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: If there exists a pair (i, l) in the set of inputs obtained from the guards of transition t2, such that the filter of guards of t2 that constrain the variable (I i) equals to the list containing the input (I i) with list l, and there exists an element l' in the set of list l, where i is less than the arity of transi...
The domain of the function "cr_omega" is equal to the set of all x such that x is an element of the set omega.
The lemma named "degree_map_poly_2" assumes that the function "f" applied to the leading coefficient of a polynomial "p" is not equal to zero. It shows that the degree of the polynomial obtained by mapping the function "f" to each coefficient of the polynomial "p" is equal to the degree of the original polynomial "p".
For any real number 'a' in a real division algebra, if 'a' is a non-negative real number and 'b' is a non-positive real number, then the result of 'a' divided by 'b' is a non-positive real number.
If set S is not empty, then the next function of S is greater than the maximum function of S.
If the function "f times g" is integrable over the measure space M, and both functions "f" and "g" are Borel measurable (with "f" measurable over the sigma-algebra F and "g" measurable over the measure space M), then the function "f times the real conditional expectation of g given F" is also integrable over the measur...
The projection of an empty list, regardless of the value of n, is also an empty list. This applies to both the "proj" and "proj_unl" functions.
The right ideal generated by a set S is equal to the set of all elements y, for which there exists a function f and a finite subset U of S such that the sum of the product of each element i in U and the value of function f at i equals y. This is denoted as "?S = ?B".
The lemma named "bounded_clinear_equal_ket" is defined with two functions 'f' and 'g' that map from 'a ell2 to any type. It assumes that both 'f' and 'g' are bounded and linear. It also assumes that for any 'i', the result of applying 'f' to 'ket i' is equal to the result of applying 'g' to 'ket i'. Under these assumpt...
If set A is finite, then the encoding of set A is even if and only if 0 is not an element of set A.
The value of a linear polynomial "p" at a point "v" is equal to the sum, over all variables "x" in the polynomial "p", of the product of the coefficient of "x" in "p" and the value of "x" in "v".
The coefficient of a multivariate polynomial (MPoly) is equal to the value obtained by looking up the term in the function 'f'.
The subfield formed by the image of the carrier R under the polynomial of constants function is a subfield of the polynomial ring over the carrier R.
If the input from agent A to the card A with message X is in the set of events, then the message X is in the knowledge of agent A in the set of events.
If 'n' is less than the size of 'w', then the 'n'th element of the bit list representation of 'w' is equal to the bit at the position 'size of w - (n+1)' in 'w'.
The lower bound of the maximum interval of x and y is equal to the maximum of the lower bounds of x and y.
An element "x" is in the union of "a" and "b" if and only if "x" is in "a" or "x" is in "b".
If "b_of_a" is a bijective function from type 'a to type 'b, then the evaluation of the converted forms of F and G using "b_of_a" and the set Cs is equivalent to the evaluation of F, G, and Cs themselves.
The double iteration of x is equal to the least upper bound of L and the iteration of x.
: Assuming "n" is greater than or equal to 1, "i" is less than the number of rows in the tensor product of "HId" repeated "n" times, and "j" is less than the number of columns in the tensor product of "HId" repeated "n" times, the following conditions hold: 1. If "i" and "j" are both even, then the element at the "i"...
The composition of the function 'norm_unboxed' and 'OpDyn' is equivalent to the identity function.
For any element 'y' in set 'S', the function 'fix' applied to 'R'' at 'y' is equal to the function 'e1' applied to 'fix' applied to the function 'H' applied to 'fix' applied to 'R'' at 'y'.
The number of variables in the extension of a structure 'A' by a predicate 'P' at index 'i' with order 'ord' is equal to the successor of the order 'ord' of the number of variables in the structure 'A'.
The well-structured program when compiled with respect to a program G is equivalent to the well-structured program of G itself.
If for any composition 'J', if 'J' has finite partial magma arrows, then 'J' has limits of a certain shape, and if 'I' is a finite set and 'I' is not the universal set, then 'I' has products.
: Given a time 't' in a set 'T', a compact set 'C' that contains an initial point 'x0' and is a subset of 'X', if for any function 'y' and any time 's' in the interval from 't0' to 't', where 'y' at 't0' equals 'x0' and 'y' maps the interval from 't0' to 's' into 'C', and 'y' is continuous on the interval from 't0' to...
