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The ordinal exponentiation of "r" to the power of "s", further raised to the power of "t", is ordinally equivalent to "r" raised to the power of the ordinal multiplication of "s" and "t". In other words, "(r to the power of s) to the power of t" is the same as "r to the power of (s times t)" in the context of ordinal n...
If "a" derives "u" and "b" derives "v", and both "a" and "b" are sentences, then it can be shown that "a" followed by "b" derives "u" followed by "v".
The lemma named "tauChainSum1" is defined with three fixed variables: P, P', and Q, all of which are of type ccs. It assumes that P leads to P' through a tau chain and that P is not equal to P'. Under these assumptions, it shows that the sum of P and Q leads to P' through a tau chain.
In the context of finite Red-Black trees, there is a lemma called 'ex_se_succ'. This lemma assumes that 'prb' is a Red-Black tree and 'rv' is in the set of red vertices of 'prb'. It shows that there exists a configuration 'c'' such that 'c'' is a successor of the configuration of 'prb' at 'rv' along a sequence of event...
The lemma shows that the composition, tensor product, associativity, and identity in the Free Monoidal Category induce a monoidal category.
Normalizing the concatenation of a normalized polynomial 'p' and another polynomial 'q' is equivalent to normalizing the concatenation of the original polynomial 'p' and 'q'.
The evaluation of "ex1" is defined as 0.
In the context of a valid undirected multigraph, a vertex v is adjacent to a vertex v' if and only if the vertex v' is adjacent to the vertex v.
If we define "prefixes" as the set of all sequences 't' such that there exists another sequence 't2' where 'x' is the concatenation of 't' and 't2', then the set of these prefixes is finite and the cardinality (or size) of this set is equal to the length of 'x' plus one.
The arrow map of the functor evaluation between two categories alpha and beta in a category CC is a single-valued function.
The lemma "exec_inc_imp_trace_inc" establishes a relationship between two entities A and B. It assumes that the external behavior of B is the same as that of A. Furthermore, for any execution sequence of B, there exists a corresponding execution sequence of A such that the traces of these execution sequences under the ...
The evaluation of a proposition 'p' in a given interpretation 'i' is equal to the evaluation of the double negation of 'p' in the same interpretation 'i'.
The function "ccTTree" is continuous with respect to the set "S" in the graph "G".
The linked list generated from a co-recursive stream, where the stream is defined by the head of the stream (SHD), a condition to end or continue (endORmore), the tail of the stream when it continues (STL_more), and the tail of the stream when it ends (STL_end), and an initial value (x), is equal to the co-recursive li...
The lemma "reflcl_over_single_ta_sound" assumes that the function context "C" is a subset of the finite set "F", and "q" is an element of "Q". It shows that "qf" is an element of the TA derivative of the reflexive closure over a single TA "Q", "F", "qc", and "qf", with respect to the context of "C" applied to the varia...
If the length of the list 'qs' is equal to the length of the list 'Strat', then the function Tp applied to the initial state 'init', the list 'qs', and the list 'Strat' is equal to the sum for all 'i' in the range from 0 to the length of 'qs' of the function tp applied to the steps' of the initial state 'init', the lis...
If an atom "a" is fresh (or distinct) from a set "S", then the atom "a" of type "var" is also fresh from the updated list of set "S".
, given the assumptions that "d" is a Dickson grading and the degree of "t" is less than or equal to the maximum degree of "d", if "p" belongs to the set of signatures of degree "d" with respect to "j", then the result of multiplying the monomial "p" by the constant "c" and the term "t" also belongs to the same set of ...
In the lemma from the base of a partially ordered semicategory, if J is a subset of I, then the base of the partially ordered semicategory with parameters alpha, J, and AA is shown.
The lemma assumes that "Subobjs P C (rev Cs)" and "wf_prog wf_md P" are true. It shows that in the context of program P and class C, the single-element list containing C is a subobject of the reversed list Cs.
If for every element 'x' in set 'A', the function 'f' applied to 'x' results in an integer, then the product of the function 'f' over all elements in set 'A' is also an integer.
If the variables "lbls", "pri", "pos", "stepss", "nxt", and "onxt" are equal to their respective counterparts "lbls'", "pri'", "pos'", "stepss'", "nxt'", and "onxt'", and for any elements "x" and "x'" and any value "d", if "x" is in the set "stepss" and "x'" is in the set "x", then the function "g" applied to "x'" and ...
The function "ipurge_tr_rev_aux" with inputs I, D, a single-element set containing u, and xs is equal to the function "ipurge_tr_rev" with inputs I, D, u, and xs.
If for all 'x', the proposition 'phi x' holds in 'v', then it implies that for all 'x', the proposition 'phi x' holds in 'v'.
