hcju/isabelleps huggingface dataset:
finite_abs_reach: fixes \<phi> :: "'b ltl" shows "finite (af_abs.abs_reach (Abs \<phi>))"
rel_funE: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h"   and x :: "'e"   and y :: "'f" assumes "(A ===> B) f g"   and "A x y" obtains "B (f x) (g y)"
inverse_arrowsI: fixes g :: "'a"   and f :: "'a" assumes "ide (g \<cdot> f)"   and "ide (f \<cdot> g)" shows "inverse_arrows f g"
intros_2: fixes tr :: "('state, 'act, 'out) trans list"   and trn :: "('state, 'act, 'out) trans" assumes "valid2 tr"   and "tgtOf (last tr) = srcOf trn"   and "validTrans trn" shows "valid2 (tr ## trn)"
def_lfp_unfold: fixes h :: "'c"   and f :: "'c \<Rightarrow> 'c" assumes "h \<equiv> lfp f"   and "mono f" shows "h = f h"
wpt_def: fixes x :: "'a" shows "wpt x = x * \<top> \<sqinter> (1::'a)"
lookup_fresh: fixes x :: "name"   and y :: "name"   and \<theta> :: "(name \<times> trm) list" assumes "x = y \<longrightarrow> x \<in> set (map fst \<theta>)"   and "\<forall>(y, t)\<in>set \<theta>. x \<sharp> t" shows "x \<sharp> lookup \<theta> y"
sg: fixes i :: "'a" assumes "i \<in> I" shows "subgroup (H i) (G i)"
nth_tabulate: fixes k :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'c"   and x :: "nat" assumes "k < n" shows "tabulate f x n ! k = f (x + k)"
WTrt_iffs_1: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and P :: "'c J_prog"   and h :: "'b"   and E :: "String.literal \<Rightarrow> ty option"   and v :: "'c val"   and T :: "ty" shows "J_heap_base.WTrt typeof_addr P h E (Val v) T = (heap_base.typeof_h typeof_addr h v = \<lfloor>T\<rfloor>)"
congruentI: fixes r :: "('a \<times> 'a) set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>y z. (y, z) \<in> r \<Longrightarrow> f y = f z" shows "f respects r"
genPrefix_mono: fixes r :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "r \<subseteq> s" shows "genPrefix r \<subseteq> genPrefix s"
borel_measurable_indicator_iff: fixes A :: "'a set"   and M :: "'a measure" shows "(indicator A \<in> borel_measurable M) = (A \<inter> space M \<in> sets M)"
even_odd_mirror_path_involution_2: fixes path :: "dir list" shows "odd_mirror_path (odd_mirror_path path) = path"
abstract_for_simple_firewall_def: shows "abstract_for_simple_firewall \<equiv> abstract_primitive (\<lambda>r. case r of Pos a \<Rightarrow> is_CT_State a \<or> is_L4_Flags a | Neg a \<Rightarrow> is_Iiface a \<or> is_Oiface a \<or> is_Prot a \<or> is_CT_State a \<or> is_L4_Flags a)"
normalize_bounds_1_main: fixes eps :: "rat"   and x :: "int poly \<times> rat \<times> rat" assumes "0 < eps"   and "invariant_1_2 x" shows "invariant_1_2 (normalize_bounds_1_main eps x) \<and> real_of_1 (normalize_bounds_1_main eps x) = real_of_1 x"
indep: fixes c :: "'b set" assumes "c \<in> C" shows "independent c"
simps_10: fixes x1 :: "'b" shows "\<^bold>\<langle>x1\<^bold>\<rangle> \<noteq> \<^bold>\<I>"
converse_iff: fixes a :: "'h"   and b :: "'i"   and r :: "('i \<times> 'h) set" shows "((a, b) \<in> r\<inverse>) = ((b, a) \<in> r)"
nonabort_i: shows "chaos \<sqsubseteq> (i::'a::refinement_lattice)"
valid_ref_def: fixes A :: "'a attree"   and A' :: "'a attree" shows "A \<sqsubseteq>\<^sub>V A' \<equiv> A \<sqsubseteq> A' \<and> \<turnstile>A'"
