hcju/isabelleps huggingface dataset:
independent_vnmI2: fixes C :: "'a pmf set"   and P :: "('a pmf \<times> 'a pmf) set" assumes "\<And>p q r \<alpha>. \<lbrakk>p \<in> C; q \<in> C; r \<in> C; \<alpha> \<in> {0<..1}\<rbrakk> \<Longrightarrow> p \<succeq>[P] q = mix_pmf \<alpha> p r \<succeq>[P] mix_pmf \<alpha> q r" shows "independent_vnm C P"
mult_le_0_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less_eq (times a b) zero = (less_eq zero a \<and> less_eq b zero \<or> less_eq a zero \<and> less_eq zero b)"
i_drop_0: fixes f :: "nat \<Rightarrow> 'd" shows "f \<Up> 0 = f"
Nml_HcompD_4: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Nml (t \<^bold>\<star> u)" shows "Nml u"
sin_inj_pi: fixes x :: "real"   and y :: "real" assumes "- (pi / 2) \<le> x"   and "x \<le> pi / 2"   and "- (pi / 2) \<le> y"   and "y \<le> pi / 2"   and "sin x = sin y" shows "x = y"
intro: fixes \<G> :: "'a cyclic_group" assumes "dis_log_alt \<G>"   and "Cyclic_Group.cyclic_group \<G>" shows "dis_log_alt_reductions \<G>"
s_r: shows "s r = Some (\<sigma>, \<tau>, \<E> [Assign (VE (Loc l)) (VE v)])"
max_factor_degree_def: fixes degs :: "nat list" shows "max_factor_degree degs = (let ds = sort degs in sum_list (drop (length ds div 2) ds))"
sum_list_map_dropWhile0: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" assumes "f (0::'c) = (0::'b)" shows "sum_list (map f (dropWhile ((=) (0::'c)) xs)) = sum_list (map f xs)"
Diff_insert: fixes A :: "'g set"   and a :: "'g"   and B :: "'g set" shows "A - insert a B = A - B - {a}"
emeasure_dens_ctxt_measure_insert: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and F :: "(nat \<Rightarrow> val) \<Rightarrow> val measure"   and t :: "pdf_type"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal"   and \<rho> :: "nat \<Rightarrow> val"   and X :: "(nat \<Rightarrow> val) set" assumes "density_context V V' \<Gamma> \<delta>"   and "has_parametrized_subprob_density (state_measure (V \<union> V') \<Gamma>) F (stock_measure t) f"   and "\<rho> \<in> space (state_measure (Suc ` V') (case_nat t \<Gamma>))"   and "X \<in> sets (dens_ctxt_measure (shift_var_set V, Suc ` V', case_nat t \<Gamma>, insert_dens V V' f \<delta>) \<rho>)" shows "emeasure (dens_ctxt_measure (shift_var_set V, Suc ` V', case_nat t \<Gamma>, insert_dens V V' f \<delta>) \<rho>) X = \<integral>\<^sup>+ x. insert_dens V V' f \<delta> (merge (shift_var_set V) (Suc ` V') (x, \<rho>)) * indicator X (merge (shift_var_set V) (Suc ` V') (x, \<rho>)) \<partial>state_measure (shift_var_set V) (case_nat t \<Gamma>)"
complex_cnj_neg_numeral: fixes w :: "num" shows "cnj (- numeral w) = - numeral w"
NewHeapElem_start_heap_obsD: fixes a :: "'addr"   and hT :: "htype" assumes "wf_syscls P"   and "NewHeapElem a hT \<in> set h.start_heap_obs" shows "typeof_addr a = \<lfloor>hT\<rfloor>"
inverse_norm: fixes n :: "int"   and w :: "'b word" shows "(word_of_int n = w) = (uint w = take_bit LENGTH('b) n)"
sets_lebesgue_on_refl: fixes S :: "'a set" shows "S \<in> sets (lebesgue_on S)"
simps_1: fixes vs :: "'b list"   and bs :: "'b list" shows "Ifm vs bs T = True"
config'_append2: fixes A :: "('state \<Rightarrow> 'b) \<times> ('state \<times> 'b \<Rightarrow> 'request \<Rightarrow> 'answer \<times> 'b)"   and s :: "'state \<times> 'b"   and xs :: "'request list"   and ys :: "'request list" shows "config' A s (xs @ ys) = config' A (config' A s xs) ys"
stabilizer_def: fixes G :: "('c, 'd) partial_object_scheme"   and \<phi> :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e" shows "stabilizer G \<phi> x = {g \<in> carrier G. \<phi> g x = x}"
