hcju/isabelleps huggingface dataset:
le_Inf_iff: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and b :: "'d"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq b (Inf A) = (\<forall>a\<in>A. less_eq b a)"
v1: shows "wfactors G (take i as' @ drop (Suc i) as') (aa_1 \<otimes> aa_2)"
q11_infinite_def: shows "q11 = (\<lambda>k. k\<^sup>2 mod 11) ` {0..}"
le1234_lt__lt: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B Le C D"   and "C D Lt E F" shows "A B Lt E F"
invariantD: fixes ds :: "'b set"   and P :: "'b set \<Rightarrow> 'a set \<Rightarrow> bool" assumes "invariant ds P" shows "P ds (fp_cop_F ds)"
set_case_simps_2: fixes x :: "'b"   and c_empty :: "'a"   and c_sing :: "'b \<Rightarrow> 'a"   and c_else :: "'a" shows "set_case {x} c_empty c_sing c_else = c_sing x"
graph_homo: fixes G :: "('e, 'f) labeled_graph"   and f :: "'f \<Rightarrow> 'g" assumes "graph G" shows "graph_homomorphism G (map_graph_fn G f) (on_graph G f)"
sw_inj: fixes xs :: "'b"   and x :: "'a"   and y :: "'a"   and zs :: "'b"   and z :: "'a"   and z' :: "'a" shows "(z @zs[x \<and> y]_xs = z' @zs[x \<and> y]_xs) = (z = z')"
list_update_set_def: fixes S :: "nat \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and xs :: "'b list" shows "list_update_set S f xs = list (\<lambda>i. if S i then f i else xs ! i) (length xs)"
ang_vec_a_opposite1: fixes z1 :: "complex"   and z2 :: "complex" shows "\<angle>a z1 z2 = \<angle>a (- z1) z2"
J: shows "Says n' run' 5 (User n') (Card n') (Crypt (sesK (c * f)) (Passwd n')) \<in> set evsFR5"
diagonalize_is_functor: shows "functor (\<cdot>) (\<cdot>\<^sub>S) \<F>C.D"
replicate_dotprod: fixes y :: "nat list"   and n :: "nat"   and x :: "nat" assumes "length y = n" shows "replicate n x \<bullet> y = x * sum_list y"
f9: shows "\<forall>x0 x1. (x0 (xx x0 x1) \<le> (if nofail (x0 (xx x0 x1)) then M (xx x0 x1) else FAIL)) = (if nofail (x0 (xx x0 x1)) then x0 (xx x0 x1) \<le> M (xx x0 x1) else x0 (xx x0 x1) \<le> FAIL)"
lens_insert_def: fixes x :: "'b \<Longrightarrow> 'a"   and A :: "'a scene" shows "insert\<^sub>S x A = (if \<lbrakk>x\<rbrakk>\<^sub>\<sim> \<subseteq>\<^sub>S A then \<lbrakk>x\<rbrakk>\<^sub>\<sim> \<squnion>\<^sub>S A else A)"
d2: fixes x' :: "real" assumes "x' \<in> {0..1}"   and "dist x' (1 / 2) < d2" shows "dist ((g1 +++ g2) x') (g1 1) < e / 2"
update_simps_4: fixes x :: "'h"   and opt :: "('f, 'g, 'h) env option"   and b :: "'g"   and es :: "'h \<Rightarrow> ('f, 'g, 'h) env option" shows "update [x] opt (Env b es) = Env b (es(x := opt))"
parent_child_rel_child: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and child :: "(_) node_ptr" assumes "h \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children" shows "(child \<in> set children) = ((ptr, cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child) \<in> Shadow_DOM.parent_child_rel h)"
functor_axioms: shows "functor (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) (\<lambda>f2. F (A1.cod f1, f2))"
TAbs': fixes T :: "POPLmarkRecord.type"   and t :: "trm" assumes "x = (\<lambda><:T. t)" shows "thesis"
