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If "r" is a cardinal order, then "r" is less than or equal to the cardinality of the universal set of type 'a.
The lemma named "satL_cong" assumes that for any 'x', if 'x' is in the variable set 'varsL l', then 'x' is equal in both contexts 'ξ1' and 'ξ2'. Under this assumption, it shows that the satisfaction of 'l' under context 'ξ1' is equivalent to the satisfaction of 'l' under context 'ξ2'.
The order of "le r" is equal to the order of "r".
If a set A is finite, then there exists a list xs such that the set of elements in xs is equal to A and all elements in xs are distinct.
The function 'f' is summable if and only if both the real part of the function 'f' and the imaginary part of the function 'f' are summable.
If there is a linear order on set A with relation r, then the following properties hold: the relation r is reflexive on set A, the relation r is transitive, the relation r is antisymmetric, and the relation r is total on set A.
The function "vec_to_iarray" applied to the scaled vector "a *\<^sub>R x" is equal to the scaled vector "a *\<^sub>R" applied to the function "vec_to_iarray x".
The coefficient of the monomial "monom m a" at the point "m'" is equal to "a" when "m'" is equal to "m".
The lemma named "smaller_fermat4" assumes the following conditions: 1. The sum of the fourth power of an integer 'a' and the fourth power of 'b' equals the square of 'c'. 2. The product of 'a', 'b', and 'c' is not zero. 3. 'a' is an odd number. 4. 'a' and 'b' are coprime, meaning they have no common factors other t...
Weak bisimilarity is a weak bisimulation under S-transform.
For all 'x' in 'C', if the memory 'x' is equal to the memory 'x' prime, and the types are well-formed in 'Gamma', then the security type environment of 'mds', 'Gamma', and 'mem' is equal to the security type environment of 'mds', 'Gamma', and 'mem' prime.
If for all polynomials 'p' in the set 'ps', 'p' is not equal to zero, then the order of 'x' in the product of the list 'ps' is equal to the sum of the list obtained by mapping the order of 'x' over 'ps'.
The lemma, in the context of an encoding, is named "Strong Operational Completeness if and only if Strong Reduction Simulation". It fixes a relation 'TRel' which is a set of pairs of 'procT' type. It shows that 'TRel' is strongly operationally complete and is a strong reduction simulation in the target if and only if t...
If a tree "Node r xs" is a subtree of "to_list_dtree", and "x" is an element of the set "r", and there is a directed edge from "x" to "y" in the tree "T", then there exists a set "ys" such that the node "Node [y] ys" is in the first element of the image of the finite set "xs".
The lemma named "weakenBisimWeakCoinduct" is defined with the following conditions: 1. It fixes a function 'F' of type 'b', and two processes 'P' and 'Q' of type psi, and a set 'X' of tuples containing an element of type 'b' and two processes of type psi. 2. It assumes that the tuple containing a context 'Psi', and t...
If a matrix A is upper triangular, and if j is less than the number of columns in A, and if j is less than i, and if i is less than the number of rows in A, then the element at the i-th row and j-th column of A is zero.
If there is an equivalence in a bicategory defined by parameters V, H, \<a>, \<i>, src, trg, f, g, \<eta>, and \<epsilon>, and there is an isomorphism from g to g' denoted by \<phi>, then there is also an equivalence in the bicategory for f, g', and the transformations ((\<phi> \<star> f) \<cdot> \<eta>) and (\<epsilon...
If a list "p" is a prefix of the list formed by "x" followed by "xs", then "x" is an element of the set of elements in "p".
The composition of the inverse of 'd' with itself is a subset of the inverse of 'd'.
The periodic zeta function of one half at 1 is equal to the negative natural logarithm of 2.
If the list of first elements from the pairs in 'xs' is distinct and the reverse of 'xs' is sorted with respect to relation 'R', then the map iterator generated by folding 'xs' from right to left is correctly ordered according to 'R' for the map created from 'xs'.
If the current list is not empty and popping the current list results in a pair (x, current'), then adding x to the front of the list current' is equal to the original list current.
