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The lemma nl32 assumes that four real numbers l', r', lr, and r are all greater than or equal to 1. It shows that four times the product of (l' + r'), (lr + r), and (lr + r' + r) is less than or equal to the cube of the sum of l', lr, r', and r.
If a complex number 'x' belongs to the set of integers, then the complex form of 'x' also belongs to the set of integers.
, assuming the relation between rho and sigma holds, it shows that the denotational semantics of e under rho is related to the denotational semantics of e under sigma, modified by the infinite iteration of C.
If U and V are Zariski open sets and V is a subset of U, then the morphisms of the sheaf of spectra from U to V form a ring homomorphism. This ring homomorphism is between the structure sheaf of U with its addition, multiplication, zero, and one operations, and the structure sheaf of V with its corresponding operations...
The product of "SLN" and "P" is equal to "P".
The lemma "ShiftP_Mem2" assumes that the atom 'u' is distinct from the tuple (f,k,del,a,b). It shows that if the pair (a, b) is in 'g' and 'g' is the result of shifting 'f' by 'del' at 'k', then there exists a 'u' such that 'u' is in 'k', 'a' is the result of adding 'del' to 'u', and the pair (u, b) is in 'f'.
These are lemmas about the types of certain operations in the context of directed graphs. 1. The operation "op_graph_restrict" takes two sets of type I and a simple labeled graph of type I, and returns a simple labeled graph of type I. 2. The operation "op_graph_restrict_right" takes a set of type I and a simple lab...
The function 'nat_to_string' applied to a number 'n' equals to the character '0' followed by a string 's' if and only if the number 'n' is zero and the string 's' is empty.
The function "darcs" applied to the result of the function "merge1" with input "t" is equal to the function "darcs" applied to "t".
The lemma named "ground\<^sub>l_vars\<^sub>l" assumes that "l" is a ground literal. It shows that the set of variables in "l" is empty.
The index of set B in relation to set ps is 0 if and only if for all elements b in set B, set ps is not a subset of b.
The lemma named "integral_le" is about functions mapping from a Euclidean space to real numbers. It assumes that both functions 'f' and 'g' are integrable over a set 'S'. If for every element 'x' in 'S', the value of function 'f' at 'x' is less than or equal to the value of function 'g' at 'x', then the integral of fun...
The binary relation "eq_onp P" is bi-unique.
In the context of a discrete covariant functor, the range of the object map of the discrete functor from I to F in category CC is a subset of the objects in category CC.
: - An instruction is a jump instruction if and only if its generalized form is also a jump instruction. - An instruction is a function call if and only if its generalized form is also a function call. - An instruction is a return instruction if and only if its generalized form is also a return instruction.
If the largest contained prefix of "a" in "r" is equal to some prefix "p", then the set of words in the word interval of the prefix "p" is a subset of the set of words in the word interval "r".
The restriction of a map 'm' to the domain of 'm' minus the union of sets 'S' and 'T' is equal to the restriction of the map 'm' to the domain of 'm' minus 'T', further restricted to the domain of 'm' minus 'S'.
Binding the function 'f' to 'Nothing' using the Maybe monad results in 'Nothing'.
If the following conditions are met: 1. The message "MsgPInitRes" is composed of the number "LID_M", the number "XID", the nonce "Chall_C", the nonce "Chall_M", and the certificate of agent P encrypted with EKj using only the private key of RCA. 2. The hash of "MsgPInitRes" encrypted with the private key of M is part...
If from a set of assumptions A, p implies q, and from the same set of assumptions, if q implies p then p is true, then from the set of assumptions A, q is true.
For any two timestamps 't' and 't'', if the statement 's' is valid for 't' according to the valid_sops_stmt function, and 't' is less than or equal to 't'', then the statement 's' is also valid for 't'' according to the valid_sops_stmt function.
The minimum period of a word is less than or equal to the length of the word.
The lemma named "coeff_powsum_mpoly_singleton" assumes that "X" is finite and "k" is greater than 0. It shows that the coefficient of the power sum polynomial of "X" with degree "k" at the single term "x" to the power of "k" is 1 if "x" is in "X", otherwise it is 0.
For any directed acyclic graph (dag), denoted as 'x', 'x' is less than or equal to 'x'. This is a reflection property of a partial order, stating that any element is less than or equal to itself.
For any predicate P and any function f' from a finite set to another set, if the domain of f' is the same as the domain of another function f, and for every element in the domain of f there exists a finite set L such that for any two distinct elements s and p not in L, P holds for s, p, f(l), and f'(l), then there exis...
