hcju/isabelleps huggingface dataset:
not_mod_2_eq_1_eq_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "(modulo a (numeral.numeral one plus (num.Bit0 num.One)) \<noteq> one) = (modulo a (numeral.numeral one plus (num.Bit0 num.One)) = zero)"
normF_triangle: fixes F :: "'a set"   and x :: "'a \<Rightarrow> complex"   and y :: "'a \<Rightarrow> complex" assumes "finite F" shows "normF F (\<lambda>a. x a + y a) \<le> normF F x + normF F y"
continuous_on_hurwitz_zeta': fixes A :: "'a set"   and f :: "'a \<Rightarrow> complex"   and a :: "real" assumes "continuous_on A f"   and "0 < a"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> 1" shows "continuous_on A (\<lambda>x. hurwitz_zeta a (f x))"
bit_eq_iff: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "(a = b) = (\<forall>n. bit a n = bit b n)"
rInput: fixes M :: "'a"   and K :: "'a"   and xvec :: "name list"   and N :: "'a"   and Tvec :: "'a list" assumes "\<Psi> \<turnstile> M \<leftrightarrow> K"   and "set xvec \<subseteq> supp N"   and "length xvec = length Tvec"   and "distinct xvec" shows "Prop (pInput M xvec N) (K\<lparr>N[xvec::=Tvec]\<rparr>) (P[xvec::=Tvec])"
finite_bernoulli_denom_set: fixes n :: "nat" assumes "0 < n" shows "finite {p. prime p \<and> p - 1 dvd n}"
scast_scast_id_1: fixes x :: "'b word" shows "scast (scast x) = x"
antimono_def: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e" assumes "class.order less_eq less" shows "order.antimono less_eq f = (\<forall>x y. less_eq x y \<longrightarrow> f y \<le> f x)"
append1_eq_conv: fixes xs :: "'d list"   and x :: "'d"   and ys :: "'d list"   and y :: "'d" shows "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
scc_root_is_Min_disc: fixes v :: "'v"   and scc :: "'v set" assumes "scc_root s' v scc" shows "Min (\<delta> s' ` (scc \<inter> dom (discovered s'))) = \<delta> s' v"
surjective: fixes r :: "('a, 'b) simple_packet_scheme" shows "r = \<lparr>p_iiface = p_iiface r, p_oiface = p_oiface r, p_src = p_src r, p_dst = p_dst r, p_proto = p_proto r, p_sport = p_sport r, p_dport = p_dport r, p_tcp_flags = p_tcp_flags r, p_payload = p_payload r, \<dots> = simple_packet.more r\<rparr>"
lActive_greater_active: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and n' :: "nat" assumes "dynamic_component.lActive active c t < n"   and "\<forall>n''>n'. \<not> active c (t n'')" shows "\<not> active c (t n)"
typ_ok_sig_contained_tvars_typ_ok_sig: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and ty :: "typ"   and idn :: "variable"   and S :: "String.literal set" assumes "typ_ok_sig \<Sigma> ty"   and "(idn, S) \<in> tvsT ty" shows "typ_ok_sig \<Sigma> (Tv idn S)"
elims: fixes x :: "sign"   and xa :: "'b equation"   and y :: "'b literal" assumes "mk_lit x xa = y" obtains xb :: "'b equation" where "x = pos" and "xa = xb__" and "y = Pos xb__"   | xb :: "'b equation" where "x = neg" and "xa = xb" and "y = Neg xb"
inj_Inr: fixes A :: "'c set" shows "inj_on Inr A"
dvd_triv_right: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.comm_monoid_mult times one" shows "dvd.dvd times a (times b a)"
bind_returns_heap_I2: fixes h :: "'a"   and f :: "('a, 'b, 'c) prog"   and h' :: "'a"   and g :: "'c \<Rightarrow> ('a, 'b, 'd) prog"   and h'' :: "'a" assumes "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> h' \<turnstile> g x \<rightarrow>\<^sub>h h''" shows "h \<turnstile> f \<bind> g \<rightarrow>\<^sub>h h''"
