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The lemma "Ord_succ_vsusbset_Vfrom_succ" assumes that "A" is a transitive set, "a" is an ordinal, and "a" is an element of the set "Vfrom A i". It shows that the successor of "a" is a subset of the set "Vfrom A" at the successor of "i".
In the context of a symmetric directed graph, the lemma named "graph_symmetric" demonstrates that if there is an arc from point u to point v in graph G, then there is also an arc from point v to point u in graph G.
The lemma named "node\<^sub>i_bal_up\<^sub>i" assumes that the balance property holds for the internal node "node\<^sub>i" with key "k", tree set "ts", and tree "t" after an insertion operation "bal_up\<^sub>i". It shows that the balance property also holds for the node "Node" with tree set "ts" and tree "t".
If "a" is the leftmost element at position "i", and there is a derivation from "a" to "b" represented by "D", then if "j" is less than or equal to "i", the derivation "D" is greater than or equal to "j".
If 'j' is less than 'nc', then the 'j'-th column of a matrix (which has 'nr' rows and 'nc' columns and is defined by a function 'f') is equal to a vector of length 'nr' where each element is defined by the function 'f' applied to the pair (i, j).
The affine dimension of the set obtained by subtracting a certain vector 'a' from all vectors in set 'S' is equal to the affine dimension of the original set 'S'. This is valid for 'a' being any vector in an n-dimensional Euclidean space.
The lemma named "closure_true" asserts that "true" is always valid.
The lemma "sum_mset_0_iff_ge_0" assumes that for any element x, the function f(x) is greater than or equal to 0. It shows that the sum of the function f(x) for all elements x in the multiset A is equal to 0 if and only if the function f(x) is equal to 0 for all elements x in the set of multiset A.
The sum up to the natural logarithm of a number x is equal to the natural logarithm of the factorial of the integer part of x.
The probability mass function (pmf) of the stochastic dominance strategy (sds) R13 for 'a' is 1/2, for 'b' is 0, for 'c' is 0, and for 'd' is 1/2.
If there is a relation R between p' and p, and between q' and q, and if q is a divisor of p, then there is the same relation R between the division of p' by q' and the division of p by q.
The lemma named "strand_sem_c_imp_ineq_model" assumes that a model M and a set S satisfy the strand semantics under an interpretation I, and that an inequality involving X and F is an element of the set S. Under these assumptions, it shows that the inequality model under the interpretation I holds for X and F.
The lemma named "lincomb_list_add_vec_1" assumes that the set of vectors 'us' is a subset of the carrier vector of dimension 'n', and 'x' is defined as the linear combination of the list 'lc' and 'us'. It also assumes that 'j' is less than the length of 'us', 'i' is less than the length of 'us', and 'i' is not equal to...
The lemma named "sync_syncV" assumes the following conditions: 1. Transaction 1 (trn1) is valid. 2. The source of transaction 1 is reachable. 3. Transaction 2 (trn2) is valid. 4. The source of transaction 2 is reachable. 5. Transaction 1 is a communication transaction (isCom1). 6. Transaction 2 is a communication tran...
If for all elements 'x' in the set 'xs', 'x' is lexicographically less than 0, then the sum of the list 'xs' is also lexicographically less than 0.
The lemma named "DynProc" makes the following assumptions and shows a result: Assumptions: 1. The set P is a subset of the set of states s such that there exists a set Z where the initial state of s is in the set P' of s and Z, and for every state t, if t is in the set Q' of s and Z, then the return state of s and t i...
If a real algebraic number "rai" satisfies the condition "invariant_1" and the degree of the polynomial "poly_real_alg_1" of "rai" is greater than 1, then "rai" also satisfies the condition "invariant_1_2". This is assumed to be true.
If a poset 'P' has a bottom element, 'x' is an element of 'P', 'y' is pseudominimal in 'P', and 'y' is less than or equal to 'x', then 'y' is pseudominimal in the set of elements in 'P' that are less than or equal to 'x'.
If x is less than or equal to y with respect to a left pre-order, then there exists exactly one z such that z divides x and y equals the product of z and x.
