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The subsequences of the mapped function 'f' applied to a list 'xs' is equal to the mapping of the function 'f' applied to the subsequences of the list 'xs'.
"it is always the case that the strong fairness of A with respect to v is equal to the strong fairness of A with respect to v".
For any time points t1 and t2 within a given interval I, the condition "P at t1 and not Q at t1 weakly until Q at t2" is equivalent to "P at t1 weakly until Q at t2".
A function 'f' is Lipschitz continuous on a set 'X' with Lipschitz constant 'L' if and only if the function '-f' (the negative of 'f') is also Lipschitz continuous on 'X' with the same Lipschitz constant 'L'. This is true for any function 'f' that maps from a metric space to a real normed vector.
If a set S is finite, then the function "to_nat_on S" is a bijection between the set S and the set of natural numbers less than the cardinality of S.
If the evaluation of conditions c1 and c2 in the context of i both yield True, then the evaluation of the conjunction (C_conj) of c1 and c2 in the same context i will also yield True.
If "k" is an element in the set derived from the stack "i₀", then "k" is less than "i₀".
: given two binary search trees 'l' and 'r', if every element 'x' in tree 'l' is less than every element 'y' in tree 'r', then the result of joining 'l' and 'r' is also a binary search tree.
If class 'x' is less than or equal to class 'y' within a certain class set 'cs', and class 'y' is less than or equal to class 'z' within the same class set 'cs', then class 'x' is less than or equal to class 'z' within the class set 'cs'. This is the principle of transitivity applied to class hierarchy.
In the lemma within the context of a functor, given that the triple [a, b', f] is an object in the comma category of the functor F down b, and b is an object in the category B, it can be shown that a is an object in the category A, b' equals 0, and f is a morphism from the object map of the functor F at a to b in the c...
If a table "tab" updated at "r" with "w" and then at "u" with "v", when accessed at "vn", returns some value "z", then there exists a value "z" such that if the table "tab" is first updated at "u" with "v" and then at "r" with "w", when accessed at "vn", it will return some value "z".
The evaluation of a monomial with a coefficient 'c' and a list of exponents 'e' followed by 'es', on a list of basis elements 'b' followed by other basis elements, is a function 'x' that evaluates the monomial with the same coefficient 'c' and the rest of the exponents 'es' on the rest of the basis elements, multiplied...
The application of the "Store" function with parameters "idx", "P", "pc", "mxs", "T\<^sub>r", and "s" is true if and only if there exist such "ts", "ST", and "LT" that "s" equals the pair "(ts#ST,LT)" and "idx" is less than the length of "LT".
If x is less than y (where x and y are integers), then x is not equal to y.
A set S satisfies a property φ under propositional entailment if and only if the intersection of S and the propositions of φ also satisfies φ under propositional entailment.
If a variable is not in the set of variables of any real multivariate polynomial f(i) for all i, then this variable is not in the set of variables of the sum of f(i) for i in the range from 0 to n-1, where n is a natural number.
For any polynomial 'p' and any natural number 'm' in a ring, the power 'm' of the polynomial map of 'p' at a point 'x' is equal to the polynomial map of 'p' at the same point 'x' raised to the power 'm'.
For all 'x' and 'y', if 'h x y' is greater than or equal to zero and there exists a constant 'c' such that the absolute value of 'f x y' is less than or equal to 'c' times 'h x y' for all 'x' and 'y', then the function that maps 'x' to the sum of 'f x y' for all 'y' in 'A x' is of the same order of growth (Big O) as th...
The set of variables in the head of the term zeta is finite.
If property P holds for an initial state s0, and for any state s, request q, and action a, if property P holds for state s then it also holds for the step from state s with request q and action a, then property P holds for the first element of all elements in the set defined by the probability mass function of the rand...
If "s" is a renaming of "d", then "s" is equal to "d".
The lemma named "list_greater_non_memb" is defined for an object identifier 'oid' of any linearly ordered type. It assumes two conditions: for every element 'x' in the set 'xs', 'x' is less than 'oid', and 'oid' is an element of the set 'xs'. If both these conditions are met, it concludes that this situation is impossi...
If the output "Outpts C A X" is in the set of events "evs" and the mode is insecure, then "X" is known to the Spy in the events "evs".
The lemma named "insert_after_ref" assumes that the list formed by concatenating 'xs', 'ref', and 'ys' contains distinct elements. It shows that the result of the 'insert_spec' function, when applied to this list and a pair consisting of 'oid' and 'Some ref', is equal to the list formed by concatenating 'xs', 'ref', 'o...
If a tuple (ta, xcp', h', (stk', loc', C, M, pc') # frs) is in the set of instructions executed by the program P at time t with heap h, stack stk, local variables loc, class C, method M, program counter pc, and frame stack frs, then the same tuple appended with an additional frame stack frs' is also in the set of instr...
