hcju/isabelleps huggingface dataset:
cot_altdef: fixes x :: "'b" shows "cot x = inverse (tan x)"
boundD_carrier: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and m :: "nat" assumes "bound \<zero> n f"   and "n < m" shows "f m \<in> carrier R"
full_abstraction_impl_trans_relation_contains_SRel_iff_TRel: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set"   and Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set" assumes "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)"   and "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel"   and "trans (Rel \<union> _Collect (P, Q) (\<exists>S. \<lbrakk>S\<rbrakk> \<in>T P \<and> S \<in>S Q))" shows "(\<forall>S1 S2. ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel) = ((TargetTerm (\<lbrakk>S1\<rbrakk>), TargetTerm (\<lbrakk>S2\<rbrakk>)) \<in> Rel)) = (SRel = _Collect (S1, S2) ((SourceTerm S1, SourceTerm S2) \<in> Rel))"
inorder_upto_split_4: fixes l :: "int tree"   and k :: "int"   and r :: "int tree"   and i :: "int"   and j :: "int" assumes "inorder \<langle>l, k, r\<rangle> = [i..j]" shows "k \<le> j"
frag_cmul_zero: fixes x :: "'b \<Rightarrow>\<^sub>0 int" shows "frag_cmul 0 x = 0"
mult_right_dist_sup: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.non_associative_left_semiring one times sup less_eq less bot" shows "times (sup x y) z = sup (times x z) (times y z)"
gFreshGVar_def: fixes MOD :: "('c, 'd, 'varSort, 'e, 'f, 'var, 'g, 'h, 'i) Recursion.model_scheme" shows "gFreshGVar MOD \<equiv> \<forall>ys y xs x. ys \<noteq> xs \<or> y \<noteq> x \<longrightarrow> gFresh MOD ys y (Var xs x) (gVar MOD xs x)"
finite_part: fixes I :: "'a set"   and P :: "'a set set" assumes "finite I"   and "part I P" shows "finite P"
product_nn_integral_insert_rev: fixes I :: "nat set"   and i :: "nat"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal" assumes "finite I"   and "i \<notin> I"   and "f \<in> borel_measurable (Pi\<^sub>M (insert i I) (\<lambda>y. stock_measure (\<Gamma> y)))" shows "integral\<^sup>N (Pi\<^sub>M (insert i I) (\<lambda>y. stock_measure (\<Gamma> y))) f = \<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x(i := y)) \<partial>Pi\<^sub>M I (\<lambda>y. stock_measure (\<Gamma> y)) \<partial>stock_measure (\<Gamma> i)"
LT1_def: fixes ars :: "('a \<times> 'b \<times> 'a) set"   and r :: "('b \<times> 'b) set"   and \<beta> :: "'a \<times> ('b \<times> 'a) list"   and \<alpha> :: "'a \<times> ('b \<times> 'a) list"   and \<sigma>1 :: "'a \<times> (bool \<times> 'b \<times> 'a) list"   and \<sigma>2 :: "'a \<times> ('b \<times> 'a) list"   and \<sigma>3 :: "'a \<times> (bool \<times> 'b \<times> 'a) list" shows "LT1 ars r \<beta> \<alpha> \<sigma>1 \<sigma>2 \<sigma>3 = (local_triangle1 ars \<beta> \<alpha> \<sigma>1 \<sigma>2 \<sigma>3 \<and> LD_1' r (hd (labels \<beta>)) (hd (labels \<alpha>)) (labels_conv \<sigma>1) (labels \<sigma>2) (labels_conv \<sigma>3))"
assms_4: shows "e \<le> v * - v\<^sup>T \<sqinter> - - g"
IH: assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> t" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>process_ret rs, Undecided\<rangle> \<Rightarrow> t"
to_set_correct_1: fixes l :: "'b list" shows "rs.\<alpha> (rs.from_list l) = set l"
