hcju/isabelleps huggingface dataset:
wtL_tL: fixes l :: "('fsym, 'psym) lit" assumes "Ik.wtL l" shows "TE.wtL (tL l)"
out213__sams: fixes D :: "'p"   and E :: "'p"   and F :: "'p"   and B :: "'p"   and A :: "'p"   and C :: "'p" assumes "D \<noteq> E"   and "E \<noteq> F"   and "B Out A C" shows "SAMS A B C D E F"
ring_tOp_closed: fixes R :: "('c, 'd) Ring_scheme"   and x :: "'c"   and y :: "'c" assumes "Ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> y \<in> carrier R"
N_def: shows "N = {x1. p_fix1 x1 = 0}"
split_part: fixes P :: "bool"   and Q :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "(\<lambda>(a, b). P \<and> Q a b) = (\<lambda>ab. P \<and> (case ab of (x, xa) \<Rightarrow> Q x xa))"
logical_instr_privilege: fixes s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "s' = snd (fst (logical_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
cong_dvd_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a" shows "dvd.dvd times a b = dvd.dvd times a c"
rel_G_mono: fixes L1 :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and L1' :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and L2 :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and L2' :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and Co1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co1' :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co2 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co2' :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Contra1' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Contra1 :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Contra2' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra2 :: "'t \<Rightarrow> 'u \<Rightarrow> bool" assumes "L1 \<le> L1'"   and "L2 \<le> L2'"   and "Co1 \<le> Co1'"   and "Co2 \<le> Co2'"   and "Contra1' \<le> Contra1"   and "Contra2' \<le> Contra2" shows "rel_G L1 L2 Co1 Co2 Contra1 Contra2 \<le> rel_G L1' L2' Co1' Co2' Contra1' Contra2'"
Re_complex_of_real: fixes z :: "real" shows "Re (complex_of_real z) = z"
fls_subdegree_mult: fixes f :: "'a fls"   and g :: "'a fls" assumes "f \<noteq> 0"   and "g \<noteq> 0" shows "fls_subdegree (f * g) = fls_subdegree f + fls_subdegree g"
lex_ext_iff: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and m :: "nat"   and xs :: "'a list"   and ys :: "'a list" shows "lex_ext f m xs ys = ((length xs = length ys \<or> length ys \<le> m) \<and> ((\<exists>i<length xs. i < length ys \<and> (\<forall>j<i. snd (f (xs ! j) (ys ! j))) \<and> fst (f (xs ! i) (ys ! i))) \<or> (\<forall>i<length ys. snd (f (xs ! i) (ys ! i))) \<and> length ys < length xs), (length xs = length ys \<or> length ys \<le> m) \<and> ((\<exists>i<length xs. i < length ys \<and> (\<forall>j<i. snd (f (xs ! j) (ys ! j))) \<and> fst (f (xs ! i) (ys ! i))) \<or> (\<forall>i<length ys. snd (f (xs ! i) (ys ! i))) \<and> length ys \<le> length xs))"
path_connectedin_topspace: fixes X :: "'c topology" shows "path_connectedin X (topspace X) = path_connected_space X"
hyps: fixes G :: "('a, 'b) graph" assumes "finite (edges G)" shows "finite (edges (rem_unPath xs G))"
elims: fixes x :: "'a"   and xa :: "nat"   and xb :: "nat"   and y :: "('a \<times> nat) convnet" assumes "shallow_model' x xa xb = y" obtains Z :: "'a" and M :: "nat" where "x = Za__" and "xa = Ma__" and "xb = 0" and "y = Conv (Za__, Ma__) (Input Ma__)"   | Z :: "'a" and M :: "nat" and N :: "nat" where "x = Z" and "xa = M" and "xb = Suc N" and "y = Pool (shallow_model' Z M 0) (shallow_model' Z M N)"
