hcju/isabelleps huggingface dataset:
induct: fixes x :: "fol fm"   and P :: "fol fm \<Rightarrow> bool" assumes "\<turnstile> x"   and "\<And>s s'. \<lbrakk>\<turnstile> concl s; P (concl s); \<turnstile> concl s'; P (concl s')\<rbrakk> \<Longrightarrow> P (concl (modusponens s s'))"   and "\<And>s uu_. \<lbrakk>\<turnstile> concl s; P (concl s)\<rbrakk> \<Longrightarrow> P (concl (gen uu_ s))"   and "\<And>uv_ uw_. P (concl (axiom_addimp uv_ uw_))"   and "\<And>ux_ uy_ uz_. P (concl (axiom_distribimp ux_ uy_ uz_))"   and "\<And>va_. P (concl (axiom_doubleneg va_))"   and "\<And>vb_ vc_ vd_. P (concl (axiom_allimp vb_ vc_ vd_))"   and "\<And>ve_ vf_. P (concl (axiom_impall ve_ vf_))"   and "\<And>vg_ vh_. P (concl (axiom_existseq vg_ vh_))"   and "\<And>vi_. P (concl (axiom_eqrefl vi_))"   and "\<And>vj_ vk_ vl_. P (concl (axiom_funcong vj_ vk_ vl_))"   and "\<And>vm_ vn_ vo_. P (concl (axiom_predcong vm_ vn_ vo_))"   and "\<And>vp_ vq_. P (concl (axiom_iffimp1 vp_ vq_))"   and "\<And>vr_ vs_. P (concl (axiom_iffimp2 vr_ vs_))"   and "\<And>vt_ vu_. P (concl (axiom_impiff vt_ vu_))"   and "P (concl axiom_true)"   and "\<And>vv_. P (concl (axiom_not vv_))"   and "\<And>vw_ vx_. P (concl (axiom_and vw_ vx_))"   and "\<And>vy_ vz_. P (concl (axiom_or vy_ vz_))"   and "\<And>wa_ wb_. P (concl (axiom_exists wa_ wb_))" shows "P x"
dt_2: fixes y :: "'a" assumes "y \<in> ball l d" shows "flow0 y (t y) \<in> {a<--<b}"
group_part_def: fixes p :: "'a"   and xs :: "'a list" shows "group_part p xs = group_part_aux (f p) xs ([], p, [])"
fac_acc_body2_def: shows "fac_acc_body2 \<equiv> \<Lambda> r n. if n = 0 then n2a\<cdot>1 else wrapA\<cdot>r\<cdot>(n -\<^sub># 1) >>= (\<Lambda> res. n2a\<cdot>n >=> n2a\<cdot>res)"
assms_2: shows "monom 1 1 [^]\<^bsub>R\<^esub> CARD('a) ^ m' = monom 1 1"
toplevel_summands_lderiv: fixes as :: "'a"   and r :: "'b rexp" shows "toplevel_summands (lderiv as r) = (\<Union>s\<in>toplevel_summands r. toplevel_summands (lderiv as s))"
bdd_above_bot: fixes A :: "'a multiset set" shows "subset_mset.bdd_below A"
Domain_iff: fixes a :: "'d"   and r :: "('d \<times> 'e) set" shows "(a \<in> Domain r) = (\<exists>y. (a, y) \<in> r)"
prime_set_ass: shows "prime_set \<subseteq> {x. prime x \<and> 2 < x}"
containID: fixes i :: "nat"   and \<Gamma> :: "'a form multiset"   and \<Delta> :: "'a form multiset"   and R :: "('a sequent list \<times> 'a sequent) set" assumes "At i \<in># \<Gamma> \<and> At i \<in># \<Delta>"   and "Ax \<subseteq> R" shows "( \<Gamma> \<Rightarrow>* \<Delta>, 0) \<in> derivable R*"
A: fixes as :: "('b poly \<times> nat) set"   and p :: "'b poly"   and n :: "nat" assumes "monic_factorization as p" shows "monic_factorization.A p n = (\<Prod>(a, i)\<in>as \<inter> UNIV \<times> {n}. a)"
ladj_adj: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" assumes "f \<stileturn> g" shows "f = ladj g"
runit_def: fixes f :: "'a" shows "\<r>[f] \<equiv> THE \<mu>. \<guillemotleft>\<mu> : R f \<Rightarrow>\<^sub>S f\<guillemotright> \<and> R \<mu> = (f \<star> \<iota>) \<cdot>\<^sub>S \<a>[f, a, a]"
ih: fixes s :: "('s, 'v) tm" assumes "s \<in> set ss" shows "zhmset_of \<epsilon>\<^sub>h \<le> eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt s))"
