hcju/isabelleps huggingface dataset:
nth_least_less_esize_less: fixes n :: "nat"   and A :: "nat set"   and k :: "enat" shows "(enat n < esize A \<and> enat (nth_least A n) < k) = (enat n < esize {i \<in> A. enat i < k})"
DSourcesA7_L0: shows "DSources level0 sA7 = {sA6}"
dependent_imp_non_trivial_combine: fixes K :: "'a set"   and Us :: "'a list" assumes "subfield K R"   and "set Us \<subseteq> carrier R"   and "local.dependent K Us" obtains Ks :: "'a list" where "length Ks = length Us" and "combine Ks Us = \<zero>" and "set Ks \<subseteq> K" and "set Ks \<noteq> {\<zero>}"
assms: fixes s :: "'a"   and x :: "'b" shows "interaction_any_bounded_by (conv s x) n"
dH_equal_def: fixes d :: "'d"   and H :: "('d \<times> 'exp) set"   and m :: "'id \<Rightarrow> 'val"   and m' :: "'id \<Rightarrow> 'val" shows "m \<sim>\<^bsub>d,H\<^esub> m' \<equiv> m =\<^bsub>d\<^esub> m' \<and> (\<forall>(d', e)\<in>H. d' \<le> d \<longrightarrow> E e m = E e m')"
simps_1: fixes x :: "complex" shows "Re (- x) = - Re x"
bisimSubstParAssoc: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<parallel> Q \<parallel> R \<sim>\<^sub>s P \<parallel> (Q \<parallel> R)"
correctCompositionDiffLevelsA23: shows "correctCompositionDiffLevels sA23"
long_division_zero_2: fixes K :: "('c multiset \<Rightarrow> 'a) set"   and q :: "('c multiset \<Rightarrow> 'a) list" assumes "subfield K L"   and "q \<in> carrier (K [X]\<^bsub>L\<^esub>)" shows "L.pmod [] q = []"
elims: fixes x :: "'b \<times> sort_dir"   and y :: "'b \<times> nat" assumes "embed_dir x = y" obtains xa :: "'b" where "x = (xa__, sort_dir.Left)" and "y = (xa__, 0)"   | xa :: "'b" where "x = (xa, sort_dir.Right)" and "y = (xa, Suc (Suc 0))"
zero_order_3: fixes n :: "'b multiset" shows "\<not> n \<subset># {#}"
quotient_of_coprime: fixes r :: "rat"   and p :: "int"   and q :: "int" assumes "quotient_of r = (p, q)" shows "algebraic_semidom_class.coprime p q"
ord_strict_p_rec: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "(p \<prec>\<^sub>p q) = (q \<noteq> 0 \<and> (p = 0 \<or> (let v1 = lt p; v2 = lt q in v1 \<prec>\<^sub>t v2 \<or> v1 = v2 \<and> lookup p v1 = lookup q v2 \<and> lower p v1 \<prec>\<^sub>p lower q v2)))"
conforms_localD: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lenv" assumes "(x, s)\<Colon>\<preceq>(G, L)" shows "G,s\<turnstile>locals s[\<sim>\<Colon>\<preceq>]L"
distrib_inf_le: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<squnion> x \<sqinter> z \<le> x \<sqinter> (y \<squnion> z)"
dominate_sdominate: fixes n1 :: "nat"   and n2 :: "nat" assumes "dominate n1 n2"   and "n1 \<noteq> n2" shows "strict_dominate n1 n2"
B: fixes cfs :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list" assumes "P x \<Turnstile> cfs\<box>" shows "(case cfs ! 0 of (pc, s, stk) \<Rightarrow> \<lambda>(pc', s', stk'). pc' = pc + size (P x) + I s \<and> Q s s' \<and> stk' = F s stk) (cfs ! (length cfs - 1))"
aff_dim_affine_hull: fixes S :: "'a set" shows "aff_dim (affine hull S) = aff_dim S"
cong_dvd_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a" shows "dvd.dvd times a b = dvd.dvd times a c"
