hcju/isabelleps huggingface dataset:
swapBinp_preserves_wls: fixes delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "wlsBinp delta binp" shows "wlsBinp delta (binp %%[z1 \<and> z2]_zs)"
domains_def: fixes f :: "'a1" shows "C1.domains f \<equiv> {a. C1.ide a \<and> f \<cdot>\<^sub>1 a \<noteq> C1.null}"
seq_INF_distrib: fixes X :: "'b::type set"   and c :: "'a::refinement_lattice"   and d :: "'b::type \<Rightarrow> 'a::refinement_lattice" assumes "(X::'b::type set) \<noteq> {}" shows "(c::'a::refinement_lattice) ; \<Sqinter> ((d::'b::type \<Rightarrow> 'a::refinement_lattice) ` (X::'b::type set)) = (\<Sqinter>x::'b::type\<in>X. c ; d x)"
hcomplex_of_hypreal_eq: fixes r :: "real star" shows "hcomplex_of_hypreal r = HComplex r 0"
Bang_1: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "P \<parallel> !P \<Longrightarrow>a<\<nu>x> \<prec> P'" shows "!P \<Longrightarrow>a<\<nu>x> \<prec> P'"
compare_Tuple3_def: shows "compare = (\<Lambda> \<langle>x1, y1, z1\<rangle> \<langle>x2, y2, z2\<rangle>. thenOrdering\<cdot>(compare\<cdot>x1\<cdot>x2)\<cdot>(thenOrdering\<cdot>(compare\<cdot>y1\<cdot>y2)\<cdot>(compare\<cdot>z1\<cdot>z2)))"
dg_10_def: fixes \<aa> :: "V" shows "dg_10 \<aa> = [ZFC_in_HOL.set {\<aa>}, []\<^sub>\<circ>, []\<^sub>\<circ>, []\<^sub>\<circ>]\<^sub>\<circ>"
distinct_15: fixes x21 :: "floatarith"   and x22 :: "floatarith"   and x23 :: "form"   and x41 :: "floatarith"   and x42 :: "floatarith" shows "Assign x21 x22 x23 \<noteq> LessEqual x41 x42"
Max_Un: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and B :: "'b set" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}"   and "finite B"   and "B \<noteq> {}" shows "linorder.Max less_eq (A \<union> B) = ord.max less_eq (linorder.Max less_eq A) (linorder.Max less_eq B)"
distinct_1: fixes x2 :: "nat" shows "0 \<noteq> Suc x2"
PL_Indis_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set" shows "PL_Indis cval indis \<equiv> PL cval \<and> PL_Indis_axioms indis"
power_one_right: fixes a :: "'a" shows "a\<^bsup>1\<^esup> = a"
satisfies_properties_sys2: shows "satisfies_properties_R p qs2 subsets2 signs2 matrix2"
Inf_from_mono: fixes N :: "('f \<times> 'l) set"   and N' :: "('f \<times> 'l) set" assumes "N \<subseteq> N'" shows "Inf_from N \<subseteq> Inf_from N'"
B_inst: shows "has_all_wt_instances_of \<Gamma> (set B') (set (B \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>))"
cMatch: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and C :: "'a"   and b :: "name" assumes "P \<longmapsto> a<x> \<prec> P'"   and "\<And>C. F C P a x P'" shows "F C ([b\<frown>b]P) a x P'"
case_split_right: fixes w :: "'a"   and z :: "'a"   and x :: "'a"   and y :: "'a" assumes "reflexive (w \<squnion> z)"   and "x * w \<le> y"   and "x * z \<le> y" shows "x \<le> y"
b: shows "(k dvd l) = (Idl\<^bsub>\<Z>\<^esub> {l} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {k})"
