hcju/isabelleps huggingface dataset:
to_nat_0: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "mod_type.to_nat Rep zero = 0"
measure_m_c: fixes X :: "char list set" assumes "finite X" shows "{(c, c \<nabla>\<^sub>c c') |c c'. strip c' = strip c \<and> c \<in> Com X \<and> c' \<in> Com X \<and> \<not> c' \<sqsubseteq> c}\<inverse> \<subseteq> measure (m_c (m_o (m_st m_ivl) (2 * card X)))"
bounded_gen_check_def: fixes C :: "nat list \<Rightarrow> nat \<Rightarrow> bool"   and B :: "nat" shows "bounded_gen_check C B \<equiv> (\<forall>x xs s. B < x \<longrightarrow> C (x # xs) s = False) \<and> (\<forall>x x' xs s s'. C (x # xs) s \<longrightarrow> x' \<le> x \<longrightarrow> s' \<le> s \<longrightarrow> C (x' # xs) s')"
reindex_bij_betw: fixes h :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "bij_betw h S T" shows "(\<Prod>x\<in>S. g (h x)) = prod g T"
rep_eq: fixes x :: "('c, 'd) uexpr"   and xa :: "'d \<Longrightarrow> 'b" shows "\<lbrakk>x \<oplus>\<^sub>p xa\<rbrakk>\<^sub>e = (\<lambda>b. \<lbrakk>x\<rbrakk>\<^sub>e (get\<^bsub>xa\<^esub> b))"
inner_prod_with_itself_Re_non0: fixes u :: "complex Matrix.vec" assumes "u \<noteq> 0\<^sub>v (dim_vec u)" shows "0 < Re \<langle>u|u\<rangle>"
lens_override_def: fixes X :: "'d \<Longrightarrow> 'c"   and S\<^sub>1 :: "'c"   and S\<^sub>2 :: "'c" shows "S\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> S\<^sub>2 = put\<^bsub>X\<^esub> S\<^sub>1 (get\<^bsub>X\<^esub> S\<^sub>2)"
add_mset_diff_bothsides: fixes a :: "'k"   and M :: "'k multiset"   and A :: "'k multiset" shows "add_mset a M - add_mset a A = M - A"
r_one: fixes x :: "'c" assumes "x \<in> carrier H" shows "x \<otimes>\<^bsub>H\<^esub> \<one>\<^bsub>H\<^esub> = x"
inangle__suma: fixes P :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p" assumes "P InAngle A B C" shows "A B P P B C SumA A B C"
length_terminals: fixes p :: "(symbol \<times> character list) list" shows "length (terminals p) = length p"
r_pdec1: fixes e :: "nat" shows "eval r_pdec1 [e] \<down>= pdec1 e"
conjugate_minus: fixes conjugate :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.conjugatable_ring conjugate plus zero minus uminus times" shows "conjugate (minus x y) = minus (conjugate x) (conjugate y)"
reflect_poly_simps_8: fixes xs :: "'a poly list" shows "reflect_poly (prod_list xs) = prod_list (map reflect_poly xs)"
rep_eq: fixes x :: "'a fset set" shows "fset (Sup x) = (if finite (\<Union> (fset ` x)) then \<Union> (fset ` x) else {})"
discrim_def: fixes a :: "real"   and b :: "real"   and c :: "real" shows "discrim a b c \<equiv> b\<^sup>2 - 4 * a * c"
spies_Notes_rev: fixes evs :: "event list"   and A :: "agent"   and X :: "msg" shows "knows Spy (evs @ [Notes A X]) = (if A \<in> bad then insert X (knows Spy evs) else knows Spy evs)"
a: shows "Expr2ZF ` WellFormedToSet aS\<^bsub>T\<^esub> n \<in> range explode"
crossproduct_noteq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.comm_semiring_1_cancel_crossproduct plus minus zero times one" shows "(a \<noteq> b \<and> c \<noteq> d) = (plus (times a c) (times b d) \<noteq> plus (times a d) (times b c))"
