hcju/isabelleps huggingface dataset:
update_accessor_congruence_unfoldE: fixes upd :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and ac :: "'b \<Rightarrow> 'a"   and r :: "'b"   and r' :: "'b"   and v' :: "'a"   and f :: "'a \<Rightarrow> 'a"   and f' :: "'a \<Rightarrow> 'a" assumes "iso_tuple_update_accessor_cong_assist upd ac"   and "r = r'"   and "ac r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "upd f r = upd f' r'"
sv_round_refines: fixes r :: "nat"   and v :: "val"   and S :: "process set"   and r_decisions :: "process \<Rightarrow> val option" shows "{Id} v_round r (const_map v S) r_decisions, sv_round r S v r_decisions {> Id}"
enat_le_imp_minus_le: fixes k :: "enat"   and m :: "enat"   and n :: "enat" assumes "k \<le> m" shows "k - n \<le> m"
c_process_process: fixes step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g" shows "(c_failures step out s\<^sub>0, {}) \<in> process_set"
assms: assumes "x \<noteq> v" shows "e1 x \<sqsubseteq> e2 x"
abs_eq: fixes xa :: "nat"   and x :: "nat \<Rightarrow> 'a" shows "vec xa x = Abs_vec (xa, mk_vec xa x)"
isometry_not_elliptic_has_attracting_fixed_point: fixes f :: "'b \<Rightarrow> 'b" assumes "isometry f"   and "\<not> elliptic_isometry f" shows "\<exists>xi\<in>Gromov_boundary. Gromov_extension f xi = xi \<and> additive_strength f xi \<le> 0"
get_set_eq: fixes r :: "'a ref"   and x :: "'a"   and h :: "heap" shows "Ref.get (Ref.set r x h) r = x"
IS_PRES_EQ_def: fixes R :: "('a \<times> 'b) set" shows "IS_PRES_EQ R \<equiv> ((=), (=)) \<in> R \<rightarrow> R \<rightarrow> bool_rel"
image_mset_ordering: fixes M1 :: "'b multiset"   and f1 :: "'c \<Rightarrow> 'b"   and L :: "'c multiset"   and M2 :: "'b multiset"   and f2 :: "'c \<Rightarrow> 'b"   and r :: "('b \<times> 'b) set" assumes "M1 = image_mset f1 L"   and "M2 = image_mset f2 L"   and "\<forall>u. u \<in># L \<longrightarrow> (f1 u, f2 u) \<in> r \<or> f1 u = f2 u"   and "\<exists>u. u \<in># L \<and> (f1 u, f2 u) \<in> r"   and "irrefl r" shows "(M1, M2) \<in> mult r"
non_empty_cycle_root_rtc_start_points: fixes r :: "'a"   and x :: "'a" assumes "non_empty_cycle_root r x" shows "x\<^sup>\<star> ; r = x ; 1"
intv_CL_glb: fixes A :: "'a set"   and a :: "'a"   and b :: "'a"   and S :: "'a set" assumes "\<And>S. S \<subseteq> A \<Longrightarrow> a \<in> A"   and "\<And>S. S \<subseteq> A \<Longrightarrow> b \<in> A"   and "\<And>S. S \<subseteq> A \<Longrightarrow> interval r a b \<noteq> {}"   and "\<And>S. S \<subseteq> A \<Longrightarrow> S \<subseteq> interval r a b"   and "S \<subseteq> A"   and "A = pset \<lparr>pset = interval r a b, order = induced (interval r a b) r\<rparr>" shows "\<exists>G. isGlb S \<lparr>pset = interval r a b, order = induced (interval r a b) r\<rparr> G"
assms: shows "apply_cltn2_line l C = apply_cltn2_line m C"
wit: shows "wit = map (\<mu> i) [0..<i]"
remdups_adj_set: fixes xs :: "'a list" shows "set (remdups_adj xs) = set xs"
rTau: assumes "\<Psi> \<rhd> P \<sim> P'" shows "Prop pTau (\<tau>) P'"
\<delta>ssI: fixes f :: "'L"   and ss :: "'Q set list" assumes "A f = Some (length ss)"   and "ss \<in> lists {s. s \<subseteq> Q}" shows "\<delta>ss_lhs f ss \<rightarrow> f ss \<in> \<delta>ss"
assms_3: shows "all_same_arities (Fubx_get F) (Fubx_get (Fubx_add F f fd))"
