hcju/isabelleps huggingface dataset:
split_asm: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'b"   and f3 :: "'c \<Rightarrow> 'b"   and match_compress :: "'c match_compress" shows "P (case match_compress of CannotMatch \<Rightarrow> f1 | MatchesAll \<Rightarrow> f2 | MatchExpr x \<Rightarrow> f3 x) = (\<not> (match_compress = CannotMatch \<and> \<not> P f1 \<or> match_compress = MatchesAll \<and> \<not> P f2 \<or> (\<exists>x3. match_compress = MatchExpr x3 \<and> \<not> P (f3 x3))))"
\<ll>_agreement: shows "\<ll> = (\<lambda>f. if arr f then f \<cdot> \<l>[local.dom f] else null)"
degree_sum_unique': fixes I :: "'b set"   and i :: "'b"   and p :: "'b \<Rightarrow> 'c poly" assumes "finite I"   and "i \<notin> I"   and "\<And>j. j \<in> I \<Longrightarrow> degree (p j) < degree (p i)" shows "degree (sum p (insert i I)) = degree (p i)"
gg: shows "length ((swap 0 ^^ ALG_P ss x y s1) s2) = 2"
SUP_cong_simp: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b" assumes "A = B"   and "\<And>x. x \<in> B =simp=> C x = D x" shows "Sup (C ` A) = Sup (D ` B)"
l8_16_1: fixes A :: "'p"   and B :: "'p"   and X :: "'p"   and U :: "'p"   and C :: "'p" assumes "Col A B X"   and "Col A B U"   and "A B Perp C X" shows "\<not> Col A B C \<and> Per C X U"
k': shows "\<forall>a. coprime a p \<longrightarrow> [a ^ (2 ^ k' * m) = 1] (mod p)"
prime_int_iff': fixes p :: "int" shows "prime p = (1 < p \<and> (\<forall>n\<in>{2..<p}. \<not> n dvd p))"
star_unfoldr_eq: fixes x :: "'a" shows "1' + x\<^sup>\<star> ; x = x\<^sup>\<star>"
nlistToList_app: fixes xs :: "('c, 'd) Node list"   and ys :: "('c, 'd) Node list" shows "nlistToList (xs @ ys) = nlistToList xs @ nlistToList ys"
1: fixes s :: "('fun, 'var) Term.term"   and t :: "('fun, 'var) Term.term" assumes "s \<in> SMP (pair ` setops\<^sub>s\<^sub>s\<^sub>t S) - range Var"   and "t \<in> SMP (trms\<^sub>s\<^sub>s\<^sub>t S) - range Var"   and "\<exists>\<delta>. Unifier \<delta> s t" shows "s \<in> SMP (trms\<^sub>s\<^sub>s\<^sub>t S) - range Var"
inj_map: fixes f :: "'a \<Rightarrow> 'b" assumes "inj f" shows "inj (map_bset f)"
simps: fixes P :: "'a prog"   and ad :: "'addr"   and al :: "addr_loc" shows "vs_type_all P (ad, al) = {v. \<exists>T. P \<turnstile> ad@al : T \<and> P \<turnstile> v :\<le> T}"
eq_fst_iff: fixes a :: "'g"   and p :: "'g \<times> 'h" shows "(a = fst p) = (\<exists>b. p = (a, b))"
sum_list_nonpos: fixes xs :: "'b multiset list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x \<subseteq># {#}" shows "monoid_add.sum_list (+) {#} xs \<subseteq># {#}"
integrable_const_bound: fixes f :: "'b \<Rightarrow> 'c"   and B :: "real" assumes "AE x in M \<bind> f. norm (f x) \<le> B"   and "f \<in> borel_measurable (M \<bind> f)" shows "integrable (M \<bind> f) f"
intro: fixes rules :: "'d stream"   and eff :: "'d \<Rightarrow> 'e \<Rightarrow> 'e fset \<Rightarrow> bool"   and "structure" :: "'f set"   and sat :: "'f \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>r s sl. \<lbrakk>r \<in> i.R rules; eff r s sl; \<And>s'. s' |\<in>| sl \<Longrightarrow> \<forall>S\<in>structure. sat S s'\<rbrakk> \<Longrightarrow> \<forall>S\<in>structure. sat S s" shows "Soundness eff rules structure sat"
compact_imp_countably_compact: fixes U :: "'a set" assumes "compact U" shows "countably_compact U"
