hcju/isabelleps huggingface dataset:
abs_eq: fixes xa :: "'c Bonk_Schramm_extension_unfolded"   and x :: "'c Bonk_Schramm_extension_unfolded" assumes "xa \<in> extended_distance_set \<and> xa \<in> extended_distance_set \<and> extended_distance xa xa = 0"   and "x \<in> extended_distance_set \<and> x \<in> extended_distance_set \<and> extended_distance x x = 0" shows "dist (abs_Bonk_Schramm_extension xa) (abs_Bonk_Schramm_extension x) = extended_distance xa x"
typ_of1_eta_norm: fixes Ts :: "typ list"   and t :: "Core.term"   and ty :: "typ" assumes "typ_of1 Ts t = Some ty" shows "typ_of1 Ts (eta_norm t) = Some ty"
MaxR_opt_is_greatest: fixes A :: "'a set"   and x :: "'a"   and y :: "'a" assumes "MaxR_opt A = Some x"   and "y \<in> A \<inter> Field r" shows "(y, x) \<in> r"
\<pi>_smallest_prime_beyond: fixes m :: "nat" shows "\<pi> (real (smallest_prime_beyond m)) = \<pi> (real (m - 1)) + 1"
assms_4: shows "fst (cs (S t i) cid) \<noteq> fst (cs (S t j) cid)"
elims: fixes x :: "formula"   and y :: "formula" assumes "remove_neg x = y" obtains \<phi> :: "formula" where "x = Neg \<phi>" and "y = \<phi>"   | v :: "String.literal" and va :: "trm list" where "x = Pred v__ va__" and "y = Pred v__ va__"   | v :: "String.literal" and va :: "formula" and vb :: "formula" where "x = formula.Let vb__ vaa__ vba__" and "y = formula.Let vb__ vaa__ vba__"   | v :: "trm" and va :: "trm" where "x = formula.Eq vc__ vab__" and "y = formula.Eq vc__ vab__"   | v :: "trm" and va :: "trm" where "x = Less vd__ vac__" and "y = Less vd__ vac__"   | v :: "trm" and va :: "trm" where "x = LessEq ve__ vad__" and "y = LessEq ve__ vad__"   | v :: "formula" and va :: "formula" where "x = Or vf__ vae__" and "y = Or vf__ vae__"   | v :: "formula" and va :: "formula" where "x = And vg__ vaf__" and "y = And vg__ vaf__"   | v :: "formula list" where "x = Ands vh__" and "y = Ands vh__"   | v :: "formula" where "x = Exists vi__" and "y = Exists vi__"   | v :: "nat" and va :: "agg_type \<times> event_data" and vb :: "nat" and vc :: "trm" and vd :: "formula" where "x = Agg vj__ vag__ vbb__ vc vd" and "y = Agg vj__ vag__ vbb__ vc vd"   | v :: "\<I>" and va :: "formula" where "x = Prev vk__ vah__" and "y = Prev vk__ vah__"   | v :: "\<I>" and va :: "formula" where "x = Next vl__ vai__" and "y = Next vl__ vai__"   | v :: "formula" and va :: "\<I>" and vb :: "formula" where "x = Since vm__ vaj__ vbc__" and "y = Since vm__ vaj__ vbc__"   | v :: "formula" and va :: "\<I>" and vb :: "formula" where "x = Until vn__ vak__ vb" and "y = Until vn__ vak__ vb"   | v :: "\<I>" and va :: "formula Regex.regex" where "x = MatchF vo__ val__" and "y = MatchF vo__ val__"   | v :: "\<I>" and va :: "formula Regex.regex" where "x = MatchP v va" and "y = MatchP v va"
fun_chain_iff: fixes S :: "nat \<Rightarrow> 'b \<Rightarrow> 'c" shows "chain S = (\<forall>x. chain (\<lambda>i. S i x))"
terminal_char2: fixes t :: "'s" shows "terminal t = (ide t \<and> local.set t = {t})"
lim_Ln_over_n: shows "(\<lambda>n. Ln (of_nat n) / of_nat n) \<longlonglongrightarrow> 0"
negation_implies: fixes x :: "'c"   and y :: "'c" shows "- (if x \<le> y then top else y) = - - x \<sqinter> - y"
lnth_ldrop: fixes n :: "enat"   and m :: "nat"   and xs :: "'b llist" assumes "n + enat m < llength xs" shows "lnth (ldrop n xs) m = lnth xs (m + the_enat n)"