factorization_m_def: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "pl.factorization_m f cfs \<equiv> case cfs of (c, fs) \<Rightarrow> pl.eq_m f (smult c (\<Prod>\<^sub># fs)) \<and> (\<forall>f\<in>#fs. pl.irreducible\<^sub>d_m f \<and> monic (pl.Mp f))"
scaleR_sum_right: fixes a :: "real"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "a *\<^sub>R sum f A = (\<Sum>x\<in>A. a *\<^sub>R f x)"
comp_right_one: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv" shows "times x one = x"
retract_of_space_topspace: fixes X :: "'c topology" shows "topspace X retract_of_space X"
ipassmt_sanity_nowildcards_match_iface: fixes ipassmt :: "iface \<Rightarrow> ('a word \<times> nat) list option"   and ifce2 :: "char list"   and ifce :: "iface"   and a :: "('a word \<times> nat) list" assumes "ipassmt_sanity_nowildcards ipassmt"   and "ipassmt (Iface ifce2) = None"   and "ipassmt ifce = Some a" shows "\<not> match_iface ifce ifce2"
lookup_bulkload: fixes xs :: "('a \<times> 'b) list" shows "RBT.lookup (RBT.bulkload xs) = map_of xs"
lsc_hull_of_convex_agrees_onRI: fixes f :: "'b \<Rightarrow> ereal" assumes "Lower_Semicontinuous.convex_on UNIV f" shows "\<forall>x\<in>rel_interior (domain f). f x = lsc_hull f x"
predicate_eq_eq: fixes eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "eq = (=)" shows "predicate_eq eq = (=)"
size_gen_4: fixes x41 :: "stmt"   and x42 :: "stmt" shows "size_stmt (Seq x41 x42) = size_stmt x41 + size_stmt x42 + Suc 0"
alpha_qVar_iff: fixes X :: "('a, 'b, 'c, 'd, 'e) qTerm"   and xs' :: "'c"   and x' :: "'d" shows "(X #= qVar xs' x') = (X = qVar xs' x')"
longest_path: fixes n :: "'v" assumes "finite E"   and "n \<in> V" shows "\<exists>v max_path. is_trail v max_path n \<and> (\<forall>v'. \<forall>e\<in>E. \<not> is_trail v' (e # max_path) n)"
trace_reject: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and t :: "(String.literal \<times> value list) list" shows "(\<not> recognises_execution e s r ((l, i) # t)) = (possible_steps e s r l i = {||} \<or> (\<forall>(s', T)|\<in>|possible_steps e s r l i. \<not> recognises_execution e s' (evaluate_updates T i r) t))"
cond_spmf_try2: fixes p :: "'h spmf"   and A :: "'h set"   and q :: "'h spmf" assumes "set_spmf p \<inter> A = {}" shows "cond_spmf (TRY p ELSE q) A = (if lossless_spmf p then return_pmf None else cond_spmf q A)"
c2_2: fixes n :: "nat" assumes "1 \<le> n" shows "norm (g (real n) - g (real (Suc n))) \<le> c2 * norm (h (real n))"
precondition_def: fixes kprecondition :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'a"   and a :: "'b option" shows "Kernel.precondition kprecondition s a \<equiv> a \<rightharpoonup> kprecondition s"
intros: fixes xs :: "nat list"   and ys :: "nat list"   and a :: "nat list"   and as :: "nat list list"   and b :: "nat list"   and bs :: "nat list list"   and ka :: "nat"   and kb :: "nat"   and c :: "nat list"   and d :: "nat list"   and zs :: "nat list" assumes "length xs < length ys"   and "xs = concat (a # as)"   and "ys = concat (b # bs)"   and "a # as \<in> lists (- {[]})"   and "b # bs \<in> lists (- {[]})"   and "length (a # as) = ka"   and "length (b # bs) = kb"   and "c = acc_lengths 0 (a # as)"   and "d = acc_lengths 0 (b # bs)"   and "zs = concat [c, a, d, b] @ interact as bs"   and "strict_sorted zs" shows "Form_Body ka kb xs ys zs"
le_trans_def: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and u :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "le_trans t u \<equiv> \<forall>P. sound P \<longrightarrow> t P \<tturnstile> u P"
INF_greatest: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'d set"   and u :: "'c"   and f :: "'d \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq u (f i)" shows "less_eq u (Inf (f ` A))"
