hcju/isabelleps huggingface dataset:
OUren: fixes ODE :: "('sf, 'sz) ODE"   and I :: "('sf, 'a, 'sz) interp"   and x :: "'sz"   and y :: "'sz"   and \<nu> :: "(real, 'sz) vec" assumes "ORadmit ODE" shows "ODE_sem I (OUrename x y ODE) \<nu> = RSadj x y (ODE_sem I ODE (RSadj x y \<nu>))"
all_def: fixes P :: "'c \<Rightarrow> 'b \<Rightarrow> bool" shows "\<^bold>\<forall>x. P x \<equiv> \<lambda>A. \<forall>x. P x A"
winningAllocationPossible: fixes b :: "'c \<times> 'd set \<Rightarrow> 'e"   and G :: "'d set"   and N :: "'c set" shows "winningAllocationRel N G (\<subseteq>) b (allAllocations N G)"
setmult_subset_G: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and K :: "'c set" assumes "Group.group G"   and "H \<subseteq> carrier G"   and "K \<subseteq> carrier G" shows "H <#>\<^bsub>G\<^esub> K \<subseteq> carrier G"
scaleC_one: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<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.complex_vector scaleR scaleC plus zero minus uminus" shows "scaleC 1 x = x"
Cons_3: shows "\<exists>n st. 0 < n \<and> sl = map Idle (replicate n st) \<or> (\<exists>st1 u1 c1 sl1. sl = map Idle (replicate n st) @ [State st1 u1 c1] @ sl1)"
section_is_mono: fixes g :: "'a" assumes "B.section g" shows "B.mono g"
vector_derivative_circlepath: fixes z :: "complex"   and r :: "real"   and x :: "real" shows "vector_derivative (circlepath z r) (at x) = complex_of_real (2 * pi) * \<i> * complex_of_real r * exp (2 * complex_of_real pi * \<i> * complex_of_real x)"
winning_region_in_V: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player" assumes "ParityGame G" shows "ParityGame.winning_region G p \<subseteq> V\<^bsub>G\<^esub>"
distinct_175: fixes x71 :: "Syntax.term"   and x72 :: "Syntax.term"   and x81 :: "Syntax.term"   and x82 :: "Syntax.var"   and x83 :: "Syntax.term" shows "Syntax.Pair x71 x72 \<noteq> Syntax.Let x81 x82 x83"
has_subprob_density_embed_measure'': fixes f :: "'a \<Rightarrow> 'b"   and N :: "'a measure"   and f' :: "'b \<Rightarrow> 'a"   and M :: "'b measure"   and \<delta> :: "'b \<Rightarrow> ennreal" assumes "inj f"   and "\<And>x. x \<in> space N \<Longrightarrow> f' (f x) = x"   and "has_subprob_density M (embed_measure N f) \<delta>" shows "has_subprob_density (distr M N f') N (\<delta> \<circ> f)"
star_right_induct_equal: fixes z :: "'a"   and y :: "'a"   and x :: "'a" assumes "z \<squnion> y * x = y" shows "z * x\<^sup>\<star> \<le> y"
rref_upt_condition1: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat" assumes "reduced_row_echelon_form_upt_k A k" shows "\<forall>i. is_zero_row_upt_k i k A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row_upt_k j k A)"
intro: fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>x y. f (x + y) = f x + f y" shows "additive f"
arev_eq: fixes a :: "'a \<times> 'a" assumes "a \<notin> arcs (with_proj G)" shows "rev_G a = a"
map_custom_empty_axioms: shows "map_custom_empty op_hm_empty"
subpath_compose: fixes a :: "real"   and b :: "real"   and \<gamma> :: "real \<Rightarrow> 'a" shows "subpath a b \<gamma> = \<gamma> \<circ> (\<lambda>x. (b - a) * x + a)"
