hcju/isabelleps huggingface dataset:
simps_1: fixes rel :: "(nat \<times> nat) set" shows "steps rel 0 = rel"
suc_pc_l: shows "Suc pc + length ls = length \<tau>s"
line_integral_exists_on_degenerate_path: fixes basis :: "'b set"   and F :: "'b \<Rightarrow> 'b"   and c :: "'b" assumes "finite basis" shows "line_integral_exists F basis (\<lambda>x. c)"
sink_out: fixes y :: "'v" shows "\<not> edge \<Delta> (sink \<Delta>) y"
Interleaves_equal_nil_all: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list"   and zs :: "'b list" assumes "xs \<cong> {[], zs, P}" shows "xs = zs"
prime_factors_multiplicity: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and n :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "set_mset (factorial_semiring.prime_factorization zero times one normalize n) = {p. normalization_semidom.prime zero times one normalize p \<and> 0 < factorial_semiring.multiplicity times one p n}"
matrix_construction_is_kronecker_product: fixes subs1 :: "nat list list"   and n1 :: "nat"   and signs1 :: "rat list list"   and signs2 :: "rat list list"   and subs2 :: "nat list list" assumes "\<And>l i. \<lbrakk>l \<in> set subs1; i \<in> set l\<rbrakk> \<Longrightarrow> i < n1"   and "\<And>j. j \<in> set signs1 \<Longrightarrow> length j = n1" shows "M_mat (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) = kronecker_product (M_mat signs1 subs1) (M_mat signs2 subs2)"
prems_1: fixes xs :: "'a list" assumes "xs \<in> set (xs__ # xss)" shows "length xs = d"
ucast_or_distrib: fixes x :: "'c word"   and y :: "'c word" shows "UCAST('c \<rightarrow> 'b) (x OR y) = UCAST('c \<rightarrow> 'b) x OR UCAST('c \<rightarrow> 'b) y"
add_diff_add: fixes a :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" shows "a + c - (b + d) = a - b + (c - d)"
order_refl: fixes less_eq :: "'p \<Rightarrow> 'p \<Rightarrow> bool"   and less :: "'p \<Rightarrow> 'p \<Rightarrow> bool"   and x :: "'p" assumes "class.preorder less_eq less" shows "less_eq x x"
IH_1: fixes Rs :: "('a \<times> 'a) set set" assumes "finite Rs" shows "card ((\<lambda>R. eval_sorter R (f True)) ` Rs) \<le> 2 ^ count_wc_queries Rs (f True)"
case_eq_if: fixes f :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and prod :: "'f \<times> 'g" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (fst prod) (snd prod)"
mprod_expR_mem: fixes e :: "nat \<Rightarrow> nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "e \<in> {j. j \<le> n} \<rightarrow> {j. 0 \<le> j}"   and "f \<in> {j. j \<le> n} \<rightarrow> carrier R" shows "mprod_expR R e f n \<in> carrier R"
prime_elem_multiplicity_prod_mset_distrib: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and A :: "'a multiset" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero p"   and "zero \<notin># A" shows "factorial_semiring.multiplicity times one p (comm_monoid_mult.prod_mset times one A) = \<Sum>\<^sub># (image_mset (factorial_semiring.multiplicity times one p) A)"
Some_le: fixes x :: "'a"   and u :: "'a option" shows "(Some x \<le> u) = (\<exists>y. u = Some y \<and> x \<le> y)"
