hcju/isabelleps huggingface dataset:
l_get_root_node_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors_si :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_si_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node_si :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_si_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_root_node_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_host get_host_locs get_ancestors_si get_ancestors_si_locs get_root_node_si get_root_node_si_locs \<equiv> l_get_parent type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs \<and> l_get_host get_host get_host_locs \<and> l_get_root_node_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_parent get_parent_locs get_host get_host_locs get_ancestors_si get_ancestors_si_locs get_root_node_si get_root_node_si_locs"
assms: shows "t\<^sub>1 \<le> t\<^sub>2"
guaranteeStrip_def: fixes f :: "'f"   and g :: "'d set"   and c :: "('d, 'e, 'f) com" shows "guaranteeStrip f g c = Guard f g c"
A: fixes xs' :: "'a list"   and zs' :: "'a list" shows "xs' \<in> traces P \<and> ys \<in> traces P \<and> ipurge_tr_rev_aux I D (D ` (X \<union> set zs')) xs' = ipurge_tr_rev_aux I D (D ` (X \<union> set zs')) ys \<longrightarrow> (xs' @ zs', X) \<in> failures P \<longrightarrow> (ys @ zs', X) \<in> failures P"
S_def: shows "rev.S \<equiv> segment G.face_cycle_succ a' a' \<union> segment G.face_cycle_succ rev.a' rev.a'"
hd0_imp_R_lin_dependent: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and ms :: "'b list" assumes "RModule R smult M" shows "\<not> R_scalar_mult.R_lin_independent R smult ((0::'b) # ms)"
unique_factorization_m_zero: fixes f :: "int poly"   and c :: "int"   and fs :: "int poly multiset" assumes "p.unique_factorization_m f (c, fs)" shows "p.M c \<noteq> 0"
Skip: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "\<turnstile>\<^sub>t {P} SKIP {P}"
not_choice_simps_7: fixes l :: "'d"   and fmsg :: "'a \<Rightarrow> 'b"   and p :: "('a, 'b, 'c, 'd) seqp" shows "not_choice ({l}send(fmsg) . p)"
SUP_constant: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and c :: "'b"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup ((\<lambda>y. c) ` A) = (if A = {} then bot else c)"
Collect_mono_iff: fixes P :: "'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool" shows "(Collect P \<subseteq> Collect Q) = (\<forall>x. P x \<longrightarrow> Q x)"
leq_pos_imp_abs_leq: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) \<le> a"   and "a \<le> b" shows "\<bar>a\<bar> \<le> \<bar>b\<bar>"
eigen_vector_hom: fixes A :: "((real, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and v :: "(real, 'b) Finite_Cartesian_Product.vec"   and x :: "real" shows "eigen_vector (HMA_Connect.map_matrix of_real A) (map_vector of_real v) (of_real x) = eigen_vector A v x"
add_size_rel_def: shows "add_size_rel = br fst (\<lambda>(l, n). n = Max (insert 0 V))"
1: shows "\<forall>j. (f \<odot> mtop) (i, j) \<noteq> bot \<and> (\<forall>k. (f \<odot> mtop) (i, j) = (f \<odot> mtop) (i, k)) \<and> (\<forall>k. i \<noteq> k \<longrightarrow> (f \<odot> mtop) (k, j) = bot)"
