hcju/isabelleps huggingface dataset:
splitAt_distinct_b: fixes vs :: "'a list"   and a :: "'a list"   and b :: "'a list"   and ram :: "'a" assumes "distinct vs"   and "(a, b) = splitAt ram vs" shows "distinct b"
bounded: fixes i :: "'a" shows "\<exists>K. \<forall>x. norm (f i x) \<le> norm x * K"
asympartp_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" shows "asympartp r x y \<equiv> r x y \<and> \<not> r y x"
case_prodE: fixes c :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and p :: "'d \<times> 'e" assumes "case p of (a, b) \<Rightarrow> c a b" obtains x :: "'d" and y :: "'e" where "p = (x, y)" and "c x y"
Sbis_pres_discr_L: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and c :: "('c, 'd, 'a) cmd"   and d :: "('c, 'd, 'a) cmd"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis"   and "(c, d) \<in> PL_Indis.Sbis aval tval cval indis"   and "PL_Indis.discr aval tval cval indis d" shows "PL_Indis.discr aval tval cval indis c"
range_h: fixes i :: "nat" assumes "i < l + 1" shows "h i \<in> {a..b}"
asymp_powser_coinduct': fixes P :: "(real \<Rightarrow> real) \<Rightarrow> real msstream \<Rightarrow> bool"   and x1 :: "real \<Rightarrow> real"   and x2 :: "real msstream" assumes "P x1 x2"   and "\<And>x1 x2. P x1 x2 \<Longrightarrow> \<exists>f cs c. x2 = MSSCons c cs \<and> P f cs \<and> x1 \<in> O[F](\<lambda>_. 1) \<and> (\<forall>\<^sub>F x in F. x1 x = c + f x * h x)" shows "local.asymp_powser x1 x2"
stmt_step_tmps_count_mono: fixes \<theta> :: "nat \<Rightarrow> nat option"   and s :: "stmt"   and t :: "nat"   and s' :: "stmt"   and t' :: "nat"   and "is" :: "instr list" assumes "\<theta>\<turnstile> (s, t) \<rightarrow>\<^sub>s ((s', t'), is)" shows "t \<le> t'"
cases: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a set"   and a2 :: "'a set" assumes "max_extp R a1 a2" obtains X :: "'a set" and Y :: "'a set" where "a1 = X" and "a2 = Y" and "finite X" and "finite Y" and "Y \<noteq> {x. \<bottom> x}" and "\<And>x. x \<in> X \<Longrightarrow> Bex Y (R x)"
makeFaceFinal_def: fixes f :: "face"   and g :: "graph" shows "makeFaceFinal f g \<equiv> Graph (makeFaceFinalFaceList f (faces g)) (countVertices g) (map (makeFaceFinalFaceList f) (faceListAt g)) (heights g)"
assms: shows "MGU \<theta> (Fun f X) (Fun g Y)"
not_zero_on_unit_circle: shows "\<not> on_circline unit_circle 0\<^sub>h"
twiddleVal_inverse: fixes \<beta> :: "(Location \<times> Location) set"   and w :: "Val"   and v :: "Val" assumes "(\<beta>, w, v) \<in> twiddleVal" shows "(Pbij_inverse \<beta>, v, w) \<in> twiddleVal"
guard_def: fixes S :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "guard S = {. grd S .} \<circ> S"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "norm x = onorm (cblinfun_apply x)"
ap_tree_K_tree: fixes u :: "'b tree"   and v :: "'c tree" shows "pure (\<lambda>x y. x) \<diamondop> u \<diamondop> v = u"
Merge_cong_start: fixes x :: "'a\<^sub>m"   and A :: "'a\<^sub>m set"   and y :: "'a\<^sub>m" assumes "x \<in> A"   and "y \<in> A"   and "finite A" shows "Finite_Set.fold (\<lambda>x yo. yo \<bind> m x) (Some x) A = Finite_Set.fold (\<lambda>x yo. yo \<bind> m x) (Some y) A"
stirling_integral_bound_aux_integral1: fixes a :: "real"   and b :: "real"   and c :: "real"   and n :: "nat"   and l :: "real"   and r :: "real" assumes "0 \<le> a"   and "0 < b"   and "0 \<le> c"   and "1 < n"   and "l < a - b"   and "a + b < r" shows "((\<lambda>x. c / max b \<bar>x - a\<bar> ^ n) has_integral 2 * c * (real n / real (n - 1)) / b ^ (n - 1) - c / real (n - 1) * (1 / (a - l) ^ (n - 1) + 1 / (r - a) ^ (n - 1))) {l..r}"