If a cryptographic message containing two nonces (Nb and M), repeated twice, and the agents A and B, encrypted with B's shared key, is part of what the Spy knows in a given event sequence (evs), and if B is not in the set of compromised agents, and the event sequence is part of the Otway-Rees protocol, then there exist...
The lemma "run_one_step_basic_tee_local_result" assumes that running one step of the function with inputs d, i, (s,vs,ves,$Tee_local x16) results in (s', vs', res). It shows that the result, res, is either a normal result, represented as RSNormal r for some r, or a crash, represented as RSCrash e for some e.
, given two groups G and H, and two elements g and h where g belongs to the carrier of G and h belongs to the carrier of H, the inverse of the direct product of G and H with respect to the pair (g, h) is equal to the pair consisting of the inverse of g in G and the inverse of h in H.
For a function 'f' mapping from a complete lattice 'a' to another complete lattice 'b', if 'f' preserves the supremum in the dual order, then 'f' also preserves the least element in the dual order.
For any 'x', the sum of the star-transform of function 'f' at 'x' and the star-transform of function 'g' at 'x' is equal to the star-transform of the function that adds the results of 'f' and 'g' at 'x'.
The lemma named "incidence_mat_non_empty_blocks" assumes that "j" is less than "b". It shows that "1" is an element in the column "j" of the matrix "N".
The product of the complement of 'x' and the bottom element is less than or equal to the complement of the product of 'x' and the bottom element.
If sigma is not unprotected, then for all x in the set of non-variable labels of l, the type of x is not equal to sigma.
The lemma named "circline_type_x_axis" demonstrates that the type of the circular line along the x-axis is equal to -1.
Any hypercomplex number raised to the power of zero equals one.
If "w" is an element of "W", then the set containing "w" is an element of "P".
If the analysis of a term 't' equals to the pair (K,T), and 'k' is an element in the set K, then 'k' is a subterm of 't'.
The lemma named "MkIde_Cod" assumes that "a" is an arrow. It shows that making "a" an identity with the codomain of "a" is equivalent to the codomain of "a".
If x is greater than or equal to x0, then the product of gc2 and f2' x is less than or equal to f x.
Mapping a homomorphism over the division of two polynomials p and q is equivalent to dividing the mappings of the homomorphism over the polynomials p and q.
If F follows from f to g, and F also follows from g to h, then it must be the case that F follows from f to h.
For a given automaton A and a sequence of steps w, the pair (True#start A, rr) is in the steps of the star of A if and only if there exist sequences us and v such that w is the concatenation of us and v, every sequence in the set us is accepted by A, and there exists a sequence r such that the pair (start A, r) is in t...
If a match "m" is found for a given pair of parameters (\<beta>, \<alpha>) and a packet "p" in an access control list "a", then the following two conditions hold: 1. If the evaluation of the ternary match tactic with parameters \<beta>, p, and m results in an unknown value, and \<alpha> is a valid access control list ...
If the formulas phi1, phi2, chi1, and chi2 are all in the set of formulas, and if it is provable that phi1 implies chi1 and phi2 implies chi2, then it is also provable that the disjunction of phi1 and phi2 implies the disjunction of chi1 and chi2.
"x is a prefix of the reverse of y if and only if x is a suffix of y".
The lemma named "abss_reps_id'" is noted with the deletion of the simplification rule "R_G.cfg_onD_state". It assumes that the configuration "cfg" is a valid configuration in "R_G" and that the state "s" is in the set defined by the probability mass function of the action of "cfg". It shows that the abstraction of the ...
The lemma named "lookup_if2" assumes that the well-formedness condition "wfG P \<B> \<Gamma>" holds and that the tuple "(x,b,c)" is in the set representation of "\<Gamma>", with the additional condition that for all other tuples "(x,b',c')" in the set, "b'" must equal "b" and "c'" must equal "c". Under these assumption...
The lemma named "finite_ta_RegionsD" assumes that "A" is a finite timed automaton. It then provides or obtains a function "k" mapping from type 'b to natural numbers, and variables "v", "n", and "x". Under these conditions, it holds that "Regions'" is a function of the clock set of "A", "v", "n", and "x", the abstracti...
The lemma named "is_starfun_starext" assumes that "F" is a star-extension of "f". It shows that "F" is equal to the star-transform of "f".
If all polynomials in 't' and 's' are non-polynomial, then all polynomials in the term addition of 't' and 's' (denoted as 'tmadd t s') are also non-polynomial.
The action "m2" is enabled with respect to the variables x and y.
The lemma named "the_left_leI" assumes two conditions. First, for all 'k', if 'j' is less than 'k' and 'k' is less than 'i', then the 'k'-th element of 'xs' is not less than the 'i'-th element of 'xs'. Second, 'i' has a left element. Under these assumptions, it shows that the left element of 'i' is less than or equal t...
