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For a function 'f' that maps a natural number to an additive commutative monoid, the sum of 'f' applied to each element in the set from 'a' to 'a+b' (exclusive), where each element is subtracted by 'a', is equal to the sum of 'f' applied to each element in the set from 0 to 'b' (exclusive).
If the program counter 'pc' is less than the length of the compiled expression 'e', then the execution move of 'e' being an instance of 'T' is equivalent to the execution move of 'e' itself, given the same context including the code index 'ci', program 'P', thread 't', heap 'h', stack 'stk', local variables 'loc', prog...
The lemma named "terminates_drop_raw" assumes that a function "g" terminates. It shows that the function "drop_raw g" also terminates.
The lemma named "dot2_rearrange_2" is defined with two fixed variables, u and x, both of which are 2-dimensional real vectors. It assumes that the dot product of u and x equals zero and that the second component of x is not zero. It then shows that u is equal to the negative of the first component of u divided by the s...
The list set of A is not equal to the empty set.
The sequential composition of a relation Q with the infimum (greatest lower bound) of a set of relations S is equal to the infimum of the set of all sequential compositions of Q with each relation P in S.
If a list 'xs' is not finite, then the infinite list generated by the nth element of 'xs' is equal to 'xs' itself.
The lemma named "finite_intersection_is_semialgebraic" assumes that "Xs" is a finite set. It shows that if "Xs" is a subset of the semialgebraic sets of "n" and "Xs" is not an empty set, then the intersection of "Xs" is semialgebraic of "n".
The monitored program location of the mapped non-local alpha function of a flowgraph "fg" and a sequence "w" is equal to the union of the monitored location of the flowgraph "fg" and sequence "w" and the monitored environment of the flowgraph "fg" and sequence "w".
The partial correspondence relation for finite sets, denoted as "pcr_fset", is right total when the equality relation is used. This is a rule for transfer in Isabelle.
If set A is equal to set B, then the membership of an element x in set A is equivalent to the membership of x in set B.
The lemma "get_root_node_is_strongly_dom_component_safe_step" assumes that the heap "h" is well-formed, the type of "h" is well-formed, and "h" contains known pointers. It also assumes that the DOM component of the pointer "ptr" in heap "h" is "c", and the root node of the pointer "ptr'" in heap "h" is "root". It shows...
The natural number constant 1, when interpreted as an irrational number in the context of variables "vs", equals 1.
If a statement A is provable from the assumptions that x is an ordinal, y is an ordinal, and a set of other assumptions H, then A is also provable from the assumption that x is not equal to y and the same set of other assumptions H.
The lemma named "graph_homomorphism_def2" demonstrates that a graph homomorphism is an embedding as described in the paper. It shows that a graph homomorphism from graph G1 to graph G2 using function f is equivalent to the following conditions: the vertices of G1 are the domain of function f, function f is univalent, G...
The strong normalization property of the composition of relations R and S is equivalent to the strong normalization property of the composition of relations S and R.
The lemma named "in_small_setD" assumes that "x" is an element of the small set "A". It shows that "x" is an element of "A".
If the invariant "ts_inv_ok" holds for the thread states "ts" and the invariant "I", then a thread "t" is not present in "ts" if and only if it is not present in "I".
If R commutes with S, then S commutes with R.
The lemma named "sterm_induct" with case names "Sconst", "Svar", "Sabs", and "Sapp" assumes the following: 1. For any "x", the property "P" holds for "Sconst x". 2. For any "x", the property "P" holds for "Svar x". 3. For any "cs", if for every pair "(pat, t)" in the set "cs" the property "P" holds for "t", then the p...
If functor FF maps category BB' to category BB, functor GG maps category CC' to category CC, and functor SS maps the product category of BB and CC to category DD, then the range of the object map of the bifunctor composition of SS, FF, and GG is a subset of the objects of category DD.
The lemma named "grid_part" assumes the following conditions: 1. "d" is less than the length of "b". 2. "p" is an element of the grid defined by "b" and "d". 3. "p'" is also an element of the same grid. 4. The level of "p'" at "d" is greater than or equal to the level of "p" at "d". 5. The index of "p'" at "d" is less...
If "xa" belongs to "F_inj", then "strict_from_inj n xa" belongs to "F_strict". This is for any "xa".
If a tree 't' is a red-black tree and 't' is not a leaf, then the minimum value obtained from the function 'pst_getmin' applied to 't' is the minimum value in the set of all elements in the tree 't'.
A set B is a proper subset of the image of a function f applied to a set A if and only if there exists a set AA such that AA is a proper subset of A and B is the image of the function f applied to AA.
The lemma named "htt_elim" assumes that the program "c" transitions from precondition "P" to postcondition "Q" in time "b". It shows that the time bound of program "c" with precondition "P" is "b".
