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The remainder when the sum of twice a number 'x' and the length of 'mopup_n_tm' of 'n' is divided by 2, is always zero. This implies that the sum is always an even number.
The count of any element 'a' in an empty signed multiset is zero.
The lemma named "observable_language_next" assumes the following conditions: 1. The input-output pair 'io' followed by the sequence 'ios' is in the language of the machine 'M' starting from the source of the transition 't'. 2. The machine 'M' is observable. 3. The transition 't' is in the set of transitions of the mac...
The lemma named "auxiliary_ineq" establishes a rule for a fixed natural number 'x'. It assumes that 'x' is greater than or equal to 2. It shows that 'x' plus 1 is less than 2 times 'x'.
If the equality of two boolean values A and B is false, then the logical disjunction of the negation of the logical disjunction of A and B, or the negation of the logical disjunction of the negation of A and the negation of B, is also false.
The lemma named "quad_part_2_deg" assumes that a natural number variable "var" is not in the variables of a square-free multivariate polynomial "sq" with real coefficients. It shows that the degree of the quadratic part 2 of the polynomial "p" with respect to the variable "var" is less than or equal to 1.
If functions 'g' and 'h' are both in the set of primitive recursive functions of one variable, then the unary recursive operation on 'g' and 'h' is also in the set of primitive recursive functions of one variable.
Rejecting an element 'a' based on a condition 'P a' from a null bag will result in an invalid operation.
, given the assumptions that "alpha1" and "beta" are ordinal numbers, "alpha2" is an element of the set of elements of "alpha1", "beta" is less than omega raised to the power of "alpha2", and both "m1" and "m2" are greater than zero, it shows that omega raised to the power of "alpha2" multiplied by the ordinal number o...
The lemma named "equal_mod_timpls_finite" assumes that "t" is a well-formed term (as indicated by "wf\<^sub>t\<^sub>r\<^sub>m t"). It shows that the set of all "s" such that "s" is equal to "t" modulo term implications (as indicated by "equal_mod_timpls TI t s") is finite.
The inverse image of a set A under a relation r is equal to the image of A under the inverse of the relation r.
The function "f" is equal to the dual of function "g" if and only if the dual of function "f" is equal to "g".
The set of Dirichlet characters of an integer 'n' is finite.
The transformation of 0 by T equals 0.
In the context of a univariate polynomial over a commutative ring, if S is a subring of R, then the carrier of the univariate polynomial over the ring R with carrier S is a subset of the carrier of the univariate polynomial over the ring R.
The lemma named "prime_elem_gauss_int_iff" is about a fixed Gaussian integer 'z'. It defines 'n' as the norm of this Gaussian integer 'z'. It shows that 'z' is a prime element if and only if 'n' is a prime number or there exists a prime number 'p' such that 'n' is equal to the square of 'p' and 'p' is congruent to 3 mo...
The lemma "Loss_lengthOut2" assumes that for all instances 't', the length of 'a t' is less than or equal to 1. It also assumes that the loss function 'Loss' with parameters 'lose', 'a', 'i2', 'x', and 'i' holds true. Under these assumptions, it shows that for all instances 't', the length of 'x t' is less than or equa...
If the function "tameEnumFilter" applied to a parameter "p" equals to some "t", then the set of tries "t" is isomorphic to the function "fgraph" applied to the Tame Enumeration of "p".
The tree automaton determined by "ps_ta" is deterministic.
The feedback function of a variable assignment (where 'x' is assigned the value of 'e') in a predicate 'Q' is the set of all states 's' such that the state resulting from the assignment of 'x' with the value of 'e' in 's' belongs to 'Q'.
If the real part of a sequence of complex intervals (represented by "C") tends to the real part of a complex number "c", and the imaginary part of the same sequence of complex intervals also tends to the imaginary part of "c", then the sequence of complex intervals itself tends to "c".
Negative one as a 2-bit word is equal to three.
If the condition in an if-then-else (ifte) statement is false (represented by the bottom element, \<bottom>), then the result is the 'else' part, denoted by 'y'.
The lemma named "median_const" assumes that "k" is greater than 0. It shows that the median of "k" for any "i" in the set from 0 to less than "k", where each "i" maps to "a", is equal to "a".
Applying the function "KM_map" to the identity function "ID" results in the identity function "ID".
If the following conditions are met: 1. Agent A inputs to their own card the set consisting of Agent B, Nonce Na, Nonce Nb, Nonce Pk, Cert1, Cert2, and Cert3, and this input is part of the set of events 'evs'. 2. Agent A is not a Spy. 3. The set of events 'evs' is part of the Shoup-Rubin-Bella protocol 'srb'. Then, i...
