hcju/isabelleps huggingface dataset:
eq_fin_prefix_fininf_transp: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list"   and w\<^sub>3 :: "'item stream" assumes "w\<^sub>1 =\<^sub>F w\<^sub>2"   and "w\<^sub>2 \<le>\<^sub>F\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>3"
\<delta>: shows "0 < \<delta> \<and> \<delta> < \<epsilon> \<and> ereal \<delta> < fps_conv_radius f"
mult_1_right_octo: fixes x :: "octo" shows "x * 1 = x"
shiftpath_alt_def: fixes a :: "real"   and f :: "real \<Rightarrow> 'b" shows "shiftpath a f = (\<lambda>x. if x \<le> 1 - a then f (a + x) else f (a + x - 1))"
nth_default_nth: fixes n :: "nat"   and xs :: "'a list"   and dflt :: "'a" assumes "n < length xs" shows "nth_default dflt xs n = xs ! n"
assms: shows "ocircline_set H = {0\<^sub>h}"
arc_to_ends_to_block: fixes e :: "'b"   and x :: "'a"   and y :: "'a" assumes "arc_to_block e = {x, y}" shows "arc_to_ends G e = (x, y) \<or> arc_to_ends G e = (y, x)"
that: fixes h :: "'a \<Rightarrow> 'b" assumes "bij_betw h B D"   and "\<And>x. x \<in> A \<Longrightarrow> h x = f x"   and "\<And>x. x \<in> B - A \<Longrightarrow> h x = g x" shows "thesis"
downset_prop: fixes x :: "'a" shows "ord.downset (\<lambda>x y. y \<le> x) x = {y. x \<le> y}"
inv_closed: fixes G :: "('f, 'g) monoid_scheme"   and x :: "'f" assumes "Group.group G"   and "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> x \<in> carrier G"
D_invar'_def: fixes Q :: "'v \<Rightarrow> nat option"   and V :: "'v \<Rightarrow> nat option" shows "D_invar' Q V \<equiv> \<exists>D S. coupling Q V D S \<and> D_invar D S"
param_sum_1: fixes Rl :: "('a \<times> 'c) set"   and Rr :: "('b \<times> 'd) set" shows "(Inl, Inl) \<in> Rl \<rightarrow> \<langle>Rl, Rr\<rangle>sum_rel"
combine_dist: fixes ps :: "(int \<times> int) list"   and ps\<^sub>L :: "(int \<times> int) set"   and ps\<^sub>R :: "(int \<times> int) set"   and l :: "int"   and p\<^sub>0\<^sub>L :: "int \<times> int"   and p\<^sub>1\<^sub>L :: "int \<times> int"   and p\<^sub>0\<^sub>R :: "int \<times> int"   and p\<^sub>1\<^sub>R :: "int \<times> int"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int" assumes "sorted_snd ps"   and "set ps = ps\<^sub>L \<union> ps\<^sub>R"   and "\<forall>p\<in>ps\<^sub>L. fst p \<le> l"   and "\<forall>p\<in>ps\<^sub>R. l \<le> fst p"   and "sparse (dist (case p\<^sub>0\<^sub>L of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1\<^sub>L of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) ps\<^sub>L"   and "sparse (dist (case p\<^sub>0\<^sub>R of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1\<^sub>R of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) ps\<^sub>R"   and "(c\<^sub>0, c\<^sub>1) = combine (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps" shows "sparse (dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) (set ps)"
wary_args: fixes s :: "('a, 'b) tm"   and t :: "('a, 'b) tm"   and arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat" assumes "s \<in> set (args t)"   and "arity.wary arity_sym arity_var t" shows "arity.wary arity_sym arity_var s"
