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A branch 'b' is a part of the branching structure consisting of 'l' and 'r' if and only if there exists a pair 'a' and 'b'' such that 'b' is the list consisting of 'a' followed by 'b'', and if 'a' is true then 'b'' is a branch of 'l', and if 'a' is not true then 'b'' is a branch of 'r'.
If the operations on a heap are deterministic, and if writing a value 'v' at address 'ad' and location 'al' in heap 'h' results in heaps 'h'' and 'h''', then 'h'' and 'h''' must be the same.
For a prime element 'p' and a non-zero element 'x', the multiplicity of 'p' in 'x' raised to the power 'n' is equal to 'n' times the multiplicity of 'p' in 'x'.
If the restriction of function F to set A is equal to the restriction of function F' to set A, and if x is an element of set A, then the value of function F at x is equal to the value of function F' at x.
The lemma Gateway_L5: assumes that "Gateway req dt a stop lose d ack i vc" is true, and that the message at time (Suc 0) for req is true, and that the message at time (Suc 0) for a is true, and that the message at time (Suc 0) for stop is true, and that the timestamp for lose is true, an...
If for all natural numbers 'n', the function 'f' applied to 'n' and some element 'ω' is measurable with respect to the measure space 'M' and the measure 'N' applied to 'n', then the function 'f' applied to 'n' and 'ω' in any order is also measurable with respect to the measure space 'M' and the product measure 'PiM' ap...
: given an irreducible element 'a' in a group 'G' and a well-factored element 'b' in 'G' with its well-formed factors 'fb', if 'a' and 'b' are in the carrier of 'G' and the set of 'fb' is a subset of the carrier of 'G', then the well-formed factors of the product of 'a' and 'b' in 'G' are 'a' followed by the factors 'f...
The lemma named "some_src_dom" assumes that "\<mu>" is an arrow. It shows that the source of the domain of "\<mu>" is the same as the source of "\<mu>".
The set of integers from a lower bound 'l' to an upper bound 'u' (not inclusive of 'u') is finite.
The lemma "one_before_two_aux" assumes that there exists a path, a reward, and an index such that "j" equals the next state after making a full observation of the drinks with a state of "Some 1", the reward "r", and the path "p". It shows that always, if the next state equals "Some 2", then the current state must be "S...
The lemma "list_all2_nths" assumes that for two lists 'xs' and 'ys', the property 'P' holds for all corresponding pairs of elements in the sublists of 'xs' and 'ys' obtained by taking the elements at the indices in set 'A' and the complement of 'A'. If these conditions are met, it shows that the property 'P' holds for ...
The lemma named "card_map_fset_le" shows that the cardinality (or size) of the finite set obtained by mapping a function 'f' over a finite set 'xs' is less than or equal to the cardinality of the original finite set 'xs'.
The operation "op_arl_empty_sz$N" is equivalent to the unprotected version of the operation "op_arl_empty_sz N".
If the modulus of 'n' and 'm' equals zero for natural numbers, then the modulus of the product of 'k' and 'n' with 'm' will also equal zero.
Under the assumption that there is an edge from the "Exit" node to any other node 'n' in the program 'prog', this is false. This means that there should be no outgoing edges from the "Exit" node in the program.
Multiplying by 1 is the same as the identity function in the context of a ring. This applies to any element of the ring.
A single element 'x' is not equal to the bottom element.
The lemma named "fimage_filter" is about a function 'f' and two structures 'S' and 'T', and a defined term 'fimage'. It assumes that 'f' is a function from 'S' to 'T' and defines 'fimage' as the image of 'f' in 'T'. It also considers a filter 'F' on 'S'. The lemma then shows that the image of 'F' under 'f' is a filter ...
The lemma named "LFP_same" assumes that the function "f" is monotonic. It then shows that the fixed point of the function "f" in the class of complete partial orders (ccpo) is equal to the least fixed point (LFP) of the function "f".