\<F>\<G>_semantics': fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" shows "\<F>\<G> \<phi> w = subformulas\<^sub>\<nu> \<phi> \<inter> {\<psi>. w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>)}"
contract_inclusion: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A" shows "K \<inter> (K \<^bold>* .\<not> \<phi>) \<subseteq> K"
simps_1: fixes n1 :: "nat"   and n2 :: "nat" shows "(I n1 < R n2) = True"
step1_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "step1 r = (\<lambda>ys xs. \<exists>us z z' vs. xs = us @ z # vs \<and> r z' z \<and> ys = us @ z' # vs)"
purged_relation_def: fixes u :: "'dom_t"   and execs1 :: "'dom_t \<Rightarrow> 'action_t list list"   and execs2 :: "'dom_t \<Rightarrow> 'action_t list list" shows "purged_relation u execs1 execs2 \<equiv> \<forall>d. ifp\<^sup>*\<^sup>* d u \<longrightarrow> execs1 d = execs2 d"
cBang: fixes Rs :: "residual"   and C :: "'a" assumes "P \<parallel> !P \<longmapsto> Rs"   and "\<And>C. F C (P \<parallel> !P) Rs" shows "F C (!P) Rs"
axioms_1: fixes get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog 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 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 adopt_node :: "(_) document_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and adopt_node_locs :: "(_) object_ptr option \<Rightarrow> (_) document_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and 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 known_ptrs :: "(_) heap \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set" assumes "l_adopt_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_owner_document get_parent get_parent_locs remove_child remove_child_locs get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs adopt_node adopt_node_locs known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs set_child_nodes set_child_nodes_locs remove heap_is_wellformed parent_child_rel" shows "l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_owner_document get_parent get_parent_locs remove_child remove_child_locs get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs adopt_node adopt_node_locs known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs set_child_nodes set_child_nodes_locs remove"
p: fixes as :: "('a poly \<times> nat) set"   and p :: "'a poly" assumes "monic_factorization as p" shows "p = (\<Prod>(a, i)\<in>as. a ^ Suc i)"
core_iff: fixes a :: "'a set" shows "(a open) = (core a = a)"
dirichlet_prod_Suc_0: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" shows "dirichlet_prod f g (Suc 0) = f (Suc 0) * g (Suc 0)"
N_le_C: fixes A :: "'s set" assumes "countable A" shows "emeasure (measure_pmf N) A \<le> emeasure (stat C) A"
distincts_grouped_by_fst: fixes xs :: "('b \<times> 'c) list" assumes "distinct xs" shows "distincts (grouped_by_fst xs)"
list_all2_lengthD: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list" assumes "list_all2 P xs ys" shows "length xs = length ys"
pfinite_converter_5: shows "a \<in> outs_\<I> \<I>1"
eq_\<I>_gpv_right_gpv_cong: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<I>' :: "('c, 'd) \<I>"   and gpv :: "('a, 'c, 'd) gpv"   and gpv' :: "('b, 'c, 'd) gpv"   and \<I> :: "('e, 'f) \<I>" assumes "eq_\<I>_gpv A \<I>' gpv gpv'" shows "eq_\<I>_gpv A (\<I> \<oplus>\<^sub>\<I> \<I>') (right_gpv gpv) (right_gpv gpv')"
rep_def: shows "Word.rep \<equiv> quot_type.rep Word.word.Rep_word"
union_set_fold: fixes xs :: "'a list"   and A :: "'a set" shows "set xs \<union> A = fold insert xs A"
