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If the list "xs" is not empty, then the first element of the concatenation of the list "xs" and the function "f" is equal to the head of the list "xs".
If the step relation is acyclic, then the goal is achieved.
The lemma named "cos_aux" demonstrates two things: Firstly, it shows that the lower bound of the auxiliary function for sine and cosine, given a certain precision, a number 'n', and the square of a number 'x', is less than or equal to the sum from 'i=0' to 'n-1' of the product of (-1) raised to the power of 'i', the r...
The shared state of the start state, which is determined by the function f, program P, class C, method M, and variables vs, is equal to the start heap.
If a real number 'a' is greater than 1 and another real number 'x' is greater than 0, then 'a' raised to the power of 'x' is greater than 1.
: Assuming 'i' is within the set of agents, it shows that 'p' is stochastically dominant over 'q' if and only if 'p' and 'q' are both in the set of lotteries on alternatives, and for every 'x' in the set of alternatives, the probability measure of 'p' for the preferred alternatives of 'i' over 'x' is greater than or eq...
For any set I, if the cardinality of I is greater than 1 or if I is infinite, then the nth element of the set I with its minimum element removed is equal to the (n+1)th element of the original set I.
If "i" is not equal to "j", then the list consisting of "i", "j", and the list obtained by removing cycles between "i" and "j" from "xs", is distinct.
The natural number 1705, when right shifted by 3 places, equals 213.
The lemma named "coinitial_iff_WE" shows that transition systems t and u are coinitial if and only if both t and u are arrays and the source of t is the same as the source of u.
If for every element 'y' in set 'S', 'y' is less than or equal to 'u', then 'u' is an upper bound of set 'S' in the universal set.
An empty set applied to a predicate P is equivalent to the conjunction of P and the identity relation.
A topological space X is a T1 space if and only if for all subsets U of X and all elements x, if U is an open set in X, then the set obtained by removing x from U is also an open set in X.
Binding a stochastic probability monad function "p" to a function that binds a stochastic probability monad function "q" to a function "f" of "x", is equivalent to binding a pair of stochastic probability monad functions "p" and "q" to a function "f" of "x" and "y".
The function "check" preserves the freshness of abstractions in the context of the module "MOD", even in the presence of potential module errors represented by "errMOD MOD".
If a real chop operation is performed on 'r', 's', and 't', and the absolute value of 's' is zero, then 'r' is equal to 't'.
If the conditions "InvRecordCount c", "InvJustifiedGII c", and "InvRecordsNonneg c" are met for a given 'c', then it follows that "InvGlobNonposEqVacant c" is also true for that 'c'.
: assuming "u" compiles to "u'" and "u" is closed, it shows that the rule set "rs" transforms the term "u" to the term "u'" when both are converted from nterm to term'.
For a given set S in a Euclidean space of any dimension 'n', the affine hull of S is closed.
For any two extended real numbers x and y, if x is positive infinity then y is not positive infinity, and if x is negative infinity then y is not negative infinity. Under these conditions, the negative of the difference between x and y is equal to the difference between y and x.
The lemma named "sup_gmctxt_least" assumes that the pair (D, E) is in the composition of gmctxt, and both D and E are less than or equal to C. It shows that the supremum (or least upper bound) of D and E is also less than or equal to C.
The inner product of two pairs of complex numbers, (a, b) and (c, d), is equal to the sum of the inner product of a and c, and the inner product of b and d.
An element x belongs to the image of set A under function AS if and only if x belongs to set A.
A method M with parameters of types Ts and returning type T, seen as native in class D by the compiled program (compP f P), is equivalent to the method being seen as native in class D by the original program P.
The set of elements in a vector "v" is equal to the multiset of elements in the same vector "v".
If the multiset of variables in 's' is a subset of the multiset of variables in 't', then the multiset of variables in the substitution of 's' with 'x' at index 'i' is also a subset of the multiset of variables in the substitution of 't' with 'x' at the same index 'i'.
If "x" is an element of the restriction of relation "r" to set "A", then there exist elements "a" and "b" such that "x" is the ordered pair of "a" and "b", "b" is an element of set "A", and the ordered pair of "a" and "b" is an element of relation "r".
If the following conditions are met: - For any predicate 'p' and any set of variables 'vs', 'P' holds for the positive atomic formula consisting of 'p' and 'vs'. - For any predicate 'p' and any set of variables 'vs', 'P' holds for the negative atomic formula consisting of 'p' and 'vs'. - For any formulas 'A' and 'B', ...
