hcju/isabelleps huggingface dataset:
rel_fundefs_rewrite_both': fixes fd1 :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef"   and fd2 :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef"   and y :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr"   and x :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr"   and f :: "'fun"   and pc :: "nat" assumes "rel_fundef (\<lambda>x y. x = norm_instr y) fd1 fd2"   and "norm_instr y = x" shows "rel_fundefs (Finca_get (Finca_add F1 f (rewrite_fundef_body fd1 pc x))) (Fubx_get (Fubx_add F2 f (rewrite_fundef_body fd2 pc y)))"
neighborhoods_def: fixes x :: "'a set" shows "top.neighborhoods x \<equiv> {U. is_zariski_open U \<and> x \<in> U}"
simps_2: fixes v :: "'b"   and va :: "'b heap list" shows "merge heap.Empty (Hp v va) = Hp v va"
C: fixes x :: "'d" assumes "x \<in> S" shows "norm (f x) \<le> C"
ldrop_LNil: fixes i :: "nat" shows "LNil \<up> i = LNil"
fps_lr_inverse_one_minus_fps_X_1: shows "fps_left_inverse (1 - fps_X) (1::'a) = Abs_fps (\<lambda>n. 1::'a)"
fv\<^sub>s\<^sub>s\<^sub>t_unlabel_dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_eq: fixes A :: "('g strand_label \<times> ('f, 'e) stateful_strand_step) list" shows "fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A) = fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
assms: shows "run_one_step d i (s, vs, ves, e) = (s', vs', RSReturn res)"
better_ImpL_substn: fixes y :: "name"   and M :: "trm"   and x :: "name"   and N :: "trm"   and a :: "coname"   and c :: "coname"   and P :: "trm" assumes "y \<sharp> (M, [x].N)" shows "ImpL <a>.M x.N y{y:=<c>.P} = fresh_fun (\<lambda>z'. Cut <c>.P z'.ImpL <a>.M x.N z')"
lcos_mult_one: fixes M :: "'a set" assumes "M \<subseteq> carrier H'" shows "\<one>\<^bsub>H'\<^esub> <#\<^bsub>H'\<^esub> M = M"
prime_power_not_one: fixes p :: "'b"   and k :: "nat" assumes "prime p"   and "0 < k" shows "p ^ k \<noteq> (1::'b)"
preserves_dom: fixes f :: "'a" assumes "B.arr f" shows "local.dom (UP f) = UP (B.dom f)"
c2dc_dc2c: shows "c2dc (dc2c \<chi>) = \<chi>"
downset_set_prop: shows "ord.downset_set (\<lambda>x y. y \<le> x) = \<Union> \<circ> (`) (ord.downset (\<lambda>x y. y \<le> x))"
simps_2: fixes x :: "bool"   and xs :: "bool list" shows "flip 0 (x # xs) = (\<not> x) # xs"
zero_min: fixes x :: "'c" shows "(0::'c) \<le> x"
ServiceCenter_def: fixes i :: "nat \<Rightarrow> ECall_Info list"   and a :: "nat \<Rightarrow> aType list" shows "ServiceCenter i a \<equiv> \<forall>t. a 0 = [] \<and> a (Suc t) = (if i t = [] then [] else [sc_ack])"
h: shows "\<not> eval (split_p var p) (xs @ x # \<Gamma>) \<and> eval (elimVar var (map Eq (p # A') @ L) F (Eq p)) (xs @ x # \<Gamma>) \<or> eval (split_p var p) (xs @ x # \<Gamma>) \<and> eval (qe_eq_repeat_helper var A' L F) (xs @ x # \<Gamma>)"
valid_reads_nth_update: fixes i :: "nat"   and \<O> :: "nat set"   and sb :: "'a memref list"   and p :: "'a"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and \<D> :: "bool"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "reads_consistent False \<O> m\<^sub>s\<^sub>b sb" shows "valid_reads m\<^sub>s\<^sub>b (ts\<^sub>s\<^sub>b[i := (p, is, xs, sb, \<D>, \<O>, \<R>)])"