ih_1: fixes p :: "nat" assumes "n \<noteq> 0"   and "e < p" shows "local.encode 0 e < local.encode 0 p"
typeII_cube_line_integral_exists_boundary: shows "\<forall>(k, \<gamma>)\<in>two_chain_boundary two_chain. line_integral_exists F {j} \<gamma>"
path_connected_def: fixes S :: "'c set" shows "path_connected S = (\<forall>x\<in>S. \<forall>y\<in>S. \<exists>g. path g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y)"
not_sub: fixes A :: "'e set"   and B :: "'e set" assumes "\<not> A \<subseteq> B" shows "\<exists>a. a \<in> A \<and> a \<notin> B"
mask_bl: fixes n :: "nat" shows "mask n = of_bl (replicate n True)"
all_envirE: fixes p :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "all_envir p" obtains "p x"
simps_1: shows "rbt_has_next ([], rbt.Empty) = False"
star_plus_one: fixes x :: "'b" shows "(1::'b) + x\<^sup>\<star> = x\<^sup>\<star>"
preservation2: fixes y :: "'a"   and x :: "'a" assumes "y \<cdot> x \<le> y" shows "(x \<cdot> y)\<^sup>\<dagger> \<cdot> x \<le> x \<cdot> y\<^sup>\<dagger>"
lookup_default_update: fixes d :: "'b"   and k :: "'c"   and v :: "'b"   and m :: "('c, 'b) mapping" shows "lookup_default d (update k v m) k = v"
eSuc_ne_0: fixes n :: "enat" shows "eSuc n \<noteq> 0"
csrp_fn_mem: fixes R :: "('e, 'f) Ring_scheme"   and I :: "'e set"   and x :: "'e set" assumes "Ring R"   and "ideal R I"   and "x \<in> carrier (R /\<^sub>r I)" shows "csrp_fn R I x \<in> carrier R"
combine_per_def: fixes P :: "'b \<Rightarrow> bool"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "combine_per P R = (\<lambda>x y. P x \<and> P y) \<sqinter> R"
scale_cancel_right: fixes a :: "'a"   and x :: "'a"   and b :: "'a" shows "(a * x = b * x) = (a = b \<or> x = (0::'a))"
lemma2_6_5_a: fixes r :: "('a \<times> 'a) set"   and S :: "'a set"   and M :: "'a multiset"   and N :: "'a multiset" assumes "trans r"   and "r \<down>s S \<subseteq> S"   and "(M, N) \<in> mul_eq r" shows "(M -s S, N -s S) \<in> mul_eq r"
paval_paval': fixes a :: "aexp"   and ps :: "char list \<Rightarrow> int option"   and v :: "int" assumes "paval' a ps = Some v" shows "paval a ps = v"
zero_hdiff: fixes x :: "hf" shows "0 - x = 0"
expandExecutionStep: fixes exec' :: "('p, 'v, 's) configuration list"   and msg :: "('p, 'v) message"   and cMsg :: "('p, 'v, 's) configuration"   and trace' :: "('p, 'v) message list" assumes "last exec' \<turnstile> msg \<mapsto> cMsg"   and "execution trans sends start exec' trace'" shows "\<exists>exec'' trace''. execution trans sends start exec'' trace'' \<and> prefixList exec' exec'' \<and> prefixList trace' trace'' \<and> last exec'' = cMsg \<and> last trace'' = msg"
foundation9': fixes \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> not x" shows "\<tau> |\<noteq> x"
right_diff_distrib: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.ring plus zero minus uminus times" shows "times a (minus b c) = minus (times a b) (times a c)"
