hcju/isabelleps huggingface dataset:
span_add_eq2: fixes y :: "'b"   and S :: "'b set"   and x :: "'b" assumes "y \<in> local.span S" shows "(x + y \<in> local.span S) = (x \<in> local.span S)"
countably_additiveI_finite: fixes M :: "'d measure"   and \<mu> :: "'d set \<Rightarrow> ennreal" assumes "finite (space M)"   and "positive (sets M) \<mu>"   and "Measure_Space.additive (sets M) \<mu>" shows "countably_additive (sets M) \<mu>"
mset_mergesort_by_rel_merge: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" shows "mset (mergesort_by_rel_merge R xs ys) = mset xs + mset ys"
silent_moves_into_RedT_\<tau>_inv: fixes t :: "'t"   and x :: "'x2"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and x' :: "'x2"   and m' :: "'m2" assumes "r2.silent_moves t (x, shr s) (x', m')"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "wset s t = None" shows "\<tau>mRed2 s (redT_upd_\<epsilon> s t x' m')"
extf_compat_list: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> 'c"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and y :: "('b, 'a) tm"   and x :: "('b, 'a) tm"   and gt :: "('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool"   and f :: "'b"   and xs :: "('b, 'a) tm list"   and xs' :: "('b, 'a) tm list" assumes "kbo_std_basis ground_heads_var gt_sym arity_sym arity_var wt_sym \<epsilon> \<delta> extf"   and "y \<noteq> x"   and "gt y x" shows "extf f gt (xs @ y # xs') (xs @ x # xs')"
lepoll_PiE: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set"   and C :: "'a \<Rightarrow> 'c set" assumes "\<And>i. i \<in> A \<Longrightarrow> B i \<lesssim> C i" shows "Pi\<^sub>E A B \<lesssim> Pi\<^sub>E A C"
mult_botr: fixes x :: "'b \<times> 'c" shows "times_prod' x bot_prod = bot_prod"
simps_1: fixes s :: "'b set" shows "partList3 s [] = []"
sg_def: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set" shows "G \<guillemotright> H  = (H \<noteq> {} \<and> H \<subseteq> carrier G \<and> (\<forall>a\<in>H. \<forall>b\<in>H. a \<cdot>\<^bsub>G\<^esub> \<rho>\<^bsub>G\<^esub> b \<in> H))"
distinct_1: fixes x2 :: "'b set" shows "FAILi \<noteq> RES x2"
unfold: fixes g :: "'f \<Rightarrow> ('e, 'f) step"   and s :: "'f"   and a :: "'e"   and s' :: "'f" assumes "g s = Yield a s'"   and "terminates_onp g s'" shows "terminates_onp g s"
functor_def: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e" shows "functor A B F \<equiv> category A \<and> category B \<and> functor_axioms A B F"
arr_dom: fixes f :: "'c" assumes "C'.arr f" shows "C'.arr (C'.dom f)"
intros: fixes a :: "'a"   and b :: "'b"   and r :: "('a \<times> 'b) set"   and c :: "'c"   and s :: "('b \<times> 'c) set" assumes "(a, b) \<in> r"   and "(b, c) \<in> s" shows "(a, c) \<in> r ; s"
cond2b: shows "\<forall>b. k < mod_type_class.to_nat b \<longrightarrow> A $ mod_type_class.from_nat k $ b = (0::'a)"
homotopic_with: fixes X :: "'b topology"   and P :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and Y :: "'c topology"   and p :: "'b \<Rightarrow> 'c"   and q :: "'b \<Rightarrow> 'c" assumes "\<And>h k. (\<And>x. x \<in> topspace X \<Longrightarrow> h x = k x) \<Longrightarrow> P h = P k" shows "homotopic_with P X Y p q = (\<exists>h. continuous_map (prod_topology (top_of_set {0..1}) X) Y h \<and> (\<forall>x\<in>topspace X. h (0, x) = p x) \<and> (\<forall>x\<in>topspace X. h (1, x) = q x) \<and> (\<forall>t\<in>{0..1}. P (\<lambda>x. h (t, x))))"
Imz: shows "Im z = - sqrt ((1 + ((Im z)\<^sup>2 + cmod (1 - z\<^sup>2)) - (Re z)\<^sup>2) / 2)"