complete_update: fixes t :: "('c \<times> 'd) tree23"   and x :: "'c"   and y :: "'d" assumes "complete t" shows "complete (update x y t)"
MProf: fixes n :: "nat" shows "(\<lambda>i. if h i < n then Top S (P i) else Pa i) \<in> Prof"
distinct_disc_20: fixes event :: "('a, 'b, 'c) event" assumes "isRecvMarker event" shows "\<not> isSnapshot event"
encR: fixes S :: "'procS"   and TRel :: "('procT \<times> 'procT) set" shows "SourceTerm S \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> TargetTerm (\<lbrakk>S\<rbrakk>)"
INFV_def: shows "INFV = \<Sqinter> (range V)"
injectivity: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "injective f"   and "f x = f y" shows "x = y"
Gets_A_knows_K: fixes A :: "agent"   and Tk :: "nat"   and B :: "agent"   and K :: "nat"   and Ticket :: "msg"   and evs :: "event list" assumes "Gets A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, Ticket\<rbrace>) \<in> set evs"   and "evs \<in> bankerb_gets" shows "Key K \<in> analz (knows A evs)"
analz_knows_absorb: fixes K :: "nat"   and evs :: "event list"   and H :: "nat set" assumes "Key K \<in> analz (knows Spy evs)" shows "analz (Key ` insert K H \<union> knows Spy evs) = analz (Key ` H \<union> knows Spy evs)"
0: shows "rawpsubst (rawpsubst (eql t1 t2) (zip (map Var us) (map snd txs))) (zip (map fst txs) us) = eql (rawpsubstT (rawpsubstT t1 (zip (map Var us) (map snd txs))) (zip (map fst txs) us)) (rawpsubstT (rawpsubstT t2 (zip (map Var us) (map snd txs))) (zip (map fst txs) us))"
less_eq_bot_Bot_is_Bot: fixes x :: "'a bot" assumes "x \<le> Bot" shows "x = Bot"
set_mset_add_mset_insert: fixes a :: "'f"   and A :: "'f multiset" shows "set_mset (add_mset a A) = insert a (set_mset A)"
idom_ops_poly: shows "idom_ops (poly_ops ops) poly_rel"
Goal: fixes \<Psi>' :: "'b"   and Pa :: "('a, 'b, 'c) psi"   and Qa :: "('a, 'b, 'c) psi" assumes "(\<Psi>', Pa, Qa) \<in> {(\<Psi>, P, Q) |\<Psi> P Q. \<exists>P' Q'. \<Psi> \<rhd> P \<approx> P' \<and> (\<Psi>, P', Q') \<in> X \<and> \<Psi> \<rhd> Q' \<sim> Q}" shows "\<Psi>' \<rhd> Pa \<leadsto><{(\<Psi>, P, Q) |\<Psi> P Q. \<exists>P' Q'. \<Psi> \<rhd> P \<approx> P' \<and> (\<Psi>, P', Q') \<in> X \<and> \<Psi> \<rhd> Q' \<sim> Q}> Qa"
components_spoly_subset: fixes p :: "'t \<Rightarrow>\<^sub>0 'd"   and q :: "'t \<Rightarrow>\<^sub>0 'd" shows "component_of_term ` keys (spoly p q) \<subseteq> component_of_term ` Keys {p, q}"
abs_sgn: fixes a :: "'b" shows "sgn \<bar>a\<bar> = of_bool (a \<noteq> (0::'b))"
imp_simps_1: fixes Q :: "fm" shows "imp F Q = T"
simps_1: fixes Obj :: "'d set"   and Mor :: "'e set"   and Dom :: "'e \<Rightarrow> 'd"   and Cod :: "'e \<Rightarrow> 'd"   and Id :: "'d \<Rightarrow> 'e"   and Comp :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and more :: "'f" shows "obj\<^bsub>\<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr>\<^esub> = Obj"
sifum_security_def: 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" shows "sifum_security dma \<C>_vars eval \<equiv> sifum_security_init dma \<C>_vars eval"
subt_decompose: fixes s :: "'b trm" shows "\<forall>t1 t2. t1 \<cdot> t2 \<prec> s \<longrightarrow> t1 \<prec> s \<and> t2 \<prec> s"