cSup_atMost: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup" shows "Sup (ord.atMost less_eq x) = x"
simps: fixes x :: "nat"   and y :: "nat"   and u :: "nat" shows "loR [x, y, u] = (x mod y ^ u = 0)"
0: shows "FvarsT \<chi> = FvarsT r - {x} \<union> (if x \<in> FvarsT r then FvarsT t else {})"
fmdom_restrict_fset: fixes A :: "'a fset"   and m :: "('a, 'b) fmap" shows "fmdom (fmrestrict_fset A m) |\<subseteq>| A"
mult_rat_1_pos: fixes r1 :: "rat"   and y :: "int poly \<times> rat \<times> rat" assumes "0 < r1"   and "invariant_1 y" shows "invariant_2 (mult_rat_1_pos r1 y) \<and> real_of_2 (mult_rat_1_pos r1 y) = real_of_rat r1 * real_of_1 y"
bounded_linear_right: fixes a :: "'b blinop" shows "bounded_linear (blinop_apply a)"
distinct_9: fixes x2 :: "('b, 'c, 'd) Tree"   and x41 :: "'d"   and x42 :: "('b, 'c, 'd) Tree" shows "tNot x2 \<noteq> tAct x41 x42"
assms_2: shows "(s, (\<alpha>, t) # ss) \<in> seq lrs"
alw: fixes \<phi> :: "'c stream \<Rightarrow> bool"   and xs :: "'c stream" assumes "\<phi> xs"   and "alw \<phi> (stl xs)" shows "alw \<phi> xs"
csupport_on_def: fixes X :: "'d set"   and S :: "'d \<Rightarrow> 'e" shows "csupport_on X S = closure (support_on X S)"
find_closest_bf_eq_val_find_closest_bf_tm: fixes p :: "int \<times> int"   and ps :: "(int \<times> int) list" shows "Time_Monad.val (find_closest_bf_tm p ps) = find_closest_bf p ps"
finite_nat_minimal_witness: fixes P :: "'a \<Rightarrow> nat \<Rightarrow> bool" assumes "\<forall>i. \<exists>N. \<forall>n\<ge>N. P i n" shows "\<exists>N. \<forall>i n. N \<le> n \<longrightarrow> P i n"
wr_com_choose: fixes c1 :: "command"   and c2 :: "command" shows "wr_com (Choose c1 c2) = wr_com c1 \<union> wr_com c2"
totient_eq: shows "totient (nat (int p)) = card (Units R)"
tab_simps_2: shows "src\<^sub>D (D.inv (\<Phi> (r, f)) \<cdot>\<^sub>D F \<rho>) = src\<^sub>D (F f)"
fun_quotient3: fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and abs1 :: "'a \<Rightarrow> 'b"   and rep1 :: "'b \<Rightarrow> 'a"   and R2 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs2 :: "'c \<Rightarrow> 'd"   and rep2 :: "'d \<Rightarrow> 'c" assumes "Quotient3 R1 abs1 rep1"   and "Quotient3 R2 abs2 rep2" shows "Quotient3 (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
arctan_approx_def': fixes n :: "nat"   and x :: "real" shows "arctan_approx n (1 / x) = (\<Sum>k<n. inverse (real (2 * k + 1) * (- x\<^sup>2) ^ k)) / x"
assms_2: shows "pre_adjust \<langle>l, (a, lv), r'\<rangle>"
assms_3: shows "AE x in M. \<bar>g x\<bar> \<le> \<bar>f x\<bar>"
assms: shows "h \<turnstile> remove_child ptr' child \<rightarrow>\<^sub>h h'"
star_trans: fixes x :: "'a" shows "qstar x \<cdot> qstar x \<le> qstar x"
subst_clause_min_set: fixes \<Phi> :: "'b ltln fset"   and m :: "'b ltln \<Rightarrow> 'b ltln option" shows "min_set (subst_clause \<Phi> m) = subst_clause \<Phi> m"
ipresOp_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'e"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'f"   and MOD :: "('index, 'bindex, 'g, 'h, 'opSym, 'i, 'e, 'f, 'j) model_scheme" shows "ipresOp h hA MOD \<equiv> \<forall>delta inp binp. wlsInp delta inp \<and> wlsBinp delta binp \<longrightarrow> h (Op delta inp binp) = igOp MOD delta (lift h inp) (lift hA binp)"
