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The lemma named "sum_Inr_prs" in the context of quotient sums, assumes two conditions: "Quotient3 R1 Abs1 Rep1" and "Quotient3 R2 Abs2 Rep2". It shows that applying the function "Rep2" followed by "map_sum Abs1 Abs2" to "Inr" results in "Inr".
If "p" is an element of the monad probability space over "X", and "f" is a quasi-Borel measurable function from the quasi-Borel product space of "X" and "Y" to the nonnegative quasi-Borel real numbers, then the function that maps each "y" in "Y" to the quasi-Borel nonnegative extended real-valued integral of the functi...
, assuming 'F' is a register, the product of 'F a' and 'F b' and 'c' is equal to 'F' applied to the composition of 'a' and 'b', then multiplied by 'c'.
If a graph H is a subdivision pair of a graph G, and if graph G is a K33 graph (a complete bipartite graph with 3 vertices in each set), then the contracted graph of H is also a K33 graph.
If a state 's' is within reach, and a pair of keys (k, k') is in the set of cards 's' for guest 'g', then key 'k' must be in the set of issued keys in state 's'.
The lemma named "dirichlet_prod_neutral_intro" establishes a relationship between a function S, which is defined for natural numbers and yields complex numbers, and another function f, which takes two natural numbers and yields a complex number. The function S is defined as the sum of the function f applied to all natu...
"evalS is equal to eval_ww".
The lemma named "Standard Standard Development" demonstrates that the standard development of any term 't' is standard.
For any fixed list of terms 'ts', 'ts' can transition to itself through zero or more steps.
If a sequence 'X' is a Cauchy sequence and it does not vanish (i.e., it does not converge to zero), then the sequence formed by taking the inverse of each term in 'X' is also a Cauchy sequence.
If the function 'f' is eventually non-negative with respect to the filter 'F', then 'f' belongs to the small omega of 'F' for the constant function 1 (of real numbers) if and only if the limit of 'f' approaches positive infinity as it is filtered by 'F'.
Appending a bottom element (undefined or error value) with any lazy list "ys" results in a bottom element.
If a subcommand is a transformation of a command 'c' with a temporary variable 'temp', and 'c' is a new name for 'temp', then the variable 'V' is not in the right-hand side of the expression 'e'.
If the tail of the list 'ls' does not contain the key 'key', then the list 'ls' itself does not contain the key 'key'.
The lemma named "arg_complex_of_real_times_i_negative" assumes that "k" is less than zero. It shows that the argument of the complex number resulting from "k" times the imaginary unit is equal to negative pi divided by 2.
The lemma DJ: It assumes that for all 'p' and 'q', the interpretation of 'f' applied to the disjunction of 'p' and 'q' is equal to the interpretation of the disjunction of 'f' applied to 'p' and 'f' applied to 'q'. It also assumes that applying 'f' to 'F' results in 'F'. It shows that the interpretation of 'DJ' app...
If b is greater than 2 and n is greater than 0, then the product of alpha of b and n minus the product of alpha of b and the successor of n and alpha of b and the predecessor of n equals 1.
If the end of path p is the start of path q, and the end of path q is the start of path r, then the path formed by concatenating p with the path formed by concatenating q and r is equivalent to the path formed by concatenating the path formed by concatenating p and q with r.
The lemma named "extends_tgt_out_edges" assumes that "g" is extended by "e" to "g'". It shows that the set of outgoing edges from the target of "e" in "g'" is empty.
The superterm relation is transitive.
The lemma "dist_fs_infinite2" shows that the hyperbolic distance from infinity to a complex number z1 is equal to 2 divided by the square root of 1 plus the square of the absolute value of z1.
If a finite group representation is not irreducible for a group G with scalar multiplication smult on a vector space V, then there exist two subspaces U and W such that U and W are both non-zero and distinct from V, and V is the direct sum of U and W.