bot_lifted_boolean_algebra_def: shows "bot \<equiv> Abs_lifted_boolean_algebra (\<lambda>f. bot)"
gcsort_aux_set: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and p :: "nat"   and t :: "nat \<times> nat list \<times> 'c list" shows "gcsort_aux index key p t \<in> gcsort_set index key p t"
compact_path_image: fixes g :: "real \<Rightarrow> 'a" assumes "path g" shows "compact (path_image g)"
seqE: fixes g :: "'a"   and f :: "'a" assumes "Right.seq g f" obtains "Right.arr f" and "Right.arr g" and "Right.dom g = Right.cod f"
cases: fixes a1 :: "'t"   and a2 :: "(status \<times> 'x) \<times> 'm"   and a3 :: "('l, 't, status \<times> 'x, 'm, 'w, 'o action) thread_action"   and a4 :: "(status \<times> 'x) \<times> 'm" assumes "a1 \<turnstile> a2 -a3\<rightarrow>i a4" obtains t :: "'t" and x :: "'x" and m :: "'m" and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action" and x' :: "'x" and m' :: "'m" where "a1 = ta__" and "a2 = ((Running, xa__), ma__)" and "a3 = convert_TA_initial (convert_obs_initial ta)" and "a4 = ((Running, x'), m')" and "ta__ \<turnstile> \<langle>xa__, ma__\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   | t :: "'t" and x :: "'x" and m :: "'m" where "a1 = tb__" and "a2 = ((PreStart, xb__), mb__)" and "a3 = \<lbrace>InitialThreadAction\<rbrace>" and "a4 = ((Running, xb__), mb__)"   | x :: "'x" and t :: "'t" and m :: "'m" where "a1 = t" and "a2 = ((Running, x), m)" and "a3 = \<lbrace>ThreadFinishAction\<rbrace>" and "a4 = ((Finished, x), m)" and "final x"
intro: fixes U\<^sub>1 :: "'a set"   and le\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and U\<^sub>2 :: "'b set"   and le\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ls\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "order_ow U\<^sub>1 le\<^sub>1 ls\<^sub>1"   and "order_ow U\<^sub>2 le\<^sub>2 ls\<^sub>2" shows "order_pair_ow U\<^sub>1 le\<^sub>1 ls\<^sub>1 U\<^sub>2 le\<^sub>2 ls\<^sub>2"
runit_naturality: fixes a :: "'a" assumes "ide a" shows "\<r>[cod a] \<cdot> (a \<otimes> \<one>) = a \<cdot> \<r>[local.dom a]"
N: fixes n :: "nat" assumes "N \<le> n" shows "prob {x \<in> space M. real n \<le> \<bar>f x\<bar>} < alpha"
fresh_star_Pair: fixes as :: "atom set"   and x :: "'d"   and y :: "'e" shows "as \<sharp>* (x, y) = (as \<sharp>* x \<and> as \<sharp>* y)"
rOfFax_def: fixes f :: "'fsym" shows "rOfFax f = Fn (Oldf f) (TE.getTvars (arOf f))"
normf1: shows "1 \<le> \<parallel>factor\<parallel>\<^sup>2"
interior_convex_hull_explicit: fixes S :: "'a set" assumes "\<not> affine_dependent S" shows "interior (convex hull S) = (if card S \<le> DIM('a) then {} else {y. \<exists>u. (\<forall>x\<in>S. 0 < u x \<and> u x < 1) \<and> sum u S = 1 \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = y})"
