hcju/isabelleps huggingface dataset:
isTrue_neg: fixes \<phi> :: "fm" assumes "\<phi> \<in> UNIV"   and "Fvars \<phi> = {}" shows "eval_fm e0 \<phi> \<or> eval_fm e0 (neg \<phi>)"
inst_fun_pcpo: shows "\<bottom> = (\<lambda>x. \<bottom>)"
necessitation_averse_closures_universal: fixes \<phi> :: "'a \<Rightarrow> \<o>" assumes "\<And>x. [\<phi> x]" shows "[\<^bold>\<forall>x. \<phi> x]"
CAs_in: shows "set CAs \<subseteq> set (prems_of \<iota>\<^sub>0)"
bisim1Val2: fixes pc :: "nat"   and e :: "'addr expr1"   and P :: "'a prog"   and h :: "'heap"   and v :: "'addr val"   and xs :: "'addr val list" assumes "pc = length (compE2 e)" shows "P,e,h \<turnstile> (Val v, xs) \<leftrightarrow> ([v], xs, pc, None)"
enum_all_sum_def: fixes P :: "'a + 'b \<Rightarrow> bool" shows "enum_all P = (enum_all (\<lambda>x. P (Inl x)) \<and> enum_all (\<lambda>x. P (Inr x)))"
observable_moves_preds_transfers_2: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) =as\<Rightarrow> (ms',s')" shows "transfers (map f as) s = s'"
path_empt: fixes SES :: "('a, 'b) SES_rec"   and s1 :: "'a" shows "s1 []\<Longrightarrow>\<^bsub>SES\<^esub> s1"
stable_def: fixes \<S> :: "'Var set \<times> 'Var set"   and x :: "'Var" shows "stable \<S> x \<equiv> x \<in> fst \<S> \<union> snd \<S>"
freshSets_5: fixes x :: "name"   and xvec :: "name list"   and C :: "'a" shows "(x # xvec) \<sharp>* C = (x \<sharp> C \<and> xvec \<sharp>* C)"
predict_imp_adapt: fixes X :: "nat \<Rightarrow> 'a \<Rightarrow> 'd"   and N :: "'d measure" assumes "predict_stoch_proc F X N" shows "adapt_stoch_proc F X N"
mem_box_componentwiseI: fixes S :: "'b set"   and x :: "'b" assumes "is_interval S"   and "\<And>i. i \<in> Basis \<Longrightarrow> x \<bullet> i \<in> (\<lambda>x. x \<bullet> i) ` S" shows "x \<in> S"
sup_left_commute: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" shows "sup_prod x (sup_prod y z) = sup_prod y (sup_prod x z)"
finprod_closed: fixes f :: "'c \<Rightarrow> 'a set"   and A :: "'c set" assumes "f \<in> A \<rightarrow> carrier (ideals_set R)" shows "finprod (ideals_set R) f A \<in> carrier (ideals_set R)"
uminus_add_conv_diff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.ab_group_add plus zero minus uminus" shows "plus (uminus a) b = minus b a"
comp_in_homI': fixes f :: "'a"   and g :: "'a"   and a :: "'a"   and c :: "'a" assumes "S.arr f"   and "S.arr g"   and "S.dom f = a"   and "S.cod g = c"   and "S.dom g = S.cod f" shows "\<guillemotleft>g \<cdot>\<^sub>S f : a \<Rightarrow>\<^sub>S c\<guillemotright>"
in_parts_Says_imp_used: fixes K :: "nat"   and X :: "msg"   and A :: "agent"   and B :: "agent"   and evs :: "event list" assumes "Key K \<in> parts {X}"   and "Says A B X \<in> set evs" shows "Key K \<in> used evs"
word_log2_maximum: assumes "bit w i" shows "i \<le> word_log2 w"
mono: fixes x :: "('a, 'b, 'c) Resumption.resumption" shows "set.mono_body (\<lambda>f. F f x)"
INFM_Suc: fixes P :: "nat \<Rightarrow> bool" shows "(\<exists>\<^sub>\<infinity>i. P (Suc i)) = (\<exists>\<^sub>\<infinity>i. P i)"
adds_except_fun: fixes s :: "'a \<Rightarrow> 'b"   and t :: "'a \<Rightarrow> 'b"   and V :: "'a set" shows "(s adds t) = (except_fun s V adds except_fun t V \<and> except_fun s (- V) adds except_fun t (- V))"
