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For any two states 's' and 's'', if a well-formed Java program 'P' transitions from 'e' in state 's' to 'e'' in state 's'', and 'e' is in an initial configuration with the shape of 's', and 'e' and 's' are type-checked under 'P' and environment 'E', then 's'' is also type-checked under 'P' and 'E'.
The list of free variables in a first-order formula is sorted and distinct.
The function "choose_arc" applied to any input "x" is regular.
If the condition "rel_S_neg_distr_cond'" holds true for these fixed elements, then the condition "rel_S_neg_distr_cond" also holds true for these elements.
The function "close_eq" applied to the state "s" and the equality "Eq a b" is true if and only if "a" is equal to "b".
The lemma valid_UV_lists_not_empty_r: Assuming that "xs" is a finite set, and the set of elements in the combined list of "U" and "V" is a subset of "xs", and that the combined list of "U" and "V" contains distinct elements. Also, assuming that the first element of "U" is "r" or "r" is not in the set of "U" or "V", and...
If all networks are distinct in a given policy "p", then all networks will also be distinct in the separated policy "p".
If "x" is greater than 0, then the function "ln_upper_5 x - ln x" has a field derivative at "x" which is equal to "diff_delta_ln_upper_5 x".
The lemma named "cauchy_not_vanishes" assumes that "X" is a Cauchy sequence and that "X" does not vanish. It shows that there exists a positive number "b" and a number "k" such that for all numbers "n" greater than or equal to "k", "b" is less than the absolute value of the nth term of sequence "X".
The reflexive transitive closure of a relation "r" from "a" to "a'" over an empty list is true if and only if "a" is equal to "a'".
"The proposition that 'x is necessary' is true."
If a formula "neg \<phi>" is provable from a set of formulas F, and the formula "\<phi>" is also provable from F, and if F is a subset of the set of all formulas, and F is finite, and both "\<phi>" and "\<psi>" are in the set of all formulas, then the formula "fls" is provable from F.
The lemma named "inv_all_diff_ev_uppertD" assumes that the matrix A satisfies the conditions "inv_all diff_ev A" and "inv_all uppert A". It also assumes that "i" and "j" are less than "n", and that the element at position (i, i) in matrix A is not equal to the element at position (j, j) in the same matrix. Under these ...
The lemma is defined with fixed integers m and e. We define f as the float of (m times 2 to the power of e). We assume that m is not divisible by 2. The lemma shows that the mantissa of f is equal to m and if m is not equal to 0, then the exponent of f is equal to e.
If the variables "lbls", "pri", "pos", "steps", "nxt", "onxt", and "iB" are equal to their corresponding primed versions, and for every element "x" in the set "steps", the function "g" applied to "x" and "d" is equal to the function "g'" applied to "x" and "d", then the function "step_fold" applied to "g", "steps", "lb...
The lemma named "float_dense_in_real" is about a fixed real number 'x'. It assumes that 'x' is less than 'y'. It shows that there exists a floating point number 'r' such that 'x' is less than 'r' and 'r' is less than 'y'.
The lemma "mat_of_row_mult_append_rows" assumes that "y1" is a vector in the carrier space with dimension "nr1", "y2" is a vector in the carrier space with dimension "nr2", "A1" is a matrix in the carrier space with dimensions "nr1" by "nc", and "A2" is a matrix in the carrier space with dimensions "nr2" by "nc". It sh...
If set A is not empty, and it is a subset of set B, and set B is finite, then the minimum value of set B is less than or equal to the minimum value of set A.
The lemma named "lex_trans" assumes that if "x" is lexicographically less than "y" and "y" is lexicographically less than "z", then it shows that "x" is lexicographically less than "z".
The lemma "insert_before_heap_is_wellformed_preserved" assumes that: - The heap 'h' is well-formed, - The operation 'insert_before' is performed on the heap 'h' with the pointer 'ptr', the node 'node', and the child 'child', resulting in a new heap 'h'', - All pointers in the heap 'h' are known, and - The heap 'h' is ...
The square-free part of any number 'n' is itself square-free.
A property P holds for the floor of a number t if and only if for all integers i, if t is greater than or equal to i and less than i plus 1, then the property P holds for i.
The lemma named "pruned_inv" assumes that a graph "g" is pruned. It shows that the graph "g" is also pruned in the context of a step.
If a trace 't' belongs to the set of traces 'jpTraces' of a joint protocol 'jp' and the length of this trace 't' is the successor of 'n', then there exists another trace 't'' and a state 's' such that 't' is the result of appending 's' to 't'', 't'' is also a trace in 'jpTraces' of 'jp', the length of 't'' is 'n', and ...
