hcju/isabelleps huggingface dataset:
has_contour_integral_join: fixes f :: "complex \<Rightarrow> complex"   and i1 :: "complex"   and g1 :: "real \<Rightarrow> complex"   and i2 :: "complex"   and g2 :: "real \<Rightarrow> complex" assumes "(f has_contour_integral i1) g1"   and "(f has_contour_integral i2) g2"   and "valid_path g1"   and "valid_path g2" shows "(f has_contour_integral i1 + i2) (g1 +++ g2)"
l_dec: fixes x :: "'a" shows "l x \<le> l (T x)"
triv_exI: fixes Q :: "'a \<Rightarrow> assn"   and x :: "'a" shows "Q x \<Longrightarrow>\<^sub>A \<exists>\<^sub>Ax. Q x"
lowEquivalence_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and H :: "'c set"   and L :: "'c set"   and High :: "'b"   and Low :: "'b"   and s :: "('c \<Rightarrow> 'd option) list"   and s' :: "('c \<Rightarrow> 'd option) list" assumes "NonInterferenceInterGraph sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses H L High Low" shows "NonInterferenceInterGraph.lowEquivalence L s s' \<equiv> \<forall>V\<in>L. hd s V = hd s' V"
zip_append2: fixes xs :: "'g list"   and ys :: "'h list"   and zs :: "'h list" shows "zip xs (ys @ zs) = zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
indicator_inter_arith: fixes A :: "'d set"   and B :: "'d set"   and x :: "'d" shows "indicator (A \<inter> B) x = indicator A x * indicator B x"
mem_box_1: fixes x :: "'b"   and a :: "'b"   and b :: "'b" shows "(x \<in> box a b) = (\<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i)"
1: shows "finite (Outside' {i} (Outside' X a)) \<and> finite ({i} \<times> Outside' X a `` {i}) \<and> Outside' {i} (Outside' X a) \<inter> {i} \<times> Outside' X a `` {i} = {} \<and> finite ({i} \<times> {\<Union> (a `` (X \<union> {i}))}) \<and> Outside' {i} (Outside' X a) \<inter> {i} \<times> {\<Union> (a `` (X \<union> {i}))} = {}"
claim'': fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and i :: "nat"   and i' :: "nat"   and j :: "nat"   and j' :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "\<pi> i = \<pi>' i'"   and "\<pi> j = \<pi>' j'"   and "\<forall>k. \<not> j cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   and "\<forall>k. \<not> i' cd\<^bsup>\<pi>'\<^esup>\<rightarrow> k"   and "\<pi> i \<noteq> local.return"   and "i < j" shows "i' < j'"
subspace_scale: fixes S :: "('b \<times> 'c) set"   and x :: "'b \<times> 'c"   and c :: "'a" assumes "p.subspace S"   and "x \<in> S" shows "local.scale c x \<in> S"
induct: fixes P1 :: "'c exp \<Rightarrow> bool"   and P2 :: "'c exp list \<Rightarrow> bool"   and exp :: "'c exp"   and exp_list :: "'c exp list" assumes "\<And>x. P1 (new x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (Cast x1 x2)"   and "\<And>x. P1 (Val x)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x3\<rbrakk> \<Longrightarrow> P1 (x1 \<guillemotleft>x2\<guillemotright> x3)"   and "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (x1:=x2)"   and "\<And>x1 x2 x3. P1 x1 \<Longrightarrow> P1 (x1\<bullet>x2{x3})"   and "\<And>x1 x2 x3. P1 (x1\<bullet>\<^sub>sx2{x3})"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P1 x4\<rbrakk> \<Longrightarrow> P1 (x1\<bullet>x2{x3} := x4)"   and "\<And>x1 x2 x3 x4. P1 x4 \<Longrightarrow> P1 (x1\<bullet>\<^sub>sx2{x3} := x4)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P2 x3\<rbrakk> \<Longrightarrow> P1 (x1\<bullet>x2(x3))"   and "\<And>x1 x2 x3. P2 x3 \<Longrightarrow> P1 (x1\<bullet>\<^sub>sx2(x3))"   and "\<And>x1 x2 x3. P1 x3 \<Longrightarrow> P1 {x1:x2; x3}"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1;; x2)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x2; P1 x3\<rbrakk> \<Longrightarrow> P1 (if (x1) x2 else x3)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (while (x1) x2)"   and "\<And>x. P1 x \<Longrightarrow> P1 (throw x)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P1 x4\<rbrakk> \<Longrightarrow> P1 (try x1 catch(x2 x3) x4)"   and "\<And>x1 x2 x3 x4. P1 x4 \<Longrightarrow> P1 (INIT x1 (x2,x3) \<leftarrow> x4)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x2; P1 x4\<rbrakk> \<Longrightarrow> P1 (RI (x1,x2) ; x3 \<leftarrow> x4)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P1 exp \<and> P2 exp_list"
subst_Pair: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d"   and a :: "'c \<times> 'd" assumes "P x y"   and "a = (x, y)" shows "P (fst a) (snd a)"
zero_spaceN_subset_spaceN: fixes N :: "'b quasinorm" shows "zero_space\<^sub>N N \<subseteq> space\<^sub>N N"
model_to_dimacs_model_sound_max_var: fixes f :: "nat formula"   and M :: "nat \<Rightarrow> bool" assumes "\<forall>v\<in>atoms f. 0 < v"   and "is_cnf f"   and "M \<Turnstile> f" shows "dimacs_model (model_to_dimacs_model M [1..<find_max (formula_vars f) + 2]) (cnf_to_dimacs f)"
image_mset_is_empty_iff: fixes f :: "'d \<Rightarrow> 'c"   and M :: "'d multiset" shows "(f `# M = {#}) = (M = {#})"
q11_array_def: shows "q11_array = IArray [True, True, False, True, True, True, False, False, False, True, False]"
antisym_conv2: fixes x :: "'b zmultiset"   and y :: "'b zmultiset" assumes "x \<subseteq>#\<^sub>z y" shows "(\<not> x \<subset>#\<^sub>z y) = (x = y)"
II2: shows "II2 \<in> local.shift cl n2 ` P n2"
all_envir_def: fixes p :: "'d \<Rightarrow> bool" shows "all_envir p = (\<forall>x. p x)"
term_cases: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and t :: "'c" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" obtains name :: "name" where "t = free name__"   | name :: "name" where "t = const name"   | u\<^sub>1 :: "'c" and u\<^sub>2 :: "'c" where "t = app u\<^sub>1 u\<^sub>2"   | "unfree t = None" and "unapp t = None" and "unconst t = None"
M4_def: shows "M4 \<equiv> ((\<forall>x y. P x y = (PP x y \<or> x = y)) \<and> (\<forall>x y. O x y = (\<exists>z. P z x \<and> P z y))) \<and> (\<forall>x y. PP x y \<longrightarrow> \<not> PP y x) \<and> (\<forall>x y z. PP x y \<longrightarrow> PP y z \<longrightarrow> PP x z)"
intersection_num_non_neg: fixes b1 :: "'a set"   and b2 :: "'a set" shows "0 \<le> b1 |\<inter>| b2"
in_idealE_rep_list: fixes p :: "'a \<Rightarrow>\<^sub>0 'b" assumes "p \<in> ideal (set fs)" obtains r :: "'t \<Rightarrow>\<^sub>0 'b" where "p = rep_list r" and "r \<in> sig_inv_set' (length fs)"
arrow: fixes D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'d"   and e :: "'d" assumes "arrow_from_functor D C F x y e" shows "partial_magma.ide D x \<and> partial_magma.in_hom C e (F x) y"
hd_AE: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and e :: "expr"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal"   and t :: "pdf_type"   and f' :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal" assumes "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d e \<Rightarrow> f"   and "\<Gamma> \<turnstile> e : t"   and "\<And>\<rho>. \<rho> \<in> space (state_measure V' \<Gamma>) \<Longrightarrow> AE x in stock_measure t. f \<rho> x = f' \<rho> x"   and "(\<lambda>(x, y). f' x y) \<in> borel_measurable (state_measure V' \<Gamma> \<Otimes>\<^sub>M stock_measure t)" shows "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d e \<Rightarrow> f'"
