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If we apply the function "right_gpv" to the "left_gpv" of a general probabilistic value (gpv), and then map this result using the identity function as the first argument, "lsumr" as the second argument, and "rsuml" as the third argument, we will get the same result as if we first apply the "right_gpv" function to the g...
The lemma named "better_nrename_Cut" assumes that "x" is fresh for the pair "(u,v)". It shows that the renaming operation on the cut operation "Cut <a>.M (x).N" with "u" renamed to "v" is equal to the cut operation with "a" and the renamed versions of "M" and "N" where "u" is replaced with "v".
The set of all differentiable functions at level k on the given charts forms a subspace.
Applying the "lists_succ" function to an empty list results in the identity function.
An empty set is less than set A if and only if set A is not an empty set.
The right-hand side of the initial item of a rule 'r' at position 'k' is equal to the second element of the rule 'r'.
The multiplication of x and y is commutative, meaning that the order in which they are multiplied does not change the result.
The normalization of a constant polynomial is equal to a constant polynomial with a normalized constant.
The lemma named "mapping_of_pmf_pmf_of_list" assumes that for any element 'x' in the second element of the set 'xs', 'x' is greater than 0, and the sum of the list obtained by mapping the second element of 'xs' equals 1. Under these assumptions, it shows that the mapping of the probability mass function of the list 'xs...
If a string 'v' is equal to the concatenation of string 'u' and string 'z', and string 'z' is not an empty string, then 'u' is a strict prefix of 'v'.
The lemma named "wfG_b_weakening" is about well-formedness in the context of a type system. It takes a fixed context Gamma. It assumes that the set of type variables B is a subset of B' and that under the context of a set of type assumptions Theta and B, Gamma is well-formed. It shows that under the context of Theta an...
If the lengths of list xs and list ys are equal, then the lengths of the results of substituting elements from xs and ys into source z and destination z of query Q are also equal.
"x" is not before "y" in the list that starts with "y" followed by the elements of list "xs".
If a complex expression "e" is non-negative under the variables "V" and the context "Γ", and if a state "σ" is in the space of the state measure with respect to "V" and "Γ", then the real part extracted from the semantics of the complex expression "e" under the state "σ" is greater than or equal to zero.
The function 'pair_of_term' is injective. This means that if two different terms 'u' and 'v' are input into the function 'pair_of_term' and the output is the same, then 'u' and 'v' must actually be the same term. In other words, no two different terms can produce the same output when input into the function 'pair_of_te...
If the function "f" applied to "x" is less than or equal to the function "f'" applied to "x'" under a certain relation "R", then the "Let" construct with "x" and "f" is also less than or equal to the "Let" construct with "x'" and "f'" under the same relation "R".
If a list 'xs' is finite, then the terminal of the time-logical list derived from the list 'xs' and a value 'b' is equal to 'b'.
The lemma "Ide_Subst" shows that if "t" and "v" are ideals, then the substitution of "v" into "t" at position "n" is also an ideal.
Applying the function "$ S" to an empty list results in an empty list.
: if for every element 'x' in the space 'M', the function 'f' of 'x' is in the space 'N', and for every set 'A' in the sets of 'N', the preimage of 'A' under 'f' intersected with the space 'M' is in the sets of 'M', then the function 'f' is in the measurable space from 'M' to 'N'.
: Assuming a trace from initial to final state, and for all events 'i' and above, there are no prerecording events, and for all events less than 'i', there are no postrecording events, and 'i' is less than or equal to the length of the trace, it can be shown that the snapshot at time 'i' is equal to the final state.
If a natural tiny transformation 'NN' exists from functor 'FF' to functor 'GG' within the context of tiny categories 'AA' and 'BB' under a fixed universe 'alpha', then the following conditions hold: 1. There exists a natural transformation 'NN' from functor 'FF' to functor 'GG' within the context of categories 'AA' an...
If "t" is a directed acyclic graph (DAG) with root "p" and functions "low" and "high", and "n" is an element in the set of "t", and "nt" is a DAG with root "n" and functions "low" and "high", then the set of "nt" is a subset of the set of "t".
