hcju/isabelleps huggingface dataset:
distance_cross_ratio_real_positive: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "u \<noteq> v" shows "\<forall>i1 i2. ideal_points (poincare_line u v) = {i1, i2} \<longrightarrow> cross_ratio u i1 v i2 \<noteq> \<infinity>\<^sub>h \<and> is_real (to_complex (cross_ratio u i1 v i2)) \<and> 0 < Re_cross_ratio u i1 v i2"
"1_2": shows "l < length (snd (Strat ! x))"
iIN_set_UN_set_eq: shows "iIN_set = iIN_UN_set"
enum_finite_3_def: shows "enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3]"
inv_f_f: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "injective f" shows "inv f (f x) = x"
tangent_space_def: fixes p :: "'a" shows "src.tangent_space p = {X. src.is_derivation X p \<and> extensional0 src.diff_fun_space X}"
vimage_algebra_def: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and M :: "'c measure" shows "vimage_algebra X f M = sigma X {f -` A \<inter> X |A. A \<in> sets M}"
l_m: fixes A :: "('d, 'e) aGroup_scheme"   and a :: "'d" assumes "aGroup A"   and "a \<in> carrier A" shows "-\<^sub>a\<^bsub>A\<^esub> a \<plusminus>\<^bsub>A\<^esub> a = \<zero>\<^bsub>A\<^esub>"
many_single_reduction: fixes \<A> :: "unit \<Rightarrow> (unit, 'x + 'x \<times> 'hash, 'hash + unit) gpv"   and q :: "nat" assumes "\<And>c_a c_o s. (c_o, c_a, s) \<in> set_spmf (map_spmf (\<lambda>key. (key, (), {})) key_gen) \<Longrightarrow> interaction_bounded_by' (Not \<circ> isl) (\<A> c_a) q"   and "\<And>c_a c_o s s' x. (c_o, c_a, s) \<in> set_spmf (map_spmf (\<lambda>key. (key, (), {})) key_gen) \<Longrightarrow> lossless_spmf (oracle_hash c_o s' x)"   and "\<And>c_a c_o s s' guess. (c_o, c_a, s) \<in> set_spmf (map_spmf (\<lambda>key. (key, (), {})) key_gen) \<Longrightarrow> lossless_spmf (case guess of (x, h) \<Rightarrow> return_spmf (h = hash c_o x \<and> x \<notin> s'))" shows "advantage_multi \<A> \<le> advantage_single (reduction q \<A>) * real q"
REP_CLS: fixes f :: "'a" assumes "is_left_adjoint f" shows "Maps.REP \<lbrakk>\<lbrakk>f\<rbrakk>\<rbrakk> \<cong> f"
emeasure_cond_pmf: fixes p :: "'c pmf"   and A :: "'c set"   and B :: "'c set" assumes "set_pmf p \<inter> A \<noteq> {}" shows "emeasure (measure_pmf (cond_pmf p A)) B = emeasure (measure_pmf p) (A \<inter> B) / emeasure (measure_pmf p) A"
these_not_empty_eq: fixes B :: "\<langle>'d\<rangle>\<^sub>\<bottom> set" shows "(Option.these B \<noteq> {}) = (B \<noteq> {} \<and> B \<noteq> {None})"
runit_hcomp_1: fixes f :: "'d"   and g :: "'d" assumes "D.ide f"   and "D.ide g"   and "src\<^sub>D f = trg\<^sub>D g" shows "\<r>\<^sub>D[f \<star>\<^sub>D g] = (f \<star>\<^sub>D \<r>\<^sub>D[g]) \<cdot>\<^sub>D \<a>\<^sub>D[f, g, src\<^sub>D g]"
assms_2: assumes "\<nexists>f. f 0 \<in> A \<and> (\<forall>i. (f i, f (Suc i)) \<in> r)" shows "P"
exec_step_ind_IAdd: fixes P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" shows "exec_step_ind (StepI IAdd) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (Intg (the_Intg (hd (tl stk)) + the_Intg (hd stk)) # tl (tl stk), loc, C\<^sub>0, M\<^sub>0, Suc pc, ics) # frs, sh)"
eqvtUnion: fixes Rel :: "'d set"   and Rel' :: "'d set" assumes "eqvt Rel"   and "eqvt Rel'" shows "eqvt (Rel \<union> Rel')"
