hcju/isabelleps huggingface dataset:
cv_inv_version3: fixes S :: "(real, 'a) vec set"   and g :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and g' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and T :: "(real, 'a) vec set"   and h :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and h' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec"   and b :: "(real, 'b) vec" assumes "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"   and "\<And>y. y \<in> T \<Longrightarrow> (h has_derivative h' y) (at y within T)"   and "\<And>x. x \<in> S \<Longrightarrow> g x \<in> T \<and> h (g x) = x"   and "\<And>y. y \<in> T \<Longrightarrow> h y \<in> S \<and> g (h y) = y"   and "\<And>y. y \<in> T \<Longrightarrow> h' y \<circ> g' (h y) = id" shows "((\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)) absolutely_integrable_on S \<and> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)) = b) = (f absolutely_integrable_on T \<and> integral T f = b)"
bound_cmono: fixes X :: "'c set"   and Y :: "'c set" assumes "X \<subseteq> Y" shows "bound Y \<le> bound X"
mem_Collect_eq: fixes a :: "'q"   and P :: "'q \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
geq_refl: fixes x :: "'a" assumes "x \<in> carrier R" shows "x \<succeq> x"
simps_1: fixes uu :: "real_min_top" shows "(uu \<le> PInfty) = True"
find_Some_iff: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b" shows "(find P xs = Some x) = (\<exists>i<length xs. P (xs ! i) \<and> x = xs ! i \<and> (\<forall>j<i. \<not> P (xs ! j)))"
HSem_ind: fixes P :: "('var \<Rightarrow> 'value) \<Rightarrow> bool"   and \<rho> :: "'var \<Rightarrow> 'value"   and \<Gamma> :: "('var \<times> 'exp) list" assumes "adm P"   and "P \<bottom>"   and "\<And>\<rho>'. P \<rho>' \<Longrightarrow> P (\<rho> ++\<^bsub>domA \<Gamma>\<^esub> \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<rho>'\<^esub>)" shows "P (\<lbrace> \<Gamma> \<rbrace>\<rho>)"
dvd_mult_imp_div: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times (times a c) b" shows "dvd.dvd times a (divide b c)"
FinGroupRepresentation_reducible': fixes G :: "'a set"   and fgsmult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and n :: "nat" assumes "FinGroupRepresentation G fgsmult V \<and> n = R_scalar_mult.dim_R UNIV (aezfun_scalar_mult.fsmult fgsmult) V" shows "\<exists>Us. Ball (set Us) (IrrFinGroupRepresentation G fgsmult) \<and> 0 \<notin> set Us \<and> V = (\<Oplus>U\<leftarrow>Us. U)"
fold_fun_left_comm: fixes A :: "'a set"   and x :: "'a"   and z :: "'b" assumes "finite A" shows "f x (Finite_Set.fold f z A) = Finite_Set.fold f (f x z) A"
eq: fixes x :: "'a" assumes "x \<in> A" shows "(\<Sum>a | x = a \<and> a \<in> A \<and> f a \<noteq> (0::'b). f a) = sum f {x}"
intros: fixes x :: "'x"   and m :: "'m"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m" assumes "\<tau>move (x, m) ta (x', m')" shows "init_fin_\<tau>move ((Running, x), m) (convert_TA_initial (convert_obs_initial ta)) ((Running, x'), m')"
Q_div_k_def: shows "Q_div_k = addrow_mat (dim_col A) (- k) 0 2"
fair_run_axioms: shows "fair_run s\<^sub>0 lstep s"
