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If "f" raised to the power of "m" is in "F", then "f" is in the radical of "F".
The lemma named "uTextPost_out" assumes that a step from state 's' with action 'a' results in an output 'ou' and a new state 's'', and that the action 'a' is a user action of posting by user 'uid' with password 'p' on post 'PID' with post state 'pst', and that the output 'ou' is 'outOK'. It shows that the user 'uid' is...
The lemma named "Alice_case" assumes that "phi" is a state 1 and "q" is a state 3, and that the pair (p, q) is a member of the measurement by Alice on "phi". If these conditions are met, it shows that the position of Alice on "phi" is "q".
The pair consisting of a list of all elements 't' that satisfy the condition 'True', and the abstract data space of the universal set, belongs to the set of all pairs '(L, D)' such that 'L' and 'D' satisfy the condition 'Data L D'.
If matrix A and matrix B are both in the carrier matrix of dimensions nr by nc, and matrix A is greater than or equal to matrix B, then the maximum matrix of A and B is A.
The lemma named "spr_simRep_sim_simps" contains two statements: 1. The image of the set "T" under the function "spr_sim", and then under the function "spr_repSim", is equal to the image of the set "T" under the function "tLast". 2. The result of applying the function "spr_sim" to the object "t", and then applying the...
: Assuming that a program P declares that class C has a static field F of type t in class D, and there is no static field state for class D that is done, and the initialization of class D from a unit state to a state with value v' is a valid sequence of steps in program P, and the static field state for class D in the...
The lemma named "hgt_Subst" shows that if "u" is an arrow, then the height of the substitution of "u" into "t" at position "k" is equal to the sum of the height of "t" and the product of the height of "u" and the multiplicity of "k" in "t".
The lemma PK_ImpI: Assuming that from A and B, p leads to q in the context of G, it can be shown that in the context of G, from A and B, p implies q.
The lemma "vec_first_add" assumes that the dimensions of vector x and vector y are greater than or equal to n. It shows that the first n elements of the sum of vectors x and y are equal to the sum of the first n elements of vector x and the first n elements of vector y.
If, under the assumption of a certain condition Φ, binding the assumption of Φ to a monadic operation M is less than or equal to another monadic operation M', then binding the assumption of Φ to the monadic operation M is indeed less than or equal to M'.
: given the assumption 'wtm' which asserts that a method is well-typed in the context of a program 'P', a class 'C', a boolean 'b', a list of types 'Ts', a return type 'T\<^sub>r', a maximum stack size 'mxs', an initial local variable size 'mxl\<^sub>0', an exception table 'is', and a list of type states 'xt', it can b...
The lemma named "slpra_induct" assumes a major premise that "xs" and "ys" are on the same level path in reverse auxiliary. It also assumes four rules: 1. For any "cs", the property "P" holds for "cs" and an empty list. 2. For any "cs", "a", and "as", if "a" is of intra-kind, "cs" and "as" are on the same level path in...
There is no microstep from the groupcast of a set containing "l" with parameters "sips" and "smsg" to "q" under the context "Gamma".
The fold operation with the "maybe or" function applied to a list that starts with an element 'a' and is followed by list 'l', with 'b' as the initial value, is equivalent to 'a' "maybe or"-ed with the fold operation of the "maybe or" function applied to list 'l' with 'b' as the initial value.
The lemma cf_prod_2_Rel_components shows the following: 1. The object mapping of the product of two categories in the category of relations is a function that takes a pair of objects in the product of two categories and maps it to the product of the two objects. 2. The arrow mapping of the product of two categories i...
The application of a complex bounded linear function 'A' is a complex linear function.
If a list composed of 'xs' followed by 'U' followed by 'ys' is distinct (meaning it has no repeated elements), and 'U' is not an empty list, and this list is equal to another list composed of 'as' followed by 'U' followed by 'bs', then 'as' must be equal to 'xs'.
The function 'ucast', which casts an 8-bit word to a 32-bit word, is injective. This means that different 8-bit words will always be cast to different 32-bit words, ensuring there are no collisions or overlaps.
If a list composed of "A" followed by "B" is a sublist of "C", then both the list containing only "A" and the list "B" are also sublists of "C".
The lemma named "subsumee_not_marked" assumes that "prb" is a RedBlack tree and "sub" is a subtree of "prb". It shows that the subsumee of "sub" is not marked in "prb".
Getting an axiom 'a' is functionally safe.
The property of being coprime is commutative. That is, if p and q are coprime, then q and p are also coprime.
A complex number x is a real number if and only if x is less than or equal to zero or x is greater than or equal to zero.
The valuation of the derivative of function f at point a is not infinity.
