hcju/isabelleps huggingface dataset:
wt_bij_finite_subst_exists: fixes S :: "'var set"   and T :: "('fun, 'var) Term.term set" assumes "finite S"   and "finite T"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s T" shows "\<exists>\<sigma>. subst_domain \<sigma> = S \<and> bij_betw \<sigma> (subst_domain \<sigma>) (subst_range \<sigma>) \<and> subterms\<^sub>s\<^sub>e\<^sub>t (subst_range \<sigma>) \<subseteq> {t. {} \<turnstile>\<^sub>c t} - T \<and> (\<forall>s\<in>subst_range \<sigma>. \<forall>u\<in>subst_range \<sigma>. (\<exists>v. v \<sqsubseteq> s \<and> v \<sqsubseteq> u) \<longrightarrow> s = u) \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<sigma> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<sigma>)"
context_equiv_refl: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and P :: "'BExp set" shows "\<Gamma> =:\<^sub>P \<Gamma>"
getMinTree_min_tree: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list" assumes "t \<in> set bq" shows "BinomialHeapStruc.prio (BinomialHeapStruc.getMinTree bq) \<le> BinomialHeapStruc.prio t"
simps_2: fixes k :: "nat" shows "ket_zero_k (Suc k) = partial_state2.ptensor_vec dims {0..<Suc k} {Suc k} (ket_zero_k k) ket_zero"
dom_eqI: fixes a :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "CC.ide a"   and "CC.seq f a" shows "CC.dom f = a"
vsubst_psubst_compose: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "good X"   and "goodEnv rho" shows "X #[rho] #[y1 // y]_ys = X #[rho &[y1 // y]_ys]"
relative_to_relative_to: fixes P :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and T :: "'c set" shows "P relative_to S relative_to T = P relative_to S \<inter> T"
hyps_4: fixes s :: "char list \<Rightarrow> int option"   and n :: "nat" assumes "Q (s, n)" shows "Q' (s, n)"
Sup_set_def: fixes A :: "'d set set" shows "\<Union> A = {x. \<Squnion> ((\<in>) x ` A)}"
integrable_of_real: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> real" assumes "integrable M f" shows "integrable M (\<lambda>x. of_real (f x))"
dur_reset: fixes xs :: "('c \<Rightarrow> real) stream"   and x :: "'c"   and i :: "nat" assumes "PTA_Reachability.pairwise eq_elapsed xs"   and "pred_stream (\<lambda>u. u \<in> V) xs"   and "zero x ([xs !! Suc i]\<^sub>\<R>)"   and "x \<in> \<X>" shows "dur xs (Suc i) - dur xs i = 0"
power2_less_0: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "\<not> less (power.power one times a 2) zero"
atLeast_Suc_lessThan: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'c" assumes "m < n" shows "sum g {m..<n} = g m + sum g {Suc m..<n}"
add_increasing: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "bot \<le> a"   and "b \<le> c" shows "b \<le> max a c"
sinks_aux_less_single_dom: fixes I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and u :: "'b"   and xs :: "'c list" shows "sinks_aux_less I D {u} xs = sinks I D u xs"
transfer: shows "rel_fun (rel_gpv (BNF_Def.Grp UNIV f) (BNF_Def.Grp UNIV g)) (rel_gpv (BNF_Def.Grp UNIV f) (rel_sum (BNF_Def.Grp UNIV g) (BNF_Def.Grp UNIV h))) left_gpv left_gpv"
igSwapIGAbsSTR_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) model_scheme" shows "igSwapIGAbsSTR MOD \<equiv> \<forall>zs z1 z2 xs x X. igSwapAbs MOD zs z1 z2 (igAbs MOD xs x X) = igAbs MOD xs (x @xs[z1 \<and> z2]_zs) (igSwap MOD zs z1 z2 X)"