bind_RES_RETURN_eq: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'a" shows "RES X \<bind> (\<lambda>x. RETURN (f x)) = SPEC (\<lambda>uu_. \<exists>x. uu_ = f x \<and> x \<in> X)"
word_enum_decomp_set: fixes x :: "'b word"   and y :: "'b word"   and as :: "'b word list"   and a :: "'b word"   and bs :: "'b word list" assumes "[x .e. y] = as @ a # bs" shows "a \<notin> set as"
IH_2: assumes "Broadcast m2 \<in> set (history i)"   and "Broadcast m3 \<in> set (history i)" shows "Broadcast m2 \<sqsubset>\<^sup>i Broadcast m3"
segment_face_cycle_x_x_eq: fixes x :: "'b" shows "segment G.face_cycle_succ x x = G.face_cycle_set x - {x}"
one_plus_of_int_in_nonpos_Ints_iff: fixes n :: "int" shows "((1::'a) + of_int n \<in> \<int>\<^sub>\<le>\<^sub>0) = (n \<le> - 1)"
inner_vec_def: fixes inner_field :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inner :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec \<Rightarrow> 'a"   and x :: "('a, 'b) vec"   and y :: "('a, 'b) vec" assumes "vec_real_inner inner_field inner" shows "inner x y = (\<Sum>i\<in>UNIV. inner_field (x $ i) (y $ i))"
f_ind: fixes ap :: "abc_inst list"   and arity :: "nat"   and fp :: "nat"   and anything :: "nat list" assumes "rec_ci f = (ap, arity, fp)" shows "{\<lambda>nl. nl = xs @ 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = xs @ rec_exec f xs # 0 \<up> (fp - Suc arity) @ anything}"
option_rel_Some2: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a option"   and y :: "'b" shows "rel_option A x (Some y) = (\<exists>x'. x = Some x' \<and> A x' y)"
numeral_less_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and m :: "num"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less (numeral.numeral one plus m) (numeral.numeral one plus n) = (m < n)"
frag_extend_eq: fixes c :: "'c \<Rightarrow>\<^sub>0 int"   and g :: "'c \<Rightarrow> 'd \<Rightarrow>\<^sub>0 int"   and h :: "'c \<Rightarrow> 'd \<Rightarrow>\<^sub>0 int" assumes "\<And>f. f \<in> Poly_Mapping.keys c \<Longrightarrow> g f = h f" shows "frag_extend g c = frag_extend h c"
connected_refl: fixes c :: "nat \<times> nat \<Rightarrow> 'c"   and v :: "nat" shows "Graph.connected c v v"
sets_stat: fixes C :: "'s set" shows "sets (stat C) = sets (count_space UNIV)"
2: shows "can f (Trg f \<^bold>\<star> f) = \<l>[\<lbrace>f\<rbrace>]"
conjug_eqE: fixes x :: "'b list"   and z :: "'b list"   and y :: "'b list" assumes "x \<cdot> z = z \<cdot> y"   and "x \<noteq> \<epsilon>" obtains u :: "'b list" and v :: "'b list" and k :: "nat" where "u \<cdot> v = x" and "v \<cdot> u = y" and "(u \<cdot> v) \<^sup>@ k \<cdot> u = z" and "v \<noteq> \<epsilon>"
db_defs_1: fixes ic :: "state \<Rightarrow> 'b"   and q :: "state \<Rightarrow> 'b list"   and oc :: "state \<Rightarrow> 'b" shows "BInit ic q oc \<equiv> $q = #[] \<and> $ic = $oc"
simps_2: fixes i :: "int" shows "val_type (IntVal i) = INTEG"
add_mono_thms_linordered_semiring_2: fixes i :: "'c"   and j :: "'c"   and k :: "'c"   and l :: "'c" assumes "i = j \<and> k \<le> l" shows "i + k \<le> j + l"
INT_extend_simps_5: fixes a :: "'c"   and B :: "'d \<Rightarrow> 'c set"   and C :: "'d set" shows "insert a (\<Inter> (B ` C)) = (\<Inter>x\<in>C. insert a (B x))"