class_add_oconf: fixes P :: "jvm_prog"   and h :: "heap"   and obj :: "obj"   and C :: "char list"   and cdec :: "jvm_method class" assumes "P,h \<turnstile> obj \<surd>"   and "\<not> is_class P C"   and "\<And>D'. P \<turnstile> fst obj \<preceq>\<^sup>* D' \<Longrightarrow> D' \<noteq> C" shows "class_add P (C, cdec),h \<turnstile> obj \<surd>"
bags_in_V: fixes G :: "('c, 'd) Graph_scheme"   and T :: "('e, 'f) Graph_scheme"   and bag :: "'e \<Rightarrow> 'c set"   and t :: "'e" assumes "TreeDecomposition G T bag"   and "t \<in> V\<^bsub>T\<^esub>" shows "bag t \<subseteq> V\<^bsub>G\<^esub>"
other: fixes j :: "'a" assumes "j \<notin> I" shows "Q (\<sigma> j) (\<sigma>' j)"
reducedly_statically_complete_calculus_axioms: shows "reducedly_statically_complete_calculus Bot Inf (\<Turnstile>) Red_I Red_F"
E_validD_1: fixes v :: "'v"   and e :: "'w"   and v' :: "'v" assumes "(v, e, v') \<in> E" shows "v \<in> V"
Entry_no_call_source: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "sourcenode a = Entry" shows "False"
map_ident: fixes t :: "('a, 'b) xstate" shows "map_xstate (\<lambda>x. x) (\<lambda>x. x) t = t"
a: shows "abc_notfinal (abc_steps_l (0, lm) A n) A"
False: shows "fst (memory_read 10 address s1) \<noteq> None"
assms_3: shows "edge_preserving R\<^sub>2 E\<^sub>2 G"
R'_def: shows "R' \<equiv> lift_pref_profile agents alts agents' alts' R"
orthogonal_def: fixes x :: "'b"   and y :: "'b" shows "orthogonal x y = (inner x y = (0::'a))"
assms_2: fixes b1 :: "'t \<Rightarrow>\<^sub>0 'b"   and b2 :: "'t \<Rightarrow>\<^sub>0 'b" assumes "b1 \<in> B"   and "b2 \<in> B" shows "b1 + b2 \<in> B"
eint_defs_3: fixes m :: "eint"   and n :: "eint" shows "m + n = (case m of eint m \<Rightarrow> case n of eint n \<Rightarrow> eint (m + n) | \<infinity> \<Rightarrow> \<infinity> | \<infinity> \<Rightarrow> \<infinity>)"
ideals_set_is_semiring: shows "semiring (ideals_set R)"
map_image_def: fixes f :: "'g \<Rightarrow> 'f"   and m :: "'e \<Rightarrow> 'g option" shows "map_image f m = (\<lambda>k. map_option f (m k))"
lookupEvalHeap: fixes v :: "'a"   and h :: "('a \<times> 'b) list"   and f :: "'b \<Rightarrow> 'c" assumes "v \<in> domA h" shows "evalHeap h f v = f (the (map_of h v))"
right_idem: fixes a :: "'e"   and b :: "'e" shows "inf (inf a b) b = inf a b"
iso_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "span_vertical_category C" shows "category.iso (span_vertical_category.vcomp C) \<mu> = (partial_magma.arr (span_vertical_category.vcomp C) \<mu> \<and> category.iso C (Chn \<mu>))"
indaut_aut: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "ChamberComplexAutomorphism X (OpposedThinChamberComplexFoldings.induced_automorphism X f g)"
a1_map_lookup: fixes as :: "event_data option list"   and tp' :: "nat" assumes "Mapping.lookup a1_map as = Some tp'" shows "tp' < tp"
f2: shows "fst (memory_read 10 (address + 4) (snd (fst (write_reg w curr_win (rd AND 30) s1)))) \<noteq> None"
coeff_minus: fixes p :: "'a poly"   and n :: "nat" shows "poly.coeff (- p) n = - poly.coeff p n"
type_wf_defs: fixes h :: "(_) heap" shows "DocumentClass.type_wf h = (CharacterDataClass.type_wf h \<and> (\<forall>document_ptr\<in>fset (document_ptr_kinds h). get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr h \<noteq> None))"
sort_conv_fold: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list" assumes "class.linorder less_eq less" shows "linorder.sort_key less_eq (\<lambda>x. x) xs = fold (linorder.insort_key less_eq (\<lambda>x. x)) xs []"
