hcju/isabelleps huggingface dataset:
t1_space_closedin_singleton: fixes X :: "'c topology" shows "t1_space X = (\<forall>x\<in>topspace X. closedin X {x})"
chainI: fixes C :: "'a set set"   and A :: "'a set set" assumes "C \<subseteq> A"   and "\<And>x y. \<lbrakk>x \<in> C; y \<in> C\<rbrakk> \<Longrightarrow> (\<subset>)\<^sup>=\<^sup>= x y \<or> (\<subset>)\<^sup>=\<^sup>= y x" shows "subset.chain A C"
af_F_prefix_lang: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and ys :: "'b set list"   and xs :: "'b set list" assumes "w \<Turnstile>\<^sub>n af (F\<^sub>n \<phi>) ys" shows "w \<Turnstile>\<^sub>n af (F\<^sub>n \<phi>) (xs @ ys)"
emeasure_eq_measure: fixes i :: "'a"   and A :: "'b set" shows "emeasure (M i) A = ennreal (M.prob i A)"
mult_2_right: fixes z :: "'a" shows "z * (2::'a) = z + z"
1: shows "Maps.PRJ\<^sub>0 \<lbrakk>\<lbrakk>r\<^sub>0\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>s\<^sub>1\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>r\<^sub>0s\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk> \<and> Maps.PRJ\<^sub>1 \<lbrakk>\<lbrakk>r\<^sub>0\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>s\<^sub>1\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>r\<^sub>0s\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>"
skip_ra_def: fixes v :: "'c \<Longrightarrow> 'b" shows "II\<^bsub>v\<^esub> = ($v\<acute> =\<^sub>u $v)"
simps: fixes a :: "'node"   and S :: "'node set" shows "(a \<in> wod_backward_slice S) = ((\<exists>n. a = n \<and> valid_node n \<and> n \<in> S) \<or> (\<exists>n' n\<^sub>1 n\<^sub>2. a = n' \<and> n' \<longrightarrow>\<^sub>w\<^sub>o\<^sub>d n\<^sub>1,n\<^sub>2 \<and> n\<^sub>1 \<in> wod_backward_slice S \<and> n\<^sub>2 \<in> wod_backward_slice S) \<or> (\<exists>n' V n''. a = n' \<and> n' influences V in n'' \<and> n'' \<in> wod_backward_slice S))"
Min_in: fixes A :: "'t set" assumes "finite A"   and "A \<noteq> {}" shows "ord_term_lin.Min A \<in> A"
implicates_def: fixes S :: "'at Literal set set" shows "implicates S = {C. entails S C}"
strict_coboundedI2: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "b < c" shows "a \<sqinter> b < c"
scale_zero_right: fixes a :: "'b" shows "smult a 0 = 0"
hr_compI: fixes b :: "'a"   and a :: "'b"   and R2 :: "('a \<times> 'b) set"   and R1 :: "'a \<Rightarrow> 'c \<Rightarrow> assn"   and c :: "'c" assumes "(b, a) \<in> R2" shows "R1 b c \<Longrightarrow>\<^sub>A hr_comp R1 R2 a c"
find_ce_spec_def: fixes G :: "('h, 'i) igb_graph_rec_scheme" shows "find_ce_spec G \<equiv> ASSERT (igb_graph G) \<bind> (\<lambda>_. ASSERT (finite ((g_E G)\<^sup>* `` g_V0 G)) \<bind> (\<lambda>_. SPEC (\<lambda>res. case res of None \<Rightarrow> \<forall>r. \<not> igb_graph.is_acc_run G r | Some None \<Rightarrow> \<exists>r. igb_graph.is_acc_run G r | Some (Some r) \<Rightarrow> igb_graph.is_acc_run G r)))"
uexpr_eq_iff: fixes e :: "('c, 'd) uexpr"   and f :: "('c, 'd) uexpr" shows "(e = f) = (\<forall>b. \<lbrakk>e\<rbrakk>\<^sub>e b = \<lbrakk>f\<rbrakk>\<^sub>e b)"
ND0aux1: fixes x :: "'b"   and y :: "'b"   and b :: "('b, 'c) Combinators list" assumes "DenyAllFromTo x y \<in> set b" shows "x \<in> set (net_list_aux b)"
rtranclE: fixes a :: "'ap"   and b :: "'ap"   and r :: "('ap \<times> 'ap) set" assumes "(a, b) \<in> r\<^sup>*" obtains "a = b"   | y :: "'ap" where "(a, y) \<in> r\<^sup>*" and "(y, b) \<in> r"
