hcju/isabelleps huggingface dataset:
supdate_glob_def: fixes s :: "s"   and i :: "nat"   and j :: "nat"   and v :: "v" shows "supdate_glob s i j v = (let k = sglob_ind s i j in supdate_glob_s s k v)"
angle_altdef: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "angle a b c = arccos ((a - b) \<bullet> (c - b) / (dist a b * dist c b))"
enc_satisfies_pred_iff_source_target_satisfies_pred_1: fixes Pred :: "('procS, 'procT) Proc \<times> ('procS, 'procT) Proc \<Rightarrow> bool" shows "(\<forall>S. Pred (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>))) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> (\<forall>(P, Q)\<in>Rel. Pred (P, Q)))"
inner_Basis: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and Basis :: "'e set"   and u :: "'e"   and v :: "'e" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "u \<in> Basis"   and "v \<in> Basis" shows "inner u v = (if u = v then 1 else 0)"
reflp_inf: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "reflp r"   and "reflp s" shows "reflp (inf r s)"
func_space_neg: fixes f :: "'e \<Rightarrow> 'a"   and S :: "'e set" assumes "f \<in> carrier (func_space S)" shows "\<ominus>\<^bsub>func_space S\<^esub> f = (\<lambda>v. if v \<in> S then \<ominus> f v else undefined)"
vempty_is_zet: shows "[]\<^sub>\<circ> \<in>\<^sub>\<circ> Vset \<alpha>"
simps_1: fixes edge_rev :: "'e \<Rightarrow> 'e"   and edge_succ :: "'e \<Rightarrow> 'e"   and more :: "'f" shows "edge_rev \<lparr>edge_rev = edge_rev, edge_succ = edge_succ, \<dots> = more\<rparr> = edge_rev"
hh': shows "\<forall>y\<in>carrier C. h (h' y) = y"
items_mono: fixes G :: "msg set"   and H :: "msg set" assumes "G \<subseteq> H" shows "items G \<subseteq> items H"
pcs: shows "pcs xt \<subseteq> {..<length ins}"
op_map_pick_remove_alt: fixes m :: "'e \<Rightarrow> 'f option"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> ('e \<Rightarrow> 'f option) \<Rightarrow> 'd nres" shows "op_map_pick_remove m \<bind> (\<lambda>((k, v), m). f k v m) = SPEC (\<lambda>(k, v). m k = Some v) \<bind> (\<lambda>(k, v). Let (m |` (- {k})) (f k v))"
Increasing_union: fixes F :: "'b program"   and f :: "'b \<Rightarrow> 'c multiset"   and g :: "'b \<Rightarrow> 'c multiset" assumes "F \<in> Increasing f"   and "F \<in> Increasing g" shows "F \<in> Increasing (\<lambda>s. f s + g s)"
Sup_fin_Sup: 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 bot :: "'a"   and top :: "'a"   and A :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "finite A"   and "A \<noteq> {}" shows "semilattice_sup.Sup_fin sup A = Sup A"
x32: shows "\<bar>x\<bar> ^ 3 = 2"
sff_3: fixes a :: "int poly"   and i :: "nat"   and b :: "int poly"   and j :: "nat" assumes "(a, i) \<in> set qis"   and "(b, j) \<in> set qis"   and "(a, i) \<noteq> (b, j)" shows "algebraic_semidom_class.coprime a b"
morph_rev_len: fixes u :: "'a list" shows "\<^bold>|f (rev u)\<^bold>| = \<^bold>|f u\<^bold>|"
discr_eff_indis: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "('d, 'c, 'a) cmd"   and i :: "nat"   and s :: "'b" assumes "PL_Indis cval indis"   and "PL_Indis.discr aval tval cval indis c"   and "i < PL.brn c" shows "(s, PL.eff aval tval cval c s i) \<in> indis"
