hcju/isabelleps huggingface dataset:
residual_network_sel_1: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a"   and y :: "'a" assumes "countable_network \<Delta>" shows "edge (countable_network.residual_network \<Delta> f) x y = (edge \<Delta> x y \<or> edge \<Delta> y x \<and> y \<noteq> source \<Delta>)"
nhdE: fixes U :: "'a llist set"   and x :: "'a llist" assumes "U \<in> nhds x" obtains m :: "'a llist set" where "U \<subseteq> carrier" and "m open" and "x \<in> m" and "m \<subseteq> U"
Cons_3: shows "x \<in> set (take n (y # ys))"
\<kappa>\<^sub>i_def: shows "M.sumlist (map (\<lambda>j. - \<mu> i j \<cdot>\<^sub>v gso j) [0..<l]) = M.sumlist (map (\<lambda>j. \<kappa>\<^sub>i j \<cdot>\<^sub>v fs ! j) [0..<l])"
Cons_replicate_eq: fixes x :: "'f"   and xs :: "'f list"   and n :: "nat"   and y :: "'f" shows "(x # xs = replicate n y) = (x = y \<and> 0 < n \<and> xs = replicate (n - 1) x)"
R_pow_int: fixes n :: "int"   and m :: "nat" shows "(n, 0) [^]\<^bsub>R\<^esub> m = (n ^ m, 0)"
lim_3: fixes x :: "'a \<times> 'b" assumes "x \<in> space (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>i. s i x) \<longlonglongrightarrow> f x"
closed_\<G>\<^sub>F\<^sub>G: fixes \<phi> :: "'b ltl"   and w :: "nat \<Rightarrow> 'b set" shows "closed (\<G>\<^sub>F\<^sub>G \<phi> w) w"
MtransC_Refl: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and cf :: "('a, 'c) com \<times> 'b" shows "PL.MtransC tval aval cf cf"
ratrel_def: fixes x :: "int \<times> int"   and y :: "int \<times> int" shows "(x \<approx> y) = (snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x)"
FDOM_pre_subset_FDOM_valid_states: fixes p :: "('a, 'b) fmap"   and e :: "('a, 'c) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) set"   and s :: "('a, 'd) fmap" assumes "(p, e) \<in> PROB"   and "s \<in> valid_states PROB" shows "fmdom' p \<subseteq> fmdom' s"
msgrelp_msgrel_eq: shows "msgrelp = (\<sim>)"
fp_cop_F_rejected_inv: fixes ds :: "'b set" shows "cop_F_rejected_inv ds (fp_cop_F ds)"
rel_spmf_mono_strong: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c spmf"   and g :: "'d spmf"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_spmf A f g"   and "\<And>x y. \<lbrakk>A x y; x \<in> set_spmf f; y \<in> set_spmf g\<rbrakk> \<Longrightarrow> B x y" shows "rel_spmf B f g"
finite_entropy_integrable: fixes S :: "'d measure"   and X :: "'a \<Rightarrow> 'd"   and Px :: "'d \<Rightarrow> real" assumes "finite_entropy S X Px" shows "integrable S (\<lambda>x. Px x * log b (Px x))"
ptr_children: shows "h \<turnstile> get_child_nodes parent \<rightarrow>\<^sub>r ptr_children"
distinct_24: fixes x71 :: "trm"   and x72 :: "char list"   and x21 :: "POPLmarkRecord.type"   and x22 :: "trm" shows "x71..x72 \<noteq> (\<lambda>:x21. x22)"
ih_17: shows "getSATFlag state' = UNDEF \<longrightarrow> InvariantEquivalentZL (getF state') (getM state') F0'"
sw_sym: fixes xs :: "'c"   and x :: "'b"   and y :: "'b"   and zs :: "'c"   and z :: "'b" shows "z @zs[x \<and> y]_xs = z @zs[y \<and> x]_xs"
hom: shows "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
ltlc_to_ltln_atoms: fixes \<phi> :: "'a ltlc" shows "atoms_ltln (ltlc_to_ltln \<phi>) = atoms_ltlc \<phi>"
