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If a function F leads from set A to set A' and concurrently from set B to set B', then F also leads from the intersection of sets B' and A to the union of the intersection of sets B and A' and the difference of sets B' and B.
The product of "x" and the star of "x" is less than or equal to the star of "x".
The theorem named "discr_ZObis" assumes that "c" and "d" are proper and discrete. It shows that "c" is approximately equal to "d" in the zero-one sense.
The lemma "FAss_invers" assumes that in the environment E, the field assignment "a..fn:=v" is well-typed with type T'. It shows that there exists a type T such that in the environment E, the field access "a..fn" is of type T and the value "v" is of type T', and in the program E, type T' is a subtype of T.
If set Q' is a subset of set Q, then the epsilon closure of set Q' is also a subset of the epsilon closure of set Q.
A set A is infinite if and only if the hypernatural number representation of set A is less than the nonstandard extension of set A.
If "mu" and "nu" are isomorphisms and the source of "nu" is the same as the target of "mu", then the horizontal composition of "nu" and "mu" is also an isomorphism.
The lemma named "while_false_trel" assumes that the evaluation of "b" under the state "σ" is false. It shows that the transition from the state "σ" and the while-loop "while b do P od" under the unifying theories of programming (UTP) semantics is to the state "σ" and the identity action "II".
The lemma named "empty_or_last_of_suffix" assumes two conditions: first, that the sequence 'q' is equal to the sequence 'q'' followed by the element 't'; and second, that the sequence 'q' is also equal to the sequence 'p' followed by the sequence 'ts'. Given these assumptions, it shows that either the sequence 'ts' is ...
The lemma named "assign_simultaneous" assumes that "y" is a very well-behaved lens and "x" is independent of "y". It shows that assigning a pair "(x, y)" to another pair "(e, the value of y)" is equivalent to just assigning "x" to "e".
If the type of a variable 'v' is 'Some x', then 'v' cannot be an address.
For all elements 'x' in the finite set 'X', if a function 'f' applied to 'x' is true, then filtering 'X' with 'f' will result in the same set 'X'.
The lemma named "ins_list_split" assumes that the result of splitting a tree "ts" at a point "x" yields two trees "ls" and "rs". It also assumes that the inorder traversal of a node constructed from "ts" and "t" is sorted in ascending order. Under these assumptions, it shows that inserting "x" into the inorder traversa...
If S is an integral domain, t is an element of the carrier of S and not equal to zero, and the polynomial ring R' is over the quotient ring of S by the principal ideal generated by t, then the homomorphism from R to R' that sends each element of S to its equivalence class in the quotient ring, when applied to t, yields...
The lemma named "not_span_independent_insert" is about a fixed vector 'v' in the context of real vectors. It assumes that a set 'S' is independent and that 'v' is not in the span of 'S'. Under these assumptions, it shows that the set 'S' remains independent even after the vector 'v' is inserted into it.
Lemma: A subset of a convex set between two points is also convex. Let C be a set of real numbers. Assume that C is convex, and x and y are elements of C such that x is less than y. Then, the set of all elements between x and y, inclusive, is a subset of C.
The function 'f' applied to the product of function 'g' over set 'A' is equal to the product of function 'f' applied to each element of 'g' over set 'A'.
An undirected edge from a point 'a' to itself is simply the set containing 'a'.
The measure of the entire universe (all possible outcomes) for a given probability mass function 'p' is equal to 1.
The trace at a state 's' and a position 'j' in a sequence 'omega' is equal to the first occurrence of a condition where 'j' is less than the first element of the sequence head 'x'. If this first occurrence is infinity, the result is undefined. Otherwise, if the first occurrence is a natural number 'i', the result is th...
"for all x and y, x explains y if and only if y depends on x."
If a function 'f' satisfies the following conditions: 1. 'f' of true is true, 2. 'f' of false is false, 3. for any two propositions 'phi' and 'psi', 'f' of ('phi' and 'psi') is equal to 'f' of 'phi' and 'f' of 'psi', 4. for any two propositions 'phi' and 'psi', 'f' of ('phi' or 'psi') is equal to 'f' of 'phi' or 'f...
