hcju/isabelleps huggingface dataset:
vquot_fm_def: fixes A :: "fm"   and V :: "name set" shows "\<lfloor>A\<rfloor>V = vquot_dbfm V (trans_fm [] A)"
fps_inverse_fps_X_plus1': assumes "inverse (1::'a) = (1::'a)" shows "inverse (1 + fps_X) = Abs_fps ((^) (- (1::'a)))"
proj2_incident_iff_Col: fixes p :: "proj2"   and q :: "proj2"   and l :: "proj2_line"   and r :: "proj2" assumes "p \<noteq> q"   and "proj2_incident p l"   and "proj2_incident q l" shows "proj2_incident r l = proj2_Col p q r"
that: fixes env :: "(name, 'a) fmap" assumes "subst t env = u" shows "thesis"
InitBlockRedsFinal: fixes e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and V :: "char list"   and v' :: "val"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty"   and v :: "val"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P,E(V \<mapsto> T) \<turnstile> \<langle>e,(h, l(V \<mapsto> v'))\<rangle> \<rightarrow>* \<langle>e',(h', l')\<rangle>"   and "final e'"   and "P \<turnstile> T casts v to v' "   and "wf_prog wf_md P" shows "P,E \<turnstile> \<langle>{V:T; V:=Val v;; e},(h, l)\<rangle> \<rightarrow>* \<langle>e',(h', l'(V := l V))\<rangle>"
comp_left_increasing_sup: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<le> (x \<squnion> z) \<sqinter> y"
integral_tweak_ends: fixes a :: "real"   and b :: "real"   and e :: "real"   and d :: "real" assumes "a < b"   and "0 < e" obtains f :: "real \<Rightarrow> real" where "continuous_on {a..b} f" and "f a = d" and "f b = 0" and "l2norm {a..b} f < e"
sin_angle_nonneg: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "0 \<le> sin (angle a b c)"
cases: fixes a :: "('b, 'c) Kruskal_Examples.term"   and F :: "('b \<times> nat) set" assumes "a \<in> gterms F" obtains f :: "'b" and n :: "nat" and ts :: "('b, 'c) Kruskal_Examples.term list" where "a = Fun f ts" and "(f, n) \<in> F" and "length ts = n" and "\<forall>s\<in>set ts. s \<in> gterms F"
perm_bij: fixes pi :: "(name \<times> name) list"   and x :: "'d"   and y :: "'d" shows "(pi \<bullet> x = pi \<bullet> y) = (x = y)"
hyps_2: shows "\<Sigma>1 = norm_stack \<Sigma>2"
fibfib: fixes h :: "nat" shows "fib' h + 1 = fib (Suc (Suc h))"
3: shows "(\<lambda>i. interval_lebesgue_integral lborel (ereal (l i)) (ereal (u i)) f) \<longlonglongrightarrow> B - A"
R_lin_independent_Cons_imp_independent_RSpans: fixes m :: "'m"   and ms :: "'m list" assumes "m \<in> M"   and "R_lin_independent (m # ms)" shows "add_independentS [RSpan [m], RSpan ms]"
quantale_closed_maps_def: shows "quantale_closed_maps = {f. \<forall>x y. f x \<cdot> f y \<le> f (x \<cdot> y)}"
plus_1: fixes x :: "'b" shows "sin (x + (2::'b) * of_real pi) = sin x"
post_chainmap: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and \<Pi> :: "('e, 'f) chain" shows "post (chainmap f g \<Pi>) = f (post \<Pi>)"
BinOp2Reds: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and v :: "val"   and bop :: "bop" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>Val v \<guillemotleft>bop\<guillemotright> e,s,b\<rangle> \<rightarrow>* \<langle>Val v \<guillemotleft>bop\<guillemotright> e',s',b'\<rangle>"
UN_subset_iff: fixes f :: "'b \<Rightarrow> 'a set"   and I :: "'b set"   and B :: "'a set" shows "(\<Union> (f ` I) \<subseteq> B) = (\<forall>i\<in>I. f i \<subseteq> B)"
sync_tLength_eq_trc: fixes t :: "'s Trace"   and t' :: "'s Trace"   and T :: "'s Trace set"   and as :: "'a set" assumes "(t, t') \<in> (\<Union> (relations (mkM T) ` as))\<^sup>*" shows "tLength t = tLength t'"
assms: shows "x \<in> xs \<or> (\<exists>xs\<in>set xss. x \<in> xs)"
