hcju/isabelleps huggingface dataset:
F: shows "(Asset n, Crypt (Auth_ShaKey n) (PriKey S)) \<in> s'"
proj2_line_through_incident_2: fixes q :: "proj2"   and p :: "proj2" shows "proj2_incident q (proj2_line_through p q)"
fs_def: shows "fs = (\<lambda>k x. (- 1) ^ k * fact (Suc k) / x ^ (k + 2))"
UN_eqpoll_UN: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'd set"   and C :: "'c \<Rightarrow> 'e set" assumes "\<And>x. x \<in> A \<Longrightarrow> B x \<approx> C x"   and "pairwise (\<lambda>x y. disjnt (B x) (B y)) A"   and "pairwise (\<lambda>x y. disjnt (C x) (C y)) A" shows "\<Union> (B ` A) \<approx> \<Union> (C ` A)"
src_vcomp: fixes \<nu> :: "'d"   and \<mu> :: "'d" assumes "D.seq \<nu> \<mu>" shows "src\<^sub>D (\<nu> \<cdot>\<^sub>D \<mu>) = src\<^sub>D \<nu>"
sup_aci_4: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.semilattice_sup sup less_eq less" shows "sup x (sup x y) = sup x y"
fuv: fixes x :: "real" assumes "x \<in> {u..<v}" shows "f x \<noteq> f 1"
assms_4: shows "der \<in> borel_measurable (G matur)"
CSHORun_def: fixes A :: "('d, 'e, 'f, 'g) CHOAlgorithm_scheme"   and rho :: "nat \<Rightarrow> 'd \<Rightarrow> 'e"   and HOs :: "nat \<Rightarrow> 'd \<Rightarrow> 'd set"   and SHOs :: "nat \<Rightarrow> 'd \<Rightarrow> 'd set"   and coords :: "nat \<Rightarrow> 'd \<Rightarrow> 'd" shows "CSHORun A rho HOs SHOs coords \<equiv> CHOinitConfig A (rho 0) (coords 0) \<and> (\<forall>r. CSHOnextConfig A r (rho r) (HOs r) (SHOs r) (coords (Suc r)) (rho (Suc r)))"
abstract_primitive_in_doubt_deny_Deny: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and m :: "'b common_primitive match_expr"   and p :: "('b, 'c) tagged_packet_scheme"   and disc :: "'b common_primitive negation_type \<Rightarrow> bool" assumes "primitive_matcher_generic \<beta>"   and "normalized_nnf_match m"   and "matches (\<beta>, in_doubt_deny) m Drop p" shows "matches (\<beta>, in_doubt_deny) (abstract_primitive disc m) Drop p"
rep_proper_id: fixes x :: "\<nu>" shows "rep (x\<^sup>P) = x"
B: fixes x :: "'a" assumes "x \<in> K" shows "norm x \<le> B"
list_app_singletonE: fixes rs\<^sub>1 :: "'b list"   and rs\<^sub>2 :: "'b list"   and x :: "'b" assumes "rs\<^sub>1 @ rs\<^sub>2 = [x]" obtains "rs\<^sub>1 = [x]" and "rs\<^sub>2 = []"   | "rs\<^sub>1 = []" and "rs\<^sub>2 = [x]"
of_nat_eq_1_iff: fixes x :: "nat" shows "(of_nat x = (1::'c)) = (x = 1)"
iso_rews_4: shows "Box_rep\<cdot>\<bottom> = \<bottom>"
point_equations_3: fixes p :: "'a" assumes "is_point p" shows "p\<^sup>T ; 1 = 1"
q_def: shows "q = (1 + r - d) / (u - d)"
fr_nStore_nReturn2: fixes n :: "nat"   and nSend :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and recv :: "nat \<Rightarrow> 'a Frame list"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nC :: "nat \<Rightarrow> Config"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list"   and t :: "nat"   and k :: "nat"   and j :: "nat" assumes "Cable n nSend recv"   and "\<forall>i<n. FlexRayController (nReturn i) recv (nC i) (nStore i) (nSend i) (nGet i)"   and "DisjointSchedules n nC"   and "IdenticCycleLength n nC"   and "t mod cycleLength (nC k) mem schedule (nC k)"   and "k < n"   and "j < n"   and "j \<noteq> k" shows "nStore j t = nReturn k t"
