hcju/isabelleps huggingface dataset:
snd_scaleC: fixes r :: "complex"   and A :: "'a \<times> 'b" shows "snd (r *\<^sub>C A) = r *\<^sub>C snd A"
sets_le_imp_space_le: fixes A :: "'c measure"   and B :: "'c measure" assumes "sets A \<subseteq> sets B" shows "space A \<subseteq> space B"
distinct_21: fixes x5 :: "char list" shows "Drop \<noteq> Call x5"
is_tree_def: fixes g :: "('a, 'b) rgraph_scheme" shows "is_tree g \<equiv> \<forall>l\<in>vertices g. \<exists>!p. path g p l"
circ_circ_sub_mult: fixes x :: "'a" shows "(1::'a)\<^sup>\<circ> * x\<^sup>\<circ> \<le> x\<^sup>\<circ>\<^sup>\<circ>"
simps_1: fixes A :: "'a mat" shows "swap_row_to_front A 0 = A"
0: shows "(\<Sum>(k, \<gamma>)\<leftarrow>l. real_of_int k * line_integral F basis \<gamma>) = one_chain_line_integral F basis one_chain"
ms3: fixes e1 :: "lam"   and Es1 :: "ctx list"   and e2 :: "lam"   and Es2 :: "ctx list"   and e3 :: "lam"   and Es3 :: "ctx list" assumes "<e1,Es1> \<mapsto>* <e2,Es2>"   and "<e2,Es2> \<mapsto>* <e3,Es3>" shows "<e1,Es1> \<mapsto>* <e3,Es3>"
dual_ex: fixes P :: "'b dual \<Rightarrow> bool" shows "(\<exists>x. P (dual x)) = (\<exists>x'. P x')"
deriv_deg_0: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "f \<in> carrier P"   and "deg R f = 0"   and "a \<in> carrier R" shows "deriv f a = \<zero>"
card_carrier_irr: shows "card carrier_irr = CARD('a) ^ degree f"
bigomega_f_aux: fixes A :: "real" obtains a :: "real" where "A \<le> a" and "\<forall>a'\<ge>a. a' \<in> \<nat> \<longrightarrow> f \<in> \<Omega>(\<lambda>x. real x powr p * (1 + integral (\<lambda>u. g' u / u powr (p + 1)) a' (real x)))"
poly_Nil: fixes x :: "'a" shows "poly [] x = (0::'a)"
pairwise_imp_mult: fixes N :: "'a multiset"   and M :: "'a multiset"   and r :: "('a \<times> 'a) set" assumes "N \<noteq> {#}"   and "\<forall>x\<in>#M. \<exists>y\<in>#N. (x, y) \<in> r" shows "(M, N) \<in> mult r"
ospec: fixes A :: "'b option"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "\<forall>x\<in>set_option A. P x"   and "A = \<lfloor>x\<rfloor>" shows "P x"
ord_Zp_ultrametric_diff: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> nonzero Zp"   and "y \<in> nonzero Zp"   and "x \<noteq> y" shows "min (ord_Zp x) (ord_Zp y) \<le> ord_Zp (x \<ominus> y)"
l_inv_ex: fixes x :: "'a" assumes "x \<in> carrier G" shows "\<exists>y\<in>carrier G. y \<oplus> x = \<zero>"
igWlsInp_def: fixes MOD :: "('b, 'c, 'd, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'g option" shows "igWlsInp MOD delta inp \<equiv> wlsOpS delta \<and> sameDom (arOf delta) inp \<and> liftAll2 (igWls MOD) (arOf delta) inp"
cramer_lemma_mat: fixes A :: "'a mat"   and n :: "nat"   and x :: "'a vec"   and k :: "nat" assumes "A \<in> carrier_mat n n"   and "x \<in> carrier_vec n"   and "k < n" shows "det (replace_col A (A *\<^sub>v x) k) = x $ k * det A"
factor_def: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" shows "a divides\<^bsub>G\<^esub> b = (\<exists>c\<in>carrier G. b = a \<otimes>\<^bsub>G\<^esub> c)"
sub_2: shows "subdivision (K, rev_K) (G0, rev_G0)"
assms_4: fixes a :: "'t \<Rightarrow>\<^sub>0 'b" assumes "a \<in> A"   and "a \<noteq> 0"   and "lt a adds\<^sub>t u" shows "rword (spp_of a) (spp_of p)"