antisymD: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a" assumes "antisym r"   and "(a, b) \<in> r"   and "(b, a) \<in> r" shows "a = b"
main: fixes n :: "nat" shows "\<forall>D. finite D \<and> card D = n \<longrightarrow> nat_to_set (set_to_nat D) = D"
upset_def: shows "\<up> = \<Up> \<circ> \<eta>"
row_not_zero_iff_pivot_fun: fixes A :: "'c mat"   and f :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "pivot_fun A f (dim_col A)"   and "i < dim_row A" shows "(row A i \<noteq> 0\<^sub>v (dim_col A)) = (f i < dim_col A)"
ifex_var_noinfluence: fixes x :: "'a"   and b :: "'a ifex"   and ass :: "'a \<Rightarrow> bool"   and val :: "bool" assumes "x \<notin> ifex_var_set b" shows "val_ifex b (ass(x := val)) = val_ifex b ass"
hom_int_pow: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and n :: "int" assumes "x \<in> carrier (reduced_homology_group (int p) (nsphere p))" shows "f (x [^]\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> n) = f x [^]\<^bsub>reduced_homology_group (1 + int p) (nsphere (Suc p))\<^esub> n"
not_free_iff_lifted: fixes s :: "dB"   and i :: "nat" shows "(\<not> free s i) = (\<exists>t. s = lift t i)"
root_1: assumes "p \<noteq> 0" shows "card (proots_within p B) = card (proots_within (fcompose p q1 q2) A)"
nat_power_real_def: fixes a :: "nat"   and b :: "real" shows "nat_power a b = real a powr b"
assms_5: fixes f :: "floatarith" assumes "f \<in> Mapping.keys M" shows "interpret_slp slp xs ! slp_index_lookup slp M f = interpret_floatarith f xs"
exhaust: fixes y :: "dbtm" obtains "y = DBZero"   | x2 :: "name" where "y = DBVar x2"   | x3 :: "nat" where "y = DBInd x3"   | x41 :: "dbtm" and x42 :: "dbtm" where "y = DBEats x41 x42"
arg_0_iff: fixes z :: "complex" shows "(z \<noteq> 0 \<and> arg z = 0) = (is_real z \<and> 0 < Re z)"
finite_vars\<^sub>s\<^sub>t\<^sub>p: fixes x :: "('d, 'c) strand_step" shows "finite (vars\<^sub>s\<^sub>t\<^sub>p x)"
INF_UNION: fixes f :: "'b \<Rightarrow> 'a"   and B :: "'c \<Rightarrow> 'b set"   and A :: "'c set" shows "\<Sqinter> (f ` \<Union> (B ` A)) = (\<Sqinter>x\<in>A. \<Sqinter> (f ` B x))"
is_ground_atm_ground: fixes t :: "('a, 'b) Term.term" shows "is_ground_atm t = ground t"
eq_\<rho>: fixes U :: "'a set"   and V :: "'a set"   and W :: "'a set"   and s :: "'b"   and t :: "'b"   and W' :: "'a set" assumes "is_open U"   and "is_open V"   and "is_open W"   and "W \<subseteq> U \<inter> V"   and "s \<in> \<FF> U"   and "t \<in> \<FF> V"   and "\<rho> U W s = \<rho> V W t"   and "is_open W'"   and "W' \<subseteq> W" shows "\<rho> U W' s = \<rho> V W' t"
elt3: shows "z \<in> set (fst (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i)))"
group_axioms: shows "Group_Theory.group stx.carrier_direct_lim stx.add_rel (stx.class_of X (zero_str\<^sub>X X))"
xor_closed: fixes L :: "('c, 'd) boolean_algebra_scheme"   and x :: "'c"   and y :: "'c" assumes "boolean_algebra L"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<oplus>\<^bsub>L\<^esub> y \<in> carrier L"
mtf2_moves_to_front: fixes xs :: "'c list"   and q :: "'c" assumes "distinct xs"   and "q \<in> set xs" shows "index (mtf2 (length xs) q xs) q = 0"
distinct_1: fixes x2 :: "'b" shows "None \<noteq> \<lfloor>x2\<rfloor>"