map_id: fixes t :: "'b \<Rightarrow> 'c" shows "id \<circ> t = t"
psubset_trans: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set" assumes "A \<subset> B"   and "B \<subset> C" shows "A \<subset> C"
c: shows "steps0 (1, l, r) (tp @ shift (mopup n) off) stpa = (s', l', r')"
ordered_map_nonempty: fixes S :: "('a \<times> 'b) set" assumes "is_map S"   and "ordered_map S \<noteq> []"   and "finite S" shows "S \<noteq> {}"
g2: shows "rec_exec rec_halt [code tp, bl2wc (<lm>)] = stpa"
letter_less_def: fixes R :: "('b \<times> 'b) set" shows "letter_less R = {(a, b). (snd a, snd b) \<in> R}"
insert_code_1: fixes x :: "'b"   and xs :: "'b list" shows "insert x (set xs) = set (List.insert x xs)"
tauLaw3PsiCong: fixes \<Psi> :: "'b"   and \<alpha> :: "'a prefix"   and P :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> \<alpha>\<cdot>(\<tau>.P) \<doteq> \<alpha>\<cdot>P"
analytic_continuation': fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and g :: "complex \<Rightarrow> complex"   and U :: "complex set"   and \<xi> :: "complex"   and w :: "complex" assumes "f holomorphic_on S"   and "g holomorphic_on S"   and "open S"   and "connected S"   and "U \<subseteq> S"   and "\<xi> \<in> S"   and "\<xi> islimpt U"   and "\<And>z. z \<in> U \<Longrightarrow> f z = g z"   and "w \<in> S" shows "f w = g w"
simps_10: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_option R None None"
extremum_unique: fixes a :: "'d multiset" shows "(a \<subseteq># {#}) = (a = {#})"
has_integral_alt': fixes f :: "'b \<Rightarrow> 'c"   and i :: "'c"   and s :: "'b set" shows "(f has_integral i) s = ((\<forall>a b. (\<lambda>x. if x \<in> s then f x else (0::'c)) integrable_on cbox a b) \<and> (\<forall>e>0. \<exists>B>0. \<forall>a b. ball (0::'b) B \<subseteq> cbox a b \<longrightarrow> norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else (0::'c)) - i) < e))"
in_hom_def: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright> \<equiv> a \<in> C.domains f \<and> b \<in> C.codomains f"
ab: assumes "rebase_dvd TYPE('a) TYPE('b)" shows "CARD('b) \<le> CARD('a)"
h: shows "is_final (steps0 (1, l, r) (tp @ shift (mopup n) off) stp)"
reachable_in_verts_2: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>* v" shows "v \<in> verts G"
Gets_imp_Says: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> set_mr" shows "\<exists>A. Says A B X \<in> set evs"
coprime_commute: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and 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 b :: "'c"   and a :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one b a = algebraic_semidom.coprime times one a b"
one_index: fixes i :: "'b" shows "1 $ i = (1::'a)"
a0: fixes i :: "nat" assumes "p < i" shows "a i = 0"
deg_mult: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<noteq> \<zero>\<^bsub>P\<^esub>"   and "q \<noteq> \<zero>\<^bsub>P\<^esub>"   and "p \<in> carrier P"   and "q \<in> carrier P" shows "deg R (p \<otimes>\<^bsub>P\<^esub> q) = deg R p + deg R q"
subtopology_eq_discrete_topology: fixes S :: "'c set"   and X :: "'c topology" assumes "S \<subseteq> topspace X \<and> S \<inter> X derived_set_of S = {}" shows "subtopology X S = discrete_topology S"