to_list_ga_rec_rule: fixes is_set :: "'a set \<Rightarrow> 'b \<Rightarrow> assn"   and is_it :: "'a set \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'c \<Rightarrow> assn"   and it_init :: "'b \<Rightarrow> 'c Heap"   and it_has_next :: "'c \<Rightarrow> bool Heap"   and it_next :: "'c \<Rightarrow> ('a \<times> 'c) Heap"   and is_list :: "'a list \<Rightarrow> 'd \<Rightarrow> assn"   and l_prepend :: "'a \<Rightarrow> 'd \<Rightarrow> 'd Heap"   and it :: "'a set"   and s :: "'a set"   and si :: "'b"   and iti :: "'c"   and l :: "'a list"   and li :: "'d" assumes "imp_set_iterate is_set is_it it_init it_has_next it_next"   and "imp_list_prepend is_list l_prepend"   and "finite it" shows "<is_it s si it iti * is_list l li> to_list_ga_rec it_has_next it_next l_prepend iti li <\<lambda>r. \<exists>\<^sub>Al'. is_set s si * is_list l' r * \<up> (set l' = set l \<union> it)>\<^sub>t"
rev_curr_def: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set"   and t :: "('f \<times> 'f) set"   and f :: "'g \<times> 'f \<Rightarrow> 'h"   and b :: "'f" assumes "Well_order r"   and "Well_order s"   and "Well_order t" shows "Ordinal_Arithmetic.rev_curr t f b = (if b \<in> Field t then \<lambda>a. f (a, b) else undefined)"
FGModuleHomSet_def: fixes G :: "'g set"   and fgsmult :: "('h, 'g) aezfun \<Rightarrow> 'e \<Rightarrow> 'e"   and V :: "'e set"   and fgsmult' :: "('h, 'g) aezfun \<Rightarrow> 'f \<Rightarrow> 'f"   and W :: "'f set" shows "FGModuleHomSet G fgsmult V fgsmult' W \<equiv> {T. FGModuleHom G fgsmult V fgsmult' T} \<inter> {T. T ` V \<subseteq> W}"
set_conv_nth: fixes xs :: "'c list" shows "set xs = {xs ! i |i. i < length xs}"
distinct_7: fixes x2 :: "('a, 'b) Term.term"   and x31 :: "poscheckvariant"   and x32 :: "('a, 'b) Term.term"   and x33 :: "('a, 'b) Term.term" shows "receive\<langle>x2\<rangle>\<^sub>s\<^sub>t \<noteq> \<langle>x31: x32 \<doteq> x33\<rangle>\<^sub>s\<^sub>t"
simps_4: fixes f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'e list \<Rightarrow> 'd" shows "(case [] of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f1"
C\<^sub>i\<^sub>i\<^sub>i: fixes y :: "i" shows "((\<^bold>\<forall>x. cod y \<cdot> x \<cong> x \<^bold>\<leftarrow> E (cod y \<cdot> x)) \<^bold>\<and> (\<^bold>\<forall>x. x \<cdot> cod y \<cong> x \<^bold>\<leftarrow> E (x \<cdot> cod y))) \<^bold>\<and> cod y \<cdot> y \<cong> y \<^bold>\<leftarrow> E y"
keys_focus: fixes X :: "'a set"   and p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "keys (focus X p) = (\<lambda>t. except t (- X)) ` keys p"
mk_eqcl_fst_snd: fixes zs :: "nat list"   and xs :: "nat option list"   and ii :: "nat"   and i :: "nat"   and T :: "bool list list" assumes "\<And>i. i < length zs \<Longrightarrow> zs ! i < length xs + ii \<and> (ii \<le> zs ! i \<longrightarrow> xs ! (zs ! i - ii) = Some i)"   and "\<And>j1 j2. \<lbrakk>j1 < j2; j2 < length zs\<rbrakk> \<Longrightarrow> zs ! j1 < zs ! j2"   and "\<And>z. z \<in> set zs \<Longrightarrow> z < ii"   and "i < length (snd (mk_eqcl xs zs ii T))"   and "length xs + ii \<le> length T + 1" shows "snd (mk_eqcl xs zs ii T) ! i < length (fst (mk_eqcl xs zs ii T)) + ii \<and> (ii \<le> snd (mk_eqcl xs zs ii T) ! i \<longrightarrow> fst (mk_eqcl xs zs ii T) ! (snd (mk_eqcl xs zs ii T) ! i - ii) = i)"
R': shows "graph_defs.is_run (degeneralize_ext m) r'"
nop_onto_vdomain: fixes A :: "V"   and n :: "V"   and f :: "V" assumes "nop_onto A n f" shows "\<D>\<^sub>\<circ> f = A ^\<^sub>\<times> n"
