hcju/isabelleps huggingface dataset:
parity_cases: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_parity plus minus zero times one divide modulo" obtains "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" and "modulo a (numeral.numeral one plus (num.Bit0 num.One)) = zero"   | "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" and "modulo a (numeral.numeral one plus (num.Bit0 num.One)) = one"
subspace_with: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd" assumes "module scale" shows "module.subspace scale = subspace_with (+) (0::'d) scale"
count_mset_0_iff: fixes xs :: "'k list"   and x :: "'k" shows "(count (mset xs) x = 0) = (x \<notin> set xs)"
sublist_EQNS_1: fixes l :: "'a list" shows "subseq [] l = True"
compat_add_triple: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and p :: "'c point" assumes "ell_field R"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "ell_field.nonsingular R a b"   and "ell_field.on_curve R a b p"   and "p \<noteq> ell_field.opp R p"   and "ell_field.add R a p p \<noteq> ell_field.opp R p" shows "ell_field.add R a (ell_field.add R a p p) (ell_field.opp R p) = p"
run_get_state: fixes f :: "'b \<Rightarrow> ('b, 'a) stateT"   and s :: "'b" shows "run_state (get f) s = run_state (f s) s"
filterlim_inverse_at_bot: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter" assumes "(f \<longlongrightarrow> 0) F"   and "\<forall>\<^sub>F x in F. f x < 0" shows "LIM x F. inverse (f x) :> at_bot"
IH: fixes Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "(X, Y) \<in> swapped" shows "\<forall>p. phi Y p"
sc_completion_def: fixes s :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and vs :: "'addr \<times> addr_loc \<Rightarrow> ('addr val \<times> bool) option" shows "sc_completion s vs = (\<forall>ttas s' t x ta x' m'. s -\<triangleright>ttas\<rightarrow>* s' \<longrightarrow> ta_seq_consist P vs (llist_of (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas))) \<longrightarrow> thr s' t = \<lfloor>(x, no_wait_locks)\<rfloor> \<longrightarrow> t \<turnstile> \<langle>x, shr s'\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle> \<longrightarrow> actions_ok s' t ta \<longrightarrow> (\<exists>ta' x'' m''. t \<turnstile> \<langle>x, shr s'\<rangle> -ta'\<rightarrow> \<langle>x'', m''\<rangle> \<and> actions_ok s' t ta' \<and> ta_seq_consist P (mrw_values P vs (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas))) (llist_of \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub>)))"
iFROM_iMOD_neq: fixes m :: "nat"   and n :: "nat"   and r :: "nat" assumes "m \<noteq> Suc 0" shows "[n\<dots>] \<noteq> [ r, mod m ]"
wf: shows "length Ts = length pns \<and> distinct pns \<and> this \<notin> set pns \<and> (\<forall>T\<in>set Ts. is_type P T)"
zip_with_map: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'c"   and xs :: "'e list"   and h :: "'f \<Rightarrow> 'd"   and ys :: "'f list" shows "zip_with f (map g xs) (map h ys) = zip_with (\<lambda>x y. f (g x) (h y)) xs ys"
minus_eq: fixes x :: "int"   and y :: "int" shows "x \<ominus> y = x \<oplus> \<ominus> y"
simps_3: fixes ev_counter' :: "nat \<Rightarrow> nat"   and ev_counter :: "nat"   and more :: "'a" shows "ev_counter_update ev_counter' \<lparr>ev_counter = ev_counter, \<dots> = more\<rparr> = \<lparr>ev_counter = ev_counter' ev_counter, \<dots> = more\<rparr>"
