hcju/isabelleps huggingface dataset:
unlabel_Cons_1: fixes l :: "'f"   and a :: "'e"   and A :: "('f \<times> 'e) list" shows "unlabel ((l, a) # A) = a # unlabel A"
E_TAbs: fixes X :: "tyvrs"   and T\<^sub>11 :: "ty"   and T\<^sub>2 :: "ty"   and t\<^sub>12 :: "trm" assumes "X \<sharp> (T\<^sub>11, T\<^sub>2)" shows "(\<lambda>X<:T\<^sub>11. t\<^sub>12) \<cdot>\<^sub>\<tau> T\<^sub>2 \<longmapsto> t\<^sub>12[X \<mapsto>\<^sub>\<tau> T\<^sub>2]"
knowledge_equiv_eq_old: fixes A :: "agent"   and evs :: "event list"   and B :: "agent" shows "knows A evs \<union> {Key (priEK B), Key (priSK B), Key (shrK B)} \<union> {X. Notes B X \<in> set evs} = knows B evs \<union> {Key (priEK A), Key (priSK A), Key (shrK A)} \<union> {X. Notes A X \<in> set evs}"
inject: fixes x2 :: "'a"   and y2 :: "'a" shows "(Normal x2 = Normal y2) = (x2 = y2)"
hom_1_iff: fixes x :: "'a" shows "(hom x = (1::'b)) = (x = (1::'a))"
dvdE: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "dvd.dvd times b a" obtains k :: "'b" where "a = times b k"
uint_sdiv_def: fixes x :: "uint"   and y :: "uint" shows "uint_sdiv x y = (if y = 0 then undefined (div) x 0 else Abs_uint (Rep_uint x sdiv Rep_uint y))"
arith_mult_neq_nzero2: fixes i :: "nat"   and n :: "nat"   and t :: "nat"   and q :: "nat" assumes "i < n"   and "0 < i" shows "n * t + i \<noteq> n * q"
simulation_def: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<leadsto>[Rel] Q \<equiv> \<forall>\<alpha> Q'. \<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q' \<longrightarrow> bn \<alpha> \<sharp>* \<Psi> \<longrightarrow> bn \<alpha> \<sharp>* P \<longrightarrow> (\<exists>P'. \<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P' \<and> (\<Psi>, P', Q') \<in> Rel)"
update_convs_1: fixes c_records' :: "'d zmultiset \<Rightarrow> 'd zmultiset"   and c_records :: "'d zmultiset"   and c_temp :: "'c \<Rightarrow> 'd zmultiset"   and c_msg :: "'c \<Rightarrow> 'c \<Rightarrow> 'd zmultiset list"   and c_glob :: "'c \<Rightarrow> 'd zmultiset"   and more :: "'e" shows "c_records_update c_records' \<lparr>c_records = c_records, c_temp = c_temp, c_msg = c_msg, c_glob = c_glob, \<dots> = more\<rparr> = \<lparr>c_records = c_records' c_records, c_temp = c_temp, c_msg = c_msg, c_glob = c_glob, \<dots> = more\<rparr>"
Update_def: fixes U :: "'b data \<Rightarrow> 'b data" shows "Update U = (\<forall>d. Data.DataSpace d = Data.DataSpace (U d))"
stable_unless: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "(\<box>(\<lambda>s. P s \<longrightarrow> (\<circle>(\<lambda>s. P s \<or> Q s)) s)) \<sigma>" shows "(\<box>(\<lambda>s. P s \<longrightarrow> (P \<W> Q) s)) \<sigma>"
WT_gpv_parametric': fixes C :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and A :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "bi_unique C" shows "rel_fun (rel_\<I> C R) (rel_fun (rel_gpv'' A C R) (=)) WT_gpv WT_gpv"
intros_1: fixes G :: "nat graph_rec"   and E :: "(nat \<times> nat) set" assumes "cfg_doms G" shows "path_entry E [] 0"
closure_invariant_closure: fixes ws :: "('a list \<times> 's \<times> 's) list"   and ps :: "('s \<times> 's) list"   and N :: "('s \<times> 's) set" shows "map_option (\<lambda>(ws, ps, N). (map snd ws, N)) (closure_invariant (ws, ps, N)) = closure (map snd ws, N)"
