hcju/isabelleps huggingface dataset:
suffix_stutter_equiv: fixes f :: "nat \<Rightarrow> nat"   and \<sigma> :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "stutter_sampler f \<sigma>" shows "\<sigma> [f n..] \<approx> \<sigma> \<circ> f [n..]"
simps_2: fixes cts :: "ct list"   and cons :: "ct list" shows "consume (TopType cts) cons = (if ct_suffix cons cts then TopType (take (length cts - length cons) cts) else if ct_suffix cts cons then TopType [] else Bot)"
some_basis_subset: fixes V :: "'b set" shows "some_basis V \<subseteq> V"
assms: shows "\<D>\<^sub>\<bullet> r \<inter>\<^sub>\<circ> \<D>\<^sub>\<bullet> s = []\<^sub>\<circ>"
coeff_0_snd_quot_of_ratfps_nonzero: fixes x :: "'b ratfps" shows "constant_term (snd (quot_of_ratfps x)) \<noteq> (0::'b)"
mset_factors_def: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and F :: "'a multiset"   and p :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.mset_factors times one zero F p \<equiv> F \<noteq> {#} \<and> (\<forall>f. f \<in># F \<longrightarrow> comm_semiring_1.irreducible times one zero f) \<and> p = comm_monoid_mult.prod_mset times one F"
type_definition_def: fixes Rep :: "'c \<Rightarrow> 'd"   and Abs :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "type_definition Rep Abs A \<equiv> (\<forall>x. Rep x \<in> A) \<and> (\<forall>x. Abs (Rep x) = x) \<and> (\<forall>y. y \<in> A \<longrightarrow> Rep (Abs y) = y)"
complete2: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B"   and "B \<in> null_sets M" shows "A \<in> null_sets M"
foundation22: fixes \<tau> :: "'d state \<times> 'd state"   and X :: "'d state \<times> 'd state \<Rightarrow> 'e"   and Y :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "(\<tau> \<Turnstile> X \<triangleq> Y) = (X \<tau> = Y \<tau>)"
guard_subset_subsumption_def: fixes t1 :: "transition"   and t2 :: "transition" shows "guard_subset_subsumption t1 t2 = (Label t1 = Label t2 \<and> Arity t1 = Arity t2 \<and> set (Guards t1) \<subseteq> set (Guards t2) \<and> Outputs t1 = Outputs t2 \<and> Updates t1 = Updates t2)"
fields_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list" shows "fields P C \<equiv> THE FDTs. P \<turnstile> C has_fields FDTs"
exhaust: fixes y :: "'b extB" obtains "y = extB.Bot"   | x2 :: "'b" where "y = extB.Val x2"
fun_complete': fixes a :: "'s"   and b :: "'s"   and F :: "'s \<Rightarrow> 's" assumes "S.ide a"   and "S.ide b"   and "F \<in> S.hom S.unity a \<rightarrow> S.hom S.unity b" shows "\<exists>!f. \<guillemotleft>f : a \<rightarrow> b\<guillemotright> \<and> (\<forall>x. \<guillemotleft>x : S.unity \<rightarrow> a\<guillemotright> \<longrightarrow> f \<cdot> x = F x)"
range_eqI: fixes b :: "'c"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d" assumes "b = f x" shows "b \<in> range f"
less_le_not_le: fixes x :: "'a"   and y :: "'a" shows "(x \<prec> y) = strict (\<preceq>) x y"
oldUses_correct: fixes n :: "'node"   and g :: "'g" assumes "n \<in> set (\<alpha>n g)" shows "uses g n = var g ` uses' g n"
ret_exivl: shows "return_time {x \<in> S. s x = 0} x \<in> existence_ivl0 x"
uw\<theta>: shows "ide \<pi>.chine \<and> \<guillemotleft>\<pi>.the_\<theta> : t\<^sub>0 \<star> \<pi>.chine \<Rightarrow> s\<^sub>0\<guillemotright>"
