hcju/isabelleps huggingface dataset:
extension_lemma_submanifoldE: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'e"   and B :: "'c set" assumes "submanifold charts k S"   and "diff_fun k (manifold.charts_submanifold charts S) f"   and "closed B"   and "B \<subseteq> manifold.carrier (manifold.charts_submanifold charts S)" obtains f' :: "'c \<Rightarrow> 'e" where "diff_fun k charts f'" and "\<And>x. x \<in> B \<Longrightarrow> f' x = f x" and "csupport_on (manifold.carrier charts) f' \<inter> manifold.carrier charts \<subseteq> manifold.carrier (manifold.charts_submanifold charts S)"
gen_step'_def: fixes s :: "'s" shows "gen_step' s \<equiv> ASSERT (gen_rwof s) \<bind> (\<lambda>_. if gds_is_empty_stack gds s then SPEC (\<lambda>v0. v0 \<in> V0 \<and> \<not> gds_is_discovered gds v0 s) \<bind> (\<lambda>v0. gds_new_root gds v0 s) else let u = hd (stack s) in SELECT (\<lambda>v. (u, v) \<in> pending s) \<bind> (\<lambda>Vs. choose_pending u Vs s \<bind> (\<lambda>s. case Vs of None \<Rightarrow> gds_finish gds u s | Some v \<Rightarrow> if gds_is_discovered gds v s then if gds_is_finished gds v s then gds_cross_edge gds u v s else gds_back_edge gds u v s else gds_discover gds u v s)))"
simps_4: fixes n :: "nat"   and y :: "'a"   and x :: "'a" assumes "n \<noteq> 1"   and "odd n" shows "local.efficient_power y x n = local.efficient_power (f x y) (f x x) (n div 2)"
m_lcomm: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
enat_add_mono: fixes x :: "nat"   and y :: "enat"   and z :: "enat" shows "(enat x + y < enat x + z) = (y < z)"
SINK_0: shows "SINK (\<lambda>_. 0) = UNIV"
cinner_add_left: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner (plus x y) z = cinner x z + cinner y z"
funion_finsert_right: fixes A :: "'m fset"   and a :: "'m"   and B :: "'m fset" shows "A |\<union>| finsert a B = finsert a (A |\<union>| B)"
ik_eq: shows "ik\<^sub>e\<^sub>s\<^sub>t (A \<cdot>\<^sub>e\<^sub>s\<^sub>t \<I>) = ik\<^sub>e\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>"
hyps: fixes x :: "real \<times> real" assumes "ccw'.sortedP 0 zs"   and "ccw'.sortedP x (map snd (polychain_of x zs))" shows "ccw'.sortedP (snd (last (polychain_of x zs))) (map snd (butlast (polychain_of x zs)))"
church_rosser: fixes y :: "'a"   and x :: "'a" assumes "y\<^sup>\<star> ; x\<^sup>\<star> \<le> x\<^sup>\<star> ; y\<^sup>\<star>" shows "(x + y)\<^sup>\<star> = x\<^sup>\<star> ; y\<^sup>\<star>"
rbt_greater_simps_2: fixes k :: "'a"   and c :: "color"   and lt :: "('a, 'c) rbt"   and kt :: "'a"   and v :: "'c"   and rt :: "('a, 'c) rbt" shows "(k \<guillemotleft>| Branch c lt kt v rt) = (k < kt \<and> k \<guillemotleft>| lt \<and> k \<guillemotleft>| rt)"
bounded: fixes n :: "nat" shows "prod a {..n} \<le> B"
transp_def: fixes r :: "'l \<Rightarrow> 'l \<Rightarrow> bool" shows "transp r = (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)"
nsum_eq: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "\<forall>j\<le>n. f j \<in> carrier A"   and "\<forall>j\<le>n. g j \<in> carrier A"   and "\<forall>j\<le>n. f j = g j" shows "\<Sigma>\<^sub>e A f n = \<Sigma>\<^sub>e A g n"