redT_thread_not_disappear: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and convert_RA :: "'d \<Rightarrow>f nat \<Rightarrow> 'f list"   and s :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and t :: "'b"   and ta :: "('d, 'b, 'a, 'c, 'e, 'f) thread_action"   and s' :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and t' :: "'b" assumes "multithreaded_base.redT final r convert_RA s (t, ta) s'"   and "thr s' t' = None" shows "thr s t' = None"
pred_cong: fixes x :: "'c option"   and ya :: "'c option"   and P :: "'c \<Rightarrow> bool"   and Pa :: "'c \<Rightarrow> bool" assumes "x = ya"   and "\<And>z. z \<in> set_option ya \<Longrightarrow> P z = Pa z" shows "pred_option P x = pred_option Pa ya"
list_set_rel_def: fixes R :: "('d \<times> 'e) set" shows "\<langle>R\<rangle>list_set_rel \<equiv> \<langle>R\<rangle>list_rel O br set distinct"
that: fixes g :: "'a \<Rightarrow> 'b" assumes "continuous_on (\<Union> \<F>) g"   and "g ` \<Union> \<F> \<subseteq> rel_frontier T"   and "\<And>x. x \<in> \<Union> \<G> \<Longrightarrow> g x = f x" shows "thesis"
fps_const_1_eq_1: shows "fps_const (1::'a) = 1"
reds: shows "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e'',s''\<rangle>"
C1_differentiable_on_def: fixes f :: "real \<Rightarrow> 'a"   and S :: "real set" shows "(f C1_differentiable_on S) = (\<exists>D. (\<forall>x\<in>S. (f has_vector_derivative D x) (at x)) \<and> continuous_on S D)"
rep_eq: shows "pdevs_apply One_pdevs = One_pdevs_raw"
prob_indep_random_variable: fixes N :: "'b measure"   and X :: "'a \<Rightarrow> 'b"   and Y :: "'a \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" assumes "indep_var N X N Y"   and "A \<in> sets N"   and "B \<in> sets N" shows "prob {x \<in> space M. X x \<in> A \<and> Y x \<in> B} = prob {x \<in> space M. X x \<in> A} * prob {x \<in> space M. Y x \<in> B}"
all_comm: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "(\<forall>x y. P x y) = (\<forall>y x. P x y)"
axioms_3: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "evaluation_functor A B" shows "functor_category A B"
nat_power_eq_Suc_0_iff: fixes x :: "nat"   and m :: "nat" shows "(x ^ m = Suc 0) = (m = 0 \<or> x = Suc 0)"
hcmod_mult_hcnj: fixes z :: "complex star" shows "hcmod (z * hcnj z) = (hcmod z)\<^sup>2"
list_emb_appendD: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "list_emb P (xs @ ys) zs" shows "\<exists>us vs. zs = us @ vs \<and> list_emb P xs us \<and> list_emb P ys vs"
inpt_str_comp: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and u :: "nat \<Rightarrow> 'a \<times> 'd"   and x :: "nat \<Rightarrow> 'b"   and r' :: "'d \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> 'e \<Rightarrow> bool" shows "(prec_st (inpt_st r) r (fst \<circ> u) x \<and> (\<forall>y. (\<box> lft_rel_st r) (fst \<circ> u) x y \<longrightarrow> prec_st (inpt_st r') r' (snd \<circ> u) y)) = (\<box> lft_pred_st (inpt_st (rel_st r r'))) u x"
cones_map_induced_arrow_1: fixes a' :: "'c" shows "induced_arrow a' \<in> D.cones a' \<rightarrow> C.hom a' a"
dual_order_iff: fixes A :: "('d, 'e) gorder_scheme" shows "partial_order (inv_gorder A) = partial_order A"
analz_insert_freshK2: fixes evs :: "event list"   and servK :: "nat"   and K :: "nat" assumes "evs \<in> kerbV"   and "servK \<notin> authKeys evs"   and "servK \<notin> range shrK"   and "K \<in> symKeys" shows "(Key K \<in> analz (insert (Key servK) (knows Spy evs))) = (K = servK \<or> Key K \<in> analz (knows Spy evs))"