Var_left: fixes x :: "'c"   and t :: "('d, 'c) Term.term"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "x \<notin> vars_term t" shows "UNIF1 (subst x t) (add_mset (Var x, t) E) (subst_mset (subst x t) E)"
image_mset_mset_set: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "inj_on f A" shows "f `# mset_set A = mset_set (f ` A)"
step_2_unique_continuous_unfolding: fixes p :: "'a pmf"   and q :: "'a pmf"   and r :: "'a pmf" assumes "p \<succeq>[\<R>] q"   and "q \<succeq>[\<R>] r"   and "p \<succeq>[\<R>] r \<and> (r, p) \<notin> \<R>" shows "\<exists>!\<alpha>. \<alpha> \<in> {0..1} \<and> q \<approx>[\<R>] mix_pmf \<alpha> p r"
hb_broadcast: fixes m1 :: "'a"   and i :: "nat"   and m2 :: "'a" assumes "Broadcast m1 \<sqsubset>\<^sup>i Broadcast m2" shows "hb m1 m2"
alt_nil: fixes ys :: "'a list"   and zs :: "'a list" assumes "xa = []"   and "xb = ys"   and "xc = zs"   and "ys = zs" shows "thesis"
M2L_def: shows "M2L = (\<exists>\<^sub>2 \<forall>\<^sub>1 x\<^sub>0 \<in> X\<^bsub>0\<^esub>)"
ucast_mask_eq: fixes n :: "nat" shows "ucast (mask n) = mask (min LENGTH('b) n)"
finite_f_1: fixes y :: "'v" shows "f (x, y) \<noteq> \<top>"
Smith_normal_form_of_JNF_def: fixes diagonalize_JNF :: "'b mat \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b) \<Rightarrow> 'b mat \<times> 'b mat \<times> 'b mat"   and A :: "'b mat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b" assumes "Diagonalize.diagonalize diagonalize_JNF" shows "diagonalize.Smith_normal_form_of_JNF diagonalize_JNF A bezout = (let (P'', D, Q'') = diagonalize_JNF A bezout; (P', S, Q') = diagonal_to_Smith_PQ_JNF D bezout in (P' * P'', S, Q'' * Q'))"
unstream_tl_trans_True: fixes g :: "('e, 'f) generator"   and s :: "'f" shows "unstream (tl_trans g) (True, s) = unstream g s"
monotone_if_bot: fixes bound :: "'a"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c"   and bot :: "'c" assumes "\<And>x y. \<lbrakk>x \<le> y; \<not> x \<le> bound\<rbrakk> \<Longrightarrow> ord (f x) (f y)"   and "\<And>x. \<not> x \<le> bound \<Longrightarrow> ord bot (f x)"   and "ord bot bot" shows "monotone (\<le>) ord (\<lambda>x. if x \<le> bound then bot else f x)"
correctCompositionKS_neg1: fixes PQ :: "specID"   and P :: "specID"   and Q :: "specID"   and ks :: "KS" assumes "subcomponents PQ = {P, Q}"   and "correctCompositionKS PQ"   and "ks \<notin> specKeysSecrets P"   and "ks \<notin> specKeysSecrets Q" shows "ks \<notin> specKeysSecrets PQ"
S_def: shows "S = {Us. set Us \<subseteq> V \<and> local.independent K Us}"
Vdsucinter_conv: fixes d :: "nat" shows "Vd (Suc d) \<inter> E `` Vd d = Vd (Suc d)"
r_def: shows "r = min (last rs) (last (butlast rs))"
lfp_def: fixes f :: "'b set acom \<Rightarrow> 'b set acom"   and i :: "com" shows "Collecting.lfp f i = lift \<Inter> i {a \<in> {c'. strip c' = i}. f a \<le> a}"
SUP_eq_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and I :: "'b set"   and c :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> less_eq c (f i)" shows "(Sup (f ` I) = c) = (\<forall>i\<in>I. f i = c)"
