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If "C" is less than or equal to the "gmctxt_of_gterm" of "t", then the length of "unfill_gholes" of "C" and "t" is equal to the number of "gholes" in "C".
If lens X is independent of lens Y, and lens X is also independent of lens Z, then lens X is independent of the sum of lens Y and lens Z.
The representation of a collineation in 2 dimensions, denoted as "cltn2_rep A", is invertible.
G is a normal series in GG.
If 'i' is less than the number of rows in matrix A, then the 'i'-th column of the transposed matrix A is equal to the 'i'-th row of matrix A.
For a map 'm' updated at 'a' with 'b', the value 'y' is associated with 'x' if and only if 'x' is equal to 'a' and 'b' is equal to 'y', or 'x' is not equal to 'a' and the original map 'm' associates 'x' with 'y'.
For any transition 't', if 't' is in the set of transitions of machine 'M', then the target of transition 't' is in the set of states of machine 'M'.
If there is a transition from (p,n) to (p',Label l') in the program and procedures, then the same transition exists when the program is followed by c₂ in the program and procedures.
If the following conditions are met: 1. The Java program P is well-formed (as per the assumption "wfP: "wf_J_prog P""), 2. The start state of the Java Memory Model for the program P, class C, method M, and variables vs is well-formed (as per the assumption "ok: "jmm_wf_start_state P C M vs"), 3. The execution (E, ws) i...
The lemma named "setdiff_irrelevant" assumes that "A" is a finite set. It shows that the function "F" applied to the function "g" and the set "A" minus the set of all elements "x" for which "g x" equals "z", is equal to the function "F" applied to the function "g" and the set "A". In other words, removing elements from...
Inserting an element 'a' into a set that contains elements 'b' and 'c' is equivalent to creating a set that contains elements 'a', 'b', and 'c'.
The lemma named "ordered_map_image" assumes that "S" is a finite set and is a map. It shows that the map of the function (which takes a pair (a, b) and returns a new pair (a, f a b)) over the ordered map of "S" is equal to the ordered map of the image of the same function applied to "S".
, given the assumptions that both formulas phi1 and phi2 belong to the set of all formulas, the second elements of the tuples in the set txs are all variables, the first elements of the tuples in the set txs are all terms, and there are no duplicate second elements in the list of tuples txs, it can be shown that the pa...
In the context of an encoding, the lemma named "indRelTEQ_impl_TRel_is_strong_reduction_simulation" is defined. It fixes a relation "TRel" which is a set of pairs of 'procT'. It assumes that "indRelTEQ TRel" is a strong reduction simulation in the STCal of Source and Target. The lemma then shows that "TRel" to the powe...
The lemma "wfT_eq_imp" is about well-formed types in the context of the MiniSail programming language. It takes as input a variable "zz" of type "x", a literal "ll" of type "l", and a type "\<tau>'". It assumes the following: 1. The base type for the literal "ll" is boolean. 2. The type "\<tau>'" is well-formed under...
The real number representation of a floating point number 'x' is within the real interval 'X' if the floating point number 'x' itself is within the interval 'X'.
If the function "f" is a response function for folding, then folding once with function "f" over a list "xs" is equivalent to folding with function "f" over the list "xs" after removing duplicates.
If a function 'f' is an element of the unit group of the function ring over the p-adic numbers with 'n' variables, then the following three conditions hold: 1. The product of 'f' and the inverse of 'f' in the function ring over the p-adic numbers is the multiplicative identity of the function ring over the p-adic numb...
The function "Un_set_drop_impl" applied to an index "i" and a set "A" is less than or equal to the specification of a function "r" such that "r" is equal to the union of the set obtained by dropping the first "i" elements from "A".
The lemma shows that the permutation of a function, which takes a boolean B and returns the implication of A and the equality of B and C, equals to foo.
If the r-th element of the outputs of transition t is equal to L v, and the value of p in context c is Some v, then the generalised output of transition t at position p and r subsumes the context c in transition t.
The lower limit (infimum) of the state 'Sts' is equal to the lower limit (infimum) of the state 'Sts''.
If you take the first "length fas" elements from the interpretation of the straight line program (slp) of "fas" with respect to "xs", it will be equal to the interpretation of the floating point arithmetic operations "fas" with respect to "xs".
The lemma named "resDerivative" is defined with the following parameters: two processes P and Q, a subject a, two names x and y, and two relations Rel and Rel'. It assumes the following conditions: 1. P is a derivative of Q with respect to the subject a and the name x under the relation Rel (denoted by "derivative P...
If I and J are subrings of R, then the intersection of I and J is also a subring of R.
