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The set of free variables in a list that starts with 'x' and continues with 'xs' is the union of the set of free variables in 'x' and the set of free variables in 'xs'.
For every element 'x' in the set 'cl_var_ty_list', if the function 'find_type_f' applied to 'P', 'ctx', and 'cl_k' (where 'cl_k', 'var_k', and 'ty_k' are the components of 'x') equals 'Some ty_k', then applying the function 'lift_opts' to the mapped list (where each element is transformed by first applying the function...
The transitive closure set of term implications, where each pair (a, b) is in the transitive closure of c and a is not equal to b, is equal to the term implication closure set of M and c.
The lemma named "lcm_proj1_if_dvd" assumes that "b" is a divisor of "a". It shows that the least common multiple of "a" and "b" is equal to the normalized value of "a".
If F is less than or equal to F', and function f tends towards limit l under F', then function f also tends towards limit l under F.
The unstream function, when applied to the Cons_trans function with parameters x and g and a tuple (True, s), is equivalent to the list that starts with x and is followed by the result of applying the unstream function to g and s.
The modulus of the sum of an arbitrary number 'm' and the extended natural number 'n', divided by the extended natural number 'n', is equal to the modulus of the arbitrary number 'm' divided by the extended natural number 'n'.
: Given a function 'f' that maps from any type 'a' to real numbers, if for any set 'A' in the sets of measure 'M', where the extended measure of 'A' is greater than 0 and less than infinity, and for any real numbers 'a' and 'b' where 'a' is less than 'b', the minimum of the outer measure of 'M' for the set of elements...
The lemma "leof_use_spec_rule" assumes two conditions. First, it assumes that "m" is less than or equal to "SPEC Psi" in the order of refinement. Second, it assumes that "m" is less than or equal to "SPEC" of a function that returns "Psi s" implies "Phi s". If these two conditions are met, it shows that "m" is less tha...
The initial state of model 2 is a subset of the state where the invariant "m2_inv5_ikk_sv" holds.
The lemma named "wf_converse_widen1" assumes that the program P is well-formed according to the well-formedness condition wfmc (as stated by "wf_prog wfmc P"). It shows that the converse of the function "widen1 P" is also well-formed.
The invariant "inv" holds for an empty set and a set U.
For all functions 'f' and 'h', if 'f' is a function from the set of all integers less than or equal to 'n+1' to the carrier 'A', and 'h' is a function from the set of all integers less than or equal to 'n+1' to itself that is also injective on this set, then the sum over the Abelian group 'A' of the composition of 'f' ...
The concatenation of the mapped function, which applies an empty list to an empty list and a list consisting of the tail of the list to a non-empty list, over a list of lists "xss", is equal to the mapping of a function that returns the tail of a list over the filtered list of lists "xss" where the lists are not empty.
The orbits of a group action on a set form a partition of that set.
The sum of the product of sets A and B with the product of sets C and D is equal to the product of the sum of sets A and C with the sum of sets B and D.
The lemma named "eqgf_comp_gmctxt" assumes that "s" is equivalent to the pair "(C, ss)" under the relation "Gf" and "s" is also equivalent to the pair "(D, ts)" under the same relation "Gf". Under these assumptions, it shows that the pair "(C, D)" is in the composition of the relation "gmctxt".
The identity operation "ID_OP" applied to the operation "f" with interface "I" is equal to the operation "f" with the same interface "I".
The lemma named "distinct_sorted_list_lem2" makes several assumptions: 1. The list "xs" is sorted. 2. The list "ys" is sorted. 3. The elements in the list "xs" are distinct. 4. The elements in the list "ys" are distinct. 5. The element "e" is less than the element "e'". 6. For every element "x" in the list "xs", "x" i...
Casting a variable (Var xs x) as a sort (asSort xs) using the function "castVar" will result in the original variable and sort pair (xs, x).
Function 'f' maps from 'Y' to 'X' in the opposite category 'Op C' if and only if function 'f' maps from 'X' to 'Y' in the category 'C'.