cons_cluster_has_ne_slices: fixes slices :: "'b \<Rightarrow> 'b set set"   and W :: "'b set"   and C :: "'b set"   and p :: "'b"   and Sl :: "'b set" assumes "stellar slices W"   and "with_w.is_cons_cluster (stellar.quorum_of slices W) W C"   and "p \<in> C"   and "Sl \<in> slices p" shows "Sl \<noteq> {}"
trss_base: fixes u :: "'a"   and a :: "'c"   and v :: "'a"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and r :: "'a list"   and c :: "'a list multiset" assumes "(u, Base a, v) \<in> edges fg" shows "((u # r, c), LBase a, v # r, c) \<in> trss fg"
right_neutral: fixes a :: "'a" shows "a \<oplus> \<zero> = a"
pF: shows "mset_factors (image_mset poly_x_minus_y F) (poly_x_minus_y p)"
if_then_else_Error_1: fixes a :: "bool"   and b :: "'m"   and c :: "('l, 'm) result"   and d :: "'l" shows "((if a then Ok b else c) = Error d) = (\<not> a \<and> c = Error d)"
m1_inv2r_servI: fixes x :: "'a m1r_state_scheme" assumes "\<And>A B Rb Kab Ts nlb. \<lbrakk>B \<notin> bad; runs x Rb = Some (Resp, [A, B], aKey Kab # aNum Ts # nlb)\<rbrakk> \<Longrightarrow> \<exists>Rs Na. Kab = sesK (Rs $ sk) \<and> runs x Rs = Some (Serv, [A, B], [aNon Na, aNum Ts])" shows "x \<in> m1_inv2r_serv"
refl: fixes a :: "'b \<times> 'c" shows "less_eq_prod a a"
gval_foldr_gOr_true: fixes l :: "'a gexp list"   and g :: "'a gexp"   and s :: "'a \<Rightarrow> value option" shows "(gval (fold gOr l g) s = true) = ((\<exists>g'\<in>set (g # l). gval g' s = true) \<and> (\<forall>g'\<in>set (g # l). gval g' s \<noteq> invalid))"
sympartp_equivpartp: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "sympartp (equivpartp r) = equivpartp r"
coeff_mult_degree_sum: fixes p :: "'b poly"   and q :: "'b poly" shows "coeff (p * q) (degree p + degree q) = lead_coeff p * lead_coeff q"
empty_tokens_is_filter: fixes T :: "(symbol \<times> character list) set" shows "empty_tokens T \<subseteq> T"
fin_dim_def: shows "fin_dim = (\<exists>A. finite A \<and> A \<subseteq> carrier_vec n \<and> local.span A = carrier_vec n)"
crel_vs_lookup: fixes param :: "'k" shows "crel_vs (\<lambda>v v'. case v' of None \<Rightarrow> True | Some v' \<Rightarrow> v = v' \<and> v = dp param) (dp param) (lookup param)"
get_ACS_def: fixes M :: "'a SecurityInvariant list" shows "TopoS_Composition_Theory_impl.get_ACS M \<equiv> filter (\<lambda>m. \<not> implc_isIFS m) M"
lideal: fixes r :: "('a \<times> 'a) set"   and a :: "('a \<times> 'a) set" assumes "r \<in> carrier_local_ring_at"   and "a \<in> \<ww>" shows "mult_local_ring_at r a \<in> \<ww>"
r_consistent_diverg: fixes r_psi :: "recf"   and e :: "nat"   and i :: "nat" assumes "recfn 2 r_psi"   and "\<exists>k<e_length e. eval r_psi [i, k] \<up>" shows "eval (r_consistent r_psi) [i, e] \<up>"
h3: shows "distinct (get_signs (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2))))"
