hcju/isabelleps huggingface dataset:
A91_Acc: shows "Acc level1 sA91 = {sA81, sA91, sA92}"
Card_order_omax: fixes R :: "('a \<times> 'a) set set" assumes "finite R"   and "R \<noteq> {}"   and "\<forall>r\<in>R. Card_order r" shows "Card_order (omax R)"
lunit_coherence: fixes f :: "'c" assumes "C.ide f" shows "\<l>\<^sub>D[F f] = F \<l>\<^sub>C[f] \<cdot>\<^sub>D \<Phi> (trg\<^sub>C f, f) \<cdot>\<^sub>D (unit (trg\<^sub>C f) \<star>\<^sub>D F f)"
D: shows "discrete_diagram J (\<cdot>) D \<and> Collect D.J.arr = I"
cone_axioms: shows "Limit.cone J C D' a \<tau>o\<chi>.map"
M: shows "mutual_information b S T X Y = KL_divergence b (density (S \<Otimes>\<^sub>M T) (\<lambda>x. ennreal (Px (fst x) * Py (snd x)))) (density (S \<Otimes>\<^sub>M T) (\<lambda>x. ennreal (Pxy x)))"
equiintegrable_add: fixes F :: "('a \<Rightarrow> 'b) set"   and I :: "'a set"   and G :: "('a \<Rightarrow> 'b) set" assumes "F equiintegrable_on I"   and "G equiintegrable_on I" shows "(\<Union>f\<in>F. \<Union>g\<in>G. {\<lambda>x. f x + g x}) equiintegrable_on I"
distinct_3: fixes x1 :: "('a, 'b, 'c, 'd) Tree set['a]"   and x31 :: "'d"   and x32 :: "'b" shows "tConj x1 \<noteq> tPred x31 x32"
prj1_simps_arr: fixes f :: "'a"   and g :: "'a" shows "arr (prj1 f g) = cospan f g"
SPEC_cons_rule: fixes m :: "'a nres"   and \<Phi> :: "'a \<Rightarrow> bool"   and \<Psi> :: "'a \<Rightarrow> bool" assumes "m \<le> SPEC \<Phi>"   and "\<And>x. \<Phi> x \<Longrightarrow> \<Psi> x" shows "m \<le> SPEC \<Psi>"
mtf2_moves_to_front: fixes xs :: "'b list"   and q :: "'b" assumes "distinct xs"   and "q \<in> set xs" shows "index (mtf2 (length xs) q xs) q = 0"
cases: fixes x :: "frame" obtains stk :: "val list" and loc :: "val list" and C :: "char list" and M :: "char list" and pc :: "nat" and ics :: "init_call_status" where "x = (stk, loc, C, M, pc, ics)"
e1_2: shows "cball x e1 \<subseteq> T"
d: fixes a :: "real"   and b :: "real" assumes "a \<le> x"   and "x \<le> b"   and "b - a < d" shows "P a b"
ide_dom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.ide C (partial_magma.dom C f)"
fun_eqvtI: fixes f :: "'b \<Rightarrow> 'c"   and p :: "perm" assumes "\<And>p x. p \<bullet> f x = f (p \<bullet> x)" shows "p \<bullet> f = f"
set_encode_eq: fixes A :: "nat set"   and B :: "nat set" assumes "finite A"   and "finite B" shows "(set_encode A = set_encode B) = (A = B)"
zero_ext_h: fixes y :: "'a" assumes "y \<in> GH - supp h" shows "f (x - y) * h y = (0::'b)"
one_is_normal: shows "{\<one>\<^bsub>add_monoid G\<^esub>} \<lhd> add_monoid G"
strict_boundedE: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" assumes "a \<subset># b \<inter># c" obtains "a \<subset># b" and "a \<subset># c"
ge: shows "\<forall>n\<ge>N. C \<le> f n"
zero_not_one: shows "(0::'a) \<noteq> (1::'a)"
Inf_M_Suc_ge: fixes k :: "nat" shows "\<Sqinter> (M k) \<le> \<Sqinter> (M (Suc k))"
split_sel_asm: fixes P :: "'f \<Rightarrow> bool"   and f1 :: "'f"   and f2 :: "'g \<Rightarrow> 'f"   and option :: "'g option" shows "P (case option of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = (\<not> (option = None \<and> \<not> P f1 \<or> option = \<lfloor>the option\<rfloor> \<and> \<not> P (f2 (the option))))"
