hcju/isabelleps huggingface dataset:
executable_def: fixes ss :: "(integer \<times> edge list) iarray iarray"   and g :: "gState\<^sub>I" shows "executable ss g = (let procs = procs g in nfoldli procs (\<lambda>_. True) (\<lambda>p E. if exclusive g = 0 \<or> exclusive g = pid p then let (min_pri, edges) = ss !! pState.idx p !! pc p in ASSERT (set edges \<subseteq> edgeSet (ss !! pState.idx p)) \<bind> (\<lambda>_. (if min_pri = 0 then WHILE\<^sub>T (\<lambda>(E, brk, uu_). E = [] \<and> brk = 0) (\<lambda>(uu_, uu_, ELSE). let g = g\<lparr>else := ELSE\<rparr>; E = getChoices g p edges in if E = [] then if \<not> ELSE then RETURN (E, 0, True) else RETURN (E, 1, False) else RETURN (E, 1, ELSE)) ([], 0, False) else let min_pri = nat_of_integer \<bar>min_pri\<bar>; pri_edges = sort_by_pri min_pri edges in ASSERT (\<forall>es\<in>set pri_edges. set es \<subseteq> edgeSet (ss !! pState.idx p)) \<bind> (\<lambda>_. let pri_edges = IArray pri_edges in WHILE\<^sub>T (\<lambda>(E, pri, uu_). E = [] \<and> pri \<le> min_pri) (\<lambda>(uu_, pri, ELSE). let es = pri_edges !! pri; g = g\<lparr>else := ELSE\<rparr>; E = getChoices g p es in if E = [] then if \<not> ELSE then RETURN (E, pri, True) else RETURN (E, pri + 1, False) else RETURN (E, pri, ELSE)) ([], 0, False))) \<bind> (\<lambda>(E', uu_, uu_). RETURN (E' @ E))) else RETURN E) [])"
suffs_cond2I: fixes aaa :: "nat list"   and a :: "nat list"   and b :: "nat list" assumes "\<forall>y\<in>set aaa. y \<le> Max (set a)"   and "length aaa = length b"   and "subdprodr_impl a b aaa" shows "suffs (cond2 a b) b (aaa, b \<bullet> aaa)"
Greatest_is_minus_1: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "\<forall>a. less_eq a (uminus one)"
BH_3: shows "B =m \<Prod>\<^sub># (factors_of_factor_tree r)"
field_differentiable_power: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and S :: "'a set"   and n :: "nat" assumes "f field_differentiable at a within S" shows "(\<lambda>z. f z ^ n) field_differentiable at a within S"
traversed_states_rem_condless_act: fixes s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "traversed_states s (rem_condless_act s [] as) = traversed_states s as"
of_real_in_Ints_iff: fixes x :: "real" shows "(of_real x \<in> \<int>) = (x \<in> \<int>)"
AXIOMS_imply_SNa_2: fixes x :: "name"   and M :: "trm"   and B :: "ty" assumes "x:M \<in> AXIOMSn B" shows "SNa M"
Deq_visible: fixes ic :: "state \<Rightarrow> 'b"   and q :: "state \<Rightarrow> 'b list"   and oc :: "state \<Rightarrow> 'b" shows "\<turnstile> <Deq ic q oc>_(ic, q, oc) = Deq ic q oc"
superclasses_rec: fixes G :: "prog"   and C :: "qtname"   and c :: "class" assumes "class G C = Some c"   and "ws_prog G" shows "superclasses G C = (if C = Object then {} else insert (super c) (superclasses G (super c)))"
surjective: fixes r :: "'a decl_scheme" shows "r = \<lparr>access = access r, \<dots> = decl.more r\<rparr>"
split_minus: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and x :: "'b \<times> 'c"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'a" shows "(case x of (x, k) \<Rightarrow> f x k) - (case x of (x, k) \<Rightarrow> g x k) = (case x of (x, k) \<Rightarrow> f x k - g x k)"
