hcju/isabelleps huggingface dataset:
lemma_3_12_i_ii: fixes H :: "'a set"   and x :: "'a"   and a :: "'a" assumes "H \<in> normalfilters"   and "H \<in> ultrafilters"   and "x \<notin> H" shows "\<exists>n. x ^ n l\<rightarrow> a \<in> H"
eventually_mono: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter"   and Q :: "'a \<Rightarrow> bool" assumes "eventually P F"   and "\<And>x. P x \<Longrightarrow> Q x" shows "eventually Q F"
commutative_squareE: fixes f :: "(bool, bool list) concrete_category.arr"   and g :: "(bool, bool list) concrete_category.arr"   and h :: "(bool, bool list) concrete_category.arr"   and k :: "(bool, bool list) concrete_category.arr" assumes "J.commutative_square f g h k" obtains "J.arr f" and "J.arr g" and "J.arr h" and "J.arr k" and "J.cod f = J.cod g" and "J.dom h = J.dom k" and "J.dom f = J.cod h" and "J.dom g = J.cod k" and "f \<cdot>\<^sub>J h = g \<cdot>\<^sub>J k"
prems: shows "\<S>\<bullet>\<C> \<turnstile> vs @ [Label n es' lfilledk] @ es'' : tsa__ _> ts'"
iLastStrong_True: fixes t0 :: "nat"   and I :: "nat set" shows "(\<ominus>\<^sub>S t t0 I. True) = (I \<down>< t0 \<noteq> {} \<and> t0 \<in> I)"
finite_dimensional_vector_space_pair_1_on_def: fixes S1 :: "'c set"   and S2 :: "'d set"   and scale1 :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and scale2 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and Basis1 :: "'c set" shows "finite_dimensional_vector_space_pair_1_on S1 S2 scale1 scale2 Basis1 \<equiv> finite_dimensional_vector_space_on S1 scale1 Basis1 \<and> vector_space_on S2 scale2"
tendsto_PInfty_eq_at_top: fixes f :: "'b \<Rightarrow> real"   and F :: "'b filter" shows "((\<lambda>z. ereal (f z)) \<longlongrightarrow> \<infinity>) F = filterlim f at_top F"
inv: shows "\<forall>ip dip. let nhip = the (nhop (rt (\<sigma> ip)) dip) in dip \<in> vD (rt (\<sigma> ip)) \<inter> vD (rt (\<sigma> nhip)) \<and> nhip \<noteq> dip \<longrightarrow> rt (\<sigma> ip) \<sqsubset>\<^bsub>dip\<^esub> rt (\<sigma> nhip)"
simps: fixes m :: "nat"   and k :: "nat"   and acc :: "nat"   and n :: "nat" shows "nth_root_nat_aux m k acc n = (let acc' = (k + 1) ^ m in if n \<le> k \<or> n < acc' then k else nth_root_nat_aux m (k + 1) acc' n)"
simps_3: shows "X4d X4.Xd2 = D2"
assms_3: shows "fmrestrict_set vs (snd a) = fmrestrict_set vs (snd a')"
codomains_char: fixes f :: "'a" shows "C.codomains f = (if C.arr f then {C.cod f} else {})"
m_lcomm: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int"   and z :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "z \<in> carrier Zp" shows "x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
eq: shows "(\<lambda>x. (f x, indicator A x)) ` space M \<inter> {x. snd x = 1} = (\<lambda>x. (f x, 1)) ` A"
reduction_subsets_is_get_subsets_R: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list"   and subsets :: "(nat list \<times> nat list) list"   and m :: "rat mat" shows "reduction_subsets_R p qs signs subsets m = get_subsets_R (reduce_system_R p (qs, m, subsets, signs))"
translate: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" shows "ccw' a (a + b) (a + c) = ccw' 0 b c"
carrier_vec_dim_vec: fixes v :: "'b Matrix.vec" shows "v \<in> carrier_vec (dim_vec v)"