n2p_is: shows "N2p = N1p \<union> Mp"
finsum_rdistr: fixes A :: "'c set"   and a :: "'a"   and f :: "'c \<Rightarrow> 'a" assumes "finite A"   and "a \<in> carrier R"   and "f \<in> A \<rightarrow> carrier R" shows "a \<otimes> finsum R f A = (\<Oplus>i\<in>A. a \<otimes> f i)"
bit_Suc: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'c"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit a (Suc n) = bit (divide a (numeral.numeral one plus (num.Bit0 num.One))) n"
empty: shows "\<forall>r'\<ge>next_round s. votes s r' = Map.empty"
except_fun_def: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" shows "except_fun f S = (\<lambda>x. f x when x \<notin> S)"
dvd_diff_nat: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k dvd m"   and "k dvd n" shows "k dvd m - n"
Polygamma_real_strict_antimono: fixes x :: "real"   and y :: "real"   and n :: "nat" assumes "0 < x"   and "x < y"   and "odd n" shows "Polygamma n y < Polygamma n x"
case_sum_expand_Inr': fixes f :: "'a + 'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'b" assumes "f \<circ> Inl = g" shows "(h = f \<circ> Inr) = (case_sum g h = f)"
Nmlize_Src_2: fixes t :: "'a local.term" assumes "Arr t" shows "\<^bold>\<lfloor>Src t\<^bold>\<rfloor> = Src t"
prems_2: fixes x :: "'a"   and y :: "'b" assumes "x \<in> insert x__ F"   and "y \<in> D" shows "f x y \<in> D"
cd_path_shift0: fixes \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat"   and k :: "nat" assumes "is_path \<pi>" shows "(i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k) = (i - k cd\<^bsup>\<pi> \<guillemotleft> k\<^esup>\<rightarrow> 0)"
mHom_inv: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and N :: "('i, 'f, 'j) Module_scheme"   and m :: "'e"   and f :: "'e \<Rightarrow> 'i" assumes "R module M"   and "R module N"   and "m \<in> carrier M"   and "f \<in> mHom R M N" shows "f (-\<^sub>a\<^bsub>M\<^esub> m) = -\<^sub>a\<^bsub>N\<^esub> f m"
compile_mn_correct: fixes xs :: "nat list"   and n :: "nat"   and f :: "recf"   and r :: "nat"   and ap :: "abc_inst list"   and arity :: "nat"   and fp :: "nat"   and anything :: "nat list" assumes "length xs = n"   and "terminate f (xs @ [r])"   and "\<And>ap arity fp anything. rec_ci f = (ap, arity, fp) \<Longrightarrow> {\<lambda>nl. nl = xs @ r # 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = xs @ r # 0 # 0 \<up> (fp - Suc arity) @ anything}"   and "rec_exec f (xs @ [r]) = 0"   and "\<forall>i<r. terminate f (xs @ [i]) \<and> (\<forall>x xa xb. rec_ci f = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ i # 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ i # rec_exec f (xs @ [i]) # 0 \<up> (xb - Suc xa) @ xc})) \<and> 0 < rec_exec f (xs @ [i])"   and "rec_ci (Mn n f) = (ap, arity, fp)" shows "{\<lambda>nl. nl = xs @ 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = xs @ rec_exec (Mn n f) xs # 0 \<up> (fp - Suc arity) @ anything}"
