hcju/isabelleps huggingface dataset:
S_def: shows "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R fact n"
s_refl: fixes CT :: "nat \<Rightarrow> classDef option"   and C :: "nat" shows "CT \<turnstile> C <: C"
det_one: fixes n :: "nat" shows "Determinant.det (1\<^sub>m n) = (1::'b)"
enum_ex_finite_3_def: fixes P :: "finite_3 \<Rightarrow> bool" shows "enum_ex P = (P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3)"
a_repr_independence': fixes y :: "int"   and x :: "int" assumes "y \<in> Idl\<^bsub>\<Z>\<^esub> {m} +>\<^bsub>\<Z>\<^esub> x"   and "x \<in> UNIV" shows "Idl\<^bsub>\<Z>\<^esub> {m} +>\<^bsub>\<Z>\<^esub> x = Idl\<^bsub>\<Z>\<^esub> {m} +>\<^bsub>\<Z>\<^esub> y"
e_append_zeros_length: fixes b :: "nat"   and z :: "nat" shows "e_length (e_append_zeros b z) = e_length b + z"
is_Array_conv: fixes T :: "ty" shows "is_Array T = (\<exists>U. T = U\<lfloor>\<rceil>)"
vec_to_iarray_inner: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec" shows "a \<bullet> b = vec_to_iarray a \<bullet>i vec_to_iarray b"
cball_times_subset: fixes a :: "'b"   and c :: "real"   and b :: "'c" shows "cball a (c / 2) \<times> cball b (c / 2) \<subseteq> cball (a, b) c"
LstSeqP_Mem: fixes H :: "fm set"   and r :: "tm"   and k :: "tm"   and y :: "tm"   and k' :: "tm"   and z :: "tm" assumes "H \<turnstile> LstSeqP r k y"   and "H \<turnstile> HPair k' z IN r"   and "H \<turnstile> k' IN k" shows "H \<turnstile> LstSeqP r k' z"
poly_power: fixes p :: "'a poly"   and n :: "nat"   and x :: "'a" shows "poly (p ^ n) x = poly p x ^ n"
numer_cf5_deriv: fixes x :: "real" shows "(numer_cf5 has_real_derivative numer_cf5_deriv x) (at x)"
up: fixes n :: "nat" shows "A n \<subseteq> A (Suc n)"
eq_Integer_numeral_simps_1: shows "eq\<cdot>0\<cdot>0 = TT"
valid_plan_valid_head: fixes h :: "'b"   and as :: "'b list"   and PROB :: "'b set" assumes "h # as \<in> valid_plans PROB" shows "h \<in> PROB"
DynCom: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "'e" assumes "\<Gamma>\<turnstile>c s \<down> Normal s" shows "\<Gamma>\<turnstile>DynCom c \<down> Normal s"
nat_pow_consistent: fixes n :: "nat"   and x :: "'a"   and H :: "'a set" shows "[n] \<cdot> x = x [^]\<^bsub>add_monoid R\<lparr>carrier := H\<rparr>\<^esub> n"
map_def: fixes f :: "'a \<times> 'a" shows "\<sigma>.map f = (if CC.arr f then \
[fst (CC.cod f), snd (CC.cod f)] \<cdot> T f else null)"
ordertype_VWF_finite_nat: fixes A :: "V set" assumes "finite A" shows "tp A = ord_of_nat (card A)"
ord_strict_higher: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and q :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "ordered_term.ord_strict_p ord_term_strict p q = (\<exists>v. lookup p v = (0::'f) \<and> lookup q v \<noteq> (0::'f) \<and> ordered_term.higher ord_term p v = ordered_term.higher ord_term q v)"
overlap_ordering: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "ord a b c"   and "ord b c d"   and "ord a b d"   and "ord a c d"   and "a \<noteq> b \<and> a \<noteq> c \<and> a \<noteq> d \<and> b \<noteq> c \<and> b \<noteq> d \<and> c \<noteq> d" shows "\<exists>f. TernaryOrdering.ordering f ord {a, b, c, d}"
simps_1: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a" shows "foldr f [] = id"
IfTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and a :: "'a expr"   and b :: "'a expr"   and c :: "'a expr"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E expr.If a b c : \<tau>" obtains \<sigma> :: "'a OCL_Types.type" and \<rho> :: "'a OCL_Types.type" where "\<tau> = \<sigma> \<squnion> \<rho>" and "\<Gamma> \<turnstile>\<^sub>E a : Boolean[1]" and "\<Gamma> \<turnstile>\<^sub>E b : \<sigma>" and "\<Gamma> \<turnstile>\<^sub>E c : \<rho>"
phase_invL_def: shows "gc.phase_invL = (\<lambda>s. (atS gc gc.idle_locs s \<longrightarrow> gc_phase s\<down> = ph_Idle) \<and> (atS gc gc.init_locs s \<longrightarrow> gc_phase s\<down> = ph_Init) \<and> (atS gc gc.mark_locs s \<longrightarrow> gc_phase s\<down> = ph_Mark) \<and> (atS gc gc.sweep_locs s \<longrightarrow> gc_phase s\<down> = ph_Sweep) \<and> (atS gc gc.no_pending_phase_locs s \<longrightarrow> filter is_mw_Phase (sys_mem_store_buffers gc s\<down>) = []))"
index_together_alt_ss: fixes ps :: "'a set"   and G :: "'a set" assumes "ps \<subseteq> G"   and "G \<in> \<G>"   and "card ps = 2" shows "\<B> index ps = 0"
assms_1: shows "Polynomial.lead_coeff p = Polynomial.lead_coeff q"
ReadMem: fixes ad :: "'a"   and al :: "addr_loc"   and v :: "'a val"   and v' :: "'a val" shows "NormalAction (ReadMem ad al v) \<approx> NormalAction (ReadMem ad al v')"
rel_return_pmf: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x1 :: "'d"   and x2 :: "'e" shows "rel_pmf R (return_pmf x1) (return_pmf x2) = R x1 x2"
inv_eq_1_iff: fixes x :: "'a" assumes "x \<in> carrier G" shows "(\<ominus> x = \<zero>) = (x = \<zero>)"
transitionI_1: fixes P :: "pi"   and Rs :: "residual" assumes "P \<Longrightarrow>\<^sub>l Rs" shows "P \<Longrightarrow>\<^sub>l\<^sup>^Rs"
max_def: fixes a :: "'t \<Rightarrow>\<^sub>0 'd"   and b :: "'t \<Rightarrow>\<^sub>0 'd" shows "local.max a b = (if a \<preceq>\<^sub>p b then b else a)"
Collect_cong: fixes P :: "'g \<Rightarrow> bool"   and Q :: "'g \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "{x. P x} = {x. Q x}"
uniformity_dist_class_def: shows "OFCLASS('a, uniformity_dist_class) \<equiv> (OFCLASS('a, dist_class) &&& OFCLASS('a, uniformity_class) &&& class.uniformity_dist dist uniformity)"
left_neutral: fixes a :: "'k" shows "(0::'k) + a = a"
lattice_is_span: fixes xs :: "int vec list" shows "lattice_of xs = span_list xs"
not_neutral_obtains_not_neutral: fixes g :: "'c \<Rightarrow> 'a" assumes "Sum_any g \<noteq> (0::'a)" obtains a :: "'c" where "g a \<noteq> (0::'a)"
simps_1: fixes bs :: "'b list"   and c :: "'b" shows "Ipoly bs (c)\<^sub>p = c"
sorted_inorder_induct_subtree: fixes ls :: "('b btree \<times> 'b) list"   and sub :: "'b btree"   and sep :: "'b"   and rs :: "('b btree \<times> 'b) list"   and t :: "'b btree" assumes "sorted_less (inorder (Node (ls @ (sub, sep) # rs) t))" shows "sorted_less (inorder sub)"
assms_2: shows "s1 \<subseteq>\<^sub>f s2"
subformulas\<^sub>\<mu>_subset: fixes \<psi> :: "'a ltln"   and \<phi> :: "'a ltln" assumes "\<psi> \<in> subfrmlsn \<phi>" shows "subformulas\<^sub>\<mu> \<psi> \<subseteq> subformulas\<^sub>\<mu> \<phi>"
