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The sum of all elements 'n' in the set 'labels t', each incremented by 2, is greater than or equal to 2.
'p' is prime if and only if 'p' is greater than zero and 'p' is a prime element.
The lemma named "update_parametric" which includes lifting syntax shows that the function "update" from the module "monad_state_base" is preserved under a certain transformation. This transformation is described by a complex function type that involves the type variables S and M. Specifically, the transformation takes ...
For a formal Laurent series 'a' over a division ring, the derivative of the inverse of 'a' is equal to the negative of the inverse of 'a' times the derivative of 'a' times the inverse of 'a'.
The lemma "VarNonOccForm_imp_dbfm_fresh" assumes that variable "v" does not occur in form "x". It shows that there exists a well-formed difference-bound matrix "A" such that "x" is equal to the evaluation of the quotation of "A" and the atom decoded from variable "v" is fresh in "A".
If the satisfaction of an if-expression "i" is None, then the value of the if-expression "i" with an assignment "ass" is False.
The supremum of a and b is equal to the supremum of b and a.
The lemma named "snd_set_split" assumes that "ps" is in the second element of the set "pth". It then obtains "pth1", "pth2", and "e" such that "pth" is equal to the concatenation of "pth1", "e", and "pth2", the second element of "e" is "ps", and "ps" is not in the second element of the set "pth1".
If a certain event 'evs' is in 'p1' and 'A' is not in 'bad', then if the encryption of the hash of 'X' with 'A's private key is in the parts that spies can see in 'evs', it implies that there exists 'B' and 'Y' such that 'A' says to 'B' something 'Y' in the set of 'evs' and the signature of 'A' on 'X' is in the parts o...
The dimension of the row of the matrix Uswap is 4.
The image of a set A under a function f is the set of all elements y such that there exists an element x in A for which y is the result of applying function f to x.
The lemma named "commute_rrstep_intro" assumes that for any states 's' and 't', if the pair (s, t) is in the composition of the rrstep relation with respect to 'F', the inverse of 'R', and 'S', then the redp commute with respect to 'F', 'R', and 'S' for states 's' and 't'. Under this assumption, it shows that the srste...
The set of elements in a binary tree after an element 'x' is inserted is equal to the union of the set of elements in the original tree and the set containing the inserted element 'x'.
The lemma named "polyfun_det_deep_model" demonstrates that the determinant of the submatrix, derived from a function 'f', is a polynomial function within the dimension of the weight space.
For any list "xs", the proposition "P" holds if one of the following conditions is met: 1. The list "xs" is empty (denoted as "Nil"). 2. The list "xs" contains exactly one element "x". 3. The list "xs" contains exactly two elements "x" and "y". 4. The list "xs" contains three or more elements, where the first three el...
The function "mdinit" is less than or equal to the function "dinit" when "dinit" is transformed by the relation "build_rel" with parameters "αs" and "dinvarm".
Merging list m1 with the result of merging lists m2 and m3, and then mapping the result, is equivalent to merging list m1 with m2 first, then merging the result with m3, and mapping the final result. This demonstrates the associative property of the merge operation in the context of these lists.
: 1. If a term 't' is in the set of terms of a strand 'S', then the term 't' substituted by a substitution 'delta' is in the set of terms of the strand 'S' substituted by 'delta'. Alternatively, there exists a set 'X' and a function 'F' such that the inequality of 'X' and 'F' is in the set 'S' and the term 't' substit...
The natural logarithm of the factorial of a number is equal to the sum up to that number of natural logarithms.
Given the assumption "A" which is defined by the predicate "offs_pred" with parameters "ns", "ub", "xs", "index", "key", "mi", and "ma", if "i" is less than the length of "ns", then for all "j" and "k", if "k" is less than the length of "ns", "i" is less than or equal to "j", "j" is less than "k", the offs_num function...
, assuming 'v' is not equal to zero, the projection of 'v' in the second model of the Klein disk model (K2) is true if and only if the dot product of 'v' and the result of matrix 'M' multiplied by 'v' is less than zero.
For any real number 'x', if 'x' is greater than or equal to 0 and less than or equal to 1, then the sequence of 'x' raised to the power of 'n' (where 'n' is each natural number in the sequence) is a bounded sequence.
If F and G are groups, and f is a group homomorphism from F to G, and if x and y are elements in the kernel of the group homomorphism f, then the product of x and y (in the group F) is also in the kernel of the group homomorphism f.
If x is greater than or equal to x1, and i is less than k, then the i-th element of bs multiplied by x plus the i-th element of hs evaluated at x is greater than 1.
"not" is a negation normal form (nNor).
The lemma named "one_test" asserts that "1" is a test.
