hcju/isabelleps huggingface dataset:
right_residual: fixes x :: "'a"   and a :: "'a"   and b :: "'a" shows "(x * a \<le> b) = (x \<le> a l\<rightarrow> b)"
prod_simps_3: fixes f0 :: "'a"   and f1 :: "'a" assumes "arr f0"   and "arr f1" shows "cod (local.prod f0 f1) = local.prod (cod f0) (cod f1)"
sum_distr: fixes X :: "'b set"   and y :: "'b" assumes "finite X" shows "\<Sum> X \<cdot> y = \<Sum> {x \<cdot> y |x. x \<in> X}"
inv_at_def: fixes n :: "nat"   and p :: "'b mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'b mat"   and m :: "nat" shows "inv_at n p A m \<equiv> \<forall>i<n. p A i m"
simps: fixes uu :: "String.literal \<Rightarrow> typ option"   and arf :: "String.literal \<Rightarrow> nat option"   and uv :: "(String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" shows "type_arity (uu, arf, uv) = arf"
one_closed: shows "\<one>\<^bsub>H\<^esub> \<in> carrier H"
map_of_eq_Some_iff: fixes xys :: "('b \<times> 'c) list"   and x :: "'b"   and y :: "'c" assumes "distinct (map fst xys)" shows "(map_of xys x = Some y) = ((x, y) \<in> set xys)"
min_product_set_empty_1: shows "\<Otimes>\<^sub>m {} = {{||}}"
fds_converges_altdef: fixes f :: "'b fds"   and s :: "'b" shows "fds_converges f s = (\<lambda>n. fds_nth f (Suc n) / nat_power (Suc n) s) sums eval_fds f s"
simulation_trans: fixes R1 :: "('h \<times> 'i) set"   and A :: "('h, 'j) graph_rec_scheme"   and B :: "('i, 'k) graph_rec_scheme"   and R2 :: "('i \<times> 'l) set"   and C :: "('l, 'm) graph_rec_scheme" assumes "simulation R1 A B"   and "simulation R2 B C" shows "simulation (R1 O R2) A C"
axioms_2: fixes R :: "('a, 'b) ring_scheme" assumes "cring R" shows "Group.comm_monoid R"
scalar_prod_smult_left: fixes w :: "'a vec"   and v :: "'a vec"   and k :: "'a" assumes "dim_vec w = dim_vec v" shows "(k \<cdot>\<^sub>v w) \<bullet> v = k * (w \<bullet> v)"
swap_Snapshot_Trans: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and d :: "('p, 's, 'm) configuration"   and ev' :: "('p, 's, 'm) event"   and e :: "('p, 's, 'm) configuration"   and d' :: "('p, 's, 'm) configuration"   and e' :: "('p, 's, 'm) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isSnapshot ev"   and "isTrans ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
curryD: fixes f :: "'e \<times> 'f \<Rightarrow> bool"   and a :: "'e"   and b :: "'f" assumes "curry f a b" shows "f (a, b)"
profile_mono_scheme_defined_def: fixes f :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "profile_mono_scheme_defined f g \<equiv> \<forall>x. f x = (\<lambda>\<tau>. if (\<delta> x) \<tau> = true \<tau> then g (x \<tau>) else invalid \<tau>)"
unifiers_occur_left_Fun: fixes x :: "'c"   and ts :: "('d, 'c) Term.term list"   and f :: "'d"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) set" assumes "x \<in> \<Union> (vars_term ` set ts)" shows "unifiers (insert (Var x, Fun f ts) E) = {}"
exec_gpv_oracle_bisim: fixes X :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and s1 :: "'a"   and s2 :: "'b"   and oracle1 :: "'a \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'a) spmf"   and oracle2 :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b) spmf"   and gpv :: "('e, 'c, 'd) gpv"   and R :: "'e \<times> 'a \<Rightarrow> 'e \<times> 'b \<Rightarrow> bool" assumes "X s1 s2"   and "\<And>s1 s2 x. X s1 s2 \<Longrightarrow> rel_spmf (\<lambda>(a, s1') (b, s2'). a = b \<and> X s1' s2') (oracle1 s1 x) (oracle2 s2 x)"   and "\<And>x s1' s2'. \<lbrakk>X s1' s2'; (x, s1') \<in> set_spmf (exec_gpv oracle1 gpv s1); (x, s2') \<in> set_spmf (exec_gpv oracle2 gpv s2)\<rbrakk> \<Longrightarrow> R (x, s1') (x, s2')" shows "rel_spmf R (exec_gpv oracle1 gpv s1) (exec_gpv oracle2 gpv s2)"