Given a number 'a' is in the carrier of Qp (the p-adic numbers), and 'a' is not equal to zero, and a natural number 'k' is greater than zero, then the product of 'k' and 'a' is not equal to zero.
The language of automaton A is a subset of the streams of the alphabet of automaton A.
The composition of the inverse of a relation 'r' with 'r' itself is equal to the identity on the domain of 'r'.
If 'f' is a morphism from 'a' to 'b' in category 'B', then the composition of the functor 'F' applied to the comma category object 'b' and the functor 'F' applied to the arrow 'f' in the comma category is equal to the functor 'F' applied to the comma category object 'a'.
If angle ABC is not congruent to angle A'B'C' and angle ABC is acute, then angle a is not congruent to angle A'B'C'.
The lemma lt1123 assumes that B is not equal to C. It shows that A is less than B and C.
If you update the nth element of the infinite list produced by zipping together two other infinite lists (f and g) with a pair of elements (xy), it is equivalent to updating the nth element of each of the original lists (f and g) with the first and second elements of the pair (xy) respectively, and then zipping the upd...
If "k" is less than the length of list "l", then the function "ias_of_list" applied to the list "l" with the "k"th element set to "False" is equal to the function "ias_of_list" applied to the list "l" minus the set containing "k".
If "ann_matches" holds for the parameters Gamma, Theta, a, and c, and if a' is in the pre-set of a, then "assertionsR" holds for the parameters Gamma, Theta, Q, A, a, c, and a'.
S_r is equal to the difference of S1 and S_e.
A relation P holds for all elements in two constant streams (streams made of repeating a single element) if and only if P holds for the individual elements of those streams.
The lemma "Exit_does_not_control" assumes that "Exit" controls "n'". If this is the case, it shows that this is false. This means that "Exit" cannot control "n'".
Zero exclusive OR with any value x equals x.
If a tree 't' is balanced and the inorder traversal of 't' is sorted, then the inorder traversal of the tree resulting from deleting 'x' from 't' is equal to the result of deleting 'x' from the inorder traversal of 't'.
The lemma named "arg_uminus_opposite_sign" assumes that "z" is not equal to zero. It shows that the argument of "z" is greater than zero if and only if the argument of "-z" is not greater than zero.
If the set of points 'ps' is a subset of the universe set 'U', then the index of 'ps' in set 'A' is less than or equal to the index of 'ps' in set 'B'.
The lemma named "combine_code_eq" assumes the following: 1. The distance delta_L' is the coded distance between points p0_L and p1_L. 2. The distance delta_R' is the coded distance between points p0_R and p1_R. 3. The list of points 'ps' is sorted based on their second element. 4. The pair (c0, c1) is the result of th...
The lemma named "tab_def_alt" demonstrates two things: 1. The "tab" is equal to the composition of the tensor product of the representation of r and tab-sub-0 of r, and the transformation of tab-sub-0 of r applied to the tensor product of tab-sub-1 of r and the star of tab-sub-0 of r. 2. The composition of the tensor...
Zero is less than one for a 3-bit word.
The truncation of an empty set in ell2 space for any element x is zero.
For any word 'x' of arbitrary length 'a', the bitwise AND of 'x' and the bitwise NOT of the mask 'n', when added to (2 to the power of 'n' minus 1), is equal to the bitwise OR of 'x' and the mask 'n'.
The principle of explosion, applied to the propositions "p" and "q", is not valid.
The lemma named "inversion_of_complex" assumes that a complex number 'z' is not equal to zero. It shows that the inversion of the complex representation of 'z' is equal to the complex representation of the reciprocal of the conjugate of 'z'.
If you first convert a quantity 'x' using a conversion factor 'c' and then convert it back using the inverse of 'c', you will get the original quantity 'x' back.
The lemma named "fill_count_none" assumes that the index is less than the key, as stated by "index_less index key". It shows that if every element 'x' in the set 'xs' has a key that is within the range from 'mi' to 'ma', and if 'ns' is not an empty list, and if the predicate 'offs_pred' holds true for 'ns', 'ub', 'xs',...
The lemma named "fresh_psubst" is defined with a fixed variable 'x'. It assumes that the atom 'x' is fresh in 'e' and 'vs'. It shows that the atom 'x' is also fresh in the term 'e' after parallel substitution with 'vs'.