diag_four_block_mat: fixes A :: "'b mat"   and n1 :: "nat"   and D :: "'b mat"   and n2 :: "nat"   and B :: "'b mat"   and C :: "'b mat" assumes "A \<in> carrier_mat n1 n1"   and "D \<in> carrier_mat n2 n2" shows "diag_mat (four_block_mat A B C D) = diag_mat A @ diag_mat D"
corec_code: fixes p :: "'c \<Rightarrow> bool"   and g1 :: "'c \<Rightarrow> 'b"   and g21 :: "'c \<Rightarrow> 'a"   and q22 :: "'c \<Rightarrow> bool"   and g221 :: "'c \<Rightarrow> ('a, 'b) tllist"   and g222 :: "'c \<Rightarrow> 'c"   and a :: "'c" shows "corec_tllist p g1 g21 q22 g221 g222 a = (if p a then TNil (g1 a) else TCons (g21 a) (if q22 a then g221 a else corec_tllist p g1 g21 q22 g221 g222 (g222 a)))"
hom_component_of_homogeneous: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and n :: "nat" assumes "homogeneous p" shows "hom_component p n = (p when n = poly_deg p)"
rep_eq: fixes x :: "uint16"   and xa :: "uint16" shows "Rep_uint16 (x div xa) = Rep_uint16 x div Rep_uint16 xa"
mHom_0: fixes N :: "('e, 'b, 'f) Module_scheme"   and f :: "'a \<Rightarrow> 'e" assumes "R module N"   and "f \<in> mHom R M N" shows "f \<zero> = \<zero>\<^bsub>N\<^esub>"
add_mset_disjoint_2: fixes a :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" shows "(\<Empt> = add_mset a A \<inter># B) = (a \<notin># B \<and> \<Empt> = A \<inter># B)"
simps_6: fixes v :: "'a rexp" shows "pnPlus (Star v) Zero = Star v"
multiplicity_geI: fixes n :: "nat" assumes "p ^ n dvd x" shows "n \<le> multiplicity p x"
not_below2not_eq: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.po below"   and "\<not> below x y" shows "x \<noteq> y"
two_chain_boundary_is_boundary_chain: fixes twoChain :: "(real \<times> real \<Rightarrow> real \<times> real) set" shows "boundary_chain (two_chain_boundary twoChain)"
it_paths_simps: fixes a2 :: "nat list"   and a1 :: "'a rose_tree" shows "(a2 \<in> it_paths a1) = ((\<exists>t. a1 = t \<and> a2 = []) \<or> (\<exists>i t t' is. a1 = t \<and> a2 = i # is \<and> i < length (children t) \<and> children t ! i = t' \<and> is \<in> it_paths t'))"
1: shows "v\<^sub>1 @ l =\<^sub>F w\<^sub>1"
assms: fixes x :: "'a" assumes "x \<in> A" shows "\<bar>f x\<bar> \<noteq> \<infinity>"
ceq: shows "subgroup_generated (reduced_homology_group (int p) (nsphere p)) {c} = reduced_homology_group (int p) (nsphere p)"
x: fixes y :: "real" assumes "y \<in> {0..1}" shows "norm (bernpoly n y) \<le> norm (bernpoly n x)"
Rep: shows "Rep = (\<lambda>b'. SOME b. A b b')"
fold_code_1: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c" shows "RBT_Impl.fold f rbt.Empty x = x"
tendsto_inverse: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and F :: "'a filter" assumes "(f \<longlongrightarrow> a) F"   and "a \<noteq> (0::'b)" shows "((\<lambda>x. inverse (f x)) \<longlongrightarrow> inverse a) F"
uncurry_pair: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and x :: "'e"   and y :: "'f" shows "cuncurry f (cpair x y) = f x y"
max_bag_card_in_bag_cards: assumes "V \<noteq> {}" shows "max_bag_card \<in> bag_cards"
blinding_of_F_def: fixes bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bL :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "blinding_of_F bo bL = rel_sum (=) (rel_prod bo bL)"
simps_6: fixes f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'e \<Rightarrow> 'c"   and x2 :: "'e" shows "(case Inr x2 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f2 x2"
target: fixes T1 :: "'procT"   and T2 :: "'procT"   and TRel :: "('procT \<times> 'procT) set"   and SRel :: "('procS \<times> 'procS) set" assumes "(T1, T2) \<in> TRel" shows "TargetTerm T1 \<sim>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> TargetTerm T2"