normal_exponent_bounds_real_2: fixes x :: "('a, 'b) IEEE.float" assumes "is_normal x" shows "real_of_int (normal_exponent x) \<le> 2 ^ (LENGTH('a) - 1) - real LENGTH('b) - 1"
dg_prod_3_ArrE: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and hgf :: "V" assumes "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>"   and "digraph \<alpha> \<CC>"   and "hgf \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G\<^sub>3 \<BB> \<times>\<^sub>D\<^sub>G\<^sub>3 \<CC>)\<lparr>Arr\<rparr>" obtains h :: "V" and g :: "V" and f :: "V" where "hgf = [h, g, f]\<^sub>\<circ>" and "h \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" and "g \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>" and "f \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"
simps_2: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list"   and k :: "nat"   and u :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "LocalLexing.\<Z> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k (Suc u) = LocalLexing.\<Y> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc (LocalLexing.\<Z> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k u) (LocalLexing.\<P> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k u) k"
inducts: fixes P :: "tyS \<Rightarrow> bool"   and tyS :: "tyS" assumes "\<And>ty. P (Ty ty)"   and "\<And>tvar tyS. P tyS \<Longrightarrow> P (\<forall>[tvar].tyS)" shows "P tyS"
subclsSD: fixes C :: "char list"   and D :: "char list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" assumes "P \<turnstile> C \<prec>\<^sub>S D" shows "\<exists>fs ms Bs. class P C = \<lfloor>(Bs, fs, ms)\<rfloor> \<and> Shares D \<in> set Bs"
leg_simps_4: shows "C.dom \<eta>.cod.leg1 = \<eta>.cod.apex"
ide_char: fixes f :: "(bool, 'arr list) concrete_category.arr" shows "ide f = (f = Zero \<or> f = One)"
scaleR_scaleC: fixes scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and scaleC :: "complex \<Rightarrow> 'd \<Rightarrow> 'd"   and r :: "real" assumes "class.scaleC scaleR scaleC" shows "scaleR r = scaleC (complex_of_real r)"
U2: shows "U2 \<equiv> Ut \<times> Unt"
rStatImp: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "(\<Psi>, P, Q) \<in> X" shows "\<Psi> \<rhd> P \<lessapprox><X> Q"
eq: fixes x :: "real" shows "\<bar>\<alpha>\<bar> * normal_density (\<beta> + \<alpha> * \<mu>) (\<bar>\<alpha>\<bar> * \<sigma>) (x * \<alpha> + \<beta>) = normal_density \<mu> \<sigma> x"
upper_triangular_def: fixes A :: "(('a, 'b) vec, 'b) vec" shows "upper_triangular A = (\<forall>i j. j < i \<longrightarrow> A $ i $ j = (0::'a))"
test_distrib_left: fixes p :: "'a"   and q :: "'a"   and r :: "'a" assumes "test p"   and "test q"   and "test r" shows "p \<cdot> (q + r) = p \<cdot> q + p \<cdot> r"
measurable_restrict_space_iff: fixes \<Omega> :: "'a set"   and M :: "'a measure"   and c :: "'b"   and N :: "'b measure"   and f :: "'a \<Rightarrow> 'b" assumes "\<Omega> \<inter> space M \<in> sets M"   and "c \<in> space N" shows "(f \<in> restrict_space M \<Omega> \<rightarrow>\<^sub>M N) = ((\<lambda>x. if x \<in> \<Omega> then f x else c) \<in> M \<rightarrow>\<^sub>M N)"
plus_ms_aux_MSLCons: fixes x :: "'a \<times> real"   and xs :: "('a \<times> real) msllist"   and y :: "'a \<times> real"   and ys :: "('a \<times> real) msllist" shows "plus_ms_aux (MSLCons x xs) (MSLCons y ys) = (if snd y < snd x then MSLCons x (plus_ms_aux xs (MSLCons y ys)) else if snd x < snd y then MSLCons y (plus_ms_aux (MSLCons x xs) ys) else MSLCons (fst x + fst y, snd x) (plus_ms_aux xs ys))"
