hcju/isabelleps huggingface dataset:
mk_rec_env_def: fixes css :: "(name, (Term_Class.term \<times> sterm) list) fmap"   and \<Gamma>' :: "(name, value) fmap" shows "mk_rec_env css \<Gamma>' = fmmap_keys (\<lambda>name cs. Vrecabs css name \<Gamma>') css"
bst_delete: fixes t :: "'b tree"   and x :: "'b" assumes "bst t" shows "bst (delete x t)"
arith_simps_2: fixes n :: "num" shows "num.One + num.Bit0 n = num.Bit1 n"
Polys_closed_times: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and X :: "'c set"   and q :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "p \<in> P[X]"   and "q \<in> P[X]" shows "p * q \<in> P[X]"
iso_is_arr: fixes f :: "'a \<times> 'a" assumes "VoV.iso f" shows "VoV.arr f"
eAbs_invert: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'c) model"   and xs :: "'varSort"   and x :: "'var"   and eX :: "'d withERR"   and A :: "'c" assumes "eAbs MOD xs x eX = OK A" shows "\<exists>s X. eX = OK X \<and> isInBar (xs, s) \<and> A = igAbs MOD xs x X \<and> igWls MOD s X"
inner_dirsum_Nil: shows "(\<Oplus>A\<leftarrow>[]. A) = 0"
simps_1: fixes v :: "nat"   and f :: "face" shows "natToVertexList v f [] = []"
l_set_disconnected_nodes_get_ancestors_si_def: fixes set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_ancestors_si_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_disconnected_nodes_get_ancestors_si set_disconnected_nodes_locs get_ancestors_si_locs \<equiv> \<forall>ptr h h'. \<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_ancestors_si_locs. r h h')"
bres_galois: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> y \<le> z) = (y \<le> proto_near_quantale.bres Sup (\<le>) (\<sqinter>) x z)"
conc_def: fixes A :: "'b list set"   and B :: "'b list set" shows "A @@ B = {xs @ ys |xs ys. xs \<in> A \<and> ys \<in> B}"
term_ok'_subst_bv: fixes \<Sigma> :: "(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)"   and T :: "typ"   and t :: "Core.term"   and x :: "variable" assumes "term_ok' \<Sigma> (Abs T t)" shows "term_ok' \<Sigma> (subst_bv (Fv x T) t)"
Par: fixes S :: "pi"   and T :: "pi"   and U :: "pi" assumes "(S, T) \<in> Rel" shows "(S \<parallel> U, T \<parallel> U) \<in> Rel'"
poly_subst_minus: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_subst f (p - q) = poly_subst f p - poly_subst f q"
assert_Inf: fixes A :: "'c Assertion set" shows "{\<cdot> \<Sqinter>A } = \<Sqinter>(assert ` A) \<sqinter> (1::'c)"
prv_pls_suc: fixes t :: "'trm"   and t' :: "'trm" assumes "t \<in> atrm"   and "t' \<in> atrm" shows "prv (eql (pls t (suc t')) (suc (pls t t')))"
split_abrupt_if: fixes P :: "abopt \<Rightarrow> bool"   and c :: "bool"   and x' :: "abopt"   and x :: "abopt" shows "P (abrupt_if c x' x) = ((c \<and> x = None \<longrightarrow> P x') \<and> (\<not> (c \<and> x = None) \<longrightarrow> P x))"
fil_seq_pred: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and s' :: "nat \<Rightarrow> nat \<Rightarrow> int"   and P :: "(nat \<Rightarrow> int) \<Rightarrow> bool"   and m :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "s' = filtered_seq s P"   and "\<forall>n. \<exists>m>n. P (s m)" shows "P (s' m)"
lower_def: shows "lower \<equiv> cindex_pathE (part_circlepath z r pi (2 * pi)) z0"
