hcju/isabelleps huggingface dataset:
holomorphic_on_If_Un: fixes f :: "complex \<Rightarrow> complex"   and A :: "complex set"   and g :: "complex \<Rightarrow> complex"   and B :: "complex set" assumes "f holomorphic_on A"   and "g holomorphic_on B"   and "open A"   and "open B"   and "\<And>z. \<lbrakk>z \<in> A; z \<in> B\<rbrakk> \<Longrightarrow> f z = g z" shows "(\<lambda>z. if z \<in> A then f z else g z) holomorphic_on A \<union> B"
hyps_3: fixes k :: "nat" shows "card {P. partition_on A P \<and> card P = k} = Stirling (card A) k"
CATCH_refine': fixes mi :: "('g + 'h) nres"   and E' :: "('g \<times> 'i) set"   and R :: "('h \<times> 'j) set"   and m :: "('i + 'j) nres"   and hi :: "'g \<Rightarrow> ('k + 'h) nres"   and E :: "('k \<times> 'l) set"   and h :: "'i \<Rightarrow> ('l + 'j) nres" assumes "mi \<le> \<Down>\<^sub>E E' R m"   and "\<And>ei e. \<lbrakk>(ei, e) \<in> E'; inres mi (Inl ei); inres m (Inl e); nofail mi; nofail m\<rbrakk> \<Longrightarrow> hi ei \<le> \<Down>\<^sub>E E R (h e)" shows "CATCH mi hi \<le> \<Down>\<^sub>E E R (CATCH m h)"
enn2real_def: fixes x :: "ennreal" shows "enn2real x = real_of_ereal (enn2ereal x)"
simps: fixes a :: "'a"   and p :: "'a \<times> 'b" shows "(a \<in> Basic_BNFs.fsts p) = (a = fst p)"
intro: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "ThinChamberComplex X"   and "ChamberComplexFolding X f"   and "ChamberComplexFolding X g"   and "OpposedThinChamberComplexFoldings_axioms X f g C0" shows "OpposedThinChamberComplexFoldings X f g C0"
ext_field_simp: fixes f :: "qtname \<times> 'a decl_scheme" shows "(declclass f, fld f) = f"
functors_preserve_homsets: fixes AA :: "('b, 'c, 'd) category_scheme"   and BB :: "('e, 'f, 'g) category_scheme"   and F :: "('b, 'c, 'e, 'f) functor"   and A :: "'b"   and B :: "'b"   and f :: "'c" assumes "Functor F : AA \<longrightarrow> BB"   and "A \<in> Ob\<^bsub>AA\<^esub>"   and "B \<in> Ob\<^bsub>AA\<^esub>"   and "f \<in> Hom\<^bsub>AA\<^esub> A B" shows "F \<^bsub>\<a>\<^esub> f \<in> Hom\<^bsub>BB\<^esub> F \<^bsub>\<o>\<^esub> A F \<^bsub>\<o>\<^esub> B"
simps_2: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and ct :: "'t conditional_action"   and cts :: "'t conditional_action list" shows "activate_cond_actions1 s1 s2 (ct # cts) = activate_cond_actions1 (activate_cond_action1 s1 s2 ct) s2 cts"
indicator_single_Some: fixes x :: "'c"   and y :: "'c" shows "indicator {Some x} (Some y) = indicator {x} y"
le_fin_transp: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list"   and w\<^sub>3 :: "'item list" assumes "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2"   and "w\<^sub>2 \<preceq>\<^sub>F w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>3"
right_neutral: fixes a :: "'b hrel" shows "a ;; II = a"
full2: fixes \<sigma> :: "'c"   and d :: "univ" shows "intT \<sigma> d"
assms_1: shows "f $ 0 * x = (1::'a)"
\<psi>D_VectorSpaceHom: fixes T :: "'v \<Rightarrow> 'w" assumes "T \<in> GRepHomSet (\<star>) W" shows "VectorSpaceHom (\<currency>\<currency>) indV (\<sharp>\<star>) (\<psi> T)"
lbk_borel: fixes T :: "nat" shows "lbk_option T \<in> borel_measurable (G T)"
inf_concat_add: fixes n :: "nat \<Rightarrow> nat"   and k :: "nat"   and i :: "nat"   and j :: "nat"   and m :: "nat" assumes "inf_concat n k = (i, j)"   and "j + m < n i" shows "inf_concat n (k + m) = (i, j + m)"
