hcju/isabelleps huggingface dataset:
corec_code: fixes p :: "'b \<Rightarrow> bool"   and g21 :: "'b \<Rightarrow> 'a"   and q22 :: "'b \<Rightarrow> bool"   and g221 :: "'b \<Rightarrow> 'a llist"   and g222 :: "'b \<Rightarrow> 'b"   and a :: "'b" shows "corec_llist p g21 q22 g221 g222 a = (if p a then LNil else LCons (g21 a) (if q22 a then g221 a else corec_llist p g21 q22 g221 g222 (g222 a)))"
vsv_vinsert_match_appI: fixes a :: "V"   and b :: "V" assumes "a \<notin>\<^sub>\<circ> \<D>\<^sub>\<circ> xs" shows "vinsert \<langle>a, b\<rangle> xs\<lparr>a\<rparr> = b"
prepend_at_less: fixes n :: "nat"   and vs :: "nat list"   and f :: "nat \<Rightarrow> nat option" assumes "n < length vs" shows "(vs \<odot> f) n \<down>= vs ! n"
sinit_prefix: fixes i :: "nat"   and j :: "nat"   and xs :: "'b seq" assumes "i \<le> j" shows "prefix (sinit i xs) (sinit j xs)"
in_star_iff_concat: fixes w :: "'b list"   and A :: "'b list set" shows "(w \<in> Regular_Set.star A) = (\<exists>ws. set ws \<subseteq> A \<and> w = concat ws)"
assms_2: assumes "\<not> \<Phi>" shows "(msgi, msg) \<in> E"
bot: fixes x :: "'a" assumes "\<not> leq x bound" shows "ord bot (f x)"
safe_distance_normal_axioms: shows "safe_distance_normal a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta>"
less_3_cases: fixes n :: "nat" assumes "n < 3" shows "n = 0 \<or> n = Suc 0 \<or> n = Suc (Suc 0)"
c: fixes n :: "'a" shows "continuous_on {a..b} (f n)"
\<II>\<nn>\<ll>_\<II>\<ss>\<II>\<nn>\<ll>: fixes op :: "'op"   and xs :: "'dyn list"   and opinl :: "'opinl" assumes "\<II>\<nn>\<ll> op xs = Some opinl" shows "\<II>\<ss>\<II>\<nn>\<ll> opinl xs"
map_invar_impl_def: fixes m :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b option"   and s :: "'a" shows "map_invar_impl m s = (\<forall>ii ti ei ri. m (ii, ti, ei) = Some ri \<longrightarrow> (\<exists>i t e. (ri, ifex_ite_opt i t e) \<in> R s \<and> (ii, i) \<in> R s \<and> (ti, t) \<in> R s \<and> (ei, e) \<in> R s))"
map_cong: fixes x :: "'g list"   and ya :: "'g list"   and f :: "'g \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'h" assumes "x = ya"   and "\<And>z. z \<in> set ya \<Longrightarrow> f z = g z" shows "map f x = map g ya"
GrD2: fixes x :: "'d"   and fx :: "'e"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "(x, fx) \<in> BNF_Def.Gr A f" shows "f x = fx"
point_has_unique_group: fixes x :: "'a \<times> 'b" assumes "x \<in> bl \<times> I" shows "\<exists>!G. x \<in> G \<and> G \<in> {{x} \<times> I |x. x \<in> bl}"
is_rwof_invar_def: fixes m0 :: "'b nres"   and cond :: "'b \<Rightarrow> bool"   and step :: "'b \<Rightarrow> 'b nres"   and I :: "'b \<Rightarrow> bool" shows "is_rwof_invar m0 cond step I \<equiv> m0 \<le>\<^sub>n SPEC I \<and> (\<forall>s. rwof m0 cond step s \<and> I s \<and> cond s \<longrightarrow> step s \<le>\<^sub>n SPEC I)"
list_rel_comp_list_wset_rel: fixes R :: "('c \<times> 'd) set"   and S :: "('d \<times> 'e) set" assumes "single_valued R" shows "\<langle>R\<rangle>list_rel O \<langle>S\<rangle>list_wset_rel = \<langle>R O S\<rangle>list_wset_rel"
