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A position 'p' is less than or equal to itself.
The set of states resulting from mapping function 'f' onto the actions in a policy 'phi' is equal to the set of states in the original policy 'phi'.
If the Lb value of v is less than or equal to v, then the optimal b value of nu is less than or equal to v.
The function "(z,x,y) maps to P of x times (x times y times z) to the power of n times (1 minus y) to the power of n divided by (1 minus (1 minus x times y) times z) to the power of (n plus 1)" is set integrable over the Lebesgue measure on the set D'.
If the set Qs is equal to the set Qs', then the disjunction (logical OR operation) of all elements in Qs, with a base case of "False", is equivalent to the disjunction of all elements in Qs', with the same base case of "False".
The start symbol is a nonterminal symbol.
The set of elements related to a single element 'a' under a relation 'r' is equal to the set of elements related to the set containing only 'a' under the same relation 'r'.
If "a1" is in the carrier of the initial object "dummy" in category 1, and "a2" is in the carrier of the initial object "dummy" in category 2, and if there are morphisms from the initial objects in both categories to objects B1 and B2 with structure maps s1 and s2 and functions f1 and f2, and similarly, there are morph...
The steps of a relation at the successor of a number n is equal to the steps of the relation at n union with the set of pairs (x, y) where x is in the predecessors of the steps of the relation at n and y is in the successors of the steps of the relation at n.
The lemma named "inter_ABC_1" assumes that the line AB is not equal to the line CA. It shows that the intersection of line AB and line CA is point A.
The probability mass function (pmf) of the stochastic dominance strategy (sds) R23 for outcome 'b' is 0, for outcome 'c' is also 0, and for outcome 'd' is 1 minus the pmf of the sds R23 for outcome 'a'.
, in a system A, if a list of propositions (ps) led by a proposition (p) implies p, then the proposition (p) is true.
For the ShadowRootClass, if the type is well-formed, the known pointer is defined, and the known pointers are defined, then the function 'to_tree_order' and 'get_root_node' from the Shadow DOM maintain the well-formedness of the heap.
The lemma named "step_friends12_\<phi>" assumes that a step from state 's' to state 's'' with action 'a' results in an output 'ou'. It also assumes that the set of friends in state 's' is not equal to the set of friends in state 's''. Under these assumptions, it shows that the function 'phi' applied to the transition f...
Applying the function f to the result of applying the function g to x and the natural number n (converted to the same type as x) is equal to applying the function f to x.
The negation of "p" and the left shift of the negation of "p" by the negation of "q" is equal to the bottom element.
The dual of a closure operator is in the Partial Order.
The function which takes an input 'y' and returns the supremum (or least upper bound) of 'x' and 'y' is monotone continuous with respect to the supremum operation and the less than or equal to relation.
The lemma "represents_in_span" assumes that "r" represents "x" in the basis "B". It shows that "x" is in the span of "B".
If the reduction of a single term "p" to "q" is possible through a function "f" and a term "t", then one of the following three conditions must hold: 1. The reduction of the term "p + r" to "q + r" is possible through the same function "f" and term "t". 2. The reduction of the term "q + r" to "p + r" is possible throu...
The size of a red-black tree obtained by applying the "rbt_baldL" function on trees t1 and t2, with elements a and b, is equal to the successor of the sum of the sizes of t1 and t2.
The theorem named "wls_vsubstAbs_compose1" assumes that "A" is a well-sorted abstraction with sort "(us, s)". It shows that substituting "y1" for "y" in "ys" in "A", and then substituting "y2" for "y" in "ys" in the result, is equal to substituting "(y1 @ys[y2 / y]_ys)" for "y" in "ys" in "A".
"x is a part of y if and only if for all z, if z overlaps with x then z overlaps with y".
A graph 'g' is cycle-free if and only if for all edges 'e' in the graph 'g', the edge 'e' is not in the transitive closure of the edges of the graph 'g' restricted by removing the edge 'e' and its swapped product.
: Assuming that the semicategory morphism (semifunctor) "GG" maps from semicategory "BB" to semicategory "CC", and the semicategory morphism (semifunctor) "FF" maps from semicategory "AA" to semicategory "BB", it can be shown that the composition of "GG" and "FF" is a semicategory morphism (semifunctor) from semicate...
The set of all labels 'v' such that there exists an edge from 'u' to 'v' in the residual graph 'cf.E' is finite.
If a list 'xs' is not empty, then the tail of 'xs' (i.e., the list 'xs' without its first element) is a strict suffix of 'xs'.
The lemma "T_in_A" assumes that "p" is a policy. It shows that for almost every trajectory "t" in the set of trajectories "T" generated by policy "p" from the initial state "S0", the second element of the "n"-th step of the trajectory is in the action set "A" corresponding to the first element of the "n"-th step of the...
