hcju/isabelleps huggingface dataset:
ipurge_ref_aux_single_dom: fixes I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and u :: "'b"   and xs :: "'a list"   and X :: "'a set" shows "ipurge_ref_aux I D {u} xs X = ipurge_ref I D u xs X"
B: shows "((F \<circ> real_of_ereal) \<longlongrightarrow> B) (at_left b)"
is_partition_of_def: fixes P :: "'c set set"   and A :: "'c set" shows "P partitions A = is_partition_of' P A"
sim_range_def: fixes M :: "('e, 'f, 'g) KripkeStructure"   and M' :: "('e, 'f, 'h) KripkeStructure"   and f :: "'g \<Rightarrow> 'h" shows "sim_range M M' f \<equiv> worlds M' = f ` worlds M \<and> (\<forall>a. relations M' a \<subseteq> worlds M' \<times> worlds M')"
cospoly_def: fixes n :: "nat" shows "cospoly n = (\<lambda>x. \<Sum>k<n. cos_coeff k * x ^ k)"
intro: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and S' :: "'d set"   and is_open' :: "'d set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" assumes "topological_space S is_open"   and "topological_space S' is_open'"   and "Set_Theory.map f S S'"   and "continuous_map_axioms S is_open is_open' f" shows "continuous_map S is_open S' is_open' f"
syncHd_Tl: fixes a :: "'a event list"   and t :: "'a event list"   and A :: "'a event set"   and u :: "'a event list" assumes "a setinterleaves ((t, u), A) \<and> hd t \<in> A \<and> hd u \<notin> A" shows "hd a = hd u \<and> tl a setinterleaves ((t, tl u), A)"
cfs_minus: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "p \<in> carrier P"   and "q \<in> carrier P" shows "(p \<ominus>\<^bsub>P\<^esub> q) n = p n \<ominus> q n"
ex_toCard_pred: fixes A :: "'a set"   and r :: "('b \<times> 'b) set" assumes "|A| \<le>o r"   and "Card_order r" shows "\<exists>f. toCard_pred A r f"
rep_eq: shows "Rep_coreflexive 1 = (1::'b)"
cccFix_eq: fixes cccExp :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and i :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls" shows "CoCallArityAnalysis.cccFix cccExp \<Gamma>\<cdot>i = (ArityAnalysis.Afix (CoCallArityAnalysis.Aexp cccExp) \<Gamma>\<cdot>(fst i \<squnion> (\<lambda>_. up\<cdot>0) f|` thunks \<Gamma>), CoCallArityAnalysis.CCfix cccExp \<Gamma>\<cdot>(ArityAnalysis.Afix (CoCallArityAnalysis.Aexp cccExp) \<Gamma>\<cdot>(fst i \<squnion> (\<lambda>_. up\<cdot>0) f|` thunks \<Gamma>), snd i))"
nested_prop_atoms\<^sub>\<mu>_finite: fixes \<phi> :: "'b ltln"   and X :: "'b ltln set" shows "finite (nested_prop_atoms\<^sub>\<mu> \<phi> X)"
partition_on_no_partition_outside_carrier: fixes A :: "'c set"   and P :: "'c set set"   and x :: "'c" assumes "partition_on A P"   and "x \<notin> A" shows "{y. \<exists>X\<in>P. x \<in> X \<and> y \<in> X} = {}"
Un_open: fixes A :: "'a set"   and B :: "'a set" assumes "A open"   and "B open" shows "A \<union> B open"
yun_monic_factorization: fixes hom :: "'c \<Rightarrow> 'd"   and f :: "'c poly" assumes "field_hom_0' hom"   and "monic f" shows "yun_gcd.yun_monic_factorization gcd (map_poly hom f) = map (\<lambda>(f, i). (map_poly hom f, i)) (yun_gcd.yun_monic_factorization gcd f)"
poly_eval_monomial: fixes a :: "'b \<Rightarrow> 'a"   and t :: "'b \<Rightarrow>\<^sub>0 nat"   and c :: "'a" shows "poly_eval a (monomial c t) = c * (\<Prod>x\<in>keys t. a x ^ lookup t x)"
