hcju/isabelleps huggingface dataset:
subfieldE_2: fixes K :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "subfield K R" shows "subcring K R"
graph_ext_def: fixes nodes :: "'c set"   and edges :: "('c \<times> 'c) set"   and more :: "'d" shows "\<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons graph_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple nodes edges) more"
poly_deg_map_key_PP: fixes p :: "('x, nat) pp \<Rightarrow>\<^sub>0 'b" shows "poly_deg (Poly_Mapping.map_key PP p) = poly_deg_pp p"
descartes_sign_rule_aux: fixes p :: "real poly" assumes "p \<noteq> 0" shows "count_roots_with ((<) 0) p \<le> sign_changes (coeffs p) \<and> even (sign_changes (coeffs p) - count_roots_with ((<) 0) p)"
ins_edge_def: fixes k :: "lit"   and v :: "lit"   and m :: "(lit, lit list) mapping" shows "ins_edge k v m = (case Mapping.lookup m k of None \<Rightarrow> Mapping.update k [v] m | Some l \<Rightarrow> Mapping.update k (remdups (v # l)) m)"
DOM_in_dom: fixes f :: "'c local.term set" assumes "arr f" shows "DOM f \<in> local.dom f"
evaluate_list_singleton_valE: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and s' :: "'b state"   and vs :: "v list" assumes "evaluate_list ck env s [e] (s', Rval vs)" obtains v :: "v" where "vs = [v]" and "evaluate ck env s e (s', Rval v)"
sp: shows "splay x CD = \<langle>C, c, D\<rangle>"
pdouble_closed: fixes a :: "int"   and b :: "int"   and p :: "int \<times> int \<times> int" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "on_curvep a b p" shows "on_curvep a b (pdouble a p)"
RedAAccBounds: fixes s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and a :: "'addr"   and T :: "ty"   and n :: "nat"   and i :: "32 word"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id" assumes "typeof_addr (hp s) a = \<lfloor>Array_type T n\<rfloor>"   and "i <s 0 \<or> int n \<le> sint i" shows "extTA,P,t \<turnstile> \<langle>addr a\<lfloor>Val (Intg i)\<rceil>,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt ArrayIndexOutOfBounds),s\<rangle>"
LT31: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and N :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" assumes "|~ P \<and> N \<longrightarrow> \<circle>P \<or> \<circle>Q"   and "|~ P \<and> Unchanged v \<longrightarrow> \<circle>P" shows "\<turnstile> \<box>N \<longrightarrow> P \<longrightarrow> \<box>P \<or> \<diamond>Q"
bind_gpv_cong: fixes gpv :: "('a, 'b, 'c) gpv"   and gpv' :: "('a, 'b, 'c) gpv"   and f :: "'a \<Rightarrow> ('d, 'b, 'c) gpv"   and g :: "'a \<Rightarrow> ('d, 'b, 'c) gpv" assumes "gpv = gpv'"   and "\<And>x. x \<in> results'_gpv gpv' \<Longrightarrow> f x = g x" shows "bind_gpv gpv f = bind_gpv gpv' g"
scalar_product_distributivity: fixes n :: "nat" shows "\<forall>v1 v2 w1 w2. length v1 = length v2 \<and> length v1 = n \<and> length w1 = length w2 \<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)"
mertens_third_theorem_asymp_equiv: shows "(\<lambda>x. \<Prod>p\<in>{p. prime p \<and> real p \<le> x}. 1 - 1 / real p) \<sim>[at_top] (\<lambda>x. third_mertens_const / ln x)"
hyps: assumes "Subobjs P (last Cs) (last Cs # rev Ds')" shows "Subobjs P C (Cs @ tl (last Cs # rev Ds'))"
neighboring: shows "\<forall>k. i < k \<and> k < j \<longrightarrow> \<not> regular_event (t ! k)"
interval_lebesgue_integrable_bigo_right: fixes f :: "real \<Rightarrow> real"   and B :: "real"   and g :: "real \<Rightarrow> real"   and A :: "real" assumes "f \<in> O[at_left B](g)"   and "continuous_on {A..<B} f"   and "set_borel_measurable borel {A<..<B} f"   and "interval_lebesgue_integrable lborel (ereal A) (ereal B) g"   and "A < B" shows "interval_lebesgue_integrable lborel (ereal A) (ereal B) f"
