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If for all functions 'f' and 'f'', when 'f' is a substitution function of 'sigma' equals 'f'', 'f'' is dsafe, and if for all functions 'f' and 'f'', when 'f' is a predicate substitution of 'sigma' equals 'f'', 'f'' is fsafe, and if 'nu' and 'omega' agree on the free variable set of 'sigma', then the adjoint of 'I' and ...
If r is related to r' under the reflexive transitive closure of the negation relation, then the concatenation of r and s is related to the concatenation of r' and s under the same relation.
The lemma named "less_one_multI" establishes that for any real numbers 'e' and 'x', if 'e' is less than or equal to 1, 'x' is greater than 0, and 'x' is less than 1, then the product of 'e' and 'x' is less than 1.
If a list 'xs' is not finite, then the reverse of 'xs' is undefined or bottom.
If there are outstanding volatile writes that are not owned by others in the list that starts with 't' and continues with 'ts', then there are also outstanding volatile writes that are not owned by others in the list 'ts'.
The lemma named "aristo_conversion3" assumes that "A Q B" and "B is not an empty set". If these conditions are met, it shows that "B I A".
For any time point 't' and time interval 'I', the condition 'True' is released by 'Q t' at time 't'' if and only if the interval 'I' is empty or 'Q' is true at the minimum time point in 'I'.
The lemma named "fair_surj" assumes that the function 'f' is surjective. It shows that the stream obtained by applying the function 'f' to the fair natural numbers stream is also fair.
If x is less than or equal to the extended non-negative real number representation of y, then the extended non-negative real number representation of x is less than or equal to y.
If "m'" is greater than zero, then the interval from "r'" modulo "m'" is not a subset of the interval up to "n".
The lemma named "eval_VcompNml" assumes that "t" and "u" are normal and that "t" is vertically sequential with "u". It shows that the evaluation of the vertical composition of "t" and "u" is equal to the product of the evaluations of "t" and "u".
The domain of the function f, when applied to the chamber system, is a subset of the codomain of the chamber system.
If "A" is an ideal in a commutative ring "R" with operations "add", "mult", "zero", and "unit", then the element "zero" is in "A".
If "c" is an element of the carrier "R", then for any polynomial "P" in the polynomial ring set "Pring_set R" over an index set "I", the scalar multiplication of "c" and "P" in the ring "R" is equal to the product of the indexed constant "c" and the polynomial "P".
If "t" is an element of the carrier "L", then the greatest fixed point of the function that maps every "x" to "t" is "t" itself.
If a system is a ground system and a pair (s, t) is in the single reduction relation step of the system, then the function symbols in term s and term t are a subset of the function symbols in the relation of the system.
The lemma named "tensor_op_almost_injective" is defined with four variables: 'a' and 'c' are linear operators on finite-dimensional Hilbert spaces 'a' and 'b', and 'b' and 'd' are linear operators on finite-dimensional Hilbert spaces 'c' and 'd'. The lemma assumes that the tensor product of 'a' and 'b' is equal to the ...
The lemma named "ord_fract" assumes that "a" and "b" are both in the set of non-zero elements of the p-adic numbers. It shows that the order of the fraction "a divided by b" is equal to the order of "a" minus the order of "b".
The image of the set of all extended real numbers greater than a certain value 'a' under the unary minus function is equal to the set of all extended real numbers less than the negative of 'a'.
If a function 'f' belongs to the set of permutations 'perm', then the set of all elements 'a' for which 'f(a)' is not equal to 'a' is finite.
The lemma named "discrete_part" assumes that "X" is discrete. It shows that "X" is a part.
The lemma named "delete_ran_inj_on" assumes two conditions: 1) The variable "j" is an element of the set that includes all integers from 0 to the successor of "n" (exclusive). 2) The image of the function "f" over the set of all integers from 0 to "n" (exclusive) equals the set of all integers from 0 to the successor...
