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The lemma named "law_restrict_iso_imp_eq" assumes that the identity function is a ring isomorphism between the law restrictions of two sets A and B, and that both A and B are rings. It then shows that the law restriction of A is equal to the law restriction of B.
: Assuming the following conditions: - For every transaction T in the set P, T is an admissible transaction. - For every pair (a,b) and (c,d) in the set TI, if b equals c and a is not equal to d, then the pair (a,d) is in the set TI. Also, for every pair (a,b) in the set TI, a is not equal to b. - For every term t in ...
The lemma named "mat_det_adj" shows that the determinant of the adjugate of a matrix A is equal to the conjugate of the determinant of matrix A.
The lemma named "no_primitives_matchNot" assumes that "m" does not have a primitive. It shows that "m" matches the negation of a match if and only if "m" does not match "a" and "p".
If for all 'k' less than or equal to 'n', 'J k' is an ideal of Ring 'R', and 'a' is in the ideal n-product of Ring 'R' and 'J', then for all 'k' less than or equal to 'n', 'a' is in 'J k'.
If a function 'f' is equal to an undefined function with 'Y' set to 'y', then 'y' is equal to the value of function 'f' at 'Y'.
The lemma named "L_eps_free_parallel_closure_reg" assumes that the transition system "ta" of "A" is epsilon-free. It shows that the language of the transitive closure of the epsilon-free regular expression of the parallel closure of "F" and "A" is equal to the language of the parallel closure of "F" and "A".
The lemma named "executable_edgeSet'" assumes that the program "prog" maintains the global state invariant "gState_inv" and the program invariant "program_inv". It shows that for any executable state "g" of the program "prog", it is less than or equal to the specification function "SPEC" that for all pairs of edge "e" ...
If you filter a list that starts with an element 'a' followed by a list 'xs' based on a predicate 'P', it will return a list that starts with 'a' followed by the filtered list of 'xs' if 'a' satisfies the predicate 'P'. Otherwise, it will just return the filtered list of 'xs'.
If a pair "ab" is in the set produced by the function "polychain_of" applied to "x" and "xs", then the difference of the second element and the first element of the pair "ab" is in the set "xs".
The failure of mapping an entry in the array at a given index 'i' with a function 'x' on a heap 'h' using the Array_Time structure is equivalent to the condition where 'i' is not less than the length of the heap 'h' at position 'p' in the Array_Time structure.
If a relation "R" is a congruence, then the infimum of the relation "R" and its retraction is also a congruence.
The expected value of a constant function, when mapped over a probability mass function of real numbers, is equal to the constant itself.
Addition is associative, meaning that the sum of X with the sum of Y and Z is equal to the sum of the sum of X and Y with Z.
The lemma named "inversion_id_iff_on_unit_circle" shows that the inversion of a point 'a' is equal to 'a' if and only if 'a' is on the unit circle.
: Assuming that the computation tree (cpt) of the free variables (FV) of a formula phi in the environment env and path list pi_l exists, and the computation tree of the free variables of the same formula phi in a different environment env1 and path list pi_l1 also exists. Also assuming that the environments and path l...
The lemma named "list_of_map" assumes that the domain of the substitution (subs) is finite. It shows that there exists a list (l) such that the map of this list is equal to the substitution.
The length of the second element of the pair returned by the function "simple_cg_closure_phase_2_helper" with inputs x1 and xs is less than or equal to the length of xs.
The lemma named "r_inv_of_ZX" shows that the product of ZX and the conjugate transpose of ZX equals the 2x2 identity matrix.
If a polynomial 'a' is congruent to a polynomial 'b' modulo 'm', then the result of multiplying a scalar 'k' with 'a' is congruent to the result of multiplying 'k' with 'b' modulo 'm'. This is under the condition that 'b' is a field with a greatest common divisor (gcd).
The natural number division of 'a' by 'b' is equal to the maximum of 0 and the floor of the division of 'a' by 'b' when both 'a' and 'b' are considered as real numbers.
The equivalence check between the first-order existential quantification of a Presburger arithmetic equation (with no specific variable, comparing 0 and 42) and the existential quantification of an equation (comparing the first element of a list with 42 and 0) is true, given an abstract index of (0, 0).
The lemma named "typeof_f32" assumes that the type of variable v is T_f32. It shows that there exists a constant c such that the variable v is equal to the constant float 32 c.
A number 'r' is less than or equal to zero if and only if for all 'n', the binary expansion of 'r' in the range 0 to 1 equals zero.
The lemma named "ncoplanar_perm_22" assumes that points A, B, C, and D are not coplanar. It shows that points D, C, A, and B are also not coplanar.
The sum of zero and zero equals zero, in the context of hereditarily finite sets.