foldr_aux_2_1: fixes ns :: "'node list"   and g :: "'g"   and u' :: "('node, ('var \<times> 'node \<times> Def) set) mapping"   and p' :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping"   and n :: "'node" assumes "set ns \<subseteq> set (\<alpha>n g)"   and "distinct ns"   and "foldr (aux_2 g) ns (Mapping.empty, Mapping.empty) = (u', p')"   and "n \<in> set ns" shows "uses' g n = lookup_multimap u' n"
Suc: shows "(\<Sum>i = 0..<n. \<bar>f i\<bar>)\<^sup>2 = (\<Sum>i = 0..<n. f i * f i) + (\<Sum>i = 0..<n. \<Sum>j = 0..<n. if i = j then 0::'a else \<bar>f i\<bar> * \<bar>f j\<bar>)"
unity_def: shows "\<I> \<equiv> C.cod \<iota>"
chain_mono: fixes Y :: "nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat" assumes "chain Y"   and "i \<le> j" shows "Y i \<sqsubseteq> Y j"
hom_induced_vector: shows "BaseRep.GRepHom (\<currency>) induced_vector"
divides_aux_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'c \<times> 'c"   and divmod_step :: "num \<Rightarrow> 'c \<times> 'c \<Rightarrow> 'c \<times> 'c"   and q :: "'c"   and r :: "'c" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "unique_euclidean_semiring_numeral.divides_aux zero (q, r) = (r = zero)"
awalk_app_isoI: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and hom :: "('a, 'b, 'e, 'f) digraph_isomorphism" assumes "awalk u p v"   and "digraph_isomorphism hom" shows "pre_digraph.awalk (app_iso hom G) (iso_verts hom u) (map (iso_arcs hom) p) (iso_verts hom v)"
image_poly_def: shows "image_poly = image_ring (eval_pmod M (P, i) Q) (poly_ring M)"
inv_hcomp: fixes \<nu> :: "'d"   and \<mu> :: "'d" assumes "D.iso \<nu>"   and "D.iso \<mu>"   and "src\<^sub>D \<nu> = trg\<^sub>D \<mu>" shows "D.inv (\<nu> \<star>\<^sub>D \<mu>) = D.inv \<nu> \<star>\<^sub>D D.inv \<mu>"
eval_app_lam: fixes v :: "exp"   and x :: "nat"   and e :: "exp" assumes "is_val v" shows "E (EApp (ELam x e) v) = E (subst x v e)"
type: shows "type_definition Rep Abs {a. sort_of a \<in> range sort_fun}"
elims: fixes x :: "pratt"   and y :: "real" assumes "Pratt_Certificate.size_pratt x = y" obtains p :: "nat" where "x = Prime p__" and "y = log 2 (real p__)"   | p :: "nat" and a :: "nat" and xa :: "nat" where "x = Triple p a xa" and "y = log 2 (real p) + log 2 (real a) + log 2 (real xa)"
uses_suc: fixes n :: "'n"   and \<sigma> :: "'var \<Rightarrow> 'val"   and \<sigma>' :: "'var \<Rightarrow> 'val" assumes "\<forall>v\<in>local.reads n. \<sigma> v = \<sigma>' v" shows "suc n \<sigma> = suc n \<sigma>'"
RETURN_None_ph_relI: fixes y :: "'e"   and x :: "'f nres"   and B :: "('e \<times> 'f) set" assumes "(RETURN y, x) \<in> \<langle>B\<rangle>nres_rel" shows "(RETURN None, x) \<in> \<langle>\<langle>B\<rangle>phantom_rel\<rangle>nres_rel"
wf_osig_imp_exe_wf_osig: fixes oss :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list" assumes "wf_osig (translate_osig oss)" shows "exe_wf_osig oss"
3: shows "card (\<Union>i\<in>{3..<n}. (\<lambda>l. replicate i R @ B # l) ` {l. length l = n - i - 1 \<and> valid l}) = (\<Sum>i = 3..<n. lcount (n - i - 1))"
cond_idem: fixes b :: "'b upred"   and P :: "('a, 'b) uexpr" shows "P \<triangleleft> b \<triangleright> P = P"
the_brp_\<alpha>_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "the_brp_\<alpha> R \<equiv> \<lambda>x. SOME y. R x y"