compose_id_inv_into: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and B :: "'a set" assumes "f ` A = B" shows "f \<circ> restrict (inv_into A f) B \<down> B = identity B"
USdBeqlen: fixes V :: "'com list"   and V' :: "'com list"   and d :: "'d" assumes "V \<approx>\<^bsub>d\<^esub> V'" shows "length V = length V'"
popE: fixes s :: "'s" assumes "invar s"   and "\<alpha> s \<noteq> Map.empty" obtains e :: "'e" and a :: "'a" and s' :: "'s" where "pop s = (e, a, s')" and "invar s'" and "\<alpha> s' = (\<alpha> s)(e := None)" and "\<alpha> s e = Some a" and "\<forall>y\<in>ran (\<alpha> s). a \<le> y"
illformed_osig_not_wf_osig: shows "\<not> wf_osig ({}, [STR ''A'' \<mapsto> [STR ''A'' \<mapsto> [insert STR ''A'' full_sort]]])"
thetaIf_def: shows "thetaIf \<equiv> {(if tst then c1 else c2, if tst then d1 else d2) |tst c1 c2 d1 d2. compatTst tst \<and> c1 \<approx>s d1 \<and> c2 \<approx>s d2}"
find_fund_sol_correct: fixes D :: "nat" shows "find_fund_sol D = (if is_square D then (0, 0) else pell.fund_sol D)"
span_image: fixes S :: "'b set" shows "m2.span (f ` S) = f ` m1.span S"
env_restr_useless: fixes m :: "'d \<Rightarrow> 'e"   and S :: "'d set" assumes "edom m \<subseteq> S" shows "m f|` S = m"
a: shows "(c::'a::refinement_lattice)\<^sup>\<omega> ; ((d::'a::refinement_lattice) ; (c \<sqinter> d)\<^sup>\<omega> \<sqinter> nil) \<sqsubseteq> (c \<sqinter> d)\<^sup>\<omega>"
ereal_infty_less_1: fixes x :: "ereal" shows "(x < \<infinity>) = (x \<noteq> \<infinity>)"
that: fixes N :: "nat set" assumes "N \<subseteq> M"   and "infinite N"   and "decides_subsets \<F> N" shows "thesis"
that: fixes m :: "nat \<Rightarrow>\<^sub>0 nat" assumes "p = monom m (coeff p m)" shows "thesis"
split_list: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = append ys (Cons x zs)"
sum_bounded_above: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and K :: "'d" assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<le> K" shows "sum f A \<le> of_nat (card A) * K"
tautologyNotFalse: fixes clause :: "Literal list"   and valuation :: "Literal list" assumes "clauseTautology clause"   and "consistent valuation" shows "\<not> clauseFalse clause valuation"
slp_of_fa_rev_un_def: fixes Unop :: "floatarith \<Rightarrow> floatarith"   and a :: "floatarith"   and M :: "(floatarith, nat) mapping"   and slp :: "floatarith list"   and slpl :: "nat"   and M1 :: "(floatarith, nat) mapping"   and slp1 :: "floatarith list"   and slpl1 :: "nat" shows "slp_of_fa_rev_un Unop a M slp slpl M1 slp1 slpl1 = (case Mapping.lookup M (Unop a) of None \<Rightarrow> (Mapping.update (Unop a) slpl1 M1, Unop (Var (slp_indexl_lookup slpl1 M1 a)) # slp1, Suc slpl1) | Some i \<Rightarrow> (Mapping.update (Unop a) slpl M, Var (slp_indexl slpl i) # slp, Suc slpl))"
hyps: assumes "\<forall>a\<in>set as. 1 \<le> a" shows "1 \<le> prod_list as"
intros_8: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs" assumes "P \<parallel> !P \<longmapsto>\<alpha> \<prec> P'" shows "!P \<longmapsto>\<alpha> \<prec> P'"
coprime_rat_poly_real_poly: fixes p :: "rat poly"   and q :: "rat poly" assumes "comm_monoid_mult_class.coprime p q" shows "comm_monoid_mult_class.coprime (real_of_rat_poly p) (real_of_rat_poly q)"
