hcju/isabelleps huggingface dataset:
type_definition_Domainp: fixes Rep :: "'d \<Rightarrow> 'e"   and Abs :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and T :: "'e \<Rightarrow> 'd \<Rightarrow> bool" assumes "type_definition Rep Abs A"   and "T \<equiv> \<lambda>x y. x = Rep y" shows "Domainp T = (\<lambda>x. x \<in> A)"
exhaust: fixes y :: "color" obtains "y = color.R"   | "y = color.B"
sl4: shows "2 * e * b * d + - 2 * a * e\<^sup>2 + - 2 * d * f \<in> radical_sqrt"
elims: fixes x :: "'b \<Rightarrow> 'b"   and xa :: "'b tree"   and y :: "'b tree" assumes "update_rightmost x xa = y" obtains uu_ :: "'b \<Rightarrow> 'b" where "x = uu_" and "xa = Leaf" and "y = Leaf"   | f :: "'b \<Rightarrow> 'b" and i :: "nat" and j :: "nat" and a :: "'b option" and t :: "'b tree" and u :: "'b tree" where "x = f" and "xa = Node i j a t u" and "y = Node i j (map_option f a) t (update_rightmost f u)"
invar_reachable: fixes I :: "nat \<times> nat \<times> (nat \<Rightarrow> thread) \<Rightarrow> bool"   and s :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)" assumes "A.is_invar I"   and "A.reachable s" shows "I s"
as_proj_pair_2: fixes vs :: "'d set" shows "as_proj [] vs = []"
disjoint_def: shows "disjoint TYPE('a) TYPE('b) \<equiv> (\<forall>pi x. pi \<bullet> x = x) \<and> (\<forall>pi x. pi \<bullet> x = x)"
disc_1: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "lnull xs"   and "lnull ys" shows "lnull (lappend xs ys)"
monotoneI: fixes orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x y. orda x y \<Longrightarrow> ordb (f x) (f y)" shows "monotone orda ordb f"
round_up_float: fixes prec :: "int"   and x :: "real" shows "round_up prec x \<in> float"
monomial_inj: fixes s :: "'a"   and c :: "'b"   and t :: "'a"   and d :: "'b" assumes "monomial c s = monomial d t" shows "c = (0::'b) \<and> d = (0::'b) \<or> c = d \<and> s = t"
openin_trans_full: fixes X :: "'a topology"   and U :: "'a set"   and S :: "'a set" assumes "openin (subtopology X U) S"   and "openin X U" shows "openin X S"
space: shows "space N = (\<Pi>\<^sub>E i\<in>I. space (M i))"
mult_isol_var: fixes u :: "'a"   and x :: "'a"   and v :: "'a"   and y :: "'a" assumes "u \<le> x"   and "v \<le> y" shows "v \<cdot> u \<le> y \<cdot> x"
iter_bind_iter: fixes guard :: "'a \<Rightarrow> bool"   and body :: "'a \<Rightarrow> 'a spmf"   and m :: "nat"   and s :: "'a"   and n :: "nat" shows "loop_spmf.iter guard body m s \<bind> loop_spmf.iter guard body n = loop_spmf.iter guard body (m + n) s"
that: fixes p :: "'a list"   and s :: "'a list" assumes "p \<cdot> u \<cdot> s = v" shows "thesis"
mult_scalar_distrib_right: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and p :: "'b \<Rightarrow>\<^sub>0 'd"   and q :: "'b \<Rightarrow>\<^sub>0 'd"   and r :: "'a \<Rightarrow>\<^sub>0 'd" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.mult_scalar pair_of_term term_of_pair (p + q) r = term_powerprod.mult_scalar pair_of_term term_of_pair p r + term_powerprod.mult_scalar pair_of_term term_of_pair q r"
elims: fixes x :: "'a Point"   and xa :: "'a Point"   and y :: "'a Vector" assumes "from x to xa = y" obtains p :: "'a Point" and q :: "'a Point" where "x = p" and "xa = q" and "y = \<lparr>tdir = tval q - tval p, xdir = xval q - xval p, ydir = yval q - yval p, zdir = zval q - zval p\<rparr>"