decomposition: fixes g :: "'b \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'b" shows "\<exists>X. X = - g ` (- f ` X)"
cart_prod_singleton_left_gen: fixes a :: "'a"   and B :: "'b multiset"   and P :: "'a \<times> 'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>x. x \<in># {#a#} \<times># B \<Longrightarrow> P x = Q (snd x)" shows "filter_mset P ({#a#} \<times># B) = {#a#} \<times># filter_mset Q B"
Unifier_mgt: fixes t :: "('c, 'd) Term.term"   and \<delta> :: "'d \<Rightarrow> ('c, 'd) Term.term"   and u :: "('c, 'd) Term.term"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" assumes "Unifier \<delta> t u"   and "\<delta> \<preceq>\<^sub>\<circ> \<theta>" shows "Unifier \<theta> t u"
llast_simps_1: shows "LList2.llast LNil = undefined"
permutation_inverse: fixes x :: "'b permutation" shows "Abs_permutation ((\<rightarrow>) x) = x"
diff_sign: shows "sgn (ipoly p l) \<noteq> sgn (ipoly p r)"
elims: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a0 :: "'b"   and a1 :: "'b"   and x :: "(bool, unit) concrete_category.arr"   and y :: "'b" assumes "binary_product_diagram C a0 a1"   and "binary_product_diagram.map C a0 a1 x = y" obtains "x = concrete_category.MkArr False False ()" and "y = a0"   | "x = concrete_category.MkArr True True ()" and "y = a1"   | "x = concrete_category.Null" and "y = partial_magma.null C"   | vb :: "unit" where "x = concrete_category.MkArr True False vb__" and "y = partial_magma.null C"   | vb :: "unit" where "x = concrete_category.MkArr False True vb" and "y = partial_magma.null C"
bind_return'': fixes M :: "'d measure"   and N :: "'d measure" assumes "sets M = sets N" shows "M \<bind> return N = M"
prems_2: fixes x :: "('a, 'b) fmap" assumes "x \<in> set (x__ # xs)" shows "fmpred P x"
k_def: shows "llength (\<pi>\<^bsub>c\<^esub>inf_llist t) = enat k"
cmult_Cons: fixes c :: "'a"   and h :: "'a"   and t :: "'a list" shows "c %* (h # t) = c * h # c %* t"
berlekamp_monic_factorization_def: fixes d :: "nat"   and f :: "'b mod_ring poly" shows "berlekamp_monic_factorization d f = (let vs = berlekamp_basis f; n = length vs; fs = berlekamp_factorization_main d [f] vs n in fs)"
natural_transformation_def: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c"   and \<tau> :: "'b \<Rightarrow> 'c" shows "natural_transformation A B F G \<tau> \<equiv> (category A \<and> category B) \<and> functor A B F \<and> functor A B G \<and> natural_transformation_axioms A B F G \<tau>"
random_member_nonempty: fixes s :: "'a fset" shows "(s \<noteq> {||}) = (random_member s \<noteq> None)"
circle_1: shows "x1' \<in> circle Ze1 Re1 \<inter> circle Ze2 Re2"
same_level_path_upd_cs_callstack_Append: fixes cs :: "'edge list"   and as :: "'edge list"   and cs' :: "'edge list"   and cs'' :: "'edge list" assumes "same_level_path_aux cs as"   and "upd_cs cs as = cs'" shows "upd_cs (cs @ cs'') as = cs' @ cs''"
r_smult: fixes x :: "(real, 'n) Finite_Cartesian_Product.vec"   and a :: "real" assumes "x \<in> X"   and "0 < a" shows "r (a *s x) = r x"
assms: shows "\<forall>x\<in>X. f x \<in> x"
parts_UN_subset2: fixes H :: "'a \<Rightarrow> msg set"   and A :: "'a set" shows "parts (\<Union> (H ` A)) \<subseteq> (\<Union>x\<in>A. parts (H x))"