filter_compliant_stateful_ACS_maximal: fixes M :: "'b SecurityInvariant_configured list"   and V :: "'b set"   and E :: "('b \<times> 'b) set"   and edgesList :: "('b \<times> 'b) list"   and stateful :: "('b \<times> 'b) set" assumes "valid_reqs (get_ACS M)"   and "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "set edgesList = E"   and "stateful = set (filter_compliant_stateful_ACS \<lparr>nodes = V, edges = E\<rparr> M edgesList)" shows "\<forall>e\<in>E - (stateful \<union> {e \<in> E. e \<in> backflows E}). \<not> \<Union> (get_offending_flows (get_ACS M) (stateful_policy_to_network_graph \<lparr>hosts = V, flows_fix = E, flows_state = stateful \<union> {e}\<rparr>)) \<subseteq> backflows (filternew_flows_state \<lparr>hosts = V, flows_fix = E, flows_state = stateful \<union> {e}\<rparr>)"
suc2plus_inj_on: shows "inj_on word_of_nat {0..unat (- 1)}"
op_ivl_Suc: fixes g :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "prod g {m..<Suc n} = (if n < m then 1::'b else prod g {m..<n} * g n)"
finite_linorder_max_induct: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "class.linorder less_eq less"   and "finite A"   and "P {}"   and "\<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. less a b; P A\<rbrakk> \<Longrightarrow> P (insert b A)" shows "P A"
simps_1: shows "sinits\<cdot>[::] = [:[::]:]"
GSMP_mono: fixes N :: "('fun, 'var) Term.term set"   and M :: "('fun, 'var) Term.term set" assumes "N \<subseteq> M" shows "GSMP N \<subseteq> GSMP M"
Collect_cong: fixes P :: "'h \<Rightarrow> bool"   and Q :: "'h \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "{x. P x} = {x. Q x}"
classes_above_class: fixes C :: "char list"   and P :: "'b prog"   and P' :: "'b prog"   and C' :: "char list" assumes "classes_above P C \<inter> classes_changed P P' = {}"   and "P \<turnstile> C \<preceq>\<^sup>* C'" shows "class P C' = class P' C'"
pfa_append_extend_grow_def: shows "pfa_append_extend_grow \<equiv> \<lambda>(a, n) x (b, m). array_ensure a (n + m + 1) default \<bind> (\<lambda>a'. Array.upd n x a' \<bind> (\<lambda>a''. blit b 0 a'' (n + 1) m \<bind> (\<lambda>_. return (a'', n + m + 1))))"
D: shows "LeftDerives1 (a @ c) i r (b @ c)"
src_trg: fixes \<mu> :: "'a arrow_of_spans_data" assumes "arr \<mu>" shows "src (trg \<mu>) = trg \<mu>"
simps_1: fixes v :: "bool"   and res :: "bool"   and S :: "'av st option" shows "inv_bval' (Bc v) res S = (if v = res then S else None)"
writes_subset: fixes S :: "('a, 'b, 'c) prog set"   and f :: "('a, 'b, 'd) prog"   and h :: "'a"   and h' :: "'a"   and S' :: "('a, 'b, 'c) prog set" assumes "writes S f h h'"   and "S \<subseteq> S'" shows "writes S' f h h'"
precedesReflexivity: fixes a :: "'a"   and l :: "'a list" assumes "a \<in> set l" shows "precedes a a l"
order_iff: fixes a :: "'d"   and b :: "'d" shows "(a \<le> b) = (a = min a b)"
node_deliver_messages_def: fixes cs :: "('msgid \<times> 'op) event list" shows "node_deliver_messages cs \<equiv> List.map_filter (\<lambda>e. case e of Broadcast x \<Rightarrow> Map.empty x | Deliver m \<Rightarrow> Some m) cs"
g_not_r: fixes z :: "complex" assumes "z \<in> S" shows "g__ z \<notin> ball (- g__ a) r"
partitions_parts_bounded: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "p partitions n" shows "sum p {..n} \<le> n"