comp_null_2: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "f \<cdot> null = null"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "nat \<Rightarrow>\<^sub>0 'b" shows "lookup (higherPowers x xa xb) = (\<lambda>xc. lookup xb (inc_above x xa xc))"
coplanar_perm_8: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet B C A D"
measure_Un_AE: fixes A :: "'b set"   and B :: "'b set"   and M :: "'b measure" assumes "AE x in M. x \<notin> A \<or> x \<notin> B"   and "A \<in> fmeasurable M"   and "B \<in> fmeasurable M" shows "Sigma_Algebra.measure M (A \<union> B) = Sigma_Algebra.measure M A + Sigma_Algebra.measure M B"
hoare_triple_def: fixes p :: "'b"   and c :: "'b"   and q :: "'b" shows "\<lbrace>p\<rbrace> c \<lbrace>q\<rbrace> \<equiv> p \<cdot> c \<le> c \<cdot> q"
simps_5: fixes e :: "nat \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a list \<Rightarrow> bool"   and p :: "('b, 'c) form"   and q :: "('b, 'c) form" shows "eval e f g (Or p q) = (eval e f g p \<or> eval e f g q)"
restrict_modes_length: fixes mdss :: "(Mode \<Rightarrow> 'Var\<^sub>C set) list"   and X :: "'Var\<^sub>C set" shows "length (conc.restrict_modes mdss X) = length mdss"
integral_bounded_linear': fixes T :: "'c \<Rightarrow> 'd"   and T' :: "'d \<Rightarrow> 'c"   and M :: "'e measure"   and f :: "'e \<Rightarrow> 'c" assumes "bounded_linear T"   and "bounded_linear T'"   and "\<not> (\<forall>x. T x = (0::'d)) \<Longrightarrow> \<forall>x. T' (T x) = x" shows "LINT x|M. T (f x) = T (integral\<^sup>L M f)"
prime_dvd_prod_mset_primes_iff: fixes p :: "'a"   and A :: "'a multiset" assumes "prime p"   and "\<And>q. q \<in># A \<Longrightarrow> prime q" shows "(p dvd \<Prod>\<^sub># A) = (p \<in># A)"
norm_bound_elements_mat: fixes A :: "'b mat"   and b :: "real" shows "norm_bound A b = (\<forall>x\<in>elements_mat A. norm x \<le> b)"
if_then_Some_else_None_eq_2: fixes a :: "bool"   and b :: "'a" shows "((if a then Some b else None) = None) = (\<not> a)"
commutative_monoid_axioms: shows "commutative_monoid stx.carrier_direct_lim stx.add_rel (stx.class_of X (zero_str\<^sub>X X))"
dim_insert: fixes x :: "('a, 'b) vec"   and S :: "('a, 'b) vec set" shows "vec.dim (insert x S) = (if x \<in> vec.span S then vec.dim S else vec.dim S + 1)"
psubstInp_preserves_wls: fixes delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" assumes "wlsInp delta inp"   and "wlsEnv rho" shows "wlsInp delta inp %[rho]"
triangle_left: shows "(\<epsilon>' \<star> f) \<cdot> (f \<star> \<eta>') = f"
rel_FGco_neg_distr_cond_def: fixes Co1 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Co1' :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Co2 :: "'y \<Rightarrow> 'z \<Rightarrow> bool"   and Co2' :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Co3 :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and Co3' :: "'ac \<Rightarrow> 'ad \<Rightarrow> bool"   and Co4 :: "'ae \<Rightarrow> 'af \<Rightarrow> bool"   and Co4' :: "'af \<Rightarrow> 'ag \<Rightarrow> bool"   and Co5 :: "'ah \<Rightarrow> 'ai \<Rightarrow> bool"   and Co5' :: "'ai \<Rightarrow> 'aj \<Rightarrow> bool"   and Co6 :: "'ak \<Rightarrow> 'al \<Rightarrow> bool"   and Co6' :: "'al \<Rightarrow> 'am \<Rightarrow> bool"   and Contra1 :: "'an \<Rightarrow> 'ao \<Rightarrow> bool"   and Contra1' :: "'ao \<Rightarrow> 'ap \<Rightarrow> bool"   and Contra2 :: "'aq \<Rightarrow> 'ar \<Rightarrow> bool"   and Contra2' :: "'ar \<Rightarrow> 'as \<Rightarrow> bool"   and Contra3 :: "'at \<Rightarrow> 'au \<Rightarrow> bool"   