hcju/isabelleps huggingface dataset:
A_trusts_OR4: fixes B' :: "agent"   and A :: "agent"   and NA :: "msg"   and B :: "agent"   and K :: "nat"   and evs :: "event list" assumes "Says B' A (Crypt (shrK A) \<lbrace>NA, Agent A, Agent B, Key K\<rbrace>) \<in> set evs"   and "A \<notin> bad"   and "A \<noteq> B"   and "evs \<in> otway" shows "\<exists>NB. Says Server B \<lbrace>Crypt (shrK A) \<lbrace>NA, Agent A, Agent B, Key K\<rbrace>, Crypt (shrK B) \<lbrace>NB, Agent A, Agent B, Key K\<rbrace>\<rbrace> \<in> set evs"
rel_interior_rel_open: fixes S :: "'a set" assumes "convex S" shows "rel_open (rel_interior S)"
Throw: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "'e"   and n :: "nat" shows "\<Gamma>\<turnstile> \<langle>Throw,Normal s\<rangle> =n\<Rightarrow> Abrupt s"
idem: fixes a :: "'c \<times> 'd" shows "sup_prod a a = a"
new_element_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_element_ptr :: "(_) element_ptr"   and ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'o, 'r option) RElement_scheme \<Rightarrow> 'u" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"   and "ptr \<noteq> new_element_ptr" shows "preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
bst_paint: fixes c :: "color"   and t :: "('b \<times> color) tree" shows "Tree2.bst (paint c t) = Tree2.bst t"
csets_infinite: fixes A :: "'c set" assumes "infinite A" shows "infinite (csets A)"
G_face_cycle_succ_a_no_loop: assumes "head G a \<noteq> tail G a" shows "G.face_cycle_succ a = a'"
B: shows "\<forall>x. \<exists>xb. xa (x + xb)"
fps_X_power_nth: fixes k :: "nat"   and n :: "nat" shows "fps_X ^ k $ n = (if n = k then 1::'b else (0::'b))"
LSM_sub_M: fixes M :: "('e, 'a, 'f) Module_scheme"   and T :: "'e set" assumes "R module M"   and "T \<subseteq> carrier M" shows "LSM\<^bsub>R\<^esub> M T \<subseteq> carrier M"
reflect_reflect_ident: fixes t :: "'c bt" shows "reflect (reflect t) = t"
lset_into_lsetp: fixes x :: "'a"   and xs :: "'a llist" assumes "x \<in> lset xs" shows "Coinductive_List.lsetp xs x"
le_add_diff_inverse: fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and minus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and zero :: "'f"   and times :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and one :: "'f"   and b :: "'f"   and a :: "'f" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq b a" shows "plus b (minus a b) = a"
typing_alt_simps_41: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and x :: "'a expr"   and xs :: "'a expr list"   and \<pi> :: "'a OCL_Types.type list" shows "(\<Gamma> \<turnstile>\<^sub>L x # xs : \<pi>) = (\<exists>\<tau> \<pi>'. \<pi> = \<tau> # \<pi>' \<and> \<Gamma> \<turnstile>\<^sub>E x : \<tau> \<and> \<Gamma> \<turnstile>\<^sub>L xs : \<pi>')"
drop_map: fixes n :: "nat"   and f :: "'h \<Rightarrow> 'g"   and xs :: "'h list" shows "drop n (map f xs) = map f (drop n xs)"
PO_m1_secrecy: shows "reach m1 \<subseteq> m1_secrecy"
type_equiv_subset: fixes t :: "'BExp set"   and P :: "'BExp set"   and t' :: "'BExp set"   and P' :: "'BExp set" assumes "t =:\<^sub>P t'"   and "P \<subseteq> P'" shows "t =:\<^sub>P' t'"
quorum_def: fixes Q :: "'node set" shows "quorum Q \<equiv> \<forall>p\<in>Q \<inter> W. \<exists>Sl\<in>slices p. Sl \<subseteq> Q"
BNTr4_1: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and W1 :: "'a set"   and W2 :: "'a set"   and g :: "'a \<Rightarrow> 'a" assumes "f \<in> carrier D \<rightarrow> carrier D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "\<forall>x\<in>carrier D. x \<preceq> f x"   and "W1 \<in> WWa D f a"   and "W2 \<in> WWa D f a"   and "ord_isom (Iod D W1) (Iod D (segment (Iod D W2) b)) g" shows "\<forall>x\<in>W1. g x = x"
