hcju/isabelleps huggingface dataset:
ImplE: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> \<^bold>\<rightarrow> \<psi> in v]"   and "[\<phi> in v]" shows "[\<psi> in v]"
db\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('a, 'b) stateful_strand_step list"   and I :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "db\<^sub>s\<^sub>s\<^sub>t S I \<equiv> db'\<^sub>s\<^sub>s\<^sub>t S I []"
assms_2: shows "ring_hom_ring (K [X]) A h"
tan_periodic_nat: fixes x :: "real"   and n :: "nat" shows "tan (x + real n * pi) = tan x"
fls_base_factor_to_fps_subdegree: fixes f :: "'a fls" shows "subdegree (fls_base_factor_to_fps f) = 0"
module: fixes R :: "('b, 'c) ring_scheme"   and N :: "'d set"   and M :: "('b, 'd, 'e) module_scheme" assumes "LinearCombinations.submodule R N M" shows "module R M"
I_equiv: fixes x :: "dB" shows "\<I> \<degree> x \<leftrightarrow> x"
strict_domE: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and n :: "'b"   and m :: "'b"   and g :: "'a" assumes "graph_Entry \<alpha>e \<alpha>n invar inEdges' Entry"   and "n \<noteq> m \<and> graph_Entry_base.dominates \<alpha>n invar inEdges' Entry g n m" obtains "m \<in> set (\<alpha>n g)" and "\<And>ns. graph_path_base.path2 \<alpha>n invar inEdges' g (Entry g) ns m \<Longrightarrow> n \<in> set (butlast ns)"
is_zero_row_upt_k_suc: fixes i :: "'d"   and k :: "nat"   and A :: "(('e, 'f) vec, 'd) vec" assumes "is_zero_row_upt_k i k A"   and "A $ i $ mod_type_class.from_nat k = (0::'e)" shows "is_zero_row_upt_k i (Suc k) A"
exp_cf6_def: shows "exp_cf6 \<equiv> \<lambda>x. numer_cf6 x / numer_cf6 (- x)"
intruder_synth_mod_timpls_is_synth_timpl_closure_set: fixes TI' :: "('sets set \<times> 'sets set) list"   and TI :: "('sets set \<times> 'sets set) list"   and M :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term list"   and t :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term" assumes "set TI' = {(a, b). (a, b) \<in> (set TI)\<^sup>+ \<and> a \<noteq> b}" shows "intruder_synth_mod_timpls M TI' t = (timpl_closure_set (set M) (set TI) \<turnstile>\<^sub>c t)"
axioms_2: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name known_ptr type_wf heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs" shows "l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_shadow_root get_shadow_root_locs get_host get_host_locs type_wf"
insert_eq_iff: fixes a :: "'c"   and A :: "'c set"   and b :: "'c"   and B :: "'c set" assumes "a \<notin> A"   and "b \<notin> B" shows "(insert a A = insert b B) = (if a = b then A = B else \<exists>C. A = insert b C \<and> b \<notin> C \<and> B = insert a C \<and> a \<notin> C)"
minus: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "additive f" shows "f (- x) = - f x"
uniformI1: fixes e :: "real"   and S :: "'c set" assumes "0 < e"   and "\<And>x y. \<lbrakk>x \<in> S; y \<in> S; dist x y < e\<rbrakk> \<Longrightarrow> x = y" shows "uniform_discrete S"
fully_unsorted_poly_rel_diff: fixes v :: "char list list \<times> int"   and v' :: "int mpoly"   and r :: "(char list list \<times> int) list"   and r' :: "int mpoly" assumes "([v], v') \<in> fully_unsorted_poly_rel O mset_poly_rel"   and "(r, r') \<in> fully_unsorted_poly_rel O mset_poly_rel" shows "(v # r, v' + r') \<in> fully_unsorted_poly_rel O mset_poly_rel"
disc_11: fixes x11 :: "'a"   and x12 :: "'b"   and x13 :: "'b" shows "\<not> isRecv (Trans x11 x12 x13)"
