hcju/isabelleps huggingface dataset:
frontier_properties: fixes S :: "complex set" assumes "SC_Chain S"   and "simply_connected S" shows "if bounded S then connected (frontier S) else \<forall>C\<in>components (frontier S). \<not> bounded C"
sel_4: fixes x4 :: "'d" shows "occurs_on (Snapshot x4) = x4"
to_tree_order_wf_get_root_node_wf_is_l_to_tree_order_wf_get_root_node_wf: shows "l_to_tree_order_wf_get_root_node_wf ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM.to_tree_order Shadow_DOM.get_root_node Shadow_DOM.heap_is_wellformed"
simps_1: fixes xs :: "'h list" shows "??.Nested_Environment.lookup_option None xs = None"
IH: assumes "wf_infpath f"   and "n\<^sub>1 \<le> n\<^sub>2" shows "\<exists>a. f n\<^sub>1 @ a = f n\<^sub>2"
f_join_def: fixes xs :: "'b list"   and I :: "nat set" shows "xs \<Join>\<^sub> I \<equiv> f_join_aux xs 0 I"
gram_schmidt0_result_3: fixes ws :: "'b vec list"   and n :: "nat" assumes "set ws \<subseteq> carrier_vec n" shows "corthogonal (gram_schmidt0 n ws)"
step_3: obtains p' :: "'t \<Rightarrow>\<^sub>0 'b" and q' :: "'t \<Rightarrow>\<^sub>0 'b" where "(red F)\<^sup>*\<^sup>* p p'" and "(red F)\<^sup>*\<^sup>* q q'" and "y = p' - q'"
simps_8: shows "eval_ternary_And TernaryUnknown TernaryTrue = TernaryUnknown"
enum_Suc: fixes i :: "nat" assumes "i < n" shows "s.enum (Suc i) = (s.enum i)(u_s i := Suc (s.enum i (u_s i)))"
gate_on_state_is_state: fixes n :: "nat"   and A :: "complex Matrix.mat"   and v :: "complex Matrix.mat" assumes "gate n A"   and "state n v" shows "state n (A * v)"
in_set_member: fixes x :: "'f"   and xs :: "'f list" shows "(x \<in> set xs) = List.member xs x"
compactin_discrete_topology: fixes X :: "'c set"   and S :: "'c set" shows "compactin (discrete_topology X) S = (S \<subseteq> X \<and> finite S)"
neg_inf_eq_sup: fixes a :: "'a"   and b :: "'a" shows "- inf a b = sup (- a) (- b)"
sol: shows "\<exists>x\<in>carrier_vec nc. A *\<^sub>v x \<le> b"
disjoint_image_subset: fixes \<A> :: "'c set set"   and f :: "'c set \<Rightarrow> 'c set" assumes "disjoint \<A>"   and "\<And>X. X \<in> \<A> \<Longrightarrow> f X \<subseteq> X" shows "disjoint (f ` \<A>)"
lossless: fixes x :: "'call"   and s :: "'s" assumes "x \<in> outs_\<I> \<I>"   and "I s" shows "lossless_spmf (the_gpv (callee s x))"
ltake_conv_fixp: shows "ltake \<equiv> curry (llist.fixp_fun (\<lambda>ltake (n, xs). case xs of LNil \<Rightarrow> LNil | LCons x xs \<Rightarrow> case n of 0 \<Rightarrow> LNil | eSuc n \<Rightarrow> LCons x (curry ltake n xs)))"
polynomial_of_mset_Cons: fixes \<phi> :: "char list \<Rightarrow> nat"   and x :: "char list multiset \<times> int"   and ys :: "(char list multiset \<times> int) multiset" assumes "poly_embed \<phi>" shows "poly_embed.polynomial_of_mset \<phi> (add_mset x ys) = Const (snd x) * poly_embed.poly_of_vars \<phi> (fst x) + poly_embed.polynomial_of_mset \<phi> ys"
triv_admI: fixes P :: "'a \<Rightarrow> bool" assumes "\<forall>x. P x" shows "adm P"
n_empty: shows "N_of_state (Liminf_state Sts) = {}"
select: fixes T :: "'a set" assumes "finite T"   and "T \<subset> S"   and "P T" shows "\<exists>s\<in>S - T. P (insert s T)"
wff_nonintDfmla: shows "wff nonintDfmla"
leg1_simps_4: shows "trg r\<^sub>1 = trg r"