ord_iff_deg_le_univariate: fixes s :: "'x \<Rightarrow>\<^sub>0 nat"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "s \<in> .[X]"   and "t \<in> .[X]" shows "(s \<preceq> t) = (deg_pm s \<le> deg_pm t)"
igFreshIGOp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) Iteration.model_scheme" shows "igFreshIGOp MOD \<equiv> \<forall>ys y delta inp binp. igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igFreshInp MOD ys y inp \<and> igFreshBinp MOD ys y binp \<longrightarrow> igFresh MOD ys y (igOp MOD delta inp binp)"
pequivI: fixes prog :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and prog' :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "\<And>P. sound P \<Longrightarrow> wp prog P = wp prog' P" shows "prog \<simeq> prog'"
eval_closed_lcl_unchanged: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option" assumes "P \<turnstile> \<langle>e,(h, l)\<rangle> \<Rightarrow> \<langle>e',(h', l')\<rangle>"   and "fv e = {}" shows "l' = l"
aff_dim_geq: fixes V :: "'b set" shows "- 1 \<le> aff_dim V"
alw_holds2: fixes P :: "'c \<Rightarrow> bool"   and ss :: "'c stream" shows "alw (holds P) ss = (P (shd ss) \<and> alw (holds P) (stl ss))"
zero_prod_def: shows "0 = (0::'c, 0::'d)"
independent_vnmD2: fixes \<alpha> :: "real"   and p :: "'a pmf"   and C :: "'a pmf set"   and q :: "'a pmf"   and r :: "'a pmf"   and P :: "('a pmf \<times> 'a pmf) set" assumes "\<alpha> \<in> {0<..1}"   and "p \<in> C"   and "q \<in> C"   and "r \<in> C"   and "independent_vnm C P"   and "p \<succeq>[P] q" shows "mix_pmf \<alpha> p r \<succeq>[P] mix_pmf \<alpha> q r"
cases: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and a1 :: "'a literal multiset list"   and a2 :: "'a literal multiset"   and a3 :: "'a multiset list"   and a4 :: "'a list"   and a5 :: "'s"   and a6 :: "'a literal multiset" assumes "ord_resolve_rename S a1 a2 a3 a4 a5 a6" obtains CAs :: "'a literal multiset list" and n :: "nat" and AAs :: "'a multiset list" and As :: "'a list" and DA :: "'a literal multiset" and \<rho> :: "'s" and \<rho>s :: "'s list" and \<sigma> :: "'s" and E :: "'a literal multiset" where "a1 = CAs" and "a2 = DA" and "a3 = AAs" and "a4 = As" and "a5 = \<sigma>" and "a6 = E" and "length CAs = n" and "length AAs = n" and "length As = n" and "\<forall>i<n. poss (AAs ! i) \<subseteq># CAs ! i" and "negs (mset As) \<subseteq># DA" and "\<rho> = hd (renamings_apart (DA # CAs))" and "\<rho>s = tl (renamings_apart (DA # CAs))" and "ord_resolve S (CAs \<cdot>\<cdot>cl \<rho>s) (DA \<cdot> \<rho>) (AAs \<cdot>\<cdot>aml \<rho>s) (As \<cdot>al \<rho>) \<sigma> E"
assms: fixes m :: "nat"   and n :: "nat" assumes "m < length cs"   and "n < m" shows "cs ! n \<sqsubset>\<^sup>i cs ! m"
one_type_greater: shows "(\<exists>f. inj f) \<or> (\<exists>g. inj g)"
ipcidr_to_interval: fixes base :: "'a word"   and len :: "nat"   and s :: "'a word"   and e :: "'a word" assumes "ipcidr_to_interval (base, len) = (s, e)" shows "ipset_from_cidr base len = {s..e}"
Loopo_undef: fixes \<alpha> :: "game option" shows "(Loopo \<alpha> = undefg) = (\<alpha> = undefg)"
