hcju/isabelleps huggingface dataset:
disjoint_family_disjointed: fixes A :: "nat \<Rightarrow> 'b set" shows "disjoint_family (disjointed A)"
p': shows "\<forall>X\<in>P. bij_betw (p' X) X (bij X)"
fps_add_assoc: fixes f :: "('c, 'd) fps"   and g :: "('c, 'd) fps"   and h :: "('c, 'd) fps" shows "f + g + h = f + (g + h)"
col_inter: fixes A :: "Points"   and C :: "Points"   and B :: "Points"   and l :: "Lines" assumes "A \<noteq> C"   and "B \<noteq> C"   and "col A B C" shows "Pascal_Property.inter l (line B C) = Pascal_Property.inter l (line A C)"
assertion_prop: 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 dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "inf (times x top) one = x"
powser_ms_aux'_MSSCons: fixes c :: "real"   and cs :: "real msstream"   and xs :: "('a \<times> real) msllist" shows "powser_ms_aux' (MSSCons c cs) xs = MSLCons (const_expansion c, 0) (times_ms_aux xs (powser_ms_aux' cs xs))"
sim_first: fixes s :: "nat \<Rightarrow> 'a"   and t :: "nat \<Rightarrow> 'a" assumes "s \<approx> t" shows "first s = first t"
open_ereal_vimage: fixes S :: "ereal set" assumes "open S" shows "open (ereal -` S)"
seq_monosub: fixes s :: "nat \<Rightarrow> 'a" shows "\<exists>f. strict_mono f \<and> monoseq (\<lambda>n. s (f n))"
bi_unique_vec_rel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "bi_unique R" shows "bi_unique (vec_rel R)"
det_mat_delete_make_mat: fixes k :: "nat"   and n :: "nat"   and B :: "'a mat" assumes "k < n" shows "Determinant.det (mat_delete (mat_delete (make_mat n k B) 0 k) (k - 1) 0) = (1::'a)"
simps_4: fixes x2 :: "'c"   and x1 :: "'c" shows "Lt x2 \<noteq> Le x1"
hyps_3: shows "fst agQ \<in> alphabet (complement A) \<and> snd agQ \<in> transition (complement A) (fst agQ) fP"
union_vars_monom_def: fixes xs :: "char list list"   and \<V> :: "char list set" shows "union_vars_monom xs \<V> = fold insert xs \<V>"
entropy_le_card: fixes X :: "'key \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> real" assumes "simple_distributed (point_measure keys (\<lambda>x. ennreal (K x))) X f" shows "K.entropy_Pow X \<le> log b (real (card (X ` space (point_measure keys (\<lambda>x. ennreal (K x))))))"
id_ord_isom: fixes D :: "('c, 'd) Order_scheme" assumes "Order D" shows "ord_isom D D (idmap (carrier D))"
derived_block_num: assumes "bl \<in># \<B>" shows "int (size \<B>\<^sup>D) = \<b> - 1"
gdpr_AT: shows "\<exists>A. \<turnstile>A \<and> attack A = (Igdpr, sgdpr)"
map_conv_bind_option: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b option" shows "map_option f x = Option.bind x (pure \<circ> f)"
of_int_moebius_mu: fixes n :: "nat" shows "of_int (moebius_mu n) = moebius_mu n"
MM3_def: shows "MM3 \<equiv> ((\<forall>x y. P x y = (PP x y \<or> x = y)) \<and> (\<forall>x y. O x y = (\<exists>z. P z x \<and> P z y))) \<and> (\<forall>x y z. PP x y \<longrightarrow> PP y z \<longrightarrow> PP x z) \<and> (\<forall>y x. PP y x \<longrightarrow> (\<exists>z. P z x \<and> \<not> O z y))"
82: fixes x :: "'a" shows "top \<squnion> - x = top"
subst_fv_dom_img_single_1: fixes v :: "'a"   and t :: "('b, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "v \<notin> fv t"   and "\<sigma> v = t"   and "\<And>w. v \<noteq> w \<Longrightarrow> \<sigma> w = Var w" shows "subst_domain \<sigma> = {v}"