streams_stl: fixes s :: "'d stream"   and A :: "'d set" assumes "s \<in> streams A" shows "stl s \<in> streams A"
suc: fixes X :: "'b set set"   and A :: "'b set set" assumes "X \<in> subset.suc_Union_closed A" shows "subset.suc A X \<in> subset.suc_Union_closed A"
transpose_zero: fixes A :: "(('a, 'c) vec, 'b) vec" shows "(Finite_Cartesian_Product.transpose A = 0) = (A = 0)"
of_complex_in_Ints_iff: fixes x :: "complex" shows "(of_complex x \<in> \<int>) = (x \<in> \<int>)"
cut_greater_Diff_conv: fixes I :: "'a set"   and t :: "'a" shows "I \<down>> t = I - {..t}"
Output: fixes a :: "name"   and b :: "name"   and P :: "pi" shows "a{b}.P \<longmapsto> a[b] \<prec> P"
ag_diff_minus: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier M"   and "b \<in> carrier M"   and "c \<in> carrier M"   and "a \<plusminus> -\<^sub>a b = c" shows "b \<plusminus> -\<^sub>a a = -\<^sub>a c"
terminatesD: fixes g :: "'c \<Rightarrow> ('d, 'c) step"   and s :: "'c" assumes "terminates g" shows "s \<in> terminates_on g"
ODnum_subTr: fixes D :: "'c Order"   and x :: "'c Order set"   and y :: "'c Order set"   and Y :: "'c Order" assumes "Worder D"   and "x = ordinal_number D"   and "y \<in> ODnums"   and "y \<sqsubset> x"   and "Y \<in> y" shows "\<exists>c\<in>carrier D. ord_equiv Y (Iod D (segment D c))"
normalize_ports_generic_def: fixes normalize_pos :: "'b common_primitive match_expr \<Rightarrow> 'b common_primitive match_expr list"   and rewrite_neg :: "'b common_primitive match_expr \<Rightarrow> 'b common_primitive match_expr"   and m :: "'b common_primitive match_expr" shows "normalize_ports_generic normalize_pos rewrite_neg m = concat (map normalize_pos (normalize_match (rewrite_neg m)))"
prim_pyth_tripleI2: fixes v :: "nat"   and u :: "nat" assumes "0 < v"   and "v < u"   and "coprime u v"   and "\<not> (odd u \<and> odd v)" shows "prim_pyth_triple (u\<^sup>2 - v\<^sup>2) (2 * u * v) (u\<^sup>2 + v\<^sup>2)"
REPI: fixes \<nu> :: "fin_string + fin_string \<Rightarrow> 32 Word.word"   and \<nu>' :: "fin_string \<Rightarrow> real" assumes "\<And>x. (\<nu> (Inl x) \<equiv>\<^sub>L \<nu>' x) \<and> (\<nu> (Inr x) \<equiv>\<^sub>U \<nu>' x)" shows "\<nu> REP \<nu>'"
set_reverse_iterateoi_I: fixes invar :: "'c \<Rightarrow> bool"   and itoi :: "'c \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and \<alpha> :: "'c \<Rightarrow> 'e set" assumes "\<And>s. invar s \<Longrightarrow> set_iterator_rev_linord (itoi s) (\<alpha> s)" shows "set_reverse_iterateoi \<alpha> invar itoi"
simps_1: fixes p :: "perm"   and \<alpha> :: "'a" shows "p \<bullet> L_action.Act \<alpha> = L_action.Act (p \<bullet> \<alpha>)"
outgoing_def: fixes v :: "nat" shows "outgoing v \<equiv> {(v, u) |u. (v, u) \<in> E}"
assms_1: fixes p :: "'node" assumes "p \<in> S" shows "\<exists>Q\<subseteq>S. quorum_of p Q"
Pair_bottom_iff: fixes x :: "'b"   and y :: "'c" shows "((x, y) = \<bottom>) = (x = \<bottom> \<and> y = \<bottom>)"
map_cong: fixes x :: "'c option"   and ya :: "'c option"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "x = ya"   and "\<And>z. z \<in> set_option ya \<Longrightarrow> f z = g z" shows "map_option f x = map_option g ya"