mult_row_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and a :: "'d"   and q :: "'b" shows "mult_row A a q = (\<chi>i j. if i = a then q * A $ a $ j else A $ i $ j)"
orientable_12_implies_11: fixes x :: "'a" assumes "asymmetric x \<and> strict_linear x" shows "(x \<squnion> (1::'a)) \<sqinter> (x \<squnion> (1::'a))\<^sup>T = (1::'a) \<and> linear (x \<squnion> (1::'a))"
irreducible\<^sub>d_mD_1: fixes f :: "int poly" assumes "q.irreducible\<^sub>d_m f" shows "0 < q.degree_m f"
arr_Par_arr_RelE: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Par \<alpha> T" obtains "arr_Rel \<alpha> T" and "vsv (T\<lparr>ArrVal\<rparr>)"
least_pr_weak': fixes w :: "'b \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and pr_strict :: "'b \<times> nat \<Rightarrow> 'b \<times> nat \<Rightarrow> bool"   and pr_weak :: "'b \<times> nat \<Rightarrow> 'b \<times> nat \<Rightarrow> bool"   and least :: "'b \<Rightarrow> bool"   and scf :: "'b \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and f :: "'b"   and g :: "'b \<times> nat" assumes "admissible_kbo w w0 pr_strict pr_weak least scf"   and "least f" shows "admissible_kbo.pr_weak' pr_weak least g (f, 0)"
scale: fixes r :: "real"   and b :: "'b" shows "g (r *\<^sub>R b) = r *\<^sub>R g b"
float_normed_cases: fixes f :: "float" obtains "f = 0"   | m :: "int" and e :: "int" where "real_of_float f = real_of_int m * 2 powr real_of_int e" and "odd m" and "f \<noteq> 0"
assms: fixes c\<^sub>1 :: "nat"   and c\<^sub>2 :: "nat"   and n :: "nat" assumes "0 < c\<^sub>1"   and "0 < c\<^sub>2" shows "\<exists>m>n. c\<^sub>1 * f m < g m \<or> c\<^sub>2 * g m < f m"
divide_nonneg_nonneg: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq zero x"   and "less_eq zero y" shows "less_eq zero (divide x y)"
get_exp_fulfills: fixes a :: "'a" assumes "a \<in> carrier G" shows "a = gen [^] get_exp gen a"
s1_dist: shows "s1 = (x * y + x * (q - b) + (q - r)) mod q"
bin_nth_Suc_BIT: fixes w :: "int"   and b :: "bool"   and n :: "nat" shows "bit (w BIT b) (Suc n) = bit w n"
gen_lossless_bind_gpv: fixes co :: "bool"   and \<I> :: "('a, 'b) \<I>"   and gpv :: "('d, 'a, 'b) gpv"   and f :: "'d \<Rightarrow> ('c, 'a, 'b) gpv" shows "gen_lossless_gpv co \<I> (gpv \<bind> f) = (gen_lossless_gpv co \<I> gpv \<and> (\<forall>x\<in>results_gpv \<I> gpv. gen_lossless_gpv co \<I> (f x)))"
delete_edges_list_set: fixes G :: "'a graph"   and E :: "('a \<times> 'a) list" shows "delete_edges_list G E = delete_edges G (set E)"
remove_def: fixes x :: "'e"   and A :: "'e set" shows "Set.remove x A = A - {x}"
sum_commute: fixes x :: "'a"   and y :: "'a" shows "sum_class.sum x + sum_class.sum y = sum_class.sum y + sum_class.sum x"
compute_float_plus_up: fixes p :: "nat"   and x :: "float"   and y :: "float" shows "float_plus_up p x y = - float_plus_down p (- x) (- y)"
prob_sum_is_one: fixes n :: "nat"   and v :: "complex Matrix.mat"   and i :: "nat" assumes "state n v" shows "prob1 n v i + prob0 n v i = 1"
QQ5_a_def: shows "QQ5_a \<equiv> [: p, t, left, right, atom, mrk \<leadsto> p', t', left', right', atom', mrk' . p \<noteq> nil \<and> \<not> gg mrk atom left p \<and> \<not> gg mrk atom right p \<and> t \<noteq> nil \<and> \<not> atom t \<and> p' = t \<and> t' = left t \<and> mrk' = mrk \<and> left' = left(t := p) \<and> right' = right \<and> atom' = atom :]"