dgrad_sig_set_closed_minus: fixes r :: "'t \<Rightarrow>\<^sub>0 'b"   and j :: "nat"   and d :: "'a \<Rightarrow> nat"   and s :: "'t \<Rightarrow>\<^sub>0 'b" assumes "r \<in> dgrad_sig_set' j d"   and "s \<in> dgrad_sig_set' j d" shows "r - s \<in> dgrad_sig_set' j d"
restrictto_compose: fixes X :: "(variable \<Rightarrow> real) set"   and V :: "variable set"   and W :: "variable set" shows "restrictto (restrictto X V) W = restrictto X (V \<inter> W)"
gcd_greatest_mult: fixes c :: "'a"   and a :: "'a"   and d :: "'a"   and b :: "'a" assumes "c dvd a * d"   and "c dvd b * d" shows "c dvd gcd a b * d"
domains_subsume_def: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and ls :: "('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr" assumes "substitution subst" shows "substitution.domains_subsume ls = (LID\<^sub>L ls \<subseteq> doms ls)"
monomial_equation: fixes c :: "nat \<Rightarrow> int"   and n :: "nat" assumes "c \<in> carrier Zp" shows "monomial c n = c \<odot>\<^bsub>function_ring (carrier Zp) Zp\<^esub> ring_id [^]\<^bsub>function_ring (carrier Zp) Zp\<^esub> n"
in_keys_lookup_in_range: fixes k :: "'b"   and f :: "'b \<Rightarrow>\<^sub>0 'c" assumes "k \<in> keys f" shows "lookup f k \<in> Poly_Mapping.range f"
inf_last_ti_nonempty_k: fixes dt :: "nat \<Rightarrow> 'd list"   and t :: "nat"   and k :: "nat" assumes "inf_last_ti dt t \<noteq> []" shows "inf_last_ti dt (t + k) \<noteq> []"
withAs: fixes P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real"   and s :: "'b" assumes "unitary P"   and "unitary Q"   and "G s"   and "P \<tturnstile> wp A Q" shows "(P \<circ> \<phi>) s \<le> wp B (Q \<circ> \<phi>) s"
return_node_def: fixes n :: "'node" shows "return_node n \<equiv> \<exists>a a'. valid_edge a \<and> n = targetnode a \<and> valid_edge a' \<and> a \<in> get_return_edges a'"
WT_elim_cases_4: fixes 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 E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "expr"   and bop :: "bop"   and e\<^sub>2 :: "expr"   and T :: "ty" assumes "P,E \<turnstile> e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2 :: T" obtains T\<^sub>1 :: "ty" and T\<^sub>2 :: "ty" where "P,E \<turnstile> e\<^sub>1 :: T\<^sub>1" and "P,E \<turnstile> e\<^sub>2 :: T\<^sub>2" and "case bop of Eq \<Rightarrow> T\<^sub>1 = T\<^sub>2 \<and> T = Boolean | Add \<Rightarrow> T\<^sub>1 = Integer \<and> T\<^sub>2 = Integer \<and> T = Integer"
assms_2: shows "finite XS \<or> |XS| <o |UNIV|"
cod_trg_eq_dom_trg: shows "assoc'_fgh.cod.trg = assoc'_fgh.dtrg"
wary_num_args_le_arity_head\<^sub>h: fixes s :: "('s, 'v) tm" assumes "wary s" shows "of_nat (num_args s) \<le> arity_hd\<^sub>h (head s)"
ex_rcoset_replist_hd0: shows "\<exists>gs. Supgroup.is_rcoset_replist G ((0::'g) # gs)"
i_aggregate_i_append_mult: fixes xs :: "'d list"   and m :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'd"   and ag :: "'d list \<Rightarrow> 'd" assumes "length xs = m * k" shows "i_aggregate (xs \<frown> f) k ag = f_aggregate xs k ag \<frown> i_aggregate f k ag"
dcharacter_axioms: shows "dcharacter n \<chi>\<^sub>0"
tendsto_inverse: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'c"   and F :: "'b filter" assumes "(f \<longlongrightarrow> a) F"   and "a \<noteq> (0::'c)" shows "((\<lambda>x. inverse (f x)) \<longlongrightarrow> inverse a) F"
