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If the system follows the Johansson's Minimal Logic (JML), then the law of the excluded middle (TND) does not hold for negation.
The polynomial of a vector, where the vector is a function of 'n', is equal to the polynomial of the reverse of the map of the function 'f' over the range from 0 to 'n' (exclusive).
"Throwable" is a class in the program P. This is a basic fact and can be introduced automatically.
The function "add0" with parameters k, y, and t is equal to the function "gtree" with parameters resulting from the function "add" with parameters k, y, and t.
The function "consistent" is true when applied to the function that maps each variable "v" to the symbol of that variable in the store, and the symbol of the store itself.
The lemma named "diff_solve_axiom2" establishes that for a fixed constant 'c' in a Heine-Borel and Banach space, the differential of 'x' being equal to 'c' and 'G' implies 'Q' is equivalent to the statement that for all non-negative 't', if 'G' holds for all 'τ' in the range from 0 to 't' when applied to 's' plus 'τ' t...
The right-sided element "x" is equal to the implication of "x" implying "x" being equal to the top element.
The lemma "learn_cons_closed_subseteq" assumes that the function "learn_cons" with parameters "psi", "U", and "s" is true, and that set "V" is a subset of set "U". Under these assumptions, it shows that the function "learn_cons" with parameters "psi", "V", and "s" is also true.
The lemma named "hfref_cons" makes several assumptions and provides a conclusion based on these assumptions. The assumptions are as follows: 1. The pair (f,g) is in the relation from the set of elements satisfying predicate P to the set S. 2. For any element x, if P' of x is true, then P of x is also true. 3. For any...
There is an infinite sequence of small-step transitions in the context of a program, starting from a state "s" and a command "c" followed by a list of commands "cs", and resulting in a list of command stacks "css". The second condition, named "cases", is a disjunction of two cases. The first case states that if there i...
If the valuation of K by v is valid, the polynomial ring R over the valuation ring Vr of K by v and X is valid, the nth element of F is in the carrier of R, and the degree of the nth element of F in the polynomial ring R over the valuation ring Vr of K by v and X is less than or equal to the successor of d, then the nt...
The lemma named "from_FSM_observable" assumes that the finite state machine "M" is observable. It shows that the finite state machine derived from "M" at state "q" is also observable.
The lemma "phase_cases" assumes that a step from state 's' with action 'a' results in output 'ou' and state 's''. It then establishes three possible cases: 1. The case 'noPH': Either the conference ID 'cid' does not exist in the set of conference IDs of state 's', or the phase of the conference 'cid' in state 's' is '...
There exist elements 'l' and 'u' in the set 'S' such that for any element 'x' in 'S', the value of the function 'f' at 'x' lies within the interval from 'f' at 'l' to 'f' at 'u'.
The negative assumption of the negative assumption of a proposition 'p', when combined with the top element using the operation '^ o', is equal to the negative assumption of the proposition 'p' combined with the bottom element.
"v" is an element of "P".
If the count of element "a" in list "B" is the successor of "m" (which means "m+1"), then the element "a" is in the set of elements of list "B".
The lemma for the Hamming stream in Linear Temporal Logic is equal to the interleaved merging of the Hamming stream multiplied by 2, with the interleaved merging of the Hamming stream multiplied by 3 and the Hamming stream multiplied by 5.
The polynomial of the root of unity 'n' for 'x' equals zero if and only if 'x' raised to the power of 'n' equals one.
The domain of the relation "MV_Rel" is such that for any function 'f', it is always true.
A set P is a neighborhood base of a topological space X if and only if the set of all subsets t of the topological space X that also belong to P is a neighborhood base of X.
: 1. If b' is equal to b and a' is equal to a, then the supremum of a and b is equal to the supremum of b' and a'. 2. If b' is equal to b and a' is equal to a, then the supremum of a and b' is equal to the supremum of b and a'. 3. If b' is equal to b and a' is equal to a, then the supremum of a' and b is equal to the ...
A multivariate polynomial (represented as "mp") is equal to the sum of the product of the isolation of variable 'x' in the multivariate polynomial at degree 'i' and the variable 'x' raised to the power of 'i', where 'i' is a natural number less than or equal to the degree of the multivariate polynomial with respect to ...
The union of set Q is a subset of the union of the partial join of sets P and Q.
Merging any list "xs" with an empty list results in the original list "xs".
The theorem "alw2" assumes two hypotheses: 1. "F" is always true. 2. If "F" and "Unchanged v" are both true, then "F" will be true in the next state. Given these assumptions, it shows that "F" will always be true.
