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The lemma named "UNIF1_unifless" assumes that "UNIF1" is a function that unifies "E" and "E'". It shows that the pair "(E', E)" is in the relation "unifless".
If 'x' is a natural number, then the real number equivalent of the floor value of 'x' is equal to 'x' itself.
For any set X, if each element x in X satisfies the condition that a function a of x is greater than or equal to zero, then the product of a(x) raised to the power of e (for each x in X) is equal to the product of a(x) (for each x in X) raised to the power of e. This is true for real numbers.
The lemma named "s_mul_ext_IdI" assumes the following conditions: 1. The multiset X is not empty. 2. M is the union of multisets X and Z. 3. N is the union of multisets Y and Z. 4. For every element y in the multiset Y, there exists an element x in the multiset X such that the pair (x, y) belongs to the relation R. I...
The lemma named "invimage_vconst_on_vempty" assumes that the constant "c" is not an element of the set "B". It shows that the inverse image of the constant function on set "A" with value "c" over the set "B" is the empty set.
For any given variables z, i, j, and n, the adverse function of z, i, and j when right restricted to n is equal to the list encoding of the prefixes of z, i, j, and n.
A block of code with initialization "init", body "bdy", return "return", and continuation "c", when executed in a normal state "s" for "n" steps, results in a state "t". The "Normal" assumption states that if the body of the code, when executed in the state resulting from the initialization of "s", for "n" steps, resu...
, assuming "a" is an identity, the evaluation of the inverse left unit of "a" in bold brackets equals the inverse left unit of the evaluation of "a".
If a matrix "m" is cycle-free up to "n", and for all "i" and "j" less than or equal to "n", the function "D" applied to "m", "i", "j", and "n" equals the function "m'" applied to "i" and "j", and if "i" and "j" are less than or equal to "n", and the set "xs" is a subset of the set of integers from 0 to "n", then there ...
: 1. The root of the difference between tree 't' and tree 't'' is equal to the difference between the root of tree 't' and the root of tree 't''. 2. The left child of the difference between tree 't' and tree 't'' is equal to the difference between the left child of tree 't' and the left child of tree 't''. 3. The right...
The lemma named "not_invertible" assumes that "x" is an element of "I". It shows that "x" is not invertible in the context of commutative multiplication.
: 1. The root of the tree that results from unfolding 'a' is equal to the result of applying function 'g1' to 'a'. 2. The left subtree of the tree that results from unfolding 'a' is equal to the tree that results from unfolding the result of applying function 'g22' to 'a'. 3. The right subtree of the tree that results ...
If you apply a function 'f' to an inserted element 'x' at position 'i' in a list 'xs', it is equivalent to inserting the result of applying 'f' to 'x' at the same position 'i' in the list obtained by applying 'f' to all elements of 'xs'.
If X is a finite set, then the sum of the function f(x) plus a constant c for each element x in X is equal to the sum of the function f(x) for each element x in X plus the cardinality (or size) of X times the constant c.
If two states s1 and s2 are memory equal at a physical address 'pa', then the alternative memory values at the 10th index of these states remain unchanged even after the 10th and 11th indices of the memory of these states are updated. Specifically, the 10th index is updated with the memory value at the 10th index with ...
The lemma named "con_imp_common_source" assumes that "t" is concatenated with "u". It shows that the intersection of the sources of "t" and the sources of "u" is not empty.
If b is an element of B G, then b is also an element of G.
If the following conditions are met: 1) there is a valid auxiliary path from 'cs' to 'as', 2) there is a valid auxiliary path from the updated 'cs' to 'cs'', 3) there is a path from 'n' to 'n'' following 'as', 4) 'cs' is a valid call list for 'n', and 5) 'cs'' is a valid return list for 'n'', then it implies that...
A principal ideal of a set S is equivalent to the existence of an element 'a' such that the left ideal generated by the set containing 'a' is equal to S.
If 'h' is an element of the carrier 'M' and 'x' is an element of the finitely generated substructure of 'R' and 'M' with 'h' as the generator, then there exists an 'a' in the carrier 'R' such that 'x' is equal to the scalar multiplication of 'a' and 'h'.
The set of all tree automaton rules 'TA_rule' with function 'f', state sequence 'qs', and state 'q', where 'TA_rule f qs q' is an element of 'X', is equal to 'X'.
If the output from the card A to A with value X is in the set of events 'evs' and 'evs' is in 'srb', then the use of card A is legal.
If "A" is a cut and "r" is an element of "A", then "r" is greater than zero.
