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The Bochner integral over the Lebesgue measure (lborel) of the function, which is the product of x raised to the power k, the exponential of negative x, and the indicator function of the set of all positive real numbers x, is equal to the factorial of k.
The well-formedness of a justification in the context of a program P composed with a function f is equivalent to the well-formedness of the justification in the context of the original program P.
If S is a relation that holds between s and t, then S also holds between the contexts C of s and t.
If the range of a function G is a set containing only the element y, and x is in the domain of G, then the value of G at x is y.
The lemma "weakFreshDerivative" is defined with the following parameters: a fixed 'b type 'Psi', a psi type 'Q', a psi type 'P', an 'a action' type 'alpha', a psi type 'P'', and a name 'x'. It assumes the following conditions: 1. 'Psi' enables a transition from 'Q' to 'P' under the action 'alpha', resulting in 'P''....
If X is a good quasi-term, rho is a good quasi-environment, z is fresh for X with respect to zs, and z is fresh for rho with respect to zs, then z remains fresh for the quasi-substitution of X with rho with respect to zs.
The elements of the cardinal multiplication of kappa-prime and kappa are equivalent to the Cartesian product of the elements of kappa-prime and kappa.
The lemma named "from_nat_neq_cols" is defined for a matrix A with arbitrary number of columns and rows. It assumes that 'i' and 'k' are less than the number of columns in A and that 'i' is not equal to 'k'. It shows that the natural number conversion of 'i' is not equal to the natural number conversion of 'k' in the c...
The lemma establishes two facts for a fixed point 'a' in a Euclidean space. First, it shows that the closed box with 'a' as both the lower and upper bounds is a set containing only 'a'. Second, it shows that the open box with 'a' as both the lower and upper bounds is an empty set.
The lemma named "sub_polyfun_alt" establishes a relationship for a fixed 'x' which is a member of a commutative ring and a multiplicative monoid. It shows that the difference between the sum (from i=0 to n) of the product of 'a' at index 'i' and 'x' raised to the power of 'i', and the sum (from i=0 to n) of the product...
If for any elements 'b' and 'c', the pair (a, b) is in the reflexive transitive closure of relation 'r' and the pair (a, c) is also in the reflexive transitive closure of relation 'r', then the pair (b, c) is in the join of relation 'r'. If this condition holds, then the Church-Rosser property holds on relation 'r' for...
If a fixpoint of a function 'f' with inputs 'm' and 'x' equals 'y', then there exists a natural number 'n' such that the composition of the function 'f' raised to the power 'n' with 'x' equals 'y'.
If 'n' is less than or equal to the length of the list 'xs', then the absolute value of the array resulting from pushing an element 'x' onto the array '(xs, n)' is equal to the absolute value of the array '(xs, n)' concatenated with the element 'x' in a list.
The lemma named "line_integral_degenerate_chain" assumes that for every pair (k, γ) in the chain, γ is a point path. It also assumes that the basis is finite. Under these assumptions, it shows that the line integral of the one-chain over the given basis and chain equals zero.
If the sparse vector 'b' is sorted, then the result of the function 'le_spvec' with 'b' and an empty list as arguments is equivalent to whether the sparse row vector of 'b' is less than or equal to zero.
The composition of a relation 'r' with the insertion of an ordered pair (a, b) into a set 's' is equal to the union of the composition of 'r' with the set containing the ordered pair (a, b) and the composition of 'r' with 's'.
The imaginary part of the scalar product of a vector with itself is zero.
For all natural numbers m and n, if n equals 0, then the quotient of m divided by n equals 0 and the remainder of m divided by n equals m. Otherwise, m equals the product of the quotient of m divided by n and n plus the remainder of m divided by n, and the remainder of m divided by n is less than n.
If 'j' is a non-negative integer and less than 'n', then the functions 'Mul_Propagate_Black' with parameter 'n' and 'Mul_Color_Target' with parameters 'j' and 'n' are interference-free.
"u" is not compatible with "v" if and only if "u" is parallel to "v".
