hcju/isabelleps huggingface dataset:
erlang_distributed_variance: fixes l :: "real"   and X :: "'a \<Rightarrow> real"   and k :: "nat" assumes "0 < l"   and "distributed M lborel X (\<lambda>x. ennreal (erlang_density k l x))" shows "expectation (\<lambda>x. (X x - expectation X)\<^sup>2) = real (k + 1) / l\<^sup>2"
fun_evaluate_1: fixes env :: "v sem_env"   and s :: "'b state"   and v :: "v"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v" shows "evaluate_match True env s v pes err_v (map_prod id (map_result hd id) (fun_evaluate_match s env v pes err_v))"
cpop_min_def: fixes \<sigma> :: "'qw \<times> 'mr" shows "cpop_min \<sigma> \<equiv> let (wl, res) = \<sigma>; (v, w, wl') = qw.pop wl in RETURN (v, w, wl', res)"
eval_primfuns_def: fixes fs :: "(primfun \<times> real) list"   and x :: "real" shows "eval_primfuns fs x = (\<Prod>f\<leftarrow>fs. eval_primfun f x)"
s_eq_s_stop: fixes x :: "real" assumes "NO_MATCH ego.t_stop x"   and "ego.t_stop \<le> x" shows "ego.s x = ego.s_stop"
DomInterAllowsMT_Ports: fixes p :: "protocol \<times> int"   and po :: "protocol \<times> int"   and a :: "(int \<times> int \<times> protocol) set set"   and b :: "(int \<times> int \<times> protocol) set set"   and c :: "(int \<times> int \<times> protocol) set set"   and d :: "(int \<times> int \<times> protocol) set set" assumes "p \<noteq> po" shows "dom (Cp (AllowPortFromTo a b p)) \<inter> dom (Cp (AllowPortFromTo c d po)) = {}"
prems_2: shows "(r - 1) ^ Suc k * fact (Suc k) < card F"
outside_no_overlap: fixes S :: "'b set" shows "outside S \<inter> S = {}"
g_2: shows "AE x in M. 0 \<le> g x"
assms_1: shows "isLastAssertedLiteral l (oppositeLiteralList c) (elements M)"
slide'_def: fixes as :: "'b list"   and t :: "'b tree"   and w :: "nat \<times> nat" shows "slide' as t w = (let ts = atomics' as (max (fst w) (Suc (r t))) (snd w) (Suc (r t)); ts' = reusables t w in fold combine (ts @ ts') Leaf)"
outstanding_volatile_refs_conv: fixes sb :: "'b memref list" shows "outstanding_refs is_volatile sb = outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb \<union> outstanding_refs is_volatile_Read\<^sub>s\<^sub>b sb"
set_integrable_bound: fixes M :: "'a measure"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" assumes "set_integrable M A f"   and "set_borel_measurable M A g"   and "AE x\<in>A in M. norm (g x) \<le> norm (f x)" shows "set_integrable M A g"
DERIV_chain: fixes f :: "'b \<Rightarrow> 'b"   and Da :: "'b"   and g :: "'b \<Rightarrow> 'b"   and x :: "'b"   and Db :: "'b"   and s :: "'b set" assumes "(f has_field_derivative Da) (at (g x))"   and "(g has_field_derivative Db) (at x within s)" shows "(f \<circ> g has_field_derivative Da * Db) (at x within s)"
to_fract_def: fixes x :: "'a" shows "to_fract x = Fraction_Field.Fract x (1::'a)"
tail_arev: fixes a :: "'b" assumes "a \<in> arcs H" shows "tail H (rev_H' a) = head H a"
coboundedI2: fixes b :: "'k multiset"   and c :: "'k multiset"   and a :: "'k multiset" assumes "b \<subseteq># c" shows "a \<inter># b \<subseteq># c"
