hcju/isabelleps huggingface dataset:
lconv_append: fixes L :: "'c \<Rightarrow> ('a \<times> 'a) set"   and R :: "'c \<Rightarrow> ('a \<times> 'a) set"   and E :: "'c \<Rightarrow> ('a \<times> 'a) set"   and xs :: "(accent \<times> 'c) list"   and ys :: "(accent \<times> 'c) list" shows "lconv L R E (xs @ ys) = lconv L R E xs O lconv L R E ys"
map_upd_triv: fixes t :: "'e \<Rightarrow> \<langle>'d\<rangle>\<^sub>\<bottom>"   and k :: "'e"   and x :: "'d" assumes "t k = \<lfloor>x\<rfloor>" shows "t(k \<mapsto> x) = t"
RSQ_def: shows "RSQ = cbox (case (l, snd p) of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (real_of_int l + \<delta>, real_of_int (snd p) + \<delta>)"
g1: fixes x :: "'a" assumes "x \<in> T"   and "b \<bullet> x = 0" shows "norm (g x) = 1"
ctxt_consn: fixes n :: "nat"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes " \<Turnstile>\<^sub>n G  "   and " \<Turnstile>\<^sub>n Call : A " shows " \<Turnstile>\<^sub>n {A} \<union> G  "
non_vol: shows "non_volatile_owned_or_read_only True \<S> \<O> sb"
synth_idem_payload: fixes X :: "msg set"   and Y :: "msg set" assumes "X \<subseteq> synth Y \<union> - payload" shows "synth X \<subseteq> synth Y \<union> - payload"
f_expand_gr0_f_expand_Suc: fixes k :: "nat"   and xs :: "'d message_af list" assumes "0 < k" shows "xs \<odot> k = xs \<odot>\<^bsub>fSuc\<^esub> (k - Suc 0)"
assms_2: fixes s :: "'a" assumes "P' s" shows "Case_Labeling.VC ((''weaken'', IC, [Case_Labeling.VAR s]) # CT) (P s)"
axioms_2: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'e set"   and scale\<^sub>1 :: "'f \<Rightarrow> 'd \<Rightarrow> 'd"   and scale\<^sub>2 :: "'f \<Rightarrow> 'e \<Rightarrow> 'e" assumes "module_pair_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>1 scale\<^sub>2" shows "module_on U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2"
before_fun2: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and x :: "'a \<Rightarrow> 'c"   and y :: "'a \<Rightarrow> 'd" shows "\<turnstile> $f<x, y> = f<$x, $y>"
ts_context: fixes c :: "'form" assumes "c |\<in>| conc_forms" shows "fst (fst (tree.root (ts c))) |\<subseteq>| ass_forms"
cl: shows "closure as ([(norm r, norm s)], []) = Some ([], ps)"
contr: shows "(\<forall>x. 0 < (if e = x then 0 else index [a] x + 1)) = False"
simps_1: fixes x :: "'b" shows "ground (Var x) = False"
a_gla1: fixes x :: "'a"   and y :: "'a" assumes "ad x \<cdot> y = (zero_class.zero::'a)" shows "ad x \<le> ad y"
verit_eq_simplify_7: shows "(0::'e) \<noteq> (1::'e)"
sum_eq: fixes x :: "'b"   and t :: "'b" shows "x + t = (0::'b) \<equiv> x = - t"
Beta_eqvt: fixes x1 :: "Lambda_Z.term"   and x2 :: "Lambda_Z.term"   and p :: "perm" assumes "x1 \<rightarrow>\<^sub>\<beta> x2" shows "p \<bullet> x1 \<rightarrow>\<^sub>\<beta> p \<bullet> x2"
v0_nonadj_v1: shows "(v0, v1) \<notin> E"
LLq_num: fixes \<phi> :: "'fmla"   and q :: "'trm" assumes "\<phi> \<in> fmla"   and "q \<in> num"   and "Fvars \<phi> = {zz}"   and "\<forall>p\<in>num. bprv (subst \<phi> p zz)" shows "prv (all zz (imp (LLq (Var zz) q) \<phi>))"
rep_eq: fixes x :: "('a, 'b) RBT.rbt" shows "keys x = RBT_Impl.keys (impl_of x)"
limit_in_range: fixes r :: "nat \<Rightarrow> 'c" shows "limit r \<subseteq> range r"
le_Suc_base: fixes a :: "nat \<Rightarrow> nat"   and j :: "nat" assumes "a \<in> s" shows "a j \<le> Suc (base j)"
