hcju/isabelleps huggingface dataset:
explicit_sctn_set_def: fixes po :: "'e \<Rightarrow> 'f set"   and X0 :: "'f set" shows "explicit_sctn_set po X0 = isEmpty_spec X0 \<bind> (\<lambda>e. WHILE\<^bsup>\<lambda>(e, X, Y). (e \<longrightarrow> X = {}) \<and> X0 = X \<union> (\<Union>(sctn, IS)\<in>Y. IS) \<and> (\<forall>(sctn, IS)\<in>Y. IS \<subseteq> po sctn)\<^esup> (\<lambda>(e, X, Y). \<not> e) (\<lambda>(e, X, Y). split_with_invar po X \<bind> (\<lambda>((S, sctn), X'). isEmpty_spec X' \<bind> (\<lambda>e. RETURN (e, X', insert (sctn, S) Y)))) (e, X0, {}) \<bind> (\<lambda>(uu_, uu_, Xis). RETURN Xis))"
AxiomP_def: fixes x :: "tm" shows "AxiomP x \<equiv> x EQ \<guillemotleft>extra_axiom\<guillemotright> OR SentP x OR Equality_axP x OR HF_axP x OR Special_axP x OR Induction_axP x"
basis_list_basis_3: fixes fs :: "'a vec list" assumes "basis_list fs" shows "basis (set fs)"
Sbis_finished_discr_R: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and c :: "('c, 'd, 'a) cmd"   and d :: "('c, 'd, 'a) cmd"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis"   and "proper c"   and "finished c"   and "(c, d) \<in> PL_Indis.Sbis aval tval cval indis" shows "PL_Indis.discr aval tval cval indis d"
TESL_interp_unfold_stepwise_weakly_precedes: fixes K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock" shows "\<lbrakk> K\<^sub>1 weakly precedes K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<Inter> {Y. \<exists>n. Y = \<lbrakk> K\<^sub>1 weakly precedes K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup>}"
insert_Diff_same: fixes B :: "'b set"   and C :: "'b set"   and a :: "'b" shows "B - C \<subseteq> insert a (B - insert a C)"
preserves_cod: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c"   and \<tau> :: "'b \<Rightarrow> 'c"   and f :: "'b" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A f" shows "partial_magma.cod B (\<tau> f) = G (partial_magma.cod A f)"
assms_2: shows "\<forall>\<^sub>F n in G. P (g n)"
from_bool_1: fixes P :: "bool" shows "(from_bool P = 1) = P"
continuous_on_componentwise: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "continuous_on S f = (\<forall>i\<in>Basis. continuous_on S (\<lambda>x. f x \<bullet> i))"
qr: shows "(ms, q) \<in> reachable qmsg (recvmsg (R \<sigma>))"
restrict_shared_in: fixes a :: "nat"   and L :: "nat set"   and m :: "nat \<Rightarrow> bool option"   and A :: "nat set" assumes "a \<in> L" shows "(m \<ominus>\<^bsub>A\<^esub> L) a = None"
CallNull: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and ps :: "char list exp list"   and vs :: "val list"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and M :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>ps,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>map Val vs,s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>M(ps),s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NullPointer,s\<^sub>2\<rangle>"
scale_minus_left: fixes a :: "'a"   and x :: "('a, 'b) Finite_Cartesian_Product.vec" shows "- a *s x = - (a *s x)"
inject: fixes x21 :: "tree"   and x22 :: "tree"   and y21 :: "tree"   and y22 :: "tree" shows "(Branching x21 x22 = Branching y21 y22) = (x21 = y21 \<and> x22 = y22)"
irreducible\<^sub>d_int_rat: fixes p :: "int poly" assumes "irreducible\<^sub>d p" shows "irreducible\<^sub>d (of_int_poly p)"