For any three hyperdual numbers a, b, and c, where a, b, and c are elements of a set that has the properties of inverse, commutative ring, and division ring, the result of dividing the sum of a and b by c is equal to the sum of a divided by c and b divided by c.
The lemma "get_root_node_ok" assumes that the heap "h" is well-formed, the pointers in "h" are known, and the type of "h" is well-formed. Additionally, it assumes that the pointer "ptr" is in the set of object pointer kinds of "h". Given these assumptions, it shows that the operation "get_root_node" on the pointer "ptr...
The lemma named "from_nat_eq_imp_eq" assumes that the function "from_nat" applied to "x" equals "from_nat" applied to "xa" for some type 'a. It also assumes that "x" and "xa" are both less than the cardinality of the type 'a. Under these assumptions, it shows that "x" must be equal to "xa".
There do not exist any integer values for x, y, and z such that the equation 4x - 6y equals 1.
The lemma named "jordan_matrix_poly_bound2" is defined with a fixed list of natural numbers and 'a' type elements, denoted as "n_as". It assumes two conditions: 1. For any natural number 'n' and element 'a' in the list "n_as", if 'n' is greater than 0, then the norm of 'a' is less than or equal to 1. 2. For any nat...
The star-transform of the complex modulus function is equal to the hypercomplex modulus function.
Evaluating the function "evalO" with parameters "b" and "((S ^^ n) Z)" equals "n".
If we have a category 'BB' and a functor 'GG' from 'AA' to 'CC', and if 'b' is an object in 'BB' and 'a' is an object in 'AA', then the object map of the right composition of 'BB', 'SS', and 'GG' applied to the pair 'b, a' is equal to the object map of 'SS' applied to the pair 'b' and the object map of 'GG' applied to ...
If the relation "r" is stable for the step function, the state sequence "τs", and the program counter "pc", and the successor of "pc" is less than the size of the state sequence "τs", then the well-typedness condition "wtc" for the code "c", the program counter "pc", and the state at "pc" in the state sequence "τs" is ...
If a function 'f' is a homomorphism on a graph 'S' with respect to an element 'x' and 'a', then 'x' is an element of the set of formulas of 'S'.
The comparison of "x" and "y" using the function "compare_3" is equivalent to comparing the real number equivalents of "x" and "y" using the "compare" function.
The derivative of the formal Laurent series of the inverse of X is equal to the negative of the square of the formal Laurent series of the inverse of X.
The lemma named "init_eqI" assumes that for any "x" less than or equal to "n", the function "f" of "x" equals the function "g" of "x". It shows that the initial segment of "f" up to "n" equals the initial segment of "g" up to "n".
The lemma named "cmax1" assumes that "r" and "s" are cardinal orders and "s" is less than or equal to "r". It shows that the maximum cardinality between "r" and "s" is "r".
If "DenyAll" is in the set converted from policy p to a list, and all elements in list l are in the list converted from policy p, and all networks in the list converted from policy p are distinct, then the function C applied to the firewall policy converted from the sorted list (which is obtained by removing shadow rul...
The lemma named "map_mmupdE" assumes that the function "map_mmupd" applied to the map "m", the set "K", the value "v", and the key "k" equals "Some x". From this assumption, it concludes two possible scenarios: either "k" is not in the set "K" and the map "m" applied to "k" equals "Some x", or "k" is in the set "K" and...
The lemma named "cat_smc_eqI" assumes that both "\<AA>" and "\<BB>" are categories in the context of some set "\<alpha>". It also assumes that the identity morphism of "\<AA>" is the same as the identity morphism of "\<BB>", and that the semicategory of "\<AA>" is the same as the semicategory of "\<BB>". If all these c...
If the number of holes in the context C is zero, then filling the holes in context C with an empty list results in the term of the multi-hole context C.
The interpretation function "Ifm" applied to a boolean list "bs" and a conjunction of a list of propositions "ps" is equivalent to stating that for every proposition "p" in the set of propositions "ps", the interpretation function "Ifm" applied to the boolean list "bs" and proposition "p" holds true.
The lemma named "inj2_distinct_concat_map" assumes that the function "f" is injective on two arguments. It shows that if the lists "ys" and "xs" are distinct, then the list resulting from applying the function "f" to each pair of elements from "xs" and "ys" is also distinct.
The delta function of "b" and "a" belongs to the set of almost everywhere zero functions.
The infinite iteration of any element 'x' multiplied by the top element equals the top element.
If 't' is less than or equal to 'k' and 't' is greater than or equal to 1, and for all subsets 'ps' of 'V', if the cardinality of 'ps' is 't', then the index of 'ps' in 'B' is less than or equal to 'Lambda_t', then 't' is a packing design of 'V', 'B', 'k', and 'Lambda_t'.