dep_var_set_def: fixes PROB :: "(('b, 'c) fmap \<times> ('b, 'd) fmap) set"   and vs1 :: "'b set"   and vs2 :: "'b set" shows "dep_var_set PROB vs1 vs2 \<equiv> disjnt vs1 vs2 \<and> (\<exists>v1 v2. v1 \<in> vs1 \<and> v2 \<in> vs2 \<and> dep PROB v1 v2)"
has_contour_integral_winding_number: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex" assumes "valid_path \<gamma>"   and "z \<notin> path_image \<gamma>" shows "((\<lambda>w. 1 / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z) \<gamma>"
NF_NT: fixes s :: "'b event list"   and P :: "'b process" assumes "(s, {}) \<notin> F P" shows "s \<notin> T P"
Prop4A: fixes c :: "IMP"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and R :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and S :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes " \<Turnstile> c : Sec A R S \<Phi> " shows "ARSsecure A R S c"
intros_1: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.valid_node sourcenode targetnode valid_edge n" shows "CFG.path sourcenode targetnode valid_edge n [] n"
finite_lists_distinct_length_eq: fixes A :: "'a set"   and n :: "nat" assumes "finite A" shows "finite {xs. length xs = n \<and> distinct xs \<and> set xs \<subseteq> A}"
open_closed_segment: fixes u :: "'a"   and w :: "'a"   and z :: "'a" assumes "u \<in> open_segment w z" shows "u \<in> closed_segment w z"
kth_obs_stable: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and l :: "nat"   and j :: "nat"   and k :: "nat" assumes "IFC prob"   and "IFC_def.is_kth_obs prob \<pi> l j"   and "k < l" shows "\<exists>i. IFC_def.is_kth_obs prob \<pi> k i"
tb: shows "tighten_poly_bounds_epsilon p eps l r (sgn (ipoly p r)) = (l', r', sr')"
carrier_DirProd: fixes G :: "('a, 'c) monoid_scheme"   and H :: "('b, 'd) monoid_scheme" shows "carrier (G \<times>\<times> H) = carrier G \<times> carrier H"
sig_red_single_tail_lc_rep_list: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "sig_red_single sing_reg (\<prec>) p q f t" shows "punit.lc (rep_list q) = punit.lc (rep_list p)"
intuitionistic_sep_impl_sep_true_simp: fixes P :: "'a \<Rightarrow> bool" assumes "intuitionistic P" shows "((\<lambda>s. True) \<longrightarrow>* P) = P"
add_mono_thms_linordered_semiring_1: fixes i :: "'c"   and j :: "'c"   and k :: "'c"   and l :: "'c" assumes "i \<le> j \<and> k \<le> l" shows "i + k \<le> j + l"
isnonconstant_def: fixes p :: "'b poly" shows "isnonconstant p = (\<not> isconstant p)"
zeta_partial_sum_le_pos': fixes s :: "real" assumes "0 < s"   and "s \<noteq> 1" shows "\<exists>c>0. \<forall>x\<ge>1. \<bar>sum_upto (\<lambda>x. real x powr - s) x - x powr (1 - s) / (1 - s)\<bar> \<le> c"
Obj: shows "X \<in> obj\<^bsub>NTCatDom \<eta>\<^esub>"
simps_1: fixes k :: "'c"   and t :: "('c, 'd) rbt"   and kts :: "('c \<times> ('c, 'd) rbt) list" shows "rbt_keys_next ((k, t) # kts, rbt.Empty) = (k, kts, t)"
p'_3: fixes x :: "'n"   and K :: "'n set" assumes "(x, K) \<in> p" shows "K \<subseteq> cbox a b"
lsetp_into_lset: fixes xs :: "'a llist"   and x :: "'a" assumes "Coinductive_List.lsetp xs x" shows "x \<in> lset xs"
