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If the following conditions are met: 1. k is greater than 0, 2. k is less than m, 3. the remainder of m divided by k is greater than 0, 4. k is less than or equal to c, then the interval from r to c with step size of the remainder of m divided by k, when divided by k, does not belong to the set of intervals.
If you apply the 'undual' operation to a value 'x' and then apply the 'dual' operation to the result, you will get back the original value 'x'.
The well-founded partial order is on the measure on function 'f' over set 'A'.
The lemma named "set_tag_name_get_child_nodes_is_l_set_tag_name_get_child_nodes" is an instance of the "l_set_tag_name_get_child_nodes" type, which is well-formed. This lemma involves the functions "set_tag_name", "set_tag_name_locs", "known_ptr", "get_child_nodes", and "get_child_nodes_locs".
The function "fun_evaluate_decs" with parameters "mn", "state", "env", and a list containing the result of "compile_group" function with parameters "all_consts" and "rs", equals a tuple. This tuple consists of "state" and the result of "Rval" function with the parameter "as_sem_env env".
If a term 't' is not beta reducible, then the term 't' decreased by 'n' is also not beta reducible.
The lemma named "split_by_degree_correct" is defined with fixed variables x, which is a list of real numbers, and p, which is a polynomial of floating point numbers. It assumes that the pair (l, r) is equal to the result of splitting the polynomial p by degree ord. It shows four things: 1. The maximum degree of l is l...
, under the assumptions that a store operation on a heap 'h' with a capability 'c' and a value 'val' results in a successful new heap 'h'', and that the value 'val' is a capability value 'v', it can be shown that there exists a mapping 'm' such that the lookup of the block id of 'c' in the heap map of 'h'' equals 'm'. ...
If the function 'f' returns 0 when applied to 0, and if 'f' is associative, then the function 'combine_matrix' with 'f' as an argument is also associative.
The product of the function A over the empty set is equal to the set containing only the element 0.
For every element 'x' in the set defined by the probability mass function 'config_rand' with parameters 'A', 'init', and 'qs', the length of the first component of 'x' is equal to the length of 'init'.
If the star of y times the star of x is less than or equal to the star of x times the star of y, then the star of the sum of x and y is equal to the star of x times the star of y.
The quantifier elimination in dense linear orders (qe_dlo) of the statement "for all values, for all values, there exists a value such that the first value is less than zero and zero is less than the second value" is equivalent to false.
The lemma "observable_transition_target_language_eq" assumes that: 1. The language of the machine M at the source state of transition t1 is the same as the language of the machine M at the source state of transition t2. 2. Transition t1 is in the set of transitions of machine M. 3. Transition t2 is in the set of transi...
For any 'x', if 'P x' holds true then for any 'x'', 'P x'' also holds true and 'x'' equals 'x'. The lemma then shows that the function "Eps_Opt P" equals "Some x" if and only if 'P x' holds true.
Getting the negation from the mapping of "NegPos_map" with "X" and "src" as parameters is equivalent to mapping "X" with the negation of "src".
If x is not equal to zero, then the product of the inverse of x and x itself is equal to one, in the context of octonions.
If 'i' is an ordinal number, then the set of all sets with rank less than the successor of 'i' is equal to the power set of the set of all sets with rank less than 'i'.
If sets A and B are finite, then the cardinality of set A is equivalent to the cardinality of set B if and only if the number of elements in set A is equal to the number of elements in set B.
If a pair (s', s) is in the relation "gstate_rel", then the function "gselect_edge" applied to s' is less than or equal to the function "select_edge" applied to s, under the relation that is the product of the identity relation on options and the "gstate_rel" relation.
For any word 'a' of any length, the operation of bitwise AND between 'a' and 'b' followed by a right shift by 'c' is equal to the operation of right shifting 'a' and 'b' by 'c' first and then performing a bitwise AND operation.
The carrier of multiple calls to 'x' is equal to the set containing 'x'.
The lemma "Ord_ex1_Limit_plus_in_omega" is based on Theorem 8.13 in Takeuti's "Introduction to Axiomatic Set Theory". It assumes that "alpha" is an ordinal and "omega" is a subset of "alpha". It shows that there exists a unique "beta" and a unique "n" such that "n" is an element of "omega", "beta" is a limit ordinal, a...
The lemma named "ground_gctxt_of_ctxt_apply" assumes that the context "C" and the term "t" are both grounded. It shows that the term of the grounded term resulting from applying the grounded context "C" to the grounded term "t" is equal to the result of applying the context "C" to the term "t".
The lemma named "identity2" establishes a property of a function "f" that maps natural numbers to natural numbers. It assumes that the function applied to a natural number "k" equals "k" itself and that "k" is greater than or equal to 2. It also assumes that for any two natural numbers "m" and "n", the function applied...
