hcju/isabelleps huggingface dataset:
antisym_conv1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.order less_eq less"   and "\<not> less x y" shows "less_eq x y = (x = y)"
power_int_diff: fixes x :: "'a"   and m :: "int"   and n :: "int" assumes "x \<noteq> (0::'a) \<or> m \<noteq> n" shows "x powi (m - n) = x powi m / x powi n"
invalidate_rtsf_left: fixes dests :: "nat \<Rightarrow> nat option"   and dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt' :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dests dip = None" shows "(invalidate rt dests \<sqsubset>\<^bsub>dip\<^esub> rt') = (rt \<sqsubset>\<^bsub>dip\<^esub> rt')"
i: shows "integrable (\<lambda>t. f t - g t) M"
simps: fixes l :: "nat"   and r :: "linear_poly" shows "lhs (l, r) = l"
Solutions_snd_not_0: fixes a :: "nat list"   and b :: "nat list"   and x :: "nat list"   and y :: "nat list" assumes "hlde a b"   and "(x, y) \<in> Solutions a b"   and "nonzero x" shows "nonzero y"
plus_p32_def: fixes p :: "uint32"   and x :: "uint32"   and y :: "uint32" shows "plus_p32 p x y \<equiv> let z = x + y in if p \<le> z then z - p else z"
fv_subst_obtain_var: fixes x :: "'a"   and t :: "('b, 'a) Term.term"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "x \<in> fv (t \<cdot> \<delta>)" shows "\<exists>y\<in>fv t. x \<in> fv (\<delta> y)"
unit_simps_4: shows "src \<eta>\<^sub>0 = src e\<^sub>0"
assms_3: shows "u \<longlonglongrightarrow> l"
inv_is_inverse: fixes f :: "'a" assumes "L.iso f" shows "L.inverse_arrows f (L.inv f)"
pw_leI: fixes S' :: "'a nres"   and S :: "'a nres" assumes "nofail S' \<longrightarrow> nofail S \<and> (\<forall>x. inres S x \<longrightarrow> inres S' x)" shows "S \<le> S'"
inessential_imp_continuous_logarithm_circle: fixes S :: "'a set"   and f :: "'a \<Rightarrow> complex"   and a :: "complex" assumes "homotopic_with_canon (\<lambda>h. True) S (sphere 0 1) f (\<lambda>t. a)" obtains g :: "'a \<Rightarrow> complex" where "continuous_on S g" and "\<And>x. x \<in> S \<Longrightarrow> f x = exp (g x)"
abs_eq: fixes x :: "real_alg_3" shows "real_of (Real_Alg_Quotient x) = real_of_3 x"
rank_le_nrows: fixes A :: "(('b, 'c) vec, 'd) vec" shows "rank A \<le> nrows A"
treap_of_bst_of_list: fixes p :: "'c \<Rightarrow> 'd"   and zs :: "'c list" assumes "inj_on p (set zs)" shows "map_tree fst (treap_of (set (map (\<lambda>x. (x, p x)) zs))) = bst_of_list (sort_key p zs)"
fd: shows "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable at x"
bin_sc_nth: fixes n :: "nat"   and w :: "int" shows "bin_sc n (bit w n) w = w"
principal_not_primitive: shows "\<not> primitive_dchar n (principal_dchar n)"
computation2: shows "rat_poly_plus (rat_poly_times A (rat_poly_times (B - rat_poly_times (rat_poly_times A A) A) A)) (rat_poly_times (B - rat_poly_times (rat_poly_times A A) A) (rat_poly_times B (B - rat_poly_times (rat_poly_times A A) A))) = rat_poly_times B (rat_poly_times (B - rat_poly_times (rat_poly_times A A) A) B)"
foldr_cong: fixes a :: "'g"   and b :: "'g"   and l :: "'h list"   and k :: "'h list"   and f :: "'h \<Rightarrow> 'g \<Rightarrow> 'g"   and g :: "'h \<Rightarrow> 'g \<Rightarrow> 'g" assumes "a = b"   and "l = k"   and "\<And>a x. x \<in> set l \<Longrightarrow> f x a = g x a" shows "foldr f l a = foldr g k b"