If we have a right total function G, and if G preserves the equality of several pairs of functions (specifically, \<alpha>e1 and \<alpha>e2, \<alpha>n1 and \<alpha>n2, invar1 and invar2, inEdges1 and inEdges2, Entry1 and Entry2, defs1 and defs2, uses1 and uses2), then if a SSA_CFG.CFG (Static Single Assignment Control ...
The lemma named "Abs1_eq" establishes the following: It fixes two variables x and y of a type 'a which is a finite set (fs). It then shows three conditions: 1. The set containing the atom 'a' in 'x' is equal to the set containing the atom 'a' in 'y' if and only if 'x' is equal to 'y'. 2. The restriction of 'x' to th...
If there are relations between L1 and L1', L2 and L2', Co1 and Co1', Co2 and Co2', Contra1' and Contra1, Contra2' and Contra2, then there is also a relation between the structure S with these parameters and the structure S' with the primed parameters. This relation is preserved when applying the function map_S to both ...
The function "to_fun" applied to the zero element of the set P is equal to the zero element of the set of functions from R.
The negation of z is an element of the set of all integers, including infinity.
If "DenyAll" is an element in the set converted from the policy list "p", and all elements in list "l" are in the policy list "p", and all networks in the policy list "p" are distinct, then the function "C" applied to the firewall policy list obtained by sorting the list after removing shadow rules, removing duplicate ...
The implementation of Prim's algorithm is as follows: First, initialize the priority queue (Q) by setting the root node (r) to 0 and the parent map (π) to an empty map. Then, enter the main loop which continues until the priority queue is empty. In each iteration of the main loop: - Extract the node (u) with the min...
If the compatibility equivalence "CE" holds between the single-rooted steps "srstep" of two relations "R" and "S" under a set "F", then for any pair of elements "s" and "t", if the pair is in the single-rooted steps with root step "R" or "S" under "F", then the pair is also in the reflexive and transitive closure of th...
The lemma "is_binqueue_add_Some" assumes that "xs" is a binomial queue of length "l", "t" is a binary tree of length "l", and "t" is a heap. It shows that adding "t" to "xs" results in a binomial queue of length "l".
"infinity is not less than any number x", which is false.
The lemma states the distributive law for integer multiplication over integer addition. It assumes that x, y, and z are integers. It shows that the product of x and the sum of y and z is equal to the sum of the product of x and y and the product of x and z.
The set "JF1set" of an element "a" is equal to the union of the set "F1set1" of the "dtor1" of "a" and the union of the set "JF1set" of all elements "b" in the set "F1set2" of the "dtor1" of "a", and the union of the set "JF2set" of all elements "b" in the set "F1set3" of the "dtor1" of "a".
If a cryptographic message containing the agent S, a number AO, a key K, agent R, and a hash value is used in an event sequence, and the key K is not in the analyzed spy events, and the event sequence is part of certified mail, then there exists a message, a context, and a nonce such that the hash value equals the hash...
: - The prime of a constant c is equal to the constant c itself. - For any function f, the prime of f applied to x is equal to f applied to the prime of x. - For any function f, the prime of f applied to x and y is equal to f applied to the prime of x and the prime of y. - For any function f, the prime of f applied to ...
The supremum of an empty set is zero in the context of complex conjugate subspace.
The transpose of a column vector "v" is equal to the row vector "v".
The lemma named "weakCongReflexive" establishes that for any given Psi (represented as \<Psi>) of type 'b and any given P (represented as P) of type psi (represented as "('a, 'b, 'c) psi"), P is weakly congruent to P.
The lemma named "sumAssoc" establishes that for any three processes P, Q, and R in the pi calculus, the association of their sum does not affect their equivalence. That is, summing P and Q first and then summing the result with R is equivalent to summing Q and R first and then summing the result with P.
The lemma l4_14R1 assumes that point B is between point A and C, and the distance between point A and B is congruent to the distance between point A' and B'. It shows that there exists a point C' such that the distances between A, B, and C are congruent to the distances between A', B', and C' respectively.
If the function sig_red_single is applied to the single regular sequence sing_reg, the top_tail, and the polynomials p, q, f, and t, then the single regular sequence of the sum of t and the leading term of f is the leading term of p.
The lemma in the context of a topological space shows that the identity function on a set S is a homeomorphism, given that S is open.
The lemma named "wt_term_code" assumes that the conditions for "exe_osig" are met for the sorts in the signature "Sigma". It shows that the term is okay in the translated signature "Sigma" if and only if the term is okay in the original "Sigma".
The initial state of the union of F and G is the intersection of the initial state of F and the initial state of G.