dord_fun_refl: fixes ord :: "('a \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'c) \<Rightarrow> bool"   and s :: "'a \<Rightarrow> 'c" assumes "ord s s" shows "dord_fun ord s s"
isnormNum_int_1: shows "isnormNum 0\<^sub>N"
Dg: shows "(g has_derivative g') (at (f x) within f ` S)"
mat_mult_assoc: fixes f :: "'b \<Rightarrow> 'e \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and h :: "'f \<Rightarrow> 'c \<Rightarrow> 'd" shows "f \<otimes> (g \<otimes> h) = f \<otimes> g \<otimes> h"
measurable_restrict_space2: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and \<Omega> :: "'e set"   and N :: "'e measure" assumes "f \<in> space M \<rightarrow> \<Omega>"   and "f \<in> M \<rightarrow>\<^sub>M N" shows "f \<in> M \<rightarrow>\<^sub>M restrict_space N \<Omega>"
PA2: fixes i :: "'a"   and p :: "'a pfm"   and q :: "'a pfm" shows "\<turnstile>\<^sub>! (K\<^sub>! i p \<^bold>\<and>\<^sub>! K\<^sub>! i (p \<^bold>\<longrightarrow>\<^sub>! q) \<^bold>\<longrightarrow>\<^sub>! K\<^sub>! i q)"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat" assumes "\<And>l i. (\<And>x. \<lbrakk>x = i - 1; l < x\<rbrakk> \<Longrightarrow> P l x) \<Longrightarrow> P l i" shows "P a0 a1"
strict_coboundedI1: fixes c :: "'a multiset"   and a :: "'a multiset"   and b :: "'a multiset" assumes "c \<subset># a" shows "c \<subset># a \<union># b"
simps_1: fixes c :: "('d, 'e, 'f) com" shows "guards [] c = c"
eucl_down_real_def: fixes prec :: "int"   and b :: "real" shows "eucl_down prec b = round_down prec b"
open_delete: fixes s :: "'a set"   and x :: "'a" assumes "open s" shows "open (s - {x})"
igSubstBinp_def: fixes MOD :: "('d, 'e, 'f, 'g, 'h, 'i, 'j, 'c, 'k) model_scheme"   and ys :: "'f"   and Y :: "'j"   and y :: "'i"   and binp :: "'b \<Rightarrow> 'c option" shows "igSubstBinp MOD ys Y y binp \<equiv> lift (igSubstAbs MOD ys Y y) binp"
int_eq_iff: fixes m :: "nat"   and z :: "int" shows "(int m = z) = (m = nat z \<and> 0 \<le> z)"
res_def: shows "\<forall>c. re c = Abs_nat_int {0}"
gtrp2gtr_s: fixes s :: "'a"   and c :: "'a multiset"   and e :: "'b el_step"   and s' :: "'a"   and c' :: "'a multiset"   and T :: "(('a \<times> 'a multiset) \<times> 'b \<times> 'a \<times> 'a multiset) set" assumes "((s, c), e, s', c') \<in> gtrp T" shows "(add_mset s c, le_rem_s e, add_mset s' c') \<in> gtr T"
assigns_r_uinj: fixes f :: "'b \<Rightarrow> 'b" assumes "inj f" shows "uinj \<langle>f\<rangle>\<^sub>a"
nat_pow_closed: fixes x :: "'a list"   and n :: "nat" assumes "x \<in> carrier (poly_ring R)" shows "x [^]\<^bsub>poly_ring R\<^esub> n \<in> carrier (poly_ring R)"
det_selection_fun_def: fixes f :: "(nat \<times> 'b) list \<Rightarrow> nat" shows "det_selection_fun f = (\<forall>xs. xs \<noteq> [] \<longrightarrow> f xs \<in> fst ` set xs)"
symmetric_mpoly_symmetric_sum: fixes g :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and A :: "nat set"   and f :: "'b \<Rightarrow> 'c mpoly" assumes "g permutes X"   and "\<And>x \<pi>. \<lbrakk>x \<in> X; \<pi> permutes A\<rbrakk> \<Longrightarrow> mpoly_map_vars \<pi> (f x) = f (g x)" shows "symmetric_mpoly A (sum f X)"