psubsetI: fixes A :: "'g set"   and B :: "'g set" assumes "A \<subseteq> B"   and "A \<noteq> B" shows "A \<subset> B"
uint_lt: fixes w :: "'a Word.word" shows "uint w < 2 ^ LENGTH('a)"
index_1_imp_mult_1: fixes bl :: "'a set" assumes "1 = 1"   and "bl \<in># \<B>"   and "2 \<le> card bl" shows "multiplicity bl = 1"
extract_def: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "List.extract P xs = (case dropWhile (Not \<circ> P) xs of [] \<Rightarrow> None | y # ys \<Rightarrow> Some (takeWhile (Not \<circ> P) xs, y, ys))"
weight_is_arity_bound: fixes t :: "('f, 'c) Term.term"   and b :: "nat" assumes "weight t \<le> b" shows "bounded_arity b (funas_term t)"
well_def_signs: shows "well_def_signs (length qs) [[1], [0], [- 1]]"
simps_4: fixes x :: "((int \<times> int) set set, int) Combinators"   and y :: "((int \<times> int) set set, int) Combinators" shows "C (x \<oplus> y) = C y \<Oplus> C x"
AbstrLevels_S14_A82: fixes i :: "AbstrLevelsID" assumes "sS14 \<in> AbstrLevel i" shows "sA82 \<notin> AbstrLevel i"
sublist_trans: fixes l1 :: "'d list"   and l2 :: "'d list"   and l3 :: "'d list" assumes "subseq l1 l2"   and "subseq l2 l3" shows "subseq l1 l3"
mult_left_zero: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "mbot \<odot> x = mbot"
le_neq_trans: fixes a :: "'b zmultiset"   and b :: "'b zmultiset" assumes "a \<subseteq>#\<^sub>z b"   and "a \<noteq> b" shows "a \<subset>#\<^sub>z b"
map\<^sub>0_simps_3: fixes a :: "'c" assumes "C.obj a" shows "trg\<^sub>D (\<sigma>\<^sub>0 a) = G.map\<^sub>0 a"
I1: fixes bs :: "'b list"   and cs :: "'c list" assumes "list_all2 P1 xs bs"   and "list_all2 P2 bs cs" shows "list_all2 P3 xs cs"
invar: fixes k :: "'b"   and v :: "'c" assumes "m k = Some v" shows "invar v"
Ass: fixes G :: "'b triple set"   and P :: "'b assn"   and X :: "vname"   and a :: "state \<Rightarrow> nat" shows "G|-{\<lambda>Z s. P Z (s[X::=a s])}. X:==a .{P}"
supset: shows "set (reduction_signs_R p qs signs subsets (M_mat_R signs subsets)) \<subseteq> set (characterize_consistent_signs_at_roots p qs)"
antisym_order_consistent_self: fixes r :: "('a \<times> 'a) set" assumes "antisym r" shows "order_consistent r r"
find_csas_lemma_nozeros: fixes qs :: "rat poly list"   and fs :: "rat poly list"   and data :: "(rat \<times> (nat \<times> nat) list) list"   and csa :: "rat list" assumes "factorize_polys qs = (fs, data)"   and "fs \<noteq> []" shows "(csa \<in> consistent_sign_vectors fs UNIV \<and> has_no_zeros csa) = (csa \<in> set (find_consistent_signs_at_roots (coprime_r (cast_rat_list fs)) (cast_rat_list fs)))"
OclAnd_commute: fixes X :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Y :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(X and Y) = (Y and X)"
monotone_on_UNIV: shows "monotone_on UNIV = monotone"
Max_insert: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "A \<noteq> {}" shows "Max (insert x A) = max x (Max A)"
not_initial_imp_not_eq: fixes n :: "nat"   and f :: "nat \<Rightarrow> nat option"   and g :: "nat \<Rightarrow> nat option" assumes "\<And>x. x < Suc n \<Longrightarrow> f x \<down>"   and "\<not> is_init_of (f \<triangleright> n) g" shows "f \<noteq> g"