R: fixes T :: "ty"   and \<Delta> :: "(name \<times> ty) list" assumes "\<Gamma> \<turnstile> t : T"   and "\<turnstile> p : T \<Rightarrow> \<Delta>"   and "\<Delta> @ \<Gamma> \<turnstile> u : U" shows "P"
intro: fixes G :: "('c, 'd) monoid_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"   and "\<And>x y z. \<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"   and "\<one>\<^bsub>G\<^esub> \<in> carrier G"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"   and "\<And>x. x \<in> carrier G \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = x" shows "Group.monoid G"
\<eta>_is_natural_transformation: shows "natural_transformation (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) B.map FG.GF.map FG.\<eta>"
exhaust_sel: fixes prod :: "'r \<times> 's" obtains "prod = (prod.fst prod, prod.snd prod)"
Abinds_Afix_below: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "ABinds \<Gamma>\<cdot>(Afix \<Gamma>\<cdot>ae) \<sqsubseteq> Afix \<Gamma>\<cdot>ae"
bchoice: fixes S :: "'a set"   and Q :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>x\<in>S. \<exists>y. Q x y" shows "\<exists>f. \<forall>x\<in>S. Q x (f x)"
assms_3: shows "interior s1 \<inter> interior s2 = {}"
H_awalkI: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "G.awalk u p v"   and "{a, a'} \<inter> set p = {}" shows "H.awalk u p v"
less_degree_aform_aform_err_to_aformD: fixes i :: "nat"   and X :: "(real \<times> real pdevs) \<times> real"   and n :: "nat" assumes "i < degree_aform (aform_err_to_aform X n)" shows "i < max (Suc n) (degree_aform_err X)"
homologous_rel_refl: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "homologous_rel p X S c c"
varsAppendValuation: fixes valuation1 :: "Literal list"   and valuation2 :: "Literal list" shows "vars (valuation1 @ valuation2) = vars valuation1 \<union> vars valuation2"
is_weak_ranking_Cons_empty: fixes xs :: "'b set list" shows "\<not> is_weak_ranking ({} # xs)"
Un_open_segment: fixes b :: "'b"   and a :: "'b"   and c :: "'b" assumes "b \<in> {a<--<c}" shows "{a<--<b} \<union> {b} \<union> {b<--<c} = {a<--<c}"
lget_return_edges: shows "lget_return_edges \<equiv> lift_get_return_edges get_return_edges valid_edge sourcenode targetnode kind"
topological_tendstoD: fixes f :: "'b \<Rightarrow> 'a"   and l :: "'a"   and F :: "'b filter"   and S :: "'a set" assumes "(f \<longlongrightarrow> l) F"   and "open S"   and "l \<in> S" shows "\<forall>\<^sub>F x in F. f x \<in> S"
ex_poincare_line_one_point: fixes z :: "complex_homo" shows "\<exists>l. is_poincare_line l \<and> z \<in> circline_set l"
that: fixes P' :: "ccs" assumes "P \<Longrightarrow>\<alpha> \<prec> P'"   and "(P', Q') \<in> Rel" shows "thesis"
lenlex_append1: fixes us :: "'a list"   and xs :: "'a list"   and R :: "('a \<times> 'a) set"   and vs :: "'a list"   and ys :: "'a list" assumes "(us, xs) \<in> lenlex R"   and "length vs = length ys" shows "(us @ vs, xs @ ys) \<in> lenlex R"
ipresIGWls_def: fixes h :: "'x \<Rightarrow> 'y"   and MOD :: "('z, 'aa, 'ab, 'ac, 'ad, 'ae, 'x, 'af, 'ag) model_scheme"   and MOD' :: "('ah, 'ai, 'aj, 'ac, 'ak, 'al, 'y, 'am, 'an) model_scheme" shows "ipresIGWls h MOD MOD' \<equiv> \<forall>s X. igWls MOD s X \<longrightarrow> igWls MOD' s (h X)"
subsetI: fixes A :: "'k set"   and B :: "'k set" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<in> B" shows "A \<subseteq> B"
