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The lemma named "the_coext_unique" assumes that there is an arrow from functor D to C, denoted by F, between x' and y with respect to function f, and that g is a co-extension of x' and f. It then shows that g is equal to the co-extension of x' and f.
: Given a fixed 'Psi' of type 'b', 'P' and 'Q' of type psi (which is a tuple of 'a, 'b, 'c'), and 'xvec' which is a list of names. Assuming that 'Psi' weakly bisimulates 'P' and 'Q', and 'xvec' is fresh for 'Psi'. It shows that 'Psi' weakly bisimulates the restriction of 'P' and 'Q' by 'xvec'.
If 'i' is less than 'n', then the 'i'-th element of the IArray created from function 'f' with length 'n' is equal to the result of function 'f' applied to 'i'.
The reciprocal Gamma function is continuous on the set A.
The lemma "exists_pos_meas_subset" assumes that "E" is a member of the set "M", the absolute value of the measure of "E" is less than infinity, and the measure of "E" is greater than zero. It shows that there exists a subset "A" of "E" such that "A" is a positive measure set and the measure of "A" is greater than zero.
The sign of a rational number, represented as a fraction "a/b", is equal to the product of the signs of the numerator "a" and the denominator "b", converted to a rational number.
If the function "top_on_opt" applied to set S and X holds true, and the variables of b are a subset of the complement of X, then the function "top_on_opt" applied to the inverse of b value prime, r, and S also holds true for X.
Swapping elements at index 'i' in a list 'xs' does not change the distinctness of the list. In other words, if a list 'xs' has distinct elements, then after swapping any two elements in 'xs', the new list will also have distinct elements.
A property P eventually holds at a neighborhood of a point a if and only if there exists an open set S such that a is in S and the property P holds for all points in S.
If "z" is implied by the supremum of "x" and "y", then "z" is also implied by "x".
The lemma named "chromatic_lb" assumes that the graph G is well-formed, finite, and non-empty. It shows that the cardinality (or size) of the vertices of G divided by the independence number of G is less than or equal to the chromatic number of G.
The set of elements in the concatenated list "xss" is a subset of the union of the set of elements in each list within "xss".
If 'V' is a finite set, then the simultaneous substitution of 't' in the set obtained by inserting 'i' into 'V' with respect to 'F' is equal to: if 'i' is in 'V', then the simultaneous substitution of 't' in 'V' with respect to 'F'; otherwise, the substitution of 'i' with 'F i' in the simultaneous substitution of 't' i...
The length of the power polynomial with variables x, y, z, and n is equal to n.
The product of two multisets, each containing a single element "a" and "b" respectively, is equal to a multiset containing a single pair "(a, b)".
The lemma named "removeCombinable" assumes the following conditions: 1. Rule 1 (r1) is defined as a pair consisting of a list with 'p' as the first element and 'ps' as the rest, and 'c'. Also, r1 is an element of the set of upward rules (upRules). 2. Rule 2 (r2) is defined as a pair consisting of a list with 'q' as th...
Lemma: not_essential_transform Assumptions: 1. g is not essential at z. 2. For almost all w in the neighborhood of z, g(w) = f(w). Conclusion: f is not essential at z.
The total weakest precondition of an if-then-else command is a function that, given a state, evaluates the boolean condition and returns the total weakest precondition of the command that the boolean condition selects.
If B is a very well-behaved lens, A is independent of B, and x is a sub-lens of B, then the parallel substitution of A and B in sigma and rho, with rho updated at x with value v, is equal to the parallel substitution of A and B in sigma and rho, with the result updated at x with value v.
If "A" is a measurable set in the measure space "M", then the return time function of "A" is integrable over the measure space "M".
If for every element x, x being an element of set a implies that x is also an element of set b, then set a is a subset of or equal to set b.
"R1_a" is in "Apply.Disjunctive".
The set of paths in a tree 't' being empty is equivalent to 'False'. In other words, the set of paths in a tree 't' can never be empty.
If a function 'f' is in set 'A', then the result of the function 'f' for any input 'x' is in the range of set 'A'.
The parts of a key-free set are a subset of the key-free set itself.
The lemma named "fin_make_untimed_inf_truncate_Nonempty_all" assumes that the element "z" at index "k" is not an empty list. It shows that for all indices "i" such that "i" is greater than or equal to "k", the function "fin_make_untimed" applied to the "inf_truncate" of "z" at index "i" will not result in an empty list...
