hcju/isabelleps huggingface dataset:
elims_2: fixes x :: "atom fm"   and xa :: "real list" assumes "eval x xa" obtains a :: "atom" and \<Gamma> :: "real list" where "x = fm.Atom a" and "xa = \<Gamma>__" and "aEval a \<Gamma>__"   | uu_ :: "real list" where "x = TrueF" and "xa = uu_"   | \<phi> :: "atom fm" and \<psi> :: "atom fm" and \<Gamma> :: "real list" where "x = And \<phi>__ \<psi>__" and "xa = \<Gamma>'__" and "eval \<phi>__ \<Gamma>'__ \<and> eval \<psi>__ \<Gamma>'__"   | \<phi> :: "atom fm" and \<psi> :: "atom fm" and \<Gamma> :: "real list" where "x = Or \<phi>'__ \<psi>" and "xa = \<Gamma>''__" and "eval \<phi>'__ \<Gamma>''__ \<or> eval \<psi> \<Gamma>''__"   | \<phi> :: "atom fm" and \<Gamma> :: "real list" where "x = Neg \<phi>''__" and "xa = \<Gamma>'''__" and "\<not> eval \<phi>''__ \<Gamma>'''__"   | \<phi> :: "atom fm" and \<Gamma> :: "real list" where "x = ExQ \<phi>'''__" and "xa = \<Gamma>''''__" and "\<exists>x. eval \<phi>'''__ (x # \<Gamma>''''__)"   | \<phi> :: "atom fm" and \<Gamma> :: "real list" where "x = AllQ \<phi>''''__" and "xa = \<Gamma>'''''__" and "\<forall>x. eval \<phi>''''__ (x # \<Gamma>'''''__)"   | i :: "nat" and \<phi> :: "atom fm" and \<Gamma> :: "real list" where "x = AllN i__ \<phi>'''''__" and "xa = \<Gamma>''''''__" and "\<forall>l. length l = i__ \<longrightarrow> eval \<phi>'''''__ (l @ \<Gamma>''''''__)"   | i :: "nat" and \<phi> :: "atom fm" and \<Gamma> :: "real list" where "x = ExN i \<phi>" and "xa = \<Gamma>" and "\<exists>l. length l = i \<and> eval \<phi> (l @ \<Gamma>)"
simps_2: fixes h :: "nat" shows "U (Suc h) = N1 ` B h"
simps_1: fixes v :: "nat \<Rightarrow> 'a"   and x :: "nat"   and c :: "'a" shows "v \<Turnstile>\<^sub>a Leq x c = (v x \<le> c)"
intro: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set" assumes "calculus Bot Inf entails Red_I Red_F"   and "statically_complete_calculus_axioms Bot Inf entails Red_I" shows "statically_complete_calculus Bot Inf entails Red_I Red_F"
fresh_Conj: fixes a :: "atom"   and A :: "fm"   and B :: "fm" shows "a \<sharp> A AND B = (a \<sharp> A \<and> a \<sharp> B)"
bigstep: shows "(C1;; C2, s) \<Rightarrow> p1 + p2 \<Down> t2"
diff_floatarith_conv_add_uminus: fixes a :: "floatarith"   and b :: "floatarith" shows "a - b = a + - b"
con_rews_2: fixes a :: "'b"   and llist :: "'b llist" shows "a :@ llist \<noteq> \<bottom>"
isInBar_def: fixes xs_s :: "'varSort \<times> 'sort" shows "isInBar xs_s \<equiv> \<exists>delta i. wlsOpS delta \<and> barOf delta i = Some xs_s"
multiplicative_function_axioms: shows "multiplicative_function (principal_dchar n)"
map_filter_None: fixes P :: "'d \<Rightarrow> 'c option"   and x :: "'d" assumes "P x = None" shows "List.map_filter P [x] = []"
simps_18: fixes x2 :: "'d" shows "vars_hd (Sym x2) = {}"
env_delete_fun_upd: fixes x :: "'a"   and m :: "'a \<Rightarrow> 'b"   and v :: "'b" shows "env_delete x (m(x := v)) = env_delete x m"
1: shows "cospan (ECC.prod f1 (local.dom f0)) (ECC.prod f0 (local.dom f0))"
step: fixes l :: "nat" assumes "l < Suc k" shows "c (0 + l) \<Rightarrow>\<^sub>d c (0 + Suc l)"
