hcju/isabelleps huggingface dataset:
exec_move_def: fixes addr2thread_id :: "'b \<Rightarrow> 'c"   and thread_id2addr :: "'c \<Rightarrow> 'b"   and spurious_wakeups :: "bool"   and empty_heap :: "'a"   and allocate :: "'a \<Rightarrow> htype \<Rightarrow> ('a \<times> 'b) set"   and typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and heap_read :: "'a \<Rightarrow> 'b \<Rightarrow> addr_loc \<Rightarrow> 'b val \<Rightarrow> bool"   and heap_write :: "'a \<Rightarrow> 'b \<Rightarrow> addr_loc \<Rightarrow> 'b val \<Rightarrow> 'a \<Rightarrow> bool"   and ci :: "('b, 'a) check_instr"   and P :: "'b J1_prog"   and t :: "'c"   and e :: "'b expr1" shows "JVM_heap_base.exec_move addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P t e \<equiv> JVM_heap_base.exec_meth addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci (compP2 P) (compE2 e) (compxE2 e 0 0) t"
unstream_mapS: fixes a :: "('b, 'c) Stream"   and f :: "'b \<rightarrow> 'd" assumes "a \<noteq> \<bottom>" shows "unstream\<cdot>(mapS\<cdot>f\<cdot>a) = mapL\<cdot>f\<cdot>(unstream\<cdot>a)"
ipurge_tr_rev_aux_single_dom: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and xs :: "'c list" shows "ipurge_tr_rev_aux I D {u} xs = ipurge_tr_rev I D u xs"
inf_assoc: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" shows "inf_prod (inf_prod x y) z = inf_prod x (inf_prod y z)"
proper_part_asymmetry: fixes x :: "i"   and y :: "i" assumes "M4"   and "PP x y" shows "\<not> PP y x"
preserves_reflects_arr: fixes f :: "'a arrow_of_spans_data" shows "arr (trg f) = arr f"
gb_schema_dummy_Nil: fixes data :: "nat \<times> nat \<times> 'd"   and D :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" shows "gb_schema_dummy data D bs [] = (gs @ bs, D)"
wp_validD: fixes P :: "'b \<Rightarrow> real"   and prog :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real" assumes "\<lbrace>P\<rbrace> prog \<lbrace>Q\<rbrace>p" shows "P \<tturnstile> wp prog Q"
vertex_retraction: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and v :: "'b" assumes "ChamberComplexRetraction X f"   and "v \<in> f ` \<Union> X" shows "f v = v"
in_set_conv_decomp_last: fixes x :: "'c"   and xs :: "'c list" shows "(x \<in> set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
rA': shows "local.sub3 d l (r, A) = (r', A')"
inject_2: fixes x2 :: "'c"   and y2 :: "'c" shows "(In2 x2 = In2 y2) = (x2 = y2)"
alternative: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "(\<Turnstile>\<^sub>3 {P} c { Q}) = (\<forall>ps n. P (ps, n) \<longrightarrow> (\<exists>ps' t n'. (c, ps) \<Rightarrow>\<^sub>A t \<Down> ps' \<and> n = n' + t \<and> Q (ps', n')))"
transition_\<X>: fixes l :: "'s"   and g :: "('c, real) cconstraint"   and \<mu> :: "('c set \<times> 's) pmf"   and X :: "'c set"   and l' :: "'s" assumes "A \<turnstile> l \<longrightarrow>\<^bsup>g,\<mu>,X\<^esup> l'" shows "X \<subseteq> \<X>"
self_codomain_iff_ide: fixes a :: "'a" shows "(a \<in> A.codomains a) = A.ide a"
filter_shuffles_disjoint2_1: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "set xs \<inter> set ys = {}"   and "zs \<in> shuffles xs ys" shows "filter (\<lambda>x. x \<in> set ys) zs = ys"
