hcju/isabelleps huggingface dataset:
markedElementsToArePrefixOfMarkedElements: fixes e :: "'b"   and M :: "('b \<times> bool) list" shows "isPrefix (markedElementsTo e M) (markedElements M)"
simps: fixes p :: "nat"   and q :: "nat"   and r :: "nat" shows "trpl p q r = Pi 0 ^ p * Pi 1 ^ q * Pi 2 ^ r"
h0: shows "satisfies_properties_R p (qs1 @ qs2) (get_subsets_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))) (get_signs_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))) (get_matrix_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2))))"
card_before: fixes xs :: "'c list"   and x :: "'c" assumes "distinct xs"   and "x \<in> set xs" shows "card (before x xs) = index xs x"
implies_galois: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> y \<le> z) = (x \<le> - y \<squnion> z)"
Collect_all_eq: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
var_app_types: fixes e :: "nat \<Rightarrow> LambdaType.type"   and i :: "nat"   and ts :: "dB list"   and us :: "dB list"   and T :: "LambdaType.type"   and Ts :: "LambdaType.type list"   and U :: "LambdaType.type" assumes "e \<turnstile> Var i \<degree>\<degree> ts \<degree>\<degree> us : T"   and "e \<tturnstile> ts : Ts"   and "e \<turnstile> Var i \<degree>\<degree> ts : U" shows "\<exists>Us. U = Us \<Rrightarrow> T \<and> e \<tturnstile> us : Us"
A: shows "uniform_limit S (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
mk_poincare_line_cmat_def: fixes A :: "real"   and B :: "complex" shows "mk_poincare_line_cmat A B = (cor A, B, cnj B, cor A)"
F_D_FD: fixes P :: "'b process"   and Q :: "'b process" assumes "P \<sqsubseteq>\<^sub>F Q"   and "P \<sqsubseteq>\<^sub>D Q" shows "P \<sqsubseteq>\<^sub>F\<^sub>D Q"
SubstForm_imp_subst_dbfm_lemma: fixes v :: "hf"   and e :: "(name, hf) finfun"   and u :: "dbtm"   and x :: "hf"   and x' :: "hf" assumes "SubstForm v \<lbrakk>quot_dbtm u\<rbrakk>e x x'" shows "\<exists>A. x = \<lbrakk>quot_dbfm A\<rbrakk>e \<and> x' = \<lbrakk>quot_dbfm (subst_dbfm u (decode_Var v) A)\<rbrakk>e"
measure_pmf_in_subprob_space: fixes M :: "'d pmf" shows "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
simps_5: fixes f1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and f2 :: "'d set \<Rightarrow> 'e \<Rightarrow> 'c"   and x11 :: "'d"   and x12 :: "'e" shows "(case Add x11 x12 of Add x xa \<Rightarrow> f1 x xa | Rem x xa \<Rightarrow> f2 x xa) = f1 x11 x12"
simps_2: fixes a :: "'a"   and w :: "nat \<Rightarrow> 'a"   and i :: "nat" shows "(a ## w) (Suc i) = w i"
program_typedef_5: fixes F :: "'c program" shows "Init F \<equiv> case Rep_Program F of (init, acts, allowed) \<Rightarrow> init"
A_nonzero_fixed_j: fixes j :: "'n" shows "\<exists>i. A $h i $h j \<noteq> 0"
concat_map_all_distinct: fixes f :: "'b \<Rightarrow> 'a list"   and xs :: "'b list"   and x :: "'b" assumes "distinct (concat (map f xs))"   and "x \<in> set xs" shows "distinct (f x)"