has_as_product_def: fixes J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'b \<Rightarrow> nat"   and a :: "nat" shows "J.has_as_product J D a \<equiv> \<exists>\<pi>. product_cone J J.comp D a \<pi>"
v2': shows "v = punit.lt (q g2) \<oplus> lt g2"
integrable_reflect: fixes f :: "'c \<Rightarrow> 'd"   and b :: "'c"   and a :: "'c" shows "((\<lambda>x. f (- x)) integrable_on cbox (- b) (- a)) = (f integrable_on cbox a b)"
Cons_8: shows "getConflictFlag stateA = getConflictFlag stateB"
Lambert_W_exp_times_self: fixes x :: "real" assumes "- 1 \<le> x" shows "Lambert_W (x * exp x) = x"
simps_6: fixes f1 :: "'a"   and f2 :: "(nat \<Rightarrow> InfTree \<times> 'a) \<Rightarrow> 'a" shows "rec_InfTree f1 f2 InfTree.Leaf = f1"
reflexive_axioms: shows "reflexive A (\<lambda>x y. y \<sqsubseteq> x)"
rm_vars_apply': fixes \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and vs :: "'c set" assumes "subst_domain \<delta> \<inter> vs = {}" shows "rm_vars vs \<delta> = \<delta>"
list_of_oalist_tc_of_list_id: fixes xs :: "('a \<times> 'b) list" assumes "tc.oalist_inv_raw xs" shows "list_of_oalist_tc (OAlist_tc xs) = xs"
pmf_K2: fixes s :: "'s"   and t :: "'s" assumes "\<And>n. prob {\<omega> \<in> space M. X n \<omega> = s} = 0" shows "pmf (K s) t = indicat_real {t} s"
fset_of_fmap_iff: fixes a :: "'c"   and b :: "'d"   and m :: "('c, 'd) fmap" shows "((a, b) |\<in>| fset_of_fmap m) = (fmlookup m a = Some b)"
real_cond_exp_add: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "integrable M f"   and "integrable M g" shows "AE x in M. real_cond_exp M F (\<lambda>x. f x + g x) x = real_cond_exp M F f x + real_cond_exp M F g x"
commut_normal: fixes H :: "'a set"   and N :: "'a set" assumes "subgroup H G"   and "N \<lhd> G" shows "H <#> N = N <#> H"
simps_2: fixes x :: "dB" shows "norm (Pure x) = Pure x"
conc_Un_distrib_1: fixes A :: "'a list set"   and B :: "'a list set"   and C :: "'a list set" shows "A @@ (B \<union> C) = A @@ B \<union> A @@ C"
times_fun_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "f * g = (\<lambda>x. f x * g x)"
non_irreducible_mat_split: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "\<not> irreducible_mat A"   and "1 < n" shows "\<exists>n1 n2 B B1 B2 B4. similar_mat A B \<and> elements_mat A = elements_mat B \<and> B = four_block_mat B1 B2 (0\<^sub>m n2 n1) B4 \<and> B1 \<in> carrier_mat n1 n1 \<and> B2 \<in> carrier_mat n1 n2 \<and> B4 \<in> carrier_mat n2 n2 \<and> 0 < n1 \<and> n1 < n \<and> 0 < n2 \<and> n2 < n \<and> n1 + n2 = n"
deg_comp_prod_def: fixes cmp :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> order" shows "deg_comp cmp = (\<lambda>x y. case comparator_of (deg' (fst x)) (deg' (fst y)) of Eq \<Rightarrow> cmp x y | Lt \<Rightarrow> Lt | Gt \<Rightarrow> Gt)"
q_3: shows "poly (of_int_poly q) (exp 1) = 0"
map_def: fixes f :: "'d" shows "\<epsilon>'F_F\<eta>.map f = (if D.arr f then (\<epsilon>'.map \<circ> F) (D.cod f) \<cdot>\<^sub>C (F \<circ> \<eta>) f else C.null)"
Hi': shows "(\<lambda>u. snd (blinfun_apply Hi (u, 0::'c))) = blinfun_apply (- T o\<^sub>L f' (x, y) o\<^sub>L embed1_blinfun)"
pair_in_Id_conv: fixes a :: "'b"   and b :: "'b" shows "((a, b) \<in> Id) = (a = b)"
assms: shows "h \<turnstile> get_root_node_si ptr \<rightarrow>\<^sub>r root"
ereal_tendsto_simps2_2: fixes f :: "'b \<Rightarrow> real"   and F :: "'b filter" shows "((ereal \<circ> f) \<longlongrightarrow> \<infinity>) F = filterlim f at_top F"
