hcju/isabelleps huggingface dataset:
cases: fixes x :: "'b ltlr" obtains \<mu> :: "'b ltlr" and \<psi> :: "'b ltlr" where "x = \<mu>__ and\<^sub>r \<psi>'__"   | \<mu> :: "'b ltlr" and \<psi> :: "'b ltlr" where "x = \<mu>'__ U\<^sub>r \<psi>''__"   | \<mu> :: "'b ltlr" and \<psi> :: "'b ltlr" where "x = \<mu>''__ R\<^sub>r \<psi>'''__"   | \<mu> :: "'b ltlr" and \<psi> :: "'b ltlr" where "x = \<mu> or\<^sub>r \<psi>"   | "x = true\<^sub>r"   | "x = false\<^sub>r"   | v :: "'b" where "x = prop\<^sub>r(v__)"   | v :: "'b" where "x = nprop\<^sub>r(va__)"   | v :: "'b ltlr" where "x = X\<^sub>r v"
bind_option_cong: fixes x :: "'m option"   and y :: "'m option"   and f :: "'m \<Rightarrow> 'n option"   and g :: "'m \<Rightarrow> 'n option" assumes "x = y"   and "\<And>z. z \<in> set_option y \<Longrightarrow> f z = g z" shows "Option.bind x f = Option.bind y g"
rec_5: shows "Keys (set fs) = set ks \<union> set vs"
diff_diff_right: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" assumes "a \<subseteq># b" shows "c - (b - a) = c + a - b"
f: shows "f \<in> {f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B}"
list_of_eucl_nth_if: fixes x :: "'a"   and n :: "nat" shows "list_of_eucl x ! n = (if n < DIM('a) then x \<bullet> Basis_list ! n else [] ! (n - DIM('a)))"
Int_def: fixes A :: "'g set"   and B :: "'g set" shows "A \<inter> B = {x \<in> A. x \<in> B}"
lookup_uminus: fixes f :: "'c \<Rightarrow>\<^sub>0 'b"   and k :: "'c" shows "lookup (- f) k = - lookup f k"
t\<^sub>1\<^sub>2: shows "instance_of\<^sub>t t\<^sub>1 t\<^sub>2"
content_def: fixes p :: "'a poly" shows "content p = gcd_list (coeffs p)"
arr_cod_iff_arr: fixes f :: "'a1" shows "A1.arr (A1.cod f) = A1.arr f"
y: shows "\<not> is_zero_row_upt_k y (Suc k) A"
case_3: fixes f1 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f2 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f3 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f4 :: "'e list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> 'c"   and x31 :: "poscheckvariant"   and x32 :: "('d, 'e) Term.term"   and x33 :: "('d, 'e) Term.term" shows "(case \<langle>x31: x32 \<doteq> x33\<rangle>\<^sub>s\<^sub>t of send\<langle>x\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f1 x | receive\<langle>x\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f2 x | \<langle>x: xa \<doteq> xb\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f3 x xa xb | \<forall>x\<langle>\<or>\<noteq>: xa\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f4 x xa) = f3 x31 x32 x33"
8: shows "eint 2 * val_Zp (pderiv f \<bullet> \<one>) < val_Zp (f \<bullet> \<one>)"
get_ancestors_di_ptr: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list" assumes "h \<turnstile> get_ancestors_di ptr \<rightarrow>\<^sub>r ancestors" shows "ptr \<in> set ancestors"
midMultiset: fixes \<Gamma> :: "'a multiset"   and A :: "'a"   and \<Gamma>' :: "'a multiset"   and B :: "'a" assumes "\<Gamma> \<oplus> A = \<Gamma>' \<oplus> B"   and "A \<noteq> B" shows "\<exists>\<Gamma>'''. \<Gamma> = \<Gamma>''' \<oplus> B \<and> \<Gamma>' = \<Gamma>''' \<oplus> A"