If node n'' postdominates node n and node n' postdominates node n'', then node n' postdominates node n.
The lemma named "typing_ctxt_correct2" shows that if a context consisting of Gamma and Delta correctly types expression E as type T derived from type U, and if the same context correctly types some term r as type U, then the same context correctly types the substitution of term r into expression E as type T.
If the run "Rb" is equal to some response with agents A and B and no keys or numbers, and the domain of the run is finite, then updating the run "Rb" to include a key "Kab" and numbers "Ts" and "Ta" in the response will result in the function "ir_runs2sigs" being equal to the function "ir_runs2sigs" applied to the run,...
The octonion conjugate of the negation of x is equal to the negation of the octonion conjugate of x.
"x" is upward closed.
The left-sided upper bound of the top element is the top element itself.
The lemma AE_bind: Assuming that "N" is a measurable function from "M" to the subprobability algebra of "B", and "P" is a measurable predicate on "B", it shows that "for almost every 'x' in 'M' bound to 'N', 'P' of 'x' holds" if and only if "for almost every 'x' in 'M', for almost every 'y' in 'N' of 'x', 'P' of 'y' ho...
The lemma named "block_ex" assumes that the function g applied to u is morphically equal to the function h applied to v. It shows that the predicate "blockP" holds true for the head of the list u.
The lemma named "approachable_sequenceE" is defined with a function 'f' that maps real numbers to a metric space 'a. It assumes that for any real number 't' greater than or equal to zero and any positive real number 'e', there exists a real number 'tt' greater than or equal to 't' such that the distance between 'f(tt)'...
: Assuming that there is a path from node 'n' to node 'n'' denoted by 'as', and 'V' is used in node 'n'', and the predicates of the kinds of 'as' hold for states 's' and 's'', and the value of 'V' in states 's' and 's'' is the same, and the value of 'V' in the state resulting from transferring the kinds of 'as' to 's'...
The signed value of an unsigned cast of a word 'w' of any length 'c' is equal to the signed value taken from the bit length of 'c' minus 1 of the unsigned value of 'w'. This is applicable for any word 'w' of any length 'b'.
If a converter 'conv' is well-typed with respect to interfaces 'I' and 'I'', and a resource 'res' is well-typed with respect to interface 'I'', then attaching the converter 'conv' to the resource 'res' is well-typed with respect to interface 'I'.
If 'n' is less than or equal to 'm', then shifting the mask 'n' to the right by 'm' positions will result in zero for any word length 'a'.
The lemma named "lessThan_conv" assumes that the cardinality of set A is n and every element x in set A is less than n. It shows that set A is equal to the set of all elements less than n.
The lemma "SubsetGuards_sound" assumes that command 'c' is a subset of command 'c'' with respect to guards. It also assumes that for all 'n', under the context of 'Γ' and 'Θ', the Hoare triple 'P c' Q,A' is valid. Under these assumptions, it shows that for all 'n', under the same context, the Hoare triple 'P c Q,A' is ...
If "k" is greater than 0, then the "f" expansion of "xs" by "k" is equal to the "f" expansion of "xs" by "k" minus 1.
If the identity function I is defined as a function that returns its input x, then the union function is equivalent to a function that applies the union operation to the result of applying the identity function I to its input x.
The lemma "alw_mono" assumes that the property "phi" always holds for the stream "xs" and that for any stream "xs", if "phi" holds then "psi" also holds. It then shows that the property "psi" always holds for the stream "xs".
If x is positive and y is positive, then the product of x and y is also positive.
The statements are lemmas about the relationship between functions and their arguments in a certain context. They can be translated into English as follows: 1. Lemma tagged_fun_relD_both: If a pair of functions (f, f') is in the relation from A to B, and a pair of arguments (x, x') is in A, then the pair formed by app...
For a given formula 'F', a literal trail 'M', and another literal trail 'p', if 'F' and 'M' maintain the invariant reason clauses, 'M' is unique, and 'p' is a prefix of 'M', then 'F' and 'p' also maintain the invariant reason clauses.
For any 'x' such that 'x' is not in the set 'xs', the function 'stkOk' with parameters 'c', 'l', '(r(x := g))', 'iL', 'iR', and '(Ref x)', 'xs' is equal to the function 'stkOk' with parameters 'c', 'l', 'r', 'iL', 'iR', '(Ref x)', 'xs'.
If the implementation of the queue dequeue operation on a queue Q results in a pair (x, Q'), and the queue Q is not empty, then x is equal to the head of the queue Q and the queue Q' is equal to the tail of the queue Q.