scf_d_polTr: fixes p :: "'a"   and d :: "nat" assumes "p \<in> carrier R"   and "deg R S X p \<le> an d" shows "scf_cond R S X p d (scf_d R S X p d)"
update_convs_2: fixes refs' :: "(typerep \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> typerep \<Rightarrow> nat \<Rightarrow> nat"   and arrays :: "typerep \<Rightarrow> nat \<Rightarrow> nat list"   and refs :: "typerep \<Rightarrow> nat \<Rightarrow> nat"   and lim :: "nat"   and more :: "'a" shows "refs_update refs' \<lparr>arrays = arrays, refs = refs, lim = lim, \<dots> = more\<rparr> = \<lparr>arrays = arrays, refs = refs' refs, lim = lim, \<dots> = more\<rparr>"
case_1: fixes f1 :: "fdecl \<Rightarrow> 'a"   and f2 :: "mdecl \<Rightarrow> 'a"   and x1 :: "fdecl" shows "(case fdecl x1 of fdecl x \<Rightarrow> f1 x | mdecl x \<Rightarrow> f2 x) = f1 x1"
zer_list_prop: shows "zer_list = sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0}"
h3g1: assumes "MPoly_Type.degree p2 var = 1" shows "p2 = (\<Sum>i\<le>1. isolate_variable_sparse p2 var i * Var var ^ i)"
non_neg_mat_def: fixes A :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" shows "non_neg_mat A \<equiv> \<forall>a\<in>elements_mat_h A. (0::'d) \<le> a"
u_eq: shows "\<integral>\<^sup>+ x. u x * indicator {x \<in> space M. u x \<noteq> 0} x \<partial>M = integral\<^sup>N M u"
gf: fixes x :: "'n" assumes "x \<in> S" shows "g (f x) = x"
zero_not_basic: fixes leq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and seq_comp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and neutral :: "'c"   and deactivation :: "'c"   and basic :: "'c set" assumes "class.stimuli leq add seq_comp neutral deactivation basic" shows "deactivation \<notin> basic"
complete_digraph_are_iso: fixes n :: "nat"   and G :: "('c, 'd) pre_digraph"   and H :: "('e, 'f) pre_digraph" assumes "K\<^bsub>n\<^esub> G"   and "K\<^bsub>n\<^esub> H" shows "digraph_iso G H"
\<psi>_floor_cancel: fixes y :: "real" shows "\<psi> (real_of_int \<lfloor>y\<rfloor>) = \<psi> y"
meta_spec: fixes P :: "'e \<Rightarrow> prop"   and x :: "'e" assumes "\<And>x. PROP P x" shows "PROP P x"
suf_not_idem: fixes z :: "'b list"   and x :: "'b list"   and k :: "nat" assumes "z \<noteq> \<epsilon>"   and "z \<noteq> x" shows "x \<^sup>@ k \<cdot> z \<noteq> x"
Gcd_minors_PAQ_dvd_Gcd_minors_A: fixes A :: "'b mat"   and m :: "nat"   and n :: "nat"   and P :: "'b mat"   and Q :: "'b mat"   and k :: "nat" assumes "A \<in> carrier_mat m n"   and "P \<in> carrier_mat m m"   and "Q \<in> carrier_mat n n"   and "invertible_mat P"   and "invertible_mat Q" shows "Gcd (minors (P * A * Q) k) dvd Gcd (minors A k)"
Un_tablesD: fixes x :: "'c"   and ts :: "('d \<Rightarrow> 'c set) set"   and k :: "'d" assumes "x \<in> Un_tables ts k" shows "\<exists>t. t \<in> ts \<and> x \<in> t k"
pr_id3_3': shows "pr_conv_3_to_1 (\<lambda>x y z. z) \<in> PrimRec1'"
bound: fixes \<eta> :: "nat" shows "interaction_any_bounded_by (\<A> \<eta>) (bound \<eta>)"
small_finite_digraphs: fixes \<alpha> :: "V" shows "small {\<CC>. finite_digraph \<alpha> \<CC>}"
functor_axioms: shows "functor CC.comp (\<cdot>) local.Prod"
c'_def: shows "c' = complex_of_real (root n (cmod c)) * cis (arg c / real n)"
