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The lemma named "lookup_if1" is defined with respect to a fixed context Γ. It assumes that Γ is well-formed under the given typing context Θ and base types B, and that the result of looking up x in Γ is Some (b,c). It then shows that the tuple (x,b,c) is in the set representation of Γ and that for all other tuples (x,b...
If the following conditions are met: 1) If P holds for a heap h, then Q also holds for the heap h, 2) P is pure, 3) Q is pure, then it implies that if P holds, then Q also holds in the context of separation logic for the heap h. Here, "pure" refers to a property of predicates in separation logic that do not depend o...
Scanning an empty set with any key 'k' and set 'I' will result in the set 'I'.
If the function 'f' applied to two zeros equals zero, then the result of the function 'combine_matrix' applied to 'f', zero, and zero is also zero.
The lemma named "resChainPres" is defined with the following parameters: a fixed 'Psi' of type 'b, 'P' of type psi (which is a tuple of 'a, 'b, 'c), 'Rel' which is a set of tuples consisting of 'b and two psi types, 'Q' of type psi, and 'xvec' which is a list of names. The lemma assumes that 'Psi' leads to 'P' in rela...
In the context of a univariate polynomial domain, there is a lemma called "lcf_pow". Given the assumption that "p" is an element of the carrier "P", it shows that the leading coefficient function (lcf) of "p" raised to the power of a natural number "n" in "P" is equal to the leading coefficient of "p" raised to the pow...
The lemma var_set_in_fun_ring_car shows that the variable set R of size n is a subset of the carrier of the function ring of size n over R.
"For all x of type 'a option, property P holds for x."
"thetaSeq" is a subset of "Sretr" of the union of "thetaSeq" and "Sbis".
If paths p and q are joinable in graph G, then the joined path from p to q is included in the set of all paths in graph G from the head of p to the last element of q.
The lemma named "real_hyp2_C_hyp2_cltn2" assumes that "J" is a K2-isometry. It shows that the hyperbolic distance between points "a" and "b" is congruent to the hyperbolic distance between the images of "a" and "b" under the K2-isometry "J".
If the well-formedness of the formula "FEx b \<phi>" with respect to an index "idx" holds, then the well-formedness of the formula "nFEx b \<phi>" with respect to the same index "idx" also holds.
The lemma named "sigma_algebra_intersection" assumes that both A and B are sigma algebras on a set Ω. It shows that the intersection of A and B is also a sigma algebra on the same set Ω.
If the event "ya2' B' A B NA NB" is in the set of events "evs" and "evs" is in "ya", and "B" is not in the set of bad agents, then the pair "Agent A, Nonce NA" is in the set spied by the agents in "evs".
If function 'f' is a ring isomorphism between ring R and ring S, and for any element 'r' in the carrier of ring R, 'f' at 'r' equals 'g' at 'r', then function 'g' is also a ring isomorphism between ring R and ring S.
If an integer "n" is not equal to "m", then the set of pairs "(a,b)" where "a" equals "n" is not equal to the set of pairs "(a,b)" where "a" equals "m".
Scaling a list of partial deviations by a factor 'r' before converting it to a partial deviation is equivalent to converting the list to a partial deviation first and then scaling the resulting partial deviation by 'r'.
If for any three elements p, q, and r from a set C and any real number α between 0 and 1 (exclusive), the preference of p over q is equivalent to the preference of a mixture of p and r over a mixture of q and r (where the mixtures are determined by α), then the preference relation P on C is von Neumann-Morgenstern inde...
The lemma named "chart_compose_lemma" is defined with two fixed variables, c1 and c2. It also defines "U" as the domain of c1 and "V" as the domain of c2. The lemma assumes that the intersection of U and V is a subset of the carrier. It also assumes that for any chart c in the set of charts, c1 and c2 are k-smooth comp...
The sequence of norms of X converges to 0 in the nonstandard sense if and only if X itself converges to 0 in the nonstandard sense.
The lemma EF_lem2a assumes that "x" is an element of "f". It shows that "x" is also an element of "EF f".
If "r" is an element of "S", then for all "C" in the set "Cs" and for all pairs "(f,g)" in "fundfoldpairs", if "r" equals the abstract induced automorphism of "f" and "g" and "C" is in the chamber system of "f", then the first element of the "Spair" of "r" satisfies "Cs" equals "Cs".
If the "nrec_pos" condition holds for a state 's' and there is a next state to 's', then the "nrec_pos" condition will also hold for this next state.
Okamoto's sigma commitment scheme provides perfect hiding against chosen plaintext attacks for an abstract commitment parameter 'n' and an adversary 'A' dependent on 'n'.
If "r" is an element of the vertices of "T", and "t" is equal to the auxiliary function "to_dtree_aux" applied to "r", then "t" is well-formed according to the "wf_darcs" function.