get_shadow_root_def: shows "get_shadow_root \<equiv> local.a_get_shadow_root"
map_simp: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>C.\<rho>' f = \<F>C.inv (\<F>C.\<rho> (\<F>C.cod f)) \<cdot> \<F>C.map f"
le_finE: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list" assumes "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2" obtains v\<^sub>1 :: "'action list" where "w\<^sub>1 @ v\<^sub>1 =\<^sub>F w\<^sub>2"
complete_space_with_middle_axioms: shows "complete_space_with_middle TYPE('a)"
list_all2_Nil: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and ys :: "'c list" shows "list_all2 P [] ys = (ys = [])"
the_deriv: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "(f has_derivative F) (at x)" shows "(THE G. (f has_derivative G) (at x)) = F"
project_act_mono: fixes D :: "'d set"   and C :: "'d set"   and h :: "'e \<times> 'f \<Rightarrow> 'd"   and act :: "('d \<times> 'd) set" assumes "D \<subseteq> C" shows "project_act h (Restrict D act) \<subseteq> project_act h (Restrict C act)"
all_relative_to: fixes P :: "'b set \<Rightarrow> bool"   and U :: "'b set"   and Q :: "'b set \<Rightarrow> bool" shows "(\<forall>S. (P relative_to U) S \<longrightarrow> Q S) = (\<forall>S. P S \<longrightarrow> Q (U \<inter> S))"
length_takeWhile_less_P_nth: fixes j :: "nat"   and P :: "'g \<Rightarrow> bool"   and xs :: "'g list" assumes "\<And>i. i < j \<Longrightarrow> P (xs ! i)"   and "j \<le> length xs" shows "j \<le> length (takeWhile P xs)"
simps_1: fixes uu :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'c)"   and uv :: "'c \<Rightarrow> bool" shows "normalized_n_primitive uu uv MatchAny = True"
sset_subset_stream_pred: fixes w :: "'b stream"   and A :: "'b set" shows "(sset w \<subseteq> A) = pred_stream (\<lambda>a. a \<in> A) w"
Fvars_fmla_simps_6: fixes A :: "fmla" shows "Fvars (Robinson_Arithmetic.neg A) = Fvars A"
l2_inv6_def: shows "l2_inv6 \<equiv> {s. \<forall>Ra K. role (guessed_runs Ra) = Init \<longrightarrow> run_ended (progress s Ra) \<longrightarrow> guessed_frame Ra xsk = Some K \<longrightarrow> Aenc K (epubKF (Ra $ kE)) \<in> extr (bad s) (ik s) (chan s)}"
lfinite_lfilterI: fixes xs :: "'a llist"   and P :: "'a \<Rightarrow> bool" assumes "lfinite xs" shows "lfinite (lfilter P xs)"
path_weight_path: fixes s :: "'sum"   and loc1 :: "'loc"   and loc2 :: "'loc" assumes "s \<in>\<^sub>A path_summary loc1 loc2" obtains xs :: "('loc \<times> 'sum \<times> 'loc) list" where "path loc1 loc2 xs \<and> (\<forall>ys. path loc1 loc2 ys \<longrightarrow> \<not> sum_weights (map (\<lambda>(s, l, t). l) ys) < sum_weights (map (\<lambda>(s, l, t). l) xs))" and "distinct xs" and "sum_weights (map (\<lambda>(s, l, t). l) xs) = s"
size_fold: fixes A :: "'b multiset" shows "size A = fold_mset (\<lambda>_. Suc) 0 A"
execn_Normal_elim_cases_7: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and r :: "('e \<times> 'e) set"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Spec r,Normal s\<rangle> =n\<Rightarrow> t" obtains ta :: "'e" where "t = Normal ta" and "(s, ta) \<in> r"   | "t = Stuck" and "\<forall>t. (s, t) \<notin> r"
stl_decr_shd: fixes js :: "'ord stream" assumes "alw (\<lambda>js. (shd (stl js), shd js) \<in> Ord) js"   and "alw (ev (\<lambda>js. shd (stl js) \<noteq> shd js)) js" shows "\<exists>js1 js2. js = js1 @- js2 \<and> set js1 \<subseteq> {shd js} \<and> (shd js2, shd js) \<in> Ord \<and> shd js2 \<noteq> shd js \<and> shd (decr (shd js) js) = shd js2 \<and> stl (decr (shd js) js) = decr (shd js2) js2"
trancl_rtrancl_absorb: fixes R :: "('a \<times> 'a) set" shows "(R\<^sup>+)\<^sup>* = R\<^sup>*"
IdTr1: fixes F :: "('e, 'f) Group_scheme"   and x :: "'e" assumes "Group F"   and "x \<in> carrier F" shows "I\<^bsub>F\<^esub> x = x"