l_type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e_axioms: shows "l_type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e type_wf"
eval\<Pi>\<^sub>1_inject: fixes x :: "\<Pi>\<^sub>1"   and y :: "\<Pi>\<^sub>1" shows "(eval\<Pi>\<^sub>1 x = eval\<Pi>\<^sub>1 y) = (x = y)"
sasp_op_to_strips_def: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and op :: "('c, 'd) sas_plus_operator" shows "\<phi>\<^sub>O \<Psi> op \<equiv> let pre = sas_plus_operator.precondition_of op; add = effect_of op in Let (concat (map (\<lambda>(v, a). map (Pair v) (filter ((\<noteq>) a) (the (range_of \<Psi> v)))) (effect_of op))) (operator_for pre add)"
p: fixes ps :: "char list \<Rightarrow> int option"   and n :: "nat" assumes "P (ps, n)" shows "wp\<^sub>3\<^sub>' c (\<lambda>(ps, n). (Q \<and>* (\<lambda>s. True)) (ps, n div k)) (ps, k * n)"
wf_syscls_is_class_Throwable: fixes P :: "'b prog" assumes "wf_syscls P" shows "is_class P Throwable"
prems_1: shows "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)"
eq_Integer_numeral_simps_4: shows "eq\<cdot>1\<cdot>1 = TT"
pickT_pred: fixes II :: "nat set"   and cl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set" assumes "II \<in> UNpart cl P" shows "\<exists>n. pickT_pred cl P II n"
simps_3: fixes x2 :: "int" shows "Num_arc x2 \<noteq> MinInfty"
ball_cong: fixes A :: "'c set"   and B :: "'c set"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> P x = Q x" shows "(\<forall>x\<in>A. P x) = (\<forall>x\<in>B. Q x)"
regionp_region_eq: fixes X :: "'d set"   and I :: "'d \<Rightarrow> intv"   and r :: "('d \<times> 'd) set" shows "regionp (\<lambda>x. x \<in> X) I (\<lambda>x xa. (x, xa) \<in> r) = (\<lambda>x. x \<in> region X I r)"
all_is_block_nominals: fixes p :: "('b, 'c) fm"   and ps :: "('b, 'c) fm list"   and xs :: "nat set"   and i :: "'c" assumes "all_is p ps xs" shows "nominals p \<union> block_nominals (omit xs ps, i) = nominals p \<union> block_nominals (ps, i)"
preserves_reflects_arr: fixes f :: "'a \<times> 'a \<times> 'a" shows "B.arr (B.HoVH f) = B.VVV.arr f"
card_vimage_inter: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and X :: "'d set" assumes "inj_on f A"   and "X \<subseteq> f ` A" shows "card (f -` X \<inter> A) = card X"
min_elems2: fixes s :: "'a event list"   and P :: "'a process"   and c :: "'a event"   and S :: "'a process"   and Q :: "'a process" assumes "s \<notin> D P"   and "s @ [c] \<in> D P"   and "P \<sqsubseteq> S"   and "Q \<sqsubseteq> S" shows "(s @ [c], {}) \<in> F Q"
is_processT9: fixes P :: "'c process" shows "\<forall>s. s @ [tick] \<in> D P \<longrightarrow> s \<in> D P"
set_cases: fixes e :: "'b"   and a :: "'b rexp" assumes "e \<in> atoms a" obtains "a = Atom e"   | z1 :: "'b rexp" and z2 :: "'b rexp" where "a = Plus z1__ z2__" and "e \<in> atoms z1__"   | z1 :: "'b rexp" and z2 :: "'b rexp" where "a = Plus z1a__ z2a__" and "e \<in> atoms z2a__"   | z1 :: "'b rexp" and z2 :: "'b rexp" where "a = Times z1b__ z2b__" and "e \<in> atoms z1b__"   | z1 :: "'b rexp" and z2 :: "'b rexp" where "a = Times z1 z2" and "e \<in> atoms z2"   | z :: "'b rexp" where "a = Star z" and "e \<in> atoms z"
