hcju/isabelleps huggingface dataset:
summable_comparison_test: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> real" assumes "\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n"   and "summable g" shows "summable f"
class_add_sees_methods_rev: fixes C :: "char list"   and cdec :: "jvm_method class"   and P :: "jvm_prog"   and D :: "char list"   and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> jvm_method) \<times> char list) option" assumes "class_add P (C, cdec) \<turnstile> D sees_methods Mm"   and "\<And>D'. P \<turnstile> D \<preceq>\<^sup>* D' \<Longrightarrow> D' \<noteq> C" shows "P \<turnstile> D sees_methods Mm"
finite_supp_map_bset_tau_steps: fixes \<tau> :: "'b"   and satisfies :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and transition :: "'c \<Rightarrow> ('b, 'c) residual \<Rightarrow> bool"   and x :: "('a, 'd, 'b) formula" assumes "indexed_weak_nominal_ts TYPE('a) \<tau> satisfies transition" shows "finite (supp (map_bset (indexed_weak_nominal_ts.tau_steps \<tau> x) (Abs_bset UNIV)))"
bourbaki_witt_fixpoint_complete_latticeI: fixes f :: "'d \<Rightarrow> 'd" assumes "\<And>x. x \<le> f x" shows "bourbaki_witt_fixpoint Sup {(x, y). x \<le> y} f"
nprv_allE: fixes F :: "'fmla set"   and x :: "'var"   and \<phi> :: "'fmla"   and t :: "'trm"   and \<psi> :: "'fmla" assumes "nprv F (all x \<phi>)"   and "nprv (insert (subst \<phi> t x) F) \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var"   and "\<psi> \<in> fmla" shows "nprv F \<psi>"
icsSp_Append_cdep: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n'' :: "'node SDG_node"   and n' :: "'node SDG_node" assumes "n ics-ns\<rightarrow>\<^sub>d* n''"   and "n'' s\<longrightarrow>\<^bsub>cd\<^esub> n'" shows "n ics-ns @ [n'']\<rightarrow>\<^sub>d* n'"
coincidence_sterm: fixes \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and \<nu>' :: "(real, 'a) vec \<times> (real, 'a) vec"   and \<theta> :: "('b, 'a) trm"   and I :: "('b, 'c, 'a) interp" assumes "Vagree \<nu> \<nu>' (FVT \<theta>)" shows "sterm_sem I \<theta> (fst \<nu>) = sterm_sem I \<theta> (fst \<nu>')"
not_neutral_contains_not_neutral: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and g :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "comm_monoid_set f z"   and "comm_monoid_set.F f z g A \<noteq> z" obtains a :: "'d" where "a \<in> A" and "g a \<noteq> z"
assms_3: shows "p + q \<le> \<^bold>|w\<^bold>|"
is_NoMsg_message_af_conv2: fixes m :: "'b message_af" shows "is_NoMsg m = (m = NoMsg)"
f_Exec_Stream_Acc_Output_one: fixes k :: "nat"   and output_fun :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'f message_af"   and c :: "'e" assumes "0 < k" shows "f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun [x] c = [last_message (map output_fun (f_Exec_Comp_Stream trans_fun (x # NoMsg\<^bsup>k - Suc 0\<^esup>) c))]"
reachNT_stateO_aux: fixes s :: "'state" assumes "reachNT s" shows "s = istate \<or> (\<exists>sh a ou. reach sh \<and> step sh a = (ou, s) \<and> \<not> T (Trans sh a ou s))"
Super: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn" shows "G,A\<turnstile>{Normal (\<lambda>s.. P\<leftarrow>In1 (val_this s))} Super-\<succ> {P}"