elims_2: fixes x :: "t"   and xa :: "t"   and xb :: "sx option" assumes "convert_cond x xa xb" obtains t1 :: "t" and t2 :: "t" and sx :: "sx option" where "x = t1" and "xa = t2" and "xb = sx" and "t1 \<noteq> t2 \<and> (sx = None) = (is_float_t t1 \<and> is_float_t t2 \<or> is_int_t t1 \<and> is_int_t t2 \<and> t_length t1 < t_length t2)"
split_list_last_prop: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. P x" shows "\<exists>ys x zs. xs = ys @ x # zs \<and> P x \<and> (\<forall>z\<in>set zs. \<not> P z)"
pnormal_length: fixes p :: "'b list" assumes "pnormal p" shows "0 < length p"
ltake_is_lprefix: fixes n :: "enat"   and xs :: "'c llist" shows "lprefix (ltake n xs) xs"
fixed_point_congruence: fixes p :: "nat"   and a :: "nat" assumes "order H' = p ^ a"   and "prime p"   and "finite (rcosets P)" shows "card (rcosets P) mod p = card H'act.fixed_points mod p"
simps_2: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and u :: "nat"   and v :: "nat"   and w :: "nat"   and ws :: "nat list" shows "len m u v (w # ws) = m u w + len m w v ws"
Init_def: shows "Init \<equiv> {(X, mrk). finite (- mrk) \<and> mrk = {}}"
epi_def: fixes G :: "('a, 'c) monoid_scheme"   and H :: "('b, 'd) monoid_scheme" shows "epi G H = {f \<in> hom G H. f ` carrier G = carrier H}"
Rep_perm_inject: fixes x :: "'b perm"   and y :: "'b perm" shows "(Rep_perm x = Rep_perm y) = (x = y)"
pos_divideR_le_eq: fixes c :: "real"   and b :: "'a"   and a :: "'a" assumes "0 < c" shows "(b /\<^sub>R c \<le> a) = (b \<le> c *\<^sub>R a)"
ntr_reverse_split: fixes c :: "'n list multiset"   and w :: "('p, 'ba) label list list"   and s' :: "'n list"   and ce' :: "'n list multiset" assumes "(c, w, {#s'#} + ce') \<in> trcl (ntr fg)"   and "valid fg c" shows "\<exists>s ce w1 w2 ce1' ce2'. c = {#s#} + ce \<and> ce' = ce1' + ce2' \<and> w \<in> w1 \<otimes>\<^bsub>\<alpha>n fg\<^esub> w2 \<and> mon_s fg s \<inter> (mon_c fg ce \<union> mon_ww fg w2) = {} \<and> mon_c fg ce \<inter> (mon_s fg s \<union> mon_ww fg w1) = {} \<and> ({#s#}, w1, {#s'#} + ce1') \<in> trcl (ntr fg) \<and> (ce, w2, ce2') \<in> trcl (ntr fg)"
minus_minus: fixes G :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "abelian_group G"   and "x \<in> carrier G" shows "\<ominus>\<^bsub>G\<^esub> (\<ominus>\<^bsub>G\<^esub> x) = x"
trss_mon_e_no_ctx: fixes s :: "'f list"   and c :: "'f list multiset"   and e :: "('g, 'h) label"   and s' :: "'f list"   and c' :: "'f list multiset"   and fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme" assumes "((s, c), e, s', c') \<in> trss fg" shows "mon_e fg e \<inter> mon_c fg c = {}"
has_integral_alt: fixes f :: "'a \<Rightarrow> 'b"   and y :: "'b"   and i :: "'a set" shows "(f has_integral y) i = (if \<exists>a b. i = cbox a b then (f has_integral y) i else \<forall>e>0. \<exists>B>0. \<forall>a b. ball (0::'a) B \<subseteq> cbox a b \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f x else (0::'b)) has_integral z) (cbox a b) \<and> norm (z - y) < e))"
coord_def: fixes X :: "(('a, 'b) vec, 'b) vec"   and v :: "('a, 'b) vec" shows "coord X v = (\<chi>i. (THE f. v = (\<Sum>x\<in>UNIV. f x *s X $ x)) i)"