under_ofilter: fixes a :: "'b" shows "s.ofilter (s.under a)"
abs_eq: fixes x :: "('d \<times> 'e) list" shows "abs_slist_if x = Abs_slist (if distinct (map fst x) \<and> rsorted (map fst x) then x else [])"
inverse_positive_imp_positive: fixes a :: "'a" assumes "(0::'a) < inverse a"   and "a \<noteq> (0::'a)" shows "(0::'a) < a"
trs_refinement: fixes r :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and r' :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "(trs r \<le> trs r') = ((\<forall>x. inpt r x \<longrightarrow> inpt r' x) \<and> (\<forall>x. inpt r x \<longrightarrow> r' x \<le> r x))"
card_geq_3_iff_contains_3_elems: fixes A :: "'a set" shows "(3 \<le> card A) = (finite A \<and> (\<exists>x y z. x \<noteq> y \<and> x \<noteq> z \<and> y \<noteq> z \<and> x \<in> A \<and> y \<in> A \<and> z \<in> A))"
invar_def: fixes istate :: "'a"   and step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and \<phi> :: "'a \<Rightarrow> bool" shows "IO_Automaton.invar istate step \<phi> \<equiv> \<forall>s a. IO_Automaton.reach istate step s \<and> \<phi> s \<longrightarrow> \<phi> (snd (step s a))"
triv_measurable_cst: fixes N :: "'c measure"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "space N = space M"   and "space M \<noteq> {}"   and "sets N = {{}, space M}"   and "f \<in> borel_measurable N" shows "\<exists>c. \<forall>x\<in>space N. f x = c"
intros_8: fixes c1 :: "('a, 'b, 'c) com"   and c2 :: "('a, 'b, 'c) com" shows "com_relp c1 (Catch c1 c2)"
wf_theory_imp_wf_sig: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set" assumes "wf_theory \<Theta>" shows "wf_sig (sig \<Theta>)"
strict_order_OrdSum_def: fixes a :: "('a, 'b) OrdSum"   and b :: "('a, 'b) OrdSum" shows "a < b \<equiv> a \<le> b \<and> \<not> b \<le> a"
are_identities_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data" assumes "two_composable_identity_arrows_of_spans C prj0 prj1 \<mu> \<nu>" shows "partial_magma.ide (span_vertical_category.vcomp C) \<mu>"
fst_sym_preproc_aux_sorted_wrt: fixes ks :: "'t list"   and vs :: "'t list"   and gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "sorted_wrt (\<succ>\<^sub>t) ks"   and "\<And>k v. \<lbrakk>k \<in> set ks; v \<in> set vs\<rbrakk> \<Longrightarrow> v \<prec>\<^sub>t k" shows "sorted_wrt (\<succ>\<^sub>t) (fst (sym_preproc_aux gs ks (vs, fs)))"
curryI: fixes f :: "'d \<times> 'e \<Rightarrow> bool"   and a :: "'d"   and b :: "'e" assumes "f (a, b)" shows "curry f a b"
inverse_unit: shows "quotient.inverse (Class \<one>) = Class \<one>"
simps: fixes a1 :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and a2 :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and TI :: "('c set \<times> 'c set) set" shows "((a1, a2) \<in> timpl_closure'_step TI) = (\<exists>a b t s. a1 = t \<and> a2 = s \<and> (a, b) \<in> TI \<and> term_variants_pred ((\<lambda>_. [])(Abs a := [Abs b])) t s)"
filterlim_principal: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'c set"   and F :: "'b filter" shows "filterlim f (principal S) F = (\<forall>\<^sub>F x in F. f x \<in> S)"
