hcju/isabelleps huggingface dataset:
pre_well_ordered_set_def: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "pre_well_ordered_set A less_eq \<equiv> semiattractive A less_eq \<and> well_related_set A less_eq"
exec_gpv_restrict_gpv_invariant: fixes gpv :: "('d, 'a, 'b) gpv"   and s :: "'s1" assumes "\<I> \<turnstile>g gpv \<surd>"   and "I1 s" shows "exec_gpv oracle1 (restrict_gpv \<I> gpv) s = exec_gpv oracle1 gpv s"
assms_3: fixes U :: "'a set"   and V :: "'a set"   and x :: "'b" assumes "U \<in> I"   and "V \<in> I"   and "V \<subseteq> U"   and "x \<in> \<FF> U" shows "(\<psi> V \<circ> \<rho> U V) x = \<psi> U x"
omax_ordLeq: fixes R :: "('c \<times> 'c) set set"   and p :: "('d \<times> 'd) set" assumes "finite R"   and "R \<noteq> {}"   and "\<forall>r\<in>R. r \<le>o p" shows "omax R \<le>o p"
cupcake_pmatch_list_preserve: fixes ps :: "pat list"   and vs :: "v list"   and cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and env :: "(char list \<times> v) list" assumes "\<And>p v env. p \<in> set ps \<and> v \<in> set vs \<longrightarrow> list_all (is_cupcake_value \<circ> snd) env \<longrightarrow> if_match (list_all (is_cupcake_value \<circ> snd)) (cupcake_pmatch cenv p v env)"   and "list_all (is_cupcake_value \<circ> snd) env" shows "if_match (list_all (\<lambda>a. is_cupcake_value (snd a))) (Matching.fold2 (\<lambda>p v m. case m of No_match \<Rightarrow> No_match | Match_type_error \<Rightarrow> Match_type_error | Match env \<Rightarrow> cupcake_pmatch cenv p v env) Match_type_error ps vs (Match env))"
simp: shows "B \<bullet> a = subst_bv a (B $ Bv 0)"
arc_eq_1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and conv :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv"   and "(less_eq (times (times x top) (conv (times x top))) one \<and> times top (times x top) = top) \<and> less_eq (times (times (conv x) top) (conv (times (conv x) top))) one \<and> times top (times (conv x) top) = top" shows "x = times (times x (conv x)) x"
cases: fixes a1 :: "'c"   and a2 :: "'c list" assumes "ListMem a1 a2" obtains x :: "'c" and xs :: "'c list" where "a1 = xa__" and "a2 = xa__ # xsa__"   | x :: "'c" and xs :: "'c list" and y :: "'c" where "a1 = x" and "a2 = y # xs" and "ListMem x xs"
elem_min_elems: fixes c :: "'c"   and B :: "'c set" assumes "c \<in> min_elems B" shows "c \<in> B"
is_inf_greatest: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and i :: "'a"   and z :: "'a" assumes "dlat le"   and "dpo.is_inf le x y i"   and "le z x"   and "le z y" shows "le z i"
f_eq: shows "mod_type_class.from_nat x = mod_type_class.from_nat xa"
one_cinner_a_scaleC_one: fixes a :: "'b" shows "((1::'b) \<bullet>\<^sub>C a) *\<^sub>C (1::'b) = a"
IH: fixes i :: "nat" shows "take_nths i k xs = nths xs (\<Union>n. {n * 2 ^ k + i})"
length_bv_prepend: fixes n :: "nat"   and b :: "'a"   and l :: "'a list" shows "length (bv_prepend n b l) = n + length l"
assms: shows "\<exists>k. P k \<and> (\<forall>k'. P k' \<longrightarrow> k' \<le> k)"
IN: fixes \<Gamma> :: "('k, 'l) web_scheme"   and x :: "'k"   and f :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "weight \<Gamma> x \<le> d_IN f x" shows "x \<in> SAT \<Gamma> f"
assms_1: shows "run (Array.len a) \<sigma> \<sigma>' r"
simps_1: fixes k :: "int"   and ks :: "int list"   and x :: "nat"   and xs :: "nat list" shows "eval_dioph (k # ks) (x # xs) = k * int x + eval_dioph ks xs"
insertion_Const: fixes f :: "nat \<Rightarrow> 'a"   and c :: "'a" shows "insertion f (Const c) = c"