exec_Normal_elim_cases_10: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and b :: "'e set"   and c :: "('e, 'd, 'f) com"   and s :: "'e"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>While b c,Normal s\<rangle> \<Rightarrow> t" obtains s' :: "('e, 'f) xstate" where "s \<in> b" and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> s'" and "\<Gamma>\<turnstile> \<langle>While b c,s'\<rangle> \<Rightarrow> t"   | "t = Normal s" and "s \<notin> b"
zero_order_1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq zero x"
epsclo_mono: fixes Q' :: "hf set"   and Q :: "hf set" assumes "Q' \<subseteq> Q" shows "epsclo Q' \<subseteq> epsclo Q"
n_val_valuationTr1: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "\<forall>x\<in>carrier K. n_val K v x \<in> Z\<^sub>\<infinity>"
coprime_r_roots_prop: fixes qs :: "real poly list" assumes "\<forall>q1 q2. q1 \<noteq> q2 \<and> List.member qs q1 \<and> List.member qs q2 \<longrightarrow> comm_monoid_mult_class.coprime q1 q2" shows "\<forall>x1 x2. x1 < x2 \<and> (\<exists>q1\<in>set qs. poly q1 x1 = 0) \<and> (\<exists>q2\<in>set qs. poly q2 x2 = 0) \<longrightarrow> (\<exists>q>x1. q < x2 \<and> poly (coprime_r qs) q = 0)"
p_prime: fixes p :: "nat"   and a :: "int" assumes "GAUSS p a" shows "prime p"
code: fixes x :: "complex"   and y :: "complex" shows "x + y = Complex (Re x + Re y) (Im x + Im y)"
psi_def: shows "psi \<equiv> \<lambda>n x. if x = n then Some 1 else Some 0"
u_simps_3: shows "trg (tab\<^sub>0 f) = src f"
Pure: fixes x :: "'a"   and gpv :: "('a, 'b, 'c) gpv"   and \<Gamma> :: "('b, 'c) \<I>" assumes "Pure x \<in> set_spmf (the_gpv gpv)" shows "x \<in> results_gpv \<Gamma> gpv"
hyps: fixes k :: "'a"   and t :: "('a, 'b) trie" assumes "(k, t) \<in> set kts" shows "finite (dom (lookup_trie t))"
intrel_iff: fixes x :: "nat"   and y :: "nat"   and u :: "nat"   and v :: "nat" shows "intrel (x, y) (u, v) = (x + v = u + y)"
Ndet_commute: fixes P :: "'c process"   and Q :: "'c process" shows "(P |-| Q) = (Q |-| P)"
coinduct: fixes X :: "'a llist \<Rightarrow> bool"   and x :: "'a llist" assumes "X x"   and "\<And>x. X x \<Longrightarrow> x = LNil \<or> (\<exists>xa. x = LCons xa LNil) \<or> (\<exists>xa y xs. x = LCons xa (LCons y xs) \<and> xa \<le> y \<and> (X (LCons y xs) \<or> lsorted (LCons y xs)))" shows "lsorted x"
Body_no_break: fixes G :: "prog"   and s0 :: "st"   and D :: "qtname"   and s1 :: "state"   and c :: "stmt"   and s2 :: "state"   and C :: "qtname"   and L :: "lenv"   and d :: "class" assumes "G\<turnstile>Norm s0 \<midarrow>Init D\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>c\<rightarrow> s2"   and "jumpNestingOkS {Ret} c"   and "\<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile>c\<Colon>\<surd>"   and "class G D = Some d"   and "wf_prog G" shows "\<forall>l. fst s2 \<noteq> Some (Jump (Break l)) \<and> fst s2 \<noteq> Some (Jump (Cont l))"
parallel_oracle_split: fixes P :: "(('a + 'b) \<times> 'c \<times> 'd) spmf \<Rightarrow> bool"   and left :: "'c \<Rightarrow> 'e \<Rightarrow> ('a \<times> 'c) spmf"   and right :: "'d \<Rightarrow> 'f \<Rightarrow> ('b \<times> 'd) spmf"   and s1s2 :: "'c \<times> 'd"   and lr :: "'e + 'f" shows "P (parallel_oracle left right s1s2 lr) = ((\<forall>s1 s2 x. s1s2 = (s1, s2) \<longrightarrow> lr = Inl x \<longrightarrow> P (map_spmf (map_prod Inl (\<lambda>s1'. (s1', s2))) (left s1 x))) \<and> (\<forall>s1 s2 y. s1s2 = (s1, s2) \<longrightarrow> lr = Inr y \<longrightarrow> P (map_spmf (map_prod Inr (Pair s1)) (right s2 y))))"