iFROM_cut_less1: fixes n :: "nat"   and t :: "nat" assumes "n < t" shows "[n\<dots>] \<down>< t = [n\<dots>,t - Suc n]"
confluent_CR: shows "is_confluent = CR {(x, y). x \<rightarrow> y}"
a: shows "Sup (emeasure M ` {Q \<in> sets M. emeasure N Q \<noteq> \<infinity>}) = Sup (range Q'')"
bvs_def: shows "bvs = drop (Suc n) vs"
nn_integral_measure_spmf: fixes p :: "'c spmf"   and f :: "'c \<Rightarrow> ennreal" shows "integral\<^sup>N (measure_spmf p) f = (\<Sum>\<^sup>+ x. ennreal (spmf p x) * f x)"
depth_app_2: fixes B :: "'b trm"   and A :: "'b trm" shows "depth B < depth (App A B)"
TermP_quot_dbtm: fixes u :: "dbtm" assumes "wf_dbtm u" shows "{} \<turnstile> TermP (quot_dbtm u)"
legs_form_cospan_1: shows "C.cospan doms.\<mu>.leg0 doms.\<nu>.leg1"
CompDiag_TensorDiag: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term"   and w :: "'a local.term" assumes "Diag t"   and "Diag u"   and "Diag v"   and "Diag w"   and "Seq t v"   and "Seq u w" shows "(t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u) \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> (v \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> w) = t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> v \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> w"
pre_oldest_configuration_Min_in: fixes q :: "'b"   and n :: "nat" assumes "pre_oldest_tokens q n \<noteq> {}" shows "Min (pre_oldest_tokens q n) \<in> pre_oldest_tokens q n"
reflection_of_barbs_in_barbed_encoding: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and a :: "'barbs" assumes "rel_reflects_binary_pred Rel (\<lambda>b. hasBarb b (STCalWB SWB TWB))"   and "(P, Q) \<in> Rel"   and "(\<exists>S. S \<in>S Q \<and> S\<down><SWB>a) \<or> (\<exists>T. T \<in>T Q \<and> T\<down><TWB>a)" shows "(\<exists>S. S \<in>S P \<and> S\<down><SWB>a) \<or> (\<exists>T. T \<in>T P \<and> T\<down><TWB>a)"
dom_simp: fixes f :: "'a" assumes "R.arr f" shows "R.dom f = local.dom f"
dual_join: fixes x :: "'b"   and y :: "'b" shows "dual (x \<squnion> y) = dual x \<sqinter> dual y"
maximal_set_idealTr: fixes S :: "'a set"   and mx :: "'a set" assumes "maximal_set {I. ideal R I \<and> S \<inter> I = {}} mx" shows "ideal R mx"
quasi_inverse_simps_4: fixes f :: "'a" assumes "equivalence_map f" shows "trg f\<^sup>~ = src f"
elims_2: fixes x :: "protocol"   and xa :: "8 word" assumes "match_proto x xa" obtains uu_ :: "8 word" where "x = ProtoAny" and "xa = uu_"   | p :: "8 word" and p_p :: "8 word" where "x = Proto p" and "xa = p_p" and "p_p = p"
get_disconnected_nodes_impl: shows "get_disconnected_nodes = a_get_disconnected_nodes"
a0n_trans_def: shows "a0n_trans \<equiv> ((\<Union>A B. \<Union> (range (a0i_running A B))) \<union> (\<Union>A B. \<Union> (range (a0n_commit A B))))\<^sup>="