sb: shows "g1 ` {0..1} \<inter> g2 ` {0..1} \<subseteq> {g1 0, g2 0}"
arc_conv_closed: fixes x :: "'a" assumes "arc x" shows "arc (x\<^sup>T)"
subst_dom_insert_finite: fixes s :: "'a \<Rightarrow> ('b, 'a) Term.term"   and v :: "'a"   and t :: "('b, 'a) Term.term" shows "finite (subst_domain s) = finite (subst_domain (s(v := t)))"
disjoint_def: fixes A :: "'b set"   and B :: "'b set" shows "disjoint A B \<equiv> A \<inter> B = {}"
select_convs_3: fixes role :: "role_t"   and owner :: "agent"   and partner :: "agent"   and more :: "'a" shows "partner \<lparr>role = role, owner = owner, partner = partner, \<dots> = more\<rparr> = partner"
integr: shows "integrable M (\<lambda>w. expl_cond_expect M Y (\<lambda>w. max 0 (X w)) w - expl_cond_expect M Y (\<lambda>w. max 0 (- X w)) w)"
measurable_snd'': fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and N :: "'e measure"   and P :: "'f measure" assumes "f \<in> M \<rightarrow>\<^sub>M N" shows "(\<lambda>x. f (snd x)) \<in> P \<Otimes>\<^sub>M M \<rightarrow>\<^sub>M N"
orthogonal_clauses_10: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and a :: "'a"   and y :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner"   and "real_inner.orthogonal inner x a"   and "real_inner.orthogonal inner y a" shows "real_inner.orthogonal inner (minus x y) a"
intros_1: fixes s :: "nat"   and e :: "((nat \<times> nat) \<times> transition) fset"   and r :: "nat \<Rightarrow>f value option" shows "visits s e s r []"
Fun_pr0: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "Fun (hfsetcat.pr0 a b) = (\<lambda>x\<^bold>\<in>local.set (hfsetcat.prod a b). UP (hsnd (DOWN x)))"
image_mset_eq_implies_bij_betw: fixes A :: "'c set"   and A' :: "'d set"   and f :: "'c \<Rightarrow> 'e"   and f' :: "'d \<Rightarrow> 'e" assumes "finite A"   and "finite A'"   and "image_mset f (mset_set A) = image_mset f' (mset_set A')" obtains bij :: "'c \<Rightarrow> 'd" where "bij_betw bij A A'" and "\<forall>x\<in>A. f x = f' (bij x)"
indRelSTEQ_trans: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" shows "trans (indRelSTEQ SRel TRel)"
mat_mult_left_right_inverse: fixes A :: "'b mat"   and n :: "nat"   and B :: "'b mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "A * B = 1\<^sub>m n" shows "B * A = 1\<^sub>m n"
sum_mat_mono: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'b mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc"   and "A \<ge>\<^sub>m B" shows "sum_mat B \<le> sum_mat A"
l2_obs_partial_secrecy: shows "oreach l2 \<subseteq> l2_partial_secrecy"
pfun_of_inject: fixes x :: "('a, 'b) ffun"   and y :: "('a, 'b) ffun" shows "(pfun_of x = pfun_of y) = (x = y)"
p_minus_1_int: shows "int (p - 1) = int p - 1"
linorder_cases: fixes x :: "'a"   and y :: "'a" obtains "x < y"   | "x = y"   | "y < x"
section_retraction_char: fixes g :: "'s"   and f :: "'s" shows "S.ide (g \<cdot>\<^sub>S f) = (S.antipar f g \<and> compose (S.Dom f) (S.Fun g) (S.Fun f) = (\<lambda>x\<in>S.Dom f. x))"
dbl_dec_simps_2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_dec minus plus one zero = uminus one"