r1: fixes f :: "'a" assumes "f \<in> M"   and "k \<in> M"   and "is_arg_min (\<lambda>x. dist x h) (\<lambda>x. x \<in> M) k" shows "2 * Re \<langle>h - k, f\<rangle> \<le> \<parallel>f\<parallel>\<^sup>2"
awalkE': fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "S.awalk u p v" obtains "set (pawalk_verts u p) \<subseteq> pverts (subdivide G e w)" and "set p \<subseteq> parcs (subdivide G e w)" and "pcas u p v" and "hd (pawalk_verts u p) = u" and "last (pawalk_verts u p) = v" and "u \<in> pverts (subdivide G e w)" and "v \<in> pverts (subdivide G e w)"
filter_generator_empty_conv: fixes g :: "('a, 'c) generator"   and s :: "'c" shows "list.has_next (local.filter_generator g) s = (\<exists>x\<in>set (list.unfoldr g s). P x)"
apart: shows "vars\<^sub>l\<^sub>s C \<inter> vars\<^sub>l\<^sub>s D = {}"
inner_eq_zero_iff: fixes x :: "'a" shows "(x \<bullet> x = 0) = (x = (0::'a))"
that_2: fixes p\<^sub>1 :: "('a \<times> 'a) list"   and u\<^sub>1 :: "'a"   and v\<^sub>1 :: "'a"   and p\<^sub>2 :: "('a \<times> 'a) list"   and u\<^sub>2 :: "'a"   and v\<^sub>2 :: "'a"   and p\<^sub>3 :: "('a \<times> 'a) list" assumes "p = p\<^sub>1 @ [(u\<^sub>1, v\<^sub>1)] @ p\<^sub>2 @ [(u\<^sub>2, v\<^sub>2)] @ p\<^sub>3"   and "P u\<^sub>1"   and "\<not> P v\<^sub>1"   and "\<not> P u\<^sub>2"   and "P v\<^sub>2" shows "thesis"
phiDefNodes_def: fixes g :: "'a"   and v :: "'var" shows "i.phiDefNodes g v \<equiv> fold (\<union>) (concat (map (\<lambda>n. if v \<in> uses2 g n then [i.phiDefNodes_aux g v (\<alpha>n2 g) n] else []) (\<alpha>n2 g))) {}"
H_def: shows "H = {x. coprime x n \<and> euler_liar (int x) n \<and> x \<in> {1..<n}}"
L_in_interval: fixes b :: "'a"   and S :: "'a set"   and a :: "'a"   and L :: "'a" assumes "b \<in> A"   and "S \<subseteq> interval r a b"   and "isLub S cl L"   and "S \<noteq> {}" shows "L \<in> interval r a b"
collapse: fixes stream :: "'d stream" shows "shd stream ## stl stream = stream"
option_rel_Some2: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a option"   and y :: "'b" shows "rel_option A x \<lfloor>y\<rfloor> = (\<exists>x'. x = \<lfloor>x'\<rfloor> \<and> A x' y)"
lift_Use_Low: fixes V :: "'f"   and L :: "'f set"   and X :: "'g"   and Use :: "'g \<Rightarrow> 'f set"   and E :: "'g"   and H :: "'f set" assumes "V \<in> L" shows "(Node X, V) \<in> lift_Use_set Use E X H L"
k: shows "hom_induced p (subtopology X T) {} X S id \<in> Group.iso (homology_group p (subtopology X T)) (relative_homology_group p X S)"
mask_plus_1: fixes n :: "nat" shows "mask n + 1 = 2 ^ n"
aless_le_suc: fixes a :: "ant"   and b :: "ant" assumes "a < b" shows "a + 1 \<le> b"
simps_1: fixes \<A> :: "'b ltln set" shows "\<A> \<Turnstile>\<^sub>P true\<^sub>n = True"
arcsin_arccos_eq: fixes x :: "real" assumes "- 1 \<le> x"   and "x \<le> 1" shows "arcsin x = pi / 2 - arccos x"
WT_gpv_full: fixes gpv :: "('f, 'd, 'e) gpv" shows "\<I>_full \<turnstile>g gpv \<surd>"
pe_pa_sim: fixes R :: "('a \<times> 'a set) set"   and S :: "('a \<times> 'a set) set" shows "R \<cdot> S \<parallel> Multirelations.U = R \<otimes> (S \<parallel> Multirelations.U)"
permutes_in_image: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and x :: "'b" assumes "p permutes S" shows "(p x \<in> S) = (x \<in> S)"
