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The function 'r_tuple_encode' for any natural number 'n' is a primitive recursive function of 'n+1'.
The cardinality (or the number of elements) of a set containing only one element 'a' is equal to the successor of 0 (which is 1).
The lemma named "ferrack" assumes that the proposition "p" is quantifier-free (as stated by "qfree p"). It shows that the function "ferrack" applied to "p" is also quantifier-free and that the interpretation function "Ifm" applied to the boolean list "bs" and "ferrack p" is equivalent to there existing an "x" such that...
The addition set of a function 'f' and the augmented plus-minus set 'aug_pm_set' with parameters 'z', 'i', and 'A' for a given number 'n' is a subset of the addition set of the same function 'f' and the augmented plus-minus set with parameters 'z', 'i', and the singleton set 's_set' with parameters 'R' and 's' and 'A' ...
The lemma named "sym_block_mult_one" assumes that "bl" is an element of "B". It shows that the multiplicity of "bl" is 1.
The lemma named "submachine_deadlock" assumes that "A" is a submachine of "B" and that there is a deadlock state "q" in machine "B". It shows that the same deadlock state "q" also exists in the submachine "A".
If zero is less than x, then the multiplication by x is normal.
If polynomial 'p' and polynomial 'q' are both in the carrier 'P', then the coefficient of the product of 'p' and 'q' at position 'n' is equal to the sum (for all 'i' from 0 to 'n') of the product of the coefficient of 'p' at position 'i' and the coefficient of 'q' at position 'n-i'.
The extended integer representation of a numeral 'k' is equal to the numeral 'k' itself. This is marked as a code abbreviation.
The lemma named "rjoin_epsilon_deterministic" assumes two conditions: 1. The state 's' of revision 'r' is defined as a tuple consisting of sigma, tau, and E applied to Rjoin of VE of Rid of r'. 2. The state 's' of revision 'r'' is undefined or None. Under these assumptions, it shows that a step in the revision 'r' fr...
In the context of a semifunctor that is fully linked, given the assumptions that the semicategory 'A' is equivalent to the digraph 'A' and the semicategory 'B' is equivalent to the digraph 'B', it can be shown that the digraph morphism of the semifunctor maps 'A' to 'B' in a fully linked manner.
For any word 'w' and any LTL formula 'φ', 'w' satisfies 'φ' in the semantics of linear temporal logic over finite words if and only if 'w' satisfies 'φ' in the semantics of linear temporal logic over infinite words.
The lemma "ntcf_ntsmcf_components" shows the following: 1. The natural transformation map of the natural transformation to the semi-category functor of N is equal to the natural transformation map of N. This is a simplification rule. 2. The domain of the natural transformation to the semi-category functor of N is equ...
The unsigned natural number representation of negative one in any word length type is greater than zero.
If lens 'x' is a bijective lens, then lens 'x' is also a very well-behaved lens.
: Assuming the following conditions: 1. The Java Virtual Machine (JVM) program G is well-typed with respect to the type mapping phi. 2. The method identified by the signature sig in the class C of the program G is defined and has the properties (C,rT,maxs,maxl,ins,et). 3. The instruction at the program counter pc is G...
If a transition "t" is concatenated with itself, then "t" is an arrow. This is assumed under the condition that "t" is concatenated with itself.
The stereographic projection function (st_proj2) is continuous on the universe set excluding the bottom sphere.
"n of x" is less than or equal to "n of y" if and only if the product of "n of x" and "n of y" equals "n of x".
If 'i' is an element in the set from 1 to 'k', and the intersection of 'S i' and 'R' is not empty, then the cost of 'R' with 'min_arg R k' is less than or equal to the cost of 'R' with 'i'.
Choosing "k+1" elements from a set of "a+1" elements is equal to choosing "k" elements from a set of "a" elements multiplied by the ratio of "(a+1)" to the natural number representation of "(k+1)".
