hcju/isabelleps huggingface dataset:
of_list_repr: fixes i :: "nat"   and ba :: "'a list \<times> 'a list"   and xs :: "'a list" assumes "list_repr i ba xs" shows "of_list_repr ba = xs"
weak_is_transitive: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "rational_preference carrier relation"   and "x \<in> carrier \<and> y \<in> carrier \<and> z \<in> carrier"   and "x \<succeq>[relation] y"   and "y \<succeq>[relation] z" shows "x \<succeq>[relation] z"
filtercomap_filtercomap: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c"   and F :: "'c filter" shows "filtercomap f (filtercomap g F) = filtercomap (\<lambda>x. g (f x)) F"
subset_mset_def: fixes A :: "'a multiset"   and B :: "'a multiset" shows "(A \<subset># B) = (A \<subseteq># B \<and> A \<noteq> B)"
m_lcomm: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "Group.comm_monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z) = y \<otimes>\<^bsub>G\<^esub> (x \<otimes>\<^bsub>G\<^esub> z)"
R_meet: fixes x :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and y :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "x is R healthy"   and "y is R healthy" shows "x \<and> y is R healthy"
lasso_rel_ext_internal_def: fixes Re :: "('f \<times> 'h) set"   and R :: "('e \<times> 'g) set" shows "lasso_rel_ext Re R \<equiv> {(\<lparr>lasso_reach = r', lasso_va = va', lasso_cysfx = cysfx', \<dots> = m'\<rparr>, \<lparr>lasso_reach = r, lasso_va = va, lasso_cysfx = cysfx, \<dots> = m\<rparr>) |r' r va' va cysfx' cysfx m' m. (r', r) \<in> \<langle>R\<rangle>list_rel \<and> (va', va) \<in> R \<and> (cysfx', cysfx) \<in> \<langle>R\<rangle>list_rel \<and> (m', m) \<in> Re}"
weakly_preserves_trg: fixes \<mu> :: "'b" assumes "B.arr \<mu>" shows "C.isomorphic (F (trg\<^sub>B \<mu>)) (trg\<^sub>C (F \<mu>))"
interaction_any_bounded_converter_parallel_converter: fixes conv1 :: "('a, 'b, 'c, 'd) converter"   and n :: "enat"   and conv2 :: "('e, 'f, 'c, 'd) converter" assumes "interaction_any_bounded_converter conv1 n"   and "interaction_any_bounded_converter conv2 n" shows "interaction_any_bounded_converter (parallel_converter conv1 conv2) n"
conf_def: fixes P :: "'a prog"   and h :: "heap"   and v :: "val"   and T :: "ty" shows "P,h \<turnstile> v :\<le> T \<equiv> \<exists>T'. typeof\<^bsub>h\<^esub> v = \<lfloor>T'\<rfloor> \<and> subtype P T' T"
dghm_singleton_ArrMap_vrange: fixes j :: "V"   and \<CC> :: "V" shows "\<R>\<^sub>\<circ> (dghm_singleton j \<CC>\<lparr>ArrMap\<rparr>) = (\<Prod>\<^sub>D\<^sub>Gj\<in>\<^sub>\<circ>ZFC_in_HOL.set {j}. \<CC>)\<lparr>Arr\<rparr>"
mu_bound: fixes j :: "nat" assumes "j < i" shows "(fs.gs.\<mu> i j)\<^sup>2 \<le> rat_of_nat (N ^ (m - 1) * m)"
no_empty_in_non_overlapping: fixes p :: "'c set set" assumes "is_non_overlapping p" shows "{} \<notin> p"
inj_one_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "inj_one_hom hom \<equiv> one_hom hom \<and> injective hom"
mult_oner: fixes x :: "'a list set" shows "p_prod x p_one = x"
trans: shows "\<forall>z\<in>set xs. \<forall>y\<in>set xs. \<forall>x\<in>set xs. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"