igWlsAbsDisj_def: fixes MOD :: "('b, 'c, 'varSort, 'sort, 'd, 'e, 'f, 'g, 'h) model_scheme" shows "igWlsAbsDisj MOD \<equiv> \<forall>xs s xs' s' A. isInBar (xs, s) \<and> isInBar (xs', s') \<and> igWlsAbs MOD (xs, s) A \<and> igWlsAbs MOD (xs', s') A \<longrightarrow> xs = xs' \<and> s = s'"
mt: fixes K :: "'a set" assumes "K \<in> \<D>" shows "content K \<noteq> 0"
consts_list_comb: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and abs_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a"   and xs :: "'a list" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "consts (term.list_comb app f xs) = consts f |\<union>| ffUnion (fset_of_list (map consts xs))"
prefix_map_inj_eq: fixes f :: "'b \<Rightarrow> 'c"   and xs :: "'b list"   and ys :: "'b list" assumes "inj_on f (ran\<^sub>l xs \<union> ran\<^sub>l ys)" shows "prefix (map f xs) (map f ys) = prefix xs ys"
l_get_dom_component_get_scdom_component_def: fixes get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_scdom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_dom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog" shows "l_get_dom_component_get_scdom_component get_owner_document heap_is_wellformed type_wf known_ptr known_ptrs get_scdom_component get_dom_component \<equiv> l_known_ptrs known_ptr known_ptrs \<and> l_get_dom_component_get_scdom_component_axioms get_owner_document heap_is_wellformed type_wf known_ptrs get_scdom_component get_dom_component"
convolution_emeasure': fixes A :: "'a set"   and M :: "'a measure"   and N :: "'a measure" assumes "A \<in> sets borel"   and "finite_measure M"   and "finite_measure N"   and "sets N = sets borel"   and "sets M = sets borel" shows "emeasure (M \<star> N) A = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator A (x + y) \<partial>N \<partial>M"
e_2: shows "{x \<in> space M. e \<le> f x} \<notin> null_sets M"
insert_absorb: fixes a :: "'b"   and A :: "'b set" assumes "(a::'b) \<in> (A::'b set)" shows "(a::'b) \<triangleright> (A::'b set) = A"
x0_in_dom: shows "x0 \<in> valid_pub0"
assms_3: shows "Marker \<noteq> last (msgs (S t i) cid)"
in_keys_mult_scalar_ge: fixes v :: "'t"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys (p \<odot> q)" shows "punit.tt p \<oplus> tt q \<preceq>\<^sub>t v"
set_sort_oalist_aux: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and xs :: "('b \<times> 'c) list \<times> 'a"   and ko :: "'a" assumes "oalist_raw.oalist_inv rep_key_order xs" shows "set (oalist_raw.sort_oalist_aux rep_key_order ko xs) = set (fst xs)"
gcd_mult_distrib_nat: fixes k :: "nat"   and m :: "nat"   and n :: "nat" shows "k * gcd m n = gcd (k * m) (k * n)"
simps: fixes P :: "'a \<Rightarrow> bool" shows "index P = (if Alm_all P then Some (LEAST i. \<forall>j\<ge>i. P j) else None)"
IH: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "set xs \<noteq> {}"   and "x \<in> set xs" shows "map_of (map (\<lambda>x. (x, f x)) xs) x = Some (f x)"
eval_fds_def: fixes f :: "'b fds"   and s :: "'b" shows "eval_fds f s = (\<Sum>n. fds_nth f n / nat_power n s)"
differentiable_def: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "(f differentiable F) = (\<exists>D. (f has_derivative D) F)"
split_snoc: fixes pat :: "[:'r:]"   and xs :: "[:'r:]"   and x :: "'r" shows "split\<cdot>pat\<cdot>(xs :@ [:x:]) = split\<cdot>pat\<cdot>(cfst\<cdot>(split\<cdot>pat\<cdot>xs) :@ [:x:])"
the_cm_def: fixes c :: "('loc, 't) Propagate.configuration"   and loc :: "'loc"   and t :: "'t"   and n :: "int" shows "the_cm c loc t n = (THE c'. next_change_multiplicity' c c' loc t n)"
