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The weakest precondition of a conditional statement (if p then x else y) for a postcondition q is equal to the sum of the atomic diagram of p and the weakest precondition of x for q, all multiplied by the sum of the dual of p and the weakest precondition of y for q.
The class declaration of "Main" in the program "tprg" is well-formed with respect to the "Object" class.
"is_NoMsg" is true for three instances of "NoMsg" of type "nat message_af".
If the server sends a message to agent A, which is encrypted with key K' and contains a number Tk, agent B, key K, and a ticket, and this event is part of the banker's gets events, then the following conditions must be true: 1. The key K' is the shared key of agent A. 2. The key K is not in the range of shared keys. 3...
For any two hyperdual numbers 'a' and 'b', where 'a' and 'b' are of a type that supports inverse, commutative ring, and division ring operations, the result of dividing the negation of 'a' by the negation of 'b' is equal to the result of dividing 'a' by 'b'.
For a lattice L, the closure of L with respect to the union of the set r indexed by I is equal to the union of the closure of L with respect to each individual set r indexed by I.
The lemma named "holomorphic_convex_primitive" is defined with a fixed function 'f' that maps complex numbers to complex numbers. It assumes that a set 'S' is convex, a set 'K' is finite, and the function 'f' is continuous on set 'S'. It also assumes that for any element 'x' in the interior of 'S' excluding 'K', 'f' is...
If the function "strip_all_single_var" applied to a variable B returns None, then the function "strip_all_single_body" applied to the same variable B will return B unchanged.
Theorem 6.2, named "loose_linkable", assumes that a graph Γ is loose. It shows that this graph Γ is also linkable.
The lemma named "conj_simps" states the following simplifications: 1. The conjunction of F and Q is F. 2. The conjunction of P and F is F. 3. The conjunction of T and Q is Q. 4. The conjunction of P and T is P. 5. The conjunction of P and P is P. 6. If P is not equal to T, P is not equal to F, Q is not equal to T, Q i...
If clause C is productive, then it is not the case that the interpretation of clause C satisfies clause C.
Given the assumptions that "Ns" is a chain with respect to the reduction relation "rhd" and the entailment relation "Turnstile", the limit inferior of "Ns" entails "Bot" if and only if the head of "Ns" entails "Bot".
The lemma named "valid_index_nths" assumes that "is" is a valid index of "ds". It shows that the nth elements of "is" in set "A" are also a valid index of the nth elements of "ds" in set "A".
If we apply the 'map_gpv' function with 'id', 'swap_sum', and 'swap_sum' as arguments to the 'right_gpv' of a given 'gpv', it is equivalent to the 'left_gpv' of the same 'gpv'.
If "p" is in the set of Test expressions and "q" is also in the set of Test expressions, then the disjunction (union) of "p" and "q" is also in the set of Test expressions.
If a transition system 'TS' is in the set of Hierarchically Partitioned Transition systems of a statechart 'ST', and 't' is in 'TS', and the target of 't' is in the states of a statechart 'A', and 'A' is in the set of Statechart Assemblies of the Hierarchical Automaton of 'ST', then the source of 't' is in the states o...
If a heap 'h' is preallocated and 'C' is in system exceptions, then there exists a state 'S' such that the heap at the address of the system exception 'C' is equal to the pair (C, S).
If for all natural numbers 'n', the norm of the function 'f' applied to 'n' is less than or equal to 'g' applied to 'n', and if 'g' is summable, then the norm of the sum to infinity of 'f' is less than or equal to the sum to infinity of 'g'. This is under the condition that 'a' is a Banach space, which is a complete no...
The set of solutions 'a' and 'b' is equal to the set of minimal solutions.
The ordinal exponentiation operation is a partial order.
If an empty list is obtained from the variables s, r, e, s', and r', then s is equal to s' and r is equal to r'.
For any fixed elements a and b of the ordered sum type ('a, 'b) OrdSum, the left residual of a and b multiplied by a is equal to a multiplied by the right residual of a and b.