unbal_ins_Unbal: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree"   and t' :: "'a tree" assumes "ins n d x t = Unbal t'" shows "\<not> bal_i (n + 1) (height t' + d)"
index_minus_vec_1: fixes i :: "nat"   and v\<^sub>2 :: "'a vec"   and v\<^sub>1 :: "'a vec" assumes "i < dim_vec v\<^sub>2" shows "(v\<^sub>1 - v\<^sub>2) $ i = v\<^sub>1 $ i - v\<^sub>2 $ i"
Gr_def: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "BNF_Def.Gr A f = {(a, f a) |a. a \<in> A}"
omega_def: shows "\<omega> = star_n (\<lambda>n. real (Suc n))"
abs_fun_fresh: fixes c :: "'b"   and a :: "'b"   and b :: "'b"   and x :: "'a"   and y :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "c \<noteq> a"   and "c \<noteq> b"   and "c \<sharp> x"   and "c \<sharp> y" shows "([a].x = [b].y) = ([(a, c)] \<bullet> x = [(b, c)] \<bullet> y)"
assms_6: shows "regular ((v \<squnion> e\<^sup>T * top) * (v \<squnion> e\<^sup>T * top)\<^sup>T)"
prev_in_basics_3: fixes u :: "'a" assumes "u \<in> {a<..b}" shows "A \<inter> {prev_in A u<..<u} = {}"
assms_1: shows "other U (net_tree_ips (p\<^sub>1 \<parallel> p\<^sub>2)) \<sigma> \<sigma>'"
adal_r'': shows "(ad, al) \<in> action_loc P E' r'"
has_vector_derivative_pair_within: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and x :: "real"   and S :: "real set"   and g :: "real \<Rightarrow> 'b"   and g' :: "'b" assumes "\<And>u. u \<in> Basis \<Longrightarrow> ((\<lambda>x. f x \<bullet> u) has_vector_derivative f' \<bullet> u) (at x within S)"   and "\<And>u. u \<in> Basis \<Longrightarrow> ((\<lambda>x. g x \<bullet> u) has_vector_derivative g' \<bullet> u) (at x within S)" shows "((\<lambda>x. (f x, g x)) has_vector_derivative (f', g')) (at x within S)"
divide_nonneg_nonpos: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq zero x"   and "less_eq y zero" shows "less_eq (divide x y) zero"
one_def: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and conv :: "'c \<Rightarrow> 'c" assumes "class.aggregation_algebra one times uminus inf less_eq less sup bot top plus conv" shows "one = top"
softSubst_implies_prv_SS: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {xx}" shows "bprv (SS \<langle>\<phi>\<rangle> \<langle>softSubst \<phi> \<langle>\<phi>\<rangle> xx\<rangle>)"
dagger_unfoldr_distr: fixes y :: "'a"   and x :: "'a" shows "y + x\<^sup>\<star> ; x ; y = x\<^sup>\<star> ; y"
inject_3: fixes x51 :: "nat"   and x52 :: "recf"   and x53 :: "recf"   and y51 :: "nat"   and y52 :: "recf"   and y53 :: "recf" shows "(Pr x51 x52 x53 = Pr y51 y52 y53) = (x51 = y51 \<and> x52 = y52 \<and> x53 = y53)"
simps_10: fixes x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list"   and y101 :: "expr"   and y102 :: "char list option"   and y103 :: "char list"   and y104 :: "expr list" shows "(Call x101 x102 x103 x104 = Call y101 y102 y103 y104) = (x101 = y101 \<and> x102 = y102 \<and> x103 = y103 \<and> x104 = y104)"
Int_left_absorb: fixes A :: "'c set"   and B :: "'c set" shows "A \<inter> (A \<inter> B) = A \<inter> B"
UNIF_subst_domain_subset: fixes ss :: "('c \<Rightarrow> ('d, 'c) Term.term) list"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset"   and E' :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "UNIF ss E E'" shows "subst_domain (compose ss) \<subseteq> vars_mset E"
S3safe: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> $S3 rmhist p \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<longrightarrow> [UNext memCh mm (resbar rmhist) p]_(rtrner memCh!p, resbar rmhist!p)"