less_multiset_ext\<^sub>D\<^sub>M_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and M :: "'c multiset"   and N :: "'c multiset" shows "less_multiset_ext\<^sub>D\<^sub>M R M N = (\<exists>X Y. X \<noteq> {#} \<and> X \<subseteq># N \<and> M = N - X + Y \<and> (\<forall>k. k \<in># Y \<longrightarrow> (\<exists>a. a \<in># X \<and> R k a)))"
power_commuting_commutes: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x \<cdot> y = y \<cdot> x" shows "x\<^bsup>n\<^esup> \<cdot> y = y \<cdot> x\<^bsup>n\<^esup>"
image2pI: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" assumes "R x y" shows "image2p f g R (f x) (g y)"
map_cong0: fixes x :: "('l, 'k, 'm) F1"   and f1 :: "'k \<Rightarrow> 'n"   and g1 :: "'k \<Rightarrow> 'n"   and f2 :: "'m \<Rightarrow> 'o"   and g2 :: "'m \<Rightarrow> 'o" assumes "\<And>z1. z1 \<in> F1set1 x \<Longrightarrow> f1 z1 = g1 z1"   and "\<And>z2. z2 \<in> F1set2 x \<Longrightarrow> f2 z2 = g2 z2" shows "F1map f1 f2 x = F1map g1 g2 x"
summable': shows "(\<lambda>(x, y). f y x) abs_summable_on B \<times> A"
complete_game_return_true: fixes h :: "'a"   and w :: "'b"   and e :: "'e" assumes "(h, w) \<in> Rel1"   and "\<Sigma>1.completeness"   and "\<forall>h w. lossless_spmf (init1 h w)"   and "\<forall>r w e. lossless_spmf (response1 r w e)"   and "e \<in> carrier L" shows "\<Sigma>1.completeness_game h w e = return_spmf True"
Dpoincare_map'_def: fixes i :: "'a"   and c :: "real"   and S :: "'a set"   and x :: "'a" shows "Dpoincare_map' i c S x = Dflow x (return_time {x \<in> S. x \<bullet> i - c = 0} x) - (blinfun_scaleR_left (f (poincare_map {x \<in> S. x \<bullet> i = c} x)) o\<^sub>L (blinfun_scaleR_left (inverse (f (poincare_map {x \<in> S. x \<bullet> i = c} x) \<bullet> i)) o\<^sub>L (blinfun_inner_left i o\<^sub>L Dflow x (return_time {x \<in> S. x \<bullet> i - c = 0} x))))"
exec_move_CASI1: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and D :: "String.literal"   and F :: "String.literal"   and e' :: "'addr expr1"   and e'' :: "'addr expr1" assumes "exec_move ci P t e h s ta h' s'" shows "exec_move ci P t (e\<bullet>compareAndSwap(D\<bullet>F, e', e'')) h s ta h' s'"
hyps_1: shows "\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'"
select_convs_5: fixes Obj :: "'a set"   and Mor :: "'b set"   and Dom :: "'b \<Rightarrow> 'a"   and Cod :: "'b \<Rightarrow> 'a"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and more :: "'c" shows "Category.Id \<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr> = Id"
stable_def: fixes r :: "('b \<times> 'b) set" shows "stable r \<equiv> \<forall>A F. |A| <o r \<and> (\<forall>a\<in>A. |F a| <o r) \<longrightarrow> |Sigma A F| <o r"
fmap_freshness_lemma: fixes h :: "('a, 'b) fmap" assumes "\<exists>a. atom a \<sharp> (h, h $$ a)" shows "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h $$ a = x"
partial_equiv_trans: fixes eqv :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.partial_equiv eqv"   and "eqv x y"   and "eqv y z" shows "eqv x z"
Least_Suc2: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat"   and Q :: "nat \<Rightarrow> bool"   and m :: "nat" assumes "P n"   and "Q m"   and "\<not> P 0"   and "\<forall>k. P (Suc k) = Q k" shows "Least P = Suc (Least Q)"