word_combine_masks: fixes w :: "'b word"   and m :: "'b word"   and z :: "'b word"   and m' :: "'b word"   and z' :: "'b word" assumes "w AND m = z"   and "w AND m' = z'" shows "w AND (m OR m') = z OR z'"
simps_2: fixes x :: "'b"   and xs :: "'b list" shows "butlast (x # xs) = (if xs = [] then [] else x # butlast xs)"
less_Sup_iff: 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"   and S :: "'c set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less a (Sup S) = (\<exists>x\<in>S. less a x)"
openI: fixes m :: "'a llist set" assumes "m \<in> T" shows "m open"
simps: fixes p :: "nat"   and n :: "nat"   and ps :: "nat list" shows "remove_prime_factor p n ps = (case divmod_nat n p of (n', m) \<Rightarrow> if m = 0 then remove_prime_factor p n' (p # ps) else (n, ps))"
lsc_hull_expl: fixes g :: "'b \<Rightarrow> ereal"   and f :: "'b \<Rightarrow> ereal" shows "(g = lsc_hull f) = (Epigraph UNIV g = closure (Epigraph UNIV f))"
dist_Gromov_completion_def: shows "dist = turn_into_distance extended_predist"
assms_2: fixes x :: "nat" assumes "of_nat x = (0::'a)" shows "c dvd x"
retract_fixpoint_property: fixes T :: "'a set"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'a" assumes "T retract_of S"   and "\<And>f. \<lbrakk>continuous_on S f; f ` S \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>x\<in>S. f x = x"   and "continuous_on T g"   and "g ` T \<subseteq> T" obtains y :: "'a" where "y \<in> T" and "g y = y"
def: shows "snd (usubstappp \<sigma> U \<alpha>**) \<noteq> undefg"
globally_consistent: shows "globally_consistent A mds mem\<^sub>1 mem\<^sub>2"
consistent_def: fixes x :: "real \<Rightarrow> 'b"   and t :: "real"   and T :: "real"   and B :: "real"   and p :: "nat"   and incr :: "real \<Rightarrow> real \<Rightarrow> 'b \<Rightarrow> 'b" shows "consistent x t T B p incr = (\<forall>h\<ge>0. t + h \<le> T \<longrightarrow> dist (x (t + h)) (discrete_evolution incr (t + h) t (x t)) \<le> B * h ^ (p + 1))"
is_Arg_exp_diff_2pi: fixes z :: "complex"   and \<theta> :: "real" assumes "is_Arg (exp z) \<theta>" shows "\<exists>k. Im z - real_of_int k * (2 * pi) = \<theta>"
intros_5: 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 Cs :: "char list list"   and C :: "char list"   and Cs' :: "char list list"   and Ds :: "char list list"   and E :: "char list \<Rightarrow> ty option"   and a :: "nat"   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 "P \<turnstile> Path last Cs to C via Cs' "   and "Ds = Cs @\<^sub>p Cs'" shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>ref (a, Cs),s\<rangle> \<rightarrow> \<langle>ref (a, Ds),s\<rangle>"
order_iff: fixes b :: "'x \<Rightarrow>\<^sub>0 nat"   and a :: "'x \<Rightarrow>\<^sub>0 nat" shows "(b \<preceq> a) = (a = ordered_powerprod_lin.max a b)"
mult_left_le: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and c :: "'e"   and a :: "'e" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "less_eq c one"   and "less_eq zero a" shows "less_eq (times a c) a"
simps_2: fixes v :: "'a"   and va :: "'a list"   and p2 :: "'a list" shows "poly_mult (v # va) p2 = poly_add (map ((\<otimes>) (lead_coeff (v # va))) p2 @ replicate (degree (v # va)) \<zero>) (poly_mult (tl (v # va)) p2)"