enum_mono: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and i :: "nat"   and j :: "nat" assumes "kuhn_simplex p n base upd s"   and "i \<le> n"   and "j \<le> n" shows "(kuhn_simplex.enum base upd i \<le> kuhn_simplex.enum base upd j) = (i \<le> j)"
arr_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "VVV.arr f = (B.arr (fst f) \<and> VV.arr (snd f) \<and> src (fst f) = trg (fst (snd f)))"
CompFun_Int_disjoint: fixes S :: "'a"   and T :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "S \<noteq> T"   and "S \<in> HAStates A"   and "T \<in> HAStates A" shows "the (CompFun A T) \<inter> the (CompFun A S) = {}"
reflection: shows "rel_reflects_binary_pred (indRelLSTPO SRel TRel) (\<lambda>b. hasBarb b (STCalWB SWB TWB))"
unroll_between_next2: fixes f :: "face"   and u :: "nat"   and v :: "nat" assumes "distinct (vertices f)"   and "u \<in> \<V> f"   and "v \<in> \<V> f"   and "u \<noteq> v" shows "between (vertices f) u (f \<bullet> v) = between (vertices f) u v @ [v]"
hyps_2: shows "Fun g U = \<Gamma> (Var x)"
scale_zero_left: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "module scale" shows "scale (0::'a) x = (0::'b)"
suc_n_length: shows "enat (Suc n) < llength Ns"
naturality: fixes f :: "'a" assumes "arr f" shows "R (cod f) \<cdot> R f = R f \<cdot> R (local.dom f)"
uniform_limit1': fixes x :: "real"   and y :: "real" assumes "- 2 < x" shows "uniform_limit {z. Re z \<in> {x..y}} (\<lambda>n z. complex_interval_lebesgue_integral lborel (ereal ((1 / 2) ^ Suc n)) (ereal (exp (- 1))) (f z)) (\<lambda>z. complex_interval_lebesgue_integral lborel 0 (ereal (exp (- 1))) (f z)) sequentially"
S5: fixes z :: "nat" shows "(univ_for_pr (c_fst z) = c_snd z) = (\<exists>y. p z y = 0)"
simps: fixes x :: "'c" shows "enum_pos x = enum_pos' enum_class.enum x"
G: shows "Key (sesK (c * f)) \<notin> analz (A \<union> spies evsFR5)"
Vinv1_finite_map_graph: fixes s :: "'b v_state_scheme" assumes "s \<in> Vinv1" shows "finite (map_graph (\<lambda>(x, y). votes s x y))"
cpx_length_of_vec_of_list: fixes l :: "complex list" shows "\<parallel>vec_of_list l\<parallel> = sqrt (\<Sum>i<length l. (cmod (l ! i))\<^sup>2)"
IfFalse: fixes b :: "state \<Rightarrow> bool"   and s :: "state"   and c1 :: "com"   and c2 :: "com"   and cs :: "com list" assumes "\<not> b s" shows "((IF b THEN c1 ELSE c2) # cs, s) \<rightarrow> (c2 # cs, s)"
a_rcos_module_imp: fixes x :: "'a"   and x' :: "'a" assumes "x \<in> carrier R"   and "x' \<in> I +> x" shows "x' \<oplus> \<ominus> x \<in> I"
csij: shows "cs\<^bsup>\<pi>'\<^esup> i' = cs\<^bsup>\<pi>'\<^esup> j'"
simps_3: fixes a :: "floatarith"   and b :: "floatarith"   and vs :: "real list" shows "interpret_form (Less a b) vs = (interpret_floatarith a vs < interpret_floatarith b vs)"
rsa_bij_betw: fixes e :: "nat"   and P :: "nat"   and Q :: "nat" assumes "coprime e ((P - 1) * (Q - 1))"   and "prime P"   and "prime Q"   and "P \<noteq> Q" shows "bij_betw (F (P * Q, e)) (local.range (P * Q, e)) (local.range (P * Q, e))"
split_asm: fixes P :: "'e \<Rightarrow> bool"   and f1 :: "'e"   and f2 :: "'f \<Rightarrow> 'e"   and option :: "'f 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))))"
integral_restrict_Int: fixes T :: "'b set"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'a" shows "integral T (\<lambda>x. if x \<in> S then f x else (0::'a)) = integral (S \<inter> T) f"