Lt_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" shows "A B Lt C D \<equiv> A B Le C D \<and> \<not> Cong A B C D"
asIMOD_igSwapAbs: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and A :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> ('varSort, 'b) sAbs" shows "igSwapAbs (asIMOD SEM) zs z1 z2 A = (\<lambda>val. if sWlsVal SEM val then A (val ^[z1 \<and> z2]_zs) else undefined)"
monom_eq_iff: fixes a :: "'b"   and n :: "nat"   and b :: "'b" shows "(monom a n = monom b n) = (a = b)"
simps_2: fixes x21 :: "'a"   and x22 :: "'a FOL_Fitting.term list"   and y21 :: "'a"   and y22 :: "'a FOL_Fitting.term list" shows "(App x21 x22 = App y21 y22) = (x21 = y21 \<and> x22 = y22)"
star_of_approx_inverse: fixes x :: "'a star"   and xa :: "'a" assumes "x \<approx> star_of xa"   and "xa \<noteq> (0::'a)" shows "inverse x \<approx> inverse (star_of xa)"
concurrent_store_delete_independent: fixes i :: "'a"   and e1 :: "'b"   and mo :: "'a"   and ir :: "'a"   and "is" :: "'a set"   and e2 :: "'b"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat" assumes "\<not> hb (i, Store e1 mo i) (ir, Delete is e2)"   and "\<not> hb (ir, Delete is e2) (i, Store e1 mo i)"   and "xs prefix of j"   and "(i, Store e1 mo i) \<in> set (node_deliver_messages xs)"   and "(ir, Delete is e2) \<in> set (node_deliver_messages xs)" shows "i \<notin> is"
IH: assumes "wf\<^sub>s\<^sub>t (V \<union> fv t) S"   and "\<Union> (fv\<^sub>r\<^sub>c\<^sub>v ` set S') \<union> \<Union> (fv_r\<^sub>e\<^sub>q assign ` set S') \<subseteq> wfvarsoccs\<^sub>s\<^sub>t S \<union> (V \<union> fv t)" shows "wf\<^sub>s\<^sub>t (V \<union> fv t) (S @ S')"
fmadd_assoc: fixes m :: "('c, 'd) fmap"   and n :: "('c, 'd) fmap"   and p :: "('c, 'd) fmap" shows "m ++\<^sub>f (n ++\<^sub>f p) = m ++\<^sub>f n ++\<^sub>f p"
no_infinite_computation_implies_wf: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate" assumes "\<not> \<Gamma>\<turnstile> (c, s) \<rightarrow> \<dots>(\<infinity>)" shows "wf {(c2, c1). \<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* c1 \<and> \<Gamma>\<turnstile> c1 \<rightarrow> c2}"
add_arc_in: fixes a :: "'b" assumes "a \<in> arcs G" shows "add_arc a = G"
distr_id: fixes N :: "'b measure" shows "distr N N (\<lambda>x. x) = N"
oVeblen_def: fixes F :: "ordinal \<Rightarrow> ordinal" shows "oVeblen F = (\<lambda>x. OrdinalVeblen.ordering (critical_set (range F) x))"
c6: assumes "ip \<in> kD rt"   and "\<pi>\<^sub>2 r \<le> sqn rt ip"   and "\<pi>\<^sub>3 r = kno"   and "sqn rt ip = \<pi>\<^sub>2 r \<Longrightarrow> the (dhops rt ip) \<le> \<pi>\<^sub>5 r \<and> the (flag rt ip) = val" shows "P (rt(ip \<mapsto> addpre (the (rt ip)) (\<pi>\<^sub>7 r)))"
connected_component_of_eq_empty: fixes X :: "'b topology"   and x :: "'b" shows "(connected_component_of_set X x = {}) = (x \<notin> topspace X)"
split_stream_first: fixes A :: "'a set"   and xs :: "'a stream" assumes "A \<inter> sset xs \<noteq> {}" obtains ys :: "'a list" and a :: "'a" and zs :: "'a stream" where "xs = ys @- a ## zs" and "A \<inter> set ys = {}" and "a \<in> A"
eventually_frequently_const_simps_4: fixes C :: "bool"   and P :: "'c \<Rightarrow> bool"   and F :: "'c filter" shows "(\<forall>\<^sub>F x in F. C \<or> P x) = (C \<or> eventually P F)"
