hcju/isabelleps huggingface dataset:
binomial_deriv2: fixes n :: "nat"   and a :: "real"   and b :: "real" shows "(\<Sum>k\<le>n. real k * real (k - 1) * real (n choose k) * a ^ (k - 2) * b ^ (n - k)) = real n * real (n - 1) * (a + b) ^ (n - 2)"
Limit_def: fixes i :: "V" shows "Limit i \<equiv> Ord i \<and> 0 \<in> elts i \<and> (\<forall>y. y \<in> elts i \<longrightarrow> ZFC_in_HOL.succ y \<in> elts i)"
simps_33: fixes vd :: "'b"   and ve :: "bexp"   and vf :: "'b acom"   and vg :: "'b"   and v :: "bexp"   and va :: "'b acom"   and vb :: "'b acom"   and vc :: "'b" shows "({vd} WHILE ve DO vf {vg} \<sqsubseteq> IF v THEN va ELSE vb {vc}) = False"
scalar_prod_right_unit: fixes i :: "nat"   and n :: "nat"   and v :: "'b vec" assumes "i < n" shows "v \<bullet> unit_vec n i = v $ i"
app: fixes s :: "dB"   and s' :: "dB"   and t :: "dB"   and t' :: "dB" assumes "s => s'"   and "t => t'" shows "s \<degree> t => s' \<degree> t'"
loop_backward_forward: fixes x :: "'a" shows "x\<^sup>T \<le> - 1' + x"
CH_range: fixes A :: "'a set" shows "CH A \<subseteq> A"
diff_closed: fixes g :: "'g"   and h :: "'g" assumes "g \<in> H"   and "h \<in> H" shows "g - h \<in> H"
cross3_simps_8: fixes a :: "'d"   and b :: "'d" shows "a + b = b + a"
exH_k_eq_H_k: fixes k :: "nat" assumes "k < n" shows "exH_k k = partial_state2.pmat_extension dims {0..<Suc k} {Suc k..<n} (H_k k)"
his_determ_ex: fixes n' :: "nat"   and nid' :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid" assumes "(n', nid') \<in> his t n nid" shows "\<exists>!nid'. (n', nid') \<in> his t n nid"
step: shows "(s, s') \<in> v_round r v_f d_f"
subst_apply_fv_unfold: fixes t :: "('d, 'e) Term.term"   and \<delta> :: "'e \<Rightarrow> ('d, 'c) Term.term" shows "fv (t \<cdot> \<delta>) = fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` fv t)"
measurable_distr2: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and L :: "'c measure"   and M :: "'d measure"   and N :: "'e measure"   and g :: "'c \<Rightarrow> 'd measure" assumes "(\<lambda>(x, y). f x y) \<in> L \<Otimes>\<^sub>M M \<rightarrow>\<^sub>M N"   and "g \<in> L \<rightarrow>\<^sub>M subprob_algebra M" shows "(\<lambda>x. distr (g x) N (f x)) \<in> L \<rightarrow>\<^sub>M subprob_algebra N"
less_minus_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less a (uminus b) = less b (uminus a)"
ALGxy_def2: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and qs :: "'c list"   and init :: "'c list"   and x :: "'c"   and y :: "'c" shows "ALGxy A qs init x y = (\<Sum>i | i < length qs \<and> qs ! i \<in> {y, x}. ALG' A qs init i y + ALG' A qs init i x)"
subgroupE_2: fixes H :: "'a set" assumes "subgroup H (add_monoid G)" shows "H \<noteq> {}"
assms_3: shows "\<Y>\<^bsub>F\<^esub> = \<X>\<^bsub>G\<^esub>"
IdNatTrans_defs_1: fixes F :: "('g, 'h, 'i, 'j, 'k, 'l) Functor" shows "IdNatTrans F \<equiv> MakeNT (IdNatTrans' F)"
simps_2: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and v :: "'b rule"   and va :: "'b rule list"   and X :: "final_decision" shows "approximating_bigstep_fun \<gamma> p (v # va) (Decision X) = Decision X"
sgn_one: shows "sgn (1::'a) = (1::'a)"