induct: fixes P :: "Formula.formula \<Rightarrow> bool"   and a0 :: "Formula.formula" assumes "\<And>t1 t2. P (formula.Eq t1 t2)"   and "\<And>t1 t2. P (formula.Less t1 t2)"   and "\<And>t1 t2. P (formula.LessEq t1 t2)"   and "\<And>t1 t2. P (formula.Neg (formula.Eq t1 t2))"   and "\<And>t1 t2. P (formula.Neg (formula.Less t1 t2))"   and "\<And>t1 t2. P (formula.Neg (formula.LessEq t1 t2))"   and "\<And>v va. P (formula.Pred v va)"   and "\<And>v va vb. P (formula.Let v va vb)"   and "\<And>va vb. P (formula.Neg (formula.Pred va vb))"   and "\<And>va vb vc. P (formula.Neg (formula.Let va vb vc))"   and "\<And>va. P (formula.Neg (formula.Neg va))"   and "\<And>va vb. P (formula.Neg (formula.Or va vb))"   and "\<And>va vb. P (formula.Neg (formula.And va vb))"   and "\<And>va. P (formula.Neg (formula.Ands va))"   and "\<And>va. P (formula.Neg (formula.Exists va))"   and "\<And>va vb vc vd ve. P (formula.Neg (formula.Agg va vb vc vd ve))"   and "\<And>va vb. P (formula.Neg (formula.Prev va vb))"   and "\<And>va vb. P (formula.Neg (formula.Next va vb))"   and "\<And>va vb vc. P (formula.Neg (formula.Since va vb vc))"   and "\<And>va vb vc. P (formula.Neg (formula.Until va vb vc))"   and "\<And>va vb. P (formula.Neg (formula.MatchF va vb))"   and "\<And>va vb. P (formula.Neg (formula.MatchP va vb))"   and "\<And>v va. P (formula.Or v va)"   and "\<And>v va. P (formula.And v va)"   and "\<And>v. P (formula.Ands v)"   and "\<And>v. P (formula.Exists v)"   and "\<And>v va vb vc vd. P (formula.Agg v va vb vc vd)"   and "\<And>v va. P (formula.Prev v va)"   and "\<And>v va. P (formula.Next v va)"   and "\<And>v va vb. P (formula.Since v va vb)"   and "\<And>v va vb. P (formula.Until v va vb)"   and "\<And>v va. P (formula.MatchF v va)"   and "\<And>v va. P (formula.MatchP v va)" shows "P a0"
tendsto_zero_norm_bound: fixes G :: "'b \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'd"   and net :: "'b filter" assumes "\<forall>x. \<parallel>G x\<parallel> \<le> \<parallel>F x\<parallel>"   and "(F \<longlongrightarrow> (0::'d)) net" shows "(G \<longlongrightarrow> (0::'c)) net"
getPos_set: fixes a :: "'a"   and x :: "'a negation_type list" shows "(Pos a \<in> set x) = (a \<in> set (getPos x))"
arcs_blocks_ne: assumes "arcs G \<noteq> {}" shows "arcs_blocks \<noteq> {#}"
multiple_is_design: fixes n :: "nat" shows "design \<V> (multiple_blocks n)"
of_int_complex_interval: fixes i :: "int" shows "of_int i \<in>\<^sub>c of_int_complex_interval i"
A_3: shows "\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y"
quad_form_vec_cnj_mat_cnj: fixes H :: "complex \<times> complex \<times> complex \<times> complex"   and z :: "complex \<times> complex" assumes "hermitean H" shows "quad_form (vec_cnj z) (mat_cnj H) = quad_form z H"
times_divide_eq_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "times a (divide b c) = divide (times a b) c"
xa_1: shows "p \<in> \<omega>_limit_set x"
hdomain_shift_disjoint: fixes delta :: "hf"   and f :: "hf" shows "delta \<sqinter> hdomain (shift f delta) = 0"
slist_apply_list_of_slist_eq: fixes a :: "('e, 'd) slist"   and i :: "'e" shows "slist_apply a i = the_default (0::'d) (map_of (list_of_slist a) i)"
