hcju/isabelleps huggingface dataset:
ide_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "CCC.ide f = (ide (fst f) \<and> CC.ide (snd f))"
elims_3: fixes \<Delta>1 :: "nat"   and \<Delta>2 :: "nat"   and \<Gamma>1 :: "nat"   and \<Gamma>2 :: "nat"   and x :: "('b \<times> nat) tree"   and xa :: "('b \<times> nat) tree" assumes "WBT \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2"   and "\<not> WBT.balanced1 \<Delta>1 \<Delta>2 x xa" obtains t1 :: "('b \<times> nat) tree" and t2 :: "('b \<times> nat) tree" where "x = t1" and "xa = t2" and "\<not> \<Delta>2 * (size_wbt t2 + 1) \<le> \<Delta>1 * (size_wbt t1 + 1)"
bv_prepend_append: fixes a :: "nat"   and b :: "'a"   and x :: "'a list" shows "bv_prepend a b x = bv_prepend a b [] @ x"
SeqSum_def: fixes f :: "nat \<Rightarrow> ZF" shows "SeqSum f \<equiv> HOLZF.Sum (Repl HOLZF.Nat (f \<circ> Nat2nat))"
\<Gamma>_sub_sem: fixes \<Gamma>1 :: "('a, 'b, 'c) formula list"   and \<Gamma>2 :: "('a, 'b, 'c) formula list"   and \<nu> :: "(real, 'c) vec \<times> (real, 'c) vec"   and I :: "('a, 'b, 'c) interp" assumes "sublist \<Gamma>1 \<Gamma>2"   and "\<nu> \<in> fml_sem I (foldr (&&) \<Gamma>2 TT)" shows "\<nu> \<in> fml_sem I (foldr (&&) \<Gamma>1 TT)"
inverse_sgn: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "inverse (sgn a) = sgn a"
0: shows "Ik_intT (resOf f) (I_intF f (map (I.int \<xi>) Tl))"
distrib_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<parallel> (b + c) = a \<parallel> b + a \<parallel> c"
add_eq_conv_diff: fixes a :: "'c"   and M :: "'c multiset"   and b :: "'c"   and N :: "'c multiset" shows "(add_mset a M = add_mset b N) = (M = N \<and> a = b \<or> M = add_mset b (N - {#a#}) \<and> N = add_mset a (M - {#b#}))"
Acts_def: fixes F :: "'c program" shows "Acts F \<equiv> case Rep_Program F of (init, acts, allowed) \<Rightarrow> acts"
assoc_simps_1: fixes f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data"   and h :: "'a arrow_of_spans_data" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "arr \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc (\<cdot>) prj0 prj1 f g h, Dom = Dom ((f \<star> g) \<star> h), Cod = Cod (f \<star> g \<star> h)\<rparr>"
path_independent_on_Chernoff_on: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set" assumes "path_independent_on A f"   and "f_range_on A f" shows "Chernoff_on A f"
red_self: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0" shows "red {p} p 0"
self_append_conv2: fixes ys :: "'a list"   and xs :: "'a list" shows "(ys = xs @ ys) = (xs = [])"
ur2: shows "0 < real_of_rat r - real_of_1 (p, l, r)"
err_semilat_Product_esl: fixes L\<^sub>1 :: "'b set \<times> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'b \<Rightarrow> 'b err)"   and L\<^sub>2 :: "'c set \<times> ('c \<Rightarrow> 'c \<Rightarrow> bool) \<times> ('c \<Rightarrow> 'c \<Rightarrow> 'c err)" assumes "err_semilat L\<^sub>1"   and "err_semilat L\<^sub>2" shows "err_semilat (Product.esl L\<^sub>1 L\<^sub>2)"
smult_is_continuous: fixes f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "is_Zp_continuous f"   and "a \<in> carrier Zp" shows "is_Zp_continuous (a \<odot>\<^bsub>function_ring (carrier Zp) Zp\<^esub> f)"
simps_3: fixes tst :: "'a"   and c1 :: "('a, 'atom) com"   and c2 :: "('a, 'atom) com" shows "SC_discr (if tst then c1 else c2) = (SC_discr c1 \<and> SC_discr c2)"
