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"0 is less than the product of m and n" if and only if "0 is less than m and 0 is less than n", where m and n are extended natural numbers.
The correlation between the environment transformer "EnvT f" and the state transformer "StateT g" with a state "s" and a monad "M" is equivalent to the application of the monad "M" to the result of applying "f" to the state "s" and the result of applying "g" to the same state "s".
The function al_consr preserves the invariance and abstraction properties of the finite tree ft.
If 'a' and 'b' are elements of the carrier 'R' and 'p' is on the curve defined by 'a' and 'b', then the result of multiplying the projective point 'p' by 'n' and then converting to affine coordinates is the same as first converting 'p' to affine coordinates and then multiplying the point by 'n'.
The lemma named "slistmem_scons" shows that an element "x" is a member of a strict list that starts with "y" and is followed by the strict list "ys" if and only if either "x" is equal to "y" and "x" is not the bottom element and "ys" is not the bottom element, or "x" is a member of the strict list "ys" and "y" is not t...
Subtracting a number 'n' from an interval 'I' results in a value that is within the shifted interval 'I'.
If the pairs (fi,i), (ft,t), and (fe,e) are in the relation "bf_ifex_rel", then the pair consisting of the boolean function "bf_ite" applied to fi, ft, and fe, and the optimized if-then-else expression "ifex_ite_opt" applied to i, t, and e, is also in the relation "bf_ifex_rel".
The lemma is about induction on a tree structure, specifically a tree of type alpha (Tree\<^sub>\<alpha>). It fixes a tree of type alpha (t\<^sub>\<alpha>) and assumes the following: 1. For any set of type alpha (tset\<^sub>\<alpha>), if a property P holds for all elements in the set, then the property P also holds fo...
The lemma named "type_update_top_top" assumes two conditions. First, that the function "type_update" when applied to the "TopType" of "ts", the list "cons" converted to a list of computational types, and the "Type" of "prods", results in a new "TopType" of "ts'". Second, that the function "c_types_agree" when applied t...
An incidence system remains valid when a block "b" is strongly deleted from the set of points "\<V>".
The lemma named "cas_simp" assumes that the edge set "es" is not empty. It shows that a conditionally arc sequence (cas) from node "u" to node "v" through the edge set "es" is equivalent to the tail of the graph "G" at the head of the edge set "es" being equal to "u" and a conditionally arc sequence (cas) from the head...
If a function 'f' is a complex linear transformation, if 'f' is injective on the complex span of a set 'S', and if the image of 'S' under 'f' equals 'T', then the complex dimension of 'S' is equal to the complex dimension of 'T'.
The lemma named "noStuck_startn" assumes that given a program 'c' and a state 's', after 'n' steps of execution, the program reaches a state 't'. It also assumes that the state 't' is not a stuck state. Under these assumptions, it shows that the initial state 's' is also not a stuck state.
The diamond of the product of x and y is equal to the diamond of x followed by the diamond of y.
If from a set of hypotheses H, it can be inferred that A is equivalent to A', and B is equivalent to B', then from the same set of hypotheses H, it can be inferred that the equivalence of A and B is equivalent to the equivalence of A' and B'.
The length of the list obtained by flipping the element at position 'i' in list 'xs' is equal to the length of the original list 'xs'.
"equal_3" of x and y is true if and only if the real number representation of x is equal to the real number representation of y.
The lemma assumes that a set V of type ('\<AA>,Void\<^sub>b\<^sub>a\<^sub>s\<^sub>e) is defined in the state \<tau>. It shows that in the state \<tau>, the set V is either equivalent to the null set or the empty set.
The lemma named "scene_inter_union_distrib" assumes that "x", "y", and "z" are elements of the scene space. It shows that the intersection of "x" with the union of "y" and "z" is equal to the union of the intersection of "x" and "y" with the intersection of "x" and "z".
The lemma named "strand_sem_subst_d" assumes that the intersection of the substitution domain of delta and the range variables of delta with the bound variables of strand S is an empty set. It shows that if the strand semantics of M and S under the composition of delta and theta hold, then the strand semantics of M and...