inverse_nonzero_iff_nonzero: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "(inverse a = zero) = (a = zero)"
of_circline_clmat_def': fixes H :: "circline_mat" shows "of_circline_clmat H = (if pos_oriented_clmat H then H else opposite_ocircline_clmat H)"
iIN_iMOD_neq: fixes m :: "nat"   and n :: "nat"   and d :: "nat"   and r :: "nat" assumes "0 < m" shows "[n\<dots>,d] \<noteq> [ r, mod m ]"
is_path_simps_2: fixes v :: "'v"   and w :: "'w"   and v' :: "'v" shows "is_path v [(v, w, v')] v' = ((v, w, v') \<in> E)"
cod_simp: fixes f :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" assumes "S.VV.arr f" shows "S.VV.cod f = S.VxV.cod f"
MN_refines_eq_app_right: fixes L :: "'b list set"   and R :: "('b list \<times> 'b list) set" assumes "MyhillNerode L R" shows "R \<subseteq> eq_app_right L"
ordertype_Times: fixes A :: "'a set"   and B :: "'b set"   and r :: "('a \<times> 'a) set"   and s :: "('b \<times> 'b) set" assumes "small A"   and "small B"   and "wf r"   and "trans r"   and "total_on A r"   and "wf s"   and "trans s"   and "total_on B s" shows "ordertype (A \<times> B) (r <*lex*> s) = ordertype B s * ordertype A r"
ax_Lit2: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and v :: "val" shows "G,A\<turnstile>{Normal P} Lit v-\<succ> {Normal (P\<down>=In1 v)}"
sub_branch_repl: fixes j :: "'b"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and i :: "'b" assumes "j \<notin> branch_nominals branch" shows "i \<notin> branch_nominals (sub_branch (id(i := j, j := i)) branch)"
GF_advice_congruent_axioms: shows "GF_advice_congruent (\<sim>\<^sub>P) id"
netmap_subnets_samef: fixes s1 :: "'c net_state"   and f :: "nat \<Rightarrow> 'c option"   and s2 :: "'c net_state" assumes "netmap s1 = f"   and "netmap s2 = f" shows "netmap (SubnetS s1 s2) = f"
compatible_def: fixes G :: "('e, 'f) pre_digraph"   and H :: "('e, 'f) pre_digraph" shows "compatible G H \<equiv> tail G = tail H \<and> head G = head H"
False: shows "\<not> (d = 0 \<and> e = 0)"
adds: fixes xs :: "('a \<times> bool) list"   and ys :: "('a \<times> bool) list" assumes "proper_signed_list xs"   and "proper_signed_list ys" shows "k (prappend_signed_list xs ys) = k xs + k ys"
S_eq: fixes G :: "real \<Rightarrow> real"   and n :: "nat"   and x :: "real" assumes "G 1 = 1"   and "\<And>x. 0 < x \<Longrightarrow> G (x + 1) = x * G x"   and "\<And>x. 0 < x \<Longrightarrow> 0 < G x"   and "convex_on {0<..} (ln \<circ> G)"   and "2 \<le> n" shows "??.Gamma_Function.S G (real n) (real n + x) = (ln (G (real n + x)) - ln (fact (n - 1))) / x"
assn_supd_def: fixes P :: "'b assn"   and f :: "state \<Rightarrow> state" shows "(P ;. f) = (\<lambda>Y s' Z. \<exists>s. P Y s Z \<and> s' = f s)"
assms_5: shows "h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>h h'"
insertI: fixes A :: "'a set"   and a :: "'a" assumes "finite A" shows "finite (Set.insert a A)"
lossless_spmf_spmf_of_spmf: fixes p :: "'c pmf" shows "lossless_spmf (spmf_of_pmf p)"
asIMOD_igSubst: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom"   and ys :: "'varSort"   and Y :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'b"   and y :: "'var"   and X :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'b" shows "igSubst (asIMOD SEM) ys Y y X = (\<lambda>val. if sWlsVal SEM val then X (val (y := Y val)_ys) else undefined)"