alphaAbs_ex_distinct_qAFresh_def: fixes xs :: "'c"   and x :: "'d"   and X :: "('e, 'f, 'c, 'd, 'g) qTerm"   and xs' :: "'c"   and x' :: "'d"   and X' :: "('e, 'f, 'c, 'd, 'g) qTerm" shows "alphaAbs_ex_distinct_qAFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<exists>y. y \<notin> {x, x'} \<and> qAFresh xs y X \<and> qAFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
pm_of_poly_of_pm: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and x :: "'a" assumes "p \<in> P[{x}]" shows "pm_of_poly x (poly_of_pm x p) = p"
freeword_funlift_Abs_freeletter: fixes f :: "'b \<Rightarrow> 'a"   and s :: "'b" shows "freeword_funlift f (Abs_freeletter s) = f s"
add_increasing2: fixes c :: "'c multiset"   and b :: "'c multiset"   and a :: "'c multiset" assumes "{#} \<subseteq># c"   and "b \<subseteq># a" shows "b \<subseteq># a + c"
not_less: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.linorder less_eq less" shows "(\<not> less x y) = less_eq y x"
conjugation_is_size_invariant: fixes P :: "'a set"   and p :: "nat"   and g :: "'a" assumes "finite (carrier G)"   and "P \<in> subgroups_of_size p"   and "g \<in> carrier G" shows "conjugation_action p g P \<in> subgroups_of_size p"
coset_join3: fixes x :: "'a"   and H :: "'a set" assumes "x \<in> carrier G"   and "subgroup H G"   and "x \<in> H" shows "x <# H = H"
state_successors_def: fixes PROB :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) set"   and s :: "('d, 'e) fmap" shows "state_successors PROB s \<equiv> state_succ s ` PROB - {s}"
14: shows "\<exists>i. g (Suc (nat (padic_val p f + padic_val p g))) = i * p ^ nat (padic_val p g) \<and> \<not> p dvd int (nat i)"
ldropn_eq_LConsD: fixes n :: "nat"   and xs :: "'c llist"   and y :: "'c"   and ys :: "'c llist" assumes "ldropn n xs = LCons y ys" shows "enat n < llength xs"
add_poly_l_add_poly: fixes x :: "(char list list \<times> int) list \<times> (char list list \<times> int) list" shows "add_poly_l x = RETURN (add_poly_l' x)"
gallery_map: fixes Cs :: "'a set list" assumes "domain.gallery Cs" shows "codomain.gallery (f \<Turnstile> Cs)"
Int_space_eq2: fixes x :: "'a set" assumes "x \<in> sigma_sets \<Omega> E" shows "x \<inter> \<Omega> = x"
lookup_single_eq: fixes k :: "'b"   and v :: "'a" shows "lookup (monomial v k) k = v"
leD: fixes y :: "'c Word.word"   and x :: "'c Word.word" assumes "y \<le>s x" shows "\<not> x <s y"
view_leq_len_leq: fixes v :: "view"   and v' :: "view"   and ts :: "traffic"   and c :: "cars" shows "ext v \<le> ext v' \<and> own v = own v' \<and> 0 < \<parallel>len v ts c\<parallel> \<longrightarrow> len v ts c \<le> len v' ts c"
has_bochner_integral_iff: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and x :: "'b" shows "has_bochner_integral M f x = (integrable M f \<and> integral\<^sup>L M f = x)"
two_comparisons_into_case_order_12: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a"   and P :: "'b"   and Q :: "'b"   and R :: "'b" assumes "comparator comp" shows "(if x = y then P else if le_of_comp comp x y then Q else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