The lemma "wt_jvm_prog_states_NonStatic" assumes that the Java Virtual Machine program P is well-formed according to the typing rules specified by Φ. It also assumes that the program P sees a method M in class C, which is non-static, takes arguments of types Ts and returns a type T. This method is represented by a quad...
A new element's tag name is equivalent to the tag name obtained from the function "l_new_element_get_tag_name", given the conditions of type well-formedness and the locations of the get_tag_name function. This is an instance of a more general rule or pattern.
If you rename the function 'h' in the 'whileAnno' loop with conditions 'b', invariant 'I', variant 'V', and command 'c', it is equivalent to having a 'whileAnno' loop with conditions 'b', invariant 'I', variant 'V', and the renamed function 'h' in command 'c'.
If the lookup action operation equals some abstract syntax tree operation, then this abstract syntax tree operation is in the set of abstract syntax tree delta.
If H is a subset of G, then H is equal to the left coset of H with respect to 0.
If the successor of an index 'i' is less than the length of the list obtained by removing adjacent duplicates from list 'xs', then the element at index 'i' in the de-duplicated list is not equal to the element at the successor of index 'i' in the same list.
Converting 0 to boolean is not true.
If 'b' is left-implicated by the product of 'a' and 'b', then it equals 'a'.
The fraction of (a, numeral b) divided by the fraction of (c, numeral d) is equal to the fraction of (a times numeral d, numeral b times c).
The lemma named "triple_val_ineq_set_semialg''" demonstrates that the set of 3-tuples in the carrier of the 3-dimensional p-adic numbers, where the valuation of the second element is greater than the valuation of the first element, is semialgebraic.
If set S is not empty, then for any element x in set S, a certain condition R holds. Therefore, the condition R holds.
Twice the cosine of a real number 'a' multiplied by the cosine of 'a' equals the cosine of twice 'a' plus one.
One can obtain a function A, which maps natural numbers to sets of type 'a. The range of this function A is a subset of the sets in measure space M. The union of all sets produced by function A equals the entire space of measure M. Furthermore, for any natural number i, the measure of the set produced by function A is ...
For a given structure R, if I is an additive subgroup of R, and R is a commutative ring, and for any elements a and x in I and R respectively, the product of x and a, and the product of a and x, both belong to I, and I is not equal to the carrier of R, and for any elements a and b in the carrier of R, if the product of...
If a program 'P' shows that the list of types 'LT' is less than or equal to the list of types 'LT'', and 'C' is not a class in 'P', then adding a class 'C' with class declaration 'cdec' to the program 'P' will still show that 'LT' is less than or equal to 'LT''.
If "x" is in the set of iterates above "a", and "a" is in the field of the less than or equal to relation, then "x" is also in the field of the less than or equal to relation.
The lemma named "gaugeD" assumes that "gamma" is a gauge. It shows that "x" is an element of "gamma x" and that "gamma x" is open.
The conjunction (and operation) of the list containing elements x, y, and z is equivalent to the conjunction of x, y, and z individually.
The lemma to_A_rule_set: Assuming that "TA_rule f qs q" is an element of the rule set A, and "q" equals "p" or the pair "(q, p)" is an element of the transitive closure of the epsilon relation of A, and "p" is not an element of the set Q_B, it can be shown that "TA_rule f qs q" is an element of the set Delta_A, and "q"...
The lemma "Generic_offending_list_correct" assumes that a list graph "G" is well-formed (valid). It also assumes that for any list graph "G" and network policy "nP", if "G" is well-formed, then the security invariant according to the specification (sinvar_spec) applied to the graph converted from the list graph and the...
Lifting a subprobability monad function on the left side of a generalized probabilistic value is equivalent to just lifting the subprobability monad function.
The lemma named "delivered_to_node" assumes that the state 's' is reachable in the parameterized network 'pnet' with parameters 'np', 'ii', and 'Ri' under the transition relation 'TT'. It also assumes that there is a transition from state 's' to state 's'' in the same parameterized network, where the transition is labe...
If "P" is a process and the sequence "s" followed by a "tick" with an empty set is in the failures of "P", then the sequence "s" with the set "X" excluding "tick" is also in the failures of "P".
The function "sorted_list_of_set" is injective (or one-to-one) on the collection of finite sets.
The coefficient of the integer of the product of the sign of x and a function f at index i is equal to the product of the sign of x and the coefficient of the integer of the function f at index i.