simps_2: fixes n :: "nat"   and s :: "'state_t"   and execs :: "'dom_t \<Rightarrow> 'action_t list list" assumes "interrupt (Suc n)" shows "run_total (Suc n) s execs = run_total n (cswitch (Suc n) s) execs"
similar_mat_def: fixes A :: "'c mat"   and B :: "'c mat" shows "similar_mat A B = (\<exists>P Q. similar_mat_wit A B P Q)"
V_in_carrier: shows "B \<subseteq> carrier (vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u})"
that_1: shows "set_of A \<subseteq> set_of A'"
assms: shows "prefix (h' # as') (rem_condless_act s [h] as)"
wtL_glitOfC: fixes l' :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit"   and c :: "('fsym, 'psym) lit list" assumes "l' \<in> glitOfC c" shows "GE.wtL l'"
f1: shows "rk {A', B', P, R} \<le> 4"
atLeastatMost_empty': fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.order less_eq less"   and "\<not> less_eq a b" shows "ord.atLeastAtMost less_eq a b = {}"
ttree_restr_noop: fixes t :: "'b ttree"   and S :: "'b set" assumes "carrier t \<subseteq> S" shows "ttree_restr S t = t"
D2: fixes x :: "'b"   and x' :: "'a" assumes "bind_ref_tag x m" shows "hn_refine (\<Gamma>1 * hn_ctxt Rh x x') (f' x') (\<Gamma>2 x x') R (f x)"
\<gamma>_rep_def: fixes p :: "int extended \<times> int extended" shows "\<gamma>_rep p = (let (l, h) = p in {i. l \<le> Fin i \<and> Fin i \<le> h})"
NSBseq_isUb: fixes X :: "nat \<Rightarrow> real" assumes "NSBseq X" shows "\<exists>U. isUb UNIV {x. \<exists>n. X n = x} U"
distinct_3: fixes x3 :: "qtname" shows "NullT \<noteq> ClassT x3"
Lxy_set_filter: fixes xs :: "'a list"   and S :: "'a set" shows "set (Lxy xs S) = S \<inter> set xs"
document_ptr_kinds_commutes: fixes document_ptr :: "(_) document_ptr"   and h :: "(_) heap" shows "(cast document_ptr |\<in>| object_ptr_kinds h) = (document_ptr |\<in>| document_ptr_kinds h)"
Reverse_Reverse_ident: fixes fsm :: "'b fsm" shows "Reverse (Reverse fsm) = fsm"
NullT: assumes "statT = NullT"   and "None = x" shows "thesis"
distinct_27: fixes x41 :: "expr"   and x42 :: "expr"   and x61 :: "Unary_PCF.type"   and x62 :: "expr" shows "x41 ? x42 \<noteq> \<Lambda>\<langle>x61\<rangle> x62"
induct: fixes x :: "nat set"   and P :: "nat set \<Rightarrow> bool" assumes "x \<in> i_set_ind"   and "\<And>n. P [n\<dots>]"   and "\<And>n. P [\<dots>n]"   and "\<And>n d. P [n\<dots>,d]"   and "\<And>r m. P [ r, mod m ]"   and "\<And>r m c. P [ r, mod m, c ]" shows "P x"
lift_and_com: fixes A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool" shows "\<turnstile> (A \<and> B) = (B \<and> A)"
split_length_lt: fixes pat :: "[:'r:]"   and xs :: "[:'r:]" assumes "pat \<noteq> \<bottom>"   and "xs \<noteq> \<bottom>" shows "lt\<cdot>(slength\<cdot>(prod.fst (split\<cdot>pat\<cdot>xs)))\<cdot>(slength\<cdot>xs + 1) = TT"
split: fixes P :: "'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> nat \<Rightarrow> 'b"   and tm :: "'c tm" shows "P (case tm of ??.Time_Monad.tm.TM x xa \<Rightarrow> f x xa) = (\<forall>x1 x2. tm = ??.Time_Monad.tm.TM x1 x2 \<longrightarrow> P (f x1 x2))"