time_closest_pair_rec_tm_simps_2: fixes xs :: "(int \<times> int) list" assumes "\<not> length xs \<le> 3" shows "time (closest_pair_rec_tm xs) = 1 + (let (xs\<^sub>L, xs\<^sub>R) = Time_Monad.val (split_at_tm (length xs div 2) xs); (ys\<^sub>L, p\<^sub>L) = Time_Monad.val (closest_pair_rec_tm xs\<^sub>L); (ys\<^sub>R, p\<^sub>R) = Time_Monad.val (closest_pair_rec_tm xs\<^sub>R); ys = Time_Monad.val (merge_tm snd ys\<^sub>L ys\<^sub>R) in time (length_tm xs) + time (split_at_tm (length xs div 2) xs) + time (closest_pair_rec_tm xs\<^sub>L) + time (closest_pair_rec_tm xs\<^sub>R) + time (merge_tm snd ys\<^sub>L ys\<^sub>R) + t_combine p\<^sub>L p\<^sub>R (fst (hd xs\<^sub>R)) ys)"
valid: shows "\<forall>M s. list_all (semantics M s) G \<longrightarrow> M, s \<Turnstile> p"
vec_mat_Tensor_matrix_id: fixes v :: "'a list" shows "vec_mat_Tensor v [[id]] = [v]"
vec_to_iarray_scaleR: fixes a :: "real"   and x :: "('a, 'b) vec" shows "vec_to_iarray (a *\<^sub>R x) = a *\<^sub>R vec_to_iarray x"
of_bits_nat_def: shows "of_bits = horner_sum of_bool 2"
m1_inv3r_initI: fixes x :: "'a m1_state_scheme" assumes "\<And>A B Rb Kab Ts Ta nlb. \<lbrakk>B \<notin> bad; A \<notin> bad; Kab \<notin> Domain (m1_state.leak x); runs x Rb = Some (Resp, [A, B], aKey Kab # aNum Ts # aNum Ta # nlb)\<rbrakk> \<Longrightarrow> \<exists>Ra nla. runs x Ra = Some (Init, [A, B], aKey Kab # aNum Ts # aNum Ta # nla)" shows "x \<in> m1_inv3r_init"
net_par_reachable_is_subnet: fixes st :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and p1 :: "net_tree"   and p2 :: "net_tree"   and I :: "'d node_action \<Rightarrow> bool" assumes "st \<in> reachable (pnet np (p1 \<parallel> p2)) I" shows "\<exists>s t. st = SubnetS s t"
distinct_3: shows "a\<^sub>c \<noteq> c\<^sub>c"
rep_eq: fixes x :: "'c" shows "blinfun_apply (blinfun_inner_right x) = (\<bullet>) x"
call_access_ok: fixes G :: "prog"   and statM :: "mhead"   and e :: "expr"   and invC :: "qtname"   and statT :: "ref_ty"   and C :: "qtname"   and L :: "lenv"   and statDeclT :: "ref_ty"   and accC :: "qtname"   and sig :: "sig"   and s :: "st"   and a :: "val" assumes "G\<turnstile>invmode statM e\<rightarrow>invC\<preceq>statT"   and "wf_prog G"   and "\<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile>e\<Colon>-RefT statT"   and "(statDeclT, statM) \<in> mheads G accC statT sig"   and "invC = invocation_class (invmode statM e) s a statT" shows "\<exists>dynM. dynlookup G statT invC sig = Some dynM \<and> G \<turnstile>Methd sig dynM in invC dyn_accessible_from accC"
fst_def: fixes prod :: "'c \<times> 'd" shows "fst prod = (case prod of (x1, x2) \<Rightarrow> x1)"
simps_39: fixes x5 :: "msg"   and x4 :: "key" shows "Hash x5 \<noteq> Key x4"
eventually_periodic: fixes C :: "'s set"   and x :: "'s" assumes "C \<in> UNIV // communicating"   and "0 < period C"   and "x \<in> C" shows "\<forall>\<^sub>F m in sequentially. 0 < p x x (m * period C)"
lens_equiv_trans: fixes X :: "'c \<Longrightarrow> 'd"   and Y :: "'e \<Longrightarrow> 'd"   and Z :: "'f \<Longrightarrow> 'd" assumes "X \<approx>\<^sub>L Y"   and "Y \<approx>\<^sub>L Z" shows "X \<approx>\<^sub>L Z"
left_neutral: fixes a :: "nat" shows "gcd 0 a = a"
trivial_combine_imp_independent: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and Us :: "'a list" assumes "ring R"   and "subfield K R"   and "set Us \<subseteq> carrier R"   and "\<And>Ks. \<lbrakk>set Ks \<subseteq> K; ring.combine R Ks Us = \<zero>\<^bsub>R\<^esub>\<rbrakk> \<Longrightarrow> set (take (length Us) Ks) \<subseteq> {\<zero>\<^bsub>R\<^esub>}" shows "ring.independent R K Us"