edge_preserving_subset: fixes R\<^sub>1 :: "('a \<times> 'b) set"   and R\<^sub>2 :: "('a \<times> 'b) set"   and E\<^sub>1 :: "('c \<times> 'a \<times> 'a) set"   and E\<^sub>2 :: "('c \<times> 'a \<times> 'a) set"   and G :: "('c \<times> 'b \<times> 'b) set" assumes "R\<^sub>1 \<subseteq> R\<^sub>2"   and "E\<^sub>1 \<subseteq> E\<^sub>2"   and "edge_preserving R\<^sub>2 E\<^sub>2 G" shows "edge_preserving R\<^sub>1 E\<^sub>1 G"
g_3_def: fixes x :: "int \<times> int" shows "g_3 x = fst x + (snd x - 1) * int p"
sel_rews_3: fixes cfun :: "'a \<rightarrow> ('b \<times> 'a)\<cdot>'c" shows "runStateT\<cdot>(StateT\<cdot>cfun) = cfun"
rep_eq: fixes x :: "mini_alg_unique"   and xa :: "mini_alg_unique" shows "Rep_mini_alg_unique (mau_plus x xa) = ma_plus (Rep_mini_alg_unique x) (Rep_mini_alg_unique xa)"
assms_1: shows "a \<in>\<^sub>\<circ> A"
liveVal'I: fixes g :: "'g"   and v :: "'val" assumes "liveVal g v" obtains vs :: "'val list" where "liveVal' g (v # vs)"
comp_in_homE: fixes g :: "'b"   and f :: "'b"   and a :: "'b"   and c :: "'b" assumes "\<guillemotleft>g \<cdot>\<^sub>D f : a \<Rightarrow>\<^sub>D c\<guillemotright>" obtains b :: "'b" where "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>" and "\<guillemotleft>g : b \<Rightarrow>\<^sub>D c\<guillemotright>"
F: shows "(xs @ y # ys, Y) \<in> seq_comp_failures P Q"
mem_correct_rbt_empty_mapping: shows "mem_correct_empty (\<lambda>k. State_Monad.get \<bind> (\<lambda>m. State_Monad.return (Mapping.lookup m k))) (\<lambda>k v. State_Monad.get \<bind> (\<lambda>m. State_Monad.set (Mapping.update k v m))) (\<lambda>_. True) Mapping.empty"
simps_3: fixes I :: "('b, 'c, 'a) interp"   and f :: "'b"   and args :: "'a \<Rightarrow> ('b, 'a) trm" shows "dterm_sem I ($f f args) = (\<lambda>v. Functions I f (\<chi>i. dterm_sem I (args i) v))"
continuous_on_product_then_coordinatewise: fixes S :: "'d set"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and i :: "'e" assumes "continuous_on S f" shows "continuous_on S (\<lambda>x. f x i)"
Compl_Int: fixes A :: "'c set"   and B :: "'c set" shows "- (A \<inter> B) = - A \<union> - B"
simps_2: fixes l :: "'a btree"   and a :: "'a"   and r :: "'a btree" shows "tree\<^sub>i (Up\<^sub>i l a r) = Node [(l, a)] r"
intro: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c" assumes "Group.group G"   and "Group.group H"   and "group_hom_axioms G H h" shows "group_hom G H h"
partitioning_foldr: fixes X :: "'b set list"   and B :: "'b set set" shows "partitioning X B = foldr addSubsetSet X B"
flow_undefined0: fixes t :: "real"   and x :: "'a" assumes "t \<notin> rev.existence_ivl0 x" shows "rev.flow0 x t = (0::'a)"
post_del_def: fixes t :: "('c \<times> nat) tree"   and t' :: "('d \<times> nat) tree" shows "post_del t t' \<equiv> invar t' \<and> (lvl t' = lvl t \<or> lvl t' + 1 = lvl t) \<and> (lvl t' = lvl t \<and> sngl t \<longrightarrow> sngl t')"
snd_Gauss_Jordan_in_ij_iarrays_PA: fixes P :: "'d iarray iarray"   and A :: "'d iarray iarray"   and i :: "nat"   and j :: "nat" shows "snd (Gauss_Jordan_in_ij_iarrays_PA (P, A) i j) = Gauss_Jordan_in_ij_iarrays A i j"