sop_closedTr0: fixes s :: "'a \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d set"   and i :: "'d \<Rightarrow> 'd"   and z :: "'d"   and f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and r :: "'a"   and x :: "'d" assumes "ipp_cond1 (s_set R s A) i"   and "ipp_cond2 z (s_set R s A) i f"   and "ipp_cond3 z i"   and "inv_ipp z i f (s_set R s A)"   and "zeroA z i f (s_set R s A) z"   and "sop_distr2 R s f i (s_set R s A) z"   and "sop_assoc R s (aug_pm_set z i (s_set R s A))"   and "sop_inv R s i (s_set R s A)"   and "sop_one R s (aug_pm_set z i (s_set R s A))"   and "sop_z R s z"   and "r \<in> carrier R"   and "x \<in> aug_pm_set z i (s_set R s A)" shows "r \<^sub>s\<cdot> x \<in> aug_pm_set z i (s_set R s A)"
remove_child_removes_child: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and remove :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and h :: "(_) heap"   and ptr' :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "l_remove_child_wf2 type_wf known_ptr known_ptrs remove_child heap_is_wellformed get_child_nodes remove"   and "heap_is_wellformed h"   and "h \<turnstile> remove_child ptr' child \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "known_ptrs h"   and "type_wf h" shows "child \<notin> set children"
restr_stack_cong: fixes S :: "stack_elem list"   and V :: "Vars.var set"   and V' :: "Vars.var set" assumes "\<And>x. x \<in> upds S \<Longrightarrow> (x \<in> V) = (x \<in> V')" shows "restr_stack V S = restr_stack V' S"
assms_1: shows "set (xs @ [x]) = set (ys @ [x] @ zs)"
sfg_G_inc_a: fixes a :: "'a" assumes "sfg A a" shows "a \<in> carrier A"
poly_compat: fixes p1 :: "('c monom \<times> 'a) list"   and p2 :: "('c monom \<times> 'a) list"   and p3 :: "('c monom \<times> 'a) list" assumes "p1 \<ge>p p2"   and "p2 >p p3" shows "p1 >p p3"
H2list_def: shows "H2list \<equiv> \<Lambda> f. f\<cdot>lnil"
subset_refl: fixes A :: "'a set" shows "A \<subseteq> A"
embed_measure_IntPairVal: shows "stock_measure (PRODUCT INTEG INTEG) = count_space (range IntPairVal)"
lenext_compat_cons: fixes ys :: "'b list"   and xs :: "'b list"   and gts :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and x :: "'b" assumes "\<lbrakk>length ys = length xs; gts ys xs\<rbrakk> \<Longrightarrow> gts (x # ys) (x # xs)"   and "lenext gts ys xs" shows "lenext gts (x # ys) (x # xs)"
Iff_MP_left': fixes H :: "fm set"   and A :: "fm"   and B :: "fm"   and C :: "fm" assumes "H \<turnstile> A IFF B"   and "insert B H \<turnstile> C" shows "insert A H \<turnstile> C"
pP: shows "bprv (PPf n \<langle>\<phi>\<rangle>)"
face_cycle_succ_cyclic: fixes a :: "'b" shows "cyclic_on H.face_cycle_succ (H.face_cycle_set a)"
A: shows "f \<in> \<omega>[F](\<lambda>_. 1)"
convex_intermediate_ball: fixes a :: "'c"   and r :: "real"   and T :: "'c set" assumes "ball a r \<subseteq> T"   and "T \<subseteq> cball a r" shows "convex T"
assms: shows "\<not> summable (\<lambda>n. f n * z ^ n)"
Gamma_poles: fixes n :: "nat" shows "filterlim Gamma at_infinity (at (- of_nat n))"
hyps: assumes "\<And>v. v \<in> set vs \<Longrightarrow> v \<in> carrier_vec n" shows "M.sumlist vs \<bullet> w = (\<Sum>v\<leftarrow>vs. v \<bullet> w)"