cross_minus_left: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec" shows "(- x) \<times> y = - (x \<times> y)"
simps_92: fixes x41 :: "'b rexp"   and x42 :: "'b rexp" shows "atoms (Plus x41 x42) = atoms x41 \<union> atoms x42"
antisym: fixes b :: "'a"   and a :: "'a" assumes "orda b a"   and "orda a b" shows "a = b"
arg: fixes x :: "real" shows "(\<Sum>n\<in>basis. (v - u) * (f (g ((v - u) * x + u)) \<bullet> n) * (vector_derivative g (at ((v - u) * x + u) within {0..1}) \<bullet> n)) = (\<Sum>b\<in>basis. f (g ((v - u) * x + u)) \<bullet> b * (v - u) * (vector_derivative g (at ((v - u) * x + u) within {0..1}) \<bullet> b))"
simps_5: fixes x21 :: "'a"   and x22 :: "'a" shows "DenyAllFromTo x21 x22 \<noteq> DenyAll"
distrib_3: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.distrib_lattice inf less_eq less sup" shows "inf x (sup y z) = sup (inf x y) (inf x z)"
convex_hull_def: fixes n :: "nat"   and Vs :: "'b vec set" shows "gram_schmidt.convex_hull n Vs = {x. \<exists>Ws c. finite Ws \<and> Ws \<subseteq> Vs \<and> gram_schmidt.convex_lincomb n c Ws x}"
list_emb_trans_right: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" assumes "list_emb P xs ys"   and "list_emb (\<lambda>y z. P y z \<and> (\<forall>x. P x y \<longrightarrow> P x z)) ys zs" shows "list_emb P xs zs"
dense_linorder_ow_def: fixes U :: "'c set"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ls :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "dense_linorder_ow U le ls \<equiv> linorder_ow U le ls \<and> dense_order_ow U le ls"
frameNilStatEq: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and \<Psi> :: "'d"   and \<Psi>' :: "'d" assumes "assertionAux SCompose SImp SBottom SChanEq" shows "assertionAux.FrameStatEq SImp (\<langle>\<epsilon>, \<Psi>\<rangle>) (\<langle>\<epsilon>, \<Psi>'\<rangle>) = assertionAux.AssertionStatEq SImp \<Psi> \<Psi>'"
cn_norm: shows "\<forall>q\<in>set (coefficients p). isnpolyh q n0"
zmod_le_nonneg_dividend: fixes m :: "int"   and k :: "int" assumes "0 \<le> m" shows "m mod k \<le> m"
action_abscD: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg"   and s :: "'s \<times> ('c \<Rightarrow> real)" assumes "cfg \<in> MDP.cfg_on s" shows "action (absc cfg) \<in> \<K> (abss s)"
u_simps_4: shows "local.dom r\<^sub>0s\<^sub>1.p\<^sub>0 = r\<^sub>0s\<^sub>1.p\<^sub>0"
data_dependency_consistent_instrs: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'p memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "data_dependency_consistent_instrs (dom \<theta>) is"
Cod_Ide: fixes t :: "'c local.term" assumes "Ide t" shows "Cod t = t"
par_step_no_change_on_send_or_receive: fixes \<sigma> :: "nat \<Rightarrow> state"   and s :: "(state, msg, pseqp, pseqp label) seqp \<times> msg list \<times> (msg list, msg, unit, unit label) seqp"   and a :: "msg seq_action"   and \<sigma>' :: "nat \<Rightarrow> state"   and s' :: "(state, msg, pseqp, pseqp label) seqp \<times> msg list \<times> (msg list, msg, unit, unit label) seqp"   and i :: "nat" assumes "((\<sigma>, s), a, \<sigma>', s') \<in> oparp_sos i (oseqp_sos \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V i) (seqp_sos \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G)"   and "a \<noteq> \<tau>" shows "\<sigma>' i = \<sigma> i"
