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The finest part of a set X is the set of all singleton sets containing each element x that belongs to X. |
If the composition of the function "norm" with itself equals "norm", then the composition of "fr" with "norm" equals the composition of "norm" with "fr" if and only if the composition of "norm" with "fr" with "norm" equals the composition of "fr" with "norm" and the composition of "norm" with "fr" with "norm" equals th... |
The lemma named "lNact_active" is defined with fixed variables cid, t, and n. It assumes that for all k less than n, cid is parallel at time t of k. It shows that the left action of cid on t at n is equal to n. |
In the TESL language, "K1 strictly precedes K2" is equivalent to the intersection of all sets Y, where Y is defined as "K1 strictly precedes K2" for some natural number n or greater. |
If a state 'q' is in the derivative of a tree automaton 'A' with respect to a tree 't', and 'q' is also in the productive states of 'A' with respect to a set 'P', then 'q' is in the derivative of the tree automaton that only includes productive states with respect to 'A' and 'P', for the same tree 't'. |
If an element 'a' is associated with an element 'b', and 'b' is a unit in the group 'G', and 'a' is a member of the carrier of 'G', then 'a' is also a unit in the group 'G'. |
The lemma assumes that "y" is not equal to the empty word and "k" is not equal to zero. It shows that the kth power of "y" is not equal to the empty word. |
The lemma named "min_maxsimpchain_rev" assumes that "xs" is a minimal maximal simplicial chain. It shows that the reverse of "xs" is also a minimal maximal simplicial chain. |
The star of the set containing only the empty list is equal to the set containing only the empty list. |
Updating the columns of matrix M with a set that includes element i and set I using function f is equivalent to updating column i of the matrix M updated with set I using function f. |
Dividing the base factor of f by the base factor of g is equal to the regular part of the formal Laurent series obtained by shifting the result of dividing f by g by the difference of their subdegrees. |
The lemma named "fun_upd_smaller_oexp" assumes that "f" is a finite function, "x" is in the field of "s", "y" is in the field of "r", and the pair "(y, f x)" is in "r". It shows that the pair consisting of the function "f" updated at "x" with "y", and "f" itself, is in the set "oexp". |
A literal, represented by 'b0' through 'b6', is equal to the string literal of ASCII characters. This is obtained by folding the list of 'b' values, where each 'b' is multiplied by 2 and added to the next 'b' value, starting from 0. This result is then added to 's'. |
The size of the block set B is equal to the size of the block image of B under the function f. |
The evaluation of the ternary implication of the ternary conversion of X1 and X2 is equal to the ternary conversion of the logical implication from X1 to X2. |
The set of integers starting from zero and less than a certain integer 'u' is finite. |
If from a set of hypotheses H, we can deduce A OR C, and also from H, we can deduce A implies B, then from H, we can deduce B OR C. |
The degree of a polynomial represented by an ordered associative list (PP_oalist xs) is equal to the sum of the list obtained by mapping the second element of each pair in the list of the ordered associative list (list_of_oalist_tc xs). |
The first step of the m1 protocol for the actors Ra, A, and B with the nonce Na, when paired with the identity function, refines the relation R_a0iim1_ir. |
If a set M rejects a set S under a family of sets F, and if F is a thin set, and if F is an initial segment of S, and if F is an element of the family of sets F, then this leads to a contradiction. |
If the tail of an arc 'a' is a vertex in graph 'G' and the head of the arc 'a' is also a vertex in graph 'G', then deleting the arc 'a' after adding it to the graph is equivalent to just deleting the arc 'a'. |
