hcju/isabelleps huggingface dataset:
of_rat_def: shows "field_char_0.of_rat (/) (1::'a) (+) (0::'a) (-) \<equiv> map_fun Rep_Rat id (\<lambda>x. of_int (fst x) / of_int (snd x))"
stable_def: fixes A :: "'b set" shows "stable A \<equiv> A co A"
assms: shows "\<rho> f|` fv e \<sqsubseteq> \<rho>' f|` fv e"
gsplit: shows "infsetsum g A = Complex (\<Sum>\<^sub>ax\<in>A. Re (g x)) (\<Sum>\<^sub>ax\<in>A. Im (g x))"
snd_mismatch_def: shows "i.c\<^sub>1 = (y \<cdot> x) ! \<^bold>|i.\<alpha>\<^bold>|"
\<phi>_in_hom_2: shows "\<guillemotleft>g\<^sub>0h\<^sub>1.\<phi> : tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>"
sep_conj_sep_emptyE: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and s :: "'a"   and Q :: "'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj"   and "P s"   and "sep_algebra.sep_conj plus sep_disj P (sep_algebra.sep_empty zero) s \<Longrightarrow> sep_algebra.sep_conj plus sep_disj Q R s" shows "sep_algebra.sep_conj plus sep_disj Q R s"
preorder_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "class.preorder less_eq less \<equiv> (\<forall>x y. less x y = strict less_eq x y) \<and> (\<forall>x. less_eq x x) \<and> (\<forall>x y z. less_eq x y \<longrightarrow> less_eq y z \<longrightarrow> less_eq x z)"
induct: fixes x1 :: "char list list"   and x2 :: "char list list"   and 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 C :: "char list"   and Pa :: "char list list \<Rightarrow> char list list \<Rightarrow> bool" assumes "P,C \<turnstile> x1 \<sqsubset>\<^sup>1 x2"   and "\<And>Cs Ds. \<lbrakk>Subobjs P C Cs; Subobjs P C Ds; Cs = butlast Ds\<rbrakk> \<Longrightarrow> Pa Cs Ds"   and "\<And>Cs D. \<lbrakk>Subobjs P C Cs; P \<turnstile> last Cs \<prec>\<^sub>S D\<rbrakk> \<Longrightarrow> Pa Cs [D]" shows "Pa x1 x2"
c: assumes "x \<noteq> y"   and "\<not> rel x y" shows "filter P (merge_wrt rel (x # xs) ys) = filter P (x # xs)"
wellformed_items_natUnion: fixes I :: "nat \<Rightarrow> item set" assumes "\<And>k. wellformed_items (I k)" shows "wellformed_items (natUnion I)"
add_left_comm: fixes y :: "'b \<times> 'c"   and x :: "'b \<times> 'c"   and z :: "'b \<times> 'c" shows "sup_prod y (sup_prod x z) = sup_prod x (sup_prod y z)"
homogeneous_def: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "homogeneous p = (\<forall>s\<in>keys p. \<forall>t\<in>keys p. deg_pm s = deg_pm t)"
open_halfspace_component_gt: fixes a :: "real"   and i :: "'b" shows "open {x. a < x \<bullet> i}"
set_iterator_genord_I: fixes l0 :: "'c list"   and S :: "'c set"   and R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and iti :: "('d \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd" assumes "distinct l0"   and "S = set l0"   and "sorted_wrt R l0"   and "iti = foldli l0" shows "set_iterator_genord iti S R"
sub_inserted1: fixes Y :: "'h set"   and a :: "'h"   and X :: "'h set" assumes "Y \<subseteq> insert a X"   and "\<not> Y \<subseteq> X" shows "a \<notin> X \<and> a \<in> Y"
bd_F_Card_order: fixes r1 :: "(bd_type_F1 \<times> bd_type_F1) set"   and r2 :: "(bd_type_F2 \<times> bd_type_F2) set" shows "Card_order (dir_image (r1 +c r2) Abs_bd_type_F)"
face_cycle_succ_closed: fixes a :: "'b" shows "(G.face_cycle_succ a \<in> arcs G) = (a \<in> arcs G)"
le_type_scheme_free_tv: fixes sch :: "type_scheme"   and sch' :: "type_scheme" assumes "sch \<le> sch'" shows "free_tv sch' \<subseteq> free_tv sch"
deriv_eq: shows "(\<lambda>h. \<nu> \<bullet> 0 + h \<bullet> axis x (1::'a)) = (\<lambda>v'. v' \<bullet> (\<chi>i. if i = x then 1::'a else (0::'a)))"