hyps_1: shows "e \<turnstile> s : T \<Rightarrow> U"
add_less_same_cancel1: fixes b :: "'d multiset"   and a :: "'d multiset" shows "(b + a \<subset># b) = (a \<subset># {#})"
rep_eq: fixes x :: "'b fset" shows "fcard x = card (fset x)"
winning_path_LCons: fixes p :: "Player"   and P :: "'a llist"   and v :: "'a" assumes "winning_path p P"   and "\<not> lnull P" shows "winning_path p (LCons v P)"
assms_3: shows "(s + t) +^ k = (0::'a)"
F1: shows "\<forall>v u. u \<in> O(v) \<longrightarrow> O(u) \<subseteq> O(v)"
g_def: shows "g = Cn 1 r_ifeq [c, r_length, m, Cn 1 r_ifz [p, at1, at0]]"
a_Inj2_inv_I: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty" assumes "\<Gamma> \<turnstile> Inj2 e, eP, eV : Syntax.Sum \<tau>\<^sub>1 \<tau>\<^sub>2" obtains eP :: "Syntax.term" and eV :: "Syntax.term" where "ePa__ = Inj2 eP" and "eVa__ = Inj2 eV" and "\<Gamma> \<turnstile> e, eP, eV : \<tau>\<^sub>2"
known_ptrs_new_ptr: fixes h' :: "(_) heap"   and h :: "(_) heap"   and new_ptr :: "(_) object_ptr" assumes "object_ptr_kinds h' = object_ptr_kinds h |\<union>| {|new_ptr|}"   and "ShadowRootClass.known_ptr new_ptr"   and "ShadowRootClass.known_ptrs h" shows "ShadowRootClass.known_ptrs h'"
hol: shows "(\<lambda>w. (w - z) * f w) holomorphic_on S"
pred_inject_1: fixes P1 :: "'d \<Rightarrow> bool"   and P2 :: "'e \<Rightarrow> bool"   and P3 :: "'f \<Rightarrow> bool"   and a :: "'d" shows "pred_generat P1 P2 P3 (Pure a) = P1 a"
fps_ginv_def: fixes b :: "'a fps"   and a :: "'a fps" shows "fps_ginv b a = Abs_fps (gcompinv b a)"
linear_injective_ker_0: fixes A :: "(('b, 'c) vec, 'd) vec" shows "inj ((*v) A) = ({x. A *v x = 0} = {0})"
eq_intro: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool" assumes "\<phi> \<pi>l \<Longrightarrow> \<psi> \<pi>l"   and "\<psi> \<pi>l \<Longrightarrow> \<phi> \<pi>l" shows "eq \<phi> \<psi> \<pi>l"
comp_ide_arr: fixes b :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VV.ide b"   and "VV.seq b f" shows "VV.comp b f = f"
polynomial_rupture: fixes K :: "('c multiset \<Rightarrow> 'a) set"   and p :: "('c multiset \<Rightarrow> 'a) list" assumes "subring K L"   and "p \<in> carrier (K [X]\<^bsub>L\<^esub>)" shows "ring.eval (Rupt\<^bsub>L\<^esub> K p) (map (L.rupture_surj K p \<circ> L.poly_of_const) p) (L.rupture_surj K p X\<^bsub>L\<^esub>) = \<zero>\<^bsub>Rupt\<^bsub>L\<^esub> K p\<^esub>"
aug_zero_unique: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and A :: "'e set"   and z1 :: "'e" assumes "commute_bpp f (aug_pm_set z i A)"   and "z1 \<in> addition_set f (aug_pm_set z i A)"   and "zeroA z i f A z"   and "zeroA z i f A z1" shows "z = z1"
le_bot: fixes a :: "'a" assumes "a \<le> (zero_class.zero::'a)" shows "a = (zero_class.zero::'a)"
map_iterator_rule_insert_P: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and m :: "'e \<Rightarrow> 'f option"   and I :: "'e set \<Rightarrow> 'd \<Rightarrow> bool"   and \<sigma>0 :: "'d"   and c :: "'d \<Rightarrow> bool"   and f :: "'e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd"   and P :: "'d \<Rightarrow> bool" assumes "map_iterator it m"   and "I {} \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> dom m - it; m k = Some v; it \<subseteq> dom m; I it \<sigma>\<rbrakk> \<Longrightarrow> I (insert k it) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I (dom m) \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> dom m; \<not> c \<sigma>; I it \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
