hcju/isabelleps huggingface dataset:
h_def: shows "h \<equiv> \<lambda>z. if z \<in> \<int> then 0 else complex_of_real pi * cot (complex_of_real pi * z) + Digamma z - Digamma (1 - z)"
coprime_r_coprime_prop: fixes ps :: "rat poly list"   and fs :: "rat poly list"   and data :: "(rat \<times> (nat \<times> nat) list) list"   and fss :: "real poly list"   and f :: "real poly" assumes "factorize_polys ps = (fs, data)"   and "fss = map real_of_rat_poly fs"   and "f \<in> set fss" shows "comm_monoid_mult_class.coprime f (coprime_r fss)"
shift_bounds_Suc_ivl: fixes g :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" shows "sum g {Suc m..<Suc n} = (\<Sum>i = m..<n. g (Suc i))"
isCont_Arcsin: fixes z :: "complex" assumes "Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1" shows "isCont Arcsin z"
ideal_explicit2: fixes S :: "'a set" assumes "finite S" shows "ideal_generated S = {y. \<exists>f. (\<Sum>i\<in>S. f i * i) = y}"
inorder_update: fixes t :: "(('c \<times> 'd) \<times> color) tree"   and x :: "'c"   and y :: "'d" assumes "sorted1 (Tree2.inorder t)" shows "Tree2.inorder (update x y t) = upd_list x y (Tree2.inorder t)"
gen_length_fusion_def: fixes g :: "('c, 'd) generator"   and n :: "nat"   and s :: "'d" shows "gen_length_fusion g n s = n + length (list.unfoldr g s)"
unsatisfiable_mono: fixes CC :: "'b literal multiset set"   and DD :: "'b literal multiset set" assumes "CC \<subseteq> DD"   and "\<not> satisfiable CC" shows "\<not> satisfiable DD"
argmin_forall: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "xs \<noteq> []"   and "\<And>x. x \<in> set xs \<Longrightarrow> P x" shows "P (argmin f xs)"
integrable_discrete_difference: fixes X :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "countable X"   and "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"   and "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"   and "\<And>x. \<lbrakk>x \<in> space M; x \<notin> X\<rbrakk> \<Longrightarrow> f x = g x" shows "integrable M f = integrable M g"
iIN_div_mod_eq_0: fixes k :: "nat"   and n :: "nat"   and d :: "nat" assumes "0 < k"   and "n mod k = 0" shows "[n\<dots>,d] \<oslash> k = [n div k\<dots>,d div k]"
perp_not_col: 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 P :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B P A" shows "\<not> Tarski_neutral_dimensionless.Col Bet A B P"
X: fixes i :: "'l"   and j :: "'k" assumes "(i, j) \<in> A" shows "G i j \<in> sets (sigma UNIV (range F))"
scaleRat_right_distrib: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleRat :: "rat \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "rat"   and x :: "'b"   and y :: "'b" assumes "class.rational_vector plus zero minus uminus scaleRat" shows "scaleRat a (plus x y) = plus (scaleRat a x) (scaleRat a y)"
LIM_fun_less_zero: fixes f :: "real \<Rightarrow> real"   and l :: "real"   and c :: "real" assumes "f \<midarrow>c\<rightarrow> l"   and "l < 0" shows "\<exists>r>0. \<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x < 0"
ts_ord_add_isol: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "n x \<sqsubseteq> n y" shows "n z \<oplus> n x \<sqsubseteq> n z \<oplus> n y"
