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: Assuming for any x, if x is greater than or equal to the integer value of a, then f(x) equals g(x), and also assuming that n equals n' and a equals a', it can be shown that the Euler-Maclaurin remainder when using n, f, and a converges to the same value as the Euler-Maclaurin remainder when using n', g, and a'.
Given distinct prime divisors (P) of a field (K) of size n, an ideal (I) of the ring of integers (O) of K at P and n, and an element (x) in I, if j is less than or equal to n and I is not equal to the zero element of the ring of integers of K at P and n, then the negation of the least element (mL) of K, P, I, and j is ...
: 1. The relation "R" between the nonempty bounded set "nebsingleton x1" and any set "X" is true if and only if for every element "x2" in the set "X", the relation "R" holds between "x1" and "x2". 2. The relation "R" between any set "X" and the nonempty bounded set "nebsingleton x2" is true if and only if for every e...
If a term 't' is of type 'T', then opening variable 'xT' in the closed variable 'xT' of term 't' is also of type 'T'. This lemma is marked as an introduction rule.
The intersection of the set of interpretations of each element in the set Gamma is equal to the interpretation of the set Gamma itself.
The SK function, when applied to the concatenation of two lists 'xs' and 'ys', and then applied to 's' and 'f', is equal to the SK function applied to 'xs' and 's', and then applied to the SK function applied to 'ys', 's', and 'f'.
The lemma tttt_4 assumes that the dimension of vector x is 4 and that the first, second, third, and fourth elements of vector x are all true. It shows that the zeros of the boolean input x is an empty set.
If 'x' is in the open ball centered at 'y' with radius 'e', and 'e' is less than or equal to 'f', then 'x' is also in the open ball centered at 'y' with radius 'f'.
If "a" is equal to zero, then applying the object map "LK23" to "a" is the same as applying the object map "FF" to zero.
The lemma named "wf_Fields_Ex" assumes that the program P is well-formed according to the metadata well-formedness condition "wf_md", and that C is a class in program P. It shows that there exists a set of field data types such that the program P confirms that the class C has these fields.
The lemma named "escape_path_Hintikka" assumes that "steps" is an escape path and that "steps" is saturated. It shows that the set of formulas in the tree "steps" satisfies the Hintikka property.
The lemma named "matrix'" assumes that both "set_of_vector X" and "set_of_vector Y" are bases. It shows that the function "f" applied to the "i"th element of "X" is equal to the sum of the product of the "j"th element of the matrix' X Y f and the "j"th element of "Y", for all "j" in the universal set.
If "i" is an ordinal number, then "j" is an element of the successor of "i" in the ZFC in HOL if and only if "j" is an ordinal number and "j" is less than or equal to "i".
If a system is secure, and both the context and its body are of low integrity, and if the system state 's' is equivalent to 'ss', and if both 's' and 'ss' can transition to 't' and 'tt' respectively when filled with context 'C' and 'I', and if the body is filled with context body 'I', and if the context result is low, ...
If a natural number 's' is greater than zero and the modulus of another natural number 'l' with 's' is less than 's' minus one, then the successor of the modulus of 'l' with 's' is equal to the modulus of the successor of 'l' with 's'.
If "x" is a transitive set, then the successor of "x" is also a transitive set. This is assumed under the condition that "x" is a transitive set.
If a monad resumption exists for the return function, the bind function (which is a pair of 'a and a list of 'w, and 'm), and the pause function (which involves 'o, 'i, and 'm), then a monad resumption also exists for the return function, the bind function (which involves 'a, and a writerT of 'w, 'a, and 'm), and the p...
If the diamond of p equals p, then the box of x applied to the diamond of x applied to p is less than or equal to p.
The lemma in the Fac_impl context shows that for any program Γ, it holds that if we call the Fac function with N, and then call the Fac function with M, and then set R to be the sum of the results, the final state will be such that R is equal to the factorial of N plus the factorial of M.
The function "list_comb" applied to "f" is injective.
