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If "C" is less than or equal to "D", and for all "E" in the set of super ground meta-context arguments of "C" and "D", "E" equals "GMHole" or the number of ground holes in "E" is zero, then the set of possible ground holes in "D" is a subset of the set of possible ground holes in "C".
The lemma named "lookup_pair_sort_oalist'" assumes that the first elements of each pair in the list 'xs' are distinct. It shows that the function 'lookup_pair' applied to the sorted version of 'xs' is equal to the function 'lookup_dflt' applied to 'xs'.
A set X is a multiset set if and only if for every element x in X, the count of x in X is equal to 1.
The opposite of a semi-functor maps from the opposite of a semi-category to the opposite of another semi-category.
Lemma lborelD_Collect (measurable (raw)): If the set of all x in the space of the Borel measure such that P(x) holds is in the sets of the Borel measure, then the set of all x in the space of the Lebesgue measure such that P(x) holds is in the sets of the Lebesgue measure.
The function 'rand_list' in both the abstract and the concrete program is related by a function mapping from natural numbers to a list of natural numbers.
: Assuming that "u" is in the unit disc and "u" is not zero. If for any "phi" and "u", when "u" is in the unit disc and "u" is not zero, the property "P" holds for the point "u" under a Moebius rotation by "phi", then the property "P" holds for "u". Also, for any real number "x", if "x" is positive and less than 1, ...
If the instruction at a given function descriptor, location, and program counter is some instruction, then there exists a sequence of instructions such that the mapping of the function descriptor's body at the location is equal to this sequence of instructions, the program counter is less than the length of this sequen...
If a trie "t" maintains the invariant "mp_trie_invar", and if a sequence "xs'" is in the set of paths of "t", then there does not exist a non-empty sequence "xs''" such that the concatenation of "xs'" and "xs''" is in the set of paths of "t".
If X is a safety property and Y is a safety property, then the intersection of X and Y is also a safety property.
If the index 'i' is less than the length of the mode list 'mdss', and 'x' is an element of 'X', and 'x' is an element of the 'i'-th mode of 'mdss' for a given 'm', then 'x' is also an element of the 'i'-th mode of the restricted modes of 'mdss' with respect to 'X' for the same 'm'.
The lookup function applied to the bulkload of a list "xs" is equivalent to the map_of function applied to the list "xs", where "xs" is a list of pairs of 'a and 'b.
For any two functions 'u' and 'v' that map a natural number to an extended non-negative real number, if for every natural number 'n', 'v' at 'n' is less than or equal to 'u' at 'n', then the limit inferior (liminf) of the difference between 'u' and 'v' is less than or equal to the difference between the limit superior ...
Removing duplicates from the list that results from removing all instances of an element 'x' from a list 'l' is the same as removing all instances of 'x' from the list that results from removing duplicates from 'l'.
The lemma named "is_le_rel_le" assumes that "rel" is a less than or equal to relation. It shows that if "rel" holds between "x" and "y", then "x" is less than or equal to "y".
: Assuming "A" is defined as the row concatenation of "A''" and "D" multiplied by the identity matrix of size "n", and "A''" is a matrix in the carrier of size "m" by "n", and "m" is greater than or equal to "n", and "D" is greater than 0, it can be shown that the function "FindPreHNF" with parameters "abs_flag", "D", ...
The lemma named "ped_inv_mapping" assumes that a natural number "a" is less than "q", and "m" is not congruent to 0 modulo "q". It shows that the probability mass function obtained by mapping each element "d" from a uniform sample of "q" to "(d + a * m) mod q" is equal to the probability mass function obtained by mappi...
: Given a real polynomial 'p', a list of real polynomials 'qs', a list of subsets (each subset is a pair of lists of natural numbers), and a list of lists of rational numbers 'signs', under the following conditions: 1. The polynomial 'p' is not zero. 2. The 'signs' are well-defined for the length of 'qs'. 3. The 'sig...
A map 'm' is empty if and only if for all elements in the map 'm', a function that always returns false is true.
The real part of the quotient of complex numbers 'a' and 'b' is less than or equal to zero if and only if the real part of the product of 'a' and the conjugate of 'b' is less than or equal to zero.
If a polynomial 'f' is reducible by a set of polynomials 'F' (as per assumption 'a1') and if a polynomial 'p' can be reduced to 'q' by the set of polynomials 'F' excluding 'p' (as per assumption 'a2'), then 'f' is still reducible by the set of polynomials 'F' where 'p' has been replaced by 'q'.
The property P holds for a Node that is a function mapping any number n to a Leaf.
If n is greater than 1, then it is not the case that n is less than or equal to the successor of 0.
If a set 's' is sequentially compact and another set 't' is closed, then the intersection of 's' and 't' is also sequentially compact.