LLL_f_bnd: fixes i :: "nat"   and j :: "nat" assumes "i < m"   and "j < n" shows "\<bar>fs ! i $ j\<bar> \<le> int (f_bnd outside)"
zero: fixes f :: "'c \<Rightarrow> 'd" assumes "Modules.additive f" shows "f (0::'c) = (0::'d)"
Pj_mem: fixes G :: "('e, 'f) Group_scheme"   and N :: "'e set"   and x :: "'e" assumes "Group G"   and "G \<triangleright> N"   and "x \<in> carrier G" shows "Pj G N x = N \<bullet>\<^bsub>G\<^esub> x"
supp_sum_def: fixes f :: "'d \<Rightarrow> 'c"   and S :: "'d set" shows "supp_sum f S = sum f (support_on S f)"
prems_3: shows "(p, n') \<noteq> n\<^sub>2'"
keys_tabulate: fixes ks :: "'a list"   and f :: "'a \<Rightarrow> 'b" shows "Mapping.keys (Mapping.tabulate ks f) = set ks"
"4_2": assumes "comp kx ky = Lt" shows "prod_ord_pair P xs ((ky, vy) # ys) = (lex_ord_pair (\<lambda>k x y. if P k x y then Some Eq else None) xs ((ky, vy) # ys) = Some Eq)"
diff_single_trivial: fixes x :: "'b"   and M :: "'b multiset" assumes "x \<notin># M" shows "remove1_mset x M = M"
mult1E: fixes N :: "'b multiset"   and M :: "'b multiset"   and r :: "('b \<times> 'b) set" assumes "(N, M) \<in> mult1 r" obtains a :: "'b" and M0 :: "'b multiset" and K :: "'b multiset" where "M = add_mset a M0" and "N = M0 + K" and "\<And>b. b \<in># K \<Longrightarrow> (b, a) \<in> r"
segment1_empty: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "segment f x (f x) = {}"
as_2: shows "simple_linux_router rt fw mlf ifl pi = None"
wrp: shows "wstate_of_dstate St0 \<leadsto>\<^sub>w\<^sup>* wstate_of_dstate (llast Sts)"
mkMinj_correct: fixes l :: "'a list"   and i :: "nat"   and M :: "mon" shows "Imon l (mkMinj i M) = Imon l (Minj i M)"
segment1_empty: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "segment f x (f x) = {}"
fl2: fixes p :: "('b, 'c) Combinators list" shows "firstList (separate p) = firstList p"
llexord_refl: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c llist" shows "llexord r xs xs"
pm_of_poly_zero: fixes x :: "'a" shows "pm_of_poly x 0 = 0"
len_lexext_compat_snoc: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list"   and x :: "'b" assumes "len_lexext gt ys xs" shows "len_lexext gt (ys @ [x]) (xs @ [x])"
tail_arev: fixes a :: "'b" assumes "a \<in> arcs G" shows "tail G (arev a) = head G a"
rotate_3: assumes "round_robin_reschedule t0 (queue @ [t]) n0 s = None" shows "set (takeWhile (\<lambda>t'. t' \<noteq> t0) (queue @ [t])) \<inter> active_threads s = {}"
rep_nat_nat_def: shows "rep_nat = (\<lambda>x. x)"
Abs: fixes T :: "POPLmarkRecord.type"   and t :: "trm" shows "(\<lambda>:T. t) \<in> value"
sufficient_iterated_singular_subdivision_exists: fixes \<C> :: "'c set set"   and X :: "'c topology"   and p :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "\<And>U. U \<in> \<C> \<Longrightarrow> openin X U"   and "topspace X \<subseteq> \<Union> \<C>"   and "singular_chain p X c" obtains n :: "nat" where "\<And>m f. \<lbrakk>n \<le> m; f \<in> Poly_Mapping.keys ((singular_subdivision p ^^ m) c)\<rbrakk> \<Longrightarrow> \<exists>V\<in>\<C>. f ` standard_simplex p \<subseteq> V"