simps: fixes x :: "'a ext"   and y :: "'a ext" shows "x * y = x \<sqinter> y"
tensor_preserves_ide: fixes a :: "'c local.term set"   and b :: "'c local.term set" assumes "\<F>C.ide a"   and "\<F>C.ide b" shows "\<F>C.ide (a \<otimes> b)"
is_closedI: fixes A :: "'a set" assumes "\<And>x y. \<lbrakk>x .= y; x \<in> A; y \<in> carrier L\<rbrakk> \<Longrightarrow> y \<in> A"   and "A \<subseteq> carrier L" shows "is_closed A"
of_int_less_0_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (ring_1.of_int one plus zero minus z) zero = (z < 0)"
uniform_limit_cong: fixes X :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and F :: "'b filter"   and h :: "'c \<Rightarrow> 'd"   and i :: "'c \<Rightarrow> 'd" assumes "\<forall>\<^sub>F y in F. \<forall>x\<in>X. f y x = g y x"   and "\<And>x. x \<in> X \<Longrightarrow> h x = i x" shows "uniform_limit X f h F = uniform_limit X g i F"
ExistsI: fixes G :: "('a, 'b) form list"   and a :: "('a, 'b) form"   and t :: "'a FOL_Fitting.term" assumes "G \<turnstile> a[t/0]" shows "G \<turnstile> Exists a"
le_mid2__le13: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a"   and A :: "'a"   and B :: "'a"   and M' :: "'a"   and A' :: "'a"   and B' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong M A B"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong M' A' B'"   and "Tarski_neutral_dimensionless.Le Bet Cong A M A' M'" shows "Tarski_neutral_dimensionless.Le Bet Cong A B A' B'"
x: shows "x \<in> {(ram2, ram1), (ram1, hd vs1), (last vs1, ram2)} \<union> Edges vs1"
abelian_groupI: fixes R :: "('h, 'i) ring_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"   and "\<zero>\<^bsub>R\<^esub> \<in> carrier R"   and "\<And>x y z. \<lbrakk>x \<in> carrier R; y \<in> carrier R; z \<in> carrier R\<rbrakk> \<Longrightarrow> x \<oplus>\<^bsub>R\<^esub> y \<oplus>\<^bsub>R\<^esub> z = x \<oplus>\<^bsub>R\<^esub> (y \<oplus>\<^bsub>R\<^esub> z)"   and "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> x \<oplus>\<^bsub>R\<^esub> y = y \<oplus>\<^bsub>R\<^esub> x"   and "\<And>x. x \<in> carrier R \<Longrightarrow> \<zero>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> x = x"   and "\<And>x. x \<in> carrier R \<Longrightarrow> \<exists>y\<in>carrier R. y \<oplus>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>" shows "abelian_group R"
eval_wp_Apply: fixes f :: "'a \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> real" shows "wp (Apply f) P = P \<circ> f"
7: shows "{e. f e \<noteq> bot \<and> f e + bot = fe + bot} \<noteq> {}"
Field_osum: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" shows "Field (r +o r') = Inl ` Field r \<union> Inr ` Field r'"
fin_1: shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M \<noteq> \<infinity>"
left_neutral: fixes a :: "'a" shows "- (1::'a) AND a = a"
\<Sigma>_protocol_def: shows "\<Sigma>_protocol \<equiv> completeness \<and> special_soundness \<and> HVZK"
mult_zero_sup_circ: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y * bot)\<^sup>\<star> = x\<^sup>\<star> * (y * bot)\<^sup>\<star>"
count_sum_mset: fixes M :: "'a multiset multiset"   and b :: "'a" shows "count (\<Sum>\<^sub># M) b = (\<Sum>P\<in>#M. count P b)"
fresh_aux: fixes c :: "name"   and a :: "name"   and x :: "'d"   and b :: "name" assumes "c \<noteq> a"   and "a \<sharp> x"   and "c \<sharp> x" shows "c \<sharp> [(a, b)] \<bullet> x"