append_prefixD: fixes xs :: "[:'r:]"   and ys :: "[:'r:]"   and zs :: "[:'r:]" assumes "prefix\<cdot>(xs :@ ys)\<cdot>zs = TT" shows "prefix\<cdot>xs\<cdot>zs = TT"
SA_refl_TVar: fixes \<Gamma> :: "binding list"   and i :: "nat" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f"   and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f TVar i" shows "\<Gamma> \<turnstile> TVar i <: TVar i"
iter1: fixes c :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<omega> \<sqsubseteq> c"
image_Fpow_surjective: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and B :: "'a set" assumes "f ` A = B" shows "(`) f ` Fpow A = Fpow B"
chain2: shows "Complete_Partial_Order.chain (ord_spmf (=)) (snd ` Y)"
Le_Lt_dbm_lt_D: fixes a :: "'c"   and b :: "'c" assumes "Le a \<prec> Le b" shows "a < b"
only_one_L_per_bin: fixes P :: "'a set set"   and B :: "'a set" assumes "bp P"   and "B \<in> P" shows "\<forall>x\<in>B. \<forall>y\<in>B. x \<noteq> y \<longrightarrow> x \<notin> L \<or> y \<notin> L"
filter_M_empty_I: fixes P :: "'a \<Rightarrow> ('b, 'c, bool) prog"   and h :: "'b"   and xs :: "'a list" assumes "\<And>x. pure (P x) h"   and "\<forall>x\<in>set xs. h \<turnstile> P x \<rightarrow>\<^sub>r False" shows "h \<turnstile> filter_M P xs \<rightarrow>\<^sub>r []"
dom_cod: fixes f :: "'a \<times> 'a" assumes "VV.arr f" shows "VV.dom (VV.cod f) = VV.cod f"
errMOD_igOpIPresIGWlsSTR: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) model" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igOpIPresIGWls Delta MOD" shows "FixSyn.igOpIPresIGWlsSTR Delta (FixSyn.errMOD Delta MOD)"
gcd_exists: fixes a :: "'a"   and b :: "'a" shows "\<exists>x. is_gcd x a b"
assms_8: shows "r \<cdot> (0::'a) = (0::'a)"
correctCompositionDiffLevelsA42: shows "correctCompositionDiffLevels sA42"
times_fds_def: fixes f :: "'b fds"   and g :: "'b fds" shows "f * g = fds (($) f \<star> ($) g)"
tuple_simps_2: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "C.commutative_square f g h k" shows "C.dom \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = C.dom h"
psubset_2: fixes g :: "'a ugraph" assumes "edges g \<subset> edges ga__"   and "Undirected_Graph.connected g" shows "Ex (is_spanning_tree g)"
ide_in_hom: fixes a :: "'b" shows "Adj.D.ide a = \<guillemotleft>a : a \<rightarrow>\<^sub>D a\<guillemotright>"
g: shows "arr (mkArr (Cod f) (Dom f) (\<lambda>y. if y \<in> Img f then SOME x. x \<in> Dom f \<and> Fun f x = y else SOME x. x \<in> Dom f))"
wf_f: shows "\<not> {- \<infinity>, \<infinity>} \<subseteq> range f"
church_rosser_to_confluence: fixes x :: "'a"   and y :: "'a" assumes "(x + y)\<^sup>\<star> \<le> x\<^sup>\<star> \<cdot> y\<^sup>\<star>" shows "y\<^sup>\<star> \<cdot> x\<^sup>\<star> \<le> x\<^sup>\<star> \<cdot> y\<^sup>\<star>"
setnMT: fixes a :: "'b list"   and b :: "'b list" assumes "set a = set b"   and "a \<noteq> []" shows "b \<noteq> []"
discounted_cong: fixes X :: "nat \<Rightarrow> 'c \<Rightarrow> real"   and Y :: "nat \<Rightarrow> 'c \<Rightarrow> real"   and r :: "real" assumes "\<forall>n w. X n w = Y n w" shows "\<forall>n w. discounted_value r X n w = discounted_value r Y n w"
b_dvd_a2: shows "gcd a1 a2 dvd a2"
put_is_create: fixes x :: "'c \<Longrightarrow> 'd"   and \<sigma> :: "'d"   and v :: "'c" assumes "pbij_lens x" shows "put\<^bsub>x\<^esub> \<sigma> v = create\<^bsub>x\<^esub> v"