ne_zero: fixes b :: "'b" assumes "f b \<noteq> (0::'a)" shows "b \<in> basis"
finite_Fvars_trm: fixes t :: "trm" shows "finite (Fvars t)"
exhaust: fixes y :: "('c, 'd) RHS" obtains x11 :: "'c" and x12 :: "'c" where "y = Branch x11 x12"   | x2 :: "'d" where "y = Leaf x2"
conjugate_unit_disc: shows "conjugate ` unit_disc = unit_disc"
skew_invar: fixes t :: "('c \<times> nat) tree" assumes "invar t" shows "skew t = t"
csum_absorb1: fixes r2 :: "('c \<times> 'c) set"   and r1 :: "('d \<times> 'd) set" assumes "Cinfinite r2"   and "r1 \<le>o r2" shows "r2 +c r1 =o r2"
assms_5: shows "mat_of_rows n fs_init = U1 * mat_of_rows n gs"
Z_good_fun_def: shows "Z_good_fun = {f \<in> {0..<n} \<rightarrow> {0..<m}. (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) \<and> inj_on f {0..<n} \<and> f ` {0..<n} = g ` {0..<n}}"
vec_lambda_cases: fixes x :: "('b, 'a) vec" obtains y :: "'a \<Rightarrow> 'b" where "x = vec_lambda y" and "y \<in> UNIV"
cases: fixes a1 :: "classes1"   and a2 :: "classes1" assumes "subclass1 a1 a2" obtains c :: "classes1" where "a1 = c" and "a2 = Object" and "c \<noteq> Object"   | "a1 = Employee" and "a2 = Person"   | "a1 = Customer" and "a2 = Person"
add_le_less_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.ordered_cancel_ab_semigroup_add plus minus less_eq less"   and "less_eq a b"   and "less c d" shows "less (plus a c) (plus b d)"
assms_4: shows "\<forall>j'<k. n cd\<^bsup>\<pi>\<^esup>\<rightarrow> j' \<longrightarrow> j' \<le> j"
divide_by_norm_code: fixes A :: "(('b, 'c) vec, 'a) vec" shows "($) (divide_by_norm A) = divide_by_norm_row A"
bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p_NegChecks: fixes a :: "('c, 'd) stateful_strand_step" assumes "\<not> is_NegChecks a" shows "bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p a = []"
linord_of_list_trans: fixes xs :: "'b list" assumes "distinct xs" shows "trans (linord_of_list xs)"
akra_bazzi_light_nat: fixes a :: "real"   and c :: "real"   and C\<^sub>1 :: "nat"   and n\<^sub>0 :: "nat"   and f :: "nat \<Rightarrow> nat"   and b :: "real"   and d :: "real"   and C\<^sub>2 :: "nat" assumes "0 < a"   and "0 < c"   and "a + c < 1"   and "0 \<le> C\<^sub>1"   and "\<forall>n>n\<^sub>0. f n = f (nat \<lceil>a * real n + b\<rceil>) + f (nat \<lceil>c * real n + d\<rceil>) + C\<^sub>1 * n + C\<^sub>2" shows "\<exists>C\<^sub>3 C\<^sub>4. \<forall>n. f n \<le> C\<^sub>3 * n + C\<^sub>4"
rt_strictly_fresher_eqI: fixes i :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "i \<in> kD rt1"   and "i \<in> kD rt2"   and "nsqn rt1 i = nsqn rt2 i"   and "\<pi>\<^sub>5 (the (rt2 i)) < \<pi>\<^sub>5 (the (rt1 i))" shows "rt1 \<sqsubset>\<^bsub>i\<^esub> rt2"
emp_act: fixes act :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and p :: "'d" assumes "unital_quantale_module act" shows "act (\<Squnion> {}) p = \<bottom>"
AssertionStatEqRefl: fixes \<Psi> :: "'b" shows "\<Psi> \<simeq> \<Psi>"
bre_while_algo: fixes \<delta> :: "('d, 'e) ta_rule set"   and Qi :: "'d set" assumes "finite \<delta>" shows "while_algo (bre_algo Qi \<delta>)"
