hcju/isabelleps huggingface dataset:
signed_take_bit_signed_take_bit: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat"   and a :: "'c" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not m (ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n a) = ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not (min m n) a"
sup_state_opt_any_None: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and s :: "(ty list \<times> ty err list) option" shows "P \<turnstile> s \<le>' None = (s = None)"
IH: shows "\<forall>ys. |ys| < |ps0| \<longrightarrow> set (map fst ys) \<subseteq> \<V> g \<longrightarrow> distinct (map fst ys) \<longrightarrow> ExcessNotAtRec ys g = Max {sum snd P |P. P \<subseteq> set ys \<and> separated g (fst ` P)}"
map_comp: fixes g :: "'e \<Rightarrow> 'b"   and f :: "'f \<Rightarrow> 'e"   and v :: "'a \<Rightarrow> 'f" shows "g \<circ> (f \<circ> v) = g \<circ> f \<circ> v"
ordertype_inc_le: fixes A :: "'a set"   and B :: "'b set"   and r :: "('a \<times> 'a) set"   and \<pi> :: "'a \<Rightarrow> 'b"   and s :: "('b \<times> 'b) set" assumes "small A"   and "small B"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; (x, y) \<in> r\<rbrakk> \<Longrightarrow> (\<pi> x, \<pi> y) \<in> s"   and "wf r"   and "total_on A r"   and "wf s"   and "trans s"   and "\<pi> ` A \<subseteq> B" shows "ordertype A r \<le> ordertype B s"
wl'_Inr: fixes ra :: "'a"   and s :: "'s" assumes "(Some (ra, s), None) \<in> whileLoop_results C (\<lambda>a. gets_the (B a))" shows "(Some ra, None) \<in> option_while' (\<lambda>a. C a s) (\<lambda>a. B a s)"
rep_eq: fixes x :: "'a" shows "pdevs_apply (singleton_pdevs x) = (\<lambda>i. if i = 0 then x else (0::'a))"
nop_vrange: shows "\<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A"
coboundedI2: fixes c :: "'a"   and b :: "'a"   and a :: "'a" assumes "c \<le> b" shows "c \<le> sup a b"
igConsIPresIGWlsSTR_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" shows "igConsIPresIGWlsSTR MOD \<equiv> igVarIPresIGWls MOD \<and> igAbsIPresIGWlsSTR MOD \<and> igOpIPresIGWlsSTR MOD"
ltl_prop_implies_def: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "\<phi> \<longrightarrow>\<^sub>P \<psi> \<equiv> \<forall>\<A>. \<A> \<Turnstile>\<^sub>P \<phi> \<longrightarrow> \<A> \<Turnstile>\<^sub>P \<psi>"
assms: shows "aff_dim S \<noteq> int DIM('n)"
SSpec_derivable: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP" assumes "G \<rhd> Call : SSpec Call" shows "G \<rhd> c : SSpec c"
hom_deg_set_closed_scalar: fixes A :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and c :: "'c"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and z :: "nat" assumes "\<And>a. a \<in> A \<Longrightarrow> c \<cdot> a \<in> A"   and "p \<in> hom_deg_set z A" shows "c \<cdot> p \<in> hom_deg_set z A"
objI_trg: fixes a :: "'a" assumes "B.arr a"   and "src a = a" shows "B.obj a"
rphil_phil: fixes e :: "dining_event"   and i :: "nat"   and s :: "nat" assumes "e \<in> Tr\<^sub>r\<^sub>p i s" shows "phil e = i"
ereal_Liminf_Sup_monoset: fixes net :: "'b filter"   and f :: "'b \<Rightarrow> ereal" shows "Liminf net f = Sup {l. \<forall>S. open S \<longrightarrow> mono_set S \<longrightarrow> l \<in> S \<longrightarrow> (\<forall>\<^sub>F x in net. f x \<in> S)}"
quotient_web_sel_3: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" shows "A (quotient_web \<Gamma> f) = \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) - (B \<Gamma> - A \<Gamma>)"
