hcju/isabelleps huggingface dataset:
pderiv_pCons: fixes a :: "'b"   and p :: "'b poly" shows "pderiv (pCons a p) = p + pCons (0::'b) (pderiv p)"
axiom_iffimp1_def: fixes p :: "fol fm"   and q :: "fol fm" shows "axiom_iffimp1 p q \<equiv> Thm (Imp (Iff p q) (Imp p q))"
rng_rng_of_frac: shows "ring rec_rng_of_frac"
induced_id_Abs_freelist_conv_sum_list: fixes ss :: "'g list" assumes "ss \<in> lists S" shows "GroupByPresentationInducedFun.induced_hom S P id (BinOpSetGroup.Abs_G_perm (FreeGroup S // Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` P\<rangle>) (FreeGroup S)) (+) (Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` P\<rangle>) (FreeGroup S) `` {Abs_freelist ss})) = sum_list ss"
A3_reversed: fixes a :: "'p"   and b :: "'p"   and c :: "'p" assumes "a a \<equiv> b c" shows "b = c"
leg0_is_map: shows "is_left_adjoint r\<^sub>0s\<^sub>1.p\<^sub>0"
assms_1: shows "(\<lambda>n. f (real n)) \<longlonglongrightarrow> c"
assms_1: shows "Quotient3 R1 Abs1 Rep1"
suc_ofilter_in: fixes A :: "'a set"   and b :: "'a" assumes "local.ofilter A"   and "local.AboveS A \<noteq> {}"   and "(b, local.suc A) \<in> r"   and "b \<noteq> local.suc A" shows "b \<in> A"
closed_subdiagonal: shows "closed {(x, y) |x y. x \<le> y}"
redT_updW_simps_3: fixes t :: "'b"   and ws :: "'b \<Rightarrow> 'c wait_set_status option"   and w :: "'c"   and ws' :: "'b \<Rightarrow> 'c wait_set_status option" shows "redT_updW t ws (Suspend w) ws' = (ws' = ws(t \<mapsto> InWS w))"
generate_pow: fixes G :: "('e, 'f) monoid_scheme"   and a :: "'e" assumes "Group.group G"   and "a \<in> carrier G" shows "generate G {a} = {a [^]\<^bsub>G\<^esub> k |k. k \<in> UNIV}"
CallNone\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and a :: "nat"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and ps :: "nat exp list"   and vs :: "val list"   and h\<^sub>2 :: "heap"   and ls\<^sub>2 :: "val list"   and sh\<^sub>2 :: "sheap"   and C :: "char list"   and fs :: "Objects.fields"   and M :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>ps,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>map Val vs,(h\<^sub>2, ls\<^sub>2, sh\<^sub>2)\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "\<nexists>b Ts T body D. P \<turnstile> C sees M, b :  Ts\<rightarrow>T = body in D" shows "P \<turnstile>\<^sub>1 \<langle>e\<bullet>M(ps),s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NoSuchMethodError,(h\<^sub>2, ls\<^sub>2, sh\<^sub>2)\<rangle>"
elims_2: fixes x :: "nat"   and xa :: "form"   and xb :: "nat"   and xc :: "nat"   and xd :: "float interval"   and xe :: "float interval option list"   and xf :: "nat list" assumes "approx_form' x xa xb xc xd xe xf" obtains prec :: "nat" and f :: "form" and n :: "nat" and ivl :: "float interval" and bs :: "float interval option list" and ss :: "nat list" where "x = prec__" and "xa = f__" and "xb = 0" and "xc = n__" and "xd = ivl__" and "xe = bs__" and "xf = ss__" and "approx_form prec__ f__ (bs__[n__ := Some ivl__]) ss__"   | prec :: "nat" and f :: "form" and s :: "nat" and n :: "nat" and ivl :: "float interval" and bs :: "float interval option list" and ss :: "nat list" where "x = prec" and "xa = f" and "xb = Suc s" and "xc = n" and "xd = ivl" and "xe = bs" and "xf = ss" and "let (ivl1, ivl2) = split_float_interval ivl in if approx_form' prec f s n ivl1 bs ss then approx_form' prec f s n ivl2 bs ss else False"