g_1: fixes x :: "'a" assumes "x \<in> X" shows "g x = f x"
entails_trans_strong: fixes N1 :: "'a set"   and N2 :: "'a set"   and N3 :: "'a set" assumes "N1 \<Turnstile> N2"   and "N1 \<union> N2 \<Turnstile> N3" shows "N1 \<Turnstile> N3"
Mp_rel_i_Mp_to_int_poly_i: fixes ops :: "'a arith_ops_record"   and m :: "int"   and f :: "'a list"   and F :: "int poly" assumes "arith_ops_record.Mp_rel_i ops m f F" shows "poly_mod.Mp m (to_int_poly_i ops f) = to_int_poly_i ops f"
restriction_consec_res: fixes v :: "view"   and u :: "view"   and w :: "view"   and ts :: "traffic"   and c :: "cars" shows "v=u--w \<longrightarrow> restrict u (res ts) c = bot \<or> restrict w (res ts) c = bot \<or> consec (restrict u (res ts) c) (restrict w (res ts) c)"
pair_spmf_def: fixes p :: "'c spmf"   and q :: "'d spmf" shows "pair_spmf p q = pair_pmf p q \<bind> (\<lambda>xy. case xy of (None, b) \<Rightarrow> return_pmf None | (Some x, None) \<Rightarrow> return_pmf None | (Some x, Some y) \<Rightarrow> return_spmf (x, y))"
assms: shows "v = |\<beta>\<^sub>0\<^sub>0\<rangle> \<or> v = |\<beta>\<^sub>0\<^sub>1\<rangle> \<or> v = |\<beta>\<^sub>1\<^sub>0\<rangle> \<or> v = |\<beta>\<^sub>1\<^sub>1\<rangle>"
distinct_disc_2: fixes status :: "status" assumes "is_success status" shows "\<not> is_failed status"
assms: shows "x \<in> zero_space\<^sub>N N"
intros_4: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and t :: "pdf_type"   and oper :: "pdf_operator"   and t' :: "pdf_type" assumes "\<Gamma> \<turnstile>\<^sub>c e : t"   and "op_type oper t = Some t'" shows "\<Gamma> \<turnstile>\<^sub>c oper $$\<^sub>c e : t'"
quot_subst_eq: fixes A :: "fm"   and i :: "name"   and t :: "tm" shows "\<guillemotleft>A(i::=t)\<guillemotright> = quot_dbfm (subst_dbfm (trans_tm [] t) i (trans_fm [] A))"
usubst_upd_idem: fixes x :: "'e \<Longrightarrow> 'f"   and \<sigma> :: "'g \<Rightarrow> 'f"   and u :: "('e, 'g) uexpr"   and v :: "('e, 'g) uexpr" assumes "mwb_lens x" shows "\<sigma>(x \<mapsto>\<^sub>s u, x \<mapsto>\<^sub>s v) = \<sigma>(x \<mapsto>\<^sub>s v)"
lower_comp: fixes x :: "'c" assumes "x \<in> carrier \<X>\<^bsub>F\<^esub>" shows "\<pi>\<^sup>*\<^bsub>F\<^esub> (\<pi>\<^sub>*\<^bsub>F\<^esub> (\<pi>\<^sup>*\<^bsub>F\<^esub> x)) = \<pi>\<^sup>*\<^bsub>F\<^esub> x"
M0_mult_ket_k_R: shows "M0 *\<^sub>v ket_k R = ket_k R"
sup_union_distrib_left: fixes A :: "'k multiset"   and B :: "'k multiset"   and C :: "'k multiset" shows "A \<union># B + C = (A + C) \<union># (B + C)"
E_of_succ_def: fixes succ :: "'b \<Rightarrow> 'c list" shows "E_of_succ succ \<equiv> {(u, v). v \<in> set (succ u)}"
KerT': shows "RModule FG (\<cdot>) KerT \<and> KerT \<subseteq> V"
plus_list_lub: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list"   and n :: "nat" assumes "set xs \<subseteq> A"   and "set ys \<subseteq> A"   and "set zs \<subseteq> A"   and "length xs = n \<and> length ys = n"   and "xs [\<sqsubseteq>\<^bsub>r\<^esub>] zs \<and> ys [\<sqsubseteq>\<^bsub>r\<^esub>] zs" shows "xs [\<squnion>\<^bsub>f\<^esub>] ys [\<sqsubseteq>\<^bsub>r\<^esub>] zs"