subst_compose: fixes \<sigma> :: "'g \<Rightarrow> ('e, 'h) Term.term"   and \<tau> :: "'h \<Rightarrow> ('e, 'f) Term.term"   and x :: "'g" shows "(\<sigma> \<circ>\<^sub>s \<tau>) x = \<sigma> x \<cdot> \<tau>"
assms: shows "\<forall>\<^sub>F x in F. g x \<noteq> (0::'b)"
inv_poly_y_x: shows "inv poly_y_x = poly_y_x"
floor_log_div: fixes b :: "nat"   and x :: "nat" assumes "1 < b"   and "0 < x"   and "0 < x div b" shows "\<lfloor>log (real b) (real x)\<rfloor> = \<lfloor>log (real b) (real (x div b))\<rfloor> + 1"
scalar_matrix_assoc: fixes k :: "real"   and A :: "(('a, 'd) vec, 'c) vec"   and B :: "(('a, 'b) vec, 'd) vec" shows "k *\<^sub>R (A ** B) = k *\<^sub>R A ** B"
sets_trivial_measure: fixes X :: "'b set" shows "sets (trivial_measure X) = {{}, X}"
maps_simps_2: fixes f :: "'b \<Rightarrow> 'a list" shows "List.maps f [] = []"
degree_rebase_poly_eq: fixes p :: "'a mod_ring poly" shows "degree (#p) = degree p"
get_scdom_component_ptrs_same_scope_component: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "ptr' \<in> set sc" shows "h \<turnstile> get_scdom_component ptr' \<rightarrow>\<^sub>r sc"
lemma_tan_total: fixes y :: "real" assumes "0 < y" shows "\<exists>x>0. x < pi / 2 \<and> y < tan x"
mcont_ennreal_of_enat: shows "mcont Sup (\<le>) Sup (\<le>) ennreal_of_enat"
HoHV_def: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data \<times> ('a, 'a set) Maps.arr arrow_of_spans_data \<times> ('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.HoHV \<mu> \<equiv> if Span.VVV.arr \<mu> then (fst \<mu> \<circ> fst (snd \<mu>)) \<circ> snd (snd \<mu>) else Span.null"
bounded_iff: fixes a :: "'d"   and b :: "'d"   and c :: "'d" shows "(a \<le> inf b c) = (a \<le> b \<and> a \<le> c)"
assms: shows "x \<in> space_as_set (eigenspace e A)"
v11_eq_iff: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"   and "b \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" shows "(\<FF>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> = \<FF>\<lparr>ArrMap\<rparr>\<lparr>b\<rparr>) = (a = b)"
L: fixes a :: "'a"   and x :: "'a"   and b :: "'a" shows "(a * x \<le> b) = (x \<le> a r\<rightarrow> b)"
insert_Collect: fixes a :: "'g"   and P :: "'g \<Rightarrow> bool" shows "insert a (Collect P) = {u. u \<noteq> a \<longrightarrow> P u}"
plus_freeword_abs_eq: fixes xs :: "('a \<times> bool) list"   and ys :: "('a \<times> bool) list" assumes "proper_signed_list xs"   and "proper_signed_list ys" shows "Abs_freeword xs + Abs_freeword ys = Abs_freeword (prappend_signed_list xs ys)"
map_update_all_def: fixes f :: "nat \<Rightarrow> 'b"   and M :: "(nat, 'b) Mapping_Str.map" shows "map_update_all f M = Map (\<lambda>i. if i \<in> keys_of M then Some (f i) else M\<langle>i\<rangle>)"
F: fixes f :: "'trm \<Rightarrow> 'trm"   and n :: "'trm" assumes "f \<in> F"   and "n \<in> num" shows "f n \<in> num"
exhaust: fixes y :: "lock_action" obtains "y = Lock"   | "y = Unlock"   | "y = UnlockFail"   | "y = ReleaseAcquire"
f_nth_derivative_lt_0: fixes k :: "nat" shows "higher_differentiable_on {..<0} f k \<and> (\<forall>t<0. nth_derivative k f t 1 = 0)"
