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The lemma named "encode12_inv1" assumes that "i" is less than "d1" and "j" is less than "d2". It shows that if you first apply the function "encode12" to the pair (i, j) and then apply the function "encode1" to the result, you will get back "i".
If for all 'x', 'P1' is not true and 'P1 x' is true implies there exists a 'u' in 'U' such that 'u' is less than or equal to 'x', and if for all 'x', 'P2' is not true and 'P2 x' is true implies there exists a 'u' in 'U' such that 'u' is less than or equal to 'x', then for all 'x', neither 'P1' nor 'P2' is true and eith...
If the input "inp" is well-sorted with respect to "delta" and the binary input "binp" is also well-sorted with respect to "delta", then the skeleton of the operation "Op" with "delta", "inp", and "binp" as parameters is equal to a branch with the skeleton of "inp" and the skeleton of "binp".
The theorem of machine type preservation: Assuming that a machine transitions from state 't' to state 't'' without any change in the stack, and assuming that 't' is of type 'T' under context 'Γ', it can be shown that 't'' is also of type 'T' under the same context 'Γ'.
Given the assumptions that "LeftDerivationFix" holds for alpha, i, D, j, and gamma, and that "derivation_ge" holds for D and d, it can be shown that either i equals j, or i is greater than d and j is greater than or equal to d.
Inserting the result of the expression "(r + m * the successor of c)" into the set consisting of "r", "the modulus of m", and "c", is equivalent to the set consisting of "r", "the modulus of m", and "the successor of c".
The lemma "proj_list_Cons" consists of three parts: 1. The projection of the list that starts with a pair consisting of the label "ln i" and some data "d", followed by the rest of the list "D", is equal to the pair ("ln i", "d") followed by the projection of the rest of the list "D". 2. The projection of the list tha...
The lemma named "primroot_root" assumes that "u" is not equal to the empty word (denoted by "\<epsilon>") and "u" is in the closure of "q" under the star operation (denoted by "q*"). It then shows that the primitive root of "q" is equal to the primitive root of "u".
If b is greater than 2, then the function A of b and n is equal to the matrix power of n and the function B of b.
If a measure 'M', a random variable 'X', and a probability 'Px' are simply distributed, then 'M' and 'X' form a simple function.
The property of being epsilon-free for a ground term tree (gtt) remains unchanged after the relabeling operation. In other words, if a ground term tree is epsilon-free before relabeling, it will still be epsilon-free after relabeling, and vice versa.
If the intersection of a set S with a property P is finite, and the intersection of a set T with the same property P is also finite, then the intersection of the sets S and T with the property P is also finite.
The exponential of a complex number, which is formed by a real part 'r' and an imaginary part 't', is equal to the product of the exponential of the real part 'r' and a complex number formed by the cosine of 't' and the sine of 't'.
The lemma named "perm_vars_inj_on_nth" assumes that "vs" is a permutation of variables up to "n". It shows that the function "nth vs" is injective (or one-to-one) on the set of integers from 0 to "n" (exclusive).
If for all elements 'x' in the set 'xs', the pure function 'P' applied to 'x' is true and 'h' is okay with 'P' applied to 'x', and if the filter function applied to 'h' and 'P' applied to 'x' equals 'ys', then 'h' applied to the filter function 'M' and 'P' applied to 'xs' results in 'ys'.
The lemma named "document_ptr_kinds_small" assumes that for any object pointer, the "get_M_object" function with parameters "object_ptr", "RObject.nothing", "h", and "h'" is preserved. It then shows that the kinds of document pointers in "h" are equal to the kinds of document pointers in "h'".
If the sets {a,b} and {c,d} are both elements of the set l, and {a,b} is not equal to {c,d}, and y is defined as "DenyAllFromTo a b", then if "AllowPortFromTo c d p" is smaller than y in the set l, it is not possible for y to be smaller than "AllowPortFromTo c d p" in the set l.
The path component set of S at point x is a subset of the connected component set of S at point x.
The domain of the EmptyMap function applied to a set S is equal to S itself.
If the lift of k and v equals to C_U with parameters nm and vs', then there exists a vs such that v equals to C_U with parameters nm and vs, and vs' equals to the map of the lift of k and vs.
The lemma named "not_conga" assumes that triangle ABC is congruent to triangle A'B'C' and that triangle ABC is not congruent to triangle DEF. It shows that triangle A'B'C' is also not congruent to triangle DEF.
The lemma named "commute_multiples_identity" assumes that a matrix A, which belongs to the carrier matrix of size n by n, is given. It shows that the multiplication of matrix A and the scalar multiple of the identity matrix of size n by a scalar k is equal to the multiplication of the scalar multiple of the identity ma...