ntrunc_one_In0: fixes M :: "('a, 'b) node set" shows "ntrunc (Suc 0) (In0 M) = {}"
starlike_def: fixes S :: "'a set" shows "starlike S = (\<exists>a\<in>S. \<forall>x\<in>S. closed_segment a x \<subseteq> S)"
algebraic_add_cancel_right: fixes y :: "'b"   and x :: "'b" assumes "algebraic y" shows "algebraic (x + y) = algebraic x"
sup_ge1: fixes x :: "'c"   and y :: "'c" shows "x \<le> x + y"
gseq_iff: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b"   and A :: "'b set" shows "((f, g) \<in> mbs.gseq A) = (f \<in> SEQ A \<and> g \<in> SEQ A \<and> (\<exists>i. size (g i) < size (f i) \<and> (\<forall>j<i. f j = g j)))"
get_ancestors_ptr: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list" assumes "h \<turnstile> get_ancestors_si ptr \<rightarrow>\<^sub>r ancestors" shows "ptr \<in> set ancestors"
is_Prog\<^sub>s\<^sub>b_def: fixes r :: "'b memref" shows "is_Prog\<^sub>s\<^sub>b r = (case r of Prog\<^sub>s\<^sub>b x xa xb \<Rightarrow> True | _ \<Rightarrow> False)"
ipresSubstAbs_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 "ipresSubstAbs h hA MOD \<equiv> \<forall>ys Y y us s A. wls (asSort ys) Y \<and> wlsAbs (us, s) A \<longrightarrow> hA (A $[Y / y]_ys) = igSubstAbs MOD ys (h Y) y (hA A)"
linear_fm_helper: fixes C :: "real"   and var :: "nat"   and b :: "real mpoly"   and c :: "real mpoly"   and z :: "nat"   and L :: "real list"   and B :: "real"   and F :: "atom fm"   and v :: "real" assumes "C \<noteq> 0"   and "var \<notin> vars b"   and "var \<notin> vars c"   and "insertion (nth_default 0 ((drop z L)[var := B / C])) b = B"   and "insertion (nth_default 0 ((drop z L)[var := B / C])) c = C"   and "var + z < length L" shows "eval F (L[var + z := B / C]) = eval (linear_substitution_fm_helper var b c F z) (L[var + z := v])"
last_plus: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'b" assumes "m \<le> n" shows "prod g {m..n} = g n * prod g {m..<n}"
assms_1: shows "split ts x = (ls, (sub, sep) # rs)"
orbit_size: fixes m :: "'a set" assumes "finite (carrier G)"   and "m \<in> subgroups_of_size (p ^ a)" shows "order G = card (conj.orbit m) * card (conj.stabilizer m)"
usubst_upd_idem: fixes x :: "'b \<Longrightarrow> 'c"   and \<sigma> :: "'d \<Rightarrow> 'c"   and u :: "('b, 'd) uexpr"   and v :: "('b, 'd) uexpr" assumes "mwb_lens x" shows "\<sigma>(x \<mapsto>\<^sub>s u, x \<mapsto>\<^sub>s v) = \<sigma>(x \<mapsto>\<^sub>s v)"
take_indices_def: fixes subsets :: "'b list"   and indices :: "nat list" shows "take_indices subsets indices = map ((!) subsets) indices"
strictly_subsume_def: fixes Ds :: "'a literal list list"   and C :: "'a literal list" shows "strictly_subsume Ds C = (\<exists>D\<in>set Ds. strictly_subsumes (mset D) (mset C))"
subcls_is_class: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog" assumes "G\<turnstile>C\<prec>\<^sub>C D" shows "is_class G C"
induct: fixes P :: "int \<times> int \<Rightarrow> (int \<times> int) list \<Rightarrow> bool"   and a0 :: "int \<times> int"   and a1 :: "(int \<times> int) list" assumes "\<And>p. P p []"   and "\<And>p p\<^sub>0. P p [p\<^sub>0]"   and "\<And>p p\<^sub>0 v va. P p (v # va) \<Longrightarrow> P p (p\<^sub>0 # v # va)" shows "P a0 a1"
simps: fixes pre :: "'a word"   and len :: "nat" shows "ipcidr_to_interval_start (pre, len) = (let netmask = mask len << LENGTH('a) - len; network_prefix = pre && netmask in network_prefix)"