trnl\<^sub>\<eta>_eq: fixes u :: "'a"   and v :: "'a"   and \<mu> :: "'a" assumes "ide u"   and "ide v"   and "src f = trg v"   and "src g = trg u"   and "\<mu> \<in> hom (f \<star> v) u" shows "A.trnl\<^sub>\<eta> v \<mu> = (g \<star> \<mu>) \<cdot> (\<eta> \<star> v)"
unknown_set_def: fixes S :: "state" shows "unknown_set S = keys_of (heap S)"
ego_other2'_def: fixes t :: "real" shows "ego_other2' t = other.s' t - u' t"
field_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and F :: "char list" shows "field P C F \<equiv> THE (D, T). P \<turnstile> C sees F:T in D"
intros: 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 C :: "char list"   and Bs :: "base list"   and rest :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list"   and D :: "char list" assumes "class P C = \<lfloor>(Bs, rest)\<rfloor>"   and "Repeats D \<in> set Bs" shows "P \<turnstile> C \<prec>\<^sub>R D"
cop_F_rejected_invI: fixes fp :: "'a set"   and ds :: "'b set" assumes "\<And>x. \<lbrakk>x \<in> fp; x \<notin> CH fp\<rbrakk> \<Longrightarrow> \<exists>fp'\<subseteq>fp. x \<in> fp' \<and> above (Pd (Xd x)) x \<subseteq> fp' \<and> Xd x \<notin> Xd ` CH fp'" shows "cop_F_rejected_inv ds fp"
cod_comp: fixes g :: "'c \<times> 'c"   and f :: "'c \<times> 'c" assumes "CopxC.seq g f" shows "CopxC.cod (g \<odot> f) = CopxC.cod g"
surjective: fixes r :: "'b member_scheme" shows "r = \<lparr>access = access r, static = static r, \<dots> = member.more r\<rparr>"
simps_25: fixes x51 :: "Core.term"   and x52 :: "Core.term"   and x41 :: "typ"   and x42 :: "Core.term" shows "x51 $ x52 \<noteq> Abs x41 x42"
extendCommute: fixes S :: "'b sequent"   and R :: "'b sequent"   and c :: "'b sequent" shows "extend S (extend R c) = extend R (extend S c)"
abs_eq: fixes x :: "real_alg_3" shows "inverse (Real_Alg_Quotient x) = Real_Alg_Quotient (inverse_3 x)"
3: shows "j \<sqinter> - choose_component (forest_components h) j \<le> forest_components h * - choose_component (forest_components h) j"
NetsCollectedSorted: fixes p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "noDenyAll1 p"   and "all_in_list p l"   and "singleCombinators p"   and "FWNormalisationCore.sorted p l" shows "NetsCollected p"
dim_zero_vs: shows "vectorspace.dim class_ring (vs (local.span {})) = 0"
mapmeasure_def: fixes n :: "nat"   and M :: "('i \<Rightarrow> 'e) measure"   and N :: "nat \<Rightarrow> 'e measure" shows "mapmeasure n M N = distr M (Pi\<^sub>F (Collect finite) N) (fm n)"
not_zero_ia_H: shows "\<not> is_zero_row ia H"
fgs_def: fixes R :: "('i, 'j) Ring_scheme"   and M :: "('h, 'i, 'k) Module_scheme"   and A :: "'h set" shows "fgs R M A = linear_span R M (carrier R) A"
assms_4: shows "coreflexive (v * z\<^sup>T)"
add_cancel_end: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> E"   and "y \<in> E"   and "z \<in> E" shows "(x + (y + z) = y) = (x = - z)"
finite_supp_implies_hereditarily_fs_Conj\<^sub>\<alpha>: fixes xset :: "('a, 'b, 'c) formula set['a]" assumes "finite (supp xset)" shows "hereditarily_fs (Conj\<^sub>\<alpha> (map_bset Rep_formula xset))"
diffE: fixes x :: "'c" assumes "x \<in> f.dest.carrier" obtains c1 :: "('c, 'd) chart" and c2 :: "('e, 'f) chart" where "c1 \<in> f.dest.atlas" and "c2 \<in> g.dest.atlas" and "x \<in> domain c1" and "g ` domain c1 \<subseteq> domain c2" and "k-smooth_on (codomain c1) (apply_chart c2 \<circ> g \<circ> inv_chart c1)"
