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Statement: stringlengths 7 24.3k
The public signature key of a cardholder C is not a key encryption key for any key K in any event sequence evs.
If the set of edges E is finite, then the graph c is a finite graph.
For any given number 'n', if 'n' is an eigenvalue block of matrix 'A', and 'i' and 'j' are less than 'n', then the element at position (i,i) in matrix 'A' is equal to the element at position (j,j) in matrix 'A'.
The lemma named "iota_in_hom" shows that "iota" is a morphism from the tensor product of "I" with itself to "I".
If a set of points 'ps' is a subset of the complement of a block 'bl', then 'ps' is also a subset of the set of all points 'V'.
P' is less than or equal to the tensor product of post and the projection of R on k, in the L-order.
If the constant term of a polynomial 'p' is not zero, then the polynomial 'p' is not empty.
The function "cosine of f(x)" is differentiable at a point x within a set X if the function f(x) is differentiable at the same point x within the same set X. Here, f(x) is a real-valued function.
In a given context G, if a statement s evaluates to a pair (w, s') in n steps via the In2 function applied to t, then there exists a value vf such that w equals In2 applied to vf and in the context G, the statement s evaluates to vf in n steps.
If a real number 'r' is an element of the set 'R' and the set of 'ms' is a subset of 'M', then the set resulting from mapping the scalar multiplication of 'r' to each element in 'ms' is also a subset of 'M'.
For any word length 'a, 2 to the power of n minus 1 is less than 2 to the power of n if and only if n is less than the length of 'a.
If the maximum degree of a node (with root r and children xs) is greater than the finite cardinality of xs, then the image of the maximum degree under the function that takes the first element of the finite set xs is not an empty set.
If "c" is a component of "s", then the set "s" minus the set "c" is equal to the union of the components of "s" excluding "c".
The variables of the resultant of a multivariate polynomial and a polynomial "p" and "q" with respect to "x" are a subset of the variables of polynomial "p" excluding "x".
If 'i' is less than or equal to the absolute value of 'x' times 'i', and 'j' is less than or equal to the absolute value of 'x' times 'j', then the sum of 'i' and 'j' is less than or equal to the absolute value of 'x' times the sum of 'i' and 'j'.
A total preorder exists on the preimage of the carrier under function f, when the relation is mapped by function f and the less than or equal to relation.
The set of free variables in term 't' is finite.
The lemma in the Deutsch theory shows that the dimension of the row of Uf is 4 and the dimension of the column of Uf is 4.
The locks of the result of the function "activate_cond_action1" with inputs "s1", "s2", and "ct" are equal to the locks of "s1". This is a simplification rule in Isabelle.
If a configuration 'cfg' is in the set of valid configurations and the state of 'cfg' is in the set Sₑ, then the value of 'cfg' is 0.
The set of elements in the intersection of multisets A and B is equal to the intersection of the set of elements in multiset A and the set of elements in multiset B.
For a given Van Emde Boas tree 't' and a tree index 'ti', the maximum tree index 'vebt_maxti' is equivalent to the maximum tree 'vebt_maxt' of 't'. This is under the condition that the Van Emde Boas tree 't' and the tree index 'ti' are associated as per 'vebt_assn_raw'.
The range of the function, which maps each time 't' to the probability of 't' in the decision based on reward 'r' for policy 'p', is bounded.
If for all 'a' greater than or equal to 0, the value of 'ms' at 'a' is 'i' if 'a' is the address of 'x' in 'xs', otherwise it's the value of 'ms' at 'a', and 'x' is in the set 'xs', then the state transformation of 'xs' with 'ms' replaced by 'i' at 'x' is equal to the state transformation of 'xs' with 'ms', with 'x' re...
The lemma named "similar_triangle_aa" assumes that points b1 and c1 are not the same, and points b2 and c2 are not the same. It also assumes that the angle at a1 between b1 and c1 is equal to the angle at a2 between b2 and c2, and that the angle at b1 between c1 and a1 is equal to the angle at b2 between c2 and a2. Giv...
The lemma named "swap_disjunct_emb_step_at" assumes that if the length of sequence p is less than or equal to the length of sequence q, then the first few elements of sequence q (up to the length of p) are not the same as sequence p. Similarly, if the length of sequence q is less than or equal to the length of sequence...
The set of parameters of the extension of a structure S by a function f and a natural number n, minus the set of parameters of S, is finite.
The lemma named "share_append_commute'" assumes that the union of all shared, unshared, and acquired elements in two lists, xs and ys, have no common elements. Under this assumption, it shows that the sharing operation on the concatenation of ys and xs is the same as the sharing operation on the concatenation of xs and...
The set of first elements from the pair in "d_states" of a machine "M" and a state "q", is a subset of the states of the machine "M" excluding the state "q".