zero_neq_one_ow_def: fixes U :: "'a set"   and one :: "'a"   and zero :: "'a" shows "zero_neq_one_ow U one zero \<equiv> zero_ow U zero \<and> one_ow U one \<and> zero_neq_one_ow_axioms one zero"
apex_def: shows "f.dom.apex \<equiv> C.dom f.dom.leg0"
norm_lemma_xy: fixes b :: "real"   and y :: "'b"   and a :: "real"   and x :: "'b" assumes "\<bar>b\<bar> + 1 \<le> norm y - a"   and "norm x \<le> a" shows "b \<le> norm (x + y)"
linerr_le: shows "\<bar>inverse (aform_val e X) - alpha * aform_val e X - zeta\<bar> \<le> delta"
single_valuedI: fixes r :: "('g \<times> 'h) set" assumes "\<And>x y z. \<lbrakk>(x, y) \<in> r; (x, z) \<in> r\<rbrakk> \<Longrightarrow> y = z" shows "single_valued r"
guessed_6: shows "sfilter P zs' = ys @- zs"
steps_append': fixes as :: "'a list"   and bs :: "'a list"   and xs :: "'a list" assumes "steps as"   and "steps bs"   and "last as = hd bs"   and "as @ tl bs = xs" shows "steps xs"
denote_dim: fixes dims :: "nat list"   and S :: "com"   and \<rho> :: "complex mat" assumes "state_sig.well_com dims S"   and "\<rho> \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"   and "partial_density_operator \<rho>" shows "state_sig.denote dims S \<rho> \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"
ldistinct_lhdD: fixes xs :: "'c llist" assumes "ldistinct xs"   and "\<not> lnull xs" shows "lhd xs \<notin> lset (ltl xs)"
incl: fixes i :: "'a"   and I :: "'a set"   and P :: "'a set set" assumes "i \<in> I" shows "i \<in> gen P I"
schroeder_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x * y \<sqinter> z = bot) = (x\<^sup>T * z \<sqinter> y = bot)"
isolated_singularity_at_inverse: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex" assumes "isolated_singularity_at f z"   and "not_essential f z" shows "isolated_singularity_at (\<lambda>w. inverse (f w)) z"
assms_1: shows "f = {(tr, ref). tr = [] \<and> ref \<inter> ev ` A = {}} \<union> {(tr, ref). tr \<noteq> [] \<and> hd tr \<in> ev ` A \<and> (\<exists>a. ev a = hd tr \<and> (tl tr, ref) \<in> F (P a))}"
assms_2: shows "constructive_security_obsf real2 ideal2 sim2 \<I>_real2 \<I>_inner2 \<I>_common2 (absorb \<A> (obsf_converter (parallel_wiring \<odot> (converter_of_resource real1 |\<^sub>\<propto> 1\<^sub>C)))) adv2"
sort_edges_correct: fixes l :: "('a \<times> 'b \<times> 'a) list" shows "sorted_wrt edges_less_eq (quicksort_by_rel edges_less_eq [] l)"
prems_1: shows "(lc\<^sub>0, lc\<^sub>0') \<in> \<R>\<^sub>A"
ex_eval_fm_iff_exists_tm: fixes e :: "(name, hf) finfun"   and k :: "name"   and A :: "fm" shows "eval_fm e (SyntaxN.Ex k A) = (\<exists>t. eval_fm e (A(k::=t)) \<and> ground t)"
gt_left_comm0: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Gt C D" shows "B A Gt C D"
intro: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.comm_monoid G"   and "Group.group G" shows "comm_group G"
hinter_hinsert_right: fixes B :: "hf"   and x :: "hf"   and A :: "hf" shows "B \<sqinter> (A \<triangleleft> x) = (if x \<^bold>\<in> B then B \<sqinter> A \<triangleleft> x else B \<sqinter> A)"
assms: shows "check_single \<C> e (Type tn) = TopType tm"
bad: fixes s :: "'s"   and x :: "'a" assumes "I s"   and "\<not> bad s"   and "x \<in> outs_\<I> \<I>" shows "spmf (map_spmf (bad \<circ> snd) (callee s x)) True \<le> k"
