hcju/isabelleps huggingface dataset:
Init_simps_2: fixes P :: "state \<Rightarrow> bool" shows "Init $P \<equiv> Init P"
js_partition_order: fixes js :: "nat list"   and k :: "nat"   and xs :: "nat list"   and ys :: "nat list"   and i :: "nat"   and j :: "nat" assumes "js <~~> [0..<k + 1]"   and "\<And>i xs ys. \<lbrakk>i < k; js = xs @ ys; xs \<noteq> []; i = last xs\<rbrakk> \<Longrightarrow> i + 1 \<in> set ys"   and "js = xs @ ys"   and "i \<in> set xs"   and "j \<in> set ys" shows "i \<le> j"
g_nz: fixes z :: "complex" shows "g z \<noteq> 0"
finsum_def: fixes G :: "('d, 'e) ring_scheme" shows "finsum G = finprod (add_monoid G)"
Rep_bword: fixes x :: "bword" shows "Rep_bword x \<in> {n. sint NEG_INF \<le> sint n \<and> sint n \<le> sint POS_INF}"
has_fields_b_fields: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> ty) list" assumes "P \<turnstile> C has_fields FDTs" shows "fields P C = FDTs"
induct: fixes P :: "(char list list \<times> int) list \<Rightarrow> bool"   and a0 :: "(char list list \<times> int) list" assumes "P []"   and "\<And>xs n. P [(xs, n)]"   and "\<And>xs n ys m p. \<lbrakk>\<lbrakk>xs = ys; n + m \<noteq> 0\<rbrakk> \<Longrightarrow> P ((xs, n + m) # p); \<lbrakk>xs = ys; \<not> n + m \<noteq> 0\<rbrakk> \<Longrightarrow> P p; xs \<noteq> ys \<Longrightarrow> P ((ys, m) # p)\<rbrakk> \<Longrightarrow> P ((xs, n) # (ys, m) # p)" shows "P a0"
comp_def: fixes Obj :: "'b set"   and Null :: "'b"   and y :: "'b"   and x :: "'b" assumes "discrete_category Obj Null" shows "discrete_category.comp Obj Null y x \<equiv> if x \<in> Obj \<and> x = y then x else Null"
frp_pref_dfs: fixes T1 :: "('d, 'f) tree_automaton_rec"   and T2 :: "('e, 'f) tree_automaton_rec" shows "wa_precise_refine (frp_algo T1 T2) (dfs_algo (ta_initial T1 \<times> ta_initial T2) (f_succ (\<delta>_prod (ta_rules T1) (ta_rules T2)))) frp_\<alpha>"
cycle_free_alt_def: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat" shows "cycle_free M n = cycle_free_up_to M n n"
vbrelation_vinsertI: fixes r :: "V"   and a :: "V"   and b :: "V" assumes "vbrelation r" shows "vbrelation (vinsert \<langle>a, b\<rangle> r)"
fun_sum_list_commute: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" assumes "f (0::'d) = (0::'c)"   and "\<And>x y. f (x + y) = f x + f y" shows "f (sum_list xs) = sum_list (map f xs)"
srf: shows "singular_relcycle 0 X {} (frag_of f)"
prod_dfs_eq_rtrancl: fixes x :: "nat \<times> nat"   and y :: "nat \<times> nat" assumes "prod_is_node A B x"   and "prod_is_node A B y" shows "prod_memb y (prod_dfs A B x) = ((x, y) \<in> (succsr (prod_succs A B))\<^sup>*)"
abs_typeE: fixes e :: "nat \<Rightarrow> LambdaType.type"   and t :: "dB"   and T :: "LambdaType.type" assumes "e \<turnstile> dB.Abs t : T" obtains U :: "LambdaType.type" and V :: "LambdaType.type" where "e\<langle>0:U\<rangle> \<turnstile> t : V"
split_sel_asm: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and prod :: "'b \<times> 'c" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (\<not> (prod = (fst prod, snd prod) \<and> \<not> P (f (fst prod) (snd prod))))"
gfp_f_doctor_optimal: fixes ds :: "'b set"   and XD_XH :: "'a set \<times> 'a set"   and x :: "'a" assumes "stable_pair_on ds XD_XH"   and "x \<in> match XD_XH" shows "\<exists>y\<in>match (gfp_F ds). (x, y) \<in> Pd (Xd x)"
wf_digraph: shows "wf_digraph (app_iso iso_f G)"