jkbpCn_step_inv: fixes t :: "('a, 'es, 'as) BEState Trace"   and s :: "('a, 'es, 'as) BEState"   and n :: "nat" assumes "t \<leadsto> s \<in> SPR.jkbpCn (Suc n)" shows "t \<in> SPR.jkbpCn n"
has_integral_combine: fixes a :: "real"   and c :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'b"   and i :: "'b"   and j :: "'b" assumes "a \<le> c"   and "c \<le> b"   and "(f has_integral i) {a..c}"   and "(f has_integral j) {c..b}" shows "(f has_integral i + j) {a..b}"
V_reset_closed': fixes u :: "'c \<Rightarrow> real"   and r :: "'c list" assumes "u \<in> V" shows "[r\<rightarrow>0]u \<in> V"
supp_Not\<^sub>\<alpha>: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>" shows "supp (Not\<^sub>\<alpha> t\<^sub>\<alpha>) = supp t\<^sub>\<alpha>"
inner_dirsum_def: fixes As :: "'a set list" shows "inner_dirsum As = (if add_independentS As then \<Sum>A\<leftarrow>As. A else 0)"
Bernstein_coeffs_01_def: fixes p :: "nat"   and P :: "real poly" shows "Bernstein_coeffs_01 p P = map (\<lambda>j. inverse (real (p choose j)) * coeff (reciprocal_poly p P \<circ>\<^sub>p [:1, 1:]) (p - j)) [0..<p + 1]"
sel_4: fixes x1 :: "'a"   and x2 :: "'b"   and x3 :: "nat"   and x4 :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.children (SkewBinomialHeapStruc.Node x1 x2 x3 x4) = x4"
less_trans: fixes y :: "'a list"   and x :: "'a list"   and z :: "'a list" assumes "y <lex x"   and "z <lex y" shows "z <lex x"
group_strategy_proofD: fixes ds :: "'b set"   and \<phi> :: "('b \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> ('c \<Rightarrow> 'a set \<Rightarrow> 'a set) \<Rightarrow> 'b set \<Rightarrow> 'a set"   and Pd :: "'b \<Rightarrow> ('a \<times> 'a) set"   and Ch :: "'c \<Rightarrow> 'a set \<Rightarrow> 'a set"   and ds' :: "'b set"   and Pd' :: "'b \<Rightarrow> ('a \<times> 'a) set" assumes "Y.group_strategy_proof ds \<phi>"   and "ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh Pd Ch"   and "ds' \<subseteq> ds"   and "ds' \<noteq> {}"   and "ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh (override_on Pd Pd' ds') Ch" shows "\<exists>d\<in>ds'. \<forall>y\<in>\<phi> (override_on Pd Pd' ds') Ch ds. y \<notin> AboveS (Pd d) (Y.dX (\<phi> Pd Ch ds) d)"
NoCycles_HA: fixes HA :: "('a, 'b, 'c) hierauto" shows "NoCycles (SAs HA) (CompFun HA)"
WR_succ: fixes i :: "hf"   and y :: "hf" assumes "Ord i" shows "WR (succ i) (q_Eats y y) = WR i y"
AssumeH_3: fixes A1 :: "fm"   and A2 :: "fm"   and A3 :: "fm"   and H :: "fm set" shows "insert A1 (insert A2 (insert A3 H)) \<turnstile> A3"
UN_least: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'd set"   and C :: "'d set" assumes "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C" shows "\<Union> (B ` A) \<subseteq> C"