set_tag_name_get_child_nodes: fixes ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) object_ptr" shows "\<forall>w\<in>set_tag_name_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_child_nodes_locs ptr'. r h h')"
lex_refl: fixes p :: "real \<times> real" shows "Counterclockwise_2D_Arbitrary.lex p p"
simps_3: fixes \<alpha> :: "V"   and \<alpha>s :: "V list"   and m :: "nat"   and ms :: "nat list" shows "Cantor_sum (\<alpha> # \<alpha>s) (m # ms) = \<omega> \<up> \<alpha> * ord_of_nat m + Cantor_sum \<alpha>s ms"
simps_3: fixes v :: "val" shows "eval_unop UBitNot v = Intg 42"
length_Basis_list: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and abs :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Basis_list :: "'a list"   and eucl_down :: "int \<Rightarrow> 'a \<Rightarrow> 'a"   and eucl_truncate_down :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and eucl_truncate_up :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" assumes "class.executable_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less Basis_list eucl_down eucl_truncate_down eucl_truncate_up" shows "length Basis_list = card Basis"
axioms_1: fixes G :: "('c, 'd) monoid_scheme" assumes "comm_group G" shows "Group.comm_monoid G"
nn_integral_stream_space: fixes f :: "'a stream \<Rightarrow> ennreal" assumes "stream.random_variable borel f" shows "integral\<^sup>N (stream_space M) f = \<integral>\<^sup>+ x. \<integral>\<^sup>+ X. f (x ## X) \<partial>stream_space M \<partial>M"
add_rel_well_defined: fixes x :: "'a set \<times> 'b"   and y :: "'a set \<times> 'b"   and z :: "'a set"   and z' :: "'a set" assumes "x \<in> Sigma I \<FF>"   and "y \<in> Sigma I \<FF>"   and "z \<in> I"   and "z \<subseteq> fst x"   and "z \<subseteq> fst y"   and "z' \<in> I"   and "z' \<subseteq> fst x"   and "z' \<subseteq> fst y" shows "\<lfloor>z, +\<^bsub>z\<^esub> (\<rho> (fst x) z (snd x)) (\<rho> (fst y) z (snd y))\<rfloor> = \<lfloor>z', +\<^bsub>z'\<^esub> (\<rho> (fst x) z' (snd x)) (\<rho> (fst y) z' (snd y))\<rfloor>"
long_chain_betw: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and X :: "'a set" assumes "[[.. a .. b .. c ..]X]" shows "[[a b c]]"
phi_backimg_s': shows "\<phi> -` {0..1} \<subseteq> {0..1}"
the_elem_image_unique: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "A \<noteq> {}"   and "\<And>y. y \<in> A \<Longrightarrow> f y = f x" shows "the_elem (f ` A) = f x"
valid_W_invD0_1: fixes r :: "'e"   and s :: "'g process_name \<Rightarrow> ('f, 'g, 'h, 'e) Model.local_state"   and p :: "'g process_name"   and q :: "'g process_name" assumes "r \<in> W (s p)"   and "valid_W_inv s"   and "p \<noteq> q" shows "r \<notin> WL q s"
acute_out2__acute: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a"   and A' :: "'a"   and A :: "'a"   and C' :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Out Bet B A' A"   and "Tarski_neutral_dimensionless.Out Bet B C' C"   and "Tarski_neutral_dimensionless.Acute Bet Cong A B C" shows "Tarski_neutral_dimensionless.Acute Bet Cong A' B C'"