extreme_point_of_convex_hull_convex_independent: fixes S :: "'b set"   and x :: "'b" assumes "compact S"   and "\<And>a. a \<in> S \<Longrightarrow> a \<notin> convex hull (S - {a})" shows "(x extreme_point_of convex hull S) = (x \<in> S)"
foldl_def: fixes f :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'a"   and \<sigma> :: "'a"   and t :: "('b, 'c) FingerTree" shows "FingerTree.foldl f \<sigma> t \<equiv> FingerTreeStruc.foldl f \<sigma> (Rep_FingerTree t)"
intS: shows "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S"
fixespointwiseD: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and a :: "'b" assumes "fixespointwise f A"   and "a \<in> A" shows "f a = id a"
inv_comp_left: fixes f :: "'a \<Rightarrow> 'a" assumes "bijection f" shows "inv f \<circ> f = id"
lens_plus_swap: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e" assumes "X \<bowtie> Y" shows "swap\<^sub>L ;\<^sub>L (X +\<^sub>L Y) = Y +\<^sub>L X"
select_convs_1: fixes nodes :: "'c set"   and edges :: "('c \<times> 'c) set"   and more :: "'d" shows "nodes \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = nodes"
restrict_star: fixes t :: "'c list"   and g :: "'c \<times> 'c \<Rightarrow> 'd" shows "t \<langle>star_matrix' t g\<rangle> t = star_matrix' t g"
dyn_data_dependence_def: fixes n :: "'node"   and V :: "'var"   and n' :: "'node"   and as :: "'edge list" shows "n influences V in n' via as \<equiv> V \<in> Def n \<and> V \<in> Use n' \<and> n -as\<rightarrow>* n' \<and> (\<exists>a' as'. as = a' # as' \<and> (\<forall>n''\<in>set (sourcenodes as'). V \<notin> Def n''))"
relcomp_def: shows "(;) \<equiv> \<lambda>r s. {(xa, x). ((\<lambda>x xa. s2p r (x, xa)) OO (\<lambda>x xa. s2p s (x, xa))) xa x}"
fork_pseudodeterministic: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and e :: "('r, 'l, 'v) expr"   and s' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "s r = Some (\<sigma>, \<tau>, \<E> [Rfork e])" shows "revision_step r s s' = (\<exists>r'. r' \<notin> RID\<^sub>G (s(r \<mapsto> (\<sigma>, \<tau>, \<E> [Rfork e]))) \<and> s' = s(r \<mapsto> (\<sigma>, \<tau>, \<E> [VE (Rid r')]), r' \<mapsto> (\<sigma>;;\<tau>, \<epsilon>, e)))"
int_of_uint64_def: shows "int_of_uint64 \<equiv> map_fun Rep_uint64 id uint"
cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_def: fixes obj :: "(_) Object" shows "cast obj = (case RObject.more obj of Inl x \<Rightarrow> Map.empty x | Inr (Inl x) \<Rightarrow> Map.empty x | Inr (Inr (Inl shadow_root)) \<Rightarrow> Some (RObject.extend (RObject.truncate obj) shadow_root) | Inr (Inr (Inr x)) \<Rightarrow> Map.empty x)"
simps_16: fixes x6 :: "'b ptrm" shows "PFst x6 \<noteq> PUnit"
assms_3: shows "C \<in> cnf (encode_all_operator_preconditions \<Pi> (\<Pi>\<^sub>\<O>) (length \<pi>))"
resPres: fixes P :: "pi"   and Q :: "pi"   and x :: "name" assumes "P \<sim>\<^sup>s Q" shows "<\<nu>x>P \<sim>\<^sup>s <\<nu>x>Q"
bind_apply_cong: fixes f :: "'e \<Rightarrow> ('d \<times> 'e) \<times> bool"   and s :: "'e"   and f' :: "'e \<Rightarrow> ('d \<times> 'e) \<times> bool"   and s' :: "'e"   and g :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'e) \<times> bool"   and g' :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'e) \<times> bool" assumes "f s = f' s'"   and "\<And>rv st. (rv, st) = fst (f' s') \<Longrightarrow> g rv st = g' rv st" shows "(f >>= g) s = (f' >>= g') s'"