cases: fixes t :: "'b"   and ws :: "'b \<Rightarrow> 'c wait_set_status option"   and a1 :: "('b, 'c) wait_set_action"   and a2 :: "'b \<Rightarrow> 'c wait_set_status option" assumes "redT_updW t ws a1 a2" obtains t' :: "'b" and w :: "'c" where "a1 = Notify wa__" and "a2 = ws(t'__ \<mapsto> PostWS WSNotified)" and "ws t'__ = \<lfloor>InWS wa__\<rfloor>"   | w :: "'c" where "a1 = Notify wb__" and "a2 = ws" and "\<And>t'. ws t' \<noteq> \<lfloor>InWS wb__\<rfloor>"   | w :: "'c" where "a1 = NotifyAll wc__" and "a2 = (\<lambda>t. if ws t = \<lfloor>InWS wc__\<rfloor> then \<lfloor>PostWS WSNotified\<rfloor> else ws t)"   | w :: "'c" where "a1 = Suspend wd__" and "a2 = ws(t \<mapsto> InWS wd__)"   | t' :: "'b" and w :: "'c" and WSInterrupted :: "wake_up_status" where "a1 = WakeUp t'a__" and "a2 = ws(t'a__ \<mapsto> PostWS WSInterrupted)" and "ws t'a__ = \<lfloor>InWS w\<rfloor>"   | t' :: "'b" where "a1 = WakeUp t'" and "a2 = ws" and "\<And>w. ws t' \<noteq> \<lfloor>InWS w\<rfloor>"   | "a1 = Notified" and "a2 = ws(t := None)"   | "a1 = WokenUp" and "a2 = ws(t := None)"
Infinitesimal_minus_iff: fixes x :: "'a star" shows "(- x \<in> Infinitesimal) = (x \<in> Infinitesimal)"
empty_correct: fixes q :: "('a, 'b) SkewBinomialHeap" shows "(SkewBinomialHeap_loc.to_mset q = {#}) = (q = SkewBinomialHeap_loc.empty)"
fun_cong: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "f = g" shows "f x = g x"
p_5: fixes K1 :: "'a set"   and K2 :: "'a set" assumes "K1 \<in> p"   and "K2 \<in> p"   and "K1 \<noteq> K2" shows "interior K1 \<inter> interior K2 = {}"
injim: fixes A :: "'a set set" shows "inj_on ((`) h) A"
assms_2: fixes d :: "int"   and gs :: "int poly multiset" assumes "factorization_m f (d, gs)" shows "Mf (d, gs) = Mf (c, fs)"
intf_of_assnI: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn" shows "intf_of_assn R TYPE('c)"
list_comb_cases': fixes f :: "'a"   and xs :: "'a list" obtains "is_app (list_comb f xs)"   | "list_comb f xs = f" and "xs = []"
val_surj_n_valTr: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v"   and "\<exists>x\<in>carrier K. v x = 1" shows "Lv K v = 1"
assms_1: shows "\<forall>i\<le>n. \<forall>j\<le>n. fw M n a n n i j \<noteq> \<infinity> \<longrightarrow> get_const (fw M n a n n i j) \<in> \<int>"
simps_2: fixes r :: "'d set"   and e :: "('d, 'e, 'f) ann" shows "pre (AnnRec r e) = r"
smult_smult: fixes a :: "'d"   and b :: "'d"   and p :: "'d poly" shows "smult a (smult b p) = smult (a * b) p"
divisionTr1: fixes g :: "'a"   and f :: "'a" assumes "Corps S"   and "g \<in> carrier R"   and "g \<noteq> \<zero>"   and "0 < deg_n R S X g"   and "f \<in> carrier R"   and "f \<noteq> \<zero>"   and "deg_n R S X g \<le> deg_n R S X f" shows "f \<plusminus> -\<^sub>a lcf R S X f \<cdot>\<^sub>r\<^bsub>S\<^esub> lcf R S X g\<^bsup>\<hyphen> S\<^esup> \<cdot>\<^sub>r X^\<^bsup>R (deg_n R S X f - deg_n R S X g)\<^esup> \<cdot>\<^sub>r g = \<zero> \<or> deg_n R S X (f \<plusminus> -\<^sub>a lcf R S X f \<cdot>\<^sub>r\<^bsub>S\<^esub> lcf R S X g\<^bsup>\<hyphen> S\<^esup> \<cdot>\<^sub>r X^\<^bsup>R (deg_n R S X f - deg_n R S X g)\<^esup> \<cdot>\<^sub>r g) < deg_n R S X f"
sectionE: fixes f :: "'a" assumes "A.section f" obtains g :: "'a" where "A.ide (g \<cdot>\<^sub>A f)"
monom_sum: fixes a :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and n :: "nat" shows "monom (sum a A) n = (\<Sum>x\<in>A. monom (a x) n)"
mat_inv_inv: fixes M :: "complex \<times> complex \<times> complex \<times> complex" assumes "mat_det M \<noteq> 0" shows "mat_inv (mat_inv M) = M"
trace_def: fixes A :: "'a mat" shows "trace A = (\<Sum>i = 0..<dim_row A. A $$ (i, i))"