zero_no_ev: fixes v :: "(real, 'n) Finite_Cartesian_Product.vec" assumes "v \<in> local.S" shows "A *v (\<chi>i. complex_of_real (v $h i)) \<noteq> 0"
removeProcs_subset': fixes ps :: "'b pState_scheme list" shows "set (fst (snd (snd (removeProcs ps)))) \<subseteq> set ps"
PiF_gen_subset: fixes I :: "'d set set"   and M :: "'d \<Rightarrow> 'e measure" shows "{Pi' J X |X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))} \<subseteq> Pow (\<Union>J\<in>I. \<Pi>' j\<in>J. space (M j))"
mat_row_last_dim_1: fixes A :: "'b mat"   and n :: "nat" shows "dim_row (mat_row_last A n) = n"
sup_flat_complete_lattice_def: fixes x :: "'a flat_complete_lattice"   and y :: "'a flat_complete_lattice" shows "sup x y = (case x of flat_complete_lattice.Value v1 \<Rightarrow> case y of flat_complete_lattice.Value v2 \<Rightarrow> if v1 = v2 then x else flat_complete_lattice.Top | flat_complete_lattice.Bot \<Rightarrow> x | flat_complete_lattice.Top \<Rightarrow> flat_complete_lattice.Top | flat_complete_lattice.Bot \<Rightarrow> y | flat_complete_lattice.Top \<Rightarrow> flat_complete_lattice.Top)"
p_p_grid: shows "p' \<in> grid (child p_p dir d) {d}"
lnull_def: fixes llist :: "'b llist" shows "lnull llist = (llist = <>)"
cmp_props_4: shows "tab\<^sub>0 (r \<star> s) \<star> rs.cmp \<cong> tab\<^sub>0 s \<star> rs.p\<^sub>0"
upper_triangular_def: fixes A :: "'b mat" shows "upper_triangular A \<equiv> \<forall>i<dim_row A. \<forall>j<i. A $$ (i, j) = (0::'b)"
line_through_K2_intersect_S_twice: fixes p :: "proj2"   and l :: "proj2_line" assumes "p \<in> hyp2"   and "proj2_incident p l" shows "\<exists>q r. q \<noteq> r \<and> q \<in> S \<and> r \<in> S \<and> proj2_incident q l \<and> proj2_incident r l"
append_eq_Cons_conv: fixes ys :: "'f list"   and zs :: "'f list"   and x :: "'f"   and xs :: "'f list" shows "(ys @ zs = x # xs) = (ys = [] \<and> zs = x # xs \<or> (\<exists>ys'. ys = x # ys' \<and> ys' @ zs = xs))"
simps_2: fixes p :: "(nat \<times> int) list"   and ds :: "nat set"   and l :: "nat" shows "gridgen p ds (Suc l) = (let sub = \<lambda>d. gridgen (child p left d) {d' \<in> ds. d' \<le> d} l @ gridgen (child p right d) {d' \<in> ds. d' \<le> d} l in p # concat (map sub (filter (\<lambda>d. d \<in> ds) [0..<length p])))"
pCons_as_add: fixes a :: "'a"   and p :: "'a poly" shows "pCons a p = [:a:] + Polynomial.monom (1::'a) 1 * p"
ordinal_mod_1: fixes x :: "ordinal" shows "x mod oSuc 0 = 0"
nn_integral_lborel_eq_integral: fixes f :: "'c \<Rightarrow> real" assumes "f \<in> borel_measurable borel"   and "\<And>x. 0 \<le> f x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel < \<infinity>" shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel = ennreal (integral UNIV f)"
sets_Collect_conj: fixes M :: "'d measure"   and P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "{x \<in> space M. P x} \<in> sets M"   and "{x \<in> space M. Q x} \<in> sets M" shows "{x \<in> space M. Q x \<and> P x} \<in> sets M"
measurable_snd: fixes M1 :: "'b measure"   and M2 :: "'c measure" shows "snd \<in> M1 \<Otimes>\<^sub>M M2 \<rightarrow>\<^sub>M M2"