simps_1: fixes uu :: "('d \<times> 'd) set"   and uv :: "'c \<Rightarrow> 'd"   and uw :: "'d" shows "ipurge_tr uu uv uw [] = []"
set_mult_def: fixes A :: "('a \<Rightarrow> 'b) set"   and B :: "('a \<Rightarrow> 'b) set" shows "set_mult A B = {\<lambda>x. f x * g x |f g. f \<in> A \<and> g \<in> B}"
prv_imp_exi: fixes x :: "'var"   and \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "x \<notin> Fvars \<phi>"   and "prv (exi x (imp \<phi> \<chi>))" shows "prv (imp \<phi> (exi x \<chi>))"
transitive_coinduct_weak: fixes P :: "pi"   and Q :: "pi"   and X :: "(pi \<times> pi) set" assumes "(P, Q) \<in> X"   and "eqvt X"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> P \<leadsto><(Strong_Early_Bisim.bisim O X O Strong_Early_Bisim.bisim)> Q"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> (Q, P) \<in> Strong_Early_Bisim.bisim O X O Strong_Early_Bisim.bisim" shows "P \<approx> Q"
poly_neighbourhood_same_sign: fixes p :: "real poly"   and x\<^sub>0 :: "real" assumes "poly p x\<^sub>0 \<noteq> 0" shows "\<forall>\<^sub>F x in at x\<^sub>0. sgn (poly p x) = sgn (poly p x\<^sub>0)"
upto_enum_step_shift: fixes p :: "'b word"   and n :: "nat"   and m :: "nat" assumes "is_aligned p n" shows "[p , p + 2 ^ m .e. p + 2 ^ n - 1] = map ((+) p) [0 , 2 ^ m .e. 2 ^ n - 1]"
ennreal_add_bot: fixes x :: "ennreal" shows "bot + x = x"
collinear_ex_real: fixes z1 :: "complex"   and z2 :: "complex"   and z3 :: "complex" shows "collinear z1 z2 z3 = (\<exists>k. z1 = z2 \<or> z3 - z1 = cor k * (z2 - z1))"
add_rng_of_frac_zero: shows "(\<zero> |\<^bsub>rel\<^esub> \<one>) \<in> set_class_of\<^bsub>rel\<^esub>"
q: shows "q = p + monom_mult (- lookup (rep_list p) (t + punit.lt (rep_list f)) / punit.lc (rep_list f)) t f"
graph_def: fixes \<alpha> :: "'e \<Rightarrow> ('f, 'g) graph"   and invar :: "'e \<Rightarrow> bool" shows "graph \<alpha> invar \<equiv> (\<forall>g. invar g \<longrightarrow> finite (nodes (\<alpha> g))) \<and> (\<forall>g. invar g \<longrightarrow> finite (edges (\<alpha> g))) \<and> (\<forall>g. invar g \<longrightarrow> valid_graph (\<alpha> g))"
convergent_powser'_exp: shows "convergent_powser' (msllist_of_msstream exp_series_stream) exp"
r_state_at_0: fixes i :: "nat" shows "eval r_state [0, i] \<down>= prod_encode (Partial_Recursive.list_encode [i, 0], 1)"
bacc_subseteq_acc: fixes r :: "('a \<times> 'a) set"   and n :: "nat" shows "bacc r n \<subseteq> Wellfounded.acc r"
dense_sup_p: fixes x :: "'a" shows "dense (x \<squnion> - x)"
listrel_eq_len: fixes xs :: "'a list"   and ys :: "'b list"   and r :: "('a \<times> 'b) set" assumes "(xs, ys) \<in> listrel r" shows "length xs = length ys"
23: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y \<squnion> z) \<squnion> (x \<sqinter> y \<squnion> x \<sqinter> z) = x \<sqinter> (y \<squnion> z)"
subset_plus_right: fixes G :: "'a set"   and A :: "'a set" assumes "AbGroup G" shows "A \<subseteq> G + A"
H_Ph_H: shows "exexH_k (n - 1) * tensor_P mat_Ph (1\<^sub>m K) * exexH_k (n - 1) = 2 \<cdot>\<^sub>m tensor_P proj_psi (1\<^sub>m K) - 1\<^sub>m d"