reduce_def: fixes s :: "nat \<Rightarrow> nat"   and n :: "nat" shows "reduce s n = (SOME r'. strict_mono r' \<and> P n (s \<circ> r'))"
vcomp_char: shows "(\<bullet>) = (\<lambda>\<nu> \<mu>. if V.seq \<nu> \<mu> then \<lparr>Chn = Chn \<nu> \<cdot> Chn \<mu>, Dom = Dom \<mu>, Cod = Cod \<nu>\<rparr> else V.null)"
vlepoll_vempty_iff: fixes A :: "V" shows "(A \<lesssim>\<^sub>\<circ> 0) = (A = 0)"
rel_frontier_nonempty_interior: fixes S :: "'a set" assumes "interior S \<noteq> {}" shows "rel_frontier S = frontier S"
is_weakly_dom_component_safe_def: shows "is_weakly_dom_component_safe \<equiv> l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_weakly_dom_component_safe get_root_node to_tree_order"
vD_flag_val: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dip \<in> vD rt" shows "the (flag rt dip) = val"
sup1_def: fixes a :: "'a"   and b :: "'a" shows "pseudo_hoop_algebra.sup1 (\<sqinter>) (r\<rightarrow>) (l\<rightarrow>) a b = ((a r\<rightarrow> b) l\<rightarrow> b) \<sqinter> ((b r\<rightarrow> a) l\<rightarrow> a)"
permBottom: fixes p :: "(name \<times> name) list" shows "p \<bullet> \<bottom> = \<bottom>"
enum_ex_option_def: fixes P :: "'a option \<Rightarrow> bool" shows "enum_ex P = (P None \<or> enum_ex (\<lambda>x. P (Some x)))"
hs'_real: fixes i :: "nat"   and x :: "nat" assumes "i < k" shows "(hs' ! i) (real x) = (hs ! i) x"
preserves_hom: fixes f :: "'a1"   and a :: "'a1"   and b :: "'a1" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>A\<^sub>1 b\<guillemotright>" shows "\<guillemotleft>local.curry F F F f : local.curry F F F a \<rightarrow>\<^sub>[\<^sub>A\<^sub>2\<^sub>,\<^sub>B\<^sub>] local.curry F F F b\<guillemotright>"
append_cols_mult_right_id2: fixes A :: "'a mat"   and n :: "nat"   and a :: "nat"   and B :: "'a mat"   and b :: "nat"   and C :: "'a mat"   and D :: "'a mat" assumes "A \<in> carrier_mat n a"   and "B \<in> carrier_mat n b"   and "C = four_block_mat D (0\<^sub>m a b) (0\<^sub>m b a) (1\<^sub>m b)"   and "D \<in> carrier_mat a a" shows "(A @\<^sub>c B) * C = A * D @\<^sub>c B"
obj_ty_eq: fixes oi :: "obj_tag"   and x :: "vn \<Rightarrow> val option"   and y :: "vn \<Rightarrow> val option" shows "obj_ty \<lparr>tag = oi, values = x\<rparr> = obj_ty \<lparr>tag = oi, values = y\<rparr>"
congruent_P2R: fixes \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" shows "f respects P2R \<phi> = f respectsP \<phi>"
exp_sq_mtx_diag: fixes f :: "'a \<Rightarrow> real" shows "exp (sq_mtx_diag f) = (\<d>\<i>\<a>\<g> i. exp (f i))"
assms: shows "strict_monoidal_extension_to_free_monoidal_category C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V F"
P2_OT_14_TT: fixes m0 :: "bool"   and m1 :: "bool"   and m2 :: "bool"   and m3 :: "bool" shows "R2_14 (m0, m1, m2, m3) (True, True) = funct_OT_14 (m0, m1, m2, m3) (True, True) \<bind> (\<lambda>(out1, out2). S2_14 (True, True) out2)"
expand_set: fixes g :: "'d \<Rightarrow> 'c" shows "Prod_any g = prod g {a. g a \<noteq> (1::'c)}"
sum: fixes g :: "'b \<Rightarrow> ('a, 'n) vec"   and S :: "'b set" shows "f (sum g S) = (\<Sum>a\<in>S. f (g a))"