assms_4: shows "the (dhops rt2 dip) \<le> the (dhops rt1 dip)"
index_mat_multrow_2: fixes i :: "nat"   and A :: "'c mat"   and j :: "nat"   and mul :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "i < dim_row A"   and "j < dim_col A" shows "mat_multrow_gen mul i a A $$ (i, j) = mul a (A $$ (i, j))"
assms: shows "run_one_step d i (s, vs, ves, $Get_global x17) = (s', vs', res)"
top_def: shows "top = \<Squnion>\<^sub>f\<^sub>i\<^sub>n UNIV"
ipurge_tr_rev_t_aux_set: fixes X :: "('c, 'd) ipurge_rec" shows "ipurge_tr_rev_t_aux X \<in> ipurge_tr_rev_t_set X"
case_distrib: fixes h :: "'d \<Rightarrow> 'c"   and f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'd"   and option :: "'e option" shows "h (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (case option of None \<Rightarrow> h f1 | Some x \<Rightarrow> h (f2 x))"
test_de_morgan: fixes x :: "'a"   and y :: "'a" shows "at x + at y = at (at (at x) \<cdot> at (at y))"
ly_def: shows "ly = y + (l - (x + y)) / 2"
sd_E_def: shows "sd_E = {x. fst x \<in> E \<and> snd x \<in> E}"
latin_rect_card_num: fixes s :: "(latin_type \<Rightarrow> nat) set"   and m :: "nat"   and n :: "nat"   and x :: "nat" assumes "latin_rect s m n"   and "x < n" shows "card {e Num |e. e \<in> s \<and> e Col = x} = m"
is_acc_iface_is_iface: fixes G :: "prog"   and P :: "pname"   and I :: "qtname" assumes "is_acc_iface G P I" shows "is_iface G I"
not_zero_i: shows "\<not> is_zero_row_upt_k i (Suc k) B"
cases: fixes x :: "('r, 's, 't, 'u, 'v, 'w, 'x) rest_scheme \<times> ('y \<times> 'r) \<times> (('z + 't) + 'aa + 'u)" obtains rest :: "('r, 's, 't, 'u, 'v, 'w, 'x) rest_scheme" and state :: "'y \<times> 'r" and iadv_core :: "'z" where "x = (resta__, state__, Inl (Inl iadv_core))"   | rest :: "('r, 's, 't, 'u, 'v, 'w, 'x) rest_scheme" and state :: "'y \<times> 'r" and iadv_rest :: "'t" where "x = (restb__, statea__, Inl (Inr iadv_rest))"   | rest :: "('r, 's, 't, 'u, 'v, 'w, 'x) rest_scheme" and state :: "'y \<times> 'r" and iusr_core :: "'aa" where "x = (restc__, stateb__, Inr (Inl iusr_core))"   | rest :: "('r, 's, 't, 'u, 'v, 'w, 'x) rest_scheme" and state :: "'y \<times> 'r" and iusr_rest :: "'u" where "x = (rest, state, Inr (Inr iusr_rest))"
A3a: fixes x :: "i" shows "x \<cdot> (dom x) \<cong> x"
M_def: shows "M \<equiv> \<chi>i j. f j $ i - x0 $ i"
disc_I: fixes H :: "ocircline"   and z :: "complex_homo" assumes "in_ocircline H z" shows "z \<in> disc H"
bigo_sum5: fixes f :: "'c \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd"   and l :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and k :: "'e \<Rightarrow> 'f \<Rightarrow> 'c"   and A :: "'e \<Rightarrow> 'f set" assumes "f \<in> O(h)"   and "\<forall>x y. (0::'d) \<le> l x y"   and "\<forall>x. (0::'d) \<le> h x" shows "(\<lambda>x. \<Sum>y\<in>A x. l x y * f (k x y)) \<in> O(\<lambda>x. \<Sum>y\<in>A x. l x y * h (k x y))"
fps_const_neg: fixes c :: "'a" shows "- fps_const c = fps_const (- c)"
INF_top_conv_1: fixes 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 B :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(Inf (B ` A) = top) = (\<forall>x\<in>A. B x = top)"