1: shows "simple_linux_router_nol12 rt fw p = Some (p\<lparr>p_oiface := oif\<rparr>)"
nat_pow_Suc: fixes x :: "int"   and n :: "nat" shows "x [^]\<^bsub>int_group\<^esub> Suc n = x [^]\<^bsub>int_group\<^esub> n \<otimes>\<^bsub>int_group\<^esub> x"
dg_GRPH_def: fixes \<alpha> :: "V" shows "dg_GRPH \<alpha> = [ZFC_in_HOL.set {\<CC>. digraph \<alpha> \<CC>}, all_dghms \<alpha>, \<lambda>\<FF>\<in>\<^sub>\<circ>all_dghms \<alpha>. \<FF>\<lparr>HomDom\<rparr>, \<lambda>\<FF>\<in>\<^sub>\<circ>all_dghms \<alpha>. \<FF>\<lparr>HomCod\<rparr>]\<^sub>\<circ>"
zero_less_Suc: fixes n :: "nat" shows "0 < Suc n"
n_t_closed: fixes x :: "'a" shows "n t x = n x"
infdist_eq_setdist: fixes x :: "'b"   and A :: "'b set" shows "infdist x A = setdist {x} A"
pr_Hom_pres_face_nodes: fixes \<phi> :: "'a \<Rightarrow> 'b"   and Fs\<^sub>1 :: "'a list set"   and Fs\<^sub>2 :: "'b list set" assumes "is_pr_Hom \<phi> Fs\<^sub>1 Fs\<^sub>2" shows "(\<Union>F\<in>Fs\<^sub>1. {\<phi> ` set F}) = (\<Union>F\<in>Fs\<^sub>2. {set F})"
\<alpha>edges_sym: fixes g :: "'m" assumes "invar g" shows "sym (\<alpha>edges_aux g)"
Quotient3_rel_rep: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and a :: "'b"   and b :: "'b" assumes "Quotient3 R Abs Rep" shows "R (Rep a) (Rep b) = (a = b)"
negation_free_or: fixes \<phi> :: "atom fm"   and \<psi> :: "atom fm" assumes "negation_free \<phi>"   and "negation_free \<psi>" shows "negation_free (or \<phi> \<psi>)"
snd_safe_distance_def: shows "delayed_safe_distance.snd_safe_distance = (other.s' \<delta> - v\<^sub>e)\<^sup>2 / (2 * (a\<^sub>o - a\<^sub>e))"
IH: assumes "us \<in> lists \<langle>G\<rangle>" shows "concat (Ref G us) = concat us"
coprime_absorb_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and y :: "'a"   and x :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times y x" shows "algebraic_semidom.coprime times one x y = dvd.dvd times y one"
finprod_one: fixes A :: "'b set" shows "(\<Prod>i\<in>A. (1::'a)) = (1::'a)"
closed_segment_neq_empty: fixes a :: "'b"   and b :: "'b" shows "closed_segment a b \<noteq> {}"
jb_imp_uppert: fixes A :: "'a mat"   and i :: "nat"   and j :: "nat" assumes "local.jb A i j" shows "local.uppert A i j"
OV_inv2_finite_map_graph: fixes s :: "'b obsv_state_scheme" assumes "s \<in> OV_inv2" shows "finite (map_graph (\<lambda>(x, y). obsv_state.obs s x y))"
unpermute_def: fixes p :: "perm" shows "unpermute p = permute (- p)"
imageE: fixes b :: "'a"   and f :: "'c \<Rightarrow> 'a"   and A :: "'c set" assumes "b \<in> f ` A" obtains x :: "'c" where "b = f x" and "x \<in> A"
op_ivl_of_ivl_coll_def: fixes XS :: "'b set" shows "op_ivl_of_ivl_coll XS = sets_of_coll XS \<bind> (\<lambda>XS. FORWEAK XS (RETURN (0::'b, 0::'b)) ivl_rep (\<lambda>(i, s) (i', s'). RETURN (inf i i' ::: lv_rel, sup s s' ::: lv_rel)) \<bind> (\<lambda>(l, u). RETURN (op_atLeastAtMost_ivl l u)))"
continuous_constant_on_closure: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and a :: "'c"   and x :: "'b" assumes "continuous_on (closure S) f"   and "\<And>x. x \<in> S \<Longrightarrow> f x = a"   and "x \<in> closure S" shows "f x = a"