If a list "L" is in the set of valid lists for parameters "A", "n", and "B", then the head of the list "L" is not equal to the chain of the next shop of the head of "L", with parameters "ofr", "A", "L", and "C".
Dropping three bits from the successor of zero results in zero.
: "If an event is in the public namespace, then the private encryption key of A is part of what the spies can see if and only if A is in the set of compromised entities."
If "a" is greater than zero, then the real power of "a" to the natural number "n" is equal to "a" raised to the power of "n".
If the following conditions are met: 1) The requirements obtained from the get_IFS function applied to M are valid, 2) The graph G is a well-formed graph, and 3) The set of edgesList is equal to the edges of the graph G, then, for the stateful set defined as the set of edges that do not violate the IFS when filtere...
Lemma: Nonnegative Integral with Count Space Indicator Assuming that there is no match between the universal set (of type 'a) and the set X (of type 'a), the nonnegative integral of the function f with respect to the count space X is equal to the nonnegative integral of the function f times the indicator function of X...
The multiset of elements in a list after performing an insertion sort is equal to the multiset of elements in the original list.
: Assuming "A" and "B" are finite sets, and "F" is a function from "A" to "B" that is invariant under permutations of the domain "A" and range "B". If "F" is injective on "A" (i.e., each element of "A" is mapped to a unique element of "B" by "F"), then for all "n", if "n" is in the number partition of "A", "B", and "F"...
The image of C0 under the function s is equal to D0.
A point 'x' is in the open segment between points 'a' and 'b' if and only if there exists a 'c' in the open interval between 0 and 1 such that 'a' is not equal to 'b' and 'x' is on the line between 'a' and 'b' at the position determined by 'c'.
The Pochhammer symbol for negative of a natural number 'n' and 'n' is equal to the product of negative one raised to the power 'n' and the factorial of 'n'. This is valid in the context of a type 'a' that is a commutative ring with unity, a semiring with no zero divisors, and a semiring with characteristic zero.
The lemma, in the context of encoding with respect to barbs, is named "OC_SS_DR_wrt_preorder_iff_source_target_rel". It fixes a success of type 'barbs' and a TRel of type set of pairs of 'procT'. It shows that the operational correspondence of TRel, the preorder of TRel, the weak reduction bisimulation of TRel and Targ...
The lemma, within the context of the Cox-Ross-Rubinstein (CRR) market model, shows that the discretized function of the projected stochastic process of the geometric process at time 'n' is indeed a discretized function.
For all real numbers x and y, there exists a real number z such that if x is less than y, then x is less than z and z is less than y.
Applying the function "items_le" with parameter "k" to the result of applying the same function with the same parameter to "I" is equivalent to just applying the function once to "I". In other words, the function "items_le" is idempotent with respect to parameter "k" and input "I".
The lemma ft_app_impl asserts that the function al_app maintains the invariance and the abstract state of the finite tree, as defined by ft_invar and ft_α, respectively, when applied.
If a set 'm' is a subset of the carrier, then the set difference of the carrier and 'm' being closed is equivalent to 'm' being open.
The function Hv applied to x equals the function Hv applied to y if and only if x equals y. This implies that the function Hv is injective.
If the following conditions are met: 1. Agent B sends a message to Agent A that includes a cryptographic public key of A, along with two nonces (random numbers used only once) NA and NB, and the identity of B, and this message is in the set of events 'evs'. 2. Some agent (possibly different from A) sends a message to ...
The number of face cycle sets in graph H is equal to the number of face cycle sets in graph G minus the number of face cycle sets in graph G for vertices a and a', plus one if set S is not empty, otherwise plus zero.
The theorem named "semantics_stateful_vs_tagged" assumes that for all matchers 'm', states 'σ', and packets 'p', the stateful matcher 'm' matches the packet 'p' in the state 'σ' if and only if the tagged stateful matcher 'm' matches the packet tagged by 'σ'. Given this assumption, it shows that the stateful semantics o...
The lemma Mα_Rel_components shows the following: 1. The natural transformation map of Mα_Rel in a category is a function that takes a triple of objects in the tripled category and returns the Mα_Rel_arrow_lr of the three objects. 2. The domain of the natural transformation of Mα_Rel in a category is the left binary c...
Given a fixed strips problem 'Pi', if 'k' is less than 't', and two operators 'op1' and 'op2' are in the set of operators of 'Pi', and the index of 'op1' in the set of operators of 'Pi' is not equal to the index of 'op2' in the same set, and 'op1' and 'op2' are interfering operators, then the set containing the inverse...