select_convs_2: fixes statename :: "nat option"   and datastate :: "nat \<Rightarrow>f value option"   and action :: "String.literal \<times> value list"   and "output" :: "value option list"   and more :: "'a" shows "datastate \<lparr>statename = statename, datastate = datastate, action = action, output = output, \<dots> = more\<rparr> = datastate"
bigo_const_inverse: fixes f :: "'b \<Rightarrow> real"   and F :: "'b filter"   and c :: "real" assumes "filterlim f at_top F"   and "F \<noteq> bot" shows "((\<lambda>_. c) \<in> O[F](\<lambda>x. inverse (f x))) = (c = 0)"
exec_step_ind_InitThrow_End: fixes a :: "nat"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" shows "exec_step_ind (StepT [] a) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (\<lfloor>a\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, No_ics) # frs, sh)"
insert_subset: fixes x :: "'b::type"   and A :: "'b::type set"   and B :: "'b::type set" shows "(insert (x::'b::type) (A::'b::type set) \<subseteq> (B::'b::type set)) = (x \<in> B \<and> A \<subseteq> B)"
Rep_Abs_equiv: fixes \<phi> :: "'b ltl" shows "Rep (Abs \<phi>) \<equiv>\<^sub>P \<phi>"
K_meas: fixes p :: "nat"   and n :: "nat" shows "K p n \<in> sets M"
p_comp_pp: fixes uminus :: "'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 one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.pd_allegory uminus inf less_eq less sup bot top one times conv" shows "uminus (times x (uminus (uminus y))) = uminus (times x y)"
min_simps_3: fixes t :: "('c, 'd) bintree"   and xs :: "('c, 'd) bintree option list" shows "Binomial_Queue.min (Some t # xs) = least (Some (priority t)) (Binomial_Queue.min xs)"
\<K>_rw2: fixes k :: "nat" assumes "0 < k" shows "(\<Sum>m = 0..<Suc (k * max_len). real (card (set_of_k_words_length_m k m)) / b ^ m) \<le> real (k * max_len)"
pr_simps_5: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "local.dom (pr1 a b) = local.prod a b"
union_part_equivp: fixes R :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b" assumes "part_equiv R" shows "part_equiv (per_union R a b)"
homotopic_with_symD: fixes P :: "('c \<Rightarrow> 'd) \<Rightarrow> bool"   and X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "homotopic_with P X Y f g" shows "homotopic_with P X Y g f"
set_map: fixes f :: "'f \<Rightarrow> 'e"   and v :: "'f pmf" shows "set_pmf (map_pmf f v) = f ` set_pmf v"
Join_left_commute: fixes A :: "'b program"   and B :: "'b program"   and C :: "'b program" shows "A \<squnion> (B \<squnion> C) = B \<squnion> (A \<squnion> C)"
int_f: shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>join M = \<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. ennreal (f x) \<partial>M' \<partial>M"
is_arg_minD2: fixes m :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "is_arg_min m P x"   and "P y" shows "\<not> m y < m x"
HTuple_eqvt: fixes p :: "perm"   and n :: "nat" shows "p \<bullet> HTuple n = HTuple (p \<bullet> n)"
extremum_uniqueI: fixes a :: "'d" assumes "\<top> \<le> a" shows "a = \<top>"
assms_5: fixes y :: "'b" assumes "y \<in> T" shows "f (g y) = y"
assms: shows "\<Gamma> \<turnstile> Roll e', eP, eV : Mu \<alpha> \<tau>"
lsum_mono2: fixes bl :: "'a list"   and f :: "'a \<Rightarrow> nat"   and al :: "'a list" assumes "\<And>b. b \<in> set bl \<Longrightarrow> 0 < f b"   and "al < bl" shows "lsum f al < lsum f bl"
