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The last program of the head program of 'p' with respect to 'xs' is equal to the last program of 'p' with respect to 'xs'.
The lemma named "suff_appE" assumes that "r" is an element of the reflexive transitive closure of "A" and "t" is an element of the suffix of "A" and "r". It then obtains a "s" such that "s" is an element of the infinite closure of "A" and "t" is equal to the concatenation of "r" and "s".
, assuming "r" is a transitive and irreflexive relation, and given two decreasing diagrams "DD" with respect to "r" and "ars", where the first diagram is from "tau" to "sigma" to "sigma prime" to "tau prime", and the second diagram is from "tau prime" to "upsilon" to "upsilon prime" to "tau double prime", then there ex...
: Given the assumption that 's' is equal to the tuple (\<sigma>, \<tau>, e), it shows that: - If 'r' is in RID_S of \<sigma>, then 'r' is in RID_L of 's'. - If 'r' is in RID_S of \<tau>, then 'r' is in RID_L of 's'. - If 'r' is in RID_E of 'e', then 'r' is in RID_L of 's'.
The lemma named "lower_bound" is defined with a fixed natural number 'b'. It assumes that 'b' is greater than 0. It shows that for all integers 'c' that are greater than or equal to negative two to the power of 'b-1', the value of negative 'c' plus two to the power of 'b-1' minus 1 is less than two to the power of 'b'.
The lemma named "indecomposable_ordertype_eq" assumes that alpha is indecomposable, the order type of set A under the well-founded relation VWF is alpha, and set B is a subset of set A which is small. It shows that either the order type of set B under the well-founded relation VWF is alpha, or the order type of the dif...
The lemma named "bulk_update_loan_closed_form" assumes the following conditions: 1. The interest rate at time 'k' is not equal to 1. 2. The interest rate at time 'k' is greater than 0. 3. The interest rate at time 0 is 0. 4. The interest rate 'i' is greater than or equal to 0. Under these assumptions, it shows that t...
The composition operation in the functor category "smc_Funct" is a single-valued function.
The lemma named "phl_n_while" assumes that for a given condition "n x" and "n y", a statement "z" ensures the condition "n x". It shows that for the same condition "n x", an infinite repetition of the sequence "n y" and "z", followed by a termination "t y", ensures the condition "n x" and termination "t y".
The set of free variables in the initial knowledge of a stateful strand S is a subset of the variables in the stateful strand S.
Lemma combine_rbt_greater [simplified]: Fix a key 'k' of type 'a'. Assume that 'k' is less than all keys in the left subtree 'l' and the right subtree 'r'. Then, 'k' is also less than all keys in the combined tree of 'l' and 'r'.
If the range of the coefficients of a polynomial 'p' is a subset of the set 'P[Y]', then the flat polynomial map of 'p' with respect to 'x' is an element of the set 'P' with 'x' inserted into 'Y'.
The lemma named "outer_prod_le_one" assumes that vector "v" is in the carrier space of dimension "n" and the inner product of vector "v" with itself is less than or equal to 1. Under these assumptions, it shows that the outer product of vector "v" with itself is less than or equal to the identity matrix of dimension "n...
If "isTable E vs ((a,b)#ps)" is true, then "isTable E vs (deleteAround g a ps)" is also true. This means if a table with elements E and vertices vs exists for the pair (a,b) and the list of pairs ps, then a table with the same elements E and vertices vs also exists for the list of pairs ps after performing the "deleteA...
For a given state transition relation "r" and "r'", and a state "u", "u'", and an input "x", if the input state of "r" from "u" to "u'" is "x" and for all outputs "y" such that "r" from "u" to "u'" with input "x" gives output "y" implies that the input state of "r'" from the second component of "u" to the second compon...
If the composition of two functions 's' and 'r' at 'a' equals 'c', and 'r' and 's' are both very smooth functions (vsv), and 'a' is in the domain of 'r', and the value of 'r' at 'a' is in the domain of 's', then there exists a 'b' such that 'r' at 'a' equals 'b' and 's' at 'b' equals 'c'.
For all variables nu and omega, and for all sets A and B, if A is a subset of B and nu agrees with omega on B, then nu also agrees with omega on A.
If zero is less than m, then infinity multiplied by the additive inverse of m equals infinity.
For any formal Laurent series 'f' with elements from a division ring, dividing 'f' by the power of the formal Laurent series 'fls_X' raised to 'n' is equal to shifting the series 'f' by 'n' positions.
The up-filter of the absolute dense value of the join of x and the negation of x is less than or equal to the stone_phi of the absolute regular value of the double negation of x.
If the initial state of a transition system T is a subset of a property P, and if this property P is preserved by the transitions of T, then all states reachable from the initial state in T are also a subset of P.