The length of the prefix slice of a sequence S and a path π is equal to the length of the path π.
The theorem "isomorphic_free_groups_bases" assumes that "i" is an isomorphism between the free groups of "X" and "Y". It shows that there exists a function "f" that is a bijective correspondence between "X" and "Y".
The domain of the contravariant Hom-functor with respect to an object X in a category C is the opposite of C.
If 'xs' and 'ys' are tangle diagrams, then the tensor product of 'xs' and 'ys' is also a tangle diagram. Furthermore, if 'x * xs' and 'y * ys' are tangle diagrams, then the tensor product of 'x * xs' and 'y * ys' is also a tangle diagram.
The length of the sequence 'xs' is infinite if and only if 'xs' is not a finite sequence.
The lemma named "xfer_holds" assumes that "sim_rel s cs" is true. It shows that "tts cs t" is in a hold state if and only if "tts s t" is also in a hold state.
The lemma introduces a fail-safe converter that is perfectly lossless at the bottom level of system I'.
The reverse of the projection of list 'l' onto set 'X' is equal to the projection of the reverse of list 'l' onto set 'X'.
The lemma named "Phi equals zero" assumes that there exists a 'd' less than the length of 'p' such that 'x d' is less than zero or greater than one, and 'p' belongs to the sparse grid with parameters 'dm' and 'lm'. Under these assumptions, it shows that the function 'Phi' of 'p' and 'x' equals zero.
The lemma named "ffold_fimage" assumes that the function "g" is injective on the finite set "A". It shows that the fold of function "f" and base case "z" over the image of "A" under "g" is equal to the fold of the composition of functions "f" and "g" with base case "z" over "A".
The lemma named 'a_l_coset_def'' establishes that for a given structure G, the left coset of 'a' and 'H' is equivalent to the union of 'a' added to 'h' for every 'h' in 'H'.
The lemma of allocation monotonicity assumes that set N1 is a subset of set N2. It shows that all allocations of set N1 and set Omega is a subset of all allocations of set N2 and set Omega.
The p-residue of any two integers m and n falls within the range from 0 (inclusive) to p raised to the power of m (exclusive).
The lemma named "linear_plus_1_le_power" is defined for a real number 'x'. Given the assumption that 'x' is greater than or equal to zero, it shows that the product of a real number 'n' and 'x' plus 1 is less than or equal to the 'n'th power of the sum of 'x' and 1.
The function "T_on'" applied to a set A, a pair (s0,x), and a sequence qs is equal to the function "T_on_rand'" applied to the embedded set A, the returned probability mass function of the pair (s0,x), and the sequence qs.
A set X is an upclosed set if and only if the upward closure of X is equal to X itself.
If the following conditions are met: 1. The support of function 'f' is a subset of 'FH'. 2. The range of function 'f' is a subset of 'V'. 3. For all 'x' and 'y' in 'FH', the value of function 'f' at the point 'x+y' is equal to the sum of the values of function 'f' at points 'x' and 'y'. 4. For all real numbers 'a', fo...
The theorem named "effective_matrix_Tensor_elements2" is defined with two fixed matrices M1 and M2. It assumes that M1 and M2 are both matrices with their respective row lengths and lengths. The theorem shows that for all i less than the product of the row lengths of M1 and M2, and for all j less than the product of th...
If a list composed of the list 'xs' followed by the element 'x' is sorted, then deleting the element 'x' from the list composed of 'xs' followed by 'ys' is equivalent to the list 'xs' followed by the list obtained by deleting 'x' from 'ys'.
If the following conditions are met: 1. Alpha is a member of the set of integers. 2. NN is a sequence of variable functions. 3. FF is a semicategory morphism from AA to BB. 4. GG is a semicategory morphism from AA to BB. 5. The cardinality of NN is equal to 5. 6. The domain of NN is FF. 7. The codomain of NN is GG. 8....
If a string 's' is a natural number, then showing (or converting) the reading (or interpretation) of 's' as a natural number will result in 's' itself.
If 'p' is a prime element, then 'p' is irreducible.
The normalization of a while-loop annotation with guards "gs", condition "b", invariant "I", variant "V", and command "c", is equivalent to the guards "gs" applied to a while-loop with condition "b" and a sequence of commands that is the result of flattening the normalization of command "c" and the flattening of guards...
For all x in a linearly ordered field, there exists a y that is greater than the absolute value of 23 times x minus 9. For all z that is greater than the absolute value of 3 times y minus 19 times the absolute value of x, the sum of x and z is greater than 2 times y.
The lemma named "reindex_bij_witness" is based on two sets of assumptions, "witness" and "eq". The "witness" assumptions are: 1. For any element 'a' in set 'S', applying function 'j' to 'a' and then function 'i' to the result gives back 'a'. 2. For any element 'a' in set 'S', the result of applying function 'j' to 'a...