neg_I: fixes A :: "fmla"   and H :: "fmla set" assumes "insert A H \<turnstile> fls" shows "H \<turnstile> neg A"
complex_powr_add_minus_powr_asymptotics: fixes a :: "real"   and s :: "complex" assumes "0 < a"   and "Re s < 1" shows "((\<lambda>x. complex_of_real (x + a) powr s - complex_of_real x powr s) \<longlongrightarrow> 0) at_top"
assms_1: shows "m \<noteq> m' \<and> valid_msg m \<and> valid_msg m'"
p_nS12: fixes s :: "'s" assumes "s \<in> S"   and "s \<notin> S1"   and "s \<notin> S2" shows "p s = 0"
less_asym: fixes x :: "char"   and y :: "char"   and P :: "bool" assumes "less_char x y"   and "\<not> P \<Longrightarrow> less_char y x" shows "P"
scaleR_iarray_def: fixes k :: "real"   and A :: "'a iarray" shows "k *\<^sub>R A = IArray.of_fun (\<lambda>i. k *\<^sub>R A !! i) (IArray.length A)"
fls_integral_X_inv_power': fixes n :: "nat" assumes "1 \<le> n" shows "fls_integral (fls_X_inv ^ n) = - fls_const (inverse (of_nat (n - 1))) * fls_X_inv ^ (n - 1)"
simps: fixes vars :: "(char list \<times> nat option \<times> char list list) list"   and init :: "nat list"   and goal :: "(nat \<times> nat) list"   and ops :: "(char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "rem_implicit_pres_ops (vars, init, goal, ops) = (vars, init, goal, map rem_implicit_pres ops)"
exhaust_sel: fixes quat :: "quat" obtains "quat = Quat (quat.Re quat) (Im1 quat) (Im2 quat) (Im3 quat)"
abs_eq: fixes xa :: "'c set"   and x :: "'c set" assumes "eq_onp ord_class.filter xa xa"   and "eq_onp ord_class.filter x x" shows "Filters.filter.Abs_filter xa \<squnion> Filters.filter.Abs_filter x = Filters.filter.Abs_filter (filter_sup xa x)"
sateq: shows "x \<Turnstile>\<^sub>c EQ (vec_to_lpoly (0\<^sub>v (dim_col A) @\<^sub>v row A\<^sup>T (i - dim_vec b))) (c $ (i - dim_vec b))"
corec_1: fixes p :: "'a \<Rightarrow> bool"   and a :: "'a"   and g1 :: "'a \<Rightarrow> 'c"   and g21 :: "'a \<Rightarrow> 'b"   and q22 :: "'a \<Rightarrow> bool"   and g221 :: "'a \<Rightarrow> ('b, 'c) tllist"   and g222 :: "'a \<Rightarrow> 'a" assumes "p a" shows "corec_tllist p g1 g21 q22 g221 g222 a = TNil (g1 a)"
index_add_vec_1: fixes i :: "nat"   and v\<^sub>2 :: "'a Matrix.vec"   and v\<^sub>1 :: "'a Matrix.vec" assumes "i < dim_vec v\<^sub>2" shows "(v\<^sub>1 + v\<^sub>2) $ i = v\<^sub>1 $ i + v\<^sub>2 $ i"
decr_grading_p_mult_scalar: fixes d :: "'a \<Rightarrow> nat"   and n :: "nat"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "hom_grading d" shows "decr_grading_p d n (p \<odot> q) = local.punit.decr_grading_p d n p \<odot> decr_grading_p d n q"
that: assumes "prefix xs ys"   and "suffix xs ys" shows "thesis"
open_map_inclusion_eq: fixes X :: "'c topology"   and S :: "'c set" shows "open_map (subtopology X S) X id = openin X (topspace X \<inter> S)"
ifmi'_spec: fixes s :: "bdd"   and ni1 :: "nat"   and ni2 :: "nat"   and v :: "nat" assumes "bdd_sane s"   and "bdd_node_valid s ni1"   and "bdd_node_valid s ni2" shows "ospec (ifmi' v ni1 ni2 s) (\<lambda>r. r = ifmi v ni1 ni2 s)"
