hcju/isabelleps huggingface dataset:
set_map0_1: fixes f1 :: "'l \<Rightarrow> 'n"   and f2 :: "'m \<Rightarrow> 'o" shows "F2set1 \<circ> F2map f1 f2 = (`) f1 \<circ> F2set1"
assms: fixes x :: "'a"   and y :: "'b" assumes "fmlookup m x = Some y" shows "P x"
in_othrow_eq: fixes e :: "'d"   and s :: "'f"   and v :: "'d + 'e" shows "(othrow e s = Some v) = (v = Inl e)"
Sup_llist_def: fixes Xs :: "'b set llist" shows "Sup_llist Xs = \<Union> (lnth Xs ` {i. enat i < llength Xs})"
WT_gpv_outs_gpv: fixes \<I> :: "('y, 'z) \<I>"   and gpv :: "('aa, 'y, 'z) gpv" assumes "\<I> \<turnstile>g gpv \<surd>" shows "outs_gpv \<I> gpv \<subseteq> outs_\<I> \<I>"
transp_less: fixes ko :: "'b key_order" shows "transp (ko.lt ko)"
restrictA_noop: fixes \<Gamma> :: "('a \<times> 'b) list"   and S :: "'a set" assumes "domA \<Gamma> \<subseteq> S" shows "restrictA S \<Gamma> = \<Gamma>"
all_in_list_sublist: fixes xs :: "('b, 'c) Combinators list"   and l :: "'b set list"   and P :: "('b, 'c) Combinators \<Rightarrow> bool" assumes "singleCombinators xs"   and "all_in_list xs l" shows "all_in_list (qsort (filter P xs) l) l"
embed_Field_ofilter: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" assumes "Well_order r"   and "Well_order r'"   and "embed r r' f" shows "ofilter r' (f ` Field r)"
v_def: shows "v \<equiv> case bezout (A $ from_nat i $ from_nat i) (A $ from_nat j $ from_nat j) of (p, q, u, v, d) \<Rightarrow> v"
WT_real: fixes \<eta> :: "nat" shows "\<I>_real \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta> \<turnstile>res real \<eta> \<surd>"
lookup_focus_in: fixes g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "g \<in> G" shows "lookup (focus X g) t \<in> P[{x}]"
language_def: shows "N.language \<equiv> {xs. N.nextl (nfa.init \<lparr>nfa.states = {0, 1}, init = {0}, final = {1}, nxt = \<lambda>q x. if q = 0 \<and> x = a then {1} else {}, eps = {}\<rparr>) xs \<inter> nfa.final \<lparr>nfa.states = {0, 1}, init = {0}, final = {1}, nxt = \<lambda>q x. if q = 0 \<and> x = a then {1} else {}, eps = {}\<rparr> \<noteq> {}}"
split_asm: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b"   and option :: "'c option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (\<not> (option = None \<and> \<not> P f1 \<or> (\<exists>x2. option = Some x2 \<and> \<not> P (f2 x2))))"
FF_def: shows "F.FF \<equiv> \<lambda>\<mu>\<nu>. if C.VV.arr \<mu>\<nu> then (F (fst \<mu>\<nu>), F (snd \<mu>\<nu>)) else D.VV.null"
aff_dim_le_DIM: fixes S :: "'a set" shows "aff_dim S \<le> int DIM('a)"
fund_sol_is_nontriv_solution: shows "nontriv_solution fund_sol"
fls_lr_inverse_subdegree_2: fixes x :: "'a"   and f :: "'a fls" assumes "x \<noteq> (0::'a)" shows "fls_subdegree (fls_right_inverse f x) = - fls_subdegree f"
llcp_LNil2: fixes xs :: "'c llist" shows "llcp xs LNil = 0"
b: fixes x :: "'b list"   and a :: "nat"   and y :: "'b list"   and b :: "nat"   and z :: "'b list"   and c :: "nat" assumes "LS x a y b z c" shows "2 \<le> b"
untime_not_empty_conv: fixes s :: "'b message_af list" shows "(untime s \<noteq> []) = (\<exists>n<length s. s ! n \<noteq> NoMsg)"
ipresFreshAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('f, 'g, 'varSort, 'h, 'i, 'var, 'd, 'e, 'j) model_scheme" shows "ipresFreshAll h hA MOD \<equiv> ipresFresh h MOD \<and> ipresFreshAbs hA MOD"