of_int_add: fixes one :: "'q"   and times :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and plus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and zero :: "'q"   and minus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and uminus :: "'q \<Rightarrow> 'q"   and w :: "int"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (w + z) = plus (ring_1.of_int one plus zero minus w) (ring_1.of_int one plus zero minus z)"
Max_exact_ge: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat" assumes "valid_decomp X qs"   and "standard_decomp k qs" shows "Max (poly_deg ` fst ` set qs) \<le> Max (poly_deg ` fst ` set (exact k qs))"
linear_bounded_pos: fixes f :: "'a \<Rightarrow> 'b" assumes "linear f" obtains B :: "real" where "0 < B" and "\<And>x. norm (f x) \<le> B * norm x"
assms_3: shows "t1 \<oplus> lt f1 \<prec>\<^sub>t t2 \<oplus> lt f2"
interp_rga_reorder: fixes pre :: "('a \<times> 'a option) list"   and suf :: "('a \<times> 'a option) list"   and oid :: "'a"   and ref :: "'a option" assumes "rga_ops (pre @ suf @ [(oid, ref)])"   and "\<And>i r. (i, Some r) \<in> set suf \<Longrightarrow> r \<noteq> oid"   and "\<And>r. ref = Some r \<Longrightarrow> r \<notin> fst ` set suf" shows "interp_rga (pre @ (oid, ref) # suf) = interp_rga (pre @ suf @ [(oid, ref)])"
nodes_semi_def: shows "nodes_semi \<equiv> ((rev \<circ> sorted_list_of_set) ` Pow (set nodes), nodes_le, nodes_sup)"
Lcm_fin_0_iff: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "finite A" shows "(semiring_gcd.Lcm_fin one zero lcm A = zero) = (zero \<in> A)"
nat_le_power_trans: fixes n :: "nat"   and m :: "nat"   and k :: "nat" assumes "n \<le> 2 ^ (m - k)"   and "k \<le> m" shows "2 ^ k * n \<le> 2 ^ m"
m_const_def: fixes x :: "const" shows "m_const x = (if x = Any then 0 else 1)"
path_components_of_path_connected_space: fixes X :: "'b topology" assumes "path_connected_space X" shows "path_components_of X = (if topspace X = {} then {} else {topspace X})"
get_mode_ptr_in_heap: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and children :: "shadow_root_mode" assumes "h \<turnstile> get_mode shadow_root_ptr \<rightarrow>\<^sub>r children" shows "shadow_root_ptr |\<in>| shadow_root_ptr_kinds h"
iso_is_section: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "category.iso C f" shows "category.section C f"
elims_2: fixes x :: "('fsym, 'tp) efsym trm" assumes "GE.wt x" obtains xa :: "Preliminaries.var" where "x = Var xa"   | f :: "('fsym, 'tp) efsym" and Tl :: "('fsym, 'tp) efsym trm list" where "x = Fn f Tl" and "GE_wtFsym f \<and> list_all GE.wt Tl \<and> GE_arOf f = map GE.tpOf Tl"
exhaust: fixes y :: "'e list" obtains "y = []"   | x21 :: "'e" and x22 :: "'e list" where "y = x21 # x22"
\<a>: fixes k :: "nat"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "standard_decomp k ps" shows "standard_decomp (\<a> ps) ps"
landau_product_preprocess_4: shows "num.One + num.One = num.Bit0 num.One"
inv_closed: fixes x :: "'b" assumes "x \<in> carrier (G i)" shows "inv\<^bsub>G i\<^esub> x \<in> carrier (G i)"
Cauchy_iff: fixes X :: "nat \<Rightarrow> 'a" shows "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
pivotal_is_dictator: fixes i :: "indi"   and b :: "alt"   and a :: "alt"   and j :: "indi" assumes "pivotal i b"   and "a \<noteq> b"   and "j dictates a,b" shows "i = j"
is_derivationI: fixes X :: "('a \<Rightarrow> real) \<Rightarrow> real"   and p :: "'a" assumes "linear_on diff_fun_space UNIV (*\<^sub>R) (*\<^sub>R) X"   and "\<And>f g. \<lbrakk>f \<in> diff_fun_space; g \<in> diff_fun_space\<rbrakk> \<Longrightarrow> X (f * g) = f p * X g + g p * X f" shows "is_derivation X p"