select_convs_4: fixes states :: "'b \<Rightarrow> 'c"   and msgs :: "nat \<Rightarrow> 'a message list"   and process_snapshot :: "'b \<Rightarrow> 'c option"   and channel_snapshot :: "nat \<Rightarrow> 'a list \<times> recording_state"   and more :: "'d" shows "cs \<lparr>states = states, msgs = msgs, process_snapshot = process_snapshot, channel_snapshot = channel_snapshot, \<dots> = more\<rparr> = channel_snapshot"
independentD: fixes s :: "'c set"   and t :: "'c set"   and u :: "'c \<Rightarrow> 'a"   and v :: "'c" assumes "m2.independent s"   and "finite t"   and "t \<subseteq> s"   and "(\<Sum>v\<in>t. u v *b v) = (0::'c)"   and "v \<in> t" shows "u v = (0::'a)"
TBasic: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<phi> \<^bold>\<rightarrow> \<^bold>\<diamond>\<phi> in v]"
finite_occurrences_val_expr_2: fixes e :: "('r, 'l, 'v) expr" shows "finite (RID\<^sub>E e)"
seq_inj: fixes x :: "'a"   and i :: "nat"   and j :: "nat"   and p :: "nat"   and n :: "nat"   and q :: "nat" assumes "seq' x i = seq' x j"   and "p = i + n"   and "q = j + n" shows "seq' x p = seq' x q"
inject_1: fixes x3 :: "'a"   and y3 :: "'a" shows "(fm.Atom x3 = fm.Atom y3) = (x3 = y3)"
angleI: fixes A :: "state \<times> state \<Rightarrow> bool"   and s :: "state"   and t :: "state"   and v :: "state \<Rightarrow> 'a" assumes "A (s, t)"   and "v t \<noteq> v s" shows "(s, t) \<Turnstile> <A>_v"
width_V_empty: fixes G :: "('a, 'b) Graph_scheme"   and T :: "('c, 'd) Graph_scheme"   and bag :: "'c \<Rightarrow> 'a set" assumes "TreeDecomposition G T bag"   and "V\<^bsub>G\<^esub> = {}" shows "TreeDecomposition.width T bag = 0"
pinduct: fixes a0 :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "recfun_aux_dom a0"   and "\<And>x. \<lbrakk>recfun_aux_dom x; \<not> Q x \<Longrightarrow> P (h x)\<rbrakk> \<Longrightarrow> P x" shows "P a0"
rep_eq: fixes x :: "'a poly"   and xa :: "'a poly" shows "coeff (x + xa) = (\<lambda>n. coeff x n + coeff xa n)"
case_prodE': fixes c :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and p :: "'c \<times> 'd"   and z :: "'e" assumes "(case p of (a, b) \<Rightarrow> c a b) z" obtains x :: "'c" and y :: "'d" where "p = (x, y)" and "c x y z"
Form_Body_unique: fixes ka :: "nat"   and kb :: "nat"   and xs :: "nat list"   and ys :: "nat list"   and zs :: "nat list"   and zs' :: "nat list" assumes "Form_Body ka kb xs ys zs"   and "Form_Body ka kb xs ys zs'"   and "kb \<le> ka"   and "ka \<le> Suc kb" shows "zs' = zs"
Valid_def: fixes p :: "'a set"   and c :: "'a com"   and q :: "'a set" shows "{p}   c   {q} \<equiv> \<forall>s s'. Sem c s s' \<longrightarrow> s \<in> Some ` p \<longrightarrow> s' \<in> Some ` q"
snoc_eq_iff_butlast: fixes xs :: "'b list"   and x :: "'b"   and ys :: "'b list" shows "(xs @ [x] = ys) = (ys \<noteq> [] \<and> butlast ys = xs \<and> last ys = x)"
