hcju/isabelleps huggingface dataset:
simps_1: shows "vfsequence_of_vlist [] = []\<^sub>\<circ>"
completion_def: fixes \<gamma> :: "'a set \<Rightarrow> 'a \<Rightarrow> 'a set set"   and A :: "'a set"   and \<phi> :: "'a" shows "* \<gamma> A \<phi> \<equiv> if A .\<bottom>. \<phi> = {} then {A} else {B \<in> A .\<bottom>. \<phi>. \<Inter> (\<gamma> A \<phi>) \<subseteq> B}"
map_closed: fixes a :: "'a" assumes "a \<in> S" shows "orbit.Class a \<in> orbit.Partition"
emb_hsize_neq: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" assumes "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"   and "t \<noteq> s" shows "hsize s < hsize t"
exists_map_of: fixes m :: "'c \<Rightarrow> 'd option" assumes "finite (dom m)" shows "\<exists>xs. map_of xs = m"
comp_comp_fun_commute: fixes g :: "'c \<Rightarrow> 'a" shows "comp_fun_commute (f \<circ> g)"
commute: fixes f :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and a :: "'b::type"   and b :: "'b::type" assumes "abel_semigroup (f::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type)" shows "(f::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (a::'b::type) (b::'b::type) = f b a"
Abs_lst1_fcb2: fixes a :: "atom"   and x :: "'e"   and b :: "atom"   and y :: "'e"   and f :: "atom \<Rightarrow> 'e \<Rightarrow> 'g \<Rightarrow> 'f"   and c :: "'g" assumes "[[a]]lst. x = [[b]]lst. y"   and "a \<sharp> f a x c"   and "{a, b} \<sharp>* c"   and "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> f a x c = f (p \<bullet> a) (p \<bullet> x) c"   and "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> f b y c = f (p \<bullet> b) (p \<bullet> y) c" shows "f a x c = f b y c"
flush_buffered_val_conv: fixes sb :: "'b memref list"   and m :: "nat \<Rightarrow> nat"   and a :: "nat" shows "flush sb m a = (case buffered_val sb a of None \<Rightarrow> m a | Some v \<Rightarrow> v)"
inner_prod_with_itself_real: fixes u :: "complex Matrix.vec" shows "\<langle>u|u\<rangle> \<in> \<real>"
exists_fresh: fixes x :: "'a" assumes "finite (supp x)" obtains c :: "name" where "c \<sharp> x"
coeff_plus: fixes lp1 :: "linear_poly"   and lp2 :: "linear_poly"   and var :: "nat" shows "Abstract_Linear_Poly.coeff (lp1 + lp2) var = Abstract_Linear_Poly.coeff lp1 var + Abstract_Linear_Poly.coeff lp2 var"
bounded_scaling: fixes S :: "'a set"   and c :: "real" assumes "bounded S" shows "bounded ((*\<^sub>R) c ` S)"
inf_add_K_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.join_semilattice plus less_eq less" shows "less_eq a (plus a b)"
in_idem: fixes A :: "'t set"   and x :: "'t" assumes "finite A"   and "x \<in> A" shows "ord_term_lin.max x (ord_term_lin.Max A) = ord_term_lin.Max A"
c_def: fixes p :: "PrIds" shows "c p = (cst!p, rtrner memCh!p, caller crCh!p)"
bind_rule: fixes M :: "'e nres"   and f :: "'e \<Rightarrow> 'f nres"   and \<Phi> :: "'f \<Rightarrow> bool" assumes "M \<le> SPEC (\<lambda>x. f x \<le> SPEC \<Phi>)" shows "M \<bind> f \<le> SPEC \<Phi>"
Vr_potent_eqTr2: fixes v :: "'a \<Rightarrow> ant"   and n :: "nat"   and m :: "nat" assumes "valuation K v"   and "vp K v \<^bsup>\<diamondsuit>Vr K v n\<^esup> = vp K v \<^bsup>\<diamondsuit>Vr K v m\<^esup>" shows "n = m"
