hcju/isabelleps huggingface dataset:
cmheads_def: fixes G :: "prog"   and S :: "qtname"   and C :: "qtname" shows "cmheads G S C = (\<lambda>sig. (\<lambda>(Cls, mthd). (ClassT Cls, mhead mthd)) ` set_option (accmethd G S C sig))"
that: fixes S :: "'a spmf \<Rightarrow> 'h spmf \<Rightarrow> bool" assumes "S p q"   and "\<And>p q ia. \<lbrakk>S p q; ia \<in> IA\<rbrakk> \<Longrightarrow> p \<bind> (\<lambda>s1. map_spmf fst (rfunc_adv rest1 s1 ia)) = q \<bind> (\<lambda>s2. map_spmf fst (rfunc_adv rest2 s2 ia))"   and "\<And>p q ia oa. \<lbrakk>S p q; ia \<in> IA\<rbrakk> \<Longrightarrow> S (cond_spmf_fst (p \<bind> (\<lambda>s1. rfunc_adv rest1 s1 ia)) oa) (cond_spmf_fst (q \<bind> (\<lambda>s2. rfunc_adv rest2 s2 ia)) oa)"   and "\<And>p q iu. \<lbrakk>S p q; iu \<in> IU\<rbrakk> \<Longrightarrow> p \<bind> (\<lambda>s1. map_spmf fst (rfunc_usr rest1 s1 iu)) = q \<bind> (\<lambda>s2. map_spmf fst (rfunc_usr rest2 s2 iu))"   and "\<And>p q iu ou. \<lbrakk>S p q; iu \<in> IU\<rbrakk> \<Longrightarrow> S (cond_spmf_fst (p \<bind> (\<lambda>s1. rfunc_usr rest1 s1 iu)) ou) (cond_spmf_fst (q \<bind> (\<lambda>s2. rfunc_usr rest2 s2 iu)) ou)" shows "thesis"
tm_append: fixes n :: "nat"   and tps :: "'a list list"   and tp :: "'a list" assumes "n < length tps"   and "tp = tps ! n" shows "\<exists>tp1 tp2. concat tps = tp1 @ tp @ tp2 \<and> tp1 = concat (take n tps) \<and> tp2 = concat (drop (Suc n) tps)"
sources_determine_composability: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and \<tau> :: "'b"   and \<mu> :: "'b" assumes "regular_weak_composition V H"   and "a \<in> weak_composition.sources V H \<tau>" shows "(H \<tau> \<mu> \<noteq> partial_magma.null V) = (H a \<mu> \<noteq> partial_magma.null V)"
in_prime_factors_imp_dvd: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "p \<in># factorial_semiring.prime_factorization zero times one normalize x" shows "dvd.dvd times p x"
tau_func: fixes M :: "('g, 'a, 'h) Module_scheme"   and N :: "('i, 'a, 'h) Module_scheme"   and P :: "('g \<times> 'i, 'a, 'h) Module_scheme" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "tau R M N P \<in> M \<times>\<^sub>c N \<rightarrow> carrier P"
f_def: fixes X :: "'a" shows "f X = e (the (map_of (zip xs [0..<length xs]) X))"
arrow: shows "C.ide (Fo y) \<and> \<guillemotleft>\<eta>o y : y \<rightarrow>\<^sub>D G (Fo y)\<guillemotright>"
so_2: shows "\<not> P,E \<turnstile> r' \<le>so w'"
Rep_unit_inject: fixes x :: "unit"   and y :: "unit" shows "(Rep_unit x = Rep_unit y) = (x = y)"
Suc_times_binomial_eq: fixes n :: "nat"   and k :: "nat" shows "Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
add_diff_cancel_left': fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a b) a = b"
capacity_const: shows "\<forall>e. (0::'a) \<le> f' e \<and> f' e \<le> c e"
Collect_conv_if2: fixes a :: "'b"   and P :: "'b \<Rightarrow> bool" shows "{x::'b. (a::'b) = x \<and> (P::'b \<Rightarrow> bool) x} = (if P a then {a} else {})"
inj_on_empty: fixes f :: "'c \<Rightarrow> 'd" shows "inj_on f {}"