convex_polygon_segments_of_aform: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "1 < length (half_segments_of_aform X)" shows "convex_polygon (segments_of_aform X)"
poly2_poly_x_mult_y: fixes p :: "'a poly"   and x :: "'a"   and y :: "'a" shows "poly2 (poly_x_mult_y p) x y = poly p (x * y)"
wf_merge: fixes n :: "nat"   and I :: "nat set"   and t1 :: "'b option list"   and J :: "nat set"   and t2 :: "'b option list"   and V :: "nat set"   and t :: "'b option list" assumes "wf_tuple n I t1"   and "wf_tuple n J t2"   and "V = I \<union> J"   and "t = New_max.merge t1 t2" shows "wf_tuple n V t"
assms_2: shows "(\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)) \<and> (\<forall>c\<in>clk_set A. v c \<le> n) \<and> (\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k))"
B: shows "sp_impl_subj_obj s1 (Step.partition tid) (PAGE page) (case dir of SEND \<Rightarrow> READ | RECV \<Rightarrow> WRITE) = sp_impl_subj_obj s2 (Step.partition tid) (PAGE page) (case dir of SEND \<Rightarrow> READ | RECV \<Rightarrow> WRITE)"
simps_1: fixes P :: "complex mat" shows "wlp SKIP P = P"
codomains_comp: fixes g :: "'a"   and f :: "'a" assumes "L.seq g f" shows "L.codomains (L.comp g f) = L.codomains g"
cr: fixes l :: "'V Standard_Constant"   and v :: "'V + 'V'" assumes "(l, v) \<in> (\<lambda>c. (S_Const c, Inl c)) ` C" shows "getRel l G = {(v, v)}"
monoseq_convergent: fixes X :: "nat \<Rightarrow> real"   and B :: "real" assumes "monoseq X"   and "\<And>i. \<bar>X i\<bar> \<le> B" obtains L :: "real" where "X \<longlonglongrightarrow> L"
is_refT_def: fixes T :: "ty" shows "is_refT T \<equiv> T = NT \<or> (\<exists>C. T = Class C)"
Gauss_Jordan_column_k_det_P_def: fixes A' :: "'d \<times> nat \<times> (('d, 'e) vec, 'f) vec"   and k :: "nat" shows "Gauss_Jordan_column_k_det_P A' k = (let det_P = fst A'; i = fst (snd A'); A = snd (snd A'); from_nat_i = mod_type_class.from_nat i; from_nat_k = mod_type_class.from_nat k in if (\<forall>m\<ge>from_nat_i. A $ m $ from_nat_k = (0::'d)) \<or> i = nrows A then (det_P, i, A) else let gauss = Gauss_Jordan_in_ij_det_P A from_nat_i from_nat_k in (fst gauss * det_P, i + 1, snd gauss))"
Ints_sum: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>a. a \<in> A \<Longrightarrow> f a \<in> \<int>" shows "sum f A \<in> \<int>"
induct: fixes g :: "('c \<Rightarrow> bool) \<times> ('c \<Rightarrow> 'a \<times> 'c)"   and x :: "'c"   and Pa :: "'c \<Rightarrow> bool" assumes "local.filter_has_next g x"   and "\<And>s. \<lbrakk>fst g s; P (fst (snd g s))\<rbrakk> \<Longrightarrow> Pa s"   and "\<And>s. \<lbrakk>fst g s; \<not> P (fst (snd g s)); local.filter_has_next g (snd (snd g s)); Pa (snd (snd g s))\<rbrakk> \<Longrightarrow> Pa s" shows "Pa x"
f_def: fixes a :: "'a"   and b :: "'b" shows "f a b = cconstruct basisA (\<lambda>x. if x = a then b else (0::'b))"
lookup_higher: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and u :: "'t"   and v :: "'t" shows "lookup (higher p u) v = (if u \<prec>\<^sub>t v then lookup p v else (0::'c))"