D_base_finite': fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and i :: "nat"   and j :: "nat"   and k :: "nat" shows "finite {linordered_monoid.len m i j xs |xs. set xs \<subseteq> {0..k} \<and> distinct (i # j # xs)}"
fork_2: shows "s r = Some (\<sigma>, \<tau>, \<E> [Rfork e])"
complete_split_min: fixes t :: "'c tree23"   and x :: "'c"   and t' :: "'c upD" assumes "split_min t = (x, t')"   and "complete t"   and "0 < height t" shows "complete (treeD t')"
ide_cod: fixes f :: "(bool, 'arr list) concrete_category.arr" assumes "arr f" shows "ide (cod f)"
prefix_fininf_length: fixes u :: "'a list"   and w :: "'a stream"   and v :: "'a list" assumes "u \<le>\<^sub>F\<^sub>I w"   and "v \<le>\<^sub>F\<^sub>I w"   and "length u \<le> length v" shows "u \<le> v"
sots1_aux: fixes k :: "nat"   and n :: "nat" assumes "prime (4 * k + 3)"   and "odd (multiplicity (4 * k + 3) n)" shows "\<not> is_sum2sq_nat n"
interior_ball: fixes x :: "'a"   and e :: "real" shows "interior (ball x e) = ball x e"
set_prime_factorization_nat': fixes n :: "nat" shows "set (prime_factorization_nat' n) = (\<lambda>p. (p, multiplicity p n - 1)) ` prime_factors n"
intro: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "semiattractive A less_eq"   and "attractive_axioms A less_eq" shows "attractive A less_eq"
mbisim_actions_ok_bisim_no_join_21: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and ta2 :: "('l, 't, 'x2, 'm2, 'w, 'b) thread_action"   and ta1 :: "('l, 't, 'x1, 'm1, 'w, 'b) thread_action"   and t :: "'t" assumes "s1 \<approx>m s2"   and "collect_cond_actions \<lbrace>ta2\<rbrace>\<^bsub>c\<^esub> = {}"   and "ta1 \<sim>m ta2"   and "r2.actions_ok s2 t ta2" shows "r1.actions_ok s1 t ta1"
abs_summable_on_reindex_iff: fixes g :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and f :: "'b \<Rightarrow> 'c" assumes "inj_on g A" shows "((\<lambda>x. f (g x)) abs_summable_on A) = (f abs_summable_on g ` A)"
distributed_distr_eq_density: fixes M :: "'d measure"   and N :: "'e measure"   and X :: "'d \<Rightarrow> 'e"   and f :: "'e \<Rightarrow> ennreal" assumes "distributed M N X f" shows "distr M N X = density N f"
indep_in_subI: fixes \<E> :: "'a set"   and X :: "'a set"   and \<E>' :: "'a set" assumes "indep_in \<E> X" shows "indep_in \<E>' (X \<inter> \<E>')"
same_gen_iapath_by_common_arc: fixes V :: "'a set"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and w :: "'a"   and q :: "('a \<times> 'a) list"   and x :: "'a"   and e :: "'a \<times> 'a" assumes "verts3 (with_proj G) \<subseteq> V"   and "V \<subseteq> pverts G"   and "gen_iapath V u p v"   and "gen_iapath V w q x"   and "e \<in> set p"   and "e \<in> set q" shows "p = q"
andalso_thms_1: fixes y :: "tr" shows "(TT andalso y) = y"
i_cut_subset_mono_disj: fixes cut_op :: "'a set \<Rightarrow> 'a \<Rightarrow> 'a set"   and A :: "'a set"   and B :: "'a set"   and t :: "'a" assumes "cut_op = (\<down><) \<or> cut_op = (\<down>\<le>) \<or> cut_op = (\<down>>) \<or> cut_op = (\<down>\<ge>)"   and "A \<subseteq> B" shows "cut_op A t \<subseteq> cut_op B t"
hyps_1: shows "a \<noteq> 0 \<or> p \<noteq> 0"