Int_ac_1: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "A \<inter> B \<inter> C = A \<inter> (B \<inter> C)"
UN_extend_simps_5: fixes A :: "'c set"   and B :: "'d \<Rightarrow> 'c set"   and C :: "'d set" shows "A \<inter> \<Union> (B ` C) = (\<Union>x\<in>C. A \<inter> B x)"
graeffe_0: fixes f :: "'a poly"   and c :: "'a"   and as :: "'a list" assumes "f = Polynomial.smult c (\<Prod>a\<leftarrow>as. [:- a, 1::'a:])" shows "graeffe_poly c as 0 = f"
sdrop_add: fixes n :: "nat"   and m :: "nat"   and s :: "'c stream" shows "sdrop n (sdrop m s) = sdrop (m + n) s"
bf: shows "isolate_variable_sparse p var (Suc 0) = B"
Wbis_transT: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and s' :: "'state" assumes "c \<approx>w d"   and "s \<approx> t"   and "(c, s) \<rightarrow>t s'" obtains t' :: "'state" where "(d, t) \<rightarrow>*t t'__" and "s' \<approx> t'__"   | d' :: "('test, 'atom) com" and t' :: "'state" where "(d, t) \<rightarrow>*c (d', t')" and "s' \<approx> t'" and "discr d'"
False: shows "\<not> \<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A"
single_not_bot: fixes x :: "'a" shows "single x \<noteq> \<bottom>"
comp_cod_arr: fixes f :: "'a"   and b :: "'a" assumes "Fop.Bop.arr f"   and "Fop.Bop.cod f = b" shows "Fop.Bop.comp b f = f"
coeff_finite_fourier_poly': fixes n :: "nat"   and k :: "nat"   and ws :: "nat \<Rightarrow> complex" assumes "n < k" shows "coeff (finite_fourier_poly' ws k) n = 1 / of_nat k * (\<Sum>m<k. ws m * unity_root k (- int n * int m))"
assms_1: shows "if bad2 s2 then X_bad s1 s2 else X s1 s2"
cond: shows "(\<forall>x. m (f x) (g x) \<noteq> None) = (\<forall>x. m' (f' x) (g' x) \<noteq> None)"
mtf2xy: fixes A :: "'a list"   and x :: "'a"   and y :: "'a"   and r :: "'a"   and a :: "nat" assumes "A \<in> {[x, y], [y, x]}"   and "r \<in> {x, y}" shows "mtf2 a r A \<in> {[x, y], [y, x]}"
H: shows "\<Gamma> \<turnstile> (\<lambda>x:S. s) : T"
f_pos: fixes i :: "'b" assumes "i \<in> S" shows "0 < e i / a i"
eWls_invert: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and s :: "'sort"   and eX :: "'c withERR" assumes "eWls MOD s eX" shows "\<exists>X. eX = OK X \<and> igWls MOD s X"
iIN_append_union_Suc: fixes n :: "nat"   and d :: "nat"   and d' :: "nat" shows "[n\<dots>,d] \<union> [Suc (n + d)\<dots>,d'] = [n\<dots>,Suc (d + d')]"
transpose_mult_vec_canon_row: fixes A :: "'a sq_mtx"   and i :: "'a" shows "A\<^sup>\<dagger> *\<^sub>V \<e> i = \<r>\<o>\<w> i A"
Bot_map: fixes B :: "'f" assumes "B \<in> Bot_F" shows "\<G>_F B \<subseteq> Bot_G"
iwlsFSb_iterAll_termFSbImorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSb MOD" shows "termFSbImorph (iter MOD) (iterAbs MOD) MOD"
simps_4: fixes wf_atom :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and a :: "'b" shows "alphabet.wf wf_atom n (Atom a) = wf_atom n a"
derived_self_is_normal: shows "derived G (carrier G) \<lhd> G"
assms_2: shows "c_types_agree (TopType ts') t_ag"
