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The lemma "well_analyzed_inv" assumes that the term "t" is well analyzed when it is decomposed and appended to "A". It shows that "t" is in the set of subterms of the intruder knowledge of "A" union the assignment right-hand side of "A", excluding the image of the variable set.
If every element 'l' in the set 'ls' is a literal or its negation, then the disjunction (logical OR) of all elements in 'ls' is in conjunctive normal form.
The permutation of the range of a function Y is equal to the range of the permutation of the function Y.
Marking guards with function "f" on a raised function "g" is equivalent to the raised function "g" itself.
The empty set is less than or approximately equal to any element 'd' of type 'c in the abstract domain.
If the evaluation of "opubx" with "xs" equals "Some z", then the evaluation of "deubx opubx" with the normalized unboxed version of "xs" equals the normalized unboxed version of "z".
If a value 'x' belongs to the set of Hyperfinite numbers, then applying the standard part function 'stc' twice on 'x' is equivalent to applying it once.
The language of the epsilon-free regular relation R is equal to the language of R.
Renaming a clause 'C' with a selected literal 'L' removed is equivalent to renaming the clause 'C' first and then removing the renamed literal 'L'.
The reciprocal of cd is greater than zero.
The source of a packet, represented by the tuple (a,e), is within a certain subnet if and only if the predicate P holds true for a and e.
"unlr of x is less than or equal to unlr of y" if and only if "x is less than or equal to y".
The lemma named "filterlim_at_top_at_top_within_bounded_right" is defined with respect to two fixed elements 'a' and 'b' of a type that has a dense order and a linear order topology, and a function 'f' that maps from this type to real numbers. The lemma assumes that 'a' is less than 'b'. It shows that the limit of the ...
The lemma named "open_exclude_setI" assumes that there exists a set U such that U is open and the intersection of U and A is the set containing x. Under this assumption, it shows two things: first, the set obtained by excluding x from the open set A is open; second, the intersection of this set obtained by excluding x ...
: assuming the infinity hyperbolic point is not in the set of the oriented circline H, then H is positively oriented if and only if the infinity hyperbolic point is not in the disc of H.
The lemma "sets_embed_measure" assumes that the function "f" is injective. It shows that the set of all sets in the embedded measure of "M" under "f" is equal to the set of all images of "A" under "f", where "A" is in the set of all sets in "M".
If proposition 'p' is quantifier-free and proposition 'q' is quantifier-free, then the proposition "if and only if 'p' then 'q'" is also quantifier-free.
If a state 's' is reachable through a set of events 'e', then the state 's' is also reachable through the set of events 'e' when a new event, represented as a pair ((aa, ba), b), is added to the set.
Mapping a function 'f' over a lazy list 'Lazy_llist xs' is equivalent to creating a new lazy list where 'f' is applied to each element of the original list 'xs', with the mapping operation being optional.
The lemma "emb_step_at_chop" assumes the following conditions: 1. The embedding step at position 'p' to the right in 't' equals 's' (emb_step_at: "emb_step_at p Right t = s"). 2. The position of 't' is at 'p' appended with 'Right' (pos:"position_of t (p @ [Right])"). 3. All elements in the list 'p' are 'Left' (all_Lef...
"ns" is well-defined.
The lemma inv1E: assumes that "inv1 P1 P2 B1 B2 V" holds, shows that "bp (P1 union wrap B1 union P2 union wrap B2 union {v | v is in V})" is true, and "the union of (P1 union wrap B1 union P2 union wrap B2) equals U minus V", and "B1 is not in (P1 union P2 union wrap B2)", and "B2 is not in (P1 union wr...
For all 'y', there exists an 'i' such that 'i' is an identity and the composition of 'i' and 'y' is isomorphic to 'y'.
The lemma named "segment_overlapping" assumes that "x" is in the orbit of "f" for both "a" and "b", and that "f" is a bijection. It shows that either the segment of "f" from "a" to "x" is a subset of the segment of "f" from "b" to "x", or the segment of "f" from "b" to "x" is a subset of the segment of "f" from "a" to ...
The lemma named "redT_upd_inv_ext" is defined with a fixed I of type "'t maps to 'i". It shows that if s transitions to s' with thread t and trace ta, and the thread state invariant holds for the thread state of s and I, then I is a submap of the updated invariants I, P, and ta for thread t.