lt__le: 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.Lt Bet Cong A B C D" shows "Tarski_neutral_dimensionless.Le Bet Cong A B C D"
hyps_1: assumes "ordered lt var"   and "not \<le> lt" shows "ordered not var"
case_on: shows "is_arith_prog_on (l + 1) h_start h_step a b"
wf_cdecl_impD: fixes G :: "prog"   and C :: "qtname"   and c :: "class"   and I :: "qtname" assumes "wf_cdecl G (C, c)"   and "I \<in> set (superIfs c)" shows "is_acc_iface G (pid C) I \<and> (\<forall>s. \<forall>im\<in>DeclConcepts.imethds G I s. ? cm:methd G C s: G\<turnstile>resTy cm\<preceq>resTy im \<and> \<not> is_static cm \<and> accmodi im \<le> accmodi cm)"
sel_2: fixes x1 :: "'b"   and x2 :: "'a"   and x3 :: "nat"   and x4 :: "('b, 'a) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.prio (BinomialHeapStruc.Node x1 x2 x3 x4) = x2"
smult_add_left: fixes a :: "'c"   and b :: "'c"   and p :: "'c poly" shows "smult (a + b) p = smult a p + smult b p"
add_strict_right_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.ordered_cancel_ab_semigroup_add plus minus less_eq less"   and "less a b" shows "less (plus a c) (plus b c)"
distinct_5: fixes x1 :: "'b"   and x41 :: "('a, 'b) process_term"   and x42 :: "('a, 'b) process_term" shows "VAR x1 \<noteq> CH x41 x42"
closed_2: shows "is_TComp_var_instance_closed \<Gamma> M"
get_disconnected_nodes_new_character_data: fixes h :: "(_) heap"   and new_character_data_ptr :: "(_) character_data_ptr"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and ptr' :: "(_) document_ptr" assumes "h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr"   and "h \<turnstile> new_character_data \<rightarrow>\<^sub>h h'"   and "r \<in> get_disconnected_nodes_locs ptr'" shows "r h h'"
simplify_preserves_equivalence: fixes T :: "'at Literal set set"   and S :: "'at Literal set set" assumes "T = simplify S"   and "all_fulfill finite S" shows "equivalent S T"
Loebian_def: fixes x :: "'a" shows "Loebian x = (\<forall>y.  |x\<rangle> y \<le>  |x\<rangle> \<Omega> x y)"
dotprod_le_drop: fixes a :: "nat list"   and b :: "nat list"   and k :: "nat" assumes "length a = length b"   and "k \<le> length a" shows "drop k a \<bullet> drop k b \<le> a \<bullet> b"
map_poly_smult: fixes f :: "'c \<Rightarrow> 'b"   and c :: "'c"   and p :: "'c poly" assumes "f (0::'c) = (0::'b)"   and "\<And>c x. f (c * x) = f c * f x" shows "map_poly f (smult c p) = smult (f c) (map_poly f p)"
simps_1: fixes carrier :: "'a set"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and more :: "'b" shows "(\<otimes>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr>\<^esub>) = mult"
size_0: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size" shows "euclidean_size zero = 0"
antipar_2: shows "src\<^sub>D g' = trg\<^sub>D (F f)"
set_7: fixes x31 :: "poscheckvariant"   and x32 :: "('d, 'c) Term.term"   and x33 :: "('d, 'c) Term.term" shows "vars\<^sub>s\<^sub>t\<^sub>p \<langle>x31: x32 \<doteq> x33\<rangle>\<^sub>s\<^sub>t = fv x32 \<union> fv x33"
filter_sup: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and F :: "'b set"   and G :: "'b set" assumes "class.semilattice_inf inf less_eq less"   and "ord.filter less_eq F"   and "ord.filter less_eq G" shows "ord.filter less_eq (semilattice_inf.filter_sup inf less_eq F G)"
