hcju/isabelleps huggingface dataset:
rec_quo_def: shows "rec_quo = (let rR = Cn (Suc (Suc (Suc 0))) rec_conj [Cn (Suc (Suc (Suc 0))) rec_le [Cn (Suc (Suc (Suc 0))) rec_mult [recf.id (Suc (Suc (Suc 0))) (Suc 0), recf.id (Suc (Suc (Suc 0))) (Suc (Suc 0))], recf.id (Suc (Suc (Suc 0))) 0], Cn (Suc (Suc (Suc 0))) rec_noteq [recf.id (Suc (Suc (Suc 0))) (Suc 0), Cn (Suc (Suc (Suc 0))) (constn 0) [recf.id (Suc (Suc (Suc 0))) 0]]] in Cn (Suc (Suc 0)) (rec_maxr rR)) [recf.id (Suc (Suc 0)) 0, recf.id (Suc (Suc 0)) (Suc 0), recf.id (Suc (Suc 0)) 0]"
tendsto_ine_in: fixes u :: "nat \<Rightarrow> 'b"   and N :: "'b quasinorm"   and x :: "'b" assumes "\<And>n. u n \<in> space\<^sub>N N"   and "x \<in> space\<^sub>N N" shows "tendsto_ine\<^sub>N N u x = tendsto_in\<^sub>N N u x"
sums_def: fixes f :: "nat \<Rightarrow> 'c"   and s :: "'c" shows "f sums s = (\<lambda>n. sum f {..<n}) \<longlonglongrightarrow> s"
nipd: shows "ipd (\<pi> j) \<noteq> \<pi> m"
forest_alt2: fixes F :: "'b uprod set" assumes "\<And>e. e \<in> F \<Longrightarrow> proper_uprod e"   and "\<And>a b. Upair a b \<in> F \<Longrightarrow> (a, b) \<notin> uconnected (F - {Upair a b})" shows "forest F"
maxBound_is_bound: fixes enum :: "'b list"   and enum_all :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and x :: "'b" assumes "class.enum enum enum_all enum_ex" shows "enum.fromEnum enum x \<le> enum.fromEnum enum (enum.maxBound enum)"
span_zero: fixes S :: "'a set" shows "(0::'a) \<in> ideal S"
runit_tensor: fixes a :: "'d"   and b :: "'d" assumes "ide a"   and "ide b" shows "\<r>[a \<otimes> b] = (a \<otimes> \<r>[b]) \<cdot> \<a>[a, b, \<I>]"
OIOAC: fixes \<phi> :: "c \<Rightarrow> w \<Rightarrow> bool" shows "\<forall>c. \<lfloor>\<^bold>O\<^sub>i\<phi>\<rfloor>\<^sub>c \<longrightarrow> ob (pv (World c)) ((\<^bold>\<diamond>\<^sub>a\<phi>) c) \<and> \<I>pv (World c) \<sqinter> (\<^bold>\<not>\<^bold>\<diamond>\<^sub>a\<phi>) c"
Scheduler_L3: fixes C :: "Config"   and activation :: "nat \<Rightarrow> nat list"   and t :: "nat" assumes "Scheduler C activation"   and "t mod cycleLength C mem schedule C" shows "activation t \<noteq> []"
dghm_in_Vset: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<beta> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "\<FF> \<in>\<^sub>\<circ> Vset \<beta>"
simps_3: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and v :: "'c match_expr"   and rs :: "'c rule list" shows "no_matching_Goto \<gamma> p (Rule v Accept # rs) = no_matching_Goto \<gamma> p rs"
gb_rename_ecnv_def: fixes ecnv :: "('c, 'd) gb_graph_rec_scheme \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'a"   and G :: "('c, 'd) gb_graph_rec_scheme" shows "gb_rename_ecnv ecnv f G \<equiv> \<lparr>gbg_F = {f ` A |A. A \<in> gbg_F G}, \<dots> = ecnv G\<rparr>"
gbm_add_edge_def: fixes v :: "'V"   and e :: "'W"   and v' :: "'V"   and g :: "'m1" shows "gbm_add_edge v e v' g \<equiv> let g = case m1.lookup v' g of None \<Rightarrow> m1.update v' (m2.empty ()) g | Some x \<Rightarrow> g in case m1.lookup v g of None \<Rightarrow> m1.update v (m2.sng v' (s3.sng e)) g | Some m2 \<Rightarrow> case m2.lookup v' m2 of None \<Rightarrow> m1.update v (m2.update v' (s3.sng e) m2) g | Some s3 \<Rightarrow> m1.update v (m2.update v' (s3.ins e s3) m2) g"
of_int_mod_ring_to_int_mod_ring: fixes x :: "'b mod_ring" shows "of_int_mod_ring (to_int_mod_ring x) = x"
real_arch_invD: fixes e :: "real" assumes "0 < e" shows "\<exists>n. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e"
