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If a map 'm'' is not empty, then for a hash map iterator 'it' over a hash map 'm' with a hash table 'ht' and a map 'm'', the operation 'hm_it_next it' results in a pair ((k,v),it') and a state where the iterator 'it' is over a hash map 'm' with a hash table 'ht' and a map 'm'' restricted to the complement of the set co...
The excess of 'f' prime at 'x' is equal to the following conditions: - If 'x' equals 'u', then it is the excess of 'f' at 'u' minus 'Delta'. - If 'x' equals 'v', then it is the excess of 'f' at 'v' plus 'Delta'. - Otherwise, it is simply the excess of 'f' at 'x'.
The Pochhammer symbol of a number 'z' and the successor of a number 'n' is equal to the sum of 'z' and the natural number representation of 'n', multiplied by the Pochhammer symbol of 'z' and 'n'.
If a value 'x' is in the set 'P' and 'x' is less than 'n', then the maximum value of the filtered set 'P' (where the filter condition is that elements are less than 'n') is also less than 'n'.
If "DenyAllFromTo b a" is an element in the set p, then a is an element in the set of the network list of p.
The cardinality (or size) of the set I divided by zero equals zero if the set I is empty; otherwise, it equals one.
If polynomials 'p' and 'q' are elements of the polynomial ring 'R', then if 'x' is a root of the product of 'p' and 'q' in the polynomial ring 'R', it implies that 'x' is either a root of 'p' or a root of 'q'.
The edges of a type I two-cube are valid paths. It assumes that 'twoC' is a type I two-cube and that the pair (k, γ) is in the boundary of 'twoC'. It then shows that γ is a valid path.
Flushing a variable 'x' is equivalent to first applying the 'flush' function to 'x' and then applying the identity function.
A real number 'x' is an algebraic integer if and only if its real representation is also an algebraic integer. This is true in a field with characteristic 0 and where 1 is a real algebraic number.
A property P relative to the universal set applied to a set S is equivalent to the property P applied to the set S.
If the supremum of z and the product of y and x is less than or equal to y, then the product of z and the star of x is less than or equal to y.
The function "kad" applied to "f" under the transformation "\<R>" is equal to the "ad_rel" function applied to the transformation of "f" under "\<R>".
The lemma "lActive_greater_active_all" assumes that for all 'n'' greater than 'n', 'c' is not parallel at 't n''. It shows that there does not exist an 'n' greater than the pair of 'c' and 't' such that 'c' is parallel at 't n'.
The lemma named "gta_langE" is an elimination rule. It assumes that a tree "t" is in the language of a generalized tree automaton "Q" with respect to some alphabet "A". From this assumption, it obtains a state "q" such that "q" is in "Q" and "q" is in the derivative of the tree automaton "A" with respect to the term of...
If a list 'xs' is a suffix of another list 'ys', then the set of all elements in 'xs' excluding the first element is a subset of the set of all elements in 'ys' excluding the first element.
The first and second elements of the fixed point of the function "ValD_copy_rec" when applied to the bottom element are both the bottom element.
The carrier of the zeroth power of the p-adic number field is the set containing the empty list.
: Assuming "z" is not in the path image of the linepath from "a" to "b", the complex index of the path from "a" to "b" at "z" is defined as follows: First, calculate "c1" as the real part of "a" minus the real part of "z", "c2" as the real part of "b" minus the real part of "z", and "c3" as the imaginary part of "a" ti...
The intersection operation (\<inter>) is respected by the relation R on sets of values (rel_vset R), meaning that the intersection operation is compatible with the equivalence relation R.
The lemma named "NPath_char" shows that "T" is a normal path in "P" if and only if "T" is an identity in "P".
"a" is upright if and only if for all "t", if the "zcount" of "a" at "t" is greater than 0, then "a" is strongly supported at "t".
The function "F" totalized is always in set "A" if and only if the function "F" itself is always in set "A".
The key relation function, when applied to any two elements and always returns true, is equivalent to a function that takes any two elements and always returns true.
The lemma named "subseq_Cons2'" assumes that if a sequence starting with 'x' followed by 'xs' is a subsequence of another sequence starting with the same 'x' followed by 'ys', then 'xs' is a subsequence of 'ys'.
If there is a transition from the state "s" with the command "SKIP" to the state "t" with the command "c" through zero or more steps, then the command "c" must be "SKIP".