le: shows "adjoint (tensor_P (1\<^sub>m N) (mat_incr K)) * (tensor_P (1\<^sub>m N) (mat_incr K) * Q2 * adjoint (tensor_P (1\<^sub>m N) (mat_incr K))) * tensor_P (1\<^sub>m N) (mat_incr K) \<le>\<^sub>L adjoint (tensor_P (1\<^sub>m N) (mat_incr K)) * (P' + Q) * tensor_P (1\<^sub>m N) (mat_incr K)"
hom_deg_setD_2: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and z :: "nat"   and A :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" assumes "p \<in> hom_deg_set z A"   and "p \<noteq> 0" shows "poly_deg p = z"
lsc_hull_liminf_at: fixes f :: "'b \<Rightarrow> ereal" shows "\<forall>x. lsc_hull f x = min (f x) (Liminf (at x) f)"
rhs: shows "S (fst y) = spmf_of_set (range (fst y))"
sup_state_refl: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and s :: "ty list \<times> ty err list" shows "P \<turnstile> s \<le>\<^sub>i s"
transD: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "trans r"   and "(x, y) \<in> r"   and "(y, z) \<in> r" shows "(x, z) \<in> r"
arr_inv: fixes f :: "'a \<times> 'a" assumes "S.VV.iso f" shows "S.VV.arr (S.VV.inv f)"
path_entry_reachable: fixes G :: "nat graph_rec"   and p :: "nat list"   and n :: "nat" assumes "cfg_doms G"   and "path_entry (g_E G) p n" shows "cfg_doms.reachable G n"
maintainedD2: fixes A :: "('a, 'b) labeled_graph"   and B :: "('a, 'b) labeled_graph"   and G :: "('a, 'c) labeled_graph"   and f :: "('b \<times> 'c) set" assumes "maintained (A, B) G"   and "graph_homomorphism A G f" obtains g :: "('b \<times> 'c) set" where "graph_homomorphism B G g" and "f \<subseteq> g"
times_scalar_id: fixes v :: "'a list" shows "local.times id v = v"
vars_A\<^sub>0: shows "\<Union> (vars ` set_mset A\<^sub>0) \<subseteq> \<V>\<^sub>0"
poly_map_poly: fixes p :: "rat poly"   and x :: "rat" shows "poly (map_poly of_rat p) (of_rat x) = of_rat (poly p x)"
iMODb_union: fixes r :: "nat"   and r' :: "nat"   and m :: "nat"   and c :: "nat"   and c' :: "nat" assumes "r \<le> r'"   and "r mod m = r' mod m"   and "r' \<le> r + m * c"   and "r + m * c \<le> r' + m * c'" shows "[ r, mod m, c ] \<union> [ r', mod m, c' ] = [ r, mod m, r' div m - r div m + c' ]"
rep_eq: fixes x :: "'b"   and xa :: "'b antichain" shows "(x \<in>\<^sub>A xa) = (x \<in> set_antichain xa)"
homotopic_nearby_loops: fixes g :: "real \<Rightarrow> 'a"   and S :: "'a set" assumes "path g"   and "open S"   and "path_image g \<subseteq> S"   and "pathfinish g = pathstart g" shows "\<exists>e>0. \<forall>h. path h \<and> pathfinish h = pathstart h \<and> (\<forall>t\<in>{0..1}. norm (h t - g t) < e) \<longrightarrow> homotopic_loops S g h"
neqs1_2: shows "(\<lambda>y. (- diamond_y (diamond_x y), diamond_x y)) \<noteq> (\<lambda>y. (diamond_y (diamond_x y), diamond_x y))"
bij_restrict: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and P :: "'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "bij_betw f A B"   and "\<forall>a\<in>A. P a = Q (f a)" shows "bij_betw f {a \<in> A. P a} {b \<in> B. Q b}"