A given element 'p' is equal to a five-element tuple. This tuple consists of the first element of 'p', the first element of the second element of 'p', the first element of the second element of the second element of 'p', the first element of the second element of the second element of the second element of 'p', and the...
For any set X of type "part_witness", the property "part" holds for X.
If the following conditions are met: 1. The abstraction function 'abs' strongly maintains a low bisimulation relation, 2. There is a secure refinement between the relations \<R>\<^sub>A and \<R> under the condition P, 3. The condition P is symmetric, then the concrete function 'conc' also strongly maintains a low bisim...
The lemma for the nth term of a formal power series logarithm is as follows: if n equals 0, then the nth term is 0. Otherwise, the nth term is equal to 1 divided by c times negative 1 raised to the power of n minus 1, all divided by n.
The function "Vf" applied to the function "Tf" on a list that has a pair (v, v') as its head and "t" as its tail, is equal to a list where the head is a pair consisting of the function "V" applied to the function "T" on "v" and the function "V" applied to the function "T" on "v'", and the tail is the function "Vf" appl...
The identity lens is a very well-behaved lens.
Mapping any function 'f' to the polynomial '0' results in '0'.
: Assuming the following conditions: - The lengths of Vs, Ts, and vs are all equal. - The elements in Vs are distinct. - There is a reduction sequence from the expression e in the initial state (h,l(Vs [mapped to] vs),sh),b to the expression e' in the final state (h',l',sh'),b'. - The expression e' is final. - l'' is t...
The function "gs_iteration" with parameters "eps" and "v" is equal to the following: if "eps" is less than or equal to 0, then the function "GS_rec_fun_b" with parameter "v" is used, otherwise the function "gs_iteration_partial" with parameters "eps" and "v" is used.
If the valuation functions 'ν' and 'ν'' agree on the free variable differences of the sum of terms 't1' and 't2', then 'ν' and 'ν'' also agree on the free variable differences of term 't2'.
If x is less than or equal to x', and the product of x' and y is less than or equal to z, then the product of x and y is less than or equal to z.
The set of all tuples (x, y, z) where x is an element of the list xs, is equal to the set of all tuples (x, y', z') where y equals y', z equals z', and x is an element of the list xs.
If we have two continuous functions 'f' and 'g' from a T2 space to a linearly ordered topology, then the function that maps each point 'x' to the maximum of 'f(x)' and 'g(x)' is also continuous.
: 1. There exists an 'i' such that the maximum value of the range of the function 'mt_pos' applied to 'cm' is equal to the value of 'mt_pos' applied to 'cm' at 'i'. 2. The range of the function 'mt_pos' applied to 'cm' (where 'cm' is a multitrack configuration of any type 'a', state 'q', and tape 'k') is finite. 3. Th...
If Zeta is a member of Beta and Alpha is a subset of Beta, then the functor maps from category A to category B within the context of Beta.
Any ordinal number raised to the power of 1 is equal to the original ordinal number.
If the trace of callee1 is equivalent to the trace of callee2 for a given set A and probabilities p and q, and if the trace of callee2 is equivalent to the trace of callee3 for the same set A and probabilities q and r, then the trace of callee1 is equivalent to the trace of callee3 for the set A and probabilities p and...
If h0 is disjoint from h1 and h0' is a submap of h0, then the map resulting from the addition of h0' and h1 is a submap of the map resulting from the addition of h0 and h1.
If the following conditions are met: 1. The tag of a capability 'c' is true. 2. The load permission of 'c' is true. 3. The offset of 'c' plus the size of a type 't' is less than or equal to the base of 'c' plus the length of 'c'. 4. The offset of 'c' is greater than or equal to the base of 'c'. 5. The offset of 'c' is...
The lemma "ev_alw_impl_ev" assumes that for a stream 'xs', the event '<φ>' is eventually true and it is always the case that '<φ>' implies the event '<ψ>' is eventually true. Under these assumptions, it shows that the event '<ψ>' is eventually true for the stream 'xs'.
The lemma named "asymp_equiv_at_top_transfer" is defined with two functions, f and g, that map to real numbers. It assumes that f is asymptotically equivalent to g at filter F and that f approaches infinity at filter F. The lemma then shows that g also approaches infinity at filter F.
The function pair (gi_V_update, g_V_update) is in the relation of function set relations with respect to Rv, mapped to the extended graph implementation relation with respect to Rm and Rv. This means that if gi_V_update is related to g_V_update in the function set relation with respect to Rv, then they are also related...
The lemma named "cf_maps_subset_Vset" assumes that beta is a member of the class of all ordinals, and alpha is an element of beta. It shows that the set of all functor maps from category A to category B, where the functor is indexed by alpha, is a subset of the elements of the set V indexed by beta.