case_2: fixes f1 :: "nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'a"   and x2 :: "nat" shows "(case R x2 of I x \<Rightarrow> f1 x | R x \<Rightarrow> f2 x) = f2 x2"
ws_cdecl_def: fixes G :: "prog"   and C :: "qtname"   and sc :: "qtname" shows "ws_cdecl G C sc = (C \<noteq> Object \<longrightarrow> is_class G sc \<and> (sc, C) \<notin> (subcls1 G)\<^sup>+)"
normal_vector_3: fixes W :: "'b vec set"   and n :: "nat"   and w :: "'b vec" assumes "finite W"   and "Suc (card W) = n"   and "\<not> module.lin_dep class_ring (module_vec TYPE('b) n) W"   and "W \<subseteq> carrier_vec n"   and "w \<in> W" shows "w \<bullet> gram_schmidt.normal_vector n W = (0::'b)"
eq1: shows "{(h', U'). (h', U') \<in> set (ps\<^sub>+) \<and> poly_deg h' = poly_deg h} = {(h0, U0)}"
mono_mprefix_DT_process: fixes P :: "'a \<Rightarrow> 'a process"   and Q :: "'a \<Rightarrow> 'a process"   and A :: "'a set" assumes "\<forall>x. P x \<sqsubseteq>\<^sub>D\<^sub>T Q x" shows "Mprefix A P \<sqsubseteq>\<^sub>D\<^sub>T Mprefix A Q"
wlp_congruence_rule: fixes l1 :: "'d list"   and l2 :: "'d list"   and R1 :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x1 :: "'d"   and R2 :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x2 :: "'d"   and w1 :: "'d \<Rightarrow> 'e"   and w2 :: "'d \<Rightarrow> 'e"   and f1 :: "'e \<Rightarrow> 'e \<Rightarrow> 'f"   and f2 :: "'e \<Rightarrow> 'e \<Rightarrow> 'f"   and g1 :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and g2 :: "'f \<Rightarrow> 'e \<Rightarrow> 'e" assumes "l1 = l2"   and "\<forall>y. ListMem y l2 \<longrightarrow> R1 x1 y = R2 x2 y"   and "\<forall>y. ListMem y l2 \<longrightarrow> R1 y x1 = R2 y x2"   and "w1 x1 = w2 x2"   and "\<forall>y1 y2. y1 = y2 \<longrightarrow> f1 (w1 x1) y1 = f2 (w2 x2) y2"   and "\<forall>y1 y2 z1 z2. y1 = y2 \<and> z1 = z2 \<longrightarrow> g1 (f1 (w1 x1) y1) z1 = g2 (f2 (w2 x2) y2) z2"   and "\<forall>x y. ListMem x l2 \<and> ListMem y l2 \<longrightarrow> R1 x y = R2 x y"   and "\<forall>x. ListMem x l2 \<longrightarrow> w1 x = w2 x"   and "\<forall>x y z. ListMem x l2 \<longrightarrow> g1 (f1 (w1 x) y) z = g2 (f2 (w2 x) y) z"   and "\<forall>x y. ListMem x l2 \<longrightarrow> f1 (w1 x) y = f2 (w1 x) y" shows "wlp R1 w1 g1 f1 x1 l1 = wlp R2 w2 g2 f2 x2 l2"
points_index_empty: fixes ps :: "'b set" shows "{#} index ps = 0"
lowIn_def: shows "lowIn = low_in \<tau>\<^sub>2"
horizontal_chop_assoc2: fixes v :: "view"   and v1 :: "view"   and v2 :: "view"   and v3 :: "view"   and v4 :: "view" shows "v=v1\<parallel>v2 \<and> v1=v3\<parallel>v4 \<longrightarrow> (\<exists>v'. v=v3\<parallel>v' \<and> v'=v4\<parallel>v2)"
foreach_impl_correct_4: fixes Qi :: "'q"   and \<pi>i :: "'m"   and u :: "'v"   and Qi' :: "'q"   and \<pi>i' :: "'m" assumes "foreach_impl Qi \<pi>i u (G_adj g u) = (Qi', \<pi>i')"   and "prim_invar_impl Qi \<pi>i" shows "M_lookup \<pi>i' = \<pi>' (Q_\<alpha> Qi) (M_lookup \<pi>i) u"
sig_inv_set'_def: fixes j :: "nat" shows "sig_inv_set' j = {r. keys (vectorize_poly r) \<subseteq> {0..<j}}"
fix_least_below: fixes F :: "'a \<rightarrow> 'a"   and x :: "'a" assumes "F\<cdot>x \<sqsubseteq> x" shows "fix\<cdot>F \<sqsubseteq> x"