Abs_eqD: fixes x :: "'a"   and y :: "'a" assumes "Abs x = Abs y"   and "x \<in> A"   and "y \<in> A" shows "x = y"
constructive_security_obsf_lifting_usr: fixes real_resource :: "('m + 'n, 'o + 'p) resource"   and ideal_resource :: "('q + 'n, 'r + 'p) resource"   and sim :: "('m, 'o, 'q, 'r) converter"   and \<I>_real :: "('m, 'o) \<I>"   and \<I>_ideal :: "('q, 'r) \<I>"   and \<I>_common :: "('n, 'p) \<I>"   and adv :: "real"   and \<I>_common' :: "('s, 't) \<I>"   and \<A> :: "('m + 's, 'o + 't) distinguisher_obsf"   and conv :: "('s, 't, 'n, 'p) converter" assumes "constructive_security_aux_obsf real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common adv"   and "exception_\<I> (\<I>_real \<oplus>\<^sub>\<I> \<I>_common') \<turnstile>g \<A> \<surd> \<Longrightarrow> constructive_security_sim_obsf real_resource ideal_resource sim \<I>_real \<I>_common (absorb \<A> (obsf_converter (1\<^sub>C |\<^sub>= conv))) adv"   and "\<I>_common', \<I>_common \<turnstile>\<^sub>C conv \<surd>"   and "pfinite_converter \<I>_common' \<I>_common conv" shows "constructive_security_obsf (1\<^sub>C |\<^sub>= conv \<rhd> real_resource) (1\<^sub>C |\<^sub>= conv \<rhd> ideal_resource) sim \<I>_real \<I>_ideal \<I>_common' \<A> adv"
reflect_poly_const: fixes a :: "'b" shows "reflect_poly [:a:] = [:a:]"
monom_mult_assoc: fixes c :: "'c"   and s :: "'b"   and d :: "'c"   and t :: "'b"   and p :: "'b \<Rightarrow>\<^sub>0 'c" shows "MPoly_Type_Class.punit.monom_mult c s (MPoly_Type_Class.punit.monom_mult d t p) = MPoly_Type_Class.punit.monom_mult (c * d) (s + t) p"
filter_single_zmset: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b" shows "filter_zmset P {#x#}\<^sub>z = (if P x then {#x#}\<^sub>z else {#}\<^sub>z)"
transaction_updates_def: fixes prot_transaction :: "('f, 'g, 'h, 'e) prot_transaction" shows "transaction_updates prot_transaction = (case prot_transaction of Transaction x1 x2 x3 x4 x5 x6 \<Rightarrow> x5)"
solve_ent_preprocess_simps_1: fixes P :: "assn"   and Q :: "assn"   and b :: "bool" shows "(P \<Longrightarrow>\<^sub>A Q * \<up> b) = ((\<forall>h. h \<Turnstile> P \<longrightarrow> b) \<and> (P \<Longrightarrow>\<^sub>A Q))"
closure_shalfspace_1: fixes sctn :: "'c sctn" assumes "normal sctn \<noteq> (0::'c)" shows "closure (sbelow_halfspace sctn) = below_halfspace sctn"
asymp_equiv_ln_fact: shows "(\<lambda>n. ln (fact n)) \<sim>[sequentially] (\<lambda>n. real n * ln (real n))"
free_Abelian_group_universal: fixes f :: "'d \<Rightarrow> 'a"   and S :: "'d set" assumes "f ` S \<subseteq> carrier G" obtains h :: "('d \<Rightarrow>\<^sub>0 int) \<Rightarrow> 'a" where "h \<in> hom (free_Abelian_group S) G" and "\<And>x. x \<in> S \<Longrightarrow> h (frag_of x) = f x"
prj_in_hom_5: shows "\<guillemotleft>\<nu>\<pi>\<rho>.Prj\<^sub>1\<^sub>0 : H\<nu>H\<pi>\<rho>.chine \<rightarrow>\<^sub>C \<pi>.apex\<guillemotright>"
iso: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah" assumes "isomorphic_categories A B" shows "\<exists>F G. inverse_functors A B F G"
"4_2": fixes t :: "('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (receive\<langle>t\<rangle>) = fv t"
diff_delta_lnexp_cf5_def: shows "diff_delta_lnexp_cf5 \<equiv> \<lambda>x. x ^ 10 / (numer_cf5 (- x) * numer_cf5 x)"