case_prod_const: fixes c :: "'d" shows "(\<lambda>(i, j). c) = (\<lambda>_. c)"
prf_encrypt_advantage: fixes \<A> :: "((bool list \<times> bool list) \<times> 'b, bool list, bool list \<times> bool list) gpv \<times> (bool list \<times> bool list \<Rightarrow> 'b \<Rightarrow> (bool, bool list, bool list \<times> bool list) gpv)"   and q :: "nat"   and \<eta> :: "nat" assumes "local.ind_cpa.ibounded_by \<A> (enat q)"   and "lossless_gpv \<I>_full (fst \<A>)"   and "\<And>cipher \<sigma>. lossless_gpv \<I>_full (snd \<A> cipher \<sigma>)" shows "local.ind_cpa.advantage \<eta> \<A> \<le> local.prf.advantage \<eta> (prf_adversary \<eta> \<A>) + real q / 2 ^ len \<eta>"
send_insert_id_2: fixes i :: "'a \<times> nat"   and n :: "nat"   and m :: "('a \<times> nat, 'b) insert_message" assumes "state i \<bind> (\<lambda>s. create_delete s n) = Inr (Insert m)" shows "False"
size_mset_add_mset: fixes x :: "'c \<times> 'd"   and M :: "('c \<times> 'd) multiset" shows "size_mset (add_mset x M) = pair_size x + size_mset M"
viewLn_inv: fixes t :: "('a, 'b) FingerTreeStruc"   and n :: "nat"   and nd :: "('a, 'b) Node"   and s :: "('a, 'b) FingerTreeStruc" assumes "FingerTreeStruc.is_measured_ftree t"   and "FingerTreeStruc.is_leveln_ftree n t"   and "FingerTreeStruc.viewLn t = Some (nd, s)" shows "FingerTreeStruc.is_measured_ftree s \<and> FingerTreeStruc.is_measured_node nd \<and> FingerTreeStruc.is_leveln_ftree n s \<and> FingerTreeStruc.is_leveln_node n nd"
OR2_analz_knows_Spy: fixes B :: "agent"   and N :: "msg"   and A :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B \<lbrace>N, Agent A, Agent B, X\<rbrace> \<in> set evs"   and "evs \<in> otway" shows "X \<in> analz (knows Spy evs)"
upper_rank_of_def: fixes carrier' :: "'a set" shows "\<E>'\<^sub>1.upper_rank_of carrier' \<equiv> Max {card B |B. \<E>'\<^sub>1.basis_in carrier' B}"
free_monoidal_category_axioms: shows "free_monoidal_category C"
beta_norm_imp_beta_reds: fixes t :: "Core.term"   and t' :: "Core.term" assumes "beta_norm t = Some t'" shows "t \<rightarrow>\<^sub>\<beta>\<^sup>* t'"
prems: shows "\<forall>a\<in>set (z # leq). case a of (d, e, f) \<Rightarrow> \<exists>y'>q. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<le> 0"
flowD_capacity_OUT: fixes \<Delta> :: "('k, 'l) network_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" assumes "flow \<Delta> f" shows "d_OUT f x \<le> d_OUT (capacity \<Delta>) x"
len1: shows "length (takeWhile ((\<noteq>) n1) (rev xs)) < length (rev xs)"
ideal_generated_triple_pair_rewrite: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a"   and d' :: "'a" assumes "ideal_generated {a, b, c} = ideal_generated {d}"   and "ideal_generated {a, b} = ideal_generated {d'}" shows "ideal_generated {d', c} = ideal_generated {d}"
measure_UNION: fixes A :: "nat \<Rightarrow> 'c set"   and M :: "'c measure" assumes "range A \<subseteq> sets M"   and "disjoint_family A"   and "emeasure M (\<Union> (range A)) \<noteq> \<infinity>" shows "(\<lambda>i. Sigma_Algebra.measure M (A i)) sums Sigma_Algebra.measure M (\<Union> (range A))"
Vr_1_f_1: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "1\<^sub>r\<^bsub>Vr K v\<^esub> = 1\<^sub>r"
