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train · 499k rows

Statement: stringlengths 7 24.3k
The lemma named "hd_last_sorted" assumes that a list "xs" is sorted in a certain order ("sortedP xs") and that the length of this list is greater than 1 ("length xs > 1"). It shows that the first element of the list ("hd xs") is less than or equal to the last element of the list ("last xs").
The lemma "grdOf" assumes that "c'" is in "gPB", "l'" is in the set "c'", "x" is in "GE.nv2L l'", and "x" is not in "infTp (tpOfV x)". It shows that "grdOf c' l' x" is in the set "c'" and "x" is a guard of "grdOf c' l' x" according to "GE.isGuard".
: 1. The fmap of a function f and the bottom element of the error type is the bottom element. 2. The fmap of a function f and an error e is the same error e. 3. The fmap of a function f and a successful result x is a successful result with the function f applied to x.
If you update a function 'f' at a point 'n' using a destructor 'destr' and a constructor 'constr' on a state 's' and apply it to 'x', it is equivalent to checking if 'x' is equal to 'n'. If 'x' is equal to 'n', then apply the constructor to the result of applying 'f' to the result of applying the destructor to 's' at '...
The lemma named "funcsetI_permutes" assumes that "f" is a permutation of "S". It shows that "f" is a function from "S" to "S".
The recursive function "word_rec" with base case "z", function "id" (which is the identity function), and input "n" is equal to "z".
The set of all real numbers is infinite.
The lemma named "is_filter_rel_witness" assumes that F is a filter, G is a filter, the relation predicate of the composition of C and C' related to F and G, and the condition of negative distribution of relation filter for C and C' holds. Under these assumptions, it shows that the relation predicate witness for C and C...
If 'p' is a power of a prime number, then 'p' is not a unit.
If a function 'f' is monotonic, and if set 'A' is a subset of set 'B', then if an element 'x' is in the image of 'A' under 'f', it must also be in the image of 'B' under 'f'.
The lemma named GrpE, which is a case predicate, assumes that "BNF_Def.Grp A f x y" is true. If this is the case, it obtains that "y" is equal to the function "f" applied to "x", and "x" is an element of "A".
The lemma "eval_monom_mono" assumes that for all x, the function f (which is an assignment from a linearly ordered set 'v to a polynomial carrier 'a) is greater than or equal to the function g at x. It also assumes that for all x, the function g at x is greater than or equal to 0. Given these assumptions, it shows that...
The lemma named "pi_iam" asserts that the "iam_iterateoi" function is a proper iterator.
If the following conditions are met: 1) there is a continuous map from space X to space Y, 2) space X is compact, 3) space Y is a Hausdorff space (a type of topological space with certain properties), and 4) the image of the topological space X under function f is the topological space Y, then it can be concluded ...
The weakest liberal precondition of the function "fb" applied to the differential equation "x' = f" with the guard "G" on the time set "T" and the state set "S" at time 0, with respect to the postcondition "Q", is the set of all states "s" such that if "s" is in "S", then for all times "t" in "T", if for all times "τ" ...
Applying the SK function to the addition of x and y is equivalent to adding the results of applying the SK function to x and y separately.
"$s^{m}_n equals (1+i)^n times $a^{m}_n" if "m is not equal to 0". This is applicable for natural numbers n and m.
The low point 'p' is less than or equal to the high point 'p'.
The lemma named "eq_almost_full_on_finite_set" assumes that "A" is a finite set. It then shows that the set "A" is almost full on the equivalence relation.
If "g" is an element of "G" and "T" of "g" equals "h", then the inverse "T" of "h" equals "g".
, assuming for any alpha and beta, if alpha is in the quasi-Borel space of X and beta is in the quasi-Borel space of Y, then the composition of function f and the function that maps r to the pair (alpha of r, beta of r) is in the quasi-Borel space of Z. Under these assumptions, it shows that function f is a morphism fr...
A record with 'xc' equal to 'x' and 'yc' equal to 'y' is equal to a record 'p' updated such that 'xc' is 'x' and 'yc' is 'y'.
If "fs" are factors of "a" in the group "G", and "f" is an element of the set "fs", and the set "fs" is a subset of the carrier "G", then "f" divides "a".
The lemma named "moebius_pt_moebius_rotation_zero" shows that the point 0 under a Möbius transformation represented by a rotation of angle phi remains at point 0.
For any elements 'x' and 'y', the product of 'n x' and the sum of 'n x' and 'n y' is equal to 'n x'.
If a proposition p is derivable from a set of propositions G in the S5 system, then under the condition of equivalence, the set G entails the proposition p.
In the context of a closed ring, there is a lemma about the coefficients of polynomial multiplication being closed. If for any index 'i', the 'i'th coefficient of polynomial 'p' and the 'i'th coefficient of polynomial 'q' are in set 'A', then it shows that the 'i'th coefficient of the product of polynomials 'p' and 'q'...