rule_trans: fixes u :: "'a"   and xs :: "'b list"   and ys :: "'b list"   and P :: "'b process"   and I :: "('a \<times> 'a) set"   and D :: "'b \<Rightarrow> 'a"   and zs :: "'b list" assumes "(u, xs, ys) \<in> rel_induct_aux P I D"   and "(u, ys, zs) \<in> rel_induct_aux P I D" shows "(u, xs, zs) \<in> rel_induct_aux P I D"
int_of_integer_pow: fixes x :: "integer"   and n :: "nat" shows "int_of_integer (x ^ n) = int_of_integer x ^ n"
finter_fminus: fixes A :: "'c fset"   and B :: "'c fset"   and C :: "'c fset" shows "A |\<inter>| B |-| C = A |\<inter>| (B |-| C)"
sel: fixes x2 :: "'c" shows "the (Some x2) = x2"
Image_def: fixes r :: "('d \<times> 'c) set"   and s :: "'d set" shows "r `` s = {y. \<exists>x\<in>s. (x, y) \<in> r}"
p: shows "t \<^sup>@ i \<cdot> r = p"
x0_hb_bound8': fixes x :: "real"   and i :: "nat" assumes "x\<^sub>1 \<le> x"   and "i < k" shows "x\<^sub>0 < bs ! i * x + hb * x / ln x powr (1 + e)"
linear_map_def: fixes K :: "('g, 'h) ring_scheme"   and V :: "('g, 'i, 'j) module_scheme"   and W :: "('g, 'k, 'l) module_scheme"   and T :: "'i \<Rightarrow> 'k" shows "linear_map K V W T \<equiv> vectorspace K V \<and> vectorspace K W \<and> mod_hom K V W T"
fs_cont: fixes k :: "nat" assumes "k \<le> N" shows "continuous_on {real_of_int a..real_of_int b} (fs k)"
assms: fixes t :: "real"   and x :: "'a"   and y :: "'a" assumes "0 < t"   and "t < 1"   and "x \<in> A"   and "y \<in> A" shows "f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
TC_induct_down_lemma: fixes a :: "V"   and b :: "V"   and d :: "V" assumes "a \<sqsubset> b"   and "b \<le> d"   and "\<And>y z. \<lbrakk>y \<sqsubset> b; y \<in> elts d; z \<in> elts y\<rbrakk> \<Longrightarrow> z \<in> elts d" shows "a \<in> elts d"
rreq_rrep_fresh_1: fixes crt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and hops :: "nat"   and rreqid :: "nat"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and oip :: "nat"   and osn :: "nat"   and sip :: "nat" shows "rreq_rrep_fresh crt (Rreq hops rreqid dip dsn dsk oip osn sip) = (sip \<noteq> oip \<longrightarrow> oip \<in> kD crt \<and> (osn < sqn crt oip \<or> sqn crt oip = osn \<and> the (dhops crt oip) \<le> hops \<and> the (flag crt oip) = val))"
simps_3: fixes x :: "'a itrm"   and uv :: "'a itrm" shows "CF_pure (x \<diamondop> uv) = CF_pure x"
ainvf_l: fixes E :: "('c, 'd) aGroup_scheme"   and F :: "('e, 'f) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e"   and x :: "'c" assumes "aGroup E"   and "aGroup F"   and "bijec\<^bsub>E,F\<^esub> f"   and "x \<in> carrier E" shows "(ainvf\<^bsub>E,F\<^esub> f) (f x) = x"
irreducibleI: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and p :: "'b" assumes "class.comm_semiring_1 times one plus zero"   and "p \<noteq> zero"   and "\<not> dvd.dvd times p one"   and "\<And>a b. p = times a b \<Longrightarrow> dvd.dvd times a one \<or> dvd.dvd times b one" shows "comm_semiring_1.irreducible times one zero p"
t_def: shows "t = (\<lambda>e. if e = 0 then Some 0 else s e)"