echelon_form_of_column_k_invertible: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and A :: "(('d, 'f) vec, 'e) vec"   and i :: "nat"   and k :: "nat" assumes "is_bezout_ext bezout" shows "\<exists>P. invertible P \<and> P ** A = fst (echelon_form_of_column_k bezout (A, i) k)"
map_of_append: fixes xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'b) list" shows "table_of (xs @ ys) = table_of ys ++ table_of xs"
filter_union_mset: fixes P :: "'a \<Rightarrow> bool"   and M :: "'a multiset"   and N :: "'a multiset" shows "filter_mset P (M + N) = filter_mset P M + filter_mset P N"
not_le: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k" assumes "class.linorder less_eq less" shows "(\<not> less_eq x y) = less y x"
hyps_1: shows "((v\<^sub>1, p\<^sub>1), v\<^sub>2, p\<^sub>2) \<in> edges"
comparator_nat_term_compare: fixes to :: "'b nat_term_order" shows "comparator (nat_term_compare to)"
comp_def: shows "comp = (\<lambda>x y. if (x, y) \<in> r then if (y, x) \<in> r then Eq else Lt else Gt)"
eq_map_tree_Leaf: fixes f :: "'d \<Rightarrow> 'c"   and t :: "'d tree" shows "(map_tree f t = \<langle>\<rangle>) = (t = \<langle>\<rangle>)"
trans1: fixes x :: "nat"   and n1 :: "nat"   and y :: "nat"   and n2 :: "nat"   and z :: "nat" assumes "x + n1 \<le> y"   and "y + n2 \<le> z" shows "x + (n1 + n2) \<le> z"
conv_radius_leI_ex': fixes R :: "ereal"   and f :: "nat \<Rightarrow> 'a" assumes "0 \<le> R"   and "\<And>r. \<lbrakk>0 < r; R < ereal r\<rbrakk> \<Longrightarrow> \<not> summable (\<lambda>n. f n * of_real r ^ n)" shows "conv_radius f \<le> R"
weakCongI: fixes P :: "pi"   and Q :: "pi" assumes "P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> Q"   and "Q \<leadsto>\<guillemotleft>weakBisim\<guillemotright> P" shows "P \<simeq> Q"
proves_eq_ext_rule: 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"   and f :: "Core.term"   and \<tau> :: "typ"   and \<tau>' :: "typ"   and g :: "Core.term"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> f"   and "typ_of f = Some (\<tau> \<rightarrow> \<tau>')"   and "term_ok \<Theta> g"   and "typ_of g = Some (\<tau> \<rightarrow> \<tau>')"   and "\<Theta>,\<Gamma> \<turnstile> Ct STR ''Pure.all'' ((\<tau> \<rightarrow> propT) \<rightarrow> propT) $ Abs \<tau> (mk_eq' \<tau>' (f $ Bv 0) (g $ Bv 0))"   and "finite \<Gamma>"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq f g"
simps: fixes xs :: "('c, 'd, 'e, 'f) Std.instr list"   and ar :: "nat" shows "compile_fundef (Fundef xs ar) = Fundef (map compile_instr xs) ar"
Let_def: fixes s :: "'b"   and f :: "'b \<Rightarrow> 'a" shows "HOL.Let s f \<equiv> f s"
lParamUses: shows "lParamUses \<equiv> lift_ParamUses ParamUses"
Zero_In_OrdP: fixes x :: "tm" shows "{OrdP x} \<turnstile> x EQ Zero OR Zero IN x"
trans: fixes b :: "'c"   and a :: "'c"   and c :: "'c" assumes "b \<le>\<^sub>\<K> a"   and "c \<le>\<^sub>\<K> b" shows "c \<le>\<^sub>\<K> a"
WHATWHERE_empty: fixes SR :: "(('a \<times> ('b \<Rightarrow> 'c)) \<times> 'a list \<times> 'a option \<times> ('b \<Rightarrow> 'c)) set"   and E :: "'e \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and pp :: "'a \<Rightarrow> nat"   and DA :: "'b \<Rightarrow> 'f"   and lH :: "('f \<times> 'e \<times> nat) set" shows "WHATWHERE.WHATWHERE_Secure SR E pp DA lH []"