pseudo_ordered_set_def: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "pseudo_ordered_set A less_eq \<equiv> reflexive A less_eq \<and> antisymmetric A less_eq"
lim_h_minus_OUT_IN: fixes x :: "'v" assumes "cycles \<Delta> \<noteq> {}" shows "KIR (\<lambda>e. \<Squnion>n. h_minus n e) x"
interval_subset_is_interval: fixes S :: "'a set"   and a :: "'a"   and b :: "'a" assumes "is_interval S" shows "(cbox a b \<subseteq> S) = (cbox a b = {} \<or> a \<in> S \<and> b \<in> S)"
preservation_and_closures_2: fixes Rel :: "('c \<times> 'c) set"   and Pred :: "'c \<Rightarrow> bool" assumes "rel_preserves_pred Rel Pred" shows "rel_preserves_pred (Rel\<^sup>+) Pred"
ring_l_one: fixes R :: "('e, 'f) Ring_scheme"   and x :: "'e" assumes "Ring R"   and "x \<in> carrier R" shows "1\<^sub>r\<^bsub>R\<^esub> \<cdot>\<^sub>r\<^bsub>R\<^esub> x = x"
ibounded_by_def: shows "ibounded_by = (\<lambda>(\<A>1, \<A>2) q. \<exists>q1 q2. interaction_any_bounded_by \<A>1 q1 \<and> (\<forall>cipher \<sigma>. interaction_any_bounded_by (\<A>2 cipher \<sigma>) q2) \<and> q1 + q2 \<le> q)"
bempty_not_binsert_1: fixes x :: "'c"   and X :: "'c set['d]" shows "bempty \<noteq> binsert x X"
bi_total_alt_def2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "bi_total R = ((R ===> (=)) ===> (=)) All All"
left_diff_repeat_zmset_distrib': fixes j :: "nat"   and i :: "nat"   and u :: "'b zmultiset" assumes "j \<le> i" shows "repeat_zmset (i - j) u = repeat_zmset i u - repeat_zmset j u"
inc_partial_density_operator_converge: fixes dim :: "nat"   and f :: "nat \<Rightarrow> complex mat"   and i :: "nat"   and j :: "nat" assumes "matrix_seq dim f"   and "i \<in> {0..<dim}"   and "j \<in> {0..<dim}" shows "convergent (\<lambda>n. f n $$ (i, j))"
coeff_pcompose_linear: fixes p :: "'b poly"   and c :: "'b"   and i :: "nat" shows "coeff (p \<circ>\<^sub>p [:0::'b, c:]) i = c ^ i * coeff p i"
end2end: fixes c :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and c' :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ce :: "Vars.var \<Rightarrow> one\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and r :: "Vars.var list" assumes "c \<Rightarrow>\<^sup>* c'"   and "\<not> boring_step c'"   and "heap_upds_ok_conf c"   and "consistent (ae, ce, a, as, r) c" shows "\<exists>ae' ce' a' as' r'. consistent (ae', ce', a', as', r') c' \<and> conf_transform (ae, ce, a, as, r) c \<Rightarrow>\<^sub>G\<^sup>* conf_transform (ae', ce', a', as', r') c'"
x0_hb_bound4: fixes as :: "real list"   and bs :: "real list"   and hs :: "(real \<Rightarrow> real) list"   and k :: "nat"   and x\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and hb :: "real"   and e :: "real"   and p :: "real"   and x :: "real"   and i :: "nat" assumes "akra_bazzi_real_recursion as bs hs k x\<^sub>0 x\<^sub>1 hb e p"   and "x\<^sub>1 \<le> x"   and "i < k" shows "bs ! i / 2 < bs ! i + (hs ! i) x / x"
known_ptrs_subset: fixes h' :: "(_) heap"   and h :: "(_) heap" assumes "object_ptr_kinds h' |\<subseteq>| object_ptr_kinds h"   and "ShadowRootClass.known_ptrs h" shows "ShadowRootClass.known_ptrs h'"
simps: fixes k :: "'a"   and f :: "'c \<Rightarrow> 'c"   and xs :: "('a \<times> 'c) list"   and ko :: "'o" shows "update_by_fun_gr_raw k f (xs, ko) = (update_by_fun_gr_pair ko k f xs, ko)"
neg_le_minus_divide_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less c zero" shows "less_eq a (uminus (divide b c)) = less_eq (uminus b) (times a c)"
assms_5: shows "(\<lambda>(x, y). g x y) \<in> borel_measurable (M \<Otimes>\<^sub>M N)"
del_list_sorted2: fixes xs :: "'b list"   and a :: "'b"   and ys :: "'b list"   and x :: "'b" assumes "sorted_less (xs @ a # ys)"   and "x < a" shows "del_list x (xs @ a # ys) = del_list x xs @ a # ys"
