hcju/isabelleps huggingface dataset:
run_reachable': fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c stream"   and x\<^sub>0 :: "'c" assumes "Graph_Defs.run E (x ## xs)"   and "E\<^sup>*\<^sup>* x\<^sub>0 x" shows "pred_stream (E\<^sup>*\<^sup>* x\<^sub>0) xs"
erf_cnj: fixes z :: "complex" shows "erf (cnj z) = cnj (erf z)"
DBFair_1a: shows "\<turnstile> \<box>[DBNext]_(inp, mid, out, q1, q2) \<and> WF(DBPass)_(inp, mid, out, q1, q2) \<longrightarrow> (qc \<noteq> #[] \<and> q2 = #[] \<leadsto> q2 \<noteq> #[])"
exponential_density_def: fixes l :: "real"   and x :: "real" shows "exponential_density l x = (if x < 0 then 0 else l * exp (- x * l))"
maplets_lookup: fixes xs :: "'c list"   and ys :: "'d list"   and x :: "'c"   and y :: "'d" assumes "length xs = length ys"   and "distinct xs"   and "[xs [\<mapsto>] ys] x = Some y" shows "y \<in> set ys"
H_def: shows "H \<equiv> \<lambda>(x, y). (x, f (x, y))"
fbm: shows "(\<lambda>x. f (t + x) + f (t - x) - 2 * l) absolutely_integrable_on {- pi..pi}"
bind_SE_def: fixes f :: "'d \<Rightarrow> ('f \<times> 'd) option"   and g :: "'f \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'd) option" shows "bind\<^sub>S\<^sub>E f g = (\<lambda>\<sigma>. case f \<sigma> of None \<Rightarrow> None | Some (out, \<sigma>') \<Rightarrow> g out \<sigma>')"
typeof_lit_conf: fixes v :: "val"   and T :: "ty"   and 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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" assumes "typeof v = \<lfloor>T\<rfloor>" shows "P,h \<turnstile> v :\<le> T"
smult_fixed_length_sublist: fixes xs :: "'b list"   and l :: "nat"   and c :: "nat"   and i :: "nat"   and \<alpha> :: "'b" assumes "length xs = l * c"   and "i < c" shows "fixed_length_sublist (vec_smult \<alpha> xs) l i = vec_smult \<alpha> (fixed_length_sublist xs l i)"
pentagon': fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "ide k"   and "src f = trg g"   and "src g = trg h"   and "src h = trg k" shows "((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> k) \<cdot> \<a>\<^sup>-\<^sup>1[f, g \<star> h, k]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, k]) = \<a>\<^sup>-\<^sup>1[f \<star> g, h, k] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, h \<star> k]"
VDMGet: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and x :: "Var"   and y :: "Var"   and F :: "Field" shows "G \<rhd> Get x y F : \<lambda>s t. \<exists>l C Flds v. fst s y = RVal (Loc l) \<and> lookup (snd s) l = Some (C, Flds) \<and> lookup Flds F = Some v \<and> t = (update (fst s) x v, snd s)"
trans_MapOps_lookup_def: shows "trans_MapOps_lookup \<equiv> \<lambda>m k. Option.bind (trie_odlist_lookup m (fst k)) (\<lambda>m'. Mapping.lookup m' (snd k))"
cod_dom: fixes f :: "'j" assumes "J.arr f" shows "J.cod (J.dom f) = J.dom f"
dom_simp: fixes f :: "'a \<times> 'a" assumes "B.VxV.arr f" shows "B.VxV.dom f = (B.dom (fst f), B.dom (snd f))"
strictly_maximal_wrt_def: fixes A :: "'a"   and C :: "'a literal multiset" shows "strictly_maximal_wrt A C \<equiv> \<forall>B\<in>atms_of C. A \<noteq> B \<and> \<not> less_atm A B"
simps_1: shows "r_tuple_encode 0 = recf.Id 1 0"