le: shows "adjoint M * M \<le>\<^sub>L 1\<^sub>m n"
inv_equality: fixes y :: "'a \<Rightarrow> 'b"   and x :: "'a \<Rightarrow> 'b" assumes "y \<otimes>\<^bsub>DirProds G I\<^esub> x = \<one>\<^bsub>DirProds G I\<^esub>"   and "x \<in> carrier (DirProds G I)"   and "y \<in> carrier (DirProds G I)" shows "inv\<^bsub>DirProds G I\<^esub> x = y"
normalize_simps_4: fixes c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "Language.normalize (Seq c\<^sub>1 c\<^sub>2) = sequence Seq (flatten (Language.normalize c\<^sub>1) @ flatten (Language.normalize c\<^sub>2))"
vsv_appI: fixes a :: "V"   and b :: "V"   and A :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<circ> A" shows "(r \<restriction>\<^sup>l\<^sub>\<circ> A)\<lparr>a\<rparr> = b"
set_equality_technical: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c"   and ys :: "'c list" assumes "{x} \<union> set xs = {y} \<union> set ys"   and "x \<notin> set xs"   and "y \<notin> set ys"   and "y \<in> set xs" shows "{x} \<union> (set xs - {y}) = set ys"
uminus_bit1_def: fixes x :: "'a bit1" shows "- x = Abs_bit1' (- Rep_bit1 x)"
C_is: shows "C = {(Operator k (index (\<Pi>\<^sub>\<O>) op))\<inverse>, (State k (index (\<Pi>\<^sub>\<V>) v))\<^sup>+}"
C: shows "enat (nat (2 * X) * nat (2 * X)) = 4 * enat (nat (X * X))"
last_tl: fixes xs :: "'d list" assumes "xs = [] \<or> tl xs \<noteq> []" shows "last (tl xs) = last xs"
rf: shows "Rf (Sup_llist Ns - Rf (Sup_llist Ns)) \<subseteq> Rf (Liminf_llist Ns)"
rep_eq: fixes x :: "'c ell2" shows "norm x = ell2_norm (Rep_ell2 x)"
get_root_node_si_locs_def: shows "get_root_node_si_locs \<equiv> l_get_root_node_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_root_node_si_locs Shadow_DOM.get_parent_locs get_host_locs"
simps_16: fixes \<C> :: "t_context"   and tn :: "t list"   and tm :: "t list"   and es :: "b_e list"   and ts :: "checker_type" shows "check_single \<C> (Loop (tn _> tm) es) ts = (if b_e_type_checker (\<C>\<lparr>label := [tn] @ label \<C>\<rparr>) es (tn _> tm) then type_update ts (to_ct_list tn) (Type tm) else Bot)"
v11_vconverse: shows "v11 (r\<inverse>\<^sub>\<circ>)"
map\<^sub>0_in_hhom: fixes a :: "'b" assumes "B.obj a" shows "\<guillemotleft>\<tau>\<^sub>0 a : src\<^sub>C (F a) \<rightarrow>\<^sub>C src\<^sub>C (G a)\<guillemotright>"
subst_subtype: fixes \<Delta> :: "binding list"   and V :: "POPLmarkRecord.type"   and \<Gamma> :: "binding list"   and T :: "POPLmarkRecord.type"   and U :: "POPLmarkRecord.type" assumes "\<Delta> @ VarB V \<Colon> \<Gamma> \<turnstile> T <: U" shows "\<down>\<^sub>e 1 0 \<Delta> @ \<Gamma> \<turnstile> \<down>\<^sub>\<tau> 1 \<parallel>\<Delta>\<parallel> T <: \<down>\<^sub>\<tau> 1 \<parallel>\<Delta>\<parallel> U"
simps: fixes a1 :: "Javatype"   and a2 :: "Javatype" shows "treach a1 a2 = ((\<exists>U T. a1 = T \<and> a2 = U \<and> U \<le> T) \<or> (\<exists>T S f U. a1 = T \<and> a2 = U \<and> treach T S \<and> S \<le> dtype f \<and> U \<le> rtype f) \<or> (\<exists>AT. a1 = ArrT AT \<and> a2 = IntgT) \<or> (\<exists>AT. a1 = ArrT AT \<and> a2 = at2jt AT) \<or> (\<exists>T U V. a1 = T \<and> a2 = V \<and> treach T U \<and> treach U V))"
fresh_3: fixes a :: "atom"   and fmla :: "fmla" shows "a \<sharp> neg fmla = a \<sharp> fmla"
MVOKBARFnotNR: fixes x :: "Vals" assumes "MVOKBARF x" shows "x \<noteq> NotAResult"