sin_arccos: fixes x :: "real" assumes "- 1 \<le> x"   and "x \<le> 1" shows "sin (arccos x) = sqrt (1 - x\<^sup>2)"
ltlc_to_pltl_semantics: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc" shows "w \<Turnstile>\<^sub>p ltlc_to_pltl \<phi> = w \<Turnstile>\<^sub>c \<phi>"
st_6: shows "\<not> t \<le> Suc (2 * n)"
compare_EQ_dest: fixes eq :: "'d \<rightarrow> 'd \<rightarrow> tr"   and compare :: "'d \<rightarrow> 'd \<rightarrow> Ordering"   and x :: "'d"   and y :: "'d" assumes "class.Ord_linear eq compare"   and "compare\<cdot>x\<cdot>y = EQ" shows "x = y"
wf_lang_wf_word: fixes n :: "nat"   and r :: "'b rexp" assumes "local.wf n r" shows "\<forall>w\<in>lang n r. wf_word n w"
fun_ran_mwb_lens: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "('c \<Rightarrow> 'b) \<Longrightarrow> 'd" assumes "mwb_lens X"   and "mwb_lens Y" shows "mwb_lens (fun_ran_lens X Y)"
lincomb_def: fixes rs :: "'f list"   and ms :: "'v list" shows "lincomb' rs ms = sum_list (map2 (\<cdot>) rs ms)"
poly_smult_zero_iff: fixes a :: "'a"   and p :: "'a poly"   and x :: "'a" shows "(poly (Polynomial.smult a p) x = (0::'a)) = (a = (0::'a) \<or> poly p x = (0::'a))"
select_convs_4: fixes Chn :: "'c"   and Dom :: "'c span_data"   and Cod :: "'c span_data"   and more :: "'b" shows "arrow_of_spans_data.more \<lparr>Chn = Chn, Dom = Dom, Cod = Cod, \<dots> = more\<rparr> = more"
domain_def: fixes R :: "('a, 'b) ring_scheme" shows "domain R \<equiv> cring R \<and> domain_axioms R"
take_0: fixes xs :: "'e list" shows "take 0 xs = []"
Act_eq_iff: fixes \<alpha>1 :: "'g"   and x1 :: "('e, 'f, 'g) formula"   and \<alpha>2 :: "'g"   and x2 :: "('e, 'f, 'g) formula" shows "(Act \<alpha>1 x1 = Act \<alpha>2 x2) = (Act\<^sub>\<alpha> \<alpha>1 (Rep_formula x1) = Act\<^sub>\<alpha> \<alpha>2 (Rep_formula x2))"
conj: shows "local.bad (local.worst_chain (Suc i)) \<and> p (local.worst_chain i) (local.worst_chain (Suc i)) \<and> (\<forall>z. local.bad z \<and> p (local.worst_chain i) z \<longrightarrow> \<not> gt (local.worst_chain (Suc i)) z)"
intro_elim_4_c: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> \<^bold>\<or> \<psi> in v]"   and "[\<^bold>\<not>\<psi> in v]" shows "[\<phi> in v]"
subspace_subset: fixes U :: "'c set"   and V :: "'c set" assumes "Subspace.subspace U V" shows "U \<subseteq> V"
eq: shows "ell_field_class.add a p\<^sub>1 p\<^sub>2 = p\<^sub>3"
cspan_singleton_scaleC: fixes a :: "complex"   and \<psi> :: "'d" assumes "a \<noteq> 0" shows "cspan {a *\<^sub>C \<psi>} = cspan {\<psi>}"
post_del_adjR: fixes l :: "('c \<times> nat) tree"   and a :: "'c"   and lv :: "nat"   and r :: "('c \<times> nat) tree"   and r' :: "('c \<times> nat) tree" assumes "invar \<langle>l, (a, lv), r\<rangle>"   and "pre_adjust \<langle>l, (a, lv), r'\<rangle>"   and "post_del r r'" shows "post_del \<langle>l, (a, lv), r\<rangle> (adjust \<langle>l, (a, lv), r'\<rangle>)"
assms: shows "homotopic_with (\<lambda>h. h ` S \<subseteq> T) X Y f g"
oinvariant_weakenD: fixes A :: "('a \<times> 'b, 'c) automaton"   and S' :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and U' :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<times> 'b \<Rightarrow> bool"   and \<sigma> :: "'a"   and p :: "'b"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "A \<Turnstile> (S', U' \<rightarrow>) P"   and "(\<sigma>, p) \<in> oreachable A S U"   and "\<And>s s' a. S s s' a \<Longrightarrow> S' s s' a"   and "\<And>s s'. U s s' \<Longrightarrow> U' s s'" shows "P (\<sigma>, p)"