If three conditions are met: 1) k is greater than 0, 2) the leaves of tree t are sorted in ascending order, and 3) the root order of tree t with respect to k is maintained, then the root order with respect to k is still maintained after inserting an element x into the tree t at position k.
, assuming for every agent 'i' in the set of all agents, the local non-satiation condition holds for the consumption set and the preference relation of 'i', and assuming that a competitive equilibrium with endowments 'P', 'X', and 'E' exists, then the sum of the dot product of 'P' and 'X i' for all agents 'i' minus the...
If function 'f' is 'L'-Lipschitz on set 'A' and function 'g' is 'M'-Lipschitz on set 'A', then the function that maps 'a' to the pair '(f a, g a)' is Lipschitz on set 'A' with Lipschitz constant being the square root of the sum of the squares of 'L' and 'M'.
If there is a transition from state "l" with clock valuation "u" to state "l'" with clock valuation "u'" in automaton "A", and if "u" belongs to the set "Z", then there exists a set "Z'" such that there is a transition from state "l" with clock zone "Z" to state "l'" with clock zone "Z'" in automaton "A", and "u'" belo...
The lemma named "per_col_mult_unitary" is defined as follows: Given a complex matrix A, assuming that A is an n x n matrix, A is unitary, D is an n x n matrix, D is a diagonal matrix, n is greater than 0, and there is a bijective function f between the set of all numbers less than n and itself. It is shown that the ...
The lemma named "is_partition_inj_map" assumes that "xs" is a partition and "f" is an injective function on the union of the sets in "xs". It shows that the result of mapping "f" over "xs" is also a partition.
A character 'c' is a digit3 if and only if the ASCII value of 'c' is also a digit3.
The function 'nat_mirror' is injective on the set from zero to the sum of 'l' and 'r'.
If "Racc" and "Rm" are single-valued, then the relation extended by "Rm" and "Racc" in "gen_igbg_impl_rel_eext" is also single-valued.
The lemma named "special_cosets_have_glbs" assumes that "X" and "Y" are elements of a set "P". It shows that there exists a "B" such that "B" is a greatest lower bound of "X" and "Y" in the superset of "P".
The lemma "comm2Aux" is defined with several fixed variables, including Psi, Psi_Q, R, K, N, R', A_R, Psi_R, P, M, xvec, P', A_P, Psi_P, and A_Q. It assumes the following conditions: 1. RTrans: Psi combined with Psi_Q leads to R transitioning to R' with label K carrying N. 2. FrR: The frame extracted from R equals a...
If a property Q holds for a natural number x, then the property Q also holds for the unique element that satisfies Q.
P_k is a path.
If the sum of p and q is reducible by a set F, then either p or q is reducible by the same set F.
In the context of an injective semiring homomorphism, if the homomorphism of vector v is equal to the homomorphism of vector w, then vector v is equal to vector w.
If the variable 'y' is not in the first element of the free variables of 't', and the pair '(b, S)' is not in the type variables of 't' or the type variables of 'τ', then binding the free variable 'y' to the type substitution of 'τ' with '(a, S)' replaced by 'Tv b S' in the i-th position of the type substitution of 't'...
: Assuming that: - B is defined as a list containing the decrement of the successor of arity by the sum of the length of fap and 5, the decrement of the successor of arity by the sum of the length of fap and 3, the goto of the successor of the length of fap, the increment of arity, and the goto of 0. - ft is defined a...
The lemma named "ntrs_stack_decomp_s" assumes two conditions. The first condition, labeled as "A", is that the triple ((u#s@r,c),ee,(s',c')) is in the set "ntrs" of the flowgraph "fg". The second condition, labeled as "EX", is a function that takes three arguments: v, u', and p. This function asserts that if s' equals ...
The error function of a real number x is less than or equal to zero if and only if x is less than or equal to zero.
If there is a binary relation A that relates the identity element of a multiplication operation in two structures, and if this relation A also preserves the multiplication operation in these structures, then this relation A preserves the product of lists in these structures. This is a rule for transferring properties b...
