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The function "snocs" applied to two lists "xs" and "ys" is equivalent to appending list "ys" to list "xs".
If 'a' is greater than or equal to zero, then the real power of 'a' to the negative one is equal to the inverse of 'a'.
The initial state "s0" is reachable.
The lgrid of point 'p', direction 'ds', and level 'p' is an empty set.
The lemma named "list_of'_Cons" assumes that a list in memory heap 'h' starting at reference 'q' represents the list '(x#xs)'. It then concludes that there exists a reference 'n' such that the value at reference 'q' in heap 'h' is a node containing 'x' and 'n', and the list in memory heap 'h' starting at reference 'n' ...
If an expression 'e' is final, then there is no small step from the state consisting of 'e', 's', and 'b' to another state consisting of 'e'', 's'', and 'b''.
Adding a formal power series constant "c" (where "c" is an element of an additive monoid) to a formal power series "f" is equivalent to creating a new formal power series where, for each term, if the term's index is zero then the value is "c" plus the zeroth term of "f", otherwise the value is the term of "f" at that i...
The degree of a power product of the first element of the representation of a natural term, which is created from the expressions "a", "b", and "i", is equal to the sum of "a" and "b". This is valid in the context of a natural term comparison.
If zero is an element of set S, then zero is also an element of the minimal antichain of set S.
The lemma named "upper_asymptotic_density_finite" assumes that "A" is finite. It shows that the upper asymptotic density of "A" equals zero.
For any real number 'x' that is greater than or equal to 1, the difference between the Mertens function of 'x' and the natural logarithm of 'x' is within the range from -2 to 25/18.
If B is a finite set, and if x is fresh for all elements in B (in other words, x does not occur in any element of B), and if A is a subset of B, then x is also fresh for all elements in A. Here, 'a::at_base set' indicates that B is a set of elements of some type 'a, which belongs to the type class at_base.
The integer root ceiling of 'p' and 'x' is equal to the ceiling of the root 'p' of the integer 'x'.
Lemma 2.3: Assuming that the expression 'e' with proof environment 'eP' and value environment 'eV' is well-typed under the context 'Γ' with type 'τ', it can be shown that the expression 'e' is well-typed under the context obtained by erasing 'Γ', with the type obtained by erasing 'τ'.
For two finite state machines M1 and M2, where M1 is of type ('a::linorder,'b::linorder,'c::linorder) and M2 is of type ('d,'b,'c), given that both M1 and M2 are observable and minimal, the size of M1 is less than or equal to m, the size of M2 is less than or equal to m, and the inputs and outputs of M2 are equal to th...
"x is greater than y if and only if Some x is greater than Some y in the option order".
In homogeneous coordinates, one is not equal to infinity, zero is not equal to infinity, zero is not equal to one, one is not equal to zero, infinity is not equal to zero, and infinity is not equal to one.
The join of any element 'x' and the bottom element is 'x', and the join of the bottom element and any element 'x' is also 'x'.
The lemma named "eventually_periodic" assumes that: 1. C is a communicating class in the set of all possible classes. 2. The period of C is greater than 0. 3. x is an element of C. It then shows that eventually, for some integer m, the probability of transitioning from state x to state x in m times the period of C is ...
If there does not exist a value 'v2' such that the result of decoding instruction 'v1' is a valid instruction 'v2', then there must exist an exception 'e' such that the result of decoding instruction 'v1' is this exception 'e'.
If a function 'f' is injective, then the index of the element 'f y' in the list obtained by mapping 'f' over 'xs' is the same as the index of 'y' in the original list 'xs'.
The lemma named "reset_vote" assumes that there is a run of the Heard-Of model with the UV algorithm, denoted as "HORun UV_M rho HOs", and that the step "r'" equals 0, denoted as "step r' = 0". It shows that the vote at the process "p" at the round "r'" is None.
The error 'err' of a set 'A' is defined as the insertion of an error 'Err' into a set that consists of elements 'x', for which there exists an element 'y' in 'A' such that 'x' is equal to 'OK y'.
If a string 'a' derives to a string 'b', and 'w' is a word that is a prefix of 'a', then the string obtained by dropping the length of 'w' from 'a' derives to the string obtained by dropping the length of 'w' from 'b'.
The set of pending elements 's' is a subset of the set 'E'.
If a separation state "y" is disjoint from a separation state "z", and a separation state "x" is disjoint from the addition of separation states "y" and "z", then the separation state "x" is disjoint from the separation state "y".
The function "check" preserves the swapping of interpretation graphs (IG) in the context of a module (MOD) that may contain errors.
If a property P holds for an element x and an empty list, then the interleaving of lists xs, ys, and zs, where the property P is applied to an element w and the concatenation of lists ws and [x], is equivalent to the concatenation of lists xs and [x] interleaved with the concatenation of lists ys and [x], zs, where the...