qFreshAbs_preserves_alphaAbs: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and A' :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and zs :: "'varSort"   and z :: "'var" assumes "qGoodAbs A \<or> qGoodAbs A'"   and "A $= A'" shows "qFreshAbs zs z A = qFreshAbs zs z A'"
ZObis_coind2: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set" assumes "theta \<subseteq> ZOretr (theta \<union> ZObis)"   and "theta\<inverse> \<subseteq> ZOretr (theta\<inverse> \<union> ZObis)" shows "theta \<subseteq> ZObis"
LHCut: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset" assumes "F, \<Gamma> \<Rightarrow>\<^sub>n"   and "\<^bold>\<not> F, \<Gamma> \<Rightarrow>\<^sub>n" shows "\<Gamma> \<Rightarrow>\<^sub>n"
uniformity_fbint_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
deriv_of_real: fixes x :: "real"   and A :: "real set" assumes "at x within A \<noteq> bot" shows "vector_derivative of_real (at x within A) = (1::'a)"
id_2: fixes I :: "'b list"   and s :: "('b, 'c) state" shows "\<nabla> (set_unsat I s) = \<nabla> s"
bounded: shows "\<forall>(q, \<tau>')\<in>set (step (hd b) (a ! hd b)). q < n"
Opair_in_games_lfp: fixes l :: "ZF"   and r :: "ZF" assumes "explode l \<subseteq> games_lfp"   and "explode r \<subseteq> games_lfp" shows "Opair l r \<in> games_lfp"
subst_fv_dom_ground_if_ground_img': fixes t :: "('b, 'a) Term.term"   and s :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "fv t \<subseteq> subst_domain s"   and "\<And>x. x \<in> subst_domain s \<Longrightarrow> fv (s x) = {}" shows "fv (t \<cdot> s) = {}"
cases: fixes eff :: "'d \<Rightarrow> 'e \<Rightarrow> 'e fset \<Rightarrow> bool"   and rules :: "'d stream"   and a :: "('e \<times> 'd) tree" assumes "RuleSystem_Defs.wf eff rules a" obtains t :: "('e \<times> 'd) tree" where "a = t" and "snd (tree.root t) \<in> i.R rules" and "eff (snd (tree.root t)) (fst (tree.root t)) ((fst \<circ> tree.root) |`| cont t)" and "\<And>t'. t' |\<in>| cont t \<Longrightarrow> RuleSystem_Defs.wf eff rules t'"
Red_F_def: fixes N :: "'f set" shows "no_labels.Red_F_\<G>_empty N = (\<Inter>q\<in>Q. no_labels.Red_F_\<G>_empty_q q N)"
adds_poly_mapping: fixes s :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 'd" shows "(s adds t) = (lookup s \<le> lookup t)"
onorm_one_dim_iso: shows "onorm one_dim_iso = 1"
simps_2: fixes x :: "three" shows "f2 x One = One"
arr_eqI: fixes t :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and t' :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.arr t"   and "J_C.arr t'"   and "J_C.Dom t = J_C.Dom t'"   and "J_C.Cod t = J_C.Cod t'"   and "J_C.Map t = J_C.Map t'" shows "t = t'"
l_create_character_data\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool" shows "l_create_character_data\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr \<equiv> known_ptr = ShadowRootClass.known_ptr"
fref_param1: fixes R :: "('e \<times> 'g) set"   and S :: "('f \<times> 'h) set" shows "R \<rightarrow> S = R \<rightarrow>\<^sub>f S"
supp_map_transform: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "supp (map_transform Aeta_expand ae \<Gamma>) \<subseteq> supp \<Gamma>"
dvd_power_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.algebraic_semidom divide plus minus zero times one"   and "x \<noteq> zero" shows "dvd.dvd times (power.power one times x m) (power.power one times x n) = (dvd.dvd times x one \<or> m \<le> n)"
empty_ran: fixes f :: "'d \<Rightarrow> 'c option" shows "(ran f = {}) = (\<forall>x. f x = None)"
exec_instr: shows "exec_instr i P h stk loc C M pc ics frs sh = (xp', h', frs', sh')"