old_fin_chain_finite: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and X :: "'a set" assumes "[[x .. y .. z]X]" shows "finite X"
strong_induct: fixes P :: "'a \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'a"   and ty :: "ty" assumes "\<And>tyvrs z. P z (Tvar tyvrs)"   and "\<And>z. P z Top"   and "\<And>ty1 ty2 z. \<lbrakk>\<And>z. P z ty1; \<And>z. P z ty2\<rbrakk> \<Longrightarrow> P z (ty1 \<rightarrow> ty2)"   and "\<And>tyvrs ty1 ty2 z. \<lbrakk>tyvrs \<sharp> z; tyvrs \<sharp> ty2; \<And>z. P z ty1; \<And>z. P z ty2\<rbrakk> \<Longrightarrow> P z (\<forall>tyvrs<:ty2. ty1)" shows "P z ty"
sdrop_simps_1: fixes n :: "nat"   and s :: "'c stream" shows "shd (sdrop n s) = s !! n"
finite_code: fixes A :: "'d set" assumes "class.finite TYPE('d)" shows "finite A = True"
wtE: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "M.wtE \<xi>" shows "U.wtE \<xi>"
tan_mono_le: fixes x :: "real"   and y :: "real" assumes "- (pi / 2) < x"   and "x \<le> y"   and "y < pi / 2" shows "tan x \<le> tan y"
set_of_positions_is_finite: fixes t :: "'b trm" shows "finite (pos_of t)"
length_map: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "length (map f xs) = length xs"
I_integral: fixes s :: "complex" assumes "a < Re s" shows "(I s \<longlongrightarrow> integral {0..} (laplace_integrand f s)) at_top"
prems_8: shows "InvariantWatchesDiffer (getF state') (getWatch1 state') (getWatch2 state')"
monotone_rel_prodD1: fixes orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ordc :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<times> 'c \<Rightarrow> 'd"   and b :: "'c" assumes "monotone (rel_prod orda ordb) ordc f"   and "class.preorder ordb (mk_less ordb)" shows "monotone orda ordc (\<lambda>a. f (a, b))"
well_order_on_extension: fixes p :: "('a \<times> 'a) set"   and A :: "'a set" assumes "wf p"   and "Field p \<subseteq> A" shows "\<exists>w. p \<subseteq> w \<and> well_order_on A w"
Inf_insert_finite: fixes S :: "'a set"   and x :: "'a" assumes "finite S" shows "Inf (insert x S) = (if S = {} then x else min x (Inf S))"
r: shows "r = (\<lambda>x y. S (f x) (f y))"
ODNmap_mem1: fixes D :: "'c Order"   and y :: "'c Order set" assumes "Worder D"   and "y \<in> segment ODnods (ordinal_number D)" shows "ODNmap D y \<in> carrier D"
disc_23: fixes x31 :: "nat"   and x32 :: "'a"   and x33 :: "'a"   and x34 :: "'b"   and x35 :: "'b"   and x36 :: "'c" shows "\<not> isRecvMarker (Recv x31 x32 x33 x34 x35 x36)"
sum_bounded_above_divide: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and K :: "'d" assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<le> K / of_nat (card A)"   and "finite A"   and "A \<noteq> {}" shows "sum f A \<le> K"
strict_order_iff: fixes a :: "'b"   and b :: "'b" shows "(a < b) = (a = inf a b \<and> a \<noteq> b)"
DERIV_neg_imp_decreasing_at_top: fixes b :: "real"   and f :: "real \<Rightarrow> real"   and flim :: "real" assumes "\<And>x. b \<le> x \<Longrightarrow> \<exists>y. (f has_real_derivative y) (at x) \<and> y < 0"   and "(f \<longlongrightarrow> flim) at_top" shows "flim < f b"