: Assuming the intersection of sets 'A' and 'B' is an empty set, and 'A' is a subset of 'Q_A', and 'B' is a subset of 'Q_B', it can be shown that the language of the generalized tree automaton (gta) for the union of 'A' and 'B' in the tree automaton union of 'A' and 'B' is equal to the union of the gta language of 'A' ...
If a list formed by appending list 'ys' to list 'xs' is "happens-before consistent" (hb_consistent), then it can be inferred that the original list 'xs' is also "happens-before consistent".
The lemma named "euler_genus_rev_swap" assumes that the arc 'a' and the arc 'b' in graph 'G' have the same endpoints, and both 'a' and 'b' are arcs in graph 'G'. It shows that the Euler genus of the graph 'G' after swapping 'a' and 'b' is equal to the original Euler genus.
The lemma named "componentsE" assumes that "S" is a component of "U". It then concludes that there exists an "x" such that "x" is an element of "U" and "S" is the connected component set of "U" at "x".
"x is equivalent to x".
If 'u' is an element of 'U' and 'v' is an element of 'V', then the sum of 'u' and 'v' is an element of the sum of 'U' and 'V'.
The power set of X is equal to the set containing only Y if and only if X is equal to 0 and Y is also equal to 0.
The lemma eqvtI: It fixes P and Q as processes in the pi calculus, and perm as a permutation of names. It assumes that P is weakly late bisimilar to Q. It shows that applying the permutation to P results in a process that is weakly late bisimilar to the result of applying the same permutation to Q.
Appending an empty list to any list "xs" results in the original list "xs".
The lemma "get_dom_component_ok" assumes that the heap "h" is well-formed, the type within the heap "h" is well-formed, and the heap "h" contains known pointers. It also assumes that the pointer "ptr" is in the set of object pointer kinds of heap "h". Given these assumptions, it shows that the heap "h" successfully exe...
The polynomial resulting from lifting a univariate polynomial 'p' and evaluating it at 'x' and 'y' is equivalent to evaluating the original polynomial 'p' at 'y'.
If we have an "If" statement with conditions x, y, and z, it will remain the same as an "If" statement with the same conditions x, y, and z.
If 'm' is greater than zero, then the natural power of 'm' raised to 'k' to the power of 'n' is equal to the natural power of 'm' to the power of 'n' raised to 'k'.
The lemma named "rightInvertible" is defined with the following conditions: - It fixes a multiset of formulas, denoted as Γ. - It assumes a set of rules, denoted as R', which is a subset of upRules, and R is the union of Ax and R'. - It assumes that the pair (Γ implies Compound F Fs, n) is derivable from R. - It assum...
If the function 'f' has a field derivative 'Da' at a point 'x' within a set 's', and the function 'g' has a field derivative 'Db' at the same point 'x' within the same set 's', then the function that is the product of 'f' and 'g' has a field derivative that is the sum of 'Da' times 'g' at 'x' and 'Db' times 'f' at 'x',...
If the variable 'runz' is equal to an empty map, then the function 'is_runs2sigs' applied to 'runz' will return a function that gives 0 for any input 's'.
If a problem 'prob' is well-formed, and for every operator 'op' in the set of operators of 'prob', the operator 'op' is consistently preserved, and every operator in the set of operators of 'prob' is a standard operator, then let 'cnf_formula' be the conversion of 'prob' to DIMACS format using the function 'SASP_to_DIM...
If set S is negligible and set T is also negligible, then the union of set S and set T is also negligible.
The lemma "lock_ok_las'_collect_locks'_may_lock" assumes that "lock_ok_las'" holds for a lock set "ls", a thread "t", and a lock action sequence "las". It also assumes that for every lock "l" in the collection of locks from "las", the thread "t" may lock "ls" at "l". Given these assumptions, if a lock "l" is in the col...
The function "start_sttp_resp" is responsive to the "storeST" function with parameters "i" and "tp".