snth_smap: fixes f :: "'e \<Rightarrow> 'd"   and s :: "'e stream"   and n :: "nat" shows "smap f s !! n = f (s !! n)"
perp_col: fixes A :: "'p"   and E :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A \<noteq> E"   and "A B Perp C D"   and "Col A B E" shows "A E Perp C D"
o_2: shows "\<forall>a b. wp\<^sub>3\<^sub>' (strip C) (\<lambda>(ps, n). (Q \<and>* (\<lambda>s. True)) (ps, n div k)) (a, b) \<longrightarrow> pre C (\<lambda>(ps, n). (Q \<and>* (\<lambda>s. True)) (ps, n div k)) (a, b)"
tseytin_assmt_new_atoms: fixes k :: "'a"   and l :: "'a formula"   and F :: "'a formula" assumes "(k, l) \<in> set (tseytin_assmt F)" shows "k \<notin> atoms F"
positive_imp_inverse_positive: fixes a :: "'a" assumes "(0::'a) < a" shows "(0::'a) < inverse a"
sum_moebius_mu_divisors': fixes n :: "nat" shows "(\<Sum>d | d dvd n. moebius_mu d) = (if n = 1 then 1::'b else (0::'b))"
scaleR_one: fixes x :: "'a" shows "1 *\<^sub>R x = x"
set_plus_mono3: fixes a :: "'b"   and C :: "'b set"   and D :: "'b set" assumes "a \<in> C" shows "a +o D \<subseteq> C + D"
OrdP_HPairE: fixes x :: "tm"   and y :: "tm"   and H :: "fm set"   and A :: "fm" shows "insert (OrdP (HPair x y)) H \<turnstile> A"
simps_2: shows "unbox Ubx2 = map_option OpUbx2 \<circ> unbox_ubx2"
wls_psubst_Op_simp: fixes delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" assumes "wlsInp delta inp"   and "wlsBinp delta binp"   and "wlsEnv rho" shows "Op delta inp binp #[rho] = Op delta inp %[rho] binp %%[rho]"
less_div_Suc_mult: fixes m :: "nat"   and n :: "nat" assumes "0 < m" shows "n < Suc (n div m) * m"
exhaust: fixes y :: "atom" obtains x1 :: "real mpoly" where "y = Less x1"   | x2 :: "real mpoly" where "y = Eq x2"   | x3 :: "real mpoly" where "y = Leq x3"   | x4 :: "real mpoly" where "y = Neq x4"
prod_algebra_sets_into_space: fixes I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure" shows "prod_algebra I M \<subseteq> Pow (\<Pi>\<^sub>E i\<in>I. space (M i))"
when_cong: fixes P :: "bool"   and Q :: "bool"   and a :: "'b"   and b :: "'b"   and zero :: "'b" assumes "P = Q"   and "Q \<Longrightarrow> a = b" shows "zero.when zero a P = zero.when zero b Q"
tl_lemma: fixes u\<^sub>1 :: "32 Word.word"   and r1 :: "real"   and u\<^sub>2 :: "32 Word.word"   and r2 :: "real"   and l\<^sub>1 :: "32 Word.word"   and l\<^sub>2 :: "32 Word.word" assumes "u\<^sub>1 \<equiv>\<^sub>U r1"   and "u\<^sub>2 \<equiv>\<^sub>U r2"   and "l\<^sub>1 \<equiv>\<^sub>L r1"   and "l\<^sub>2 \<equiv>\<^sub>L r2" shows "Interval_Word32.tl l\<^sub>1 u\<^sub>1 l\<^sub>2 u\<^sub>2 \<equiv>\<^sub>L r1 * r2"
SCallParamsThrow: fixes P :: "(char list list \<times> char list exp) prog"   and es :: "char list exp list"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and vs :: "val list"   and ex :: "char list exp"   and es' :: "char list exp list"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and C :: "char list"   and M :: "char list" assumes "P \<turnstile> \<langle>es,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>map Val vs @ throw ex # es',s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(es),s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw ex,s\<^sub>2\<rangle>"