wpc_least: fixes t :: "'e itself"   and Rs :: "(('f, 'g) labeled_graph \<times> ('f, 'g) labeled_graph) set"   and S :: "nat \<Rightarrow> ('f, 'h) labeled_graph" assumes "WPC t Rs S" shows "least t Rs (S 0) (chain_sup S)"
inj_mirror_point: fixes x :: "'a" shows "inj (mirror_point x)"
fvi_trm_Suc: fixes x :: "nat"   and b :: "nat"   and t :: "'b trm" shows "(x \<in> fvi_trm (Suc b) t) = (Suc x \<in> fvi_trm b t)"
analz_image_keys: fixes evs :: "event list"   and KK :: "nat set"   and N :: "nat" assumes "evs \<in> tls"   and "KK \<subseteq> range sessionK" shows "(Nonce N \<in> analz (Key ` KK \<union> knows Spy evs)) = (Nonce N \<in> analz (knows Spy evs))"
max_def_raw: shows "max \<equiv> \<lambda>a b. if a \<le> b then b else a"
upd_2: fixes i :: "nat"   and m :: "nat"   and M1 :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and f :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and n :: "nat"   and N :: "nat"   and j :: "nat" assumes "i \<le> m" shows "DBM_upd M1 f (Suc m) n N i j = DBM_upd M1 f (Suc m) 0 N i j"
CN_2: fixes l :: "float poly"   and r :: "float poly"   and ord :: "nat" assumes "(l, r) = split_by_degree ord p" shows "(maxdegree l \<le> ord \<and> Ipoly x (map_poly real_of_float p) = Ipoly x (map_poly real_of_float l) + Ipoly x (map_poly real_of_float r)) \<and> num_params l \<le> num_params p \<and> num_params r \<le> num_params p"
mkPoint_img_1: fixes a :: "'s" shows "S.img \<in> S.hom S.unity a \<rightarrow> S.set a"
cases: fixes x :: "'b tree" obtains hsx :: "'b tree" and xa :: "'b" and hs :: "'b tree" where "x = \<langle>hsx, xa, hs\<rangle>"   | "x = \<langle>\<rangle>"
equiv_appL: fixes s :: "dB"   and s' :: "dB"   and t :: "dB" assumes "s \<leftrightarrow> s'" shows "s \<degree> t \<leftrightarrow> s' \<degree> t"
h1: shows "FlexRayController (nReturn k) recv (nC k) (nStore k) (nSend k) (nGet k)"
subsetD: fixes A :: "'a set"   and B :: "'a set"   and c :: "'a" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
submodule_of_mdl: fixes H :: "'a set"   and N :: "'a set" assumes "submodule R M H"   and "submodule R M N"   and "H \<subseteq> N" shows "submodule R (mdl M N) H"
select_convs_1: fixes vars :: "char list list"   and body :: "stmt"   and more :: "'a" shows "vars \<lparr>vars = vars, body = body, \<dots> = more\<rparr> = vars"
lfp_gen_induct: fixes f :: "'c \<Rightarrow> 'c"   and m0 :: "'c"   and P :: "'c" assumes "mono f"   and "m0 \<le> P"   and "\<And>m. \<lbrakk>m \<le> lfp (\<lambda>s. sup m0 (f s)); m \<le> P; f m \<le> lfp (\<lambda>s. sup m0 (f s))\<rbrakk> \<Longrightarrow> f m \<le> P" shows "lfp (\<lambda>s. sup m0 (f s)) \<le> P"
den_conv: shows "den = pure snd \<diamondop> stern_brocot_recurse"
to_nat_less_card: fixes a :: "'a" shows "to_nat a < CARD('a)"
prod_coprime: fixes A :: "'b set"   and g :: "'b \<Rightarrow> nat" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<noteq> y\<rbrakk> \<Longrightarrow> coprime (g x) (g y)" shows "f (prod g A) = (\<Prod>x\<in>A. f (g x))"
simps_187: fixes x101 :: "Term.var"   and x102 :: "expr" shows "x101:=x102 \<noteq> Super"
expand_finfun_eq: fixes f :: "'a \<Rightarrow>f 'b"   and g :: "'a \<Rightarrow>f 'b" shows "(f = g) = (($) f = ($) g)"
mk_v_agree: fixes I :: "('b, 'c, 'a) interp"   and ODE :: "('b, 'a) ODE"   and \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and sol :: "(real, 'a) vec" shows "Vagree (mk_v I ODE \<nu> sol) \<nu> (- semBV I ODE) \<and> Vagree (mk_v I ODE \<nu> sol) (mk_xode I ODE sol) (semBV I ODE)"
