hcju/isabelleps huggingface dataset:
sset_range: fixes s :: "'a stream" shows "sset s = range ((!!) s)"
cgenideal_eq_genideal: fixes R :: "('c, 'd) ring_scheme"   and i :: "'c" assumes "cring R"   and "i \<in> carrier R" shows "PIdl\<^bsub>R\<^esub> i = Idl\<^bsub>R\<^esub> {i}"
length_filter_mono: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and xs :: "'b list" assumes "\<And>x. P x \<Longrightarrow> Q x" shows "length (filter P xs) \<le> length (filter Q xs)"
cases: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "wave \<Gamma> f" obtains "separating \<Gamma> (TER\<^bsub>\<Gamma>\<^esub> f)" and "\<And>x. x \<notin> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) \<Longrightarrow> d_OUT f x = 0"
lemma_3_5_i_1: fixes a :: "'a"   and b :: "'a" shows "(d1 a b = (1::'a)) = (a = b)"
eq: fixes numnoabs :: "Ferrack.num \<Rightarrow> bool"   and t :: "Ferrack.num"   and bs :: "real list" assumes "numnoabs t" shows "Ifm bs (case rsplit0 t of (x, xa) \<Rightarrow> eq x xa) = Ifm bs (Eq t) \<and> isrlfm (case rsplit0 t of (x, xa) \<Rightarrow> eq x xa)"
distinct_1: fixes x1 :: "'a" shows "Fin x1 \<noteq> \<infinity>"
is_min_sig_GB_def: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set" shows "is_min_sig_GB d G = (G \<subseteq> dgrad_sig_set' (length fs) d \<and> (\<forall>u. d (pp_of_term u) \<le> dgrad_max d \<longrightarrow> component_of_term u < length fs \<longrightarrow> is_sig_GB_in d G u) \<and> (\<forall>g\<in>G. \<not> is_sig_red (\<preceq>\<^sub>t) (=) (G - {g}) g))"
half_nonnegative_int_iff: fixes k :: "int" shows "(0 \<le> k div 2) = (0 \<le> k)"
Syntax_with_Connectives_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c" shows "Syntax_with_Connectives var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi \<equiv> Generic_Syntax var trm fmla Var FvarsT substT Fvars subst \<and> Syntax_with_Connectives_axioms var trm fmla FvarsT substT Fvars subst eql cnj imp all exi"
nolb_Or: fixes f :: "atom fm"   and g :: "atom fm"   and xs :: "real list"   and l :: "real"   and x :: "real" shows "nolb (Or f g) xs l x = (nolb f xs l x \<and> nolb g xs l x)"
hom_0_iff: fixes x :: "'a"   and d :: "nat" shows "(monom x d = 0) = (x = (0::'a))"
complex_roots_2: fixes p :: "complex poly" shows "length (complex_roots_complex p) = degree p"
ex_star_eq: fixes P :: "'b star \<Rightarrow> bool" shows "(\<exists>x. P x) = (\<exists>X. P (star_n X))"
assms: shows "k \<in>\<^sub>\<circ> \<omega>"
LPDs_Test_2: fixes \<phi> :: "nat" shows "LPDs (MTestPos \<phi>) \<subseteq> {MTestPos \<phi>}"
run_state_set: fixes s' :: "'b"   and s :: "'b" shows "run_state (local.set s') s = ((), s')"
dom_simp: fixes f :: "'a \<times> 'a" assumes "VxV.arr f" shows "VxV.dom f = (local.dom (fst f), local.dom (snd f))"
assms_1: shows "\<forall>x\<ge>1. f x = dirichlet_prod' ainv g x"
v_cons: fixes \<Gamma> :: "(name \<times> ty) list"   and x :: "name"   and T :: "ty" assumes "valid \<Gamma>"   and "x \<sharp> \<Gamma>" shows "valid ((x, T) # \<Gamma>)"
set_quicksort_by_rel: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sl :: "'b list"   and xs :: "'b list" shows "set (quicksort_by_rel R sl xs) = set (xs @ sl)"