If a regular expression "r" is in the set of modified regular expressions "mrexps" of a string "s", then reading a character "a" from the regular expression "r" results in a regular expression that is also in the set of modified regular expressions "mrexps" of the same string "s".
If 'p' is a prime element and 'p' does not divide 'a', then 'a' and the power of 'p' to 'm' are coprime.
If the unfolding of the folding of a list 'ls' using its length as a parameter equals the list 'ls' itself, and the list 'ls' is composed of an element 'a' followed by a list 'ls1', then the unfolding of the folding of a list composed of an element 'x' followed by the list 'ls', using its length as a parameter, will eq...
If the natural transformation "eta1" is naturally isomorphic to "eta2", then the composition of "eta1" and "eta2" is a natural transformation.
It is not the case that, given a context G and a state s where some error e has occurred, the catch operation on term tn is valid.
The negation of the selection-like operation on a set X, a valuation function 'v', and a set V is equal to the union of the negation of set X and the set of all outcomes 'w' such that 'w' does not agree with the valuation function 'v' on set V.
The lemma named "sub_block_fresh" assumes that "i" is not in the nominals of the branch and that the block is in the branch. It shows that substituting "i" with "j" in the block using function "f" is the same as substituting the block with function "f" without any specific substitution.
The weight of a semantically secure probabilistic message function (spmfs) made lossless is equal to 0 if the probabilistic message function is equivalent to returning a probability mass function of None. Otherwise, the weight is 1.
The lemma "deleteMin_mset" assumes that the queue "q" maintains the invariant "queue_invar". It also assumes that the multiset representation of the queue "q" is not an empty set. Under these assumptions, it shows that the multiset representation of the queue after deleting the minimum element is equal to the multiset ...
The set difference of two word intervals, when converted to a set, is equal to the set difference of the sets of the two word intervals.
The lemma "GSMP_disjoint_snd_specific_not_fst_specific" assumes that the set of terms projected from a stateful protocol \<A> onto a list l is disjoint from the set of terms projected onto a different list l', both with respect to a security property Sec. It also assumes that the message m is specific to the projection...
The base-2 logarithm of an integer 'a' is equal to the integer of the base-2 logarithm of 'a'.
The lemma named "non_extensible_Borsuk_map" is defined for a fixed 'a' which is a type of Euclidean space. It assumes that 's' is compact, 'c' is a component of the complement of 's', 'c' is bounded, and 'a' is an element of 'c'. It shows that there does not exist a function 'g' that is continuous on the union of 's' a...
If a path is constructed inversely from point q to x and y in a domain D and a set W, and q is not equal to x, then a path can be constructed inversely from the chosen point in the domain multiplied by q to x and y in the domain D and the set W plus the chosen point in the domain multiplied by the transpose of q.
If there is a heap node refinement from gamma to gamma prime with relation R and abstract value a, then there is also a heap node refinement from gamma to either gamma prime or gamma double prime with the same relation R and abstract value a. This is assumed under the condition that the computation c refines the abstra...
The lemma named "tendsto_Ln" is about limits in complex analysis. It takes two fixed parameters, L and F. It assumes that the function f tends to L under F, and that L is not a non-positive real number. Under these assumptions, it shows that the natural logarithm of the function f (Ln (f x)) tends to the natural logari...
"x is less than or equal to y in the L-ordering" if and only if "x is less than or equal to y in the H-ordering".
The lemma named "skel_swap" assumes that "X" is good. It shows that the skeleton of "X" after swapping "x1" and "x2" in "xs" is equal to the original skeleton of "X".
If the function "wcode_on_left_moving_2" holds true for the inputs "ires", "rs", and the pair consisting of "b" and a list starting with "Bk" followed by "list", then it also holds true for the inputs "ires", "rs", and the pair consisting of the tail of "b" and a list starting with the head of "b", followed by "Bk" and...
"zero is not a period of any word w".
The lemma named "strong_delete_point_groups_index_zero" assumes that "G" is a set obtained by removing "x" from all sets "b" that are elements of "B" and contain "x". It also assumes that "xa" and "y" are elements of "G" and that "xa" is not equal to "y". It then shows that the index of the set containing "xa" and "y" ...
