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The lemma named "obtains_dbm_le" is defined with a fixed integer "m". It assumes that "x" is an element of "X" and "m" is less than or equal to "k x". It then obtains "M" such that "vabstr {u in V. u x is less than or equal to m} M" and "M" is normalized.
A pair of less than or equal to operations is in a relation that maps from string relations to boolean relations.
In the context of ordered terms, the leading term of the sum of a function over a set A is less than or equal to the maximum leading term of the function for any element in set A.
The lemma named "n2l_l2n_map_id" assumes that for any element 'x' in the set 'zs', 'x' is not an empty list. It shows that if you map the composition of functions 'n2l' and 'l2n' over the set 'zs', you will get 'zs' itself.
P is a non-zero element of the set of p-adic integers.
The lemma named "rwp_on_Field" assumes that the function 'f' has its range on set 'A'. It then shows that the field of the relation 'rwp_on' with respect to set 'A' and function 'f' is a subset of 'A'.
The intersection of the union of 'x' and the negation of 'y', with the union of 'x' and 'y', is equal to 'x'.
If for all 'x', 'P of x' and either 'Q of x' or 'R of x' implies 'S of x', and if for all 'x', 'S of x' and 'R of x' implies 'L of x', and if for all 'x', 'M of x' implies 'R of x', then for all 'x', 'P of x' and 'M of x' implies 'L of x'.
The function "isolate_variable_sparse" applied to a constant polynomial (of type real multivariate polynomial) with respect to a variable 'x' and the successor of an integer 'i' equals zero.
The lemma "case_sum_preserves_morphisms" assumes that there is a morphism 'f' from 'X' to 'Z' and a morphism 'g' from 'Y' to 'Z'. It shows that the case sum of 'f' and 'g' is a morphism from the binary coproduct of 'X' and 'Y' to 'Z'.
The lemma "phiDefNode_aux_is_join_node" assumes that "n" is in the auxiliary definition nodes of "phi" for a graph "g", a variable "v", an undefined value "un", and a map "m". It shows that the length of the predecessors of "n" in the graph "g" is not equal to 1.
The transformation of a matrix 'A' with respect to a set of variables 'vars1' is equal to the transformation of the tensor product of matrix 'A' and the identity matrix of size 'K'.
The pair (delta_neg, delta_pos) is equal to the limit function "f_lim".
An if-then-else construct with conditions "b", "c1" and "c2" applied to a set "S" is less than or equal to a constant "c" if and only if there exist constants "c1'", "c2'" and a set "S'" such that "c" is equal to an if-then-else construct with conditions "b", "c1'" and "c2'" applied to "S'", and "c1" is less than or eq...
The lemma named "solve_subset_Minimal_Solutions" in the context of "hlde" shows that the set resulting from the function "solve" with parameters "a" and "b" is a subset of "Minimal_Solutions".
"if P is true, then P is necessarily true in all accessible worlds".
If Cl_2 is a subset of C, then the double dual of B is equivalent to C.
If x is an element of set X, then the image of x under function f is an element of set Y.
The lemma "get_component_to_tree_order_subset" assumes that the heap "h" is well-formed, has a well-defined type, and has known pointers. It also assumes that the tree order of the pointer "ptr" in heap "h" results in "nodes", and that getting the component of the pointer "ptr" in heap "h" results in "c". The lemma the...
The lemma assumes that the property P holds for the number 3 (expressed as 2 + 1). It then shows that for any natural numbers x and y, the property P also holds for the number 3 (expressed as 1 + 2).
The "return" and "bind" operations commute in the monad for the type 'a probability.
If an encrypted message Y with key K is in the guardK function with parameters n and Ks, and the key K is not in the inverse of Ks, then Y is in the guardK function with parameters n and Ks.
If a predicate P holds for all pairs of elements in lists xs and ys, and if y is an element of list ys, then there exists an element x in list xs such that the predicate P holds for the pair (x, y).
The lemma named "indexed_const_P_mult_eq" assumes that "a" and "b" are elements in the carrier set "R". It shows that the product of the indexed constants of "a" and "b" is equal to the indexed constant of the product of "a" and "b".
The statements are from the IL_Interval.thy file in the Natural Interval Logic section. 1. If x is in the interval starting from n, then x plus any number k is also in the interval starting from n. 2. If x is in the interval starting from n, then the successor of x (x+1) is also in the interval starting from n. 3. If ...