The relation "l is similar to r" is equivalent to "r is similar to l". |
If a program Γ can transition from executing the body "bdy" in a normal state initialized with "init s" to a normal state "t", and if the same program Γ can transition from executing "c s t" in a normal state returned with "return s t" to a state "u", then the program Γ can transition from executing a block composed of... |
The application of the "IfFalse" function with parameters b, P, pc, mxs, Tr, and s is true if and only if there exist such ST and LT that s equals to the pair (Boolean#ST, LT) and the integer value of pc plus b is greater than or equal to zero. |
The lemma named "pure_state_nested" assumes that the registers F and G are compatible, H is a regular register, and H is an isolated register. It shows that the pure state of the composition of F and G, when applied to the tensor product of the pure state of H with h and g, is equal to the pure state of the composition... |
If for all elements 'a', the equivalence relation holds between the worlds and the relations of 'a' in model 'M', then model 'M' satisfies the S5n property. |
In the context of Probabilistic Timed Automaton Regions, there is a lemma called compatible_stream. This lemma assumes two conditions. The first condition is a function φ that for any two elements x and y, if x is in set S and x is equivalent to y, then the function φ applied to x is equivalent to the function φ applie... |
If sets A and B are finite, then the minimum product of sets A and B is also finite. |
The size of a bag, represented as "Bag ms", is equal to the result of folding the multiset "ms" with a function that adds each element to a running total, starting from zero. This essentially means that the size of the bag is the sum of its elements. |
If the correspondence "abc_list_crsp" holds true for two lists "lma" and "lmb", and if "m" is less than the length of both lists, then the correspondence "abc_list_crsp" will still hold true even if the "m"th element of both lists is replaced with "n". |
The lemma L6_Controller_Zero:
Assuming that the function "Controller_L" takes the parameters "y", the finite or infinite append of "Zero" to "l", "l", and "z",
and that the successor of "t" in "l" equals "Zero",
and that the successor of "t" in "z" equals an empty list,
it can be shown that "t" in "l" equals "Zero". |
If the set of vectors 'ms' is a subset of 'M', 'ms' is linearly independent, and 'n' is in the span of 'ms', then there exists a unique set of scalars 'rs' such that 'rs' is a subset of the real numbers 'R', the length of 'rs' is equal to the length of 'ms', and 'n' is the dot product of 'rs' and 'ms'. |
The strongly normalizing property on a relation 'r' for the image of a set 'A' under 'r' is equivalent to the strongly normalizing property on 'r' for the set 'A' itself. |
For all 'a', if the value of 'a' in the register 'r' is not None, then the value of 'a' in the register 'r' is equal to the value of 'a' in the register 'r''. If the guards 'G' can be applied to the join of the input 'i' and the register 'r', then the guards 'G' can also be applied to the join of the input 'i' and the ... |
The negation of a predicate P with variables x and y, after a substitution of (x, y) with v, is equivalent to the negation of the result of the same substitution applied directly to the predicate P with variables x and y. |
The Gamma function is asymptotically equivalent to the function that takes a complex number 's' and returns the square root of 2π times 's' divided by 'e' to the power of 's', all divided by 's' to the power of 1/2, as 's' approaches infinity. |
The constant 0 is a lemma. |
The lemma named "value_cases" is defined as follows:
Given a fixed value 'v', one of the following cases must be true:
1. (PVal) There exists a post 'pst' such that 'v' is equal to the left injection of 'PVal' from 'Post_ISSUER'.
