hcju/isabelleps huggingface dataset:
all_CAND: fixes \<Gamma> :: "(name \<times> ty) list"   and M :: "trm"   and \<Delta> :: "(coname \<times> ty) list"   and \<theta>_n :: "(name \<times> coname \<times> trm) list"   and \<theta>_c :: "(coname \<times> name \<times> trm) list" assumes "\<Gamma> \<turnstile> M \<turnstile> \<Delta>"   and "\<theta>_n ncloses \<Gamma>"   and "\<theta>_c ccloses \<Delta>" shows "SNa (\<theta>_n,\<theta>_c<M>)"
final: shows "final Accessible_dfa \<subseteq> states Accessible_dfa"
preserves_arr: fixes f :: "'c" assumes "C.arr f" shows "arr (E f)"
ex_pos_a: fixes x\<^sub>0 :: "nat"   and x\<^sub>1 :: "nat"   and k :: "nat"   and as :: "real list"   and bs :: "real list"   and ts :: "(nat \<Rightarrow> nat) list"   and f :: "nat \<Rightarrow> real"   and g :: "nat \<Rightarrow> real" assumes "master_theorem_function x\<^sub>0 x\<^sub>1 k as bs ts f g" shows "\<exists>a\<in>set as. 0 < a"
bit_signed_take_bit_iff: fixes m :: "nat"   and a :: "'a"   and n :: "nat" shows "bit (signed_take_bit m a) n = ((2::'a) ^ n \<noteq> (0::'a) \<and> bit a (min m n))"
heterogeneous_supterm: fixes t :: "('fun, 'var) Term.term"   and \<A> :: "('a strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and t' :: "('fun, 'var) Term.term" assumes "heterogeneous\<^sub>l\<^sub>s\<^sub>t t \<A> Sec"   and "t \<sqsubseteq> t'" shows "heterogeneous\<^sub>l\<^sub>s\<^sub>t t' \<A> Sec"
square_diff_square_factored: fixes x :: "'a"   and y :: "'a" shows "x * x - y * y = (x + y) * (x - y)"
rep_eq: fixes x :: "'a fract"   and xa :: "'a fract" shows "(x \<le> xa) = (fst (rep_fract x) * snd (rep_fract xa) * (snd (rep_fract x) * snd (rep_fract xa)) \<le> fst (rep_fract xa) * snd (rep_fract x) * (snd (rep_fract x) * snd (rep_fract xa)))"
simps: fixes uu :: "'c \<Rightarrow> 'd list"   and XLF :: "'f \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm set list"   and uv :: "'c \<times> 'f \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qAbs set list"   and uw :: "('c \<Rightarrow> 'd \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm set option) list"   and s :: "'f" shows "termsOfS (ParS uu XLF uv uw) s = set (XLF s)"
seq_inj_def: shows "seq_inj \<equiv> map_fun (map_fun id Sequence.seq.Rep_seq \<circ> Sequence.seq.Rep_seq) Sequence.seq.Abs_seq (\<lambda>f i. f (fst (prod_decode i)) (snd (prod_decode i)))"
represents_add: fixes p :: "int poly"   and x :: "'a"   and q :: "int poly"   and y :: "'a" assumes "p represents x"   and "q represents y" shows "poly_add p q represents x + y"
iso_tuple_update_accessor_cong_assist_id: fixes upd :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and ac :: "'b \<Rightarrow> 'a" assumes "iso_tuple_update_accessor_cong_assist upd ac" shows "upd id = id"
1: shows "Cring_Poly.compose R p q = finsum P (rev_compose R q \<circ> g) {..n}"
extend_evals: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and es :: "expr list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and es'' :: "expr list"   and s'' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and E :: "char list \<Rightarrow> ty option"   and es' :: "expr list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "wwf_prog P"   and "P,E \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>]* \<langle>es'',s''\<rangle>"   and "P,E \<turnstile> \<langle>es'',s''\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>" shows "P,E \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>"