mem_closed_if_AE_lebesgue_open: fixes S :: "'b set"   and C :: "'b set"   and x :: "'b" assumes "open S"   and "closed C"   and "AE x\<in>S in lebesgue. x \<in> C"   and "x \<in> S" shows "x \<in> C"
conjugate_sprod_vec: fixes v :: "'c vec"   and n :: "nat"   and w :: "'c vec" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "conjugate (v \<bullet> w) = inner_prod w (conjugate v)"
absorb1: fixes b :: "'c"   and a :: "'c" assumes "b \<le> a" shows "a \<squnion> b = a"
le_infI1: fixes x :: "'c"   and a :: "'c"   and b :: "'c" assumes "x \<le> a" shows "x \<le> a \<squnion> b"
h'_lf: fixes h' :: "'c \<Rightarrow> real"   and x0 :: "'c"   and H :: "'c set"   and h :: "'c \<Rightarrow> real"   and xi :: "real"   and H' :: "'c set"   and E :: "'c set" assumes "\<And>x. h' x = (let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi)"   and "H' = H + lin x0"   and "Subspace.subspace H E"   and "linearform H h"   and "x0 \<notin> H"   and "x0 \<in> E"   and "x0 \<noteq> (0::'c)"   and "vectorspace E" shows "linearform H' h'"
app_iso_def: fixes hom :: "('g, 'h, 'e, 'f) digraph_isomorphism"   and G :: "('g, 'h) pre_digraph" shows "app_iso hom G \<equiv> \<lparr>verts = iso_verts hom ` verts G, arcs = iso_arcs hom ` arcs G, tail = iso_tail hom, head = iso_head hom\<rparr>"
progressing_def: fixes p :: "('b \<times> 'b) list" shows "progressing p \<equiv> \<forall>xs x y ys. p \<noteq> xs @ (x, y) # (y, x) # ys"
ln_x_over_x_mono: fixes x :: "real"   and y :: "real" assumes "exp 1 \<le> x"   and "x \<le> y" shows "ln y / y \<le> ln x / x"
ternary_lift4: fixes tv :: "ternaryvalue" shows "(eval_ternary_Not tv = TernaryTrue) = (tv = TernaryFalse)"
l3_step3_def: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and Nr :: "msg"   and gny :: "msg" shows "l3_step3 Ra A B Nr gny \<equiv> {(s, s'). guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> progress s Ra = Some {xnx, xni, xgnx} \<and> sklvl1.guessed_frame Ra xgny = Some gny \<and> sklvl1.guessed_frame Ra xnr = Some Nr \<and> sklvl1.guessed_frame Ra xsk = Some (Exp gny (NonceF (Ra $ nx))) \<and> implConfid B A Nr \<in> ik s \<and> implInsec B A \<langle>gny, hmac \<langle>Number 0, Exp Gen (NonceF (Ra $ nx)), gny, Messages.Agent B, Messages.Agent A\<rangle> (hmac (NonceF (Ra $ ni)) Nr)\<rangle> \<in> ik s \<and> s' = s\<lparr>progress := progress s(Ra \<mapsto> {xnx, xni, xnr, xgnx, xgny, xsk, xEnd}), ik := {implInsec A B (hmac \<langle>Number 1, gny, Exp Gen (NonceF (Ra $ nx)), Messages.Agent A, Messages.Agent B\<rangle> (hmac (NonceF (Ra $ ni)) Nr))} \<union> ik s, secret := {x. x = Exp gny (NonceF (Ra $ nx)) \<and> Ra = test} \<union> secret s, signalsInit := if can_signal s A B then addSignal (signalsInit s) (Commit A B \<langle>NonceF (Ra $ ni), Nr, Exp gny (NonceF (Ra $ nx))\<rangle>) else signalsInit s, signalsInit2 := if can_signal s A B then addSignal (signalsInit2 s) (Commit A B (Exp gny (NonceF (Ra $ nx)))) else signalsInit2 s, signalsResp := if can_signal s A B then addSignal (signalsResp s) (Running A B \<langle>NonceF (Ra $ ni), Nr, Exp gny (NonceF (Ra $ nx))\<rangle>) else signalsResp s, signalsResp2 := if can_signal s A B then addSignal (signalsResp2 s) (Running A B (Exp gny (NonceF (Ra $ nx)))) else signalsResp2 s\<rparr>}"