If a rule "r" takes a state "s" to a state "t" under the context of "\<Gamma>,\<gamma>,p" in the semantics of Goto, and if the state "s" is a decision state "X", then the state "t" must also be the decision state "X".
The floor logarithm of 'x' base 'n' is less than or equal to 'n'.
A list named "stack" is empty if and only if the stack is empty.
If "k" is greater than 0, then the weak since temporal operator applied to the interval "I" multiplied by "k" and the propositions "P" and "Q" at times "t1" and "t2" respectively, is equivalent to the weak since temporal operator applied to the interval "I" and the propositions "P" and "Q" at times "t1 * k" and "t2 * k...
If a τ-execution move is possible from state 's' to state 's'' in a given context 'ci', program 'P', thread 't', expression 'e', and heap 'h', then a τ-execution move is also possible in the same context, program, thread, and heap for the while-loop of the expression 'e' and another expression 'e''.
If the determinant of a matrix A is not equal to zero and the product of matrix A and matrix B equals matrix C, then matrix B is equal to the product of the inverse of matrix A and matrix C.
The function "list_all2" with the flipped predicate "P" applied to lists "xs" and "ys" is equivalent to the function "list_all2" with the predicate "P" applied to lists "ys" and "xs".
The lemma named "no_step" assumes that the image of the set containing 'a' under the relation 'A' is an empty set. It shows that 'a' is in the set of normal forms of 'A'.
The lemma named "interp_alt_factorization_reflect" assumes that zero is not in the first element of the set of the second element of "fctrs". It shows that the reflection of the polynomial interpreted by the alternative factorization of "fctrs" is equal to the interpretation of the factorization of "fctrs". Here, 'a is...
If there is a directed path from x to y in a transition system T, and ys is a forward list, and the intersection of the sets xs and ys is empty, and x is an element of the set xs, and y is an element of the set ys, then there exists an element y' in the set ys such that there is a directed path from x to the head of ys...
"x divided by 1 is less than or equal to x".
The integer value of 'x' raised to the power of 'i' is equal to the integer value of 'x' raised to the power of 'i'.
Substituting a variable 'x' with 'y' in the eta expansion of 'e' with 'n' is equivalent to the eta expansion of 'n' in the expression obtained by substitifying 'x' with 'y' in 'e'.
In the context of a directed graph, the lemma "dg_finite_pdigraph_dg_singleton" assumes that "j" is an element of the set "Vset" indexed by "α". It shows that there exists a finite partially directed graph indexed by "α", consisting of the set containing only "j", and with each element "i" mapped to the directed graph ...
The complex version of an integer power of a number 'x' raised to the power 'n' is equal to the integer power of the complex version of 'x', in a division ring that also supports complex division algebra.
The number of all edges between set X and set Y is equal to the number of all edges between set Y and set X.
The lemma named "rtype_simps" states the following: - The return type of "CounterImpl'value" is an integer. - The return type of "UndoCounter'save" is an integer. - The return type of "Dummy'dummy" is null. - The return type of "Counter'dummy" is null.
The function which takes 'x' and returns the inner product of 'x' with 'n' at index 'i' is linear.
If an operation 'f' is normally valid with precondition 'P' and postcondition 'Q', then the operation 'K_bind f x' is also normally valid with the same precondition 'P' and postcondition 'Q'.
The leading term of the polynomial mapping created from a list "xs" is equal to the maximum value in the list obtained by mapping the first element of each pair in the list "xs" after removing any duplicate or zero-value pairs.
The convex hull of the sum of sets B_i for all i in set A is equal to the sum of the convex hulls of the sets B_i for all i in set A.
This lemma fixes polynomials F, B, G, H and a natural number J. It defines df as the degree of F, dg as the degree of G, dh as the degree of H, and db as the degree of B. It also defines n as the sum of df and dg minus twice J, and f, b, g, h as the coefficients of F, B, G, H respectively. The lemma assumes that F plus...