Max_eq_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and m :: "'b" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "(linorder.Max less_eq A = m) = (m \<in> A \<and> (\<forall>a\<in>A. less_eq a m))"
rell_F_def: fixes L1 :: "'p \<Rightarrow> 'z \<Rightarrow> bool"   and L2 :: "'q \<Rightarrow> 'aa \<Rightarrow> bool"   and L3 :: "'r \<Rightarrow> 'ab \<Rightarrow> bool" shows "rell_F L1 L2 L3 = rel_F L1 L2 L3 (=) (=) (=) (=) (=) (=)"
hom_diff: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b formula"   and y :: "'b formula" shows "hom f (x - y) = hom f x - hom f y"
prob_space_distr: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and M' :: "'b measure" assumes "prob_space M"   and "f \<in> M \<rightarrow>\<^sub>M M'" shows "prob_space (distr M M' f)"
SDG_return_edge_CFG_node: fixes n :: "'node SDG_node"   and p :: "'pname"   and n' :: "'node SDG_node" assumes "n -p\<rightarrow>\<^bsub>ret\<^esub> n'" shows "\<exists>m. n = CFG_node m"
LAT2: fixes p1 :: "L"   and p2 :: "L"   and p3 :: "L" assumes "LEQ p1 p2"   and "LEQ p2 p3" shows "LEQ p1 p3"
equation_1: fixes x :: "'a prog"   and xa :: "ty"   and xb :: "ty" shows "widen_i_i_i x xa xb = sup (Predicate.single (x, xa, xb) \<bind> (\<lambda>x. case x of (P_, T_, Ta_) \<Rightarrow> if T_ = Ta_ then Predicate.single () else bot)) (sup (Predicate.single (x, xa, xb) \<bind> (\<lambda>x. case x of (P_, Class C_, Class D_) \<Rightarrow> rtranclp_FioB_i_i (subcls1_i_i_o P_) C_ D_ \<bind> (\<lambda>x. case x of () \<Rightarrow> Predicate.single ()) | (P_, Class C_, _) \<Rightarrow> bot | (P_, _, ba) \<Rightarrow> bot)) (sup (Predicate.single (x, xa, xb) \<bind> (\<lambda>x. case x of (P_, NT, Class C_) \<Rightarrow> Predicate.single () | (P_, NT, _) \<Rightarrow> bot | (P_, _, ba) \<Rightarrow> bot)) (sup (Predicate.single (x, xa, xb) \<bind> (\<lambda>x. case x of (P_, NT, A_\<lfloor>\<rceil>) \<Rightarrow> Predicate.single () | (P_, NT, _) \<Rightarrow> bot | (P_, _, ba) \<Rightarrow> bot)) (sup (Predicate.single (x, xa, xb) \<bind> (\<lambda>x. case x of (P_, A_\<lfloor>\<rceil>, Class C_) \<Rightarrow> eq_i_i C_ Object \<bind> (\<lambda>x. case x of () \<Rightarrow> Predicate.single ()) | (P_, A_\<lfloor>\<rceil>, _) \<Rightarrow> bot | (P_, _, ba) \<Rightarrow> bot)) (Predicate.single (x, xa, xb) \<bind> (\<lambda>x. case x of (P_, A_\<lfloor>\<rceil>, B_\<lfloor>\<rceil>) \<Rightarrow> widen_i_i_i P_ A_ B_ \<bind> (\<lambda>x. case x of () \<Rightarrow> Predicate.single ()) | (P_, A_\<lfloor>\<rceil>, _) \<Rightarrow> bot | (P_, _, ba) \<Rightarrow> bot))))))"
scalar_prod_lincomb_gso: fixes k :: "nat"   and g :: "nat \<Rightarrow> 'a"   and h :: "nat \<Rightarrow> 'a" assumes "k \<le> length fs" shows "M.sumlist (map (\<lambda>i. g i \<cdot>\<^sub>v gso i) [0..<k]) \<bullet> M.sumlist (map (\<lambda>i. h i \<cdot>\<^sub>v gso i) [0..<k]) = (\<Sum>i\<leftarrow>[0..<k]. g i * h i * (gso i \<bullet> gso i))"
Sup_upper: fixes x :: "V"   and A :: "V set" assumes "x \<in> A"   and "small A" shows "x \<subseteq>\<^sub>\<circ> \<Squnion> A"
r_one: fixes G :: "('f, 'g) monoid_scheme"   and x :: "'f" assumes "Group.monoid G"   and "x \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = x"