If from the conjunction of Gamma and P, Q can be inferred, then from Gamma, it can be inferred that P implies Q.
In the context of a univariate polynomial ring (UP_cring), the lemma named "to_fun_nat_pow" is stated. It assumes that "h" is an element of the carrier of the univariate polynomial ring (UP R) and "a" is an element of the carrier R. It shows that the function "to_fun" applied to the nth power of "h" in the univariate p...
The lemma named "wf_plan_act_is_sound" assumes that the plan action "PAction" with parameters "n" and "args" is well-formed. It shows that the operation is sound when an action schema is instantiated with the resolution of the action schema "n" and the parameters "args". This is also true when the PDDL operation is con...
If a list 't' is an infinite suffix of a set 'A' and 'r' is in the reflexive transitive closure of 'A', then 'r' is a finite prefix of 'A' in the list 't'.
The lemma named "Span_eq_combine_set_length_version" assumes that the set "Us" is a subset of the carrier "R". It shows that the span of "K" and "Us" is equal to the set of combinations of "Ks" and "Us", where the length of "Ks" is equal to the length of "Us" and the set "Ks" is a subset of "K".
If "p" is a list in the heap "h" and "y" is not an element in the list at location "p" in the heap "h", then updating the heap "h" at location "y" with "q" does not change the list at location "p".
The sum of the function 'f' of 'n' times 0 raised to the power of 'n', for all 'n', is equal to the function 'f' of 0. This is true for the function 'f' that maps a natural number to a real normed algebra 1.
If the rate is greater than zero, then the predicate function P applied to the Poisson probability mass function with that rate is true for all P.
If the function 'f' has a product 'a' and the function 'g' has a product 'b', and both 'a' and 'b' are not zero, then the function that multiplies 'f' and 'g' for each 'n' has a product that is the product of 'a' and 'b'.
The lemma named "binopE" assumes that "f" is a binary operation on set "A". From this assumption, it can be obtained that "f" is a nullary operation on set "A" with arity 2.
"x" is in the set "set1_blindable_m" of "y" if and only if "y" is equal to "Unblinded x".
The lemma named "hv_zk" assumes that "h" is equal to "g" raised to the power of "x1" multiplied by "g prime" raised to the power of "x2". It shows that the relation "R" in the base of Sigma protocols with parameters "h", "(x1, x2)", and "c" is equal to the function "S" in the base of Sigma protocols with parameters "h"...
If the function "step_thread" with parameters "update_state", "s", and "t'" equals the floor function of the tuple "(t, None, σ')", then there exists "x", "ln", and "n" such that the thread "t" in "s" equals the floor function of the tuple "(x, ln)", and "ln" times "n" is greater than 0, and the thread "t" is not waiti...
The sine of (x plus the successor of m times pi divided by 2) is equal to the cosine of (x plus m times pi divided by 2).
: Assuming the following conditions: 1. The evaluation of a single cupcake (a term in the CakeML language) using a semantic environment results in a return value 'ml_v'. 2. The term 't' is well-defined. 3. The term 't' is closed (it has no free variables). 4. The term 't' does not shadow any constants. 5. The term 't' ...
For all elements 'j' in the set obtained from the subarray of 'a' from index 'l' to 'r' in the heap 'h', property 'P' holds for 'j' if and only if for all 'k' such that 'k' is greater than or equal to 'l' and less than 'r' and less than the length of array 'a' in heap 'h', property 'P' holds for the element at index 'k...
The theorem embedI: Assumes that r is a well-order and s is a well-order. And for any element a in the field of r, the function f maps a to the field of s and the image of the function f under the relation r for a is a subset of the image of the function f under the relation s for a. It shows that there exists a functi...
The set A is less than or equal to the set of all lists of A in terms of cardinality.
Folding function f from initial value b over array a and folding function g from initial value c over array a is equivalent to folding a function over array a that applies f and g to each element of the array, starting from initial values b and c respectively.