and Contra3' :: "'au \<Rightarrow> 'av \<Rightarrow> bool"   and Contra4 :: "'aw \<Rightarrow> 'ax \<Rightarrow> bool"   and Contra4' :: "'ax \<Rightarrow> 'ay \<Rightarrow> bool"   and uu :: "('az \<times> 'ba \<times> 'bb \<times> 'bc \<times> 'bd) itself" shows "rel_FGco_neg_distr_cond Co1 Co1' Co2 Co2' Co3 Co3' Co4 Co4' Co5 Co5' Co6 Co6' Contra1 Contra1' Contra2 Contra2' Contra3 Contra3' Contra4 Contra4' uu = (\<forall>L1 L1'. rel_FGco (L1 OO L1') (Co1 OO Co1') (Co2 OO Co2') (Co3 OO Co3') (Co4 OO Co4') (Co5 OO Co5') (Co6 OO Co6') (Contra1 OO Contra1') (Contra2 OO Contra2') (Contra3 OO Contra3') (Contra4 OO Contra4') \<le> rel_FGco L1 Co1 Co2 Co3 Co4 Co5 Co6 Contra1 Contra2 Contra3 Contra4 OO rel_FGco L1' Co1' Co2' Co3' Co4' Co5' Co6' Contra1' Contra2' Contra3' Contra4')"
posaux6: fixes a :: "'b"   and l :: "'b list"   and b :: "'b" assumes "a \<in> set l"   and "b \<in> set l"   and "a \<noteq> b" shows "position a l \<noteq> position b l"
relcompp_witness_2: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'c"   and y :: "'e" assumes "(A OO B) x y" shows "B (relcompp_witness A B (x, y)) y"
real_root_le_mono: fixes n :: "nat"   and x :: "real"   and y :: "real" assumes "0 < n"   and "x \<le> y" shows "root n x \<le> root n y"
not_ordLess_iff_ordLeq: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Well_order r"   and "Well_order r'" shows "((r', r) \<notin> ordLess) = (r \<le>o r')"
surjective: fixes r :: "('a, 'b, 'c) conf_scheme" shows "r = \<lparr>store = store r, pred = pred r, \<dots> = more r\<rparr>"
path_imp_reversepath: fixes g :: "real \<Rightarrow> 'c" assumes "path g" shows "path (reversepath g)"
skip_doesnt_access: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and lc :: "(('a, 'c, 'd) Stmt \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)"   and annos :: "'a ModeUpd list"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "sifum_modes ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars"   and "lc \<in> sifum_security_init.loc_reach dma (sifum_lang_no_dma.eval\<^sub>w ev\<^sub>A ev\<^sub>B) ((sifum_modes.annotate Skip annos, mds), mem)"   and "lc = ((c', mds'), mem')" shows "sifum_security_init.doesnt_read_or_modify \<C>_vars (sifum_lang_no_dma.eval\<^sub>w ev\<^sub>A ev\<^sub>B) c' x \<and> sifum_security_init.doesnt_modify dma (sifum_lang_no_dma.eval\<^sub>w ev\<^sub>A ev\<^sub>B) c' x"
\<omega>_limit_point_def: fixes x :: "'a"   and p :: "'a" shows "rev.\<omega>_limit_point x p = ({0..} \<subseteq> rev.existence_ivl0 x \<and> (\<exists>s. s \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity> \<and> (rev.flow0 x \<circ> s) \<longlonglongrightarrow> p))"
square_part_ge_Suc_0: fixes n :: "nat" assumes "0 < n" shows "Suc 0 \<le> square_part n"
coeffs_Poly: fixes as :: "'a list" shows "coeffs (Poly as) = strip_while ((=) (0::'a)) as"
length_drop_bl: fixes bs :: "bool list" shows "length (dropWhile Not (to_bl (of_bl bs))) \<le> length bs"
inducts: fixes P1 :: "'e \<Rightarrow> bool"   and P2 :: "'f \<Rightarrow> bool"   and x :: "'e \<times> 'f"   and P :: "'e \<times> 'f \<Rightarrow> bool" assumes "pred_prod P1 P2 x"   and "\<And>a b. \<lbrakk>P1 a; P2 b\<rbrakk> \<Longrightarrow> P (a, b)" shows "P x"
is_FunE: fixes t :: "('c, 'd) Term.term" assumes "is_Fun t" obtains f :: "'c" and ts :: "('c, 'd) Term.term list" where "t = Fun f ts"
power: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat"   and m :: "nat" assumes "completely_multiplicative_function f" shows "f (n ^ m) = f n ^ m"