Cons_2: shows "\<forall>a\<in>set (U # Us). FGModule G (*) a"
abs_eq: fixes x :: "('a, 'b, 'c) Tree" shows "Not\<^sub>\<alpha> (abs_Tree\<^sub>\<alpha> x) = abs_Tree\<^sub>\<alpha> (tNot x)"
connect_def: fixes d :: "('a, 'b) distinguisher"   and res :: "('a, 'b) resource" shows "connect d res = run_gpv run_resource d res"
Arity_total: fixes x :: "Arity"   and y :: "Arity" shows "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
abs_fun_def: fixes R :: "('c \<times> 'b) set"   and m :: "'c nres" shows "\<Up> R m \<equiv> case m of FAILi \<Rightarrow> FAIL | RES X \<Rightarrow> if X \<subseteq> Domain R then RES (R `` X) else FAIL"
subalgebras_filtration: fixes M :: "'b measure"   and F :: "'c \<Rightarrow> 'b measure"   and G :: "'c \<Rightarrow> 'b measure" assumes "Filtration.filtration M F"   and "\<forall>t. subalgebra (F t) (G t)"   and "\<forall>s t. s \<le> t \<longrightarrow> subalgebra (G t) (G s)" shows "Filtration.filtration M G"
assoc_unit_r: fixes a :: "'a"   and b :: "'a" assumes "a \<in> Units G"   and "a \<sim> b"   and "b \<in> carrier G" shows "b \<in> Units G"
assumptions_closed: fixes a :: "'form" assumes "a \<in> set assumptions" shows "closed a"
res_to_cong_simps_8: fixes m :: "int"   and a :: "int"   and b :: "int" assumes "residues m" shows "(a mod m = b mod m) = [a = b] (mod m)"
Maclaurin_lemma2: fixes n :: "nat"   and h :: "real"   and diff :: "nat \<Rightarrow> real \<Rightarrow> real"   and k :: "nat"   and B :: "real" assumes "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> h \<longrightarrow> (diff m has_real_derivative diff (Suc m) t) (at t)"   and "n = Suc k" shows "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> h \<longrightarrow> ((\<lambda>t. diff m t - ((\<Sum>p<n - m. diff (m + p) 0 / fact p * t ^ p) + B * (t ^ (n - m) / fact (n - m)))) has_real_derivative diff (Suc m) t - ((\<Sum>p<n - Suc m. diff (Suc m + p) 0 / fact p * t ^ p) + B * (t ^ (n - Suc m) / fact (n - Suc m)))) (at t)"
Always_Un: fixes F :: "'b program"   and f' :: "'b \<Rightarrow> 'c set"   and f :: "'b \<Rightarrow> 'c set"   and g' :: "'b \<Rightarrow> 'c set"   and g :: "'b \<Rightarrow> 'c set" assumes "F \<in> Always {s. f' s \<subseteq> f s}"   and "F \<in> Always {s. g' s \<subseteq> g s}" shows "F \<in> Always {s. f' s \<union> g' s \<subseteq> f s \<union> g s}"
tautologies_are_valid: fixes atom_ordering :: "('a \<times> 'a) set"   and C :: "'a Literal set"   and I :: "'a set" assumes "propositional_atoms atom_ordering"   and "tautology C" shows "propositional_atoms.validate_clause I C"
"3_1": fixes v :: "'a option list" assumes "ys = map (MFOTL.eval_trm (map the v)) ts"   and "wf_tuple n (\<Union> (fv_trm ` set ts)) v"   and "\<forall>t\<in>set ts. \<forall>x\<in>fv_trm t. x < n" shows "\<exists>f. match ts (map (MFOTL.eval_trm (map the v)) ts) = Some f \<and> v = tabulate f 0 n"
60: fixes g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "\<And>f. f \<in> F \<Longrightarrow> homogeneous f"   and "g \<in> punit.reduced_GB F" shows "poly_deg g \<le> \<b> qs' 0"
\<rho>': fixes x :: "val" assumes "x \<in> type_universe (\<Gamma> v)" shows "\<rho>(v := x) \<in> space (state_measure (insert v V') \<Gamma>)"
lists_eq_set: fixes A :: "'d set" shows "lists A = {xs. set xs \<subseteq> A}"
split_mult_neg_le: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq zero a \<and> less_eq b zero \<or> less_eq a zero \<and> less_eq zero b" shows "less_eq (times a b) zero"