pr_dup_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "elementary_category_with_binary_products C pr0 pr1"   and "partial_magma.ide C a" shows "C (pr0 a a) (elementary_category_with_binary_products.tuple C pr0 pr1 a a) = a"
initmissing_def': shows "initmissing = apfst (Lib.default someinit)"
induct: fixes P :: "('b \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b nres) \<Rightarrow> 'b \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'b \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b nres"   and a1 :: "'b"   and a2 :: "nat"   and a3 :: "nat"   and a4 :: "nat" assumes "\<And>f a n. P f a n 0 0"   and "\<And>f a n i. P f a n i n \<Longrightarrow> P f a n (Suc i) 0"   and "\<And>f a n i j. P f a n i j \<Longrightarrow> P f a n i (Suc j)" shows "P a0 a1 a2 a3 a4"
smult_add_right: fixes a :: "'a"   and p :: "'a poly"   and q :: "'a poly" shows "Polynomial.smult a (p + q) = Polynomial.smult a p + Polynomial.smult a q"
epsclo_Un: fixes Q1 :: "hf set"   and Q2 :: "hf set" shows "epsclo (Q1 \<union> Q2) = epsclo Q1 \<union> epsclo Q2"
rbt_sorted_entries: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "('c, 'd) rbt" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t" shows "linorder.sorted less_eq (map fst (RBT_Impl.entries t))"
stronger_PiE_finprod_imp: fixes A :: "'a set" assumes "A \<subseteq> carrier G"   and "\<forall>f\<in>\<Pi>\<^sub>E a\<in>A. generate G {a}. finprod G f A = \<one> \<longrightarrow> (\<forall>a\<in>A. f a = \<one>)" shows "\<forall>f\<in>Pi\<^sub>E ((\<lambda>a. generate G {a}) ` A) id. finprod G f ((\<lambda>a. generate G {a}) ` A) = \<one> \<longrightarrow> (\<forall>H\<in>(\<lambda>a. generate G {a}) ` A. f H = \<one>)"
rtrancl_into_trancl1: fixes a :: "'g"   and b :: "'g"   and r :: "('g \<times> 'g) set"   and c :: "'g" assumes "(a, b) \<in> r\<^sup>*"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>+"
mat_row_carrierI: fixes nr :: "nat"   and nc :: "nat"   and r :: "nat \<Rightarrow> 'b vec" shows "mat\<^sub>r nr nc r \<in> carrier_mat nr nc"
Deterministic_trans_fun_imp_acc_trans_fun: fixes trans_fun :: "'d message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and localState :: "'e \<Rightarrow> 'f"   and k :: "nat"   and pointwise_shrink :: "'e list \<Rightarrow> 'e" assumes "Deterministic_Trans_Fun trans_fun localState" shows "Deterministic_Trans_Fun (Acc_Trans_Fun_Step k trans_fun pointwise_shrink) localState"
m_o2: fixes h :: "nat"   and m :: "'a \<Rightarrow> nat"   and X :: "char list set"   and o1 :: "(char list \<Rightarrow> 'a) option"   and o2 :: "(char list \<Rightarrow> 'a) option" assumes "Measure_fun h m"   and "finite X"   and "Measure_fun.top_on_opt o1 (- X)"   and "Measure_fun.top_on_opt o2 (- X)"   and "o1 < o2" shows "Measure1_fun.m_o m h o2 X < Measure1_fun.m_o m h o1 X"
grey_protects_white_def: fixes g :: "'d"   and w :: "'d" shows "g grey_protects_white w = (\<lambda>s. grey g s \<and> (g has_white_path_to w) s)"
T1: fixes x :: "'a"   and y :: "'a" assumes "R1 x y" shows "T1 x (Abs1 y)"
simps_2: fixes p :: "'a \<times> 'b"   and ps :: "('a \<times> 'b) list" shows "AList.clearjunk (p # ps) = p # AList.clearjunk (AList.delete (fst p) ps)"
