hcju/isabelleps huggingface dataset:
borel_measurable_sum: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M" shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
wf_and_no_antichain_imp_qo_extension_wf: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "wfp_on (strict P) A"   and "\<nexists>f. antichain_on P f A"   and "\<forall>x\<in>A. \<forall>y\<in>A. P x y \<longrightarrow> Q x y"   and "qo_on Q A" shows "wfp_on (strict Q) A"
intros_15: fixes x :: "name"   and y :: "name"   and P :: "pi" shows "<\<nu>x>x<y>.P \<equiv>\<^sub>e \<zero>"
lfinite_lset: fixes xs :: "'c llist" assumes "lfinite xs"   and "\<not> lnull xs" shows "llast xs \<in> lset xs"
Limsup_eq: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and net :: "'a filter" assumes "\<forall>\<^sub>F x in net. f x = g x" shows "Limsup net f = Limsup net g"
pred_set_fmap_def: fixes P :: "('c \<Rightarrow> 'd option) \<Rightarrow> bool" shows "pred_set_fmap P = (\<lambda>S. P (\<lambda>x. if x \<in> fst ` S then THE y. \<exists>z. y = Some z \<and> (x, z) \<in> S else None))"
qfree_lift_eq_qe: fixes aneg :: "'c \<Rightarrow> 'c fm"   and anormal :: "'c \<Rightarrow> bool"   and I\<^sub>a :: "'c \<Rightarrow> 'd list \<Rightarrow> bool"   and depends\<^sub>0 :: "'c \<Rightarrow> bool"   and decr :: "'c \<Rightarrow> 'c"   and solvable\<^sub>0 :: "'c \<Rightarrow> bool"   and trivial :: "'c \<Rightarrow> bool"   and subst\<^sub>0 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and qe :: "'c list \<Rightarrow> 'c fm"   and as :: "'c list" assumes "ATOM_EQ aneg anormal I\<^sub>a depends\<^sub>0 decr solvable\<^sub>0 trivial subst\<^sub>0"   and "\<And>as. \<forall>a\<in>set as. depends\<^sub>0 a \<Longrightarrow> qfree (qe as)"   and "\<forall>a\<in>set as. depends\<^sub>0 a" shows "qfree (ATOM_EQ.lift_eq_qe solvable\<^sub>0 trivial subst\<^sub>0 qe as)"
minus_linear_poly_def: fixes lp1 :: "linear_poly"   and lp2 :: "linear_poly" shows "lp1 - lp2 = lp1 + - lp2"
sendmsg_def: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b seq_action" shows "sendmsg P a \<equiv> case a of send m \<Rightarrow> P m | _ \<Rightarrow> True"
atom_map_def: fixes x :: "'a" shows "atom_map x = {y. atom y \<and> y \<le> x}"
inter_eqvt: fixes p :: "perm"   and A :: "'e set"   and B :: "'e set" shows "p \<bullet> (A \<inter> B) = p \<bullet> A \<inter> p \<bullet> B"
current_weight_mono: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and \<Gamma>' :: "('a, 'c) web_scheme" assumes "current \<Gamma> f"   and "edge \<Gamma> = edge \<Gamma>'"   and "A \<Gamma> = A \<Gamma>'"   and "B \<Gamma> = B \<Gamma>'"   and "\<And>x. weight \<Gamma> x \<le> weight \<Gamma>' x" shows "current \<Gamma>' f"
I_minWeightBasisG: fixes carrier :: "'c set"   and indep :: "'c set \<Rightarrow> bool"   and weight :: "'c \<Rightarrow> 'd"   and T :: "'c set"   and E :: "'c set" assumes "weighted_matroid carrier indep"   and "weighted_matroid.I_minWeightBasis carrier indep weight (T, E)" shows "weighted_matroid.no_smallest_element_skipped carrier indep weight E T"
distinct_1: fixes assertion :: "'d"   and name' :: "name"   and frame' :: "'d frame" shows "\<langle>\<epsilon>, assertion\<rangle> \<noteq> \<lparr>\<nu>name'\<rparr>frame'"
simps_4: fixes m1 :: "'b match_expr"   and m2 :: "'b match_expr" shows "normalize_match (MatchNot (MatchAnd m1 m2)) = normalize_match (MatchNot m1) @ normalize_match (MatchNot m2)"