suc_n_kess_a: shows "Suc n < mod_type_class.to_nat a"
psimps: fixes k :: "nat"   and xs :: "'a list" assumes "mom_select_dom (k, xs)" shows "mom_select k xs = (if length xs \<le> 20 then slow_select k xs else let M = mom_select (((length xs + 4) div 5 - 1) div 2) (map slow_median (chop 5 xs)); (ls, es, gs) = partition3 M xs in if k < length ls then mom_select k ls else if k < length ls + length es then M else mom_select (k - length ls - length es) gs)"
avg_aux_def: fixes s :: "'s" shows "avg_aux s \<equiv> iterate s (\<lambda>x (c, s). (c + 1, s + x)) (0, 0)"
sigma_finite_measure_distr: fixes M :: "'d measure"   and N :: "'c measure"   and f :: "'d \<Rightarrow> 'c" assumes "sigma_finite_measure (distr M N f)"   and "f \<in> M \<rightarrow>\<^sub>M N" shows "sigma_finite_measure M"
ClassCastC_def: shows "ClassCastC \<equiv> (ClassCast, [], [], [])"
b: fixes a :: "bit list" shows "length (remzero a) \<le> length a"
well_foundedI_extremal: assumes "\<And>X. \<lbrakk>X \<subseteq> A; X \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists>x\<in>X. \<forall>y\<in>X. \<not> y \<sqsubset> x" shows "well_founded A (\<sqsubset>)"
hyps_thms_if: fixes p :: "'b pl"   and tt :: "'b set" shows "hyps p tt |- (if tt[[p]] then p else p -> false)"
id_act: fixes G :: "('d, 'e) monoid_scheme"   and action :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and x :: "'f" assumes "orbit_stabiliser G action" shows "action \<one>\<^bsub>G\<^esub> x = x"
lookup_eq_value: fixes v :: "'b"   and xs :: "'x"   and k :: "'a" assumes "v \<noteq> (0::'b)" shows "(lookup xs k = v) = ((k, v) \<in> set (fst (list_of_oalist xs)))"
Ge_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" shows "A BGe C D \<equiv> C D Le A B"
fmapU_eq_bindU: fixes fmapU :: "(udom \<rightarrow> udom) \<rightarrow> udom\<cdot>'a \<rightarrow> udom\<cdot>'a"   and returnU :: "udom \<rightarrow> udom\<cdot>'a"   and bindU :: "udom\<cdot>'a \<rightarrow> (udom \<rightarrow> udom\<cdot>'a) \<rightarrow> udom\<cdot>'a"   and f :: "udom \<rightarrow> udom"   and xs :: "udom\<cdot>'a" assumes "class.monad fmapU returnU bindU" shows "fmapU\<cdot>f\<cdot>xs = bindU\<cdot>xs\<cdot>(\<Lambda> x. returnU\<cdot>(f\<cdot>x))"
check_in_heap_def: fixes ptr :: "(_) object_ptr" shows "check_in_heap ptr = Heap_Error_Monad.bind get_heap (\<lambda>h. if ptr |\<in>| object_ptr_kinds h then return () else error SegmentationFault)"
simps_16: fixes x2 :: "'a" shows "syms_hd (Sym x2) = {x2}"
prf_key_gen_lossless: shows "lossless_spmf prf_key_gen"
row_bra: fixes v :: "complex vec" shows "row \<langle>|v\<rangle>| 0 = vec (dim_vec v) (\<lambda>i. cnj (v $ i))"
coeff_smult: fixes a :: "'d"   and p :: "'d poly"   and n :: "nat" shows "coeff (smult a p) n = a * coeff p n"
weak_transition_stepI: fixes P :: "'state"   and \<alpha> :: "('act, 'pred) S_action"   and P' :: "'state" assumes "P \<Rightarrow>{\<alpha>}\<^sub>S P'" shows "P \<Rightarrow>\<langle>\<alpha>\<rangle>\<^sub>S P'"
subprob_spaceI: fixes M :: "'c measure" assumes "emeasure M (space M) \<le> 1"   and "space M \<noteq> {}" shows "subprob_space M"
