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If two functions are equal (denoted by "rel_fun (=)"), then the corresponding finite sets of these functions (denoted by "pcr_fset (=)") also preserve the same relation for "srules" and "rules". This is a transfer rule used in the Isabelle theorem prover for transferring properties between different representations of ...
Under certain conditions, the cost of reversing the concatenation of lists A, U, V, and B is less than or equal to the cost of reversing the concatenation of lists A, V, U, and B if and only if the rank of the reversed list U is less than or equal to the rank of the reversed list V. The conditions are as follows: 1....
A state sigma satisfies the return of a proposition P if and only if P is true.
If B gets a message containing a nonce M and a cryptic message encrypted with B's shared key, which itself contains an arbitrary value X, a nonce Nb, and a key K, and this event is in the set of all events, then either the key K is not in the range of shared keys and there exists some A and Na such that X is a cryptic ...
: Assuming that z1, z2, and z3 are distinct complex numbers and f is a Möbius transformation such that f(z1) equals 0, f(z2) equals 1, and f(z3) equals infinity, then f is equal to the cross ratio of z, z1, z2, and z3.
If w is an element of the tensor product of w1 and w2 with respect to alpha, then the mon_ww of w with respect to fg is equal to the union of the mon_ww of w1 and the mon_ww of w2 with respect to fg.
The product of an ordinal number 'x' and the limit ordinal of a function 'f' is equal to the limit ordinal of the function that multiplies 'x' with the output of function 'f' at any given input 'n'.
The multiset of pairs (m) from the Cartesian product of multiset M and the multiset formed by adding element b to multiset N, where the first element of the pair is in the second element of the pair, is equal to the sum of the multiset of pairs (m) from the Cartesian product of multiset M and N, where the first element...
Multiplying any value 'r' by zero results in zero.
If a condition 'b' chooses between 'P' and 'P', it is equivalent to just 'P'.
"H of p, 0, and q holds true".
If action 'a' is well-defined, then repeating action 'a' 'n' times is also well-defined.
If 'x' is in the Standard set, then 'x' raised to the power of 'n' is also in the Standard set.
: given a time 't' that belongs to the set 'T', and assuming that 'x' solves the ordinary differential equation 'f' over the closed interval from 't0' to 't' (as per the assumption 'sol'), and that the value of 'x' at 't0' is 'x0' (as per the assumption 'iv'), it can be shown that the value of 'x' at time 't' belongs t...
"the function h of t equals 0 if t is greater than or equal to 2".
If a tau-execution move from state 's' to state 's'' is possible in a program 'P' with thread 't', execution context 'ci', and expression 'e' in a heap 'h', then a tau-execution move is also possible in the same program, thread, and execution context with the expression 'throw e'.
If list L is in the set of valid lists for A, n, and B, then there exists some M and some other list L', such that L is the concatenation of M and L'.
The imaginary part of the division of complex numbers 'a' and 'b' is zero if and only if the imaginary part of the product of 'a' and the conjugate of 'b' is zero.
The lemma named "rel_S_witness'2" is defined with several fixed variables: L1, L2, Co1, Co1', Co2, Co2', Contra1, Contra1', Contra2, Contra2', tytok, and x. These variables represent different types and relations between these types. The lemma assumes two conditions: 1. The relation "rel_S" holds between x and y unde...
If the following conditions are met: 1. The server sends a message to agent A, which is encrypted with key K' and contains a number Tk, agent B, key K, and a ticket, and this action is in the set of events; 2. The spy does not note down the number Tk and key K, and this action is not in the set of events; 3. Agent A i...
If "Z" is a well-behaved (mwb) lens, then the composition of "X" and "Z" is independent of the composition of "Y" and "Z" if and only if "X" is independent of "Y".
For a function 'f' that maps a list of type 'a to a type 'c (where 'c is a Banach space with a second countable topology), 'f' is integrable with respect to measure 'M'.
If for every pair of key and value (k, v) in the set derived from the list of ordered associative list with type constraints (xs), the function f applied to k and v equals the function g applied to k and v, then the result of mapping the function f over the values in the ordered associative list with type constraints (...
The lemma named "exp_split_high_low" assumes that "x" is less than 2 raised to the power of the sum of "n" and "m", and both "n" and "m" are greater than 0. It then shows that "high x n" is less than 2 raised to the power of "m", and "low x n" is less than 2 raised to the power of "n".
The lemma named "sum_sum_eq_sum_Un" is defined with a fixed function A that maps a natural number to a set of natural numbers. It assumes two conditions: 1. For all natural numbers x and y in the range from 1 to m, if x is not equal to y, then the intersection of the set A(x) and the set A(y) is empty. 2. For all nat...