src_vcomp: fixes \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.seq \<nu> \<mu>" shows "Span.src (\<nu> \<bullet> \<mu>) = Span.src \<nu> \<bullet> Span.src \<mu>"
scale_left_diff_distrib: fixes a :: "'a"   and b :: "'a"   and x :: "'b" shows "(a - b) *s x = a *s x - b *s x"
coerce_LNil: shows "COERCE('b\<cdot>llist, 'a\<cdot>llist)\<cdot>LNil = LNil"
take_subseq: fixes n :: "nat"   and xs :: "'a list" shows "take n xs \<preceq> xs"
case_2: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> ('c, 'b) Term.term list \<Rightarrow> 'a"   and x21 :: "'c"   and x22 :: "('c, 'b) Term.term list" shows "(case Fun x21 x22 of Var x \<Rightarrow> f1 x | Fun x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
disc_16: fixes x11 :: "'a"   and x12 :: "'b"   and x13 :: "'b" shows "\<not> isSnapshot (Trans x11 x12 x13)"
ph: assumes "w1 = POS_INF \<and> w2 \<noteq> POS_INF \<and> 0 <s w2" shows "P w1 w2"
in_inv_image: fixes x :: "'c"   and y :: "'c"   and r :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" shows "((x, y) \<in> inv_image r f) = ((f x, f y) \<in> r)"
balanced_Node_if_wbal1: fixes l :: "'b kdt"   and r :: "'b kdt"   and k :: "'b"   and v :: "real" assumes "balanced l"   and "balanced r"   and "KD_Tree.size_kdt l = KD_Tree.size_kdt r + 1" shows "balanced (Node k v l r)"
assms: shows "integrable (M1 \<Otimes>\<^sub>M M2) f"
tabulation_axioms: shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg r r.tab (tab\<^sub>0 r) (tab\<^sub>1 r)"
assms_1: shows "\<forall>m<n. time (Rep_run \<rho> m K) < \<tau>"
remove_elt_pairs_map: fixes ys :: "('a \<times> 'b) list"   and x :: "'a" assumes "Lists_Ex.strict_sorted (map fst ys)" shows "map_of_alist (remove_elt_pairs x ys) = delete_map x (map_of_alist ys)"
truncate_up_less_zero_iff: fixes p :: "nat"   and x :: "real" shows "(truncate_up p x < 0) = (x < 0)"
ennreal_top_divide: fixes x :: "ennreal" shows "\<top> / x = (if x = \<top> then 0 else \<top>)"
iso_runit: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "category.iso V (bicategory.runit V H \<a> \<i> src f)"
ipassmt_iface_replace_dstip_mexpr_def: fixes ipassmt :: "iface \<Rightarrow> ('b word \<times> nat) list option"   and ifce :: "iface" shows "ipassmt_iface_replace_dstip_mexpr ipassmt ifce \<equiv> case ipassmt ifce of None \<Rightarrow> Match (OIface ifce) | Some ips \<Rightarrow> match_list_to_match_expr (map (Match \<circ> Dst) (map (uncurry IpAddrNetmask) ips))"
mem_simps_9: fixes A :: "'h"   and C :: "'h set set" shows "(A \<in> \<Union> C) = (\<exists>X\<in>C. A \<in> X)"
f1: fixes x :: "'a"   and y :: "'a" shows "ad x \<le> ad (ad y \<cdot> x)"
dvd_add_right_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.comm_semiring_1_cancel plus minus zero times one"   and "dvd.dvd times a b" shows "dvd.dvd times a (plus b c) = dvd.dvd times a c"
three: shows "3\<^sub>\<nat> = ZFC_in_HOL.set {[]\<^sub>\<circ>, 1\<^sub>\<nat>, 2\<^sub>\<nat>}"
b0: fixes x :: "'b list"   and a :: "nat"   and y :: "'b list"   and b :: "nat"   and z :: "'b list"   and c :: "nat" assumes "LS x a y b z c" shows "b \<noteq> 0"
comp_fun_idem_sup: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.semilattice_sup sup less_eq less" shows "comp_fun_idem sup"
nop_bij_n: shows "n \<in>\<^sub>\<circ> \<omega>"
zero_min: fixes x :: "'a" shows "(0::'a) \<le> x"
\<PP>_covers_\<Q>: fixes k :: "nat" shows "\<Q> k \<subseteq> \<PP>"
hyp_port_prefix: fixes f :: "'form"   and t :: "('form, 'rule, 'subst, 'var) itnode rose_tree"   and "is" :: "nat list" assumes "f \<in> hyps_along t is" shows "prefix (hyp_port_path_for t is f @ [hyp_port_i_for t is f]) is"