seq_concat_helper: fixes s :: "'a"   and ls :: "('b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set"   and ss2 :: "'a \<times> ('b \<times> 'a) list" assumes "(s, ls) \<in> seq ars"   and "ss2 \<in> seq ars"   and "lst (s, ls) = fst ss2" shows "(s, ls @ snd ss2) \<in> seq ars \<and> lst (s, ls @ snd ss2) = lst ss2"
surjective: fixes r :: "('e, 'f, 'g, 'h) FSM_scheme" shows "r = \<lparr>succ = succ r, inputs = inputs r, outputs = outputs r, initial = initial r, \<dots> = more r\<rparr>"
comp_wiring_simps: fixes f :: "'a \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'd"   and f' :: "'e \<Rightarrow> 'b"   and g' :: "'c \<Rightarrow> 'f" shows "(f, g) \<circ>\<^sub>w (f', g') = (f' \<circ> f, g \<circ> g')"
a4: shows "tiTable_SampleT req x stop lose (fin_inf_append [init_state] st) (fin_inf_append [[]] buffer) ack i1 vc st"
FactGroup_DirProd_multiplication_iso_set: fixes K :: "('e, 'f) monoid_scheme"   and H :: "'a set"   and N :: "'e set" assumes "Group.group K"   and "H \<lhd> G"   and "N \<lhd> K" shows "(\<lambda>(X, Y). X \<times> Y) \<in> Group.iso ((G Mod H) \<times>\<times> (K Mod N)) (G \<times>\<times> K Mod H \<times> N)"
br_list_rel: fixes a :: "'a \<Rightarrow> 'b"   and I :: "'a \<Rightarrow> bool" shows "\<langle>br a I\<rangle>list_rel = br (map a) (list_all I)"
n: shows "nsphere n = subtopology (subtopology (nsphere (Suc n)) {x. 0 \<le> x (Suc n)}) {x. x (Suc n) = 0}"
guha_unmatched: fixes ft :: "('c, 'd) flow_entry_match list"   and \<gamma> :: "'c set \<Rightarrow> 'e \<Rightarrow> bool"   and p :: "'e" assumes "\<forall>fe\<in>set ft. \<gamma> (ofe_fields fe) p = False" shows "guha_table_semantics \<gamma> ft p None"
simps_2: shows "P_D2_H2_d D2 = linord_of_list [(D2, H1), (D2, H2)]"
reflpD: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "reflp r" shows "r x x"
ifps_Xa0: shows "fps_ginv fps_X a $ 0 = (0::'a)"
RootEx_HA: fixes HA :: "('a, 'b, 'c) hierauto" shows "RootEx (SAs HA) (CompFun HA)"
homeomorphic_punctured_sphere_affine_gen: fixes S :: "'a set"   and a :: "'a"   and T :: "'b set" assumes "convex S"   and "bounded S"   and "a \<in> rel_frontier S"   and "affine T"   and "aff_dim S = aff_dim T + 1" shows "rel_frontier S - {a} homeomorphic T"
in_restrict_rel_iff: fixes x :: "'b"   and y :: "'b"   and A :: "'b set"   and R :: "('b \<times> 'b) set" shows "((x, y) \<in> restrict_rel A R) = ((x, y) \<in> R \<and> x \<in> A \<and> y \<in> A)"
synPrefix1: fixes ta :: "'c event list"   and s :: "'c event list"   and t :: "'c event list"   and A :: "'c event set"   and u :: "'c event list" assumes "ta = []" shows "\<exists>t1 u1. s @ t setinterleaves ((ta, u), A) \<longrightarrow> t1 \<le> ta \<and> u1 \<le> u \<and> s setinterleaves ((t1, u1), A)"
i_shrink_nth: fixes f :: "nat \<Rightarrow> 'b message_af"   and k :: "nat"   and n :: "nat" shows "(f \<div> k) n = last_message (f \<Up> (n * k) \<Down> k)"
induces_monoidal_category: shows "monoidal_category (\<cdot>) T\<^sub>F\<^sub>M\<^sub>C \<alpha>\<^sub>F\<^sub>M\<^sub>C \<iota>\<^sub>F\<^sub>M\<^sub>C"
inc_i: fixes i :: "'b word"   and m :: "'b word" assumes "1 \<le> i"   and "i < m" shows "1 \<le> i + 1 \<and> i + 1 \<le> m"
refl: fixes a :: "'c \<times> 'd" shows "less_eq_prod a a"
