hcju/isabelleps huggingface dataset:
distinct_insort: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "distinct (linorder.insort_key less_eq f x xs) = (x \<notin> set xs \<and> distinct xs)"
jordan_matrix_dim_1: fixes n_as :: "(nat \<times> 'b) list" shows "dim_row (jordan_matrix n_as) = sum_list (map fst n_as)"
\<phi>_simps_2: shows "src f\<^sub>0g\<^sub>1.\<phi> = src f\<^sub>0g\<^sub>1.p\<^sub>0"
no_kth_obs_missing_cs: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and k :: "nat"   and i :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "is_kth_obs \<pi> k i"   and "\<nexists>i'. is_kth_obs \<pi>' k i'" obtains l :: "nat" and j :: "nat" where "is_kth_obs \<pi> l j" and "\<nexists>j'. cs\<^bsup>\<pi>\<^esup> j = cs\<^bsup>\<pi>'\<^esup> j'"
PQ: fixes t :: "'a \<times> 'b \<times> 'a" assumes "P t" shows "Q t"
pol_deg_eq_c_max: fixes p :: "'a"   and c :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "p \<in> carrier R"   and "pol_coeff S c"   and "p = polyn_expr R X (fst c) c" shows "deg_n R S X p = c_max S c"
rel_pred_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'c set"   and B :: "'d set" shows "rel_pred R A B = rel_fun R (=) (\<lambda>x. x \<in> A) (\<lambda>y. y \<in> B)"
move_defensive_by_Odd_def: fixes m :: "'a"   and p :: "'a list" shows "move_defensive_by_Odd m p \<equiv> odd (length p) \<longrightarrow> \<not> winning_position_Even (p @ [m])"
assms_2: shows "h \<turnstile> create_character_data document_ptr text \<rightarrow>\<^sub>h h'"
norm_pre_pure_iff: fixes P :: "assn"   and b :: "bool"   and f :: "'a Heap"   and Q :: "'a \<Rightarrow> assn" shows "<P * \<up> b> f <Q> = (b \<longrightarrow> <P> f <Q>)"
con_elim: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<psi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<pi>l :: "('a \<times> 'b set) stream list" assumes "Shallow S s0 \<delta> AP L AP'"   and "Shallow.con \<phi> \<psi> \<pi>l" obtains "\<phi> \<pi>l" and "\<psi> \<pi>l"
otherassms_8: shows "(\<xi>', q) \<in> reachable A I"
sgn_abs: fixes a :: "'a" shows "\<bar>sgn a\<bar> = of_bool (a \<noteq> (0::'a))"
rel_refl: fixes Ra :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c option" assumes "\<And>x. Ra x x" shows "rel_option Ra x x"
resultant_as_poly: fixes p :: "'b poly"   and q :: "'b poly" assumes "0 < degree p"   and "0 < degree q" shows "\<exists>p' q'. degree p' < degree q \<and> degree q' < degree p \<and> [:resultant p q:] = p' * p + q' * q"
cancels_to_self: fixes l :: "(bool \<times> 'c) list" shows "cancels_to l l"
basis_reduction_step_def: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list" shows "basis_reduction_step upw i fs = (if i = 0 then (True, Suc i, fs) else let fs' = basis_reduction_add_rows upw i fs in if \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs') (i - 1)\<parallel>\<^sup>2 \<le> \<alpha> * \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs') i\<parallel>\<^sup>2 then (True, Suc i, fs') else basis_reduction_swap i fs')"
word_and_1_shiftl: fixes x :: "'a word"   and n :: "nat" shows "x AND (1 << n) = (if x !! n then 1 << n else 0)"
inv_fg_closure2: fixes l :: "(bool \<times> 'd) list"   and gens :: "'d set" assumes "l \<in> lists (UNIV \<times> gens)" shows "inv_fg l \<in> lists (UNIV \<times> gens)"
h_def: fixes a :: "nat"   and b :: "nat"   and c :: "nat" shows "h a b c = p a c + b"
dX_Int_Field_Pd: fixes X :: "'a set"   and d :: "'b" assumes "Pdds'.dX X d \<subseteq> Field (Pd_shuffle_to_top (insert d__ ds') Y d)" shows "X \<inter> Field (Pd_shuffle_to_top (insert d__ ds') Y d) = Pdds'.dX X d"