map_iterator_abs_def: fixes \<alpha> :: "'e \<Rightarrow> 'f"   and invar :: "'e \<Rightarrow> bool"   and it :: "('g \<Rightarrow> bool) \<Rightarrow> ('h \<times> 'e \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g"   and m :: "'h \<Rightarrow> 'f option" shows "map_iterator_abs \<alpha> invar it m \<equiv> map_iterator_abs_genord \<alpha> invar it m (\<lambda>_ _. True)"
pCons_induct: fixes P :: "'a poly \<Rightarrow> bool"   and p :: "'a poly" assumes "P 0"   and "\<And>a p. \<lbrakk>a \<noteq> (0::'a) \<or> p \<noteq> 0; P p\<rbrakk> \<Longrightarrow> P (pCons a p)" shows "P p"
sup_left_divisibility: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "(x \<preceq> y) = (\<exists>z. x \<oplus> z = y)"
boundedE: fixes b :: "'d"   and c :: "'d"   and a :: "'d" assumes "b \<squnion> c \<le> a" obtains "b \<le> a" and "c \<le> a"
CallParamsThrow: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and v :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and es :: "char list exp list"   and vs :: "val list"   and ex :: "char list exp"   and es' :: "char list exp list"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and M :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>map Val vs @ throw ex # es',s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>M(es),s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw ex,s\<^sub>2\<rangle>"
final_i_def: shows "final_i \<equiv> SOME i. enat i \<le> llength t \<and> (\<forall>p. ps (coS t i) p \<noteq> None) \<and> (\<forall>c. Marker \<notin> set (msgs (coS t i) c))"
grand_parent_upper_bound: fixes l :: "int"   and c :: "int" shows "(parent l (parent l c) < c) = (l \<le> c)"
to_tree_order_def: shows "to_tree_order \<equiv> l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_to_tree_order get_child_nodes"
diff_inv_eq_inv_set: fixes I :: "'c \<Rightarrow> bool"   and f :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and U :: "'c \<Rightarrow> real set"   and S :: "'c set"   and t\<^sub>0 :: "real"   and G :: "'c \<Rightarrow> bool" shows "diff_invariant I f U S t\<^sub>0 G = (\<forall>s. I s \<longrightarrow> x\<acute>=f & G on U S @ t\<^sub>0 s \<subseteq> {s. I s})"
set_relator_eq_onp: fixes P :: "'b \<Rightarrow> bool" shows "rel_set (eq_onp P) = eq_onp (\<lambda>A. Ball A P)"
sum_le_prod: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'f" assumes "\<And>x. x \<in> A \<Longrightarrow> (0::'f) \<le> f x" shows "sum f A \<le> (\<Prod>x\<in>A. (1::'f) + f x)"
simpl_ports_conjunct_not_UNIV: fixes x :: "16 word \<times> 16 word"   and p1 :: "16 word \<times> 16 word"   and p2 :: "16 word \<times> 16 word" assumes "Collect (simple_match_port x) \<noteq> UNIV"   and "x = simpl_ports_conjunct p1 p2" shows "Collect (simple_match_port p1) \<noteq> UNIV \<or> Collect (simple_match_port p2) \<noteq> UNIV"
exec: shows "\<Gamma>\<turnstile> \<langle>dynCall init p return c,Normal s\<rangle> \<Rightarrow> t"
whn_not_Nats: shows "whn \<notin> \<nat>"
conditionalize: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "finite A" shows "sum g A = (\<Sum>a. if a \<in> A then g a else (0::'c))"
rec_stat_def: shows "rec_stat = Cn 1 rec_lo [recf.id 1 0, constn (Pi 1)]"
le_LNil: fixes s :: "'a llist" shows "(s \<le> LNil) = (s = LNil)"
add_uminus_minus_mat: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'a mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "A + - B = A - B"
