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Split-Isa-AF-MMA

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: Assuming that the states 's' and 's'' are equivalent with respect to 'Low', 'High' is not in the backward slice of 'S', 'Low' is in 'S', there is a path from 'Entry' to 'Low' through 'as' that is valid in state 's', and there is a similar path through 'as'' that is valid in state 's'', then the states resulting from ...
Lemma 2: "The union of Gamma and AX10 proves that F or not F is true."
The inverse of a non-negative extended real number 'a' equals infinity if and only if 'a' equals zero.
The set of all states 'q' for which there exists a function 'f', a state 'p', and a set of states 'ps' such that 'f ps' transitions to 'p' according to the rules of automaton 'A' and 'p' is either equal to 'q' or 'p' is transitively related to 'q' through the epsilon relation of 'A', is finite. In other words, this lem...
Multiplying the number 1 with any element 'a' from a generic group 'G' will result in the element 'a' itself.
The set of items less than or equal to 'k' in 'I' is a subset of 'I'.
Any quaternion 'q' can be expressed as the sum of its real part and its three imaginary parts, each multiplied by the corresponding imaginary unit (i, j, k).
If r is equivalent to s, then r is approximately equal to s.
For a finite state machine with precomputations, the implementation of the finite state machine with precomputations derived from a list is equivalent to the implementation of the finite state machine with precomputations from the same list.
The lemma named "farkas_coefficients" assumes that "cs" is finite. It shows that there exists a "C" such that "C" is a Farkas coefficient of "cs" if and only if there does not exist a "v" that satisfies "cs" under the constraint semantics.
If set B is a subset of set A, then the cardinality (or size) of the difference between set A and set B is equal to the cardinality of set A minus the cardinality of set B.
If 'm' is greater than zero, then the product of the antipode of 'm' and infinity equals infinity.
A cone in a category with respect to a covariant functor is equal to the composition of the hom-functor of the category of functors and the opposite of the delta functor.
The delay bisimulation divergence of two transition systems (trsys2 and trsys1) with the bisimulation and thread-local simulation relations flipped, and the tau-moves also flipped, is equal to the delay bisimulation divergence of the two transition systems (trsys1 and trsys2) with the original bisimulation and thread-l...
The lemma named "abort_parametric" includes lifting syntax and shows that if two instances of the Code.abort function are equal, and if applying these functions to two equal arguments results in a value of type A, then these two applications are also equal.
: Assuming that the function 'f' is defined on the set 'A' (as indicated by "f_range_on A f"), and for any subsets 'B' of 'A' and any elements 'x' and 'y' in 'B', if 'y' is in the function 'f' applied to 'B' and both 'x' and 'y' are in 'B', then 'y' is in the function 'f' applied to the set containing 'x' and 'y' (as ...
: Assuming the following conditions: 1. If the gas in state 'st' is less than or equal to 'c', then throw 'Gas st', otherwise subtract 'c' from the gas in state 's' and return the result in state 'st'. This operation results in a normal state 's''. 2. The expression 'e' with parameters 'ad', 'e\<^sub>p', 'e', 'cd', and...
The function "chop_sup" is commutative, meaning that the order of the arguments "a" and "b" does not affect the result. In other words, "chop_sup" of "a" and "b" is equal to "chop_sup" of "b" and "a".
The intersection of a set "bl" with itself is equal to the size of the set "bl".
"it is necessarily true that if 'F' is always eventually true, then 'F' is always true in the next state, given the variable 'v'."
The set of all elements greater than or equal to a certain value 'l' is not empty.
The lemma named "has_vector_derivative_withinD" is about a function 'f' that maps real numbers to a Euclidean space. It assumes that 'f' has a vector derivative 'f'' at a point 'x0' within a set 'S'. The lemma then shows that the function which maps 'x' to the result of '(f x - f x0) / (x - x0)' tends to 'f'' at 'x0' w...
The signature of the product of two tree automata A and B is a subset of the union of the signatures of A and B.
The limit superior of the sequence obtained by shifting the sequence u by one position to the right is equal to the limit superior of the original sequence u.
The lemma of the ceiling function of a rational number is equal to the ceiling of the rational number itself.
Z is a 1-dimensional gate.
The graph of a deterministic Rabin automaton (DRA) A is indeed a graph.
The cardinality (or the number of elements) of the set of partitions 'p' of a number 'n' is equal to the cardinality of the set of counts 'N' that partition the number 'n'.
The extended class is a subclass of the base class.