cons2: shows "evaluate_dec ck mn env s1 d (s2, Rval new_env) \<and> evaluate_decs ck mn (extend_dec_env new_env env) s2 ds (s3, r) \<and> (\<forall>extra count count' s' r'. clock s2 = count + extra \<and> clock s' = count' + extra \<and> (s3, r) = (s', r') \<and> ck = True \<longrightarrow> evaluate_decs ck mn (extend_dec_env new_env env) (update_clock (\<lambda>_. count) s2) ds (update_clock (\<lambda>_. count') s', r'))"
False: shows "\<not> (x \<in> Gromov_boundary \<or> y \<in> Gromov_boundary \<or> z \<in> Gromov_boundary)"
markedElementsAreElements: fixes m :: "'b"   and M :: "('b \<times> bool) list" assumes "m \<in> set (markedElements M)" shows "m \<in> set (elements M)"
comm_scalar_prod: fixes v\<^sub>1 :: "'c vec"   and n :: "nat"   and v\<^sub>2 :: "'c vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n" shows "v\<^sub>1 \<bullet> v\<^sub>2 = v\<^sub>2 \<bullet> v\<^sub>1"
r_neg: fixes x :: "'a list" assumes "x \<in> carrier (K [X])" shows "x \<oplus>\<^bsub>K [X]\<^esub> \<ominus>\<^bsub>K [X]\<^esub> x = \<zero>\<^bsub>K [X]\<^esub>"
isomorphic_to_identity_is_equivalence: fixes \<phi> :: "'a \<Rightarrow> 'a" assumes "natural_isomorphism (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) B.L B.map \<phi>" shows "equivalence_functor (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) B.L"
sigma_algebra_def: fixes \<Omega> :: "'c set"   and M :: "'c set set" shows "sigma_algebra \<Omega> M \<equiv> algebra \<Omega> M \<and> sigma_algebra_axioms M"
one_divide_eq_0_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "(divide one a = zero) = (a = zero)"
tuple_simps_1: fixes f :: "'a"   and g :: "'a" assumes "span f g" shows "arr \<langle>f, g\<rangle>"
span_finite: fixes S :: "'b set" assumes "finite S" shows "local.span S = range (\<lambda>u. \<Sum>v\<in>S. u v *s v)"
simple_snd_is_var: fixes t :: "('c, 'd) Term.term"   and S :: "('c, 'd) strand_step list" assumes "send\<langle>t\<rangle>\<^sub>s\<^sub>t \<in> set S"   and "simple S" shows "\<exists>v. t = Var v"
real_powr_at_bot_neg: fixes a :: "real" assumes "0 < a"   and "a < 1" shows "filterlim ((powr) a) at_top at_bot"
rep_eq: fixes x :: "perm"   and xa :: "('e, 'f, 'g) formula" shows "Rep_formula (x \<bullet> xa) = x \<bullet> Rep_formula xa"
fmdom'_fmupd: fixes a :: "'c"   and b :: "'d"   and m :: "('c, 'd) fmap" shows "fmdom' (fmupd a b m) = insert a (fmdom' m)"
cases: fixes a :: "('i \<Rightarrow> 'a) set" assumes "generatorp a" obtains J :: "'i set" and X :: "('i \<Rightarrow> 'a) set" where "a = emb I J X" and "finite J" and "J \<subseteq> I" and "X \<in> sets (Pi\<^sub>M J M)"
apply_power_mod_order_eq: fixes f :: "'a perm"   and n :: "nat"   and a :: "'a" shows "(f ^ (n mod order f a)) \<langle>$\<rangle> a = (f ^ n) \<langle>$\<rangle> a"
inv_into_f_eq: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and x :: "'c"   and y :: "'d" assumes "inj_on f A"   and "x \<in> A"   and "f x = y" shows "inv_into A f y = x"
Can_norm_iff_Ide_Diagonalize: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Arr C t" shows "monoidal_language.Can C (free_monoidal_category.norm C t) = monoidal_language.Ide C (monoidal_language.Diagonalize C t)"