mult2_alt_addL: fixes x :: "'b"   and X :: "'b multiset"   and Y :: "'b multiset"   and b :: "bool"   and ns :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" shows "((add_mset x X, Y) \<in> mult2_alt b ns s) = ((\<exists>y. y \<in># Y \<and> (x, y) \<in> s \<and> ({#x \<in># X. (x, y) \<notin> s#}, Y - {#y#}) \<in> mult2_alt_ns ns s) \<or> (\<exists>y. y \<in># Y \<and> (x, y) \<in> ns \<and> (x, y) \<notin> s \<and> (X, Y - {#y#}) \<in> mult2_alt b ns s))"
D_1: fixes m :: "nat"   and n :: "nat" shows "compact (D m n)"
mpdiv_vals_nonequiv: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set" assumes "distinct_pds K n P" shows "vals_nonequiv K n (\<lambda>j. \<nu>\<^bsub>K P j\<^esub>)"
mn: shows "n \<le> 2 \<or> m \<le> 2"
rtrancl_fin: fixes a :: "'a" shows "finite {b. (a, b) \<in> R\<^sup>*}"
is_heap_merge: fixes t1 :: "('c, 'd) bintree"   and t2 :: "('c, 'd) bintree" assumes "is_heap_list (priority t1) (children t1)"   and "is_heap_list (priority t2) (children t2)" shows "is_heap_list (priority (merge t1 t2)) (children (merge t1 t2))"
is_subseqI: fixes f :: "nat \<Rightarrow> nat"   and s' :: "nat \<Rightarrow> 'a"   and s :: "nat \<Rightarrow> 'a" assumes "is_increasing f"   and "s' = take_subseq s f" shows "is_subseq_of s s'"
derived_set_of_Union: fixes \<F> :: "'c set set"   and X :: "'c topology" assumes "finite \<F>" shows "X derived_set_of \<Union> \<F> = \<Union> ((derived_set_of) X ` \<F>)"
summable_erf: fixes z :: "'b" shows "summable (\<lambda>n. of_real (erf_coeffs n) * z ^ n)"
SNa_Redu_def: shows "SNa_Redu \<equiv> A_Redu_set \<inter> UNIV \<times> SNa_set"
coeff_mult_degree_sum: fixes p :: "'a poly"   and q :: "'a poly" shows "coeff (p * q) (degree p + degree q) = lead_coeff p * lead_coeff q"
R: shows "(succi, E) \<in> \<langle>Rv\<rangle>succg_rel"
conjugate_unique_p: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and h :: "'a \<Rightarrow> 'a" assumes "bounded_lattice.conjugate (\<squnion>) top f g"   and "bounded_lattice.conjugate (\<squnion>) top f h" shows "uminus \<circ> g = uminus \<circ> h"
v11_vrange_ex1_eq: fixes b :: "V" shows "(b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vrat_inverse) = (\<exists>!a. a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vrat_inverse \<and> a\<inverse>\<^sub>\<rat> = b)"
size_multiset_eq_0_iff_empty: fixes f :: "'k \<Rightarrow> nat"   and M :: "'k multiset" shows "(size_multiset f M = 0) = (M = {#})"
reads_obsp: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat"   and \<sigma>' :: "'var \<Rightarrow> 'val"   and k' :: "nat" assumes "path \<sigma> k = path \<sigma>' k'"   and "\<sigma>\<^bsup>k\<^esup> \<restriction> local.reads (path \<sigma> k) = \<sigma>'\<^bsup>k'\<^esup> \<restriction> local.reads (path \<sigma> k)" shows "obsp \<sigma> k = obsp \<sigma>' k'"
simps_4: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and x :: "Vars.var"   and S :: "stack_elem list" shows "ArityAnalysis.AEstack Aexp as (Arg x # S) = esing x\<cdot>(up\<cdot>0) \<squnion> ArityAnalysis.AEstack Aexp as S"
pmaxsimpchain_map: fixes xs :: "'a set list" assumes "domain.pgallery xs" shows "SimplicialComplex.pmaxsimpchain (f \<turnstile> X) (f \<Turnstile> xs)"
refl_onD2: fixes A :: "'d set"   and r :: "('d \<times> 'd) set"   and x :: "'d"   and y :: "'d" assumes "refl_on A r"   and "(x, y) \<in> r" shows "y \<in> A"
