hcju/isabelleps huggingface dataset:
conga_sym_equiv: 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 A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C' = Tarski_neutral_dimensionless.CongA Bet Cong A' B' C' A B C"
measurable_uniform_int: shows "uniform_int \<in> stock_measure (PRODUCT INTEG INTEG) \<rightarrow>\<^sub>M subprob_algebra (stock_measure INTEG)"
convex_sum: fixes S :: "'a set"   and C :: "'b set"   and a :: "'a \<Rightarrow> real"   and y :: "'a \<Rightarrow> 'b" assumes "finite S"   and "convex C"   and "sum a S = 1"   and "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> a i"   and "\<And>i. i \<in> S \<Longrightarrow> y i \<in> C" shows "(\<Sum>j\<in>S. a j *\<^sub>R y j) \<in> C"
is_canon_rewriterD4: fixes rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and A :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and a :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_canon_rewriter rword A u p"   and "a \<in> A"   and "a \<noteq> 0"   and "lt a adds\<^sub>t u" shows "rword (spp_of a) (spp_of p)"
fcomp_id: fixes f :: "'a \<Rightarrow> 'b" shows "fcomp f id = f"
Re_complex_div_gt_0: fixes a :: "complex"   and b :: "complex" shows "(0 < Re (a / b)) = (0 < Re (a * cnj b))"
silent_move_equal_length_2: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a :: "'edge"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) -a\<rightarrow>\<^sub>\<tau> (ms',s')" shows "length ms' = length s'"
word_rsplit_def: fixes w :: "'d word" shows "word_rsplit w = map word_of_int (bin_rsplit LENGTH('c) (LENGTH('d), uint w))"
insert_monoD: fixes W :: "'b set"   and W' :: "'b set"   and a :: "'b" assumes "W \<subseteq> W'" shows "insert a W \<subseteq> insert a W'"
mulexp_zero: fixes p :: "'a list"   and q :: "'a list" shows "mulexp 0 p q = q"
swap_pair_fun: fixes C :: "'a \<times> 'b \<Rightarrow> real \<times> real" shows "(\<lambda>y. prod.swap (C (y, 0::'b))) = (\<lambda>x. (C (x, 0::'b) \<bullet> j, C (x, 0::'b) \<bullet> i))"
assms_2: shows "set signs = consistent_sign_vectors qs UNIV"
subgroup_generated_def: fixes S :: "'a set" shows "additive.subgroup_generated S = additive.generate (R \<inter> S)"
primitive_root_unity_def: fixes k :: "nat"   and x :: "'b" shows "primitive_root_unity k x = (k \<noteq> 0 \<and> x ^ k = (1::'b) \<and> (\<forall>k'<k. k' \<noteq> 0 \<longrightarrow> x ^ k' \<noteq> (1::'b)))"
invertible_right_inverse: fixes u :: "'a \<Rightarrow> 'a" assumes "invertible u"   and "u \<in> S \<rightarrow>\<^sub>E S" shows "compose S u (inverse u) = identity S"
prv_eql_subst_trm3: fixes x :: "'var"   and \<phi> :: "'fmla"   and t1 :: "'trm"   and t2 :: "'trm" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "prv (eql t1 t2)"   and "prv (subst \<phi> t1 x)" shows "prv (subst \<phi> t2 x)"
if_eqvt: fixes p :: "perm"   and b :: "bool"   and x :: "'e"   and y :: "'e" shows "p \<bullet> (if b then x else y) = (if p \<bullet> b then p \<bullet> x else p \<bullet> y)"
ResB: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and y :: "name" assumes "P \<Longrightarrow>a<\<nu>x> \<prec> P'"   and "y \<noteq> a"   and "y \<noteq> x" shows "<\<nu>y>P \<Longrightarrow>a<\<nu>x> \<prec> <\<nu>y>P'"
nl41: fixes l' :: "real"   and r' :: "real"   and ll :: "real"   and r :: "real" assumes "1 \<le> l'"   and "1 \<le> r'"   and "1 \<le> ll"   and "1 \<le> r" shows "4 * (l' + ll) * (r' + r) \<le> (l' + ll + r' + r)\<^sup>2"
sep_conj_com: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "P ** Q = Q ** P"
P4: shows "\<forall>u v. nat_to_set u = D \<and> nat_to_set v = D \<longrightarrow> u = v"