schroeder_2: 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"   and y :: "'b"   and z :: "'b" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv" shows "(inf (times x y) z = bot) = (inf (times z (conv y)) x = bot)"
assms_2: shows "(\<Psi>, P, Q) \<in> Rel"
Field_def: fixes r :: "('a \<times> 'a) set" shows "Field r = Domain r \<union> Range r"
luckiestFind_eval: fixes xs :: "real list"   and var :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and \<Gamma> :: "real list" assumes "length xs = var" shows "(\<exists>x. eval (list_conj (map fm.Atom L @ F)) (xs @ x # \<Gamma>)) = (\<exists>x. eval (luckiestFind var L F) (xs @ x # \<Gamma>))"
P_1: shows "path (ins_edge (u, v) (graph_join g\<^sub>1 g\<^sub>2)) a p a"
that: assumes "u \<in> verts G"   and "v \<in> verts G" shows "thesis"
unique_connecting_path_tl: fixes v :: "nat"   and u :: "nat"   and w :: "nat" assumes "v \<in> V"   and "u \<in> set (w \<leadsto> v)"   and "u \<rightarrow> w" shows "tl (w \<leadsto> v) = u \<leadsto> v"
smap_ID': fixes xs :: "[:'a:]" shows "smap\<cdot>ID\<cdot>xs = xs"
converse_rtrancl_induct: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set"   and P :: "'b \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>*"   and "P b"   and "\<And>y z. \<lbrakk>(y, z) \<in> r; (z, b) \<in> r\<^sup>*; P z\<rbrakk> \<Longrightarrow> P y" shows "P a"
relhoare_refl: fixes R :: "('b \<times> 'b) set" shows "{Id} R, R {> Id}"
span_prj: fixes f :: "'a"   and g :: "'a" assumes "C.cospan f g" shows "C.span \<p>\<^sub>0[f, g] \<p>\<^sub>1[f, g]"
dgrad_p_set_le_subset: fixes F :: "('t \<Rightarrow>\<^sub>0 'f) set"   and G :: "('t \<Rightarrow>\<^sub>0 'f) set"   and d :: "'a \<Rightarrow> nat" assumes "F \<subseteq> G" shows "dgrad_p_set_le d F G"
word_overflow: fixes x :: "'b word" shows "x < x + 1 \<or> x + 1 = 0"
MGU_Const: fixes c :: "'b"   and d :: "'b" shows "MGU [] (Const c) (Const d) = (c = d)"
simps_1: fixes x :: "gauss_int"   and y :: "gauss_int" shows "ReZ (x * y) = ReZ x * ReZ y - ImZ x * ImZ y"
nat_to_sch_srj: fixes sch :: "PrimScheme" shows "nat_to_sch (sch_to_nat sch) = sch"
INTER_eq: fixes B :: "'d \<Rightarrow> 'c set"   and A :: "'d set" shows "\<Inter> (B ` A) = {y. \<forall>x\<in>A. y \<in> B x}"
neq: shows "uint w \<noteq> uint (2 ^ LENGTH(31))"
PQ: fixes s :: "'a \<times> 'b" assumes "P s" shows "Q s"
DataPart_def: fixes d :: "'b data"   and n :: "nat" shows "(d !P! n) = DataValue d ! n"
is_run_def: fixes r :: "nat \<Rightarrow> 'q \<times> 's" shows "prod.is_run r \<equiv> r 0 \<in> prod.V0 \<and> ipath prod.E r"
no_meet2: fixes A :: "'c set"   and B :: "'c set" assumes "A \<inter> B = {}" shows "\<forall>a\<in>B. a \<notin> A"
simps_2: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and proper_interval :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and ao :: "'a option"   and y :: "'a"   and ys :: "'a list" shows "proper_intrvl.set_less_aux_Compl less proper_interval ao [] (y # ys) = (proper_interval ao (Some y) \<or> proper_intrvl.set_less_aux_Compl less proper_interval (Some y) [] ys)"