For all elements 'x' in the set 'xs', if the first element of the function 'f' applied to 'x' is equal to the first element of 'x', then the mapping of the function 'f' applied to 'xs' with 'a' is equal to the mapping of the option of a function that takes 'b' and returns the second element of the function 'f' applied ...
The result of deleting a key 'k' from a hash map 'hm' is equivalent to the original hash map 'hm' restricted to the complement of the set containing the key 'k'.
The set resulting from removing all instances of an element 'x' from a list 'xs' is equal to the set of 'xs' minus the set containing 'x'.
: Assuming that a cone decomposition 'T' exists for the concatenation of two lists 'ps' and 'qs', and for any pair (h, U) in the set 'ps', the cone of (h, U) is a subset of P[X] and 'h' is in the ideal 'F'. Also, for any pair (h, U) in the set 'qs', the cone of (h, U) is a subset of P[X]. Furthermore, for any 'a' in t...
If the list "s # ss" is valid, then the list "ss" is also valid and the recursive function of the length of the first element of the second element of "s" is the first element of "s".
The first element (index 1) of the Dirichlet series expansion of a function 'f' is equal to the exponential of the first element of the function 'f' itself.
The conjunctive normal form (cnf) of the disjunction (logical OR operation) of a clause R is equal to the set of R.
The lemma (in the Congruence Rule context) named "Bet_end_Point" shows that "p2" is not a between point of "Se p1 p1" (the segment endpoint of p1 and p1).
If a pair (xs, s) is in the set "isInBar" and X is well-sorted in the module MOD with sort s, then the abstraction of X with variable x and sort xs in the module MOD is equal to the well-sorted abstraction of X with variable x and sort xs in the module MOD.
"for all 'y' of any type 'a', there exists a 'z' such that for all 'x', 'F x z' is equal to 'x equals y'. This implies that it is not the case that there exists a 'w' such that for all 'x', 'F x w' is equal to 'for all 'u', 'F x u' implies that there exists a 'y' such that 'F y u' and it is not the case that there exis...
The dimension of the first 'n' elements of vector 'v' is 'n'.
In the context of strong adversaries in key agreement, the initial running of protocol A0i between parties A and B with key KE and a nonce generated from Rb's secret key, combined with the identity function, refines into the first level step 2 of Rb with A, B, and KE, under the relation R01ia.
If a function 'f' is regular, then applying the 'limit' function to 'f' twice in succession on a set 'I' is the same as applying the 'limit' function to 'f' once on the set 'I'.
If a program P shows that class C has field data types FDTs, then the program P and heap h confirm that the blank state of program P and class C is valid.
, under the assumption that "ed_ou" does not hold for a set X in a topological space, it follows that the identity function I is not equal to the composition of I and a function K, and also that the function K is not equal to the composition of K and the identity function I.
: if a function 'f' is continuous at a point 'x' and the value of function 'f' at 'x' is not equal to zero, then the function that takes 'x' and returns the natural logarithm of 'f(x)' is also continuous at 'x'. Here, 'f(x)' is a real number.
The guard condition being false and any process P is equivalent to the process being stopped.
The multiset extension "mulex1" of a relation "P" between a singleton multiset containing "x" and a singleton multiset containing "y" is equivalent to the relation "P" between "x" and "y".
Folding a function 'f' over a list with an initial value 'a' (wrapped in a singleton probabilistic monad) and a list consisting of an element 'x' followed by a list 'xs', is equivalent to applying the function 'f' to 'a' and 'x', binding the result with a lambda function that continues the fold operation on the rest of...
If relations A and C are right-unique and relation R is right-total, then the relation "rel_gpv''" derived from A, C, and R is also right-unique.
If function 'f' permutes set 'A' and function 'g' is cyclic on set 'C', and if set 'A' and set 'B' have no elements in common and set 'C' is a subset of set 'B', then function 'f' composed with function 'g' is cyclic on set 'C'.
The sum of the function g applied to each edge in the outgoing set from node u is equal to the sum of the function g applied to each pair (u,v) where v is in the set of edges E that are connected to u.
If the Ticket Granting Server (Tgs) sends a message to agent A, which includes an encrypted message (using authK) containing a service key (servK), agent B, a timestamp (Ts), and a service ticket (servTicket), and this message is part of a set of events (evs), and the key authK is not in the analyzed set of what the sp...