leq_antisym: fixes x :: "'b llist"   and y :: "'b llist" assumes "lprefix x y"   and "lprefix y x" shows "x = y"
connected_edge: fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow> w" shows "v \<rightarrow>\<^sup>* w"
e_nth_def: fixes e :: "nat"   and n :: "nat" shows "e_nth e n \<equiv> if e = 0 then 0 else e_tuple_nth (pdec1 (e - 1)) (pdec2 (e - 1)) n"
split_part: fixes P :: "bool"   and Q :: "'k \<Rightarrow> 'l \<Rightarrow> bool" shows "(\<lambda>(a, b). P \<and> Q a b) = (\<lambda>ab. P \<and> (case ab of (x, xa) \<Rightarrow> Q x xa))"
I_intF: fixes f :: "'fsym"   and al :: "univ list" assumes "wtFsym f"   and "list_all2 I_intT (arOf f) al" shows "I_intT (resOf f) (I_intF f al)"
simps_2: fixes dt :: "loc \<Rightarrow> ty option"   and b :: "bool" shows "typeof dt (Bool b) = Some (PrimT Boolean)"
assoc: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a \<sqinter> b \<sqinter> c = a \<sqinter> (b \<sqinter> c)"
RETURN_as_SPEC_refine: fixes M :: "'b nres"   and a :: "'c"   and R :: "('b \<times> 'c) set" assumes "M \<le> SPEC (\<lambda>c. (c, a) \<in> R)" shows "M \<le> \<Down> R (RETURN a)"
upChar_ordst: fixes Y :: "char list" shows "ordst Y (upChar Y)"
distinct_107: fixes x91 :: "'c"   and x92 :: "'a set"   and x93 :: "('a, 'b, 'c) com"   and x111 :: "('a, 'b, 'c) com"   and x112 :: "('a, 'b, 'c) com" shows "Guard x91 x92 x93 \<noteq> Catch x111 x112"
pprod_tau_nu: fixes x :: "'a"   and y :: "'a" shows "x \<parallel> y = \<nu> x \<parallel> \<nu> y + d (\<tau> x) \<cdot> \<nu> y + d (\<tau> y) \<cdot> \<nu> x + \<tau> x \<parallel> \<tau> y"
not_before_in: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b" assumes "x \<in> set xs"   and "y \<in> set xs" shows "(\<not> x < y in xs) = (y < x in xs \<or> x = y)"
meval_ext: fixes M :: "('b, 'c) Mapping_Str.map"   and N :: "('b, 'c) Mapping_Str.map" assumes "\<forall>x. M\<langle>x\<rangle> = N\<langle>x\<rangle>" shows "M = N"
s_eq: shows "Smith_normal_form (diagonal_to_Smith A bezout) = Smith_normal_form_upt_k (diagonal_to_Smith A bezout) (Suc (min (nrows (diagonal_to_Smith A bezout)) (ncols (diagonal_to_Smith A bezout)) - 1))"
single_valuedD: fixes r :: "('d \<times> 'e) set"   and x :: "'d"   and y :: "'e"   and z :: "'e" assumes "single_valued r"   and "(x, y) \<in> r"   and "(x, z) \<in> r" shows "y = z"
distrib_sup_le: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> y \<sqinter> z \<le> (x \<squnion> y) \<sqinter> (x \<squnion> z)"
extendID: fixes S :: "('a, 'b) sequent"   and i :: "nat"   and \<Gamma> :: "('a, 'b) form multiset"   and \<Delta> :: "('a, 'b) form multiset" assumes "extend S ( \<LM> At i  \<RM> \<Rightarrow>* \<LM> At i  \<RM>) = ( \<Gamma> \<Rightarrow>* \<Delta>)" shows "At i \<in># \<Gamma> \<and> At i \<in># \<Delta>"
uncurry0_def: fixes c :: "'e" shows "uncurry0 c \<equiv> \<lambda>_. c"
solution_of_nat_of_nat: fixes a :: "nat"   and b :: "nat" shows "solution (of_nat a, of_nat b) = solution (a, b)"
cofactor_make_mat_0k: fixes Smith_1x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat"   and Smith_2x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat \<times> 'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and k :: "nat"   and n :: "nat"   and B :: "'a mat" assumes "Smith_Impl Smith_1x2 Smith_2x2 div_op"   and "k < n"   and "k \<noteq> 0"   and "1 < n" shows "Determinant.cofactor (Smith_Impl.make_mat n k B) 0 k = - B $$ (1, 0)"