axioms_1: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "qmpt M T" shows "sigma_finite_measure M"
inducts: fixes \<Delta> :: "('a, 'b) network_scheme"   and S :: "'a set" assumes "cut \<Delta> S" obtains "source \<Delta> \<in> S" and "sink \<Delta> \<notin> S"
simps_1: fixes T1 :: "'a trm"   and T2 :: "'a trm" shows "varsA (Eq T1 T2) = vars T1 \<union> vars T2"
simps_11: shows "pratt1_max \<top>\<^sub>1 \<top>\<^sub>1 = \<top>\<^sub>1"
profile_bin_scheme_axioms: shows "profile_bin_scheme valid valid f (\<lambda>x y. \<lfloor>\<lfloor>x = y\<rfloor>\<rfloor>)"
lm044: fixes a :: "('c \<times> 'd set) set"   and X :: "'c set"   and i :: "'c" assumes "a \<in> allocationsUniverse" shows "a - (X \<union> {i}) \<times> Range a \<union> ({(i, \<Union> (a `` (X \<union> {i})))} - {(i, {})}) \<in> allocationsUniverse \<and> \<Union> (Range (a - (X \<union> {i}) \<times> Range a \<union> ({(i, \<Union> (a `` (X \<union> {i})))} - {(i, {})}))) = \<Union> (Range a)"
csppa_Formal_in_notin_slice: fixes x :: "nat"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and m :: "'node"   and i :: "nat"   and S :: "'node SDG_node set" assumes "x < length fs"   and "Formal_in (m, x + i) \<notin> S" shows "csppa m S i fs ! x = Map.empty"
intro: fixes s\<^sub>0 :: "'b"   and lstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'b" assumes "df_system.is_run s\<^sub>0 lstep s" shows "run_axioms s\<^sub>0 lstep s"
sum_not_less_zero: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and s :: "'b" assumes "graph weights"   and "s < (0::'b)" shows "False"
infsetsum_Times': fixes A :: "'d set"   and B :: "'e set"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "countable A"   and "countable B"   and "(\<lambda>(x, y). f x y) abs_summable_on A \<times> B" shows "(\<Sum>\<^sub>ax\<in>A. infsetsum (f x) B) = (\<Sum>\<^sub>a(x, y)\<in>A \<times> B. f x y)"
fls_base_factor_to_fps_base_factor: fixes f :: "'a fls" shows "fls_base_factor_to_fps (fls_base_factor f) = fls_base_factor_to_fps f"
diff_add: fixes a :: "'b"   and b :: "'b" assumes "a \<le> b" shows "b - a + a = b"
powsum_split_var2: fixes x :: "'a"   and m :: "nat" shows "x\<^bsup>m\<^esup> + x\<^bsub>0\<^esub>\<^bsup>m\<^esup> = x\<^bsub>0\<^esub>\<^bsup>m\<^esup>"
pmod_def: fixes p :: "'a list"   and q :: "'a list" shows "p pmod q = (if q = [] then p else snd (long_division p q))"
v11_vcomp_vconverse': shows "\<FF>\<lparr>ArrMap\<rparr> \<circ>\<^sub>\<circ> (\<FF>\<lparr>ArrMap\<rparr>)\<inverse>\<^sub>\<circ> = vid_on (\<BB>\<lparr>Arr\<rparr>)"
distinct_17: shows "hp_InitMark \<noteq> hp_IdleMarkSweep"
ex_consume_\<phi>: fixes trn :: "('state, 'act, 'out) trans"   and vl :: "'value list" assumes "\<not> \<phi> trn" obtains vl' :: "'value list" where "consume trn vl vl'"
cbelow_on_second_below: fixes A :: "'b set"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and z :: "'b"   and rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "cbelow_on A ord z rel a b" shows "ord b z"
integral_minus_iff: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" shows "integrable M (\<lambda>x. - f x) = integrable M f"