prefix_def: fixes xs :: "'e list"   and ys :: "'e list" shows "prefix xs ys = (\<exists>zs. ys = xs @ zs)"
\<psi>_\<phi>: fixes f :: "'d"   and b :: "'d"   and a :: "'d" assumes "\<guillemotleft>f : b \<rightarrow>\<^sub>D a\<guillemotright>" shows "\<psi>D (b, a) (\<phi>D (b, a) f) = f"
IH: assumes "na__ \<le> deg R f" shows "deg R (shift na__ f) + na__ = deg R f"
netlistalternativeSeparateaux: fixes y :: "('b, 'c) Combinators"   and z :: "('b, 'c) Combinators list" shows "net_list_aux [y] @ net_list_aux z = net_list_aux (y # z)"
tendsto_norm: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and F :: "'a filter" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. norm (f x)) \<longlongrightarrow> norm a) F"
mult_isor: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.near_dioid plus times less_eq less"   and "less_eq x y" shows "less_eq (times x z) (times y z)"
csol_3: fixes t :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl t0 x0" shows "csol t0 x0 t t0 = x0"
list_emb_set: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list"   and x :: "'b" assumes "list_emb P xs ys"   and "x \<in> set xs" obtains y :: "'b" where "y \<in> set ys" and "P x y"
S_disj: shows "S \<inter> path_image \<gamma> = {}"
App_2: fixes b :: "name"   and ba :: "Lambda_Z.term" shows "subst b ba\<^sup>\<bullet> t\<^sub>2\<^sup>\<bullet> \<rightarrow>\<^sub>\<beta>\<^sup>* (subst b ba t\<^sub>2)\<^sup>\<bullet>"
subcomplex_chamber: fixes Y :: "'a set set"   and C :: "'a set" assumes "domain.ChamberSubcomplex Y"   and "SimplicialComplex.maxsimp Y C" shows "domain.chamber C"
Q: shows "((X\<^sub>r \<mu> \<in> old (fst (n, ns)) \<longrightarrow> \<mu> \<in> next (fst (n, ns))) \<and> (\<forall>nd\<in>snd (n, ns). X\<^sub>r \<mu> \<in> old nd \<and> name nd \<in> incoming (fst (n, ns)) \<longrightarrow> \<mu> \<in> old (fst (n, ns)) \<union> new (fst (n, ns)))) \<and> (\<forall>nd\<in>snd (n, ns). \<forall>nd'\<in>snd (n, ns). X\<^sub>r \<mu> \<in> old nd \<and> name nd \<in> incoming nd' \<longrightarrow> \<mu> \<in> old nd') \<and> ((\<forall>nm\<in>incoming (fst (n, ns)). nm < name (fst (n, ns))) \<and> 0 < name (fst (n, ns)) \<and> (\<forall>q\<in>snd (n, ns). name q < name (fst (n, ns)) \<and> (\<forall>nm\<in>incoming q. nm < name (fst (n, ns))))) \<and> expand_name_ident (snd (n, ns))"
ennreal_suminf_multc: fixes f :: "nat \<Rightarrow> ennreal"   and r :: "ennreal" shows "(\<Sum>i. f i * r) = (\<Sum>i. f i) * r"
hd_dropWhile: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" assumes "dropWhile P xs \<noteq> []" shows "\<not> P (hd (dropWhile P xs))"
goodAbs_imp_qGoodAbs_pick: fixes A :: "('e, 'f, 'varSort, 'var, 'g) qAbs set" assumes "goodAbs A" shows "qGoodAbs (pick A)"
le_DT_adm: fixes u :: "'a \<Rightarrow> 'b process"   and v :: "'a \<Rightarrow> 'b process" assumes "cont u"   and "monofun v" shows "adm (\<lambda>x. u x \<sqsubseteq>\<^sub>D\<^sub>T v x)"
p_fun_inj: fixes x :: "'a" shows "is_p_fun x = is_inj (x\<^sup>T)"
vertex_retraction: fixes v :: "'a" assumes "v \<in> g ` \<Union> X" shows "g v = v"