If a sequence 's' is free of ticks, then the front of sequence 's' is also free of ticks.
The intersection of 'm' and the inverse of 'ov' is an empty set.
A relation "r" is strongly confluent if for any elements x, y, and z, whenever x is related to both y and z by the relation "r", there exists an element u such that y is transitively related to u by "r" and z is transitively and reflexively related to u by "r".
The function ModPonP with parameters t, u, and v is a sigma formula.
The set of options enforced by a predicate P on an element x is equal to the set of elements in the options of x that satisfy the predicate P.
The composition of relations "s" and the inverse of "ov" is a subset of the union of the inverse of "ov", "f", and "d".
If the program P can transition from the expression 'e' in the initial state 's0' to 'null' in the state 's1', and if the program P can transition from the expressions 'es' in the state 's1' to the mapped values 'vs' in the state 's2', then the program P can transition from the method call 'e.M(es)' in the initial stat...
If "x" is a very well-behaved lens, then the conditional expression where "P" is substituted with "true" for "x" when the variable "x" is true, and "P" is substituted with "false" for "x" when the variable "x" is false, is equivalent to "P".
There exists a list of observation events, denoted as "obs", and a heap, denoted as "h'", such that the heap copies from the base heap, which is read in a typed manner according to the type of address and the Java Memory Model heap read protocol, and written according to the Java Memory Model heap write protocol, from ...
The lemma named "greatest_ge_nonzero_row'" is defined with respect to a matrix 'A' of a certain type. It assumes that the matrix 'A' is in echelon form up to column 'k', and that the row index 'i' is less than or equal to the greatest row index 'm' for which the row is not all zeros up to column 'k'. It also assumes th...
The first element of the closed segment between two points 'a' and 'b' is equal to the closed segment between the first elements of 'a' and 'b'.
If 'a' and 'b' are not in the field of 'Qeq', then the transitive symmetric closure of the set 'Qeq' with the pair (a, b) added is equal to the union of the transitive symmetric closure of 'Qeq' and the set containing the pairs (a, a), (b, b), (a, b), and (b, a).
The lemma for the main code of quadratic Hensel is defined as follows: If 'j' is less than or equal to 1, then the output is (D1, H1). If 'j' is an even number, then the output is determined by the result of the quadratic Hensel loop for 'j' divided by 2, which is then passed through the simple quadratic Hensel step. I...
The lemma states two things about octonions. First, the inner product of the unit octonion e4 and any octonion x is equal to the fourth imaginary part of x. Second, the inner product of any octonion x and the unit octonion e4 is also equal to the fourth imaginary part of x.
The insertion of function 'f' into a constant zero of type real multivariate polynomial equals zero.
The union of all open intervals from negative real number 'n' to positive real number 'n', for all 'n', equals the set of all elements (the universe).
A number 'x' is equal to the substitution of a floating point arithmetic expression 'fa' with a variable function 's x', if and only if 'fa' is equal to the number 'x'.
The lemma "fsm_from_list_simps" states the following: 1. The initial state of a finite state machine (FSM) created from a list is equal to the first state 'q' if the list is empty. If the list is not empty, the initial state is the source state of the first transition 't'. 2. The states of the FSM created from a list...
The lemma named "Combination" is about a fixed variable "bit". It assumes that "qs" is in the pattern, "a" is not equal to "b", the set of "a" and "b" is equal to the set of "x" and "y", and the set of "qs" is a subset of the set of "a" and "b". It also assumes that "inv_COMB s a [x,y]" is true. Furthermore, it assume...
The lemma named "arr_char" shows that "arr" is equivalent to "Cell" for any "mu".
The lemma named "euclid_ext_poly_dynamic" assumes the following conditions: 1. The polynomials 'f' and 'g' are coprime in the ring of polynomials modulo 'm' (denoted by "coprime_m f g"). 2. The polynomial 'f' is equal to its modulo 'p' reduction (denoted by "Mp f = f"). 3. The polynomial 'g' is equal to its modulo 'p'...
Lemma Radon_partition: Given a finite set C that is affine dependent, there exist two sets m and p such that: 1. m and p are disjoint (i.e., their intersection is empty), 2. the union of m and p is equal to C, and 3. the intersection of the convex hulls of m and p is non-empty.
If you first convert a function from a 3-argument form to a 1-argument form using the function "pr_conv_3_to_1", and then convert it back to a 3-argument form using the function "pr_conv_1_to_3", you will get the original function back. This is a property of these two conversion functions, indicating that they are inve...
The lemma named "dvd_gcd_mult" is applicable to any semiring with a greatest common divisor (gcd) operation. It assumes two conditions: that "k" divides the product of "p" and "q", and that "k" also divides the product of "p" and "r". Under these assumptions, it shows that "k" also divides the product of "p" and the gr...