The lemma named "conjugE_nemp" assumes that "u" is equivalent to "v" and "u" is not equal to the empty string. It then obtains "r" and "s" such that the concatenation of "r" and "s" equals "u", the concatenation of "s" and "r" equals "v", and "s" is not equal to the empty string.
The lemma named "output_completion_for_FSM_length" assumes that for all 'io' in set 'P', the length of 'io' is less than or equal to 'k'. It shows that for all 'io' in the output completion for FSM 'M' of set 'P', the length of 'io' is also less than or equal to 'k'.
, under the assumptions that "A" is a Hintikka set, "x" is a property at "a" in "H", the negation of "x" is a property at "i" in "H", "a" is in the assignment of "A" and "H" at "i", and "i" is in the names of "H", it leads to a contradiction.
The function "cross_alt" is commutative for the inputs p, q, a, and b. This means that swapping the order of the first two arguments (p and q) does not change the result of the function.
If for all indices 'i', the function 'f' applied to 'x' and 'i' tends towards 'a' at index 'i' as 'x' approaches a certain value (denoted by 'net'), then the function that maps each index 'i' to 'f' applied to 'x' and 'i' also tends towards the function that maps each index 'i' to 'a' at index 'i' as 'x' approaches tha...
The theorem named "subgroupI" is about a fixed group G. It assumes the following: 1. G is a subset of M. 2. The identity element is in G. 3. For any two elements g and h in G, their product is also in G. 4. Any element g in G is invertible. 5. The inverse of any element g in G is also in G. Given these assumptions, t...
If a regular expression 'r' is well-formed with respect to a natural number 'n', and a word 'ws' is a well-formed word with respect to 'n', then the language of 'n' derived from 'ws' and 'r' is equal to the left quotient of 'ws' and the language of 'n' and 'r'.
If the element "ram" is not in the set "a" and also not in the set "b", then the first element of the pair resulting from splitting the list formed by appending "ram" to the end of "a" and then appending "b", at the point "ram", is equal to "a".
An empty multiset is justified by another empty multiset.
The index of a constant is a primitive recursive function of one variable.
Mapping a function 'f' over a list 'xs' belongs to the set of lists of length 'size xs' with elements from set 'A' if and only if the image of the set 'xs' under the function 'f' is a subset of 'A'.
The lemma "replace_Goto_with_Call_in_terminal_chain_Undecided" assumes that a chain is defined as some rule set "rs" and that this rule set is a terminal chain. It shows that for a given context, packet and decision, the transition from an undecided state using a rule with a "Goto" action to a chain is equivalent to th...
"if eventually F holds, and if F always leads to G, then eventually G holds."
The set resulting from the union of two lists, "xs" and "ys", is equal to the union of the sets of "xs" and "ys".
The lemma for case two: Assuming "case_two j", it can be shown that there exists a 't' such that for all 't'' less than or equal to 't', the prefixes of 't'' and 'j' are defined, and for all 't'' greater than 't', the prefixes of 't'' and 'j' are undefined.
For a given interval from 'n' to 'd', when it is cut at a point 't' such that 't' is less than or equal to the end of the interval, the resulting interval is as follows: if 't' is less than 'n', then the resulting interval is empty. If 't' is less than or equal to 'n' plus 'd', then the resulting interval is from 'n' t...
A sequence 'xs' is accepted by the combined finite state machine (Plus fsm1 fsm2) if and only if the sequence 'xs' is accepted by either finite state machine fsm1 or finite state machine fsm2.
The lemma named "short_vector_iso_inv" assumes that the result "res" is such that "short_vector_iso" is equal to "v" and "m" is greater than 0. It shows that there exists a sequence "fs" such that "fs" satisfies the LLL weak invariant, "fs" is weakly reduced with respect to "m", and "v" is equal to the first element of...
A pair of states (c, c') is in the reflexive transitive closure of the step relation if and only if there exists a path p in the labeled transition system such that the path starts at state c and ends at state c'.