"d of x is less than or equal to d of y" if and only if "d of x equals the product of d of x and d of y".
If a path is found in a well-formed program from a context 'ctx' to a class 'cl', and another path is found from the same context to the fully qualified name of the class of a field 'f', then 'f' is in the set of fields in the first path.
The lemma for the exponential function of a formal power series when multiplied by an imaginary unit and a constant is equal to the cosine of the constant plus the imaginary unit times the sine of the constant. This is denoted as "?l = ?r".
The lemma shows four things: 1. The freshness of 'a' in relation to the sequence quote predicate of 't', 'u', 's', and 'k' is equivalent to 'a' being fresh in relation to 't', 'u', 's', and 'k' individually. 2. The sequence quote predicate of 't', 'u', 's', and 'k' is a sigma formula. 3. If the sequence quote predic...
The lemma "Dags_root_in_Nodes" assumes that "t" is in the directed acyclic graphs (Dags) of the nodes at the successor of "nb" level in the list "levellista", with "low" and "high" as parameters. It shows that there exists a "p" such that "p" is a directed acyclic graph (Dag) with "low" and "high" as parameters and "t"...
For all x in the free variables of the regular expression r, if the value of x in v is equal to the value of x in v', then the epsilon of the regular expression r with respect to the satisfaction relation, V, v, and i is equal to the epsilon of the regular expression r with respect to the satisfaction relation, V, v', ...
The lemma named "subdivision_genus_same_rev" assumes that there is a subdivision from graph G to graph H, with the reverse of G being rev_G and the edge reverse of H being edge_rev HM. It also assumes that H and HM form a directed graph map and that the Euler genus of the directed graph map of H and HM is m. Given thes...
If there is a vertex walk 'p' in graph 'G' and 'u' is a vertex in the set 'p', then 'u' is also a vertex in the graph 'G'.
, given the assumption "SubstForm v u x x'", we can obtain a term 't' and a formula 'A' such that 'u' is the evaluation of the quotation of term 't', 'x' is the evaluation of the quotation of formula 'A', and 'x'' is the evaluation of the quotation of the formula 'A' after substituting term 't' for the variable 'v'.
If p is equal to the integer n, then the reduced homology group of n in the n-sphere is isomorphic to the integer group. Otherwise, the reduced homology group of p in the n-sphere is a trivial group.
The lemma named "mat_rearrange" is defined with a fixed function 'F' that maps from types 'a, 'k1, 'k2, 'b to 'c, where 'c is a type that belongs to the class of dioid_one_zero. It assumes that the sets of all possible values ('UNIV') for types 'k1 and 'k2 are finite. It then shows that the sum of the sums of the funct...
The lemma named "closest_pair_rec_ys_eq" assumes three conditions: 1. The length of the list "xs" is greater than 1. 2. The tuple (ys, c₀, c₁) is equal to the result of the function "closest_pair_rec" applied to the list "xs". 3. The tuple (ys', δ', c₀', c₁') is equal to the result of the function "closest_pair_rec_co...
The range of the function A (with a subscript f) applied to the identity function equals "Allow".
The set interleaving of "s", "X", and "t" is equal to the set interleaving of "t", "X", and "s".
The function "Map.empty" is injective (or one-to-one) on the domain of "Map.empty". The statement "ran_empty_upto" asserts that the range of "Map.empty" is a subset of the set of all integers less than 0.
If a machine "M" is observable, then the following properties hold: 1. The initial state of the minimised machine is the set of states in "M" where the left quotient of "M" and the state is equal to the left quotient of "M" and its initial state. 2. The states of the minimised machine are the sets of states in "M" wh...
The lemma named "sub_in_const" assumes that "g" is an element of the carrier "P", "f" is also an element of the carrier "P", and the degree of "g" is zero. Under these assumptions, it shows that the degree of the composition of "f" and "g" is also zero.
The well-foundedness (wf) of the inverse image of "qTermQSwappedLess" under the function "aux_alpha_ignoreSecond" is true.
If a state "s" satisfies the invariant "I", then the resource generated by the oracle using the callee and state "s" is well-typed according to the interface "\<I>".
