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The lemma named "cardinalityIntersectionEquality" assumes that both X and Y are finite sets and that they have the same cardinality. It shows that the cardinality of the intersection of X and Y being equal to the cardinality of X implies that X is equal to Y. |
If the tuples (M,l,s,n,l1,t) and (M,l,s,m,l2,r) are both in the set "Step", then n equals m, t equals r, and l1 equals l2. |
If the second element of the result of executing an instruction on a SPARC state is false, and the trap set of the state is empty, and the execution mode value of the state is true, and the fetched instruction from the delayed pool write of the state is a right value, and the decoded instruction of this value is a left... |
An element 'x' belongs to the set of all 'a' such that the pair (a, j) is in the union of the sets 'e', the inverse of 'ov', the inverse of 's', the inverse of 'd', 'f', the inverse of 'f', and the inverse of 'm'. This is equivalent to saying that the pair (x, j) is in the union of the same sets. |
If the positive relational cardinalities of a table 't' are given by 'cf' and the selectivity is reasonable, then the cardinalities are reasonable for the table 't' with respect to 'cf' and 'sel'. |
The lemma named "of_nat_Sup_ennreal" assumes that the set A is not empty and is bounded above. It shows that the natural number representation of the supremum (least upper bound) of set A is equal to the supremum of the natural number representations of all elements in set A, where these representations are in the exte... |
If 'x' satisfies the property 'acyclic_4a', then 'x' also satisfies the property 'acyclic_4'. |
If the remainder of m divided by k equals zero, then the function that divides x by k is an injective function on the interval from r to the remainder of m. |
The last message of a list, which is the result of appending a message 'm' to a list 'xs', is equal to 'm' if 'm' is not a "NoMsg". If 'm' is a "NoMsg", then the last message of the list is the last message of 'xs'. |
The norm of a signed number represented by the sequence "\<zero>#\<one>#xs" is equal to the sequence itself. |
The constant functor from a small category to another small category, with a given object and its identity morphism, is equal to the small functor obtained from the constant functor between the two categories with the given object. |
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Assuming there is a set of processes (denoted as "qq") for which the function F returns a value "m", and the cardinality (or size) of this set is greater than a threshold "T", and there is another set of processes for which the function F returns a value "m'", and the cardinality of this set is also greater than "T",... |
The lemma in the context of the prisoner's dilemma shows that the unit vector with 4 dimensions and index 0 is a state in a 2-dimensional quantum system. |
The lemma named "subterm_cl_subst" assumes that "x" is a subterm of clause "C". It shows that the substitution of "x" with a substitution function "σ" is a subterm of the clause obtained by substitifying "C" with the same substitution function "σ". |
If a value 'v' is an element of the set 'vs', and if the pair (a, b) is equal to the result of splitting the set 'vs' at a certain point 'ram', then 'v' is either an element of the set 'a', an element of the set 'b', or equal to 'ram'. |
If U is a unitary operator, then the composition of U and its conjugate transpose (U*) is equal to the identity operator on complex bounded linear functions. |
If 'a' is less than or equal to 'b', and the function 'f' is integrable between 'a' and 'b', and for all 'x' in the range from 'a' to 'b', 'f(x)' is greater than or equal to zero, then the integral of 'f' from 'a' to 'b' is also greater than or equal to zero. |
If for every natural number 'n', there exists a natural number 'm' such that the set 'f n' is a subset of the set 'g m', then the natural union of the set 'f' is a subset of the natural union of the set 'g'. |
The lemma named "progress_unop_f" assumes the following conditions:
1. The sequence "cs" is well-typed under the context "\<S>\<bullet>\<C>", and its type is a function from an empty list to a list containing "t".
2. "t" is a floating-point type.
3. "cs" is a list of constants.
4. "e" is a unary operation on floating-... |
"C (-1, 1)" is a non-negative polynomial. |
The lemma "eqButUIDf_cong" assumes the following conditions:
1. "frds" and "frds1" are equal except for the user ID function.