fls_inverse_regpart: fixes f :: "'a fls" assumes "fls_subdegree f = 0" shows "fls_regpart (inverse f) = inverse (fls_regpart f)"
roots_of_real_poly_def: fixes p :: "real poly" shows "roots_of_real_poly p \<equiv> let (c, pis) = yun_factorization gcd p in if c \<noteq> 0 \<and> (\<forall>(p, i)\<in>set pis. degree p \<le> 2 \<or> (\<forall>x\<in>set (coeffs p). x \<in> \<rat>)) then Some (concat (map (roots_of_real_main \<circ> fst) pis)) else None"
cases: fixes a1 :: "cmd"   and a2 :: "nat"   and a3 :: "cmd" assumes "labels a1 a2 a3" obtains c :: "cmd" where "a1 = ca__" and "a2 = 0" and "a3 = ca__"   | V :: "char list" and e :: "expr" where "a1 = V:=e" and "a2 = 1" and "a3 = Skip"   | c\<^sub>1 :: "cmd" and l :: "nat" and c :: "cmd" and c\<^sub>2 :: "cmd" where "a1 = c\<^sub>1__;; c\<^sub>2__" and "a2 = l__" and "a3 = cb__;; c\<^sub>2__" and "labels c\<^sub>1__ l__ cb__"   | c\<^sub>2 :: "cmd" and l :: "nat" and c :: "cmd" and c\<^sub>1 :: "cmd" where "a1 = c\<^sub>1'__;; c\<^sub>2'__" and "a2 = la__ + #:c\<^sub>1'__" and "a3 = cc__" and "labels c\<^sub>2'__ la__ cc__"   | c\<^sub>1 :: "cmd" and l :: "nat" and c :: "cmd" and b :: "expr" and c\<^sub>2 :: "cmd" where "a1 = if (b__) c\<^sub>1''__ else c\<^sub>2''__" and "a2 = lb__ + 1" and "a3 = cd__" and "labels c\<^sub>1''__ lb__ cd__"   | c\<^sub>2 :: "cmd" and l :: "nat" and c :: "cmd" and b :: "expr" and c\<^sub>1 :: "cmd" where "a1 = if (ba__) c\<^sub>1 else c\<^sub>2" and "a2 = lc__ + #:c\<^sub>1 + 1" and "a3 = ce__" and "labels c\<^sub>2 lc__ ce__"   | c' :: "cmd" and l :: "nat" and c :: "cmd" and b :: "expr" where "a1 = while (bb__) c'__" and "a2 = l + 2" and "a3 = c;; while (bb__) c'__" and "labels c'__ l c"   | b :: "expr" and c' :: "cmd" where "a1 = while (b) c'" and "a2 = 1" and "a3 = Skip"
f_im: fixes v :: "'c" assumes "v \<in> carrier V" shows "T v \<in> carrier W"
lg_cong: fixes l :: "'p \<Rightarrow> 'p \<Rightarrow> bool"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "QCong l"   and "l A B"   and "l C D" shows "Cong A B C D"
set_iterator_image_filter_def: fixes g :: "'i \<Rightarrow> 'h option"   and it :: "('g \<Rightarrow> bool) \<Rightarrow> ('i \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g" shows "set_iterator_image_filter g it \<equiv> \<lambda>c f. it c (\<lambda>x \<sigma>. case g x of None \<Rightarrow> \<sigma> | Some x' \<Rightarrow> f x' \<sigma>)"
lfp_fusion2: fixes g :: "'d \<rightarrow> 'e"   and f :: "'d \<rightarrow> 'd"   and h :: "'e \<rightarrow> 'e" assumes "g oo f = h oo g"   and "g\<cdot>\<bottom> = \<bottom>" shows "g\<cdot>(fix\<cdot>f) = fix\<cdot>h"
rep_eq: fixes x :: "('b, 'a) ffun" shows "fran x = pran (pfun_of x)"
rep_eq: fixes x :: "'a"   and xa :: "'b" shows "lookup (monomial xa x) = (\<lambda>k'. xa when x = k')"
intro: fixes A :: "V"   and f :: "V" assumes "nop_onto A (2\<^sub>\<nat>) f" shows "binop_onto A f"
id: shows "convex_hull (convex_hull A + convex_hull B) = convex_hull A + convex_hull B"
rpo_optim_def: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat" shows "rpo_optim ground_heads_var gt_sym extf arity_sym arity_var \<equiv> rpo_basis ground_heads_var gt_sym arity_sym arity_var extf \<and> rpo_optim_axioms extf"