The lemma aux1 assumes the following: 1. For all elements 'x' in 'passed', 'a' is not a sub-element of 'x'. 2. 'Passed' is a subset of the collection of reachable elements. 3. 'a' is reachable. Under these assumptions, it shows that the tuple consisting of 'a' inserted into 'passed', 'wait'', 'brk'', along with 'passe...
The smallest closed collection of disjoint intervals in a set Omega is a closed collection of disjoint intervals in Omega.
The infimum preserves the operation K.
If it is necessarily the case that a property G holds for a certain description, then it is necessarily the case that this description, when applied to G, is equivalent to the property G itself. This description is defined as the unique x such that x has property A and for all F, the relation between x and F is equival...
The lemma standard Reduction Inference is equal to Reduction Inference with a given set of clauses.
"the function mkId with parameter h forms a perfect bijection", where Pbij refers to a perfect bijection in the context of the SIFPL theory in Isabelle.
The lemma named "vpra_Cons_Return" assumes that the kind of 'a' is a function 'f' returning from procedure 'p' denoted as "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f". It shows that if a valid reverse auxiliary path exists for 'cs' and 'as', then a valid reverse auxiliary path also exists for 'cs' and the list formed...
The set of finite shortest path costs, given a cost function 'f', is finite.
The term "domain" is equivalent to "dom".
The swap function is injective.
For any given 'p', if 'q' is not in the set 'ps', then the list 'p' with 'h' updated at 'q' with 'y' is equal to the list 'p' with 'h' and 'ps'.
The data sources at level 1 for sA72 are equal to the set containing sA6.
For a given 'z' which is a non-zero element of a real normed field and a Banach space, and a given 'n' which is greater than or equal to 2, the series formed by the inverse of '(z + k)^n' for all natural numbers 'k' is summable.
If x and y are dense, then the infimum (greatest lower bound) of x and y is also dense.
, given a fixed Dirichlet series 's' and a formal Dirichlet series 'f', if the sequence of coefficients of 'f' is bounded, then the absolute convergence abscissa of 'f' is less than or equal to 1.
The lemma named "prefix_sdrop_slength" assumes that "xs" is a prefix of "ys". It shows that if you concatenate "xs" with the result of dropping the length of "xs" elements from "ys", you will get "ys" back.
The lemma named "in_obs_valid" assumes that "ns'" is an observable node set of "ns" in "S" and every node in "ns" is a valid node. It shows that every node in "ns'" is also a valid node.
If the following two conditions are met: 1. The result of executing monad 'm'' is less than or equal to the result of executing 'ERETURN x' with respect to the relation 'R'' and exception set 'E'. 2. For any 'x'', if 'x'' is a successful result of 'm'' and the pair '(x',x)' is in the relation 'R'', then the result of ...
The lemma named "orthonormal_optimal_partial_sum" assumes that there is an orthonormal system S with function w, every function w i is square integrable over S, function f is square integrable over S, and I is a finite set. It shows that the L2 norm of the function that subtracts the sum of the product of the orthonorm...
The lemma named "map2set_delete" is assumed to be a general algorithm priority tag. It also assumes that the operation "op_map_delete" is a generic operation "d" in the relation of "Rk" to the pair of "Rk" and "Id" in the relation "R". This shows that "d" is in the relation of "Rk" to the map-to-set relation "R" of "Rk...
The lemma named "inf_concat_simple_mono" assumes that "k" is less than or equal to "k'". It shows that the first element of the infinite concatenation function "inf_concat_simple" with parameters "f" and "k" is less than or equal to the first element of the same function with parameters "f" and "k'".
The rebase function applied to the polynomial constructed by the polynomial constructor "pCons" with parameters "a" and "p" is equal to the polynomial constructed by "pCons" with parameters "a" under a certain function and the rebase function applied to "p".
The observable reach of m1 is a subset of m1_secrecy.
The function 'f' is continuous at a point 'x' within a set 's' if and only if for every open set 'B' such that 'f(x)' is in 'B', there exists an open set 'A' containing 'x' such that for every 'y' in 's' that is also in 'A', 'f(y)' is in 'B'.
If two functions 'f' and 'g' are monotone with respect to a partial order (less than or equal to), then a function that takes an input 'x' and returns the maximum of 'f(x)' and 'g(x)' is also monotone with respect to the same partial order. Here, 'enat' refers to the extended natural numbers, which includes infinity.
The substitution of a set containing a single atomic assertion with a substitution function is equal to the set containing the result of the substitution of the atomic assertion with the same substitution function.