C_SC_equiv: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "tarski_first3 C" shows "C a b c d = tarski_first3.SC C {a, b} {c, d}"
Cod_in_Obj: fixes f :: "(unit, unit) concrete_category.arr" assumes "arr f" shows "Cod f \<in> {}"
Nset_2: fixes i :: "'e"   and j :: "'e" shows "{i, j} = {j, i}"
IH: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and s' :: "('a, 'b) fmap"   and s :: "('a, 'b) fmap"   and vs :: "'a set" assumes "as \<in> valid_plans PROB"   and "s' \<in> valid_states PROB"   and "s \<in> valid_states PROB"   and "\<forall>a. ListMem a as \<longrightarrow> agree (fmrestrict_set vs (snd a)) (fmrestrict_set vs s)"   and "fmrestrict_set vs s' = fmrestrict_set vs s" shows "fmrestrict_set vs (exec_plan s' as) = fmrestrict_set vs s"
TableInv_axioms: shows "TableInv f1 f2 f1' f2' e c l0"
ih_19: shows "getSATFlag state' = FALSE \<longrightarrow> \<not> satisfiable F0'"
\<theta>fun_id_2: shows "t' \<cdot> \<theta> = t'"
sse_algo_def: fixes \<Sigma>i :: "'b set"   and R :: "('b \<times> 'b) set" shows "sse_algo \<Sigma>i R \<equiv> \<lparr>wa_cond = sse_cond, wa_step = sse_step R, wa_initial = {sse_initial \<Sigma>i}, wa_invar = sse_invar \<Sigma>i R\<rparr>"
assms_4: fixes y :: "'a" assumes "y \<in> set (local.filter_min_append xs ys)"   and "x \<notin> set (local.filter_min_append xs ys)"   and "rel y x" shows "thesis"
raw_subst_indep: fixes t :: "'a" shows "abs_pred (\<lambda>t. \<forall>env\<^sub>1 env\<^sub>2. closed_env env\<^sub>2 \<longrightarrow> fdisjnt (fmdom env\<^sub>1) (fmdom env\<^sub>2) \<longrightarrow> subst t (env\<^sub>1 ++\<^sub>f env\<^sub>2) = subst (subst t env\<^sub>2) env\<^sub>1) t"
mult_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a + b + c = a + (b + c)"
M2_def: shows "M2 = (\<lambda>i. (f i, f (2 * m - i))) ` {..<m}"
gfp_F_stable_on: fixes dsa :: "'b set"   and d :: "'b"   and ds :: "'b set" shows "Singleton_for_d.stable_on dsa d ds (Singleton_for_d.match (Singleton_for_d.gfp_F dsa d ds))"
dom_id: fixes a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "noDenyAll (a # p)"   and "separated (a # p)"   and "p \<noteq> []"   and "x \<notin> dom (Cp (list2FWpolicy p))"   and "x \<in> dom (Cp a)" shows "x \<notin> dom (Cp (list2FWpolicy (insertDenies p)))"
VDM_valid_def: fixes c :: "IMP"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" shows " \<Turnstile> c : A  = (\<forall>s t. s , c \<Down> t  \<longrightarrow> A s t)"
rel_gpv_lift_spmf1: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and p :: "'f spmf"   and gpv :: "('g, 'i, 'j) gpv" shows "rel_gpv A B (lift_spmf p) gpv = (\<exists>q. gpv = lift_spmf q \<and> rel_spmf A p q)"
lin_span_coeff_mono: fixes A :: "'b set"   and H :: "'a set" assumes "ideal R A"   and "H \<subseteq> carrier M" shows "linear_span R M A H \<subseteq> linear_span R M (carrier R) H"
assms_2: shows "inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V"
length_filter_less: fixes x :: "'a"   and xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "x \<in> set xs"   and "\<not> P x" shows "length (filter P xs) < length xs"
drlex_pm_zero_min: fixes s :: "'a \<Rightarrow>\<^sub>0 'b" shows "drlex_pm 0 s"