function_one_closed: shows "\<one>\<^bsub>F\<^esub> \<in> carrier F"
nat_pow_Suc: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and n :: "nat" shows "x [^]\<^bsub>G\<^esub> Suc n = x [^]\<^bsub>G\<^esub> n \<otimes>\<^bsub>G\<^esub> x"
qPsubst_preserves_qGood: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm option"   and X :: "('e, 'f, 'varSort, 'var, 'g) qTerm" assumes "qGoodEnv rho"   and "qGood X" shows "qGood X #[[rho]]"
nat_diff_left_cancel_le2: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k - m \<le> k - n"   and "n \<le> k" shows "n \<le> m"
l'_height: shows "height (MKT ln ll lr lh) = height (snd (delete_max (MKT ln ll lr lh))) \<or> height (MKT ln ll lr lh) = height (snd (delete_max (MKT ln ll lr lh))) + 1"
cfinite_def: fixes r :: "('d \<times> 'd) set" shows "BNF_Cardinal_Arithmetic.cfinite r = finite (Field r)"
in_inv_image: fixes x :: "'g"   and y :: "'g"   and r :: "('h \<times> 'h) set"   and f :: "'g \<Rightarrow> 'h" shows "((x, y) \<in> inv_image r f) = ((f x, f y) \<in> r)"
List_case_NIL: fixes c :: "'b"   and h :: "('c, unit) Old_Datatype.node set \<Rightarrow> ('c, unit) Old_Datatype.node set \<Rightarrow> 'b" shows "List_case c h NIL = c"
face_of_Times_decomp: fixes C :: "('b \<times> 'c) set"   and S :: "'b set"   and S' :: "'c set" shows "(C face_of S \<times> S') = (\<exists>F F'. F face_of S \<and> F' face_of S' \<and> C = F \<times> F')"
os_iterate_impl: shows "imp_list_iterate os_list os_is_it os_it_init os_it_has_next os_it_next"
vp2: shows "val_process Mkt pf2 n w = (\<Sum>x\<in>support_set pf1 \<union> support_set pf2. prices Mkt x n w * pf2 x (Suc n) w)"
coeff_0_reflect_poly: fixes p :: "'c poly" shows "coeff (reflect_poly p) 0 = lead_coeff p"
\<sigma>_p: shows "Some \<sigma> = mgu (set_mset ` set (Map2.map2 add_mset As AAs))"
vreal_uminus_transfer: shows "rel_fun cr_vreal cr_vreal vreal_uminus_app uminus"
sel_2: fixes z :: "gauss_int" shows "ImZ (gauss_cnj z) = - ImZ z"
fun_left_comm: fixes y :: "'m"   and x :: "'m"   and z :: "'m" shows "munionM y (munionM x z) = munionM x (munionM y z)"
zsplit0_I: fixes t :: "MIR.num"   and n :: "int"   and a :: "MIR.num"   and x :: "int"   and bs :: "real list" assumes "zsplit0 t = (n, a)" shows "Inum (real_of_int x # bs) (CN 0 n a) = Inum (real_of_int x # bs) t \<and> numbound0 a"
lfilter_id_conv: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "(lfilter P xs = xs) = (\<forall>x\<in>lset xs. P x)"
cinner_Pair_0_1: fixes x :: "'c \<times> 'd"   and b :: "'d" shows "\<langle>x, (0::'c, b)\<rangle> = \<langle>snd x, b\<rangle>"
hd_prefixes: fixes xs :: "'b list" shows "hd (prefixes xs) = []"
at_within_eq_bot_iff: fixes c :: "'c"   and A :: "'c set" shows "(at c within A = bot) = (c \<notin> closure (A - {c}))"
rel_mono_strong: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a fset"   and y :: "'b fset"   and Ra :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "rel_fset R x y"   and "\<And>z yb. \<lbrakk>z \<in> fset x; yb \<in> fset y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "rel_fset Ra x y"
associatedI2': fixes a :: "'a"   and b :: "'a"   and u :: "'a" assumes "a = b \<oplus> u"   and "u \<in> Units (add_monoid R)"   and "b \<in> carrier R" shows "a \<sim>\<^bsub>add_monoid R\<^esub> b"