a_get_scdom_component_def: fixes get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and ptr :: "(_) object_ptr" shows "l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_scdom_component get_owner_document get_disconnected_nodes to_tree_order ptr = Heap_Error_Monad.bind (get_owner_document ptr) (\<lambda>document. Heap_Error_Monad.bind (get_disconnected_nodes document) (\<lambda>disc_nodes. Heap_Error_Monad.bind (to_tree_order (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^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 document)) (\<lambda>tree_order. Heap_Error_Monad.bind (map_M (to_tree_order \<circ> 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) disc_nodes) (\<lambda>disconnected_tree_orders. return (tree_order @ concat disconnected_tree_orders)))))"
r_totalizer_total: fixes d :: "nat" shows "Partial_Recursive.total (r_totalizer d)"
s1_2: shows "results_in t1 s1 \<le> t2"
simps_4: fixes I :: "('e, 'f, 'd) interp"   and \<phi> :: "('e, 'f, 'd) formula"   and \<psi> :: "('e, 'f, 'd) formula" shows "fml_sem I (\<phi> && \<psi>) = fml_sem I \<phi> \<inter> fml_sem I \<psi>"
awalk_to_apath_induct: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and P :: "'b list \<Rightarrow> bool" assumes "awalk u p v"   and "\<And>p. \<lbrakk>awalk u p v; distinct (awalk_verts u p)\<rbrakk> \<Longrightarrow> P p"   and "\<And>p q r s. \<lbrakk>awalk u p v; awalk_cyc_decomp p = (q, r, s); \<not> distinct (awalk_verts u p); P (q @ s)\<rbrakk> \<Longrightarrow> P p" shows "P p"
impl_def: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "P \<longrightarrow> Q \<equiv> \<lambda>A. P A \<longrightarrow> Q A"
inner_Basis_inf_left: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and abs :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and i :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less"   and "i \<in> Basis" shows "inner (inf x y) i = inf (inner x i) (inner y i)"
zero_fract_def: shows "0 \<equiv> abs_fract (0::'b, 1::'b)"
struct_specI: fixes sel :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g" assumes "sel_spec sel"   and "ap_spec ap"   and "ab_spec ab"   and "compl_struct compl" shows "struct_spec sel ap ab compl"
f_2: shows "\<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M < \<infinity>"
ide_compE: fixes g :: "('obj, 'obj) concrete_category.arr"   and f :: "('obj, 'obj) concrete_category.arr" assumes "ide (g \<cdot> f)" obtains "seq g f" and "seq f g" and "g \<cdot> f = local.dom f" and "g \<cdot> f = cod g"
D_mono': fixes e :: "'a exp"   and A :: "'a set option"   and A' :: "'a set option" assumes "\<D> e A"   and "A \<sqsubseteq> A'" shows "\<D> e A'"
id_iso: fixes G :: "('d, 'e) monoid_scheme" shows "id \<in> Group.iso G G"
intros_1: fixes p :: "'b fm" assumes "tautology p" shows "\<turnstile> p"
subst_to_scheme_inverse: fixes n :: "nat"   and sch :: "type_scheme" assumes "new_tv n sch" shows "subst_to_scheme (\<lambda>k. if n \<le> k then BVar (k - n) else FVar k) (bound_typ_inst (\<lambda>k. TVar (k + n)) sch) = sch"
ctr_transfer_2: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_behaviour R Diverges Diverges"