aligned\<^sub>d_mono_hmset: fixes base :: "nat"   and e :: "nat"   and M :: "hmultiset"   and M' :: "hmultiset" assumes "2 \<le> base"   and "aligned\<^sub>d base e M"   and "hmsetmset M' \<subseteq># hmsetmset M" shows "aligned\<^sub>d base e M'"
Y_size: shows "card (set (take (Suc i) BL)) = Suc i"
i_Exec_Comp_Stream_Acc_Output_Init_def: fixes k :: "nat"   and output_fun :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'e" shows "i_Exec_Comp_Stream_Acc_Output_Init k output_fun trans_fun input c \<equiv> [output_fun c] \<frown> i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c"
f_Exec_State_Idle_replicate_NoMsg_ge_state: fixes localState :: "'d \<Rightarrow> 'e"   and output_fun :: "'d \<Rightarrow> 'f message_af"   and trans_fun :: "'g message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and m :: "nat"   and c :: "'d"   and n :: "nat" assumes "State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun NoMsg\<^bsup>m\<^esup> c))"   and "m \<le> n" shows "localState (f_Exec_Comp trans_fun NoMsg\<^bsup>n\<^esup> c) = localState (f_Exec_Comp trans_fun NoMsg\<^bsup>m\<^esup> c)"
matrix_A_R_def: fixes sign_list :: "rat list list"   and subset_list :: "(nat list \<times> nat list) list" shows "M_mat_R sign_list subset_list = mat_of_rows_list (length sign_list) (map (mtx_row_R sign_list) subset_list)"
dims_tensor_prod: fixes A :: "'a tensor"   and B :: "'a tensor" shows "dims (local.tensor_prod_otimes A B) = dims A @ dims B"
mkarr_Diagonalize_rep: fixes f :: "'c local.term set" assumes "arr f"   and "Diag (DOM f)"   and "Diag (COD f)" shows "mkarr \<^bold>\<lfloor>rep f\<^bold>\<rfloor> = f"
ustar_as_nu: fixes P :: "'b hrel" shows "P\<^sup>\<star> = (\<nu> X \<bullet> (P ;; X) \<sqinter> II)"
minus_Real: fixes X :: "nat \<Rightarrow> rat" assumes "Real.cauchy X" shows "- Real.Real X = Real.Real (\<lambda>n. - X n)"
ltree_insert: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "ltree t" shows "ltree (insert x t)"
iT_Plus_iAll_conv: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<box> t I \<oplus> k. P t) = (\<box> t I. P (t + k))"
fmpred_drop: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "('a, 'b) fmap"   and a :: "'a" assumes "fmpred P m" shows "fmpred P (fmdrop a m)"
Cons_3: shows "\<nexists>p. epath E2 u p v"
conjugate_exponent_real_ennreal: fixes p :: "real" assumes "1 < p" shows "conjugate_exponent (ennreal p) = ennreal (conjugate_exponent p)"
g_orbital_orbit: fixes s :: "'a"   and U :: "'a \<Rightarrow> real set"   and Y :: "real \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> bool" assumes "s \<in> S"   and "is_interval (U s)"   and "U s \<subseteq> T"   and "Y \<in> Sols (\<lambda>t. f) U S 0 s" shows "g_orbital (\<lambda>t. f) G U S 0 s = g_orbit Y G (U s)"
encode_operator_effect_def: fixes \<Pi> :: "'a strips_problem"   and t :: "nat"   and op :: "'a strips_operator" shows "encode_operator_effect \<Pi> t op \<equiv> let vs = \<Pi>\<^sub>\<V>; ops = \<Pi>\<^sub>\<O> in \<^bold>\<And>map (\<lambda>v. \<^bold>\<not> (Atom (Operator t (index ops op))) \<^bold>\<or> Atom (State (Suc t) (index vs v))) (add_effects_of op) @ map (\<lambda>v. \<^bold>\<not> (Atom (Operator t (index ops op))) \<^bold>\<or> \<^bold>\<not> (Atom (State (Suc t) (index vs v)))) (delete_effects_of op)"