modus_tollens_1: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> \<^bold>\<rightarrow> \<psi> in v]"   and "[\<^bold>\<not>\<psi> in v]" shows "[\<^bold>\<not>\<phi> in v]"
not_\<omega>\<^sub>z_lt_numeral: fixes n :: "num" shows "\<not> \<omega>\<^sub>z < numeral n"
isomorphic_symmetric: fixes f :: "'a"   and g :: "'a" assumes "C.isomorphic f g" shows "C.isomorphic g f"
verts_app_inv_iso: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism" assumes "digraph_isomorphism hom" shows "iso_verts (inv_iso hom) ` iso_verts hom ` verts G = verts G"
wf_max: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "Max x y \<in> carrier R"
simps_12: fixes n :: "nat"   and t :: "tm"   and p :: "fm" shows "subst n t (E p) = E (subst (Suc n) (incrtm0 t) p)"
fct_gen_subalgebra_sigma_sets: fixes X :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" assumes "X \<in> space M \<rightarrow> space N" shows "sets (fct_gen_subalgebra M N X) = {X -` B \<inter> space M |B. B \<in> sets N}"
kpower_inductl: fixes f :: "'a \<Rightarrow> 'a set"   and g :: "'a \<Rightarrow> 'b set"   and i :: "nat" assumes "f \<circ>\<^sub>K g \<le> g" shows "kpower f i \<circ>\<^sub>K g \<le> g"
sums_unique: fixes s :: "'a" assumes "f sums s" shows "s = suminf f"
7: shows "(H * (d \<sqinter> - a))\<^sup>\<star> * H * a * top \<le> (H * (d \<sqinter> - a \<squnion> (d \<sqinter> - a)\<^sup>T))\<^sup>\<star> * H * a * top"
path2_snoc: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and m' :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "m \<in> set (old.predecessors g m')" shows "g \<turnstile> n-ns @ [m']\<rightarrow>m'"
AssertAllState'_unsat_atoms_implies_bounds: fixes t :: "(nat \<times> linear_poly) list"   and as :: "('i \<times> 'a atom) list" assumes "\<triangle> t" shows "set as \<Turnstile>\<^sub>i \<B>\<I> (assert_all_state t as)"
True: fixes F :: "('c \<Rightarrow> bool) \<Rightarrow> bool" assumes "is_filter F" shows "F (\<lambda>x. True)"
adopt_node_get_shadow_root: fixes parent :: "(_) object_ptr option"   and owner_document :: "(_) document_ptr"   and document_ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) element_ptr" shows "\<forall>w\<in>Shadow_DOM.adopt_node_locs parent owner_document document_ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_shadow_root_locs ptr'. r h h')"
map\<^sub>0_simps_2: fixes a :: "'a" assumes "C.obj a" shows "src\<^sub>D (\<tau>\<tau>'.map\<^sub>0 a) = G.map\<^sub>0 a"
mem_fine3: fixes \<delta> :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and D :: "(real \<times> real \<times> real) list"   and u :: "real"   and z :: "real"   and v :: "real" assumes "fine \<delta> (a, b) D"   and "(u, z, v) \<in> set D" shows "v - u < \<delta> z"
prj_chine_assoc'_1: shows "Prj\<^sub>1\<^sub>1 \<cdot> chine_assoc' = Prj\<^sub>1"
ordering2: shows "\<forall>x\<in>{a, b}. \<exists>n. (finite {a, b} \<longrightarrow> n < card {a, b}) \<and> (if n = 0 then a else b) = x"
d_c_a_b: fixes c :: "'a"   and a :: "'a"   and b :: "'a" shows "d_aux c a b = d_aux a b c"
sort_of_permute: fixes p :: "perm"   and a :: "atom" shows "sort_of (p \<bullet> a) = sort_of a"