det_row_add: fixes a :: "nat \<Rightarrow> 'b vec"   and k :: "nat"   and n :: "nat"   and b :: "nat \<Rightarrow> 'b vec"   and c :: "nat \<Rightarrow> 'b vec" assumes "a k \<in> carrier_vec n"   and "b k \<in> carrier_vec n"   and "c \<in> {0..<n} \<rightarrow> carrier_vec n"   and "k < n" shows "det (mat\<^sub>r n n (\<lambda>i. if i = k then a i + b i else c i)) = det (mat\<^sub>r n n (\<lambda>i. if i = k then a i else c i)) + det (mat\<^sub>r n n (\<lambda>i. if i = k then b i else c i))"
cases: fixes x :: "('c, 'd, 'varSort, 'var, 'e) qAbs" obtains xs :: "'varSort" and xa :: "'var" and X :: "('c, 'd, 'varSort, 'var, 'e) qTerm" where "x = qAbs xs xa X"
assms_14: shows "\<forall>j\<le>2 * d + (4 + k). lose (t + j) = [False]"
multiplicity_same_power: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "p \<noteq> zero"   and "\<not> dvd.dvd times p one" shows "factorial_semiring.multiplicity times one p (power.power one times p n) = n"
clo: fixes i :: "'b" assumes "i \<in> I" shows "closedin X (T i)"
hom_into_subgroup_eq_gen: fixes G :: "('b, 'c) monoid_scheme"   and h :: "'d \<Rightarrow> 'b"   and K :: "('d, 'e) monoid_scheme"   and H :: "'b set" assumes "Group.group G" shows "(h \<in> hom K (subgroup_generated G H)) = (h \<in> hom K G \<and> h ` carrier K \<subseteq> carrier (subgroup_generated G H))"
winding_number_join: fixes \<gamma>1 :: "real \<Rightarrow> complex"   and z :: "complex"   and \<gamma>2 :: "real \<Rightarrow> complex" assumes "path \<gamma>1"   and "z \<notin> path_image \<gamma>1"   and "path \<gamma>2"   and "z \<notin> path_image \<gamma>2"   and "pathfinish \<gamma>1 = pathstart \<gamma>2" shows "winding_number (\<gamma>1 +++ \<gamma>2) z = winding_number \<gamma>1 z + winding_number \<gamma>2 z"
WHILE_eq_I_rwof: fixes m :: "'s nres"   and c :: "'s \<Rightarrow> bool" shows "m \<bind> WHILE c gen_step = m \<bind> WHILE\<^bsup>rwof m c gen_step\<^esup> c gen_step"
Basic: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and f :: "'b \<Rightarrow> 'b"   and s :: "'b" shows "\<Gamma>\<turnstile> (Basic f, Normal s) \<rightarrow> (Skip, Normal (f s))"
AKcryptSK_analz_insert: fixes K :: "nat"   and K' :: "nat"   and evs :: "event list" assumes "AKcryptSK K K' evs"   and "K \<in> symKeys"   and "evs \<in> kerbIV_gets" shows "Key K' \<in> analz (insert (Key K) (knows Spy evs))"
assms: shows "run_one_step d i (s, vs, ves, Callcl cl) = (s', vs', res)"
simps_2: fixes carrier :: "'e set"   and mult :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and more :: "'f" shows "\<one>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr>\<^esub> = one"
match_list_comb_list_comb_some_2: fixes f :: "term"   and ps :: "term list"   and g :: "'c"   and vs :: "'c list"   and env :: "(name, 'c) fmap" assumes "match (list_comb f ps) (list_comb g vs) = Some env"   and "left_nesting f = left_nesting g"   and "is_const (fst (strip_comb f))" shows "length ps = length vs"
inner_sum_left: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and y :: "'b" shows "inner (sum f A) y = (\<Sum>x\<in>A. inner (f x) y)"
Span_hom: fixes K :: "'a set"   and Us :: "'a list" assumes "K \<subseteq> carrier (L\<lparr>carrier := A\<rparr>)"   and "set Us \<subseteq> carrier (L\<lparr>carrier := A\<rparr>)" shows "Rupt.Span ((rupture_surj A P \<circ> poly_of_const) ` K) (map (rupture_surj A P \<circ> poly_of_const) Us) = (rupture_surj A P \<circ> poly_of_const) ` Hom.Span K Us"