If a command "c" and state "s" result in a program "p" that terminates in time "t", then the time "p" is greater than 0.
Marking guards with a function 'f' on a dynamic call, which is characterized by an initialization 'init', a procedure 'p', a return 'return', and a continuation 'c', is equivalent to a dynamic call with the same 'init', 'p', and 'return', but with the continuation being a function that marks guards with 'f' on the resu...
If "ly" is equal to the layout of "aprog" and fetching "as" from "aprog" results in "ins", then the configuration instruction of "ly" and the start of "ly" as "as" for "ins" is equal to the Turing machine state of "aprog" at position "as".
If vector 'v' is in the cone field of 'x' and 'y', and 'r' is greater than 0, then 'v' is also in the cone field of 'r' scaled by 'x' and 'y'.
For any set A, given that the first elements of the pairs in the list xys are distinct, the image of A under the function defined by xys is equal to the union of the image of A under the relation defined by xys and the set containing the element None, if and only if A is not a subset of the set of first elements of the...
The reflexive transitive closure of the edge relation of a non-deterministic generalized Büchi automaton A is equal to the set of all pairs (p, q) where q is reachable from p in A.
The lemma named "is_splitting_blinD" assumes that "X" is a splitting bilinear function with respect to "Q" and "R". If this is true, it shows that "X" is equal to the difference of "Q" and "R", "Q" is invertible in the context of linear functions, the inverse of "Q" is a non-negative bilinear function, and "R" is also ...
The sum of a list 'ds' appended with an element 'd', when processed through the function 'sum_list'' with precision 'p', is equal to the result of truncating upwards with precision 'p' the sum of 'd' and the result of 'sum_list'' with precision 'p' applied to the list 'ds'.
The lemma named "psubst_LLq" assumes the following conditions: 1. Both "t1" and "t2" are in the set of atomic terms "atrm", and the first element of each pair in the set "txs" is also in "atrm". 2. The second element of each pair in the set "txs" is in the set of variables "var". 3. The second elements of the pairs in...
For a given matrix Q of type 'a (where 'a is a linearly ordered integral domain), Q is an orthogonal matrix if and only if Q is either a rotation matrix or a rotoinversion matrix.
The lemma named "null_sym_diff_transitive" assumes that the symmetric difference of sets A and B, and the symmetric difference of sets B and C, both belong to the null sets of a measure space M. It also assumes that sets A and C are measurable in the measure space M. Under these assumptions, it shows that the symmetric...
If "s" is equivalent to "t", then "lift s k" is equivalent to "lift t k".
If for every element 'a' in the set 'IF1set x', the function 'f' applied to 'a' equals the function 'g' applied to 'a', then applying the function 'IF1map' to 'f' and 'x' equals applying 'IF1map' to 'g' and 'x'.
The lemma "SMP_D_aux2" is defined with a fixed term 't' of a certain type. It assumes that 't' belongs to the set SMP M, and that 't' equals some variable 'x'. It also assumes that M has a finite SMP representation with respect to arity, Ana, and Γ. The lemma shows that there exists an 'm' in M and a substitution 'δ' s...
'nub_body' prime is equal to 'nub_body' double prime.
A space X is locally path-connected if and only if there is a neighborhood base of X such that for each neighborhood U in the base, U is an open set in X and U is path-connected in X. The second part states that a space X is locally path-connected if and only if for every open set U in X and for every point x in U, th...
If the length of the list "xs" is equal to the extended natural number "n", then the actions of the list formed by appending "xs" and "ys" is equal to the union of the actions of "xs" and the actions of "ys" shifted by "n".
The function "glist_member" with equality as its argument, when paired with the membership function, belongs to the identity relation when mapped from the identity relation to the function "br set distinct" mapped to the identity relation.
: Assuming that S and T are finite sets, c is a function from the set of integers from 0 to n (exclusive) to the set of n-dimensional vectors, T is a subset of the set of integers from 0 to n (exclusive), and a is a function from T to S to the set of n-dimensional vectors. Then, the determinant of the matrix of size...