aux_lemma: fixes aa :: "'a"   and ba :: "'a"   and da :: "'a" assumes "aa \<in> carrier R"   and "ba \<in> carrier R"   and "da \<in> carrier R"   and "PIdl da = PIdl aa <+> PIdl ba" shows "da gcdof aa ba"
Nmlize_Vcomp_Cod_Arr: fixes t :: "'c local.term" assumes "Arr t" shows "\<^bold>\<lfloor>Cod t \<^bold>\<cdot> t\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
find_safe: fixes w :: "'a stream"   and A :: "('a, 'b) nba"   and V :: "(nat \<times> 'b) set" assumes "w \<notin> language A"   and "V \<noteq> {}"   and "V \<subseteq> gunodes A w"   and "\<And>v. v \<in> V \<Longrightarrow> gsuccessors A w V v \<noteq> {}" obtains v :: "nat \<times> 'b" where "v \<in> V" and "\<forall>u\<in>greachable A w V v. \<not> gaccepting A u"
add_mono_thms_linordered_field_5: fixes i :: "'c"   and j :: "'c"   and k :: "'c"   and l :: "'c" assumes "i < j \<and> k < l" shows "i + k < j + l"
trigonometric_set_1: fixes x :: "real" shows "trigonometric_set 0 x = 1 / sqrt (2 * pi)"
rw2: shows "(\<lambda>x. diamond_y_gen (x_coord x)) = diamond_y_gen \<circ> x_coord"
add_lemma: fixes x :: "nat"   and y :: "nat" shows "rec_exec rec_add [x, y] = x + y"
rtranclp_unfold: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" shows "r\<^sup>*\<^sup>* a b \<equiv> a = b \<or> r\<^sup>+\<^sup>+ a b"
analz_insert_eq: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> analz H" shows "analz (insert X H) = analz H"
induct: fixes P :: "nat \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "'b list" assumes "\<And>n. P n []"   and "\<And>xs. P 0 xs"   and "\<And>n xs. \<lbrakk>0 < n; xs \<noteq> []; P n (drop n xs)\<rbrakk> \<Longrightarrow> P n xs" shows "P a0 a1"
is_regular_spair_sym: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_regular_spair p q" shows "is_regular_spair q p"
is_scc_restrict1: fixes E :: "('b \<times> 'b) set"   and U :: "'b set" assumes "is_scc E U" shows "U \<times> U \<subseteq> (Restr E U)\<^sup>*"
pref_pref_lcp: fixes ps :: "'b list"   and xs :: "'b list"   and ys :: "'b list" assumes "ps \<le>p xs"   and "ps \<le>p ys" shows "ps \<le>p xs \<and>\<^sub>p ys"
SUCCEED_refine: fixes R :: "('b \<times> 'c) set"   and X' :: "'c nres" shows "SUCCEED \<le> \<Down> R X'"
power_le_one_iff: fixes a :: "real"   and n :: "nat" assumes "0 \<le> a" shows "(a ^ n \<le> 1) = (n = 0 \<or> a \<le> 1)"
univ: fixes Rep :: "'d \<Rightarrow> 'e"   and Abs :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "type_definition Rep Abs A" shows "UNIV = Abs ` A"
mat_inverse_same: fixes A :: "'a mat" shows "mat_inverse_var A = mat_inverse A"
support_set: fixes p :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "permutation p" shows "set (support p a) = range (\<lambda>i. (p ^^ i) a)"
len_eq0_iff: fixes n :: "nat" shows "(len n = 0) = (n = 0)"
gc_W_empty_mut_inv_def: fixes m :: "'e" shows "mut_m.gc_W_empty_mut_inv m = (\<lambda>s. sys_W s = {} \<and> sys_ghost_hs_in_sync m s \<and> WL (mutator m) s \<noteq> {} \<longrightarrow> (\<exists>x. \<not> sys_ghost_hs_in_sync x s \<and> WL (mutator x) s \<noteq> {}))"