substT_notIn: fixes x :: "'var"   and t1 :: "'trm"   and t2 :: "'trm" assumes "x \<in> var"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "x \<notin> FvarsT t1" shows "substT t1 t2 x = t1"
continuation_map_generat: fixes x :: "('g, 'h, 'i) generat"   and f :: "'g \<Rightarrow> 'k"   and g :: "'h \<Rightarrow> 'l"   and h :: "'i \<Rightarrow> 'j" assumes "\<not> is_Pure x" shows "continuation (map_generat f g h x) = h (continuation x)"
cComm2: fixes a :: "name"   and b :: "name"   and P' :: "pi"   and P'' :: "pi"   and C :: "'a" assumes "P \<longmapsto> a[b] \<prec> P'"   and "!P \<longmapsto> a<b> \<prec> P''"   and "\<And>C. F C (!P) (a<b> \<prec> P'')" shows "F C (P \<parallel> !P) (\<tau> \<prec> P' \<parallel> P'')"
AbstrLevel2: shows "AbstrLevel level2 = {sS1, sS2, sS3, sS4, sS5, sS6, sS7, sS8, sS9, sS10, sS11, sS12, sS13, sS14, sS15}"
linear: fixes x :: "real \<times> real"   and y :: "real \<times> real" shows "le x y \<or> le y x"
i: shows "\<forall>n\<ge>i. b < \<bar>X n\<bar>"
floor_floor_div_eq: fixes x :: "real"   and d :: "nat" assumes "0 \<le> x" shows "\<lfloor>real (nat \<lfloor>x\<rfloor>) / real d\<rfloor> = \<lfloor>x / real d\<rfloor>"
totient_naive: fixes k :: "nat"   and acc :: "nat"   and n :: "nat" shows "totient_naive k acc n = card {x \<in> {0<..k}. coprime x n} + acc"
inf_bot_left: fixes x :: "'a multiset" shows "\<Empt> \<inter># x = \<Empt>"
of_complex_eq_minus_of_complex_iff: fixes x :: "complex"   and y :: "complex" shows "(of_complex x = - of_complex y) = (x = - y)"
MGU_outer: shows "MGU \<theta>2 (N \<lhd> \<theta>1) (N' \<lhd> \<theta>1)"
MGFn_def: fixes n :: "nat"   and t :: "Term.term"   and G :: "prog" shows "{=:n} t\<succ> {G\<rightarrow>} = {\<doteq> \<and>. G\<turnstile>init\<le>n} t\<succ> {G\<rightarrow>}"
rw_add_def: fixes p :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)"   and q :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" shows "rw_add p q = (fst p \<preceq>\<^sub>t fst q)"
share_domain_changesX: fixes X :: "nat set"   and \<S>' :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option"   and a :: "nat"   and sb :: "'b memref list" assumes "\<forall>a\<in>X. \<S>' a = \<S> a"   and "a \<in> all_shared sb \<union> all_unshared sb \<union> X" shows "share sb \<S>' a = share sb \<S> a"
bottomI: fixes x :: "'d" assumes "x \<sqsubseteq> \<bottom>" shows "x = \<bottom>"
set_spmf_eq_empty: fixes p :: "'c spmf" shows "(set_spmf p = {}) = (p = return_pmf None)"
prod: fixes j :: "nat"   and i :: "nat" shows "ws ! j * unity_root k (- int i * int j) * unity_root k (m * int i) = ws ! j * unity_root k (int i * (m - int j))"
I_carr: shows "I \<in> {..Suc (Suc n)} \<rightarrow> carrier (ideals_set R)"
less_sets_Un1: fixes A :: "'a set"   and A' :: "'a set"   and B :: "'a set" shows "(A \<union> A' \<lless> B) = (A \<lless> B \<and> A' \<lless> B)"