For a given state sigma, message, hops, rreqid, dip, dsn, dsk, oip, osn, sip, and handled, if the rreq_rrep_fresh function applied to the routing table of sigma at sip and a Rreq message with the given parameters returns true, and the rreq_rrep_sn function applied to the same Rreq message also returns true, then the ms...
For a real number x, given that x is greater than or equal to 1, the inverse hyperbolic cosine of x equals 0 if and only if x equals 1.
The lemma "eqButUIDf_cong" assumes the following conditions: 1. The function "frds" is equivalent to "frds1" except for the user ID. 2. If the user ID is in the set of user IDs for AID', then "uu" is equal to "uu1". 3. If the user ID is not in the set of user IDs for AID', then the list "uu" is equivalent to "uu1" exc...
The function "prod.swap" is in the relation of the Cartesian product of A and B to the Cartesian product of B and A. This function swaps the elements of ordered pairs, effectively changing the order from (A, B) to (B, A).
For all 'i' less than the length of 'signs', the 'i'-th column of the alternative matrix 'A_R' with parameters 'signs' and 'subsets' is equal to a vector of length 'subsets' where each element is the result of the function 'z_R' applied to the 'j'-th element of 'subsets' and the 'i'-th element of 'signs'.
For any values 'a', 'b', and 'x', the power of 'x' to the 'a' raised to the power of 'b' is equal to 'x' raised to the power of the product of 'a' and 'b'.
The lemma named "one_le_integral_t" assumes that "x" is recurrent. It shows that "1" is less than or equal to "U prime of x and x".
"det_bound_fact" is a determinant bound.
If A and B are Polynomial Rings over Y, if f is a ring homomorphism from S to B, if p is an element of the ring R, and if the evaluation homomorphism of p under these conditions is not the zero element of A, then p is not the zero element.
The lemma named "mset_single_cases2" has two cases, "loc" and "env". It assumes that adding the multiset "s" to "c" is equal to adding the multiset "r'" to "c'". It also assumes two cases: one where "s" equals "r'" and "c" equals "c'", and another where "c'" is the result of removing "s" from "c'" and then adding "s", ...
Given the conditions: 1. The index is less than the key (as stated by "index_less index key"), and 2. For all elements x in the set xs, the key of x is within the range from mi to ma (as stated by "\<forall>x \<in> set xs. key x \<in> {mi..ma}"), Then, if i is less than the index of key x in n, mi, ma (as stated by "i...
The size of a multiset that contains only one element 'b', when a function 'f' is applied to it, is equal to the successor of the function 'f' applied to 'b'.
The last element of the sequence "kp6x8ul" is the pair (5,2).
If the category 'A' is the opposite of 'A' and the category 'B' is the opposite of 'B', then the opposite of the functor 'F' maps from the category 'A' to the category 'B' in the context of an arbitrary Grothendieck universe 'α'.
The lemma named "locate_pred_unique" assumes the following: 1. For any element 'a' in the set 'al', the function 'f' applied to 'a' is greater than 0. 2. The predicate 'locate_pred' holds for the function 'f', the set 'al', the index 'i', and the pair 'n1_j1'. 3. The same predicate 'locate_pred' also holds for the func...
A topological space X is equivalent to a discrete topology with an empty set if and only if the topological space of X is an empty set.
The lemma named "supp_union_fset" establishes that for any two given finite sets S and T of any type 'a, the support of the union of S and T is equal to the union of the support of S and the support of T.
The absolute value of the least common multiple (lcm) of two integers i and j is equal to the least common multiple of i and j. This is applicable for any two integers i and j.
If 'j' is an element of the set 'Vset' indexed by 'α' and 'f' is a mapping from 'a' to 'b' in the graph 'CC', then the set containing the pair '(j, f)' is a mapping from the set containing the pair '(j, a)' to the set containing the pair '(j, b)' in the product of the graph 'CC' over the set containing 'j'.