resolvingBidBridgingLemma: fixes N :: "'c set"   and \<Omega> :: "'d list"   and a :: "('c \<times> 'd set) set"   and b :: "'c \<times> 'd set \<Rightarrow> 'e"   and r :: "integer" assumes "0 < card N"   and "distinct \<Omega>"   and "a \<subseteq> N \<times> (Pow (set \<Omega>) - {{}})" shows "sum (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel N (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg N \<Omega>)) r) <| (N \<times> finestpart (set \<Omega>))) N (set \<Omega>)) a = sum (resolvingBidAlg N \<Omega> b r) a"
assms_1: shows "S \<sqsubseteq> (\<lceil>b\<rceil>\<^sub>< \<and> P)"
Df: shows "\<forall>\<^sub>F x in at_top. (f has_real_derivative f' x) (at x)"
vreal_fpair_in_fproduct_vreal: fixes a :: "real"   and b :: "real" shows "[a\<^sub>\<real>, b\<^sub>\<real>]\<^sub>\<circ> \<in>\<^sub>\<circ> \<real>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat>"
stutter_free_reduced: fixes \<sigma> :: "nat \<Rightarrow> 'b" assumes "stutter_free \<sigma>" shows "\<natural>\<sigma> = \<sigma>"
Atoms_pderiv: fixes a :: "'b"   and r :: "'b rexp" shows "Atoms (pderiv a r) \<subseteq> atoms r"
assms: shows "i < dim_row U\<^sub>f \<and> j < dim_col U\<^sub>f"
Image_X: fixes Y :: "'a set" assumes "countable Y" shows "countable (X `` Y)"
component_eq_subset: fixes F :: "'c program"   and G :: "'c program" shows "(F \<le> G) = (Init G \<subseteq> Init F \<and> Acts F \<subseteq> Acts G \<and> AllowedActs G \<subseteq> AllowedActs F)"
times_Integer_def: fixes a :: "Integer"   and b :: "Integer" shows "a * b = (case a of MkI\<cdot>x \<Rightarrow> \<Lambda> (MkI\<cdot>y). MkI\<cdot>(x * y))\<cdot>b"
inf_absorb1: fixes x :: "'c multiset"   and y :: "'c multiset" assumes "x \<subseteq># y" shows "x \<inter># y = x"
leg_in_hom_2: shows "\<guillemotleft>assoc\<nu>\<pi>\<rho>.cod.leg1 : assoc\<nu>\<pi>\<rho>.cod.apex \<rightarrow>\<^sub>C assoc\<nu>\<pi>\<rho>.cod.trg\<guillemotright>"
arith_series_nat: fixes a :: "nat"   and d :: "nat"   and n :: "nat" shows "(\<Sum>i = 0..n. a + i * d) = Suc n * (2 * a + n * d) div 2"
ennexp_strict_mono2: fixes x :: "ereal"   and y :: "ereal" assumes "x < y" shows "ennexp x < ennexp y"
conf_lupd: fixes G :: "prog"   and vn :: "lname"   and va :: "val"   and s :: "st"   and v :: "val"   and T :: "ty" shows "G,lupd(vn\<mapsto>va) s\<turnstile>v\<Colon>\<preceq>T = G,s\<turnstile>v\<Colon>\<preceq>T"
correctDeCompositionVAR_def: fixes S :: "CSet" shows "correctDeCompositionVAR S \<equiv> \<forall>v\<in>VAR S. \<forall>C1\<in>subcomp S. \<forall>C2\<in>subcomp S. v \<in> VAR C1 \<and> v \<in> VAR C2 \<longrightarrow> C1 = C2"
R_hits_Q: shows "Bex (set R) Q.hitting_paths"
assms_6: fixes A :: "'a set" assumes "A \<in> G" shows "(\<lambda>a. emeasure (K a) A) \<in> borel_measurable M"
monitorable_formula_code: fixes \<phi> :: "Formula.formula" shows "mmonitorable \<phi> = mmonitorable_exec \<phi>"
map_of_simp: shows "state_mem_defs.map_of lookup_trace = local.map_of \<circ> snd"
J_def: shows "J = (SIGMA (i, j):Set.filter (\<lambda>(i, j). j < i) ({..n} \<times> {..n}). {j<..i})"
hfref_mop_conv: fixes g :: "'a \<Rightarrow> 'b Heap"   and P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd nres"   and Q :: "'c \<Rightarrow> bool"   and R :: "('c \<Rightarrow> 'a \<Rightarrow> assn) \<times> ('c \<Rightarrow> 'a \<Rightarrow> assn)"   and S :: "'d \<Rightarrow> 'b \<Rightarrow> assn" shows "((g, mop P f) \<in> [Q]\<^sub>a R \<rightarrow> S) = ((g, f) \<in> [\<lambda>x. P x \<and> Q x]\<^sub>a R \<rightarrow> S)"