If j is less than k, then the pair formed by applying the function f to a, j times and k times respectively, belongs to the relation leq.
If there is a write operation "SW" from a setter that transforms a heap "h" into a heap "h'", and for all write operations "w" in "SW", if "w" transforms "h" into "h'", then all object pointers preserve the property "get_M_object RObject.nothing" between "h" and "h'", then the kinds of shadow root pointers in "h" are t...
The negation of 'p' multiplied by the result of 'x' being in the context of the negation of 'p' and 'y' is equal to the negation of 'p' multiplied by 'x'.
"A" is locally reflexive in the namespace "NS".
If for any elements 'a', 'b', and 'c' in a set 'A', where 'a' is related to 'b' and 'a' is also related to 'c' through a reflexive transitive closure of a relation 'r', then 'b' is related to 'c' through the join of relation 'r'. If this condition holds, then the relation 'r' is Church-Rosser on set 'A'.
The lemma named "bij_betw_open_intervals" is given certain fixed real numbers a, b, c, and d. It assumes that a is less than b and c is less than d. It shows that there exists a function f that is a bijection between the open interval from a to b and the open interval from c to d.
The lemma named "gf_F_nonneg" establishes that for a fixed real number 'z', if 'z' is greater than or equal to 0 and less than 1, then the function 'gf_F' with parameters 'x', 'y', and 'z' will also be greater than or equal to 0.
, assuming 'rqrm_invar rqrm' and 'rqrm_prop (ls_α δ) rqrm' are true, and 'ls_invar δ' is also true, it shows that the function 'det_wa_wa' applied to 'brc_det_algo rqrm δ' precisely refines 'br'_algo (ls_α δ)' according to 'brc_α'.
For a given restriction 'rest', a process 'p', and two traces 'tr' and 'tr'', applying the function 'trace_rest_aux' to the concatenation of 'tr' and 'tr'' is equivalent to applying 'trace_rest_aux' to 'tr' and then applying 'trace_rest_aux' to the result with 'tr''.
If the function 'h' applied to 'a' equals 'None', then 'h' is less than or equal to 'h' with 'a' mapped to 'x'.
The floor function tends towards positive infinity when the input also tends towards positive infinity.
The lemma "get_parent_is_strongly_scdom_component_safe_step" assumes that the heap "h" is well-formed, the type of "h" is well-formed, and "h" contains known pointers. It also assumes that the strongly connected component of "ptr" in "h" is "sc", and that the parent of "ptr'" in "h" is "parent". It shows that "parent" ...
The lemma "G_relpow_matpow_pos_ge" assumes that the pair (i,j) is in the m-th power of G and that n is greater than or equal to m. It shows that the n-th power of the matrix (A + the identity matrix), at the i-th row and j-th column, is greater than 0.
: 1. The sum of the reciprocals of all real numbers up to a certain number 'x' is always greater than or equal to zero. 2. Eventually, for sufficiently large 'x', the sum of the reciprocals of all real numbers up to 'x' is less than or equal to the natural logarithm of 'x' plus 13/22.
If for all elements 'i', 'f i' is greater than or equal to 0, and 'x' is also greater than or equal to 0, then the function 'f' summed over all 'i' and converted to extended nonnegative real equals 'x' converted to extended nonnegative real if and only if the sum of the function 'f' over all 'i' equals 'x'.
For all 'i', the reach of 'i' in 'q' is a subset of the reach of 'i' in 'r' union with the set containing 'k'. This is equivalent to saying that for all 'x' that is not equal to 'k', for all 'i' that is not in 'above x r', 'i' is not in 'above x q'.
A pair of finite dimensional vector spaces (s1, b1) and (s2, b2) exists if and only if there is a pair of finite dimensional vector spaces on with the universal set (UNIV), the universal set (UNIV), addition (+), subtraction (-), unary minus (uminus), zero (0), s1, b1, addition (+), subtraction (-), unary minus (uminus...