The statement consists of four parts: 1. The freshness of 'a' in the substitution term 'SubstTermP' with variables 'v', 'i', 't', 'u' is equivalent to 'a' being fresh in 'v', 'i', 't', and 'u'. 2. The substitution term 'SubstTermP' with variables 'v', 'i', 't', 'u' is a sigma formula. 3. If the substitution term 'S...
The parallel substitution of "Y" for "y" in "rho", followed by the parallel substitution of "rho'", is equal to the parallel substitution of "rho'" in "rho", followed by the parallel substitution of "Y" after the substitution of "rho'" for "y" under the sort "ys".
If an element 'x' is in the set 'xs', then there exists an index 'i' that is less than the length of 'xs', such that the element at index 'i' of 'xs' is 'x'.
The lemma named "poly_eval_index" assumes that the multiplicative identity is not equal to the additive identity and that the function 'g' is a closed function in 'R'. It shows that the evaluation of a polynomial in 'R' and 'S' with function 'g' and the multiset transformed to an indexed polynomial in 'R' with singleto...
The Laplace transform of the function that is constantly 1 has an inverse with respect to s, if the real part of s is greater than 0.
The relation NS is reflexive for any string s, meaning that the string s is related to itself under the NS relation.
If for every integer 'n', the coefficient of 'n' in polynomial 'p' is equal to the coefficient of 'n' in polynomial 'q', then polynomial 'p' is equal to polynomial 'q'.
The quantity conversion function "qconv" with the identity conversion "id\<^sub>C" applied to any quantity "x" will result in the same quantity "x".
The closure of a generalized multi-context under a relation R is equal to the generalized multi-context extension on a predicate P, where P is a function that checks if the function symbols in a context C are a subset of F. This is denoted as "?Ls = ?Rs".
, assuming "\<omega>" is in the space of the product measure "PiM" from 0 to "n" over "M", and "A" is in the sets of the product measure "PiM" from 0 to "n+1" over "M", it shows that the extended probability "eP" of "n", "\<omega>", and "A" is equal to the probability "P" of "n", "\<omega>", and the preimage of "A" und...
The lemma named "poincare_on_ray_poincare_collinear" assumes that points "p", "a", and "b" are in the unit disc and that "p" is on the Poincare ray from "a" to "b". It shows that "p", "a", and "b" are Poincare collinear.
If the following conditions are met: 1. The state is within the domain of the exhaustive unit propagation function. 2. The watch lists only contain clauses from the formula (getF state). 3. The watch lists are unique. 4. The watch lists are characterized by the first and second watch states. 5. The watch states are el...
For a given state sigma, the result of binding an IO program "ioprog" to a list of inputs "iotas" using the "mbindFailSave" function, and then checking if the length of "iotas" is equal to the length of the output list "os" and some property "P" holds for "iotas" and "os", is equivalent to the result of binding the IO ...
The lemma "timpls_transformable_to_subst_subterm" establishes a condition for a term to be transformable under a certain substitution. Specifically, it fixes two terms 's' and 't' and two substitutions 'delta' and 'sigma' of a certain type. It assumes that term 't' is transformable to another term under substitutions '...
The set of vectors negated from set A is equal to the set of vectors from A applied with the negation function.
If 'h' is in the projected probability of 'PROB' over 'vs', and 's' is in the valid states of 'PROB', then the successor state of 's' under 'h' is also in the valid states of 'PROB'.
Applying the composition function (denoted by \<circ>) to the probabilistic monads u, v, and w in a nested manner (first v and w, then the result with u) is the same as applying the composition function to u, v, and w in a sequential manner.
The lemma named "inj1_hom" is defined with respect to some fixed R, M1, and M2. It assumes that M1 is a module over R (as stated by h1) and M2 is also a module over R (as stated by h2). It then shows that there is a module homomorphism from M1 to the direct sum of M1 and M2, which is the injection function inj1 for M1 ...