cong_add: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b"   and a :: "'b"   and d :: "'b"   and e :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a"   and "unique_euclidean_semiring.cong modulo d e a" shows "unique_euclidean_semiring.cong modulo (plus b d) (plus c e) a"
R: fixes s1 :: "'a"   and s2 :: "'b"   and a :: "'e"   and b :: "'e" assumes "bad1 s1 = bad2 s2"   and "\<not> bad2 s2 \<Longrightarrow> a = b \<and> X s1 s2" shows "R (a, s1) (b, s2)"
aff_independent_finite: fixes B :: "'a set" assumes "\<not> affine_dependent B" shows "finite B"
list_neq_remove1: fixes a :: "'a"   and m :: "'a"   and xs :: "'a list" assumes "a \<noteq> m \<and> a \<in> set xs" shows "length (list_neq (remove1 a xs) m) < length (list_neq xs m)"
ell2_norm_infsetsum: fixes x :: "'c \<Rightarrow> complex" assumes "has_ell2_norm x" shows "ell2_norm x = sqrt (\<Sum>\<^sub>ai. (cmod (x i))\<^sup>2)"
is_map: shows "is_left_adjoint TTfgh_TfTgh.chine"
size_add_mset: fixes a :: "'b"   and A :: "'b multiset" shows "size (a, A) = Suc (size A)"
chine_simps_2: shows "C.dom \<nu>.chine = \<nu>.dom.apex"
lft_rel_st_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and u :: "nat \<Rightarrow> 'b"   and x :: "nat \<Rightarrow> 'c"   and y :: "nat \<Rightarrow> 'd" shows "lft_rel_st r u x y = r (u 0) (u 1) (x 0) (y 0)"
permute_list_zip: fixes f :: "nat \<Rightarrow> nat"   and A :: "nat set"   and xs :: "'c list"   and ys :: "'d list" assumes "f permutes A"   and "A = {..<length xs}"   and "length xs = length ys" shows "permute_list f (zip xs ys) = zip (permute_list f xs) (permute_list f ys)"
lcompare_double_def: fixes x :: "double"   and y :: "double" shows "lcompare_double x y = (if Code_Double.is_zero x \<and> Code_Double.is_zero y then compare_double (copysign_double 1 x) (copysign_double 1 y) else compare_double x y)"
distinct_1: fixes x1 :: "val"   and x2 :: "char list" shows "Val x1 \<noteq> Var x2"
map_of_set_def: fixes S :: "'a set"   and x :: "'a" shows "map_of_set S x \<equiv> if x \<in> S then Some () else None"
lm081: fixes X :: "'a set"   and Q :: "('a \<times> 'b) set"   and P :: "('a \<times> 'b) set" assumes "X \<inter> Domain Q = {}" shows "(P +< Q) `` X = Outside' (Domain Q) P `` X"
urel64_uminus: fixes x :: "uint64"   and y :: "int" assumes "urel64 x y" shows "urel64 (uminus_p64 pp x) (uminus_p p y)"
a3: shows "f \<subseteq> edges G \<and> (\<forall>(e1, e2)\<in>f. \<not> p e1 \<le> p e2)"
bound0_I: fixes p :: "fm"   and bbs :: "bool list"   and b :: "int"   and bs :: "int list"   and b' :: "int" assumes "bound0 p" shows "Ifm bbs (b # bs) p = Ifm bbs (b' # bs) p"
ContractsWithSubstitutesAndIRC_axioms: shows "ContractsWithSubstitutesAndIRC Xd Xh Pd Ch"
neg_numeral_less_neg_one_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (uminus (numeral.numeral one plus m)) (uminus one) = (m \<noteq> num.One)"
Uvariation_Vagree: fixes \<nu> :: "variable \<Rightarrow> real"   and \<nu>' :: "variable \<Rightarrow> real"   and V :: "variable set" shows "Uvariation \<nu> \<nu>' (- V) = Vagree \<nu> \<nu>' V"
subset_path_image_join: fixes g1 :: "real \<Rightarrow> 'b"   and s :: "'b set"   and g2 :: "real \<Rightarrow> 'b" assumes "path_image g1 \<subseteq> s"   and "path_image g2 \<subseteq> s" shows "path_image (g1 +++ g2) \<subseteq> s"
put_nondet_def: fixes put :: "'c \<Rightarrow> 'm \<Rightarrow> 'm"   and s :: "'c"   and m :: "('a, 'm) nondetT" shows "put_nondet put s m = NondetT (put s (run_nondet m))"
