hcju/isabelleps huggingface dataset:
med_progress_def: fixes r :: "fid_t \<Rightarrow> Runs.var set option" shows "med_progress r \<equiv> \<lambda>R. option_inter {xnx, xny, xgnx, xgny, xsk, xEnd} (r R)"
roolw: fixes f :: "'a" assumes "f \<in> Hom A B" shows "Id A \<in> Hom A (Dom f)"
FD_D: fixes P :: "'b process"   and Q :: "'b process" assumes "P \<sqsubseteq>\<^sub>F\<^sub>D Q" shows "P \<sqsubseteq>\<^sub>D Q"
order_iff_strict: fixes b :: "'a multiset"   and a :: "'a multiset" shows "(b \<subseteq># a) = (b \<subset># a \<or> a = b)"
add_independentS_append_conv_double_sum: fixes As :: "'a set list"   and Bs :: "'a set list" assumes "add_independentS (As @ Bs)" shows "add_independentS [\<Sum>X\<leftarrow>As. X, \<Sum>X\<leftarrow>Bs. X]"
EquivI: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> in v] \<Longrightarrow> [\<psi> in v]"   and "[\<psi> in v] \<Longrightarrow> [\<phi> in v]" shows "[\<phi> \<^bold>\<equiv> \<psi> in v]"
Cd_greatest: fixes d :: "'b"   and X :: "'a set" shows "Bossy.Cd d X = greatest (BPd d) (X \<inter> Field (BPd d))"
D: shows "prob_space (distr (uniform_measure M {\<omega> \<in> space M. X n \<omega> = s}) (count_space UNIV) (X (Suc n)))"
wf: shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f\<^sub>B B"
\<eta>\<epsilon>': shows "adjoint_equivalence_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>'"
substAbs_Abs: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and x :: "'var" assumes "good X"   and "good Y" shows "Abs xs x X $[Y / x]_xs = Abs xs x X"
simps_2: fixes r :: "ref_ty" shows "default_val (RefT r) = Null"
greedy_conforming_path_properties_4: fixes v0 :: "'a"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a" assumes "v0 \<in> V"   and "strategy p \<sigma>"   and "strategy p** \<sigma>'" shows "maximal_path (greedy_conforming_path p \<sigma> \<sigma>' v0)"
SUP_upper2: fixes Inf :: "'h set \<Rightarrow> 'h"   and Sup :: "'h set \<Rightarrow> 'h"   and inf :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and sup :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and bot :: "'h"   and top :: "'h"   and i :: "'i"   and A :: "'i set"   and u :: "'h"   and f :: "'i \<Rightarrow> 'h" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A"   and "less_eq u (f i)" shows "less_eq u (Sup (f ` A))"
delta_min_append: fixes as :: "(QDelta \<times> QDelta) list"   and bs :: "(QDelta \<times> QDelta) list" shows "\<delta>_min (as @ bs) = min (\<delta>_min as) (\<delta>_min bs)"
dests_vD_inc_sqn: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PAodv-:15, PPkt-:7, PRreq-:11, PRreq-:24, PRrep-:10} \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> vD (rt \<xi>) \<and> the (dests \<xi> ip) = inc (sqn (rt \<xi>) ip))) \<and> (l = PRerr-:1 \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> vD (rt \<xi>) \<and> sqn (rt \<xi>) ip < the (dests \<xi> ip))))"
inj_def: fixes f :: "'ag \<Rightarrow> 'ah" shows "inj f = (\<forall>x y. f x = f y \<longrightarrow> x = y)"
adjacent_nodes_ss_V: fixes u :: "nat" shows "adjacent_nodes u \<subseteq> V"