noStucknI: fixes \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'e) com option"   and c :: "('d, 'f, 'e) com"   and s :: "('d, 'e) xstate"   and n :: "nat" assumes "\<And>t. \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t \<Longrightarrow> t \<noteq> Stuck" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow>\<notin>{Stuck}"
Or_def: fixes P :: "fml"   and Q :: "fml" shows "(P || Q) = ! (! P && ! Q)"
le_top: fixes x :: "'b" shows "x \<le> \<top> \<cdot> x"
iTILL_iMOD_subset_conv: fixes n' :: "nat"   and r :: "nat"   and m :: "nat" assumes "0 < n'" shows "([\<dots>n'] \<subseteq> [ r, mod m ]) = (r = 0 \<and> m = Suc 0)"
sorted_wrt_quicksort: fixes R :: "('a \<times> 'a) set"   and xs :: "'a list" assumes "trans R"   and "total_on (set xs) R"   and "\<And>x. x \<in> set xs \<Longrightarrow> (x, x) \<in> R" shows "Linorder_Relations.sorted_wrt R (quicksort R xs)"
intro: fixes bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h :: "'a \<Rightarrow> 'b" assumes "bo \<le> vimage2p h h (=)" shows "blinding_respects_hashes h bo"
Lam_def: fixes x1 :: "name"   and x2 :: "trm" shows "Lam [x1].x2 \<equiv> Abs_trm (Lam_Rep ([x1].Rep_trm x2))"
binop_dfa_def: fixes f :: "bool \<Rightarrow> bool \<Rightarrow> bool"   and A :: "nat bdd list \<times> bool list"   and B :: "nat bdd list \<times> bool list" shows "binop_dfa f A B = (let (tab, ps) = prod_dfs A B (0, 0) in (map (\<lambda>(i, j). bdd_binop (\<lambda>k l. the (tab ! k ! l)) (fst A ! i) (fst B ! j)) ps, map (\<lambda>(i, j). f (snd A ! i) (snd B ! j)) ps))"
pow_eq_id: fixes x :: "nat"   and n :: "nat" assumes "x \<in> carrier G" shows "(x [^]\<^bsub>G\<^esub> n = \<one>\<^bsub>G\<^esub>) = (local.ord x dvd n)"
lesssub_def: fixes x :: "'a"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a" shows "(x \<sqsubset>\<^bsub>r\<^esub> y) = (x \<sqsubseteq>\<^bsub>r\<^esub> y \<and> x \<noteq> y)"
disjunctive_def: shows "disjunctive = {x. \<forall>y z. times_abc x (sup_b y z) = sup_c (times_abc x y) (times_abc x z)}"
finite_measure_subadditive: fixes A :: "real set"   and B :: "real set" assumes "A \<in> sets M"   and "B \<in> sets M" shows "Sigma_Algebra.measure M (A \<union> B) \<le> Sigma_Algebra.measure M A + Sigma_Algebra.measure M B"
assms_2: fixes t :: "'a set" assumes "t \<in> T" shows "t \<subseteq> S"
sup_mono: fixes sup :: "'x \<Rightarrow> 'x \<Rightarrow> 'x"   and less_eq :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and less :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and a :: "'x"   and c :: "'x"   and b :: "'x"   and d :: "'x" assumes "class.semilattice_sup sup less_eq less"   and "less_eq a c"   and "less_eq b d" shows "less_eq (sup a b) (sup c d)"
uminus_assertion: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<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 dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "mbt_algebra.neg_assert one times inf dual bot x \<in> mbt_algebra.assertion one times inf less_eq dual top"
vsv_vrange_vsubset: fixes A :: "V" assumes "\<And>x. x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vreal_uminus \<Longrightarrow> -\<^sub>\<real> x \<in>\<^sub>\<circ> A" shows "\<R>\<^sub>\<circ> vreal_uminus \<subseteq>\<^sub>\<circ> A"
that: assumes "a \<in>\<^sub>\<circ> A"   and "b \<in>\<^sub>\<circ> B" shows "thesis"