Optimal_def: fixes f2 :: "real"   and c :: "real"   and e :: "real"   and l0 :: "nat" shows "Optimal f2 c e l0 \<equiv> (1 < e \<and> 1 < c) \<and> 0 < f2 \<and> e * c / (f2 * (min e c - 1)) \<le> real l0"
seqE: fixes g :: "'a"   and f :: "'a" assumes "L.seq g f" obtains "L.arr f" and "L.arr g" and "L.dom g = L.cod f"
lm080: fixes P :: "('d \<times> 'c) set"   and Q :: "('d \<times> 'c) set"   and X :: "'d set" shows "(P +< Q) `` (X \<inter> Domain Q) = Q `` X"
prems_5: shows "\<delta>_states \<delta> \<subseteq> Q"
mult_strict_left_mono_neg: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b < a"   and "c < (0::'a)" shows "c * a < c * b"
span_empty: shows "local.span {} = {0\<^sub>v n}"
State_Idle_append_replicate_NoMsg_output_Msg_eq_last_message: fixes localState :: "'d \<Rightarrow> 'e"   and output_fun :: "'d \<Rightarrow> 'f message_af"   and trans_fun :: "'g message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'g message_af list"   and c :: "'d"   and n :: "nat" assumes "State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun xs c))"   and "output_fun (f_Exec_Comp trans_fun xs c) \<noteq> NoMsg"   and "xs \<noteq> []" shows "last_message (map output_fun (f_Exec_Comp_Stream trans_fun (xs @ NoMsg\<^bsup>n\<^esup>) c)) = output_fun (f_Exec_Comp trans_fun xs c)"
tl_take: fixes n :: "nat"   and xs :: "'b list" shows "tl (take n xs) = take (n - 1) (tl xs)"
lt_minus_higher: fixes v :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys p" shows "lt (p - higher p v) = v"
simps_1: fixes T1 :: "'b trm"   and T2 :: "'b trm" shows "nvA (Eq T1 T2) = nvT T1 \<union> nvT T2"
set_less_aux_rec: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B"   and "A \<noteq> {}"   and "B \<noteq> {}" shows "(A \<sqsubset>' B) = (Min B < Min A \<or> Min A = Min B \<and> A - {Min A} \<sqsubset>' B - {Min A})"
exhaust: fixes y :: "'c Tree" obtains "y = E"   | x21 :: "'c" and x22 :: "'c Tree" and x23 :: "'c Tree" where "y = T x21 x22 x23"
add_trap_set_def: fixes t :: "Trap"   and s :: "'b sparc_state" shows "add_trap_set t s \<equiv> s\<lparr>traps := traps s \<union> {t}\<rparr>"
of_bl_append: fixes xs :: "bool list"   and ys :: "bool list" shows "of_bl (xs @ ys) = of_bl xs * 2 ^ length ys + of_bl ys"
seq_char: fixes g :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "VV.seq g f = (VV.arr f \<and> VV.arr g \<and> VxV.seq g f)"
multi_reduce_pres_denot: fixes e :: "exp"   and e' :: "exp" assumes "e \<longrightarrow>* e'" shows "E e = E e'"
circ_back_loop_fixpoint: fixes z :: "'a"   and y :: "'a" shows "z * y\<^sup>\<circ> * y \<squnion> z = z * y\<^sup>\<circ>"
exists_smooth_on: fixes x :: "'a" assumes "x \<in> f.src.carrier" shows "\<exists>c1\<in>f.src.atlas. \<exists>c2\<in>f.dest.atlas. x \<in> domain c1 \<and> f ` domain c1 \<subseteq> domain c2 \<and> k-smooth_on (codomain c1) (apply_chart c2 \<circ> f \<circ> inv_chart c1)"
th0: fixes n :: "nat" shows "- ((- (1::'a)) ^ n) = (- (1::'a)) ^ Suc n"
simps_21: fixes v :: "'a"   and va :: "bexp"   and vb :: "'a acom"   and vc :: "'a"   and vd :: "bexp"   and ve :: "'a acom"   and vf :: "'a acom"   and vg :: "'a" shows "({v} WHILE va DO vb {vc} \<sqsubseteq> IF vd THEN ve ELSE vf {vg}) = False"