Substituting a type "T" with a substitution instance "inst2" and then substituting the result with another substitution instance "inst1" is equivalent to substituting the type "T" with a combined substitution instance that is created by mapping the second element of each pair in "inst2" with "inst1" and then appending ...
The lemma named "mat_minus_minus" is about matrices. It assumes that A, B, and C are matrices of the same dimensions, n by m. The lemma shows that subtracting matrix B from matrix A and then subtracting matrix C is equivalent to subtracting matrix B from matrix A and then adding matrix C.
The lemma "freshBoundOutputTransition" is defined with fixed elements P, a, x, P', and c, where P, a, x, P' are of type pi and c is of type name. It assumes that P makes a transition to a new process P' with a bound output on channel a and name x. It also assumes that c is fresh for P, meaning it does not occur in P,...
The product of zero, X squared, and Z to the power of seven, plus the product of zero, Y cubed, and Z squared, equals zero.
If the initial value of "u" is zero, and for all "i" less than "n", if "x i" is true then the next value of "u" is the successor of the current value of "u", and if "x i" is not true then the next value of "u" is the same as the current value of "u", then for all "i" less than "n", the count of "x" at "i" is equal to t...
The arbitrary intersection of an arbitrary intersection of a set P is equal to the arbitrary intersection of the set P.
The lemma l7_20: assumes that points A, M, and B are collinear and the distance from M to A is equal to the distance from M to B. It shows that either point A is the same as point B or M is the midpoint of line segment AB.
If the length of a list "xs" is less than a certain value "k", then slicing the list "xs" at the position "k" will result in an empty list.
The lemma named "and_or_s" assumes that the function "get_S" applied to "w1" equals zero and the bitwise AND operation between "w2" and the 32-bit binary number "00000000000000000000000010000000" equals zero. Under these assumptions, it shows that the function "get_S" applied to the bitwise OR operation between the bit...
If a message from S to B containing an encrypted message with B's shared key, which includes a nonce NB, agents A and B, and a key K, is in the set of events, and if B is not compromised, and A is not equal to B, and the set of events is part of the Otway-Rees protocol, then there exists a nonce NA such that a message ...
The auxiliary function for converting a binary number to a binary list, when applied to a binary number that has been truncated to 'm' bits and a binary list 'bl', with a length 'n', is equal to a list of 'False' values replicated 'n - m' times, concatenated with the result of the same auxiliary function applied to the...
Given two morphisms "f" from "a" to "b" and "g" from "c" to "d", the domain of the tensor product of "f" and "g" is the tensor product of "a" and "c".
The lemma named "sum_strict_mono_ex1_complex_lex" establishes the following: Given two functions 'f' and 'g' that map from an arbitrary type 'i' to complex numbers, and a finite set 'A'. If for every element 'x' in 'A', the value of 'f' at 'x' is less than or equal to the value of 'g' at 'x' in the complex lexicograph...
The lemma "projxx" is established under the following conditions: 1. A variable 'a' is an element of the set {x, y}. 2. 'a' is not equal to 'e'. 3. 'e' is an element of the set {x, y}. 4. A function 'f' is an element of the set that includes an empty list and a list containing 'e'. 5. The set of a list 'bs' concatenat...
The general depth-first search (gen_dfs) in context c is less than or equal to the abstract iterative depth-first search (it_dfs) in context a, when related by the simple state relation (simple_state_rel) over the set of end states (ES). This is a statement about refinement, indicating that the concrete implementation ...
Merging an empty heap with any heap 'h' results in the heap 'h'.
The lemma "c2p_mono_p2c_strict" assumes that "n" is greater than or equal to the length of the projection of the infinite list "t" onto "c" minus one, and that "n" is less than the "c"-truncated "t" at "n'". It shows that the "c"-lifted "t" at "n" is less than "n'".
If the path 'p' is not empty, 'u' is the tail of the graph 'G' at the head of 'p', 'v' is the head of 'G' at the last of 'p', and for any 'i', if 'i' plus one is less than the length of 'p', then the head of 'G' at 'p' indexed at 'i' equals the tail of 'G' at 'p' indexed at 'i' plus one, it can be shown that there is a...