sup2I1: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and y :: "'e"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "A x y" shows "sup A B x y"
map_add_Some_iff: fixes m :: "'c \<Rightarrow> 'b option"   and n :: "'c \<Rightarrow> 'b option"   and k :: "'c"   and x :: "'b" shows "((m ++ n) k = Some x) = (n k = Some x \<or> n k = None \<and> m k = Some x)"
F1: fixes F :: "nat \<Rightarrow> 'a fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'a fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'a fract"   and G1 :: "'a poly"   and G2 :: "'a poly" assumes "subresultant_prs_locale F n \<delta> f k \<beta> G1 G2" shows "F 1 = map_poly to_fract G1"
uminus: fixes F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "o[F](\<lambda>x. - g x) = o[F](g)"
in_SPRAY_path: fixes P :: "'a set"   and x :: "'a" assumes "P \<in> SPRAY x" shows "x \<in> P"
map_prod_image: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and A :: "'e set"   and B :: "'f set" shows "map_prod f g ` (A \<times> B) = f ` A \<times> g ` B"
length_filter_P_impl_Q: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and xs :: "'a list" assumes "\<And>x. P x \<Longrightarrow> Q x" shows "length (filter P xs) \<le> length (filter Q xs)"
measure_Union': fixes F :: "'c set set"   and M :: "'c measure" assumes "finite F"   and "\<And>S. S \<in> F \<Longrightarrow> S \<in> fmeasurable M"   and "disjoint F" shows "Sigma_Algebra.measure M (\<Union> F) = sum (Sigma_Algebra.measure M) F"
list_of_eucl_coord_update: fixes xs :: "nat list"   and i :: "'a"   and vs :: "real list"   and x :: "'a"   and p :: "real" assumes "length xs = DIM('a)"   and "distinct xs"   and "i \<in> Basis"   and "\<And>n. n \<in> set xs \<Longrightarrow> n < length vs" shows "list_updates xs (list_of_eucl (x + (p - x \<bullet> i) *\<^sub>R i)) vs = (list_updates xs (list_of_eucl x) vs)[xs ! index Basis_list i := p]"
game2b_c: shows "map_spmf fst game2_b = map_spmf fst game2_c"
SUP_absorb: 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 k :: "'b"   and I :: "'b set"   and A :: "'b \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "k \<in> I" shows "sup (A k) (Sup (A ` I)) = Sup (A ` I)"
update_id: shows "State_Monad.update (\<lambda>x. x) = State_Monad.return ()"
emb_at_Nil: fixes t :: "('c, 'd) tm" shows "emb_at [] t = t"
\<I>simple_closed_glob_consistent: shows "conc.closed_glob_consistent \<I>simple"
abs_eq: fixes xa :: "\<upsilon> \<Rightarrow> j \<Rightarrow> i \<Rightarrow> bool"   and x :: "\<kappa>" shows "\<lparr>make\<Pi>\<^sub>1 xa,x\<rparr> = make\<o> (\<lambda>s w. proper x \<and> xa (\<nu>\<upsilon> (rep x)) s w)"
CSP_meet: fixes x :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and y :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "is_CSP_process x"   and "is_CSP_process y" shows "is_CSP_process (x \<and> y)"
intro: fixes U\<^sub>1 :: "'a set"   and Sup\<^sub>1 :: "'a set \<Rightarrow> 'a"   and U\<^sub>2 :: "'b set"   and Sup\<^sub>2 :: "'b set \<Rightarrow> 'b" assumes "Sup_ow U\<^sub>1 Sup\<^sub>1"   and "Sup_ow U\<^sub>2 Sup\<^sub>2" shows "Sup_pair_ow U\<^sub>1 Sup\<^sub>1 U\<^sub>2 Sup\<^sub>2"
join_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "compatible a b"   and "compatible c d"   and "a \<sqsubseteq> c"   and "b \<sqsubseteq> d" shows "a \<squnion> b \<sqsubseteq> c \<squnion> d"
