hcju/isabelleps huggingface dataset:
stirling_integral_bound': fixes n :: "nat"   and \<alpha> :: "real" assumes "0 < n"   and "\<alpha> \<in> {0<..<pi}" obtains c :: "real" where "\<And>s. s \<in> complex_cone' \<alpha> - {0} \<Longrightarrow> cmod (stirling_integral n s) \<le> c / cmod s ^ n"
pr_def: shows "pr \<equiv> \<lambda>q c. \<Sum>i\<le>q. frag_cmul ((- 1) ^ i) (frag_of (simplex_map (Suc q) (\<lambda>z. h (z 0, c (z \<circ> Suc))) (simp q i)))"
sub_def: shows "sub \<equiv> \<lambda>i f. f i"
unroll_while: fixes s :: "state"   and t :: "state"   and G :: "prog"   and l :: "label"   and e :: "expr"   and c :: "stmt"   and v :: "val"   and s' :: "state"   and C :: "qtname"   and L :: "lenv"   and T :: "ty" assumes "(s, t) \<in> (unroll G l e c)\<^sup>*"   and "G\<turnstile>t \<midarrow>e-\<succ>v\<rightarrow> s'"   and "normal s' \<longrightarrow> \<not> the_Bool v"   and "\<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile>e\<Colon>-T"   and "wf_prog G" shows "G\<turnstile>s \<midarrow>l\<bullet> While(e) c\<rightarrow> s'"
mst_post: fixes r :: "'a"   and v :: "'a"   and w :: "'a"   and t :: "'a" assumes "r \<sqinter> top = r"   and "r \<sqinter> id r \<le> top"   and "id v = id r \<sqinter> top"   and "w \<sqinter> id w \<le> top \<and> w \<sqinter> top \<le> - top"   and "t \<le> w"   and "w \<le> v \<sqinter> id v" shows "w = t"
Nats_subset_Ints: shows "\<nat> \<subseteq> \<int>"
jordan_block_pow: fixes n :: "nat"   and a :: "'a"   and r :: "nat" shows "jordan_block n a ^\<^sub>m r = Matrix.mat n n (\<lambda>(i, j). if i \<le> j then of_nat (r choose (j - i)) * a ^ (r + i - j) else (0::'a))"
adaptive_atol_spec_def: shows "adaptive_atol_spec = SPEC (\<lambda>x. True)"
coplanar_perm_8: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Coplanar A B C D" shows "Coplanar B C A D"
noex: shows "\<forall>f\<in>P. |vertices f| \<le> 4"
stake_nth: fixes i :: "nat"   and j :: "nat"   and s :: "nat \<Rightarrow> 'b" assumes "i < j" shows "stake j s ! i = s i"
dual_weak_order_iff: fixes A :: "('a, 'b) gorder_scheme" shows "weak_partial_order (inv_gorder A) = weak_partial_order A"
hd_0_subst: fixes L :: "'a list"   and x :: "'a" assumes "0 < length L" shows "hd (L[0 := x]) = x"
h2: shows "\<forall>m. ks = sKS m \<longrightarrow> \<not> ine P (sE m) \<and> \<not> ine Q (sE m)"
comp_ide_self: fixes a :: "'a2" assumes "A2.ide a" shows "a \<cdot>\<^sub>A\<^sub>2 a = a"
cases: fixes x :: "('dyn, 'a, 'b, 'c, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef" obtains xs :: "('dyn, 'a, 'b, 'c, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr list" and ar :: "nat" where "x = Fundef xs ar"
comp_inf_vector: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x * (y \<sqinter> z * top) = (x \<sqinter> top * z\<^sup>T) * y"
fst_imageIcc: fixes a :: "'c \<times> 'd"   and b :: "'c \<times> 'd" shows "fst ` {a..b} = (if a \<le> b then {fst a..fst b} else {})"
face: fixes X :: "'a set"   and Y :: "'a set" assumes "X \<in> \<F>"   and "Y \<in> \<F>" shows "X \<inter> Y face_of X"
cols_def: fixes xss :: "'c list list"   and n :: "nat" shows "cols xss n = (\<forall>xs\<in>set xss. length xs = n)"
continuous_on_id': fixes s :: "'a set" shows "continuous_on s id"
fst_scaleC: fixes r :: "complex"   and A :: "'a \<times> 'b" shows "fst (r *\<^sub>C A) = r *\<^sub>C fst A"