ok3_move3: fixes r :: "'b mrexp3"   and m :: "'b"   and c :: "bool" assumes "ok3 r" shows "ok3 (move3 m r c)"
p_prod_tau_alpha_var: fixes x :: "'a"   and y :: "'a" shows "x \<parallel> y = x \<parallel> \<nu> y + \<nu> x \<parallel> y + \<tau> (x \<parallel> y)"
fps_nth: fixes x :: "('c, 'd) fps" shows "($) x \<in> {f. True}"
diff_add: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ordered_cancel_comm_monoid_diff plus zero less_eq less minus"   and "less_eq a b" shows "plus (minus b a) a = b"
addfunsetD_range: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and M :: "'b set" assumes "f \<in> addfunset A M" shows "range f \<subseteq> M"
cases: fixes x :: "'a literal" obtains t :: "'a trm" and s :: "'a trm" where "x = Pos (Eq ta__ sa__)"   | t :: "'a trm" and s :: "'a trm" where "x = Neg (Eq t s)"
r: shows "{x \<in> space N. J x = K} = space N \<inter> ({x. J x = K} \<inter> space N)"
l5_2: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A \<noteq> B"   and "Bet A B C"   and "Bet A B D" shows "Bet B C D \<or> Bet B D C"
EV_def: fixes f :: "nat \<Rightarrow> bool" shows "EV f \<equiv> \<exists>n. f n"
xb: shows "Derivation x (derivation_shift D 1 (length (snd r))) b"
leq_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "leq x y"   and "leq y z" shows "leq x z"
characteristic_iso_3: fixes f :: "'a" assumes "S.ide f" shows "\<guillemotleft>(\<iota> \<star> f) \<cdot>\<^sub>S local.inv \<a>[b, b, f] : L (L f) \<Rightarrow>\<^sub>S L f\<guillemotright>"
mapping_eqI': fixes m :: "('b, 'c) mapping"   and d :: "'c"   and m' :: "('b, 'c) mapping" assumes "\<And>x. x \<in> keys m \<Longrightarrow> lookup_default d m x = lookup_default d m' x"   and "keys m = keys m'" shows "m = m'"
zero_power: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "nat" assumes "class.semiring_1 one times plus zero"   and "0 < n" shows "power.power one times zero n = zero"
expand_expand_init_propag: fixes n' :: "('b, 'c) node_scheme"   and n_ns :: "'b node \<times> 'b node set" assumes "(\<forall>nm\<in>incoming n'. nm < name n') \<and> name n' \<le> name (fst n_ns) \<and> (incoming n' \<inter> incoming (fst n_ns) \<noteq> {} \<longrightarrow> new n' \<subseteq> old (fst n_ns) \<union> new (fst n_ns))"   and "\<forall>nd\<in>snd n_ns. \<forall>nm\<in>incoming n'. nm \<in> incoming nd \<longrightarrow> new n' \<subseteq> old nd" shows "expand n_ns \<le> SPEC (\<lambda>r. name n' \<le> fst r \<and> (\<forall>nd\<in>snd r. \<forall>nm\<in>incoming n'. nm \<in> incoming nd \<longrightarrow> new n' \<subseteq> old nd))"
norm_gt_square: fixes a :: "real"   and x :: "'a" shows "(a < norm x) = (a < 0 \<or> a\<^sup>2 < x \<bullet> x)"
lim: shows "(F \<longlongrightarrow> T) at_top"
wf_J_mdecl_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and C :: "char list" shows "wf_J_mdecl P C \<equiv> \<lambda>(M, Ts, T, pns, body). length Ts = length pns \<and> distinct pns \<and> this \<notin> set pns \<and> (\<exists>T'. P,[this \<mapsto> Class C, pns [\<mapsto>] Ts] \<turnstile> body :: T' \<and> P \<turnstile> T' \<le> T) \<and> \<D> body \<lfloor>{this} \<union> set pns\<rfloor>"
