hcju/isabelleps huggingface dataset:
rl_thm': fixes p :: "fm"   and i :: "int"   and bs :: "real list"   and a :: "real"   and R :: "(MIR.num \<times> int) set" assumes "iszlfm p (real_of_int i # bs)"   and "(\<lambda>(b, k). (Inum (a # bs) b, k)) ` R = (\<lambda>(b, k). (Inum (a # bs) b, k)) ` set (\<rho> p)" shows "(\<exists>x. Ifm (real_of_int x # bs) p) = ((\<exists>j\<in>{1..\<delta> p}. Ifm (real_of_int j # bs) (minusinf p)) \<or> (\<exists>(e, c)\<in>R. \<exists>j\<in>{1..c * \<delta> p}. Ifm (a # bs) (\<sigma> p c (Add e (C j)))))"
add_eq_0_iff2: fixes a :: "'a"   and b :: "'a" shows "(a + b = (0::'a)) = (a = - b)"
distinct_1: fixes x1 :: "'b"   and x2 :: "'c" shows "Pro x1 \<noteq> Nom x2"
hom_ps: assumes "\<And>f. f \<in> F \<Longrightarrow> homogeneous f" shows "hom_decomp ps"
range_disjointed_sets': fixes \<Omega> :: "'c set"   and M :: "'c set set"   and A :: "nat \<Rightarrow> 'c set" assumes "algebra \<Omega> M"   and "range A \<subseteq> M" shows "range (disjointed A) \<subseteq> M"
mpt_factorI': fixes proj :: "'a \<Rightarrow> 'c"   and M2 :: "'c measure"   and T2 :: "'c \<Rightarrow> 'c" assumes "proj \<in> measure_preserving M M2"   and "AE x in M. proj (T x) = T2 (proj x)"   and "sigma_finite_measure M2"   and "T2 \<in> M2 \<rightarrow>\<^sub>M M2" shows "mpt_factor proj M2 T2"
assms_3: shows "a \<otimes> f (deg R f) \<noteq> \<zero>"
quat_proj_add: fixes i :: "nat"   and x :: "quat"   and y :: "quat" assumes "i \<le> 3" shows "quat_proj (x + y) i = quat_proj x i + quat_proj y i"
d: shows "as\<^sub>2 = map (\<lambda>v. (v, False)) (delete_effects_of op)"
case_prod_Pair_iden: fixes p :: "'b \<times> 'c" shows "(case p of (x, y) \<Rightarrow> (x, y)) = p"
simps_3: fixes v :: "'c" shows "dbm_add (Lt v) \<infinity> = \<infinity>"
val_1px: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "0 \<le> v (1\<^sub>r \<plusminus> x)" shows "0 \<le> v x"
cong: fixes ops :: "'c approximate_set_ops"   and optns :: "'c numeric_options" shows "approximate_sets.above_sctn_impl ops optns \<equiv> approximate_sets.above_sctn_impl ops optns"
invert_side_of_triangle_2: fixes h :: "'a \<times> 'a"   and f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "CC.arr h"   and "CC.comp f g = h"   and "CC.iso g" shows "CC.seq h (CC.inv g) \<and> f = CC.comp h (CC.inv g)"
Union_least: fixes A :: "'b set set"   and C :: "'b set" assumes "\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C" shows "\<Union> A \<subseteq> C"
ordLeq_iff_ordLess_or_ordIso: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" shows "(r \<le>o r') = (r <o r' \<or> r =o r')"
Dmonoid_def: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" shows "Dmonoid prod one \<equiv> Dsemi prod \<and> Dmonoid_axioms prod one"
select_convs_7: fixes igWls :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and igWlsAbs :: "'b \<times> 'g \<Rightarrow> 'd \<Rightarrow> bool"   and igVar :: "'b \<Rightarrow> 'c \<Rightarrow> 'i"   and igAbs :: "'b \<Rightarrow> 'c \<Rightarrow> 'i \<Rightarrow> 'd"   and igOp :: "'h \<Rightarrow> ('e \<Rightarrow> 'i option) \<Rightarrow> ('f \<Rightarrow> 'd option) \<Rightarrow> 'i"   and igFresh :: "'b \<Rightarrow> 'c \<Rightarrow> 'i \<Rightarrow> bool"   and igFreshAbs :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool"   and igSwap :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'i \<Rightarrow> 'i"   and igSwapAbs :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'd"   and igSubst :: "'b \<Rightarrow> 'i \<Rightarrow> 'c \<Rightarrow> 'i \<Rightarrow> 'i"   and igSubstAbs :: "'b \<Rightarrow> 'i \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'd"   and more :: "'j" shows "igFreshAbs \<lparr>igWls = igWls, igWlsAbs = igWlsAbs, igVar = igVar, igAbs = igAbs, igOp = igOp, igFresh = igFresh, igFreshAbs = igFreshAbs, igSwap = igSwap, igSwapAbs = igSwapAbs, igSubst = igSubst, igSubstAbs = igSubstAbs, \<dots> = more\<rparr> = igFreshAbs"