If the function "applied_rule_rev" with inputs C, x, and p returns some value r, then r is an element of the set p.
: given the assumptions that functor FF maps category AA to category BB, functor GG maps category BB to category CC, and functor HH maps category CC to category DD, and that the composition of functors HH and GG equals Q, it can be shown that the composition of functors HH and the composition of functors GG and FF equa...
The lemma "nonregular_event_induces_snapshot" assumes that process p has not yet taken a snapshot in configuration c, event ev transitions configuration c to c', the event ev occurs on process p, and the event ev is not a regular event. It shows that if the event ev is not a regular event, then process p has taken a sn...
The absolute value of a floating point number, represented as a pair (a,b), is equal to the floating point number itself if 'a' is greater than or equal to zero. Otherwise, it is equal to the floating point number with 'a' negated.
If the evaluation of proposition P with value x is true, and the evaluation of proposition Q with value x is true for value y, then the evaluation of the combined proposition (P bound to Q) is also true for value y.
An empty set, when checked if it is empty, will indeed return true.
The set of elements in a tree (t) remains the same even after the tree is painted with a color (c).
If 'b' is less than or equal to 'a', then the sum of 'a' minus 'b' and 'c' is equal to 'a' plus 'c' minus 'b' in the context of extended nonnegative real numbers.
The lemma named "mod_ring_rel_unsafe" assumes that "x" is less than the cardinality of the set 'a'. It shows two things: 1. The modular ring relation of the integer value of "x" and the natural number value of "x" holds true. 2. If "x" is greater than 0, then the natural number value of "x" is not equal to zero in ...
The real part of a complex number is less than or equal to the modulus of the complex number.
If a state 's' is sequentially consistent with a set of values 'vs' (as per the assumption 'sc_c'), and if there is a sequence of thread actions 'ttas' that can transition the state 's' to a new state 's' (as per the assumption 'tauRed'), and if the thread actions are consistent with the set of values 'vs' (as per the ...
The lemma named "larger_heapI" assumes that for any heap location "hl" and any value "x", if the heap "h" at location "hl" equals to some value "x", then the heap "h'" at the same location also equals to the same value "x". Under this assumption, it shows that the heap "h'" is larger than the heap "h".
The lemma named "Gamma_integral_real" assumes that a real number "x" is greater than zero. It shows that the integral from zero to infinity of the function "t to the power of (x - 1) divided by e to the power of t" is equal to the Gamma function of "x".
The lemma "index_take_if_index" assumes that the index of element x in list xs is less than or equal to n. It shows that the index of x in the sublist of xs (obtained by taking the first n elements) is the same as the index of x in the original list xs.
For a function 'f' mapping from a Euclidean space 'm' to another Euclidean space 'n', if a set 'S' is in the Lebesgue sets, the dimension of 'm' is less than or equal to the dimension of 'n', and 'f' is differentiable on 'S', then the image of 'S' under 'f' is also in the Lebesgue sets.
If the pseudo-projection of 'n' on 'w' is the same as the pseudo-projection of 'n' on 'x', then the stake of 'n' on 'w' is equal to the stake of 'n' on 'x'. This is assuming that the pseudo-projection of 'n' on 'w' is equal to the pseudo-projection of 'n' on 'x'.
The antidomain of x multiplied by the antidomain of x equals the antidomain of x.
The lemma named "outer_minus_minus" is defined for a fixed complex matrix vector 'a'. It assumes that the dimension of vector 'a' is equal to the dimension of vector 'b', and the dimension of vector 'u' is equal to the dimension of vector 'v'. It shows that the outer product of the difference between vectors 'a' and 'b...
The lemma named "ide_char_SC" assumes that "f" is an arrow. It shows that "f" is an identity if and only if the domain of "f" is equal to the codomain of "f" and the function of "f" maps each element in its domain to itself.
The lemma named "CSP4_imp_right_Skip" assumes that the relation of P is CSP4 healthy. It shows that the sequential composition of P and Skip is equal to P.
The lemma named "his_le" assumes that "x" is an element of the history at time "t", node "n", and node id "nid". It shows that the first element of "x" is less than or equal to "n".