f_aggregate_assoc: fixes a :: "nat"   and ag :: "'b list \<Rightarrow> 'b"   and xs :: "'b list"   and b :: "nat" assumes "\<And>xs. length xs mod a = 0 \<Longrightarrow> ag (f_aggregate xs a ag) = ag xs" shows "f_aggregate (f_aggregate xs a ag) b ag = f_aggregate xs (a * b) ag"
finfun_apply_inject: fixes x :: "('a, 'b) finfun"   and y :: "('a, 'b) finfun" shows "(finfun_apply x = finfun_apply y) = (x = y)"
assms_5: fixes c :: "'a"   and b :: "bool" shows "P1 c (Bool b)"
unat_of_bl_length: fixes xs :: "bool list" shows "unat (of_bl xs) < 2 ^ length xs"
MapsToCompI: fixes f :: "'b"   and g :: "'b" assumes "f \<approx>> g" shows "f ;; g maps dom f to cod g"
pps_to_list_def: fixes S :: "('x \<Rightarrow>\<^sub>0 nat) set" shows "pp_pm.punit.pps_to_list S = rev (pp_pm.ordered_powerprod_lin.sorted_list_of_set S)"
rank_of_indep_in_le: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> carrier"   and "indep_in \<E> X" shows "card X \<le> rank_of \<E>"
congruent_def: fixes r :: "('a \<times> 'a) set"   and f :: "'a \<Rightarrow> 'b" shows "f respects r = (\<forall>(y, z)\<in>r. f y = f z)"
ih: fixes ta :: "('s, 'v) tm"   and sa :: "('s, 'v) tm" assumes "size ta + size sa < size t + size s" shows "(ta >\<^sub>t sa) = (ta >\<^sub>t\<^sub>o sa)"
disjE_realizer2: fixes P :: "bool"   and Q :: "'b \<Rightarrow> bool"   and x :: "'b option"   and R :: "'c \<Rightarrow> bool"   and f :: "'c"   and g :: "'b \<Rightarrow> 'c" assumes "case x of None \<Rightarrow> P | Some q \<Rightarrow> Q q"   and "P \<Longrightarrow> R f"   and "\<And>q. Q q \<Longrightarrow> R (g q)" shows "R (case x of None \<Rightarrow> f | Some q \<Rightarrow> g q)"
of_seq_rec: fixes f :: "nat \<Rightarrow> 'b" shows "of_seq f = f 0 ## of_seq (f \<circ> Suc)"
listset_doubletonI: fixes a :: "'b"   and A :: "'b set"   and b :: "'b"   and B :: "'b set"   and ys :: "'b list" assumes "a \<in> A"   and "b \<in> B"   and "ys = [a, b]" shows "ys \<in> listset [A, B]"
intros_2: fixes \<Gamma> :: "(name \<times> ty) list"   and t :: "trm"   and T :: "ty"   and u :: "trm"   and U :: "ty" assumes "\<Gamma> \<turnstile> t : T"   and "\<Gamma> \<turnstile> u : U" shows "\<Gamma> \<turnstile> \<langle>t, u\<rangle> : T \<otimes> U"
reflect_poly_Poly_nz: fixes xs :: "'b list" assumes "no_trailing ((=) (0::'b)) xs" shows "reflect_poly (Poly xs) = Poly (rev xs)"
B: fixes x :: "'a" assumes "x \<in> inv f ` Basis" shows "norm x \<le> B"
abs_idempotent: fixes abs :: "'g \<Rightarrow> 'g"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and uminus :: "'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs (abs a) = abs a"
FS: shows "set (rsplit0 (Floor a)) = (\<Union>(p, n, s)\<in>{(p, n, s). (p, n, s) \<in> set (rsplit0 a) \<and> n = 0}. {(p, 0, Floor s)}) \<union> (\<Union>(p, n, s)\<in>{(p, n, s). (p, n, s) \<in> set (rsplit0 a) \<and> 0 < n}. {(case (p, n, s) of (p, n, s) \<Rightarrow> \<lambda>j. ((case (p, n, s) of (p, n, s) \<Rightarrow> fp p n s) j, 0, Add (Floor s) (C j))) j |j. j \<in> {0..n}}) \<union> (\<Union>(p, n, s)\<in>{(p, n, s). (p, n, s) \<in> set (rsplit0 a) \<and> n < 0}. {(case (p, n, s) of (p, n, s) \<Rightarrow> \<lambda>j. ((case (p, n, s) of (p, n, s) \<Rightarrow> fp p n s) j, 0, Add (Floor s) (C j))) j |j. j \<in> {n..0}})"
dim_nonzero: fixes M :: "'b set" assumes "0 < dim_R M" shows "M \<noteq> 0"
Image_singleton_iff: fixes b :: "'a"   and r :: "('b \<times> 'a) set"   and a :: "'b" shows "(b \<in> r `` {a}) = ((a, b) \<in> r)"