assms_1: shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[r], Undecided\<rangle> \<Rightarrow> t"
valid_prefix_00: shows "valid_prefix (PrefixMatch 0 0)"
enabled_def: fixes cfg :: "('a, 'b, 'c) configuration"   and msg :: "('a, 'b) message" shows "enabled cfg msg \<equiv> msg \<in># msgs cfg"
Suc: shows "e_nth (block i t) (gap i t) = 0"
bv_smult_def: fixes w1 :: "bit list"   and w2 :: "bit list" shows "bv_smult w1 w2 = int_to_bv (bv_to_int w1 * bv_to_int w2)"
dropWhile_eq_self_iff: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "(dropWhile P xs = xs) = (xs = [] \<or> \<not> P (hd xs))"
parts_crypt_sign: fixes s :: "(agent \<times> msg) set"   and SK :: "nat option \<times> nat set \<times> nat set"   and n :: "nat"   and A :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "Crypt (SesK SK) (Sign n A) \<in> parts (used s)" shows "(Asset n, SesKey SK) \<in> s \<or> SesKey SK \<in> spied s"
distinct_9: fixes x6 :: "'a fm" shows "TrueF \<noteq> Neg x6"
pre_chainmap: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and \<Pi> :: "('e, 'f) chain" shows "pre (chainmap f g \<Pi>) = f (pre \<Pi>)"
gt_0_times_gt_0_imp: fixes a :: "zhmultiset"   and b :: "zhmultiset" assumes "0 < a"   and "0 < b" shows "0 < a * b"
vars0_def: shows "st2.vars0 = vars2 \<union> vars1"
graph_lm_1: fixes x :: "nat"   and y :: "nat"   and f :: "nat \<Rightarrow> nat" assumes "(x, y) \<in> graph f" shows "y = f x"
snth_SCons: fixes x :: "'d"   and xs :: "'d stream"   and n :: "nat" shows "(x ## xs) !! n = (case n of 0 \<Rightarrow> x | Suc n' \<Rightarrow> xs !! n')"
t1: assumes "Can t1" shows "D.ide \<lbrace>t1\<rbrace>"
env_restr_eqI: fixes S :: "'d set"   and m\<^sub>1 :: "'d \<Rightarrow> 'e"   and m\<^sub>2 :: "'d \<Rightarrow> 'e" assumes "\<And>x. x \<in> S \<Longrightarrow> m\<^sub>1 x = m\<^sub>2 x" shows "m\<^sub>1 f|` S = m\<^sub>2 f|` S"
induct: fixes P :: "'b heap \<Rightarrow> 'b heap \<Rightarrow> bool"   and a0 :: "'b heap"   and a1 :: "'b heap" assumes "\<And>h. P h heap.Empty"   and "\<And>v va. P heap.Empty (Hp v va)"   and "\<And>x hsx y hsy. P (Hp x hsx) (Hp y hsy)" shows "P a0 a1"
RS2Set: fixes p :: "('a, 'b) Combinators list" shows "set (removeShadowRules2 p) \<subseteq> set p"
of_nat_0_le_iff: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero (semiring_1.of_nat one plus zero n)"
apply_inverse_power_eq: fixes f :: "'a perm"   and n :: "nat"   and a :: "'a" shows "inverse (f ^ n) \<langle>$\<rangle> a = (f ^ (order f a - n mod order f a)) \<langle>$\<rangle> a"
well_order_on_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "well_order_on A r \<equiv> linear_order_on A r \<and> wf (r - Id)"
mat_swaprows_def: fixes k :: "nat"   and l :: "nat"   and A :: "'b mat" shows "swaprows k l A = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if k = i then A $$ (l, j) else if l = i then A $$ (k, j) else A $$ (i, j))"
f_values: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "bob_fun f n" shows "\<forall>x\<in>{i. i < 2 ^ n}. f x = 0 \<or> f x = 1"