If a state variable 'sv' is in the set of atoms resulting from encoding the effect of an operator 'op' at time 't' in a problem with no operation (noop) abstracted, and if 't' is less than 'h', and if the operator 'op' is in the set of operators of the problem with no operation abstracted, then the state variable 'sv' ...
The upper bound of the product of a floating point interval and another interval is equal to the second element of the bounds of the product of the lower and upper bounds of the two intervals.
If a natural transformation N is an isomorphism from functor F to functor G in the category of functors from category B to category C, and if there is a functor H from category A to category B, then the composition of the natural transformation N and the functor H is an isomorphism from the composition of the functors ...
If a predicate P holds for an element 'a', then taking elements from the beginning of a sequence of 'a' replicated 'x' times (denoted as a\<up>x) while the predicate P holds, will result in the original sequence of 'a' replicated 'x' times.
: Assuming that: - A' is a matrix in the carrier of size m by n, - a is less than m, - n is greater than 0, - A is defined as A' appended by D times the identity matrix of size n, - The element at the position (a,0) in matrix A is not equal to 0, - m is greater than or equal to n, - For all x in the set xs, x is less ...
The forward derivative of R with respect to X is a subset of Y if and only if X is a subset of the backward derivative of R with respect to Y.
For every natural number x, there exists a natural number y, such that if zero is less than or equal to five, then y is equal to five plus x.
The lemma named "tailrec_transform_aux2" is defined with a fixed "m" of type "'a nres". It shows that the least fixed point of a function "x" such that it is the supremum of "m" and the bind of "x" and "a", is less than or equal to the bind of "m" and the recursive function "D" such that it is the supremum of the bind ...
If X is an element of the parts of the kparts of H, then X is also an element of the parts of H.
The second parallel covariant functor from categories 'aa', 'bb', 'gg', 'ff' to 'aa'', 'bb'', 'gg'', 'ff'' is a tiny map functor in the category 'CC' with respect to 'alpha'.
If a system A with metadata security (mds) is globally consistent, and X is a subset of the domain of A, then the restriction of A to X is also globally consistent with the same metadata security.
The lemma named "coprime_h_c_poly2" is about a fixed polynomial 'h' of type 'mod_ring poly'. It assumes that the polynomial 'h' subtracted by the constant 'c1' is coprime with the polynomial 'h' subtracted by the constant 'c2'. It also assumes that the polynomial 'h' subtracted by the constant 'c1' is not a unit. Under...
The lemma named "interpretation_comp" assumes that "theta" is an interpretation in a substitution. It then shows that both the composition of "sigma" and "theta", and the composition of "theta" and "sigma" are also interpretations in a substitution.
The lemma named "random_serial_dictatorship_singleton" assumes that the set of agents and alternatives are both finite, the subset of agents is included in the set of agents, and x is an element of the set of alternatives. It shows that the random serial dictatorship of the subset of agents with x as the only alternati...
If r and s are both positive, i is less than or equal to the product of r and s minus 1, and the quotient of i divided by s is less than r minus 1, then the successor of the function "rtos" with parameters r, s, and i modulo r equals the successor of the quotient of i divided by s.
The function "show_real" applied to a rational number "x" (represented as "Ratreal x") is equal to the function "show" applied to "x".
If a graph G is combinatorially planar (as assumed), then it is also Kuratowski planar.
In the context of a cancellative monoid, the lemma named "associatedE2" is stated as follows: Assuming that 'a' is associated with 'b', and given a unit 'u' such that 'a' equals the product of 'b' and 'u' (with 'u' being a unit in the group 'G'), and assuming that both 'a' and 'b' are elements of the carrier 'G', then...
In the lemma for a conditionally complete linear order, if set A is not empty and the image of function f over set A is bounded below, then the infimum (greatest lower bound) of the image of function f over set A is less than or equal to x if and only if for all y greater than x, there exists an i in set A such that y ...
The sets of Invariants are equal to the set of all subsets A of M such that the preimage of A under transformation T intersected with the space M is equal to A.
The lemma "assign_eval_w_const_A" shows that for a given constant "c" and variable "x", the assignment of "c" to "x" in the context of "Eg1", along with the memory state before and after the assignment, is included in the weak evaluation "A.eval_w".
If the sum of 'a' and 'a' equals zero, then the sequence formed by appending 'a', 'a' to the sequence 'as' and then appending 'bs' is not a reduced word in the set 'A'.
Prepending a list 'xs' to a lazy list that starts with 'y' and continues with 'ys' is equivalent to prepending a list that is the result of appending 'y' to 'xs' to the lazy list 'ys'.
The function 'na' applied to the function 'an' of 'n' equals 'n'.
The lemma named "supapartmentD" assumes that "C" and "D" are chambers. It defines "A" as the supapartment of "C" and "D". It shows that "A" is in the set of all apartments, and both "C" and "D" are in "A".