, assuming 'R' is a preference profile and 'y' is strictly preferred over 'x' according to the Pareto preference relation of 'R', then 'x' is not in the social choice function of 'R'.
If the lazy list "t" taken to the power of "i" is not equal to an empty lazy list, then the length of the lazy list "t" dropped to "i" is equal to "i".
"is_Less a" is true if and only if there exist integers i and j such that a equals "Less i j".
The lemma named "transitive" is defined with three fixed elements: P, Q, and R, all of which are of type pi. It assumes two conditions: P is bisimilar to Q, and Q is bisimilar to R. Under these assumptions, it shows that P is bisimilar to R.
The lemma "tmneg_allpolys_npoly" assumes that 'a is a field with characteristic 0. It shows that if all polynomials are non-polynomial for a given term 't', then all polynomials are also non-polynomial for the negation of term 't'.
If the result of splitting the maximum element from a tree 't' is a pair (t',a), and if 't' is an AVL tree and 't' is not a leaf, then depending on the case of 't': - If 't' is the same as 't', then 't' is an AVL tree and the height of 't' is equal to the height of 't'. - If 't' is different from 't', then 't' is an A...
The composition of the stream processor "sp" with the stream processor "sp'" is equal to the transitive closure of the composition of "sp" with "sp'".
If X is a Lindelof space and S is a closed subset within X, then the subtopology of X restricted to S is also a Lindelof space.
The lemma named "orbit_sim_step" assumes that "s" is in the orbit of "t" under the function "f". It shows that the function "f" applied to "s" is in the orbit of the function "f" applied to "t".
If 'r' is less than the length of list 'xs', and the pair (rs, xs') is equal to the result of function 'part1' applied to 'xs', 'l', 'r', and 'a', and 'l' is less than or equal to 'i', and 'i' is less than 'rs', then the 'i'-th element of list 'xs'' is less than or equal to 'a'.
The cardinality (or size) of the set [r, mod m, c] is equal to "Suc 0" (which is 1 in Isabelle) if m equals 0. Otherwise, it is equal to "Suc c" (which is c+1).
The function which maps a real number 't' to the inverse of 't' to the power of 'k' divided by the exponential of the inverse of 't', tends to 0 as 't' approaches 0 from the right.
The supremum of an empty set is equal to an empty interpretation.
If for all indices 'i' less than the length of the list 'ts', the pair (ts[i], ts'[i]) belongs to the relation 'Red_term' raised to the power of ns[i], and if the sizes of the lists 'ts', 'ts'', and 'ns' are all equal, and if there is no match between the patterns 'ps' and the list 'ts' mapped by the function 'dterm', ...
If 'x' is not a number (NaN), then 'x' cannot be infinity.
For every element 'e' in the set of degenerate lotteries, and for every element 'm' in the set P, either 'e' is at least as preferred as 'm' according to the preference relation R, or 'm' is at least as preferred as 'e' according to the same preference relation R.
The lemma defines identity for properties. According to this definition, two properties F and G are identical if and only if, necessarily, for all individuals x, x has property F if and only if x has property G.
Subtracting the signed modulus of 'a' and 'b' from 'a' is equal to the product of the signed division of 'a' and 'b' and 'b'.
If A is an ideal of a ring R, H is a subset of the carrier of a module M, r is an element of the carrier of R, and H is not an empty set, then for any s in the set of natural numbers less than or equal to n that maps to A, and any g in the set of natural numbers less than or equal to n that maps to H, the scalar multip...
The lemma named "SELECT_pw_simps" contains three statements: 1. The SELECT operation with predicate P never fails. 2. The result of the SELECT operation with predicate P is None if and only if for all x, P x is not true. 3. The result of the SELECT operation with predicate P is Some x if and only if P x is true.
If 'p' and 'q' are elements of the set 'up R', then the function that maps each 'i' to the sum of 'p i' and 'q i' is also an element of the set 'up R'.