If a list 'ys' is an element of the set produced by the function 'n_lists' with parameters 'n' and 'xs', then the length of the list 'ys' is 'n'.
The power of a number 'p' to the multiplicity of 'p' in 'x' divides 'x'.
The value of function "u" at point "x" equals zero if "x" is a zero of the function "u" in the real numbers domain and "u" is an element of the set "V".
If the following conditions are met: 1) The Access Control System (ACS) requirements are valid for a given model M, 2) The graph with nodes V and edges E is well-formed, 3) The set of edgesList is equal to E, 4) The stateful set is equal to the set of edges that comply with the stateful ACS for the graph with nodes...
The tag "X" is not equal to the number "X'".
If N is a normal subgroup of G and a is an element of the carrier of G, then the coset product of N, N, and the right coset of N by a is equal to the right coset of N by a.
: Given the assumptions that 's' is connected to 't' in a tree 'T', 'v' is in the left part of 's' to 't', 'w' is in the left part of 't' to 's', and 'v' is connected to 'w' through a set 'xs', it can be shown that the intersection of the set 'xs' and the bags (sets of vertices) of 's' and 't' is not empty.
The full mask is a valid mask.
The lemma named "subst_lemma" assumes that "x" is not equal to "y" (assumption a) and "x" is fresh for "u" (assumption b). It then shows that substituting "x" with "v" in "s" and then substituting "y" with "u" in the result is equal to first substituting "y" with "u" in "s" and then substituting "x" with "v" where "y" ...
The function "code" is applied to the singleton set "sings B".
"x" is less than or equal to the product of "x" and the star of "x".
An encoding respects a binary predicate 'Pred' if and only if there exists a relation 'Rel' such that for all source terms 'S', the pair consisting of the source term 'S' and the target term, which is the encoding of 'S', is in 'Rel', and 'Rel' respects the binary predicate 'Pred'.
The after_summary of an empty multiset and any set S is an empty multiset.
The lemma for the domain of a finite function after an update in the code is: the domain of a finite function after an update is equal to the update of the domain of the original finite function, where the update is determined by whether the new value is not equal to the default value of the original finite function.
In the context of a binary operation, there is a lemma stating that if 'a' and 'b' are elements of set 'A', then applying the function 'fflip' to 'b' and 'a' is equivalent to applying the function 'f' to 'a' and 'b'.
If a graph 'g'' is an element of the set produced by the function 'next_plane0' applied to a graph 'g' with respect to a point 'p', and the graph 'g' has minimum graph properties, then the set of vertices of the graph 'g' is a subset of the set of vertices of the graph 'g''.
The lemma named "t_cvt" assumes that the conversion of value "v" with sign extension "sx" and type "t" equals "v'". It then shows that the type "t" is equal to the type of value "v'".
If a transition sequence is valid from a certain state and that state is reachable, and the transition sequence is not empty, then the target of the last transition in the sequence is also reachable.
The abscissa of convergence of the nth derivative of a Dirichlet series f is less than or equal to the abscissa of convergence of the original series f.
The lemma "RENAMINGI" assumes that for any given 'd', if 'RENAMING' holds true between 's' and 'd', then 'P' holds true. Based on this assumption, it shows that 'P' is true.
If X and Y are Cauchy sequences, then the real number representation of the difference between X and Y is equal to the real number representation of the sequence formed by taking the difference between the nth terms of X and Y for all n.
A function 'f' tends towards a complex number 'x' in a filter 'F' if and only if both the real part of the function 'f' tends towards the real part of 'x' in 'F' and the imaginary part of the function 'f' tends towards the imaginary part of 'x' in 'F'.
The lemma for the game semantics loop fixpoint is monotonic: "The function that maps a set Z to the union of set X and the game semantics of I and alpha applied to Z is a monotonic function."
The lemma named "prime_sqrtI" assumes that a natural number "n" is greater than or equal to 2. It also assumes that for any number "j" that is greater than or equal to 2 and less than "i", "j" does not divide "n". Additionally, it assumes that "i" squared is not less than or equal to "n". Under these assumptions, it sh...
If H is a subgroup of G, and if a is an element of the group G, and h is an element of the subgroup H, then the product of a and h is an element of the left coset of H in G with respect to a.
The dimension of the vector resulting from the partial tensor operation on vectors v1 and v2 is equal to d0.