If the set G is finite, then the condition that the set obtained by applying the Poly_Mapping.map_key PP function to each element of G forms a Groebner basis according to the pp_pm.punit.is_Groebner_basis function is equivalent to the condition that G forms a Groebner basis according to the punit.is_Groebner_basis func...
The lemma named "sectionE" assumes that "f" is a section. It then obtains a "g" such that the identity of "g" composed with "f" is true.
The probability measure on X for a constant function at a given state p, at time n, is equal to the nth power of the push-forward measure under the constant function at state p.
The greatest common divisor (gcd) of two polynomials 'p' and 'q' mapped by a function 'hom' is equal to the gcd of the individual polynomials 'p' and 'q' each mapped by the same function 'hom'.
If "a" is not equal to "b", then removing "a" from the doubly linked list after inserting "b" is the same as inserting "b" into the doubly linked list after removing "a".
If 'n' is greater than 0, if the limit of the function 'f' as 'x' approaches 'F' is negative infinity, and if 'n' is an odd number, then the limit of the function '(f x)^n' as 'x' approaches 'F' is also negative infinity.
A frame (stk,loc,C,M,pc,ics) conforms to the state (ST,LT) in the program P with heap h and static heap sh if and only if the stack stk is a subtype of ST, the local variables loc are a subtype of LT, the program counter pc is less than the size of the instruction sequence, and the instruction sequence (C,M,pc,ics) is ...
The degree of the product of two quasi-reduced polynomials 'f' and 'g' is less than twice 'n'.
The sum of the function 'f' applied to all elements in set 'X', where 'f' is zero for elements not in 'X', is equal to the sum of the function 'f' applied directly to all elements in set 'X'.
If we have a pure function 'f' applied to 'h', and 'h' successfully returns 'x' when applied to 'f', but 'h' returns an error 'e' when applied to 'g x', then applying 'h' to the bind of 'f' and 'g' will also return the same error 'e'. This is under the assumption that the bind operation is defined in the context of mon...
If i is not equal to j, then the nth' element of the list obtained by augmenting the list σ at index i with value v, at index j, is equal to the nth' element of the original list σ at index j.
If an element 'x' belongs to the 'i'th position of a set 'B', and for any function 'f' that belongs to 'B' where 'f' at position 'i' equals 'x', it implies a statement 'Q', then 'Q' must be true.
If the following conditions are met: 1. The tag of capability 'c' is true. 2. The load permission of capability 'c' is true. 3. The sum of the offset of capability 'c' and the size of type 't' is less than or equal to the sum of the base of capability 'c' and its length. 4. The offset of capability 'c' is greater than...
A set of events F is independent if the following conditions are met: 1. For any element i in set I, F(i) is a subset of all possible events. 2. For any set A and set J, where J is not empty, J is a subset of I, J is finite, and for all elements j in J, A(j) is in F(j), the probability of the intersection of all A(j) ...
The lemma named "ce" is false.
If command 1 is weakly bisimilar to command 2, command 1 is weakly bisimilar to itself, and command 2 is weakly bisimilar to itself, then the "If" statement with the test condition and command 1 and command 2 is weakly bisimilar to the "If" statement with the same test condition and command 1 and command 2.
The lemma named "dilating_fun_image" assumes that the function "f" is a dilating function for a run "r". It shows that the set of all "k" such that "f m" is less than or equal to "k", "k" is less than or equal to "f n", and "hamlet" applied to the "k"-th element of the run "r" at a clock "c" is true, is equal to the im...
If the following conditions are met: 1. The precondition "gen_dfs_pre" is true for a graph E, a set U, a set S, a set V, and a node u0. 2. The invariant "gen_dfs_fe_inv" is true for the graph E, the set U, the set S with u0 inserted, the set V with u0 inserted, the node u0, an empty set, a set V', and a boolean value "...
If the boolean values 'b' and 'c' are equal, and the values 'x' and 'u' are equal, and the values 'y' and 'v' are equal, then the result of the conditional statement 'if b then x else y' will be equal to the result of the conditional statement 'if c then u else v'.
If 'm' is an element of the first part of 'e', and 'x' is an element of either the first part of 'e', the second part of 'e', or 'mon_pl' of 'w', then 'x' is an element of 'alpha ah' of the list consisting of 'e' and 'w' with respect to 'm'.
The theorem "push_pres_Labeling" assumes the precondition "push_precond" for the functions f, l, and e. It shows that the "Labeling" function, when applied to the parameters c, s, t, and the effect of the "push" function on f and e, is true.
The lemma named "sat_slice_iff" assumes that "v" is an element of "S". It shows that the satisfaction of a formula "phi" at a point "i" in a signal "sigma" under valuation "v" in the context of Metric First-Order Temporal Logic (MFOTL) is equivalent to the satisfaction of the same formula "phi" at the same point "i" un...
