hcju/isabelleps huggingface dataset:
H_weak_loop: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "mono f"   and "x \<le> f x" shows "x \<le> fiter_id f x"
assms_2: shows "canon_ang_P \<alpha> \<alpha>\<^sub>2"
mat_of_cols_index: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and ws :: "'a vec list" assumes "i < n"   and "j < length ws" shows "mat_of_cols n ws $$ (i, j) = ws ! j $ i"
exhaust_sel: fixes optionT :: "('d, 'e) optionT" obtains "optionT = OptionT (run_option optionT)"
termMOD_igFreshIGOp: shows "igFreshIGOp termMOD"
CJ_14p: fixes A :: "c \<Rightarrow> w \<Rightarrow> bool"   and B :: "c \<Rightarrow> w \<Rightarrow> bool" shows "\<forall>c. (\<lambda>w. ob (A c) (B c) \<and> (\<^bold>\<box>\<^sub>pA) c w \<and> (\<^bold>\<diamond>\<^sub>pB) c w \<and> \<I>pv w \<sqinter> (\<^bold>\<not>B) c) \<sqsubseteq> (\<^bold>O\<^sub>iB) c"
std_dual_def: fixes B :: "'c set"   and S :: "'c set"   and a :: "'c" assumes "independent B"   and "span B = S" shows "std_dual B S a = restrict0 S (restrict0 S (inner_Basis B a))"
lists_def2: fixes A :: "'a set" shows "lists A = {l. set l \<subseteq> A}"
underS_Field: fixes r :: "('a \<times> 'a) set"   and a :: "'a" shows "underS r a \<subseteq> Field r"
ZObis_def: shows "ZObis \<equiv> gfp ZOretr"
AB: shows "A * B = mat n n (\<lambda>(i, j). if i = j then A $$ (i, i) * B $$ (i, i) else 0)"
is_sentence_item_\<alpha>: fixes x :: "item" assumes "wellformed_item x" shows "is_sentence (item_\<alpha> x)"
intros_7: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and v :: "val"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" shows "P \<turnstile> \<langle>Val v,s\<rangle> \<Rightarrow> \<langle>Val v,s\<rangle>"
unique_map_inj: fixes l :: "('a \<times> 'b) list"   and f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'd" assumes "unique l"   and "inj f" shows "unique (map (\<lambda>(k, x). (f k, g k x)) l)"
mpj_mHom: fixes H :: "'a set" assumes "submodule R M H" shows "mpj M H \<in> mHom R M (M /\<^sub>m H)"
s1: shows "s1 = (x * ((y + (q - b)) mod q) + (q - r)) mod q"
ennreal_zero_neq_top: shows "0 \<noteq> \<top>"
strict_coboundedI2: fixes b :: "'c"   and c :: "'c"   and a :: "'c" assumes "b < c" shows "min a b < c"
vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p_cases_6: fixes c :: "poscheckvariant"   and t :: "('d, 'c) Term.term"   and s :: "('d, 'c) Term.term" shows "vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<langle>c: t \<in> s\<rangle> = fv t \<union> fv s"
assms: shows "A \<lesssim>\<^sub>\<circ> B"
P2_def: shows "P2 = {#p \<in># P. 0 < Polynomial.degree p#}"
poly_mult_def: fixes p :: "'a poly"   and q :: "'a poly" shows "poly_mult p q \<equiv> poly_div p (reflect_poly q)"
length_induct: fixes P :: "'b list \<Rightarrow> bool"   and xs :: "'b list" assumes "\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs" shows "P xs"
strong_confluentpI: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>r x y; r x z\<rbrakk> \<Longrightarrow> \<exists>u. r\<^sup>*\<^sup>* y u \<and> r\<^sup>=\<^sup>= z u" shows "strong_confluentp r"
f4: fixes A :: "'a set"   and Aa :: "'a set" shows "A - - (- Aa) = A - Aa"
dagger_unfoldl_distr: fixes y :: "'a"   and x :: "'a" shows "y + x\<^sup>+ ; y = x\<^sup>\<star> ; y"
