function "T" is "A", it can be shown that "T" is a monomorphism from "A" to "B" in the category of sets. This is a reference ...
The lemma named "obj_UP\<^sub>0" assumes that "a" is an object in category B. It shows that "a" is also an object when it is transformed by the operation "UP\<^sub>0".
A function 'f' is continuous at a point 'x' within a set 's' if and only if for every positive number 'e', there exists a positive number 'd' such that the image of the intersection of the ball centered at 'x' with radius 'd' and the set 's' under the function 'f' is a subset of the ball centered at 'f(x)' with radius ...
The set of input/output targets in machine M starting from state q is a subset of the states in machine M.
The lemma named "add_anno_acq_AsmNoWrite" assumes that the variable "v" is not in the domain of the context "Γ". It shows that if "v" is not in the set "C", then adding the annotation "Acq v AsmNoWrite" to the context "Γ" results in the same context but with "v" mapped to its dma_type.
The lemma named "nAct_strict_mono_back" assumes that the nth occurrence of action c in trace t is less than the nth' occurrence of action c in trace t. It shows that n is less than n'.
Operation A is equivalent to the negation of the dual of operation A being approximately equal to the top element.
The inclusion relation forms a partial order on a given set S.
The composition of the functions "induced" and "contained.Class" on the group G is equal to "eta".
If F is a function that constrains A to A', then F is a function that Constrains A to A'.
If the set K is the intersection of the consequence sets of A and B, then K is equal to its own consequence set.
"F is a frame satisfying the third frame condition".
The set of all 4-tuples (x, y, z, zs) where x, y, z, and zs are elements in the list 'as' in that order, is equal to the set of all 4-tuples (x, y, z, zs) where the list consisting of x, y, z, and zs is an element of the set of all lists 'as'.
: if a number 'q' belongs to the set of real numbers, and for any real number 'r', a property 'P' holds for 'r' when it is considered as an element of the set 'Reals', then the same property 'P' also holds for 'q'.
If two temporal linked lists "xs" and "ys" satisfy the properties P and Q respectively, then "xs" is empty if and only if "ys" is empty.
In the context of a ring, if A, B, and C are ideals of R, and A is a subset of B, then the product of C and A is a subset of the product of C and B.
If interval I is a subset of the interval up to k, then subtracting from k the result of subtracting I from k equals I.
If the size of "is" is greater than 0, then the "start" is in the "nlists" of the size of "is" in set A.
For any function 'f' from type 'a' to a set of type 'b', the supremum of 'f' and 'f' is 'f' itself.
If 'x' adheres to 'A', then the following holds: if 'U' is not in the neighborhood of 'x', then 'R' is true; and if the intersection of 'U' and 'A' is not empty, then 'R' is true. This implies 'R'.
If we have a valuation v on a field K, a polynomial ring R over the valuation ring (Vr K v) with indeterminate X, and two polynomials p and q in the carrier of R, such that the degree of both p and q in the ring R with respect to the valuation ring and indeterminate X is less than or equal to the successor of some inte...
: "For all n greater than or equal to N, if the norm of Xn is less than or equal to K, then X is a bounded sequence."
If the sequence of the "Skip" command followed by a second command "c2" in a state "s" can reduce to "Skip" in a new state "s'", then the second command "c2" in the state "s" can also reduce to "Skip" in the new state "s'".
If the predicate "P" holds for "m" and "n", and "m" and "n" are elements of set "A", and "mulex_on" holds for "P", "A", "M", and "N", then "mulex_on" also holds for "P", "A", and the multisets resulting from adding "m" to "M" and "n" to "N".
The lemma named "unit_char'" assumes that "a" is an object of "S". It shows that the unit of "a" is equal to "a".
: 1. If the top-level evaluation of a program 'tp' in a given environment 'env' and state 's' results in a new state 's'' and a result 'r', then the result 'r' is not a timeout error and the clock value of the initial state 's' is equal to the clock value of the final state 's''. 2. The top-level evaluation of a prog...
: Given a transition system "TS" that belongs to the set of high-level Petri nets of a statechart "ST", a state "S" that belongs to the step configuration of the hierarchical automaton of "ST" and the configuration of "ST" with respect to "TS", an automaton "A" that belongs to the set of state automata of the hierarchi...
If a function 'f' is raised to the power of the product of the order of 'f' at 'a' and any integer 'n', and this powered function is applied to 'a', the result will be 'a'.
The lemma "dense_run_dil_inverse_only" assumes that function 'f' is dilating over a subset 'sub' of a run 'r', function 'g' is contracting over the run 'r' and the subset 'sub' of function 'f', and the run 'sub' is dense. Under these assumptions, it shows that function 'g' is equal to the inverse dilation of function '...