nAct_strict_mono_back: fixes tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and n :: "enat"   and t :: "cnf llist"   and n' :: "enat" assumes "dynamic_component.nAct tCMP active c n t < dynamic_component.nAct tCMP active c n' t" shows "n < n'"
a: shows "p_dst p \<in> ipcidr_union_set (set i_ips)"
SD_inefficientI: fixes q :: "'alt pmf"   and p :: "'alt pmf"   and i :: "'agent" assumes "q \<in> lotteries_on alts"   and "\<And>i. i \<in> agents \<Longrightarrow> SD (R i) p q"   and "i \<in> agents"   and "p \<prec>[SD (R i)] q" shows "\<not> SD_efficient R p"
46: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and u :: "'a" shows "x \<sqinter> y \<squnion> (z \<squnion> u \<sqinter> y) = z \<squnion> (x \<squnion> u) \<sqinter> y"
methd_Object_SomeD: fixes G :: "prog"   and sig :: "sig"   and m :: "qtname \<times> methd" assumes "wf_prog G"   and "methd G Object sig = Some m" shows "declclass m = Object"
test_restrictl: fixes p :: "'a"   and x :: "'a" assumes "test p" shows "p \<cdot> x \<le> x"
mono_inv: fixes f :: "'c \<Rightarrow> 'd" assumes "mono f"   and "bij f" shows "mono (inv f)"
Subspace_Span: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and ms :: "'b list" assumes "VectorSpace smult V"   and "set ms \<subseteq> V" shows "RModule UNIV smult (R_scalar_mult.RSpan UNIV smult ms) \<and> R_scalar_mult.RSpan UNIV smult ms \<subseteq> V"
mgu_new: fixes t1 :: "typ"   and t2 :: "typ"   and u :: "nat \<Rightarrow> typ"   and n :: "nat" assumes "mgu t1 t2 = Some u"   and "new_tv n t1"   and "new_tv n t2" shows "new_tv n u"
nfa_of_dfa_def: shows "nfa_of_dfa = (\<lambda>(bdd, as). (map (bdd_map (\<lambda>q. (replicate (length bdd) False)[q := True])) bdd, as))"
one_sheaf_spec_in_sheaf_spec: fixes U :: "'a set set" assumes "is_zariski_open U" shows "one_sheaf_spec U \<in> \<O> U"
length_finite_permutations_of_set: fixes xs :: "'b list"   and A :: "'b set" assumes "xs \<in> permutations_of_set A" shows "length xs = card A"
run_update: fixes p :: "'d ref"   and x :: "'d"   and \<sigma> :: "heap option"   and \<sigma>' :: "heap option"   and r :: "unit" assumes "run (p := x) \<sigma> \<sigma>' r"   and "\<not> is_exn \<sigma>" obtains "\<sigma>' = Some (Ref.set p x (the_state \<sigma>))" and "r = ()"
lg_null_instance: fixes l :: "'p \<Rightarrow> 'p \<Rightarrow> bool"   and A :: "'p" assumes "QCongNull l" shows "l A A"
butlast_tail: fixes Xs :: "'a list"   and X :: "'a"   and Y :: "'a" shows "butlast (Xs @ [X, Y]) = Xs @ [X]"
shows_list_mat_def: fixes As :: "'a mat list" shows "shows_list As \<equiv> shows_sep shows_mat (shows '', '') As"
that: fixes n :: "int" assumes "x = of_int n"   and "n \<le> 0" shows "thesis"
cod_char: fixes f :: "'a \<times> 'a" shows "S.VV.cod f = (if S.VV.arr f then S.VxV.cod f else S.VxV.null)"
float_le: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" assumes "is_finite a"   and "is_finite b" shows "(a \<le> b) = (valof a \<le> valof b)"
mcont2mcont': fixes lub' :: "'e set \<Rightarrow> 'e"   and ord' :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'a"   and lub :: "'d set \<Rightarrow> 'd"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and t :: "'d \<Rightarrow> 'e" assumes "\<And>x. mcont lub' ord' Sup (\<le>) (f x)"   and "\<And>y. mcont lub ord Sup (\<le>) (\<lambda>x. f x y)"   and "mcont lub ord lub' ord' t" shows "mcont lub ord Sup (\<le>) (\<lambda>x. f x (t x))"
induct: fixes P :: "('b word \<times> 'b word) list \<Rightarrow> bool"   and a0 :: "('b word \<times> 'b word) list" assumes "P []"   and "\<And>s e. P [(s, e)]"   and "\<And>s e v va. P (v # va) \<Longrightarrow> P ((s, e) # v # va)" shows "P a0"