: Assuming a path 'p' exists from state 'q' in machine 'M', and the target state of path 'p' from state 'q' is 'q' itself, it shows two things: 1. A path exists in machine 'M' from the target state of the first 'i' steps of path 'p' from state 'q', which is followed by the remaining steps of path 'p' after the first 'i...
If a program P sees a field F with type T and property b in class D, and the same program P sees the same field F with type T' and property b' in class D', then property b must be equal to property b', type T' must be equal to type T, and class D' must be equal to class D.
If a list "xs" is represented by "i" and "ba", and "i" is not equal to 0, then the function "get_nth_im1" applied to "ba" will return the element at the position "i - 1" in the list "xs".
If a function 'f' applied to every element of a list 'l' results in a distinct list, then the original list 'l' must also be distinct.
If for every 'i' in set 'S', the function 'f' applied to 'i' is continuous, then the function 'F' applied to the product of 'f' applied to each 'i' in set 'S' is also continuous.
If two sets 'as' and 'bs' are essentially equal in the group 'G', and if 'bs' is a well-defined factor of 'b' in 'G', and if 'b', 'as', and 'bs' are all elements of the carrier of 'G', then 'as' is also a well-defined factor of 'b' in 'G'.
The set of continuous functions (cont) applied to the function H with input n is equal to the union of the identity function (id) and the composition of H with the root function, both applied to the set of continuous functions (cont) with input pick n.
Given a set "S" and an element "xs" in "S", if "S" is finite, then there exists an element "ys" such that "xs" concatenated with "ys" is in "S" and there does not exist a non-empty element "zs" such that "xs" concatenated with "ys" and "zs" is in "S". This essentially means that "ys" is a maximal extension of "xs" in "...
The lemma named "open_unresolvable_eqs" assumes that "X" is an open unresolvable set in a topological space. It shows that the composition of the identity function "I" and the Kuratowski closure operator "K" applied twice is equal to the composition of "I" and "K". Similarly, the composition of "K" and "I" applied twic...
The function "inv_loop5_exit" when applied to the variables "x" and a pair consisting of "b" and a list starting with "Oc", will always return False.
The result of the proof for the differential of a constant is correct. This is demonstrated by showing that the differential of a constant zero equals to a constant zero, with no assumptions needed.
If group G acts on groups H and K, and if elements a and b belong to the intersection of groups H and K, then the product of a and b in the intersection of H and K is equal to the product of a and b in group K.
If you divide 'a' by 'b' using signed division (sdiv), multiply the result by 'b', and then add the remainder of 'a' divided by 'b' using signed modulus (smod), the result will be equal to 'a'.
For any postcondition "post", an empty precondition set for the actions "Ra" and "Rc" always holds. The comment suggests not to declare this lemma as an introduction rule.
The constraint of C on the intersection of A and B is equal to the intersection of the constraint of C on A and the constraint of C on B.
If in the program 'prog', there is a control edge from node 'n' to node 'n' with properties (p,es,rets), then there exists a label 'l' such that node 'n' is labeled 'l' and node 'n' is labeled the successor of 'l'.
If a set H is a subset of the carrier G and an element h is in the generated group of G and H, then the inverse of h is also in the generated group of G and H.
If a complex number 'z' is not infinity, then the product of zero and 'z' is zero.
If a sequence of observations "obs" is similar to another sequence of observations "obs'", then "obs'" is also similar to "obs". This is assuming the property of symmetry.
If "n" is greater than or equal to "k" and "k" is greater than or equal to 2, then the probability of the set of edge spaces in the probability space "P" where "k" is less than or equal to the chromatic number of the undirected edge graph of the edge space, is less than or equal to the combination of "n" and "k" times ...
If the program P shows that the main class "ClassMain P" sees a method M with types Ts returning type T as method body mb in class C, then this is false. This implies that there are no method calls from the main class in program P.
The order of a stochastic process monad function (spmf) "R" with parameters "p" and "return_spmf y" is equivalent to the condition where for all "x" in the set of spmf "p", "R x y" holds true.
The feedback of a relation R preserves the infimum.
The lemma named "symmetric_second_derivative" assumes that "a" is in the interior of "G". It shows that the second derivative of "f" with respect to "i" and "j" is equal to the second derivative of "f" with respect to "j" and "i".
If a positive integer w is given, and i is less than or equal to j and i is also greater than or equal to zero, then i is less than or equal to the product of j and w.
A string 'w' is in the language of a regular expression 's' if and only if there exists a regular expression 'r' in the top-level summands of 's' such that 'w' is in the language of 'r'.
The polynomial constructed with coefficients 0 and 1 is not equal to the polynomial with only coefficient 1, in the field of integer polynomials.
