hcju/isabelleps huggingface dataset:
rcosetsI: fixes H :: "nat set"   and x :: "nat" assumes "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "H #>\<^bsub>G\<^esub> x \<in> rcosets\<^bsub>G\<^esub> H"
abs_ge_zero: fixes abs :: "'d \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq zero (abs a)"
simps: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a llist" shows "lfilter P xs = (case xs of []\<^sub>l \<Rightarrow> []\<^sub>l | x #\<^sub>l xs' \<Rightarrow> if P x then x #\<^sub>l lfilter P xs' else lfilter P xs')"
\<iota>_def: shows "\<iota> \<equiv> \<l>[\<I>]"
L0_def: fixes x :: "nat"   and y :: "nat" shows "L0 x y = {(x + 1, y), (x, y + 1), (x + 1, y + 1)}"
sum_card_RF': fixes i :: "nat" assumes "i < m" shows "(\<Sum>j<ka. card (RF j i)) = enum (DF k i) (ka - Suc 0)"
intra_SDG_path_is_SDG_path: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "n i-ns\<rightarrow>\<^sub>d* n'" shows "n is-ns\<rightarrow>\<^sub>d* n'"
c_ipurge_tr_ipurge: fixes D :: "'f \<Rightarrow> 'e"   and x :: "'f"   and I :: "('e \<times> 'e) set"   and u :: "'e"   and xs :: "'f list" assumes "D x \<notin> c_sources I D u (x # xs)" shows "ipurge_tr I D (D x) (c_ipurge I D u xs) = c_ipurge I D u xs"
min_less_iff_conj: fixes z :: "'t"   and x :: "'t"   and y :: "'t" shows "(z \<prec>\<^sub>t ord_term_lin.min x y) = (z \<prec>\<^sub>t x \<and> z \<prec>\<^sub>t y)"
a_de_morgan: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "rel_ad (rel_ad x ; rel_ad y) = rdom (x \<union> y)"
set_sorted_list_of_set: fixes A :: "'a Word.word set" assumes "finite A" shows "set (signed.sorted_list_of_set A) = A"
set: fixes A :: "'a set" assumes "A \<in> sets M" shows "P (indicat_real A)"
shift_degree: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> carrier P"   and "n \<le> deg R f" shows "deg R (shift n f) + n = deg R f"
prime_sum_div: fixes n :: "nat" assumes "prime n" shows "Esigma n = n + 1"
DERIV_def: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a" shows "(f has_field_derivative D) (at x) = (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0::'a\<rightarrow> D"
disjoint_sets_nres: fixes X :: "'b set"   and Y :: "'b set" shows "ivl_rep X \<bind> (\<lambda>(iX, sX). ivl_rep Y \<bind> (\<lambda>(iY, sY). RETURN (list_ex (\<lambda>i. sX \<bullet> i < iY \<bullet> i \<or> sY \<bullet> i < iX \<bullet> i) Basis_list))) \<le> disjoint_sets X Y"
translation_bijective: fixes a :: "'a" assumes "a \<in> G" shows "bij_betw (a)\<^sub>R G G"
assms: shows "A \<Turnstile> (act TT, U \<rightarrow>) P"
simps: fixes i :: "abc_inst" shows "length_of i = (case i of Inc n \<Rightarrow> 2 * n + 9 | Dec n e \<Rightarrow> 2 * n + 16 | Goto n \<Rightarrow> 1)"
saction_simps_4: fixes P :: "'a prog"   and a :: "'b"   and hT :: "htype" shows "saction P (NormalAction (NewHeapElem a hT)) = True"
not_squarefreeI: fixes x :: "'a"   and n :: "'a" assumes "x\<^sup>2 dvd n"   and "\<not> x dvd (1::'a)" shows "\<not> squarefree n"
