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The lemma named "AbsV_aux" assumes that the function "eta" maps the value function "f" applied to a value to the unit continuation applied to the value continuation of "f'". It shows that "eta" also maps the value function applied to the strictified version of "f" to the unit continuation applied to the value continuat...
The lemma "lift_instr_local_var_in_range" assumes that the function "lift_instr" with parameters F, L, ret, N, x, and initial state Sigma_i, returns a pair consisting of y and the final state Sigma_o. Under this assumption, it shows that the local variable y is in the range N in the Subx programming language.
The lemma named "asymp_equiv_sandwich''" is defined for a function 'f' that maps an arbitrary type 'a to a real number. It assumes three conditions: 1. For any real number 'c'' in the open interval between 'l' and 1, it is eventually true that the value of function 'f' at any point 'x' is greater than or equal to 'c''...
If a list 'xs' is a strict suffix of another list 'ys', then dropping 'n' elements from the list 'xs' will still result in a list that is a strict suffix of 'ys'.
The projection of the dual of a labeled stateful strand "A" with respect to a label "l" is equal to the dual of the projection of "A" with respect to the same label "l".
Lemma well-founded H: Assuming that the relation tr0 is well-founded and n is an element of the transitive closure of tr0 over the universal set, we can show that the relation H n is also well-founded.
The function "map_of" applied to a list "xs" is equal to the function "Map.map_of" applied to the same list "xs".
If a relation 'w' is acyclic, then it is also asymmetric.
For all sets S and T, if S and T are both arbitrary intersections of P, then their union is also an arbitrary intersection of P. This is equivalent to saying that for all sets S and T, if P holds for S and T, then their union is an arbitrary intersection of P.
The composition of any substitution 's' with an empty list is an empty list.
The lemma named "resultant_smult_right" assumes that a scalar "c" (from an integral domain) is not equal to zero. It shows that the resultant of a polynomial "f" and another polynomial "g" scaled by "c" is equal to "c" raised to the degree of "f" times the resultant of "f" and "g".
If in a given context G and a set of triples A, a term t is valid with respect to the precondition P and the postcondition that is a normal state where for all addresses a, if a new address can be allocated in the heap of the state, then the postcondition Q holds for the value of the address, the normal state after ini...
"n" is less than the successor of the factorial of "n".
Reversing the path twice will result in the original path.
If a function 'A' relates to a sub-probability monad function (rel_spmf) that further relates to a product function (rel_prod) of 'B' and a resource function (rel_resource) of 'A' and 'B', then this function 'A' relates to a resource function of 'A' and 'B'. This is denoted using the 'Resource' keyword. This lemma is m...
In the lemma named "elem_in_nhd" within the context of a carrier, if U is an element of the neighborhood of x, then x is an element of U.
The application of the composition of two functions 'g' and 'f', where 'f' is a finite function, is equivalent to the composition of function 'g' and the application of function 'f'.
The length of the result of multiplying a constant 'k' with a polynomial 'p' and then adding a term 'h' to the front of 'p', is equal to the successor (or one more than) the length of the original polynomial 'p'.
The lemma named "poly_roots_set_same" establishes that for any real numbers a, b, and c, the set of real numbers x that satisfy the equation "a * x^2 + b * x + c = 0" is the same as the set of real numbers x that make the polynomial with coefficients c, b, and a equal to zero when x is substituted into it.
In the context of p-adic integers, there is a lemma named "ord_Zp_p_int_unit". It assumes that the integer n is not a multiple of p. It shows that the p-adic order of the product of the integer n and the multiplicative identity is zero.
, under the context Γ, if the initial state of N is a, then after the operation of incrementing N by 1, the final state of N will be a + 1.
, given the assumptions that "xe" is in the set generated by the function "compxEs2" with parameters "es", "pc", and "d", and that "pc'" is either less than "pc" or greater than or equal to the sum of "pc" and the size of the set generated by "compEs2" with parameter "es", it follows that "xe" does not match the except...
The lemma named "factor_mset_prod" assumes that "A" is a finite set and the image of "A" under function "f" is a subset of the carrier "G". It shows that the factor multiset of the product of "f a" for all "a" in "A" in group "G" is equal to the sum of the factor multisets of "f a" for all "a" in "A" in group "G".
A vote 'v' is safe for a round 'r' and a state 's' if and only if for all rounds 'r'' that are less than 'r', for all quorums 'Q' and votes 'w', if 'Q' is a quorum for 'w' in the votes of state 's' in round 'r'', then 'v' must be equal to 'w'.
: if under the condition K, the predicate function P(x) is a member of the set M, then the predicate function that asserts both K and P(x) is also a member of the set M.