fps_integral_conv_plus_const: fixes a :: "'a fps"   and a0 :: "'a" shows "fps_integral a a0 = fps_integral0 a + fps_const a0"
winning_strategy_def: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and v0 :: "'c" assumes "ParityGame G" shows "ParityGame.winning_strategy G p \<sigma> v0 \<equiv> \<forall>P. vmc_path G P v0 p \<sigma> \<longrightarrow> ParityGame.winning_path G p P"
simps_1: fixes ik :: "msg set"   and secret :: "msg set"   and more :: "'a" shows "secret \<lparr>ik = ik, secret = secret, \<dots> = more\<rparr> = secret"
dim_submatrix_2: fixes A :: "'b mat"   and I :: "nat set"   and J :: "nat set" shows "dim_col (submatrix A I J) = card {j. j < dim_col A \<and> j \<in> J}"
intros_2: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and m :: "'a"   and n :: "'a" assumes "IFC prob"   and "m \<in> IFC.slice prob"   and "(m, n) \<in> IFC.pdg prob" shows "n \<in> IFC.slice prob"
map_of_eq_Some_iff: fixes xys :: "('a \<times> 'b) list"   and x :: "'a"   and y :: "'b" assumes "distinct (map fst xys)" shows "(map_of xys x = \<lfloor>y\<rfloor>) = ((x, y) \<in> set xys)"
smap_sconst: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "smap f (sconst x) = sconst (f x)"
enc_prop_nec_2: fixes v :: "i"   and x :: "\<nu>" shows "[(\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<rightarrow> (\<^bold>\<exists>p. F \<^bold>= (\<^bold>\<lambda>x. p))) \<^bold>\<rightarrow> \<^bold>\<box>(\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<rightarrow> (\<^bold>\<exists>p. F \<^bold>= (\<^bold>\<lambda>x. p))) in v]"
mergeSL: fixes L :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c"   and S :: "'a \<Rightarrow> 'c \<Rightarrow> 'c" assumes "adm_woL L" shows "local.adm_wo (mergeSL S L)"
igWlsInp_def: fixes MOD :: "('i, 'j, 'k, 'sort, 'l, 'm, 'n, 'o, 'p) model_scheme"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'n option" shows "igWlsInp MOD delta inp \<equiv> wlsOpS delta \<and> sameDom (arOf delta) inp \<and> liftAll2 (igWls MOD) (arOf delta) inp"
cl_mono: fixes r :: "'b set"   and r' :: "'b set"   and L :: "'b set set" assumes "r \<subseteq> r'" shows "cl L r \<subseteq> cl L r'"
word_mult_less_mono1: fixes i :: "'b word"   and j :: "'b word"   and k :: "'b word" assumes "i < j"   and "0 < k"   and "unat j * unat k < 2 ^ LENGTH('b)" shows "i * k < j * k"
det_identical_columns: fixes A :: "'b mat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat n n"   and "i \<noteq> j"   and "i < n"   and "j < n"   and "col A i = col A j" shows "det A = (0::'b)"
l_get_dom_component_get_scdom_component_axioms: shows "l_get_dom_component_get_scdom_component get_owner_document heap_is_wellformed DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs get_scdom_component get_dom_component"
is_canon_rewriterD3: fixes rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and A :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_canon_rewriter rword A u p" shows "lt p adds\<^sub>t u"
conjug_Lyndon_ex: fixes w :: "'a list" assumes "primitive w" obtains n :: "nat" where "Lyndon (rotate n w)"
igAbsIPresIGWls_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igAbsIPresIGWls MOD \<equiv> \<forall>xs s x X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igWlsAbs MOD (xs, s) (igAbs MOD xs x X)"
drop_last_message: fixes evts :: "('a \<times> ('a, 'b) operation) event list"   and i :: "nat"   and msgs :: "('a \<times> ('a, 'b) operation) list"   and last_msg :: "'a \<times> ('a, 'b) operation" assumes "evts prefix of i"   and "node_deliver_messages evts = msgs @ [last_msg]" shows "\<exists>pre. pre prefix of i \<and> node_deliver_messages pre = msgs"
assms_6: shows "h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>r Some parent"