is_projection_on_image: fixes \<pi> :: "'a \<Rightarrow> 'a"   and M :: "'a set" assumes "is_projection_on \<pi> M" shows "range \<pi> = M"
Dag_subdag: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and t :: "dag"   and s :: "dag" assumes "Dag p l r t"   and "subdag t s" shows "\<exists>q. Dag q l r s"
l_insert_before_wf2_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and insert_before :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> (_) node_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool" shows "l_insert_before_wf2 type_wf known_ptr known_ptrs insert_before heap_is_wellformed \<equiv> l_known_ptrs known_ptr known_ptrs \<and> l_insert_before_wf2_axioms type_wf known_ptrs insert_before heap_is_wellformed"
wf\<^sub>s\<^sub>s\<^sub>tI: fixes S :: "('d, 'c) stateful_strand_step list"   and V :: "'c set" assumes "wfrestrictedvars\<^sub>s\<^sub>s\<^sub>t S \<subseteq> V" shows "wf'\<^sub>s\<^sub>s\<^sub>t V S"
zero_less_divide_1_iff: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sgn :: "'e \<Rightarrow> 'e"   and abs :: "'e \<Rightarrow> 'e"   and a :: "'e" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less zero (divide one a) = less zero a"
simps: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a"   and a2 :: "'a" shows "r\<^sup>+\<^sup>+ a1 a2 = ((\<exists>a b. a1 = a \<and> a2 = b \<and> r a b) \<or> (\<exists>a b c. a1 = a \<and> a2 = c \<and> r\<^sup>+\<^sup>+ a b \<and> r b c))"
indets_power_subset: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and n :: "nat" shows "indets (p ^ n) \<subseteq> indets p"
elims_1: fixes x :: "'c \<times> 'd"   and xa :: "'c \<Rightarrow> 'd option"   and y :: "bool" assumes "(x \<in>\<^sub>m xa) = y" obtains k :: "'c" and v :: "'d" and m :: "'c \<Rightarrow> 'd option" where "x = (k, v)" and "xa = m" and "y = (m k = Some v)"
delayed_cond3': assumes "other.s \<delta> \<le> u_max"   and "u_max < other.s_stop" shows "delayed_safe_distance.collision {0..} = (a\<^sub>e < a\<^sub>o \<and> other.s' \<delta> < v\<^sub>e \<and> other.s \<delta> - ego.q \<delta> \<le> delayed_safe_distance.snd_safe_distance \<and> v\<^sub>e - a\<^sub>e / a\<^sub>o * other.s' \<delta> < 0)"
step: fixes conv :: "('a, 'b, 'c, 'd) converter"   and a :: "'a" assumes "X conv"   and "a \<in> outs_\<I> \<I>" shows "pfinite_gpv \<I>' (run_converter conv a) \<and> (\<forall>(b, conv')\<in>results_gpv \<I>' (run_converter conv a). X conv' \<or> pfinite_converter \<I> \<I>' conv')"
u: assumes "n < length u" shows "(u \<frown> v \<frown> w) n = ((u @ v) \<frown> w) n"
f2'_le_f: fixes x :: "nat" assumes "x\<^sub>0 \<le> x" shows "gc2 * f2' x \<le> f x"
distinct_67: fixes x31 :: "Syntax.var"   and x32 :: "Syntax.term"   and x41 :: "Syntax.var"   and x42 :: "Syntax.term" shows "Lam x31 x32 \<noteq> Rec x41 x42"
override_on_belowI: fixes S :: "'c set"   and y :: "'c \<Rightarrow> 'd"   and z :: "'c \<Rightarrow> 'd"   and x :: "'c \<Rightarrow> 'd" assumes "\<And>a. a \<in> S \<Longrightarrow> y a \<sqsubseteq> z a"   and "\<And>a. a \<notin> S \<Longrightarrow> x a \<sqsubseteq> z a" shows "x ++\<^bsub>S\<^esub> y \<sqsubseteq> z"