obtain_list_from_elements: fixes n :: "nat"   and P :: "'a \<Rightarrow> nat \<Rightarrow> bool" assumes "\<forall>i<n. \<exists>li. P li i" obtains l :: "'a list" where "length l = n" and "\<forall>i<n. P (l ! i) i"
rec_F_def: shows "rec_F = Cn (Suc (Suc 0)) rec_valu [Cn (Suc (Suc 0)) rec_right [Cn (Suc (Suc 0)) rec_conf [recf.id (Suc (Suc 0)) 0, recf.id (Suc (Suc 0)) (Suc 0), rec_halt]]]"
norm_eq: fixes x :: "'a"   and u :: "'a"   and y :: "'a"   and v :: "'a"   and x' :: "'a"   and u' :: "'a"   and y' :: "'a"   and v' :: "'a" assumes "x = norm x *\<^sub>R u"   and "y = norm y *\<^sub>R v"   and "x' = norm x *\<^sub>R u'"   and "y' = norm y *\<^sub>R v'"   and "norm u = 1"   and "norm u' = 1"   and "norm v = 1"   and "norm v' = 1"   and "norm (u' - v') = norm (u - v)" shows "norm (x' - y') = norm (x - y)"
one_uint16_def: shows "1 \<equiv> Abs_uint16 1"
get_shadow_root_is_l_get_shadow_root: shows "l_get_shadow_root ShadowRootClass.type_wf get_shadow_root get_shadow_root_locs"
simps: fixes a1 :: "'a"   and a2 :: "'a list" shows "ListMem a1 a2 = ((\<exists>x xs. a1 = x \<and> a2 = x # xs) \<or> (\<exists>x xs y. a1 = x \<and> a2 = y # xs \<and> ListMem x xs))"
conjug_rotate_iff: fixes u :: "'b list"   and v :: "'b list" shows "u \<sim> v = (\<exists>n. v = rotate n u)"
prems: shows "\<forall>x\<in>fv (Agg y \<omega> b f \<phi>). v ! x = v' ! x"
seq_compact_closed_subset: fixes s :: "'b set"   and t :: "'b set" assumes "closed s"   and "s \<subseteq> t"   and "seq_compact t" shows "seq_compact s"
returns_to_laterI: fixes P :: "'a set"   and x :: "'a"   and t :: "real" assumes "returns_to P x"   and "closed P"   and "0 < t"   and "t \<in> existence_ivl0 x"   and "\<And>s. \<lbrakk>0 < s; s \<le> t\<rbrakk> \<Longrightarrow> flow0 x s \<notin> P" shows "returns_to P (flow0 x t)"
poly_of_pm_sum: fixes x :: "'b"   and f :: "'c \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and I :: "'c set" shows "poly_of_pm x (sum f I) = (\<Sum>i\<in>I. poly_of_pm x (f i))"
order_rel_prodI: fixes orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.order orda (mk_less orda)"   and "class.order ordb (mk_less ordb)" shows "class.order (rel_prod orda ordb) (mk_less (rel_prod orda ordb))"
mat_erase_def: fixes A :: "'b mat"   and i :: "nat"   and j :: "nat" shows "mat_erase A i j = Matrix.mat (dim_row A) (dim_col A) (\<lambda>(i', j'). if i' = i \<or> j' = j then 0::'b else A $$ (i', j'))"
fresh_perm_app: fixes a :: "name"   and pi :: "(name \<times> name) list"   and x :: "'d" assumes "a \<sharp> pi"   and "a \<sharp> x" shows "a \<sharp> pi \<bullet> x"
b_meet_c_eq_d: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "d_aux a b c \<le> e_aux a b c" shows "b_aux a b c \<sqinter> c_aux a b c = d_aux a b c"
uint_bl: fixes w :: "'a word" shows "to_bl w = bin_to_bl (size w) (uint w)"
max_var_max: fixes a :: "nat"   and lp :: "linear_poly" assumes "a \<in> vars lp" shows "a \<le> Abstract_Linear_Poly.max_var lp"
has_codomain_char: fixes f :: "'a1 \<times> 'a2" shows "(A1xA2.codomains f \<noteq> {}) = (A1.arr (fst f) \<and> A2.arr (snd f))"