If we have two assumptions: 1) Under the context of program states Γ and program specifications Θ, the program c, when executed from a state satisfying precondition P, will result in a state satisfying postcondition Q, with A being the set of states where the execution can abort, and F being the set of fault states. 2...
The lemma named "lsum_list_mono" is defined with a fixed function 'f' that maps from 'a' to 'b', where 'b' is a monoid addition and an ordered additive abelian semigroup. The lemma assumes that for any 'x' in the list set 'xs', the function 'f' applied to 'x' is less than or equal to the function 'g' applied to 'x'. Th...
If a secure message containing a key K and an agent A is in the channel state s, and the state s is in the invariant m2_inv4_M3, m2_inv3_extrKey, and m2_inv2b_corrKey_leaked, and the key K is not in the domain of the leaked keys in state s, then the pair (K, B) is in the authorized keys of the run state s.
The lemma named "Cong_char" shows that the congruence relation "Cong" between terms "t" and "t'" is equivalent to the congruence relation "Cong'" between terms "t" and "t'".
The function 'ereal' is strictly monotonically increasing.
If the state 's' is error-free, then the state 's' updated by the function 'abupd' is also error-free. The function 'abupd' uses either the identity function (if 'stat' is true) or the function 'np' with argument 'a' (if 'stat' is false).
The function "frag_extend" applied to the result of "frag_cmul" function with parameters c and x is equal to the result of "frag_cmul" function with parameters c and the result of "frag_extend" function applied to x.
"TT" is valid, where "TT" is a truth value in propositional logic, representing a tautology or a statement that is always true. This is a simplification rule in Isabelle.
For all natural numbers 'm', 'm' is less than or equal to the identity of 0 if and only if 'm' is equal to the identity of 0. And for all natural numbers 'm' and 'n', 'm' is less than or equal to the successor of 'n' if and only if 'm' is equal to the successor of 'n' or 'm' is less than or equal to 'n'.
If the check of a program P and state sigma holds true, then the deterministic execution of the program P with thread t and state sigma does not result in a TypeError.
The head of the list converted from a stream 'xs' is equal to the head of the stream 'xs'.
The function 'blinding_of_on' is antisymmetric on the universal set with respect to 'h' and 'bo'.
If a list 'xs' appended with an element 'x' is sorted, then inserting 'x' into the list 'xs' will result in the list 'xs' appended with 'x'.
The lemma named "prod_acc" assumes that the range of the second element of the relation 'r' is a subset of 'sa.V'. It shows that 'r' is accepted by the product automaton if and only if the first element of 'r' is accepted by the infinite generalized Büchi automaton.
The sum of two polynomial constructors, where the first is constructed with elements 'a' and 'p' and the second with elements 'b' and 'q', is equal to a new polynomial constructor constructed with the sum of 'a' and 'b' and the sum of 'p' and 'q'.
"NullT is less than or equal to CClassT C".
The homogenization of the negation of a polynomial p with respect to a variable x is equal to the negation of the homogenization of the polynomial p with respect to the same variable x. This is true for any polynomial p in a univariate ring.
The relation "ord0rel" is an equivalence relation on the universal set.
The pell valuation of the pell power of a number 'z' raised to the power 'n' is equal to the pell valuation of 'z' raised to the power 'n'.
The lemma holdsIstate_distinct_IDs asserts that the distinct_IDs property holds in the initial state.
The product of the inverses of a sequence converges if and only if the product of the sequence itself converges.
The lemma named "ENR_openin" is defined for a set 'S' in a Euclidean space. It assumes that 'T' is an Euclidean Neighbourhood Retract (ENR) and 'S' is an open set in the topology of 'T'. Under these assumptions, it shows that 'S' is also an Euclidean Neighbourhood Retract (ENR).
If B is a subset of the field of relation r, and a is an element of the set of all elements that are greater than B, and for all elements a' in the set of all elements that are greater than B, a is related to a' by relation r, then a is equal to the supremum of B.
The lemma named "semialg_map_comp_closed" assumes two conditions: first, that "f" is a semialgebraic map from "m" to "n", and second, that "g" is a semialgebraic map from "k" to "m". Under these assumptions, it shows that the composition of "f" and "g" is a semialgebraic map from "k" to "n".
The function "compute_SCC" is less than or equal to the function "compute_SCC_spec" in terms of correctness.
Given a non-zero number 'n' and a property 'P' that holds for at least one prime factor of 'n', we can find a prime 'p', a positive integer 'k', and another number 'n'' such that 'P' holds for 'p', 'p' is a prime number that divides 'n', 'p' does not divide 'n'', 'k' is greater than zero, and 'n' is equal to 'p' raised...