unitfactor_ee: fixes u :: "'a"   and as :: "'a list" assumes "u \<in> Units G"   and "set as \<subseteq> carrier G" shows "essentially_equal G (as[0 := as ! 0 \<otimes> u]) as"
next_present: fixes r :: "'a ref"   and h' :: "heap"   and x :: "'a"   and h :: "heap" assumes "(r, h') = Ref.alloc x h" shows "Ref.present h' r"
redT_normal: fixes t :: "'thread_id"   and x :: "'x"   and s :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and ta :: "('l, 'thread_id, 'x, 'm, 'w, ('addr, 'thread_id) obs_event action) thread_action"   and x' :: "'x"   and m' :: "'m"   and s' :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set" assumes "t \<turnstile> \<langle>x, shr s\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "actions_ok s t ta"   and "redT_upd s t ta x' m' s'" shows "s -t\<triangleright>ta\<rightarrow> s'"
greedy_path_ltl: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and \<sigma>' :: "'c \<Rightarrow> 'c"   and v0 :: "'c"   and v :: "'c"   and P :: "'c llist" assumes "ParityGame G"   and "ParityGame.greedy_conforming_path G p \<sigma> \<sigma>' v0 = LCons v P" shows "P = LNil \<or> P = ParityGame.greedy_conforming_path G p \<sigma> \<sigma>' (\<sigma> v0) \<or> P = ParityGame.greedy_conforming_path G p \<sigma> \<sigma>' (\<sigma>' v0)"
lcm_right_idem: fixes a :: "'a"   and b :: "'a" shows "lcm (lcm a b) b = lcm a b"
path_component_of_sym: fixes X :: "'c topology"   and x :: "'c"   and y :: "'c" assumes "path_component_of X x y" shows "path_component_of X y x"
eq_iff: fixes x :: "'c poly"   and y :: "'c poly" shows "(map_poly to_fract x = map_poly to_fract y) = (x = y)"
simps_4: fixes \<phi> :: "atom fm"   and \<psi> :: "atom fm" shows "clearQuantifiers (And \<phi> \<psi>) = and (clearQuantifiers \<phi>) (clearQuantifiers \<psi>)"
lprefix_conv_lappend: fixes xs :: "'a llist"   and ys :: "'a llist" shows "lprefix xs ys = (\<exists>zs. ys = xs @\<^sub>l zs)"
1: shows "rat_poly_inv (rat_poly_times A (rat_poly_times B A)) = rat_poly_inv A"
Limsup_add_ereal_right: fixes F :: "'a filter"   and c :: "ereal"   and g :: "'a \<Rightarrow> ereal" assumes "F \<noteq> bot"   and "\<bar>c\<bar> \<noteq> \<infinity>" shows "Limsup F (\<lambda>n. g n + c) = Limsup F g + c"
\<eta>_leg_simps_3: shows "\<eta>.cod.leg0 = gf.leg0"
sup_compl_top: fixes x :: "'a" shows "sup_class.sup x (- x) = top_class.top"
select_convs_4: fixes nothing :: "unit"   and nothinga :: "unit"   and doctype :: "char list"   and document_element :: "(_) element_ptr option"   and disconnected_nodes :: "(_) node_ptr list"   and more :: "'p" shows "disconnected_nodes \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype, document_element = document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr> = disconnected_nodes"
infnorm_mul: fixes a :: "real"   and x :: "'b" shows "infnorm (a *\<^sub>R x) = \<bar>a\<bar> * infnorm x"
equivalent_def: fixes atom_ordering :: "('a \<times> 'a) set"   and S1 :: "'a Literal set set"   and S2 :: "'a Literal set set" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.equivalent S1 S2 = (propositional_atoms.entails_formula S1 S2 \<and> propositional_atoms.entails_formula S2 S1)"