opt_bst_simps_1: fixes j :: "int"   and i :: "int" assumes "j < i" shows "local.opt_bst i j = \<langle>\<rangle>"
\<BB>_Obj: shows "\<BB>\<lparr>Obj\<rparr> = ZFC_in_HOL.set {a}"
sum_in_spanI: fixes q :: "'a \<Rightarrow> 'a"   and B :: "'a set" shows "(\<Sum>b\<in>B. q b * b) \<in> ideal B"
eta: fixes s :: "Core.term"   and T :: "typ" assumes "\<not> is_dependent s" shows "Abs T (s $ Bv 0) \<rightarrow>\<^sub>\<eta> decr 0 s"
restrict_def: fixes R :: "('c \<times> 'd) set"   and X :: "'c set" shows "R || X = X \<times> Range R \<inter> R"
preserves_hom: fixes f :: "'c \<times> 'c"   and a :: "'c \<times> 'c"   and b :: "'c \<times> 'c" assumes "C.VV.in_hom f a b" shows "\<guillemotleft>\<Phi> f : H\<^sub>DoFF.map a \<Rightarrow>\<^sub>D FoH\<^sub>C.map b\<guillemotright>"
dependent_finite: fixes S :: "'b set" assumes "finite S" shows "dependent S = (\<exists>u. (\<exists>v\<in>S. u v \<noteq> (0::'a)) \<and> (\<Sum>v\<in>S. u v *s v) = (0::'b))"
expand: fixes gauss_int :: "gauss_int"   and gauss_int' :: "gauss_int" assumes "ReZ gauss_int = ReZ gauss_int' \<and> ImZ gauss_int = ImZ gauss_int'" shows "gauss_int = gauss_int'"
2: fixes x :: "'a" shows "(\<sqinter>xa\<in>range right \<rightarrow>  P) \<le> (right`!`x \<rightarrow> P)"
tensor_assoc: fixes f :: "'c local.term set"   and g :: "'c local.term set"   and h :: "'c local.term set" assumes "arr f"   and "arr g"   and "arr h" shows "(f \<otimes>\<^sub>S g) \<otimes>\<^sub>S h = f \<otimes>\<^sub>S g \<otimes>\<^sub>S h"
edge_preservingI: fixes R :: "('g \<times> 'f) set"   and E :: "('e \<times> 'g \<times> 'g) set"   and G :: "('e \<times> 'f \<times> 'f) set" assumes "on_triple R `` E \<subseteq> G" shows "edge_preserving R E G"
perron_frobenius_positive_ev: fixes A :: "((complex, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" assumes "real_non_neg_mat A"   and "\<And>v. v \<in> S A \<Longrightarrow> A *v (\<chi>i. complex_of_real (v $h i)) \<noteq> 0" shows "\<exists>v. eigen_vector A v (complex_of_real (HMA_Connect.spectral_radius A)) \<and> real_non_neg_vec v"
rep_eq: fixes x :: "'a prefix" shows "last_ts x = (case Rep_prefix x of [] \<Rightarrow> 0 | a # list \<Rightarrow> snd (last (Rep_prefix x)))"
deriv_ident: shows "deriv (\<lambda>w. w) = (\<lambda>z. 1::'a)"
assms_2: fixes a :: "V" assumes "P a"   and "\<forall>b. Ord b \<and> P b \<longrightarrow> a \<le> b" shows "Q a"
conga_diff2: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "A B C CongA A' B' C'" shows "C \<noteq> B"
ereal_mult_mono_strict: fixes b :: "ereal"   and c :: "ereal"   and a :: "ereal"   and d :: "ereal" assumes "0 < b"   and "0 < c"   and "a < b"   and "c < d" shows "a * c < b * d"
elims_1: fixes x :: "instr list"   and y :: "bool" assumes "distinct_load_tmps x = y" obtains "x = []" and "y"   | r :: "instr" and rs :: "instr list" where "x = r # rs" and "y = (case r of Read volatile a t \<Rightarrow> t \<notin> load_tmps rs \<and> distinct_load_tmps rs | RMW a t sop cond ret A L R W \<Rightarrow> t \<notin> load_tmps rs \<and> distinct_load_tmps rs | _ \<Rightarrow> distinct_load_tmps rs)"