If a tau-execution move from state 's' to state 's'' is possible in a context 'ci' for a program 'P' running on a thread 't' with an expression 'e' and a heap 'h', then a tau-execution move is also possible in the same context for the same program and thread with the expression 'e instanceof T' and the same heap and st...
The halting problem is undecidable. Given a function 'code' that maps natural numbers to recursive functions, and assuming that for any recursive function 'f' of one variable, there exists a natural number 'i' such that 'code i' equals 'f', it is shown that it is not possible to decide whether for a given 'x', the eval...
If a constant 'c' is not equal to zero, then the formal power series of the constant 'c' is also not equal to zero.
The lemma named "sum_image_eq" assumes that the function "f" is injective on the set "A". It shows that the sum of the image of set "A" under function "f" is equal to the sum of the function "f" applied to each element in set "A".
The lemma named "star_power_slide" assumes that "k" is less than or equal to "n". It shows that the "k"th power of "x" multiplied by the star of the "n"th power of "x" is equal to the star of the "n"th power of "x" multiplied by the "k"th power of "x".
If the condition CoP1 is true for a certain value of η, then the condition CoPw is also true for that same value of η.
: 1. If an atomic action "atm" is compatible, then it is single-input single-output (siso). 2. If command "c1" and command "c2" are both single-input single-output (siso), then their sequence (Seq c1 c2) is also siso. 3. If a test "tst" is compatible and both commands "c1" and "c2" are siso, then the conditional state...
If a policy "p" is a well-formed policy1 and "p" is equal to "DenyAll" followed by some policy set "ps", and if the rule "DenyAll" is the last rule applied to "p" for a certain "x" in the context of policy "Cp", then for any rule "r" in the policy set "ps", "x" is not in the domain of "Cp r".
A subtree of a tree T at a certain position "ds" is a subset of the union of two other subtrees: one at the position "ds" followed by "Left", and the other at the position "ds" followed by "Right", along with the set containing the position "ds" itself.
If a list "xs" is not empty and the first element of the last item in the list is "None", then the first element of the last item in the list "xs" after applying the function "lift P" to each item in the list will be "Some P".
The inference function 'infer', when applied to the parameters 'f', 'k', 'r', 'm', 'check', and 'np' with an empty set, will result in an empty set.
The lemma named "reachable_states_cases" assumes three conditions: 1. State 'q' is among the reachable states of machine 'M'. 2. Property 'P' holds for the initial state of machine 'M'. 3. For any transition 't' in machine 'M', if the source state of 't' is among the reachable states of 'M', then property 'P' holds fo...
The sublist obtained by taking elements from the list 'xs' while the predicate 'P' holds true, is a sublist of 'xs'.
If the degree of a polynomial 'p' is zero, then the polynomial 'p' is equal to the constant polynomial with coefficient as the zeroth coefficient of 'p'.
The free variable of an application of an expression 'e' and a variable 'x' is the set that includes 'x' and the free variables of 'e'.
If "u" is an element in the set "p", then there exist two sets "q" and "r" such that "p" is the union of "q" and "r", the last element of "q" is "u", the first element of "r" is "u", and both "q" and "r" are not empty sets.
The lemma named "inv_begin_steps" assumes that the function "inv_begin" with parameters n and cf is true, and that n is greater than 0. It shows that "inv_begin" with parameters n and the function "steps0" with parameters cf, tm_copy_begin_orig, and stp is also true.
The greatest common divisor (GCD) of a set of integers K is always greater than or equal to 0.
For a list starting with element 'x' followed by list 'xs', and a fill value 'fill', the operation 'takefill' with a numeral 'k' will result in a list starting with 'x' followed by the result of 'takefill' operation with the predecessor of 'k' on the list 'xs'.
The signed right shift of a negative numeral by another numeral is equal to the signed drop bit operation on the negative numeral with the second numeral as the argument.
If measure N is absolutely continuous with respect to measure M, and the sets of N and M are equal, then there exists a function f that is Borel measurable with respect to M such that the density of M with respect to f is equal to N.