assms_2: shows "\<forall>\<^sub>F n in F. x n \<in> s"
abs_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b set \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b set"   and x :: "'b" assumes "quot_type R Abs Rep" shows "quot_type.abs R Abs x = Abs (Collect (R x))"
prj_simps_5: shows "C.dom fg.prj\<^sub>0 = f.leg0 \<down>\<down> g.leg1"
properD1: fixes ct :: "'s \<Rightarrow> 's pmf" assumes "proper ct" shows "ct \<in> Pi\<^sub>E S K"
wf_map_prod_image: fixes r :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'c" assumes "wf r"   and "inj f" shows "wf (map_prod f f ` r)"
t_one_def: shows "t_one \<equiv> \<Union>p. {(p, [])}"
assms_2: shows "bij_lens (x +\<^sub>L y)"
has_fps_expansion_add: fixes f :: "'a \<Rightarrow> 'a"   and F :: "'a fps"   and g :: "'a \<Rightarrow> 'a"   and G :: "'a fps" assumes "f has_fps_expansion F"   and "g has_fps_expansion G" shows "(\<lambda>x. f x + g x) has_fps_expansion F + G"
qtable1: shows "qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) (ne + length aux) \<phi>) rel1"
dom_comp: fixes g :: "'a"   and f :: "'a" assumes "S.seq g f" shows "S.dom (g \<cdot>\<^sub>S f) = S.dom f"
in_sets: shows "{f \<in> space (lborel_f (Suc n)). insertion f p = 0} \<in> sets (lborel_f (Suc n))"
det_Gauss_Jordan_column_k_det_P: fixes det_P :: "'b"   and B :: "(('b, 'c) vec, 'c) vec"   and A :: "(('b, 'd) vec, 'd) vec"   and i :: "nat"   and k :: "nat" assumes "det_P * det B = det A" shows "fst (Gauss_Jordan_column_k_det_P (det_P, i, A) k) * det B = det (snd (snd (Gauss_Jordan_column_k_det_P (det_P, i, A) k)))"
integral_measure_pmf_real: fixes A :: "'b set"   and M :: "'b pmf"   and f :: "'b \<Rightarrow> real" assumes "finite A"   and "\<And>a. \<lbrakk>a \<in> set_pmf M; f a \<noteq> 0\<rbrakk> \<Longrightarrow> a \<in> A" shows "measure_pmf.expectation M f = (\<Sum>a\<in>A. f a * pmf M a)"
CCexp_eq: fixes e :: "exp"   and a :: "Arity" shows "CCexp e\<cdot>a = snd (cccExp e\<cdot>a)"
P_eq_val_equiv: fixes v :: "'a \<Rightarrow> ant"   and v' :: "'a \<Rightarrow> ant" assumes "valuation K v"   and "valuation K v'" shows "v_equiv K v v' = (P\<^bsub> K v\<^esub> = P\<^bsub> K v'\<^esub>)"
reflection_of_binary_predicates_and_closures_1: fixes Rel :: "('c \<times> 'c) set"   and Pred :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_reflects_binary_pred Rel Pred" shows "rel_reflects_binary_pred (Rel\<^sup>=) Pred"
Sup_empty: fixes Inf :: "'f set \<Rightarrow> 'f"   and Sup :: "'f set \<Rightarrow> 'f"   and inf :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and sup :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and bot :: "'f"   and top :: "'f" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup {} = bot"
inductive_on_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "inductive_on P A = (\<forall>Q. (\<forall>y\<in>A. (\<forall>x\<in>A. P x y \<longrightarrow> Q x) \<longrightarrow> Q y) \<longrightarrow> (\<forall>x\<in>A. Q x))"
rtranclp_idemp: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "r\<^sup>*\<^sup>*\<^sup>*\<^sup>* = r\<^sup>*\<^sup>*"
prems_1: fixes y :: "'a" assumes "y \<in> set (local.filter_min_aux (x0 # xs) ys)"   and "rel y x" shows "thesis"
ln_less_zero_iff: fixes x :: "real" assumes "0 < x" shows "(ln x < 0) = (x < 1)"
strict_trans2: fixes a :: "('a, 'b) Term.term"   and b :: "('a, 'b) Term.term"   and c :: "('a, 'b) Term.term" assumes "a \<sqsubset> b"   and "b \<sqsubseteq> c" shows "a \<sqsubset> c"
continuous_on_ln: fixes s :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "continuous_on s f"   and "\<forall>x\<in>s. f x \<noteq> 0" shows "continuous_on s (\<lambda>x. ln (f x))"