eval_single: fixes x :: "'b" shows "pred.eval (Predicate.single x) = (=) x"
that: shows "s\<^sub>0 \<rightarrow>* b"
Comp_is_extensional_3: fixes G :: "'a set"   and F :: "'a set" assumes "Comp G F \<noteq> {}" shows "F \<noteq> {}"
weakly_bisimilar_tau_simulation_step: fixes P :: "'state"   and Q :: "'state"   and P' :: "'state" assumes "P \<approx>\<cdot> Q"   and "P \<Rightarrow> P'" obtains Q' :: "'state" where "Q \<Rightarrow> Q'" and "P' \<approx>\<cdot> Q'"
inj_on_finite: fixes f :: "'q \<Rightarrow> 'r"   and A :: "'q set"   and B :: "'r set" assumes "inj_on f A"   and "f ` A \<subseteq> B"   and "finite B" shows "finite A"
Nil: fixes r :: "('a \<times> 'a) set" shows "sorted_on r []"
Y_def: fixes x :: "'a" shows "Y x = (\<Sum>i\<in>I. X i x)"
b_distinct: shows "b \<notin> lgrid (child b left d) {d} lm \<union> lgrid (child b right d) {d} lm"
upb1_def: shows "upb1 = loc_upb (c_fst (c_snd n)) x"
lgc_fair: fixes Ns :: "('f inference set \<times> ('f \<times> 'l) set) llist" assumes "chain (\<leadsto>LGC) Ns"   and "active_subset (snd (lhd Ns)) = {}"   and "passive_subset (Liminf_llist (lmap snd Ns)) = {}"   and "\<forall>\<iota>\<in>Inf_F. prems_of \<iota> = [] \<longrightarrow> \<iota> \<in> fst (lhd Ns)"   and "Liminf_llist (lmap fst Ns) = {}" shows "empty_ord.fair (lmap snd Ns)"
U_def: shows "U \<equiv> {if learn_lim \<phi> {f k} (\<phi> k) then f' k else f k |k. k \<in> UNIV}"
dist_suc: fixes v :: "nat"   and d :: "nat"   and v' :: "nat"   and v'' :: "nat" assumes "local.dist v d v'"   and "(v', v'') \<in> E" shows "local.dist v (Suc d) v''"
division_ring_divide_zero: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide a zero = zero"
outstanding_non_volatile_refs_owned_or_read_only_def: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" shows "outstanding_non_volatile_refs_owned_or_read_only \<S> ts \<equiv> \<forall>i is \<O> \<R> \<D> \<theta> sb p. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> non_volatile_owned_or_read_only False \<S> \<O> sb"
dim_sums_Int: fixes S :: "('b \<times> 'c) set"   and T :: "('b \<times> 'c) set" assumes "p.subspace S"   and "p.subspace T" shows "p.dim {x + y |x y. x \<in> S \<and> y \<in> T} + p.dim (S \<inter> T) = p.dim S + p.dim T"
Inl_Rep_not_Inr_Rep: fixes a :: "'a"   and b :: "'b" shows "Inl_Rep a \<noteq> Inr_Rep b"
L4_Controller_One: fixes y :: "nat \<Rightarrow> nat list"   and l :: "nat \<Rightarrow> bit"   and z :: "nat \<Rightarrow> bit list"   and t :: "nat" assumes "Controller_L y (fin_inf_append [Zero] l) l z"   and "l (Suc t) = One" shows "z (Suc t) = [] \<and> l t = One \<or> z (Suc t) = [One] \<and> l t = Zero"
new_tstp_lt_isl_2: shows "isl new_tstp = (0 < left I)"
assms: shows "obj_initial (smc_SemiCAT \<alpha>) \<AA>"
finite_UNIV_expr_def: shows "finite_UNIV = Phantom(expr) False"
6: shows "\<Sqinter> insert s ss = s \<sqinter> \<Sqinter> ss"
invariance_of_dimension_closedin_Euclidean_space: fixes n :: "nat"   and S :: "(nat \<Rightarrow> real) set" assumes "closedin (Euclidean_space n) S" shows "(subtopology (Euclidean_space n) S homeomorphic_space Euclidean_space n) = (S = topspace (Euclidean_space n))"
Sup_insert: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and a :: "'b"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (insert a A) = sup a (Sup A)"
centi_def: shows "centi = (1::'a) / (10::'a)\<^sup>2"