OrE: fixes G :: "('a, 'b) form list"   and a :: "('a, 'b) form"   and b :: "('a, 'b) form"   and c :: "('a, 'b) form" assumes "G \<turnstile> Or a b"   and "a # G \<turnstile> c"   and "b # G \<turnstile> c" shows "G \<turnstile> c"
mapl_F_def: fixes l1 :: "'t \<Rightarrow> 'ad"   and l2 :: "'u \<Rightarrow> 'ae"   and l3 :: "'v \<Rightarrow> 'af" shows "mapl_F l1 l2 l3 = map_F l1 l2 l3 id id id id id id"
simps_1: fixes \<theta> :: "trm"   and \<eta> :: "trm" shows "Timeso (Aterm \<theta>) (Aterm \<eta>) = Aterm (Times \<theta> \<eta>)"
assms: shows "\<forall>M s. M, s \<Turnstile> p"
tM'T: shows "(\<Union> (range (relations M')))\<^sup>* `` {t} \<subseteq> T"
arev_eq_iff: fixes x :: "'b"   and y :: "'b" shows "(rev_H' x = rev_H' y) = (x = y)"
whisker_left: fixes f :: "'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "ide f"   and "seq \<nu> \<mu>" shows "f \<star> \<nu> \<cdot> \<mu> = (f \<star> \<nu>) \<cdot> (f \<star> \<mu>)"
countably_additive_def: fixes M :: "'b set set"   and f :: "'b set \<Rightarrow> ennreal" shows "countably_additive M f = (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> \<Union> (range A) \<in> M \<longrightarrow> (\<Sum>i. f (A i)) = f (\<Union> (range A)))"
d_rest_ax: fixes x :: "'a" shows "d x \<cdot> x = x"
shext_upd_obj: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and i :: "init_state"   and i' :: "init_state"   and sfs' :: "sfields" assumes "sh C = \<lfloor>(sfs, i)\<rfloor>"   and "i \<le>\<^sub>i i'" shows "sh \<unlhd>\<^sub>s sh(C \<mapsto> (sfs', i'))"
space_closed: fixes i :: "nat" shows "A i \<subseteq> Pow (space M)"
arr_dom_iff_arr: fixes f :: "'b" shows "B.arr (B.dom f) = B.arr f"
longest_common_prefix: fixes xs :: "'c list"   and ys :: "'c list" shows "\<exists>ps xs' ys'. xs = ps @ xs' \<and> ys = ps @ ys' \<and> (xs' = [] \<or> ys' = [] \<or> hd xs' \<noteq> hd ys')"
P_def: shows "P = (\<lambda>t. Re (g t - z) = 0)"
step_None: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list" shows "(step e s r l i = None) = (possible_steps e s r l i = {||})"
fls_lr_inverse_X_intpow_1: fixes i :: "int"   and x :: "'a" shows "fls_left_inverse (fls_X_intpow i) x = fls_shift i (fls_const x)"
qTermLess_modulo_def: fixes rel :: "(('d, 'e, 'f, 'g, 'h) qTerm \<times> ('d, 'e, 'f, 'g, 'h) qTerm) set" shows "qTermLess_modulo rel \<equiv> {uu_. \<exists>X delta inp binp i. uu_ = (termIn X, termIn (qOp delta inp binp)) \<and> inp i = Some X} \<union> {uu_. \<exists>A delta inp binp j. uu_ = (absIn A, termIn (qOp delta inp binp)) \<and> binp j = Some A} \<union> {(termIn Y, absIn (qAbs xs x X)) |X Y xs x. (X, Y) \<in> rel}"
il_unit_bot: fixes x :: "'a" shows "bot \<squnion> x = x"
pi: fixes e :: "real" assumes "0 < e" shows "\<exists>p. winding_number_prop \<gamma> z e p n"
not_knows_elem_not_knows_conc: fixes P :: "specID"   and a :: "Expression"   and e :: "Expression list" assumes "\<not> knows P [a]" shows "\<not> knows P (a # e)"
l_type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_axioms: shows "l_type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a type_wf"
is_fair'_def: fixes s :: "nat \<Rightarrow> 'a" shows "is_fair' s \<equiv> \<forall>l i. \<exists>j\<ge>i. \<not> can_step l (s j) \<or> rstep l s j"