preserves_section_retraction: fixes e :: "'a"   and m :: "'a" assumes "ide (e \<cdot> m)" shows "ide (src e \<cdot> src m)"
generates_max_ideal_cases: fixes X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and F' :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" assumes "generates_max_ideal X F"   and "F' \<subseteq> P[X]"   and "ideal F \<subseteq> ideal F'" obtains "ideal F = ideal F'"   | "ideal F' = UNIV"
eventually_compose_filterlim: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter"   and f :: "'c \<Rightarrow> 'b"   and G :: "'c filter" assumes "eventually P F"   and "filterlim f F G" shows "\<forall>\<^sub>F x in G. P (f x)"
vec_of_basis_enum_to_inverse: fixes \<psi> :: "'b" shows "vec_of_basis_enum (inverse \<psi>) = vec_of_list [inverse (vec_of_basis_enum \<psi> $ 0)]"
continuous_on_path: fixes f :: "real \<Rightarrow> 'c"   and t :: "real set" assumes "path f"   and "t \<subseteq> {0..1}" shows "continuous_on t f"
S5Basic_7: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<box>\<phi> \<^bold>\<equiv> \<^bold>\<box>\<^bold>\<box>\<phi> in v]"
apply_pair: fixes f :: "V"   and A :: "V"   and B :: "V \<Rightarrow> V"   and x :: "V" assumes "f \<in> elts (VPi A B)"   and "x \<in> elts A" shows "\<langle>x, app f x\<rangle> \<in> elts f"
QUPU: fixes s :: "'a"   and s' :: "'a" assumes "QU s s'" shows "PU s s'"
is_rat: fixes x :: "'a" shows "is_rat x = (x \<in> \<rat>)"
sublist_snoc: fixes xs :: "'c list"   and ys :: "'c list"   and y :: "'c" shows "Sublist.sublist xs (ys @ [y]) = (suffix xs (ys @ [y]) \<or> Sublist.sublist xs ys)"
prime_power_dvd_cancel_right: fixes p :: "'a"   and b :: "'a"   and n :: "nat"   and a :: "'a" assumes "prime p"   and "\<not> p dvd b"   and "p ^ n dvd a * b" shows "p ^ n dvd a"
pos_less_divide_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less a (divide b c) = less (times a c) b"
a_comm: fixes x :: "'a set"   and y :: "'a set" assumes "x \<in> carrier (R Quot I)"   and "y \<in> carrier (R Quot I)" shows "x \<oplus>\<^bsub>R Quot I\<^esub> y = y \<oplus>\<^bsub>R Quot I\<^esub> x"
r_prefixes_0: fixes s :: "nat \<Rightarrow> nat option"   and j :: "nat" assumes "r1_bc s" shows "eval (r1_bc.r_prefixes s) [0, j] \<down>= singleton_encode j"
lossless_try_spmf: fixes p :: "'a spmf"   and q :: "'a spmf" shows "lossless_spmf (TRY p ELSE q) = (lossless_spmf p \<or> lossless_spmf q)"
2: shows "{(e, f e) |e. f e \<noteq> bot} \<noteq> {}"
order_iff: fixes a :: "'c"   and b :: "'c" shows "(a \<le> b) = (a = a \<sqinter> b)"
Basic: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'e \<Rightarrow> 'e"   and s :: "'e"   and n :: "nat" shows "\<Gamma>\<turnstile> \<langle>Basic f,Normal s\<rangle> =n\<Rightarrow> Normal (f s)"
child_lv_other: fixes d' :: "nat"   and d :: "nat"   and p :: "(nat \<times> int) list"   and dir :: "dir" assumes "d' \<noteq> d" shows "lv (child p dir d') d = lv p d"
dim_solution_set_homogeneous_eq_dim_null_space: fixes A :: "(('d, 'e) vec, 'f) vec" shows "vec.dim (solution_set A 0) = vec.dim (null_space A)"
nth_default_append: fixes dflt :: "'f"   and xs :: "'f list"   and ys :: "'f list"   and n :: "nat" shows "nth_default dflt (xs @ ys) n = (if n < length xs then xs ! n else nth_default dflt ys (n - length xs))"
collect_clkt_def: fixes S :: "('e \<times> ('c, 'd) cconstraint \<times> 'f \<times> 'c list \<times> 'e) set" shows "collect_clkt S = \<Union> {collect_clock_pairs (fst (snd t)) |t. t \<in> S}"
yuniq: shows "\<forall>y'. \<langle>k, y'\<rangle> \<^bold>\<in> s \<longrightarrow> y' = y"
