hcju/isabelleps huggingface dataset:
loxodromic_isometry_inv: fixes f :: "'b \<Rightarrow> 'b" assumes "loxodromic_isometry f" shows "loxodromic_isometry (inv f)"
targets_eq_codomains: fixes \<mu> :: "'a" shows "targets \<mu> = H.codomains \<mu>"
t: shows "t \<in> set (unlabel (transaction_strand TT))"
strict_boundedE: fixes a :: "nat"   and b :: "nat"   and c :: "nat" assumes "a dvd gcd b c \<and> a \<noteq> gcd b c" obtains "a dvd b \<and> a \<noteq> b" and "a dvd c \<and> a \<noteq> c"
con_comp_sinks_range: fixes u :: "'e option"   and xs :: "'f list"   and p :: "'g \<Rightarrow> 'f"   and q :: "'h \<Rightarrow> 'f"   and I :: "('e \<times> 'e) set"   and D :: "'g \<Rightarrow> 'e"   and E :: "'h \<Rightarrow> 'e" assumes "u \<in> range Some"   and "set xs \<subseteq> range p \<union> range q" shows "sinks (con_comp_pol I) (con_comp_map D E p q) u xs \<subseteq> range Some"
skelAll_vsubstAll: fixes P :: "('d, 'e, 'varSort, 'var, 'f) param"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set" assumes "goodPar P" shows "(good X \<longrightarrow> (\<forall>y1 y2. {y1, y2} \<subseteq> varsOf P \<longrightarrow> skel (X #[y1 // y2]_ys) = skel X)) \<and> (goodAbs A \<longrightarrow> (\<forall>y1 y2. {y1, y2} \<subseteq> varsOf P \<longrightarrow> skelAbs (A $[y1 // y2]_ys) = skelAbs A))"
ccw_contr_on_line_right: fixes a :: "real"   and b :: "real"   and x :: "real"   and c :: "real"   and d :: "real" assumes "0 \<le> det3 (a, b) (x, c) (x, d)"   and "a < x" shows "c \<le> d"
convex_translation_eq: fixes a :: "'b"   and s :: "'b set" shows "convex ((+) a ` s) = convex s"
y_1: shows "y \<in> inside (path_image (p +++ linepath a b))"
intro: fixes \<Omega> :: "'c set"   and M :: "'c set set" assumes "subset_class \<Omega> M"   and "Dynkin_system_axioms \<Omega> M" shows "Dynkin_system \<Omega> M"
CD_on_Xd: fixes x :: "'a"   and ds :: "'b set"   and A :: "'a set" assumes "x \<in> ManiPref'.CD_on ds A" shows "Xd x \<in> ds"
cfun_join_def: shows "local.cfun_join = (\<Lambda> f g x. f\<cdot>x \<squnion> g\<cdot>x)"
g_orbital_def: fixes f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> bool"   and U :: "'b \<Rightarrow> real set"   and S :: "'b set"   and t\<^sub>0 :: "real"   and s :: "'b" shows "g_orbital f G U S t\<^sub>0 s = \<Union> {\<gamma> X G (U s) |X. X \<in> Sols f U S t\<^sub>0 s}"
preservation: shows "rel_weakly_preserves_barbs (indRelSTEQ SRel TRel) (STCalWB SWB TWB)"
is_natural_1: fixes f :: "'a \<times> 'a \<times> 'a" assumes "VVV.arr f" shows "HoHV f \<cdot> \<alpha>' (VVV.dom f) = \<alpha>' f"
l_set_child_nodes_get_disconnected_nodes_axioms: shows "l_set_child_nodes_get_disconnected_nodes type_wf set_child_nodes set_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs"
axioms_2: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'e set"   and scale\<^sub>1 :: "'f \<Rightarrow> 'd \<Rightarrow> 'd"   and scale\<^sub>2 :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and basis1 :: "'d set"   and basis2 :: "'e set" assumes "finite_dimensional_vector_space_pair_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>1 scale\<^sub>2 basis1 basis2" shows "finite_dimensional_vector_space_on U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2 basis2"
lprefix_refl: fixes xs :: "'b llist" shows "lprefix xs xs"
spec_imp_inv1: fixes s :: "('c, 'd) configuration stream" assumes "spec s" shows "alw (holds InvGlobVacantUptoImpliesNrec) s"
Pi_eq_empty: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set" shows "(Pi A B = {}) = (\<exists>x\<in>A. B x = {})"
phis'_def: shows "i.phis' \<equiv> \<lambda>g (n, v, m, def). if m = n \<and> n \<in> i.phiDefNodes g v \<and> v \<in> i.vars g \<and> def = PhiDef then Some (map (\<lambda>m. i.lookupDef g m v) (i.predecessors g n)) else None"
cmp_img_def: fixes f :: "'b \<Rightarrow> 'c"   and cmp :: "'c \<Rightarrow> 'c \<Rightarrow> 'a"   and a :: "'b"   and b :: "'b" shows "cmp_img f cmp a b \<equiv> cmp (f a) (f b)"
intros: fixes x :: "nat"   and y :: "nat" assumes "equals (x, y) (0, 0)" shows "zerozero' (x, y)"
cupcake_v_update_preserve: fixes env :: "v sem_env"   and f :: "(char list, char list, v) namespace \<Rightarrow> (char list, char list, v) namespace" assumes "is_cupcake_all_env env"   and "is_cupcake_ns (f (sem_env.v env))" shows "is_cupcake_all_env (update_v f env)"