image_Int_subset: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and B :: "'c set" shows "order.greater_eq (f ` A \<inter> f ` B) (f ` (A \<inter> B))"
fid_on_fimage: fixes A :: "V"   and B :: "V" shows "fid_on A `\<^sub>\<bullet> B = A \<inter>\<^sub>\<circ> B"
case_3: fixes f1 :: "'e \<Rightarrow> 'd"   and f2 :: "nat set \<Rightarrow> 'e \<Rightarrow> 'd"   and f3 :: "nat \<Rightarrow> 'e \<Rightarrow> 'd"   and f4 :: "nat \<Rightarrow> 'd"   and f5 :: "'e \<Rightarrow> 'd"   and f6 :: "nat \<Rightarrow> 'd"   and f7 :: "'e \<Rightarrow> 'd"   and f8 :: "'d"   and x31 :: "nat"   and x32 :: "'e" shows "(case unicast x31 x32 of broadcast x \<Rightarrow> f1 x | groupcast x xa \<Rightarrow> f2 x xa | unicast x xa \<Rightarrow> f3 x xa | \<not>unicast x \<Rightarrow> f4 x | send x \<Rightarrow> f5 x | deliver x \<Rightarrow> f6 x | receive x \<Rightarrow> f7 x | \<tau>\<^sub>s \<Rightarrow> f8) = f3 x31 x32"
set_filter_diff_card: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" shows "card ({a \<in> A. P a} - {x}) = card {a \<in> A - {x}. P a}"
wf_prefix: fixes V :: "'d set"   and S :: "('e, 'd) strand_step list"   and S' :: "('e, 'd) strand_step list" assumes "wf\<^sub>s\<^sub>t V (S @ S')" shows "wf\<^sub>s\<^sub>t V S"
while_proof: fixes Q :: "('Q, unit) HashMap.hashmap \<times> 'Q list \<times> (('Q, 'L) ta_rule, nat) HashMap.hashmap \<Rightarrow> bool" assumes "\<And>s. \<lbrakk>s \<in> dwa_invar (brc_det_algo (build_rqrm \<delta>) \<delta>); \<not> dwa_cond (brc_det_algo (build_rqrm \<delta>) \<delta>) s\<rbrakk> \<Longrightarrow> Q s" shows "Q loop"
sum: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b"   and A :: "'d set" assumes "additive f" shows "f (sum g A) = (\<Sum>x\<in>A. f (g x))"
Fs_dim_1: shows "dim_row (mat_of_rows n (map gso [0..<length fs])) = length fs"
dghm_ArrMap_app_in_HomCod_Arr: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> op_dg \<AA>\<lparr>Arr\<rparr>" shows "\<FF>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"
Xm0: shows "\<forall>y\<in>X. y $ m = (0::'a)"
poincare_distance_eq_ex_moebius: fixes u :: "complex_homo"   and v :: "complex_homo"   and u' :: "complex_homo"   and v' :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "u' \<in> unit_disc"   and "v' \<in> unit_disc"   and "poincare_distance u v = poincare_distance u' v'" shows "\<exists>M. unit_disc_fix M \<and> moebius_pt M u = u' \<and> moebius_pt M v = v'"
partial_order_def: fixes L :: "('e, 'f) gorder_scheme" shows "partial_order L \<equiv> weak_partial_order L \<and> partial_order_axioms L"
proper_implies_part: fixes x :: "i"   and y :: "i" assumes "M"   and "PP x y" shows "P x y"
rel_pmf_bind_reflI: fixes p :: "'d pmf"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e pmf"   and g :: "'d \<Rightarrow> 'f pmf" assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> rel_pmf R (f x) (g x)" shows "rel_pmf R (p \<bind> f) (p \<bind> g)"
WT_callee: fixes s :: "'s2" assumes "Ir s" shows "\<I>r \<turnstile>c right s \<surd>"
simps_1: fixes xs :: "[:'r:]" shows "prefix\<cdot>xs\<cdot>\<bottom> = \<bottom>"