signed_list_lconjaction_snd: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and ss :: "'b list"   and x :: "'b \<times> bool" assumes "class.group_add minus plus zero uminus"   and "\<forall>s\<in>set ss. plus s s = zero" shows "snd (foldr (signed_funaction (\<lambda>x y. minus (plus x y) x)) ss x) = (if gcd_nat.greater_eq (count_list (group_add.lconjseq minus plus (rev ss)) (fst x)) 2 then snd x else \<not> snd x)"
distinct_pds_valuation2: fixes j :: "nat"   and n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set" assumes "j \<le> n"   and "distinct_pds K n P" shows "valuation K (\<nu>\<^bsub>K P j\<^esub>)"
power_div_nat: fixes c :: "nat"   and b :: "nat"   and a :: "nat" assumes "c \<le> b"   and "0 < a" shows "a ^ b div a ^ c = a ^ (b - c)"
rep_eq: fixes x :: "nat" shows "bounds_of_interval (pi_float_interval x) = (lb_pi x, ub_pi x)"
eq: fixes i :: "'a" assumes "i \<in> I" shows "sets (M i) = sets (N i)"
rtl_complement_closed: fixes L :: "'a tree set"   and A :: "'a \<Rightarrow> nat option" assumes "L \<in> regular_languages A" shows "ranked_trees A - L \<in> regular_languages A"
assms_3: fixes n :: "nat" assumes "2 \<le> n" shows "\<pi> (real n) < 1 / c * (real n / ln (real n))"
synth_simps_3: fixes X :: "msg"   and H :: "msg set" shows "(Hash X \<in> synth H) = (Hash X \<in> H \<or> X \<in> synth H)"
bBkA: shows "(\<lambda>v. v @\<^sub>v 0\<^sub>v m) ` baseB \<subseteq> mat_kernel A"
compl_inf: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "\<^bold>- (x \<sqinter> y) = \<^bold>- x \<squnion> \<^bold>- y"
dgrad_setI: fixes d :: "'b \<Rightarrow> nat"   and t :: "'b"   and m :: "nat" assumes "d t \<le> m" shows "t \<in> dgrad_set d m"
cases: fixes a1 :: "nat \<times> nat \<Rightarrow> 'capacity"   and a2 :: "nat \<Rightarrow> nat"   and a3 :: "nat \<times> nat \<Rightarrow> 'capacity"   and a4 :: "nat \<Rightarrow> nat" assumes "((a1, a2), a3, a4) \<in> gap_algo_rel" obtains f :: "nat \<times> nat \<Rightarrow> 'capacity" and l :: "nat \<Rightarrow> nat" and e :: "nat \<times> nat" where "a1 = push_effect f__ e" and "a2 = l__" and "a3 = f__" and "a4 = l__" and "Height_Bounded_Labeling c s t f__ l__" and "push_precond f__ l__ e"   | f :: "nat \<times> nat \<Rightarrow> 'capacity" and l :: "nat \<Rightarrow> nat" and u :: "nat" where "a1 = f" and "a2 = gap_relabel_effect f l u" and "a3 = f" and "a4 = l" and "Height_Bounded_Labeling c s t f l" and "relabel_precond f l u"
sorted_upd_until: shows "sorted (map fst (update_until args rel1 rel2 nt auxlist))"
gker_normal: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "Group F"   and "Group G"   and "f \<in> gHom F G" shows "F \<triangleright> gker\<^bsub>F,G\<^esub> f"
ins_list_sorted2: fixes xs :: "'b list"   and a :: "'b"   and x :: "'b"   and ys :: "'b list" assumes "sorted_less (xs @ [a])"   and "x < a" shows "ins_list x (xs @ a # ys) = ins_list x xs @ a # ys"
eq_def: fixes p :: "tm" shows "eq p = (case p of CP (C c) \<Rightarrow> if c = 0\<^sub>N then T else F | CP _ \<Rightarrow> Eq p | _ \<Rightarrow> Eq p)"
eq_iff: fixes a :: "'c \<times> 'd"   and b :: "'c \<times> 'd" shows "(a = b) = (less_eq_prod b a \<and> less_eq_prod a b)"
preserves_reflects_arr: fixes f :: "'b" shows "arr (\<ll>'.map f) = arr f"