The sum of the function 'f' applied to each pair (x, y) in the list 'xys', when further applied to 'z', is equal to the sum of the function 'f' applied to each pair (x, y) and 'z' in the list 'xys'.
Dropping an empty set from a finite map leaves the map unchanged.
If the conditions Fr_1, Fr_2, and Fr_3 hold for a frame \<F>, then for all propositions 'a' and 'b', if 'a' implies 'b' and 'b' is not true, then 'a' is not true is equivalent to a tautology.
: Assuming that "f" is in the carrier of the ultra product of the semialgebraic set "m", "c" is in the carrier of the semialgebraic set "m", "A" is a semialgebraic set of "m", and for any integer "i", "A" is a subset of the zero set of the Taylor expansion of "f" at "c" in the semialgebraic set "m" or "A" is a subset ...
The lemma named "compactD" assumes that a set 'A' is compact in a metric space and the range of a function 'f' is a subset of 'A'. It shows that there exists a strictly monotonically increasing function 'h' and a limit 'l' such that the composition of 'f' and 'h' converges to 'l' in the long run.
If A is an ideal of a ring R, H is a subset of the carrier M, f is a function from the set of natural numbers less than or equal to n to H, and s is a function from the set of natural numbers less than or equal to n to A, then the additive inverse of the linear combination of R, M, n, s, and f is equal to the linear co...
For all variable declarations (vds), variables (vars), modified variables (vars'), and values (vs), if the length of these lists is equal to the length of the list of pairs (y_ty_list), then mapping the function that extracts the first element of each quintuple formed by zipping together the first elements of y_ty_list...
: - The hashcode of 'None' is 0. - The hashcode of 'Some a' is 1 plus the hashcode of 'a'.
If "u" is an element in the set of vertices of graph "G" and "e" is an element in the set of in-out arcs of graph "G" at vertex "u", then "u" is adjacent to the opposite of "e" at "u" in graph "G".
If the implementation of the matrix of rows list for a given integer 'n' and list 'rs' equals to 'Some A', then the implementation of the matrix 'A' is equal to the matrix of rows list for 'n' and 'rs'.
If Fr_4 holds for a set F, then for all subsets A of F, the closure of A is in F.
In the context of a non-primitive Dirichlet character, the conductor is less than the modulus 'n'.
The lemma named "convex_comb_dets" assumes that the determinant of a 3x3 matrix composed of points p, q, and r is greater than 0. It then shows that the point s can be expressed as a convex combination of points p, q, and r, where the weights are the determinants of the matrices obtained by replacing one column with s,...
If a property P holds for a heap h and P is intuitionistic, then the implication from the separation logic truth to P holds for the heap h.
The range of the set of all pairs (x, y) such that x and y are nodes connected in graph G is equal to the set V.
Theorem TBtheorem5a_empty: Assuming that either "P" or "Q" is an output event in "E", and that the subcomponents of "PQ" are "P" and "Q", and that "PQ" is a correct composition output, and that the location of "PQ" is empty, it can be shown that "PQ" is an output event in "E".
If x is less than or equal to y, then the forest components of x are less than or equal to the forest components of y.
When looking up an empty list in a trie (which is a type of search tree), if the trie has a value at the root (represented by "vo"), it will return that value (represented by "vs"). If the trie does not have a value at the root (represented by "None"), it will return an empty list.
"n of x" is less than or equal to "n of y" if and only if there exists a "z" such that "n of x" equals "n of y" multiplied by "n of z".
Given the following assumptions: 1. The selection function S is stable under renaming, meaning that for any renaming function ρ and clause C, applying S to the renamed clause C·ρ results in the same as applying S to C and then renaming the result. 2. There exists an ordered resolution of a set of clauses CAs, a claus...
The lemma is_map_to_sorted_listE: Assuming that "is_map_to_sorted_list ordR Rk Rv Rm tsl" and "(m,m') is in the relation Rm over Rk and Rv", we can obtain a list l' such that "(tsl m,l') is in the list relation over the product relation of Rk and Rv" and "the return of l' is less than or equal to the sorted list conve...
If the inorder traversal of a tree 't' is sorted, and if splaying element 'x' in tree 't' results in a node with left subtree 'l', value 'a', and right subtree 'r', then the inorder traversal of the left subtree 'l', followed by element 'x', followed by the inorder traversal of the right subtree 'r' is also sorted.
If sets A and B are countable, and for every element m in set B there exists an element n in set A such that the function of n is less than or equal to the function of m, then the infimum (greatest lower bound) of the function of n in set A is less than or equal to the infimum of the function of n in set B.
If a tree 't' is complete, then the tree resulting from deleting element 'x' from 't' (denoted as treeD(del x t)) is also complete.
Multiplying any polynomial 'p' by zero results in zero.
The operation "op_smc" applied to the zero semicategory "smc_0" equals the zero semicategory "smc_0".