diff_less_self: fixes i :: "nat"   and j :: "nat" assumes "0 < i"   and "0 < j" shows "i - j < i"
simps: fixes x :: "'a"   and y :: "'b Filters.filter" shows "rho_pair (x, y) = (SOME z. up_filter z = phi x \<sqinter> y)"
H2_def: shows "H2 = addcol (- k) 2 0 H"
loop: fixes vs :: "e list"   and n :: "nat"   and t1s :: "t list"   and t2s :: "t list"   and m :: "nat"   and es :: "b_e list" assumes "const_list vs"   and "length vs = n"   and "length t1s = n"   and "length t2s = m" shows "\<lparr>vs @ [$Loop (t1s _> t2s) es]\<rparr> \<leadsto> \<lparr>[Label n [$Loop (t1s _> t2s) es] (vs @ ($* es))]\<rparr>"
classical_logic_is_not_usable: fixes p :: "fm" shows "valid_boole (Imp (if [Con' (Con' (Pro ''p'') (Pro ''q'')) (Neg' (Pro ''r'')), Imp' (Con' (Pro ''p'') (Pro ''q'')) (Pro ''r'')] = [] then Truth else fold Con' (butlast [Con' (Con' (Pro ''p'') (Pro ''q'')) (Neg' (Pro ''r'')), Imp' (Con' (Pro ''p'') (Pro ''q'')) (Pro ''r'')]) (last [Con' (Con' (Pro ''p'') (Pro ''q'')) (Neg' (Pro ''r'')), Imp' (Con' (Pro ''p'') (Pro ''q'')) (Pro ''r'')])) p)"
simps_8: fixes I :: "('b, 'c, 'a) interp"   and ODE :: "('b, 'a) ODE"   and \<phi> :: "('b, 'c, 'a) formula" shows "prog_sem I (EvolveODE ODE \<phi>) = {(\<nu>, mk_v I ODE \<nu> (sol t)) |\<nu> sol t. 0 \<le> t \<and> (sol solves_ode (\<lambda>_. ODE_sem I ODE)) {0..t} {x. mk_v I ODE \<nu> x \<in> fml_sem I \<phi>} \<and> sol 0 = fst \<nu>}"
cases: fixes a :: "'c llist" assumes "lfinite a" obtains "a = LNil"   | xs :: "'c llist" and x :: "'c" where "a = LCons x xs" and "lfinite xs"
Blinfun_inverse: fixes y :: "'a \<Rightarrow> 'b" assumes "y \<in> {f. bounded_linear f}" shows "(*\<^sub>v) (Blinfun y) = y"
Units_r_inv_ex: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.monoid G"   and "x \<in> Units G" shows "\<exists>y\<in>carrier G. x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub>"
4: shows "Left.in_hom (Left.L \<mu>) (trg \<mu> \<star> local.dom \<mu>) (trg \<mu> \<star> cod \<mu>)"
20: shows "f \<squnion> f\<^sup>T \<le> (w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T) \<sqinter> - ((w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T) \<sqinter> - forest_components f * selected_edge h j g * top \<sqinter> top * selected_edge h j g\<^sup>T * forest_components f) \<sqinter> - ((w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T) \<sqinter> - forest_components f * selected_edge h j g * top \<sqinter> top * selected_edge h j g\<^sup>T * forest_components f)\<^sup>T \<squnion> (w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T)\<^sup>T \<sqinter> - ((w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T) \<sqinter> - forest_components f * selected_edge h j g * top \<sqinter> top * selected_edge h j g\<^sup>T * forest_components f) \<sqinter> - ((w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T) \<sqinter> - forest_components f * selected_edge h j g * top \<sqinter> top * selected_edge h j g\<^sup>T * forest_components f)\<^sup>T"
lex: shows "snd (lex_ext_unbounded kbo (ss1 @ s # ss2) (ss1 @ t # ss2))"
assms_5: shows "g \<in> borel_measurable (nat_filtration n)"