collapse: fixes envT :: "('a, 'b) envT" shows "EnvT (run_env envT) = envT"
inj_vint_of_int: shows "inj vint_of_int"
hcomp_def: fixes \<nu> :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data" shows "\<nu> \<star> \<mu> \<equiv> if arr \<mu> \<and> arr \<nu> \<and> src \<nu> = trg \<mu> then \<lparr>Chn = chine_hcomp \<nu> \<mu>, Dom = \<lparr>Leg0 = Leg0 (Dom \<mu>) \<cdot> \<p>\<^sub>0[Leg0 (Dom \<nu>), Leg1 (Dom \<mu>)], Leg1 = Leg1 (Dom \<nu>) \<cdot> \<p>\<^sub>1[Leg0 (Dom \<nu>), Leg1 (Dom \<mu>)]\<rparr>, Cod = \<lparr>Leg0 = Leg0 (Cod \<mu>) \<cdot> \<p>\<^sub>0[Leg0 (Cod \<nu>), Leg1 (Cod \<mu>)], Leg1 = Leg1 (Cod \<nu>) \<cdot> \<p>\<^sub>1[Leg0 (Cod \<nu>), Leg1 (Cod \<mu>)]\<rparr>\<rparr> else null"
y_def: shows "y = f\<lparr>x\<rparr>"
trnr\<^sub>\<epsilon>_def: fixes u :: "'a"   and \<nu> :: "'a" shows "r.T0.trnr\<^sub>\<epsilon> u \<nu> \<equiv> \<r>[u] \<cdot> (u \<star> r.\<epsilon>) \<cdot> \<a>[u, tab\<^sub>0 r, (tab\<^sub>0 r)\<^sup>*] \<cdot> (\<nu> \<star> (tab\<^sub>0 r)\<^sup>*)"
uI_proj: shows "uI = (\<lambda>n x. uJ n (projI x))"
simps_1: fixes l :: "'b \<Rightarrow> 'b"   and r :: "'b \<Rightarrow> 'b"   and s :: "'b" shows "root (tree_iterate l r s) = s"
that: fixes j' :: "nat" assumes "k < j'"   and "j' mod i = j" shows "thesis"
symKey_neq_priK: fixes K :: "key"   and A :: "agent" assumes "K \<in> symKeys" shows "K \<noteq> priK A"
map_append_poly_mult_def: fixes x :: "char list list \<times> int" shows "map_append_poly_mult x = map_append (mult_monomials x)"
sentences_def: fixes P :: "'c option process" shows "sentences P \<equiv> {xs. xs @ [None] \<in> traces P}"
l_cancel_one': fixes x :: "'a list"   and a :: "'a list" assumes "x \<in> carrier (poly_ring R)"   and "a \<in> carrier (poly_ring R)" shows "(x = x \<oplus>\<^bsub>poly_ring R\<^esub> a) = (a = \<zero>\<^bsub>poly_ring R\<^esub>)"
prems_1: fixes n :: "nat" assumes "n < length (x # xs) - 1" shows "((x # xs) ! Suc n, (x # xs) ! n) \<in> R"
simps_1: fixes A :: "char list set option" shows "\<D>s [] A = True"
chop_fun: fixes t :: "('c, 'd) tm" assumes "is_App t"   and "is_App (fun t)" shows "App (chop (fun t)) (arg t) = chop t"
linepath_in_convex_hull: fixes a :: "'a"   and S :: "'a set"   and b :: "'a"   and x :: "real" assumes "a \<in> convex hull S"   and "b \<in> convex hull S"   and "0 \<le> x"   and "x \<le> 1" shows "linepath a b x \<in> convex hull S"
divide_strict_right_mono_neg: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b < a"   and "c < (0::'a)" shows "a / c < b / c"
primes_M_def: shows "\<MM> = prime_sum_upto (\<lambda>p. ln (real p) / real p)"
exhaust: fixes y :: "('a, 'b) envT" obtains x :: "'a \<Rightarrow> 'b" where "y = EnvT x"
dist_mult: fixes x :: "real"   and y :: "real"   and z :: "real" shows "dist (x * y) (x * z) = \<bar>x\<bar> * dist y z"
hyps: fixes ptr :: "(_) object_ptr" assumes "h \<turnstile> get_ancestors_di ptr \<rightarrow>\<^sub>r ancestors"   and "ptr' \<in> set ancestors" shows "ptr' |\<in>| object_ptr_kinds h"
inject_3: fixes x1 :: "('a, 'b, 'c) psiCase"   and y1 :: "('a, 'b, 'c) psiCase" shows "(Case x1 = Case y1) = (x1 = y1)"