simps_2: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'a"   and x1 :: "'c"   and x2 :: "'d" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
fe_inv_def: shows "fe_inv = (\<lambda>S V0 u0 it (V, cyc). gen_dfs_fe_inv E U S V0 u0 it V (cyc \<noteq> None) \<and> (case cyc of None \<Rightarrow> E `` V \<inter> onstack = {} | Some (p, v) \<Rightarrow> v \<in> onstack \<and> p \<noteq> [] \<and> path E u0 p v))"
mult2_s_def: fixes ns :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" shows "mult2_s ns s \<equiv> multpw ns O mult s"
rotate_min_def: fixes vs :: "nat list" shows "rotate_min vs \<equiv> rotate_to vs (ListAux.min_list vs)"
group_closure_add: fixes s :: "'a"   and S :: "'a set"   and t :: "'a" assumes "s \<in> group_closure S"   and "t \<in> group_closure S" shows "s + t \<in> group_closure S"
CR_imp_UNF: fixes r :: "('b \<times> 'b) set" assumes "CR r" shows "UNF r"
some_child_sub_1: fixes sub :: "'b"   and sep :: "'c"   and t :: "('b \<times> 'c) list" assumes "(sub, sep) \<in> set t" shows "sub \<in> set (subtrees t)"
assms_1: fixes u :: "nat \<Rightarrow> 'a"   and b :: "'a" assumes "\<forall>n. u n \<in> S" shows "u \<longlonglongrightarrow> b = (\<lambda>n. f (u n)) \<longlonglongrightarrow> f b"
comp_cod_arr: fixes f :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "CC.arr f"   and "CC.cod f = b" shows "CC.comp b f = f"
iff_simps_4: fixes p :: "fm" shows "Ferrack.iff p F = not p"
spec_def: fixes s :: "('c, 'd) configuration stream" shows "spec s = (holds init_config s \<and> alw next s)"
Diagonal_to_Smith_row_i_PQ_def: fixes i :: "nat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and P :: "(('b, 'c) vec, 'c) vec"   and A :: "(('b, 'd) vec, 'c) vec"   and Q :: "(('b, 'd) vec, 'd) vec" shows "Diagonal_to_Smith_row_i_PQ i bezout (P, A, Q) = diagonal_to_Smith_i_PQ [i + 1..<min (nrows A) (ncols A)] i bezout (P, A, Q)"
"2_2": assumes "0 < zcount (c_imp c loc1) t" shows "0 < zcount M (results_in t (sum_weights (map (\<lambda>a. case a of (s, l, t) \<Rightarrow> l) xs)))"
advantage_def: fixes \<A> :: "(bool, '\<beta>, '\<gamma>) gpv" shows "prf'.advantage \<A> = \<bar>spmf (prf'.game_0 \<A>) True - spmf (prf'.game_1 \<A>) True\<bar>"
iT_Mult_iAll_conv: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<box> t I \<otimes> k. P t) = (\<box> t I. P (t * k))"
machin_Euler: shows "5 * arctan (1 / 7) + 2 * arctan (3 / 79) = pi / 4"
finlsts_rev_cases: fixes t :: "'b llist"   and A :: "'b set" assumes "t \<in> A\<^sup>\<star>" obtains "t = LNil"   | a :: "'b" and l :: "'b llist" where "l \<in> A\<^sup>\<star>" and "a \<in> A" and "t = l @@ a ## LNil"
lang_deriv: fixes c :: "'a"   and r :: "'a rexp" shows "lang (deriv c r) = Deriv c (lang r)"
pow_mat_dim_2: fixes A :: "'a mat"   and k :: "nat" shows "dim_col (A ^\<^sub>m k) = (if k = 0 then dim_row A else dim_col A)"
assms: shows "{x \<in> space (stream_space M). P x} \<in> S.events"
set_of_insert: fixes t :: "'b tree"   and x :: "'b" assumes "avl t" shows "set_of (AVL.insert x t) = Set.insert x (set_of t)"
carrier_submanifold: shows "a.sub.carrier = domain c \<inter> carrier"
simple_ruleset_def: fixes rs :: "'a rule list" shows "simple_ruleset rs \<equiv> \<forall>r\<in>set rs. get_action r = Accept \<or> get_action r = Drop"
step': shows "rev.flow0 (flow0 x t2) (t2 - t) \<notin> {a<--<rev.flow0 (flow0 x t2) (t2 - t1)}"