If x is not equal to y, then the assignment of a value to x does not modify y, even when annotations are applied.
The lemma named "powsum_mpoly_recurrence'" assumes that "k" is greater than the cardinality of "X" and "X" is finite. It shows that the power sum polynomial of "X" and "k" in a commutative ring is equal to the negative of the sum from 1 to the cardinality of "X" of the product of (-1) raised to the power of "i", the sy...
The lemma named "intersects_x_axis_mk_circline" assumes that "A" is a real number and that either "A" is not zero or "B" is not zero. It shows that a circline, created with parameters A, B, the conjugate of B, and A, intersects the x-axis if and only if "A" is zero or the square of the real part of "B" is greater than ...
The lemma named "filter_fast_forward" assumes two conditions. First, that "n+1" is less than or equal to "n'". Second, that for all "k", if "k" is greater than "n" and less than "n'", then "P k" is not true. Given these assumptions, it shows that the filter function applied to the list starting from "MkI (n+1)" with th...
The lemma named "Wbis_transT2" assumes three conditions and two cases, and it proves a thesis. The conditions are: "c" is weakly bisimilar to "d", "s" is similar to "t", and "d" and "t" can transition to "t'". The first case is when "c" and "s" can transition to "s'" and "s'" is similar to "t'", then the thesis holds. ...
In the context of a programming language, a variable V of type T in an expression e under the environment E, heap h, and static heap sh is well-typed as T' if and only if the expression e is well-typed as T' under the same environment E where variable V is mapped to type T, and the same heap h and static heap sh.
The lemma named "poly_add" is about polynomials in a commutative ring. It takes two fixed polynomials, p and q. It assumes that q is not equal to zero, that the polynomial p evaluated at x equals zero, and that the polynomial q evaluated at y equals zero. It then shows that the polynomial resulting from the addition of...
The lemma "assigned_nodes_is_component_unsafe" assumes that the heap "h" is well-formed, has a valid type, and has known pointers. It also assumes that the assigned nodes of the element pointer in the heap "h" are equal to "nodes", and that the node pointer is in the set of "nodes". It shows that the intersection of th...
For all values of 'k', the product of the 'k'th bit of 'a' and the 'k'th bit of 'b' is zero if and only if for all values of 'k', the binary carry of 'a' and 'b' at the 'k'th position is zero.
The Stone phi function of the absolute regular value of negative x intersected with the upward filter of the absolute dense value of the union of y and negative y is less than or equal to the upward filter of the absolute dense value of the union of y, negative y, and x.
The lemma named "max0_id_pos" assumes that "x" is greater than or equal to zero and "x" is within the carrier "R". It shows that the maximum value between zero and "x" is "x".
The lemma named "weight_le_mono_one" assumes that the weight of 's' is less than or equal to the weight of 't' (denoted as "weight_le s t"). It shows that the weight of a function 'f' applied to a list composed of 'ss1', 's', and 'ss2' is less than or equal to the weight of the same function 'f' applied to a list compo...
The dimension of the vector obtained by applying the "lift_01_vec" function to a vector "v" is equal to the dimension of the original vector "v". Furthermore, if "i" is less than the dimension of vector "v", then the "i"-th element of the vector obtained by applying the "lift_01_vec" function to "v" is equal to the r...
The multiset representation of a skew-linked tree, which is created by skew-linking elements 'e' and 'a' with trees 't1' and 't2', is equal to the multiset containing the pair (e,a) added to the multiset representations of 't1' and 't2'.
If "c" is greater than 0, then scaling the Dirichlet series by "c" and the integer "n" is equivalent to the integer "n".
The lemma named "measurable_envelopeD1" assumes that "E" is a measurable envelope of "A" with respect to measure "M", and "F" is a set in the sigma-algebra of "M" that is a subset of the difference between "E" and "A". Under these assumptions, it shows that the outer measure of "F" with respect to "M" is zero.