The lemma named "integral_component_eq" is about a function 'f' that maps from one Euclidean space to another. It assumes that 'f' is integrable over a set 'S'. The lemma shows that the integral over 'S' of the function 'f' projected onto a component 'k' is equal to the projection onto 'k' of the integral over 'S' of t...
The lemma named "representation_bound" is defined with a fixed set B of type 'N::real_inner. It assumes that B is finite, B is independent, b is an element of B, and B is pairwise orthogonal. It then obtains a value m where m is greater than 0 and for any x in the span of B, the absolute value of the representation of ...
For any hypermultisets k, m, and n, the product of m and k is less than or equal to the product of n and k if and only if k is greater than 0 and m is less than or equal to n.
"for all sets S, there exists a set X that is the infimum of S".
For a given commutative ring 'a and mod types 'nc and 'nr, the function "diagonal_step_JNF" applied to a matrix A and indices i and j is equivalent to the function "diagonal_step" applied to a matrix B and the same indices i and j, under the condition that matrix A and B are connected by the HMA_M relation from the Mod...
If a function "f" is residuated, then applying "f" to the result of applying the residual of "f" to the result of applying "f" to the residual of "f" applied to "x", is equal to applying "f" to the residual of "f" applied to "x".
The program invariant holds for the first element of the pair resulting from setting up the abstract syntax tree.
N of 2 is equal to 1 divided by q squared times (q minus 1).
The lemma "SubstTerm_subst_dbtm" assumes that "u" is a term. It shows that the substitution of a variable "v" in a term "u" with a term "t" in the SubstTerm function is equivalent to the term resulting from the substitution of "v" with "t" in "u".
If the difference between 2 raised to the power of the length of 'a minus 'n' and 1 is less than or equal to a boundary 'bnd', and 'n' is greater than 0, then the unsigned natural number representation of the right shift of 'x' by 'n' places is less than or equal to the boundary 'bnd'. Here, 'a is a type with a length,...
The lemma named "sum_bij" assumes that "F" is a bijective function and for all "x" in the set "s", the function "f" of "x" equals the function "g" of "F(x)". It shows that for any set "t" such that "t" is the image of "s" under the function "F", the sum of the function "f" over the set "s" equals the sum of the functio...
The lemma named "differenceset_translate_eq_left" assumes that both sets A and B are subsets of set G and that element x is in set G. It shows that the difference set of {x} and A equals the difference set of {x} and B if and only if set A equals set B.
The function which maps a state-configuration pair (s,t) to the variable (b,Φ,s,t) is monotone.
The lemma named "conv_chop" assumes that the pair "(s,ss1@ss2)" is in the conversion of the abstract rewriting system (ars). It shows that the pair "(s,ss1)" is in the conversion of the abstract rewriting system and the pair "(lst_conv (s,ss1),ss2)" is also in the conversion of the abstract rewriting system. Here, "@ss...
A polynomial 'p' is irreducible if and only if the absolute value of the integer polynomial of 'p' is irreducible.
A pseudo allocation 'a' is equal to the set of all pairs (x, {y}), where y is an element of the union of the image of x under 'a' and x is in the domain of 'a'. This is denoted as "?L = ?R".
The lemma named "set_integrable_bigo" is defined with the following conditions: It fixes two functions 'f' and 'g' that map real numbers to a type 'a'. This type 'a' is a Banach space, a real normed field, and has a second countable topology. It assumes that 'f' is a member of the Big O notation of 'g', and that 'g' ...
For any two extended non-negative real numbers 'a' and 'b', if 'a' is less than or equal to 'b', then subtracting 'a' from 'b' and then adding 'a' results in 'b'.
The lemma named "is_alignedE'" is about a fixed word 'w' of any length 'a'. It assumes that 'w' is aligned at 'n'. It then concludes that there exists a natural number 'q' such that 'w' is equal to the result of pushing the bit 'n' of the word representation of 'q', and 'q' is less than 2 raised to the power of the dif...
The guard function, when given the value "True", will return "Some ()".
If the function "find_handler" with parameters "(P\<^bsub>wf\<^esub>)", "a", "h", and a zipped list of "stkss P cs stk" and another zipped list of "locss P cs loc" and "cs", equals to "(xf, h', frs)", then updating the location "loc" with "frs" will result in the same location "loc".
If the following conditions are met: 1. The authentication request data is a set containing the merchant's local identifier (LID_M), transaction identifier (XID), other identification data (HOIData), and other data (HOD). 2. The signature of the merchant (M) on the authentication request data and the hash of the purch...
The value of the function X applied to the product of functions f and g equals zero. This is under the conditions that X is an element of the tangent space at point p, both f and g are elements of the differentiable function space, and the values of both f and g at point p are zero.