assms_4: shows "interval_lebesgue_integrable lborel (ereal A) (ereal B) g"
surj_f_inv_f: fixes f :: "'i \<Rightarrow> 'h"   and y :: "'h" assumes "surj f" shows "f (inv f y) = y"
append_in_starI: fixes u :: "'a list"   and A :: "'a list set"   and v :: "'a list" assumes "u \<in> A\<star>"   and "v \<in> A\<star>" shows "u @ v \<in> A\<star>"
M_Problem_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set" shows "M_Problem wtFsym wtPsym arOf resOf parOf \<Phi> \<equiv> M.Problem wtFsym wtPsym arOf resOf parOf \<Phi>"
cobounded1: fixes a :: "'b"   and b :: "'b" shows "order.greater_eq a (inf a b)"
multiple_is_wellformed: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and n :: "nat" assumes "incidence_system point_set block_collection" shows "incidence_system point_set (repeat_mset n block_collection)"
check_poly_weak_mono_all: fixes p :: "('c monom \<times> 'd) list" assumes "check_poly_weak_mono_all p" shows "poly_weak_mono_all p"
PreFunctorFunctor: fixes F :: "('g, 'h, 'i, 'j, 'k, 'l, 'm) Functor_scheme" assumes "Functor F" shows "PreFunctor F"
FlexRayArchitecture_def: fixes n :: "nat"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nC :: "nat \<Rightarrow> Config"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list" shows "FlexRayArchitecture n nReturn nC nStore nGet \<equiv> CorrectSheaf n \<and> (\<exists>nSend recv. Cable n nSend recv \<and> (\<forall>i<n. FlexRayController (nReturn i) recv (nC i) (nStore i) (nSend i) (nGet i)))"
bind_split: fixes P :: "'c option \<Rightarrow> bool"   and m :: "'d option"   and f :: "'d \<Rightarrow> 'c option" shows "P (Option.bind m f) = ((m = None \<longrightarrow> P None) \<and> (\<forall>v. m = Some v \<longrightarrow> P (f v)))"
C1_differentiable_imp_piecewise: fixes f :: "real \<Rightarrow> 'b"   and S :: "real set" assumes "f C1_differentiable_on S" shows "f piecewise_C1_differentiable_on S"
times_lepoll_mono: fixes A :: "'a set"   and C :: "'b set"   and B :: "'c set"   and D :: "'d set" assumes "A \<lesssim> C"   and "B \<lesssim> D" shows "A \<times> B \<lesssim> C \<times> D"
map_of_inject_set: fixes xs :: "('d \<times> 'e) list"   and ys :: "('d \<times> 'e) list" assumes "distinct (map fst xs)"   and "distinct (map fst ys)" shows "(map_of xs = map_of ys) = (set xs = set ys)"
Rep_Abs_prop_entailment: fixes A :: "'b ltl set"   and \<phi> :: "'b ltl" shows "A \<Turnstile>\<^sub>P Rep (Abs \<phi>) = A \<Turnstile>\<^sub>P \<phi>"
bounded: shows "\<exists>K. \<forall>a b. norm (blinfun_apply a b) \<le> norm a * norm b * K"
lpi: shows "real_of_float (float_round_down prec (lb_pi prec)) \<le> pi"
Un_least: fixes A :: "'g set"   and C :: "'g set"   and B :: "'g set" assumes "A \<subseteq> C"   and "B \<subseteq> C" shows "A \<union> B \<subseteq> C"
class_add_has_fields: fixes P :: "jvm_prog"   and D :: "char list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and C :: "char list"   and cdec :: "jvm_method class" assumes "P \<turnstile> D has_fields FDTs"   and "\<not> is_class P C" shows "class_add P (C, cdec) \<turnstile> D has_fields FDTs"
index1: assumes "\<And>G. G \<in> \<G> \<Longrightarrow> \<not> ps \<subseteq> G" shows "\<B> index ps = 1"
tensor_vec_eval: fixes i :: "nat"   and v1 :: "'b vec"   and v2 :: "'b vec" assumes "i < d" shows "tensor_vec v1 v2 $ i = v1 $ encode1 i * v2 $ encode2 i"
comm_monoid_axioms: shows "comm_monoid (\<sqinter>) \<top>"
simps_6: shows "J.TT \<noteq> J.Null"
atLeastLessThan_empty_iff2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.preorder less_eq less" shows "({} = ord.atLeastLessThan less_eq less a b) = (\<not> less a b)"