rep_eq: fixes x :: "'c \<Rightarrow> ('b, 'a) uexpr"   and xa :: "('c, 'a) uexpr" shows "\<lbrakk>msubst x xa\<rbrakk>\<^sub>e = (\<lambda>b. \<lbrakk>x (\<lbrakk>xa\<rbrakk>\<^sub>e b)\<rbrakk>\<^sub>e b)"
1: assumes "ord s t"   and "ord t s" shows "lookup s = lookup t"
wf_if_measure: fixes P :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> nat"   and g :: "'b \<Rightarrow> 'b" assumes "\<And>x. P x \<Longrightarrow> f (g x) < f x" shows "wf {(y, x). P x \<and> y = g x}"
s\<^sub>2: shows "s\<^sub>2 = s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>(l \<mapsto> v), \<E> [VE (Loc l)]))"
nub_def: shows "nub \<equiv> \<mu> nub. \<Lambda> v. Fixrec.run\<cdot>(match_lnil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>lnil) +++ match_lcons\<cdot>v\<cdot>(\<Lambda> x xs. Fixrec.succeed\<cdot>(x :@ nub\<cdot>(lfilter\<cdot>(neg oo Abs_cfun ((=\<^sub>B) x))\<cdot>xs))))"
f_1: fixes x :: "real" assumes "x \<in> {a..b}" shows "(F has_real_derivative f x) (at x)"
T2: fixes w :: "i"   and x :: "\<kappa>"   and F :: "\<Pi>\<^sub>1" shows "[\<lbrace>x,F\<rbrace> in w] = (\<exists>r o\<^sub>1. Some r = d\<^sub>1 F \<and> Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> o\<^sub>1 \<in> en r)"
IH: shows "\<forall>ys. length ys < length xs \<longrightarrow> total_on (set ys) R \<longrightarrow> (\<forall>x\<in>set ys. (x, x) \<in> R) \<longrightarrow> (\<forall>x xa. (x, xa) \<in> set_pmf (rquicksort R ys) \<longrightarrow> Linorder_Relations.sorted_wrt R x \<and> mset x = mset ys)"
assms_4: shows "pseudominimal_in (P .\<^bold>\<le> x) w"
axioms_1: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and alt :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_state_alt return bind get put alt" shows "monad_state return bind get put"
starfun2_star_of: fixes f :: "'d \<Rightarrow> 'b \<Rightarrow> 'c"   and x :: "'d" shows "(*f2* f) (star_of x) = *f* f x"
init_fin_simps_3: fixes t :: "'t"   and x :: "'x2"   and m :: "'m2"   and ta :: "('l, 't, status \<times> 'x2, 'm2, 'w, 'o action) thread_action"   and xm' :: "(status \<times> 'x2) \<times> 'm2" shows "r2.init_fin t ((Running, x), m) ta xm' = ((\<exists>tab. ta = convert_TA_initial (convert_obs_initial tab) \<and> (\<exists>x' m'. xm' = ((Running, x'), m') \<and> r2.r_syntax t x m tab x' m')) \<or> ta = \<lbrace>ThreadFinishAction\<rbrace> \<and> xm' = ((Finished, x), m) \<and> final2 x)"
simplex_unsat0: fixes cs :: "('a \<times> constraint) list"   and I :: "'a list" assumes "solve_exec_code cs = Inl I" shows "minimal_unsat_core (set I) cs"
odd_p: fixes p :: "nat"   and a :: "int" assumes "GAUSS p a" shows "odd p"
GSMP_disjoint_def: fixes P1 :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set"   and P2 :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set"   and Secrets :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set" shows "GSMP_disjoint P1 P2 Secrets \<equiv> GSMP P1 \<inter> GSMP P2 \<subseteq> Secrets \<union> {m. {} \<turnstile>\<^sub>c m}"
