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Statement: stringlengths 7 24.3k
If an integer i is greater than zero and another integer j is less than zero, then the product of i and j is less than zero.
The theorem named "reachable_s_finite_thm" is defined for a state 's' of any type 'a'. It assumes that the problem, denoted as 'PROB', is finite and that the state 's' is a valid state in the problem 'PROB'. Under these assumptions, it shows that the set of states reachable from 's' in the problem 'PROB' is finite.
If "n" is not equal to infinity, then the natural number equivalent of the successor of "n" is equal to the successor of the natural number equivalent of "n".
The transitive closure of a single pair set {(a,b)} is equal to the set itself {(a,b)}.
If the first Fréchet condition is satisfied by a filter and the second Fréchet condition is also satisfied by the same filter, then the Cantor intersection property holds.
If A, B, and Y form a Polynomial Ring, h is a ring homomorphism from S to B, f is a polynomial homomorphism from R to S with respect to X, A, B, and Y, and for all elements in the carrier of S, f equals h, and if c is a polynomial coefficient of S, then the first element of the coefficient function of f with respect to...
The lemma named "autoref_ahm_isSng" under the category of "autoref_rules" assumes that "is_bounded_hashcode Rk eq bhc" is a side effect free algorithm, as defined by "autoref_tag_defs". It shows that the function pair "(ahm_isSng, op_map_isSng)" is included in the relation "ahm_rel bhc" with respect to "Rk" and "Rv", a...
A member of the product of two red-black trees (rbt1 and rbt2) is equivalent to a pair (ab) of type 'a and 'b that belongs to the Cartesian product of the set of members of rbt1 and the set of members of rbt2.
Any number 'n' can be decomposed into the product of its squarefree part and the square of its square part.
If the following conditions are met: 1. The pair (efn, fd) is in the set of fields of the superclass of c in the program G. 2. The field class fc is in the set of class fields of c. 3. The program G is a well-structured program. 4. The field name of efn is the same as the field name of fc. 5. The declaring class of ef...
If there exists an element 'x' in set 'A' and set 'A' is finite, then the maximum value of set 'A' is also an element of set 'A'.
A sequence 's' is a strict prefix of a sequence 'w' if and only if the reverse of 's' is a strict suffix of the reverse of 'w'.
The lemma named "Gamma_Gamma_Beta" assumes that the sum of x and y is not a non-positive integer. It shows that the product of the Gamma function of x and the Gamma function of y equals the product of the Beta function of x and y and the Gamma function of the sum of x and y.
The lemma "weakly_preferred_by_men_strictly_preferred_by_a_man" assumes two matches, "match µ" and "match µ'". It shows that "match µ" is weakly preferred by men over "match µ'" if and only if it is not the case that "match µ'" is strictly preferred by a man over "match µ".
The lemma named "wfG_intros2" assumes that a well-formed constant "c" exists in the context of a well-formed boolean "b", a variable "x", and a context "Gamma" under a given set of parameters "P" and a base type "B". It shows that a well-formed context "Gamma" exists with the same variable "x", boolean "b", and constan...
If set A is finite, then there exists an element x such that the FoldSet function with parameters h, z, and A results in x.
The shortest path from a point P through a sequence of two commands c1 and c2 to a point t is equivalent to the shortest path from the shortest path from P through command c1 to command c2 and then to t.
The lemma named "higher_deriv_cmult" assumes that a function "f" is holomorphic on a set "A", "x" is an element of "A", and "A" is an open set. It shows that the j-th derivative of the function that multiplies "c" and "f" at "x" is equal to "c" times the j-th derivative of "f" at "x".
For any given fill, the operation of taking and filling up to the sum of a certain number 'm' and the length of a list 'xs', applied to the concatenation of 'xs' and another list 'w', is equivalent to the concatenation of 'xs' and the operation of taking and filling up to 'm' applied to 'w'.
If the cardinality of set S is greater than or equal to 1, then there exists an element 'y' and a set 'S'' such that set S is the union of the set containing 'y' and set 'S'', and 'y' is not an element of set 'S''.
If a vertex 'v' is in the set of vertices 'G', then for any edge 'e' in the incoming arcs of 'v' in 'G', there exists an edge 'e'' in the outgoing arcs of 'v' in 'G' such that the head of 'e'' is the tail of 'e'.
If a list 'xs' is finite, then the llist (lazy list) created from the concatenation of the list 'xs' and any other list 'ys', is equal to the lazy list created by appending 'xs' and the lazy list of 'ys'.