real_tendsto_sandwich: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real"   and net :: "'a filter"   and h :: "'a \<Rightarrow> real"   and c :: "real" assumes "\<forall>\<^sub>F n in net. f n \<le> g n"   and "\<forall>\<^sub>F n in net. g n \<le> h n"   and "(f \<longlongrightarrow> c) net"   and "(h \<longlongrightarrow> c) net" shows "(g \<longlongrightarrow> c) net"
healthy_intros_10: fixes f :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and g :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "nearly_healthy (wlp f)"   and "nearly_healthy (wlp g)" shows "nearly_healthy (wlp (f \<Sqinter> g))"
qmsg_queue_contents: shows "qmsg \<TTurnstile>\<^sub>A (\<lambda>((msgs, q), a, msgs', q'). case a of receive m \<Rightarrow> set msgs' \<subseteq> set (msgs @ [m]) | _ \<Rightarrow> set msgs' \<subseteq> set msgs)"
eval_var_eq: fixes G :: "prog"   and s :: "state"   and t :: "Term.var"   and w :: "vals"   and s' :: "state" shows "G\<turnstile>s \<midarrow>In2 t\<succ>\<rightarrow> (w, s') = (\<exists>vf. w = In2 vf \<and> G\<turnstile>s \<midarrow>t=\<succ>vf\<rightarrow> s')"
properfactor_of_zero_2: fixes b :: "'a" assumes "b \<in> carrier R" shows "properfactor R b \<zero> = (b \<noteq> \<zero>)"
termination_0_1_immediate: fixes p :: "real"   and s :: "'a" assumes "\<And>s. guard s \<Longrightarrow> p \<le> spmf (map_spmf guard (iter (N s) s)) False"   and "0 < p"   and "\<And>s. guard s \<Longrightarrow> lossless_spmf (iter (N s) s)" shows "lossless_spmf (fuse.while s)"
det_not_inj_on: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat"   and B :: "'b mat" assumes "\<not> inj_on f {0..<n}" shows "Determinant.det (mat\<^sub>r n n (\<lambda>i. Matrix.row B (f i))) = (0::'b)"
us_le_mono: fixes i :: "nat"   and j :: "nat"   and y :: "nat list" assumes "i < j" shows "\<^bold>u y i \<le>\<^sub>v \<^bold>u y j"
gen_binomial_complex: fixes z :: "complex"   and a :: "complex" assumes "cmod z < 1" shows "(\<lambda>n. (a gchoose n) * z ^ n) sums (1 + z) powr a"
Gram_Schmidt_column_k_iarrays_efficient2_def: fixes A :: "real iarray iarray"   and k :: "nat" shows "Gram_Schmidt_column_k_iarrays_efficient2 A k = tabulate2 (nrows_iarray A) (ncols_iarray A) (let col_k = column_iarray k A; col = col_k - (\<Sum>x\<leftarrow>map (\<lambda>n. column_iarray n A) [0..<k]. (col_k \<bullet>i x / (x \<bullet>i x)) *\<^sub>R x) in (\<lambda>a b. (if b = k then col else column_iarray b A) !! a))"
extended_distance_set_def: shows "extended_distance_set = {z. extended_distance z z = 0}"
DIGraxiom_def: shows "DIGraxiom = (($\<phi> vid1 local.empty \<rightarrow> [[EvolveODE (OVar vid1) ($\<phi> vid1 local.empty)]]Geq (Differential (f1 fid1 vid1)) (Differential (f1 fid2 vid1))) \<rightarrow> ($\<phi> vid1 local.empty \<rightarrow> Greater (f1 fid1 vid1) (f1 fid2 vid1)) \<rightarrow> [[EvolveODE (OVar vid1) ($\<phi> vid1 local.empty)]]Greater (f1 fid1 vid1) (f1 fid2 vid1))"
A: fixes x :: "'x"   and s :: "'a" shows "f x s \<noteq> dSUCCEED"
vcard_mult: fixes x :: "V"   and y :: "V" shows "vcard (x * y) = vcard x \<otimes> vcard y"
monoE: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "class.order less_eq less"   and "order.mono less_eq f"   and "less_eq x y" obtains "f x \<le> f y"