lookup_OAlist_tc_map2_val_neutr: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xs :: "('b, 'c) oalist_tc"   and ys :: "('b, 'c) oalist_tc"   and k :: "'b" assumes "\<And>k x. f k x (0::'c) = x"   and "\<And>k x. f k (0::'c) x = x" shows "OAlist_tc_lookup (OAlist_tc_map2_val_neutr f xs ys) k = f k (OAlist_tc_lookup xs k) (OAlist_tc_lookup ys k)"
inj_atom_map: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<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" assumes "class.atomic_boolean_algebra minus uminus inf less_eq less sup bot top" shows "inj (bounded_lattice.atom_map less_eq less bot)"
derives1_implies_Derives1: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and u :: "symbol list"   and v :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.derives1 \<NN> \<TT> \<RR> u v" shows "\<exists>i r. CFG.Derives1 \<NN> \<TT> \<RR> u i r v"
shd_sset: fixes a :: "'c stream" shows "shd a \<in> sset a"
vonStaudt_Clausen_lemma: fixes n :: "nat"   and p :: "nat" assumes "0 < n"   and "prime p" shows "[\<Sum>m<p. (- 1) ^ m * int (p - 1 choose m) * int m ^ (2 * n) = if p - 1 dvd 2 * n then - 1 else 0] (mod int p)"
proj2_abs_mult: fixes c :: "real"   and v :: "(real, 3) vec" assumes "c \<noteq> 0" shows "proj2_abs (c *\<^sub>R v) = proj2_abs v"
Range_rtrancl: fixes R :: "('a \<times> 'a) set" shows "Range (R\<^sup>*) = UNIV"
finite_nat_set_bounded_iff2: fixes N :: "nat set" shows "finite N = (\<exists>n. \<forall>i\<in>N. i \<le> n)"
simps_5: fixes x :: "'d"   and \<phi> :: "('b + 'e, 'c, 'd) formula"   and \<sigma> :: "'e \<Rightarrow> ('b, 'd) trm" shows "FsubstFO (Exists x \<phi>) \<sigma> = Exists x (FsubstFO \<phi> \<sigma>)"
iterator_to_ordered_iterator_def: fixes sort_fun :: "'e list \<Rightarrow> 'd list"   and it :: "('e list \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'e list \<Rightarrow> 'e list) \<Rightarrow> 'e list \<Rightarrow> 'e list" shows "iterator_to_ordered_iterator sort_fun it = foldli (sort_fun (iterate_to_list it))"
simps_2: fixes e :: "expr"   and es :: "expr list" shows "fvs (e # es) = fv e \<union> fvs es"
intro: fixes L :: "('a, 'b) gorder_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier L; y \<in> carrier L\<rbrakk> \<Longrightarrow> x \<sqsubseteq>\<^bsub>L\<^esub> y \<or> y \<sqsubseteq>\<^bsub>L\<^esub> x" shows "weak_total_order_axioms L"
vec_simps_24: fixes aj :: "'b"   and ch :: "'b"   and bj :: "'b" shows "(aj < ch - bj) = (aj + bj < ch)"
LLq: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm" shows "LLq t1 t2 \<in> fmla"
NonUniformImpliesNotDecided: fixes c :: "('p, 'v, 's) configuration"   and v :: "bool" assumes "\<forall>cfg. reachable c cfg \<longrightarrow> agreement cfg"   and "\<And>cc Q. terminationPseudo 1 cc Q"   and "nonUniform c"   and "initReachable c \<and> <\<bottom>, outM v> \<in># msgs c" shows "False"
r_univuniv_recfn: shows "recfn 2 r_univuniv"
diff_add_assoc2: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" assumes "a \<subseteq># b" shows "b + c - a = b - a + c"
Sadmit_def: fixes \<sigma> :: "('sf, 'sc, 'sz) subst"   and S :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" shows "Sadmit \<sigma> S = ((\<forall>i\<ge>0. i < length (fst S) \<longrightarrow> Fadmit \<sigma> (fst S ! i)) \<and> (\<forall>i\<ge>0. i < length (snd S) \<longrightarrow> Fadmit \<sigma> (snd S ! i)))"
simps: fixes a1 :: "'b"   and a2 :: "'b list" shows "ListMem a1 a2 = ((\<exists>x xs. a1 = x \<and> a2 = x # xs) \<or> (\<exists>x xs y. a1 = x \<and> a2 = y # xs \<and> ListMem x xs))"