Pareto_iff: fixes x :: "'b"   and y :: "'b" shows "Pareto R x y = (\<forall>i\<in>dom. R i x y)"
Quotient_mpoly: shows "Quotient (=) MPoly mapping_of cr_mpoly"
same_face: shows "G.face_cycle_set a' = G.face_cycle_set a"
ladder_early_stage_3: fixes index :: "nat" assumes "index + 1 < length L" shows "take (ladder_n L' index) E = take (ladder_n L index) D"
span_clauses_1: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and S :: "'c set" assumes "module scale"   and "a \<in> S" shows "a \<in> module.span scale S"
assms: shows "h \<turnstile> f \<bind> g \<rightarrow>\<^sub>r y"
bin_sign_and: fixes i :: "int"   and j :: "int" shows "bin_sign (i AND j) = - (bin_sign i * bin_sign j)"
is_of_type_var_conv: fixes x :: "variable"   and T :: "PDDL_STRIPS_Semantics.type" shows "is_of_type (ty_term Q objT) (term.VAR x) T = is_of_type Q x T"
braun_list_eq: fixes t :: "'b tree"   and xs :: "'b list" shows "braun_list t xs = (braun t \<and> xs = list t)"
simps: fixes p :: "('b, 'c) fm"   and ps :: "('b, 'c) fm list"   and i :: "'c" shows "(p on (ps, i)) = (p \<in>. ps \<or> p = Nom i)"
in_arcs_G: shows "{uv, swap_in (parcs I') uv} \<subseteq> arcs (with_proj I')"
mop_alt_unfolds_7: fixes m :: "'e nres" shows "ASSERT True \<bind> (\<lambda>_. m) = m"
norm: fixes v :: "int poly" assumes "v \<in> set vs" shows "Mp v = v"
simps_1: fixes P :: "'b prog"   and C :: "char list"   and Cs :: "char list list" shows "supercls_lst P (C # Cs) = ((\<forall>C'\<in>set Cs. P \<turnstile> C' \<preceq>\<^sup>* C) \<and> supercls_lst P Cs)"
ideD_1: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a" shows "arr a"
2: shows "natural_isomorphism (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (\<Phi>'.Y a') (\<Phi>'.Y a) \<Phi>'\<Psi>.map"
otherassms_8: shows "(\<sigma>', q) \<in> oreachable A S U"
filter_def: fixes F :: "'a set" shows "ord_class.filter F \<equiv> F \<noteq> {} \<and> (\<forall>x\<in>F. \<forall>y\<in>F. \<exists>z\<in>F. z \<le> x \<and> z \<le> y) \<and> is_up_set F"
inter_restrict: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and B :: "'c set" assumes "finite A" shows "F g (A \<inter> B) = F (\<lambda>x. if x \<in> B then g x else \<^bold>1) A"
PRES: shows "\<forall>x\<in>A. f x \<in> B"
ideD_2: fixes a :: "'a \<times> 'a" assumes "VV.ide a" shows "VV.dom a = a"
consistent_def: shows "B.consistent \<equiv> \<not> bprv fls"
p_less: shows "p - 1 = 2 ^ k' * m'"
sums_infsetsum_nat': fixes f :: "nat \<Rightarrow> 'a" assumes "f abs_summable_on UNIV" shows "f sums infsetsum f UNIV"
inorder_balance_list: fixes xs :: "'a list" shows "inorder (balance_list xs) = xs"
More: fixes f :: "'a\<cdot>'f resT \<rightarrow> 'a\<cdot>'f resT"   and m :: "'a\<cdot>'f resT\<cdot>'f" assumes "\<And>r. P (f\<cdot>r)" shows "P (More\<cdot>(fmap\<cdot>f\<cdot>m))"
Lim_inject: fixes f :: "'g \<Rightarrow> ('f, 'g) Old_Datatype.node set"   and g :: "'g \<Rightarrow> ('f, 'g) Old_Datatype.node set" assumes "Old_Datatype.Lim f = Old_Datatype.Lim g" shows "f = g"