wset_Suspend_okD1: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('c \<times> 'a \<Rightarrow>f nat) option) \<times> 'd) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and final :: "'c \<Rightarrow> bool"   and r :: "('a, 'b, 'c, 'd, 'e, 'f) semantics"   and convert_RA :: "'a \<Rightarrow>f nat \<Rightarrow> 'f list"   and I :: "(('a, 'b) locks \<times> (('b \<Rightarrow> ('c \<times> 'a \<Rightarrow>f nat) option) \<times> 'd) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set) set" assumes "s \<in> multithreaded_base.wset_Suspend_ok final r convert_RA I" shows "s \<in> I"
IH: fixes gpv :: "('c, 'a, 'b) gpv"   and s :: "'s" assumes "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "exec_gpv' (restrict_gpv \<I> gpv) s = exec_gpv' gpv s"
assms_1: shows "spmf.admissible (\<lambda>f. P (\<lambda>c s. f (c, s)))"
assms_2: fixes s :: "'a" assumes "P s" shows "Q s"
simps_9: fixes v :: "'b poly"   and vc :: "nat"   and vb :: "'b poly" shows "Polynomial_Expression.degree (CN v (Suc vc) vb) = 0"
dual_conjunctive: fixes x :: "'a" assumes "x \<in> conjunctive.conjunctive (\<sqinter>) (\<sqinter>) (*)" shows "x ^ o \<in> disjunctive.disjunctive (\<squnion>) (\<squnion>) (*)"
ss: shows "{a. \<exists>b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r} \<subseteq> (\<lambda>x. x\<lparr>0\<^sub>\<nat>\<rparr>) ` elts r"
\<Phi>_def: shows "dcharacter d \<Phi> \<and> (\<forall>n. \<chi> n = \<Phi> n * \<chi>\<^sub>1 n)"
min_le_iff_disj: fixes less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'h"   and y :: "'h"   and z :: "'h" assumes "class.linorder less_eq less" shows "less_eq (ord.min less_eq x y) z = (less_eq x z \<or> less_eq y z)"
viewR_def: fixes t :: "('c, 'd) FingerTreeStruc" shows "viewR t = (case viewRn t of None \<Rightarrow> None | Some (a, t2) \<Rightarrow> Some (n_unwrap a, t2))"
filter_mset_inequality: fixes f :: "'a \<Rightarrow> bool"   and xs :: "'a multiset" assumes "filter_mset f xs \<noteq> xs" shows "\<exists>x\<in>#xs. \<not> f x"
Fun_eval: fixes b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide b"   and "ide c" shows "Fun (eval b c) = (\<lambda>x\<^bold>\<in>local.set (HF'.prod (exp b c) b). UP (happ (hfst (DOWN x)) (hsnd (DOWN x))))"
inf_sup_ord_1: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "class.lattice inf less_eq less sup" shows "less_eq (inf x y) x"
size_sift_down: fixes l :: "'b tree"   and a :: "'b"   and r :: "'b tree" assumes "braun \<langle>l, a, r\<rangle>" shows "size (sift_down l a r) = size l + size r + 1"
r_coset_is_image: fixes H :: "nat set"   and a :: "nat" shows "H #>\<^bsub>G\<^esub> a = (\<lambda>x. x \<otimes>\<^bsub>G\<^esub> a) ` H"
evalS_body_evalK_body: shows "unwrap oo evalS_body oo wrap = evalK_body oo unwrap oo wrap"
interior_cball: fixes x :: "'a"   and e :: "real" shows "interior (cball x e) = ball x e"
lookup_some_upper: fixes env :: "('a, 'b, 'c) env"   and xs :: "'c list"   and y :: "'c"   and ys :: "'c list"   and e :: "('a, 'b, 'c) env" assumes "lookup env (xs @ y # ys) = Some e" shows "\<exists>b' es' env'. lookup env xs = Some (Env b' es') \<and> es' y = Some env' \<and> lookup env' ys = Some e"
nat_pow_0: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d" shows "x [^]\<^bsub>G\<^esub> 0 = \<one>\<^bsub>G\<^esub>"
IH: shows "unlabel (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>) = unlabel A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>"
fail_sys_def: fixes init :: "(nat \<Rightarrow> 'a) set"   and p :: "'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "nat \<Rightarrow> 'b" shows "fail_sys init p r x = (\<exists>n u y. u \<in> init \<and> (\<forall>i<n. r (u i) (u (Suc i)) (x i) (y i)) \<and> \<not> p (u n) (u (Suc n)) (x n))"