limsup_infinity_imp_Inf_many: fixes f :: "nat \<Rightarrow> ereal" assumes "limsup f = \<infinity>" shows "\<forall>m. \<exists>\<^sub>\<infinity>i. ereal m < f i"
Chains_def: fixes r :: "('a \<times> 'a) set" shows "Chains r = {C. \<forall>a\<in>C. \<forall>b\<in>C. (a, b) \<in> r \<or> (b, a) \<in> r}"
ccw'_sortedP_polychain_of01_1: fixes u :: "real \<times> real"   and x0 :: "real \<times> real" assumes "ccw'.sortedP 0 [u]" shows "ccw'.sortedP x0 (map snd (polychain_of x0 [u]))"
f_the_inv_into_f: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and y :: "'b" assumes "inj_on f A"   and "y \<in> f ` A" shows "f (the_inv_into A f y) = y"
assms_3: shows "p \<equiv> Poly (map (\<lambda>k. c (N - k)) [0..<Suc N])"
lprod_list: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ah :: "'b list"   and at :: "'b list"   and bh :: "'b list"   and bt :: "'b list"   and a :: "'b"   and b :: "'b" assumes "lprodp R (ah @ at) (bh @ bt)"   and "R a b \<or> a = b" shows "lprodp R (ah @ a # at) (bh @ b # bt)"
ab_spec_add_basis_sorted: fixes rel :: "nat \<times> 'h \<Rightarrow> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> bool" shows "ab_spec (add_basis_sorted rel)"
mid_le_upper_2: fixes x :: "float interval" shows "real_of_float (mid x) \<le> real_of_float (upper x)"
mwb_UNIV_src_is_vwb_lens: fixes X :: "'a \<Longrightarrow> 'b" assumes "mwb_lens X"   and "\<S>\<^bsub>X\<^esub> = UNIV" shows "vwb_lens X"
simps_1: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and h :: "'a"   and a :: "'b" shows "heap_base.typeof_h typeof_addr h (Addr a) = map_option ty_of_htype (typeof_addr h a)"
f_fi: fixes y :: "'v" assumes "y \<in> f ` V" shows "f (fi y) = y"
leD: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and y :: "'f"   and x :: "'f" assumes "class.order less_eq less"   and "less_eq y x" shows "\<not> less x y"
r: shows "set (remove1 \<one> gs) = set gs - {\<one>}"
pre_if_regular_and_not_post: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat" assumes "regular_event (t ! i)"   and "\<not> postrecording_event t i"   and "i < length t" shows "prerecording_event t i"
match_raw_ternary: fixes \<beta> :: "'a \<Rightarrow> 'b \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> 'b \<Rightarrow> bool"   and expr :: "'a"   and a :: "action"   and p :: "'b" shows "matches (\<beta>, \<alpha>) (Match expr) a p = (case \<beta> expr p of TernaryTrue \<Rightarrow> True | TernaryFalse \<Rightarrow> False | TernaryUnknown \<Rightarrow> \<alpha> a p)"
eigenvalue_root_char_poly: fixes A :: "'c mat"   and n :: "nat"   and k :: "'c" assumes "A \<in> carrier_mat n n" shows "eigenvalue A k = (poly (char_poly A) k = (0::'c))"
tt_plus_eqI: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0"   and "punit.tt p \<prec> punit.tt q" shows "punit.tt (p + q) = punit.tt p"
arc_to_block_is_inj_out_arcs: fixes G :: "('a, 'b) pre_digraph"   and v :: "'a" assumes "graph G" shows "inj_on (graph.arc_to_block G) (out_arcs G v)"
idplus: fixes m :: "'n"   and n :: "'n"   and c :: "real" assumes "m \<noteq> n" shows "P (\<chi>i j. if i = m \<and> j = n then c else of_bool (i = j))"
\<phi>\<psi>: shows "meta_adjunction C D F G \<phi> \<psi>"