maintained_holds_iff: fixes G :: "('a, 'b) labeled_graph"   and e\<^sub>L :: "'a allegorical_term"   and e\<^sub>R :: "'a allegorical_term" assumes "graph G" shows "maintained (translation e\<^sub>L, translation (A_Int e\<^sub>L e\<^sub>R)) G = G \<Turnstile> e\<^sub>L \<sqsubseteq> e\<^sub>R"
rp: shows "squash (rpoly p x) = squash c * (\<Prod>ai\<leftarrow>fs. case ai of (a, i) \<Rightarrow> squash (rpoly a x) ^ Suc i)"
S6: shows "pr_index_enumerator n m < c_pair 4 (c_pair index_of_id (pr_index_enumerator n m))"
nn_integral_count_space_nn_integral: fixes I :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> ennreal"   and M :: "'c measure" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. \<integral>\<^sup>+ i. f i x \<partial>count_space I \<partial>M = \<integral>\<^sup>+ i. integral\<^sup>N M (f i) \<partial>count_space I"
select_append_1: fixes ys :: "'a list"   and zs :: "'a list"   and k :: "nat" assumes "\<forall>y\<in>set ys. \<forall>z\<in>set zs. y \<le> z"   and "k < length ys" shows "select k (ys @ zs) = select k ys"
filterlim_at_bot_dense: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "filterlim f at_bot F = (\<forall>Z. \<forall>\<^sub>F x in F. f x < Z)"
assms_1: shows "line_integral_exists F basis (rec_join ((- 1, \<gamma>) # l))"
intro: fixes L :: "('c, 'd) gorder_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier L; y \<in> carrier L\<rbrakk> \<Longrightarrow> \<exists>s. is_lub L s {x, y}" shows "weak_upper_semilattice_axioms L"
IdI: fixes a :: "'c" shows "(a, a) \<in> Id"
Diff_insert_absorb: fixes x :: "'c"   and A :: "'c set" assumes "x \<notin> A" shows "insert x A - {x} = A"
opp_closed: fixes a :: "'b"   and b :: "'b"   and p :: "'b point" assumes "on_curve a b p" shows "on_curve a b (opp p)"
Suc_diff_diff: fixes m :: "nat"   and n :: "nat"   and k :: "nat" shows "Suc m - n - Suc k = m - n - k"
unit_disc_fix_rotation: fixes \<phi> :: "real" shows "unit_disc_fix (moebius_rotation \<phi>)"
match_ex_table_shift_pc_None: fixes n :: "nat"   and pc :: "nat"   and P :: "'a prog"   and C :: "String.literal"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list" assumes "n \<le> pc" shows "(match_ex_table P C pc (shift n xt) = None) = (match_ex_table P C (pc - n) xt = None)"
Re_double: fixes x :: "complex" shows "Re (2 * x) = 2 * Re x"
closed_compactin: fixes X :: "'c topology"   and K :: "'c set"   and C :: "'c set" assumes "compactin X K"   and "C \<subseteq> K"   and "closedin X C" shows "compactin X C"
LeftDerivationIntrosAt_ignore_appendix: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and a :: "symbol list"   and index :: "nat"   and D' :: "(nat \<times> symbol \<times> symbol list) list"   and L' :: "(nat \<times> nat \<times> nat) list" assumes "is_ladder D L"   and "LeftDerivationIntrosAt a D L index"   and "1 \<le> index"   and "index < length L" shows "LeftDerivationIntrosAt a (D @ D') (L @ L') index"
trans_ParTL: fixes c1 :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state"   and c2 :: "('test, 'atom) com" assumes "(c1, s) \<rightarrow>t s'" shows "(c1 | c2, s) \<rightarrow>c (c2, s')"
bounded_imp_bdd_above: fixes S :: "real set" assumes "bounded S" shows "bdd_above S"
successive_ends_1: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list"   and x :: "'d"   and ys :: "'d list" assumes "successive P xs"   and "\<not> P x"   and "successive P ys" shows "successive P (xs @ x # ys)"