snd_synthetic_divmod: fixes p :: "'b poly"   and c :: "'b" shows "snd (synthetic_divmod p c) = poly p c"
left_step_still_available_case: assumes "s\<^sub>2' \<noteq> \<epsilon>"   and "s\<^sub>2' r = s\<^sub>1 r"   and "s\<^sub>2' r'' = None" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> (revision_step r' s\<^sub>2 s\<^sub>3 \<or> s\<^sub>2 = s\<^sub>3) \<and> (revision_step r s\<^sub>2' s\<^sub>3' \<or> s\<^sub>2' = s\<^sub>3')"
complex_of_real_i: fixes r :: "real" shows "cor r * \<i> = Complex 0 r"
y_axis_integral_measurable: shows "(\<lambda>x. integral UNIV (\<lambda>y. f (y, x))) \<in> borel_measurable lborel"
triv_admI: fixes P :: "'e \<Rightarrow> bool" assumes "\<forall>x. P x" shows "adm P"
not_in_segment2: fixes y :: "'b"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "y \<notin> segment f x y"
hyps: fixes k :: "nat"   and p :: "'b" shows "path A (stake (length r) (sdrop k w) || r) p = gupath A w ([Suc k..<Suc k + length r] || r) (k, p)"
splitFace_holds_facesAt_eq: fixes g' :: "graph"   and v :: "nat"   and a :: "nat"   and f' :: "face"   and n :: "nat"   and g'' :: "graph" assumes "pre_splitFace g' v a f' [countVertices g'..<countVertices g' + n]"   and "minGraphProps g'"   and "g'' = snd (snd (splitFace g' v a f' [countVertices g'..<countVertices g' + n]))" shows "facesAt_eq g''"
deterministic_map: fixes f :: "'a \<Rightarrow> 'b"   and R :: "('a \<times> 'c) set"   and g :: "'c \<Rightarrow> 'd" assumes "inj_on f (fst ` R)"   and "deterministic R" shows "deterministic (map_prod f g ` R)"
bound_Cauchy_to_lim: fixes y :: "nat \<Rightarrow> 'c"   and x :: "'c"   and c :: "real"   and n :: "nat" assumes "y \<longlonglongrightarrow> x"   and "\<And>n. norm (y (Suc n) - y n) \<le> c ^ n"   and "y 0 = (0::'c)"   and "c < 1" shows "norm (x - y (Suc n)) \<le> c / (1 - c) * c ^ n"
x_def: fixes xs :: "'b list"   and i :: "nat" shows "x xs i = xs ! i"
ih_1: assumes "n \<noteq> 0" shows "local.decode 0 (local.encode 0 e) = e"
axioms_2: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_linear f" shows "bounded_linear_axioms f"
Argument_exists: fixes z :: "complex"   and R :: "real set"   and r :: "real" assumes "z \<noteq> 0"   and "R = {r - pi<..r + pi}" obtains s :: "real" where "is_Arg z s" and "s \<in> R"
fundamental_theorem_of_calculus: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'b" assumes "a \<le> b"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> (f has_vector_derivative f' x) (at x within {a..b})" shows "(f' has_integral f b - f a) {a..b}"
gso_connect: fixes n :: "nat"   and us :: "'a vec list" shows "snd (gram_schmidt.main n us) = gram_schmidt n us"
trie_update_with_def: fixes ks :: "'d list"   and v :: "'e"   and f :: "'e \<Rightarrow> 'e" shows "trie_update_with ks v f \<equiv> update_with_trie ks (\<lambda>vo. f (case vo of None \<Rightarrow> v | Some v \<Rightarrow> v))"
Nml_HcompD_6: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml (t \<^bold>\<star> u)" shows "\<^bold>\<langle>src (un_Prim t)\<^bold>\<rangle>\<^sub>0 = Trg u"
inner_Basis_eq_representation: fixes i :: "'a"   and x :: "'a" assumes "i \<in> B"   and "finite B" shows "local.inner_Basis i x = representation B x i"
gen_valid: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and x :: "nat" assumes "cf.isPath u p x" shows "relabel_effect f l u u \<le> relabel_effect f l u x + length p"
antisym_converse: fixes r :: "('a \<times> 'a) set" shows "antisym (r\<inverse>) = antisym r"