The lemma "has_sum_mono_neutral" is defined with respect to a function 'f' that maps from a generic type 'a' to another generic type 'b'. The type 'b' is assumed to be an ordered commutative monoid with addition and a linear order topology. The lemma assumes that the function 'f' has a sum 'A' equal to 'a' and anothe...
If 'i' is less than 'j', then the function 'sliden i' maps any element from the set of natural numbers between 'i' and 'j' to the set of natural numbers 'k' such that 'k' is less than or equal to the difference of 'j' and 'i'.
The set of elements x, for which the difference between the values of functions M and N at x is greater than 0, is finite if the set of elements x, for which the value of function M at x is greater than 0, is finite. This is true for any functions M and N that map from any type 'a to natural numbers.
The application of the "Ifcmpeq" function with parameters b, G, pc, maxs, rT, and a pair (ST, LT) is true if and only if there exist types T1 and T2 and a stack ST' such that ST is the stack obtained by pushing T1 and T2 onto ST', b plus the integer value of pc is non-negative, and either T1 is a primitive type equal t...
The lemma "larger_implies_larger_heap" assumes that "a" is greater than or equal to "b". It shows that the larger heap is obtained from "a" compared to "b".
If a cryptographic message containing the agents A and Tgs_B, the session key SesKey, and a number T, encrypted with the shared key of Tgs_B, is part of what the spies can see in a given event sequence in the Kerberos IV protocol, and Tgs_B is not compromised, then the session key SesKey is not in the range of the shar...
If the following conditions are met: 1. For any value 'h' that is less than the length of array 'a', the bucket is okay with respect to the bucket hash code 'bhc', the length of array 'a', 'h', and the value at index 'h' in array 'a'. 2. For any value 'h' that is less than the length of array 'a', the list map is inva...
If "D" is a well-ordered set and "x" is the ordinal number of "D", then the restriction of the function "ODNmap D" to the segment "ODnods" of the ordinal number of "D" is a function from the segment "ODnods" of the ordinal number of "D" to the carrier of "D".
The opposite of the composition of a natural transformation and a semicategory morphism is equal to the composition of the opposite of the natural transformation and the opposite of the semicategory morphism.
The set of unsigned integers of length 'n' is equal to the range of the function that takes a number 'w' and returns the remainder of 'w' divided by 2 to the power of 'n'.
The lemma named "seq_lemma" assumes that there are two proven triples: the first one consists of p1, c1, and q1, and the second one consists of p2, c2, and q2. It also assumes that q1 is equal to p2. Under these assumptions, it shows that there is a proven triple consisting of p1, the sequence of c1 and c2, and q2.
The box cut iteration of "x" unioned with the star of "x", applied to "n(y)", is equal to the box cut iteration of "n(y)" multiplied by "x", unioned with the star of "n(y)" multiplied by "x", applied to "n(y)".
The result of the counterexample proof, "CEProof", is sound.
, assuming the list of ordered associative lists with total correctness (oalist_tc) "xs" is not empty, the second element of the minimum key-value pair in "xs" is equal to the value associated with the minimum key in "xs".
If 'y' is an element of the set obtained from the list 'xs' with 'x' added to the front, then 'y' is either equal to 'x' or 'y' is an element of the set obtained from the list 'xs'.
The fold function, when applied to the reverse of a list Ts and a type T, equals the Boolean type if and only if the list Ts is empty and the type T is the Boolean type. This is true in both directions, meaning that the Boolean type equals the fold function applied to the reverse of a list Ts and a type T if and only i...
The lemma "no_state_change_if_no_event" assumes that an event 'ev' causes the system to transition from configuration 'c' to configuration 'c'', and that the event 'ev' does not occur on process 'p'. It shows that the state of process 'p' in configuration 'c' is the same as the state of process 'p' in configuration 'c'...
The lemma establishes two things for a given expression 'e' of a certain type and a list of expressions 'es' of the same type. Firstly, it shows that the jump_ok function, when applied to the compiled version of 'e' and the integers 'n' and 'n' incremented by one, holds true. Secondly, it shows that the same holds true...