winning_path_def: fixes p :: "Player"   and P :: "'a llist" shows "winning_path p P \<equiv> \<not> lfinite P \<and> (\<exists>a\<in>path_inf_priorities P. (\<forall>b\<in>path_inf_priorities P. a \<le> b) \<and> (if p = Even then even else odd) a) \<or> \<not> lnull P \<and> lfinite P \<and> llast P \<in> VV p** \<or> lnull P \<and> p = Even"
cil_last_case_split: fixes w :: "'d list"   and e1 :: "'d"   and w1 :: "'d list"   and \<alpha> :: "'d \<Rightarrow> 'e set \<times> 'e set"   and e2 :: "'d"   and w2 :: "'d list" assumes "w \<in> e1 # w1 \<otimes>\<^bsub>\<alpha>\<^esub> e2 # w2" obtains w' :: "'d list" where "w = e1 # w'a__" and "w'a__ \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> e2 # w2" and "fst (\<alpha> e1) \<inter> mon_pl (map \<alpha> (e2 # w2)) = {}"   | w' :: "'d list" where "w = e2 # w'" and "w' \<in> e1 # w1 \<otimes>\<^bsub>\<alpha>\<^esub> w2" and "fst (\<alpha> e2) \<inter> mon_pl (map \<alpha> (e1 # w1)) = {}"
prod_pos: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<forall>a\<in>A. (0::'b) < f a" shows "(0::'b) < prod f A"
real_alg_2_list_convert_id_def: shows "real_alg_2_list_convert_id \<equiv> (Rep_real_alg_2_list ---> map Real_Alg_Invariant) id"
sees_method_decl_above: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list" assumes "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D" shows "P \<turnstile> C \<preceq>\<^sup>* D"
infinite_super: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T"   and "infinite S" shows "infinite T"
ct_prefix_code_2: fixes x :: "ct"   and xs :: "ct list" shows "ct_prefix (x # xs) [] = False"
trms\<^sub>s\<^sub>s\<^sub>t_cons: fixes a :: "('a, 'b) stateful_strand_step"   and A :: "('a, 'b) stateful_strand_step list" shows "trms\<^sub>s\<^sub>s\<^sub>t (a # A) = trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p a \<union> trms\<^sub>s\<^sub>s\<^sub>t A"
\<R>\<^sub>1_elim': fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and c\<^sub>1 :: "('b, 'a, 'c) Stmt"   and mds :: "Mode \<Rightarrow> 'b set"   and mem\<^sub>1 :: "'b \<Rightarrow> 'd"   and c\<^sub>2 :: "('b, 'a, 'c) Stmt"   and mds\<^sub>2 :: "Mode \<Rightarrow> 'b set"   and mem\<^sub>2 :: "'b \<Rightarrow> 'd"   and dma :: "'b \<Rightarrow> Sec"   and \<Gamma> :: "'b \<Rightarrow> Sec option" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"   and "(((c\<^sub>1, mds), mem\<^sub>1), (c\<^sub>2, mds\<^sub>2), mem\<^sub>2) \<in> sifum_types.\<R>\<^sub>1 aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>" obtains \<Gamma>' :: "'b \<Rightarrow> Sec option" where "c\<^sub>2 = c\<^sub>1" and "mds\<^sub>2 = mds" and "sifum_types.has_type aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>' c\<^sub>1 \<Gamma>" and "sifum_types.mds_consistent dma mds \<Gamma>'" and "sifum_types.tyenv_eq dma \<Gamma>' mem\<^sub>1 mem\<^sub>2"
that: fixes j :: "nat"   and r :: "'a list" assumes "j < \<^bold>|ws\<^bold>|"   and "r \<le>np ws ! j"   and "concat (take j ws) \<cdot> r = u" shows "thesis"
map_pmf_comp: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and M :: "'e pmf" shows "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M"
Mut_init_def: shows "Mut_init \<equiv> \<lambda>s. T \<in> Reach (E s) \<and> R < length (E s) \<and> T < length (M s)"
projection_commute: fixes l :: "'a list"   and X :: "'a set"   and Y :: "'a set" shows "l \<upharpoonleft> X \<upharpoonleft> Y = l \<upharpoonleft> Y \<upharpoonleft> X"
midpoints_in_convex_hull: fixes x :: "'b"   and s :: "'b set"   and y :: "'b" assumes "x \<in> convex hull s"   and "y \<in> convex hull s" shows "midpoint x y \<in> convex hull s"