associatedI: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times a b"   and "dvd.dvd times b a" shows "normalize a = normalize b"
card_ge_4E: fixes A :: "'a set" assumes "finite A"   and "4 \<le> card A" obtains a :: "'a" and b :: "'a" and c :: "'a" and d :: "'a" where "distinct [a, b, c, d]" and "{a, b, c, d} \<subseteq> A"
simps_2: fixes x :: "(int \<times> int \<times> protocol) set set"   and y :: "(int \<times> int \<times> protocol) set set" shows "Cp (DenyAllFromTo x y) = deny_all_from_to x y"
fst_sup: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "fst (x \<squnion> y) = fst x \<squnion> fst y"
scaleC_nonpos_nonneg: fixes a :: "complex"   and x :: "'b" assumes "a \<le> 0"   and "(0::'b) \<le> x" shows "a *\<^sub>C x \<le> (0::'b)"
mono_lookup: fixes F :: "'a Abs_State.st"   and F' :: "'a Abs_State.st"   and x :: "char list" assumes "F \<sqsubseteq> F'" shows "lookup F x \<sqsubseteq> lookup F' x"
False_step_conc: fixes p :: "bool list"   and q :: "bool list"   and L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and a :: "'b option" shows "((False # p, q) \<in> step (RegExp2NAe.conc L R) a) = (\<exists>r. q = False # r \<and> (p, r) \<in> step R a)"
class_add_hconf_wf: fixes wf_md :: "jvm_prog \<Rightarrow> char list \<Rightarrow> jvm_method mdecl \<Rightarrow> bool"   and P :: "jvm_prog"   and h :: "heap"   and C :: "char list"   and cdec :: "jvm_method class" assumes "wf_prog wf_md P"   and "P \<turnstile> h \<surd>"   and "\<not> is_class P C"   and "\<And>a obj. h a = \<lfloor>obj\<rfloor> \<Longrightarrow> fst obj \<noteq> C" shows "class_add P (C, cdec) \<turnstile> h \<surd>"
mod_Integer_def: shows "mod = (\<Lambda> (MkI\<cdot>x) (MkI\<cdot>y). MkI\<cdot>(modulo x y))"
RSpan_Cons: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "'b"   and ms :: "'b list" assumes "R_scalar_mult R" shows "R_scalar_mult.RSpan R smult (m # ms) = R_scalar_mult.RSpan R smult [m] + R_scalar_mult.RSpan R smult ms"
isometry_on_homeomorphism_2: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "isometry_on X f" shows "homeomorphism_on X f"
prems: fixes i :: "'c" assumes "i \<in> insert j I" shows "f i \<in> hom_deg_set z A"
partial_order_onD_1: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "partial_order_on A r" shows "refl_on A r"
le_trans_le_utrans: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "le_trans t u" shows "le_utrans t u"
simps_6: fixes x :: "'a rexp" shows "lang (x\<^sup>\<star>\<^sub>r) = lang x\<^sup>\<star>"
nfv_simps_4: fixes I :: "\<I>"   and \<phi> :: "'c MFOTL.formula" shows "MFOTL.nfv (formula.Next I \<phi>) = MFOTL.nfv \<phi>"
DEFL_eq_maybe: shows "DEFL('c\<cdot>maybe) = DEFL(one \<oplus> 'c\<^sub>\<bottom>)"
weak_sharing_consistent_preserves_distinct: fixes \<O> :: "nat set"   and sb :: "'b memref list"   and \<S> :: "nat \<Rightarrow> bool option" assumes "weak_sharing_consistent \<O> sb"   and "\<O> \<inter> read_only \<S> = {}" shows "acquired True sb \<O> \<inter> read_only (share sb \<S>) = {}"