bot_eq_sup_iff: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and bot :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "(bot = sup x y) = (x = bot \<and> y = bot)"
cases: fixes a1 :: "real \<Rightarrow> real"   and a2 :: "'a"   and a3 :: "(real \<Rightarrow> real) list" assumes "(a1 expands_to a2) a3" obtains F :: "'a" and basis :: "(real \<Rightarrow> real) list" and f :: "real \<Rightarrow> real" where "a1 = f" and "a2 = F" and "a3 = basis" and "is_expansion F basis" and "\<forall>\<^sub>F x in at_top. eval F x = f x"
tensors_from_net_eqI: fixes m1 :: "real mat convnet"   and m2 :: "real mat convnet" assumes "valid_net' m1"   and "valid_net' m2"   and "input_sizes m1 = input_sizes m2"   and "\<And>inputs. input_sizes m1 = map dim_vec inputs \<Longrightarrow> evaluate_net m1 inputs = evaluate_net m2 inputs" shows "tensors_from_net m1 = tensors_from_net m2"
vifintersection_vrange_VLambda: fixes I :: "V"   and f :: "V \<Rightarrow> V" shows "\<Inter>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I) = \<Inter>\<^sub>\<circ> (\<R>\<^sub>\<circ> (VLambda I f))"
path_trans_single: fixes SES :: "('a, 'b) SES_rec"   and s :: "'a"   and t :: "'b list"   and s' :: "'a"   and e :: "'b"   and sn :: "'a" assumes "s t\<Longrightarrow>\<^bsub>SES\<^esub> s'"   and "s' e\<longrightarrow>\<^bsub>SES\<^esub> sn" shows "s (t @ [e])\<Longrightarrow>\<^bsub>SES\<^esub> sn"
prob_space_axioms: fixes i :: "'i" shows "prob_space (distr M (M' i) (X i))"
leadsTo_cancel_Diff1: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set"   and A' :: "'c set"   and B' :: "'c set" assumes "F \<in> A \<longmapsto> B \<union> A'"   and "F \<in> B - A' \<longmapsto> B'" shows "F \<in> A \<longmapsto> B' \<union> A'"
subset_inj_on: fixes f :: "'b \<Rightarrow> 'c"   and B :: "'b set"   and A :: "'b set" assumes "inj_on f B"   and "A \<subseteq> B" shows "inj_on f A"
sqrt_int_code: fixes x :: "int" shows "sqrt_int x = (if x < 0 then [] else case sqrt_int_main x of (y, True) \<Rightarrow> if y = 0 then [0] else [y, - y] | (y, False) \<Rightarrow> [])"
standard_uniqueness_isomorphs: fixes X :: "'b set set"   and W :: "'c set set"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and C :: "'c set" assumes "ThinishChamberComplex X"   and "ChamberComplexIsomorphism W X f"   and "ChamberComplexIsomorphism W X g"   and "SimplicialComplex.maxsimp W C"   and "fun_eq_on f g C" shows "fun_eq_on f g (\<Union> W)"
step_node_tau: shows "\<tau> \<equiv> \<tau>\<^sub>n"
is_sup_binary: fixes x :: "'b"   and y :: "'b" shows "is_sup x y (\<Squnion>{x, y})"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "8 word \<Rightarrow> 'a"   and protocol :: "protocol" shows "P (case protocol of ProtoAny \<Rightarrow> f1 | Proto x \<Rightarrow> f2 x) = ((protocol = ProtoAny \<longrightarrow> P f1) \<and> (\<forall>x2. protocol = Proto x2 \<longrightarrow> P (f2 x2)))"
smallest_prime_beyond_le: fixes n :: "nat" shows "n \<le> smallest_prime_beyond n"
am_d_def: fixes antidomain_op :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.antidomain_left_monoid antidomain_op one times" shows "antidomain_left_monoid.am_d antidomain_op x = antidomain_op (antidomain_op x)"