measurable_add_dim: fixes i :: "'a"   and I :: "'a set"   and M :: "'a \<Rightarrow> 'b measure" shows "(\<lambda>(f, y). f(i := y)) \<in> Pi\<^sub>M I M \<Otimes>\<^sub>M M i \<rightarrow>\<^sub>M Pi\<^sub>M (insert i I) M"
invmode_IntVir_eq: fixes m :: "'b member_scheme"   and e :: "expr" shows "(invmode m e = IntVir) = (\<not> is_static m \<and> e \<noteq> Super)"
CollectD: fixes a :: "'g"   and P :: "'g \<Rightarrow> bool" assumes "a \<in> {x. P x}" shows "P a"
set_val_multiset: fixes t :: "'c Tree"   and v' :: "'c" assumes "t \<noteq> E" shows "Heap.multiset (set_val t v') + {#val t#} = {#v'#} + Heap.multiset t"
g: shows "\<guillemotleft>g : G a \<rightarrow>\<^sub>D G a'\<guillemotright>"
intro: fixes Q :: "value \<Rightarrow> value \<Rightarrow> bool" assumes "\<And>t\<^sub>1 t\<^sub>2. Q t\<^sub>1 t\<^sub>2 \<Longrightarrow> veq_structure t\<^sub>1 t\<^sub>2"   and "\<And>name ts name' us. Q (Vconstr name ts) (Vconstr name' us) = (name = name' \<and> list_all2 Q ts us)" shows "value_struct_rel Q"
finite_supp: fixes permute :: "perm \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e" assumes "class.fs permute" shows "finite (pt.supp permute x)"
RInitRed: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and Cs :: "char list list"   and e\<^sub>0 :: "char list exp" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>RI (C,e) ; Cs \<leftarrow> e\<^sub>0,s,b\<rangle> \<rightarrow> \<langle>RI (C,e') ; Cs \<leftarrow> e\<^sub>0,s',b'\<rangle>"
Lim_withinI: fixes S :: "'a set"   and a :: "'a"   and f :: "'a \<Rightarrow> 'b"   and l :: "'b" assumes "\<And>e. 0 < e \<Longrightarrow> \<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l \<le> e" shows "(f \<longlongrightarrow> l) (at a within S)"
strict_monoD: fixes F :: "ordinal \<Rightarrow> ordinal"   and x :: "ordinal"   and y :: "ordinal" assumes "normal F"   and "x < y" shows "F x < F y"
sublens_iff_sublens': fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b" assumes "vwb_lens X"   and "vwb_lens Y" shows "X \<subseteq>\<^sub>L Y = X \<subseteq>\<^sub>L' Y"
coplanar_perm_19: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet D A C B"
IH: shows "\<forall>m<n. \<forall>\<Gamma> \<Delta>. ( \<Gamma> \<oplus> F \<nabla> [x].A \<Rightarrow>* \<Delta>, m) \<in> derivable  R* \<longrightarrow> (\<forall>r'\<in>R. leftPrincipal r' (F \<nabla> [x].A) \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')) \<longrightarrow> \<not> multSubst \<Gamma>' \<and> \<not> multSubst \<Delta>' \<longrightarrow> (\<exists>m'\<le>m. ( \<Gamma> + \<Gamma>' \<Rightarrow>* \<Delta> + \<Delta>', m') \<in> derivable  R*)"
qtable_union: fixes n :: "nat"   and A :: "nat set"   and P :: "'b option list \<Rightarrow> bool"   and Q1 :: "'b option list \<Rightarrow> bool"   and X :: "'b option list set"   and Q2 :: "'b option list \<Rightarrow> bool"   and Y :: "'b option list set"   and Q :: "'b option list \<Rightarrow> bool" assumes "qtable n A P Q1 X"   and "qtable n A P Q2 Y"   and "\<And>x. \<lbrakk>wf_tuple n A x; P x\<rbrakk> \<Longrightarrow> Q x = (Q1 x \<or> Q2 x)" shows "qtable n A P Q (X \<union> Y)"
upd_commute: fixes a :: "'a"   and a' :: "'a"   and b :: "'b"   and b' :: "'b"   and c :: "'c" assumes "a \<noteq> a'" shows "upd a b (upd a' b' c) = upd a' b' (upd a b c)"
merge0_inv: shows "\<forall>I I'. oneone I \<and> oneone I' \<longrightarrow> PlaneGraphIso.compat I' I \<longrightarrow> oneone (merge0 I' I)"
