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If 'i' is not equal to 'i1' and 'i' is not equal to 'i2', then the function 'f' with 'i1' replaced by 'v1' and 'i2' replaced by 'v2', evaluated at 'i', is equal to the function 'f' evaluated at 'i'.
: Assuming that 'nu' is an arrow in category D, it shows that 'G nu' is an arrow in category C. It also shows that the source of 'G nu' in category C is equal to 'G0' applied to the source of 'nu' in category D, and the target of 'G nu' in category C is equal to 'G0' applied to the target of 'nu' in category D. Further...
The property of having distinct IDs is invariant in the context of the system.
: Assuming that the heap 'h' is well-formed, has a valid type, and contains known pointers. Also, assuming that the component 'c' can be retrieved from 'h' using the pointer 'ptr', and the tree order 'to'' can be retrieved from 'h' using the pointer 'ptr''. If 'ptr'' is in the set 'c', and the component 'c'' can be re...
The lemma named "nonneg_infsum_le_0D_complex" is defined with respect to a function 'f' that maps an arbitrary type 'a to complex numbers. It assumes that the infinite sum of the function 'f' over a set 'A' is less than or equal to zero. It also assumes that the absolute sum of 'f' over 'A' is summable and that for any...
The fraction with numerator 'a' and denominator 'numeral b' is less than the fraction with numerator 'c' and denominator 'numeral d' if and only if 'a' multiplied by 'numeral d' is less than 'c' multiplied by 'numeral b'.
The glued generators of a word 'w' and a set 'C' is the set of all elements formed by concatenating the 'k'-th power of 'w' with 'g', where 'g' is an element of 'C' and 'g' is not equal to 'w'.
The lemma named "Re_det_sgn_congruence" assumes that the determinant of a matrix M is not equal to zero. It shows that the sign of the real part of the determinant of the congruence transformation of M and another matrix H is equal to the sign of the real part of the determinant of H.
If the image of a set A under a function h is a subset of B, then h is a homomorphism between the subgroup generated by A in group G and the subgroup generated by B in group H.
The lemma named "restrict_endo" assumes that "Y" is a Chamber Subcomplex and "f" maps "Y" to a subset of "Y". It shows that restricting "f" to the union of "Y" results in a Chamber Complex Endomorphism of "Y".
The constant polynomial of 1 is equal to 1.
The set of all integer numbers less than a given integer 'n' is infinite.
The statement consists of five parts: 1. The freshness of 'a' in the sequence of coded terms 'SeqCTermP' with variables 'vf', 's', 'k', and 't' is equivalent to 'a' being fresh in 's', 'k', and 't'. 2. The evaluation of the formula 'SeqCTermP' with variables 'vf', 's', 'k', and 't' in the environment 'e' is equivale...
The function f converges nonstandardly to L at point a if and only if the function (λh. f (a + h)) converges nonstandardly to L at point 0.
The lemma "obtain_fresh_variable'" assumes that the set "Gamma" is finite. It then obtains a variable "x" such that the pair "(x, tau)" is not in the union of the free variables of "t", the free variables of "u", and the free variables of "Gamma".
A set I of real numbers is bounded above if it is an interval, and there exist real numbers a and b such that a is less than or equal to b, a is in the set I, but b is not in the set I.
The lemma named "subst_apply_fv_empty" assumes that the set of free variables in term 't' is empty. It then shows that the set of free variables in the term resulting from the application of substitution 'σ' to 't' is also empty.
If "C" is not equal to "GMHole", then the mapping of "f" onto "C" will also not be equal to "GMHole".
If set A is a subset of a set that only contains the element x, then the sum of the function f over set A equals to the value of function f at x if x is in set A, otherwise it equals to 0.
If the heap S is not an empty map and the invariant G holds for S, then for a Dijkstra state with a priority queue pq, the implementation of the Dijkstra step for the graph G and the Dijkstra state results in a state S' such that the Dijkstra state of S' is a result and it is a valid Dijkstra step from S to S' in the g...