In the context of a ring, the lemma named "simple_extension_incl" assumes that "K" is a subset of the carrier "R" and "x" is an element of the carrier "R". It shows that "K" is a subset of the simple extension of "K" with "x".
For a fixed function 'm' mapping from a finite set 'n' to real numbers, the image of the function that multiplies 'm' at 'k' with 'x' at 'k' over the closed interval box from 'a' to 'b' is equal to the following: if the closed interval box from 'a' to 'b' is empty, then the result is also an empty set; otherwise, the r...
If S is a bounded set, then the supremum (the least upper bound) of the set that results from inserting a number x into S is equal to x if S is an empty set; otherwise, it is equal to the maximum of x and the supremum of S.
Given the assumptions that "q1" and "q2" are in the homogenous components of "p", and "t1" and "t2" are in the keys of "q1" and "q2" respectively, it shows that the degree of the polynomial map of "t1" is equal to the degree of the polynomial map of "t2" if and only if "q1" is equal to "q2".
The uncurried version of "marl_set" is analogous to the uncurried version of the function that first applies the "op_list_set" operation and then returns the result. This analogy holds under the assumption that for any input ((l,i),_), the index i is less than the length of the list l. The input is assumed to be a depe...
The pochhammer of 0 and any natural number n equals 0, if n is greater than 0.
X is equal to the reverse permutation of y by pi. The lemma then shows that the permutation of x by pi is equal to y.
The repetition function of 0 with any function 'f' is equal to 0.
The lemma demonstrates that for a given precedence 'd' and XML data 'xml', appending two lists 'r' and 's' and then applying the 'shows_prec' function is equivalent to applying the 'shows_prec' function on 'xml' and 'r' first, and then appending 's'.
The lemma named "pivot_eq_satisfies_eq" assumes that "y" is in the set of right-hand side variables of equation "e". It shows that a valuation "v" satisfies equation "e" if and only if "v" satisfies the equation obtained by pivoting "e" on "y".
If the natural number order is order-isomorphic to the cardinality of set A, then set A is not finite.
, under the assumption that the pair (f, length of ss) and the pair (g, length of ts) both belong to the set F, the function "mk" applied to f and ss equals the function "mk" applied to g and ts if and only if f equals g and ss equals ts.
The lemma named "disct_fct_point_measurable" assumes that "f" is a discrete function and "f" is point measurable in the space of "N" under measure "M". Under these assumptions, it shows that "f" is measurable under measures "M" and "N".
The lemma "gbm_invar_split" assumes that "gbm_invar g" holds true. It shows four things: 1. "m1.invar g" is true. 2. For any "v" and "m2", if "m1.\<alpha> g v" equals "Some m2", then "m2.invar m2" is true. 3. For any "v", "m2", "v'" and "s3", if "m1.\<alpha> g v" equals "Some m2" and "m2.\<alpha> m2 v'" equals "Some s...
For a given term 't', if 't' is locally closed (as per the assumption "lc t"), then 't' can be reduced to itself through parallel beta reduction.
If a filter F satisfies the condition FF, then it satisfies the condition CoP2 for negation.
In the context of a pre-digraph base, there is a lemma stating that if Z is a member of beta and alpha is a subset of beta, then the product of the objects of A indexed by I is a subset of the set V of beta.
Mapping a function 'f' over a command 'c' is equivalent to the sequence of 'c1' and 'c2' if and only if there exist commands 'c1' and 'c2' such that 'c' is the sequence of 'c1' and 'c2' and the map of 'f' over 'c1' equals 'c1' and the map of 'f' over 'c2' equals 'c2'.
The set of nodes is equal to the union of the set of all reachable nodes from the initial nodes.
For any fixed variable 'x', and for any processes 'P', 'Q', 'M', and state transformation 'σ', the following two conditions hold: 1. The state transformation 'σ' with 'x' substituted by a literal value 'v', applied to the parallel composition of 'P' and 'Q' under 'M', is equal to the state transformation 'σ' applied t...
The lemma named "ordertype_VWF_\<omega>" assumes that "A" is a finite set. It shows that the order type of set "A" under the well-founded relation "VWF" is an element of the set of natural numbers, denoted as "ω".
If two lists 'l' and 'l'' have equal length, then the semantics of the implication chain formed by zipping these two lists and folding them with the implication operator 'Imp', and a proposition 'p', is equivalent to the condition that the semantics of the two lists are equal, implying the semantics of the proposition ...
A bit set at position 'm' in a number 'a' is even if and only if the number 'a' is even and 'm' is not equal to zero.