If process P is less than or equal to process P', then the synchronization of process P with process Q over a set of events A is also less than or equal to the synchronization of process P' with process Q over the same set of events A.
The lemma named "minus_vars_is_semialg" assumes that "i" is less than "n" and "a" is in the carrier of the nth power of the p-adic numbers. It shows that the function "minus_vars i" is a semi-algebraic function of "n".
The validity of lists in set S and graph G is equivalent to the validity of lists in set S and the graph obtained by restricting G to S.
If P is like a simplex, then P has a bottom.
A Turing Machine Configuration (TMC) has a numerical result list without initial blanks if and only if the TMC has a numerical result list after adding blanks to the initial left and right tape. Specifically, if the tape is initially empty and contains a list of natural numbers, then after running the program, there ex...
If "it" is a valid iterator over a flattened list "l'", originating from list "l" and pointer "p", then the function "flatten_it_has_next" applied to "it" will return a result "r" such that "r" is true if and only if the list "l'" is not empty. This is guaranteed under the separation logic heap condition "is_flatten_it...
The lemma named "matchPres" is defined with fixed variables P and Q of type pi, and a and b of type name. It assumes that P is weakly early bisimilar to Q. It shows that if P is weakly early bisimilar to Q, then the process of matching a with b in P is also weakly early bisimilar to the process of matching a with b in ...
A pair of constant finite functions, one with value 'b' and the other with value 'c', is equivalent to a single constant finite function with a pair value '(b, c)'.
The lemma named "rep_in_terms_of_tab" demonstrates that the representation of 'r' is equal to the transformation of 'r' with respect to 'tab' in the context of the initial object 'T0'.
: 1. An empty list is equivalent to the operation "op_custom_empty". 2. The operation "op_list_empty" is equivalent to the operation "op_custom_empty". 3. The monadic operation "mop_list_empty" returns the operation "op_custom_empty".
For any tree 't', the set of fresh elements at an empty path is an empty set.
The lemma named "h_plus_OUT_le_IN" assumes that "x" is not equal to the source of "Delta". It shows that the outgoing flow of "h_plus i" at "x" is less than or equal to the incoming flow of "h_plus i" at "x".
"if it is necessarily the case that proposition A is equivalent to proposition B, then the obligation of A is equivalent to the obligation of B".
The theorem named "induced_Fiber_simp" assumes that "a" is an element of set "S". It shows that the induced class of "a" is equal to "alpha of a".
For any set A, if A is finite and the cardinality of A is the successor of n, then there exists a function f that maps from the set of all i less than or equal to n to A and is surjective to A.
The lemma named "rel_fun_Grp_copy_Abs" includes lifting syntax. Assuming that there is a type definition "Rep Abs A", it shows that the relation function of the group "A Abs" and the group "B g" is equal to the group of functions "f" such that the image of "f" in "A" is a subset of "B", with "Rep" mapping to "g".
The lemma for the next probability to End is defined as follows: if the state is Mix j, the next probability to End is 1 minus p_f. If the state is End, the next probability to End is 1. For all other states, the next probability to End is 0.
If "i" is an element of "I", and "f" is measurable in the context of "M i" and "N", and for any "m" and "n" that are elements of "I", the space of "M m" is equal to the space of "M n", then "f" is measurable in the supremum of "M i" over "I" and "N".
The union of two sets, s and t, is negligible if and only if both s and t are negligible.
The lemma named "wfPhi_f_poly_supp_c" is about well-formedness in the MiniSail language. It takes as input a type 'tau', a context 'Theta', a list of function definitions 'Phi', and a function type 'ft'. It assumes that the function 'f' with a certain function type (which includes a bound variable 'bv', a type 'x', a b...
A set S is open in the topology generated by the arbitrary union of the finite intersection of set B relative to set U if and only if S is an arbitrary union of the finite intersection of set B relative to set U.