If the leading term of a polynomial p is less than the leading term of the sum of polynomials p and q, then the leading term of p is less than the leading term of q. This is assumed under the condition that the polynomials are ordered.
If a function h is in the set of two-argument primitive recursive functions, then the operation PrimRecOp1 with a natural number a and the function h is in the set of one-argument primitive recursive functions.
If a list "xs" is not empty, then the nth element of the tail of "xs" is equal to the (n+1)th element of "xs".
The lemma named "less_and_not_mask_eq" is defined with a fixed variable 'i' of any length word. It assumes that 'r2' is less than or equal to 'r1' and that the bitwise AND of 'i' and the bitwise NOT of the mask of 'r2' is equal to the bitwise AND of 'x' and the bitwise NOT of the mask of 'r2'. It shows that the bitwise...
The operation "plus trilean" is commutative, meaning that the order in which the elements 'a' and 'b' are processed does not change the result. In other words, 'a' plus 'b' is equal to 'b' plus 'a'.
The lemma named "ta_union_der_disj_states'" assumes that the intersection of the states of automaton A and automaton B is an empty set. It shows that the derivative of the union of automaton A and B with respect to a tree t is equal to the union of the derivative of automaton A with respect to t and the derivative of a...
If "Only_G \<G>" holds true and "\<G> satisfies \<phi> under path semantics", then "\<G> also satisfies Unf_G \<phi> under path semantics".
The lemma "WeakConjunctive_Refinement" assumes that "HC" is weakly conjunctive. It shows that "P" is less than or equal to "HC(P)".
The lemma named "defined_by_str" assumes that a string "s" is an element of a set "X", and that "X" is an equivalence class in the universe of all sets under the equivalence relation "A". It then shows that "X" is equal to the equivalence class of the set containing only "s" under the equivalence relation "A".
If the following conditions are met: 1) A program 'G' satisfies the assertion '{A} C,m,l {B} I', 2) The method state table of 'C,m' equals 'Some(Mspec,Minv,Anno)', 3) The annotation of 'l' equals 'Some Q', 4) The assertion 'A' holds for states 's0' and 's', then the assertion 'Q' must also hold for states 's0' and...
The lemma named "substc_rename2" assumes a condition "a" where "y" is fresh for the pair "(N,x)". It then shows that the substitution of "a" with "N" under the scope of "x" in "M" is equal to the substitution of "a" with "N" under the scope of "y" in "M", where "N" is permuted by swapping "y" and "x".
Lemma: holomorphic_on_diff [holomorphic_intros]: If function f is holomorphic on set s and function g is also holomorphic on set s, then the function (lambda z. f(z) - g(z)) is holomorphic on set s.
: Assuming a function 'f' is in the set of unique runs of the converse, and the intersection of set 'A' and the range of function 'f' is an empty set, it can be shown that the set of 'f' union with the pair (x, y) for every 'y' in set 'A' is a subset of the set of unique runs of the converse.
If Q is a subset of Q', and if under the contexts Gamma and Theta, the Hoare triple {P} c {Q} is valid with respect to the fault set F and the abort set A, then under the same contexts, the Hoare triple {P} c {Q'} is also valid with respect to the same fault set F and abort set A.
The lemma named "mono2_Step" is fixing C1 and C2 as elements of a semilattice sup acom. It assumes two conditions: first, for any x, e, S1, and S2, if S1 is less than or equal to S2, then the function f applied to x, e, and S1 is less than or equal to the function f applied to x, e, and S2. Second, for any b, S1, and S...
If 'c' is a singular chain of dimension 'p' in a topological space 'X', then the chain map of 'c' with the identity function 'id' is equal to 'c' itself.
If the road is free, then the width of the road is equal to 1.
: Assuming the following conditions: 1. The effect of the function 'rev' applied to 'q' and 'p' transforms the heap 'h' into 'h'' and returns 'v'. 2. The function 'list_of'' applied to 'h' and 'q' returns 'qs'. 3. The function 'list_of'' applied to 'h' and 'p' returns 'ps'. 4. For all 'qrs' and 'prs', if 'refs_of'' ap...
The sum decode of x is equal to the sum decode of y if and only if x is equal to y.
This is a lemma stating that the theorem "Thm" holds for the type "(bool * bool)" with 0 and "\<Phi>1".
The lemma named "oinvariant_anyact" assumes that "A satisfies the condition that action TT leads to P under U". It shows that "A also satisfies the condition that any action S leads to P under U".
If the conditions Gamma and (h1,h2 implies h0) and ((A implies B) h2) imply Delta, then the following two conditions hold: 1. Either Gamma and (h1,h2 implies h0) and ((A implies B) h2) imply (A h1) or Delta. 2. If Gamma and (h1,h2 implies h0) and ((A implies B) h2) and (B h0) are true, then Delta is implied.