If the degree of a polynomial p is 4, then there exist coefficients a, b, c, d, and e such that the polynomial p can be expressed as [: e, d, c, b, a :] and the coefficient a is not equal to zero.
The variable from the correct system holds true.
The lemma named "rawpsubstT_suc" assumes that "r" is an element of "trm", the second element of each pair in the set "txs" is a subset of "var", and the first element of each pair in the set "txs" is a subset of "trm". It shows that the raw term substitution of the successor of "r" with "txs" is equal to the successor ...
If alpha is a limit ordinal, I is an element of the set Vset alpha, for every element i in I, A i is an element of the set Vset alpha, and I is a finite set, then the product of A i for each i in I is an element of the set Vset alpha.
If 's' is the supremum of 'x' and 'y', and both 'x' and 'y' are less than or equal to 'z', then 's' is also less than or equal to 'z'.
The sum group of integer groups over an index set I is isomorphic to the free Abelian group over the same index set I.
The lemma named "lunit_coherence1" assumes that "f" is an identity of category "C". It shows that the left unit of the functor applied to "f" composed with the inverse of the unit of the target of "f" tensor product with the functor applied to "f" is equal to the left unit of "f".
If U is a subspace, then U has a finite dimension.
If transition t2 subsumes transition t1, and if transition t1 can take a certain input i under a specific condition r, then the evaluation of updates on t1 with input i and condition r for a certain variable a equals some value x. This implies that the evaluation of updates on t2 with the same input i, condition r and ...
The function "gx" applied to the pair "(xcpt, s)" equals "xcpt".
The N function of the product of x and the maximum element is equal to the intersection of the product of x and the maximum element with the identity element.
If the projection of a set C through a function h extended by a function F is in the transient state of the intersection of the projection set of C through h and a set D, then the function F is in the transient state of the intersection of the projection set of C through h and the set D.
If the following conditions are met: 1. The function f commutes with the augmented power set of z, i, and A, 2. The function f is associative with the augmented power set of z, i, and A, 3. The first condition for the indexed power product is satisfied for A and i, 4. The second condition for the indexed power product...
If a state 's' is an element of the concretization of 'S', and the abstract value of 'e' in state 's' is an element of the concretization of 'a', then the state 's' is an element of the concretization of the abstract filter of 'e', 'a', and 'S'.
For any two rational numbers 'a' and 'b', if the sign of 'a' is 1 and 'a' is less than or equal to 'b', then the sign of 'b' is also 1.
In the context of a ring, if A, B, and C are ideals of R, then the product of A with the sum of B and C is equal to the sum of the product of A with B and the product of A with C.
The star of the sum of x and y is equal to the star of the sum of the star of x and the star of y.
If 'as' is an element of the carrier of the nth power of the p-adic numbers, 'bs' is an element of the carrier of the kth power of the p-adic numbers, 'x' is equal to the concatenation of 'as' and 'bs', and 'x' is an element of the partial pullback of 'n', 'f', 'k', and 'S', then the result of applying 'f' to 'as' and ...
For any ordinal number alpha, the ordinal exponentiation of 0 to the power of alpha equals 1 if alpha is 0, otherwise it equals 0.
If a word "w" is not in the set "us", then the result of the function "glue" applied to "w" and "us" is equal to "us".
The cardinality of the Cartesian product of a set A and a singleton set {x} is equal to the cardinality of set A.
A plan is valid if and only if there exists a model such that the plan is a path of actions from the initial state in this model and this model satisfies the goal condition.
The result type of a method (represented by "m" of any type 'a in the mhead_scheme) is equal to the result type of "m".
The theorem in the Second Sylow theory, named "p_sylow_mod_p", shows that the cardinality (or the number of elements) of the subgroups of size "p to the power of a" when divided by p gives a remainder of 1.
If 'x' is less than or equal to 'y' under the relation 'R1' and the exception 'E1', and 'y' is less than or equal to 'z' under the relation 'R2' and the exception 'E2', then 'x' is less than or equal to 'z' under the combined relation 'R1 O R2' and the combined exception 'E1 O E2'.