dg_prod_3_is_arrI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and a :: "V"   and b :: "V"   and f :: "V"   and a' :: "V"   and b' :: "V"   and f' :: "V"   and a'' :: "V"   and b'' :: "V"   and f'' :: "V" assumes "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>"   and "digraph \<alpha> \<CC>"   and "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b"   and "f' : a' \<mapsto>\<^bsub>\<BB>\<^esub> b'"   and "f'' : a'' \<mapsto>\<^bsub>\<CC>\<^esub> b''" shows "[f, f', f'']\<^sub>\<circ> : [a, a', a'']\<^sub>\<circ> \<mapsto>\<^bsub>\<AA> \<times>\<^sub>D\<^sub>G\<^sub>3 \<BB> \<times>\<^sub>D\<^sub>G\<^sub>3 \<CC>\<^esub> [b, b', b'']\<^sub>\<circ>"
arity_hd_head: fixes s :: "('s, 'v) tm" assumes "wary s" shows "arity_hd (head s) = arity s + enat (num_args s)"
funcset_compose: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and C :: "'d set" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> C" shows "compose A g f \<in> A \<rightarrow> C"
ord_term_minimum_dgrad_set: fixes d :: "'a \<Rightarrow> nat"   and v :: "'t"   and V :: "'t set"   and m :: "nat" assumes "dickson_grading d"   and "v \<in> V"   and "pp_of_term ` V \<subseteq> dgrad_set d m" obtains u :: "'t" where "u \<in> V" and "\<And>w. w \<prec>\<^sub>t u \<Longrightarrow> w \<notin> V"
pathin_def: fixes X :: "'c topology"   and g :: "real \<Rightarrow> 'c" shows "pathin X g \<equiv> continuous_map (top_of_set {0..1}) X g"
pair_map_pmf2: fixes A :: "'c pmf"   and f :: "'e \<Rightarrow> 'd"   and B :: "'e pmf" shows "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)"
cet_eq: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and t :: "pdf_type"   and e2 :: "cexpr" assumes "\<Gamma> \<turnstile>\<^sub>c e1 : t"   and "\<Gamma> \<turnstile>\<^sub>c e2 : t" shows "\<Gamma> \<turnstile>\<^sub>c e1 =\<^sub>c e2 : BOOL"
simps_3: fixes r :: "real"   and x :: "quat" shows "Im2 (r *\<^sub>R x) = r * Im2 x"
False: shows "\<not> (N = 0 \<or> m = 0)"
lim_ln1_over_ln: shows "(\<lambda>n. ln (real (Suc n)) / ln (real n)) \<longlonglongrightarrow> 1"
cases: fixes a1 :: "pi"   and a2 :: "pi" assumes "a1 \<sim>\<^sub>l a2" obtains P :: "pi" and Q :: "pi" where "a1 = P" and "a2 = Q" and "P \<leadsto>\<^sub>l[Strong_Late_Bisim.bisim] Q" and "Q \<sim>\<^sub>l P"
is_prefix'_def: fixes xs :: "'d list"   and ys :: "'d list" shows "is_prefix' xs ys \<equiv> CATCH (ebind (check_prefix xs ys) (\<lambda>_. ERETURN True)) (\<lambda>_. ERETURN False)"
a_d_zero: fixes x :: "'a" shows "ad x \<cdot> d x = 0"
Abs_counterModel_inverse: fixes f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list" shows "Rep_model (Abs_model (counterM f, counterEvalP f)) = (counterM f, counterEvalP f)"
axioms_1: fixes x :: "'d \<Longrightarrow> 'e" assumes "vwb_lens x" shows "wb_lens x"
P_conforms: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c" assumes "vmc_path G P v0 p \<sigma>" shows "ParityGame.path_conforms_with_strategy G p P \<sigma>"
p\<^sub>0_in_hom: shows "\<guillemotleft>p\<^sub>0 : src p\<^sub>0 \<rightarrow> src s\<^sub>1\<guillemotright>"
r1_7: fixes K1 :: "'a set"   and K2 :: "'a set" assumes "K1 \<in> r1"   and "K2 \<in> r1"   and "K1 \<noteq> K2" shows "interior K1 \<inter> interior K2 = {}"
e'_eq: shows "ball x e' = ball x e \<inter> Y"
matcheq_matchNone_not_matches: fixes m :: "'c match_expr"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d" assumes "matcheq_matchNone m" shows "\<not> matches \<gamma> m p"