countable_set_pmf: fixes p :: "'c pmf" shows "countable (set_pmf p)"
is_derivation_scaleR: fixes x :: "('a \<Rightarrow> real) \<Rightarrow> real"   and p :: "'a"   and c :: "real" assumes "is_derivation x p" shows "is_derivation (c *\<^sub>R x) p"
convex_polygon_def: fixes xs :: "((real \<times> real) \<times> real \<times> real) list" shows "convex_polygon xs = (convex_polychain xs \<and> (xs \<noteq> [] \<longrightarrow> fst (hd xs) = snd (last xs)))"
counterexample: fixes r :: "'b rexp"   and s :: "'b rexp"   and ws :: "('a list \<times> 's \<times> 's) list"   and ps :: "('s \<times> 's) list"   and N :: "('s \<times> 's) set" assumes "closure_invariant ([([], init r, init s)], [], {(post (init r), post (init s))}) = Some (ws, ps, N)"   and "local.wf n r"   and "local.wf n s"   and "ws \<noteq> []" shows "(rev (fst (hd ws)) \<in> lang n r) = (rev (fst (hd ws)) \<notin> lang n s)"
simps: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and cms :: "('a \<times> (Mode \<Rightarrow> 'b set)) list"   and mem :: "'b \<Rightarrow> 'c" assumes "sifum_security stop eval some_val some_val'" shows "sifum_security.sound_mode_use eval (cms, mem) = (list_all (\<lambda>cm. sifum_security.locally_sound_mode_use eval (cm, mem)) cms \<and> sifum_security.globally_sound_mode_use eval (cms, mem))"
ordinal_minus_eq_0: fixes x :: "ordinal"   and y :: "ordinal" assumes "x \<le> y" shows "x - y = 0"
aff_dim_zero: fixes S :: "'c set" assumes "(0::'c) \<in> affine hull S" shows "aff_dim S = int (dim S)"
simps: fixes A :: "'b mat"   and b :: "'b vec" shows "mat_append_col A b = mat_of_cols (dim_row A) (cols A @ [b])"
t_times_pow_pos: fixes p :: "int"   and f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int"   and n :: "nat" assumes "hensel p f a" shows "0 < eint (2 ^ n) * hensel.hensel_factor p f a"
reduced_run_decompose: fixes ex :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and en :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and init :: "'b \<Rightarrow> bool"   and int :: "'b \<Rightarrow> 'c"   and ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and src :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ren :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and q :: "'b"   and v\<^sub>1 :: "'a list"   and v\<^sub>2 :: "'a stream"   and l :: "'a list"   and w :: "'a list"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list"   and u :: "'a stream" assumes "ample_abstract ex en init int ind src ren"   and "ample_abstract.reduced_run ex en int ind ren q v\<^sub>1 v\<^sub>2 l w w\<^sub>1 w\<^sub>2 u" shows "(traces.eq_swap ind)\<^sup>*\<^sup>* w (w\<^sub>1 @ w\<^sub>2)"
succ_as_E: fixes succ :: "'c \<Rightarrow> 'b list"   and x :: "'c" shows "set (succ x) = E_of_succ succ `` {x}"
card_minus_fMin: fixes s :: "'a fset" assumes "s \<noteq> {||}" shows "card (fset s - {fMin s}) < card (fset s)"
lcoset_closed: fixes G :: "'b set"   and a :: "'b"   and A :: "'b set" assumes "Group G"   and "a \<in> G"   and "order.greater_eq G A" shows "order.greater_eq G (a +o A)"
Min_le_iff: fixes A :: "'a word set"   and x :: "'a word" assumes "finite A"   and "A \<noteq> {}" shows "(signed.Min A \<le>s x) = (\<exists>a\<in>A. a \<le>s x)"