cart_eq_inner_axis: fixes a :: "(real, 'b) vec"   and i :: "'b" shows "a $ i = a \<bullet> axis i 1"
simps_37: fixes vd :: "bexp"   and ve :: "'b acom"   and vf :: "'b acom"   and vg :: "'b"   and v :: "'b"   and va :: "bexp"   and vb :: "'b acom"   and vc :: "'b" shows "(IF vd THEN ve ELSE vf {vg} \<sqsubseteq> {v} WHILE va DO vb {vc}) = False"
tuple_prj_spc: fixes f :: "'a"   and g :: "'a" assumes "cospan f g" shows "\<langle>\<p>\<^sub>1[f, g] \<lbrakk>f, g\<rbrakk> \<p>\<^sub>0[f, g]\<rangle> = f \<down>\<down> g"
minimal_antichain_def: fixes A :: "'a set" shows "minimal_antichain A = {x \<in> A. \<not> (\<exists>y\<in>A. y < x)}"
Prefixes_subset_paths_le: fixes p :: "(symbol \<times> character list) list"   and P :: "(symbol \<times> character list) list set" assumes "Prefixes p \<subseteq> P" shows "Prefixes p \<subseteq> paths_le (charslength p) P"
trivial_IrrFinGroupRepI: fixes smult :: "('a, 'g) aezfun \<Rightarrow> 'b \<Rightarrow> 'b" assumes "of_nat (card G) \<noteq> (0::'a)"   and "\<forall>a\<in>ActingGroup.group_ring. smult a (0::'b) = (0::'b)" shows "IrrFinGroupRepresentation G smult 0"
finfun_Ball_except_def: fixes xs :: "'a list"   and A :: "'a \<Rightarrow>f bool"   and P :: "'a \<Rightarrow> bool" shows "finfun_Ball_except xs A P = (\<forall>a. A $ a \<longrightarrow> a \<in> set xs \<or> P a)"
domain: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "MaxR r" shows "MaxR.MaxR_opt r (A \<inter> Field r) = MaxR.MaxR_opt r A"
rep_eq: fixes x :: "'a sq_mtx"   and xa :: "'a sq_mtx" shows "to_vec (x * xa) = to_vec x ** to_vec xa"
phis_phi: fixes g :: "'g"   and n :: "'node"   and v :: "'val"   and vs :: "'val list" assumes "Mapping.lookup (phis g) (n, v) = Some vs" shows "phi g v = Some vs"
sign_normal_of_Float: fixes x :: "Float.float" shows "sign (normal_of_Float x) = (if 0 < x then 0 else 1)"
in_small_setD: fixes x :: "V"   and A :: "V set" assumes "x \<in>\<^sub>\<circ> ZFC_in_HOL.set A"   and "small A" shows "x \<in> A"
map2_val_neutr_def: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'x"   and ys :: "'x" shows "map2_val_neutr f xs ys = oalist_of_list (map2_val_raw f id id (list_of_oalist xs) (list_of_oalist ys))"
ideI: fixes a :: "'a" assumes "Arr a"   and "B.ide a" shows "S.ide a"
openin_subtopology_Int2: fixes X :: "'c topology"   and T :: "'c set"   and S :: "'c set" assumes "openin X T" shows "openin (subtopology X S) (S \<inter> T)"
rotate_drop_take: fixes n :: "nat"   and xs :: "'b list" shows "rotate n xs = drop (n mod \<^bold>|xs\<^bold>|) xs \<cdot> take (n mod \<^bold>|xs\<^bold>|) xs"
fps_inverse_0_iff': fixes f :: "'a fps" shows "(fps_nth (inverse f) 0 = (0::'a)) = (inverse (fps_nth f 0) = (0::'a))"
gcsrp_def: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and S :: "'e set" shows "gcsrp G H S \<equiv> \<exists>f. bij_to f (set_rcs G H) S"
t2_def: shows "t2 = TaylorModel p2 e2"
sum_mono_inv: fixes f :: "'e \<Rightarrow> 'd"   and I :: "'e set"   and g :: "'e \<Rightarrow> 'd"   and i :: "'e" assumes "sum f I = sum g I"   and "\<And>i. i \<in> I \<Longrightarrow> f i \<le> g i"   and "i \<in> I"   and "finite I" shows "f i = g i"