The lemma named "cdens_ctxt_invarD" assumes that "cdens_ctxt_invar" holds for variables "vs" and "vs'", and context "\<Gamma>", and density "\<delta>". If this assumption is true, it shows the following: 1. The concatenation of "vs" and "vs'" contains distinct elements. 2. The free variables in "\<delta>" are a subset...
The lemma named "ref_WHILEI_invarI" assumes that if the invariant "I" holds for the state "s", then "M" is less than or equal to the refinement of "WHILEI I b f s" under relation "R". It shows that "M" is less than or equal to the refinement of "WHILEI I b f s" under relation "R".
The lemma named "IP_to_UP_var" demonstrates that the function "IP_to_UP" applied to the index "i" and the multiset "mset_to_IP" of the ring "R" with the singleton set containing "i", is equal to the polynomial "X_poly" of the ring "R".
"z times pi-one is less than or equal to x times pi-one and z times pi-one is less than or equal to y times pi-one if and only if z times pi-one is less than or equal to the greatest lower bound of x times pi-one and y times pi-one."
The lemma named "qbs_morphism_dest" assumes that the function 'f' is a mapping from 'X' to the quasi-Borel space of 'Y' under the measure. It shows that 'f' is also a mapping from the measure of the quasi-Borel space of 'X' to 'Y'.
If the Ackermann function domain includes the pair (ack m n, l), then it also includes the pair (n, m, l).
If "d" is false, then "d" cannot be true.
If the product of assumptions A and B is invalid for both p and p', then this implies that the context of the product of invalid assumptions A and B is also applicable for both p and p'.
The argument maximum (argmax) of a function 'f' over a set 'A' is equal to the set of all elements 'x' in 'A' for which the value of the function 'f' at 'x' is equal to the maximum value of the function 'f' over the set 'A'.
The lemma named "ai_1_gt" assumes that for any natural number 'n', 'a n' is either 0 or 1, and 'a i' equals 1. It shows that the value of one-half raised to the power of the successor of 'i' is less than or equal to the sum, over all natural numbers 'n', of the real number conversion of 'a' at the index 'n+i' multiplie...
If beta is a member of the class of ordinals and alpha is an element of beta, then the codomain of the graph is an element of the set Vset beta.
The successor of a number x is in the image of the function "Suc" applied to a set X if and only if x is in X.
When the condition is True, the function "when" applied to True and any value X is equal to X itself.
If M is a fixed point on the unit circle and H is a Poincare line, then the Moebius transformation of the circline H by M results in another Poincare line.
Applying the "cp" function twice to a relation "Q" is equivalent to applying the "cp" function once to "Q".
"not not p is equivalent to p".
In the context of encoding with respect to barbs, we have a lemma named "indRelRTPO_impl_TRel_reflects_barbs". This lemma takes a fixed relation "TRel" which is a set of pairs of 'procT'. It assumes that the relation reflects barbs (interactions) in the induced relation "indRelRTPO TRel" in the source and target calcul...
An empty list is a reduced word for 0 in the set A.
The lemma named "trace_measure2_id" assumes that M0 and M1 are n by n matrices, and that the sum of the product of the adjoint of M0 and M0 and the product of the adjoint of M1 and M1 equals the n by n identity matrix. It also assumes that A is an n by n matrix. Under these assumptions, it shows that the sum of the tra...
If a function 'f' is signum-able at 'F', then 'f' has the signum of 'f' at 'F'.
The lemma "step_preserves_SN_on" assumes two conditions: 1. The ordered pair (a, b) is in the relation r. 2. The relation r is strongly normalizing on the set containing only a. Given these assumptions, it shows that the relation r is also strongly normalizing on the set containing only b.
If node 'n' influences variable 'V' in node 'n' through a path starting with 'a' followed by 'as', then the source node of 'a' is 'n'.
The image of the quotient group G/L under the induced function is equal to the image of the group G under the function eta.
"Tb' of n" is less than or equal to five times "cnt'" of the buildup of Van Emde Boas tree of n.
, assuming "v" is an element of set "V", the result of multiplying 0 with "v" is equal to "v".
The first successor (or the second element) of the sequence generated by the constant Dirichlet series of any constant "c" is equal to "c" itself.
If the following conditions are met: 1. The polynomial D1S + H1T is congruent to 1 modulo p. 2. The polynomial D1*H1 is congruent to C modulo p. 3. D1 is a monic polynomial. 4. The polynomials D1 and H1 are congruent to themselves modulo p. 5. The result of the function "linear_hensel_main" with input n is the pair ...