elims: fixes x :: "('e, 'f) atm"   and y :: "Preliminaries.var set" assumes "varsA x = y" obtains T1 :: "'e trm" and T2 :: "'e trm" where "x = Eq T1 T2" and "y = vars T1 \<union> vars T2"   | p :: "'f" and Tl :: "'e trm list" where "x = Pr p Tl" and "y = \<Union> (set (map vars Tl))"
euler_char_iso: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism" assumes "digraph_isomorphism hom" shows "pre_digraph_map.euler_char (app_iso hom G) (map_iso hom) = euler_char"
second_countable_topology_axioms_def: fixes "open" :: "'c set \<Rightarrow> bool" shows "class.second_countable_topology_axioms open \<equiv> \<exists>B. countable B \<and> open = generate_topology B"
assms: shows "\<forall>n\<in>{..length ys}. \<exists>Y. (ipurge_tr I (c_dom D) u (c_tr step out (foldl step s\<^sub>0 xs) (take n ys)), Y) \<in> futures (c_process step out s\<^sub>0) (c_tr step out s\<^sub>0 xs)"
hom_add: fixes x :: "'a"   and y :: "'a" shows "[:x + y:] = [:x:] + [:y:]"
r_parallel_def: shows "r_parallel \<equiv> Cn 3 r_both [Mn 3 (Cn 4 r_le [Cn 4 r_pdec1 [r_both], r_constn 3 1]), recf.Id 3 0, recf.Id 3 1, recf.Id 3 2]"
restriction_disj: fixes v :: "view"   and u :: "view"   and w :: "view"   and f :: "cars \<Rightarrow> nat_int"   and c :: "cars" shows "v=u--w \<longrightarrow> restrict u f c \<sqinter> restrict w f c = bot"
is_red_def: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and a :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "punit.is_red F a = (\<not> relation.is_final (punit.red F) a)"
igFreshIGAbs1_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Iteration.model_scheme" shows "igFreshIGAbs1 MOD \<equiv> \<forall>ys y s X. isInBar (ys, s) \<and> igWls MOD s X \<longrightarrow> igFreshAbs MOD ys y (igAbs MOD ys y X)"
cod_cod: fixes f :: "'a" assumes "L'.C.arr f" shows "L'.C.cod (L'.C.cod f) = L'.C.cod f"
section_retraction_char: fixes g :: "'s"   and f :: "'s" shows "ide (g \<cdot> f) = (antipar f g \<and> compose (Dom f) (Fun g) (Fun f) = (\<lambda>x\<in>Dom f. x))"
isCont_ln: fixes x :: "real" assumes "x \<noteq> 0" shows "isCont ln x"
is_red_singleton_not_0: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow>\<^sub>0 'f"   and p :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "ordered_term.is_red pair_of_term term_of_pair ord_term {f} p" shows "f \<noteq> 0"
CDPlayer_CTRL_States_def: shows "CDPlayer_CTRL_States = {''ReadTracks'', ''CDFull'', ''CDEmpty''}"
ujk: shows "unat j + unat k < 2 ^ LENGTH('a)"
power_one_over: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "power.power one times (divide one a) n = divide one (power.power one times a n)"
natpermute_finite: fixes n :: "nat"   and k :: "nat" shows "finite (natpermute n k)"
le: fixes n :: "nat" shows "sum f {..<n} \<le> x"
right_invariant_eq_nextl: shows "right_invariant eq_nextl"
Abs_lst_eq_flipI: fixes b :: "'b"   and t :: "'a"   and a :: "'b" assumes "atom b \<sharp> t" shows "[[atom a]]lst. t = [[atom b]]lst. (a \<leftrightarrow> b) \<bullet> t"
l11_36_aux1: fixes A :: "'p"   and B :: "'p"   and A' :: "'p"   and D :: "'p"   and E :: "'p"   and D' :: "'p"   and C :: "'p"   and F :: "'p" assumes "A \<noteq> B"   and "A' \<noteq> B"   and "D \<noteq> E"   and "D' \<noteq> E"   and "Bet A B A'"   and "Bet D E D'"   and "A B C LeA D E F" shows "D' E F LeA A' B C"