distrib_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a \<squnion> b) \<cdot> c = a \<cdot> c \<squnion> b \<cdot> c"
measure_unat: fixes p :: "'a word" assumes "p \<noteq> 0" shows "unat (p - 1) < unat p"
heap_is_wellformed_def: shows "heap_is_wellformed = (\<lambda>h. heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h \<and> acyclic (parent_child_rel h \<union> local.a_host_shadow_root_rel h \<union> local.a_ptr_disconnected_node_rel h) \<and> local.a_all_ptrs_in_heap h \<and> local.a_distinct_lists h \<and> local.a_shadow_root_valid h)"
continuous_map_product_then_coordinatewise_1: fixes T1 :: "'c topology"   and T :: "'d \<Rightarrow> 'e topology"   and I :: "'d set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and i :: "'d" assumes "continuous_map T1 (product_topology T I) f"   and "i \<in> I" shows "continuous_map T1 (T i) (\<lambda>x. f x i)"
poly_y_x_map_poly_pCons_0: fixes p :: "'a poly poly" shows "poly_y_x (map_poly (pCons (0::'a)) p) = pCons 0 (poly_y_x p)"
assms_9: shows "g \<circ> h \<circ> u \<le>\<le> u \<circ> h \<circ> g"
card_UN_disjoint: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set" assumes "finite I"   and "\<forall>i\<in>I. finite (A i)"   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}" shows "card (\<Union> (A ` I)) = (\<Sum>i\<in>I. card (A i))"
map_simp_2: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and H :: "'d \<Rightarrow> 'e"   and \<sigma> :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and f :: "'d" assumes "vertical_composite A B F G H \<sigma> \<tau>"   and "partial_magma.arr A f" shows "vertical_composite.map A B \<sigma> \<tau> f = B (\<tau> f) (\<sigma> (partial_magma.dom A f))"
simps_7: fixes C :: "'c"   and \<phi> :: "('b + 'e, 'c, 'd) formula"   and \<sigma> :: "'e \<Rightarrow> ('b, 'd) trm" shows "FsubstFO (InContext C \<phi>) \<sigma> = InContext C (FsubstFO \<phi> \<sigma>)"
assms_3: shows "\<forall>s\<in>pair ` setops\<^sub>s\<^sub>s\<^sub>t S. \<forall>t\<in>pair ` setops\<^sub>s\<^sub>s\<^sub>t S. (\<exists>\<delta>. Unifier \<delta> s t) \<longrightarrow> \<Gamma> s = \<Gamma> t"
projs_1: fixes dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat" shows "\<pi>\<^sub>2 (dsn, dsk, flag, hops, nhip) = dsn"
Cons_nth_drop_Suc: fixes i :: "nat"   and xs :: "'e list" assumes "i < length xs" shows "xs ! i # drop (Suc i) xs = drop i xs"
nth_swap: fixes i :: "nat"   and xs :: "'a list"   and j :: "nat"   and k :: "nat" assumes "i < length xs"   and "j < length xs" shows "swap xs i j ! k = (if k = i then xs ! j else if k = j then xs ! i else xs ! k)"
open_map_def: fixes X1 :: "'a topology"   and X2 :: "'b topology"   and f :: "'a \<Rightarrow> 'b" shows "open_map X1 X2 f \<equiv> \<forall>U. openin X1 U \<longrightarrow> openin X2 (f ` U)"
rep_eq: fixes x :: "\<o>" shows "eval\<o> \<lparr>x\<rparr> = eval\<o> x"
cases_1: assumes "v \<in> verts G" shows "P 0 v"
norm_v_ge_0: fixes v :: "(complex, 'n) Finite_Cartesian_Product.vec" shows "le_vec 0 (map_vector cmod v)"