strict_trans1: fixes a :: "'g"   and b :: "'g"   and c :: "'g" assumes "a \<le> b"   and "b < c" shows "a < c"
map_def: fixes f :: "'a" shows "A.map f = (if A.arr f then f else A.null)"
cut_le_Min_empty: fixes t :: "'a"   and I :: "'a set" assumes "t < iMin I" shows "I \<down>\<le> t = {}"
start_def: fixes A :: "'a \<times> 'b \<times> 'c" shows "start A = fst A"
hm_it_next_rule': fixes l' :: "('a \<times> 'b) list"   and l :: "('a \<times> 'b) list list"   and ht :: "('a, 'b) hashtable"   and it :: "nat \<times> ('a \<times> 'b) list \<times> ('a, 'b) hashtable" assumes "l' \<noteq> []" shows "<hm_is_it' l ht l' it> hm_it_next it <\<lambda>((k, v), it'). hm_is_it' l ht (butlast l') it' * \<up> (last l' = (k, v) \<and> distinct (map fst l'))>"
alw_holds_smap_conv_comp: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and s :: "'b stream" shows "alw (holds P) (smap f s) = alw (\<lambda>s. (P \<circ> f) (shd s)) s"
prv_exi_imp_gen: fixes x :: "'var"   and \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "x \<notin> Fvars \<chi>"   and "bprv (imp \<phi> \<chi>)" shows "bprv (imp (exi x \<phi>) \<chi>)"
euler_slp_impl_def: fixes ode_ops :: "'c \<times> ('d \<times> 'e \<times> 'a \<times> 'f \<times> 'g) \<times> 'h \<times> 'i" shows "euler_slp_impl ode_ops = (case ode_ops of (xa, (xb, xc, x, xd, xe), xf, xg) \<Rightarrow> x)"
simps: fixes x :: "real \<times> real"   and a :: "(real \<times> real) list" shows "ccw'.sortedP x a = (a = [] \<or> (\<exists>ys x. a = x # ys \<and> (\<forall>y\<in>set ys. ccw' x x y) \<and> ccw'.sortedP x ys))"
simps: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_backto_standard_pos ires rs (l, r) = (wcode_backto_standard_pos_B ires rs (l, r) \<or> wcode_backto_standard_pos_O ires rs (l, r))"
convex_hull_in_cone: fixes n :: "nat"   and C :: "'b vec set" shows "gram_schmidt.convex_hull n C \<subseteq> gram_schmidt.cone n C"
neq_iff: fixes x :: "'f"   and y :: "'f" shows "(x \<noteq> y) = (x < y \<or> y < x)"
simps_1: fixes uu :: "'a simple_packet set" shows "rmshadow [] uu = []"
vmc_path_lnull_ltl_no_deadend: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c" assumes "vmc_path G P v0 p \<sigma>"   and "\<not> lnull (ltl P)" shows "vmc_path_no_deadend G P v0 p \<sigma>"
prems_6: shows "\<alpha> = \<omega>_sum \<beta>s"
fmeasurableI: fixes A :: "'c set"   and M :: "'c measure" assumes "A \<in> sets M"   and "emeasure M A < \<infinity>" shows "A \<in> fmeasurable M"
is_weak_ranking_iff: fixes xs :: "'b set list" shows "is_weak_ranking xs = (distinct xs \<and> disjoint (set xs) \<and> {} \<notin> set xs)"
atLeastatMost_empty_iff: fixes a :: "'b multiset"   and b :: "'b multiset" shows "(subset_mset.atLeastAtMost a b = {}) = (\<not> a \<subseteq># b)"
step: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and ys :: "'b list"   and z :: "'b" assumes "r x y"   and "rtrancl_path r y ys z" shows "rtrancl_path r x (y # ys) z"
map_simp: fixes f :: "'a \<times> 'a" assumes "VV.arr f" shows "VV.map f = f"
bij_betw_tan: shows "bij_betw tan {- pi / 2<..<pi / 2} UNIV"
safe_roomk_currk: fixes s :: "event list"   and r :: "room" assumes "hotel s"   and "safe\<^sub>0 s r" shows "roomk s r = currk s r"