contour_integral_zero: fixes g :: "real \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex" assumes "valid_path g"   and "path_image g \<subseteq> S"   and "pathfinish g = pathstart g"   and "f holomorphic_on S"   and "\<And>\<gamma> z. \<lbrakk>path \<gamma>; path_image \<gamma> \<subseteq> S; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> S\<rbrakk> \<Longrightarrow> winding_number \<gamma> z = 0" shows "(f has_contour_integral 0) g"
top_ivl_def: shows "\<top> \<equiv> [-\<infinity>, \<infinity>]"
obj_def: fixes a :: "'a" shows "obj a \<equiv> arr a \<and> I.map a = a"
vars\<^sub>s\<^sub>s\<^sub>t_unlabel_dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_eq: fixes A :: "('g strand_label \<times> ('f, 'e) stateful_strand_step) list" shows "vars\<^sub>l\<^sub>s\<^sub>s\<^sub>t (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A) = vars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
inclosed_closed: fixes np :: "nat \<Rightarrow> ((nat \<Rightarrow> 'a) \<times> 'b, 'c seq_action) automaton"   and p :: "net_tree"   and U :: "(nat \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> bool"   and P :: "(nat \<Rightarrow> 'a) \<times> 'b net_state \<Rightarrow> bool" assumes "opnet np p \<Turnstile> (otherwith (=) (net_tree_ips p) inoclosed, U \<rightarrow>) P" shows "oclosed (opnet np p) \<Turnstile> (\<lambda>_ _ _. True, U \<rightarrow>) P"
monoD: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'h"   and x :: "'g"   and y :: "'g" assumes "class.order less_eq less"   and "order.mono less_eq f"   and "less_eq x y" shows "f x \<le> f y"
f_Exec_Stream_append_last_Cons: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and ys :: "'e list"   and c :: "'d" shows "f_Exec_Comp_Stream trans_fun (xs @ ys) c = f_Exec_Comp_Stream trans_fun xs c @ f_Exec_Comp_Stream trans_fun ys (last (c # f_Exec_Comp_Stream trans_fun xs c))"
collapse_2: fixes generat :: "('d, 'e, 'f) generat" assumes "\<not> is_Pure generat" shows "IO (generat.output generat) (continuation generat) = generat"
reindex_bij_betw_not_neutral: fixes S' :: "'a set"   and T' :: "'b set"   and h :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "finite S'"   and "finite T'"   and "bij_betw h (S - S') (T - T')"   and "\<And>a. a \<in> S' \<Longrightarrow> g (h a) = (0::'c)"   and "\<And>b. b \<in> T' \<Longrightarrow> g b = (0::'c)" shows "(\<Sum>x\<in>S. g (h x)) = sum g T"
del_emp_concat: fixes us :: "'b list list" shows "concat us = concat (filter (\<lambda>x. x \<noteq> \<epsilon>) us)"
vec_of_basis_enum_inverse: fixes w :: "'b" shows "basis_enum_of_vec (vec_of_basis_enum w) = w"
gen_model_tc_rels_rev: fixes M :: "('e, 'f, 'g) KripkeStructure"   and w' :: "'g"   and w'' :: "'g"   and as :: "'e set"   and w :: "'g" assumes "kripke M"   and "(w', w'') \<in> (\<Union> (relations M ` as))\<^sup>+"   and "w' \<in> worlds (gen_model M w)" shows "(w', w'') \<in> (\<Union> (relations (gen_model M w) ` as))\<^sup>+"
dim_image_le: fixes f :: "('b, 'c) vec \<Rightarrow> ('b, 'd) vec"   and S :: "('b, 'c) vec set" assumes "Vector_Spaces.linear (*s) (*s) f" shows "vec.dim (f ` S) \<le> vec.dim S"
strip_def: shows "strip = (\<lambda>(a, b). State_Monad.update ((##) b) \<bind> (\<lambda>_. pure a))"
insert_ident: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "x \<notin> A"   and "x \<notin> B" shows "(x \<triangleright> A = x \<triangleright> B) = (A = B)"