The comparison of 'b' to 'a' is equivalent if and only if the comparison of 'a' to 'b' is equivalent.
For any real number 'x' that is greater than or equal to zero, the square root of 'x' multiplied by itself equals 'x'.
If the set of states 's' where 'I s' holds is less than or equal to the forward box differential equation 'x' equals 'f' and a condition that is always true on 'U S' at time 't0', then the set of states 's' where 'I s' holds is also less than or equal to the forward box differential equation 'x' equals 'f' and any cond...
The length of the ladder after shifting by N is equal to the original length of the ladder.
The list resulting from applying the quicksort algorithm to any list 'xs' is sorted.
Scaling a scaled probability mass function (SPMF) by a factor of 'r' is equivalent to scaling the original SPMF by the product of 'r' and the maximum of 0 and the minimum of the inverse of the weight of the SPMF and 'r''. In other words, scaling twice in succession is the same as scaling once with the combined scale fa...
Composing a list "xs" with a list "ys" after deleting an element "k" from "xs" is equivalent to deleting "k" from the result of composing "xs" with "ys".
The error function complement (erfc) is holomorphic (complex differentiable) on the set A.
The lemma "replace_subterm_is_a_function" demonstrates that for any terms 't', 'u', and 'v', if 'u' is a subterm of 't' at position 'p', then there exists a term 's' such that 'v' can replace 'u' in 't' at position 'p' to get 's'.
The lemma "justified_write_seen_hb_read_committed" assumes that a program P with execution E and write sequence ws is justified by a justification J. It also assumes that a read action r is in the set of read actions of the justifying execution of J at step n, and that r is in the set of committed actions at step n of ...
The elements of x are equal to the empty set if and only if x is equal to zero.
The previous interval "iprev" of a time "t" in the range "[r, mod m]" is equal to "t - m" if "t" is in the range "[r, mod m]" and "r" is less than "t", otherwise it is equal to "t".
The lemma named "RDirProd_list_truncate" demonstrates that the truncation of the direct product of a list of rings "Rs" in a monoid is equal to the direct product of the list of rings "Rs".
If the lookup of polynomial 'p' at 'v' equals zero and the higher order part of 'p' at 'v' equals zero, and if polynomial 'q' is less than or equal to 'p' in the polynomial order, then the lookup of 'q' at 'v' equals zero and the higher order part of 'q' at 'v' equals zero.
The lemma named "terminal_char1" shows that a terminal object 't' is equivalent to an identity object 't' that has exactly one element 'x' in its set.
The lemma named "piecewise_C1_differentiable_D2" is about a function g2 that maps real numbers to a real normed field. The lemma assumes that the function formed by concatenating g1 and g2 is piecewise C1 differentiable on the interval from 0 to 1, and that the end of the path defined by g1 is the same as the start of ...
The successor of an extended natural number 'n' is equal to the successor of 'n' in the extended natural number system.
If a function 'f' is continuous at every point 'x' in a set 's', then 'f' is continuous on the set 's'.
The lemma "class_add_is_relevant_class_rev" assumes the following: 1. "irc": Class C0 is relevant in the context of the program P with the addition of the class C and its declaration cdec at index i. 2. "ncp": For any class declaration cd and class D' in the set of program P, it is not the case that the first element...
An atom of type "tvar" denoted as "x" is fresh in a term denoted as "t".
If a function 'f' is monotonic, and 'x' is less than or equal to the value of function 'f' at 'x', then 'x' is also less than or equal to the value of the iterated function 'f' at 'x'.
The lemma named "unfoldI" is defined with two fixed variables, P and Q, both of type pi. It assumes that for all substitutions 's', P substituted with 's' is weakly bisimilar to Q substituted with 's', and vice versa. If this assumption holds, it shows that P is strongly bisimilar to Q.
The lemma named "translate_collision_range" assumes that the initial speed is less than or equal to the maximum speed, and that the maximum speed is less than the stopping speed of the other vehicle. It shows that a collision occurring at a delayed safe distance is equivalent to a collision reaction happening after a c...
The lemma named "dverts_merge_sub" assumes that for all 't' in the first element of the image of the function 'fset' applied to 'sucs t0', the maximum degree of 't' is less than or equal to 1. It shows that the set of directed vertices of 'merge t0' is a subset of the set of directed vertices of 't0'.
The Pochhammer symbol of the negative of a natural number 'n' (in a domain that is an integral domain and has characteristic 0) equals zero if and only if 'k' is greater than 'n'.
If the nth element of the list 'qs' is not in the initial set, then the function t_BIT of n is less than or equal to the successor of the size of the initial set.