If the following conditions are met: 1) v is a trivial code with respect to vs, 2) the image of the set vs excluding v under the function f is not equal to the set containing only v, and 3) the function f applied to v equals v, then it can be inferred that v is a trivial code with respect to the list obtained by ap...
The lemma named "exists_leI" assumes a hypothesis that for all natural numbers 'n'' less than 'n', the property 'P' does not hold for 'n''. Based on this assumption, it shows that there exists a natural number 'n'' that is less than or equal to 'n' for which the property 'P' holds.
The uncurried function of "return" composed with "op_list_append" is in the relation of the list assertion "A" in the doubly-underlined form and "A" in the doubly-underlined form, mapped to the list assertion "A". This is a rule for refining the function that appends two lists in the Isabelle Refinement Framework.
The value of an affine form after truncation is equal to the sum of the original affine form value and the value of the truncation error of the affine form.
The lemma named "lm04_20_zero_definition" is defined as follows: Let 'c' be a natural number, 'l' be a register, 'ic' be a configuration, 'p' be a program, and 'q' be a natural number. Define 'b' as the result of function B applied to 'c'. Define 'f' as the result of function F applied to 'q', 'c', and 'b'. Define 'd...
1 is not irreducible.
The result of the function "remove_sel" with arguments x and e is greater than 0.
The operation "set_bit" on a variable "x" at position "n" with a boolean value "b" is equivalent to the operation "uint32_set_bit" on "x" at the integer value of "n" with "b", if "n" is less than 32. Otherwise, it simply returns "x".
The lemma named "fresh_fun_simp_OrL" assumes that the variable "z'" is fresh for "P", "M", "N", "u", and "x". It shows that applying the "fresh_fun" function to the expression "Cut <c>.P (z').OrL (x).M (u).N z'" results in the same expression "Cut <c>.P (z').OrL (x).M (u).N z'". Here, "fresh" means that "z'" does not a...
The lemma named "span_prj" assumes that "f" and "g" form a cospan. It shows that the projections "p0" and "p1" of "f" and "g" form a span.
If "l" and "r" are height-balanced trees, and the height of "l" minus a certain value "m" is less than or equal to the height of "r" and the height of "r" is less than or equal to the height of "l" plus "m" plus 1, then the tree resulting from balancing "l" and "r" with "a" is also a height-balanced tree.
The double complement of a proposition phi is equal to the proposition phi itself.
If "x" is less than or equal to "y" in the right order, then "x" is related to "z" and the sum of "x" and "z" equals "y" if and only if "z" equals the right quotient of "y" and "x".
The lemma named "shiftl_shiftr2" is about a fixed word 'a' of any length. It shows that if 'b' is less than 'c', then shifting 'a' to the left by 'b' bits and then to the right by 'c' bits is equal to shifting 'a' to the right by the difference of 'c' and 'b' bits and then performing a bitwise AND operation with a mask...
The update function, when applied to a function that does not change the input (i.e., a function that returns the input as it is), remains the same.
The Hadamard matrix is Hermitian.
If the following conditions are met: 1. The function "robdd_invar_ext" holds for the set "bs2", the integer "n", and the element "b" (as stated by the assumption "pre"). 2. For any element "b2" in the set "bs1", if "b" is in the set of sub-robdds of "bs2", then there exists an element "b1" in "bs2" such that "b2" is i...
The m-th power of the double-prime a function for m periods and n years is equal to the sum, for k less than n times m, of the m-th root of v raised to the k-th power, multiplied by (k+1) and divided by the square of m, given that m is not equal to zero. This is for m as a natural number.
The lemma named "aff_dim_affine_hull2" assumes that the affine hull of set S is equal to the affine hull of set T. Under this assumption, it shows that the affine dimension of set S is equal to the affine dimension of set T.
The lemma named "times_X_not_zero" assumes that "p" is an element of the carrier "P" and that "p" is not equal to zero in "P". It shows that the product of "X" and "p" in "P" is not equal to zero in "P".
The lemma "r_find_k" demonstrates two things: 1. The evaluation of "r_find_k" with the input [b] is defined. 2. If we let k be the result of the evaluation of "r_find_k" with the input [b], then the evaluation of the function "phi" with the inputs being the result of the function "s" applied to the result of appending...
The lemma "leq_implies_path" assumes that there is a less than or equal to relation from C to D in the context of P, C is a class in P, and the program P is well-formed with respect to wf_md. Under these assumptions, it shows that there exists a path from C to D via some Cs in the context of P.
The lemma "terms_reach_Bot" assumes that the set of function symbols in the generalized term "t" is a subset of the set "F". It shows that the bottom element is in the set of transition actions of the pattern automaton for the set "F" and relation "R" on the term derived from the generalized term "t".