The lemma named "refines_asym1" assumes that "P refines A and is refined by Q" and "Q refines A and is refined by P". Under these assumptions, it shows that "P is a subset of Q".
: Assuming "ys" is a subset of the list "xs", and there exists a bijective function "f" between the set of all indices of "xs" and the set of all indices of "xs'", and for all indices "i" less than the length of "xs", the "i"th element of "xs'" is equal to the "f(i)"th element of "xs". Then, there exists a list "ys'" s...
The lemma named "assoc_agreement" assumes that "a", "b", and "c" are identities. It shows that the association of "a", "b", and "c" in category "C" is equal to the association of "a", "b", and "c".
The lemma "nort_null_comp" assumes the following conditions: 1. "pret" is a directed acyclic graph (DAG) with root "p" and edges "low" and "high". 2. The binary decision tree (bdt) of "pret" with variable "var" is "prebdt". 3. "nort" is a DAG with root "repc no" and edges "repb applied to low" and "repb applied to hig...
The "set_iterator_emp" is a valid set iterator for an empty set.
The lemma "elem_index_less_varnum" assumes that "x" is an element of the keys of "t". It then concludes that either "x" is an element of "X" or the element index of "x" is less than the variable number of "X" in "t".
The lemma "GF_advice_sync" is defined with two fixed variables: "index" which is a function mapping natural numbers to natural numbers, and "formula" which is a function mapping natural numbers to a less than or equal to normal form (ltln) of any type 'a. The lemma assumes that for any natural number 'i' less than 'n...
If the following conditions are met: the well-formed program 'wfmd P' is a well-formed program, 'A' is widened to 'B' in program 'P', 'A' is not a null type, and 'A' is a type in program 'P', then the pair (A, B) is in the reflexive transitive closure of the relation 'widen1 P'.
The lemma named "psi_naturality" assumes three conditions: 1. There is a morphism 'f' from 'x' to 'x' prime in category 'C'. 2. There is a morphism 'g' from 'y' prime to 'y' in category 'D'. 3. There is a morphism 'h' from 'y' to 'Gx' in category 'D'. It then shows that the composition of 'f' and 'psi x h' in categor...
, given the assumptions that a complex number 'u' is in the unit disc and is a real number, and that for any complex number 'a' and 'u' in the unit disc, if 'a' is real and its absolute value is less than 1, then the property 'P' holds for the point 'u' transformed by the Blaschke product of 'a', then the property 'P' ...
The lemma named "chinese_remainder_nat" is defined with three functions: A, which is a set of type 'a; m, which is a function that takes an element of type 'a and returns a natural number; and u, which is a function that takes an element of type 'a and returns a natural number. The lemma assumes two conditions: first, ...
The lemma named "mem_set_1" assumes that "a" is a member of the list "l". It shows that "a" is in the set of elements that make up the list "l".
If the conditions for "exe_osig" are met for the pair (sub, tcs), then if "exehas_sort" is true for the pair (sub, tcs), type T, and sort S, it implies that "has_sort" is also true for the translated "osig" of the pair (sub, tcs), type T, and sort S.
The lemma named "points_index_0_left_imp" assumes that the index of B in ps is 0 and that b is an element of B. It shows that ps is not a subset of b.
The function "f3" when applied to an empty list or a list containing "A" returns an empty tuple.
A string "u" belongs to the set of all strings composed solely of the element "a" if and only if the set of elements in "u" is a subset of the set containing only "a".
If the list "al" is not empty and "s1" is equal to the target of the last element in the trace of "s" and "al", then the trace of "s" and the concatenation of "al" and "al1" is equal to the concatenation of the trace of "s" and "al" with the trace of "s1" and "al1".
If the length of the list "xs" is equal to the length of the list "XS", then the list formed by appending "xs" to the list consisting of "x" followed by "ys" is in the set "Joints" of the list formed by appending "XS" to the list consisting of "X" followed by "YS" if and only if the list formed by appending "xs" to the...
The measure of the image of a function "f" applied to a probability measure "p" is equal to the measure of the preimage of "f" under the same probability measure "p".
An empty list is not parallel to any list 'x'.
The lemma named "ENR_linear_image_eq" is about a function 'f' that maps from one Euclidean space to another. It assumes that 'f' is a linear function and is injective (i.e., it never maps distinct elements to the same element). The lemma shows that the Euclidean Neighborhood Retract (ENR) of the image of 'f' applied to...
In the context of a category, the lemma "cat_CId_is_iso_arr'" can be introduced with the following conditions: - "a" is an object in the category "CC" - "CC'" is identical to "CC" - "b" is identical to "a" - "c" is identical to "a" Under these conditions, it can be shown that the identity morphism of "a" in the categ...
The insertion of a function 'f' into a multivariate polynomial 'mpoly_of_poly' with parameters 'i' and 'p' is equal to the polynomial 'p' with the function 'f' applied to 'i'.