set_sortQ: fixes xs :: "('a, 'b) Combinators list"   and l :: "'a set list" shows "set (qsort xs l) = set xs"
var_not_app: fixes x :: "'b"   and A :: "'b trm"   and B :: "'b trm" shows "Var x \<noteq> App A B"
FCIA_def: fixes \<rho> :: "'b V_rec \<Rightarrow> 'b set"   and \<Gamma> :: "'b Gamma"   and \<V> :: "'b V_rec"   and Tr :: "'b list set" shows "FCIA \<rho> \<Gamma> \<V> Tr \<equiv> \<forall>\<alpha> \<beta>. \<forall>c\<in>C\<^bsub>\<V>\<^esub> \<inter> \<Upsilon>\<^bsub>\<Gamma>\<^esub>. \<forall>v\<in>V\<^bsub>\<V>\<^esub> \<inter> \<nabla>\<^bsub>\<Gamma>\<^esub>. \<beta> @ [v] @ \<alpha> \<in> Tr \<and> \<alpha> \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [] \<and> Adm \<V> \<rho> Tr \<beta> c \<longrightarrow> (\<exists>\<alpha>' \<delta>'. set \<delta>' \<subseteq> N\<^bsub>\<V>\<^esub> \<inter> \<Delta>\<^bsub>\<Gamma>\<^esub> \<and> \<beta> @ [c] @ \<delta>' @ [v] @ \<alpha>' \<in> Tr \<and> \<alpha>' \<upharpoonleft> V\<^bsub>\<V>\<^esub> = \<alpha> \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> \<alpha>' \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [])"
firstPosLast: fixes l :: "'b list"   and x :: "'b" assumes "l \<noteq> []"   and "uniq l" shows "(firstPos x l = length l - 1) = (x = last l)"
comp_arr_dom: fixes f :: "'a arrow_of_spans_data"   and a :: "'a arrow_of_spans_data" assumes "V.arr f"   and "V.dom f = a" shows "f \<bullet> a = f"
induct: fixes x1 :: "char list \<Rightarrow> Types.ty"   and x2 :: "aexp"   and x3 :: "Types.ty"   and P :: "(char list \<Rightarrow> Types.ty) \<Rightarrow> aexp \<Rightarrow> Types.ty \<Rightarrow> bool" assumes "x1 \<turnstile> x2 : x3"   and "\<And>\<Gamma> i. P \<Gamma> (Ic i) Types.ty.Ity"   and "\<And>\<Gamma> r. P \<Gamma> (Rc r) Types.ty.Rty"   and "\<And>\<Gamma> x. P \<Gamma> (V x) (\<Gamma> x)"   and "\<And>\<Gamma> a1 \<tau> a2. \<lbrakk>\<Gamma> \<turnstile> a1 : \<tau>; P \<Gamma> a1 \<tau>; \<Gamma> \<turnstile> a2 : \<tau>; P \<Gamma> a2 \<tau>\<rbrakk> \<Longrightarrow> P \<Gamma> (Plus a1 a2) \<tau>" shows "P x1 x2 x3"
irrefl: fixes a :: "'e list" shows "\<not> strict_prefix a a"
drop_replicate: fixes i :: "nat"   and k :: "nat"   and x :: "'c" shows "drop i (replicate k x) = replicate (k - i) x"
runit_in_hom: fixes a :: "'a" assumes "ide a" shows "\<guillemotleft>\<r>[a] : a \<otimes> \<I> \<rightarrow> a\<guillemotright>"
singular_relcycle_minus: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "singular_relcycle p X S (- c) = singular_relcycle p X S c"
pdevs_val_trunc_err_pdevs: fixes e :: "nat \<Rightarrow> real"   and p :: "nat"   and X :: "'a pdevs" shows "pdevs_val e (trunc_err_pdevs p X) = pdevs_val e (trunc_pdevs p X) - pdevs_val e X"
x: shows "x \<in> {h. [h ^ CARD('a) = h] (mod u) \<and> degree h < degree u}"
spec_ops_rem_last: fixes xs :: "('b \<times> 'c) list"   and x :: "'b \<times> 'c"   and deps :: "'c \<Rightarrow> 'b set" assumes "spec_ops (xs @ [x]) deps" shows "spec_ops xs deps"
card_of_empty_ordIso: shows "|{}| =o |{}|"
Range_Collect_case_prod: fixes P :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "Range {(x, y). P x y} = {y. \<exists>x. P x y}"
simps: fixes xs :: "('dyn, 'd, 'e, 'f, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr list"   and ar :: "nat" shows "Subx.generalize_fundef (Fundef xs ar) = Fundef (map Subx.generalize_instr xs) ar"