Sum1: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs"   and Q :: "ccs" assumes "P \<longmapsto>\<alpha> \<prec> P'" shows "P \<oplus> Q \<longmapsto>\<alpha> \<prec> P'"
iso_cmp: fixes \<mu>\<nu> :: "'a \<times> 'a" assumes "S.VV.ide \<mu>\<nu>" shows "B.iso (cmp \<mu>\<nu>)"
vimage_vsingleton_iff: fixes b :: "V"   and r :: "V"   and a :: "V" shows "(b \<in>\<^sub>\<circ> r `\<^sub>\<circ> set {a}) = (\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r)"
serverK_not_spied: fixes A :: "agent"   and B :: "agent"   and PMS :: "nat"   and evs :: "event list"   and Na :: "nat"   and Nb :: "nat"   and NA :: "nat"   and NB :: "nat" assumes "Notes A \<lbrace>Agent B, Nonce PMS\<rbrace> \<in> set evs"   and "Says B Spy (Key (serverK (Na, Nb, PRF (PMS, NA, NB)))) \<notin> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> tls" shows "Key (serverK (Na, Nb, PRF (PMS, NA, NB))) \<notin> parts (knows Spy evs)"
fls_base_factor_mult_base_factor_2: fixes f :: "'a fls"   and g :: "'a fls" shows "fls_base_factor (fls_base_factor f * g) = fls_base_factor (f * g)"
emptyNotInRange: fixes a :: "('a \<times> 'b set) set"   and G :: "'b set"   and N :: "'a set" assumes "a \<in> allAllocations N G" shows "{} \<notin> Range a"
region_integral_is_one_twoD_integral: shows "LBINT x. LBINT y. f (x, y) = integral\<^sup>L (lborel \<Otimes>\<^sub>M lborel) f"
reduced_word_for_eq_length: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and A :: "'b set"   and a :: "'b"   and as :: "'b list"   and bs :: "'b list" assumes "class.monoid_add plus zero"   and "monoid_add.reduced_word_for plus zero A a as"   and "monoid_add.reduced_word_for plus zero A a bs" shows "length as = length bs"
P_t_app: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and t :: "'c"   and u\<^sub>1 :: "'d"   and u\<^sub>2 :: "'d" assumes "term_struct_rel P"   and "P t (app u\<^sub>1 u\<^sub>2)" shows "\<exists>t\<^sub>1 t\<^sub>2. t = app t\<^sub>1 t\<^sub>2 \<and> P t\<^sub>1 u\<^sub>1 \<and> P t\<^sub>2 u\<^sub>2"
cross7_zero_right: fixes x :: "(real, 7) vec" shows "x \<times>\<^sub>7 0 = 0"
simps: fixes uu :: "'a"   and uv :: "'b"   and q :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" shows "BsSkewBinomialHeapStruc.level (BsSkewBinomialHeapStruc.Element uu uv q) = Suc (BsSkewBinomialHeapStruc.queue_level q)"
IH: assumes "bvars\<^sub>s\<^sub>s\<^sub>t S \<inter> subst_domain \<theta> = {}" shows "trms\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>) = trms\<^sub>s\<^sub>s\<^sub>t S \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>"
lsu_range_fix: shows "range \<nu>\<^sup>\<natural> = Fix \<nu>\<^sup>\<natural>"
simps_1: fixes xs :: "('a, 'b) Node list"   and t :: "('a, 'b) FingerTreeStruc" shows "FingerTreeStruc.app3 Empty xs t = FingerTreeStruc.lconsNlist xs t"
binomial_maximum': fixes n :: "nat"   and k :: "nat" shows "2 * n choose k \<le> 2 * n choose n"