count: fixes x :: "'d multiset" shows "count x \<in> multiset"
get_ancestors_si_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"
finite_inf_closed: fixes A :: "'a set" assumes "finite A"   and "A \<subseteq> carrier (division_rel G)"   and "A \<noteq> {}" shows "\<Sqinter>\<^bsub>division_rel G\<^esub>A \<in> carrier (division_rel G)"
ID_distr_local_1: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and \<tau> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and e :: "('b, 'a, 'c) expr" shows "LID\<^sub>L (\<sigma>, \<tau>, e) = LID\<^sub>S \<sigma> \<union> LID\<^sub>S \<tau> \<union> LID\<^sub>E e"
inline_sel: fixes callee :: "'b \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'b, 'c, 'd) gpv"   and c :: "('a, 'e, 'f) gpv"   and s :: "'b" shows "the_gpv (inline callee c s) = map_spmf (\<lambda>result. case result of Inl xs \<Rightarrow> Pure xs | Inr (out, oracle, rpv) \<Rightarrow> IO out (\<lambda>input. bind_gpv (oracle input) (\<lambda>(x, s'). inline callee (rpv x) s'))) (inline1 callee c s)"
scale_shift_ms_aux_def: shows "scale_shift_ms_aux = (\<lambda>(a, b). mslmap (map_prod ((*) a) ((+) b)))"
ln_272_gt_1: shows "1 < ln (272 / 100)"
lossless_spmf_def: fixes p :: "'a spmf" shows "lossless_spmf p = (weight_spmf p = 1)"
swaps_eq_Nil_iff: fixes acts :: "nat list"   and xs :: "'b list" shows "(swaps acts xs = []) = (xs = [])"
monomial_decomp_shift_list: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and hU :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set"   and x :: "'x" assumes "monomial_decomp ps"   and "hU \<in> set ps" shows "monomial_decomp (shift_list hU x ps)"
less_asym: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and P :: "bool" assumes "class.preorder less_eq less"   and "less x y"   and "\<not> P \<Longrightarrow> less y x" shows "P"
rel_refl: fixes R1a :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and R2a :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "('a, 'b, 'c) gpv" assumes "\<And>x. R1a x x"   and "\<And>x. R2a x x" shows "rel_gpv R1a R2a x x"
gcd_proj2_iff: fixes m :: "'c"   and n :: "'c" shows "(normalization_euclidean_semiring_class.gcd m n = normalize n) = (n dvd m)"
compos_aI_r: fixes A :: "('c, 'd) aGroup_scheme"   and B :: "('e, 'f) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "aGroup A"   and "aGroup B"   and "f \<in> aHom A B" shows "compos A f aI\<^bsub>A\<^esub> = f"
map_comp0: fixes g1 :: "'i \<Rightarrow> 'e"   and f1 :: "'c \<Rightarrow> 'i"   and g2 :: "'j \<Rightarrow> 'h"   and f2 :: "'g \<Rightarrow> 'j"   and g3 :: "'k \<Rightarrow> 'f"   and f3 :: "'d \<Rightarrow> 'k" shows "map_pre_G\<^sub>M (g1 \<circ> f1) (g2 \<circ> f2) (g3 \<circ> f3) = map_pre_G\<^sub>M g1 g2 g3 \<circ> map_pre_G\<^sub>M f1 f2 f3"
linepath_inner: fixes a :: "'a"   and b :: "'a"   and x :: "real"   and v :: "'a" shows "linepath a b x \<bullet> v = linepath (a \<bullet> v) (b \<bullet> v) x"
generate_subgroup_incl: fixes G :: "('a, 'b) monoid_scheme"   and K :: "'a set"   and H :: "'a set" assumes "Group.group G"   and "K \<subseteq> H"   and "subgroup H G" shows "generate G K \<subseteq> H"