WL_def: fixes p :: "'h process_name" shows "WL p = (\<lambda>s. W (s p) \<union> ghost_honorary_grey (s p))"
path_component_in_topspace: fixes X :: "'b topology"   and x :: "'b"   and y :: "'b" assumes "path_component_of X x y" shows "x \<in> topspace X \<and> y \<in> topspace X"
c_is_sub_fun_def: fixes ls :: "nat"   and f :: "nat \<Rightarrow> nat" shows "c_is_sub_fun ls f = (\<forall>x. c_assoc_have_key ls x = 0 \<longrightarrow> c_assoc_value ls x = f x)"
individual_weight_less_eq_lp: fixes g :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and w :: "'a \<Rightarrow> nat"   and x :: "'a"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and l :: "'a list" assumes "geq_arg g" shows "w x \<le> wlp R w g f x l"
norm_eq_sqrt_inner: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "norm x = sqrt (inner x x)"
a: shows "liminf (u \<circ> r) \<noteq> - \<infinity>"
hom_nat_pow: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e"   and x :: "'c"   and n :: "nat" assumes "ring_hom_ring R S h"   and "x \<in> carrier R" shows "h (x [^]\<^bsub>R\<^esub> n) = h x [^]\<^bsub>S\<^esub> n"
Gromov_hyperbolic_ineq_not_distinct: fixes x :: "'b"   and y :: "'b"   and z :: "'b"   and t :: "'b" assumes "x = y \<or> x = z \<or> x = t \<or> y = z \<or> y = t \<or> z = t" shows "dist x y + dist z t \<le> max (dist x z + dist y t) (dist x t + dist y z)"
processedI: fixes a :: "'e"   and xs :: "'e list"   and b :: "'e"   and ps :: "('e \<times> 'e) list" assumes "a \<in> set xs"   and "b \<in> set xs"   and "(a, b) \<notin>\<^sub>p set ps" shows "processed (a, b) xs ps"
swap_identical_heads: fixes i :: "nat"   and j :: "nat"   and t :: "'d list" assumes "i < j"   and "j < length t" shows "take i t = take i (swap_events i j t)"
thetaAtm_Sbis: fixes atm :: "'atom" assumes "compatAtm atm" shows "thetaAtm atm \<subseteq> Sbis"
shift_instr_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (shift_instr instr s1)) \<and> t2 = snd (fst (shift_instr instr s2))" shows "low_equal t1 t2"
simps: fixes G :: "game"   and H :: "game" shows "plus_game G H = Game (zunion (zimage (\<lambda>g. plus_game g H) (left_options G)) (zimage (plus_game G) (left_options H))) (zunion (zimage (\<lambda>g. plus_game g H) (right_options G)) (zimage (plus_game G) (right_options H)))"
a: fixes a :: "nat"   and b :: "nat" shows "(a < b) = (\<not> b \<le> a)"
ns_mul_ext_refl: fixes ns :: "('a \<times> 'a) set"   and A :: "'a multiset"   and s :: "('a \<times> 'a) set" assumes "refl ns" shows "(A, A) \<in> ns_mul_ext ns s"
type_stable_is_sufficient': fixes t :: "'BExp set"   and \<S> :: "'Var set \<times> 'Var set" assumes "type_stable \<S> t" shows "\<forall>mem mem'. (\<forall>x. stable \<S> x \<longrightarrow> mem x = mem' x) \<longrightarrow> type_max t mem = type_max t mem'"
f4: shows "\<forall>n. mapping_of (isolate_variable_sparse mpa va n) = 0"
FAccRedsNull: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>null,s',b'\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>F{D},s,b\<rangle> \<rightarrow>* \<langle>THROW NullPointer,s',b'\<rangle>"