is_renaming_inv_renaming_cancel_atm_list: fixes \<rho> :: "'s"   and As :: "'a list" assumes "is_renaming \<rho>" shows "As \<cdot>al \<rho> \<cdot>al inv_renaming \<rho> = As"
carrier: shows "(\<Oplus>\<^bsub>V\<^esub>u'\<in>set us. - (w \<bullet>c u') / (u' \<bullet>c u') \<cdot>\<^sub>v u') \<in> carrier_vec n"
w\<^sub>h'_def: shows "w\<^sub>h' = g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine"
distinct_mset_set_mset_ident: fixes M :: "'c multiset" assumes "distinct_mset M" shows "mset_set (set_mset M) = M"
aa: fixes u :: "'a event list"   and t :: "'a event list"   and f :: "nat \<Rightarrow> 'a event list"   and xa :: "nat" assumes "front_tickFree u \<and> tickFree t \<and> x = trace_hide t (ev ` A) @ u \<and> isInfHiddenRun f (Y xa) A \<and> (\<forall>i. f i \<notin> D (Y xa)) \<and> t \<in> range f" shows "\<exists>t u. front_tickFree u \<and> tickFree t \<and> x = trace_hide t (ev ` A) @ u \<and> ((\<forall>x. t \<in> D (Y x)) \<or> (\<exists>f. strict_mono f \<and> (\<forall>i x. f i \<in> T (Y x)) \<and> (\<forall>i. trace_hide (f i) (ev ` A) = trace_hide (f 0) (ev ` A)) \<and> t \<in> range f))"
\<phi>_simps_2: shows "src \<phi> = src p\<^sub>0"
fold_congs_10: fixes r :: "'a State_scheme"   and r' :: "'a State_scheme"   and v' :: "Literal \<Rightarrow> nat list"   and f :: "(Literal \<Rightarrow> nat list) \<Rightarrow> Literal \<Rightarrow> nat list"   and f' :: "(Literal \<Rightarrow> nat list) \<Rightarrow> Literal \<Rightarrow> nat list" assumes "r = r'"   and "getWatchList r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "getWatchList_update f r = getWatchList_update f' r'"
rel_cases: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a stream"   and b :: "'b stream" assumes "stream_all2 R a b" obtains x1a :: "'a" and x2a :: "'a stream" and y1a :: "'b" and y2a :: "'b stream" where "a = x1a ## x2a" and "b = y1a ## y2a" and "R x1a y1a" and "stream_all2 R x2a y2a"
is_arrD_2: fixes \<CC> :: "V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" shows "\<CC>\<lparr>Dom\<rparr>\<lparr>f\<rparr> = a"
Diff_iff: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" shows "((c::'b) \<in> (A::'b set) - (B::'b set)) = (c \<in> A \<and> c \<notin> B)"
Span_contains_spanset_append_right: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and ms :: "'b list"   and ns :: "'b list" assumes "Module smult M"   and "set ms \<subseteq> M"   and "set ns \<subseteq> M" shows "set ns \<subseteq> R_scalar_mult.RSpan UNIV smult (ms @ ns)"
intros_1: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "r a b" shows "r\<^sup>+\<^sup>+ a b"
vNM_utilty_implies_continuity: fixes outcomes :: "'b set"   and \<R> :: "('b pmf \<times> 'b pmf) set" assumes "finite outcomes"   and "outcomes \<noteq> {}"   and "\<exists>u. ordinal_utility (\<P> outcomes) \<R> (\<lambda>x. measure_pmf.expectation x u)" shows "continuous_vnm (\<P> outcomes) \<R>"
h_4: fixes l :: "'V Standard_Constant"   and x :: "'V'"   and y :: "'V'" shows "((l, x, y) \<in> edges G') = ((l, h x, h y) \<in> edges G')"
f_Exec_Stream_Previous_f_Exec_Stream_Init: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and c :: "'d"   and n :: "nat" shows "f_Exec_Comp_Stream_Init trans_fun xs c ! n = (f_Exec_Comp_Stream trans_fun xs c)\<^bsup>\<leftarrow>' c\<^esup> n"