The lemma named "alpha_equiv_suff" is defined with three fixed natural numbers a, b, and c. It assumes that there exist natural numbers r, s, t, u, v, w, x, and y such that they satisfy the alpha equations with a, b, and c. If this assumption holds, it shows that b is greater than 3 and the integer value of a is equal ...
If a pair of sets, where the first set contains only the element 'a' and the second set contains only the element 'b', belongs to the set relation 'R', then the pair (a, b) is in the relation 'R'.
For any i, j, x1, and x2, if i is in the signature of x1 and j is in the set determined by the sigma function applied to i (which returns the free variable terms of x if x exists, otherwise it returns an empty set), then the left injection of i is in the intersection of the set of all left injections of x where x is in...
The lemma named "complements_complement_pair'" assumes that F and G are compatible. It shows that G and the complement of the composition of F and G are complements.
"H" is a subset of "analz H".
The lemma named "app_invT" assumes that "r" is an element of the transitive closure of "A". It shows that if the concatenation of "r" and "s" is an element of the omega closure of "A", then "s" is also an element of the omega closure of "A".
The lemma named "siso_While" assumes that the test "tst" is compatible and "c" is a single-input single-output system. It shows that the while loop of the test "tst" and "c" is also a single-input single-output system.
The security invariant with offending flows is reflexive with respect to the "sinvar" function and the subset relation.
The lemma named "insert_sorted_split_list_right" assumes that the result of splitting the list "ts" at element "x" is a pair of lists "(ls, sep#rs)", the list "ts" is sorted in ascending order, and "x" is not equal to "sep". Under these assumptions, it shows that inserting "x" into the list "(sep#rs)" results in the li...
Given a state sigma and three variables I, M, and N, if I is assigned the value of M and then, in the state tau where I equals the value of M in state sigma, M is assigned the value of N and N is assigned the value of I, then in state tau, M equals the value of N and N equals the value of I. Furthermore, in state sigma...
A callee invariant holds for a function that returns a sub-probability monadic function (return_spmf) applied to another function 'f' and its inputs 's' and 'x', if and only if for all 's' and 'x', if the invariant 'I' holds for 's', then 'I' also holds for the second element of the pair resulting from applying 'f' to ...
If a set 'W' is a subset of 'V', and a strategy 'σ' attracts player 'p' from set 'A' to 'W', and if the set of all elements in path 'P' that are also in 'A' is not empty, then the set of all elements in path 'P' that are also in 'W' cannot be empty.
If 'b' is an object in category B, and if BCG is a morphism from B to C in the comma category of 'b' and functor F, and if ABF is a morphism from A to B in the same comma category, then the composition of BCG and ABF is a morphism from A to C in the comma category of 'b' and functor F.
If a list "xs" is distinctly infinite (ldistinct), then any list derived from "xs" by dropping "n" elements from the beginning (ldrop n xs) is also distinctly infinite.
For any set, if an element 'a' is added to an empty set, then the 'any' operation on the set will return the element 'a'.
The function "tabulate_class" is equivalent to the function "Mapping.of_alist".
If an acyclic relation R exists on the range of the function zm, and if ym is a strict subtuple of xm under the relation R, and if zm is a strict subtuple of ym under the transitive closure of the relation R, then zm is a strict subtuple of xm under the transitive closure of the relation R.
The set of "ebounds" for a given list "as" is equal to the set of tuples (r/c, (-1/c) * cs) where r, c, and cs are such that the equation r equals the list formed by c and cs is in the set "as" and c is not equal to zero.
"parse_attributes" is a parser.
If "C" is a chamber and "z" is a panel of "C", then the adjacent chamber to "C" through "z" is also a chamber.
Given the following conditions: 1. "i" is less than "j", 2. "j" is less than the length of "t", 3. For all "k", if "i" is less than "k" and "k" is less than "j", then "k" is not a regular event in "t", 4. "i" is a post-recording event in "t", 5. "j" is a pre-recording event in "t", 6. The trace from "init" to "t" to "...
A class hierarchy in a program P is acyclic if and only if for every class in the program P, represented by the tuple (C, D, fs, ml), if the class C is not the Object class, then class D is not a subclass of class C.
If the following conditions are met: 1. The assertion A is provable when we assume that x is equal to Zero and this assumption is added to the set of hypotheses H. 2. The assertion A is also provable when we assume that x is equal to the successor of some variable k and this assumption, along with the assumption that ...
The lemma named "swap_cols_rows_block_dims" states three things: 1. The number of rows in the matrix obtained by swapping the i-th and j-th columns and rows of matrix A is equal to the number of rows in the original matrix A. 2. The number of columns in the matrix obtained by swapping the i-th and j-th columns and ro...