chain_boundary_boundary: fixes p :: "nat"   and X :: "'b topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "singular_chain p X c" shows "chain_boundary (p - Suc 0) (chain_boundary p c) = 0"
st_SReal: fixes x :: "real star" assumes "x \<in> HFinite" shows "st x \<in> \<real>"
app6: fixes env :: "v sem_env"   and es :: "exp0 list"   and rs :: "(v, v) result list"   and err :: "v error_result"   and op0 :: "op0" assumes "list_all2_shortcircuit (cupcake_evaluate_single env) (rev es) rs"   and "sequence_result rs = Rerr err" shows "cupcake_evaluate_single env (App op0 es) (Rerr err)"
mult_fract_cancel: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c \<noteq> (0::'a)" shows "Fraction_Field.Fract (c * a) (c * b) = Fraction_Field.Fract a b"
algebra_Pow: fixes sp :: "'c set" shows "algebra sp (Pow sp)"
Least_eq: shows "(LEAST ka. Gauss_Jordan_in_ij A (0::'rows) (mod_type_class.from_nat k) $ i $ ka \<noteq> (0::'a)) = mod_type_class.from_nat k"
invertible_Gauss_Jordan: fixes A :: "(('a, 'c) vec, 'b) vec" shows "\<exists>P. invertible P \<and> Gauss_Jordan A = P ** A"
o_eq_dest: fixes a :: "'f \<Rightarrow> 'e"   and b :: "'d \<Rightarrow> 'f"   and c :: "'g \<Rightarrow> 'e"   and d :: "'d \<Rightarrow> 'g"   and v :: "'d" assumes "a \<circ> b = c \<circ> d" shows "a (b v) = c (d v)"
no_clm_before_res: fixes c :: "cars" shows "\<forall>ts v. \<nexists>va u. v=va\<parallel>u \<and> (0 < \<parallel>ext va\<parallel> \<and> len va ts c = ext va \<and> restrict va (clm ts) c = lan va \<and> |lan va| = 1) \<and> 0 < \<parallel>ext u\<parallel> \<and> len u ts c = ext u \<and> restrict u (res ts) c = lan u \<and> |lan u| = 1"
P: shows "P = (\<lambda>X. X \<inter> C) ` P'"
prec_frame: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and R1 :: "assn"   and R2 :: "assn"   and h :: "heap \<times> nat set"   and x :: "'a"   and p :: "'b"   and F1 :: "assn"   and y :: "'a"   and F2 :: "assn" assumes "precise P"   and "h \<Turnstile> R1 \<and>\<^sub>A R2"   and "R1 \<Longrightarrow>\<^sub>A P x p * F1"   and "R2 \<Longrightarrow>\<^sub>A P y p * F2" shows "x = y"
c1_2: fixes x :: "real" assumes "1 \<le> x" shows "sum_upto f x - g x \<le> c1 * h x"
test_multipliciativer: fixes R :: "('a \<times> 'a set) set"   and S :: "('a \<times> 'a set) set"   and T :: "('a \<times> 'a set) set" assumes "R \<subseteq> 1\<^sub>\<sigma>"   and "S \<subseteq> 1\<^sub>\<sigma>" shows "R \<inter> S \<cdot> T = R \<cdot> T \<inter> (S \<cdot> T)"
naturality: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "\<sigma>\<^sub>1 (C.cod \<mu>) \<cdot>\<^sub>D (G \<mu> \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>)) = (\<sigma>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D \<sigma>\<^sub>1 (C.dom \<mu>)"
sq_norm_poly_ge_0: fixes p :: "'d poly" shows "(0::'d) \<le> \<parallel>p\<parallel>\<^sup>2"
qSwapAll_simps_1: fixes zs :: "'f"   and x :: "'g"   and y :: "'g"   and zs' :: "'f"   and z :: "'g" shows "qVar zs' z #[[x \<and> y]]_zs = qVar zs' (z @zs'[x \<and> y]_zs)"
transf_def: fixes G :: "nat graph_rec"   and n :: "nat"   and input :: "nat list" assumes "cfg_doms G" shows "transf n input \<equiv> n # input"
lim: shows "(f \<longlongrightarrow> C) at_top"