iter_induct_eq: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice" assumes "iteration_finite_conjunctive (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "(seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (finite_or_infinite_iteration.iter seq (nil::'b::refinement_lattice) (c::'b::refinement_lattice)) (d::'b::refinement_lattice) = lfp (\<lambda>x::'b::refinement_lattice. d \<sqinter> seq c x)"
bilinear_sum: fixes h :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'a"   and S :: "'d set"   and g :: "'e \<Rightarrow> 'b"   and T :: "'e set" assumes "bilinear h" shows "h (sum f S) (sum g T) = (\<Sum>(i, j)\<in>S \<times> T. h (f i) (g j))"
bottom_left_bottom: fixes a :: "'b" shows "gcd (1::'b) a = (1::'b)"
deriv: shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
pos_le_divide_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less_eq a (divide b c) = less_eq (times a c) b"
limitin_def: fixes X :: "'d topology"   and f :: "'e \<Rightarrow> 'd"   and l :: "'d"   and F :: "'e filter" shows "limitin X f l F \<equiv> l \<in> topspace X \<and> (\<forall>U. openin X U \<and> l \<in> U \<longrightarrow> (\<forall>\<^sub>F x in F. f x \<in> U))"
split_list_first_prop: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. P x" shows "\<exists>ys x zs. xs = ys @ x # zs \<and> P x \<and> (\<forall>y\<in>set ys. \<not> P y)"
UNlift01_def: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set"   and II :: "nat set" shows "UNlift01 cl dl P F II \<equiv> if II = BrnFT cl dl then BrnFT dl cl else local.shift dl (pickT cl P II) (F (pickT cl P II) (back cl (pickT cl P II) II))"
UN_Un_distrib: fixes A :: "'b \<Rightarrow> 'a set"   and B :: "'b \<Rightarrow> 'a set"   and I :: "'b set" shows "\<mu> (\<P> (\<lambda>i. A i \<union> B i) I) = \<mu> (\<P> A I) \<union> \<mu> (\<P> B I)"
r_maxletr: fixes x\<^sub>0 :: "nat"   and x\<^sub>1 :: "nat" shows "eval r_maxletr [x\<^sub>0, x\<^sub>1] \<down>= (GREATEST y. y \<le> x\<^sub>0 \<and> triangle y \<le> x\<^sub>1)"
as2: fixes n :: "nat" assumes "mining (Suc n)" shows "bc n < bc (Suc n)"
Standard_Ifun: fixes f :: "('b \<Rightarrow> 'c) star"   and x :: "'b star" assumes "f \<in> Standard"   and "x \<in> Standard" shows "f \<star> x \<in> Standard"
stasbs_1: shows "ss = as @ pair_relator_halflist s t @ bs"
is_rewrite_ord_finite_canon_rewriterE: fixes rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and A :: "('t \<Rightarrow>\<^sub>0 'b) set"   and a :: "'t \<Rightarrow>\<^sub>0 'b"   and u :: "'t" assumes "is_rewrite_ord rword"   and "finite A"   and "a \<in> A"   and "a \<noteq> 0"   and "lt a adds\<^sub>t u" obtains p :: "'t \<Rightarrow>\<^sub>0 'b" where "is_canon_rewriter rword A u p"
hyps: fixes us :: "'a vec list"   and us' :: "'a vec list" assumes "gram_schmidt_sub0 n us ws = us'"   and "set ws \<subseteq> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "distinct us"   and "lin_indpt (set us)"   and "corthogonal us" shows "set us' \<subseteq> carrier_vec n \<and> distinct us' \<and> corthogonal us' \<and> local.span (set (us @ ws)) = local.span (set us')"
closed_except_self: fixes t :: "'a" shows "closed_except t (frees t)"
env_delete_fun_upd2: fixes x :: "'c"   and m :: "'c \<Rightarrow> 'd"   and v :: "'d" shows "(env_delete x m)(x := v) = m(x := v)"
valuate_with_dim_poly: fixes z :: "linear_poly"   and X :: "nat \<Rightarrow> rat" shows "z \<lbrace> X \<rbrace> = (\<Sum>i = 0..<dim_poly z. Abstract_Linear_Poly.coeff z i * X i)"