disj_com: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b"   and A :: "'b set" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "(disj_PL p q \<in> Cn A) = (disj_PL q p \<in> Cn A)"
sumlist_map_as_finsum: fixes f :: "'b \<Rightarrow> 'a vec"   and xs :: "'b list" assumes "f \<in> set xs \<rightarrow> carrier_vec n"   and "distinct xs" shows "M.sumlist (map f xs) = finsum V f (set xs)"
subseq_filter_left: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "subseq (filter P xs) xs"
primepow_gt_0_nat: fixes n :: "nat" assumes "primepow n" shows "0 < n"
bintrunc_shiftl: fixes n :: "nat"   and m :: "int"   and i :: "nat" shows "take_bit n (m << i) = take_bit (n - i) m << i"
IH: fixes S :: "variable set"   and s :: "variable" assumes "finite S" shows "\<forall>s\<in>set (fst (variant_variables n s S)). s \<notin> S"
welldefined_subsets1: shows "all_list_constr_R subsets1 (length qs1)"
pathfinish_compose: fixes f :: "'d \<Rightarrow> 'c"   and p :: "real \<Rightarrow> 'd" shows "pathfinish (f \<circ> p) = f (pathfinish p)"
power_inductr: fixes z :: "'a list set"   and y :: "'a list set"   and x :: "'a list set"   and n :: "nat" assumes "z \<union> p_prod y x \<subseteq> y" shows "p_prod z (path_dioid.power x n) \<subseteq> y"
le_refl: fixes x :: "'a" shows "x \<sqsubseteq> x"
degeneralize_language: fixes A :: "('a, 'b) ngbta" shows "NBTA.language (degeneralize A) = NGBTA.language A"
encR: fixes S :: "'procS"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" shows "SourceTerm S \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> TargetTerm (\<lbrakk>S\<rbrakk>)"
norm_scaleR: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "real"   and x :: "'a" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (scaleR a x) = \<bar>a\<bar> * norm x"
lunit_hcomp_1: fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "src f = trg g" shows "\<l>[f \<star> g] \<cdot> \<a>[trg f, f, g] = \<l>[f] \<star> g"
independent_insert: fixes a :: "'b"   and S :: "'b set" shows "independent (insert a S) = (if a \<in> S then independent S else independent S \<and> a \<notin> span S)"
simps_2: fixes v :: "'b"   and va :: "'b"   and y :: "('b, 'c) Combinators"   and z :: "('b, 'c) Combinators list" shows "separate (DenyAllFromTo v va # y # z) = (if first_bothNet (DenyAllFromTo v va) = first_bothNet y then separate (DenyAllFromTo v va \<oplus> y # z) else DenyAllFromTo v va # separate (y # z))"
a: shows "abc_steps_l (0, args) A stp = (s, nl)"
wf_fdecl_def: 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" shows "wf_fdecl P \<equiv> \<lambda>(F, T). is_type P T"
singular_chain_singleton: fixes X :: "'c topology"   and a :: "'c"   and p :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "topspace X = {a}" shows "singular_chain p X c = (\<exists>b. c = frag_cmul b (frag_of (\<lambda>x\<in>standard_simplex p. a)))"
eq_sym: fixes eq :: "'d \<rightarrow> 'd \<rightarrow> tr"   and x :: "'d"   and y :: "'d" assumes "class.Eq_sym eq" shows "eq\<cdot>x\<cdot>y = eq\<cdot>y\<cdot>x"
ennreal_Sup: fixes A :: "real set" assumes "\<Squnion> (ennreal ` A) \<noteq> \<top>"   and "A \<noteq> {}" shows "ennreal (\<Squnion> A) = \<Squnion> (ennreal ` A)"
add_mono_neutr: fixes b :: "'a"   and a :: "'a" assumes "\<one> \<le> b" shows "a \<le> a + b"
axioms: fixes S1 :: "'c set"   and S2 :: "'d set"   and s1 :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and s2 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd" assumes "linear_on S1 S2 s1 s2 f" shows "module_hom_on S1 S2 s1 s2 f"
lookup_empty: shows "RBT.lookup RBT.empty = Map.empty"