2: fixes l :: "'lbl"   and A :: "('lbl strand_label \<times> ('fun, ('fun, 'atom) Term.term \<times> nat) stateful_strand_step) list"   and l' :: "'lbl" assumes "l \<notin> set (L A)" shows "GSMP (N2 (proj_unl l A)) \<subseteq> GSMP (N2 (proj_unl l' A))"
healthy_wp_PC: fixes f :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and g :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "healthy (wp f)"   and "healthy (wp g)"   and "unitary P" shows "healthy (wp (f \<^bsub>P\<^esub>\<oplus> g))"
i_join_def: fixes f :: "nat \<Rightarrow> 'b"   and I :: "nat set" shows "f \<Join>\<^sub> I \<equiv> \<lambda>n. f (I \<rightarrow> n)"
hyps: shows "a \<in> set R \<longrightarrow> b \<in> set R \<longrightarrow> c \<in> set R \<longrightarrow> a \<noteq> b \<longrightarrow> position a R \<le> position b R \<longrightarrow> position b R \<le> position c R \<longrightarrow> a \<noteq> c"
inside_outside_intersect_connected: fixes T :: "'a set"   and S :: "'a set" assumes "connected T"   and "inside S \<inter> T \<noteq> {}"   and "outside S \<inter> T \<noteq> {}" shows "S \<inter> T \<noteq> {}"
le_funE: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "f \<le> g" obtains "f x \<le> g x"
node_rel_def: fixes Re :: "('f \<times> 'h) set"   and R :: "('e \<times> 'g) set" shows "\<langle>Re, R\<rangle>node_rel = {(\<lparr>name_impl = namei, incoming_impl = inci, new_impl = newi, old_impl = oldi, next_impl = nexti, \<dots> = morei\<rparr>, \<lparr>name = name, incoming = inc, new = new, old = old, next = next, \<dots> = more\<rparr>) |namei name inci inc newi new oldi old nexti next morei more. (namei, name) \<in> nat_rel \<and> (inci, inc) \<in> \<langle>nat_rel\<rangle>dflt_rs_rel \<and> (newi, new) \<in> \<langle>\<langle>R\<rangle>ltlr_rel\<rangle>lss.rel \<and> (oldi, old) \<in> \<langle>\<langle>R\<rangle>ltlr_rel\<rangle>lss.rel \<and> (nexti, next) \<in> \<langle>\<langle>R\<rangle>ltlr_rel\<rangle>lss.rel \<and> (morei, more) \<in> Re}"
basis_expansion_unique: fixes f :: "'b \<Rightarrow> 'a"   and x :: "('a, 'b) vec" shows "((\<Sum>i\<in>UNIV. f i *s axis i (1::'a)) = x) = (\<forall>i. f i = x $ i)"
E_sup_def: fixes K :: "'a \<Rightarrow> 'a pmf set"   and s :: "'a"   and f :: "'a stream \<Rightarrow> ennreal" assumes "Markov_Decision_Process K" shows "Markov_Decision_Process.E_sup K s f = (\<Squnion>cfg\<in>Markov_Decision_Process.cfg_on K s. integral\<^sup>N (Markov_Decision_Process.T cfg) f)"
B_def: shows "B = - 1\<^sub>m nc @\<^sub>r A @\<^sub>r - A"
hyp: fixes cs2 :: "('a, 'b, 'c) com list"   and css2 :: "(('a, 'b, 'c) com list \<times> ('a, 'b, 'c) com list) list"   and s2 :: "('a, 'c) xstate" assumes "\<Gamma>\<turnstile> (cs1, css1, s1) \<rightarrow> (cs2, css2, s2)" shows "\<Gamma>\<turnstile>cs2,css2 \<Down> s2"
sw_simps2: fixes xs :: "'c"   and y :: "'b"   and x :: "'b" shows "x @xs[y \<and> x]_xs = y"
min_correct: fixes T :: "nat \<Rightarrow> nat" shows "\<forall>x\<in>{1..m}. T (min\<^sub>k T m) \<le> T x"
eucl_less_def: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "euclidean_space.eucl_less inner Basis a b = (\<forall>i\<in>Basis. inner a i < inner b i)"
arr_dom_iff_arr: fixes f :: "'a1" shows "A1.arr (A1.dom f) = A1.arr f"