prefix_closed_def: fixes A :: "'b llist set" shows "prefix_closed A \<equiv> \<forall>t\<in>A. \<forall>s\<le>t. s \<in> A"
MkArr_in_hom': fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "is_left_adjoint f"   and "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" shows "Maps.in_hom (Maps.MkArr a b \<lbrakk>f\<rbrakk>) (Maps.MkIde a) (Maps.MkIde b)"
ord_eq_le_trans: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'x \<Rightarrow>\<^sub>0 nat" assumes "a = b"   and "b \<preceq> c" shows "a \<preceq> c"
smc_0_components_1: shows "smc_0\<lparr>Obj\<rparr> = []\<^sub>\<circ>"
map_fst_zip: fixes xs :: "'a list"   and ys :: "'b list" assumes "length xs = length ys" shows "map fst (zip xs ys) = xs"
bij_betw_id: fixes A :: "'e set" shows "bij_betw id A A"
adds_triv_right: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.comm_monoid_add plus zero" shows "plus.adds plus a (plus b a)"
assms_3: fixes a :: "'a" assumes "a \<in> X" shows "f a \<le> SPEC (\<lambda>x. WHILE (\<lambda>(uu_, X). X \<noteq> {}) (\<lambda>(b, X). ASSERT (X \<noteq> {}) \<bind> (\<lambda>_. op_set_npick_remove X \<bind> (\<lambda>(a, X). f a \<bind> (\<lambda>b'. c b b' \<bind> (\<lambda>b. RETURN (b, X)))))) (x, X) \<le> SPEC (\<lambda>(b, uu_). RETURN b \<le> SPEC P))"
cSup_atLeastAtMost: fixes y :: "'b multiset"   and x :: "'b multiset" assumes "y \<subseteq># x" shows "Sup (subset_mset.atLeastAtMost y x) = x"
subterms_Unop: fixes a :: "floatarith" shows "subterms (Unop a) = insert (Unop a) (subterms a)"
add: fixes x :: "'a"   and y :: "'a" assumes "x \<in> V"   and "y \<in> V" shows "f (x + y) = f x + f y"
wf_mdeclE: fixes G :: "prog"   and C :: "qtname"   and sig :: "sig"   and m :: "methd" assumes "wf_mdecl G C (sig, m)" obtains "wf_mhead G (pid C) sig (mhead m)" and "unique (lcls (mbody m))" and "\<forall>pn\<in>set (pars m). table_of (lcls (mbody m)) pn = None" and "\<forall>(vn, T)\<in>set (lcls (mbody m)). is_acc_type G (pid C) T" and "jumpNestingOkS {Ret} (stmt (mbody m))" and "is_class G C" and "\<lparr>prg = G, cls = C, lcl = callee_lcl C sig m\<rparr>\<turnstile>stmt (mbody m)\<Colon>\<surd>" and "\<exists>A. \<lparr>prg = G, cls = C, lcl = callee_lcl C sig m\<rparr>\<turnstile> parameters m \<guillemotright>\<langle>stmt (mbody m)\<rangle>\<guillemotright> A \<and> Result \<in> nrm A"
val_Integer_simps_2: shows "\<lbrakk>0\<rbrakk> = 0"
minor_2: fixes x :: "'a \<Rightarrow>\<^sub>0 int"   and y :: "'a \<Rightarrow>\<^sub>0 int" assumes "Poly_Mapping.keys x \<subseteq> S"   and "Poly_Mapping.keys y \<subseteq> S"   and "P x"   and "P y" shows "P (x - y)"
binomial_ring: fixes a :: "'b"   and b :: "'b"   and n :: "nat" shows "(a + b) ^ n = (\<Sum>k\<le>n. of_nat (n choose k) * a ^ k * b ^ (n - k))"