statClosed: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and \<Psi> :: "'d"   and \<phi> :: "'e"   and p :: "(name \<times> name) list" assumes "assertionAux SCompose SImp SBottom SChanEq"   and "SImp \<Psi> \<phi>" shows "SImp (p \<bullet> \<Psi>) (p \<bullet> \<phi>)"
subprob_density_distr_real_inverse: fixes M :: "real measure"   and f :: "real \<Rightarrow> ennreal" assumes "has_subprob_density M lborel f" shows "has_subprob_density (distr M borel inverse) lborel (\<lambda>x. f (inverse x) * ennreal (inverse (x * x)))"
apply_inverse_self_in_orbit: fixes f :: "'a perm"   and a :: "'a" shows "inverse f \<langle>$\<rangle> a \<in> orbit f a"
simps: fixes St :: "('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> nat" shows "deterministic_RP St = (if is_final_dstate St then let (uu_, uu_, Q, uu_) = St in Some (map fst Q) else deterministic_RP (deterministic_RP_step St))"
execs_drop_suffix: fixes f :: "nat \<Rightarrow> com list \<times> state"   and c :: "com"   and cs :: "com list"   and s :: "state"   and k :: "nat"   and p :: "nat \<Rightarrow> com list" assumes "f 0 = (c # cs, s)"   and "\<forall>i. f i \<rightarrow> f (Suc i)"   and "\<forall>i<k. p i \<noteq> [] \<and> fst (f i) = p i @ cs"   and "fst (f k) = p k @ cs" shows "([c], s) \<rightarrow>\<^sup>* (p k, snd (f k))"
neq_commute: fixes a :: "'c"   and b :: "'c" shows "(a \<noteq> b) = (b \<noteq> a)"
basic_identity\<^sub>1_def: shows "(\<^bold>=\<^sub>1) \<equiv> \<lambda>F G. \<^bold>\<box>(\<^bold>\<forall>x. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<lbrace>x\<^sup>P,G\<rbrace>)"
power_smallo_exponential: fixes b :: "real"   and n :: "real" assumes "1 < b" shows "(\<lambda>x. x powr n) \<in> o((powr) b)"
gga_edges_list_it_def: fixes G :: "'G" shows "gga_edges_list_it G \<equiv> set_iterator_product (nodes_it G) (succ_it G)"
min_cost_le_walk_cost: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c"   and c :: "'d \<Rightarrow> real" assumes "wf_digraph G"   and "pre_digraph.awalk G u p v" shows "wf_digraph.\<mu> G c u v \<le> ereal (wf_digraph.awalk_cost c p)"
prems_4: shows "lsum f (take n1 (al @ [a])) + j1 = lsum f (take n2 (al @ [a])) + j2"
digit_encode_drop: fixes n :: "nat"   and ds :: "nat list"   and a :: "nat" shows "drop n (digit_encode ds a) = digit_encode (drop n ds) (a div prod_list (take n ds))"
mat_empty_column_length: fixes nr :: "nat"   and nc :: "nat"   and M :: "'a list list" assumes "mat nr nc M"   and "M = []" shows "nc = 0"
uniformity_blinop_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
vconverse_atI: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "r\<lparr>a\<rparr> = b" shows "\<langle>b, a\<rangle> \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<circ>"
one_power2: shows "(1::'a)\<^sup>2 = (1::'a)"
assms_3: fixes k :: "'u"   and it :: "'u set"   and \<sigma> :: "'\<sigma>" assumes "k \<in> it"   and "\<forall>j\<in>it. j \<le> k"   and "\<forall>j\<in>\<alpha> m - it. k \<le> j"   and "it \<subseteq> \<alpha> m"   and "I it \<sigma>" shows "I (it - {k}) (f k \<sigma>)"
1: fixes u :: "'x \<Rightarrow>\<^sub>0 nat" assumes "u \<in> keys g" shows "deg_pm u = deg_pm (lpp g)"
strip_comb_simps_1: fixes t :: "'a"   and u :: "'a" shows "strip_comb (app t u) = (let (f, args) = strip_comb t in (f, args @ [u]))"
sel_2: fixes z :: "gauss_int" shows "Im (gauss2complex z) = real_of_int (ImZ z)"