eqvtUnion: fixes Rel :: "('a \<times> 'a) set"   and Rel' :: "('a \<times> 'a) set" assumes "eqvt Rel"   and "eqvt Rel'" shows "eqvt (Rel \<union> Rel')"
IH: assumes "\<not> beta_reducible t" shows "beta_norm s = Some t"
smooth_on_chart_inv2: fixes k :: "enat"   and c :: "('c, 'c) chart"   and X :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "k-smooth_on (apply_chart c ` X) (f \<circ> inv_chart c)"   and "c \<in> manifold_eucl.atlas k"   and "open X"   and "X \<subseteq> domain c" shows "k-smooth_on X f"
Block_self: fixes a :: "'c set \<times> 'd"   and A :: "('c set \<times> 'd) set" assumes "a \<in> A"   and "A \<in> eqY.Partition" shows "eqY.Class a = A"
join_ti_Suc: fixes s :: "nat \<Rightarrow> 'a list"   and x :: "nat"   and i :: "nat" shows "join_ti s x (Suc i) = join_ti s x i @ s (x + Suc i)"
wt_stmt_eq: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and t :: "stmt"   and U :: "tys" shows "(E,dt\<Turnstile>In1r t\<Colon>U) = (U = Inl (PrimT Void) \<and> E,dt\<Turnstile>t\<Colon>\<surd>)"
error_def: fixes exception :: "'e" shows "error exception = Prog (\<lambda>h. Inl exception)"
apply_cltn2_line_compose: fixes l :: "proj2_line"   and A :: "cltn2"   and B :: "cltn2" shows "apply_cltn2_line (apply_cltn2_line l A) B = apply_cltn2_line l (cltn2_compose A B)"
field_cont_on_bot_edge_image: shows "continuous_on {p. \<exists>x. x \<in> cbox a b \<and> p = (x, g2 x)} F"
rtrancl_eq_or_trancl: fixes x :: "'g"   and y :: "'g"   and R :: "('g \<times> 'g) set" shows "((x, y) \<in> R\<^sup>*) = (x = y \<or> x \<noteq> y \<and> (x, y) \<in> R\<^sup>+)"
trans_def: fixes r :: "('c \<times> 'c) set" shows "trans r = (\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r)"
fpxs_as_fls': fixes f :: "'b fpxs"   and d :: "nat" assumes "fpxs_root_order f dvd d"   and "0 < d" obtains f' :: "'b fls" where "f = fpxs_compose_power (fpxs_of_fls f') (1 / rat_of_nat d)"
l11_60_bis: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and P :: "'p" assumes "\<not> Col A B C"   and "D \<noteq> P"   and "Coplanar A B C D"   and "Per A D P"   and "Per B D P"   and "Per C D P" shows "D OrthAt A B C D P"
conjugate_def: fixes u :: "'b list"   and v :: "'b list" shows "u \<sim> v \<equiv> \<exists>r s. r \<cdot> s = u \<and> s \<cdot> r = v"
NoWrite\<^sub>A_implies_NoWrite\<^sub>C: fixes x :: "Eg1.var"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set" assumes "x \<in> mds\<^sub>A_of mds\<^sub>C AsmNoWrite" shows "Eg2_var\<^sub>C_of_Eg1 x \<in> mds\<^sub>C AsmNoWrite"
less_gexp_trans: fixes x :: "'b gexp"   and y :: "'b gexp"   and z :: "'b gexp" assumes "x < y"   and "y < z" shows "x < z"
odd_card: shows "odd (card {f. (\<exists>s\<in>simplices. face f s) \<and> rl ` f = {..n} \<and> bnd f})"
\<pi>_functional: fixes k :: "nat"   and T :: "(symbol \<times> character list) set" shows "\<pi> k T = limit (Scan T k \<circ> Complete k \<circ> Predict k)"
Events_LabelFunSem: fixes S :: "'e"   and ST :: "('d, 'e, 'f) status" shows "(EN S \<in> the (LabelFunSem (HA ST) ST)) = (S \<in> Events ST)"
CLS_eqI: fixes f :: "'a"   and g :: "'a" assumes "ide f" shows "(\<lbrakk>\<lbrakk>f\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>g\<rbrakk>\<rbrakk>) = (f \<cong> g)"
elements_mat_permutes: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat"   and A :: "'a mat"   and B :: "'a mat" assumes "p permutes {..<n}"   and "A \<in> carrier_mat n n"   and "B = Matrix.mat n n (\<lambda>(i, j). A $$ (p i, p j))" shows "elements_mat A = elements_mat B"