2. (PValS) There exists an 'aid' and a post 'pst' such that 'v' is equal to the left injec... |
The interpretation of the symbolic run, when the context Γ1 is concatenated with the concatenation of Γ1 and Γ2, is equal to the interpretation of the symbolic run when Γ1 is concatenated with Γ2. |
The lemma named "Gmult_add_diff" assumes that "a" is an element of the group "G". It shows that the group multiplication of "a" with the sum of "n" and "k", subtracted by the group multiplication of "a" with "n", equals the group multiplication of "a" with "k". |
The sum of 'a' and 'b' with upward rounding at precision 'p' is less than or equal to the sum of 'c' and 'd' with upward rounding at the same precision, if the sum of 'a' and 'b' is less than or equal to the sum of 'c' and 'd'. |
The composition of the relation "rel_gpv'" with respect to A and the relation "rel_gpv'" with respect to B is equal to the relation "rel_gpv'" with respect to the composition of A and B. |
The representation of an unsigned integer with overflow is equal to 2 raised to the power of (default size - 1). |
If the valuation (vv) on the field (K) is not equivalent to the successor of a natural number (n), then for all natural numbers (j) less than or equal to the successor of n, the Ostrowski base of K, vv, the successor of n, and j is an element of the carrier of K. |
The decomposition of the rest of a list (S@S') is equal to the concatenation of the decomposition of the rest of the list S and the decomposition of the rest of the list S'. |
The lemma named "homeomorphic_hyperplanes_eq" is defined with two fixed variables: 'a' of any Euclidean space 'M' and 'c' of any Euclidean space 'N'. It assumes that 'a' is not equal to 0 and 'c' is not equal to 0. It shows that the set of all 'x' such that 'a' dotted with 'x' equals 'b' is homeomorphic (topologically ... |
The bitwise AND operation between a word (a fixed-size piece of data) of an integer 'i' and a mask 'n' is equal to a word of an integer that takes the bit at the minimum position between the length of 'a' and 'n'. Here, 'a' is a type that has a length. |
The term function of the variable x in the context Gamma is well-formed. |
, under the assumptions that the nodes of A are finite, w is not in the language of A, v is in the generalized unambiguous nodes of A with respect to w, and u is in the generalized unambiguous successors of A with respect to w and v, it shows that the rank of u in A with respect to w is less than or equal to the rank o... |
Interpreting the floatarith of the radian value of 'w' with respect to 'xs' is equal to the radian value of the interpretation of the floatarith of 'w' with respect to 'xs'. |
For any variable 'x' of a certain type, the inverse of 'x' is equal to the derivative of 'x'. |
The lemma named "ta_hb_consistent_coinduct_append" assumes two things. The first assumption, referred to as "major", is that X is an element of tobs. The second assumption, referred to as "step", is a function that takes two arguments, E and tobs. If X is an element of tobs, then either tobs is empty, or there exist tw... |
A hyperreal number 'x' belongs to the set of infinitesimals if and only if its hyperreal representation also belongs to the set of infinitesimals. This is applicable in the context of a one-dimensional real normed algebra. |
If "S" is an Integral Domain, "t" is an element of the carrier of "S", "g" is an element of the carrier of "R", and there exists an "h" in the carrier of "R" such that "g" equals the product of "t" and "h", then "g" is a principal ideal of "S" modulo "t". |
The number of roots in the upper half plane of a polynomial 'p' is equal to zero if 'p' is zero. Otherwise, it is calculated as follows:
First, the polynomial 'p' is divided by the greatest common divisor of 'p' and its derivative to get 'pf'. Then, 'pf' is scaled by the inverse of its leading coefficient to get 'pp'... |
:
Assuming
- "\<NN>" is a digraph homomorphism from "\<FF>" to "\<GG>", which are digraphs from "\<AA>" to "\<BB>" in a category with objects of type "\<alpha>",
- "\<NN>'" is a digraph homomorphism from "\<FF>'" to "\<GG>'", which are digraphs from "\<AA>'" to "\<BB>'" in a category with objects of type "\<alpha>",
-... |
If "d" is greater than 0, then eventually there will be a point "t" that lies within the open ball centered at "z" with radius "d", and this point "t" is not equal to "z" and is within the set "A". This event occurs at "z" within the set "A". |