homomorphism_thm_order: fixes G :: "('a, 'b) monoid_scheme"   and h :: "'a \<Rightarrow> 'a" assumes "group_hom G G h" shows "order (G\<lparr>carrier := h ` carrier G\<rparr>) * card (kernel G (G\<lparr>carrier := h ` carrier G\<rparr>) h) = order G"
Sup_insert: 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 a :: "'a"   and A :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (insert a A) = sup a (Sup A)"
\<gamma>_diff: shows "\<gamma> C1_differentiable_on {0..1} - spikes"
take_Suc_conv_app_nth: fixes i :: "nat"   and xs :: "'c list" assumes "i < length xs" shows "take (Suc i) xs = take i xs @ [xs ! i]"
mono_neutral_cong_right: fixes T :: "'c set"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and h :: "'c \<Rightarrow> 'a" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (1::'a)"   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "prod g T = prod h S"
snoc_3: shows "\<S>\<bullet>\<C> \<turnstile> xs @ [x] : ts _> ts'"
Ord_TC_less_iff: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "Ord \<alpha>"   and "Ord \<beta>" shows "(\<beta> \<sqsubset> \<alpha>) = (\<beta> < \<alpha>)"
zero_order_5: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "'a" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "(\<not> less zero n) = (n = zero)"
tab\<^sub>0_simps_2: fixes r :: "'a" assumes "ide r" shows "is_left_adjoint (tab\<^sub>0 r)"
ODord: fixes D :: "'c Order"   and d :: "'c" assumes "Worder D"   and "d \<in> carrier D" shows "ordinal_number (Iod D (segment D d)) \<sqsubset> ordinal_number D"
Dirichlet_L_conv_hurwitz_zeta_nonprincipal: fixes s :: "complex"   and n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex" assumes "s \<noteq> 1" shows "Dirichlet_L n \<chi> s = of_nat n powr - s * (\<Sum>k = 1..n. \<chi> k * hurwitz_zeta (real k / real n) s)"
not_and_lta: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "\<not> (Tarski_neutral_dimensionless.LtA Bet Cong A B C D E F \<and> Tarski_neutral_dimensionless.LtA Bet Cong D E F A B C)"
proj_image: fixes \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" shows "proj \<phi> ` Collect P = Collect (P /// \<phi>)"
w: shows "\<integral>\<^sup>+ x. ennreal (w x) \<partial>M < \<infinity>"
zero_fun_apply: fixes x :: "'j" shows "0 x = (0::'i)"
Gp_mult_induced: fixes L :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<guillemotright> L "   and "a \<in> L"   and "b \<in> L" shows "a \<cdot>\<^bsub>\<natural>L\<^esub> b = a \<cdot> b"
simps: fixes a1 :: "letter list"   and a2 :: "letter list" shows "emb a1 a2 = ((\<exists>bs. a1 = [] \<and> a2 = bs) \<or> (\<exists>as bs b. a1 = as \<and> a2 = b # bs \<and> emb as bs) \<or> (\<exists>as bs a. a1 = a # as \<and> a2 = a # bs \<and> emb as bs))"
dg_prod_ArrD_1: fixes f :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V" assumes "f \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Arr\<rparr>" shows "vsv f"
group_def: fixes G :: "('d, 'e) monoid_scheme" shows "Group.group G \<equiv> Group.monoid G \<and> Group.group_axioms G"
content_le_lead_coeff: fixes f :: "int poly" shows "\<bar>content f\<bar> \<le> \<bar>lead_coeff f\<bar>"