The lemma named "wp_g_dyn" is defined as follows: Given a function 'phi' that maps from a preorder 'a' to 'b', the weakest precondition of the evolution of 'phi' over 'G' and 'U' is equal to the predicate that for all 's', for all 't' in 'U' of 's', if for all 'tau' in the downward closure of 'U' of 's' at 't', 'G' ho...
For any types Ts, values vs, locations l and l', if the size of parameters ps is equal to the size of Ts and the size of ps is also equal to the size of vs, and if the program P evaluates the blocks of ps, Ts, vs, and e in the state (h, l, sh) to the state (e', h', l', sh'), then there exists a location l'' such that t...
: Given an interpretation 'I' of first-order logic, and assuming that 'I' interprets the substitutions of 't1' and 't2' by a substitution function 'σ', and given a literal 'L' oriented by an instance of 't1', 's', and a polarity 'polarity' with a substitution function 'σ'', if the literal 'L' substituted by 'σ' is not ...
The lemma named "dtree_cong" assumes that the root of tree "tr" is equal to the root of tree "tr'" and the content of tree "tr" is equal to the content of tree "tr'". Under these assumptions, it shows that tree "tr" is equal to tree "tr'".
The difference between the formal power series of a constant c and the formal power series of a constant d is equal to the formal power series of the difference between c and d. Here, c and d belong to an additive group.
The binary concatenation of a list of words "wl" each of length "n" is equal to the binary conversion of the concatenation of the list obtained by mapping each word in "wl" to a binary list of length "n".
The integer value of the successor of a natural number 'n' is equal to the integer value of 'n' plus one.
The leaf nodes of a tree 't' are not empty.
In the protocol AODV, at any given instance 'i', if any message 'm' is not a packet, then the sender of the message 'm' must be 'i'.
The function "map_of" applied to the result of updating the associative list "al" with key "k" and value "v", is equivalent to the function "map_of" applied to "al" where the key "k" is mapped to the value "v".
The lemma named "le_mult_floor_Ints" assumes that "a" is a non-negative integer. It shows that the product of the floor of "a" and the floor of "b" is less than or equal to the floor of the product of "a" and "b" in the context of a linearly ordered integral domain.
: 1. If 'i' is less than the number of rows in matrix B and 'j' is less than the number of columns in matrix B, then the element at position (i,j) in the sum of matrices A and B is equal to the sum of the elements at position (i,j) in matrices A and B. 2. The number of rows in the sum of matrices A and B is equal to ...
There exists an element 'y' in set T such that applying function 'g' to '
In the context of a callee invariant, the lemma for executing a generalized probabilistic value (gpv) with a lossless gpv is stated. It assumes that the gpv is valid under the interface \<I> and the state 's' satisfies the invariant 'I'. It shows that executing the gpv with a callee using a lossless gpv (with responses...
The lemma named "Lcm_eq_Max_nat" is defined with a fixed natural number set "M". It assumes that "M" is finite, not empty, and does not contain zero. It also assumes that for any two numbers "m" and "n" in "M", their least common multiple (lcm) is also in "M". Under these assumptions, it shows that the least common mul...
The lemma named "ctxt_subst" assumes that under the context of Gamma and Delta, the expression E is of type U given T. It shows that the context substitution of E in the mu-expression of T strongly reduces to the mu-expression of U in the context where the 0th element is replaced with the 0th element of the lifted cont...
The center of x, y, and z is on the geodesic between x and y, x and z, y and z, y and x, z and x, and z and y.
There exists a finite and deterministic Minsky machine U, such that 0 and 1 are not in the first component of any pair in U. Furthermore, the set of all x such that there exists a sequence of steps in U starting from (2, a function that returns x if the input is 0 and 0 otherwise) and ending in (0, any state), is recur...
If the first elements of the pairs in list "ys" are distinct, then the difference between the count of element "x" in list "xs" and the count of element "x" in list "ys" is equal to the count of element "x" in the list resulting from subtracting entries of "ys" from "xs" in a raw manner.
The lemma named "r_dummy_append" assumes that the function 'f' is a recursive function with the length of the list 'xs' and the length of the list 'ys' is 'n'. It shows that the evaluation of the function 'r_dummy' with 'n' and 'f' as parameters on the concatenation of the lists 'xs' and 'ys' is equal to the evaluation...
The field of a class C with field F in a program P is defined as the unique result of binding the predicate "sees_field_i_i_i_o_o_o" with the parameters P, C, and F to a lambda function. This lambda function takes a tuple of a type T, a field map fm, and a class D, and returns a single predicate with the values D, T, a...
An allocation 'a' belongs to the set of all possible allocations (allocationsUniverse) if and only if 'a' belongs to the set of all allocations (allAllocations'') for the domain of 'a' and the union of the range of 'a'.