simps: fixes a1 :: "'a list set"   and a2 :: "'a list set" shows "bisimilar a1 a2 = (\<exists>K L. a1 = K \<and> a2 = L \<and> ([] \<in> K) = ([] \<in> L) \<and> (\<forall>x. bisimilar (Deriv x K) (Deriv x L)))"
cancel_helper: shows "matr_option (dim_row (M_mat_R signs subsets)) (mat_inverse (M_mat_R signs subsets)) * M_mat_R signs subsets *\<^sub>v w_vec_R p qs signs = 1\<^sub>m (length signs) *\<^sub>v w_vec_R p qs signs"
upright_sum_upright: fixes X :: "'c set"   and A :: "'c \<Rightarrow> 'd zmultiset" assumes "finite X"   and "\<forall>x. upright (A x)" shows "upright (sum A X)"
lfinite_conv_llength_enat: fixes xs :: "'b llist" shows "lfinite xs = (\<exists>n. llength xs = enat n)"
Spy_see_shrK: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> recur" shows "(Key (shrK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
span_scale: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and S :: "'b set"   and c :: "'a" assumes "module scale"   and "x \<in> module.span scale S" shows "scale c x \<in> module.span scale S"
insert_type: fixes t :: "'a bro"   and h :: "nat"   and x :: "'a" assumes "t \<in> B h" shows "insert x t \<in> B h \<union> B (Suc h)"
clopen: fixes S :: "'a set" shows "(closed S \<and> open S) = (S = {} \<or> S = UNIV)"
min_extend_val': shows "sint (- 0x7FFFFFFF) = - 2147483647"
simps_2: fixes xs :: "'a list"   and n :: "nat" shows "peval xs (PExpr1 (PVar n)) = nth_el xs n"
value_Zl_mI: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set"   and j :: "nat" assumes "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "j \<le> n" shows "Zl_mI K P I j \<in> I \<and> (\<nu>\<^bsub>K P j\<^esub>) (Zl_mI K P I j) = LI K (\<nu>\<^bsub>K P j\<^esub>) I"
Syntax_PseudoOrder_axioms: shows "Syntax_PseudoOrder var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num Lq"
mult_1_right: fixes a :: "'a" shows "a \<sqinter> top = a"
split_aforms_largest_uncond_take_def: fixes n :: "nat"   and X :: "('a \<times> 'a pdevs) list" shows "split_aforms_largest_uncond_take n X = (let (i, x) = max_pdev (abssum_of_pdevs_list (map snd (take n X))) in split_aforms X i)"
llist_less_def: fixes s :: "'b llist"   and t :: "'b llist" shows "(s < t) = (s \<le> t \<and> s \<noteq> t)"
mod_le_dividend: fixes m :: "nat"   and n :: "nat" shows "m mod n \<le> m"
funpow_increasing: fixes m :: "nat"   and n :: "nat"   and f :: "'a \<Rightarrow> 'a" assumes "m \<le> n"   and "mono f" shows "(f ^^ n) top \<le> (f ^^ m) top"
asm: shows "v = Tensor.mat_of_cols_list 8 [[0, 0, 0, 0, \<alpha> / complex_of_real (sqrt 2), - \<beta> / complex_of_real (sqrt 2), - \<beta> / complex_of_real (sqrt 2), \<alpha> / complex_of_real (sqrt 2)]]"
simps_9: fixes \<phi> :: "'b ltl"   and w :: "nat \<Rightarrow> 'b set" shows "\<G>\<^sub>F\<^sub>G (X \<phi>) w = \<G>\<^sub>F\<^sub>G \<phi> w"
partition_on_transform: fixes A :: "'b set"   and P :: "'b set set"   and F :: "'b set \<Rightarrow> 'c set" assumes "partition_on A P"   and "\<Union> (F ` P) = F (\<Union> P)"   and "\<And>p q. \<lbrakk>p \<in> P; q \<in> P; disjnt p q\<rbrakk> \<Longrightarrow> disjnt (F p) (F q)" shows "partition_on (F A) (F ` P - {{}})"
m_upd_lookup: fixes xs :: "'a option list"   and tstp :: "nat + nat" assumes "Mapping.lookup m_upd xs = Some tstp" shows "tstp_lt tstp (cur - (left I - 1)) tp \<and> isl tstp = (0 < left I)"