M0_def: shows "M0 = (\<lambda>i. (f (2 * i), f (Suc (2 * i)))) ` {..<m}"
Pxy_nn_1: fixes x :: "'b \<times> 'c" assumes "x \<in> space (S \<Otimes>\<^sub>M T)" shows "0 \<le> Pxy x"
retraction_imp_quotient_map: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" assumes "retraction_map X Y f" shows "quotient_map X Y f"
exhaust_sel: fixes lit :: "lit" obtains "lit = Lit (pos lit) (var lit)"
case_optionE: fixes P :: "bool"   and Q :: "'a \<Rightarrow> bool"   and x :: "'a option" assumes "case x of None \<Rightarrow> P | Some y \<Rightarrow> Q y" obtains "x = None" and "P"   | y :: "'a" where "x = Some y" and "Q y"
ineq_model_singleE: fixes \<I> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and X :: "'a list"   and t :: "('b, 'a) Term.term"   and t' :: "('b, 'a) Term.term" assumes "ineq_model \<I> X [(t, t')]" shows "\<forall>\<delta>. subst_domain \<delta> = set X \<and> ground (subst_range \<delta>) \<longrightarrow> t \<cdot> \<delta> \<cdot> \<I> \<noteq> t' \<cdot> \<delta> \<cdot> \<I>"
above_ord_nonzero: fixes x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "x \<in> carrier Zp"   and "x \<noteq> \<zero>"   and "ord_Zp x < int n" shows "x n \<noteq> 0"
lfp_fixpoint: fixes f :: "'c \<Rightarrow> 'c" assumes "mono f" shows "f (lfp f) = lfp f"
case2: shows "queue_invar (Node e1 a1 r ts1 # ts2)"
CHECK_monadic_def: fixes c :: "('c + bool) nres"   and e :: "'c" shows "CHECK_monadic c e \<equiv> ebind c (\<lambda>b. CHECK b e)"
rep_in_iso_class: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b set" assumes "category C"   and "category.is_iso_class C F" shows "iso_class_rep F \<in> F"
rep_pt: shows "\<forall>pt\<in>set_of rt. rep pt = pt"
uniformly_continuous_on_sequentially: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "uniformly_continuous_on s f = (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and> (\<lambda>n. dist (x n) (y n)) \<longlonglongrightarrow> 0 \<longrightarrow> (\<lambda>n. dist (f (x n)) (f (y n))) \<longlonglongrightarrow> 0)"
assms_2: shows "\<Gamma>(chain \<mapsto> Rule m a # rs),\<gamma>,p\<turnstile> \<langle>rs', s\<rangle> \<Rightarrow> t"
mem_list1: shows "\<forall>r\<in>set (characterize_root_list_p p). consistent_sign_vec qs1 r \<in> set (map (consistent_sign_vec qs1) (characterize_root_list_p p))"
node_not_in_ancestors: shows "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node \<notin> set ancestors"
smooth_on_inner: fixes k :: "enat"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "k-smooth_on S f"   and "k-smooth_on S g"   and "open S" shows "k-smooth_on S (\<lambda>x. f x \<bullet> g x)"
nth_gen_akiyama_tanigawa_row: fixes i :: "nat"   and u :: "nat"   and l :: "nat"   and f :: "nat \<Rightarrow> real"   and n :: "nat" assumes "i < u - l" shows "gen_akiyama_tanigawa_row f n l u ! i = gen_akiyama_tanigawa f n (i + l)"
commute: fixes a :: "'a\<^sub>m"   and b :: "'a\<^sub>m" shows "m a b = m b a"
list_of_pdevs_def: fixes x :: "'d pdevs" shows "list_of_pdevs x = map (\<lambda>i. (i, pdevs_apply x i)) (rev (sorted_list_of_set (pdevs_domain x)))"