The angular component of one is equal to one in the p-adic integers.
The lemma named "oprod_wf_Id" assumes that "r" is a total relation, "r'" is also a total relation, "r - Id" is a well-founded relation, and "r' - Id" is a well-founded relation. It shows that the ordinal product of "r" and "r'", subtracted by the identity relation "Id", is a well-founded relation.
If a tree 't' is less than or equal to 'w', and 't' is also less than or equal to the product of vector 'v' and its transpose, given that 'v' is a vector, then the union of 't' and 'e' is less than or equal to the prim_W function of 'w', 'v', and 'e'.
The lemma named "split_block_hermitian_4" assumes that matrix A is Hermitian and the dimension of its row is greater than or equal to n. If the matrices A1, A2, A3, and A4 are the result of splitting the matrix A at the nth row and nth column, then it shows that the matrix A4 is also Hermitian.
The lemma named "integral_component_ubound" is about a function 'f' that maps from one Euclidean space to another. It assumes that the function 'f' is integrable on the closed box defined by 'a' and 'b'. It also assumes that for all 'x' in this closed box, the dot product of 'f' and 'x' with respect to 'k' is less than...
The lemma named "cos_diff_limit_1" assumes that the cosine of the difference between two sequences, theta and Theta, tends towards 1 as the index j goes to infinity. Under this assumption, it is possible to obtain a sequence k such that the difference between the sequence theta and the product of the sequence k and 2pi...
An element 'x' is in the multiset union of 'MM' if and only if there exists a multiset 'M' such that 'M' is in 'MM' and 'x' is in 'M'.
If for all pairs of elements x and y, if Q holds for x and P holds for the pair x and y, then Q holds for y and f of y equals f of x. Then, if Q holds for x and a binary relation chain P holds for the list formed by appending y to the list that starts with x and continues with elements from zs, then f of y equals f of ...
The lemma named "merge_length_le_Suc" assumes that "as" and "bs" are merged into "us" and "vs". It shows that the length of "us" is less than or equal to the successor of the length of "vs".
If "Some f" equals the result of the function "match" applied to "ts" and "xs", then the function "wf_tuple" applied to "n", the union of the set "ts" with the function "MFOTL.fv_trm t" applied to each element "t" in "ts", and the function "tabulate" applied to "f", "0", and "n" is true.
The lemma "deriv_ln_Gamma_complex" assumes that "z" is not a non-positive real number. It shows that the derivative of the natural logarithm of the Gamma function at "z" is equal to the Digamma function of "z", where "z" is a complex number.
If the current list is not empty, then the list resulting from dropping the first element of the current list is equal to the tail of the current list.
The product of 'x' and 'L' is equal to the least upper bound of the product of 'x' and 'bot' and the product of the negation of 'x' and 'L' with 'L'.
If the following conditions are met: 1. S1 is the top element on the optional set excluding X, 2. S2 is also the top element on the optional set excluding X, 3. X is a finite set, 4. S2 is less than or equal to S1, 5. The symmetric difference of S1 and S2 is less than S1, Then, the function n_o applied to the sy...
If the string "u" is not empty, then the first element of the concatenation of "u" and "v" is the same as the first element of "u".
The lemma "resolve2_no_heap" assumes that the effect of the function "resolve2" with parameters "l", "xs", "ys" transforms heap "h" into heap "h'" and results in "r". It shows that heap "h" is equal to heap "h'".
If 'p' is a prime number, then 'p' divides the product of 'a' and 'b' if and only if 'p' divides 'a' or 'p' divides 'b'.
The lemma named "neg_equal_zero" assumes that "x" is an element of the carrier "R". It shows that the negation of "x" equals "x" if and only if "x" equals zero.
If a function 'f' is continuous on a set 'A', then the function that applies the exponential function to 'f' (denoted as 'eexp (f x)') is also continuous on 'A'.