FormP_graph_aux_def: shows "FormP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>FormP_aux k s y. x1 = y \<and> x2 = SyntaxN.Ex k (SyntaxN.Ex s (SeqFormP (Var s) (Var k) y)) \<and> atom k \<sharp> (s, y) \<and> atom s \<sharp> y)"
IdentityFunctor'_def: fixes C :: "('a, 'b, 'c) Category_scheme" shows "FId' C \<equiv> \<lparr>CatDom = C, CatCod = C, MapM = \<lambda>f. f\<rparr>"
hunion_hempty_left: fixes A :: "hf" shows "0 \<squnion> A = A"
orderI: fixes a :: "'a"   and b :: "'a" assumes "a = inf a b" shows "a \<le> b"
left_idem: fixes a :: "'a multiset"   and b :: "'a multiset" shows "a \<inter># (a \<inter># b) = a \<inter># b"
\<phi>\<psi>: shows "inverse_functors J.comp J' \<phi> \<psi>"
simps_2: fixes v :: "'b hp" shows "merge None (Some v) = Some v"
of_nat_1: shows "of_nat 1 = (1::'b)"
inc_dot_dot: fixes d :: "nat"   and x :: "item" shows "item_dot (inc_dot d x) = item_dot x + d"
op_list_singleton_def: fixes x :: "'a" shows "op_list_singleton x \<equiv> [x]"
prod_zero_iff: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "finite A" shows "(prod f A = (0::'a)) = (\<exists>a\<in>A. f a = (0::'a))"
simps: fixes M1 :: "('e, 'f, 'g) FSM"   and M2 :: "('e, 'f, 'g) FSM"   and xs :: "('e \<times> 'f) list" shows "sequence_to_failure M1 M2 xs = (butlast xs \<in> L M2 \<inter> L M1 \<and> xs \<in> L M1 - L M2)"
assms_3: shows "q \<le> x\<^sup>\<star> ; r"
linear_zero: shows "linear_on diff_fun_space UNIV (*\<^sub>R) (*\<^sub>R) (\<lambda>x. 0)"
rec_noteq_def: shows "rec_noteq = CN rec_not [rec_eq]"
d_add_ax: fixes x :: "'a"   and y :: "'a" shows "d (x + y) = d x + d y"
comp1_reachable: fixes C1 :: "('e set \<times> 'e set \<times> 'e set) \<times> 'f set \<times> ('f \<times> 'e \<times> 'f) set"   and C2 :: "('e set \<times> 'e set \<times> 'e set) \<times> 'g set \<times> ('g \<times> 'e \<times> 'g) set"   and s :: "'f \<times> 'g" assumes "reachable (C1 || C2) s" shows "reachable C1 (fst s)"
cod_eqI': fixes a :: "'s"   and f :: "'s" assumes "a \<in> codomains f" shows "a = cod f"
eq: shows "(\<lambda>b. {x \<in> A. f x = b}) ` B - {{}} = P"
\<phi>: shows "\<phi> = Vx : A \<turnstile> e\<equiv>d : B"
mult_single: fixes k :: "'a"   and a :: "'b"   and l :: "'a"   and b :: "'b" shows "Poly_Mapping.single k a * Poly_Mapping.single l b = Poly_Mapping.single (k + l) (a * b)"
vector_complement_closed: fixes uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and conv :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.pd_allegory uminus inf less_eq less sup bot top one times conv"   and "times x top = x" shows "times (uminus x) top = uminus x"
sum_portfolio: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and pf2 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real" assumes "portfolio pf1"   and "portfolio pf2" shows "portfolio (qty_sum pf1 pf2)"
is_semi: fixes G :: "nat graph_rec" assumes "dom_sl G" shows "semilat ((rev \<circ> sorted_list_of_set) ` Pow (set (cfg_doms.nodes G)), nodes_le, nodes_sup)"
prime_square_sum_nat_decomp_def: fixes p :: "nat" shows "prime_square_sum_nat_decomp p = (if prime p \<and> (p = 2 \<or> [p = 1] (mod 4)) then THE (x, y). x \<le> y \<and> x\<^sup>2 + y\<^sup>2 = p else (0, 0))"
1: shows "rawpsubstT (rawpsubstT t1 (zip (map Var us) (map snd txs))) (zip (map fst txs) us) = rawpsubstT (rawpsubstT t1 (zip (map Var vs1) (map snd txs))) (zip (map fst txs) vs1)"