i_Exec_Comp_Stream_Acc_Output__eq_Msg_State_Idle_conv: fixes k :: "nat"   and localState :: "'a \<Rightarrow> 'b"   and output_fun :: "'a \<Rightarrow> 'c message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'd message_af"   and c :: "'a"   and t :: "nat"   and m :: "'c message_af"   and t0 :: "nat"   and s :: "nat \<Rightarrow> 'a" assumes "Suc 0 < k"   and "State_Idle localState output_fun trans_fun (i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun input c t)"   and "m \<noteq> NoMsg"   and "t0 = t * k"   and "s = i_Exec_Comp_Stream trans_fun (input \<odot> k) c" shows "(i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c t = m) = (\<not> State_Idle localState output_fun trans_fun (localState (s t2)). t2 \<U> t1 [0\<dots>] \<oplus> t0. output_fun (s t1) = m \<and> (State_Idle localState output_fun trans_fun (localState (s t1)) \<or> (\<circle> t3 t1 [0\<dots>]. output_fun (s t4) = NoMsg. t4 \<U> t5 [0\<dots>] \<oplus> t3. output_fun (s t5) = NoMsg \<and> State_Idle localState output_fun trans_fun (localState (s t5)))))"
preprm_unit_involution: fixes a :: "'b"   and b :: "'b" shows "preprm_compose (preprm_unit a b) (preprm_unit a b) =p preprm_id"
Cons_eq_appendI: fixes x :: "'e"   and xs1 :: "'e list"   and ys :: "'e list"   and xs :: "'e list"   and zs :: "'e list" assumes "x # xs1 = ys"   and "xs = xs1 @ zs" shows "x # xs = ys @ zs"
ipresSubst_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'varSort, 'k, 'l, 'var, 'h, 'm, 'n) model_scheme" shows "ipresSubst h MOD \<equiv> \<forall>ys Y y s X. wls (asSort ys) Y \<and> wls s X \<longrightarrow> h (X #[Y / y]_ys) = igSubst MOD ys (h Y) y (h X)"
tm_exhaust_apps: fixes s :: "('a, 'b) tm" obtains \<zeta> :: "('a, 'b) hd" and ss :: "('a, 'b) tm list" where "s = apps (Hd \<zeta>) ss"
assms_4: shows "homotopic_with (\<lambda>k. True) Y Y (f \<circ> g) id"
img_is_subring: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e"   and K :: "'c set" assumes "ring_hom_ring R S h"   and "subring K R" shows "subring (h ` K) S"
facet_unique_other_chamber: fixes B :: "'a set"   and z :: "'a set"   and C :: "'a set"   and D :: "'a set" assumes "chamber B"   and "z \<lhd> B"   and "chamber C"   and "z \<lhd> C"   and "chamber D"   and "z \<lhd> D"   and "C \<noteq> B"   and "D \<noteq> B" shows "C = D"
starI: fixes x :: "'a"   and u :: "real" assumes "x \<in> S"   and "0 \<le> u"   and "u < 1" shows "u *\<^sub>R x \<in> rel_interior S"
size_if_splay_max: fixes t :: "'b tree"   and l :: "'b tree"   and u :: "'b"   and r :: "'b tree" assumes "splay_max t = \<langle>l, u, r\<rangle>" shows "size t = size l + size r + 1"
add_less_imp_less_left: fixes c :: "'a multiset"   and a :: "'a multiset"   and b :: "'a multiset" assumes "c + a \<subset># c + b" shows "a \<subset># b"
numeral_neq_fps_zero: fixes f :: "num" shows "numeral f \<noteq> 0"
weakCongTrans_def: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs" shows "P \<Longrightarrow>\<alpha> \<prec> P' \<equiv> \<exists>P'' P'''. P \<Longrightarrow>\<^sub>\<tau> P'' \<and> P'' \<longmapsto>\<alpha> \<prec> P''' \<and> P''' \<Longrightarrow>\<^sub>\<tau> P'"
word_le_rbl: fixes x :: "'a word"   and y :: "'a word" shows "(x \<le> y) = rev_bl_order True (rev (to_bl x)) (rev (to_bl y))"
FF_def: shows "FF = Exists (Neg (formula.Eq (Var 0) (Var 0)))"
elims: fixes x :: "'b literal"   and xa :: "('b \<times> 'b trm) list"   and y :: "'b literal" assumes "subst_lit x xa = y" obtains e :: "'b equation" and s :: "('b \<times> 'b trm) list" where "x = Pos e__" and "xa = s__" and "y = Pos (subst_equation e__ s__)"   | e :: "'b equation" and s :: "('b \<times> 'b trm) list" where "x = Neg e" and "xa = s" and "y = Neg (subst_equation e s)"