The lemma named "next_fresh" assumes that a pair (r, h') is the result of allocating x in h. It shows that r is not present in h.
If F and G are groups, f is a group homomorphism from F to G, and X is in the right coset space of F with respect to the kernel of f in F and G, then the image of X under the restriction of f to F and the image of f is equal to the image of X under the restriction of f to F and G.
, under certain conditions, the function L_10_5_τ is a category cone. The conditions are as follows: - The functor KK maps category BB to category CC. - The functor TT maps category BB to category AA. - The object c is an element of the set of objects in category CC. - The natural transformation upsilon' is equal to...
The inorder traversal of the mirror image of a tree 't' is equal to the reverse of the inorder traversal of the original tree 't'.
: If the following conditions are met: 1. B says to A an encrypted message containing two nonces (NA and NB) using A's public encryption key, and this communication is part of the set of events 'evs'. 2. For all entities C, A does not say to C an encrypted message containing the nonce NB using C's public encryption ke...
If the following conditions are met: 1. The set 'p' is a subset of the set of states 's' such that if 's' is in 'b' then 's' is in 'w', and if 's' is not in 'b' then 's' is in 'w''. 2. The parallel composition of 'w' and 'c1' leads to 'q'. 3. The parallel composition of 'w'' and 'c2' also leads to 'q'. Then, the pa...
The lemma named "next_candidates" assumes that the result of the function "next_candidates" with input "n" is "(m,ps)", and that "n" satisfies the "candidate_invariant". It shows that: 1. "m" also satisfies the "candidate_invariant". 2. The list "ps" is sorted. 3. The set of all "i" that are prime, greater than or equ...
For any string "w", "w" is accepted by the non-deterministic finite automaton (constructed from the regular expression "r") if and only if "w" belongs to the language defined by the regular expression "r".
In the context where 'is_functor' holds, there is a lemma named 'cf_HOM_snd_is_functor'[cat_cs_intros]. Given the assumptions that 'beta' is a member of the class of all sets, 'alpha' is an element of 'beta', the category 'CC'' is the opposite category of 'BB', and 'DD' is the category of functors from 'AA' to the cate...
If a function 'Y' maps from the space 'M' to the space 'N', and 'Y' is a discrete function, and for every 'r' in the intersection of the range of 'Y' and space 'N', there exists a set 'A' in the sets of 'N' such that the intersection of the range of 'Y' and 'A' is equal to the set containing 'r', then the explicit cond...
If the invariant "S" holds, then the function "iterateoi S" is a linear order set iterator for the abstract state "S".
The lemma named "integrableI_bounded_set" is about a function 'f' from 'a' to 'b', where 'b' is a Banach space with a second countable topology. It assumes the following: 1. 'A' is a measurable set in the measure space 'M'. 2. The function 'f' is Borel measurable in 'M'. 3. The measure of 'A' in 'M' is finite. 4. Almo...
The default matrix of dimensions d by n belongs to the carrier matrix of dimensions n by n.
The cost of the pair (a,c) is c and the result of the pair (a,c) is a.
The lemma named "stereographic_circline_inv_stereographic_circline" shows that the composition of the function "stereographic_circline" and its inverse "inv_stereographic_circline" is the identity function.
The omega power of a variable 'x' multiplied by the star power of the same variable 'x' is equal to the omega power of the variable 'x'.
The nonstandard version of a number 'k' is approximately equal to zero if and only if 'k' is equal to zero.
The interval of the sum of 'a' and 'b' is equal to the sum of the interval of 'a' and the interval of 'b'.
The pre-post condition from -p to -r is less than or equal to the composition of the pre-post condition from -p to -q and the pre-post condition from -q to -r.
: assuming the first element of the pair 's' is equal to 'x'', it can be inferred that there exists an 'x' and 's' such that 's' is equal to the pair '(x, s)' and 'x' is equal to 'x''.
If term 's' is okay according to set 'Σ' and term 'u' is okay according to set 'Σ', then the term resulting from the substitution operation 'subst_bv2' on 's' with level 'lev' and 'u' is also okay according to set 'Σ'.
The mask function of a number 'n' is equal to the integer representation of the mask function of 'n'.
The identity relation is less than or equal to the upward closure.
The inverse of the swap operation is the swap operation itself.
For all lists y_ty_list, vds, vars, vs, and vars', if the lengths of y_ty_list, vds, vars, and vs are equal to the length of vars', then mapping a function that extracts the fourth element from each quintuple in a list, which is created by zipping together the first elements of y_ty_list, the results of applying a func...
Lemma: measurable_abs_powr is measurable. Fix a real number p. Assume that the function f is borel measurable on the space M. Then, the function that maps x to the absolute value of f(x) raised to the power of p is also borel measurable on the space M.