The lemma named "circline_type_pos_card_eq0" assumes that the type of a given circline, H, is greater than 0. It shows that the set of points on this circline, H, is empty.
The lemma named "free_in_quad_fm_helper" assumes that a variable is not in the variables of a, b, c, and d. It shows that this variable plus z is free in the function "quadratic_sub_fm_helper" with parameters var, a, b, c, d, F, and z.
The lemma named "onorm_scaleR_left" assumes that "r" is a bounded linear function. It shows that the operator norm of the function that multiplies "r" by "f" is equal to the product of the operator norm of "r" and the norm of "f".
The lemma named "domain_zero_sum" assumes that the sum of the domain block of x and the domain block of y equals zero. Under this assumption, it shows that both the domain block of x and the domain block of y must be zero.
The function "arl_length" corresponds to the function "RETURN o op_list_length" under the relation "is_array_list" to the power of "k", and this correspondence is in the arrow direction towards "nat_assn".
In the lemma for a cyclic group, assuming "a" is an element of the group "G", it shows that "a" is equal to the generator raised to the power of the exponent of the generator and "a".
The lemma "prv_dsj_cnj_distrib1" assumes that phi, chi1, and chi2 are formulas. It shows that if there is a proof of the implication of the disjunction of phi and the conjunction of chi1 and chi2, then there is also a proof of the conjunction of the disjunction of phi and chi1 and the disjunction of phi and chi2. This ...
If a natural number 'n' is greater than 0, then the division of the successor of 'm' by 'n' is equal to the successor of the division of 'm' by 'n' if and only if the successor of the modulus of 'm' and 'n' is equal to 'n'. Otherwise, it is equal to the division of 'm' by 'n'.
There exists a set F of natural number sets such that F is finite and the set shattered by F is infinite.
For any state 's', the local conformance of a list 'l' to a list 'L' in a context 'G' with local variables set to 'x' is equivalent to the local conformance of 'l' to 'L' in the same context 'G' with the same state 's'.
If a tau-execution move is valid for a certain configuration in a program P, thread t, expression e, heap h, and states s and s', then the same tau-execution move is also valid for the expression "e instanceof T" with the same configuration, program, thread, heap, and states.
If the following conditions are met: 1. A is an n x n matrix. 2. k is less than n. 3. The pair (lb, l) is in the set produced by the function "identify_blocks" applied to A and k. 4. The function "inv_upto" applied to jb, A, and k holds true. 5. The function "inv_from" applied to "uppert", A, and k holds true. 6. The ...
The lemma "list_map_autoref_update_dj" in the context of autoref rules, assumes the following: 1. The priority tag optimization is in effect. 2. As a side precondition, the key 'k' is not in the domain of the map 'm'. 3. The pair (k, k') is in the relation Rk and the pair (v, v') is in the relation Rv. 4. The pair (l,...
If the set V is a subset of Vs, the support of alpha and beta are subsets of the set of atoms of Vs, and if the hypotheses H entail the formula PfP (ssubst floor beta floor V V F) and the quote_all of p and V is a subset of H, then the hypotheses H entail the formula PfP (ssubst floor alpha OR beta floor V V F).
If S is a cring (a commutative ring), and if phi is a ring homomorphism from R to S, and if a is an element of the carrier of R, then the polynomial lift homomorphism from R to S applied to the polynomial X minus a raised to the power of a natural number n in the polynomial ring over R, is equal to the polynomial X min...
For any two structures R and S, if R is a ring and S is a commutative ring, and if a and b are elements of the carrier set of R, and c and d are elements of the carrier set of S, then the sum of a and the additive inverse of the sum of a and the additive inverse of b equals b, and the product of c and d in S equals the...
There is a reflexive transitive closure from g to h if and only if the pair (g, h) is in the reflexive transitive closure of the relation succs. This is denoted as "?L = ?R".