The well-formedness of a structure 's' with respect to a natural number 'n' is equivalent to the well-formedness of all elements 'r' in the top-level intersections of 's' with respect to 'n'.
"final" is equivalent to the intersection over all v in V and all w in V of the set of states s such that v is reachable from the root and the set of states s such that either v is not connected to w by an edge in E or there are no messages between v and w.
For all x and y, if the function of x equals the function of y, then x equals y. Given this, if the table of the map that applies the function to each pair (k, x) in table t equals Some (function of x) for a certain key k, then the table of t for the same key k equals Some x.
In the context of a sigma finite measure, the lemma "measurable_emeasure" is defined as follows: Assuming that for any element 'x' in the space 'N', the set 'A x' is a subset of the space 'M', and that the set of all 'x' in the product space of 'N' and 'M' such that the second element of 'x' is in 'A' of the first ele...
If the differential invariant "I" holds for the function "f" over the time interval "T", the state space "S", the initial time "t₀", and the guard "G", and if the predicate "P" implies the invariant "I", and if the conjunction of the invariant "I" and the guard "G" implies the predicate "Q", then the differential equat...
: Assuming for any term 't', the first element of 'A' derived with respect to the term of 't' is equal to the first element of 'B' derived with respect to the term of 't', and the second element of 'A' derived with respect to the term of 't' is equal to the second element of 'B' derived with respect to the term of 't',...
If multiset M is less than multiset N, then if it is not the case that P, and N is less than multiset M, then P must be true. This is under the condition that 'a is a preorder multiset.
The initial segment of a list containing only the element 'x' is an empty list.
If P and Q are coefficients in a carrier, then the product of P and Q in the ring R is also a coefficient in the same carrier.
The set of states 's' where 'r' squared equals the sum of the squares of the first and second components of 's', is less than or equal to the forward box dynamic of the evolution of a function 'phi' under a function 'G' over a time 'T', that also results in a set of states 's' where 'r' squared equals the sum of the sq...
The lemma named "check_bool_options" assumes that under the context of type environment Theta, variable environment B, and local typing context Gamma, the value v is inferred to have the type of a boolean with the constraint that it is always true, and that the support of v is empty. It then shows that the value v is e...
: Assuming that: 1. For any x, y, and z, if Q holds between x and y, and P holds between y and z, then P also holds between x and z. 2. For any f and g, if f is an element of the set of faces of g and f is not final, then P holds between g and the result of making f final in g. 3. For any g, u, v, f, and vs, if the pr...
Either "t" is lexicographically before "q" and "t" is not equal to "q", or "q" is lexicographically before "t" and "t" is not equal to "q", or "t" is equal to "q".
The time taken by the function "Array_Time.make" with parameters 'n' and 'f', given a heap 'h', is equal to 'n+1'.
The lemma named "simple_cg_lookup_invar_with_conv_eq" assumes that "M1" and "M2" are observable. It shows that the function "convergence_graph_lookup_invar" when applied to "M1", "M2", "simple_cg_lookup_with_conv", and "G" equals the function "convergence_graph_lookup_invar" when applied to "M1", "M2", "simple_cg_looku...
The lemma named "lift_root_Any_Single_eq" demonstrates that the lift root step of "Any" and "Single" with respect to a relation R is equal to the union of R and the lift root step of "NonRoot" and "Single" with respect to the same relation R.
The second element of the pair resulting from the function "takedropWhile_queue" applied to a function "f" and a queue "q" is equal to the result of applying the function "takeWhile" to the function "f" and the linearization of the queue "q".
If a binary operation 'op' has the same boolean type for two different results, represented by ρ₁ and ρ₂, when applied to the same types τ and σ, then ρ₁ and ρ₂ must be equal.
If for all indices in set I, R is a module M i, and x and y are in the direct sum of the carrier sets of I and M, and for all indices in set I, x j equals y j, then x equals y.
: Assuming that S is a convex and open set, and a, b, c are elements in S. Also, d is not equal to zero and the dot product of d and a, and d and b are less than or equal to k. Furthermore, the function f is holomorphic on the set of elements z in S such that the dot product of d and z is less than k, and also holomorp...
The lemma "approximation_preproc_nat" in the "Approximation" theory in "Decision_Procs" states the following: 1. The real number representation of 0 is equal to the floating point representation of 0. 2. The real number representation of 1 is equal to the floating point representation of 1. 3. The real number represen...
If a matrix A is an n by n matrix, and if l is not equal to i, and if l and i are both greater than or equal to k minus 1, and if l and i are both less than n, then the matrix resulting from the function "step_3_c_inner_loop" with inputs val, l, i, k, and A is similar to the matrix A.
