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The "new_last" has hitting paths.
The lemma named "oreachable_other'" can be eliminated. It assumes that the pair "(σ, p)" is in the set "oreachable" defined by parameters A, S, and U. It also assumes that U maps σ to σ'. It shows that the pair "(σ', p)" is also in the set "oreachable" defined by parameters A, S, and U.
If R is a subring of S and x is an element of the carrier S, then the additive inverse of x in S is equal to the additive inverse of x in the ring.
If the function "f" applied to "x" equals "y", then the function "iso_tuple_surjective_proof_assist" applied to "x", "y", and "f" is true.
The sequence of processes P, Q, and S is associative. This means that the sequence of P followed by the sequence of Q and S is equivalent to the sequence of P and Q followed by S.
The properties of mu are such that the product of mu and omega equals 1, and mu is not equal to 1.
The lemma named "nset_image_obtains" assumes that "X" is in the k-th power set of the image of "A" under function "f" and that "f" is an injective function on "A". It then obtains a set "Y" such that "Y" is in the k-th power set of "A" and "X" is the image of "Y" under function "f".
A set S is a subset of the topological space X and the closure of set S in X is a subset of S if and only if S is a closed set in X.
The lemma named "pivot_bounds_id'" assumes that "s" is a tableau, "x_i" is in the left variables of "s", and "x_j" is in the right variables of the left variable "x_i" in "s". It shows that the bounds and indices of the pivot operation on "x_i" and "x_j" in "s" are equal to the bounds and indices of "s".
The function "qps" applied to an empty set results in an empty set.
Assuming an integer 'n' is not equal to zero, the product of 'n' and the multiplicative identity (one) is not equal to the additive identity (zero).
The function f is in the set of Landau symbols for the function f itself.
: Given the assumptions that the maximum variable in the floatarith function 'f' is less than or equal to 1, 'a' is a subset of all elements in 'I', 'x' is in the range of the Taylor model 'tg' evaluated at 'e', the computation of the Taylor model by composition 't' is defined, the derivative of 'f' at 'x' exists when...
The lemma named "bindET_unitET_right_counterexample" is defined with a fixed 'm' of type "'a\<cdot>('m::monad,'e) errorT". It assumes that 'm' is equal to "ErrorT\<cdot>(return\<cdot>\<bottom>)" and that "return\<cdot>\<bottom>" is not equal to "\<bottom>" of type "('a\<cdot>'e error)\<cdot>'m". It shows that "bindET\<...
The function "redT_updT'" when applied to the "init_fin_descend_thr" of thread states "ts" and thread "nt" and time "t" equals None if and only if the function "redT_updT'" applied to the thread states "ts", thread "nt" and time "t" equals None.
For every element 'y', if 'F' leads to 'CU' within the slice of 'B' at 'y', then 'F' also leads to 'CU' within the projection of 'B' through 'h'.
If the length of 'a is less than or equal to n, then the unsigned cast of x (where x is a word of length 'a) is less than or equal to the mask of n.
The lemma named "set_of_variables_is_finite_eq" shows that the set of variables in an equation 'e' is finite.
The lemma "ipresCons_imp_ipresSubst" assumes the following: 1. "ipresCons h hA MOD" which means that the function h preserves the consistency of hA in the module MOD. 2. "igSubstCls MOD" which means that the substitution class is well-defined in the module MOD. 3. "igConsIPresIGWls MOD" which means that the consistenc...
The discrete category functor for a given category and its opposite category are the same.
The lemma named "fps_ln_fps_exp_inv" is about a fixed element 'a' from a field with characteristic 0. Given that 'a' is not equal to 0, it shows that the formal power series logarithm of 'a' is equal to the inverse of the formal power series exponential of 'a' minus 1.
In the context of a program G, executing a "Skip" command for n steps results in the same state s.
If the sum of p and q is less than or equal to the length of w, then q is less than or equal to the length of w after dropping p elements.
If 'i' and 'j' are less than the size of the list 'xs', then swapping the elements at positions 'i' and 'j' in the list 'xs' does not change the distinctness of the list. In other words, if the list 'xs' had distinct elements before the swap, it will still have distinct elements after the swap, and vice versa.
Two real numbers represented as "Ratreal x" and "Ratreal y" are equal in the HOL (Higher Order Logic) system if and only if "x" and "y" are equal in the same system.