comp_assoc: fixes f :: "'aj \<Rightarrow> 'ah"   and g :: "'ai \<Rightarrow> 'aj"   and h :: "'ag \<Rightarrow> 'ai" shows "f \<circ> g \<circ> h = f \<circ> (g \<circ> h)"
sym: fixes cmp :: "'b comparator"   and b :: "'b"   and a :: "'b" shows "(compare cmp b a = Equiv) = (compare cmp a b = Equiv)"
equalityCE: fixes A :: "'f set"   and B :: "'f set"   and c :: "'f" assumes "A = B" obtains "c \<in> A" and "c \<in> B"   | "c \<notin> A" and "c \<notin> B"
commute: fixes a :: "'a"   and b :: "'a" shows "inf a b = inf b a"
mult_scaleR_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and a :: "real"   and y :: "'c" assumes "class.real_algebra plus zero minus uminus scaleR times" shows "times x (scaleR a y) = scaleR a (times x y)"
f4: assumes "triangle_circumscribes_triangle A B C A' B' C'" shows "col M N P"
d: fixes f :: "'a fps" assumes "f \<in> A" shows "f dvd d"
seqE: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" assumes "CCC.seq g f" obtains "seq (fst g) (fst f)" and "CC.seq (snd g) (snd f)"
code: fixes p :: "'d spmf" shows "lift_spmf p = GPV (map_spmf Pure p)"
exec_move_CastI: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and T :: "ty" assumes "exec_move ci P t e h s ta h' s'" shows "exec_move ci P t (Cast T e) h s ta h' s'"
FGModule_axioms: shows "FGModule G (\<cdot>) V"
ladder_n_simp2: fixes L :: "(nat \<times> nat \<times> nat) list"   and d :: "nat \<times> nat \<times> nat" shows "ladder_n (L @ [d]) (length L) = deriv_n d"
seqI': fixes f :: "'b"   and a :: "'b"   and b :: "'b"   and g :: "'b"   and c :: "'b" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>"   and "\<guillemotleft>g : b \<Rightarrow>\<^sub>D c\<guillemotright>" shows "D.seq g f"
simps_1: fixes xs :: "'b list"   and k :: "nat" shows "list_slice_aux xs k 0 = []"
lemma_R_for_BC: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and i :: "nat"   and d :: "nat" assumes "learn_bc \<psi> U (\<phi> i)" shows "learn_bc \<psi> U (totalizer d i)"
simps_1: fixes f :: "'b \<Rightarrow> 'b" shows "orbits_list f [] = []"
wellformed_def: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp" shows "wellformed \<Gamma> = wf {(q, p). p \<leadsto>\<^bsub>\<Gamma>\<^esub> q}"
N: shows "nfa_is_node N (nfa_steps (quantify_nfa v N) q xs)"
lunit\<^sub>F\<^sub>M\<^sub>C_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term set" assumes "free_monoidal_category C" shows "free_monoidal_category.lunit\<^sub>F\<^sub>M\<^sub>C C a = free_monoidal_category.mkarr C \<^bold>\<l>\<^bold>[free_monoidal_category.rep C a\<^bold>]"
g_1: shows "graph_homomorphism (LG e\<^sub>2 v) G g"
star_class_defs_3: shows "(+) \<equiv> *f2* (+)"
simple_def: fixes S :: "('c, 'd) strand_step list" shows "simple S \<equiv> list_all simple\<^sub>s\<^sub>t\<^sub>p S"
simps_3: fixes X :: "'b set set"   and x :: "'b set"   and xs :: "'b set list"   and y :: "'b set" assumes "SimplicialComplex X" shows "SimplicialComplex.min_maxsimpchain X (x # xs @ [y]) = (x \<noteq> y \<and> is_arg_min length (\<lambda>zs. SimplicialComplex.maxsimpchain X (x # zs @ [y])) xs)"
run_bind_writer: fixes return :: "'f \<times> 'e list \<Rightarrow> 'd"   and bind :: "'d \<Rightarrow> ('f \<times> 'e list \<Rightarrow> 'd) \<Rightarrow> 'd"   and m :: "('e, 'f, 'd) writerT"   and f :: "'f \<Rightarrow> ('e, 'f, 'd) writerT" shows "run_writer (bind_writer return bind m f) = bind (run_writer m) (\<lambda>(a, ws). bind (run_writer (f a)) (\<lambda>(b, ws'). return (b, ws @ ws')))"