The function "set_iterator_filter" with parameters P and it is equivalent to a function that takes parameters c and f, and applies the function it to c and a function that takes parameters x and σ. If P x is true, then the function f is applied to x and σ, otherwise σ is returned.
There exists a number 'x' such that φ holds for the list consisting of 'x' and the elements of 'xs' if and only if there exists a number 'j' in the set from 0 to 'd' - 1 such that either the infimum of φ holds for the list consisting of 'j' and the elements of 'xs', or there exists a pair (i, ks) in the set of lower bo...
The right element of a triple consisting of "l", "st", and "r" is "r".
For a given graph 'g', the set of phi nodes obtained from the uninst_code's SSA (Static Single Assignment) using the constant function of 'phis' of 'g' is equal to the set of phi nodes of 'g' obtained from the SSA.
The function "FWbisimulation_base_aux" with parameters "final2", "r2", "final1", and "r1" is equal to the same function with parameters "final1", "r1", "final2", and "r2". In other words, the order of the parameters does not affect the outcome of the function.
The lemma named "bet_out__bet" assumes that point P is between points A and C, and point P is oriented away from point A towards point B. Under these assumptions, it shows that point P is between points B and C.
The lemma named "env_app_ok" is about two environments, e1 and e2, both of type 'a environment. It assumes that both e1 and e2 are okay, and that the intersection of the domains of e1 and e2 is empty. Under these assumptions, it shows that the combination of e1 and e2 (e1+e2) is also okay.
The block matrix created by the function "make_vert_block" with input "n" is equal to the result of the function "make_vert_equiv" with the same input "n".
The lemma named "check_funtyp_subst_b" is defined with a fixed variable "b'" of type "b". It assumes that the function type "check_funtyp" with parameters "Theta", "Phi", "bv", "AF_fun_typ x b c tau s" is true, and that "Theta" with an empty set is well-formed with "b'". It shows that the function type "check_funtyp" w...
The lemma named "iwf_globalize'" is based on three assumptions. The first assumption is that the local incredible well-formed tree "t" is associated with the entry "ent". The second assumption is that for any element "x" in the first part of the entry "ent", "x" is closed. The third assumption is that the second part o...
A word is accepted if and only if it is accepted by the Rabin acceptance condition.
If Q is a Boolean, then the copy of Q is equal to Q itself.
If a sequence of events 'evs' is in 'sr', then what the Spy knows after the event "Inputs Spy C X" is added to 'evs' is the same as what the Spy knew before, plus the new information 'X'.
For any shared state 'S', any set of resources 'R', and any set 'L', given the following conditions: 1. The ownership of threads in 'ts' is distinct. 2. The sharing is consistent in 'ts' with respect to 'S'. 3. For all threads 'i' with properties 'p', 'is', 'O', 'R', 'D', 'theta', and 'sb', if 'i' is less than the len...
The value range of the relation 'r' minus the value range of the relation 's' is a subset of the value range of the relation 'r' minus 's'.
Applying the fls_base_factor function twice on a formal Laurent series 'f' of elements of type 'a' with zero as an identity element, is the same as applying the fls_base_factor function once on 'f'.
A triple (u, a#xs, v) is in the transitive closure of a relation R if and only if there exists an intermediate element 'uh' such that (u, a, uh) is in R and (uh, xs, v) is in the transitive closure of R. Here, 'a#xs' represents a list where 'a' is the first element and 'xs' is the rest of the list.
For a quantum program represented by a while loop with predicate P, matrices M0 and M1, and a density operator D, if the initial state satisfies the condition that it is a superposition of the state after executing M0 and the state after executing M1, weighted by the adjoints of exM0 and exM1 respectively, then after t...
If a relation Ra is related, then the option of that relation Ra is also related.
The lemma shows that "map" is a functor from category A to category B.
The lemma named "product_qbs_canonical1" assumes that for any 'i' in 'I', 'f i' is a quasi-Borel measurable function from 'Y' to 'X i'. And for any 'i' not in 'I', 'f i' is an undefined function. Under these assumptions, it shows that the function 'f' that takes 'y' and 'i' and returns 'f i y' is a quasi-Borel measurab...
The step function applied to any input 'x' does not result in an empty set.
The lemma named "pderivs_lang_Star" shows that the set of partial derivatives of the language "Star r" with respect to the universal set "UNIV1" is a subset of the product of the set of partial derivatives of the language "r" with respect to "UNIV1" and the language "Star r".
If the lemma DM4 holds for a certain eta, then the lemma iDM4 also holds for that eta.
