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If functor G maps category A to category C and functor H maps category B to category C, and if ABF is an arrow from object A to object B in the comma category of G and H, then the arrow map of the opposite comma functor of G and H applied to ABF is an arrow from the flipped object of A to the flipped object of B in the...
If P and Q are elements of the polynomial ring set over R and I, and if g is a function that is closed over R, then the evaluation of the product of P and Q under the polynomial function g is equal to the product of the evaluations of P and Q under the same function g.
The lemma "count_roots_interval" assumes that the count of roots in an interval for a polynomial p is equal to (cr, nr), and that the polynomial p is square-free. It shows that if a is less than or equal to b, then the count of roots between a and b is equal to the cardinality of the set of x such that a is less than o...
For any automaton 'A', if 'p' is a final state of 'A', then the pair (True#p, True#start A) is an element of the epsilon closure of the star of 'A'.
If a guard 'g' is in the set 'G', and if 'g' constrains the variable 'V' of the index 'I i', and if the index 'I i' is restricted once in 'G', and if the guards 'G' are applied to the join of 'iaa' and 'c', then the guards of the set 'G' with 'g' removed can be applied to the join of 'iaa' with 'ia' at index 'i' and 'c...
The infimum distance between a point 'x' and a set 'A' is equal to the set distance between the set containing 'x' and 'A'.
The lemma named "locate_unique" makes the following assumptions: 1. For any element 'a' in the set 'al', the function 'f' applied to 'a' is greater than 0. 2. 'n' is less than the length of 'al'. 3. 'j' is less than the function 'f' applied to the 'n'th element of 'al'. 4. 'i' is equal to the sum of the function 'f' a...
The set of ordered pairs {a, b}, where both a and b are elements of set A and the ordered pair {a, b} is an element of relation r, is small.
The lemma named "convex_cone" assumes that set C is a subset of the carrier vector of dimension n. It shows that the cone of set C is convex.
: if a condition 'p' is a subset of a set of states 's' such that if 's' is in 'b' then 's' is in 'w', and if 's' is not in 'b' then 's' is in 'w'', and if it is valid that 'w' leads to 'q' through 'c1', and it is also valid that 'w'' leads to 'q' through 'c2', then it is valid that 'p' leads to 'q' through the conditi...
The lemma "osubst_preserves_safe" assumes that the substitution "sigma" is state-safe. It shows that if the ordinary differential equation (ODE) is operationally safe and the substitution "sigma" is admissible for the ODE with respect to a set U, then the ODE obtained by substituting "sigma" into the original ODE is al...
A run "r" is an accepting run if and only if there exists another run "r'" such that the first element of "r'" is "r" and "r'" is an accepting run in the degenerate case with transition system "T" and acceptance condition "m".
If Fr_6 holds for a frame F, then MT1 holds for the complement of a proposition.
The lemma named "partial_heap_added_first_el" assumes that "l" is less than or equal to "m", "m" is less than or equal to "r", and the function "is_partial_heap" with parameters "cmp", "heap", "l", "m + 1", and "r" is true. Under these assumptions, it shows that the function "is_partial_heap_except_down" with parameter...
The lemma named "frameIntComposition" is defined with the following parameters: - two elements, Psi and Psi', of an arbitrary type 'b - a list of names, AF - another element, PsiF, of the same arbitrary type 'b The lemma assumes that Psi is equivalent to Psi'. It then shows that the frame composed of AF and the tens...
A trivial interaction (denoted by \<I>_trivial) of the sum (denoted by \<oplus>\<^sub>\<I>) of two interactions (\<I>\<^sub>1 and \<I>\<^sub>2) is equivalent to each of the two interactions being trivial.
The hyperdual extension of the sine function, when applied to the product of a variable 'x' and the hyperdual number 'e2', is equal to the product of 'e2' and 'x'.