2: shows "(\<lambda>x. if \<exists>xK\<in>p. f xK = x then (\<gamma> \<circ> fst \<circ> the_inv_into p f) x else \<gamma> x) fine {(f xK, snd xK) |xK. xK \<in> p}"
Un_assoc: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set" shows "A \<union> B \<union> C = A \<union> (B \<union> C)"
coclop_idem: fixes f :: "'b \<Rightarrow> 'b" assumes "coclop f" shows "f \<circ> f = f"
has_ivl_integral_nonneg: fixes f :: "real \<Rightarrow> real"   and i :: "real"   and a :: "real"   and b :: "real" assumes "(f has_ivl_integral i) a b"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> 0 \<le> f x"   and "\<And>x. \<lbrakk>b \<le> x; x \<le> a\<rbrakk> \<Longrightarrow> f x \<le> 0" shows "0 \<le> i"
extremum_uniqueI: fixes a :: "'a" assumes "\<top> \<le> a" shows "a = \<top>"
add_mult_distrib: fixes x :: "V"   and y :: "V"   and z :: "V" shows "x * (y + z) = x * y + x * z"
eventually_sequentially_Suc: fixes P :: "nat \<Rightarrow> bool" shows "(\<forall>\<^sub>F i in sequentially. P (Suc i)) = eventually P sequentially"
sinit_linear_asym_lemma2: fixes R :: "('b \<times> 'b) set"   and i :: "nat"   and xs :: "'b seq"   and ys :: "'b seq"   and j :: "nat" assumes "asym R"   and "(sinit i xs, sinit i ys) \<in> lexord R"   and "(sinit j ys, sinit j xs) \<in> lexord R" shows "False"
poly_shift_monom: fixes n :: "nat"   and c :: "'b"   and m :: "nat" shows "poly_shift n (monom c m) = (if n \<le> m then monom c (m - n) else 0)"
ent_mp: fixes P :: "assn"   and Q :: "assn" shows "P * (P -* Q) \<Longrightarrow>\<^sub>A Q"
a2: shows "\<Gamma>1 \<subseteq> \<Gamma>2"
prems: shows "c\<^sub>1 \<le> mds\<^sub>1"
lvivl_rel_br: shows "lvivl_rel = br (\<lambda>(x, y). set_of_ivl (eucl_of_list x, eucl_of_list y)) (\<lambda>(x, y). length x = DIM('c) \<and> length y = DIM('c))"
skl1_implements_l1: shows "implements med11 l1 skl1"
assms_2: shows "unitary11_gen (k *\<^sub>s\<^sub>m M)"
simps_12: fixes a :: "'c local.term"   and b :: "'c local.term" shows "(a \<^bold>\<otimes> b) \<^bold>\<Down> \<^bold>\<I> = \<^bold>\<r>\<^bold>[a \<^bold>\<otimes> b\<^bold>]"
is_processT6_S2: fixes X :: "'c event set"   and P :: "'c process" assumes "tick \<notin> X"   and "[tick] \<in> T P" shows "([], X) \<in> F P"
rec: fixes n :: "nat" assumes "length fs + 1 - length cs \<le> n" shows "(\<Sum>k<length cs. cs ! k * f (n + k)) = eval_polyexp g (n + length cs - 1)"
simps_2: fixes M :: "('a, 'b) message set"   and uu :: "'c" shows "is_certified_associated_string M (Inl uu) = False"
norm_scaleR: fixes a :: "real"   and x :: "'a" shows "norm (a *\<^sub>R x) = \<bar>a\<bar> * norm x"
ctxt_1: shows "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"
real_poly_power: fixes p :: "'a poly"   and n :: "nat" assumes "set (coeffs p) \<subseteq> \<real>" shows "set (coeffs (p ^ n)) \<subseteq> \<real>"
eval_eqI: fixes t :: "'b EV.term"   and u :: "'b EV.term" assumes "EV.VPar t u"   and "EV.Nmlize t = EV.Nmlize u" shows "\<lbrace>t\<rbrace> = \<lbrace>u\<rbrace>"