match_ex_table_append_not_pcs: fixes pc :: "nat"   and xt0 :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and P :: "'a prog"   and C :: "String.literal"   and xt1 :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list" assumes "pc \<notin> pcs xt0" shows "match_ex_table P C pc (xt0 @ xt1) = match_ex_table P C pc xt1"
meridian_reflect_K2_isometry: shows "is_K2_isometry meridian_reflect"
Ln_of_real: fixes z :: "real" assumes "0 < z" shows "Ln (complex_of_real z) = complex_of_real (ln z)"
map_simp: fixes f :: "'a" assumes "A.arr f" shows "A.map f = f"
assms_4: shows "X \<in> P1 \<union> P2"
map_simp_ide: fixes a :: "'c" assumes "Cop.ide a" shows "\<Phi>'\<Psi>.map a = \<Psi>.map a \<cdot>\<^sub>S \<Phi>' a"
assms: shows "\<forall>i. f (Suc i) \<le> f i"
1: shows "finite (supp_fun s \<union> supp_fun t)"
connected_def: fixes u :: "nat"   and v :: "nat" shows "local.connected u v \<equiv> \<exists>p. isPath u p v"
fds_norm_def: fixes f :: "'b fds" shows "fds_norm f = fds (\<lambda>n. of_real (norm (fds_nth f n)))"
simps_2: fixes x :: "'e"   and xs :: "'e list" shows "butlast (x # xs) = (if xs = [] then [] else x # butlast xs)"
prod_mset_def: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.comm_monoid_mult times one" shows "comm_monoid_mult.prod_mset times one \<equiv> comm_monoid_mset.F times one"
sin_add: fixes x :: "'b"   and y :: "'b" shows "sin (x + y) = sin x * cos y + cos x * sin y"
concl_of_in_n_un_rf_n: shows "concl_of \<gamma> \<in> N \<union> Rf N"
less_le_not_le: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.preorder less_eq less" shows "less x y = (less_eq x y \<and> \<not> less_eq y x)"
D_def: shows "D \<equiv> EM_remainder' N (fs N) (real_of_int a) (real_of_int b)"
forall_\<Pi>\<^sub>1_def: shows "forall \<equiv> forall\<^sub>1"
R: shows "\<forall>i\<in>Basis. ((\<lambda>x. f x \<bullet> i) \<longlongrightarrow> l \<bullet> i) F"
in_orbitI: fixes f :: "'a perm"   and n :: "nat"   and a :: "'a"   and b :: "'a" assumes "(f ^ n) \<langle>$\<rangle> a = b" shows "b \<in> orbit f a"
False: shows "degree F - J + (degree G - J) \<noteq> 0"
circ_mult_upper_bound: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x \<le> z\<^sup>\<star>"   and "y \<le> z\<^sup>\<star>" shows "x * y \<le> z\<^sup>\<star>"
approx_emptyD: fixes prec :: "nat"   and arith :: "floatarith"   and ivl :: "float interval" assumes "approx prec arith [] = Some ivl" shows "interpret_floatarith arith [] \<in>\<^sub>r ivl"
count_in_diffI: fixes N :: "'k multiset"   and x :: "'k"   and M :: "'k multiset" assumes "\<And>n. count N x = n + count M x \<Longrightarrow> False" shows "x \<in># M - N"
order_tendstoI: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and y :: "'b"   and f :: "'c \<Rightarrow> 'b"   and F :: "'c filter" assumes "class.order_topology less_eq less open"   and "\<And>a. less a y \<Longrightarrow> \<forall>\<^sub>F x in F. less a (f x)"   and "\<And>a. less y a \<Longrightarrow> \<forall>\<^sub>F x in F. less (f x) a" shows "filterlim f (topological_space.nhds open y) F"
geq: shows "0 \<le> inner_prod v ((B - A) *\<^sub>v v)"
typ_of1_incr_bv: fixes Ts :: "typ list"   and t :: "Core.term"   and ty :: "typ"   and Ts' :: "typ list" assumes "typ_of1 Ts t = Some ty"   and "0 \<le> length Ts" shows "typ_of1 (take 0 Ts @ Ts' @ drop 0 Ts) (incr_bv (length Ts') 0 t) = Some ty"
