hcju/isabelleps huggingface dataset:
tendsto_add_smallo_iff: fixes f2 :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and f1 :: "'b \<Rightarrow> 'c"   and a :: "'c" assumes "f2 \<in> o[F](f1)" shows "(f1 \<longlongrightarrow> a) F = ((\<lambda>x. f1 x + f2 x) \<longlongrightarrow> a) F"
ih_16: shows "InvariantGetReasonIsReason (getReason state') (getF state') (getM state') (set (getQ state'))"
sent_messages_consistent: shows "consistent {m. \<exists>i. event_at i (Send m)}"
word_sub_le_iff: fixes x :: "'b word"   and y :: "'b word" shows "(x - y \<le> x) = (y \<le> x)"
normalize_def: fixes x :: "'b" shows "Projections.normalize x = (1 / norm x) *\<^sub>R x"
pOp_cancel_l: fixes A :: "('c, 'd) aGroup_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A"   and "c \<plusminus>\<^bsub>A\<^esub> a = c \<plusminus>\<^bsub>A\<^esub> b" shows "a = b"
rel_simps_46: shows "(0::'b) \<le> (0::'b)"
fresh_fNext_def: shows "fresh_fNext \<equiv> fresh.fNext next default"
assms_3: shows "length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) \<le> l"
vconn_implies_hpar_2: fixes \<mu> :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data" assumes "\<guillemotleft>\<mu> : f \<Rightarrow> g\<guillemotright>" shows "trg \<mu> = trg f"
lm075: fixes XX :: "'a set set" shows "finestpart (\<Union> XX) \<subseteq> \<Union> (finestpart ` XX)"
degree_prod_eq: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c poly" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> 0" shows "degree (prod f A) = (\<Sum>x\<in>A. degree (f x))"
ctr: fixes rs :: "'rule stream"   and s :: "'state" shows "mkTree rs s = Node (s, shd (trim rs s)) (mkTree (stl (trim rs s)) |`| pickEff (shd (trim rs s)) s)"
set_delete_aux: fixes xs :: "('a \<times> 'b) list"   and k :: "'a" assumes "distinct (map fst xs)" shows "list.set (delete_aux k xs) = list.set xs - {k} \<times> UNIV"
rm_set_lookup: fixes k :: "'k" shows "set (rm_set_lookup rm k) = {d \<in> data. key d = k}"
conjugation_is_Bij: fixes g :: "'a"   and p :: "nat" assumes "finite (carrier G)"   and "g \<in> carrier G" shows "conjugation_action p g \<in> Bij (subgroups_of_size p)"
conv_compl: fixes x :: "'a" shows "(- x)\<^sup>\<smile> = - x\<^sup>\<smile>"
cong_ex_bigtheta: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f2 :: "'a \<Rightarrow> 'b"   and g1 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<in> \<Theta>[F](f2)"   and "g1 \<in> \<Theta>[F](g2)" shows "(f1 \<in> L F g1) = (f2 \<in> L F g2)"
atLeastAtMost_rev: fixes g :: "nat \<Rightarrow> 'b"   and n :: "nat"   and m :: "nat" shows "prod g {n..m} = (\<Prod>i = n..m. g (m + n - i))"
invfun_mem: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and b :: "'d" assumes "f \<in> A \<rightarrow> B"   and "inj_on f A"   and "surj_to f A B"   and "b \<in> B" shows "(f\<inverse>\<^bsub>B,A\<^esub>) b \<in> A"
induct: fixes P :: "'a common_primitive match_expr \<Rightarrow> bool"   and a0 :: "'a common_primitive match_expr" assumes "P MatchAny"   and "\<And>uu_ uv_. P (Match (Src (IpAddrRange uu_ uv_)))"   and "\<And>uw_. P (Match (Src (IpAddr uw_)))"   and "\<And>ux_ uy_. P (Match (Src (IpAddrNetmask ux_ uy_)))"   and "\<And>v. P (Match (Dst v))"   and "\<And>v. P (Match (IIface v))"   and "\<And>v. P (Match (OIface v))"   and "\<And>v. P (Match (Prot v))"   and "\<And>v. P (Match (Src_Ports v))"   and "\<And>v. P (Match (Dst_Ports v))"   and "\<And>v. P (Match (MultiportPorts v))"   and "\<And>v. P (Match (L4_Flags v))"   and "\<And>v. P (Match (CT_State v))"   and "\<And>v. P (Match (Extra v))"   and "\<And>va_. P (MatchNot (Match (Src va_)))"   and "\<And>v. P (MatchNot (Match (Dst v)))"   and "\<And>v. P (MatchNot (Match (IIface v)))"   and "\<And>v. P (MatchNot (Match (OIface v)))"   and "\<And>v. P (MatchNot (Match (Prot v)))"   and "\<And>v. P (MatchNot (Match (Src_Ports v)))"   and "\<And>v. P (MatchNot (Match (Dst_Ports v)))"   and "\<And>v. P (MatchNot (Match (MultiportPorts v)))"   and "\<And>v. P (MatchNot (Match (L4_Flags v)))"   and "\<And>v. P (MatchNot (Match (CT_State v)))"   and "\<And>v. P (MatchNot (Match (Extra v)))"   and "\<And>m1 m2. \<lbrakk>P m1; P m2\<rbrakk> \<Longrightarrow> P (MatchAnd m1 m2)"   and "\<And>vc_ vd_. P (MatchNot (MatchAnd vc_ vd_))"   and "\<And>ve_. P (MatchNot (MatchNot ve_))"   and "P (MatchNot MatchAny)" shows "P a0"
DirProd_commute_iso_set: fixes G :: "('e, 'g) monoid_scheme"   and H :: "('f, 'h) monoid_scheme" shows "(\<lambda>(x, y). (y, x)) \<in> Group.iso (G \<times>\<times> H) (H \<times>\<times> G)"
same_ODE_same_sol: fixes F :: "real \<Rightarrow> variable \<Rightarrow> real"   and x :: "char"   and I :: "interp"   and \<theta> :: "trm"   and J :: "interp"   and \<eta> :: "trm" assumes "\<And>\<nu>. Uvariation \<nu> (F 0) {RVar x, DVar x} \<Longrightarrow> term_sem I \<theta> \<nu> = term_sem J \<eta> \<nu>" shows "solves_ODE I F x \<theta> = solves_ODE J F x \<eta>"
component_of_syzygy_module_list: fixes s :: "'t \<Rightarrow>\<^sub>0 'c"   and bs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "s \<in> syzygy_module_list bs" shows "component_of_term ` keys s \<subseteq> {0..<length bs}"
mini_to_mini: fixes F :: "'a formula" assumes "is_mini_formula F" shows "to_mini_formula F = F"
int_add_floor: fixes z :: "int"   and x :: "'b" shows "z + \<lfloor>x\<rfloor> = \<lfloor>of_int z + x\<rfloor>"
at_left_eq: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" assumes "class.linorder_topology less_eq less open"   and "less y x" shows "topological_space.at_within open x (ord.lessThan less x) = (INF a\<in>ord.lessThan less x. principal (ord.greaterThanLessThan less a x))"
fresh_def: fixes xs :: "'a"   and x :: "'b"   and X :: "('c, 'd, 'a, 'b, 'e) qTerm set" shows "fresh xs x X \<equiv> qFresh xs x (pick X)"
assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "a \<squnion> b \<squnion> c = a \<squnion> (b \<squnion> c)"
m_inv_def: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b" shows "inv\<^bsub>G\<^esub> x = (THE y. y \<in> carrier G \<and> x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub> \<and> y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>)"
r': shows "Seq r' c\<^sub>2 \<in> redexes c''"
diff_single_eq_union: fixes x :: "'a"   and M :: "'a multiset"   and N :: "'a multiset" assumes "x \<in># M" shows "(M \<ominus> x = N) = (M = add_mset x N)"
fst_snd_rw: shows "fst (snd g) = fst (snd f)"
sorted_append: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (xs @ ys) = (linorder.sorted less_eq xs \<and> linorder.sorted less_eq ys \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. less_eq x y))"
u_4: assumes "g \<noteq> h" shows "is_Abs g \<and> is_Abs h \<and> (the_Abs g, the_Abs h) \<in> (set TI)\<^sup>+"
Fract_def: shows "Fraction_Field.Fract \<equiv> map_fun id (map_fun id abs_fract) (\<lambda>a b. if b = (0::'b) then (0::'b, 1::'b) else (a, b))"
compose_list_refl: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'b list" assumes "equivp R" shows "(list_all2 R OOO (\<approx>)) r r"
mem_val_alt_8_unchanged2: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36"   and addr :: "word36"   and val :: "word8" assumes "mem_equal s1 s2 pa" shows "mem_val_alt 8 ((pa AND 68719476732) + 2) (s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr>) = mem_val_alt 8 ((pa AND 68719476732) + 2) (s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>)"