parallel_converter2_parametric: shows "rel_fun (rel_converter (BNF_Def.Grp UNIV f)\<inverse>\<inverse> (BNF_Def.Grp UNIV g) (BNF_Def.Grp UNIV h) (BNF_Def.Grp UNIV k)\<inverse>\<inverse>) (rel_fun (rel_converter (BNF_Def.Grp UNIV f')\<inverse>\<inverse> (BNF_Def.Grp UNIV g') (BNF_Def.Grp UNIV h') (BNF_Def.Grp UNIV k')\<inverse>\<inverse>) (rel_converter (rel_sum (BNF_Def.Grp UNIV f)\<inverse>\<inverse> (BNF_Def.Grp UNIV f')\<inverse>\<inverse>) (rel_sum (BNF_Def.Grp UNIV g) (BNF_Def.Grp UNIV g')) (rel_sum (BNF_Def.Grp UNIV h) (BNF_Def.Grp UNIV h')) (rel_sum (BNF_Def.Grp UNIV k)\<inverse>\<inverse> (BNF_Def.Grp UNIV k')\<inverse>\<inverse>))) parallel_converter2 parallel_converter2"
diff_union_swap2: fixes y :: "'a"   and M :: "'a multiset"   and x :: "'a" assumes "y \<in># M" shows "add_mset x M \<ominus> y = add_mset x (M \<ominus> y)"
sound_bounded: fixes P :: "'c \<Rightarrow> real" assumes "sound P" shows "Expectations.bounded P"
Suc_le: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and a :: "'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "mod_type.to_nat Rep a + 1 < CARD('a)" shows "less a (plus a one)"
ord_equiv_sym: fixes E :: "('e, 'f) Order_scheme" assumes "Order E"   and "ord_equiv D E" shows "ord_equiv E D"
r: shows "Sup ({norm (f x) / norm x |x. x \<noteq> (0::'a)} \<union> {norm (f x) / norm x |x. x = (0::'a)}) = Sup {norm (f x) / norm x |x. True}"
match_raw_ternary: fixes \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> 'd \<Rightarrow> bool"   and expr :: "'c"   and a :: "action"   and p :: "'d" shows "Matching_Ternary.matches (\<beta>, \<alpha>) (Match expr) a p = (case \<beta> expr p of TernaryTrue \<Rightarrow> True | TernaryFalse \<Rightarrow> False | TernaryUnknown \<Rightarrow> \<alpha> a p)"
c_def: shows "c = 28274063397213534906669125255762067746830085389618481175335056 / 337877029279505250241149903214554249587517250716358486542628059"
flows_reverse: fixes t :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "t - t0 \<in> existence_ivl0 x0" shows "flow0 (flow0 x0 (t - t0)) (t0 - t) = x0"
psitau_def: fixes b :: "nat"   and x :: "nat" shows "\<psi>\<^sup>\<tau> b x \<equiv> if x < e_length b then Some (e_nth b x) else \<tau> (e_hd b) x"
cod_char: fixes f :: "'a" shows "cod f = (if arr f then C.cod f else C.null)"
coretract_inverse: fixes y :: "'b" assumes "y \<in> carrier \<Y>" shows "\<pi>\<^sup>* (\<pi>\<^sub>* y) = y"
Rep_uint_inject: fixes x :: "uint"   and y :: "uint" shows "(Rep_uint x = Rep_uint y) = (x = y)"
that: fixes a :: "'a \<times> bool" assumes "a \<in> set as"   and "a = (v, False)" shows "thesis"
basic_trans_rules_24: fixes x :: "'c"   and y :: "'c" assumes "x \<le> y"   and "y \<le> x" shows "x = y"
to_bool_and_1: fixes x :: "'b word" shows "to_bool (x AND 1) = x !! 0"
mssnth_ln_series_stream: fixes n :: "nat" shows "mssnth ln_series_stream n = (- 1) ^ Suc n / real n"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and f4 :: "expr \<Rightarrow> 'b"   and f5 :: "varRef \<Rightarrow> expr \<Rightarrow> 'b"   and f6 :: "procVarDecl \<Rightarrow> 'b"   and f7 :: "String.literal \<Rightarrow> expr list \<Rightarrow> 'b"   and f8 :: "chanRef \<Rightarrow> expr list \<Rightarrow> bool \<Rightarrow> 'b"   and f9 :: "chanRef \<Rightarrow> recvArg list \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> 'b"   and edgeEffect :: "edgeEffect" shows "P (case edgeEffect of EEEnd \<Rightarrow> f1 | EEId \<Rightarrow> f2 | EEGoto \<Rightarrow> f3 | EEAssert x \<Rightarrow> f4 x | EEAssign x xa \<Rightarrow> f5 x xa | EEDecl x \<Rightarrow> f6 x | EERun x xa \<Rightarrow> f7 x xa | EESend x xa xb \<Rightarrow> f8 x xa xb | EERecv x xa xb xc \<Rightarrow> f9 x xa xb xc) = ((edgeEffect = EEEnd \<longrightarrow> P f1) \<and> (edgeEffect = EEId \<longrightarrow> P f2) \<and> (edgeEffect = EEGoto \<longrightarrow> P f3) \<and> (\<forall>x4. edgeEffect = EEAssert x4 \<longrightarrow> P (f4 x4)) \<and> (\<forall>x51 x52. edgeEffect = EEAssign x51 x52 \<longrightarrow> P (f5 x51 x52)) \<and> (\<forall>x6. edgeEffect = EEDecl x6 \<longrightarrow> P (f6 x6)) \<and> (\<forall>x71 x72. edgeEffect = EERun x71 x72 \<longrightarrow> P (f7 x71 x72)) \<and> (\<forall>x81 x82 x83. edgeEffect = EESend x81 x82 x83 \<longrightarrow> P (f8 x81 x82 x83)) \<and> (\<forall>x91 x92 x93 x94. edgeEffect = EERecv x91 x92 x93 x94 \<longrightarrow> P (f9 x91 x92 x93 x94)))"
inv_equality: fixes g :: "'e \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> 'e" assumes "\<And>x. g (f x) = x"   and "\<And>y. f (g y) = y" shows "inv f = g"
take_in_listsetI: fixes xs :: "'a list"   and XS :: "'a set list"   and n :: "nat" assumes "xs \<in> listset XS" shows "take n xs \<in> listset (take n XS)"
type_copy_Rep_o_Abs: fixes Rep :: "'c \<Rightarrow> 'd"   and Abs :: "'d \<Rightarrow> 'c" assumes "type_definition Rep Abs UNIV" shows "Rep \<circ> Abs = id"
adjoint_pair_antipar_2: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.adjoint_pair f g" shows "Span.ide g"
exec: shows "exec_move ci P t body h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')"
borel_measurableI: fixes f :: "'d \<Rightarrow> 'c"   and M :: "'d measure" assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M" shows "f \<in> borel_measurable M"
emep1: fixes n :: "int"   and d :: "int" assumes "even n"   and "even d"   and "0 \<le> d" shows "(n + 1) mod d = n mod d + 1"
hom_imp_img_comm_group: fixes h :: "'a \<Rightarrow> 'e"   and H :: "('e, 'f) monoid_scheme" assumes "h \<in> hom (add_monoid R) H" shows "comm_group (H\<lparr>carrier := h ` carrier R, one := h \<zero>\<rparr>)"
gen_dfs_post_imp_fe_inv: fixes E :: "('a \<times> 'a) set"   and U :: "'a set"   and S :: "'a set"   and V :: "'a set"   and u0 :: "'a"   and it :: "'a set"   and V' :: "'a set"   and t :: "'a"   and V'' :: "'a set"   and cyc :: "bool" assumes "gen_dfs_pre E U S V u0"   and "gen_dfs_fe_inv E U (insert u0 S) (insert u0 V) u0 it V' False"   and "t \<in> it"   and "it \<subseteq> E `` {u0}"   and "t \<notin> V'"   and "gen_dfs_post E U (insert u0 S) V' t V'' cyc" shows "gen_dfs_fe_inv E U (insert u0 S) (insert u0 V) u0 (it - {t}) V'' cyc"
C: shows "ipurge_tr_rev_aux I D (sources_aux I D U (x # zs)) ys = []"
ipurge_ref_aux_less_nil: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and xs :: "'b list"   and X :: "'b set" shows "ipurge_ref_aux_less I D U xs (ipurge_ref_aux I D U [] X) = ipurge_ref_aux I D U xs X"
floor_less_zero: fixes x :: "'b" shows "(\<lfloor>x\<rfloor> < 0) = (x < (0::'b))"