fmrel_on_fsetD: fixes x :: "'d"   and S :: "'d fset"   and P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and m :: "'d \<rightharpoonup>\<^sub>f 'e"   and n :: "'d \<rightharpoonup>\<^sub>f 'f" assumes "x |\<in>| S"   and "fmrel_on_fset S P m n" shows "rel_option P (fmlookup m x) (fmlookup n x)"
inv_xa_P_init: shows "invertible_mat (addrow_mat (dim_row H) (- div_op (H $$ (i, 0)) (H $$ (0, 0))) i 0 * P_init)"
acquired_mono: fixes A :: "nat set"   and B :: "nat set"   and pending_write :: "bool"   and xs :: "'b memref list" assumes "A \<subseteq> B" shows "acquired pending_write xs A \<subseteq> acquired pending_write xs B"
tr_par_wf': fixes D :: "('lbl strand_label \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and X :: "'var set"   and A' :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list" assumes "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (unlabel D) \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {}"   and "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (unlabel D) \<subseteq> X"   and "wf'\<^sub>s\<^sub>s\<^sub>t X (unlabel A)"   and "fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {}"   and "A' \<in> set (tr\<^sub>p\<^sub>c A D)" shows "wf\<^sub>l\<^sub>s\<^sub>t X A'"
Spy_see_priK: fixes evs :: "event list"   and b :: "keymode"   and A :: "agent" assumes "evs \<in> tls" shows "(Key (privateKey b A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
sat_primalD: fixes x :: "'a vec"   and A :: "'a mat"   and b :: "'a vec" assumes "x \<in> sat_primal A b" shows "[A*\<^sub>vx]\<le>b"
full_rank: shows "vec_space.rank (dim_row (submatrix A I J)) (submatrix A I J) = dim_row (submatrix A I J)"
consistentDP_def: fixes P :: "heap \<Rightarrow> bool"   and Q :: "heap \<Rightarrow> bool"   and lookup :: "'c \<Rightarrow> 'd option Heap"   and lookup_st :: "'c \<Rightarrow> (heap, 'd option) state"   and update :: "'c \<Rightarrow> 'd \<Rightarrow> unit Heap"   and update_st :: "'c \<Rightarrow> 'd \<Rightarrow> (heap, unit) state"   and dp :: "'c \<Rightarrow> 'd" assumes "dp_heap P Q lookup lookup_st update update_st" shows "dp_heap.consistentDP P Q dp lookup_st \<equiv> rel_fun (=) (dp_heap.crel_vs P Q dp lookup_st (=)) dp"
eval_in_hom: fixes b :: "'a"   and c :: "'a" assumes "ide b"   and "ide c" shows "\<guillemotleft>eval b c : exp b c \<otimes> b \<rightarrow> c\<guillemotright>"
g: shows "\<forall>\<^sub>F x in F. norm (g x) \<le> B"
plus_vec_def: fixes v\<^sub>1 :: "'a vec"   and v\<^sub>2 :: "'a vec" shows "v\<^sub>1 + v\<^sub>2 \<equiv> vec (dim_vec v\<^sub>2) (\<lambda>i. v\<^sub>1 $ i + v\<^sub>2 $ i)"
safeD_2: fixes x :: "'c" assumes "x \<in> Csafe" shows "isFDERIV DIM('c) [0..<DIM('c)] ode_e (list_of_eucl x)"
set_locals_id: fixes s :: "st" shows "set_locals (locals s) s = s"
iso_hcomp: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "local.iso \<mu>"   and "local.iso \<nu>"   and "src \<nu> = trg \<mu>" shows "local.iso (\<nu> \<star> \<mu>)"
add_block_rep_number_in: fixes x :: "'a"   and b :: "'a set" assumes "x \<in> b" shows "add_block b rep x = \<B> rep x + 1"