heap_is_wellformed_def: fixes 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"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name known_ptr type_wf heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs" shows "heap_is_wellformed = (\<lambda>h. heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h \<and> acyclic (parent_child_rel h \<union> l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_host_shadow_root_rel get_shadow_root h) \<and> l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_all_ptrs_in_heap get_shadow_root h \<and> l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_distinct_lists get_shadow_root h \<and> l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_shadow_root_valid get_shadow_root get_tag_name h)"
CatIdInMor: fixes X :: "'a" assumes "X \<in> obj" shows "id X \<in> mor"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>\<^sub>D) (\<cdot>\<^sub>D) (G \<circ> F) D.map \<eta>_inv.map"
disjoint_submaps_exist: fixes h :: "'a" shows "\<exists>h\<^sub>0 h\<^sub>1. h\<^sub>0 \<oplus> h\<^sub>1 = Some h"
Hilbert_fun_X: fixes z :: "nat" assumes "poly_deg f \<le> z" shows "Hilbert_fun P[X] z = z - poly_deg f + (card X - 1) choose (card X - 1) + Hilbert_fun P z + Hilbert_fun N z"
gen_dfs_var_def: fixes U :: "'b set" shows "gen_dfs_var U \<equiv> finite_psupset U"
finite_bit_word: fixes w :: "'b word" shows "finite {n. bit w n}"
rbt_lookup_keys: fixes t :: "('a, 'c) rbt" assumes "rbt_sorted t" shows "dom (rbt_lookup t) = set (keys t)"
\<iota>_FL_def: shows "\<iota>_FL = Infer (zip (prems_of \<iota>) Ll) (concl_of \<iota>, L0)"
standard_open_is_zariski_open: fixes x :: "('a \<times> 'a) set" assumes "x \<in> local.carrier_quotient_ring" shows "local.local.is_zariski_open (local.local.standard_open x)"
algebraic_inverse: fixes x :: "'b" assumes "algebraic x" shows "algebraic (inverse x)"
fminus_filter_singleton: fixes l :: "'a list"   and x :: "'a" shows "fset_of_list l |-| {|x|} = fset_of_list (filter (\<lambda>e. e \<noteq> x) l)"
Abs: fixes x :: "name"   and T :: "ty"   and \<Gamma> :: "(name \<times> ty) list"   and t :: "trm"   and U :: "ty" assumes "(x, T) # \<Gamma> \<turnstile> t : U" shows "\<Gamma> \<turnstile> (\<lambda>x:T. t) : T \<rightarrow> U"
permute_cfun_def: fixes p :: "perm"   and f :: "'c \<rightarrow> 'd" shows "p \<bullet> f = (\<Lambda> x. p \<bullet> f\<cdot>(- p \<bullet> x))"
le_trans_equiv_rsp: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and u :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and v :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "equiv_trans t u" shows "le_trans t v = le_trans u v"
C1: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi" assumes "(\<Psi>, P, Q) \<in> Rel"   and "\<Psi> \<rhd> Q \<sim> R" shows "(\<Psi>, P, R) \<in> Rel"
FullSpec_impl_ISpec: shows "\<turnstile> FullSpec \<longrightarrow> ISpec inp qc out"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs"   and "l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs" shows "l_set_disconnected_nodes_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs get_tag_name get_tag_name_locs"
f'_def: shows "f' = (\<lambda>x. - s * x powr (- s - 1))"