worklist_aux_unfold: fixes succs :: "'d \<Rightarrow> 'c \<Rightarrow> 'c list"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and ws :: "'c list"   and s :: "'d" shows "worklist_aux succs f (ws, s) = (case ws of [] \<Rightarrow> Some ([], s) | x # ws' \<Rightarrow> worklist_aux succs f (succs s x @ ws', f x s))"
cases: fixes a1 :: "'b com option"   and a2 :: "'b"   and a3 :: "'b com option"   and a4 :: "'b" assumes "(a1, a2) -e\<rightarrow> (a3, a4)" obtains P :: "'b com option" and s :: "'b" and t :: "'b" where "a1 = P" and "a2 = s" and "a3 = P" and "a4 = t"
size_rbl_succ: fixes bl :: "bool list" shows "length (rbl_succ bl) = length bl"
oracle1: fixes s1 :: "'s1"   and s2 :: "'s2"   and x :: "'a" assumes "X s1 s2"   and "x \<in> outs_\<I> \<I>" shows "oracle1 s1 x = map_spmf fst (joint_oracle s1 s2 x)"
is_empty_conv: fixes ts :: "('c, 'd) trie" shows "is_empty_trie ts = (ts = Trie None [])"
f: shows "finite (insert (u * x) ((*) u ` S))"
AA15: fixes P :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool"   and v :: "'c \<Rightarrow> 'd" shows "\<turnstile> \<diamond>\<langle>P \<and> Q\<rangle>_v \<longrightarrow> \<diamond>\<langle>P\<rangle>_v"
val_map_tree_neq_None: fixes t :: "'b tree"   and f :: "'b \<Rightarrow> 'c" assumes "val t \<noteq> None" shows "val (map_tree f t) \<noteq> None"
assms_1: shows "\<turnstile> A \<longrightarrow> B"
minus_eq_nat_subst_order: fixes B :: "nat"   and C :: "nat"   and D :: "nat"   and A :: "nat" assumes "0 < B - C - D"   and "A = B - C - D + B" shows "A = 2 * B - C - D"
differing_vars_elim: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and x :: "'b"   and mem\<^sub>1 :: "'b \<Rightarrow> 'c"   and mem\<^sub>2 :: "'b \<Rightarrow> 'c"   and mems :: "(('b \<Rightarrow> 'c) \<times> ('b \<Rightarrow> 'c)) list"   and i :: "nat" assumes "sifum_security stop eval some_val some_val'"   and "x \<in> sifum_security.differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i" shows "fst (mems ! i) x \<noteq> mem\<^sub>1 x \<or> snd (mems ! i) x \<noteq> mem\<^sub>2 x"
i_cut_not_in_imp_2: fixes x :: "'a"   and I :: "'a set"   and t :: "'a" assumes "x \<notin> I" shows "x \<notin> I \<down>< t"
MorphismsPreserved: fixes I :: "('a, 'b, 'c, 'd) Interpretation"   and A :: "'a"   and e :: "('a, 'b) Expression"   and B :: "'a"   and i :: "('c, 'd) IType" assumes "Interpretation I"   and "L\<lbrakk>Vx : A \<turnstile> e : B\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> i" shows "\<exists>g. i = IMor g \<and> g maps\<^bsub>iC\<^bsub>I\<^esub>\<^esub> Ty\<lbrakk>A\<rbrakk>\<^bsub>I\<^esub> to Ty\<lbrakk>B\<rbrakk>\<^bsub>I\<^esub>"
atLeastAtMost_singleton_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "({a..b} = {c}) = (a = b \<and> b = c)"
FMIN: fixes y :: "('a, 'b) BsSkewElem" assumes "y \<in># image_mset fst (SkewBinomialHeapStruc.queue_to_multiset (map bsmapt q))" shows "eprio (findMin q) \<le> eprio y"
subst_lt: fixes j :: "nat"   and i :: "nat"   and u :: "dB" assumes "j < i" shows "Var j[u/i] = Var j"
tJ_def: fixes \<psi> :: "'fmla" shows "tJ \<psi> \<equiv> encF (ssap \<psi>) (tenc (encF (ssap \<psi>)))"