iteratei_rule_P: fixes iti :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S0 :: "'e set"   and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and I :: "'e set \<Rightarrow> 'd \<Rightarrow> bool"   and \<sigma>0 :: "'d"   and c :: "'d \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and P :: "'d \<Rightarrow> bool" assumes "set_iterator_genord iti S0 R"   and "I S0 \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>c \<sigma>; x \<in> S; I S \<sigma>; S \<subseteq> S0; \<forall>y\<in>S - {x}. R x y; \<forall>y\<in>S0 - S. R y x\<rbrakk> \<Longrightarrow> I (S - {x}) (f x \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> {}; \<not> c \<sigma>; I S \<sigma>; \<forall>x\<in>S. \<forall>y\<in>S0 - S. R y x\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (iti c f \<sigma>0)"
le_iff_join: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c" assumes "lattice L"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "(x \<sqsubseteq>\<^bsub>L\<^esub> y) = (x = x \<sqinter>\<^bsub>L\<^esub> y)"
PiE: fixes f :: "'f \<Rightarrow> 'g"   and A :: "'f set"   and B :: "'f \<Rightarrow> 'g set"   and x :: "'f" assumes "f \<in> Pi A B" obtains "f x \<in> B x"   | "x \<notin> A"
1: shows "Fvars (rawpsubst \<chi> txs) = Fvars \<chi> - snd ` set txs \<union> \<Union> {if x \<in> Fvars \<chi> then FvarsT t else {} |t x. (t, x) \<in> set txs}"
set_subset_Cons: fixes xs :: "'e list"   and x :: "'e" shows "set xs \<subseteq> set (x # xs)"
divide_strict_right_mono_neg: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less b a"   and "less c zero" shows "less (divide a c) (divide b c)"
atfront: fixes e :: "'a" assumes "e \<in> S"   and "e \<noteq> a" shows "index qs a \<le> index qs e"
obj_has_symmetric_tab: 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 r :: "'b" assumes "identity_in_bicategory_of_spans V H \<a> \<i> src trg r"   and "horizontal_homs.obj V src r" shows "category.isomorphic V (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg r) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg r)"
polychain_map_pairself: fixes xs :: "('a \<times> 'a) list"   and f :: "'a \<Rightarrow> 'b" assumes "polychain xs" shows "polychain (map (pairself f) xs)"
insert_DiffM: fixes x :: "'a"   and M :: "'a multiset" assumes "x \<in># M" shows "add_mset x (M - \<LM> x  \<RM>) = M"
qp_insert: fixes i :: "name"   and i' :: "name" assumes "atom i \<sharp> p'"   and "atom i' \<sharp> (i, p')" shows "quote_perm ((atom i \<rightleftharpoons> atom i') + p') (insert i V')"
diff_id: shows "diff k (charts_submanifold S) (charts_submanifold S) (\<lambda>x. x)"
conjugate_cvec_def: fixes z :: "complex \<times> complex" shows "conjugate_cvec z = vec_cnj z"
unit_cpx_vec_length: fixes i :: "nat"   and n :: "nat" assumes "i < n" shows "\<parallel>unit_vec n i\<parallel> = 1"
member: shows "GEN_OP mem2 (\<in>) (A \<rightarrow> \<langle>A\<rangle>Rs2 \<rightarrow> bool_rel)"
append_take_drop_id: fixes n :: "nat"   and xs :: "'b list" shows "take n xs \<cdot> drop n xs = xs"
fst_conv: fixes x1 :: "'r"   and x2 :: "'s" shows "prod.fst (x1, x2) = x1"
rel_eq_onp: fixes P :: "'c \<Rightarrow> bool" shows "rel_pmf (eq_onp P) = eq_onp (pred_pmf P)"
diff_0_right: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.group_add minus plus zero uminus" shows "minus a zero = a"