join2_def: fixes l :: "('a \<times> 'b) tree"   and r :: "('a \<times> 'b) tree" shows "join2 l r = (if r = \<langle>\<rangle> then l else let (m, r') = split_min r in join l m r')"
bdd_exhaust: fixes a :: "bdd"   and b :: "bdd" assumes "dpm a = dpm b"   and "dcl a = dcl b" shows "a = b"
getFrN_length: fixes xs :: "'var list"   and ts :: "'trm list"   and \<phi>s :: "'fmla list"   and n :: "nat" assumes "set xs \<subseteq> var"   and "set ts \<subseteq> trm"   and "set \<phi>s \<subseteq> fmla" shows "length (getFrN xs ts \<phi>s n) = n"
les_def: fixes R :: "'a \<Rightarrow> ('b \<times> 'c ifex) set"   and s :: "'a"   and s' :: "'a" shows "bdd_impl_pre.les R s s' \<equiv> \<forall>ni n. (ni, n) \<in> R s \<longrightarrow> (ni, n) \<in> R s'"
all_distinct_right: fixes l :: "'b tree"   and x :: "'b"   and b :: "bool"   and r :: "'b tree" assumes "all_distinct (Node l x b r)" shows "all_distinct r"
left_gpv_def: shows "left_gpv \<equiv> corec_gpv (\<lambda>gpv. map_spmf (map_generat id Inl (\<lambda>uu input. case input of Inl input' \<Rightarrow> Inr (uu input') | Inr b \<Rightarrow> Inl Fail)) (the_gpv gpv))"
rank_Some_time: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat"   and j :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w x n = Some j" shows "x \<le> n"
set_plus_def: fixes A :: "'d set"   and B :: "'d set" shows "A + B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a + b}"
mult_one_left: fixes n :: "eint" shows "1 * n = n"
le_less: fixes x :: "'t"   and y :: "'t" shows "(x \<preceq>\<^sub>t y) = (x \<prec>\<^sub>t y \<or> x = y)"
fmlookup_filter: fixes P :: "'b \<Rightarrow> bool"   and m :: "('b, 'a) fmap"   and x :: "'b" shows "fmlookup (fmfilter P m) x = (if P x then fmlookup m x else None)"
measurable_pred_stream: fixes P :: "'a \<Rightarrow> bool" shows "Measurable.pred (stream_space (count_space UNIV)) (pred_stream P)"
list_gcd: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "gcd_list xs dvd x"
run_step_break_imp_lfilled: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and es :: "e list"   and s' :: "s"   and vs' :: "v list"   and n :: "nat"   and res :: "v list" assumes "run_step d i (s, vs, es) = (s', vs', RSBreak n res)" shows "s = s' \<and> vs = vs' \<and> (\<exists>n' lfilled es_c. n \<le> n' \<and> Lfilled_exact (n' - n) lfilled (vs_to_es res @ [$Br n'] @ es_c) es)"
is_none_bind: fixes f :: "'e option"   and g :: "'e \<Rightarrow> 'd option" shows "Option.is_none (f \<bind> g) = (Option.is_none f \<or> Option.is_none (g (the f)))"
intros_2: fixes slices :: "'b \<Rightarrow> 'b set set"   and W :: "'b set"   and x :: "'c" assumes "elementary slices W" shows "elementary.path [x]"
equality: fixes r :: "('e, 'f) monoid_scheme"   and r' :: "('e, 'f) monoid_scheme" assumes "carrier r = carrier r'"   and "(\<otimes>\<^bsub>r\<^esub>) = (\<otimes>\<^bsub>r'\<^esub>)"   and "\<one>\<^bsub>r\<^esub> = \<one>\<^bsub>r'\<^esub>"   and "monoid.more r = monoid.more r'" shows "r = r'"
interaction_any_bounded_converter_mono: fixes conv :: "('a, 'b, 'c, 'd) converter"   and n :: "enat"   and m :: "enat" assumes "interaction_any_bounded_converter conv n"   and "n \<le> m" shows "interaction_any_bounded_converter conv m"
assms_3: fixes h1 :: "'a"   and h2 :: "'a" assumes "h1 \<in> H"   and "h2 \<in> H"   and "h1 \<otimes> h2 = \<zero>" shows "h1 = \<zero> \<or> h2 = \<zero>"
err_def: fixes A :: "'a set" shows "err A = insert Err {OK x |x. x \<in> A}"