indaut_eq_foldch_fg: fixes C :: "'a set" assumes "C \<in> f \<turnstile> folding_g.\<C>" shows "fun_eq_on \
 g C"
poly_prod_mset: fixes p :: "'c \<Rightarrow> 'b poly"   and A :: "'c multiset"   and y :: "'b" shows "poly (\<Prod>\<^sub># (image_mset p A)) y = (\<Prod>x\<in>#A. poly (p x) y)"
R_def: shows "R \<equiv> \<Sum>(p, i)\<in>{(p, i). prime p \<and> 1 < i \<and> real (p ^ i) \<le> x}. ln (real p) / real (p ^ i)"
split_min: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and i :: "'b"   and j :: "'b" assumes "class.linorder less_eq less" shows "P (ord.min less_eq i j) = ((less_eq i j \<longrightarrow> P i) \<and> (\<not> less_eq i j \<longrightarrow> P j))"
div_gcd_dvd2: fixes b :: "'a"   and a :: "'a" shows "b div gcd a b dvd b"
meval_intro: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and n :: "nat"   and mem :: "'Var \<Rightarrow> 'Val"   and cm' :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "(cms ! n, mem) \<leadsto> (cm', mem')"   and "n \<sqsubset> length cms" shows "(cms, mem) \<rightarrow> (cms[n := cm'], mem')"
disc_34: fixes x61 :: "'a OCL_Types.type"   and x62 :: "String.literal"   and x63 :: "'a expr list" shows "\<not> is_MetaOperationCall (StaticOperationCall x61 x62 x63)"
"case": fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'a"   and x1 :: "'c"   and x2 :: "'d" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
drefinesI: fixes A :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and G :: "'d \<Rightarrow> bool"   and \<phi> :: "'d \<Rightarrow> 'c"   and B :: "bool \<Rightarrow> ('d \<Rightarrow> real) \<Rightarrow> 'd \<Rightarrow> real" assumes "\<And>P Q. \<lbrakk>unitary P; unitary Q; P \<tturnstile> wp A Q\<rbrakk> \<Longrightarrow> \<guillemotleft> G \<guillemotright> && (P \<circ> \<phi>) \<tturnstile> wp B (Q \<circ> \<phi>)" shows "drefines \<phi> G A B"
b'': fixes j :: "nat" assumes "j < n" shows "b j \<in> Basis"
wt_jvm_prog_impl_wt_instr: fixes \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and P :: "jvm_prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and ins :: "bytecode"   and xt :: "ex_table"   and pc :: "nat" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = (mxs, mxl\<^sub>0, ins, xt) in C"   and "pc < length ins" shows "P,T,mxs,length ins,xt \<turnstile> ins ! pc,pc :: \<Phi> C M"
simps_2: fixes p :: "nat"   and x :: "real"   and n :: "nat" shows "power_up p x (Suc n) = (if odd n then truncate_up p ((power_up p x (Suc n div 2))\<^sup>2) else truncate_up p (x * power_up p x n))"
ipurge_ref_def: fixes I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and u :: "'b"   and xs :: "'a list"   and X :: "'a set" shows "ipurge_ref I D u xs X \<equiv> {x \<in> X. (u, D x) \<notin> I \<and> (\<forall>v\<in>sinks I D u xs. (v, D x) \<notin> I)}"
equality: fixes r :: "('c, 'd, 'e) pre_digraph_scheme"   and r' :: "('c, 'd, 'e) pre_digraph_scheme" assumes "verts r = verts r'"   and "arcs r = arcs r'"   and "tail r = tail r'"   and "head r = head r'"   and "pre_digraph.more r = pre_digraph.more r'" shows "r = r'"
emeasure_lborel_cball_finite: fixes c :: "'b"   and r :: "real" shows "emeasure lborel (cball c r) < \<infinity>"
step_4: shows "h \<turnstile> get_root_node_si c \<rightarrow>\<^sub>r 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 child"
finite_cartesian_product: fixes A :: "'a set"   and B :: "'b set" assumes "finite A"   and "finite B" shows "finite (A \<times> B)"
canonize_idempotent: fixes A :: "('b \<Rightarrow> bool) set"   and s :: "'b" shows "canonize A (canonize A s) = canonize A s"