emeasure_T': fixes X :: "'s stream set"   and I :: "'s pmf" assumes "X \<in> sets (stream_space (count_space UNIV))" shows "emeasure (T' I) X = \<integral>\<^sup>+ s. emeasure (T s) {\<omega> \<in> space (stream_space (count_space UNIV)). s ## \<omega> \<in> X} \<partial>measure_pmf I"
zero: fixes A :: "(('b, 'd) vec, 'c) vec" shows "A *v 0 = 0"
ty_vrs_prm_simp: fixes pi :: "(vrs \<times> vrs) list"   and S :: "ty" shows "pi \<bullet> S = S"
env_bigger2: fixes a :: "char list"   and e :: "'b environment"   and b :: "char list"   and x :: "char list"   and X :: "'b"   and Y :: "'b" assumes "a \<notin> env_dom e"   and "b \<notin> env_dom e"   and "x \<in> env_dom e"   and "a \<noteq> b" shows "x \<in> env_dom (e\<lparr>a:X\<rparr>\<lparr>b:Y\<rparr>)"
bad_def: fixes p :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" shows "bad p x = (\<exists>f. inf_chain p f \<and> f 0 = x)"
simps: fixes x :: "'a"   and xs :: "'a Stream" shows "i_th\<cdot>(x && xs) = Nat_case\<cdot>x\<cdot>(i_th\<cdot>xs)"
nomatch: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and a :: "action" assumes "\<not> matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m a], Undecided\<rangle> \<Rightarrow> Undecided"
nn_integral_0_iff: fixes u :: "'a \<Rightarrow> ennreal"   and M :: "'a measure" assumes "u \<in> borel_measurable M" shows "(integral\<^sup>N M u = 0) = (emeasure M {x \<in> space M. u x \<noteq> 0} = 0)"
inj_zero_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "inj_zero_hom hom \<equiv> zero_hom hom \<and> injective hom"
empty_sets: fixes M :: "'b measure" shows "{} \<in> sets M"
int_of_signed_def: fixes i :: "32 word" shows "int_of_signed i \<equiv> sint i"
I_maximal: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme"   and J :: "'a set" assumes "maximalideal I R"   and "ideal J R"   and "I \<subseteq> J"   and "J \<subseteq> carrier R" shows "J = I \<or> J = carrier R"
unl_Rex: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and w :: "'b" shows "(\<exists>x. A x) w \<equiv> \<exists>x. A x w"
ltake_all: fixes xs :: "'a llist"   and m :: "enat" assumes "llength xs \<le> m" shows "ltake m xs = xs"
sum_bounds_lt_plus1: fixes f :: "nat \<Rightarrow> 'c"   and mm :: "nat" shows "(\<Sum>k<mm. f (Suc k)) = sum f {1..mm}"
has_vderiv_on_cong: fixes S :: "real set"   and f :: "real \<Rightarrow> 'b"   and g :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'b"   and g' :: "real \<Rightarrow> 'b"   and T :: "real set" assumes "\<And>x. x \<in> S \<Longrightarrow> f x = g x"   and "\<And>x. x \<in> S \<Longrightarrow> f' x = g' x"   and "S = T" shows " D f = f' on S =  D g = g' on T"
Myhill_Nerode1: fixes A :: "'b list set" assumes "finite (UNIV // \<approx>A)" shows "\<exists>r. A = lang r"
atMost_def: fixes u :: "'a Word.word" shows "signed.atMost u \<equiv> {x. x \<le>s u}"
flip: shows "bounded_bilinear (\<lambda>x y. y ** x)"
intro: fixes rs :: "nat list"   and y :: "nat" assumes "deep_model_correct_params rs"   and "deep_model_correct_params_y_axioms rs y" shows "deep_model_correct_params_y rs y"
gc_W_empty_mut_inv_load_W: fixes s :: "'e process_name \<Rightarrow> ('f, 'e, 'g, 'h) Model.local_state" assumes "\<forall>m. mut_m.gc_W_empty_mut_inv m s"   and "\<forall>m. sys_ghost_hs_in_sync m s"   and "WL gc s = {}"   and "WL sys s = {}" shows "no_grey_refs s"