create_character_data_preserves_wellformedness_3: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and "text" :: "char list"   and h' :: "(_) heap" assumes "heap_is_wellformed h"   and "h \<turnstile> create_character_data document_ptr text \<rightarrow>\<^sub>h h'"   and "type_wf h"   and "known_ptrs h" shows "known_ptrs h'"
p2: shows "prod_list (map (sq_norm \<circ> row (sylvester_mat f g)) [degree g..<degree f + degree g]) = \<parallel>g\<parallel>\<^sup>2 ^ n"
neq0_conv: fixes n :: "nat" shows "(n \<noteq> 0) = order.greater n 0"
simps_1: fixes f :: "'a \<Rightarrow> bool"   and env :: "'a" shows "if_match f (Match env) = f env"
qfree_lift_dnfeq_qe: fixes qe :: "atom list \<Rightarrow> atom fm"   and \<phi> :: "atom fm" assumes "\<And>as. \<forall>a\<in>set as. depends\<^sub>d\<^sub>l\<^sub>o a \<Longrightarrow> qfree (qe as)" shows "qfree (DLO\<^sub>e.lift_dnfeq_qe qe \<phi>)"
rec_mult_def: shows "rec_mult = Pr 1 z (Cn 3 rec_add [recf.id 3 0, recf.id 3 2])"
simps: fixes s :: "'a + 'b"   and a :: "'a" shows "setlp s a = (\<exists>x. a = x \<and> s = Inl x)"
comp_tuple_arr: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "span f g"   and "arr h"   and "local.dom f = cod h" shows "tuple f g \<cdot> h = tuple (f \<cdot> h) (g \<cdot> h)"
simps_3: fixes p :: "int \<times> int"   and p\<^sub>0 :: "int \<times> int"   and v :: "int \<times> int"   and va :: "(int \<times> int) list" shows "find_closest_bf_code p (p\<^sub>0 # v # va) = (let (\<delta>\<^sub>1, p\<^sub>1) = find_closest_bf_code p (v # va); \<delta>\<^sub>0 = dist_code p p\<^sub>0 in if \<delta>\<^sub>0 < \<delta>\<^sub>1 then (\<delta>\<^sub>0, p\<^sub>0) else (\<delta>\<^sub>1, p\<^sub>1))"
nz': fixes b :: "nat" shows "(0::'a, b) \<notin> set cs"
betw_b_in_path: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and ac :: "'b set" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "betw a b c"   and "ac \<in> \<P> \<and> a \<in> ac \<and> c \<in> ac \<and> a \<noteq> c" shows "b \<in> ac"
wf': shows "is_pref_profile (permute_profile \<sigma> R2)"
pfsubset_trans: fixes A :: "'b fset"   and B :: "'b fset"   and C :: "'b fset" assumes "A |\<subset>| B"   and "B |\<subset>| C" shows "A |\<subset>| C"
set_1: fixes x :: "('c \<times> ('c, 'd) rose_tree\<^sub>m list, 'd \<times> 'd rose_tree\<^sub>h list) blindable\<^sub>m" shows "set1_rose_tree\<^sub>m (Tree\<^sub>m x) = (\<Union>uu\<in>set1_blindable\<^sub>m x. fsts uu \<union> (\<Union>uu\<in>snds uu. \<Union> (set1_rose_tree\<^sub>m ` set uu)))"
P_accD_1: fixes x1 :: "'a"   and x2 :: "'b"   and y1 :: "'a"   and y2 :: "'b" assumes "((x1, x2), y1, y2) \<in> (SIGMA x:UNIV. set_pmf (Kp x))\<^sup>*" shows "(x1, y1) \<in> K1.acc"
nn_integral_disjoint_family: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and B :: "nat \<Rightarrow> 'b set" assumes "f \<in> borel_measurable M"   and "\<And>n. B n \<in> sets M"   and "disjoint_family B" shows "set_nn_integral M (\<Union> (range B)) f = (\<Sum>n. set_nn_integral M (B n) f)"