strict_lunit': fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.ide f" shows "S.lunit' f = f"
encode_cfun_map: fixes f :: "'e \<rightarrow> 'g"   and g :: "'h \<rightarrow> 'f"   and x :: "'g\<^sub>\<bottom> \<rightarrow>! 'h" shows "encode_cfun\<cdot>(cfun_map\<cdot>f\<cdot>g\<cdot>(decode_cfun\<cdot>x)) = sfun_map\<cdot>(u_map\<cdot>f)\<cdot>g\<cdot>x"
simps_48: fixes \<phi> :: "Formula.formula"   and I :: "\<I>"   and \<psi> :: "Formula.formula" shows "safe_formula (formula.Since \<phi> I \<psi>) = (fv \<phi> \<subseteq> fv \<psi> \<and> (safe_formula \<phi> \<or> (case \<phi> of formula.Neg x \<Rightarrow> safe_formula x | _ \<Rightarrow> False)) \<and> safe_formula \<psi>)"
sys_xcpts_defs_3: shows "OutOfMemory \<equiv> ''OutOfMemory''"
assms_1: shows "\<forall>\<^sub>F x in at_top. sgn (f x) \<le> u x"
less_llength_ltake: fixes i :: "enat"   and k :: "enat"   and Xs :: "'b llist" shows "(i < llength (ltake k Xs)) = (i < k \<and> i < llength Xs)"
l_set_tag_name_get_disconnected_nodes_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_tag_name_get_disconnected_nodes type_wf set_tag_name set_tag_name_locs get_disconnected_nodes get_disconnected_nodes_locs \<equiv> l_set_tag_name type_wf set_tag_name set_tag_name_locs \<and> l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs \<and> l_set_tag_name_get_disconnected_nodes_axioms set_tag_name_locs get_disconnected_nodes_locs"
quot_def: shows "quot = A $$ (i_end, k) / A $$ (l, k)"
is_extensional: fixes f :: "'a" assumes "\<not> A.arr f" shows "A.map f = A.null"
Ndet_commute: fixes P :: "'a process"   and Q :: "'a process" shows "(P |-| Q) = (Q |-| P)"
inverse_aform_float: fixes p :: "nat"   and x :: "real \<times> real pdevs"   and a :: "real"   and bb :: "real pdevs"   and baa :: "real" assumes "inverse_aform p x = Some ((a, bb), baa)" shows "a \<in> float"
pred_mono_strong: fixes P :: "'g \<Rightarrow> bool"   and x :: "'g list"   and Pa :: "'g \<Rightarrow> bool" assumes "list_all P x"   and "\<And>z. \<lbrakk>z \<in> set x; P z\<rbrakk> \<Longrightarrow> Pa z" shows "list_all Pa x"
ANDRIGHT_eqvt_name: fixes pi :: "(name \<times> name) list"   and A :: "ty"   and B :: "ty"   and X :: "ctrm set"   and Y :: "ctrm set" shows "pi \<bullet> ANDRIGHT (A AND B) X Y = ANDRIGHT (A AND B) (pi \<bullet> X) (pi \<bullet> Y)"
reslI: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<cdot> y \<le> z" shows "x \<le> inv_resl z y"
ABind_nonrec_eqvt: fixes \<pi> :: "perm"   and x :: "Vars.var"   and e :: "exp" shows "\<pi> \<bullet> ABind_nonrec x e = ABind_nonrec (\<pi> \<bullet> x) (\<pi> \<bullet> e)"
smult_closed: fixes R :: "('h, 'i) ring_scheme"   and M :: "('h, 'j, 'k) module_scheme"   and a :: "'h"   and x :: "'j" assumes "module R M"   and "a \<in> carrier R"   and "x \<in> carrier M" shows "a \<odot>\<^bsub>M\<^esub> x \<in> carrier M"
ST: fixes u :: "'a"   and v :: "'a" assumes "(x, u) \<in> E\<^sup>+"   and "(u, v) \<in> E"   and "P x u" shows "P x v"