unit_circle_fix_moebius_uminus: fixes M :: "moebius" assumes "unit_circle_fix M" shows "unit_circle_fix (- M)"
weakCongSync: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs"   and Q :: "ccs"   and Q' :: "ccs" assumes "P \<Longrightarrow>\<alpha> \<prec> P'"   and "Q \<Longrightarrow>coAction \<alpha> \<prec> Q'"   and "\<alpha> \<noteq> \<tau>" shows "P \<parallel> Q \<Longrightarrow>\<tau> \<prec> P' \<parallel> Q'"
order_antisym: fixes x :: "'g"   and y :: "'g" assumes "x \<le> y"   and "y \<le> x" shows "x = y"
add_strict_left_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.ordered_cancel_ab_semigroup_add plus minus less_eq less"   and "less a b" shows "less (plus c a) (plus c b)"
mset_nondetMs_4: fixes return :: "'a multiset \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a multiset \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_commute return bind" shows "monad_fail_alt (nondetM_base.return_nondet return (\<lambda>x. {#x#})) (nondetM_base.bind_nondet bind (mmerge return bind)) (nondetM_base.fail_nondet return {#}) (nondetM_base.alt_nondet return bind (+))"
hbt_insert_auto: fixes m :: "nat"   and t :: "('b \<times> nat) tree"   and x :: "'b" assumes "HBT m"   and "HBT.hbt m t" shows "HBT.hbt m (HBT.insert m x t) \<and> height (HBT.insert m x t) \<in> {height t, height t + 1}"
sem: shows "sterm_sem (adjointFO I \<sigma> \<nu>) \<theta> = sterm_sem (adjointFO I \<sigma> \<omega>) \<theta>"
non_terminating_FD: fixes P :: "'b process" shows "non_terminating P = CHAOS UNIV \<sqsubseteq>\<^sub>F\<^sub>D P"
ring_epimorphism_axioms: shows "ring_epimorphism f A addA multA zeroA oneA B addB multB zeroB oneB"
eq: shows "\<Union> (G ` K) = \<Union> (G ` S)"
2: fixes n :: "int" assumes "homologous_rel_set 0 X {} (frag_cmul n (frag_of f)) = \<one>\<^bsub>homology_group 0 X\<^esub>" shows "n = 0"
ln_exp_cf6_upper_bound_neg: fixes x :: "real" assumes "x \<le> 0" shows "x \<le> ln (exp_cf6 x)"
add_neg_numeral_special_3: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "plus (numeral.numeral one plus m) (uminus one) = neg_numeral.sub minus plus one m num.One"
emeasure_lborel_countable: fixes A :: "'a set" assumes "countable A" shows "emeasure lborel A = 0"
assms_3: shows "(i, Create i e1) \<in> set (node_deliver_messages xs)"
eq_iff: fixes a :: "'b word"   and b :: "'b word" shows "(a = b) = (b \<le>s a \<and> a \<le>s b)"
scopeExtLeft: fixes x :: "name"   and P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs" assumes "x \<sharp> P"   and "\<And>y R T. y \<sharp> R \<Longrightarrow> (\<lparr>\<nu>y\<rparr>(R \<parallel> T), R \<parallel> \<lparr>\<nu>y\<rparr>T) \<in> Rel" shows "\<lparr>\<nu>x\<rparr>(P \<parallel> Q) \<leadsto>[Rel] P \<parallel> \<lparr>\<nu>x\<rparr>Q"
fun_upd_image: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d"   and y :: "'c"   and A :: "'d set" shows "f(x := y) ` A = (if x \<in> A then insert y (f ` (A - {x})) else f ` A)"
homotopy_equivalence_relative_homology_group_isomorphism: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and T :: "'c set"   and g :: "'c \<Rightarrow> 'b"   and p :: "int" assumes "continuous_map X Y f"   and "f ` S \<subseteq> T"   and "continuous_map Y X g"   and "g ` T \<subseteq> S"   and "homotopic_with (\<lambda>h. h ` S \<subseteq> S) X X (g \<circ> f) id"   and "homotopic_with (\<lambda>k. k ` T \<subseteq> T) Y Y (f \<circ> g) id" shows "hom_induced p X S Y T f \<in> Group.iso (relative_homology_group p X S) (relative_homology_group p Y T)"