Match_2: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and a :: "name" assumes "P \<Longrightarrow>\<alpha> \<prec> P'" shows "[a\<frown>a]P \<Longrightarrow>\<alpha> \<prec> P'"
other: fixes \<sigma> :: "'a"   and s :: "'b"   and \<sigma>' :: "'a" assumes "(\<sigma>, s) \<in> oreachable A S U"   and "U \<sigma> \<sigma>'"   and "P (\<sigma>, s)" shows "P (\<sigma>', s)"
wf_while_option_Some: fixes P :: "'b \<Rightarrow> bool"   and b :: "'b \<Rightarrow> bool"   and c :: "'b \<Rightarrow> 'b"   and s :: "'b" assumes "wf {(t, s). (P s \<and> b s) \<and> t = c s}"   and "\<And>s. \<lbrakk>P s; b s\<rbrakk> \<Longrightarrow> P (c s)"   and "P s" shows "\<exists>t. while_option b c s = Some t"
finite_ir_commit: fixes runz :: "fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option"   and A :: "agent"   and B :: "agent"   and Kab :: "key"   and Nb :: "fresh_t" assumes "finite (dom runz)" shows "finite (ri_running runz A B Kab Nb)"
asm: shows " N \<longrightarrow>\<^sub>1 N'"
Ri_def: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist"   and N :: "'a literal multiset set" shows "src.Ri Sts N = {\<gamma> \<in> gd.ord_\<Gamma> Sts. src.redundant_infer N \<gamma>}"
v11_vrange_ex1_eq: fixes b :: "V" shows "(b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vint_uminus) = (\<exists>!a. a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vint_uminus \<and> -\<^sub>\<int> a = b)"
f3: shows "\<forall>x y. x \<in> carrier R \<and> y \<in> carrier R \<longrightarrow> rng_to_rng_of_frac (x \<oplus> y) = rng_to_rng_of_frac x \<oplus>\<^bsub>rec_rng_of_frac\<^esub> rng_to_rng_of_frac y"
assms_4: shows "(c, s) \<rightarrow>c (c', s')"
ssubst: fixes t :: "'a"   and s :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "t = s"   and "P s" shows "P t"
mann2: fixes y :: "'a"   and x :: "'a" shows "y = - (- (- (- x \<squnion> y) \<squnion> x \<squnion> y \<squnion> y) \<squnion> - (- x \<squnion> y))"
\<omega>_limit_set_empty: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and x :: "'b" assumes "auto_ll_on_open f X"   and "\<not> {0..} \<subseteq> auto_ll_on_open.existence_ivl0 f X x" shows "auto_ll_on_open.\<omega>_limit_set f X x = {}"
proj_append_1: fixes l :: "'e"   and A' :: "('e strand_label \<times> 'f) list"   and B' :: "('e strand_label \<times> 'f) list" shows "proj l (A' @ B') = proj l A' @ proj l B'"
upd_gobj_def: fixes r :: "oref"   and n :: "vn"   and v :: "val" shows "upd_gobj r n v = case_st (\<lambda>g. st (chg_map (upd_obj n v) r g))"
lift_inference: fixes S :: "'a eclause set" assumes "inference_saturated S" shows "ground_inference_saturated S"
analz_cut: fixes Y :: "msg"   and X :: "msg"   and H :: "msg set" assumes "Y \<in> analz (insert X H)"   and "X \<in> analz H" shows "Y \<in> analz H"
subst_unrest: fixes x :: "'a \<Longrightarrow> 'b"   and P :: "('c, 'b) uexpr"   and \<sigma> :: "'d \<Rightarrow> 'b"   and v :: "('a, 'd) uexpr" assumes "x \<sharp> P" shows "\<sigma>(x \<mapsto>\<^sub>s v) \<dagger> P = \<sigma> \<dagger> P"
gt_chopI: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "is_App t"   and "chop t \<ge>\<^sub>t s" shows "gt_chop t s"