not_Infinitesimal_mult: fixes x :: "'b star"   and y :: "'b star" assumes "x \<notin> Infinitesimal"   and "y \<notin> Infinitesimal" shows "x * y \<notin> Infinitesimal"
collapse: fixes prod :: "'c \<times> 'd" shows "(fst prod, snd prod) = prod"
exhaust: fixes y :: "'f list" obtains "y = []"   | x21 :: "'f" and x22 :: "'f list" where "y = x21 # x22"
ass_adv_14_OT: fixes msgs :: "bool \<times> bool \<times> bool \<times> bool"   and D :: "'v_14_OT1 \<Rightarrow> bool spmf"   and c0 :: "bool"   and c1 :: "bool" shows "\<bar>spmf (S1_14_OT msgs () \<bind> D) True - spmf (R1_14_OT msgs (c0, c1) \<bind> D) True\<bar> \<le> adv_14_OT"
simps_4: fixes x2 :: "wake_up_status"   and x1 :: "'b" shows "PostWS x2 \<noteq> InWS x1"
assms: shows "Complete_Partial_Order.chain (ord_spmf (=)) Y"
assms_3: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and c :: "'a"   and exp :: "exp" assumes "atom ` domA \<Gamma> \<sharp>* c"   and "\<not> nonrec \<Gamma>"   and "\<And>c x e. (x, e) \<in> set \<Gamma> \<Longrightarrow> P c e"   and "\<And>c. P c exp" shows "P c (Let \<Gamma> exp)"
closed_orbit_subset_\<omega>_limit_set: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and x :: "'b" assumes "auto_ll_on_open f X"   and "auto_ll_on_open.closed_orbit f X x" shows "range (auto_ll_on_open.flow0 f X x) \<subseteq> auto_ll_on_open.\<omega>_limit_set f X x"
comp_cod_arr: fixes f :: "'c local.term set"   and b :: "'c local.term set" assumes "\<F>C.arr f"   and "\<F>C.cod f = b" shows "b \<cdot> f = f"
Gr_domain_1: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "Domain (BNF_Def.Gr A f) = A"
ide_in_hom: fixes a :: "'c" shows "C.ide a = \<guillemotleft>a : a \<rightarrow>\<^sub>C a\<guillemotright>"
codomains_simp: fixes f :: "(bool, 'arr list) concrete_category.arr" assumes "arr f" shows "codomains f = {cod f}"
profile_non_empty: fixes A :: "'c set"   and Is :: "'d set"   and P :: "'d \<Rightarrow> ('c \<times> 'c) set" assumes "profile A Is P" shows "Is \<noteq> {}"
nth_equalityI: fixes xs :: "'b list"   and ys :: "'b list" assumes "length xs = length ys"   and "\<And>i. i < length xs \<Longrightarrow> xs ! i = ys ! i" shows "xs = ys"
inject_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(Opaque x1 = Opaque y1) = (x1 = y1)"
append_assoc: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" shows "(xs @ ys) @ zs = xs @ ys @ zs"
hbt_delete_2: fixes m :: "nat"   and t :: "('b \<times> nat) tree"   and x :: "'b" assumes "HBT m"   and "HBT.hbt m t" shows "height t \<in> {height (HBT.delete m x t), height (HBT.delete m x t) + 1}"
CFG_SSA_wf_axioms_def: fixes \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and phis :: "'a \<Rightarrow> 'b \<times> 'd \<Rightarrow> 'd list option" shows "CFG_SSA_wf_axioms \<alpha>n invar inEdges' Entry defs uses phis \<equiv> (\<forall>v g n. v \<in> CFG_SSA_base.allUses \<alpha>n inEdges' uses phis g n \<longrightarrow> n \<in> set (\<alpha>n g) \<longrightarrow> CFG_SSA_base.defAss \<alpha>n invar inEdges' Entry defs phis g n v) \<and> (\<forall>g v. phis g (Entry g, v) = None)"
apf_def: fixes x :: "'c \<Rightarrow> 'e \<Rightarrow> 'd"   and y :: "'c \<Rightarrow> 'e" shows "x \<diamondop> y = (\<lambda>z. x z (y z))"