run_bind_option: fixes return :: "'e option \<Rightarrow> 'd"   and bind :: "'d \<Rightarrow> ('e option \<Rightarrow> 'd) \<Rightarrow> 'd"   and x :: "('e, 'd) optionT"   and f :: "'e \<Rightarrow> ('e, 'd) optionT" shows "run_option (bind_option return bind x f) = bind (run_option x) (\<lambda>x. case x of None \<Rightarrow> return None | Some y \<Rightarrow> run_option (f y))"
abs_triangle_half_r: fixes y :: "'a"   and x1 :: "'a"   and e :: "'a"   and x2 :: "'a" assumes "\<bar>y - x1\<bar> < e / (2::'a)"   and "\<bar>y - x2\<bar> < e / (2::'a)" shows "\<bar>x1 - x2\<bar> < e"
count_nodes_eq_0_iff: fixes t :: "tree" shows "(count_nodes t = 0) = (t = Leaf)"
simps_4: fixes c :: "int"   and e :: "Cooper.num" shows "iszlfm (NEq (CN 0 c e)) = (0 < c \<and> numbound0 e)"
compl_fam_generate_subset: fixes gs :: "'a set"   and A :: "'a set" assumes "compl_fam (\<lambda>g. generate G {g}) gs"   and "gs \<subseteq> carrier G"   and "A \<subseteq> gs" shows "compl_fam (\<lambda>g. generate G {g}) A"
a_in_lcs: fixes H :: "'a set"   and a :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G" shows "a \<in> a \<diamondsuit> H"
exec_gpv_mk_lossless_gpv: fixes callee :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a) spmf"   and I :: "'a \<Rightarrow> bool"   and \<I> :: "('b, 'c) \<I>"   and gpv :: "('d, 'b, 'c) gpv"   and s :: "'a"   and x :: "'d" assumes "callee_invariant_on callee I \<I>"   and "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "exec_gpv callee (mk_lossless_gpv (responses_\<I> \<I>) x gpv) s = exec_gpv callee gpv s"
table_fvi_tl: fixes n :: "nat"   and b :: "nat"   and \<phi> :: "Formula.formula"   and X :: "'a option list set" assumes "table (Suc n) (Formula.fvi b \<phi>) X" shows "table n (Formula.fvi (Suc b) \<phi>) (tl ` X)"
simps_1: fixes psubst :: "variable \<Rightarrow> object"   and x :: "variable" shows "subst_term psubst (term.VAR x) = psubst x"
field_differentiable_def: fixes f :: "'b \<Rightarrow> 'b"   and F :: "'b filter" shows "f field_differentiable F \<equiv> \<exists>f'. (f has_field_derivative f') F"
nprv_addImp2LemmaI: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<phi>3 :: "'fmla"   and F :: "'fmla set" assumes "prv (imp \<phi>1 (imp \<phi>2 \<phi>3))"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<phi>3 \<in> fmla"   and "nprv F \<phi>1"   and "nprv F \<phi>2" shows "nprv F \<phi>3"
count_mset_set_1: fixes A :: "'c set"   and x :: "'c" assumes "finite A"   and "x \<in> A" shows "count (mset_set A) x = 1"
decisive_on: fixes d :: "'d" shows "decisive_on (Field (Pd d)) (Cd d)"
refine: fixes D :: "nat"   and preci :: "nat"   and prec :: "nat"   and Zi :: "(real \<times> real pdevs) list"   and Z :: "'a \<times> 'a pdevs"   and sctni :: "real list sctn"   and sctn :: "'a sctn" assumes "DIM_precond TYPE('a) D"   and "(preci, prec) \<in> nat_rel"   and "(Zi, Z) \<in> lv_aforms_rel"   and "(sctni, sctn) \<in> \<langle>lv_rel\<rangle>sctn_rel" shows "(nres_of (inter_aform_plane_lv D preci Zi sctni), inter_aform_plane prec Z sctn) \<in> \<langle>lv_aforms_rel\<rangle>nres_rel"
vsubset_vinsert_leftD: fixes a :: "V"   and A :: "V"   and B :: "V" assumes "vinsert a A \<subseteq>\<^sub>\<circ> B" shows "A \<subseteq>\<^sub>\<circ> B"