problem_plan_bound_works: fixes PROB :: "(('c, bool) fmap \<times> ('c, bool) fmap) set"   and s :: "('c, bool) fmap"   and as :: "(('c, bool) fmap \<times> ('c, bool) fmap) list" assumes "finite PROB"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' \<le> problem_plan_bound PROB"
unit_inv_inv: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "Dmonoid prod one"   and "Dmonoid.unit prod one x" shows "Dmonoid.inv prod one (Dmonoid.inv prod one x) = x"
F: fixes t :: "('sequent \<times> 'rule) tree"   and S :: "'structure" assumes "local.wf t"   and "good t"   and "S \<in> structure"   and "\<not> sat S (fst (tree.root t))" shows "False"
arr_cod_iff_arr: fixes f :: "(bool, bool list) concrete_category.arr" shows "J.arr (J.cod f) = J.arr f"
elims: fixes x :: "'a ifex list"   and y :: "'a option" assumes "lowest_tops x = y" obtains "x = []" and "y = None"   | v :: "'a" and uu_ :: "'a ifex" and uv_ :: "'a ifex" and r :: "'a ifex list" where "x = IF v uu_ uv_ # r__" and "y = Some (case lowest_tops r__ of None \<Rightarrow> v | Some u \<Rightarrow> min u v)"   | r :: "'a ifex list" where "x = Trueif # ra__" and "y = lowest_tops ra__"   | r :: "'a ifex list" where "x = Falseif # r" and "y = lowest_tops r"
conjugate_exponent_real_4: fixes p :: "real" assumes "1 < p" shows "(p - 1) * conjugate_exponent p = p"
distinct_1: fixes x21 :: "'a"   and x22 :: "'a" shows "DenyAll \<noteq> DenyAllFromTo x21 x22"
simps_1: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and x :: "'c"   and xs :: "'c list"   and y :: "'d"   and ys :: "'d list" shows "zip_with f (x # xs) (y # ys) = f x y # zip_with f xs ys"
stut_const: fixes c :: "'b" shows "STUTINV #c"
inf_bot_bot: fixes x :: "'c" shows "x \<sqinter> \<bottom> = \<bottom>"
quat_inner_k_left: fixes x :: "quat" shows "\<k> \<bullet> x = Im3 x"
coeff_inverse: fixes x :: "'b poly" shows "Abs_poly (coeff x) = x"
simp_el_indecomp_2: fixes b :: "'b list"   and G :: "'b list set" assumes " b \<in>B G " shows "b \<noteq> \<epsilon>"
t_fusion_def: fixes x :: "'a \<times> ('b \<times> 'a) list"   and y :: "'a \<times> ('b \<times> 'a) list" shows "t_fusion x y \<equiv> if Dioid_Models.last x = first y then (fst x, snd x @ snd y) else undefined"
rel_fun_mono: fixes X :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and Y :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_fun X A f g"   and "\<And>x y. Y x y \<longrightarrow> X x y"   and "\<And>x y. A x y \<Longrightarrow> B x y" shows "rel_fun Y B f g"
ContractsWithIRC_axioms: shows "ContractsWithIRC Xd Xh (TruePref.Pd_shuffle_to_top ds' (ManiPref.cop ds)) Ch"
ring_def: fixes R :: "('a, 'b) ring_scheme" shows "ring R \<equiv> abelian_group R \<and> Group.monoid R \<and> ring_axioms R"
normal_axioms_def: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" shows "normal_axioms H G \<equiv> \<forall>x\<in>carrier G. H #>\<^bsub>G\<^esub> x = x <#\<^bsub>G\<^esub> H"
simps_1: fixes \<mu> :: "'a" shows "Src \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0 = \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0"