Arr_implies_Ide_Dom: fixes t :: "'a local.term" assumes "Arr t" shows "Ide (Dom t)"
assms_2: assumes "\<And>Y. Y \<^bold>\<in> B \<Longrightarrow> (\<exists>!Z. \<langle>Y, Z\<rangle> \<^bold>\<in> F) \<and> (\<forall>Z. \<langle>Y, Z\<rangle> \<^bold>\<in> F \<longrightarrow> Z \<^bold>\<in> C)" shows "T"
hmsetmset_less: fixes M :: "hmultiset"   and N :: "hmultiset" shows "(hmsetmset M < hmsetmset N) = (M < N)"
rep_eq: fixes x :: "'e \<Rightarrow> 'd"   and xa :: "('e, 'c) uexpr" shows "\<lbrakk>uop x xa\<rbrakk>\<^sub>e = (\<lambda>b. x (\<lbrakk>xa\<rbrakk>\<^sub>e b))"
dim0I: assumes "gen_set {\<zero>\<^bsub>V\<^esub>}" shows "dim = 0"
isolate_subset: fixes x :: "'c"   and S :: "'c set"   and T :: "'c set" assumes "x isolate S"   and "T \<subseteq> S"   and "x \<in> T" shows "x isolate T"
Act_Basic_10: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<A>(\<phi> \<^bold>\<or> \<psi>) \<^bold>\<equiv> \<^bold>\<A>\<phi> \<^bold>\<or> \<^bold>\<A>\<psi> in v]"
distinct_202: fixes x11 :: "Syntax.term"   and x81 :: "Syntax.term"   and x82 :: "Syntax.var"   and x83 :: "Syntax.term" shows "Prj1 x11 \<noteq> Syntax.Let x81 x82 x83"
cong_exp_iff_simps_13: 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 modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'c \<times> 'c"   and divmod_step :: "num \<Rightarrow> 'c \<times> 'c \<Rightarrow> 'c \<times> 'c"   and m :: "num"   and q :: "num"   and n :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "(modulo (numeral.numeral one plus (num.Bit1 m)) (numeral.numeral one plus (num.Bit0 q)) = modulo (numeral.numeral one plus (num.Bit1 n)) (numeral.numeral one plus (num.Bit0 q))) = (modulo (numeral.numeral one plus m) (numeral.numeral one plus q) = modulo (numeral.numeral one plus n) (numeral.numeral one plus q))"
support_flow_conv: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal" shows "support_flow f = {e. 0 < f e}"
hfun_to_arr_in_hom: fixes B :: "hf"   and C :: "hf"   and F :: "hf" assumes "hfun B C F" shows "\<guillemotleft>hfun_to_arr B C F : hf_to_ide B \<rightarrow> hf_to_ide C\<guillemotright>"
abs_ln_one_plus_x_minus_x_bound_nonpos: fixes x :: "real" assumes "- (1 / 2) \<le> x"   and "x \<le> 0" shows "\<bar>ln (1 + x) - x\<bar> \<le> 2 * x\<^sup>2"
success_shiftr_p: fixes h :: "heap"   and i :: "nat ref"   and a :: "'b array"   and key :: "'b" assumes "Ref.get h i < Array.length h a" shows "success (shiftr_p a key i) h"
subst_tm_commute2: fixes j :: "name"   and t :: "tm"   and i :: "name"   and u :: "tm"   and tm :: "tm" assumes "atom j \<sharp> t"   and "atom i \<sharp> u"   and "i \<noteq> j" shows "subst j u (subst i t tm) = subst i t (subst j u tm)"
initialproc_def: fixes fg :: "('c, 'd, 'e, 'f, 'g) flowgraph_rec_scheme"   and p :: "'d" shows "initialproc fg p \<equiv> p = main fg \<or> (\<exists>u v. (u, Spawn p, v) \<in> edges fg)"
hcomp_arr_obj_1: fixes \<mu> :: "'a"   and a :: "'a" assumes "arr \<mu>"   and "obj a"   and "src \<mu> = a" shows "\<mu> \<star> a = \<r>\<^sup>-\<^sup>1[cod \<mu>] \<cdot> \<mu> \<cdot> \<r>[local.dom \<mu>]"