inv_unique: fixes y :: "nat \<Rightarrow> 'a"   and x :: "nat \<Rightarrow> 'a"   and y' :: "nat \<Rightarrow> 'a" assumes "y \<otimes>\<^bsub>P\<^esub> x = \<one>\<^bsub>P\<^esub>"   and "x \<otimes>\<^bsub>P\<^esub> y' = \<one>\<^bsub>P\<^esub>"   and "x \<in> carrier P"   and "y \<in> carrier P"   and "y' \<in> carrier P" shows "y = y'"
measurable_marg_dens': fixes x :: "nat" assumes "x \<in> V" shows "(\<lambda>(xa, y). marg_dens \<Y> x xa y) \<in> borel_measurable (state_measure V' \<Gamma> \<Otimes>\<^sub>M stock_measure (\<Gamma> x))"
resolution_superset: fixes C\<^sub>1 :: "fterm literal set"   and C\<^sub>2 :: "fterm literal set"   and L\<^sub>1 :: "fterm literal set"   and L\<^sub>2 :: "fterm literal set"   and \<sigma> :: "char list \<Rightarrow> fterm" shows "mresolution C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma> \<subseteq> resolution C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma>"
rtrancl_induct: fixes a :: "'e"   and b :: "'e"   and r :: "('e \<times> 'e) set"   and P :: "'e \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>*"   and "P a"   and "\<And>y z. \<lbrakk>(a, y) \<in> r\<^sup>*; (y, z) \<in> r; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
redT_preserves_wset_thread_ok: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s -t\<triangleright>ta\<rightarrow> s'"   and "wset_thread_ok (wset s) (thr s)" shows "wset_thread_ok (wset s') (thr s')"
lev_def: fixes I :: "int set" shows "lev I = Zleast {n \<in> I. 0 < n}"
LENGTH_statelist': fixes s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "length (statelist' s as) = length as + 1"
f: fixes A :: "'b set" assumes "A \<in> sets M" shows "f -` A \<inter> X \<in> sets N"
approximating_bigstep_to_undecided: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state" assumes "\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow>\<^sub>\<alpha> Undecided" shows "s = Undecided"
from_hma\<^sub>v_def: fixes v :: "('b, 'c) Finite_Cartesian_Product.vec" shows "Mod_Type_Connect.from_hma\<^sub>v v = Matrix.vec CARD('c) (\<lambda>i. v $h mod_type_class.from_nat i)"
rec: shows "basis_reduction_add_rows_loop_cost st i (j - 1) fs = (res, c2)"
intro: fixes s :: "nat"   and k :: "nat set"   and t :: "nat" assumes "s \<in> k"   and "t \<notin> k" shows "NCut_axioms s t k"
z_non_zero_proj2_pt: fixes v :: "(real, 2) vec" shows "z_non_zero (proj2_pt v)"
mmu_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state" assumes "low_equal s1 s2" shows "mmu s1 = mmu s2"
right_unit: fixes a :: "'a" assumes "a \<in> R" shows "a + \<zero> = a"
isCont_def: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'd" shows "isCont f a = f \<midarrow>a\<rightarrow> f a"
quotl_unique: fixes x :: "'a"   and y :: "'a" assumes "x \<preceq>\<^sub>L y" shows "\<exists>!z. D z x \<and> y = z \<oplus> x"
label_stab_map_chamber_map: fixes \<phi> :: "'a \<Rightarrow> 'a"   and C :: "'a set" assumes "label_wrt C0 \<phi>"   and "fixespointwise \<phi> C0"   and "chamber C" shows "SimplicialComplex.maxsimp (CoxeterComplex.TheComplex (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})) ((\<lambda>v. {w \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>. w \<rightarrow> \<phi> v = v}) ` C)"
set_add_closed: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> carrier R"   and "B \<subseteq> carrier R" shows "A <+> B \<subseteq> carrier R"