is_done_orefine: fixes s :: "'a \<Rightarrow> node_state option"   and v :: "'a" assumes "oGS_invar s" shows "is_done_oimpl v s = (v \<in> oGS_\<alpha> s)"
fps_lr_inverse_mult_ring1_2: fixes x :: "'a"   and f :: "'a fps"   and y :: "'a"   and g :: "'a fps" assumes "x * f $ 0 = (1::'a)"   and "f $ 0 * x = (1::'a)"   and "y * g $ 0 = (1::'a)"   and "g $ 0 * y = (1::'a)" shows "fps_right_inverse (f * g) (y * x) = fps_right_inverse g y * fps_right_inverse f x"
contlubE: fixes f :: "'b \<Rightarrow> 'c"   and Y :: "nat \<Rightarrow> 'b" assumes "contlub f"   and "chain Y" shows "f (\<Squnion>i. Y i) = (\<Squnion>i. f (Y i))"
path_weightp_distinct_member: fixes s :: "'lbl"   and l1 :: "'vtx"   and l2 :: "'vtx" assumes "s \<in> {s. path_weightp l1 l2 s}" shows "\<exists>u. u \<in> {s. path_weightp_distinct l1 l2 s} \<and> u \<le> s"
eq_E_simple_1: fixes v :: "i"   and x :: "\<kappa>"   and y :: "\<kappa>" shows "[x \<^bold>=\<^sub>E y \<^bold>\<equiv> \<lparr>O!,x\<rparr> \<^bold>& \<lparr>O!,y\<rparr> \<^bold>& \<^bold>\<box>(\<^bold>\<forall>F. \<lparr>F,x\<rparr> \<^bold>\<equiv> \<lparr>F,y\<rparr>) in v]"
is_gcd_is_bezout_ext: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b" assumes "class.bezout_ring times one plus zero minus uminus"   and "bezout_ring.is_bezout_ext times plus uminus bezout" shows "GCD_ring.is_gcd times (\<lambda>a b. case bezout a b of (x, xa, u, v, gcd') \<Rightarrow> gcd')"
diff_left: fixes a :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and a' :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and b :: "'b" shows "(a - a') *\<^sub>V b = a *\<^sub>V b - a' *\<^sub>V b"
is_SDG_path_Append: fixes n'' :: "'node SDG_node"   and ns' :: "'node SDG_node list"   and n' :: "'node SDG_node"   and n :: "'node SDG_node"   and ns :: "'node SDG_node list" assumes "n'' is-ns'\<rightarrow>\<^sub>d* n'"   and "n is-ns\<rightarrow>\<^sub>d* n''" shows "n is-ns @ ns'\<rightarrow>\<^sub>d* n'"
set_plus_closed: fixes A :: "'g set"   and B :: "'g set" assumes "A \<subseteq> G"   and "B \<subseteq> G" shows "A + B \<subseteq> G"
chain'': fixes x :: "'b" shows "Complete_Partial_Order.chain (\<le>) ((\<lambda>f. f x) ` Y)"
Inf_eq_bot_iff: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<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 A :: "'c set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "(Inf A = bot) = (\<forall>x. less bot x \<longrightarrow> (\<exists>i\<in>A. less i x))"
ndepth_Push_Node_Inl: fixes a :: "'g"   and n :: "('f, 'g) Old_Datatype.node" shows "ndepth (Push_Node (Inl a) n) = Suc (ndepth n)"
c_iop_welldef: fixes G :: "('c, 'd) Group_scheme"   and N :: "'c set"   and a :: "'c" assumes "Group G"   and "G \<triangleright> N"   and "a \<in> carrier G" shows "c_iop G N (N \<bullet>\<^bsub>G\<^esub> a) = N \<bullet>\<^bsub>G\<^esub> \<rho>\<^bsub>G\<^esub> a"
prems_3: fixes i :: "int" assumes "k < i"   and "\<And>j. \<lbrakk>k < j; j < i\<rbrakk> \<Longrightarrow> P j" shows "P i"