A set S is a retract of a topological space X if and only if S is a subset of the topological space X and the identity map is a section map from the subtopology of X restricted to S to X.
The intersection of scene X with itself is equal to X.
The theorem named "psubstt_id" establishes that for any term 't' and any list of terms 'ts', applying the identity function 'id' to 't' through 'psubstt' results in 't' itself, and applying the identity function to 'ts' through 'psubstts' results in 'ts' itself.
The lemma l7_25: assumes that the distance from C to A is congruent to the distance from C to B, it shows that there exists a point X that is the midpoint of A and B.
The theorem STL5_2: assumes that if F holds and f does not change, then F will hold in the next state (h1), and if G holds and g does not change, then G will hold in the next state (h2), it shows that "always (F and G)" is equivalent to "(always F) and (always G)".
The lemma "no_writes_unchanged0" assumes that for all "l" less than "k", "v" is not in the set of writes of the path "σ" at "l". It shows that the value of "v" at "k" in the path "σ" is equal to the value of "v" in "σ".
The lemma named "dims_tensor_deep_model" assumes that when weights are removed from the model 'm', it results in a deep model 'l' with parameters 'rs'. It then shows that the dimensions of the tensor derived from the network model 'm' at position 'y' are equal to a replication of the last parameter in 'rs', and this re...
For any two nested multisets X and Y, if X is less than Y and Y is less than X, then this is a contradiction.
The composition of the relation "R" with the pure function "S" under the higher-order relation "hr_comp" is equivalent to the pure function of the composition of "R" and "S".
If two states 's' and 's1' are equal excluding the PID (Paper ID), then the following conditions hold: 1. The conference IDs of 's' and 's1' are equal. 2. The configurations of 's' and 's1' are equal. 3. The user IDs of 's' and 's1' are equal. 4. The passwords of 's' and 's1' are equal. 5. The users of 's' and 's1' ar...
The most significant bit of the negation of an integer 'x' is the logical negation of the most significant bit of 'x'.
If for every set A in a collection S, F leads to B within the context of A, then F leads to B within the context of the union of all sets in S.
The lemma "Vset_mono_strict" assumes that "m" is an ordinal number and "n" is an element of "m". It shows that the Vset of "n" is strictly less than the Vset of "m".
The function 'flip' is injective on the set 'A'.
The lemma named "sum_over_subset_pmf_support" assumes that "T" is a finite set and the set of probability mass function "p" is a subset of "T". It shows that the sum of the product of "a", the probability mass function "p" at "i", and "f" at "i" over all "i" in "T" is equal to the sum of the product of "a", the probabi...
The lemma named "substAbs_compose1" assumes that "A" is a good abstraction, and "Y1" and "Y2" are good. It shows that substituting "Y1" for "y" in "A", and then substituting "Y2" for "y" in the result, is equal to substituting "Y1" with "Y2" substituted for "y" in "A".
The lemma named "vfsequence_vcpower" assumes that "n" is a natural number and "xs" is an element of the Cartesian power of "A" with "n". It shows that "xs" is a variable finite sequence.
: Given a real polynomial 'p', a list of real polynomials 'qs', a list of subsets (each subset is a pair of lists of natural numbers), and a list of lists of rational numbers 'signs', under the following conditions: 1. The length of 'subsets' is equal to the length of 'signs'. 2. The matrix 'A_R' formed from 'signs' ...
The lemma "obtain_block_index_map_block_set" assumes that a block "bl" is in the multiset created by mapping each column "c" of matrix "M" to a block. It then obtains an index "j" such that "j" is less than the number of columns in matrix "M" and the block "bl" is equal to the block obtained by mapping the j-th column ...
If a set S is a subset of A, and L is an element of A, and for all elements x in S, the ordered pair (x, L) is in the relation r, then the ordered pair (least upper bound of S in the complete lattice, L) is in the relation r.
If the pair (rho, eta) is well-formed according to the context Gamma, then Gamma is a good context.