sets_into_space: fixes x :: "'a set"   and i :: "nat" assumes "x \<in> A i" shows "x \<subseteq> space M"
two_comparisons_into_case_order_11: fixes x :: "'a"   and y :: "'a"   and P :: "'d"   and R :: "'d"   and Q :: "'d" shows "(if x = y then P else if le y x then R else Q) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
vsv_VLambda_cong: fixes r :: "V"   and f :: "V \<Rightarrow> V" assumes "vsv r"   and "\<And>a. a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r \<Longrightarrow> r\<lparr>a\<rparr> = f a" shows "VLambda (\<D>\<^sub>\<circ> r) f = r"
stabilizer_subgroup: fixes G :: "('d, 'e) monoid_scheme"   and E :: "'f set"   and \<phi> :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and x :: "'f" assumes "group_action G E \<phi>"   and "x \<in> E" shows "subgroup (stabilizer G \<phi> x) G"
sv_trans_def: shows "sv_trans = (\<Union>r S v. \<Union> (range (sv_round r S v)))\<^sup>="
generalizes_def: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" shows "generalizes C D = (\<exists>\<sigma>. C \<cdot> \<sigma> = D)"
bij_fv_UV_1: shows "bij_betw fv GU HU"
simps: fixes args :: "Monitor.args"   and nt :: "nat"   and t :: "nat"   and gc :: "nat"   and maskL :: "bool list"   and maskR :: "bool list"   and data_prev :: "(nat \<times> 'a option list set) queue"   and data_in :: "(nat \<times> 'a option list set) queue"   and tuple_in :: "('a option list, nat) mapping"   and tuple_since :: "('a option list, nat) mapping" shows "add_new_ts_mmsaux' args nt (t, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since) = (let I = args_ivl args; (data_prev, move) = takedropWhile_queue (\<lambda>(t, X). left I \<le> nt - t) data_prev; data_in = fold (\<lambda>(t, X). append_queue (t, X)) move data_in; tuple_in = fold (\<lambda>(t, X) tuple_in. upd_set tuple_in (\<lambda>_. t) {as \<in> X. valid_tuple tuple_since (t, as)}) move tuple_in in (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since))"
edge_obs_subset: fixes a :: "'edge"   and S :: "'node set" assumes "valid_edge a"   and "sourcenode a \<notin> S" shows "obs (targetnode a) S \<subseteq> obs (sourcenode a) S"
fcomp_fconst_on_fid_on: fixes A :: "V"   and c :: "V" shows "fconst_on A c \<circ>\<^sub>\<bullet> fid_on A = fconst_on A c"
llist_inf_eq_LNil: fixes xs :: "'a llist"   and ys :: "'a llist" shows "lnull (xs \<sqinter> ys) = (xs \<noteq> LNil \<longrightarrow> ys \<noteq> LNil \<longrightarrow> lhd xs \<noteq> lhd ys)"
lcm_neg2: fixes a :: "'a"   and b :: "'a" shows "lcm a (- b) = lcm a b"
bits_of_1_nat: shows "bits_of 1 = [True]"
ev_mono: fixes \<phi> :: "'d stream \<Rightarrow> bool"   and xs :: "'d stream"   and \<psi> :: "'d stream \<Rightarrow> bool" assumes "ev \<phi> xs"   and "\<And>xs. \<phi> xs \<Longrightarrow> \<psi> xs" shows "ev \<psi> xs"
consistentI: fixes xs :: "'a list"   and ys :: "'a list"   and z :: "'a" assumes "hb_consistent (xs @ ys)"   and "\<forall>x\<in>set (xs @ ys). \<not> z \<prec> x" shows "hb_consistent (xs @ ys @ [z])"
hyps: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) \<times> 'x set) list" assumes "valid_decomp X qs"   and "standard_decomp k qs"   and "exact_decomp m qs"   and "hom_decomp qs" shows "hom_decomp (exact_aux k m qs)"