If there exists a positive integer 'k' such that for all 'l' and 's', if 'P l s' holds then 'e' prime of 's' is less than or equal to 'k' times 'e of s', and if 'c' is a command that, when executed in a state satisfying 'P', terminates in a state where 'e' prime is less than or equal to 'Q', then 'c' is also a command ...
The lemma named "collision_within_p" assumes that the stopping distance of the ego vehicle is less than or equal to the stopping distance of the other vehicle, and that the stopping distance of the ego vehicle is less than the stopping distance of the other vehicle. It shows that a collision occurs over the time interv...
The inner product of the Euclidean representation of two lists "xs" and "ys" is equal to the inner product of these two lists in the context of a lower value relation. This is true if the lengths of both lists "xs" and "ys" are equal to the dimension of the executable Euclidean space 'a.
The product of the function "n" of "x" and the greatest element ("top") meet "L" is equal to the product of the function "n" of "x" and "L".
If 'a' is an object in the category, then the identity morphism of 'a' is a monic (or injective) arrow from 'a' to 'a' in the category.
The lemma named "alphaBoundOutput" is defined with four fixed terms: 'a', 'x', 'P', and 'x'', where 'a' and 'x' are names, 'P' is a pi calculus process, and 'x'' is another name. The lemma assumes that 'x'' is fresh for 'P', meaning 'x'' does not occur in 'P'. The lemma then shows that the bound output process with 'a'...
If the relational incorrectness is valid for two pairs of preconditions and postconditions (P1, Q1) and (P2, Q2) with the same commands c and c', then the relational incorrectness is also valid for the disjunction of these preconditions and the disjunction of these postconditions with the same commands c and c'.
The lemma named "compress_normalize_output_interfaces_None" assumes that a generic primitive matcher "beta" exists, the match "m" is in normalized negation normal form, and the function "compress_normalize_output_interfaces" applied to "m" returns None. Under these assumptions, it shows that the match "m" does not matc...
Given a positive integer "m" and a power residue class "c" of "m", there exists an element "x" that belongs to both "c" and the ring of p-adic integers.
For a fixed polynomial 'f' from an ordered additive abelian group with absolute value, the infinity norm of 'f' is greater than 0 if and only if 'f' is not equal to 0.
The function "check_generate" with parameters a and b is sorted with respect to the relation "<\<^sub>r\<^sub>l\<^sub>e\<^sub>x\<^sub>2".
The lemma "path_conforms_with_strategy_irrelevant_updates" assumes that a path 'p' conforms with a strategy 'σ' on a set 'P'. It also assumes that for any element 'v' in the set 'P', the strategy 'σ' at 'v' is equal to the strategy 'σ'' at 'v'. Under these assumptions, it shows that the path 'p' also conforms with the ...
The set obtained from converting a list 'l' using the 'lm.to_list' function is equal to the set obtained from the 'Assoc_List.set' function applied to 'l'.
Evaluating the function "r_e2ls" with a list containing "e" as an argument is equal to the function "e2ls" applied to "e".
If 'b' is a non-zero element in the fraction field of 'R', then the product of 'b' and the inverse of 'b' in the fraction field of 'R' is equal to the multiplicative identity in the fraction field of 'R'. Similarly, the product of the inverse of 'b' and 'b' in the fraction field of 'R' is also equal to the multiplicati...
For any numerals 'l' and 'k', the signed take bit operation on 'l' and the negative of 'k' (where 'k' is a numeral with an additional bit 1) is equal to the signed take bit operation on the predecessor of 'l' and the negative of 'k' minus 1, all multiplied by 2 and added to 1. This is true for all integers.
The lemma named "same_carrierI" makes the following assumptions: 1. The set C is finite. 2. The sets A and B are subsets of C. 3. For any element 'a' in the set C but not in A, the function 'g' applied to 'a' equals 1. Similarly, for any element 'b' in the set C but not in B, the function 'h' applied to 'b' equals 1. ...