T2I: fixes T :: "'b set set" assumes "topology T"   and "\<And>x y. \<lbrakk>x \<in> carrier\<^bsub>T\<^esub>; y \<in> carrier\<^bsub>T\<^esub>; x \<noteq> y\<rbrakk> \<Longrightarrow> \<exists>u\<in>nhds\<^bsub>T\<^esub> x. \<exists>v\<in>nhds\<^bsub>T\<^esub> y. u \<inter> v = {}" shows "T2 T"
fls_residue_deriv_times_lr_inverse_eq_subdegree_1: fixes y :: "'a"   and f :: "'a fls" assumes "y * fls_nth f (fls_subdegree f) = (1::'a)"   and "fls_nth f (fls_subdegree f) * y = (1::'a)" shows "fls_residue (fls_deriv f * fls_right_inverse f y) = of_int (fls_subdegree f)"
bex2I: fixes a :: "'d"   and b :: "'e"   and S :: "('d \<times> 'e) set"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "(a, b) \<in> S"   and "(a, b) \<in> S \<Longrightarrow> P a b" shows "\<exists>a b. (a, b) \<in> S \<and> P a b"
semigroup_add_ow_axioms: shows "semigroup_add_ow U (+\<^sub>o\<^sub>w)"
nfresh_isfresh: fixes S :: "'a set"   and n :: "nat" assumes "finite S" shows "set (nfresh S n) \<inter> S = {}"
Int_assoc: fixes A :: "'k set"   and B :: "'k set"   and C :: "'k set" shows "A \<inter> B \<inter> C = A \<inter> (B \<inter> C)"
P_subset: shows "P x \<subseteq> \<Union> (set (map P S)) \<union> V"
time_closest_pair_conv_closest_pair_recurrence: fixes ps :: "(int \<times> int) list" shows "real (time (closest_pair_tm ps)) \<le> closest_pair_time (length ps)"
le: fixes i :: "'n" shows "a * (HMA_Connect.spectral_radius A * v $h i) + b * (HMA_Connect.spectral_radius A * w $h i) \<le> a * (local.B *v v) $h i + b * (local.B *v w) $h i"
prod_list_prod_nth: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "'b list" assumes "class.monoid_mult one times" shows "prod_list xs = prod ((!) xs) {..<length xs}"
is_zero_def: fixes a :: "('a, 'b) IEEE.float" shows "is_zero a = (IEEE.exponent a = 0 \<and> fraction a = 0)"
sup_left_idem: fixes x :: "'c multiset"   and y :: "'c multiset" shows "x \<union># (x \<union># y) = x \<union># y"
2: fixes k :: "real" shows "set_borel_measurable lebesgue s (f k)"
indets_homogenize_subset: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "indets (homogenize x p) \<subseteq> insert x (indets p)"
TESL_interp_stepwise_sporadicon_coind_unfold: fixes K\<^sub>1 :: "clock"   and \<tau> :: "'b tag_const"   and K\<^sub>2 :: "clock"   and n :: "nat" shows "\<lbrakk> K\<^sub>1 sporadic \<tau> on K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup> = \<lbrakk> K\<^sub>1 \<Up> n \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<inter> \<lbrakk> K\<^sub>2 \<Down> n @ \<tau> \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<union> \<lbrakk> K\<^sub>1 sporadic \<tau> on K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> Suc n\<^esup>"
apath_simps_2: fixes G :: "('b, 'c) pre_digraph"   and u :: "'b"   and e :: "'c"   and es :: "'c list"   and w :: "'b" assumes "wf_digraph G" shows "pre_digraph.apath G u (e # es) w = (e \<in> arcs G \<and> tail G e = u \<and> pre_digraph.apath G (head G e) es w \<and> tail G e \<notin> set (pre_digraph.awalk_verts G (head G e) es))"
inj_on_insert: fixes f :: "'k \<Rightarrow> 'l"   and a :: "'k"   and A :: "'k set" shows "inj_on f (insert a A) = (inj_on f A \<and> f a \<notin> f ` (A - {a}))"
prems_2: shows "l \<subseteq>\<^sub>m [Vs [\<mapsto>] ls]"
simps_1: fixes l :: "lambda"   and \<beta> :: "nat \<Rightarrow> contour option" shows "benv_in_d (DC (l, \<beta>)) = {\<beta>}"
l4_3: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C"   and "Bet A' B' C'"   and "Cong A C A' C'"   and "Cong B C B' C'" shows "Cong A B A' B'"