assms_1: shows "\<lbrace>c\<rbrace> P \<lbrace>d\<^sub>1\<rbrace>\<^sub>u"
sin_cos_squared_add: fixes x :: "'b" shows "(sin x)\<^sup>2 + (cos x)\<^sup>2 = (1::'b)"
fv_subset: shows "\<Union> (set A_neg) \<subseteq> \<Union> (set A_pos)"
B: shows "is_tangle_diagram (basic (make_vert_block (nat (codomain_block x)))) \<and> is_tangle_diagram ys \<longrightarrow> is_tangle_diagram (basic (make_vert_block (nat (codomain_block x))) \<otimes> ys)"
section_is_mono: fixes g :: "'s" assumes "section g" shows "local.mono g"
vec_of_scal_def: fixes x :: "'b" shows "vec_of_scal x \<equiv> vec 1 (\<lambda>i. x)"
add_diff_inverse: fixes a :: "'b"   and b :: "'b" assumes "a \<le> b" shows "a + (b - a) = b"
embedS_or_iso: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Well_order r"   and "Well_order r'" shows "(\<exists>g. embedS r r' g) \<or> (\<exists>h. embedS r' r h) \<or> (\<exists>f. iso r r' f)"
in_homE: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" obtains "C.arr f" and "C.dom f = a" and "C.cod f = b"
mgenerator2Tr2: fixes n :: "nat"   and j :: "nat"   and k :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and m :: "int" assumes "0 < n"   and "j \<le> n"   and "k \<le> n"   and "distinct_pds K n P" shows "(\<nu>\<^bsub>K P j\<^esub>) (mprod_exp K (K_gamma k) (Kb\<^bsub>K n P\<^esub>) n\<^bsub>K\<^esub>\<^bsup>m\<^esup>) = ant (m * (\<gamma>\<^bsub>k j\<^esub>))"
diff_minus_eq_add: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.group_add minus plus zero uminus" shows "minus a (uminus b) = plus a b"
primI: fixes u :: "'b list" assumes "\<And>r k. r \<^sup>@ k = u \<Longrightarrow> k = 1" shows "primitive u"
bot_join: fixes x :: "'a" assumes "x \<in> carrier L" shows "\<bottom> \<squnion> x = x"
inner_zero_right: fixes x :: "'a" shows "inner_field x (0::'a) = (0::'a)"
mC_C_def: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" shows "mC_C theta c d s t P F \<equiv> mC_C_part c d P F \<and> inj_on F P \<and> mC_C_wt c d s t P F \<and> mC_C_eff_cont theta c d s t P F"
assms: shows "\<forall>op\<in>set ops. v \<notin> set (add_effects_of op) \<and> v \<notin> set (delete_effects_of op)"
measurable_restrict_subset: fixes J :: "'d set"   and L :: "'d set"   and M :: "'d \<Rightarrow> 'e measure" assumes "J \<subseteq> L" shows "(\<lambda>f. restrict f J) \<in> Pi\<^sub>M L M \<rightarrow>\<^sub>M Pi\<^sub>M J M"
sdnets_charn: fixes a :: "('b, 'c) Combinators" assumes "onlyTwoNets a"   and "\<not> member DenyAll a" shows "sdnets a = {(first_srcNet a, first_destNet a)} \<or> sdnets a = {(first_srcNet a, first_destNet a), (first_destNet a, first_srcNet a)}"
commute: fixes a :: "'e"   and b :: "'e" shows "a * b = b * a"
simps_1: fixes xs :: "'a list" shows "vec_mat_Tensor xs [] = []"
simps_3: fixes C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom" shows "kdeps (C\<^sub>1;; C\<^sub>2) = kdeps C\<^sub>1 \<union> fune C\<^sub>1 (kdeps C\<^sub>2)"