Cons_2: shows "subpath g v1 (e' # es) v2"
assms_3: shows "P \<subseteq> {q. irreducible q \<and> monic q}"
match: fixes P :: "'c \<Rightarrow> bool"   and pat :: "term"   and t :: "'c"   and env :: "(name, 'c) fmap" assumes "simple_syntactic_and P"   and "match pat t = Some env"   and "P t" shows "fmpred (\<lambda>_. P) env"
right_idem: fixes a :: "'g"   and b :: "'g" shows "a \<sqinter> b \<sqinter> b = a \<sqinter> b"
hcomp_in_vhom: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and h :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and k :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.in_hom \<mu> f g"   and "Span.in_hom \<nu> h k"   and "Span.src h = Span.trg f" shows "Span.in_hom (\<nu> \<circ> \<mu>) (h \<circ> f) (k \<circ> g)"
sat_slice_iff: fixes v :: "'c list"   and S :: "'c list set"   and \<sigma> :: "(char list \<times> 'c list) trace"   and i :: "nat"   and \<phi> :: "'c MFOTL.formula" assumes "v \<in> S" shows "MFOTL.sat \<sigma> v i \<phi> = MFOTL.sat (map_\<Gamma> (\<lambda>D. D \<inter> relevant_events \<phi> S) \<sigma>) v i \<phi>"
inf_prod_def: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "inf x y = (inf (fst x) (fst y), inf (snd x) (snd y))"
good_item_simps_40: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y :: "'var"   and xsa :: "'varSort"   and xa :: "'var" assumes "good X" shows "freshAbs ys y (Abs xsa xa X) = (ys = xsa \<and> y = xa \<or> fresh ys y X)"
lcs_adds_plus: fixes s :: "'a"   and t :: "'a" shows "lcs s t adds s + t"
simps_4: fixes p :: "'a poly"   and q :: "'a poly" shows "polynate (Mul p q) = polynate p *\<^sub>p polynate q"
prems_2: shows "instrs ((r # sb) @ sb') @ is\<^sub>s\<^sub>b = is @ prog_instrs ((r # sb) @ sb')"
idem: fixes a :: "'d" shows "inf_class.inf a a = a"
a3: assumes "processed (a, b) xs ((p, q) # ps)" shows "thesis"
Fishburn_liftable: fixes alts' :: "'a set"   and alts :: "'a set" assumes "alts' \<subseteq> alts" shows "liftable_set_extension alts' alts Fishb"
eventually_at_ball': fixes d :: "real"   and z :: "'a"   and A :: "'a set" assumes "0 < d" shows "\<forall>\<^sub>F t in at z within A. t \<in> ball z d \<and> t \<noteq> z \<and> t \<in> A"
g: fixes j :: "nat"   and r :: "nat"   and e :: "nat" shows "eval g [j, r, e] \<down>= (if 2 \<le> e_nth e j then e_nth e j else r)"
HK_3: shows "H \<inter> K \<subseteq> {\<one>\<^bsub>homology_group p (subtopology X S)\<^esub>}"
pred_set: fixes P :: "'b \<Rightarrow> bool" shows "pred_pmf P = (\<lambda>x. Ball (set_pmf x) P)"
map_comp: fixes g :: "'d \<Rightarrow> 'c"   and f :: "'e \<Rightarrow> 'd"   and v :: "'b \<Rightarrow> 'e" shows "g \<circ> (f \<circ> v) = g \<circ> f \<circ> v"
l_set_disconnected_nodes_get_child_nodes_axioms: shows "l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes_locs get_child_nodes_locs"
r_discharge: fixes a :: "'b tree" shows "r (discharge a) = r a"
DERIV_power: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a"   and s :: "'a set"   and n :: "nat" assumes "(f has_field_derivative D) (at x within s)" shows "((\<lambda>x. f x ^ n) has_field_derivative of_nat n * (D * f x ^ (n - Suc 0))) (at x within s)"
synth_simp_intro: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> synth H"   and "X \<in> range Nonce \<union> range Key \<union> range Passwd \<union> range Pri_AgrK \<union> range Pub_AgrK" shows "X \<in> H"
partList4_empty: fixes ts :: "'a set list"   and s :: "'a set" assumes "{} \<notin> set ts" shows "{} \<notin> set (partList4 s ts)"
timpl_closure_fv_eq: fixes s :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and T :: "('c set \<times> 'c set) set" assumes "s \<in> timpl_closure t T" shows "fv s = fv t"