subst_atms_id_subst: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and AA :: "'b set" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms" shows "substitution_ops.subst_atms subst_atm AA id_subst = AA"
f0: shows "U\<^sup>\<dagger> * U = 1\<^sub>m (dim_col U)"
product_part_in_first_factor: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "CM"   and "O x y"   and "P z (x \<otimes> y)" shows "P z x"
irreducible_def: shows "local.irreducible = (UNIV \<subseteq> G\<^sup>+)"
apply_guards_cons: fixes a :: "vname gexp"   and G :: "vname gexp list"   and c :: "vname \<Rightarrow> value option" shows "apply_guards (a # G) c = (gval a c = true \<and> apply_guards G c)"
atLeastLessThan_def: fixes l :: "'a"   and u :: "'a" shows "{l..<u} \<equiv> {l..} \<inter> {..<u}"
zorder_exist_pole: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set"   and z :: "complex" assumes "f holomorphic_on s - {z}"   and "open s"   and "z \<in> s"   and "is_pole f z" shows "zorder f z < 0 \<and> zor_poly f z z \<noteq> 0 \<and> (\<exists>r>0. cball z r \<subseteq> s \<and> zor_poly f z holomorphic_on cball z r \<and> (\<forall>w\<in>cball z r - {z}. f w = zor_poly f z w / (w - z) ^ nat (- zorder f z) \<and> zor_poly f z w \<noteq> 0))"
word_le_nat_alt: fixes a :: "'b word"   and b :: "'b word" shows "(a \<le> b) = (unat a \<le> unat b)"
bchoice: fixes S :: "'e set"   and Q :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "\<forall>x\<in>S. \<exists>y. Q x y" shows "\<exists>f. \<forall>x\<in>S. Q x (f x)"
basis_orthogonal': fixes xs :: "('a, 'b) vec list" shows "length (Gram_Schmidt xs) = length xs \<and> span (set (Gram_Schmidt xs)) = span (set xs) \<and> pairwise orthogonal (set (Gram_Schmidt xs))"
inv_imp_plus_work_nonneg_def: fixes c :: "('a, 'b, 'c) configuration_scheme" shows "inv_imp_plus_work_nonneg c \<equiv> \<forall>loc t. 0 \<le> zcount (c_imp c loc) t + zcount (c_work c loc) t"
not_interp_to_interp_imp_less: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and A :: "'b"   and N :: "'b literal multiset set"   and C :: "'b literal multiset"   and D :: "'b literal multiset" assumes "ground_resolution_with_selection S"   and "A \<notin> ground_resolution_with_selection.interp S N C"   and "A \<in> ground_resolution_with_selection.interp S N D" shows "C < D"
assms_3: shows "freshAbs zs z (A $[Y / y]_ys)"
ihyp: fixes y :: "game" assumes "zin y (options x)" shows "eq_game (plus_game y (neg_game y)) zero_game"
relcomp_distrib: fixes R :: "('e \<times> 'g) set"   and S :: "('g \<times> 'f) set"   and T :: "('g \<times> 'f) set" shows "R O (S \<union> T) = R O S \<union> R O T"
closest_pair_bf_code_dist_eq: fixes ps :: "(int \<times> int) list"   and \<delta> :: "int"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int" assumes "1 < length ps"   and "(\<delta>, c\<^sub>0, c\<^sub>1) = closest_pair_bf_code ps" shows "\<delta> = dist_code c\<^sub>0 c\<^sub>1"
chambercomplex_fold_A_im: shows "ChamberComplex (fold_A \<turnstile> A)"