If "b" is an element of the arcs of "H" but not an element of the union of the face cycle set of "a" and the face cycle set of "a'" in "G", then the face cycle set of "b" in "H" is an element of the face cycle sets of "G" excluding the face cycle set of "a" and the face cycle set of "a'".
Extending a binary vector 'w' with 'n' number of zeros does not change its natural number equivalent.
The product of "d" of "x" and "x" is equal to "x".
If "r" is not zero, then zero raised to the power of "r" is equivalent to zero.
For any two extended real numbers 'a' and 'b', the product of 'a' and 'b' is greater than or equal to zero if and only if either both 'a' and 'b' are greater than or equal to zero, or both 'a' and 'b' are less than or equal to zero.
The lemma named "list_succ_nth" assumes that the list "xs" contains distinct elements and that "n" is less than the length of the list "xs". It shows that the successor of the nth element in the list "xs" is equal to the element at the position "(n+1) mod length of xs" in the list "xs".
The lemma named "b_e_type_br" assumes that for a given context "\<C>", the expression "e" of type "Br i" is well-typed, transforming a sequence of types "ts" into another sequence of types "ts'". Under these assumptions, it shows two things: 1. The index "i" is less than the length of the label of the context "\<C>"....
The lemma "refrel_helper_If_Reg_exI" assumes two conditions: 1. The evaluation of the boolean expression "Neq (Load x) (Const 0)" in the context of the memory state "mem2C" after applying the function "sifum_refinement.memA_of varC_of" is equal to the evaluation of the boolean expression "Neq (Load reg3) (Const 0)" in...
: Assuming that "S" is a chain, and "f" is a graph homomorphism from "S 0" to "C", and for any "f" and "i", if "f" is a graph homomorphism from "S i" to "C", then there exists a "h" such that "h" is a graph homomorphism from "S (Suc i)" to "C" and "f" and "h" agree on "S i". Under these assumptions, it can be shown ...
A function 'f' is constant on the union of two sets 'S' and 'S'' with a value 'y' if and only if 'f' is constant on 'S' with a value 'y' and 'f' is constant on 'S'' with a value 'y'.
The lemma "rev_map_invar_implies_invar_ids_equal" assumes that the reverse map is invariant for a given binary string "bs" and "rev_map". It shows that the invariant of the binary string "bs" is equal to the ids of the reduced ordered binary decision diagram.
The skew of a tree "t" is determined by the following conditions: If "t" is a node with a left child that is also a node (with its own left child "t1", value "y", and level "lvy"), and the original node has a value "x", level "lvx", and right child "t3", then if the levels "lvx" and "lvy" are equal, the result is a n...
If the atom "y" is fresh for the formula "phi", then there exists an "x" such that "phi" is equal to the existence of "y" where "x" is substituted by the variable "y" in "phi".
The lemma named "skelInp_def2" assumes that the input "inp" is good. It shows that the skeleton of the input "inp" is equal to the lifting of the skeleton applied to the input "inp".
The lemma named "mod_plus_minus_range" assumes that "b" is greater than 0. It shows that the modulus of "y" plus or minus "b" is within the range from negative half of "b" rounded down to positive half of "b" rounded down.
The lemma named "option_quotient" assumes that there is a 3-argument Quotient relation R with corresponding abstraction function Abs and representation function Rep. It shows that there is also a 3-argument Quotient relation for the option of R (rel_option R) with corresponding abstraction function (map_option Abs) and...
The lemma named "decrtm0" assumes a condition "tmbound0 t". It shows that the interpretation of term 't' with respect to variable list 'vs' and a list 'bs' with 'x' at the front is equal to the interpretation of the term 't' after applying 'decrtm0' to it, with respect to 'vs' and 'bs'.
"my tautology test for boolean is applied to the pigeonhole principle for 6 elements."
For any extended integers 'a' and 'b', subtracting 'b' from 'a' is equivalent to adding the negative of 'b' to 'a'.
The theorem named "limit_is_suffix" assumes that the range of function x is finite. It shows that there exists a number k such that the limit of function x equals the range of the suffix of function x at point k.
If the term of a ground term 's' is equal to the context 'C' applied to a list 'l' and a substitution 'σ', and if 'x' is in the variables of the term 'l', then 'x' is in the ground of the substitution 'σ'.
The lemma named "inv_st_exec2" assumes that the state "s" satisfies the invariant "inv_st". It shows that after executing the function "exec" twice on the state "s", the resulting state still satisfies the invariant "inv_st".
If the result of running the Writer Transformer on 'a' is equal to the result of running the Writer Transformer on 'b', then 'a' is equal to 'b'.