take_columns_def: fixes A :: "'b matrix"   and c :: "nat" shows "take_columns A c \<equiv> Abs_matrix (\<lambda>j i. if i < c then Rep_matrix A j i else (0::'b))"
Cl2Equiv: fixes T :: "(ZF, ZF) Axioms"   and A :: "ZF"   and e :: "(ZF, ZF) Expression"   and B :: "ZF"   and d :: "(ZF, ZF) Expression" assumes "ZFAxioms T"   and "Sig aS\<^bsub>T\<^esub> \<triangleright> Vx : A \<turnstile> e : B"   and "[A,e,B]\<^bsub>T\<^esub> = [A,d,B]\<^bsub>T\<^esub>" shows "Vx : A \<turnstile> e\<equiv>d : B \<in> Axioms.Theory T"
ffilter_fBall: fixes xs :: "'a fset"   and P :: "'a \<Rightarrow> bool" shows "fBall xs P = (ffilter P xs = xs)"
count_roots_between: fixes a :: "real"   and b :: "real" assumes "p \<noteq> 0"   and "a \<le> b" shows "sign_changes ps a - sign_changes ps b = card {x. a < x \<and> x \<le> b \<and> poly p x = 0}"
B: shows "pairwise (\<lambda>x y. disjnt (B x) (B y)) A"
is_infI: fixes i :: "'a"   and x :: "'a"   and y :: "'a" assumes "i \<sqsubseteq> x"   and "i \<sqsubseteq> y"   and "\<And>z. \<lbrakk>z \<sqsubseteq> x; z \<sqsubseteq> y\<rbrakk> \<Longrightarrow> z \<sqsubseteq> i" shows "is_inf x y i"
y_def: shows "y = pbasepoint pl \<leadsto> direction1 pl"
map_simp_ide: fixes a :: "'a" assumes "A.ide a" shows "\<sigma>'\<sigma>.map a = B (\<sigma> a) (\<sigma>' a)"
simps_1: fixes i :: "nat"   and v :: "'c" shows "[][i := v] = []"
prems_6: shows "(r # rs) \<bullet>\<cdot> ms = (s # ss) \<bullet>\<cdot> ms"
nn_integral_zero': fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "AE x in M. f x = 0" shows "integral\<^sup>N M f = 0"
dagger_of_Z: shows "Z\<^sup>\<dagger> = Z"
h'_g: fixes y :: "'v"   and n :: "nat" assumes "y \<in> Y n" shows "(\<Sum>\<^sup>+ x\<in>A_n' n. h' n x y) = ennreal (g n y)"
hyps_3: shows "P \<parallel> Q \<Longrightarrow>\<^sub>\<tau> P \<parallel> Q''"
HEndPhase0_HInv4a1_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase0 s s' p"   and "Inv2a s'"   and "Inv2c_inner s p"   and "HInv4d s p" shows "HInv4a1 s' p"
Rep_ivl_inverse: fixes x :: "'b ivl" shows "Abs_ivl (Rep_ivl x) = x"
ahm_update_correct_2: fixes hm :: "('a, 'b) ArrayHashMap_Impl.hashmap"   and k :: "'a"   and v :: "'b" assumes "ArrayHashMap_Impl.ahm_invar hm" shows "ArrayHashMap_Impl.ahm_\<alpha> (ArrayHashMap_Impl.ahm_update k v hm) = ArrayHashMap_Impl.ahm_\<alpha> hm(k \<mapsto> v)"
closed_sum_def: fixes A :: "'a set"   and B :: "'a set" shows "A +\<^sub>M B = closure (A + B)"
set_of_plus: fixes A :: "'b interval"   and B :: "'b interval" shows "set_of (A + B) = set_of A + set_of B"
lossless_R1_14: fixes S1_OT12 :: "bool \<times> bool \<Rightarrow> unit \<Rightarrow> 'g spmf"   and R1_OT12 :: "bool \<times> bool \<Rightarrow> bool \<Rightarrow> 'g spmf"   and adv_OT12 :: "real"   and S2_OT12 :: "bool \<Rightarrow> bool \<Rightarrow> 'h spmf"   and R2_OT12 :: "bool \<times> bool \<Rightarrow> bool \<Rightarrow> 'h spmf"   and protocol_OT12 :: "bool \<times> bool \<Rightarrow> bool \<Rightarrow> (unit \<times> bool) spmf"   and msgs :: "bool \<times> bool \<times> bool \<times> bool"   and C :: "bool \<times> bool" assumes "ot14_base S1_OT12 R1_OT12 adv_OT12 S2_OT12 R2_OT12 protocol_OT12" shows "lossless_spmf (ot14_base.R1_14 R1_OT12 msgs C)"