SD_preorderD_1: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and q :: "'b pmf"   and p :: "'b pmf" assumes "Order_Predicates.preorder_on carrier le"   and "SD le q p" shows "p \<in> lotteries_on carrier"
vsv_dg_prod_Dom: shows "vsv (dg_prod I \<AA>\<lparr>Dom\<rparr>)"
reflpI: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. r x x" shows "reflp r"
hyp_equiv: shows "set (map set (all_partitions_list xs)) = all_partitions (set xs)"
rec: fixes f1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and a :: "'c"   and b :: "'d" shows "old.rec_prod f1 (a, b) = f1 a b"
comp_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and g :: "'b local.term set" assumes "free_monoidal_category C" shows "free_monoidal_category.comp C f g \<equiv> if free_monoidal_category.ARR C f \<and> free_monoidal_category.ARR C g \<and> monoidal_language.Dom C (free_monoidal_category.rep C f) = monoidal_language.Cod C (free_monoidal_category.rep C g) then free_monoidal_category.mkarr C (free_monoidal_category.rep C f \<^bold>\<cdot> free_monoidal_category.rep C g) else {}"
chine_composite_1: shows "\<mu>\<nu>_\<pi>.chine = \<nu>.leg0 \<cdot> \<mu>\<nu>.prj\<^sub>0 \<down>\<down> \<pi>.leg1"
H_QR: shows "Hermite_reduce_above (row_add A (mod_type_class.from_nat n) i ((res (A $ i $ j) (A $ mod_type_class.from_nat n $ j) - A $ mod_type_class.from_nat n $ j) div A $ i $ j)) n i j res = Q ** row_add A (mod_type_class.from_nat n) i ((res (A $ i $ j) (A $ mod_type_class.from_nat n $ j) - A $ mod_type_class.from_nat n $ j) div A $ i $ j)"
switch_def: fixes v :: "view"   and c :: "cars"   and w :: "view" shows "v = c > w \<equiv> ext v = ext w \<and> lan v = lan w \<and> own w = c \<and> more v = more w"
rotate_id: fixes ls :: "'a list"   and n :: "nat" shows "rotate (|ls| - n mod |ls|) (rotate n ls) = ls"
g: shows "(\<Sum>x\<in>UNIV. g x *s X $ x) = v"
insert_iff: fixes a :: "'f"   and b :: "'f"   and A :: "'f set" shows "(a \<in> insert b A) = (a = b \<or> a \<in> A)"
inf_continuous_def: fixes F :: "'d \<Rightarrow> 'e" shows "inf_continuous F = (\<forall>M. decseq M \<longrightarrow> F (Inf (range M)) = (INF i. F (M i)))"
PSQ: shows "Smith_normal_form_of_JNF A bezout = (P, S, Q)"
uint_sint: fixes w :: "'b word" shows "uint w = take_bit LENGTH('b) (sint w)"
Diagonalize_DOM: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<^bold>\<lfloor>\<F>C.DOM f\<^bold>\<rfloor> = Dom \<^bold>\<lfloor>\<F>C.rep f\<^bold>\<rfloor>"
rinss_bst_of_list: shows "distr (rinss xs \<langle>\<rangle> A) (tree_sigma (count_space A)) (map_tree fst) = distr (Pi\<^sub>M (set xs) (\<lambda>x. \<U> 0 1)) (tree_sigma (count_space A)) (\<lambda>p. bst_of_list (sort_key p xs))"
abs_eq: fixes x :: "'b \<Longrightarrow> 'a" shows "\<lbrakk>x\<rbrakk>\<^sub>\<sim> = Abs_scene (\<lambda>s\<^sub>1 s\<^sub>2. if mwb_lens x then s\<^sub>1 \<triangleleft>\<^bsub>x\<^esub> s\<^sub>2 else s\<^sub>1)"
CD_on_Xd: fixes x :: "'a"   and ds :: "'b set"   and A :: "'a set" assumes "x \<in> Pds'.CD_on ds A" shows "Xd x \<in> ds"