image_vimage_eq: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'a set" shows "\<P> f (f -` A) = A \<inter> range f"
intro: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and m :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option" assumes "blinding_of_on UNIV h bo"   and "merge_on_axioms A h bo m" shows "merge_on A h bo m"
assms_4: shows "index (mtf2 n q xs) x < index xs q - n"
conjugate_square_greater_0_vec: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "((0::'b) < v \<bullet>c v) = (v \<noteq> 0\<^sub>v n)"
G: shows "(\<forall>r\<in>ls.\<alpha> \<delta>. hm.\<alpha> (brc_rcm_init \<delta>) r = Some (card (set (rhsq r)))) \<and> hm.invar (brc_rcm_init \<delta>)"
outside_connected_component_lt: fixes S :: "'a set" shows "outside S = {x. \<forall>B. \<exists>y. B < norm y \<and> connected_component (- S) x y}"
has_derivative_norm: fixes x :: "'b" assumes "x \<noteq> (0::'b)" shows "(norm has_derivative (\<lambda>h. h \<bullet> sgn x)) (at x)"
mono_seq_D_right: fixes S :: "'a process"   and S' :: "'a process"   and P :: "'a process" assumes "S \<sqsubseteq>\<^sub>D S'" shows "(P `;` S) \<sqsubseteq>\<^sub>D (P `;` S')"
Auth_Fst: fixes x :: "rat"   and y :: "rat"   and H :: "msg set" assumes "Auth_Data x y \<in> items H"   and "Pri_AgrK y \<in> items H" shows "Pri_AgrK x \<in> items H"
acyclic_3a_implies_4b: fixes x :: "'a" assumes "acyclic_3a x" shows "acyclic_4b x"
asymp_powser_o: fixes h :: "real \<Rightarrow> real"   and F :: "real filter"   and f :: "real \<Rightarrow> real"   and cs :: "real msstream"   and g :: "real \<Rightarrow> real"   and G :: "real filter" assumes "asymp_powser h F f cs"   and "filterlim g F G" shows "asymp_powser (h \<circ> g) G (f \<circ> g) cs"
assms_2: shows "\<forall>n\<^sub>0. (\<forall>i<k. P (n\<^sub>0 + i)) \<longrightarrow> P (n\<^sub>0 + k)"
get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'p, 'r option) RElement_scheme \<Rightarrow> 'w" assumes "h \<turnstile> ok get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter" shows "ptr |\<in>| element_ptr_kinds h"
da_elim_cases_19: fixes Env :: "env"   and B :: "lname set"   and c1 :: "stmt"   and c2 :: "stmt"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>c1 Finally c2\<rangle>\<guillemotright> A" obtains C1 :: "assigned" and C2 :: "assigned" where "Env\<turnstile> B \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1" and "Env\<turnstile> B \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2" and "nrm A = nrm C1 \<union> nrm C2" and "brk A = (brk C1 \<Rightarrow>\<union>\<^sub>\<forall> nrm C2) \<Rightarrow>\<inter>  brk C2"
rightUniquePair: fixes f :: "('a \<times> 'b) set"   and x :: "'a"   and y :: "'b" assumes "runiq f"   and "(x, y) \<in> f" shows "y = f ,, x"
assms_1: fixes n :: "nat" shows "S n \<in> lmeasurable"
add: shows "mset_set (set_mset M) \<subseteq># M"
restricted_deduct_if_deduct: fixes M :: "('fun, 'var) Term.term set"   and P :: "('fun, 'var) Term.term \<Rightarrow> bool"   and m :: "('fun, 'var) Term.term" assumes "\<forall>m\<in>M. \<forall>f T. Fun f T \<sqsubseteq> m \<longrightarrow> P (Fun f T)"   and "\<forall>f T t. M \<turnstile> Fun f T \<longrightarrow> P (Fun f T) \<longrightarrow> t \<in> set T \<longrightarrow> P t"   and "\<forall>t K T k. M \<turnstile> t \<longrightarrow> P t \<longrightarrow> Ana t = (K, T) \<longrightarrow> M \<turnstile> k \<longrightarrow> k \<in> set K \<longrightarrow> P k"   and "M \<turnstile> m"   and "P m" shows "\<langle>M;P\<rangle> \<turnstile>\<^sub>r m"