DN_in_hom_2: fixes \<mu> :: "('a E.term, 'a) arr" assumes "arr \<mu>" shows "\<guillemotleft>DN \<mu> : DN (local.dom \<mu>) \<Rightarrow>\<^sub>B DN (cod \<mu>)\<guillemotright>"
sethi_instr_result: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "fst i = sethi_type SETHI" shows "snd (sethi_instr i s) = False"
set_map: fixes f :: "'b \<Rightarrow> 'a"   and v :: "'b option" shows "set_option (map_option f v) = f ` set_option v"
mul_two_principal_idealsTr: fixes a :: "'a"   and b :: "'a"   and x :: "'a"   and y :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "x \<in> R \<diamondsuit>\<^sub>p a"   and "y \<in> R \<diamondsuit>\<^sub>p b" shows "\<exists>r\<in>carrier R. x \<cdot>\<^sub>r y = r \<cdot>\<^sub>r (a \<cdot>\<^sub>r b)"
mon_c_union_conc: fixes fg :: "('g, 'h, 'i, 'f, 'j) flowgraph_rec_scheme"   and a :: "'g list multiset"   and b :: "'g list multiset" shows "mon_c fg (a + b) = mon_c fg a \<union> mon_c fg b"
cs_path_swap: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and k :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "\<pi> =\<^bsub>k\<^esub> \<pi>'" shows "cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k"
extract_var_def: fixes p :: "'a mpoly"   and v :: "nat" shows "extract_var p v = (\<Sum>m. MPoly_Type.monom (remove_key v m) (MPoly_Type.monom (Poly_Mapping.single v (lookup m v)) (MPoly_Type.coeff p m)))"
homeomorphism_def: fixes s :: "'a set"   and t :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" shows "homeomorphism s t f g = ((\<forall>x\<in>s. g (f x) = x) \<and> f ` s = t \<and> continuous_on s f \<and> (\<forall>y\<in>t. f (g y) = y) \<and> g ` t = s \<and> continuous_on t g)"
f6: shows "\<not> (\<exists>(a', b', c')\<in>set b. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
map_upds_Cons: fixes m :: "'a \<Rightarrow> 'b option"   and a :: "'a"   and as :: "'a list"   and b :: "'b"   and bs :: "'b list" shows "m(a # as [\<mapsto>] b # bs) = m(a \<mapsto> b, as [\<mapsto>] bs)"
simps_5: fixes sp_impl_subj_subj :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and sp_impl_subj_obj :: "partition_id_t \<Rightarrow> obj_id_t \<Rightarrow> mode_t \<Rightarrow> bool"   and current :: "thread_id_t"   and obj :: "obj_id_t \<Rightarrow> obj_t"   and thread :: "thread_id_t \<Rightarrow> thread_t"   and more :: "'a" shows "thread \<lparr>sp_impl_subj_subj = sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current, obj = obj, thread = thread, \<dots> = more\<rparr> = thread"
distinct_1: fixes x1 :: "'b"   and x2 :: "'b nmultiset multiset" shows "Elem x1 \<noteq> MSet x2"
distinct_55: fixes name1 :: "name"   and name2 :: "name"   and pi :: "pi"   and pi1' :: "pi"   and pi2' :: "pi" shows "name1<name2>.pi \<noteq> pi1' \<parallel> pi2'"
kE: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and x :: "Preliminaries.var" shows "kE \<xi> x = fromFull (tpOfV x) (\<xi> x)"
f_2: fixes x :: "'a" shows "Im (f x) = 0"
calM_def: shows "calM \<equiv> {s. s \<subseteq> carrier G \<and> card s = p ^ a}"