fundamental_property: fixes n :: "'node"   and c :: "'com"   and cf :: "'var \<Rightarrow> 'val option"   and c' :: "'com"   and s' :: "('var \<Rightarrow> 'val option) list"   and cfs' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "n \<triangleq> c"   and "\<langle>c,[cf]\<rangle> \<Rightarrow> \<langle>c',s'\<rangle>" shows "\<exists>n' as. n -as\<rightarrow>\<^sub>\<surd>* n' \<and> n' \<triangleq> c' \<and> preds (kinds as) [(cf, undefined)] \<and> transfers (kinds as) [(cf, undefined)] = cfs' \<and> map fst cfs' = s'"
trans_def: fixes r :: "('a \<times> 'a) set" shows "Relation.trans r = (\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r)"
content_eq_fold_coeffs: fixes p :: "'b poly" shows "content p = fold_coeffs gcd p (0::'b)"
replicate_Suc_iff_anywhere: fixes x :: "'a"   and b :: "nat"   and ys :: "'a list" shows "x # x \<up> b @ ys = x \<up> Suc b @ ys"
higher_eq_zero_iff': fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'a" shows "(punit.higher p v = 0) = (punit.lt p \<preceq> v)"
ring_iso_sym: fixes R :: "('e, 'f) ring_scheme"   and S :: "('g, 'h) ring_scheme" assumes "ring R"   and "R \<simeq> S" shows "S \<simeq> R"
red_preserves_hconf: fixes e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "P \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>"   and "P,E,h \<turnstile> e : T"   and "P \<turnstile> h \<surd>" shows "P \<turnstile> h' \<surd>"
union_restrict: fixes I :: "nat set"   and z1 :: "'b option list"   and z2 :: "'b option list"   and J :: "nat set" assumes "restrict I z1 = restrict I z2"   and "restrict J z1 = restrict J z2" shows "restrict (I \<union> J) z1 = restrict (I \<union> J) z2"
fdisj: fixes n :: "nat"   and m :: "nat"   and f :: "form"   and g :: "form"   and xs :: "(nat \<times> form) list"   and nfs :: "(nat \<times> form) list" assumes "(n, (m, FDisj f g) # xs) \<in> deriv nfs"   and "\<not> is_axiom (s_of_ns ((m, FDisj f g) # xs))" shows "(Suc n, xs @ [(0, f), (0, g)]) \<in> deriv nfs"
qGood_qGoodInp: fixes delta :: "'f"   and qinp :: "'d \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and qbinp :: "'e \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qAbs option" shows "qGood (qOp delta qinp qbinp) = (qGoodInp qinp \<and> qGoodBinp qbinp)"
s2: shows "Partial_Cost_Model.config' (rTS h0) ([x, y], h) u = ([x, y], x # y # h)"
divide_eq2: fixes h :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. h x \<noteq> (0::'b)" shows "((\<lambda>x. f x / h x) \<in> o[F](g)) = (f \<in> o[F](\<lambda>x. g x * h x))"
smallo_1_tendsto_0: fixes f :: "'a \<Rightarrow> real" assumes "(f \<longlongrightarrow> 0) at_top" shows "f \<in> o(\<lambda>x. 1)"
upper_triangular_four_block: fixes A :: "'b mat"   and n :: "nat"   and D :: "'b mat"   and m :: "nat"   and B :: "'b mat" assumes "A \<in> carrier_mat n n"   and "D \<in> carrier_mat m m"   and "upper_triangular A"   and "upper_triangular D" shows "upper_triangular (four_block_mat A B (0\<^sub>m m n) D)"
wf_trms_subst_rm_vars': fixes \<delta> :: "'b \<Rightarrow> ('fun, 'b) Term.term"   and X :: "'b set" assumes "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range (rm_vars X \<delta>))"