R2: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and x :: "'c"   and xs :: "'c list"   and ys :: "'c list"   and X :: "('c, 'd) ipurge_rec" assumes "\<lparr>Pol = I, Map = D, Dom = u, In = x # xs, Out = ys\<rparr> \<in> ipurge_tr_rev_t_set X"   and "D x \<notin> sources I D u (x # xs)" shows "\<lparr>Pol = I, Map = D, Dom = u, In = ipurge_tr I D (D x) xs, Out = ys\<rparr> \<in> ipurge_tr_rev_t_set X"
Pow_mono: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B" shows "Pow A \<subseteq> Pow B"
1: fixes y :: "'a list"   and z :: "'b list" assumes "length y = length z" shows "PROP P (y || z)"
prefs_wf'_1: fixes i :: "'agent" assumes "i \<in> agents" shows "total_preorder_on alts (R2 i)"
\<tau>move1SeqThrow: fixes P :: "'b prog"   and h :: "'heap"   and a :: "'addr"   and e :: "('c, 'd, 'addr) exp" shows "\<tau>move1 P h (Throw a;; e)"
obj_ty_def: fixes obj :: "obj" shows "obj_ty obj \<equiv> Class (fst obj)"
D_if_big_step: fixes c :: "com"   and s :: "char list \<Rightarrow> int"   and t :: "char list \<Rightarrow> int" assumes "(c, s) \<Rightarrow> t" shows "(s, t) \<in> D c"
cmp: shows "comparator (pot_comp (\<lambda>x y. lex_comp y x))"
smult_diff_left: fixes a :: "'a"   and b :: "'a"   and p :: "'a poly" shows "Polynomial.smult (a - b) p = Polynomial.smult a p - Polynomial.smult b p"
bdia_seq_var: fixes x :: "'a"   and p :: "'a"   and p' :: "'a"   and y :: "'a"   and q :: "'a" assumes "\<langle>x| p \<le> p'"   and "\<langle>y| p' \<le> q" shows "\<langle>x \<cdot> y| p \<le> q"
inv_implications_nonneg_def: fixes c :: "('a, 'b, 'c) configuration_scheme" shows "inv_implications_nonneg c = (\<forall>loc t. 0 \<le> zcount (c_imp c loc) t)"
less_1: fixes y :: "nat" assumes "y < i" shows "local.bad (local.worst_chain (Suc y))"
mpt_power: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "mpt M T" shows "mpt M (T ^^ n)"
finite_UNIV_num1_def: shows "finite_UNIV = Phantom(num1) True"
linear_span_sc_closed: fixes A :: "'b set"   and H :: "'a set"   and r :: "'b"   and x :: "'a" assumes "ideal R A"   and "H \<subseteq> carrier M"   and "r \<in> carrier R"   and "x \<in> linear_span R M A H" shows "r \<cdot>\<^sub>s x \<in> linear_span R M A H"
injectivity_1: fixes X :: "'a set"   and Y :: "'a set"   and k :: "nat"   and n :: "nat"   and k' :: "nat"   and xs :: "'a list"   and xs' :: "'a list"   and ys :: "'a list"   and ys' :: "'a list"   and zs :: "'a list"   and zs' :: "'a list" assumes "X \<inter> Y = {}"   and "k \<in> {0..n} \<and> k' \<in> {0..n}"   and "(length xs = k \<and> distinct xs \<and> set xs \<subseteq> X) \<and> length xs' = k' \<and> distinct xs' \<and> set xs' \<subseteq> X"   and "(length ys = n - k \<and> distinct ys \<and> set ys \<subseteq> Y) \<and> length ys' = n - k' \<and> distinct ys' \<and> set ys' \<subseteq> Y"   and "interleavings xs ys zs \<and> interleavings xs' ys' zs'"   and "zs = zs'" shows "k = k'"
UNIV_rw: shows "UNIV = insert i (UNIV - {i})"
3: shows "Cop_S.iso (Cop_S.MkArr (\<Phi>'.Y a) (\<Phi>'.Y a') \<Phi>\<Psi>'.map)"
cong_def: fixes a :: "'c"   and c :: "'c"   and b :: "'c" shows "cong a c b = (\<exists>k. a - c = b * k)"
ground_range_vars: fixes s :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "ground (subst_range s)" shows "range_vars s = {}"
bound: fixes s :: "'a" assumes "guard s" shows "f s \<le> bound"
fsmult_smult_domain_comm: fixes r :: "('f, 'g) aezfun"   and v :: "'v"   and a :: "'f" assumes "r \<in> FG"   and "v \<in> V" shows "a \<sharp>\<cdot> r \<cdot> v = r \<cdot> a \<sharp>\<cdot> v"