The subterms of the term 't' abstracted by 'α' are equal to the subterms of 't' abstracted in the set by 'α'.
If a relation "R" is a left-linear rewriting system, then for any pair "(l, r)" in "R", "r" is a ground term.
If a list "as" contains distinct elements, then the function "lists_fcs" applied to "as" will also produce a list with distinct elements.
If a is greater than or equal to 1, then the sequence defined by raising a to the power of the reciprocal of each natural number n is a bounded sequence.
: Assuming that the word 'w' is a periodic extension of both 'r' and 's', and the sum of the lengths of 's' and 'r' minus their greatest common divisor is less than or equal to the length of 'w', it can be shown that the concatenation of 's' and 'r' is equal to the concatenation of 'r' and 's'.
The lemma named "pow_res_is_semialg" assumes that "x" is an element of the carrier of Qp (the p-adic numbers). It shows that the image of the function "pow_res n x" under "to_R1" is a semialgebraic set of dimension 1.
If for any path 'p' from the source to the sink in a network 'Δ', there exists a node 'z' in the path 'p' that belongs to a set 'S', or the source of the network 'Δ' is in the set 'S', then 'S' is a separating network for 'Δ'.
The lemma named "unitary_diag_spectrum_eq'" is about a complex matrix A. It assumes that A is an n x n matrix and that A is unitarily similar to a diagonal matrix B through a unitary matrix U. It shows that the spectrum of A is equal to the diagonal elements of B.
If an agent A receives some information X in a sequence of events, the knowledge of the Spy is the same as before this event occurred.
The lemma "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p_NegCheck" states the following four conditions: 1. The free variables of a stateful strand step that checks if none of the variables in set X are in pairs F or not in pairs G, are the free variables of pairs F union the free variables of pairs G minus the set X. 2. The fr...
The left quotient of a character 'c' and the star of a set 'A' is equal to the concatenation of the left quotient of 'c' and 'A' with the star of 'A'.
The set of edges in the policy made for subnet hosts with the model [Subnets_m] is a subset of the set of edges in the policy made for subnet hosts with the model SubnetsInGW_ACL_ms.
If the index 'i' is less than the length of the list 'Vs' and the element at index 'i' in 'Vs' is not in the free variables of the expression 'e', then the function 'unmod' applied to the compilation of the expression 'e' with respect to 'Vs' at index 'i' holds true. The second proposition, fvs_unmods_compEs1, states ...
The left restriction of the intersection of two relations 'r' and 's' over a set 'A' is equal to the intersection of the left restrictions of 'r' and 's' over the same set 'A'.
If you drop bits from an integer x at the 0th position, the result will be 0.
A product topology of spaces X and Y is a Hausdorff space if and only if the topological space of the product topology of X and Y is empty, or both X and Y are Hausdorff spaces.
The lemma "le_imp_Di_subset" assumes that vector "v" is less than or equal to vector "y". It shows that the Diophantine set of "i" and "v" is a subset of the Diophantine set of "i" and "y".
: Given a tagged division 'd' of a set 'i', and two distinct pairs (x1, K1) and (x2, K2) that are elements of 'd', if the intersection of K1 and the set of all 'x' such that 'x' dot product with 'k' is less than or equal to 'c' is equal to the intersection of K2 and the same set, then the interior of the intersection o...
The post-measurement of the Bell state "beta_11" at position 2 is equal to the unit vector at position 1 in a 4-dimensional space.
: - "x equals DONE" is equivalent to "x is DONE" - "DONE equals x" is also equivalent to "x is DONE"
The lemma TS_pairwise' assumes that 'qs' is a subset of the set 'init', the pair (x, y) is in the set where x and y are both in the set 'init' and x is not equal to y, x is not equal to y, and 'init' contains distinct elements. It shows that the probability of x occurring before y in the embedded random transposition s...
If a set 'us' is corthogonal, then the elements of 'us' are distinct.
The lemma named "Gromov_product_at_quasi_isometry" is about a function 'f' that maps from one Gromov hyperbolic space to another. The lemma assumes that 'f' is a lambda C-quasi-isometry. The lemma then shows two things: 1. The Gromov product at the images of three points under 'f' is greater than or equal to the Gro...
If 'n' is an element in the set of nodes 'g' (represented as "\<alpha>n g"), then the uses of 'n' in 'g' is equal to the image of 'n' under the function 'var g' applied to the uses of 'n' in 'g' (represented as "uses' g n").
The lemma named "Con_single_ide_iff" assumes that "a" is an identity of "R". It shows that the concatenation of the star of "a" and any transition system "T" is equivalent to "T" being an arrow and "a" being in the sources of "T".
There exists a set G such that G is an ultra filter and the top element is in G.