balance_inv1: fixes l :: "('a, 'b) rbt"   and r :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b" assumes "inv1l l"   and "inv1l r" shows "inv1 (balance l k v r)"
Cd_singleton: fixes x :: "'a"   and d :: "'b"   and X :: "'a set" shows "(x \<in> Bossy.Cd d X) = (Bossy.Cd d X = {x})"
subint1_def2: fixes G :: "prog" shows "subint1 G = (SIGMA I:{I. is_iface G I}. set (isuperIfs (the (iface G I))))"
callee_invariant: fixes y :: "bool list"   and s' :: "'b \<Rightarrow> bool list option"   and s :: "'b \<Rightarrow> bool list option"   and x :: "'b" assumes "(y, s') \<in> set_spmf (prf.random_oracle s x)"   and "finite (dom s)"   and "x \<in> outs_\<I> \<I>_full" shows "finite (dom s')"
Join_welldef_D2: fixes F :: "'b program"   and G :: "'b program" assumes "F \<squnion> G \<in> welldef" shows "G \<in> welldef"
comp_eq_dest_lhs: fixes a :: "'s \<Rightarrow> 'r"   and b :: "'q \<Rightarrow> 's"   and c :: "'q \<Rightarrow> 'r"   and v :: "'q" assumes "a \<circ> b = c" shows "a (b v) = c v"
abs_eq: fixes x :: "moebius_mat" shows "unit_disc_fix (abs_moebius x) = unit_disc_fix_mmat x"
continuous_on_at_left: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on {a..b} f"   and "a < b" shows "continuous (at_left b) f"
NSDERIVD2: fixes g :: "'a \<Rightarrow> 'a"   and x :: "'a"   and Db :: "'a"   and y :: "'a star" assumes "NSDERIV g x :> Db"   and "y \<in> Infinitesimal"   and "y \<noteq> 0" shows "((*f* g) (star_of x + y) - star_of (g x)) / y \<approx> star_of Db"
assms_1: shows "mat_det A1 \<noteq> 0"
Stuck: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com" shows "\<Gamma>\<turnstile>c \<down> Stuck"
inverse_positive_iff_positive: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less zero (inverse a) = less zero a"
just: fixes v :: "'a" assumes "v \<in> verts G"   and "v \<noteq> s"   and "num v \<noteq> \<infinity>" shows "\<exists>e\<in>arcs G. v = head G e \<and> dist v = dist (tail G e) + ereal (c e) \<and> num v = num (tail G e) + enat 1"
mk_ps_chain_cons: fixes \<Pi> :: "(node \<rightharpoonup>\<^sub>f topbot, node + node fset \<times> command_gadget \<times> node fset) chain"   and x :: "node + node fset \<times> command_gadget \<times> node fset"   and \<pi> :: "(node + node fset \<times> command_gadget \<times> node fset) list" shows "mk_ps_chain \<Pi> (x # \<pi>) = mk_ps_chain (cSnoc \<Pi> x (next_ps (post \<Pi>) x)) \<pi>"
AA16: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<turnstile> \<diamond>\<langle>P \<and> Q\<rangle>_v \<longrightarrow> \<diamond>\<langle>Q\<rangle>_v"
mono_ndet_FD: fixes P :: "'a process"   and P' :: "'a process"   and S :: "'a process"   and S' :: "'a process" assumes "P \<le> P'"   and "S \<le> S'" shows "(P |-| S) \<le> (P' |-| S')"
OF_match_fields_alt: fixes m :: "of_match_field set"   and p :: "(32, 'a) simple_packet_ext_scheme" shows "OF_match_fields m p = (if \<exists>f\<in>m. \<not> prerequisites f m then None else if \<forall>f\<in>m. match_no_prereq f p then Some True else Some False)"
process_call_split: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs1 :: "'c rule list"   and rs2 :: "'c rule list" shows "process_call \<Gamma> (rs1 @ rs2) = process_call \<Gamma> rs1 @ process_call \<Gamma> rs2"