comp_substs_def: fixes comp_subst :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<sigma>s :: "'b list"   and \<tau>s :: "'b list" shows "substitution_ops.comp_substs comp_subst \<sigma>s \<tau>s = Map2.map2 comp_subst \<sigma>s \<tau>s"
inter_uniqueness_not_par: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p"   and D :: "'p" assumes "\<not> Col A B C"   and "Col A B P"   and "Col C D P" shows "\<not> A B Par C D"
path_via': shows "P \<turnstile> Path last Cs to C via Cs'' "
abs_prm_def: shows "abs_prm \<equiv> quot_type.abs (=p) Abs_prm"
Cons_pathI: fixes t :: "'b ttree"   and x :: "'b"   and t' :: "'b ttree"   and xs :: "'b list" assumes "possible t x = possible t' x"   and "\<lbrakk>possible t x; possible t' x\<rbrakk> \<Longrightarrow> (xs \<in> paths (nxt t x)) = (xs \<in> paths (nxt t' x))" shows "(x # xs \<in> paths t) = (x # xs \<in> paths t')"
clock_simAbs_refl: fixes t :: "'s Trace"   and ec :: "'s odlist \<times> 's odlist"   and a :: "'a" assumes "t \<in> Clock.jkbpC"   and "clock_simAbs ec = Clock.sim_equiv_class a t" shows "clock_sim t \<in> clock_simAbs ec"
map_map_idx: fixes f :: "'f \<Rightarrow> 'e"   and g :: "'g \<Rightarrow> nat \<Rightarrow> 'f"   and xs :: "'g list"   and n :: "nat" shows "map f (map_idx g xs n) = map_idx (\<lambda>x i. f (g x i)) xs n"
assms_3: shows "\<forall>b\<^sub>\<sigma> n. local.etp.HCP_adv n \<A> m2 b\<^sub>\<sigma> D \<le> etp_advantage n"
IH_1: shows "singular_subdivision p (chain_map p g a) = chain_map p g (singular_subdivision p a)"
bconf_Block: fixes P :: "'a prog"   and sh :: "sheap"   and V :: "'b"   and T :: "ty"   and e :: "'b exp"   and b :: "bool" shows "(P,sh \<turnstile>\<^sub>b ({V:T; e},b) \<surd>) = (P,sh \<turnstile>\<^sub>b (e,b) \<surd>)"
elims_3: fixes x :: "'a"   and xa :: "'b list"   and xb :: "'a" assumes "\<not> cas x xa xb" obtains u :: "'a" and v :: "'a" where "x = ub__" and "xa = []" and "xb = va__" and "ub__ \<noteq> va__"   | u :: "'a" and e :: "'b" and es :: "'b list" and v :: "'a" where "x = u" and "xa = e # es" and "xb = v" and "\<not> (tail G e = u \<and> cas (head G e) es v)"
to_int_mod_ring_of_int_mod_ring: fixes x :: "int" assumes "0 \<le> x"   and "x < int CARD('a)" shows "to_int_mod_ring (of_int_mod_ring x) = x"
Bound: fixes x :: "name"   and a :: "name"   and Q' :: "pi" assumes "x \<sharp> C"   and "Q \<longmapsto> a<\<nu>x> \<prec> Q'" shows "\<exists>P'. P \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel"
telescope_sums': fixes f :: "nat \<Rightarrow> 'a"   and c :: "'a" assumes "f \<longlonglongrightarrow> c" shows "(\<lambda>n. f n - f (Suc n)) sums (f 0 - c)"
assms_2: shows "\<R>\<^sub>\<circ> r = set {a}"
sep_disj_addD2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.sep_algebra plus zero sep_disj"   and "sep_disj x (plus y z)"   and "sep_disj y z" shows "sep_disj x z"
measurable_eval_cexpr: fixes t :: "pdf_type"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and V :: "nat set" assumes "case_nat t \<Gamma> \<turnstile>\<^sub>c e : REAL"   and "free_vars e \<subseteq> shift_var_set V" shows "(\<lambda>(x, y). eval_cexpr e x y) \<in> borel_measurable (state_measure V \<Gamma> \<Otimes>\<^sub>M stock_measure t)"