return_def: fixes result :: "'f" shows "return result = Prog (\<lambda>h. Inr (result, h))"
le_post: fixes c :: "'b acom"   and d :: "'b acom" assumes "c \<le> d" shows "post c \<le> post d"
hbexI: fixes P :: "hf \<Rightarrow> bool"   and x :: "hf"   and A :: "hf" assumes "P x"   and "x \<^bold>\<in> A" shows "HBex A P"
le_trans_antisym: fixes x :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and y :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "le_trans x y"   and "le_trans y x" shows "equiv_trans x y"
assms_1: shows "check_single \<C> e ctn = ctm"
I: fixes l :: "(int \<times> (real \<Rightarrow> real \<times> real)) list"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "distinct l"   and "rec_join l = \<gamma>"   and "valid_chain_list l" shows "chain_subdiv_path \<gamma> (set l)"
simps_8: fixes I :: "('e, 'f, 'd) interp"   and ODE :: "('e, 'd) ODE"   and \<phi> :: "('e, 'f, 'd) formula" shows "prog_sem I (EvolveODE ODE \<phi>) = {(\<nu>, mk_v I ODE \<nu> (sol t)) |\<nu> sol t. 0 \<le> t \<and> (sol solves_ode (\<lambda>_. ODE_sem I ODE)) {0..t} {x. mk_v I ODE \<nu> x \<in> fml_sem I \<phi>} \<and> sol 0 = fst \<nu>}"
nstut_all: fixes F :: "'a \<Rightarrow> (nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "\<And>x. NSTUTINV F x" shows "NSTUTINV (\<forall>x. F x)"
simps_1: fixes r :: "'c rexp"   and s :: "'c rexp" shows "Inters (rexp.Inter r s) = zip_with_option merge_distinct (Inters r) (Inters s)"
objE: fixes a :: "'a arrow_of_spans_data" assumes "obj a" obtains "ide a" and "src a = a" and "trg a = a"
ub': shows "(\<forall>z. x < z \<and> z < ub' \<longrightarrow> 0 < poly q z) \<or> (\<forall>z. x < z \<and> z < ub' \<longrightarrow> poly q z < 0)"
inline_assoc: fixes callee1 :: "'c \<Rightarrow> 'f \<Rightarrow> ('g \<times> 'c, 'd, 'e) gpv"   and callee2 :: "'b \<Rightarrow> 'h \<Rightarrow> ('i \<times> 'b, 'f, 'g) gpv"   and gpv :: "('a, 'h, 'i) gpv"   and s2 :: "'b"   and s1 :: "'c" shows "inline callee1 (inline callee2 gpv s2) s1 = map_gpv (\<lambda>(r, s2, s1). ((r, s2), s1)) id (inline (\<lambda>(s2, s1) c2. map_gpv (\<lambda>((r, s2), s1). (r, s2, s1)) id (inline callee1 (callee2 s2 c2) s1)) gpv (s2, s1))"
zip_map_fst_snd: fixes zs :: "('c \<times> 'd) list" shows "zip (map fst zs) (map snd zs) = zs"
inj_on_DirProd_carrier: fixes G :: "('c, 'd) monoid_scheme"   and Gs :: "('c, 'e) monoid_scheme list" shows "inj_on (\<lambda>(g, gs). g # gs) (carrier (G \<times>\<times> DirProd_list Gs))"
hd_R': shows "hd R' = hd (tl P_k)"
assms_3: shows "Sig aS\<^bsub>T\<^esub> \<triangleright> Vx : B \<turnstile> d' : C"
min_less_iff_conj: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and z :: "'k"   and x :: "'k"   and y :: "'k" assumes "class.linorder less_eq less" shows "less z (ord.min less_eq x y) = (less z x \<and> less z y)"
Arity_above_up_top: fixes a :: "Arity\<^sub>\<bottom>" shows "(up\<cdot>0 \<sqsubseteq> a) = (a = up\<cdot>0)"
simps_1: fixes P :: "'g \<Rightarrow> bool" shows "takeWhile P [] = []"