The lemma named "unique_factorization_m_factor_partition" assumes the following conditions: 1. l is not equal to 0. 2. There exists a unique factorization of f under modulo p^l, where the factors are the leading coefficient of f and the multiset gs. 3. f is the product of f1 and f2. 4. The leading coefficient of f is ...
The lemma named "clamp_empty_interval" assumes that "i" is in the basis and that the dot product of "a" and "i" is greater than the dot product of "b" and "i". It shows that the clamp function of "a" and "b" is equal to a function that always returns "a".
The lemma named "padic_one_id" assumes that "p" is a prime number and "x" is an element of the carrier of the p-adic integers. It shows that the multiplicative identity (one) of the p-adic integers, when multiplied with "x", equals "x".
If V is equal to the set containing only i, then the function "singleton_table" with parameters n, i, and z is a table with parameters n, V.
If X is a contractible space, S is a subset of the top space of X, and S is not an empty set, then the homology group of the subtopology of X and S at 0 is isomorphic to the direct product of the integer group and the relative homology group of X and S at 1.
For any sharing relation and any ownership relation, if the sharing relation is consistent with the ownership relation and the store buffer, then the read-only part of the shared store buffer and the sharing relation is a subset of the union of the read-only part of the sharing relation, the ownership relation, and all...
If a sequence 's' belongs to the set of closed sequences 'R', then the composition of the ring identity function and 's' is equal to 's'.
, given an empty set, the Herbrand Representation Property (HRP) of the order of 'i' is equivalent to the order of 'i'.
If we have distinct prime divisors K and n for P, and x and y are elements of the carrier of the ring of integers of K at P and n, and for all j less than or equal to n, the valuation of x at P and j is equal to the valuation of y at P and j, then the ideal generated by x in the ring of integers of K at P and n is equa...
The lemma named "clique_exists_gt0" assumes that the set of vertices 'uverts G' in a graph 'G' is finite and the cardinality (or the number of elements) of 'uverts G' is greater than 0. It shows that there exists a clique 'C' and a number 'p' such that 'C' is a clique in 'G' with 'p' vertices, 'p' is less than or equal...
If vector v is the horizontal concatenation of vectors u and w, then the left boundary of the extension of vector v is less than or equal to the right boundary of the extension of vector u.
For any X, if a' fits T in the context of G and s, then either there exists a primitive type pt such that T equals pt, or there exists a type t such that T equals the reference type of t and a' equals Null, or there exists a type t such that T equals the reference type of t and a' does not equal Null and the object typ...
Applying the right associative wiring to the combination of "res1", "res2", and "res3" in the given order is equivalent to the direct combination of "res1", "res2", and "res3".
If "x" is not equal to "DenyAll", then the result of applying the function "removeShadowRules1_alternative" to the list "xs" appended with "x", is equal to the result of applying the same function to the list "xs" alone and then appending "x".
If relation R1 and relation R2 are both left total, then the relation of their resumption is also left total.
If the formula psi can be proven from the set of formulas phi1, phi2, and phi3, and each of phi1, phi2, phi3, and phi4 are valid formulas, and psi is also a valid formula, then psi can also be proven from the set of formulas phi1, phi2, phi3, and phi4.
The lemma "degree_det_le" assumes that for any indices i and j, if i and j are less than n, then the degree of the element at position (i,j) in matrix A is less than or equal to k. It also assumes that A is an n by n matrix. Under these assumptions, it shows that the degree of the determinant of matrix A is less than o...
Repeating an empty list "n" times results in an empty list.
Lemma 6.3.1.a: Assuming that v0 is an element of the bounded function set, it shows that the uniform limit over the entire set of the function sequence, which is the nth composition of the function that maps v to the Bellman operator of the bounded function v, starting from v0, is the optimal value function, as n goes ...
For any real numbers 'a', 'x', and 'y', if 'x' and 'y' are both greater than or equal to zero, then the hyperreal power of the quotient of 'x' and 'y' by 'a' is equal to the quotient of the hyperreal power of 'x' by 'a' and the hyperreal power of 'y' by 'a'.
For any two words 'a' and 'b' of any length, the list of words from 'a' to 'b' is distinct.