monoE: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "class.order less_eq less"   and "order.mono less_eq f"   and "less_eq x y" obtains "f x \<le> f y"
theI2: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c"   and Q :: "'c \<Rightarrow> bool" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
p_max_def: shows "ego.p_max = ego.p ego.t_stop"
well_def: fixes j :: "nat" assumes "j \<in> set (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))" shows "j < length (cols (M_mat_R signs subsets))"
preprm_disagreement_composition: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<noteq> b"   and "b \<noteq> c"   and "a \<noteq> c" shows "{x. preprm_apply (preprm_compose (preprm_unit a b) (preprm_unit b c)) x \<noteq> preprm_apply (preprm_unit a c) x} = {a, b}"
assms: shows "\<forall>b\<in>B. b \<in> ideal_generated A"
assms_5: shows "ordinal_utility (\<P> outcomes) \<R> (\<lambda>x. measure_pmf.expectation x u)"
1: shows "\<mu>.cod.leg0 \<cdot> \<mu>.chine \<cdot> doms.Prj\<^sub>1\<^sub>1 = \<nu>.cod.leg1 \<cdot> \<nu>.chine \<cdot> doms.Prj\<^sub>0\<^sub>1"
sup_rep_def: fixes p1 :: "int extended \<times> int extended"   and p2 :: "int extended \<times> int extended" shows "sup_rep p1 p2 = (if is_empty_rep p1 then p2 else if is_empty_rep p2 then p1 else let (l1, h1) = p1; (l2, h2) = p2 in (min l1 l2, max h1 h2))"
plusU_LNil_right: fixes xs :: "udom\<cdot>llist" shows "plusU\<cdot>xs\<cdot>LNil = xs"
Re_interval_def: fixes complex_interval :: "complex_interval" shows "Re_interval complex_interval = (case complex_interval of Complex_Interval x1 x2 \<Rightarrow> x1)"
M_dim_1: shows "dim_row (M (length fs)) = length fs"
le_INF_iff: fixes u :: "'al"   and U\<^sub>2 :: "'c set"   and f :: "'c \<Rightarrow> 'al"   and A :: "'c set" assumes "u \<in> U"   and "\<forall>x\<in>U\<^sub>2. f x \<in> U"   and "A \<subseteq> U\<^sub>2" shows "(u \<le>\<^sub>o\<^sub>w \<Sqinter>\<^sub>o\<^sub>w (f ` A)) = (\<forall>x\<in>A. u \<le>\<^sub>o\<^sub>w f x)"
equiv_transI: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>P. sound P \<Longrightarrow> t P = u P" shows "equiv_trans t u"
ereal_image_uminus_shift: fixes X :: "ereal set"   and Y :: "ereal set" shows "(uminus ` X = Y) = (X = uminus ` Y)"
k_frontier_is_node: shows "list_all k_isNode (k_frontier a)"
gf_id: fixes x :: "'V'" assumes "x \<in> vertices G'" shows "(g \<circ> f) (h x) = h x"
env_restr_fun_upd_other: fixes x :: "'c"   and S :: "'c set"   and m1 :: "'c \<Rightarrow> 'd"   and v :: "'d" assumes "x \<notin> S" shows "m1(x := v) f|` S = m1 f|` S"
qmdecl_resTy_simp: fixes sig :: "sig"   and m :: "qtname \<times> methd" shows "resTy (qmdecl sig m) = resTy m"
G_lower: fixes x :: "real"   and n :: "nat" assumes "0 < x"   and "1 \<le> n" shows "Gamma_series x n \<le> G x"
cDiff_cempty: fixes A :: "'f cset" shows "cDiff A cempty = A"
cut_le_mem_iff: fixes x :: "'b"   and I :: "'b set"   and t :: "'b" shows "(x \<in> I \<down>\<le> t) = (x \<in> I \<and> x \<le> t)"
assms: shows "rel_spmf R (p \<upharpoonleft> A) (q \<upharpoonleft> B)"