small_asymmetric: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> o[F](g)"   and "g \<in> o[F](f)" shows "\<forall>\<^sub>F x in F. f x = (0::'c)"
sas_plus_equivalent_to_strips_i_a_I: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and ops' :: "('c \<times> 'd) strips_operator list"   and s :: "'c \<Rightarrow> 'd option"   and op :: "('c, 'd) sas_plus_operator" assumes "is_valid_problem_sas_plus \<Psi>"   and "set ops' \<subseteq> set ((\<phi> \<Psi> )\<^sub>\<O>)"   and "are_all_operators_applicable (\<phi>\<^sub>S \<Psi> s) ops'"   and "op \<in> set (map (strips_op_to_sasp \<Psi>) ops')" shows "map_of (sas_plus_operator.precondition_of op) \<subseteq>\<^sub>m \<phi>\<^sub>S\<inverse> \<Psi> \<phi>\<^sub>S \<Psi> s"
ChiRel_HAStates2: fixes a :: "'a"   and b :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "(a, b) \<in> ChiRel A" shows "b \<in> HAStates A"
is_monom_def: fixes l :: "linear_poly" shows "is_monom l = (length (vars_list l) = 1)"
fpty_dclI: fixes P :: "cld list"   and ctx :: "ctx"   and dcl :: "char list"   and ctxclds_opt :: "(ctx \<times> cld) list option" assumes "find_path P ctx (cl_fqn (fqn_def dcl)) ctxclds_opt" shows "find_path_ty P (ty_def ctx dcl) ctxclds_opt"
subdomainE_1: fixes H :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "subdomain H R" shows "H \<subseteq> carrier R"
reindex: fixes F :: "nat \<Rightarrow> 'a" shows "(\<Sum>i | 1 \<le> i \<and> i \<le> n. F i) = (\<Sum>m | m \<le> n \<and> odd m. \<Sum>i | \<exists>k. i = 2 ^ k * m \<and> i \<le> n. F i)"
connected_Union: fixes S :: "'a set set" assumes "\<And>s. s \<in> S \<Longrightarrow> connected s"   and "\<Inter> S \<noteq> {}" shows "connected (\<Union> S)"
strict_mono_enumerate: fixes S :: "nat set" assumes "infinite S" shows "strict_mono (wellorder_class.enumerate S)"
neq_nb: fixes t :: "tm" assumes "tmbound0 t" shows "bound0 (neq t)"
cop_stable_no_blocking_on: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set" assumes "ContractsWithIRC Xd Xh Pd Ch" shows "Contracts.stable_no_blocking_on Pd Ch ds (Contracts.CH Ch (Contracts.fp_cop_F Pd Ch ds))"
finite_rtrancl_delta_Image: fixes r :: "'s1"   and s :: "'s2"   and R :: "'a1 \<Rightarrow> 'a2 \<Rightarrow> bool" assumes "wellformed1 r"   and "wellformed2 s" shows "finite ({((r, s), delta1 a r, delta2 b s) |a b r s. a \<in> set alphabet1 \<and> b \<in> set alphabet2 \<and> R a b}\<^sup>* `` {(r, s)})"
f_inv_on_f: fixes y :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "y \<in> f ` A" shows "f (inv_on f A y) = y"
GroupG: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "FGModule G smult V" shows "Group G"
pull_posneg_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "pull_posneg w1 w2" shows "kauff_mat w1 = kauff_mat w2"
FORKs\<^sub>n\<^sub>o\<^sub>r\<^sub>m_1: fixes N :: "nat"   and fs :: "nat list" assumes "DiningPhilosophers N"   and "0 < length fs" shows "DiningPhilosophers.FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m N 0 (fs ! 0) = DiningPhilosophers.FORKs\<^sub>n\<^sub>o\<^sub>r\<^sub>m N 1 fs"