If alpha-prime is equal to alpha and I-prime is equal to I, then alpha-prime, I-prime, and AA form a base pcategory.
Two done general probabilistic values (gpv) are equivalent under a certain interface \<I> and relation A if and only if the relation A holds for the results x and y of these gpvs.
The lemma "weakTransitionFrameImp" is defined with the following parameters: a fixed 'Psi' of type 'b, 'Q', 'P', 'P'', and 'R' of type psi (which is a tuple of 'a, 'b, 'c), 'M' and 'N' of type 'a, and 'xvec' which is a list of names. The lemma assumes that 'Psi' weakly transitions from 'Q' to 'P' with an action 'alpha...
The natural number representation of the unsigned version of a word 'w' is equal to the unsigned version of the word 'w'.
For the lemma "Interleaves_suffix_aux": If for all 'n' less than the length of list 'ws', a property 'P' holds for the 'n'th element of 'ws' and the list obtained by dropping the first 'n+1' elements of 'ws', then if list 'ws' interleaves with list 'zs' under property 'P' after prepending element 'x' and appending lis...
For the implementation of the H-method using lists to be complete, the following conditions must be met: 1. Both the model M and the implementation I are observable. 2. The model M is minimal. 3. The size of the implementation I is less than or equal to m. 4. m is greater than or equal to the size of the model M. 5. T...
, under the assumption that "n" is not equal to "m", it shows that the multiset representation of the natural number "n" at index "i" is not equal to the multiset representation of the natural number "m" at the same index "i".
The lemma named "noFault_startn" assumes that a given execution "execn" from state "s" to state "t" under a program "c" after "n" steps, and that the state "t" is not a fault "f". It then shows that the initial state "s" is also not a fault "f".
Zipping a list "xs" with the same list "xs" after applying a substitution operation "delta" is equivalent to mapping a function over the list "xs" where the function takes an element "t" and returns a pair consisting of "t" and "t" after applying the substitution "delta".
The lemma named "sequence_to_failure_alt_def" assumes the following conditions: 1. The set of input/output targets of the product machine (PM) from its initial state using the input/output sequence (io) is equal to the failure state (FAIL). 2. Both M1 and M2 are observable finite state machines (OFSMs). 3. The product...
If "B" gets "X" in the set of events "evs" and "evs" is in "orb", then "X" is in the knowledge of "B" in the set of events "evs".
Sorting a list 'xs' does not change its distinctness. In other words, if the elements in the list 'xs' are distinct before sorting, they will remain distinct after sorting, and vice versa.
The property "P" always holds for the stream composed of the head "h" and the tail "t" if and only if the property "P" holds for the head "h" and the property "P" always holds for the tail "t".
Two nodes x and y are connected in a graph where an edge between nodes a and b is added to graph H if and only if the pair (x, y) is in the per_union of the set of all pairs (x, y) where x and y are nodes connected in graph H, and a and b are nodes. This is under the condition that graph H is a subgraph of graph G, the...
If the equality relation is less than or equal to R1 and the equality relation is also less than or equal to R2, then the relation of sp_mu with respect to R1 and R2 for any x is equal to x itself.
The "less than" relation is equivalent to the relation obtained by taking the "less than or equal to" relation on terms and removing the identity relation.
If H is a subset of the carrier G, then the derived group of G and H is a subgroup of G.
The lemma named "param_nfoldli" shows that the function "nfoldli" is parameterized over two relations, Ra and Rb. This function takes four arguments: a list of elements related by Ra, a function related to the identity function by Rb, a function that takes two arguments related by Ra and Rb and returns a result related...
If a nonce 'n' is not equal to 'ofr', and 'I', 'J' are in 'agl', and 'L' is in the guard of nonce 'n' with private key 'A', then the promise 'B' of 'ofr' to 'A' with respect to 'r', 'I', 'L', 'J', and 'C' is in the guard of nonce 'n' with private key 'A'.
The lemma named "algebraic_div" assumes that both "x" and "y" are algebraic numbers. It shows that the result of dividing "x" by "y" is also an algebraic number.