list_digraph_ext_def: fixes ext :: "'c"   and G_list :: "'b list \<times> ('b \<times> 'b) list" shows "list_digraph_ext ext G_list \<equiv> \<lparr>pverts = set (fst G_list), parcs = set (snd G_list), \<dots> = ext\<rparr>"
vector_transpose_matrix: fixes x :: "('a, 'c) vec"   and A :: "(('a, 'c) vec, 'b) vec" shows "x v* Finite_Cartesian_Product.transpose A = A *v x"
nth_default_append: fixes dflt :: "'b"   and xs :: "'b list"   and ys :: "'b list"   and n :: "nat" shows "nth_default dflt (xs @ ys) n = (if n < length xs then xs ! n else nth_default dflt ys (n - length xs))"
join_meet_distrib1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<squnion> y \<sqinter> z = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
complement_2: fixes x :: "'b"   and y :: "'b" shows "! (d x) = ! (d (! (d (! (d x) \<squnion> ! (d y))) \<squnion> ! (d (! (d x) \<squnion> ! (d (! (d y)))))))"
winding_number_ivt_pos: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex"   and w :: "real" assumes "valid_path \<gamma>"   and "z \<notin> path_image \<gamma>"   and "0 \<le> w"   and "w \<le> Re (winding_number \<gamma> z)" shows "\<exists>t\<in>{0..1}. Re (winding_number (subpath 0 t \<gamma>) z) = w"
lipschitz_on_continuous_on: fixes L :: "real"   and X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "L-lipschitz_on X f" shows "continuous_on X f"
simps: fixes a :: "'a set"   and A :: "'a set set" shows "(a \<in> tiling A) = (a = {} \<or> (\<exists>aa t. a = aa \<union> t \<and> aa \<in> A \<and> t \<in> tiling A \<and> aa \<subseteq> - t))"
le_supI: fixes a :: "'b"   and x :: "'b"   and b :: "'b" assumes "a \<le> x"   and "b \<le> x" shows "a + b \<le> x"
simps: 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 C :: "char list"   and a :: "char list list \<times> (char list \<Rightarrow> val option)" shows "init_obj P C a = (\<exists>Cs. a = (Cs, init_class_fieldmap P (last Cs)) \<and> Subobjs P C Cs)"
Eps_Opt_eq_Some_implies: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "Eps_Opt P = Some x" shows "P x"
intra_SDG_path_valid_SDG_node_2: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "n i-ns\<rightarrow>\<^sub>d* n'" shows "valid_SDG_node n'"
mbisim_K_flip: fixes bisim :: "'a \<Rightarrow> 'd \<times> 'e \<Rightarrow> 'b \<times> 'c \<Rightarrow> bool"   and c :: "bool"   and s1 :: "('f, 'a) locks \<times> (('a \<Rightarrow> ('b \<times> 'f \<Rightarrow>f nat) option) \<times> 'c) \<times> ('a \<Rightarrow> 'g wait_set_status option) \<times> 'a set"   and s2 :: "('f, 'a) locks \<times> (('a \<Rightarrow> ('d \<times> 'f \<Rightarrow>f nat) option) \<times> 'e) \<times> ('a \<Rightarrow> 'g wait_set_status option) \<times> 'a set" shows "FWbisimulation_base.mbisim (\<lambda>t. flip (bisim t)) (\<lambda>x1 x2. c) s1 s2 = FWbisimulation_base.mbisim bisim (\<lambda>x1 x2. c) s2 s1"
valid_indexE: fixes "is" :: "nat list"   and ds :: "nat list" assumes "is \<lhd> ds" obtains "isa__ = []" and "ds = []"   | "is" :: "nat list" and dsa :: "nat list" and i :: "nat" and d :: "nat" where "isa__ = i # is" and "ds = d # dsa" and "is \<lhd> dsa" and "i < d"
continuous_map_atin: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" shows "continuous_map X Y f = (\<forall>x\<in>topspace X. limitin Y f (f x) (atin X x))"
rep_eq: fixes x :: "rat" shows "rep_real_alg_3 (of_rat_3 x) = of_rat_2 x"
carrier_f_nxt: fixes f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set"   and x :: "'b"   and x' :: "'b" shows "carrier (f_nxt f T x x') \<subseteq> carrier (f x')"
is_tm_dghm_axioms: shows "\<FF> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<CC>"