AE_iff_null: fixes M :: "'b measure"   and P :: "'b \<Rightarrow> bool" assumes "{x \<in> space M. \<not> P x} \<in> sets M" shows "almost_everywhere M P = ({x \<in> space M. \<not> P x} \<in> null_sets M)"
assms_2: shows "analytically_valid (cubeImage diamond_cube) (\<lambda>x. F x \<bullet> j) i"
semidecidable_decidable: fixes X :: "nat set" assumes "semidecidable X"   and "semidecidable (- X)" shows "decidable X"
derf: fixes x :: "'a" assumes "x \<in> U" shows "(f has_derivative blinfun_apply (f' x)) (at x)"
support_on_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and S :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "class.monoid_add plus zero" shows "monoid_add.support_on zero S f = {x \<in> S. f x \<noteq> zero}"
leD: fixes y :: "'e word"   and x :: "'e word" assumes "y \<le>s x" shows "\<not> x <s y"
norm_octo_def: fixes x :: "octo" shows "norm x = sqrt ((Ree x)\<^sup>2 + (Im1 x)\<^sup>2 + (Im2 x)\<^sup>2 + (Im3 x)\<^sup>2 + (Im4 x)\<^sup>2 + (Im5 x)\<^sup>2 + (Im6 x)\<^sup>2 + (Im7 x)\<^sup>2)"
diff_strict_left_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c"   and a :: "'c"   and c :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less b a" shows "less (minus c a) (minus c b)"
ex_gt_Y: fixes xa :: "'a" assumes "count (mset (y # ys)) xa < count (mset (x # xs)) xa" shows "\<exists>ya. gt ya xa \<and> count (mset (x # xs)) ya < count (mset (y # ys)) ya"
exhaust: fixes y :: "('c, 'd) lit" obtains x1 :: "('c, 'd) atm" where "y = Pos x1"   | x2 :: "('c, 'd) atm" where "y = Neg x2"
rel_funI: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" shows "rel_fun A B f g"
hyps_2: shows "(y, z) \<in> {(x, y). x \<rightarrow>\<^sub>i y}"
fundantivertex_surj_on: shows "fundantivertex ` (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) = C0"
connected_path_image: fixes g :: "real \<Rightarrow> 'b" assumes "path g" shows "connected (path_image g)"
pair_pmf_def: fixes A :: "'d pmf"   and B :: "'e pmf" shows "pair_pmf A B = A \<bind> (\<lambda>x. B \<bind> (\<lambda>y. return_pmf (x, y)))"
eval_fps_def: fixes f :: "'b fps"   and z :: "'b" shows "eval_fps f z = (\<Sum>n. fps_nth f n * z ^ n)"
RUN_all_tickfree_traces2: fixes s :: "'b event list" assumes "tickFree s" shows "s \<in> T (RUN UNIV)"
init_bounds_1: fixes init :: "(nat \<times> linear_poly) list \<Rightarrow> ('a, 'b) state"   and t :: "(nat \<times> linear_poly) list" assumes "Init init" shows "\<B>\<^sub>i\<^sub>l (init t) = Mapping.empty"
restr_empty_1: fixes al :: "('b \<times> 'c) list" shows "restrictA {} al = []"
widest_spread_spec: fixes k :: "'a"   and s :: "real"   and ks :: "'a list"   and ps :: "(real, 'a) vec list" assumes "(k, s) = widest_spread ks ps" shows "widest_spread_axis k (set ks) (set ps)"
history_consistent_hd_prog_p: fixes \<theta> :: "nat \<Rightarrow> nat option"   and p :: "'p"   and xs :: "'p memref list" assumes "history_consistent \<theta> p xs" shows "p = hd_prog p xs"
Beta: fixes x :: "name"   and s :: "lam"   and ss :: "lam list"   and t :: "lam"   and r :: "lam" assumes "x \<sharp> (s, ss, t)"   and "r[x::=s] \<degree>\<degree> ss \<rightarrow>\<^sub>s t" shows "(Lam [x].r) \<degree> s \<degree>\<degree> ss \<rightarrow>\<^sub>s t"