coneI: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and a :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" assumes "p = a * h"   and "a \<in> P[U]" shows "p \<in> cone (h, U)"
proper_ideal: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set" assumes "Ring R"   and "ideal R I"   and "1\<^sub>r\<^bsub>R\<^esub> \<notin> I" shows "I \<noteq> carrier R"
P_mod_diff: fixes P :: "'a set"   and R' :: "('a set, 'e) Ring_scheme"   and Y :: "'a set"   and g :: "'a"   and h :: "'a" assumes "Idomain S"   and "ideal S P"   and "PolynRg R' (S /\<^sub>r P) Y"   and "g \<in> carrier R"   and "h \<in> carrier R" shows "(erH R S X R' (S /\<^sub>r P) Y (pj S P) g = erH R S X R' (S /\<^sub>r P) Y (pj S P) h) = P_mod R S X P (g \<plusminus> -\<^sub>a h)"
not_less_empty: fixes M :: "'b multiset"   and r :: "('b \<times> 'b) set" shows "(M, {#}) \<notin> mult1 r"
has_bochner_integral_real_indicator: fixes A :: "'c set"   and M :: "'c measure" assumes "A \<in> sets M"   and "emeasure M A < \<infinity>" shows "has_bochner_integral M (indicat_real A) (Sigma_Algebra.measure M A)"
hd_conv_nth: fixes xs :: "'c list" assumes "xs \<noteq> []" shows "hd xs = xs ! 0"
assms_2: shows "\<not> (\<exists>i'\<ge>n. \<parallel>cid\<parallel>\<^bsub>t i'\<^esub>)"
marker_in_set: shows "Marker \<in> set (msgs (S t (Suc j)) cid)"
same_prefix: shows "take (j - 1) (t[j - 1 := t ! j, j := t ! (j - 1)]) = take (j - 1) t"
equals0D: fixes A :: "'d set"   and a :: "'d" assumes "A = {}" shows "a \<notin> A"
le_comm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Le Bet Cong A B C D" shows "Tarski_neutral_dimensionless.Le Bet Cong B A D C"
sorted_del_list: fixes xs :: "'b list"   and x :: "'b" assumes "sorted xs" shows "sorted (del_list x xs)"
rind_hom_def: fixes A :: "('c, 'e) aGroup_scheme"   and B :: "('d, 'f) aGroup_scheme"   and f :: "'c \<Rightarrow> 'd" shows "f\<degree>\<^bsub>A,B\<^esub> = (\<lambda>X\<in>set_ar_cos A (ker\<^bsub>A,B\<^esub> f). f (SOME x. x \<in> X))"
poset_to_PosetComplex_OrderingSetMap: fixes P :: "'a set" assumes "\<And>x. x \<in> P \<Longrightarrow> simplex_like (P .\<^bold>\<le> x)" shows "OrderingSetMap (\<^bold>\<le>) (\<^bold><) dual_order.greater_eq dual_order.greater P (poset_simplex_map P)"
intros_5: fixes \<tau> :: "'a OCL_Types.type" shows "element_type (Sequence \<tau>) \<tau>"
long_divides_def: fixes p :: "'a list list"   and q :: "'a list list"   and t :: "'a list list \<times> 'a list list" shows "UP.long_divides p q t = (t \<in> carrier (poly_ring (poly_ring R)) \<times> carrier (poly_ring (poly_ring R)) \<and> p = q \<otimes>\<^bsub>poly_ring (poly_ring R)\<^esub> fst t \<oplus>\<^bsub>poly_ring (poly_ring R)\<^esub> snd t \<and> (snd t = [] \<or> degree (snd t) < degree q))"
Int_less_eq: fixes n :: "'b"   and m :: "'b" assumes "n \<le> m" shows "\<tau>\<^sub>c\<^sub>s\<^sub>t n \<le> \<tau>\<^sub>c\<^sub>s\<^sub>t m"
COPY_def: fixes x :: "'a" shows "COPY x \<equiv> x"
eq: shows "mbuf2t_take f z (mbuf2_add xs ys buf) nts = (z', buf', nts')"
natLeq_on_Well_order: fixes n :: "nat" shows "Well_order (natLeq_on n)"