Keys_badE: fixes IK :: "msg set"   and Bad :: "agent set" assumes "Keys_bad IK Bad" obtains "range pubK \<subseteq> IK" and "IK \<inter> range LtK \<subseteq> range pubK \<union> \<Union> (keys_of ` Bad)"
psubst_subst_psubst: fixes x :: "name"   and \<theta> :: "(name \<times> trm) list"   and e' :: "trm"   and e :: "trm" assumes "x \<sharp> \<theta>" shows "\<theta><e>[x::=e'] = ((x, e') # \<theta>)<e>"
prems: fixes p :: "'a" assumes "p \<in># add_mset p__ A" shows "prime p"
subset_UNIV: fixes A :: "'d set" shows "A \<subseteq> UNIV"
depth_pair_1: fixes A :: "'b trm"   and B :: "'b trm" shows "depth A < depth (SimplyTyped.Pair A B)"
Quotient_alt_def3: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Abs :: "'d \<Rightarrow> 'e"   and Rep :: "'e \<Rightarrow> 'd"   and T :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "Quotient R Abs Rep T = ((\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> (\<forall>x y. R x y = (\<exists>z. T x z \<and> T y z)))"
parTrace_imp_MtransC: fixes cfl :: "(('test, 'atom) com \<times> 'state) list" assumes "parTrace cfl" shows "hd cfl \<rightarrow>*c last cfl"
set_category_def: fixes S :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and \<AA> :: "('b \<times> 'b) set" shows "set_category S \<AA> \<equiv> category S \<and> set_category_axioms S \<AA>"
assms_2: fixes z :: "'a" assumes "z \<in> cball a r" shows "f z \<noteq> 0"
ProcRec1: fixes \<Gamma> :: "'c \<Rightarrow> ('a, 'c, 'd) com option"   and \<Theta> :: "('a set \<times> 'c \<times> 'a set \<times> 'a set) set"   and P :: "'b \<Rightarrow> 'a set"   and p :: "'c"   and r :: "(('a \<times> 'c) \<times> 'a \<times> 'c) set"   and Q :: "'b \<Rightarrow> 'a set"   and A :: "'b \<Rightarrow> 'a set"   and F :: "'d set" assumes "\<forall>\<sigma> Z. \<Gamma>,\<Theta> \<union> (\<Union>Z. {(P Z \<inter> _Collect s (((s, p), \<sigma>, p) \<in> r), p, Q Z, A Z)})\<turnstile>\<^sub>t\<^bsub>/F\<^esub> ({\<sigma>} \<inter> P Z) the (\<Gamma> p) (Q Z),(A Z)"   and "wf r"   and "p \<in> dom \<Gamma>" shows "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P Z) Call p (Q Z),(A Z)"
bin_last_def: fixes w :: "int" shows "odd w = (w mod 2 = 1)"
rref_A: shows "reduced_row_echelon_form_upt_k A k"
change_ord_def: fixes to :: "'b nat_term_order" shows "change_ord to = (\<lambda>x. x)"
columns_image_basis: fixes A :: "((real, 'b) vec, 'a) vec" shows "columns A = (*v) A ` range (\<lambda>i. axis i 1)"
alpha_equiv_refl: fixes \<Gamma> :: "(name, name) fmap"   and t :: "nterm" assumes "fmpred (=) \<Gamma>" shows "alpha_equiv \<Gamma> t t"
filter_min_def: fixes xs :: "'a list" shows "local.filter_min xs = local.filter_min_aux xs []"