nec_imp_act: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<A>\<phi> in v]"
bounded_iff: fixes b :: "'g multiset"   and c :: "'g multiset"   and a :: "'g multiset" shows "(b \<union># c \<subseteq># a) = (b \<subseteq># a \<and> c \<subseteq># a)"
simps: fixes a1 :: "'d"   and a2 :: "'d" assumes "three TYPE('d)" shows "three.rel_12_23 a1 a2 = (a1 = three.three\<^sub>1 \<and> a2 = three.three\<^sub>2 \<or> a1 = three.three\<^sub>2 \<and> a2 = three.three\<^sub>3)"
finite_nat_ex_max: fixes N :: "nat set" assumes "finite N" shows "\<exists>m. \<forall>n\<in>N. n < m"
intro: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and u_block_size :: "int"   and index :: "int" assumes "t_design point_set block_collection u_block_size 2 index" shows "bibd point_set block_collection u_block_size index"
endpowSuc: fixes n :: "nat" shows "endpow (Suc n) = T \<circ> endpow n"
assoc_p_subid: fixes x :: "('a \<times> 'a set) set"   and y :: "('a \<times> 'a set) set"   and z :: "('a \<times> 'a set) set" shows "x \<cdot> y \<cdot> (z \<cdot> 1\<^sub>\<pi>) = x \<cdot> (y \<cdot> (z \<cdot> 1\<^sub>\<pi>))"
tl_remdups_adj: fixes ys :: "'g list" assumes "ys \<noteq> []" shows "tl (remdups_adj ys) = remdups_adj (dropWhile (\<lambda>x. x = hd ys) (tl ys))"
one_carrier: shows "1\<^sub>m 0 \<in> carrier_mat 0 0"
distinct1_rotate: fixes xs :: "'b list" shows "distinct (rotate1 xs) = distinct xs"
basic_trans_rules_20: fixes a :: "'a"   and b :: "'a"   and P :: "bool" assumes "a < b"   and "b < a" shows "P"
sweep_loop_invL_def: shows "gc.sweep_loop_invL = (\<lambda>s. (at gc gc.sweep_loop_check s \<longrightarrow> (gc_mark s\<down> \<noteq> None \<longrightarrow> obj_at (\<lambda>obj. Some (obj_mark obj) = gc_mark s\<down>) (gc_tmp_ref s\<down>) s\<down>) \<and> (gc_mark s\<down> = None \<and> obj_at (\<lambda>s. True) (gc_tmp_ref s\<down>) s\<down> \<longrightarrow> marked (gc_tmp_ref s\<down>) s\<down>)) \<and> (at gc gc.sweep_loop_free s \<longrightarrow> gc_mark s\<down> \<noteq> None \<and> (the \<circ> gc_mark) s\<down> \<noteq> gc_fM s\<down> \<and> obj_at (\<lambda>obj. Some (obj_mark obj) = gc_mark s\<down>) (gc_tmp_ref s\<down>) s\<down>) \<and> (at gc gc.sweep_loop_ref_done s \<longrightarrow> obj_at (\<lambda>s. True) (gc_tmp_ref s\<down>) s\<down> \<longrightarrow> marked (gc_tmp_ref s\<down>) s\<down>) \<and> (atS gc gc.sweep_loop_locs s \<longrightarrow> (\<forall>x. x \<notin> gc_refs s\<down> \<and> obj_at (\<lambda>s. True) x s\<down> \<longrightarrow> marked x s\<down>)) \<and> (atS gc gc.black_heap_locs s \<longrightarrow> (\<forall>x. obj_at (\<lambda>s. True) x s\<down> \<longrightarrow> marked x s\<down>)) \<and> (atS gc gc.sweep_loop_not_choose_ref_locs s \<longrightarrow> gc_tmp_ref s\<down> \<in> gc_refs s\<down>))"
assms_2: shows "\<forall>n. prices Mkt asset n w \<noteq> 0"
occs_Fv_eq_elem_fv: fixes v :: "variable"   and S :: "typ"   and t :: "Core.term" shows "occs (Fv v S) t = ((v, S) \<in> fv t)"
supt_supteq_not_supteq: fixes s :: "('d, 'e) Term.term"   and t :: "('d, 'e) Term.term" shows "s \<rhd> t = (s \<unrhd> t \<and> (t, s) \<notin> {\<unrhd>})"
le_neq_implies_less': fixes m :: "nat"   and n :: "nat" assumes "m \<noteq> n"   and "m \<le> n" shows "m < n"
append_append_commute: fixes i1 :: "'d"   and e1 :: "'c"   and i2 :: "'d"   and e2 :: "'c" shows "\<langle>Append i1 e1\<rangle> \<rhd> \<langle>Append i2 e2\<rangle> = \<langle>Append i2 e2\<rangle> \<rhd> \<langle>Append i1 e1\<rangle>"