carrier_Fexp: fixes Texp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var ttree"   and Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and e :: "exp"   and a :: "Arity" assumes "TTreeAnalysisCarrier Texp Aexp" shows "carrier (Texp e\<cdot>a) = edom (Aexp e\<cdot>a)"
UINF_ind_Healthy: fixes P :: "'b \<Rightarrow> 'a hrel" assumes "\<And>i. P i is \<H>" shows "(\<Sqinter> i \<bullet> P i) is \<H>"
list_to_lpoly_def: fixes cs :: "rat list" shows "list_to_lpoly cs = sum_list (map2 (\<lambda>x y. lp_monom y x) [0..<length cs] cs)"
2: shows "cone (monomial (1::'a) t', V) \<subseteq> normal_form F ` P[X]"
dc2c_def: fixes \<chi> :: "nat \<Rightarrow> complex" shows "dc2c \<chi> = (\<lambda>x. if x < n then \<chi> x else 0)"
sqninv: fixes dests :: "nat \<Rightarrow> nat option"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and ip :: "nat"   and rsn :: "nat" assumes "\<forall>ip\<in>dom dests. ip \<in> kD rt \<and> sqn rt ip \<le> the (dests ip)"   and "dests ip = Some rsn" shows "sqn (invalidate rt dests) ip \<le> rsn"
Max_in: fixes A :: "'c DBMEntry set" assumes "finite A"   and "A \<noteq> {}" shows "linordered_monoid.Max A \<in> A"
simps_6: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and p :: "float poly"   and t :: "taylor_model" shows "eval_poly_at_tm prec ord I a (Neg p) t = tm_neg (eval_poly_at_tm prec ord I a p t)"
rel_carrier_Eps_in_1: fixes X :: "('a set \<times> 'b) set" assumes "X \<in> stk1.carrier_direct_lim" shows "(SOME x. x \<in> X) \<in> X"
vp1: shows "cls_val_process Mkt pf1 (Suc n) w = (\<Sum>x\<in>support_set pf1 \<union> support_set pf2. prices Mkt x (Suc n) w * pf1 x (Suc n) w)"
wordinterval_lowest_element_set_eq: fixes r :: "'b wordinterval"   and x :: "'b word" assumes "\<not> wordinterval_empty r" shows "(wordinterval_lowest_element r = Some x) = is_lowest_element x (wordinterval_to_set r)"
field_def2: fixes P :: "'b prog"   and C :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list" assumes "P \<turnstile> C sees F,b:T in D" shows "field P C F = (D, b, T)"
simps_2: fixes cs :: "(Term_Class.term \<times> sterm) list"   and \<Gamma> :: "(name, value) fmap" shows "vwelldefined' (Vabs cs \<Gamma>) = (pred_fmap id (fmmap vwelldefined' \<Gamma>) \<and> list_all (\<lambda>(pat, t). consts t |\<subseteq>| fmdom \<Gamma> |\<union>| C) cs \<and> fdisjnt C (fmdom \<Gamma>))"
comp_in_homI: fixes f :: "'d"   and a :: "'d"   and b :: "'d"   and g :: "'d"   and c :: "'d" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>"   and "\<guillemotleft>g : b \<Rightarrow>\<^sub>D c\<guillemotright>" shows "\<guillemotleft>g \<cdot>\<^sub>D f : a \<Rightarrow>\<^sub>D c\<guillemotright>"
Diagonalize_Cod: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Arr C t" shows "monoidal_language.Diagonalize C (monoidal_language.Cod C t) = monoidal_language.Cod C (monoidal_language.Diagonalize C t)"
sq_mtx_norm_1: shows "\<parallel>1\<parallel> = 1"
tag_name_eq2_h2: fixes ptr' :: "(_) element_ptr" assumes "ptr' \<noteq> new_element_ptr" shows "|h2 \<turnstile> get_tag_name ptr'|\<^sub>r = |h3 \<turnstile> get_tag_name ptr'|\<^sub>r"