divide_eq_eq_numeral1_1: fixes b :: "'a"   and w :: "num"   and a :: "'a" shows "(b / numeral w = a) = (if numeral w \<noteq> (0::'a) then b = a \<cdot> numeral w else a = (0::'a))"
orthogonal_matrix_def: fixes Q :: "(('a, 'b) vec, 'b) vec" shows "orthogonal_matrix Q = (Finite_Cartesian_Product.transpose Q ** Q = mat (1::'a) \<and> Q ** Finite_Cartesian_Product.transpose Q = mat (1::'a))"
pseudo_proj_True_singleton: fixes w :: "bool stream"   and n :: "nat" assumes "w = pseudo_proj_True n w" shows "pseudo_proj_True n -` {w} \<inter> space (bernoulli_stream p) \<in> sets (bernoulli_stream p)"
rotate_drop_take: fixes n :: "nat"   and xs :: "'b list" shows "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
sup_Some: fixes x :: "'b"   and y :: "'b" shows "Some x \<squnion> Some y = Some (x \<squnion> y)"
simps_4: fixes Obj :: "'e set"   and Mor :: "'d set"   and Dom :: "'d \<Rightarrow> 'e"   and Cod :: "'d \<Rightarrow> 'e"   and Id :: "'e \<Rightarrow> 'd"   and Comp :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and more :: "'f" shows "Cod \<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr> = Cod"
union: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "A \<noteq> {}"   and "finite B"   and "B \<noteq> {}" shows "\<Sqinter> (A \<union> B) = \<Sqinter> A \<sqinter> \<Sqinter> B"
IH: fixes s :: "('a, 'b) fmap" shows "length as = length (state_list s as) - 1"
ProcDynModifyReturnNoAbrSameFaults: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and init :: "'e \<Rightarrow> 'e"   and p :: "'e \<Rightarrow> 'd"   and return' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and Modif :: "'e \<Rightarrow> 'e set"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P dynCall init p return' c Q,A"   and "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s\<in>P. \<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>{\<sigma>} Call (p s) (Modif \<sigma>),{}" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P dynCall init p return c Q,A"
component_eq_subset: fixes F :: "'d program"   and G :: "'d program" shows "(F \<le> G) = (Init G \<subseteq> Init F \<and> Acts F \<subseteq> Acts G \<and> AllowedActs G \<subseteq> AllowedActs F)"
is_natural_transformation: shows "natural_transformation (\<cdot>) (\<cdot>) (\<lambda>f. local.Prod (cod \<iota>, f)) local.map \<l>.map"
shadows_consts_def: fixes t :: "'b" shows "shadows_consts t = (\<not> fdisjnt all_consts (all_frees t))"
tllist_ord_TNil_TCons: fixes b :: "'f"   and b1 :: "'f"   and y :: "'g"   and ys :: "('g, 'f) tllist" shows "tllist_ord b (TNil b1) (TCons y ys) = (b1 = b)"
rbt_sorted_entries: fixes t :: "('a, 'c) rbt" assumes "rbt_sorted t" shows "sorted (map fst (RBT_Impl.entries t))"
arcs_not_in: shows "{a, a'} \<inter> arcs H = {}"
diff_0_right: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e" assumes "class.group_add minus plus zero uminus" shows "minus a zero = a"
corec_prod_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" shows "corec_prod f g = (\<lambda>s. (f s, g s))"
take_max_eqD: fixes a :: "nat"   and b :: "nat"   and xs :: "'a list"   and ys :: "'a list" assumes "take (max a b) xs = take (max a b) ys" shows "take a xs = take a ys \<and> take b xs = take b ys"
restriction_to_subgroup_is_module: fixes H :: "'g set" assumes "ActingGroup.Subgroup H" shows "FGModule H (\<cdot>) V"
sum_confine_eq_Int: fixes I :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "finite I" shows "sum (confine f A) I = sum f (I \<inter> A)"