The lemma named "put_eq" assumes that the "get" function applied to sigma equals k, and that the "put" function applied to sigma with u equals the "put" function applied to rho with v. It shows that the "put" function applied to rho with k equals sigma.
If polynomial 'p' and 'q' are both in the carrier set 'P', then the degree of the sum of 'p' and 'q' in ring 'R' is less than or equal to the maximum of the degree of 'p' and the degree of 'q' in ring 'R'.
If "False" is assumed, then "FAIL" is less than or equal to "SPEC Phi".
A vertex 'u' is reachable from itself if and only if 'u' is a vertex in graph 'G'.
The representation of the intersection of two formulas 'x' and 'y' is equal to the intersection of the representations of 'x' and 'y'.
The lemma named "list_segment_subset" assumes that "j" is less than or equal to "k". It shows that the set of list segments from "i" to "j" in "as" is a subset of the set of list segments from "i" to "k" in "as".
If there are two morphisms 'f' and 'g' from a chamber complex 'W' to another chamber complex 'X', and if these two morphisms are functionally equivalent on a chamber 'C' and there exists a gallery from 'C' to another chamber 'D' in 'W', and the images of this gallery under 'f' and 'g' are proper galleries, then 'f' and...
The set of operations in a stateful strand "A" is a subset of the set of operations in the stateful strand that results from adding an element "a" to "A".
If a set 'mx' is the maximal set among all sets 'I' that are ideals of the ring 'R' and have no intersection with set 'S', then 'mx' is also an ideal of the ring 'R'.
If 'm' is less than infinity and 'n' is less than infinity, then the product of the difference of 'm' and 'n' with 'l' is equal to the difference of the product of 'm' and 'l' and the product of 'n' and 'l'.
The function "bind" is parameterized over two relations, Ra and Rb. It maps from a non-deterministic result relation over Ra to a function from Ra to a non-deterministic result relation over Rb, and finally to a non-deterministic result relation over Rb.
If "b" is an object in the category "CC" and "f" is a morphism from "a" to "a'" in the opposite category of "CC", then the application of the arrow map of the Hom functor from "CC" to "b" on "f" is equal to the composition of the Hom functor of "CC" with the pair consisting of "f" and the identity morphism of "b" in "C...
If a type check in a given environment G, with maximum stack size mxs, maximum number of registers mxr, and state s is successful, and if mxs is less than or equal to a new maximum stack size mxs', then the type check will also be successful in the same environment G, with the new maximum stack size mxs', the same maxi...
The lemma named "term_of_uint32_code" is defined as follows: It introduces two abbreviations: "TR" stands for "typerep.Typerep" and "bit0" stands for "Numeral_Type.bit0". It then shows that the term representation of any instance 'x' of the class 'term_of' is equivalent to applying the function 'Uint32.uint32.Abs_uin...
Applying a function 'f' to 'x' and then iterating that function 'n' times is equivalent to iterating the function 'f' 'n' times on 'x' and then applying the function 'f'.
If "g" and "g'" are elements of the carrier of group G, and the homomorphism of "g" equals the homomorphism of "g'", then the coset of the kernel of the homomorphism "h" with respect to "g" is a subset of the coset of the kernel of the homomorphism "h" with respect to "g'".
The lemma named "fk_update_2_distr" assumes that the list "as" is not empty. It shows that the result of folding the function "fk_update_2" over the list "as", starting with the return of the probability mass function (pmf) of the tuple (0,0,0), is equal to the pmf of the set of all numbers less than the length of "as"...
If A, B, and Y form a Polynomial Ring, and if function 'f' is a polynomial homomorphism from R to S with respect to X, A, B, and Y, and if 'p' is an element of the carrier set R, then the image of 'p' under function 'f' is an element of the carrier set A.
The domain of the function "ufa_\<alpha>" applied to a list "l" is the set of all integers from 0 to the length of the list "l" (exclusive).
The interpretation function "Ifm" applied to the simplified form of a proposition "p" is equivalent to the interpretation function applied directly to the proposition "p", given the same boolean sequence "bs".
The lemma named "lex_ext_stri_imp_nstri" assumes that the first element of the lexicographic extension function "lex_ext" with inputs "f", "m", "xs", and "ys" is true. It then shows that the second element of the same function is also true.
The intersection of the TESL interpretation of a single event specification φ and the TESL interpretation of a list of event specifications Φ is equal to the TESL interpretation of the list formed by prepending φ to Φ.
The nth power of the formal Laurent series X at the kth position is equal to 1 if k is equal to n, and 0 otherwise.
: 1. For any goal state and start state, if the goal state follows the start state, then the goal state succeeds the start state. 2. For any goal state, intermediate state, and start state, if the goal state succeeds the intermediate state and the intermediate state succeeds the start state, then the goal state succee...