invimage_vid_on: fixes A :: "V"   and B :: "V" shows "vid_on A -`\<^sub>\<circ> B = A \<inter>\<^sub>\<circ> B"
ChamberComplexEndomorphism_axioms_def: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" shows "ChamberComplexEndomorphism_axioms X f \<equiv> \<forall>v. v \<notin> \<Union> X \<longrightarrow> f v = v"
hyp: shows "\<forall>n. f ^ x $ n mod p = g ^ x $ n mod p"
discount_factor_times_rfr: fixes r :: "real"   and n :: "nat"   and w :: "'b" assumes "- 1 < r" shows "(1 + r) * discount_factor r (Suc n) w = discount_factor r n w"
take_butlast: fixes n :: "nat"   and xs :: "'d list" assumes "n < length xs" shows "take n (butlast xs) = take n xs"
span_base: fixes Sa :: "(('c \<Rightarrow> real) \<Rightarrow> real) set"   and p :: "'c"   and a :: "('c \<Rightarrow> real) \<Rightarrow> real" assumes "Sa \<subseteq> manifold_eucl.dest.tangent_space p"   and "a \<in> Sa" shows "a \<in> span Sa"
nda: fixes a :: "('c, 'd) Combinators"   and p :: "('c, 'd) Combinators list" assumes "singleCombinators (a # p)"   and "noDenyAll p" shows "noDenyAll1 (a # p)"
irreducible\<^sub>dD_1: fixes p :: "'c poly" assumes "irreducible\<^sub>d p" shows "0 < degree p"
sim: shows "\<forall>x\<le>l. \<forall>ts\<^sub>x \<S>\<^sub>x m\<^sub>x ts\<^sub>x' \<S>\<^sub>x' m\<^sub>x'. c (n + x) = (ts\<^sub>x, m\<^sub>x, \<S>\<^sub>x) \<longrightarrow> c' (n + x) = (ts\<^sub>x', m\<^sub>x', \<S>\<^sub>x') \<longrightarrow> (\<forall>j<length ts\<^sub>x. j \<noteq> i \<longrightarrow> ts\<^sub>x' ! j = ts\<^sub>x ! j) \<and> (\<forall>a. a \<notin> u_owns \<longrightarrow> a \<notin> owned (ts ! i) \<longrightarrow> \<S>\<^sub>x' a = \<S>\<^sub>x a) \<and> (\<forall>a. a \<notin> u_owns \<longrightarrow> m\<^sub>x' a = m\<^sub>x a)"
h7a: assumes "Av < 0"   and "0 < Bv"   and "Av < - Bv * sqrt Cv" shows "Bv\<^sup>2 * Cv < Av\<^sup>2"
g_def: shows "g = (\<lambda>x. if x \<in> S then f x else 0)"
cont: shows "continuous_on (S \<union> \<Union> G \<union> (S \<union> UF)) (\<lambda>x. if x \<in> S \<union> \<Union> G then f x else g x)"
elims: fixes x :: "'a decision decision"   and y :: "'a decision" assumes "flat_orD x = y" obtains ya :: "'a" where "x = allow (allow ya__)" and "y = allow ya__"   | ya :: "'a" where "x = allow (deny yaa__)" and "y = deny yaa__"   | ya :: "'a" where "x = deny (allow yab__)" and "y = deny yab__"   | ya :: "'a" where "x = deny (deny ya)" and "y = deny ya"
d_b_c_a: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "d_aux b c a = d_aux a b c"
po_refl: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "porder le" shows "le x x"
assms_4: fixes S :: "ccs"   and T :: "ccs" assumes "(S, T) \<in> Rel" shows "S \<leadsto>\<^sup>^<Rel> T"
trancl_induct: fixes a :: "'e"   and b :: "'e"   and r :: "('e \<times> 'e) set"   and P :: "'e \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>+"   and "\<And>y. (a, y) \<in> r \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>(a, y) \<in> r\<^sup>+; (y, z) \<in> r; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
m'_def: shows "m' = (\<lambda>i. if i < \<phi> 0 then m i else t (\<phi> (i - \<phi> 0)))"