SN_on_mono: fixes r :: "('b \<times> 'b) set"   and A :: "'b set"   and s :: "('b \<times> 'b) set" assumes "SN_on r A"   and "s \<subseteq> r" shows "SN_on s A"
m_defsAll_2: fixes c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" shows "mC_C_part c d P F \<equiv> {} \<notin> P \<and> {} \<notin> F ` P \<and> part {..<brn c} P \<and> part {..<brn d} (F ` P)"
dens: shows "dens_ctxt_\<alpha> (vs, vs', \<Gamma>, \<delta>) \<turnstile>\<^sub>d e \<Rightarrow> (\<lambda>\<rho> x. ennreal (eval_cexpr f \<rho> x))"
fold_congs_6: fixes r :: "'a t_context_scheme"   and r' :: "'a t_context_scheme"   and v' :: "t list"   and f :: "t list \<Rightarrow> t list"   and f' :: "t list \<Rightarrow> t list" assumes "r = r'"   and "local r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "local_update f r = local_update f' r'"
borel_def: fixes "open" :: "'a set \<Rightarrow> bool" assumes "class.topological_space open" shows "topological_space.borel open = sigma UNIV {S. open S}"
is_universal: fixes a' :: "'b"   and \<chi>' :: "'j \<Rightarrow> 'b" assumes "\<chi>a.D.cone a' \<chi>'" shows "\<exists>!f. \<guillemotleft>f : a' \<rightarrow>\<^sub>B l a\<guillemotright> \<and> \<chi>a.D.cones_map f (\<chi> a) = \<chi>'"
monotone_epred: shows "monotone (\<le>) (\<le>) epred"
\<alpha>_def: shows "\<alpha> = the_default 0 \<circ>\<circ> map_of"
exponent_denormal_of_Float: fixes x :: "Float.float" shows "IEEE.exponent (denormal_of_Float x) = 0"
to_F_in: shows "to_F \<iota> \<in> no_labels.Red_I_\<G> (fst ` N)"
seq_conv_if: fixes x :: "'c" shows "seq\<cdot>x = (if x = \<bottom> then \<bottom> else ID)"
poly_mult_in_carrier: fixes p1 :: "'a list"   and p2 :: "'a list" assumes "set p1 \<subseteq> carrier R"   and "set p2 \<subseteq> carrier R" shows "set (poly_mult p1 p2) \<subseteq> carrier R"
poincare_distance_refl: fixes u :: "complex_homo" assumes "u \<in> unit_disc" shows "poincare_distance u u = 0"
hyps_1: fixes p :: "'f poly" assumes "n = degree p"   and "p \<noteq> 0"   and "\<not> inj_on (polymap p) V" shows "\<exists>c. \<not> inj_on (polymap [:- c, 1::'f:]) V"
termination_variant: fixes guard :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> nat"   and bound :: "nat"   and p :: "real"   and body :: "'a \<Rightarrow> 'a spmf"   and s :: "'a" assumes "\<And>s. guard s \<Longrightarrow> f s \<le> bound"   and "\<And>s. guard s \<Longrightarrow> p \<le> spmf (map_spmf (\<lambda>s'. f s' < f s) (body s)) True"   and "0 < p"   and "\<And>s. guard s \<Longrightarrow> lossless_spmf (body s)" shows "lossless_spmf (loop_spmf.while guard body s)"
unit_simps_2: fixes a :: "'a" assumes "S.obj a" shows "S.src \<i>[a] = a"
sel_4: fixes x1 :: "real"   and x2 :: "real"   and x3 :: "real"   and x4 :: "real" shows "Im3 (Quat x1 x2 x3 x4) = x4"
function_zero_eval: fixes a :: "'a" assumes "a \<in> carrier R" shows "\<zero>\<^bsub>function_ring (carrier R) R\<^esub> a = \<zero>"
star_inductl_var_equiv: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>\<star> \<cdot> y \<le> y) = (x \<cdot> y \<le> y)"
intros_13: fixes \<AA> :: "'b formula"   and \<Gamma> :: "'b formula list"   and \<Theta> :: "'b formula list"   and \<BB> :: "'b formula" assumes "\<AA> # \<Gamma> \<Rightarrow> \<Theta>"   and "\<BB> # \<Gamma> \<Rightarrow> \<Theta>" shows "\<AA> \<^bold>\<or> \<BB> # \<Gamma> \<Rightarrow> \<Theta>"
Confidentiality_Kas: fixes A :: "agent"   and Ka :: "nat"   and authK :: "nat"   and Ta :: "nat"   and authTicket :: "msg"   and evs :: "event list" assumes "Says Kas A (Crypt Ka \<lbrace>Key authK, Agent Tgs, Number Ta, authTicket\<rbrace>) \<in> set evs"   and "\<not> expiredAK Ta evs"   and "A \<notin> bad"   and "evs \<in> kerbIV" shows "Key authK \<notin> analz (knows Spy evs)"