a_3: fixes v :: "'variable" assumes "v \<notin> set (add_effects_of op) \<and> v \<notin> set (delete_effects_of op)" shows "(s ++ map_of (effect_to_assignments op)) v = s v"
E_TApp: fixes t :: "trm"   and t' :: "trm"   and T :: "ty" assumes "t \<longmapsto> t'" shows "t \<cdot>\<^sub>\<tau> T \<longmapsto> t' \<cdot>\<^sub>\<tau> T"
transfer_prover_start: fixes x :: "'e"   and x' :: "'e"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and y :: "'f" assumes "x = x'"   and "Transfer.Rel R x' y" shows "Transfer.Rel R x y"
scale_scale: fixes a :: "real"   and b :: "real"   and x :: "'a" shows "a *\<^sub>R b *\<^sub>R x = (a * b) *\<^sub>R x"
wp_mod_def: fixes \<pi> :: "char list \<Rightarrow> com option"   and vs :: "char list set"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "wp_mod \<pi> vs c Q s = wp \<pi> c (\<lambda>s'. modifies vs s' s \<and> Q s') s"
ds_ord_def: fixes x :: "'a"   and y :: "'a" shows "domain_semigroup.ds_ord d (\<cdot>) x y = (x = d x \<cdot> y)"
Red_I_to_Inf: fixes N :: "'g set" shows "Red_I_G N \<subseteq> Inf_G"
typing_det: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and expr :: "'a expr"   and \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E expr : \<tau>"   and "\<Gamma> \<turnstile>\<^sub>E expr : \<sigma>" shows "\<tau> = \<sigma>"
sup_inf_distrib1: fixes x :: "'a multiset"   and y :: "'a multiset"   and z :: "'a multiset" shows "x \<union># (y \<inter># z) = x \<union># y \<inter># (x \<union># z)"
prems: shows "{a. case a of (t, X) \<Rightarrow> (t, X) \<in> set (x # xs) \<and> as \<in> Z X t} = {}"
hB: shows "\<forall>x. insertion (nth_default 0 (L[var := x])) B = Bv"
nn_integral_null_set: fixes N :: "'b set"   and M :: "'b measure"   and u :: "'b \<Rightarrow> ennreal" assumes "N \<in> null_sets M" shows "set_nn_integral M N u = 0"
subst_all_ground_ident: fixes M :: "('b, 'a) Term.term set"   and s :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "ground M" shows "M \<cdot>\<^sub>s\<^sub>e\<^sub>t s = M"
Ain: shows "A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b)) \<noteq> (0::'b)"
rel_refl_strong: fixes x :: "'b ref"   and Ra :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>z. z \<in> set_ref x \<Longrightarrow> Ra z z" shows "rel_ref Ra x x"
column_Gram_Schmidt_upt_k: fixes k :: "nat"   and A :: "((real, 'a) vec, 'b) vec" assumes "k < ncols A" shows "column (mod_type_class.from_nat k) (Gram_Schmidt_upt_k A k) = column (mod_type_class.from_nat k) A - (\<Sum>x\<in>{column i (Gram_Schmidt_upt_k A k) |i. i < mod_type_class.from_nat k}. (x \<bullet> column (mod_type_class.from_nat k) A / (x \<bullet> x)) *\<^sub>R x)"
sup_state_conv2: fixes P :: "'b prog"   and s1 :: "ty list \<times> ty err list"   and s2 :: "ty list \<times> ty err list" shows "P \<turnstile> s1 \<le>\<^sub>i s2 = (P \<turnstile> fst s1 [\<le>] fst s2 \<and> P \<turnstile> snd s1 [\<le>\<^sub>\<top>] snd s2)"
DD: shows "DD ars r (\<tau>, \<sigma>, \<sigma>', \<tau>')"
Legendre_mod: fixes p :: "int"   and n :: "int"   and a :: "int" assumes "p dvd n" shows "Legendre (a mod n) p = Legendre a p"
pos_1: fixes x :: "'a" assumes "x \<in> space M" shows "0 < h x \<and> h x < \<infinity>"
netglobal_def: fixes P :: "(nat \<Rightarrow> state) \<Rightarrow> bool" shows "opq.netglobal P \<equiv> \<lambda>s. P (fst (opq.initmissing (netgmap (\<lambda>(p, q). (fst (Fun.id p), snd (Fun.id p), q)) s)))"
that: fixes L :: "real" assumes "X \<longlonglongrightarrow> L"   and "\<forall>i. X i \<le> L" shows "thesis"