In the context of a functor, if there exists a universal arrow from an object 'c' to a functor 'F' at an object 'r' with a morphism 'u', and there is a morphism 'f' from 'r' to 'r'' in a category 'A', and 'u'' is the morphism obtained by applying the umap of 'F' at 'c', 'r', 'u', and 'r'' to 'f', then there exists a un...
The well-founded relation is an element of the set ZF.
The function "get_shadow_root" for a given element pointer reads from the locations specified by "get_shadow_root_locs" for the same element pointer, without modifying the state between the initial state 'h' and the final state 'h''.
Whatever the Spy knows from a list of events is a subset of what the Spy knows from a list of events where a communication from A to B with message X is added.
If a tuple (v,w,v') is an element of set E, then there is a distinct path or trail from v to v' that includes the edge (v,w,v').
If the tree 't' satisfies the invariant 'inv_max_hi', and if 'a' is an element in the set represented by the tree 't', then the high value of 'a' is less than or equal to the maximum high value in the tree 't'.
The bit count of the unary representation of a natural number 'n' is 'n' plus one.
A list with an added element at the beginning (LCons x xs) is finite if and only if the original list (xs) is finite.
The inner product of vector 'a' and the product of a real number 'm' and vector 'b' is equal to 'm' times the inner product of vector 'a' and vector 'b'.
The lemma named "ordered_fmap_distinct" assumes that "S" is a finite map. It shows that the list of first elements (keys) in the ordered finite map "S" is distinct.
The Bernstein polynomial of "j" and "p" between 0 and 1 is equal to the Bernstein polynomial of "j" and "p" with parameters 01.
If a coverage "cov" of "x" over "Q" and "G" holds, and if a satisfaction "sat" of the negation of the disjunction between the disjunction of "qps G" and the disjunction of the equivalence relation between "x" and "y" in the set of equations "eqs" of "x" and "G" holds for an interpretation "I" and a state "σ", then the ...
The boolean value of the disjunctive normal form (DNF) when prepending a list with element 'a' to 'ds' is equivalent to the boolean value of the DNF of a list containing only 'a' and the boolean value of the DNF of 'ds'.
: Assuming that the domain of \<pi> is a subset of the globally true propositions of \<phi>, and for every proposition \<chi> in the domain of \<pi>, the value of \<pi> at \<chi> is less than the maximum rank of \<Sigma> at \<chi>. Also assuming that the pair (delta \<Sigma>, initial \<phi>, (M_fin \<pi>, UNIV)) is ac...
If a certain move "exec_move" is executable with parameters ci, P, t, e, h, s, ta, h', and s', then the same move is also executable with an if-else condition where "e" is the condition and "e1" is the action when the condition is true, while "e2" is the action when the condition is false, with the rest of the paramete...
If "S" is an element of the states of a hierarchical automaton "A" and "SA" is an element of the composition function of "A" and "S", then the initial state of "SA" is an element of the initial states of the hierarchical automaton "A".
If a hypernatural number 'N' belongs to the set of infinitely large hypernatural numbers (HNatInfinite), then the star-embedding of 'N' in any linearly ordered semidomain is greater than zero.
The lemma "NilTrans" shows the following: 1. If the null process transitions to any process P' via any action alpha, then this is false. 2. If the process P prefixed with an input action b transitions to any process P' via an output action c, then this is false. 3. If the process P prefixed with an input action b tran...
If a set S is semialgebraic in the context of (n+n+l), then the intersection of the preimage of S under the function "augment n" and the carrier of the power (n+l) of the p-adic numbers is also semialgebraic.
If a set S is a subset of A, then the greatest lower bound (glb) of the set S in the complete lattice is an element of A.
The list obtained from the heap of a set of elements 'xs' is sorted.
: if for any positive number 'e', there exists a natural number 'M' such that for all natural numbers 'm' and 'n' greater than or equal to 'M', the norm of the difference between 'X m' and 'X n' is less than 'e', then 'X' is a Cauchy sequence. Here, 'X' is a function from natural numbers to a real normed vector.