dvd_power_le: fixes a :: "nat"   and y :: "nat"   and l :: "nat" assumes "Suc 0 < a"   and "a ^ y dvd a ^ l" shows "y \<le> l"
6: shows "sset (gtrace r v) \<subseteq> gunodes A w"
iso_imp_img_group: fixes h :: "'a \<Rightarrow> 'd"   and H :: "('d, 'e) monoid_scheme" assumes "h \<in> Group.iso G H" shows "Group.group (H\<lparr>one := h \<one>\<rparr>)"
primitive_root_unity_simple_computation: fixes k :: "nat"   and x :: "'a" shows "primitive_root_unity k x = (if k = 0 then False else x ^ k = (1::'a) \<and> (\<forall>i\<in>{1..<k}. x ^ i \<noteq> (1::'a)))"
Array_Array_widen: fixes P :: "'a prog"   and T :: "ty"   and U :: "ty" assumes "P \<turnstile> T\<lfloor>\<rceil> \<le> U\<lfloor>\<rceil>" shows "P \<turnstile> T \<le> U"
weak_convex1D: fixes x :: "'a"   and y :: "'a"   and u :: "real"   and v :: "real" assumes "weak_convex_pref relation"   and "x \<succeq> y"   and "0 \<le> u"   and "0 \<le> v"   and "u + v = 1" shows "u *\<^sub>R x + v *\<^sub>R y \<succeq> y"
cases: fixes P :: "'a stream \<Rightarrow> bool"   and a1 :: "nat"   and a2 :: "'a stream" assumes "ev_at P a1 a2" obtains \<omega> :: "'a stream" where "a1 = 0" and "a2 = \<omega>__" and "P \<omega>__"   | \<omega> :: "'a stream" and n :: "nat" where "a1 = Suc n" and "a2 = \<omega>" and "\<not> P \<omega>" and "ev_at P n (stl \<omega>)"
cut_greater_Max_not_empty: fixes I :: "'a set"   and t :: "'a" assumes "I \<noteq> {}"   and "finite I"   and "t < Max I" shows "I \<down>> t \<noteq> {}"
QQ8_a_def: shows "QQ8_a \<equiv> [: p, t, left, right, atom, mrk \<leadsto> p', t', left', right', atom', mrk' . p = nil \<and> p' = p \<and> t' = t \<and> mrk' = mrk \<and> left' = left \<and> right' = right \<and> atom' = atom :]"
assms: shows "\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
axioms_2: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and plus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>M\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'f set"   and plus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>M\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f" assumes "module_pair_ow U\<^sub>M\<^sub>_\<^sub>1 plus\<^sub>M\<^sub>_\<^sub>1 zero\<^sub>M\<^sub>_\<^sub>1 minus\<^sub>M\<^sub>_\<^sub>1 uminus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 plus\<^sub>M\<^sub>_\<^sub>2 zero\<^sub>M\<^sub>_\<^sub>2 minus\<^sub>M\<^sub>_\<^sub>2 uminus\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2" shows "module_ow U\<^sub>M\<^sub>_\<^sub>2 plus\<^sub>M\<^sub>_\<^sub>2 zero\<^sub>M\<^sub>_\<^sub>2 minus\<^sub>M\<^sub>_\<^sub>2 uminus\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2"
hom1: shows "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f1 g1"
exec_instr_Push: fixes v :: "val"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" shows "exec_instr (Push v) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh = (None, h, (v # stk, loc, C\<^sub>0, M\<^sub>0, Suc pc, ics) # frs, sh)"
Beta_implies_One_star: fixes t1 :: "lam"   and t2 :: "lam" assumes " t1 \<longrightarrow>\<^sub>\<beta> t2" shows " t1 \<longrightarrow>\<^sub>1\<^sup>* t2"
dom_eq_empty_conv: fixes f :: "'a \<Rightarrow> 'b option" shows "(dom f = {}) = (f = Map.empty)"
LowIn3: shows "the (\<sigma>\<^sub>3 ''lowIn'') = lowIn"
strict_implies_order: fixes b :: "('a, 'b) Term.term"   and a :: "('a, 'b) Term.term" assumes "b \<sqsubset> a" shows "b \<sqsubseteq> a"