face_of_Inter: fixes A :: "'b set set"   and S :: "'b set" assumes "A \<noteq> {}"   and "\<And>T. T \<in> A \<Longrightarrow> T face_of S" shows "\<Inter> A face_of S"
nth_root_nat_nth_power: fixes k :: "nat"   and n :: "nat" assumes "0 < k" shows "nth_root_nat k (n ^ k) = n"
consistentI: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "a_consistent (ae, a, as) (\<Gamma>, e, S)"   and "\<And>x. x \<in> thunks \<Gamma> \<Longrightarrow> ae x = up\<cdot>0" shows "consistent (ae, a, as) (\<Gamma>, e, S)"
preserves_ide: fixes a :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "VV.ide a" shows "ide (fst a \<star> snd a)"
I_intF_def: fixes f :: "'fsym"   and al :: "univ list" shows "I_intF f al \<equiv> intF f (map2 proj (arOf f) al)"
monic: shows "monic (Mp (\<Prod>\<^sub># fs))"
oinvariantD: fixes A :: "('e \<times> 'f, 'g) automaton"   and S :: "'e \<Rightarrow> 'e \<Rightarrow> 'g \<Rightarrow> bool"   and U :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and P :: "'e \<times> 'f \<Rightarrow> bool"   and s :: "'e \<times> 'f" assumes "A \<Turnstile> (S, U \<rightarrow>) P"   and "s \<in> oreachable A S U" shows "P s"
number_partition_of_functions_of: fixes A :: "'c set"   and B :: "'d set"   and N :: "nat multiset" assumes "finite A"   and "finite B"   and "number_partition (card A) N"   and "size N \<le> card B" shows "number_partition_of A B (Twelvefold_Way_Entry10.functions_of A B N) = N"
oexp_def: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set" assumes "wo_rel2 r s" shows "r ^o s = {(f, g). f \<in> FinFunc r s \<and> g \<in> FinFunc r s \<and> ((let m = wo_rel.max_fun_diff s f g in (f m, g m) \<in> r) \<or> f = g)}"
is_Times_def: fixes rexp :: "'b rexp" shows "is_Times rexp = (\<exists>x51 x52. rexp = Times x51 x52)"
linorder_neqE_nat: fixes x :: "nat"   and y :: "nat" assumes "x \<noteq> y" obtains "order.greater y x"   | "order.greater x y"
DBM_reset'_neg_diag_preservation': fixes k :: "nat"   and n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and v :: "'d \<Rightarrow> nat"   and cs :: "'d list"   and d :: "'c" assumes "k \<le> n"   and "M k k < \<one>"   and "\<forall>c. 0 < v c"   and "\<forall>c\<in>set cs. v c \<le> n" shows "reset' M n cs v d k k < \<one>"
S\<^sub>1_in_hom: shows "Maps.in_hom rs.S\<^sub>1 (Maps.MkIde (src (tab\<^sub>0 s))) (Maps.MkIde (trg s))"
wlp_while_def: fixes M0 :: "complex mat"   and M1 :: "complex mat"   and WS :: "complex mat \<Rightarrow> complex mat"   and P :: "complex mat" shows "wlp_while M0 M1 WS P = (THE Q. limit_mat (\<lambda>n. wlp_while_n M0 M1 WS n P) Q d)"
HStartBallot_HInv1: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HInv1 s"   and "HStartBallot s s' p" shows "HInv1 s'"
neg_equal_zero_eucl: fixes a :: "'a" shows "(- a = a) = (a = (0::'a))"
tendsto_norm_zero_iff: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F = (f \<longlongrightarrow> (0::'c)) F"
reflp_on_def: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" shows "reflp_on P A = (\<forall>a\<in>A. P a a)"
hyp: fixes M' :: "trm" assumes "SNa M'"   and "\<forall>M''. M' \<longrightarrow>\<^sub>a M'' \<longrightarrow> P M''" shows "P M'"