If you flatten the sequence "call init p return result", the result will be the same sequence "call init p return result".
A single polynomial with variable x and degree 0 is equal to 0.
If a variable "q" is in the derivative of a tree automaton "A" with respect to a string "s" according to the function "ta_der'", then "q" is also in the derivative of the tree automaton "A" with respect to the same string "s" according to the function "ta_der".
The lemma named "tmsub_allpolys_npoly" assumes that 'a is a field with no characteristic (i.e., 'a::field_char_0). It shows that if all polynomials in 't' and 's' are non-polynomial (i.e., allpolys isnpoly t and allpolys isnpoly s), then all polynomials in the term substitution of 't' and 's' (i.e., tmsub t s) are also...
If a well-quasi-order is defined on a set A with property P, then for every index i, if the function f at index i is in set A, it implies that the function f is good with respect to property P.
The product case of Poisson density is in the Borel measurable space, which is the product of the Borel space and the count space over the universal set.
: given two tiny functors "GG" and "HH" from categories "AA" and "BB" to a category "CC" respectively, the comma category of "GG" and "HH" is also a tiny category.
A 3x3 matrix (consisting of vectors that represent the projective points p, q, and r) is not invertible if and only if the set of projective points {p, q, r} is collinear.
A process is maintained if the following conditions are met: 1. The pair (t, X) is in the process P's failures F, and the event sequence t is free of ticks, even when a tick is added to the set X. 2. The pair (t, X) exists such that t can be split into two sequences t1 and t2, where t1 followed by a tick is in the tr...
The natural order relation "natLeq" is an order filter for the set of natural numbers from 0 to less than n.
If a heap 'h' copies from address 'a' to address 'a'' through a list 'als' to produce a new heap 'h'' and a set of observations 'obs', and if a new heap element 'NewHeapElem' with address 'ad' and class type 'CTn' is in the set of observations, and the type of the address 'ad' is not 'None', then the type of the addres...
The theorem "red_dfs_correct" is defined with fixed variables v0 and u0 of type 'v. It assumes the following preconditions: 1. The red depth-first search invariant holds for the set E, U, reds, and onstack. 2. u0 is an element of U. 3. u0 is not an element of reds. The theorem shows that the red depth-first search o...
If an element "t" is in the multiset "S", then the union of multiset "T" and the difference of "S" and the singleton multiset of "t" is equal to the difference of the union of "T" and "S" and the singleton multiset of "t".
The lookup of a permutation of a polynomial mapping at a point x is equal to the lookup of the original mapping at the point f(x) if f is a bijection, otherwise it is equal to the lookup of the original mapping at x.
The lemma "Normalize Domain" assumes that "t" is an arrow. It shows that the domain of the normalized "t" is equal to the normalization of the domain of "t".
The lemma named "has_sum_superconst_infinite_enat" is about a function 'f' that maps any type 'a' to an extended natural number (enat). It assumes that for any 'x' in set 'S', the value of function 'f' at 'x' is greater than or equal to 'b'. It also assumes that 'b' is greater than 0 and that the set 'S' is infinite. U...
If the list "p" is not empty, "p" is a position of "t", and there is an embedding step at "p" from "t" to "s", then the priority of the embedding position "prio_emb_pos prio t s" is less than or equal to the priority of "p".
The lemma named "components_red_subset" assumes that "p" is reduced to "q" by the set "F". It shows that the set of term components of the keys of "q" is a subset of the union of the set of term components of the keys of "p" and the set of term components of the keys of "F".
: Assuming the following conditions: - The length of list L is greater than the variable 'var'. - The constant 'C' is not equal to zero. - The variable 'var' is not in the variables of 'b'. - The variable 'var' is not in the variables of 'c'. - The insertion of the default nth element of the list L updated with 'B/C' ...
The range of the composition function applied to a pseudo-hybrid automaton with statechart SA and data D is an empty set.