unrest_expr_alt_def: fixes x :: "'a \<Longrightarrow> 'b"   and P :: "('c, 'b) uexpr" assumes "weak_lens x" shows "(x \<sharp> P) = (\<forall>b b'. \<lbrakk>P\<rbrakk>\<^sub>e (b \<triangleleft>\<^bsub>x\<^esub> b') = \<lbrakk>P\<rbrakk>\<^sub>e b)"
H'_def: shows "H' = inv_into H (ordermap H r) ` elts (\<alpha>' ! i)"
Lindelof_space_subtopology: fixes X :: "'a topology"   and S :: "'a set" shows "Lindelof_space (subtopology X S) = (\<forall>\<U>. (\<forall>U\<in>\<U>. openin X U) \<and> topspace X \<inter> S \<subseteq> \<Union> \<U> \<longrightarrow> (\<exists>V. countable V \<and> V \<subseteq> \<U> \<and> topspace X \<inter> S \<subseteq> \<Union> V))"
cr_prod1_def: fixes c' :: "'f"   and A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "cr_prod1 c' A = (\<lambda>a (b, c). A a b \<and> c' = c)"
Par1B: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi" assumes "P \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e P'"   and "x \<sharp> Q" shows "P \<parallel> Q \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e P' \<parallel> Q"
assms_1: shows "\<AA> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
equiv_trans_trans: fixes x :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and y :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and z :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "equiv_trans x y"   and "equiv_trans y z" shows "equiv_trans x z"
less_add: fixes N :: "'b multiset"   and a :: "'b"   and M0 :: "'b multiset"   and r :: "('b \<times> 'b) set" assumes "(N, add_mset a M0) \<in> mult1 r" shows "(\<exists>M. (M, M0) \<in> mult1 r \<and> N = add_mset a M) \<or> (\<exists>K. (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r) \<and> N = M0 + K)"
closedin_limpt: fixes T :: "'c set"   and S :: "'c set" shows "closedin (top_of_set T) S = (S \<subseteq> T \<and> (\<forall>x. x islimpt S \<and> x \<in> T \<longrightarrow> x \<in> S))"
apos_na_pos: fixes n :: "ant" assumes "n \<noteq> \<infinity>"   and "0 \<le> n" shows "0 \<le> na n"
insert_commute: fixes x :: "'c"   and y :: "'c"   and A :: "'c set" shows "??.Set.insert x (??.Set.insert y A) = ??.Set.insert y (??.Set.insert x A)"
simps_1: fixes \<tau> :: "'b"   and satisfies :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and transition :: "'c \<Rightarrow> ('b, 'c) residual \<Rightarrow> bool"   and x :: "('a, 'd, 'b) formula" assumes "indexed_weak_nominal_ts TYPE('a) \<tau> satisfies transition" shows "indexed_weak_nominal_ts.tau_steps \<tau> x 0 = x"
fold_insert: fixes A :: "'b set"   and x :: "'b"   and f :: "'b \<Rightarrow> 'c set"   and z :: "'c set" assumes "finite A"   and "x \<notin> A" shows "Finite_Set.fold (\<lambda>\<phi>. (\<union>) (f \<phi>)) z (insert x A) = f x \<union> Finite_Set.fold (\<lambda>\<phi>. (\<union>) (f \<phi>)) z A"
a4: fixes x :: "'a"   and y :: "'a" shows "ad (x + y) = ad x \<cdot> ad y"
distinct_13: fixes x1 :: "'e" shows "broadcast x1 \<noteq> \<tau>\<^sub>s"
elims_2: fixes x :: "('fsym, 'tp) efsym" assumes "GE_wtFsym x" obtains f :: "'fsym" where "x = Oldf f" and "wtFsym f"   | \<sigma> :: "'tp" where "x = Wit \<sigma>" and "\<not> isRes \<sigma> \<or> protCl \<sigma>"
last_remdups_adj: fixes xs :: "'g list" shows "last (remdups_adj xs) = last xs"