intros_2: fixes v :: "'a" assumes "deadend v" shows "maximal_path (LCons v LNil)"
measure_eqI_generator_eq: fixes E :: "'b set set"   and \<Omega> :: "'b set"   and M :: "'b measure"   and N :: "'b measure"   and A :: "nat \<Rightarrow> 'b set" assumes "Int_stable E"   and "E \<subseteq> Pow \<Omega>"   and "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"   and "sets M = sigma_sets \<Omega> E"   and "sets N = sigma_sets \<Omega> E"   and "range A \<subseteq> E"   and "\<Union> (range A) = \<Omega>"   and "\<And>i. emeasure M (A i) \<noteq> \<infinity>" shows "M = N"
non_empty_projection_on_subset: fixes X :: "'b set"   and Y :: "'b set"   and l\<^sub>1 :: "'b list"   and l\<^sub>2 :: "'b list" assumes "X \<subseteq> Y \<and> l\<^sub>1 \<upharpoonleft> Y = l\<^sub>2 \<upharpoonleft> Y" shows "l\<^sub>1 \<upharpoonleft> X = l\<^sub>2 \<upharpoonleft> X"
a_consistent_alts_on_stack: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and b :: "bool"   and e1 :: "exp"   and e2 :: "exp"   and S :: "stack_elem list" assumes "a_consistent (ae, a, as) (\<Gamma>, Bool b, Alts e1 e2 # S)" obtains a' :: "Arity" and as' :: "Arity list" where "as = a' # as'" and "a = 0"
map_nth_Joints': fixes xs :: "'a list"   and XS :: "('a \<times> 'a pdevs) list"   and "is" :: "nat list" assumes "xs \<in> Joints XS"   and "\<And>i. i \<in> set is \<Longrightarrow> i < length XS" shows "map ((!) xs) is \<in> Joints (map ((!) XS) is)"
act_inv_iff: fixes g :: "cltn2"   and b :: "hyp2"   and c :: "hyp2" assumes "g \<in> carrier \<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>" shows "(hyp2_cltn2 b (inv\<^bsub>\<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> g) = c) = (b = hyp2_cltn2 c g)"
cblinfun_of_mat_adjoint: fixes M :: "complex mat" assumes "M \<in> carrier_mat (length canonical_basis) (length canonical_basis)" shows "cblinfun_of_mat (mat_adjoint M) = cblinfun_of_mat M*"
r_c_qt_imp_cf: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" assumes "finite A"   and "complete A r"   and "quasi_trans r"   and "refl_on A r" shows "choiceFn A r"
reds_preserves_iconf: fixes P :: "(char list list \<times> char list exp) prog"   and es :: "char list exp list"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and es' :: "char list exp list"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>es,(h, l, sh),b\<rangle> [\<rightarrow>] \<langle>es',(h', l', sh'),b'\<rangle>"   and "iconfs sh es" shows "iconfs sh' es'"
ide_in_hom: fixes a :: "'a \<times> 'a" shows "CC.ide a = CC.in_hom a a a"
OF_match_fields_safe_def: fixes m :: "of_match_field set" shows "OF_match_fields_safe m \<equiv> the \<circ> OF_match_fields m"
graph_homomorphism_Id: fixes a :: "('b, 'c) labeled_graph" shows "graph_homomorphism (restrict a) (restrict a) (Id_on (vertices a))"
of_nat_less_numeral_power_cancel_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "nat"   and i :: "num"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less (semiring_1.of_nat one plus zero x) (power.power one times (numeral.numeral one plus i) n) = (x < numeral i ^ n)"