The lemma l11_18_2 assumes that point B is between points C and D, and the angle ABC is congruent to the angle ABD. Under these assumptions, it shows that angle ABC is a right angle.
In the context of a pair of sigma-finite measures, we have a lemma about almost everywhere (AE) conditions in the pair measure. If we assume that the set of all x in the space of the product measure of M1 and M2, for which property P holds, is in the sets of the product measure of M1 and M2, and we also assume that alm...
For any natural number n, if n is greater than 1, then the sum of n and n is less than or equal to the square of n.
The lemma "det_swap_final_cols" assumes that a matrix A is an m by m matrix, and that m is equal to the sum of l, k, and n. It shows that the determinant of A is equal to (-1) raised to the power of the product of k and n, times the determinant of a new m by m matrix. This new matrix is defined such that its (i, j) ent...
If the key "K" is not used in the event sequence "evs", then the card key of the card "C" is not equal to "K".
If the "invar" property holds for a set of states, then the size of the set of states after a step operation is the same as the size of the original set of states, assuming that the size is small.
The supremum (or least upper bound) of n(x) and the product of n(x) and n(y) is equal to n(x).
The function "f" applied to "k" is equal to the unique list "xs" such that the length of "xs" is "k" and there exists a function "g" where the concatenation of "xs" and "g" equals "f".
The lemma "join_time_empty1" assumes that "i" is less than "n" and that the join time of "s", "n", and "t" equals an empty list. It shows that the function "s" applied to "n*t + i" equals an empty list.
The computation of the likelihood for a given state 's' and observations 'os' equals 'x' if and only if 's' is an element of the state space 'S' and 'x' equals the likelihood of state 's' given the observations 'os'.
A 2x2 square matrix with 1s (of type 2 sq_mtx) is equal to a 2x2 diagonal matrix with both diagonal elements as 1.
A function that takes a pair and applies another function 'f' to the second element of the pair is equivalent to a function that directly takes the second element of the pair and applies the function 'f' to it.
The parallel composition of processes P and Q is commutative, meaning that P parallel to Q is equal to Q parallel to P.
The length of the result of the function "update_until" with parameters "pos", "I", "rel1", "rel2", "nt", and "aux" is equal to the successor of the length of "aux".
An element "x" is a member of the union of two set-dlists "xs1" and "xs2" if and only if "x" is a member of "xs1" or "xs2".
The supremum function F_sup is supremum continuous.
The function "bl_bin" applied to any list "xs" is equal to the function "bl2wc" applied to the same list "xs".
The lemma named "nn_integral_erlang_ith_moment" is defined with fixed natural numbers k and i, and a real number l. It assumes that l is greater than 0. The lemma shows that the non-negative integral of the product of the Erlang density (with parameters k and l) and x raised to the power of i, with respect to the Lebes...
"There exists, in an epistemic sense, a God."
If the control variable of the first memory is zero, and the buffer is not in the set of memory locations that the assembly language cannot read or write, and the first and second memories are equivalent with respect to a low-level security policy, then the buffer contents in the first and second memories are equal.
For all k, if i is less than or equal to k and k is less than j, then the function interfree_aux_right with parameters Gamma, Theta, F, and the tuple (q, c k, a k) holds true. This implies that the function interfree_aux_right with parameters Gamma, Theta, F, and the tuple (q, Parallel (map c from i to less than j), An...
An element 'a' is in the difference of multisets 'M' and 'N' if and only if the count of 'a' in 'N' is less than the count of 'a' in 'M'.
In a linear order, if we have two set iterators 'it_a' and 'it_b' that are correctly mapping over sets 'S_a' and 'S_b' respectively in a linear order, and if for any pair of elements 'kv' from 'S_a' and 'kv'' from 'S_b', the first element of 'kv' is less than the first element of 'kv'', then the union of these two set ...
For any elements 'x' and 'y', the sum of 'n of x' and the product of 'n of x' and 'n of y' is equal to 'n of x'.
: assuming for all 'a' and 'b', if 'a' is in the range of function 'f' and 'b' is in the range of function 'g', then the result of function 'm' applied to 'a' and 'b' is equal to the result of function 'm'' applied to 'a' and 'b'. Under this assumption, it shows that the merge function of 'm', 'f', and 'g' is equal to ...
The uncurried function of "swap_args2 append" and the uncurried function of "RETURN composed with list append operation" are in the relation of "is_list" in the domain and "is_list" in the codomain, with the domain being a dependent product of "is_list" and the Kleisli extension of the identity relation.
Adding zero to any number "x" will result in the same number "x".
If "w" is a Lyndon word and "s" is a suffix of "w" that is less than or equal to "w" in the shortlex order, then "w" is less than or equal to "s" in the lexicographic order.