simp: shows "(if i \<le> limit \<and> 2 \<le> i then if i dvd n then let n' = n div i in if i dvd n' then let n'' = n' div i in prime_product_factor_main (factor_sq * i) factor_pr (nat (root_nat_floor 3 n'')) n'' i else case sqrt_nat n' of [] \<Rightarrow> prime_product_factor_main factor_sq (factor_pr * i) (nat (root_nat_floor 3 n')) n' (Suc i) | sn # x \<Rightarrow> (factor_sq * sn, factor_pr * i) else prime_product_factor_main factor_sq factor_pr limit n (Suc i) else (factor_sq, factor_pr * n)) = (sq, p)"
neg_inf_eq_sup: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "uminus (inf a b) = sup (uminus a) (uminus b)"
c_sig: shows "c_sig \<le> 4 * j + 1"
SeqQuoteP_Mem_imp_QMem_and_Subset: fixes i :: "name"   and j :: "name"   and j' :: "name"   and i' :: "name"   and si :: "name"   and ki :: "tm"   and sj :: "name"   and kj :: "tm" assumes "atom i \<sharp> (j, j', i', si, ki, sj, kj)"   and "atom i' \<sharp> (j, j', si, ki, sj, kj)"   and "atom j \<sharp> (j', si, ki, sj, kj)"   and "atom j' \<sharp> (si, ki, sj, kj)"   and "atom si \<sharp> (ki, sj, kj)"   and "atom sj \<sharp> (ki, kj)" shows "{SeqQuoteP (Var i) (Var i') (Var si) ki, SeqQuoteP (Var j) (Var j') (Var sj) kj} \<turnstile> (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j'))) AND (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j')))"
vec_cnj_vec_cnj: fixes v :: "complex \<times> complex" shows "vec_cnj (vec_cnj v) = v"
elims: fixes x :: "rat mat"   and xa :: "rat vec"   and xb :: "rat vec"   and y :: "constraint list" assumes "mat_leqb_eqc x xa xb = y" obtains A :: "rat mat" and b :: "rat vec" and c :: "rat vec" where "x = A" and "xa = b" and "xb = c" and "y = (let lst = matrix_to_lpolies (two_block_non_interfering A A\<^sup>T) in map (\<lambda>i. LEQ (lst ! i) (b $ i)) [0..<dim_vec b] @ map (\<lambda>i. EQ (lst ! i) ((b @\<^sub>v c) $ i)) [dim_vec b..<dim_vec (b @\<^sub>v c)])"
strong_induct: fixes x1 :: "binding list"   and x2 :: "trm"   and x3 :: "ty"   and P :: "'a \<Rightarrow> binding list \<Rightarrow> trm \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'a" assumes "x1 \<turnstile> x2 : x3"   and "\<And>x T \<Gamma> y. \<lbrakk>VarB x T \<in> set \<Gamma>; \<turnstile> \<Gamma> ok\<rbrakk> \<Longrightarrow> P y \<Gamma> (Var x) T"   and "\<And>\<Gamma> t\<^sub>1 T\<^sub>1 T\<^sub>2 t\<^sub>2 y. \<lbrakk>\<Gamma> \<turnstile> t\<^sub>1 : T\<^sub>1 \<rightarrow> T\<^sub>2; \<And>z. P z \<Gamma> t\<^sub>1 (T\<^sub>1 \<rightarrow> T\<^sub>2); \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>1; \<And>z. P z \<Gamma> t\<^sub>2 T\<^sub>1\<rbrakk> \<Longrightarrow> P y \<Gamma> (t\<^sub>1 \<cdot> t\<^sub>2) T\<^sub>2"   and "\<And>x T\<^sub>1 \<Gamma> t\<^sub>2 T\<^sub>2 y. \<lbrakk>x \<sharp> y; VarB x T\<^sub>1 # \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>2; \<And>z. P z (VarB x T\<^sub>1 # \<Gamma>) t\<^sub>2 T\<^sub>2\<rbrakk> \<Longrightarrow> P y \<Gamma> (\<lambda>x:T\<^sub>1. t\<^sub>2) (T\<^sub>1 \<rightarrow> T\<^sub>2)"   and "\<And>\<Gamma> t S T y. \<lbrakk>\<Gamma> \<turnstile> t : S; \<And>z. P z \<Gamma> t S; \<Gamma>\<turnstile>S<:T\<rbrakk> \<Longrightarrow> P