hcju/isabelleps huggingface dataset:
assms: shows "is_Hd (subst \<rho> (Hd (head s)))"
remainder_expansion: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and K :: "'b set"   and A :: "'b set"   and \<psi> :: "'b"   and B :: "'b set"   and \<phi> :: "'b" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL"   and "K = Cn A"   and "\<psi> \<in> Cn K"   and "\<psi> \<notin> Cn B"   and "B \<in> Tarskian_logic.remainder Cn K \<phi>" shows "Cn (B \<union> {\<psi>}) = K"
"1_1": assumes "n \<noteq> 0"   and "n \<noteq> 1"   and "a \<noteq> 1"   and "n < 0" shows "Jacobi a (- n) = jacobi_code a (- n)"
collapse_2: fixes llist :: "'c llist" assumes "\<not> lnull llist" shows "LCons (lhd llist) (ltl llist) = llist"
finite_measure_count_space: fixes A :: "'c set" assumes "finite A" shows "finite_measure (count_space A)"
\<omega>_sum_0E: fixes \<alpha>s :: "V list" assumes "\<omega>_sum \<alpha>s = 0"   and "list.set \<alpha>s \<subseteq> ON" shows "\<alpha>s = []"
root_iff: shows "\<forall>x. (a * x\<^sup>2 + b * x + c = 0) = (x = - r)"
Abrupt: fixes G :: "prog"   and xc :: "abrupt"   and s :: "st"   and t :: "Term.term" shows "G\<turnstile>(Some xc, s) \<midarrow>t\<succ>\<rightarrow> (undefined3 t, (Some xc, s))"
is_type_bool: fixes P :: "'b prog" shows "is_type P Boolean = True"
Bs_3: shows "B4 \<in> carrier_mat n2 n2"
dim_subadditive: fixes K :: "('b, 'c) ring_scheme"   and V :: "('b, 'd, 'e) module_scheme"   and W1 :: "'d set"   and W2 :: "'d set" assumes "vectorspace K V"   and "subspace K W1 V"   and "vectorspace.fin_dim K (V\<lparr>carrier := W1\<rparr>)"   and "subspace K W2 V"   and "vectorspace.fin_dim K (V\<lparr>carrier := W2\<rparr>)" shows "vectorspace.dim K (V\<lparr>carrier := module.submodule_sum V W1 W2\<rparr>) \<le> vectorspace.dim K (V\<lparr>carrier := W1\<rparr>) + vectorspace.dim K (V\<lparr>carrier := W2\<rparr>)"
runit_eqI: fixes f :: "'a"   and a :: "'a" assumes "\<guillemotleft>f : C.tensor a C.unity \<rightarrow> a\<guillemotright>"   and "C.tensor f C.unity = C.tensor a \<iota> \<cdot> C.assoc a C.unity C.unity" shows "f = C.runit a"
join_eq_bind_pmf: fixes M :: "'b pmf pmf" shows "join_pmf M = M \<bind> id"
fls_pow_base: fixes f :: "'a fls"   and n :: "nat" shows "fls_nth (f ^ n) (int n * fls_subdegree f) = fls_nth f (fls_subdegree f) ^ n"
right_impl_times: fixes a :: "'a"   and b :: "'a" shows "a * (a r\<rightarrow> b) = b * (b r\<rightarrow> a)"
corresponding_union_find_def: fixes uf :: "('a \<times> 'a) set"   and T :: "'edge set" shows "corresponding_union_find uf T \<equiv> \<forall>a\<in>V. \<forall>b\<in>V. per_compare uf a b = ((a, b) \<in> connected T)"
hvzk_h_sub2_rewrite: fixes h' :: "'a"   and w :: "nat"   and z :: "nat"   and c :: "nat" assumes "h' = g' [^] w"   and "z < order \<G>" shows "g' [^] (z + order \<G> * w * c - w * c) = g' [^] z \<otimes> inv (h' [^] c)"
trail_def: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" shows "trail u p v \<equiv> awalk u p v \<and> distinct p"
set_intros_3: fixes x3 :: "'a" shows "x3 \<in> LID\<^sub>V (Loc x3)"
insert: fixes A :: "'k set"   and x :: "'k" assumes "finite A"   and "x \<notin> A" shows "mset_set (insert x A) = add_mset x (mset_set A)"
rcos_ring_hom: shows "(+>) (I 0) \<in> ring_hom R (R Quot I 0)"
1: shows "\<guillemotleft>\<l>[S.cod \<mu>] \<cdot>\<^sub>S L \<mu> : L (S.dom \<mu>) \<Rightarrow>\<^sub>S S.cod \<mu>\<guillemotright>"
semiring_of_sets_axioms: shows "semiring_of_sets \<Omega> M"