Pasch_nondeg: fixes x :: "complex_homo"   and y :: "complex_homo"   and z :: "complex_homo"   and u :: "complex_homo"   and v :: "complex_homo" assumes "x \<in> unit_disc"   and "y \<in> unit_disc"   and "z \<in> unit_disc"   and "u \<in> unit_disc"   and "v \<in> unit_disc"   and "distinct [x, y, z, u, v]"   and "\<not> poincare_collinear {x, y, z}"   and "poincare_between x u z"   and "poincare_between y v z" shows "\<exists>a. a \<in> unit_disc \<and> poincare_between u a y \<and> poincare_between x a v"
AbstrLevels_A4_A41: fixes i :: "AbstrLevelsID" assumes "sA4 \<in> AbstrLevel i" shows "sA41 \<notin> AbstrLevel i"
simps_125: fixes f1 :: "char list \<Rightarrow> 'b"   and f2 :: "48 word \<Rightarrow> 'b"   and f3 :: "48 word \<Rightarrow> 'b"   and f4 :: "16 word \<Rightarrow> 'b"   and f5 :: "16 word \<Rightarrow> 'b"   and f6 :: "16 word \<Rightarrow> 'b"   and f7 :: "32 prefix_match \<Rightarrow> 'b"   and f8 :: "32 prefix_match \<Rightarrow> 'b"   and f9 :: "8 word \<Rightarrow> 'b"   and f10 :: "16 word \<Rightarrow> 16 word \<Rightarrow> 'b"   and f11 :: "16 word \<Rightarrow> 16 word \<Rightarrow> 'b"   and x4 :: "16 word" shows "(case EtherType x4 of IngressPort x \<Rightarrow> f1 x | EtherSrc x \<Rightarrow> f2 x | EtherDst x \<Rightarrow> f3 x | EtherType x \<Rightarrow> f4 x | VlanId x \<Rightarrow> f5 x | VlanPriority x \<Rightarrow> f6 x | IPv4Src x \<Rightarrow> f7 x | IPv4Dst x \<Rightarrow> f8 x | IPv4Proto x \<Rightarrow> f9 x | L4Src x xa \<Rightarrow> f10 x xa | L4Dst x xa \<Rightarrow> f11 x xa) = f4 x4"
vars_mult_monom: fixes p :: "int mpoly"   and x' :: "nat" shows "vars (p * MPoly_Type.monom (monomial (Suc 0) x') 1) = (if p = 0 then {} else insert x' (vars p))"
Measurement_System_ext_def: fixes unit_sys :: "'a"   and more :: "'b" shows "\<lparr>unit_sys = unit_sys, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons Measurement_System_ext_Tuple_Iso unit_sys more"
LtKeys_parts_implSecureSet: fixes Ag :: "(agent \<times> agent) set" shows "parts {implSecure A B M |A B M. (A, B) \<in> Ag \<and> M \<in> payload} \<inter> range LtK = {}"
assms_2: fixes a :: "'a"   and b :: "'a" assumes "x * a = x * b"   and "x \<noteq> (0::'a)" shows "a = b"
zero_iff: fixes x :: "'a" assumes "x \<in> E" shows "(norm x = 0) = (x = (0::'a))"
D_def: fixes n :: "nat" shows "D n = inverse (real (n + 1)) + ln (real (n + 1)) - ln (real (n + 2))"
Bex_set: fixes xs :: "'e list"   and P :: "'e \<Rightarrow> bool" shows "Bex (set xs) P = list_ex P xs"
path1: fixes u :: "'b"   and v :: "'b"   and E :: "('b \<times> 'b) set" assumes "(u, v) \<in> E" shows "path E u [u] v"
init_segment_refl: fixes S :: "nat set" shows "init_segment S S"
IH2: fixes s :: "'a net_state"   and s' :: "'a net_state" assumes "wf_net_tree p2"   and "s \<in> reachable (pnet np p2) TT"   and "(s, i:deliver(d), s') \<in> automaton.trans (pnet np p2)" shows "(\<forall>j. j \<noteq> i \<longrightarrow> netmap s' j = netmap s j) \<and> net_ip_action np (i:deliver(d)) i p2 s s'"
eq: fixes ys :: "('a \<times> 'b) list"   and x :: "'a"   and y :: "'b" assumes "strict_prefix ys xs"   and "x \<in> outs_\<I> \<I>" shows "spmf (trace_callee run_resource p ys x) y = spmf (trace_callee run_resource q ys x) y"
2: shows "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
prems_1: shows "abc_final (abc_steps_l (0, lm) p (Suc n)) p"