elem_set_implies_elem_tl_app_cons: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list"   and y :: "'a" assumes "x \<in> set xs" shows "x \<in> set (tl (ys @ y # xs))"
emeasure_density: fixes f :: "'c \<Rightarrow> ennreal"   and M :: "'c measure"   and A :: "'c set" assumes "f \<in> borel_measurable M"   and "A \<in> sets M" shows "emeasure (density M f) A = set_nn_integral M A f"
lower_inv_eq: fixes x :: "'c" assumes "x \<in> carrier \<X>\<^bsub>F\<^esub>" shows "\<pi>\<^sub>*\<^bsub>F\<^esub> (\<pi>\<^sup>*\<^bsub>F\<^esub> x) = x"
ide_src: fixes \<mu> :: "'b" assumes "D.arr \<mu>" shows "D.ide (src\<^sub>D \<mu>)"
StableNoDecomp_XD_XH: fixes XD :: "X4 set"   and XH :: "X4 set" shows "StableNoDecomp.stable_pair_on UNIV (XD, XH) = (XD = {} \<and> XH = {Xd1, Xd1', Xd2, Xd2'})"
H: shows "\<forall>l\<in>L. \<forall>u\<in>U. l < u"
matrix_to_iarray_nth: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and j :: "'b" shows "matrix_to_iarray A !! mod_type_class.to_nat i !! mod_type_class.to_nat j = A $ i $ j"
2: fixes aa :: "ereal"   and ca :: "ereal"   and ba :: "ereal" assumes "interval_lebesgue_integrable lborel aa ca f"   and "aa \<le> ba"   and "ba \<le> ca" shows "interval_lebesgue_integral lborel ba ca f + interval_lebesgue_integral lborel aa ba f = interval_lebesgue_integral lborel aa ca f"
set_seq_def: fixes s :: "'b option set" shows "set_seq s \<equiv> if \<forall>x\<in>s. x \<noteq> None then Some (the ` s) else None"
order_less_irrefl: fixes x :: "'g" shows "\<not> x < x"
cancel_semigroup_axioms: shows "cancel_semigroup (\<oplus>\<^sub>a)"
order_pcompose: fixes p :: "'b poly"   and q :: "'b poly"   and x :: "'b" assumes "p \<circ>\<^sub>p q \<noteq> 0" shows "order x (p \<circ>\<^sub>p q) = order x (q - [:poly q x:]) * order (poly q x) p"
seq_char: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "C.CC.seq g f = (C.seq (fst g) (fst f) \<and> C.seq (snd g) (snd f))"
1: shows "arr_to_hfun (hfun_to_arr A C (hcomp G F)) = arr_to_hfun (local.comp (hfun_to_arr B C G) (hfun_to_arr A B F))"
consistency_onD: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set"   and B :: "'d set"   and C :: "'d set" assumes "consistency_on A f"   and "B \<subseteq> A"   and "C \<subseteq> B"   and "f B \<subseteq> C" shows "f B = f C"
segment_vertical: fixes a :: "(real, 2) vec"   and b :: "(real, 2) vec"   and x :: "(real, 2) vec" assumes "a $ 1 = b $ 1" shows "(x \<in> closed_segment a b) = (x $ 1 = a $ 1 \<and> x $ 1 = b $ 1 \<and> (a $ 2 \<le> x $ 2 \<and> x $ 2 \<le> b $ 2 \<or> b $ 2 \<le> x $ 2 \<and> x $ 2 \<le> a $ 2))"
that: fixes A :: "'a multiset"   and C :: "'a zmultiset"   and B :: "'a multiset" assumes "M = zmset_of A + C"   and "N = zmset_of B + C"   and "A \<subseteq># B" shows "thesis"
rogers_fixed_point_theorem: fixes f :: "recf"   and k :: "nat" assumes "recfn 1 f"   and "Partial_Recursive.total f" shows "\<exists>n\<ge>k. \<forall>x. eval r_phi [n, x] = eval r_phi [the (eval f [n]), x]"
search_l: shows "search l x = has_overlap (Tree2.set_tree l) x"