equiv_Eps_in: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and X :: "'a set" assumes "equiv A r"   and "X \<in> A // r" shows "(SOME x. x \<in> X) \<in> X"
finite_UNIV_prod_def: shows "finite_UNIV = Phantom('a \<times> 'b) (of_phantom finite_UNIV \<and> of_phantom finite_UNIV)"
nat_term_compI: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "\<And>u v. \<lbrakk>snd (rep_nat_term u) = snd (rep_nat_term v); fst (rep_nat_term u) = 0\<rbrakk> \<Longrightarrow> cmp u v \<noteq> Gt"   and "\<And>u v. \<lbrakk>fst (rep_nat_term u) = fst (rep_nat_term v); snd (rep_nat_term u) < snd (rep_nat_term v)\<rbrakk> \<Longrightarrow> cmp u v = Lt"   and "\<And>t u v. cmp u v = Lt \<Longrightarrow> cmp (splus t u) (splus t v) = Lt"   and "\<And>u v a b. \<lbrakk>fst (rep_nat_term u) = fst (rep_nat_term a); fst (rep_nat_term v) = fst (rep_nat_term b); snd (rep_nat_term u) = snd (rep_nat_term v); snd (rep_nat_term a) = snd (rep_nat_term b); cmp a b = Lt\<rbrakk> \<Longrightarrow> cmp u v = Lt" shows "nat_term_comp cmp"
minimal: fixes x :: "'a"   and Q :: "'a \<Rightarrow> bool" assumes "x \<in> A"   and "Q x" shows "\<exists>y\<in>A. P\<^sup>=\<^sup>= y x \<and> Q y \<and> (\<forall>z\<in>A. P z y \<longrightarrow> \<not> Q z)"
r_distr: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "z \<otimes>\<^bsub>R\<^esub> (x \<oplus>\<^bsub>R\<^esub> y) = z \<otimes>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> z \<otimes>\<^bsub>R\<^esub> y"
semidecisive_def: fixes scf :: "('a \<Rightarrow> ('b \<times> 'b) set) \<Rightarrow> ('b \<times> 'b) set"   and A :: "'b set"   and Is :: "'a set"   and C :: "'a set"   and x :: "'b"   and y :: "'b" shows "semidecisive scf A Is C x y \<equiv> C \<subseteq> Is \<and> (\<forall>P. profile A Is P \<and> (\<forall>i\<in>C. x \<^bsub>(P i)\<^esub>\<prec> y) \<and> (\<forall>i\<in>Is - C. y \<^bsub>(P i)\<^esub>\<prec> x) \<longrightarrow> x \<^bsub>(scf P)\<^esub>\<prec> y)"
qiterl_inductl: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x \<squnion> y \<cdot> z \<le> z" shows "qiterl x y \<le> z"
get_M_defs: shows "get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t \<equiv> \<lambda>ptr getter. Heap_Error_Monad.bind get_heap (\<lambda>h. case get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr h of None \<Rightarrow> error SegmentationFault | Some res \<Rightarrow> return (getter res))"
active_has_cf_outgoing: fixes u :: "nat" assumes "(0::'a) < excess fo u" shows "lo.cf.outgoing u \<noteq> {}"
map_snd_enumerate: fixes n :: "nat"   and xs :: "'b list" shows "map snd (List.enumerate n xs) = xs"
in_arcs_in_arcs: fixes x :: "'a"   and G :: "('b, 'a) pre_digraph"   and u :: "'b" assumes "x \<in> in_arcs G u" shows "x \<in> arcs G"
assms_7: shows "has_least_fixpoint (g \<circ> h)"
loc_reach_mem_diff_subset: fixes mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and c' :: "'Com"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and c :: "'Com" assumes "\<forall>x. x \<in> mds AsmNoWrite \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x"   and "\<langle>c', mds', mem'\<rangle> \<in> loc_reach \<langle>c, mds, mem\<^sub>1\<rangle>" shows "\<langle>c', mds', mem'\<rangle> \<in> loc_reach \<langle>c, mds, mem\<^sub>2\<rangle>"
eucl_of_list_eq_takeI: fixes xs :: "real list"   and x :: "'a" assumes "eucl_of_list (take DIM('a) xs) = x" shows "eucl_of_list xs = x"