ord_eq_less_trans: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "a = b"   and "c <lex b" shows "c <lex a"
is_sentence_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and s :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.is_sentence \<NN> \<TT> s = list_all (CFG.is_symbol \<NN> \<TT>) s"
Liminf_llist_imp_exists_index: fixes x :: "'a"   and Xs :: "'a set llist" assumes "x \<in> Liminf_llist Xs" shows "\<exists>i. enat i < llength Xs \<and> x \<in> lnth Xs i"
encR: fixes S :: "'a"   and Enc :: "'a \<Rightarrow> 'b" shows "(SourceTerm S, TargetTerm (Enc S)) \<in> encoding.indRelRPO Enc"
b: fixes x :: "real" assumes "b \<le> x" shows "\<bar>f x\<bar> \<le> c * \<bar>g x\<bar>"
f_borel_Suc_real_cond_exp: fixes f :: "bool stream \<Rightarrow> real"   and n :: "nat"   and g :: "bool stream \<Rightarrow> 'a"   and N :: "'a measure" assumes "f \<in> borel_measurable (F (Suc n))"   and "g \<in> F n \<rightarrow>\<^sub>M N"   and "set_discriminating n g N"   and "\<forall>w. g -` {g w} \<in> sets (F n)"   and "\<forall>r\<in>range g \<inter> space N. \<exists>A\<in>sets N. range g \<inter> A = {r}"   and "\<forall>y z. g y = g z \<and> y !! n = z !! n \<longrightarrow> f y = f z"   and "0 < p"   and "p < 1" shows "AE w in M. real_cond_exp M (fct_gen_subalgebra M N g) f w = p * f (pseudo_proj_True n w) + (1 - p) * f (pseudo_proj_False n w)"
max_tstp_intro'''': fixes tstp :: "nat + nat"   and tstp' :: "nat + nat"   and ts' :: "nat"   and tp' :: "nat" assumes "isl tstp = isl tstp'"   and "ts_tp_lt' ts' tp' tstp'" shows "ts_tp_lt' ts' tp' (max_tstp tstp tstp')"
allUses_def: fixes g :: "'g"   and n :: "'node" shows "allUses g n \<equiv> uses g n \<union> phiUses g n"
in_carrier1: shows "(g' [^] x2) [^] c \<in> carrier \<G>"
h1: shows "\<Gamma> \<turnstile> s is t : T\<^sub>1\<rightarrow>T\<^sub>2"
simps_9: fixes f :: "'f \<Rightarrow> 'e"   and x2 :: "'f" shows "map_option f (Some x2) = Some (f x2)"
pseudo_proj_False_preimage: fixes g :: "bool stream \<Rightarrow> 'b"   and n :: "nat"   and N :: "'b measure"   and z :: "bool stream" assumes "g \<in> F n \<rightarrow>\<^sub>M N"   and "set_discriminating n g N" shows "pseudo_proj_False n -` g -` {g z} = pseudo_proj_False n -` pseudo_proj_False n ` g -` {g z}"
WP1rd: fixes p :: "('b, 'c) Combinators list" assumes "wellformed_policy1_strong p" shows "wellformed_policy1_strong (remdups p)"
scale_scale: fixes a :: "real"   and b :: "real"   and x :: "'c" shows "a *\<^sub>R b *\<^sub>R x = (a * b) *\<^sub>R x"
is_arg_max_linorder: fixes f :: "'a \<Rightarrow> 'b"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" shows "is_arg_max f P x = (P x \<and> (\<forall>y. P y \<longrightarrow> f y \<le> f x))"
type_less_x_ObjectType_intro: fixes \<tau> :: "'a basic_type"   and \<C> :: "'a"   and \<D> :: "'a" assumes "\<tau> = \<langle>\<C>\<rangle>\<^sub>\<T>"   and "\<C> < \<D>" shows "\<tau> < \<langle>\<D>\<rangle>\<^sub>\<T>"
concurrent_ops_commute_def: fixes xs :: "('a \<times> ('a, 'b) operation) list" shows "hb.concurrent_ops_commute xs \<equiv> \<forall>x y. {x, y} \<subseteq> set xs \<longrightarrow> hb.concurrent x y \<longrightarrow> interp_msg x \<rhd> interp_msg y = interp_msg y \<rhd> interp_msg x"
ground_subset: fixes M :: "('a, 'b) Term.term set"   and N :: "('a, 'b) Term.term set" assumes "M \<subseteq> N"   and "ground N" shows "ground M"
