hcju/isabelleps huggingface dataset:
gcd_list_greatest: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and bs :: "'b list"   and a :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "\<And>b. b \<in> set bs \<Longrightarrow> dvd.dvd times a b" shows "dvd.dvd times a (semiring_gcd.Gcd_fin one zero gcd (set bs))"
simps_9: fixes vk :: "'u \<Rightarrow> 'v fundef option"   and op :: "'e"   and \<Sigma> :: "Unboxed.type option list" shows "sp_instr vk (IOp op) \<Sigma> = sp_gen_pop_push (replicate (\<AA>\<rr>\<ii>\<tt>\<yy> op) None, None) \<Sigma>"
arg_min_list_in: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'c" assumes "xs \<noteq> []" shows "arg_min_list f xs \<in> set xs"
hurwitz_zeta_formula: fixes a :: "real"   and s :: "complex" assumes "a \<in> {0<..1}"   and "s \<noteq> 0"   and "a \<noteq> 1 \<or> s \<noteq> 1" shows "rGamma s * hurwitz_zeta a (1 - s) = complex_of_real (2 * pi) powr - s * (\<i> powr - s * perzeta a s + \<i> powr s * perzeta (- a) s)"
update_convs_1: fixes carrier' :: "'a set \<Rightarrow> 'a set"   and carrier :: "'a set"   and more :: "'b" shows "carrier_update carrier' \<lparr>carrier = carrier, \<dots> = more\<rparr> = \<lparr>carrier = carrier' carrier, \<dots> = more\<rparr>"
merge_in_state_measure: fixes x :: "nat \<Rightarrow> val"   and A :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and y :: "nat \<Rightarrow> val"   and B :: "nat set" assumes "x \<in> space (state_measure A \<Gamma>)"   and "y \<in> space (state_measure B \<Gamma>)" shows "merge A B (x, y) \<in> space (state_measure (A \<union> B) \<Gamma>)"
r2s2p: shows "r2p = s2p \<circ> r2s"
DERIV_real_root: fixes n :: "nat"   and x :: "real" assumes "0 < n"   and "0 < x" shows "(root n has_real_derivative inverse (real n * root n x ^ (n - Suc 0))) (at x)"
interior_closed_Un_empty_interior: fixes S :: "'a set"   and T :: "'a set" assumes "closed S"   and "interior T = {}" shows "interior (S \<union> T) = interior S"
OrderingSetMap_axioms: shows "OrderingSetMap (\<^bold>\<le>) (\<^bold><) (\<^bold>\<le>*) (\<^bold><*) P f"
right_cancel: fixes x :: "'a list"   and y :: "'a list"   and z :: "'a list" assumes "x \<in> carrier (K [X])"   and "y \<in> carrier (K [X])"   and "z \<in> carrier (K [X])" shows "(y \<oplus>\<^bsub>K [X]\<^esub> x = z \<oplus>\<^bsub>K [X]\<^esub> x) = (y = z)"
i: shows "i < IArray.length (matrix_to_iarray (Gram_Schmidt_column_k A k))"
fst_row_add_iterate_PA_eq_row_add: fixes a :: "'a"   and i :: "'a"   and n :: "nat"   and A :: "(('b, 'c) vec, 'a) vec"   and P :: "(('b, 'a) vec, 'a) vec"   and j :: "'c"   and b :: "'a" assumes "a \<noteq> i"   and "n < nrows A"   and "mod_type_class.to_nat a \<le> n" shows "fst (row_add_iterate_PA (P, A) n i j) $ a $ b = row_add P a i (- A $ a $ j) $ a $ b"
poly_shift_def: fixes f :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "poly_shift f n = f (Suc n)"
nn_integral_FTC_Icc: fixes f :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and F :: "real \<Rightarrow> real" assumes "f \<in> borel_measurable borel"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> (F has_real_derivative f x) (at x)"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> 0 \<le> f x"   and "a \<le> b" shows "\<integral>\<^sup>+x\<in>{a..b}. ennreal (f x)\<partial>lborel = ennreal (F b - F a)"
integral_empty: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "space M = {}" shows "integral\<^sup>L M f = (0::'c)"
D_2: shows "\<forall>D'\<in>decomps\<^sub>e\<^sub>s\<^sub>t A N \<I>. ik\<^sub>e\<^sub>s\<^sub>t D' \<subseteq> ik\<^sub>e\<^sub>s\<^sub>t D"
es_outputs_are_events_def: fixes ES :: "'c ES_rec" shows "es_outputs_are_events ES \<equiv> O\<^bsub>ES\<^esub> \<subseteq> E\<^bsub>ES\<^esub>"