non_terminating_F: fixes P :: "'b process" shows "non_terminating P = CHAOS UNIV \<sqsubseteq>\<^sub>F P"
ext_compat_cons_axioms_def: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" shows "ext_compat_cons_axioms ext \<equiv> \<forall>gt ys xs x. ext gt ys xs \<longrightarrow> ext gt (x # ys) (x # xs)"
lazy_sequence_eq_iff: fixes xq :: "'a lazy_sequence"   and yq :: "'a lazy_sequence" shows "(xq = yq) = (list_of_lazy_sequence xq = list_of_lazy_sequence yq)"
base_case_satisfy_equation_R: fixes p :: "real poly"   and q :: "real poly" assumes "p \<noteq> 0" shows "satisfy_equation_R p [q] [([], []), ([0], []), ([], [0])] [[1], [0], [- 1]]"
finite_vrs_1: fixes x :: "binding" shows "finite (tyvrs_of x)"
base: fixes x :: "variable"   and V :: "variable set" assumes "x \<notin> V"   and "FVG \<alpha> \<subseteq> V" shows "{x} \<in> Cignorabimus \<alpha> V"
many_strongly_connected_iff_10: fixes x :: "'a" assumes "is_p_fun x" shows "many_strongly_connected x = (x ; x\<^sup>T ; x\<^sup>\<star> \<le> x\<^sup>+)"
subo': shows "Subobjs\<^sub>R P C' (D # D' # Cs')"
f: fixes i :: "nat" assumes "i < length ss2" shows "ss1 ! i = ss2 ! f i"
expand_fls_eq: fixes f :: "'a fls"   and g :: "'a fls" shows "(f = g) = (\<forall>n. fls_nth f n = fls_nth g n)"
cd_not_ret: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "IFC prob"   and "IFC_def.is_cdi prob i \<pi> k" shows "\<pi> k \<noteq> IFC_def.return prob"
fine_Inter: fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b set"   and s :: "'c set"   and p :: "('a \<times> 'b set) set" shows "((\<lambda>x. \<Inter> {f d x |d. d \<in> s}) fine p) = (\<forall>d\<in>s. f d fine p)"
no_zero_divisors: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.semiring_no_zero_divisors plus zero times"   and "a \<noteq> zero"   and "b \<noteq> zero" shows "times a b \<noteq> zero"
gamma1_smooth: shows "gamma1 piecewise_C1_differentiable_on {0..1}"
Diff_insert0: fixes x :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "x \<notin> A" shows "A - insert x B = A - B"
leqprop: shows "\<exists>y'>k. \<forall>(d, e, f)\<in>set c. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f \<le> 0"
induced_automorphism_fg_symmetric: shows "\
 = OpposedThinChamberComplexFoldings.induced_automorphism X g f"
cs_cases: fixes \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat" obtains "cs\<^bsup>\<pi>\<^esup> i = [\<pi> i]" and "\<forall>k. \<not> i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   | k :: "nat" where "cs\<^bsup>\<pi>\<^esup> i = cs\<^bsup>\<pi>\<^esup> k @ [\<pi> i]" and "i icd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
fixp_preserves_mono: fixes F :: "('d \<Rightarrow> 'a) \<Rightarrow> 'd \<Rightarrow> 'a"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x. monotone (fun_ord (\<le>)) (\<le>) (\<lambda>f. F f x)"   and "\<And>f. monotone ordb (\<le>) f \<Longrightarrow> monotone ordb (\<le>) (F f)" shows "monotone ordb (\<le>) (ccpo.fixp (fun_lub Sup) (fun_ord (\<le>)) F)"
all_zero: shows "\<forall>j<ia. \<not> vec_to_iarray A !! (rev [0..<IArray.length (vec_to_iarray A)] ! j) \<noteq> (0::'b)"