no_recording_cs_if_not_snapshotted: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat"   and p :: "'d"   and cid :: "nat"   and q :: "'d" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "\<not> ps (distributed_system.s channel trans send recv init t i) p \<noteq> None"   and "channel cid = Some (q, p)" shows "cs (distributed_system.s channel trans send recv init t i) cid = cs init cid"
assms_3: shows "c \<in> {real_of_float cl..real_of_float cu}"
ih: fixes n' :: "nat"   and \<pi>' :: "nat \<Rightarrow> 'n" assumes "n' < n"   and "is_path \<pi>'"   and "\<pi>' 0 = x"   and "\<pi>' n' = local.return" shows "thesis"
eps_fv_cong: fixes r :: "'b Regex.regex"   and test :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and test' :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'a" assumes "\<And>i x. x \<in> regex.atms r \<Longrightarrow> test i x = test' i x" shows "Regex.eps test i r = Regex.eps test' i r"
nullhomotopic_from_sphere_extension: fixes a :: "'b"   and r :: "real"   and S :: "'a set"   and f :: "'b \<Rightarrow> 'a" shows "(\<exists>c. homotopic_with_canon (\<lambda>x. True) (sphere a r) S f (\<lambda>x. c)) = (\<exists>g. continuous_on (cball a r) g \<and> g ` cball a r \<subseteq> S \<and> (\<forall>x\<in>sphere a r. g x = f x))"
eval_trm_cong: fixes t :: "'a MFOTL.trm"   and v :: "'a list"   and v' :: "'a list" assumes "\<forall>x\<in>fv_trm t. v ! x = v' ! x" shows "MFOTL.eval_trm v t = MFOTL.eval_trm v' t"
initial_ps_def: fixes G :: "diagram" shows "initial_ps G \<equiv> [ initials G |=> Top ]"
simps_1: fixes ParamUses :: "'h \<Rightarrow> 'g set list"   and n :: "'h" shows "lift_ParamUses ParamUses (Node n) = ParamUses n"
quote_all_PfP_ssubst: fixes \<beta> :: "fm"   and V :: "name set" assumes "{} \<turnstile> \<beta>"   and "V \<subseteq> Vs"   and "supp \<beta> \<subseteq> atom ` Vs" shows "quote_all p V \<turnstile> PfP (ssubst \<lfloor>\<beta>\<rfloor>V V F)"
inv_f_f: fixes x :: "'b" shows "inv (inv (inv (inv hom))) (inv (inv (inv hom)) x) = x"
higher_def: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and t :: "'c" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "ordered_term.higher ord_term p t = except p {s. ord_term s t}"
preserves_hom: fixes f :: "'a \<times> 'a \<times> 'a"   and a :: "'a \<times> 'a \<times> 'a"   and b :: "'a \<times> 'a \<times> 'a" assumes "VVV.in_hom f a b" shows "\<guillemotleft>\<a>[fst f, fst (snd f), snd (snd f)] : HoHV a \<Rightarrow> HoVH b\<guillemotright>"
Residue_theorem: fixes s :: "complex set"   and pts :: "complex set"   and f :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex" assumes "open s"   and "connected s"   and "finite pts"   and "f holomorphic_on s - pts"   and "valid_path g"   and "pathfinish g = pathstart g"   and "path_image g \<subseteq> s - pts"   and "\<forall>z. z \<notin> s \<longrightarrow> winding_number g z = 0" shows "contour_integral g f = complex_of_real (2 * pi) * \<i> * (\<Sum>p\<in>pts. winding_number g p * residue f p)"
ln_sum_bigo_ln: fixes c :: "real" shows "(\<lambda>x. ln (x + c)) \<in> O(ln)"
assms_1: fixes x :: "'a" assumes "x \<in> Mapping.keys m" shows "Mapping.lookup_default 0 m x = pmf p x"
E_def: shows "E = (SIGMA s:UNIV. \<Union> (set_pmf ` K s))"
rhs: shows "init_segment S T \<and> n \<in> T"