subst_preserves_beta2': fixes r :: "dB"   and s :: "dB"   and t :: "dB"   and i :: "nat" assumes "r \<rightarrow>\<^sub>\<beta>\<^sup>* s" shows "t[r/i] \<rightarrow>\<^sub>\<beta>\<^sup>* t[s/i]"
fgroup_byE: fixes c :: "'d"   and cs :: "'d fset"   and b :: "'e"   and f :: "'f \<Rightarrow> 'e \<times> 'd"   and as :: "'f fset" assumes "c |\<in>| cs"   and "(b, cs) |\<in>| fgroup_by f as" obtains a :: "'f" where "a |\<in>| as" and "f a = (b, c)"
terminates_Skip': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "('e, 'f) xstate" shows "\<Gamma>\<turnstile>Skip \<down> s"
d_a_export: fixes antidomain_op :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.antidomain_left_monoid antidomain_op one times" shows "antidomain_left_monoid.am_d antidomain_op (times (antidomain_op x) y) = times (antidomain_op x) (antidomain_left_monoid.am_d antidomain_op y)"
hom_induced_vector: shows "GRepHom (\<currency>) induced_vector"
valid_cfg_action: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg"   and s :: "'s \<times> ('c \<Rightarrow> real)" assumes "cfg \<in> MDP.valid_cfg"   and "s \<in> set_pmf (action cfg)" shows "s \<in> S"
open_openin: fixes S :: "'c set" shows "open S = openin euclidean S"
f1: shows "dim_row (v \<Otimes> w) = dim_row u"
gcd_non_0_int: fixes y :: "int"   and x :: "int" assumes "0 < y" shows "gcd x y = gcd y (x mod y)"
simps_2: fixes z :: "nat"   and i :: "nat"   and j :: "nat"   and x :: "nat" shows "local.prefixes z i j (Suc x) = local.prefixes z i j x @ [if x = 0 then j else if s (Partial_Recursive.list_encode (local.prefixes z i j x @ [0])) \<noteq> s (Partial_Recursive.list_encode (local.prefixes z i j x)) then 0 else if s (Partial_Recursive.list_encode (local.prefixes z i j x @ [1])) \<noteq> s (Partial_Recursive.list_encode (local.prefixes z i j x)) then 1 else z]"
fmap_ext: fixes m :: "'e \<rightharpoonup>\<^sub>f 'f"   and n :: "'e \<rightharpoonup>\<^sub>f 'f" assumes "\<And>x. fmlookup m x = fmlookup n x" shows "m = n"
cinner_add_right: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner x (plus y z) = cinner x y + cinner x z"
vD_or_iD: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and P :: "(nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option) \<Rightarrow> nat \<Rightarrow> bool" assumes "ip \<in> kD rt"   and "ip \<in> vD rt \<Longrightarrow> P rt ip"   and "ip \<in> iD rt \<Longrightarrow> P rt ip" shows "P rt ip"
append_vec_def: fixes v :: "'b vec"   and w :: "'b vec" shows "v @\<^sub>v w \<equiv> let n = dim_vec v; m = dim_vec w in vec (n + m) (\<lambda>i. if i < n then v $ i else w $ (i - n))"
less_imp_not_eq: fixes x :: "'a"   and y :: "'a" assumes "x < y" shows "(x = y) = False"
conjLeft: fixes phi1 :: "bool"   and phi2 :: "bool" assumes "phi1 \<and> phi2" obtains "phi1"
add: fixes x :: "'a"   and y :: "'a" assumes "x \<in> F"   and "y \<in> F" shows "f (x + y) = f x + f y"
sublist_altdef: fixes xs :: "'c list"   and ys :: "'c list" shows "sublist xs ys = (\<exists>ys'. prefix ys' ys \<and> suffix xs ys')"
range_eq_M: shows "image_mset f (mset_set A) = M"