is_interval_co: fixes r :: "real"   and s :: "real" shows "is_interval {r..<s}"
Polys_alt: fixes X :: "'b set" shows "P[X] = {p. indets p \<subseteq> X}"
walk_decomp_2: fixes G :: "('a, 'b) Graph_scheme"   and xs :: "'a list"   and ys :: "'a list" assumes "Digraph G"   and "Digraph.walk G (xs @ ys)" shows "Digraph.walk G ys"
inf_greatest: fixes x :: "'b upred"   and y :: "'b upred"   and z :: "'b upred" assumes "y \<sqsubseteq> x"   and "z \<sqsubseteq> x" shows "(y \<and> z) \<sqsubseteq> x"
H_fcs_eq_G_fcs: fixes b :: "'b" assumes "b \<in> arcs G"   and "{b, G.face_cycle_succ b} \<inter> {a, a'} = {}" shows "H.face_cycle_succ b = G.face_cycle_succ b"
enabled_mono2: fixes F :: "state \<times> state \<Rightarrow> bool"   and s :: "state"   and G :: "state \<times> state \<Rightarrow> bool" assumes "s \<Turnstile> Enabled F"   and "\<And>t. F (s, t) \<Longrightarrow> G (s, t)" shows "s \<Turnstile> Enabled G"
simps_14: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b lambda \<Rightarrow> 'b lambda \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f3 :: "('b \<Rightarrow> 'b lambda \<times> 'a) \<Rightarrow> 'a"   and x21 :: "'b lambda"   and x22 :: "'b lambda" shows "rec_lambda f1 f2 f3 (App x21 x22) = f2 x21 x22 (rec_lambda f1 f2 f3 x21) (rec_lambda f1 f2 f3 x22)"
oLimit_shift_mono: fixes f :: "nat \<Rightarrow> ordinal"   and k :: "nat" assumes "mono f" shows "oLimit (\<lambda>n. f (n + k)) = oLimit f"
exhaust: fixes y :: "'d net_state" obtains x11 :: "nat" and x12 :: "'d" and x13 :: "nat set" where "y = NodeS x11 x12 x13"   | x21 :: "'d net_state" and x22 :: "'d net_state" where "y = SubnetS x21 x22"
Un_type_wellformed: fixes ts :: "'BExp set set" assumes "Ball ts type_wellformed" shows "type_wellformed (\<Union> ts)"
least_def: fixes a :: "hf" shows "least a \<equiv> THE x. x \<^bold>\<in> a \<and> (\<forall>y. y \<^bold>\<in> a \<longrightarrow> Rep_hf x \<le> Rep_hf y)"
transP_tranclp: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "transp r\<^sup>+\<^sup>+"
one_side_chara: fixes P :: "'p"   and Q :: "'p"   and A :: "'p"   and B :: "'p" assumes "P Q OS A B" shows "\<forall>X. Col X P Q \<longrightarrow> \<not> Bet A X B"
rep_eq: fixes x :: "('b, 'c) mapping" shows "Mapping.keys x = dom (mapping.rep x)"
splay_Leaf_iff: fixes x :: "'c"   and t :: "('c \<times> 'd) tree" shows "(Splay_Map.splay x t = \<langle>\<rangle>) = (t = \<langle>\<rangle>)"
top_empty_eq2: shows "\<top> = (\<lambda>x y. (x, y) \<in> UNIV)"
ireturn_rule: fixes \<Phi> :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "\<Phi> x" shows "RETURN x \<le> SPEC \<Phi>"
two: shows "Missing_List.min_list (y # ys) = min y (Missing_List.min_list ys)"
poly_add_closed: fixes K :: "'a set"   and p1 :: "'a list"   and p2 :: "'a list" assumes "subring K R"   and "polynomial K p1"   and "polynomial K p2" shows "polynomial K (poly_add p1 p2)"
n_mult_monom: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "n_mult (monom P a (Suc n)) = monom P ([Suc n] \<cdot> a) (Suc n)"
assms: shows "low_equal s1 s2 \<and> t1 = snd (fst (shift_instr instr s1)) \<and> t2 = snd (fst (shift_instr instr s2))"
pi: fixes f :: "complex \<Rightarrow> complex" assumes "f holomorphic_on S" shows "contour_integral g f = contour_integral p f"