denotational_semantics_similar: fixes \<rho> :: "Vars.var \<Rightarrow> Value"   and \<sigma> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue"   and e :: "exp" assumes "\<rho> \<triangleleft>\<triangleright>\<^sup>* \<sigma>" shows "\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub> \<triangleleft>\<triangleright> (\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<sigma>\<^esub>)\<cdot>C\<^sup>\<infinity>"
Grp_fst_snd: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "(BNF_Def.Grp {(x, y). R x y} fst)\<inverse>\<inverse> OO BNF_Def.Grp {(x, y). R x y} snd = R"
less_divide_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less a (divide b c) = (if less zero c then less (times a c) b else if less c zero then less b (times a c) else less a zero)"
dom_comp: fixes g :: "'s"   and f :: "'s" assumes "S.seq g f" shows "S.dom (g \<cdot> f) = S.dom f"
retraction_is_epi: fixes g :: "'b" assumes "B.retraction g" shows "B.epi g"
\<pi>_lower_bound_ge_strong: fixes n :: "nat" assumes "1 \<le> n" shows "ln 2 / 2 * real (2 * n) / ln (real (2 * n)) \<le> \<pi> (real (2 * n))"
hyps: assumes "a \<in> outstanding_refs is_Write\<^sub>s\<^sub>b xs" shows "\<exists>sop volatile v ys zs A L R W. xs = ys @ Write\<^sub>s\<^sub>b volatile a sop v A L R W # zs"
simps_11: fixes t1 :: "Syntax.term"   and t2 :: "Syntax.term"   and t3 :: "Syntax.term"   and t' :: "Syntax.term"   and x :: "Syntax.var" shows "Case t1 t2 t3[t' / x] = Case (t1[t' / x]) (t2[t' / x]) (t3[t' / x])"
proots_within_pCons_1_iff_1: fixes a :: "'b"   and s :: "'b set" shows "proots_within [:- a, 1::'b:] s = (if a \<in> s then {a} else {})"
stutter_equiv_refl: fixes \<sigma> :: "nat \<Rightarrow> 'f" shows "\<sigma> \<approx> \<sigma>"
del_edge_undirected_degree_plus': fixes g :: "('a, 'b) graph"   and v :: "'a"   and e :: "'b"   and v' :: "'a" assumes "finite (edges g)"   and "(v, e, v') \<in> edges g"   and "(v', e, v) \<in> edges g" shows "degree v' (del_unEdge v e v' g) + 1 = degree v' g"
summable_comparison_test_bigo': fixes g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'b" assumes "summable (\<lambda>n. norm (g n))"   and "(\<lambda>n. norm (f n)) \<in> O(\<lambda>n. norm (g n))" shows "summable f"
gpv'_def: shows "gpv' = map_gpv f id gpv"
eucl_le: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and 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 inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and abs :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less" shows "less_eq x y = (\<forall>i\<in>Basis. inner x i \<le> inner y i)"
extend_1: shows "reduced_run q v\<^sub>1 ([a] @- v\<^sub>2) l w w\<^sub>1 w\<^sub>2 u"
simps: fixes a :: "'a literal multiset"   and CC :: "'a literal multiset set" shows "(a \<in> saturate CC) = ((\<exists>C. a = C \<and> C \<in> CC) \<or> (\<exists>CC' D E. a = E \<and> Infer CC' D E \<in> \<Gamma> \<and> (\<forall>x. x \<in># CC' \<longrightarrow> x \<in> saturate CC) \<and> D \<in> saturate CC))"
notempty: shows "\<forall>e\<in>S0. 0 < length e"
simps: fixes x :: "ordinal" shows "to_cnf x = (if x = 0 then [] else oLog \<omega> x # to_cnf (x - \<omega> ** oLog \<omega> x))"