fps_shift_altdef: fixes n :: "nat"   and f :: "'a fps" shows "fps_shift n f = f / fps_X ^ n"
WTrt_elim_cases_7: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and P :: "'c J_prog"   and h :: "'b"   and E :: "String.literal \<Rightarrow> ty option"   and D :: "ty"   and e :: "'c expr"   and T :: "ty" assumes "J_heap_base.WTrt typeof_addr P h E (Cast D e) T" obtains T :: "ty" where "Ta__ = D" and "J_heap_base.WTrt typeof_addr P h E e T" and "is_type P D"
prems_1: shows "length xs = card {a \<in> A. a < length xs + length ys}"
closed_transdb_inv_aux: fixes e :: "db"   and k :: "nat"   and \<Gamma> :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "\<forall>v. freedb e v \<longrightarrow> v < k"   and "\<forall>v. freedb e v \<longrightarrow> \<Gamma> v = k - v - 1" shows "(i \<in> set (free (transdb_inv e \<Gamma> 0 k))) = (i < k \<and> freedb e (k - i - 1))"
fset_member_def: shows "fset_member \<equiv> (rep_fset ---> id) List.member"
abs_reach_def: fixes f :: "'b ltl \<Rightarrow> 'c \<Rightarrow> 'b ltl"   and \<Phi> :: "'b ltl_prop_equiv_quotient" assumes "lift_ltl_transformer f" shows "lift_ltl_transformer.abs_reach f \<Phi> = {foldl (lift_ltl_transformer.f_abs f) \<Phi> w |w. True}"
rename_increasing: fixes h :: "'c \<Rightarrow> 'd"   and F :: "'c program"   and func :: "'d \<Rightarrow> 'e" assumes "bij h" shows "(rename h F \<in> increasing func) = (F \<in> increasing (func \<circ> h))"
Input: fixes a :: "name"   and x :: "name"   and P :: "pi"   and u :: "name" shows "a<x>.P \<Longrightarrow>a<u> \<prec> P[x::=u]"
power_mod_div: fixes x :: "nat"   and n :: "nat"   and m :: "nat" shows "x mod 2 ^ n div 2 ^ m = x div 2 ^ m mod 2 ^ (n - m)"
vec_hom_zero: fixes n :: "nat" shows "vec\<^sub>h (0\<^sub>v n) = 0\<^sub>v n"
eq_iff_5: fixes dbfm :: "dbfm"   and dbfma :: "dbfm" shows "(DBEx dbfm = DBEx dbfma) = (dbfm = dbfma)"
plusl_infty_zero: shows "\<infinity>\<^sub>1 +\<^sub>1 [0]\<^sub>1 = \<infinity>\<^sub>1"
eq_iff: fixes x :: "'d list"   and y :: "'d list" shows "(x = y) = (prefix x y \<and> prefix y x)"
round_robin_reschedule_Some_NoneD: fixes t0 :: "'t"   and queue :: "'t list"   and n0 :: "nat"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and \<sigma>' :: "'t list \<times> nat" assumes "round_robin_reschedule t0 queue n0 s = \<lfloor>(t, None, \<sigma>')\<rfloor>"   and "t0 \<in> set queue" shows "\<exists>x ln n. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> 0 < ln $ n \<and> \<not> waiting (wset s t) \<and> may_acquire_all (locks s) t ln"
in_vsingleton: fixes f :: "V"   and a :: "V" assumes "f = a" shows "f \<in>\<^sub>\<circ> ZFC_in_HOL.set {a}"
lunit_char_1: fixes f :: "'c" assumes "C.ide f" shows "\<guillemotleft>\<l>\<^sub>C[f] : C.L f \<Rightarrow>\<^sub>C f\<guillemotright>"
pochhammer_rec: fixes a :: "'a"   and n :: "nat" shows "pochhammer a (Suc n) = a * pochhammer (a + (1::'a)) n"
is_initial: fixes y' :: "'c"   and g :: "'d" assumes "arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G y y' g" shows "\<exists>!f. \<eta>y.is_ext y' g f"
f_antecedent_def: fixes r :: "prop_rule" shows "predicate.f_antecedent r = fset_from_list (antecedent r)"
hyps: fixes x2a :: "('f, 'a) Term.term" assumes "x2a \<in> set ts"   and "weight x2a \<le> b" shows "bounded_arity b (funas_term x2a)"
elims: fixes x :: "'c"   and xa :: "'c list"   and y :: "nat option" assumes "mem_idx x xa = y" obtains uu_ :: "'c" where "x = uu_" and "xa = []" and "y = None"   | xb :: "'c" and a :: "'c" and as :: "'c list" where "x = xb" and "xa = a # as" and "y = (if xb = a then Some 0 else map_option Suc (mem_idx xb as))"