subgroup_generated_by_image: fixes S :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set set" assumes "S \<subseteq> carrier (relative_homology_group (int n) (lsphere n) (equator n))" shows "carrier (subgroup_generated (relative_homology_group (int n) (nsphere n) (upper n)) (hom_induced (int n) (lsphere n) (equator n) (nsphere n) (upper n) id ` S)) = hom_induced (int n) (lsphere n) (equator n) (nsphere n) (upper n) id ` carrier (subgroup_generated (relative_homology_group (int n) (lsphere n) (equator n)) S)"
Minimal_Truth_Soundness_axioms: shows "Minimal_Truth_Soundness var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num bprv isTrue"
has_contour_integral_shiftpath_D: fixes f :: "complex \<Rightarrow> complex"   and i :: "complex"   and a :: "real"   and g :: "real \<Rightarrow> complex" assumes "(f has_contour_integral i) (shiftpath a g)"   and "valid_path g"   and "pathfinish g = pathstart g"   and "a \<in> {0..1}" shows "(f has_contour_integral i) g"
eSuc_epred: fixes n :: "enat" assumes "n \<noteq> 0" shows "eSuc (epred n) = n"
comp_apply_eq: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and x :: "'c"   and h :: "'d \<Rightarrow> 'a"   and k :: "'c \<Rightarrow> 'd" assumes "f (g x) = h (k x)" shows "(f \<circ> g) x = (h \<circ> k) x"
monom_rep_X_pow: fixes a :: "nat \<Rightarrow> int"   and n :: "nat" assumes "a \<in> carrier Zp" shows "monom (UP Zp) a n = a \<odot>\<^bsub>UP Zp\<^esub> X_poly Zp [^]\<^bsub>UP Zp\<^esub> n"
img_point_elem_set: fixes x :: "'s"   and a :: "'s" assumes "\<guillemotleft>x : S.unity \<rightarrow> a\<guillemotright>" shows "S.img x \<in> S.set a"
P1: shows "p_residue (Suc (nat (ord_Zp x))) (x n) = x (Suc (nat (ord_Zp x)))"
IH: fixes Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" assumes "vc C Q" shows "\<turnstile> {pre C Q} strip C {Q}"
inverse_gf_U'_tendsto: fixes K :: "'a \<Rightarrow> 'a pmf"   and y :: "'a" assumes "MC_syntax.recurrent K y" shows "((\<lambda>x. - 1 / - MC_syntax.gf_U' K y y x) \<longlongrightarrow> enn2real (1 / MC_syntax.U' K y y)) (at_left 1)"
zip_commute: fixes xs :: "'a list"   and ys :: "'b list" shows "zip xs ys = map2 (\<lambda>x y. (y, x)) ys xs"
insert_is_Un: fixes a :: "'b"   and A :: "'b set" shows "Set.insert a A = {a} \<union> A"
cases: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'b"   and a2 :: "'b list"   and a3 :: "'b" assumes "rtrancl_path r a1 a2 a3" obtains x :: "'b" where "a1 = xa__" and "a2 = []" and "a3 = xa__"   | x :: "'b" and y :: "'b" and ys :: "'b list" and z :: "'b" where "a1 = x" and "a2 = y # ys" and "a3 = z" and "r x y" and "rtrancl_path r y ys z"
Sup_subset_mono: fixes A :: "('b \<times> 'c) set"   and B :: "('b \<times> 'c) set" assumes "A \<subseteq> B" shows "less_eq_prod (Sup_prod A) (Sup_prod B)"
sum_overlap: fixes w :: "i"   and x :: "i"   and y :: "i" shows "O w (x \<oplus> y) = (O w x \<or> O w y)"
mbind_eq_sexec': fixes f :: "'f \<Rightarrow> 'e \<Rightarrow> ('d \<times> 'e) option"   and a :: "'f"   and \<sigma> :: "'e"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> ('g \<times> 'e) option"   and P' :: "'d \<Rightarrow> 'e \<Rightarrow> ('g \<times> 'e) option" assumes "\<And>b \<sigma>''. f a \<sigma> = Some (b, \<sigma>'') \<Longrightarrow> P b \<sigma>'' = P' b \<sigma>''" shows "bind\<^sub>S\<^sub>E (f a) P \<sigma> = bind\<^sub>S\<^sub>E (f a) P' \<sigma>"