RA4_analz_spies: fixes C' :: "agent"   and B :: "agent"   and K :: "nat"   and X :: "msg"   and X' :: "msg"   and RA :: "msg"   and evs :: "event list" assumes "Says C' B \<lbrace>Crypt K X, X', RA\<rbrace> \<in> set evs" shows "RA \<in> analz (knows Spy evs)"
silent_moves_called_node_in_slice1_nodestack_in_slice1: fixes S :: "'node SDG_node set"   and m :: "'node"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and m' :: "'node"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and nx :: "'node SDG_node"   and rs :: "'edge list"   and cs :: "'edge list" assumes "S,kind \<turnstile> (m # ms,s) =as\<Rightarrow>\<^sub>\<tau> (m' # ms',s')"   and "valid_node m"   and "CFG_node m' \<in> sum_SDG_slice1 nx"   and "nx \<in> S"   and "\<forall>mx\<in>set ms. \<exists>mx'. call_of_return_node mx mx' \<and> mx' \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "\<forall>i<length rs. rs ! i \<in> get_return_edges (cs ! i)"   and "ms = targetnodes rs"   and "valid_return_list rs m"   and "length rs = length cs" shows "\<forall>mx\<in>set ms'. \<exists>mx'. call_of_return_node mx mx' \<and> mx' \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"
finsert_fminus_if: fixes x :: "'a"   and A :: "'a fset"   and B :: "'a fset" shows "finsert x A |-| B = (if x |\<in>| B then A |-| B else finsert x (A |-| B))"
funpow_dist1_le_self: fixes m :: "nat"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "(f ^^ m) x = x"   and "0 < m"   and "y \<in> orbit f x" shows "funpow_dist1 f x y \<le> m"
that: fixes y :: "'b" assumes "y |\<in>| N"   and "P x y" shows "thesis"
Quotient_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "Quotient R Abs Rep T = ((\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. R (Rep a) (Rep a)) \<and> (\<forall>r s. R r s = (R r r \<and> R s s \<and> Abs r = Abs s)) \<and> T = (\<lambda>x y. R x x \<and> Abs x = y))"
in_set_half_segments_of_aform_aform_valE: fixes x2 :: "real \<times> real"   and y2 :: "real \<times> real"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "(x2, y2) \<in> set (half_segments_of_aform X)" obtains e :: "nat \<Rightarrow> real" where "y2 = aform_val e X" and "e \<in> UNIV \<rightarrow> {- 1..1}"
rel_interior_ball: fixes S :: "'b set" shows "rel_interior S = {x \<in> S. \<exists>e>0. ball x e \<inter> affine hull S \<subseteq> S}"
associatedE1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalize a = normalize b" obtains u :: "'b" where "dvd.dvd times u one" and "a = times u b"
max_word_not_0: shows "- 1 \<noteq> 0"
simps_3: fixes a :: "('a, 'b) Node"   and b :: "('a, 'b) Node"   and c :: "('a, 'b) Node" shows "FingerTreeStruc.digitToList (Three a b c) = FingerTreeStruc.nodeToList a @ FingerTreeStruc.nodeToList b @ FingerTreeStruc.nodeToList c"
fds_abs_converges_norm_power: fixes f :: "'a fds"   and s :: "real"   and n :: "nat" assumes "fds_abs_converges (fds_norm f) s" shows "fds_abs_converges (fds_norm (f ^ n)) s"
catalan_closed_form: fixes n :: "nat" shows "catalan n = (2 * n choose n) div Suc n"