v11_vrange_ex1_eq: fixes b :: "V" shows "(b \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>) = (\<exists>!a. a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr> \<and> \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> = b)"
IHP_2: shows "((s, c), ee # ww, s', c') \<in> trcl (ntrs fg)"
closed_Collect_le: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "continuous_on UNIV f"   and "continuous_on UNIV g" shows "closed {x. f x \<le> g x}"
rank_Gauss_Jordan: fixes A :: "(('b, 'c) vec, 'd) vec" shows "rank A = rank (Gauss_Jordan A)"
Fract_of_int_quotient: fixes k :: "int"   and l :: "int" shows "Fract k l = rat_of_int k / rat_of_int l"
long_division_closed_1: fixes K :: "'a set"   and p :: "'a list"   and q :: "'a list" assumes "subfield K R"   and "p \<in> carrier (K [X])"   and "q \<in> carrier (K [X])" shows "p pdiv q \<in> carrier (K [X])"
assms_5: shows "A \<union> \<Union> (block_nominals ` S) \<subseteq> used"
bst_of_list_append: fixes xs :: "'a list"   and ys :: "'a list" shows "bst_of_list (xs @ ys) = fold Tree_Set.insert ys (bst_of_list xs)"
cBex_cong: fixes A :: "'d cset"   and B :: "'d cset"   and P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "A = B"   and "\<And>x. cin x B \<Longrightarrow> P x = Q x" shows "cBex A P = cBex B Q"
starlike_imp_path_connected: fixes S :: "'b set" assumes "starlike S" shows "path_connected S"
fpxs_root_order_of_fls: fixes f :: "'b fls" shows "fpxs_root_order (fpxs_of_fls f) = 1"
assms_1: shows "p \<in> hyp2 \<union> S"
character_ext: fixes G :: "('c, 'd) monoid_scheme"   and \<chi> :: "'c \<Rightarrow> complex"   and \<chi>' :: "'c \<Rightarrow> complex" assumes "character G \<chi>"   and "character G \<chi>'"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<chi> x = \<chi>' x" shows "\<chi> = \<chi>'"
inext_nth_card_Max_aux: fixes I :: "nat set"   and n :: "nat" assumes "card I = Suc n" shows "I \<rightarrow> n = Max I"
norm_of_real_add1: fixes x :: "real" shows "norm (of_real x + (1::'a)) = \<bar>x + 1\<bar>"
foundation3: fixes \<tau> :: "'a state \<times> 'a state" shows "\<tau> |\<noteq> invalid"
lookupa_csubst: fixes a :: "coname"   and \<theta>_c :: "(coname \<times> name \<times> trm) list"   and y :: "name"   and x :: "name"   and P :: "trm" assumes "a \<sharp> \<theta>_c" shows "Cut <a>.Ax y a x.P = lookupa y a \<theta>_c{a:=x.P}"
uniform_distributed_bounds: fixes X :: "'a \<Rightarrow> real"   and a :: "real"   and b :: "real" assumes "distributed M lborel X (\<lambda>x. ennreal (indicat_real {a..b} x / content {a..b}))" shows "a < b"
ds: shows "matrix_sum d (\<lambda>k. tensor_P (1\<^sub>m N) (proj_k k)) R \<in> carrier_mat d d"
pderiv_pCons: fixes a :: "'a"   and p :: "'a poly" shows "pderiv (pCons a p) = p + pCons (0::'a) (pderiv p)"
map_poly_hom_coeff_lift: fixes a :: "'a poly" shows "map_poly (pCons (0::'a)) [:a:] = [:pCons (0::'a) a:]"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "\<And>xs gt. \<forall>x\<in>set xs. \<not> gt x x \<Longrightarrow> \<not> ext gt xs xs" shows "ext_irrefl_axioms ext"
nasty_arithmetic1: fixes T :: "real"   and t :: "real" assumes "0 < T"   and "0 \<le> t" obtains q :: "nat" and r :: "real" where "t = real q * T + r" and "0 \<le> r" and "r < T"
the_g: shows "(THE f. \<forall>i. v $ i = (\<Sum>x\<in>UNIV. f x * X $ x $ i)) = g"