dist_le_max_dist_triangle: fixes G :: "'a set"   and x :: "'a"   and y :: "'a"   and m :: "'a"   and z :: "'a" assumes "geodesic_segment_between G x y"   and "m \<in> G" shows "dist m z \<le> max (dist x z) (dist y z) + deltaG TYPE('a)"
distinct_27: fixes x11 :: "floatarith"   and x12 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Add x11 x12 \<noteq> Power x151 x152"
chine_assoc'_props_4: shows "fgh.Prj\<^sub>0 \<cdot> fgh.chine_assoc' = fgh.Prj\<^sub>0\<^sub>0"
empty': shows "\<forall>i p is xs sb \<D> \<O> \<R>. i < length ts\<^sub>h \<longrightarrow> ts\<^sub>h ! i = (p, is, xs, sb, \<D>, \<O>, \<R>) \<longrightarrow> sb = []"
rcoset_mult_add: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "ideal I R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "[mod I:] (I +>\<^bsub>R\<^esub> x) \<Otimes>\<^bsub>R\<^esub> (I +>\<^bsub>R\<^esub> y) = I +>\<^bsub>R\<^esub> x \<otimes>\<^bsub>R\<^esub> y"
ordertype_infinite_\<omega>: fixes A :: "V set" assumes "A \<subseteq> elts \<omega>"   and "infinite A" shows "tp A = \<omega>"
Times_mset_monoL: fixes M :: "'a multiset"   and N :: "'a multiset"   and Z :: "'b multiset" assumes "M < N"   and "Z \<noteq> {#}" shows "M \<times># Z < N \<times># Z"
simps: fixes t1 :: "t"   and t2 :: "t"   and sx :: "sx option" shows "convert_cond t1 t2 sx = (t1 \<noteq> t2 \<and> (sx = None) = (is_float_t t1 \<and> is_float_t t2 \<or> is_int_t t1 \<and> is_int_t t2 \<and> t_length t1 < t_length t2))"
zs_def: shows "zs = filter (\<lambda>y. \<not> y \<le> x) xs"
s_eq: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set" assumes "kuhn_simplex p n base upd s" shows "s = kuhn_simplex.enum base upd ` {..n}"
tensor_gate: fixes m :: "nat"   and G1 :: "complex Matrix.mat"   and n :: "nat"   and G2 :: "complex Matrix.mat" assumes "gate m G1"   and "gate n G2" shows "gate (m + n) (G1 \<Otimes> G2)"
simps_1: fixes step :: "'a \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and A :: "('a \<Rightarrow> 'b) \<times> ('a \<times> 'b \<Rightarrow> 'c \<Rightarrow> 'd \<times> 'b)"   and s :: "'a \<times> 'b" shows "On_Off.config' step A s [] = s"
rtrancl_eta_Abs: fixes s :: "dB"   and s' :: "dB" assumes "s \<rightarrow>\<^sub>\<eta>\<^sup>* s'" shows "Abs s \<rightarrow>\<^sub>\<eta>\<^sup>* Abs s'"
base: fixes u :: "real"   and d :: "real"   and v :: "real" shows "geom_rand_walk u d v 0 = (\<lambda>w. v)"
monotone_def: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and r :: "('b \<times> 'b) set" shows "Tarski.monotone f A r \<equiv> \<forall>x\<in>A. \<forall>y\<in>A. (x, y) \<in> r \<longrightarrow> (f x, f y) \<in> r"
iTILL_Max: fixes n :: "nat" shows "Max [\<dots>n] = n"
assms_1: shows "x * f $ 0 = (1::'a)"
map_apfst_remdups': fixes f :: "'d \<Rightarrow> 'c"   and xs :: "('d \<times> 'e) list" shows "map (f \<circ> fst) (remdups' snd xs) = map fst (remdups' snd (map (apfst f) xs))"
mat_to_cols_list_def: fixes A :: "complex Matrix.mat" shows "mat_to_cols_list A = map (\<lambda>j. map (\<lambda>i. A $$ (i, j)) [0..<dim_row A]) [0..<dim_col A]"
irrefl: fixes a :: "'a list" shows "\<not> strict_prefix a a"
bdd_above_top: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and top :: "'a"   and A :: "'a set" assumes "class.order_top less_eq less top" shows "preorder.bdd_above less_eq A"