S4bReturn_successors: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> ($(S4 rmhist p \<and> ires!p \<noteq> #NotAResult) \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> (\<forall>l. $MemInv mm l)) \<and> <MemReturn rmCh ires p>_(m p) \<longrightarrow> S5 rmhist p$"
Ints_of_int: fixes z :: "int" shows "of_int z \<in> \<int>"
fork: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and e :: "('r, 'l, 'v) expr"   and r' :: "'r" assumes "s r = Some (\<sigma>, \<tau>, \<E> [Rfork e])"   and "r' \<notin> RID\<^sub>G s" shows "revision_step r s (s(r \<mapsto> (\<sigma>, \<tau>, \<E> [VE (Rid r')]), r' \<mapsto> (\<sigma>;;\<tau>, \<epsilon>, e)))"
lemma2: fixes a :: "letter"   and vs :: "letter list list"   and ws :: "letter list list" assumes "R a vs ws"   and "good vs" shows "good ws"
nth_default_def: fixes dflt :: "'b"   and xs :: "'b list"   and n :: "nat" shows "nth_default dflt xs n = (if n < length xs then xs ! n else dflt)"
norm_v_X: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec"   and v :: "(complex, 'a) Finite_Cartesian_Product.vec"   and \<alpha> :: "complex" assumes "perron_frobenius A"   and "eigen_vector (HMA_Connect.map_matrix complex_of_real A) v \<alpha>" shows "map_vector cmod v \<in> perron_frobenius.X"
ocircline_set_moebius_ocircline: fixes M :: "moebius"   and H :: "ocircline" shows "ocircline_set (moebius_ocircline M H) = moebius_pt M ` ocircline_set H"
functor_axioms: shows "functor A C (K \<circ> H)"
minus_Mp_1: fixes f :: "int poly"   and g :: "int poly" shows "Mp f - g =m f - g"
is_word_Derivation: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list" assumes "is_word a"   and "Derivation a D b" shows "D = []"
rel_funI: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'i" assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" shows "rel_fun A B f g"
assms_2: shows "Deliver (i', Insert e' n) \<in> set es"
connected_component_in: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "connected_component S x y" shows "x \<in> S \<and> y \<in> S"
sa1_tb1: shows "s * a1 + t * b1 = (1::'a)"
satisfies_T0: shows "is_left_adjoint fg\<^sub>0h\<^sub>1.p\<^sub>0"
eq_iff: fixes a :: "'b"   and b :: "'b" shows "(a = b) = (ord' a b \<and> ord' b a)"
sel_12: fixes x31 :: "bool"   and x32 :: "'b" shows "nul (Atom2 x31 x32) = False"
Algorithm_axioms: shows "Algorithm jkbp envInit envAction envTrans envVal jview envObs jviewInit jviewIncr simf simRels simVal simAbs simObs simInit simTrans simAction aOps tOps"
relcomp_unfold: fixes r :: "('c \<times> 'e) set"   and s :: "('e \<times> 'd) set" shows "r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}"
p4: shows "{OrdP (Var k)} \<turnstile> (SyntaxN.All j (SyntaxN.All j' (SyntaxN.All sj (SyntaxN.All lj (SeqQuoteP (Var i) (Var i') (Var si) (Var li) IMP SeqQuoteP (Var j) (Var j') (Var sj) (Var lj) IMP HaddP (Var li) (Var lj) (Var k) IMP (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j'))) AND (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j'))))))))(li::=ki)"
lookup: fixes x :: "'a \<Rightarrow>\<^sub>0 'b" shows "poly_mapping.lookup x \<in> {f. finite {x. f x \<noteq> (0::'b)}}"
surjective: fixes r :: "('a, 'b, 'c, 'd) bar_scheme" shows "r = \<lparr>bar1 = bar1 r, bar2 = bar2 r, bar3 = bar3 r, bar21 = bar21 r, bar32 = bar32 r, bar31 = bar31 r, \<dots> = bar.more r\<rparr>"