parallel_fix_ind: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and F :: "'g \<rightarrow> 'g"   and G :: "'h \<rightarrow> 'h" assumes "adm (\<lambda>x. P (fst x) (snd x))"   and "P \<bottom> \<bottom>"   and "\<And>x y. P x y \<Longrightarrow> P (F\<cdot>x) (G\<cdot>y)" shows "P (fix\<cdot>F) (fix\<cdot>G)"
col_def: fixes m :: "'b list list"   and i :: "nat" shows "col m i \<equiv> m ! i"
det_interchange_columns: fixes A :: "(('c, 'd) vec, 'd) vec"   and i :: "'d"   and j :: "'d" shows "det (interchange_columns A i j) = of_int (if i = j then 1 else - 1) * det A"
bsok_simps_9: fixes a :: "'a expr1"   and i :: "'a expr1"   and n :: "nat" shows "bsok (a\<lfloor>i\<rceil>) n = (bsok a n \<and> bsok i n)"
axioms_1: fixes f :: "'c \<Rightarrow> 'd" assumes "bounded_clinear f" shows "clinear f"
finish_simps_2: fixes u :: "'c"   and s :: "('c, 'd) state_scheme" shows "discovered (param_DFS_defs.finish u s) = discovered s"
shiftr_bl: fixes x :: "'b word"   and n :: "nat" shows "x >> n \<equiv> of_bl (take (LENGTH('b) - n) (to_bl x))"
le_add_same_cancel1: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq a (plus a b) = less_eq zero b"
fmlookup_ran_iff: fixes y :: "'c"   and m :: "('d, 'c) fmap" shows "(y |\<in>| fmran m) = (\<exists>x. fmlookup m x = Some y)"
agree_plus1: fixes \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec"   and \<nu>' :: "(real, 'd) vec \<times> (real, 'd) vec"   and t1 :: "('e, 'd) trm"   and t2 :: "('e, 'd) trm" assumes "Vagree \<nu> \<nu>' (FVDiff (Plus t1 t2))" shows "Vagree \<nu> \<nu>' (FVDiff t1)"
usubstappf_and_usubstappp_antimon_2: fixes V :: "variable set"   and U :: "variable set"   and \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and \<alpha> :: "game" assumes "V \<subseteq> U"   and "snd (usubstappp \<sigma> U \<alpha>) \<noteq> undefg" shows "snd (usubstappp \<sigma> U \<alpha>) = snd (usubstappp \<sigma> V \<alpha>)"
resample_conv_while: shows "resample A B = spmf_of_set A \<bind> local.while"
\<K>_def: fixes kripke :: "('b, 'c) kripke" shows "\<K> kripke = (case kripke of Kripke x1 x2 \<Rightarrow> x2)"
cod_eqI: fixes b :: "'a1"   and f :: "'a1" assumes "C1.ide b"   and "C1.seq b f" shows "C1.cod f = b"
fmap_induct: fixes P :: "('a, 'b) fmap \<Rightarrow> bool"   and m :: "('a, 'b) fmap" assumes "P {$$}"   and "\<And>x y m. \<lbrakk>P m; m $$ x = None\<rbrakk> \<Longrightarrow> P m(x $$:= y)" shows "P m"
simps_5: fixes va :: "('a, 'b) Term.term"   and vb :: "('a, 'b) Term.term"   and S :: "('a, 'b) strand_step list" shows "assignment_rhs\<^sub>s\<^sub>t (\<langle>check: va \<doteq> vb\<rangle>\<^sub>s\<^sub>t # S) = assignment_rhs\<^sub>s\<^sub>t S"
rational_root_test_main_1: fixes df :: "int \<Rightarrow> int list"   and dp :: "int \<Rightarrow> int list"   and p :: "rat poly"   and x :: "rat" assumes "rational_root_test_main df dp p = Some x" shows "poly p x = 0"
finite_dimensional_vector_space_pair_with_def: fixes plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f"   and scale\<^sub>1 :: "'g \<Rightarrow> 'f \<Rightarrow> 'f"   and basis\<^sub>1 :: "'f set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h"   and scale\<^sub>2 :: "'g \<Rightarrow> 'h \<Rightarrow> 'h"   and basis\<^sub>2 :: "'h set" shows "finite_dimensional_vector_space_pair_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 basis\<^sub>2 \<equiv> finite_dimensional_vector_space_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 \<and> finite_dimensional_vector_space_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 basis\<^sub>2"