comp_inv_arr': fixes f :: "('a, 'a set) Maps.arr" assumes "Maps.iso f" shows "Maps.inv f \<odot> f = Maps.dom f"
sel_13: fixes x51 :: "nat"   and x52 :: "'a"   and x53 :: "'a" shows "partner (RecvMarker x51 x52 x53) = x53"
assms_2: shows "c \<in> manifold_eucl.atlas k"
GEN_ALGO_tag_def: fixes P :: "'a" shows "GEN_ALGO_tag P \<equiv> P"
fps_divide_subdegree_ge: fixes f :: "'a fps"   and g :: "'a fps" assumes "f / g \<noteq> 0" shows "subdegree f - subdegree g \<le> subdegree (f / g)"
simps_1: fixes \<O> :: "nat set" shows "read_only_reads \<O> [] = {}"
redT_updTs_new_thread: fixes ts :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option"   and tas :: "('c, 'a, 'd) new_thread_action list"   and t :: "'c"   and x :: "'a"   and w :: "'b \<Rightarrow>f nat" assumes "redT_updTs ts tas t = \<lfloor>(x, w)\<rfloor>"   and "thread_oks ts tas"   and "ts t = None" shows "\<exists>m. NewThread t x m \<in> set tas \<and> w = no_wait_locks"
Ord_in_Ord: fixes k :: "V"   and m :: "V" assumes "Ord k"   and "m \<in> elts k" shows "Ord m"
rel_dds'_rel_dds: fixes B :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_dds' B = rel_dds (=) B"
B: shows "B \<in> N \<rightarrow>\<^sub>M prob_algebra L"
Lambert_W_def: fixes x :: "real" shows "Lambert_W x = (if x < - exp (- 1) then - 1 else THE w. - 1 \<le> w \<and> w * exp w = x)"
sup_left_idem: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.semilattice_sup (sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool)" shows "(sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (x::'b) (sup x (y::'b)) = sup x y"
simps_2: fixes n :: "nat"   and rai :: "int poly \<times> rat \<times> rat" shows "real_of_2 (Irrational n rai) = real_of_1 rai"
multpw_splitR: fixes X1 :: "'b multiset"   and X2 :: "'b multiset"   and Y :: "'b multiset"   and ns :: "('b \<times> 'b) set" assumes "(X1 + X2, Y) \<in> multpw ns" obtains Y1 :: "'b multiset" and Y2 :: "'b multiset" where "Y = Y1 + Y2" and "(X1, Y1) \<in> multpw ns" and "(X2, Y2) \<in> multpw ns"
transfer: shows "(rel_set pcr_CoCalls ===> pcr_CoCalls ===> (=)) (\<lambda>S x. x = \<Union> S) is_cc_lub"
not_less: fixes xa :: "'o"   and x :: "'a"   and y :: "'a" shows "(\<not> lt xa x y) = le xa y x"
remdups_adj_Cons_alt: fixes x :: "'g"   and xs :: "'g list" shows "x # tl (remdups_adj (x # xs)) = remdups_adj (x # xs)"
numbound0_I: fixes a :: "Ferrack.num"   and b :: "real"   and bs :: "real list"   and b' :: "real" assumes "numbound0 a" shows "Inum (b # bs) a = Inum (b' # bs) a"
one_cblinfun_apply_one: shows "1 *\<^sub>V (1::'b) = (1::'a)"
zmod_zminus1_not_zero: fixes k :: "int"   and l :: "int" assumes "- k mod l \<noteq> 0" shows "k mod l \<noteq> 0"
isometryD_3: fixes f :: "'c \<Rightarrow> 'd" assumes "isometry f" shows "surj f"
nat_pow_eone: fixes x :: "'a" assumes "x \<in> carrier R" shows "x [^] 1 = x"
set: fixes x :: "'c" shows "set_dual (dual x) = {x}"
ABinds_delete_below: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "ABinds (delete x \<Gamma>)\<cdot>ae \<sqsubseteq> ABinds \<Gamma>\<cdot>ae"
cdeclaredfield_def: fixes G :: "prog"   and C :: "qtname" shows "cdeclaredfield G C = (case class G C of None \<Rightarrow> Map.empty | Some c \<Rightarrow> table_of (cfields c))"