lfp_trans_def: fixes T :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "lfp_trans T = Inf_trans {t. (\<forall>P. sound P \<longrightarrow> sound (t P)) \<and> le_trans (T t) t}"
assms: shows "x \<in> frontier {y. infdist y S \<le> r}"
fes_4: shows "\<forall>fe'\<in>set ft. ofe_prio fe1 < ofe_prio fe' \<longrightarrow> \<not> \<gamma> (ofe_fields fe') p"
group_hom_axioms: shows "group_hom (relative_homology_group (int n) (lsphere n) (equator n)) (relative_homology_group (int n) (nsphere n) (upper n)) (hom_induced (int n) (lsphere n) (equator n) (nsphere n) (upper n) id)"
index_zero_mat_2: fixes nr :: "nat"   and nc :: "nat" shows "dim_row (0\<^sub>m nr nc) = nr"
Infinitesimal_hIm: fixes x :: "complex star" assumes "x \<in> Infinitesimal" shows "hIm x \<in> Infinitesimal"
weight_simp: fixes w :: "'b \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and scf :: "'b \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and f :: "'b"   and ts :: "('b, 'c) Term.term list" shows "weight_fun.weight w w0 scf (Fun f ts) = w (f, length ts) + sum_list (map (weight_fun.weight w w0 scf) (scf_list (scf (f, length ts)) ts))"
set_bd_2: fixes x :: "('l, 'm, 'k) F1" shows "|F1set2 x| \<le>o bd_F1"
f': shows "f \<in> count_space A \<rightarrow>\<^sub>M count_space B"
Skip_is_CSP3: shows "action.relation_of Skip is CSP3 healthy"
OUTfromVCorrect1_data12: shows "OUTfromVCorrect1 data12"
len_stable_up: fixes v :: "view"   and v1 :: "view"   and v2 :: "view"   and ts :: "traffic"   and c :: "cars" shows "v=v1--v2 \<longrightarrow> len v ts c = len v2 ts c"
Vinv2_event_pres: fixes r :: "nat"   and v_f :: "process \<Rightarrow> val option"   and d_f :: "process \<Rightarrow> val option" shows "{Vinv2 \<inter> Vinv1} v_round r v_f d_f {> Vinv2}"
subset_insertI: fixes B :: "'g set"   and a :: "'g" shows "B \<subseteq> insert a B"
rbt_comp_insert_with_key_def: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'a"   and v :: "'c"   and t :: "('a, 'c) rbt" shows "local.rbt_comp_insert_with_key f k v t = paint color.B (local.rbt_comp_ins f k v t)"
gt: shows "row A k \<bullet> col B j \<succ> row A k \<bullet> col C j"
nextVertex_inj: fixes f :: "face"   and v :: "nat"   and i :: "nat"   and a :: "nat" assumes "distinct (vertices f)"   and "v \<in> \<V> f"   and "i < |vertices f|"   and "a < |vertices f|"   and "f\<^bsup>a\<^esup> \<bullet> v = f\<^bsup>i\<^esup> \<bullet> v" shows "i = a"
equiv_lift: fixes s :: "dB"   and t :: "dB"   and k :: "nat" assumes "s \<leftrightarrow> t" shows "lift s k \<leftrightarrow> lift t k"
all_formulas_entailed: fixes N2 :: "'f set"   and N1 :: "'f set" assumes "\<forall>C\<in>N2. N1 \<Turnstile> {C}" shows "N1 \<Turnstile> N2"
assms_3: shows "\<forall>T' A'. lb T' A' j \<longrightarrow> makespan T \<le> 2 * makespan T'"
abs_eq: fixes x :: "int" shows "Word.of_int x = Word.Word (take_bit LENGTH('k) x)"
span_eq_iff: fixes s :: "'c set" shows "(span s = s) = subspace s"
simps_115: fixes x15 :: "expr"   and x4 :: "val" shows "throw x15 \<noteq> Val x4"
simps: fixes f :: "val \<Rightarrow> 'a"   and s :: "val \<Rightarrow> 'b"   and x :: "val"   and y :: "val" shows "extract_pair' f s <|x, y|> = (f x, s y)"
assms: fixes S :: "'a ttree set" shows "f (Either S) = Either (f ` S)"