gen_gens: fixes x :: "'c"   and gens :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "x \<in> gens" shows "x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>S b\<guillemotright>" shows "\<guillemotleft>R f : R a \<Rightarrow>\<^sub>S R b\<guillemotright>"
Un_insert_left: fixes a :: "'c"   and B :: "'c set"   and C :: "'c set" shows "Set.insert a B \<union> C = Set.insert a (B \<union> C)"
\<phi>_simps_2: shows "src t\<^sub>0u\<^sub>1.\<phi> = src t\<^sub>0u\<^sub>1.p\<^sub>0"
nats_of_boolss_insertll: fixes n :: "nat"   and bss :: "bool list list"   and bs :: "bool list"   and i :: "nat" assumes "list_all (is_alph n) bss"   and "length bs = length bss"   and "i \<le> n" shows "nats_of_boolss (Suc n) (insertll i bs bss) = insertl i (nat_of_bools bs) (nats_of_boolss n bss)"
atLeastatMost_subset_iff: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset"   and d :: "'c multiset" shows "(subset_mset.atLeastAtMost a b \<subseteq> subset_mset.atLeastAtMost c d) = (\<not> a \<subseteq># b \<or> c \<subseteq># a \<and> b \<subseteq># d)"
simps: fixes p :: "char list"   and ts :: "hterm list" shows "fatom_of_hatom (p, ts) = (p, fterms_of_hterms ts)"
f'_def: shows "f' = (\<lambda>f. (\<lambda>a. f (Inl a), \<lambda>b. f (Inr b)))"
simps_2: shows "Cod \<^bold>\<I> = \<^bold>\<I>"
sum_list_prod_map_aezfun: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> ('b, 'a) aezfun"   and xys :: "('c \<times> 'd) list" shows "aezfun (\<Sum>(x, y)\<leftarrow>xys. f x y) = (\<Sum>(x, y)\<leftarrow>xys. aezfun (f x y))"
ContraProve: fixes H :: "fmla set"   and B :: "fmla"   and A :: "fmla" assumes "H \<turnstile> B" shows "insert (neg B) H \<turnstile> A"
DAimpliesMR_E: fixes p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "DenyAll \<in> set p" shows "\<exists>r. applied_rule_rev C x p = \<lfloor>r\<rfloor>"
eint_mult_mono': fixes c :: "eint"   and n :: "eint"   and k :: "eint" assumes "0 \<le> c \<and> c \<noteq> \<infinity>"   and "n < k" shows "n * c \<le> k * c"
in_gamma_meet: fixes x :: "int"   and a1 :: "'av"   and a2 :: "'av" assumes "x \<in> \<gamma> a1"   and "x \<in> \<gamma> a2" shows "x \<in> \<gamma> (a1 \<sqinter> a2)"
drop_in_carrier: fixes R :: "('c, 'd) ring_scheme"   and xs :: "'c list"   and n :: "nat" assumes "cring R"   and "cring.in_carrier R xs" shows "cring.in_carrier R (drop n xs)"
all_paths_def: shows "all_paths \<equiv> {xs |xs. path xs}"
label_stab_map_iso_to_coxeter_complex: fixes \<phi> :: "'a \<Rightarrow> 'a" assumes "label_wrt C0 \<phi>"   and "fixespointwise \<phi> C0" shows "ChamberComplexIsomorphism X (CoxeterComplex.TheComplex (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})) (\<lambda>v. {w \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>. w \<rightarrow> \<phi> v = v})"
carrier_vec_dim_vec: fixes v :: "'b vec" shows "v \<in> carrier_vec (dim_vec v)"
antisymmetric_axioms: shows "antisymmetric A (\<sqsubset>)"
Pair_def: fixes a :: "'d"   and b :: "'e" shows "(a, b) = Abs_prod (Pair_Rep a b)"
bind_distr: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and X :: "'b measure"   and N :: "'b \<Rightarrow> 'c measure"   and K :: "'c measure" assumes "f \<in> M \<rightarrow>\<^sub>M X"   and "N \<in> X \<rightarrow>\<^sub>M subprob_algebra K"   and "space M \<noteq> {}" shows "distr M X f \<bind> N = M \<bind> (\<lambda>x. N (f x))"