splay_bstR: fixes t :: "'b tree"   and a :: "'b"   and l :: "'b tree"   and e :: "'b"   and r :: "'b tree"   and x :: "'b" assumes "bst t"   and "splay a t = \<langle>l, e, r\<rangle>"   and "x \<in> set_tree r" shows "a < x"
assms_5: shows "h \<turnstile> get_component ptr' \<rightarrow>\<^sub>r c'"
a: shows "Vagree \<omega> \<omega>' (FVF \<phi>)"
get_min_min: fixes h :: "'b hp option"   and x :: "'b" assumes "h \<noteq> None"   and "invar h"   and "x \<in> set_hp (the h)" shows "get_min h \<le> x"
measurable_count_space_extend: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and A :: "'c set"   and B :: "'c set" assumes "f \<in> M \<rightarrow>\<^sub>M count_space A"   and "A \<subseteq> B" shows "f \<in> M \<rightarrow>\<^sub>M count_space B"
out341__inangle: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p" assumes "A \<noteq> B"   and "B Out C P" shows "P InAngle A B C"
ag_pOp_add_r: fixes A :: "('e, 'f) aGroup_scheme"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A"   and "a = b" shows "a \<plusminus>\<^bsub>A\<^esub> c = b \<plusminus>\<^bsub>A\<^esub> c"
induced_signed_lconjperm_inv_induced_id_sum_list: fixes ss :: "'w list" assumes "ss \<in> lists S" shows "GroupByPresentationInducedFun.induced_hom S (map (charpair S) ` R) signed_lconjpermutation (the_inv_into (Group.quotient_group (FreeGroup S) \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>) (GroupByPresentationInducedFun.induced_hom S (map (charpair S) ` R) id) (sum_list ss)) = sum_list (map signed_lconjpermutation ss)"
add_right_cancel: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and a :: "'e"   and c :: "'e" assumes "class.cancel_semigroup_add plus" shows "(plus b a = plus c a) = (b = c)"
alpha_qSwap_preserves_qGood: fixes zs :: "'varSort"   and y :: "'var"   and x :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and zs' :: "'varSort"   and y' :: "'var"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "X #[[y \<and> x]]_zs #= X' #[[y' \<and> x']]_zs'" shows "qGood X = qGood X'"
rep_eq: fixes x :: "'b prefix"   and xa :: "'b trace" shows "prefix_of x xa = (stake (length (Rep_prefix x)) (Rep_trace xa) = Rep_prefix x)"
add_less_same_cancel1: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c"   and a :: "'c" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less (plus b a) b = less a zero"
ovalidNF_assume_pre: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b option"   and Q :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>s. P s \<Longrightarrow> ovalidNF P f Q" shows "ovalidNF P f Q"
Suc_unat_diff_1: fixes x :: "'b word" assumes "1 \<le> x" shows "Suc (unat (x - 1)) = unat x"
lclss_of_state_def: fixes St :: "'a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set" shows "lclss_of_state St = (\<lambda>C. (C, New)) ` N_of_state St \<union> (\<lambda>C. (C, Processed)) ` P_of_state St \<union> (\<lambda>C. (C, Old)) ` Q_of_state St"
select_convs_2: fixes id :: "nat"   and no :: "nat"   and nhid :: "nat"   and msg :: "msg"   and num :: "nat"   and sid :: "nat"   and more :: "'a" shows "no \<lparr>state.id = id, no = no, nhid = nhid, msg = msg, num = num, sid = sid, \<dots> = more\<rparr> = no"
Cons_lenlex_iff: fixes m :: "'a"   and ms :: "'a list"   and n :: "'a"   and ns :: "'a list"   and r :: "('a \<times> 'a) set" shows "((m # ms, n # ns) \<in> lenlex r) = (length ms < length ns \<or> length ms = length ns \<and> (m, n) \<in> r \<or> m = n \<and> (ms, ns) \<in> lenlex r)"
less_pred_def: fixes P :: "'a pred"   and Q :: "'a pred" shows "(P < Q) = (eval P < eval Q)"
monomial_not_0: fixes p :: "'c \<Rightarrow>\<^sub>0 'd" assumes "is_monomial p" shows "p \<noteq> 0"