The lemma named "proj2_set_Col_coeff" assumes that a set S is collinear in a projective plane (as indicated by "proj2_set_Col S"), and that the set {p,q,r} is a subset of S. It also assumes that point p is not equal to point q, and point r is not equal to point p. Under these assumptions, it shows that point r is equal...
The lemma named "del_list_split_right" assumes that a node "Node ts t" is left-aligned with "u", the leaves of "Node ts t" are sorted in ascending order, and the split of "ts" at "x" results in a tuple "(ls, (sub,sep)#rs)". Under these assumptions, it shows that deleting "x" from the list of leaves obtained by appendin...
If the action "HEndPhase1" transitions the system from state "s" to state "s'" for a process "p", and if the invariant "HInv4a" holds for the system state "s" and process "p", and if the invariant "Inv2a" holds for the system state "s", then the invariant "HInv4a" will also hold for the system state "s'" and process "p...
If "h a" is a list in the heap "h", then the list in the heap "h" at the reference "Ref a" is the list that starts with "a" and is followed by the list in the heap "h" at "h a".
The lemma "cross7_basis" states the following: 1. The seven-fold cross product of axis 1 with itself equals axis 4. 2. The seven-fold cross product of axis 2 with axis 1 equals the negative of axis 4. 3. The seven-fold cross product of axis 2 with axis 3 equals axis 5. 4. The seven-fold cross product of axis 3 with ax...
The embedding of G evaluated at state s is less than or equal to 1.
If two conditions "c1" and "c2" have the same states, and if "c1" and "c2" both undergo the same symbolic execution "se" with a given list "l" to become "c1'" and "c2'" respectively, then the states of "c1'" and "c2'" are equal.
If "r" is a reducible expression (redex), then the decomposition of expression "e" at "r" is equivalent to the statement that the context of "r" equals "e".
If a function 'f' is injective on a set 'A', then the image of 'A' under 'f' is infinite if and only if 'A' is infinite.
The function "internal_iface_name_wildcard_longest" when applied to the arguments "eth+" and "e+" returns "eth+".
The dimension of the row and the column of the product of a matrix N and its transpose is v.
An object 'P' at reference 'r' in a system 's' where the heap of the system has been updated such that the reference 'r'' is now None, is equivalent to an object 'P' at reference 'r' in the original system 's' and 'r' is not equal to 'r''.
If "F" is in "A Ensures B", then "F" is in "A LeadsTo B".
The variables agree on the sets V and W if and only if the variables agree on the union of sets V and W.
The minimum disjunctive normal form of the representation of a propositional linear temporal logic formula, after it has been abstracted, is equal to the minimum disjunctive normal form of the original formula.
If you delete an empty set of edges from a graph G, the graph remains unchanged.
The lemma named "in_set_replicate" assumes that "x" is an element in the set that is created by replicating "y" "n" times. It shows that "x" is equal to "y".
If initial condition I holds, then the mutator m will preserve this condition.
The elements "a" and "b" in the list are distinct.
If group G acts on groups H and N, and H is a subset of N, then the kernel of the quotient map from G/H to G/N is equal to the carrier of the quotient group Gp G N divided by H.
The lemma "the_ntcf_rKe_components" shows the following: 1. The natural transformation map of "the_ntcf_rKe" with parameters alpha, TT, KK, and lim_Obj is equal to a function that, for each object 'c' in the domain of the hom-set of TT, maps 'c' to the natural transformation map of the universal arrow of the limit obj...
The lemma named "set_to_list_length" assumes that the set "S" is finite. It shows that the length of the list obtained from the set "S" is equal to the cardinality of the set "S".
The transaction positive checks component of a variable 'x' in a set 'C' is equal to the set of all 's' such that there exists an 'a' where 'a' is a variable 'x' in the function of the set 's' with no arguments, and this is included in the set 'C'.
The norm of the result of applying a bounded function 'f' to 'x' is less than or equal to the norm of the function 'f' itself.
If the function "fetch" with inputs "tp", "0", and "b" returns a pair of values "(nact, ns)", then applying the function "action_map" to "nact" will yield the result "4".
The lemma Perp_in_perm: assumes that "X is perpendicular at A to line segment BC and line segment AD" shows that "X is perpendicular at A to line segment BC and line segment AD, and X is perpendicular at B to line segment AC and line segment AD, and X is perpendicular at A to line segment BD and line segment AC, an...
The lemma named "par_id_3" assumes that line segment AB is parallel to line segment AC. It shows that points A, C, and B are collinear.