nn_integral_embed_measure': fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and g :: "'b \<Rightarrow> ennreal" assumes "inj_on f (space M)"   and "g \<in> borel_measurable (embed_measure M f)" shows "integral\<^sup>N (embed_measure M f) g = \<integral>\<^sup>+ x. g (f x) \<partial>M"
asm: shows "\<not> fair_state_seq (lmap state_of_wstate Sts)"
to_fun_closed: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "f \<in> carrier P"   and "x \<in> carrier R" shows "to_fun f x \<in> carrier R"
vectorspace_def: fixes K :: "('a, 'b) ring_scheme"   and V :: "('a, 'c, 'd) module_scheme" shows "vectorspace K V \<equiv> Module.module K V \<and> field K"
sum_eq: shows "real_of_float (Float m1 e1) + real_of_float (Float m2 e2) = (real_of_int (m1 * 2 ^ Suc k1) + real_of_int m2 * 2 powr (real_of_int e2 - real_of_int e1 + real k1 + 1)) * 2 powr real_of_int (e1 - int k1 - 1)"
assms_2: shows "\<forall>\<^sub>F x in at_top. l x \<le> f x"
iso_group_mul_eq: shows "((\<lambda>(x, y). x \<otimes> y) \<in> Group.iso (subgroup_generated G A \<times>\<times> subgroup_generated G B) G) = (A \<inter> B = {\<one>} \<and> A <#> B = carrier G \<and> A \<lhd> G \<and> B \<lhd> G)"
MapO_def: fixes F :: "('e, 'd, 'f, 'g, 'h, 'i, 'j) Functor_scheme"   and X :: "'e" shows "F @@ X \<equiv> THE Y. Y \<in> obj\<^bsub>CatCod F\<^esub> \<and> F ## id\<^bsub>CatDom F\<^esub> X = id\<^bsub>CatCod F\<^esub> Y"
is_abelian_group: shows "abelian_group (R Quot I)"
j_le: shows "j < dim_vec (tensors_from_net (Conv (id_matrix r0 M) (Input M)))"
induct: fixes g :: "'g"   and x :: "'node list"   and P :: "'node list \<Rightarrow> bool" assumes "local.path g x"   and "\<And>n. \<lbrakk>n \<in> set (\<alpha>n g); invar g\<rbrakk> \<Longrightarrow> P [n]"   and "\<And>ns n'. \<lbrakk>local.path g ns; P ns; n' \<in> set (predecessors g (hd ns))\<rbrakk> \<Longrightarrow> P (n' # ns)" shows "P x"
pow_simps_3: fixes n :: "nat"   and P :: "pol" shows "Commutative_Ring.pow (Suc (2 * n)) P = P \<langle>*\<rangle> Commutative_Ring.pow n (sqr P)"
Min_gr_iff: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "(x < Min A) = (\<forall>a\<in>A. x < a)"
subgroup_generated_is_subgroup: fixes H :: "'a set" shows "subgroup (additive.subgroup_generated H) R (+) \<zero>"
sxalloc_elim_cases_1: fixes G :: "prog"   and s :: "st"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>sxalloc\<rightarrow> s'" obtains "s' = Norm s"
differentMarkedElementsHaveDifferentLevels: fixes M :: "('b \<times> bool) list"   and a :: "'b"   and b :: "'b" assumes "uniq (elements M)"   and "a \<in> set (markedElements M)"   and "b \<in> set (markedElements M)"   and "a \<noteq> b" shows "elementLevel a M \<noteq> elementLevel b M"
sigma_le_sets: fixes A :: "'c set set"   and X :: "'c set"   and N :: "'c measure" assumes "A \<subseteq> Pow X" shows "(sets (sigma X A) \<subseteq> sets N) = (X \<in> sets N \<and> A \<subseteq> sets N)"
wt_ge_0: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and c :: "('c, 'd, 'a) cmd"   and i :: "nat"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b" assumes "PL cval"   and "proper c"   and "i < PL.brn c" shows "0 \<le> PL.wt aval tval cval c s i"
l2_def: shows "l2 \<equiv> \<lparr>init = l2_init, trans = l2_trans, obs = id\<rparr>"
load_tmps_fresh_def: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" shows "load_tmps_fresh ts \<equiv> \<forall>\<R> i p is \<O> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> load_tmps is \<inter> dom \<theta> = {}"
aval_take: fixes x :: "vname aexp"   and a :: "nat"   and i :: "value list"   and r :: "nat \<Rightarrow>f value option" assumes "max_input x < Some a" shows "aval x (join_ir i r) = aval x (join_ir (take a i) r)"