If the universe of a reference set is not finite, then for variables xs, elem, next, alloc, p, and q, if Xs equals xs and p equals Null, then while xs is not an empty list, the following invariant holds: the list next p is a valid list and the set of the list next p is a subset of the set alloc, and the map of elem on ...
Updating the k-th element of a list 'xs' with a value 'x' and then removing the last element of the updated list is equivalent to two cases. If 'k' is equal to the size of the list 'xs' minus one, then it is the same as removing the last element of the original list 'xs'. Otherwise, it is the same as first removing the...
For all a and b, if a strictly implies b is approximately equal to the top element, then a is less than or equal to b.
If a block "b" is an element of the dual blocks of "V" and "Bs", then "b" is a subset of the set that includes all numbers from 0 to the length of "Bs" minus one.
The image of a path under a translation function, which adds a constant 'a' to each point 'x' on the path defined by 'g', is equal to the image of the original path image under the same translation function.
The distribution of permutations for "s_A n" is equivalent to the distribution of permutations for "s_mtf n".
Enforcing an interface \<I> on the binding of a generative probabilistic value (gpv) with a function f is equivalent to binding the enforcement of the interface \<I> on the gpv with the enforcement of the interface \<I> on the function f.
For any element "x" of a multiplication monoid, multiplying "x" by the multiplicative identity "1" results in "x".
If for any two processes P and Q, if P is weakly bisimilar to Q, then P satisfies a property x if and only if Q also satisfies x. Given that P is weakly bisimilar to Q, and P satisfies the action alpha followed by property x, then Q also satisfies the action alpha followed by property x.
A word "w" is accepted by the Rabin acceptance condition (represented by "accept_R") on a transition system with transition function "delta", initial state "q0", and acceptance pairs "alpha", if and only if there exists an acceptance pair (Fin, Inf) in "alpha" such that the limit set of the run of the transition system...
The lemma named "scopeExtPar'" is defined with fixed elements P and Q of type pi, and x of type name. It assumes that x is fresh for Q. It shows that a new name x in the parallel composition of P and Q is weakly bisimilar to the parallel composition of a new name x in P and Q.
If a program c, under the context Γ, totally and correctly satisfies the precondition P, the postcondition Q, and the abort condition A, without any faults F, then the program c, under the same context Γ and with an empty set of auxiliary variables, also totally and correctly satisfies the precondition P, the postcondi...
The LLL bound invariant holds true for the initial state of the LLL algorithm, given by "fs_init", at the 0th index.
The minimum set of the union of set A and set B is equal to the union of set A and set B.
If we have a Hoare triple "{P}c{Q}" that is valid, and for all states 's', 'P' implies 'P'', then we can infer that the Hoare triple "{P'}c{Q}" is also valid.
The lemma named "t_stop_zero" assumes that "t_stop" is less than or equal to "x" and "x" is less than or equal to "delta". Under these assumptions, it shows that the velocity of the entity "e" is zero.
The lemma "init_fin_Trsys_Running_InitialThreadAction" assumes the following: 1. There is a transition system "redT" from state "s" to state "s'" with a set of thread actions "ttas". 2. In state "s", thread "t" is not in the "Running" state with any value "x" and lock state "ln". 3. In state "s'", thread "t" is in t...
The identity function is a unitary function in the context of complex bounded linear functions.
If for every element 'i' in a set 'I', 'i' satisfies the safety property with respect to a function 'X', then the intersection of all 'X i' for each 'i' in 'I' also satisfies the safety property.
The ideal core is well-typed with respect to the adversary core interface and the user core interface, given the invariant "invar_ideal'".
In the lemma within the Polynomial Ring, if the polynomial coefficient S is related to c, then for all j less than or equal to the first element of c, the product of the second element of c and the j-th power of X is in the carrier of the ring R.
The subclass relationship in a given class structure G is defined as the set of all pairs (C, D) where C is a class in G, C is not the Object class, and D is the superclass of C.
The set of all boolean lists of length 'm' is finite.
The length of an alternating list is equal to half of the original list's length plus one, rounded down to the nearest whole number.
'F' is a face of the set of all 'x' such that 'a' dotted with 'x' is less than or equal to 'b' if and only if 'F' is either an empty set, or 'F' is the set of all 'x' such that 'a' dotted with 'x' equals 'b', or 'F' is the set of all 'x' such that 'a' dotted with 'x' is less than or equal to 'b'.