simps_126: fixes f1 :: "char list \<Rightarrow> 'b"   and f2 :: "48 word \<Rightarrow> 'b"   and f3 :: "48 word \<Rightarrow> 'b"   and f4 :: "16 word \<Rightarrow> 'b"   and f5 :: "16 word \<Rightarrow> 'b"   and f6 :: "16 word \<Rightarrow> 'b"   and f7 :: "32 prefix_match \<Rightarrow> 'b"   and f8 :: "32 prefix_match \<Rightarrow> 'b"   and f9 :: "8 word \<Rightarrow> 'b"   and f10 :: "16 word \<Rightarrow> 16 word \<Rightarrow> 'b"   and f11 :: "16 word \<Rightarrow> 16 word \<Rightarrow> 'b"   and x5 :: "16 word" shows "(case VlanId x5 of IngressPort x \<Rightarrow> f1 x | EtherSrc x \<Rightarrow> f2 x | EtherDst x \<Rightarrow> f3 x | EtherType x \<Rightarrow> f4 x | VlanId x \<Rightarrow> f5 x | VlanPriority x \<Rightarrow> f6 x | IPv4Src x \<Rightarrow> f7 x | IPv4Dst x \<Rightarrow> f8 x | IPv4Proto x \<Rightarrow> f9 x | L4Src x xa \<Rightarrow> f10 x xa | L4Dst x xa \<Rightarrow> f11 x xa) = f5 x5"
sccs_altdef2: shows "sccs = {H. max_subgraph strongly_connected H}"
divide_by_norm_row_code: fixes A :: "(('d, 'c) vec, 'e) vec"   and a :: "'e" shows "($) (divide_by_norm_row A a) = (\<lambda>b. ((1 / norm (column b A)) *\<^sub>R column b A) $ a)"
ep_pair_ssum: shows "ep_pair ssum_emb ssum_prj"
root_equals_start_points: fixes r :: "'a"   and x :: "'a" assumes "backward_terminating_path_root r x"   and "x \<noteq> 0" shows "r = start_points x"
mono_seq_DT_left: fixes P :: "'b process"   and P' :: "'b process"   and S :: "'b process" assumes "P \<sqsubseteq>\<^sub>D\<^sub>T P'" shows "(P `;` S) \<sqsubseteq>\<^sub>D\<^sub>T (P' `;` S)"
simps_17: fixes prec :: "nat"   and x :: "nat"   and a :: "floatarith"   and n :: "nat"   and vs :: "float interval option list" shows "isDERIV_approx prec x (Power a (Suc n)) vs = isDERIV_approx prec x a vs"
sg_m_i_closed: fixes H :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<guillemotright> H "   and "a \<in> H"   and "b \<in> H" shows "a \<cdot> \<rho> b \<in> H"
PSimQ: fixes R :: "pi"   and S :: "pi" assumes "(R, S) \<in> Rel" shows "R \<leadsto><Rel> S"
rotate_pow_mod: fixes n :: "nat"   and u :: "'b list"   and k :: "nat" shows "rotate n (u \<^sup>@ k) = rotate (n mod \<^bold>|u\<^bold>|) (u \<^sup>@ k)"
cases: fixes x :: "example_peers" obtains "x = PeerA"   | "x = PeerB"   | "x = PeerC"
power_poly_f_mod_def: fixes modulus :: "'b poly" shows "power_poly_f_mod modulus = (\<lambda>a n. a ^ n mod modulus)"
assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<sqinter> b \<sqinter> c = a \<sqinter> (b \<sqinter> c)"
emeasure_T': fixes X :: "'s stream set"   and I :: "'s pmf" assumes "X \<in> sets (stream_space (count_space UNIV))" shows "emeasure (K.T' I) X = \<integral>\<^sup>+ s. emeasure (K.T s) {\<omega> \<in> space (stream_space (count_space UNIV)). s ## \<omega> \<in> X} \<partial>measure_pmf I"
lim_gf_G: fixes x :: "'s"   and y :: "'s" shows "((\<lambda>z. ennreal (gf_G x y z)) \<longlongrightarrow> G x y) (at_left 1)"
cong_def: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and b :: "'c"   and c :: "'c"   and a :: "'c" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b c a = (modulo b a = modulo c a)"
bi_unique_mod_ring_rel_integer_1: shows "bi_unique mod_ring_rel_integer"
rw_rat_alt: shows "rw_rat = (\<lambda>p q. \<not> rw_rat_strict q p)"
measurable_component_singleton: fixes i :: "'b"   and I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure" assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> Pi\<^sub>M I M \<rightarrow>\<^sub>M M i"