wf\<^sub>t\<^sub>r\<^sub>m_code: fixes t :: "('fun, 'b) Term.term" shows "wf\<^sub>t\<^sub>r\<^sub>m t = wf\<^sub>t\<^sub>r\<^sub>m' arity t"
simps: fixes head :: "'a"   and tail :: "'a iarray"   and prevs :: "(nat \<times> 'b) list" shows "local.next_subseqs (head, tail, prevs) = next_subseqs1 head tail [] [] prevs"
sub2: fixes B :: "'a"   and k :: "'a"   and b :: "'a"   and a :: "'a" assumes "B \<equiv> k + b" shows "a - B \<equiv> - k + (a - b)"
Un_upper1_approx: fixes A :: "((nat \<times> (nat \<Rightarrow> 'a option)) \<times> 'a \<aproc>) set"   and B :: "((nat \<times> (nat \<Rightarrow> 'a option)) \<times> 'a \<aproc>) set" shows "A \<lessapprox> A \<union> B"
path_to_eq_from: fixes loc1 :: "'loc"   and loc2 :: "'loc"   and xs :: "('loc \<times> 'sum \<times> 'loc) list"   and i :: "nat" assumes "path loc1 loc2 xs"   and "i + 1 < length xs" shows "(case xs ! (i + 1) of (s, l, t) \<Rightarrow> s) = (case xs ! i of (s, l, t) \<Rightarrow> t)"
times_right_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and z :: "'b"   and y :: "'b" assumes "SN_Orders.class.ordered_semiring_0 plus zero times less_eq"   and "less_eq zero x"   and "less_eq z y" shows "less_eq (times x z) (times x y)"
strict_trans1: fixes b :: "'b"   and a :: "'b"   and c :: "'b" assumes "b \<le> a"   and "c < b" shows "c < a"
expand: fixes gpv :: "('a, 'b, 'c) gpv"   and gpv' :: "('a, 'b, 'c) gpv" assumes "the_gpv gpv = the_gpv gpv'" shows "gpv = gpv'"
comp_arr_ide: fixes a :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VV.ide a"   and "VV.seq f a" shows "VV.comp f a = f"
swap_msgs_Trans_Send: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isTrans ev"   and "isSend ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
prefix_definedD: fixes xs :: "[:'r:]"   and ys :: "[:'r:]" assumes "prefix\<cdot>xs\<cdot>ys = TT" shows "xs \<noteq> \<bottom> \<and> ys \<noteq> \<bottom>"
single_nz_val_access2: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and v :: "'a" assumes "i < n"   and "j < n"   and "i \<noteq> j" shows "single_nz_val n i v $ j = (0::'a)"
SeqQuoteP_imp_LstSeqP: fixes t :: "tm"   and u :: "tm"   and s :: "tm"   and k :: "tm" shows "{SeqQuoteP t u s k} \<turnstile> LstSeqP s k (HPair t u)"
inj_on_inverseI: fixes A :: "'d set"   and g :: "'e \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> 'e" assumes "\<And>x. x \<in> A \<Longrightarrow> g (f x) = x" shows "inj_on f A"
pcompose_idR: fixes p :: "'b poly" shows "p \<circ>\<^sub>p [:0::'b, 1::'b:] = p"
Theorem_10_fp_cop_F: fixes dsa :: "'b set"   and da :: "'b"   and ds :: "'b set"   and X :: "'a set"   and x :: "'a"   and d :: "'b" assumes "Singleton_for_d.stable_on dsa da ds X"   and "x \<in> X" shows "\<exists>y\<in>Singleton_for_d.CH (Contracts.fp_cop_F (Singleton_for_d.Pd_singletons_for_ds dsa da X {d}) Ch ds). (x, y) \<in> Singleton_for_d.Pd_singletons_for_ds dsa da X {d} (Xd x)"
lspasl_c: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Gamma :: "bool"   and h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and h3 :: "'a"   and Delta :: "bool" assumes "class.heap_sep_algebra plus zero sep_disj"   and "Gamma \<and> heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> h2 = h3 \<longrightarrow> Delta" shows "Gamma \<and> heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> heap_sep_algebra.tern_rel plus sep_disj h1 h3 h0 \<longrightarrow> Delta"