fps_integral0_linear2: fixes f :: "'a fps"   and a :: "'a"   and g :: "'a fps"   and b :: "'a" shows "fps_integral0 (f * fps_const a + g * fps_const b) = fps_integral0 f * fps_const a + fps_integral0 g * fps_const b"
obj_def: fixes a :: "'b" shows "B.obj a \<equiv> B.arr a \<and> src\<^sub>B a = a"
cont': shows "continuous_on UNIV (\<lambda>v. sterm_sem I \<theta>\<^sub>1 v *\<^sub>R blin_frechet (good_interp I) (simple_term \<theta>\<^sub>2) v + sterm_sem I \<theta>\<^sub>2 v *\<^sub>R blin_frechet (good_interp I) (simple_term \<theta>\<^sub>1) v)"
rep_eq: fixes x :: "'d ccsubspace"   and xa :: "'d ccsubspace" shows "(x \<le> xa) = (space_as_set x \<subseteq> space_as_set xa)"
thetaIfWT_WretrT: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis" shows "PL_Indis.thetaIfWT tval aval indis \<subseteq> PL_Indis.WretrT tval aval indis (PL_Indis.thetaIfWT tval aval indis \<union> PL_Indis.WbisT tval aval indis)"
closedin_subtopology_refl: fixes U :: "'a topology"   and X :: "'a set" shows "closedin (subtopology U X) X = (X \<subseteq> topspace U)"
intro\<^sub>3: fixes lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and lc' :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "lc \<R>\<^sup>3\<^bsub>\<Gamma>,\<S>,P\<^esub> lc'" shows "\<R>p \<Gamma> \<S> P lc lc'"
inside_3: shows "blinfun_apply (Ds x) (f x) < 0"
poly_mult_lead_coeff: fixes K :: "'a set"   and p1 :: "'a list"   and p2 :: "'a list" assumes "subring K R"   and "polynomial K p1"   and "polynomial K p2"   and "p1 \<noteq> []"   and "p2 \<noteq> []" shows "lead_coeff (poly_mult p1 p2) = lead_coeff p1 \<otimes> lead_coeff p2"
tr'_par_iff_unlabel_tr: fixes A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and D :: "('lbl strand_label \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and B :: "('fun, 'var) strand_step list" assumes "\<forall>(i, p)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<union> set D. \<forall>(j, q)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<union> set D. p = q \<longrightarrow> i = j" shows "(\<exists>C\<in>set (tr'\<^sub>p\<^sub>c A D). B = unlabel C) = (B \<in> set (tr (unlabel A) (unlabel D)))"
perm_fresh_fresh: fixes a :: "name"   and x :: "'a"   and b :: "name" assumes "a \<sharp> x"   and "b \<sharp> x" shows "[(a, b)] \<bullet> x = x"
antisym: fixes b :: "'c Word.word"   and a :: "'c Word.word" assumes "b \<le>s a"   and "a \<le>s b" shows "a = b"
esssup_non_measurable: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure" assumes "f \<notin> borel_measurable M" shows "esssup M f = top"
assms_2: shows "Nom a at b in (ps, a) # branch"
bounded_hashcode_nat_def: fixes n :: "nat"   and x :: "'a" shows "bounded_hashcode_nat n x = nat_of_hashcode (hashcode x) mod n"
proots_within_times: fixes p :: "'b poly"   and q :: "'b poly"   and s :: "'b set" shows "proots_within (p * q) s = proots_within p s \<union> proots_within q s"
mtx_nonzeroN: shows "mtx_nonzero c \<subseteq> {0..<N} \<times> {0..<N}"
trg_vcomp: fixes \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.seq \<nu> \<mu>" shows "Span.trg (\<nu> \<bullet> \<mu>) = Span.trg \<nu> \<bullet> Span.trg \<mu>"