coprime_prod: fixes A :: "'a mod_ring set"   and g :: "'a mod_ring \<Rightarrow> 'a mod_ring poly"   and a :: "'a mod_ring" assumes "\<forall>x\<in>A. comm_monoid_mult_class.coprime (g a) (g x)" shows "comm_monoid_mult_class.coprime (g a) (prod g A)"
balance_bd: fixes l :: "('a, 'b) rbt"   and r :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b" assumes "bd_inv l"   and "bd_inv r"   and "black_depth l = black_depth r" shows "black_depth (balance l k v r) = black_depth l + 1"
shift_bounds_nat_ivl: fixes g :: "nat \<Rightarrow> 'a"   and m :: "nat"   and k :: "nat"   and n :: "nat" shows "prod g {m + k..<n + k} = (\<Prod>i = m..<n. g (i + k))"
Union_quotient_restricted: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> bool" assumes "equiv A r"   and "P respects r" shows "\<Union> ({x \<in> A. P x} // r) = {x \<in> A. P x}"
simps_1: fixes f :: "'d \<Rightarrow> 'd"   and x :: "'d" shows "shd (siterate f x) = x"
plussub_def: fixes x :: "'d"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and y :: "'e" shows "x \<squnion>\<^bsub>f\<^esub> y = f x y"
cInf_eq_Min: fixes X :: "'c set" assumes "finite X"   and "X \<noteq> {}" shows "Inf X = Min X"
assms_3: fixes k :: "'a" assumes "k \<in> fst ` set (fst xs) \<union> fst ` set (fst ys)" shows "lookup_raw xs k = lookup_raw ys k"
KBasic2_1: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<box>\<phi> \<^bold>\<equiv> \<^bold>\<box>\<^bold>\<not>\<^bold>\<not>\<phi> in v]"
eval_Sup: fixes A :: "'a pred set" shows "eval (\<Squnion> A) = \<Squnion> (eval ` A)"
assoc_add_vec: fixes v\<^sub>1 :: "'a vec"   and n :: "nat"   and v\<^sub>2 :: "'a vec"   and v\<^sub>3 :: "'a vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n"   and "v\<^sub>3 \<in> carrier_vec n" shows "v\<^sub>1 + v\<^sub>2 + v\<^sub>3 = v\<^sub>1 + (v\<^sub>2 + v\<^sub>3)"
absorb2: fixes b :: "'a"   and a :: "'a" assumes "b \<le> a" shows "min a b = b"
minus_in_Ints_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> '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 x :: "'b" assumes "class.ring_1 one times plus zero minus uminus" shows "(uminus x \<in> ring_1.Ints one plus zero minus) = (x \<in> ring_1.Ints one plus zero minus)"
triangulation_def: fixes \<T> :: "'b set set" shows "triangulation \<T> \<equiv> finite \<T> \<and> (\<forall>T\<in>\<T>. \<exists>n. n simplex T) \<and> (\<forall>T T'. T \<in> \<T> \<and> T' \<in> \<T> \<longrightarrow> T \<inter> T' face_of T)"
get_shadow_root_def: shows "get_shadow_root \<equiv> l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_shadow_root"
assms: shows "\<forall>i>n. \<exists>j>i. P (f i) (f j)"
a: shows "i - j cd\<^bsup>\<pi> \<guillemotleft> j\<^esup>\<rightarrow> k - j"
L_alt: fixes idx :: "'n"   and I :: "'i set" shows "L idx I = to_language {xs. \<exists>\<AA>\<in>I. \<exists>\<BB>. \<AA> = fold CONS (rev xs) \<BB> \<and> Length \<BB> = 0 \<and> #\<^sub>V \<BB> = idx \<and> (\<forall>x\<in>set xs. size x = idx)}"
in_quotient_imp_in_rel: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and X :: "'a set"   and x :: "'a"   and y :: "'a" assumes "equiv A r"   and "X \<in> A // r"   and "{x, y} \<subseteq> X" shows "(x, y) \<in> r"