The double application of the function 'd' on the product of 'x' and 'y' is equal to the product of the function 'd' applied to 'x' and 'y' separately.
If a matrix A is in the carrier matrix of size m by 1, and A is in echelon form, and for all x in the set xs, x is less than m, and xs is a list of integers from 0 to i (exclusive), and i is greater than 0, then the Hermite normal form of the matrix A, given the range of the Euclidean associative function and the range...
The lemma named "has_fps_expansion_divide'" is about formal power series (fps) in the context of Banach spaces and real normed fields. It takes two formal power series, F and G. It assumes that a function f has a formal power series expansion F, a function g has a formal power series expansion G, and that the zeroth te...
The natural numbers 'm' and 'n' are equal if and only if the natural number representation of 'm' is equal to the natural number representation of 'n'.
The lemma named "DERIV_chain3" assumes that the derivative of function g at any point x is g'(x), and the derivative of function f at point x is f'. Under these assumptions, it shows that the derivative of the function g(f(x)) at point x is the product of f' and g'(f(x)).
The lemma l10_4 assumes that point P is the reflection of point P' across line AB. It shows that point P' is also the reflection of point P across the same line AB.
The lemma named "bal_list_merge" assumes that the balance condition is upheld when an element 'x' is inserted between two nodes, 'Node as a' and 'Node bs b', using the 'bal_up' function. It shows that the balance condition is also upheld when the same element 'x' is inserted into a new node created by merging 'Node as ...
If a sequence 's' is a Cauchy sequence in the p-adic integers, then 's' is in the set of closed sequences of p-adic integers.
The lemma "nearly_healthy_wlp_AC" is about a fixed program 'f'. It assumes that the weakest liberal precondition of 'f' is nearly healthy and that the weakest liberal precondition of 'g' is nearly healthy. It shows that the weakest liberal precondition of the least upper bound of 'f' and 'g' is also nearly healthy.
If a measure 'M' is simply distributed with respect to a random variable 'X' and a probability function 'Px', then 'M' is also distributed over the count space of the image of 'M' under 'X', with respect to 'X' and 'Px'.
The circular composition of "x" and "L", when multiplied by "y", is less than or equal to the join of "y" and the product of "x" and "L".
If the restriction of two states, s1 and s2, to a set S are equal, then s1 and s2 are equal on the set S.
If an element 'a' is in the set 'xs' and the set 'xs' is a subset of the set 'Vars', then 'a' is a subset of the union of 'xs'.
The multiplication of a polynomial 'p' with the negation of another polynomial 'q' is equal to the negation of the multiplication of 'p' and 'q'.
The lemma named "proj_pick" assumes two conditions: first, that phi is an equivalence relation with respect to predicate P, and second, that X is the quotient set of P by phi. It then shows that the projection of the chosen element from X under phi is equal to X.
The lemma named "replay'_code" assumes that the theory 'thy' is well-formed according to 'exe_wf_theory'. It shows that the replay function, when applied to the translated theory 'thy', the variables 'vs', the names 'ns', the hypotheses 'Hs', and the proposition 'P', is equal to the execution of the replay function on ...
The lemma named "cprank_shallow_model" assumes that "m" is equal to the insertion of weights "s" into the shallow model with parameters "Y", "Z", "M", and "N", and that "y" is less than "Y". It shows that the CP-rank of the tensor derived from the network "m" at the position "y" is less than or equal to "Z".
The set of unit vectors of length 'n' forms a basis.
The evaluation of the predicate "exec_1" with parameters P, t, and sigma is equivalent to a lambda function that takes a tuple of "ta" and "sigma'" and applies the "execute.exec_1" function with parameters P, t, sigma, ta, and sigma'.
If "Some f" equals the match of "ts" and "xs", then for all "t" in the set "ts", "x" is not in the function "MFOTL.fv_trm t". This implies that the function "f" of "x" equals "None".
The lemma named "tripleI" is defined with two fixed variables, T and U. It assumes that the first, second, and third elements of T are equal to the first, second, and third elements of U, respectively. If these conditions are met, it concludes that T is equal to U.
The function f is less than or equal to the function that is the sum of functions f and g.
For any term "t" and term implication "TI", "t" is equal to itself modulo term implications.
If the interval I is not empty, then the difference between the minimum element of I and the interval I itself equals the set containing only the element 0.
The show function for an integer "n" is a string representation of an integer.
Lemma update [simp]: Assuming "the set of elements 'a' such that 'g a' is not equal to the identity element is finite" and "g a is equal to the identity element", we can show that "the group operation of 'G' applied to the function 'g' with 'a' updated to 'b' is equal to 'b' multiplied by the group operation of '...