subspace_is_derivation: fixes p :: "'a" shows "subspace {X. is_derivation X p}"
parent_in_V: fixes G :: "('a, 'b) Graph_scheme"   and root :: "'a"   and v :: "'a" assumes "RootedTree G root"   and "v \<in> V\<^bsub>G\<^esub>"   and "v \<noteq> root" shows "RootedTree.parent G root v \<in> V\<^bsub>G\<^esub>"
union_genideal: fixes I :: "'a set"   and J :: "'a set" assumes "ideal I R"   and "ideal J R" shows "Idl (I \<union> J) = I <+> J"
prefix_same_length_eq: fixes a :: "'c list"   and b :: "'c list" shows "(prefix a b \<and> length a = length b) = (a = b)"
ZassenhausTr1: fixes H :: "'a set"   and H1 :: "'a set"   and K :: "'a set"   and K1 :: "'a set" assumes "G \<guillemotright> H "   and "G \<guillemotright> H1 "   and "G \<guillemotright> K "   and "G \<guillemotright> K1 "   and "\<natural>H \<triangleright> H1"   and "\<natural>K \<triangleright> K1" shows "H1 \<diamondop> H \<inter> K1 = H \<inter> K1 \<diamondop> H1"
pmdl_closed_trd: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "p \<in> pmdl B"   and "set fs \<subseteq> pmdl B" shows "trd fs p \<in> pmdl B"
spmf_of_pmf_return_pmf: fixes x :: "'a" shows "spmf_of_pmf (return_pmf x) = return_spmf x"
tensor_prod_unit_vec: fixes M :: "nat"   and j :: "nat" shows "unit_vec M j \<otimes> unit_vec M j = tensor_from_lookup [M, M] (\<lambda>is. if is = [j, j] then 1::'a else (0::'a))"
minus_fpxs_def: fixes f :: "'a fpxs"   and g :: "'a fpxs" shows "f - g = f + - g"
rsp: shows "((\<approx>) ===> (\<approx>) ===> (\<approx>)) (@) (@)"
unfold: fixes g :: "('a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'b \<times> 'a)"   and s :: "'a" assumes "fst g s"   and "snd (snd g s) \<in> terminates_on g" shows "s \<in> terminates_on g"
cardinalityPreservation: fixes XX :: "'a set set" assumes "\<forall>X\<in>XX. finite X"   and "is_non_overlapping XX" shows "card (\<Union> XX) = sum card XX"
simps: fixes a1 :: "('p, 's, 'm) configuration"   and a2 :: "('p, 's, 'm) event list"   and a3 :: "('p, 's, 'm) configuration" shows "trace a1 a2 a3 = ((\<exists>c. a1 = c \<and> a2 = [] \<and> a3 = c) \<or> (\<exists>c ev c' t c''. a1 = c \<and> a2 = ev # t \<and> a3 = c'' \<and> c \<turnstile> ev \<mapsto> c' \<and> trace c' t c''))"
intros: fixes \<Delta> :: "('a, 'b) network_scheme"   and S :: "'a set" assumes "source \<Delta> \<in> S"   and "sink \<Delta> \<notin> S" shows "cut \<Delta> S"
preserves_reflects_arr: fixes f :: "'c local.term set" shows "\<F>C.arr (\<F>C.\<rho> f) = \<F>C.arr f"
topological_basisE: fixes "open" :: "'c set \<Rightarrow> bool"   and B :: "'c set set"   and O' :: "'c set"   and x :: "'c" assumes "class.topological_space open"   and "topological_space.topological_basis open B"   and "open O'"   and "x \<in> O'" obtains B' :: "'c set" where "B' \<in> B" and "x \<in> B'" and "B' \<subseteq> O'"
ind_in_set_iff: fixes x :: "nat"   and A :: "nat set"   and B :: "nat set" assumes "x \<in> A"   and "B \<subseteq> A" shows "(ind_in_set A x \<in> ind_in_set A ` B) = (x \<in> B)"
subset_code_1: fixes xs :: "'b list"   and B :: "'b set" shows "order.greater_eq B (set xs) = (\<forall>x\<in>set xs. x \<in> B)"
power2_abs: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "power.power one times (abs a) 2 = power.power one times a 2"
intro: fixes \<eta> :: "'c \<Rightarrow> 'd"   and G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and G' :: "'d set"   and composition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit' :: "'d" assumes "group_homomorphism \<eta> G composition unit G' composition' unit'"   and "monoid_epimorphism \<eta> G composition unit G' composition' unit'" shows "group_epimorphism \<eta> G composition unit G' composition' unit'"