inf_prefix_append: fixes zs :: "'b list"   and xs :: "'b list"   and ys :: "'b list" shows "(zs @ xs) \<sqinter> (zs @ ys) = zs @ xs \<sqinter> ys"
store_sub2_low_equal_sub4: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and addr :: "word32"   and instr :: "sparc_operation \<times> inst_operand list"   and curr_win :: "'b word"   and rd :: "word5"   and y :: "'b sparc_state"   and ya :: "'b sparc_state"   and yb :: "'b sparc_state"   and yc :: "'b sparc_state" assumes "low_equal s1 s2 \<and> memory_write 10 addr (st_byte_mask instr addr) (st_data0 instr curr_win rd addr s2) s1 = Some y \<and> memory_write 10 addr (st_byte_mask instr addr) (st_data0 instr curr_win rd addr s2) s2 = Some ya \<and> memory_write 10 (addr + 4) 15 (user_reg_val curr_win (rd OR 1) y) y = Some yb \<and> memory_write 10 (addr + 4) 15 (user_reg_val curr_win (rd OR 1) ya) ya = Some yc" shows "low_equal yb yc"
new_tv_subst_var: fixes n :: "nat"   and m :: "nat"   and S :: "nat \<Rightarrow> typ" assumes "n < m"   and "new_tv m S" shows "new_tv m (S n)"
iterate_size_abort_correct: fixes it :: "(nat \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat"   and S0 :: "'a set"   and n :: "nat" assumes "set_iterator it S0" shows "iterate_size_abort it n = min n (card S0) \<and> finite S0"
diff_diff_eq2: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.group_add minus plus zero uminus" shows "minus a (minus b c) = minus (plus a c) b"
SourcesS12_L2: shows "Sources level2 sS12 = {sS9, sS10}"
intros_4: fixes u :: "'d \<Rightarrow> 'c"   and c :: "'d"   and d :: "'c" assumes "u c = d" shows "u \<turnstile> EQ c d"
preL: fixes b :: "'a \<times> 'b" shows "pre_adjust \<langle>AA_Map.delete x l, (b, lv), r\<rangle>"
iso_inv_iso: fixes f :: "'a \<times> 'a" assumes "VoV.iso f" shows "VoV.iso (VoV.inv f)"
\<sigma>_eq: shows "\<sigma> = Finite_Set.fold f \<sigma>0 S"
image_UN: fixes f :: "'e \<Rightarrow> 'd"   and B :: "'f \<Rightarrow> 'e set"   and A :: "'f set" shows "f ` \<Union> (B ` A) = (\<Union>x\<in>A. f ` B x)"
closed_connected_component: fixes S :: "'b set"   and x :: "'b" assumes "closed S" shows "closed (connected_component_set S x)"
simps_2: fixes lformula0 :: "'a \<Rightarrow> bool"   and a :: "'a" shows "Formula_Operations.lformula lformula0 (FBase a) = lformula0 a"
prefix_def: fixes l1 :: "'b list"   and l2 :: "'b list" shows "l1 \<preceq> l2 \<equiv> \<exists>l3. l1 @ l3 = l2"
BinOpThrow1: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e :: "expr"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and bop :: "bop"   and e\<^sub>2 :: "expr" assumes "P,E \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>1\<rangle>" shows "P,E \<turnstile> \<langle>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>1\<rangle>"
memop_step: shows "(is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is', \<theta>', sb', m', \<D>', \<O>', \<R>', \<S>')"
subspace_imp_affine: fixes s :: "'a set" assumes "subspace s" shows "affine s"
assms_3: fixes x :: "'a"   and y :: "'a" shows "u \<cdot> (x + y) \<le> u \<cdot> x + u \<cdot> y"
intro: fixes \<Omega> :: "'c set"   and M :: "'c set set" assumes "ring_of_sets \<Omega> M"   and "algebra_axioms \<Omega> M" shows "algebra \<Omega> M"
path_inv_Branching: fixes p :: "bool list"   and l :: "tree"   and r :: "tree" shows "path p (Branching l r) = (p = [] \<or> (\<exists>a p'. p = a # p' \<and> (a \<longrightarrow> path p' l) \<and> (\<not> a \<longrightarrow> path p' r)))"
rebase_poly_id: fixes p :: "'a mod_ring poly" assumes "rebase_ge TYPE('a) TYPE('b)" shows "##p = #p"