subgraph_in_degree: fixes H :: "('a, 'a \<times> 'a) pre_digraph"   and v :: "'a" assumes "subgraph H (with_proj slim)" shows "in_degree H v \<le> in_degree (with_proj slim) v"
ClassCastC_def: shows "ClassCastC \<equiv> (ClassCast, Object, [], [])"
inv_def: fixes G :: "graph"   and S :: "state" shows "Dijkstra.inv G S = (let V = known_set S; W = unknown_set S; M = heap S in length (est S) = Dijkstra.size G \<and> known_dists G V \<and> keys_of M \<subseteq> verts G \<and> (\<forall>i\<in>W. M\<langle>i\<rangle> = Some (est S ! i)) \<and> (\<forall>i\<in>V. est S ! i = dist G 0 i) \<and> (\<forall>i\<in>verts G. est S ! i = dist_on G 0 i V))"
exact_seq_solvable_imp: fixes G3 :: "'b monoid"   and G2 :: "'b monoid"   and G1 :: "'b monoid"   and g1 :: "'b \<Rightarrow> 'b"   and g2 :: "'b \<Rightarrow> 'b" assumes "exact_seq ([G1], [])  \<longlongrightarrow>\<^bsub>g1\<^esub> G2  \<longlongrightarrow>\<^bsub>g2\<^esub> G3"   and "inj_on g1 (carrier G1)"   and "g2 ` carrier G2 = carrier G3"   and "solvable G2" shows "solvable G1 \<and> solvable G3"
of_complex_hcoords_def: shows "of_complex_hcoords \<equiv> map_fun id Abs_complex_homo_coords of_complex_cvec"
intros_2: shows "rel_12_23 \<^bold>2 \<^bold>3"
ts_tuple_rel_ext': fixes tas :: "nat \<times> 'a option list"   and nt :: "nat"   and X :: "'a option list set"   and tass :: "(nat \<times> 'a option list set) list"   and Y :: "'a option list set" assumes "tas \<in> ts_tuple_rel (set ((nt, X) # tass))" shows "tas \<in> ts_tuple_rel (set ((nt, X \<union> Y) # tass))"
ts''_def_1: shows "\<C> \<turnstile> [C v1, C v2] : ts _> ts''"
config'_embed: fixes A :: "('state \<Rightarrow> 'b) \<times> ('state \<times> 'b \<Rightarrow> 'request \<Rightarrow> 'answer \<times> 'b)"   and s0 :: "'state \<times> 'b"   and qs :: "'request list" shows "config'_rand (local.embed A) (return_pmf s0) qs = return_pmf (config' A s0 qs)"
star_elim: fixes x :: "'a"   and X :: "'a set" shows "(x \<in> X\<^sup>\<star>) = (\<exists>k. x \<in> X ^ k)"
not_singleton_existence: fixes t :: "'c" shows "\<exists>x. x \<noteq> t"
index_lfp: fixes f :: "'a \<Rightarrow> 'a" shows "local.lfp f \<in> L"
sort_eqv_trans: fixes cs :: "(String.literal \<times> String.literal) set"   and x :: "String.literal set"   and y :: "String.literal set"   and z :: "String.literal set" assumes "wf_subclass_loc cs"   and "sort_eqv cs x y"   and "sort_eqv cs y z" shows "sort_eqv cs x z"
fundamental_theorem_of_calculus_interior_stronger': fixes S :: "real set"   and a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'b" assumes "finite S"   and "a \<le> b"   and "\<And>x. x \<in> {a<..<b} - S \<Longrightarrow> (f has_vector_derivative f' x) (at x within {a..b} - S)"   and "continuous_on {a..b} f" shows "(f' has_integral f b - f a) {a..b}"
wfI: fixes r :: "('b \<times> 'b) set"   and A :: "'b set"   and B :: "'b set" assumes "r \<subseteq> A \<times> B"   and "\<And>x P. \<lbrakk>\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x; x \<in> A; x \<in> B\<rbrakk> \<Longrightarrow> P x" shows "wf r"
count_list_append: fixes xs :: "'a list"   and ys :: "'a list"   and a :: "'a" shows "count_list (xs @ ys) a = count_list xs a + count_list ys a"