tm_shift_eq_step: fixes s :: "nat"   and l :: "cell list"   and r :: "cell list"   and A :: "(action \<times> nat) list"   and s' :: "nat"   and l' :: "cell list"   and r' :: "cell list"   and off :: "nat" assumes "step0 (s, l, r) A = (s', l', r')"   and "s' \<noteq> 0" shows "step (s + off, l, r) (shift A off, off) = (s' + off, l', r')"
\<beta>i: fixes i :: "nat" assumes "4 \<le> i"   and "i \<le> Suc k" shows "\<beta> i = (- 1) ^ (\<delta> (i - 2) + 1) * f (i - 2) * h (i - 2) ^ \<delta> (i - 2)"
assms: shows "(\<lambda>n. to_Gromov_completion (v n)) \<longlonglongrightarrow> eta"
Cons_in_subseqsD: fixes y :: "'c"   and ys :: "'c list"   and xs :: "'c list" assumes "y # ys \<in> set (subseqs xs)" shows "ys \<in> set (subseqs xs)"
weakly_logically_equivalent_def: fixes \<tau> :: "'b"   and satisfies :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and transition :: "'c \<Rightarrow> ('b, 'c) residual \<Rightarrow> bool"   and P :: "'c"   and Q :: "'c" assumes "indexed_weak_nominal_ts TYPE('a) \<tau> satisfies transition" shows "indexed_weak_nominal_ts.weakly_logically_equivalent TYPE('a) \<tau> satisfies transition P Q \<equiv> \<forall>x. indexed_weak_nominal_ts.weak_formula \<tau> x \<longrightarrow> nominal_ts.valid satisfies transition P x = nominal_ts.valid satisfies transition Q x"
assms_5: fixes I :: "'a set"   and a :: "'a"   and b :: "'a"   and J :: "'a set"   and c :: "'a"   and d :: "'a" assumes "Q I a b"   and "Q J c d"   and "I \<subseteq> A"   and "J \<subseteq> A"   and "[[a b c d]] \<or> [[a c b d]] \<or> [[a c d b]]" shows "P I J"
prob_space_uniform_measure: fixes M :: "'b measure"   and A :: "'b set" assumes "emeasure M A \<noteq> 0"   and "emeasure M A \<noteq> \<infinity>" shows "prob_space (uniform_measure M A)"
ennreal_le_iff2: fixes x :: "real"   and y :: "real" shows "(ennreal x \<le> ennreal y) = (0 \<le> y \<and> x \<le> y \<or> x \<le> 0 \<and> y \<le> 0)"
simply_connected_sphere_gen: fixes S :: "'a set" assumes "convex S"   and "bounded S"   and "3 \<le> aff_dim S" shows "simply_connected (rel_frontier S)"
Standard_starfun: fixes x :: "'b star"   and f :: "'b \<Rightarrow> 'c" assumes "x \<in> Standard" shows "(*f* f) x \<in> Standard"
distinct_mset_set_mset_ident: fixes M :: "'a multiset" assumes "distinct_mset M" shows "mset_set (set_mset M) = M"
nonzeros_msum_pdevs_raw: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" shows "{i. msum_pdevs_raw n f g i \<noteq> (0::'b)} = {0..<n} \<inter> {i. f i \<noteq> (0::'b)} \<union> (+) n ` {i. g i \<noteq> (0::'b)}"
prodag_carrier: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "carrier (a\<Pi>\<^bsub>I\<^esub> A) = carr_prodag I A"
less_asym: fixes y :: "'a list"   and x :: "'a list"   and P :: "bool" assumes "y <lex x"   and "\<not> P \<Longrightarrow> x <lex y" shows "P"
sign_inverse: fixes p :: "'b \<Rightarrow> 'b" assumes "permutation p" shows "sign (inv p) = sign p"
prems_7: shows "leftderives [\<SS>] (terminals (take r p) @ [N] @ \<gamma>)"
frp_invar_add_def: fixes \<delta>1 :: "('a, 'c) ta_rule set"   and \<delta>2 :: "('b, 'c) ta_rule set" shows "frp_invar_add \<delta>1 \<delta>2 \<equiv> {(Q, W, \<delta>d). \<delta>d = {r \<in> \<delta>_prod \<delta>1 \<delta>2. lhs r \<in> Q - set W}}"
m_imp_m: fixes n :: "nat" assumes "length (f n) \<le> m" shows "length (f (Suc n)) \<le> m"