eq_self_neq_FF: fixes x :: "'a" shows "eq\<cdot>x\<cdot>x \<noteq> FF"
LLq_def: fixes t1 :: "'trm"   and t2 :: "'trm" shows "LLq t1 t2 = psubst Lq [(t1, zz), (t2, yy)]"
assms: shows "a \<noteq> (0::'a) \<or> b \<noteq> (0::'a)"
ex_se_succ: fixes rv :: "'a \<times> nat"   and l :: "('b, 'c) label" assumes "RedBlack prb"   and "rv \<in> red_vertices prb" shows "\<exists>c'. se (confs prb rv) l c'"
Cons_4: shows "Span K (u # Us) \<inter> Span K Vs = {\<zero>}"
prv_eql_pls_plus: fixes m :: "nat"   and n :: "nat" shows "prv (eql (pls (Num m) (Num n)) (Num (m + n)))"
intro: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "known_ptr = DocumentClass.known_ptr"   and "type_wf = DocumentClass.type_wf"   and "get_child_nodes = l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_child_nodes"   and "get_child_nodes_locs = l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_child_nodes_locs" shows "l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs"
simps_6: fixes i :: "'b"   and p :: "('c, 'b) fm" shows "nominals (\<^bold>@ i p) = {i} \<union> nominals p"
density_RN_deriv: fixes N :: "'a measure" assumes "absolutely_continuous (density M (\<lambda>x. 1 + h x)) N"   and "sets N = sets (density M (\<lambda>x. 1 + h x))" shows "density (density M (\<lambda>x. 1 + h x)) (RN_deriv (density M (\<lambda>x. 1 + h x)) N) = N"
bind_returns_result_E4: fixes h :: "'x"   and f :: "('x, 'y, 'aa) prog"   and g :: "'aa \<Rightarrow> ('x, 'y, 'z) prog"   and y :: "'z"   and x :: "'aa" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"   and "h \<turnstile> f \<rightarrow>\<^sub>r x" obtains h' :: "'x" where "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> g x \<rightarrow>\<^sub>r y"
Says_Kas_message_form: fixes A :: "agent"   and K :: "nat"   and authK :: "nat"   and Peer :: "agent"   and Ta :: "nat"   and authTicket :: "msg"   and evs :: "event list" assumes "Says Kas A (Crypt K \<lbrace>Key authK, Agent Peer, Number Ta, authTicket\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV_gets" shows "K = shrK A \<and> Peer = Tgs \<and> authK \<notin> range shrK \<and> authK \<in> authKeys evs \<and> authK \<in> symKeys \<and> authTicket = Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>"
null_sets_density_iff: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and A :: "'b set" assumes "f \<in> borel_measurable M" shows "(A \<in> null_sets (density M f)) = (A \<in> sets M \<and> (AE x\<in>A in M. f x = 0))"
sep_reorder_dollar_3: fixes X :: "nat"   and Y :: "nat" shows "($ X \<and>* $ Y) = $ (X + Y)"
simps_5: fixes e :: "(name, nat) finfun"   and t :: "trm"   and u :: "trm" shows "eval_trm e (tms t u) = eval_trm e t * eval_trm e u"
restrict_UNIV: fixes f :: "'d \<Rightarrow> 'e" shows "restrict f UNIV = f"
bigo_const_mult1: fixes c :: "'b"   and f :: "'a \<Rightarrow> 'b" shows "(\<lambda>x. c * f x) \<in> O(f)"
P'''Trans: shows "P''' \<longmapsto> b<x> \<prec> P''"
preserves_reflects_arr: fixes f :: "'c" shows "C.arr (\<I>\<^sub>C \<otimes>\<^sub>C f) = C.arr f"
msum_aforms'_refine_raw: fixes x :: "(real \<times> real pdevs) list"   and y :: "(real \<times> real pdevs) list" shows "(msum_aforms' x y, {map2 (+) a b |a b. a \<in> Joints x \<and> b \<in> Joints y}) \<in> aforms_rel"
roots_of_unity_2: fixes n :: "nat" assumes "n \<noteq> 0" shows "{x. poly (root_unity n) x = 0} = {x. x ^ n = 1}"