fall_elim: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and F :: "('a \<times> 'b set) stream \<Rightarrow> ('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<pi>l :: "('a \<times> 'b set) stream list" assumes "Shallow S s0 \<delta> AP L AP'"   and "Shallow.fall S s0 \<delta> AP L AP' F \<pi>l" obtains "\<And>\<pi>. \<lbrakk>Kripke.wfp S \<delta> AP L AP' \<pi>; \<pi>l \<noteq> [] \<Longrightarrow> stateOf \<pi> = stateOf (last \<pi>l); \<pi>l = [] \<Longrightarrow> stateOf \<pi> = s0\<rbrakk> \<Longrightarrow> F \<pi> \<pi>l"
weak: shows "\<forall>x xs. inDed xs \<longrightarrow> inDed (x # xs)"
match_ex_table_shift: fixes P :: "'a prog"   and C :: "String.literal"   and pc :: "nat"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and pc' :: "nat"   and d :: "nat"   and n :: "nat" assumes "match_ex_table P C pc xt = \<lfloor>(pc', d)\<rfloor>" shows "match_ex_table P C (n + pc) (shift n xt) = \<lfloor>(n + pc', d)\<rfloor>"
DirProd_group: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme" assumes "Group.group G"   and "Group.group H" shows "Group.group (G \<times>\<times> H)"
assms_2: shows "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^sub>.\<^sub>w\<^sub>i\<^sub>d\<^sub>e\<^bsub>\<alpha>\<^esub> \<AA>"
natural_transformation_axioms: shows "natural_transformation D C F Adj.FGF.map (F \<circ> \<eta>)"
finite_valid_event_ids: shows "finite (Collect is_valid_event_id)"
component_secure_prot_stateful_def: fixes n :: "'a"   and P :: "('a strand_label \<times> ('fun, 'var) stateful_strand_step) list set"   and Sec :: "('fun, 'var) Term.term set"   and attack :: "'fun" shows "component_secure_prot_stateful n P Sec attack \<equiv> \<forall>\<A>\<in>P. suffix [\<langle>n, send\<langle>Fun attack []\<rangle>\<rangle>] \<A> \<longrightarrow> (\<forall>\<I>\<^sub>\<tau>. (subst_domain \<I>\<^sub>\<tau> = \<V> \<and> ground (subst_range \<I>\<^sub>\<tau>)) \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I>\<^sub>\<tau> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<I>\<^sub>\<tau>) \<longrightarrow> \<not> \<lbrakk>{}; {}; proj_unl n \<A>\<rbrakk>\<^sub>s \<I>\<^sub>\<tau> \<and> (\<forall>\<A>'. prefix \<A>' \<A> \<longrightarrow> (\<forall>t\<in>Sec - declassified\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A>' \<I>\<^sub>\<tau>. \<not> \<lbrakk>{}; {}; proj_unl n \<A>' @ [send\<langle>t\<rangle>]\<rbrakk>\<^sub>s \<I>\<^sub>\<tau>)))"
rows_A_n: shows "{row A i |i. i < nr} \<subseteq> carrier_vec n"
closure_eq_empty: fixes S :: "'b set" shows "(closure S = {}) = (S = {})"
subclseq_ObjectD: fixes C :: "qtname"   and G :: "prog" assumes "G\<turnstile>Object\<preceq>\<^sub>C C" shows "C = Object"
ln_lower_5_def: shows "ln_lower_5 \<equiv> \<lambda>x. - ln_upper_5 (inverse x)"
inf_continuousD: fixes F :: "'b \<Rightarrow> 'c"   and M :: "nat \<Rightarrow> 'b" assumes "inf_continuous F"   and "decseq M" shows "F (Inf (range M)) = (INF i. F (M i))"
i_aggregate_i_drop_mod: fixes n :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'b"   and ag :: "'b list \<Rightarrow> 'b" assumes "n mod k = 0" shows "i_aggregate (f \<Up> n) k ag = i_aggregate f k ag \<Up> (n div k)"
thing4: shows "- (2 ^ 31) \<le> sint (SCAST(32 \<rightarrow> 64) w2)"