mon_ah_subset: fixes w :: "('b set \<times> 'b set) list" shows "mon_ah (\<alpha>ah w) \<subseteq> mon_pl w"
nth_Cons_MkI: fixes i :: "int"   and a :: "'b"   and xs :: "['b]" assumes "0 < i" shows "(a : xs) !! MkI\<cdot>i = xs !! MkI\<cdot>(i - 1)"
topspace_euclidean_subtopology: fixes S :: "'c set" shows "topspace (top_of_set S) = S"
simps_1: fixes m :: "nat \<Rightarrow> nat" shows "volatile_reads_consistent m [] = True"
finfun_leI: fixes A :: "'a \<Rightarrow>f bool"   and B :: "'a \<Rightarrow>f bool" assumes "\<And>x. A $ x \<Longrightarrow> B $ x" shows "A \<le> B"
UpdateApply_def: fixes U :: "'b update"   and D :: "'b data" shows "U !!! D \<equiv> Rep_update U D"
REFSTEP: fixes it :: "'S set"   and it' :: "'Sa set"   and \<sigma> :: "'\<sigma>"   and \<sigma>' :: "'\<sigma>a"   and x' :: "'Sa"   and x :: "'S" assumes "it \<subseteq> S"   and "it' \<subseteq> S'"   and "\<Phi> it \<sigma>"   and "\<Phi>' it' \<sigma>'"   and "x' = \<alpha> x"   and "it' = \<alpha> ` it"   and "((it, \<sigma>), it', \<sigma>') \<in> R"   and "x \<in> it"   and "x' \<in> it'" shows "f x \<sigma> \<le> \<Down> {(\<sigma>, \<sigma>'). ((it - {x}, \<sigma>), it' - {x'}, \<sigma>') \<in> R} (f' x' \<sigma>')"
unlr_leq: fixes x :: "'a admS"   and y :: "'a admS" shows "(unlr x \<subseteq> unlr y) = (x \<le> y)"
simps_2: fixes z :: "octo" shows "Im1 (Octonions.cnj z) = - Im1 z"
simps_4: fixes M :: "('fun, 'var) Term.term set"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) set"   and uu :: "poscheckvariant"   and t :: "('fun, 'var) Term.term"   and t' :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) stateful_strand_step list" shows "\<lbrakk>M; D; \<langle>uu: t \<doteq> t'\<rangle> # S\<rbrakk>\<^sub>s = (\<lambda>\<I>. Unifier \<I> t t' \<and> \<lbrakk>M; D; S\<rbrakk>\<^sub>s \<I>)"
SmallerMultipleStepsWithLimit: fixes limit :: "nat"   and A :: "nat \<Rightarrow> nat"   and k :: "nat" assumes "\<forall>n\<ge>limit. A (Suc n) < A n" shows "\<forall>n\<ge>limit. A (n + k) \<le> A n - k"
lower_closed: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme"   and x :: "'d" assumes "connection G"   and "x \<in> carrier \<X>\<^bsub>G\<^esub>" shows "\<pi>\<^sup>*\<^bsub>G\<^esub> x \<in> carrier \<Y>\<^bsub>G\<^esub>"
lprefix_lsetD: fixes xs :: "'a llist"   and ys :: "'a llist" assumes "lprefix xs ys" shows "lset xs \<subseteq> lset ys"
intro: fixes G :: "'a set" assumes "Group G"   and "FinGroup_axioms G" shows "FinGroup G"
1: shows "is_poincare_line (moebius_circline (moebius_rotation (- pi / 2)) H)"
entries_impl_of: fixes t :: "('c, 'd) mapping_rbt" shows "RBT_Impl.entries (impl_of t) = entries t"
vrat_uminus_transfer: shows "rel_fun cr_vrat cr_vrat vrat_uminus_app uminus"
has_bochner_integral_cong: fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "M = N"   and "\<And>x. x \<in> space N \<Longrightarrow> f x = g x"   and "x = y" shows "has_bochner_integral M f x = has_bochner_integral N g y"
simps_13: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and f4 :: "'b" shows "(case Private of Private \<Rightarrow> f1 | Package \<Rightarrow> f2 | Protected \<Rightarrow> f3 | Public \<Rightarrow> f4) = f1"
poly_mult_const_1: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p :: "'c list"   and a :: "'c" assumes "domain R"   and "subring K R"   and "polynomial\<^bsub>R\<^esub> K p"   and "a \<in> K - {\<zero>\<^bsub>R\<^esub>}" shows "ring.poly_mult R [a] p = map ((\<otimes>\<^bsub>R\<^esub>) a) p"