lazy_channel_recv'_def: shows "lazy_channel_recv' = (\<lambda>s q. case s of (flg, cstate.Collect m, None, as) \<Rightarrow> return_spmf (Some m, flg, cstate.Fail, None, as) | (flg, cstate.Collect m, Some (a', m'), as) \<Rightarrow> case as m' of None \<Rightarrow> spmf_of_set (rnd \<eta>) \<bind> (\<lambda>a. return_spmf (if a = a' then Some m' else None, flg \<or> a = a', cstate.Fail, None, as(m' \<mapsto> a))) | Some a \<Rightarrow> return_spmf (if a = a' then Some m' else None, flg, cstate.Fail, None, as) | (flg, _, None, as) \<Rightarrow> return_spmf (None, s) | (flg, _, Some (a', m'), as) \<Rightarrow> case as m' of None \<Rightarrow> spmf_of_set (rnd \<eta>) \<bind> (\<lambda>a. return_spmf (if a = a' then Some m' else None, flg \<or> a = a', cstate.Fail, None, as(m' \<mapsto> a))) | Some a \<Rightarrow> return_spmf (if a = a' then Some m' else None, flg, cstate.Fail, None, as))"
stake_run: fixes r :: "'action stream"   and p :: "'state" assumes "\<And>k. R.path (stake k r) p" shows "R.run r p"
v11_vrange_ex1_eq: fixes b :: "V" shows "(b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vrat_uminus) = (\<exists>!a. a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vrat_uminus \<and> -\<^sub>\<rat> a = b)"
lnths_upt_eq_ltake: fixes xs :: "'b llist"   and n :: "nat" shows "lselect {..<n} xs = ltake (enat n) xs"
HBL1_rev_prv: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and num :: "'b set"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and S :: "'c"   and P :: "'c"   and \<phi> :: "'c" assumes "Goedel_Form_Classical_HBL1_rev_prv var trm fmla Var num FvarsT substT Fvars subst eql cnj imp all exi fls prv bprv enc S P"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "prv (HBL1.PP var subst P (enc \<phi>))" shows "prv \<phi>"
l_set_disconnected_nodes_get_tag_name_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_disconnected_nodes_get_tag_name type_wf set_disconnected_nodes set_disconnected_nodes_locs get_tag_name get_tag_name_locs \<equiv> l_set_disconnected_nodes type_wf set_disconnected_nodes set_disconnected_nodes_locs \<and> l_get_tag_name type_wf get_tag_name get_tag_name_locs \<and> l_set_disconnected_nodes_get_tag_name_axioms set_disconnected_nodes_locs get_tag_name_locs"
fls_X_intpow_times_conv_shift_2: fixes f :: "'a fls"   and i :: "int" shows "f * fls_X_intpow i = fls_shift (- i) f"
f_nth_derivative_cases: fixes k :: "nat" shows "higher_differentiable_on UNIV f k \<and> (\<forall>t\<le>0. nth_derivative k f t 1 = 0) \<and> (\<exists>p. real_polynomial_function p \<and> (\<forall>t>0. nth_derivative k f t 1 = p t / t ^ (2 * k) * exp (- inverse t)))"
S2: fixes y :: "i" shows "E y \<^bold>\<leftarrow> E (cod y)"
map_vec_to_iarray_rw: fixes A :: "(('b, 'c) vec, 'd) vec"   and i :: "'d" shows "map (\<lambda>x. vec_to_iarray (A $ mod_type_class.from_nat x)) [0..<CARD('d)] ! mod_type_class.to_nat i = vec_to_iarray (A $ i)"
singleton_subset_conv: fixes a :: "'a"   and A :: "'a set" shows "({a} \<subseteq> A) = (a \<in> A)"
dsumag_pOp: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "(\<plusminus>\<^bsub>a\<Oplus>\<^bsub>I\<^esub> A\<^esub>) = prod_pOp I A"
prj'_joint_monic: fixes h :: "'a"   and a :: "'a"   and h' :: "'a" assumes "\<guillemotleft>h : a \<rightarrow>\<^sub>C H\<nu>H\<pi>\<rho>.chine\<guillemotright>"   and "\<guillemotleft>h' : a \<rightarrow>\<^sub>C H\<nu>H\<pi>\<rho>.chine\<guillemotright>"   and "\<nu>\<pi>\<rho>.Prj\<^sub>1 \<cdot> h = \<nu>\<pi>\<rho>.Prj\<^sub>1 \<cdot> h'"   and "\<nu>\<pi>\<rho>.Prj\<^sub>1\<^sub>0 \<cdot> h = \<nu>\<pi>\<rho>.Prj\<^sub>1\<^sub>0 \<cdot> h'"   and "\<nu>\<pi>\<rho>.Prj\<^sub>0\<^sub>0 \<cdot> h = \<nu>\<pi>\<rho>.Prj\<^sub>0\<^sub>0 \<cdot> h'" shows "h = h'"