The lemma named "bangPres" is defined with respect to a process P, a relation Rel, and another process Q in the context of Calculus of Communicating Systems (CCS). It assumes that the pair (P, Q) is in the relation Rel. It also assumes three conditions: 1. For any processes R and S, if the pair (R, S) is in the rela...
The inverse of relation b composed with the inverse of relation d is a subset of the inverse of relation b.
The theorem named "Pluennecke_Ruzsa_ineq_alt_2" assumes that "A" is a finite set and is a subset of "G". It also assumes that the cardinality (or size) of the difference set of "A" with itself is less than or equal to "K" times the real number representation of the cardinality of "A". Additionally, it assumes that "r" ...
If "x" is an element in the multiset of vertices of a node (Node r xs), then there exists a pair (t,e) in the finite set "xs" such that "x" is an element in the multiset of vertices of "t", or "x" is equal to "r".
The lemma "ik_assignment_rhs_decomp_fv" assumes that a term "t" is in the subterms of the intruder knowledge or the right-hand side of an assignment in a stateful protocol "A". It shows that the set of free variables in the stateful protocol "A" appended with the decomposition of term "t" is equal to the set of free va...
The lemma endofunctor_R shows that R is an endofunctor on V.
For a program where the variable 'x' is incremented by -1, if initially 'v' is greater than 0 and the cost is 2*v and 'x' points to the integer value of 'v', then after the execution, 'v' is still greater than 0, the cost is 2*v - 1, and 'x' points to the integer value of 'v' - 1.
If a sequence "xs" is in the concurrent composition divergences of P and Q with respect to p and q, then the pair consisting of the sequence "xs" and a set "X" is in the concurrent composition failures of P and Q with respect to p and q.
"P is less than or equal to the weakest precondition of the sum of X and Y given Q" if and only if "P is less than or equal to the weakest precondition of X given Q" and "P is less than or equal to the weakest precondition of Y given Q".
The n-box of y with respect to x is equal to the product of n(y) with respect to x and L.
If a matrix is formed by appending matrix M2 to matrix M1, and the row length and length of the resulting matrix are equal to the row length and length of the combined matrices M1 and M2, then a matrix can also be formed with the row length and length of matrix M2.
If a ghost state is defined as a tuple consisting of four elements (\<D>, \<O>, \<R>, \<S>), and a new ghost state is defined as a tuple where the second element is the union of \<O> and A minus R, and the fourth element is the symmetric difference of \<S> and R minus L with respect to W, then there is a valid transiti...
The lemma named "dim_span_le_n" assumes that "W" is a subset of the vector carrier "n". It shows that the dimension of the span of "W" is less than or equal to "n".
The lemma l11_29_b: assumes "There exists a point Q such that C is in the angle A B Q and the angle A B Q is congruent to the angle D E F" shows "The angle A B C is less than or equal to the angle D E F"
The theorem of integral substitution states the following: Assuming that the function f is set integrable over the interval from g(a) to g(b) with respect to the Lebesgue measure, and that the function g has a real derivative g'(x) at each point x in the interval from a to b. Also, assuming that the derivative functio...
If the target "B" has limits for an undefined type 'j', then it shows that there are limits for the undefined type 'j'.
The lemma named "range_inj_infinite" is defined with a fixed function 'f' that maps natural numbers to some type 'a'. If it is assumed that the function 'f' is injective, it is shown that the range of the function 'f' is infinite.
, in any possible world, the proposition "x is identical to the unique entity such that the property phi holds for that entity" is logically equivalent to the conjunction of the propositions "the property phi holds for x" and "for all entities z, if the property phi holds for z, then z is identical to x".
For a matrix A of any semiring, if A is within the carrier matrix of dimensions nr by nc, then multiplying A by the identity matrix of size nc on the right will result in A itself.
The lemma proves that the product of two numbers, m and n, in a formal system is equal to the product of m and n in the natural number system.