If 'x' overlaps with 'y', and 'x' is a part of 'v' and 'y' is a part of 'z', then the product of 'x' and 'y' is a part of the product of 'v' and 'z'.
If the following conditions are met: 1. k is greater than 0, 2. the length of the list ys is equal to k, 3. the length of the list xs is divisible by k with no remainder, then the function f_aggregate applied to the concatenation of the lists xs and ys, with parameters k and ag, is equal to the concatenation of th...
For any given predicate P, the function "pred_stream P" is measurable in the stream space of the count space of the universal set.
The domain of a finite map remains the same even after a function is mapped over the finite map.
The lemma named "mat_gt_arc_imp_mat_ge" assumes that matrix A is greater than matrix B in terms of arc (as per the condition "mat_gt_arc A B"), and both matrices A and B are in the carrier matrix of dimensions nr by nc (as per the conditions "A \<in> carrier_mat nr nc" and "B \<in> carrier_mat nr nc"). Under these assu...
Mapping a polynomial 'p' raised to the power 'n' to real numbers is equivalent to mapping the polynomial 'p' to real numbers first and then raising it to the power 'n'.
The lemma "acyclic_paths_from_reachable_states" assumes that: 1. The machine M is acyclic, meaning it does not contain any cycles. 2. There is a path in machine M from its initial state to some state p'. 3. The target of the initial state of machine M through path p' is state q. 4. There is a path in machine M from sta...
Updating the nth element of the partial deviation (pdev) of a list of elements (xs) with a new value (x) is equivalent to updating the nth element of the list with the new value and then computing the partial deviation.
The intersection of the ceiling of P and the ceiling of Q is equal to the ceiling of the function that takes a state s and returns the logical conjunction (AND operation) of P of s and Q of s.
"x is less than or equal to y if and only if x implies y is less than or equal to x".
If for all keys 'k' and values 'v', when 'm1' at key 'k' equals 'v', then applying function 'f1' to 'k' and 'v' equals applying function 'f2' to 'k' and 'v', and if 'm1' equals 'm2', then the result of applying the function 'mapCollect' to 'f1' and 'm1' equals the result of applying 'mapCollect' to 'f2' and 'm2'.
If a state 's' can reach a state 's'' and if state 's' is reachable, then state 's'' is also reachable.
The theorem named "If_ZObis" assumes that the test "tst" is compatible and that "c1" is approximately equal to "d1" and "c2" is approximately equal to "d2". It shows that if these conditions are met, then the "If" statement with "tst", "c1", and "c2" is approximately equal to the "If" statement with "tst", "d1", and "d...
For a given system P, P implies that the concatenation of lists xs and ys is less than or equal to Ts if and only if there exist lists Ts1 and Ts2 such that Ts is the concatenation of Ts1 and Ts2, the length of xs is equal to the length of Ts1, the length of ys is equal to the length of Ts2, and P implies that xs is le...
If 'p' is a permutation of the set of all numbers less than 'n', and 'i' is less than 'n', then the 'i'-th row of the permutation matrix 'n' by 'p' is equal to the unit vector 'n' of the inverse of 'p' at 'i'.
The lemma "S_transition_eqvt" assumes that if there is a transition from state P to state P' under the action alpha in the system S, then there is also a transition from the permuted state (p bullet P) to the permuted state (p bullet P') under the permuted action (p bullet alpha) in the same system S.
The Markov chain B is aperiodic over the entire universe of discourse.
The lemma is in the context of an encoding. It fixes three relations: 'Rel' which is a set of pairs of processes, 'SRel' which is a set of pairs of source processes, and 'TRel' which is a set of pairs of target processes. The lemma assumes four conditions: 1. The encoding is fully abstract with respect to 'SRel' and...
The product of the floor values of two numbers 'a' and 'b' is less than or equal to the floor value of the product of 'a' and 'b'. Here, the floor value is the greatest integer less than or equal to a given number.