If there is a left derivation from the start symbol to a string consisting of a nonterminal N followed by a string v, and the start symbol is not equal to N, then there exists a number n, a nonterminal M, strings a, a1, a2, and w such that n is less than the length of D, (M, a) is a rule in the grammar, there is a left...
For any three variables ve1, ve2, and ve3 of type "'c \<avenv>", if ve1 is approximately less than or equal to ve2 and ve2 is approximately less than or equal to ve3, then it follows that ve1 is approximately less than or equal to ve3. This is a statement of transitivity for the approximate less than or equal relation.
"pvalid p x" is true if and only if there exists some "u" and "γ" such that "pvalid_with p x u γ" is true.
The set of free variables in the result of integrating the variable 'v' into the expression 'e' under the context 'Γ' is equal to the set of free variables in 'e' minus the set containing 'v'.
For a function 'f' mapping any type 'a' to a complex number, 'f' is summable on a set 'A' if and only if 'f' is absolutely summable on 'A'.
If an expression "ξ" is well-typed according to the typing rules of system "I", then the transformed expression "transE ξ" is also well-typed according to the typing rules of system "Ik".
For every natural number 'x' less than the absolute value of 'm', a property 'P' holds for 'x' if and only if for every natural number 'n' less than 'm', the same property 'P' holds for the absolute value of 'n'.
The modulus of an integer m and an integer n is equal to the integer of the modulus of m and n.
The lemma named "phiArg_trancl_same_var" assumes that there is a transitive closure of the phi argument of g from phi to n. It shows that the variable of g at phi is equal to the variable of g at n.
If the accessibility relation holds for all pairs of elements (a, a') in a preordered set, where a is less than or equal to a', then the accessibility relation also holds for all pairs of elements (c, c') in a set of abstract commands, where c is less than or equal to c'.
"smc_0" is a semicategory within the alpha domain.
The previous interval of 'n' in the product of interval 'I' and integer 'k' is equal to: if 'n' modulo 'k' equals zero, then the product of the previous interval of the quotient of 'n' and 'k' in interval 'I' and 'k', otherwise it is 'n'.
If functions 'f' and 'g' are both Borel measurable with respect to the Lebesgue measure, and almost everywhere in the extended interval from the minimum of 'a' and 'b' to the maximum of 'a' and 'b', 'f' equals 'g', then the Lebesgue integrability of 'f' over the interval from 'a' to 'b' is equivalent to the Lebesgue in...
The star of x and y, star with the product of the star of x and 1, and z, is less than or equal to the star of x and the star of the product of y and the star of x and 1, and z.
The sparse row vector of the sum of two sparse vectors 'a' and 'b' is equal to the sum of the sparse row vectors of 'a' and 'b'.
The lemma "language_state_for_inputs_prefix" assumes that the concatenation of "vs" and "xs" is in the input language of machine M1 for the set containing the concatenation of "vs'" and "xs'", and that the length of "vs" is equal to the length of "vs'". It shows that "vs" is in the input language of machine M1 for the ...
"if it is possible that A, B, and C are all true, and if it is obligatory that if B then C, then it is obligatory that if A and B then C."
The lemma named "non_empty_cycle_root_start_end_points_plus" assumes that "r" is a non-empty root cycle starting at "x". It shows that the relation from "x" to itself with at least one intermediate step is a subset of the transitive closure of "x".
An empty list is in the possible outcomes of tree 't'.
If "p" is less than the head of "ws", and "s" is less than the last element of "ws", and the concatenation of "p", "u", and "s" equals the concatenation of "ws", then "p", "u", and "s" are interpreted as equivalent to "ws".
The nonce (a random or pseudo-random number that is generated for a specific use, typically for authentication purposes) of a chain involving entities B, A, L, and C is equal to the nonce of the entity ofr.
If the triple (ta, va, h') is in the set of aggregated external reductions of a program P with thread t, method M, arguments vs, and heap h, and if a new heap element with address ad and class type CTn is in the set of object-created actions of ta, and the type of the address ad is not None, then the type of the addres...
The variables of 0 are an empty set.
The polynomial of 0 for any x is 0.
The lemma lm045 assumes that the intersection of the domain of 'a' and 'X' is not empty and 'a' belongs to the allocations universe. It shows that the union of the image of 'X' under 'a' is not empty.
The lemma named "not_fic_subst1" assumes that "a" is not a fixed point of "M". It shows that "a" is also not a fixed point of the substitution of "M" with "y" replaced by "<c>.P".