rank_ins: fixes bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" assumes "SkewBinomialHeapStruc.rank_invar bq" shows "SkewBinomialHeapStruc.rank_invar (SkewBinomialHeapStruc.ins t bq)"
gSubstBinpPresGWlsBinp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) Recursion.model_scheme" shows "gSubstBinpPresGWlsBinp MOD \<equiv> \<forall>ys y Y' Y delta binp' binp. wls (asSort ys) Y' \<and> gWls MOD (asSort ys) Y \<and> wlsBinp delta binp' \<and> gWlsBinp MOD delta binp \<longrightarrow> gWlsBinp MOD delta (gSubstBinp MOD ys Y' Y y binp' binp)"
order_iff: fixes b :: "'k multiset"   and a :: "'k multiset" shows "(b \<subseteq># a) = (a = a \<union># b)"
eqOnUOut_def: fixes i :: "nat"   and i' :: "nat"   and u :: "'U"   and ou :: "'Out" shows "eqOnUOut i i' u ou \<equiv> eq (atom (Obs u ou) i) (atom (Obs u ou) i')"
prems: shows "valid_unMultigraph.is_trail G1 v (x # xs) v'"
mon_s_unconc: fixes fg :: "('g, 'h, 'i, 'f, 'j) flowgraph_rec_scheme"   and a :: "'g list"   and b :: "'g list" shows "mon_s fg (a @ b) = mon_s fg a \<union> mon_s fg b"
int_pow_eq: fixes x :: "'a"   and m :: "int"   and n :: "int" assumes "x \<in> carrier G" shows "(x [^] m = x [^] n) = (int (ord x) dvd n - m)"
upper: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a" assumes "Turn_into_distance f" shows "turn_into_distance f x y \<le> f x y"
HInfinite_HFinite_not_Infinitesimal_mult: fixes x :: "'a star"   and y :: "'a star" assumes "x \<in> HInfinite"   and "y \<in> HFinite - Infinitesimal" shows "x * y \<in> HInfinite"
terminates_noFault_strip_guards: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "'e"   and F :: "'f set" assumes "\<Gamma>\<turnstile>c \<down> Normal s"   and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow>\<notin>Fault ` F" shows "\<Gamma>\<turnstile>strip_guards F c \<down> Normal s"
subset_le_dim: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> vs1.span T" shows "vs1.dim S \<le> vs1.dim T"
fsubset_antisym: fixes m :: "('c, 'd) fmap"   and n :: "('c, 'd) fmap" assumes "m \<subseteq>\<^sub>f n"   and "n \<subseteq>\<^sub>f m" shows "m = n"
prod_lessThan_split: fixes g :: "nat \<Rightarrow> real"   and n :: "nat"   and m :: "nat" shows "prod g {..<n + m} = prod g {..<n} * (\<Prod>x<m. g (x + n))"
cil_cons2: fixes w :: "'b list"   and wa :: "'b list"   and \<alpha> :: "'b \<Rightarrow> 'c set \<times> 'c set"   and wb :: "'b list"   and e :: "'b" assumes "w \<in> wa \<otimes>\<^bsub>\<alpha>\<^esub> wb"   and "fst (\<alpha> e) \<inter> mon_pl (map \<alpha> wa) = {}" shows "e # w \<in> wa \<otimes>\<^bsub>\<alpha>\<^esub> e # wb"
moebius_similarity_comp: fixes a :: "complex"   and c :: "complex"   and b :: "complex"   and d :: "complex" assumes "a \<noteq> 0"   and "c \<noteq> 0" shows "moebius_comp (moebius_similarity a b) (moebius_similarity c d) = moebius_similarity (a * c) (a * d + b)"
\<pi>_mono: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "primes_pi x \<le> primes_pi y"
one_diff_power_eq: fixes x :: "'b"   and n :: "nat" shows "(1::'b) - x ^ n = ((1::'b) - x) * sum ((^) x) {..<n}"