nat_abs_mult_distrib: fixes w :: "int"   and z :: "int" shows "nat \<bar>w * z\<bar> = nat \<bar>w\<bar> * nat \<bar>z\<bar>"
coprime_power_right_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and n :: "nat" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "algebraic_semidom.coprime times one a (power.power one times b n) = (algebraic_semidom.coprime times one a b \<or> n = 0)"
inf_sup_aci_1: fixes x :: "'b"   and y :: "'b" shows "x \<sqinter> y = y \<sqinter> x"
terminatess_impl_no_inf_chain: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile>cs,css \<Down> s" shows "\<nexists>f. f 0 = (cs, css, s) \<and> (\<forall>i. \<Gamma>\<turnstile> f i \<rightarrow>\<^sup>+ f (Suc i))"
substL_conc: fixes L1 :: "nat list set"   and L2 :: "nat list set"   and \<sigma> :: "nat \<Rightarrow> nat rexp option" shows "L (substL (L1 @@ L2) \<sigma>) = L (concS (substL L1 \<sigma>) (substL L2 \<sigma>))"
quicksort_correct: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd"   and lo0 :: "nat"   and hi0 :: "nat"   and xs0 :: "'c list" assumes "\<And>x y z. \<lbrakk>R (h x) (h y); R (h y) (h z)\<rbrakk> \<Longrightarrow> R (h x) (h z)"   and "\<And>x y. x \<noteq> y \<Longrightarrow> R (h x) (h y) \<or> R (h y) (h x)"   and "lo0 \<le> hi0"   and "hi0 < length xs0" shows "WB_Sort.quicksort R h (lo0, hi0, xs0) \<le> \<Down> Id (SPEC (quicksort_post R h lo0 hi0 xs0))"
intensional_simps_17: fixes y :: "'a \<Rightarrow> bool" shows "#True \<and> y \<equiv> y"
impl_blue_dfs_ahs_def: shows "impl_blue_dfs_ahs \<equiv> \<lambda>x. NDFS_SI_Statistics.start_nres \<bind> (\<lambda>xa. LIST_FOREACH' (RETURN (id (gi_V0 x))) (\<lambda>(xb, xc, xd). xd = NO_CYC) (\<lambda>xb (xc, xd, xe). if \<not> ahs.memb xb xc then REC\<^sub>T (\<lambda>xf (xg, xh, xi, xj). let y' = ahs.ins xj xg; y'a = ahs.ins xj xi; y'b = bgi_F x xj in NDFS_SI_Statistics.vis_blue_nres \<bind> (\<lambda>xk. LIST_FOREACH' (RETURN (id (gi_E x xj))) (\<lambda>(xl, xm, xn, xo). xo = NO_CYC) (\<lambda>xl (xm, xn, xo, xp). if ahs.memb xl xo \<and> (y'b \<or> bgi_F x xl) then RETURN (xm, xn, xo, init_wit_blue_early xj xl) else if \<not> ahs.memb xl xm then xf (xm, xn, xo, xl) \<bind> (\<lambda>(xq, xr, xs, xt). RETURN (xq, xr, xs, prep_wit_blue xj xt)) else NDFS_SI_Statistics.match_blue_nres \<bind> (\<lambda>xq. RETURN (xm, xn, xo, xp))) (y', xh, y'a, NO_CYC) \<bind> (\<lambda>(xl, xm, xn, xo). (if xo = NO_CYC \<and> y'b then impl_red_dfs_ahs (gi_E x) xn xm xj \<bind> (\<lambda>(xp, xq). RETURN (xp, init_wit_blue xj xq)) else RETURN (xm, xo)) \<bind> (\<lambda>(xp, xq). let y'c = ahs.delete xj xn in RETURN (xl, xp, y'c, xq))))) (xc, xd, ahs.empty (), xb) \<bind> (\<lambda>(xf, xg, xh, xi). RETURN (xf, xg, xi)) else RETURN (xc, xd, NO_CYC)) (ahs.empty (), ahs.empty (), NO_CYC) \<bind> (\<lambda>(xb, xc, xd). NDFS_SI_Statistics.stop_nres \<bind> (\<lambda>xe. RETURN (extract_res xd))))"
perm_supp_eq: fixes p :: "perm"   and x :: "'c" assumes "supp p \<sharp>* x" shows "p \<bullet> x = x"
inf_continuous_compose: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'c" assumes "inf_continuous f"   and "inf_continuous g" shows "inf_continuous (\<lambda>x. f (g x))"