has_derivative_zero_unique_strong_interval: fixes k :: "real set"   and a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and y :: "'a"   and x :: "real" assumes "finite k"   and "continuous_on {a..b} f"   and "f a = y"   and "\<And>x. x \<in> {a..b} - k \<Longrightarrow> (f has_derivative (\<lambda>h. 0::'a)) (at x within {a..b})"   and "x \<in> {a..b}" shows "f x = y"
subcocycle_lim_birkhoff_2: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and u :: "'b \<Rightarrow> real" assumes "fmpt M T"   and "integrable M u" shows "AE x in M. fmpt.subcocycle_lim M T (qmpt.birkhoff_sum T u) x = real_cond_exp M (qmpt.Invariants M T) u x"
IN_le_OUT: fixes x :: "'v" assumes "x \<notin> B \<Gamma>" shows "d_IN f' x \<le> d_OUT f' x"
open_sums: fixes S :: "'a set"   and T :: "'a set" assumes "open S \<or> open T" shows "open (\<Union>x\<in>S. \<Union>y\<in>T. {x + y})"
Derives1_split: fixes u :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and v :: "symbol list" assumes "Derives1 u i r v" shows "\<exists>x y. u = x @ [fst r] @ y \<and> v = x @ snd r @ y \<and> length x = i"
mat_gtD_1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sd :: "nat"   and A :: "'b mat"   and B :: "'b mat" assumes "ord.mat_gt less_eq gt sd A B" shows "ord.mat_ge less_eq A B"
STEP: fixes s :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "I s" shows "if bval b s then wlp \<pi> c I s else Q s"
distinct_sorted_list_of_set: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and A :: "'g set" assumes "class.linorder less_eq less" shows "distinct (linorder.sorted_list_of_set less_eq A)"
simps_1: fixes Y :: "'d list" shows "injectionsAlg [] Y = [{}]"
inf_less_eq: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.linear_lattice inf less_eq less sup" shows "less_eq (inf x y) z = (less_eq x z \<or> less_eq y z)"
in_nres_rel_iff: fixes a :: "'a nres"   and b :: "'b nres"   and R :: "('a \<times> 'b) set" shows "((a, b) \<in> \<langle>R\<rangle>nres_rel) = (a \<le> \<Down> R b)"
set_map: fixes f :: "'e \<Rightarrow> 'c"   and v :: "'e set['d]" shows "set_bset (map_bset f v) = f ` set_bset v"
assms_1: shows "ab = [a, b]\<^sub>\<circ>"
min_list_conv_Min: fixes xs :: "nat list" assumes "xs \<noteq> []" shows "ListAux.min_list xs = Min (set xs)"
subadditive_un_le_nu1: fixes u :: "nat \<Rightarrow> real"   and n :: "nat" assumes "subadditive u"   and "0 < n" shows "u n \<le> real n * u 1"
one_le_card_finite: shows "Suc 0 \<le> CARD('b)"
interleaves_equal_nil_all: fixes P :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"   and xs :: "'a list"   and zs :: "'a list" assumes "xs \<simeq> {[], zs, P}" shows "xs = zs"
maxcoeff_ismaxcoeff: fixes t :: "Ferrack.num" shows "ismaxcoeff t (maxcoeff t)"
H'_1: shows "A \<in> f \<turnstile> \<C>"
charact_cond_exp: fixes Y :: "'a \<Rightarrow> 'c"   and X :: "'a \<Rightarrow> real"   and N :: "'c measure" assumes "disc_fct Y"   and "integrable M X"   and "point_measurable M (space N) Y"   and "Y \<in> space M \<rightarrow> space N"   and "\<forall>r\<in>range Y \<inter> space N. \<exists>A\<in>sets N. range Y \<inter> A = {r}" shows "AE w in M. real_cond_exp M (fct_gen_subalgebra M N Y) X w = expl_cond_expect M Y X w"
ars_r_def: fixes x :: "'b ppath set" shows "rdom x = pp_a (pp_a x)"