simps_2: fixes d :: "nat" shows "beta_interp.intv_const (Regions_Beta.intv.Intv d) = d"
span_mono: fixes A :: "('t \<Rightarrow>\<^sub>0 'd) set"   and B :: "('t \<Rightarrow>\<^sub>0 'd) set" assumes "A \<subseteq> B" shows "pmdl A \<subseteq> pmdl B"
H1: shows "ExcessNotAtRec (deleteAround g a ps) g = ListSum (ExcessNotAtRecList (deleteAround g a ps) g) E"
push_bit_take_bit: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat"   and a :: "'c" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "push_bit m (take_bit n a) = take_bit (m + n) (push_bit m a)"
mono_iff_le_Suc: fixes f :: "nat \<Rightarrow> 'c" shows "incseq f = (\<forall>n. f n \<le> f (Suc n))"
quasi_inverses_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b" assumes "bicategory V H \<a> \<i> src trg" shows "bicategory.quasi_inverses V H \<a> \<i> src trg f g \<equiv> \<exists>\<phi> \<psi>. equivalence_in_bicategory V H \<a> \<i> src trg f g \<phi> \<psi>"
iii: shows "continuous_on ({0..1} - s' \<inter> {0..1}) (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b))"
case2: shows "direction lineC = 0 \<longrightarrow> lineA \<parallel>  lineC"
pcomp_shorter: fixes v :: "'b list"   and w :: "'b list" assumes "v \<bowtie> w"   and "\<^bold>|v\<^bold>| \<le> \<^bold>|w\<^bold>|" shows "v \<le>p w"
sum_measure_le_measure_inter3: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A \<in> PS.events"   and "B \<in> PS.events"   and "C \<in> PS.events" shows "PS.prob A + PS.prob B + PS.prob C \<le> 2 + PS.prob (A \<inter> B \<inter> C)"
nth_merges: fixes p :: "nat"   and ss :: "'a list"   and n :: "nat"   and ps :: "(nat \<times> 'a) list" assumes "p < length ss"   and "ss \<in> list n A"   and "\<forall>(p, t)\<in>set ps. p < n \<and> t \<in> A" shows "merges f ps ss ! p = map snd (filter (\<lambda>(p', t'). p' = p) ps) \<Squnion>\<^bsub>f\<^esub> ss ! p"
subst_atm_mset_list_nth: fixes i :: "nat"   and AAs :: "'a multiset list"   and \<eta> :: "'s" assumes "i < length AAs" shows "(AAs \<cdot>aml \<eta>) ! i = AAs ! i \<cdot>am \<eta>"
max2_equals2: fixes a :: "'a"   and b :: "'a" assumes "a \<in> Field r"   and "b \<in> Field r" shows "(max2 a b = b) = ((a, b) \<in> r)"
VSLucky_def: shows "VSLucky = opt \<circ> QE_dnf opt (\<lambda>amount. luckyFind') \<circ> VSLuckiest \<circ> opt"
fresh_def: fixes permute :: "perm \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "atom"   and x :: "'e" assumes "class.pt permute" shows "pt.fresh permute a x \<equiv> a \<notin> pt.supp permute x"
SeqFormP_Eq: fixes s :: "name"   and k :: "name"   and x :: "tm"   and y :: "tm" assumes "atom s \<sharp> (k, x, y)"   and "atom k \<sharp> (x, y)" shows "{TermP x, TermP y} \<turnstile> SyntaxN.Ex k (SyntaxN.Ex s (SeqFormP (Var s) (Var k) (Q_Eq x y)))"
h2: shows "sortedTree h t2 \<longrightarrow> setOf (remove h e t2) = setOf t2 - eqs h e"