box_stack_Cons: fixes f :: "'a"   and x21 :: "('a, ('dyn, 'ubx1, 'ubx2) unboxed) frame"   and x22 :: "('a, ('dyn, 'ubx1, 'ubx2) unboxed) frame list" shows "Subx.box_stack f (x21 # x22) = Subx.box_frame f x21 # Subx.box_stack f x22"
equiv_refl: fixes eqv :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d" assumes "class.equiv eqv" shows "eqv x x"
summable_def: fixes f :: "nat \<Rightarrow> 'b" shows "summable f = (\<exists>s. f sums s)"
empty_fold_graphE: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and z :: "'f"   and x :: "'f" assumes "fold_graph f z {} x" obtains "x = z"
finite_nonempty_carrier_has_maximum: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set" assumes "rational_preference carrier relation"   and "finite carrier"   and "carrier \<noteq> {}" shows "\<exists>e\<in>carrier. \<forall>m\<in>carrier. e \<succeq>[relation] m"
valid_path_cons_simp: fixes x :: "'a"   and xs :: "'a llist" shows "valid_path (LCons x xs) = (xs = LNil \<and> x \<in> V \<or> x \<in> V \<and> lhd xs \<in> V \<and> x \<rightarrow> lhd xs \<and> valid_path xs \<and> \<not> lnull xs)"
map: fixes f :: "'d \<Rightarrow> 'c"   and x1 :: "'d"   and x2 :: "'d stream" shows "smap f (x1 ## x2) = f x1 ## smap f x2"
distinct_85: fixes x61 :: "'d set"   and x62 :: "('d, 'e, 'f) com"   and x91 :: "'f"   and x92 :: "'d set"   and x93 :: "('d, 'e, 'f) com" shows "While x61 x62 \<noteq> Guard x91 x92 x93"
minimal: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b"   and Q :: "'b \<Rightarrow> bool" assumes "minimal_element P A"   and "x \<in> A"   and "Q x" shows "\<exists>y\<in>A. P\<^sup>=\<^sup>= y x \<and> Q y \<and> (\<forall>z\<in>A. P z y \<longrightarrow> \<not> Q z)"
advantage_nonneg: fixes \<eta> :: "nat"   and \<A> :: "(bool, bool list, bool list) gpv" shows "0 \<le> local.prf.advantage \<eta> \<A>"
bot_fun_def: shows "UML_Types.bot_class.bot \<equiv> \<lambda>x. UML_Types.bot_class.bot"
p1p2: shows "p1 \<noteq> p2 \<and> p1 < length csa \<and> p2 < length csa \<and> csa ! p1 = 0 \<and> csa ! p2 = 0"
Push_Node_def: shows "Push_Node \<equiv> \<lambda>n x. Abs_Node (apfst (Push n) (Rep_Node x))"
typing_alt_simps_10: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and x :: "String.literal \<times> 'a OCL_Types.type option \<times> 'a expr"   and xs :: "(String.literal \<times> 'a OCL_Types.type option \<times> 'a expr) list"   and \<tau> :: "'a OCL_Types.type" shows "(\<Gamma> \<turnstile>\<^sub>E Literal (TupleLiteral (x # xs)) : \<tau>) = (\<exists>\<xi> \<tau>' \<sigma>. \<tau> = Tuple (\<xi>(tuple_element_name x \<mapsto>\<^sub>f \<sigma>)) \<and> \<Gamma> \<turnstile>\<^sub>E Literal (TupleLiteral xs) : Tuple \<xi> \<and> \<Gamma> \<turnstile>\<^sub>E tuple_element_expr x : \<tau>' \<and> tuple_element_type x = Some \<sigma> \<and> \<tau>' \<le> \<sigma>)"
that: fixes j :: "nat" assumes "j + 1 < length xs"   and "remdups_adj xs ! i = xs ! j"   and "remdups_adj xs ! (i + 1) = xs ! (j + 1)" shows "thesis"
conga_line: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "A \<noteq> B"   and "B \<noteq> C"   and "A' \<noteq> B'"   and "B' \<noteq> C'"   and "Bet A B C"   and "Bet A' B' C'" shows "A B C CongA A' B' C'"