Goal: fixes \<Psi>' :: "'b"   and Pa :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P'a :: "('a, 'b, 'c) psi"   and xa :: "name" assumes "\<Psi>' \<rhd> Pa \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'a"   and "xa \<sharp> \<Psi>'"   and "xa \<sharp> M"   and "xa \<sharp> xvec"   and "xa \<sharp> N" shows "\<Psi>' \<rhd> \<lparr>\<nu>xa\<rparr>Pa \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> \<lparr>\<nu>xa\<rparr>P'a"
chine_assoc'_props_3: shows "doms.Prj\<^sub>0\<^sub>1 \<cdot> doms.chine_assoc' = doms.Prj\<^sub>1\<^sub>0"
some_third_chamberD_simplex: fixes C :: "'a set"   and z :: "'a set"   and D :: "'a set" assumes "chamber C"   and "z \<lhd> C" shows "some_third_chamber C D z \<in> X"
exhaust: fixes y :: "nat" obtains "y = 0"   | nat :: "nat" where "y = Suc nat"
assms: shows "(augment_list \<sigma> ^^ n) p = []"
gcd_Esigma_mult: fixes m :: "nat"   and n :: "nat" assumes "gcd m n = 1" shows "Esigma (m * n) = Esigma m * Esigma n"
map_graph_minus: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" shows "map_graph (f -- g) = map_graph f - map_graph g"
pathstart_compose: fixes f :: "'c \<Rightarrow> 'b"   and p :: "real \<Rightarrow> 'c" shows "pathstart (f \<circ> p) = f (pathstart p)"
indicator_scaleR_eq_if: fixes S :: "'e set"   and x :: "'e"   and f :: "'e \<Rightarrow> 'd" shows "indicat_real S x *\<^sub>R f x = (if x \<in> S then f x else (0::'d))"
GEM_def: shows "GEM \<equiv> GM \<and> EM \<and> GEM_axioms"
len_gt1: shows "1 < length qs \<longrightarrow> p \<noteq> 0 \<and> 0 < length qs \<longrightarrow> satisfies_properties_R p qs (get_subsets_R (calculate_data_R p qs)) (get_signs_R (calculate_data_R p qs)) (get_matrix_R (calculate_data_R p qs))"
obtain_comp_block_orig: fixes bl1 :: "'a set" assumes "bl1 \<in># des.complement_blocks" obtains bl2 :: "'a set" where "bl2 \<in># \<B>\<^sup>C" and "bl1 = bl2\<^sup>c"
leD: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and y :: "'g"   and x :: "'g" assumes "class.order less_eq less"   and "less_eq y x" shows "\<not> less x y"
indRelTEQ_impl_TRel_preserves_barbs: fixes TRel :: "('procT \<times> 'procT) set" assumes "rel_preserves_binary_pred (indRelTEQ TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))" shows "rel_preserves_binary_pred TRel (\<lambda>a. hasBarb a TWB)"
assms_2: fixes b :: "'a" assumes "ide b" shows "\<exists>!f. \<guillemotleft>f : a \<rightarrow> b\<guillemotright>"
scale_eq_0_iff: fixes a :: "'b"   and x :: "'b poly" shows "(smult a x = 0) = (a = (0::'b) \<or> x = 0)"
le_fin_prefix_transp: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list"   and w\<^sub>3 :: "'item list" assumes "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2"   and "w\<^sub>2 \<le> w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>3"
dfs_match: fixes xs :: "('c, 'd) bintree option list"   and f :: "('c, 'd) bintree \<Rightarrow> 'e"   and t :: "('c, 'd) bintree"   and a :: "'c" assumes "distinct (prios xs)"   and "distinct (dfs f xs)"   and "Some t \<in> set xs"   and "priority t = a" shows "set (dfs f (map (match a) xs)) = set (dfs f xs) - set (bt_dfs f t)"
strict_boundedE: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a < inf b c" obtains "a < b" and "a < c"