If a vector 'a' is not in the span of a set 'S' and if 'S' is an independent set, then the set 'S' remains independent even after the vector 'a' is inserted into it.
If a sequence 'C' appended to 'Gamma' is valid, and if the negation of sequence 'C' appended to 'Delta' is also valid, then the sequence formed by appending 'Gamma' and 'Delta' is valid.
If the function "gram_schmidt_sub" is applied to the vectors "us" and "ws" with the result being "us'", and if the set of vectors "ws" and "us" are both subsets of the set of n-dimensional vectors, and if the vectors "us" and "ws" are distinct and not linearly dependent, and if the vectors "us" are corthogonal (orthogo...
: Assuming that the store 's' is correctly typed according to the typing scheme 'S', and that the index 'i' is less than the length of the instances in 'S', and that the memory index of the store 's' at 'i' equals 'j'. It can be shown that 'j' is less than the length of the memories in 'S', and that the memory at in...
The lemma named "Par2B" is defined with fixed elements Q, a, x, Q', and P, where Q, Q', and P are processes in the pi calculus, and a and x are names. It assumes that there is a transition from process Q to process Q' with a bound output of name 'a' and a new name 'x'. It also assumes that the name 'x' is fresh for p...
If a pair (cl, f) is in the set of all pairs (cl, f) such that cl is in CL and f is a function from the power set of cl to the power set of cl, then f is a function from the power set of cl to the power set of cl.
The domain of the non-deterministic choice between processes P and Q is equal to the union of the domain of process P and the domain of process Q.
Given a Landau symbol L that relates to L' and Lr, and given that the function f is in Big Theta notation of the constant c1 times the evaluation of the primary functions fs, and the function g is in Big Theta notation of the constant c2 times the evaluation of the primary functions gs, then the function f is in the La...
The generic depth-first search (gen_dfsT) is less than or equal to the iterated depth-first search (it_dfsT) when transformed down by the simple state relation (simple_state_rel) on the set of end states (ES). This is in the context of refinement, meaning that the generic depth-first search is a more abstract represent...
If "k" is greater than zero, then the finiteness of the interval "I" multiplied by "k" is equivalent to the finiteness of the interval "I" itself.
In the context of a commutative group, the lemma named "IDirProds_Un" assumes that for every element 'i' in set 'A', 'S i' is a subgroup of 'G', and for every element 'j' in set 'B', 'S j' is also a subgroup of 'G'. Under these assumptions, it shows that the internal direct product of 'G' and 'S' over the union of sets...
If a word 'w' has a period 'n', then 'w' is not equal to the empty word.
Within the context of strong adversaries in key agreement, the second step of the SKL1 protocol involving parties A and B, a nonce Ni, and a generator gnx, refines the initial state of the protocol, represented by "a0i_skip_r". This is denoted by the relation "R0sk1iar". The "Id" represents the identity function, imply...
If p holds before x and q holds after x, then executing x in state p will result in state q.
The filter of a function 'f' applied to the union of two sets 'x' and 'y' is equal to the union of the filter of function 'f' applied to set 'x' and the filter of function 'f' applied to set 'y'.
The lemma named "ortho_scalprod0" assumes that "z1" is not equal to zero and "z2" is not equal to zero. It shows that the angle between "z1" and "z2" is either pi/2 or -pi/2 if and only if the scalar product of "z1" and "z2" is zero.
If a certain infinite natural number 'i' is less than another infinite natural number 'j', then 'i' is also less than the sum of 'm' and 'j'.
The split of a tree 't' is defined as follows: If 't' is a node with three sub-nodes 't1', 't2', 't3' and a fourth tree 't4', each with associated values 'x', 'y', 'z' and levels 'lvx', 'lvy', 'lvz' respectively, then if the levels 'lvx', 'lvy' and 'lvz' are all equal, the result is a new tree where 't1' and 't2' are...
For any element 'x' of a type that is a bounded lattice with a bottom element, the infimum (greatest lower bound) of 'x' and the bottom element is the bottom element.
: Assuming that "a" is a subset of "I", the approximation of the Taylor model "t" for a function "f" with precision "prec" and order "ord" in the interval "I" and environment "env" is defined, all elements of "xs" are within the range of the Taylor models "tms" evaluated at "e" in the environment "env", "e" develops a...
If "r" is a cardinal order on set "B" and there is a bijective function "f" between set "A" and set "B", then "r" is order-isomorphic to the cardinality of set "A".
The roots of a polynomial "p" within an empty set are also an empty set.
The integer 1 is equal to the V-set 1.
If "I" is an ideal of ring "R" and "x" is an element of the quotient ring "R / I", then applying the projection function "pj" to the result of the "csrp_fn" function applied to "x" will yield "x" itself.