elims_3: fixes x :: "iface"   and xa :: "char list" assumes "\<not> match_iface x xa" obtains i :: "char list" and p_iface :: "char list" where "x = Iface i" and "xa = p_iface" and "\<not> internal_iface_name_match i p_iface"
simps_2: fixes c :: "'b"   and s :: "'b list"   and r :: "'b rexp" shows "derivs (c # s) r = derivs s (deriv c r)"
the_coext_unique: fixes x' :: "'d"   and f :: "'c"   and g :: "'d" assumes "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F x' (C.cod f') f"   and "x''\<epsilon>.is_coext x' f g" shows "g = x''\<epsilon>.the_coext x' f"
list_Previous_0: fixes xs :: "'d list"   and init :: "'d" shows "xs\<^bsup>\<leftarrow>' init\<^esup> 0 = init"
Rep_List_inverse: fixes x :: "'b SList.list" shows "Abs_List (Rep_List x) = x"
not_fresher_trans: fixes x :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and y :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and z :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" assumes "\<not> x \<sqsubseteq> y"   and "\<not> z \<sqsubseteq> x" shows "\<not> z \<sqsubseteq> y"
isomorphic_implies_hpar_2: fixes f :: "'d"   and f' :: "'d" assumes "D.isomorphic f f'" shows "D.ide f'"
elims: fixes x :: "'a list"   and xa :: "'a"   and y :: "'a list" assumes "insert_body x xa = y" obtains e :: "'a" where "x = []" and "xa = e__" and "y = [e__]"   | xb :: "'a" and xs :: "'a list" and e :: "'a" where "x = xb # xs" and "xa = e" and "y = (if xb < e then e # xb # xs else xb # insert_body xs e)"
assms_3: shows "x \<in> carrier - (E - {e}) - insert e F"
sorted_le2lt: fixes xs :: "'a list" assumes "linorder_class.sorted xs"   and "distinct xs" shows "Sorted_Less.sorted xs"
case_prodI2: fixes p :: "'a \<times> 'c"   and c :: "'a \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>a b. p = (a, b) \<Longrightarrow> c a b" shows "case p of (a, b) \<Rightarrow> c a b"
collect_clks_id: fixes cc :: "('e, 'f) cconstraint" shows "collect_clks cc = fst ` collect_clock_pairs cc"
span_li_not_depend_1: fixes S :: "'c set"   and N :: "'c set" assumes "S \<subseteq> N"   and "LinearCombinations.submodule R N M" shows "LinearCombinations.module.span R (md N) S = local.span S"
in_cong_bigtheta: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "f \<in> \<Theta>[F](g)" shows "(f \<in> \<omega>[F](h)) = (g \<in> \<omega>[F](h))"
L6_Controller_One: fixes y :: "nat \<Rightarrow> nat list"   and l :: "nat \<Rightarrow> bit"   and z :: "nat \<Rightarrow> bit list"   and t :: "nat" assumes "Controller_L y (fin_inf_append [Zero] l) l z"   and "l (Suc t) = One"   and "z (Suc t) = []" shows "l t = One"
ord_p_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "p \<preceq>\<^sub>p q \<equiv> p \<prec>\<^sub>p q \<or> p = q"
PiE_E: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'b \<Rightarrow> 'c set"   and x :: "'b" assumes "f \<in> Pi\<^sub>E A B" obtains "x \<in> A" and "f x \<in> B x"   | "x \<notin> A" and "f x = undefined"
dghm_dag_Rel_ArrMap_app: fixes \<alpha> :: "V"   and x :: "V" assumes "arr_Rel \<alpha> x" shows "\<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha>\<lparr>ArrMap\<rparr>\<lparr>x\<rparr> = x\<inverse>\<^sub>R\<^sub>e\<^sub>l"
ord_isom_self_id: fixes f :: "'a \<Rightarrow> 'a" assumes "ord_isom D D f" shows "f = idmap (carrier D)"
subset_states_finite: fixes Q :: "hf set" assumes "Q \<subseteq> nfa.states M" shows "finite Q"
fin: shows "\<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M < \<infinity>"
St_def: shows "St \<equiv> quot_type.abs eq_st Abs_st"