sortedP_right_of_last: fixes x :: "real \<times> real"   and zs :: "(real \<times> real) list"   and y :: "real \<times> real" assumes "ccw'.sortedP x zs"   and "y \<in> set zs"   and "y \<noteq> last zs" shows "ccw' x y (last zs)"
euler_genus_da: shows "rev.H.euler_genus = G.euler_genus"
\<Z>_Limit_\<alpha>\<omega>: fixes \<alpha> :: "V" assumes "\<Z> \<alpha>" shows "Limit (\<alpha> + \<omega>)"
nth_in: fixes xs :: "'b list"   and n :: "nat"   and A :: "'b set"   and i :: "nat" assumes "length xs = n"   and "set xs \<subseteq> A"   and "i < n" shows "xs ! i \<in> A"
extensionalityI: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> extensional A"   and "g \<in> extensional A"   and "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "f = g"
multiplicity_one: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "factorial_semiring.multiplicity times one p one = 0"
nf_ACI_D_2: fixes \<phi> :: "('a, 'b) aformula" assumes "nf_ACI \<phi>" shows "sorted (conjuncts_list \<phi>)"
la_ex_pres_ls_vars: fixes ls :: "local_state"   and gs :: "global_state"   and a :: "action"   and ls' :: "local_state"   and gs' :: "global_state" assumes "la_ex (ls, gs) a = Some (ls', gs')" shows "ls_vars ls' = ls_vars ls"
swap_Snapshot_Send: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isSnapshot ev"   and "isSend ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
assms_3: fixes \<psi> :: "'a ltl"   and i :: "nat" assumes "G \<psi> \<in> \<G>"   and "i \<le> j" shows "S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (w [i \<rightarrow> j]))"
subS01: shows "(\<lambda>x. x /\<^sub>R norm x) ` (S - {0::'a}) \<subseteq> sphere (0::'a) 1 \<inter> S"
new'_def: shows "Array.new' = Array.new \<circ> nat_of_integer"
deg_mult_ring: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P" shows "deg R (p \<otimes>\<^bsub>P\<^esub> q) \<le> deg R p + deg R q"
inj_comp_end: fixes S :: "'v \<Rightarrow> 'v"   and T :: "'v \<Rightarrow> 'v" assumes "VEnd S"   and "inj_on S V"   and "VEnd T"   and "inj_on T V" shows "inj_on (S \<circ> T) V"
AE_bind: fixes N :: "'c \<Rightarrow> 'd measure"   and M :: "'c measure"   and B :: "'d measure"   and P :: "'d \<Rightarrow> bool" assumes "N \<in> M \<rightarrow>\<^sub>M subprob_algebra B"   and "Measurable.pred B P" shows "almost_everywhere (M \<bind> N) P = (AE x in M. almost_everywhere (N x) P)"
adjacent_nodes_finite: fixes u :: "nat" shows "finite (adjacent_nodes u)"
card: assumes "rebase_ge TYPE('a) TYPE('b)" shows "CARD('a) \<le> CARD('b)"
Nats_def: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d" assumes "class.semiring_1 one times plus zero" shows "semiring_1.Nats one plus zero = range (semiring_1.of_nat one plus zero)"
unitor_coincidence_2: shows "\<r>[\<I>] = \<iota>\<^sub>F\<^sub>M\<^sub>C"
filterlim_transform_within: fixes g :: "'b \<Rightarrow> 'c"   and G :: "'c filter"   and x :: "'b"   and S :: "'b set"   and F :: "'c filter"   and d :: "real"   and f :: "'b \<Rightarrow> 'c" assumes "filterlim g G (at x within S)"   and "G \<le> F"   and "0 < d"   and "\<And>x'. \<lbrakk>x' \<in> S; 0 < dist x' x; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'" shows "filterlim f F (at x within S)"
partList0_set_equi: fixes s :: "'b set"   and ts :: "'b set list" shows "set (partList0 s ts) = (\<inter>) s ` set ts \<union> (\<lambda>x. x - s) ` set ts"
eval_nat: fixes \<rho> :: "(nat \<times> bval) list"   and n :: "nat" shows "\<rho> \<turnstile> ENat n \<Down> BNat n"