induct: fixes x1 :: "nat"   and x2 :: "('c, 'd) bintree list"   and P :: "nat \<Rightarrow> ('c, 'd) bintree list \<Rightarrow> bool" assumes "is_bintree_list x1 x2"   and "P 0 []"   and "\<And>l ts t. \<lbrakk>is_bintree_list l ts; P l ts; is_bintree_list l (children t); P l (children t)\<rbrakk> \<Longrightarrow> P (Suc l) (t # ts)" shows "P x1 x2"
simps_27: fixes f1 :: "'a"   and f2 :: "bool \<Rightarrow> 'a"   and f3 :: "int \<Rightarrow> 'a"   and f4 :: "real \<Rightarrow> 'a"   and f5 :: "val \<Rightarrow> val \<Rightarrow> 'a"   and x3 :: "int" shows "(case IntVal x3 of UnitVal \<Rightarrow> f1 | BoolVal x \<Rightarrow> f2 x | IntVal x \<Rightarrow> f3 x | RealVal x \<Rightarrow> f4 x | <|x, xa|> \<Rightarrow> f5 x xa) = f3 x3"
times_right_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and z :: "'a"   and y :: "'a" assumes "SN_Orders.class.ordered_semiring_0 plus zero times less_eq"   and "less_eq zero x"   and "less_eq z y" shows "less_eq (times x z) (times x y)"
Rep_inject: fixes Rep :: "'f \<Rightarrow> 'g"   and Abs :: "'g \<Rightarrow> 'f"   and A :: "'g set"   and x :: "'f"   and y :: "'f" assumes "type_definition Rep Abs A" shows "(Rep x = Rep y) = (x = y)"
supp_def: fixes x :: "'e" shows "supp x = {a. infinite {b. [(a, b)] \<bullet> x \<noteq> x}}"
rel_spmf_weightD: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c spmf"   and q :: "'d spmf" assumes "rel_spmf A p q" shows "weight_spmf p = weight_spmf q"
is_path_undir_split: fixes G :: "('c, 'd) graph"   and v :: "'c"   and p1 :: "('c \<times> 'd \<times> 'c) list"   and p2 :: "('c \<times> 'd \<times> 'c) list"   and v' :: "'c" assumes "valid_graph G" shows "is_path_undir G v (p1 @ p2) v' = (\<exists>u. is_path_undir G v p1 u \<and> is_path_undir G u p2 v')"
nu_meet: fixes x :: "'a"   and y :: "'a" shows "\<nu> (x \<sqinter> y) = \<nu> x \<sqinter> \<nu> y"
counit_simps_2: shows "local.dom \<psi> = f \<star> g"
underS_Field: fixes i :: "'c"   and R :: "('c \<times> 'c) set"   and j :: "'c" assumes "i \<in> underS R j" shows "i \<in> Field R"
6: shows "top * e \<le> top * e * (w\<^sup>T \<sqinter> - v * - v\<^sup>T \<sqinter> w\<^sup>\<star> * e\<^sup>T * top)\<^sup>\<star>"
assms_4: shows "\<not> (\<exists>v\<in>V. degree v G = 2)"
interleave_rev2: fixes w :: "'b list"   and w1 :: "'b list"   and w2 :: "'b list" assumes "rev w \<in> rev w1 \<otimes> rev w2" shows "w \<in> w1 \<otimes> w2"
B_2: shows "\<forall>x. norm (g' x) \<le> norm x * B"
sum_eq_0_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and F :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "finite F" shows "(comm_monoid_add.sum plus zero f F = zero) = (\<forall>a\<in>F. f a = zero)"
wf_types_clinit: fixes wf_md :: "(char list list \<times> char list exp) prog \<Rightarrow> char list \<Rightarrow> (char list list \<times> char list exp) mdecl \<Rightarrow> bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and a :: "(char list list \<times> char list exp) class"   and sh :: "sheap"   and sfs :: "sfields"   and h :: "heap"   and E :: "char list \<Rightarrow> ty option" assumes "wf_prog wf_md P"   and "class P C = \<lfloor>a\<rfloor>"   and "sh C = \<lfloor>(sfs, Processing)\<rfloor>" shows "P,E,h,sh \<turnstile> C\<bullet>\<^sub>sclinit([]) : Void"
exr: shows "x \<in> set (map (consistent_sign_vec_copr (qs1 @ qs2)) (characterize_root_list_p p)) \<longrightarrow> (\<exists>r\<in>set (characterize_root_list_p p). x = consistent_sign_vec_copr (qs1 @ qs2) r)"