CQ2Concl_def: shows "CQ2Concl = (Geq ($' vid1) (Differential (Const 0)) \<leftrightarrow> Geq ($' vid1) (Const 0))"
intra_path_def: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" shows "n -as\<rightarrow>\<^sub>\<iota>* n' \<equiv> n -as\<rightarrow>* n' \<and> (\<forall>a\<in>set as. intra_kind (kind a))"
foldl_rule_P: fixes I :: "'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and \<sigma>0 :: "'a"   and l0 :: "'b list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> bool" assumes "I \<sigma>0 [] l0"   and "\<And>l1 l2 x \<sigma>. \<lbrakk>l0 = l1 @ x # l2; I \<sigma> l1 (x # l2)\<rbrakk> \<Longrightarrow> I (f \<sigma> x) (l1 @ [x]) l2"   and "\<And>\<sigma>. I \<sigma> l0 [] \<Longrightarrow> P \<sigma>" shows "P (foldl f \<sigma>0 l0)"
take_Cons_numeral: fixes v :: "num"   and x :: "'a"   and xs :: "'a list" shows "take (numeral v) (x # xs) = x # take (numeral v - 1) xs"
infinite_int_iff_unbounded_le: fixes S :: "int set" shows "infinite S = (\<forall>m. \<exists>n. m \<le> \<bar>n\<bar> \<and> n \<in> S)"
last_leq_paths: fixes Cs :: "char list list"   and Ds :: "char list list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "(Cs, Ds) \<in> (leq_path1 P C)\<^sup>+"   and "wf_prog wf_md P" shows "(last Cs, last Ds) \<in> (subcls1 P)\<^sup>+"
step_sim: fixes a :: "'b"   and a' :: "'b"   and b :: "'e" assumes "(a, a') \<in> s1.b.E"   and "(a, b) \<in> R2" shows "\<exists>b'. (b, b') \<in> s2.b.E \<and> (a', b') \<in> R2"
NWL_MIN: shows "\<forall>v\<in>V - wl. \<forall>p. is_path v0 p v \<longrightarrow> path_weight' (res v) \<le> Num (path_weight p)"
sel_1: fixes x21 :: "'b"   and x22 :: "'b list" shows "hd (x21 # x22) = x21"
composition_1: shows "\<C> \<turnstile> es : t1s _> t2s"
p_eq_sum_p_u: fixes x :: "'s"   and y :: "'s"   and n :: "nat" shows "p x y (Suc n) = (\<Sum>i\<le>n. p y y (n - i) * u x y i)"
assms: shows "w\<^sub>1 =\<^sub>F w\<^sub>2"
jumpNestingOk_mono: fixes jmps' :: "jump set"   and t :: "Term.term"   and jmps :: "jump set" assumes "jumpNestingOk jmps' t"   and "jmps' \<subseteq> jmps" shows "jumpNestingOk jmps t"
mopup_false2: fixes s :: "nat"   and n :: "nat"   and RR :: "bool" assumes "0 < s"   and "s \<le> 2 * n"   and "s mod 2 = Suc 0"   and "Suc s \<noteq> 2 * n"   and "\<not> Suc (Suc s) \<le> 2 * n" shows "RR"
le_max_iff_disj: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and z :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less" shows "less_eq z (ord.max less_eq x y) = (less_eq z x \<or> less_eq z y)"
module_ow_axioms: shows "module_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 (+\<^sub>V\<^sub>S\<^sub>_\<^sub>2) 0\<^sub>V\<^sub>S\<^sub>_\<^sub>2 (-\<^sub>V\<^sub>S\<^sub>_\<^sub>2) uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 (*s\<^sub>o\<^sub>w\<^sub>_\<^sub>2)"
geq: fixes x :: "'a \<times> real" assumes "x \<in> C" shows "g x = f x"
put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def: fixes document_ptr :: "(_) document_ptr"   and document :: "('n, 'l, 'm, 'u option) RDocument_scheme" shows "put document_ptr document = put (cast document_ptr) (cast document)"
restrict_in: fixes x :: "'b"   and A :: "'b set"   and m :: "'b \<Rightarrow> 'c option" assumes "x \<in> A" shows "(m |` A) x = m x"
not_prime_unit: fixes x :: "'a" assumes "is_unit x" shows "\<not> prime x"
l_prop: shows "l < j \<and> k \<le> l \<and> C \<in> lnth Ns l"
filterlim_floor_sequentially: shows "filterlim floor at_top at_top"
poly_map_poly: fixes p :: "int poly"   and x :: "int" shows "poly (of_int_poly p) (of_int x) = of_int (poly p x)"