2. If "uid" equals "UID1", then the lists "uu" and "uu1" are equal except for "UID2".
3. If "uid" equals "UID2", then the lists "uu" and "uu1" are equal except for "UID1".
4. If "uid" does not ... |
The strict constructor of a sequence with the bottom element is equal to the bottom element. |
The lemma named "ex_suma" assumes that A is not equal to B, B is not equal to C, D is not equal to E, and E is not equal to F. It shows that there exist points G, H, and I such that the sum of the angles ABC and DEF equals the angle GHI. |
If the product of z and x is less than or equal to the product of y and the star of y and z, then the product of z and the star of x and w is less than or equal to the star of y and the product of z and w. |
For any additive abelian group, the difference between 'a' and the sum of 'x' and 'b' is equal to the negation of 'x' plus the difference between 'a' and 'b'. |
If a formula "phi" is well-formed at a given index "idx", and for all "k" in the set "ks", for all "v" in the set of "vs" at "k", "k" is less than "v" and "v" is less than "idx", then the formula "phi" restricted to the variables "ks" and "vs" is also well-formed at the index "idx". |
The lemma named "left_right_simps" consists of three parts:
1. The left part of the application of 'l' and 'r' is 'l'.
2. The right part of the application of 'l' and 'r' is 'r'.
3. If 't' is an application, then applying the left part of 't' to the right part of 't' equals 't'. |
The lemma "oelimder_bcast" assumes that "p" is equal to the broadcast of "fmsg" at location "l" followed by "qq", "l'" is in the set of labels of "q" in the context of "Γ", and the triple "((σ, p), a, (σ', q))" is in the indexed operational semantic transition relation "oseqp_sos" for "Γ" and "i". It then concludes tha... |
The function f_expand commutes, meaning that the order in which two elements 'a' and 'b' are processed by the function does not change the result when applied to a stream 'xs'. |
The weakest precondition of the differential equation where 'x' changes over time according to function 'f' and condition 'G' holds on the trajectory from time set 'T' to state set 'S' at time 0, given postcondition 'Q', is equivalent to the condition that for any state 's' in 'S', if for all time 't' in 'T', condition... |
The lemma named "akra_bazzi_term_ceiling_subtract" in the Akra_Bazzi theory, assumes the following conditions:
1. A real number 'b' is greater than 0.
2. 'b' is less than 1.
3. The real value of 'x₀' is less than or equal to 'b' times the real value of 'x₁' minus 'c'.
4. 1 is less than or equal to 'c' plus the product... |
If a function 'f' maps from set 'A' to set 'B', and 'A1' is a subset of 'A', then the restriction of function 'f' to 'A1' also maps from 'A1' to 'B'. |
"The least prefixpoint of the index 'i' in relation to 'r' is a subset of the least fixed point of the function 'addLPD' applied to 'r'." |
If "a" divides "b", then the greatest common divisor of "a" and the sum of "b" and "c" is equal to the greatest common divisor of "a" and "c". |
If the second element of the af-sub-nu function of X, with parameters xi and zeta, and w, is equivalent to false-sub-n, then the second element of the af-sub-nu function of X, with parameters xi and zeta, and the concatenation of w and nu, is equal to the nu-subset of the normalisation of the af function of xi with the... |
If the action "HStartBallot" transitions the system from state "s" to state "s'" for a process "p", and if the inner invariant "HInv5" holds for the system state "s" and a process "q", and if "p" is not equal to "q", then the inner invariant "HInv5" also holds for the system state "s'" and the process "q". |
If a is equal to b, and a is less than c, then b is less than c. |
A total preorder on a set of alternatives 'alts' is finite if and only if it is a total preorder on 'alts'. |
The norm1 of the scaled vector (r * v) is equal to the absolute value of the scalar (r) times the norm1 of the vector (v). |