CARD_length_enum: shows "CARD('b) = length enum_class.enum"
scale_zero_left: fixes x :: "'b" shows "(0::'a) *s x = (0::'b)"
hypnat_le_add1: fixes x :: "nat star"   and n :: "nat star" shows "x \<le> x + n"
g_rw: shows "g = (fst g, fst (snd g), snd (snd g))"
p2: shows "lead_coeff p2 \<in> carrier R - {\<zero>}"
simps_6: fixes v :: "String.literal" shows "is_Array (Class v) = False"
ordered_fmap_distinct: fixes S :: "('a \<times> 'b) fset" assumes "is_fmap S" shows "distinct (map fst (ordered_fmap S))"
numer_cf3_mono: fixes y :: "real"   and x :: "real" assumes "y \<le> x" shows "numer_cf3 y \<le> numer_cf3 x"
Vfbasis_V: shows "set Vfbasis \<subseteq> V"
insert_Diff_single: fixes a :: "'b"   and A :: "'b set" shows "a \<triangleright> A - {a} = a \<triangleright> A"
pbernpoly_0: fixes n :: "nat" shows "pbernpoly n 0 = bernoulli n"
sorted_on_sort_key_on: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" assumes "f ` set xs \<subseteq> Field r" shows "sorted_on r (map f (sort_key_on r f xs))"
sublist_el': fixes i :: "nat"   and j :: "nat"   and xs :: "'c list"   and x :: "'c" assumes "i \<le> j"   and "j < length xs" shows "x \<in> set (WB_Sort.sublist xs i j) \<equiv> \<exists>k\<ge>i. k \<le> j \<and> xs ! k = x"
T: fixes t' :: "trm" assumes "t \<mapsto> t'"   and "r = t' \<star> [y]n\<ggreater>L" shows "P"
nth_in_Union_over_set: fixes i :: "nat"   and xs :: "'c set list" assumes "i < length xs" shows "xs ! i \<subseteq> \<Union> (set xs)"
call_indirect: fixes i :: "nat"   and \<C> :: "t_context"   and t1s :: "t list"   and t2s :: "t list" assumes "i < length (types_t \<C>)"   and "types_t \<C> ! i = t1s _> t2s"   and "table \<C> \<noteq> None" shows "\<C> \<turnstile> [Call_indirect i] : t1s @ [T_i32] _> t2s"
empty_not_in_map_cone: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" shows "{} \<notin> set (map cone ps)"
Broadcast_Expunge_Deliver_prefix_closed: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and i :: "'a"   and e :: "'b"   and mo :: "'a"   and j :: "nat" assumes "xs @ [Broadcast (i, Expunge e mo i)] prefix of j" shows "Deliver (mo, Append mo e) \<in> set xs \<or> (\<exists>mo2. Deliver (mo, Store e mo2 mo) \<in> set xs)"
simps_5: fixes f1 :: "'h \<Rightarrow> 'g"   and f2 :: "'i \<Rightarrow> 'g"   and x1 :: "'h" shows "(case Inl x1 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f1 x1"
upper_triangular_inverse: fixes A :: "(('a, 'b) vec, 'b) vec" assumes "upper_triangular A"   and "invertible A" shows "upper_triangular (matrix_inv A)"
widen_acom_def: shows "(\<nabla>) = map2_acom (\<nabla>)"
dfa_def: fixes M :: "'b dfa" shows "dfa M \<equiv> (dfa.init M \<in> dfa.states M \<and> dfa.final M \<subseteq> dfa.states M) \<and> (\<forall>q x. q \<in> dfa.states M \<longrightarrow> dfa.nxt M q x \<in> dfa.states M) \<and> finite (dfa.states M)"
delete_simps_1: fixes k :: "'a" shows "delete k [] = []"
insert_disjoint_2: fixes a :: "'a"   and A :: "'a set"   and B :: "'a set" shows "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"
simp_2: shows "mk_rtrancl_main todo fin = (if set todo = {} then fin else let (a, tod) = case todo of h # t \<Rightarrow> (h, t) in if \<exists>b\<in>fin. subsumes b a then mk_rtrancl_main tod fin else mk_rtrancl_main (r a @ tod) (insert a fin))"