If "x" is less than or equal to "y", then the star operation applied to "y" and the star operation applied to "x" and "z" is equal to the star operation applied to "y" and "z".
If the following conditions are met: 1. A variable 'x' is equal to another variable 'x''. 2. For any 'a' and 't', if 'x' is equal to an array 'STArray' with elements 'a' and 't', then the function 'f' applied to 'a', 't', and 's' is equal to the function 'f'' applied to 'a', 't', and 's'. 3. For any 'a' and 't', if 'x...
If the generalized depth-first search (gen_dfs) does not fail (as per assumption NF) and if the generalized reachability condition (gen_rwof') holds for a state s (as per assumption A), then the generalized reachability condition (gen_rwof) also holds for the state s.
The lemma named "prv_suc_mono_LLq" assumes that "t1" and "t2" are arithmetic terms. It shows that if "t1" is less than or equal to "t2", then the successor of "t1" is less than or equal to the successor of "t2", and this statement is provable.
If a set V is not connected, then the radius is infinity.
The lemma "scc_root_unique_root" assumes two conditions: "scc_root s v scc" and "scc_root s v' scc". If these two conditions are met, it shows that "v" is equal to "v'". This means that in the context of this lemma, if "v" and "v'" are both roots of the same strongly connected component (scc) in a given state (s), then...
If variable 'v' equals 1 when condition 'P' is true and 0 otherwise, and variable 'w' equals 1 when condition 'Q' is true and 0 otherwise, then the product of 'v' and 'w' equals 1 if both conditions 'P' and 'Q' are true, and 0 otherwise.
If the set U is a subset of the unaffected domains of I, D, and the image of the set ys under D in the sequence zs, then the reverse auxiliary function of ipurge_tr for the sequence xs appended with zs is equal to the reverse auxiliary function of ipurge_tr for the sequence xs appended with ys and the auxiliary functio...
: the union of the negation of x and the complement of y equals 1, and the intersection of the negation of x and the complement of y equals the bottom element if and only if the complement of x equals the complement of y.
The substitution of a polynomial function 'f' at the sum of 's' and 't' is equal to the product of the substitution of 'f' at 's' and the substitution of 'f' at 't'.
If a tuple (evs, S, A, U) is in the protocol, then the symmetric key 'n' is in U.
In the context of a network, the lemma named "am_s_is_incoming" assumes that "am" is an adjacency map. It shows that the image of the source node "s" under the edge relation "E" is equal to the set of the adjacency map "am" at "s".
The lemma named "contains_distinguishing_trace_correctness" assumes that "M" is observable and "q1" and "q2" are states in "M". It shows that "contains_distinguishing_trace M t q1 q2" is equal to the existence of an "io" such that "io" is in "t" and "io" distinguishes "M q1 q2".
The scalar multiple of a constant "c" and a polynomial "p" divides 1 if and only if both "c" divides 1 and "p" divides 1.
If X and Y are elements of a power set, and the simple map of X is a subset of the simple map of Y, then X is a superset of Y.
The lemma named "Hintikka_Extend" is defined with a fixed set H of formulas. It assumes that this set H is consistent, maximal, and saturated. The lemma then shows that the set H is a Hintikka set.
If a property P holds for a binary element 'a' if and only if it holds for '1-a', and if the property P holds for 'a', then the property P also holds for any binary element 'x'.
The lemma named "le_of_comp_of_ords_linorder" assumes that "le" and "lt" form a linear order according to the class "linorder". It shows that the "le" (less than or equal to) relation derived from the comparison function "comp_of_ords" applied to "le" and "lt" is equal to the original "le" relation.
The ordinal of a singleton hashed multiset is equal to omega multiplied by the ordinal of the hashed multiset k.
The lemma named "inv_image_partition" assumes that for any element x in the set xs, P x holds true, and for any element y in the set ys, P y does not hold true. It shows that the inverse image of the partition P mapped to the pair (xs, ys) is equal to the shuffles of xs and ys.