exhaust: fixes y :: "msg" obtains x1 :: "agent" where "y = msg.Agent x1"   | x2 :: "nat" where "y = Number x2"   | x3 :: "fresh_t" where "y = Nonce x3"   | x4 :: "key" where "y = Key x4"   | x5 :: "msg" where "y = Hash x5"   | x61 :: "msg" and x62 :: "msg" where "y = \<lbrace>x61, x62\<rbrace>"   | x71 :: "key" and x72 :: "msg" where "y = Crypt x71 x72"
h8: shows "(\<exists>(a', b', c')\<in>set c. (a' = 0 \<and> b' \<noteq> 0) \<and> (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0)) \<longrightarrow> (\<exists>x. (\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and> (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and> (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0))"
model_existence: fixes V :: "'b fm set"   and p :: "'b fm" assumes "V \<in> mk_finite_char (close {set G |G. \<not> \<turnstile> imply G \<^bold>\<bottom>})"   and "maximal V (mk_finite_char (close {set G |G. \<not> \<turnstile> imply G \<^bold>\<bottom>}))" shows "(p \<in> V) = (Kripke pi (reach (mk_finite_char (close {set G |G. \<not> \<turnstile> imply G \<^bold>\<bottom>}))), V \<Turnstile> p) \<and> ((\<^bold>\<not> p) \<in> V) = (Kripke pi (reach (mk_finite_char (close {set G |G. \<not> \<turnstile> imply G \<^bold>\<bottom>}))), V \<Turnstile> \<^bold>\<not> p)"
better_LAxL_intro: fixes M :: "trm"   and x :: "name"   and a :: "coname"   and y :: "name" assumes "fin M x" shows "Cut <a>.Ax y a x.M \<longrightarrow>\<^sub>l M[x\<turnstile>n>y]"
dvd_mod_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.semidom_modulo divide plus minus zero times one modulo"   and "dvd.dvd times c b" shows "dvd.dvd times c (modulo a b) = dvd.dvd times c a"
Neg: fixes a :: "'d"   and ps :: "('c, 'd) fm list"   and branch :: "(('c, 'd) fm list \<times> 'd) list"   and p :: "('c, 'd) fm"   and A :: "'d set"   and n :: "nat" assumes "(\<^bold>\<not> \<^bold>\<not> p) at a in (ps, a) # branch"   and "new p a ((ps, a) # branch)"   and "A, Suc n \<turnstile> (p # ps, a) # branch" shows "A, n \<turnstile> (ps, a) # branch"
dbl_simps_2: shows "neg_numeral_class.dbl (0::'a) = (0::'a)"
imod_by_0_enat: fixes a :: "enat" shows "a mod enat 0 = a"
periodN_fac': fixes v :: "'b list"   and p :: "nat"   and u :: "'b list" assumes "periodN v p"   and "u \<le>f v"   and "u \<noteq> \<epsilon>" shows "periodN u p"
splay_bstL: fixes t :: "'a tree"   and a :: "'a"   and l :: "'a tree"   and e :: "'a"   and r :: "'a tree"   and x :: "'a" assumes "bst t"   and "splay a t = \<langle>l, e, r\<rangle>"   and "x \<in> set_tree l" shows "x < a"
wuntil_def: fixes \<gamma>' :: "(nat \<Rightarrow> 'KS) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma> :: "(nat \<Rightarrow> 'KS) \<Rightarrow> nat \<Rightarrow> bool" shows "\<gamma>' \<WW>\<^sub>b \<gamma> \<equiv> \<gamma>' \<UU>\<^sub>b \<gamma> \<or>\<^sup>b \<box>\<^sub>b\<gamma>'"
2: fixes T :: "'a set" assumes "T \<in> insert (- K) (F ` \<C>)" shows "open T"
simps_2: fixes xc :: "'a"   and xm :: "'a"   and xs :: "'a list" shows "polychain_of xc (xm # xs) = (xc, xc + xm) # polychain_of (xc + xm) xs"
emeasure_spmf_single: fixes p :: "'d spmf"   and x :: "'d" shows "emeasure (measure_spmf p) {x} = ennreal (spmf p x)"
ltake_ldrop: fixes n :: "enat"   and m :: "enat"   and xs :: "'b llist" shows "ltake n (ldrop m xs) = ldrop m (ltake (n + m) xs)"