, under the assumption that there exists a reflexive and transitive closure of a signature reduction from 'p' to 'q' with respect to a single regular sequence 'F' and a well-founded relation, the leading coefficient of the representative list of 'q' is equal to the leading coefficient of the representative list of 'p'.
If for all x, y, and z, the product of x and z is less than or equal to the product of z and y implies that the circular product of x and z is less than or equal to the circular product of z and y, then 1 is equal to the bottom element.
For any four entities alpha1, alpha2, alpha3, and alpha4, the assertion that the statement phi of alpha1, alpha2, alpha3, and alpha4 is necessarily true is equivalent to the statement phi of alpha1, alpha2, alpha3, and alpha4 itself, in the context of a given universe of discourse v.
: 1. The matrix created from a list of vectors 'vs' with 'n' rows is in the carrier of matrices with 'n' rows and the number of columns equal to the length of 'vs'. 2. The number of rows in the matrix created from a list of vectors 'vs' with 'n' rows is 'n'. 3. The number of columns in the matrix created from a list of...
The lemma named "pair_at_lang_fun_states" assumes three conditions: 1. The function 'f' is a finite injection on the set of states of the first component of 'G'. 2. The function 'g' is a finite injection on the set of states of the second component of 'G'. 3. The set 'Q' is a subset of the Cartesian product of the set...
The lemma is from the Polynomials theory in Algebra. In English: In the context of a ring, we have a lemma called "const_term_zero". Assuming that "K" is a subring of "R", "p" is a polynomial over "K", "p" is not an empty set, and the constant term of "p" is zero, we can obtain a "p'" such that "p'" is a polynomial ove...
The lemma named "submap_imp_state_succ_submap_a" assumes that if state s1 is a submap of state s2 and state s2 is a submap of state s3, then it shows that state s1 is a submap of state s3.
The lemma "remove_is_empty_imp_mul" assumes that subtracting the set "dm r {#\<alpha>#}" from the set "M" results in an empty set. It shows that the pair "(M,{#\<alpha>#})" is in the multiplication of the relation "r".
The lemma named "at_least2_Suc" assumes that "k" is greater than or equal to 2. It then obtains a "k'" such that "k" is equal to the successor of the successor of "k'".
For any sequence of the form "0^n" (where n is a natural number) in a commutative ring with a topological space, the sequence is summable.
If the set of codomains of a function 'f' is not empty, then the codomain of 'f' is an element of the set of codomains of 'f'.
If a program P and environment E type-check an expression e to a type T₁, then for any other type T₂, if P and E also type-check e to T₂, then T₁ must be equal to T₂. Similarly, the lemma WTs₁_unique states that if a program P and environment E type-check a list of expressions es to a list of types Ts₁, then for any o...
The lemma "dens_ctxt_measure_empty_bind" assumes that a state "rho" is in the space of the state measure of "V'" and "Gamma". It also assumes that a function "f" is measurable in the state measure of "V'" and "Gamma" with respect to the subprobability algebra "N". The lemma shows that the density context measure of an ...
If a function 'f' restricts a set, then the function applied to the intersection of two sets 'A' and 'B' is equal to the intersection of the function applied to set 'A' and the function applied to set 'B'.
The lemma named "b_e_type_empty1" assumes that, under the context "\<C>", an empty list is of type that transforms "ts" into "ts'". It shows that "ts" must be equal to "ts'".
The set of all ordered pairs [a, b], where a and b are elements of the set of rational numbers and the rational number representation of a is less than the rational number representation of b, is small.
If the length of list xs is not equal to the length of list ys, then the function "matchs" applied to xs and ys will return None.
If a program P determines that a class C with a class hierarchy Cs selects a method M, which is defined as a tuple of types Ts, a type T, and a method m, via a class hierarchy Cs', and if the program P is well-formed with respect to the method declaration wf_md, then the type T is a valid type in the program P.
If 'x' is not in the field of 's', then the constant 'x' is undefined.
The join of functions 'f' and 'g' applied to 'x' is equal to the join of the results of applying 'f' and 'g' separately to 'x'.
The Discrete Fourier Transform (DFT) of a sequence "a" of length "2*m" at the index "i" is equal to the sum of the DFT of the even-indexed elements of "a" at the index "i" and the product of the "i"-th power of the "2*m"-th root of unity and the DFT of the odd-indexed elements of "a" at the index "i".
If set A is not finite, then the symmetric polynomial of set A with degree k equals zero.
If x is not equal to y, then the base dimension of x is not equal to the base dimension of y.