The lemma l11_31_2: assumes that A is not equal to B, C is not equal to B, D is not equal to E, F is not equal to E, and D, E, F are collinear (in that order). It shows that the angle ABC is less than or equal to the angle DEF.
The lemma named "find_set_function" assumes that the precondition for the function "find_set" with parameters "p" and "x" holds true, and that "y" is equal to the result of the function "find_set" with parameters "p" and "x". Under these assumptions, it shows that the postcondition for the function "find_set" with para...
The lemma AbstrLevels_S11opt_A93 assumes that sS11opt is in the abstraction level i. It shows that sA93 is not in the abstraction level i.
In the context of a variable finite sequence, the variable sequence 'xs' is finite.
The lemma named "nofail_simps" which is a simplification and refinement power simplification, states the following: - The "nofail" of "FAIL" is equivalent to "False". - The "nofail" of "RES X" is equivalent to "True". - The "nofail" of "RETURN x" is equivalent to "True". - The "nofail" of "SUCCEED" is equivalent to "T...
The lemma named "BIT_inv_initial" assumes that "x" is not equal to "y" (where "x" and "y" are natural numbers). It shows that the "BIT_inv" function, when applied to the pair "[x, y]" mapped through a probability mass function (pmf), and given "x" and the list "[x, y]" as arguments, holds true.
The set of properties of the state, which is the last element of the list obtained by mapping the function "state_of_wstate" over the list "Sts", is empty.
The lemma named "next_plane0_incr_except" assumes that "g'" is an element in the set produced by the function "next_plane0" with parameter "p" applied to "g", "g" is invertible, and "v" is an element in the vertex set of "g". It shows that the "except" function applied to "g" and "v" is less than or equal to the "excep...
Theorem Knaster_Tarski_dual: Assuming that for any x and y, if x is less than or equal to y then the function of x is less than or equal to the function of y, we can obtain a value 'a' from a complete lattice such that the function of 'a' equals 'a' and for any 'a' prime, if the function of 'a' prime equals 'a' prime t...
The segment end in the Gabow's Strongly Connected Components (SCC) algorithm, represented by "GS.seg_end (S,B,I',P')", is equal to "seg_end".
If the following conditions are met: - "s" is transitive - "ns" is transitive - "ns" is compatible with "s" on the left - "ns" is compatible with "s" on the right - "ns" is reflexive - The pair (A, B) is in the multiset extension of "ns" and "s" - The pair (B, C) is in the multiset extension of "s" and "ns" Then, it c...
The lemma named "a_consistent_lamvar" assumes three conditions: 1. The function "a_consistent" is true when applied to the tuple (ae, a, as) and the tuple (\<Gamma>, Var x, S). 2. The function "map_of" applied to \<Gamma> and x returns Some e. 3. The function "ae" applied to x equals "up" applied to u. Under these as...
The value of function v at 0 and x equals the maximum of the value of function u at 0 and x, and 0.
The multiple blocks of the successor of a number n is equal to the multiple blocks of n plus B.
The inverse of the relation "ov" and the inverse of the relation "f" are disjoint, meaning they have no elements in common.
The lemma named "tm_dither_loops''" shows that for any tape configuration where there exists some integers k and l such that the tape is in the state of having k blank spaces, followed by a single occurrence of the symbol Oc, followed by l blank spaces, the Turing machine "tm_dither" will halt.
Applying the function "fmapU" with function "f" to the result of binding "m" and "k" using "bindU" is equivalent to binding "m" with a lambda function that applies "fmapU" with "f" to the result of applying "k" to "x".
A random binary search tree that is empty is equivalent to returning a probability mass function of a leaf.
If a function 'f' is strictly monotonically increasing on a set 'A', and 'I' is a subset of 'A', and 'n' is an element of 'A', then the image of 'I' under 'f' with all elements strictly greater than 'f(n)' is equal to the image of 'I' under 'f' with all elements strictly greater than 'n'.