The lemma named "effect_tapE" in the context of the Heap_Monad theory in Imperative_HOL states the following: Given the assumption that the effect of applying function 'f' to 'h' using 'tap' results in 'h'' and 'r', it can be concluded that 'h'' is equal to 'h' and 'r' is equal to the result of applying function 'f' to...
The lemma named "mmulti_join_link" assumes that the list "A_pos" is not empty and that for all pairs (A, X) in the combined list of "A_pos" and "A_neg", the table of n elements with A and X is valid and the set with n elements and A is well-formed. Under these assumptions, it shows that the result of the multi-join ope...
The lemma "eeqButPID_sym" assumes that "psts" is equivalent to "psts1" except for the PID. It shows that "psts1" is also equivalent to "psts" except for the PID.
The set of all x such that the dot product of a and x is greater than b is empty if and only if a equals zero and b is greater than or equal to zero.
The lemma "row_space_subset_phull" assumes that the set of keys of "ps" is a subset of the set "ts". It shows that the image of the row space of the matrix formed by "polys_to_mat ts ps" under the function "row_to_poly ts" is a subset of the convex hull of the set "ps".
Taking the first "n+1" elements from the infinite list "f" is equivalent to taking the first "n" elements from "f" and appending the "n-th" element of "f".
The lemma "fresh_smallstep_I" is defined for a variable 'x'. It assumes that a small step transition from the pair (\<pi>, e) to the pair (\<pi>', e') exists and that the atom 'x' is fresh in 'e'. It then shows that the atom 'x' is also fresh in 'e''.
The lemma "valid_plan_valid_tail" assumes that if a plan consisting of "h" followed by "as" is a valid plan in the problem "PROB", then it shows that "as" is also a valid plan in the problem "PROB".
The second-order relation of a set relation parameterized by A is equivalent to the set relation of the second-order relation of A.
The phase at time 0 is 0.
The theorem "missing_start_no_insertion" assumes the following: 1. "ops" are insert operations. 2. "xs" is an insert sequence that starts with "start", and "xs" are insert operations. 3. "ys" is another insert sequence that also starts with "start", and "ys" are insert operations. 4. The set of "xs" is a subset of the...
This is a lemma for strong induction on expressions. It assumes the following: 1. For any variable and context, the property P holds for the variable. 2. For any expression, variable, and context, if the property P holds for the expression, then the property P holds for the application of the expression to the variabl...
If "c" is proper, then the branching number of "c" is greater than 0.
In the lemma of an additive subgroup, H is a subset of the carrier G.
If 'i' and 'j' are less than the length of list 'l', then the assertion that list 'l' and list 'li' are associated according to 'A' implies that after swapping the elements at positions 'i' and 'j' in both lists, the lists remain associated according to 'A'. This is always true.
: Assuming "t" is within the reach of the transition function "delta\<^sub>C" of the "ltl_to_rabin_base_code_def" when applied to "af", "af\<^sub>G", "Abs", and "Sigma", starting from the initial state defined by "Abs", "Abs", and "phi". And assuming the domain of "pi" is a subset of the globally true subformulas of "p...
The lemma named "equal_ket" is defined with two fixed variables A and B, which are complex bounded linear functions. The lemma assumes that for any 'x', the result of applying A to the ket vector of 'x' is equal to the result of applying B to the same ket vector. Under this assumption, the lemma shows that A is equal t...
The lemma named "ordLeq_natLeq_on_imp_finite" assumes that the cardinality of set A is less than or equal to the natural order on n. It shows that set A is finite.
The lemma named "logical_instr_sub1_privilege" assumes that the state 's' is the second element of the first pair resulting from the function 'logical_instr_sub1' applied to the instruction name, result, and a SPARC state. It also assumes that the 'get_S' function applied to the PSR CPU register value of state 's' equa...
The lemma, in the context of a flowgraph, is named 'ntrp_unsplit'. It assumes the following conditions: 1. 'w' belongs to the tensor product of 'wa' and the mapping of 'ENV wb' over the normal language of the flowgraph. 2. The triple '((s,ca),wa,(s',ca'))' is in the transitive closure of the non-terminal relation of t...
If the variables of "a" are a subset of the domain of "ps", then the partial boolean value of "a" with respect to "ps" is equal to the boolean value of "a" with respect to "ps".
If set A is a superset of set B, and the valuation functions ν and ν' agree on set A, then the valuation functions ν and ν' also agree on set B.
The lemma named "conga__acute" assumes that the angle ABC is congruent to the angle ACB. It shows that the angle ABC is acute.
A field "F" of type "T" in class "D" is accessible in the program "P" compiled with function "f", if and only if the field "F" of type "T" in class "D" is accessible in the original program "P".