The multiset image of the list of elements in set A is equal to the multiset of set A.
The index of the pair (a, b) in the list created by zipping the list 'as' with a list of 'd' repetitions of 'b', is equal to the index of 'a' in the list 'as'. This is true if 'd' is equal to the length of the list 'as'.
If a triple (c,e,c') is in the set gtr T, and for all states s, ce, s', and ce', where c is the addition of s to ce and c' is the addition of s' to ce', and the quadruple ((s,ce),e,(s',ce')) is in the set T, then P holds. This implies that P holds under these conditions.
What agent A knows from a list of events that starts with "ev" followed by other events "evs" is the same as what agent A knows from the event "ev" alone combined with what agent A knows from the other events "evs".
The inner product is linear. It fixes a complex vector 'u', a function 'v' that maps natural numbers to complex vectors, and a function 'l' that maps natural numbers to complex numbers. It assumes that for all 'i' in the set {0, 1}, the dimension of vector 'u' is equal to the dimension of vector 'v' at 'i'. It shows th...
The list obtained by removing duplicate elements from the right in a list 'xs' is distinct.
If the control flow stack (cfs) is not empty, and if the program P is valid for the control flow stack, and if the predicate mpred holds for the program P, the control flow stack, and the indices 0 and length of cfs - 1, then the following holds: In the case where the first and last elements of the control flow stack ...
If a list "xs" is not empty, then the set that consists of the first element of "xs" combined with the set of the rest of the elements in "xs" is equal to the set of all elements in "xs".
For any integers x, y, and a, the absolute value of a times y divides x if and only if a times y divides x.
"y plus the infinite power of x times x times y" is equal to "the infinite power of x times y".
: Assuming "call1 (obj\<bullet>M(pns))" equals "\<lfloor>(a, M', vs)\<rfloor>", we can obtain one of the following three cases: 1. (CallObj) "call1 obj" equals "\<lfloor>(a, M', vs)\<rfloor>" 2. (CallParams) There exists a value "v" such that "obj" equals "Val v" and "calls1 pns" equals "\<lfloor>(a, M', vs)\<rfloor>" ...
The lemma fixes a function 'x' that maps from a state to a natural number. Assuming that 'x' is a base variable, the lemma shows that this assumption leads to a contradiction or an impossibility.
A sequence 'S' is less than or equal to the concatenation of a list 'L' and the sequence 'S'.
If F is a register, then the function F applied to 'Some' equals 'Some'.
A vector x is a zero vector if and only if the negation of vector x is a one vector.
The lemma named "mkarr_memb_ARR" assumes that "f" is an arrow and "t" is an element of "f". It shows that making an arrow from "t" results in "f".
If under the program P and environment E, the evaluation of expression e0 from state s0 leads to a value v and state s1, and the evaluation of expression e1 from state s1 leads to expression e2 and state s2, then under the same program P and environment E, the evaluation of the sequence of expressions e0 followed by e1...
The lemma named "not_Greater_bounded" assumes the following conditions: 1. The interval function I applied to x is not equal to the function Greater applied to the function k applied to x. 2. x is an element of the set X. 3. The region r is a valid region with respect to the set X, the function k, and the interval fun...
If the invariant "invar" holds for a state named "small" of any type 'a, then the size of the state after a step is equal to the size of the state before the step.
If "x" is in the set of variables and "t" is in the set of terms, then substituting "x" with term "t" in the number "n" will result in the number "n" itself. This is a simplification rule.
The lemma named "dghm_const_components" demonstrates the following: 1. The object map of a constant digraph homomorphism from digraph \<CC> to digraph \<DD> with object 'a' and arrow 'f' is a constant function on the objects of \<CC> with value 'a'. 2. The arrow map of a constant digraph homomorphism from digraph \<C...