minus_mult_cancel: fixes V :: "'b set"   and x :: "'b"   and a :: "real" assumes "vectorspace V"   and "x \<in> V" shows "- a \<cdot> - x = a \<cdot> x"
delta_gt_SN: fixes \<delta> :: "'a" assumes "(0::'a) < \<delta>" shows "SN {(x, y). (0::'a) \<le> y \<and> delta_gt \<delta> x y}"
BisT_transT: fixes indis :: "('a \<times> 'a) set"   and c :: "('b, 'c) com"   and d :: "('b, 'c) com"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'a"   and t :: "'a"   and s' :: "'a" assumes "PL_Indis indis"   and "(c, d) \<in> PL_Indis.BisT tval aval indis"   and "PL.mustT tval aval c s"   and "PL.mustT tval aval d t"   and "(s, t) \<in> indis"   and "PL.transT tval aval (c, s) s'" obtains t' :: "'a" where "PL.MtransT tval aval (d, t) t'" and "(s', t') \<in> indis"
add_less_zeroD: 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 sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less (plus x y) zero" shows "less x zero \<or> less y zero"
Lindelof_space_proper_map_preimage: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "proper_map X Y f"   and "Lindelof_space Y" shows "Lindelof_space X"
isMaxim_max_fun_diff: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" assumes "f \<noteq> g"   and "f \<in> FINFUNC"   and "g \<in> FINFUNC" shows "s.isMaxim {a \<in> Field s. f a \<noteq> g a} (s.max_fun_diff f g)"
fromFull: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<sigma> :: "'c"   and a :: "univ" assumes "InfModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP" shows "intT \<sigma> (InfModel.fromFull intT \<sigma> a)"
Pi_pmf_default_swap: fixes A :: "'b set"   and dflt' :: "'c"   and dflt :: "'c"   and p :: "'b \<Rightarrow> 'c pmf" assumes "finite A" shows "map_pmf (\<lambda>f x. if x \<in> A then f x else dflt') (Pi_pmf A dflt p) = Pi_pmf A dflt' p"
succ_eq_add1: fixes x :: "V" shows "ZFC_in_HOL.succ x = x + 1"
mult_less_cancel_right2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (times a c) c = ((less_eq zero c \<longrightarrow> less a one) \<and> (less_eq c zero \<longrightarrow> less one a))"
mult1_def: fixes r :: "('a \<times> 'a) set" shows "mult1 r = {(N, M). \<exists>a M0 K. M = add_mset a M0 \<and> N = M0 + K \<and> (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r)}"
simps_2: fixes y :: "'b option" shows "None \<squnion> y = y"
path_invar: fixes g :: "'g"   and ns :: "'node list" assumes "old.path g ns" shows "invar g"
surjective: fixes r :: "'b control_state_scheme" shows "r = \<lparr>break_status = break_status r, return_status = return_status r, \<dots> = more r\<rparr>"
ind: fixes lm2 :: "nat list"   and lm3 :: "nat list"   and lm4 :: "nat list" assumes "n \<le> aa - ba"   and "ba < aa"   and "length lm1 = ba"   and "length lm2 = aa - ba - n"   and "length lm3 = n" shows "{\<lambda>nl. nl = lm1 @ 0 \<up> n @ lm2 @ lm3 @ lm4} mv_boxes aa ba n {\<lambda>nl. nl = lm1 @ lm3 @ lm2 @ 0 \<up> n @ lm4}"
\<R>_bisim: fixes \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set" shows "strong_low_bisim_mm (\<R> \<Gamma>' \<S>' P')"
leg1_is_map: shows "is_left_adjoint (tab\<^sub>1 (r \<star> s))"
test_one: shows "n n (1::'a) = (1::'a)"
simps_1: fixes x :: "gauss_int"   and y :: "gauss_int" shows "ReZ (x + y) = ReZ x + ReZ y"
in_set_conv_decomp_last: fixes x :: "'a"   and xs :: "'a list" shows "(x \<in> list.set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> list.set zs)"