For all 'x', 'p a' and if 'p x' then 'p of f x' implies 'p of f of f x' is equivalent to for all 'x', either 'p a' is false or 'p x' is true or 'p of f of f x' is true, and either 'p a' is false or 'p of f x' is false or 'p of f of f x' is true.
In the context of a group, if H and K are subgroups of G and H is a subset of K, then the group operation of H and K in G equals K.
If a sequence 'ss' is in the list of sequences 'S' and the sequence 'ss' is not distinct when left-conjugated, then there exist elements 'a', 'b' and sequences 'as', 'bs', 'cs' such that 'ss' equals the concatenation of 'as', 'a', 'bs', 'b', and 'cs', and the sum of the elements in 'ss' equals the sum of the elements i...
If 'x' overlaps 'y', then 'y' is a part of the product of 'x' and 'y'.
The representation of the absolute value of the modulus of 'm' and the integer cardinality of 'a' is equal to the modulus of 'm' and the integer cardinality of 'a'.
If the analysis of set G is a subset of the analysis of set G', and the analysis of set H is a subset of the analysis of set H', then the analysis of the union of sets G and H is a subset of the analysis of the union of sets G' and H'.
If a property P is bounded and non-negative, then P is sound.
The reach of l1 is a subset of l1_iagreement_Resp in the context of strong adversaries in key agreement.
There exists a function from one set to another that is injective (each element in the first set is mapped to a unique element in the second set) and not surjective (not all elements in the second set have a corresponding element in the first set).
The lemma lt2 has two parts: 1. For any integers k1, k2, j1, j2, if k1 is less than j1, then the pair (k1, k2) is less than the pair (j1, j2) in the lt2 relation. 2. For any integers k1, k2, j1, j2, if k1 is less than or equal to j1 and k2 is less than j2, then the pair (k1, k2) is less than the pair (j1, j2) in the ...
The transformation of a lazy list 'Lazy_llist xs' with terminal 'b' is equivalent to a lazy terminal list 'Lazy_tllist' where, if the lazy list 'xs' is empty (None), the result is a terminal 'b', otherwise (Some), the result is a pair consisting of the first element 'x' and the transformation of the rest of the list 'y...
The lemma named "eqf_MFunI" assumes that the lengths of the lists "sss" and "Cs" are equal, and the length of the list "ts" is also equal to the length of "Cs". Furthermore, for any index "i" that is less than the length of "Cs", the "i"-th element of "ts" is functionally equivalent to the pair consisting of the "i"-th...
The first elements of the pairs in the second element of the result of the function "factor_complex_main" applied to a polynomial "p" are equal to the roots of the complex polynomial "p".
If "x" is an indexed member of set "s", then "x" is also a member of set "s".
The Dirichlet product of the Dirichlet product of functions f and g with function h is equal to the Dirichlet product of function f with the Dirichlet product of functions g and h. This is the associativity property of the Dirichlet product.
For a relation W that maps to a monad M, and a state transition relation S, the function "tell_state" preserves this relation. This is expressed in terms of a transfer rule, which is a tool used in Isabelle to express that certain properties or structures are preserved under some transformation or relation.
"it is not the case that the social choice function of R11 is strictly preferred, according to the Fishburn preference relation on the set A2, to the social choice function of R13."
The rotation of Nibble2 is not equal to Nibble3.
The set of free variables in the relational calculus expression Q, where x is bottom, is a subset of the set of free variables in Q excluding x.
The lemma named "map_set_parametric" that includes lifting syntax shows that if a function maps elements from set A to set B, then the image of this function also maps the relation of set A to the relation of set B.
If you replace a subterm at position 'p' in a ground term 's' with the subterm at the same position 'p' in 's', the ground term remains unchanged.
A set S cannot be a facet of itself.
The sum of the function "f" over the set "X" is an element of the set "S" if for every element "x" in the set "X", the function "f" of "x" is an element of the set "S".
The function "alw (always)" relates to a property P in a sequence s if and only if P relates to the first element and the second element of the sequence s, and "alw" relates to P in the rest of the sequence s.
The length of the array obtained by getting the array at index 'r' from the updated heap is equal to the length of the array obtained by getting the array at index 'r' from the original heap. Here, the heap is updated by replacing the element at index 'i' in array 'a' with 'x'.
If 'x' is a vector, then 'x' is down-closed.
: If the function 'f' applied to the state 'Σ' yields some value 'n', and if the function 'MST' applied to the pair (first of 'f', second of 'f') yields a specification 'mkSPEC' with 'Cachera n' and 'Anno2', and if there exists parameters 'par', body 'body', and location 'l0' such that 'mbody_is' holds for (first of '...