primeideal_def: fixes I :: "'e set"   and R :: "('e, 'f) ring_scheme" shows "primeideal I R \<equiv> ideal I R \<and> cring R \<and> primeideal_axioms I R"
degree_m_eq: fixes f :: "int poly"   and m :: "int" assumes "lead_coeff f mod m \<noteq> 0"   and "1 < m" shows "degree (poly_mod.Mp m f) = degree f"
abs_eq: fixes x :: "'d \<Rightarrow> 'd \<Rightarrow> 'c" shows "to_vec (of_fun x) = (\<chi>i. vec_lambda (x i))"
comm: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_ab_monoid_add plus less_eq less neutral" shows "plus a b = plus b a"
leastP_def: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" shows "leastP P x = (P x \<and> x <=* Collect P)"
upd_cs_snoc_Return_Cons: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and f :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option"   and cs :: "'a list"   and as :: "'a list"   and c' :: "'a"   and cs' :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"   and "CFG.upd_cs kind cs as = c' # cs'" shows "CFG.upd_cs kind cs (as @ [a]) = cs'"
fromEnum_def: fixes enum :: "'b list"   and enum_all :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and x :: "'b" assumes "class.enum enum enum_all enum_ex" shows "enum.fromEnum enum x \<equiv> the_index enum x"
less_kbo_def: fixes s :: "('c, 'd) Term.term"   and t :: "('c, 'd) Term.term" shows "less_kbo s t = fst (kbo t s)"
inv_union: fixes t1 :: "('a \<times> 'b) tree"   and t2 :: "('a \<times> 'b) tree" assumes "inv t1"   and "inv t2" shows "inv (local.union t1 t2)"
simps_1: fixes P :: "jvm_prog"   and h :: "heap"   and sh :: "sheap"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and C' :: "char list"   and Cs :: "char list list" shows "(P,h,sh \<turnstile>\<^sub>i (C, M, pc, Calling C' Cs)) = (let ins = instrs_of P C M in Called_context P (last (C' # Cs)) (ins ! pc) \<and> is_class P C')"
subst_syntactic_and_def: fixes P :: "'c \<Rightarrow> bool" shows "subst_syntactic_and P \<equiv> simple_syntactic_and P \<and> subst_syntactic_and_axioms P"
t_subset_Nodes: shows "set_of t \<subseteq> Nodes (Suc nb) levellista"
con: shows "contour_integral (circlepath z r) (\<lambda>u. f u / (u - w) ^ Suc (Suc k)) = X"
x'_def: shows "I x = I x' \<and> P x = P x' \<and> S x = S x' \<and> x' \<in> insert_messages M"
igSubstIPresIGWls_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igSubstIPresIGWls MOD \<equiv> \<forall>ys Y y s X. igWls MOD (asSort ys) Y \<and> igWls MOD s X \<longrightarrow> igWls MOD s (igSubst MOD ys Y y X)"
filterlim_at_withinI: fixes "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and c :: "'a"   and F :: "'b filter"   and A :: "'a set" assumes "class.topological_space open"   and "filterlim f (topological_space.nhds open c) F"   and "\<forall>\<^sub>F x in F. f x \<in> A - {c}" shows "filterlim f (topological_space.at_within open c A) F"
subst_sterm: fixes \<sigma> :: "('sf, 'sc, 'sz) subst"   and \<theta> :: "('sf, 'sz) trm"   and I :: "('sf, 'sc, 'sz) interp"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "TadmitF \<sigma> \<theta>"   and "\<And>i f'. SFunctions \<sigma> i = Some f' \<Longrightarrow> dfree f'" shows "sterm_sem I (Tsubst \<theta> \<sigma>) (fst \<nu>) = sterm_sem (local.adjoint I \<sigma> \<nu>) \<theta> (fst \<nu>)"