compute_exponent: fixes m :: "int"   and e :: "int" shows "Float.exponent (Float m e) = (if m = 0 then 0 else if even m then Float.exponent (normfloat (Float m e)) else e)"
nonneg: shows "nonneg_cexpr (shift_var_set V) (case_nat t \<Gamma>) e"
transfer_connect: fixes s :: "'s net_state"   and i :: "nat"   and i' :: "nat"   and s' :: "'s net_state"   and n :: "net_tree"   and \<sigma> :: "nat \<Rightarrow> 'g"   and \<zeta> :: "'l net_state" assumes "(s, connect(i, i'), s') \<in> automaton.trans (pnet np n)"   and "s \<in> reachable (pnet np n) TT"   and "netgmap sr s = netmask (net_tree_ips n) (\<sigma>, \<zeta>)"   and "wf_net_tree n" obtains \<sigma>'' :: "nat \<Rightarrow> 'g" and \<zeta>'' :: "'l net_state" where "((\<sigma>, \<zeta>), connect(i, i'), \<sigma>'', \<zeta>'') \<in> automaton.trans (opnet onp n)" and "\<forall>j. j \<notin> net_ips \<zeta> \<longrightarrow> \<sigma>'' j = \<sigma> j" and "netgmap sr s' = netmask (net_tree_ips n) (\<sigma>'', \<zeta>'')"
bal_delete: fixes t :: "('c \<times> 'd) tree234"   and x :: "'c" assumes "bal t" shows "bal (Tree234_Map.delete x t)"
pre_zeta_bound: fixes s :: "complex"   and a :: "real" assumes "0 < Re s"   and "0 < a" shows "cmod (pre_zeta a s) \<le> (1 + cmod s / Re s) / 2 * a powr - Re s"
l_get_parent_wf_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog" shows "l_get_parent_wf type_wf known_ptr known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_parent \<equiv> l_known_ptrs known_ptr known_ptrs \<and> l_get_parent_wf_axioms type_wf known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_parent"
partition_on_cart_prod: fixes I :: "'b set"   and A :: "'c set"   and G :: "'c set set" assumes "0 < card I"   and "A \<noteq> {}"   and "G \<noteq> {}"   and "partition_on A G" shows "partition_on (A \<times> I) {g \<times> I |g. g \<in> G}"
sorted_list_of_set_range: fixes m :: "nat"   and n :: "nat" shows "sorted_list_of_set {m..<n} = [m..<n]"
K2_imp_M_neg: fixes v :: "(real, 3) vec" assumes "v \<noteq> 0"   and "proj2_abs v \<in> hyp2" shows "v \<bullet> (M *v v) < 0"
less_eq_prod_def: fixes p :: "'a \<times> 'b"   and q :: "'a \<times> 'b" shows "(p \<le> q) = prod_le (\<le>) (\<le>) p q"
a5: shows "all_security_requirements_fulfilled M \<lparr>nodes = V, edges = E'\<rparr>"
moebius_inversion: fixes g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "\<And>n. 0 < n \<Longrightarrow> g n = (\<Sum>d | d dvd n. f d)"   and "0 < n" shows "f n = dirichlet_prod moebius_mu g n"
"0_7": shows "\<forall>(i, p)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A. \<forall>(j, q)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A. i \<noteq> j \<longrightarrow> mgu (pair p) (pair q \<cdot> var_rename (max_var_set (fv (pair p)))) = None"
equal: fixes equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.equal equal" shows "equal = (=)"
Int_mono: fixes A :: "'c set"   and C :: "'c set"   and B :: "'c set"   and D :: "'c set" assumes "A \<subseteq> C"   and "B \<subseteq> D" shows "A \<inter> B \<subseteq> C \<inter> D"
rtrancl_path_distinct: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list"   and y :: "'a" assumes "rtrancl_path r x xs y" obtains xs' :: "'a list" where "rtrancl_path r x xs' y" and "distinct (x # xs')" and "set xs' \<subseteq> set xs"
drop_perp_perp: fixes l :: "proj2_line"   and p :: "proj2" shows "M_perp l (drop_perp p l)"
not_nonnegative_int_iff: fixes k :: "int" shows "(0 \<le> NOT k) = (k < 0)"