irreducible_preservation: fixes p :: "int poly"   and x :: "'a"   and q :: "int poly"   and y :: "'b"   and f :: "int poly \<Rightarrow> int poly" assumes "irreducible p"   and "p represents x"   and "q represents y"   and "degree q \<le> degree p"   and "\<And>q. q represents y \<Longrightarrow> f q represents x \<and> degree (f q) \<le> degree q"   and "primitive q" shows "irreducible q"
carrier_vec_def: fixes n :: "nat" shows "carrier_vec n = {v. dim_vec v = n}"
assms_2: fixes \<psi> :: "'fmla" assumes "\<psi> \<in> set \<psi>s" shows "nprv (insert \<psi> F) \<chi>"
embed_le_1: fixes G :: "'a \<Rightarrow> bool"   and s :: "'a" shows "\<guillemotleft> G \<guillemotright> s \<le> 1"
args_to_set_alt2: fixes 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 ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" assumes "ap_spec ap"   and "ab_spec ab"   and "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)" shows "args_to_set (gs, ab gs bs hs data, ap gs bs (ps -- sps) hs data) = fst ` (set gs \<union> set bs \<union> set hs)"
equivalent_objects_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "bicategory V H \<a> \<i> src trg" shows "bicategory.equivalent_objects V H \<a> \<i> src trg a b \<equiv> \<exists>f. horizontal_homs.in_hhom V src trg f a b \<and> bicategory.equivalence_map V H \<a> \<i> src trg f"
fv_def: fixes x :: "'a" shows "fv x = (if x \<in> GU then fv_U x else fv_V x)"
exhaust: fixes y :: "'a error_result" obtains x1 :: "'a" where "y = Rraise x1"   | x2 :: "abort" where "y = Rabort x2"
sign_vec_index_of: fixes f :: "rat poly"   and ftrs :: "rat poly list"   and x :: "real" assumes "f \<in> set ftrs" shows "sign_vec ftrs x ! index_of ftrs f = squash (rpoly f x)"
foo98: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and aa :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and r :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "applied_rule_rev Cp x (aa # p) = \<lfloor>a\<rfloor>"   and "x \<in> dom (Cp r)"   and "r \<in> set p" shows "a \<in> set p"
simps_1: fixes N :: "nat" shows "freenonces (NONCE N) = {N}"
neq_Nil_conv2: fixes xs :: "'c list" shows "(xs \<noteq> []) = (\<exists>x xs'. xs = xs' @ [x])"
pcompose_smult: fixes a :: "'a"   and p :: "'a poly"   and r :: "'a poly" shows "Polynomial.smult a p \<circ>\<^sub>p r = Polynomial.smult a (p \<circ>\<^sub>p r)"
sets_restrict_space_iff: fixes \<Omega> :: "'c set"   and M :: "'c measure"   and A :: "'c set" assumes "\<Omega> \<inter> space M \<in> sets M" shows "(A \<in> sets (restrict_space M \<Omega>)) = (A \<subseteq> \<Omega> \<and> A \<in> sets M)"
1: shows "arr f \<and> arr (B.\<rr> (B.dom f))"
sound_mode_use_def: shows "sound_mode_use \<equiv> \<lambda>uu. case uu of (cms, mem) \<Rightarrow> list_all (\<lambda>cm. locally_sound_mode_use (cm, mem)) cms \<and> globally_sound_mode_use (cms, mem)"
unifiers_subst_set: fixes \<tau> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set" shows "(\<tau> \<in> unifiers (subst_set \<sigma> E)) = (\<sigma> \<circ>\<^sub>s \<tau> \<in> unifiers E)"
\<i>_simps_1: fixes a :: "'a" assumes "B.obj a" shows "seq (UP \<i>\<^sub>B[a]) (cmp\<^sub>U\<^sub>P (a, a))"
pareto_separable_onI: fixes A :: "'x set" assumes "\<And>B C a b. \<lbrakk>B \<subseteq> A; C \<subseteq> A; a \<in> A; b \<in> A; a \<noteq> b; Xd a = Xd b; Xh a = Xh b; a \<in> Ch (Xh b) (insert a (insert b B))\<rbrakk> \<Longrightarrow> b \<notin> Ch (Xh b) (insert a (insert b C))" shows "pareto_separable_on A"