input_not_constrained_aval_swap_inputs: fixes a :: "vname aexp"   and v :: "nat"   and i :: "value list"   and c :: "nat \<Rightarrow>f value option"   and x :: "value" assumes "\<not> aexp_constrains a (V (I v))" shows "aval a (join_ir i c) = aval a (join_ir (i[v := x]) c)"
assms_1: fixes b :: "'a set" assumes "b \<in># \<B>" shows "b \<subseteq> V"
Extend_satisfies_bounded_decr: fixes k :: "'k"   and i :: "nat"   and \<phi> :: "('a, 'k) aformula"   and \<AA> :: "'i"   and P :: "'v" assumes "\<not> local.find k i \<phi>"   and "LEQ k i (#\<^sub>V \<AA>)"   and "len P \<le> Length \<AA>" shows "(Extend k i \<AA> P \<Turnstile>\<^sub>b \<phi>) = (\<AA> \<Turnstile>\<^sub>b decr k i \<phi>)"
gba_to_idx_ext_correct: fixes G :: "('b, 'c, 'd) gba_rec_scheme"   and ecnv :: "('b, 'c, 'd) gba_rec_scheme \<Rightarrow> 'e" assumes "gba G"   and "\<And>A. A \<in> gbg_F G \<Longrightarrow> finite A" shows "gba_to_idx_ext ecnv G \<le> SPEC (\<lambda>G'. igba.accept G' = gba.accept G \<and> g_V G' = g_V G \<and> g_E G' = g_E G \<and> g_V0 G' = g_V0 G \<and> igba_rec.more G' = ecnv G \<and> igba G')"
differentiable_norm_compose_at: fixes g :: "'c \<Rightarrow> 'b"   and x :: "'c"   and s :: "'c set" assumes "g x \<noteq> (0::'b)"   and "g differentiable at x within s" shows "(\<lambda>x. norm (g x)) differentiable at x within s"
full_functor_axioms: shows "full_functor (\<cdot>\<^sub>S) (\<cdot>\<^sub>S) L"
non_empty_keys: fixes t :: "('b, 'c) RBT.rbt" assumes "t \<noteq> RBT.empty" shows "RBT.keys t \<noteq> []"
simps: fixes args :: "Monitor.args"   and cur :: "nat"   and nt :: "nat"   and gc :: "nat"   and maskL :: "bool list"   and maskR :: "bool list"   and data_prev :: "(nat \<times> 'a option list set) queue"   and data_in :: "(nat \<times> 'a option list set) queue"   and tuple_in :: "('a option list, nat) mapping"   and tuple_since :: "('a option list, nat) mapping"   and ys :: "(nat \<times> 'a option list set) list" shows "valid_mmsaux args cur (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since) ys = (args_L args \<subseteq> args_R args \<and> maskL = join_mask (args_n args) (args_L args) \<and> maskR = join_mask (args_n args) (args_R args) \<and> (\<forall>(t, y)\<in>set ys. table (args_n args) (args_R args) y) \<and> table (args_n args) (args_R args) (Mapping.keys tuple_in) \<and> table (args_n args) (args_R args) (Mapping.keys tuple_since) \<and> Ball (\<Union> (snd ` set (linearize data_prev))) (wf_tuple (args_n args) (args_R args)) \<and> cur = nt \<and> ts_tuple_rel (set ys) = {tas \<in> ts_tuple_rel (set (linearize data_prev) \<union> set (linearize data_in)). valid_tuple tuple_since tas} \<and> sorted (map fst (linearize data_prev)) \<and> (\<forall>t\<in>fst ` set (linearize data_prev). t \<le> nt \<and> nt - t < left (args_ivl args)) \<and> sorted (map fst (linearize data_in)) \<and> (\<forall>t\<in>fst ` set (linearize data_in). t \<le> nt \<and> left (args_ivl args) \<le> nt - t) \<and> (\<forall>as. Mapping.lookup tuple_in as = safe_max (fst ` {tas \<in> ts_tuple_rel (set (linearize data_in)). valid_tuple tuple_since tas \<and> as = snd tas})) \<and> (\<forall>as\<in>Mapping.keys tuple_since. case Mapping.lookup tuple_since as of Some t \<Rightarrow> t \<le> nt))"