direct_decomp_direct_decomp: fixes A :: "'c set"   and s :: "'c set"   and ss :: "'c set list"   and rs :: "'c set list" assumes "direct_decomp A (s # ss)"   and "direct_decomp s rs" shows "direct_decomp A (ss @ rs)"
less_sets_trans: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A \<lless> B"   and "B \<lless> C"   and "B \<noteq> {}" shows "A \<lless> C"
ennreal_SUP_eq_top: fixes I :: "'b set"   and f :: "'b \<Rightarrow> ennreal" assumes "\<And>n. \<exists>i\<in>I. of_nat n \<le> f i" shows "Sup (f ` I) = top"
Implies_def: fixes P :: "('a, 'b, 'c) formula"   and Q :: "('a, 'b, 'c) formula" shows "(P \<rightarrow> Q) = (Q || ! P)"
scale_cancel_left: fixes a :: "'d"   and x :: "'d"   and y :: "'d" shows "(a * x = a * y) = (x = y \<or> a = (0::'d))"
triangular_columnI: fixes j :: "nat"   and A :: "'a mat" assumes "\<And>i. \<lbrakk>j < i; i < dim_row A\<rbrakk> \<Longrightarrow> A $$ (i, j) = (0::'a)" shows "local.triangular_column j A"
ndec_seq_mem: fixes a :: "nat"   and A :: "nat set"   and n :: "nat" assumes "a \<in> A"   and "\<nexists>m. m \<in> A \<and> (\<forall>x\<in>A. m \<le> x)" shows "ndec_seq A a n \<in> A"
delete_id_if_wbt_notin: fixes \<Delta>1 :: "nat"   and \<Delta>2 :: "nat"   and \<Gamma>1 :: "nat"   and \<Gamma>2 :: "nat"   and t :: "('b \<times> nat) tree"   and x :: "'b" assumes "WBT \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2"   and "WBT.wbt \<Delta>1 \<Delta>2 t"   and "\<not> isin t x" shows "WBT.delete \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2 x t = t"
one_le_\<omega>: shows "1 \<le> \<omega>"
pure_universal_suffix_closed: fixes \<nu> :: "'a ltln"   and u :: "'a set list"   and w :: "nat \<Rightarrow> 'a set" assumes "pure_universal \<nu>"   and "(u \<frown> w) \<Turnstile>\<^sub>n \<nu>" shows "w \<Turnstile>\<^sub>n \<nu>"
all_nth_imp_all_set: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "\<forall>i<length xs. P (xs ! i)"   and "x \<in> set xs" shows "P x"
OV_inv4D: fixes x :: "'c obsv_state_scheme"   and r :: "nat"   and p :: "process"   and v :: "val"   and q :: "process"   and w :: "val" assumes "x \<in> OV_inv4"   and "obsv_state.obs x r p = Some v \<and> obsv_state.obs x r q = Some w" shows "w = v"
4: shows "mat (row_length (B1 \<circ> B2)) (length (B1 \<circ> B2)) (B1 \<circ> B2)"
path_image_linepath: fixes a :: "'b"   and b :: "'b" shows "path_image (linepath a b) = {a--b}"
assms_5: shows "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)"
simps_2: fixes \<mu> :: "'a" shows "Can \<^bold>\<langle>\<mu>\<^bold>\<rangle> = ide \<mu>"
sub_composition_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> K"   and "b \<in> K" shows "composition a b \<in> K"
r_a_def: shows "r_a = {(x, 0) |x. x \<in> A}"
outer_prod_def: fixes v :: "'c vec"   and w :: "'c vec" shows "outer_prod v w = mat (dim_vec v) 1 (\<lambda>(i, j). v $ i) * mat 1 (dim_vec w) (\<lambda>(i, j). conjugate w $ j)"
source_def: fixes t :: "'d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd" shows "source t = fst t"
subset_refl: fixes A :: "'h set" shows "A \<subseteq> A"