descendants_oob_head': fixes k :: "'b"   and i :: "'b"   and ps :: "('c, 'b) fm list"   and a :: "'b"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and xs :: "(nat \<times> nat) set"   and m :: "nat" assumes "descendants k i ((ps, a) # branch) xs" shows "(length branch, m + length ps) \<notin> xs"
if_true: fixes mem :: "'Var \<Rightarrow> 'Val"   and b :: "'BExp"   and t :: "('Var, 'AExp, 'BExp) Stmt"   and e :: "('Var, 'AExp, 'BExp) Stmt" assumes "ev\<^sub>B mem b" shows "eval\<^sub>w_simple_abv (Stmt.If b t e, mem) (t, mem)"
wp_UN_eq: fixes act :: "('b \<times> 'b) set"   and I :: "'c set"   and A :: "'c \<Rightarrow> 'b set" assumes "single_valued act"   and "I \<noteq> {}" shows "wp act (\<Union> (A ` I)) = (\<Union>i\<in>I. wp act (A i))"
suminf_add_ereal: fixes f :: "nat \<Rightarrow> ereal"   and g :: "nat \<Rightarrow> ereal" assumes "\<And>i. 0 \<le> f i"   and "\<And>i. 0 \<le> g i" shows "(\<Sum>i. f i + g i) = suminf f + suminf g"
rcsTr8: fixes H :: "'a set"   and a :: "'a"   and h :: "'a"   and x :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "h \<in> H"   and "x \<in> H \<bullet> a" shows "h \<cdot> x \<in> H \<bullet> a"
is_infinity_def: fixes a :: "('c, 'd) IEEE.float" shows "is_infinity a = (IEEE.exponent a = emax TYPE(('c, 'd) IEEE.float) \<and> fraction a = 0)"
open_segment_of_real: fixes x :: "real"   and y :: "real" shows "open_segment (of_real x) (of_real y) = of_real ` open_segment x y"
fresh_eta_expand: fixes a :: "atom"   and n :: "nat"   and e :: "exp" shows "a \<sharp> eta_expand n e = a \<sharp> e"
assms_3: shows "\<guillemotleft>h : local.dom h \<rightarrow> local.prod a b\<guillemotright>"
pw_ESPEC_2: fixes \<Phi> :: "'c \<Rightarrow> bool"   and \<Psi> :: "'d \<Rightarrow> bool"   and e :: "'c" shows "inres (ESPEC \<Phi> \<Psi>) (Inl e) = \<Phi> e"
simps: fixes x :: "'c regular"   and y :: "'c dense Filters.filter"   and z :: "'c regular"   and w :: "'c dense Filters.filter" shows "stone_phi.pairs_sup (x, y) (z, w) = (x \<squnion> z, y \<sqinter> w)"
empty_fv_exists_fun: fixes t :: "('b, 'a) Term.term" assumes "fv t = {}" shows "\<exists>f X. t = Fun f X"
unfold_congs_2: fixes r :: "('a, 'b) simple_packet_scheme"   and r' :: "('a, 'b) simple_packet_scheme"   and v' :: "char list"   and f :: "char list \<Rightarrow> char list"   and f' :: "char list \<Rightarrow> char list" assumes "r = r'"   and "p_oiface r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "p_oiface_update f r = p_oiface_update f' r'"
term_ok_vars_typ_ok: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and t :: "Core.term"   and x :: "variable"   and \<tau> :: "typ" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> t"   and "(x, \<tau>) \<in> fv t" shows "typ_ok \<Theta> \<tau>"
in_remove1_mset_neq: fixes a :: "'c"   and b :: "'c"   and C :: "'c multiset" assumes "a \<noteq> b" shows "(a \<in># remove1_mset b C) = (a \<in># C)"
imp_elim: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<psi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<pi>l :: "('a \<times> 'b set) stream list" assumes "Shallow S s0 \<delta> AP L AP'"   and "Shallow.imp \<phi> \<psi> \<pi>l"   and "\<phi> \<pi>l" obtains "\<psi> \<pi>l"
wf_dbfm_imp_is_fm: fixes x :: "dbfm" assumes "wf_dbfm x" shows "\<exists>A. x = trans_fm [] A"