satisfies_state_def: fixes v :: "nat \<Rightarrow> 'b"   and s :: "('c, 'b) state" shows "v \<Turnstile>\<^sub>s s \<equiv> v \<Turnstile>\<^sub>b \<B> s \<and> v \<Turnstile>\<^sub>t \<T> s"
ysq_dom_y: fixes b :: "real" shows "\<exists>w. \<forall>y<w. b * y < y\<^sup>2"
lvl_split: fixes t :: "('b \<times> nat) tree" shows "lvl (split t) = lvl t \<or> lvl (split t) = lvl t + 1 \<and> sngl (split t)"
set_plus_image: fixes S :: "'a set"   and T :: "'a set" shows "S + T = (\<lambda>(x, y). x + y) ` (S \<times> T)"
width_add1: fixes x :: "nat"   and y :: "nat" shows "\<forall>ts v. (\<exists>va u. v=va--u \<and> |lan va| = x \<and> |lan u| = y) \<longrightarrow> |lan v| = x + y"
states_alt_def: fixes r :: "(('e \<times> 'f) \<times> 'd) list"   and p :: "'d" shows "states r p = map snd r"
irrefl_distinct: fixes r :: "('b \<times> 'b) set" shows "irrefl r = (\<forall>(a, b)\<in>r. a \<noteq> b)"
assert_top: shows "{\<cdot> \<top> } = (1::'c)"
\<theta>: shows "\<theta> \<equiv> \<lambda>y. if x = y then t else \<sigma> y"
universe_eq: shows "u = (THE x. \<forall>y. P y x)"
contractible_space_singleton: fixes X :: "'b topology"   and a :: "'b" assumes "topspace X = {a}" shows "contractible_space X"
suminf_PInfty_fun: fixes f :: "nat \<Rightarrow> ereal" assumes "\<And>i. 0 \<le> f i"   and "suminf f \<noteq> \<infinity>" shows "\<exists>f'. f = (\<lambda>x. ereal (f' x))"
homotopy_eqv_contractible_sets: fixes S :: "'b set"   and T :: "'c set" assumes "contractible S"   and "contractible T"   and "(S = {}) = (T = {})" shows "S homotopy_eqv T"
process_ret_split_obvious: fixes rs\<^sub>1 :: "'c rule list"   and rs\<^sub>2 :: "'c rule list" shows "process_ret (rs\<^sub>1 @ rs\<^sub>2) = process_ret rs\<^sub>1 @ add_missing_ret_unfoldings rs\<^sub>1 (process_ret rs\<^sub>2)"
butlast_tl: fixes xs :: "'c list" shows "butlast (tl xs) = tl (butlast xs)"
implements_actJP_jAction: fixes jp :: "'a \<Rightarrow> ('cobs \<times> 'as option, 'aAct, 'b) Protocol"   and t :: "('a, 'es, 'as) BEState Trace"   and n :: "nat"   and a :: "'a" assumes "SPR.implements jp"   and "t \<in> SPR.jkbpCn n" shows "set (SPR.actJP jp t a) = set (jAction (SPR.MCn n) t a)"
vconst_on_vempty: fixes c :: "V" shows "vconst_on []\<^sub>\<circ> c = []\<^sub>\<circ>"
Abs_stone_phi_pair_inject: fixes x :: "'c regular \<times> 'c dense Filters.filter"   and y :: "'c regular \<times> 'c dense Filters.filter" assumes "x \<in> stone_phi.pairs"   and "y \<in> stone_phi.pairs" shows "(Abs_stone_phi_pair x = Abs_stone_phi_pair y) = (x = y)"
rmshadow_sound: fixes P :: "'i simple_packet set"   and rs :: "'i simple_rule list" assumes "p \<in> P" shows "simple_fw (rmshadow rs P) p = simple_fw rs p"
clof: fixes T :: "'a set" assumes "T \<in> \<F>" shows "compact T"
weak_meet_assoc_lemma: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<sqinter> (y \<sqinter> z) .= \<Sqinter>{x, y, z}"
prems: shows "a # A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta> = B1 @ B2"
rel_cong: fixes x :: "'c list"   and ya :: "'c list"   and y :: "'d list"   and xa :: "'d list"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and Ra :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "x = ya"   and "y = xa"   and "\<And>z yb. \<lbrakk>z \<in> set ya; yb \<in> set xa\<rbrakk> \<Longrightarrow> R z yb = Ra z yb" shows "list_all2 R x y = list_all2 Ra ya xa"