1: shows "e * top \<le> w\<^sup>T\<^sup>\<star> * d\<^sup>T * top"
is_dghmD_12: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<D>\<^sub>\<circ> (\<FF>\<lparr>ArrMap\<rparr>) = \<AA>\<lparr>Arr\<rparr>"
fin_dim_span_cols: fixes A :: "'a mat"   and nc :: "nat" assumes "A \<in> carrier_mat n nc" shows "vectorspace.fin_dim class_ring (vs (local.span (set (cols A))))"
set_eq_subset: fixes A :: "'f set"   and B :: "'f set" shows "(A = B) = (A \<subseteq> B \<and> B \<subseteq> A)"
swap_comp_swap: shows "prod.swap \<circ> prod.swap = id"
disjoint: shows "{0..<Suc k} \<inter> {Suc (Suc k)..<n} = {}"
simps_1: 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" shows "is_subobj P (C, []) = False"
extensional_arb: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and x :: "'d" assumes "f \<in> extensional A"   and "x \<notin> A" shows "f x = undefined"
in_open_segment_iff_rot: fixes a :: "'a"   and b :: "'a"   and x :: "'a" assumes "a \<noteq> b" shows "(x \<in> {a<--<b}) = ((x - a) \<bullet> rot (b - a) = 0 \<and> x \<bullet> (b - a) \<in> {a \<bullet> (b - a)<..<b \<bullet> (b - a)})"
inducts: fixes P :: "'a option \<Rightarrow> bool"   and option :: "'a option" assumes "P None"   and "\<And>x. P (Some x)" shows "P option"
Abs_dataspace_inverse: fixes y :: "'b set list" assumes "y \<in> dataspace" shows "Rep_dataspace (Abs_dataspace y) = y"
sel: fixes x :: "'a \<Rightarrow> bool" shows "eval (Pred x) = x"
assms_4: fixes S :: "('a, 'b) strand_step list"   and v :: "'b" assumes "wf\<^sub>s\<^sub>t V S"   and "simple S"   and "P S" shows "P (S @ [send\<langle>Var v\<rangle>\<^sub>s\<^sub>t])"
simps_1: shows "funC (C []) = (\<lambda>i. [C []])"
insert_Diff_if: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" shows "Set.insert x A - B = (if x \<in> B then A - B else Set.insert x (A - B))"
disjointI: fixes A :: "'a set set" assumes "\<And>a b. \<lbrakk>a \<in> A; b \<in> A; a \<noteq> b\<rbrakk> \<Longrightarrow> a \<inter> b = {}" shows "disjoint A"
distinct_1: fixes x1 :: "'a" shows "ev x1 \<noteq> tick"
reduced_GB_is_reduced_GB_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "finite (local.punit.component_of_term ` Keys F)"   and "F \<subseteq> punit.dgrad_p_set d m" shows "punit.is_reduced_GB (punit.reduced_GB F)"
assms_5: shows "(G ===> (=)) inEdges inEdges2"
start_points_in_root_aux: fixes r :: "'a"   and x :: "'a" assumes "backward_finite_path_root r x" shows "x ; 1 \<le> x\<^sup>T\<^sup>\<star> ; r"
set_intros: fixes x2 :: "'b" shows "x2 \<in> set_err (OK x2)"
hyps_2: shows "lookup (cMethods CDef) (\<lambda>md. mName md = m) = None"
S_transform_eqvt: fixes p :: "perm"   and x :: "('a, 'c, 'b) formula" shows "p \<bullet> S_transform x = S_transform (p \<bullet> x)"