If "C" is a class in "P" and the size of "ST" is less than "mxs", then a new instance of class "C" can be type-checked with the initial type state "ST" and the resulting type state is "Class C" prepended to "ST". Both initial and resulting type states have the same exception table "E" and the same set of locked objects...
If a function 'f' has a field derivative 'y' at 'F', then 'f' also has a vector derivative 'y' at 'F'.
The lemma for the measure of a set in the SPMF (Stochastic Process Monadic Formalism) is defined as follows: If the set A is finite and not empty, then the measure of the SPMF of the set A is equal to the measure of the PMF (Probability Mass Function) of the set A. Otherwise, the measure of the SPMF of the set A is equ...
The precondition of "q" for the bottom element is equal to the bottom element.
If a matrix A is in reduced row echelon form up to column k (as per the assumption "r"), then for any row i, if that row is not a zero row up to column k, the first non-zero entry in that row (i.e., the entry in the column which is the least k such that A's entry in row i and column k is not zero) is 1.
If a set 'x' can be arranged in a linear order of three elements, then it can be oriented in a sequence of one or two elements.
In the context of a group, the lemma "generate_incl" assumes that "H" is a subset of the carrier "G". It then shows that the generated group from "H" in "G" is also a subset of the carrier "G".
The lemma "emeasure_join" assumes that the set of events in probability space M is equal to the set of events in the subprobability space N, and that A is an event in the probability space N. It shows that the outer measure of event A in the joined probability space of M is equal to the integral of the outer measure of...
If a certificate "cert" passes the check under the conditions of a certain environment "G", maximum stack "mxs", maximum registers "mxr", and a certain number "n", then the certificate "cert" is considered okay under the conditions of the environment "G", maximum stack "mxs", maximum registers "mxr", with an error stat...
If the following conditions are met: 1. The strong tricolour invariant holds for a state 's'. 2. For all roots 'r' in the mutator roots of 's', 'r' is marked in 's'. 3. The mutator's ghost honorary root in 's' is empty. 4. There are no pending mutations for the mutator 'm' in 's'. 5. The valid references invariant hol...
The lemma named "slpa_cases" consumes 4 assumptions and has two case names: "intra_path" and "return_intra_path". It assumes that "as" is a same level path auxiliary of "cs" and that updating "cs" with "as" results in an empty list. It also assumes that every element "c" in the set "cs" is a valid edge and every elemen...
The lemma named "inputPres" is defined with fixed elements P and Q of type pi, a and x of type name, and Rel of type set of pairs of pi. It assumes that for every element y, the pair formed by replacing x with y in P and Q is in Rel, and that Rel is equivariant. It shows that the late weak simulation from "a input on x...
The set of atoms of a clause C is empty if and only if the clause C itself is empty.
If 'x' is in the set of all acquired elements in the sequence 'sb' up to the point where predicate 'P' holds, then 'x' is in the set of all acquired elements in the entire sequence 'sb'.
Scaling a value 'x' by a real number 'r' is equivalent to multiplying 'x' by the octonion representation of the real number 'r'.
Hiding a process P with a set of communication events cs is equivalent to first hiding the process P with the set of communication events cs and then applying the relation of the Skip process.
The lemma named "cl_altdef" assumes that "X" is a subset of "carrier". It shows that the closure of "X" is equal to the union of all subsets "Y" of the power set of "carrier" such that "X" is a subset of "Y" and the rank of "Y" is equal to the rank of "X".
For any integer 'x', the division of the sum of 1 and twice 'x' by 2 equals 'x', and the division of the sum of 0 and twice 'x' by 2 also equals 'x'.
A tree "t" is fully zero if and only if "t" is empty.
The lemma named "lossless_gpv_inline1" assumes the following conditions: 1. A tuple consisting of an output, a response process value, and another response process value is in the set of sub-probability mass functions generated by inlining a callee with a generative probabilistic value and a state. 2. The input is in...
The lemma named "toVariable_channels_inv'" demonstrates that if the function "toVariable" with parameters g, p, and v equals the tuple (a, b, c), then for every element x in the set c, the function "channel_inv" applied to x is true.
The computational relation of the composition of R1 and R2, and the composition of R0 and R1, is equivalent to the computational relation of the composition of R0 and R2, for the standard composition function and the monadic composition function comp\<^sub>T.
The lemma "WHILEI_weaken" assumes that for any "x", if "I x" holds, then "I' x" also holds. It shows that the while-loop with invariant "I'" and body "b" and function "f" applied to "x" is less than or equal to the while-loop with invariant "I", body "b" and function "f" applied to "x".
If the initial state has 'N' equal to 'a', then after the operation of incrementing 'N' by 1, the final state will have 'N' equal to 'a + 1'.