intro_elim_3_b: fixes v :: "i"   and \<psi> :: "\<o>"   and \<phi> :: "\<o>" assumes "[\<psi> in v]" shows "[\<phi> \<^bold>\<or> \<psi> in v]"
rev_replicate: fixes n :: "nat"   and x :: "'a" shows "rev (replicate n x) = replicate n x"
simps: fixes r :: "(bool \<times> 'b) rexp"   and b :: "bool" shows "L_b (r, b) = Lm r \<union> (if b then {[]} else {})"
eq: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq s t"
lhd_lzip: fixes xs :: "'c llist"   and ys :: "'d llist" assumes "\<not> lnull xs"   and "\<not> lnull ys" shows "lhd (lzip xs ys) = (lhd xs, lhd ys)"
eventually_nonneg_at_top: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "filterlim f at_top F" shows "eventually_nonneg F f"
in_on_graph: fixes x :: "'a"   and G :: "('b, 'a) labeled_graph"   and a :: "'a \<Rightarrow> 'c"   and y :: "'d"   and b :: "('c \<times> 'd) set" assumes "x \<in> vertices G"   and "(a x, y) \<in> b" shows "(x, y) \<in> on_graph G a O b"
B: shows "B \<in> folding.\<C> - f \<turnstile> folding.\<C>"
set_plus_mono: fixes C :: "'c set"   and D :: "'c set"   and a :: "'c" assumes "C \<subseteq> D" shows "a +o C \<subseteq> a +o D"
abs_eq: fixes x :: "(Vars.var \<times> Vars.var) set"   and xb :: "Vars.var"   and xa :: "Vars.var" assumes "eq_onp sym x x" shows "xb--xa\<in>Abs_CoCall x = ((xb, xa) \<in> x)"
abs_eq: fixes x :: "nat \<Rightarrow> rat" assumes "realrel x x" shows "- Real x = Real (\<lambda>n. - x n)"
cases: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" assumes "a1 >\<^sub>t a2" obtains ti :: "('s, 'v) tm" and t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = tb__" and "a2 = sb__" and "ti \<in> set (args tb__)" and "ti \<ge>\<^sub>t sb__"   | t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = tc__" and "a2 = sc__" and "head tc__ >\<^sub>h\<^sub>d head sc__" and "chkvar tc__ sc__" and "chkargs (>\<^sub>t) tc__ sc__"   | t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = t" and "a2 = s" and "head t = head s" and "chkargs (>\<^sub>t) t s" and "\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s)"
Gmult_flincomb_comm: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and g :: "'a"   and vs :: "'c list"   and as :: "'b list" assumes "FGModule G smult V"   and "g \<in> G"   and "set vs \<subseteq> V" shows "aezfun_scalar_mult.Gmult smult g (scalar_mult.lincomb (aezfun_scalar_mult.fsmult smult) as vs) = scalar_mult.lincomb (aezfun_scalar_mult.fsmult smult) as (map (aezfun_scalar_mult.Gmult smult g) vs)"
isDiagonal_upt_k_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" shows "isDiagonal_upt_k A k = (\<forall>a b. to_nat a \<noteq> to_nat b \<and> (to_nat a < k \<or> to_nat b < k) \<longrightarrow> A $ a $ b = (0::'a))"
reduced_GB_D4: fixes G :: "('a \<Rightarrow>\<^sub>0 'c) set"   and g :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.is_reduced_GB G"   and "g \<in> G" shows "g \<noteq> 0"
id1: fixes pmod :: "'a poly" shows "(if even d1 then - pmod else pmod) = (- 1) ^ (d1 + 1) * pmod"
is_group: shows "Group.group (G\<lparr>carrier := S\<rparr>)"
fmrestrict_set_twice: fixes S :: "'a set"   and T :: "'a set"   and m :: "('a, 'b) fmap" shows "fmrestrict_set S (fmrestrict_set T m) = fmrestrict_set (S \<inter> T) m"