simps_2: fixes A :: "'a set"   and As :: "'a set list" shows "listset (A # As) = set_Cons A (listset As)"
restrict_map_def: fixes m :: "'g \<Rightarrow> 'h option"   and A :: "'g set" shows "m |` A = (\<lambda>x. if x \<in> A then m x else None)"
hom_induced_hom: shows "GroupHom (Group.quotient_group (FreeGroup S) \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) F"
trimmingD_\<E>: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "trimming \<Gamma> f g" shows "\<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> g) - A \<Gamma> = \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) - A \<Gamma>"
l3_defs_6: fixes implem :: "chan \<Rightarrow> msg" assumes "sklvl3 implem" shows "l3_init \<equiv> {\<lparr>ik = ik_init, secret = {}, progress = Map.empty, signalsInit = \<lambda>x. 0, signalsResp = \<lambda>x. 0, signalsInit2 = \<lambda>x. 0, signalsResp2 = \<lambda>x. 0, l3_state.bad = bad_init\<rparr>}"
nat_even_induct: fixes n :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "gcd_nat.greater_eq n 2"   and "P 0"   and "\<And>m. \<lbrakk>gcd_nat.greater_eq m 2; P m\<rbrakk> \<Longrightarrow> P (Suc (Suc m))" shows "P n"
zs_gt_ys: shows "msetext_dersh gt zs ys"
roots3_generic_def: fixes depressed_solver :: "'a \<Rightarrow> 'a \<Rightarrow> 'a list"   and p :: "'a poly" shows "roots3_generic depressed_solver p = (let cs = coeffs p; a = cs ! 3; b = cs ! 2; c = cs ! 1; d = cs ! 0; a3 = (3::'a) * a; ba3 = b / a3; b2 = b * b; b3 = b2 * b; e = (c - b2 / a3) / a; f = (d + (2::'a) * b3 / ((27::'a) * a\<^sup>2) - b * c / a3) / a in Let (depressed_solver e f) (map (\<lambda>y. y - ba3)))"
OF_match_fields_def: fixes m :: "of_match_field set"   and p :: "(32, 'b) simple_packet_ext_scheme" shows "OF_match_fields m p = map_option all_true (set_seq ((\<lambda>f. match_prereq f m p) ` m))"
nonzero_append: fixes xs :: "'a list"   and ps :: "pexpr list"   and qs :: "pexpr list" shows "nonzero xs (ps @ qs) = (nonzero xs ps \<and> nonzero xs qs)"
le_multiset_right_total: fixes M :: "'a multiset"   and x :: "'a" shows "M < add_mset x M"
continuous_map_axioms_def: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and is_open' :: "'e set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e" shows "continuous_map_axioms S is_open is_open' f \<equiv> \<forall>U. is_open' U \<longrightarrow> is_open (f \<^sup>\<inverse> S U)"
eq: shows "complex_of_real (cos (2 * a)) = complex_of_real (cos a) * complex_of_real (cos a) - complex_of_real (sin a) * complex_of_real (sin a)"
ring_simprules_10: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y = y \<oplus>\<^bsub>R\<^esub> x"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>) (\<cdot>) F (F \<circ> F) (F \<circ> \<phi>'.map)"
UNlift01_inj_on_UNpart1: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set" assumes "length cl = length dl"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "\<And>n. n < length dl \<Longrightarrow> part {..<brn (dl ! n)} (F n ` P n) \<and> {} \<notin> F n ` P n"   and "\<And>n. n < length cl \<Longrightarrow> inj_on (F n) (P n)" shows "inj_on (UNlift01 cl dl P F) (UNpart1 cl dl P)"
subset_image_iff: fixes B :: "'b set"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "order.greater_eq (f ` A) B = (\<exists>AA. order.greater_eq A AA \<and> B = f ` AA)"