A8: shows "\<forall>TQ TR. TQ \<in>T Q \<and> TR \<in>T R \<longrightarrow> (TQ, TR) \<in> TRel\<^sup>+"
lhsnot0: shows "\<Lambda> * (\<v> - 1) div (\<k> - 1) * (\<k> - 1) \<noteq> 0"
assms: fixes J :: "'a set" assumes "J \<in> I"   and "finite J" shows "A \<in> Pi\<^sub>F {J} M \<rightarrow>\<^sub>M N"
is_valid_problem_sas_plus_then_strips_transformation_too: fixes \<Psi> :: "('a, 'b) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "is_valid_problem_strips (\<phi> \<Psi> )"
eval_app2_6: fixes te :: "ExVar \<Rightarrow> Ty option"   and t :: "Ty" assumes "evm(xm \<mapsto> v2) hastyenv te "   and "te \<turnstile> em ===> t" shows "v hasty t"
r_alt_def': fixes i :: "nat" shows "fst (fst (r i)) = Abs (af \<phi> (prefix i w))"
circle_cube_def: shows "circle_cube = (\<lambda>(x, y). ((x - 1 / 2) * d, (2 * y - 1) * d * sqrt (1 / 4 - (x - 1 / 2) * (x - 1 / 2))))"
vars_set_def: fixes s :: "Literal set" shows "vars s \<equiv> {vbl. \<exists>l. l \<in> s \<and> var l = vbl}"
dual_upset: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "ord.upset less_eq = ord.downset (\<lambda>x y. less_eq y x)"
VLambda_vinsert: fixes a :: "V"   and A :: "V"   and f :: "V \<Rightarrow> V" shows "VLambda (vinsert a A) f = VLambda (set {a}) f \<union>\<^sub>\<circ> VLambda A f"
indep_vars_subset: fixes M' :: "'b \<Rightarrow> 'c measure"   and X :: "'b \<Rightarrow> (nat \<Rightarrow> real) \<Rightarrow> 'c"   and I :: "'b set"   and J :: "'b set" assumes "P.indep_vars M' X I"   and "J \<subseteq> I" shows "P.indep_vars M' X J"
soundness': fixes A :: "'b set"   and n :: "nat"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and M :: "('d, 'c) model"   and g :: "'b \<Rightarrow> 'd" assumes "A, n \<turnstile> branch"   and "M, g \<Turnstile>\<^sub>\<Theta> branch" shows "False"
Builds_insf: fixes B :: "hf \<Rightarrow> bool"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and s :: "hf"   and l :: "hf"   and k :: "hf"   and z :: "hf"   and y :: "hf" assumes "Builds B C s l"   and "LstSeq s k z"   and "l \<^bold>\<in> succ k" shows "Builds B C (insf s (succ k) y) l"
exp_base_append: fixes x :: "'a"   and n :: "nat"   and m :: "nat" shows "exp_base x (n + m) = map (([^]) x) (rev [n..<n + m]) @ exp_base x n"
partition_onD1: fixes A :: "'a set"   and P :: "'a set set" assumes "partition_on A P" shows "A = \<Union> P"
Sup_subset_mono: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "\<Squnion>A \<le> \<Squnion>B"
cart_basis'_def: shows "cart_basis' = (\<chi>i. axis i (1::'a))"
rep_eq: fixes x :: "\<kappa>" shows "Semantics.d\<^sub>\<kappa> x = eval\<kappa> x"
rev_rev_ident: fixes xs :: "'b list" shows "rev (rev xs) = xs"
simps_3: fixes f :: "'a"   and v :: "'a" shows "\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<Down> \<^bold>\<langle>v\<^bold>\<rangle> = \<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>v\<^bold>\<rangle>"
simps_1: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and S :: "'b SDG_node set"   and cs :: "'a list" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses" shows "SDG.slice_edges sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S cs [] = []"