The homogenization of a polynomial 'p' with respect to a variable 'x' is equal to the sum of the monomial multiplication of each homogeneous component 'q' of 'p' by 1 and a single polynomial mapping of 'x' to the difference of the degrees of 'p' and 'q'.
The right-associative fold function "foldri" over a list "l" is a proper iterator. This is indicated by the "proper_it" function, which checks if a given function is a proper iterator. The function is applied to the same "foldri l" twice.
If the following conditions are met: the mutator store instruction's mark object invariant for a mutator m, the mark object invariant, the handshake invariant, the tso lock invariant, and the logical state transition predicate (LSTP) which includes the handshake phase invariant, the valid write-set invariant, the tso s...
Type 'a is a permutation type of type 'x, and "at" which states that type 'x is an atom type. The lemma then shows that the empty permutation of type 'x applied to the set "Xs" equals the set "Xs" itself.
The size of the tree after the minimum element has been deleted is equal to the size of the original tree minus one.
If the closure of a set phi under operator 1 exists, then the boundary of the set phi under operator 1 exists in the context of the basic topological operators.
The lemma shows that if 'a' is fresh for 'x1' and 'a' is fresh for 'x2', then 'a' is fresh for the pair '(x1, x2)'.
Inserting a node into an empty tree (Leaf) results in two possible outcomes. If the balance index of the tree after insertion (calculated by the function "bal_i" with parameters "n+1" and "d+1") is true, then the result is a balanced tree (Bal2) with the inserted node as the root and two empty subtrees. If the balance ...
The node properties of the Security Invariant Non-Interference with default node properties is equal to the node properties of the Network Model.
The lemma named "mono2mono_lzip1" shows that the function "lzip" with the first argument as "xs" and the second argument as a fixed "ys" is monotone. This means that if one list "xs" is less than or equal to another list (in terms of some ordering represented by "⊑"), then the result of zipping these lists with "ys" wi...
The vector of ketplus in ell2 space is equal to the vector ketplus.
For any list "xs" of type 'a, the filter of "xs" using a predicate "P" is equal to the map of the nth element of "xs" over the indices found by applying the predicate "P" to "xs".
The lemma named "ufa_compress_correct" assumes that "ufa_invar l" holds true and "x" is less than the length of "l". It shows that the function "ufa_α" applied to the list "l" with the "x" element replaced by the representative of "x" in "l" is equal to "ufa_α" applied to the list "l".
If x and y are elements of the carrier ODnods, and x is strictly less than y, then x is not equal to y.
If "B t" holds, then there exists a sequence "s" and a number "k" such that the "BuildSeq" of "B", "C", "s", "k", and "t" is true.
If x is less than or equal to y, then the post-precondition of y with respect to -q is less than or equal to the post-precondition of x with respect to -q.
The lemma "transC_mayDiverge" assumes that if there is a transition from state "s" to state "s'" under command "c" resulting in command "c'", and it is possible for command "c'" to diverge in state "s'", then it is also possible for command "c" to diverge in state "s".
The lemma named "formula_strong_induct_aux" is defined with a variable "x". It assumes the following conditions: 1. For any set "xset" and any element "c", if "xset" is a finite set, and for any element "x" in "xset", "P c x" holds, then "P c (Conj xset)" also holds. 2. For any formula and any element "c", if "P c fo...
The lemma named "outer_prod_add_right" is about a fixed vector 'v' of complex numbers. It assumes that the dimension of vector 'w' is equal to the dimension of vector 'x'. It shows that the outer product of 'v' and the sum of 'w' and 'x' is equal to the sum of the outer product of 'v' and 'w' and the outer product of '...
The lemma named "mkArr_restrict_eq" assumes that the function "mkArr" with inputs A, B, and F is an arrow (or morphism in category theory). It shows that restricting the function F to the set A and then applying "mkArr" to A, B, and the restricted function results in the same output as applying "mkArr" to A, B, and F w...
Given a well-formed heap 'h' with known pointers and correct types, if 'h' can get a root node 'root_ptr' from a pointer 'ptr' and 'h' can get a tree order 'to' from 'root_ptr', and if another pointer 'ptr'' is in the set 'to', then 'h' can get the same root node 'root_ptr' from 'ptr''.