reducible\<^sub>dI: fixes p :: "'d poly" assumes "0 < degree p \<Longrightarrow> \<exists>q r. degree q < degree p \<and> degree r < degree p \<and> p = q * r" shows "\<not> irreducible\<^sub>d p"
language_state_for_inputs_union: fixes M :: "('d, 'e, 'f) FSM"   and q :: "'f"   and T1 :: "'d list set"   and T2 :: "'d list set" shows "LS\<^sub>i\<^sub>n M q T1 \<union> LS\<^sub>i\<^sub>n M q T2 = LS\<^sub>i\<^sub>n M q (T1 \<union> T2)"
mult_scaleR_right: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and a :: "real"   and y :: "'d" assumes "class.real_algebra plus zero minus uminus scaleR times" shows "times x (scaleR a y) = scaleR a (times x y)"
oseqp_sos_resp_local_steps: fixes \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and i :: "nat" shows "local_steps (oseqp_sos \<Gamma> i) {i}"
A2: fixes f :: "nat \<Rightarrow> real" shows "map dim_vec (take (length (input_sizes (insert_weights s m1 (\<lambda>i. f (i + a))))) inputs) = input_sizes m1"
simps_1: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "R ^^ 0 = (=)"
b0k: shows "bezout_iterate (interchange_rows A (0::'rows) (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'a))) (nrows A - Suc 0) (0::'rows) (mod_type_class.from_nat k) bezout $ (0::'rows) $ mod_type_class.from_nat k \<noteq> (0::'a)"
nonzero_Complexs_divide: fixes a :: "'b"   and b :: "'b" assumes "a \<in> \<complex>"   and "b \<in> \<complex>"   and "b \<noteq> (0::'b)" shows "a / b \<in> \<complex>"
"1_3": shows "v \<preceq>\<^sub>I w"
assms_3: shows "abs_conv_abscissa (fds_deriv f / f) \<le> s0"
in_arcsD1: fixes u :: "'a"   and v :: "'a" assumes "(u, v) \<in> parcs G" shows "u \<in> pverts G"
set_pmf_code: fixes p :: "'b pmf" shows "set_pmf p = Mapping.keys (mapping_of_pmf p)"
negGorbit_list_V: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and gs :: "'a list"   and T :: "'d \<Rightarrow> 'c"   and as :: "'d list" assumes "FGModule G smult V"   and "set gs \<subseteq> G"   and "T ` set as \<subseteq> V" shows "set (concat (aezfun_scalar_mult.negGorbit_list smult gs T as)) \<subseteq> V"
acc91: shows "Acc level1 sA91 = {sA81} \<union> Acc level1 sA81"
one_step_implies_mult_eq: fixes r :: "('a \<times> 'a) set"   and K :: "'a multiset"   and J :: "'a multiset"   and I :: "'a multiset" assumes "trans r"   and "set_mset K \<subseteq> r \<down>m J" shows "(I + K, I + J) \<in> mult_eq r"
ses_inputs_outputs_disjoint_def: fixes SES :: "('c, 'd) SES_rec" shows "ses_inputs_outputs_disjoint SES \<equiv> I\<^bsub>SES\<^esub> \<inter> O\<^bsub>SES\<^esub> = {}"
fun_type_arg_cong_2: fixes f :: "'fun"   and g :: "'fun"   and x :: "('fun, 'var) Term.term"   and X :: "('fun, 'var) Term.term list"   and y :: "('fun, 'var) Term.term"   and Y :: "('fun, 'var) Term.term list" assumes "f \<in> \<Union> (range \<Sigma>n) - \<C>"   and "g \<in> \<Union> (range \<Sigma>n) - \<C>"   and "\<Gamma> (Fun f (x # X)) = \<Gamma> (Fun g (y # Y))" shows "\<Gamma> (Fun f X) = \<Gamma> (Fun g Y)"
process_prop_2_def: fixes P :: "('a list \<times> 'a set) set \<times> 'a list set" shows "process_prop_2 P \<equiv> \<forall>xs x X. (xs @ [x], X) \<in> fst P \<longrightarrow> (xs, {}) \<in> fst P"