flat_ordI: fixes x :: "'c"   and a :: "'c"   and y :: "'c" assumes "x \<noteq> a \<Longrightarrow> x = y" shows "flat_ord a x y"
filter_join_def: fixes pos :: "bool"   and X :: "'c set"   and m :: "('c, 'd) mapping" shows "filter_join pos X m = Mapping.filter (join_filter_cond pos X) m"
reals_zero_comparable_iff: fixes x :: "complex" shows "(x \<in> \<real>) = (x \<le> 0 \<or> 0 \<le> x)"
lift_bin_aux: fixes a :: "float interval option"   and b :: "float interval option"   and f :: "float interval \<Rightarrow> float interval \<Rightarrow> float interval option"   and ivl :: "float interval" assumes "lift_bin a b f = Some ivl" obtains ivl1 :: "float interval" and ivl2 :: "float interval" where "a = Some ivl1" and "b = Some ivl2" and "f ivl1 ivl2 = Some ivl"
dom_eqI: fixes a :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VV.ide a"   and "VV.seq f a" shows "VV.dom f = a"
preserves_dom: fixes f :: "'d" assumes "D.arr f" shows "C.dom (\<tau> f) = F' (D.dom f)"
rev_rev_ident: fixes xs :: "'g list" shows "rev (rev xs) = xs"
distinct_zipI2: fixes ys :: "'c list"   and xs :: "'d list" assumes "distinct ys" shows "distinct (zip xs ys)"
borel_measurable_suminf: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and M :: "'b measure" assumes "\<And>i. f i \<in> borel_measurable M" shows "(\<lambda>x. \<Sum>i. f i x) \<in> borel_measurable M"
assms_2: shows "M \<longrightarrow>\<^sub>a M'"
swap_sym: fixes b :: "'b"   and a :: "'b" shows "swap b a = swap a b"
finite_measure: fixes M :: "'b measure" assumes "prob_space M" shows "finite_measure M"
length_filter_le: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "length (filter P xs) \<le> length xs"
subprob_algebra_cong: fixes M :: "'c measure"   and N :: "'c measure" assumes "sets M = sets N" shows "subprob_algebra M = subprob_algebra N"
iff: shows "AE x in M. (x \<in> A) = (x \<in> B)"
simps_10: fixes i :: "int"   and c :: "int"   and e :: "Cooper.num" shows "mirror (NDvd i (CN 0 c e)) = NDvd i (CN 0 c (Neg e))"
deadlock_no_red: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s -1-t\<triangleright>ta\<rightarrow> s'"   and "r1.deadlock s" shows "False"
permute_fun_comp: fixes p :: "perm"   and f :: "'b \<Rightarrow> 'c" shows "p \<bullet> f = permute p \<circ> f \<circ> permute (- p)"
freshInp_def: fixes xs :: "'a"   and x :: "'b"   and inp :: "'c \<Rightarrow> ('d, 'e, 'a, 'b, 'f) qTerm set option" shows "freshInp xs x inp \<equiv> liftAll (fresh xs x) inp"
ent_frame_fwd: fixes P :: "assn"   and R :: "assn"   and Ps :: "assn"   and F :: "assn"   and Q :: "assn" assumes "P \<Longrightarrow>\<^sub>A R"   and "Ps \<Longrightarrow>\<^sub>A P * F"   and "R * F \<Longrightarrow>\<^sub>A Q" shows "Ps \<Longrightarrow>\<^sub>A Q"
acyclic_3a_def: fixes x :: "'a" shows "acyclic_3a x \<equiv> \<forall>y. y \<le> x \<and> x \<le> y\<^sup>\<star> \<longrightarrow> y = x"