The lemma provides simplification rules for binding over lazy lists: 1. Binding a bottom (undefined) lazy list with any function results in a bottom (undefined) value. 2. Binding an empty lazy list with any function results in an empty lazy list. 3. Binding a non-empty lazy list (consisting of an element and the rest ...
The set of free variables in the intruder knowledge of a stateful strand S is a subset of the well-formed restricted variables of the stateful strand S.
If x is greater than or equal to zero, then the result of dividing x by the square root of x is equal to the square root of x.
The lemma named "contraction_dist" is defined with a fixed function 'C' that maps from a complete space to itself. It assumes that for any two elements 'v' and 'u' in this space, the distance between 'Cv' and 'Cu' is less than or equal to 'c' times the distance between 'v' and 'u'. It also assumes that 'c' is a non-neg...
In the context of well-founded sub-relations, there is a lemma that states: if a function 'g' extends 'subs' to 'subs'', then 'subs'' is a well-founded sub-relation of 'g'. This is assumed by the statement "extends g subs sub subs'".
The function 'nat_to_string' when applied to any natural number 'n', produces a string that is a valid representation of a natural number.
The lemma named "ex_cross_ratio" assumes that "u" is not equal to "v", "u" is not equal to "w", and "v" is not equal to "w". It shows that there exists a "z" such that the cross ratio of "z", "u", "v", and "w" equals "c".
If there is an adjacent path from vertex v to a list of vertices ps appended with vertex q, and if the list of vertices ps is not empty, then the last vertex in ps is adjacent to vertex q.
The lemma named "ordertype_omega_omega" is about a function F that maps natural numbers to sets of lists of natural numbers. It assumes that for any natural number j, the order type of the set F(j) under the length-lexicographic order is omega raised to the power of j. Under this assumption, it shows that the order typ...
The lemma named "swapcols_is_transp_swap_rows" assumes that matrix A is an n by m matrix, and that k and l are less than m. It shows that swapping the kth and lth columns of matrix A is equivalent to transposing matrix A, swapping the kth and lth rows of the transposed matrix, and then transposing the result.
If in the map 'm', 'a' maps to 'Some b', then changing the map 'm' at 'a' with the function 'f' applied to 'b' results in the same map 'm' where 'a' is mapped to the result of 'f' applied to 'b'.
The function "mirror_point" is injective when applied to any real vector 'x'.
If a function f is primitive recursive, then there exists a number k such that for all numbers l, the value of function f at l is less than the Ackermann function of k and the sum of the list l.
If a transformer "t" is healthy, then the less than or equal to transition function applied to a function "P" and a state "s", which always returns 0, is less than or equal to the transformer "t".
The relation of function definitions is empty when both functions return None.
: Assuming that "d1" is in the atlas of the source manifold and "d2" is in the atlas of the destination manifold, and that "y" is in the domain of "d1", the carrier of the source manifold, and the image of "f" at "y" is in the domain of "d2", then there exists a set "X" such that: 1. The composition of the applicatio...
If for all elements x and y in a set S, if relation RR holds between x and y, then relation RR' also holds between x and y (as per the assumption RRREF), then the result of the function FOREACHoci with parameters RR, I, S, c, f, and initial state σ0 is less than or equal to the result of the same function with paramete...
The lemma named "conn2_bb" establishes a relationship between a blackboard operation and a knowledge source operation. It takes two fixed parameters: "k" and "kid" of type 'kid. It assumes that "kid" is parallel in the context of "k". It then shows that the blackboard operation on the state of "the_bb" in the context o...
If proposition 'p' satisfies both the axioms AxPT and AxP4, and if 'M' is a reflexive and transitive model, and 'w' is an element of the set of possible worlds in 'M', then 'p' is true in the model 'M' at the world 'w'.
If a pair (start A, q) is in the steps of A following a non-empty string u, and A ends at p, then the pair (p, q) is in the steps of the sum of A following the same string u.