The AF letter of the unfolding of a formula phi and a letter nu is equivalent to the AF letter of the formula phi and the letter nu.
The opposite of a tiny semicategory is also a tiny semicategory.
The lemma "augment_cf_impl_simps" states two conditions: 1. If the function "augment_cf_impl" is called with an empty list for its second argument, it will simply return the first argument "cf" without any changes. 2. If the function "augment_cf_impl" is called with a list that has at least one element (represented b...
, starting from a state where the proposition 'True' holds (which is always the case), if we initialize a variable 'S' to 0 and a variable 'I' to 1, and then enter a loop where, as long as 'I' is not equal to 'n', we increment 'S' by 'I' and increment 'I' by 1, then after the loop finishes, 'S' will be equal to the sum...
The lemma named "foldl_rule_aux_P" is defined with respect to a function 'I' that takes a state and a list of elements of type 'a. It assumes three conditions: 1. The initial condition: 'I' holds for the initial state 'σ0' and the initial list 'l0'. 2. The step condition: For any list 'l1', 'l2', element 'x', and stat...
The lemma "PosOrd_StarsE2" assumes that the sequence "Stars" with an element "v" followed by a sequence "vs1" is strictly less than the sequence "Stars" with the same element "v" followed by a sequence "vs2". It shows that the sequence "Stars" with "vs1" is strictly less than the sequence "Stars" with "vs2".
The function "list_concat" when applied to two arguments (made possible by uncurrying) in the context of a monadic return operation, is refined by the same function in the context of a non-monadic return operation. This is true when the function is applied to two arguments that are both lists of elements of type A, and...
The lemma establishes that for any expressions E and Es, if the inline call of expression e and E equals to Throw a, and the call of E equals to aMvs, then e equals to Throw a or e equals to addr a.
If "x" and "m" are coprime (i.e., their greatest common divisor is 1) and "m" is greater than 1, then the product of the modular inverse of "x" with respect to "m" and "x", when taken modulo "m", equals 1.
If a transition system 'TS' is in the Hierarchical Protocol Theory of a statechart 'ST', and a state 'S' is in the configuration of 'ST' and also in the ChiPlus relation of the Hierarchical Automaton of 'ST' applied to the target of 'TS', then 'S' is in the ChiStar relation of the Hierarchical Automaton of 'ST' applied...
The reverse of word w concatenated with the reverse of word v is equivalent to word w being a suffix of word v.
The lemma "subst_infer_check_s" is about variable substitution in type checking. It has three parts, each with different conditions and results. 1. Given a variable v, a statement s, a context \<Gamma>, and a type \<tau>', if the context \<Gamma> is formed by appending a tuple (x, b, c) to \<Gamma>\<^sub>1, where c is...
"for all 'x', if 'P' is not true for 'x', then the epsilon of 'P' equals 'y'".
The substitution of a clause with an inserted formula phi into a set of formulas Phi, given a mapping m, is equal to the minimum disjunctive normal form of the substitution of the formula phi with the mapping m, tensor multiplied with the substitution of the clause with the set of formulas Phi and the mapping m.
For a deterministic fold map operation on a set X, if each element in the set satisfies a certain property P, then the entire map operation on the set also satisfies the property P.
: given the assumptions that the domain of function map 's2' is a subset of 'vs2' and the domain of function map 's1' is a subset of 'vs1', it shows that if the restriction of 's1' to 'vs2' is equal to the restriction of 's2' to 'vs1', then 's1' and 's2' agree.
If a pair (a, a') is in the set K and another pair (aa, a'a) is in the fmap_rel of sets K and V, then the pair formed by looking up a in aa and a' in a'a is in the option_rel of set V.
If Q is true, then D and F are both true and at the same time, D and F are both not true.
The function "traverse" applied to an argument "f" is equivalent to the composition of the function "un_B" and the recursive reverse function "recurse_rev" applied to the composition of "B" and "f". This is denoted as "?lhs = ?rhs", where "?lhs" represents the left-hand side of the equation and "?rhs" represents the ri...
If the return time to a set P is continuous at a point x, and x returns to P, and if P is a closed set, then the Poincare map is continuous at x.
The lemma named "real_alg_2'" assumes three conditions: "unique_root (q,l,r)", "poly_cond q", and "root_info_cond ri q". It shows that the function "real_alg_2'" when applied to the parameters "ri", "q", "l", and "r", satisfies the "invariant_2" condition. Moreover, the real number derived from the function "real_alg_2...
For a function 'f' mapping from a Euclidean space 'n' to a Banach space 'a', the function 'f' is integrable over the set 's' if and only if the function that equals 'f' for elements in 's' and equals zero otherwise, is integrable over the entire universe set.