heap_is_wellformed_children_disc_nodes: fixes 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 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 heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and h :: "(_) heap"   and node_ptr :: "(_) node_ptr" assumes "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel"   and "heap_is_wellformed h"   and "node_ptr |\<in>| node_ptr_kinds h"   and "\<not> (\<exists>parent\<in>fset (object_ptr_kinds h). node_ptr \<in> set |h \<turnstile> get_child_nodes parent|\<^sub>r)" shows "\<exists>document_ptr\<in>fset (document_ptr_kinds h). node_ptr \<in> set |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r"
WhileF\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and c :: "nat exp" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>false,s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>while (e) c,s\<^sub>0\<rangle> \<Rightarrow> \<langle>unit,s\<^sub>1\<rangle>"
X_XD_smallest_XH_largest: shows "X = XD_smallest \<inter> XH_largest"
lunit_coherence2: fixes f :: "'c" assumes "C.ide f" shows "lF f = F \<l>\<^sub>C[f] \<cdot>\<^sub>D \<Phi> (trg\<^sub>C f, f)"
axioms_1: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "ChamberComplexEndomorphism X f" shows "ChamberComplexMorphism X X f"
thm_2_3_7_indets: fixes fs :: "(('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) list"   and b :: "nat" assumes "extended_ord.is_GB_cofactor_bound (set fs) b" shows "(1 \<in> ideal (set fs)) = (1 \<in> set (extended_ord.punit.Macaulay_list (extended_ord.deg_shifts (\<Union> (indets ` set fs)) b fs)))"
empty: shows "GEN_OP emp op_map_empty (\<langle>A, B\<rangle>Rm)"
ftimesI2: fixes a :: "V"   and A :: "V"   and b :: "V"   and B :: "V" assumes "a \<in>\<^sub>\<circ> A"   and "b \<in>\<^sub>\<circ> B" shows "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> A \<times>\<^sub>\<bullet> B"
takeWhile_dropWhile_id: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "takeWhile P xs @ dropWhile P xs = xs"
minus_left: fixes a :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and b :: "'b" shows "- a *\<^sub>V b = - (a *\<^sub>V b)"
map_of_zip_inject: fixes ys :: "'a list"   and xs :: "'b list"   and zs :: "'a list" assumes "length ys = length xs"   and "length zs = length xs"   and "distinct xs"   and "map_of (zip xs ys) = map_of (zip xs zs)" shows "ys = zs"
is_gcd_connect: fixes a :: "'a"   and b :: "'a"   and x :: "'a" assumes "a \<noteq> (0::'a)"   and "b \<noteq> (0::'a)" shows "x gcdof\<^bsub>mk_monoid\<^esub> a b = is_gcd x a b"
is_balanced_inter: fixes A :: "nat set"   and B :: "nat set" assumes "\<forall>x\<in>A. f x = 0"   and "\<forall>x\<in>B. f x = 1" shows "A \<inter> B = {}"
fst_eq_Domain: fixes R :: "('b \<times> 'c) set" shows "fst ` R = Domain R"
cblinfun_compose_zero_right: fixes U :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "U o\<^sub>C\<^sub>L 0 = 0"
rot_circle_cube_vertical_boundary_explicit: shows "vertical_boundary rot_circle_cube = {rot_circle_right_edge, rot_circle_left_edge}"
disc_48: fixes x61 :: "'a OCL_Types.type"   and x62 :: "String.literal"   and x63 :: "'a expr list" shows "\<not> is_Call (StaticOperationCall x61 x62 x63)"
hyps_2: shows "P \<turnstile> \<langle>e1,s1,b1\<rangle> \<rightarrow>* \<langle>Val v',s\<^sub>1,b\<^sub>1\<rangle>"