The lemma named "decr_grading_zero" assumes that "d" is a homogenous grading. It shows that decreasing the grading of "d" by "n" times 0 equals 0, in the context of a cancelable commutative monoid with addition.
The polymap of the product of two elements p and q is equal to the composition of the polymap of p and the polymap of q.
Attaching a callee to the fusion of two converters (one composed of an adversary core and the rest, the other composed of a user core and the rest) and a fused resource (composed of a core and the rest) is equivalent to applying a state isomorphism (iso_trisplit) to the fusion of an attached core (composed of the adver...
The lemma "lhopital_left_at_top_at_bot" is about two real-valued functions, f and g. It assumes that as x approaches a from the left, f(x) tends to positive infinity and g(x) tends to negative infinity. It also assumes that the derivatives of f and g exist and are denoted by f' and g' respectively, and that the limit o...
The lemma named "mpt_factor_projection" in the context of measure-preserving transformations (mpt) states the following: Given a function 'f' from some set 'a' to a set 'b' which has a second countable topology, and assuming that 'f' is Borel measurable with respect to measure 'M', and that the sigma-finite measure of...
If a program P type checks the list of types E appended with T, denoted as "E@[T]", under the environment A, then it also type checks the list of types E under the same environment A.
If a formula "psi" is not in the mu-subformulas of "phi", then the nu-substitution of "phi" with the set "X" minus "psi" is equal to the nu-substitution of "phi" with the set "X".
If H is a member of filters and the right congruence of H is equal to the left congruence of H, then H is a member of normal filters.
The span of the list consisting of the element 'm' followed by the list 'ms' is equal to the sum of the span of the list containing 'm' and the span of the list 'ms'.
Looking up a key 'k' in a mapping 'm' that has had a function 'f' applied to its values, is equivalent to looking up the key 'k' in the original mapping 'm' and then applying the function 'f' to the result, if the result exists.
The Mangoldt function of 0 is equal to 0.
If 'c' is an object in category 'B' and 'r' is an object in category 'A', then the following conditions hold: 1. The natural transformation map of the universal arrow from 'c' to 'r' under functor 'F' and object 'u' is equal to the function that maps each object 'd' in category 'A' to the universal map from 'c' to 'r'...
A word 'w' is sign extended to 'n' if and only if the result of sign extending 'w' to 'n' is equal to 'w'.
The key "K" is in the guard of level "n" and set of keys "Ks".
The external choice operation on the disjunction of two predicates P and Q with respect to a variable x is equivalent to the disjunction of the external choice operations on the individual predicates P and Q with respect to the same variable x.
The lemma named "mpoly_of_poly_pCons" is defined for a polynomial 'p' in a semiring. It shows that the multivariate polynomial of a polynomial constructed with 'pCons' (a function that adds a constant term to a polynomial) is equal to the constant plus the variable times the multivariate polynomial of the original poly...
The size of the result of the function "remove_neg" applied to a formula phi is less than or equal to the size of the original formula phi.
If a matching exists in set A with a preference match in set A, then the set A is stable.
If a term 't' is in the intruder knowledge of a strand 'S', then there exists a term sequence 'ts' such that 't' is in the set of 'ts' and 'Receive ts' is in the set of 'S'.
There exists a value 'M'' such that the integral 'I' has an exponential order 'M'' with the constant 'c'.
If a monad reader is defined with the return function, a bind function of type ('a \<times> 's, 'm), and an ask function of type ('r, 'm), then a monad reader state can be defined with the return function, a bind function of type ('a, ('s, 'm) stateT), an ask function of type ('r, ('s, 'm) stateT), a get_state function...
The nth element of the list "xs" with "init" as the initial element is equal to the nth element of the list that starts with "init" followed by the elements of "xs".
The initial state of m3 is a subset of the state m3_inv3_sesK_compr.
If a certain extended natural number 'i' is less than another extended natural number 'j', and a third extended natural number 'k' is less than a fourth extended natural number 'l', then the sum of 'i' and 'k' is less than the sum of 'j' and 'l'.