counit_simps_5: shows "trg \<psi> = trg f"
ii_notin_disc_o_x_axis: shows "ii\<^sub>h \<in> disc_compl o_x_axis"
distributed_simple_function_superset: fixes X :: "'a \<Rightarrow> 'c"   and P :: "'c \<Rightarrow> real"   and A :: "'c set" assumes "simple_function M X"   and "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = prob (X -` {x} \<inter> space M)"   and "X ` space M \<subseteq> A"   and "finite A" shows "distributed M (count_space A) X (\<lambda>x. ennreal (if x \<in> X ` space M then P x else 0))"
counit_simps_3: fixes a :: "'a" assumes "C.obj a" shows "src\<^sub>D (\<sigma>.counit a) = G.map\<^sub>0 a"
Rep_Abs_1: shows "Rep_bit1 (Abs_bit1 1) = 1"
make_mat_carrier: fixes n :: "nat"   and k :: "nat"   and B :: "'a mat" shows "make_mat n k B \<in> carrier_mat n n"
refine: fixes Xi :: "'b list list"   and X :: "'c set"   and bi :: "real list"   and b :: "'c"   and yi :: "real"   and y :: "real" assumes "(Xi, X) \<in> clw_rel appr_rel"   and "(bi, b) \<in> lv_rel"   and "(yi, y) \<in> rnv_rel" shows "(nres_of (project_set_clw_impl Xi bi yi), project_set_clw X b y) \<in> \<langle>clw_rel appr_rel\<rangle>nres_rel"
add_edge_is_connected_2: fixes G :: "('c, 'd) graph"   and a :: "'c"   and b :: "'d"   and c :: "'c" assumes "valid_graph G" shows "nodes_connected (add_edge a b c G) c a"
b: shows "\<forall>\<^sub>F x in at_top. 1 < b x"
assms_3: shows "matcher_agree_on_exact_matches (\<gamma>::'i common_primitive \<Rightarrow> ('i, 'pkt_ext) tagged_packet_scheme \<Rightarrow> bool) common_matcher"
square_eta: shows "square (\<rightarrow>\<^sub>\<eta>) (\<rightarrow>\<^sub>\<eta>) (\<rightarrow>\<^sub>\<eta>\<^sup>=) (\<rightarrow>\<^sub>\<eta>\<^sup>=)"
r: shows "\<forall>a\<in>A. f a = a [^] r a"
rev_shiftr: fixes w :: "'b word"   and n :: "nat" shows "word_reverse w >> n = word_reverse (w << n)"
assms_2: assumes "arr f"   and "arr g"   and "arr h"   and "arr k"   and "cod f = cod g"   and "local.dom h = local.dom k"   and "local.dom f = cod h"   and "local.dom g = cod k"   and "f \<cdot> h = g \<cdot> k" shows "T"
Polygamma_converges: fixes z :: "'a"   and n :: "nat"   and d :: "real" assumes "z \<noteq> (0::'a)"   and "2 \<le> n" shows "uniformly_convergent_on (ball z d) (\<lambda>k z. \<Sum>i<k. inverse ((z + of_nat i) ^ n))"
fun_dom_scene_compl: fixes A :: "'a set" shows "fds (- A) = - fds A"
no_multi_arcs: fixes G :: "('a, 'b) pre_digraph"   and e1 :: "'b"   and e2 :: "'b" assumes "nomulti_digraph G"   and "e1 \<in> arcs G"   and "e2 \<in> arcs G"   and "arc_to_ends G e1 = arc_to_ends G e2" shows "e1 = e2"
lproject_prefix_limit: fixes w :: "'b llist"   and A :: "'b set"   and x :: "'b llist" assumes "\<And>v. \<lbrakk>v \<le> w; lfinite v\<rbrakk> \<Longrightarrow> lproject A v \<le> x" shows "lproject A w \<le> x"
map\<^sub>0_in_hom_1: fixes a :: "'c" assumes "C.obj a" shows "\<guillemotleft>\<epsilon>\<^sub>0 a : FG.map\<^sub>0 a \<rightarrow>\<^sub>C I\<^sub>C.map\<^sub>0 a\<guillemotright>"
rep_eq: fixes x :: "'d \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'c"   and xa :: "('d, 'b) uexpr"   and xb :: "('e, 'b) uexpr"   and xc :: "('f, 'b) uexpr" shows "\<lbrakk>trop x xa xb xc\<rbrakk>\<^sub>e = (\<lambda>b. x (\<lbrakk>xa\<rbrakk>\<^sub>e b) (\<lbrakk>xb\<rbrakk>\<^sub>e b) (\<lbrakk>xc\<rbrakk>\<^sub>e b))"