fin1_atoms: fixes x :: "'b"   and mr :: "'b mrexp3"   and r :: "'b rexp" assumes "x \<in> fin1 mr"   and "mr \<in> mrexps3 r" shows "x \<in> atoms r"
RRR_interior_closure: shows "interior (insert 3 ({0..2} \<union> {5..7})) = {0<..<2} \<union> {5<..<7}"
eq_dff_refl: fixes x :: "'b fract" shows "x =dff x"
new_element_child_nodes: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_element_ptr :: "(_) element_ptr" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr" shows "h' \<turnstile> get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t new_element_ptr child_nodes \<rightarrow>\<^sub>r []"
trans_reach: fixes T :: "('a, 'b) TS_scheme" shows "{reach T} TS.trans T {> reach T}"
that: fixes a :: "'n" assumes "a \<noteq> (0::'n)"   and "\<And>S. f ` S \<subseteq> {x. a \<bullet> x = 0}" shows "thesis"
BIND_def: fixes name :: "char list"   and inp :: "nat"   and P :: "'a" shows "BIND name inp P \<equiv> P"
open_UN: fixes "open" :: "'c set \<Rightarrow> bool"   and A :: "'d set"   and B :: "'d \<Rightarrow> 'c set" assumes "class.topological_space open"   and "\<forall>x\<in>A. open (B x)" shows "open (\<Union> (B ` A))"
min_def: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" shows "ord.min less_eq a b = (if less_eq a b then a else b)"
IH1: fixes \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "Vagree \<nu> \<omega> (\<Union>y\<in>{y. Inl (Inr y) \<in> SIGF x2}. FVT (\<sigma> y))"   and "fsafe x2" shows "fml_sem (adjointFO I \<sigma> \<nu>) x2 = fml_sem (adjointFO I \<sigma> \<omega>) x2"
fct_gen_subalgebra_info: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and x :: "'b"   and w :: "'b"   and A :: "'b set" assumes "f \<in> space M \<rightarrow> space N"   and "x \<in> space M"   and "w \<in> space M"   and "f x = f w"   and "A \<in> sets (fct_gen_subalgebra M N f)" shows "(x \<in> A) = (w \<in> A)"
finite_induct: fixes P :: "C \<Rightarrow> bool"   and n :: "nat"   and x :: "C" assumes "P \<bottom>"   and "\<And>Ca. P Ca \<Longrightarrow> P (C\<cdot>Ca)" shows "P (C_take n\<cdot>x)"
bres_def: fixes x :: "'a"   and z :: "'a" shows "x \<rightarrow> z = \<Squnion>{y. x \<cdot> y \<le> z}"
induct: fixes P :: "bexp \<Rightarrow> bexp \<Rightarrow> bool"   and a0 :: "bexp"   and a1 :: "bexp" assumes "\<And>b. P (Bc True) b"   and "P (Bc False) (Bc True)"   and "\<And>v. P (bexp.Not v) (Bc True)"   and "\<And>v va. P (And v va) (Bc True)"   and "\<And>v va. P (Less v va) (Bc True)"   and "P (Bc False) (Bc False)"   and "\<And>v. P (Bc False) (bexp.Not v)"   and "\<And>v va. P (Bc False) (And v va)"   and "\<And>v va. P (Bc False) (Less v va)"   and "\<And>v. P (bexp.Not v) (Bc False)"   and "\<And>v va. P (And v va) (Bc False)"   and "\<And>v va. P (Less v va) (Bc False)"   and "\<And>v va. P (bexp.Not v) (bexp.Not va)"   and "\<And>v va vb. P (bexp.Not v) (And va vb)"   and "\<And>v va vb. P (bexp.Not v) (Less va vb)"   and "\<And>v va vb. P (And v va) (bexp.Not vb)"   and "\<And>v va vb vc. P (And v va) (And vb vc)"   and "\<And>v va vb vc. P (And v va) (Less vb vc)"   and "\<And>v va vb. P (Less v va) (bexp.Not vb)"   and "\<And>v va vb vc. P (Less v va) (And vb vc)"   and "\<And>v va vb vc. P (Less v va) (Less vb vc)" shows "P a0 a1"
ereal_liminf_limsup_minus: fixes u :: "nat \<Rightarrow> ereal"   and v :: "nat \<Rightarrow> ereal" shows "liminf (\<lambda>n. u n - v n) \<le> limsup u - limsup v"