inf_idem: fixes x :: "'a multiset" shows "x \<inter># x = x"
fold_pair_size_plus: fixes M :: "('c \<times> 'd) multiset"   and n :: "nat" shows "fold_mset ((+) \<circ> pair_size) 0 M + n = fold_mset ((+) \<circ> pair_size) n M"
HA_InitStatus: fixes A :: "('d, 'e, 'f) hierauto" shows "HA (InitStatus A) = A"
covering_space_lift_path_strong: fixes C :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set"   and a :: "'c"   and g :: "real \<Rightarrow> 'd" assumes "covering_space C p S"   and "a \<in> C"   and "path g"   and "path_image g \<subseteq> S"   and "pathstart g = p a" obtains h :: "real \<Rightarrow> 'c" where "path h" and "path_image h \<subseteq> C" and "pathstart h = a" and "\<And>t. t \<in> {0..1} \<Longrightarrow> p (h t) = g t"
fset_to_fset: fixes A :: "'c set" assumes "finite A" shows "fset (the_inv fset A) = A"
filterlim_at_top_imp_at_infinity: fixes f :: "'b \<Rightarrow> real"   and F :: "'b filter" assumes "filterlim f at_top F" shows "filterlim f at_infinity F"
elims_3: fixes x :: "nat"   and xa :: "cell list \<times> cell list" assumes "\<not> inv_end0 x xa" obtains n :: "nat" and l :: "cell list" and r :: "cell list" where "x = n" and "xa = (l, r)" and "\<not> (0 < n \<and> (l, r) = ([Bk], Oc \<up> n @ Bk # Oc \<up> n))"
closed_csubspace_def: fixes A :: "'d set" shows "closed_csubspace A \<equiv> csubspace A \<and> closed A"
intros: fixes S :: "'procS" shows "indRelLp (TargetTerm (\<lbrakk>S\<rbrakk>)) (SourceTerm S)"
Throw_\<tau>execI: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "\<tau>exec_move ci P t e h s s'" shows "\<tau>exec_move ci P t (throw e) h s s'"
Cons_3: shows "ccw'.sortedP 0 (P # Q # R # Rs)"
SModule_restrict_scalars: fixes S :: "('f, 'g) aezfun set" assumes "Ring1 S \<and> S \<subseteq> FG" shows "RModule S (\<cdot>) V"
inverse_aform_err_float_1: fixes p :: "nat"   and x :: "(real \<times> real pdevs) \<times> real"   and a :: "real"   and b :: "real pdevs"   and ba :: "real" assumes "inverse_aform_err p x = Some ((a, b), ba)" shows "a \<in> float"
ord_mod: fixes n :: "'a"   and k :: "'a" shows "Pocklington.ord n (k mod n) = Pocklington.ord n k"
x: fixes \<psi> :: "'c" assumes "\<psi> \<in> space_as_set INFUV" shows "(U o\<^sub>C\<^sub>L Uinv) *\<^sub>V \<psi> = \<psi>"
TauR_def: shows "\<tau> \<equiv> Early_Semantics.freeRes.Abs_freeRes Early_Semantics.freeRes_Rep.TauR_Rep"
r_append_prim: shows "prim_recfn 2 r_append"
sum_negf: fixes f :: "'r \<Rightarrow> 'q"   and A :: "'r set" shows "(\<Sum>x\<in>A. - f x) = - sum f A"
fls_divide_shift_denom_nonzero: fixes g :: "'a fls"   and f :: "'a fls"   and m :: "int" assumes "g \<noteq> 0" shows "f / fls_shift m g = fls_shift (- m) (f / g)"
relpow_commute: fixes R :: "('a \<times> 'a) set"   and n :: "nat" shows "R ; R ^^ n = R ^^ n ; R"
\<phi>_im: shows "\<phi> ` HRepHomSet (\<star>) W \<subseteq> GRepHomSet (\<star>) W"
div: shows "pdivmod_monic (Mp (T * U)) D = (Q, R)"
True_eqvt: fixes p :: "perm" shows "p \<bullet> True = True"
to_fun_mult: fixes g :: "nat \<Rightarrow> nat \<Rightarrow> int"   and f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and x :: "nat \<Rightarrow> int" assumes "g \<in> carrier (UP Zp)"   and "f \<in> carrier (UP Zp)"   and "x \<in> carrier Zp" shows "f \<otimes>\<^bsub>UP Zp\<^esub> g \<bullet> x = f \<bullet> x \<otimes> (g \<bullet> x)"