The function "next_plane" preserves the invariant "inv" for a given plane "p".
If a node "r" with a set of children "xs" equals to "t", and "x" is in the image of the first element of the finite set "xs", then the set of descendant vertices of "x" is a proper subset of the set of descendant vertices of "t".
Special soundness is equivalent to the following condition: for all h, a, e, z, e', and z', if e and e' are in the challenge space, h is in the valid public, and e is not equal to e', then if the check of h, a, e, z and the check of h, a, e', z' both hold, the binding of the special soundness adversary function with h,...
The function "incr_list1" is refined by the function "incr_list" in the identity relation. This means that for every input, "incr_list1" and "incr_list" produce the same output.
The composition of the 'dual' function followed by the 'undual' function is equivalent to the identity function.
If we have a distinct list obtained by mapping a function 'f' over a list 'l', and if 'f x' equals 'f y' where 'x' and 'y' are elements in the list 'l', then 'x' must be equal to 'y'.
The lemma about the components of the opposite semicategory states the following: - The set of objects in the opposite semicategory is the same as the set of objects in the original semicategory. - The set of arrows in the opposite semicategory is the same as the set of arrows in the original semicategory. - The domain...
The application of function "f" to "x" equals "Some y" if and only if there exist "f'" and "x'" such that "f" equals "Some f'", "x" equals "Some x'", and "y" equals the result of applying "f'" to "x'".
The lemma "inputPushResLeft" is defined with the fixed elements: Psi of type 'b, x of type name, M of type 'a, xvec of type "name list", N of type 'a, and P of type psi, which is a tuple of 'a, 'b, 'c. The lemma assumes that the relation "Rel" is equivariant, and that x is fresh for Psi, M, xvec, and N. It also assume...
The lemma "new_element_ptr_in_heap" assumes that a new element is created in heap 'h' resulting in a new heap 'h'', and that the new element is associated with a pointer 'new_element_ptr'. It shows that the 'new_element_ptr' is in the set of element pointers in the new heap 'h''.
The intersection of 'x' and 'nc' added to the product of 'x' and zero equals 'x'.
If a program 'P' is well-formed according to the metadata well-formedness condition 'wf_md', then there exists a class 'C', fields 'fs', and methods 'ms' such that the class 'Object' in program 'P' is defined as a tuple of 'C', 'fs', and 'ms'.
The lemma is named "dominated_convergence_at_top". It fixes a function 'f' that maps a real number to a Euclidean space. It assumes that for any real number 'k', the function 'f' at 'k' is integrable over a set 's'. It also assumes that another function 'h' is integrable over the same set 's'. Furthermore, it assumes t...
The field of the set "rel" is equal to the set of the "exefield" of "rel".
For any evaluation function 'eval' with parameters 'e', 'f', and 'g', the evaluation of substituting constants 'cs' in the conjunction of a set 'G' is equal to the evaluation of the conjunction of the set obtained by mapping the substitution of constants 'cs' over 'G'.
The lemma "path_is_trancl" assumes that there is a path "l" from node "u" to node "v" in the edge relation "E" and that this path is not empty. It shows that the pair "(u, v)" is in the transitive closure of the edge relation "E".
A Braun tree "t" corresponds to a list "xs" if and only if "t" is a Braun tree and "xs" is the list representation of "t".
The identity function is idempotent.
If c0 is less than or equal to c1, then the sequence of d followed by c0 is less than or equal to the sequence of d followed by c1.
The lemma named "mat_adj_inv" assumes that the determinant of matrix M is not equal to 0. It shows that the adjugate of the inverse of matrix M is equal to the inverse of the adjugate of matrix M.
An omega pair is equal to the set of the first element of the pair cross-multiplied with the set of elements 'y' such that 'y' is an element of the second element of the pair.