subst_equiv_simp: fixes p :: "('a, 'b, 'c) formula"   and q :: "('a, 'b, 'c) formula"   and \<sigma> :: "('a, 'b, 'c) subst" shows "Fsubst (p \<leftrightarrow> q) \<sigma> = (Fsubst p \<sigma> \<leftrightarrow> Fsubst q \<sigma>)"
dominated_convergence_2: fixes f :: "nat \<Rightarrow> 'c \<Rightarrow> 'd"   and S :: "'c set"   and h :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>k. f k integrable_on S"   and "h integrable_on S"   and "\<And>k x. x \<in> S \<Longrightarrow> norm (f k x) \<le> h x"   and "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>k. f k x) \<longlonglongrightarrow> g x" shows "(\<lambda>k. integral S (f k)) \<longlonglongrightarrow> integral S g"
component_le_onorm: fixes f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec"   and i :: "'b"   and j :: "'a" assumes "linear f" shows "\<bar>matrix f $ i $ j\<bar> \<le> onorm f"
normalized_not_None: fixes xs :: "('c, 'd) bintree option list" assumes "normalized xs" shows "set xs \<noteq> {None}"
fail_def: shows "fail \<equiv> \<lambda>s. ({}, True)"
while_eval_elim: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and e :: "'c"   and c :: "('b, 'a, 'c) Stmt"   and annos :: "'b ModeUpd list"   and mds :: "Mode \<Rightarrow> 'b set"   and mem :: "'b \<Rightarrow> 'd"   and c' :: "('b, 'a, 'c) Stmt"   and mds' :: "Mode \<Rightarrow> 'b set"   and mem' :: "'b \<Rightarrow> 'd" assumes "sifum_modes aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"   and "(((sifum_modes.annotate (While e c) annos, mds), mem), (c', mds'), mem') \<in> sifum_lang.eval\<^sub>w ev\<^sub>A ev\<^sub>B" shows "c' = Stmt.If e (c ;; While e c) Stop \<and> mds' = sifum_modes.update_annos mds annos \<and> mem' = mem"
prog_configs_append: fixes xs :: "'a memref list"   and ys :: "'a memref list" shows "prog_configs (xs @ ys) = prog_configs xs \<union> prog_configs ys"
power_sq_power: fixes x :: "'a"   and n :: "nat" shows "proto_dioid.power_sq 1\<^sub>\<sigma> (\<cdot>) sup x n = proto_dioid.p_power 1\<^sub>\<sigma> (\<cdot>) (sup 1\<^sub>\<sigma> x) n"
lookup_of_alist: fixes xs :: "('c \<times> 'b) list"   and k :: "'c" shows "lookup (of_alist xs) k = map_of xs k"
lead_coeff_of_int: fixes k :: "int" shows "lead_coeff (of_int k) = of_int k"
unit_simps_4: fixes a :: "'a" assumes "B.obj a" shows "B.dom \<i>[a] = a \<star>\<^sub>B a"
image_mset_map_of: fixes xs :: "('c \<times> 'd) list" assumes "distinct (map fst xs)" shows "{#the (map_of xs i). i \<in># mset (map fst xs)#} = mset (map snd xs)"
ring_simprules_3: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e" assumes "ring R"   and "x \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> x \<in> carrier R"
fundamental_theorem_of_algebra_alt: fixes p :: "complex poly" assumes "\<nexists>a l. a \<noteq> 0 \<and> l = 0 \<and> p = pCons a l" shows "\<exists>z. poly p z = 0"
inj_on_insert: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and A :: "'c set" shows "inj_on f (insert a A) = (inj_on f A \<and> f a \<notin> f ` (A - {a}))"
finsum_split: fixes A :: "'b set"   and a :: "'b"   and f :: "'b \<Rightarrow> 'a" assumes "finite A"   and "True"   and "a \<in> A" shows "sum f A = f a + sum f (A - {a})"
P: shows "P = Mod_Type_Connect.from_hma\<^sub>m P_HMA"