disjE_PL: fixes A :: "'a set"   and p :: "'a"   and q :: "'a"   and r :: "'a" assumes "A \<turnstile> p .\<or>. q"   and "A \<turnstile> p .\<longrightarrow>. r"   and "A \<turnstile> q .\<longrightarrow>. r" shows "A \<turnstile> r"
6: shows "(concat xs, concat ys) \<in> \<langle>A\<rangle>list_rel"
init_nsc_code_def: shows "init_nsc_code = Incremental_Atom_Ops_For_NS_Constraint_Ops.init_nsc init_state preprocess"
simps_9: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc" shows "\<xi> \<Turnstile>\<^sub>c F\<^sub>c \<phi> = (\<exists>i. suffix i \<xi> \<Turnstile>\<^sub>c \<phi>)"
path_connectedin_singleton: fixes X :: "'a topology"   and a :: "'a" shows "path_connectedin X {a} = (a \<in> topspace X)"
DERIV_pos_imp_increasing: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real" assumes "a < b"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> \<exists>y. (f has_real_derivative y) (at x) \<and> 0 < y" shows "f a < f b"
axioms_2: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e" assumes "functor A B F" shows "category B"
blinfun_apply_componentwise: fixes B :: "'a \<Rightarrow>\<^sub>L 'b" shows "B = (\<Sum>i\<in>Basis. blinfun_scaleR (blinfun_inner_left i) (blinfun_apply B i))"
fps_shift_altdef': fixes n :: "nat"   and f :: "'a fps" assumes "inverse (1::'a) = (1::'a)" shows "fps_shift n f = f / fps_X ^ n"
hom_zero: fixes p :: "'a poly" shows "0 \<circ>\<^sub>p p = 0"
samp_uni_add_mult: fixes x :: "nat"   and q :: "nat"   and xa :: "nat"   and ya :: "nat"   and y :: "nat" assumes "coprime x q"   and "xa < q"   and "ya < q"   and "(y + x * xa) mod q = (y + x * ya) mod q" shows "xa = ya"
upto_split3: fixes i :: "int"   and j :: "int"   and k :: "int" assumes "i \<le> j"   and "j \<le> k" shows "[i..k] = [i..j - 1] @ j # [j + 1..k]"
Cons_1: fixes acc :: "(bool \<times> 'b \<times> 'c) list" assumes "(ic, p, q) \<in> set (fold (\<lambda>a. case a of (ic, a) \<Rightarrow> case a of (q_in_bs, a) \<Rightarrow> case a of (p, q) \<Rightarrow> \<lambda>ps'. if \<not> ic \<and> c q_in_bs ps' p q then ps' else (ic, p, q) # ps') ps acc)" shows "(ic, p, q) \<in> set acc \<or> (\<exists>q_in_bs. (ic, q_in_bs, p, q) \<in> set ps)"
reindex_cong: fixes l :: "'b \<Rightarrow> 'c"   and B :: "'b set"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and h :: "'b \<Rightarrow> 'd" assumes "inj_on l B"   and "A = l ` B"   and "\<And>x. x \<in> B \<Longrightarrow> g (l x) = h x" shows "sum g A = sum h B"
transitively_orientable_orientable: fixes x :: "'a" assumes "bounded_distrib_allegory.transitively_orientable (\<sqinter>) (\<sqinter>) (\<le>) (\<squnion>) bot id x" shows "bounded_distrib_allegory.orientable (\<sqinter>) (\<squnion>) bot id x"
intros: fixes h :: "real \<Rightarrow> real"   and F :: "real filter"   and f :: "real \<Rightarrow> real"   and cs :: "real msstream"   and f' :: "real \<Rightarrow> real"   and c :: "real" assumes "asymp_powser h F f cs"   and "f' \<in> O[F](\<lambda>_. 1)"   and "\<forall>\<^sub>F x in F. f' x = c + f x * h x" shows "asymp_powser h F f' (MSSCons c cs)"
active_subset_union: fixes M :: "('f \<times> 'l) set"   and N :: "('f \<times> 'l) set" shows "active_subset (M \<union> N) = active_subset M \<union> active_subset N"
F'_iter_stable_pair_on: shows "stable_pair_on (ds - {d'}) (map_prod id undual F'_iter)"
Basis_prod_def: shows "Basis = (\<lambda>u. (u, 0::'b)) ` Basis \<union> Pair (0::'a) ` Basis"