xy_1: shows "x\<^sub>1 = l'\<^sup>2 - (2::'a) * x\<^sub>2"
hom_0: fixes x :: "'b mod_ring" assumes "to_int_mod_ring x = 0" shows "x = 0"
image_eqI: fixes b :: "'b"   and f :: "'d \<Rightarrow> 'b"   and x :: "'d"   and A :: "'d set" assumes "b = f x"   and "x \<in> A" shows "b \<in> f ` A"
j_in: shows "j \<in> set (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))"
upmost_radical_sqrt2: fixes e :: "expr" assumes "radicals e \<noteq> {}" shows "\<exists>r\<in>radicals e. \<forall>s\<in>radicals e. r \<notin> radicals s"
multiplicity_prime_power: 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 p :: "'b"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero p" shows "factorial_semiring.multiplicity times one p (power.power one times p n) = n"
int_set_bit_True_conv_OR: fixes i :: "int"   and n :: "nat" shows "set_bit_class.set_bit i n True = i OR (1 << n)"
subst_apply_extstrand_def: fixes S :: "('a, 'b) extstrand_step list"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "S \<cdot>\<^sub>e\<^sub>s\<^sub>t \<theta> \<equiv> map (\<lambda>x. x \<cdot>\<^sub>e\<^sub>s\<^sub>t\<^sub>p \<theta>) S"
intros: fixes x :: "'g"   and r :: "('g \<times> 'g) set" assumes "\<And>y. (y, x) \<in> r \<Longrightarrow> y \<in> Wellfounded.acc r" shows "x \<in> Wellfounded.acc r"
simps_2: fixes x :: "'c"   and xs :: "'c list"   and ys :: "'c list" shows "list_union (x # xs) ys = (let zs = list_union xs ys in if x \<in> set zs then zs else x # zs)"
zcount_wl: shows "zcount (c_work c loc1) t' = 0"
cases: fixes x :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" obtains A :: "'a \<Rightarrow> 'b" and uu_ :: "'a \<Rightarrow> 'b" and uv_ :: "'a \<Rightarrow> 'b" where "x = MkArr A uu_ uv_"   | "x = Null"
inf_dres_addsimps_2: fixes x :: "'b dres" shows "inf dSUCCEEDi x = dSUCCEEDi"
continuous_on_sin_real: fixes a :: "real"   and b :: "real" shows "continuous_on {a..b} sin"
image_eqI: fixes b :: "'n"   and f :: "'o \<Rightarrow> 'n"   and x :: "'o"   and A :: "'o set" assumes "b = f x"   and "x \<in> A" shows "b \<in> f ` A"
dioph_ineq_dfs_start: fixes n :: "nat"   and ks :: "int list"   and l :: "int" shows "fst (dioph_ineq_dfs n ks l) ! int_encode l = Some 0"
C_subset: fixes M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat"   and i :: "nat" shows "C M2 M1 \<Omega> V m i \<subseteq> TS M2 M1 \<Omega> V m i"
assms_8: fixes x :: "'b"   and d :: "(String.literal, nat) assoc_list \<times> 'a \<times> nat \<times> edgeIndex \<times> edgeIndex option \<times> 'c" assumes "x \<in> set steps" shows "g x d = g' x d"
hom_1: fixes x :: "'a" assumes "hom x = (1::'b)" shows "x = (1::'a)"
unfold: shows "lrev_body1 = (\<Lambda> r v. Fixrec.run\<cdot>(match_lnil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>(list2H\<cdot>lnil)) +++ match_lcons\<cdot>v\<cdot>(\<Lambda> x xs. Fixrec.succeed\<cdot>(list2H\<cdot>(wrapH\<cdot>r\<cdot>xs :++ x :@ lnil)))))"
eq: fixes t :: "'a \<Rightarrow>\<^sub>0 nat" shows "deg_pm (\<Sum>y\<in>keys t. monomial (lookup t y) (Some y)) = deg_pm t"