"case": fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and x1 :: "'e"   and x2 :: "'f" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
f_inv_into_f: fixes y :: "'e"   and f :: "'f \<Rightarrow> 'e"   and A :: "'f set" assumes "y \<in> f ` A" shows "f (inv_into A f y) = y"
get_alloc: fixes xs :: "'a list"   and h :: "heap"   and ys :: "'a list" shows "Array.get (snd (Array.alloc xs h)) (fst (Array.alloc ys h)) = xs"
cSup_least: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and X :: "'d set"   and z :: "'d" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> less_eq x z" shows "less_eq (Sup X) z"
pow_distrib2: shows "order \<G> * c * x2 - c * x2 = (order \<G> - 1) * c * x2"
term_poly_list_rel_def: shows "term_poly_list_rel = {(xs, ys). ys = mset xs \<and> distinct xs \<and> sorted_wrt var_order xs}"
pt_name1: fixes x :: "'d" shows "[] \<bullet> x = x"
left_eq: shows "{p. \<exists>y. y \<in> cbox (g2 a) (g1 a) \<and> p = (y, a)} = (\<lambda>y. twoC (y, 0)) ` {0..1}"
Sup_insert: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and a :: "'b::type"   and A :: "'b::type set" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)" shows "(Sup::'b::type set \<Rightarrow> 'b::type) (insert (a::'b::type) (A::'b::type set)) = (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) a (Sup A)"
merge_def: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'c"   and B :: "'c"   and x :: "'c"   and ss :: "(nat \<times> 'c) list"   and c :: "'c list"   and pc :: "nat" assumes "lbv A r f T B"   and "x \<in> A"   and "snd ` set ss \<subseteq> A" shows "merge c f r T pc ss x = (if \<forall>(pc', s')\<in>set ss. pc' \<noteq> pc + 1 \<longrightarrow> s' \<sqsubseteq>\<^bsub>r\<^esub> c ! pc' then map snd (filter (\<lambda>(p', t'). p' = pc + 1) ss) \<Squnion>\<^bsub>f\<^esub> x else T)"
assms_1: shows "P |h \<turnstile> f|\<^sub>r"
W1b: fixes a :: "'a" shows "(1::'a) r\<rightarrow> a = a"
vint_plus_def: shows "vint_plus = (\<lambda>x\<in>\<^sub>\<circ>\<int>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat>. (int_of_vint (x\<lparr>0\<^sub>\<nat>\<rparr>) + int_of_vint (x\<lparr>1\<^sub>\<nat>\<rparr>))\<^sub>\<int>)"
OP_def: fixes x :: "'f" shows "Autoref_Tagging.OP x \<equiv> x"
elims_3: fixes x :: "('c \<times> color) tree" assumes "\<not> invc x" obtains l :: "('c \<times> color) tree" and a :: "'c" and c :: "color" and r :: "('c \<times> color) tree" where "x = \<langle>l, (a, c), r\<rangle>" and "\<not> ((c = Red \<longrightarrow> color l = Black \<and> color r = Black) \<and> invc l \<and> invc r)"
pfxm_mask_def: fixes x :: "'a prefix_match" shows "pfxm_mask x \<equiv> mask (LENGTH('a) - pfxm_length x)"
simps: fixes fs :: "'b list"   and rs :: "'b list" shows "invar (fs, rs) = (fs = [] \<longrightarrow> rs = [])"
preferred_alts_refl: fixes x :: "'a" assumes "x \<in> carrier" shows "x \<in> preferred_alts le x"
Units_l_inv_ex: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> Units Zp" shows "\<exists>y\<in>carrier Zp. y \<otimes> x = \<one>"
validCE_not_act: fixes cid :: "'id"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'cmp"   and n :: "nat"   and \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool" assumes "\<nexists>i. \<parallel>cid\<parallel>\<^bsub>t i\<^esub>"   and "eval cid t t' n \<gamma>" shows "\<gamma> (lnth (\<pi>\<^bsub>cid\<^esub>inf_llist t @\<^sub>l inf_llist t')) n"