The lemma named "TBOUND_Let_weak" under the assumption that for any 'x' equal to 'v', 'x' is time bounded by 'f x' and 'bnd', shows that 'x' being equal to 'v' in 'f x' is also time bounded by 'bnd'.
The lemma named "landau_function_family_powr_closure" assumes that the function family "F" is not the bottom element and that the function "f" has a limit at the top in the function family "F". It shows that the function family "F" is a Landau function family for the power closure of the function "f".
The set of right-hand side variables in the tableau resulting from preprocessing the constraints 'cs' starting from 'start', is a subset of the variables in the constraints found in the flattened list of 'cs'.
If "I" is a finite set, then if for every element "i" in "I", "F" leads from "A i" to the union of "A' i" and "C", and for every element "i" in "I", "F" is co-related from "A' i" to the union of "A' i" and "C", then "F" leads from the intersection of "A i" for all "i" in "I" to the union of the intersection of "A' i" f...
If the predicate "Pred" does not hold for the pointer "ptr", then invoking the pair of "Pred" and function "f" along with the list "xs" using the pointer "ptr" and arguments "args" is equivalent to invoking the list "xs" with the same pointer "ptr" and arguments "args".
The distinguishing set reduced from a machine M forms a finite tree.
The execution of a sequence starting with "s" and followed by a concatenation of "rs" and "r" is equivalent to the execution of a sequence starting with "s" and "rs", then followed by "r".
If L1 is less than or equal to L1', L2 is less than or equal to L2', L3 is less than or equal to L3', Co1 is less than or equal to Co1', Co2 is less than or equal to Co2', Co3 is less than or equal to Co3', Co4 is less than or equal to Co4', Contra1' is less than or equal to Contra1, Contra2' is less than or equal to C...
If the values in the list 'xs' are distinct and the value 'v' is not in the set of values of 'xs', then the set resulting from the depth-first search (dfs) on the list 'alist' after inserting 'a' and 'v' into 'xs' is equal to the set resulting from converting the priority queue (pqueue) 'xs' into an association list (a...
The lemma in the Cox-Ross-Rubinstein (CRR) market viability model shows that the fair price in the market for an Asian option with a given maturity is equal to the Asian price with parameters u, d, initial value, interest rate, and maturity.
If a pair of multisets (xs, ys) belongs to the multiset extension "s_mul_ext" of a relation NS and S, then there exist lists xs1, xs2, ys1, and ys2 such that: - xs is the multiset sum of the multisets of xs1 and xs2, - ys is the multiset sum of the multisets of ys1 and ys2, - the length of xs1 is equal to the length o...
"not equal to 'a' and 'True' is equivalent to the negation of 'a'".
If a filter frame satisfies the filter property, then it also satisfies the third complementation property.
The set of complex numbers x, where the imaginary part of x is greater than or equal to a constant b, forms a convex set.
: 1. The binary decision diagram node is valid for the base case of 0. 2. The binary decision diagram node is also valid for the successor of 0.
The lower bound of the power of a floating point interval is equal to the first element of the floating point power bounds for the lower and upper bounds of x.
, if we substitute a pair (y, z) with a variable 'u' in the variable 'x' of a utp expression, the result is still the variable 'x'.
: Assuming that "y" is in the set of free variables of the raw substitution of a formula "phi" with a list of term-variable pairs "txs", and that "phi" is a formula, and that the second elements of the pairs in "txs" are all variables, and that the first elements of the pairs in "txs" are all terms, and that the secon...
In the context of a category, if 'g' is an inverse of 'f' in the category 'CC', then 'g' is also a left inverse of 'f' in the category 'CC'.
If a relation 'A' is single-valued and its inverse is also single-valued, then the pair (count, count) is in the multiset relation of 'A' mapping to the natural number relation.
Given a valuation v on a field K, and assuming that K is complete with respect to v, and given two polynomial rings R and S with R over the valuation ring of K and S over the quotient of the valuation ring by the valuation ideal, and given a non-zero polynomial f in R, and two polynomials g' and h' in S with positive d...
The Lebesgue measure on the real numbers (lborel) is equal to the interval measure with the identity function as its mapping (i.e., the measure of an interval is simply the difference between its endpoints).
If in the context G, the class C is a subtype of the class cm, then in the same context G, C is a subclass of cm.