smooth_compat_commute: fixes k :: "enat"   and x :: "('c, 'd) chart"   and y :: "('c, 'd) chart" shows "k-smooth_compat x y = k-smooth_compat y x"
at_least_two_cars_exists: shows "\<exists>c d. c \<noteq> d"
sccs_eq: shows "G.sccs_verts = (`) proj_verts_H ` H.sccs_verts"
maxcoeff_pos: fixes t :: "Ferrack.num" shows "0 \<le> maxcoeff t"
Abs_multiset_inverse: fixes y :: "'k \<Rightarrow> nat" assumes "y \<in> multiset" shows "count (Abs_multiset y) = y"
ap_stream_id: fixes x :: "'d stream" shows "pure (\<lambda>x. x) \<diamondop> x = x"
lookup_map_default: fixes x :: "'d"   and d :: "'c"   and f :: "'c \<Rightarrow> 'c"   and m :: "('d, 'c) mapping" shows "Mapping.lookup (Mapping.map_default x d f m) x = Some (f (Mapping.lookup_default d m x))"
sup_bot_right: fixes sup :: "'x \<Rightarrow> 'x \<Rightarrow> 'x"   and less_eq :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and less :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and bot :: "'x"   and x :: "'x" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "sup x bot = x"
Subtype: fixes U :: "Javatype"   and T :: "Javatype" assumes "U \<le> T" shows "treach T U"
paste_disj_domains: fixes P :: "('a \<times> 'b) set"   and Q :: "('a \<times> 'b) set" assumes "Domain P \<inter> Domain Q = {}" shows "P +* Q = P \<union> Q"
hd_def: fixes list :: "'d list" shows "hd list = (case list of x21 # x22 \<Rightarrow> x21)"
image_nonempty: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and A1 :: "'c set" assumes "f \<in> A \<rightarrow> B"   and "A1 \<subseteq> A"   and "A1 \<noteq> {}" shows "f ` A1 \<noteq> {}"
greaterThanAtMost_subseteq_atLeastAtMost_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.dense_linorder less_eq less" shows "(ord.greaterThanAtMost less_eq less a b \<subseteq> ord.atLeastAtMost less_eq c d) = (less a b \<longrightarrow> less_eq c a \<and> less_eq b d)"
prod_simps_3: fixes f0 :: "'a"   and f1 :: "'a" assumes "arr f0"   and "arr f1" shows "cod (ECC.prod f0 f1) = ECC.prod (cod f0) (cod f1)"
diff_add_zero: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and a :: "'k"   and b :: "'k" assumes "class.comm_monoid_diff plus minus zero" shows "minus a (plus a b) = zero"
errMOD_igVarIPresIGWlsSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model" assumes "igVarIPresIGWls MOD" shows "igVarIPresIGWls (errMOD MOD)"
simps: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM"   and xs :: "('d \<times> 'e) list" shows "sequence_to_failure M1 M2 xs = (butlast xs \<in> L M2 \<inter> L M1 \<and> xs \<in> L M1 - L M2)"
eq_commute: fixes a :: "'g"   and b :: "'g" shows "(a = b) = (b = a)"
semiring_1_cancel_ow_def: fixes U :: "'c set"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "semiring_1_cancel_ow U plus minus zero one times \<equiv> (semiring_ow U plus times \<and> cancel_comm_monoid_add_ow U plus minus zero) \<and> zero_neq_one_ow U one zero \<and> monoid_mult_ow U one times"
rbt_insertwk_is_rbt: fixes t :: "('a, 'c) rbt"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'a"   and x :: "'c" assumes "is_rbt t" shows "is_rbt (rbt_insert_with_key f k x t)"
cl_ident: fixes r :: "'b set"   and L :: "'b set set" assumes "r \<in> L" shows "cl L r = r"