The lemma named "subterm_funs_term_in_type" assumes that the term "t" is well-formed, the function "f" with type "T" is a subterm of "t", and the type of the function "f" with type "T" is the composition of "f" and the map of "T" under the typing function "Γ". It shows that "f" is in the set of function symbols of the ...
The lemma named "add_new_point_rep_number" assumes that "p" is not in the set "V". It shows that the repetition number of "p" in the set obtained by adding "p" to the blocks "bs" is equal to the size of the set of blocks "B" that are in "bs".
If a valid abstraction A exists for X and k, and Z is a subset of V, then if A leads from the closure of Z at state l to Z' at state l', there exists a Z'' such that A leads from Z at state l to Z'' at state l' in an alpha transition, and Z' is a subset of Z''.
If memory state m1 is a subset of memory state m2, then the read-only part of memory state m1 is also a subset of the read-only part of memory state m2.
Removing an element 'a' from the finite set created from a list 'A' is equivalent to creating a finite set from a list that is filtered to exclude the element 'a'.
If three conditions hold - the first being equality and the other two being some relation R - then these conditions are preserved under the transformation defined by the "if\<^sub>T" function in the State Monad Extension. This is a rule for transferring properties between different representations of data.
The lemma named "map_idx_nth" assumes that "i" is less than the length of the list "xs". It shows that the "i"-th element of the list obtained by mapping the function "f" over the list "xs" with starting index "n", is equal to the result of applying the function "f" to the "i"-th element of "xs" and the sum of "n" and ...
The lemma "ord_adds" assumes that "s adds t". It shows that "s is less than or equal to t".
The function "circuit_checker" for a board "b" and a sequence of positions "ps" is equivalent to the function "knights_circuit" for the same board "b" and sequence of positions "ps".
If "Ks" is good and "K" is an element of "Ks", then the agent "K" is not in "bad".
In the lemma named "pcat_prodfunctor_proj_is_functor'" within the context of a product category, given the assumptions that 1) the functor SS maps from the product category of AA over the index set I to the category DD, 2) the object c is an element of the product category of AA over the index set J, 3) the index set ...
For any complex number 'z', the conjugate of 'z' is zero if and only if 'z' itself is zero.
For all 'm' and 'a', if 'm' at 'a' equals None, then for all 'c', if 'm' at 'a' mapped to 'c' is at least free 'n', then for all 'b' and 'd', if 'a' is not equal to 'b', 'm' at 'b' mapped to 'd' is at least free 'n'.
If 'b' is an object in category B and the functor applied to 'b' equals the functor restricted to 'b', then the functor restricted to 'b' is a tm_functor from the functor applied to 'b' to category A.
The theorem 'wls_vsubstEnv_substEnv_compose1' assumes that 'rho' is a well-sorted environment and 'Y' is well-sorted with respect to the sort 'ys'. It shows that substituting 'Y' for 'y' in 'rho' according to sort 'ys', and then substituting 'y1' for 'y' according to sort 'ys', is equal to substituting 'Y' after 'y1' h...
The upper adjoint of the inverse gorder of A and the inverse gorder of B with respect to function f is equal to the lower adjoint of B and A with respect to the same function f.
The lemma named "dom_prog_ordE" assumes that "x" is dominated by "y" in the program order. It then concludes that either "x" leads to a non-termination exception when evaluated in the context "h", or "x" and "y" evaluate to the same result in the context "h".
If a relation "rel_resumption" holds between two resumptions "r1" and "r2" under conditions "A" and "B", and "r2" is not in a "Done" state, then the same relation "rel_resumption" will hold between the resumptions resulting from resuming "r1" and "r2" with the same input "inp".
The lemma named "vsubst_subst_compose1" assumes that "X" and "Y" are good. It shows that substituting "Y" for "y" in "X" under the context "ys", and then substitifying "y1" for "y" under the same context, is equivalent to substituting "Y" (which has already had "y1" substituted for "y" under the context "ys") for "y" i...
If the compilation of a command 'c' is correct, then the command 'c' is a valid predicate for the first and last elements of the command sequence 'cfs'.
The codomain of the second projection functor from the product category of categories "A" and "B" is "B".
If 'j' is less than or equal to the length of list 'xs', then the length of the sublist of 'xs' from index 'i' to 'j' is equal to 'j' minus 'i'.
The function "af" applied to a formula "phi" and a sequence "w" appended with a state "nu" is equivalent to the function "step" applied to the function "af_UU" applied to the unfolding of "phi" and the sequence "w", and the state "nu".
If set R is finite, then the set of partial derivatives of R is also finite.
The function "is_generalisation_of" in the context of "Store_Reuse" is equivalent to the function "is_generalisation_of" with the same parameters x, xa, xb, and xc.