If the term "X" is well-sorted in the model "MOD" and the sort "s", then checking if the element "y" from the set "ys" is fresh in the term "OK X" in the model "MOD" is equivalent to checking if "y" is fresh in the term "X" in the same model.
If a point 'u' is in the unit disc, and for any 'a' and 'u' such that 'u' is in the unit disc and the complex modulus of 'a' is less than 1, the property 'P' holds for the point 'u' transformed by the Blaschke factor of 'a', then the property 'P' also holds for 'u'. This is also true if the property 'P' holds for the z...
The lemma named "primepow_mult_aprimedivisorI" assumes that "n" is a power of a prime number in a factorial semiring with multiplicative operations. It shows that the product of "n" and a prime divisor of "n" is also a power of a prime number.
The first element of the pair resulting from the function "c_hupd" applied to "h" and the pair "(x, s)" is "x".
The lemma "new_character_data_new_ptr" assumes that a new character data is successfully created in the heap 'h', resulting in a new heap 'h'', and that the new character data is pointed to by 'new_character_data_ptr'. It shows that the set of object pointers in the new heap 'h'' is equal to the union of the set of obj...
The lemma named "cf_cat_prod_12_of_3_is_functor'" in the context of category theory introduces the following conditions: Assume that there are three categories, denoted as "A", "B", and "C", all of the same type "alpha". Also, assume that "A'" is the product category of "A", "B", and "C", and "B'" is the product catego...
If a function 'f' is measurable from a measure space 'M' to the Borel sigma-algebra, then the essential supremum of 'f' over 'M' is equal to the infimum of the set of all 'z' such that the measure of the set of all 'x' in the space of 'M' for which 'f(x)' is greater than 'z' is zero.
A list "xs" is not null if and only if there exists an element "x" and a list "xs'" such that "xs" is the list that results from prepending "x" to "xs'".
The value of the function "combine_tm" with inputs (p₀L, p₁L), (p₀R, p₁R), l, and ps is equal to the result of the function "combine" with the same inputs (p₀L, p₁L), (p₀R, p₁R), l, and ps.
If an adversary A is lossless in the indistinguishable chosen plaintext attack scenario, and the order of group G is greater than zero for any given η, then the adversary A is also lossless in the linear computational Diffie-Hellman scenario when using the ElGamal adversary.
If the range of a substitution 's' is ground (i.e., it contains no variables), then a variable 'x' is not in the range of variables of 's'.
If the functions 'f' and 'g' are continuous at a point 'x', then the function that applies 'f' and 'g' to 'x' and takes their pointwise OR is also continuous at 'x'. This lemma is marked for simplification and for use in proving continuity of other functions.
The lemma le_infI': Assuming that for any 'k', there exists a 'v' such that 'v' is less than or equal to 'w1' in the finite order and 'k' is less than the length of 'v' and 'v' is less than or equal to 'w2' in the finite order, it can be shown that 'w1' is less than or equal to 'w2' in the infinite order.
If a property P holds for all elements at any index i of a list xs, and if x is an element of the list xs, then the property P holds for x.
The polynomial resulting from the operation "spoly" on p and q is the negative of the polynomial resulting from the operation "spoly" on q and p.
If "A says to B message X" is in the set of events, then "X is in the maximum knowledge of A" in the set of events.
If for all elements 'z' in the set 'F1set1' of 'x', the function 'f1' applied to 'z' equals the function 'g1' applied to 'z', and if for all elements 'z' in the set 'F1set2' of 'x', the function 'f2' applied to 'z' equals the function 'g2' applied to 'z', then the mapping of 'f1' and 'f2' on 'x' using 'F1map' equals th...
For any integers x and y, the bitwise NOT of x XORed with y is equal to the bitwise NOT of x XORed with y. Similarly, x XORed with the bitwise NOT of y is also equal to the bitwise NOT of x XORed with y.
If there is a step epsilon closure from state 'q' to state 'q'' in a non-deterministic finite automaton with transition function 'bs', and there is no epsilon transition from state 'q' to any state 'q'', then state 'q' must be equal to state 'q''.