single_nz_valI: fixes v :: "'a vec"   and i :: "nat"   and val :: "'a" assumes "\<And>j. \<lbrakk>j < dim_vec v; j \<noteq> i\<rbrakk> \<Longrightarrow> v $ j = (0::'a)"   and "v $ i = val" shows "v = single_nz_val (dim_vec v) i val"
arr_dom_iff_arr: fixes f :: "'j" shows "J.arr (J.dom f) = J.arr f"
bind_Ok: fixes x :: "('a, 'b) result" shows "x \<bind> Ok = x"
poly_add_is_polynomial: fixes p1 :: "'a list"   and p2 :: "'a list" assumes "set p1 \<subseteq> K"   and "set p2 \<subseteq> K" shows "polynomial K (poly_add p1 p2)"
lconjby_rconjby: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.group_add minus plus zero uminus" shows "minus (plus x (plus (plus (uminus x) y) x)) x = y"
s0_le_s_stop: shows "s0 \<le> s_stop"
fdiv_def: fixes m :: "roundmode"   and a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "fdiv m a b = (if is_nan a \<or> is_nan b \<or> is_zero a \<and> is_zero b \<or> is_infinity a \<and> is_infinity b then some_nan else if is_infinity a \<or> is_zero b then if sign a = sign b then \<infinity> else - \<infinity> else if is_infinity b then if sign a = sign b then 0 else - 0 else zerosign (if sign a = sign b then 0 else 1) (IEEE.round m (valof a / valof b)))"
yields_isomorphic_representation_2: shows "local.iso (\<rho>'.T0.trnr\<^sub>\<epsilon> r' \<rho>')"
im_jointfunTr1: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'h"   and m :: "nat"   and g :: "nat \<Rightarrow> 'h" shows "jointfun n f m g ` {i. i \<le> n} = f ` {i. i \<le> n}"
assms_5: shows "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"
proper_rev_map_flip_signed: fixes xs :: "('a \<times> bool) list" assumes "proper_signed_list xs" shows "proper_signed_list (rev (map flip_signed xs))"
to_metric_completion_dense: fixes U :: "'b metric_completion set" assumes "open U"   and "U \<noteq> {}" shows "\<exists>x. to_metric_completion x \<in> U"
rep_eq: fixes x :: "nat"   and xa :: "float"   and xb :: "float" shows "real_of_float (float_plus_up x xa xb) = plus_up x (real_of_float xa) (real_of_float xb)"
self_codomain_iff_ide: fixes a :: "'a" shows "(a \<in> codomains a) = ide a"
valid_reads_def: fixes m :: "nat \<Rightarrow> nat"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" shows "valid_reads m ts \<equiv> \<forall>i p is \<O> \<R> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> reads_consistent False \<O> m sb"
subst_bvs_subst_bvs1': fixes s :: "Core.term list"   and t :: "Core.term" shows "subst_bvs s t = subst_bvs1' t 0 s"
lookup_map_scale: fixes c :: "'b"   and p :: "'a \<Rightarrow>\<^sub>0 'b" shows "lookup (c \<cdot> p) = (\<lambda>x. c * lookup p x)"
mat_t_mult_mm: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and B :: "complex \<times> complex \<times> complex \<times> complex" shows "mat_transpose (A *\<^sub>m\<^sub>m B) = mat_transpose B *\<^sub>m\<^sub>m mat_transpose A"
plus_perm_def: fixes p :: "perm"   and q :: "perm" shows "p + q = Abs_perm (Rep_perm p \<circ> Rep_perm q)"
simps: fixes c :: "rat vec"   and b :: "rat vec" shows "x_times_c_geq_y_times_b c b = GEQPP (vec_to_lpoly (c @\<^sub>v 0\<^sub>v (dim_vec b))) (vec_to_lpoly (0\<^sub>v (dim_vec c) @\<^sub>v b))"
AE_I2: fixes M :: "'b measure"   and P :: "'b \<Rightarrow> bool" assumes "\<And>x. x \<in> space M \<Longrightarrow> P x" shows "almost_everywhere M P"