ask_ask: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'm" shows "ask (\<lambda>r. ask (f r)) = ask (\<lambda>r. f r r)"
eval_Runit: fixes t :: "'c local.term" assumes "Arr t" shows "\<lbrace>\<^bold>\<r>\<^bold>[t\<^bold>]\<rbrace> = \<r>\<^sub>D[\<lbrace>Cod t\<rbrace>] \<cdot>\<^sub>D (\<lbrace>t\<rbrace> \<otimes>\<^sub>D \<I>\<^sub>D)"
at_least_at_most_upper: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<in> \<lbrace>a..b\<rbrace>" shows "x \<sqsubseteq> b"
ih\<^sub>1: fixes \<Delta> :: "binding list" assumes "\<turnstile> \<Delta> ok"   and "\<Delta> extends \<Gamma>" shows "\<Delta>\<turnstile>T\<^sub>1<:S\<^sub>1"
that: fixes u :: "'t"   and g :: "'t \<Rightarrow>\<^sub>0 'b" assumes "u \<prec>\<^sub>t lt p"   and "d (pp_of_term u) \<le> dgrad_max d"   and "component_of_term u < length fs"   and "\<not> is_syz_sig d u"   and "is_canon_rewriter rword G u g"   and "u = (punit.lt (rep_list p) - punit.lt (rep_list g)) \<oplus> lt g"   and "is_sig_red (\<prec>\<^sub>t) (=) {g} p" shows "thesis"
StoreElim1: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and os :: "Val list"   and S :: "(Var \<times> Val) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and x :: "Var" assumes "(M, l, (os, S, h), n, ll, t) \<in> Step"   and "get_ins M l = Some (store x)" shows "\<exists>v ops. os = v # ops \<and> n = Suc 0 \<and> t = (ops, S[x\<mapsto>v], h) \<and> ll = l + 1"
poly_Nil_ext: shows "poly [] = (\<lambda>x. 0::'a)"
comp_reduce: fixes f :: "'s"   and g :: "'s"   and k :: "'s"   and h :: "'s" assumes "f \<cdot>\<^sub>S g = k"   and "S.seq f g"   and "S.seq g h" shows "f \<cdot>\<^sub>S g \<cdot>\<^sub>S h = k \<cdot>\<^sub>S h"
Q_\<alpha>_def: fixes G_invar :: "'a \<Rightarrow> bool"   and G_adj :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) list"   and G_empty :: "'a"   and G_add_edge :: "'b \<times> 'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and M_empty :: "'c"   and M_update :: "'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and M_delete :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and M_lookup :: "'c \<Rightarrow> 'b \<Rightarrow> 'b option"   and M_invar :: "'c \<Rightarrow> bool"   and Q_empty :: "'d"   and Q_update :: "'b \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd"   and Q_delete :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and Q_invar :: "'d \<Rightarrow> bool"   and Q_lookup :: "'d \<Rightarrow> 'b \<Rightarrow> nat option"   and Q_is_empty :: "'d \<Rightarrow> bool"   and Q_getmin :: "'d \<Rightarrow> 'b \<times> nat"   and g :: "'a"   and G_\<alpha>w :: "'a \<Rightarrow> 'b set \<Rightarrow> nat"   and G_\<alpha>g :: "'a \<Rightarrow> 'b ugraph"   and Qi :: "'d" assumes "Prim_Impl G_invar G_adj G_empty G_add_edge M_empty M_update M_delete M_lookup M_invar Q_empty Q_update Q_delete Q_invar Q_lookup Q_is_empty Q_getmin g G_\<alpha>w G_\<alpha>g" shows "Prim_Impl.Q_\<alpha> Q_lookup Qi \<equiv> enat_of_option \<circ> Q_lookup Qi"
Diag_Diagonalize_2: fixes t :: "'c local.term" assumes "Arr t" shows "Dom \<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>Dom t\<^bold>\<rfloor>"
IterateTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and its :: "String.literal list"   and its_ty :: "'a OCL_Types.type option"   and res :: "String.literal"   and res_t :: "'a OCL_Types.type option"   and res_init :: "'a expr"   and body :: "'a expr"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E IterateCall src k its its_ty res res_t res_init body : \<tau>" obtains res_ta :: "'a OCL_Types.type" and \<tau>' :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" where "\<Gamma> \<turnstile>\<^sub>I (src, its, its_ty, expr.Let res (Some res_ta) res_init body) : (\<tau>', \<sigma>, \<tau>)" and "\<tau> \<le> res_ta" and "k = ArrowCall" and "res_t = Some res_ta"