The lemma "tr_pairs_index" assumes that "G" is an element of the set produced by the function "tr_pairs" on "F" and "D", and that "i" is less than the length of "F". It shows that there exists an element "d" in the set "D" such that the "i"-th element of "G" is equal to the pair of the "i"-th element of "F" and "d".
If vertex v1 is adjacent to vertex v2, then both v1 and v2 are elements of the vertex set V.
The lemma named "ht_at_phl_cond" assumes two conditions: 1. The relation between alpha times u and tau times v holds for x and z. 2. The relation between alpha times u and tau times w holds for y and z. If these two conditions are met, it shows that the relation between alpha times u and the sum of tau times v times ...
The function "back" applied to "cl" and "n" is an injective function (a function that assigns a unique value to each unique argument) on the power set of the set from "brnL cl n" to "brn (cl!n)".
If a group G acts on a subgroup H, and if "a" is an element of the carrier of group G and "h" is an element of H, then the action of H on "a" is equal to the action of H on the product of "h" and "a".
The lemma named "bangPres" is defined with fixed variables P and Q of type pi, and Rel of type set of pairs of pi. The assumptions are as follows: - P is weakly late bisimilar to Q with respect to Rel (denoted as "P \<leadsto>\<^sup>^<Rel> Q"). - The pair (P, Q) is in the set Rel. - For any P and Q, if the pair (P, Q)...
The lemma named "is_fmapI" is an introduction rule. It assumes that for any elements 'a', 'b1', and 'b2', if the pair ('a', 'b1') is in the set 'M' and the pair ('a', 'b2') is also in the set 'M', then 'b1' must be equal to 'b2'. Under this assumption, it shows that 'M' is a function map.
If for any two elements 's' and 't' from the keys of a polynomial 'p', the degree of 's' is equal to the degree of 't', then the polynomial 'p' is homogeneous.
If there is a quotient map from set X to set X' using function f, then the composition of functions g and f forms a continuous map from set X to set X'' if and only if function g forms a continuous map from set X' to set X''.
The domain of the locals of the store 's' is a subset of the domain of the locals of the store of the second element of the first element of 'avar G i a s'' when it is valued 's'.
The polynomial of the product of a list of elements 'ps' evaluated at 'x' is equal to the product of the list obtained by mapping each element 'p' in 'ps' to its polynomial evaluated at 'x'.
Taking the signed bit of negative 1705 at the position of the third successor of 0 (which is 3) results in 7, for integer numbers.
The lemma named "flip_base_eq" establishes that for any base 'b' and any two elements 'x' and 'y', flipping 'x' and 'y' in 'b' results in 'b' itself.
If a program P satisfies that boolean expression b is semantically equivalent to boolean expression b', then for any state s that P holds, the boolean value of b in state s is equal to the boolean value of b' in state s.
The lemma l10_15: Assuming "A, B, C are collinear" and "A, B, P are not collinear", it can be shown that "there exists a point Q such that line AB is perpendicular to line QC and point P is on the opposite side of line AB from point Q".
If the output function 'c' equals 'NoMsg' and the initial streams are connected through the composition of the output function and the initial execution of the component stream (which is a function of the transition function, input, and 'c'), then the composition of the output function and the initial execution of the ...
The lemma states the following conditions: 1. F is the product of G and H. 2. The reduction of the formal power series polynomial G is g. 3. The reduction of the formal power series polynomial H is h. 4. The degree of G is equal to the degree of g. 5. The degree of H is equal to the degree of h. 6. The leading coeffic...
The lemma named "prefix_finI_item" assumes that "a" is equal to "b" and "u" is less than or equal to "v". It shows that the list "a # u" is less than or equal to the list "b # v".
The lemma named "mult_less_le_imp_less" assumes four conditions: 1) 'a' is less than 'b', 2) 'c' is less than or equal to 'd', 3) 'a' is greater than or equal to zero, and 4) 'c' is greater than zero. Under these assumptions, it shows that the product of 'a' and 'c' is less than the product of 'b' and 'd'.