prems_2: fixes i :: "nat"   and j :: "nat" assumes "i \<le> Suc (Suc n)"   and "j \<le> Suc (Suc n)"   and "i \<noteq> j" shows "I i <+> I j = carrier R"
degree_diff_less: fixes p :: "'b poly"   and n :: "nat"   and q :: "'b poly" assumes "degree p < n"   and "degree q < n" shows "degree (p - q) < n"
map_iterator_rev_linord_foldli_conv: fixes iti :: "('c \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'd \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and m :: "'a \<Rightarrow> 'd option" shows "map_iterator_rev_linord iti m = (\<exists>l0. distinct (map fst l0) \<and> m = map_of l0 \<and> sorted (rev (map fst l0)) \<and> iti = foldli l0)"
uminus_zero_vec_eq: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "(- v = 0\<^sub>v n) = (v = 0\<^sub>v n)"
rebase_dvd_axioms: shows "rebase_dvd TYPE('q) TYPE('p)"
admissibleD: fixes lub :: "'f set \<Rightarrow> 'f"   and ord :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and P :: "'f \<Rightarrow> bool"   and A :: "'f set" assumes "ccpo.admissible lub ord P"   and "Complete_Partial_Order.chain ord A"   and "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> P x" shows "P (lub A)"
alg_equiv_inv_auto_2: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and T\<^sub>1 :: "ty"   and T\<^sub>2 :: "ty" assumes "\<Gamma> \<turnstile> s \<Leftrightarrow> t : T\<^sub>1\<rightarrow>T\<^sub>2" obtains x :: "name" where "x \<sharp> (\<Gamma>, s, t)" and "(x, T\<^sub>1) # \<Gamma> \<turnstile> App s (Var x) \<Leftrightarrow> App t (Var x) : T\<^sub>2"
Gets_imp_knows_Spy: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> ns_public" shows "X \<in> knows Spy evs"
rep_eq: fixes x :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xa :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and xb :: "'e \<Rightarrow>\<^sub>L 'g"   and xc :: "'f \<Rightarrow>\<^sub>L 'h" shows "rel_blinfun x xa xb xc = (x ===> xa) (blinfun_apply xb) (blinfun_apply xc)"
Dom_dom: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr f" shows "Dom (local.dom f) = Dom f"
card_le_dim_spanning: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and Basis :: "'e set"   and B :: "'e set"   and V :: "'e set" assumes "finite_dimensional_vector_space scale Basis"   and "B \<subseteq> V"   and "V \<subseteq> module.span scale B"   and "finite B"   and "card B \<le> vector_space.dim scale V" shows "\<not> module.dependent scale B"
map_def: fixes f :: "'a" shows "Aop.map f = (if Aop.arr f then f else Aop.null)"
seqI': fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and c :: "'a \<times> 'a" assumes "B.VV.in_hom f a b"   and "B.VV.in_hom g b c" shows "B.VV.seq g f"
singular_relcycle: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "singular_relcycle p X S c = (singular_chain p X c \<and> singular_chain (p - 1) (subtopology X S) (chain_boundary p c))"
l_zero: fixes x :: "'c" assumes "x \<in> carrier M" shows "\<zero>\<^bsub>M\<^esub> \<oplus>\<^bsub>M\<^esub> x = x"
vertices: fixes S :: "'b set"   and w :: "'b"   and s :: "'b" assumes "CoxeterComplex S"   and "w \<in> \<langle>S\<rangle>"   and "s \<in> S" shows "w +o \<langle>S - {s}\<rangle> \<in> \<Union> (CoxeterComplex.TheComplex S)"