fls_base_factor_to_fps_inverse: fixes f :: "'a fls" shows "fls_base_factor_to_fps (inverse f) = inverse (fls_base_factor_to_fps f)"
is_failed_is_success_completeD: fixes x :: "status" assumes "\<not> is_failed x"   and "\<not> is_success x" shows "is_found x"
bounded_partial_sums_imp_fps_converges: fixes f :: "'a fds"   and s0 :: "'a"   and s :: "'a" assumes "Bseq (\<lambda>n. \<Sum>k\<le>n. fds_nth f k / nat_power k s0)"   and "s0 \<bullet> (1::'a) < s \<bullet> (1::'a)" shows "fds_converges f s"
sort_key_simps_2: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'c"   and xs :: "'c list" shows "sort_key f (x # xs) = insort_key f x (sort_key f xs)"
finite_distinct_list: fixes A :: "'b set" assumes "finite A" shows "\<exists>xs. set xs = A \<and> distinct xs"
norm_triangle_ineq3: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "\<bar>norm a - norm b\<bar> \<le> norm (minus a b)"
bisim: fixes s1 :: "'s1"   and s2 :: "'s2"   and x :: "'a" assumes "X s1 s2"   and "x \<in> outs_\<I> \<I>"   and "I1 s1"   and "I2 s2" shows "rel_spmf (\<lambda>(a, s1') (b, s2'). bad1 s1' = bad2 s2' \<and> (if bad2 s2' then X_bad s1' s2' else a = b \<and> X s1' s2')) (oracle1 s1 x) (oracle2 s2 x)"
tab_in_vhom': shows "\<guillemotleft>\<tau>\<mu>.tab : t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<Rightarrow> (t \<star> u) \<star> u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0\<guillemotright>"
mult_left_dist_sup: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.idempotent_left_zero_semiring one times sup less_eq less bot" shows "times x (sup y z) = sup (times x y) (times x z)"
unit_in_hom_1: shows "\<guillemotleft>\<eta> : src f \<rightarrow> src f\<guillemotright>"
simps_3: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and xs :: "'d list" shows "successively P (x # y # xs) = (P x y \<and> successively P (y # xs))"
num_syms_geq_num_vars: fixes t :: "('a, 'b) Term.term" shows "num_vars t \<le> num_syms t"
combine_altdef: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t1 :: "('a, 'b) RBT.rbt"   and t2 :: "('a, 'b) RBT.rbt" shows "RBT.combine f t1 t2 = combine_with_key (\<lambda>_. f) t1 t2"
assms_2: shows "finite {b. g b \<noteq> (0::'a)}"
solution_pell_mul: fixes z1 :: "'a \<times> 'a"   and z2 :: "'a \<times> 'a" assumes "solution z1"   and "solution z2" shows "solution (pell_mul z1 z2)"
cong_exp_iff_simps_8: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and m :: "num"   and q :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "(modulo (numeral.numeral one plus (num.Bit0 m)) (numeral.numeral one plus (num.Bit0 q)) = modulo (numeral.numeral one plus num.One) (numeral.numeral one plus (num.Bit0 q))) = False"
fmfilter_cong': fixes m :: "('c, 'd) fmap"   and n :: "('c, 'd) fmap"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "m = n"   and "\<And>x. x \<in> fmdom' m \<Longrightarrow> P x = Q x" shows "fmfilter P m = fmfilter Q n"
at_is: shows "At = LessUni p \<or> At = EqUni p"
sorted_list_of_set_other_def: fixes A :: "'b set" assumes "finite A" shows "sorted_list_of_set A = (THE xs. sorted xs \<and> distinct xs \<and> ran\<^sub>l xs = A)"
A1: shows "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel"
neg_elim: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<pi>l :: "('a \<times> 'b set) stream list"   and \<chi> :: "bool" assumes "Shallow S s0 \<delta> AP L AP'"   and "Shallow.neg \<phi> \<pi>l"   and "\<phi> \<pi>l" shows "\<chi>"