If tree "t" satisfies the condition "invc2", then painting the tree "t" with color "c" will also satisfy the condition "invc2".
If a list 'xs' is equal to the concatenation of list 'ys', an element 'x', and list 'zs', and if the reverse auxiliary purge of the sources auxiliary with respect to 'I', 'D', 'U', and the list consisting of 'x' and 'zs' applied to 'ys' is an empty list, and if there exists an element 'v' in the sources auxiliary with ...
In the context of Zermelo-Fraenkel set theory, there is a lemma stating that if the function VLambda I F is an element of the set Vset alpha, then the discrete category of the transitive morphism constant functor for I and F in the category of sets with respect to alpha is valid.
The lemma named "not_wfP_silent_move_from_τdiverge" assumes that the well-founded predicate "wfP" is not true when applied to the flipped version of "silent_move_from s". If this assumption holds, it shows that "s" infinitely makes silent transitions.
The lemma named "permutation_insert_delete" assumes that "p" is a permutation of the set of numbers from 0 to n (inclusive), and "i" is a number less than or equal to n. It shows that if you insert "i" at the position "p i" in the permutation obtained by deleting "i" from "p", you get back "p". This is expressed as "?l...
"bold 1 is not equal to bold 2", "bold 1 is not equal to bold 3", and "bold 2 is not equal to bold 3".
Under the context of G and A, the term t is valid from precondition P to postcondition Q. 2. It assumes "P", which states that P holds for Y, s0, and Z. 3. It assumes "valid_A", which states that for all terms t in A, t is valid under G. 4. It assumes "conf", which states that s0 is conformant to (G,L). 5. It assumes "...
, under the assumptions that the real value of the signed integer "ra" is less than the real value of the signed integer "POS_INF" and the real value of the signed integer "NEG_INF" is less than the real value of the signed integer "ra", it can be shown that "ra" is not equal to "NEG_INF".
If S is an affine set and x, y, z are elements in S, then the result of the operation "x plus v times the vector from z to y" is also an element in S.
The complex conjugate of the negation of any complex number z is equal to the negation of the complex conjugate of z.
For a given set X and a real number r, if r is less than 1 and for all x in X the measure of Kx in X is less than or equal to r, then for almost every omega in Tx, the event of holding in the complement of X occurs.
The domain of the object map of the Hom-functor in a category is equal to the product of the opposite category and the category itself.
: - Positive infinity is not equal to zero. - Zero is not equal to positive infinity. - Negative infinity is not equal to zero. - Zero is not equal to negative infinity.
Evaluating the "and" operation of 'a' and 'b' in a list 'L' is equivalent to evaluating 'a' in list 'L' and evaluating 'b' in list 'L'.
: 1. Casting an unsigned 64-bit word representation of a natural number 'x' to a signed 64-bit word representation results in the same natural number 'x'. 2. Casting an unsigned 64-bit word representation of a natural number 'x' to a 16-bit signed word representation results in the same natural number 'x'. 3. Casting a...
The lemma shows that the head strategy is a reduction strategy.
Applying the inverse function of 'f' to the result of function 'f' applied to 'x' gives 'x' back.
In the lemma for group homomorphism, if 'x' is an element of the carrier 'G', then the homomorphism of the inverse of 'x' is an element of the carrier 'H'.
The set of natural number permutations of 'n' taken 'k' at a time is finite.
"kruskal1" is related to "kruskal0" in the non-deterministic result relation, where the relation is the identity relation.
In the lemma from the Valuation2 theory in Valuation, within the Corps, if distinct polynomials K and n exist for any P, then the valuation of K for any P is not equivalent for any n.
If a pair of states (s,t) satisfies the action A, then the pair of states (s,t) also satisfies the action A in the context of variable v.
The lemma named "admissible_transactions_fv_bvars_disj" assumes that for all transactions T in the set P, T is an admissible transaction. It shows that the union of the free variables of all transactions in the set P and the union of the bound variables of all transactions in the set P are disjoint, meaning they have n...
If it is valid that for any inputs 'a' and 'c', 'P a c' and 'bval b a' hold, then it is also valid that for any input 'P', if 'b' is true then 'c1' is executed, else 'c2' is executed.
If in a subgraph of G induced by S, there is a path from vertex u to vertex v, and S is a subset of the vertices of G, then there is a path from u to v in the graph G.
The union of sets, where each set is defined by the condition that the function 't' applied to 's' equals 'v' and 'v' is less than 'w', is equal to the set of all 's' for which the function 't' applied to 's' is less than 'w'.