: "The list of successors 'xss' contains 'x' if and only if 'x' is in the union of the sets of all lists in 'xss'."
The binding of a list 'xs' with a function 'f' equals an empty list if and only if for all elements 'x' in the set of 'xs', the function 'f' applied to 'x' equals an empty list.
For any natural number 'n' that is less than the length of the list 'ws', if the property 'P' does not hold for the 'n'th element of 'ws' and the list obtained by dropping the first 'n+1' elements of 'ws' and appending 'xs', then the interleaving of 'xs' with the set '{ys, zs, P}' is equivalent to the interleaving of '...
: Assuming 'rqrm_invar rqrm' and 'rqrm_prop \<delta> rqrm' are true, and 'hs_invar Qi' is a simplifying assumption, if the following conditions are met: 1. '\<Sigma>' is in 'brec_invar_add' of 'hs_\<alpha> Qi', 2. 'brec_\<alpha> \<Sigma>' is in 'brw_invar \<delta>', and 3. 'brec_cond \<Sigma>' is true, then 'brec_step ...
The infimum (greatest lower bound) of "C" and "GMHole" is "GMHole".
For any message 'm', the transition from learning 'm' at level 1 to faking 'm' at level 2 refines the relation 'R12s', given that 'R12s' intersects with the universal set and the intersection of 'l2_inv3' and 'l2_inv7'.
The conjunction of P with the conjunction of Q and R is equal to the conjunction of Q with the conjunction of P and R. This is a statement about the commutativity of conjunctions in a separation algebra.
For any states 's' and 's'', and any commands 'c' and 'c'', if the tuple ((s,c),w,(s',c')) is in the transitive closure of the transition relation of a flow graph 'fg', then for any command sequence 'csp' such that 'c'' equals 'csp' plus 'c', if for all states 's' in 'csp' there exists a process 'p' and edges 'u' and '...
: 1. The almost fairness of the conjunction of two properties, phi and psi, in a world w, is equivalent to the conjunction of the almost fairness of phi in w and the almost fairness of psi in w. 2. The almost fairness of the disjunction of two properties, phi and psi, in a world w, is equivalent to the disjunction of t...
If a command list "cl" is proper and a command "c" is in the set of "cl", then the command "c" is proper.
The universal set of the type ('a, 'b) foo is equal to the union of the range of the set containing "Foo", the range of the range of the set containing "Bar", the range of the range of the range of the set containing "FooBar", and the range of the set containing the function that maps any unit to "Stop".
The lemma "outputPushResLeft" is defined with the fixed variables Psi of type 'b, x of type name, M of type 'a, N of type 'a, and P of type psi, which is a tuple of 'a, 'b, 'c. The lemma assumes that the relation Rel is equivariant, x is fresh for Psi, x is fresh for M, x is fresh for N, and for any Q, the tuple (Psi...
If a memory is the result of applying the function "mem_empty" with the size of an index to an empty heap, then the map of the heap of this memory is a submap of an empty map. This is expressed in the context of a heap monad, where 'v is an optional array type.
A tree with subset "gamma" is equal to the insertion of the pair (0, gamma) into the union of "delta" in the subset of "gamma" and the image of the tree with subset "delta" under the function "incLevel".
If 'i' is an element of set 'A' and 'i' is less than the length of list 'v', then the 'i'-th element of the list obtained by restricting list 'v' to set 'A' is equal to the 'i'-th element of list 'v'.
"thetaZOParT" is a subset of "ZOretr" of "thetaZOParT".
The lemma "obtain_inverse_matrix" assumes that a matrix A is a square matrix of size n (i.e., "A" is in the set of n by n matrices) and that matrix A is invertible. It then concludes that there exists a matrix B such that B is the inverse of A (i.e., the product of A and B is the identity matrix), A is the inverse of B...
If "k" is not less than the length of list "l", then "k" is not in the integer array set of the list "l".
The maximum degree of the function "ikkbz_sub" applied to "t1" is less than or equal to 1.
If H holds for p, x, and q, then x is less than or equal to the reference of p and q.
The theorem named "wls_swap_psubst" assumes that "X" is a well-sorted term of sort "s" and "rho" is a well-sorted environment. It shows that swapping "z1" and "z2" in the term obtained by substituting "rho" in "X", is the same as substituting in "X" the environment obtained by swapping "z1" and "z2" in "rho", and then ...