consistent_map_lists_def: fixes xs1 :: "('a \<times> 'b) list"   and xs2 :: "('a \<times> 'a) list" shows "consistent_map_lists xs1 xs2 \<equiv> \<forall>(x1, x2)\<in>set xs1. \<forall>(y1, y2)\<in>set xs2. x1 = y1 \<longrightarrow> x1 = y2"
lim_imp_Limsup: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and f0 :: "'b" assumes "F \<noteq> bot"   and "(f \<longlongrightarrow> f0) F" shows "Limsup F f = f0"
bl1_Normal_1: shows "frs = (stk, loc, C, M, pc) # FRS"
suffix_ConsD': fixes x :: "'c"   and xs :: "'c list"   and ys :: "'c list" assumes "suffix (x # xs) ys" shows "strict_suffix xs ys"
linear: fixes xa :: "'o"   and x :: "'a"   and y :: "'a" shows "le xa x y \<or> le xa y x"
type_ground_inhabited: fixes t :: "('fun, 'var) Term.term" shows "\<exists>t'. fv t' = {} \<and> \<Gamma> t = \<Gamma> t'"
3: shows "chine_hcomp (\<nu>' \<bullet> \<nu>) (\<mu>' \<bullet> \<mu>) = \<langle>Chn (\<nu>' \<bullet> \<nu>) \<cdot> \<p>\<^sub>1[\<nu>.dom.leg0, \<mu>.dom.leg1] \<lbrakk>\<nu>'.cod.leg0, \<mu>'.cod.leg1\<rbrakk> Chn (\<mu>' \<bullet> \<mu>) \<cdot> \<p>\<^sub>0[\<nu>.dom.leg0, \<mu>.dom.leg1]\<rangle>"
univ_poly_consistent: fixes K :: "'a set" assumes "subring K R" shows "univ_poly (R\<lparr>carrier := K\<rparr>) = univ_poly R"
projection_on_union: fixes l :: "'b list"   and Y :: "'b set"   and X :: "'b set" assumes "l \<upharpoonleft> Y = []" shows "l \<upharpoonleft> (X \<union> Y) = l \<upharpoonleft> X"
TensorDiag_in_Hom: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Diag t"   and "Diag u" shows "t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u \<in> Hom (Dom t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> Dom u) (Cod t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> Cod u)"
g'_def: shows "g' = snd (snd (splitFace g v w f [countVertices g..<countVertices g + n]))"
r: fixes i :: "nat" assumes "i < length xs" shows "xs ! i = r i *\<^sub>R sum_list xs"
assms_2: shows "n < \<^bsub>c\<^esub>\<up>\<^bsub>t\<^esub>n'"
rank_Gauss_Jordan_code: fixes A :: "(('a, 'b) vec, 'c) vec" shows "rank A = (if A = 0 then 0 else let A' = Gauss_Jordan A in mod_type_class.to_nat (GREATEST a. row a A' \<noteq> 0) + 1)"
add_diff_eq_iff_ennreal: fixes x :: "ennreal"   and y :: "ennreal" shows "(x + (y - x) = y) = (x \<le> y)"
TunerMode_CTRL_Init_def: shows "TunerMode_CTRL_Init = ''1''"
offs_pred_next: 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" assumes "offs_pred ns ub xs index key mi ma"   and "i < length ns"   and "0 < offs_num (length ns) xs index key mi ma i" shows "ns ! i < offs_next ns ub xs index key mi ma i"
inverse_unique: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and g :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "inverse_arrows f g" shows "local.inv f = g"
homotopic_paths_reparametrize: fixes p :: "real \<Rightarrow> 'b"   and s :: "'b set"   and f :: "real \<Rightarrow> real"   and q :: "real \<Rightarrow> 'b" assumes "path p"   and "path_image p \<subseteq> s"   and "continuous_on {0..1} f"   and "f ` {0..1} \<subseteq> {0..1}"   and "f 0 = 0"   and "f 1 = 1"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> q t = p (f t)" shows "homotopic_paths s p q"
Iff_MP_same: fixes H :: "fm set"   and A :: "fm"   and B :: "fm" assumes "H \<turnstile> A IFF B"   and "H \<turnstile> A" shows "H \<turnstile> B"
h: shows "map (\<lambda>i. s i * nth_default (0::'a) (a # b # xs) i) [0..<n] = s 0 * a # s 1 * b # map (\<lambda>i. s (i + 2) * nth_default (0::'a) xs i) [0..<m]"