The lemma named "next_action_consistent" demonstrates that for all states 's' and 't', and all executions 'execs', if the current state 's' is equivalent to state 't' and for all data 'd' involved in the next action of state 's' with respect to 'execs', 'd' is equivalent in states 's' and 't', and if the current state ...
The lemma "transp_rel_generatI" assumes that "A", "B", and "C" are transitive relations. It shows that the relation generated by "A", "B", and "C" is also a transitive relation.
If the precondition P is a subset of the set of states s such that the function f applied to s is in the postcondition Q, then under the context of program \<Gamma> and proof environment \<Theta>, the Hoare triple {P} f {Q} is valid with respect to the fault set F and the assertion set A.
: Assuming - "s" is succeeded by "s'" - "x_i" is in the set of left variables of the tableau of "s" - "x_i" is in the set of right variables of the tableau of "s'" - the direction is either Positive or Negative It shows that - if the lower bound of "x_i" in "s" is not greater than or equal to the value of "x_i" in "...
The lemma named "termination_0_1_invar" is defined with a fixed predicate 'I' that takes a state 's' and returns a boolean. It assumes the following conditions: 1. For any state 's', if the guard condition is true and the predicate 'I' is true for 's', then 'p' is less than or equal to the weight of the loop (while th...
The lemma named "sdp_repeat" is defined with a fixed program 'a'. It assumes that 'a' is sub-distributive over probabilistic conjunction, the weakest precondition of 'a' is healthy, and the weakest liberal precondition of 'a' is nearly healthy. Under these assumptions, it shows that the repeat of 'a' for 'n' times is a...
The theorem named "cop_correct_nterm" in IFC_Tracking from IFC.thy in Isabelle states the following: Assuming that the observation of sigma at k is not equal to the observation of sigma prime at k, the observation of sigma at k is not None, and the observation of sigma prime at k is not None. Also, assuming that sigma...
The lemma 'line_integral_exists_on_typeII_Cube_boundaries' assumes that the pair (k, γ) is in the boundary of the two-dimensional cube. It shows that the line integral exists for the function F on the path γ with respect to the set {j}.
The lemma named "subset_mono" assumes that "N" is a subset of "M". It shows that the image of "N" under the function "insert_messages" is also a subset of the image of "M" under the same function "insert_messages".
The lemma named "subst_b_fresh_u" is about a fixed atom 'x' of type 'u'. It shows that if 'x' is fresh for 'v', then 'x' is also fresh for 'v' after the substitution of 'bv' with 'b' in 'v'. The same holds for 'ce', 'e', 'c', 't', 'd', 'g', and 's'. If 'x' is fresh for any of these, then 'x' is also fresh for them afte...
The lemma "Con_implies_Arr" assumes that there is a concatenation of T and U. It shows that both T and U are arrows.
The public key of 'b' for 'A' is equal to the public key of 'b'' for 'A'' if and only if 'b' is equal to 'b'' and 'A' is equal to 'A''.
The lemma "chain_append_at_right_edge" assumes that there is a long chain from 'f' to 'Y' between 'a1', 'a', and 'an', and that 'a1', 'an', and 'b' are in a certain relation 'Yb'. It defines a function 'g' such that if 'j' is less than or equal to the cardinality of 'Y' minus one, then 'g' equals 'f j', otherwise 'g' e...
The lemma named "isometry_inverse" assumes that "f" is an isometry. It then shows that the inverse of "f" is also an isometry and that "f" is a bijection.
The lemma "trancl_steps_relpow" assumes that "a" is a subset of the transitive closure of "b". It shows that if the pair "(x,y)" is in the nth relational power of "a", then there exists a number "m" such that "m" is greater than or equal to "n" and the pair "(x,y)" is in the mth relational power of "b".
The list up to a certain point, defined by the function 'upto' applied to the constructed integers 'MkI' of 'm' and 'n', is a finite list. This is denoted as "?P m n".