verticesFrom_nextElem_eq: fixes f :: "face"   and v :: "nat"   and u :: "nat" assumes "distinct (vertices f)"   and "v \<in> \<V> f"   and "u \<in> \<V> f" shows "nextElem (verticesFrom f v) (hd (verticesFrom f v)) u = nextElem (vertices f) (hd (vertices f)) u"
"with": shows "module.dependent \<equiv> dependent.with (0::'d) (0::'e) (+)"
map_family_aHom: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and S :: "('f, 'g) aGroup_scheme"   and g :: "'c \<Rightarrow> 'f \<Rightarrow> 'd" assumes "\<forall>j\<in>I. aGroup (A j)"   and "aGroup S"   and "\<forall>j\<in>I. g j \<in> aHom S (A j)" shows "(\<lambda>y\<in>carrier S. \<lambda>j\<in>I. g j y) \<in> aHom S (a\<Pi>\<^bsub>I\<^esub> A)"
geom_lgeom: fixes p :: "real"   and M :: "bool stream measure"   and u :: "real"   and d :: "real"   and v :: "real"   and n :: "nat"   and w :: "bool stream" assumes "infinite_coin_toss_space p M" shows "geom_rand_walk u d v n w = lgeom_proc u d v (stake n w)"
semiring_normalization_rules_12: fixes a :: "'a" shows "a * (1::'a) = a"
terminatess_Stuck: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" shows "\<Gamma>\<turnstile>cs,css \<Down> Stuck"
in_num: fixes n :: "'trm" assumes "n \<in> num" shows "n \<in> trm"
lhd_wit: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist" assumes "chain R\<^sup>+\<^sup>+ xs" shows "lhd (wit R xs) = lhd xs"
pr0_simps_1: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "arr (pr0 a b)"
nonpos_Reals_mult_I1: fixes a :: "'a"   and b :: "'a" assumes "a \<in> \<real>\<^sub>\<ge>\<^sub>0"   and "b \<in> \<real>\<^sub>\<le>\<^sub>0" shows "a * b \<in> \<real>\<^sub>\<le>\<^sub>0"
union: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "A \<noteq> {}"   and "finite B"   and "B \<noteq> {}" shows "Max (A \<union> B) = max (Max A) (Max B)"
path_connected_space_subconnected: fixes X :: "'a topology" shows "path_connected_space X = (\<forall>x\<in>topspace X. \<forall>y\<in>topspace X. \<exists>S. path_connectedin X S \<and> x \<in> S \<and> y \<in> S)"
proj_LCons: fixes tCMP :: "'b \<Rightarrow> cnf \<Rightarrow> 'a"   and active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and i :: "'b"   and x :: "cnf"   and xs :: "cnf llist" shows "dynamic_component.proj tCMP active i (x #\<^sub>l xs) = (if active i x then tCMP i x #\<^sub>l dynamic_component.proj tCMP active i xs else dynamic_component.proj tCMP active i xs)"
AX: fixes G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and C :: "Class"   and m :: "Method"   and l :: "int"   and A :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and B :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and I :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and Mspec :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and Minv :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and Anno :: "(int \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list" assumes "G\<down>(C, m, l) = Some (A, B, I)"   and "MST\<down>(C, m) = Some (Mspec, Minv, Anno)"   and "\<forall>Q. Anno\<down>l = Some Q \<longrightarrow> (\<forall>s0 s. A s0 s \<longrightarrow> Q s0 s)"   and "\<forall>s0 s. A s0 s \<longrightarrow> I s0 s (heap s)" shows "G \<rhd> \<langle> A \<rangle> C,m,l \<langle> B \<rangle> I"