The lemma named "prm_set_unit_action" assumes that "a" is an element of set "S" and "b" is not an element of set "S". It shows that swapping "a" and "b" in set "S" results in a set that is the union of the set "S" without "a" and the set containing "b".
The function "ensure_pre_insertion_validity" with parameters "node", "ptr", and "child" is pure with respect to the heap "h". This means that the function does not have any side effects and its output is solely determined by its input parameters.
The lemma named "simply_connected_retraction_gen" shows that if the following conditions are met: 1) Set S is simply connected, 2) Function h is continuous on set S, 3) The image of set S under function h equals set T, 4) Function k is continuous on set T, 5) The image of set T under function k is a subset of set ...
If a greedy conforming path "p" from "σ" to "σ'" starting at "v0" is equal to the list consisting of "v" followed by "P", then "P" is an empty list if and only if "v0" is a dead end.
Taking the first 'n' elements from a function 'f' and then dropping the first 'm' elements is equivalent to first dropping the first 'm' elements from the function 'f' and then taking the 'n - m' elements.
The lemma named "fvar_upd_no_jump" assumes two conditions: 1. "upd" is defined as the second element of the first element of the function "fvar" with parameters "statDeclC", "stat", "fn", "a", and "s'". 2. The "abrupt" function applied to "s" does not equal to "Some (Jump j)". Under these assumptions, it shows that t...
: 1. The successor of a numeral word 'w' is equal to the numeral 'w' plus one. 2. The predecessor of a numeral word 'w' is equal to the numeral 'w' minus one. 3. The successor of the negative of a numeral word 'w' is equal to the negative of the numeral 'w' plus one. 4. The predecessor of the negative of a numeral word...
"E is a sub-event-system of itself under the identity relation".
If a list of words 'ws' is in the set of all lists over a set 'A', then applying a function 'f' to the concatenation of 'ws' is equal to the concatenation of the list obtained by applying 'f' to each word in 'ws'.
If "t" matches relations, then the cost of output for "cf'" and "sf" with "t" is equal to the cost of output for "cf", "sf", and "t".
The sine of any natural number multiplied by pi is equal to zero.
The start heap P is preallocated.
Zero is less than the extended non-negative real number representation of x if and only if zero is less than x.
The lemma named "stoch_non_neg_vec_norm1" assumes that "v" is a stochastic vector and a non-negative vector in a real n-dimensional space. It shows that the 1-norm of "v" is equal to 1.
If the length of a list 'xs' is equal to the product of 'm' and 'k', then the function 'f' applied to the concatenation of 'xs' and 'ys' with 'k' as an argument is equal to the function 'f' applied to 'xs' with 'k' as an argument concatenated with the function 'f' applied to 'ys' with 'k' as an argument.
: assuming that the function "find0" with parameters "k", "i", and "a" returns false, "i" is less than the successor of the length of the vector "\<AA>", and either "a" is a level 0 formula or the length of "P" is less than or equal to the length of "\<AA>", it follows that the formula "a" is satisfied by the extension...
The function "h" applied to a machine "M" and a pair "(q,x)" is equal to the following: First, we define "m" as the set transformed into a map image by applying the function that takes a quadruple "(q,x,y,q')" and returns a pair "((q,x),y,q')" to the transitions of the machine "M". Then, if "m" applied to the pair "(q,...
The pair "(n_ns', n_ns)" is in the relation that is the product of the node relation over the unit relation and "R", and the list set relation over the node relation over the unit relation and "R". 3. The third assumption, named "B", is a side precondition. It states that for a pair "(n, ns)" in "n_ns", the function "n...
If "a" is an element of group "G", then the result of the operation "Gmult" of "a" and any integer "n" is also an element of group "G".
If "r" is a vertex in graph "G", then the cost of "r" in the reverse of "ikkbz_sub r" is equal to the cost on the left of "ikkbz_sub r".
The inverse of a constant Dirichlet series is equal to the Dirichlet series of the inverse of the constant. This is applicable in a field.
If for every element X in set P, X is less than or equal to Y, and set P is not empty, then the supremum of set P is less than or equal to Y.
If a heap is well-formed and has a well-defined type, and if a known pointer points to a parent, then if the parent-child relationship exists in the heap, the child pointer is included in the set of object pointers in the heap.
The strict lower set of a relation 'r' on a set 'A' is a subset of the lower set of the same relation 'r' on the same set 'A'.
"unwind_cont Δ1" is true for the set that includes Δ1, Δ2, and Δe.
The property l1_inv5 is preserved under the transition relation l1. In other words, if l1_inv5 holds before the transition l1, it will still hold after the transition.