assms_2: shows "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"
induct: fixes P :: "('a, 'b) Combinators list \<Rightarrow> bool"   and a0 :: "('a, 'b) Combinators list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x v va. P (v # va) \<Longrightarrow> P (x # v # va)" shows "P a0"
simps_2: fixes k :: "nat"   and acc :: "nat"   and n :: "nat" shows "totient_naive (Suc k) acc n = (if coprime (Suc k) n then totient_naive k (acc + 1) n else totient_naive k acc n)"
iaction_def: fixes \<Delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool"   and s :: "'c"   and vl :: "'d"   and s1 :: "'state"   and vl1 :: "'value list" shows "iaction \<Delta> s vl s1 vl1 \<equiv> \<exists>a1 ou1 s1' vl1'. let trn1 = Trans s1 a1 ou1 s1' in validTrans trn1 \<and> \<phi> trn1 \<and> consume trn1 vl1 vl1' \<and> \<not> \<gamma> trn1 \<and> \<Delta> s vl s1' vl1'"
le_infI2: fixes b :: "'b"   and x :: "'b"   and a :: "'b" assumes "b \<le> x" shows "a \<sqinter> b \<le> x"
P1: shows "Cring_Poly.compose R f g = Cring_Poly.compose R (Cring_Poly.truncate R f) g \<oplus>\<^bsub>P\<^esub> Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) g"
fbd_fbb_demorgan: fixes f :: "'c \<Rightarrow> 'd set" shows "\<partial> \<circ> bd\<^sub>\<F> f = bb\<^sub>\<F> f \<circ> \<partial>"
intro: fixes is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e" assumes "\<And>x U. \<lbrakk>x \<in> U; is_open U\<rbrakk> \<Longrightarrow> stalk.is_local is_open \<FF> \<rho> add_str mult_str zero_str one_str (topological_space.neighborhoods is_open x) x U" shows "locally_ringed_space_axioms is_open \<FF> \<rho> add_str mult_str zero_str one_str"
ccpo_Sup_upper: fixes Sup :: "'b set \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b" assumes "class.ccpo Sup less_eq less"   and "Complete_Partial_Order.chain less_eq A"   and "x \<in> A" shows "less_eq x (Sup A)"
diagonal_to_Smith_i_dvd2: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and i :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and xs :: "nat list"   and c :: "nat"   and a :: "'c"   and b :: "'b" assumes "is_bezout_ext bezout"   and "i < min (nrows A) (ncols A)"   and "\<forall>x. x \<in> set xs \<longrightarrow> x < min (nrows A) (ncols A)"   and "\<forall>a b. to_nat a \<in> insert i (set xs) \<and> to_nat a = to_nat b \<longrightarrow> A $ from_nat c $ from_nat c dvd A $ a $ b"   and "c \<notin> set xs"   and "c < min (nrows A) (ncols A)"   and "distinct xs"   and "to_nat a = to_nat b"   and "to_nat a \<in> insert i (set xs)" shows "A $ from_nat c $ from_nat c dvd diagonal_to_Smith_i xs A i bezout $ a $ b"
distinct_append: fixes xs :: "'g list"   and ys :: "'g list" shows "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
assms_2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> F"   and "y \<in> F" shows "N (x + y) \<le> ennreal C * N x + ennreal C * N y"
sup_loc_refl: fixes P :: "'b prog"   and LT :: "ty err list" shows "P \<turnstile> LT [\<le>\<^sub>\<top>] LT"
eval_fds_logderiv_zeta_1: fixes s :: "complex" assumes "1 < Re s" shows "(\<lambda>p. complex_of_real (ln (real p)) / (of_nat p powr s - 1)) abs_summable_on {p. prime p}"
Ivl_simps_1: fixes a :: "'a"   and b :: "'a" shows "lower (Ivl a b) = min a b"
interaction_bound_def: fixes "consider" :: "'e \<Rightarrow> bool" shows "interaction_bound consider \<equiv> lfp.fixp_fun (\<lambda>interaction_bound gpv. \<Squnion>generat\<in>set_spmf (the_gpv gpv). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound (c input)) else \<Squnion>input. interaction_bound (c input))"
Rep_perm_swap: fixes a :: "atom"   and b :: "atom" shows "Rep_perm (a \<rightleftharpoons> b) = (if sort_of a = sort_of b then \<lambda>c. if a = c then b else if b = c then a else c else id)"