The lemma named "chain_sym_obtain" assumes that there is a function 'f' leading to a set 'X' in the order 'a' to 'b' to 'c'. It then concludes that there exists another function 'g' that leads to the same set 'X' but in the reverse order 'c' to 'b' to 'a'. Furthermore, this function 'g' is defined as 'f' applied to the...
The lemma "reachable_imp" assumes that "p" and "q" are points. If the transitive closure of "p" followed by "q" is less than or equal to the transitive closure of the transpose of "p" followed by "p", then "p" is less than or equal to the transitive closure of "p" followed by "q".
The 7-fold cross product of x, y, and z, plus the 7-fold cross product of y, z, and x, plus the 7-fold cross product of z, x, and y, when passed through the function Hv, is equal to the negative three-halves times the difference between the product of Hv of x, Hv of y, and Hv of z, and the product of Hv of x and the pr...
The lemma named "type_copy_id" defines a type using the identity function "id" for both the injection and projection functions, and the universe "UNIV" as the underlying set.
The lemma "obtain_fac_interpret" assumes that the concatenation of "pu", "u", and "su" equals the concatenation of the list "ws", and that "u" is not an empty string. It then obtains "ps", "ss", "p", "s", and "vs" such that "p u s" is equivalent to "vs" under some relation, the concatenation of "ps", "vs", and "ss" equ...
The derivative of the function that adds a constant 'a' to 'x' is always 1, regardless of the value of 'z'.
If the function "redT_updTs" applied to the thread states "ts", thread actions "tas", and thread "t" equals the pair "(x, w)", and if the thread states and actions are valid (as indicated by "thread_oks ts tas"), and if there is no thread state for thread "t" (as indicated by "ts t = None"), then there exists a memory ...
For any program P and heap h, replicating a value v n times is conformant to replicating an error n times.
If 'x' is an element of set 'S', then updating the map 'ρ' with the map 'ρ' where 'x' is mapped to 'z', restricted to 'S', is equal to updating the map 'ρ' with the map 'ρ' restricted to the set 'S' without 'x', and then mapping 'x' to 'z'.
For any program, command, postcondition, and state, the weakest precondition of the modified program with the left-hand side variable of the command is equal to the weakest precondition of the original program.
The norm of x is less than or equal to the norm of y if and only if the complex inner product of x with itself is less than or equal to the complex inner product of y with itself.
The step function in the uninst_code module, when applied to the constant values of u and p and the graph g, is equal to the same step function applied to the result of the function u_g (with inputs g and u), the function phis (with g replaced by p), and the graph g.
The box operator applied to the product of the function 'a' applied to 'y' and 'a' applied to 'z', with 'x' as the parameter, is equal to the product of the box operator applied to the function 'a' applied to 'y' with 'x' as the parameter and the box operator applied to the function 'a' applied to 'z' with 'x' as the p...
"a is less than or equal to b if and only if the left complement of a with respect to b equals 1."
The range of the function 'f' starting from the 'n'th element is a subset of the range of the function 'f'.
The root state of a statechart is not in the range of the function "ChiPlus" applied to the hierarchical automaton of the statechart.
The lemma named "comOf_invV" assumes that a transition "trn" is valid for a node "nid", the source of the transition "trn" for the node "nid" is reachable, and a property "\<phi>" holds for the node "nid" and the transition "trn". It shows that the communication partner of the node "nid" for the inverse of a function "...
In the context of a ring, the lemma named "indexed_padd_index_free" assumes that both polynomials P and Q are index-free with respect to index i. It then shows that the sum of P and Q (denoted as P \<Oplus> Q) is also index-free with respect to the same index i.
The set of nested propositional atoms of the first element of the function "af\<^sub>\<nu>" applied to X, p, and w, is a subset of the set of nested propositional atoms of the first element of p with respect to X.
If the function "get_disconnected_nodes" when applied to a document pointer in a heap 'h' returns an 'ok' status, then the document pointer is in the set of document pointer kinds in the heap 'h'.
For any configuration 'cfg' that is a valid configuration, the following two conditions are equivalent: 1. For all configurations 'cfg'' that are in the K_cfg of 'cfg', the first element of the state of 'cfg'' is equal to the first element of the state of 'cfg'. 2. For all configurations 'cfg'' that are in the K_cfg o...
If function 'f' is a submap of function 'g' and function 'g' is a submap of function 'f', then function 'f' is equal to function 'g'.
The congruence of the product of "k" and "a" to the product of "k" and "b" modulo "m" is equivalent to the congruence of "a" to "b" modulo "m", given that "k" and "m" are coprime. This is valid for any "a", "k", and "m" in a unique Euclidean ring with a ring greatest common divisor.
The composition of two epsilon-free ground term tables, denoted as \<G>\<^sub>1 and \<G>\<^sub>2, is also an epsilon-free ground term table.