If "n" is greater than 0, then the list that starts with "f 0" and continues with the mapping of the function "f" over the list of integers from 1 to "n" (exclusive) is equal to the mapping of the function "f" over the list of integers from 0 to "n" (exclusive).
If "n" is greater than 1, "f" is a holomorphic function on set "A", and "χ" is not equal to the principal Dirichlet character of "n" and is a Dirichlet character of "n", or "χ" is equal to the principal Dirichlet character of "n" and for all "x" in "A", "f" of "x" is not equal to 1, then the function "Dirichlet_L" of "...
The lemma "reduce_below_abs" assumes that matrix A is in the carrier matrix of dimensions m by n. It shows that the function "reduce_below_abs" with parameters a, xs, D, and A results in a matrix that is also in the carrier matrix of dimensions m by n.
For any ownership set, snapshot, and pending write, if the pending write is non-volatile and is owned or read-only, then the outstanding references that are non-volatile writes in the store buffer are a subset of the union of the ownership set and all acquired in the store buffer.
The lemma named "sequence_unique_limpt" is defined for a function 'f' that maps natural numbers to a space 'a' which is a T2 space (also known as a Hausdorff space). The lemma assumes that the function 'f' converges to a limit 'l' in a sequential manner and that 'l'' is a limit point of the range of function 'f'. The l...
If 'z' is not a non-positive integer, then the Gamma function has a field derivative at 'z' within a set 'A', and this derivative is the product of the Gamma function at 'z' and the Digamma function at 'z'.
The lemma named "mat_mod_eq_cond" assumes that all elements of the matrix M are in the set of non-negative integers less than m. Under this assumption, it shows that the modular matrix of M is equal to M itself.
The lemma named "diamond_cube_is_only_horizontal_div_of_rot" demonstrates that the only horizontal division of the boundary of the diamond cube is the rotated diamond cube.
If two functions 'f' and 'g' are continuous on a set 'A', and for all 'x' in 'A', the absolute value of 'f(x)' and 'g(x)' is not infinity, then the function that is the difference between 'f' and 'g' is also continuous on 'A'. This is in the context of extended real numbers.
If a function 'f' is strictly monotonically increasing, then the limit of the function 'f' along a subsequence is also a subsequence.
The lemma named "two_reduced_heads_imp_nzero_relfun" assumes that "s" is not equal to "t", "s" is the head of a reduced word for "S" and "w" in the sequence "as", and "t" is the head of a reduced word for "S" and "w" in the sequence "bs". Under these assumptions, it shows that the relational function of "s" and "t" is ...
If you map the second element (snd) over the list produced by enumerating "xs" starting from "n", you will get back the original list "xs".
An index 'i' is in the set resulting from finding indices of 'x' in 'xs' if and only if 'i' is less than the length of 'xs' and the element at index 'i' of 'xs' is equal to 'x'.
The lemma named "observable_io_targets" assumes that "M" is observable and "io" is in the language of the system "M" starting from state "q". It then obtains a state "q'" such that the set of states reachable from state "q" in system "M" via "io" is only "q'".
If the composition of 'u' and 'ctor1' equals the composition of 'ctor1' and 'F1map f u v', and the composition of 'v' and 'ctor2' equals the composition of 'ctor2' and 'F2map f u v', then 'u' equals 'IF1map f' and 'v' equals 'IF2map f'.
The lemma named "resolvent_upon_is_resolvent" assumes that a positive instance of A is in P1 and a negative instance of A is in P2. It shows that the resolvent of P1 and P2 is the resolvent upon P1, P2, and A.
The lemma named "dlex_fun_trans" assumes that if the function "dlex_fun" holds true for the pair (s, t) and the pair (t, u), then it must also hold true for the pair (s, u).
The left restriction of a relation 'r' to a set 'A' is a subset of the field pairs of the relation 'r'.