set_integral_diff_1: fixes M :: "'c measure"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "set_integrable M A f"   and "set_integrable M A g" shows "set_integrable M A (\<lambda>x. f x - g x)"
IH: shows "smult (lead_coeff p) (\<Prod>z\<in>{z. poly p z = 0}. [:- z, 1:] ^ order z p) = p"
simps_2: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" shows "takeWhile P (x # xs) = (if P x then x # takeWhile P xs else [])"
cont: shows "(\<lambda>x. indicat_real (- \<real>\<^sub>\<le>\<^sub>0) x *\<^sub>R Ln x) \<in> borel_measurable borel"
euclid_Col_cart2_incident: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2"   and l :: "proj2_line" assumes "z_non_zero p"   and "z_non_zero q"   and "z_non_zero r"   and "p \<noteq> q"   and "proj2_incident p l"   and "proj2_incident q l"   and "real_euclid.Col (cart2_pt p) (cart2_pt q) (cart2_pt r)" shows "proj2_incident r l"
N'_def: shows "N' = (\<lambda>i. if i \<in> I then M' i else M' j)"
refl_onD1: fixes A :: "'e set"   and r :: "('e \<times> 'e) set"   and x :: "'e"   and y :: "'e" assumes "refl_on A r"   and "(x, y) \<in> r" shows "x \<in> A"
le_Max_insertI2: fixes A :: "'a Word.word set"   and x :: "'a Word.word"   and b :: "'a Word.word" assumes "finite A"   and "A \<noteq> {}"   and "x \<le>s signed.Max A" shows "x \<le>s signed.Max (insert b A)"
distinct_59: fixes x3 :: "db"   and x10 :: "nat" shows "DBAbsN x3 \<noteq> DBNum x10"
order_iff_strict: fixes b :: "'d"   and a :: "'d" shows "(b \<le> a) = (b < a \<or> a = b)"
cross_skew: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec" shows "x \<times> y = - (y \<times> x)"
in_set_spmf_iff_spmf: fixes x :: "'c"   and p :: "'c spmf" shows "(x \<in> set_spmf p) = (spmf p x \<noteq> 0)"
split_with_info_def: fixes X :: "'f set" shows "split_with_info X = SPEC (\<lambda>(S, Y, YS). X = Y \<union> YS)"
A_set_def: shows "A_set = {1, 1 / M} \<union> {\<bar>x' - x\<bar> |x'. x' \<in> roots}"
inject: fixes x2 :: "'g"   and y2 :: "'g" shows "(Some x2 = Some y2) = (x2 = y2)"
IH: fixes s :: "'a" shows "<I (Suc k) s> imp_for' (Suc k) u f s <I u>"
encode_problem_parallel_complete_vi_f: fixes \<Pi> :: "'b strips_problem"   and \<pi> :: "'b strips_operator list list"   and l :: "nat"   and v :: "'b" assumes "is_valid_problem_strips \<Pi>"   and "is_parallel_solution_for_problem \<Pi> \<pi>"   and "length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) \<le> l"   and "l < length \<pi> + 1" shows "valuation_for_plan \<Pi> \<pi> (State l (index (\<Pi>\<^sub>\<V>) v)) = valuation_for_plan \<Pi> \<pi> (State (length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1) (index (\<Pi>\<^sub>\<V>) v))"
simps_2: fixes u :: "'b"   and v :: "'b"   and w :: "'b"   and e :: "'b \<times> 'b"   and es :: "('b \<times> 'b) list" shows "sd_path (u, v) w (e # es) = (if e = (u, v) then [(u, w), (w, v)] else if e = (v, u) then [(v, w), (w, u)] else [e]) @ sd_path (u, v) w es"
assms_1: fixes S :: "'a set" assumes "S \<in> \<F>" shows "closedin (top_of_set (\<Union> \<F>)) S"
cring_def: fixes R :: "('e, 'f) ring_scheme" shows "cring R \<equiv> ring R \<and> Group.comm_monoid R"
euler_genus_def: shows "rev.H.euler_genus \<equiv> (int (2 * card rev.H.sccs) - int (card rev.H.isolated_verts) - rev.H.euler_char) div 2"