The lemma fixes functions f and F. It defines F as a function that takes a number n and a list xs, and checks if xs is an empty list (LNil). If xs is empty, it returns an empty list. If xs is not empty, it checks if n is 0. If n is 0, it returns an empty list. If n is not 0, it returns a list with the first element of ...
If "P" is a process, then for all "s", "X", and "Y", either the pair (s, Y) is not in the failures of "P", or "X" is not a subset of "Y", or the pair (s, X) is in the failures of "P".
If "k" is less than "n", then the difference between "k" and the interval from "n" to "d" is an empty set.
"0" is a test.
The lemma named "invar_children" assumes that a node (with elements e, a, r, ts) in a skew binomial tree maintains the tree invariant. It shows that the queue invariant is maintained in the tree sequence 'ts'. Here, 'a' is of a linear order type.
The lemma "Calling_collects" assumes the following conditions: 1. P is a subtype of the Object C. 2. The pair (sigma-prime, cset-prime) is in the big-step semantics of the JVM smart collection for P and sigma. 3. Sigma is not in the JVM end set. 4. The instruction counter of the head of the frame stack of sigma is cal...
, in the absence of any specific context, if a list of "Err" values of length "n" is less than or equal to "LT" according to the partial order defined by "P", then "LT" must be a list of "Err" values of length "n".
The composition of the substitution function "substPB" with parameters "pi1" and "pi2" is equal to the substitution function "substPB" with the composition of the substitution "subst pi1" and "pi2" as its parameter.
The lemma named "continuous_sum" is about continuous functions. It fixes a function 'f' that maps from type 'a to a type 'b in a T2 space to a type 'c in a topological commutative monoid with addition. It shows that if for every 'i' in 'I', the function 'f i' is continuous at 'F', then the function that maps 'x' to the...
If a function "f" belongs to the set "Ex1", then "f" is a bijective function.
, assuming "a" is an identity, the codomain of the right unit of "a" is equal to "a".
The lemma Z_I_unitary shows that Z_I is unitary.
The lemma named "bin_packing_lower_bound_card" assumes that "S" is an empty set, "inv\<^sub>2 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L" holds, and "bp P" is true. It shows that the cardinality of the union of "P\<^sub>1", the wrap of "B\<^sub>1", "P\<^sub>2", the wrap of "B\<^sub>2", and the set of all single-elemen...
The theorem named "extends_to_equivalence_of_bicategories" shows that "equivalence_of_bicategories" holds for the given parameters: VD, HD, aD, iD, srcD, trgD, VC, HC, aC, iC, srcC, trgC, F, Phi, right_map, right_cmp, unit0, unit1, counit0, counit1.
If a command 'c' belongs to the security level 'Sec' of a policy 'Phi', then the command 'c' is secure.
If "d" is an element of "D", then the norm of the inverse of "Q" of "d" composed with "R" of "d" is less than one.
The function type of the rewritten function definition body (with parameters fd, l, pc, instr') is equal to the function type of the original function definition (fd).
If a list "xs" appended with a single element "x" is equal to a list of integers from "a" up to but not including "n_vars", and "a" is less than "n_vars", then the following conditions must hold: the list "xs" is equal to a list of integers from "a" up to but not including "n_vars - 1", the element "x" is equal to "n_v...
For any given 7-element set 'x', 'x' is equal to 1, or 'x' is equal to 2, or 'x' is equal to 3, or 'x' is equal to 4, or 'x' is equal to 5, or 'x' is equal to 6, or 'x' is equal to 7.
The lemma named "tensor_P_adjoint_left_right" assumes that "m1", "m2", "m3", and "m4" are square matrices of dimensions N by N and K by K respectively. It shows that the adjoint of the tensor product of "m1" and "m2", multiplied by the tensor product of "m3" and "m4", and then multiplied by the tensor product of "m1" a...
In the context of a factorial monoid, the division relation on G forms a weak lattice.
The lemma named "quotient_is_simulation" demonstrates that "quot" is a simulation of the residuated transition system "resid" under the condition of "Resid".
: If the following conditions are met: - The axiom EQU001_0_ax of equality is satisfied, - The axiom CAT003_0_ax of category theory is satisfied for the composition of functions f1, codomain, domain, equality, existence, and equivalence, - The equality of the composition of functions f1, codomain, domain, equivalence,...
The lemma "Pr_before_C" assumes that "L" is a subset of "H". It shows that the probability of "before_C L" occurring in a probability space, given that "hit_C" occurs, is equal to the cardinality of "L" multiplied by "p_j" and "p_f", plus the sum of "p_i l" for all "l" in "L", multiplied by "1" minus "p_H" multiplied b...