inc_def: shows "inc = (\<Lambda> x. inc_Arity x)"
indexed_padd_zero_2: fixes P :: "'d multiset \<Rightarrow> 'c multiset \<Rightarrow> 'a" assumes "L.carrier_coeff P" shows "L.indexed_padd (L.indexed_const \<zero>\<^bsub>L\<^esub>) P = P"
case_7: fixes f1 :: "bool \<Rightarrow> 'a"   and f2 :: "int \<Rightarrow> 'a"   and f3 :: "int \<Rightarrow> 'a"   and f4 :: "int \<Rightarrow> 'a"   and f5 :: "CTypeId \<Rightarrow> nat \<Rightarrow> 'a"   and f6 :: "Arraytype \<Rightarrow> nat \<Rightarrow> 'a"   and f7 :: "'a" shows "(case nullV of boolV x \<Rightarrow> f1 x | intgV x \<Rightarrow> f2 x | shortV x \<Rightarrow> f3 x | byteV x \<Rightarrow> f4 x | objV x xa \<Rightarrow> f5 x xa | arrV x xa \<Rightarrow> f6 x xa | nullV \<Rightarrow> f7) = f7"
part_extensionality: fixes x :: "i"   and y :: "i" shows "(x = y) = (\<forall>z. P z x = P z y)"
mult_aform'E: fixes e :: "nat \<Rightarrow> real"   and x :: "real"   and X :: "(real \<times> real pdevs) \<times> real"   and y :: "real"   and Y :: "(real \<times> real pdevs) \<times> real"   and p :: "nat" assumes "e \<in> UNIV \<rightarrow> {- 1..1}"   and "x \<in> aform_err e X"   and "y \<in> aform_err e Y" shows "x * y \<in> aform_err e (mult_aform' p X Y)"
perp_col4: fixes P :: "'p"   and Q :: "'p"   and R :: "'p"   and S :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "P \<noteq> Q"   and "R \<noteq> S"   and "Col A B P"   and "Col A B Q"   and "Col C D R"   and "Col C D S"   and "A B Perp C D" shows "P Q Perp R S"
intro: fixes invar :: "'a \<Rightarrow> bool"   and \<alpha> :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and update_dj :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "\<And>m k v. \<lbrakk>invar m; k \<notin> dom (\<alpha> m)\<rbrakk> \<Longrightarrow> \<alpha> (update_dj k v m) = \<alpha> m(k \<mapsto> v)"   and "\<And>m k v. \<lbrakk>invar m; k \<notin> dom (\<alpha> m)\<rbrakk> \<Longrightarrow> invar (update_dj k v m)" shows "map_update_dj \<alpha> invar update_dj"
Picard_step_ivl_def: fixes ode_ops :: "ode_ops"   and X0 :: "'c set"   and t0 :: "real"   and h :: "real"   and X :: "'c set" shows "Picard_step_ivl ode_ops X0 t0 h X = ASSERT (0 \<le> h) \<bind> (\<lambda>_. ASSERT (wd ode_ops TYPE('c)) \<bind> (\<lambda>_. let H = lv_ivl [0] [h]; D = DIM('c); env = concat ` listset [list_of_eucl ` X0, H, list_of_eucl ` X] in approx_slp_spec (euler_incr_fas' ode_ops D) D (euler_incr_slp ode_ops) env \<bind> (\<lambda>env. case env of None \<Rightarrow> RETURN None | Some env \<Rightarrow> op_ivl_rep_of_set (eucl_of_list ` env) \<bind> (\<lambda>(l, u). ASSERT (l \<le> u) \<bind> (\<lambda>_. safe_set ode_ops {l..u} \<bind> (\<lambda>s. if s then mk_safe ode_ops ({l..u} ::: appr_rel) \<bind> (\<lambda>r. RETURN (Some (r ::: appr_rel))) else RETURN None))))))"