Acts_eq: fixes init :: "'c set"   and acts :: "('c \<times> 'c) set set"   and allowed :: "('c \<times> 'c) set set" shows "Acts (mk_program (init, acts, allowed)) = insert Id acts"
sas_plus_parallel_plan_to_strips_parallel_plan_def: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and \<psi> :: "('a, 'b) sas_plus_operator list list" shows "\<phi>\<^sub>P \<Psi> \<psi> \<equiv> map (map (sasp_op_to_strips \<Psi>)) \<psi>"
base_in_s: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set" assumes "kuhn_simplex p n base upd s" shows "base \<in> s"
mix_pmf_partition: fixes y :: "'b"   and p :: "'b pmf" assumes "y \<in> set_pmf p"   and "set_pmf p - {y} \<noteq> {}" obtains a :: "real" and q :: "'b pmf" where "a \<in> {0<..<1}" and "set_pmf q = set_pmf p - {y}" and "p = mix_pmf a q (return_pmf y)"
success_of_listI: fixes xs :: "'b list"   and h :: "heap" shows "success (Array.of_list xs) h"
fg_start_round_strict_mono: fixes A :: "('f, 'g, 'h, 'i) CHOAlgorithm_scheme"   and rho :: "nat \<Rightarrow> ('g, 'f, 'h) config"   and HOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and SHOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and coords :: "nat \<Rightarrow> 'f \<Rightarrow> 'f"   and p :: "'f" assumes "fg_run A rho HOs SHOs coords" shows "strict_mono (fg_start_round rho p)"
rec_2: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and x2 :: "'b" shows "rec_option f1 f2 (Some x2) = f2 x2"
u1u2_2: shows "t1 \<in> {t. linepath a b t \<bullet> c = d \<and> 0 \<le> t \<and> t \<le> 1}"
assms_5: shows "cbelow_on (dgrad_p_set d m) (\<prec>\<^sub>p) z (\<lambda>a b. red F a b \<or> red F b a) p q"
star_left_induct_equal: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<squnion> x * y = y" shows "x\<^sup>\<star> * z \<le> y"
mod_ring_rel64_def: fixes x :: "uint64"   and y :: "'a mod_ring" shows "mod_ring_rel64 x y = (\<exists>z. urel64 x z \<and> mod_ring_rel z y)"
distinct_1: fixes x1 :: "'k"   and x2 :: "'l" shows "Error x1 \<noteq> Ok x2"
constant_in_extended_predist_pos_1: shows "0 < epsilonG TYPE('a)"
cod_mkArr: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's" assumes "S.arr (S.mkArr A B F)" shows "S.cod (S.mkArr A B F) = S.mkIde B"
arg_min_nat_le: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and m :: "'a \<Rightarrow> nat" assumes "P x" shows "m (arg_min m P) \<le> m x"
that: fixes a' :: "'edge"   and as' :: "'edge list" assumes "as = a' # as'"   and "n = sourcenode a'"   and "valid_edge a'"   and "targetnode a' -as'\<rightarrow>* n'" shows "thesis"
low_indistinguishable_sym: fixes mds :: "Mode \<Rightarrow> 'Var set"   and lc :: "('Var, 'AExp, 'BExp) Stmt"   and lc' :: "('Var, 'AExp, 'BExp) Stmt" assumes "lc \<sim>\<^bsub>mds\<^esub> lc'" shows "lc' \<sim>\<^bsub>mds\<^esub> lc"
Cons: fixes xs :: "char list exp list"   and ys :: "val list"   and zs :: "char list exp list"   and y :: "val" assumes "map_val2 xs ys zs" shows "map_val2 (Val y # xs) (y # ys) zs"
C''_def: shows "C'' = complex_of_real a powr - s"
mkPinj_ci: fixes l :: "'a list"   and a :: "nat"   and B :: "pol" shows "Ipol l (mkPinj a B) = Ipol l (Pinj a B)"
Max_wrt_among_subset_2: fixes A :: "'alt set" shows "Max_wrt_among (R j) A \<subseteq> A"