divide_strict_right_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a < b"   and "(0::'a) < c" shows "a / c < b / c"
Int: fixes a :: "'b set"   and M :: "'b measure"   and b :: "'b set" assumes "a \<in> fmeasurable M"   and "b \<in> fmeasurable M" shows "a \<inter> b \<in> fmeasurable M"
intros_3: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> nat"   and s :: "('b, 'a) tm"   and t :: "('b, 'a) tm" assumes "kbo_std ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym"   and "vars_mset s \<subseteq># vars_mset t"   and "kbo_std.wt ground_heads_var \<delta> arity_sym wt_sym t = kbo_std.wt ground_heads_var \<delta> arity_sym wt_sym s"   and "ground_heads.gt_hd gt_sym ground_heads_var (head t) (head s)" shows "kbo_std.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym t s"
splus_prod_def: fixes t :: "'b \<times> 'c"   and u :: "'b \<times> 'c" shows "splus t u = pprod.splus (fst t) u"
A_def: shows "A \<equiv> A_abs\<cdot>ONE"
refl: fixes a :: "'a list" shows "a \<preceq> a"
R5_1: shows "pmf (sds R5) a = 1 / 2"
newton_step_closed: shows "newton_step a \<in> carrier Zp"
simps_4: fixes v :: "'b"   and va :: "('b, 'a) Term.term list"   and x :: "'a"   and E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and bs :: "('a \<times> ('b, 'a) Term.term) list" shows "unify ((Fun v va, Var x) # E) bs = (if x \<in> fv (Fun v va) then None else unify (subst_list (subst x (Fun v va)) E) ((x, Fun v va) # bs))"
rel_Grp: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'e" shows "rel_bset (BNF_Def.Grp A f) = BNF_Def.Grp {x. set_bset x \<subseteq> A} (map_bset f)"
is_sup_def: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and s :: "'a" assumes "dpo le" shows "dpo.is_sup le x y s = (le x s \<and> le y s \<and> (\<forall>z. le x z \<and> le y z \<longrightarrow> le s z))"
ZeroHeightMultiElim: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and ll :: "int"   and r :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" assumes "(M, l, s, 0, ll, r) \<in> MStep" shows "r = s \<and> ll = l"
const_map_in_set: fixes x :: "'a"   and A :: "'a set"   and v :: "'b" assumes "x \<in> A" shows "const_map v A x = Some v"
comp_simp: fixes g :: "'a"   and f :: "'a" assumes "S.seq g f" shows "S.comp g f = g \<cdot>\<^sub>B f"
is_interval_cc: fixes b :: "'b"   and a :: "'b" shows "is_interval {b..a}"
simps_1: fixes l :: "label" shows "(Old \<sqsubset>l l) = (l \<noteq> Old)"
iso_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool"   and f :: "'b" assumes "subcategory C Arr" shows "category.iso (subcategory.comp C Arr) f = (category.iso C f \<and> partial_magma.arr (subcategory.comp C Arr) f \<and> partial_magma.arr (subcategory.comp C Arr) (category.inv C f))"
f_def: shows "f = (\<lambda>n. \<sigma> n - ln (real n) - 2 * C)"
assms_1: shows "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r"
a: shows "idn \<Gamma> a nmaps x to Some (b, M)"
f2: shows "rk {A, B, A', B', P} = 3"
simps_7: fixes c :: "int"   and v :: "nat"   and Q :: "pol" shows "isnorm (PX (Pc c) (Suc v) Q) = (c \<noteq> 0 \<and> isnorm Q)"
unit_factor_snd_quot_of_fract: fixes x :: "'b fract" shows "unit_factor (denominator x) = (1::'b)"
terminal_char2: fixes t :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "terminal t = (ide t \<and> local.set t = {t})"
store_sops_le: shows "\<forall>i\<in>\<Union> (fst ` store_sops is). i < t"
abs_pos: fixes abs :: "'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.partial_abs_if abs uminus zero less_eq"   and "less_eq zero a" shows "abs a = a"
norm_conv_dist: fixes x :: "'a" shows "\<parallel>x\<parallel> = dist x (0::'a)"
Comp_eq_iso_class_memb: fixes h :: "'a"   and G :: "'a set"   and F :: "'a set" assumes "h \<in> Comp G F" shows "Comp G F = \<lbrakk>h\<rbrakk>\<^sub>B"