morphism_presheaves_of_rings_def: fixes X :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and \<FF>' :: "'d set \<Rightarrow> 'f set"   and \<rho>' :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'f \<Rightarrow> 'f"   and b' :: "'f"   and add_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str' :: "'d set \<Rightarrow> 'f"   and one_str' :: "'d set \<Rightarrow> 'f"   and fam_morphisms :: "'d set \<Rightarrow> 'e \<Rightarrow> 'f" shows "morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms \<equiv> presheaf_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<and> presheaf_of_rings X is_open \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' \<and> morphism_presheaves_of_rings_axioms is_open \<FF> \<rho> add_str mult_str zero_str one_str \<FF>' \<rho>' add_str' mult_str' zero_str' one_str' fam_morphisms"
landau_trans_1: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> O[F](g)"   and "g \<in> \<Theta>[F](h)" shows "f \<in> O[F](h)"
f_def: shows "f = (\<lambda>t. geod (N t) (K t))"
simps_17: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and f3 :: "'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and f4 :: "('b, 'c) Combinators \<Rightarrow> ('b, 'c) Combinators \<Rightarrow> 'a"   and x21 :: "'b"   and x22 :: "'b" shows "(case DenyAllFromTo x21 x22 of DenyAll \<Rightarrow> f1 | DenyAllFromTo x xa \<Rightarrow> f2 x xa | AllowPortFromTo x xa xb \<Rightarrow> f3 x xa xb | x \<oplus> xa \<Rightarrow> f4 x xa) = f2 x21 x22"
ntrp_valid_preserve_s: fixes s :: "'n list"   and c :: "'n list multiset"   and e :: "('p, 'ba) label list el_step"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "((s, c), e, s', c') \<in> ntrp fg"   and "valid fg (add_mset s c)" shows "valid fg (add_mset s' c')"
NO_MATCH_def: fixes pat :: "'c"   and val :: "'d" shows "NO_MATCH pat val \<equiv> True"
module_hom_scale_self: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and c :: "'b" assumes "module scale" shows "module_hom scale scale (scale c)"
last_ps_sgn_const: fixes x :: "real"   and y :: "real" shows "sgn (poly (last (p # ps)) x) = sgn (poly (last (p # ps)) y)"
fold_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "('b, 'c) RBT_Impl.rbt" shows "RBT_Impl.fold f t = List.fold (\<lambda>(x, y). f x y) (RBT_Impl.entries t)"
quadratic_hensel_step_code: fixes q :: "int"   and S :: "int poly"   and T :: "int poly"   and D :: "int poly"   and H :: "int poly" shows "local.quadratic_hensel_step q S T D H = (let dupe = dupe_monic_dynamic q D H S T; U = poly_mod.Mp q (sdiv_poly (C - D * H) q); (A, B) = dupe U; D' = D + Polynomial.smult q B; H' = H + Polynomial.smult q A; U' = poly_mod.Mp q (sdiv_poly (S * D' + T * H' - 1) q); (A', B') = dupe U'; q' = q * q; S' = poly_mod.Mp q' (S - Polynomial.smult q A'); T' = poly_mod.Mp q' (T - Polynomial.smult q B') in (S', T', D', H'))"
continuous_map_image_closure_subset: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "continuous_map X Y f" shows "f ` (X closure_of S) \<subseteq> Y closure_of f ` S"
det_state_cover_d_reachable: fixes M :: "('d, 'e, 'f) FSM"   and V :: "'d list set"   and v :: "'d list" assumes "is_det_state_cover M V"   and "v \<in> V" obtains q :: "'f" where "d_reaches M (initial M) v q"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>\<And>x xa. P p; \<And>x xa. P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>\<And>x xa. P p; \<And>x xa. P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>c e. P (Eq (CN 0 c e))"   and "\<And>c e. P (NEq (CN 0 c e))"   and "\<And>c e. P (Lt (CN 0 c e))"   and "\<And>c e. P (Le (CN 0 c e))"   and "\<And>c e. P (Gt (CN 0 c e))"   and "\<And>c e. P (Ge (CN 0 c e))"   and "P T"   and "P F"   and "\<And>va. P (Lt (C va))"   and "\<And>va. P (Lt (Bound va))"   and "\<And>vd vb vc. P (Lt (CN (Suc vd) vb vc))"   and "\<And>va. P (Lt (Neg va))"   and "\<And>va vb. P (Lt (Add va vb))"   and "\<And>va vb. P (Lt (Sub va vb))"   and "\<And>va vb. P (Lt (Mul va vb))"   and "\<And>va. P (Lt (Floor va))"   and "\<And>va vb vc. P (Lt (CF va vb vc))"   and "\<And>va. P (Le (C va))"   and "\<And>va. P (Le (Bound va))"   and "\<And>vd vb vc. P (Le (CN (Suc vd) vb vc))"   and "\<And>va. P (Le (Neg va))"   and "\<And>va vb. P (Le (Add va vb))"   and "\<And>va vb. P (Le (Sub va vb))"   and "\<And>va vb. P (Le (Mul va vb))"   and "\<And>va. P (Le (Floor va))"   and "\<And>va vb vc. P (Le (CF va vb vc))"   and "\<And>va. P (Gt (C va))"   and "\<And>va. P (Gt (Bound va))"   and "\<And>vd vb vc. P (Gt (CN (Suc vd) vb vc))"   and "\<And>va. P (Gt (Neg va))"   and "\<And>va vb. P (Gt (Add va vb))"   and "\<And>va vb. P (Gt (Sub va vb))"   and "\<And>va vb. P (Gt (Mul va vb))"   and "\<And>va. P (Gt (Floor va))"   and "\<And>va vb vc. P (Gt (CF va vb vc))"   and "\<And>va. P (Ge (C va))"   and "\<And>va. P (Ge (Bound va))"   and "\<And>vd vb vc. P (Ge (CN (Suc vd) vb vc))"   and "\<And>va. P (Ge (Neg va))"   and "\<And>va vb. P (Ge (Add va vb))"   and "\<And>va vb. P (Ge (Sub va vb))"   and "\<And>va vb. P (Ge (Mul va vb))"   and "\<And>va. P (Ge (Floor va))"   and "\<And>va vb vc. P (Ge (CF va vb vc))"   and "\<And>va. P (Eq (C va))"   and "\<And>va. P (Eq (Bound va))"   and "\<And>vd vb vc. P (Eq (CN (Suc vd) vb vc))"   and "\<And>va. P (Eq (Neg va))"   and "\<And>va vb. P (Eq (Add va vb))"   and "\<And>va vb. P (Eq (Sub va vb))"   and "\<And>va vb. P (Eq (Mul va vb))"   and "\<And>va. P (Eq (Floor va))"   and "\<And>va vb vc. P (Eq (CF va vb vc))"   and "\<And>va. P (NEq (C va))"   and "\<And>va. P (NEq (Bound va))"   and "\<And>vd vb vc. P (NEq (CN (Suc vd) vb vc))"   and "\<And>va. P (NEq (Neg va))"   and "\<And>va vb. P (NEq (Add va vb))"   and "\<And>va vb. P (NEq (Sub va vb))"   and "\<And>va vb. P (NEq (Mul va vb))"   and "\<And>va. P (NEq (Floor va))"   and "\<And>va vb vc. P (NEq (CF va vb vc))"   and "\<And>v va. P (Dvd v va)"   and "\<And>v va. P (NDvd v va)"   and "\<And>v. P (NOT v)"   and "\<And>v va. P (Imp v va)"   and "\<And>v va. P (Iff v va)"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
t1: shows "f abs_summable_on S \<and> (\<Sum>\<^sub>ax\<in>S. norm (f x)) \<le> B"
x_1: shows "(\<lambda>xa. ereal (birkhoff_sum f xa x)) \<longlonglongrightarrow> - \<infinity>"
gap_pres_Labeling: fixes k :: "nat" assumes "gap_precond l k" shows "Labeling c s t f (gap_effect l k)"
distinct_mset_inter_mset_1: fixes C :: "'a multiset"   and D :: "'a multiset" assumes "distinct_mset C" shows "distinct_mset (C \<inter># D)"
full_Suc_Node2_iff: fixes n :: "nat"   and l :: "'b tree234"   and p :: "'b"   and r :: "'b tree234" shows "full (Suc n) \<langle>l, p, r\<rangle> = (full n l \<and> full n r)"
comm_mult_strict_left_mono: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a < b"   and "(0::'b) < c" shows "c * a < c * b"