0: shows "\<forall>(k, \<gamma>)\<in>one_chain. line_integral_exists F basis \<gamma>"
equal_pair_count_maximality1b: fixes ps :: "('b \<times> 'b) list" assumes "equal_pair_count ps \<noteq> length ps" shows "\<not> equal_pair (ps ! equal_pair_count ps)"
assms_2: shows "P' \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P''"
df_system_axioms: shows "df_system as\<^sub>0 alstep"
cproper_interval_finite_1_def: shows "cproper_interval = proper_interval"
simps_2: fixes x2 :: "nat"   and y2 :: "nat" shows "(Number x2 = Number y2) = (x2 = y2)"
Suc: fixes p :: "(nat \<times> int) list"   and ds :: "nat set" shows "distinct (gridgen p ds l)"
lens_indep_overrideI: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b" assumes "vwb_lens X"   and "vwb_lens Y"   and "\<And>s\<^sub>1 s\<^sub>2 s\<^sub>3. s\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>2 \<triangleleft>\<^bsub>Y\<^esub> s\<^sub>3 = s\<^sub>1 \<triangleleft>\<^bsub>Y\<^esub> s\<^sub>3 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>2" shows "X \<bowtie> Y"
tendsto_at_topI_sequentially: fixes f :: "real \<Rightarrow> 'a"   and y :: "'a" assumes "\<And>X. filterlim X at_top sequentially \<Longrightarrow> (\<lambda>n. f (X n)) \<longlonglongrightarrow> y" shows "(f \<longlongrightarrow> y) at_top"
PROTECT_def: fixes x :: "'e" shows "PROTECT x \<equiv> x"
closed_orbit_subset_\<omega>_limit_set: fixes x :: "'a" assumes "closed_orbit x" shows "range (flow0 x) \<subseteq> \<omega>_limit_set x"
inter_aform_plane_ortho_nres_def: fixes p :: "nat"   and Z :: "'a \<times> 'a pdevs"   and n :: "'a"   and g :: "real" shows "inter_aform_plane_ortho_nres p Z n g = RETURN (inter_aform_plane_ortho p Z n g)"
def_prg_AllowedActs: fixes F :: "'b program"   and init :: "'b set"   and acts :: "('b \<times> 'b) set set"   and allowed :: "('b \<times> 'b) set set" assumes "F = mk_total_program (init, acts, allowed)" shows "AllowedActs F = insert Id allowed"
split_aform_def: fixes X :: "'b \<times> 'b pdevs"   and i :: "nat" shows "split_aform X i = (let xi = pdevs_apply (snd X) i /\<^sub>R 2 in ((fst X - xi, pdev_upd (snd X) i xi), fst X + xi, pdev_upd (snd X) i xi))"
iT_Mult_Div: fixes d :: "nat"   and k :: "nat"   and I :: "nat set" assumes "0 < d"   and "k mod d = 0" shows "I \<otimes> k \<oslash> d = I \<otimes> k div d"
access_def: fixes root :: "('a att_scheme \<times> 'b, 'a att_scheme, 'c) env"   and path :: "'c list"   and uid :: "nat"   and perms :: "perm set" shows "access root path uid perms = (case lookup root path of None \<Rightarrow> None | Some file \<Rightarrow> if uid = 0 \<or> uid = owner (attributes file) \<or> perms \<subseteq> others (attributes file) then Some file else None)"
hd_Cons_tl: fixes xs :: "'d list" assumes "xs \<noteq> []" shows "hd xs # tl xs = xs"
ct_closed: fixes s :: "'s"   and t :: "'s" assumes "s \<in> S"   and "t \<in> set_pmf (ct s)" shows "t \<in> S"
simps_2: fixes E :: "nat \<Rightarrow> 'a"   and F :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and G :: "'c \<Rightarrow> 'a list \<Rightarrow> bool"   and P :: "'c"   and ts :: "'b tm list" shows "\<lbrakk>E, F, G\<rbrakk> (\<^bold>\<ddagger> P ts) = G P (map \<lparr>E, F\<rparr> ts)"