matrix_equation_hyp: shows "M_mat_R signs subsets *\<^sub>v w_vec_R p qs signs = v_vec_R p qs subsets"
two_calc_ideal_points: fixes H :: "circline" assumes "is_poincare_line H" shows "card (calc_ideal_points H) = 2"
thin3: fixes A1 :: "fm"   and A2 :: "fm"   and H :: "fm set"   and B :: "fm"   and A3 :: "fm" assumes "insert A1 (insert A2 H) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 H)) \<turnstile> B"
xgs_def: shows "xgs = (replicate (length Gs) (0::'a))[n := x]"
freshTauTransition: fixes P :: "pi"   and P' :: "pi"   and c :: "name" assumes "P \<longmapsto> \<tau> \<prec> P'"   and "c \<sharp> P" shows "c \<sharp> P'"
ocallT: fixes \<sigma> :: "nat \<Rightarrow> 'e"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and pn :: "'g"   and a :: "'f seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and p' :: "('e, 'f, 'g, 'h) seqp"   and i :: "nat" assumes "((\<sigma>, \<Gamma> pn), a, \<sigma>', p') \<in> oseqp_sos \<Gamma> i" shows "((\<sigma>, call(pn)), a, \<sigma>', p') \<in> oseqp_sos \<Gamma> i"
simps_2: fixes SES :: "('c, 'd) SES_rec"   and s1 :: "'c"   and e :: "'d"   and t :: "'d list" shows "path SES s1 (e # t) = (if \<exists>s2. s1 e\<longrightarrow>\<^bsub>SES\<^esub> s2 then path SES (the (T\<^bsub>SES\<^esub> s1 e)) t else None)"
l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.type_wf set_val set_val_locs"
simps_5: fixes uv :: "'b ext" shows "ext.Top + uv = ext.Top"
LNil: assumes "xs = LNil"   and "ys = LNil" shows "thesis"
simps_2: fixes scf :: "'a \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and f :: "'a"   and ts :: "('a, 'b) Term.term list" shows "scf_term scf (Fun f ts) = Fun f (scf_list (scf (f, length ts)) (map (scf_term scf) ts))"
D: shows "\<forall>x\<in>{0..}. (\<lambda>k. if x \<in> {0..real k} then complex_of_real x powr (z - 1) * (1 - complex_of_real x / of_nat k) ^ k else 0) \<longlonglongrightarrow> complex_of_real x powr (z - 1) / complex_of_real (exp x)"
resumptionp_resumption_eq: shows "resumptionp = (\<lambda>x. x \<in> resumption)"
in_G_fcs_in_H_fcs: fixes a :: "'b"   and x :: "'b" assumes "a \<in> arcs G"   and "x \<in> GM.face_cycle_set a" shows "x \<in> proj_arcs_H ` H.face_cycle_set (proj_arcs_G a)"
flow_undefined_1: fixes x0 :: "'a" assumes "0 \<notin> UNIV" shows "flow0 x0 = (\<lambda>_. 0::'a)"
not_choice_def: fixes p :: "('a, 'b, 'c, 'd) seqp" shows "not_choice p \<equiv> \<forall>p1 p2. p \<noteq> p1 \<oplus> p2"
hyps_2: fixes s :: "'a" assumes "\<Gamma>\<turnstile>merge_guards c2 \<down> Normal s" shows "\<Gamma>\<turnstile>c2 \<down> Normal s"
comp_id: fixes f :: "'a \<Rightarrow> 'b" shows "f \<circ> id = f"
ext_inject: fixes store :: "'c \<Rightarrow> nat"   and pred :: "(('c \<times> nat \<Rightarrow> 'd) \<Rightarrow> bool) set"   and more :: "'e"   and store' :: "'c \<Rightarrow> nat"   and pred' :: "(('c \<times> nat \<Rightarrow> 'd) \<Rightarrow> bool) set"   and more' :: "'e" shows "(\<lparr>store = store, pred = pred, \<dots> = more\<rparr> = \<lparr>store = store', pred = pred', \<dots> = more'\<rparr>) = (store = store' \<and> pred = pred' \<and> more = more')"
KC2_secrecy: fixes B :: "agent"   and K :: "nat"   and C :: "agent"   and N :: "nat"   and KC2 :: "nat"   and X :: "msg"   and Y :: "msg"   and evs :: "event list" assumes "Gets B \<lbrace>Crypt K \<lbrace>Agent C, Nonce N, Key KC2, X\<rbrace>, Y\<rbrace> \<in> set evs"   and "Nonce N \<notin> analz (knows Spy evs)"   and "KC2 \<in> symKeys"   and "evs \<in> set_cr" shows "Key KC2 \<notin> analz (knows Spy evs)"