below_antisym: fixes below :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.po below"   and "below x y"   and "below y x" shows "x = y"
Transaction'_def: shows "Transaction' = (\<lambda>((cm, pm), views). Transaction cm pm views)"
pick_injective_good: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" assumes "good X"   and "good Y" shows "(pick X = pick Y) = (X = Y)"
associative_weak_composition_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "associative_weak_composition V H \<a> \<equiv> weak_composition V H \<and> associative_weak_composition_axioms V H \<a>"
lsided_times_top: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.ab_quantale times Inf Sup inf less_eq less sup bot top" shows "quantale.lsd times less_eq top (times top x)"
arccos_cos: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> pi" shows "arccos (cos x) = x"
rep_eq: fixes P :: "'b \<Rightarrow> bool"   and x :: "('b, 'a) generator" shows "generator (filter_generator P x) = (filter_has_next P (generator x), filter_next P (generator x))"
star_n_mult: fixes X :: "nat \<Rightarrow> 'b"   and Y :: "nat \<Rightarrow> 'b" shows "star_n X * star_n Y = star_n (\<lambda>n. X n * Y n)"
inject: fixes nat :: "nat"   and nat' :: "nat" shows "(Suc nat = Suc nat') = (nat = nat')"
hasD: shows "P \<turnstile> D has F:T (fm) in E"
more_arith_simps_1: fixes b :: "'a"   and a :: "'a" shows "(- b \<le> - a) = (a \<le> b)"
is_character_data_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes ptr :: "(_) node_ptr" shows "is_character_data_ptr_kind ptr = (case cast ptr of None \<Rightarrow> False | Some x \<Rightarrow> True)"
run_put_option: fixes put :: "'f \<Rightarrow> 'd \<Rightarrow> 'd"   and s :: "'f"   and m :: "('e, 'd) optionT" shows "run_option (put_option put s m) = put s (run_option m)"
intros_1: fixes g :: "('a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'b \<times> 'a)"   and s :: "'a" assumes "\<not> fst g s" shows "s \<in> terminates_on g"
Conf_HAStates: fixes ST :: "('d, 'e, 'f) status" shows "Conf ST \<subseteq> HAStates (HA ST)"
Ord_ord_of_nat: fixes k :: "nat" shows "Ord (k\<^sub>\<nat>)"
infinite_countable_subset: fixes S :: "'a set" assumes "infinite S" shows "\<exists>f. inj f \<and> range f \<subseteq> S"
has_ivl_integral_def: fixes f :: "real \<Rightarrow> 'a"   and y :: "'a"   and a :: "real"   and b :: "real" shows "(f has_ivl_integral y) a b = (if a \<le> b then (f has_integral y) {a..b} else (f has_integral - y) {b..a})"
hull_induct: fixes x :: "'b list"   and G :: "'b list set"   and P :: "'b list \<Rightarrow> bool" assumes "x \<in> \<langle>G\<rangle>"   and "P \<epsilon>"   and "\<And>w. w \<in> G \<Longrightarrow> P w"   and "\<And>w1 w2. \<lbrakk>w1 \<in> \<langle>G\<rangle>; P w1; w2 \<in> \<langle>G\<rangle>; P w2\<rbrakk> \<Longrightarrow> P (w1 \<cdot> w2)" shows "P x"
snd_Gauss_Jordan_upt_k_det_P_is_snd_Gauss_Jordan_upt_k_PA: fixes A :: "(('b, 'c) vec, 'c) vec"   and k :: "nat" shows "snd (Gauss_Jordan_upt_k_det_P A k) = snd (Gauss_Jordan_upt_k_PA A k)"
dB': fixes t :: "real"   and x :: "'a" assumes "(t, x) \<in> cball (t0, x0) (dB / 2)" shows "norm (f t x) \<le> B"