WTrt_elim_cases_11: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and P :: "'c J_prog"   and h :: "'b"   and E :: "String.literal \<Rightarrow> ty option"   and a :: "'c expr"   and T :: "ty" assumes "J_heap_base.WTrt typeof_addr P h E (a\<bullet>length) T" obtains T :: "ty" where "Ta__ = Integer" and "J_heap_base.WTrt typeof_addr P h E a (T\<lfloor>\<rceil>)"   | "J_heap_base.WTrt typeof_addr P h E a NT"
msetext_huet_mono_strong: fixes ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and gt' :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>y\<in>set ys. \<forall>x\<in>set xs. gt y x \<longrightarrow> gt' y x"   and "msetext_huet gt ys xs" shows "msetext_huet gt' ys xs"
horizontal_chop_width_stable: fixes v :: "view"   and u :: "view"   and w :: "view" shows "v=u\<parallel>w \<longrightarrow> |lan v| = |lan u| \<and> |lan v| = |lan w|"
nn_integral_density: fixes f :: "'c \<Rightarrow> ennreal"   and M :: "'c measure"   and g :: "'c \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "integral\<^sup>N (density M f) g = \<integral>\<^sup>+ x. f x * g x \<partial>M"
upset_faithful: fixes x :: "'a"   and y :: "'a" assumes "\<up> x \<subseteq> \<up> y" shows "y \<le> x"
assms_2: fixes i :: "'b"   and j :: "'c"   and k :: "'d"   and l :: "'e"   and m :: "'f" assumes "(i, j, k, l, m) \<in> A" shows "P i j k l m is H"
countable_INT'': fixes I :: "'d set"   and F :: "'d \<Rightarrow> 'c set" assumes "UNIV \<in> sigma_sets UNIV (range greaterThan)"   and "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sigma_sets UNIV (range greaterThan)" shows "\<Inter> (F ` I) \<in> sigma_sets UNIV (range greaterThan)"
S2: shows "c_unfold k (c_snd u) \<noteq> []"
hseqE: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "S.hseq \<nu> \<mu>" obtains "S.arr \<mu>" and "S.arr \<nu>" and "S.src \<nu> = S.trg \<mu>"
equality: fixes r :: "'a v_state_scheme"   and r' :: "'a v_state_scheme" assumes "next_round r = next_round r'"   and "votes r = votes r'"   and "decisions r = decisions r'"   and "v_state.more r = v_state.more r'" shows "r = r'"
vertical_composite_def: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah"   and G :: "'ag \<Rightarrow> 'ah"   and H :: "'ag \<Rightarrow> 'ah"   and \<sigma> :: "'ag \<Rightarrow> 'ah"   and \<tau> :: "'ag \<Rightarrow> 'ah" shows "vertical_composite A B F G H \<sigma> \<tau> \<equiv> (category A \<and> category B \<and> functor A B F) \<and> (functor A B G \<and> functor A B H) \<and> natural_transformation A B F G \<sigma> \<and> natural_transformation A B G H \<tau>"
fresh_finfun_update: fixes a :: "atom"   and f :: "('c, 'd) finfun"   and x :: "'c"   and y :: "'d" assumes "a \<sharp> f"   and "a \<sharp> x"   and "a \<sharp> y" shows "a \<sharp> finfun_update f x y"
i_expand_i_take_mult: fixes n :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'a message_af" shows "f \<odot> k \<Down> (n * k) = f \<Down> n \<odot> k"
term_rawInduct: fixes phi :: "('a, 'b, 'varSort, 'var, 'c) qTerm set \<Rightarrow> bool"   and phiAbs :: "('a, 'b, 'varSort, 'var, 'c) qAbs set \<Rightarrow> bool"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set" assumes "\<And>xs x. phi (Var xs x)"   and "\<And>delta inp binp. \<lbrakk>goodInp inp; goodBinp binp; liftAll phi inp; liftAll phiAbs binp\<rbrakk> \<Longrightarrow> phi (Op delta inp binp)"   and "\<And>xs x X. \<lbrakk>good X; phi X\<rbrakk> \<Longrightarrow> phiAbs (Abs xs x X)" shows "(good X \<longrightarrow> phi X) \<and> (goodAbs A \<longrightarrow> phiAbs A)"