reflects_adjunction: fixes f :: "'a"   and g :: "'a"   and \<eta> :: "'a"   and \<epsilon> :: "'a" assumes "C.ide f"   and "C.ide g"   and "\<guillemotleft>\<eta> : src\<^sub>C f \<Rightarrow>\<^sub>C g \<star>\<^sub>C f\<guillemotright>"   and "\<guillemotleft>\<epsilon> : f \<star>\<^sub>C g \<Rightarrow>\<^sub>C src\<^sub>C g\<guillemotright>"   and "adjunction_in_bicategory (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (F f) (F g) (D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)) (D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g))" shows "adjunction_in_bicategory (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C f g \<eta> \<epsilon>"
s_top_def: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and K :: "'e set" shows "H \<diamondop>\<^bsub>G\<^esub> K = {z. \<exists>x\<in>H. \<exists>y\<in>K. x \<cdot>\<^bsub>G\<^esub> y = z}"
ball_conj_distrib: fixes A :: "'d set"   and P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" shows "(\<forall>x\<in>A. P x \<and> Q x) = ((\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>A. Q x))"
g_impl_rel_ext_internal_def: fixes Rm :: "('b \<times> 'd) set"   and Rv :: "('a \<times> 'c) set" shows "g_impl_rel_ext Rm Rv \<equiv> \<langle>Rm, \<langle>Rv\<rangle>fun_set_rel, \<langle>Rv\<rangle>slg_rel, \<langle>Rv\<rangle>list_set_rel\<rangle>gen_g_impl_rel_ext"
rel_fundefs_generalize: fixes F1 :: "'fenv_inca"   and F2 :: "'fenv_ubx"   and f :: "'fun"   and fd2 :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef" assumes "rel_fundefs (Finca_get F1) (Fubx_get F2)"   and "Fubx_get F2 f = Some fd2" shows "rel_fundefs (Finca_get F1) (Fubx_get (Fubx_add F2 f (Subx.generalize_fundef fd2)))"
B': fixes x :: "real" assumes "1 \<le> x" shows "\<bar>F x\<bar> \<le> B * sqrt x"
choose_approx: fixes n :: "nat"   and N :: "nat"   and k :: "nat" assumes "n \<le> N" shows "n choose k \<le> N choose N div 2"
composite_simps_1: shows "src (g \<star> h) = src h"
CB_below: fixes b :: "bool discr"   and v :: "CValue" shows "(CB\<cdot>b \<sqsubseteq> v) = (v = CB\<cdot>b)"
induct: fixes x1 :: "'a"   and x2 :: "'a"   and r :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "(x1, x2) \<in> r\<^sup>*"   and "\<And>a. P a a"   and "\<And>a b c. \<lbrakk>(a, b) \<in> r\<^sup>*; P a b; (b, c) \<in> r\<rbrakk> \<Longrightarrow> P a c" shows "P x1 x2"
semiring_ow_axioms_def: fixes U :: "'b set"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "semiring_ow_axioms U plus times \<equiv> (\<forall>a b c. a \<in> U \<longrightarrow> b \<in> U \<longrightarrow> c \<in> U \<longrightarrow> times (plus a b) c = plus (times a c) (times b c)) \<and> (\<forall>a b c. a \<in> U \<longrightarrow> b \<in> U \<longrightarrow> c \<in> U \<longrightarrow> times a (plus b c) = plus (times a b) (times a c))"
offs_none_def: fixes ns :: "nat list"   and ub :: "nat"   and xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and mi :: "'d"   and ma :: "'d"   and i :: "nat" shows "offs_none ns ub xs index key mi ma i \<equiv> (\<exists>j<length ns. 0 < offs_num (length ns) xs index key mi ma j \<and> i \<in> {ns ! j + offs_num (length ns) xs index key mi ma j..<offs_next ns ub xs index key mi ma j}) \<or> offs_num (length ns) xs index key mi ma 0 = 0 \<and> i < offs_next ns ub xs index key mi ma 0 \<or> 0 < offs_num (length ns) xs index key mi ma 0 \<and> i < ns ! 0"