The lemma named "Chernoff_a" assumes that the function 'f' is defined on the range 'A'. It shows that the Chernoff condition holds on 'A' for the function 'f' if and only if for all subsets 'B' and 'C' of 'A', the function 'f' applied to the union of 'B' and 'C' is a subset of the union of the function 'f' applied to '... |
If a program G is well-formed (as indicated by "wf_prog G"), then the fields of the standard exception (SXcpt xn) in program G are empty (represented by "[]"). |
A property P holds for a parallel oracle with states s1s2 and input lr if and only if the following two conditions are met:
1. For all states s1, s2 and input x, if s1s2 is the pair (s1, s2) and lr is the left input x, then the property P holds for the mapped stochastic process monad (map_spmf) that applies a function... |
The composition of "p" with an empty set using the "o_orD" operation results in an empty set. |
If a list "xs" is distinct (meaning it has no duplicate elements), then removing adjacent duplicates from "xs" will result in the same list "xs". |
If design U is not equal to design V, then U is a proper sub-design of V with respect to sets A and B. |
The shared key of agent A (denoted as "Key (shrK A)") is in the initial state of agent A (denoted as "initState A"). |
The function "tak" with inputs x, y, and z is defined as follows: if x is less than or equal to y, then the output is y; otherwise, if y is less than or equal to z, then the output is z; otherwise, the output is x. |
The probability mass function mapped over the composition of the first and second elements of the configuration randomly generated by the BIT_init and BIT_step functions with initial state s0 and queries qs, is equal to the bit vector of the length of the initial state s0. |
"x plus k" is an element of the interval "I plus k" if and only if "x" is an element of the interval "I". |
If 'x' is a limit point of the set of all limit points of 'S', then 'x' is also a limit point of 'S'. This is applicable for 'x' in any metric space. |
The lemma named "gfp_pulldown" is defined with a fixed expectation 'P'. It assumes the following:
1. For any 'u' and 'P', if 'P' is unitary, then the transformation 'T' applied to 'u' and 'P' is equal to the transformation 't' applied to 'P' and 'u' applied to 'P'. This is referred to as the "tailcall" assumption.
2.... |
The lemma "attracted_strategy_step" assumes that sigma is a strategy for player p and that this strategy attracts a set A to a set W. It also assumes that v0 is not a dead end, that v0 is in set A but not in set W, and that v0 is in the set of vertices for player p. Given these assumptions, it shows that the strategy s... |
The hyperbolic sine of a real number x is less than the hyperbolic cosine of x. |
If "a" is not equal to "b", and the set containing "a" and "b" is equal to the set containing "x" and "y", and the BIT_inv function applied to "s", "a", and the list containing "x" and "y" is true, and the set of "qs" is a subset of the set containing "a" and "b", and "qs" is in the language defined by the sequence of ... |
The lemma named "Clean_PPf_encPf" assumes that a formula phi belongs to the set of all formulas, that phi has no free variables, that a number p belongs to the set of all numbers, and that p is not in the image of the proof encoding function encPf. Under these assumptions, it shows that it can be formally proven in bou... |
If for all 'n', 'a n' is less than or equal to 'b n' and 'b n' is less than or equal to 'c n', and if 'a' and 'c' both converge to a limit 'L' (which is a real number), then 'b' also converges to the same limit 'L'. |
For any polynomial 'p' in a commutative ring with unity, the resultant of 1 and 'p' is 1, and the resultant of 'p' and 1 is also 1. |
The composition of a constant function on a set A with a relation r is equal to the Cartesian product of the preimage of A under r and the set containing the constant c. |
Filtering a list 'l' using a function 'f' in List.filter is equivalent to using filter_code with function 'f' on list 'l'. |
:
1. The merge status of 'a' and 'st' is failed if and only if 'a' is failed or 'st' is failed.
2. The merge status of 'a' and 'st' is found if and only if 'a' is not failed, 'st' is not failed, and either 'a' is found or 'st' is found.