wt_cont_eff_Par_1: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and i :: "nat"   and s :: "'state" assumes "properL cl"   and "n < length cl"   and "i < brn (cl ! n)" shows "wt (Par cl) s (brnL cl n + i) = 1 / real (length cl) * wt (cl ! n) s i"
Empty_is_rbt: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "ord.is_rbt less rbt.Empty"
step'_Com: fixes num' :: "int \<Rightarrow> 'a"   and plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and test_num' :: "int \<Rightarrow> 'a \<Rightarrow> bool"   and filter_plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"   and filter_less' :: "bool \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"   and \<gamma> :: "'a \<Rightarrow> int set"   and S :: "'a Abs_State.st option"   and X :: "char list set"   and c :: "'a Abs_State.st option acom" assumes "Abs_Int1 num' plus' test_num' filter_plus' filter_less' \<gamma>"   and "domo S \<subseteq> X"   and "vars (strip c) \<subseteq> X"   and "c \<in> Com X" shows "Abs_Int1.step' num' plus' test_num' filter_plus' filter_less' S c \<in> Com X"
proj_set_subset: fixes n :: "'a"   and A :: "('a strand_label \<times> 'b) list" shows "set (proj n A) \<subseteq> set A"
scale_cancel_right: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'a"   and x :: "'b"   and b :: "'a" assumes "vector_space scale" shows "(scale a x = scale b x) = (a = b \<or> x = (0::'b))"
chain: shows "Complete_Partial_Order.chain (rel_prod orda ordb) Y"
gr_implies_not_zero: fixes m :: "'m multiset"   and n :: "'m multiset" assumes "m \<subset># n" shows "n \<noteq> {#}"
greaterThanAtMost_iff: fixes i :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and l :: "'g"   and u :: "'g" shows "(i \<in> ord.greaterThanAtMost less_eq less l u) = (less l i \<and> less_eq i u)"
vertical_composite_axioms: shows "vertical_composite A B F G H \<rho> \<sigma>"
in_hhomE: fixes \<mu> :: "('a S.E.term, 'a) S.arr"   and a :: "('a S.E.term, 'a) S.arr"   and b :: "('a S.E.term, 'a) S.arr" assumes "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>S b\<guillemotright>" obtains "S.arr \<mu>" and "S.obj a" and "S.obj b" and "S.src \<mu> = a" and "S.trg \<mu> = b"
nth_subdegree_nonzero: fixes f :: "'a fps" assumes "f \<noteq> 0" shows "f $ subdegree f \<noteq> (0::'a)"
interaction_bounded_by: fixes "consider" :: "'h \<Rightarrow> bool"   and gpv :: "('i, 'h, 'j) gpv"   and n :: "enat" assumes "interaction_bound consider gpv \<le> n" shows "interaction_bounded_by consider gpv n"
psubstAll_idEnv: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set" shows "(good X \<longrightarrow> X #[idEnv] = X) \<and> (goodAbs A \<longrightarrow> A $[idEnv] = A)"
STA_drop_block: fixes ps :: "('b, 'c) fm list"   and ps' :: "('b, 'c) fm list"   and a :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and A :: "'c set"   and n :: "nat" assumes "set ps \<subseteq> set ps'"   and "(ps', a) \<in>. branch"   and "A, n \<turnstile> (ps, a) # branch" shows "A, Suc n \<turnstile> branch"
r'_1: shows "r' = Resumption.resumption.resume \<r>' inp"
is_top_sorted_cons: fixes R :: "('a \<times> 'a) set"   and x :: "'a"   and l :: "'a list" shows "is_top_sorted R (x # l) = ({x} \<times> set l \<inter> (R\<^sup>*)\<inverse> = {} \<and> is_top_sorted R l)"
assms_5: fixes q :: "int poly"   and ri :: "root_info" assumes "(q, ri) \<in> set todo \<union> set old" shows "root_info_cond ri q"
almost_full_on_lists: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "almost_full_on P A" shows "almost_full_on (list_emb P) (lists A)"