The lemma named "mono2mono_ldropWhile" shows that the function "ldropWhile P" is monotone with respect to the partial order relation "sqsubseteq". This lemma is used to introduce continuity and is also used for simplification.
The lemma named "comp_red_basis_adds" assumes that a polynomial 'p' is an element of the set 'xs' and that 'p' is not equal to zero. It then obtains another polynomial 'q' such that 'q' is an element of the set resulting from the computation of the reduction basis of 'xs' and the leading term of 'q' adds to the leading...
The paths of the call-control tree (ccTTree) for a set S and a graph G are equivalent to the valid lists for the same set S and graph G.
"if the composition of x and y exists, then both x and y exist."
If 'a' is less than or equal to the sum of 'a' and 'b', then for all 'n', the sum of the 'n'-th bit of 'a' and the 'n'-th bit of 'b' is less than or equal to 1. This is represented as "?P implies ?Q".
The lemma assumes that the Java program P is well-formed. It shows two things: 1. If a thread t executes an expression e in state s and transitions to expression e' in state s' with thread action ta, and if a new thread t'' with expression e'' and state x'' is in the set of thread actions of t, then the expression loc...
The logical 'or' operation between 'null' and 'true' always equals 'true'.
A list "llist_of xs" is equal to a list constructed by "LCons y ys" if and only if there exists a list "xs'" such that "xs" is equal to the list formed by appending "y" to "xs'" and "ys" is equal to the list "llist_of xs'".
The product of S' and Z equals the bottom element.
: If n is greater than or equal to m, then lifting the term 't' twice, first by 'n' and then by 'm', is equivalent to lifting the term 't' first by 'm' and then by 'n+1'. Similarly, if n is greater than or equal to m, then lifting the command 'c' twice, first by 'n' and then by 'm', is equivalent to lifting the comm...
The lemma named "row_mat_plus" assumes that "m1" and "m2" are matrices with "nr" rows and "nc" columns. It also assumes that "i" is less than the number of rows "nr". It shows that the "i"-th row of the matrix resulting from the addition of "m1" and "m2" (using the addition operation "pl") is equal to the vector result...
The sum of the replacements in set A with the function "Singleton composed with f" is equal to the replacements in set A with the function f.
If a heap is valid, an index is valid for the heap, and the priority of a value is less than or equal to the priority of the heap at the index, then the operation of decreasing the key at the index in the heap will maintain the validity of the heap.
For a graph that is not connected, the eccentricity of any vertex in the graph is infinity.
The length of the result of the addition operation on three variables a, b, and c is always 7.
If a function 'f' is integrable over a set 'S', then the negative of the function 'f' is also integrable over the same set 'S'.
If a natural number "t" is greater than zero, then the integer division of "t" and "n" added to "t", all divided by 2, is equal to the floor function of the division of "t" and "n" added to "t", all divided by 2, converted to a natural number.
The power of p to the ll is not equal to 0.
The substitution of "grounding_\<rho>" in "s" results in a ground term.
In the context of a commutative ring, the projection equality is reflexive, meaning that a point is equal to itself.
The lemma named "pt_eqvt_fun1a" is defined with a fixed function 'f' that maps from type 'a' to type 'b'. It assumes that 'a' is a permutation type with respect to 'x', 'b' is also a permutation type with respect to 'x', 'x' is an atom type, and the support of function 'f' is an empty set of type 'x'. Under these assum...
For all x, if x is greater than a and less than or equal to b, then the polynomial p evaluated at x is greater than 0. This is equivalent to either a being greater than or equal to b, or the polynomial p not being 0 and the polynomial p evaluated at b being greater than 0 and the count of roots of the polynomial p betw...
The universal intersection of an empty set and any set x is an empty set.
Given a binary code, if the following conditions are met: 1. The variable 'us' is either empty or its last element is not 'x'. 2. The sequence of 'n' 'x's is not a prefix of 'us'. 3. 'us' is a list of elements 'x' and 'y'. Then, the result of gluing 'x' to 'us' is a list of sequences where each sequence is 'k' 'x's f...
If the side precondition is that 's' is in the right part of a sum type, and 's' is related to 's'' through a sum relation of 'Ra' and 'Rb', then the projection of 's'' to the right part of the sum type is related to the operation of projecting 's' to the right part of the sum type through the relation 'Rb'.