The lemma named "equivalence_functor_R" demonstrates that "R" is an equivalence functor from category "V" to itself.
The roots of a polynomial "q" are equal to the sum of the roots of its prime factors.
If a certain weak simulation relationship holds, it will still hold even after we permute the elements involved in the relationship.
In the context of a vector space, the lemma named "sum_map_linear" is defined. It fixes two variables, V1 and V2. It assumes that both V1 and V2 are subspaces of V in the field K. The lemma then shows that the function which maps a vector v to the direct sum of its first and second components is a linear map from the d...
: assuming A, B, C, and D are coplanar (lie on the same plane), C is not equal to D, and C, D, and E are collinear (lie on the same line), it can be shown that A, B, C, and E are also coplanar.
The non-deterministic choice of P with itself is equal to P.
If I is an element of the set i_set0, then the result of the operation k minus I is also an element of the set i_set0.
The abstract game representation of the equivalence class of the game "g" is equal to the abstract game representation of the equivalence class of the game "h" if and only if the game "g" is equivalent to the game "h".
The lemma "repeatAmountOfQuantifiers_helper_eval" assumes that for all lists 'xs' and functions 'F', the evaluation of 'F' with 'xs' is equal to the evaluation of the step of 'F' with 'xs'. Under this assumption, it shows that the evaluation of 'F' with 'xs' is equal to the evaluation of the function 'repeatAmountOfQua...
The lemma named "sum_count_vec" assumes that the set of vector 'v' is finite. It shows that the sum of the count of vector 'v' for each element 'i' in the set of vector 'v' is equal to the dimension of vector 'v'.
The lemma named "decide_exis_lem_helper_R" is defined for a fixed polynomial "fml" of real numbers. It assumes that the pair "(fml_struct,polys)" is equal to the conversion of "fml". It then shows that there exists a real number "x" such that the semantic lookup of "fml_struct" in the map of "polys" to "x" is equivalen...
The norm of a projection P is less than or equal to 1, given that P is a projection, for P as a continuous linear operator on a complex Hilbert space.
If "v" is a valuation of "K" and "x" is an element of the carrier of "Vr K v", then "v x" equals zero if and only if "Rxa (Vr K v) x" equals the carrier of "Vr K v".
If the binary search trees "l" and "r" satisfy the black depth invariant and "l" satisfies the color invariant, and if the black depth of "l" is one more than the black depth of "r", then the tree resulting from balancing "l" and "r" with "k" and "v" also satisfies the black depth invariant, and its black depth is equa...
The lemma named "cart_prod_singleton_right_gen" assumes that for any element 'x' in the Cartesian product of multiset 'A' and a singleton multiset containing 'b', the property 'P' holds for 'x' if and only if the property 'Q' holds for the first element of 'x'. It then shows that the multiset of all 'x' in the Cartesia...
If 'p' is a prime number, then 'p' is greater than 1 in the set of integers.
The open ball, which is the set of all points 'y' at a distance less than 'd' from a point 'x', is open.
The lemma establishes a relationship for a function phi, which is a mapping from a preorder 'a to 'b. It shows that the relational composition of rel_R with the predicate Q and a lambda function, which asserts that for all 't' in the set U of 's', if for all 'tau' in the downward closure of U of 's' at 't', G holds for...
The lemma named "typ_ok_code" assumes that the theory 'Theta' is well-formed according to the function 'exe_wf_theory'. It shows that the type 'ty' is okay in the translated theory 'Theta' according to the function 'typ_ok' if and only if the type 'ty' is okay in the original theory 'Theta' according to the function 'e...
If the stripped versions of two commands C1 and C2 are equal, and if the top of the abstract command C1 is not in the variables of C1, and the top of the abstract command C2 is not in the variables of C2, and if C2 is less than or equal to C1, and if the narrowing of C1 and C2 is less than C1, then the narrowing of C1 ...
: Given two positive integers k1 and k2, and two computational states c1 and c2 that are equivalent under the condition "equiv_states", and for any input0 and m such that the set of input0 is a subset of A and m is an element of A, the execution equivalence "Equiv_Exec" holds for the mapped input functions input_fun1 ...
Converting a formal power series (fps) to a formal Laurent series (fls) and then applying the fls_base_factor_to_fps function results in the unit_factor of the original formal power series.
For any point 'x' in a Euclidean space, the distance between the point 'y' clamped between 'a' and 'b' and the point 'x' clamped between 'a' and 'b' is less than or equal to the distance between 'y' and 'x'.
If the certificate of entity B with public key KB is part of what the spies can see in the event sequence 'evs' in the TLS protocol, then the public key KB must be the public key of entity B.