For all 'x', if the set of 'x' is a subset of the set of 'xs', and the function 'ldeep_T' with parameters 'sf', 'cf'' and 'x' equals to the function 'ldeep_T' with parameters 'sf', 'cf' and 'x', then the function 'c_list' with parameters 'sf', 'cf'', 'h', 'r', and 'xs' equals to the function 'c_list' with parameters 's...
The lemma named "translation_composition_eq" assumes that both "a" and "b" are elements of a group "G". It shows that the composition of "a" and "b" in the group "G" is equal to the right translation of the product of "b" and "a".
The formula "pls_cong_ax" evaluates to true under the environment "e".
The lemma named "subst_idem_comp_upd_eq" assumes that "v" is not in the domain of substitution "<I>" and that the substitution "<theta>" is idempotent. It shows that the composition of "<I>" and "<theta>" is equal to the composition of the updated "<I>" with "v" replaced by "<theta> v" and "<theta>".
The next instance of time 't' in the universal set, starting from time 't0', for a given property 'P' is equivalent to the property 'P' at the time 't0 + 1'. 2. The "iNextWeak_UNIV" states that the weak next instance of time 't' in the universal set, starting from time 't0', for a given property 'P' is equivalent to t...
The lemma "HPhase1or2ReadThen_HInv4a_q" assumes three conditions: 1) The action "HPhase1or2ReadThen" transitions the system from state 's' to state 's'' with parameters 'p', 'd', and 'r'. 2) The invariant "HInv4a" holds for the system state 's' and parameter 'q'. 3) 'p' is not equal to 'q'. If these assumptions ho...
If "b" is greater than 2, then the integer "n" is less than or equal to "alpha" of "b" and "n".
The lemma named "atc_to_io_set_code" assumes that "A" is acyclic. It shows that the function "atc_to_io_set" applied to "M" and "A" is equal to the function "acyclic_language_intersection" applied to "M" and "A".
The lemma for_insert_elt_correct: Assuming that "l" is less than or equal to the length of array "a" in heap "h", and "l" is greater than or equal to 1, it shows that there exists a heap "h'" such that the effect of inserting an element "a" for a range from 1 to "l" (exclusive) in heap "h" results in heap "h'", and the...
For any real vector 'x', the operation of mirroring the point 'x' with respect to a point 'p' twice will result in the original point 'x'.
If the following conditions are met: 1. A process 'p' is in the set of all processes 'Procs'. 2. The set of all processes 'Procs' is a subset of the domain of the function 'Γ'. 3. The relation 'r' is well-founded. 4. For all processes 'p' in 'Procs', for all 'τ' and 'Z', the Hoare triple of 'Γ' and 'Θ' union the set o...
If an element 'a' is in the inverse image of 'A' under the relation 'r', then 'a' is also in the preimage of 'A' under the relation 'r'.
If set A is a superset of set B, and the valuation functions nu and nu' agree on set A, then the valuation functions nu and nu' also agree on set B.
If a function 'f' is Borel measurable in a measure space 'M', and almost everywhere in 'M', the absolute value of 'f' is less than or equal to a constant 'C', and the measure of the set of points in 'M' where 'f' is not zero is not infinite, then 'f' belongs to the space of 'p'-integrable functions in 'M'.
If 'l' is less than or equal to 'r' and 'r' is less than or equal to the length of list 'xs', then taking the first 'l' elements of 'xs', appending the sublist from 'l' to 'r' of 'xs', and appending the elements of 'xs' after 'r', will result in the original list 'xs'.
The lemma named "sub2_det" assumes that a matrix A is a square matrix of dimension d, the result of the function "sub2" applied to the parameters d, l, and the pair (r,A) is the pair (r',A'), r is not equal to 0, and l is less than d. Under these assumptions, it shows that the product of r and the determinant of A' is ...
The lemma "halloc_no_abrupt" assumes that from state s0, the heap allocation of object instance 'oi' successfully transitions to state s1 (denoted by "G\<turnstile>s0\<midarrow>halloc oi\<succ>a\<rightarrow>s1") and that state s1 is normal (denoted by "normal s1"). It shows that state s0 must also be normal (denoted by...