prob_index_0_alice: fixes \<phi> :: "complex Matrix.mat" assumes "state 1 \<phi>" shows "prob0 3 (alice \<phi>) 0 = 1 / 2 \<and> prob1 3 (alice \<phi>) 0 = 1 / 2"
Hausdorff_space_euclidean: shows "Hausdorff_space euclidean"
SDG_cdep_edge_CFG_node: fixes n :: "'node SDG_node"   and n' :: "'node SDG_node" assumes "n \<longrightarrow>\<^bsub>cd\<^esub> n'" shows "\<exists>m. n = CFG_node m"
simps: fixes a1 :: "'b formula"   and a2 :: "'b formula"   and a3 :: "'b formula" shows "Dis (a1::'b formula) (a2::'b formula) (a3::'b formula) = ((\<exists>(F::'b formula) G::'b formula. a1 = F \<^bold>\<or> G \<and> a2 = F \<and> a3 = G) \<or> (\<exists>(F::'b formula) G::'b formula. a1 = F \<^bold>\<rightarrow> G \<and> a2 = \<^bold>\<not> F \<and> a3 = G) \<or> (\<exists>(F::'b formula) G::'b formula. a1 = \<^bold>\<not> (F \<^bold>\<and> G) \<and> a2 = \<^bold>\<not> F \<and> a3 = \<^bold>\<not> G) \<or> (\<exists>F::'b formula. a1 = \<^bold>\<not> (\<^bold>\<not> F) \<and> a2 = F \<and> a3 = F))"
simps_1: fixes M :: "('fun, 'var) Term.term set" shows "\<lbrakk>M; []\<rbrakk>\<^sub>c' = (\<lambda>\<I>. True)"
setcompr_eq_image: fixes f :: "'b \<Rightarrow> 'a"   and P :: "'b \<Rightarrow> bool" shows "{uu_. \<exists>x. uu_ = f x \<and> P x} = f ` {x. P x}"
wtE_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "M.wtE \<xi> \<equiv> \<forall>x. D (\<xi> x)"
decode_plus: fixes e :: "nat"   and M :: "hmultiset"   and N :: "hmultiset" assumes "local.aligned\<^sub>d e M" shows "local.decode e (M + N) = local.decode e M + local.decode e N"
topspace_discrete_topology: fixes U :: "'d set" shows "topspace (discrete_topology U) = U"
measurable_cexpr_sem': fixes \<rho> :: "nat \<Rightarrow> val"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and V :: "nat set" assumes "\<rho> \<in> space (state_measure V' \<Gamma>)"   and "\<Gamma> \<turnstile>\<^sub>c e : REAL"   and "free_vars e \<subseteq> V \<union> V'" shows "(\<lambda>\<sigma>. extract_real (cexpr_sem (merge V V' (\<sigma>, \<rho>)) e)) \<in> borel_measurable (state_measure V \<Gamma>)"
induct: fixes P :: "vname \<Rightarrow> vname \<Rightarrow> bool"   and a0 :: "vname"   and a1 :: "vname" assumes "\<And>n1 n2. P (I n1) (R n2)"   and "\<And>n1 n2. P (R n1) (I n2)"   and "\<And>n1 n2. P (I n1) (I n2)"   and "\<And>n1 n2. P (R n1) (R n2)" shows "P a0 a1"
\<gamma>\<^sub>\<tau>'_eq: shows "\<gamma>\<^sub>\<tau>' = local.inv \<gamma>\<^sub>\<tau>"
elapsed_def: fixes u :: "'c \<Rightarrow> 'b"   and u' :: "'c \<Rightarrow> 'b" shows "elapsed u u' \<equiv> Max ({u' c - u c |c. c \<in> \<X>} \<union> {0::'b})"
finite_UNIV_finite_3_def: shows "finite_UNIV = Phantom(Enum.finite_3) True"
sync_jview_jAction_eq: fixes T :: "'s Trace set"   and n :: "nat"   and T' :: "'s Trace set"   and t :: "'s Trace" assumes "{t \<in> T. tLength t = n} = {t \<in> T'. tLength t = n}"   and "t \<in> {t \<in> T. tLength t = n}" shows "jAction (mkM T) t = jAction (mkM T') t"
C_msort_bu_def: fixes xs :: "'c list" shows "C_msort_bu xs = C_merge_all (map (\<lambda>x. [x]) xs)"
B: shows "trnr\<^sub>\<epsilon> u \<in> hom v (u \<star> f) \<rightarrow> hom (v \<star> g) u"
paths_le_continuous: fixes k :: "nat" shows "continuous (paths_le k)"
subset_diff: fixes B :: "'a set"   and A :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "B \<subseteq> A"   and "finite A" shows "prod g A = prod g (A - B) * prod g B"