inner_dirsum_def: fixes As :: "'b set list" shows "inner_dirsum As = (if add_independentS As then \<Sum>A\<leftarrow>As. A else 0)"
word_log_esimps_5: fixes x :: "'b word" shows "x XOR 0 = x"
projection_image: fixes M :: "'a set" assumes "convex M"   and "closed M"   and "M \<noteq> {}" shows "range (projection M) = M"
strong_induct: fixes x1 :: "(name \<times> ty) list"   and x2 :: "trm"   and x3 :: "ty"   and P :: "'a \<Rightarrow> (name \<times> ty) list \<Rightarrow> trm \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'a" assumes "x1 \<turnstile> x2 : x3"   and "\<And>\<Gamma> x T y. \<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P y \<Gamma> (Var x) T"   and "\<And>\<Gamma> e\<^sub>1 T\<^sub>1 T\<^sub>2 e\<^sub>2 y. \<lbrakk>\<Gamma> \<turnstile> e\<^sub>1 : T\<^sub>1\<rightarrow>T\<^sub>2; \<And>z. P z \<Gamma> e\<^sub>1 (T\<^sub>1\<rightarrow>T\<^sub>2); \<Gamma> \<turnstile> e\<^sub>2 : T\<^sub>1; \<And>z. P z \<Gamma> e\<^sub>2 T\<^sub>1\<rbrakk> \<Longrightarrow> P y \<Gamma> (App e\<^sub>1 e\<^sub>2) T\<^sub>2"   and "\<And>x \<Gamma> T\<^sub>1 e T\<^sub>2 y. \<lbrakk>x \<sharp> y; x \<sharp> \<Gamma>; (x, T\<^sub>1) # \<Gamma> \<turnstile> e : T\<^sub>2; \<And>z. P z ((x, T\<^sub>1) # \<Gamma>) e T\<^sub>2\<rbrakk> \<Longrightarrow> P y \<Gamma> (Lam [x].e) (T\<^sub>1\<rightarrow>T\<^sub>2)" shows "P z x1 x2 x3"
expandTauChainFrame: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and C :: "'e" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* C" obtains \<Psi>' :: "'b" and A\<^sub>P' :: "name list" and \<Psi>\<^sub>P' :: "'b" where "extractFrame P' = \<langle>A\<^sub>P', \<Psi>\<^sub>P'\<rangle>" and "\<Psi>\<^sub>P \<otimes> \<Psi>' \<simeq> \<Psi>\<^sub>P'" and "A\<^sub>P' \<sharp>* P'" and "A\<^sub>P' \<sharp>* C" and "distinct A\<^sub>P'"
stop_loc_reach: fixes c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c', mds', mem'\<rangle> \<in> loc_reach \<langle>Stop, mds, mem\<rangle>" shows "c' = Stop \<and> mds' = mds"
kruskal_exists_spanning: fixes g :: "'a" assumes "symmetric g" shows "\<exists>f. spanning_forest f g"
intros_1: fixes x :: "'a"   and xs :: "'a list" shows "nth_el1 (x # xs) 0 x"
RETURN_SPEC_refine: fixes x :: "'d"   and R :: "('d \<times> 'c) set"   and \<Phi> :: "'c \<Rightarrow> bool" assumes "\<exists>x'. (x, x') \<in> R \<and> \<Phi> x'" shows "RETURN x \<le> \<Down> R (SPEC \<Phi>)"
hv_zk: fixes h :: "'a"   and x :: "nat"   and c :: "nat" assumes "(h, x) \<in> R_DL" shows "Schnorr_\<Sigma>.R h x c = Schnorr_\<Sigma>.S h c"
xfer_relabel_bound: fixes f' :: "nat \<times> nat \<Rightarrow> 'a"   and l' :: "nat \<Rightarrow> nat"   and B :: "nat"   and p :: "pr_operation list" assumes "\<forall>p'. ((f, l), p', f', l') \<in> trcl pr_algo_lts' \<longrightarrow> length (filter is_RELABEL' p') \<le> B"   and "((f, l), p, f', l') \<in> trcl pr_algo_lts" shows "length (filter is_RELABEL p) \<le> B"
conj_zero_left: fixes x :: "'b" shows "(0::'b) AND x = (0::'b)"