The lemma named "eqvtTrans" is about two relations, "Rel" and "Rel'", which are sets of pairs of elements of type 'a (where 'a is a type that is a name in the pi calculus). The lemma assumes that both "Rel" and "Rel'" are equivariant (as stated by "eqvt Rel" and "eqvt Rel'"). The lemma then shows that the composition o...
The parallel composition of the non-deterministic choice between processes P and Q with process M is equal to the non-deterministic choice between the parallel composition of process P with M and the parallel composition of process Q with M.
: Assuming "timpls_transformable_to_pred A (Fun f T) t" holds, we can show that: 1. "t" is a function. 2. The length of "T" is equal to the length of the arguments of "t". 3. For any "i" less than the length of "T", "timpls_transformable_to_pred A (T ! i) (args t ! i)" holds. 4. If "f" is not an abstraction, then "f"...
The product of three categories, denoted as \<AA>, \<BB>, and \<CC>, forms a finite pcategory of order \<alpha> and size 3.
1 mile per hour is equal to 1.609344 kilometers per hour.
If "t" is an element of "local.ik_add", then there exist such "asid" and "l" that "t" equals to the Mac function applied to "macKey asid" and "l".
The composition of R2 and H2 is equal to the composition of H2 and R2.
The lemma "distinguishing_transitions_left_observable_against_distinguishing_transitions_left" assumes that transition 1 (t1) and transition 2 (t2) are both in the set of distinguishing transitions left for a machine M and states q1 and q2. Furthermore, it assumes that the source of t1 is the same as the source of t2, ...
The function "list.unfoldr" applied to the "map_generator" function with argument "g" is equivalent to the composition of the "map" function with argument "f" and the "list.unfoldr" function with argument "g". This is denoted as "?lhs = ?rhs", where "?lhs" represents the left-hand side of the equation and "?rhs" repres...
For any optional toy "i", for any "qmsg" in the set "R" subscript "i", it holds that with respect to other optional toys not including "i" (represented by "otherwith nos_inc {i} (oarrivemsg msg_ok), other nos_inc {i}"), globally, the number of the toy "i" (represented by "no (\<sigma> i)") is less than or equal to the ...
The lemma named "Non_Strict_stable2" demonstrates that the set of pairs {(M1, W3), (M2, W2), (M3, W1)} forms a stable matching under non-strict preferences.
The epsilon closure set of an empty set after a step with any string "bs" is also an empty set.
If a valuation v is applied to a field K, and if x is an element of the carrier of K, and if the valuation of x is not non-negative (i.e., it's negative), then the valuation of the inverse of x is positive.
The evaluation of the binary operation "WTrt_binop_i_i_i_i_o" with parameters P, T1, bop, T2, and T is equivalent to the statement that under the system P, the binary operation bop between T1 and T2 results in T.
'x' is isolated in set 'S' if and only if 'x' is an element of 'S' and there exists a positive number 'e' such that for all elements 'y' in 'S', if the distance between 'x' and 'y' is less than 'e', then 'y' must be equal to 'x'.
This is a lemma about the behavior of the "integer_shiftr" function, which performs a right shift operation on integers. The lemma includes several cases: 1. If the shift amount is a negative number, the result is undefined. 2. If the shift amount is zero, the result is the original integer. 3. If the original integer...
If a sequence A tends to a complex number 'a' and another sequence B tends to a complex number 'b', then the sequence formed by subtracting the corresponding terms of B from A tends to the complex number obtained by subtracting 'b' from 'a'.
If the following conditions are met: 1) Property P holds for the function f at 0, 2) For any natural number n, if property P holds for the function f at n, then it also holds for the function f at the successor of n, 3) The function f is a surjection from the set of all sets to the set I, 4) a is an element of set ...
If P is equal to Q, then P is equivalent to Q.
The size of a formula 'p' remains the same even after replacing 'm' with 't' in 'p'.
If N is a module over a ring R, and the carriers of both modules M and N are the singleton set containing only the zero element, then M is isomorphic to N over the ring R.
The diamond modality of "x" applied to the conjunction of "d" and "y" is equal to the diamond modality of "x" applied to "y".
If tree "t" is in the first element of the finite set "xs", then the node "r" with "xs" is not equal to the node "v" with the successors of "t".
If matrix A is an n by n matrix, then matrix A is similar to the matrix obtained by applying the function "step_2" to matrix A.
The function "is_float_zero" applied to a floating point number represented as "Float m e" is equivalent to the condition that the mantissa "m" equals zero.