intros: fixes K :: "'c \<Rightarrow> 'c pmf"   and \<omega> :: "'c stream"   and s :: "'c" assumes "MC_syntax.enabled K (shd \<omega>) (stl \<omega>)"   and "shd \<omega> \<in> set_pmf (K s)" shows "MC_syntax.enabled K s \<omega>"
ev_nxt: fixes \<phi> :: "'c stream \<Rightarrow> bool" shows "ev \<phi> = (\<lambda>xs. \<phi> xs \<or> nxt (ev \<phi>) xs)"
declclass_methd_Object: fixes G :: "prog"   and sig :: "sig"   and m :: "qtname \<times> methd" assumes "wf_prog G"   and "methd G Object sig = Some m" shows "declclass m = Object"
cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes node_ptr :: "(_) node_ptr" shows "cast node_ptr = (case node_ptr of node_ptr.Ext (Inl x) \<Rightarrow> Map.empty x | node_ptr.Ext (Inr (Inl character_data_ptr)) \<Rightarrow> Some character_data_ptr | node_ptr.Ext (Inr (Inr x)) \<Rightarrow> Map.empty x)"
inject: fixes x2 :: "'d"   and y2 :: "'d" shows "(\<lfloor>x2\<rfloor> = \<lfloor>y2\<rfloor>) = (x2 = y2)"
lemma_R_for_TOTAL_simple: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "learn_total \<psi> U s" shows "\<exists>s'\<in>\<R>. learn_total \<psi> U s'"
finite_neq_0: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c"   and h :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "finite {x. f x \<noteq> (0::'b)}"   and "finite {x. g x \<noteq> (0::'c)}"   and "\<And>x. h x (0::'b) (0::'c) = (0::'d)" shows "finite {x. h x (f x) (g x) \<noteq> (0::'d)}"
isodefl_ID_DEFL: shows "isodefl ID DEFL('a)"
vec_eq_iff: fixes x :: "'d Matrix.vec"   and y :: "'d Matrix.vec" shows "(x = y) = (dim_vec x = dim_vec y \<and> (\<forall>i<dim_vec y. x $v i = y $v i))"
cyc_free_not_empty: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'a DBMEntry"   and v :: "'b \<Rightarrow> nat" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs"   and "\<forall>c. v c \<le> n \<longrightarrow> 0 < v c" shows "[M]\<^bsub>v,n\<^esub> \<noteq> {}"
add2_comm: fixes a :: "'b option"   and b :: "'b option" shows "a ++ b = b ++ a"
freeA: shows "var \<notin> vars (- isolate_variable_sparse p var (Suc 0))"
exp_gt_one: fixes x :: "real" assumes "0 < x" shows "1 < exp x"
Pi_inc_gr: fixes i :: "nat"   and j :: "nat" assumes "i < j" shows "Pi i < Pi j"
simps_2: fixes c :: "color"   and uu :: "color"   and l :: "('b, 'c) RBT_Impl.rbt"   and k :: "'b"   and v :: "'c"   and r :: "('b, 'c) RBT_Impl.rbt" shows "paint c (Branch uu l k v r) = Branch c l k v r"
ranE: fixes v :: "'d"   and m :: "'e \<Rightarrow> 'd option" assumes "v \<in> ran m" obtains k :: "'e" where "m k = Some v"
m_assoc: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a"   and z :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P"   and "z \<in> carrier P" shows "x \<otimes>\<^bsub>P\<^esub> y \<otimes>\<^bsub>P\<^esub> z = x \<otimes>\<^bsub>P\<^esub> (y \<otimes>\<^bsub>P\<^esub> z)"
lam: shows "(\<lambda>i. f i (x i \<otimes>\<^bsub>A i\<^esub> y i)) \<in> I \<rightarrow> carrier G"
LEQ: shows "\<forall>p\<in>P. \<exists>r\<in>R. p <o r"