The interval from some number to 'n' going backwards from 'a' is equal to 'n' minus 'a' if 'a' is less than or equal to 'n', otherwise it is zero.
The standard model 2 is omega-consistent.
The set of final states in the machine M is finite.
If a function 'f' is equal to 'g' in the source-sink coframe, then the arrow map of the coframe applied to 'f' will yield 'g'.
"n" is greater than zero.
A set T is connected in the subtopology of a topological space X restricted to a subset S if and only if T is connected in the original topological space X and T is a subset of S.
The nth vector of the real number representation of matrix A is equal to the real number representation of the ith vector of matrix A.
If there is a left derivation introduction for the concatenation of alpha and delta at position i with rule r, index ix, derivation D, position j, and gamma, and if i is less than the length of alpha, and gamma at position j is a nonterminal, then there exists an omega such that alpha has a left derivation with the pai...
The lemma named "drop_mess" assumes that the successor (or next number) of the length of list xs0 is less than or equal to the length of list xs. It also assumes that dropping the difference between the length of xs and the successor of the length of xs0 from xs results in a list that starts with x and is followed by x...
If the vertices of a face 'f' are distinct and 'a' is the next element of 'b' in the vertices of 'f', then the pair (a, b) is included in the edges of the face 'f'.
In the context of a total order, for any two elements 'a' and 'b' in the carrier set 'D', either 'a' is less than 'b', 'a' is equal to 'b', or 'b' is less than 'a'.
The count of polynomial roots for the negation of a polynomial 'p' within a set 's' is equal to the count of polynomial roots for the polynomial 'p' within the same set 's'.
The lemma named "agg_sum_not_bot" is defined with a fixed function 'f' which is a square of 'a' (where 'a' is finite) and 'b' (where 'b' is an aggregation order). The lemma assumes that the function 'f' applied to any pair (i,j) is not equal to the lowest element (bot). It then shows that the double summation of the fu...
The total probability measure of the universe (or the entire sample space) for a given probability mass function M is equal to 1.
If we assume that relation R is a subset of relation R' and relation S is a subset of relation S', and that the relation R' is strongly normalizing with respect to S', then we can show that the relation R is strongly normalizing with respect to S.
The parameters of a term 't' are finite.
The minimum height of a k-d tree is less than or equal to the logarithm base 2 of the size of the k-d tree.
If t is not Null, then the sum of all elements n in the labels of t, each incremented by 2, is greater than or equal to 2.
The validity of excluding an element 'x' from a bag 'X' is equivalent to the definedness of 'X' and the validity of 'x'.
The error function complement (erfc) of a real number 'a' is less than or equal to the error function complement of 'b' if and only if 'a' is greater than or equal to 'b'.
If the pair (p, q) is in the transitive closure of the epsilon relation of a tree automaton A, and if q is in the set of productive states of the tree automaton A with respect to a set P, then p is also in the set of productive states of the tree automaton A with respect to the set P.
If the nth element of list l is some value x, then x is an element of the list l.
If for all 'x', the list 'g x' has distinct elements, and if the list 'xs' also has distinct elements, then the list formed by prepending each element 'x' from 'xs' to each list 'g x' will also have distinct elements.
The lemma named "aux_abs_int" establishes a rule for an integer 'c'. Assuming 'c' is not equal to zero, it shows that the absolute value of 'x' is less than or equal to the absolute value of the product of 'x' and 'c'.
The function "fm_of_nat" is surjective.
If the norm function and the subtraction function both preserve the relation A, then the distance function also preserves the relation A. This is a rule for transferring properties between different mathematical structures.
The lemma named "subrobdds_proper_simps" states two things: 1. The proper sub-robdds of a robdd leaf node, represented by "robdd_leaf f", is an empty set. 2. The proper sub-robdds of a robdd variable node, represented by "robdd_var i l v r", is the union of the proper sub-robdds of its left child "l" and right child ...
A NullPointer is not equal to an Object.
If a Hoare triple is valid for precondition 'p', statement 'S', and postcondition 'q', and if 'S' is refined to 'S1' with data refinement from 'D' to 'D1', then a new Hoare triple is valid with precondition 'wp D p', statement 'S1', and postcondition 'wp D1 q'. Here, 'wp' stands for weakest precondition.
"H" is a subset of "synth(H)".
The lemma named "ineM_ine" assumes that "ineM P M E" is true. If this assumption holds, it shows or proves that "ine P E" is also true.