The lemma named "mcont2mcont_lmap" shows that the function "lmap f" is monotone continuous with respect to the least upper bound operator "lSup" and the partial order relation "sqsubseteq". This lemma is then used to show that "llist.mcont2mcont" is also monotone continuous, and it is marked as a simplification rule an...
The lemma assumes that function 'f' is a mapping from the set of rational numbers to the set of rational numbers. It shows that the function 'λx. 2 * f x + (f x)^2' is also a mapping from the set of rational numbers to the set of rational numbers.
If a function 'f' maps from set 'A' to set 'B', and set 'B' is a subset of set 'B1', then the function 'f' also maps from set 'A' to set 'B1'.
If the valuation functions 'v' and 'v'' agree on the set 'V' and also agree on the set 'W', then 'v' and 'v'' must agree on the union of sets 'V' and 'W'.
The initialization of model 2 is a subset of the invariant "m2_inv3_extrKey".
The auxiliary function d with inputs a, b, and c is less than or equal to the auxiliary function a with the same inputs a, b, and c.
If a bisimulation exists between 'e' and 'e'' with respect to variables 'Vs' and 'xs', and 'i' is hidden with respect to 'Vs', then 'e'' is unmodified at 'i'. Similarly, the lemma "hidden_bisims_unmods" states that if bisimulations exist between 'es' and 'es'' with respect to variables 'Vs' and 'xs', and 'i' is hidde...
If a sparse vector 'v' (of a type that is a lattice ring) is sorted, then the sparse row vector of the absolute value of 'v' is equal to the absolute value of the sparse row vector of 'v'.
The lemma "ii_notin_disc_o_x_axis" shows that the hyperbolic imaginary unit "ii_h" is in the complement of the disc on the x-axis.
For an interval I and integers k and t, the cut of the sum of I and k that is less than or equal to t is equal to an empty set if t is less than k, otherwise it is equal to the cut of I that is less than or equal to the difference of t and k, plus k.
The lemma named "step_invariantE" is about a step invariant. It assumes five conditions: 1. The assertion "A" satisfies the invariant "I" implies property "P". 2. The state "s" is reachable from the initial state "I" in the system "A". 3. The transition from state "s" to state "s'" under action "a" is in the transitio...
If set A is not empty, then there exists a list 'xs' that belongs to the n-lists of set A.
The prefix of the intersection of lists "xs" and "ys" is a prefix of the list "xs".
If the references of the set resulting from setting the value 'v' at position 'p' in the heap 'h' are in 'q' and 'rs', and 'p' is not in the set 'rs', then the references of the set resulting from setting the value 'v' at position 'p' in the heap 'h' are in 'q' and 'rs' are equal to the references of the heap 'h' in 'q...
The function "word_of_int", which converts an integer to a word, is injective (or one-to-one) on the set of integers from 0 to 2 raised to the power of the length of 'a (excluding 2 raised to the power of the length of 'a itself). Here, 'a is a type with a length.
The principal part of a formal Laurent series 'f' is equal to 0 if and only if the subdegree of 'f' is greater than or equal to 0.
The lemma named "bangSC" establishes that for any process P, the replication of P (notated as "!P") is strongly early bisimilar (notated as "\<sim>\<^sub>e") to the parallel composition of P and the replication of P (notated as "P \<parallel> !P").
The lemma named "disjoint_lookup" establishes that for any lists of words 'vs' and 'ws', 'vs' and 'ws' are disjoint if and only if there is no index 'i' such that 'i' is an index in both 'vs' and 'ws'.
If function 'f' maps from set 'A' to set 'B' and is injective on 'A', and function 'g' maps from set 'B' to set 'C' and is injective on 'B', then the composition of 'g' and 'f' (first apply 'f', then 'g') is also injective on 'A'.
If a function 'f' belongs to the induced space of a group 'G' with a subset 'FH' and a set 'V', and if 'x' and 'y' are elements of 'FH', then the value of the function 'f' at the point 'x+y' is equal to the sum of the values of the function 'f' at points 'x' and 'y'.
"khl" is less than "np".
If V is approximately equal to V' with respect to d, then the length of V is equal to the length of V'.
If the evaluation of expression e1 in state s1 results in value v1 after n1 steps and ends in state t1, and the evaluation of expression e2 in state s2 results in value v2 after n2 steps and ends in state t2, then the evaluation of expression e1 in state s1 will result in value v1 after the maximum of n1 and n2 steps a...
If the height of a tree 't' is zero, then the sibling of any node 'a' in the tree 't' is 'a' itself.