The lemma named "iterative_HSem'" assumes that "x" is not in the domain of "Γ". It shows that the least fixed point of "ρ'", when combined with the domain of "Γ" and the interpretation of "Γ" under "ρ'", and then "x" is mapped to the interpretation of "e" under "ρ'", is equal to the least fixed point of "ρ'", when comb...
For all elements 'm' in the set 'ms', if 'm' is a normalized negation normal form match, then the result of applying the 'andfold_MatchExp' function to 'ms' is also a normalized negation normal form match.
The lemma named "measurable_Min_nat" is about a measurable function. It fixes a predicate 'P' that takes a natural number and an arbitrary type 'a' and returns a boolean. It assumes that for any natural number 'i', the predicate 'P' applied to 'i' is measurable with respect to a measure 'M'. The lemma then shows that t...
The lemma named "cltn2_compose_abs" assumes that both M and N are invertible. It shows that the composition of the abstraction of M and the abstraction of N (in the cltn2 space) is equal to the abstraction of the product of M and N.
If from the set H with elements A3, A1, and A2 added, we can infer B, then we can also infer B from the set H with elements A1, A2, and A3 added.
: Assuming "s'" is the result of applying operator "op" to state "s", it shows that: 1. For any variable "v", if "v" is in the set of added effects of the operator "op", then the value of "v" in the state "s'" is "True". 2. For any variable "v", if "v" is not in the set of added effects of the operator "op" and "v" ...
If a match "m" is in normalized negation normal form (as assumed by "normalized_nnf_match m"), and for all "a", "disc2" does not apply to source ports "Src_Ports a" or destination ports "Dst_Ports a" (as stated by "disc2_noSrcPorts" and "disc2_noDstPorts"), then if "m'" is an element in the set resulting from the funct...
If for every 'z', there exists a 'w' such that for every 'x', there exists a 'y' where if 'P' holds for 'x' and 'z' then 'P' holds for 'y' and 'w', and 'P' holds for 'y' and 'z', and if 'P' holds for 'y' and 'w' then there exists a 'u' for which 'Q' holds for 'u' and 'w'. And if for every 'x' and 'z', 'P' does not hold...
The lemma named "finite_strict_prefix_set" demonstrates that the set of all elements 'xa' that are strict prefixes of a given list 'x' is finite.
If the cast of a character data pointer is not equal to an object pointer, and if applying the setter function to the character data pointer in the heap 'h' results in a new heap 'h'', then the getter function for the object in the heap is preserved from 'h' to 'h''.
The function POR_base_lf_rep is monotonic.
In the context of a group, if G acts on H and a is an element of the carrier G, then a is an element of the right coset of H with respect to a.
If a tau (silent) execution move from a state (stk, loc, pc, xcp) to another state (stk', loc', pc', xcp') is possible in a program P on thread t for an expression e', then a tau execution move is also possible from a state ((stk @ [v]), loc, (length (compE2 e) + pc), xcp) to another state ((stk' @ [v]), loc', (length ...
For a function 'f' from a quantale with dual 'a' to another quantale with dual 'b', given that 'f' is in the set of quantale homomorphisms, the function 'f' applied to the composition of the right adjoint of 'f' and 'f' applied to the product of 'x' and 'y' is equal to the product of 'f' applied to 'x' and 'f' applied ...
If a parallel composition is valid for the intersection of conditions q and r, and action a leading to condition q, then the auxiliary interference freedom holds for the tuple consisting of None, condition q, and an atomic action that waits for condition r to be true for all x, followed by action a.
The lemma "phiDefNode_is_join_node" assumes that "n" is in the set of phi definition nodes of graph "g" and variable "v". It shows that the length of the list of predecessors of node "n" in graph "g" is not equal to 1.
The lemma "rev_sub_is_hom" assumes that "g" is an element of the carrier set "P". It shows that the reverse subtraction of "g" is a ring homomorphism from "P" to "P".
Given two matrices M0 and M1 of dimension d, and assuming that the sum of the adjoint of M0 multiplied by M0 and the adjoint of M1 multiplied by M1 equals the identity matrix of dimension d. Also, for any matrix ρ of dimension d that is a partial density operator, the function DS applied to ρ results in a positive matr...
The lemma named "betaupright_upright_vut" assumes three conditions: 1. "va" is beta upright to "vb", 2. "vb" is upright, 3. and there is a vacancy up to the sum of "va" and "vb" at time "t". If these conditions are met, it shows that there is a vacancy up to "va" at time "t".
The lemma named "valof_uminus" assumes that "a" is finite. It shows that the value of the negation of "a" is equal to the negation of the value of "a".
: 1. The strict drop from an undefined or bottom value results in an undefined or bottom value. 2. The strict drop of an undefined or bottom value from any integer 'i' also results in an undefined or bottom value.