l9_4_1_aux: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a"   and C :: "'a"   and R :: "'a"   and A :: "'a"   and P :: "'a"   and Q :: "'a"   and M :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Le Bet Cong S C R A"   and "Tarski_neutral_dimensionless.TS Bet P Q A C"   and "Tarski_neutral_dimensionless.Col Bet R P Q"   and "Tarski_neutral_dimensionless.Perp Bet Cong P Q A R"   and "Tarski_neutral_dimensionless.Col Bet S P Q"   and "Tarski_neutral_dimensionless.Perp Bet Cong P Q C S"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong M R S" shows "\<forall>U C'. Tarski_neutral_dimensionless.Midpoint Bet Cong M U C' \<longrightarrow> Tarski_neutral_dimensionless.Out Bet R U A = Tarski_neutral_dimensionless.Out Bet S C C'"
simps: fixes x :: "'b extN"   and y :: "'b extN" shows "x * y = x \<sqinter> y"
fundfold_trivial_f\<C>: fixes r :: "'a permutation"   and C :: "'a set" assumes "r \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "\<forall>(f, g)\<in>fundfoldpairs. r = Abs_induced_automorph f g \<longrightarrow> C \<in> f \<turnstile> \<C>" shows "fst (Spair r) ` C = C"
rep_eq: fixes x :: "('a, 'b) lifted_distrib_lattice_top"   and xa :: "('a, 'b) lifted_distrib_lattice_top" shows "Rep_lifted_distrib_lattice_top (x \<squnion> xa) = (\<lambda>f. Rep_lifted_distrib_lattice_top x f \<squnion> Rep_lifted_distrib_lattice_top xa f)"
cyclic_groupI2: fixes a :: "'a" assumes "a \<in> carrier (G\<lparr>carrier := generate G {a}\<rparr>)" shows "Finite_And_Cyclic_Groups.cyclic_group (G\<lparr>carrier := generate G {a}, carrier := generate (G\<lparr>carrier := generate G {a}\<rparr>) {a}\<rparr>) a"
Diagonalize_preserves_Ide: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Ide C a" shows "monoidal_language.Ide C (monoidal_language.Diagonalize C a)"
StrongPostdomination_axioms: shows "StrongPostdomination sourcenode targetnode kind valid_edge (_Entry_) (_Exit_)"
incl_Rel_vcard: fixes A :: "V"   and B :: "V" shows "vcard (incl_Rel A B) = 3\<^sub>\<nat>"
hcomp_obj_arr: fixes b :: "'b"   and f :: "'b" assumes "D.obj b"   and "D.arr f"   and "b = trg\<^sub>D f" shows "b \<star>\<^sub>D f = f"
get_root_node_ptr_in_heap: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) 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_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_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_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs get_disconnected_nodes get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs"   and "h \<turnstile> ok get_root_node ptr" shows "ptr |\<in>| object_ptr_kinds h"
sim_kripke: fixes M :: "('e, 'f, 'g) KripkeStructure"   and M' :: "('e, 'f, 'h) KripkeStructure"   and f :: "'g \<Rightarrow> 'h" assumes "sim M M' f"   and "kripke M" shows "kripke M'"
fixp_preserves_mcont1: fixes F :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd"   and lubb :: "'c set \<Rightarrow> 'c"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. lfp.mono_body (\<lambda>f. F f x)"   and "f \<equiv> lfp.fixp_fun F"   and "\<And>f. mcont lubb ordb Sup (\<le>) f \<Longrightarrow> mcont lubb ordb Sup (\<le>) (F f)" shows "mcont lubb ordb Sup (\<le>) f"
eval_Neg: fixes \<phi> :: "atom fm"   and L :: "real list" shows "eval (Neg \<phi>) L = eval (neg \<phi>) L"
pdevs_apply_One_slist: fixes i :: "nat" shows "pdevs_apply (Pdevs One_slist) i = (if i < length Basis_list then Basis_list ! i else (0::'d))"
eventually_sequentiallyI: fixes c :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "\<And>x. c \<le> x \<Longrightarrow> P x" shows "eventually P sequentially"