ahm_lookup_def: fixes k :: "'b"   and hm :: "('b, 'a) hashmap" shows "ahm_lookup k hm = ArrayHashMap_Impl.ahm_lookup k (impl_of hm)"
simps_2: fixes atom_ordering :: "('a \<times> 'a) set"   and A :: "'b" assumes "propositional_atoms atom_ordering" shows "atom (Neg A) = A"
beukers_integral1_conv_nn_integral: fixes r :: "nat"   and s :: "nat" shows "beukers_integral1 r s = enn2real (beukers_nn_integral1 r s)"
L2: shows "\<forall>x y. ((x, y) \<in> r) = (c_pair x y \<in> A)"
cdf_lim_at_top_prob: fixes k :: "nat" shows "(cdf (\<mu> k) \<longlongrightarrow> 1) at_top"
eq: shows "set fs \<union> set (f4_red_aux bs ps) = set fs \<union> set (Macaulay_red (Keys_to_list fs) fs)"
merge_attractor_strategies: fixes S :: "'a set"   and p :: "Player"   and W :: "'a set" assumes "S \<subseteq> V"   and "\<And>v. v \<in> S \<Longrightarrow> \<exists>\<sigma>. strategy p \<sigma> \<and> strategy_attracts_via p \<sigma> v S W" shows "\<exists>\<sigma>. strategy p \<sigma> \<and> strategy_attracts p \<sigma> S W"
gcd_eucl: shows "normalization_euclidean_semiring_class.gcd = gcd"
HS_Aux1: fixes p :: "L"   and G :: "Var \<Rightarrow> L"   and c :: "IMP"   and H :: "Var \<Rightarrow> L" assumes "(p, G, c, H) \<in> HS" shows "\<forall>x. LEQ (G x) (H x) \<or> LEQ p (H x)"
rel_spmf_reflI: fixes p :: "'a spmf"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> P x x" shows "rel_spmf P p p"
Vfrom_mono1: fixes a :: "V"   and b :: "V"   and i :: "V" assumes "a \<le> b" shows "Vfrom a i \<le> Vfrom b i"
assms_6: shows "sets (restrict_space M \<Omega>) = sigma_sets \<Omega> E"
thetaPar_def: shows "thetaPar \<equiv> {(c1 | c2, d1 | d2) |c1 c2 d1 d2. c1 \<approx>s d1 \<and> c2 \<approx>s d2}"
map_Val_nthrow_neq: fixes vs :: "val list"   and vs' :: "val list"   and ex' :: "'c exp"   and es' :: "'c exp list" assumes "map Val vs = map Val vs' @ throw ex' # es'" shows "False"
roots_within_def: fixes f :: "'a \<Rightarrow> 'b"   and s :: "'a set" shows "roots_within f s = {x \<in> s. f x = (0::'b)}"
LIMSEQ_const_mult: fixes a :: "real"   and c :: "real" assumes "f \<longlonglongrightarrow> a" shows "(\<lambda>x. c * f x) \<longlonglongrightarrow> c * a"
LT_imp_DD: fixes r :: "('a \<times> 'a) set"   and s :: "'b"   and \<beta>_step :: "'a \<times> 'b"   and \<upsilon>_step :: "('a \<times> 'b) list"   and \<alpha>_step :: "'a \<times> 'b"   and \<rho>_step :: "('a \<times> 'b) list"   and ars :: "('b \<times> 'a \<times> 'b) set"   and \<gamma>1 :: "'b \<times> (bool \<times> 'a \<times> 'b) list"   and \<gamma>2 :: "'b \<times> ('a \<times> 'b) list"   and \<gamma>3 :: "'b \<times> (bool \<times> 'a \<times> 'b) list"   and \<delta>1 :: "'b \<times> (bool \<times> 'a \<times> 'b) list"   and \<delta>2 :: "'b \<times> ('a \<times> 'b) list"   and \<delta>3 :: "'b \<times> (bool \<times> 'a \<times> 'b) list" assumes "trans r"   and "irrefl r"   and "\<And>y. \<lbrakk>(y, (s, [\<beta>_step] @ \<upsilon>_step), s, [\<alpha>_step] @ \<rho>_step) \<in> pex r; peak ars y\<rbrakk> \<Longrightarrow> \<exists>\<sigma>' \<tau>'. DD ars r (fst y, snd y, \<sigma>', \<tau>')"   and "LT ars r ((s, [\<beta>_step]), (s, [\<alpha>_step]), \<gamma>1, \<gamma>2, \<gamma>3, \<delta>1, \<delta>2, \<delta>3)" shows "\<exists>\<kappa> \<mu>. DD ars r ((s, [\<beta>_step]), (s, [\<alpha>_step]), \<kappa>, \<mu>)"