The lemma named "invertible_Hermite_of_list_of_rows_mod_type" is about a fixed integer matrix A. It assumes that A is in the carrier matrix of the cardinality of 'm and 'n, both of which are of mod_type. It also assumes that A is in echelon form. The lemma then shows that there exists a matrix P, which is in the carrie...
For the equivalence relation 'eqvt', the following holds: 1. The permutation of 'Alts' function with arguments 'e1' and 'e2' is equal to the 'Alts' function with the permutation of 'e1' and 'e2' as arguments. 2. The permutation of 'Arg' function with argument 'v' is equal to the 'Arg' function with the permutation of...
The difference sequence of the negation of a sequence "c" is equal to the negation of the difference sequence of "c".
: Assuming that "g" has minimum graph properties (mgp), "a" is an element of the vertices of "g" (a), "V" is a subset of the vertices of "g" (Vg), "g" and "V" are separated (ps), for any face "f" in the set of faces at "a", "f" does not contain "a" in "V" (s2), and for any face "f" in the set of faces at "a", if the nu...
If the following conditions are met: 1. Agent A receives a message containing a nonce Nb and a cryptographic pair of A and B, which itself contains nonce Na and Nb, from the set of events 'evs'. 2. There is no illegal use of Card B. 3. The set of events 'evs' is in the Shoup-Rubin-Bella protocol. Then it implies that...
The degree of a power product 's' is equal to zero if and only if 's' is equal to zero. This is applicable for 's' which is a function from 'a' to 'b', where 'b' is an additive linear order with a minimum value.
Converting a formal power series (represented by "Abs_fps F") to a formal Laurent series (using the function "fps_to_fls") is equivalent to creating a formal Laurent series (represented by "Abs_fls") with the function "λn. if n<0 then 0 else F (nat n)", which returns 0 for negative indices and the corresponding value o...
If the length of a list 'xs' is divisible by 'k' (i.e., the remainder when the length of 'xs' is divided by 'k' is zero), then the result of appending a function 'f' to 'xs' and then shrinking by 'k' is the same as the result of first shrinking 'xs' by 'k', appending 'f', and then shrinking 'f' by 'k'.
If functor G maps category A to category C and functor H maps category B to category C, and if A' is the opposite category of the comma category of G and H, and B' is the comma category of the opposite functor of H and the opposite functor of G, then the opposite functor of the comma functor of G and H is an isomorphis...
The lemma asso3 assumes that "a" is not disjoint with "b", and "b" combined with "c" equals "bc". It shows that "a" is not disjoint with "bc".
If a logical relation R exists, and for all elements 'xs' in R, the function 'strictify' applied to 'fs j' and 'xs j' is in R, then the function 'ValF' applied to 'strictify' and 'fs i' is also in R.
A natural number 'm' is equal to zero in the context of a word of length 'a' if and only if there exists a number 'q' such that 'm' is equal to 'q' multiplied by 2 to the power of the length of 'a'.
The weakest precondition of the sequence of two programs P and Q with respect to a postcondition r is equal to the weakest precondition of program P with respect to the weakest precondition of program Q with respect to r.
The lemma named "contour_integrable_negatepath" assumes that "gamma" is a valid path and that the function "f" with its argument negated is contour integrable on "gamma". It shows that "f" is contour integrable on the negation of "gamma".
Filling the holes in a multi-function (MFun) with a function 'f' and a list of contexts 'cs' using a list of terms 'ts', is equivalent to creating a function 'f' with a list that is obtained by filling the holes in each context in 'cs' with the corresponding partition of holes from 'ts', for each index 'i' from 0 to th...
Given the following conditions: 1. A polynomial 'f' is an element of the carrier of the ring of univariate polynomials over the p-adic numbers. 2. 'a' is an element of the ring of integers of the p-adic numbers. 3. The Gauss norm of 'f' is greater than or equal to zero. 4. The valuation of 'f' evaluated at 'a' is grea...
The allocation function Alloc for a given system G, object tag otag, predicate P, state s, and variable Z is equivalent to the statement that for all states s' and addresses a, if G transitions from state s to state s' by allocating an object with tag otag at address a, then the predicate P holds for the value at addre...