The polynomial of the number theoretic transform (NTT) coefficients of the product of two polynomials f and g is equal to the polynomial of the product of the NTT coefficients of f and g, where the coefficients are taken from the range of 0 to the integer value of n.
"S is a lower bound of the union of S".
The natural number 1705, when right shifted by three places, equals 213.
There exists a set, denoted as A, with the following properties: 1. A is countable. 2. For any subset A of the set A, the element x is in A. 3. For any subset A of the set A, A is open. 4. For any open set S, if x is in S, then there exists a subset A of the set A such that A is a subset of S. 5. For any two subsets A...
The restriction of the relation (A cross A) transitive closure to R is a subset of the transitive closure of the relation (A minus R) cross (A minus R).
The norm of a vector x is equal to the distance from x to 0.
: If a state sigma satisfies the condition that, after executing the monadic bind operation "mbindFSave" on a list (a#S) with the input-output program "ioprog", it returns a value "P s", then this is equivalent to the following: If the input-output program "ioprog" applied to "a" and "sigma" returns "None", then the ...
The lemma named "supr_Above" assumes that set B is a subset of the field of relation r (SUB) and that the set of elements that are greater than all elements of B is not empty (ABOVE). It shows that the supremum of B is an element of the set of elements that are greater than all elements of B.
The lemma named "monad_resumption" asserts that "return_resumption" and "bind_resumption" form a monad.
If x, y, and z are elements of the ring's carrier set, then the product of x and the product of y and z is equal to the product of y and the product of x and z.
The theorem named "evalOP_sound" assumes that "P" evaluates to "V". It shows that the evaluation of "P" in the environment "rho" is equal to the evaluation of "V" in the same environment "rho".
: Assuming that the tuple (V, V', Γ, δ) leads to a function 'f' from expression 'e' in a density context, and that 'e' is of type 't' under context 'Γ', and the free variables in 'e' are a subset of the union of V and V'. Also, for any state 'ρ' in the space of the state measure of V' under context 'Γ', and any 'x' in ...
The lemma named "almost_full_on_with_bot" assumes that there is an almost full relation on a set A with respect to a binary relation P. It shows that there is also an almost full relation on the set A with the bottom element added, with respect to the option order of P.
The lemma named "vinsert_set_insert_eq" assumes that "B" is a small set. It shows that the set resulting from inserting element "a" into the set that results from inserting element "b" into "B" is equal to the set that results from inserting "a" into the set that results from converting the set that results from insert...
If phi is equivalent to psi under a constant relation, then phi is equivalent to psi.
If the application "app" of instruction "i" in the context of "G" with maximum stack size "mxs", return type "rT", program counter "pc", exception table "et", and state "s" is true, and "mxs" is less than or equal to "mxs'", then the application "app" with "mxs'" instead of "mxs" is also true.
If set X is a subset of set A and x is an extreme element of set X with respect to the binary relation "less than or equal to", then y is also an extreme element of set X with respect to the same binary relation if and only if x is equal to y.
The linepath from point a to point b is continuous at point x.
: 1. A match does not occur between the source ports and the protocol and ports specified, if and only if, the protocol does not match the protocol of the packet or the source port of the packet is not in the set of ports specified. 2. A match does not occur between the destination ports and the protocol and ports sp...
The function "new_tv" with parameters n and "BVar m" always returns True.
If alpha prime is equal to alpha, A prime is equal to A, and B prime is equal to B, then the functor F maps from A prime to B prime in the category of semi-functors with respect to alpha prime.
If a function 'f' tends towards zero as its argument goes to infinity, and 'n' is a positive integer, then the function 'f' raised to the power of 'n' also tends towards zero as its argument goes to infinity.
If "y" is fresh for "t", then substituting "y" for "x" in "t" and then substituting "u" for "y" is equivalent to directly substituting "u" for "x" in "t".
The equality of the numeral 1 of type num1 to itself is true.
If "x" is an atom in an atomic boolean algebra, then the atom mapped by "x" is also an atom.
The sum up to a certain point "x" of the function that calculates the sum of "f n d" for all "d" that divide "n", is equal to the sum up to "x" of the function that calculates the sum up to "x/k" of "f (d * k) k" for each "k".
The lemma named "resPres" is defined with fixed variables P and Q of type pi, and x of type name. It assumes that P is strongly early bisimilar to Q (denoted as "P \<sim>\<^sup>s Q"). The lemma then shows that the restriction of P by x is also strongly early bisimilar to the restriction of Q by x (denoted as "<\<nu>x>P...