Lyndon_mono_fac_Lyndon_mono: fixes ps :: "'a list list"   and ws :: "'a list list" assumes "ps \<le>f ws"   and "Lyndon_mono ws" shows "Lyndon_mono ps"
zero_order_5: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "'b" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "(\<not> less zero n) = (n = zero)"
deriv_RP: shows "chain (\<leadsto>) (lmap state_of_wstate Sts)"
abs_summable_on_subset: fixes f :: "'c \<Rightarrow> 'd"   and B :: "'c set"   and A :: "'c set" assumes "f abs_summable_on B"   and "A \<subseteq> B" shows "f abs_summable_on A"
supp_Eff: fixes f :: "'b" shows "supp (Eff f) = supp f"
closed_a_consistent: fixes e :: "exp" assumes "fv e = {}" shows "a_consistent (\<bottom>, 0, []) ([], e, [])"
lemma4: fixes a :: "letter"   and ws :: "letter list list"   and zs :: "letter list list" assumes "R a ws zs"   and "ws \<noteq> []" shows "T a ws zs"
mat_of_rows_list_dim_2: fixes n :: "nat"   and vs :: "'a list list" shows "dim_row (mat_of_rows_list n vs) = length vs"
poly_eq_iff: fixes p :: "linear_poly"   and q :: "linear_poly" shows "(p = q) = (\<forall>v. Abstract_Linear_Poly.coeff p v = Abstract_Linear_Poly.coeff q v)"
SMVT_lemma4: fixes f :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> real" assumes "f integrable_on {a..b}"   and "a \<le> b"   and "\<And>x y. x \<le> y \<Longrightarrow> g x \<le> g y"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> 0 \<le> g x \<and> g x \<le> 1" obtains c :: "real" where "a \<le> c" and "c \<le> b" and "((\<lambda>x. g x *\<^sub>R f x) has_integral integral {c..b} f) {a..b}"
plus_Mp_2: fixes f :: "int poly"   and g :: "int poly" shows "r.eq_m (f + r.Mp g) (f + g)"
dvdm_add: fixes m :: "int"   and u :: "int poly"   and a :: "int poly"   and b :: "int poly" assumes "poly_mod.dvdm m u a"   and "poly_mod.dvdm m u b" shows "poly_mod.dvdm m u (a + b)"
cases: fixes a1 :: "'a list"   and a2 :: "'a list multiset"   and a3 :: "('b, 'c) label"   and a4 :: "'a list"   and a5 :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "((a1, a2), a3, a4, a5) \<in> trss fg" obtains u :: "'a" and a :: "'c" and v :: "'a" and r :: "'a list" and c :: "'a list multiset" where "a1 = u__ # r__" and "a2 = ca__" and "a3 = LBase a" and "a4 = v__ # r__" and "a5 = ca__" and "(u__, Base a, v__) \<in> edges fg"   | u :: "'a" and p :: "'b" and v :: "'a" and c :: "'a list multiset" and r :: "'a list" where "a1 = ua__ # ra__" and "a2 = cb__" and "a3 = LCall p__" and "a4 = entry fg p__ # va__ # ra__" and "a5 = cb__" and "(ua__, Call p__, va__) \<in> edges fg" and "mon fg p__ \<inter> mon_c fg cb__ = {}"   | p :: "'b" and r :: "'a list" and c :: "'a list multiset" where "a1 = return fg pa__ # rb__" and "a2 = cc__" and "a3 = LRet" and "a4 = rb__" and "a5 = cc__"   | u :: "'a" and p :: "'b" and v :: "'a" and r :: "'a list" and c :: "'a list multiset" where "a1 = u # r" and "a2 = c" and "a3 = LSpawn p" and "a4 = v # r" and "a5 = add_mset [entry fg p] c" and "(u, Spawn p, v) \<in> edges fg"
sinh_real_zero_iff: fixes x :: "real" shows "(sinh x = 0) = (x = 0)"
\<mu>LTL_simp: fixes \<phi> :: "'a ltln" shows "(\<phi> \<in> \<mu>LTL) = is_\<mu>LTL \<phi>"
lappendt_LNil: fixes tr :: "('a, 'b) tllist" shows "lappendt LNil tr = tr"
restrict_def: fixes R :: "('a \<times> 'b) set"   and X :: "'a set" shows "R || X = X \<times> Range R \<inter> R"
monom_mult_monomial: fixes c :: "'d"   and s :: "'a"   and v :: "'t"   and d :: "'d" shows "monom_mult c s (monomial d v) = monomial (c * d) (s \<oplus> v)"
cinner_gt_zero_iff: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and x :: "'c" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "(0 < cinner x x) = (x \<noteq> zero)"
s_init_def: fixes U0 :: "'c set" shows "s_init U0 \<equiv> RETURN (None, U0, \<lambda>u. if u \<in> U0 then Some [] else None)"