assms_6: shows "l \<circ> g = h \<circ> l"
out_reg_mod_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and w :: "word32"   and curr_win :: "'b word"   and rd :: "word5"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = out_reg_mod w curr_win rd s1 \<and> t2 = out_reg_mod w curr_win rd s2" shows "low_equal t1 t2"
mem_appR: fixes B :: "'b list"   and x :: "'b"   and A :: "'b list" assumes "List.member B x" shows "List.member (A @ B) x"
allE: fixes P :: "'h \<Rightarrow> bool"   and x :: "'h" assumes "\<forall>x. P x" obtains "P x"
assms: fixes n :: "nat" shows "measurable_envelope M (A n) (E n)"
STA_struct: fixes A :: "'b set"   and n :: "nat"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and branch' :: "(('c, 'b) fm list \<times> 'b) list" assumes "infinite UNIV"   and "finite A"   and "A, n \<turnstile> branch"   and "set branch \<subseteq> set branch'" shows "A \<turnstile> branch'"
sets_density: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" shows "sets (density M f) = sets M"
parts_simp_insert: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> range Agent \<union> range Number \<union> range Nonce \<union> range Key \<union> range Hash \<union> range Passwd \<union> range Pri_AgrK \<union> range Pub_AgrK" shows "parts (insert X H) = insert X (parts H)"
wfp: 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 \<pi> :: "('a \<times> 'b set) stream" assumes "Kripke S s0 \<delta> AP L" shows "Kripke.wfp S \<delta> AP L AP' \<pi> = (\<forall>i. fst (\<pi> !! i) \<in> S \<and> snd (\<pi> !! i) \<subseteq> AP' \<and> snd (\<pi> !! i) \<inter> AP = L (fst (\<pi> !! i)) \<and> fst (\<pi> !! Suc i) \<in> \<delta> (fst (\<pi> !! i)))"
simps: fixes a1 :: "'c"   and a2 :: "('c, 'd) bintree list" shows "is_heap_list a1 a2 = ((\<exists>h. a1 = h \<and> a2 = []) \<or> (\<exists>h ts t. a1 = h \<and> a2 = t # ts \<and> is_heap_list h ts \<and> is_heap_list (priority t) (children t) \<and> h \<le> priority t))"
trclp_trcl_eq: fixes t :: "('c \<times> 'd \<times> 'c) set" shows "trclp (\<lambda>x xa xb. (x, xa, xb) \<in> t) = (\<lambda>x xa xb. (x, xa, xb) \<in> trcl t)"
P_works: fixes a :: "nat" assumes "odd n"   and "2 \<le> a"   and "a < n"   and "prime n" shows "P n a"
is_aligned_beyond_length: fixes n :: "nat"   and w :: "'b word" assumes "LENGTH('b) \<le> n" shows "is_aligned w n = (w = 0)"
valid_def: fixes p :: "fm" shows "valid p \<equiv> \<forall>i. eval i p = Det True"
IfTrue: fixes tst :: "'test"   and s :: "'state"   and c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" assumes "tval tst s" shows "(if tst then c1 else c2, s) \<rightarrow>c (c1, s)"
in_lset_conv_lnth: fixes x :: "'c"   and xs :: "'c llist" shows "(x \<in> lset xs) = (\<exists>n. enat n < llength xs \<and> lnth xs n = x)"
Send: shows "\<exists>u u' m. j + 1 \<le> k \<and> k < i \<and> t ! k = Send cid p q u u' m"
MkIde_Dom': fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "('a, 'b) concrete_category.arr" assumes "concrete_category Obj Hom Id Comp"   and "partial_magma.ide (concrete_category.COMP Obj Hom Comp) a" shows "MkArr (concrete_category.Dom a) (concrete_category.Dom a) (Id (concrete_category.Dom a)) = a"