Making a tree with the successor of a number n is equivalent to making a tree with n and then combining it with another tree made with n.
The lemma defines two terms: gamma, which is a function that always returns true, and m, which is defined as MatchAny. It shows that given a list of rules for 'FORWARD', 'foo', 'bar', and 'baz', and given gamma and a packet p, if we start with the rule 'Call FORWARD' and an undecided state, we will end up with a final ...
The lemma "encode_goal_state_is_cnf" assumes that "\<Pi>" is a valid problem in STRIPS (Stanford Research Institute Problem Solver) format. It shows that the encoding of the goal state "\<Pi>" at time "t" is in conjunctive normal form (CNF).
In the evaluation of a logical expression, the implication of the existence of a universal predicate with variables 1 and 0 implies the universality of the existence of a predicate with variables 0 and 1.
If from the assumption set "H" with the addition of the assumption "A" with "i" replaced by "x", we can derive "B", then from the assumption set "H" with the addition of the assumption "for all i, A holds", we can also derive "B".
The lemma "yields_isomorphic_representation" shows that "T0.trnr\<^sub>\<epsilon> r \<rho>" is a morphism from "g star f star" to "r", and that "T0.trnr\<^sub>\<epsilon> r \<rho>" is an isomorphism.
: Given the assumptions that the approximation of a straight-line program (slp) with parameters p, order ord, dimension d, and slp VS equals some X, the slp is defined as the slp of a list of floating-point arithmetic operations (fas), the dimension d is equal to the length of fas, the error e is a function mapping an...
Dropping zero elements from any list "a" will result in the original list "a".
This is a lemma from the Stateful Protocol Model theory in Automated Stateful Protocol Verification. It consists of six parts: 1. If the arity of function 'f' is not zero, then the type of the function 'Fu f' applied to a list of terms 'T' is the type composition of 'Fu f' and the types of the terms in 'T'. 2. If the ...
The domain of a partial bijective function is equal to the range of the inverse of that partial bijective function.
The bottom element is similar to itself under the function map.
A branch is done if and only if there exists a proposition 'p' such that 'p' is in the set 'z' and the negation of 'p' is also in the set 'z'.
For any given precision, number 'n', 'i', and 'j', the lower bound of the sine-cosine auxiliary function when the input is zero is less than or equal to one.
The logical implication operator (->) is equivalent to a function that takes an argument x and applies the logical implication operator to x. Furthermore, the logical implication operator is also equivalent to a function that takes two arguments, x and y, and applies the logical implication from x to y.
If the session key is not expired at a certain timestamp in a sequence of events, and the sum of the service key's lifetime and this timestamp is less than or equal to the sum of the authentication key's lifetime and another timestamp, then the authentication key is not expired at this other timestamp in the sequence o...
If "t" is a heap and "t" has children, then the priority of "t" is less than or equal to the minimum priority among its children.
: 1. The set of depth-first search (dfs) values in 'xs' is equal to the set of first elements (fst) in the depth-first search of 'alist' in 'xs'. 2. The set of depth-first search priorities in 'xs' is equal to the set of second elements (snd) in the depth-first search of 'alist' in 'xs'.
The multiset obtained from the "siftDown" operation on a set "t" is equal to the multiset of the original set "t".
The lemma named "shuffle_to_top_range'" assumes that the pair (x, y) is in the result of the function "shuffle_to_top" with parameters Y and d. It shows that x is in the field of the preference relation for d or in the result of the function dX with parameters Y and d, and similarly for y.
The lemma named "smallD_nonneg_real" assumes that function "f" is little-o of function "g" at filter "F", that function "f" is non-negative at some point in the future according to filter "F", and that "c" is a positive real number. It shows that, at some point in the future according to filter "F", the relation "R" ho...
If there exist steps, k, and l such that the steps0 function, when applied to the tuple (1, [], <ns::nat list>) and Turing machine 'tm', equals the tuple (0, Bk raised to the power of k, <n::nat> appended with Bk raised to the power of l), then the function 'fun_of_tm' applied to the Turing machine 'tm' and 'ns' equals...
The moment of a standard normal distribution is integrable with respect to the Lebesgue measure. This is calculated as the product of the standard normal density function of x and x raised to the power of k.
The lemma named "sumAssocRight" is defined with fixed elements P, Q, and R of type pi, and Rel of type set of pairs of pi. It assumes that the identity relation is a subset of Rel. It shows that the sum of P and the sum of Q and R is related to the sum of P and Q and R under the relation Rel. This demonstrates the righ...
If x is less than or equal to 1, then the star of x equals 1.