maximal_element_def: fixes D :: "('h, 'i) Order_scheme"   and m :: "'h" shows "maximal\<^bsub>D\<^esub> m = (m \<in> carrier D \<and> (\<forall>b\<in>carrier D. m \<preceq>\<^bsub>D\<^esub> b \<longrightarrow> m = b))"
f1_Two: fixes x :: "three"   and y :: "three" assumes "x = Two \<or> y = Two" shows "f1 x y = Two"
in_star_iff_concat: fixes w :: "'a list"   and A :: "'a list set" shows "(w \<in> A\<star>) = (\<exists>ws. set ws \<subseteq> A \<and> w = concat ws)"
swap_apply_1: fixes a :: "'d"   and b :: "'d"   and f :: "'d \<Rightarrow> 'c" shows "Fun.swap a b f a = f b"
Abs_run_inverse: fixes y :: "nat \<Rightarrow> clock \<Rightarrow> bool \<times> 'b tag_const" assumes "y \<in> {\<rho>. \<forall>c. mono (\<lambda>n. time (\<rho> n c))}" shows "Rep_run (Abs_run y) = y"
h_prop: shows "(cms\<^sub>2 ! k, subst h (snd (mems ! k))) \<leadsto> \<langle>c\<^sub>h, snd (cms\<^sub>1' ! k), mem\<^sub>h\<rangle> \<and> (cms\<^sub>1' ! k, subst (g1 h) mem\<^sub>1') \<approx> \<langle>c\<^sub>h, snd (cms\<^sub>1' ! k), mem\<^sub>h\<rangle>"
add_zeros: fixes m :: "nat"   and n :: "nat" shows "foldl (+) m (replicate n 0) = m"
map_reduce: fixes m :: "'c \<Rightarrow> 'd option"   and a :: "'c"   and B :: "'c set" assumes "dom m = {a} \<union> B" shows "\<exists>m'. dom m' = B \<and> (\<forall>x\<in>B. m x = m' x)"
max_rank_def: shows "max_rank = card (reach \<Sigma> \<delta> q\<^sub>0 - Collect sink)"
welldefined: shows "all_list_constr_R subsets (length qs)"
cont2contlubE: fixes f :: "'b \<Rightarrow> 'c"   and Y :: "nat \<Rightarrow> 'b" assumes "cont f"   and "chain Y" shows "f (\<Squnion>i. Y i) = (\<Squnion>i. f (Y i))"
hermite_ring: shows "\<forall>A. admits_triangular_reduction A"
llist_of_tllist_tdropn: fixes n :: "nat"   and xs :: "('a, 'b) tllist" shows "llist_of_tllist (tdropn n xs) = ldropn n (llist_of_tllist xs)"
assms_4: shows "\<forall>x. insertion (nth_default 0 (L[var := x])) A = Av"
boundedI: fixes b :: "'c multiset"   and a :: "'c multiset"   and c :: "'c multiset" assumes "b \<subseteq># a"   and "c \<subseteq># a" shows "b \<union># c \<subseteq># a"
intra_SDG_path_intra_CFG_path: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.intra_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'" obtains as :: "'a list" where "CFG.intra_path sourcenode targetnode kind valid_edge (parent_node n) as (parent_node n')"
nb: shows "set (concat (\<Phi>\<inverse> \<Pi> \<A> t)) = (\<Union>k<t. set (decode_plan' \<Pi> \<A> k))"
norm_power_diff: fixes z :: "'b"   and w :: "'b"   and m :: "nat" assumes "norm z \<le> 1"   and "norm w \<le> 1" shows "norm (z ^ m - w ^ m) \<le> real m * norm (z - w)"
simplicial_simplex_singular_face: fixes p :: "nat"   and S :: "(nat \<Rightarrow> real) set"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real"   and k :: "nat" assumes "simplicial_simplex p S f"   and "1 \<le> p"   and "k \<le> p" shows "simplicial_simplex (p - Suc 0) S (singular_face p k f)"
mon: fixes f :: "int poly" assumes "f \<in># fs" shows "monic (q.Mp f)"
preserves_hom: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>" shows "\<guillemotleft>F f : F a \<Rightarrow>\<^sub>C F b\<guillemotright>"
Diff_eq: fixes A :: "'d set"   and B :: "'d set" shows "A - B = A \<inter> - B"
statediff_def: fixes \<nu> :: "variable \<Rightarrow> real"   and \<omega> :: "variable \<Rightarrow> real" shows "statediff \<nu> \<omega> = {x. \<nu> x \<noteq> \<omega> x}"