is_sig_red_top_tail_cases: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b" shows "is_sig_red sing_reg (\<preceq>) F p = (is_sig_red sing_reg (=) F p \<or> is_sig_red sing_reg (\<prec>) F p)"
infinite_cyclic_subgroup_order: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (reduced_homology_group (int p) (nsphere p))" shows "infinite (carrier (subgroup_generated (reduced_homology_group (int p) (nsphere p)) {x})) = (ord x = 0)"
normal_generateI: fixes H :: "'a set" assumes "H \<subseteq> carrier G"   and "\<And>h g. \<lbrakk>h \<in> H; g \<in> carrier G\<rbrakk> \<Longrightarrow> g \<otimes> h \<otimes> inv g \<in> H" shows "generate G H \<lhd> G"
v_1: shows "c \<cdot>\<^sub>v us ! j \<in> carrier_vec n"
d_OUT_def: fixes g :: "'c \<times> 'c \<Rightarrow> ennreal"   and x :: "'c" shows "d_OUT g x = (\<Sum>\<^sup>+ y. g (x, y))"
restrictedRange: fixes P :: "('b \<times> 'a) set"   and X :: "'b set" shows "Range (P || X) = P `` X"
coboundedI1: fixes c :: "'c multiset"   and a :: "'c multiset"   and b :: "'c multiset" assumes "c \<subseteq># a" shows "c \<subseteq># a \<union># b"
term_cases_def: fixes if_const :: "name \<Rightarrow> 'a"   and if_free :: "name \<Rightarrow> 'a"   and if_app :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and otherwise :: "'a"   and t :: "'b" shows "term_cases if_const if_free if_app otherwise t = (case unconst t of None \<Rightarrow> case unfree t of None \<Rightarrow> case unapp t of None \<Rightarrow> otherwise | Some (t, u) \<Rightarrow> if_app t u | Some name \<Rightarrow> if_free name | Some name \<Rightarrow> if_const name)"
rangeE: fixes b :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "b \<in> range f" obtains x :: "'b" where "b = f x"
accp_wfPI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>x. Wellfounded.accp r x" shows "wfP r"
countable_infiniteE': fixes A :: "'c set" assumes "countable A"   and "infinite A" obtains g :: "nat \<Rightarrow> 'c" where "bij_betw g UNIV A"
sup_bot_left: fixes x :: "'c \<times> 'd" shows "sup_prod bot_prod x = x"
less_add_Suc2: fixes i :: "nat"   and m :: "nat" shows "i < Suc (m + i)"
assms_1: shows "\<forall>n\<in>ns. P n"
F: fixes n :: "nat" shows "closed (F n) \<and> F n \<subseteq> S \<and> S - F n \<in> lmeasurable \<and> Sigma_Algebra.measure lebesgue (S - F n) < 1 / real (Suc n)"
typeof_lit_eq_Boolean: fixes v :: "'a val" shows "(typeof v = \<lfloor>Boolean\<rfloor>) = (\<exists>b. v = Bool b)"
Mp_rel_i_def: fixes f :: "'i list"   and F :: "int poly" shows "Mp_rel_i f F = (map (arith_ops_record.to_int ff_ops) f = coeffs (Mp F))"
length_nat_pos: fixes x :: "nat" assumes "0 < x" shows "0 < length_nat x"
vec_to_iarray_nth': fixes A :: "('a, 'b) vec"   and i :: "'b" shows "vec_to_iarray A !! mod_type_class.to_nat i = A $ i"
Inv_preserves_Can_2: fixes t :: "'a local.term" assumes "Can t" shows "Dom (Inv t) = Cod t"
rot_circle_bot_edge_def: shows "rot_circle_bot_edge = (1, \<lambda>y. (0::'b, - (d / 2)))"
mult_double_iso: fixes x :: "'a"   and y :: "'a"   and w :: "'a"   and z :: "'a" assumes "x \<le> y" shows "w \<parallel> x \<parallel> z \<le> w \<parallel> y \<parallel> z"
ipath_def: fixes E :: "('h \<times> 'h) set"   and r :: "nat \<Rightarrow> 'h" shows "ipath E r \<equiv> \<forall>i. (r i, r (Suc i)) \<in> E"
a: fixes n :: "'a"   and v :: "'b" shows "(THE k. (\<lambda>p. (fst p, snd p, fst p, PhiDef)) -` {(n, v, n, PhiDef)} = {k}) = (n, v)"
RPDi_ok: fixes m :: "nat"   and mr :: "mregex"   and ms :: "mregex"   and n :: "nat" assumes "ok m mr"   and "ms \<in> RPDi n mr" shows "ok m ms"