The lemma named "is_aligned_add_conv" is defined for a fixed offset of type "'a::len word". It assumes that "w" is aligned to "n" and that the offset is less than 2 to the power of "n". It shows that the binary list representation of the sum of "w" and the offset is equal to the concatenation of the first (LENGTH('a) -...
A pair (x, y) is in the restriction of set A to relation r if and only if the pair (x, y) is in relation r and x is in set A.
For any two elements 'a' and 'b' of the type 'star' of a real normed vector, the hyperreal norm of 'a' minus the hyperreal norm of 'b' is less than or equal to the hyperreal norm of 'a' minus 'b'.
The lemma "reachNT_non_isPC_isChair" assumes that the state "s" is reachable in a non-trivial way and that "uid" is in the set of User IDs. It shows that: 1. The user with ID "uid" is not the Nth reviewer of the paper with ID "PID" for the conference with ID "cid". 2. If the paper with ID "PID" is in the set of paper...
The function "lift_root_step" applied to the parameters "\<F>", "PAny", "EParallel", and "gcomp_rel \<F> \<R> \<S>" is equal to the composition of the function "lift_root_step" applied to the parameters "\<F>", "PAny", "EParallel", and "\<R>", and the function "lift_root_step" applied to the parameters "\<F>", "PAny", ...
The syntactic criteria bisimulation (SC_BisT) of a parallel composition of two commands (c1 and c2) is equivalent to the conjunction of the syntactic criteria bisimulation of each command individually (c1 and c2).
If the function "tinres" is applied to a list consisting of "m" instances of "Bk" followed by two "Bk" elements, then there exists an "m" such that "la" equals "m" instances of "Bk".
If a list, formed by appending lists 'rs' and 'ts', is a heap list with respect to 'l', then both 'ts' and 'rs' are also heap lists with respect to 'l'.
The lemma "HSem_subst_var_var" assumes that the atom "n" is not in "x". It shows that the denotational semantics of the environment where "x" is mapped to the variable "n", and "n" is mapped to the expression "e", in the context of "Γ" and environment "ρ", is equal to the denotational semantics of the environment where...
The set of subformulas of a given formula phi is finite.
If A and B are both groups, and f is a homomorphism from A to B, and g is a homomorphism from B to A, and the composition of g and f is the identity function on A, then f is an injective function from A to B.
The lemma "simplex_index_real" has two parts: 1. If the simplex index of a constraint set 'cs' is 'Unsat I', then the set 'I' is a subset of the first elements of the set 'cs', and there does not exist a value 'v' such that the pair (set 'I', 'v') satisfies the set 'cs'. Furthermore, if the indices in 'cs' are distinc...
If the output from card A, which includes a nonce Na and other elements, is in the set of events, then this output is unique in the set of events.
If "x" is acyclic_4, then "x" is irreflexive.
The lemma "weak_preservation_of_barbs" is defined with respect to a relation "Rel" of type set of pairs of processes and a calculus with barbs "CWB" of type process and barbs. It assumes that the relation weakly preserves barbs in the calculus. The lemma shows that the weak barbed preservation condition holds for the r...
A singular chain "c" of dimension "p" in a topological space "X" is equivalent to its negative "-c".
Mapping the direction vector function over the polychain of x and xs results in xs.
The set of multiset elements obtained from the function "darcs_mset" applied to a tree "t" is equal to the set of elements obtained from the function "darcs" applied to the same tree "t".
The lemma named "point_atomic_vector" assumes that "x" is a point, "y" is a vector, "y" is regular, and "y" is less than or equal to "x". It shows that "y" is either equal to "x" or "y" is the bottom element.
The sort of the atom of a variable of type name2 is not equal to the sort of the atom of a variable of type name.
: 1. If there are no grey references in state 's', then reference 'r' is not in the write set 'W' of process 'p' in state 's'. 2. If there are no grey references in state 's', then reference 'r' is not in the write list 'WL' of process 'p' in state 's'. 3. If there are no grey references in state 's', then reference 'r...
The function "iPrefix" with parameters c, i, j, and s is monotonic.
The set of reachable states of a machine that has been restricted to its reachable states is equal to the set of states of the same machine after it has been restricted to its reachable states.
The lemma named "vid_on_eq_atI" assumes that "a" is equal to "b" and "a" is an element of set "A". It shows that the identity function on set "A" applied to "a" is equal to "b".
If a set A is compact and for any element z in A, z is not a limit point of B, then the intersection of A and B is a finite set.
For any element 'a', the function 'cr_S' applied to the representation of 'a' equals 'a'.