w3_def: shows "w3 = geodesic_segment_param Gyz y (Gromov_product_at y x z)"
inner_zero_left: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and x :: "'e" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner zero x = 0"
MatchNot_ternary_ternary_eval: fixes \<beta> :: "'a \<Rightarrow> 'b \<Rightarrow> ternaryvalue"   and p :: "'b"   and m' :: "'a match_expr"   and m :: "'a match_expr"   and \<alpha> :: "action \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "action" assumes "ternary_ternary_eval (map_match_tac \<beta> p m') = ternary_ternary_eval (map_match_tac \<beta> p m)" shows "matches (\<beta>, \<alpha>) (MatchNot m') a p = matches (\<beta>, \<alpha>) (MatchNot m) a p"
\<delta>_etc: fixes \<zeta> :: "('s, 'v) hd"   and ss :: "('s, 'v) tm list" assumes "wary (App (apps (Hd \<zeta>) ss) t)" shows "\<delta>\<^sub>h + \<delta>\<^sub>h * (arity_sym\<^sub>h (min_ground_head \<zeta>) - of_nat (length ss) - 1) = \<delta>\<^sub>h * (arity_sym\<^sub>h (min_ground_head \<zeta>) - of_nat (length ss))"
jointfun_hom: fixes f :: "nat \<Rightarrow> 'e"   and n :: "nat"   and A :: "'e set"   and g :: "nat \<Rightarrow> 'e"   and m :: "nat"   and B :: "'e set" assumes "f \<in> {i. i \<le> n} \<rightarrow> A"   and "g \<in> {j. j \<le> m} \<rightarrow> B" shows "jointfun n f m g \<in> {j. j \<le> Suc (n + m)} \<rightarrow> A \<union> B"
vars_of_replacement_set: fixes t :: "'b trm"   and p :: "indices list"   and v :: "'b trm"   and s :: "'b trm" assumes "replace_subterm t p v s" shows "vars_of s \<subseteq> vars_of t \<union> vars_of v"
inj_index_independent_rows: fixes A :: "(('a, 'b) vec, 'c) vec"   and x :: "'c"   and y :: "'c" assumes "reduced_row_echelon_form A"   and "row x A \<in> {row i A |i. row i A \<noteq> 0}"   and "A $ x = A $ y" shows "x = y"
unsigned_or_eq: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and v :: "'b word"   and w :: "'b word" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "semiring_1.unsigned one plus zero (v OR w) = or (semiring_1.unsigned one plus zero v) (semiring_1.unsigned one plus zero w)"
admissible_disj: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "ccpo.admissible Sup (\<le>) P"   and "ccpo.admissible Sup (\<le>) Q" shows "ccpo.admissible Sup (\<le>) (\<lambda>x. P x \<or> Q x)"
Neg_atm_of_iff: fixes L :: "'b literal" shows "(Neg (atm_of L) = L) = is_neg L"
hyps_2: shows "(x, T1) # \<Gamma>1 \<turnstile> t : T2"
discr_ZObis: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" assumes "discr c"   and "discr d" shows "c \<approx>01 d"
cases: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'a cfg"   and a2 :: "'b cfg" assumes "rel_cfg P a1 a2" obtains cfg1 :: "'a cfg" and cfg2 :: "'b cfg" where "a1 = cfg1" and "a2 = cfg2" and "P (state cfg1) (state cfg2)" and "rel_pmf (\<lambda>s t. rel_cfg P (cont cfg1 s) (cont cfg2 t)) (action cfg1) (action cfg2)"
equivp_symp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "equivp R"   and "R x y" shows "R y x"
hom_det: fixes A :: "'b mat" shows "det (coeff_lift_hom.mat_hom A) = [:det A:]"
smap_simps_1: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b stream" shows "shd (smap f a) = f (shd a)"
satPB_Un: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and \<Phi>1 :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list set"   and \<Phi>2 :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list set" shows "GE.satPB \<xi> (\<Phi>1 \<union> \<Phi>2) = (GE.satPB \<xi> \<Phi>1 \<and> GE.satPB \<xi> \<Phi>2)"