Smith_1xn_aux_Q_carrier: fixes S' :: "'a mat"   and Q' :: "'a mat"   and i :: "nat"   and A :: "'a mat"   and S :: "'a mat"   and Q :: "'a mat"   and n :: "nat" assumes "(S', Q') = Smith_1xn_aux i A (S, Q)"   and "A \<in> carrier_mat 1 n"   and "Q \<in> carrier_mat n n" shows "Q' \<in> carrier_mat n n"
is_imgu_def: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set" shows "is_imgu \<sigma> E = (\<sigma> \<in> unifiers E \<and> (\<forall>\<tau>\<in>unifiers E. \<tau> = comp_subst_abbrev \<sigma> \<tau>))"
find_sgas_def: fixes ps :: "real poly list" shows "find_sgas ps = (let r = coprime_r ps in find_consistent_signs_at_roots r ps @ find_sgas_aux ps)"
right_mult_one_mat: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "A * 1\<^sub>m nc = A"
mult_left_isotone: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd"   and z :: "'c \<times> 'c \<Rightarrow> 'd" assumes "x \<preceq> y" shows "x \<odot> z \<preceq> y \<odot> z"
resultant_as_nonzero_poly: fixes p :: "'b poly"   and q :: "'b poly" assumes "0 < degree p"   and "0 < degree q" shows "\<exists>p' q'. degree p' < degree q \<and> degree q' < degree p \<and> [:resultant p q:] = p' * p + q' * q \<and> p' \<noteq> 0 \<and> q' \<noteq> 0"
lm02: fixes X :: "'a set" assumes "finite X" shows "trivial X = (card X \<le> 1)"
IH: assumes "v \<in> vars\<^sub>s\<^sub>t S" shows "\<exists>S' x S''. S = S' @ x # S'' \<and> v \<in> vars\<^sub>s\<^sub>t\<^sub>p x"
field_differentiable_bound: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'a"   and f' :: "'a \<Rightarrow> 'a"   and B :: "real"   and x :: "'a"   and y :: "'a" assumes "convex S"   and "\<And>z. z \<in> S \<Longrightarrow> (f has_field_derivative f' z) (at z within S)"   and "\<And>z. z \<in> S \<Longrightarrow> norm (f' z) \<le> B"   and "x \<in> S"   and "y \<in> S" shows "norm (f x - f y) \<le> B * norm (x - y)"
False: shows "t \<oplus> lt f \<notin> keys (p + r)"
measure_translation_subtract: fixes a :: "'a"   and S :: "'a set" shows "Sigma_Algebra.measure lebesgue ((\<lambda>x. x - a) ` S) = Sigma_Algebra.measure lebesgue S"
balanced_alt_def_all: shows "\<forall>ps\<subseteq>\<V>. int (card ps) = 2 \<longrightarrow> int (\<B> index ps) = 1"
eq_app_right_def: fixes L :: "'b list set" shows "eq_app_right L \<equiv> {(u, v). \<forall>w. (u @ w \<in> L) = (v @ w \<in> L)}"
induct: fixes P :: "('a \<Rightarrow> bool) \<Rightarrow> 'a queue \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> bool"   and a1 :: "'a queue" assumes "\<And>f q. (\<And>x y x2. \<lbrakk>(x, y) = safe_hd q; x = Some x2; f x2\<rbrakk> \<Longrightarrow> P f (tl_queue y)) \<Longrightarrow> P f q" shows "P a0 a1"
ccpo: shows "class.ccpo heap.lub_fun heap.le_fun (mk_less heap.le_fun)"
flowsfix: shows "flows_fix (generate_valid_stateful_policy_IFSACS_2 G M edgesList) = edges G"
finite_polyhedron_extreme_points: fixes S :: "'b set" assumes "polyhedron S" shows "finite {v. v extreme_point_of S}"
\<omega>_minus_of_nat: fixes n :: "nat" shows "\<omega> - of_nat n = \<omega>"
simE_1: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and a :: "name"   and x :: "name"   and Q' :: "pi" assumes "P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"   and "Q \<longmapsto> a<\<nu>x> \<prec> Q'"   and "x \<sharp> P" shows "\<exists>P'. P \<Longrightarrow>a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel"