arithmetic_simps_1: shows "num.One + num.One = num.Bit0 num.One"
f_last_message_hold_drop_mod: fixes n :: "nat"   and k :: "nat"   and xs :: "'b message_af list" assumes "n mod k = 0" shows "xs \<up> n \<longmapsto> k = xs \<longmapsto> k \<up> n"
intY1_mono: shows "Tarski.monotone (restrict f intY1) intY1 (induced intY1 r)"
w1: shows "Sup {norm (f x) / norm x |x. True} = Sup {norm (f x) |x. norm x = 1}"
noSpec_\<Gamma>: shows "\<forall>p\<in>dom \<Gamma>. noSpec (the (\<Gamma> p))"
proves_eq_symmetric_rule: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and t :: "Core.term"   and s :: "Core.term"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> t"   and "term_ok \<Theta> s"   and "typ_of s = typ_of t"   and "\<Theta>,\<Gamma> \<turnstile> mk_eq s t"   and "finite \<Gamma>"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq t s"
bounded_clinear_apply_cblinfun: fixes b :: "'a" shows "bounded_clinear (\<lambda>a. cblinfun_apply a b)"
le_ASSERTI: fixes \<Phi> :: "bool"   and M :: "'a nres"   and M' :: "'a nres" assumes "\<Phi> \<Longrightarrow> M \<le> M'" shows "M \<le> ASSERT \<Phi> \<bind> (\<lambda>_. M')"
f_expand_drop_mult: fixes n :: "nat"   and k :: "nat"   and xs :: "'b message_af list" shows "xs \<odot> k \<up> (n * k) = xs \<up> n \<odot> k"
HcompNml_Prim: fixes t :: "'a local.term"   and \<nu> :: "'a" assumes "\<not> is_Prim\<^sub>0 t" shows "\<^bold>\<langle>\<nu>\<^bold>\<rangle> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> t = \<^bold>\<langle>\<nu>\<^bold>\<rangle> \<^bold>\<star> t"
welltyped_B: fixes b :: "bool" shows "b ::: \<B>"
finite_subset: fixes A :: "'d set"   and B :: "'d set" assumes "A \<subseteq> B"   and "finite B" shows "finite A"
lemma_path_to_deadend: fixes p :: "Player" assumes "v0 \<in> attractor p (deadends p**)" shows "\<exists>p \<sigma>. strategy p \<sigma> \<and> winning_strategy p \<sigma> v0"
listset_Nil_mem_conv: fixes AVS :: "'a set list" shows "([] \<in> listset AVS) = (AVS = [])"
of_complex_inner_1': fixes x :: "complex" shows "(1::'b) \<bullet>\<^sub>C of_complex x = x"
accp_acc_eq: fixes r :: "('a \<times> 'a) set" shows "Wellfounded.accp (\<lambda>x xa. (x, xa) \<in> r) = (\<lambda>x. x \<in> Wellfounded.acc r)"
simps_3: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Ide (t \<^bold>\<star> u) = (Ide t \<and> Ide u \<and> Src t = Trg u)"
multiset_of_sort: fixes xs :: "'a list" shows "mset (local.sort xs) = mset xs"
holomorphic_on_linear: fixes c :: "complex"   and s :: "complex set" shows "(*) c holomorphic_on s"
Domain_Id_on: fixes A :: "'d set" shows "Domain (Id_on A) = A"
pwd_spy_i_rel: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "s\<^sub>0 \<Turnstile> fst (pwd_spy_i n)"
distinct_11: fixes list :: "char list"   and var :: "NominalSequents.var"   and form :: "form" shows "list \<nabla> [var].form \<noteq> ff"
split_asm: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "nat \<Rightarrow> 'c"   and nat :: "nat" shows "P (case nat of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = (\<not> (nat = 0 \<and> \<not> P f1 \<or> (\<exists>x2. nat = Suc x2 \<and> \<not> P (f2 x2))))"
sel_3: fixes x11 :: "'a"   and x12 :: "nat"   and x13 :: "'a interval" shows "op_idx (INS x11 x12 x13) = x12"
hyp: fixes env :: "('a, 'b, 'c) env"   and e :: "('a, 'b, 'c) env" assumes "lookup env xs = Some e" shows "lookup (update (xs @ y # ys) opt env) xs = Some (update (y # ys) opt e)"
cases: fixes a1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a2 :: "'a \<Rightarrow> bool" assumes "invariant a1 a2" obtains rel :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and I :: "'a \<Rightarrow> bool" where "a1 = rel" and "a2 = I" and "\<And>x y. \<lbrakk>rel x y; I x\<rbrakk> \<Longrightarrow> I y"