Un_iff: fixes c :: "'f"   and A :: "'f set"   and B :: "'f set" shows "(c \<in> A \<union> B) = (c \<in> A \<or> c \<in> B)"
diff_right_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less_eq a b" shows "less_eq (minus a c) (minus b c)"
cexp_cprod_ordLeq: fixes r1 :: "('d \<times> 'd) set"   and r2 :: "('e \<times> 'e) set"   and r3 :: "('f \<times> 'f) set" assumes "Card_order r1"   and "Cinfinite r2"   and "(r3, czero) \<notin> ordIso \<and> Card_order r3"   and "r3 \<le>o r2" shows "(r1 ^c r2) ^c r3 =o r1 ^c r2"
intros_9: fixes fop :: "binop_f"   and c1 :: "f32"   and c2 :: "f32"   and c :: "f32" assumes "app_binop_f fop c1 c2 = Some c" shows "\<lparr>[$C ConstFloat32 c1, $C ConstFloat32 c2, $Binop_f T_f32 fop]\<rparr> \<leadsto> \<lparr>[$C ConstFloat32 c]\<rparr>"
llist_less_finT: fixes r :: "'b llist"   and s :: "'b llist"   and A :: "'b set" assumes "r < s"   and "s \<in> A\<^sup>\<star>" shows "r \<in> A\<^sup>\<star>"
yun_monic_factorization_1: fixes as :: "('c poly \<times> nat) set"   and p :: "'c poly"   and bs :: "('c poly \<times> nat) list" assumes "monic_factorization as p"   and "yun_gcd.yun_monic_factorization gcd p = bs" shows "square_free_factorization p (1::'c, bs)"
repr_comb: fixes x :: "'b \<Rightarrow> real" shows "repr (comb x) = (\<lambda>b. if b \<in> B then x b else 0)"
fermat_little: fixes P :: "nat"   and x :: "nat" assumes "prime P" shows "[x ^ P = x] (mod P)"
a_suc_rw: shows "column (mod_type_class.from_nat (Suc k)) (Gram_Schmidt_upt_k A k) = column (mod_type_class.from_nat (Suc k)) A"
min_gallery_pgallery: fixes X :: "'b set set"   and xs :: "'b set list" assumes "ChamberComplex X"   and "SimplicialComplex.min_maxsimpchain X xs" shows "SimplicialComplex.pmaxsimpchain X xs"
lift_M': fixes x :: "'b word"   and y :: "'b word" shows "uint (M' x y) = L (uint x) (uint y) mod 2 ^ LENGTH('b)"
w2Geq: shows "- (2 ^ 31) \<le> sint (SCAST(32 \<rightarrow> 64) w2)"
inject: fixes x :: "'e"   and ya :: "'e" shows "(NondetT x = NondetT ya) = (x = ya)"
linear_surjective_imp_injective: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and B1 :: "'e set"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and B2 :: "'f set"   and f :: "'e \<Rightarrow> 'f" assumes "finite_dimensional_vector_space_pair s1 B1 s2 B2"   and "linear s1 s2 f"   and "surj f"   and "vector_space.dim s2 UNIV = vector_space.dim s1 UNIV" shows "inj f"
ind: fixes s' :: "nat"   and l' :: "cell list"   and r' :: "cell list" assumes "steps (s, l, r) (A, off) stp = (s', l', r')"   and "s' \<noteq> 0" shows "steps (s, l, r) (A @ B, off) stp = (s', l', r')"
domainD: fixes x :: "'b" assumes "x \<in> domain" shows "x \<sim> x"
assms: shows "\<forall>n s. numbound0 s \<longrightarrow> isrlfm (f n s)"
Suc_1: assumes "k \<le> length gs" shows "M.sumlist (map (\<lambda>i. g i \<cdot>\<^sub>v gs ! i) [0..<k]) \<bullet> M.sumlist (map (\<lambda>i. h i \<cdot>\<^sub>v gs ! i) [0..<k]) = (\<Sum>i\<leftarrow>[0..<k]. g i * h i * (gs ! i \<bullet> gs ! i))"
mult_pos_neg2: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less zero a"   and "less b zero" shows "less (times b a) zero"