map_def: fixes f :: "'j" shows "Fa.map f = (if \<chi>.J.arr f then F a else B.null)"
ExIRule: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>"   and y :: "'a" assumes "[\<phi> y in v]" shows "[\<^bold>\<exists>x. \<phi> x in v]"
inf_assoc: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.semilattice_inf inf less_eq less" shows "inf (inf x y) z = inf x (inf y z)"
iff_conv_conj_imp: fixes P :: "bool"   and Q :: "bool" shows "(P = Q) = ((P \<longrightarrow> Q) \<and> (Q \<longrightarrow> P))"
f_Exec_Stream_Init_Nil: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d" shows "f_Exec_Comp_Stream_Init trans_fun [] c = [c]"
min_less_iff_conj: fixes z :: "'b word"   and x :: "'b word"   and y :: "'b word" shows "(z <s signed.min x y) = (z <s x \<and> z <s y)"
assms_4: shows "(s, a, s') \<in> automaton.trans (pnet np n)"
t's': shows "t' \<leadsto> s' \<in> SPR.jkbpCn (tLength (t' \<leadsto> s'))"
cr_dnelist_def: shows "cr_dnelist \<equiv> \<lambda>x y. x = list_of_dnelist y"
well_def_def: fixes prog :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "well_def prog \<equiv> healthy (wp prog) \<and> nearly_healthy (wlp prog) \<and> wp_under_wlp prog \<and> sub_distrib_pconj prog \<and> sublinear (wp prog) \<and> bd_cts (wp prog)"
card1: shows "card X = 1 \<longrightarrow> (\<exists>f. TernaryOrdering.ordering f ord X)"
units_of_carrier: fixes G :: "('a, 'b) monoid_scheme" shows "carrier (units_of G) = Units G"
intros_13: fixes \<chi> :: "'a ltln"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" assumes "\<chi> \<turnstile>\<^sub>s \<phi>" shows "\<psi> R\<^sub>n \<chi> \<turnstile>\<^sub>s \<phi>"
less_finite_4_def: fixes x :: "finite_4"   and y :: "finite_4" shows "(x < y) = (case (x, y) of (a\<^sub>1, a\<^sub>1) \<Rightarrow> False | (a\<^sub>1, _) \<Rightarrow> True | (a\<^sub>4, x) \<Rightarrow> False | (_, a\<^sub>4) \<Rightarrow> True | (_, _) \<Rightarrow> False)"
obtain_qFresh: fixes V :: "'var set"   and XS :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort" assumes "finite V \<or> |V| <o |UNIV|"   and "finite XS \<or> |XS| <o |UNIV|"   and "Ball XS qGood" shows "\<exists>x. x \<notin> V \<and> (\<forall>X\<in>XS. qAFresh xs x X \<and> qFresh xs x X)"
invertible_snd_QR_decomposition: fixes A :: "((real, 'a) vec, 'b) vec" assumes "rank A = ncols A" shows "invertible (snd (QR_decomposition A))"
star_inductr_var_equiv: fixes y :: "'a"   and x :: "'a" shows "(y \<cdot> x \<le> y) = (y \<cdot> x\<^sup>\<star> \<le> y)"
2: shows "S.par (\<r>[S.cod \<mu>] \<cdot>\<^sub>S R \<mu>) (\<mu> \<cdot>\<^sub>S \<r>[S.dom \<mu>])"
simps_1: fixes xs :: "'c llist"   and ys :: "'c llist" shows "lnull (lappend xs ys) = (lnull xs \<and> lnull ys)"
two_sides_in_angle: fixes B :: "'p"   and P' :: "'p"   and P :: "'p"   and A :: "'p"   and C :: "'p" assumes "B \<noteq> P'"   and "B P TS A C"   and "Bet P B P'" shows "P InAngle A B C \<or> P' InAngle A B C"
hom_det: fixes hom :: "'b \<Rightarrow> 'c"   and A :: "'b mat" assumes "comm_ring_hom hom" shows "det (map_mat hom A) = hom (det A)"
se_star_one: fixes c1 :: "('d, 'e) conf"   and l :: "('d, 'e) label"   and c2 :: "('d, 'e) conf" shows "se_star c1 [l] c2 = se c1 l c2"