v3: fixes x :: "'a" assumes "x \<in> range ((*\<^sub>V) P)" shows "x \<in> (\<lambda>x. x - P *\<^sub>V x) -` {0::'a}"
when_simps_1: fixes zero :: "'b"   and a :: "'b" shows "zero.when zero a True = a"
symp_circline_eq_cmat: shows "symp circline_eq_cmat"
emeasure_density_add: fixes X :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal"   and g :: "'b \<Rightarrow> ennreal" assumes "X \<in> sets M"   and "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "emeasure (density M f) X + emeasure (density M g) X = emeasure (density M (\<lambda>x. f x + g x)) X"
split_face_edges_or: fixes f12 :: "face"   and f21 :: "face"   and oldF :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and newVertexList :: "nat list"   and a :: "nat"   and b :: "nat" assumes "(f12, f21) = split_face oldF ram1 ram2 newVertexList"   and "pre_split_face oldF ram1 ram2 newVertexList"   and "(a, b) \<in> \<E> oldF" shows "(a, b) \<in> \<E> f12 \<or> (a, b) \<in> \<E> f21"
union_inter_neutral: fixes A :: "'a set"   and B :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite A"   and "finite B"   and "\<forall>x\<in>A \<inter> B. g x = (0::'b)" shows "sum g (A \<union> B) = sum g A + sum g B"
Rep_combine_matrix: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and A :: "'b matrix"   and B :: "'c matrix"   and j :: "nat"   and i :: "nat" assumes "f (0::'b) (0::'c) = (0::'a)" shows "Rep_matrix (combine_matrix f A B) j i = f (Rep_matrix A j i) (Rep_matrix B j i)"
cases: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and a1 :: "char list"   and a2 :: "char list" assumes "subcls1p P a1 a2" obtains C :: "char list" and D :: "char list" and rest :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list" where "a1 = C" and "a2 = D" and "class P C = \<lfloor>(D, rest)\<rfloor>" and "C \<noteq> Object"
resample_def: shows "resample \<equiv> curry (spmf.fixp_fun (\<lambda>resample (A, B). spmf_of_set A \<bind> (\<lambda>x. if x \<in> B then return_spmf x else curry resample A B)))"
chop_singleton_left: shows "\<forall>r. \<exists>s. R_Chop(r,s,r)"
phiDefNodes_aux_single_pred: fixes g :: "'g"   and n :: "'node"   and m :: "'node"   and v :: "'var"   and un :: "'node list" assumes "predecessors g n = [m]" shows "phiDefNodes_aux g v (removeAll n un) m = phiDefNodes_aux g v un m"
upt_eq_lel_conv: fixes l :: "nat"   and h :: "nat"   and is1 :: "nat list"   and i :: "nat"   and is2 :: "nat list" shows "([l..<h] = is1 @ i # is2) = (is1 = [l..<i] \<and> is2 = [Suc i..<h] \<and> l \<le> i \<and> i < h)"
assms_1: shows "A \<union> B = C \<union> D"
power_eq_0_iff: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and a :: "'d"   and n :: "nat" assumes "class.semiring_1_no_zero_divisors one times plus zero" shows "(power.power one times a n = zero) = (a = zero \<and> 0 < n)"
BoolVal_in_type_universe: fixes v :: "bool" shows "BoolVal v \<in> type_universe BOOL"
assms: shows "(Re b - Re z) * (Im z - Im a) \<noteq> (Im b - Im z) * (Re z - Re a)"
assms_2: shows "typing_cond\<^sub>s\<^sub>s\<^sub>t (unlabel A)"
conga_sym: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'" shows "Tarski_neutral_dimensionless.CongA Bet Cong A' B' C' A B C"
reachable_append: fixes g :: "(node \<times> node) list"   and xs :: "node list"   and ys :: "node list" shows "reachable g (xs @ ys) = reachable g xs \<union> reachable g ys"
degree_Poly': fixes xs :: "'c list" assumes "xs \<noteq> []" shows "degree (Poly xs) < length xs"
CASE: fixes s' :: "'s"   and ce' :: "'s multiset"   and ee :: "'l el_step" assumes "c' = add_mset s' ce'"   and "e = le_rem_s ee"   and "((s, c), ee, s', ce') \<in> gtrp gtrs" shows "P"