H3: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c"   and R :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "\<turnstile> \<box>[P \<longrightarrow> Q]_v"   and "\<turnstile> \<box>[Q \<longrightarrow> R]_v" shows "\<turnstile> \<box>[P \<longrightarrow> R]_v"
gc_to_red: fixes Ns :: "('f \<times> 'l) set llist" assumes "chain (\<leadsto>GC) Ns" shows "chain (\<rhd>L) Ns"
comp_char: fixes g :: "'a"   and f :: "'a" shows "S.comp g f = (if S.arr f \<and> S.arr g \<and> seq g f then g \<cdot> f else null)"
coprime_fib_Suc_nat: fixes n :: "nat" shows "coprime (fib n) (fib (Suc n))"
intF_def: 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 infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and eintT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and eintF :: "('a, 'c) efsym \<Rightarrow> univ list \<Rightarrow> univ"   and eintP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and f :: "'a"   and al :: "univ list" assumes "ProblemIkTpart_TEModel wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw eintT eintF eintP" shows "ProblemIkTpart_TEModel.intF arOf resOf prot protFw eintT eintF f al \<equiv> if ProblemIkTpart.unprot prot protFw (resOf f) \<or> protFw (resOf f) then eintF (Oldf f) (map2 (ProblemIkTpart_TEModel.ntsem prot protFw eintT eintF) (arOf f) al) else eintF (Tag (resOf f)) [eintF (Oldf f) (map2 (ProblemIkTpart_TEModel.ntsem prot protFw eintT eintF) (arOf f) al)]"
rtrancl_idemp_self_comp: fixes R :: "('a \<times> 'a) set" shows "R\<^sup>* O R\<^sup>* = R\<^sup>*"
assms_5: shows "b\<^sup>\<infinity> = b\<^sup>\<star>"
rbt_comp_insert_with_key: fixes c :: "'b \<Rightarrow> 'b \<Rightarrow> order" assumes "comparator c" shows "RBT_Comparator_Impl.rbt_comp_insert_with_key c = ord.rbt_insert_with_key (lt_of_comp c)"
replicate_mset_0: fixes x :: "'a" shows "replicate_mset 0 x = {#}"
CollectI: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "P a" shows "s2p {x. P x} a"
K2_isometry_exp_2dist: fixes a :: "proj2"   and b :: "proj2"   and J :: "cltn2" assumes "a \<in> hyp2"   and "b \<in> hyp2"   and "is_K2_isometry J" shows "exp_2dist (apply_cltn2 a J) (apply_cltn2 b J) = exp_2dist a b"
ordinal_utility_def: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and u :: "'b \<Rightarrow> real" shows "ordinal_utility carrier relation u \<equiv> (\<forall>x y. x \<in> carrier \<longrightarrow> y \<in> carrier \<longrightarrow> x \<succeq>[relation] y = (u y \<le> u x)) \<and> (\<forall>x y. x \<succeq>[relation] y \<longrightarrow> x \<in> carrier) \<and> (\<forall>x y. x \<succeq>[relation] y \<longrightarrow> y \<in> carrier)"
assms_4: shows "t \<le> v ; v\<^sup>T"
interleave_valid_list': fixes xs :: "Vars.var list"   and S :: "Vars.var set"   and S' :: "Vars.var set"   and G :: "CoCalls" assumes "xs \<in> valid_lists (S \<union> S') G" shows "\<exists>ys zs. xs \<in> ys \<otimes> zs \<and> ys \<in> valid_lists S G \<and> zs \<in> valid_lists S' G"
Pax_def: shows "Pax = Pos ''a'' [\<epsilon> ''x'']"