comp_lassocr\<^sub>C: fixes conv1 :: "('h, 'k, 'n, 'q) converter"   and conv2 :: "('i, 'l, 'o, 'r) converter"   and conv3 :: "('j, 'm, 'p, 's) converter" shows "((conv1 |\<^sub>= conv2) |\<^sub>= conv3) \<odot> lassocr\<^sub>C = lassocr\<^sub>C \<odot> (conv1 |\<^sub>= conv2 |\<^sub>= conv3)"
asym_cases: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> y" obtains "x \<sqsubset> y"   | "x \<sim> y"
lfp_lowerbound: fixes a :: "'b acom"   and c :: "com"   and f :: "'b acom \<Rightarrow> 'b acom" assumes "a \<in> {C. strip C = c}"   and "f a \<le> a" shows "Collecting.lfp c f \<le> a"
IH: fixes s :: "'s"   and gpv :: "('c, 'a, 'b) gpv" assumes "I s"   and "\<I> \<turnstile>g gpv \<surd>" shows "set_spmf (exec_gpv_ gpv s) \<subseteq> {a. case a of (x, a) \<Rightarrow> I a}"
euclidean_size_lcm_less1: fixes b :: "'a"   and a :: "'a" assumes "b \<noteq> (0::'a)"   and "\<not> b dvd a" shows "euclidean_size a < euclidean_size (lcm a b)"
better_CutR_intro: fixes N :: "trm"   and N' :: "trm"   and a :: "coname"   and M :: "trm"   and x :: "name" assumes "N \<longrightarrow>\<^sub>a N'" shows "Cut <a>.M x.N \<longrightarrow>\<^sub>a Cut <a>.M x.N'"
start_points_path_aux: fixes r :: "'a"   and x :: "'a" assumes "backward_finite_path_root r x"   and "start_points x \<noteq> 0" shows "x ; r = 0"
update_im1_def: fixes ba :: "'a list \<times> 'a list"   and x :: "'a" shows "update_im1 ba x = (x # tl (fst ba), snd ba)"
induct: fixes P :: "'c set \<Rightarrow> ('d, 'c) stateful_strand_step list \<Rightarrow> bool"   and a0 :: "'c set"   and a1 :: "('d, 'c) stateful_strand_step list" assumes "\<And>V. P V []"   and "\<And>V t S. P V S \<Longrightarrow> P V (receive\<langle>t\<rangle> # S)"   and "\<And>V t S. P (V \<union> fv t) S \<Longrightarrow> P V (send\<langle>t\<rangle> # S)"   and "\<And>V t t' S. P (V \<union> fv t) S \<Longrightarrow> P V (\<langle>t := t'\<rangle> # S)"   and "\<And>V uu_ uv_ S. P V S \<Longrightarrow> P V (\<langle>uu_ == uv_\<rangle> # S)"   and "\<And>V t s S. P V S \<Longrightarrow> P V (insert\<langle>t,s\<rangle> # S)"   and "\<And>V uw_ ux_ S. P V S \<Longrightarrow> P V (delete\<langle>uw_,ux_\<rangle> # S)"   and "\<And>V t s S. P (V \<union> fv t \<union> fv s) S \<Longrightarrow> P V (select\<langle>t,s\<rangle> # S)"   and "\<And>V uy_ uz_ S. P V S \<Longrightarrow> P V (\<langle>uy_ in uz_\<rangle> # S)"   and "\<And>V va_ vb_ vc_ S. P V S \<Longrightarrow> P V (\<forall>va_\<langle>\<or>\<noteq>: vb_ \<or>\<notin>: vc_\<rangle> # S)" shows "P a0 a1"
gbinomial_absorption': fixes k :: "nat"   and a :: "'a" assumes "0 < k" shows "a gchoose k = a / of_nat k * (a - (1::'a) gchoose (k - 1))"
ordIso_refl_on: shows "refl_on {r. Well_order r} ordIso"
eo: shows "q = a \<or> q \<in> set qs"
fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_subst_subset: fixes F :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('c, 'a) Term.term" assumes "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>) \<subseteq> subst_domain \<sigma>" shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<subseteq> subst_domain \<sigma> \<union> subst_domain \<delta>"