evnodd_subset: fixes A :: "(nat \<times> nat) set"   and b :: "nat" shows "evnodd A b \<subseteq> A"
leg1_uniquely_isomorphic_2: shows "\<exists>!\<nu>. \<guillemotleft>\<nu> : local.dom \<omega> \<Rightarrow> s\<^sub>1 \<star> w\<guillemotright>"
set_sort_oalist: fixes xs :: "('a \<times> 'c) list" assumes "distinct (map fst xs)" shows "set (cmp2.sort_oalist xs) = {kv \<in> set xs. snd kv \<noteq> (0::'c)}"
pred_map: fixes Q :: "'c \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'c"   and x :: "('d, 'e) fmap" shows "pred_fmap Q (fmmap f x) = pred_fmap (Q \<circ> f) x"
perzeta_conv_hurwitz_zeta_multiplication': assumes "0 < 2"   and "s \<noteq> 1" shows "perzeta (real_of_int 1 / real 2) s = of_nat 2 powr - s * (\<Sum>n = 1..2. exp (complex_of_real (2 * pi * real n * real_of_int 1 / real 2) * \<i>) * hurwitz_zeta (real n / real 2) s)"
hyps_10: assumes "list_all is_Update (unlabel A)" shows "list_all is_Update (unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A))"
sorted_wrt_dist_nbors: fixes p :: "(real, 'b) vec"   and ps :: "(real, 'b) vec list"   and n :: "nat"   and q :: "(real, 'b) vec" assumes "sorted_wrt_dist p ps" shows "sorted_wrt_dist p (upd_nbors n p q ps)"
totient_gt_0_iff: fixes n :: "nat" shows "(0 < totient n) = (0 < n)"
poly_shift_add: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P"   and "g \<in> carrier P" shows "poly_shift (f \<oplus>\<^bsub>P\<^esub> g) = poly_shift f \<oplus>\<^bsub>P\<^esub> poly_shift g"
1: shows "S.arr m \<and> S.set (S.dom m) \<subseteq> S.set (S.cod m) \<and> m = S.mkArr (S.set (S.dom m)) (S.set (S.cod m)) (\<lambda>x. x)"
2: shows "{(x, {(x, Y) |Y x. Y \<in> finestpart (a ,, x) \<and> x \<in> Domain a} ,,, x) |x. x \<in> Domain {(x, Y) |Y x. Y \<in> finestpart (a ,, x) \<and> x \<in> Domain a}} \<subseteq> a"
blocking_on_Field: fixes ds :: "'d set"   and X :: "'x set"   and h :: "'h"   and X'' :: "'x set"   and d :: "'d" assumes "blocking_on ds X h X''" shows "dX X'' d \<subseteq> Field (Pd d)"
borel_measurable_ln: fixes f :: "'c \<Rightarrow> real"   and M :: "'c measure" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. ln (f x)) \<in> borel_measurable M"
assms: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "Ex (extreme_bound A r {x, y})"
simps_3: fixes t :: "'c local.term"   and u :: "'c local.term" shows "Ide (t \<^bold>\<star> u) = (Ide t \<and> Ide u \<and> Src t = Trg u)"
cols_length: fixes A :: "'b mat" shows "length (cols A) = dim_col A"
simps_2: fixes c1 :: "'b com"   and c2 :: "'b com"   and s :: "'b"   and s' :: "'b" shows "Sem (c1; c2) s s' = (\<exists>s''. Sem c1 s s'' \<and> Sem c2 s'' s')"
assms: shows "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>L<SRel,TRel> Q"
cyc_free_subs_diag: fixes n :: "nat"   and I :: "nat set"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and i :: "nat" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> I \<longrightarrow> (0::'a) \<le> len m i i xs"   and "i \<le> n" shows "(0::'a) \<le> m i i"
span: shows "LinearCombinations.module.span K (vs (local.module.span S)) U = local.module.span S"
rA'2: shows "local.sub3 d d (1::'a, A) = (r', A')"
zero_less_mult_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less zero (times a b) = (less zero a \<and> less zero b \<or> less a zero \<and> less b zero)"