extendCommute: fixes S :: "'a sequent"   and R :: "'a sequent"   and c :: "'a sequent" shows "extend S (extend R c) = extend R (extend S c)"
strict_trans1: fixes a :: "('d, 'e) Term.term"   and b :: "('d, 'e) Term.term"   and c :: "('d, 'e) Term.term" assumes "a \<unrhd> b"   and "b \<rhd> c" shows "a \<rhd> c"
Gram_Schmidt_matrix_efficient: fixes A :: "((real, 'a) vec, 'b) vec" shows "Gram_Schmidt_matrix A = foldl Gram_Schmidt_column_k_efficient A [0..<ncols A]"
le_Suc: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "less a b" shows "less_eq (plus a one) b"
UNIV: fixes S :: "'f set set" shows "generate_topology S UNIV"
LIMSEQ_power_zero: fixes x :: "'c" assumes "norm x < 1" shows "(^) x \<longlonglongrightarrow> (0::'c)"
elims_2: fixes x :: "int \<times> int"   and xa :: "int \<times> int" assumes "le_rat_raw x xa" obtains a :: "int" and b :: "int" and c :: "int" and d :: "int" where "x = (a, b)" and "xa = (c, d)" and "a * d * (b * d) \<le> c * b * (b * d)"
assms_1: shows "local.factorize_int_last_nz_poly f = (c, fs)"
greatest: fixes A :: "'a measure set"   and x :: "'a measure" assumes "\<And>z. z \<in> A \<Longrightarrow> x \<le> z" shows "x \<le> Inf A"
poly_MVT: fixes a :: "real"   and b :: "real"   and p :: "real poly" assumes "a < b" shows "\<exists>x>a. x < b \<and> poly p b - poly p a = (b - a) * poly (pderiv p) x"
associated_trans: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Group.monoid G"   and "a \<sim>\<^bsub>G\<^esub> b"   and "b \<sim>\<^bsub>G\<^esub> c"   and "a \<in> carrier G"   and "c \<in> carrier G" shows "a \<sim>\<^bsub>G\<^esub> c"
fresh_star_perm_set_conv: fixes as :: "atom set"   and p :: "perm" assumes "as \<sharp>* p"   and "finite as" shows "supp p \<sharp>* as"
simps_10: fixes v :: "cmsg"   and vb :: "cmsg" shows "catomic (cSign v vb) = False"
infinite_countable_subset: fixes S :: "'b set" assumes "infinite S" shows "\<exists>f. inj f \<and> Set.range f \<subseteq> S"
ext_inject: fixes g_V :: "'b set"   and g_E :: "('b \<times> 'b) set"   and g_V0 :: "'b set"   and more :: "'d"   and g_V' :: "'b set"   and g_E' :: "('b \<times> 'b) set"   and g_V0' :: "'b set"   and more' :: "'d" shows "(\<lparr>g_V = g_V, g_E = g_E, g_V0 = g_V0, \<dots> = more\<rparr> = \<lparr>g_V = g_V', g_E = g_E', g_V0 = g_V0', \<dots> = more'\<rparr>) = (g_V = g_V' \<and> g_E = g_E' \<and> g_V0 = g_V0' \<and> more = more')"
power_minus_odd: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and n :: "nat"   and a :: "'c" assumes "class.ring_1 one times plus zero minus uminus"   and "odd n" shows "power.power one times (uminus a) n = uminus (power.power one times a n)"
EASSUME_simps_1: shows "EASSUME True = ERETURN ()"
less_lsum_unique: fixes al :: "'b list"   and f :: "'b \<Rightarrow> nat"   and n1 :: "nat"   and j1 :: "nat"   and n2 :: "nat"   and j2 :: "nat" assumes "\<And>a. a \<in> set al \<Longrightarrow> 0 < f a"   and "n1 < length al \<and> j1 < f (al ! n1)"   and "n2 < length al \<and> j2 < f (al ! n2)"   and "lsum f (take n1 al) + j1 = lsum f (take n2 al) + j2" shows "n1 = n2 \<and> j1 = j2"