plus_1_eq_Suc: shows "(+) 1 = Suc"
complex_cnj_of_nat: fixes n :: "nat" shows "cnj (of_nat n) = of_nat n"
nn_cond_exp_charact: fixes f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "\<And>A. A \<in> sets G \<Longrightarrow> set_nn_integral M A f = set_nn_integral M A g"   and "f \<in> borel_measurable M"   and "g \<in> borel_measurable G" shows "AE x in M. g x = nn_cond_exp M G f x"
\<Delta>_simps_4: shows "local.dom ((\<a>[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> TTfgh.tab) = (tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1"
saturated_def: fixes Bot :: "'b set"   and Inf :: "'b inference set"   and entails :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Red_I :: "'b set \<Rightarrow> 'b inference set"   and Red_F :: "'b set \<Rightarrow> 'b set"   and N :: "'b set" assumes "calculus Bot Inf entails Red_I Red_F" shows "calculus.saturated Inf Red_I N = (inference_system.Inf_from Inf N \<subseteq> Red_I N)"
const_ss_8: fixes X :: "'a state \<times> 'a state \<Rightarrow> 'b" assumes "const X" shows "const (\<upsilon> X)"
abupd_abrupt_if_False: fixes xo :: "abopt"   and s :: "state" shows "abupd (abrupt_if False xo) s = s"
simps: fixes a1 :: "'a sequent"   and a2 :: "nat"   and R :: "('a sequent list \<times> 'a sequent) set" shows "((a1, a2) \<in> derivable R) = ((\<exists>C. a1 = C \<and> a2 = 0 \<and> ([], C) \<in> R) \<or> (\<exists>r m. a1 = snd r \<and> a2 = m + 1 \<and> r \<in> R \<and> fst r \<noteq> [] \<and> (\<forall>p\<in>set (fst r). \<exists>n\<le>m. (p, n) \<in> derivable R)))"
cfs_closed: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c"   and n :: "nat" assumes "UP_ring R"   and "f \<in> carrier (UP R)" shows "f n \<in> carrier R"
partition_on_mset_sum_size_eq: fixes A :: "'a multiset"   and P :: "'a multiset multiset" assumes "partition_on_mset A P" shows "\<Sum>\<^sub># (image_mset size P) = size A"
proto: shows "closed (pnet (\<lambda>i. paodv i \<langle>\<langle> qmsg) n) \<TTurnstile> netglobal (\<lambda>\<sigma>. \<forall>i\<in>net_tree_ips n. \<forall>dip. let nhip = the (nhop (rt (\<sigma> i)) dip) in dip \<in> vD (rt (\<sigma> i)) \<inter> vD (rt (\<sigma> nhip)) \<and> nhip \<noteq> dip \<longrightarrow> rt (\<sigma> i) \<sqsubset>\<^bsub>dip\<^esub> rt (\<sigma> nhip))"
assms_1: shows "arr \<mu> \<and> a \<in> sources \<mu> \<and> b \<in> targets \<mu>"
A0: shows "A \<ge>\<^sub>m 0\<^sub>m n n"
fg_symmetric: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "OpposedThinChamberComplexFoldings X g f (g ` C0)"
open_vimage_vec_nth: fixes S :: "'a set"   and i :: "'b" assumes "open S" shows "open ((\<lambda>x. x $ i) -` S)"
lins2D_4: fixes \<pi> :: "(node + node fset \<times> command_gadget \<times> node fset) list"   and S :: "node fset"   and G :: "diagram"   and j :: "nat"   and k :: "nat"   and e :: "node fset \<times> command_gadget \<times> node fset"   and w :: "node" assumes "\<pi> \<in> lins2 S G"   and "j < length \<pi>"   and "k < length \<pi>"   and "\<pi> ! j = Inr e"   and "\<pi> ! k = Inl w"   and "w |\<in>| thd3 e" shows "j < k"
l: fixes v :: "nat"   and f :: "face"   and a :: "nat"   and b :: "nat" assumes "v \<in> \<V> f" shows "is_nextElem (vertices f) a b = is_nextElem (verticesFrom f v) a b"
horner_eval_Suc: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a"   and n :: "nat" shows "horner_eval f x (Suc n) = horner_eval f x n + f n * x ^ n"
A_1: shows "\<R> = {region X I r |I r. valid_region X k I r}"
simps_3: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and as :: "'b negation_type list" shows "NegPos_map f (Neg a # as) = Neg (f a) # NegPos_map f as"