The lemma named "poly_maps_memI" assumes three conditions: 1. "fs" is a polynomial tuple of length "n", 2. the length of "fs" is "m", 3. "F" is the polynomial map of "n" and "fs". If these conditions are met, it shows that "F" is in the set of polynomial maps of "n" and "m".
The lemma named "image_Fpow_surjective" assumes that the image of a function 'f' applied to a set 'A' equals to set 'B'. It shows that the image of function 'f' applied to the function power of 'A' equals to the function power of 'B'.
If the finite cardinality (or size) of a set 'xs' is less than or equal to 1, and an element 'x' belongs to the finite set 'xs', then the finite cardinality of the set 'xs' must be 1.
If polynomial 'p' is not equal to zero, then the polynomial '[-a, 1:]' raised to the power of 'order a p' divides 'p' and the polynomial '[-a, 1:]' raised to the power of 'Successor (order a p)' does not divide 'p'.
The lemma named "rx_Max" assumes that "x" is an element of "X". It shows that "r x" is equal to the supremum of the set of all "ρ" such that "ρ" times "x" is less than or equal to "A" times "x".
The lemma named "not_proj2_set_Col_iff_span" assumes that the cardinality of set S is 3. It shows that set S is not a collinear set of projective points if and only if the span of the image of set S under the projective representation equals the universal set.
The lemma named "lselect_discard_end" assumes that for any element 'i' in set 's', 'i' is less than 'k'. It then shows that selecting elements from the list obtained by taking the first 'k' elements of list 'w' according to set 's' is the same as selecting elements from list 'w' according to set 's'.
If the functions 'f' and 'g' are both differentiable in the field 'F', then the function that is the difference of 'f' and 'g' (i.e., 'f' minus 'g') is also differentiable in the field 'F'.
The normal form of the sum of the normal forms of two elements p and q, with respect to a set F, is equal to the sum of the normal forms of the elements p and q, with respect to the same set F.
For a red-black tree "t" at a pointer "p", after the insertion of a key-value pair "k, v" at pointer "p", the resulting tree is a binary tree with the key-value pair "k, v" inserted into "t". This is proven under the Hoare triple logic, which is a form of axiomatic semantics used for proving the correctness of computer...
If 'a' is not equal to zero, 'a' is not a unit, and for any 'b' that divides 'a', 'a' divides 'b' or 'b' is a unit, then 'a' is irreducible.
For every 'w' in the set of disconnected nodes locations pointed by 'ptr', if 'h' transitions to 'h'' under 'w', then for every 'r' in the parent locations, 'r' also transitions from 'h' to 'h''.
If deleting a shadow root pointer from a heap results in a new heap, and an object pointer is not equal to the cast of the shadow root pointer, then getting the object from the new heap using the object pointer is the same as getting the object from the original heap using the same object pointer.
If the sum of the distance from point x to point z and the distance from point y to point z is less than or equal to e, then the distance from point x to point y is less than or equal to e.
For a given set 'S' in a Euclidean space, if 'S' is an Absolute Neighborhood Retract (ANR), then 'S' is locally connected.
In the context of a complex vector space, the lemma named "vec_of_basis_enum_cspan" is defined. It fixes a set X of type 'a, where 'a is an enumerated basis. It assumes that the length of the canonical basis, represented as a list of type 'a, is equal to n. The lemma then shows that the image of the complex span of X u...
The evaluation function is single-valued.
The lemma named "expands_to_basic_powr" assumes that the basis is well-formed with "b" as the first element of the basis list and the length of the basis is equal to the expansion level of the type 'a multiserie. It shows that the function which raises "b" to the power "e" expands to a multiseries with a constant expan...
The lemma named "step_z_dbm_sound" assumes that there is a transition from state "l" with DBM "D" to state "l'" with DBM "D'" under the automaton "A" with clock numbering "v" and "n". It also assumes that the global clock numbering of the automaton "A" is "v" and "n". Under these assumptions, it shows that there is a t...