The omega power of x joined with the star power of x after an action y is equal to an action y joined with the omega power of x joined with the star power of x after the product of the normal form of y and the action of x after y.
An error "e" does not read from any location in the heaps "h" and "h'".
If a Turing machine program "tp" is equivalent to an Abacus program "ap", and if the Abacus program "ap" loops on a set of natural numbers "ns", then the Turing machine configuration (after applying the "mopup" function to the Turing machine program "tp" and a register "r") also loops on the same set of natural numbers...
Appending an empty namespace to any namespace 'e' results in 'e'. Similarly, appending any namespace 'e' to an empty namespace also results in 'e'.
If "x" is an element of set "A", then the value of "x" in the restriction of map "m" to set "A" is the same as the value of "x" in map "m".
If the environment "e" is okay, then the domain of the environment "e" is finite.
The lemma named "fresh_star_finite_insert" is defined with a fixed set S of type 'a, where 'a is a finite set (fs). It shows that if S is finite, then 'a is fresh for the insertion of 'x' into set S if and only if 'a' is fresh for 'x' and 'a' is fresh for set S.
The lemma named "feasible_fixes_bot" assumes that "t" is feasible. It shows that applying "t" to a function that always returns 0, results in a function that always returns 0.
The integral of a formal Laurent series obtained from a formal power series is equal to the formal Laurent series obtained from the integral of the original formal power series.
If the well-formed dual of 'n' and 'r' holds, then the language dual of the normalized dual of 'r' is equal to the language dual of 'n' and 'r'.
The lemma named "enum_rec_disjointify_disjoint" assumes that "k" is less than "m". It shows that the intersection of the function "enum_rec_disjointify" applied to "f" and "m" and the same function applied to "f" and "k" is an empty set. This means that the two sets are disjoint.
The S_transform function is equivariant, meaning that permuting the input 'x' with 'p' and then applying the S_transform function is the same as first applying the S_transform function to 'x' and then permuting the result with 'p'.
The residuated product of x and y, right-residuated by 1, is equal to the product of y right-residuated by 1 and x right-residuated by 1.
The lemma named "normalize_positive_src_ports_normalized_n_primitive" assumes that a match "m" is in normalized negation normal form (nnf), and that "m" does not have a negated source port. It then shows that for all "m'" in the set resulting from normalizing the positive source ports of "m", "m'" has normalized source...
The lemma named "normal_set_mult_subgroup" shows that the set multiplication of H and S is a subgroup of G.
The lemma named "find_non0" assumes that "sel_fun" is a determinant selection function, and that the result of the function "find_non0" with inputs "l" and "A" is "m". It shows that the element at the position (m,l) in matrix A is not zero, "l" is less than "m", and "m" is less than the number of rows in matrix A.
The function "cogen" is a parameterized function. It maps from a function space (from A to a predicate relation B) to another function space (from a list relation of A to a predicate relation B).
The set of elements 'xs' is a subset of the set resulting from the depth-first search (dfs) on graph 'g' with 'xs' and 'ys'.
If the initial message in the global exchange configuration "c0" is valid (as per "cri.InvMsgInGlob"), if the initial configuration "c0" is safe (as per "inv_init_imp_prop_safe"), if the global timestamps in the initial configuration "c0" are equal (as per "InvGlobPointstampsEq"), and if the next received update from "...
The post-measurement of the first Bell state at the 0th position with 2 as the measurement outcome is equal to the negative of the unit vector at the 2nd position in a 4-dimensional space.
For all 'b' in the backflows of the filtered new flows state of 'T', 'b' is not in the fixed flows of 'T'.
If history 'h' is less than or equal to history 'h'', if 'F' is equal to 'F'', and if 'M' is a subset of 'M'', then the acquisition history update of 'h', 'F', and 'M' is less than or equal to the acquisition history update of 'h'', 'F'', and 'M''.
Zero divided by any ordinal number is zero.
For a set S that belongs to the Lebesgue measure sets, S is negligible if and only if the exterior measure of S in the Lebesgue measure is equal to zero.
The lemma named "mult_smult_assoc_mat" assumes that matrix A is a member of the set of matrices with "nr" rows and "n" columns, and matrix B is a member of the set of matrices with "n" rows and "nc" columns. It shows that the scalar multiplication of a scalar "k" and matrix A, followed by the matrix multiplication with...
If a regular expression 'r' is well-formed for the successor of 'n', then the prefix normal form of projection of 'r' is well-formed for 'n'.
The lemma "Max_lt_imp_lt_mset" assumes that the set N is not empty (N ≠ {#}) and that the maximum element of set M is less than the maximum element of set N (Max_mset M < Max_mset N). Under these assumptions, it shows that set M is less than set N (M < N).