lprefix_trans: fixes xs :: "'c llist"   and ys :: "'c llist"   and zs :: "'c llist" assumes "xs \<sqsubseteq> ys"   and "ys \<sqsubseteq> zs" shows "xs \<sqsubseteq> zs"
assms_2: shows "p \<in> punit_dgrad_max_set d"
star_of_approx_one_iff: fixes k :: "'a" shows "(star_of k \<approx> 1) = (k = (1::'a))"
hrcis_zero_arg: fixes r :: "real star" shows "hrcis r 0 = hcomplex_of_hypreal r"
recurrence_series_infinite: fixes f :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M" shows "AE x in M. 0 < f x \<longrightarrow> (\<Sum>n. f ((T ^^ n) x)) = \<infinity>"
norm_infsetsum_bound: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "norm (infsetsum f A) \<le> (\<Sum>\<^sub>ax\<in>A. norm (f x))"
assms_2: fixes x :: "'a"   and x' :: "'b" assumes "(x, x') \<in> R'" shows "f x \<le> \<Down>\<^sub>E E R (f' x')"
sup_measure_def: fixes A :: "'a measure"   and B :: "'a measure" shows "sup A B = sup_lexord A B space (sigma (space A \<union> space B) {}) (sup_lexord A B sets (sigma (space A) (sets A \<union> sets B)) (sup_measure' A B))"
casexxy: fixes \<sigma> :: "nat list"   and as :: "nat list"   and x :: "nat"   and bs :: "nat list"   and cs :: "nat list"   and init :: "nat list"   and i :: "nat"   and h :: "nat list" assumes "\<sigma> = as @ [x] @ bs @ [x] @ cs"   and "x \<notin> set cs"   and "set cs \<subseteq> set init"   and "x \<in> set init"   and "distinct init"   and "x \<notin> set bs"   and "set as \<subseteq> set init"   and "set bs \<subseteq> set init" shows "i < length cs \<longrightarrow> (\<forall>j<i. cs ! j \<noteq> cs ! i) \<longrightarrow> cs ! i \<noteq> x \<longrightarrow> cs ! i \<notin> set bs \<longrightarrow> x < cs ! i in s_TS init h \<sigma> (length (as @ [x] @ bs @ [x]) + i + 1)"
domIff: fixes a :: "'d"   and m :: "'d \<Rightarrow> \<langle>'e\<rangle>\<^sub>\<bottom>" shows "(a \<in> dom m) = (m a \<noteq> None)"
set_map_to_operator_atom: fixes t :: "nat"   and \<Pi> :: "('b, 'c) strips_problem_scheme"   and v :: "'d"   and vs :: "'d list" shows "set (map (\<lambda>op. Atom (Operator t (index (\<Pi>\<^sub>\<O>) op))) (filter (\<lambda>op. ListMem v vs) (\<Pi>\<^sub>\<O>))) = {Atom (Operator t (index (\<Pi>\<^sub>\<O>) op)) |op. op \<in> set (\<Pi>\<^sub>\<O>) \<and> v \<in> set vs}"
set_list_diff: fixes xs :: "'c list"   and ys :: "'c list" shows "set (list_diff xs ys) = set xs - set ys"
effect_success: fixes c :: "'a Heap"   and h :: "heap"   and h' :: "heap"   and r :: "'a" assumes "effect c h h' r" shows "success c h"
invertible_matrix_imp_invertible_lf: fixes A :: "(('a, 'b) vec, 'b) vec" assumes "invertible A" shows "invertible_lf (*s) (*s) ((*v) A)"
add_commute: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd" shows "a \<otimes> b = b \<otimes> a"
execute_fin_word_invisible: fixes w :: "'action list"   and p :: "'state" assumes "path w p"   and "set w \<subseteq> invisible" shows "int (target w p) = int p"
rel_gpv''_lift_spmf: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and p :: "'d spmf"   and q :: "'e spmf" shows "rel_gpv'' A C R (lift_spmf p) (lift_spmf q) = rel_spmf A p q"
intro: fixes s :: "'a"   and S :: "('a, 'b) ring_scheme" assumes "s \<in> carrier S" shows "UP_univ_prop_axioms S s"
DS: fixes \<rho> :: "complex mat" assumes "\<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>" shows "positive (denote S \<rho>) \<and> trace (denote S \<rho>) \<le> trace \<rho> \<and> denote S \<rho> \<in> carrier_mat d d"