that_3: assumes "h1 \<in> \<Theta>[F](h2)" shows "thesis"
assms_2: shows "\<forall>\<^sub>F x in F. g x \<noteq> (0::'b)"
domo_bfilter: fixes num' :: "int \<Rightarrow> 'b"   and plus' :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and test_num' :: "int \<Rightarrow> 'b \<Rightarrow> bool"   and filter_plus' :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b"   and filter_less' :: "bool \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b"   and \<gamma> :: "'b \<Rightarrow> int set"   and b :: "bexp"   and X :: "char list set"   and S :: "'b Abs_State.st option"   and bv :: "bool" assumes "Abs_Int1 num' plus' test_num' filter_plus' filter_less' \<gamma>"   and "vars b \<subseteq> X"   and "domo S \<subseteq> X" shows "domo (Abs_Int1.bfilter num' plus' test_num' filter_plus' filter_less' b bv S) \<subseteq> X"
lub_prod: fixes S :: "nat \<Rightarrow> 'a \<times> 'b" assumes "chain S" shows "(\<Squnion>i. S i) = (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
assms_1: shows "\<forall>x\<in>set xs. P x"
dist_perm_eta: fixes n :: "nat"   and vs :: "nat list" assumes "perm_vars n vs" obtains vs' :: "nat list" where "\<And>t. \<forall>i\<in>frees t. n \<le> i \<Longrightarrow> (Abs ^^ n) (var_dist vs' ((Abs ^^ n) (var_dist vs (liftn n t 0)))) \<leftrightarrow> strip_context n t 0"
splitAt_no_ram: fixes ram :: "'c"   and vs :: "'c list" assumes "ram \<notin> set vs" shows "splitAt ram vs = (vs, [])"
right_minus: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.group_add minus plus zero uminus" shows "plus a (uminus a) = zero"
normalize_poly_spec_def: fixes p :: "int mpoly" shows "normalize_poly_spec p = SPEC (\<lambda>r. p - r \<in> More_Modules.ideal polynomial_bool \<and> vars r \<subseteq> vars p)"
RS2_NMT: fixes p :: "('a, 'b) Combinators list" assumes "p \<noteq> []" shows "removeShadowRules2 p \<noteq> []"
trpl_lemma: fixes p :: "nat"   and q :: "nat"   and r :: "nat" shows "rec_exec rec_trpl [p, q, r] = trpl p q r"
Cond: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and F :: "'g set"   and P :: "'f set"   and b :: "'f set"   and c\<^sub>1 :: "('f, 'e, 'g) com"   and Q :: "'f set"   and A :: "'f set"   and c\<^sub>2 :: "('f, 'e, 'g) com" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P \<inter> b) c\<^sub>1 Q,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P \<inter> - b) c\<^sub>2 Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Cond b c\<^sub>1 c\<^sub>2 Q,A"
get_dom_component_ptr: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   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_dom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list" assumes "l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c" shows "ptr \<in> set c"
nn_integral_divide: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and c :: "ennreal" assumes "f \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. f x / c \<partial>M = integral\<^sup>N M f / c"
lift_f_trivial_lconjby_P': fixes p :: "'a freeword"   and w :: "'a freeword" assumes "p \<in> P'" shows "lift_f (lconjby w p) = (0::'b)"
less_trans: fixes x :: "('a, 'b) Term.term"   and y :: "('a, 'b) Term.term"   and z :: "('a, 'b) Term.term" assumes "x \<sqsubset> y"   and "y \<sqsubset> z" shows "x \<sqsubset> z"