ring_simprules_16: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "ring R"   and "x \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
HStartBallot_HInv5_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HStartBallot s s' p"   and "HInv5_inner s q"   and "p \<noteq> q" shows "HInv5_inner s' q"
above_def: fixes r :: "('d \<times> 'd) set"   and a :: "'d" shows "above r a \<equiv> {b. (a, b) \<in> r}"
assms_5: shows "proj2_incident q (apply_cltn2_line m C)"
mon_w_single: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and e :: "('c, 'f) label" shows "mon_w fg [e] = mon_e fg e"
put_closure: fixes x :: "'d \<Longrightarrow> 'e"   and \<sigma> :: "'e"   and v :: "'d" assumes "weak_lens x" shows "put\<^bsub>x\<^esub> \<sigma> v \<in> \<S>\<^bsub>x\<^esub>"
line_assm_def: fixes \<A> :: "'b \<Rightarrow> bool" shows "line_assm \<A> \<equiv> (`) (\<lambda>k. turn_true \<A> (Atom k))"
prv_subst_sdsj: fixes F :: "'fmla set"   and t :: "'trm"   and x :: "'var" assumes "F \<subseteq> fmla"   and "finite F"   and "t \<in> trm"   and "x \<in> var" shows "prv (eqv (subst (sdsj F) t x) (sdsj ((\<lambda>\<phi>. subst \<phi> t x) ` F)))"
vconst_onD: fixes a :: "V"   and c :: "V"   and A :: "V" assumes "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> vconst_on A c" shows "a \<in>\<^sub>\<circ> A"
a_d_closed: fixes x :: "('b \<times> 'b) set" shows "rel_antirange_kleene_algebra.ardual.apd_d (rel_ar x) = rel_ar x"
Min_in: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "local.Min A \<in> A"
labels_conv_def: fixes c :: "'b \<times> (bool \<times> 'a \<times> 'b) list" shows "labels_conv c = map (\<lambda>q. fst (snd q)) (snd c)"
facet_map: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and C :: "'b set"   and z :: "'b set" assumes "ChamberComplexEndomorphism X f"   and "SimplicialComplex.maxsimp X C"   and "z \<lhd> C" shows "f ` z \<lhd> f ` C"
rf_stock: fixes Mkt :: "('c, 'd) discrete_market"   and r :: "real"   and risk_free_asset :: "'d" assumes "rsk_free_asset Mkt r risk_free_asset" shows "risk_free_asset \<in> stocks Mkt"
preprm_unit_def: fixes a :: "'b"   and b :: "'b" shows "preprm_unit a b \<equiv> [(a, b)]"
subclsSI: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Bs :: "base list"   and rest :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list"   and D :: "char list" assumes "class P C = \<lfloor>(Bs, rest)\<rfloor>"   and "Shares D \<in> set Bs" shows "P \<turnstile> C \<prec>\<^sub>S D"
hn_refineD: fixes \<Gamma> :: "assn"   and c :: "'a Heap"   and \<Gamma>' :: "assn"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and m :: "'b nres" assumes "hn_refine \<Gamma> c \<Gamma>' R m"   and "nofail m" shows "<\<Gamma>> c <\<lambda>r. \<Gamma>' * (\<exists>\<^sub>Ax. R x r * \<up> (RETURN x \<le> m))>\<^sub>t"
CDPlayer_CTRL_Labels_def: shows "CDPlayer_CTRL_Labels = {(En ''LastTrack'', defaultguard, CDPlayer_CTRL_Action1), (En ''NewTrack'', defaultguard, CDPlayer_CTRL_Action2), (And (En ''CDEject'') (In ''On''), defaultguard, CDPlayer_CTRL_Action3), (En ''CDIn'', defaultguard, defaultaction)}"
is_ground_atm_def: fixes A :: "'a" shows "is_ground_atm A = (\<forall>\<sigma>. A = A \<cdot>a \<sigma>)"
uses'_def: fixes g :: "'g"   and n :: "'node" shows "uses' g n \<equiv> substNext g ` uses g n"