points_index_ps_nin: fixes ps :: "'a set" assumes "\<not> ps \<subseteq> \<V>" shows "\<B> index ps = 0"
le_iff_inf: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.semilattice_inf inf less_eq less" shows "less_eq x y = (inf x y = x)"
insert: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "A \<noteq> {}" shows "join.Sup_fin (insert x A) = x + join.Sup_fin A"
\<psi>_dim_2: shows "dim_vec \<psi> = N"
progress_mono: fixes j :: "nat"   and j' :: "nat"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and \<phi> :: "Formula.formula" assumes "j \<le> j'" shows "Monitor.progress \<sigma> P \<phi> j \<le> Monitor.progress \<sigma> P \<phi> j'"
"1_3": shows "length as' \<le> S vs lss PROB (fmrestrict_set vs s)"
absOfS_preserves_wls: fixes P :: "('index, 'bindex, 'varSort, 'var, 'opSym, 'sort) paramS"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and us :: "'varSort"   and s :: "'sort" assumes "wlsPar P"   and "A \<in> absOfS P (us, s)" shows "wlsAbs (us, s) A"
short_ch_def: fixes X :: "'a set" shows "short_ch X \<equiv> \<exists>x\<in>X. \<exists>y\<in>X. (\<exists>Q. path Q x y) \<and> \<not> (\<exists>z\<in>X. z \<noteq> x \<and> z \<noteq> y)"
closest_pair_recurrence: shows "closest_pair_recurrence \<in> \<Theta>(\<lambda>x. real x * ln (real x))"
linorder_not_le: fixes x :: "'k"   and y :: "'k" shows "(\<not> x \<le> y) = (y < x)"
add_missing_ret_unfoldings_def: fixes rs1 :: "'c rule list"   and rs2 :: "'c rule list" shows "add_missing_ret_unfoldings rs1 rs2 \<equiv> foldr (\<lambda>rf. (\<circ>) (add_match (MatchNot (get_match rf)))) (filter (\<lambda>r. get_action r = Return) rs1) id rs2"
fa: shows "f \<star> a \<noteq> null \<and> (f \<star> a) \<star> a \<noteq> null \<and> ((f \<star> a) \<star> a) \<star> a \<noteq> null"
VcompNml_assoc: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term" assumes "Nml t"   and "Nml u"   and "Nml v"   and "Dom t = Cod u"   and "Dom u = Cod v" shows "(t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u) \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> v = t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> v"
split_label: fixes s :: "'a"   and l :: "String.literal"   and i :: "value list"   and r :: "nat \<Rightarrow>f value option"   and e :: "(('a \<times> 'b) \<times> 'c transition_scheme) fset" shows "ffilter (\<lambda>((origin, dest), t). origin = s \<and> Label t = l \<and> can_take_transition t i r) e = ffilter (\<lambda>((origin, dest), t). origin = s \<and> can_take_transition t i r) (ffilter (\<lambda>((origin, dest), t). Label t = l) e)"
decomps_2: shows "Cs @ [C] = As @ B # Bs"
closed_translation: fixes S :: "'a set"   and a :: "'a" assumes "closed S" shows "closed ((+) a ` S)"
lfilter_code_1: fixes P :: "'b \<Rightarrow> bool" shows "lfilter P LNil = LNil"
isEmpty_def: fixes q :: "('c, 'd) BinomialHeap" shows "isEmpty q \<equiv> Rep_BinomialHeap q = []"
simps_17: fixes vb :: "'a"   and vc :: "value list"   and v :: "'a"   and va :: "value list" shows "local.less_gexp_aux (In vb vc) (In v va) = (vb < v \<or> vb = v \<and> vc < va)"
nres_inequalities_3: fixes x :: "'c" shows "FAIL \<noteq> RETURN x"
bernoulli'_fps_aux: shows "(fps_exp (1::'b) - 1) * Abs_fps (\<lambda>n. of_real (bernoulli' n) / fact n) = fps_exp (1::'b) * fps_X"
IHP_2: shows "((u # r, c), e # w, r', c') \<in> trcl (trss fg)"