Liminf_def: fixes F :: "'d filter"   and f :: "'d \<Rightarrow> 'c" shows "Liminf F f = (SUP P\<in>{P. eventually P F}. Inf (f ` {x. P x}))"
tn_cond: shows "3 \<le> length tn \<and> tn ! (length tn - 2) = tn ! (length tn - 3)"
wf_invalid_point: fixes x :: "'a"   and b :: "'a set" assumes "x \<notin> bl"   and "b \<in># \<B>\<^sup>D" shows "x \<notin> b"
simps_1: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list" shows "NetsCollected2 (x # xs) = (xs = [] \<or> first_bothNet x \<noteq> firstList xs \<and> NetsCollected2 xs)"
iso_tuple_surjective_proof_assist_def: fixes x :: "'a"   and y :: "'b"   and f :: "'a \<Rightarrow> 'b" shows "iso_tuple_surjective_proof_assist x y f = (f x = y)"
dom_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "local.dom \<a>[a, b, c] = (a \<otimes> b) \<otimes> c"
IH: fixes vs :: "'a set"   and s :: "('a, 'b) fmap"   and s' :: "('a, 'b) fmap" assumes "vars_change as vs s = []"   and "sat_precond_as s as"   and "ListMem b as"   and "fmrestrict_set vs s = fmrestrict_set vs s'" shows "fmrestrict_set vs (fst b) \<subseteq>\<^sub>f fmrestrict_set vs s'"
A5: shows "Esigma (3\<^sup>2) = 13"
Sigma_cong: fixes A :: "'e set"   and B :: "'e set"   and C :: "'e \<Rightarrow> 'f set"   and D :: "'e \<Rightarrow> 'f set" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sigma A C = Sigma B D"
assms: shows "(f, h) \<in> [comp_PRE S1 Q P X]\<^sub>a R \<rightarrow> S"
assms_2: shows "(red F)\<^sup>*\<^sup>* p q"
c1: shows "higher_differentiable_on (S \<times> cbox a b) (\<lambda>(x, t). f x t) k"
selected_part_def: fixes Sel :: "'a set"   and C :: "'a Literal set" shows "selected_part Sel C = {L \<in> C. \<exists>A\<in>Sel. L = Neg A}"
permutes_not_in: fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and x :: "'b" assumes "f permutes S"   and "x \<notin> S" shows "f x = x"
sbelow_sctns_nres: fixes sctns :: "'c sctn set"   and X :: "'c set" assumes "finite sctns" shows "FOREACH\<^bsup>\<lambda>it b. b \<longrightarrow> (\<forall>sctn\<in>sctns - it. X \<subseteq> sbelow_halfspace sctn)\<^esup> sctns (\<lambda>sctn b. sbelow_sctn X sctn \<bind> (\<lambda>b'. RETURN (b' \<and> b))) True \<le> sbelow_sctns X sctns"
adds_canc: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and s :: "'a"   and u :: "'a"   and t :: "'a" assumes "class.comm_powerprod plus minus zero" shows "plus.adds plus (plus s u) (plus t u) = plus.adds plus s t"
card_UNIV_finfun_def: shows "card_UNIV_class.card_UNIV = Phantom('a \<Rightarrow>f 'b) (let ca = of_phantom card_UNIV_class.card_UNIV; cb = of_phantom card_UNIV_class.card_UNIV in if ca \<noteq> 0 \<and> cb \<noteq> 0 \<or> cb = 1 then cb ^ ca else 0)"
slength_sfoldl_aux: fixes z :: "Integer"   and xs :: "[:'a:]" shows "sfoldl\<cdot>(\<Lambda> i uu_. i + 1)\<cdot>z\<cdot>xs = z + slength\<cdot>xs"
tendsto_cong: fixes "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and F :: "'b filter"   and c :: "'a" assumes "class.topological_space open"   and "\<forall>\<^sub>F x in F. f x = g x" shows "filterlim f (topological_space.nhds open c) F = filterlim g (topological_space.nhds open c) F"
intros_2: fixes i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and v :: "'var"   and k :: "nat" assumes "i dd\<^bsup>\<pi>,v\<^esup>\<rightarrow> k" shows "(\<pi> k, \<pi> i) \<in> pdg"