step: fixes c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and c'' :: "('Var, 'AExp, 'BExp) Stmt"   and mds'' :: "Mode \<Rightarrow> 'Var set"   and mem'' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c', mds', mem'\<rangle> \<in> loc_reach lc"   and "\<langle>c', mds', mem'\<rangle> \<leadsto> \<langle>c'', mds'', mem''\<rangle>" shows "\<langle>c'', mds'', mem''\<rangle> \<in> loc_reach lc"
subdg_trans: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<CC> :: "V" assumes "\<AA> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>" shows "\<AA> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
RPDs_def: fixes r :: "mregex" shows "RPDs r = (\<Union>i. RPDi i r)"
restrict_chart_carrier: fixes x :: "('b, 'b) chart" assumes "x \<in> manifold_eucl.dest.charts_submanifold (codomain c)" shows "restrict_chart a.dest.carrier x = x"
continuous_Dirichlet_L: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and s :: "complex"   and A :: "complex set" assumes "1 < n"   and "\<chi> \<noteq> principal_dchar n \<and> dcharacter n \<chi> \<or> \<chi> = principal_dchar n \<and> s \<noteq> 1" shows "continuous (at s within A) (Dirichlet_L n \<chi>)"
outer_loop_rel_def: fixes c :: "nat \<times> nat \<Rightarrow> 'd"   and src :: "nat" shows "Graph.outer_loop_rel c src \<equiv> inv_image (less_than_bool <*lex*> greater_bounded (Graph.max_dist c src + 1) <*lex*> finite_psubset) (\<lambda>(f, PRED, C, N, d). (\<not> f, d, C))"
distinct_1: fixes x21 :: "'b tree"   and x22 :: "'b"   and x23 :: "'b tree" shows "\<langle>\<rangle> \<noteq> \<langle>x21, x22, x23\<rangle>"
simps_1: fixes uu :: "('a, 'b) hd" shows "args (Hd uu) = []"
cases: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b \<times> 'b" assumes "class.normalization_euclidean_semiring divide plus minus zero times one modulo euclidean_size unit_factor normalize" obtains a :: "'b" and b :: "'b" where "x = (a, b)"
weak_partial_order_def: fixes L :: "('a, 'b) gorder_scheme" shows "weak_partial_order L \<equiv> equivalence L \<and> weak_partial_order_axioms L"
assms: fixes x :: "'a" shows "UNIV \<noteq> {x}"
symvars_of_adapt_aexp: fixes e :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd"   and s :: "'c \<Rightarrow> nat" shows "Aexp.vars (adapt_aexp e s) \<subseteq> Store.symvars s"
assms_2: shows "\<Theta>,\<Gamma> \<turnstile> Ct STR ''Pure.all'' ((\<tau> \<rightarrow> propT) \<rightarrow> propT) $ B"
l_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M create_document"
red_single_trans: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and p0 :: "'t \<Rightarrow>\<^sub>0 'c"   and f :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a"   and g :: "'t \<Rightarrow>\<^sub>0 'c" assumes "red_single p p0 f t"   and "lt g adds\<^sub>t lt f"   and "g \<noteq> 0" obtains p1 :: "'t \<Rightarrow>\<^sub>0 'c" where "red_single p p1 g (t + (lp f - lp g))"
outgoing'_def: fixes k :: "nat set" shows "cf.outgoing' k \<equiv> {(v, u) |u v. u \<notin> k \<and> v \<in> k \<and> (v, u) \<in> cf.E}"
map_append_alt_def2: fixes f :: "'b \<Rightarrow> 'a"   and b :: "'a list"   and xs :: "'b list" shows "(RETURN \<circ>\<circ>\<circ> map_append) f b xs = REC\<^sub>T (\<lambda>g xs. case xs of [] \<Rightarrow> RETURN b | x # xs \<Rightarrow> g xs \<bind> (\<lambda>y. RETURN (f x # y))) xs"
path_rev_sym: fixes g :: "'b ugraph"   and v :: "'b"   and p :: "('b \<times> 'b) list"   and u :: "'b" assumes "path g v p u" shows "path g u (revp p) v"
bdd_above_multiset_imp_bdd_above_count: fixes A :: "'k multiset set"   and x :: "'k" assumes "subset_mset.bdd_above A" shows "bdd_above ((\<lambda>X. count X x) ` A)"