If 'p' is a prime number and 'm' divides 'p', then 'm' must be either 1 or 'p'.
The left universal variable 'x' is independent of the right universal variable 'y'.
The second element of the result from the function "store_sub1" with parameters "instr", "rd", "s_val", and "s" is False.
If "i" is less than or equal to "j" and a sorted sparse vector is formed with "j" and "y" as the first element followed by vector "v", then the sparse row vector "v" and the singleton matrix at position (0,i) with value "x" are disjoint matrices.
If a block 'bl' is an element of the set of blocks 'B', and the cardinality (or size) of 'bl' is greater than or equal to 2, then the multiplicity of 'bl' is equal to 1.
If the inorder traversal of a tree 't' is sorted (according to the function 'sorted1'), then the inorder traversal of the updated tree (where 'x' is replaced with 'y') is equal to the updated list obtained by replacing 'x' with 'y' in the inorder traversal of the original tree.
If there is a path from state 's' to state 's'' following a certain path 'p', then state 's'' is reachable from state 's'.
There is a bijective function between the set of equivalence relations on a set A with k equivalence classes and the set of partitions of A into k parts. This function maps each equivalence relation to its set of equivalence classes.
The inverse of "thetaParZOL2" is a subset of the retracement of "thetaParZOL2" inverse union "ZObis".
In the context of an infinite coin toss space, there is a lemma regarding the natural filtration of a Borel measurable simple function. Given a function 'f' that maps a boolean stream to a 'b' in a T2 space, if 'f' belongs to the Borel measurable set under the natural filtration 'n', then 'f' is a simple function in th...
If a tree 't' is in the set of trees derived from a source 's' according to the tree automaton 'A', and if the source 's' is ground (i.e., it contains no variables), then for all trees 'u' in the set of source arguments of 't' and 's', 'u' is also ground.
The constant function on the set difference of A and B with value c is equal to the set difference of the constant function on A with value c and the constant function on B with value c.
For any fixed formula 'A', the floor of 'A' with an empty set is equal to 'A' enclosed in guillemotless and guillemotright quotation marks.
The lemma named "measure_lex2_induct" is defined with two fixed functions, "h1" and "h2", which map from some types 'a1 and 'a2 to natural numbers respectively. The lemma assumes that for any pair of elements x1 and x2, if the following two conditions hold: 1) For any pair of elements y1 and y2, if the value of h1 at...
If "r" is an element of set "R" and "m" is an element of set "M", then the product of the negative of "r" and "m" is equal to the negative of the product of "r" and "m".
If a function 'f' is residuated, then the residual of 'f' with respect to 'y' is equal to the supremum (or greatest element) of the set of all 'x' such that 'f' of 'x' is less than or equal to 'y'.
The lemma named "deleteMin_queue_invar" assumes that the queue invariant holds for a queue 'q' and that 'q' is not empty. It shows that the queue invariant still holds after the minimum element is deleted from 'q'.
The product of the modulus of two integers 'a' and 'b' is equal to the product of the modulus of integer 'a' and the modulus of integer 'b'.
The successor of 0 left shifted by 3 equals 8.
For any natural number 'n', if 'n' is odd, then the result of 'n' divided by 2 is less than 'n'. This lemma is used for termination simplification.
For all elements 'x' in set 'A', if process 'P' of 'x' is less than or equal to process 'P' prime of 'x', then the mixed prefix of 'A' and 'P' is less than or equal to the mixed prefix of 'A' and 'P' prime.
If the function "flat_1" applied to "x" equals "allow y", then "x" must be either "allow(allow y)" or "allow(deny y)".
The lemma named "actions_mono" assumes that phi is less than or equal to psi. It shows that the actions of phi are also less than or equal to the actions of psi.
If a key K is in the analysis of the union of the image of key nE and H, then K is in nE or the key K is in the analysis of H. This implies that the key K being in the analysis of the union of the image of key nE and H is equivalent to K being in nE or the key K being in the analysis of H.
The ground meta-context is derived from the ground generalized meta-context 'C'.
The lemma 'progress' consists of two parts: 1. If a term 't' of type 'T' is lambda-closed and there is no 's' such that 't' can be reduced to 's', then 't' is in normal form. 2. If a command 'c' is lambda-closed and 'c' is equal to the term 't' under a context 'beta', and there is no 'd' such that 't' can be reduced ...
The lemma named "set_object_value_consistent" assumes that under the equivalence relation "vpeq" for a certain user "u", two states "s1" and "s2" are equivalent. It shows that if an object value "y" is set for an object "x" in both states "s1" and "s2", the resulting states are still equivalent under the same equivalen...