For a function "spmf_of_prob_optionT", if it transforms an element according to the relation "cr_spmf_prob_optionT A", then it also transforms the same element according to the relation "rel_spmf A". Here, "x" is any arbitrary element and "A" is any arbitrary set.
"diamond_cube" is equal to "diamond_cube_gen".
The lemma named "pt_prod_inst" assumes that for any types 'a and 'x, 'a is a permutation type of 'x, and for any types 'b and 'x, 'b is also a permutation type of 'x. It shows that for any types 'a, 'b, and 'x, the product type of 'a and 'b is a permutation type of 'x.
For a fixed matrix 'A' of any size, which elements are from a Bezout ring, there exists an index 'm' greater than 'i' such that the element of 'A' at the position (m, k) is not zero if and only if the vector obtained from the k-th column of 'A' starting from the index 'i+1' is not an all-zero vector.
If M is an element of set A and N is an element of set B, then the sequence consisting of M followed by N is an element of the unordered product of sets A and B.
The lemma "indep_in_subI_subset" assumes that set E' is a subset of set E and that X is independent in set E'. Under these assumptions, it shows that X is also independent in set E.
The function "reverse" is related to the function "RETURN o op_list_rev" through a relation that transforms a list (is_list) into another list (is_list).
The lemma named "Close1" is defined with the following parameters: P, y, P'', a, x, P', Q, and Q' where P, P'', P', Q, and Q' are processes in the pi calculus, and y, a, and x are names in the pi calculus. The lemma assumes the following: 1. P can weakly transition to P' by outputting x on channel a and continuing as ...
: Assuming "Substable_Cond" is a substable condition for a statement "phi", it shows that if for all "v", "psi" is equivalent to "chi", and if "Theta" holds for "phi" and "psi" in "v", then "Theta" also holds for "phi" and "chi" in "v".
The hashcode of a given element 'a' is equal to the result of walking through an iarray (an immutable array) with a function that multiplies the current hash 'h' by 33 and adds the hashcode of the current value 'v', starting from the 0th index.
If a list 'xs' is finite, then the sum of the list that starts with an element 'x' and is followed by the list 'xs' is equal to 'x' plus the sum of the list 'xs'.
For any two ordinal numbers alpha and beta, the pair (alpha, beta) is in the Von Neumann universe if and only if alpha is less than beta.
: Assuming that the machine M2 is observable and minimal, and the size of M2 is less than or equal to the size of the prime version of M1 plus some additional states represented as a natural number. Also, assuming that the inputs and outputs of M2 are the same as those of M1. If M1 is already in its prime form, then M...
The lemma named "poincare_line" assumes that "z" is not equal to "w". It shows that "z" is on the Poincare line between "z" and "w", and "w" is also on the Poincare line between "z" and "w".
If a tableau "s" is normalized (denoted by "\<triangle> (\<T> s)"), and if "x\<^sub>i" is in the set of left variables of "s" (denoted by "x\<^sub>i \<in> lvars (\<T> s)"), and "x\<^sub>j" is in the set of right variables corresponding to the left variable "x\<^sub>i" in "s" (denoted by "x\<^sub>j \<in> rvars_of_lvar (...
The support of the zero function is an empty set.
The system satisfies IL in the precondition state.
The application of the function "map_add" on two maps "m1" and "m2" at a point "x" is equal to the supremum (or greatest element) of the options resulting from applying "m1" and "m2" separately at the same point "x".
If "r" is a prefix of "u", and "v" is not an empty string, then "r" is a proper prefix of the concatenation of "u" and "v".
The lemma named "expands_to_max_eq" assumes that a function "f" expands to "F" on a certain basis, and that eventually for all large enough "x", "f(x)" equals "g(x)". Under these assumptions, it shows that the function that takes "x" and returns the maximum of "f(x)" and "g(x)" also expands to "F" on the same basis.
If "x" and "y" are both co-tests, then the product of "x" and "y" is also a co-test.
Theorem T6: "If always F holds, then always the next state of F holds with respect to variable v."
The list derived from inserting an element 'x' into a distinct list 'xs' is equal to: if 'xs' already contains 'x', then it is the list derived from 'xs' with duplicate elements removed; otherwise, it is 'x' followed by the list derived from 'xs' with duplicate elements removed.
The set of all elements 'x' of a certain type 'a' in a T1 space, where 'x' is not equal to 'c', is open.
The lemma named "sigma_sets_mono'" assumes that set A is a subset of set B. It shows that the sigma set of X and A is a subset of the sigma set of X and B.