A function 'g' is measurable with respect to the point measure of a set 'A' and a function 'f' if and only if 'g' maps elements from set 'A' to the space of measure 'M'.
"Thread" is not equal to "Object", "Object" is not equal to "Thread", "Object" is not equal to "Throwable", "Throwable" is not equal to "Object", "Thread" is not equal to "Throwable", and "Throwable" is not equal to "Thread".
The set of a sub-branch derived from a function 'f' and a branch is equal to the image of the sub-block derived from the same function 'f' over the set of the branch.
If for all elements 'i' in the set 'bs', 'i' is not true, and the length of 'bs' is equal to the length of 'as', then the projection of tuple 'bs' onto 'as' is equal to a replication of 'None' for the length of 'bs'.
The theorem "wls_fresh_psubst" assumes that "X" is a well-sorted term of sort "s" and "rho" is a well-sorted environment. It shows that "z" is fresh for the parallel substitution of "X" by "rho" if and only if for all "y" and "ys", "y" is fresh for "X" or "z" is fresh for the image of "rho" at "ys" and "y".
The lemma named "weak_mat_gt_both_mono" assumes that the set of matrix pairs ABs is a subset of the set of all nxn matrices. It also assumes that for every pair of matrices (A, B) in the set ABs, the matrix A is weakly greater than the matrix B. Under these assumptions, the lemma shows that there exists a relation "gt"...
The type environment derived from a mode state is secure with respect to a given memory state.
The lemma named "restrict_index_out" assumes that "i" is less than the length of "X" and "i" is not in "I". It shows that the "i"-th element of the restriction of "I" on "X" is "None".
The Cartesian basis forms a basis when considered as a set of vectors.
For any list 's', condition 'c', function 'f', and state 'σ', the monadic n-fold left-inclusive iteration of 's' with condition 'c', function 'f', and state 'σ' is equivalent to the binding of the evaluation of 's' to a function that binds the evaluation of 'σ' to the same monadic n-fold left-inclusive iteration.
If x is not in the free variable term set of t, then substituting x with t in y equals t.
The nonstandard derivative of a function 'f' at a point 'x' is 'D' if and only if the function that maps 'h' to the difference quotient of 'f' at 'x' and 'h' tends to 'D' as 'h' approaches zero in a nonstandard way.
If a set "S" is an element of a multiset "\<P>", then "S" is a distinct multiset.
The lemma "weakenStatImpE" is defined with the following parameters: a fixed 'b type variable named Psi, a psi type variable P with three type parameters 'a, 'b, 'c, a set named Rel containing tuples of 'b and two psi type variables, a psi type variable Q with three type parameters 'a, 'b, 'c, and a fixed 'b type varia...
The lemma named "Natural Number Predicate Tipping Point" is defined with two fixed elements, n2 and Pr. It assumes three conditions: 1. n2 is greater than 0. 2. The predicate Pr holds true for 0. 3. The predicate Pr does not hold true for n2. The lemma then shows that there exists a natural number n, less than n2, fo...
"blinding_of_list" is equivalent to "list_all2".
If both "phi" and "psi" are formulas, the second element of each pair in the set "txs" is a variable, the first element of each pair in the set "txs" is a term, and "phi" is provably equivalent to "psi", then it is provable that the result of the raw substitution of "phi" and "psi" with "txs" are equivalent.
If a generalized probabilistic value (gpv) is lossless and the interaction with the interface is valid, then the gpv is also probability-lossless.
: Assuming that "p" belongs to the monadP_qbs_Px of "X", "q" belongs to the monadP_qbs_Px of "Y", "f" is a quasi-Borel measurable function from the quasi-Borel product space of "X" and "Y" to the quasi-Borel real numbers, and "q" is quasi-Borel integrable with respect to the function that maps "y" to the quasi-Borel in...
If a function 'f' is increasing and 's'' is a subsequence of 's' taken according to 'f', then 's'' is a subsequence of 's'.
If there is a minimal sequence to failure extending V from M1 to M2 with inputs vs and xs, and M1 and M2 are both observable finite state machines (OFSMs), and there are test tools for M2 and M1 with failure, PM, V, and Omega, and V'' is in the set N of the concatenation of vs and xs' for M1 and V, and xs' is a prefix ...
If E is less than or equal to E' and A is a subset of A', then in the context of P, the type inference function ty\<^sub>i' with parameters ST, E', and A' is less than or equal to the type inference function ty\<^sub>i' with parameters ST, E, and A.
Given two conditions: the length of 'w' is equal to the length of 'r1', and the length of 'r1' is equal to the length of 'r2', it shows that a path exists in the product of A and B, with the path defined by the concatenation of 'w', 'r1', and 'r2' starting from the point '(p1, p2)', if and only if a path exists in A de...