congs_trans: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "xs \<cong> ys"   and "ys \<cong> zs" shows "xs \<cong> zs"
bl_bin_nonzero: fixes args :: "nat list" assumes "args \<noteq> []" shows "0 < bl_bin (<args>)"
equals0D: fixes A :: "'c set"   and a :: "'c" assumes "A = {}" shows "a \<notin> A"
pterm_to_sterm_sterm_to_pterm: fixes t :: "pterm" assumes "pre_strong_term_class.wellformed t" shows "sterm_to_pterm (pterm_to_sterm t) = t"
Fubini_set_integral: fixes A :: "'a set"   and B :: "'b set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "set_integrable (lborel \<Otimes>\<^sub>M lborel) (A \<times> B) (\<lambda>(x, y). f x y)" shows "LBINT y:B. (LBINT x:A. f x y) = LBINT x:A. set_lebesgue_integral lborel B (f x)"
ordertype_map_image: fixes B :: "V set"   and A :: "V set" assumes "B \<subseteq> A"   and "small A" shows "ordertype (ordermap A VWF ` A - ordermap A VWF ` B) VWF = ordertype (A - B) VWF"
pdivmod2_def: fixes f :: "int poly"   and g :: "int poly" shows "pdivmod2 f g = (if Mp g = 0 then (0, f) else let ilc = inverse_p m (lead_coeff (Mp g)); h = Polynomial.smult ilc (Mp g); (q, r) = pseudo_divmod (Mp f) (Mp h) in (Polynomial.smult ilc q, r))"
dlist_induct: fixes P :: "'b dlist \<Rightarrow> bool"   and dl :: "'b dlist" assumes "P DList.empty"   and "\<And>a dl. P dl \<Longrightarrow> P (DList.insert a dl)" shows "P dl"
ldropn_ldropn: fixes n :: "nat"   and m :: "nat"   and xs :: "'c llist" shows "ldropn n (ldropn m xs) = ldropn (n + m) xs"
sum_var_list_not: fixes x :: "'a"   and m :: "('a \<times> nat) list" assumes "x \<notin> fst ` set m" shows "sum_var_list m x = 0"
c: shows "AE x in MA. (\<lambda>n. A.birkhoff_sum (\<lambda>x. real (phiA x)) n x / real n) \<longlonglongrightarrow> real_cond_exp MA A.Invariants (\<lambda>x. real (phiA x)) x"
fv_compE1: fixes e :: "'a expr"   and Vs :: "String.literal list" assumes "fv e \<subseteq> set Vs" shows "fv (compE1 Vs e) = index Vs ` fv e"
Sup_trans_least: fixes S :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) set"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<forall>t\<in>S. le_utrans t u"   and "\<And>P. unitary P \<Longrightarrow> unitary (u P)" shows "le_utrans (Sup_trans S) u"
s_def: shows "s = complex_of_real (c - \<epsilon> / 2)"
cases: fixes a1 :: "Val list"   and a2 :: "Var list"   and a3 :: "(Var \<times> Val) list"   and a4 :: "Val list" assumes "(a1, a2, a3, a4) \<in> Frame" obtains oo :: "Val list" and ops :: "Val list" where "a1 = ops__" and "a2 = []" and "a3 = emp" and "a4 = oo__" and "oo__ = ops__"   | oo :: "Val list" and par :: "Var list" and S :: "(Var \<times> Val) list" and ops :: "Val list" and R :: "(Var \<times> Val) list" and x :: "Var" and v :: "Val" where "a1 = v # oo" and "a2 = x # par" and "a3 = R" and "a4 = ops" and "(oo, par, S, ops) \<in> Frame" and "R = S[x\<mapsto>v]"
ang_vec_a_scale2: fixes k :: "real"   and z1 :: "complex"   and z2 :: "complex" assumes "k \<noteq> 0" shows "\<angle>a z1 (cor k * z2) = \<angle>a z1 z2"
tllist_ord_simps_3: fixes b :: "'f"   and x :: "'g"   and xs :: "('g, 'f) tllist"   and b2 :: "'f" shows "tllist_ord b (TCons x xs) (TNil b2) = False"
hyps: fixes ys :: "'a list" shows "set (local.filter_min_aux xs ys) \<subseteq> set xs \<union> set ys"