lipschitz_on_constant: fixes U :: "'b set"   and c :: "'c" shows "0-lipschitz_on U (\<lambda>x. c)"
zout: shows "z \<in> outside (path_image c \<union> path_image c2)"
strongly_strategyproof_sds_axioms_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" shows "strongly_strategyproof_sds_axioms agents alts sds \<equiv> \<forall>R i Ri'. pref_profile_wf agents alts R \<longrightarrow> i \<in> agents \<longrightarrow> total_preorder_on alts Ri' \<longrightarrow> social_decision_scheme.strongly_strategyproof_profile sds R i Ri'"
line_def: fixes z1 :: "complex"   and z2 :: "complex" shows "line z1 z2 = {z. collinear z1 z2 z}"
KGsiKGigroup: shows "Group.group (G\<lparr>carrier := K \<inter> \<GG> ! (i + 1)\<rparr> Mod K \<inter> \<GG> ! i)"
cases: fixes x :: "'b tree \<times> 'b tree" obtains t :: "'b tree" where "x = (\<langle>\<rangle>, t__)"   | t :: "'b tree" where "x = (t, \<langle>\<rangle>)"   | l1 :: "'b tree" and a1 :: "'b" and r1 :: "'b tree" and l2 :: "'b tree" and a2 :: "'b" and r2 :: "'b tree" where "x = (\<langle>l1, a1, r1\<rangle>, \<langle>l2, a2, r2\<rangle>)"
split: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and prod :: "'b \<times> 'c" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (\<forall>x1 x2. prod = (x1, x2) \<longrightarrow> P (f x1 x2))"
bot_less: fixes a :: "'a" shows "(a \<noteq> \<bottom>) = (\<bottom> < a)"
paths_mono: fixes t :: "'b ttree"   and t' :: "'b ttree" assumes "t \<sqsubseteq> t'" shows "paths t \<sqsubseteq> paths t'"
wf_extend_oprod1: fixes r :: "('d \<times> 'd) set"   and A :: "'e set" assumes "wf r" shows "wf {((x, y1), x, y2). x \<in> A \<and> (y1, y2) \<in> r}"
remove_cycles_one: fixes as :: "'b list"   and x :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "remove_cycles (as @ x # xs) x ys = remove_cycles (x # xs) x ys"
set_map_pmf: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'b pmf" shows "set_pmf (map_pmf f M) = f ` set_pmf M"
k': shows "(spr_simAbs (fst k'), snd k') \<in> (\<Union>a. SPRsingle.sim_equiv_class a ` SPR.jkbpC) \<times> UNIV"
lbrules_3: shows "\<forall>x\<in>{1..m}. sum t (A x) = T x"
image_cong_simp: fixes M :: "'c set"   and N :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "M = N"   and "\<And>x. s2p N x =simp=> f x = g x" shows "\<P> f M = \<P> g N"
eq_1: shows "return_pmf x = map_pmf fst pq"
simps_4: fixes f1 :: "int \<Rightarrow> 'a"   and f2 :: "'a"   and int :: "int" shows "(case eint int of eint x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2) = f1 int"
compose_extensional: fixes A :: "'c set"   and f :: "'e \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" shows "compose A f g \<in> extensional A"
of_nats: shows "word_of_nat x = word_of_nat y"
right_idem: fixes a :: "'a"   and b :: "'a" shows "a \<cdot> b \<cdot> b = a \<cdot> b"
preorder_on_map: fixes f :: "'c \<Rightarrow> 'b" shows "Order_Predicates.preorder_on (f -` f ` carrier) (map_relation f (map_relation (inv f) le))"
scene_override_overshadow_left: fixes S\<^sub>1 :: "'a"   and S\<^sub>2 :: "'a"   and X :: "'a scene"   and S\<^sub>3 :: "'a" shows "S\<^sub>1 \<oplus>\<^sub>S S\<^sub>2 on X \<oplus>\<^sub>S S\<^sub>3 on X = S\<^sub>1 \<oplus>\<^sub>S S\<^sub>3 on X"