chkchop_mono: fixes gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and gt' :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool" assumes "gt \<le> gt'" shows "chkchop gt \<le> chkchop gt'"
minus_flip: fixes a :: "'b"   and b :: "'b" shows "- (a \<leftrightarrow> b) = (a \<leftrightarrow> b)"
matchIdLeft: fixes Rel :: "(pi \<times> pi) set"   and a :: "name"   and P :: "pi" assumes "Id \<subseteq> Rel" shows "[a\<frown>a]P \<leadsto>[Rel] P"
tab_simps_1: shows "seq (\<phi> \<star> f) \<rho>"
PNT1_imp_PNT2: assumes "\<pi> \<sim>[at_top] (\<lambda>x. x / ln x)" shows "(\<lambda>x. \<pi> x * ln (\<pi> x)) \<sim>[at_top] (\<lambda>x. x)"
simps_1: fixes ys :: "[:'a:]" shows "[::] :@ ys = ys"
shift_simps_1: fixes xs :: "'d list"   and s :: "'d stream" shows "shd (xs @- s) = (if xs = [] then shd s else hd xs)"
assms_4: shows "R \<subseteq> V \<times> V"
elims: fixes x :: "nat set \<times> 'a option list set"   and xa :: "nat set \<times> 'a option list"   and y :: "nat set \<times> 'a option list set" assumes "semiJoin x xa = y" obtains s :: "nat set" and tab :: "'a option list set" and st :: "nat set" and tup :: "'a option list" where "x = (s, tab)" and "xa = (st, tup)" and "y = (s, Set.filter (isSameIntersection tup (s \<inter> st)) tab)"
minGraphProps3: fixes g :: "graph"   and f :: "face" assumes "minGraphProps g"   and "f \<in> \<F> g" shows "distinct (vertices f)"
dom_char: fixes f :: "'b1 \<times> 'b2" shows "B1xB2.dom f = (if B1.arr (fst f) \<and> B2.arr (snd f) then (B1.dom (fst f), B2.dom (snd f)) else (B1.null, B2.null))"
is_functor: shows "functor (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>A\<^sub>1) A1.map"
m'_eq: shows "m' = map_option \<alpha> \<circ> m"
Special_Solutions_def: fixes a :: "nat list"   and b :: "nat list" assumes "hlde a b" shows "hlde.Special_Solutions a b = {sij a b i j |i j. i < length a \<and> j < length b}"
takeq: shows "w \<le>p take q w \<cdot> w"
has_integral_on_superset: fixes f :: "'c \<Rightarrow> 'd"   and i :: "'d"   and S :: "'c set"   and T :: "'c set" assumes "(f has_integral i) S"   and "\<And>x. x \<notin> S \<Longrightarrow> f x = (0::'d)"   and "S \<subseteq> T" shows "(f has_integral i) T"
wp2_is_weakestprePotential2: fixes c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and P :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<forall>s. wp\<^sub>2 c Q s \<le> P s" shows "\<Turnstile>\<^sub>2 {P} c {Q}"
qml_2: fixes \<phi> :: "\<o>" shows "[[\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<phi>]]"
pdg_digraph_dg_prod: shows "digraph \<alpha> (\<Prod>\<^sub>D\<^sub>Gi\<in>\<^sub>\<circ>3\<^sub>\<nat>. if i = []\<^sub>\<circ> \<Rightarrow> \<AA> | i = 1\<^sub>\<nat> \<Rightarrow> \<BB> | otherwise \<Rightarrow> \<CC>)"
map_add_None: fixes m :: "'b \<Rightarrow> 'a option"   and n :: "'b \<Rightarrow> 'a option"   and k :: "'b" shows "(Map.map_add m n k = None) = (n k = None \<and> m k = None)"