If the relation 'r' is reflexive, and the pair (xs, ys) is in the generalized prefix of 'r', and the lengths of xs and ys are equal, then the pair formed by appending zs to both xs and ys is also in the generalized prefix of 'r'.
This lemma, named "find_lasso_spec_autoref", is about the automatic refinement of a function to find a lasso in a graph. It fixes a relation "Re" and a node relation, which is a set of pairs of 'vi' and 'v'. The lemma assumes the following: 1. The graph 'G' is a well-formed graph according to the "igb_fr_graph" predi...
The radical of 0 is equal to 0.
If the optional map of a function 'f' applied to a list 'l' equals to some list 'l'', then the length of list 'l'' is equal to the length of list 'l'.
An abelian semigroup addition on a set S with a certain addition operation is equivalent to a semigroup addition on the same set S with the same addition operation.
If a signed cast (sc) is equivalent to a signed cast operation, and the source size of the signed cast is equal to the target size of the signed cast plus a certain number (n), then the bit list of the signed cast of a word (w) is equal to the bit list of the word with the first n bits dropped.
For any state 'tau', if 'X' of 'tau' is not equal to true of 'tau' and 'X' of 'tau' is not equal to bottom of 'tau', then the logical OR operation between null and 'X' of 'tau' is equal to null of 'tau'.
"x is less than or equal to y" if and only if "x is less than y" or "x is equal to y".
: Given the assumptions that 1) delta is less than or equal to other's stopping time, 2) other's distance at time delta is less than the velocity of entity e, and 3) it is not the case that the acceleration of other is greater than the acceleration of entity e and other's distance at time delta is less than the veloci...
The set LPO_S is a subset of the set LPO_NS.
If a valuation v is applied to the field K, then the product of the valuation ring (Vr) of K under v and the principal ideal generated by the valuation group (Pg) of K under v is equal to the valuation ideal (vp) of K under v.
If the server says to agent A that there is a set of events which includes an encrypted message (using the shared key of A) containing the agent B, a key K, a nonce na, and another nonce NB, then the key K is associated with the nonce NB in the set of events.
If an array A is a subtype of T according to a given program P, then there exists an array B such that T is equal to array B and A is a subtype of B according to the same program P, or T is equal to the class Object.
The lemma for the totient of the primorial' of k equals the primorial' of k times the product of (1 - 1 divided by the i-th prime) for all i less than k.
List "ls" is equal to the result of permuting list "f" with the concatenation of two lists: the first list is obtained by mapping the function that gets the element at a certain index in list "ls" to the list of indices that are not in the set of "ss", and the second list is obtained by mapping the same function to the...
The number of face cycle sets in graph H is equal to the number of face cycle sets in graph G.
The theorem named "siso_While" assumes that the test "tst" is compatible and the command "c" is single-input single-output (siso). It shows that the while loop with the test "tst" and the command "c" is also single-input single-output.
The theorem named "grow_mining" assumes that the length of the blockchain at a certain point 'n' is less than the length of the blockchain at the next point 'Suc n'. If this is true, it shows that mining has occurred at the next point 'Suc n'.
If the m-th power of a string t is not empty, then the string t itself is not empty.
If the set "Us" is a subset of the carrier "R", then the combination of "Ks" and the replication of "n" zeros is equal to the combination of "Ks" and "Us".
The lemma named "cmult_le_self" assumes that "kappa" is a cardinal number and "alpha" is not zero. It shows that "kappa" is less than or equal to the product of "kappa" and "alpha".
If "x" is not in the mode state "mds_A" of "mds_C m", then the variable "C" of "x" is not in the mode state "mds_C m".
The lemma named "d_IN_current_of_bipartite" assumes that "f" is a current of a bipartite web of "Gamma". It shows that the inward degree of "current_of_bipartite f" at a point "x" is equal to the inward degree of "f" at the point "Inr x" minus "f" at the point pair "(Inl x, Inr x)".