r'_def: shows "r' = (\<lambda>x. integral {2..x} r)"
finite_incidence_system_axioms: shows "finite_incidence_system \<V> (add_block bl)"
ge_phi': fixes \<phi> :: "real" assumes "\<phi>' \<le> \<phi>"   and "\<phi> \<le> pi / 2" shows "d / l \<le> sin \<phi>"
seqE: fixes g :: "'c local.term set"   and f :: "'c local.term set" assumes "\<F>C.seq g f" obtains "\<F>C.arr f" and "\<F>C.arr g" and "\<F>C.dom g = \<F>C.cod f"
is_process6: fixes P :: "('a event list \<times> 'a event set) set \<times> 'a event list set" assumes "is_process P" shows "\<forall>s X. (s @ [tick], {}) \<in> FAILURES P \<longrightarrow> (s, X - {tick}) \<in> FAILURES P"
decomp_closure_def: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and public :: "'a \<Rightarrow> bool"   and M :: "('a, 'b) Term.term set"   and M' :: "('a, 'b) Term.term set" assumes "intruder_model arity Ana" shows "intruder_model.decomp_closure arity public Ana M M' \<equiv> \<forall>t. (intruder_model.intruder_deduct arity public Ana M t \<and> (\<exists>t'\<in>M. t \<sqsubseteq> t')) = (t \<in> M')"
winning_region_in_V: fixes p :: "Player" shows "winning_region p \<subseteq> V"
emeasure_eq_measure: fixes A :: "nat set set set" shows "emeasure P A = ennreal (prob A)"
ordRels_def_3: shows "ordIso = {(r, r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. iso r r' f)}"
simps_1: fixes t :: "'b \<Rightarrow> exp \<Rightarrow> exp"   and e :: "exp" shows "lift_transform t Ibottom e = e"
hlog_one: fixes a :: "real star" shows "hlog a 1 = 0"
divide_nonneg_nonneg: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and 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 sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq zero x"   and "less_eq zero y" shows "less_eq zero (divide x y)"
closed_hterm_2: fixes hts :: "'a hterm list" shows "closedts 0 (terms_of_hterms hts)"
ide_cod: fixes f :: "'a \<times> 'a" assumes "VV.arr f" shows "VV.ide (VV.cod f)"
elem_of_symcl_2: fixes a :: "'b"   and b :: "'b"   and Rel :: "('b \<times> 'b) set" assumes "(a, b) \<in> Rel" shows "(b, a) \<in> symcl Rel"
fv_wellformed_transaction_unfold: fixes T :: "('e, 'f, 'g, 'h) prot_transaction" assumes "wellformed_transaction T" shows "fv_transaction T = fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_receive T) \<union> fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_selects T) \<union> set (transaction_fresh T)"
fps_integral0_fps_const_mult_right: fixes a :: "'a fps"   and c :: "'a" shows "fps_integral0 (a * fps_const c) = fps_integral0 a * fps_const c"
runit_closed': fixes f :: "'a" assumes "Arr f" shows "Arr (B.\<rr> f)"
null_space_orthogonal_complement_row_space: fixes A :: "((complex, 'a) vec, 'b) vec" shows "null_space A = complex_matrix.orthogonal_complement (row_space (\<chi>i j. cnj_class.cnj (A $ i $ j)))"
count_eq_zero_iff: fixes M :: "'e multiset"   and x :: "'e" shows "(count M x = 0) = (x \<notin># M)"
sound_modes_no_read: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and cms :: "('a \<times> (Mode \<Rightarrow> 'b set)) list"   and mem :: "'b \<Rightarrow> 'c"   and x :: "'b"   and i :: "nat" assumes "sifum_security stop eval some_val some_val'"   and "sifum_security.sound_mode_use eval (cms, mem)"   and "x \<in> (map snd cms ! i) GuarNoRead"   and "i \<sqsubset> length cms" shows "sifum_security.doesnt_read eval (fst (cms ! i)) x"