If the Basic Step Integrity Axiom (BSIA) holds for a modified view (represented by \<rho>' \<V>' Tr\<^bsub>ES\<^esub>), and if this modified view is defined as the union of the original view and a set N (with N being empty and C remaining the same), and if the original view and the modified view are equivalent, then th...
It is not the case that a value 'v' in a given state 's' and a boolean 'b' under a certain proposition 'P' can make a small step transition to an expression 'e'' in a state 's'' and a boolean 'b''.
The lemma T_L assumes that the knowledge of all agents (AxT) is less than or equal to a given knowledge A. It shows that if a proposition p is derivable from the knowledge A, then agent i knows the proposition p.
The reach of skl1 is a subset of skl1_inv2.
The ordered entries of a mapping, when applied to a mapping of 'xs', is equivalent to sorting the key-value pairs in 'xs' after removing any duplicate keys.
The count of an element 'x' in the sum of a function 'f' applied to a set 'A' is equal to the sum of the counts of 'x' in the function 'f' applied to each element 'a' in the set 'A'.
If the absolute freelist of 'ss' is in 'Q', and 'ss' and 'ts' are flip alternate sublists adjacent to each other, then the absolute freelist of 'ts' is also in 'Q'.
: Given the following conditions: 1. A Java program G transitions from a state with no exception, a heap 'hp', and local variables 'loc', via the execution of an expression 'ex', to a state with no exception, a new heap 'hp', and new local variables 'loc'. 2. G is a well-formed Java program. 3. The signature 'S' of a ...
If a tau-execution move is valid for a given configuration in a program P, thread t, expression e, and states h, s, and s', then a tau-execution move is also valid for the same configuration but with a new array expression "newA T\<lfloor>e\<rceil>".
If the following conditions are met: 1. NRO is equal to the encryption of a set containing a number f_nro, an agent B, a nonce L, and a value C, using the private key of agent A. 2. NRO is a part of the information that the spies have in the event sequence evs. 3. Agent A is not in the set of compromised agents. 4. Th...
: Assuming that: - 'xs' is a finite set, - for any distinct elements 'x1' and 'x2' in 'xs', and any elements 'y1' in 'f(x1)' and 'y2' in 'f(x2)', the intersection of 'g(y1)' and 'g(y2)' is empty, - for any element 'x1' in 'xs', and any distinct elements 'y1' and 'y2' in 'f(x1)', the intersection of 'g(y1)' and 'g(y2)'...
If P and Q are both intuitionistic, then the conjunction of P and Q (P and Q) is also intuitionistic.
If a non-zero and non-infinite extended real number 'c' is given, then 'a' divided by 'c' is less than 'b' if and only if 'a' is less than 'b' multiplied by 'c'.
The function "indexed_const" is a ring homomorphism from the ring R to the polynomial ring Pring R I.
If "c" is a co-trace of "t", then there exists a unique "c'" such that "c'" is a trace of the length "i" prefix of "t" in "c".
The lemma "is_state_separator_from_canonical_separator_initial" assumes that a state separator is derived from a canonical separator for states q1 and q2 in machine M, and that both q1 and q2 are states in machine M. Under these assumptions, it shows that the initial state of A is the ordered pair (q1, q2) in the left ...
The span of a set 'b' is equal to the set of all linear combinations of vectors in 'b', where the coefficients are not zero. This is represented by the sum of the product of the coefficient 'f v' and the vector 'v'. The function 'f' is such that the set of vectors for which 'f v' is not zero is finite, and for all vect...
The application of the composition of two bounded linear functions "f" and "g" to a pair "b" is equal to the sum of applying "f" to the first element of "b" and applying "g" to the second element of "b".
The degree of a polynomial P is zero if and only if there exists a constant a such that P is equal to the polynomial that is constantly a.
The lemma "rel_mset_Plus" assumes that the relation R holds between elements a and b, and also that the multiset relation R holds between multisets M and N. It then shows that the multiset relation R also holds between the multisets obtained by adding element a to multiset M and element b to multiset N.
: If for all states 's' and 'ss', 's' is equivalent to 'ss' implies that the evaluation of expression 'e' in state 's' is equal to the evaluation of expression 'e' in state 'ss', then 'G' proves that the assignment of expression 'e' to variable 'x' has a security level defined by the function 'Sec' which takes a pair ...
The set of variables "\<C>_vars" of any element "x" is finite.
If the function "find_handler" with parameters P, a, h, frs, and sh returns a tuple (Some x, h, frs', sh'), then frs' must be an empty list.
For any two vectors 'v' and 'w' of a type 'a' that is a conjugatable ring and a commutative ring, if 'v' and 'w' are both n-dimensional vectors, then the scalar product of 'v' and 'w' is equal to the scalar product of the conjugate of 'w' and 'v'.
For any word 'a' of any length, if 'a' is less than or equal to 'a' minus one, then 'a' must be equal to zero.