If 'n' is less than or equal to the length of the list 'ss', then for a hash table 'src' represented by 'ss' and a hash table 'dst' represented by 'ds', the operation 'ht_copy' that copies 'n' elements from 'src' to 'dst' results in a state where 'src' remains a hash table represented by 'ss' and 'r' is a hash table re...
Any element "x" multiplied by the identity element "one" equals "x" itself.
The comparison of natural term according to a given order is equivalent to the comparison of keys according to the key order derived from the same natural term order.
The duration of "xs" at index "i" is greater than or equal to zero.
The application of a bounded continuous function 'x' is continuous on any set 'T'. Additionally, the range of the application of this bounded continuous function 'x' is also bounded.
The range of the rcut is a subset of the power set of the rational numbers.
The lemma "create_gba_acc" shows that for the function "create_gba" applied to any formula "phi", it is less than or equal to the specification that for all sequences "xi", the generalized Büchi automaton "A" accepts "xi" if and only if "xi" satisfies the formula "phi" in the real world semantics.
If a term 't' is not in normal form (as per the assumption "\<not> NF t") and 't' is an ideal (as per the assumption "Ide t"), then there exists a term 'u' such that 't' and 'u' are coinitial (meaning they share the same initial state or point) and 'u' is not an ideal.
, given the assumptions that "a" and "b" are elements of the carrier "R", "p1", "p2", and "p3" are on the curve defined by "a" and "b", "p1" and "p2" are generic, "p2" and "p3" are generic, the addition of "p1" and "p2" with "a" is generic with "p3", and "p1" is generic with the addition of "p2" and "p3" with "a", it c...
The row of the minor matrix M at index i, j is equal to the axis at index j with a value of 1.
The function 'fun_evaluate_prog' with parameters 'state', 'env', and 'compile rs' is equal to a tuple. This tuple consists of 'state' and the result of the function 'combine_dec_result' with parameters 'as_sem_env env' and 'Rval' with 'sem_env.v' and 'sem_env.c' both being equal to 'nsEmpty'. This lemma is simplified a...
The lemma named "flrestriction_vsingleton_nin" assumes that "a" is not an element of the set "A". It shows that the left restriction of the set containing the ordered pair (a, b) to "A" is an empty set.
A zero vector of size "n+1" is equivalent to a vector that starts with zero and is followed by a zero vector of size "n".
The well-sortedness of an input in the term model "termMOD" is equivalent to the well-sortedness of the input in the general model "delta".
The lemma l9_19: Assuming "Point P lies on the line through points X and Y" and "Point P lies on the line through points A and B", it shows "Point X and Y are on the same side of line AB if and only if Point P is outside of line AB and Point X and Y do not lie on the same line as Point A".
If variable 'x' does not affect 'u' and 'x' does not affect 'v', then 'x' will not affect the division of 'u' by 'v'.
The lemma named "mul_eq_singleton" assumes that the pair (M, {#α#}) belongs to the multiplicative equality relation 'r'. It shows that either 'M' is equal to the multiset {#α#}, or the set of elements in 'M' is a subset of the downward closure of 'r' with respect to the multiset {#α#}.
The lemma named "STA_Dup" assumes that a branch is derivable from a set A and a natural number n, and that a duplication of an element q at position i exists in the branch with respect to a list xs. It shows that the branch, with the duplicated element omitted, is still derivable from the set A and the natural number n...
The lemma states the uniqueness of polynomial equality. It fixes two polynomials 'f' and 'g' in a factorial ring with gcd and semiring with gcd and multiplication normalization. It assumes that 'f' and 'g' are coprime, and that either 'B' is zero or its degree is less than the degree of 'f', and similarly for 'B''. It ...
: 1. If a term is transformable to a variable 'x' under a set of term implications 'TI', then the term is the variable 'x'. 2. If a variable 'x' is transformable to a term under a set of term implications 'TI', then the term is the variable 'x'. 3. If a term is transformable to a variable 'x' under a set of term impli...