nDASCaux: fixes p :: "('c, 'd) Combinators list"   and r :: "('c, 'd) Combinators" assumes "DenyAll \<notin> set p"   and "singleCombinators p"   and "r \<in> set p" shows "\<not> member DenyAll r"
plus_unit_def: shows "(+) \<equiv> map_fun Linear_Aggregation_Algebras.unit.Rep_unit (map_fun Linear_Aggregation_Algebras.unit.Rep_unit Linear_Aggregation_Algebras.unit.Abs_unit) local.tl"
unsigned_greater_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'b \<times> 'b"   and divmod_step :: "num \<Rightarrow> 'b \<times> 'b \<Rightarrow> 'b \<times> 'b"   and w :: "'c word" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "less_eq zero (semiring_1.unsigned one plus zero w)"
fin_sat_insert: fixes a :: "'b formula"   and G :: "'b formula set" assumes "fin_sat (a \<triangleright> G)" shows "fin_sat G"
psi'_l_def: fixes l :: "nat" shows "psi'_l l = alpha_l l \<cdot>\<^sub>v \<alpha> - beta_l l \<cdot>\<^sub>v \<beta>"
deadlock_implies_deadlocked': fixes 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" assumes "deadlock s" shows "deadlocked' s"
add_epsilon: fixes f :: "'a \<Rightarrow> real" assumes "\<And>x y. \<lbrakk>le x y; le y x\<rbrakk> \<Longrightarrow> f x = f y" shows "\<exists>\<epsilon>>0. vnm_utility carrier le (\<lambda>x. u x + \<epsilon> * f x)"
diff_zero: fixes x :: "'a" assumes "x \<in> V" shows "x - (0::'a) = x"
Seq: fixes P\<^sub>1 :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c\<^sub>1 :: "com"   and P\<^sub>2 :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c\<^sub>2 :: "com"   and P\<^sub>3 :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<turnstile>\<^sub>2\<^sub>' {P\<^sub>1} c\<^sub>1 {P\<^sub>2}"   and "\<turnstile>\<^sub>2\<^sub>' {P\<^sub>2} c\<^sub>2 {P\<^sub>3}" shows "\<turnstile>\<^sub>2\<^sub>' {P\<^sub>1} c\<^sub>1;; c\<^sub>2 {P\<^sub>3}"
setle_def: fixes S :: "'a set"   and x :: "'a" shows "S *<= x = (\<forall>y\<in>S. y \<le> x)"
map_sq_matrix_mult: fixes f :: "'b \<Rightarrow> 'c"   and A :: "('b, 'd) sq_matrix"   and B :: "('b, 'd) sq_matrix" assumes "\<And>a b. f (a + b) = f a + f b"   and "\<And>a b. f (a * b) = f a * f b"   and "f (0::'b) = (0::'c)" shows "map_sq_matrix f (A * B) = map_sq_matrix f A * map_sq_matrix f B"
eval_fm_AxiomP: fixes e :: "(name, hf) finfun"   and x :: "tm" shows "eval_fm e (AxiomP x) = (\<lbrakk>x\<rbrakk>e \<in> Axiom)"
surj_iff: fixes f :: "'b \<Rightarrow> 'a" shows "surj f = (f \<circ> inv f = id)"
ones_def: shows "ones \<equiv> Abs_fps (\<lambda>n. 1::'a)"
sngl_NodeI: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and lv :: "nat"   and r :: "('b \<times> nat) tree"   and l' :: "('b \<times> nat) tree"   and a' :: "'b" assumes "sngl \<langle>l, (a, lv), r\<rangle>" shows "sngl \<langle>l', (a', lv), r\<rangle>"
scc_ofI_reachable': fixes G :: "('c, 'd) pre_digraph"   and v :: "'c"   and u :: "'c" assumes "sym_digraph G"   and "v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> u" shows "u \<in> pre_digraph.scc_of G v"
strong_fermat_witness_iff: shows "local.strong_fermat_witness = (\<exists>k m. odd m \<and> n - 1 = 2 ^ k * m \<and> [a ^ m \<noteq> 1] (mod n) \<and> (\<forall>i\<in>{0..<k}. [a ^ (2 ^ i * m) \<noteq> n - 1] (mod n)))"
resolution_class_union: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and S :: "'b set set" assumes "incidence_system point_set block_collection"   and "incidence_system.resolution_class point_set block_collection S" shows "\<Union> S = point_set"