ZObis_finished_discr_L: fixes c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd" assumes "c \<approx>01 d"   and "proper d"   and "finished d" shows "discr c"
hd_\<beta>2a_vs_\<beta>1a: shows "head_\<omega> \<beta>2a < head_\<omega> \<beta>1a \<or> \<beta>2a = 0 \<and> \<beta>1a = 0"
leq_refl: fixes x :: "'f option" shows "option_ord x x"
sturm_gcd: fixes r :: "real poly"   and p :: "real poly" assumes "r \<in> set (sturm p)" shows "gcd p (pderiv p) dvd r"
prefix_split_trace: fixes c :: "('p, 's, 'm) configuration"   and t :: "('p, 's, 'm) event list"   and t' :: "('p, 's, 'm) event list" assumes "\<exists>c'. trace c t c'"   and "prefix t' t" shows "\<exists>c''. trace c t' c''"
pred_def: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b measure" shows "pred M P = ({x \<in> space M. P x} \<in> sets M)"
residue_factor_exists: fixes m :: "nat"   and x :: "nat \<Rightarrow> int"   and k :: "nat" assumes "m = nat (ord_Zp x)"   and "0 < k"   and "x \<in> carrier Zp"   and "x \<noteq> \<zero>" obtains u :: "int" where "u \<in> carrier (residue_ring (p ^ k)) \<and> u * p ^ m = x (m + k)"
Polygamma_def: fixes n :: "nat"   and z :: "'b" shows "Polygamma n z = (if n = 0 then (\<Sum>k. inverse (of_nat (Suc k)) - inverse (z + of_nat k)) - euler_mascheroni else (- (1::'b)) ^ Suc n * fact n * (\<Sum>k. inverse ((z + of_nat k) ^ Suc n)))"
C: fixes self :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "\<AA> state \<times> \<AA> state" shows "\<lambda>_. self \<tau>.boss.contents()->including\<^sub>S\<^sub>e\<^sub>t(\<lambda>_. self \<tau>.salary) \<tau> = self.boss.contents()->including\<^sub>S\<^sub>e\<^sub>t(self.salary) \<tau>"
ex_fold_CONS: fixes \<AA> :: "'i" shows "\<exists>xs \<BB>. \<AA> = fold CONS xs \<BB> \<and> Length \<BB> = 0 \<and> Length \<AA> = length xs \<and> #\<^sub>V \<BB> = #\<^sub>V \<AA> \<and> (\<forall>x\<in>set xs. size x = #\<^sub>V \<AA>)"
integrable_restrict_UNIV: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "((\<lambda>x. if x \<in> s then f x else (0::'d)) integrable_on UNIV) = (f integrable_on s)"
p_not_sys: fixes p :: "'e process_name" shows "(p \<noteq> sys) = (p = gc \<or> (\<exists>m. p = mutator m))"
\<psi>': shows "adjoint_equivalence_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg e e' \<phi> \<psi>'"
ereal_right_distrib: fixes a :: "ereal"   and b :: "ereal"   and r :: "ereal" assumes "0 \<le> a"   and "0 \<le> b" shows "r * (a + b) = r * a + r * b"
strict_boundedE: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "semilattice_order f less_eq less"   and "less a (f b c)" obtains "less a b" and "less a c"
proj_along_geodesic_contraction: fixes G :: "'a set"   and px :: "'a"   and x :: "'a"   and py :: "'a"   and y :: "'a" assumes "geodesic_segment G"   and "px \<in> proj_set x G"   and "py \<in> proj_set y G" shows "dist px py \<le> max (5 * deltaG TYPE('a)) (dist x y - dist px x - dist py y + 10 * deltaG TYPE('a))"
card_M1_le_NVL1: fixes V :: "'b set"   and E :: "('b \<times> 'b) set"   and M :: "('b \<times> 'b) set"   and L :: "'b \<Rightarrow> nat" assumes "matching V E M"   and "OSC L E" shows "card (matching_i 1 V E M L) \<le> N V L 1"
enf_m: shows "\<forall>G. c_sinvar m G = (\<forall>e\<in>edges G. P e)"
struct_specD3: fixes sel :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ap :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g" assumes "punit.struct_spec sel ap ab compl" shows "ab_spec ab"
4: shows "w\<^sub>1 = w\<^sub>1 @ v\<^sub>1"