two_cases: fixes P :: "bool"   and Q :: "bool" obtains "P" and "Q"   | "P" and "\<not> Q"   | "\<not> P" and "Q"   | "\<not> P" and "\<not> Q"
n_left_distrib: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "!x \<cdot> (y + z) = !x \<cdot> y + !x \<cdot> z"
set_cases: fixes e :: "'b"   and a :: "'b llist" assumes "e \<in> lset a" obtains z2 :: "'b llist" where "a = LCons e z2__"   | z1 :: "'b" and z2 :: "'b llist" where "a = LCons z1 z2" and "e \<in> lset z2"
listrel_iff_nth: fixes xs :: "'g list"   and ys :: "'h list"   and r :: "('g \<times> 'h) set" shows "((xs, ys) \<in> listrel r) = (length xs = length ys \<and> (\<forall>n<length xs. (xs ! n, ys ! n) \<in> r))"
surjective_st: fixes s :: "st" shows "st (globs s) (locals s) = s"
Sec_def: fixes \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and s :: "Var \<Rightarrow> Val"   and t :: "Var \<Rightarrow> Val" shows "Sec \<Phi> s t = ((\<forall>r.  s \<approx> r  \<longrightarrow> \<Phi> (t, r)) \<and> (\<forall>r. \<Phi> (r, s) \<longrightarrow>  r \<approx> t ))"
Le_enat_def: shows "Le_enat \<equiv> {(xa, x). Le_enatp xa x}"
has_integral_spike_interior: fixes f :: "'c \<Rightarrow> 'd"   and y :: "'d"   and a :: "'c"   and b :: "'c"   and g :: "'c \<Rightarrow> 'd" assumes "(f has_integral y) (cbox a b)"   and "\<And>x. x \<in> box a b \<Longrightarrow> g x = f x" shows "(g has_integral y) (cbox a b)"
impl_safe_def: fixes c :: "('loc, 't) configuration" shows "impl_safe c \<equiv> \<forall>loc1 loc2 t s. 0 < zcount (c_imp c loc1) t \<and> s \<in>\<^sub>A path_summary loc1 loc2 \<longrightarrow> (\<exists>t'. t' \<in>\<^sub>A frontier (c_imp c loc2) \<and> t' \<le> results_in t s)"
1: shows "multiplicative_function (\<lambda>p. h p / f p)"
dprodE: fixes c :: "('a, 'b) node set \<times> ('a, 'b) node set"   and r :: "(('a, 'b) node set \<times> ('a, 'b) node set) set"   and s :: "(('a, 'b) node set \<times> ('a, 'b) node set) set" assumes "c \<in> dprod r s" obtains x :: "('a, 'b) node set" and y :: "('a, 'b) node set" and x' :: "('a, 'b) node set" and y' :: "('a, 'b) node set" where "(x, x') \<in> r" and "(y, y') \<in> s" and "c = (Scons x y, Scons x' y')"
infsetsum'_converges_add: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "infsetsum'_converges f A"   and "infsetsum'_converges g A" shows "infsetsum'_converges (\<lambda>x. f x + g x) A"
is_prefix_append: fixes a :: "'b list"   and b :: "'b list"   and D :: "'b list" shows "is_prefix (a @ b) D = (is_prefix a D \<and> is_prefix b (drop (length a) D))"
arg_max_natI: fixes P :: "'a \<Rightarrow> bool"   and k :: "'a"   and f :: "'a \<Rightarrow> nat"   and b :: "nat" assumes "P k"   and "\<forall>y. P y \<longrightarrow> f y < b" shows "P (arg_max f P)"
r_dovedelay: fixes z :: "nat"   and i :: "nat"   and y :: "nat" shows "eval r_dovedelay [z, i, y] \<down>= (if 0 < the (eval r_result1 [pdec2 z, i, pdec1 z]) \<and> y \<le> z then 0 else 1)"
add0_def: fixes k :: "int"   and y :: "'b"   and t :: "'b tree23" shows "add0 k y t = (case add k y t of Stay t' \<Rightarrow> t' | Sprout l p r \<Rightarrow> Branch2 l p r)"