mono_neutral_cong_left: fixes T :: "'c set"   and S :: "'c set"   and h :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. h i = (1::'a)"   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "prod g S = prod h T"
insert_absorb2: fixes x :: "'g"   and A :: "'g set" shows "insert x (insert x A) = insert x A"
l_count_app: fixes x :: "'b"   and ys :: "'b list"   and zs :: "'b list" shows "l_count x (ys @ zs) = l_count x ys + l_count x zs"
any_le_inf: fixes x :: "'d DBMEntry" shows "x \<le> \<infinity>"
closed_orbitI: fixes t :: "real"   and t' :: "real"   and y :: "'a" assumes "t \<noteq> t'"   and "t \<in> existence_ivl0 y"   and "t' \<in> existence_ivl0 y"   and "flow0 y t = flow0 y t'" shows "closed_orbit y"
height_gt_0_imp_mergeSibling_right: fixes t\<^sub>2 :: "'a tree"   and w :: "nat"   and t\<^sub>1 :: "'a tree"   and a :: "'a" assumes "0 < height t\<^sub>2" shows "mergeSibling (Node w t\<^sub>1 t\<^sub>2) a = Node w (mergeSibling t\<^sub>1 a) (mergeSibling t\<^sub>2 a)"
mk_next_pow_atoms: fixes n :: "nat"   and \<phi> :: "'a ltln" shows "atoms_ltln (X\<^sub>n' n \<phi>) = atoms_ltln \<phi>"
ln_primorial'_asymp_equiv: shows "(\<lambda>k. ln (real (primorial' k))) \<sim>[sequentially] (\<lambda>x. real x * ln (real x))"
left_tree_initial': fixes G :: "('a, 'b) Graph_scheme"   and s :: "'a"   and t :: "'a" assumes "Tree G"   and "s \<in> V\<^bsub>G\<^esub>"   and "t \<in> V\<^bsub>G\<^esub>"   and "s \<noteq> t" shows "t \<notin> Tree.left_tree G s t"
x_axis_cmat_def: shows "x_axis_cmat = (0, \<i>, - \<i>, 0)"
add_anno_def: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and upd :: "'Var ModeUpd" shows "\<Gamma> \<oplus>\<^sub>\<S> upd = (\<lambda>x. Some (to_total \<Gamma> x)) |` add_anno_dom \<Gamma> \<S> upd"
has_codomain_char: fixes f :: "'a \<times> 'a" shows "(CC.codomains f \<noteq> {}) = (arr (fst f) \<and> arr (snd f))"
canonical_retraction_comp_endomorphism: fixes A :: "'a set set"   and B :: "'a set set"   and C :: "'a set"   and D :: "'a set" assumes "A \<in> \<A>"   and "B \<in> \<A>"   and "chamber C"   and "chamber D"   and "C \<in> A"   and "D \<in> B" shows "ChamberComplexEndomorphism X (canonical_retraction A C \<circ> canonical_retraction B D)"
O_assoc: fixes R :: "('a \<times> 'd) set"   and S :: "('d \<times> 'c) set"   and T :: "('c \<times> 'b) set" shows "(R O S) O T = R O S O T"
justified_action_committedD: fixes \<E> :: "('a \<times> ('b, 'a) obs_event action) llist set"   and P :: "'c prog"   and E :: "('d \<times> ('e, 'd) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat"   and J :: "nat \<Rightarrow> ('e, 'd) justifying_execution"   and a :: "nat" assumes "drf \<E> P"   and "P \<turnstile> (E, ws) weakly_justified_by J"   and "a \<in> actions E" obtains n :: "nat" and a' :: "nat" where "a = action_translation (J n) a'" and "a' \<in> committed (J n)"
halfspace_simps_4: fixes sctn :: "'c sctn" shows "sbelow_halfspace sctn = Collect (lt_halfspace sctn)"
run_bind: fixes m :: "'a + 'b"   and f :: "'b \<Rightarrow> 'a + 'c" assumes "isOK m" shows "projr (m \<bind> f) = projr (f (projr m))"
assms_4: shows "h \<turnstile> get_disconnected_nodes owner_document \<rightarrow>\<^sub>r disc_nodes"