If the predicate "mpred" holds for the sequence "P", the control flow sequence "cfs", and the integers "k" and "m", and if "mpred" also holds for "P", "cfs", "m", and "n", then "mpred" must also hold for "P", "cfs", "k", and "n".
The last message of a function mapped stream "xs" with respect to "k" is equal to the last message of the stream "xs" itself.
The set of all polynomials over a field K forms an abelian monoid.
If the tuple (b, A, Φ, s, t) is in the set var, and for all s and t, if Φ(s, t) is true then Ψ(s, t) is also true, then the tuple (b, A, Ψ, s, t) is also in the set var.
The lemma named "measurable_finite" is about a function 'S' that maps from any type 'a' to a set of natural numbers. It assumes that for any natural number 'i', the set of all elements 'x' in the space 'M' such that 'i' is in the image of 'x' under 'S', is a measurable set in 'M'. Under this assumption, it shows that t...
The multiplication of a diagonal vector "x" and a matrix "A" is equal to the function that multiplies the "i"th element of "x" and the element at the "i"th row and "j"th column of "A". This is denoted as "?A = ?B".
In the lemma for a weak lower semilattice, if x and y are elements of the carrier L, then the meet of x and y is less than or equal to x.
The application of "IAdd" in the context of G, maxs, rT, pc, et, and some state s, is equivalent to the existence of some ST and LT such that s is equal to the pair (PrimT Integer#PrimT Integer#ST,LT). This is also expressed as "?app s = ?P s".
The lemma named "prv_eql_subst_trm_rev2" assumes that "x" is a variable, "\<phi>" is a formula, "t1" and "t2" are terms, and that it is provable that "t1" is equal to "t2". Under these assumptions, it shows that it is provable that if "\<phi>" is substituted with "t2" for "x", then "\<phi>" is substituted with "t1" for...
If a property P holds for the last element of list xs, then either there exists an element x such that when the function combine_lists_P is applied to P, y, and xs, the result is a list containing only x, or the property P holds for the last element of the list resulting from applying the function combine_lists_P to P,...
The remainder of an integer 'a' when divided by 2 is less than 2.
The multiset of the range of a function 'a' for a single element set '{i}' is equal to the multiset containing the value of the function 'a' at 'i'.
: Given a functor F, categories \<DD>, \<FF>, and \<GG>, a natural transformation \<epsilon>\<^sub>m, and a natural transformation \<epsilon> defined as a 5-tuple consisting of \<epsilon>\<^sub>m, the composition of \<FF> and \<GG>, the identity functor on \<DD>, and two instances of \<DD>. Assuming that \<FF> is a ...
The uncurrying of function f and the uncurrying of function g are both in the set of constant pairs.
If you put an element into a heap at a position specified by an element pointer, and then get the element at the same element pointer, you will retrieve the same element.
The function 'take_bit LENGTH('a)' when partially applied to a word of type 'a, when transferred via the word partial correspondence relation (pcr_word), corresponds to the function 'ucast' that casts a word of type 'a to a word of type 'b. Both 'a and 'b are types with a length.
If the maximum number of variables in expression 'e' is zero, then the maximum number of variables in the expression resulting from the function 'extRet2J1' applied to 'e' and 'va' is also zero.
If "M" is an element of "sexp", then "In0(M)" is also an element of "sexp".
If for all elements 'X' in a set 'A', 'X' is a subset of 'C', then the union of 'A' is also a subset of 'C'.
If set S is not empty, then the disjunction of P and the infimum of the function F(Q) for all Q in set S is equal to the infimum of the disjunction of P and F(Q) for all Q in set S.
The image filter operation is correctly implemented in the sorted list set, preserving the abstraction function and the invariant.
If a function M preserves the unit disc, then the image of the unit disc under the function M is the unit disc itself.
The lemma named "zero_search" declares variables A and i. It starts with the condition that is always true. It sets i to 0. Then, it enters a loop that continues as long as i is less than the length of A and the i-th element of A is not equal to the key. The invariant of this loop is that for all j less than i, the j-t...
The refinement relation "R_a0m1a_is" intersected with the universal set and "m1a_inv0_fin" refines the medium "med_a0m1a_is" from the abstract model "a0n" to the concrete model "m1a".
The lemma named "effect_freezeI" assumes that "h'" is equal to "h", "r" is the result of getting "h" at index "a", and "n" is the length of "h" at index "a". Under these assumptions, it shows that the effect of freezing "a" on "h" results in "h'", "r", and "n+1".