The exponential of x is less than or equal to the exponential of y if and only if x is less than or equal to y.
The lookup of the product of two functions 'f' and 'g' at a point 'k' is equal to the sum over 'l' of the product of the lookup of 'f' at 'l' and the sum over 'q' of the lookup of 'g' at 'q' when 'k' equals 'l' plus 'q'.
The star of a singleton set containing 'x' is equal to the set of 'm' replications of 'x', where 'm' can be any true statement.
: Assuming that a substitution 'sigma' extends to both substitutions 'tau' and 'rho', and the domain of 'tau' and 'rho' is the union of the variables of term 't' and the domain of 'sigma', and term 't' substituted with 'rho' is equal to term 't' substituted with 'tau', then it shows that 'rho' is equal to 'tau'.
If 'p' is a prime element in a factorial semiring, then 'p' is square-free.
Taking the successor of any number x from an empty list will result in an empty list.
"if it is possible that P and Q are true for some variable v, then it is possible that P is true for the same variable v."
If "a" is a proper subset of "b", then the rank of "a" is less than the rank of "b".
If the lengths of two lists, xs and ys, are equal, then the existence of a substitution of xs with ys in a relation Q is equivalent to the existence of a bounded substitution of xs with ys in Q, where the source and destination of the substitution are also substituted in Q.
"it is not the case that the sum of x and y is less than x".
In the context of a complete lattice (CL), if 'L' is the least upper bound (Lub) of a set 'S' and 'y' is an element of 'S', then the pair '(y, L)' is in the relation 'r'.
If a new character data is derived from 'h' to 'h'', then the 'get_MShadowRoot' function for a given pointer and getter is preserved between 'h' and 'h''.
If we have a finite, non-empty set 'I' of independent variables 'X' that are normally distributed with mean 'µ' and standard deviation 'σ' (where 'σ' is greater than 0 for all 'i' in 'I'), then the sum of these variables is also normally distributed. The mean of this sum is the sum of the means of the individual variab...
The lemma named "is_gcd_some_gcdI" assumes that there exists a number 'x' which is the greatest common divisor (gcd) of 'a' and 'b'. It shows that the greatest common divisor of 'a' and 'b' is the same as the greatest common divisor of 'a' and 'b' obtained by the function 'some_gcd'.
The lemma named "state_rank_distinct" assumes that "p" is not equal to "q". It also assumes that the rank of state "p" at position "n" is "i" and the rank of state "q" at position "n" is "j". Under these assumptions, it shows that "i" is not equal to "j".
If the expression "e2" with state "xs" τ-reduces in zero or more steps to the expression "e2'" with state "xs'" in the thread "t" with heap "h" under the program "P" and thread actions "extTA", then the expression "Val v bop e2" with state "xs" τ-reduces in zero or more steps to the expression "Val v bop e2'" with stat...
If a state 's' is an element of the concretization of 'S' under the function 'γf', then the abstract value of 'a' under the state 's' is an element of the concretization of the abstract value of 'a' under 'S'.
Replicating "R" "m" times results in a valid sequence.
The substitution domain of the variable "Var" is an empty set.
If function 'g' is a bounded linear function, then the function that applies 'f' to the output of 'g' (represented as 'blinfun_apply f (g x)') is also a bounded linear function.
The lemma named "Cong\<^sub>0_append_Arr_NPath" assumes that the list "T" is not empty, "T" appended with "U" forms an arrow, and "U" forms a normal path. Under these assumptions, it shows that "T" appended with "U" is congruent to "T" under the relation "Cong\<^sub>0".
The lemma named "fps_inverse_one_minus_fps_X" is defined for any division ring 'a. It introduces a formal power series 'ones' which is defined as the absolute value of the function that maps every natural number 'n' to 1. The lemma then shows that the inverse of (1 - fps_X) is equal to 'ones'.
The identity function is an isomorphic register.
Running the state transformer on the result of applying a function 'f' to a monad 'm' and a state 's' is equivalent to applying a function that takes a pair of a value 'x' and a state 's'' and returns a pair of the result of applying 'f' to 'x' and 's'' to the result of running the state transformer on 'm' and 's'.
The hyperbolic sine of x equals the hyperbolic sine of y if and only if x equals y, where x and y are real numbers.
If a system "s" completes a happens-before relation "E" (as assumed by "hb_completion s E"), and if the system "s" transitions to a new state "s'" with thread "t" and thread action "ta" (as indicated by "s -t\<triangleright>ta\<rightarrow> s'"), and if the system is non-speculative with respect to a certain set of writ...
If "P" implies "A" and not "P" implies "B", then if "P" is true, "A" is true, otherwise "B" is true.