The lemma "sum_list_us_gt0" assumes that the sum of the list "y" is not equal to zero. It shows that the sum of the list resulting from the function "u" applied to "y" and "i" is greater than zero.
If x is identical to y, then y is identical to x, in the context of v.
The lemma "is_red_addsE'" assumes that "p" is reducible by the set "F". It shows that there exists a non-zero "f" in "F" and a "v" in the keys of "p" such that the leading term of "f" divides "v".
The lemma named "ind_in_set_less" establishes that for a given natural number 'x' and a set 'A' of natural numbers, if 'A' is finite and 'x' is an element of 'A', then the index of 'x' in 'A' is less than the cardinality (or size) of 'A'.
If the following conditions are met: 1. The function 'f' commutes with the augmented power monoid set of 'z', 'i', and the set containing 'a'. 2. The augmented power monoid set of 'z', 'i', and the set containing 'a' is associative with the function 'f'. 3. The first condition of the indexed power product is satisfied...
The lemma belongs to a module and is named "finite_in_span". It fixes a set S and a vector v. It assumes that the set S is finite, that S is a subset of the carrier M, and that the vector v is in the span of S. It shows that there exists a function a such that a maps from S to the carrier R and the linear combination o...
If there is a linear order on a set D with a comparison function 'cmp', and if there is a subset D' of D, then there is also a linear order on the subset D' with the same comparison function 'cmp'.
The non-negative bounded function "bfun_nonneg f" is greater than or equal to 0.
The function "map_converter" with parameters "id", "b", "out", "id", and "res" is equal to the function "map_converter'" with parameters "b", "out", and "res".
The lemma onormI assumes that for all x, the norm of the function f(x) is less than or equal to b times the norm of x. It also assumes that x is not equal to zero and that the norm of the function f(x) equals b times the norm of x. Under these assumptions, it shows that the operator norm of the function f equals b.
If a tree is aligned with lower bound 'l' and upper bound 'u', then all elements 'x' in the set of leaves of the tree are greater than 'l' and less than or equal to 'u'.
The lemma named "ofsm_table_fix_partition_fixpoint_trivial_partition" assumes that "q" is in the states of "M". It shows that the function "ofsm_table_fix" applied to "M", a function that maps "q" to the states of "M", 0, and "q" is equal to the function "ofsm_table" applied to "M", a function that maps "q" to the stat...
If the function "f" applied to the successor of "n" is not a terminal in the substitution "subs", and if "f" is a branch of "gamma" under the substitution "subs", then the function "f" applied to "n" is not a terminal in the substitution "subs".
If the function "dec" is defined as a function that takes an input "w" and checks if there exists a "t" in set "A" such that "t" encoded equals "w", then it returns the unique "t" that satisfies this condition; otherwise, it returns some "t" in set "A". Then, "dec" is equivalent to a function that takes an input "w" an...
The lemma "reduce_simple_not_nil" assumes that a transition from "es" to "es'" exists. It shows that "es" is not an empty list.
The lemma "reduce_simple_not_value" assumes that "es" leads to "es'". It shows that "es" is not equal to the sequence of values "vs".
If one is equal to negative one, then m equals two.
If a natural transformation 'N' is a tiny isomorphism from functor 'F' to functor 'G' in the context of tiny categories 'A' and 'B' with a fixed universe 'alpha', then the following are true: 1. 'N' is an isomorphism from 'F' to 'G' in the context of categories 'A' and 'B' with a fixed universe 'alpha'. 2. 'F' is a fu...
"L" is an instance of itself.
If a formula "psi" is provable from a set of formulas "F", and if "F" is a subset of the set of all formulas and is finite, and if "phi" and "psi" are both formulas, then the disjunction of "phi" and "psi" is provable from "F".
The lemma named "wfG_dom_supp" is about a fixed variable "x". It assumes that "G" is a well-formed context under the given "P" and "\<B>". It shows that the atom of "x" is in the atomic domain of "G" if and only if the atom of "x" is in the support of "G".
"ex_TCons Y" is true if and only if there exists an element "xs" in "Y" such that "xs" is not "TNil".
Converting a disjunctive normal form (DNF) to a boolean value with a list prepended with elements 'a' and 'b' is equivalent to either converting a DNF to a boolean value with a list containing 'a' and the original list, or converting a DNF to a boolean value with a list containing 'b' and the original list.
If a match "m" is in normalized negation normal form, and the IP assignment "ipassmt" satisfies both the "no wildcards" and "disjoint" sanity checks, then there exists a set of IP addresses "p_ips" such that the IP assignment of the interface "p_iiface p" equals "p_ips" and the source IP "p_src p" is in the union set o...

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