The square root of 1 in the complex number system is 1.
For a property P that applies to words of a certain length 'a, if P holds for 0 and if P holding for any word n implies that P also holds for the word (1 + n), then P holds for any word m.
The identity function on a set of type 'a is not None if and only if the identity function on type 'a, which has a computable equality, is not None.
The integral of a formal Laurent series multiplied by a constant is equal to the constant multiplied by the integral of the formal Laurent series, where the formal Laurent series has elements from a division ring.
If "n" is equal to the sum of "k" and "m", then taking "k" elements from the list obtained by filling "w" with "fill" until it has "n" elements is equivalent to filling "w" with "fill" until it has "k" elements.
If all elements of the list 'oft' satisfy all prerequisites of 'ofe_fields', then the linear match of 'OF_match_fields_unsafe' with 'oft' is equal to the linear match of 'OF_match_fields_safe' with 'oft'.
If a command 'c1' is valid from precondition 'P' to postcondition 'Q', and a command 'c2' is valid from precondition 'Q' to postcondition 'R', then the sequence of commands 'c1' followed by 'c2' is valid from precondition 'P' to postcondition 'R'.
If the set of states 's' for which 'G s' holds is a subset of 'Q', then 'P' is a subset of the weakest liberal forward box of the differential equation 'x' equals 'f' and 'G' on 'T S' at time 't0' with respect to 'Q'.
The difference between two real numbers represented as Ratreal x and Ratreal y is equal to the Ratreal representation of the difference between x and y.
If a function 'p' is univalent (i.e., for every input, there is exactly one output), then the image of the roots of 'p' under 'p' is equal to the roots of 'p'.
The lemma named "assoc_naturality" assumes that "\<mu>", "\<nu>", and "\<tau>" are arrows, the source of "\<mu>" is the target of "\<nu>", and the source of "\<nu>" is the target of "\<tau>". It shows that the associator of the codomain of "\<mu>", the codomain of "\<nu>", and the codomain of "\<tau>", composed with th...
The lemma named "power_odd_inj" is defined for any two elements 'a' and 'b' of a linearly ordered integral domain. It assumes that 'k' is an odd number and that 'a' raised to the power of 'k' equals 'b' raised to the power of 'k'. Under these conditions, it shows that 'a' must be equal to 'b'.
The function "e2rv" applied to the encoding of a configuration consisting of "ss" and "rv" equals 0 if "rv" is undefined, otherwise it equals the successor of the value of "rv".
For any real normed field or Banach space 'x', the function 'p' that sums up to 'n' such that if 'p' is even and 'n' is odd, then the result is negative of the expression ((-1) raised to the power of (p divided by 2) times (p choose n) divided by the factorial of p) times the real number 'x' raised to the power of 'n' ...
If an atom 'a' does not occur free in term 'tm', then substituting 'a' with any term 'x' in 'tm' will result in 'tm' itself.
For any integers k and l, the bitwise OR operation on k and l is greater than or equal to k, given that l is greater than or equal to 0.
The lemma assumes that the concatenation of p, u, v, u, and q equals the concatenation of u, v, v, and u, and that p and q are not empty. It shows that the concatenation of u and v equals the concatenation of v and u.
If the composition of functions f and g is equal to the composition of functions f and h, and if function f is injective (or one-to-one) on the union of the ranges of functions g and h, then function g is equal to function h.
In the context of a group, if H and J are subgroups of G and the cardinalities (sizes) of H and J are coprime (i.e., their greatest common divisor is 1), then H and J are complementary.
If H is a submodule of M over a ring R, and a and b are elements of the ring R, and X is an element of the set of cosets of M by H, then the scalar product of the product of a and b with X in the set of cosets is equal to the scalar product of a with the scalar product of b and X in the set of cosets.
The lemma named "order_charpoly_smult" is about a fixed square matrix A of complex numbers. It assumes that a scalar k is not equal to zero. It shows that the order of the characteristic polynomial of the matrix scaled by k is equal to the order of the characteristic polynomial of the original matrix A, with the variab...
If a function "c" with no arguments is a subterm of "t", then this function "c" is also a subterm of "t" after applying a substitution "σ".
The lemma named "inductive_unwinding_1" assumes two conditions: 1. The property P is closed under the union of refinements, denoted by "ref_union_closed P". 2. The property P is secure with respect to the interference relation I and the domain map D, denoted by "secure P I D". Under these assumptions, the lemma shows...