The set of all elements 'x' such that 'x' is in set 'A' but not in set 'B' is small.
The lemma "trace_and_start_determines_end" shows that if there is a trace from configuration 'c' to 'c' prime through a series of events 't', and there is also a trace from configuration 'c' to 'd' prime through the same series of events 't', then 'c' prime must be equal to 'd' prime.
If two states 's' and 's1' are equal except for their PID (Process ID), then the function 'sharedWith' applied to these states and their respective PIDs are also equal except for their F (possibly a specific field or function).
In the maximally entangled case, if Alice plays the "miracle move" and Bob plays the classical half move, given that gamma equals pi/2, then Alice's payoff and Bob's payoff will both be 1 if Alice's phi equals pi/2, Alice's theta equals pi/2, Bob's phi equals 0, and Bob's theta equals pi/2.
If the length of the list "xs" is not equal to "n", then the list "xs" is not in the set of probability mass functions of "bv n".
If omega-consistency is true, then omega-consistency standard 2 is also true.
If a relation R1 is a subset of another relation R2, and R2 is compatible with R1, then R1 is a transitive relation.
The dual of y being less than the dual of x is equivalent to x being less than y.
In the context of a Category, the lemma named "iso2Inv" assumes that "f" is an isomorphism. It then shows that the inverse of "f" is a valid inverse under the operation "cinv".
The lemma named "msb_min_neg" is defined for a fixed word "w". It assumes two conditions: the most significant bit of the negation of "w" (msb1) and the most significant bit of "w" (msb2). If these conditions are met, it shows that the unsigned integer representation of "w" is equal to 2 raised to the power of the leng...
In the context of a pre-digraph, there is a lemma about the range of the arrow map of an upward digraph homomorphism. If for any index 'i' in the set 'I', the function 'phi' maps 'i' from the digraph 'CC' to the digraph 'AA' at index 'i', then the range of the arrow map of the upward digraph homomorphism constructed fr...
The function "phi" applied to the bottom element equals the absolute filter of the set containing the top element.
If for all alpha, phi of alpha is equivalent to psi of alpha, and for all alpha, psi of alpha is equivalent to chi of alpha, then it follows that for all alpha, phi of alpha is equivalent to chi of alpha, in the context of v.
The function "next_plane" with respect to a given parameter "p" preserves the invariant condition that a generator "g" is both an involution (i.e., its own inverse) and not tame7.
If function 'f' and function 'g' both have set restrictions, then the composition of 'f' and 'g' applied to a set 'I' is equal to the composition of 'g' and 'f' applied to the same set 'I'. In other words, 'f' and 'g' commute under the operation of function composition when applied to the set 'I'.
The lemma demonstrates that the function which takes a pair of extended natural numbers (x, y) and returns their maximum is monotone. This means that if x is less than or equal to y, then the maximum of x and y is also less than or equal to the maximum of x and y.
For any given problem 'PROB', assuming 'PROB' is finite, it shows that the problem plan bound by Charles for 'PROB' is less than or equal to 2 raised to the power of the cardinality of the problem domain of 'PROB' minus 1.
The factorial of 'n' divided by the factorial of 'm' is equal to the falling factorial of 'n' subtract 'm' and 'n', given the condition that 'm' is less than or equal to 'n'.
The class of the list named "list_name" in environment E is equal to "list_class".
For any set 'gamma', if 'gamma' is not a terminal subset, then for every subset 'sigma' of 'gamma', if 'sigma' is a sequent in the deductions of CutFreePC, then 'gamma' is also a sequent in the deductions of CutFreePC.
The supremum of an empty set is an empty list.
The proposition "The Q is not equal to x, where x is a natural number."
If the rank invariant holds for a list consisting of an element 't' and a binomial queue 'bq', and the rank of an element 't'' is less than or equal to the rank of 't', then the rank skew invariant holds for a list consisting of 't'', 't', and 'bq'.
The real number one is equal to the V-set representation of one.
: assuming there is no equivalent equation existing for terms 't' and 's' in the clause 'C' within the set of integer clauses 'S' under the substitution 'σ' and literal 'L1', it shows that the maximal literal is unique for the substituted terms 't' and 's' in the clause 'C', literal 'L1', set 'S', under the substitutio...
The lemma named "ide_img" assumes that "f" is an arrow. It shows that the image of "f" is an identity.