map_spmf_bind_spmf: fixes f :: "'s \<Rightarrow> 'r"   and p :: "'t spmf"   and g :: "'t \<Rightarrow> 's spmf" shows "map_spmf f (p \<bind> g) = p \<bind> map_spmf f \<circ> g"
preorder_refl: fixes x :: "'a" shows "x \<le> x"
find_lambda_def: fixes f :: "'b \<Rightarrow> 'b" shows "fx0.find_lambda f \<equiv> (\<lambda>s. s\<lparr>l := 1, hare := f (tortoise s)\<rparr>) ;; while (\<lambda>s. hare s \<noteq> tortoise s) (\<lambda>s. s\<lparr>hare := f (hare s), l := l s + 1\<rparr>)"
intro: fixes \<U> :: "'b set"   and \<A> :: "'b set multiset"   and \<V> :: "'b set"   and \<B> :: "'b set multiset" assumes "sub_set_system \<U> \<A> \<V> \<B>"   and "incidence_system \<U> \<A>" shows "sub_incidence_system \<U> \<A> \<V> \<B>"
lm073: fixes X :: "'c set"   and Y :: "'d set" shows "injections X Y = totalRels X Y \<inter> injectionsUniverse"
starP2_star_n: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'a"   and Y :: "nat \<Rightarrow> 'b" shows "(*p2* P) (star_n X) (star_n Y) = (\<forall>\<^sub>F n in \<U>. P (X n) (Y n))"
distinct_mapI: fixes f :: "'h \<Rightarrow> 'g"   and l :: "'h list" assumes "distinct (map f l)" shows "distinct l"
ilist_i_take_eq_conv: fixes f :: "nat \<Rightarrow> 'd"   and g :: "nat \<Rightarrow> 'd" shows "(f = g) = (\<forall>n. f \<Down> n = g \<Down> n)"
rbt_delete_is_rbt: fixes t :: "('a, 'c) rbt"   and k :: "'a" assumes "is_rbt t" shows "is_rbt (rbt_delete k t)"
alluopairs_set1: fixes xs :: "'b list" shows "set (alluopairs xs) \<subseteq> {(x, y). x \<in> set xs \<and> y \<in> set xs}"
abs_numeral: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and n :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs (numeral.numeral one plus n) = numeral.numeral one plus n"
clock_simTrans_def: fixes agents :: "'b odlist"   and jkbp :: "'b \<Rightarrow> ('b, 'd, 'e) GC list"   and envAction :: "'c \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('b \<Rightarrow> 'e) \<Rightarrow> 'c \<Rightarrow> 'c"   and envVal :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and envObs :: "'b \<Rightarrow> 'c \<Rightarrow> 'g" shows "ClockView.clock_simTrans agents jkbp envAction envTrans envVal envObs \<equiv> \<lambda>a (Y, X). let X' = ClockView.clock_trans agents jkbp envAction envTrans envVal envObs Y X; Y' = ClockView.clock_trans agents jkbp envAction envTrans envVal envObs Y Y in map (\<lambda>obs. clock_mkSuccs (envObs a) obs Y') (map (envObs a) (toList X'))"
simple_labels_in_sterms: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and p :: "('f, 'g, 'e, 'h) seqp"   and l :: "'h" assumes "simple_labels \<Gamma>"   and "wellformed \<Gamma>"   and "\<exists>pn. p \<in> subterms (\<Gamma> pn)"   and "l \<in> labels \<Gamma> p" shows "\<forall>p'\<in>sterms \<Gamma> p. l \<in> labels \<Gamma> p'"
DAinRotate: fixes p :: "('a, 'b) Combinators" assumes "DenyAll \<in> set (policy2list p)" shows "DenyAll \<in> set (policy2list (rotatePolicy p))"
path_in_condensation_impl_path: fixes a :: "'val set"   and b :: "'val set"   and g :: "'g"   and P :: "'val set"   and \<phi>\<^sub>a :: "'val"   and \<phi>\<^sub>b :: "'val" assumes "(a, b) \<in> (condensation_edges g P)\<^sup>+"   and "\<phi>\<^sub>a \<in> a"   and "\<phi>\<^sub>b \<in> b" shows "(\<phi>\<^sub>a, \<phi>\<^sub>b) \<in> (induced_phi_graph g P)\<^sup>*"
smalleraux3: fixes x :: "'b set"   and l :: "'b set list"   and y :: "'b set"   and a :: "('b, 'c) Combinators"   and b :: "('b, 'c) Combinators" assumes "x \<in> set l"   and "y \<in> set l"   and "x \<noteq> y"   and "x = bothNet a"   and "y = bothNet b"   and "smaller a b l"   and "singleCombinators [a]"   and "singleCombinators [b]" shows "\<not> smaller b a l"
basic1: shows "\<forall>u v. positionVector (u \<leadsto> v) = positionVector u \<oplus> v"