obj_fields_marked_locs_def: shows "obj_fields_marked_locs = {mark_loop_mark_object_loop, mark_loop_mark_choose_field, mark_loop_mark_deref, mark_loop_mark_field_done, mark_loop_blacken} \<union> mark_loop_mo_locs"
simps_7: fixes X :: "WS1S_Nameful.so"   and \<phi> :: "WS1S_Nameful.ws1s" shows "fos (Ex2 X \<phi>) = fos \<phi>"
subset_implies_mult: fixes A :: "'a multiset"   and B :: "'a multiset"   and r :: "('a \<times> 'a) set" assumes "A \<subset># B" shows "(A, B) \<in> mult r"
llist_all2_ldropnI: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist"   and n :: "nat" assumes "llist_all2 P xs ys" shows "llist_all2 P (ldropn n xs) (ldropn n ys)"
ascP_f_Cons: fixes f :: "'b list \<Rightarrow> 'b list"   and x :: "'b"   and xs :: "'b list" assumes "ascP f" shows "f (x # xs) = f [] @ x # xs"
Can_rep_can: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "free_monoidal_category.can C f" shows "monoidal_language.Can C (free_monoidal_category.rep C f)"
Reals_divide: fixes a :: "'a"   and b :: "'a" assumes "a \<in> \<real>"   and "b \<in> \<real>" shows "a / b \<in> \<real>"
div_by_1: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and 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 a :: "'c" assumes "class.semidom_divide divide plus minus zero times one" shows "divide a one = a"
rep_eq: fixes x :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'b"   and xa :: "('c, 'a) uexpr"   and xb :: "('d, 'a) uexpr"   and xc :: "('e, 'a) uexpr" shows "\<lbrakk>trop x xa xb xc\<rbrakk>\<^sub>e = (\<lambda>b. x (\<lbrakk>xa\<rbrakk>\<^sub>e b) (\<lbrakk>xb\<rbrakk>\<^sub>e b) (\<lbrakk>xc\<rbrakk>\<^sub>e b))"
inangle3123: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A \<noteq> B"   and "C \<noteq> B" shows "C InAngle A B C"
fps_shift_code: fixes n :: "nat"   and x :: "'b ratfps" shows "fps_shift n (fps_of_ratfps x) = fps_of_ratfps (ratfps_shift n x)"
ex_simps_6: fixes P :: "bool"   and Q :: "'d \<Rightarrow> bool" shows "(\<exists>x. P \<longrightarrow> Q x) = (P \<longrightarrow> (\<exists>x. Q x))"
reduced_word_problem_ConsCons_step: fixes w :: "'w"   and a :: "'w"   and as :: "'w list"   and b :: "'w"   and bs :: "'w list" assumes "\<And>y ss ts. \<lbrakk>order.greater (S_length w) (S_length y); y \<noteq> (0::'w); S_reduced_for y ss; S_reduced_for y ts\<rbrakk> \<Longrightarrow> \<exists>xss. flip_altsublist_chain (ss # xss @ [ts])"   and "S_reduced_for w (a # as)"   and "S_reduced_for w (b # bs)"   and "a \<noteq> b" shows "\<exists>xss. flip_altsublist_chain ((a # as) # xss @ [b # bs])"
CESem_simps_2: fixes e :: "exp"   and x :: "Vars.var"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" shows "\<N>\<lbrakk> App e x \<rbrakk>\<^bsub>\<rho>\<^esub> = (\<Lambda> (C\<cdot>r). ((\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub>)\<cdot>r \<down>CFn \<rho> x|\<^bsub>r\<^esub>)\<cdot>r)"
exhaust: fixes y :: "prim_ty" obtains "y = Void"   | "y = Boolean"   | "y = Integer"