prikey_mpair_fst: fixes K :: "nat"   and Y :: "msg"   and s :: "(agent \<times> msg) set" assumes "(Spy, \<lbrace>AgrKey (PriK K), Y\<rbrace>) \<in> s"   and "insert (Spy, AgrKey (PubK {})) (insert (Spy, AgrKey (PubK {Rev_PriK})) ({Spy} \<times> ((\<lambda>x. EncKey (SigK x)) ` bad_sigk \<union> (\<lambda>x. AgrKey (PriK (Sec_PriK x))) ` bad_sec_prik \<union> (\<lambda>x. AgrKey (PriK (Tok_PriK x))) ` bad_tok_prik \<union> range (\<lambda>x. EncKey (VerK x)) \<union> range (\<lambda>x. AgrKey (PubK {Sec_PriK x, Rev_PriK})) \<union> range (\<lambda>x. AgrKey (PubK {Tok_PriK x})) \<union> range (\<lambda>(n, m). AgrKey (PubK {Tok_PriK n, Sec_PriK m, Rev_PriK})) \<union> range (\<lambda>(n, m). Hash (AgrKey (PubK {Tok_PriK n, Sec_PriK m, Rev_PriK}))) \<union> range (\<lambda>x. \<langle>x, AgrKey (PubK {Tok_PriK x})\<rangle>) \<union> {\<langle>n, AgrKey (PriK (Tok_PriK n))\<rangle> |n. n \<in> bad_tok_prik} \<union> {\<langle>n, AgrKey (PubK {Tok_PriK n, Sec_PriK m, Rev_PriK})\<rangle> |n m. (n, m) \<in> bad_id} \<union> {\<langle>n, Hash (AgrKey (PubK {Tok_PriK n, Sec_PriK m, Rev_PriK}))\<rangle> |n m. (n, m) \<in> bad_id}))) \<Turnstile> s" shows "(Spy, AgrKey (PriK K)) \<in> s"
M: fixes a :: "nat"   and b :: "nat" shows "a < Suc b \<equiv> a = b \<or> a < b"
diagseq_def: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool"   and i :: "nat" assumes "Diagonal_Subsequence.subseqs P" shows "subseqs.diagseq P i = subseqs.seqseq P i i"
verit_comp_simplify1_1: fixes a :: "'b" shows "(a < a) = False"
irreducible\<^sub>dI: fixes p :: "'a poly" assumes "0 < degree p"   and "\<And>q r. \<lbrakk>0 < degree q; degree q < degree p; 0 < degree r; degree r < degree p; p = q * r\<rbrakk> \<Longrightarrow> False" shows "irreducible\<^sub>d p"
differing_vars_neg_intro: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and mem\<^sub>1 :: "'a \<Rightarrow> 'b"   and x :: "'a"   and mems :: "(('a \<Rightarrow> 'b) \<times> ('a \<Rightarrow> 'b)) list"   and i :: "nat"   and mem\<^sub>2 :: "'a \<Rightarrow> 'b" assumes "sifum_security_init dma \<C>_vars eval"   and "mem\<^sub>1 x = fst (mems ! i) x"   and "mem\<^sub>2 x = snd (mems ! i) x" shows "x \<notin> sifum_security_init.differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i"
more_defined_def: fixes p :: "'c \<Rightarrow> 'd decision option"   and q :: "'c \<Rightarrow> 'd decision option" shows "more_defined p q = (dom q \<subseteq> dom p)"
Const_add: fixes a :: "'a"   and b :: "'a" shows "Const (a + b) = Const a + Const b"
coeff_map_poly_hom: fixes hom :: "'a \<Rightarrow> 'b"   and p :: "'a poly"   and i :: "nat" assumes "zero_hom hom" shows "coeff (map_poly hom p) i = hom (coeff p i)"
step: fixes n :: "nat" assumes "i \<le> n"   and "n < j"   and "\<forall>n'>n. n' \<le> j \<longrightarrow> P n'" shows "P n"
rtrancl3p_trans: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and bs :: "'b list"   and a' :: "'a"   and bs' :: "'b list"   and a'' :: "'a" assumes "rtrancl3p r a bs a'"   and "rtrancl3p r a' bs' a''" shows "rtrancl3p r a (bs @ bs') a''"
sim_fD: fixes u :: "'e"   and v :: "'e"   and M :: "('f, 'g, 'e) KripkeStructure"   and a :: "'f"   and M' :: "('f, 'g, 'h) KripkeStructure"   and f :: "'e \<Rightarrow> 'h" assumes "(u, v) \<in> relations M a"   and "sim M M' f" shows "(f u, f v) \<in> relations M' a"
trace_simps_6: fixes sig :: "'j signature" shows "trace sig \<equiv> filter (\<lambda>a. a \<in> externals sig) \<circ> schedule"