If a state is invariant and the size is okay for a given state and number 'n', then the size will also be okay after performing a step on the state for the same number 'n'.
The lemma named "modular_1" assumes that "f" and "g" form a conjugation. Under this assumption, it shows that the product of "f x" and "y" is equal to the product of "f" applied to the product of "x" and "g y", and "y".
The lemma named "g_rep_neg" makes the following assumptions and shows the following results: Assumptions: 1. A is an element of the sets in M. 2. A is not a positive measure set. Results: 1. The representative negative set of A is an element of the sets in M. 2. The representative negative set of A is a subset of A. ...
The theorem, assuming that the program P is well-formed (wwf_prog P), shows progress in two parts: 1. If an expression 'e' of type 'T' is well-typed in the context of program 'P', environment 'E', and heap 'h' (P,E,h ⊢ e : T), then for any location 'l', given that the heap 'h' and the environment 'E' are well-formed u...
A proposition p is a logical consequence in the system K5 of a set of premises G if and only if p can be derived from G in the system K5.
: Assuming the following conditions hold: 1. The left subtree (l) satisfies the invariant 1 property. 2. The right subtree (r) satisfies the invariant 1 property for left-leaning trees. 3. The black height of the left subtree (l) is equal to the black height of the right subtree (r) plus 1. 4. The left subtree (l) sati...
If the rank of an element 'x' in a list 'xs' is 'i', then 'i' is less than the length of the list 'xs'.
If there is a path from vertex 'v' to vertex 'w' through a sequence of vertices that includes 'w' followed by 'ys', then 'ys' must be empty.
: The pair (D x, u) is in the set I, or there exists a v in the set of sources I D u xs such that the pair (D x, v) is in the set I, if and only if D x is in the set of sources I D u with x added to the front of xs.
: Assuming that the control flow graph (CFG) node "Low" is in set S, if the following conditions are met: 1. The auxiliary paths 'as' and 'as'' are at the same level. 2. The update of the control stack 'cs' with 'as' results in an empty list. 3. All elements 'c' in the control stack 'cs' are valid edges. 4. There are...
The definedness of Boolean is true.
The lemma named "bisim_J0_J1_start" assumes that the Java program P is well-formed (as stated by "wf_J_prog P") and that the start state of the program, with class C, method M, and variables vs, is also well-formed (as stated by "wf_start_state P C M vs"). Given these assumptions, it shows that the start state of the p...
For every index 'i', if 'i' is less than the length of vector 'v', then the 'i'-th element of the product of a scalar 'a' and vector 'v' is equal to the result of function 'f' applied to scalar 'a' and the 'i'-th element of vector 'v'.
The signed hereditary multiset of the sum of a list of multisets is equal to the sum of the list obtained by mapping the signed hereditary multiset function over the original list of multisets.
The lemma "s_states_properties" makes several assertions about the properties of the function "s_states" applied to a finite state machine M and two states q1 and q2 from M: 1. The first element of each pair in the list produced by "s_states M q1 q2" is unique. 2. For any pair (qq, x) in the list produced by "s_state...
A function 'f' is C-Lipschitz on an empty set if and only if 'C' is greater than or equal to zero.
For all elements 'x' in the set 'xs', if the second element of 'x' is not infinity, then for all elements 'y' in the multiset formed by mapping the function 'p_unwrap' over the second elements of 'xs', the second element of the result of applying 'p_unwrap' to the sum of the second elements of 'xs' is less than or equa...
For a fixed constant 'c' of type 'real_normed_field', the function 'q(t) * c' is differentiable at a point 't' if and only if 'c' is zero or the function 'q(t)' is differentiable at 't'.
The lemma "Trgs_con_closed" shows that if "b" is an element of the transition targets of "T", "b'" is an ideal element, and "b" is concatenated with "b'", then "b'" is also an element of the transition targets of "T".
In the context of a group, the lemma named "derived_set_consistent" assumes that "K" is a subset of "H" and "H" is a subgroup of "G". It shows that the derived set of "G" with "H" as the carrier is equal to the derived set of "G" with "K".
"kp5x7ur" is a knight's path on a 5x7 board.
The function 'compP' when applied to 'f' and 'P' preserves the property 'distinct_fst' of 'P'. In other words, if the pairs in 'P' have distinct first elements, then the pairs in the result of 'compP f P' also have distinct first elements.
The product of negative infinity and negative infinity equals positive infinity.
For all indices 'i', if the 'i'-th component of vector 'f' equals the result of function 'g' applied to 'i', then it is equivalent to saying that the vector obtained by applying function 'g' to each index is equal to vector 'f'.