sorted_wrt_dist_def: fixes p :: "(real, 'b) vec" shows "sorted_wrt_dist p \<equiv> sorted_wrt (\<lambda>p\<^sub>0 p\<^sub>1. dist p\<^sub>0 p \<le> dist p\<^sub>1 p)"
elims: fixes x :: "'b tree"   and y :: "'b tree" assumes "mirror x = y" obtains "x = \<langle>\<rangle>" and "y = \<langle>\<rangle>"   | l :: "'b tree" and xa :: "'b" and r :: "'b tree" where "x = \<langle>l, xa, r\<rangle>" and "y = \<langle>mirror r, xa, mirror l\<rangle>"
nolub_def: fixes f :: "atom fm"   and xs :: "'a list"   and l :: "'a"   and x :: "'a"   and u :: "'a" shows "nolub f xs l x u = ((\<forall>y\<in>{l<..<x}. y \<notin> LB f xs) \<and> (\<forall>y\<in>{x<..<u}. y \<notin> UB f xs))"
composition_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> A"   and "b \<in> A" shows "mult a b \<in> A"
proots_withinI: fixes p :: "'a poly"   and x :: "'a"   and s :: "'a set" assumes "poly p x = (0::'a)"   and "x \<in> s" shows "x \<in> proots_within p s"
\<I>\<^sub>\<tau>': shows "\<lbrakk>{}; {}; unlabel \<A>\<rbrakk>\<^sub>s \<I>\<^sub>\<tau>"
right_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier C"   and "y \<in> carrier C"   and "z \<in> carrier C" shows "(y \<otimes>\<^bsub>C\<^esub> x = z \<otimes>\<^bsub>C\<^esub> x) = (y = z)"
left_gpv_map: fixes f :: "'m \<Rightarrow> 'h"   and g :: "'n \<Rightarrow> 'i"   and gpv :: "('m, 'n, 'k) gpv"   and h :: "'o \<Rightarrow> 'j" shows "left_gpv (map_gpv f g gpv) = map_gpv f (map_sum g h) (left_gpv gpv)"
tassoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "a \<cdot> b \<cdot> c = a \<cdot> (b \<cdot> c)"
prime_field: fixes a :: "nat" assumes "a < order \<G>"   and "a \<noteq> 0" shows "coprime a (order \<G>)"
inl_invertible: fixes op :: "op"   and xs :: "dynamic list"   and opinl :: "opinl" assumes "inl op xs = Some opinl" shows "deinl opinl = op"
assms: shows "\<forall>xs A. (\<forall>X\<in>A. (xs, X) \<in> failures P) \<longrightarrow> (xs, \<Union>X\<in>A. X) \<in> failures P"
intros_6: shows "CK_nf (C \<circ> K \<circ> C)"
tab_in_vhom': shows "\<guillemotleft>D.inv (\<Phi> (r, f)) \<cdot>\<^sub>D F \<rho> : F g \<Rightarrow>\<^sub>D F r \<star>\<^sub>D F f\<guillemotright>"
bin_nth_uint': fixes w :: "'b word"   and n :: "nat" shows "bit (uint w) n = (rev (bin_to_bl (size w) (uint w)) ! n \<and> n < size w)"
disj_def: shows "(\<^bold>\<or>) \<equiv> \<lambda>x. (\<^bold>\<rightarrow>) (\<^bold>\<not>x)"
assigns_subst: fixes \<sigma> :: "'b \<Rightarrow> 'b"   and \<rho> :: "'b \<Rightarrow> 'b" shows "\<lceil>\<sigma>\<rceil>\<^sub>s \<dagger> \<langle>\<rho>\<rangle>\<^sub>a = \<langle>\<rho> \<circ> \<sigma>\<rangle>\<^sub>a"
prems_2: shows "\<Union> (fv\<^sub>r\<^sub>c\<^sub>v ` set S') \<union> \<Union> (fv_r\<^sub>e\<^sub>q assign ` set S') \<subseteq> wfvarsoccs\<^sub>s\<^sub>t (\<langle>assign: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S) \<union> V"
le_mod_geq: fixes n :: "nat"   and m :: "nat" assumes "n \<le> m" shows "m mod n = (m - n) mod n"
continuous_on_sin: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. sin (f x))"
monad_prob_optionT: fixes return :: "'a option \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a option \<Rightarrow> 'b) \<Rightarrow> 'b"   and sample :: "'c pmf \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_prob return bind sample" shows "monad_prob (return_option return) (bind_option return bind) (sample_option sample)"
rr: shows "\<rho> \<sqsubseteq> \<rho>'"
read_before: fixes r :: "nat" assumes "r \<in> read_actions E" shows "P,E \<turnstile> ws r \<le>hb r"