try_bind_assert_gpv: fixes b :: "bool"   and f :: "unit \<Rightarrow> ('a, 'b, 'c) gpv"   and gpv :: "('a, 'b, 'c) gpv" shows "TRY assert_gpv b \<bind> f ELSE gpv = (if b then TRY f () ELSE gpv else gpv)"
foreach_impl_correct_1: fixes Qi :: "'q"   and \<pi>i :: "'m"   and u :: "'v"   and Qi' :: "'q"   and \<pi>i' :: "'m" assumes "foreach_impl Qi \<pi>i u (G_adj g u) = (Qi', \<pi>i')"   and "prim_invar_impl Qi \<pi>i" shows "Q_invar Qi'"
extf_singleton: fixes y :: "('s, 'v) tm"   and x :: "('s, 'v) tm"   and f :: "'s"   and gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool" assumes "y \<noteq> x" shows "extf f gt [y] [x] = gt y x"
countable_PiE: fixes I :: "'d set"   and F :: "'d \<Rightarrow> 'e set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> countable (F i)" shows "countable (Pi\<^sub>E I F)"
ext_hd_or_tl_axioms: shows "ext_hd_or_tl len_lexext"
R_Int: fixes a :: "(int, 'n) vec"   and b :: "(int, 'n) vec" assumes "a \<noteq> b" shows "R a \<inter> R b \<in> null_sets lebesgue"
inverse_distributive_1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> R"   and "b \<in> R"   and "c \<in> R" shows "a \<cdot> (b - c) = a \<cdot> b - a \<cdot> c"
remdups_adj_map_injective: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list" assumes "inj f" shows "remdups_adj (map f xs) = map f (remdups_adj xs)"
sorted_wrt_take: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and n :: "nat" assumes "sorted_wrt f xs" shows "sorted_wrt f (take n xs)"
chain_nontrivE: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and xs :: "'b llist" assumes "chain R (LCons x (LCons y xs))" obtains "chain R (LCons y xs)" and "R x y"
elims_3: fixes x :: "'b formula" assumes "\<not> is_disj x" obtains F :: "'b formula" and G :: "'b formula" where "x = F \<^bold>\<or> G" and "\<not> (is_lit_plus F \<and> is_disj G)"   | v :: "'b" where "x = Atom v__" and "\<not> is_lit_plus (Atom v__)"   | "x = \<bottom>" and "\<not> is_lit_plus \<bottom>"   | v :: "'b formula" where "x = \<^bold>\<not> va__" and "\<not> is_lit_plus (\<^bold>\<not> va__)"   | v :: "'b formula" and va :: "'b formula" where "x = vb__ \<^bold>\<and> vaa__" and "\<not> is_lit_plus (vb__ \<^bold>\<and> vaa__)"   | v :: "'b formula" and va :: "'b formula" where "x = v \<^bold>\<rightarrow> va" and "\<not> is_lit_plus (v \<^bold>\<rightarrow> va)"
get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'o, 'r option) RElement_scheme \<Rightarrow> 'u" assumes "l_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas type_wf"   and "type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter"
smult_map_poly: fixes a :: "'a" shows "Polynomial.smult a = map_poly ((*) a)"
var_rename_fv_disjoint: fixes s :: "('b, 'a \<times> nat) Term.term"   and t :: "('c, 'a \<times> nat) Term.term" shows "fv s \<inter> fv (t \<cdot> var_rename (max_var_set (fv s))) = {}"
sum_impl_def: fixes g :: "'b \<Rightarrow> 'a nres"   and S :: "'b set" shows "sum_impl g S \<equiv> foreach S (\<lambda>x a. g x \<bind> (\<lambda>b. return (a + b))) (0::'a)"
extensional_def: fixes A :: "'a set" shows "extensional A = {f. \<forall>x. x \<notin> A \<longrightarrow> f x = undefined}"
ncols_def: fixes A :: "(('a, 'b) vec, 'c) vec" shows "ncols A = CARD('b)"