The modulus of x and y is equal to x if y is zero, otherwise it is equal to the uint16_mod function of x and y.
If propositions 'p' and 'q' are both bound to zero, then the logical equivalence (iff) of 'p' and 'q' is also bound to zero.
: Assuming a run "rho" of the Ate_M algorithm with a given set of higher order oracles "HOs" and "SHOs", and assuming that these oracles satisfy the communication per round property for the Ate_M algorithm. If process "p" does not decide on value "v" at round "r" but decides on "v" at the next round, and if another pro...
The lemma named "max_shift" is defined with a function 'f' that maps natural numbers to clock time and a set 'S'. It assumes that the set 'S' is not empty and is finite. It shows that the maximum value of the function 'f' applied to the set 'S' plus a value 'x' is equal to the maximum value of the function 'f' applied ...
The least upper bound of a set containing only the probability measure 'p' in the subprobability monad is 'p' itself.
The lemma named "os2__inangle" assumes that point A is on the same side of line segment BC as point P, and point C is on the same side of line segment AB as point P. Under these assumptions, it shows that point P is in the angle ABC.
The lemma named "mono2mono_lfilter" shows that the function "lfilter P" is monotone with respect to the partial order relation "sqsubseteq". This lemma is used to introduce continuity and is also used for simplification.
If "x" is not equal to the maximum element (often denoted as "top" in lattice theory), then the infimum (greatest lower bound) of "x" and "y" is also not equal to the maximum element.
Three plus one equals four in the context of integers.
For any two formal power series 'f' and 'g', if 'n' is less than the difference between the subdegrees of 'f' and 'g', then the nth coefficient of the quotient of 'f' divided by 'g' is zero. Here, 'f' and 'g' are of a type that supports addition, unary minus, multiplication with zero, and inversion operations.
"It is necessarily not the case that A is true in all possible worlds".
For all function 'f' and list 'xs', it is equivalent to fold left injective over the list 'xs' with identity function and function 'f' starting with the initial value as True.
The lemma named "leastI" assumes two conditions: 1. "S" is a subgraph of a labeled graph "G". 2. For any labeled graph "C", if "C" is a consequence graph of "Rs", then "C" is maintained by the pair "S" and "G". If these two conditions are met, it shows that "G" is the least graph of type "'x" with respect to "Rs" a...
If H is a subgraph of G, then H is also a finite directed graph.
The antidomain of x multiplied by the sum of the antidomain of y and the antidomain of z is equal to the sum of the antidomain of x multiplied by the antidomain of y and the antidomain of x multiplied by the antidomain of z.
The separation conjunction of any proposition 'P' with 'sep_false' is equivalent to 'sep_false'.
The lemma "insert_index" has two conditions: 1. If 'i' is greater than 'i'', then the result of the function 'insert_index' with inputs 'i' and 'i'' is 'i''. 2. If 'i' is less than or equal to 'i'', then the result of the function 'insert_index' with inputs 'i' and 'i'' is the successor of 'i''.
If X is in the used' of events (evs), then if Y is in the parts of the set containing X, it implies that Y is also in the used' of events (evs).
For a function 'f' that maps integers to integers, the condition that for all integers 'a' and 'b', the value of 'f' at '2a' plus twice the value of 'f' at 'b' equals the value of 'f' at the sum of 'a' and 'b' is equivalent to either of the following conditions: for all integers 'x', the value of 'f' at 'x' is zero, or...
The polynomial "p" is divisible by the polynomial "[:-a, 1:] raised to the power of the order of "a" in "p".
If we have a map iterator "it" over a map "m" with a relation "R", an initial state "I" with an empty set and an initial state "σ0", and a property "P", then the property "P" holds for the state resulting from applying a function "f" to each element in the map "m" in the order defined by the relation "R". This is und...
If the list of first elements (fst) of the pairs in the list xs is distinct, then the mapping of the reversed list xs is equal to the mapping of the original list xs.
, given the assumptions that the function "Jcc_pieces" with parameters P, E, C, M, h0, vs, ls0, pc, ics, frs, sh0, I, h2, ls2, sh2, v, xa, and (e.F{D}:=e') returns a tuple (True, frs0, (xp',h',(v#vs',ls',C0,M',pc',ics')#frs',sh'), err), it can be shown that the function "Jcc_pieces" with updated parameters P, E, C, M, ...
The invariant m2_inv9_M5 holds after the execution of step 1 of protocol m2, involving parties A and B and a nonce Na, given that it held before this step.