assms_4: shows "x \<notin> at_least_as_good y carrier relation"
add_power_prime_poly_mod_ring: fixes x :: "'b mod_ring poly"   and y :: "'b mod_ring poly"   and n :: "nat" shows "(x + y) ^ CARD('b) ^ n = x ^ CARD('b) ^ n + y ^ CARD('b) ^ n"
xt1_12: fixes a :: "'c"   and b :: "'c" assumes "a \<noteq> b"   and "b \<le> a" shows "b < a"
rightreg_scalar_multD2: fixes x :: "'r"   and r :: "'r"   and f :: "'r \<Rightarrow> 'a" assumes "x \<notin> R" shows "(r \<currency> f) x = (0::'a)"
PK_H: shows "(P, K) = local.reduce_column_aux div_op xs H (P_init, K_init)"
r_pdec22_def: shows "r_pdec22 \<equiv> Cn 1 r_pdec2 [r_pdec2]"
refines_def: fixes x :: "'a"   and y :: "'a" shows "refines x y \<equiv> d y \<sqsubseteq> d x \<and> d y \<cdot> x \<sqsubseteq> y"
makespan_correct_1: fixes m :: "nat"   and T :: "nat \<Rightarrow> nat" assumes "LoadBalancing m" shows "\<forall>x\<in>{1..m}. T x \<le> f_Max\<^sub>0 T m"
frontier_of_complement: fixes X :: "'c topology"   and S :: "'c set" shows "X frontier_of (topspace X - S) = X frontier_of S"
fun_upd_idem_iff: fixes f :: "'f \<Rightarrow> 'g"   and x :: "'f"   and y :: "'g" shows "(f(x := y) = f) = (f x = y)"
zero_closed: fixes H :: "'c set"   and G :: "('c, 'd) ring_scheme" assumes "additive_subgroup H G" shows "\<zero>\<^bsub>G\<^esub> \<in> H"
strict_order_iff: fixes a :: "'b"   and b :: "'b" shows "(a < b) = (a = a \<sqinter> b \<and> a \<noteq> b)"
not_isAbrD: fixes s :: "('d, 'e) xstate" assumes "\<not> isAbr s" shows "(\<exists>s'. s = Normal s') \<or> s = Stuck \<or> (\<exists>f. s = Fault f)"
leq: shows "(lang r = lang s) = (\<forall>(r', s')\<in>{((r0, s0), delta a r0, delta a s0) |r0 s0 a. a \<in> set as}\<^sup>* `` {(init r, init s)}. final r' = final s')"
rel_fun_def: fixes A :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_fun A B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
all_prime_simps_2: fixes p :: "nat"   and ps :: "nat list" shows "all_prime (p # ps) = (prime p \<and> all_prime ps)"
is_ortho_set_cindependent: fixes A :: "'b set" assumes "is_ortho_set A" shows "cindependent A"
elims_1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b list"   and y :: "bool" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq x = y" obtains "xa__ = []" and "y"   | x :: "'b" and ys :: "'b list" where "xa__ = x # ys" and "y = (Ball (ran\<^sub>l ys) (less_eq x) \<and> linorder.sorted less_eq ys)"
dimension_is_inj: fixes K :: "'a set"   and n :: "nat"   and E :: "'a set"   and m :: "nat" assumes "subfield K R"   and "dimension n K E"   and "dimension m K E" shows "n = m"
erfc_analytic_on: fixes A :: "complex set" shows "erfc analytic_on A"
case_nat_case_nat: fixes x :: "'a"   and n :: "nat \<Rightarrow> 'a"   and y :: "nat"   and m :: "nat \<Rightarrow> nat"   and i :: "nat" shows "(case case i of 0 \<Rightarrow> y | Suc x \<Rightarrow> m x of 0 \<Rightarrow> x | Suc x \<Rightarrow> n x) = (case i of 0 \<Rightarrow> case y of 0 \<Rightarrow> x | Suc x \<Rightarrow> n x | Suc i' \<Rightarrow> case m i' of 0 \<Rightarrow> x | Suc x \<Rightarrow> n x)"
embed_underS: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd"   and a :: "'c" assumes "Well_order r"   and "embed r r' f"   and "a \<in> Field r" shows "bij_betw f (underS r a) (underS r' (f a))"