The lemma named "mirror2_aux_knights_path" assumes that there is a knight's path on board 'b' following the path 'ps'. It shows that there is also a knight's path on the mirrored version of board 'b' (created by the function 'mirror2_board' with parameter 'n') following the mirrored version of path 'ps' (created by the...
The lemma "msubstleneg" assumes that the polynomial "c" evaluated with variables "vs" is less than zero (assumption "nz") and that the linear expression "Le (CNP 0 a b)" is a linear expression (assumption "l"). It shows that the interpretation function "Ifm" with variables "vs", the list "x#bs", and the result of the f...
If 'x' is an element of the carrier of the fraction field 'R', 'n' is greater than 0, and 'x' raised to the power of 'n' (in the fraction field 'R') is an element of the non-zero elements of the fraction field 'R', then 'x' is also an element of the non-zero elements of the fraction field 'R'.
The lemma named "WHILEIT_pat" in the "def_pat_rules" contains two statements. The first statement says that "WHILEIT$I" is equivalent to unprotected version of "WHILEIT I". The second statement says that "WHILET" is equivalent to a protected constant version of "WHILEIT" where the function always returns true.
The mirror element of 'x' in the interval from 'n' to 'd' is equal to 'n' plus 'n' plus 'd' minus 'x'. 2. The second lemma states that the mirror element of 'x' in the interval up to 'n' is equal to 'n' minus 'x'. 3. The third lemma states that the mirror element of 'x' in the interval defined by 'r', 'mod m', and 'c' ...
If A is a finite set and f is a function from A to the set of n-dimensional vectors, then the dimension of the finite sum of the function f over the set A is n.
If there is an ATC I/O reduction on sets from M1 to M2, then the union of the input language of M1 and the set of all I/O behaviors of M1 is a subset of the union of the input language of M2 and the set of all I/O behaviors of M2.
For any set S of real normed vectors, the interior of the set obtained by adding a vector 'a' to each element of S is equal to the set obtained by adding 'a' to each element of the interior of S.
If the first element of a pair P reduces to M' according to beta reduction, then either there exists a P' such that P reduces to P' according to beta reduction and M' is the first element of P', or there exist elements A and B such that P is the pair of A and B and M' is A.
The term of the bottom gterm of 's' is equal to the term of the bottom gterm of 's'.
The lemma named "isomorphic_src" assumes that "a" is an object in category "C". It shows that in category "D", the source of "F a" is isomorphic to "F a".
If a lazy list "xs" is null, then the set of elements in "xs" is empty.
The lemma named "dom_char" shows that the domain of function f is equal to the domain of function f.
If the cast shadow root pointer is not equal to the document pointer, and if applying the setter to the shadow root pointer in the heap 'h' results in a new heap 'h'', then the getter for the document pointer is preserved between heap 'h' and heap 'h''.
If H holds for the combination of t p and t r leading to q, and if H also holds for the combination of t p and at r leading to q, then H will hold for p leading to either x or y depending on the condition r, and finally leading to q.
For any list of formulas (which are pairs of natural numbers and some type 'b'), it is logically implied that if there exists a case where for all instances a certain predicate 'p' holds for a pair of variables (1 and 0), then for all instances there exists a case where the same predicate 'p' holds for the swapped pair...
The lemma named "graph_memE" makes the following assumptions and shows the following results: Assumptions: 1. The function 'f' maps from the carrier of the m-th power of the p-adic numbers to the carrier of the p-adic numbers. 2. 'x' is an element of the graph of the function 'f' with respect to 'm'. Results: 1. The ...
The first element of the head (first element) of a list containing a single pair (a, b) is equal to a. The second lemma states that the second element of the head (first element) of a list containing a single pair (a, b) is equal to b.