, assuming two functors "GG" and "HH" from categories "AA" and "BB" to a category "CC" respectively, the range of the domain of the comma category formed by "GG" and "HH" is a subset of the objects of the same comma category. |
The lemma named "subst_c_eval_v" is given certain fixed variables 'v' and 'c'. Assuming that 'v' evaluates to 's' under 'i', 'c' with 'v' substituted for 'x' evaluates to 's1' under 'i', and 'c' evaluates to 's2' under 'i' with 's' substituted for 'x', it shows that 's1' is equal to 's2'. |
The lemma named "safe" asserts that the property "Safe P" ensures safety. |
The relation of probability mass functions (pmf) where the relation function always returns False, for any x and y, is also False. |
The "blue_basic_invar" is an invariant. |
The lemma named "supp_weakening" assumes that the set G is a subset of the set G'. It shows that the support of G is a subset of the support of G'. |
:
1. If a term 't' of type 'T' is well-typed in a context that extends 'Δ' with a variable of type 'U' and 'Γ', and a term 'u' of type 'U' is well-typed in the context 'Γ', then after substituting the term 'u' for the variable in 't' and shifting the context 'Δ' down by one, the resulting term is well-typed in the con... |
The function "Abs" is a bijection between the set of integers from 0 to the cardinality of the set 'a (not inclusive) and the universal set of 'a. |
If the series formed by the product of a constant "c" and a function "f" is summable (i.e., the series converges), and the constant "c" is not equal to zero, then the series formed by the function "f" alone is also summable. |
Given the assumption that "M" is a chain, one of the following three cases must be true:
1. "M" is a subset of the set containing only "dSUCCEED", and the supremum of "M" is "dSUCCEED".
2. There exists a "r" such that "M" is a subset of the set containing "dSUCCEED" and "dRETURN r", "dRETURN r" is an element of "M", a... |
A list "xs" is equal to an empty list if and only if "xs" is null. It also states that an empty list is equivalent to null. |
The lemma named "standard_rules_edges" assumes that a pair (lhs, rhs) is in the set of standard rules for C and L, and that a triple (l, x, y) is in the set of edges for rhs. It shows that if l equals S_Bot, S_Top, S_Idt, is in the set of S_Const images of C, or is in L, then the thesis holds. |
The lemma named "dim_row_matrix_A_R" shows that the number of rows in the matrix "A_R", which is determined by the signs and subsets, is equal to the length of the subsets. |
In the context of a ring, the lemma named "set_add_comm" assumes that both "I" and "J" are subsets of the carrier "R". It then shows that the set addition of "I" and "J" is equal to the set addition of "J" and "I". This demonstrates the commutative property of set addition in this context. |
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Assuming that points A and A' are not the same, and points A, B, and C are all distinct. Also, points A, A', B, and B' are coplanar (lie on the same plane), and the angle BAA' is a right angle, as well as the angle B'A'A. Furthermore, points A, A', C, and C' are coplanar, and the angles CAA' and BAC are right angles.... |
The lemma named "psign_diff_cancel" assumes that the polynomial "r" evaluated at "x" is not equal to zero. Under this assumption, it shows that the difference in the sign of the polynomial "r*p" and "r*q" evaluated at "x" is equal to the difference in the sign of the polynomial "p" and "q" evaluated at "x". |
The disjunction (logical "or") of two predicates P and Q is commutative. That is, "P or Q" is equivalent to "Q or P". |
The sum of the function f(l+c) for all natural numbers l from a to b is equal to the sum of the function f(l) for all natural numbers l from a+c to b+c. |
The lemma named "zero_rel_carrier" assumes that "U" is an element of "I". It shows that the ordered pair consisting of "U" and the zero element of "U" is in the carrier of the direct limit. |
If "x" is an element of "While_program", and "p", "q", "r", and "s" are elements of "Pre_expression", then the function "tfun2" with parameters "p", "q", "x", "r", and "s" is also an element of "Pre_expression". |