extend_evals: fixes P :: "(char list list \<times> char list exp) prog"   and es :: "char list exp list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and es'' :: "char list exp list"   and s'' :: "heap \<times> sfields \<times> sheap"   and b'' :: "bool"   and es' :: "char list exp list"   and s' :: "heap \<times> sfields \<times> sheap" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>es,s,b\<rangle> [\<rightarrow>]* \<langle>es'',s'',b''\<rangle>"   and "P \<turnstile> \<langle>es'',s''\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>"   and "iconfs (shp s) es"   and "P,shp s \<turnstile>\<^sub>b (es,b) \<surd>" shows "P \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>"
agree_DRESTRICT: fixes s1 :: "('d, 'e) fmap"   and s2 :: "('d, 'e) fmap"   and vs :: "'d set" assumes "agree s1 s2" shows "agree (fmrestrict_set vs s1) (fmrestrict_set vs s2)"
nat_seg_image_imp_finite: fixes A :: "'b set"   and f :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "A = f ` {i. i < n}" shows "finite A"
bot_errorable_def: shows "\<bottom> \<equiv> Abs_errorable None"
nolub_Or: fixes f :: "atom fm"   and g :: "atom fm"   and xs :: "'a list"   and l :: "'a"   and x :: "'a"   and u :: "'a" shows "nolub (Or f g) xs l x u = (nolub f xs l x u \<and> nolub g xs l x u)"
assms_4: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i \<le> j"   and "j \<le> k"   and "(0::'a) < coeff p i"   and "(0::'a) < coeff p k" shows "(0::'a) < coeff p j"
intro: fixes lookup :: "'a \<Rightarrow> ('c, 'b option) state"   and empty :: "'c"   and P :: "'c \<Rightarrow> bool" assumes "state_mem_defs.map_of lookup empty \<subseteq>\<^sub>m Map.empty"   and "P empty" shows "mem_correct_empty_axioms lookup P empty"
case_prod_curry: fixes f :: "'c \<times> 'd \<Rightarrow> 'e" shows "(\<lambda>(x, y). curry f x y) = f"
assms_1: shows "split_aforms XS i = (YS, ZS)"
cdSp_Append_cdep: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n'' :: "'node SDG_node"   and n' :: "'node SDG_node" assumes "n cd-ns\<rightarrow>\<^sub>d* n''"   and "n'' \<longrightarrow>\<^bsub>cd\<^esub> n'" shows "n cd-ns @ [n'']\<rightarrow>\<^sub>d* n'"
strip_while_Nil: fixes P :: "'b \<Rightarrow> bool" shows "strip_while P [] = []"
in_closure_connected_component: fixes x :: "'a"   and S :: "'a set"   and y :: "'a" assumes "x \<in> S"   and "open S" shows "(x \<in> closure (connected_component_set S y)) = (x \<in> connected_component_set S y)"
predE: fixes cid :: "'kid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'KS"   and n :: "nat"   and P :: "bool" assumes "sb.eval cid t t' n (sb.pred P)" shows "P"
domD: fixes x :: "'a" assumes "local.dom x"   and "\<not> Q x" shows "local.dom (h x)"
inject_4: fixes x4 :: "'a \<times> nat"   and y4 :: "'a \<times> nat" shows "(Actual_in x4 = Actual_in y4) = (x4 = y4)"
anormal_atoms_qelim: fixes qe :: "'a list \<Rightarrow> 'a fm"   and as :: "'a list"   and a :: "'a" assumes "\<And>as. \<forall>a\<in>set as. depends\<^sub>0 a \<and> anormal a \<Longrightarrow> normal (qe as)"   and "\<forall>a\<in>set as. anormal a"   and "a \<in> atoms (qelim qe as)" shows "anormal a"