less_trans: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.preorder less_eq less"   and "less x y"   and "less y z" shows "less x z"
Dirichlet_L_nonprincipal_mangoldt_aux_bound: shows "(\<lambda>x. L * sum_upto (\<lambda>k. \<chi> k * mangoldt k / of_nat k) x - sum_upto (\<lambda>k. \<chi> k * complex_of_real (ln (real k)) / of_nat k) x) \<in> O(\<lambda>_. 1)"
split_Nil_iff: fixes xs :: "'a list"   and ys :: "'a list" shows "(splice xs ys = []) = (xs = [] \<and> ys = [])"
xn_cases: fixes xn :: "xname" shows "xn = Throwable \<or> xn = NullPointer \<or> xn = OutOfMemory \<or> xn = ClassCast \<or> xn = NegArrSize \<or> xn = IndOutBound \<or> xn = ArrStore"
permutation_delete_permutes: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat"   and i :: "nat" assumes "p permutes {0..<Suc n}"   and "i < Suc n" shows "permutation_delete p i permutes {0..<n}"
hyps_3: assumes "f \<in> S \<rightarrow> mat_kernel A" shows "finsum VK f S = finsum NC.V f S"
while_option_stop2: fixes b :: "'b \<Rightarrow> bool"   and c :: "'b \<Rightarrow> 'b"   and s :: "'b"   and t :: "'b" assumes "while_option b c s = Some t" shows "\<exists>k. t = (c ^^ k) s \<and> \<not> b t"
eval_simps'_4: fixes f :: "'a local.term" assumes "Arr f" shows "local.dom \<lbrace>f\<rbrace> = \<lbrace>Dom f\<rbrace>"
"2_2": fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and a :: "'c"   and q :: "'a \<Rightarrow>\<^sub>0 'b"   and b :: "'c" assumes "((p, i, a), q, j, b) \<in>\<^sub>p snd ` set ((c, (p__, i', a__), q__, j', b__) # ps)" shows "thesis"
\<omega>consistent_impliesStd1: assumes "\<omega>consistent" shows "\<omega>consistentStd1"
USdB_def: fixes SR :: "(('a \<times> ('b \<Rightarrow> 'c)) \<times> 'a list \<times> ('b \<Rightarrow> 'c)) set"   and DA :: "'b \<Rightarrow> 'e"   and d :: "'e" shows "Strong_Security.USdB SR DA d \<equiv> \<Union> {r. Strong_Security.Strong_d_Bisimulation SR DA d r}"
attach_id_converter: fixes res :: "('i, 'j) resource" shows "1\<^sub>C \<rhd> res = res"
hom_induced_compose': fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'b set"   and Z :: "'c topology"   and g :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and p :: "int"   and x :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set" assumes "continuous_map X Y f"   and "f ` S \<subseteq> T"   and "continuous_map Y Z g"   and "g ` T \<subseteq> U" shows "hom_induced p Y T Z U g (hom_induced p X S Y T f x) = hom_induced p X S Z U (g \<circ> f) x"
negGorbit_Cons: fixes g :: "'g"   and gs :: "'g list"   and T :: "'b \<Rightarrow> 'v"   and as :: "'b list" shows "negGorbit_list (g # gs) T as = map ((*\<cdot>) (- g) \<circ> T) as # negGorbit_list gs T as"
closure_iff_nhds_not_empty: fixes x :: "'b"   and X :: "'b set" shows "(x \<in> closure X) = (\<forall>A S. S \<subseteq> A \<longrightarrow> open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"
l3_defs_8: shows "l3_trans \<equiv> \<Union>M X Rb Ra A B K. (l3_step1 Ra A B \<union> l3_step2 Rb A B X \<union> l3_step3 Ra A B X \<union> l3_step4 Rb A B X \<union> l3_dy M \<union> l3_lkr_others A \<union> l3_lkr_after A \<union> l3_skr Ra K)\<^sup>="