The lemma named "scene_space_ub" assumes that both "a" and "b" are elements of the scene space. It shows that "a" is a subset of the union of "a" and "b" in the scene space.
If "t" and "u" are equivalent under uninterpreted transformations, then "t" is less than or equal to "v" under uninterpreted transformations if and only if "u" is less than or equal to "v" under uninterpreted transformations.
If time bound "TBOUND" applies to a combination of constant "c" and time "t", and constant "c" refines to constant "d", then time bound "TBOUND" also applies to the combination of constant "d" and time "t".
The lemma named "Dup_head_oob" assumes that a duplication "Dup" exists for a certain parameter "p" at index "i" in a list that consists of a pair "(ps, a)" and a branch. It then shows that the pair consisting of the length of the branch and the length of "ps" is not in the set "xs".
The lemma "mbindFailSave versus mbindFailStop" assumes that for all inputs and states, the input-output program does not equal None. It shows that the state satisfies the property P after executing the monadic bind operation on the input-output program with fail-save semantics is equivalent to the state satisfying the ...
The function 'atom_map' is injective.
Lemma ex_gt_imp_less_multiset: "If there exists an element 'y' in multiset N such that for all elements 'x' in multiset M, 'x' is less than 'y', then multiset M is less than multiset N."
The lemma le_fininfI[intro 0] assumes that v2 is less than or equal to w2 under the fininf order, and w1 is less than or equal to v2 under the F order. Under these assumptions, it shows that w1 is less than or equal to w2 under the fininf order.
Running the environment function 'ask_env' with a function 'f' and a parameter 'r' is equivalent to running the environment function with 'f r' and 'r'.
If the index of element x in list xs is greater than the index of element y in list xs, then the index of element x in list xs is equal to the successor of the index of element x in the list obtained by removing the first occurrence of element y from list xs.
If "b" is not equal to zero, then the Euclidean size of the greatest common divisor (gcd) of "a" and "b" is less than or equal to the Euclidean size of "b".
If the functions f and g are both differentiable at a point z, and if g at point z is not equal to zero, then the derivative of the function that is the quotient of f and g at point z is equal to the difference of the product of the derivative of f at point z and g at point z and the product of f at point z and the der...
The zipping of two streams, where the first stream starts with element 'a' followed by stream 's1' and the second stream starts with element 'b' followed by stream 's2', is equal to a new stream that starts with the pair '(a, b)' followed by the zipping of the streams 's1' and 's2'.
The lemma "add_transitions_simps" states the following: 1. The initial state of the machine after adding transitions is the same as the initial state of the machine before adding transitions. 2. The states of the machine after adding transitions are the same as the states of the machine before adding transitions. 3. T...
If a well-formed tuple "v" exists for the union of the free variables of terms "t1" and "t2", and if either "t1" or "t2" is a constant, and if the evaluation of "t1" and "t2" (after mapping "v" to them) are equal, then "v" is in the equivalence relation of "n", "t1", and "t2".
If the left tree (lt) and the right tree (rt) are both red-black trees, then the union of these two trees (rbt_union lt rt) is also a red-black tree.
If 'ok' is greater than 0 and less than or equal to the length of list 'f', then the reverse of taking 'ok' elements from the list that starts with 'x' and is followed by 'f', is equal to the tail of the reverse of taking 'ok' elements from 'f', appended with 'x'.
If a function F leads from set A to set A' and F is stable in set B, then F leads from the intersection of sets A and B to the intersection of sets A' and B.
The possibility of a property P holding at some time t' in a singleton time set {t} is equivalent to the property P holding at time t.
If 'a' is in the domain of map 'm', then there exists a 'b' such that the map 'm' at 'a' equals 'Some b'.
There is a relation between the set "Basis" and the set "Basis_S" through the correspondence relation "cr_S". This is a transfer rule used for transferring properties between these two sets.
The sequential composition of "STOP" and any process "P" is equivalent to "STOP".
The lemma named "brouwer_compactness_lemma" is about a function 'f' that maps from a topological space 'a' to a real normed vector 'b'. It assumes that the set 's' is compact, the function 'f' is continuous on 's', and there is no element 'x' in 's' such that 'f' of 'x' equals zero. Under these assumptions, it is possi...
The residue of an integer is equal to the integer modulo m.
"if 'x' is a 'Just' value, then the result is the 'Just' value of 'x', otherwise the result is 'y'. This is equivalent to applying the 'fromMaybe' function to 'y' and 'x'."
A node in a weight balanced tree (represented as "Node l x r") is considered balanced if and only if the following conditions are met: 1. The ratio of the size of the left subtree (denoted as "size1 l") to the total size of the left and right subtrees (denoted as "size1 l + size1 r") is greater than or equal to alpha ...
For any real number 'x', the function 'numer_cf6' applied to 'x' will always be greater than zero.