relpow_E: fixes x :: "'a"   and z :: "'a"   and n :: "nat"   and R :: "('a \<times> 'a) set" assumes "(x, z) \<in> R ^^ n" obtains "n = 0" and "x = z"   | y :: "'a" and m :: "nat" where "n = Suc m" and "(x, y) \<in> R ^^ m" and "(y, z) \<in> R"
U'W': shows "U' \<union> W' \<subseteq> carrier V"
eq_Max_iff: fixes A :: "'a Word.word set"   and m :: "'a Word.word" assumes "finite A"   and "A \<noteq> {}" shows "(m = signed.Max A) = (m \<in> A \<and> (\<forall>a\<in>A. a \<le>s m))"
take_Cons_numeral: fixes v :: "num"   and x :: "'c"   and xs :: "'c list" shows "take (numeral v) (x # xs) = x # take (numeral v - 1) xs"
tendsto_le: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and "open" :: "'d set \<Rightarrow> bool"   and F :: "'e filter"   and f :: "'e \<Rightarrow> 'd"   and x :: "'d"   and g :: "'e \<Rightarrow> 'd"   and y :: "'d" assumes "class.linorder_topology less_eq less open"   and "F \<noteq> bot"   and "filterlim f (topological_space.nhds open x) F"   and "filterlim g (topological_space.nhds open y) F"   and "\<forall>\<^sub>F x in F. less_eq (g x) (f x)" shows "less_eq y x"
simps_13: fixes vs :: "'b list"   and bs :: "'b list"   and p :: "fm" shows "Ifm vs bs (A p) = (\<forall>x. Ifm vs (x # bs) p)"
subdivision_gen_contr: fixes V :: "'a set" assumes "sG.is_slim V" shows "subdivision_pair (gen_contr_graph (with_proj slim) V) slim"
F_4: fixes x :: "real" shows "\<bar>F x\<bar> \<le> 1"
hcomp_in_hhomE: fixes \<nu> :: "'a"   and \<mu> :: "'a"   and a :: "'a"   and c :: "'a" assumes "\<guillemotleft>\<nu> \<star>\<^sub>C \<mu> : a \<rightarrow>\<^sub>C c\<guillemotright>" obtains "C.arr \<mu>" and "C.arr \<nu>" and "src\<^sub>C \<nu> = trg\<^sub>C \<mu>" and "src\<^sub>C \<mu> = a" and "trg\<^sub>C \<nu> = c"
InvariantWatchListsUniq_def: fixes Wl :: "Literal \<Rightarrow> nat list" shows "InvariantWatchListsUniq Wl = (\<forall>l. uniq (Wl l))"
cnj_E: fixes A :: "fmla"   and B :: "fmla"   and H :: "fmla set"   and C :: "fmla" assumes "insert A (insert B H) \<turnstile> C" shows "insert (A AND B) H \<turnstile> C"
hyps_1: fixes i :: "nat" assumes "n = j - i"   and "trace init t final"   and "i \<le> j"   and "j \<le> length t"   and "msgs (S t i) cid = []"   and "ps (S t i) p \<noteq> None"   and "channel cid = Some (p, q)"   and "\<forall>k. i \<le> k \<and> k < j \<and> regular_event (t ! k) \<longrightarrow> occurs_on (t ! k) \<noteq> p" shows "msgs (S t j) cid = []"
bs_def: shows "bs = [ram1] @ between vs ram1 ram2 @ [ram2]"
SD_efficient_singleton_iff: fixes x :: "'alt" assumes "x \<in> alts" shows "SD_efficient R (return_pmf x) = (x \<notin> pareto_losers R)"
mat_mult: fixes m :: "(('e, 'g) vec, 'd) vec"   and m' :: "(('e, 'f) vec, 'g) vec" shows "($) (m ** m') = mat_mult_row m m'"
accepts_def: fixes A :: "'c \<times> ('d option \<Rightarrow> 'c \<Rightarrow> 'c set) \<times> ('c \<Rightarrow> bool)"   and w :: "'d list" shows "NAe.accepts A w = (\<exists>q. (start A, q) \<in> NAe.steps A w \<and> fin A q)"
smooth_on_def: fixes k :: "enat"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "k-smooth_on S f = (\<forall>n. enat n \<le> k \<longrightarrow> higher_differentiable_on S f n)"