relabel_pres_height_bound: fixes u :: "nat" assumes "relabel_precond f l u" shows "Height_Bounded_Labeling c s t f (relabel_effect f l u)"
SeqFormP_graph_aux_def: shows "SeqFormP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>SeqFormP_aux l s k t sl m n sm sn. x1 = (s, k, t) \<and> x2 = LstSeqP s k t AND All2 n (SUCC k) (SyntaxN.Ex sn (HPair (Var n) (Var sn) IN s AND (AtomicP (Var sn) OR SyntaxN.Ex m (SyntaxN.Ex l (SyntaxN.Ex sm (SyntaxN.Ex sl (Var m IN Var n AND Var l IN Var n AND HPair (Var m) (Var sm) IN s AND HPair (Var l) (Var sl) IN s AND MakeFormP (Var sn) (Var sm) (Var sl)))))))) \<and> atom l \<sharp> (s, k, t, sl, m, n, sm, sn) \<and> atom sl \<sharp> (s, k, t, m, n, sm, sn) \<and> atom m \<sharp> (s, k, t, n, sm, sn) \<and> atom n \<sharp> (s, k, t, sm, sn) \<and> atom sm \<sharp> (s, k, t, sn) \<and> atom sn \<sharp> (s, k, t))"
LAM_strict: shows "(\<Lambda> x. \<bottom>) = \<bottom>"
restrict_upd_same: fixes m :: "'a \<Rightarrow> 'b option"   and x :: "'a"   and y :: "'b" shows "m(x \<mapsto> y) |` (- {x}) = m |` (- {x})"
ts: shows "t \<leadsto> s \<in> SPR.jkbpCn (tLength (t \<leadsto> s))"
induced_automorphism_halfmorphism_fimage_to_fopp: shows "ChamberComplexMorphism (f \<turnstile> X) folding_f.Y \
"
phiArg_exI: fixes m :: "'node"   and g :: "'g"   and v :: "'val" assumes "m \<in> set (old.predecessors g (defNode g v))"   and "phi g v \<noteq> None"   and "v \<in> allVars g" obtains v' :: "'val" where "v' \<in> phiUses g m" and "phiArg g v v'"
diagonal_step_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat"   and j :: "nat"   and d :: "'a"   and v :: "'a" shows "diagonal_step A i j d v = (\<chi>a b. if a = from_nat i \<and> b = from_nat i then d else if a = from_nat j \<and> b = from_nat j then v * A $ from_nat j $ from_nat j else A $ a $ b)"
lhs: shows "id_func AA \<^bsub>\<o>\<^esub> (Dom f) = Dom f"
e_tuple_nth_elementary: fixes k :: "nat"   and a :: "nat"   and i :: "nat" assumes "k \<le> a" shows "e_tuple_nth a i k = (if a = k then iterate k pdec2 i else pdec1 (iterate k pdec2 i))"
ord_le_eq_trans: fixes less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and a :: "'a::type"   and b :: "'a::type"   and c :: "'a::type" assumes "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (a::'a::type) (b::'a::type)"   and "(b::'a::type) = (c::'a::type)" shows "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (a::'a::type) (c::'a::type)"
wt_top: fixes \<Gamma> :: "(nat \<times> ty) list"   and x :: "nat"   and e :: "exp" shows "\<Gamma> \<turnstile> ELam x e : TFun \<top>"
nat_lemma2: fixes b :: "nat"   and m :: "nat"   and n :: "nat"   and t :: "nat"   and a :: "nat"   and s :: "nat"   and u :: "nat" assumes "b = m * n + t"   and "a = s * n + u"   and "t = u"   and "b - a < n" shows "m \<le> s"
decision: fixes \<Gamma> :: "char list \<Rightarrow> 'a rule list option"   and \<gamma> :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b"   and rs :: "'a rule list"   and X :: "final_decision" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Decision X\<rangle> \<Rightarrow> Decision X"
a_zero_def: fixes antidomain_op :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "class.antidomain_left_monoid antidomain_op one times" shows "antidomain_left_monoid.a_zero antidomain_op one = antidomain_op one"
lincomb_replicate0_left: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and ms :: "'b list"   and k :: "nat" assumes "VectorSpace smult V"   and "set ms \<subseteq> V" shows "scalar_mult.lincomb smult (replicate k (0::'a)) ms = (0::'b)"