The lemma is named "is_SNF_Smith_mxn_HMA". It fixes a matrix A of type 'a, where 'a is a commutative ring. The dimensions of the matrix are 'n and 'm, which are modular types. It assumes that the tuple (P,S,Q) is equal to the result of the function Smith_mxn_HMA applied to Smith_1x2, Smith_2x2, div_op, and A. It also a...
If x is greater than pi/2 and less than or equal to pi, then the cosine of x is less than zero.
The lemma named "order_divides_expdiff" establishes a condition for two natural numbers 'n' and 'a'. It assumes that 'n' and 'a' are coprime, meaning they have no common factors other than 1. The lemma then shows that 'a' raised to the power of 'd' is congruent to 'a' raised to the power of 'e' modulo 'n' if and only i...
The lemma "msg_after_marker_and_nonempty_implies_postrecording_event" assumes that: 1. There is a trace from the initial state to the final state. 2. A marker is in the set of messages on the channel identified by 'cid' at time 't' for process 'i'. 3. The marker is not the last message on the channel identified by 'ci...
If I is an ideal of ring R, and if a and b are elements of the carrier of R such that the sum of a and b is in I, and a is in I, then b must also be in I.
The lemma named "unsatReport" is defined with respect to a formula 'F', a literal trail 'M', and another formula 'F0'. It assumes that the 'InvariantImpliedLiterals' holds for 'F' and 'M', and that 'F' is equivalent to 'F0' according to the 'InvariantEquivalent'. It also assumes that the decisions for 'M' are empty and...
If the function "inv_tm_erase_right_then_dblBk_left_erp" holds true for the parameters CL, CR, and cf, and if there are no double blanks in CL and CR, then the function "inv_tm_erase_right_then_dblBk_left_erp" will also hold true for the parameters CL, CR, and the result of the function "step0" applied to cf and "tm_er...
The relation "cr_vrat" is right total. This is a rule for transfer in Isabelle.
A tree with vertices 'v', parent 'p', and path 'pth' is finite if and only if the edge tree with vertices 'v' and parent 'p' is finite.
The lemma "lselect_discard_start" assumes that for any index 'i' in set 's', 'i' is greater than or equal to 'k'. It shows that selecting elements from the list 'w' dropped by 'k' elements at the start, where the indices of the selected elements are those in 's' but shifted by 'k', is the same as selecting elements fro...
The lemma named "list_to_lpoly_coeff_nth" assumes that "i" is less than the length of the list "cs". It shows that the coefficient of the linear polynomial converted from the list "cs" at index "i" is equal to the element at index "i" in the list "cs".
If under the program P and environment E, the expression 'e' with state 's' can be reduced to a reference 'ref(a, Cs)' with state 's'', and if the program P allows a path from the last class in 'Cs' to class 'C' via 'Cs'', and 'Ds' is the concatenation of 'Cs' and 'Cs'', then under the same program P and environment E,...
The lemma named "monom_term_car" assumes that "c" and "x" are elements of the ring "R". It shows that the product of "c" and "x" raised to the power of a natural number "n" is also an element of the ring "R".
The lemma "MFR_in_msgs" assumes that the domain of messages is not empty and that "MFR msgs v" holds. It shows that there exists a "q" in the domain of messages such that "v" equals the message "q".
If a relation R is related and "mi" is less than or equal to the downward refinement of "m" with respect to R under E, then "mi" is less than or equal to the downward refinement of "m" with respect to R under E.
Binding a set "A" with a single element is equal to the set "A" itself.
The lemma named "resCases" has two case names, "cOpen" and "cRes". It fixes the variables Psi, x, P, alpha, P', and C. The lemma assumes the following: 1. Psi is in a transition relation with a restricted process P, which results in an action alpha and a process P'. 2. x is fresh for Psi. 3. x is fresh for alpha. 4. ...
If the graph of a function 'f' over a set 'X' is a runiq (a function with a unique range value for each domain element), then the domain of this graph is equal to 'X'.