quasi_trans_def: fixes r :: "('b \<times> 'b) set" shows "quasi_trans r \<equiv> \<forall>x y z. x \<^bsub>r\<^esub>\<prec> y \<and> y \<^bsub>r\<^esub>\<prec> z \<longrightarrow> x \<^bsub>r\<^esub>\<prec> z"
Collect_subset: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "rel_ext (\<lambda>x. x \<in> A \<and> P x) \<subseteq> A"
rcs_tool1: fixes H :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "b \<in> carrier G"   and "b \<cdot> \<rho> a \<in> H" shows "a \<cdot> \<rho> b \<in> H"
complex_scaleR: fixes r :: "real"   and a :: "real"   and b :: "real" shows "r *\<^sub>R Complex a b = Complex (r * a) (r * b)"
sig_red_single_lt: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "sig_red_single sing_reg top_tail p q f t" shows "lt q \<preceq>\<^sub>t lt p"
interval_lebesgue_integrable_mult_right: fixes c :: "'b"   and M :: "real measure"   and a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'b" assumes "c \<noteq> (0::'b) \<Longrightarrow> interval_lebesgue_integrable M a b f" shows "interval_lebesgue_integrable M a b (\<lambda>x. c * f x)"
Map_comp: fixes t' :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and t :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and a' :: "'a"   and a :: "'a" assumes "seq t' t"   and "A.seq a' a" shows "Map (t' \<cdot> t) (a' \<cdot>\<^sub>A a) = Map t' a' \<cdot>\<^sub>B Map t a"
fpxs_eq_iff: fixes f :: "'b fpxs"   and g :: "'b fpxs" shows "(f = g) = (\<forall>r. fpxs_nth f r = fpxs_nth g r)"
lcdm: shows "l' cd\<^bsup>\<pi>'\<^esup>\<rightarrow> m"
gcd_le1_nat: fixes a :: "nat"   and b :: "nat" assumes "a \<noteq> 0" shows "gcd a b \<le> a"
assms: fixes x :: "'a \<times> 'b" assumes "x \<in> S" shows "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S"
simps: fixes i :: "int"   and j :: "int" shows "local.min_wpl i j = (if j < i then 0 else (MIN k\<in>{i..j}. local.min_wpl i (k - 1) + local.min_wpl (k + 1) j) + w i j)"
freshChainPermSimp_2: fixes xvec :: "name list"   and p :: "(name \<times> name) list" assumes "xvec \<sharp>* p" shows "rev p \<bullet> xvec = xvec"
exe_wf_tcsigs_def: fixes cs :: "(String.literal \<times> String.literal) list"   and arss :: "('a \<times> (String.literal \<times> String.literal set list) list) list" shows "exe_wf_tcsigs cs arss \<equiv> exe_coregular_tcsigs cs arss \<and> exe_complete_tcsigs cs arss \<and> exe_consistent_length_tcsigs arss \<and> exe_all_normalized_and_ex_tcsigs cs arss"
f_last_message_hold_nth: fixes n :: "nat"   and xs :: "'b message_af list"   and k :: "nat" assumes "n < length xs" shows "xs \<longmapsto> k ! n = last_message (xs \<up> (n div k * k) \<down> Suc (n mod k))"
assms_2: shows "lin_ts_mmuaux aux = ts # tss''"
exponent_p_a_m_k_equation: fixes m :: "nat"   and k :: "nat"   and p :: "nat"   and a :: "nat" assumes "0 < m"   and "0 < k"   and "p \<noteq> 0"   and "k < p ^ a" shows "multiplicity p (p ^ a * m - k) = multiplicity p (p ^ a - k)"
2810: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - ((y \<squnion> top) \<sqinter> x) = - ((y \<squnion> top) \<sqinter> x)"