simps_7: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c list \<Rightarrow> 'b \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "'c list" shows "rec_list f1 f2 (x21 # x22) = f2 x21 x22 (rec_list f1 f2 x22)"
R'_1: shows "hd (tl R'_walk) \<leadsto>R'\<leadsto> z"
that: fixes s :: "'t \<Rightarrow>\<^sub>0 'b" assumes "(sig_red sing_reg (\<preceq>) F)\<^sup>*\<^sup>* r s"   and "rep_list s = 0" shows "thesis"
hyps: shows "strip_while P (dropWhile Q xs) = dropWhile Q (strip_while P xs)"
enn2real_top: shows "enn2real \<top> = 0"
conv_order: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (id x \<le> id y)"
infinite_part2: fixes V :: "'a set" assumes "infinite V" shows "infinite (part2 V)"
dom_kildall_stables: shows "stables r step (dom_kildall start)"
walk_in_V: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list" assumes "Graph G"   and "Graph.walk G xs" shows "set xs \<subseteq> V\<^bsub>G\<^esub>"
l_disjoint: fixes G :: "('a, 'b) graph_rec_scheme"   and l :: "'a set list"   and D :: "'a set"   and i :: "nat"   and j :: "nat" assumes "cscc_invar_ext G l D"   and "i < j"   and "j < length l" shows "l ! i \<inter> l ! j = {}"
idem: fixes d :: "'a defl"   and x :: "'a" shows "cast\<cdot>d\<cdot>(cast\<cdot>d\<cdot>x) = cast\<cdot>d\<cdot>x"
postListpostSet2: fixes xs :: "('a \<times> ('b \<Rightarrow> 'c) \<times> 'd) list"   and l :: "'a \<Rightarrow> 'c"   and s :: "'b" shows "postList xs l s = postSet (set xs) l s"
SemWhileF: fixes b :: "BExpr"   and s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and c :: "OBJ" assumes "\<not> evalB b (fst s)"   and "t = s" shows " s , While b c \<rightarrow>\<^sub>1  t "
stabilizer_m_inv_closed: fixes x :: "'c"   and g :: "'a" assumes "x \<in> E"   and "g \<in> stabilizer G \<phi> x" shows "inv g \<in> stabilizer G \<phi> x"
join_isol: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "z + x \<le> z + y"
to_F_def: fixes \<iota>\<^sub>F\<^sub>L :: "('f \<times> 'l) inference" shows "to_F \<iota>\<^sub>F\<^sub>L = Infer (map fst (prems_of \<iota>\<^sub>F\<^sub>L)) (fst (concl_of \<iota>\<^sub>F\<^sub>L))"
bexE: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "\<exists>x::'b\<in>A::'b set. (P::'b \<Rightarrow> bool) x" obtains x :: "'b" where "(x::'b) \<in> (A::'b set)" and "(P::'b \<Rightarrow> bool) (x::'b)"
parts_msg_key: fixes K :: "key" shows "parts_msg (Key K) = {Key K}"
intT_proj: fixes \<sigma> :: "'tp"   and a :: "univ" shows "intT \<sigma> (proj \<sigma> a)"
unat_mod_distrib: fixes v :: "'a word"   and w :: "'a word" shows "unat (v mod w) = unat v mod unat w"
comp_null_2: fixes f :: "(bool, bool list) concrete_category.arr" shows "f \<cdot>\<^sub>J J.null = J.null"
closed_segment_half_open_segment_subsetI: fixes x :: "real"   and t0 :: "real"   and t1 :: "real" assumes "x \<in> {t0--<t1}" shows "{t0--x} \<subseteq> {t0--<t1}"
at_within_nhd: fixes "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and S :: "'c set"   and T :: "'c set"   and U :: "'c set" assumes "class.topological_space open"   and "x \<in> S"   and "open S"   and "T \<inter> S - {x} = U \<inter> S - {x}" shows "topological_space.at_within open x T = topological_space.at_within open x U"
rreq_oip: shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PRreq-:3, PRreq-:4, PRreq-:15, PRreq-:27} \<longrightarrow> oip \<xi> \<in> kD (rt \<xi>) \<and> (osn \<xi> < sqn (rt \<xi>) (oip \<xi>) \<or> sqn (rt \<xi>) (oip \<xi>) = osn \<xi> \<and> the (dhops (rt \<xi>) (oip \<xi>)) \<le> Suc (hops \<xi>) \<and> the (flag (rt \<xi>) (oip \<xi>)) = val))"
ennreal_minus_top: fixes a :: "real" shows "ennreal a - top = 0"