residual_network_sel_4: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" shows "sink (res.residual_network f) = sink (residual_network f)"
ans4: fixes x :: "'a"   and y :: "'a" shows "ad (x + y) = ad x \<cdot> ad y"
simps_1: fixes uu :: "'d \<Rightarrow> 'c"   and x :: "'b" shows "sub uu (Pro x) = Pro x"
zhmsetmset_inverse: fixes x :: "zhmultiset" shows "ZHMSet (zhmsetmset x) = x"
trivial_RModuleHom: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c" assumes "RModule R smult M"   and "\<forall>r\<in>R. smult' r (0::'c) = (0::'c)" shows "RModuleHom R smult M smult' 0"
distinct_17: fixes x11 :: "'d"   and x12 :: "'a \<Rightarrow> 'a set"   and x13 :: "('a, 'b, 'c, 'd) seqp"   and x10 :: "'c" shows "{x11}\<langle>x12\<rangle> x13 \<noteq> call(x10)"
atLeastAtMost_iff: fixes i :: "'b"   and l :: "'b"   and u :: "'b" shows "(i \<in> {l..u}) = (l \<le> i \<and> i \<le> u)"
OclIncluding_def: fixes x :: "'d state \<times> 'd state \<Rightarrow> Bag('e)"   and y :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "x->including\<^sub>B\<^sub>a\<^sub>g(y) = (\<lambda>\<tau>. if (\<delta> x) \<tau> = true \<tau> \<and> (\<upsilon> y) \<tau> = true \<tau> then Abs_Bag\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<lfloor>\<lfloor>\<lceil>\<lceil>Rep_Bag\<^sub>b\<^sub>a\<^sub>s\<^sub>e (x \<tau>)\<rceil>\<rceil>(y \<tau> := \<lceil>\<lceil>Rep_Bag\<^sub>b\<^sub>a\<^sub>s\<^sub>e (x \<tau>)\<rceil>\<rceil> (y \<tau>) + 1)\<rfloor>\<rfloor> else invalid \<tau>)"
put_closure: fixes x :: "'e \<Longrightarrow> 'f"   and \<sigma> :: "'f"   and v :: "'e" assumes "weak_lens x" shows "put\<^bsub>x\<^esub> \<sigma> v \<in> \<S>\<^bsub>x\<^esub>"
Abs_eint_induct: fixes P :: "eint \<Rightarrow> bool"   and x :: "eint" assumes "\<And>y. y \<in> UNIV \<Longrightarrow> P (Abs_eint y)" shows "P x"
relcomp_fold: fixes R :: "('b \<times> 'c) set"   and S :: "('c \<times> 'd) set" assumes "finite R"   and "finite S" shows "R O S = Finite_Set.fold (\<lambda>(x, y) A. Finite_Set.fold (\<lambda>(w, z) A'. if y = w then insert (x, z) A' else A') A S) {} R"
state_equal_to_snapshot_def: fixes c :: "('d, 'e, 'f, 'g) configuration_scheme"   and c' :: "('d, 'e, 'f, 'h) configuration_scheme" shows "state_equal_to_snapshot c c' \<equiv> ps_equal_to_snapshot c c' \<and> cs_equal_to_snapshot c c'"
F2set2_image_Lev1: fixes xe :: "(bd_type_F + bd_type_F) list"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and xc :: "'d"   and xd :: "bd_type_F"   and x :: "'e" assumes "xe \<in> Lev1 s1 s2 n xc"   and "xe @ [Inl xd] \<in> Lev1 s1 s2 (Suc n) xc"   and "rv1 s1 s2 xe xc = Inr x" shows "xd \<in> tobd_F22 s2 x ` F2set2 (s2 x)"
nearly_healthy_wlp_Apply: fixes f :: "'a \<Rightarrow> 'a" shows "nearly_healthy (wlp (Apply f))"
nprv_clear5_2: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and \<phi>1 :: "'c"   and \<phi>3 :: "'c"   and \<phi>4 :: "'c"   and \<phi>5 :: "'c"   and \<psi> :: "'c"   and \<phi>2 :: "'c" assumes "Deduct_with_False_Disj var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv"   and "Deduct_with_False_Disj.nprv cnj imp fls prv {\<phi>1, \<phi>3, \<phi>4, \<phi>5} \<psi>"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<phi>3 \<in> fmla"   and "\<phi>4 \<in> fmla"   and "\<phi>5 \<in> fmla"   and "\<psi> \<in> fmla" shows "Deduct_with_False_Disj.nprv cnj imp fls prv {\<phi>1, \<phi>2, \<phi>3, \<phi>4, \<phi>5} \<psi>"