orient_lit_inst_subterms: fixes L :: "'a literal"   and t :: "'a trm"   and s :: "'a trm"   and polarity :: "sign"   and \<sigma> :: "('a \<times> 'a trm) list"   and u :: "'a trm" assumes "orient_lit_inst L t s polarity \<sigma>"   and "u \<in> subterms_of t" shows "u \<in> subterms_of_lit L"
group_isomorphisms_imp_iso: fixes G :: "('d, 'e) monoid_scheme"   and H :: "('f, 'g) monoid_scheme"   and f :: "'d \<Rightarrow> 'f"   and g :: "'f \<Rightarrow> 'd" assumes "group_isomorphisms G H f g" shows "f \<in> Group.iso G H"
simps_2: fixes pending_write :: "bool"   and x :: "'b memref"   and xs :: "'b memref list"   and A :: "nat set" shows "acquired_reads pending_write (x # xs) A = (case x of Write\<^sub>s\<^sub>b volatile x xa xb A' L R W \<Rightarrow> if volatile then acquired_reads True xs (if pending_write then A \<union> A' - R else A' - R) else acquired_reads pending_write xs A | Read\<^sub>s\<^sub>b volatile a t v \<Rightarrow> if pending_write \<and> \<not> volatile \<and> a \<in> A then insert a (acquired_reads pending_write xs A) else acquired_reads pending_write xs A | Prog\<^sub>s\<^sub>b p1 p2 list \<Rightarrow> acquired_reads pending_write xs A | Ghost\<^sub>s\<^sub>b A' L R W \<Rightarrow> acquired_reads pending_write xs (A \<union> A' - R))"
imp_igWlsInp: fixes MOD :: "('o, 'p, 'q, 'sort, 'r, 's, 't, 'u, 'v) model_scheme"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 't option"   and h :: "'t \<Rightarrow> 'w"   and MOD' :: "('x, 'y, 'z, 'sort, 'aa, 'ab, 'w, 'ac, 'ad) model_scheme" assumes "igWlsInp MOD delta inp"   and "ipresIGWls h MOD MOD'" shows "igWlsInp MOD' delta (lift h inp)"
map_map: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "map f (map g xs) = map (f \<circ> g) xs"
croots_def: fixes p :: "complex poly" shows "croots p = (if p = 0 \<or> 2 < degree p then [] else if degree p = 0 then [] else if degree p = 1 then [roots1 p] else croots2 p)"
simps_1: fixes s :: "'a" shows "s \<star> [] = s"
continuous_map_alt: fixes T1 :: "'d topology"   and T2 :: "'e topology"   and f :: "'d \<Rightarrow> 'e" shows "continuous_map T1 T2 f = ((\<forall>U. openin T2 U \<longrightarrow> openin T1 (f -` U \<inter> topspace T1)) \<and> f ` topspace T1 \<subseteq> topspace T2)"
assms_3: shows "rel_interior S \<inter> T \<noteq> {}"
TR1_sub_carr: fixes M :: "('f, 'a, 'g) Module_scheme"   and N :: "('h, 'a, 'g) Module_scheme"   and P :: "('f \<times> 'h, 'a, 'g) Module_scheme" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "TR1 R M N P \<subseteq> carrier P"
gt_iff_sub_diff_same: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" shows "(t >\<^sub>t s) = (gt_sub t s \<or> gt_diff t s \<or> gt_same t s)"
simps_2: fixes C :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option" shows "not_MT C [] = False"
iD_flag_is_inv: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "ip \<in> iD rt" shows "the (flag rt ip) = Aodv_Basic.inv"
cycle_free_diag_equiv: fixes n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a" shows "(\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> (0::'a) \<le> len m i i xs) = cycle_free m n"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>D) (\<cdot>\<^sub>D) ((\<otimes>\<^sub>D) \<I>\<^sub>D) D.map D.\<ll>"