open_Compl: fixes "open" :: "'d set \<Rightarrow> bool"   and S :: "'d set" assumes "class.topological_space open"   and "topological_space.closed open S" shows "open (- S)"
position_of_right: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm"   and ds :: "dir list" shows "position_of (App t s) (dir.Right # ds) = position_of s ds"
inversion_noteq_unit_disc: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc" shows "inversion u \<noteq> v"
coeff_zero: fixes R :: "('c, 'd) ring_scheme"   and n :: "nat" assumes "UP_ring R" shows "coeff (UP R) \<zero>\<^bsub>UP R\<^esub> n = \<zero>\<^bsub>R\<^esub>"
must_lock_d_eq_must_lock: fixes \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list"   and t :: "'thread_id"   and xcp :: "'addr option"   and h :: "'heap"   and frs :: "'addr frame list" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "\<Phi> \<turnstile> t:(xcp, h, frs) \<surd>" shows "execd_mthr.must_sync P t (xcp, frs) h = exec_mthr.must_sync P t (xcp, frs) h"
divide_less_0_1_iff: fixes a :: "'a" shows "((1::'a) / a < (0::'a)) = (a < (0::'a))"
h0: shows "\<forall>i<length (take_indices subsets S). row (alt_matrix_A_R signs (take_indices subsets S)) i = row (take_rows_from_matrix (alt_matrix_A_R signs subsets) S) i"
mono_deductions: fixes A :: "(formula list \<times> formula list set) set"   and B :: "(formula list \<times> formula list set) set" assumes "A \<subseteq> B" shows "deductions A \<subseteq> deductions B"
bet_per2__suma: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p" assumes "A \<noteq> B"   and "B \<noteq> C"   and "D \<noteq> E"   and "E \<noteq> F"   and "G \<noteq> H"   and "H \<noteq> I"   and "Per A B C"   and "Per D E F"   and "Bet G H I" shows "A B C D E F SumA G H I"
Exy: shows "infdist (f w) {f x--f y} \<le> E"
closed_orbit_period_zero_fixed_point: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and x :: "'b" assumes "auto_ll_on_open f X"   and "auto_ll_on_open.closed_orbit f X x"   and "auto_ll_on_open.period f X x = 0" shows "f x = (0::'b)"
simps_6: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "restricted_subformulas_inner (\<phi> W\<^sub>n \<psi>) = restricted_subformulas_inner \<phi> \<union> restricted_subformulas_inner \<psi>"
ldropn_LCons: fixes n :: "nat"   and x :: "'c"   and xs :: "'c llist" shows "ldropn n (LCons x xs) = (case n of 0 \<Rightarrow> LCons x xs | Suc n' \<Rightarrow> ldropn n' xs)"
card_Diff1_less: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "x \<in> A" shows "card (A\<setminus>{x}) < card A"
finite_union_of_Un: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and T :: "'b set" assumes "(finite union_of P) S"   and "(finite union_of P) T" shows "(finite union_of P) (S \<union> T)"
P4: shows "Bet B' C0 C' \<and> Cong B C B' C0"
\<tau>move2_IfCondD: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and e :: "'addr expr1"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option" assumes "\<tau>move2 P h stk (if (e) e1 else e2) pc xcp"   and "pc < length (compE2 e)" shows "\<tau>move2 P h stk e pc xcp"
RModuleIso_def: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c"   and N :: "'c set" shows "RModuleIso R smult M smult' T N \<equiv> RModuleHom R smult M smult' T \<and> RModuleIso_axioms M T N"
distinct_11: fixes x3 :: "fmla"   and x41 :: "name"   and x42 :: "fmla" shows "Robinson_Arithmetic.neg x3 \<noteq> exi x41 x42"
shows_quote_def: fixes s :: "char list \<Rightarrow> char list" shows "shows_quote s = shows CHR 0x27 \<circ> s \<circ> shows CHR 0x27"
matchPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and a :: "name"   and b :: "name" assumes "P \<leadsto>[Rel] Q"   and "Rel \<subseteq> Rel'" shows "[a\<frown>b]P \<leadsto>[Rel'] [a\<frown>b]Q"