intro: fixes f :: "'b \<Rightarrow> 'c" assumes "clinear f"   and "bounded_clinear_axioms f" shows "bounded_clinear f"
INTR: fixes S1 :: "'a set"   and S2 :: "'a set"   and \<sigma> :: "'b" assumes "S\<^sub>0 = S1 \<union> S2"   and "I S1 S2 \<sigma>"   and "\<not> c \<sigma>" shows "\<Phi> \<sigma>"
arr_iff_in_hom: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "arr f = \<guillemotleft>f : local.dom f \<rightarrow> cod f\<guillemotright>"
widen_refT: fixes T :: "ty"   and P :: "'a prog"   and U :: "ty" assumes "is_refT T"   and "P \<turnstile> U \<le> T" shows "is_refT U"
Un_upper2: fixes B :: "'x set"   and A :: "'x set" shows "B \<subseteq> A \<union> B"
mset_subset_eqD: fixes A :: "'k multiset"   and B :: "'k multiset"   and x :: "'k" assumes "A \<subseteq># B"   and "x \<in># A" shows "x \<in># B"
inf_absorb1: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "inf x y = x"
P_sup_def: fixes s :: "'s"   and P :: "'s stream \<Rightarrow> bool" shows "P_sup s P = (\<Squnion>cfg\<in>cfg_on s. emeasure (T cfg) {x \<in> space St. P x})"
simps_15: fixes x2 :: "'f" shows "set_option (Some x2) = {x2}"
mat_inv_mult_mm: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and B :: "complex \<times> complex \<times> complex \<times> complex" assumes "mat_det A \<noteq> 0"   and "mat_det B \<noteq> 0" shows "mat_inv (A *\<^sub>m\<^sub>m B) = mat_inv B *\<^sub>m\<^sub>m mat_inv A"
UNION_singleton_eq_range: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "(\<Union>x\<in>A. {f x}) = f ` A"
distinct_189: fixes x7 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Abs x7 \<noteq> x151 ^\<^sub>e x152"
lappend_LNil2: fixes xs :: "'c llist" shows "lappend xs LNil = xs"
at3: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "at TYPE('a)" shows "swap (a, b) c = (if a = c then b else if b = c then a else c)"
simps_3: fixes x :: "'a"   and y :: "'a" shows "(extC.Val x \<le> extC.Val y) = (x \<le> y)"
powerset_lift_singleton: fixes f :: "'d \<rightarrow> 'c set"   and x :: "'d" shows "\<^ps> f\<cdot>{x} = f\<cdot>x"
primitive_extractor_negation_type_matching1: fixes disc :: "'a \<Rightarrow> bool"   and sel :: "'a \<Rightarrow> 'b"   and C :: "'b \<Rightarrow> 'a"   and m :: "'a match_expr"   and as :: "'b negation_type list"   and rest :: "'a match_expr"   and \<gamma> :: "('a \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and a :: "action"   and p :: "'c" assumes "wf_disc_sel (disc, sel) C"   and "normalized_nnf_match m"   and "primitive_extractor (disc, sel) m = (as, rest)"   and "matches \<gamma> m a p" shows "(\<forall>m\<in>set (map C (getPos as)). matches \<gamma> (Match m) a p) \<and> (\<forall>m\<in>set (map C (getNeg as)). matches \<gamma> (MatchNot (Match m)) a p)"
runit_simps_1: fixes f :: "'a arrow_of_spans_data" assumes "ide f" shows "arr \<r>[f]"
last: shows "\<forall>ax ax'. ax \<in> set as' \<and> sourcenode ax = sourcenode ax' \<and> valid_edge ax' \<longrightarrow> n postdominates targetnode ax'"
partial_order_axioms_def: fixes L :: "('a, 'b) gorder_scheme" shows "partial_order_axioms L \<equiv> (.=\<^bsub>L\<^esub>) = (=)"
assms_6: shows "4 \<le> rk {A', A, C', a}"
domain_eq: shows "Domainp (pcr_alist (=) (=)) = distinct \<circ> list.map fst"