isom_gch_units_transp: fixes E :: "('e, 'f) Group_scheme"   and n :: "nat"   and g :: "nat \<Rightarrow> ('g set, 'h) Group_scheme"   and h :: "nat \<Rightarrow> ('g set, 'h) Group_scheme"   and i :: "nat"   and j :: "nat" assumes "Ugp E"   and "Gchain n g"   and "Gchain n h"   and "i \<le> n"   and "j \<le> n"   and "i < j"   and "isom_Gchains n (transpos i j) g h" shows "card {i. i \<le> n \<and> g i \<cong> E} = card {i. i \<le> n \<and> h i \<cong> E}"
aprimedivisor_nat_2: fixes n :: "nat" assumes "n \<noteq> Suc 0" shows "aprimedivisor n dvd n"
paths_stay_in_winning_region: fixes \<sigma>' :: "'a \<Rightarrow> 'a" assumes "strategy p \<sigma>'"   and "winning_strategy p \<sigma>' v0"   and "\<And>va. va \<in> winning_region p \<Longrightarrow> \<sigma>' va = (\<sigma>(v := w)) va" shows "lset P \<subseteq> winning_region p"
dual_space_eq: fixes S :: "'b set" shows "dual_space S = {E. linear_fun_on S E} \<inter> {E. extensional0 S E}"
Un_Int_crazy: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" shows "A \<inter> B \<union> B \<inter> C \<union> C \<inter> A = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"
run_one_step_basic_br_table_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and js :: "nat list"   and j :: "nat"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Br_table js j) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
many_strongly_connected_iff_8: fixes x :: "'a" assumes "is_p_fun x" shows "many_strongly_connected x = (x ; x\<^sup>T \<le> x\<^sup>+)"
Zfun_def: fixes f :: "'b \<Rightarrow> 'd"   and F :: "'b filter" shows "Zfun f F = (\<forall>r>0. \<forall>\<^sub>F x in F. norm (f x) < r)"
hyps: assumes "\<And>i j. i < length xs \<and> j < length xs \<and> i \<noteq> j \<and> xs ! i = xs ! j \<longrightarrow> f (xs ! i) = (0::'a) \<and> f (xs ! j) = (0::'a)" shows "sum_list (map f (remdups xs)) = sum_list (map f xs)"
bmod_comp_def: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'c"   and x :: "'d" shows "\<otimes> R f g x = \<Squnion>{f y \<oplus> g z |y z. R x y z}"
factorize_rat_poly_monic_def: fixes p :: "rat poly" shows "factorize_rat_poly_monic p = (let (c, fs) = factorize_rat_poly p; lcs = \<Prod>(f, i)\<leftarrow>fs. lead_coeff f ^ Suc i in Let (map (\<lambda>(f, i). (normalize f, i)) fs) (Pair (c * lcs)))"
hyps: assumes "find_adds fs u = None"   and "f \<in> set fs" obtains "f = 0"   | "f \<noteq> 0" and "\<not> lt f adds\<^sub>t u"
integrable_fg: shows "set_integrable lborel (einterval a b) (\<lambda>x. f (g x) * g' x)"
NF_ND: fixes s :: "'b event list"   and X :: "'b event set"   and P :: "'b process" assumes "(s, X) \<notin> F P" shows "s \<notin> D P"
subgroup_imp_normal: fixes A :: "'a set" assumes "subgroup A (add_monoid G)" shows "A \<lhd> add_monoid G"
le_list_def: fixes s :: "'a list"   and t :: "'a list" shows "(s \<le> t) = (\<exists>r. s @ r = t)"
simps_2: fixes uv :: "'b list" shows "\<epsilon> \<and>\<^sub>p uv = \<epsilon>"
rel_spmf_simps: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'a spmf"   and q :: "'b spmf" shows "rel_spmf R p q = (\<exists>pq. (\<forall>(x, y)\<in>set_spmf pq. R x y) \<and> map_spmf fst pq = p \<and> map_spmf snd pq = q)"