Assuming that A, B, C, and D are true, we can show that (A and B) and (C and D) are also true.
If there are two morphisms, 'f' from 'a' to 'b' and 'g' from 'b' to 'a' in the category, and if the composition of 'g' and 'f' equals the identity morphism at 'a' and the composition of 'f' and 'g' equals the identity morphism at 'b', then 'f' is an isomorphism from 'a' to 'b' and 'g' is an isomorphism from 'b' to 'a' ...
The lemma named "lin_comb_imp_lin_dep_fin" is about a set A of vectors. It assumes that A is finite and is a subset of the set of all vectors of length n. It also assumes that the linear combination of the vectors in A, using coefficients from c, equals the zero vector of length n, and that there exists a vector a in A...
Zero is less than or equal to the real number representation of the upper bound of pi for any given precision p.
"N is not equal to 0".
If 'm' is a Moebius transformation, then the inverse of 'm' is also a Moebius transformation.
"n to the power of n is less than or equal to the square of the factorial of n".
The lemma named "val_ultrametric_noteq''" assumes that "x" and "y" are elements of the carrier of the p-adic numbers. If the valuation of "x" is greater than the valuation of "y", then the valuation of the difference "y - x" is equal to the valuation of "y".
If a given environment 'G' and state 's' confirm that a location 'l' is conformant to a location type 'L' where 'vn' is mapped to 'T', and 'vn' is not defined in 'L', then 'G' and 's' confirm that 'l' is conformant to 'L'.
There exist some 'c1' and 'c2' such that 'c' is equal to the catch of 'c1' and 'c2', and 'c1' is a subset of or equal to 'c1' and 'c2' is a subset of or equal to 'c2', given that 'c' is a subset of or equal to the catch of 'c1' and 'c2'.
The lemma named "spmf_of_pmf_parametric", which includes lifting syntax, shows that if there is a relation A that holds for a probability mass function (pmf), then the same relation A holds for the sub-probability mass function (spmf) when the function "spmf_of_pmf" is applied.
The lemma provides simplification rules for the function "Fvars" applied to terms in the Goedel Hereditarily Finite Set theory: 1. The set of free variables in the term "Zero" is an empty set. 2. The set of free variables in the term "Var a" is a set containing only "a". 3. The set of free variables in the term "Eats ...
The lemma named "n_SPRAY_intro" assumes the following conditions: 1. Set S is a subset of SPRAY x. 2. The cardinality of set S is the successor of n. 3. Set S is an independent set. 4. For every point P in SPRAY x, P is a dependent path of S. If these conditions are met, it shows that x is an n-SPRAY.
For a matrix A of type 'a, which is a real normed algebra, and a vector v of type 'n and 'k, the function (*v) applied to A tends to the result of the matrix A multiplied by the vector v, as v approaches a certain value within a set S.
The lemma provides simplification rules for binding operations on binary trees: 1. Binding an undefined binary tree with any function results in an undefined binary tree. 2. Binding a leaf node with a function applies the function to the value of the leaf node. 3. Binding a node with a function applies the binding ope...
If a state 's' is not reachable with the function 'finsert' applied to the pair '((aa, ba), b)' and 'e', then the state 's' is also not reachable with 'e'.
The cardinality (or size) of the universal set of unordered pairs of elements from a set 'a is equal to the product of the cardinality of the universal set of 'a and the result of adding 1 to the cardinality of the universal set of 'a, all divided by 2. This is symbolically represented as "?UPROD = ?A * _ div _".
A function 'f' is Borel measurable with respect to a measure space 'M' if and only if for every basis vector 'i' and every real number 'a', the set of all elements 'w' in the space 'M' such that the inner product of 'f(w)' and 'i' is less than or equal to 'a' belongs to the sets of 'M'. Here, 'f' maps elements from som...
If the set of chambers 'Cs' is a subset of the set of chambers transformed by function 'f', then the sequence of chambers 'Cs' under the building relation 's' is equal to the sequence of chambers 'Cs' under the building relation 'g'.