The lemma named "rot_circle_eq_circle" assumes that "x" and "y" are both greater than or equal to 0 and less than or equal to 1. It shows that the rotation of a circle in a cube, given the coordinates (x, y), is equal to the circle in a cube with the rotation of x and y.
For any given function 'v' that maps from a float to a real number, and any set 's' of floats, assuming that the set 's' is finite and not empty, there exists an element 'a' such that 'a' is the closest to 'x' according to the function 'v'.
The lemma fixes a variable 'pre' of type 'ipv4addr'. It shows that the set of IP addresses from a CIDR block, defined by the prefix 'pre' and length 'len', is equal to the set of all IP addresses that can be obtained by performing a bitwise AND operation between 'pre' and the result of shifting the mask 'len' left by '...
The lemma "inj_on_imp_Ker0" assumes that the function T is injective on the set G. It then shows that the kernel of this function is equal to 0.
A singleton set containing a real number is an element of the Cartesian product of the set of real numbers and the set containing only the natural number 1.
: Assuming 1) "FlexRayArch" holds for 'n', 'nReturn', 'nC', 'nStore', and 'nGet', 2) for all 'i' less than 'n', the message at time 'Suc 0' is 'nReturn i', 3) 'n' and 'nC' have disjoint schedules, 4) 'n' and 'nC' have identical cycle lengths, it can be shown that "FrameTransmission" holds for 'n', 'nStore', 'nRe...
If 'b' is an element of the smallest countably complete distributive intersection sets of 'M' over a set 'Ω', then if 'a' is also an element of these smallest sets, the intersection of 'a' and 'b' will also be an element of these smallest sets.
The function "K" applied to the pair (t, s) is equal to the distribution of the pair (t', s) under the exponential function with escape rate s and the function J s, where the pair (t', s) is mapped to the pair (t + t', s).
Executing a generative probabilistic value (gpv) that has been mapped with a function 'f' and the identity function, on a given oracle and state, is equivalent to mapping a state-probability mass function (spmf) with a function that applies 'f' to the first element of a pair, on the execution of the original gpv on the...
If tree "t1" is in the first component of the finite set "xs", then the count of "x" in the multiset of vertices of "t1" is less than or equal to the count of "x" in the multiset of vertices of the node "r" with children "xs".
If set S is countable, then the finite power set of S is also countable.
The function "join_option" applied to "x" equals "Some y" if and only if "x" equals "Some (Some y)".
The rank of any single-element set P is always 1.
If an element 'a' is in vector 'v', then the image of 'a' under function 'f' is in the vector obtained by mapping 'f' over 'v'.
If a function 'f' belongs to the finite support 'z' of set 'A', then the support 'z' of set 'A' for function 'f' is finite.
If a function 'f' maps from the set containing only the natural number 0 to the carrier of a group 'G', then the finite product of 'f' over the range from 0 to 0 equals to 'f' at 0.
If 'u' and 'v' are elements of set 'V', then the set of connecting paths between 'u' and 'v' is not empty.
If a type operation 'op' with a certain kind 'k' maps a type 'τ' and a type 'σ' to a type 'ρ1', and the same type operation 'op' with the same kind 'k' also maps the same type 'τ' and type 'σ' to another type 'ρ2', then 'ρ1' must be equal to 'ρ2'. This means the type operation is deterministic.
: a set 's' is a subset of the domain of the function 'f' applied to 'A', and 'A' is equal to the restriction of the function 'f' applied to 'A' over the complement of 's', if and only if 'A' is a submap of the function 'f' applied to 'A', and 's' is equal to the difference between the domain of the function 'f' applie...
"f is an arrow from a to b in the digraph dg_1 with parameters aa and ff if and only if a is equal to aa, b is equal to aa, and f is equal to ff."
The lemma named "MinPredicate2" is defined as follows: Let P be a predicate that applies to a natural number. If there exists a natural number that satisfies the predicate P, then there exists a natural number n0 such that P is true for n0 and either n0 is equal to 0 or P is not true for the natural number one less th...