cdlemma: fixes xs' :: "'a list"   and ys' :: "'a list"   and r :: "('a \<times> 'a) set" assumes "(xs', ys') \<in> genPrefix r" shows "\<forall>x xs. xs' = x # xs \<longrightarrow> (\<exists>y ys. ys' = y # ys \<and> (x, y) \<in> r \<and> (xs, ys) \<in> genPrefix r)"
bigtheta_trans1: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> o[F](g)"   and "g \<in> \<Theta>[F](h)" shows "f \<in> o[F](h)"
assms_2: shows "AE x in M. \<forall>\<^sub>F n in sequentially. P n x"
elem_in_nhd: fixes U :: "'a set"   and x :: "'a" assumes "U \<in> nhds x" shows "x \<in> U"
simps: fixes u :: "name"   and p :: "tm"   and q :: "tm"   and v :: "name" assumes "atom u \<sharp> (p, q, v)"   and "atom v \<sharp> (p, q)" shows "SubstP p q = SyntaxN.Ex v (SyntaxN.Ex u (SubstFormP (Var v) (Var u) p q))"
intro: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>G :: "'c \<times> 'c \<Rightarrow> 'd"   and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd" assumes "bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C"   and "bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D"   and "pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F"   and "pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G"   and "pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1" shows "converse_pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
suppD: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "x \<in> supp f" shows "f x \<noteq> (0::'d)"
rtranclp_invariant_induct: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and I :: "'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* a b"   and "invariant r I"   and "I a"   and "P a"   and "\<And>y z. \<lbrakk>r\<^sup>*\<^sup>* a y; r y z; I y; I z; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
nextss_Cons: fixes g :: "(node \<times> node) list"   and x :: "node"   and xs :: "node list" shows "nextss g (x # xs) = set (nexts g x) \<union> nextss g xs"
ipresIGFreshAbs_def: fixes hA :: "'h \<Rightarrow> 'i"   and MOD :: "('j, 'k, 'l, 'm, 'n, 'o, 'p, 'h, 'q) model_scheme"   and MOD' :: "('r, 's, 'l, 't, 'u, 'o, 'v, 'i, 'w) model_scheme" shows "ipresIGFreshAbs hA MOD MOD' \<equiv> \<forall>ys y us s A. igWlsAbs MOD (us, s) A \<longrightarrow> igFreshAbs MOD ys y A \<longrightarrow> igFreshAbs MOD' ys y (hA A)"
root_exp_cancel': fixes xa :: "real" assumes "0 \<le> xa" shows "root n (xa ^ n) = xa"
conv_backward_terminating_path: fixes x :: "'a" shows "backward_terminating_path x = forward_terminating_path (x\<^sup>T)"
triangle_in_hom_1: shows "\<guillemotleft>(\<rho>.\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, f\<^sup>*, f] \<cdot> (f \<star> \<rho>.\<eta>) : f \<star> src f \<Rightarrow> trg f \<star> f\<guillemotright>"
distinctI: fixes xs :: "'b list" assumes "\<And>i j. \<lbrakk>i < j; i < length xs; j < length xs\<rbrakk> \<Longrightarrow> xs ! i \<noteq> xs ! j" shows "distinct xs"
set_less_aux_Min_antimono: fixes A :: "'a set"   and B :: "'a set" assumes "Min A < Min B"   and "finite A"   and "finite B"   and "A \<noteq> {}" shows "B \<sqsubset>' A"
set3_FGl_def: fixes x :: "(('q, 'r, 's, 't, 'u, 'v, 'w) G, 'q, 'p, 's, 'x, 'y, 'u, 'z, 'aa, 'ab) F" shows "set3_FGl x = set3_F x"
unit_is_iso: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b" assumes "equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "category.iso V \<eta>"
mem: shows "\<alpha> \<in> set (remove1 \<beta> xs) \<and> \<beta> \<in> set (remove1 \<alpha> xs) \<and> remove1 \<alpha> xs \<noteq> [] \<and> remove1 \<beta> xs \<noteq> []"