open_countable_basisE: fixes p :: "'d set \<Rightarrow> bool"   and B :: "'d set set"   and X :: "'d set" assumes "countable_basis p B"   and "p X" obtains B' :: "'d set set" where "B' \<subseteq> B" and "X = \<Union> B'"
succ_in: fixes i :: "'a" assumes "local.aboveS i \<noteq> {}" shows "(i, succ i) \<in> r"
1: shows "tfr\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>s\<^sub>t A')"
insertion_aux_def: shows "insertion_aux \<equiv> map_fun id (map_fun lookup id) insertion_fun"
dec_clock_monotone: fixes ck :: "bool"   and mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and d :: "dec"   and res :: "'b state \<times> (v sem_env, v) result" assumes "evaluate_dec ck mn env s d res"   and "ck = True" shows "clock (fst res) \<le> clock s"
cbv_star_eval: fixes t1 :: "lam"   and t2 :: "lam"   and t3 :: "lam" assumes " t1 \<longrightarrow>cbv* t2"   and "t2 \<Down> t3" shows "t1 \<Down> t3"
var_infinite_def: fixes uu :: "'a" shows "var_infinite uu \<equiv> infinite UNIV"
hausdorff_distance_points: fixes x :: "'b"   and y :: "'b" shows "hausdorff_distance {x} {y} = dist x y"
rv_le_rv_measurable: fixes f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and g :: "'b \<Rightarrow> real" assumes "f \<in> rv M"   and "g \<in> rv M" shows "{w. f w \<le> g w} \<in> measurable_sets M"
convert_substNegInfinity: fixes var :: "nat"   and A :: "atom"   and xs' :: "real list"   and x :: "real"   and xs :: "real list"   and A' :: "atomUni" assumes "convert_atom var A (xs' @ x # xs) = Some A'"   and "length xs' = var" shows "eval (substNegInfinity var A) (xs' @ x # xs) = evalUni (substNegInfinityUni A') x"
tauChainAddTau_1: fixes P :: "pi"   and P' :: "pi"   and P'' :: "pi" assumes "P \<Longrightarrow>\<^sub>\<tau>\<^sub>l P'"   and "P' \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l P''" shows "P \<Longrightarrow>\<^sub>\<tau>\<^sub>l P''"
a_set_disconnected_nodes_def: fixes document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" shows "local.a_set_disconnected_nodes document_ptr disc_nodes = put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr disconnected_nodes_update disc_nodes"
r_append_def: shows "r_append \<equiv> let f = recf.Id 2 0; g = Cn 4 r_snoc [recf.Id 4 1, Cn 4 r_nth [recf.Id 4 3, recf.Id 4 0]] in Cn 2 (Pr 2 f g) [Cn 2 r_length [recf.Id 2 1], recf.Id 2 0, recf.Id 2 1]"
P_integrable: fixes a :: "real" assumes "0 < a" shows "P integrable_on {a..}"
bit_1_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit one n = (one \<noteq> zero \<and> n = 0)"
fv_ik_subset_fv_st': fixes S :: "('b, 'a) strand_step list" shows "fv\<^sub>s\<^sub>e\<^sub>t (ik\<^sub>s\<^sub>t S) \<subseteq> fv\<^sub>s\<^sub>t S"
eventually_inf_principal: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter"   and s :: "'a set" shows "eventually P (inf F (principal s)) = (\<forall>\<^sub>F x in F. x \<in> s \<longrightarrow> P x)"
WT_nsub_RI: fixes P :: "(char list list \<times> char list exp) prog"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty" assumes "P,E \<turnstile> e :: T" shows "\<not> sub_RI e"
labeled_bot_entailed: shows "entails_\<G> (snd (lhd Ns)) {(B, active)}"
pairwise_rel_prime_hyp: shows "\<forall>q. List.member qs q \<longrightarrow> coprime p q"
hom_minus: fixes x :: "'a poly"   and y :: "'a poly"   and p :: "'a poly" shows "(x - y) \<circ>\<^sub>p p = x \<circ>\<^sub>p p - y \<circ>\<^sub>p p"
getEnv_updEnv1: fixes ys :: "'varSort"   and xs :: "'varSort"   and y :: "'var"   and x :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" assumes "ys \<noteq> xs \<or> y \<noteq> x" shows "(rho [x \<leftarrow> X]_xs) ys y = rho ys y"