If a τ-execution move is possible in a given context "ci" for a program "P" by a thread "t" from an expression "e" with a heap "h" and states "s" to "s'", then a τ-execution move is also possible for the same context, program, and thread from the expression "e" with field "F" of class "D" being assigned the value of ex...
: Assuming that "Ri'" is a total preorder on the alternatives "alts", "i" is an agent, the set of agents is equal to "agents", the agents are finite, the alternatives are finite and not empty. Let "w'" be defined as the winners of the Random Serial Dictatorship (RSD) when "Ri'" is used as the preference relation for...
If H is a submodule of the module M over the ring R, then the kernel of the map from M to the quotient module M/H is equal to H.
The lemma named "red_wf_dgrad_p_set" assumes that "d" is a Dickson grading and "F" is a subset of the set of polynomials of degree "m" with respect to "d". It shows that the inverse of the reduction relation with respect to "F" is a well-founded relation.
The lemma named "hermitian_decomp_dim_carrier" assumes a Hermitian decomposition of matrices A, B, and U. It shows that matrix B is in the carrier matrix of the dimension of the row of A and the dimension of the column of A.
The lemma "FOREACHci_refine" is about refining a certain type of loop, specifically a "foreach" loop with a condition and an invariant. It establishes a relationship between two such loops, one operating on a set 'S' and the other on a set 'S'', with the elements of 'S'' being a transformation (denoted by alpha) of the...
The last element in the list of coefficients of a polynomial 'p' is equal to the leading coefficient of the same polynomial 'p'.
If a list 'xs' is sorted and contains distinct elements, and if 'i' is less than 'j' and 'j' is less than the length of the list 'xs', then the element at position 'i' in the list 'xs' is less than the element at position 'j' in the same list.
If we assume that phi, chi, and psi are formulas, and if we can prove that phi implies chi (1) and chi implies psi (2), then we can also prove that phi implies psi.
If a relation 'r' is a subset of the precondition of 'a', and for all states 's' in 'r', the pair of contexts 'Γ' and 'Θ' proves that 'a' and the command 'c' derived from 's' lead to postcondition 'Q' and assertion 'A' under some fairness constraint 'F', then the pair of contexts 'Γ' and 'Θ' also proves that the annota...
If "r" is not less than or equal to "l", and the length of the list "xs" is not less than or equal to "r", and "x" is equal to the partition of the list "xs" from "l" to "r", and the pair "(p, xs1)" is equal to "x", then "p" minus the successor of "l" is less than "r" minus "l".
If F is a subset of E, then the relation {(x, y) \|x y. nodes_connected (ind F) x y} is transitive. This relation represents pairs of nodes (x, y) that are connected in the induced subgraph of F.
For any comparable element 'c', if the identity of the comparable element equals to some 'c', then 'c' is a comparator.
A sequence "w" satisfies the unf operator applied to a logical statement "phi" if and only if "w" satisfies "phi".
The dimension of the vector c is equal to the dimension of the column A.
The left fold operation over a list "l" with the union operation and starting with an empty set is equal to the union of all elements in the set "l".
If Turing machine A at position 0 and Turing machine B at position 0 are both composable, then the Turing machine formed by composing A and B at position 0 is also composable.
The union over all i of the downward closure of a set L at i is equal to the union over all i of the set L at i.
The set of all labels 'l' such that there exists a command 'c' for which 'l' labels 'prog' in 'c', is finite.
The lemma named "dom_trm_vars" assumes that the term "t" dominates the clause "C". It shows that the variables of the term "t" are a subset of the variables of the clause "C".
An empty concurrent implementation is equivalent to an empty monadic implementation in the binary decision diagram.
The sum of an ordinal number 'x' and an ordinal limit of a function 'f' is equal to the ordinal limit of a new function that adds 'x' to the output of function 'f' for each natural number 'n'.
If a function symbol 'f' is well-typed and the result of 'f' is either protected or forward-protected, then the well-typedness condition holds for the function application expression 'fax f' in the ground expressions.
The identity function and the function "l3_dy" applied to "LtK ltk" both refine to "R23s".
The class unit is a normal subgroup in the context of group theory. Specifically, it asserts that the unit element, denoted as \<one>, forms a normal subgroup when combined with the group operation (\<cdot>) and the group G under the normality condition Normal.
If F is a member of filters, and if a is congruent to b with respect to F, and b is congruent to c with respect to F, then a is congruent to c with respect to F. This is a statement of the transitivity property in the context of these filters and congruence relations.
There exists exactly one "i" in "I" such that "P i" holds for "x", is also a measurable predicate in measure space "M".

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