last_conv_nth_default: fixes xs :: "'b list"   and dflt :: "'b" assumes "xs \<noteq> []" shows "last xs = nth_default dflt xs (length xs - 1)"
verticesFrom_in: fixes v :: "nat"   and f :: "face"   and a :: "nat" assumes "v \<in> \<V> f"   and "a \<in> \<V> f" shows "a \<in> set (verticesFrom f v)"
normalise_language_equivalent: fixes eq :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> bool"   and normalise :: "'b ltln \<Rightarrow> 'b ltln"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" assumes "GF_advice_congruent eq normalise" shows "w \<Turnstile>\<^sub>n normalise \<phi> = w \<Turnstile>\<^sub>n \<phi>"
induct: fixes x1 :: "char list \<Rightarrow> ty"   and x2 :: "bexp"   and P :: "(char list \<Rightarrow> ty) \<Rightarrow> bexp \<Rightarrow> bool" assumes "x1 \<turnstile> x2"   and "\<And>\<Gamma> v. P \<Gamma> (Bc v)"   and "\<And>\<Gamma> b. \<lbrakk>\<Gamma> \<turnstile> b; P \<Gamma> b\<rbrakk> \<Longrightarrow> P \<Gamma> (bexp.Not b)"   and "\<And>\<Gamma> b1 b2. \<lbrakk>\<Gamma> \<turnstile> b1; P \<Gamma> b1; \<Gamma> \<turnstile> b2; P \<Gamma> b2\<rbrakk> \<Longrightarrow> P \<Gamma> (And b1 b2)"   and "\<And>\<Gamma> a1 \<tau> a2. \<lbrakk>\<Gamma> \<turnstile> a1 : \<tau>; \<Gamma> \<turnstile> a2 : \<tau>\<rbrakk> \<Longrightarrow> P \<Gamma> (Less a1 a2)" shows "P x1 x2"
assms: shows "crdt_ops (xs @ [(oid, oper)]) deps"
wp1_auxa: fixes p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "wellformed_policy1_strong p" shows "\<exists>r. applied_rule_rev C x p = \<lfloor>r\<rfloor>"
is_extensional: fixes f :: "'a" assumes "\<not> S.arr f" shows "\<rr> f = S.null"
circ_left_top: fixes x :: "'a" shows "surjective (x\<^sup>\<circ>)"
last_tl: fixes l :: "'b list" assumes "1 < length l" shows "last (tl l) = last l"
restrict_index_out: fixes i :: "nat"   and X :: "'b option list"   and I :: "nat set" assumes "i < length X"   and "i \<notin> I" shows "restrict I X ! i = None"
cases: fixes x :: "char list \<times> char list" obtains "end" :: "char list" and t :: "char" and ts :: "char list" where "x = (enda__, t # ts)"   | "end" :: "char list" where "x = (end, [])"
add_pow_ldistr: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and k :: "nat" assumes "semiring R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "[k] \<cdot>\<^bsub>R\<^esub> a \<otimes>\<^bsub>R\<^esub> b = [k] \<cdot>\<^bsub>R\<^esub> (a \<otimes>\<^bsub>R\<^esub> b)"
vsv_Limit_vrange_in_VsetI: fixes \<alpha> :: "V" assumes "Limit \<alpha>"   and "\<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> Vset \<alpha>"   and "vfinite (\<D>\<^sub>\<circ> f)" shows "\<R>\<^sub>\<circ> f \<in>\<^sub>\<circ> Vset \<alpha>"
DERIV_ln_divide: fixes x :: "real" assumes "0 < x" shows "(ln has_real_derivative 1 / x) (at x)"
tranclp_trancl_eq: fixes r :: "('a \<times> 'a) set" shows "(\<lambda>x xa. (x, xa) \<in> r)\<^sup>+\<^sup>+ = (\<lambda>x xa. (x, xa) \<in> r\<^sup>+)"
ipresWlsAbs_def: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'c"   and MOD :: "('d, 'e, 'varSort, 'sort, 'f, 'g, 'h, 'c, 'i) Iteration.model_scheme" shows "ipresWlsAbs hA MOD \<equiv> \<forall>us s A. wlsAbs (us, s) A \<longrightarrow> igWlsAbs MOD (us, s) (hA A)"