conditional_entropy_nonneg: fixes X :: "'key \<Rightarrow> 'a"   and Y :: "'key \<Rightarrow> 'b" assumes "simple_function (point_measure keys (\<lambda>x. ennreal (K x))) X"   and "simple_function (point_measure keys (\<lambda>x. ennreal (K x))) Y" shows "0 \<le> K.conditional_entropy_Pow X Y"
one_integer_group: shows "\<one>\<^bsub>integer_group\<^esub> = 0"
octo_distrib_left: fixes a :: "octo"   and b :: "octo"   and c :: "octo" shows "a * (b + c) = a * b + a * c"
inf_sup_aci_2: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "class.lattice inf less_eq less sup" shows "inf (inf x y) z = inf x (inf y z)"
gcd_rat_poly_unfold: shows "gcd = gcd_rat_poly"
not_refTE: fixes T :: "ty" assumes "\<not> is_refT T" obtains "T = Void \<or> T = Boolean \<or> T = Integer"
cod_dom: fixes f :: "'b" assumes "C'.arr f" shows "C'.cod (C'.dom f) = C'.dom f"
c_assoc_have_key_pos: fixes ls :: "nat"   and x :: "nat" assumes "c_assoc_have_key ls x = 0" shows "0 < ls"
assms_4: shows "awalk u1 (e1 # p) v"
tm_append_shift_append_steps: fixes st :: "nat"   and l :: "cell list"   and r :: "cell list"   and tp :: "(action \<times> nat) list"   and stp :: "nat"   and st' :: "nat"   and l' :: "cell list"   and r' :: "cell list"   and tp1 :: "(action \<times> nat) list"   and tp2 :: "(action \<times> nat) list" assumes "steps0 (st, l, r) tp stp = (st', l', r')"   and "0 < st'"   and "is_even (length tp1)" shows "steps0 (st + length tp1 div 2, l, r) (tp1 @ shift tp (length tp1 div 2) @ tp2) stp = (st' + length tp1 div 2, l', r')"
DmdDmd: fixes F :: "'a \<Rightarrow> bool" shows "\<turnstile> (\<diamond>\<diamond>F) = (\<diamond>F)"
tendsG: fixes x :: "'a" assumes "x \<notin> NG" shows "(\<lambda>n. G n x) \<longlonglongrightarrow> (if x \<in> S then g x else (0::'c))"
rep_eq: fixes x :: "perm"   and xa :: "'b fs_set" shows "Rep_fs_set (x \<bullet> xa) = x \<bullet> Rep_fs_set xa"
compress_prule_set_def: shows "compress_prule_set = fcompress \<circ> (|`|) (map_prod id compress_pterm)"
Y_def: shows "Y \<equiv> mat 2 2 (\<lambda>(i, j). if i = j then 0 else if i = 0 then - \<i> else \<i>)"
bind_returns_result_I: fixes h :: "'a"   and f :: "('a, 'b, 'c) prog"   and x :: "'c"   and h' :: "'a"   and g :: "'c \<Rightarrow> ('a, 'b, 'd) prog"   and y :: "'d" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> g x \<rightarrow>\<^sub>r y" shows "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"
dense_interval: fixes L :: "'a set"   and l :: "'a"   and x :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "finite L"   and "l \<in> L"   and "l < x"   and "P x"   and "\<And>y l. \<lbrakk>\<forall>y\<in>{l<..<x}. y \<notin> L; l < x; l < y; y \<le> x\<rbrakk> \<Longrightarrow> P y" shows "\<exists>l\<in>L. l < x \<and> (\<forall>y\<in>{l<..<x}. y \<notin> L) \<and> (\<forall>y. l < y \<and> y \<le> x \<longrightarrow> P y)"
eq_onp_to_Domainp: fixes P :: "'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "Quotient (eq_onp P) Abs Rep T" shows "Domainp T = P"
simps_6: fixes v :: "'b rexp" shows "le_rexp (Star v) Zero = False"
assms_2: fixes t :: "'a ttree"   and t' :: "'a ttree"   and x :: "'a" assumes "P t t'" shows "possible t x = possible t' x"