normalize_bezout_gcd: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and p :: "'b"   and q :: "'b"   and u :: "'b"   and v :: "'b"   and d :: "'b"   and a :: "'b"   and b :: "'b" assumes "is_bezout_ext bezout"   and "(p, q, u, v, d) = bezout a b" shows "normalize d = gcd a b"
simps_1: fixes A :: "'f set option" shows "\<D>s [] A = True"
br_list_rel: fixes f :: "'c \<Rightarrow> 'd"   and P :: "'c \<Rightarrow> bool" shows "\<langle>br f P\<rangle>list_rel = br (map f) (list_all P)"
respection: shows "rel_respects_binary_pred (indRelRTPO TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))"
rel_refl: fixes x :: "'a \<times> 'a" assumes "x \<in> R \<times> S" shows "x \<sim> x"
g2D: fixes x :: "real" assumes "x \<in> {0..1} - insert 0 ((\<lambda>x. 2 * x - 1) ` S)" shows "g2 differentiable at x"
preserves_arr: fixes f :: "'a" assumes "B.arr f" shows "B.arr (trg\<^sub>B f)"
im_sheaf_morphisms_def: fixes U :: "'a set set"   and V :: "'a set set" shows "im_sheaf_morphisms U V \<equiv> \<rho> (f \<^sup>\<inverse> X U) (f \<^sup>\<inverse> X V)"
lincomb_empty: fixes a :: "'a Matrix.vec \<Rightarrow> 'a" shows "NC.lincomb a {} = 0\<^sub>v n"
elims_3: fixes x :: "'a extC"   and xa :: "'a extC" assumes "\<not> x < xa" obtains x :: "'a extC" and y :: "'a extC" where "xa__ = x" and "xa = y" and "\<not> (x \<le> y \<and> \<not> y \<le> x)"
simps_1: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and x :: "Preliminaries.var" shows "Ik.int \<xi> (Var x) = \<xi> x"
0: shows "exi out (cnj \<sigma> \<phi>) = exi z (subst (cnj \<sigma> \<phi>) (Var z) out)"
s_ce_is_pr: shows "s_ce \<in> PrimRec2"
oalist_inv_map2_val_raw: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and ys :: "('a \<times> 'd) list \<times> 'o"   and g :: "('a \<times> 'c) list \<times> 'o \<Rightarrow> ('a \<times> 'e) list \<times> 'o"   and h :: "('a \<times> 'd) list \<times> 'o \<Rightarrow> ('a \<times> 'e) list \<times> 'o"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "oalist_inv xs"   and "oalist_inv ys"   and "map2_val_compat' g"   and "map2_val_compat' h" shows "oalist_inv (map2_val_raw f g h xs ys)"
mHom_eq: fixes N :: "('i, 'b, 'j) Module_scheme"   and f :: "'a \<Rightarrow> 'i"   and g :: "'a \<Rightarrow> 'i" assumes "R module N"   and "f \<in> mHom R M N"   and "g \<in> mHom R M N"   and "\<forall>m\<in>carrier M. f m = g m" shows "f = g"
structure_is_trivial: shows "D.\<phi> = D.FoT\<^sub>C.map"
pdivides_imp_splitted: fixes p :: "'a list"   and q :: "'a list" assumes "p \<in> carrier (poly_ring L)"   and "q \<in> carrier (poly_ring L)"   and "q \<noteq> []"   and "splitted q"   and "p pdivides q" shows "splitted p"
rotate_add: fixes m :: "nat"   and n :: "nat" shows "rotate (m + n) = rotate m \<circ> rotate n"
intros_1: fixes uu :: "'c" shows "Lt uu \<prec> \<infinity>"
singletonI: fixes a :: "'a" shows "a \<in> {a}"
c_hd_is_pr: shows "c_hd \<in> PrimRec1"
measurable_translation: fixes S :: "'c set"   and a :: "'c" assumes "S \<in> lmeasurable" shows "(+) a ` S \<in> lmeasurable"
induct: fixes P :: "exp \<Rightarrow> bool"   and exp :: "exp" assumes "\<And>x. P (EVar x)"   and "\<And>x. P (ENat x)"   and "\<And>x1a x2a. P x2a \<Longrightarrow> P (ELam x1a x2a)"   and "\<And>x1a x2a. \<lbrakk>P x1a; P x2a\<rbrakk> \<Longrightarrow> P (EApp x1a x2a)"   and "\<And>x1a x2a x3. \<lbrakk>P x2a; P x3\<rbrakk> \<Longrightarrow> P (EPrim x1a x2a x3)"   and "\<And>x1a x2a x3. \<lbrakk>P x1a; P x2a; P x3\<rbrakk> \<Longrightarrow> P (EIf x1a x2a x3)" shows "P exp"