If 'x' is not in the set of the root of 't1' and the successors of 't1' is the set containing the pair '(t2,e2)', then the union of the set of the root of 't1' and the path vertices of 't2' with 'x' is equal to the path vertices of 't1' with 'x'.
If two states, s1 and s2, are equal on the variables of boolean expression b, then the boolean value of b evaluated at s1 is equal to the boolean value of b evaluated at s2.
The minimum function is defined as a function that takes two natural numbers, x and y. If x is less than or equal to y, then the function returns x, otherwise it returns y.
If the Fr_2 property holds for a set F, then for all sets A and B, if the operation Op is applied to B, then the logical composition of A and B under the operation implies the negation of C.
If a program P can transition from an expression e with state s and heap b to a value v with a new state (h', l', sh') and heap b', then the program P can also transition from assigning the value of expression e to variable V with the original state s and heap b to a unit with the new state (h', l' with V mapped to v, ...
The difference in the number of sign changes in the list formed by 'x', 'y', and 'xs' and the list formed by the sum of 'x' and 'y', and 'xs' is greater than or equal to the difference in the number of sign changes in the partial sums of the list formed by 'x', 'y', and 'xs' and the partial sums of the list formed by t...
The lemma named "acyclic_dt_d_in_forest_modulo_equivalence" shows that if a forest modulo equivalence holds for x and d, then d transposed is acyclic.
If a function 'f' is primitive recursive and the length of a list 'xs' is equal to 'x', then the function 'f' will terminate when applied to the list 'xs'.
The lemma named "discounted_adapted" assumes that the stochastic process X is adapted to the Borel sigma-algebra F. It shows that the discounted value of the stochastic process X is also adapted to the Borel sigma-algebra F.
If D is an S-inductive set, and X1 and X2 are chains in D such that X1 is a subset of X2, then the supremum of X1 is less than or equal to the supremum of X2.
The signed bit representation of negative one (-1) remains negative one (-1) regardless of the number of bits (n) taken.
The sum of the function 'f' from 'k' to 'j', added to the sum of the function 'f' from 'l' to 'k', is equal to the sum of the function 'f' from 'l' to 'k', added to the sum of the function 'f' from 'k' to 'j'. This essentially asserts the commutative property of addition for these sums.
The lemma named "pderiv_deg_bound" assumes that "p" is an element of the carrier "P" and the degree of "p" in ring "R" is less than or equal to "d+1". It shows that the degree of the derivative of "p" in ring "R" is less than or equal to "d".
If the end of phase 1 in states s and s' for a process q is true, and the left side of invariant 3 for states s', processes p and q, and disk d is true, then the right side of invariant 3 for states s', processes p and q, and disk d is also true.
For any 'i' belonging to 'I', the value range of the object map of the projection of 'i' in the digraph 'I' and 'A' is a subset of the objects of 'A' at 'i'.
If 'r' is an element of the reflexive transitive closure of set 'A', then 'r' is also an element of the reflexive transitive closure of the universal set.
If vertex 'v' or vertex 'u' is not in the set 'V', then the shortest path from 'u' to 'v' is infinity.
If "sz" is less than the length of a certain type 'a, then the result of 2 to the power of "sz" minus 1 is less than 2 to the power of "sz". This is a simplification rule.
For any fixed values of 'v' and 'pre', the fourth projection of the addition of 'pre' to 'v' is equal to the fourth projection of 'v'.
If the product of x and z is less than or equal to the product of z and the transitive closure of y, then the product of the transitive closure of x and z is less than or equal to the union of the product of z and the transitive closure of y, and the product of the transitive closure of x and the bottom element.
In a program P with shared state sh, the bytecode expression "INIT C (Cs,b') ← e,b" is in a well-formed state if and only if b is false.
In the context of an additive group, if A is a coset of H and a is an element of the carrier set A, then a is an element of the sum of a and H in the group A.