select_convs_3: fixes ISignature :: "('a, 'c) Signature"   and ICategory :: "('b, 'd) Category"   and ITypes :: "'a \<Rightarrow> 'b"   and IFunctions :: "'c \<Rightarrow> 'd"   and more :: "'e" shows "ITypes \<lparr>ISignature = ISignature, ICategory = ICategory, ITypes = ITypes, IFunctions = IFunctions, \<dots> = more\<rparr> = ITypes"
get_tag_name_is_l_get_tag_name: shows "l_get_tag_name DocumentClass.type_wf get_tag_name get_tag_name_locs"
ext_inject: fixes carrier :: "'e set"   and more :: "'f"   and carrier' :: "'e set"   and more' :: "'f" shows "(\<lparr>carrier = carrier, \<dots> = more\<rparr> = \<lparr>carrier = carrier', \<dots> = more'\<rparr>) = (carrier = carrier' \<and> more = more')"
arrI: fixes f :: "'a \<times> 'a" assumes "fst f \<star> snd f \<noteq> null" shows "VoV.arr f"
hyps_3: assumes "S \<noteq> {}" shows "\<exists>m\<in>S. \<forall>s\<in>S. f m \<le> f s"
path_weight_refl: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and loc :: "'a" assumes "graph weights" shows "(0::'b) \<in>\<^sub>A graph.path_weight weights loc loc"
sorted01: fixes xs :: "'a list" assumes "length xs \<le> 1" shows "sorted xs"
assms_3: shows "AEeq M f (\<lambda>x. c)"
H2: shows "d = {d. d \<noteq> [] \<and> hd d \<in> ev ` A \<and> (\<exists>a. ev a = hd d \<and> tl d \<in> D (P a))}"
has_parametrized_subprob_densityD_1: fixes M :: "'a measure"   and N :: "'a \<Rightarrow> 'b measure"   and R :: "'b measure"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> ennreal"   and x :: "'a" assumes "has_parametrized_subprob_density M N R f"   and "x \<in> space M" shows "N x = density R (f x)"
intro: fixes I :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "I \<noteq> R"   and "\<And>\<aa>. \<lbrakk>ideal \<aa> R addition multiplication zero unit; \<aa> \<noteq> R; I \<subseteq> \<aa>\<rbrakk> \<Longrightarrow> I = \<aa>" shows "max_ideal_axioms R I addition multiplication zero unit"
complete_c_K5: shows "K\<^bsub>5\<^esub> (with_proj c_K5)"
Fpow_not_empty: fixes A :: "'a set" shows "Fpow A \<noteq> {}"
bij: shows "bij_betw (\<lambda>(x, y). (y, x)) (B \<times> A) (A \<times> B)"
idem2: fixes x :: "'a"   and y :: "'a" shows "c (b x y) y"
History: shows "\<turnstile> Implementation \<longrightarrow> (\<exists>\<exists> rmhist. Hist rmhist)"
card_bij: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'a" assumes "f \<in> A \<rightarrow> B"   and "inj_on f A"   and "g \<in> B \<rightarrow> A"   and "inj_on g B"   and "finite A"   and "finite B" shows "card A = card B"
nextV2: fixes f :: "face"   and v :: "nat" shows "f\<^bsup>2\<^esup> \<bullet> v = f \<bullet> (f \<bullet> v)"
norm_le_square: fixes x :: "'a"   and a :: "real" shows "(norm x \<le> a) = (0 \<le> a \<and> x \<bullet> x \<le> a\<^sup>2)"
nvC_varsC: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and x :: "Preliminaries.var"   and c :: "('d, 'e) lit list" assumes "ProblemIk wtFsym wtPsym arOf resOf parOf \<Phi> infTp"   and "x \<in> ProblemIk.nvC c" shows "x \<in> varsC c"