prod_primeTr1: fixes n :: "nat"   and P :: "nat \<Rightarrow> 'a set"   and A :: "'a set"   and i :: "nat" assumes "\<forall>k\<le>Suc n. prime_ideal R (P k)"   and "ideal R A"   and "\<forall>l\<le>Suc n. \<not> A \<subseteq> P l"   and "\<forall>k\<le>Suc n. \<forall>l\<le>Suc n. k = l \<or> \<not> P k \<subseteq> P l"   and "i \<le> Suc n" shows "\<forall>l\<le>n. ppa R P A i l \<in> A \<and> ppa R P A i l \<in> P (skip i l) \<and> ppa R P A i l \<notin> P i"
emeasure_lub_spmf: fixes Y :: "'a spmf set"   and A :: "'a set" assumes "Complete_Partial_Order.chain (ord_spmf (=)) Y"   and "Y \<noteq> {}" shows "emeasure (measure_spmf (lub_spmf Y)) A = (\<Squnion>y\<in>Y. emeasure (measure_spmf y) A)"
l_set_attribute_get_attribute_axioms_def: fixes get_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, char list option) prog"   and set_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> char list option \<Rightarrow> ((_) heap, exception, unit) prog" shows "l_set_attribute_get_attribute_axioms get_attribute set_attribute \<equiv> \<forall>h ptr k v h'. h \<turnstile> set_attribute ptr k v \<rightarrow>\<^sub>h h' \<longrightarrow> h' \<turnstile> get_attribute ptr k \<rightarrow>\<^sub>r v"
circline_set_moebius_circline_iff: fixes M :: "moebius"   and z :: "complex_homo"   and H :: "circline" shows "(moebius_pt M z \<in> circline_set (moebius_circline M H)) = (z \<in> circline_set H)"
zcf_degree_zero: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P"   and "deg R f = 0" shows "f (deg R f) = zcf f"
has_derivative_scaleR: fixes f :: "'a \<Rightarrow> real"   and f' :: "'a \<Rightarrow> real"   and x :: "'a"   and s :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and g' :: "'a \<Rightarrow> 'b" assumes "(f has_derivative f') (at x within s)"   and "(g has_derivative g') (at x within s)" shows "((\<lambda>x. f x *\<^sub>R g x) has_derivative (\<lambda>h. f x *\<^sub>R g' h + f' h *\<^sub>R g x)) (at x within s)"
d\<mu>_def: fixes i :: "nat"   and j :: "nat" shows "fs.d\<mu> i j = int_of_rat (rat_of_int (fs'.d fs (Suc j)) * fs.gs.\<mu> i j)"
plus_zero_current: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal" shows "plus_current f (\<lambda>_. 0) = f"
division_ofD_5: fixes s :: "'a set set"   and i :: "'a set"   and K1 :: "'a set"   and K2 :: "'a set" assumes "s division_of i"   and "K1 \<in> s"   and "K2 \<in> s"   and "K1 \<noteq> K2" shows "interior K1 \<inter> interior K2 = {}"
loopfp\<tau>_def: fixes \<alpha> :: "game"   and I :: "interp"   and X :: "(variable \<Rightarrow> real) set" shows "loopfp\<tau> \<alpha> I X = lfp (\<lambda>Z. X \<union> game_sem I \<alpha> Z)"
\<sigma>_\<sigma>2: fixes v :: "'a" assumes "v \<in> V'" shows "\<sigma> v = \<sigma>2 v"
mod_ring_normalize_integer: shows "(mod_ring_rel_integer ===> mod_ring_rel_integer) (\<lambda>x. if x = 0 then 0 else 1) normalize"
distinct_5: fixes x2 :: "nat"   and x3 :: "nat" shows "Intv x2 \<noteq> Greater x3"
UN_extend_simps_4: fixes A :: "'b \<Rightarrow> 'a set"   and C :: "'b set"   and B :: "'a set" shows "\<Union> (A ` C) \<inter> B = (\<Union>x\<in>C. A x \<inter> B)"