space_measure_of_conv: fixes \<Omega> :: "'d set"   and A :: "'d set set"   and \<mu> :: "'d set \<Rightarrow> ennreal" shows "space (measure_of \<Omega> A \<mu>) = \<Omega>"
powhr_add: fixes x :: "real star"   and a :: "real star"   and b :: "real star" shows "x powhr (a + b) = x powhr a * x powhr b"
IH: fixes y :: "V"   and b :: "V" assumes "y \<in> elts x"   and "rank b \<sqsubset> y" shows "b \<in> elts (Vset y)"
Diff_Int_distrib: fixes C :: "'h set"   and A :: "'h set"   and B :: "'h set" shows "C \<inter> (A - B) = C \<inter> A - C \<inter> B"
order_pq: fixes x :: "'a" shows "order x p = order x q + min (order x p) (order x (pderiv p))"
currentD_IN_eq_0: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" assumes "current \<Gamma> f" shows "(d_IN f x = 0) = (\<forall>y. f (y, x) = 0)"
simps: fixes ins :: "instr array"   and s :: "integer"   and pc :: "nat" shows "exec' ins s pc = (if pc < array_length ins then case array_get ins pc of AssI ns bs \<Rightarrow> [pc] | TestI b d \<Rightarrow> if bval b s then exec' ins s (pc + 1) else let pc' = nat (int (pc + 1) + d) in if pc < pc' then exec' ins s pc' else [pc'] | ChoiceI bis \<Rightarrow> let succs = concat (map (\<lambda>(b, i). if bval b s then [nat (int (pc + 1) + i)] else []) bis) in if succs = [] then [pc] else concat (map (\<lambda>pc'. if pc < pc' then exec' ins s pc' else [pc']) succs) | GotoI d \<Rightarrow> let pc' = nat (int (pc + 1) + d) in if pc < pc' then exec' ins s pc' else [pc'] else [pc])"
apsnd_apfst_commute: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'c"   and p :: "'f \<times> 'e" shows "apsnd f (apfst g p) = apfst g (apsnd f p)"
flow_initial_time: fixes x0 :: "'a" assumes "0 \<in> T"   and "x0 \<in> S" shows "flow 0 x0 0 = x0"
strong_sum_eq: fixes x :: "i"   and y :: "i" shows "x \<oplus> y = (THE z. (P x z \<and> P y z) \<and> (\<forall>w. P w z \<longrightarrow> O w x \<or> O w y))"
simps_2: fixes x :: "'b" shows "New_max.merge_option (Some x, None) = Some x"
last_message_NoMsg_conv: fixes xs :: "'b message_af list" shows "(last_message xs = NoMsg) = (\<forall>i<length xs. xs ! i = NoMsg)"
par_strict_col_par_strict: fixes C :: "'p"   and E :: "'p"   and A :: "'p"   and B :: "'p"   and D :: "'p" assumes "C \<noteq> E"   and "A B ParStrict C D"   and "Col C D E" shows "A B ParStrict C E"
real_alg_2_list_length: fixes xs :: "real_alg_2_list" assumes "\<not> real_alg_2_list_nil xs" shows "real_alg_2_list_length (real_alg_2_list_tl xs) < real_alg_2_list_length xs"
zero_cong: fixes m :: "int" assumes "residues m" shows "\<zero>\<^bsub>residue_ring m\<^esub> = 0"
rev_bexI: fixes x :: "'c"   and A :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "x \<in> A"   and "P x" shows "\<exists>x\<in>A. P x"
ab_semigroup_mult_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "ab_semigroup_mult_hom hom \<equiv> semigroup_mult_hom hom"
suminf_ennreal_eq: fixes f :: "nat \<Rightarrow> real"   and x :: "real" assumes "\<And>i. 0 \<le> f i"   and "f sums x" shows "(\<Sum>i. ennreal (f i)) = ennreal x"
JVM_le_unfold: fixes P :: "'a prog"   and m :: "nat"   and n :: "nat" shows "JVM_SemiType.le P m n \<equiv> Err.le (Opt.le (Product.le (Listn.le (subtype P)) (Listn.le (Err.le (subtype P)))))"