exception_\<I>_def: fixes \<I> :: "('y, 'z) \<I>" shows "exception_\<I> \<I> = map_\<I> id exception_of_option (stop_\<I> \<I>)"
dim_update_vec: fixes v :: "'c vec"   and i :: "nat"   and a :: "'c" shows "dim_vec (v |\<^sub>v i \<mapsto> a) = dim_vec v"
Ix: shows "M \<le> Gromov_product_at px x y"
pseudo_ordering_axioms: shows "pseudo_ordering UNIV (\<le>) (<)"
dom_src: fixes \<mu> :: "'a arrow_of_spans_data" assumes "arr \<mu>" shows "local.dom (src \<mu>) = src \<mu>"
pgalleryD_distinct: fixes xs :: "'a set list" assumes "pgallery xs" shows "distinct xs"
IH_1: fixes t :: "'a bro"   and a :: "'a"   and t' :: "'a bro" assumes "t \<in> B h"   and "split_min t = Some (a, t')" shows "t' \<in> T h"
c_futures_failures: fixes yps :: "('a \<times> 'b) list"   and Y :: "('a \<times> 'b) set"   and step :: "'c \<Rightarrow> 'a \<Rightarrow> 'c"   and out :: "'c \<Rightarrow> 'a \<Rightarrow> 'b"   and s\<^sub>0 :: "'c"   and xps :: "('a \<times> 'b) list" shows "((yps, Y) \<in> futures (c_process step out s\<^sub>0) xps) = ((xps @ yps, Y) \<in> c_failures step out s\<^sub>0)"
spawn_no_mon: fixes u :: "'n"   and p :: "'p"   and v :: "'n" assumes "(u, Spawn p, v) \<in> edges fg" shows "mon fg p = {}"
2: fixes x :: "nat"   and y :: "nat" assumes "x < y" shows "(0 ## (Suc \<circ> f)) x < (0 ## (Suc \<circ> f)) y"
\<delta>1''': shows "\<delta>1''' \<in> seq ars"
loc2_axioms_def: fixes s :: "(nat \<times> form) list"   and mo :: "U set \<times> (nat \<Rightarrow> U list \<Rightarrow> bool)" shows "loc2_axioms s mo \<equiv> mo = model s"
symclp_greater: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "r \<le> symclp r"
cong_nat_imp_eq: fixes m :: "nat"   and x :: "nat"   and a :: "nat"   and y :: "nat" assumes "0 < m"   and "x \<in> {a..<a + m}"   and "y \<in> {a..<a + m}"   and "[x = y] (mod m)" shows "x = y"
h1: shows "set (characterize_consistent_signs_at_roots_copr p (qs1 @ qs2)) \<subseteq> set (get_signs (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2))))"
vec_elements_h_def': fixes v :: "('b, 'c) Finite_Cartesian_Product.vec" shows "vec_elements_h v = {v $h i |i. True}"
generate_img: fixes K :: "'a set" assumes "K \<subseteq> carrier G" shows "generate H (h ` K) = h ` generate G K"
lisetI: fixes i :: "nat"   and w :: "'a llist"   and A :: "'a set" assumes "enat i < llength w"   and "w ?! i \<in> A" shows "i \<in> liset A w"
Tn_integral_preserving_1: fixes f :: "'a \<Rightarrow> 'c"   and n :: "nat" assumes "integrable M f" shows "integrable M (\<lambda>x. f ((T ^^ n) x))"
mem_set_indexed_members: fixes i :: "nat"   and x :: "'b"   and s :: "'b fset" shows "((i, x) \<in> set (indexed_members s)) = (x |\<in>|\<^bsub>i\<^esub> s)"
stored_reused_aux_is_generalised_output_of: fixes t' :: "transition"   and t :: "transition"   and mr :: "nat"   and mp :: "nat"   and p :: "nat"   and r :: "nat" assumes "stored_reused_aux t' t mr mp = Some (p, r)" shows "is_generalised_output_of t' t p r"
Rep_cfun_inverse: fixes x :: "'e \<rightarrow> 'f" shows "Abs_cfun (Rep_cfun x) = x"
downset_set_idem: shows "\<Down> \<circ> \<Down> = \<Down>"
inject_2: fixes x21 :: "'f"   and x22 :: "'e trm list"   and y21 :: "'f"   and y22 :: "'e trm list" shows "(Pr x21 x22 = Pr y21 y22) = (x21 = y21 \<and> x22 = y22)"
mono: shows "SemilatAlg.mono r step (length \<tau>s) A"