coeff_smult: fixes a :: "'a"   and p :: "'a poly"   and n :: "nat" shows "poly.coeff (Polynomial.smult a p) n = a * poly.coeff p n"
has_greatest_postfixpoint_def: fixes f :: "'a \<Rightarrow> 'a" shows "has_greatest_postfixpoint f \<equiv> \<exists>x. is_greatest_postfixpoint f x"
op_set_ndelete_spec: fixes x :: "'a"   and X :: "'a set" shows "op_set_ndelete x X = SPEC (\<lambda>R. R = X - {x} \<or> R = X)"
induct: fixes P :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree \<Rightarrow> 'c \<times> 'd \<Rightarrow> (('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree \<Rightarrow> bool"   and a0 :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a1 :: "'c \<times> 'd"   and a2 :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" assumes "\<And>t1 a1 uu_ uv_ t2 a2 t3 a3 t4. P t1 a1 (R uu_ (R uv_ t2 a2 t3) a3 t4)"   and "\<And>t1 a1 a2 uw_ t3 a3 ux_ t4. P t1 a1 (R uw_ \<langle>\<rangle> a2 (R ux_ t3 a3 t4))"   and "\<And>t1 a1 v vc vf vb a2 uw_ t3 a3 ux_ t4. P t1 a1 (R uw_ (B vf v vc vb) a2 (R ux_ t3 a3 t4))"   and "\<And>t1 a. P t1 a \<langle>\<rangle>"   and "\<And>t1 a v ve vb. P t1 a (B ve \<langle>\<rangle> v vb)"   and "\<And>t1 a va. P t1 a \<langle>\<langle>\<rangle>, va, \<langle>\<rangle>\<rangle>"   and "\<And>t1 a va v ve vh vd. P t1 a \<langle>\<langle>\<rangle>, va, B vh v ve vd\<rangle>"   and "\<And>t1 a vc vf vi ve v vh vb. P t1 a (B vh (B vi vc vf ve) v vb)"   and "\<And>t1 a vc vf vi ve va. P t1 a \<langle>B vi vc vf ve, va, \<langle>\<rangle>\<rangle>"   and "\<And>t1 a vc vf vi ve va v vh vl vg. P t1 a \<langle>B vi vc vf ve, va, B vl v vh vg\<rangle>"   and "\<And>t1 a v vc vf vb. P t1 a (B vf v vc vb)" shows "P a0 a1 a2"
hyps_1: fixes xs :: "'a rexp list" shows "(Pluss r = Some xs) = (Full \<notin> set xs \<and> xs = sorted_list_of_set (toplevel_summands r - {Zero}))"
pfun_sum_def: fixes f :: "('b, 'a) pfun" shows "pfun_sum f = sum (pfun_app f) (pdom f)"
extend_guarantees_imp_guarantees: fixes F :: "'a program"   and X :: "'a program set"   and Y :: "'a program set" assumes "extend h F \<in> extend h ` X guarantees extend h ` Y" shows "F \<in> X guarantees Y"
dghm_id_ArrMap_vrange: fixes \<CC> :: "V" shows "\<R>\<^sub>\<circ> (dghm_id \<CC>\<lparr>ArrMap\<rparr>) = \<CC>\<lparr>Arr\<rparr>"
word_n1_ge: fixes y :: "'a word" shows "y \<le> - 1"
AE_lborel_inner_neq: fixes k :: "'a"   and c :: "real" assumes "k \<in> Basis" shows "AE x in lborel. x \<bullet> k \<noteq> c"
distinct_1: fixes x1 :: "('a, 'b) hd"   and x21 :: "('a, 'b) tm"   and x22 :: "('a, 'b) tm" shows "Hd x1 \<noteq> App x21 x22"
that: fixes w :: "complex" assumes "w \<noteq> 0"   and "z = w ^ n" shows "thesis"
h2c: shows "all_list_constr (subsets_smash (length qs1) subsets1 subsets2) (length (qs1 @ qs2))"
simps: fixes a1 :: "(char list \<times> trm) list"   and a2 :: "(char list \<times> trm) list" shows "(a1 [\<longmapsto>] a2) = ((\<exists>t t' l fs. a1 = (l, t) \<Colon> fs \<and> a2 = (l, t') \<Colon> fs \<and> t \<longmapsto> t') \<or> (\<exists>v fs fs' l. a1 = (l, v) \<Colon> fs \<and> a2 = (l, v) \<Colon> fs' \<and> v \<in> value \<and> fs [\<longmapsto>] fs'))"