rk2_step_def: fixes ode_ops :: "ode_ops"   and X0 :: "'c set"   and h :: "real" shows "rk2_step ode_ops X0 h = one_step ode_ops X0 h (\<lambda>X0 hl hu CX. let H = lv_ivl [min hl hu] [max hl hu] in rk2_param_spec \<bind> (\<lambda>rps. let rkp = lv_ivl [rps] [rps]; s2 = lv_ivl [0] [1] in mk_safe ode_ops CX \<bind> (\<lambda>_. ASSUME (ncc CX) \<bind> (\<lambda>_. let env = concat ` listset [list_of_eucl ` X0, list_of_eucl ` CX, rkp, H, s2] in approx_slp_spec (rk2_fas' ode_ops DIM('c)) (2 * DIM('c)) (rk2_slp ode_ops) env \<bind> (\<lambda>env. case env of None \<Rightarrow> RETURN None | Some env \<Rightarrow> let res' = take DIM('c) ` env in ASSERT (env_len res' DIM('c)) \<bind> (\<lambda>_. let res = eucl_of_list ` res' in ASSUME (ncc res) \<bind> (\<lambda>_. let err' = drop DIM('c) ` take (DIM('c) * 2) ` env in ASSERT (env_len err' DIM('c)) \<bind> (\<lambda>_. let err = eucl_of_list ` err' in default_reduce_argument_spec \<bind> (\<lambda>ra. reduce_spec ra res \<bind> (\<lambda>res. ASSUME (ncc res) \<bind> (\<lambda>_. safe_set ode_ops res \<bind> (\<lambda>s. if s then mk_safe ode_ops res \<bind> (\<lambda>res. RETURN (Some (res, err))) else RETURN None))))))))))))"
equator_reflect_def: shows "equator_reflect \<equiv> cltn2_abs rep_equator_reflect"
nonzero_mult_div_cancel_left: 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 a :: "'a"   and b :: "'a" assumes "class.semidom_divide divide plus minus zero times one"   and "a \<noteq> zero" shows "divide (times a b) a = b"
intro: fixes G :: "('c, 'd) monoid_scheme"   and gen :: "'c" assumes "Group.group G"   and "cyclic_group_axioms G gen" shows "Finite_And_Cyclic_Groups.cyclic_group G gen"
deg_le_sect_pp_aux_1_2: fixes t :: "('x, nat) pp"   and xs :: "'x list"   and n :: "nat" assumes "t \<in> keys (deg_le_sect_pp_aux xs n)" shows "keys_pp t \<subseteq> set xs"
d: shows "d = nat (fst (bezw e ((P - 1) * (Q - 1))) mod int ((P - 1) * (Q - 1)))"
h7b: shows "(\<Sum>i = 0..MPoly_Type.degree p x - 1. isolate_variable_sparse p x (i + 1) * Var x ^ i * Const (real (i + 1))) = sum f {0..MPoly_Type.degree p x - 1}"
id_take_nth_drop: fixes i :: "nat"   and xs :: "'b list" assumes "i < length xs" shows "xs = take i xs @ xs ! i # drop (Suc i) xs"
simps_13: fixes v :: "'a local.term"   and va :: "'a local.term" shows "Diag (v \<^bold>\<cdot> va) = False"
less_le_not_le: fixes x :: "'b"   and y :: "'b" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
f8: fixes n :: "nat"   and ss :: "symbol list" shows "n = length (take n ss) \<or> length ss < n"
assert_SE_def: fixes P :: "'a \<Rightarrow> bool" shows "assert\<^sub>S\<^sub>E P = (\<lambda>\<sigma>. if P \<sigma> then Some (True, \<sigma>) else None)"
error_free_abrupt_if: fixes s :: "state"   and x :: "abrupt"   and p :: "bool" assumes "error_free s"   and "\<nexists>err. x = Error err" shows "error_free (abupd (abrupt_if p (Some x)) s)"
differentiable_at_withinI: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and s :: "'b set" assumes "f differentiable at x" shows "f differentiable at x within s"
simps_3: fixes x21 :: "'g"   and x22 :: "'g list" shows "x21 # x22 \<noteq> []"