dom_simp: fixes f :: "'c \<times> 'c" assumes "C'.VV.arr f" shows "C'.VV.dom f = C.VxV.dom f"
assms_2: fixes f :: "floatarith" assumes "f \<in> Mapping.keys M" shows "subterms f \<subseteq> Mapping.keys M"
pos: fixes x :: "nat \<times> nat" shows "0 \<le> e x"
interaction_bounded_by_exec_gpv_count': fixes "consider" :: "'a \<Rightarrow> bool"   and gpv :: "('d, 'a, 'b) gpv"   and n :: "enat"   and count :: "'s \<Rightarrow> enat"   and s :: "'s" assumes "interaction_bounded_by consider gpv n"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (callee s x); I s; consider x; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> count s' \<le> eSuc (count s)"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (callee s x); I s; \<not> consider x; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> count s' \<le> count s"   and "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "set_spmf (exec_gpv callee gpv s) \<subseteq> {(x, s'). count s' \<le> n + count s}"
pderiv_add: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P" shows "pderiv (p \<oplus>\<^bsub>P\<^esub> q) = pderiv p \<oplus>\<^bsub>P\<^esub> pderiv q"
revis_closure: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A" shows "K \<^bold>* \<phi> = Cn (K \<^bold>* \<phi>)"
comp_in_homI': fixes f :: "'a"   and g :: "'a"   and a :: "'a"   and c :: "'a" assumes "arr f"   and "arr g"   and "local.dom f = a"   and "cod g = c"   and "local.dom g = cod f" shows "\<guillemotleft>g \<cdot> f : a \<rightarrow> c\<guillemotright>"
assms_1: shows "A \<Delta> B \<in> null_sets M"
Abs_formula_eqvt: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c) Formula.Tree\<^sub>\<alpha>"   and p :: "perm" assumes "Formula.hereditarily_fs t\<^sub>\<alpha>" shows "p \<bullet> Formula.formula.Abs_formula t\<^sub>\<alpha> = Formula.formula.Abs_formula (p \<bullet> t\<^sub>\<alpha>)"
translate_origin: fixes p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real" assumes "NO_MATCH (0::'a) p" shows "ccw' p q r = ccw' 0 (q - p) (r - p)"
wt_subst: fixes \<rho> :: "'v \<Rightarrow> ('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "wary_subst \<rho>"   and "wary s" shows "wt (subst \<rho> s) = wt s + extra_wt \<rho> s"
beh_prefix_closed: fixes c :: "'e list"   and b :: "'e list"   and T :: "('e, 'f) TS_scheme" assumes "c @ b \<in> beh T" shows "b \<in> beh T"
bound: fixes w :: "'domain"   and w' :: "'domain" shows "Sigma_Algebra.measure (measure_spmf seed_gen) {x. hash x w = hash x w' \<and> w \<noteq> w'} \<le> \<epsilon>_uh"
XU_orig: shows "excess f u \<noteq> (0::'capacity)"
cinfinite_mono: fixes r1 :: "('c \<times> 'c) set"   and r2 :: "('d \<times> 'd) set" assumes "r1 \<le>o r2"   and "cinfinite r1" shows "cinfinite r2"
le_plus_2: fixes s' :: "'sum"   and s :: "'sum" shows "s' \<le> followed_by s s'"
None_in_map_option_image: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d option set" shows "(None \<in> map_option f ` A) = (None \<in> A)"
content_ff_map_poly_to_fract: fixes p :: "'b poly" shows "content_ff (map_poly to_fract p) \<in> range to_fract"
norm_ge_zero: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "0 \<le> norm x"
sublens_greatest: fixes X :: "'e \<Longrightarrow> 'f" assumes "vwb_lens X" shows "X \<subseteq>\<^sub>L \<Sigma>"