sub: fixes gpv' :: "('a, 'b, 'c) gpv" assumes "gpv' \<in> sub_gpvs \<I> gpv" shows "lossless_gpv \<I> gpv'"
holomorphic_on_balls_imp_entire': fixes f :: "complex \<Rightarrow> complex"   and c :: "complex"   and B :: "complex set" assumes "\<And>r. 0 < r \<Longrightarrow> f holomorphic_on ball c r" shows "f holomorphic_on B"
a1: shows "s' = snd (fst (logical_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
dnf_finite: fixes \<phi> :: "'b ltln" shows "finite (dnf \<phi>)"
Ide_case: fixes a :: "'c local.term" assumes "Ide a" shows "F (mkarr a) = E.map (mkarr a)"
simps: fixes key :: "'b \<Rightarrow> 'c"   and xs :: "'b list" shows "Mergesort_Complexity.c_msort key xs = Mergesort_Complexity.c_sequences key xs + Mergesort_Complexity.c_merge_all key (sequences key xs)"
atLeast_le_enum: fixes N :: "nat set"   and x :: "nat"   and i :: "nat" assumes "N \<subseteq> {x..}"   and "infinite N" shows "x \<le> enum N i"
M_Nil: fixes fs :: "(char list \<times> trm) list" shows "\<turnstile> [] [\<rhd>] fs \<Rightarrow> []"
suffix_length_le: fixes xs :: "'a list"   and ys :: "'a list" assumes "suffix xs ys" shows "length xs \<le> length ys"
partition_aux_body_def: shows "partition_aux_body \<equiv> \<lambda>r (xxs, ecs). case xxs of [] \<Rightarrow> ([], []) | x # xs \<Rightarrow> let (xxs', xec) = partition_split r x xs in (xxs', (x # xec) # ecs)"
measure_eqI_finite: fixes M :: "'b measure"   and A :: "'b set"   and N :: "'b measure" assumes "sets M = Pow A"   and "sets N = Pow A"   and "finite A"   and "\<And>a. a \<in> A \<Longrightarrow> emeasure M {a} = emeasure N {a}" shows "M = N"
link_def: fixes g :: "'v \<times> 'v \<Rightarrow> ennreal" shows "link g = (if \<exists>z. z \<notin> A \<Gamma> \<and> z \<notin> B \<Gamma> \<and> d_IN g z < d_OUT g z then let z = SOME z. z \<notin> A \<Gamma> \<and> z \<notin> B \<Gamma> \<and> d_IN g z < d_OUT g z; factor = d_IN g z / d_OUT g z in (\<lambda>(x, y). (if x = z then factor else 1) * g (x, y)) else g)"
shiftr_def: fixes a :: "'b array"   and key :: "'b"   and i :: "nat ref" shows "shiftr a key i = while (shiftr_p a key i) (shiftr_f a i)"
locally_path_connected_Euclidean_space: fixes n :: "nat" shows "locally_path_connected_space (Euclidean_space n)"
min_elems_def: fixes X :: "'c set" shows "min_elems X = {s \<in> X. \<forall>t. t \<in> X \<longrightarrow> \<not> t < s}"
mappend_assoc: fixes below :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and liftemb :: "'d\<^sub>\<bottom> \<rightarrow> udom\<^sub>\<bottom>"   and liftprj :: "udom\<^sub>\<bottom> \<rightarrow> 'd\<^sub>\<bottom>"   and liftdefl :: "'d itself \<Rightarrow> (udom\<^sub>\<bottom>) defl"   and emb :: "'d \<rightarrow> udom"   and prj :: "udom \<rightarrow> 'd"   and defl :: "'d itself \<Rightarrow> udom defl"   and mempty :: "'d"   and mappend :: "'d \<rightarrow> 'd \<rightarrow> 'd"   and xs :: "'d"   and ys :: "'d"   and zs :: "'d" assumes "class.monoid below liftemb liftprj liftdefl emb prj defl mempty mappend" shows "mappend\<cdot>(mappend\<cdot>xs\<cdot>ys)\<cdot>zs = mappend\<cdot>xs\<cdot>(mappend\<cdot>ys\<cdot>zs)"
simps_3: fixes uu :: "'a" shows "(Some uu \<le> None) = False"