returns_result_eq: fixes h :: "'e"   and f :: "('e, 'f, 'g) prog"   and y :: "'g"   and y' :: "'g" assumes "h \<turnstile> f \<rightarrow>\<^sub>r y"   and "h \<turnstile> f \<rightarrow>\<^sub>r y'" shows "y = y'"
mismatch_ext_left: fixes u :: "'b list"   and v :: "'b list"   and p :: "'b list" assumes "\<not> u \<bowtie> v" shows "mismatch_pair u v = mismatch_pair (p \<cdot> u) (p \<cdot> v)"
axioms_2: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and alt :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_state_alt return bind get put alt" shows "monad_alt return bind alt"
axioms_4: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs get_disconnected_nodes get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs" shows "l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs"
detTAG_def: fixes x :: "'c" shows "detTAG x \<equiv> x"
polyhedron_imp_convex: fixes S :: "'b set" assumes "polyhedron S" shows "convex S"
compl_fam_imp_generate_inj: fixes gs :: "'a set" assumes "gs \<subseteq> carrier G"   and "compl_fam (\<lambda>g. generate G {g}) gs" shows "inj_on (\<lambda>g. generate G {g}) gs"
d: fixes k :: "nat" assumes "k < n" shows "f k \<in> carrier_mat d d"
QgrpUnit_4: fixes G :: "('e, 'f) Group_scheme"   and E :: "('g, 'h) Group_scheme"   and H :: "'e set"   and H1 :: "'e set" assumes "Group G"   and "Ugp E"   and "G \<guillemotright> H "   and "G \<guillemotright> H1 "   and "\<natural>\<^bsub>G\<^esub>H \<triangleright> H1"   and "\<not> (\<natural>\<^bsub>G\<^esub>H / H1) \<cong> E" shows "H \<noteq> H1"
map_simp: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>C.\<ll>' f = \<F>C.inv (\<F>C.\<ll> (\<F>C.cod f)) \<cdot> \<F>C.map f"
Weierstrass_m_test: fixes A :: "'b set"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and M :: "nat \<Rightarrow> real" assumes "\<And>n x. x \<in> A \<Longrightarrow> norm (f n x) \<le> M n"   and "summable M" shows "uniform_limit A (\<lambda>n x. \<Sum>i<n. f i x) (\<lambda>x. \<Sum>i. f i x) sequentially"
continuous_on_g: fixes s :: "real set"   and n :: "nat" shows "continuous_on s (g n)"
algebraic_inverse: fixes x :: "'a" assumes "algebraic x" shows "algebraic (inverse x)"
nilTrans_2: fixes \<Psi> :: "'b"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P :: "('a, 'b, 'c) psi"   and K :: "'a"   and yvec :: "name list"   and N' :: "'a"   and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> M\<lparr>\<lambda>*xvec N\<rparr>.P \<longmapsto> K\<lparr>\<nu>*yvec\<rparr>\<langle>N'\<rangle> \<prec> P'" shows "False"
einterval_iff: fixes x :: "real"   and a :: "ereal"   and b :: "ereal" shows "(x \<in> einterval a b) = (a < ereal x \<and> ereal x < b)"
foundation3: fixes \<tau> :: "'d state \<times> 'd state" shows "\<tau> |\<noteq> invalid"
load_tmps_read_tmps_distinct_def: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" shows "load_tmps_read_tmps_distinct ts \<equiv> \<forall>i p is \<O> \<R> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> load_tmps is \<inter> read_tmps sb = {}"
assms_4: shows "n = vcard (xs #\<^sub>\<circ> x)"
moebius_rotation_def: fixes \<phi> :: "real" shows "moebius_rotation \<phi> = moebius_similarity (cis \<phi>) 0"
embed_measure_count_space: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "inj f" shows "embed_measure (count_space A) f = count_space (f ` A)"