vimage_Union: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'d set set" shows "f -` \<Union> A = \<Union> ((-`) f ` A)"
levelsCounter_def: fixes t :: "('b \<times> bool) list" shows "levelsCounter t = levelsCounter_aux t [0]"
degree_linear_power: fixes a :: "'a"   and n :: "nat" shows "degree ([:a, 1::'a:] ^ n) = n"
sum_plus: fixes x :: "'a"   and y :: "'a" shows "sum_class.sum x + sum_class.sum y = sum_class.sum (x \<squnion> y) + sum_class.sum (x \<sqinter> y)"
that: fixes x :: "'a" assumes "xs = butlast xs @ [x]"   and "ys = butlast ys @ [x]" shows "thesis"
assms_2: shows "R.dimension m K (a_kernel (R\<lparr>carrier := E\<rparr>) S h)"
assms: shows "\<Gamma>,High \<turnstile> if (b) c1 else c2"
sum_list_zero_nreduced: fixes as :: "'a list"   and A :: "'a set" assumes "as \<noteq> []"   and "sum_list as = (0::'a)" shows "\<not> reduced_word A as"
simps: fixes L :: "'a list"   and x :: "'a" shows "close L x = filter (\<lambda>y. y \<noteq> x) L"
T_def: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg" shows "MDP.T cfg = distr (MDP.MC.T cfg) MDP.St (smap state)"
rep_eq: fixes x :: "('a, 'b) lifted_distrib_lattice_top"   and xa :: "('a, 'b) lifted_distrib_lattice_top" shows "(x \<le> xa) = (\<forall>f. Rep_lifted_distrib_lattice_top x f \<le> Rep_lifted_distrib_lattice_top xa f)"
divide_less_eq_1_pos: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sgn :: "'e \<Rightarrow> 'e"   and abs :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less (divide b a) one = less b a"
insert: fixes A :: "'b set"   and x :: "'b"   and g :: "'b \<Rightarrow> 'a" assumes "finite A"   and "x \<notin> A" shows "prod g (insert x A) = g x * prod g A"
insertion_mult: fixes f :: "nat \<Rightarrow> 'a"   and p :: "'a mpoly"   and q :: "'a mpoly" shows "insertion f (p * q) = insertion f p * insertion f q"
reach_trans: fixes tr :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and p :: "'a"   and a :: "'b list"   and r :: "'a"   and b :: "'b list"   and q :: "'a" assumes "reach tr p a r"   and "reach tr r b q" shows "reach tr p (a @ b) q"
N: fixes n :: "nat" assumes "N \<le> n" shows "C + deltaG TYPE('a) \<le> Gromov_product_at a (u n) (v n)"
lm078: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and Y :: "'c set" assumes "finite X" shows "sum f X = sum f (X - Y) + sum f (X \<inter> Y)"
vimage_Collect: fixes P :: "'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e"   and Q :: "'d \<Rightarrow> bool" assumes "\<And>x. P (f x) = Q x" shows "f -` Collect P = Collect Q"
start_of_all_le: fixes ap :: "abc_inst list"   and as :: "nat" shows "start_of (layout_of ap) as \<le> start_of (layout_of ap) (length ap)"
bangRelp_bangRel_eq: fixes Rel :: "(pi \<times> pi) set" shows "bangRelp (\<lambda>x xa. (x, xa) \<in> Rel) = (\<lambda>x xa. (x, xa) \<in> bangRel Rel)"
check_poly_gt: fixes p :: "('b monom \<times> 'a) list"   and q :: "('b monom \<times> 'a) list" assumes "check_poly_gt (\<succ>) p q" shows "p >p q"
msaux_axioms: shows "msaux valid_mmsaux init_mmsaux add_new_ts_mmsaux gc_join_mmsaux add_new_table_mmsaux result_mmsaux"
finite_dom_rbt_lookup: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('b, 'c) rbt" assumes "class.linorder less_eq less" shows "finite (dom (ord.rbt_lookup less t))"