tt_skip: fixes p :: "'a" shows "\<lparr>p\<rparr>1::'a\<lparr>p\<rparr>"
pure_def: fixes f :: "('d, 'e, 'f) prog"   and h :: "'d" shows "pure f h = (h \<turnstile> ok f \<longrightarrow> h \<turnstile> f \<rightarrow>\<^sub>h h)"
ladder_\<gamma>_simp2: fixes u :: "nat"   and L :: "(nat \<times> nat \<times> nat) list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and a :: "symbol list"   and D' :: "(nat \<times> symbol \<times> symbol list) list" assumes "u < length L"   and "is_ladder D L" shows "ladder_\<gamma> a (D @ D') L u = ladder_\<gamma> a D L u"
monic_factorization: fixes p :: "'c poly" assumes "monic p" shows "\<exists>as. monic_factorization as p"
supd_abrupt_invariant: fixes f :: "st \<Rightarrow> st"   and s :: "state" shows "fst (supd f s) = fst s"
diff: shows "n\<^sub>k = \<delta>k + Suc n"
NSLIM_at0_iff: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and L :: "'b" shows "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L = (\<lambda>h. f (a + h)) \<midarrow>0::'a\<rightarrow>\<^sub>N\<^sub>S L"
si_2: shows "case si of None \<Rightarrow> True | Some ssi \<Rightarrow> ssi \<in> set (wordinterval_CIDR_split_prefixmatch (uncurry WordInterval (sports r)))"
weak_lattice_def: fixes L :: "('c, 'd) gorder_scheme" shows "weak_lattice L \<equiv> weak_upper_semilattice L \<and> weak_lower_semilattice L"
assms: shows "(\<Delta>1, \<Delta>2) \<in> allConsec \<Delta>s"
induct: fixes P :: "char list \<Rightarrow> char list \<Rightarrow> bool"   and a0 :: "char list"   and a1 :: "char list" assumes "P [] []"   and "\<And>i is. P (i # is) []"   and "\<And>uu_ uv_. P [] (uu_ # uv_)"   and "\<And>i is p_i p_is. (\<not> (i = CHR ''+'' \<and> is = []) \<Longrightarrow> P is p_is) \<Longrightarrow> P (i # is) (p_i # p_is)" shows "P a0 a1"
remove_cycles_removes: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "cnt x (remove_cycles xs x ys) \<le> max 1 (cnt x ys)"
reach_def: fixes nil :: "'a"   and "next" :: "('a \<times> 'a) set"   and x :: "'a" assumes "graph nil next" shows "graph.reach nil next x \<equiv> {y. (x, y) \<in> next\<^sup>* \<and> y \<noteq> nil}"
convergent_def: fixes "open" :: "'c set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'c" assumes "class.topological_space open" shows "topological_space.convergent open X = (\<exists>L. filterlim X (topological_space.nhds open L) sequentially)"
zcount_union: fixes M :: "'c zmultiset"   and N :: "'c zmultiset"   and a :: "'c" shows "zcount (M + N) a = zcount M a + zcount N a"
compress_irule_set_def: shows "compress_irule_set = fcompress \<circ> (|`|) (map_prod id (fcompress \<circ> (|`|) (map_prod id compress_pterm)))"
d'_Field: shows "PdXXX.dX X d' \<subseteq> Field (Pd' d')"
concurrent_append_expunge_independent_technical: fixes i :: "'a"   and mo :: "'a"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and e :: "'b"   and r :: "'a" assumes "i = mo"   and "xs prefix of j"   and "(i, Append i e) \<in> set (node_deliver_messages xs)"   and "(r, Expunge e mo r) \<in> set (node_deliver_messages xs)" shows "hb (i, Append i e) (r, Expunge e mo r)"