Evaluating the function "evalO" with parameters "b" and "omega" equals "b".
The lemma "valid_action_valid_succ" assumes that "h" is in the set of valid probabilities "PROB" and "s" is in the set of valid states of "PROB". It shows that the successor state of "s" under the action "h" is also in the set of valid states of "PROB".
The lemma named "finite_ord_FO_resolution_inferences_between" assumes that the set "CC" is finite. It shows that the set of ordered first-order resolution inferences between "CC" and "C" is also finite.
If x is greater than or equal to zero, then the value of the function "exp_cf2" at x is less than or equal to the value of the exponential function at x.
The lemma named "dterms_not_call" assumes that the term "Gamma" is well-formed and that "q" is in the derived terms of "Gamma" and "p". It shows that "q" is not a call.
, assuming "x" is not equal to "y", "v" is in the language generated by the sequence of the product of the atom "y" and the atom "x", followed by the star of the product of the atom "y" and the atom "x", and then the atom "x", and "h" is either a list consisting of two "x"s followed by some list "hs", a list with a sin...
If we assume that the category 'CC' prime is the opposite of the category 'CC', then the opposite of the natural transformation 'pi' is a functor from the product of 'P' and the set {a, b} to the category 'CC' prime in the 2-category with objects as categories.
The lemma lm043 assumes that the union of set XX is a subset of X, x is an element of XX, and x is not an empty set. It shows that the intersection of x and X is not an empty set.
The encryption of some value X' with a key K is not equal to the hash of the pair (X, Y).
Applying a function 'f' to the first 'm' elements of an infinite list and then dropping the first 'n' elements is equivalent to mapping the function 'f' over the list of integers from 'm' to 'm+n'.
If "n" is less than or equal to "r'", then the interval from "r'" with a step size of "m'" is a subset of the interval starting from "n".
In the context of a valid unMultigraph, the lemma named "trail_bound" assumes that the set of edges 'E' is finite and there is a trail from vertex 'v' to vertex 'v'' following the path 'ps'. It then shows that the length of the path 'ps' is less than or equal to the cardinality (or size) of the set of edges 'E'.
If 'u' is an element of set 'V', then a connecting walk from 'u' to 'u' that only contains 'u' is true.
If the initial state of 'n' is greater than or equal to 0, then the weakest liberal precondition of the program 'exp_count_up1' ensures that the final state of 'a' is equal to 2 raised to the power of the natural number equivalent of the initial state of 'n'.
The lemma named "completion_eq" assumes two conditions: "redex r" and "redex r'". It shows that the equality of the expressions "\<E>[r]" and "\<E>'[r']" is equivalent to the equality of the expressions "\<E>" and "\<E>'" and the equality of "r" and "r'".
For a relation R0 and a relation R1, if a function respects the relation R0 and another function respects the relation from R0 to a value-space relation R1, then the bind operation respects the value-space relation R1. This is done by applying the function to a value.
Inserting an element 'x' into a set monad 'Set_Monad xs' is equivalent to creating a new set monad with 'x' as the first element followed by the elements of 'xs'.
The lemma named "isomorphic_cards" assumes that graph G1 is isomorphic to graph G2. It shows that the cardinality (or the number of elements) of the vertex set of G1 is equal to the cardinality of the vertex set of G2, and the cardinality of the edge set of G1 is equal to the cardinality of the edge set of G2.
The lemma "RETURN_RES_refine" assumes that there exists an 'x prime' such that 'x prime' is related to 'x' by relation 'R' and 'x prime' is in set 'X'. It shows that the return of 'x' is less than or equal to the refinement of the set 'X' by relation 'R'.
The lemma named "wf_not_in_prefix" assumes that under the context of a well-formedness judgement, where "\<Theta>" and "B" are some context parameters and "\<Gamma>'@(x,b1,c1) #\<^sub>\<Gamma>\<Gamma>" is a well-formed context. It shows that "x" is not in the first component of the set obtained by converting "\<Gamma>'...