A: shows "par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t A Sec"
gen_build_invar: fixes As :: "'s \<times> natural \<times> natural"   and bound :: "natural"   and n :: "nat" assumes "invar (fst As)" shows "invar (fst (local.gen_build bound n As))"
closed_cbox: fixes a :: "'c"   and b :: "'c" shows "closed (cbox a b)"
tt_cons2: fixes x :: "'a"   and q :: "'a"   and p :: "'a" shows "(\<exists>p'. \<lparr>p'\<rparr>x\<lparr>q\<rparr> \<and> p \<le> p') = \<lparr>p\<rparr>x\<lparr>q\<rparr>"
T_intersec_meas_3: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> sets M"   and "B \<in> sets M" shows "T -` A \<inter> B \<in> sets M"
map_snd_pair_pmf: fixes A :: "'c pmf"   and B :: "'b pmf" shows "map_pmf snd (pair_pmf A B) = B"
list_to_nat_th: fixes u :: "nat" shows "list_to_nat (nat_to_list u) = u"
incomplete_alt_size: fixes bl :: "'a set" assumes "incomplete_block bl" shows "int (card bl) < \<v>"
CallNull\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and es :: "nat exp list"   and vs :: "val list"   and s\<^sub>2 :: "heap \<times> val list \<times> sheap"   and M :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>map Val vs,s\<^sub>2\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e\<bullet>M(es),s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NullPointer,s\<^sub>2\<rangle>"
map_add_dom_app_simps_2: fixes m :: "'d"   and l1 :: "'d \<Rightarrow> 'e option"   and l2 :: "'d \<Rightarrow> 'e option" assumes "m \<notin> dom l1" shows "(l1 ++ l2) m = l2 m"
MonotProblem_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set" shows "MonotProblem wtFsym wtPsym arOf resOf parOf \<Phi> \<equiv> Problem wtFsym wtPsym arOf resOf parOf \<Phi> \<and> MonotProblem_axioms wtFsym wtPsym arOf resOf parOf \<Phi>"
lfp_mono: fixes f :: "'c \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'c" assumes "\<And>Z. f Z \<le> g Z" shows "lfp f \<le> lfp g"
eventually_top: fixes P :: "'c \<Rightarrow> bool" shows "eventually P top = (\<forall>x. P x)"
GSubspace_def: shows "GSubspace = (\<lambda>N. RModule (aezfun_setspan G) fgsmult N \<and> N \<subseteq> V)"
permute_Lam: fixes f :: "'d \<Rightarrow> 'e"   and p :: "perm" assumes "cont f" shows "p \<bullet> (\<Lambda> x. f x) = (\<Lambda> x. (p \<bullet> f) x)"
funas_term_subst: fixes t :: "('c, 'e) Term.term"   and \<sigma> :: "'e \<Rightarrow> ('c, 'd) Term.term" shows "funas_term (t \<cdot> \<sigma>) = funas_term t \<union> \<Union> (funas_term ` \<sigma> ` vars_term t)"
representsI: fixes p :: "int poly"   and x :: "'b" assumes "ipoly p x = (0::'b)"   and "p \<noteq> 0" shows "p represents x"
induct: fixes g :: "'g"   and x :: "'val"   and P :: "'val \<Rightarrow> bool" assumes "liveVal g x"   and "\<And>n val. \<lbrakk>n \<in> set (\<alpha>n g); val \<in> uses g n\<rbrakk> \<Longrightarrow> P val"   and "\<And>v v'. \<lbrakk>liveVal g v; P v; phiArg g v v'\<rbrakk> \<Longrightarrow> P v'" shows "P x"
tendsto_Lim: fixes net :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and l :: "'c" assumes "net \<noteq> bot"   and "(f \<longlongrightarrow> l) net" shows "Lim net f = l"
simps_19: fixes v :: "'a exp" shows "init_exp_of (throw v) = None"