Gsk': shows "Gsk\<^sup>T \<in> carrier_mat n k"
compatible_modes_new_vars: fixes mdss :: "(Mode \<Rightarrow> 'Var\<^sub>C set) list" assumes "length mdss = length cms"   and "modes_respect_priv mdss" shows "conc.compatible_modes (conc.restrict_modes mdss (- range var\<^sub>C_of))"
cases: fixes x :: "'e list" obtains xa :: "'e" and xs :: "'e list" where "x = xa # xs"   | "x = []"
circle_cube_boundary_to_polarcircle_def: fixes \<gamma> :: "int \<times> (real \<Rightarrow> real \<times> real)" shows "circle_cube_boundary_to_polarcircle \<gamma> \<equiv> if \<gamma> = circle_top_edge then {subcube 0 (1 / 4) (1, circle_polar), subcube (1 / 4) (1 / 2) (1, circle_polar)} else if \<gamma> = circle_bot_edge then {subcube (1 / 2) (3 / 4) (1, circle_polar), subcube (3 / 4) 1 (1, circle_polar)} else {}"
lens_override_plus: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e"   and S\<^sub>1 :: "'e"   and S\<^sub>2 :: "'e" assumes "X \<bowtie> Y" shows "S\<^sub>1 \<triangleleft>\<^bsub>X +\<^sub>L Y\<^esub> S\<^sub>2 = S\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> S\<^sub>2 \<triangleleft>\<^bsub>Y\<^esub> S\<^sub>2"
col_polys_to_mat: fixes j :: "nat"   and ts :: "'d list"   and ps :: "('d \<Rightarrow>\<^sub>0 'e) list" assumes "j < length ts" shows "col (polys_to_mat ts ps) j = vec_of_list (map (\<lambda>p. lookup p (ts ! j)) ps)"
all_partitions_def: fixes A :: "'b set" shows "all_partitions A = {P. P partitions A}"
nonzero_divide_eq_eq: fixes c :: "'a"   and b :: "'a"   and a :: "'a" assumes "c \<noteq> (0::'a)" shows "(b / c = a) = (b = a * c)"
try_bind_assert_spmf: fixes b :: "bool"   and f :: "unit \<Rightarrow> 'a spmf"   and q :: "'a spmf" shows "TRY assert_spmf b \<bind> f ELSE q = (if b then TRY f () ELSE q else q)"
onode_sos_net_states: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and s :: "'e net_state"   and a :: "'f node_action"   and \<sigma>' :: "nat \<Rightarrow> 'd"   and s' :: "'e net_state"   and S :: "(((nat \<Rightarrow> 'd) \<times> 'e) \<times> 'f seq_action \<times> (nat \<Rightarrow> 'd) \<times> 'e) set" assumes "((\<sigma>, s), a, \<sigma>', s') \<in> onode_sos S" shows "\<exists>i \<zeta> R \<zeta>' R'. s = NodeS i \<zeta> R \<and> s' = NodeS i \<zeta>' R'"
induct: fixes x1 :: "'node SDG_node set"   and x2 :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and x3 :: "'node list"   and x4 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and x5 :: "'edge list"   and x6 :: "'node list"   and x7 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and P :: "'node SDG_node set \<Rightarrow> ('edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind) \<Rightarrow> 'node list \<Rightarrow> (('var \<Rightarrow> 'val option) \<times> 'ret) list \<Rightarrow> 'edge list \<Rightarrow> 'node list \<Rightarrow> (('var \<Rightarrow> 'val option) \<times> 'ret) list \<Rightarrow> bool" assumes "x1,x2 \<turnstile> (x3,x4) =x5\<Rightarrow> (x6,x7)"   and "\<And>S f ms s as ms' s' a ms'' s''. \<lbrakk>S,f \<turnstile> (ms,s) =as\<Rightarrow>\<^sub>\<tau> (ms',s'); S,f \<turnstile> (ms',s') -a\<rightarrow> (ms'',s'')\<rbrakk> \<Longrightarrow> P S f ms s (as @ [a]) ms'' s''" shows "P x1 x2 x3 x4 x5 x6 x7"