If there does not exist any element 'u' in 'U' such that the pair '(u, D x)' belongs to 'I', then the function 'ipurge_ref_aux_less' with parameters 'I', 'D', 'U', the list consisting of 'x' followed by 'xs', and 'X' is equal to the function 'ipurge_ref_aux_less' with parameters 'I', 'D', 'U', 'xs', and 'X'.
The lemma named "add_mset_le_le_le" is defined for a variable 'a' of any linearly ordered type. It assumes two conditions: 'a' is less than or equal to 'b', and 'A' is less than or equal to 'B'. Under these assumptions, it shows that the multiset obtained by adding 'a' to 'A' is less than or equal to the multiset obtai...
If the length of the list "xs" is the successor of the extended natural number "n", then the last element of the list "xs" is the "n"-th element of the list "xs".
The lemma named "blinfun_mono_norm_pos" is defined with a fixed function 'f' that maps from a bounded linear function of 'c' to real numbers, to a bounded linear function of 'd' to real numbers. It assumes that for any bounded linear function 'v' of 'c' to real numbers, if 'v' is greater than or equal to zero, then the...
The lemma "InvariantNoDecisionsWhenConflictNorUnitAfterUnitPropagate" assumes the following conditions: 1. The state of the system maintains the invariant of watches, which is defined by the function "InvariantWatchesEl" on the formula (getF state), and the two watch variables (getWatch1 state) and (getWatch2 state). ...
The most significant bit (msb) of a numeral word 'w' of any length 'a' is equal to the bit at the position (LENGTH('a) - 1) of the numeral 'w' when it is considered as an integer.
If a partial state "part s" can transition to "part s'" in "m" steps under the command "C", then the full state "s" can also transition to "s'" in "m" steps under the same command "C".
The lemma named "expectation_gpv_top_pfinite" assumes that the generalized probabilistic verifier (gpv) is almost surely terminating (pfinite_gpv). It shows that the expected value of the gpv, when the interpretation function (\<I>) and the reward function (\<lambda>_. \<top>) are both the top element (\<top>), is also...
If a state 's' transitions to a state 't' through a while loop 'w' with 'n' steps, then for all conditions 'b' and commands 'c', if 'w' is a while loop with condition 'b' and command 'c', and predicate 'P' holds for state 's', and for all states 's' and 's'', if predicate 'P' holds for state 's' and condition 'b' is tr...
The length of the sequence of initial segments of a sequence 'xs' is equal to the length of 'xs' plus one.
The lemma named "protectImp" assumes that if proposition P implies proposition Q, then it shows that if the pure proposition P implies the pure proposition Q.
The function "fun_evaluate_match" is equivalent to a case match on the result of "match_result". If "match_result" returns an error, then "fun_evaluate_match" also returns the same error. If "match_result" returns a value, then "fun_evaluate_match" evaluates a list with the updated environment. The function "fun_evalua...
The lemma "bisimilarity implies equivalence for actions" assumes the following: 1. For any two processes P and Q, if P is bisimilar to Q, then P satisfies a property x if and only if Q also satisfies x. 2. Process P is bisimilar to process Q. 3. Process P satisfies the action α with property x. Under these assumptions...
A left-deep tree "t" is equal to a right-deep tree when "t" is mirrored.
The lemma named "map_poly_of_real_Re" assumes that the set of coefficients of a polynomial 'p' is a subset of the real numbers. It shows that mapping the polynomial 'p' first by the real part function and then by the 'of_real' function results in the original polynomial 'p'.
If 'a' is a prime number and 'a' to the power of 'm' equals 'a' to the power of 'n', then 'm' must be equal to 'n'.
For any 'x' of type 'a word (where 'a is a length), it is not true that zero is less than or equal to the unsigned integer representation of 'x' minus two to the power of the length of 'a'.
: 1. The state 'q' is in the set of accepting states 'faccs' for a transition function 'delta' and a natural number 'n' if and only if 'q' is an accepting state for 'delta', 'n', and 'q'. This is denoted as ?T1. 2. The mapping of the function 'faccs' with 'delta' over a list of states 'ts' is equal to the mapping of ...
A full node of a balanced binary tree with key 'k', value 'a', and height 'h' is balanced.
Converting a concatenated list of 'xs' and 'ys' to a normal list is equivalent to converting 'xs' to a normal list, concatenating it with the normal list form of 'ys'.
If you select like a set from set X with respect to an empty set, you will get the original set X.
The "happens_before" relation for the program obtained by compiling P using function f is the same as the "happens_before" relation for the original program P.
The field operations of a finite field with integer elements, as defined by "finite_field_ops_integer pp", are related to the modular ring relation for integers, "mod_ring_rel_integer".
The start state type response is consistent with the compiled type expression for a given Java micro benchmark, global environment, and expression.