inverse_composition_commute: fixes x :: "'a"   and y :: "'a" assumes "invertible x"   and "invertible y"   and "x \<in> G"   and "y \<in> G" shows "local.inverse (x \<cdot> y) = local.inverse y \<cdot> local.inverse x"
monad_prob_stateT': assumes "monad_prob Monad_Overloading.return Monad_Overloading.bind sample" shows "monad_prob Monad_Overloading.return Monad_Overloading.bind sample"
perp_up_degenerate_in_S_incident_1: fixes a :: "proj2"   and l :: "proj2_line" assumes "a \<in> hyp2"   and "proj2_incident a l" shows "perp_up a l \<in> S"
cases: fixes a1 :: "(Location \<times> Location) set"   and a2 :: "Val"   and a3 :: "Val" assumes "(a1, a2, a3) \<in> twiddleVal" obtains \<beta> :: "(Location \<times> Location) set" where "a1 = \<beta>__" and "a2 = RVal Nullref" and "a3 = RVal Nullref"   | l1 :: "Location" and l2 :: "Location" and \<beta> :: "(Location \<times> Location) set" where "a1 = \<beta>'__" and "a2 = RVal (Loc l1)" and "a3 = RVal (Loc l2)" and "(l1, l2) \<in> \<beta>'__"   | i1 :: "int" and i2 :: "int" and \<beta> :: "(Location \<times> Location) set" where "a1 = \<beta>" and "a2 = IVal i1" and "a3 = IVal i2" and "i1 = i2"
assms_1: fixes tset\<^sub>\<alpha> :: "('a, 'b, 'c) Tree\<^sub>\<alpha> set['a]" assumes "\<And>x. x \<in> set_bset tset\<^sub>\<alpha> \<Longrightarrow> P x" shows "P (Conj\<^sub>\<alpha> tset\<^sub>\<alpha>)"
set_cases_1: fixes e :: "'a"   and a :: "('a, 'b) rbt" assumes "e \<in> set1_rbt a" obtains z1 :: "color" and z2 :: "('a, 'b) rbt" and z3 :: "'a" and z4 :: "'b" and z5 :: "('a, 'b) rbt" where "a = Branch z1__ z2__ z3__ z4__ z5__" and "e \<in> set1_rbt z2__"   | z1 :: "color" and z2 :: "('a, 'b) rbt" and z4 :: "'b" and z5 :: "('a, 'b) rbt" where "a = Branch z1a__ z2a__ e z4a__ z5a__"   | z1 :: "color" and z2 :: "('a, 'b) rbt" and z3 :: "'a" and z4 :: "'b" and z5 :: "('a, 'b) rbt" where "a = Branch z1 z2 z3 z4 z5" and "e \<in> set1_rbt z5"
g2: fixes z :: "real" assumes "1 < z * 2"   and "z \<le> 1"   and "z * 2 - 1 \<notin> s2" shows "vector_derivative (\<lambda>x. if x * 2 \<le> 1 then g1 (2 * x) else g2 (2 * x - 1)) (at z) = 2 *\<^sub>R vector_derivative g2 (at (z * 2 - 1))"
assms_2: shows "Complete_Partial_Order.chain (\<le>) B"
evals_cases_2: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and es :: "expr list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>e # es,s\<rangle> [\<Rightarrow>] \<langle>e',s'\<rangle>" obtains v :: "val" and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" and es' :: "expr list" where "e' = Val v # es'" and "P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>" and "P,E \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>"   | e'a :: "expr" where "e' = throw e'a # es" and "P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>throw e'a,s'\<rangle>"
unif_rtrancl_mono: fixes E :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset"   and E' :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset"   and M :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset" assumes "(E, E') \<in> unif\<^sup>*" shows "\<exists>\<sigma>. (E + M, E' + subst_mset \<sigma> M) \<in> unif\<^sup>*"
OAlist_tc_except_min_def: fixes xs :: "('c, 'd) oalist_tc" shows "OAlist_tc_except_min xs = OAlist_tc (tl (list_of_oalist_tc xs))"