Class_widen: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and T :: "ty"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P \<turnstile> Class C \<le> T"   and "wf_prog wf_md P"   and "is_class P C" shows "\<exists>D. T = Class D \<and> P \<turnstile> Path C to D unique"
S12r: fixes x :: "'a" shows "(1::'a) + x\<^sup>\<star> \<cdot> x = x\<^sup>\<star>"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc"   and TRel :: "('procT \<times> 'procT) set" shows "a1 \<R>\<lbrakk>\<cdot>\<rbrakk>T<TRel> a2 = ((\<exists>S. S \<in>S a1 \<and> \<lbrakk>S\<rbrakk> \<in>T a2) \<or> (\<exists>S. \<lbrakk>S\<rbrakk> \<in>T a1 \<and> S \<in>S a2) \<or> (\<exists>T1 T2. T1 \<in>T a1 \<and> T2 \<in>T a2 \<and> (T1, T2) \<in> TRel))"
le_sup_iff: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "(x \<squnion> y \<le> z) = (x \<le> z \<and> y \<le> z)"
\<tau>\<^sub>0_def: shows "\<tau>\<^sub>0 \<equiv> (\<lparr>heap = Map.empty, assocs = Map.empty\<rparr>, \<lparr>heap = Map.empty, assocs = Map.empty\<rparr>)"
delete_id: fixes k :: "'a"   and al :: "('a \<times> 'b) list" assumes "k \<notin> fst ` set al" shows "AList.delete k al = al"
mtx_vec_mult_canon: fixes A :: "'a sq_mtx"   and i :: "'a" shows "A *\<^sub>V \<e> i = \<c>\<o>\<l> i A"
lookup_atomize_poly: fixes p :: "'k \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "lookup (atomize_poly p) v = lookup (lookup p (component_of_term v)) (pp_of_term v)"
abs_sum_abs: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "\<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar> = (\<Sum>a\<in>A. \<bar>f a\<bar>)"
collapse: fixes option :: "'d option" assumes "option \<noteq> None" shows "Some (the option) = option"
star_subdist_var_3: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and star :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.left_pre_kleene_algebra plus times one less_eq less star" shows "less_eq (times (star x) (star y)) (star (plus x y))"
assms: shows "ae f|` domA \<Gamma> = ae' f|` domA \<Gamma>"
\<alpha>_build: fixes n :: "nat"   and seed :: "natural \<times> natural" shows "apfst \<alpha> (local.build n seed) = Benchmark_Comparison.build n seed"
posl: fixes x :: "'c"   and y :: "'c" assumes "D x y"   and "x \<oplus> y \<in> E" shows "x \<in> E"
H_3: shows "finite {i. X i \<noteq> topspace euclidean}"
Pi_def: fixes A :: "'e set"   and B :: "'e \<Rightarrow> 'f set" shows "Pi A B = {f. \<forall>x. x \<in> A \<longrightarrow> f x \<in> B x}"
borel_sigma_sets_Ioc: shows "borel = sigma UNIV (range (\<lambda>(a, b). {a<..b}))"
inf_sup_ord_2: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y \<le> y"
pairwise_property_lemma: fixes A :: "('a list \<Rightarrow> 'b pmf) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> ((nat \<times> nat list) \<times> 'b) pmf)" assumes "\<And>init qs x y. \<lbrakk>distinct init; qs \<in> {xs. set xs \<subseteq> set init}; (x, y) \<in> {(x, y). x \<in> set init \<and> y \<in> set init \<and> x \<noteq> y}; x \<noteq> y\<rbrakk> \<Longrightarrow> Pbefore_in x y A qs init = Pbefore_in x y A (Lxy qs {x, y}) (Lxy init {x, y})"   and "\<And>xa r. \<forall>z\<in>set_pmf (snd A xa r). snd (fst z) = []" shows "List_Factoring.pairwise A"