If a non-zero vector "x" is scaled by two different scalars "a" and "b" and the results are equal, then the scalars "a" and "b" must be equal.
For any two predicates P and Q of type 'alpha upred, the infimum (greatest lower bound) of P and Q is equal to the logical disjunction (OR operation) of P and Q.
"d of x is less than or equal to d of y" if and only if "d of x is less than or equal to d of y".
The location 'l' is in the set 'L' if the pair '(l, R)' is in the set 'S'.
The image of the set "ns" under the function "old_next_pair" is equal to the image of the set "(upd_incoming n ns)" under the same function "old_next_pair". In other words, applying the function "old_next_pair" to both "ns" and "(upd_incoming n ns)" results in the same set.
The lemma named "disjoint_dlverts_subset" assumes that the set "xs" is a subset of the set "ys" and that the set "ys" has disjoint dlverts. Under these assumptions, it shows that the set "xs" also has disjoint dlverts.
For a set N that belongs to the sets of M, N belongs to the null sets of the completion of M if and only if N belongs to the null sets of M.
The lemma named "ide_char" shows that "t" is an identity if and only if "t" is an arrow and "t" is an identity in "R".
The lemma named "card_inj_on_subset_funcset" assumes that both sets B and C are finite, and that set A is a subset of set B. It shows that the cardinality of the set of all functions from B to C that are injective on A is equal to the cardinality of C raised to the power of the difference between the cardinalities of B...
If there is a digraph homomorphism from digraph B to digraph C denoted by GG, and another digraph homomorphism from digraph A to digraph B denoted by FF, then there is a digraph homomorphism from digraph A to digraph C denoted by the composition of GG and FF. This is a reference to section 1.2 in the book 'Categories a...
The lemma named "list_eq_transfer" assumes that "A" is bi-unique. It shows that if two lists are equivalent under the relation "A", then the equality of the function "list_eq" applied to these two lists is preserved.
The conditional probability in model M for events A and B is greater than or equal to zero.
The peek operation on the conjunction of P and p is equivalent to the conjunction of the peek operation on P and p.
If 'p' and 'q' are elements of the carrier 'R', and the degree of 'p' and 'q' with respect to 'R', 'S', and 'X' is less than or equal to the successor of 'd', then the sum of the highest degree part of 'p' and 'q' with respect to 'R', 'S', 'X', and the successor of 'd' is equal to the highest degree part of the sum of ...
If a formula p is provable in a set of formulas G, then p is also valid in G under the equivalence relation.
The lemma named "s_delayed_has_real_derivative" in the context of Safe Distance Reaction, assumes that delta is less than or equal to time t. It shows that the composition of the ego2's position function and the time delay function has a field derivative at time t within the range starting from delta. This derivative i...
, in the context of a certain variable 'v', the proposition "If phi then (if psi then chi)" is logically equivalent to the proposition "If psi then (if phi then chi)".
"REL_OF_INTF_P J S" and "REL_OF_INTF I R".
If a set Us is independent over a field K, then the dimension of the span of Us over K is equal to the length of Us.
If the atoms of a regular expression 'r' are a subset of a set 'A', then the language of 'r' is a subset of the lists of 'A'.
The lemma named "cong_reflexive" assumes that "t" is an arrow. It shows that "t" is similar to itself.
The function rv of vector v equals zero if and only if the vector v itself equals zero.
If an extended natural number "i" is less than or equal to another extended natural number "j", then the product of "k" and "i" is also less than or equal to the product of "k" and "j".
If there is an undirected path in graph G from vertex v to vertex v' following path p, then both v and v' are elements of the vertex set V.
If an expression 'e' is typable in a type environment 'E' with a type 'D'', and for all 'd'' in 'D'', 'd'' is less than or equal to 'd', then 'e' is indistinguishable at level 'd'.
: Assuming that the relation "rs" and the environment "Γ" semantically evaluate the term "t" to "u", the term "t" is closed except for the free variables in the domain of "Γ", and the environment "Γ" is closed. It shows that the relation "rs" semantically evaluates the substitution of term "t" with environment "Γ" to "...
The semidirect product of any element 'x' and the semidirect unit is equal to 'x' itself.
The result of applying the function "inline11" to the bind of gpv and f, with s as an argument, is equal to the bind of the result of applying the function "inline12" to gpv and s, with a function that takes a result and returns either the application of "inline13" to f and a new state, if the result is of the form Inl...
: If for any predicate P, whenever P holds eventually for filter F, x is less than or equal to the supremum of the image of f over the set of elements satisfying P, and for any y, if y is less than or equal to the supremum of the image of f over the set of elements satisfying P whenever P holds eventually for filter F,...