height_balL2: fixes l :: "('b \<times> nat) tree"   and r :: "('b \<times> nat) tree"   and a :: "'b" assumes "hbt l"   and "hbt r"   and "height l \<noteq> height r + m + 1" shows "height (balL l a r) = 1 + max (height l) (height r)"
unapp_trm_def: fixes t :: "'b trm" shows "unapp t = (case t of Var x \<Rightarrow> Map.empty x | trm.Const x \<Rightarrow> Map.empty x | t \<cdot> u \<Rightarrow> Some (t, u))"
in_implicit_pres: fixes x :: "nat"   and a :: "nat"   and effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list" assumes "(x, a) \<in> set (implicit_pres effs)" shows "\<exists>epres v vp. (epres, x, vp, v) \<in> set effs \<and> vp = Some a"
lnull_lappend: fixes xs :: "'b llist"   and ys :: "'b llist" shows "lnull (xs $ ys) = (lnull xs \<and> lnull ys)"
Lambert_W_gtI: fixes y :: "real"   and x :: "real" assumes "y * exp y < x" shows "y < Lambert_W x"
prod_rel_comp_ivl_rel: fixes A :: "('c \<times> 'd) set"   and B :: "('d \<times> 'e) set" assumes "single_valued A"   and "single_valued B" shows "(A \<times>\<^sub>r A) O \<langle>B\<rangle>ivl_rel = \<langle>A O B\<rangle>ivl_rel"
r_one: fixes x :: "int vec" assumes "x \<in> carrier_vec n" shows "x + 0\<^sub>v n = x"
less_le: fixes x :: "'b list"   and y :: "'b list" shows "strict_prefix x y = (prefix x y \<and> x \<noteq> y)"
isomorphic_sym: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and W :: "'d set" assumes "FGModule G smult V"   and "\<exists>T. FGModuleIso G smult V smult' T W" shows "\<exists>T. FGModuleIso G smult' W smult T V"
that: fixes f :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<And>f'. \<guillemotleft>f' : a \<rightarrow> b\<guillemotright> \<Longrightarrow> f' = f" shows "thesis"
F_set_conv_fold: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'b list" assumes "semilattice_set f"   and "xs \<noteq> []" shows "semilattice_set.F f (set xs) = Finite_Set.fold f (hd xs) (set (tl xs))"
finite_subsets_at_top_minus: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B" shows "finite_subsets_at_top (B - A) \<le> filtermap (\<lambda>F. F - A) (finite_subsets_at_top B)"
ex_ex1I: fixes P :: "'h \<Rightarrow> bool" assumes "\<exists>x. P x"   and "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> x = y" shows "\<exists>!x. P x"
arrI: fixes f :: "'a" assumes "\<guillemotleft>f : trg e \<rightarrow> trg (trg f)\<guillemotright>" shows "hom.arr f"
RedCastFail: fixes h :: "heap"   and a :: "nat"   and D :: "char list"   and fs :: "Objects.fields"   and C :: "char list"   and P :: "(char list list \<times> char list exp) prog"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool" assumes "h a = \<lfloor>(D, fs)\<rfloor>"   and "(D, C) \<notin> (subcls1 P)\<^sup>*" shows "P \<turnstile> \<langle>Cast C (addr a),(h, l, sh),b\<rangle> \<rightarrow> \<langle>THROW ClassCast,(h, l, sh),b\<rangle>"
encode_problem_parallel_correct_ii_1: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat"   and k :: "nat" assumes "is_valid_problem_strips \<Pi>"   and "\<A> \<Turnstile> \<Phi> \<Pi> t"   and "k < length (\<Phi>\<inverse> \<Pi> \<A> t)" shows "are_all_operators_applicable (\<Phi>\<^sub>S\<inverse> \<Pi> \<A> k) ((\<Phi>\<inverse> \<Pi> \<A> t) ! k)"
Inf_inner_def: fixes X :: "'b set"   and y :: "'b" shows "Inf_inner X y = SPEC (\<lambda>r. \<forall>x\<in>X. r \<le> x \<bullet> y)"
distinct_21: shows "cospan_shape.BB \<noteq> cospan_shape.AT"
division_doublesplit: fixes p :: "'a set set"   and a :: "'a"   and b :: "'a"   and k :: "'a"   and c :: "real"   and e :: "real" assumes "p division_of cbox a b"   and "k \<in> Basis" shows "(\<lambda>l. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> e}) ` {l \<in> p. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> e} \<noteq> {}} division_of cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> e}"