insert_ops_split: fixes ops :: "('a \<times> 'a option) list"   and oid :: "'a"   and ref :: "'a option" assumes "insert_ops ops"   and "(oid, ref) \<in> set ops" shows "\<exists>pre suf. ops = pre @ [(oid, ref)] @ suf \<and> (\<forall>i\<in>set (map fst pre). i < oid) \<and> (\<forall>i\<in>set (map fst suf). oid < i)"
has_bochner_integral_restrict_space: fixes \<Omega> :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and i :: "'c" assumes "\<Omega> \<inter> space M \<in> sets M" shows "has_bochner_integral (restrict_space M \<Omega>) f i = has_bochner_integral M (\<lambda>x. indicat_real \<Omega> x *\<^sub>R f x) i"
weaken_post: fixes P :: "state \<Rightarrow> bool"   and c :: "com"   and Q :: "state \<Rightarrow> bool"   and Q' :: "state \<Rightarrow> bool" assumes "\<turnstile> {P} c {Q}"   and "\<forall>s. Q s \<longrightarrow> Q' s" shows "\<turnstile> {P} c {Q'}"
standard_maintains_constant_rules: fixes C :: "'b set"   and G :: "('b Standard_Constant, 'b + 'c) labeled_graph"   and R :: "('b Standard_Constant, nat) labeled_graph \<times> ('b Standard_Constant, nat) labeled_graph" assumes "standard' C G"   and "R \<in> constant_rules C" shows "maintained R G"
eFreshAbs_nchotomy: fixes eA :: "'c withERR"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'c) model"   and ys :: "'varSort"   and y :: "'var" shows "(\<exists>us s A. eA = OK A \<and> igWlsAbs MOD (us, s) A) \<or> eFreshAbs MOD ys y eA"
snth_of_seq: fixes f :: "nat \<Rightarrow> 'c" shows "(!!) (of_seq f) = f"
eval_nat_numeral_3: fixes n :: "num" shows "numeral (num.Bit1 n) = Suc (numeral (num.Bit0 n))"
vfsequence_vempty: shows "vfsequence []\<^sub>\<circ>"
TESL_interp_formula_stuttering: fixes \<phi> :: "'b TESL_atomic"   and \<Phi> :: "'b TESL_atomic list" assumes "\<phi> \<in> set \<Phi>" shows "\<lbrakk>\<lbrakk> \<phi> # \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
fold_matches_first_step_not_None: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a option"   and x :: "'b"   and xs :: "'b list"   and y :: "'c"   and ys :: "'c list"   and subs :: "'a"   and subs' :: "'a" assumes "fold (\<lambda>(T, U) subs. Option.bind subs (f T U)) (zip (x # xs) (y # ys)) (Some subs) = Some subs'" obtains point :: "'a" where "f x y subs = Some point" and "fold (\<lambda>(T, U) subs. Option.bind subs (f T U)) (zip xs ys) (Some point) = Some subs'"
Gauss_Jordan_def: fixes A :: "(('a, 'b) vec, 'c) vec" shows "Gauss_Jordan A = Gauss_Jordan_upt_k A (ncols A - 1)"
adjoint_pair_antipar_3: fixes f :: "'a"   and g :: "'a" assumes "C.adjoint_pair f g" shows "src\<^sub>C f = trg\<^sub>C g"
word_le_minus_mono_left: fixes y :: "'b word"   and z :: "'b word"   and x :: "'b word" assumes "y \<le> z"   and "x \<le> y" shows "y - x \<le> z - x"
ratfps_closed_form_exists: fixes q :: "complex poly"   and p :: "complex poly" assumes "constant_term q \<noteq> 0" obtains r :: "complex poly" and rs :: "complex \<Rightarrow> complex poly" where "\<And>n. (fps_of_poly p / fps_of_poly q) $ n = coeff r n + (\<Sum>c | poly (reflect_poly q) c = 0. poly (rs c) (of_nat n) * c ^ n)" and "\<And>z. poly (reflect_poly q) z = 0 \<Longrightarrow> degree (rs z) \<le> order z (reflect_poly q) - 1"
\<psi>_def: shows "\<psi> \<equiv> softSubst \<chi> \<langle>\<chi>\<rangle> xx"
iT_Plus_neg_image_conv: fixes I :: "nat set"   and k :: "nat" shows "I \<oplus>- k = (\<lambda>n. n - k) ` I \<down>\<ge> k"
is_red_adds_iff: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "is_red F p = (\<exists>f\<in>F. \<exists>v\<in>keys p. f \<noteq> 0 \<and> lt f adds\<^sub>t v)"