meval_intro: fixes dma :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> Sec"   and \<C>_vars :: "'c \<Rightarrow> 'c set"   and eval :: "((('e \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('e \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) set"   and cms :: "('e \<times> (Mode \<Rightarrow> 'c set)) list"   and n :: "nat"   and mem :: "'c \<Rightarrow> 'd"   and cm' :: "'e \<times> (Mode \<Rightarrow> 'c set)"   and mem' :: "'c \<Rightarrow> 'd" assumes "sifum_security_init dma \<C>_vars eval"   and "((cms ! n, mem), cm', mem') \<in> eval"   and "n < length cms" shows "((cms, mem), n, cms[n := cm'], mem') \<in> sifum_security_init.meval eval"
foldr_max_sorted: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list"   and y :: "'g" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq (rev xs)" shows "foldr (ord.max less_eq) xs y = (if xs = [] then y else ord.max less_eq (xs ! 0) y)"
lalternate_ltake: fixes n :: "nat"   and xs :: "'b llist" shows "ltake (enat n) (lalternate xs) = lalternate (ltake (enat (2 * n)) xs)"
invar2_def: shows "invar2 \<equiv> \<lambda>(c, n, ts). \<forall>k. c \<le> k \<and> k < n \<longrightarrow> (\<exists>t<N. has_ticket (ts t) k)"
FalschA: shows "[\<bottom>] \<Rightarrow> []"
outer_remains_def: fixes xs :: "'b list"   and xs' :: "'b list"   and l :: "nat"   and r :: "nat" shows "outer_remains xs xs' l r = (length xs = length xs' \<and> (\<forall>i. i < l \<or> r < i \<longrightarrow> xs ! i = xs' ! i))"
measurable_compose_n: fixes T :: "'b \<Rightarrow> 'b"   and M :: "'b measure"   and n :: "nat" assumes "T \<in> M \<rightarrow>\<^sub>M M" shows "T ^^ n \<in> M \<rightarrow>\<^sub>M M"
asIMOD_wlsFSb: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "wlsSEM SEM" shows "iwlsFSb (asIMOD SEM)"
defs_uses_disjoint: fixes n :: "'b"   and g :: "'a" assumes "n \<in> set (\<alpha>n g)" shows "defs g n \<inter> lookup_multimap (uses g) n = {}"
that: fixes s :: "'t \<Rightarrow>\<^sub>0 'b" assumes "(red F)\<^sup>*\<^sup>* p s"   and "(red F)\<^sup>*\<^sup>* q s" shows "thesis"
False: shows "\<not> \<bar>f p - f r\<bar> \<le> \<Delta>"
incseq_SucI: fixes X :: "nat \<Rightarrow> 'd" assumes "\<And>n. X n \<le> X (Suc n)" shows "incseq X"
extending_Constrains: fixes F :: "'a program"   and A :: "'a set"   and B :: "'a set" shows "extending (\<lambda>G. reachable (extend h F \<squnion> G)) h F (extend_set h A Co extend_set h B) (A Co B)"
simple_match_dst_alt: fixes r :: "('b, 'c) simple_match_scheme"   and p :: "'b word" assumes "simple_match_valid r" shows "simple_match_ip (dst r) p = prefix_match_semantics (PrefixMatch (fst (dst r)) (snd (dst r))) p"
one_power2: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e" assumes "class.semiring_1 one times plus zero" shows "power.power one times one 2 = one"
mustT_If_R: fixes tst :: "'test"   and s :: "'state"   and c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" assumes "\<not> tval tst s"   and "mustT (if tst then c1 else c2) s" shows "mustT c2 s"
lift_f_Ker_Q: shows "order.greater_eq (ker lift_f \<inter> FreeGroup S) \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>"
nth_append_first: fixes i :: "nat"   and l :: "'a list"   and l' :: "'a list" assumes "i < length l" shows "(l @ l') ! i = l ! i"
subset_antisym: fixes A :: "'a set"   and B :: "'a set" assumes "(A::'a set) \<subseteq> (B::'a set)"   and "(B::'a set) \<subseteq> (A::'a set)" shows "(A::'a set) = (B::'a set)"
isolated_singularity_at_def: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex" shows "isolated_singularity_at f z = (\<exists>r>0. f analytic_on ball z r - {z})"
sup_commute: fixes x :: "'c"   and y :: "'c" shows "sup x y = sup y x"
bounded_by_bound_of: fixes P :: "'b \<Rightarrow> real" assumes "Expectations.bounded P" shows "bounded_by (bound_of P) P"