The successors of the abstract compilation of a given variable 'a' starting at position 'n' is the set of positions from 'n + 1' to 'n' plus the size of the abstract compilation of 'a'.
"there exists a step, left tape and right tape such that P is true for the left tape and right tape and the Turing machine reaches a final state after a certain number of steps" implies that "the Turing machine, when started in a state satisfying P, will eventually reach a state satisfying Q". This is equivalent to "fo...
If in a program P, the sequence Ss is a subtype of the sequence Ts, and the sequence Ts is a subtype of the sequence Us, then it follows that in the program P, the sequence Ss is a subtype of the sequence Us. This is a statement of transitivity in the subtype relation.
The product of the property "p" and the bottom element, unioned with the identity element, equals the property "p".
The lemma, named "Call_lemma", assumes two conditions: 1. For all 'q' in the domain of 'Γ', and for all 'Z', 'Γ' and 'Θ' totally prove that if a state 's' equals 'Z' and 'Γ' does not lead the call of 'q' from a normal state 's' to either a stuck state or a fault state not in 'F', and 'Γ' leads the call of 'q' from a n...
If H is a submodule of M in a ring R, and if m is an element of the carrier M, then the coset operation of m added to H in M is equal to m added to H in M.
If the pair (a, b) is in the set xs and the pair (a, b) is not in the set that results from updating the list xs with the value x at position n, then a must be equal to n.
The lemma named "optimize_matches_a_simplers" assumes that a ruleset 'rs' is simple and for all 'a' and 'm', if 'a' is either 'Accept' or 'Drop', then the match result of applying function 'f' to 'a' and 'm' under a certain condition 'γ' is the same as the match result of 'm' and 'a' under the same condition 'γ'. Under...
A property X holds if and only if for all finite sets GG, if GG intersects with X and GG is acceptable under the identity function, then the supremum of all elements in GG belongs to X.
If 'f' is a morphism from 'a' to 'b' in the product digraph over a set 'I' of digraphs 'A_i', then the following conditions hold: 1. 'f' is a very simple function (vsv). 2. The domain of 'f' is 'I'. 3. 'a' is a very simple function (vsv). 4. The domain of 'a' is 'I'. 5. 'b' is a very simple function (vsv). 6. The doma...
A singleton matrix with elements j, i, and x is greater than or equal to zero if and only if the element x is greater than or equal to zero. This is applicable for any 'a' that is an order and zero.
The lemma "acute2_suma__nbet" assumes that angle ABC is acute, angle DEF is acute, and the sum of angles ABC and DEF equals angle GHI. It shows that point H is not between points G and I.
Setting the nth bit of a word 'a' to a value 'x' is equivalent to converting the result of a binary operation 'bin_sc' with inputs 'n', 'x', and the unsigned integer representation of 'a' back into a word.
If a list starting with "x" followed by "xs" is normalized, then the list "xs" is also normalized.
The lemma "encode_operators_is_cnf" assumes that "\<Pi>" is a valid problem in STRIPS (Stanford Research Institute Problem Solver). It shows that the encoding of the operators of "\<Pi>" at time "t" is in conjunctive normal form.
The result type of a pair (x, m) is equal to the result type of m.
There exists an 'x' such that for any 'y' of type boolean 'one_or_two', 'x' is not equal to 'y'.
The probability mass function of returning a value 'x' is 'x' itself.
The function 'h' is injective on the carrier of group 'G' if and only if the kernel of the function 'h' from group 'G' to group 'H' is trivial, i.e., it only contains the identity element.
If the first elements of the list "ms" are distinct, then the first elements of the list resulting from applying the "methods" function to "C" and "ms" are also distinct.
The first element of a pair in a unary expression is equal to the first element of the pair itself.
If a regular expression "from_mregex mr \<phi>s" is safe in the past and strict, then for any "ms" in "RPDi n mr", the set of free variables in the regular expression "from_mregex ms \<phi>s" is equal to the set of free variables in the regular expression "from_mregex mr \<phi>s".
The lemma named "wellformed_\<R>" in the context of semi-mojmir shows that the set of all states reachable from the initial state through the transition function "step" over the alphabet "\<Sigma>" is finite.
The lemma named "simp_in_frac" assumes that the pair (r, s) is in the carrier of the relation and that s' is in S. It shows that the relation of r and s is equal to the relation of the product of s' and r and the product of s' and s.
The lemma named "amalgamation_V01_R1" assumes that "f" is an element of "V01". It shows that the amalgamation of the first and second elements of "f" is an element of "R".
Any assertion P implies truth.
If a binary search tree (bst) is formed with a node containing a pair (a, n) and left subtree l and right subtree r, then a binary search tree is also formed when joining the left subtree l, the element a, and the right subtree r.