intro: fixes ops :: "('e, 'f, 'd, 'g) graph_ops_scheme" assumes "graph (gop_\<alpha> ops) (gop_invar ops)"   and "graph_empty (gop_\<alpha> ops) (gop_invar ops) (gop_empty ops)"   and "graph_add_node (gop_\<alpha> ops) (gop_invar ops) (gop_add_node ops)"   and "graph_delete_node (gop_\<alpha> ops) (gop_invar ops) (gop_delete_node ops)"   and "graph_add_edge (gop_\<alpha> ops) (gop_invar ops) (gop_add_edge ops)"   and "graph_delete_edge (gop_\<alpha> ops) (gop_invar ops) (gop_delete_edge ops)"   and "graph_from_list (gop_\<alpha> ops) (gop_invar ops) (gop_from_list ops)"   and "graph_to_list (gop_\<alpha> ops) (gop_invar ops) (gop_to_list ops)"   and "graph_nodes_it (gop_\<alpha> ops) (gop_invar ops) (gop_nodes_list_it ops)"   and "graph_edges_it (gop_\<alpha> ops) (gop_invar ops) (gop_edges_list_it ops)"   and "graph_succ_it (gop_\<alpha> ops) (gop_invar ops) (gop_succ_list_it ops)" shows "StdGraph ops"
recurrent_subset_rel_incl_2: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "recurrent_subset_infty A \<subseteq> recurrent_subset_infty B"
prod_mem_prod_ideals: fixes R :: "('e, 'f) Ring_scheme"   and I :: "'e set"   and J :: "'e set"   and i :: "'e"   and j :: "'e" assumes "Ring R"   and "ideal R I"   and "ideal R J"   and "i \<in> I"   and "j \<in> J" shows "i \<cdot>\<^sub>r\<^bsub>R\<^esub> j \<in> I \<diamondsuit>\<^sub>r\<^bsub>R\<^esub> J"
measure_spmf_bind_pmf: fixes p :: "'d pmf"   and f :: "'d \<Rightarrow> 'c spmf" shows "measure_spmf (p \<bind> f) = measure_pmf p \<bind> measure_spmf \<circ> f"
productive_onI: fixes g :: "'c \<Rightarrow> ('d, 'c) step"   and s :: "'c" assumes "productive g" shows "s \<in> productive_on g"
idx'_correct: fixes xs :: "'a list"   and ctr :: "nat"   and y :: "'a"   and n :: "nat" assumes "distinct xs" shows "(idx' ctr xs y = Some n) = (ctr \<le> n \<and> n < length xs + ctr \<and> xs ! (n - ctr) = y)"
cases: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and a1 :: "'c rule list"   and a2 :: "state"   and a3 :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>a1, a2\<rangle> \<Rightarrow>\<^sub>s a3" obtains t :: "state" where "a1 = []" and "a2 = t__" and "a3 = t__"   | m :: "'c match_expr" and rs :: "'c rule list" where "a1 = Rule m__ Accept # rs__" and "a2 = Undecided" and "a3 = Decision FinalAllow" and "matches \<gamma> m__ p"   | m :: "'c match_expr" and rs :: "'c rule list" where "a1 = Rule ma__ Drop # rsa__" and "a2 = Undecided" and "a3 = Decision FinalDeny" and "matches \<gamma> ma__ p"   | m :: "'c match_expr" and rs :: "'c rule list" where "a1 = Rule mb__ Reject # rsb__" and "a2 = Undecided" and "a3 = Decision FinalDeny" and "matches \<gamma> mb__ p"   | m :: "'c match_expr" and rs :: "'c rule list" and t :: "state" where "a1 = Rule mc__ Log # rsc__" and "a2 = Undecided" and "a3 = ta__" and "matches \<gamma> mc__ p" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rsc__, Undecided\<rangle> \<Rightarrow>\<^sub>s ta__"   | m :: "'c match_expr" and rs :: "'c rule list" and t :: "state" where "a1 = Rule md__ Empty # rsd__" and "a2 = Undecided" and "a3 = tb__" and "matches \<gamma> md__ p" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rsd__, Undecided\<rangle> \<Rightarrow>\<^sub>s tb__"   | m :: "'c match_expr" and rs :: "'c rule list" and t :: "state" and a :: "action" where "a1 = Rule me__ a # rse__" and "a2 = Undecided" and "a3 = tc__" and "\<not> matches \<gamma> me__ p" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rse__, Undecided\<rangle> \<Rightarrow>\<^sub>s tc__"   | m :: "'c match_expr" and chain :: "char list" and rs\<^sub>1 :: "'c rule list" and m' :: "'c match_expr" and