For a given infinite stream 'z', if we truncate it at index 'i' and append a list containing element 'a', then make it untimed, the last element of the resulting list will be 'a'.
If function 'h' is injective, then the composition of the size function of a finite map 'f' and 'g' with the finite map 'h' is equal to the size function of 'f' composed with the function 'g' and 'h'.
If the saturation push precondition for f, l, and the pair (u,v) holds, then \<Delta> is equal to the capacity function of (u,v).
If a word (a string of binary digits) 'x' is less than or equal to a mask 'n', then the bitwise AND operation between 'x' and the mask 'n' is equal to 'x'. This is applicable for any length of the word 'x'.
: Assuming that one_chain1 is a reparameterization of one_chain2, every pair (k2, γ2) in one_chain2 is piecewise C1 differentiable on the interval from 0 to 1, and for every pair (k2, γ2) in one_chain2, for every basis element b, the function F is continuous on the path image of γ2. Also, the basis is finite. Additi...
The most significant bit (msb) of the result of pushing a bit with a given position 'n' into a word 'w' is true if and only if 'n' is less than the word length of the type 'a' and the bit at the position (word_length of type 'a' - (n+1)) in the word 'w' is true.
The Herbrand domain inclusion of 't' and 'u' is a Sigma formula.
The "tm" function of a state sigma, where "sqsum" is replaced by "s", is equal to the "tm" function of the original state sigma.
The optimal binary search tree's weighted path length from 'i' to 'j' is equal to the pair consisting of the optimal binary search tree from 'i' to 'j' and the weighted path length from 'i' to 'j' of the optimal binary search tree from 'i' to 'j'.
The lemma named "nth_prime_partition" assumes that "x" is greater than or equal to 2. It shows that there exists a "k" such that "x" is in the range from the "kth" prime number to the next prime number (not inclusive of the next prime number).
A queue is empty if and only if the queue is equivalent to an empty queue.
The filtered list using any predicate 'P' on an empty lazy list 'LNil' results in an empty lazy list 'LNil'.
The lemma fixes hypermultisets alpha1, alpha2, beta1, beta2, gamma, and delta. It assumes that the sum of alpha2 and the product of beta2 and gamma is less than the sum of alpha1 and the product of beta1 and gamma. It also assumes that beta2 is less than or equal to beta1 and that gamma is less than delta. Under these ...
The exponential function "eexp" is strictly increasing. This means that "eexp x" is less than "eexp y" if and only if "x" is less than "y".
The contour integral of a function 'f' over a subpath 'g' from point 'u' to the same point 'u' is equal to zero.
The evaluation of the universal quantification of a list conjunction "A" for a given element "x" is equivalent to the condition that for every element "l" in the set "A", the evaluation of "l" for "x" holds true.
The lemma named "bisim_J1_JVM_start" assumes that the program P is well-formed according to the J1 framework (as stated by "wf_J1_prog P") and that the start state of the program, given by the class C, method M, and variables vs, is also well-formed (as stated by "wf_start_state P C M vs"). Under these assumptions, the...
If T is a subset of the carrier D, and both a and b are elements of T, then the relation "a is less than b" in the strict order of D restricted to T (SIod D T) is equivalent to the relation "a is less than b" in the strict order of D.
The "tick" event does not belong to the set of traces of any process that is the result of a generalized prefixing operation over a set A of events, each leading to a corresponding process P.
If set A is finite and the set difference between A and the range of function sigma is not empty, then the random permutation of sigma and x is a lossless stochastic process.
The generalized roofed function of a web minus a flow is equal to the generalized roofed function of the web itself.
The number 131059 is a prime number.
If a pair (A, B) is in the relation defined by the composition of "fully_unsorted_poly_rel" and "mset_poly_rel", then the set of variables in B is a subset of the image of the set of variables in A under the function φ.
The domain of the function C, when applied to the combination of "DenyAllFromTo x y", "DenyAllFromTo y x", and "AllowPortFromTo x y dn", is equal to the domain of the function C when applied to the combination of "DenyAllFromTo x y" and "DenyAllFromTo y x".
The lemma named "map_partial_pullback_carrier" assumes that the function 'f' is a mapping from the carrier of the m-th power of the p-adic numbers to the carrier of the n-th power of the p-adic numbers. It shows that the partial pullback map of 'f' with respect to 'm' and 'l' on the carrier of the (n+l)-th power of the...
The lemma named "inner_prod_expand" assumes that the dimensions of vector a are equal to the dimensions of vectors b, c, and d. It shows that the inner product of the sum of vectors a and b with the sum of vectors c and d is equal to the sum of the inner products of a with c, a with d, b with c, and b with d.