pullback_has_bottom: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "OrderingSetIso less_eq less less_eq' less' P f"   and "ordering.has_bottom less_eq' (f ` P)" shows "ordering.has_bottom less_eq P"
target: fixes T1 :: "'procT"   and T2 :: "'procT"   and TRel :: "('procT \<times> 'procT) set"   and SRel :: "('procS \<times> 'procS) set" assumes "(T1, T2) \<in> TRel" shows "TargetTerm T1 \<R>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> TargetTerm T2"
c: fixes i :: "nat" shows "coeff p i = fpxs_compose_power (fpxs_of_fls (c i)) (1 / rat_of_nat b)"
toList_cases: fixes y :: "'b list" assumes "y \<in> rel_ext (\<lambda>x. sorted x \<and> distinct x)" obtains x :: "'b odlist" where "y = toList x"
row_space_def: fixes B :: "'a mat" shows "row_space B = local.span (set (rows B))"
imonad1: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'a nres" shows "RETURN x \<bind> f = f x"
nat_filtration_not_borel_info: fixes f :: "bool stream \<Rightarrow> 'b"   and n :: "nat"   and N :: "'b measure" assumes "f \<in> nat_filtration n \<rightarrow>\<^sub>M N"   and "set_discriminating n f N" shows "f \<circ> pseudo_proj_True n = f"
mask_nonnegative_int: fixes n :: "nat" shows "0 \<le> mask n"
monad_writerT: fixes return :: "'a \<times> 'b list \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b list \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad return bind" shows "monad (return_writer return) (bind_writer return bind)"
simple_product_run: fixes \<delta>\<^sub>1 :: "'d \<Rightarrow> 'f \<Rightarrow> 'd"   and \<delta>\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'e"   and q\<^sub>1 :: "'d"   and q\<^sub>2 :: "'e"   and w :: "nat \<Rightarrow> 'f"   and i :: "nat" shows "run \<delta>\<^sub>1 \<times> \<delta>\<^sub>2 (q\<^sub>1, q\<^sub>2) w i = (run \<delta>\<^sub>1 q\<^sub>1 w i, run \<delta>\<^sub>2 q\<^sub>2 w i)"
mult_comp_val_process: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and Mkt :: "('d, 'c) discrete_market"   and qty :: "nat \<Rightarrow> 'd \<Rightarrow> real" assumes "portfolio pf1" shows "\<forall>n w. val_process Mkt (qty_mult_comp pf1 qty) n w = val_process Mkt pf1 n w * qty (Suc n) w"
2: shows "\<S>\<bullet>\<C> \<turnstile> $* es' : t2s _> t3s"
ag_p_inv: fixes A :: "('d, 'e) aGroup_scheme"   and x :: "'d"   and y :: "'d" assumes "aGroup A"   and "x \<in> carrier A"   and "y \<in> carrier A" shows "-\<^sub>a\<^bsub>A\<^esub> (x \<plusminus>\<^bsub>A\<^esub> y) = -\<^sub>a\<^bsub>A\<^esub> x \<plusminus>\<^bsub>A\<^esub> -\<^sub>a\<^bsub>A\<^esub> y"
frameChainEqLength: fixes xvec :: "name list"   and \<Psi> :: "'d"   and yvec :: "name list"   and \<Psi>' :: "'d" assumes "\<langle>xvec, \<Psi>\<rangle> = \<langle>yvec, \<Psi>'\<rangle>" shows "length xvec = length yvec"
simps_1: fixes n :: "nat"   and I :: "nat set" shows "f_join_aux [] n I = []"
p: shows "p = neville_aitken (\<lambda>i. fst (xs_ys ! i)) (\<lambda>i. snd (xs_ys ! i)) (length xs_ys - 1)"