simps: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "CardinalityArityTransformation Aexp Aheap" shows "CardinalityArityTransformation.a_transform Aheap (ae, a, as) (\<Gamma>, e, S) = (map_transform Aeta_expand ae (map_transform (AbstractTransform.transform (Rep_cfun inc) (Rep_cfun pred) (\<lambda>\<Delta> e a. (a, Aheap \<Delta> e\<cdot>a)) fst snd (\<lambda>_. 0) (\<lambda>a. Var) (\<lambda>a. App) (\<lambda>a. Lam) (\<lambda>b \<Gamma>. Terms.Let (map_transform Aeta_expand (snd b) \<Gamma>))) ae \<Gamma>), AbstractTransform.transform (Rep_cfun inc) (Rep_cfun pred) (\<lambda>\<Delta> e a. (a, Aheap \<Delta> e\<cdot>a)) fst snd (\<lambda>_. 0) (\<lambda>a. Var) (\<lambda>a. App) (\<lambda>a. Lam) (\<lambda>b \<Gamma>. Terms.Let (map_transform Aeta_expand (snd b) \<Gamma>)) a e, CardinalityArityTransformation.transform_alts Aheap as S)"
p_dvd_orbit_size: fixes p :: "nat"   and a :: "nat"   and N :: "'c set" assumes "order G = p ^ a"   and "prime p"   and "finite M"   and "N \<in> orbits"   and "1 < card N" shows "p dvd card N"
simps_5: fixes uy :: "ephkey" shows "catomic (cEphK uy) = True"
l2_when_left_child: fixes l :: "nat"   and l' :: "nat"   and i :: "int"   and i' :: "int" assumes "l < l'"   and "(i - 1) * 2 ^ (l' - l) < i'"   and "i' < i * 2 ^ (l' - l)" shows "l2_\<phi> (l', i') (l, i) = (1 + real_of_int i' / 2 ^ (l' - l) - real_of_int i) / 2 ^ (l' + 1)"
elims_3: 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 x :: "'a list"   and xa :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "\<not> CFG.valid_path_aux kind get_return_edges x xa" obtains cs :: "'a list" and a :: "'a" and as :: "'a list" where "x = cs" and "xa = a # as" and "\<not> (case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> CFG.valid_path_aux kind get_return_edges (a # cs) as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> case cs of [] \<Rightarrow> CFG.valid_path_aux kind get_return_edges [] as | c' # cs' \<Rightarrow> a \<in> get_return_edges c' \<and> CFG.valid_path_aux kind get_return_edges cs' as | _ \<Rightarrow> CFG.valid_path_aux kind get_return_edges cs as)"
SUP_sigma_sigma: fixes M :: "'d set"   and f :: "'d \<Rightarrow> 'e set set"   and \<Omega> :: "'e set" assumes "M \<noteq> {}"   and "\<And>m. m \<in> M \<Longrightarrow> f m \<subseteq> Pow \<Omega>" shows "(SUP m\<in>M. sigma \<Omega> (f m)) = sigma \<Omega> (\<Union> (f ` M))"
well_order_on_domain: fixes A :: "'d set"   and r :: "('d \<times> 'd) set"   and a :: "'d"   and b :: "'d" assumes "well_order_on A r"   and "(a, b) \<in> r" shows "a \<in> A \<and> b \<in> A"
code_simple: fixes c :: "'a list" assumes "c \<in> \<C>" shows " c \<in>B \<C> "
assms_2: fixes x :: "'a" assumes "P x" shows "Q x"
Nil: fixes \<O>s :: "nat set list"   and \<R>s :: "(nat \<Rightarrow> bool option) list"   and i :: "nat"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option" shows "\<O>s,\<R>s,i\<turnstile> ([], \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
extended_hyperb_ineq_4_points': fixes e :: "'a"   and x :: "'a Gromov_completion"   and y :: "'a Gromov_completion"   and z :: "'a Gromov_completion"   and t :: "'a Gromov_completion" shows "Min {extended_Gromov_product_at e x y, extended_Gromov_product_at e y z, extended_Gromov_product_at e z t} \<le> extended_Gromov_product_at e x t + ereal (2 * deltaG TYPE('a))"