simps_2: fixes clause :: "Literal list"   and formula :: "Literal list list"   and valuation :: "Literal list" shows "formulaFalse (clause # formula) valuation = (clauseFalse clause valuation \<or> formulaFalse formula valuation)"
M_def: fixes x :: "int" shows "q.M x = x mod p ^ n"
that: fixes x :: "'z" assumes "x \<in> X" shows "P x"
ncols_transpose: fixes A :: "'b matrix" shows "ncols (transpose_matrix A) = nrows A"
indspaceD: fixes f :: "('f, 'g) aezfun \<Rightarrow> 'v"   and G :: "'g set"   and FH :: "('f, 'g) aezfun set"   and V :: "'v set" assumes "f \<in> BaseFGMod.indspace G FH V" shows "f \<in> addfunset FH V \<inter> BaseFGMod.smultfunset G FH"
widen_null: fixes P :: "'a prog"   and C :: "char list" shows "P \<turnstile> NT \<le> Class C"
test_mult: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "rel_at (rel_at (rel_at (rel_at x) ; rel_at (rel_at y))) = rel_at (rel_at y) ; rel_at (rel_at x)"
incidence_system_axioms: shows "incidence_system \<V> (add_block bl)"
split_min: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and i :: "'c"   and j :: "'c" assumes "class.linorder less_eq less" shows "P (ord.min less_eq i j) = ((less_eq i j \<longrightarrow> P i) \<and> (\<not> less_eq i j \<longrightarrow> P j))"
prems_4: shows "s \<leadsto>(x # xs)\<leadsto>\<^bsub>T\<^esub> t"
Diagonalize_Diagonalize: fixes t :: "'c local.term" assumes "Arr t" shows "\<^bold>\<lfloor>\<^bold>\<lfloor>t\<^bold>\<rfloor>\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
conj_left_commute: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "b AND a AND c = a AND b AND c"
inf_commute: fixes x :: "'g multiset"   and y :: "'g multiset" shows "x \<inter># y = y \<inter># x"
is_extensional: fixes f :: "'j" assumes "\<not> J.arr f" shows "A'.map f = C.null"
dghm_dag_Rel_ArrMap_vsv: fixes \<alpha> :: "V" shows "vsv (\<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha>\<lparr>ArrMap\<rparr>)"
card_Suc_lemma: fixes a :: "'b"   and A :: "'b set"   and n :: "nat" assumes "card (insert a A) \<le> Suc n"   and "a \<notin> A"   and "finite A" shows "card A \<le> n"
s_2: shows "(u \<circ> r \<circ> s) \<longlonglongrightarrow> liminf (u \<circ> r)"
assms: shows "frees u |\<subseteq>| fset_of_list (common_prefix \<Gamma> \<Gamma>')"
partition_onD3: fixes A :: "'b set"   and P :: "'b set set" assumes "partition_on A P" shows "{} \<notin> P"
assms: shows "list_all (\<lambda>A. A \<noteq> {}) XS"
times_in_float_intervalI: fixes a :: "real"   and A :: "float interval"   and b :: "real"   and B :: "float interval" assumes "a \<in>\<^sub>r A"   and "b \<in>\<^sub>r B" shows "a * b \<in>\<^sub>r A * B"
Qg_carrier: fixes G :: "('e, 'f) Group_scheme"   and N :: "'e set" assumes "Group G" shows "carrier (G / N) = set_rcs G N"
one_discr_def: shows "1 \<equiv> Discr (1::'b)"
Polys_def: fixes X :: "'a set" shows "P[X] = {p. keys p \<subseteq> .[X]}"
list_set_autoref_insert: assumes "PREFER single_valued R" shows "((#), insert) \<in> R \<rightarrow> \<langle>R\<rangle>list_wset_rel \<rightarrow> \<langle>R\<rangle>list_wset_rel"
cases: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e list" assumes "class.linorder less_eq less" obtains "x = []"   | xa :: "'e" and ys :: "'e list" where "x = xa # ys"