get_ancestors_si_obtains_children_or_shadow_root: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list"   and ancestor :: "(_) object_ptr"   and thesis :: "bool" assumes "heap_is_wellformed h"   and "known_ptrs h"   and "type_wf h"   and "h \<turnstile> get_ancestors_si ptr \<rightarrow>\<^sub>r ancestors"   and "ancestor \<noteq> ptr"   and "ancestor \<in> set ancestors" shows "((\<forall>children ancestor_child. h \<turnstile> get_child_nodes ancestor \<rightarrow>\<^sub>r children \<longrightarrow> ancestor_child \<in> set children \<longrightarrow> cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ancestor_child \<in> set ancestors \<longrightarrow> thesis) \<longrightarrow> thesis) \<or> ((\<forall>ancestor_element shadow_root. ancestor = cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ancestor_element \<longrightarrow> h \<turnstile> get_shadow_root ancestor_element \<rightarrow>\<^sub>r Some shadow_root \<longrightarrow> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root \<in> set ancestors \<longrightarrow> thesis) \<longrightarrow> thesis)"
second_inner_invariant_when_e_bot: fixes h :: "'a"   and j :: "'a"   and g :: "'a"   and f :: "'a"   and d :: "'a" assumes "selected_edge h j g = bot"   and "selected_edge h j g \<le> - forest_components f"   and "boruvka_inner_invariant j f h g d" shows "boruvka_inner_invariant (j \<sqinter> - choose_component (forest_components h) j) (f \<sqinter> - selected_edge h j g\<^sup>T \<sqinter> - path f h j g \<squnion> (f \<sqinter> - selected_edge h j g\<^sup>T \<sqinter> path f h j g)\<^sup>T \<squnion> selected_edge h j g) h g (d \<squnion> selected_edge h j g)"
power_sgn: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "sgn (power.power one times a n) = power.power one times (sgn a) n"
disc_1: fixes x1 :: "'a" shows "is_pos (Pos x1)"
open_codomain: fixes c :: "('c, 'b) chart" shows "open (codomain c)"
fmember_elim: fixes x :: "'b"   and xs :: "'b fset" assumes "x |\<in>| xs" obtains "x \<in> \<langle>xs\<rangle>\<^sub>f"
induct: fixes P :: "'b poly \<Rightarrow> bool"   and a0 :: "'b poly" assumes "\<And>c. P (c)\<^sub>p"   and "\<And>c n p. \<lbrakk>P c; P p\<rbrakk> \<Longrightarrow> P (CN c n p)"   and "\<And>v. P (Bound v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v va. P (Mul v va)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Pw v va)" shows "P a0"
surj: fixes f :: "'a \<Rightarrow> 'a" assumes "bijection f" shows "surj f"
set_local: fixes i :: "nat"   and \<C> :: "t_context"   and t :: "t" assumes "i < length (local \<C>)"   and "local \<C> ! i = t" shows "\<C> \<turnstile> [Set_local i] : [t] _> []"
cases: fixes x :: "('a, 'b) lit" obtains a :: "('a, 'b) atm" where "x = Pos a__"   | a :: "('a, 'b) atm" where "x = Neg a"
infinite: fixes A :: "'c set" assumes "infinite A" shows "mset_set A = {#}"
implem_LtK2: fixes x :: "chan" shows "implem x \<notin> range LtK"
wp_star_induct_var: fixes q :: "'a"   and x :: "'a" assumes "d q \<le> wp x q" shows "d q \<le> wp (x\<^sup>\<star>) q"
prems_2: shows "mcp (vs @ xs' @ [x]) V vs"
Lyndon_fac_simp: fixes w :: "'a list" assumes "w \<noteq> \<epsilon>" shows "LynFac w = LynFac (w\<^sup><\<inverse>LynSuf w ) \<cdot> [LynSuf w]"
IH_readable_2: shows "cnf G \<union> \<Union> (cnf ` set_mset \<Gamma>) \<turnstile> \<box>"
