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The lemma named "map_simp" assumes that "f" is an arrow in the product category of A1 and A2. It shows that the mapping of "f" is equal to a pair where the first element is the second element of "f" and the second element is the first element of "f".
If "a" is approximately equal to "b" and "c" is approximately equal to "d", then "a" minus "c" is approximately equal to "b" minus "d".
For all 'y', if 'a' is transitively related to 'y' through relation 'r' and property 'P' holds for 'a', then property 'P' also holds for 'y'. Given this, the pair '(b, a)' is in the transitive closure of the set of pairs '(y, x)' where 'P' holds for 'x' and 'x' is related to 'y' through 'r', if and only if the pair '(b...
If G is a group, then the subgroup H of G is also a group.
If a heap 'h' is well-formed, and if 'h' leads to the retrieval of disconnected nodes 'disc_nodes' from a document pointer 'document_ptr', then these disconnected nodes are distinct.
If a function 'f' is regular, then 'f' is set-monotone.
If C is an element of the carrier of the function fTo D, and if Ca is a Chain in the interval order of the set of all S in the carrier of fTo D such that C is a subset of S, then there exists an X such that X is the minimum element in the interval order of the set of all S in the carrier of fTo D such that C is a subse...
: Assuming for any n, s0, L, accC, T, C, vs, s1, Y, Z, if the following conditions are met: - For all t in A, G proves n of type t - s0 is a subtype of (G,L) - s0 is normal - The program G with class accC and local L proves t is of type T - The program G with class accC and local L proves the domain of the locals of t...
"p" is a specific type of logical formula (iszlfm) with respect to the list "a#bs", and 2) "RR" which states that the image of the set "R" under the function "(lambda (b,k). (Inum (a#bs) b,k))" is equal to the image of the set "rho p" under the same function. Under these assumptions, the lemma shows that there exist...
If the function 'f' applied to 0 equals 0, then mapping the function 'f' to a polynomial monomial 'monom c n' results in a new monomial where 'f' is applied to the coefficient 'c' and the degree 'n' remains the same.
The lemma named "ncoplanar_perm_6" assumes that points A, B, C, and D are not coplanar. It shows that if you swap the positions of points A and B, the points B, A, C, and D are still not coplanar.
If the exclusive OR (xor) of two sets 'xs' and 'ys' equals a third set 'zs', and 'xs' is equal to 'ys', then 'zs' must be an empty set.
A Gaussian integer x is a unit if and only if x divides 1, which is equivalent to saying that x is a member of the set {1, -1, i, -i}. The lemma also states that a Gaussian integer x is a unit if and only if the norm of x is 1.
The third step in protocol m3 refines the third step in protocol m2. This is under the condition that the relation R23 intersects with the set of session keys in protocol m2 (m2_inv3a_sesK_compr) and the set of session keys in protocol m3 (m3_inv3_sesK_compr) that also intersect with the set of long-term keys in protoc...
In the context of a functor, the lemma shows two things: 1. The intersection of the object 'b' and the domain of the homomorphism of the functor 'F' is equal to the projection of 'b' down to the functor 'F'. 2. The intersection of the object 'b' and the codomain of the homomorphism of the functor 'F' is equal to the ...
The lemma named "rev_length" assumes that the effect of reversing array 'a' from index 'i' to index 'j' transforms heap 'h' into heap 'h'' and results in 'r'. It shows that the length of array 'a' in heap 'h' is equal to the length of array 'a' in heap 'h''.
The lemma named "subresultant_mat_dim" is defined for any values of j, p, and q. It defines S as the subresultant matrix of j, p, and q. It shows that the number of rows in S is equal to the degree of p minus j plus the degree of q minus j, and the number of columns in S is also equal to the degree of p minus j plus th...
The lemma named "exec_wf_progress" assumes that the Java Virtual Machine program P is well-formed according to the function "wf_jvm_prog" with respect to a certain type system state "Phi". It shows that the progress property holds for the final state of the JVM, the method execution of program P, and the well-formed st...
The lemma named "sshiftr_clamp_pos" assumes that the length of 'a is less than or equal to n and the signed integer of x is greater than or equal to 0. It shows that the logical right shift of x by n equals 0, where x is a word of length 'a.
The reverse of list xs is a permutation of list ys if and only if list xs is a permutation of list ys.
The lemma named "delta_gt_SN" assumes that delta is greater than 0. It shows that the set of all pairs (x, y) such that y is greater than or equal to 0 and delta_gt of delta, x, and y is strongly normalizing.