Lagrange: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec"   and z :: "(real, 3) vec" shows "x \<times> y \<times> z = (x \<bullet> z) *\<^sub>R y - (x \<bullet> y) *\<^sub>R z"
\<PP>_covers_\<P>: fixes k :: "nat"   and u :: "nat" shows "\<P> k u \<subseteq> \<PP>"
lsl_lsu_prop: shows "\<nu> \<circ> \<nu>\<^sup>\<natural> \<le> \<nu>\<^sup>\<natural>"
Fun_pr1: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "Fun (pr1 a b) = (\<lambda>x\<^bold>\<in>local.set (local.prod a b). UP (hfst (DOWN x)))"
deriv_n_ladder_shift_j: fixes index :: "nat"   and L :: "(nat \<times> nat \<times> nat) list"   and d :: "nat" assumes "index < length L" shows "deriv_n (ladder_shift_j d L ! index) = deriv_n (L ! index)"
orbit_apply: fixes f :: "'a perm"   and a :: "'a" shows "orbit f (f \<langle>$\<rangle> a) = orbit f a"
nonempty_simple_path_endless: fixes c :: "real \<Rightarrow> 'a" assumes "simple_path c" shows "path_image c - {pathstart c, pathfinish c} \<noteq> {}"
simps_2: fixes r :: "('g \<times> 'g) set"   and n :: "nat" shows "lexn r (Suc n) = map_prod (\<lambda>(x, xs). x # xs) (\<lambda>(x, xs). x # xs) ` (r <*lex*> lexn r n) \<inter> {(xs, ys). length xs = Suc n \<and> length ys = Suc n}"
ind: fixes k :: "nat" assumes "k < length (concat xs)" shows "P (concat xs ! k) (concat ys ! k)"
BPd_simps_4: fixes d :: "'b" shows "(d \<notin> {Xd d1h1, Xd d2h1, Xd d3h1}) = (BPd d = {})"
bound_of_const: fixes c :: "real" shows "bound_of (\<lambda>x. c) = c"
compact_eq_bounded_closed: fixes S :: "'c set" shows "compact S = (bounded S \<and> closed S)"
start_def: fixes A :: "'b \<times> 'c \<times> 'd" shows "start A = fst A"
the_coext_prop_2: fixes x' :: "'d"   and f :: "'c" assumes "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F x' (C.cod f') f" shows "f = \<epsilon>o (C.cod f') \<cdot>\<^sub>C F (x''\<epsilon>.the_coext x' f)"
uconnectedempty: shows "uconnected {} = {(a, a) |a. True}"
measurable_discrete_difference: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and X :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "countable X"   and "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"   and "\<And>x. x \<in> X \<Longrightarrow> g x \<in> space N"   and "\<And>x. \<lbrakk>x \<in> space M; x \<notin> X\<rbrakk> \<Longrightarrow> f x = g x" shows "g \<in> M \<rightarrow>\<^sub>M N"
fmap_eq_dom_iff: fixes A :: "('a, 'b) fmap"   and A' :: "('a, 'b) fmap" shows "(A = A') = (dom_m A = dom_m A' \<and> (\<forall>i\<in>#dom_m A. the (fmlookup A i) = the (fmlookup A' i)))"
reflexion_preserves_finiteness: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and P1 :: "'b eclause"   and C :: "'b eclause"   and \<sigma> :: "('b \<times> 'b trm) list"   and k :: "inferences"   and C' :: "'b literal set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms"   and "finite (cl_ecl P1)"   and "basic_superposition.reflexion trm_ord sel filter_trms P1 C \<sigma> k C'" shows "finite (cl_ecl C) \<and> finite C'"
continuous_on_imp_integrable_on_Pair1: fixes a :: "'a"   and c :: "'b"   and b :: "'a"   and d :: "'b"   and f :: "'a \<times> 'b \<Rightarrow> 'c"   and x :: "'a" assumes "continuous_on (cbox (a, c) (b, d)) f"   and "x \<in> cbox a b" shows "(\<lambda>y. f (x, y)) integrable_on cbox c d"