Hermite_of_upt_row_i_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and i :: "nat"   and ass :: "'b \<Rightarrow> 'b"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "Hermite_of_upt_row_i A i ass res = foldl (Hermite_of_row_i ass res) A (map mod_type_class.from_nat [0..<i])"
entails_def: fixes P :: "assn"   and Q :: "assn" shows "(P \<Longrightarrow>\<^sub>A Q) = (\<forall>h. h \<Turnstile> P \<longrightarrow> h \<Turnstile> Q)"
path_connectedin_continuous_map_image: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "continuous_map X Y f"   and "path_connectedin X S" shows "path_connectedin Y (f ` S)"
simps_3: fixes uu :: "'b" shows "(Some uu \<le> None) = False"
inverse_1: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse one = one"
V\<^sub>1: shows "\<V> f\<^sub>1 = {u, v} \<union> set (between (vertices f) u v) \<union> set vs"
ZHMSet_inverse: fixes y :: "hmultiset zmultiset" shows "zhmsetmset (ZHMSet y) = y"
non_overlapping_extension1: fixes P :: "'b set set"   and X :: "'b set" assumes "is_non_overlapping P"   and "X \<inter> \<Union> P = {}"   and "X \<noteq> {}" shows "is_non_overlapping (insert X P)"
h_def: fixes z :: "complex" shows "h z = (if z \<in> U then contour_integral \<gamma> (d z) else contour_integral \<gamma> (\<lambda>w. f w / (w - z)))"
2: fixes x1 :: "'a"   and x2 :: "'b"   and y1 :: "'a"   and y2 :: "'b" assumes "((x1, x2), y1, y2) \<in> (SIGMA x:UNIV. set_pmf (Kp x))\<^sup>*" shows "(x1, y1) \<in> K1.acc \<and> (x2, y2) \<in> K2.acc"
D_T_subset: fixes P :: "'a process" shows "D P \<subseteq> T P"
subseqs_length_def: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c"   and i :: "nat"   and xs :: "'d list" shows "subseqs_length f b i xs = (let n = length xs in if n < i then [] else subseqs_i_n_main f b xs i n)"
simps_2: fixes c :: "int"   and i :: "nat"   and P :: "pol" shows "Pc c \<langle>*\<rangle> Pinj i P = (if c = 0 then Pc 0 else mkPinj i (P \<langle>*\<rangle> Pc c))"
simps_3: fixes \<phi> :: "('a, 'b, 'c) formula"   and \<sigma> :: "('a, 'b, 'c) subst" shows "Fsubst (! \<phi>) \<sigma> = ! (Fsubst \<phi> \<sigma>)"
commutes_with_unit: shows "(g \<circ> f \<down> A) zeroA = zeroC"
sc_bounded_by: fixes P :: "'a \<Rightarrow> real"   and c :: "real" assumes "sound P"   and "0 \<le> c" shows "bounded_by (c * bound_of P) (\<lambda>x. c * P x)"
sup_eq_bot_iff: fixes x :: "'a"   and y :: "'a" shows "(x + y = 0) = (x = 0 \<and> y = 0)"
map\<^sub>0_simps_3: fixes a :: "'c" assumes "C.obj a" shows "trg\<^sub>D (\<sigma>'.map\<^sub>0 a) = F.map\<^sub>0 a"
f: shows "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"
epred_le_epredI: fixes n :: "enat"   and m :: "enat" assumes "n \<le> m" shows "epred n \<le> epred m"
general_procedure: fixes Q :: "(att \<times> char list, att, nat) env \<Rightarrow> bool"   and y :: "operation"   and users :: "nat set"   and bs :: "operation list"   and P :: "operation \<Rightarrow> bool"   and root :: "(att \<times> char list, att, nat) env" assumes "\<And>r r'. \<lbrakk>Q r; r \<midarrow>y\<rightarrow> r'\<rbrakk> \<Longrightarrow> False"   and "\<And>root. init users \<Midarrow>bs\<Rightarrow> root \<Longrightarrow> Q root"   and "\<And>r x r'. \<lbrakk>r \<midarrow>x\<rightarrow> r'; Q r; P x\<rbrakk> \<Longrightarrow> Q r'"   and "init users \<Midarrow>bs\<Rightarrow> root" shows "\<nexists>xs. (\<forall>x\<in>set xs. P x) \<and> can_exec root (xs @ [y])"