mono_D11: fixes P :: "'c process"   and Q :: "'c process"   and S :: "'c process" assumes "P \<sqsubseteq> Q" shows "D (Q `;` S) \<subseteq> D (P `;` S)"
cases: fixes a :: "'c llist" assumes "ldistinct a" obtains "a = LNil"   | x :: "'c" and xs :: "'c llist" where "a = LCons x xs" and "x \<notin> lset xs" and "ldistinct xs"
monoid_axioms: shows "monoid (\<cdot>) (1::'c)"
hom_commute: fixes h :: "'a \<Rightarrow> 'a"   and N :: "'a set" assumes "\<And>x y. h (max x y) = max (h x) (h y)"   and "finite N"   and "N \<noteq> {}" shows "h (Max N) = Max (h ` N)"
assert_tableau_valuated: fixes s :: "('i, 'a) state"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s" shows "\<nabla> (assert a s)"
start: shows "(\<Sum>bl\<in>#remove1_mset b1 \<B>. b1 |\<inter>| bl * (b1 |\<inter>| bl - 1) div 2) = \<Lambda> * (\<Lambda> - 1) div 2 * (\<v> - 1)"
leq_SubobjsR_isSubobj: fixes C :: "char list"   and C' :: "char list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and D :: "char list"   and Cs :: "char list list" assumes "P \<turnstile> C \<preceq>\<^sup>* C'"   and "P \<turnstile> C' \<prec>\<^sub>S D"   and "Subobjs\<^sub>R P D Cs" shows "is_subobj P (C, Cs)"
IH1: fixes s :: "'a net_state"   and s' :: "'a net_state" assumes "s \<in> reachable (pnet np p1) TT"   and "(s, connect(i, i'), s') \<in> automaton.trans (pnet np p1)" shows "netmap s' = netmap s"
\<W>_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list"   and P :: "(symbol \<times> character list) list set"   and k :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "LocalLexing.\<W> \<NN> \<TT> \<RR> \<SS> Lex Doc P k = {u \<in> LocalLexing.\<X> \<TT> Lex Doc k. \<exists>p\<in>by_length k P. LocalLexing.admissible \<NN> \<TT> \<RR> \<SS> (p @ [u])}"
span_insertI: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and r :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<in> pmdl B" shows "p \<in> pmdl (insert r B)"
Rep_inject: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and x :: "'a"   and y :: "'a" assumes "type_definition Rep Abs A" shows "(Rep x = Rep y) = (x = y)"
induced_play_def: fixes A :: "'c llist set"   and N :: "enat"   and \<sigma> :: "'d list \<Rightarrow> 'd" assumes "GSgame A N" shows "GSgame.induced_play N \<sigma> \<equiv> ltake (2 * N) \<circ> lmap last \<circ> ltl \<circ> strategy_progression \<sigma>"
assms: shows "\<tau> \<Turnstile> H .allInstances@pre()->includes\<^sub>S\<^sub>e\<^sub>t(x)"
sorted_list_of_set_Min_Cons: fixes A :: "'b set" assumes "finite A"   and "A \<noteq> {}" shows "sorted_list_of_set A = Min A # sorted_list_of_set (A - {Min A})"
directed_union: fixes X :: "'a set"   and Y :: "'a set" assumes "directed X (\<sqsubseteq>)"   and "directed Y (\<sqsubseteq>)"   and "\<forall>x\<in>X. \<forall>y\<in>Y. \<exists>z\<in>X \<union> Y. x \<sqsubseteq> z \<and> y \<sqsubseteq> z" shows "directed (X \<union> Y) (\<sqsubseteq>)"
CompFun_ChiRel2: fixes S :: "'a"   and T :: "'a"   and A :: "('a, 'b, 'c) hierauto"   and SA :: "('a, 'b, 'c) seqauto" assumes "(S, T) \<in> ChiRel A"   and "T \<in> States SA"   and "SA \<in> SAs A" shows "SA \<in> the (CompFun A S)"
set_mtf2: fixes n :: "nat"   and x :: "'b"   and xs :: "'b list" shows "set (mtf2 n x xs) = set xs"