rel: shows "release (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) (dom \<S>) \<R> a \<noteq> Some False"
Amicable_pair_deficient: fixes n :: "nat"   and m :: "nat" assumes "n < m"   and "m Amic n" shows "deficient_number m"
hcomplex_hcnj_minus: fixes z :: "complex star" shows "hcnj (- z) = - hcnj z"
fw_mono: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat" assumes "i \<le> n"   and "j \<le> n" shows "fw m n k i j \<le> m i j"
eval\<^sub>1_preserves_unmod: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and ls :: "val list"   and e' :: "nat exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and ls' :: "val list"   and i :: "nat" assumes "P \<turnstile>\<^sub>1 \<langle>e,(h, ls)\<rangle> \<Rightarrow> \<langle>e',(h', ls')\<rangle>"   and "unmod e i"   and "i < length ls" shows "ls ! i = ls' ! i"
Inf_union_distrib: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf (A \<union> B) = inf (Inf A) (Inf B)"
Nmlize_Dom: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Arr V src trg t" shows "bicategorical_language.Nmlize V (bicategorical_language.Dom V src trg t) = bicategorical_language.Dom V src trg (bicategorical_language.Nmlize V t)"
remove_all_rev_def: fixes x :: "'b"   and xs :: "'b list" shows "remove_all_rev x xs = (if x \<in> set xs then rev (tl (remove_cycles (rev xs) x [])) else xs)"
accept_Language: fixes s :: "'s2" assumes "wellformed2 s" shows "accept2 s = \<oo> (Language2 s)"
locals_init_obj: fixes G :: "prog"   and oi :: "obj_tag"   and r :: "oref"   and s :: "st" shows "locals (init_obj G oi r s) = locals s"
wcc_increasing: fixes x :: "'a" shows "x \<le> wcc x"
bv_to_int_Cons1: fixes bs :: "bit list" shows "bv_to_int (\<one> # bs) = - int (bv_to_nat (bv_not bs) + 1)"
52423: fixes x :: "'a"   and y :: "'a" shows "- (- x \<sqinter> - (- x \<sqinter> y)) \<sqinter> y = y"
assms: fixes x :: "'a"   and i :: "nat" assumes "x \<in> set xs" shows "c x i = d x i"
hyps_F_b: fixes ve :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option"   and b :: "contour"   and d :: "d"   and ds :: "d list"   and x :: "(nat \<times> (nat \<Rightarrow> contour option)) \<times> d" assumes "\<forall>b'\<in>contours_in_ve ve. b' < b"   and "\<forall>b'\<in>contours_in_d d. b' < b"   and "\<forall>d'\<in>set ds. \<forall>b'\<in>contours_in_d d'. b' < b"   and "x \<in> evalF\<cdot>(Discr (d, ds, ve, b))" shows "case x of (a, b) \<Rightarrow> (case a of (lab, \<beta>) \<Rightarrow> \<lambda>t. \<exists>b'. b' \<in> ran \<beta> \<and> b \<le> b') b"
set_empty: fixes xs :: "'k list" shows "(set xs = {}) = (xs = [])"
simps: fixes n :: "nat"   and j :: "nat"   and "is" :: "nat list"   and ps :: "nat list" shows "prime_factorization_nat_main n j is ps = (case is of [] \<Rightarrow> case next_candidates j of (j, is) \<Rightarrow> prime_factorization_nat_main n j is ps | i # is \<Rightarrow> case divmod_nat n i of (n', m) \<Rightarrow> if m = 0 then case remove_prime_factor i n' (i # ps) of (n', ps') \<Rightarrow> if n' = 1 then ps' else prime_factorization_nat_main n' j is ps' else if i * i \<le> n then prime_factorization_nat_main n j is ps else n # ps)"
conjugate_char_1_isotone: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "conjugate f g"   and "isotone f"   and "isotone g" shows "f (x \<sqinter> - g y) \<le> f x \<sqinter> - y \<and> g (y \<sqinter> - f x) \<le> g y \<sqinter> - x"