inf_aci_3: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.semilattice_inf inf less_eq less" shows "inf x (inf y z) = inf y (inf x z)"
USdBeqlen: fixes V :: "'a list"   and V' :: "'a list"   and SR :: "(('a \<times> ('b \<Rightarrow> 'c)) \<times> 'a list \<times> ('b \<Rightarrow> 'c)) set"   and DA :: "'b \<Rightarrow> 'e"   and d :: "'e" assumes "(V, V') \<in> Strong_Security.USdB SR DA d" shows "length V = length V'"
cong_equivp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "local.cong R" shows "equivp R"
constant_function_closed: fixes a :: "'a" assumes "a \<in> carrier R" shows "constant_function S a \<in> carrier F"
simps: fixes q :: "'e"   and l :: "'d"   and qs :: "'e list" shows "rhsl q \<rightarrow> l qs = l"
rep_eq: fixes x :: "'c"   and xa :: "'c multiset" shows "count (add_mset x xa) = (\<lambda>b. if b = x then Suc (count xa b) else count xa b)"
mk_minsky_prim_rec: fixes g :: "nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" assumes "mk_minsky1 g"   and "mk_minsky3 h" shows "mk_minsky2 (PrimRecOp g h)"
ras_top_refine: fixes l :: "'c list"   and s :: "'d array \<times> nat"   and R :: "('d \<times> 'c) set" assumes "l \<noteq> []"   and "(s, l) \<in> \<langle>R\<rangle>ras_rel" shows "(ras_top s, hd l) \<in> R"
assoc_def: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "\<a>[a, b, c] \<equiv> \<langle>\<pp>\<^sub>1[a, b] \<cdot> \<pp>\<^sub>1[a \<otimes> b, c], \<langle>\<pp>\<^sub>0[a, b] \<cdot> \<pp>\<^sub>1[a \<otimes> b, c], \<pp>\<^sub>0[a \<otimes> b, c]\<rangle>\<rangle>"
len_signed: fixes x :: "'b signed itself" shows "len_of x = LENGTH('b)"
one_app_cong: fixes t1 :: "lam"   and t2 :: "lam"   and s1 :: "lam"   and s2 :: "lam" assumes " t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* t2"   and " s1 \<longrightarrow>\<^sub>\<beta>\<^sup>* s2" shows " App t1 s1 \<longrightarrow>\<^sub>\<beta>\<^sup>* App t2 s2"
graeffe_poly_impl_main: fixes f :: "'a poly"   and c :: "'a"   and as :: "'a list"   and m :: "nat" assumes "f = Polynomial.smult c (\<Prod>a\<leftarrow>as. [:- a, 1::'a:])" shows "graeffe_poly_impl_main ((- (1::'a)) ^ degree f) f m = graeffe_poly c as m"
path_split_3: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n :: "'b"   and as :: "'a list"   and a :: "'a"   and as' :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.path sourcenode targetnode valid_edge n (as @ a # as') n'" shows "CFG.path sourcenode targetnode valid_edge (targetnode a) as' n'"
hermitian_proj_psi_l: fixes k :: "nat" assumes "k < K" shows "hermitian (proj_psi_l k)"
dom_cod: fixes f :: "'a1" assumes "C1.arr f" shows "C1.dom (C1.cod f) = C1.cod f"
problem_plan_bound_works: fixes PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set"   and s :: "('a, bool) fmap"   and as :: "(('a, bool) fmap \<times> ('a, bool) fmap) list" assumes "finite PROB"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' \<le> problem_plan_bound PROB"
in_relationsI: fixes f :: "'a \<Rightarrow> int"   and A :: "'a set" assumes "(\<Otimes>a\<in>A. a [^] f a) = \<one>"   and "f \<in> extensional A" shows "f \<in> relations A"