simps_9: fixes a :: "('f, 'g, 'h) exp"   and i :: "('f, 'g, 'h) exp"   and A :: "'f set option" shows "\<D> (a\<lfloor>i\<rceil>) A = (\<D> a A \<and> \<D> i (A \<squnion> \<A> a))"
curry_def: fixes F :: "'a1 \<times> 'a2 \<Rightarrow> 'b"   and G :: "'a1 \<times> 'a2 \<Rightarrow> 'b"   and \<tau> :: "'a1 \<times> 'a2 \<Rightarrow> 'b"   and f1 :: "'a1" shows "local.curry F G \<tau> f1 = (if A1.arr f1 then A2_B.MkArr (\<lambda>f2. F (A1.dom f1, f2)) (\<lambda>f2. G (A1.cod f1, f2)) (\<lambda>f2. \<tau> (f1, f2)) else A2_B.null)"
assms_1: fixes y :: "real" assumes "root_cond plr y" shows "x = y"
paths_disjoint: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and xs :: "'c list"   and ys :: "'c list"   and v :: "'c" assumes "DisjointPaths G v0 v1 paths"   and "xs \<in> paths"   and "ys \<in> paths"   and "xs \<noteq> ys"   and "v \<in> set xs"   and "v \<in> set ys" shows "v = v0 \<or> v = v1"
callee_invariant': fixes y :: "'b"   and s' :: "'s"   and s :: "'s"   and x :: "'a" assumes "(y, s') \<in> set_spmf (callee s x)"   and "I s"   and "x \<in> outs_\<I> \<I>" shows "I s' \<and> y \<in> responses_\<I> \<I> x"
composition_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> A"   and "b \<in> A" shows "add a b \<in> A"
reducible\<^sub>dI: fixes p :: "'c poly" assumes "0 < degree p \<Longrightarrow> \<exists>q r. degree q < degree p \<and> degree r < degree p \<and> p = q * r" shows "\<not> irreducible\<^sub>d p"
assms: shows "x \<in> (\<Sum>X\<leftarrow>A # As. X)"
assms_2: fixes \<F>' :: "'a set set" assumes "finite \<F>'"   and "\<F>' \<subseteq> \<F>" shows "\<Inter> \<F>' \<noteq> {}"
weight_random_oracle: fixes \<sigma> :: "'c \<Rightarrow> 'b option"   and x :: "'c" assumes "finite A"   and "A - ran \<sigma> \<noteq> {}" shows "weight_spmf (random_permutation \<sigma> x) = 1"
regions_2: shows "[xs !! Suc j]\<^sub>\<R> \<in> \<R>"
trans: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ordering less_eq less"   and "less_eq a b"   and "less_eq b c" shows "less_eq a c"
Body: fixes G :: "prog"   and s0 :: "st"   and D :: "qtname"   and s1 :: "state"   and c :: "stmt"   and s2 :: "state"   and s3 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>Init D\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>c\<rightarrow> s2"   and "s3 = (if \<exists>l. fst s2 = Some (Jump (Break l)) \<or> fst s2 = Some (Jump (Cont l)) then abupd (\<lambda>x. Some (Error CrossMethodJump)) s2 else s2)" shows "G\<turnstile>Norm s0 \<midarrow>Body D c-\<succ>the (locals (snd s2) Result)\<rightarrow> abupd (absorb Ret) s3"
measurable_differentiable_image_eq: fixes S :: "(real, 'a) vec set"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and f' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec" assumes "S \<in> sets lebesgue"   and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"   and "inj_on f S" shows "(f ` S \<in> lmeasurable) = ((\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S)"
DDH0_eq_ddh_0: fixes \<A> :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool spmf" shows "DDH0 \<A> = ddh_0 \<A>"