M_All: fixes P :: "M \<Rightarrow> bool" shows "(\<forall>m. P m) = (\<forall>m\<in>set [M1, M2, M3]. P m)"
DBM_le_subset: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and M' :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and u :: "'d \<Rightarrow> 'c"   and v :: "'d \<Rightarrow> nat" assumes "\<forall>i j. i \<le> n \<longrightarrow> j \<le> n \<longrightarrow> M i j \<preceq> M' i j"   and "u \<in> [M]\<^bsub>v,n\<^esub>" shows "u \<in> [M']\<^bsub>v,n\<^esub>"
delete_notin_dom: fixes k :: "'a"   and al :: "('a \<times> 'b) list" shows "k \<notin> fst ` set (delete k al)"
cycle_free_loop_dest: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "nat"   and n :: "nat"   and xs :: "nat list"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "class.linordered_ab_monoid_add plus zero less_eq less"   and "i \<le> n"   and "set xs \<subseteq> {0..n}"   and "linordered_ab_monoid_add.cycle_free plus zero less_eq m n" shows "less_eq zero (linordered_ab_monoid_add.len plus m i i xs)"
simps_3: fixes y :: "'b"   and ys :: "['b]" shows "eq\<cdot>[]\<cdot>(y : ys) = FF"
simps_3: fixes x :: "'b"   and y :: "'b" shows "(extC.Val x \<le> extC.Val y) = (x \<le> y)"
state_change_on_neither_1: fixes i :: "nat"   and t :: "nat" assumes "gap i (Suc t) = gap i t" shows "change_on_neither (block i t) (gap i t)"
simps_2: fixes b :: "float interval option"   and f :: "float interval \<Rightarrow> float interval \<Rightarrow> float interval option" shows "lift_bin None b f = None"
P_def: shows "P = (\<lambda>i. \<Prod>X\<in>{X. X \<subseteq> {..<n} \<and> card X = i}. [:- root' X, 1:])"
take_all: fixes xs :: "'e list"   and n :: "nat" assumes "length xs \<le> n" shows "take n xs = xs"
vector2_append1_def: fixes v :: "(real, 2) vec" shows "vector2_append1 v = vector [v $ 1, v $ 2, 1]"
prems_2: shows "Suc n < \<^bold>|w\<^bold>|"
xis: shows "xis = concat (map (\<lambda>(p, i). map (\<lambda>r. (r, i)) (roots_of_complex_main p)) pis)"
c\<^sub>2'_def: shows "c\<^sub>2' = c\<^sub>h"
part_equiv_def: fixes R :: "('b \<times> 'b) set" shows "part_equiv R = (sym R \<and> trans R)"
isom_gch_unitsTr4: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and E :: "('i, 'j) Group_scheme" assumes "Group F"   and "Group G"   and "Ugp E"   and "F \<cong> G"   and "F \<cong> E" shows "G \<cong> E"
FOREACHcdi_def: fixes I :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> bool" shows "FOREACHcdi I \<equiv> FOREACHcd"
word_arith_wis_5: fixes a :: "'b word" shows "word_succ a = word_of_int (uint a + 1)"
sim_step: fixes s :: "nat \<Rightarrow> 'a"   and t :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "s \<approx> t" shows "\<exists>m. s |\<^sub>s n \<approx> t |\<^sub>s m \<and> (s |\<^sub>s Suc n \<approx> t |\<^sub>s Suc m \<or> s |\<^sub>s Suc n \<approx> t |\<^sub>s m)"
non_volatile_owned_or_read_only_read_only_reads_eq': fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "non_volatile_owned_or_read_only False \<S> \<O> sb"   and "\<forall>a\<in>read_only_reads (acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<O>) (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb). a \<in> read_only \<S> \<longrightarrow> a \<in> read_only \<S>'" shows "non_volatile_owned_or_read_only False \<S>' \<O> sb"
pdigraph_axioms: shows "pdigraph \<alpha> I \<AA>"
inner_simps_6: fixes x :: "'c"   and r :: "real"   and y :: "'c" shows "x \<bullet> r *\<^sub>R y = r * (x \<bullet> y)"
right_minus_eq: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.group_add minus plus zero uminus" shows "(minus a b = zero) = (a = b)"
eta: shows "(\<lambda>t. \<chi>i. fst (mk_v I ODE (sol 0, b) (sol t)) $ i) = (\<lambda>t. fst (mk_v I ODE (sol 0, b) (sol t)))"