The lemma named "parIdLeft" is defined with a fixed process 'P' of type 'ccs' and a relation 'Rel' of type set of pairs of 'ccs'. It assumes that for any process 'Q', the pair consisting of 'Q' in parallel with the null process and 'Q' itself is in the relation 'Rel'. It then shows that the process 'P' in parallel with...
The implementation of the Knuth-Morris-Pratt algorithm (kmp_impl) is correct. Given two lists 's' and 't' at indices 'si' and 'ti' respectively, the algorithm will return a result 'r'. If 'r' is None, then there does not exist an index 'i' such that 's' is a sublist of 't' at index 'i'. If 'r' is Some 'i', then 's' is ...
The lemma named "diagonal_mat_mult_index" is about matrices. It takes a matrix B of any ring type. It assumes that matrix A is a diagonal matrix, and both matrices A and B are of size n by n. It also assumes that i and j are less than n. The lemma shows that the element at position (i,j) in the product of matrices A an...
If a polynomial 'p' is an element of the carrier 'R' and 'p' is not equal to zero, and if the degree of 'p' with respect to 'R', 'S', and 'X' is greater than zero, then the degree of the polynomial expression with respect to 'R', 'X', and the degree of 'p' minus one (for some function 'f' that is a solution to the coef...
There exists a natural number 'x' such that for any 'y' in the universe, there exists a 'z' in the universe where 'x' plus 'y' equals 'z'. And there exists a 'z' where 'x' plus 'y' equals 'z'.
If there are retraction maps from a topological space X to another topological space X' with functions r and identity, and if V is a subset of U, and the image of U under r is a subset of V, and r is homotopic with identity in X with the condition that the image of U under any homotopy h is a subset of U, then the rela...
If a sequence of events 'evs' is in the Kerberos IV protocol, and 'K' is a symmetric key that is not used in 'evs', then 'K' is not in the set of keys for which the analysis of the information spied in 'evs' is possible.
, given the following assumptions: 1. The ownership of threads 'ts' is distinct (denoted by 'dist'). 2. The sharing is consistent between the threads 'ts' and the shared memory state 'S' (denoted by 'consis'). 3. The index 'i' is less than the length of the threads 'ts' (denoted by 'i_bound'). 4. The 'i'-th thread in ...
If the remaining steps in the state of the deque (with direction 'dir' and big and small buffers 'big' and 'small') is greater than or equal to 4, and the sum of the new size of 'big', the remaining steps in the state, and 2 is less than or equal to three times the new size of 'small', then the sum of the new size of '...
If a function 'f' is distributed over two measure spaces 'M' and 'N' with a random variable 'X', and another function 'g' is measurable over the measure space 'N', and for all 'x' in the space 'N', 'f' of 'x' is greater than or equal to zero, then the function 'f' times 'g' is integrable over the measure space 'N' if a...
If 'x' is greater than or equal to -1 and less than or equal to 1, then the arcsine of 'x' is equal to pi/2 minus the arccosine of 'x'.
The lemma named "arith2" establishes a rule about three natural numbers, represented as "t", "n", and "q". It assumes that the sum of "t" and the product of "n" and "t" equals the sum of "q" and the product of "n" and "q". Under this assumption, it shows that "t" must be equal to "q".
The lemma named "is_poly_tupleE" assumes two conditions: first, that "fs" is a polynomial tuple of length "n", and second, that "j" is less than the length of "fs". If these conditions are met, it shows that the "j"-th element of "fs" is in the carrier of the polynomial ring "R" in "n" variables.
If y is equal to 1 plus the product of x and y, then y is equal to the omega power of x plus the star power of x.
For any two sets S and S', if the domain of S is equal to the domain of S', then the domain of the shared elements in S with respect to the buffer sb is equal to the domain of the shared elements in S' with respect to the same buffer sb.
If t2 is less than or equal to t1, then the union of the set I cut at a value less than or equal to t1 and the set I cut at a value greater than or equal to t2 is equal to the original set I.
The lemma named "subst_ev_flip" is defined with fixed variables e, ea, and c. It assumes that the atom c is distinct from the pair (e, ea) and the group (x, xa, e, ea), and that swapping x and c in e results in the same expression as swapping xa and c in ea. Under these assumptions, it shows that substituting x with v'...
The lemma named "two_elem_cases" assumes that "u" is an element in the set containing "x" and "y". It then concludes that either "u" equals "x" or "u" equals "y".
If a pair (a, b) is in the reflexive transitive closure of a relation A, and if a pair (b, c) is in the well-founded (strict) part of the relation A, then the pair (a, c) is also in the well-founded part of the relation A.
If 'x' is an element of the set 'p', 'x' satisfies the property of 'singleCombinators', and 'x' is not equal to 'DenyAll', then the destination network of 'x' is an element of the set produced by the function 'net_list_aux' applied to 'p'.