rawpsubstT_compose: fixes t :: "'trm"   and txs1 :: "('trm \<times> 'var) list"   and txs2 :: "('trm \<times> 'var) list" assumes "t \<in> trm"   and "snd ` set txs1 \<subseteq> var"   and "fst ` set txs1 \<subseteq> atrm"   and "snd ` set txs2 \<subseteq> var"   and "fst ` set txs2 \<subseteq> atrm" shows "rawpsubstT (rawpsubstT t txs1) txs2 = rawpsubstT t (txs1 @ txs2)"
some_eq_ex: fixes P :: "'b \<Rightarrow> bool" shows "P (SOME x. P x) = (\<exists>x. P x)"
insert_iff: fixes a :: "'g"   and b :: "'g"   and A :: "'g set" shows "(a \<in> insert b A) = (a = b \<or> a \<in> A)"
that: fixes z :: "'a" assumes "x = y \<otimes> z"   and "z \<in> kernel G H h" shows "thesis"
ord_resolve_reductive: fixes CAs :: "'a literal multiset list"   and DA :: "'a literal multiset"   and AAs :: "'a multiset list"   and As :: "'a list"   and E :: "'a literal multiset" assumes "ord_resolve CAs DA AAs As E" shows "E < DA"
l8_14_3: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and Y :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.PerpAt Bet Cong X A B C D"   and "Tarski_neutral_dimensionless.PerpAt Bet Cong Y A B C D" shows "X = Y"
pl_2: fixes \<phi> :: "\<o>"   and \<psi> :: "\<o>"   and \<chi> :: "\<o>" shows "[[\<phi> \<^bold>\<rightarrow> (\<psi> \<^bold>\<rightarrow> \<chi>) \<^bold>\<rightarrow> (\<phi> \<^bold>\<rightarrow> \<psi> \<^bold>\<rightarrow> (\<phi> \<^bold>\<rightarrow> \<chi>))]]"
R2_14_def: fixes M :: "bool \<times> bool \<times> bool \<times> bool"   and C :: "bool \<times> bool" shows "R2_14 M C = (let (m0, m1, m2, m3) = M; (c0, c1) = C in coin_spmf \<bind> (\<lambda>S0. coin_spmf \<bind> (\<lambda>S1. coin_spmf \<bind> (\<lambda>S2. coin_spmf \<bind> (\<lambda>S3. coin_spmf \<bind> (\<lambda>S4. coin_spmf \<bind> (\<lambda>S5. let a0 = S0 \<oplus> S2 \<oplus> m0; a1 = S0 \<oplus> S3 \<oplus> m1; a2 = S1 \<oplus> S4 \<oplus> m2; a3 = S1 \<oplus> S5 \<oplus> m3 in R2_OT12 (S0, S1) c0 \<bind> (\<lambda>a. R2_OT12 (S2, S3) c1 \<bind> (\<lambda>b. R2_OT12 (S4, S5) c1 \<bind> (\<lambda>c. return_spmf (C, (a0, a1, a2, a3), a, b, c)))))))))))"
ntrs_mon_s: fixes s :: "'n list"   and c :: "'n list multiset"   and e :: "('p, 'ba) label list"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "((s, c), e, s', c') \<in> ntrs fg" shows "mon_s fg s' = mon_s fg s \<union> fst (\<alpha>n fg e)"
B'_def: shows "B' = B \<union> (*\<^sub>C) \<i> ` B"
rec_eval: fixes e :: "'c"   and r :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'a"   and y :: "'c" assumes "Rec e r x y" shows "rec e r x = y"
ars_r_def: fixes x :: "'a" shows "d x = ad (ad x)"
get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_ptr_in_heap: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap"   and ptr :: "(_) shadow_root_ptr"   and getter :: "('z, 'x, 'y, 'ag option) RShadowRoot_scheme \<Rightarrow> 'aj" assumes "l_get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_lemmas type_wf"   and "h \<turnstile> ok get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr getter" shows "ptr |\<in>| shadow_root_ptr_kinds h"
s_AuthI: fixes v :: "Syntax.term"   and \<pi> :: "Syntax.term list" assumes "value v" shows "\<lless>\<pi>, Auth v\<ggreater> I\<rightarrow> \<lless>\<pi>, v\<ggreater>"
le_less_trans: fixes x :: "'c list"   and y :: "'c list"   and z :: "'c list" assumes "prefix x y"   and "strict_prefix y z" shows "strict_prefix x z"
dim_li_is_basis: fixes A :: "'a vec set" assumes "fin_dim"   and "finite A"   and "A \<subseteq> carrier_vec n"   and "lin_indpt A"   and "local.dim \<le> card A" shows "basis A"
\<omega>_limit_set_eq: fixes x :: "'a" assumes "{0..} \<subseteq> rev.existence_ivl0 x" shows "rev.\<omega>_limit_set x = (\<Inter>\<tau>\<in>{0..}. closure (rev.flow0 x ` {\<tau>..}))"