sub_set_system_def: fixes \<U> :: "'b set"   and \<A> :: "'b set multiset"   and \<V> :: "'b set"   and \<B> :: "'b set multiset" shows "sub_set_system \<U> \<A> \<V> \<B> \<equiv> incidence_system \<V> \<B> \<and> sub_set_system_axioms \<U> \<A> \<V> \<B>"
basis_subset_carrier: fixes X :: "'a set" assumes "\<E>.basis X" shows "X \<subseteq> carrier"
exec_gpv_fixp_induct: fixes P :: "(('a, 'b, 'c) gpv \<Rightarrow> 'd \<Rightarrow> ('a \<times> 'd) spmf) \<Rightarrow> bool"   and callee :: "'d \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd) spmf" assumes "spmf.admissible (\<lambda>f. P (\<lambda>c s. f (c, s)))"   and "P (\<lambda>_ _. return_pmf None)"   and "\<And>exec_gpv. P exec_gpv \<Longrightarrow> P (\<lambda>c s. the_gpv c \<bind> case_generat (\<lambda>x. return_spmf (x, s)) (\<lambda>out c. callee s out \<bind> (\<lambda>(x, y). exec_gpv (c x) y)))" shows "P (exec_gpv callee)"
Cons_1: fixes h :: "'a"   and l1 :: "'a list" assumes "subseq (h # l1) l2" shows "\<exists>l2aa l2b. l2 = l2aa @ [h] @ l2b \<and> \<not> ListMem h l2aa"
gen_length_code_1: fixes n :: "nat" shows "List.gen_length n [] = n"
ideal_eq_carrier_iff: fixes a :: "'a" assumes "a \<in> carrier R" shows "(carrier R = PIdl a) = (a \<in> Units R)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (A_B.Cod (fst Fg)) (A_B.Cod (fst Fg')) (A_B.Map (fst Fg'))"
findMin_correct_aux_1: fixes q :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" assumes "SkewBinomialHeapStruc.invar (map BsSkewBinomialHeapStruc.bsmapt q)"   and "q \<noteq> []" shows "(BsSkewBinomialHeapStruc.findMin q, BsSkewBinomialHeapStruc.eprio (BsSkewBinomialHeapStruc.findMin q)) \<in># SkewBinomialHeapStruc.queue_to_multiset (map BsSkewBinomialHeapStruc.bsmapt q)"
lcm_0_right: fixes a :: "'a" shows "lcm a (0::'a) = (0::'a)"
lspace_iff: fixes p :: "ennreal"   and S :: "'a set" assumes "p < \<infinity>"   and "0 < p" shows "lspace (lebesgue_on S) p = {f \<in> borel_measurable (lebesgue_on S). integrable (lebesgue_on S) (\<lambda>x. norm (f x) powr enn2real p)}"
fun_belowI: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "\<And>x. f x \<sqsubseteq> g x" shows "f \<sqsubseteq> g"
isMaxFlow_def: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'b" assumes "Network c s t" shows "Network.isMaxFlow c s t f \<equiv> Flow c s t f \<and> (\<forall>f'. Flow c s t f' \<longrightarrow> Flow.val c s f' \<le> Flow.val c s f)"
diget_to_char_inv: fixes n :: "nat" assumes "is_digit n" shows "char_to_digit (digit_to_char n) = n"
powsum1: fixes x :: "'a"   and n :: "nat" shows "x\<^bsub>n\<^esub>\<^bsup>0\<^esup> = x\<^bsup>n\<^esup>"
digit_encode_valid_index: fixes a :: "nat"   and ds :: "nat list" assumes "a < prod_list ds" shows "digit_encode ds a \<lhd> ds"
AsAwaitSkip: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and Q :: "'e set"   and A :: "'e set"   and r :: "'e set"   and ae :: "('e, 'd, 'f) ann"   and b :: "'e set"   and e :: "('e, 'd, 'f) com" shows "assertionsR \<Gamma> \<Theta> Q A (AnnRec r ae) (Await b e) Q"
fmdom'_restrict_set: fixes A :: "'c set"   and m :: "('c, 'd) fmap" shows "fmdom' (fmrestrict_set A m) \<subseteq> A"
hcomplex_split_polar: fixes z :: "complex star" shows "\<exists>r a. z = hcomplex_of_hypreal r * HComplex ((*f* cos) a) ((*f* sin) a)"