The lemma named "inv_in_span" assumes that the set U is a subset of the carrier vector of dimension n, and that a is in the span of U. It shows that the negative of a is also in the span of U.
If the multiset F added to Γ implies n, then the big conjunction of F and Γ also implies n.
"there exists some y such that y is identical to the unique x for which a certain condition phi holds" if and only if "it is necessary that there exists exactly one x for which the condition phi holds" in the universe of discourse v.
If the stack "stk" approximates the stack type "ST" in the context of the class hierarchy "G" and heap "hp", then the length of the stack "stk" is equal to the length of the stack type "ST".
The return probability mass function of "None" is less than or equal to "x".
The lemma named "dur_monoD" assumes that the duration of sequence "xs" at position "i" is less than the duration of the same sequence at position "j". It then shows that "i" is less than "j".
The member identifier of the field method, when given the parameters n and f, is equal to the field identifier of n.
The lemma shows that the set of non-even numbers is infinite.
"x is less than y" if and only if "x is less than or equal to y" and "x is not approximately equal to y".
The function "ValK_copy_i", which maps a natural number to a function from 'o ValK to 'o ValK, forms a chain.
The lemma named "forest_separate" assumes "x" is a forest. It shows that the star of "x" multiplied by the transpose of the star of "x", intersected with the transpose of "x" multiplied by "x", is less than or equal to 1.
A finger tree "t" is not empty if and only if the right view of the tree "t" is not None.
The Gamma function of 1 equals 1.
The union of set A and set B is equal to the union of set B and set A. This is the principle of commutativity for set union.
The lemma assumes that in the program P, class C is less than or equal to class D (denoted by "P \<turnstile> C \<preceq>\<^sup>* D") and that the program P is well-formed with respect to method declarations (denoted by "wf_prog wf_md P"). It shows that there exists a sequence of classes Cs such that in the program P, ...
For any given index 'i', the function 'deltas' applied to the concatenation of lists 'xs' and 'ys' at index 'i' is equal to the function 'deltas' applied to list 'ys' at the index resulting from the function 'deltas' applied to list 'xs' at index 'i'.
The lemma named "comp_assoc" assumes that the composition of "t" and "u" is composable with "v". It shows that the composition of "t" with the composition of "u" and "v" is equal to the composition of the composition of "t" and "u" with "v". This is the associative property of composition.
The lemma "oelimder_gcast" assumes the following conditions: 1. The process 'p' is defined as a groupcast from 'l' to 'qq' with the IPs 'fips' and the message 'fmsg'. 2. The label 'l'' is in the set of labels of the process 'q' in the system 'Γ'. 3. The triple '((σ, p), a, (σ', q))' is in the operational semantics of ...
: given two assumptions, the first one being that "Quotient3 R1 Abs1 Rep1" holds and the second one being that "Quotient3 R2 Abs2 Rep2" holds, it shows that applying the function "map_prod Rep1 Rep2" to the second element of a pair and then applying "Abs2" to the result is equivalent to just taking the second element o...
The closure of zero is zero.
The system of allocation (sysOfAlloc) is a bijection.
A system is consistent if and only if there exists a formula within the set of all formulas such that the formula is not provable.
If set I is not empty, then the nth element of I is in set I.
The distance from point a to point e is less than or equal to the sum of the distance from point a to b, the distance from point b to c, the distance from point c to d, and the distance from point d to e.
The unity root of an integer 'k' and the natural number equivalent of the modulus of 'n' and the integer 'k' is equal to the unity root of 'k' and 'n'.
For any sequence 'sb', the set of all elements acquired after dropping elements from the beginning of 'sb' that satisfy a certain property 'P', is a subset of the set of all elements acquired from 'sb'.
, for a function 'f' tending towards the negative of a value 'y' in a topological additive group, it is equivalent to the function with each of its values negated tending towards 'y'.
The lemma named "permChainFreshFresh" is defined with the following parameters: a list of names "xvec", another list of names "yvec", and a variable "p" of a type that is a name point. It assumes that "xvec" is fresh for "p", "yvec" is fresh for "p", and the lengths of "xvec" and "yvec" are equal. The lemma then sho...
Applying the function "finfun_rec" with parameters "cnst", "upd", and "(K$ c)" is equivalent to applying the function "cnst" with parameter "c".
For any value 'x', applying the star function to a constant function that always returns 'k', and then applying this to 'x', is equal to the star of 'k'.
Applying the "poly_cutoff" function to any number "n" and zero will always result in zero.
The function "d" of the intersection of "x" and "nc" is equal to the intersection of the sigma unit and the parallel of "nc" and the product of the intersection of "x" and "nc" and the pi unit.
In the type hierarchy of a given program "tprg", the set of superclasses for the base class is simply the "Object" class.
For a hyperreal number x, x is in the monad of 0 if and only if the absolute value of x is in the monad of 0.