NotInv: assumes "\<Gamma> + image_mset \<^bold>\<not> \<Delta> \<Rightarrow> {#}" shows "\<Gamma> \<Rightarrow> \<Delta>"
same_snapshot_state_implies_same_result_snapshot_state: fixes c :: "('p, 's, 'm) configuration"   and p :: "'p"   and d :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and d' :: "('p, 's, 'm) configuration" assumes "ps c p = ps d p"   and "states c p = states d p"   and "c \<turnstile> ev \<mapsto> c'"   and "d \<turnstile> ev \<mapsto> d'" shows "ps d' p = ps c' p"
chart_eqI: fixes c :: "('c, 'd) chart"   and d :: "('c, 'd) chart" assumes "domain c = domain d"   and "codomain c = codomain d"   and "\<And>x. apply_chart c x = apply_chart d x"   and "\<And>x. inv_chart c x = inv_chart d x" shows "c = d"
keys_def_alt: fixes t :: "('d, 'e) RBT.rbt" shows "RBT.keys t = map fst (RBT.entries t)"
oneTS_stepyx: fixes x :: "'a"   and y :: "'a"   and hs :: "'a list" assumes "x \<noteq> y" shows "TS_step_d ([x, y], hs) x = ((0, []), x # hs)"
adopt_node_impl: shows "adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_adopt_node Shadow_DOM.get_owner_document Shadow_DOM.get_parent Shadow_DOM.remove_child get_disconnected_nodes set_disconnected_nodes"
minimal_antichain_subset: fixes A :: "'b set" shows "minimal_antichain A \<subseteq> A"
f1: shows "3 \<le> rk {A, B, C, \<alpha>}"
fps_mult_nth_outside_subdegrees_1: fixes n :: "nat"   and f :: "'a fps"   and g :: "'a fps" assumes "n < subdegree f" shows "fps_nth (f * g) n = (0::'a)"
k_def: shows "k = max (2 * s + 1) (nat \<lceil>(2 * d * real s + 2 * K * real s) / (d / 2)\<rceil>)"
that: fixes zs :: "'a list"   and x' :: "'a"   and ys :: "'a list" assumes "xs = zs @ x' # ys"   and "P x'"   and "\<forall>z\<in>set zs. \<not> P z" shows "thesis"
agree_on_subset: fixes f :: "('d \<times> 'e) set"   and g :: "('d \<times> 'e) set"   and G :: "('f, 'd) labeled_graph" assumes "f \<subseteq> g"   and "vertices G \<subseteq> Domain f"   and "univalent g" shows "agree_on G f g"
delete_edges_list_union: fixes G :: "'c graph"   and ff :: "('c \<times> 'c) list"   and keeps :: "('c \<times> 'c) list" shows "delete_edges_list G (ff @ keeps) = delete_edges G (set ff \<union> set keeps)"
imp_self: fixes F :: "'b formula" shows "AX0 \<turnstile>\<^sub>H F \<^bold>\<rightarrow> F"
bdd_lookup_quantify_bdd_set_of_bv: fixes w :: "bool list"   and n :: "nat"   and bdd :: "bool list bdd"   and N :: "bool list bdd list \<times> bool list"   and v :: "nat" assumes "length w = n"   and "bddh (Suc n) bdd"   and "bdd_all (nfa_is_node N) bdd"   and "v \<le> n" shows "set_of_bv (bdd_lookup (quantify_bdd v bdd) w) = (\<Union>b. set_of_bv (bdd_lookup bdd (insertl v b w)))"
greatest_is_MaxR_opt: fixes r :: "('a \<times> 'a) set"   and x :: "'a"   and A :: "'a set" assumes "MaxR r"   and "x \<in> A \<inter> Field r"   and "\<forall>y\<in>A \<inter> Field r. (y, x) \<in> r" shows "MaxR.MaxR_opt r A = Some x"
vectorspace_def: fixes K :: "('e, 'f) ring_scheme"   and V :: "('e, 'g, 'h) module_scheme" shows "vectorspace K V \<equiv> module K V \<and> field K"
alt_group_is_subgroup: fixes n :: "nat" shows "subgroup (carrier (alt_group n)) (sym_group n)"