HierAuto_def: fixes D :: "'a data"   and F :: "('b, 'c, 'a) seqauto set"   and E :: "'c set"   and G :: "'b \<Rightarrow> ('b, 'c, 'a) seqauto set option" shows "HierAuto D F E G = (\<Union> (SAEvents ` F) \<subseteq> E \<and> MutuallyDistinct F \<and> finite F \<and> IsCompFun F G)"
image_smult_cbox: fixes m :: "real"   and a :: "'b"   and b :: "'b" shows "(*\<^sub>R) m ` cbox a b = (if cbox a b = {} then {} else if 0 \<le> m then cbox (m *\<^sub>R a) (m *\<^sub>R b) else cbox (m *\<^sub>R b) (m *\<^sub>R a))"
a_Lam_inv_V: fixes v :: "Syntax.term"   and vP :: "Syntax.term"   and x :: "Syntax.var"   and vV' :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty" assumes "{$$} \<turnstile> v, vP, Lam x vV' : Fun \<tau>\<^sub>1 \<tau>\<^sub>2" obtains v' :: "Syntax.term" and vP' :: "Syntax.term" where "v = Lam x v'" and "vP = Lam x vP'" and "{$$}(x $$:= \<tau>\<^sub>1) \<turnstile> v', vP', vV' : \<tau>\<^sub>2"
norm_mig_componentwise_le: fixes x :: "'c"   and a :: "'c"   and b :: "'c" assumes "x \<in> {a..b}" shows "norm (mig_componentwise a b) \<le> norm x"
pow_mat_dim_square_2: fixes A :: "'b mat"   and n :: "nat"   and k :: "nat" assumes "A \<in> carrier_mat n n" shows "dim_col (A ^\<^sub>m k) = n"
compE2_not_Return: fixes e :: "'a expr1" shows "Return \<notin> set (compE2 e)"
inorder_ins: fixes t :: "('b \<times> bal) tree"   and x :: "'b" assumes "avl t"   and "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (tree (ins x t)) = ins_list x (Tree2.inorder t)"
B1_def: shows "B1 = (\<Sum>(i, k)\<in>I. real_of_int (- 2 * (poly.coeff p i)\<^sup>2) / real_of_int (int (k ^ 3)))"
cdf_cts_step_1: fixes M :: "real measure"   and x :: "real"   and y :: "real" assumes "real_distribution M"   and "x < y" shows "cdf M x \<le> integral\<^sup>L M (cts_step x y)"
distinct_3: fixes x31 :: "'a"   and x32 :: "'a ifex"   and x33 :: "'a ifex" shows "Trueif \<noteq> IF x31 x32 x33"
prems: shows "\<exists>a\<in>set (xs @ [x]). P a"
path_append_elim: fixes r :: "'transition list"   and s :: "'transition list"   and p :: "'state" assumes "path (r @ s) p" obtains "path r p" and "path s (target r p)"
space_restrict_space: fixes M :: "'a measure"   and \<Omega> :: "'a set" shows "space (restrict_space M \<Omega>) = \<Omega> \<inter> space M"
has_fps_expansion_sin: fixes c :: "'a" shows "(\<lambda>x. sin (c * x)) has_fps_expansion fps_sin c"
typeof_simps_2: fixes i :: "int" shows "\<tau> intgV i = IntgT"
cc_restr_join: fixes S :: "Vars.var set"   and G :: "CoCalls"   and G' :: "CoCalls" shows "(G \<squnion> G') G|` S = G G|` S \<squnion> G' G|` S"
cases: fixes a1 :: "'a"   and a2 :: "'a"   and \<delta> :: "('a, 'b) ta_rule set" assumes "(a1, a2) \<in> f_succ \<delta>" obtains q :: "'a" and l :: "'b" and qs :: "'a list" and q' :: "'a" where "a1 = q" and "a2 = q'" and "q \<rightarrow> l qs \<in> \<delta>" and "q' \<in> set qs"
eq_rowI: fixes B :: "'b mat"   and A :: "'b mat" assumes "\<And>i. i < dim_row B \<Longrightarrow> row A i = row B i"   and "dim_row A = dim_row B"   and "dim_col A = dim_col B" shows "A = B"
finite_theNFTBoth: fixes cl :: "('d, 'e, 'f) cmd list"   and dl :: "('g, 'h, 'i) cmd list" shows "finite (theNFTBoth cl dl)"
noInSends: fixes p :: "'p"   and s :: "'s"   and m :: "'v messageValue"   and p2 :: "'p"   and v :: "bool" shows "sends p s m <p2, inM v> = 0"