filter_mset_mset_set: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "finite A" shows "filter_mset P (mset_set A) = mset_set {x \<in> A. P x}"
\<Phi>_p_def: fixes x :: "real" shows "\<Phi>_p x = x ^ 3 / 3 * 2 ^ (l' + 1) * 2 ^ (l + 1) + x * (real_of_int i + 1) * (real_of_int i' + 1) - x\<^sup>2 / 2 * ((real_of_int i' + 1) * 2 ^ (l + 1) + (real_of_int i + 1) * 2 ^ (l' + 1))"
sumAssocLeft: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "Id \<subseteq> Rel" shows "(P \<oplus> Q) \<oplus> R \<leadsto>[Rel] P \<oplus> Q \<oplus> R"
rewrite_X_enat_correct: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n \<phi> = w \<Turnstile>\<^sub>n to_ltln (rewrite_X_enat \<phi>)"
1: shows "finite (And\<^sub>n ` fset ` min_dnf \<phi>)"
Sup_closed: shows "\<Squnion>\<^sub>o\<^sub>w ` Pow U \<subseteq> U"
stone_phi_def: fixes x :: "'c regular" shows "stone_phi x = Filters.filter.Abs_filter (stone_phi_base x)"
f_def: fixes k :: "'b" shows "f k = (if k = a then p else if a = a \<and> k = b then q else if a = b \<and> k = a then u else if a = b \<and> k = b then v else if a = k then 1::'a else (0::'a)) * A $ k $ j"
extremum_unique: fixes a :: "'c list" shows "suffix a [] = (a = [])"
length_take: fixes n :: "nat"   and xs :: "'b list" assumes "n \<le> length xs" shows "length (take n xs) = n"
dyadic_413_cases: fixes m :: "nat"   and k :: "nat" obtains "of_nat m / (2::'b) ^ k \<in> \<nat>"   | m' :: "nat" and k' :: "nat" where "k'__ < k" and "of_nat m / (2::'b) ^ k = of_nat (4 * m'__ + 1) / (2::'b) ^ Suc k'__"   | m' :: "nat" and k' :: "nat" where "k' < k" and "of_nat m / (2::'b) ^ k = of_nat (4 * m' + 3) / (2::'b) ^ Suc k'"
bisum_is_map_zip: fixes f :: "'b \<times> 'b \<Rightarrow> 'a"   and l1 :: "'b list" shows "sum_list (map f (zip ((0::'b) # l1) (l1 @ [0::'b]))) = bisum (\<lambda>x y. f (x, y)) l1"
RModule_addfunset: fixes R :: "'a set"   and M :: "'b set" assumes "Ring1 R"   and "Group M" shows "RModule R (Ring1.rightreg_scalar_mult R) (addfunset R M)"
lessx: fixes n :: "nat" assumes "n < x" shows "\<not> P n"
Figure_3: fixes i :: "nat"   and k :: "nat" assumes "i < k" shows "foldl1 f2 (map (h2 i) [0..<k + 1]) = Two"
widen_antisym: fixes m :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and T :: "ty"   and U :: "ty" assumes "wf_prog m P"   and "P \<turnstile> T \<le> U"   and "P \<turnstile> U \<le> T" shows "T = U"
assms_1: shows "closedin (top_of_set (S \<union> T)) S"
MGF_lemma2_simult: fixes F :: "state triple set" assumes "state_not_singleton"   and "WT_bodies"   and "F \<subseteq> (\<lambda>pn. {=}.the (body pn).{->}) ` dom body" shows "(\<lambda>pn. {=}.BODY pn.{->}) ` dom body||-F"
incl_Rel_ArrVal_vsv: fixes A :: "V"   and B :: "V" shows "vsv (incl_Rel A B\<lparr>ArrVal\<rparr>)"
closest_pair_rec_code_dist_eq: fixes xs :: "(int \<times> int) list"   and ys :: "(int \<times> int) list"   and \<delta> :: "int"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int" assumes "1 < length xs"   and "(ys, \<delta>, c\<^sub>0, c\<^sub>1) = closest_pair_rec_code xs" shows "\<delta> = dist_code c\<^sub>0 c\<^sub>1"