subset_range_from_nat_into: fixes A :: "'b set" assumes "countable A" shows "A \<subseteq> range (from_nat_into A)"
r_diff_distr: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "z \<otimes> (x \<ominus> y) = z \<otimes> x \<ominus> z \<otimes> y"
next_in_def: fixes A :: "'c set"   and u :: "'c" shows "next_in A u = Min (A \<inter> {u<..})"
sip_Some: fixes ip :: "'a word \<times> nat" assumes "ipcidr_conjunct sip1 sip2 = Some ip" shows "simple_match_ip ip (p_src p) = (simple_match_ip sip1 (p_src p) \<and> simple_match_ip sip2 (p_src p))"
integrable_spike: fixes f :: "'a \<Rightarrow> 'b"   and T :: "'a set"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "f integrable_on T"   and "negligible S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x" shows "g integrable_on T"
hcomplex_defs_4: shows "hcnj = *f* cnj"
leg_simps_4: shows "C.dom src_\<mu>.cod.leg1 = src_\<mu>.cod.apex"
that: fixes g' :: "complex \<Rightarrow> complex" assumes "continuous_on (cball z r) g"   and "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative g' w) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F" shows "thesis"
assms_2: fixes r :: "'r"   and n :: "'m" assumes "r \<in> R"   and "n \<in> N" shows "r \<cdot> n \<in> N"
undeclared_not_declared: fixes G :: "prog"   and m :: "memberdecl"   and C :: "qtname" assumes "G\<turnstile> memberid m undeclared_in C" shows "\<not> G\<turnstile> m declared_in C"
subjects_reduction2: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and es :: "expr list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and es' :: "expr list"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   and Ts :: "ty list" assumes "wf_C_prog P"   and "P,E \<turnstile> \<langle>es,(h, l)\<rangle> [\<rightarrow>] \<langle>es',(h', l')\<rangle>"   and "P,E \<turnstile> (h, l) \<surd>"   and "P,E,h \<turnstile> es [:] Ts" shows "P,E,h' \<turnstile> es' [:]\<^bsub>NT\<^esub> Ts"
thin3: fixes A1 :: "fmla"   and A2 :: "fmla"   and H :: "fmla set"   and B :: "fmla"   and A3 :: "fmla" assumes "insert A1 (insert A2 H) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 H)) \<turnstile> B"
not_weakly_preferred_iff: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier"   and "b \<in> carrier" shows "(\<not> le a b) = b \<prec>[le] a"
Radon_ex_lemma: fixes c :: "'c set" assumes "finite c"   and "affine_dependent c" shows "\<exists>u. sum u c = 0 \<and> (\<exists>v\<in>c. u v \<noteq> 0) \<and> (\<Sum>v\<in>c. u v *\<^sub>R v) = (0::'c)"
trans_S: fixes S :: "('a \<times> 'a) set"   and NS :: "('a \<times> 'a) set" assumes "pre_order_pair S NS" shows "trans S"
unit_simps_1: fixes a :: "'a" assumes "B.obj a" shows "B.arr \<i>[a]"
t_Lam: fixes x :: "name"   and \<Gamma> :: "(name \<times> ty) list"   and T\<^sub>1 :: "ty"   and t :: "lam"   and T\<^sub>2 :: "ty" assumes "x \<sharp> \<Gamma>"   and "(x, T\<^sub>1) # \<Gamma> \<turnstile> t : T\<^sub>2" shows "\<Gamma> \<turnstile> Lam [x].t : T\<^sub>1 \<rightarrow> T\<^sub>2"
indV: shows "indV \<equiv> BaseFGMod.indspace G FH V"
le_imp_inverse_le_neg: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b"   and "b < (0::'a)" shows "inverse b \<le> inverse a"
dvd_Lcm: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and A :: "'a set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "a \<in> A" shows "dvd.dvd times a (Lcm A)"
distinct_1: fixes x2 :: "'c" shows "None \<noteq> Some x2"
iter_widen_def: fixes f :: "'a \<Rightarrow> 'a" shows "iter_widen f = while_option (\<lambda>x. \<not> f x \<le> x) (\<lambda>x. x \<nabla> f x)"