lcoset_refl: fixes G :: "'a set"   and a :: "'a" assumes "Group G" shows "a \<in> a +o G"
completely_multiplicative_fds_deriv: fixes f :: "nat \<Rightarrow> 'b" assumes "completely_multiplicative_function f" shows "fds_deriv (fds f) = - fds (\<lambda>n. f n * mangoldt n) * fds f"
subst_idem_def: fixes s :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "subst_idem s \<equiv> s \<circ>\<^sub>s s = s"
borel_measurable_min: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
Rep_process_cases: fixes y :: "('c list \<times> 'c set) set \<times> 'c list set" assumes "y \<in> process_set" obtains x :: "'c process" where "y = Rep_process x"
poly: fixes y :: "real" shows "(poly [:f, e, 0, 1:] y = 0) = (y ^ 3 + e * y + f = 0)"
contradiction': fixes Q_hit :: "'a list"   and Q_hit_pre :: "'a list"   and Q_hit_post :: "'a list" assumes "Q_hit \<in> Q"   and "y \<in> set Q_hit"   and "Q_hit = Q_hit_pre @ y # Q_hit_post" shows "False"
VDMPut: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and x :: "Var"   and F :: "Field"   and e :: "Expr" shows "G \<rhd> Put x F e : \<lambda>s t. \<exists>l C Flds. fst s x = RVal (Loc l) \<and> lookup (snd s) l = Some (C, Flds) \<and> t = (fst s, (l, C, (F, evalE e (fst s)) # Flds) # snd s)"
Join_SKIP_left: fixes F :: "'b program" shows "\<bottom> \<squnion> F = F"
urD_3: shows "poly (real_of_int_poly p) (real_of_1 (p, l, r)) = 0"
topspace_discrete_topology: fixes U :: "'c set" shows "topspace (discrete_topology U) = U"
simple_integral_const: fixes M :: "'b measure"   and c :: "ennreal" shows "\<integral>\<^sup>S x. c \<partial>M = c * emeasure M (space M)"
finite_pset: fixes r :: "'a rexp" shows "finite (pset r)"
s2: shows "s2 = (x * y + (q - s1)) mod q"
vec_of_dim_0: fixes v :: "'a vec" shows "(dim_vec v = 0) = (v = 0\<^sub>v 0)"
jump_poly_smult_1: fixes c :: "real"   and q :: "real poly"   and p :: "real poly"   and x :: "real" shows "jump_poly (smult c q) p x = sign c * jump_poly q p x"
brnL_cases: fixes cl :: "('test, 'atom, 'choice) cmd list"   and ii :: "nat" assumes "properL cl"   and "ii < brnL cl (length cl)" obtains n :: "nat" and i :: "nat" where "n < length cl" and "i < brn (cl ! n)" and "ii = brnL cl n + i"
Gromov_hyperbolic_closure: fixes delta :: "real"   and A :: "'b set" assumes "Gromov_hyperbolic_subset delta A" shows "Gromov_hyperbolic_subset delta (closure A)"
map_ide: fixes b :: "'c"   and a :: "'c" assumes "C.ide b"   and "C.ide a" shows "HomC.map (b, a) = S.mkIde (HomC.set (b, a))"
fv_App: fixes e :: "exp"   and x :: "Vars.var" shows "fv (App e x) = insert x (fv e)"
mset_union_diff_comm: fixes t :: "'c"   and S :: "'c multiset"   and T :: "'c multiset" assumes "t \<in># S" shows "T + (S - {#t#}) = T + S - {#t#}"
summable_finite: fixes N :: "nat set"   and f :: "nat \<Rightarrow> 'b" assumes "finite N"   and "\<And>n. n \<notin> N \<Longrightarrow> f n = (0::'b)" shows "summable f"
subterms_of_trm_ord_eq: fixes s :: "'a trm"   and t :: "'a trm" assumes "s \<in> subterms_of t" shows "s = t \<or> (s, t) \<in> trm_ord"