sorted_take_nth: fixes i :: "nat"   and xs :: "'b list"   and y :: "'b" assumes "0 < i"   and "i < length xs"   and "xs ! (i - 1) \<le> y"   and "sorted (take i xs)" shows "\<forall>x\<in>set (take i xs). x \<le> y"
k': shows "1 / 2 ^ k \<le> 2 * exp (15 / 2 / 5 * ln 2) * exp (- ((D - C / 2) * ln 2 / (5 * delta)))"
\<S>_abss_reps: fixes s :: "'s \<times> ('c \<Rightarrow> real) set" assumes "s \<in> \<S>" shows "abss (reps s) = s"
z: fixes x :: "'c" assumes "x \<in> A" shows "g ` {y \<in> A. f y = f x} = {g x}"
no_winning_strategy_on_deadends: fixes v :: "'a"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "v \<in> VV p"   and "deadend v"   and "strategy p \<sigma>" shows "\<not> winning_strategy p \<sigma> v"
prod_cont_iff: fixes f :: "'r \<times> 's \<Rightarrow> 't" shows "cont f = ((\<forall>y. cont (\<lambda>x. f (x, y))) \<and> (\<forall>x. cont (\<lambda>y. f (x, y))))"
homs_2: fixes x :: "'a permutation"   and y :: "'a permutation" assumes "x \<in> pG"   and "y \<in> pG" shows "binop (\<ii>\<pp> x) (\<ii>\<pp> y) = \<ii>\<pp> (x + y)"
assms_3: shows "snd (cs c cid) \<noteq> Done"
maxsimp_map_into_image: fixes x :: "'a set" assumes "folding_g.chamber x" shows "SimplicialComplex.maxsimp (f \<turnstile> X) (f ` x)"
A2: shows "computable {n. nat_to_ce_set n \<in> C}"
cblinfun_eq_gen_eqI: fixes G :: "'c set"   and A :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'd"   and B :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'd" assumes "\<And>x. x \<in> G \<Longrightarrow> A *\<^sub>V x = B *\<^sub>V x"   and "ccspan G = \<top>" shows "A = B"
Not_eqvt: fixes p :: "perm"   and x :: "('b, 'c, 'd, 'e) formula" shows "p \<bullet> FL_Formula.Not x = FL_Formula.Not (p \<bullet> x)"
Rep_SkewBinomialHeap_inverse: fixes x :: "('a, 'b) SkewBinomialHeap" shows "Abs_SkewBinomialHeap (Rep_SkewBinomialHeap x) = x"
abs_fset_def: shows "abs_fset \<equiv> quot_type.abs (\<approx>) Abs_fset"
gt_trans: fixes z :: "'a"   and y :: "'a"   and x :: "'a" assumes "gt z y"   and "gt y x" shows "gt z x"
converseE: fixes yx :: "'d \<times> 'e"   and r :: "('e \<times> 'd) set" assumes "yx \<in> r\<inverse>" obtains x :: "'e" and y :: "'d" where "yx = (y, x)" and "(x, y) \<in> r"
pmf_expectation_eq_infsetsum: fixes p :: "'a pmf"   and f :: "'a \<Rightarrow> real" shows "measure_pmf.expectation p f = (\<Sum>\<^sub>ax. pmf p x * f x)"
run_catch_env: fixes catch :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and m1 :: "('c, 'm) envT"   and m2 :: "('c, 'm) envT"   and r :: "'c" shows "run_env (local.catch_env catch m1 m2) r = catch (run_env m1 r) (run_env m2 r)"
top_higher: fixes x :: "'a hrel" assumes "x \<in> carrier thy_order" shows "le thy_order x \<^bold>\<top>"
arr_char: fixes f :: "'d \<times> 'd \<times> 'd" shows "D'.VVV.arr f = (D'.arr (fst f) \<and> D'.VV.arr (snd f) \<and> D'.src (fst f) = D'.trg (fst (snd f)))"
Suc: fixes j' :: "nat" shows "horner F G n ((F ^^ j') s) (f j') x = (\<Sum>j = 0..<n. (- 1) ^ j * (1 / real (f (j' + j))) * x ^ j)"
assms_11: shows "comm_monoid_mult_class.coprime lu p"
zariski_is_topological_space: shows "topological_space cring0.spectrum cring0.is_zariski_open"