Converting a formal power series with a constant coefficient "c" to a formal Laurent series results in a formal Laurent series with the same constant coefficient "c".
The lemma named "Ide_appendE" assumes that the concatenation of T and U is an identity, both T and U are not empty, and if T is an identity, U is an identity, and the target of T is the same as the source of U, then a certain statement (referred to as the "thesis") is true. Under these assumptions, the lemma shows that...
If a function 'f' is absolutely integrable on a set 'S' and 'S' is in the sets of Lebesgue measure, then 'f' is Borel measurable on the Lebesgue measure restricted to 'S'. Here, 'f' is a function from a Euclidean space to another Euclidean space.
For any polynomial 'p' in a commutative semiring, prepending '0' to 'p' is equivalent to multiplying the polynomial 'p' by the polynomial [:0,1:].
The curry function is a morphism from the exponential quasi-Borel space of the product of X and Y to Z, to the exponential quasi-Borel space of X to the exponential quasi-Borel space of Y to Z.
The lemma named "abst_action_repcs" assumes that the configuration "cfg" is a valid configuration in the probabilistic timed automata and that the abstraction of state "s" is equal to the state of the configuration "cfg". It shows that the abstraction of the action of the reconfigured state "s" with respect to the conf...
If 'x' is assumed to be a prime number, then 'x' cannot be equal to 1.
The lemma "LI_in_SMP_subset_single" assumes that there is a transition from state (S, θ) to state (S', θ'), the state (S, θ) is well-formed, the substitution θ is well-typed, the set of terms in state S is a finite set of terms, the set of terms in state S is well-formed, and all steps in state S are finite. Additional...
If a message from the Server to Agent A, which is encrypted with key K' and contains a number Tk, Agent B, a key K, and a ticket, is in the set of events, and these events are part of the Bankerberos protocol, then the following conditions must hold: 1. The key K' is the shared key of Agent A. 2. The key K is not in t...
: If a pair of abstract and concrete states, where the abstract state is the result of assigning a constant 'z' to a variable 'x' followed by some abstract tail, and the concrete state is represented by 'c2C', 'mdsC', and 'mem2C', belongs to the refinement relation 'RefRel_HighBranch', then there exists a unique concr...
If x is an element of A, then x is greater than zero.
If the pair (si, s) is in the cyc_check relation, then the break state of 'si' in the cyc_check state implementation is equal to the break state of 's' in the cyc_check state.
The lemma is about a function called "isFDERIV_FDERIV_floatariths_linear" that is related to the derivative of a function in floating point arithmetic. It fixes 'x' and 'h' as elements of an executable Euclidean space. It assumes that for any three indices 'i', 'j', and 'k', if each of them is less than the dimensi...
"x" is less than or equal to "y" if and only if the intersection of "x" and the negation of "y" is the bottom element.
If a list 'xs' is sorted according to a relation 'R', and 'R' is transitive, total on the universal set, and antisymmetric, then the list obtained by removing duplicates from the sorted list 'xs' is distinct.
If 'a' is less than or equal to 'b', both 'a' and 'b' are elements of set 'I', predicate 'P' is not true for 'a' but is true for 'b', then there exists an element 'n' in set 'I' such that 'a' is less than or equal to 'n', 'n' is less than 'b', predicate 'P' is not true for 'n', but is true for the immediate next elemen...
The lemma named "streams_stake_set" shows that if "pw" is in the stream of "A", then the set of the first "n" elements of "pw" is a subset of "A".
The multiset obtained from sorting a list is equal to the multiset of the original list.
The function "try_swap" with parameters p, c1, c2, w, x equals to the function "try_swap" with parameters p, c1, c2, w', x if and only if w x equals to w' x.
The transition function of "paodv i" is equal to the sequential process semantics of AODV.

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