cone_hull_expl: fixes S :: "'c set" shows "cone hull S = {c *\<^sub>R x |c x. 0 \<le> c \<and> x \<in> S}"
setdist_subset_right: fixes T :: "'b set"   and u :: "'b set"   and S :: "'b set" assumes "T \<noteq> {}"   and "T \<subseteq> u" shows "setdist S u \<le> setdist S T"
sorted_sorted_domain_raw: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and ko :: "'o" assumes "oalist_inv xs" shows "sorted_wrt (lt_of_key_order (rep_key_order ko)) (sorted_domain_raw ko xs)"
UNIV_eq_I: fixes A :: "'a set" assumes "\<And>x. x \<in> A" shows "UNIV = A"
cSup_upper: fixes x :: "'c multiset"   and X :: "'c multiset set" assumes "x \<in> X"   and "subset_mset.bdd_above X" shows "x \<subseteq># Sup X"
homotopic_with_refl: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"   and X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" shows "homotopic_with P X Y f f = (continuous_map X Y f \<and> P f)"
simps: fixes f :: "'a \<Rightarrow> 'a list"   and ws :: "'a list"   and Z :: "'a set" shows "While_Combinator.rtrancl_while_step f (ws, Z) = (let x = hd ws; new = remdups (filter (\<lambda>y. y \<notin> Z) (f x)) in (new @ tl ws, set new \<union> Z))"
dvd_dvd_imp_smult: fixes p :: "'b poly"   and q :: "'b poly" assumes "p dvd q"   and "q dvd p" shows "\<exists>c. p = Polynomial.smult c q"
th: shows "\<forall>j. isrlfm (MIR.disj (NEq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (NDvd i (Sub (C j) (Floor (Neg s)))))"
abs_eq: fixes xa :: "'d"   and x :: "('b, 'c, 'd) Tree" shows "Act\<^sub>\<alpha> xa (abs_Tree\<^sub>\<alpha> x) = abs_Tree\<^sub>\<alpha> (tAct xa x)"
pwD2: fixes S :: "'b nres"   and S' :: "'b nres"   and x :: "'b" assumes "S \<le> S'"   and "inres S x" shows "inres S' x"
prems: fixes x :: "'a" shows "(x \<in> fset (insert_fset x__ A)) = (x \<in> fset B)"
intros_2: fixes c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and c'' :: "('Var, 'AExp, 'BExp) Stmt"   and mds'' :: "Mode \<Rightarrow> 'Var set"   and mem'' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c', mds', mem'\<rangle> \<in> loc_reach lc"   and "\<langle>c', mds', mem'\<rangle> \<leadsto> \<langle>c'', mds'', mem''\<rangle>" shows "\<langle>c'', mds'', mem''\<rangle> \<in> loc_reach lc"
smc_Dom_vdomain: shows "\<D>\<^sub>\<circ> (\<BB>\<lparr>Dom\<rparr>) = \<BB>\<lparr>Arr\<rparr>"
horner_sum_foldr: fixes f :: "'c \<Rightarrow> 'a"   and a :: "'a"   and xs :: "'c list" shows "horner_sum f a xs = foldr (\<lambda>x b. f x + a * b) xs (0::'a)"
gcd_rat_to_gcd_int: fixes f :: "int poly"   and g :: "int poly" shows "gcd (of_int_poly f) (of_int_poly g) = Polynomial.smult (inverse (rat_of_int (lead_coeff (gcd f g)))) (of_int_poly (gcd f g))"
converse_trancl_induct: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set"   and P :: "'b \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>+"   and "\<And>y. (y, b) \<in> r \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>(y, z) \<in> r; (z, b) \<in> r\<^sup>+; P z\<rbrakk> \<Longrightarrow> P y" shows "P a"
perm_2: fixes pi :: "(vrs \<times> vrs) list"   and x2 :: "tyvrs"   and x1 :: "ty" shows "pi \<bullet> TVarB x2 x1 = TVarB (pi \<bullet> x2) (pi \<bullet> x1)"