fin2: fixes X' :: "'v set" assumes "X' \<subset> X" shows "integral\<^sup>N (count_space X') (weight \<Gamma>) \<noteq> \<top>"
antisym_mono_imp_fp_complete: fixes CC :: "'a set set" assumes "antisymmetric A (\<sqsubseteq>)"   and "CC-complete A (\<sqsubseteq>)"   and "\<forall>C\<subseteq>A. well_related_set C (\<sqsubseteq>) \<longrightarrow> C \<in> CC"   and "\<forall>X\<in>CC. \<forall>Y\<in>CC. X \<sqsubseteq>\<^sup>s Y \<longrightarrow> X \<union> Y \<in> CC"   and "monotone_on A (\<sqsubseteq>) (\<sqsubseteq>) f" shows "CC-complete {p \<in> A. f p = p} (\<sqsubseteq>)"
singleDSourceEmpty_Acc: fixes i :: "AbstrLevelsID"   and C :: "CSet"   and S :: "CSet" assumes "DAcc i C = {S}"   and "isNotDSource i S" shows "Acc i C = {S}"
e: shows "terminal_arrow_from_functor C C (\<lambda>x. local.prod x b) x c e"
strict_monoD: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "class.order less_eq less"   and "order.strict_mono less f"   and "less x y" shows "f x < f y"
unat_split: fixes P :: "nat \<Rightarrow> bool"   and x :: "'a word" shows "P (unat x) = (\<forall>n. word_of_nat n = x \<and> n < 2 ^ LENGTH('a) \<longrightarrow> P n)"
sorted_quicksort: fixes xs :: "'a list" shows "sorted (quicksort xs)"
polyadd_normh: fixes p :: "poly"   and n0 :: "nat"   and q :: "poly"   and n1 :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1" shows "isnpolyh (p +\<^sub>p q) (min n0 n1)"
tendsto_lowerbound: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and x :: "'a"   and F :: "'b filter"   and a :: "'a" assumes "class.linorder_topology less_eq less open"   and "filterlim f (topological_space.nhds open x) F"   and "\<forall>\<^sub>F i in F. less_eq a (f i)"   and "F \<noteq> bot" shows "less_eq a x"
prios_ins_special: fixes k :: "'a"   and p :: "'b"   and t :: "('a \<times> 'b) tree"   and l :: "('a \<times> 'b) tree"   and k' :: "'a"   and p' :: "'b"   and r :: "('a \<times> 'b) tree" assumes "ins k p t = \<langle>l, (k', p'), r\<rangle>"   and "p' = p"   and "p \<in> prios r \<union> prios l" shows "p \<in> prios t"
elims_2: fixes x :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> 'a option) \<times> 'b" assumes "is_std_sig x" obtains ctf :: "String.literal \<Rightarrow> typ option" and arf :: "String.literal \<Rightarrow> 'a option" and uu_ :: "'b" where "x = (ctf, arf, uu_)" and "arf STR ''fun'' = Some (2::'a) \<and> arf STR ''prop'' = Some (0::'a) \<and> arf STR ''itself'' = Some (1::'a) \<and> ctf STR ''Pure.eq'' = Some (Core.aT \<rightarrow> Core.aT \<rightarrow> propT) \<and> ctf STR ''Pure.all'' = Some ((Core.aT \<rightarrow> propT) \<rightarrow> propT) \<and> ctf STR ''Pure.imp'' = Some (propT \<rightarrow> propT \<rightarrow> propT) \<and> ctf STR ''Pure.type'' = Some (itselfT Core.aT)"
P2: shows "Bet D E D0 \<and> Cong D E E D0"
ln_mono: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "0 < y"   and "x \<le> y" shows "ln x \<le> ln y"
second_elt_is_hd_tl: fixes x :: "'a list"   and a :: "'a"   and x' :: "'a list" shows "tl x = a # x' \<longrightarrow> a = x ! 1"
finprod_split: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'c"   and a :: "'e" assumes "finite A"   and "f ` A \<subseteq> carrier M"   and "a \<in> A" shows "finsum M f A = f a \<oplus>\<^bsub>M\<^esub> finsum M f (A - {a})"
hcpow_not_zero: fixes r :: "complex star"   and n :: "nat star" assumes "r \<noteq> 0" shows "r pow n \<noteq> 0"
gallery_append_reduce2: fixes X :: "'b set set"   and xs :: "'b set list"   and ys :: "'b set list" assumes "ChamberComplex X"   and "SimplicialComplex.maxsimpchain X (xs @ ys)" shows "SimplicialComplex.maxsimpchain X ys"