AssertAllState''_sat: fixes t :: "(nat \<times> linear_poly) list"   and ats :: "('i \<times> 'a atom) list" assumes "\<triangle> t" shows "\<not> \<U> (assert_bound_loop ats (init t)) \<longrightarrow> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s (assert_bound_loop ats (init t)) \<and> \<diamond> (assert_bound_loop ats (init t))"
M_merge: shows "(\<lambda>x. merge (shift_var_set V) (Suc ` V') (x, \<rho>)) \<in> Pi\<^sub>M (insert 0 (Suc ` V)) (\<lambda>y. stock_measure (case y of 0 \<Rightarrow> t | Suc x \<Rightarrow> \<Gamma> x)) \<rightarrow>\<^sub>M state_measure (shift_var_set (V \<union> V')) (case_nat t \<Gamma>)"
norm1_scaleR: fixes r :: "real"   and v :: "('a, 'b) Finite_Cartesian_Product.vec" shows "norm1 (r *\<^sub>R v) = \<bar>r\<bar> * norm1 v"
resNilRight: fixes Rel :: "(pi \<times> pi) set"   and x :: "name" shows "\<zero> \<leadsto>[Rel] <\<nu>x>\<zero>"
insert_iff: fixes a :: "'a"   and b :: "'a"   and A :: "'a set" shows "(a \<in> {b} \<union> A) = (a = b \<or> a \<in> A)"
totalI2: fixes f :: "recf" assumes "recfn 2 f"   and "\<And>x y. eval f [x, y] \<down>" shows "Partial_Recursive.total f"
strict_implies_order: fixes ko :: "'a key_order"   and a :: "'a"   and b :: "'a" assumes "ko.lt ko a b" shows "ko.le ko a b"
assms_2: fixes i :: "nat"   and j :: "nat" assumes "i < m"   and "j < n" shows "polyfun N (\<lambda>x. A x $$ (i, j))"
uik: shows "unat i * unat k < 2 ^ LENGTH('a)"
rtrancl_fold_product1: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" shows "{(r, s). \<exists>a\<in>A. s = f a r}\<^sup>* = {(r, fold f w r) |r w. w \<in> lists A}"
right_idem: fixes a :: "'a"   and b :: "'a" shows "a \<squnion> b \<squnion> b = a \<squnion> b"
elims: fixes x :: "checker_type"   and xa :: "ct list"   and xb :: "checker_type"   and y :: "checker_type" assumes "type_update x xa xb = y" obtains curr_type :: "checker_type" and cons :: "ct list" and prods :: "checker_type" where "x = curr_type" and "xa = cons" and "xb = prods" and "y = produce (consume curr_type cons) prods"
q: shows "q = (b ^ 3 - (4::'a) * b * c + (8::'a) * d) / (8::'a)"
Max_mapping_coboundedI: fixes X :: "('a \<Rightarrow> 'b option) set"   and P :: "'a \<Rightarrow> 'b option" assumes "finite X"   and "\<forall>Q\<in>X. dom Q = dom P"   and "P \<in> X" shows "rel_mapping (\<le>) P (Max_mapping X)"
inv': shows "\<phi> \<circ> \<psi> = J'.map"
divisor_sigma_1_left_nat: fixes n :: "nat" shows "divisor_sigma (Suc 0) n = divisor_sum n"
stabinvg: shows "stabilizer m #> inv g \<in> carrier (G Mod stabilizer m)"
eigen_value_root_charpoly: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and k :: "'a" shows "eigen_value A k = (poly (charpoly A) k = (0::'a))"
restrict_Pi_cancel: fixes x :: "'h \<Rightarrow> 'i"   and I :: "'h set"   and A :: "'h \<Rightarrow> 'i set" shows "(restrict x I \<in> Pi I A) = (x \<in> Pi I A)"
P2: shows "Bet A E C \<and> (E = B \<or> B Out E D)"
locally_sound_respects_guarantees: fixes cm :: "'Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)"   and mem :: "'Var\<^sub>A \<Rightarrow> 'Val" assumes "abs.locally_sound_mode_use (cm, mem)" shows "abs.respects_own_guarantees cm"
distrib_3: fixes x :: "'b multiset"   and y :: "'b multiset"   and z :: "'b multiset" shows "x \<inter># (y \<union># z) = x \<inter># y \<union># (x \<inter># z)"
critical_set_oSuc: fixes A :: "ordinal set"   and x :: "ordinal" assumes "normal_set A" shows "critical_set A (oSuc x) = range (oDeriv (OrdinalVeblen.ordering (critical_set A x)))"