The probability mass function (pmf) of the stochastic dominance strategy (sds) R18 for 'a' is equal to the pmf of the sds R14 for 'a'. The pmf of the sds R18 for 'b' and 'd' is zero. The pmf of the sds R18 for 'c' is equal to 1 minus the pmf of the sds R14 for 'a'.
"a modulo m equals b modulo m" if and only if "a is congruent to b modulo m".
An empty set is equal to the union of sets A and B if and only if both A and B are empty sets.
If psi is a subformula of phi, then the set of all globally eventually true subformulas of psi in a world w is a subset of the set of all globally eventually true subformulas of phi in the same world w.
If a binomial queue is valid for a list consisting of an element 'x' followed by a list 'xs' at level 'l', then the binomial queue is also valid for the list 'xs' at the next level, 'l+1'.
If the list L l is in the analysis of H, and if the set of l is a subset of the analysis of H, then P is implied. Therefore, P is implied.
The element Empty is a member of the set Natural numbers.
The lemma named "closed_map_restrict" assumes the following conditions: 1. U is a closed 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, if U is a closed set in the topology induced by set S, then the image of U under function f is a closed set in...
: Given a type 'tau', a base type 'b''', and a base variable 'bv', if 'tau' is well-formed under the context of 'bv' and the context of '(x,b,c)', and if 'b''' is well-formed under the context of 'B', then the type 'tau' with 'bv' substituted by 'b''' is also well-formed under the context of 'B' and the context of '(x,...
The lemma named "simplicial_chain_simplicial_cone" assumes that "c" is a simplicial chain of dimension "p" in the set "S". It also assumes that for any "x" in "S" and any "u" between 0 and 1 inclusive, the function "(1 - u) * v i + u * x i" is in the set "T". Under these assumptions, it shows that the simplicial cone o...
The lemma assumes that for any integers x, y, and z, the sum of z and the sum of x and y is equal to the sum of z, y, and x. It then shows that for any integers x, y, and z, the sum of x, y, and z is equal to the sum of z, y, and x.
The lemma named "non_empty_cycle_root_no_start_end_points" assumes that "r" is a non-empty root cycle at "x". It shows that the first successor of "x" is the same as the first predecessor of "x".
If x is not equal to y, then the set obtained by inserting x into set A and then removing y is equal to the set obtained by first removing y from set A and then inserting x.
: 1. The dot product of a vector 'x' and the cross product of vectors 'x' and 'y' equals zero. 2. The dot product of a vector 'x' and the cross product of vectors 'y' and 'x' equals zero. 3. The dot product of the cross product of vectors 'x' and 'y' and vector 'y' equals zero. 4. The dot product of the cross product o...
If a polynomial 'p' is in the range of a function 'ffp', and 'p' is equal to the scalar multiplication of the inverse of 'x' and another polynomial 'q', and 'p'' is equal to the scalar division of another polynomial 'q'' and 'x'', and 'x' is equal to the function 'ff' of 'x'', and 'q' is equal to the function 'ffp' of ...
If the structure specification "struct_spec" is given by the selection function "sel", the approximation function "ap", the abstraction function "ab", and the completion function "compl", and if the completion function is connected ("compl_conn") and forms a complete module ("compl_pmdl"), then the polynomial ideal gen...
There exist values u and v such that the vector formed by appending u and v is less than or equal to the vector formed by appending x and y, and that the pair (u, v) is a minimal solution to the same equation.
, in Herbrand's interpretation, k is a member of the successor of k.
The remainder of any number k divided by 1 is 0.
It is not the case that interval 'u' is non-meeting with interval 'r' and there exists an interval 'p' such that 'r' is non-meeting with 'p' and 'p' is non-meeting with 'u'.
The number of rows in the matricization of a tensor T with respect to a set of modes rmodes is equal to the product of the dimensions of T at the indices specified by rmodes. Similarly, the number of columns in the matricization of T with respect to rmodes is equal to the product of the dimensions of T at the indices n...
The lemma "winning_strategy_on_deadends" assumes that "v" is in the set "VV p", "v" is a deadend, and "p" has a strategy "σ". It shows that "σ" is a winning strategy for "p" at "v".