rel_mset_Zero_iff_1: fixes rel :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Y :: "'b multiset" shows "rel_mset rel {#} Y = (Y = {#})"
linepath_of_real: fixes a :: "real"   and b :: "real"   and x :: "real" shows "linepath (of_real a) (of_real b) x = of_real ((1 - x) * a + x * b)"
ik_proj_subst_GSMP_subset: fixes I :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and n :: "'a"   and A :: "('a strand_label \<times> ('fun, 'var) strand_step) list" assumes "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t I"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range I)"   and "subst_domain I = \<V> \<and> ground (subst_range I)" shows "ik\<^sub>s\<^sub>t (proj_unl n A) \<cdot>\<^sub>s\<^sub>e\<^sub>t I \<subseteq> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj n A))"
closure_def: fixes as :: "'b list" shows "Equivalence_Checking.closure as = rtrancl_while (\<lambda>(r, s). nullable r = nullable s) (\<lambda>(r, s). map (\<lambda>a. (nderiv a r, nderiv a s)) as)"
mpoly_map_vars_diff: fixes f :: "nat \<Rightarrow> nat"   and p :: "'a mpoly"   and q :: "'a mpoly" shows "mpoly_map_vars f (p - q) = mpoly_map_vars f p - mpoly_map_vars f q"
ground_subs: fixes ts :: "fterm list"   and \<sigma> :: "char list \<Rightarrow> fterm" assumes "ground\<^sub>t\<^sub>s ts" shows "ts \<cdot>\<^sub>t\<^sub>s \<sigma> = ts"
rel_map_2: fixes Sa :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a option"   and g :: "'c \<Rightarrow> 'b"   and y :: "'c option" shows "rel_option Sa x (map_option g y) = rel_option (\<lambda>x y. Sa x (g y)) x y"
proj7_addpre: fixes v :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and npre :: "nat set" shows "\<pi>\<^sub>7 (addpre v npre) = \<pi>\<^sub>7 v \<union> npre"
homogeneous_ideal_homogenize: fixes F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and x :: "'c" assumes "\<And>f. f \<in> F \<Longrightarrow> homogeneous f"   and "p \<in> ideal F" shows "homogenize x p \<in> ideal F"
continuous_on_joinpaths: fixes g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" assumes "continuous_on {0..1} g1"   and "continuous_on {0..1} g2"   and "pathfinish g1 = pathstart g2" shows "continuous_on {0..1} (g1 +++ g2)"
Sec_def: fixes \<Phi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool"   and s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list" shows "Sec \<Phi> s t = ((\<forall>r \<beta>. s \<equiv>\<^sub>\<beta> r \<longrightarrow> \<Phi> (t, r, \<beta>)) \<and> (\<forall>r \<beta>. \<Phi> (r, s, \<beta>) \<longrightarrow> (\<exists>\<gamma>. r \<equiv>\<^sub>\<gamma> t \<and> Pbij_extends \<gamma> \<beta>)))"
cl_max'_2: fixes C :: "'b formula set set"   and F :: "'b formula"   and G :: "'b formula"   and S :: "'b formula set" assumes "pcp (C::'b formula set set)"   and "subset_closed (C::'b formula set set)"   and "(F::'b formula) \<triangleright> (G::'b formula) \<triangleright> pcp_lim (C::'b formula set set) (S::'b formula set) \<in> C" shows "(F::'b formula) \<in> pcp_lim (C::'b formula set set) (S::'b formula set) \<and> (G::'b formula) \<in> pcp_lim C S"
ccHeap_Exp: fixes e :: "exp"   and a :: "Arity"   and \<Delta> :: "(Vars.var \<times> exp) list" shows "\<G>\<^bsub>a\<^esub> e \<sqsubseteq> ccHeap \<Delta> e\<cdot>a"