A step from square s_i to square s_j is valid if and only if a step from the mirror image of square s_i to the mirror image of square s_j is also valid, where the mirror image is determined by the function "mirror1_square" with respect to a board of size n.
The lemma named "foldr_fun_upd" assumes that the function 'f' is injective on the set 'xs' and that 'x' is an element of the set 'xs'. It shows that if you fold over the set 'xs' with a function that updates an accumulator '\<A>' at the position 'f x' with the value 'g x', then the value of the accumulator at the posit...
The language of the generalized tree automaton resulting from the intersection of automaton A and automaton B is equal to the intersection of the languages of automaton A and automaton B.
The lemma named "Hoare_consequence" assumes three conditions: 1. P' implies P, 2. if P is true before execution of program p, then Q is true after execution, 3. Q implies Q'. If these conditions are met, it shows that if P' is true before execution of program p, then Q' is true after execution.
The lemma named "ltrm_inv_simp" assumes that "f" is an element of the carrier "P". It shows that applying the function "ltrm" twice to "f" is equivalent to applying it once.
The lemma named "unitary_eigenvalues_norm_square" is about a complex matrix U. It assumes that U is a unitary matrix, that U is an n by n matrix, and that k is an eigenvalue of U. It shows that the product of the conjugate of k and k itself equals 1.
There exists a sequence 'ys'' such that the function 'f' applied to the concatenation of sequences 'xs' and 'ys' equals the concatenation of the function 'f' applied to 'xs' and 'ys''.
The stake of "i" and "xs" equals bottom if and only if "i" equals bottom or "xs" equals bottom.
The lemma named "bind_simps" states two things: 1. Binding an empty list with any function results in an empty list. 2. Binding a list that has at least one element (x) followed by other elements (xs) with a function results in the function applied to the first element (x) concatenated with the result of binding the r...
In the context of an Order, if E and F are Orders, and there is an order isomorphism from D to E via function f, and an order isomorphism from E to F via function g, then there is an order isomorphism from D to F via the composition of functions g and f on the carrier of D.
Given three matrices: m1, m2, and m3, where m1 and m2 are square matrices of dimension d1 and m3 is a square matrix of dimension d2, the partial tensor product of the sum of m1 and m2 with m3 is equal to the sum of the partial tensor products of m1 with m3 and m2 with m3.
If "A" is right total and "B" is right unique, then the relation of resources "A" and "B" is also right unique.
If "g" and "g'" are elements of a group "G", then the transformation of the difference between "g" and "g'" is equal to the difference between the transformation of "g" and the transformation of "g'".
The lemma named "antiframe_hoare_r" assumes that "a" is a very well-behaved lens, "a" is unaltered by "r", "a" does not modify "q", and "P" is a program that, when run in a state satisfying precondition "p", ensures postcondition "q". Under these assumptions, it shows that if "P" is run in a state satisfying both "p" a...
The perzeta function at 0 is equal to the zeta function.
The lemma named "sum_atLeastAtMost_int_last" assumes that "a" is less than "b", where "a" and "b" are integers. It shows that the sum of the function "f" from "a" to "b" is equal to the sum of the function "f" from "a" to "b" excluding "b", plus the function "f" at "b".
If a well-formed factorization of "a" exists in the set "fs" within the group "G", and "fs" is a permutation of "fs'", and the set "fs" is a subset of the carrier of "G", then a well-formed factorization of "a" also exists in the set "fs'" within the group "G".
The lemma "sbh_step_preserves_tmps_bound'" assumes a step from the state "(is,θ,sb,m,D,O,R,S)" to the state "(is',θ',sb',m',D',O',R',S')" in the Store Buffer with History model. It shows that the union of the domain of θ and the load temporary variables in is is equal to the union of the domain of θ' and the load tempo...
If a function 'f', which maps natural numbers to elements of a ring, has an inverse, then the function that scales 'f' by a factor 'r' also has an inverse.
Let f be a function that takes two arguments, i and x, and returns a non-negative extended real number. Suppose that the set I is countable, and for each element i in I, the function f(i) is Borel measurable with respect to the measure space M. Then, the statement shows that the double non-negative Lebesgue integral of...