assms_7: fixes y :: "'a" assumes "y \<in> A_distinct" shows "x \<le> y"
intros_8: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c1 :: "('a, 'c) com"   and s :: "'b"   and s' :: "'b"   and c2 :: "('a, 'c) com" assumes "PL.transT tval aval (c1, s) s'" shows "PL.transC tval aval (c1 | c2, s) (c2, s')"
envObs_rel_equiv: fixes a :: "'a" shows "equiv UNIV (rel_ext (envObs_rel (envObs a)))"
concat_is_inverse_of_embed: fixes xs :: "'a list" shows "concat (List_Supplement.embed xs) = xs"
set_val_locs_impl: shows "set_val_locs = a_set_val_locs"
dedekind: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x * y \<sqinter> z \<le> (x \<sqinter> z * y\<^sup>T) * (y \<sqinter> x\<^sup>T * z)"
gcd_abs2_int: fixes x :: "int"   and y :: "int" shows "gcd x \<bar>y\<bar> = gcd x y"
fps_mult_nth_1': fixes f :: "'a fps"   and g :: "'a fps" shows "fps_nth (f * g) (Suc 0) = fps_nth f 0 * fps_nth g (Suc 0) + fps_nth f (Suc 0) * fps_nth g 0"
laplace_transform_zero: fixes f :: "real \<Rightarrow> complex"   and M :: "real"   and a :: "real"   and t :: "real" assumes "continuous_on {0..} f"   and "exponential_order M a f"   and "\<And>s. a < Re s \<Longrightarrow> (f has_laplace 0) s"   and "0 \<le> t" shows "f t = 0"
lookup_update': fixes k :: "'c"   and v :: "'b"   and m :: "('c, 'b) mapping"   and k' :: "'c" shows "Mapping.lookup (Mapping.update k v m) k' = (if k = k' then Some v else Mapping.lookup m k')"
span_base: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and S :: "'b set" assumes "Modules.module scale"   and "a \<in> S" shows "a \<in> module.span scale S"
disc_58: fixes x8 :: "'b ltl" shows "is_LTLGlobal (G x8)"
tfr_stp_all_same_type: fixes S :: "('fun, 'var) strand_step list"   and a :: "poscheckvariant"   and t :: "('fun, 'var) Term.term"   and t' :: "('fun, 'var) Term.term"   and S' :: "('fun, 'var) strand_step list"   and \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "list_all tfr\<^sub>s\<^sub>t\<^sub>p (S @ \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S')"   and "Unifier \<delta> t t'" shows "\<Gamma> t = \<Gamma> t'"
induct: fixes P :: "'b tree \<Rightarrow> bool"   and a0 :: "'b tree" assumes "\<And>uu_ uv_. P \<langle>uu_, uv_, \<langle>\<rangle>\<rangle>"   and "\<And>hs1 uw_ hs2 ux_. P \<langle>hs1, uw_, \<langle>hs2, ux_, \<langle>\<rangle>\<rangle>\<rangle>"   and "\<And>hs1 uy_ hs2 uz_ v va vb. P \<langle>v, va, vb\<rangle> \<Longrightarrow> P \<langle>hs1, uy_, \<langle>hs2, uz_, \<langle>v, va, vb\<rangle>\<rangle>\<rangle>"   and "P \<langle>\<rangle>" shows "P a0"
sig_trd_irred: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b" shows "\<not> is_sig_red (\<prec>\<^sub>t) (\<preceq>) (set bs) (sig_trd bs p)"
Probabilistic_Timed_Automaton_Regions_Reachability_axioms: shows "Probabilistic_Timed_Automaton_Regions_Reachability k v n not_in_X A \<phi> \<psi> s"
subgraph_isomorphic_pre_subgraph_closed: fixes G\<^sub>1 :: "nat set \<times> nat set set"   and G\<^sub>2 :: "nat set \<times> nat set set"   and G\<^sub>3 :: "nat set \<times> nat set set" assumes "uwellformed G\<^sub>1"   and "subgraph G\<^sub>1 G\<^sub>2"   and "G\<^sub>2 \<sqsubseteq> G\<^sub>3" shows "G\<^sub>1 \<sqsubseteq> G\<^sub>3"
is_tdghm_axioms: shows "\<NN> : \<HH> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<HH>' : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<DD>"