The lemma named "bij_betw_nth" assumes that a list "xs" contains distinct elements, "A" is a set of all numbers less than the length of "xs", and "B" is the set of all elements in "xs". It shows that there is a bijective (one-to-one and onto) correspondence between the sets "A" and "B" when the function that maps each ...
If a node is left-aligned with a list of tuples (ts' appended with (sub', sep') and (sub, sep)) and a tree t, and a separator sep' is aligned with a tree t' and u, then the node is left-aligned with the list of tuples (ts' appended with (sub', sep')) and the tree t'.
The alpha representation of the start state of the scheduler context, when refined with an empty resource manager, an update to the resource manager, another empty resource manager, an empty resource set, a function f, and variables P, C, M, and vs, is equal to the start state of the scheduler context with the same fun...
The lemma assumes that x and y are compatible. It then shows two things: first, that x is less than or equal to the join of x and y (join_above1), and second, that y is less than or equal to the join of x and y (join_above2).
The lemma named "distinct_join_raw" assumes that the first elements (when the elements are considered as pairs) of the list 'xs' are distinct. Under this assumption, it shows that the first elements of the list resulting from the function 'join_raw' applied to 'f', 'xs', and 'ys' are also distinct.
K is a ring.
The elements of the transposed matrix M are either 0 or 1.
The lemma named "Union_eqvt" shows that the permutation of the union of a set A is equal to the union of the permutation of set A.
If a branch is a subset of Gamma function 'f', and if the terminal is a subset of the function 'f' at 'm', then the terminal is also a subset of the function 'f' at 'm + n'.
The implementation of "atLeastLessThan" for any two natural numbers 'a' and 'b' is less than or equal to the specification that the result 'r' is a set of natural numbers starting from 'a' and less than 'b'.
The function "listwordinterval_compress" when applied to the list of 32-bit word pairs [(1,3), (8,10), (2,5), (3,7)] results in the list [(8, 10), (1, 7)].
The set generated by the probability mass function "bv" applied to a number "n" is equal to the set of boolean lists whose length is "n".
If the function "generalized_sfw" applied to the filter "f" on the firewall "fw" and packet "p" equals some pair (r,d), then the packet "p" simply matches "r" and the filter "f" applied to the pair (r,d) is true.
The lemma named "qc_wf_below" assumes that the composition of the relation 's' and 'ns' is a subset of the transitive closure of the union of 's' and 'ns' composed with 's', and that the well-foundedness below 's' holds for 'A'. Under these assumptions, it shows that the well-foundedness below the composition of the re...
If there is a list relation predicate such that for every pair of elements x and y, both R and S hold true, then there is a list relation predicate where R holds true for every pair of elements x and y.
The nth elements of a list that replicates 'x' 'n' times, where 'A' is a set of indices, is equal to a list that replicates 'x' for the count of elements 'i' that are less than 'n' and are in the set 'A'.
The lemma named "coord_ring_cfs_closed" assumes that "p" is an element of the carrier of the ring of polynomials in "n" variables. It shows that the "m"-th coefficient of the polynomial "p" is an element of the carrier of the ring "R".
If G is a group and N is a normal subgroup of G, and if 'a' is an element of the image of the set of right cosets of G and N under the gcsrp_map, then the top element of the gcsrp of G, N, and the identity element of gcsrp of G and N with 'a' is equal to 'a'.
The cosine of x is less than or equal to the cosine polynomial of x, where the degree of the polynomial is 4n+1.
The derivative bound of an empty set is 'a' for any 'n'.
If "x" is an element in the multiset of vertices of a node (with root "r" and forest "xs"), and there does not exist a tree "t1" and an edge "e1" such that "(t1,e1)" is in the finite set "xs" and "x" is in the multiset of vertices of "t1", then "x" must be equal to "r".
The encoding of a frame, which consists of a function 'f', a list 'xs', and another list 'ls', is equal to the encoding of a triple where the first element is the encoding of the function 'f', the second element is the encoding of the list 'xs', and the third element is the encoding of the list 'ls'.
The lemma named "sqrt_case" assumes that the determinant (SQ) is greater than or equal to zero. It shows that the expression "the square root of the determinant raised to the power of half of i, multiplied by the remainder of i divided by 2, multiplied by the square root of the determinant, plus the square root of the ...
The minimum product of A and B is equal to the minimum product of B and A.
The lemma named "wffp_append" assumes three conditions: "wffp sl", "wffp sl1", and "hd sl1 is in the transition function of the last element of sl". If these conditions are met, it shows that "wffp of the concatenation of sl and sl1" is true.