If the exponential of a complex number 'z' equals 'w', and the imaginary part of 'z' is greater than negative pi and less than or equal to pi, then the natural logarithm of 'w' equals 'z'.
If H is a subgroup of G, and both a and b are elements of the group G, then if the right coset of a multiplied by b is in H, it implies that the right coset of b multiplied by a is also in H.
The fbox of "x" and "y" is equal to the fbox of "x" and "d y".
Lemma well-founded pick: Assuming that the relation tr0 is well-founded and n is an element of the inductive iteration of the universal set with respect to tr0, we can show that the relation "pick n" is also well-founded.
The lemma named "deriv_to_deriv" assumes that the pair "(C,n)" is in the set that is derivable from R. It shows that "C" is in the set that is derivable from R in a different way.
If a program S satisfies the Hoare triple {P}S{Q} in the untyped setting, then the sequence of assuming P, executing S, is equivalent to the sequence of assuming P, executing S, and then assuming Q.
The lemma named "intf_of_prod_assn" assumes that the interface of the assertion A is of type 'a' and the interface of the assertion B is of type 'b'. It shows that the interface of the assertion of the product of A and B is of the type 'a * 'b'.
If the function "find_handler" with parameters P, a, h, frs, and sh equals the tuple (xp', h', frs', sh'), then h' is equal to h.
The lemma named "rep_runit" assumes that "a" is an identity. It shows that the representation of the right unit of "a" is equal to the parallel of the right unit of the representation of "a".
If a certain natural number 'a' belongs to a set 'A' of natural numbers, and there does not exist any number 'm' in the set 'A' such that 'm' is less than or equal to all other numbers in 'A', then the 'n'-th term of the non-decreasing sequence of 'A' starting from 'a' is less than or equal to the difference of 'a' and...
: 1. The remainder of the product of 'a' and 'b' when divided by 'n' is equal to 'c' modulo 'n' if and only if the product of 'a' and 'b' is equal to 'c' modulo 'n', where 'a' is first reduced modulo 'n'. 2. The remainder of the product of 'a' and 'b' when divided by 'n' is equal to 'c' modulo 'n' if and only if the ...
The lemma named "mono2mono_restrict_spmf" shows that the function which restricts a subprobability mass function (spmf) to a set A is monotone with respect to the order relation "ord_spmf (=)". This means that if one spmf is less than or equal to another (in the sense defined by "ord_spmf (=)"), then the same is true f...
The lemma named "powr_mono" is defined for a real number 'x'. It assumes that 'a' is less than or equal to 'b' and 'x' is greater than or equal to 1. Under these conditions, it shows that 'x' raised to the power of 'a' is less than or equal to 'x' raised to the power of 'b'.
If regular expression r is algebraically equivalent to regular expression s (denoted by ACI r s), then the language of r is equal to the language of s.
For a balanced incomplete block design (BIBD), the dual system is symmetric. This is defined over the set from 0 to the length of the block design, with parameters for the dual blocks, the set of points, and the replication number.
The function "list_all2" is parametric. This means that if we have a relation from A to B that is a boolean relation, then this function will preserve the relation when applied to lists of A and B, resulting in a boolean relation.
The lemma "search_converg" assumes that "s b" converges to "i" and "search b" converges (denoted by "?y"). It shows that the function "phi" of "i" and "the ?y" also converges, and "the ?y" is greater than or equal to the effective length of "b".
If "x" is not equal to the default value "dflt", then the pair (n, x) is in the set obtained by enumerating the list "xs" starting from 0 if and only if the nth element of the list "xs" (with a default value of "dflt" if "n" is out of range) is equal to "x".
The lemma named "binomial_mono" assumes that "k" is less than or equal to "k'" and "2*k'" is less than or equal to "n". It shows that the binomial coefficient of "n" and "k" is less than or equal to the binomial coefficient of "n" and "k'".
A function "lift i F" is in the stable set of "lift_set i A" if and only if the function "F" is in the stable set of "A".
Repeating a multiset A (m + n) times is equivalent to repeating the multiset A m times and adding it to the repetition of the multiset A n times.
For a semigroup, if the sum of 'b' and 'a' equals the sum of 'c' and 'a', then 'b' must be equal to 'c'.
If the following conditions are met: 1) The non-failure of S implies the non-failure of M, and 2) For any x, if M does not fail and the result of M is x, then the function f applied to x is less than or equal to S, then the binding of M and f is less than or equal to S.
The representation of the signed right shift of a 64-bit unsigned integer 'w' by 'n' places is equal to the following: If 'n' is less than 0 or greater than or equal to 64, then it is the representation of an undefined signed drop bit operation on 'w' with 'n' places. Otherwise, it is the signed drop bit operation on t...
The set of totatives of a number 'p' is equal to the set of all numbers greater than 0 and less than 'p'.