y \<Gamma> t T"   and "\<And>X T\<^sub>1 \<Gamma> t\<^sub>2 T\<^sub>2 y. \<lbrakk>X \<sharp> y; TVarB X T\<^sub>1 # \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>2; \<And>z. P z (TVarB X T\<^sub>1 # \<Gamma>) t\<^sub>2 T\<^sub>2\<rbrakk> \<Longrightarrow> P y \<Gamma> (\<lambda>X<:T\<^sub>1. t\<^sub>2) (\<forall>X<:T\<^sub>1. T\<^sub>2)"   and "\<And>X \<Gamma> t\<^sub>1 T\<^sub>2 T\<^sub>11 T\<^sub>12 y. \<lbrakk>X \<sharp> y; X \<sharp> (\<Gamma>, t\<^sub>1, T\<^sub>2); \<Gamma> \<turnstile> t\<^sub>1 : (\<forall>X<:T\<^sub>11. T\<^sub>12); \<And>z. P z \<Gamma> t\<^sub>1 (\<forall>X<:T\<^sub>11. T\<^sub>12); \<Gamma>\<turnstile>T\<^sub>2<:T\<^sub>11\<rbrakk> \<Longrightarrow> P y \<Gamma> (t\<^sub>1 \<cdot>\<^sub>\<tau> T\<^sub>2) (T\<^sub>12[X \<mapsto> T\<^sub>2]\<^sub>\<tau>)" shows "P z x1 x2 x3"
small_big_asymmetric: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> o[F](g)"   and "g \<in> O[F](f)" shows "\<forall>\<^sub>F x in F. f x = (0::'b)"
periodN_I: fixes w :: "'b list"   and n :: "nat" assumes "w \<noteq> \<epsilon>"   and "n \<noteq> 0"   and "w \<le>p take n w \<cdot> w" shows "periodN w n"
less_real_def: fixes x :: "real"   and y :: "real" shows "(x < y) = Real.positive (y - x)"
App_def: fixes f :: "'g \<Rightarrow> 'h" shows "App f \<equiv> f"
dist_norm: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a" assumes "class.dist_norm minus dist norm" shows "dist x y = norm (minus x y)"
ce_rel_lm_10: fixes x :: "nat"   and A :: "nat set" assumes "x \<in> A" shows "(c_fst x, c_snd x) \<in> ce_set_to_rel A"
zero_le: fixes s :: "'sum" shows "(0::'sum) \<le> s"
max_lit_exists: fixes C :: "'a literal set" assumes "finite C" shows "C \<noteq> {} \<longrightarrow> ground_clause C \<longrightarrow> (\<exists>L. L \<in> C \<and> maximal_literal L C)"
startlink: assumes "as \<noteq> qs" shows "\<exists>xss. flip_altsublist_chain ((a # as) # xss @ [rs])"
subset': shows "outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b sb \<subseteq> A"
degree_oneE: fixes p :: "'a list"   and K :: "'a set" assumes "p \<in> carrier (K [X])"   and "degree p = 1" obtains a :: "'a" and b :: "'a" where "a \<in> K" and "a \<noteq> \<zero>" and "b \<in> K" and "p = [a, b]"
assms_4: shows "io_targets M1 q1 io = {q1t}"
right_cancel: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int"   and z :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "z \<in> carrier Zp" shows "(y \<oplus> x = z \<oplus> x) = (y = z)"
map_fst_pair_pmf: fixes A :: "'a pmf"   and B :: "'b pmf" shows "map_pmf fst (pair_pmf A B) = A"
perm_swap_2: fixes a :: "name"   and b :: "name"   and x :: "'b" shows "[(a, b)] \<bullet> [(b, a)] \<bullet> x = x"
arr_char: fixes f :: "'a" shows "Left.arr f = left (trg \<mu>) f"
abc: shows "z div gcd x y * gcd x y = z \<and> a ^ 3 + b ^ 3 = (z div gcd x y) ^ 3 \<and> a * b * (z div gcd x y) \<noteq> 0"