The white balance prism is identity for the value 1.
If the tail of the concatenation of the list obtained by mapping a function 'f' over a list of integers from 0 to the length of list 'l', and a list containing a single element 'a', is empty, then the list 'l' must be empty.
The advantage of Party 1 in the 14th Oblivious Transfer (OT_14) is negligible, given the function R1_12_0T and inputs n, m1, m2, and D, if the advantage in the 12th Oblivious Transfer (adv_OT12) is negligible.
If there exists an 'x' such that property 'P' holds for 'x' implies property 'Q', then if property 'P' holds for any 'y', it also implies property 'Q'.
If the split interval of X at x equals the pair (A, B), then the set of X is a subset of the union of the set of A and the set of B.
The lemma named "foldl_ins_treap_of" assumes that the list "ys" contains distinct elements, the first element of each pair in the set "ys" is unique, and the second element of each pair in the set "ys" is unique. It shows that if you fold the list "ys" from the left, inserting each pair into a tree starting with an emp...
If for every pair of variables and assignments in a set, the variables are not equal or the assignments are equal, and a specific variable and assignment pair is in the set, then the mapping of the variable in the set is equal to the assignment.
If K' is a field, v is a valuation on K, v' is a valuation on K', x is an element of the carrier set of K, and the completion of K to K' with respect to v and v' exists, then the valuation of x in K is equal to the valuation of x in K'.
Applying the function 'f' to the sublist generated by applying the function 'g' to the list 'ps' is equivalent to applying the composition of functions 'f' and 'g' to the list 'ps'.
: Given a transition from (f,l) to (f',l') in the preflow-push algorithm, if the label of node u is greater than or equal to the label of node v plus one, and the edge (u,v) is not in the residual graph of f, but is in the residual graph of f', then the label of node v in the new labeling l' is greater than its label i...
The function which maps each natural number 'n' to the logarithm base 2 of the real number obtained by choosing 3 from 'n + 3' divided by 4, plus 1, is asymptotic to the function which maps each natural number 'n' to three times the natural logarithm of 'n' divided by the natural logarithm of 2.
If the ordered lists "xs" and "ys" are valid, and "aux" is not equal to "Some Eq", and if "k" is in the set of first elements of pairs in either "xs" or "ys", and "aux" is equal to the function "f" applied to "k", the pair lookup in "xs" for "k", and the pair lookup in "ys" for "k", and for any "k'" in the set of first...
The set of all 'b' such that there exists an 'a' where the ordered pair (a, b) is in the relation 'r' is small.
The lemma named "preprm_ext_reflexive" shows that "x" is a permutation of itself.
The lemma named "univ_eq_linorder" shows that the universal comparison is an equivalent linear order.
The set resulting from inserting an element 'x' into a sorted list 'xs' at a location 'l', is a subset of the set obtained by inserting 'x' into the set of 'xs'.
Applying the "xcpt_eff" function with any parameters "i", "G", "pc", and "s" to an empty list will always result in an empty list.
If there exists an 'x' such that property 'P' holds for 'x', and there exists a 'y' such that property 'Q' holds for 'y', then the statement that for all 'x', if property 'P' holds for 'x' then property 'R' holds for 'x', and for all 'y', if property 'Q' holds for 'y' then property 'S' holds for 'y', is equivalent to t...
The lemma named "encoding_compose_2" assumes two conditions: first, that "f" is an encoding function, and second, that "g" is an injective function. Under these assumptions, it shows that the composition of "f" and "g", where "g" is applied first and then "f", is also an encoding function.
For an infinite list, if you append a list 'b' to another list 'xs' and then get the element at the position which is the sum of the length of 'xs' and a number 'n', it is the same as getting the 'n'-th element from the list 'b'.
The function "as_is_empty" corresponds to the function "is_Nil" in the context of the abstract data type relation "as_rel" refined by "R", and this correspondence is a boolean relation.
The coercion of the bottom element is equal to the bottom element itself.
The extended real number representation of the sum of two natural numbers (m + n) is equal to the sum of the extended real number representations of these two natural numbers separately (m and n).
If the list 'xs' is not empty, then the tail of the concatenation of lists 'xs' and 'ys' is equal to the concatenation of the tail of 'xs' and 'ys'.
Unfolding a list using a filter generator function "g" on a list "s" is equivalent to filtering a list with predicate "P" and then unfolding it using the function "g".
The function "start_sttp_resp_cons" is responsive to the "storeST" function with parameters "i" and "tp".
The set of the sum of two intervals A and B is equal to the set of the sum of two other intervals A' and B', if and only if the set of A is equal to the set of A' and the set of B is equal to the set of B'. This is true for any A and B that are intervals in a linearly ordered additive abelian group.