objE: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "horizontal_homs V src trg"   and "horizontal_homs.obj V src a" obtains "partial_magma.ide V a" and "src a = a" and "trg a = a"
deadlocked_def: fixes final :: "'c \<Rightarrow> bool"   and r :: "('a, 'b, 'c, 'd, 'e, 'f) semantics" shows "multithreaded_base.deadlocked final r \<equiv> \<lambda>s. {x. multithreaded_base.deadlockedp final r s x}"
derives_pad: fixes \<alpha> :: "symbol list"   and \<beta> :: "symbol list"   and u :: "symbol list"   and v :: "symbol list" assumes "derives \<alpha> \<beta>"   and "is_sentence u"   and "is_sentence v" shows "derives (u @ \<alpha> @ v) (u @ \<beta> @ v)"
map_of_append: fixes xs :: "('c \<times> 'd) list"   and ys :: "('c \<times> 'd) list" shows "map_of (xs @ ys) = map_of ys ++ map_of xs"
diff_single_trivial: fixes x :: "'b"   and M :: "'b multiset" assumes "x \<notin># M" shows "M - (x, {#}) = M"
diff_fun_space_def: shows "sub.diff_fun_space = {f. diff_fun k (charts_submanifold S) f \<and> extensional0 sub.carrier f}"
smooth_compat_def: fixes k :: "enat"   and c1 :: "('c, 'd) chart"   and c2 :: "('c, 'd) chart" shows "k-smooth_compat c1 c2 = (k-smooth_on (apply_chart c1 ` (domain c1 \<inter> domain c2)) (apply_chart c2 \<circ> inv_chart c1) \<and> k-smooth_on (apply_chart c2 ` (domain c1 \<inter> domain c2)) (apply_chart c1 \<circ> inv_chart c2))"
bin_mask_p1_conv_shift: fixes n :: "nat" shows "mask n + 1 = 1 << n"
diff_less_0_iff_less: fixes a :: "'a"   and b :: "'a" shows "(a - b < (0::'a)) = (a < b)"
size_mono: fixes m' :: "('b, 'c) mapping"   and m :: "('b, 'd) mapping" assumes "finite (keys m')"   and "keys m \<subseteq> keys m'" shows "Mapping.size m \<le> Mapping.size m'"
app: fixes A :: "'b ptrm"   and B :: "'b ptrm"   and C :: "'b ptrm"   and D :: "'b ptrm" assumes "A \<approx> B"   and "C \<approx> D" shows "PApp A C \<approx> PApp B D"
take_or_pad_def: fixes a :: "'a list"   and n :: "nat" shows "take_or_pad a n = (if n \<le> length a then take n a else a @ padding (n - length a))"
\<Z>_\<omega>_\<alpha>\<omega>: fixes \<alpha> :: "V" assumes "\<Z> \<alpha>" shows "\<omega> \<in>\<^sub>\<circ> \<alpha> + \<omega>"
intros: fixes x :: "('d \<times> 'e) bset"   and R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "set_bset x \<subseteq> {(x, y). R x y}" shows "rel_bset R (map_bset fst x) (map_bset snd x)"
is_embedding: fixes f :: "'a"   and f' :: "'a" assumes "A.arr f"   and "A.arr f'"   and "F f = F f'" shows "f = f'"
exhaust: fixes y :: "'a match_result" obtains "y = No_match"   | "y = Match_type_error"   | x3 :: "'a" where "y = Match x3"
sample_parametric: fixes R :: "'p \<Rightarrow> 'p \<Rightarrow> bool" assumes "bi_unique R" shows "(rel_pmf R ===> (R ===> (=)) ===> (=)) sample sample"
ctr_2: fixes xs :: "('a, 'b) tllist" assumes "\<not> is_TNil xs" shows "llist_of_tllist xs = LCons (case xs of TCons x xs' \<Rightarrow> x) (case xs of TCons x xs' \<Rightarrow> llist_of_tllist xs')"