The lemma named "st_hypreal_sqrt" assumes that "x" belongs to the set of Hyperfinite numbers and "x" is greater than or equal to zero. It shows that the standard part of the hyperreal square root of "x" is equal to the hyperreal square root of the standard part of "x".
The lemma named "coprime_heuristic" assumes that "f" and "g" are coprime according to some heuristic, and that the content of "f" or "g" is 1. It shows that "f" and "g" are coprime.
The relation "relH" holds for the union of two sets "as" and "as'" and two elements "h" and "h'" if and only if the relation "relH" holds for both the set "as" and the set "as'" with the same two elements "h" and "h'".
The lemma named "space_bind_measurable" assumes two conditions: first, that the function 'f' is measurable with respect to the measure 'M' and the subprobability algebra 'B', and second, that the space of 'M' is not empty. Under these assumptions, it shows that the space of the bind operation on 'M' and 'f' is equal to...
The degree of the variable 'x' raised to the power 'i' in a multivariate polynomial of real numbers is equal to 'i'.
If two lists 'l' and 'r' have the same length, and 'i' is less than the length of list 'l', then the 'i'-th element of the bitwise OR operation on lists 'l' and 'r' is equal to the bitwise OR operation on the 'i'-th elements of lists 'l' and 'r'.
The lemma named "ns_mul_ext_IdE" assumes that the pair (M, N) belongs to the extension of the identity relation Id by the relation R under the operation ns_mul. Under this assumption, it is possible to obtain three multisets X, Y, and Z such that M equals the sum of X and Z, and N equals the sum of Y and Z. Furthermore...
For any two hyperdual numbers 'a' and 'b' of a ring with no zero divisors, 'a' is equal to the product of 'a' and 'b' if and only if 'a' is zero, or 'b' is one, or the base of 'a' is zero and the base of 'b' is one and the product of the first epsilon component of 'a' and the second epsilon component of 'b' is equal to...
If two pairs of lists, (xs1, xs2) and (ys1, ys2), are equal, and for all indices 'i' less than the length of xs2, the relation R holds between the 'i'th elements of xs1 and xs2 if and only if the relation R' holds between the 'i'th elements of ys1 and ys2, then the predicate "list_all2" applied to the relation R and th...
The lemma named "acyclic_reachable_points" assumes that "p" is a point, "q" is a point, "p" is less than or equal to "x;q", and "x" is acyclic. It shows that "p" is not equal to "q".
The lemma named "canonical_saturated_1" assumes the following conditions: 1. The value of "Le r" is less than or equal to the value of "M" at the position (v c1, 0). 2. The value of "Le (- r)" is less than or equal to the value of "M" at the position (0, v c1). 3. The matrix "M" is cycle-free for the value "n". 4. The...
For a function 'f' mapping from a Euclidean space to an extended real number, if 'f' is convex on the entire universe and 'f' is finite on the entire universe, then 'f' is lower semi-continuous.
The transitive closure of the restriction of a relation A to strongly normalizing elements in A is equal to the restriction of the transitive closure of A to strongly normalizing elements in A.
The universal set is equal to the set containing five elements: a1, a2, a3, a4, and a5.
If the columns of a square matrix A (of real numbers) are dependent, then the determinant of A is zero.
Lemma: image_mset_mono Assumptions: 1. mono_f: For all x in the multiset M and all y in the multiset N, if x is less than y, then f(x) is less than f(y). 2. less: M is less than or equal to N. Conclusion: The image of the multiset M under the function f is less than or equal to the image of the multiset N under the f...
: assuming function 'f' is continuous on a set 's', it shows that the function 'f' remains continuous on set 's' even when it is extended to extended real numbers.
A singleton set containing the element 't' is equivalent to a function that, for any input 'i', returns 't' if 'i' equals 'vid1', and otherwise returns the constant zero.
The function induced by the group presentation, denoted as "GroupByPresentationInducedFun", when applied to the set S, the presentation P, and the identity function "id", holds true.