gfp_trans_lemma3: fixes T :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>t u. \<lbrakk>le_utrans t u; \<And>P. unitary P \<Longrightarrow> unitary (t P); \<And>P. unitary P \<Longrightarrow> unitary (u P)\<rbrakk> \<Longrightarrow> le_utrans (T t) (T u)"   and "\<And>t P. \<lbrakk>\<And>Q. unitary Q \<Longrightarrow> unitary (t Q); unitary P\<rbrakk> \<Longrightarrow> unitary (T t P)" shows "le_utrans (T (gfp_trans T)) (gfp_trans T)"
Rep_stone_phi_pair: fixes x :: "'c stone_phi_pair" shows "Rep_stone_phi_pair x \<in> stone_phi.pairs"
simps_4: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'b"   and x1 :: "'c" shows "(case top.Value x1 of top.Value x \<Rightarrow> f1 x | top.Top \<Rightarrow> f2) = f1 x1"
simple_path_imp_arc: fixes g :: "real \<Rightarrow> 'a" assumes "simple_path g"   and "pathfinish g \<noteq> pathstart g" shows "arc g"
lunit'_in_vhom: fixes f :: "'a"   and b :: "'a" assumes "ide f"   and "trg f = b" shows "\<guillemotleft>\<l>\<^sup>-\<^sup>1[f] : f \<Rightarrow> b \<star> f\<guillemotright>"
elims_3: fixes G :: "('a, 'b) cfg"   and x :: "('c + 'b) list" assumes "\<not> \<oo>\<^sub>P G x" obtains uu_ :: "'c" and uv_ :: "('c + 'b) list" where "x = Inl uu_ # uv_"   | N :: "'b" and \<alpha> :: "('c + 'b) list" where "x = Inr N # \<alpha>" and "\<not> ([] |\<in>| cfg.prod G N \<and> \<oo>\<^sub>P G \<alpha>)"
set_mode_get_tag_name: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) element_ptr" assumes "l_set_mode_get_tag_name type_wf set_mode set_mode_locs get_tag_name get_tag_name_locs" shows "\<forall>w\<in>set_mode_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_tag_name_locs ptr'. r h h')"
l1_inv5D: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and x :: "l1_state"   and gny :: "msg" assumes "guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr>"   and "in_progressS (progress x Ra) {xnx, xgnx, xgny, xsk, xEnd}"   and "guessed_frame Ra xgny = Some gny"   and "can_signal x A B"   and "x \<in> l1_inv5" shows "0 < signalsResp x (Running A B (Exp gny (NonceF (Ra $ nx))))"
simps_2: fixes R :: "('a, 'b) ring_scheme"   and Us :: "'a list" assumes "ring R" shows "ring.combine R [] Us = \<zero>\<^bsub>R\<^esub>"
not_outside_2: fixes x :: "'a"   and y :: "'a" assumes "le x y" shows "y \<in> carrier"
fdemodalisation22: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "( |x\<rangle> y \<le> d z) = (ad z \<cdot> x \<cdot> d y = (zero_class.zero::'a))"
simps_14: fixes a :: "'a"   and x :: "'a"   and c :: "'a"   and D :: "'a tree"   and AB :: "'a tree"   and C :: "'a tree" assumes "a < x"   and "c < x"   and "D \<noteq> \<langle>\<rangle>" shows "splay x \<langle>AB, a, \<langle>C, c, D\<rangle>\<rangle> = (case splay x D of \<langle>D1, x, xa\<rangle> \<Rightarrow> \<langle>\<langle>\<langle>AB, a, C\<rangle>, c, D1\<rangle>, x, xa\<rangle>)"
Max_in: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "linorder.Max less_eq A \<in> A"
collect_locks_def: fixes las :: "'a lock_actions" shows "collect_locks las = {l. Lock \<in> set (las $ l)}"
limitin_canonical_iff: fixes f :: "'c \<Rightarrow> 'b"   and l :: "'b"   and F :: "'c filter" shows "limitin euclidean f l F = (f \<longlongrightarrow> l) F"
oinv: shows "A \<Turnstile> (act I, U \<rightarrow>) P"