append_Nil2: fixes xs :: "'c list" shows "xs \<cdot> [] = xs"
cnf_form_of_def: shows "cnf_form_of \<equiv> form_of_cnf \<circ> cnf_lists"
Cons_4: shows "lst_conv (s, x # xs) = fst ss2"
simps_1: fixes xs :: "'a list"   and c :: "int" shows "peval xs (PExpr1 (PCnst c)) = \<guillemotleft>c\<guillemotright>"
vconn_implies_hpar_4: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>B g\<guillemotright>" shows "trg\<^sub>B g = trg\<^sub>B f"
separation_by_component_open: fixes S :: "'a set" assumes "open S"   and "\<not> connected (- S)" obtains C :: "'a set" where "C \<in> components S" and "\<not> connected (- C)"
AB_4: shows "normalize (\<Prod>\<^sub># B) = normalize y"
\<Z>_srngD_5: fixes \<alpha> :: "V"   and \<SS> :: "V"   and a :: "V"   and b :: "V"   and c :: "V" assumes "\<Z>_srng \<alpha> \<SS>"   and "a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>"   and "b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>"   and "c \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>" shows "(a +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b) *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c = a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c"
snd_sum: fixes f :: "'e \<Rightarrow> 'd \<times> 'c"   and A :: "'e set" shows "snd (sum f A) = (\<Sum>x\<in>A. snd (f x))"
VVp_to_V: fixes G :: "('c, 'd) ParityGame_scheme"   and v :: "'c"   and p :: "Player" assumes "ParityGame G"   and "v \<in> (if p = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>)" shows "v \<in> V\<^bsub>G\<^esub>"
orthogonal_complement_kernel_functional: fixes f :: "'b \<Rightarrow> complex" assumes "bounded_clinear f" shows "\<exists>x. orthogonal_complement (f -` {0}) = cspan {x}"
11: shows "arc (minarc (v * - v\<^sup>T \<sqinter> g))"
learn_limE_1: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "learn_lim \<psi> U s" shows "environment \<psi> U s"
Seq_WbisT: fixes c1 :: "(test, atom) com"   and d1 :: "(test, atom) com"   and c2 :: "(test, atom) com"   and d2 :: "(test, atom) com" assumes "c1 \<approx>wT d1"   and "c2 \<approx>wT d2" shows "c1 ;; c2 \<approx>wT d1 ;; d2"
simps_2: shows "right \<langle>\<rangle> = \<langle>\<rangle>"
top_greatest: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and top :: "'e"   and a :: "'e" assumes "class.order_top less_eq less top" shows "less_eq a top"
scale_left_commute: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'b"   and b :: "'b"   and x :: "'c" assumes "module scale" shows "scale a (scale b x) = scale b (scale a x)"
zero_less_divide_iff: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less zero (divide a b) = (less zero a \<and> less zero b \<or> less a zero \<and> less b zero)"
distinct_pps_to_list: fixes S :: "'t set" shows "distinct (pps_to_list S)"
log_mult_ext2: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "0 < y" shows "x * log b (x * y) = x * log b x + x * log b y"
OclNot_not: fixes X :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "not (not X) = X"
divide_nonneg_nonpos: fixes x :: "'a"   and y :: "'a" assumes "(0::'a) \<le> x"   and "y \<le> (0::'a)" shows "x / y \<le> (0::'a)"
winning_region_def: fixes p :: "Player" shows "G'.winning_region p \<equiv> {v \<in> V\<^bsub>G'\<^esub>. \<exists>\<sigma>. G'.strategy p \<sigma> \<and> G'.winning_strategy p \<sigma> v}"
linear_surjective_imp_injective: fixes f :: "'a \<Rightarrow> 'b" assumes "clinear f"   and "surj f"   and "cdim UNIV = cdim UNIV" shows "inj f"
nat_of_name_name_eq: fixes n :: "nat" shows "nat_of_name (name_of_nat n) = n"
zless_imp_zle: fixes z :: "int"   and z' :: "int" assumes "z < z'" shows "z \<le> z'"