index_mono: fixes a :: "nat"   and a0 :: "nat"   and b :: "nat"   and b0 :: "nat" assumes "a < a0"   and "b < b0" shows "a * b0 + b < a0 * b0"
repeatM_plus: fixes n :: "nat"   and m :: "nat"   and f :: "('c, 'd) state" shows "repeatM (n + m) f = pure (@) \<diamondop> repeatM n f \<diamondop> repeatM m f"
intro: fixes F :: "('d \<Rightarrow> bool) \<Rightarrow> bool" assumes "F (\<lambda>x. True)"   and "\<And>P Q. \<lbrakk>F P; F Q\<rbrakk> \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)"   and "\<And>P Q. \<lbrakk>\<forall>x. P x \<longrightarrow> Q x; F P\<rbrakk> \<Longrightarrow> F Q" shows "is_filter F"
poly_inf_def: fixes p :: "'a poly" shows "poly_inf p \<equiv> sgn (lead_coeff p)"
multiplicity_decompose': 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 x :: "'a"   and p :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "\<not> dvd.dvd times p one" obtains y :: "'a" where "x = times (power.power one times p (factorial_semiring.multiplicity times one p x)) y" and "\<not> dvd.dvd times p y"
compose_def: fixes A :: "'b set"   and g :: "'d \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'd" shows "compose A g f = (\<lambda>x\<in>A. g (f x))"
sdrop_while_SCons: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a"   and s :: "'a stream" shows "sdrop_while P (a ## s) = (if P a then sdrop_while P s else a ## s)"
WTALength: fixes is_lub :: "ty \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> bool"   and P :: "'b J_prog"   and E :: "String.literal \<Rightarrow> ty option"   and a :: "'b expr"   and T :: "ty" assumes "is_lub,P,E \<turnstile> a :: T\<lfloor>\<rceil>" shows "is_lub,P,E \<turnstile> a\<bullet>length :: Integer"
OAlist_tc_map2_val_def: fixes f :: "'e \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> 'f"   and xs :: "('e, 'g) oalist_tc"   and ys :: "('e, 'h) oalist_tc" shows "OAlist_tc_map2_val f xs ys = OAlist_tc (map2_val_pair_tc f (map_val_pair (\<lambda>k v. f k v (0::'h))) (map_val_pair (\<lambda>k. f k (0::'g))) (list_of_oalist_tc xs) (list_of_oalist_tc ys))"
deg_comp_zero_min: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and u :: "'a"   and v :: "'a" assumes "comparator cmp"   and "snd (rep_nat_term u) = snd (rep_nat_term v)"   and "fst (rep_nat_term u) = 0" shows "deg_comp cmp u v \<noteq> Gt"
DERIV_exp: fixes x :: "'a" shows "(exp has_field_derivative exp x) (at x)"
set_iterator_product_foldli_conv: fixes as :: "'f list"   and bs :: "'f \<Rightarrow> 'g list" shows "set_iterator_product (foldli as) (\<lambda>a. foldli (bs a)) = foldli (concat (map (\<lambda>a. map (Pair a) (bs a)) as))"
neg_closed: fixes g :: "'g" assumes "g \<in> H" shows "- g \<in> H"
set_empty2: fixes xs :: "'b list" shows "({} = set xs) = (xs = \<epsilon>)"
abs_eq: fixes xa :: "'b poly"   and x :: "'b poly" shows "xa + x = Abs_poly (\<lambda>n. coeff xa n + coeff x n)"
bij_betw_imp_surj: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "bij_betw f A UNIV" shows "surj f"
assms_2: assumes "(m, m') \<in> R" shows "f m \<le> \<Down> S (f' m')"
v: fixes n :: "nat" shows "dist (f (v n)) (u n) \<le> C"
nonempty: fixes x :: "'c"   and A :: "'c set" assumes "x \<in> A" shows "A \<noteq> {}"
R: shows "(w', w'') \<in> (\<Union> (relations (gen_model M w) ` as))\<^sup>+"
sgn_scaleR: fixes r :: "real"   and x :: "'a" shows "sgn (r *\<^sub>R x) = sgn r *\<^sub>R sgn x"