total_inj_id: fixes x :: "'a" assumes "is_total x"   and "is_inj x" shows "x ; x\<^sup>\<smile> = 1'"
iLastStrong_iff: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iLastStrong t0 I P = (iLast t0 I P \<and> I \<down>< t0 \<noteq> {} \<and> t0 \<in> I)"
extended_hyperb_ineq': fixes e :: "'b"   and x :: "'b Gromov_completion"   and y :: "'b Gromov_completion"   and z :: "'b Gromov_completion" shows "min (extended_Gromov_product_at e x y) (extended_Gromov_product_at e y z) \<le> extended_Gromov_product_at e x z + ereal (deltaG TYPE('b))"
f_aggregate_nth: fixes n :: "nat"   and xs :: "'d list"   and k :: "nat"   and ag :: "'d list \<Rightarrow> 'd" assumes "n < length xs div k" shows "f_aggregate xs k ag ! n = ag (xs \<up> (n * k) \<down> k)"
wf_iff_acyclic_if_finite: fixes r :: "('b \<times> 'b) set" assumes "finite r" shows "wf r = acyclic r"
map\<^sub>1_def: fixes f :: "'a" shows "\<tau>\<tau>'.map\<^sub>1 f = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 (trg\<^sub>C f), map\<^sub>0 (trg\<^sub>C f), G f] \<cdot>\<^sub>D (\<tau>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D map\<^sub>1 f) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 (trg\<^sub>C f), F f, map\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D (\<tau>\<^sub>1 f \<star>\<^sub>D map\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 (src\<^sub>C f), map\<^sub>0 (src\<^sub>C f)]"
pseudo_complement: fixes x :: "'a"   and y :: "'a" shows "(x \<sqinter> y = bot) = (x \<le> - y)"
neq_ring: fixes I :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "max_lideal I R addition multiplication zero unit" shows "I \<noteq> R"
lepoll_imp_Card_le: fixes a :: "V"   and b :: "V" assumes "elts a \<lesssim> elts b" shows "vcard a \<le> vcard b"
parComm: fixes Rel :: "(ccs \<times> ccs) set"   and P :: "ccs"   and Q :: "ccs" assumes "\<And>R T. (R \<parallel> T, T \<parallel> R) \<in> Rel" shows "P \<parallel> Q \<leadsto>[Rel] Q \<parallel> P"
eventually_happens: fixes P :: "'b \<Rightarrow> bool"   and net :: "'b filter" assumes "eventually P net" shows "net = bot \<or> (\<exists>x. P x)"
is_SNF_A': shows "is_SNF (Matrix.mat 1 2 (\<lambda>(a, b). if b = 0 then A $$ (0, 0) else A $$ (0, Suc i))) (1\<^sub>m 1, Smith_1x2 (Matrix.mat 1 2 (\<lambda>(a, b). if b = 0 then A $$ (0, 0) else A $$ (0, Suc i))))"
l_set_attribute_get_child_nodes_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> char list option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_attribute_get_child_nodes type_wf set_attribute set_attribute_locs known_ptr get_child_nodes get_child_nodes_locs \<equiv> l_set_attribute type_wf set_attribute set_attribute_locs \<and> l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<and> l_set_attribute_get_child_nodes_axioms set_attribute_locs get_child_nodes_locs"
stabilizer_def: fixes G :: "('d, 'e) partial_object_scheme"   and \<phi> :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and x :: "'f" shows "stabilizer G \<phi> x = {g \<in> carrier G. \<phi> g x = x}"
Y_def: shows "Y \<equiv> \<lambda>i. iterate i\<cdot>(\<Lambda> x. (\<sqinter>xa\<in>A \<rightarrow>  x) |-| SKIP)\<cdot>\<bottom>"
pair_bidirected_digraph_slim: shows "pair_bidirected_digraph slim"