vec_conjugate_rat: shows "conjugate = (\<lambda>x. x)"
strict_arr_char: fixes f :: "'c local.term set" shows "\<F>\<^sub>SC.arr f = (arr f \<and> Diag (DOM f) \<and> Diag (COD f))"
0: shows "LBINT x. (F x * g x + f x * G x) * indicat_real {a..b} x = F b * G b - F a * G a"
simps_5: fixes locals' :: "locals \<Rightarrow> locals"   and heap :: "heap"   and locals :: "locals"   and more :: "'a" shows "locals_update locals' \<lparr>heap = heap, locals = locals, \<dots> = more\<rparr> = \<lparr>heap = heap, locals = locals' locals, \<dots> = more\<rparr>"
acute_lea_acute: fixes D :: "'p"   and E :: "'p"   and F :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Acute D E F"   and "A B C LeA D E F" shows "Acute A B C"
S_3: shows "S \<longlonglongrightarrow> 1"
overrides_t_SomeD: fixes x :: "'a"   and s :: "'b \<Rightarrow> 'a set"   and t :: "'b \<Rightarrow> 'a set"   and k :: "'b" assumes "x \<in> (s \<oplus>\<oplus> t) k" shows "x \<in> t k \<or> t k = {} \<and> x \<in> s k"
simps_3: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool"   and i :: "'b"   and v :: "'b"   and va :: "'b list"   and x :: "'c"   and z :: "'c" shows "rel_comp_pow r (i # v # va) x z = (\<exists>y. r i x y \<and> rel_comp_pow r (v # va) y z)"
oalist_inv_raw_update_by_pair: fixes xs :: "('a \<times> 'b) list"   and kv :: "'a \<times> 'b" assumes "tc.oalist_inv_raw xs" shows "tc.oalist_inv_raw (update_by_pair_tc kv xs)"
ide_in_hom_2: fixes f :: "'b" assumes "B.ide f" shows "\<guillemotleft>f : f \<rightarrow>\<^sub>B f\<guillemotright>"
orderI: fixes a :: "'d multiset"   and b :: "'d multiset" assumes "a = a \<union># b" shows "b \<subseteq># a"
dec_def: fixes as :: "'a list"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "dec as xs y ys \<equiv> y \<in> set xs \<and> distinct xs \<and> as = xs @ y # ys"
step: fixes x :: "'a node \<times> 'a node set" assumes "((X\<^sub>r \<mu> \<in> old (fst x) \<longrightarrow> \<mu> \<in> next (fst x)) \<and> (\<forall>nd\<in>snd x. X\<^sub>r \<mu> \<in> old nd \<and> name nd \<in> incoming (fst x) \<longrightarrow> \<mu> \<in> old (fst x) \<union> new (fst x))) \<and> (\<forall>nd\<in>snd x. \<forall>nd'\<in>snd x. X\<^sub>r \<mu> \<in> old nd \<and> name nd \<in> incoming nd' \<longrightarrow> \<mu> \<in> old nd') \<and> ((\<forall>nm\<in>incoming (fst x). nm < name (fst x)) \<and> 0 < name (fst x) \<and> (\<forall>q\<in>snd x. name q < name (fst x) \<and> (\<forall>nm\<in>incoming q. nm < name (fst x)))) \<and> expand_name_ident (snd x)" shows "f x \<le> SPEC (\<lambda>r. \<forall>nd\<in>snd r. \<forall>nd'\<in>snd r. X\<^sub>r \<mu> \<in> old nd \<and> name nd \<in> incoming nd' \<longrightarrow> \<mu> \<in> old nd')"
f7: shows "\<not> (\<exists>(a', b', c')\<in>set b. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
adh_imp_closure: fixes x :: "'b"   and T :: "'b set set"   and A :: "'b set" assumes "x \<in> carrier\<^bsub>T\<^esub>"   and "x adh\<^bsub>T\<^esub> A" shows "x \<in> closure\<^bsub>T\<^esub> A"
f: fixes i :: "'a" assumes "i \<in> I" shows "f i \<in> Group.iso (G i) (H i)"
F: fixes x :: "'a" assumes "x \<in> s" shows "norm (f x) \<le> F"