addr_loc_default_conf: fixes P :: "'a prog"   and CTn :: "htype"   and F :: "String.literal"   and T :: "ty"   and fm :: "fmod"   and C :: "String.literal"   and h :: "'heap" assumes "P \<turnstile> class_type_of CTn has F:T (fm) in C" shows "h.conf P h (addr_loc_default P CTn (CField C F)) T"
divide_right: fixes h :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. h x \<noteq> (0::'b)"   and "f \<in> o[F](g)" shows "(\<lambda>x. f x / h x) \<in> o[F](\<lambda>x. g x / h x)"
feasible_pr_0: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "feasible t" shows "t (\<lambda>x. 0) = (\<lambda>x. 0)"
bind_singleton_conv_image: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'b" shows "A \<bind> (\<lambda>x. {f x}) = f ` A"
tr_step: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and t :: "('p, 's, 'm) event list"   and c'' :: "('p, 's, 'm) configuration" assumes "c \<turnstile> ev \<mapsto> c'"   and "trace c' t c''" shows "trace c (ev # t) c''"
le_SucI: fixes m :: "nat"   and n :: "nat" assumes "order.greater_eq n m" shows "order.greater_eq (Suc n) m"
addition3: fixes A :: "('c, 'd) aGroup_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'c"   and j :: "nat" assumes "aGroup A"   and "\<forall>j\<le>Suc n. f j \<in> carrier A"   and "j \<le> Suc n"   and "j \<noteq> Suc n" shows "\<Sigma>\<^sub>e A f Suc n = \<Sigma>\<^sub>e A cmp f (\<tau>\<^bsub>j Suc n\<^esub>) Suc n"
image_set_system_wellformed: fixes f :: "'a \<Rightarrow> 'c" shows "incidence_system (f ` V) (blocks_image B f)"
wtPB_Un: fixes \<Phi>1 :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list set"   and \<Phi>2 :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list set" shows "GE.wtPB (\<Phi>1 \<union> \<Phi>2) = (GE.wtPB \<Phi>1 \<and> GE.wtPB \<Phi>2)"
Zassenhaus: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and H1 :: "'e set"   and K :: "'e set"   and K1 :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<guillemotright> H1 "   and "G \<guillemotright> K "   and "G \<guillemotright> K1 "   and "\<natural>\<^bsub>G\<^esub>H \<triangleright> H1"   and "\<natural>\<^bsub>G\<^esub>K \<triangleright> K1" shows "(\<natural>\<^bsub>G\<^esub>H1 \<diamondop>\<^bsub>G\<^esub> H \<inter> K / (H1 \<diamondop>\<^bsub>G\<^esub> H \<inter> K1)) \<cong> (\<natural>\<^bsub>G\<^esub>K1 \<diamondop>\<^bsub>G\<^esub> K \<inter> H / (K1 \<diamondop>\<^bsub>G\<^esub> K \<inter> H1))"
ball_empty: fixes P :: "'b \<Rightarrow> bool" shows "Ball {} P = True"
mk_minsky_apply3: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and t :: "nat"   and k :: "nat"   and u :: "nat"   and v :: "nat"   and w :: "nat" assumes "mk_minsky3 f"   and "t < k"   and "u < k"   and "v < k"   and "w < k" shows "mk_minsky (\<lambda>vs vs'. \<forall>x<k. vs' x = (if x = t then f (vs u) (vs v) (vs w) else vs x))"
smult_eq_iff: fixes b :: "'b"   and a :: "'b"   and p :: "'b poly"   and q :: "'b poly" assumes "b \<noteq> (0::'b)" shows "(smult a p = smult b q) = (smult (a / b) p = q)"
tauChainPar1: fixes P :: "ccs"   and P' :: "ccs"   and Q :: "ccs" assumes "P \<Longrightarrow>\<^sub>\<tau> P'" shows "P \<parallel> Q \<Longrightarrow>\<^sub>\<tau> P' \<parallel> Q"
const_def: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "deutsch f" shows "deutsch.const f n = (\<forall>x\<in>{0, 1}. f x = n)"
distinct_remove1: fixes xs :: "'a list"   and x :: "'a" assumes "distinct xs" shows "distinct (remove1 x xs)"