The lemma named "relH_subset" assumes two conditions: first, that there is a relation 'relH' between 'bs', 'h', and 'h'', and second, that 'as' is a subset of 'bs'. If these two conditions are met, it shows or concludes that there is a relation 'relH' between 'as', 'h', and 'h''.
The function g_rec belongs to the set of primitive recursive functions of two variables.
If a list 'ss' is in the set of lists that are either in 'S' or the negation of 'S', then the sum of the elements in the list 'ss' is in the generated set 'S'.
If the server sends a message to B, which includes a nonce NA, a key K encrypted with A's shared key, and a nonce NB and key K encrypted with B's shared key, and this event is in the set of events, and B is not compromised, and the set of events is part of the Otway-Rees protocol, then there exists a message X such tha...
The sign of the binary number obtained by taking the signed bit at position 'n' from the binary number 'bin' is -1 if and only if the bit at position 'n' in 'bin' is set.
If X is a very well-behaved lens, then overriding S with itself on X is equivalent to S.
A sequence of commands (c1 followed by c2) terminates in a state s0 if and only if command c1 terminates in state s0 and for all states s1 such that s0 transitions to s1 through command c1, command c2 also terminates in state s1.
If a formula 'phi' is in the model assignments of a model 'M', and if 'x' is an object in the model 'M', then the variable case of 'x' in the formula 'phi' is also in the model assignments of the model 'M'.
If "x" is an element of the set of all numbers less than zero, then the Frechet derivative of the function "f" at "x" is equal to the function that maps "x" to zero.
: Assuming that M1 and M2 are observable, and for any x2, u, and v, if x2 is in the set of list xs, and u and v are in x2, and u is in the language of M1 and M2, then u and v converge in M1 and M2. Furthermore, if x2 is in the set of the second element of the pair resulting from the function "simple_cg_closure_phase...
For any two elements x and y of a star-algebra over a real normed division algebra, the hypernorm of the product of x and y is equal to the product of the hypernorm of x and the hypernorm of y.
If a class C is equal to some m and ws_prog holds, then the field of class C is equal to an empty map if C is an Object, otherwise it is equal to the field of the superclass of m combined with the map of the fields of m.
Sorting a list after removing duplicates is the same as removing duplicates after sorting the list.
If property P always holds for a stream ω, then it is always the case that P always holds for the stream ω.
There exists a group G', a composition operation, and a unit such that G' is a transformation group of G and the group G with its operation and unit is isomorphic to the group G' with its composition operation and unit.
The lemma named "list_of_permutation_element_n_conv" assumes that "n" is not equal to zero. It shows that the set of the list of permutation elements "n" of "x" in "L" is equal to the set of lists "xs" such that for any index "i", "i" is less than the length of "xs", and for all "j" less than the length of "xs", if "j"...
If the set of states reachable by the extended system (h F union G) is a subset of C, and if z is a state in the set of states reachable by the extended system, then the state f z is in the set of states reachable by the system (F union the projection of h C G).
The lemma "wp_method_via_spy_framework_lists_completeness" is defined for two finite state machines M1 and M2. M1 is of type ('a::linorder,'b::linorder,'c::linorder) and M2 is of type ('d,'b,'c). The lemma assumes that both M1 and M2 are observable and minimal. It also assumes that the size of M1 is less than or equal ...
In the context of a commutative ring, given that "a", "b", and "c" are elements of the carrier set of the ring, it is shown that the square of the difference of "a", "b", and "c" is equal to the sum of the squares of "a", "b", and "c" minus twice the product of "a" and "b", plus twice the product of "b" and "c", minus ...
The lemma "silent_moves_into_silent_moves2" assumes that there is a silent transition from state 's' to state 's''. It shows that there exists a transition sequence 'tls' such that silent moves from state 's' to state 's'' occur according to 'tls'.
The lemma named "not_residue_primroot_1" assumes that "n" is greater than 2. It shows that "1" is not a primitive root residue of "n".
The logical 'and' operation with 'false' and any other value 'X' will always result in 'false'.
If a full chain R exists for a list xs, then a chain R also exists for the same list xs.
If the function "my_invar" holds true for a set 's', then "my_invar" also holds true for the set obtained by applying the function "filter_le_avg" to 's'. Furthermore, the transformation of the set obtained by applying "filter_le_avg" to 's' under the function "my_α" equals the set of all elements 'x' in the transforma...
: assuming that either points A, B, and C are not collinear, or points A, C, and B are not collinear, or points B, A, and C are not collinear, or points B, C, and A are not collinear, or points C, A, and B are not collinear, or points C, B, and A are not collinear, it can be shown that points A, B, and C are not collin...
If a set T is a subset of S and T forms a reduced word with ts, then ts is S-reduced.
If for all elements 'x' and 'y', where 'x' is in set 'A' and 'y' is in set 'B', 'x' is not equal to 'y', then it shows that sets 'A' and 'B' are disjoint.