rs\<^sub>2 :: "'c rule list" and rrs :: "'c rule list" and t :: "state" where "a1 = Rule mf__ (Call chain__) # rrs__" and "a2 = Undecided" and "a3 = td__" and "matches \<gamma> mf__ p" and "\<Gamma> chain__ = Some (rs\<^sub>1 @ Rule m' Return # rs\<^sub>2)" and "matches \<gamma> m' p" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow>\<^sub>s Undecided" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rrs__, Undecided\<rangle> \<Rightarrow>\<^sub>s td__"   | m :: "'c match_expr" and chain :: "char list" and rs :: "'c rule list" and X :: "final_decision" and rrs :: "'c rule list" where "a1 = Rule mg__ (Call chaina__) # rrsa__" and "a2 = Undecided" and "a3 = Decision X" and "matches \<gamma> mg__ p" and "\<Gamma> chaina__ = Some rsf__" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rsf__, Undecided\<rangle> \<Rightarrow>\<^sub>s Decision X"   | m :: "'c match_expr" and chain :: "char list" and rs :: "'c rule list" and rrs :: "'c rule list" and t :: "state" where "a1 = Rule m (Call chain) # rrs" and "a2 = Undecided" and "a3 = t" and "matches \<gamma> m p" and "\<Gamma> chain = Some rs" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>s Undecided" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rrs, Undecided\<rangle> \<Rightarrow>\<^sub>s t"
simps_6: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> ('d \<Rightarrow> ('b, 'c, 'd) env option) \<Rightarrow> 'a"   and x21 :: "'c"   and x22 :: "'d \<Rightarrow> ('b, 'c, 'd) env option" shows "(case Env x21 x22 of Val x \<Rightarrow> f1 x | Env x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
root_nat_floor_upper: fixes p :: "nat"   and x :: "nat" assumes "p \<noteq> 0" shows "int x < (root_nat_floor p x + 1) ^ p"
Diff_single_insert: fixes A :: "'c set"   and x :: "'c"   and B :: "'c set" assumes "A - {x} \<subseteq> B" shows "A \<subseteq> insert x B"
cases: fixes a :: "C"   and b :: "nat" assumes "a \<in> hbase b" obtains "a = C []"   | i :: "nat" and n :: "C" and ms :: "C list" where "a = C (replicate i n @ ms)" and "i \<noteq> 0" and "i < b" and "n \<in> hbase b" and "C ms \<in> hbase b" and "\<And>m'. m' \<in> set ms \<Longrightarrow> evalC b n < evalC b m'"
rel_fset_alt_def: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_fset R = (\<lambda>A B. (\<forall>x. \<exists>y. x |\<in>| A \<longrightarrow> y |\<in>| B \<and> R x y) \<and> (\<forall>y. \<exists>x. y |\<in>| B \<longrightarrow> x |\<in>| A \<and> R x y))"
Inc: fixes A :: "nat \<Rightarrow> 'c set"   and \<Omega> :: "'c set"   and M :: "'c set set" assumes "range A \<in> Pow (smallest_ccdi_sets \<Omega> M)"   and "A 0 = {}"   and "\<And>n. A n \<subseteq> A (Suc n)" shows "\<Union> (range A) \<in> smallest_ccdi_sets \<Omega> M"
USdB_Strong_d_Bisimulation: fixes SR :: "(('a \<times> ('b \<Rightarrow> 'c)) \<times> 'a list \<times> ('b \<Rightarrow> 'c)) set"   and DA :: "'b \<Rightarrow> 'e"   and d :: "'e" shows "Strong_Security.Strong_d_Bisimulation SR DA d (Strong_Security.USdB SR DA d)"
obj_self_composable_1: fixes a :: "'a" assumes "S.obj a" shows "S.hcomp a a \<noteq> S.null"
frac_le: fixes y :: "'d"   and x :: "'d"   and w :: "'d"   and z :: "'d" assumes "(0::'d) \<le> y"   and "x \<le> y"   and "(0::'d) < w"   and "w \<le> z" shows "x / z \<le> y / w"