add_instr_cache_def: fixes state :: "'b sparc_state"   and va :: "word32"   and word :: "word32"   and byte_mask :: "word4" shows "add_instr_cache state va word byte_mask \<equiv> let tag = ucast (va >> 12); offset0 = ucast va AND 4092; offset1 = offset0 OR 1; offset2 = offset0 OR 2; offset3 = offset0 OR 3; byte0 = ucast (word >> 24); byte1 = ucast (word >> 16); byte2 = ucast (word >> 8); byte3 = ucast word; s0 = if byte_mask AND 8 >> 3 = 1 then icache_mod (tag, offset0) byte0 state else state; s1 = if byte_mask AND 4 >> 2 = 1 then icache_mod (tag, offset1) byte1 s0 else s0; s2 = if byte_mask AND 2 >> 1 = 1 then icache_mod (tag, offset2) byte2 s1 else s1; s3 = if byte_mask AND 1 = 1 then icache_mod (tag, offset3) byte3 s2 else s2 in s3"
termMOD_iwlsFSb: shows "iwlsFSb termMOD"
bisimulation_locale: fixes t :: "'t" shows "bisimulation (r1 t) (r2 t) (bisim t) ta_bisim_bisim_syntax"
impl_of: fixes x :: "('b, 'c) mapping_rbt" shows "mapping_rbt.impl_of x \<in> {t. ord.is_rbt cless t \<or> ID ccompare = None}"
simps_6: fixes \<Gamma> :: "'b \<Rightarrow> PreSimplyTyped.type option"   and P :: "'b ptrm" shows "ptrm_infer_type \<Gamma> (PFst P) = (case ptrm_infer_type \<Gamma> P of None \<Rightarrow> None | Some TUnit \<Rightarrow> None | Some (TVar xa) \<Rightarrow> Map.empty xa | Some (TArr type1 xa) \<Rightarrow> Map.empty xa | Some (TPair \<tau> \<sigma>) \<Rightarrow> Some \<tau>)"
AbsE: fixes v :: "i"   and x :: "\<kappa>" assumes "[\<lparr>A!,x\<rparr> in v]" shows "\<exists>o\<^sub>1 y. Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> o\<^sub>1 = \<alpha>\<nu> y"
smc_SemiCAT_components_3: fixes \<alpha> :: "V" shows "smc_SemiCAT \<alpha>\<lparr>Dom\<rparr> = (\<lambda>\<FF>\<in>\<^sub>\<circ>all_smcfs \<alpha>. \<FF>\<lparr>HomDom\<rparr>)"
set_hyp: shows "list_constr I (length qs) \<longrightarrow> sign \<in> set (characterize_consistent_signs_at_roots_copr p qs)"
abs_of_nonneg: fixes abs :: "'c \<Rightarrow> 'c"   and 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 a :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less_eq zero a" shows "abs a = a"
hcomplex_defs_3: shows "iii = hcomplex_of_complex \<i>"
fin1: fixes K :: "'b \<Rightarrow> 'b DBMEntry" assumes "theLe \<circ> K = id" shows "finite ({u c' - d |c' d. 0 < v c' \<and> v c' \<le> n \<and> c \<noteq> c' \<and> M (v c') (v c) = K d} \<union> {- d |d. M 0 (v c) = K d})"
bounded_linear_transfer: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "bi_unique A"   and "bi_total B"   and "(A ===> A ===> A) (+) (+)"   and "((=) ===> A ===> A) (*\<^sub>R) (*\<^sub>R)"   and "(B ===> B ===> B) (+) (+)"   and "((=) ===> B ===> B) (*\<^sub>R) (*\<^sub>R)"   and "(A ===> (=)) norm norm"   and "(B ===> (=)) norm norm" shows "((B ===> A) ===> (=)) bounded_linear bounded_linear"
size_4: fixes x21 :: "'c tree"   and x22 :: "'c"   and x23 :: "'c tree" shows "size \<langle>x21, x22, x23\<rangle> = size x21 + size x23 + Suc 0"
exec_step_ind_cases_1: fixes n :: "nat"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap"   and \<sigma> :: "jvm_state" assumes "exec_step_ind (StepI (Load n)) P h stk loc C M pc ics frs sh \<sigma>" obtains "\<sigma> = (None, h, (loc ! n # stk, loc, C, M, Suc pc, ics) # frs, sh)"
substitute_substitute: fixes f' :: "'a \<Rightarrow> 'a ttree"   and f :: "'a \<Rightarrow> 'a ttree"   and T :: "'a set"   and t :: "'a ttree" assumes "\<And>x. const_on f' (carrier (f x)) TTree.empty" shows "substitute f T (substitute f' T t) = substitute (\<lambda>x. f x \<otimes>\<otimes> f' x) T t"