locks_init_fin_lift_state: fixes s :: "status"   and \<sigma> :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('c \<times> 'a \<Rightarrow>f nat) option) \<times> 'd) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set" shows "locks (init_fin_lift_state s \<sigma>) = locks \<sigma>"
that: fixes a :: "'a"   and v :: "'a llist" assumes "w = <a> $ v"   and "n \<le> llength v" shows "thesis"
int_simps_14: fixes P :: "'b \<Rightarrow> bool" shows "\<turnstile> (P \<longrightarrow> #False) = (\<not> P)"
coplanar_perm_12: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet C A B D"
aux: fixes sa :: "'a"   and s'a :: "'b list"   and i :: "nat"   and c :: "'d \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and \<sigma> :: "'d" assumes "(sa, s'a) \<in> arel"   and "i \<le> sz sa" shows "idx_iteratei_aux get (sz sa) i sa c f \<sigma> = foldli (drop (sz sa - i) s'a) c f \<sigma>"
admissible_targets_1: fixes l :: "'s"   and g :: "('c, real) cconstraint"   and \<mu> :: "('c set \<times> 's) pmf"   and X :: "'c set"   and l' :: "'s" assumes "(l, g, \<mu>) \<in> PTA.trans_of A"   and "(X, l') \<in> set_pmf \<mu>" shows "{v. v \<turnstile> g}\<^bsub>X \<rightarrow> 0\<^esub> \<subseteq> {v. v \<turnstile> PTA.inv_of A l'}"
Nil_is_rev_conv: fixes xs :: "'b list" shows "(\<epsilon> = rev xs) = (xs = \<epsilon>)"
inj_Inl: fixes A :: "'d set" shows "inj_on Inl A"
null_sets_def: fixes M :: "'b measure" shows "null_sets M = {N \<in> sets M. emeasure M N = 0}"
preserves_iso: fixes f :: "'b" assumes "D.iso f" shows "D.iso (trg\<^sub>D f)"
alw_inv: fixes f :: "'a stream \<Rightarrow> 'b stream"   and P :: "'b stream \<Rightarrow> bool"   and s :: "'a stream" assumes "\<And>s. f (stl s) = stl (f s)" shows "alw P (f s) = alw (\<lambda>x. P (f x)) s"
butterfly_0_right: fixes a :: "'b" shows "butterfly a (0::'a) = 0"
jumpF_eventually_const: fixes f :: "real \<Rightarrow> real"   and c :: "real"   and F :: "real filter" assumes "\<forall>\<^sub>F x in F. f x = c"   and "F \<noteq> bot" shows "jumpF f F = 0"
eq_None_iff_not_dom: fixes f :: "'c \<Rightarrow> 'b option"   and x :: "'c" shows "(f x = None) = (x \<notin> dom f)"
sum_norm_le: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> real" assumes "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> g x" shows "norm (sum f S) \<le> sum g S"
proj4_def: shows "\<pi>\<^sub>4 \<equiv> \<lambda>(uu_, uu_, flag, uu_, uu_, uu_). flag"
ordsetmap: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and a :: "'b"   and b :: "'b" assumes "OrderingSetMap less_eq less less_eq' less' P f"   and "a \<in> P"   and "b \<in> P"   and "less_eq a b" shows "less_eq' (f a) (f b)"
phl_seq: fixes p :: "'a"   and x :: "'a"   and r :: "'a"   and y :: "'a"   and q :: "'a" assumes "p \<parallel> x \<le> x \<parallel> r"   and "r \<parallel> y \<le> y \<parallel> q" shows "p \<parallel> (x \<parallel> y) \<le> x \<parallel> y \<parallel> q"
vi: fixes i :: "nat" shows "\<exists>vi. norm vi = 1 \<and> (\<forall>l\<le>i. vi \<bullet> (T ^^ l) (0::'a) \<le> (- A + eps i) * real l)"
int_pow_closed: fixes x :: "nat \<Rightarrow> int"   and i :: "int" assumes "x \<in> carrier Zp" shows "[i] \<cdot> x \<in> carrier Zp"