append_eq_map_conv: fixes ys :: "'g list"   and zs :: "'g list"   and f :: "'h \<Rightarrow> 'g"   and xs :: "'h list" shows "(ys @ zs = map f xs) = (\<exists>us vs. xs = us @ vs \<and> ys = map f us \<and> zs = map f vs)"
atLeastLessThan_concat: fixes m :: "nat"   and n :: "nat"   and p :: "nat"   and g :: "nat \<Rightarrow> 'c" assumes "m \<le> n"   and "n \<le> p" shows "sum g {m..<n} + sum g {n..<p} = sum g {m..<p}"
antecedent: shows "\<forall>z. PP z x \<longrightarrow> P z y"
try_spmf_bind_out1: fixes f :: "'a \<Rightarrow> 'b spmf"   and p :: "'a spmf"   and q :: "'a spmf" assumes "\<And>x. lossless_spmf (f x)" shows "TRY p ELSE q \<bind> f = TRY p \<bind> f ELSE (q \<bind> f)"
assms_2: shows "v * - v\<^sup>T \<sqinter> g \<noteq> bot"
AND_twice: fixes w :: "'b"   and m :: "'b" shows "(w AND m) AND m = w AND m"
dis_rews_8: fixes discr :: "bool discr" shows "is_CB\<cdot>(CB\<cdot>discr) = TT"
coeff_Const: fixes p :: "'a"   and m :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (Const p) m = (p when m = 0)"
assms_3: fixes \<rho>' :: "complex mat"   and k :: "nat" assumes "\<rho>' \<in> carrier_mat d d"   and "k < n" shows "(DS ! k) \<rho>' \<in> carrier_mat d d"
rep_mkarr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Arr C f" shows "free_monoidal_category.rep C (free_monoidal_category.mkarr C f) = free_monoidal_category.norm C f"
valid_gc_mmsaux: fixes args :: "Monitor.args"   and cur :: "nat"   and aux :: "nat \<times> nat \<times> bool list \<times> bool list \<times> (nat \<times> 'a option list set) queue \<times> (nat \<times> 'a option list set) queue \<times> ('a option list, nat) mapping \<times> ('a option list, nat) mapping"   and ys :: "(nat \<times> 'a option list set) list" assumes "valid_mmsaux args cur aux ys" shows "valid_mmsaux args cur (gc_mmsaux aux) ys"
trg_def: fixes \<mu> :: "'a" shows "S.trg \<mu> = (if S.arr \<mu> then trg\<^sub>B \<mu> else S.null)"
simps: fixes a1 :: "'a"   and a2 :: "(bool \<times> 'b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set" shows "((a1, a2) \<in> conv ars) = ((\<exists>a. a1 = a \<and> a2 = []) \<or> (\<exists>a \<alpha> b ss. a1 = a \<and> a2 = (True, \<alpha>, b) # ss \<and> (a, \<alpha>, b) \<in> ars \<and> (b, ss) \<in> conv ars) \<or> (\<exists>b \<alpha> a ss. a1 = a \<and> a2 = (False, \<alpha>, b) # ss \<and> (b, \<alpha>, a) \<in> ars \<and> (b, ss) \<in> conv ars))"
is_joinI: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and z :: "'a"   and y :: "'a" assumes "class.cpo below"   and "below x z"   and "below y z"   and "\<And>a. \<lbrakk>below x a; below y a\<rbrakk> \<Longrightarrow> below z a" shows "cpo.join below x y = z"
A: fixes W :: "'a option set" shows "W \<subseteq> X \<longrightarrow> (xs, W) \<in> seq_comp_failures P Q"
integrable_abs_cancel: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "integrable M (\<lambda>x. \<bar>f x\<bar>)"   and "f \<in> borel_measurable M" shows "integrable M f"
codomains_def: fixes f :: "Arr" shows "codomains f \<equiv> {b. ide b \<and> local.comp b f \<noteq> null}"
par_Int_bot: fixes P :: "'b process"   and A :: "'b set" shows "(P \<lbrakk>A\<rbrakk> \<bottom>) = \<bottom>"
generate_maskedDB_elim: fixes emBits :: "nat"   and x :: "bit list"   and M :: "bit list"   and y :: "bit list"   and z :: "bit list" assumes "roundup emBits 8 * 8 - emBits \<le> length x"   and "roundup emBits 8 * 8 - length (sha1 M) - 8 = length (maskedDB_zero x emBits)" shows "generate_maskedDB (maskedDB_zero x emBits @ y @ z) emBits (length (sha1 M)) = maskedDB_zero x emBits"