correct: shows "policy_axioms Policy.sp_spec_subj_obj Policy.sp_spec_subj_subj Policy.ifp"
RangeE: fixes b :: "'a"   and r :: "('b \<times> 'a) set" assumes "b \<in> Range r" obtains a :: "'b" where "(a, b) \<in> r"
Abs_splitres_inverse: fixes r :: "('a, 'b) FingerTreeStruc"   and s :: "('a, 'b) FingerTreeStruc"   and a :: "'a \<times> 'b" assumes "FingerTreeStruc.ft_invar r"   and "FingerTreeStruc.ft_invar s" shows "Rep_splitres (Abs_splitres (r, a, s)) = (r, a, s)"
principal_dchar_not_coprime: fixes k :: "nat"   and n :: "nat" assumes "\<not> coprime k n" shows "principal_dchar n k = 0"
scaleR_vector_assoc: fixes c :: "real"   and A :: "(('a, 'c) vec, 'b) vec"   and x :: "('a, 'c) vec" shows "c *\<^sub>R (A *v x) = c *\<^sub>R A *v x"
eq_self_neq_FF': fixes eq :: "'b \<rightarrow> 'b \<rightarrow> tr"   and x :: "'b" assumes "class.Eq_eq eq" shows "eq\<cdot>x\<cdot>x \<noteq> FF"
uv_1: shows "(u1, u2) \<noteq> vec_zero"
h2: shows "(\<exists>(a', b', c')\<in>set eq. (a' = 0 \<and> b' \<noteq> 0) \<and> (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<or> a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> ((\<forall>(d, e, f)\<in>set eq. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0) \<or> (\<forall>(d, e, f)\<in>set eq. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0))) \<or> (\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and> (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0) \<longrightarrow> (\<exists>x. (\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
f_def: shows "f = prime_sum_upto (\<lambda>p. 1 / real p)"
exec_gpv_oracle_bisim': fixes X :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and s1 :: "'a"   and s2 :: "'b"   and oracle1 :: "'a \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'a) spmf"   and oracle2 :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b) spmf"   and gpv :: "('e, 'c, 'd) gpv" assumes "X s1 s2"   and "\<And>s1 s2 x. X s1 s2 \<Longrightarrow> rel_spmf (\<lambda>(a, s1') (b, s2'). a = b \<and> X s1' s2') (oracle1 s1 x) (oracle2 s2 x)" shows "rel_spmf (\<lambda>(a, s1') (b, s2'). a = b \<and> X s1' s2') (exec_gpv oracle1 gpv s1) (exec_gpv oracle2 gpv s2)"
init_unsat_flag: fixes t :: "(nat \<times> linear_poly) list" shows "\<not> \<U> (init t)"
simps_3: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and xs :: "'b list" shows "binrelchain P (x # y # xs) = (P x y \<and> binrelchain P (y # xs))"
f_shrink_last_map: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list"   and k :: "nat" shows "map f xs \<div>\<^sub>l k = map f (xs \<div>\<^sub>l k)"
strict_order_iff: fixes a :: "'a"   and b :: "'a" shows "(a \<sqsubset> b) = (a = ord.min (\<sqsubseteq>) a b \<and> a \<noteq> b)"
Linear_order_Restr: fixes r :: "('d \<times> 'd) set"   and A :: "'d set" assumes "Linear_order r" shows "Linear_order (Restr r A)"
unopI: fixes A :: "V"   and f :: "V" assumes "nop A (1\<^sub>\<nat>) f" shows "unop A f"
Var: fixes xs :: "'varSort"   and x :: "'var" shows "phi (Var xs x)"