bres_iso: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b"   and z :: "'a \<times> 'b" assumes "less_eq_prod x y" shows "less_eq_prod (prod.bres z x) (prod.bres z y)"
simps: fixes l0 :: "'a"   and r0 :: "'b"   and l1 :: "'a"   and r1 :: "'b"   and a :: "'a"   and b :: "'b" shows "idx (Bound (l0, r0) (l1, r1)) (a, b) = idx0 (Bound l0 l1) a * size1 (Bound r0 r1) + idx1 (Bound r0 r1) b"
wf_Fields_Ex: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "String.literal" assumes "wf_prog wf_md P"   and "is_class P C" shows "\<exists>FDTs. P \<turnstile> C has_fields FDTs"
valid_regions_distinct: fixes X :: "'a set"   and k :: "'a \<Rightarrow> nat"   and I :: "'a \<Rightarrow> intv"   and r :: "('a \<times> 'a) set"   and I' :: "'a \<Rightarrow> intv"   and r' :: "('a \<times> 'a) set"   and v :: "'a \<Rightarrow> real" assumes "valid_region X k I r"   and "valid_region X k I' r'"   and "v \<in> region X I r"   and "v \<in> region X I' r'" shows "region X I r = region X I' r'"
zero_less_ceiling: fixes x :: "'b" shows "(0 < \<lceil>x\<rceil>) = ((0::'b) < x)"
X_def: shows "X = (\<lambda>y. if y \<in> J then V y else UNIV)"
oc': fixes F :: "char list"   and T :: "ty"   and D :: "char list" assumes "P \<turnstile> C\<^sub>0 has F,NonStatic:T in D" shows "\<exists>v. fs (F, D) = \<lfloor>v\<rfloor> \<and> P,h \<turnstile> v :\<le> T"
group_elements_finite: fixes G :: "'a set" assumes "G \<in> \<G>" shows "finite G"
osafe_OSing_simps: fixes x :: "'b"   and \<theta> :: "('a, 'b) trm" shows "osafe (OSing x \<theta>) = dfree \<theta>"
name_upd_incoming: fixes n :: "('b, 'd) node_scheme"   and ns :: "('b, 'c) node_scheme set" shows "name ` upd_incoming n ns = name ` ns"
l3_inv6I: fixes x :: "l3_state" assumes "\<And>A B M. implInsec A B M \<in> synth (analz (ik x)) \<and> M \<in> payload \<Longrightarrow> implInsec A B M \<in> ik x \<or> M \<in> synth (analz (ik x))" shows "x \<in> l3_inv6"
set_less_eq_aux''_def: shows "(\<sqsubseteq>''') = (SOME sleq. (linear_order {(x, y). x \<le> y} \<longrightarrow> linear_order_on infinite_complement_partition {(A, B). sleq A B}) \<and> order_consistent {(A, B). A \<sqsubseteq>'' B} {(A, B). sleq A B})"
comm_ring_isom_axioms: shows "comm_ring_isom poly_y_x"
map_of_delete_aux': fixes xs :: "('c \<times> 'd) list"   and k :: "'c" assumes "distinct (map fst xs)" shows "map_of (delete_aux k xs) = (map_of xs)(k := None)"
fMin_fimage_Suc: fixes x :: "nat"   and A :: "nat fset" assumes "x |\<in>| A" shows "fMin (Suc |`| A) = Suc (fMin A)"
word_bw_lcs_1: fixes y :: "'b word"   and x :: "'b word"   and z :: "'b word" shows "y AND x AND z = x AND y AND z"
islimpt_eq_infinite_ball: fixes x :: "'b"   and S :: "'b set" shows "x islimpt S = (\<forall>e>0. infinite (S \<inter> ball x e))"
orderE: fixes b :: "'k"   and a :: "'k" assumes "b \<le> a" obtains "a = max a b"
simps_1: fixes Mkt :: "('c, 'd) discrete_market"   and p :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" shows "up_cl_proc Mkt p 0 = val_process Mkt p 0"
subst_pabs_id: fixes cs :: "(Term_Class.term \<times> pterm) fset"   and env :: "(name, pterm) fmap" assumes "\<And>pat rhs. (pat, rhs) |\<in>| cs \<Longrightarrow> subst rhs (fmdrop_fset (frees pat) env) = rhs" shows "subst (Pabs cs) env = Pabs cs"