If we have two morphisms "e" from an object "E" to objects "a" and "b" in the category, and these morphisms are equal to the composition of two other morphisms "g" and "f" with "e", where "g" and "f" are morphisms from "a" to "b", then the base of the natural transformation equalizer of "a", "b", "g", "f", "E", and "e"...
The lemma named "goodstein_step" assumes that the Goodstein sequence at position "i" is greater than 0. It shows that the encoding of the Goodstein sequence at position "i" incremented by 2, with a second parameter of 0, is greater than the encoding of the Goodstein sequence at position "i" incremented by 3, with a sec...
If a number 'n' is greater than or equal to the length of a word 'w' (where 'w' is a word of any length 'a'), then right shifting the word 'w' by 'n' places will result in 0.
The function QDelta, when applied to the first and second elements of a QDelta object, returns the original QDelta object.
The lemma for the natural number monotonically increasing characterization: Given a function 'f' that maps a natural number to another natural number, and assuming that 'f' is monotonically increasing, we can obtain a natural number 'n' such that for all natural numbers 'm', if 'n' is less than or equal to 'm', then 'f...
The XOR (exclusive OR) operation between two integers 'k' and 'l' is defined as follows: If 'k' equals -1, then the result is the NOT operation on 'l'. If 'l' equals -1, then the result is the NOT operation on 'k'. If 'k' equals 0, then the result is 'l'. If 'l' equals 0, then the result is 'k'. Otherwise, the result i...
The modulus of a natural number 'a' with respect to the product of two natural numbers 'b' and 'c', further taken modulus with 'c', is equal to the modulus of 'a' with respect to 'c'.
If a set 'us' is a subset of 'V' and 'us' is linearly independent, then the length of 'us' is less than or equal to the dimension of 'V'.
If the uncurried version of two functions 'f' and 'g' are in the functional refinement of 'A' leading to 'B' in the non-deterministic result relation, then for any 'x', 'x'', 'y', 'y'', 'z', 'z'' such that 'P' holds for '((x', y'), z')' and the pair '(((x, y), z), ((x', y'), z'))' is in 'A', the result of function 'f' ...
The image of a function 'f' applied to a single-element multiset 'x' is equal to a single-element multiset of 'f' applied to 'x'.
The lemma named "simulation_comp" assumes that there is a simulation from system A to system B using function F, and a simulation from system B to system C using function G. It shows that there is a simulation from system A to system C using the composition of functions G and F.
The lemma named "eqvtSubstClosed" is defined with a fixed relation "Rel" which is a set of pairs of pi. The assumption is that the relation "Rel" is equivariant. The lemma shows that if this assumption holds, then the relation "Rel" is also substitution-closed under equivariance.
If G is a well-formed graph, then a well-formed stateful policy is defined where the hosts are the nodes of G, the fixed flows are the product of the nodes of G with themselves, and the stateful flows are also the product of the nodes of G with themselves.
If a ternary operation "op" on strings has the same types for its operands and results in two cases, then the results of these two cases must be the same.
If 'i' is less than 'n', then the 'i'-th element of a list that replicates 'x' 'n' times is 'x'.
The lemma named "cod_char" shows that the codomain in category V is defined as follows: if a morphism 'mu' is an arrow in category V, then the codomain is a structure where the chain is the apex of the codomain 'mu' in the span in category C, and both the domain and the codomain are the codomain 'mu'. Otherwise, if 'mu...
The sum of a function "f" applied to each element in a multiset "M", multiplied by a constant "c", is equal to the sum of the function "f" applied to each element in "M" multiplied by the constant "c". This is a property of distribution over multiplication in the context of multisets.
The lemma "preserve_deadlocked" assumes that the Java program P is well-formed (wf_J_prog P). It shows that the state of deadlock is preserved under certain conditions. These conditions are: synchronization expressions are okay (sync_es_ok) between threads and shared resources, locks are okay (lock_ok) between locks an...
The lemma named "sources_single_Src" assumes that "t" is an arrow in the lambda calculus. It shows that the sources of the source of "t" are the same as the sources of "t" itself.