The lemma named "canonical_separator_states" assumes that the pair (s1, s2) is in the states of the canonical separator of machine M with respect to states q1 and q2. It also assumes that q1 and q2 are states in machine M. It then shows that the pair (s1, s2) is in the states of the product of the machines resulting fr...
The intersection of an empty set and any natural number interval is an empty set.
The positivity of a literal L remains the same after a substitution operation σ is applied to it.
Adding a value "r" to itself is equal to the value "r".
If the block map of 'b' equals the scan of 'r', and the fetch of 'tp', 'st', and 'b' equals '(nact, ns)', and 'st' is less than or equal to the length of 'tp' divided by 2, then the action of the code of 'tp' and 'st' with respect to 'r' is equal to the action map of 'nact'.
The lemma named "distinct_is_map" assumes that the first elements of each pair in the list 'm' are distinct when mapped. Under this assumption, it shows that 'm' can be considered a set map.
If the initial state 's' satisfies the property HInit, then it also satisfies the property HInv5.
The negation of the union of two relations 'r' and 's' is equal to the intersection of the negations of 'r' and 's'.
The function "viterbi" with parameters "s", "t", and "os" is equivalent to the first element (or "fst") of the state resulting from running the function "viterbi_ix\<^sub>m'" with parameters "os" (converted to an immutable array), "s", "t", and "0", starting with an empty mapping.
: Assuming that for any x, y, and z, if x is transitively related to y through relation R and y is directly related to z through relation R, and if property P holds for x and z, then property P also holds for y. Furthermore, if x is transitively related to y through relation R, and y is transitively related to z throu...
The lemma named "bd" assumes the following conditions: 1. "s" is less than or equal to "t" (ie). 2. "C" has a rho bound of 1 (rb1). 3. "D" has a rho bound of 2 (rb2). 4. The difference between "D" at time "t" and "D" at time "s" is less than or equal to the difference between "C" at time "t" and "C" at time "s" (PC_ie...
A relation P is trivial if and only if its inverse is also trivial.
If "c" is a singular relative cycle of dimension "p" in the topological space "X" with sub-space "S", then the cycle obtained by multiplying "c" with a constant "k" (denoted as "frag_cmul k c") is also a singular relative cycle of the same dimension in the same topological space with the same sub-space.
The lemma named "Riemann_localization_integral_range_half" assumes that a function 'f' is absolutely integrable on the interval from negative pi to positive pi, and that 'd' is a positive number less than or equal to pi. It shows that the difference between two Lebesgue integrals, one over the interval from 0 to pi and...
A valid path is equivalent to a valid chain list. It assumes that a path is equivalent to a chain subdivision, the chain is a boundary chain, and for all elements in the chain, the path is valid. It then shows that the path is valid.
If we have two categories denoted by "\<AA>" and "\<BB>", and a natural transformation "\<NN>" from a functor "\<SS>" to another functor "\<SS>'" in the functor category of the product of "\<AA>" and "\<BB>" to "\<CC>", all in a universe "\<alpha>", and if we define "\<T>" as the bifunctor flip of "\<AA>", "\<BB>", and...
If 'x' is an element of the carrier of Zp and 'x' is not an element of the nonzero elements of Zp, then 'x' must be equal to zero.
For any polynomial 'p' in a normalization semidomain and an integral domain with division, the set of roots of the normalized version of 'p' is equal to the set of roots of 'p' itself.
The value range of a relation 'r' is a subset of the union of the union of 'r'.
If the coefficient of the zeroth term of a formal power series 'f' is not zero, then the subdegree of 'f' is zero.
The lemma named "use_premise" assumes two conditions: first, that "t" is healthy, and second, that for any state "s", if "P" holds for "s", then "t" applied to "Q" at state "s" is greater than or equal to 1. Given these assumptions, it shows that "P" entails "t" applied to "Q".
If x is independent of y, then putting u into x after putting v into y on a state sigma is the same as putting v into y after putting u into x on the same state sigma.