3. The merge status of 'a' and 'st' is success if and only if both 'a' and 'st' ar... |
If the combination of pairs is in alpha, and for all P in the set P, the accepting pair R delta, q0, P, and w holds, then the accept GR of delta, q0, alpha, and w also holds. |
The set of all prefixes of a list "xs" is a subset of the set of all prefixes of the list formed by appending any list "ys" to "xs". |
If 'r' is less than or equal to the length of list 'xs' and 'r' is also less than or equal to 'l', then the sublist of 'xs' from index 'l' to 'r' is an empty list. |
In the context of a program P and environment E, the sequence of expressions e1 followed by e2 is well-typed with type T2 if and only if there exists a type T1 such that e1 is well-typed with type T1 and e2 is well-typed with type T2 in the same context. |
This lemma, named "map_project_code", is about the behavior of the function "map_project" on different types of sets. It has three parts:
1. For a set represented as a list (Set_Monad xs), applying the function "map_project" with a function "f" is equivalent to applying the function "map_filter" with the same function... |
The type of the value 'v' is 'Some NT' if and only if 'v' is 'Null'. |
The list relation of the identity relation is equal to the identity relation itself. |
The lemma named "po_on_predecessors_eq_conv" assumes that there is a partial order 'P' on a set 'A', and 'x' and 'y' are elements of 'A'. It shows that the set of all elements 'z' in 'A' such that 'z' is related to 'x' by the reflexive transitive closure of 'P' is equal to the set of all elements 'z' in 'A' such that '... |
If the pair (v, w) is in the set REACHABLE, then the system F transitions from a state reachable from v to a state reachable from w. |
The imaginary part 1 of the sum of a function 'f' over a set 'S' is equal to the sum of the imaginary part 1 of the function 'f' applied to each element in the set 'S'. |
For all A, if A is topologically equivalent to the complement of A, then it implies that the boundary of A is the set of all points. |
The lemma named "weakBisimE" is defined with two fixed elements, P and Q, both of type pi.
It assumes that P is weakly bisimilar to Q.
It then shows two things:
1. P weakly simulates Q under the relation of weak bisimilarity.
2. Q is weakly bisimilar to P. |
:
Assuming that the expression e1 in state s0 and b0 evaluates to a value v in state s1 and b1, and the expression e2 in state s1 and b1 evaluates to a thrown exception e in state s2 and b2, then it shows that the expression e1 with field F of class D assigned the value of e2 in state s0 and b0 also evaluates to a thr... |
: if we have a normal form function 'f' for a dynamic array 'a' with a default value 'dflt', then getting an element at index 'i' from the dynamic array 'a' will result in a value 'r' such that 'r' is equal to the value of the function 'f' at index 'i'. This is under the condition that the dynamic array 'a' remains in ... |
If 'lo' is less than or equal to 'hi', 'lo' is less than 'p', 'p' is less than 'hi', and 'hi' is less than the length of the list 'xs', then the concatenation of the sublist of 'xs' from 'lo' to 'p' and the sublist of 'xs' from 'p+1' to 'hi' is equal to the sublist of 'xs' from 'lo' to 'hi'. |
For any real number 'x' that is not equal to zero, there exists another real number 'y' such that the product of 'y' and 'x' equals one. This is essentially stating the existence of a multiplicative inverse for every non-zero real number. |
If we have a category 'BB' and two functors 'GG' and 'SS' such that 'GG' maps from 'AA' to 'CC' and 'SS' maps from the product of 'BB' and 'CC' to 'DD', and if 'AA'' is the product of 'BB' and 'AA', then the right composition of 'SS' and 'GG' is a functor from 'AA'' to 'DD'. |
The lemma named "even_inter_mod_2" assumes that the cardinality of 'b, which is a prime cardinal, is 2. It also assumes that 'i' is less than 'b', 'j' is less than 'b', and 'i' is not equal to 'j'. It shows that the natural number of the intersection of 'b's at index 'i' and 'b's at index 'j' is equal to zero in the mo... |
"counterAssign" is a member of the model assignments generated by the "counterModel" of "gamma". |
The element x with indices 1,1 is equal to the element x with index 1. |
If 'x' is less than or equal to 'n' minus one, then 'x' is less than or equal to 'n'. |
The lemma named "bp_NE" assumes "bp P". It shows that "P" is not an empty set. |
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