encode_negative_transition_frame_axiom': fixes t :: "nat"   and deleting_operators :: "(nat \<times> nat) strips_operator list"   and sv :: "sat_plan_variable"   and v_idx :: "nat" assumes "t < h"   and "set deleting_operators \<subseteq> set ((\<phi> prob_with_noop abs_prob )\<^sub>\<O>)"   and "sv \<in> atoms (\<^bold>\<not> (Atom (State t v_idx)) \<^bold>\<or> (Atom (State (Suc t) v_idx) \<^bold>\<or> \<^bold>\<Or>map (\<lambda>op. Atom (Operator t (index ((\<phi> prob_with_noop abs_prob )\<^sub>\<O>) op))) deleting_operators))" shows "valid_state_var sv"
finint: shows "\<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M \<noteq> \<infinity>"
cSUP_le_iff: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and u :: "'c" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_above less_eq (f ` A)" shows "less_eq (Sup (f ` A)) u = (\<forall>x\<in>A. less_eq (f x) u)"
exhaust: fixes y :: "final_decision" obtains "y = FinalAllow"   | "y = FinalDeny"
sets_PiM: fixes I :: "'c set"   and M :: "'c \<Rightarrow> 'd measure" shows "sets (Pi\<^sub>M I M) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) (prod_algebra I M)"
at_within_Icc_at_right: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.order_topology less_eq less open"   and "less a b" shows "topological_space.at_within open a (ord.atLeastAtMost less_eq a b) = topological_space.at_within open a (ord.greaterThan less a)"
SSup_def: shows "SSup = Sup \<circ> Rep_downset"
is_rewrite_ord_canon_rewriterD1: 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"   and v :: "'t"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_rewrite_ord rword"   and "is_canon_rewriter rword A u p"   and "is_canon_rewriter rword A v q"   and "lt p adds\<^sub>t v"   and "lt q adds\<^sub>t u" shows "lt p = lt q"
case_optionE: fixes P :: "bool"   and Q :: "'r \<Rightarrow> bool"   and x :: "'r option" assumes "case x of None \<Rightarrow> P | Some y \<Rightarrow> Q y" obtains "x = None" and "P"   | y :: "'r" where "x = Some y" and "Q y"
mult_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<parallel> b \<parallel> c = a \<parallel> (b \<parallel> c)"
not_less_eq: fixes m :: "nat"   and n :: "nat" shows "(\<not> m < n) = (n < Suc m)"
simps_3: fixes ll :: "'b tree"   and a :: "'b"   and lr :: "'b tree"   and b :: "'b"   and r :: "'b tree" shows "del_min \<langle>\<langle>ll, a, lr\<rangle>, b, r\<rangle> = (if ll = \<langle>\<rangle> then \<langle>lr, b, r\<rangle> else \<langle>del_min ll, a, \<langle>lr, b, r\<rangle>\<rangle>)"
prod_assn_precise: fixes P1 :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and P2 :: "'c \<Rightarrow> 'd \<Rightarrow> assn" assumes "precise P1"   and "precise P2" shows "precise (P1 \<times>\<^sub>a P2)"
exp_exists: fixes m :: "nat" shows "m = (\<Sum>x<nlen m. m div 10 ^ x mod 10 * 10 ^ x)"
root_int_main_def: fixes p :: "nat"   and n :: "int" shows "root_int_main p n \<equiv> if p = 0 then (1, n = 1) else let pm = p - 1 in root_int_main' pm (int pm) (int p) (start_value n p) n"
map_id: fixes t :: "('d, 'e) nondetT" shows "map_nondetT' id id t = t"
r_univuniv_def: shows "r_univuniv \<equiv> Cn 2 r_phi [Cn 2 r_phi [recf.Id 2 0, recf.Id 2 0], recf.Id 2 1]"
pred_subset_eq: fixes R :: "'c set"   and S :: "'c set" shows "((\<lambda>x. x \<in> R) \<le> (\<lambda>x. x \<in> S)) = (R \<subseteq> S)"