h_less_rt: shows "return_time p1 x0 < return_time p2 x0"
assms: shows "has_parametrized_subprob_density M N R f"
state_qbit_def: fixes n :: "nat" shows "state_qbit n \<equiv> {v |v. dim_vec v = 2 ^ n \<and> \<parallel>v\<parallel> = 1}"
I1: assumes "Arr t" shows "\<guillemotleft>\<lbrace>t\<rbrace> : \<lbrace>Dom t\<rbrace> \<rightarrow> \<lbrace>Cod t\<rbrace>\<guillemotright>"
Standard_starfun: fixes x :: "'a star"   and f :: "'a \<Rightarrow> 'b" assumes "x \<in> Standard" shows "(*f* f) x \<in> Standard"
assms: shows "g \<turnstile> Entry g-ns\<rightarrow>n"
indicator: fixes x :: "real" shows "indicator {..0} (- x) = indicator {0..} x"
simps: fixes prec :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and p :: "float poly"   and e :: "float interval" shows "compute_bound_tm prec I a (TaylorModel p e) = compute_bound_poly prec (map_poly interval_of p) I a + e"
closure_delay_mono: fixes Z :: "('c \<Rightarrow> real) set" assumes "Z \<subseteq> V" shows "Closure\<^sub>\<alpha> Z\<^sup>\<up> \<subseteq> Closure\<^sub>\<alpha> Z\<^sup>\<up>"
T_Tinv_of_set: fixes A :: "'a set" assumes "invertible_qmpt"   and "A \<in> sets M" shows "T -` (Tinv -` A \<inter> space M) \<inter> space M = A"
dense_closed_pp: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top"   and "uminus x = bot" shows "uminus (uminus (uminus x)) = bot"
pvalid_next_symbol_derivable: fixes p :: "(symbol \<times> character list) list"   and x :: "item"   and s :: "symbol" assumes "pvalid p x"   and "next_symbol x = Some s" shows "\<exists>\<delta>. is_derivation (terminals p @ [s] @ \<delta>)"
count_queries_ge: fixes xs :: "'a list"   and sorter :: "'a sorter" assumes "distinct xs"   and "\<And>R. linorder_on (set xs) R \<Longrightarrow> is_sorting R xs (eval_sorter R sorter)" shows "log 2 (fact (length xs)) \<le> real (count_wc_queries {R. linorder_on (set xs) R} sorter)"
ideal_ring_multiple1: fixes I :: "'a set"   and x :: "'a"   and r :: "'a" assumes "ideal R I"   and "x \<in> I"   and "r \<in> carrier R" shows "x \<cdot>\<^sub>r r \<in> I"
ivl_rel_def: fixes S :: "('b \<times> 'c) set" shows "\<langle>S\<rangle>ivl_rel = (S \<times>\<^sub>r S) O br set_of_ivl top"
p_def: shows "p = lhr_fps_numerator m cs ((!) fs)"
assms_3: shows "(\<Union>x\<in>set E. fv (fst x) \<union> fv (snd x)) \<inter> set (map fst B) = {}"
cod_eqI: fixes b :: "'a"   and f :: "'a" assumes "R.ide b"   and "R.seq b f" shows "R.cod f = b"
set_shuffles: fixes zs :: "'a list"   and xs :: "'a list"   and ys :: "'a list" assumes "zs \<in> shuffles xs ys" shows "set zs = set xs \<union> set ys"
derived_def: fixes G :: "('a, 'b) monoid_scheme"   and H :: "'a set" shows "derived G H = generate G (derived_set G H)"
frame_rule: fixes P :: "assn"   and c :: "'b Heap"   and Q :: "'b \<Rightarrow> assn"   and R :: "assn" assumes "<P> c <Q>" shows "<P * R> c <\<lambda>x. Q x * R>"
zless_neq: fixes i :: "int"   and j :: "int" assumes "i < j" shows "i \<noteq> j"
SPEC_rule_conjI: fixes A :: "'c nres"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "A \<le> SPEC P"   and "A \<le> SPEC Q" shows "A \<le> SPEC (\<lambda>v. P v \<and> Q v)"