If B receives a message containing an encrypted block (which includes agents A and B, a key K, and a nonce NB) and neither A nor B are in the set of compromised agents, and the events are part of the Yahalom protocol, then there exists a nonce NA such that the server sends a message to A containing the nonce NB, an enc...
If 'p' is in 'nodes' and 'q' is reachable from 'p', then 'q' must also be in 'nodes'.
The lemma named "length_distincts" assumes that the list "xss" contains distinct elements. It shows that the length of this list is equal to the cardinality of the set of sets derived from "xss".
The theorem "closest_pair_recurrence" belongs to the set of functions that grow at the rate of "n times the natural logarithm of n".
The unit factor of the second element of the normalized quotient of x is equal to 1.
If xs is equal to ys, then the stream of xs is equivalent to the stream of ys.
The set of shifted variables obtained from a list of variables "vs" is equal to the result of applying the shift_var_set function to the set of variables "vs".
The type of an arithmetic expression "Const c" is empty under any context "Gamma".
The lemma named "integrable_self_fin_uvp" assumes that for every asset in the support set of the portfolio, the product of the price of the asset at time 'n' and the portfolio at time 'n+1' is integrable over the measure space 'M'. It also assumes that the portfolio is self-financing in the market. Under these assumpti...
In the context of p-adic integers, the alternative definition of p-residue for any integers m and n is that the p-residue of m and n is equivalent to the remainder of n divided by p to the power of m.
"1 is less than the ceiling of x" if and only if "1 is less than x".
The lemma "max_Some" consists of two parts: 1. If the invariant "s" holds and the maximum value of "s" under predicate "P" is some pair "(k,v)", then this pair "(k,v)" is in the relation of the abstraction of "s" under predicate "P". 2. If the invariant "s" holds, the maximum value of "s" under predicate "P" is some ...
: 1. If an expression 'e' is well-typed as 'T' in the environment 'E', and the program 'prg E' is well-structured, then 'T' is a type in the program 'prg E'. 2. If a list of expressions 'es' is well-typed as a list of types 'Ts' in the environment 'E', and the program 'prg E' is well-structured, then all types in the...
If a binary tree 't' is sorted, then the tree remains sorted even after removing an element 'x' from 't'.
The function "vprod_size" is injective on the finite set "A".
For two paths 'p' and 'q', if 'p' is a path and a complex number 'ζ' is not in the image of 'p', and 'q' is a path and 'ζ' is not in the image of 'q', and the end of path 'p' is the same as the start of 'p' and the end of path 'q' is the same as the start of 'q', then the winding number of 'p' around 'ζ' is equal to th...
The number 997 is a prime number.
If an integer 'i' is less than or equal to another integer 'j', and a third integer 'k' is less than or equal to zero, then the product of 'j' and 'k' is less than or equal to the product of 'i' and 'k'.
If beta is an ordinal and alpha is an element of beta, then the natural transformation N from functor F to functor G is a tiny natural transformation of small semicategories from category A to category B with respect to beta.
If X and Y are Borel predictable stochastic processes, then the product of X and Y (at each time point and for each outcome) is also a Borel predictable stochastic process.
The hash of X and Y equals the set containing X' and Y' if and only if X' equals the hash of the set containing X and Y, and Y' equals Y.
Subtracting an empty list from any list "xs" of elements from an additive group results in the original list "xs".
If we have two continuous maps 'p' and 'q' from a topological space 'X' to the real numbers, and two continuous maps 'f' and 'g' from subsets of 'X' to a topological space 'Y' (where the subsets are defined by the conditions 'p x ≤ q x' and 'q x ≤ p x' respectively), and if 'f' and 'g' coincide on the set where 'p' and...
If the function "raw_match_term" applied to terms "t" and "u" with substitutions "subs" results in some substitutions "subs'", then the domain of "subs'" is a subset of the set union of the term variables of "t" and the domain of "subs".
The set of all subsequences "ys" of a list "xs", where the length of "ys" is equal to the length of "xs", is equal to the set containing "xs" itself.
The partial Hoare triple using the function HT'_partial with parameters pi, P, a lambda function that returns c, and Q is equal to the partial Hoare triple using the function HT_partial with parameters pi, P, c, and Q.
The global variables in a state 's' remain the same even after a local update operation mapping variable name 'vn' to value 'v'.
If sigma is an element of ptrm n, then sigma is also an element of fmla.