orient_lit_semantics_pos: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and l :: "'b literal"   and u :: "'b trm"   and v :: "'b trm"   and \<eta> :: "('b \<times> 'b trm) list"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms"   and "fo_interpretation I"   and "basic_superposition.orient_lit_inst trm_ord l u v pos \<eta>"   and "validate_ground_lit I (subst_lit l \<sigma>)" shows "I (u \<lhd> \<sigma>) (v \<lhd> \<sigma>)"
top_join: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c" assumes "bounded_lattice L"   and "x \<in> carrier L" shows "\<top>\<^bsub>L\<^esub> \<squnion>\<^bsub>L\<^esub> x = \<top>\<^bsub>L\<^esub>"
liftn_lift_swap: fixes n :: "nat"   and t :: "dB"   and k :: "nat" shows "liftn n (lift t k) k = lift (liftn n t k) k"
bind_strict_mono_strong2: fixes m :: "('d, 'e) state"   and f :: "'e \<Rightarrow> ('d, 'f) state" assumes "strict_mono_state m"   and "\<And>x s s'. run_state m s = (x, s') \<Longrightarrow> mono_state (f x)" shows "strict_mono_state (m \<bind> f)"
less_self: fixes t :: "'a list"   and a :: "'a" shows "t < t @ [a]"
image_mono: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "A \<subseteq> B" shows "f ` A \<subseteq> f ` B"
Nil_in_conc: fixes A :: "'a list set"   and B :: "'a list set" shows "([] \<in> A @@ B) = ([] \<in> A \<and> [] \<in> B)"
that: fixes i :: "'a" assumes "i \<noteq> x" shows "pmf p i = pmf q i"
Busemann_function_xx: fixes xi :: "'b Gromov_completion"   and x :: "'b" shows "Busemann_function_at xi x x = 0"
finite_carrier: fixes carrier :: "'c set"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "finite_total_preorder_on carrier le" shows "finite carrier"
factt: fixes s :: "nat"   and n :: "nat" assumes "s \<le> n" shows "sum ((^) 2) {s..n} < 2 ^ (n + 1)"
eventually_at_bot_linorderI: fixes c :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "\<And>x. x \<le> c \<Longrightarrow> P x" shows "eventually P at_bot"
summable_suminf_not_top: fixes f :: "nat \<Rightarrow> real" assumes "\<And>i. 0 \<le> f i"   and "(\<Sum>i. ennreal (f i)) \<noteq> top_class.top" shows "summable f"
cases: fixes a1 :: "'b sequent"   and a2 :: "nat"   and R :: "('b sequent list \<times> 'b sequent) set" assumes "(a1, a2) \<in> derivable R" obtains C :: "'b sequent" where "a1 = C" and "a2 = 0" and "([], C) \<in> R"   | r :: "'b sequent list \<times> 'b sequent" and m :: "nat" where "a1 = snd r" and "a2 = m + 1" and "r \<in> R" and "fst r \<noteq> []" and "\<forall>p\<in>set (fst r). \<exists>n\<le>m. (p, n) \<in> derivable R"
induct: fixes P :: "('c \<times> ('c, 'd) RHS) list \<Rightarrow> 'd list \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "('c \<times> ('c, 'd) RHS) list"   and a1 :: "'d list"   and a2 :: "nat" assumes "\<And>G k. P G [] k"   and "\<And>G t k. P G [t] k"   and "\<And>G t v va k. (\<And>x21 x22. init_match G t = x21 # x22 \<Longrightarrow> P G (v # va) (k + 1)) \<Longrightarrow> P G (t # v # va) k" shows "P a0 a1 a2"
assms_2: shows "theta\<inverse> \<subseteq> WretrT (theta\<inverse>)"
max_tstp_intro''': fixes tstp :: "nat + nat"   and tstp' :: "nat + nat"   and ts' :: "nat"   and tp' :: "nat" assumes "isl tstp = isl tstp'"   and "ts_tp_lt' ts' tp' tstp" shows "ts_tp_lt' ts' tp' (max_tstp tstp tstp')"