: Assuming that for any index 'i' less than the length of list 'T', the 'i'th element of 'T' and the 'i'th element of 'S' are in the transitive closure 'c' of term implication 'timpl_closure' and that the lengths of 'T' and 'S' are equal. Also, either 'f' is equal to 'g' or there exist elements 'a' and 'b' such that t...
If tree 'a' satisfies the invariant 'invc', tree 'b' satisfies the invariant 'invc2', and the color of tree 'a' is Black, then the function 'baldR' applied to tree 'a', some element 'x', and tree 'b' will also satisfy the invariant 'invc'.
If the previous value of j and i is equal to some pair (j', i') and i is less than or equal to j, then i' is less than j'.
The lemma named "abs_pred_stermI" is an introduction rule. It assumes that for any list "cs", if for any pattern "pat" and term "t" in the set "cs", the predicate "P" holds for "t", then "P" also holds for the abstraction of "cs". Under this assumption, it shows that the abstraction predicate "P" holds for any term "t"...
The lemma named "gbinomial_ratio_limit" is defined for a variable 'a' of type 'real_normed_field'. It assumes that 'a' is not a natural number. The lemma shows that the limit of the sequence defined by the ratio of 'a' choose 'n' to 'a' choose 'n+1' is -1 as 'n' goes to infinity.
Lemma D_mono: If the function D holds for the set A, command c, and measure M, and A is a subset of A', then there exists a measure M' such that the function D holds for the set A', command c, and measure M', and M is less than or equal to M'.
The representation of the image of a function 'f' applied to a set 'A' in the Zermelo set theory is equal to the image of the function 'f' applied to the representation of the set 'A'.
For any two real numbers m and n, which are integers, the following holds: 1. The Lebesgue integral over the interval from -pi to pi of the function cos(m * x) * cos(n * x) equals 2 * pi if the absolute value of m equals the absolute value of n and n equals 0, pi if the absolute value of m equals the absolute value of...
For a function 'x' mapping natural numbers to extended real numbers, 'x0' is less than or equal to the limit inferior of 'x' if and only if for all sets 'S' that are open and monotonically increasing, if 'x0' is an element of 'S', then there exists a natural number 'N' such that for all natural numbers 'n' greater than...
The affine dimension of a set S translated by a vector a is equal to the affine dimension of the original set S. This is valid for any vector a in a Euclidean space of any dimension n.
For a polynomial 'p' and a variable 'x', applying the function 'R₁' to the polynomial and then evaluating it at 'R₂(x)' is the same as first applying the homomorphism 'hom' to the polynomial, evaluating it at 'x', and then applying the function 'R₂'.
The lemma SpecAnno: Assumes a consequence: "P is a subset of the set of states s such that there exists a Z where s is in P' of Z and Q' of Z is a subset of Q and A' of Z is a subset of A." Assumes a specification: "For all Z, the tuple of the program context, the proof context, the precondition P' of Z, the command c ...
For any three matrices A, B, and C, the right distribution property of matrix multiplication over matrix addition holds. That is, multiplying a matrix A by the sum of matrices B and C is the same as adding the result of multiplying matrix A by matrix B to the result of multiplying matrix A by matrix C.
If "p" and "q" are tests, then "p" times "x" equals "p" times "x" times "q" if and only if "p" times "x" is less than or equal to "x" times "q".
The trace of 'pre' equals 1.
A polynomial ideal is a full polynomial ideal if and only if the set of polynomials is a full polynomial ideal.
Updating an empty list with a None value in an environment results in the same environment.
There exists an integer x such that for all y, if x is less than y, then y is either less than 0 or greater than or equal to 0.
If a term 't' can be inferred to 'u' using rule set 'rs' and if 't' is closed, then 'u' is also closed. The assumption is that 't' can be rewritten to 'u' using the rules in 'rs' and 't' is a closed term. The conclusion is that 'u' is also a closed term.
The function "int_encode" is a bijection.
The lemma named "trace_property_rule" makes the following assumptions: 1. For any initial state s0 of event E, the property I holds for the empty trace and s0. 2. For any states s, s', and s'', any trace τ, and any event e, if the initial state of event E is s, if E transitions from state s to state s' with trace τ, i...
The lemma named "fold_max_rev_eq" is defined with a fixed list "xs" of a linearly ordered type. Assuming that "xs" is not an empty list, it shows that the result of applying the "max" function in a fold operation over the tail of "xs", starting with the head of "xs", is equal to the result of applying the "max" functio...