mat_of_cols_cons_mat_vec: fixes v :: "'b vec"   and ls :: "'b vec list"   and a :: "'b vec"   and nr :: "nat"   and m :: "'b" assumes "v \<in> carrier_vec (length ls)"   and "dim_vec a = nr" shows "mat_of_cols nr (a # ls) *\<^sub>v vCons m v = m \<cdot>\<^sub>v a + mat_of_cols nr ls *\<^sub>v v"
u': shows "- u = (\<Sum>i\<in>Basis. - (u \<bullet> i) *\<^sub>R i)"
eq_Apps_conv_9: fixes T :: "Unary_PCF.type"   and t :: "expr"   and f :: "expr"   and ts :: "expr list" shows "(\<Lambda>\<langle>T\<rangle> t = f \<bullet> ts) = (\<Lambda>\<langle>T\<rangle> t = f \<and> ts = [])"
SN: shows "SN {(s, t). fst (g s t)}"
simplex_in_max: fixes y :: "'a set" assumes "y \<in> X" shows "\<exists>x. chamber x \<and> order.greater_eq x y"
hyps: fixes t :: "('a, 'b) SkewBinomialTree" assumes "tail_invar bq" shows "tail_invar (remove1 t bq)"
check_eqv_def: fixes n :: "nat"   and \<phi> :: "'a formula"   and \<psi> :: "'a formula" shows "dual.check_eqv n \<phi> \<psi> = (wf_formula n (FOr \<phi> \<psi>) \<and> dual.check_eqvRE enum_class.enum n (Plus (rexp_of'' n (norm \<phi>)) One) (Plus (rexp_of'' n (norm \<psi>)) One))"
lucas_lehmer_hom_mult: fixes x :: "int \<times> int"   and y :: "int \<times> int" shows "lucas_lehmer_hom m (lucas_lehmer_mult' x y) = lucas_lehmer_mult m (lucas_lehmer_hom m x) (lucas_lehmer_hom m y)"
bind_spmf_pmf_assoc: fixes p :: "'j pmf"   and f :: "'j \<Rightarrow> 'i spmf"   and g :: "'i \<Rightarrow> 'h spmf" shows "p \<bind> f \<bind> g = p \<bind> (\<lambda>x. f x \<bind> g)"
hom_add: fixes hom :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "plus_hom hom" shows "hom (x + y) = hom x + hom y"
HMA_irreducible: shows "(HMA_M ===> (=)) irreducible_mat fixed_mat.irreducible"
ht_phl_export2: fixes z :: "'a"   and w :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<cdot> w \<le> w \<cdot> z \<cdot> w"   and "\<lbrace>x\<rbrace>y\<lbrace>z\<rbrace>" shows "\<lbrace>x\<rbrace>y \<cdot> w\<lbrace>z \<cdot> w\<rbrace>"
r6_trans: fixes s :: "trm"   and s' :: "trm"   and x :: "name"   and t :: "trm" assumes " s \<mapsto>\<^sup>* s' " shows " s to x in t \<mapsto>\<^sup>* s' to x in t "
in_sort_edge_distinct: fixes x :: "'a"   and L :: "'a list" assumes "x \<notin> set L"   and "distinct L" shows "distinct (in_sort_edge x L)"
assms_11: shows "InvariantNoDecisionsWhenConflict [getC state] (getM state) (getBackjumpLevel state)"
eq2: shows "card (component_of_term ` Keys (fst ` set (gs @ bs))) = fst data + card (const_lt_component ` (fst ` set (gs @ bs) - {0}) - {None})"
the_leftI: fixes xs :: "int list"   and j :: "nat" assumes "has_left xs j" shows "the_left xs j < j \<and> xs ! the_left xs j < xs ! j"
minWeightBasis_def: shows "minWeightBasis \<equiv> obtain_sorted_carrier \<bind> (\<lambda>l. ASSERT (set l = E) \<bind> (\<lambda>_. nfoldli l (\<lambda>_. True) (\<lambda>e T. ASSERT (forest T \<and> e \<in> E \<and> T \<subseteq> E) \<bind> (\<lambda>_. if forest (insert e T) then RETURN (insert e T) else RETURN T)) {} \<bind> RETURN))"
that: fixes ys :: "'a list"   and a :: "'a"   and zs :: "'a stream" assumes "xs = ys @- a ## zs"   and "A \<inter> set ys = {}"   and "a \<in> A" shows "thesis"
in_mono: fixes A :: "'x set"   and B :: "'x set"   and x :: "'x" assumes "A \<subseteq> B" shows "x \<in> A \<longrightarrow> x \<in> B"