r: shows "r = (\<Squnion>m. C\<^bsup>f m\<^esup>)"
goedel_rosser_first: assumes "consistent" shows "\<not> prv \<phi>R \<and> \<not> prv (neg \<phi>R)"
simplify_sum_of_powers: fixes x :: "nat"   and n :: "nat" shows "(x - 1) * sum ((^) x) {0..n} = x ^ (n + 1) - 1"
leq: fixes x :: "'a" assumes "x \<in> A" shows "f x \<le> g x"
rtranclp_unfold: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" shows "r\<^sup>*\<^sup>* a b \<equiv> a = b \<or> r\<^sup>+\<^sup>+ a b"
x: shows "(x, a) \<in> \<langle>A\<rangle>phantom_rel"
poly_subst_monomial: fixes f :: "'e \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and t :: "'e \<Rightarrow>\<^sub>0 nat"   and c :: "'d" shows "poly_subst f (monomial c t) = punit.monom_mult c 0 (subst_pp f t)"
t_2: shows "t \<notin> Sec - \<Union> {GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l1 \<A>)) \<inter> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l2 \<A>)) |l1 l2. l1 \<noteq> l2}"
b1: shows "q1 = 0 \<or> b1 \<in> sqrt_irrat"
is_span: shows "C.span src_f.cod.leg0 src_f.cod.leg1"
fst_add: fixes x :: "'d \<times> 'e"   and y :: "'d \<times> 'e" shows "fst (x + y) = fst x + fst y"
ODnods_def: shows "ODnods = \<lparr>carrier = ODnums, rel = ODrel\<rparr>"
crename_OrR2': fixes R :: "trm"   and a :: "coname"   and b :: "coname"   and c :: "coname"   and N :: "trm"   and d :: "coname" assumes "R[a\<turnstile>c>b] = OrR2 <c>.N d"   and "c \<sharp> (R, a, b)"   and "d \<sharp> a" shows "(\<exists>N'. R = OrR2 <c>.N' d \<and> N'[a\<turnstile>c>b] = N) \<or> (\<exists>N'. R = OrR2 <c>.N' a \<and> b = d \<and> N'[a\<turnstile>c>b] = N)"
not_sum_power2_lt_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "\<not> less (plus (power.power one times x 2) (power.power one times y 2)) zero"
ord_order_ow_def: fixes U\<^sub>2 :: "'a set"   and le\<^sub>2 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls\<^sub>2 :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "ord_order_ow U\<^sub>2 le\<^sub>2 ls\<^sub>2 \<equiv> order_ow U\<^sub>2 le\<^sub>2 ls\<^sub>2"
ag_inv_zero: shows "-\<^sub>a \<zero> = \<zero>"
ereal_add_uminus_conv_diff: fixes x :: "ereal"   and y :: "ereal" shows "- x + y = y - x"
that: fixes b :: "int"   and a :: "int" assumes "0 < b"   and "coprime a b"   and "x = of_int a / of_int b" shows "thesis"
fls_shift_fls_shift: fixes m :: "int"   and k :: "int"   and f :: "'a fls" shows "fls_shift m (fls_shift k f) = fls_shift (k + m) f"
collinear_3_eq_affine_dependent: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "collinear {a, b, c} = (a = b \<or> a = c \<or> b = c \<or> affine_dependent {a, b, c})"
xor_ac_3: fixes x :: "'e"   and y :: "'e"   and z :: "'e" assumes "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<oplus> y \<oplus> z = y \<oplus> x \<oplus> z"
DJ_def: fixes f :: "fm \<Rightarrow> fm"   and p :: "fm" shows "DJ f p \<equiv> evaldjf f (disjuncts p)"
last_in_set: fixes as :: "'a list" assumes "as \<noteq> []" shows "last as \<in> list.set as"
set_eq: shows "{x. p x = 1} = {x \<in> {..n}. p x = 1}"
gen: fixes S :: "variable set" assumes "finite S"   and "S \<subseteq> - FVT \<theta>" shows "term_sem I \<theta> \<omega>' = term_sem I \<theta> (stateinterpol \<omega> \<omega>' S)"
simps_1: fixes a :: "Arity"   and e :: "exp"   and x :: "Vars.var" shows "ccTransform a (App e x) = App (ccTransform (inc\<cdot>a) e) x"