conjugate_dist_add: fixes a :: "'b"   and b :: "'b" shows "conjugate (a + b) = conjugate a + conjugate b"
gchoose_rising_sum_nat: fixes k :: "nat"   and n :: "nat" shows "(\<Sum>j\<le>n. int j + int k gchoose k) = int n + int k + 1 gchoose Suc k"
map_eq_map_tailrec: shows "map = List.map_tailrec"
gauss_int_norm_dvd_mono: fixes a :: "gauss_int"   and b :: "gauss_int" assumes "a dvd b" shows "gauss_int_norm a dvd gauss_int_norm b"
HcompNml_Nml_Src: fixes t :: "'c local.term" assumes "Nml t" shows "t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Src t = t"
abs_eq: fixes x :: "nat set"   and xa :: "nat" assumes "eq_onp (\<lambda>S. \<exists>m n. {m..n} = S) x x" shows "(xa \<^bold>\<in> Abs_nat_int x) = (xa \<in> x)"
p: shows "\<forall>s\<in>P. \<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P' s Z) Call (p s) (Q' s Z),(A' s Z)"
a: shows "flip (Suc z') X ! Suc y = flip z' as ! y"
irreducible\<^sub>d_def: fixes p :: "'a poly" shows "irreducible\<^sub>d p = (0 < degree p \<and> (\<forall>q r. degree q < degree p \<longrightarrow> degree r < degree p \<longrightarrow> p \<noteq> q * r))"
simps_1: fixes uu :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" shows "merge_wrt uu xs [] = xs"
hyps: fixes A :: "(('a, 'b) vec, 'c) vec"   and P :: "(('a, 'c) vec, 'c) vec" assumes "n < nrows A"   and "n < mod_type_class.to_nat a" shows "fst (row_add_iterate_PA (P, A) n i j) $ a $ b = P $ a $ b"
NatTransComp_Comp2: fixes x :: "'g"   and f :: "('g, 'j, 'h, 'k, 'i, 'l) NatTrans"   and g :: "('g, 'j, 'h, 'k, 'i, 'l) NatTrans" assumes "x \<in> obj\<^bsub>NTCatDom f\<^esub>"   and "f \<approx>>\<bullet> g" shows "f \<bullet> g $$ x = f $$ x ;;\<^bsub>NTCatCod f\<^esub> (g $$ x)"
VUnion_upper: fixes x :: "V"   and A :: "V" assumes "x \<in>\<^sub>\<circ> A" shows "x \<subseteq>\<^sub>\<circ> \<Union>\<^sub>\<circ> A"
alw_alw: fixes \<phi> :: "'d stream \<Rightarrow> bool" shows "alw (alw \<phi>) = alw \<phi>"
ct_list_eq_concat: fixes xs :: "ct list"   and ys :: "ct list"   and xs' :: "ct list"   and ys' :: "ct list" assumes "ct_list_eq xs ys"   and "ct_list_eq xs' ys'" shows "ct_list_eq (xs @ xs') (ys @ ys')"
iso_\<epsilon>': shows "S.iso (UP.unit (trg f) \<cdot>\<^sub>S \<xi>' \<cdot>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (f, g)))"
mult_of_def: fixes R :: "('b, 'c) ring_scheme" shows "mult_of R \<equiv> \<lparr>carrier = carrier R - {\<zero>\<^bsub>R\<^esub>}, monoid.mult = (\<otimes>\<^bsub>R\<^esub>), one = \<one>\<^bsub>R\<^esub>\<rparr>"
simps_2: fixes p :: "int \<times> int"   and p\<^sub>0 :: "int \<times> int" shows "find_closest_bf_code p [p\<^sub>0] = (dist_code p p\<^sub>0, p\<^sub>0)"
countable_finite: fixes S :: "'g set" assumes "finite S" shows "countable S"
that: fixes \<Psi>\<^sub>F :: "'b" assumes "F = \<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>"   and "\<Psi>' = \<Psi> \<otimes> \<Psi>\<^sub>F" shows "thesis"
assms_3: shows "x' \<oplus> \<ominus> x \<in> H"
intros_2: fixes n' :: "'node"   and n\<^sub>1 :: "'node"   and n\<^sub>2 :: "'node"   and S :: "'node set" assumes "n' \<longrightarrow>\<^sub>w\<^sub>o\<^sub>d n\<^sub>1,n\<^sub>2"   and "n\<^sub>1 \<in> wod_backward_slice S"   and "n\<^sub>2 \<in> wod_backward_slice S" shows "n' \<in> wod_backward_slice S"