The lemma named "add_minus_zero_rel" assumes that "a" is a valid fraction. It shows that the relation of adding "a" and the negative of "a" equals to the zero relation.
The list set of a list "A" after taking "k" elements from a list "w" is equal to the intersection of the list set of "A" and "w" with the set of all numbers less than "k".
The lemma named "flip_u_eq" is defined with fixed variables u, u', Θ, and τ. It assumes that under the context of Θ, B, and Γ, τ is well-formed. It then shows that swapping u and u' in τ, Γ, Θ, and B does not change their values.
The length of the list obtained by merging adjacent elements in a list 'xss' is equal to the length of the original list 'xss' when both are concatenated.
The lemma named "insert_second_part" assumes the following conditions: 1. The reference is equal to some value 'r'. 2. 'r' is not an element of the set 'xs'. 3. 'r' is an element of the set 'ys'. Under these assumptions, it shows that the result of the function 'insert_spec' applied to the concatenation of 'xs' and '...
The set of all elements greater than B is monotonically increasing.
If the length of list "l" is 2, then there exist elements "a" and "b" such that the list "l" is composed of "a" and "b".
If the composition of relations P and Q is less than or equal to R, and if P is a relation mapping from P1 to P2, and Q is a relation mapping from P2 to P3, then R is a relation mapping from P1 to P3.
If four functions F1, F2, F3, and F4 are continuous with respect to four variables X1, X2, X3, and X4, then the fixed point of a function that takes a 4-tuple and applies F1, F2, F3, and F4 to its elements is equal to the result of permuting the third and fourth elements of the 4-tuple and applying the same function to...
For any given point 'a' of type 'real_normed_vector', the open segment between point 'a' and another point 'b' is path connected.
Multiplying zero with any formal power series 'f' (where 'f' is of a type 'a and 'b, with 'b' being a commutative monoid under addition and having a multiplication operation that returns zero when multiplied with zero) results in zero.
If the session key of the i-th run (R$i) is in the set of received keys (rkeys) of U, then R is in U.
If a variable 'x' does not affect the outcome of any predicate 'P(i)' and 'Q(i)' for all 'i', then 'x' also does not affect the outcome of the infimum (greatest lower bound) of all predicates 'Q(i)' that satisfy 'P(i)'.
The lemma named "subcocycle_lim_real_ereal" assumes that "u" is a subcocycle and the average of "u" in the extended real number system is greater than negative infinity. Under these assumptions, it shows that for almost every "x" in the measure space "M", the limit of "u" in the extended real number system equals the e...
If for any subtree (Node r1 {(t2,e2)}) of t0, the rank of the reverse of r1 is less than or equal to the rank of the reverse of the root of t2, and if the subtree (Node r {(t1,e1)}) is a subtree of the merge of t0, and the rank of the reverse of the root of t1 is less than the rank of the reverse of r, then the maximum...
The lemma tauLaw3SimRight: fixes a type 'b named \<Psi> and a psi type P :: "('a, 'b, 'c) psi" and a psi type Q :: "('a, 'b, 'c) psi" assumes that "Rel" is equivariant and that for any \<Psi>, xvec, and Tvec, if the length of xvec equals the length of Tvec, then the relation Rel holds between \<Psi>, P with ...
The response function, when given parameters r, w, and c, is lossless in the context of a semantically secure probabilistic multi-function (SPMF).
The binary search tree of a list created by appending list 'ys' to list 'xs' is equivalent to folding the function 'Tree_Set.insert' over list 'ys' starting with the binary search tree of list 'xs'.
If x is greater than or equal to zero and y is strictly greater than zero, then the sum of x and y is strictly greater than zero.
The ordered keys of an empty set are equal to an empty list.
: the intersection of the complements of x and y is less than or equal to the complement of z if and only if the complement of y is less than or equal to the union of the double complement of x and the complement of z.
Applying a permutation "π" to the result of the function "cccFix" from the "CoCallArityAnalysis" module with arguments "cccExp" and "Γ" is equivalent to applying the function "cccFix" to the permuted arguments "cccExp" and "Γ".
Given the assumptions that "p" is not equal to 0 and "p" is not equal to 1, if the Jacobi symbol of "n" and "p" equals 0, then "n" and "p" are not coprime.
Mapping over a list "fs" with a function that first applies a function to pair each element "F" with "D", and then applies another function to keep the structure of each element the same, is equivalent to directly mapping over the list "fs" with a function that pairs each element "F" with "D".