fix_index': fixes r :: "'c"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and i :: "nat"   and j :: "nat"   and n :: "nat"   and v :: "'d \<Rightarrow> nat" assumes "Le r \<le> M i j"   and "Le (- r) \<le> M j i"   and "cycle_free M n"   and "canonical M n"   and "i \<le> n"   and "j \<le> n"   and "i \<noteq> j" shows "(\<forall>u. u \<turnstile>\<^bsub>v,n\<^esub> \<lambda>i' j'. if i' = i \<and> j' = j then Le r else if i' = j \<and> j' = i then Le (- r) else M i' j' \<longrightarrow> u \<turnstile>\<^bsub>v,n\<^esub> M) \<and> cycle_free (\<lambda>i' j'. if i' = i \<and> j' = j then Le r else if i' = j \<and> j' = i then Le (- r) else M i' j') n"
not_shadows: shows "fBall rs (\<lambda>(uu_, rhs). \<not> shadows_consts rhs)"
that: fixes B :: "'a zmultiset" assumes "A = add_zmset a B" shows "thesis"
bottom_left_bottom: fixes a :: "'a" shows "lcm (0::'a) a = (0::'a)"
lemma_2_10_24: fixes a :: "'a"   and b :: "'a" shows "a \<le> (a l\<rightarrow> b) r\<rightarrow> b"
cross7_simps_9: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "b + (a + c) = a + (b + c)"
assms: shows "\<forall>v\<in>vars_lit L. \<sigma> v = \<tau> v"
boundedI: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<le> b"   and "a \<le> c" shows "a \<le> inf b c"
singleton_orbits: shows "\<Union> {N \<in> conjP.orbits. card N = 1} = conjP.fixed_points"
main_part_sets: fixes S :: "'b set"   and M :: "'b measure" assumes "S \<in> sets (completion M)" shows "main_part M S \<in> sets M"
that: fixes enum :: "'a list"   and enum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool" assumes "class.enum enum enum_all enum_ex \<and> n = CARD('a)" shows "thesis"
init_tr_def: shows "init_tr = (\<lambda>(bd, as). make_tr (\<lambda>i. make_tr (\<lambda>j. as ! i \<noteq> as ! j) i 0) (length bd - 1) 1)"
root_opt_bst_wpl2_bound: fixes i :: "int"   and j :: "int" assumes "opt_bst_wpl2_dom (i, j)"   and "i \<le> j" shows "root (fst (opt_bst_wpl2 i j)) \<in> {i..j}"
proots_half_def: fixes p :: "complex poly"   and a :: "complex"   and b :: "complex" shows "proots_half p a b = proots_count p {w. 0 < Im ((w - a) / (b - a))}"
isCont_scaleR: fixes a :: "'c"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> 'd" assumes "isCont f a"   and "isCont g a" shows "isCont (\<lambda>x. f x *\<^sub>R g x) a"
Fake_analz_insert: fixes X :: "msg"   and G :: "msg set"   and H :: "msg set" assumes "X \<in> synth (analz G)" shows "analz (insert X H) \<subseteq> synth (analz G) \<union> analz (G \<union> H)"
convolution_emeasure_3': fixes A :: "'a set"   and M :: "'a measure"   and N :: "'a measure"   and L :: "'a measure" assumes "A \<in> sets borel"   and "finite_measure M"   and "finite_measure N"   and "finite_measure L"   and "sets N = sets borel"   and "sets M = sets borel"   and "sets L = sets borel" shows "emeasure ((L \<star> M) \<star> N) A = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. \<integral>\<^sup>+ z. indicator A (x + y + z) \<partial>N \<partial>M \<partial>L"
ipurge_fail_aux_t_def: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and xs :: "'b list"   and X :: "'b set" shows "ipurge_fail_aux_t I D U xs X \<equiv> ipurge_fail_aux_t_out (ipurge_fail_aux_t_aux (ipurge_fail_aux_t_in I D U xs X))"
reds: shows "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>1\<rangle> \<rightarrow>* \<langle>e\<^sub>2,(h\<^sub>2, l\<^sub>2)\<rangle>"
OrdP_cases_lemma: fixes p :: "name"   and x :: "tm" assumes "atom p \<sharp> x" shows "{OrdP x, Neg (x EQ Zero)} \<turnstile> SyntaxN.Ex p (OrdP (Var p) AND x EQ SUCC (Var p))"
num_params_tm_const: fixes c :: "float" shows "num_params (tm_poly (tm_const c)) = 0"
card_of_Sigma_mono1: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set"   and B :: "'a \<Rightarrow> 'c set" assumes "\<forall>i\<in>I. |A i| \<le>o |B i|" shows "|Sigma I A| \<le>o |Sigma I B|"