eval_Mn_convergE_2: fixes n :: "nat"   and f :: "recf"   and xs :: "nat list"   and z :: "nat" assumes "recfn (length xs) (Mn n f)"   and "eval (Mn n f) xs \<down>= z" shows "eval f (z # xs) \<down>= 0"
preserves_trg: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and src\<^sub>C :: "'d \<Rightarrow> 'd"   and trg\<^sub>C :: "'d \<Rightarrow> 'd"   and D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and src\<^sub>D :: "'e \<Rightarrow> 'e"   and trg\<^sub>D :: "'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and \<mu> :: "'d" assumes "weak_arrow_of_homs C src\<^sub>C trg\<^sub>C D src\<^sub>D trg\<^sub>D F"   and "partial_magma.arr C \<mu>" shows "trg\<^sub>D (F \<mu>) = weak_arrow_of_homs.map\<^sub>0 src\<^sub>D F (trg\<^sub>C \<mu>)"
lr_of_tran_s3_Cons: fixes ifs :: "char list list"   and a :: "'b \<times> 32 simple_match \<times> char list \<times> simple_action"   and ard :: "('b \<times> 32 simple_match \<times> char list \<times> simple_action) list" shows "lr_of_tran_s3 ifs (a # ard) = concat (map (\<lambda>(p, r, c, a). map (\<lambda>b. (p, b, case a of Accept \<Rightarrow> [Forward c] | Drop \<Rightarrow> [])) (simple_match_to_of_match r ifs)) [a]) @ lr_of_tran_s3 ifs ard"
rb_aux_inv_D4_1: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list"   and j :: "nat" assumes "rb_aux_inv (bs, ss, ps)"   and "Inr j \<in> set ps" shows "j < length fs"
get_ancestors_di_def: fixes ptr :: "(_) object_ptr" shows "get_ancestors_di ptr = Heap_Error_Monad.bind (check_in_heap ptr) (\<lambda>_. Heap_Error_Monad.bind (case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of None \<Rightarrow> case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^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 ptr of None \<Rightarrow> return [] | Some shadow_root_ptr \<Rightarrow> Heap_Error_Monad.bind (get_host shadow_root_ptr) (\<lambda>host. get_ancestors_di (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host)) | Some node_ptr \<Rightarrow> Heap_Error_Monad.bind (get_parent node_ptr) (\<lambda>parent_ptr_opt. case parent_ptr_opt of None \<Rightarrow> Heap_Error_Monad.bind (get_disconnected_document node_ptr) (\<lambda>document_ptr. get_ancestors_di (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr)) | Some parent_ptr \<Rightarrow> get_ancestors_di parent_ptr)) (\<lambda>ancestors. return (ptr # ancestors)))"
ceiling_less_one: fixes x :: "'b" shows "(\<lceil>x\<rceil> < 1) = (x \<le> (0::'b))"
pderiv_mult: fixes p :: "'b poly"   and q :: "'b poly" shows "pderiv (p * q) = p * pderiv q + q * pderiv p"
Nonce_Aenc: fixes X :: "nonce_t"   and X' :: "msg"   and Y' :: "msg" shows "Nonce X \<noteq> Aenc X' Y'"
assms: shows "P,h \<turnstile> t \<surd>t"