meas_upd2: shows "(\<lambda>xa. (snd xa)(x := fst (fst xa), y := snd (fst xa))) \<in> (stock_measure (\<Gamma> x) \<Otimes>\<^sub>M stock_measure (\<Gamma> y)) \<Otimes>\<^sub>M Pi\<^sub>M (set (filter (\<lambda>z. z \<noteq> x \<and> z \<noteq> y) vs)) (\<lambda>y. stock_measure (\<Gamma> y)) \<rightarrow>\<^sub>M Pi\<^sub>M (set vs) (\<lambda>y. stock_measure (\<Gamma> y))"
T_A'_def: fixes n :: "nat" shows "T_A' n = sum t_A' {..<n}"
match_iface_refl: fixes x :: "char list" shows "match_iface (Iface x) x"
const_map_def: fixes v :: "'b"   and S :: "'a set" shows "const_map v S \<equiv> (\<lambda>_. Some v) |` S"
ins2_ins_1: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree" assumes "ins n d x t = Same" shows "ins2 n d x t = Same2"
not_bot_below_trans: fixes a :: "'d"   and b :: "'d" assumes "a \<noteq> \<bottom>"   and "a \<sqsubseteq> b" shows "b \<noteq> \<bottom>"
normalize_poly_p_normalize_poly_spec: fixes p :: "(char list multiset \<times> int) multiset"   and p' :: "int mpoly" assumes "(p, p') \<in> mset_poly_rel" shows "SPEC (normalize_poly_p\<^sup>*\<^sup>* p) \<le> \<Down> mset_poly_rel (normalize_poly_spec p')"
euler_maclaurin_nat_int_transfer: fixes f :: "real \<Rightarrow> 'a"   and a :: "nat"   and F :: "real \<Rightarrow> 'a"   and T :: "real \<Rightarrow> 'a"   and C :: "'a" assumes "(\<lambda>b. (\<Sum>k = a..b. f (real k)) - F (real b) - T (real b)) \<longlonglongrightarrow> C" shows "((\<lambda>b. (\<Sum>k = int a..b. f (real_of_int k)) - F (real_of_int b) - T (real_of_int b)) \<longlongrightarrow> C) at_top"
dep_vars_Cons_ddep: fixes V :: "'var"   and as' :: "'edge list"   and as :: "'edge list"   and n' :: "'node"   and V' :: "'var"   and a :: "'edge" assumes "(V, as', as) \<in> dependent_live_vars n'"   and "V' \<in> Use (sourcenode a)"   and "n' = last (targetnodes (a # as))"   and "sourcenode a -{V}a # as'\<rightarrow>\<^sub>d\<^sub>d last (targetnodes (a # as'))" shows "(V', [], a # as) \<in> dependent_live_vars n'"
less_enatE: fixes n :: "enat"   and m :: "nat" assumes "n < enat m" obtains k :: "nat" where "n = enat k" and "k < m"
lunit_inv: fixes a :: "'a" assumes "ide a" shows "inverse_arrows \<l>[a] \<l>\<^sup>-\<^sup>1[a]"
totient_mult: fixes a :: "nat"   and b :: "nat" shows "totient (a * b) = totient a * totient b * gcd a b div totient (gcd a b)"
ActualConjI: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<^bold>\<A>\<phi> \<^bold>& \<^bold>\<A>\<psi> in v]" shows "[\<^bold>\<A>(\<phi> \<^bold>& \<psi>) in v]"
var\<^sub>C_of_in_\<C>\<^sub>C: fixes x\<^sub>A :: "'Var\<^sub>A" assumes "x\<^sub>A \<in> \<C>\<^sub>A" shows "var\<^sub>C_of x\<^sub>A \<in> \<C>\<^sub>C"
coeff_mpoly_times: fixes p :: "'b mpoly"   and q :: "'b mpoly"   and mon :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (p * q) mon = prod_fun (MPoly_Type.coeff p) (MPoly_Type.coeff q) mon"
eq_inf: fixes a :: "(real \<times> real \<times> real) list" shows "(\<forall>(a, b, c)\<in>set a. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c = 0) = (\<forall>(a, b, c)\<in>set a. a = 0 \<and> b = 0 \<and> c = 0)"
dg_prod_2_Dom_app: fixes g :: "V"   and f :: "V" assumes "[g, f]\<^sub>\<circ> \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Arr\<rparr>" shows "(\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Dom\<rparr> \<lparr>g, f\<rparr>\<^sub>\<bullet> = [\<AA>\<lparr>Dom\<rparr>\<lparr>g\<rparr>, \<BB>\<lparr>Dom\<rparr>\<lparr>f\<rparr>]\<^sub>\<circ>"
nth_deriv_conv: fixes n :: "nat" shows "fds_abs_converges ((fds_deriv ^^ n) f) (complex_of_real a)"
vsv_vrrestriction: fixes A :: "V" shows "vsv (r\<inverse>\<^sub>\<circ> \<restriction>\<^sup>r\<^sub>\<circ> A)"
bounded_clinear_ident: shows "bounded_clinear (\<lambda>x. x)"