map_prod_vimage: fixes f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g"   and A :: "'f set"   and B :: "'g set" shows "map_prod f g -` (A \<times> B) = f -` A \<times> g -` B"
set_dual_def: shows "set_dual = \<eta> \<circ> \<partial>"
Chain_3_supset: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g"   and C :: "('a set \<times> ('a \<Rightarrow> 'g)) set"   and a1 :: "'a set"   and b1 :: "'a \<Rightarrow> 'g"   and a2 :: "'a set"   and b2 :: "'a \<Rightarrow> 'g"   and a3 :: "'a set"   and b3 :: "'a \<Rightarrow> 'g" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N"   and "C \<subseteq> fsps R M N f H"   and "C \<noteq> {}"   and "\<forall>a\<in>C. \<forall>b\<in>C. fst a \<subseteq> fst b \<or> fst b \<subseteq> fst a"   and "(a1, b1) \<in> C"   and "(a2, b2) \<in> C"   and "(a3, b3) \<in> C" shows "\<exists>(g, h)\<in>C. a1 \<subseteq> g \<and> a2 \<subseteq> g \<and> a3 \<subseteq> g"
bigo_LIMSEQ1: fixes f :: "nat \<Rightarrow> real"   and g :: "nat \<Rightarrow> real" assumes "f \<in> O(g)"   and "g \<longlonglongrightarrow> 0" shows "f \<longlonglongrightarrow> 0"
suffix_\<nu>_stable: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<forall>\<^sub>\<infinity>j. \<nu>_stable \<phi> (suffix j w)"
assms: shows "le\<cdot>(slength\<cdot>xs)\<cdot>i = TT"
neighbor_invar_axioms: shows "neighbor_invar c s t fo lo no"
order_Abs_N: fixes a :: "'b"   and b :: "'b" assumes "a \<in> N"   and "b \<in> N" shows "(a \<le> b) = (Abs_N a \<le> Abs_N b)"
trace_equivalent_def: fixes e1 :: "((nat \<times> nat) \<times> transition) fset"   and e2 :: "((nat \<times> nat) \<times> transition) fset" shows "trace_equivalent e1 e2 = (T e1 = T e2)"
assms: fixes V :: "'a set"   and x :: "'a" assumes "openin (top_of_set S) V"   and "x \<in> V" shows "\<exists>U. openin (top_of_set S) U \<and> connected U \<and> x \<in> U \<and> U \<subseteq> V"
connected_component_idemp: fixes S :: "'c set"   and x :: "'c" shows "connected_component_set (connected_component_set S x) x = connected_component_set S x"
simps_5: fixes f1 :: "'b"   and f2 :: "'b" shows "rec_T1 f1 f2 A = f1"
set_obtain_sublist: fixes S :: "'b set"   and ls :: "'b list" assumes "S \<subseteq> set ls" obtains ss :: "'b list" where "distinct ss" and "S = set ss"
norm_diff_temp_dyn: fixes a :: "real"   and s\<^sub>1 :: "(real, 4) vec"   and L :: "real"   and s\<^sub>2 :: "(real, 4) vec" assumes "0 < a" shows "\<parallel>temp_vec_field a L s\<^sub>1 - temp_vec_field a L s\<^sub>2\<parallel> = \<bar>a\<bar> * \<bar>s\<^sub>1 $ 1 - s\<^sub>2 $ 1\<bar>"
le_numeral_Suc: fixes k :: "num"   and n :: "nat" shows "(numeral k \<le> Suc n) = (pred_numeral k \<le> n)"
is_unit_mult_iff: fixes a :: "'b"   and b :: "'b" shows "(a * b dvd (1::'b)) = (a dvd (1::'b) \<and> b dvd (1::'b))"
assms_1: shows "e \<le> v * - v\<^sup>T \<sqinter> g"
Y_le_Max_X: fixes y :: "'a" assumes "y \<in> Y" shows "y \<le> Max X"
list_rel_sv_iff: fixes R :: "('a \<times> 'b) set" shows "single_valued (\<langle>R\<rangle>list_rel) = single_valued R"
eventually_frequentlyE: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter"   and Q :: "'b \<Rightarrow> bool" assumes "eventually P F"   and "\<forall>\<^sub>F x in F. \<not> P x \<or> Q x"   and "F \<noteq> bot" shows "frequently Q F"