simps: fixes a :: "nat"   and b :: "nat"   and c :: "nat"   and d :: "nat" shows "ipv4addr_of_dotdecimal (a, b, c, d) = ipv4addr_of_nat (d + 256 * c + 65536 * b + 16777216 * a)"
assms_2: shows "(r, r') \<in> sorted_poly_rel O mset_poly_rel"
assms_2: shows "pd_kleene_allegory_class.acyclic w"
L_def: shows "L \<equiv> {a \<in> A. \<forall>s\<in>A. f s \<sim> s \<or> f s = s \<longrightarrow> a \<sqsubseteq> s}"
Abss_def: fixes xTs :: "(char list \<times> Unary_PCF.type) list"   and b :: "expr" shows "\<Lambda>[xTs] b = fold (\<lambda>xT t. \<Lambda>\<langle>snd xT\<rangle> close0_Var xT t) (rev xTs) b"
interleave_append_tail: fixes s :: "'a event list"   and t1 :: "'a event list"   and C :: "'a event set"   and u :: "'a event list"   and t2 :: "'a event list" assumes "s setinterleaves ((t1, u), C)"   and "set t2 \<inter> C = {}" shows "s @ t2 setinterleaves ((t1 @ t2, u), C)"
SCall: fixes P :: "(char list list \<times> char list exp) prog"   and ps :: "char list exp list"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and vs :: "val list"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp"   and D :: "char list"   and sfs :: "sfields"   and l\<^sub>2' :: "sfields"   and e' :: "char list exp"   and h\<^sub>3 :: "heap"   and l\<^sub>3 :: "sfields"   and sh\<^sub>3 :: "sheap" assumes "P \<turnstile> \<langle>ps,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>map Val vs,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"   and "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = (pns, body) in D"   and "sh\<^sub>2 D = \<lfloor>(sfs, Done)\<rfloor> \<or> M = clinit \<and> sh\<^sub>2 D = \<lfloor>(sfs, Processing)\<rfloor>"   and "length vs = length pns"   and "l\<^sub>2' = [pns [\<mapsto>] vs]"   and "P \<turnstile> \<langle>body,(h\<^sub>2, l\<^sub>2', sh\<^sub>2)\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>3, l\<^sub>3, sh\<^sub>3)\<rangle>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(ps),s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>3, l\<^sub>2, sh\<^sub>3)\<rangle>"
ordst_antirefl: fixes X :: "char list" shows "\<not> ordst X X"
assms_1: shows "u \<longlonglongrightarrow> a"
strict_monoD: fixes x :: "ordinal"   and y :: "ordinal" assumes "x < y" shows "F x < F y"
surjective: fixes r :: "'a state_scheme" shows "r = \<lparr>owns = owns r, currk = currk r, issued = issued r, cards = cards r, roomk = roomk r, isin = isin r, safe = safe r, \<dots> = more r\<rparr>"
disj: shows "disjoint_family_on (\<lambda>n. {x \<in> space M. P n x} \<inter> S) I"
Exit_no_return_target: fixes a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and f :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option" assumes "valid_edge a"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"   and "targetnode a = Exit" shows "False"
that: fixes k :: "'a set"   and l :: "'a set" assumes "k exposed_face_of S"   and "l exposed_face_of T"   and "F = {x + y |x y. x \<in> k \<and> y \<in> l}" shows "thesis"
if_not_P: fixes P :: "bool"   and x :: "'b"   and y :: "'b" assumes "\<not> P" shows "(if P then x else y) = y"
subst_lit_is_neg: fixes L :: "('a, nat) Term.term literal"   and \<sigma> :: "nat \<Rightarrow> ('a, nat) Term.term" shows "is_neg (subst_lit L \<sigma>) = is_neg L"
prj_char_1: shows "Maps.PRJ\<^sub>0 \<lbrakk>\<lbrakk>tab\<^sub>0 f\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>"