empty_le: fixes A :: "'b multiset" shows "{#} \<subseteq># A"
gen_length_def: fixes n :: "nat"   and xs :: "'d list" shows "List.gen_length n xs = n + length xs"
prems_1: shows "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"
ifs_1: shows "p - Suc 0 \<le> lo"
indRelRTPO_impl_TRel_is_weak_barbed_contrasimulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "weak_barbed_contrasimulation (indRelRTPO TRel) (STCalWB SWB TWB)" shows "weak_barbed_contrasimulation (TRel\<^sup>+) TWB"
law_restrict_mult: fixes R :: "('c, 'd) ring_scheme" shows "(\<otimes>\<^bsub>law_restrict R\<^esub>) = (\<lambda>a\<in>carrier R. restrict ((\<otimes>\<^bsub>R\<^esub>) a) (carrier R))"
assms_3: fixes i :: "nat" assumes "i < length xs2" shows "R (xs1 ! i) (xs2 ! i) = R' (ys1 ! i) (ys2 ! i)"
a_Roll: fixes \<alpha> :: "Syntax.tvar"   and \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau> :: "ty" assumes "atom \<alpha> \<sharp> \<Gamma>"   and "\<Gamma> \<turnstile> e, eP, eV : subst_type \<tau> (Mu \<alpha> \<tau>) \<alpha>" shows "\<Gamma> \<turnstile> Roll e, Roll eP, Roll eV : Mu \<alpha> \<tau>"
snoc: shows "applied_rule_rev Cp x (a @ [b] @ xs) = \<lfloor>b\<rfloor> \<longrightarrow> r \<in> set xs \<longrightarrow> b \<notin> set xs \<longrightarrow> x \<notin> dom (Cp r)"
wf_weaken: fixes \<Delta> :: "binding list"   and \<Gamma> :: "binding list"   and T :: "POPLmarkRecord.type"   and B :: "binding" assumes "\<Delta> @ \<Gamma> \<turnstile>\<^sub>w\<^sub>f T" shows "\<up>\<^sub>e (Suc 0) 0 \<Delta> @ B \<Colon> \<Gamma> \<turnstile>\<^sub>w\<^sub>f \<up>\<^sub>\<tau> (Suc 0) \<parallel>\<Delta>\<parallel> T"
Sretr_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and theta :: "(('c, 'd, 'a) cmd \<times> ('c, 'd, 'a) cmd) set" assumes "PL_Indis cval indis" shows "PL_Indis.Sretr aval tval cval indis theta \<equiv> {(c, d). PL_Indis.matchC_C aval tval cval indis theta c d}"
phi_no_closed_loop: fixes p :: "'val"   and g :: "'g"   and vs :: "'val list" assumes "p \<in> allVars g"   and "phi g p = Some vs" shows "set vs \<noteq> {p}"
rep_eq: fixes x :: "'c \<rightharpoonup>\<^sub>f 'd" shows "lookup x = the \<circ> fmlookup x"
distinct_9: fixes x2 :: "Vars.var"   and x4 :: "Vars.var" shows "Arg x2 \<noteq> Dummy x4"
col_cop2__cop: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and U :: "'a"   and V :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and P :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "U \<noteq> V"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C U"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C V"   and "Tarski_neutral_dimensionless.Col Bet U V P" shows "Tarski_neutral_dimensionless.Coplanar Bet A B C P"
cop_F_closed_inv_def: fixes ds :: "'b set"   and fp :: "'a set" shows "cop_F_closed_inv ds fp = (\<forall>x\<in>fp. above (Pd (Xd x)) x \<subseteq> fp)"
extract_var_def: fixes p :: "'b mpoly"   and v :: "nat" shows "extract_var p v = (\<Sum>m. monom (remove_key v m) (monom (Poly_Mapping.single v (lookup m v)) (coeff p m)))"
matrix_mult_dot: fixes A :: "((real, 'b) vec, 'a) vec"   and x :: "(real, 'b) vec" shows "A *v x = (\<chi>i. A $ i \<bullet> x)"