elims_3: fixes \<Delta>1 :: "nat"   and \<Delta>2 :: "nat"   and \<Gamma>1 :: "nat"   and \<Gamma>2 :: "nat"   and x :: "('b \<times> nat) tree" assumes "WBT \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2"   and "\<not> WBT.wbt \<Delta>1 \<Delta>2 x" obtains l :: "('b \<times> nat) tree" and uu_ :: "'b" and n :: "nat" and r :: "('b \<times> nat) tree" where "x = \<langle>l, (uu_, n), r\<rangle>" and "\<not> (n = size l + size r + 1 \<and> WBT.balanced1 \<Delta>1 \<Delta>2 l r \<and> WBT.balanced1 \<Delta>1 \<Delta>2 r l \<and> WBT.wbt \<Delta>1 \<Delta>2 l \<and> WBT.wbt \<Delta>1 \<Delta>2 r)"
infsetsum_cong: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and B :: "'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x"   and "A = B" shows "infsetsum f A = infsetsum g B"
prod_set_simps_2: fixes x :: "'c"   and y :: "'b" shows "Basic_BNFs.snds (x, y) = {y}"
IH: fixes s :: "'s" shows "T s = T s \<bind> (\<lambda>\<omega>. distr (T ((s ## \<omega>) !! n)) (stream_space (count_space UNIV)) ((@-) (stake n \<omega>)))"
simps_2: fixes f :: "'d \<Rightarrow> 'd" shows "Language.normalize (Basic f) = Basic f"
weight_gpv_def: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv" shows "weight_gpv \<I> gpv = enn2real (expectation_gpv 0 \<I> (\<lambda>_. 1) gpv)"
assoc_closed': fixes \<mu>\<nu>\<tau> :: "'a \<times> 'a \<times> 'a" assumes "VVV.arr \<mu>\<nu>\<tau>" shows "Arr (\<a> (fst \<mu>\<nu>\<tau>) (fst (snd \<mu>\<nu>\<tau>)) (snd (snd \<mu>\<nu>\<tau>)))"
allAllocationsEquivalence: fixes \<Omega> :: "'c set"   and N :: "'b set" shows "allAllocations N \<Omega> = allAllocations' N \<Omega> \<and> allAllocations N \<Omega> = allAllocations'' N \<Omega>"
LAssRedsVal: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and v :: "val"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and V :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>Val v,(h', l', sh'),b'\<rangle>" shows "P \<turnstile> \<langle>V:=e,s,b\<rangle> \<rightarrow>* \<langle>unit,(h', l'(V \<mapsto> v), sh'),b'\<rangle>"
extra_wt_def: fixes \<rho> :: "'v \<Rightarrow> ('s, 'v) tm"   and s :: "('s, 'v) tm" shows "extra_wt \<rho> s = (\<Sum>x\<in>#vars_mset s. wt (\<rho> x) - wt (Hd (Var x)))"
open_segment_commute: fixes a :: "'c"   and b :: "'c" shows "open_segment a b = open_segment b a"
append_one_prefix: fixes xs :: "'e list"   and ys :: "'e list" assumes "prefix xs ys"   and "length xs < length ys" shows "prefix (xs @ [ys ! length xs]) ys"
mdc_leq_dyn_type: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr"   and T :: "ty" assumes "P,E,h \<turnstile> e : T" shows "\<forall>C a Cs D S. T = Class C \<and> e = ref (a, Cs) \<and> h a = \<lfloor>(D, S)\<rfloor> \<longrightarrow> P \<turnstile> D \<preceq>\<^sup>* C"
prime_factorization_gcd: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.factorial_semiring_gcd divide plus minus zero times one unit_factor normalize gcd lcm Gcd Lcm"   and "a \<noteq> zero"   and "b \<noteq> zero" shows "factorial_semiring.prime_factorization zero times one normalize (gcd a b) = factorial_semiring.prime_factorization zero times one normalize a \<inter># factorial_semiring.prime_factorization zero times one normalize b"
def_ass_uses: shows "\<forall>m\<in>set (gen_wf_\<alpha>n gen_cfg_wf). \<forall>v\<in>gen_wf_uses gen_cfg_wf m. gen_wf.defAss' gen_cfg_wf m v"