map: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "map_dual f (dual x) = dual (f x)"
HStartBallot_HInv3: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HStartBallot s s' p"   and "HInv3 s" shows "HInv3 s'"
gens_span_free_group: fixes gens :: "'d set" shows "\<langle>\<iota> ` gens\<rangle>\<^bsub>\<F>\<^bsub>gens\<^esub>\<^esub> = carrier \<F>\<^bsub>gens\<^esub>"
s: shows "conv_abscissa f < ereal s \<and> s < \<sigma>"
induct: fixes P1 :: "('d, 'c) Expr \<Rightarrow> bool"   and P2 :: "('d, 'c) Expr list \<Rightarrow> bool"   and Expr_list :: "('d, 'c) Expr list" assumes "\<And>x. P1 (Const x)"   and "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (Op x1 x2)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P2 Expr_list"
base: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and xs :: "'b stream" assumes "\<phi> xs" shows "ev \<phi> xs"
simps: fixes M :: "('d, 'e, 'f) FSM"   and q :: "'f"   and xs :: "'d list" shows "language_state_for_input M q xs = {xs || ys |ys. length xs = length ys \<and> xs || ys \<in> LS M q}"
mutual_information_nonneg: fixes S :: "'d measure"   and T :: "'e measure"   and X :: "'a \<Rightarrow> 'd"   and Px :: "'d \<Rightarrow> real"   and Y :: "'a \<Rightarrow> 'e"   and Py :: "'e \<Rightarrow> real"   and Pxy :: "'d \<times> 'e \<Rightarrow> real" assumes "sigma_finite_measure S"   and "sigma_finite_measure T"   and "distributed M S X (\<lambda>x. ennreal (Px x))"   and "\<And>x. x \<in> space S \<Longrightarrow> 0 \<le> Px x"   and "distributed M T Y (\<lambda>x. ennreal (Py x))"   and "\<And>y. y \<in> space T \<Longrightarrow> 0 \<le> Py y"   and "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>x. ennreal (Pxy x))"   and "\<And>x y. \<lbrakk>x \<in> space S; y \<in> space T\<rbrakk> \<Longrightarrow> 0 \<le> Pxy (x, y)"   and "integrable (S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))))" shows "0 \<le> mutual_information b S T X Y"
key_compare_inverse: fixes x :: "'a key_order" shows "Abs_key_order (key_compare x) = x"
assms_2: shows "0 \<in>\<^sub>\<circ> n"
mono_Un: fixes f :: "'b set \<Rightarrow> 'c set"   and A :: "'b set"   and B :: "'b set" assumes "mono f" shows "f A \<union> f B \<subseteq> f (A \<union> B)"
leof_add_nofailI: fixes m :: "'c nres"   and m' :: "'c nres" assumes "nofail m \<Longrightarrow> m \<le>\<^sub>n m'" shows "m \<le>\<^sub>n m'"
uninst_allVars_simps_4: fixes \<alpha>n :: "'d \<Rightarrow> 'e list"   and inEdges' :: "'d \<Rightarrow> 'e \<Rightarrow> ('e \<times> 'f) list"   and defs :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and u :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and phis :: "'d \<Rightarrow> 'e \<times> 'c \<Rightarrow> 'c list option"   and g :: "'d"   and p :: "'d \<Rightarrow> 'e \<times> 'c \<Rightarrow> 'c list option" shows "CFG_SSA_base.allVars \<alpha>n inEdges' defs u (phis(g := p g)) g = CFG_SSA_base.allVars \<alpha>n inEdges' defs u p g"
read_only_share_unowned_in: fixes \<O> :: "nat set"   and sb :: "'b memref list"   and a :: "nat"   and \<S> :: "nat \<Rightarrow> bool option" assumes "weak_sharing_consistent \<O> sb"   and "a \<in> read_only (share sb \<S>)" shows "a \<in> read_only \<S> \<union> \<O> \<union> all_acquired sb"