SeqConstP_lemma: fixes m :: "name"   and s :: "tm"   and k :: "tm"   and c :: "tm"   and n :: "name"   and sm :: "name"   and sn :: "name" assumes "atom m \<sharp> (s, k, c, n, sm, sn)"   and "atom n \<sharp> (s, k, c, sm, sn)"   and "atom sm \<sharp> (s, k, c, sn)"   and "atom sn \<sharp> (s, k, c)" shows "{SeqConstP s k c} \<turnstile> c EQ Zero OR SyntaxN.Ex m (SyntaxN.Ex n (SyntaxN.Ex sm (SyntaxN.Ex sn (Var m IN k AND Var n IN k AND SeqConstP s (Var m) (Var sm) AND SeqConstP s (Var n) (Var sn) AND c EQ Q_Eats (Var sm) (Var sn)))))"
strongly_connected_spanning_imp_strongly_connected: fixes H :: "('a, 'b) pre_digraph" assumes "spanning H (G \<restriction> verts H)"   and "strongly_connected H" shows "strongly_connected (G \<restriction> verts H)"
ld: shows "LeftDerivation \<alpha> (take (ladder_n L' (Suc index)) D') (ladder_\<gamma> \<alpha> D' L' (Suc index))"
k_frag: fixes x :: "'b" assumes "x \<in> T" shows "k (frag_of x) = (frag_of \<circ> g) x"
hyps_3: fixes c :: "('a, 'b, 'c) com" shows "\<Gamma>\<turnstile> (subst_redex c r, s) \<rightarrow>\<^sup>+ (subst_redex c r', s')"
rel_resource_neg_distr: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and A' :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and B' :: "'h \<Rightarrow> 'g \<Rightarrow> bool" assumes "left_unique A"   and "right_total A"   and "right_unique A'"   and "left_total A'" shows "rel_resource (A OO A') (B OO B') \<le> rel_resource A B OO rel_resource A' B'"
\<sigma>_2: shows "bij_betw \<sigma> (subst_domain \<sigma>) (subst_range \<sigma>)"
RedT_preserves_ts_inv_ok: fixes final :: "'g \<Rightarrow> bool"   and r :: "('j, 'h, 'g, 'i, 'k, 'l) semantics"   and convert_RA :: "'j \<Rightarrow>f nat \<Rightarrow> 'l list"   and s :: "('j, 'h) locks \<times> (('h \<Rightarrow> ('g \<times> 'j \<Rightarrow>f nat) option) \<times> 'i) \<times> ('h \<Rightarrow> 'k wait_set_status option) \<times> 'h set"   and ttas :: "('h \<times> ('j, 'h, 'g, 'i, 'k, 'l) thread_action) list"   and s' :: "('j, 'h) locks \<times> (('h \<Rightarrow> ('g \<times> 'j \<Rightarrow>f nat) option) \<times> 'i) \<times> ('h \<Rightarrow> 'k wait_set_status option) \<times> 'h set"   and I :: "'h \<Rightarrow> 'm option"   and Q :: "'m \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> 'i \<Rightarrow> bool" assumes "multithreaded_base.RedT final r convert_RA s ttas s'"   and "ts_inv_ok (thr s) I" shows "ts_inv_ok (thr s') (upd_invs I Q (concat (map (thr_a \<circ> snd) ttas)))"
nlt: fixes A :: "'p"   and B :: "'p" shows "\<not> A B Lt A B"
hbspec: fixes A :: "hf"   and P :: "hf \<Rightarrow> bool"   and x :: "hf" assumes "HBall A P"   and "x \<^bold>\<in> A" shows "P x"
assms_2: shows "\<not> valid_unMultigraph.connected (del_unEdge v w v' G)"
covering: fixes X :: "'a set"   and U :: "'a set"   and index :: "real set"   and cover :: "real \<Rightarrow> 'a set" assumes "cover_of_subset X U index cover" shows "U \<subseteq> \<Union> (cover ` index)"
wtc_less: fixes pc :: "nat" assumes "stable r step \<tau>s pc"   and "Suc pc < length \<tau>s" shows "wtc c pc (\<tau>s ! pc) \<sqsubseteq>\<^bsub>r\<^esub> \<tau>s ! Suc pc"
mcont: fixes x :: "'a" shows "mcont lSup (\<sqsubseteq>) lub ord (\<lambda>xs. f x xs (LCons x xs))"
additive_subgroup_def: fixes H :: "'c set"   and G :: "('c, 'd) ring_scheme" shows "additive_subgroup H G \<equiv> subgroup H (add_monoid G)"
partCompat_gen: fixes theta :: "('a \<times> 'a) set"   and P :: "'b set set"   and f :: "'b \<Rightarrow> 'a"   and I :: "'b set" assumes "sym theta"   and "trans theta"   and "partCompat P theta f"   and "I \<in> P" shows "Resumption_Based.compat (gen P I) theta f"
analz_synth_analz: fixes H :: "msg set" shows "analz (synth (analz H)) = synth (analz H)"