The function "x times the natural logarithm of 3x" is little-o of the function "x times the natural logarithm of x" for real numbers. This means that the growth rate of the function "x times the natural logarithm of 3x" is significantly smaller than the growth rate of the function "x times the natural logarithm of x" a...
If property P holds for zeroX, and for any value v if property P holds then it also holds for the next value of v, then property P holds for any given value v'.
Let a and b be sequences of elements in a real normed algebra that is also a Banach space, with elements indexed by natural numbers. If the series of the norms of the elements of a and b are both summable, then the series formed by the Cauchy product of a and b, i.e., the series whose k-th term is the sum of the produc...
If a while-loop with annotations (whileAnnoG gs b I V c) under the conditions Gamma and Theta, and with preconditions P and postconditions Q,A, is valid in a partial correctness setting, then a while-loop without annotations (while gs b c) under the same conditions and with the same preconditions and postconditions is ...
The matrix is an ordered incidence system for the range from 0 to the dimension of the row of the matrix M, with the mapping of each column to a block in the columns of matrix M.
The lemma named "root_cycle_converse" assumes that "r" is connected to the root "x", "r" is a point, and "r" is not equal to 0 when multiplied by "x". Under these assumptions, it shows that the transpose of "x" multiplied by "r" is not equal to 0.
The lemma epm_12: Assuming "v" is a vector, it shows that the intersection of "w" and the negation of "prim_EP w v e", further intersected with "prim_E w v e", is the bottom element.
Under the context Γ₀, the operation call of 'OclIsUndefinedOp' on 'NullLiteral' using dot notation with an empty argument list is of type 'Boolean[1]'.
For a function 'f' from type 'a' to type 'b' (where 'b' is a weak quantale), the binary modality of 'f' and 'g' composed with 'h' at 'x' is equal to the binary modality of 'f' composed with the binary modality of 'g' and 'h' at 'x'.
The lemma named "is_loopfree_step1" makes the following assumptions: 1. The graph G at the nth step is loop-free. 2. The first element of the nth arc in graph G is not equal to the second element of the nth arc in graph G. 3. The number n is less than the count of arcs in graph G. Under these assumptions, it shows tha...
A transition ((s1, s2), t) is in the set of outgoing transitions from state s in the extended finite state machine e if and only if the transition ((s1, s2), t) is in the machine e and the initial state of the transition is s.
"x is less than or equal to x".
If 'x' is an object in the category 'D', 'a' is the object mapping of functor 'F' applied to the object mapping of functor 'G' applied to 'x', 'b' is 'x', and 'D'' is 'D', then the natural transformation map of the counit of the adjunction 'Phi' applied to 'x' is a morphism from 'a' to 'b' in the category 'D''.
If a set S is connected within a topological space X, then the closure of the set S within the same topological space X is also connected.
: Assuming: 1. "\<A>" is in the set of reachable constraints of "P", 2. All transactions in "P" are well-formed, 3. For each transaction "T" in "P", every element "x" in the set of unlabelled transaction updates of "T" is an update, is a variable, is a function set, and the first element of the variable is a value ato...
The star of a formal power series applied to an empty list is equal to the star of the value of the formal power series at the empty list.
There is no failure in the operation "oreturn x" for any input, as indicated by the function that always returns true.
Casting a document pointer extension, specifically a document extension pointer, to a shadow root pointer results in None.
If 'i' is less than or equal to 'n' and 'j' is less than or equal to 'n', then the result of the function 'fw' with parameters 'm', 'n', '0', 'i', 'j', 'i', 'j' is less than or equal to the result of the function 'm' with parameters 'i', 'j'.
The lemma named "resolution_decreases_selected_part" assumes the following conditions: 1. An ordered selection resolution is performed on the propositions P1 and P2, resulting in proposition C with A being the selected part. 2. The negation of A is in P2. 3. Both P1 and P2 are finite. 4. The cardinality (or size) of t...
The length of the list "y_cl_var_var'_v_list" is equal to the length of the same list after it has been mapped to a new list that only contains the first element of each tuple in the original list.
The lemma named "qtname_declclass_def" defines that the declaration class of a qualified type name 'q' is equivalent to 'q' itself.
, under the assumptions that for almost every 'x' in measure space 'M', 'x' belongs to set 'A' if and only if 'x' belongs to set 'B', and that both 'A' and 'B' are in the sets of 'M', it can be shown that the outer measure of 'A' in 'M' is equal to the outer measure of 'B' in 'M'.
If the head of an edge 'e' in graph 'G' is the root, then 'e' is not in the set of arcs 'T'.
The lemma named "tauPres" is defined with fixed variables P and Q of type pi, and Rel and Rel' of type set of pairs of pi. It assumes that the pair (P, Q) is an element of the set Rel. The lemma then shows that the tau transition of P weakly simulates the tau transition of Q under the relation Rel.