"v" unsignedly divides "w" if the unsigned natural number of "w" is equal to the product of the unsigned natural numbers of "v" and "u".
The lemma named "errMOD_igWlsDisj" assumes that the model "MOD" satisfies the property "igWlsDisj". It shows that if this is the case, then the error model "errMOD" derived from "MOD" also satisfies the property "igWlsDisj".
A term "t" minus a term "s" equals "t" if and only if the set of keys of "t" and the set of keys of "s" have no elements in common. Here, "s" is a function mapping to a natural number.
The lemma named "bounded_nat_bit_count_2" assumes that "y" is less than "l". It shows that the bit count of the natural number "y" in base "l" is equal to the floor logarithm base 2 of "l-1".
If under program P and environment E, the expression e with state s reduces to expression e' with state s', then under the same program P and environment E, the expression "throw e" with state s will reduce to "throw e'" with state s'.
In the context of complex square matrices, given a finite set I, if for every element i in I, the norm of u(i) equals 1, the dimension of the vector u(i) equals dimR, and p(i) is greater than or equal to 0, and if the sum of all p(i) equals 1, then the sum of the matrices p(i) times the rank 1 projection of u(i) for al...
: Assuming that "t" is an element of "T", and assuming that the initial condition is that "x" at time "t0" equals "x0", and assuming that "x" is continuous on the closed segment from "t0" to "t", and assuming that "x" has a derivative on the open segment from "t0" to "t" which equals the function "f" of "s" and "y"...
The lemma named "can_assoc" assumes that "a", "b", and "c" are identities. It shows that the canonical associator of "a", "b", and "c" exists.
The default operations implement the standard set.
If we have two categories, denoted by \<CC> and \<DD>, and two functors, denoted by \<FF> and \<GG>, that map between these categories. Also, we have two natural transformations, denoted by \<eta> and \<epsilon>, that map between the identity functor on \<CC> and the composition of \<GG> and \<FF>, and between the comp...
The product of a multiset created from a list is equal to the product of the list itself.
Reversing 'n' elements of a list 'xs' and appending it to another list 'acc' is equivalent to dropping the first 'length of xs - n' elements from the reversed list 'xs' and then appending it to 'acc'.
For any word 'w', the count of trailing zeros in 'w' remains the same even after taking the bit length of 'w'.
If there is an alpha step from state (l, Z) to state (l', Z') in automaton A, and there is a sequence of zero or more alpha steps from state (l', Z') to state (l'', Z'') in automaton A, then there is a sequence of zero or more alpha steps from state (l, Z) to state (l'', Z'') in automaton A.
"x" is less than or equal to its upper bound.
If 'v' is a valuation of 'K' and 'x' is an element of the valuation ring of 'K' with respect to 'v', then the 'n'-th power of 'x' in the valuation ring is equal to the 'n'-th power of 'x' in 'K'.
The lemma named "vset_diff_disj_eff_vs" is defined with fixed variables PROB, a, and vs. It assumes that the action 'a' is applied to the variable set of 'PROB' minus 'vs'. It then shows that the intersection of the domain of the second element of 'a' and 'vs' is an empty set.
The binomial coefficient "n choose k" is equal to the factorial of n divided by the product of the factorial of k and the factorial of (n - k), if k is less than or equal to n. Otherwise, it is zero.
If the schedule "sched" evaluates from global configuration "gc" to "gc'", then the length of the first element of "gc'" is equal to the length of the first element of "gc".
If "i" is less than or equal to "j", and "A" is an n by n matrix, then the matrix "A" is similar to the matrix obtained by applying the "step_1_main" function with parameters n, i, j, and A.
The lemma "make_first_column_positive_preserves_dimensions" demonstrates that the function "make_first_column_positive" preserves the dimensions of matrix A. Specifically, it shows that the number of rows in the resulting matrix is the same as the number of rows in matrix A, and the number of columns in the resulting m...