The range of the function "af_letter" applied to a proposition "phi" is a subset of the set of all propositions "psi" such that the nested propositional atoms of "psi" are a subset of the nested propositional atoms of "phi". |
If H is a subgroup of G, then H is normal in the normalizer of H in G. |
A polynomial with coefficients listed as a sequence 'as' followed by zero is equivalent to a polynomial with coefficients listed as 'as' only. |
If "LeftDerivationFix" holds for the variables a, i, D, j, and γ, then there exists a ladder L such that "LeftDerivationLadder" holds for the variables a, D, L, and γ, and the last j value of the ladder L equals j, and the last n value of the ladder L equals the length of D. |
The domain of x is less than or equal to the domain of y combined with the domain of x multiplied by the action of y. |
The lemma named "defNode_ex1" assumes that a variable "v" is in the set of all variables of a graph "g". It shows that there exists exactly one node "n" such that "n" is in the set of nodes of the graph "g" and the variable "v" is in the set of all definitions of the graph "g" at node "n". |
If the successor of a number 'as' is less than or equal to the length of a list 'xs', then taking the successor of 'as' elements from the list 'xs' is equivalent to taking 'as' elements from the list 'xs' and appending the 'as'-th element of 'xs' to the end. |
The parallel composition of "x times 0" and "y times 0" is equal to the meet (or intersection) of "x times 0" and "y times 0". |
If the execution of protocol run "Ra" results in the initial state with actors "A" and "B" and no actions, then updating the protocol run "Ra" to include some actions "al" does not change the result of function "azC". |
The theorem named "compactness" is defined with a fixed set S of countable formulas. It demonstrates that set S is satisfiable if and only if it is finitely satisfiable. This is represented by the equivalence of "?l" and "?r". |
"it is not the case that there exists an 'x' for which 'F x' is true" is equivalent to "for all 'x', 'F x' is not true". |
If the following conditions are met:
1. There are at least 3 elements in set A (has3A).
2. The social welfare function (swf) satisfies the independence of irrelevant alternatives (iia) condition for all individuals (Is) in set A.
3. The social welfare function (swf) is a social welfare function for all individuals (Is... |
The interval from r' (inclusive) with a step size of m' and a count of c' is a subset of the interval starting from n, if and only if n is less than or equal to r'. |
If 'c' is a polynomial coefficient in ring 'S', and there exists a 'j' less than or equal to the first element of 'c' such that the second element of 'c' at index 'j' is not the zero element of 'S', then the second element of 'c' at the maximum 'c' in 'S' is not the zero element of 'S'. |
If the addition of "x" and "y" in the cdl_heap results in "None" for a given object id, then the separation heap of "y" for the same object id will also be "None". |
The lemma named 'substn_crename_comm'' assumes that 'a' is not equal to 'c' and 'a' is not in 'P'. It shows that substituting 'x' with '<c>.P' in 'M' and then renaming 'a' to 'c' in 'b' is equal to renaming 'a' to 'c' in 'b' in 'M' and then substituting 'x' with '<c>.P'. |
The ket vector of x is equal to the canonical basis at the index given by the enumeration of x. |
If a complex function 'f' is such that for every 'x' in its domain, 'f(x)' equals the inner product of 't' and 'x', and also 'f(x)' equals the inner product of 'u' and 'x', then 't' must be equal to 'u'. |
If "i" is less than "k + 1", then incrementing the bound variable "bv" by "inc2" at "k+inc1" in the term resulting from incrementing "bv" by "inc1" at "i" in term "t", is equal to incrementing "bv" by "inc1" at "i" in the term resulting from incrementing "bv" by "inc2" at "k" in term "t". |
The lemma named "ta_der_gterm_induct" with the case name "GFun" assumes the following:
1. The state 'q' is reachable from the tree automaton 'A' derived from the term of the generalized term 't'.
2. For any function 'f', list of terms 'ts', list of states 'ps', and states 'p' and 'q', if the tree automaton rule for ... |
The lemma "normalize_src_ports_normalized_n_primitive" assumes that the match "m" is in normalized negation normal form (nnf). It shows that for all "m'" in the set that results from normalizing the source ports of "m", the source ports of "m'" are also normalized. |
The lemma named "dependent_proj2_abs" assumes that neither p nor q are zero, and that either i or j is not zero. It also assumes that the sum of i times p and j times q equals zero. Under these assumptions, it shows that the absolute projection of p equals the absolute projection of q. |
The derivative of a disjunction (FOr) of two formulas (phi and psi) with respect to a list of variables (xs) and a derivative operator (deriv d0) is the same as the disjunction of the derivatives of the two formulas with respect to the same list of variables and derivative operator. |
If an element "x" is in the set "A" and another element "a" is in the set "x", then "a" is in the union of the set "A". |
If "j" is less than "i", then substituting "u" into the second substitution of the bound variable "Bv j" at position "i" results in the bound variable "Bv j". |
If "y" is an element of the carrier of "Y", then the coretract of the retract of "y" is equal to "y". |
The function which maps each element 'x' of a complete lattice to the greatest lower bound of the set of all 'y' such that 'x' is less than or equal to 'y', is a closure operator. |
If the derivative of a state 's' is infinite and the state 's' is initial, then for all 'n', if function 'f' contains 'n' in the interval (0, FEx g), there exists an 'n'' such that function 'f' contains 'n'' in the interval (m, FEx g). |
For any integer 'i' that is not equal to zero, if 'a' has no zero coefficient, then the result of multiplying 'a' by 'i' (nummul a i) will also have no zero coefficient. |
The lemma named "permutep_monom_of_set" assumes that "f" is a bijective function. It shows that the permutation of the monomial of a set "A" under the function "f" is equal to the monomial of the preimage of "A" under the function "f". |
If the domain of A is a subset of the domain of B, then the domain of the restriction of A by C with respect to x is a subset of the domain of the restriction of B by C with respect to x. |
The integral of the sum of two Dirichlet series (a + b) for the sum of two functions (f + g) is equal to the sum of the integral of the first Dirichlet series for the first function and the integral of the second Dirichlet series for the second function. |
The double application of the function fp to a proposition phi is logically equivalent to the original proposition phi. |
A certain function or operation, represented by "\<s>", is not equal to the identity function or operation, represented by "id". |
The lemma named "impl_red_dfs_ahs_autoref" is about a function "impl_red_dfs_ahs" that is related to another function "red_dfs". It assumes that a relation "R", which is a set of pairs of type 'a and 'a where 'a is hashable, is preferred to be identified. The lemma shows that the function "impl_red_dfs_ahs" and "red_df... |
If a certificate for a cardholder C and an encryption key EKi, which is only used for encryption and is signed with the private key of a Registration Certification Authority (RCA), is part of the knowledge of a Spy in a set of events 'evs' that belong to the cardholder registration set, then the encryption key EKi is e... |
If for all 'k' in the union of the first elements of the sets generated from the first elements of the ordered associative lists 'xs' and 'ys', the function 'f' applied to 'k', the lookup of 'k' in 'xs', and the lookup of 'k' in 'ys' equals 'Some Eq', then the lexicographic order of 'ko', 'f', 'xs', and 'ys' equals 'So... |
The lemma named "wf_imp_SN" assumes that the inverse of set A is well-founded. It shows that set A is strongly normalizing. |
If the length of string 'u' plus a non-negative integer 'q' is less than or equal to the length of string 'w', and 'w' is a prefix of the infinite repetition of 'u', then taking 'q' elements after dropping the length of 'u' from 'w' is equal to taking 'q' elements from 'w'. |
The proposition "P" is true for the recursive function "rec_X" with inputs "a, b, c, d, e, f, x". |
If A is not equal to infinity, then A is less than infinity. |
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