tuple_span_eq: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "span f g" shows "tuple f g = HF'.tuple f g"
resultant_1_2: fixes p :: "'a poly" shows "resultant p 1 = (1::'a)"
eval_cases_7: 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 E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>V:=e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains v :: "val" and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" and l :: "char list \<Rightarrow> val option" and T :: "ty" and v' :: "val" where "e' = Val v'" and "s' = (h, l(V \<mapsto> v'))" and "P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>Val v,(h, l)\<rangle>" and "E V = \<lfloor>T\<rfloor>" and "P \<turnstile> T casts v to v' "   | e'a :: "expr" where "e' = throw e'a" and "P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>throw e'a,s'\<rangle>"
inj_on_eq_iff: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and x :: "'b"   and y :: "'b" assumes "inj_on f A"   and "x \<in> A"   and "y \<in> A" shows "(f x = f y) = (x = y)"
simps_2: fixes A :: "'a mat"   and I :: "nat" shows "swap_col_to_front A (Suc I) = swap_col_to_front (swapcols I (Suc I) A) I"
le_supI1: fixes x :: "'b"   and a :: "'b"   and b :: "'b" assumes "x \<le> a" shows "x \<le> a \<squnion> b"
between_same: fixes x :: "'a" shows "(1::'a) / (2::'a) * (x + x) = x"
fresh_plus_perm: fixes a :: "atom"   and p :: "perm"   and q :: "perm" assumes "a \<sharp> p"   and "a \<sharp> q" shows "a \<sharp> p + q"
lookup''_tp_minus: shows "Mapping.lookup a2_map'' (tp - (len - 1)) = Some mc"
plus_of_int: fixes x :: "'a"   and n :: "int" shows "cos (x + of_int n * ((2::'a) * of_real pi)) = cos x"
eff_def: fixes i :: "instr"   and P :: "'b prog"   and pc :: "nat"   and et :: "ex_table"   and t :: "(ty list \<times> ty err list) option" shows "eff i P pc et t = (case t of None \<Rightarrow> [] | \<lfloor>\<tau>\<rfloor> \<Rightarrow> norm_eff i P pc \<tau> @ xcpt_eff i P pc \<tau> et)"
pis: fixes p :: "complex poly"   and i :: "nat" assumes "(p, i) \<in> set pis" shows "degree p \<le> 2 \<or> (\<forall>x\<in>set (coeffs p). x \<in> \<rat>)"
that: fixes S :: "'a set" assumes "a \<in> S"   and "(0::'a) \<notin> S"   and "pairwise orthogonal S"   and "independent S"   and "finite S"   and "span S = UNIV"   and "card S = DIM('a)" shows "thesis"
exhaust: fixes y :: "'a ifex" obtains "y = Trueif"   | "y = Falseif"   | x31 :: "'a" and x32 :: "'a ifex" and x33 :: "'a ifex" where "y = IF x31 x32 x33"
viewL_correct_nonEmpty: fixes t :: "('a, 'b) FingerTreeStruc" assumes "FingerTreeStruc.ft_invar t"   and "t \<noteq> Empty" obtains a :: "'a \<times> 'b" and s :: "('a, 'b) FingerTreeStruc" where "FingerTreeStruc.viewL t = Some (a, s)" and "FingerTreeStruc.ft_invar s" and "FingerTreeStruc.toList t = a # FingerTreeStruc.toList s"
sup_bot_def: fixes x :: "'a bot"   and y :: "'a bot" shows "sup x y = (case x of Value v \<Rightarrow> case y of Value v' \<Rightarrow> Value (sup v v') | Bot \<Rightarrow> x | Bot \<Rightarrow> y)"
fun_lborel_prod_integrable: shows "integrable (lborel \<Otimes>\<^sub>M lborel) f"
assms_1: shows "{m'. coeff p m' \<noteq> (0::'a)} \<subseteq> S"
simps_9: fixes v :: "'d"   and va :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and vb :: "('a, 'b, 'c, 'd) seqp" shows "stermsl ({v}receive(va) . vb) = {{v}receive(va) . vb}"
p_comp_pp: fixes x :: "'a"   and y :: "'a" shows "- (x \<sqinter> - - y) = - (x \<sqinter> y)"