If the length of the initial list "initlm" is greater than the maximum of two numbers "m" and "n", and "m" is not equal to "n", then the following conditions hold: 1. The function that takes a pair of a number and a list and returns whether the number is equal to 3, when applied to the result of the function "abc_step...
Inserting an element 'x' into a set 'X' is equivalent to the union of set 'X' and the set containing 'x'.
For any three extended non-negative real numbers x, y, and z, if y is not equal to infinity, then the result of subtracting y from the sum of y and x is equal to x.
If 'x' is in the gamma of 'a1' and 'x' is also in the gamma of 'a2', then 'x' must be in the gamma of the infimum (greatest lower bound) of 'a1' and 'a2'.
In the context of an isomorphism in a digraph homomorphism, if 'f' is a mapping from 'a' to 'b' in a digraph 'A', then the inverse of the digraph homomorphism 'F' applied to the arrow map of 'F' applied to 'f' equals 'f'.
: Assuming that a point 'x' returns to the set of points in 'S' where the dot product of 'x' and 'i' equals 'c' (denoted as "returns_to ?P _"), and 'S' is a closed set. Also, assuming that the dot product of the function 'f' evaluated at the Poincare map of 'x' and 'i' is not zero. Furthermore, assuming that for almos...
If "p" is a pointer to a flattened list "l", then initializing the flatten iterator on "p" results in a flatten iterator over the list "l" with "l" as the remaining elements to be iterated over.
If a function 'f' expands to 'F' at some base series 'bs', then it is eventually true that the value of 'f' at any 'x' is less than or equal to the value of 'f' at the same 'x', as 'x' approaches infinity.
The lemma named "ahm_\<alpha>_aux_conv_map_of_concat" assumes that "bhc" is a bounded hashcode with identity relation and equality, and that the auxiliary invariant "ahm_invar_aux" holds for "bhc", "n", and the array "xs". Under these assumptions, it shows that the auxiliary function "ahm_\<alpha>_aux" applied to "bhc"...
The lemma named "flowsto_unionE" assumes that there is a flow from set X0 over time T through set CX to the union of sets Y and Z. It then concludes that there exist two sets, X1 and X2, such that X0 is the union of X1 and X2, there is a flow from X1 over time T through set CX to Y, and there is a flow from X2 over tim...
A function that always returns a constant value P is continuous.
If there exists a triple set A and a term t such that under the context G, the assertion P' holds before the execution of t and Q' holds after the execution of t, and for all Y, s, and Z, if P holds for Y, s, and Z, then for all Y', s', if for all Y and Z', P' holds for Y, s, and Z' implies Q' holds for Y', s', and Z',...
The composition of the converse of 'x' and the complement of the composition of 'x' and 'y', all multiplied by 'y', equals zero.
The relation "R" holds between an option "x" and the result of mapping a function "f" over an option "y" if and only if the relation "R" holds between "x" and "y" where "y" is transformed by the function "f".
If the least value of function 'f' for 'x' is less than 'x', then the value of function 'f' at 'x' and the least value of 'f' for 'x' is not zero.
The integer 2 is less than or equal to the integer 3.
If "n" is greater than zero, then the set of all blocks is a subset of the set of multiple blocks of "n".
Looking up a key 'k' in an associative list 'al' that has been updated with a value 'v' using a function 'f', is equivalent to looking up the key 'k' in the original associative list 'al' and then applying the function 'f' to the result. If the key 'k' is not found in the original list, the function 'f' is applied to t...
The difference between two fiber relations 'r' and 's' is also a fiber relation.
The first encoding of a partial state with given dimensions and variables is equal to the second encoding of a partial state with the same dimensions and variables.
If 'x' is equivalent to the implication operator, then it leads to a contradiction or falsehood.
A constant function is n-times differentiable on the set S.
If "u" is a unit in the p-adic numbers, and "a" and "b" are both elements in the p-adic numbers, and the valuation of the product of "u" and "a" is less than or equal to the valuation of the product of "u" and "b", then the valuation of "a" is less than or equal to the valuation of "b".