The lemma named "mahler_measure_dvd" assumes that "f" is not equal to zero and "h" divides "f". It shows that the Mahler measure of "h" is less than or equal to the Mahler measure of "f".
The lemma named "lebesgue_measurable_diff_null" is about a function 'f' that maps from one Euclidean space to another. It assumes that 'N' is in the null sets of Lebesgue measure. It shows that 'f' is Borel measurable on the Lebesgue measure of the set difference with 'N' if and only if 'f' is Borel measurable on the L...
: assuming "X" and "X'" are well-sorted terms of sort "s", "y" is a fresh variable for "X" and "X'" with respect to variables "xs", and swapping "y" and "x" in "X" results in the same term as swapping "y" and "x'" in "X'", it can be shown that the abstraction of "X" with respect to "x" and variables "xs" is equal to th...
The change in the potential function Phi after inserting an element x into a pairing heap h is less than or equal to the logarithm base 2 of the size of the heap h plus one.
A ruleset is well-formed under conditions gamma and p with a single rule consisting of a match and an action, if and only if either the match does not occur under conditions gamma, m, a, and p, or there does not exist a chain such that the action equals a call to the chain, and the action is not a return, and there doe...
For any extended nonnegative real numbers a, b, and c, the value of a plus b minus c is less than or equal to the value of a plus the difference of b and c.
: Assuming the following conditions: 1. For any index 'i', if 'i' is less than the length of list 'zs', then the 'i'-th element of 'zs' is less than the sum of the length of list 'xs' and 'ii', and if the 'i'-th element of 'zs' is greater than or equal to 'ii', then the ('i'-th element of 'zs' minus 'ii')-th element o...
The relation "MP_Rel" is right unique. This is a rule for transfer in Isabelle.
If for all indices 'i', 'f i' and 'g i' are elements of the carrier set 'A', then the n-fold sum of the element-wise addition of 'f i' and 'g i' is equal to the addition of the n-fold sum of 'f i' and the n-fold sum of 'g i'.
If a set "SS" is a semigroup under an operation in a set "alpha", then "SS" forms a semigroup basis in "alpha". Furthermore, for any elements "a", "b", and "c" in the set "SS", the operation on "a" and "b", followed by the operation on the result and "c", is equal to the operation on "a" and the result of the operation...
"PrefixMatch with a 32-bit word of 0 and 42 is a valid prefix."
The lemma makes three assumptions based on the condition that "z" is not equal to zero. 1. The exponential of the natural logarithm of "z" is equal to "z". 2. The imaginary part of the natural logarithm of "z" is greater than negative pi. 3. The imaginary part of the natural logarithm of "z" is less than or equal to...
The alpha function applied to the union of a set "qs" in the relational space "rs" is equal to the union of the alpha function applied to each element in the set "qs".
The lemma named "sinits_foldr'" from the Theory_Of_Lists in BirdKMP, which is referenced on page 30 of "Bird:1987", demonstrates that the function "sinits" applied to a list "xs" is equal to the function "sfoldr" applied to a lambda function that prepends an element "x" to each list in "xs", with the initial list being...
If the "af" function of the second element of pair "p" and "w" is not equivalent to false, then the second element of the "af\<^sub>\<nu>" function of "X", "p", and "w" is equal to the "af" function of the second element of "p" and "w".
The lemma named "Qp_pow_ConsI" assumes that "t" is in the carrier of Qp and "x" is in the carrier of Qp raised to the power of "m". It shows that the list consisting of "t" followed by "x" is in the carrier of Qp raised to the power of "m+1".
Appending an empty list to any list results in the original list itself.
The natural logarithm of the nth prime number minus the sum of the natural logarithm of n and the natural logarithm of the natural logarithm of n is an infinitesimally small quantity when compared to the constant function 1.
The lemma "get_ancestors_si_parent_child_host_shadow_root_rel" assumes that the heap "h" is well-formed, contains known pointers, and is type well-formed. It also assumes that the function "get_ancestors_si" when applied to a child in the heap "h" returns "ancestors". Furthermore, it assumes that the pair "(ptr, child)...
Returning a value from a function "f" applied to a variable "x" within a "Let" expression is equivalent to defining a "Let" expression with variable "x" and a function that returns the value of function "f" applied to "x".
If under the program P and environment E, the expression 'e' evaluates from initial state 's0' to 'false' in the final state 's1', then under the same program P and environment E, the while loop 'while (e) c' evaluates from the initial state 's0' to 'unit' in the final state 's1'.
The result of applying a complex bounded linear function A to a vector psi is in the image of the function A when it is applied to the entire space.
The lemma named "sorted_list_lemma_var" is defined with respect to a list "l" of real numbers and a real number "x". It assumes that the length of the list "l" is greater than 1, the list "l" is strictly sorted in ascending order, the real number "x" is not an element of the list "l", "x" is greater than the first elem...