simps: fixes a1 :: "'a basic_type"   and a2 :: "'a basic_type" shows "a1 \<sqsubset>\<^sub>B a2 = (a1 = OclVoid \<and> a2 = Boolean \<or> a1 = OclVoid \<and> a2 = UnlimitedNatural \<or> a1 = OclVoid \<and> a2 = String \<or> (\<exists>\<C>. a1 = OclVoid \<and> a2 = \<langle>\<C>\<rangle>\<^sub>\<T>) \<or> (\<exists>\<E>. a1 = OclVoid \<and> a2 = Enum \<E>) \<or> a1 = UnlimitedNatural \<and> a2 = Integer \<or> a1 = Integer \<and> a2 = Real \<or> (\<exists>\<C> \<D>. a1 = \<langle>\<C>\<rangle>\<^sub>\<T> \<and> a2 = \<langle>\<D>\<rangle>\<^sub>\<T> \<and> \<C> < \<D>) \<or> a1 = Boolean \<and> a2 = OclAny \<or> a1 = Real \<and> a2 = OclAny \<or> a1 = String \<and> a2 = OclAny \<or> (\<exists>\<C>. a1 = \<langle>\<C>\<rangle>\<^sub>\<T> \<and> a2 = OclAny) \<or> (\<exists>\<E>. a1 = Enum \<E> \<and> a2 = OclAny))"
subm: shows "h\<^sub>0' ++ h\<^sub>1 \<subseteq>\<^sub>m h\<^sub>0 ++ h\<^sub>1"
basic_trans_rules_23: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
AE_BallI: fixes X :: "'a set"   and P :: "'a \<Rightarrow> bool"   and F :: "'a measure" assumes "\<forall>x\<in>X. P x" shows "set_almost_everywhere X F P"
subst_closed_id: fixes t :: "'a"   and env :: "(name, 'a) fmap" assumes "closed t" shows "subst t env = t"
simE_1: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and a :: "name"   and x :: "name"   and Q' :: "pi" assumes "P \<leadsto>\<^sup>^<Rel> Q"   and "Q \<longmapsto> a<\<nu>x> \<prec> Q'"   and "x \<sharp> P" shows "\<exists>P'. P \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel"
leadsETo_Un_duplicate2: fixes F :: "'d program"   and A :: "'d set"   and CC :: "'d set set"   and A' :: "'d set"   and C :: "'d set" assumes "F \<in> A leadsTo[CC] (A' \<union> C \<union> C)" shows "F \<in> A leadsTo[CC] (A' \<union> C)"
greatest_less_zero_row: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat"   and i :: "'d" assumes "reduced_row_echelon_form_upt_k A k"   and "is_zero_row_upt_k i k A"   and "\<not> (\<forall>a. is_zero_row_upt_k a k A)" shows "(GREATEST m. \<not> is_zero_row_upt_k m k A) < i"
complete_sc_ta_seq_consist: fixes s :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and vs :: "'addr \<times> addr_loc \<Rightarrow> ('addr val \<times> bool) option" assumes "sc_completion s vs"   and "\<And>vs ln. ta_seq_consist P vs (llist_of (convert_RA ln))" shows "ta_seq_consist P vs (lconcat (lmap (\<lambda>(t, ta). llist_of \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) (complete_sc s vs)))"
is_lasso_prpl_conv: fixes m :: "'a"   and prpl :: "'v list \<times> 'v list" shows "gb_graph.is_lasso_prpl (to_gbg_ext m) prpl = (snd prpl \<noteq> [] \<and> gb_graph.is_lasso (to_gbg_ext m) (lasso_of_prpl prpl))"
simps_2: fixes \<theta> :: "(name \<times> trm) list"   and t\<^sub>1 :: "trm"   and t\<^sub>2 :: "trm" shows "\<theta><App t\<^sub>1 t\<^sub>2> = App \<theta><t\<^sub>1> \<theta><t\<^sub>2>"
HPhase1or2Write_HInv5_p2: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "HPhase1or2Write s s' p d"   and "HInv4c s p"   and "phase s p = 2"   and "\<exists>D\<in>MajoritySet. \<exists>q. \<forall>d\<in>D. bal (dblock s p) < mbal (disk s d q) \<and> \<not> hasRead s p d q" shows "\<exists>D\<in>MajoritySet. \<exists>q. \<forall>d\<in>D. bal (dblock s' p) < mbal (disk s' d q) \<and> \<not> hasRead s' p d q"
set_contI: fixes f :: "'a \<Rightarrow> 'b set" assumes "\<And>Y. chain Y \<Longrightarrow> f (\<Squnion>i. Y i) = \<Union> (f ` range Y)" shows "cont f"
map_conv_bind_option: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b option" shows "map_option f x = x \<bind> Some \<circ> f"