args_apps: fixes s :: "('c, 'd) tm"   and ss :: "('c, 'd) tm list" shows "args (apps s ss) = args s @ ss"
pre_between_r2: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "pre_between vs ram1 ram2" shows "ram2 \<in> set vs"
subtopology_eq_discrete_topology_empty: fixes X :: "'c topology" shows "(X = discrete_topology {}) = (topspace X = {})"
mult_smult_assoc_mat: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and B :: "'b mat"   and nc :: "nat"   and k :: "'b" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "k \<cdot>\<^sub>m A * B = k \<cdot>\<^sub>m (A * B)"
preal_min_simp_3: fixes y :: "preal" shows "preal_min preal.PInfty y = y"
simple_linux_router_def: fixes rt :: "'a routing_rule list"   and fw :: "'a simple_rule list"   and mlf :: "'a word \<Rightarrow> 48 word option"   and ifl :: "interface list"   and p :: "'a Simple_Packet.simple_packet_ext" shows "simple_linux_router rt fw mlf ifl p \<equiv> iface_packet_check ifl p \<bind> (\<lambda>_. let rd = routing_table_semantics rt (p_dst p); p = p\<lparr>p_oiface := output_iface rd\<rparr>; fd = simple_fw fw p in (case fd of Decision FinalAllow \<Rightarrow> Some () | Decision FinalDeny \<Rightarrow> None) \<bind> (\<lambda>_. let nh = fromMaybe (p_dst p) (next_hop rd) in mlf nh \<bind> (\<lambda>ma. Some (p\<lparr>p_l2dst := ma\<rparr>))))"
plossless_gpv_stop: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv" assumes "\<I> \<turnstile>g gpv \<surd>" shows "plossless_gpv (stop_\<I> \<I>) (gpv_stop gpv) = plossless_gpv \<I> gpv"
simps: fixes f :: "int vec list \<times> int vec list"   and mu :: "int iarray iarray"   and d :: "int iarray" shows "inc_state (f, mu, d) = (inc_i f, mu, d)"
Rep_enat_inject: fixes x :: "enat"   and y :: "enat" shows "(Rep_enat x = Rep_enat y) = (x = y)"
assms: fixes v :: "i" assumes "[\<chi> in v]" shows "[\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<psi> in v]"
a_get_owner_document\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes shadow_root_ptr :: "(_) shadow_root_ptr" shows "local.a_get_owner_document\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr = local.CD.a_get_owner_document\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr)"
monom_mult_list_inv: fixes m1 :: "('a \<times> nat) list"   and m2 :: "('a \<times> nat) list" assumes "monom_inv m1"   and "monom_inv m2" shows "monom_inv (monom_mult_list m1 m2)"
FCIA2: shows "FCIA \<rho>2 \<Gamma>2 \<V>2 Tr\<^bsub>ES2\<^esub>"
seqSubstRes: fixes x :: "name"   and \<sigma> :: "(name \<times> name) list"   and P :: "pi" assumes "x \<sharp> \<sigma>" shows "<\<nu>x>P[<\<sigma>>] = <\<nu>x>P[<\<sigma>>]"
subo: shows "Subobjs P C' (Cs @ C # Cs')"
Sup_downset_adj_var: fixes X :: "'b set"   and y :: "'b" shows "(\<Squnion> X \<le> y) = (X \<subseteq> \<down> y)"
protocol_14_OT_def: fixes M :: "bool \<times> bool \<times> bool \<times> bool"   and C :: "bool \<times> bool" shows "protocol_14_OT M C = (let (c0, c1) = C; (m00, m01, m10, m11) = M in coin_spmf \<bind> (\<lambda>S0. coin_spmf \<bind> (\<lambda>S1. coin_spmf \<bind> (\<lambda>S2. coin_spmf \<bind> (\<lambda>S3. coin_spmf \<bind> (\<lambda>S4. coin_spmf \<bind> (\<lambda>S5. let a0 = S0 \<oplus> S2 \<oplus> m00; a1 = S0 \<oplus> S3 \<oplus> m01; a2 = S1 \<oplus> S4 \<oplus> m10; a3 = S1 \<oplus> S5 \<oplus> m11 in protocol_OT12 (S0, S1) c0 \<bind> (\<lambda>(uu_, Si). protocol_OT12 (S2, S3) c1 \<bind> (\<lambda>(uu_, Sj). protocol_OT12 (S4, S5) c1 \<bind> (\<lambda>(uu_, Sk). let s2 = Si \<oplus> (if c0 then Sk else Sj) \<oplus> (if c0 then if c1 then a3 else a2 else if c1 then a1 else a0) in return_spmf ((), s2)))))))))))"