If the lock actions are okay for a given lock 'l', thread 't', and lock actions 'las', then a property 'P' holds. Furthermore, if 'las' can be split into two lists 'xs' and 'ys' such that 'las' is the concatenation of 'xs' and 'Lock' followed by 'ys', the lock actions are okay for 'l', 't', and 'xs', and it is not poss...
In a ring, if a polynomial 'x' belongs to the polynomial ring 'R' and the coefficient of 'x' at any index 'k' greater than or equal to 'n' is zero, then 'x' belongs to the set of polynomials of degree bounded by 'n' over the ring 'R'.
If 'x' is transcendental over 'K' and 'p' is an element of the carrier of 'K[X]', and if evaluating 'p' at 'x' equals zero, then 'p' must be an empty list.
For any real numbers 'x' and 'a' that are greater than zero, the natural logarithm of 'x' plus 'a' minus the natural logarithm of 'x' is less than or equal to 'a' times the sum of the inverse of 'x' and the inverse of 'x' plus 'a', all divided by 2.
If the Justus-Meyer Logic holds, then the second law of contraposition applies to the implication and negation operators.
If a set H is a subset of the carrier of a group G, and if the group G with its carrier replaced by H forms a monoid, then H is a submonoid of G.
The lemma named "ensures" makes the following assumptions and shows the following result: Assumption 1: If the current state satisfies condition P and condition N, then the next state will satisfy either condition P or condition Q. Assumption 2: If the current state satisfies conditions P, N, and A, then the next sta...
For any given numbers 'n' and 'x', the function 'all_croots' behaves as follows: If 'n' equals zero, then it returns an empty list. If 'x' equals zero, then it returns a list containing only zero. If 'x' is an algebraic number, then it returns the genuine roots implemented by the function 'all_croots_part1' with 'n' an...
If "f" is residuated, then there exists a "g" such that "g" and "f" form a conjugation pair.
The lemma of the integer subdegree of a formal Puiseux series is equal to the product of the value of the formal power series and the root order of the formal power series.
If D is not in the set obtained by mapping the function "getbase" over the list "xs", then "Repeats D" is not in the set "xs".
The set of all tree automata rules, for any given states and symbols, is infinite.
The lemma "Pop_correct" assumes the following conditions: 1. The program P is well-formed according to the typing rules wt. 2. The program P can see a method M in class C with the signature Ts -> T, and the method's body is defined by the maximum stack size (mxs), the maximum number of local variables (mxl₀), the list...
, in the context of a given program Γ, the execution of command 'c' in a 'Stuck' state can transition to a 'Skip' command in a 'Stuck' state through zero or more steps.
The lemma named "mgu_wf_trm" assumes that the most general unifier (mgu) of terms s and t is some substitution sigma, and that both terms s and t are well-formed. Under these assumptions, it shows that the substitution sigma applied to any term v results in a well-formed term.
The set of all pairs (y) where there exists an element (x) of a certain type (a type that belongs to a T2 space) such that y is equal to the pair (x, x), is a closed set.
If "x" is not the bottom element, then the supremum (or join) of "x" and "y" is also not the bottom element.
A complex number x is equal to another complex number y if and only if the real part of x is equal to the real part of y and the imaginary part of x is equal to the imaginary part of y.
: If Fr_1 is a property of a set F, then for all elements a, b, and c, the implication of a to b and the implication of c to b is less than or equal to the implication of the disjunction of a and c to b.
The lemma named "roofed_circ_essential" establishes that for a given structure 'Gamma', the set of nodes that are essentially roofed with respect to a set 'S' is equal to the set of nodes that are roofed with respect to 'S'.
The lemma named "sort_key_eq_sort_key" assumes that the multiset of xs is equal to the multiset of ys, and that the function f is injective on the set of xs. It shows that the sorted key of xs using function f is equal to the sorted key of ys using the same function.
The lemma named "Ball_scf_diff" assumes that "R1" and "R2" are preference profiles. It shows that for every element "x" in the difference of the social choice function of "R1" and the social choice function of "R2", property "P" holds for "x" if and only if either "a" is in the social choice function of "R2", or "a" is...