The lemma "subset_with_given_card_exists" assumes that "n" is less than or equal to the cardinality of set "A". It shows that there exists a subset "B" of "A" such that the cardinality of "B" is "n".
Given a type T and a type variable X, if X is fresh for T (i.e., X does not occur in T), then substituting X with another type S in T does not change T.
The transitive closure of a relation "r" applied four times from "x" to "y" is equal to the transitive closure of the same relation "r" applied twice from "x" to "y".
The lemma named "upto_enum_red'" assumes that "X" is greater than or equal to 1. It shows that the list of 'a type words from 0 to X-1 is equal to the list obtained by mapping the function "of_nat" to the list of natural numbers from 0 to the unat of X. Here, 'a is a type that has a length, and "of_nat" is a function t...
The pair (i, j) belongs to the set "evnodd A b" if and only if the pair (i, j) belongs to the set A and the sum of i and j modulo 2 equals b.
If set B is finite and set A is a subset of B, then set A is also finite.
The lemma named "pro_path" assumes that "x" is in the prolongation of "a" and "b" (which implies that the prolongation is not empty). It then obtains a path "ab" from "a" to "b" such that the prolongation of "a" and "b" is a subset of "ab".
The vector v satisfies the constraint of being greater than or equal to zero.
If the pair (C, D) is in the transitive closure of the subclass relation "subcls1" in a given graph G, then C is a class in G.
The lemma named "phull_row_echelon" assumes that the keys of the set "ps" are a subset of the set "ts" and that "ts" is distinct. It shows that the convex hull of the set resulting from converting the row echelon form of the matrix obtained from "ps" using "ts" back into polynomials using "ts" is equal to the convex hu...
If the union of set A and set B is a subset of set C, then it implies that both set A and set B are subsets of set C.
The set of all elements 't' for which the count in 'M' is greater than zero is finite.
If the list "xs" is not empty, then the argument that minimizes the function "f" over the list "xs" is the element at the index which is the maximum of the set of indices "i" such that "i" is less than the length of "xs" and for all elements "x" in the set of "xs", the value of function "f" at the element of "xs" at in...
The image of the set "red s" under the relation "E" is a subset of "red s".
If a formula "phi" is nu-stable for a sequence "w", then the formula "phi" is also nu-stable for any suffix of the sequence "w" starting from the index "i".
For a given polynomial 'p' over a field, if 'p' is irreducible, then the degree of 'p' is greater than 0.
The lemma named "inf_step2inf_step_table_LCons" is defined for a fixed "tl". It shows that the function "inf_step2inf_step_table" applied to a state "s" and a list "LCons tl tls" is equal to a list "LCons" with the first element being a tuple of "s", "tl", and "SOME s'" such that there exists a transition system from "...
If a program P, environment E, and heap h type-check an expression e to a type T, then the program P, environment E, and heap h also type-check the expression e to a type T in a slightly different way (indicated by the prime symbol). The lemma wts_wts' states that if a program P, environment E, and heap h type-check a...
If a well-formed C++ program P, under environment E, evaluates an expression e with state s to an expression e' with state s', and if the expression e is of type T and the state s is valid under the program P and environment E, then the state s' is also valid under the program P and environment E.
If the function "f" is summable, then the sum of the negation of "f" for all "n" is equal to the negation of the sum of "f" for all "n".
A pseudo allocation of 'a' is the union of sets, where each set contains a pair. The first element of the pair is the first element of a pair from 'a', and the second element of the pair is a set containing 'y'. 'y' is an element from the second element of the pair from 'a'.
The last arc of a walk, where the walk is the concatenation of two walks 'p' and 'q' starting from a vertex 'u', is the same as the last arc of the walk 'q' starting from the last arc of the walk 'p' starting from 'u'.
If the function 'extending' holds for sets YA' and YA, and also for sets YB' and YB, with the same parameters C, h, and F, then 'extending' also holds for the intersection of sets YA' and YB', and the intersection of sets YA and YB.
It is not possible for any process 't' to be less than an empty process.
The Lucas-Lehmer transformation to a real number of the product of a list containing a natural number 'n' and a variable 'x' in the Lucas-Lehmer ring is equal to the product of the natural number 'n' and the Lucas-Lehmer transformation to a real number of 'x'.
If a list 'xs' is in the listset of a list consisting of 'A' followed by 'As', then there exists an element 'y' and a list 'ys' such that 'y' is in 'A', 'ys' is in the listset of 'As', and 'xs' is the list that starts with 'y' and continues with 'ys'.
The function 'term_of_pair' is injective. This means that if two pairs 'p' and 'q' are such that the function 'term_of_pair' applied to 'p' equals the function 'term_of_pair' applied to 'q', then 'p' must be equal to 'q'.