oalist_inv_raw_ConsD2: fixes k :: "'a"   and v :: "'c"   and xs :: "('a \<times> 'c) list" assumes "oalist_inv_raw ((k, v) # xs)" shows "v \<noteq> (0::'c)"
move_nothing: fixes ts :: "traffic"   and v :: "view" shows "move ts ts v = v"
reannotateWhileNoGuard: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and b :: "'e set"   and I :: "'e set"   and V :: "('e \<times> 'e) set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and J :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P whileAnno b I V c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P whileAnno b J V c Q,A"
Fun_mkArr: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and A :: "'b set"   and B :: "'b set"   and F :: "'b \<Rightarrow> 'b" assumes "set_category_given_img S img \<AA>"   and "partial_magma.arr S (set_category_given_img.mkArr S img \<AA> A B F)" shows "set_category_given_img.Fun S img (set_category_given_img.mkArr S img \<AA> A B F) = restrict F A"
compatible_l_def: fixes R1 :: "('b \<times> 'b) set"   and R2 :: "('b \<times> 'b) set" shows "compatible_l R1 R2 \<equiv> R1 O R2 \<subseteq> R2"
union_ac_1: fixes M :: "'b multiset"   and N :: "'b multiset"   and K :: "'b multiset" shows "M + N + K = M + (N + K)"
l_set_tag_name_get_child_nodes_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   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" shows "l_set_tag_name_get_child_nodes type_wf set_tag_name set_tag_name_locs known_ptr get_child_nodes get_child_nodes_locs \<equiv> l_set_tag_name type_wf set_tag_name set_tag_name_locs \<and> l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<and> l_set_tag_name_get_child_nodes_axioms set_tag_name_locs get_child_nodes_locs"
not_equiv_le_trans: fixes a :: "'a"   and b :: "'a" assumes "\<not> a \<approx> b"   and "a \<le> b" shows "a < b"
rel_pmf_of_setI: fixes A :: "'c set"   and B :: "'d set"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "A \<noteq> {}"   and "finite A"   and "B \<noteq> {}"   and "finite B"   and "\<And>X. X \<subseteq> A \<Longrightarrow> card B * card X \<le> card A * card {y \<in> B. \<exists>x\<in>X. R x y}" shows "rel_pmf R (pmf_of_set A) (pmf_of_set B)"
q3: shows "\<forall>x. (\<forall>i<n. x i \<le> p) \<longrightarrow> (\<forall>i<n. x i = p \<longrightarrow> (label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 1)"
unique_is_poincare_line: fixes u :: "complex_homo"   and v :: "complex_homo"   and l1 :: "circline"   and l2 :: "circline" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "u \<noteq> v"   and "is_poincare_line l1"   and "is_poincare_line l2"   and "{u, v} \<subseteq> circline_set l1 \<inter> circline_set l2" shows "l1 = l2"
finite_total_preorder_on_def: fixes carrier :: "'c set"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "finite_total_preorder_on carrier le \<equiv> total_preorder_on carrier le \<and> finite_total_preorder_on_axioms carrier"
cut_less_le_iprev_conv: fixes t :: "nat"   and I :: "nat set" assumes "t \<in> I"   and "t \<noteq> iMin I" shows "I \<down>< t = I \<down>\<le> iprev t I"
elims_3: fixes x :: "('c, 'd) strand_step" assumes "\<not> simple\<^sub>s\<^sub>t\<^sub>p x" obtains X :: "'d list" and F :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" where "x = \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t" and "\<nexists>\<I>. ineq_model \<I> X F"   | va :: "'c" and vb :: "('c, 'd) Term.term list" where "x = send\<langle>Fun va__ vb__\<rangle>\<^sub>s\<^sub>t"   | v :: "poscheckvariant" and va :: "('c, 'd) Term.term" and vb :: "('c, 'd) Term.term" where "x = \<langle>v: va \<doteq> vb\<rangle>\<^sub>s\<^sub>t"
fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_cons_subset: fixes F :: "(('b, 'a) Term.term \<times> ('c, 'a) Term.term) list"   and f :: "('b, 'a) Term.term \<times> ('c, 'a) Term.term" shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<subseteq> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (f # F)"
map_simp_ide: fixes a :: "'d \<times> 'c" assumes "DopxC.ide a" shows "\<Phi> a = \<Phi>o a"
hom_one: shows "finprod G \<one>\<^bsub>DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I\<^esub> I = \<one>\<^bsub>G\<lparr>carrier := J\<rparr>\<^esub>"
path_component_imp_homotopic_points: fixes S :: "'c set"   and a :: "'c"   and b :: "'c" assumes "path_component S a b" shows "homotopic_loops S (linepath a a) (linepath b b)"
len_lexext_rev_compat_snoc: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list"   and x :: "'b" assumes "len_lexext_rev gt ys xs" shows "len_lexext_rev gt (ys @ [x]) (xs @ [x])"
separation_by_Un_closed_pointwise: fixes S :: "'b set"   and T :: "'b set"   and a :: "'b"   and b :: "'b" assumes "closed S"   and "closed T"   and "S \<inter> T = {}"   and "connected_component (- S) a b"   and "connected_component (- T) a b" shows "connected_component (- (S \<union> T)) a b"
valid_Tree_eqvt: fixes P :: "'state"   and t :: "('b, 'pred, 'act) Tree"   and p :: "perm" assumes "valid_Tree P t" shows "valid_Tree (p \<bullet> P) (p \<bullet> t)"
iEx_True: fixes I :: "nat set" shows "(\<diamond> t I. True) = (I \<noteq> {})"
kat_4_equiv_opp: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "(y \<cdot> d x = d z \<cdot> y \<cdot> d x) = (y \<cdot> d x \<le> d z \<cdot> y)"
o_id: fixes f :: "'g \<Rightarrow> 'h" shows "f \<circ> id = f"
elims: fixes h :: "'b \<Rightarrow> 'a"   and x :: "'b rose_tree"   and y :: "'a rose_tree\<^sub>h" assumes "hash_source_tree h x = y" obtains data :: "'b" and subtrees :: "'b rose_tree list" where "x = Tree (data, subtrees)" and "y = Tree\<^sub>h (Content (h data, map (hash_source_tree h) subtrees))"
lt_spair': fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and a :: "'a"   and b :: "'a"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "rep_list p \<noteq> 0"   and "a + punit.lt (rep_list p) = b + punit.lt (rep_list q)"   and "b \<oplus> lt q \<prec>\<^sub>t a \<oplus> lt p" shows "lt (spair p q) = (a - gcs a b) \<oplus> lt p"
Cons_both_possible_leftE: fixes t :: "'b ttree"   and x :: "'b"   and xs :: "'b list"   and t' :: "'b ttree" assumes "possible t x"   and "xs \<in> paths (nxt t x \<otimes>\<otimes> t')" shows "x # xs \<in> paths (t \<otimes>\<otimes> t')"
lookup_tabulate: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "distinct xs" shows "Mapping.lookup (Mapping.tabulate xs f) x = (if x \<in> set xs then Some (f x) else None)"
preserves_seq: fixes g :: "'a"   and f :: "'a" assumes "B.seq g f" shows "B.seq (if B.arr g then g else B.null) (if B.arr f then f else B.null)"
A_or_B: shows "singleton default A = default () \<or> singleton default B = default ()"
component_of_def: fixes F :: "'c program"   and H :: "'c program" shows "F component_of H \<equiv> \<exists>G. F ok G \<and> F \<squnion> G = H"
simps_14: fixes v :: "Cooper.num" shows "qfree (Eq v) = True"
Rep_inverse': fixes a :: "'b word"   and r :: "nat" assumes "unat a = r" shows "word_of_nat r = a"
dist_midpoint_4: fixes a :: "'b"   and b :: "'b" shows "dist (midpoint a b) b = dist a b / 2"
size_neq_size_imp_neq: fixes x :: "'g"   and y :: "'g" assumes "size x \<noteq> size y" shows "x \<noteq> y"
conv: fixes x :: "real" assumes "M.prob {x} = 0" shows "(\<lambda>n. \<mu>.C n x) \<longlonglongrightarrow> M.C x"