\<Gamma>_consts_simps_7: fixes a :: "'atom"   and t :: "nat" shows "\<Gamma> (Fun (PubConstAtom a t) []) = Var (prot_atom.Atom a)"
countable_finite: fixes S :: "'a set" assumes "finite S" shows "countable S"
tendsto_mult_left: fixes f :: "'e \<Rightarrow> 'f"   and l :: "'f"   and F :: "'e filter"   and c :: "'f" assumes "(f \<longlongrightarrow> l) F" shows "((\<lambda>x. c * f x) \<longlongrightarrow> c * l) F"
bilateral_substitutes_on_def: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "Bossy'.bilateral_substitutes_on A f = (\<not> (\<exists>B\<subseteq>A. \<exists>a b. {a, b} \<subseteq> A \<and> Xd a \<notin> Xd ` B \<and> Xd b \<notin> Xd ` B \<and> b \<notin> f (B \<union> {b}) \<and> b \<in> f (B \<union> {a, b})))"
SC_ZObis_Seq: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" shows "SC_ZObis (c1 ;; c2) = (SC_siso c1 \<and> SC_ZObis c2 \<or> SC_ZObis c1 \<and> SC_discr c2)"
bounds_compare_contradictory_7: fixes c :: "'a"   and b :: "'a option" assumes "c <\<^sub>l\<^sub>b b"   and "c \<ge>\<^sub>l\<^sub>b b" shows "False"
cInf_le_finite: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "'a set"   and x :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "finite X"   and "x \<in> X" shows "less_eq (Inf X) x"
arr_inv: fixes f :: "'a1" assumes "C1.iso f" shows "C1.arr (C1.inv f)"
flat_Stream: fixes xs :: "'b list"   and ws :: "'b list stream" assumes "xs \<noteq> []" shows "flat (xs ## ws) = xs @- flat ws"
fmdrop_lookup: fixes a :: "'b"   and m :: "('b, 'a) fmap" shows "fmlookup (fmdrop a m) a = None"
lift_state_oracle_def: fixes F :: "('k \<Rightarrow> 'i \<Rightarrow> (('j \<times> 'g) \<times> 'k) spmf) \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> (('j \<times> 'g) \<times> 'h) spmf"   and "oracle" :: "'g \<times> 'k \<Rightarrow> 'i \<Rightarrow> ('j \<times> 'g \<times> 'k) spmf" shows "lift_state_oracle F oracle = (\<lambda>(t, s') a. map_spmf rprodl (F ((Pair t ---> id ---> map_spmf lprodr) oracle) s' a))"
monic_lhs: shows "monic (monom 1 CARD('a) - monom 1 1)"
open_contains_ball_eq: fixes S :: "'a set"   and x :: "'a" assumes "open S" shows "(x \<in> S) = (\<exists>e>0. ball x e \<subseteq> S)"
resumption_ord_trans: fixes r :: "('f, 'g, 'h) Resumption.resumption"   and r' :: "('f, 'g, 'h) Resumption.resumption"   and r'' :: "('f, 'g, 'h) Resumption.resumption" assumes "resumption_ord r r'"   and "resumption_ord r' r''" shows "resumption_ord r r''"
h_def: shows "h \<equiv> fps_right_inverse g y * fps_right_inverse f x"
lmap_conv_fixp: fixes f :: "'c \<Rightarrow> 'd" shows "lmap f \<equiv> llist.fixp_fun (\<lambda>lmap xs. case xs of LNil \<Rightarrow> LNil | LCons x xs \<Rightarrow> LCons (f x) (lmap xs))"
SN_rel_on_alt_s_empty: fixes R :: "('a \<times> 'a) set" shows "SN_rel_on_alt R {} = SN_on R"
compl: fixes A :: "real set" assumes "A \<in> sets borel"   and "P A" shows "P (- A)"
BOresChainSimps_2: fixes N' :: "'d"   and P' :: "('d, 'e, 'f) psi"   and xvec :: "name list"   and N :: "'d"   and P :: "('d, 'e, 'f) psi" shows "(N' \<prec>' P' = \<lparr>\<nu>*xvec\<rparr>N \<prec>' P) = (xvec = [] \<and> N = N' \<and> P = P')"
coprime_Suc_right_nat: fixes n :: "nat" shows "coprime n (Suc n)"