neg_numeral_less_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (uminus (numeral.numeral one plus n)) zero"
auxlist'_def: shows "auxlist' \<equiv> update_until args rel1 rel2 (\<tau> \<sigma> (ne + length auxlist)) auxlist"
fresh_4: fixes a :: "name"   and x1 :: "('d, 'e, 'f) psiCase" shows "a \<sharp> Case x1 = a \<sharp> x1"
f1: shows "(r'' |\<^bsub>rel\<^esub> s'') \<otimes>\<^bsub>rec_rng_of_frac\<^esub> ((r |\<^bsub>rel\<^esub> s) \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r' |\<^bsub>rel\<^esub> s')) = (r'' \<otimes> (s' \<otimes> r \<oplus> s \<otimes> r') |\<^bsub>rel\<^esub> s'' \<otimes> (s \<otimes> s'))"
is_lub_subD: fixes wf_md :: "'b prog \<Rightarrow> String.literal \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and T :: "ty"   and U :: "ty"   and V :: "ty" assumes "wf_prog wf_md P"   and "is_type P T"   and "is_type P U"   and "P \<turnstile> lub(T, U) = V" shows "SemiType.sup P T U = OK V"
simps_3: fixes x2 :: "final_decision" shows "Decision x2 \<noteq> Undecided"
prj_simps_12: shows "C.seq \<pi>\<rho>.prj\<^sub>0 \<p>\<^sub>0[\<nu>.leg0, \<pi>.leg1 \<cdot> \<pi>\<rho>.prj\<^sub>1]"
Chernoff_a: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "f_range_on A f" shows "Chernoff_on A f = (\<forall>B C. B \<subseteq> A \<and> C \<subseteq> A \<longrightarrow> f (B \<union> C) \<subseteq> f B \<union> C)"
S: fixes x :: "('f, 'v) Term.term" shows "(Fun f ss, x) \<notin> S"
integral: fixes a :: "'a"   and b :: "'a" assumes "a * b = (0::'a)"   and "a \<in> UNIV"   and "b \<in> UNIV" shows "a = (0::'a) \<or> b = (0::'a)"
wellformed_F: fixes q :: "'b"   and \<nu> :: "'a" assumes "q \<in> F" shows "\<delta> q \<nu> \<in> F"
drop_Suc_Cons: fixes n :: "nat"   and x :: "'c"   and xs :: "'c list" shows "drop (Suc n) (x # xs) = drop n xs"
HInter_hinsert: fixes A :: "hf"   and a :: "hf" assumes "A \<noteq> 0" shows "\<Sqinter>(A \<triangleleft> a) = a \<sqinter> \<Sqinter>A"
der: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b" shows "\<exists>y. (f has_real_derivative y) (at x) \<and> 0 < y"
induced_arrowI_2: fixes \<chi>' :: "'b \<Rightarrow> 'a"   and a' :: "'a" assumes "\<chi>' \<in> D0.cones a'" shows "D0.cones_map (\<pi>0.induced_arrow a' \<chi>') \<pi>0 = \<chi>'"
hyps: fixes G :: "('a, 'b) Graph_scheme" assumes "\<And>S. Separation G v0 v1 S \<Longrightarrow> n \<le> card S"   and "v0_v1_Digraph G v0 v1" shows "\<exists>paths. DisjointPaths G v0 v1 paths \<and> card paths = n"
distinct_rem_mid: fixes xs :: "'a list"   and x :: "'a"   and ys :: "'a list" assumes "distinct (xs @ [x] @ ys)" shows "distinct (xs @ ys)"
subgroup_self: shows "subgroup (carrier (DirProds G I)) (DirProds G I)"
commutators_def: fixes G :: "('e, 'f) Group_scheme" shows "commutators G = {z. \<exists>a\<in>carrier G. \<exists>b\<in>carrier G. a \<cdot>\<^bsub>G\<^esub> b \<cdot>\<^bsub>G\<^esub> \<rho>\<^bsub>G\<^esub> a \<cdot>\<^bsub>G\<^esub> \<rho>\<^bsub>G\<^esub> b = z}"
has: shows "P \<turnstile> last Cs has M = (Ts', T', pns', body') via Ds"
small_powers_of_2: assumes "3 \<le> LENGTH('a)" shows "LENGTH('a) < 2 ^ (LENGTH('a) - 1)"
embedding_map_def: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" shows "embedding_map X Y f \<equiv> homeomorphic_map X (subtopology Y (f ` topspace X)) f"