ev_False: fixes \<omega> :: "'a stream" shows "ev (\<lambda>x. False) \<omega> = False"
gen: shows "genericJoin V Q Qn = (let (I, J) = getIJ Q Qn V; Q_I_pos = projectQuery I (filterQuery I Q); Q_I_neg = filterQueryNeg I Qn; R_I = genericJoin I Q_I_pos Q_I_neg; Q_J_neg = Qn - Q_I_neg; Q_J_pos = filterQuery J Q; X = {(t, genericJoin J (newQuery J Q_J_pos (I, t)) (newQuery J Q_J_neg (I, t))) |t. t \<in> R_I} in \<Union>(t, x)\<in>X. {merge xx t |xx. xx \<in> x})"
conjugate_id: fixes conjugate :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.conjugate conjugate" shows "conjugate (conjugate a) = a"
list_succ_def: fixes xs :: "'b list"   and x :: "'b" shows "list_succ xs x = (if x \<in> set xs then xs ! ((index xs x + 1) mod length xs) else x)"
pow_assoc: shows "\<mu> \<circ> \<P> \<mu> = \<mu> \<circ> \<mu>"
search_correct: fixes t :: "('b ivl \<times> 'b) tree"   and x :: "'b ivl" assumes "inv_max_hi t"   and "Sorted_Less.sorted (Tree2.inorder t)" shows "search t x = has_overlap (Tree2.set_tree t) x"
pat_type_perm_eq: fixes pi :: "(name \<times> name) list"   and p :: "pat" shows "pat_type (pi \<bullet> p) = pat_type p"
comp_null_2: fixes f :: "'b" shows "B f \<sigma>.B.null = \<sigma>.B.null"
distinct_5: fixes x21 :: "'b"   and x22 :: "'b form list" shows "Compound x21 x22 \<noteq> ff"
length_takeWhile_le: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "length (takeWhile P xs) \<le> length xs"
assms_3: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and c :: "'a"   and exp :: "exp" assumes "atom ` domA \<Gamma> \<sharp>* c"   and "\<And>c x. x \<in> domA \<Gamma> \<Longrightarrow> P c (the (map_of \<Gamma> x))"   and "\<And>c. P c exp" shows "P c (Let \<Gamma> exp)"
intros_1: fixes n :: "'node" assumes "valid_node n" shows "n -[]\<rightarrow>* n"
fun_upd_in_PiE: fixes x :: "'c"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd set" assumes "x \<notin> S"   and "f \<in> Pi\<^sub>E (insert x S) T" shows "f(x := undefined) \<in> Pi\<^sub>E S T"
Ch_unilateral_substitutes: shows "\<forall>h. TruePref_tax.unilateral_substitutes_on UNIV (Ch h)"
bet_neq32__neq: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C"   and "C \<noteq> B" shows "A \<noteq> C"
alloc_def: fixes x :: "'a"   and h :: "heap" shows "Ref.alloc x h = (let l = lim h; r = Ref l in (r, Ref.set r x (h\<lparr>lim := l + 1\<rparr>)))"
CCons_1: assumes "gallery (J # Js)"   and "{} \<notin> set (wall_crossings (J # Js))"   and "\<not> distinct (wall_crossings (J # Js))" shows "\<exists>f g As A B Bs E F Fs. (f, g) \<in> foldpairs \<and> A \<in> f \<turnstile> \<C> \<and> B \<in> g \<turnstile> \<C> \<and> E \<in> g \<turnstile> \<C> \<and> F \<in> f \<turnstile> \<C> \<and> (J # Js = As @ [A, B, F] @ Fs \<or> J # Js = As @ [A, B] @ Bs @ [E, F] @ Fs)"
set_intros_3: fixes ya :: "'d"   and x33 :: "'d ifex"   and x31 :: "'d"   and x32 :: "'d ifex" assumes "ya \<in> set_ifex x33" shows "ya \<in> set_ifex (IF x31 x32 x33)"
saturate_code: fixes f :: "'a \<Rightarrow> 'a"   and S :: "'a" shows "saturate f S = (let S' = f S in if S' = S then S else saturate f S')"
underS_incr: fixes r :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c" assumes "trans r"   and "antisym r"   and "(a, b) \<in> r" shows "underS r a \<subseteq> underS r b"
cases: fixes x :: "fml option" obtains \<phi> :: "fml" where "x = Afml \<phi>"   | "x = undeff"
denote_while_def: fixes M0 :: "complex mat"   and M1 :: "complex mat"   and DS :: "complex mat \<Rightarrow> complex mat"   and \<rho> :: "complex mat" shows "denote_while M0 M1 DS \<rho> = matrix_inf_sum d (\<lambda>n. denote_while_n M0 M1 DS n \<rho>)"