If for any set X, the condition that set A is a subset of X and set B is a subset of the complement of X implies that X is not computable, then sets A and B are recursively inseparable.
If "k" is not equal to zero, "M" is equal to "k" times the matrix (a, b, conjugate of b, conjugate of a), "a" is not equal to zero, and the determinant of the matrix (a, b, conjugate of b, conjugate of a) is not equal to zero, then there exists a "k'", a phase "phi", and an "a'" such that "k'" is not equal to zero, "a'...
If function F follows function g, then function F is increasing with respect to function g.
: Given the following assumptions: 1. "R" is a well-founded relation (WF). 2. The precondition implies "p" (I0). 3. For any state "st", if "b" and "p" are true and "e" equals "st", then after executing "Q", "p" remains true and the pair "(e, st)" is in the relation "R" (induct_step). 4. If "b" is false and "p" is true...
The derivative of the formal Laurent series obtained from a formal power series 'f' is equal to the formal Laurent series obtained from the derivative of the formal power series 'f'.
If the delta function is applied to 'b' and 'a', and then 'a' is applied to the result, it will equal 'b'.
The lemma named "pflat_len_Stars_simps" assumes that "n" is less than the length of the list "vs". It shows that the length of the flattened list of "Stars vs" at position "n#p" is equal to the length of the flattened list of "vs" at position "n".
The function "sign_extend" is equal to the function "signed_take_bit".
If the length of the list "ps" is greater than 1 and the pair (c0, c1) is equal to the result of the function "closest_pair_bf" applied to "ps", then c0 is an element of the set "ps".
If the precondition P is a subset of the postcondition Q, then under the total correctness logic for Hoare triples, the Skip command ensures the transition from P to Q, with A as the set of assertions that may be violated.
The lemma named "is_Prog_sb_simps" states the following conditions: - The function "is_Prog_sb" returns false when applied to a "Read_sb" operation with parameters "volatile", "a", "t", "v". - The function "is_Prog_sb" returns false when applied to a "Write_sb" operation with parameters "volatile", "a", "sop", "v", "A...
The lemma "isPrefixPrefixToLevelLowerLevel" assumes that a certain "level" is less than "level'". It shows that the prefix to the "level" in a certain model "M" is a prefix to the "level'" in the same model "M".
The function 'filtermap' applied to an input 'f' is equivalent to the composition of the function 'Abs_filter' and the function 'raw_filtermap' applied to 'f', which is then composed with the function 'Rep_filter'.
If the function "find_path_f" with parameters P, ctx, and "cl_fqn fqn" returns some path, and the function "find_cld_f" with parameters P, ctx, and fqn returns some "ctxcld", then "ctxcld" is an element of the set "path".
The lemma named "domA_fv_pure" establishes that for a list "Gamma" consisting of pairs of elements, where the first element is of a base type 'a and the second element is of a pure type 'b, the set of free variables in "Gamma" is equal to the domain of "Gamma".
The domain of the composition of two multirelations R and S, each composed with the identity multirelation, is equal to the product of the domains of R and S, each composed with the identity multirelation.
The lemma named "decomposeLS_fragmentsLS" assumes that the tuple "(c, ictxt, fctxt)" is an element of the decomposition of LS 's'. It then shows that the tuple "(c, loc_compC c (fctxt c @ tl (cPGM s)))" is an element of the fragments of LS 's'.
The lemma named "ccw_half_segments_of_aform" assumes two conditions. The first condition is that the pair (xi, xj) is an element of the set produced by the function "half_segments_of_aform" applied to X. The second condition is that the pair (c, d) is also an element of the same set. Given these assumptions, the lemma ...
The value of function "u" at "y" is not equal to infinity.
For any propositions P and Q, and any variable x, the equivalence of P and Q after an alphabet extension with x is the same as the equivalence of P after an alphabet extension with x and Q after an alphabet extension with x.
For all finite functions, the property P holds if and only if the property P holds for all elements in the function.
The lemma named "funs_term_type_iff'" assumes that "M" is a well-formed term set and "f" is a function with arity greater than 0. It shows that "f" is in the union of the function terms of the typing context of "M" if and only if "f" is in the union of the function terms of "M" or there exists a variable "x" in the fre...
The inverse of the association is equal to the association prime.