If "T" is an arrow and "U" is an arrow, and the intersection of the targets of "T" and the sources of "U" is not empty, then "T" and "U" are sequential.
If the number of vertices in function f is 4, and a is an element in the vertex set of f, and the vertices in function f are distinct, then the vertex set of f is equal to the set containing a, the image of a under f, the image of the image of a under f, and the image of the image of the image of a under f.
If a relation between converters A, B, C, and R holds, then for any input A, the relation between the generalized probabilistic value (gpv) of the product of B and the converter relation (A, B, C, R), C, and R also holds when the "run_converter" function is applied.
If a variable 'x' is in the union of the set of variables in the term 't' for all 't' in the set 'ts', then the set of unifiers for the insertion of the pair (Variable 'x', Function 'f' with terms 'ts') into the set 'E' is empty.
If the width is equal to x and the width is equal to y, then the width is equal to the sum of x and y.
A property P holds eventually for all values of x and y in a filter F if and only if for all functions Y, the property P holds eventually for all values of x and the value of Y at x in the filter F.
Zero is less than one in the context of natural numbers.
If the stack 's' is not empty, then there exists an initial vertex 'v0' in the set 'V0' such that there is a path from 'v0' to the head of the stack 's', following the reverse of the tail of the stack 's' in the edge set 'E'.
The length of the interpretation of a straight-line program (slp) with a given list (xs) is equal to the sum of the length of the straight-line program and the length of the list.
If T is an element of the set of group homomorphisms from group G to group H, then T is a group homomorphism from group G.
If 'n' is an element of the set of natural numbers (denoted as 'ω'), then 'n' is an ordinal number.
"x followed by y is less than or equal to z" if and only if "y is less than or equal to the negation of the converse of x followed by the negation of z".
The lemma named "Suc_unat_diff_1" is defined for a fixed 'x' which is a word of any length 'a. It assumes that 'x' is greater than or equal to 1. It shows that the successor of the unsigned natural number representation of ('x' minus 1) is equal to the unsigned natural number representation of 'x'.
The product of 'x' and 'L' is equal to the product of 'x', 'L', 'x', and 'L' in that order.
If 'i' is less than the length of 'ts' and 'is' has distinct load_tmps, then the load_tmps remain distinct even after the 'i'-th element of 'ts' is updated to the tuple (p,is,xs,sb,\<D>,\<O>,\<R>).
If for all elements x, y, and z, if x is an element of the set xs and the pair (x,y) is in relation r and the pair (y,z) is in relation r, then the pair (x,z) is also in relation r. If this is true, then if the pair (xs,ys) is in the lexicographic order relation r and the pair (ys,zs) is in the lexicographic order rela...
The parallel composition of two derivation trees, tr1 and tr2, is finite.
The lemma named "open_map_restrict" assumes the following: 1. U is an open set in the topology induced by the intersection of set S and the preimage of set T' under function f. 2. For any set U that is open in the topology induced by set S, the image of U under function f is open in the topology induced by set T. 3....
"1 plus x times the star of x equals the star of x".
The second element of the quotient of a fraction, when the fraction is converted from 'x', is equal to 1.
If a term "t" can be deduced from a set "M", then there exists a number "k" such that the term "t" can be numerically deduced from the set "M" and the number "k".
The lemma "defAss_dominating" assumes that "n" is an element in the set "αn of g". It shows that "defAss g n v" is equivalent to the existence of an "m" in the set "αn of g" such that "m" dominates "n" and "v" is in "allDefs g m".
If 'x' is in the limit of a pcp with parameters 'C' and 'S', then there exists a 'k' such that 'x' is in the sequence of the pcp with parameters 'C', 'S', and 'k'.
If the identity of 'a' is true and the identity of 'b' is true, then the identity of 'a' and 'b' together is also true.
The theorem named "certify_characterization" assumes that "cert" is a pair pseudo graph. It shows that "cert" is certified if and only if "cert" is a subgraph of G and the set of vertices of "cert" is equal to the set of vertices of the slimmed pair pre-digraph of "cert". Additionally, there exists a graph H such that ...
The lemma named "oneodd" establishes a condition for natural numbers 'b' and 'n'. It assumes that 'b' is greater than 2. It then shows that either the function 'α' with inputs 'b' and 'n' results in an odd number, or the function 'α' with inputs 'b' and 'n+1' results in an odd number.
The lemma named "adjust_idx_nth" assumes that "i" is less than the length of the list "xs". It shows that the element at index "j" in the list "xs" with the "i"-th element removed is equal to the element at the index adjusted by "i" in the original list "xs".