simps: fixes a1 :: "('e, 'f, 'g) var0"   and a2 :: "('e, 'f, 'g) var0" shows "a1 \<sim> a2 = ((\<exists>x. a1 = x \<and> a2 = x) \<or> (\<exists>a x. a1 = PMF (return_pmf (a, x)) \<and> a2 = BPS (bsingleton (a, x))) \<or> (\<exists>a x. a1 = BPS (bsingleton (a, x)) \<and> a2 = PMF (return_pmf (a, x))))"
coerce_Nothing: shows "COERCE('d\<cdot>maybe, 'c\<cdot>maybe)\<cdot>Nothing = Nothing"
sorted_list_of_mset_set: fixes A :: "'k set" shows "sorted_list_of_multiset (mset_set A) = sorted_list_of_set A"
fair_def: fixes s :: "'c"   and t :: "'c" shows "MC_syntax.fair s t = (\<lambda>xs. alw (ev (HLD {s})) xs \<longrightarrow> alw (ev (\<lambda>xs. HLD {s} xs \<and> nxt (HLD {t}) xs)) xs)"
im_faces: fixes x :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and y :: "'b set" assumes "x \<in> f \<turnstile> X"   and "order.greater_eq x y" shows "y \<in> f \<turnstile> X"
bound_compare_defs_4: fixes lt :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "'a"   and b :: "'b option" shows "\<lhd>\<^sub>u\<^sub>b lt c b \<equiv> case b of None \<Rightarrow> True | Some b' \<Rightarrow> lt c b'"
insert_Collect: fixes a :: "'e"   and P :: "'e \<Rightarrow> bool" shows "insert a (Collect P) = {u. u \<noteq> a \<longrightarrow> P u}"
fun_evaluate_2: fixes env :: "v sem_env"   and s :: "'b state"   and es :: "exp0 list" shows "evaluate_list True env s es (fun_evaluate s env es)"
Cauchy_Schwarz_ineq2: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c"   and y :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "\<bar>inner x y\<bar> \<le> norm x * norm y"
nonzero_inverse_mult_distrib: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "a \<noteq> zero"   and "b \<noteq> zero" shows "inverse (times a b) = times (inverse b) (inverse a)"
i_one: fixes G :: "('c, 'd) Group_scheme" assumes "Group G" shows "\<rho>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = \<one>\<^bsub>G\<^esub>"
linorder_injI: fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>x y. x < y \<Longrightarrow> f x \<noteq> f y" shows "inj f"
\<beta>sON: shows "list.set \<beta>s \<subseteq> ON"
aux: fixes b :: "'b option" shows "(None \<noteq> b) = (\<exists>y. b = Some y)"
runit_char_2: fixes a :: "'c" assumes "C.ide a" shows "\<r>\<^sub>C[a] \<otimes>\<^sub>C \<I>\<^sub>C = (a \<otimes>\<^sub>C \<iota>\<^sub>C) \<cdot>\<^sub>C \<a>\<^sub>C[a, \<I>\<^sub>C, \<I>\<^sub>C]"
IH: assumes "character (G\<lparr>carrier := H', carrier := H\<rparr>) \<chi>" shows "card {\<chi>'. \<chi>' \<in> characters (G\<lparr>carrier := H'\<rparr>) \<and> (\<forall>x\<in>H. \<chi>' x = \<chi> x)} * card H = order (G\<lparr>carrier := H'\<rparr>)"
sum_nonpos: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq># {#}" shows "comm_monoid_add.sum (+) {#} f A \<subseteq># {#}"
iT_div_3: fixes k :: "nat"   and n :: "nat"   and d :: "nat" assumes "0 < k" shows "[n\<dots>,d] \<oslash> k = [n div k\<dots>,d div k + (n mod k + d mod k) div k]"
borderI: fixes x :: "'b list"   and w :: "'b list" assumes "x \<le>p w"   and "x \<le>s w"   and "x \<noteq> w"   and "x \<noteq> \<epsilon>" shows "x \<le>b w"