The function 'compress' applied to a set of elements 'xs' is equivalent to the set of 'xs' after removing duplicates.
The lemma is within the context of the Cox-Ross-Rubinstein (CRR) market model. It assumes that the parameter 'q' is defined as "(1 + r - d)/(u - d)", and that the market 'Mkt' is viable. Under these assumptions, it shows three things: 1. The parameter 'q' is greater than zero (gt_param). 2. The parameter 'q' is less ...
"a is less than or equal to b divided by c if and only if c times a is less than or equal to b", given the condition that "c is greater than 0".
, given two assumptions, a certain equality holds. The first assumption, "f_bind", asserts that for any monad 'm', any value 'x', and any function 'g', binding 'm' to a function that binds 'f x' to 'g a' is equal to binding 'f x' to a function that binds 'm' to a function 'g a b'. The second assumption, "f_get", assert...
The lemma named "psi_1_values_odd" is defined with fixed variables i, j, and n. It assumes that i is less than the number of rows in psi_1 of n, j is less than the number of columns in psi_1 of n, and i is an odd number. Under these assumptions, it shows that the element at the i-th row and j-th column of psi_1 of n is...
If a set 's' is bounded, then the set of second elements (obtained by the 'snd' function) from the set 's' is also bounded.
If a set 'a' is a subset of 'A' and 'A' is an element of the set 'Vset i', then 'a' is also an element of the set 'Vset i'.
The lemma named "assoc_list_of_array_code_induct" assumes an induction hypothesis (IH): for any integer n, if n is less than the length of array a, then the property P holds for the successor of n. Based on this assumption, it shows that the property P holds for n.
If the following conditions are met: 1. k is greater than 0, 2. the product of t and k is less than t1, 3. t1 is less than or equal to t2, 4. t2 is less than or equal to the sum of the product of t and k and k, then the interval expansion of function f by k from t1 to t2 is equal to the replication of NoMsg from ...
The predicate "P" always holds for the stream "xs" if and only if the predicate "P" is true for the stream "xs".
Looking up a value 'x' in a finite map 'm' from which a finite set 'A' has been dropped, is equivalent to: if 'x' is not in 'A', then look up 'x' in 'm', otherwise return 'None'.
If a predicate P is equal to another predicate P', and for all elements x, if P x implies that Q x is equal to Q' x, then for all elements x, P x implies Q x is equivalent to P' x implies Q' x.
If 'x' is equal to 'insec' or 'x' is equal to 'auth', then 'M' is in the set of dynamically faked messages by bad actors, given the initial knowledge 'IK' and the insertion of the channel 'Chan x A B M' into the set 'CH'.
The lemma OclNot_contrapos_nn: Assuming that A is valid for a given state tau, B is not valid for the same state tau, and if A is valid for tau then B is also valid for tau, it can be shown that A is not valid for tau.
: 1. The negative of 2 raised to the power of (the size of a 32-bit word 'x' minus 1) equals negative 2147483648. 2. 2 raised to the power of (the size of a 32-bit word 'x' minus 1), minus 1, equals 2147483647. 3. The negative of 2 raised to the power of (the size of a 32-bit signed word 'y' minus 1) equals negative 2...
The universal set of all 'univ' type elements is countable.
The lemma "Lt_cases" assumes that either "A is less than B and C is less than D", or "B is less than A and C is less than D", or "A is less than B and D is less than C", or "B is less than A and D is less than C". It shows that "A is less than B and C is less than D".
The length of the list "cl_var_ty_list" is equal to the length of the list obtained by mapping a function over "cl_var_ty_list", where the function takes a tuple of three elements (cl, var, ty) and returns the third element (ty).
If two extended finite state machines, represented by (e1, s1, r1) and (e2, s2, r2), are executionally equivalent for a given input x, then the equivalence is symmetric. That is, if machine (e1, s1, r1) is executionally equivalent to machine (e2, s2, r2) for input x, then machine (e2, s2, r2) is also executionally equi...
The lemma "find_recfun_someD" assumes that the function "find_recfun" with parameters "n" and "funs" returns a pair "(x, e)". It shows that the triplet "(n, x, e)" is an element of the set "funs".
The composition of the substitution function "substC" with parameters "pi1" and "pi2" is equal to the substitution function "substC" with the composition of the substitution "subst pi1" and "pi2" as its parameter.
Given a list of configurations (cfgList) and a list of messages (msgList), if the following conditions are met: 1. The first configuration in the list is an initial configuration (Init). 2. The last configuration in the list is non-uniform (NonUni). 3. The list of configurations and messages represents an execution of...
If process P is less than or equal to process Q in the failures-divergences model, then the process resulting from hiding actions in set A from process P is also less than or equal to the process resulting from hiding actions in set A from process Q in the failures-divergences model.