map_iterator_genord_rule_insert_P: assumes "map_iterator_genord it m (\<lambda>_ _. True)"   and "I {} \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> dom m - it; m k = Some v; it \<subseteq> dom m; I it \<sigma>; \<forall>k' v'. k' \<in> dom m - it - {k} \<and> m k' = Some v' \<longrightarrow> True; \<forall>k' v'. k' \<in> it \<and> m k' = Some v' \<longrightarrow> True\<rbrakk> \<Longrightarrow> I (insert k it) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I (dom m) \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> dom m; \<not> c \<sigma>; I it \<sigma>; \<forall>k v k' v'. k \<in> it \<and> m k = Some v \<and> k' \<notin> it \<and> m k' = Some v' \<longrightarrow> True\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
presheaf_of_rings_axioms: shows "presheaf_of_rings {} (\<lambda>U. U = {}) im0.im_sheaf im0.im_sheaf_morphisms (0::'e) im0.add_im_sheaf im0.mult_im_sheaf im0.zero_im_sheaf im0.one_im_sheaf"
wf_if_measure: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> nat"   and g :: "'a \<Rightarrow> 'a" assumes "\<And>x. P x \<Longrightarrow> f (g x) < f x" shows "wf {(y, x). P x \<and> y = g x}"
hyps_7: shows "D' = D \<union> (set hs \<times> (set gs \<union> set bs \<union> set hs) \<union> set (ps -- sps) -\<^sub>p set (ap gs bs (ps -- sps) hs data'))"
psums_nth: fixes n :: "nat"   and xs :: "'a list" assumes "n < length xs" shows "psums xs ! n = sum ((!) xs) {..n}"
arr_char: fixes f :: "'d" shows "Dop.arr f = D.arr f"
has_vderiv_on_subset: fixes f :: "real \<Rightarrow> 'c"   and f' :: "real \<Rightarrow> 'c"   and S :: "real set"   and T :: "real set" assumes "(f has_vderiv_on f') S"   and "T \<subseteq> S" shows "(f has_vderiv_on f') T"
finished_discovered: shows "dom (finished s) \<subseteq> dom (discovered s)"
ctermsl_trans: fixes q :: "('a, 'b, 'c, 'd) seqp"   and p :: "('a, 'b, 'c, 'd) seqp"   and r :: "('a, 'b, 'c, 'd) seqp" assumes "q \<in> ctermsl p"   and "r \<in> ctermsl q" shows "r \<in> ctermsl p"
numeral_le_\<omega>\<^sub>z: fixes n :: "num" shows "numeral n \<le> \<omega>\<^sub>z"
dom_comp: fixes g :: "'a"   and f :: "'a" assumes "Adj.C.seq g f" shows "Adj.C.dom (g \<cdot>\<^sub>C f) = Adj.C.dom f"
ntrs_is_trss: fixes s :: "'a list"   and c :: "'a list multiset"   and w :: "('b, 'c) label list list"   and s' :: "'a list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "((s, c), w, s', c') \<in> trcl (ntrs fg)" shows "((s, c), foldl (@) [] w, s', c') \<in> trcl (trss fg)"
Standard_divide: fixes x :: "'a star"   and y :: "'a star" assumes "x \<in> Standard"   and "y \<in> Standard" shows "x / y \<in> Standard"
fps_inverse_nth_0: fixes f :: "'a fps" shows "fps_nth (inverse f) 0 = inverse (fps_nth f 0)"
real_root_abs: fixes n :: "nat"   and x :: "real" assumes "0 < n" shows "root n \<bar>x\<bar> = \<bar>root n x\<bar>"
item_dot_finished_item: fixes \<alpha> :: "symbol list" shows "item_dot (finished_item \<alpha>) = length \<alpha>"
pairwise_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and M :: "'b set" shows "Compiler_Utils.pairwise P M = (\<forall>m\<in>M. \<forall>n\<in>M. P m n)"