SUP_least: fixes A :: "'b set"   and u :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "\<And>i. i \<in> A \<Longrightarrow> u \<le> f i" shows "u \<le> \<Sqinter> (f ` A)"
change_dyn_def: fixes ts :: "traffic"   and c :: "cars"   and f :: "real \<Rightarrow> real"   and ts' :: "traffic" shows " ts \<^bold>\<midarrow> dyn(c,f) \<^bold>\<rightarrow> ts' \<equiv> pos ts' = pos ts \<and> res ts' = res ts \<and> clm ts' = clm ts \<and> dyn ts' = (dyn ts)(c := f) \<and> physical_size ts' = physical_size ts"
convergent_imp_bounded: fixes S :: "nat \<Rightarrow> 'a"   and l :: "'a" assumes "S \<longlonglongrightarrow> l" shows "bounded (range S)"
foldli_map: fixes f1 :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and c :: "'a \<Rightarrow> bool"   and f2 :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and \<sigma> :: "'a" shows "foldli (map f1 xs) c f2 \<sigma> = foldli xs c (f2 \<circ> f1) \<sigma>"
ring_mat_simps_5: fixes ty :: "'a itself"   and n :: "nat"   and b :: "'b" shows "carrier (ring_mat ty n b) = carrier_mat n n"
f_def: shows "f = butlast (prio_emb_pos prio t s) ! length (butlast (prio_emb_pos prio (prio_emb_step prio t s) s))"
dp_sur1: fixes x :: "'a"   and y :: "'a" assumes "is_direct_product x y" shows "is_sur x"
inj_on_contraD: fixes f :: "'g \<Rightarrow> 'h"   and A :: "'g set"   and x :: "'g"   and y :: "'g" assumes "inj_on f A"   and "x \<noteq> y"   and "x \<in> A"   and "y \<in> A" shows "f x \<noteq> f y"
inject_2: fixes x2 :: "sig"   and y2 :: "sig" shows "(mid x2 = mid y2) = (x2 = y2)"
iD_flag_is_inv: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "ip \<in> iD rt" shows "the (flag rt ip) = Aodv_Basic.inv"
AssertAllState''_axioms: shows "AssertAllState'' init_state assert_bound_code check_code"
app_2: shows "linear (app t\<^sub>1 t\<^sub>2)"
neighborhoods_eq: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and I :: "'c set set"   and x :: "'c" assumes "stalk S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I x" shows "topological_space.neighborhoods is_open x = I"
mono_D1: fixes P :: "'c process"   and Q :: "'c process"   and S :: "'c process" assumes "P \<sqsubseteq> Q" shows "D (Q [+] S) \<subseteq> D (P [+] S)"
Abs_oalist_ntm_inverse: fixes y :: "('a \<times> 'b) list \<times> 'a nat_term_order" assumes "y \<in> Collect ko_ntm.oalist_inv" shows "list_of_oalist_ntm (Abs_oalist_ntm y) = y"
list_rel_congD: fixes li :: "'c list"   and l :: "'d list"   and S :: "('c \<times> 'd) set" assumes "(li, l) \<in> \<langle>S\<rangle>list_rel" shows "(li, l) \<in> \<langle>S \<inter> set li \<times> set l\<rangle>list_rel"
set_tag_name_get_disconnected_nodes_is_l_set_tag_name_get_disconnected_nodes: shows "l_set_tag_name_get_disconnected_nodes ShadowRootClass.type_wf set_tag_name set_tag_name_locs get_disconnected_nodes get_disconnected_nodes_locs"
hnorm_less_Infinitesimal: fixes e :: "real star"   and x :: "'c star" assumes "e \<in> Infinitesimal"   and "hnorm x < e" shows "x \<in> Infinitesimal"
2: shows "D'.in_hom \<psi> (local.map (trg\<^sub>C \<mu>)) (D'.trg (local.map \<mu>))"