card_permutations: fixes S :: "'c set"   and n :: "nat" assumes "card S = n"   and "finite S" shows "card {p. p permutes S} = fact n"
Si_alt_def: fixes t :: "real" shows "Si t = interval_lebesgue_integral lborel 0 (ereal t) sinc"
nilSimRight: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set" shows "P \<leadsto>[Rel] \<zero>"
simps_2: fixes x :: "'a"   and xs :: "[:'a:]" assumes "x \<noteq> \<bottom>"   and "xs \<noteq> \<bottom>" shows "snull\<cdot>(x :# xs) = FF"
\<tau>_prefix_conv: fixes p :: "'a prefix"   and s :: "'a trace"   and s' :: "'a trace"   and i :: "nat" assumes "prefix_of p s"   and "prefix_of p s'"   and "i < plen p" shows "\<tau> s i = \<tau> s' i"
ctrm_of_sub: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P"   and "g \<in> carrier P" shows "zcf (Cring_Poly.compose R f g) = to_fun f (zcf g)"
getRel_hom: fixes y :: "'b"   and z :: "'b"   and l :: "'c"   and G :: "('c, 'b) labeled_graph"   and f :: "'b \<Rightarrow> 'd" assumes "(y, z) \<in> getRel l G"   and "y \<in> vertices G"   and "z \<in> vertices G" shows "(f y, f z) \<in> getRel l (map_graph_fn G f)"
strongSimWeakCongSim: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<Psi> \<rhd> P \<leadsto>[Rel] Q"   and "Rel \<subseteq> Rel'" shows "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>Rel'\<guillemotright> Q"
finite_nat_Int_greaterThan_iff: fixes N :: "nat set"   and n :: "nat" shows "finite (N \<inter> {n<..}) = finite N"
preserves_trg: fixes \<mu> :: "'b" assumes "D.arr \<mu>" shows "trg\<^sub>C (F.right_map \<mu>) = E.G.map\<^sub>0 (trg\<^sub>D \<mu>)"
hom_closed: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "hom_boundary (int n) (nsphere n) (equator n) x \<in> carrier (homology_group (int n - 1) (subtopology (nsphere n) (equator n)))"
deg_pm_mono: fixes s :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 'd" assumes "s adds t" shows "deg_pm s \<le> deg_pm t"
dom_def: fixes m :: "'b \<Rightarrow> 'c option" shows "Map.dom m = {a. m a \<noteq> None}"
prems_3: fixes e :: "'b"   and w1 :: "'a"   and w2 :: "'a"   and es :: "'b list" assumes "e \<in> arcs G"   and "arc_to_ends G e = (w1, w2)"   and "P w2 es v" shows "P w1 (e # es) v"
integrable_Cauchy: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a" shows "(f integrable_on cbox a b) = (\<forall>e>0. \<exists>\<gamma>. gauge \<gamma> \<and> (\<forall>\<D>1 \<D>2. \<D>1 tagged_division_of cbox a b \<and> \<gamma> fine \<D>1 \<and> \<D>2 tagged_division_of cbox a b \<and> \<gamma> fine \<D>2 \<longrightarrow> norm ((\<Sum>(x, K)\<in>\<D>1. content K *\<^sub>R f x) - (\<Sum>(x, K)\<in>\<D>2. content K *\<^sub>R f x)) < e))"
finite_has_minimal2: fixes A :: "('a, 'b) Term.term set"   and a :: "('a, 'b) Term.term" assumes "finite A"   and "a \<in> A" shows "\<exists>m\<in>A. m \<sqsubseteq> a \<and> (\<forall>b\<in>A. b \<sqsubseteq> m \<longrightarrow> m = b)"
b1_def: shows "b1 = ball x e \<inter> {y. 0 < (y - a) \<bullet> rot (a - b)}"