transfer: fixes A2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and A1 :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "bi_unique A2"   and "bi_unique A1" shows "((A1 ===> A2) ===> rel_set A1 ===> rel_set A2 ===> (=)) bij_betw bij_betw"
bit_list_cases: fixes w :: "bit list"   and P :: "bit list \<Rightarrow> bool" assumes "w = [] \<Longrightarrow> P w"   and "\<And>bs. w = \<zero> # bs \<Longrightarrow> P w"   and "\<And>bs. w = \<one> # bs \<Longrightarrow> P w" shows "P w"
emeasure_lfp': fixes P :: "'b measure \<Rightarrow> bool"   and M :: "'b measure"   and F :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool" assumes "P M"   and "sup_continuous F"   and "\<And>M A. \<lbrakk>P M; \<And>N. P N \<Longrightarrow> Measurable.pred N A\<rbrakk> \<Longrightarrow> Measurable.pred M (F A)" shows "emeasure M {x \<in> space M. lfp F x} = (SUP i. emeasure M {x \<in> space M. (F ^^ i) (\<lambda>x. False) x})"
scaleR_sum_left: fixes f :: "'c \<Rightarrow> real"   and A :: "'c set"   and x :: "'b" shows "sum f A *\<^sub>R x = (\<Sum>a\<in>A. f a *\<^sub>R x)"
ideal_prod_ideal: fixes R :: "('e, 'f) Ring_scheme"   and I :: "'e set"   and J :: "'e set" assumes "Ring R"   and "ideal R I"   and "ideal R J" shows "ideal R (I \<diamondsuit>\<^sub>r\<^bsub>R\<^esub> J)"
termination_lemma: fixes ss :: "nat list list"   and qs :: "(nat \<times> nat list) list"   and w :: "nat list" assumes "ss \<in> list n A"   and "distinct (map fst qs)"   and "\<forall>(q, t)\<in>set qs. q < n \<and> t \<in> A"   and "Sorted_Less.sorted w"   and "w \<noteq> []" shows "ss [\<sqsubset>\<^bsub>r\<^esub>] merges f qs ss \<or> merges f qs ss = ss \<and> (sorted_list_of_set ({q. \<exists>t. (q, t) \<in> set qs \<and> t \<squnion>\<^bsub>f\<^esub> ss ! q \<noteq> ss ! q} \<union> set (tl w)), w) \<in> (\<lambda>x. case x of (x, y) \<Rightarrow> (sorted_list_of_set x, sorted_list_of_set y)) ` {(A, B). A \<subset> B \<and> finite B}"
WTrtFAcc: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr"   and C :: "char list"   and Cs :: "char list list"   and F :: "char list"   and T :: "ty" assumes "P,E,h \<turnstile> e : Class C"   and "Cs \<noteq> []"   and "P \<turnstile> C has least F:T via Cs" shows "P,E,h \<turnstile> e\<bullet>F{Cs} : T"
counit_simps_2: shows "local.dom \<epsilon> = f \<star> f\<^sup>*"
pid_interp_gc_def: fixes gc :: "(nat, local_state, global_state) Pid_Scheduler.pid_global_config" shows "pid_interp_gc gc \<equiv> interp_gs (pid_global_config.state gc)"
run_effectE: fixes c :: "'b Heap"   and \<sigma> :: "heap option"   and \<sigma>' :: "heap option"   and r :: "'b" assumes "run c \<sigma> \<sigma>' r"   and "\<not> is_exn \<sigma>'" obtains h :: "heap" and h' :: "heap" where "\<sigma> = Some h" and "\<sigma>' = Some h'" and "effect c h h' r"
simps_2: fixes l :: "('b \<times> 'c) tree"   and a :: "'b"   and uu :: "'c"   and r :: "('b \<times> 'c) tree" shows "Tree2.bst \<langle>l, (a, uu), r\<rangle> = ((\<forall>x\<in>Tree2.set_tree l. x < a) \<and> Ball (Tree2.set_tree r) ((<) a) \<and> Tree2.bst l \<and> Tree2.bst r)"
Ball_set_list_all: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> bool" shows "Ball (set xs) P = list_all P xs"
wp1_prec2: fixes c1 :: "com"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and l :: "char list \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int"   and x :: "char list"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat" shows "(wp\<^sub>1 c1 Q l s \<and> l x = prec c1 e s) = wp\<^sub>1 c1 (\<lambda>l s. Q l s \<and> e s = l x) l s"