foldr_aux_1_1: fixes us :: "'var list"   and g :: "'g"   and n :: "'node"   and u :: "('node, ('var \<times> 'node \<times> Def) set) mapping"   and p :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping"   and u' :: "('node, ('var \<times> 'node \<times> Def) set) mapping"   and p' :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping" assumes "set us \<subseteq> uses g n"   and "Mapping.lookup u n = None"   and "foldr (aux_1 g n) us (u, p) = (u', p')"   and "finite (Mapping.keys p)"   and "\<And>n v. (n, v) \<in> Mapping.keys p \<Longrightarrow> length (predecessors g n) \<noteq> 1"   and "\<And>v. Mapping.lookup p (Entry g, v) \<in> {None, Some []}" shows "lookupDef g n ` set us = lookup_multimap u' n"
zdiv_zminus2_eq_if: fixes b :: "int"   and a :: "int" assumes "b \<noteq> 0" shows "a div - b = (if a mod b = 0 then - (a div b) else - (a div b) - 1)"
exists_least_iff: fixes P :: "'a \<Rightarrow> bool" shows "(\<exists>n. P n) = (\<exists>n. P n \<and> (\<forall>m<n. \<not> P m))"
Deriv_def: fixes x :: "'b"   and A :: "'b list set" shows "Deriv x A = {xs. x # xs \<in> A}"
declclass_qmdecl_simp: fixes sig :: "sig"   and m :: "qtname \<times> methd" shows "declclass (qmdecl sig m) = declclass m"
Suc_length_conv: fixes n :: "nat"   and xs :: "'e list" shows "(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
induct: fixes P :: "int \<times> (int \<times> int) \<times> int \<times> int \<Rightarrow> (int \<times> int) list \<Rightarrow> bool"   and a0 :: "int \<times> (int \<times> int) \<times> int \<times> int"   and a1 :: "(int \<times> int) list" assumes "\<And>\<delta> c\<^sub>0 c\<^sub>1. P (\<delta>, c\<^sub>0, c\<^sub>1) []"   and "\<And>\<delta> c\<^sub>0 c\<^sub>1 p. P (\<delta>, c\<^sub>0, c\<^sub>1) [p]"   and "\<And>\<delta> c\<^sub>0 c\<^sub>1 p\<^sub>0 v va. \<lbrakk>\<And>x xa y. \<lbrakk>x = find_closest_code p\<^sub>0 \<delta> (v # va); (xa, y) = x; \<delta> \<le> xa\<rbrakk> \<Longrightarrow> P (\<delta>, c\<^sub>0, c\<^sub>1) (v # va); \<And>x xa y. \<lbrakk>x = find_closest_code p\<^sub>0 \<delta> (v # va); (xa, y) = x; \<not> \<delta> \<le> xa\<rbrakk> \<Longrightarrow> P (xa, p\<^sub>0, y) (v # va)\<rbrakk> \<Longrightarrow> P (\<delta>, c\<^sub>0, c\<^sub>1) (p\<^sub>0 # v # va)" shows "P a0 a1"
Entry_edge_Exit_or_High: fixes a :: "'edge" assumes "valid_edge a"   and "sourcenode a = (_Entry_)" shows "targetnode a = (_Exit_) \<or> targetnode a = (_High_)"
1: shows "(Cring_Poly.truncate R q \<otimes>\<^bsub>P\<^esub> (Cring_Poly.truncate R p \<oplus>\<^bsub>P\<^esub> monom P (p (deg R p)) (deg R p))) (deg R p + deg R q) = \<zero>"
add_im_sheaf_def: shows "add_im_sheaf \<equiv> \<lambda>V. +\<^bsub>identity S \<^sup>\<inverse> S V\<^esub>"
lub_upper: fixes M :: "'a set"   and z :: "'a" assumes "M \<in> Chains leq"   and "z \<in> M" shows "(z, lub M) \<in> leq"
Aexp_eq: fixes e :: "exp"   and a :: "Arity" shows "\<A>\<^bsub>a\<^esub> e = fst (cccExp e\<cdot>a)"
tau_eq_state': fixes j :: "nat"   and t :: "nat"   and i :: "nat" assumes "j < t + 2"   and "j \<noteq> gap i t" shows "\<tau> i j \<down>= e_nth (block i t) j"
paths_inj: fixes t :: "'b ttree"   and t' :: "'b ttree" assumes "paths t = paths t'" shows "t = t'"
Exit_Use_empty: fixes V :: "'var" assumes "V \<in> Use (_Exit_)" shows "False"
smooth_on_divide: fixes k :: "enat"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "k-smooth_on S f"   and "k-smooth_on S g"   and "open S"   and "\<And>x. x \<in> S \<Longrightarrow> g x \<noteq> (0::'c)" shows "k-smooth_on S (\<lambda>x. f x / g x)"