pred_inject_1: fixes P :: "'a \<Rightarrow> bool" shows "pred_tree P tree.Empty"
Fejer_kernel_continuous: fixes n :: "nat" shows "continuous_on {- pi..pi} (Fejer_kernel n)"
project_act_Restrict_subset_project_act: fixes C :: "'c set"   and act :: "('c \<times> 'c) set" shows "project_act h (Restrict C act) \<subseteq> project_act h act"
encode_all_operator_effects_valid: fixes t :: "nat"   and sv :: "sat_plan_variable" assumes "t < Suc h"   and "sv \<in> atoms (encode_all_operator_effects (\<phi> prob_with_noop abs_prob ) ((\<phi> prob_with_noop abs_prob )\<^sub>\<O>) t)" shows "valid_state_var sv"
Hausdorff: fixes A :: "'c set set" shows "Ex (subset.maxchain A)"
bounded_closure_image: fixes f :: "'b \<Rightarrow> 'a"   and S :: "'b set" assumes "bounded (f ` closure S)" shows "bounded (f ` S)"
iso_inv_into: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" assumes "Well_order r"   and "iso r r' f" shows "iso r' r (inv_into (Field r) f)"
top_flat_complete_lattice_def: shows "top = flat_complete_lattice.Top"
map_bind_spmf: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'e spmf"   and g :: "'e \<Rightarrow> 'd spmf" shows "map_spmf f (p \<bind> g) = p \<bind> map_spmf f \<circ> g"
ZFUnionInRangeExplode: fixes A :: "ZF set"   and B :: "ZF set" assumes "A \<in> range explode"   and "B \<in> range explode" shows "A \<union> B \<in> range explode"
is_path_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a" shows "IFC_def.is_path prob \<pi> = (\<forall>n. (\<pi> n, \<pi> (Suc n)) \<in> IFC_def.edges prob)"
abs_fresh_2: fixes x :: "'a"   and b :: "name"   and a :: "name" assumes "finite (supp x)" shows "b \<sharp> [a].x = (b = a \<or> b \<sharp> x)"
set_disconnected_nodes_locs_impl: shows "set_disconnected_nodes_locs = a_set_disconnected_nodes_locs"
group_incl_imp_subgroup: fixes H :: "'a set" assumes "H \<subseteq> carrier R"   and "Group.group (add_monoid R\<lparr>carrier := H\<rparr>)" shows "subgroup H (add_monoid R)"
symclp_def: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" shows "symclp r x y = (r x y \<or> r y x)"
finsertI1: fixes a :: "'a"   and B :: "'a fset" shows "a |\<in>| finsert a B"
trg_cod: fixes \<mu> :: "'b" assumes "D.arr \<mu>" shows "trg\<^sub>D (D.cod \<mu>) = trg\<^sub>D \<mu>"
dfs_invariant: fixes ks :: "int list"   and l :: "int"   and S :: "nat option list \<times> int list"   and xs :: "int list"   and I :: "nat option list \<times> int list \<Rightarrow> bool"   and n :: "nat" assumes "dioph_invariant ks l S"   and "list_all (dioph_is_node ks l) xs"   and "I S"   and "\<And>S x. \<lbrakk>\<not> dioph_memb x S; dioph_is_node ks l x; dioph_invariant ks l S; I S\<rbrakk> \<Longrightarrow> I (dioph_ins x S)" shows "I (dioph_ineq_dfs.dfs n ks S xs)"
Efin_imp_Vfin: assumes "finite E" shows "finite V"
filterlim_at_within_not_equal: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and s :: "'b set"   and F :: "'a filter"   and b :: "'b" assumes "filterlim f (at a within s) F" shows "\<forall>\<^sub>F w in F. f w \<in> s \<and> f w \<noteq> b"
roots_of_2_3: fixes p :: "int poly" shows "distinct (map real_of_2 (roots_of_2 p))"