If "DenyAll" is an element in the set converted from the policy list "p", then the result of the function "normalize_manual_orderPr" with inputs "p" and "l" cannot be an empty list.
Applying the function "f" to the pure stochastic process monad of "x" is equivalent to the pure stochastic process monad of the function "f" applied to "x".
If "LeftDerivationFix a i D j b" is assumed and "is_nonterminal (a ! i)" is also assumed, then there exists some U, a1, a2, b1 such that a splits at i into a1, U, a2 and b splits at j into b1, U, a2, and a LeftDerivation from a1 to b1 through D exists.
In a linear order, a list is sorted with respect to the first element of its tuples if and only if the list obtained by mapping the first element of each tuple in the original list is sorted.
If the auxiliary function "interfree_aux" is true when applied to the parameters "Any", "q", and "Some a", then it is also true when applied to the parameters "Any", "q", and "Some (AnnWhile r b i a)".
The lemma named "restrict_restrict_space" assumes that the intersection of set A with the space of measure M is in the sets of M, and the intersection of set B with the space of measure M is also in the sets of M. It then shows that the restriction of the restriction of the measure space M to set A and then to set B is...
The half size of a term 't' multiplied by 2 is equal to the size of the term 't' plus 1.
The lemma named "dim_vec_2_cases" assumes that the dimension of vector x is 2. It shows that either both elements of the vector are true, or the first element is false and the second is true, or the first element is true and the second is false, or both elements are false.
The probability mass function (spmf) of a particular event 'x' in a probability distribution 'p' is equal to zero if and only if 'x' is not in the set of events defined by the probability distribution 'p'.
If a model "A" satisfies the Tseytin transformation of a formula "F", then the model "A" also satisfies the formula "F".
For an integer k, k is less than or equal to the result of the operation "signed_take_bit" applied to n and k if and only if k is less than 2 to the power of n.
If a program P and environment E type-check an expression 'e' as class D, and if C is a class in program P, and if the static cast from class D to class C is well-typed in program P, then the program P and environment E will type-check the static cast of expression 'e' to class C.
Lemma inFr_self_hsubst: Assuming that the root of tree tr0 is an element of the set ns, we want to show that the element t is in the frontier of the set ns with tree tr0 substituted by its hole (hsubst tr0) if and only if either t is in the left inverse image of the set of continuous functions of tree tr0, or t is in t...
The lemma named "const" assumes a condition "C1" where "X" is constant. It then shows that the function "f" applied to "X" is also constant.
The third step of the m1 protocol in the Kerberos security model, involving parties Rs, A, and B, and parameters Kab, Na, and Ts, refines the initial model's step a0ii, under the relation R_a0iim1_ir. This is indicated by the identity function (Id) mapping between the two steps.
The lemma shows that there is an equivalence pseudofunctor between two bicategories.
The function "an" of 1 equals 1.
If "Rb" is not in the domain of "runz", then updating "runz" by mapping "Rb" to the tuple (Resp, [A, B], []) does not change the result of the function "ri_runs2sigs" applied to "runz".
If a filter F exists, then for all propositions a and b, the implication of a leading to either b or not b is logically equivalent to the absolute truth.
The combined control flow tree of two sets S and G, and S' and G', is less than or equal to the control flow tree of the union of S and S', and the union of G and G'.
The lemma named "eqButUID_createFriend12_friendIDs_eq" assumes the following conditions: 1. The states 's' and 's1' are equal except for the User ID. 2. The states 's' and 's1' are reachable. 3. Neither 's' nor 's1' are friends with '12'. Under these assumptions, it shows that the set of friend IDs resulting from cre...
The lemma named "image_mset_Diff" assumes that multiset B is a subset of multiset A. It shows that the image of the function f applied to the difference of A and B is equal to the difference of the image of the function f applied to A and the image of the function f applied to B.
If there is a message from Tgs to A, which is encrypted with authK and contains the key servK, the agent B, a timestamp Ts, and a service ticket, in the set of events evs that belong to the Kerberos IV protocol, then there exists a timestamp Ta such that a message from Kas to A, encrypted with A's shared key, containin...
The half segments of an affine form X form a convex polychain.
If the length of a list 'as' is 'n' and for all 'i' less than 'n', the 'i'-th element of 'as' is in the carrier 'R', then 'as' is in the carrier of the 'n'-th power of 'R'.
The cosine of the arctangent of x is not equal to zero.