antisym: fixes a :: "'b::order"   and b :: "'b::order" assumes "(a::'b::order) \<le> (b::'b::order)"   and "(b::'b::order) \<le> (a::'b::order)" shows "(a::'b::order) = (b::'b::order)"
null_char: shows "hhom\<^sub>D.null = D.null"
distinct_mset_add_mset: fixes a :: "'a"   and L :: "'a multiset" shows "distinct_mset (add_mset a L) = (a \<notin># L \<and> distinct_mset L)"
subseteq_guards_DynCom: fixes c :: "('d, 'e, 'f) com"   and C :: "'d \<Rightarrow> ('d, 'e, 'f) com" assumes "c \<subseteq>\<^sub>g DynCom C" shows "\<exists>C'. c = DynCom C' \<and> (\<forall>s. C' s \<subseteq>\<^sub>g C s)"
real_vector_space_on_axioms: shows "real_vector_space_on (dual_space S)"
f: shows "(\<lambda>t. local.monom (fst t) (snd t)) \<in> set (dense_repr p) \<rightarrow> carrier (K [X])"
SMP_wf_trm: fixes t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set" assumes "t \<in> SMP M"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s M" shows "wf\<^sub>t\<^sub>r\<^sub>m t"
IP: fixes \<sigma> :: "'\<sigma>"   and k :: "'u"   and it :: "'u set" assumes "c \<sigma>"   and "k \<in> it"   and "\<forall>j\<in>it. k \<le> j"   and "\<forall>j\<in>\<alpha> m - it. j \<le> k"   and "it \<subseteq> \<alpha> m"   and "I it \<sigma>" shows "I (it - {k}) (f k \<sigma>)"
iarray_to_matrix_def: fixes A :: "'b iarray iarray" shows "iarray_to_matrix A = (\<chi>i j. A !! mod_type_class.to_nat i !! mod_type_class.to_nat j)"
gSubstAllPresGWlsAll_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gSubstAllPresGWlsAll MOD \<equiv> gSubstPresGWls MOD \<and> gSubstAbsPresGWlsAbs MOD"
redT_updTs_invariant: fixes I :: "'t \<Rightarrow> unit option"   and ts :: "'t \<Rightarrow> ('x \<times> 'b \<Rightarrow>f nat) option"   and m :: "'m"   and t :: "'t"   and x :: "'x"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m"   and ln :: "'b \<Rightarrow>f nat"   and ln' :: "'b \<Rightarrow>f nat" assumes "ts_inv (\<lambda>_. P) I ts m"   and "t \<turnstile> \<langle>x, m\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   and "thread_oks ts \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>"   and "ts t = \<lfloor>(x, ln)\<rfloor>" shows "ts_inv (\<lambda>_. P) (upd_invs I (\<lambda>_. P) \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>) (redT_updTs ts \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>(t \<mapsto> (x', ln'))) m'"
prod_mono3: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and J :: "'e set"   and I :: "'e set"   and g :: "'e \<Rightarrow> 'd"   and f :: "'e \<Rightarrow> 'd" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "finite J"   and "I \<subseteq> J"   and "\<And>i. i \<in> J \<Longrightarrow> less_eq zero (g i)"   and "\<And>i. i \<in> I \<Longrightarrow> less_eq (g i) (f i)"   and "\<And>i. i \<in> J - I \<Longrightarrow> less_eq (g i) one" shows "less_eq (comm_monoid_mult.prod times one g J) (comm_monoid_mult.prod times one f I)"
lookup_operator_works_1: fixes op :: "(nat, nat) sas_plus_operator"   and \<pi>' :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat" assumes "lookup_action op = Some \<pi>'" shows "lookup_operator (fst \<pi>') = Some \<pi>'"
path_verts: fixes p :: "nat list"   and n :: "nat" assumes "path_entry (g_E G) p n" shows "n \<in> set (g_V G)"