cupcake_state_preserve: fixes static_cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and env :: "v sem_env"   and s :: "'a state"   and e :: "exp0"   and res :: "'a state \<times> (v, v) result" assumes "cakeml_static_env static_cenv"   and "Big_Step_Unclocked_Single.evaluate env s e res"   and "cakeml_static_env.is_cupcake_all_env static_cenv env"   and "is_cupcake_exp e" shows "s = fst res"
simps_3: fixes p :: "real \<times> real \<times> real" shows "getPoly (NeqUni p) = p"
list_of_oalist_ntm_inverse: fixes x :: "('b, 'c) oalist_ntm" shows "Abs_oalist_ntm (list_of_oalist_ntm x) = x"
monotone_insert: fixes x :: "'b" shows "monotone (\<subseteq>) (\<subseteq>) (insert x)"
comp_assoc_assoc'_1: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "B.ide f"   and "B.ide g"   and "B.ide h"   and "src\<^sub>B f = trg\<^sub>B g"   and "src\<^sub>B g = trg\<^sub>B h" shows "\<a>\<^sub>B[f, g, h] \<cdot>\<^sub>B B.\<a>' f g h = f \<star>\<^sub>B g \<star>\<^sub>B h"
minim_Under: fixes r :: "('a \<times> 'a) set"   and B :: "'a set" assumes "wo_rel r"   and "B \<subseteq> Field r"   and "B \<noteq> {}" shows "minim r B \<in> Under r B"
stutter_extend_def: fixes G :: "('a, 'b) graph_rec_scheme" shows "stutter_extend G \<equiv> \<lparr>g_V = g_V G, g_E = stutter_extend_edges (g_V G) (g_E G), g_V0 = g_V0 G, \<dots> = graph_rec.more G\<rparr>"
select_convs_1: fixes globals :: "'c"   and locals :: "'a \<Rightarrow> 'b"   and more :: "'d" shows "locals \<lparr>globals = globals, locals = locals, \<dots> = more\<rparr> = locals"
eventually_frequently_const_simps_3: fixes P :: "'c \<Rightarrow> bool"   and C :: "bool"   and F :: "'c filter" shows "(\<forall>\<^sub>F x in F. P x \<or> C) = (eventually P F \<or> C)"
Int_Union_pairwise_disjoint: fixes \<A> :: "'a set set"   and \<B> :: "'a set set" assumes "disjoint (\<A> \<union> \<B>)" shows "\<Union> \<A> \<inter> \<Union> \<B> = \<Union> (\<A> \<inter> \<B>)"
igFreshIGAbs2_def: fixes MOD :: "('d, 'e, 'varSort, 'sort, 'f, 'g, 'h, 'i, 'j) model_scheme" shows "igFreshIGAbs2 MOD \<equiv> \<forall>ys y xs x s X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igFresh MOD ys y X \<longrightarrow> igFreshAbs MOD ys y (igAbs MOD xs x X)"
ex: shows "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> P x = P1"
simps_6: fixes a :: "nat"   and b :: "nat"   and A :: "atom fm" shows "swap_fm a b (Neg A) = Neg (swap_fm a b A)"
uw\<theta>: shows "ide (src s\<^sub>0) \<and> \<guillemotleft>w.the_\<theta> : s\<^sub>0 \<star> src s\<^sub>0 \<Rightarrow> s\<^sub>0\<guillemotright>"
sorted_wrt_linorder_unique: fixes A :: "'b set"   and R :: "('b \<times> 'b) set"   and xs :: "'b list"   and ys :: "'b list" assumes "linorder_on A R"   and "mset xs = mset ys"   and "Linorder_Relations.sorted_wrt R xs"   and "Linorder_Relations.sorted_wrt R ys" shows "xs = ys"
is_equalized_by_cone: fixes a :: "'a"   and \<chi> :: "(bool, bool list) concrete_category.arr \<Rightarrow> 'a" assumes "D.cone a \<chi>" shows "D.is_equalized_by (\<chi> J.Zero)"
cltn2_abs_rep: fixes A :: "cltn2" shows "cltn2_abs (cltn2_rep A) = A"