suffixes_snoc: fixes xs :: "'c list"   and x :: "'c" shows "suffixes (xs @ [x]) = [] # map (\<lambda>ys. ys @ [x]) (suffixes xs)"
winningAllocationIsMaximalWithoutSeller: fixes \<Omega> :: "'c list"   and N :: "integer set"   and b :: "integer \<times> 'c set \<Rightarrow> 'd"   and r :: "integer" assumes "distinct \<Omega>"   and "set \<Omega> \<noteq> {}"   and "finite N" shows "Outside' {seller} (the_elem (argmax (sum (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel (N \<union> {seller}) (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg (N \<union> {seller}) \<Omega>)) r) <| ((N \<union> {seller}) \<times> finestpart (set \<Omega>))) (N \<union> {seller}) (set \<Omega>))) (maximalStrictAllocations N (set \<Omega>) b))) \<in> Outside' {seller} ` maximalStrictAllocations N (set \<Omega>) b"
real_vector_space_on_axioms: fixes p :: "'b" shows "real_vector_space_on (dest.tangent_space p)"
hyp: fixes i :: "'a" shows "continuous_map (Y i) Z (g \<circ> f i)"
weak_total_orderI: assumes "\<And>x y. \<lbrakk>x \<in> carrier L; y \<in> carrier L\<rbrakk> \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x" shows "weak_total_order L"
monoid_list_axioms: shows "monoid_list (*) (1::'a)"
star_subset_lists: fixes A :: "'c list set"   and S :: "'c set" assumes "A \<subseteq> lists S" shows "star A \<subseteq> lists S"
subst_trm_ty_fresh': fixes X :: "tyvrs"   and T :: "ty"   and t :: "trm" assumes "X \<sharp> T" shows "X \<sharp> t[X \<mapsto>\<^sub>\<tau> T]"
has_contour_integral_bound_part_circlepath_strong: fixes f :: "complex \<Rightarrow> complex"   and i :: "complex"   and z :: "complex"   and r :: "real"   and s :: "real"   and t :: "real"   and k :: "complex set"   and B :: "real" assumes "(f has_contour_integral i) (part_circlepath z r s t)"   and "finite k"   and "0 \<le> B"   and "0 < r"   and "s \<le> t"   and "\<And>x. x \<in> path_image (part_circlepath z r s t) - k \<Longrightarrow> cmod (f x) \<le> B" shows "cmod i \<le> B * r * (t - s)"
w_2: shows "f ` C = insert w (f ` C \<inter> f ` D)"
smallest_accepting_rank_def: shows "\<UU>.smallest_accepting_rank \<equiv> if \<UU>.accept then Some (LEAST i. finite \<UU>.fail \<and> finite (\<UU>.merge i) \<and> infinite (\<UU>.succeed i)) else None"
set_union_code: fixes xs :: "'c list"   and ys :: "'c list" shows "set xs \<union> set ys = set (xs @ ys)"
atom_hoare_sound: fixes p :: "'b set"   and c :: "'b com"   and q :: "'b set" assumes "\<parallel>- p c q"   and "atom_com c" shows "\<parallel>= p  c q"
SUP_eqI: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and x :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq (f i) x"   and "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> less_eq (f i) y) \<Longrightarrow> less_eq x y" shows "Sup (f ` A) = x"
the_\<nu>_props_1: shows "\<guillemotleft>\<pi>.the_\<nu> : s\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<pi>.chine\<guillemotright>"
inner_sum_left: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and y :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner (comm_monoid_add.sum plus zero f A) y = (\<Sum>x\<in>A. inner (f x) y)"
preserves_arr: fixes f :: "'a" assumes "A.arr f" shows "B.arr (F f)"
bis_defs_1: shows "Sbis \<equiv> gfp Sretr"
take_set_of_apprD: fixes xs :: "real list"   and XS :: "'b list"   and n :: "nat" assumes "xs \<in> set_of_appr XS" shows "take n xs \<in> set_of_appr (take n XS)"