U': fixes ts :: "'t list"   and f :: "'op" assumes "\<forall>s\<in>set ts. inv' s"   and "length ts = arity f" shows "U' f ts = U f (map hom ts)"
similar_mat_wit_refl: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "similar_mat_wit A A (1\<^sub>m n) (1\<^sub>m n)"
process_mru_def: fixes vs :: "'b \<Rightarrow> 'a \<Rightarrow> 'c option" shows "process_mru vs \<equiv> \<lambda>a. mru_of_set vs {a}"
ex_un_null: shows "\<exists>!n. \<forall>f. J.comp n f = n \<and> J.comp f n = n"
zero_smallest: fixes r :: "('c \<times> 'c) set"   and j :: "'c" assumes "wo_rel r"   and "j \<in> Field r" shows "(zero r, j) \<in> r"
last_index_drop: fixes xs :: "'c list"   and x :: "'c"   and i :: "nat" assumes "last_index xs x < i" shows "x \<notin> set (drop i xs)"
snapshot_algorithm_is_correct: fixes t :: "('a, 'b, 'c) event list" assumes "trace init t final" shows "\<exists>t' i. trace init t' final \<and> t' <~~> t \<and> state_equal_to_snapshot (S t' i) final \<and> i \<le> length t'"
F_IN_OUT: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" assumes "f \<in> Field leq"   and "x \<notin> B \<Gamma>" shows "d_IN f x \<le> d_OUT f x"
put\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_ptr_in_heap: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and shadow_root :: "('o, 'm, 'n, 'w option) RShadowRoot_scheme"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "put shadow_root_ptr shadow_root h = h'" shows "shadow_root_ptr |\<in>| shadow_root_ptr_kinds h'"
runit_char_4: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "prebicategory_with_homs_and_units V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "\<exists>!\<mu>. partial_magma.in_hom V \<mu> (H f (src f)) f \<and> H \<mu> (src f) = V (H f (\<i> (src f))) (\<a> f (src f) (src f))"
elims: fixes x :: "float interval option"   and xa :: "float interval option"   and xb :: "float interval \<Rightarrow> float interval \<Rightarrow> float interval option"   and y :: "float interval option" assumes "lift_bin x xa xb = y" obtains ivl1 :: "float interval" and ivl2 :: "float interval" and f :: "float interval \<Rightarrow> float interval \<Rightarrow> float interval option" where "x = Some ivl1" and "xa = Some ivl2" and "xb = f__" and "y = f__ ivl1 ivl2"   | b :: "float interval option" and f :: "float interval \<Rightarrow> float interval \<Rightarrow> float interval option" where "x = None" and "xa = b" and "xb = fa__" and "y = None"   | a :: "float interval option" and f :: "float interval \<Rightarrow> float interval \<Rightarrow> float interval option" where "x = a" and "xa = None" and "xb = f" and "y = None"
ln_lower_3_def: shows "ln_lower_3 \<equiv> \<lambda>x. - ln_upper_3 (inverse x)"
AbGroup_set_plus: fixes H :: "'b set"   and G :: "'b set" assumes "AbGroup H"   and "AbGroup G" shows "AbGroup (H + G)"
H_awalkI: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d"   and u :: "'c"   and p :: "'d list"   and v :: "'c" assumes "bidel_arc_not_biconnected G M a"   and "pre_digraph.awalk G u p v"   and "{a, edge_rev M a} \<inter> set p = {}" shows "pre_digraph.awalk (bidel_arc.H G M a) u p v"
tabulate_def: shows "Mapping.tabulate \<equiv> map_fun id (map_fun id Mapping) (\<lambda>ks f. map_of (map (\<lambda>k. (k, f k)) ks))"
sba2_bot_unit: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> - (y \<squnion> - y) = x"