Given a function 'f' that converges to a non-zero constant 'c' at a filter 'F', the following conditions hold: 1. The limit of the function 'f(x) / g(x)' as 'x' approaches 'F' is negative infinity if and only if the function 'g' converges to zero at 'F' and the sign of the function 'g(x)' is the opposite of the sign o...
The set of all elements greater than or equal to 'a' and less than 'b' is equal to the set of all elements greater than or equal to 'a' and less than or equal to 'b', excluding 'b'.
If a list 'xs' is forward and all elements in 'xs' are distinct, then there are no backward elements in 'xs'.
The lemma named "iterates_above_ge" assumes that "y" is in the set of iterates above "a" and "a" is in the field of "leq". It shows that the pair "(a, y)" is in the relation "leq".
If the set of all elements 'x' such that the function 'f' applied to 'x' is not equal to 'c' is finite, then the set of all elements 'x' such that the function 'f' applied to 'inc_above b i x' is not equal to 'c' is also finite. Here, 'inc_above' is another function.
The lemma "mem_read_delayed_write_low_equal" assumes that the states s1 and s2 are low equal and both their delayed pools are empty. It shows that the first element of the memory read from the delayed pool write of state s1 at the program counter value is equal to the first element of the memory read from the delayed p...
If the following conditions are met: 1. The method tauExec_methr with parameters ci, P, ins, xt, t, h, s, and s' is executed. 2. The condition Q is true for state s. 3. For any stack (stk), local variables (loc), program counter (pc), exception (xcp), and state (s'), if the method tauExec_methr with parameters ci, P, ...
If the condition CoP2 holds for a certain value η, then the condition iDM3 also holds for that same value η.
The preimage of a set S under the first projection function is equal to the Cartesian product of the set S and the universal set.
If set Z is a subset of set V, then the approximation of set Z with respect to beta is also a subset of set V.
"if not p then q" is equivalent to "if not q then p".
Lemma path_post: Assuming that there is a path f with (n # nl) as its nodes, and that nl is not an empty list, it can be shown that there is a path f with nl as its nodes.
Assuming that the Java program P is well-formed, the lemma "Reds_dom_lcl" shows that if P leads from the expression state (e, (h, l, sh), b) to the expression state (e', (h', l', sh'), b') after zero or more steps, then the domain of l' is a subset of the union of the domain of l and the free variables of e.
Applying a hyperdual function "f" to a hyperdual number with components "a", "b", "c", and "d" results in a new hyperdual number. The first component of this new hyperdual number is the result of applying the function "f" to "a". The second component is "b" times the derivative of the function "f" at "a". The third com...
The product of the integer k and the unit element, raised to the power of a natural number n, is equal to the product of k raised to the power of n and the unit element.
The minimum of positive infinity and any number 'y' is 'y'.
If Y1 is in the solutions of the function f with respect to the function T, the set S, the initial time t0, and the state s, and if the interval from t0 to t is a subset of T, and if set A is a subset of the interval from t0 to t, then set A is a subset of the existence interval of state s.
If there is a reflexive relation 'r' on set 'A' and a reflexive relation 's' on set 'B', then there is a reflexive relation on the intersection of sets 'A' and 'B' that holds if and only if both 'r' and 's' hold.
If the absolute value of x equals infinity if and only if the absolute value of y equals infinity, then the real number representation of the extended real number x minus the real number representation of the extended real number y equals the real number representation of the difference between the extended real number...
If the function 'lCoPw' applied to 'eta' under the operation 'bold right arrow' holds true, then the function 'lCoP3' applied to 'eta' under the same operation also holds true.
There is a refinement relation between the operation "hm_key_of_op'" and "hm_key_of_op". This relation is within the context of the heap map implementation relation "hm_impl1_rel" and the natural number relation "nat_rel". The result of this operation is also in the natural number relation.
Shifting the number 0 to the left by one bit results in 0.
For a fixed function 'rho' mapping from a type 'a' to a type 'b', an element 'xa' is in the effective domain of the permuted function 'rho' by 'p' if and only if the inverse permuted 'xa' by 'p' is in the effective domain of 'rho'. Here, 'a' is any type that is an instance of the 'at_base' class, and 'b' is any type th...
A triple (v, M, P) is in the set S_cs of the flowgraph for a given k if and only if there exists a path p, a configuration c', and a sequence of actions w such that the transition from the initial configuration ([entry of the flowgraph for path p], empty multiset) to the final configuration ([v], c') is in the transiti...
The lemma "steps_Stuck_prop" assumes that given a transition system Γ, there is a sequence of steps from state (c, s) to state (c', s'). It shows that if the initial state s is "Stuck", then the final state s' is also "Stuck".
If the sum of pc1 and the size of the compiled expression e is less than or equal to pc2, then the intersection of the pcs of the compiled expression e at pc1 with depth d1 and the pcs of the compiled expressions es at pc2 with depth d2 is an empty set.