The lemma "new_element_is_element_ptr" assumes that "new_element" when executed in the heap "h" returns a result "new_element_ptr". It shows that "new_element_ptr" is an element pointer.
The empty list is not in H1.
If a list 'xs' is a prefix of another list 'ys' and if the list 'ys' is sorted, then the list 'xs' is also sorted.
The lemma wfX_wfB2 is fixing variables of various types including \<Gamma>, \<Gamma>', v, e, c, \<tau>, ts, \<Delta>, s, b, \<B>, \<Phi>, ftq, ft, ce, td, cs, and css. It shows the following: 1. If an expression e is well-formed under contexts \<Theta>, \<Phi>, \<B>, \<Gamma>, and \<Delta> with type b, then the type b...
If "A" is equal to a record with "nrm" equal to "B" and "brk" equal to a function that maps any "l" to the universal set, then in the environment "Env", "B" is definitely assigned after the literal "v".
, under the assumption that a certain condition "possible n p w w'" holds and that "p'" is not equal to "p", it can be shown that the value of "w'" at "p'" is equal to the value of "w" at "p'".
The function "igSwap" applied to "termMOD" is equal to the function "swap".
, for any real numbers x and y, the result of adding x and y and then subtracting x is equal to y.
"R is equal to R plus the Cartesian product of the set containing x and the image of the set containing x under R."
This lemma, named "term_powerprod_gd_term", is about a certain pair_of_term function. This function takes a natural number term and returns a pair consisting of a graded Dickson power product and another natural number term. The lemma assumes several conditions: 1. The function term_powerprod is defined for pair_of_...
If for all 'x', 'x' is intuitionistic in the property 'P', then there exists an 'x' such that 'P' is intuitionistic.
The lemma named "preserves_joins" assumes that "v" is the join of "t" and "u" in system "A". It shows that in system "B", the image of "v" under function "F" is the join of the images of "t" and "u" under the same function "F".
In the context of a group, the lemma named "pow_eq_div2" is defined. It takes two natural numbers, m and n, as fixed variables. It assumes that x is an element of the carrier G and that x raised to the power of m equals x raised to the power of n. Under these assumptions, it shows that x raised to the power of (m - n) ...
If the norm of vector x plus the norm of vector y is less than or equal to a certain value e, then the norm of the difference between vector x and vector y is also less than or equal to e.
The lemma named "iso_assoc" assumes that "a", "b", and "c" are identities. It shows that the associator of "a", "b", and "c" is an isomorphism.
If for every element 'a' in the probability mass function 'M', 'f' of 'a' is not zero implies 'a' is in 'A', then the Lebesgue integral of 'f' with respect to 'M' is equal to the sum over all elements 'a' in 'A' of the product of the probability mass function 'M' at 'a' and 'f' of 'a'.
The substitution of a floatarith 's' in 'f' equals 'f' itself, given the condition that the maximum variable in the floatarith 'f' is zero.
In the lemma named "sub_nhd" within the context of a carrier, if U is an element of the neighborhood of x, then there exists a V in the neighborhood of x such that V is a subset of U and for all elements z in V, U is in the neighborhood of z.
If x11 is not in the domain of A, then dropping x11 from A will result in A itself.
The lemma named "algebraicE" assumes that "x" is algebraic. It then obtains a polynomial "p" such that for any index "i", the coefficient of "p" at "i" is an integer, "p" is not equal to zero, and the value of the polynomial "p" at "x" is zero.
The state of the dining philosophers problem with 'n' philosophers is a function of the current state 's', such that for every event 'e' in the set of trace events of 'n' philosophers in state 's', the next state of the dining philosophers problem is the updated state after event 'e'.
The theorem, named 'Morse_Gromov_theorem', is within the context of a Gromov hyperbolic space. It fixes a function 'f' that maps real numbers to an arbitrary type 'a'. The theorem assumes that this function 'f' is a lambda C-quasi-isometry on the interval from 'a' to 'b', and that there is a geodesic segment between 'f...
If the length of the list 'xs' is equal to the arity of the de-inlined operation 'opinl', then the inline evaluation of 'opinl' with 'xs' is equal to the evaluation of the de-inlined 'opinl' with 'xs'.
The lemma named PP_PPf assumes that phi is a formula. It shows that the predicate wrepr.PP of the Goedel number of phi is equal to the existence of a variable yy such that the predicate PPf of the variable yy and the Goedel number of phi holds.
The language of the normalized version of a monadic second-order logic formula is the same as the language of the original formula.
If a ruleset "rs" is good, then the ruleset obtained by normalizing "rs" in disjunctive normal form is also good.