measurable_completion2: fixes f :: "'a \<Rightarrow> 'c"   and N :: "'c measure" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "null_sets N \<subseteq> null_sets (distr M N f)" shows "f \<in> M \<rightarrow>\<^sub>M completion N"
has_integral_alt': fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and s :: "'a set" shows "(f has_integral i) s = ((\<forall>a b. (\<lambda>x. if x \<in> s then f x else (0::'b)) integrable_on cbox a b) \<and> (\<forall>e>0. \<exists>B>0. \<forall>a b. ball (0::'a) B \<subseteq> cbox a b \<longrightarrow> norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else (0::'b)) - i) < e))"
cases: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and a1 :: "('d, 'e, 'f) configuration"   and a2 :: "('d, 'e, 'f) event list"   and a3 :: "('d, 'e, 'f) configuration" assumes "distributed_system channel"   and "distributed_system.trace channel trans send recv a1 a2 a3" obtains c :: "('d, 'e, 'f) configuration" where "a1 = c__" and "a2 = []" and "a3 = c__"   | c :: "('d, 'e, 'f) configuration" and ev :: "('d, 'e, 'f) event" and c' :: "('d, 'e, 'f) configuration" and t :: "('d, 'e, 'f) event list" and c'' :: "('d, 'e, 'f) configuration" where "a1 = c" and "a2 = ev # t" and "a3 = c''" and "distributed_system.next channel trans send recv c ev c'" and "distributed_system.trace channel trans send recv c' t c''"
word_succ_p1: fixes a :: "'a Word.word" shows "word_succ a = a + 1"
j0_not_eq_j1: shows "parallel_pair.j0 \<noteq> parallel_pair.j1"
g_nonneg: fixes x :: "real" assumes "x\<^sub>0 \<le> x" shows "0 \<le> g x"
c: shows "c - a = k *\<^sub>R (b - a)"
hyps_2: shows "pi \<bullet> (\<parallel><A>\<parallel>) = \<parallel><A>\<parallel>"
step_abs: fixes s :: "'a"   and s' :: "'a" assumes "s \<in> wa_invar WC"   and "s \<in> wa_cond WC"   and "(s, s') \<in> wa_step WC" shows "(\<alpha>1 s, \<alpha>1 s') \<in> wa_step WB"
bounded_subset_ballD: fixes S :: "'b set"   and x :: "'b" assumes "bounded S" shows "\<exists>r>0. S \<subseteq> ball x r"
sympD: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b"   and a :: "'b" assumes "symp r"   and "r b a" shows "r a b"
elims_1: fixes x :: "'b literal"   and y :: "bool" assumes "positive_literal x = y" obtains xa :: "'b equation" where "x = Pos xa__" and "y"   | xa :: "'b equation" where "x = Neg xa" and "\<not> y"
get_map_def: fixes M :: "('a \<times> 'b) set"   and k :: "'a" shows "get_map M k = the_elem (Set.filter (\<lambda>(k', uu_). k = k') M)"
discount_factor_constant_time_pos: fixes r :: "real"   and n :: "nat" assumes "- 1 < r" shows "\<exists>c>0. \<forall>w\<in>space M. discount_factor r n w = c"
assms_4: shows "2 * card P\<^sub>2 \<le> card (\<Union> P\<^sub>2)"
dg_prod_3_Dom_app: fixes f :: "V"   and f' :: "V"   and f'' :: "V" assumes "[f, f', f'']\<^sub>\<circ> \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G\<^sub>3 \<BB> \<times>\<^sub>D\<^sub>G\<^sub>3 \<CC>)\<lparr>Arr\<rparr>" shows "(\<AA> \<times>\<^sub>D\<^sub>G\<^sub>3 \<BB> \<times>\<^sub>D\<^sub>G\<^sub>3 \<CC>)\<lparr>Dom\<rparr> \<lparr>f, f', f''\<rparr>\<^sub>\<bullet> = [\<AA>\<lparr>Dom\<rparr>\<lparr>f\<rparr>, \<BB>\<lparr>Dom\<rparr>\<lparr>f'\<rparr>, \<CC>\<lparr>Dom\<rparr>\<lparr>f''\<rparr>]\<^sub>\<circ>"
that: fixes x :: "'b" assumes "m = Inr x" shows "thesis"
orderI: fixes a :: "'c"   and b :: "'c" assumes "a = inf a b" shows "a \<le> b"