gallery_to_0: fixes S :: "'b set"   and w :: "'b" assumes "CoxeterComplex S"   and "w \<in> \<langle>S\<rangle>"   and "w \<noteq> (0::'b)" shows "\<exists>xs. SimplicialComplex.maxsimpchain (CoxeterComplex.TheComplex S) (dual_order.poset_simplex_map (PreCoxeterSystem.special_cosets S) {w} # xs @ [dual_order.poset_simplex_map (PreCoxeterSystem.special_cosets S) 0])"
assms_1: shows "\<D>\<^sub>\<circ> r = A"
verit_prod_simplify_2: fixes a :: "'a" shows "a * (1::'a) = a"
one_prime_factor_iff_primepow: fixes n :: "'a" shows "(card (prime_factors n) = Suc 0) = primepow (normalize n)"
nat_term_order_eq_POT_ps: fixes ps :: "bool"   and to1 :: "'a nat_term_order"   and to2 :: "'a nat_term_order"   and dg :: "bool" assumes "ps \<or> is_scalar TYPE('a)" shows "nat_term_order_eq (POT to1) to2 dg ps = nat_term_order_eq to1 to2 dg ps"
functor_def: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah" shows "functor A B F \<equiv> category A \<and> category B \<and> functor_axioms A B F"
equals0I: fixes A :: "'f set" assumes "\<And>y. y \<in> A \<Longrightarrow> False" shows "A = {}"
i: shows "\<forall>n\<ge>i. \<bar>X n\<bar> < s"
mk_poly_sub_dim_2: fixes A :: "'b poly mat"   and l :: "nat"   and j :: "nat" shows "dim_col (mk_poly_sub A l j) = dim_col A"
eFresh_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and s :: "'sort"   and X :: "'c"   and ys :: "'varSort"   and y :: "'var" assumes "igWls MOD s X" shows "eFresh MOD ys y (OK X) = igFresh MOD ys y X"
view'_def: shows "\<V>' = \<lparr>V = V\<^bsub>\<V>\<^esub> \<union> N\<^bsub>\<V>\<^esub>, N = {}, C = C\<^bsub>\<V>\<^esub>\<rparr>"
rev_fBexI: fixes x :: "'a"   and A :: "'a fset"   and P :: "'a \<Rightarrow> bool" assumes "x |\<in>| A"   and "P x" shows "fBex A P"
permutation_is_nilpotent: fixes p :: "'b \<Rightarrow> 'b" assumes "permutation p" obtains n :: "nat" where "p ^^ n = id" and "0 < n"
even_two_times_div_two: 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 a :: "'c" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment"   and "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" shows "times (numeral.numeral one plus (num.Bit0 num.One)) (divide a (numeral.numeral one plus (num.Bit0 num.One))) = a"
mult_one_left: fixes z :: "complex_homo" shows "1\<^sub>h *\<^sub>h z = z"
finite_SigmaI: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set" assumes "finite A"   and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)" shows "finite (Sigma A B)"
strongly_connectedI: fixes G :: "('c, 'd) pre_digraph" assumes "verts G \<noteq> {}"   and "\<And>u v. \<lbrakk>u \<in> verts G; v \<in> verts G\<rbrakk> \<Longrightarrow> u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v" shows "strongly_connected G"
set_empty: fixes xs :: "'b list" shows "(set xs = {}) = (xs = [])"
LFOci_refine: fixes li :: "'a list"   and l :: "'b set"   and A :: "('a \<times> 'b) set"   and R :: "('d \<times> 'c) set"   and ci :: "'d \<Rightarrow> bool"   and c :: "'c \<Rightarrow> bool"   and fi :: "'a \<Rightarrow> 'd \<Rightarrow> 'd nres"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c nres"   and s0i :: "'d"   and s0 :: "'c"   and I :: "'b set \<Rightarrow> 'c \<Rightarrow> bool" assumes "(li, l) \<in> \<langle>A\<rangle>list_set_rel"   and "\<And>s si. (si, s) \<in> R \<Longrightarrow> ci si = c s"   and "\<And>x xi s si. \<lbrakk>(xi, x) \<in> A; (si, s) \<in> R\<rbrakk> \<Longrightarrow> fi xi si \<le> \<Down> R (f x s)"   and "(s0i, s0) \<in> R" shows "nfoldli li ci fi s0i \<le> \<Down> R (FOREACH\<^sub>C\<^bsup>I\<^esup> l c f s0)"