The lemma named "older_seniors_cases_subset" has three assumptions. 1. If the set of older seniors of 'x' at 'n' is a proper subset of the set of older seniors of 'y' at 'n', then 'P' holds. 2. If the set of older seniors of 'x' at 'n' is equal to the set of older seniors of 'y' at 'n', then 'P' holds. 3. If the set o...
The lemma named "head_correct" assumes that the finger tree "t" satisfies the invariant "ft_invar" and "t" is not empty. It shows that the head of the tree "t" is equal to the head of the list representation of the tree "t".
If expression "e" types term "t" as "T", then for all "i", term "t" is not equal to bound variable "i".
The lemma named "imp_cnj_compat_eqvL" assumes that the formulas phi1, phi2, and chi are in the set of all formulas. It shows that if phi1 is equivalent to phi2, then the conjunction of phi1 and chi is equivalent to the conjunction of phi2 and chi. This is proven.
For a given domain 'd' and a variable 'z', the product of 'd' and 'z' under the operation 'fbox star' and 'fbox' with 'x' is equal to the operation 'fbox star' with 'x' on 'z'.
Applying the function "apsnd" with the function "h" to the result of the co-recursive product of functions "f", "g", and "s" is equivalent to the co-recursive product of functions "f", the composition of functions "h" and "g", and "s".
The extension of a partial matrix with the identity matrix of dimension d1 is equal to the identity matrix of dimension d0.
The function "times_ms_aux" applied to two instances of "MSLCons" (each containing a pair of elements x and y) is equal to a new instance of "MSLCons". This new instance contains a pair where the first element is the product of the first elements of x and y, and the second element is the sum of the second elements of x...
The set generated by the implementation of "gen" function with parameters "x" and "Q" is equal to the set of all "G" such that "G" is generated by the "gen" function with parameters "x" and "Q".
If a context, which is a list of constraints (\<Gamma>) with an additional constraint (\<gamma>) at the beginning, is consistent, then the original context (\<Gamma>) is also consistent. Here, \<tau> is a linearly ordered field.
The lemma named "arc_edge_1" assumes the following conditions: 1. The edge 'e' is less than or equal to the product of vector 'v' and the transpose of the negation of 'v', and this result is further intersected with 'g'. 2. 'v' is a vector. 3. The transpose of 'v' is equal to the product of the transpose of 'r' and th...
If the alpha abstraction of all distinct quasi-fresh variables xs, x, X and xs', x', X' holds, then the alpha abstraction of all distinct quasi-alpha-fresh variables xs, x, X and xs', x', X' also holds.
If "x" is an element of set "X", then "X" is equal to the set obtained by inserting "x" into the set "X" after removing "x" from it.
The function "map_to_sorted_list R" has an interface type that takes an input of a map with key-value pairs (Rk, Rv) and returns a non-deterministic result of a sorted list of these key-value pairs.
The lemma about the components of the Yoneda functor shows the following: 1. The object mapping of the Yoneda functor for a category \<DD> and a cardinal number \<alpha> is a function that maps each object r in the opposite category of \<DD> to the contravariant functor from \<DD> to the category of sets that sends ea...
The set obtained by applying the "lrange_filter" function on a list "xs" with a parameter "l" is equal to the result of applying the "Lrange" function on the set of "xs" with the same parameter "l".
If a function A is monotonically increasing over lists, then the function that applies a filter P to the list produced by A is also monotonically increasing.
The lemma "represented_at_pre_objects_defined" assumes that for a given state tau, if all instances of an object H at the previous state include x, then it shows that all instances of H at the previous state and x are defined in the state tau.
The length of the segment taken from word 'w' up to position 'p' is less than or equal to 'p'.
The function "check_acyclicity" applied to a mapping and a parameter P is equivalent to the following process: First, apply a function to each element of P. This function takes a tuple (C, D, _), and if C equals Object, it does nothing. Otherwise, it looks up D in the mapping. If D is not found in the mapping, it does...
Lemma NSLIM_LIM: Assuming f: "f converges to L in nonstandard sense at a" Shows "f converges to L at a"
If "a" is a weak unit and the composition of "mu" and "a" is not null, then "a" belongs to the sources of "mu".
The lemma named "card_code" is defined for the following variables: "dxs" which is a set represented as a doubly-linked list, "xs" which is a list, "rbt" which is a set represented as a red-black tree, and "A" which is a set. The lemma shows that: 1. The cardinality (or size) of the set represented by "dxs" is equal...
The lemma named "ofsm_table_fix_partition_fixpoint" assumes that "f" is an equivalence relation on the states of a finite state machine "M", the size of "M" is less than or equal to "m", and "q" is a state in "M". It shows that the output of the function "ofsm_table" with inputs "M", "f", and "(m - the cardinality of t...
The content of polynomial g is 1, if the content of polynomial f is 1 and g divides f, in the context of a semiring with a greatest common divisor.

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