assms_3: shows "arr = arr' on - {l\<^sub>0..<m}"
pick_equiv: fixes a :: "'d" shows "pick \<lfloor>a\<rfloor> \<sim> a"
incl_subgroup: fixes G :: "('c, 'd) monoid_scheme"   and J :: "'c set"   and I :: "'c set" assumes "Group.group G"   and "subgroup J G"   and "subgroup I (G\<lparr>carrier := J\<rparr>)" shows "subgroup I G"
dvd_defD1: fixes a :: "nat"   and b :: "nat" assumes "a dvd b" shows "\<exists>k. b = a * k"
strict_mono_to_nat: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "order.strict_mono less (mod_type.to_nat Rep)"
cr_vrat_transfer_domain_rule: shows "Domainp cr_vrat = (\<lambda>x. x \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>)"
assms_2: shows "\<guillemotleft>\<theta> : p\<^sub>0 \<star> w \<Rightarrow> u\<guillemotright>"
strict_pref_def_irrefl: fixes x :: "'a"   and r :: "('a \<times> 'a) set" shows "\<not> x \<^bsub>r\<^esub>\<prec> x"
tauChainFreshChain: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and xvec :: "name list" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"   and "xvec \<sharp>* P" shows "xvec \<sharp>* P'"
disc_iff_1: fixes sys :: "'b \<Rightarrow> ('a, 'b) process_term"   and T :: "('a, 'b) process_term" shows "isAction (solution sys T) = isACT sys T"
degree_m_smult_le: fixes c :: "int"   and f :: "int poly" shows "pl.degree_m (smult c f) \<le> pl.degree_m f"
\<Sigma>npub_unfold: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and public :: "'a \<Rightarrow> bool"   and n :: "nat" assumes "intruder_model arity Ana" shows "{f. public f} \<inter> {f. arity f = n} = {f. arity f = n \<and> public f}"
new_def: fixes A :: "'c set" shows "new A = (SOME a. a \<notin> A)"
IS_LEFT_UNIQUED: fixes r :: "('a \<times> 'b) set"   and y :: "'a"   and x :: "'b"   and z :: "'a" assumes "IS_LEFT_UNIQUE r"   and "(y, x) \<in> r"   and "(z, x) \<in> r" shows "y = z"
span_def: fixes R :: "('h, 'i) ring_scheme"   and M :: "('h, 'j, 'k) module_scheme"   and S :: "'j set" assumes "module R M" shows "module.span R M S = {module.lincomb M a A |a A. finite A \<and> A \<subseteq> S \<and> a \<in> A \<rightarrow> carrier R}"
l6_3_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Out Bet P A B" shows "A \<noteq> P \<and> B \<noteq> P \<and> (\<exists>C. C \<noteq> P \<and> Bet A P C \<and> Bet B P C)"
fun_complex_eq: fixes f :: "'c \<Rightarrow> complex" shows "f = (\<lambda>x. complex_of_real (Re (f x)) + \<i> * complex_of_real (Im (f x)))"
im_sheaf_def: fixes V :: "'a set set" shows "local.im_sheaf V \<equiv> \<O>\<^sub>X (f \<^sup>\<inverse> X V)"
nth_least_wellformed: fixes n :: "nat"   and A :: "'a set" assumes "enat n < esize A" shows "nth_least A n \<in> A"
iter_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and step :: "nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and \<tau>s :: "'b list"   and w :: "nat set" shows "iter f step \<tau>s w = while (\<lambda>(\<tau>s, w). w \<noteq> {}) (\<lambda>(\<tau>s, w). let p = SOME p. p \<in> w in propa f (step p (\<tau>s ! p)) \<tau>s (w - {p})) (\<tau>s, w)"
disjoint_witness_correct_2: fixes \<alpha>1 :: "'a \<Rightarrow> 'b set"   and invar1 :: "'a \<Rightarrow> bool"   and \<alpha>2 :: "'c \<Rightarrow> 'b set"   and invar2 :: "'c \<Rightarrow> bool"   and disjoint_witness :: "'a \<Rightarrow> 'c \<Rightarrow> 'b option"   and s1 :: "'a"   and s2 :: "'c"   and a :: "'b" assumes "set_disjoint_witness \<alpha>1 invar1 \<alpha>2 invar2 disjoint_witness"   and "invar1 s1"   and "invar2 s2"   and "disjoint_witness s1 s2 = Some a" shows "a \<in> \<alpha>1 s1 \<inter> \<alpha>2 s2"