simps_3: fixes dims :: "nat list"   and S1 :: "com"   and S2 :: "com" shows "state_sig.well_com dims (S1;; S2) = (state_sig.well_com dims S1 \<and> state_sig.well_com dims S2)"
response_def: fixes r :: "nat"   and w :: "nat"   and e :: "nat" shows "response r w e = return_spmf ((w * e + r) mod order \<G>)"
pairs_sup_dist_inf: fixes phi :: "'c \<Rightarrow> 'd Filters.filter"   and x :: "'c"   and y :: "'d Filters.filter"   and z :: "'c"   and w :: "'d Filters.filter"   and u :: "'c"   and v :: "'d Filters.filter" assumes "triple phi"   and "(x, y) \<in> triple.pairs phi"   and "(z, w) \<in> triple.pairs phi"   and "(u, v) \<in> triple.pairs phi" shows "triple.pairs_sup (x, y) (triple.pairs_inf (z, w) (u, v)) = triple.pairs_inf (triple.pairs_sup (x, y) (z, w)) (triple.pairs_sup (x, y) (u, v))"
inject: fixes x1 :: "('e \<times> 'f \<times> 'f) set"   and x2 :: "'f set"   and y1 :: "('e \<times> 'f \<times> 'f) set"   and y2 :: "'f set" shows "(LG x1 x2 = LG y1 y2) = (x1 = y1 \<and> x2 = y2)"
1: shows "arr f \<and> arr g \<and> local.dom g = cod f"
2: shows "|pickE ` Rho| <o |UNIV| \<or> finite (pickE ` Rho)"
simps_2: fixes k :: "nat"   and ts :: "('b btree \<times> 'b) list"   and t :: "'b btree" shows "root_order k (Node ts t) = (0 < length ts \<and> length ts \<le> 2 * k \<and> Ball (set (subtrees ts)) (order k) \<and> order k t)"
aPP: shows "adjoint P * P = 1\<^sub>m n"
mkarr_rep: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>C.mkarr (\<F>C.rep f) = f"
TAtom_term_cases: fixes t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and \<tau> :: "'atom" assumes "wf\<^sub>t\<^sub>r\<^sub>m t"   and "\<Gamma> t = Var \<tau>" shows "(\<exists>v. t = Var v) \<or> (\<exists>f. t = Fun f [])"
vcgaEquivalence: fixes \<Omega> :: "'c list"   and N :: "integer set"   and b :: "integer \<times> 'c set \<Rightarrow> 'd"   and r :: "integer" assumes "distinct \<Omega>"   and "set \<Omega> \<noteq> {}"   and "finite N" shows "the_elem (Outside' {seller} ` (argmax \<circ> sum) (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel (N \<union> {seller}) (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg (N \<union> {seller}) \<Omega>)) r) <| ((N \<union> {seller}) \<times> finestpart (set \<Omega>))) (N \<union> {seller}) (set \<Omega>)) ((argmax \<circ> sum) b (allAllocations (N \<union> {seller}) (set \<Omega>)))) = Outside' {seller} (the_elem (argmax (sum (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel (N \<union> {seller}) (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg (N \<union> {seller}) \<Omega>)) r) <| ((N \<union> {seller}) \<times> finestpart (set \<Omega>))) (N \<union> {seller}) (set \<Omega>))) (maximalStrictAllocations N (set \<Omega>) b)))"
nth_append: fixes xs :: "'a list"   and ys :: "'a list"   and n :: "nat" shows "(xs @ ys) ! n = (if n < length xs then xs ! n else ys ! (n - length xs))"
Up: shows "Up \<equiv> \<lambda>(s\<^sub>1, s\<^sub>2) e. (Up\<^sub>1 s\<^sub>1 e, Up\<^sub>2 s\<^sub>2 e)"
subset: shows "{x. pmf q' x < pmf p' x} \<subseteq> set_pmf p'"
assms_3: shows "\<D>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A ^\<^sub>\<times> n"
assms_2: shows "Q2 = newQuery J Q1 (I, t)"
is_increasing_imp_geq_id: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "is_increasing f" shows "n \<le> f n"
Called_context_Called_set: fixes P :: "jvm_prog"   and D :: "char list"   and i :: "instr" assumes "Called_context P D i" shows "i \<in> Called_set"
ide_cod: fixes f :: "'a" assumes "S.arr f" shows "S.ide (S.cod f)"