restrict_empty_right: fixes ks :: "'c list"   and f :: "'c \<times> 'c \<Rightarrow> 'd" shows "ks \<langle>f\<rangle> [] = mbot"
IsoCompInv: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b" assumes "Category C"   and "ciso\<^bsub>C\<^esub> f" shows "f \<approx>>\<^bsub>C\<^esub> Cinv\<^bsub>C\<^esub> f"
rel_pmf_bind_reflI: fixes p :: "'b pmf"   and R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd pmf"   and g :: "'b \<Rightarrow> 'e pmf" assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> rel_pmf R (f x) (g x)" shows "rel_pmf R (p \<bind> f) (p \<bind> g)"
ereal_mult_cancel_left: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal" shows "(a * b = a * c) = (\<bar>a\<bar> = \<infinity> \<and> 0 < b * c \<or> a = 0 \<or> b = c)"
split: fixes P :: "'d \<Rightarrow> bool"   and f1 :: "'e \<Rightarrow> 'd"   and f2 :: "'f \<Rightarrow> 'g \<Rightarrow> 'd"   and generat :: "('e, 'f, 'g) generat" shows "P (case generat of Pure x \<Rightarrow> f1 x | IO x xa \<Rightarrow> f2 x xa) = ((\<forall>x1. generat = Pure x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22. generat = IO x21 x22 \<longrightarrow> P (f2 x21 x22)))"
one_Integer_def: shows "1 = MkI\<cdot>1"
hyp2_S_points_fix_line: fixes a :: "proj2"   and p :: "proj2"   and J :: "cltn2"   and l :: "proj2_line"   and b :: "proj2" assumes "a \<in> hyp2"   and "p \<in> S"   and "is_K2_isometry J"   and "apply_cltn2 a J = a"   and "apply_cltn2 p J = p"   and "proj2_incident a l"   and "proj2_incident p l"   and "proj2_incident b l" shows "apply_cltn2 b J = b"
tranclpD: fixes R :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "R\<^sup>+\<^sup>+ x y" shows "\<exists>z. R x z \<and> R\<^sup>*\<^sup>* z y"
atom_map_atoms: shows "\<Union> (range atom_map) = Collect atom"
cls_val_process_adapted: fixes p :: "'a \<Rightarrow> nat \<Rightarrow> bool stream \<Rightarrow> real" assumes "trading_strategy p"   and "support_adapt Mkt p" shows "borel_adapt_stoch_proc G (cls_val_process Mkt p)"
JFset_induct: fixes phi1 :: "'d \<Rightarrow> 'd JF1 \<Rightarrow> bool"   and phi2 :: "'d \<Rightarrow> 'd JF2 \<Rightarrow> bool"   and a :: "'d JF1"   and b :: "'d JF2" assumes "\<And>a aa. aa \<in> F1set1 (dtor1 a) \<Longrightarrow> phi1 aa a"   and "\<And>b a. a \<in> F2set1 (dtor2 b) \<Longrightarrow> phi2 a b"   and "\<And>a a'. a' \<in> F1set2 (dtor1 a) \<Longrightarrow> a' \<in> F1set2 (dtor1 a)"   and "\<And>a a' aa. \<lbrakk>a' \<in> F1set2 (dtor1 a); aa \<in> JF1set a'; phi1 aa a'\<rbrakk> \<Longrightarrow> phi1 aa a"   and "\<And>a b'. b' \<in> F1set3 (dtor1 a) \<Longrightarrow> b' \<in> F1set3 (dtor1 a)"   and "\<And>a b' aa. \<lbrakk>b' \<in> F1set3 (dtor1 a); aa \<in> JF2set b'; phi2 aa b'\<rbrakk> \<Longrightarrow> phi1 aa a"   and "\<And>b a'. a' \<in> F2set2 (dtor2 b) \<Longrightarrow> a' \<in> F2set2 (dtor2 b)"   and "\<And>b a' a. \<lbrakk>a' \<in> F2set2 (dtor2 b); a \<in> JF1set a'; phi1 a a'\<rbrakk> \<Longrightarrow> phi2 a b"   and "\<And>b b'. b' \<in> F2set3 (dtor2 b) \<Longrightarrow> b' \<in> F2set3 (dtor2 b)"   and "\<And>b b' a. \<lbrakk>b' \<in> F2set3 (dtor2 b); a \<in> JF2set b'; phi2 a b'\<rbrakk> \<Longrightarrow> phi2 a b" shows "(\<forall>aa\<in>JF1set a. phi1 aa a) \<and> (\<forall>a\<in>JF2set b. phi2 a b)"