is_tm_dghm_axioms: shows "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<CC>"
inject: fixes x2 :: "int"   and y2 :: "int" shows "(Num_arc x2 = Num_arc y2) = (x2 = y2)"
derivable_A: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "fixed_point_proof A f"   and "fixed_point_proof.derivable A less_eq f x" shows "x \<in> A"
presWls_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'i"   and MOD :: "('j, 'k, 'l, 'sort, 'm, 'n, 'i, 'o, 'p) Recursion.model_scheme" shows "presWls h MOD \<equiv> \<forall>s X. wls s X \<longrightarrow> gWls MOD s (h X)"
extension_valid_maximal_conforming: fixes v' :: "'a" assumes "v' \<rightarrow> w0"   and "v' \<in> VV p \<Longrightarrow> \<sigma> v' = w0" shows "vmc_path G (LCons v' P) v' p \<sigma>"
matches_dest_1: fixes \<gamma> :: "('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)"   and m :: "'a match_expr"   and m' :: "'a match_expr"   and a :: "action"   and p :: "'b" assumes "matches \<gamma> (MatchAnd m m') a p"   and "\<not> matches \<gamma> m a p" shows "False"
False: shows "\<not> continuous_map (nsphere p) (nsphere p) f"
var: fixes t :: "'trm" assumes "set (map snd txs) \<subseteq> var"   and "set (map fst txs) \<subseteq> trm"   and "set [\<phi>] \<subseteq> fmla"   and "t \<in> set (map fst txs)" shows "set (getFrN (map snd txs) (map fst txs) [\<phi>] (length txs)) \<inter> set (map snd txs) = {}"
\<upsilon>_I: fixes p :: "fm"   and n :: "int"   and t :: "MIR.num"   and x :: "real"   and bs :: "real list" assumes "isrlfm p"   and "0 < real_of_int n"   and "numbound0 t" shows "Ifm (x # bs) (\<upsilon> p (t, n)) = Ifm (Inum (x # bs) t / real_of_int n # bs) p \<and> bound0 (\<upsilon> p (t, n))"
posliveness_dense_iff: fixes P :: "'a llist set" assumes "P \<subseteq> A\<^sup>\<spadesuit>" shows "posliveness A P = (P dense)"
add_zmset_eq_conv_diff: fixes a :: "'b"   and M :: "'b zmultiset"   and b :: "'b"   and N :: "'b zmultiset" shows "(add_zmset a M = add_zmset b N) = (M = N \<and> a = b \<or> M = add_zmset b (N - {#a#}\<^sub>z) \<and> N = add_zmset a (M - {#b#}\<^sub>z))"
start_lv: fixes d :: "nat"   and dm :: "nat" assumes "d < dm" shows "lv (start dm) d = 0"
trg_def: fixes \<mu> :: "'d" shows "D'.trg \<mu> = (if D'.arr \<mu> then trg\<^sub>D \<mu> else D'.null)"
outside_union_restrict: fixes P :: "('a \<times> 'b) set"   and X :: "'a set" shows "P = Outside' X P \<union> P || X"
fid_onI: fixes a :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> A" shows "[a, a]\<^sub>\<circ> \<in>\<^sub>\<circ> fid_on A"
eq_Real: fixes X :: "nat \<Rightarrow> rat"   and Y :: "nat \<Rightarrow> rat" assumes "Real.cauchy X"   and "Real.cauchy Y" shows "(Real.Real X = Real.Real Y) = Real.vanishes (\<lambda>n. X n - Y n)"
eval_fds_diff: fixes f :: "'a fds"   and s :: "'a"   and g :: "'a fds" assumes "fds_converges f s"   and "fds_converges g s" shows "eval_fds (f - g) s = eval_fds f s - eval_fds g s"
le_some_optE: fixes m :: "'b"   and x :: "'b option" assumes "Some m \<le> x" obtains m' :: "'b" where "x = Some m'" and "m \<le> m'"