LAssThrow: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and V :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>V:=e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
assms_2: shows "F ivl1 ivl2 = Some ivl"
prems_1: shows "v \<in> fv\<^sub>s\<^sub>t (x # S')"
prefixes_le_nth: fixes x :: "nat"   and j :: "nat"   and t :: "nat"   and d :: "nat" assumes "x < e_length (local.prefixes j t)" shows "e_nth (local.prefixes j t) x = e_nth (local.prefixes j (t + d)) x"
case_eq_if: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'d \<Rightarrow> 'b"   and sum :: "'c + 'd" shows "(case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = (if isl sum then f1 (projl sum) else f2 (projr sum))"
add_gr_0: fixes m :: "nat"   and n :: "nat" shows "n \<in># (+) m = (0 < m \<or> 0 < n)"
atLeastatMost_empty': fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.order less_eq less"   and "\<not> less_eq a b" shows "ord.atLeastAtMost less_eq a b = {}"
snth_SCons: fixes x :: "'b"   and xs :: "'b stream"   and n :: "nat" shows "(x ## xs) !! n = (case n of 0 \<Rightarrow> x | Suc n' \<Rightarrow> xs !! n')"
invariantsHoldInState_def: fixes state :: "State"   and F0 :: "Literal list list"   and decisionVars :: "nat set" shows "invariantsHoldInState state F0 decisionVars \<equiv> InvariantVarsM (getM state) F0 decisionVars \<and> InvariantVarsF (getF state) F0 decisionVars \<and> InvariantConsistent (getM state) \<and> InvariantUniq (getM state) \<and> InvariantReasonClauses (getF state) (getM state) \<and> InvariantEquivalent F0 (getF state) \<and> InvariantCFalse (getConflictFlag state) (getM state) (getC state) \<and> InvariantCEntailed (getConflictFlag state) (getF state) (getC state)"
it_step_insert_iff: fixes it :: "'d set"   and S :: "'d set"   and x :: "'d" assumes "it \<subseteq> S"   and "x \<in> it" shows "S - (it - {x}) = insert x (S - it)"
i_set_cont_subset_i_set: shows "i_set_cont \<subseteq> i_set"
bounded_linear_mult_const: fixes g :: "'d \<Rightarrow> 'e"   and y :: "'e" assumes "bounded_linear g" shows "bounded_linear (\<lambda>x. g x * y)"
is_fixpoint_def: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "is_fixpoint f x \<equiv> f x = x"
assms_2: shows "p \<in> dgrad_sig_set' (length fs) d"
distinct_split1: fixes xs :: "'a list"   and y :: "'a list"   and r :: "'a"   and z :: "'a list" assumes "distinct xs"   and "xs = y @ [r] @ z" shows "r \<notin> set y"
CIntegral: fixes x :: "nat"   and e :: "cexpr" shows "free_vars (cexpr_subst x e e') \<subseteq> (\<lambda>y. if x \<le> y then y + 1 else y) -` free_vars e' \<union> free_vars e"
Brouwer_degree2_unique_generator: fixes p :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real"   and a :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and d :: "int" assumes "continuous_map (nsphere p) (nsphere p) f"   and "subgroup_generated (reduced_homology_group (int p) (nsphere p)) {a} = reduced_homology_group (int p) (nsphere p)"   and "hom_induced (int p) (nsphere p) {} (nsphere p) {} f a = a [^]\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> d" shows "Brouwer_degree2 p f = d"
onode_comps_2: fixes i :: "nat"   and S :: "((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and R :: "nat set" shows "init (\<langle>i : S : R\<rangle>\<^sub>o) = {(\<sigma>, NodeS i s R) |\<sigma> s. (\<sigma>, s) \<in> init S}"