The application of the function "Trivial_Aheap" to the arguments "Gamma" and "e" multiplied by "a" is equal to the function that maps "x" to "up" multiplied by 0, restricted to the domain of "Gamma".
If tree 't' is left-aligned with 'u', then the list obtained by performing an inorder traversal on 't' and appending 'u' to the result is sorted in ascending order.
A scene A is independent of its own complement.
The lemma named "varsOppositeLiteralList" demonstrates that the variables in the list of opposite literals of a clause are the same as the variables in the original clause.
The lemma named "expands_to_min_lt" assumes that a function 'f' expands to 'F' on a certain basis and that eventually, for large values of 'x', 'f' is less than 'g'. Under these assumptions, it shows that the function that takes 'x' and returns the minimum of 'f(x)' and 'g(x)' also expands to 'F' on the same basis.
The property of being distinct for the reversed version of a list of lists (xss) is equivalent to the property of being distinct for the original list of lists.
Inserting a key K into the union of the image of a set of keys KK and a set C is equal to the union of the image of inserting K into the set KK and the set C.
If "y" is an element of the affine form error "e" with parameters "c" and "d'", and "d'" is equal to "d", then "y" is also an element of the affine form error "e" with parameters "c" and "d".
For any real number 'x', 'x' squared is less than 'y' if and only if 'y' is greater than or equal to zero and the absolute value of 'x' is less than the square root of 'y'.
The lemma named "width_lower_bound_1" assumes that there is a directed edge from vertex v to vertex w. It shows that the width is greater than or equal to 1.
If S1 and S2 are monotonic, then the function that takes F and s as arguments and returns S1 F s if b s is true, otherwise returns S2 F s, is also monotonic.
Set A is equipollent to itself.
A single-term summation is summable, where the summation function is defined as follows: for a given index r, if r is equal to i, then the function returns the value f(r), otherwise it returns 0.
The lemma proves that the Gödel sentence is equivalent to the statement "for all yy, the Peano Arithmetic does not prove the formula with variable yy and Gödel sentence as parameters".
The lemma named "word_add_le_mono1" is defined with a fixed variable 'i' of type 'word' with a length 'a'. It assumes two conditions: 'i' is less than or equal to 'j', and the unsigned natural number representation of 'j' plus the unsigned natural number representation of 'k' is less than 2 to the power of the length o...
The sum of sets A and B is a subset of the sum of sets C and D if and only if set A is a subset of set C and set B is a subset of set D.
If "n" is greater than 0, and the list "xs" is not empty, and "n" is greater than or equal to the length of the list "xs", then chopping the list "xs" at "n" will result in a list containing the list "xs" itself.
If set A is a subset of set B, then F is an element of A that leads to B under the condition CC.
The relationship "TUM to BLUBB to Leaf" is less than or equal to the relationship "TUM to Leaf".
The lemma named "ordertype_map_image" assumes that set B is a subset of set A and set A is small. It shows that the order type of the image of the set difference between set A and set B under the order map with respect to the well-founded relation VWF is equal to the order type of the set difference between set A and s...
The lemma named "norm_memb_eq_rep_ARR" assumes that "f" is an arrow and "t" is an element of "f". It shows that the norm of "t" is equal to the representative of "f".
If a primitive matcher is generic and a match is in normalized negation normal form, then if the match does not result in a drop action when in doubt, it will also not result in a drop action when the primitive is abstracted and still in doubt.
A node pointer cannot be cast to a shadow root pointer kind.
For any integer b, the modulus of (1 plus 2 times b) and (2 times 2 to the power of n) is equal to 1 plus 2 times the modulus of b and 2 to the power of n.
If for every element 'i' in set 'I', the value of function 'f' at 'i' is greater than or equal to 1, then the product of function 'f' over all elements in set 'I' is also greater than or equal to 1.
: Assuming the following conditions: 1. The state (is, θ, m, Δ, Ο, Σ) is safe under the ownership relation Οs and index i. 2. For all relations in the set of relations Ρs, the relation is equal to an empty map. 3. The length of the ownership relation Οs is equal to the length of the set of relations Ρs. 4. For all ele...
A key 'x' is in the image of set 'A' under the 'Key' function if and only if 'x' is in 'A'.
For any state of affairs theta, any sequence of operations ys, and any program p, if every operation in the sequence of operations xs is a valid operation, and if the set of temporary read operations in xs and ys are disjoint, and if the history is consistent for the state of affairs theta, the program p, and the seque...