simps: fixes i :: "nat" shows "f1.gso i = f1 ! i + M.sumlist (map (\<lambda>j. - f1.\<mu> i j \<cdot>\<^sub>v f1.gso j) [0..<i])"
mult_le_one: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "less_eq a one"   and "less_eq zero b"   and "less_eq b one" shows "less_eq (times a b) one"
okjectEqvt: fixes p :: "(name \<times> name) list"   and \<alpha> :: "'d action" shows "p \<bullet> object \<alpha> = object (p \<bullet> \<alpha>)"
adopt_node_def: shows "Shadow_DOM.adopt_node \<equiv> l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_adopt_node Shadow_DOM.get_owner_document Shadow_DOM.get_parent Shadow_DOM.remove_child get_disconnected_nodes set_disconnected_nodes"
assms_2: shows "m \<le>\<^sub>n SPEC (\<lambda>s. \<Psi> s \<longrightarrow> \<Phi> s)"
finish_simps_2: fixes u :: "'v"   and s :: "'v tarjan_state" shows "discovered (finish u s) = discovered s"
comp_arr_dom: fixes f :: "'a1"   and a :: "'a1" assumes "A1.arr f"   and "A1.dom f = a" shows "f \<cdot>\<^sub>A\<^sub>1 a = f"
replace2: fixes f' :: "'c"   and ls :: "'c list"   and fs :: "'c list" assumes "f' \<notin> set ls" shows "replace f' fs ls = ls"
validk_def: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and n :: "nat" shows "validk P c Q n = (\<exists>k>0. \<forall>s. P s \<longrightarrow> (\<exists>s' m. (c, s) \<Rightarrow> m \<Down> s' \<and> m \<le> k * n \<and> Q s'))"
abs_induct: fixes P :: "'b prm \<Rightarrow> bool"   and x :: "'b prm" assumes "\<And>y. P (abs_prm y)" shows "P x"
inner_dirsumD2: fixes Ms :: "'a set list" assumes "\<not> add_independentS Ms" shows "(\<Oplus>M\<leftarrow>Ms. M) = 0"
subgroup_auto: shows "subgroup (auto G) (BijGroup (carrier G))"
poly_cnj: fixes p :: "complex poly"   and z :: "complex" shows "cnj (poly p z) = poly (map_poly cnj p) (cnj z)"
residue_mult_assoc: fixes k :: "nat"   and x :: "int"   and y :: "int"   and z :: "int" shows "x \<otimes>\<^bsub>residue_ring (p ^ k)\<^esub> y \<otimes>\<^bsub>residue_ring (p ^ k)\<^esub> z = x \<otimes>\<^bsub>residue_ring (p ^ k)\<^esub> (y \<otimes>\<^bsub>residue_ring (p ^ k)\<^esub> z)"
constant_function_add: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "constant_function S (a \<oplus> b) = constant_function S a \<oplus>\<^bsub>F\<^esub> constant_function S b"
lookup_map2_val_rneutr: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'c"   and xs :: "('b, 'c) oalist_ntm"   and ys :: "('b, 'd) oalist_ntm"   and k :: "'b" assumes "\<And>k x. f k x (0::'d) = x" shows "OAlist_lookup_ntm (OAlist_map2_val_rneutr_ntm f xs ys) k = f k (OAlist_lookup_ntm xs k) (OAlist_lookup_ntm ys k)"
eval: shows "prg Env\<turnstile>s0 \<midarrow>e-\<succ>v\<rightarrow> s1"
correctCompositionDiffLevelsS1opt: shows "correctCompositionDiffLevels sS1opt"
greek_less_cons_mono: fixes r :: "('a \<times> 'a) set"   and s :: "(accent \<times> 'a) list"   and t :: "(accent \<times> 'a) list"   and x :: "accent \<times> 'a" assumes "trans r"   and "(s, t) \<in> greek_less r" shows "(x # s, x # t) \<in> greek_less r"
in_qle_def: fixes x :: "'a list list"   and M :: "'a list list set" shows "x \<in>\<^sub>\<simeq> M \<equiv> \<exists>y\<in>M. x \<simeq> y"
i_cut_Diff_disj: fixes cut_op :: "'a set \<Rightarrow> 'a \<Rightarrow> 'a set"   and A :: "'a set"   and B :: "'a set"   and t :: "'a" assumes "cut_op = (\<down><) \<or> cut_op = (\<down>\<le>) \<or> cut_op = (\<down>>) \<or> cut_op = (\<down>\<ge>)" shows "cut_op (A - B) t = cut_op A t - cut_op B t"
find_pred: fixes m' :: "('a, 'b) insert_message" assumes "m' \<in> insert_messages M \<union> {m}" shows "P m' \<in> set (ext_ids s)"