The lemma named "semClosed" assumes that "\<phi>" is a well-formed formula with no free variables, and that the pointer of "\<pi>l" is equal to "s0". It shows that the semantic closure of "\<phi>" is equal to the semantics of "\<phi>" in the environment "env" and path "\<pi>l".
Adding a vertex 'u' to a well-formed directed graph results in a well-formed directed graph.
If the types are well-formed according to the context Γ, and if according to the same context Γ, the expression 'e' is of type 't', then the type 't' is well-formed.
The evaluation of the function "f" applied to a set "P" is equal to the supremum (or least upper bound) of the set of all elements "x" such that the evaluation of "P" at "x" is true, where the function "f" maps "x" to "y".
'j' is either equal to 3 or there exists another integer 'j'' of 3 bits such that 'j' is equal to the integer representation of the bitwise representation of 'j''.
The set resulting from converting a word interval using the "l2wi_negation_type_union" function is equal to the union of two sets. The first set is the union of all intervals from "i" to "j" where "(i,j)" is in the set obtained from the "getPos" function applied to "l". The second set is the union of the negations of a...
If 'r' is not a pure value, then the continuation of 'r' is included in the generated continuations of 'r'.
The lemma named "uncountable_closed_segment" is about a fixed point 'a' which is a real normed vector. It assumes that 'a' is not equal to 'b'. It shows that the closed segment between 'a' and 'b' is uncountable.
The variable "x" does not affect the constant 1.
The lemma named "iso_category_refl" assumes that "\<AA>" is a category in a context "\<alpha>". It shows that the category "\<AA>" is isomorphic to itself in the context "\<alpha>".
If T satisfies the Zermelo-Fraenkel axioms, then T can be interpreted in the context of Cartesian Inference.
If a function 'f' is a wave in a network 'Γ' and 'x' is not in the reachable set of the terminal set of 'f' in 'Γ', then the outgoing flow of 'x' in 'f' is zero.
, assuming "a" and "b" are both in the non-zero elements of the ring "R", the fraction formed by the numerator of the fraction "a/b" over the denominator of the fraction "a/b" is equal to the fraction "a/b".
The carrier of the function "f_nxt" applied to the function "f", the tree "T", and the nodes "x" and "x'", is a subset of the carrier of the function "f" applied to the node "x'".
The omega power of x multiplied by the bottom element, combined with the star power of x, is equal to the omega power of x combined with the star power of x.
The lemma named "topspace_nsphere_1_eq_0" is about a function 'x' that maps natural numbers to real numbers. It assumes that 'x' is in the topological space of an n-sphere, the square of 'x' at a certain point 'k' equals 1, and 'i' is not equal to 'k'. Under these conditions, it shows that the value of 'x' at 'i' equal...
If S is an integral domain and both p and q are elements of the ring R, then the degree of the product of p and q is equal to the sum of the degrees of p and q.
The lemma named "yun_wrel_minus" assumes that the function "yun_wrel" is true for the pairs (F, c, f) and (G, c, g). It shows that "yun_wrel" is also true for the pair ((F - G), c, (f - g)).
If the list of sets (A # As) is additively independent, then the direct sum over the list (A # As) is equal to the direct sum of A and the direct sum over the list As.
If "T" is a type in the program "P", then "T" is also a type in the program obtained by adding the class "C" with the class declaration "cdec" to the program "P".
The lemma named "lookup_fix" establishes that for a fixed value 'y' and a function 'F' that maps from one function type to another, the application of function 'F' to the fixed point of 'F' at 'y' is equal to the fixed point of 'F' at 'y'.
If it is actually the case that for all 'x', the condition 'phi x' holds in 'v', then for all 'x', it is actually the case that the condition 'phi x' holds in 'v'.
The lemma "redT_ex_new_thread'" assumes that there is a transition from state 's' to state 's'' with thread 't'' and thread action 'ta', the thread 't' in state 's'' is in the state '(x, w)', and the thread 't' in state 's' is in the 'None' state. It shows that there exists a memory 'm' and a value 'x' such that the ac...
The composition of the "allocAsk" function with the "sysOfClient" function is equal to the composition of the "allocAsk" function with the "snd" function.
For a given predicate 'Pred' of a process, there exists a relation 'Rel' such that for all source terms 'S', the pair (SourceTerm S, TargetTerm of the encoding of S) is in 'Rel' and 'Rel' preserves the predicate 'Pred' if and only if there exists a relation 'Rel' such that for all source terms 'S', the pair (TargetTerm...
Updating a function 'f' after updating a function 'g' on a state 'σ' is equivalent to updating the composition of function 'f' and 'g' on the same state 'σ'.