step: fixes c' :: "'Com"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and lc :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and c'' :: "'Com"   and mds'' :: "Mode \<Rightarrow> 'Var set"   and mem'' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c', mds', mem'\<rangle> \<in> loc_reach lc"   and "\<langle>c', mds', mem'\<rangle> \<leadsto> \<langle>c'', mds'', mem''\<rangle>" shows "\<langle>c'', mds'', mem''\<rangle> \<in> loc_reach lc"
integral_pbernpoly_1_aux: fixes N :: "nat" assumes "0 < N" shows "integral {0..real N} (\<lambda>x. - complex_of_real (pbernpoly 1 x) / (complex_of_real x + s)) = approx N"
uniformly_on_def: fixes S :: "'d set"   and l :: "'d \<Rightarrow> 'e" shows "uniformly_on S l = (INF e\<in>{0<..}. principal {f. \<forall>x\<in>S. dist (f x) (l x) < e})"
simps_3: fixes G :: "prog"   and C :: "qtname"   and dynC :: "qtname"   and static_membr :: "bool" shows "G,ClassT C \<turnstile> dynC valid_lookup_cls_for static_membr = G\<turnstile>Class dynC\<preceq>Class C"
norm_mult: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.real_normed_div_algebra divide inverse one times plus zero minus uminus scaleR dist norm sgn uniformity open" shows "norm (times x y) = norm x * norm y"
sign_idempotent: fixes p :: "'b \<Rightarrow> 'b" shows "sign p * sign p = 1"
endomorph: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and T :: "'b \<Rightarrow> 'b" assumes "RModuleEnd R smult M T" shows "T ` M \<subseteq> M"
simps_2: fixes a :: "'b set"   and b :: "'b set" shows "(RES a \<le> RES b) = (a \<subseteq> b)"
cmmi: shows "m O m\<inverse> \<subseteq> f \<union> f\<inverse> \<union> e"
map_sel: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c dual" shows "\<partial>\<^sup>- (map_dual f a) = f (\<partial>\<^sup>- a)"
bin_rsplitl_def: fixes n :: "nat"   and w :: "nat \<times> int" shows "bin_rsplitl n w = bin_rsplitl_aux n (fst w) (snd w) []"
rotation_minus_pi_2_positive_y_axis: shows "moebius_pt (moebius_rotation (- pi / 2)) ` positive_y_axis = positive_x_axis"
bij_betw_cong: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and A' :: "'c set" assumes "\<And>a. a \<in> A \<Longrightarrow> f a = g a" shows "bij_betw f A A' = bij_betw g A A'"
visible_def: fixes ex :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and en :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and int :: "'b \<Rightarrow> 'c" shows "transition_system_interpreted.visible ex en int \<equiv> {a. \<exists>q. en a q \<and> int q \<noteq> int (ex a q)}"
srelated_env_unique: fixes penv :: "('a, pterm) fmap"   and senv :: "('a, sterm) fmap"   and senv' :: "('a, sterm) fmap" assumes "srelated'.P_env penv senv"   and "srelated'.P_env penv senv'" shows "senv = senv'"
arrI: fixes f :: "'a1"   and a :: "'a1"   and b :: "'a1" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>A\<^sub>1 b\<guillemotright>" shows "A1.arr f"
pred_inject_1: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" shows "pred_behaviour P (Terminates a) = P a"
assms_2: shows "finite {v. f v \<noteq> 0}"
MakeNT_cod: fixes f :: "('g, 'd, 'h, 'e, 'i, 'f) NatTrans" shows "NTCatCod f = NTCatCod (MakeNT f)"
at0: shows "eval at0 [f \<triangleright> n] \<down>= the (f 0)"
1: fixes i :: "nat" obtains j :: "nat" where "i < length (seq' j)"