valid_graph_repr_def: fixes l :: "('b \<times> 'b) list" shows "valid_graph_repr l = (\<forall>(u, v)\<in>set l. u \<noteq> v)"
not_angle: fixes A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'a" shows "\<turnstile> (\<not> <A>_v) = [\<not> A]_v"
prj_simps_3: shows "C.arr fgf.Prj\<^sub>0"
ext_inject: fixes Pol :: "('d \<times> 'd) set"   and Map :: "'c \<Rightarrow> 'd"   and Dom :: "'d"   and In :: "'c list"   and Out :: "'c list"   and more :: "'e"   and Pol' :: "('d \<times> 'd) set"   and Map' :: "'c \<Rightarrow> 'd"   and Dom' :: "'d"   and In' :: "'c list"   and Out' :: "'c list"   and more' :: "'e" shows "(\<lparr>Pol = Pol, Map = Map, Dom = Dom, In = In, Out = Out, \<dots> = more\<rparr> = \<lparr>Pol = Pol', Map = Map', Dom = Dom', In = In', Out = Out', \<dots> = more'\<rparr>) = (Pol = Pol' \<and> Map = Map' \<and> Dom = Dom' \<and> In = In' \<and> Out = Out' \<and> more = more')"
rapprox_rat: fixes x :: "int"   and y :: "int"   and prec :: "nat" shows "real_of_int x / real_of_int y \<le> real_of_float (rapprox_rat prec x y)"
monotone_onE: fixes X :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "monotone_on X r s f" obtains "\<And>x y. \<lbrakk>x \<in> X; y \<in> X; r x y\<rbrakk> \<Longrightarrow> s (f x) (f y)"
cases: fixes a1 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and a2 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "a1 \<Rightarrow>\<^sub>G a2" obtains c :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" and c' :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" where "a1 = c" and "a2 = c'" and "c \<Rightarrow> c'"   | \<Gamma> :: "(Vars.var \<times> exp) list" and e :: "exp" and x :: "Vars.var" and S :: "stack_elem list" where "a1 = (\<Gamma>, e, Upd x # S)" and "a2 = (\<Gamma>, e, S @ [Dummy x])"
tt_def: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.tt p = (if p = 0 then punit.min_term else ordered_powerprod_lin.Min (keys p))"
cases: fixes G :: "prog"   and a1 :: "ty"   and a2 :: "ty" assumes "G\<turnstile>a1\<succ>a2" obtains C :: "qtname" and D :: "qtname" where "a1 = Class D" and "a2 = Class C__" and "G\<turnstile>C__\<preceq>\<^sub>C D"   | C :: "qtname" and I :: "qtname" where "a1 = Class Ca__" and "a2 = Iface I__" and "\<not> G\<turnstile>Ca__\<leadsto>I__"   | T :: "ty" where "a1 = Class Object" and "a2 = T__.[]"   | I :: "qtname" and C :: "qtname" where "a1 = Iface Ia__" and "a2 = Class C"   | I :: "qtname" and J :: "qtname" and mrt :: "mhead \<Rightarrow> ty" where "a1 = Iface I" and "a2 = Iface J" and "imethds G I hidings imethds G J entails \<lambda>(md, mh) (md', mh'). G\<turnstile>mrt mh\<preceq>mrt mh'" and "(I, J) \<notin> (subint1 G)\<^sup>*"   | S :: "ref_ty" and T :: "ref_ty" where "a1 = RefT S.[]" and "a2 = RefT T.[]" and "G\<turnstile>RefT S\<succ>RefT T"
P_def: shows "P = (\<lambda>u. u \<in> carrier (residue_ring (p ^ k)) \<and> u * p ^ m = x (m + k))"
isSome_def: fixes opt :: "'b option" shows "isSome opt \<equiv> case opt of None \<Rightarrow> False | Some x \<Rightarrow> True"
this_loc_rpg: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and cf :: "nat \<times> nat \<Rightarrow> 'a" assumes "RGraph c s t cf" shows "RPreGraph c s t cf"