ground_emb: fixes t :: "('a, 'b) tm"   and s :: "('a, 'b) tm" assumes "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"   and "ground t" shows "ground s"
restrict_restrict: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and A1 :: "'c set" assumes "restrict f A \<in> A \<rightarrow> B"   and "A1 \<subseteq> A" shows "restrict (restrict f A) A1 = restrict f A1"
inf_compl_bot: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<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 bot :: "'d"   and top :: "'d"   and x :: "'d" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "inf x (uminus x) = bot"
function_one_def: fixes S :: "'d set"   and R :: "('e, 'f) ring_scheme" shows "function_one S R = (\<lambda>x\<in>S. \<one>\<^bsub>R\<^esub>)"
isovarsparNotIn: fixes x :: "nat"   and p :: "real mpoly" assumes "x \<notin> vars p" shows "isolate_variable_sparse p x 0 = p"
weak_group_morphism_range: fixes f :: "'a \<Rightarrow> 'd"   and H :: "'a set"   and a :: "'a" assumes "weak_group_morphism f H G"   and "a \<in> carrier G" shows "f ` (H #> a) = {f a}"
that: fixes d' :: "'a"   and d'e :: "'a set" assumes "d' \<in> ab \<and> path d'e d' e \<and> [[b d d']]" shows "thesis"
simps: fixes a1 :: "('p, 'v, 's) configuration"   and a2 :: "('p, 'v, 's) configuration" shows "reachable a1 a2 = ((\<exists>cfg1. a1 = cfg1 \<and> a2 = cfg1) \<or> (\<exists>cfg1 cfg2 msg cfg3. a1 = cfg1 \<and> a2 = cfg3 \<and> reachable cfg1 cfg2 \<and> cfg2 \<turnstile> msg \<mapsto> cfg3))"
tbisim_def: fixes nw :: "bool"   and t :: "'t"   and ts1 :: "('x1 \<times> 'l \<Rightarrow>f nat) option"   and m1 :: "'m1"   and ts2 :: "('x2 \<times> 'l \<Rightarrow>f nat) option"   and m2 :: "'m2" shows "tbisim nw t ts1 m1 ts2 m2 = (case ts1 of None \<Rightarrow> ts2 = None | \<lfloor>(x1, ln)\<rfloor> \<Rightarrow> \<exists>x2. ts2 = \<lfloor>(x2, ln)\<rfloor> \<and> t \<turnstile> (x1, m1) \<approx> (x2, m2) \<and> (nw \<or> x1 \<approx>w x2))"
c''_1: shows "c'' \<in> {a..b} \<inter> g -` {c..d}"
not_na_le_minf: fixes n :: "nat" shows "\<not> an n \<le> - \<infinity>"
simps: fixes a1 :: "('b \<times> nat) option"   and a2 :: "'b" shows "may_lock a1 a2 = ((\<exists>t. a1 = None \<and> a2 = t) \<or> (\<exists>t n. a1 = \<lfloor>(t, n)\<rfloor> \<and> a2 = t))"
rel_mono_strong: fixes R1 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R2 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'e + 'g"   and y :: "'f + 'h"   and R1a :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R2a :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "rel_sum R1 R2 x y"   and "\<And>z1 y1. \<lbrakk>z1 \<in> Basic_BNFs.setl x; y1 \<in> Basic_BNFs.setl y; R1 z1 y1\<rbrakk> \<Longrightarrow> R1a z1 y1"   and "\<And>z2 y2. \<lbrakk>z2 \<in> Basic_BNFs.setr x; y2 \<in> Basic_BNFs.setr y; R2 z2 y2\<rbrakk> \<Longrightarrow> R2a z2 y2" shows "rel_sum R1a R2a x y"
RA2_1: fixes P :: "('b, 'c) urel" shows "P ;; II = P"
cinfinite_cprod: fixes r1 :: "('k \<times> 'k) set"   and r2 :: "('l \<times> 'l) set" assumes "cinfinite r1"   and "cinfinite r2" shows "cinfinite (r1 *c r2)"
ipresSwapAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'varSort, 'g, 'h, 'var, 'c, 'd, 'i) Iteration.model_scheme" shows "ipresSwapAll h hA MOD \<equiv> ipresSwap h MOD \<and> ipresSwapAbs hA MOD"
fold_mset_fusion: fixes g :: "char list \<Rightarrow> 'b \<Rightarrow> 'b"   and h :: "'b \<Rightarrow> 'a mpoly"   and w :: "'b"   and A :: "char list multiset" assumes "comp_fun_commute g"   and "\<And>x y. h (g x y) = h y * Var (\<phi> x)" shows "h (fold_mset g w A) = fold_mset (\<lambda>a b. b * Var (\<phi> a)) (h w) A"
elims_2: fixes x :: "'a tree" assumes "is_empty x" obtains t :: "'a tree" where "x = t" and "t = \<langle>\<rangle>"
reducible_by_flipping: fixes ss :: "'w list" assumes "ss \<in> lists S"   and "\<not> S_reduced ss" shows "\<exists>xss as t bs. flip_altsublist_chain (ss # xss @ [as @ [t, t] @ bs])"
h0: fixes i :: "'a"   and x :: "'b" assumes "i \<in> I" shows "h i (0, x) = f i x"
finprod_Un_disjoint: fixes A :: "'e set"   and B :: "'e set"   and g :: "'e \<Rightarrow> 'a" assumes "finite A"   and "finite B"   and "A \<inter> B = {}"   and "g \<in> A \<rightarrow> carrier G"   and "g \<in> B \<rightarrow> carrier G" shows "finprod G g (A \<union> B) = finprod G g A \<otimes> finprod G g B"
sub_id: fixes p :: "('c, 'd) fm" shows "sub id p = p"
wf_subcls1: assumes "ws_prog" shows "wf (subcls1\<inverse>)"
filter_mono_suffix: fixes xs :: "'b list"   and ys :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "suffix xs ys" shows "suffix (filter P xs) (filter P ys)"
some_NF: fixes r :: "('b \<times> 'b) set"   and x :: "'b" assumes "SN_on r {x}" shows "(x, some_NF r x) \<in> r\<^sup>* \<and> some_NF r x \<in> NF r"
disjoint_subset: fixes A :: "'b set set"   and a :: "'b set"   and b :: "'b set" assumes "disjoint A"   and "a \<in> A"   and "b \<subseteq> a" shows "disjoint (A - {a} \<union> {b})"
subtopology_UNIV: fixes U :: "'c topology" shows "subtopology U UNIV = U"
cases: fixes x :: "('b \<times> nat) option" obtains "x = None"   | t :: "'b" and n :: "nat" where "x = \<lfloor>(t, n)\<rfloor>"
hyp2_S_not_equal: fixes a :: "proj2"   and p :: "proj2" assumes "a \<in> hyp2"   and "p \<in> S" shows "a \<noteq> p"
connected: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" assumes "Tree G"   and "v \<in> V\<^bsub>G\<^esub>"   and "w \<in> V\<^bsub>G\<^esub>" shows "Graph.connected G v w"
set_mult_inclusion: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and P :: "'c set" assumes "Group.group G"   and "subgroup H G"   and "P \<subseteq> carrier G"   and "H <#>\<^bsub>G\<^esub> P \<subseteq> H" shows "P \<subseteq> H"
simps_4: fixes uu :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and uv :: "'d" shows "matches uu MatchAny uv = True"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "(symbol \<times> character list) set" assumes "\<Z> x xa = y" obtains k :: "nat" where "x = kb__" and "xa = 0" and "y = {}"   | k :: "nat" and u :: "nat" where "x = k" and "xa = Suc u" and "y = \<Y> (\<Z> k u) (\<P> k u) k"
chain_def: fixes A :: "'f set set"   and C :: "'f set set" shows "subset.chain A C = (C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. (\<subset>)\<^sup>=\<^sup>= x y \<or> (\<subset>)\<^sup>=\<^sup>= y x))"