The lemma "state_and_event_determine_next" assumes that an event 'ev' transitions the system from state 'c' to state 'c''', and also from state 'c' to state 'c'''. It then shows that the states 'c''' and 'c''' must be the same. This means that a specific event originating from a specific state will always lead to the s...
Applying the function "fold_coeffs" with the function "f" to 0 is equivalent to the identity function.
Mapping the composition of the functions "of_nat" and "nat" over the list of natural numbers from 1 to 4 results in a list of 32-bit words [1, 2, 3, 4].
The nonstandard version of a number 'k' is approximately equal to 1 if and only if 'k' is equal to 1.
The lemma asserts that it is necessarily the case that for any property P (with no positive instances), there exists an entity x such that it is possible for some entity z to be identical to x given the property P.
The lemma named "ffact_add_nat" shows that the falling factorial of the sum of two numbers "n" and "m" with respect to "k" is equal to the sum of the product of the binomial coefficient of "k" choose "i", the falling factorial of "n" with respect to "i", and the falling factorial of "m" with respect to "k - i", where "...
The function 'ca_lr' is monotonic.
Replicating a value 'k' for the number of times 'k' appears in a multiset 'xs' is equivalent to filtering 'xs' to only include the instances of 'k'.
If "r" and "s" satisfy the ACI (Associativity, Commutativity, and Idempotence) property, then the normalized form of "r" is equal to the normalized form of "s".
A list 'cs' is in the set generated by the function 'generate_choices' applied to a list consisting of 'xys' followed by 'xyss' if and only if the following conditions are met: 1. The length of 'cs' is equal to the length of the list consisting of 'xys' followed by 'xyss'. 2. The first element of the first pair in 'cs...
If zero is less than or equal to B, and x is less than or equal to the extended nonnegative real number of B, then the extended nonnegative real number of x is less than or equal to B.
The function "subst_bvs1'" applied to the application of "s" to "t" with arguments "k" and "args" is equal to the function "subst_bvs1'" applied to "s" with arguments "k" and "args", applied to the function "subst_bvs1'" applied to "t" with arguments "k" and "args".
The interior of the product of two sets A and B is equal to the product of the interiors of sets A and B.
The image of the set B under the restriction of the relation r to the set A is equal to the intersection of the set A and the image of the intersection of sets A and B under the relation r.
The bound variables of a stateful strand "A" are a subset of the bound variables of the stateful strand that results from prepending an action "a" to "A".
Repeating a multiset (M) for the successor of a number (n) is equivalent to the multiset (M) added to the repetition of the multiset (M) for the number (n).
The lemma named "bij_betw1" assumes that the pair consisting of (N,e) and d is an element in the set defined by the function I. It shows that the function F, when applied to the pair (N, e), establishes a bijective relationship between the range of the pair (N, e) and itself.
The lemma named "L_eps_free_nhole_mctxt_closure_reg" assumes that the transition system "ta" of "A" is epsilon-free. It shows that the language of the epsilon-free non-hole multi-context closure regular of "F" and "A" is equal to the language of the non-hole multi-context closure regular of "F" and "A".
Permuting the columns of the transpose of a matrix M according to a permutation p is equivalent to transposing the matrix obtained by permuting the rows of M according to the same permutation p.
The least upper bound (lub) of the image of a set A under the function "map_generat" with parameters f, g, and h is equal to the least upper bound of the set A under the composition of lub1 with the image of f, lub2 with the image of g, and lub3 with the image of h.
The lemma l8_10: assumes that angle ABC is a right angle and the three sides of triangle ABC are congruent to the three sides of triangle A'B'C' respectively, it shows that angle A'B'C' is also a right angle.
The function "stermsl" applied to the sum of "p1" and "p2" is equal to the function "stermsl" applied to the sum of "p2" and "p1". This implies that the function "stermsl" is commutative over the operation represented by "\<oplus>".
The length of the result from the function 'fill' when applied to the parameters 'xs', 'ns', 'index', 'key', 'n', 'mi', and 'ma' is equal to 'n'.
The lemma named "shift_Nil_iff" assumes that "qs" is a valid decomposition of "X", "qs" is a standard decomposition of "k", and "qs" is an exact decomposition of "Suc m". It shows that the shift of "qs" is empty if and only if "qs" is empty.
If the second element of the record 'r' is a number 'x1', and 'x1' is not less than 100, then the possible steps for the 'drinks2' machine from state 2, with the action 'vend' and no parameters, is a set containing only the pair (3, vend).
The real number representation of the inner product of a vector 'x' with itself, when multiplied by a scalar 'A', is equal to the inner product of 'x' with itself multiplied by 'A'.
The function "step" with parameters "xs", "r", and "act" results in an empty list if and only if "xs" is an empty list.
The lemma named "prefix_definedD" assumes that the prefix of list "ys" is "xs", which is true. It then shows that neither "xs" nor "ys" can be undefined or bottom.