n: shows "n = (\<Sum>(q, ri)\<leftarrow>old. root_info.l_r ri l r)"
invar'_def: fixes m :: "'e\<cdot>('f, 'g) errorT" shows "invar' m = (fmapET\<cdot>ID\<cdot>m = m)"
infplays_augment_one_joint_2: fixes p :: "'c list"   and \<sigma>\<^sub>e :: "'c list \<Rightarrow> 'c"   and \<sigma>\<^sub>o :: "'c list \<Rightarrow> 'c" assumes "odd (length p)" shows "strategy_progression (joint_strategy \<sigma>\<^sub>e \<sigma>\<^sub>o) (augment_list \<sigma>\<^sub>o p) = strategy_progression (joint_strategy \<sigma>\<^sub>e \<sigma>\<^sub>o) p"
phis_same_var_inv: fixes n :: "'node"   and v :: "'val"   and vs :: "'val list"   and v' :: "'val" assumes "phis' g (n, v) = Some vs"   and "v' \<in> set vs" shows "var g v' = var g v"
inner_rot_neg_move_base: fixes x :: "'a"   and y :: "'a"   and a :: "'a"   and b :: "'a" assumes "(x - y) \<bullet> rot (a - b) < 0"   and "y \<in> {a--b}" shows "(x - a) \<bullet> rot (a - b) < 0"
equalityI: fixes A :: "'k set"   and B :: "'k set" assumes "A \<subseteq> B"   and "B \<subseteq> A" shows "A = B"
euler_witness_coset: fixes p :: "nat"   and x :: "nat"   and y :: "nat" assumes "euler_witness_context p"   and "euler_witness (int x) p"   and "y \<in> euler_witness_context.H p #>\<^bsub>euler_witness_context.G p\<^esub> x" shows "euler_witness (int y) p"
feq: fixes x :: "'a" assumes "x \<in> sphere a r" shows "f x = exp (g x)"
right_one_inf: fixes a :: "'a" shows "(1::'a) \<sqinter> a = a"
that: fixes d :: "real" assumes "0 < d"   and "a + d *\<^sub>R l \<in> rel_frontier S"   and "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> a + e *\<^sub>R l \<in> rel_interior S" shows "thesis"
invar_from_cscc_invarI: fixes v0 :: "'v"   and D0 :: "'v set"   and L :: "'v set list"   and PDPE :: "'v set list \<times> 'v set \<times> ('v \<times> 'v) set" assumes "cscc_invar v0 D0 (L, PDPE)" shows "invar v0 D0 PDPE"
is_sup_join: fixes x :: "'b"   and y :: "'b" shows "is_sup x y (x \<squnion> y)"
WHILE_DIAGONAL: fixes X :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP"   and p :: "L"   and G :: "Var \<Rightarrow> L"   and q :: "L"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and b :: "BExpr" assumes "X \<rhd> c : Sec (Q p G) (EQ G q) (EQ G q) \<Phi>"   and "\<not> LEQ p q" shows "X \<rhd> While b c : Sec (Q p G) (EQ G q) (EQ G q) (\<lambda>(s, t). EQ G q s t)"
col_space_eq_range: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "linear (*s) (*s) f" shows "col_space (matrix f) = range f"
prems_4: shows "pred_mapping (\<lambda>x. x \<le> plen \<pi>) P"
az: fixes n :: "nat" assumes "n < m" shows "a n = (0::'a)"
openin_Int_closure_of_eq: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "openin X S" shows "S \<inter> X closure_of T = S \<inter> X closure_of (S \<inter> T)"
simps_10: fixes e1 :: "'b aexp"   and e2 :: "'b aexp"   and v :: "'b gexp"   and va :: "'b gexp" shows "GExp_Lexorder.less_gexp_aux (Eq e1 e2) (Nor v va) = True"
tendsto_componentwise_iff: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and F :: "'b filter" shows "(f \<longlongrightarrow> l) F = (\<forall>i\<in>Basis. ((\<lambda>x. f x \<bullet> i) \<longlongrightarrow> l \<bullet> i) F)"