cases: fixes a1 :: "nat \<times> nat \<Rightarrow> 'capacity"   and a2 :: "nat \<Rightarrow> nat"   and a3 :: "pr_operation'"   and a4 :: "nat \<times> nat \<Rightarrow> 'capacity"   and a5 :: "nat \<Rightarrow> nat" assumes "pr_algo_lts'p a1 a2 a3 a4 a5" obtains f :: "nat \<times> nat \<Rightarrow> 'capacity" and l :: "nat \<Rightarrow> nat" and e :: "nat \<times> nat" where "a1 = fb__" and "a2 = lb__" and "a3 = NONSAT_PUSH'" and "a4 = push_effect fb__ e__" and "a5 = lb__" and "Height_Bounded_Labeling c s t fb__ lb__" and "nonsat_push_precond fb__ lb__ e__"   | f :: "nat \<times> nat \<Rightarrow> 'capacity" and l :: "nat \<Rightarrow> nat" and e :: "nat \<times> nat" where "a1 = fc__" and "a2 = lc__" and "a3 = SAT_PUSH' e" and "a4 = push_effect fc__ e" and "a5 = lc__" and "Height_Bounded_Labeling c s t fc__ lc__" and "sat_push_precond fc__ lc__ e"   | f :: "nat \<times> nat \<Rightarrow> 'capacity" and l :: "nat \<Rightarrow> nat" and u :: "nat" where "a1 = f" and "a2 = l" and "a3 = RELABEL'" and "a4 = f" and "a5 = relabel_effect f l u" and "Height_Bounded_Labeling c s t f l" and "relabel_precond f l u"
elims_1: fixes x :: "'v"   and xa :: "'v list"   and y :: "bool" assumes "adj_path x xa = y" obtains v :: "'v" where "x = v__" and "xa = []" and "y = (v__ \<in> V)"   | v :: "'v" and u :: "'v" and us :: "'v list" where "x = v" and "xa = u # us" and "y = (adjacent v u \<and> adj_path u us)"
sort_by_pri_length: fixes min_pri :: "nat"   and edges :: "'a edge_scheme list" shows "length (sort_by_pri min_pri edges) = min_pri + 1"
HStartBallot_allBlocks: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HStartBallot s s' p" shows "allBlocks s' \<subseteq> allBlocks s \<union> {dblock s' p}"
sgn_pos: 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 a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less zero a" shows "sgn a = one"
RedCallNull: fixes M :: "char list"   and vs :: "val list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>null\<bullet>M(map Val vs),s,b\<rangle> \<rightarrow> \<langle>THROW NullPointer,s,b\<rangle>"
assms: fixes t :: "real" assumes "a < t"   and "t < b" shows "f t = c"
map2_val_compatD1: fixes f :: "('a \<times> 'c) list \<Rightarrow> ('a \<times> 'd) list"   and zs :: "('a \<times> 'c) list" assumes "map2_val_compat f"   and "oalist_inv_raw zs" shows "oalist_inv_raw (f zs)"
continuous_componentwise: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> 'd" shows "continuous F f = (\<forall>i\<in>Basis. continuous F (\<lambda>x. f x \<bullet> i))"
integral_def: fixes i :: "'d set"   and f :: "'d \<Rightarrow> 'c" shows "integral i f = (SOME y. (f has_integral y) i \<or> \<not> f integrable_on i \<and> y = (0::'c))"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes set_child_nodes_locs set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"   and "l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs" shows "l_set_child_nodes_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes set_child_nodes_locs set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_shadow_root get_shadow_root_locs"
minus_mult_minus: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ring plus zero minus uminus times" shows "times (uminus a) (uminus b) = times a b"
IH: fixes s :: "'s" shows "snd (viterbi s t_end os) = viterbi_prob s t_end os"
leof_refl: fixes a :: "'b nres" shows "a \<le>\<^sub>n a"
sorted_is_smaller: fixes a :: "('a, 'b) Combinators"   and p :: "('a, 'b) Combinators list"   and l :: "'a set list"   and b :: "('a, 'b) Combinators" assumes "FWNormalisationCore.sorted (a # p) l"   and "in_list a l"   and "in_list b l"   and "all_in_list p l"   and "singleCombinators [a]"   and "singleCombinators p"   and "b \<in> set p" shows "smaller a b l"
primerec_accum: fixes rf :: "recf"   and n :: "nat" assumes "primerec (rec_accum rf) n" shows "primerec rf n"
b: shows "set lI = {x \<in> I. x \<notin> s} \<and> nodup_all lI"
ring_simprules_8: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<zero> \<oplus> x = x"
indRel_def: shows "indRel \<equiv> _Collect (x, y) (indRelp x y)"
lin_subspace: fixes V :: "'c set"   and x :: "'c" assumes "vectorspace V"   and "x \<in> V" shows "Subspace.subspace (lin x) V"
star_n_mult: fixes X :: "nat \<Rightarrow> 'a"   and Y :: "nat \<Rightarrow> 'a" shows "star_n X * star_n Y = star_n (\<lambda>n. X n * Y n)"
carrier_subgroup_generated_subgroup: fixes H :: "'f set"   and G :: "('f, 'g) monoid_scheme" assumes "subgroup H G" shows "carrier (subgroup_generated G H) = H"
equivalence_mapI: fixes f :: "'a"   and g :: "'a" assumes "quasi_inverses f g" shows "equivalence_map f"
vlepoll_VPow_omega_if_vreal_lepoll_real: fixes x :: "V set" assumes "x \<lesssim> UNIV" shows "ZFC_in_HOL.set x \<lesssim>\<^sub>\<circ> VPow \<omega>"
cprank_max1_prod: fixes A :: "'b tensor"   and B :: "'b tensor" assumes "cprank_max1 A"   and "cprank_max1 B" shows "cprank_max1 (A \<otimes> B)"
feq: fixes z :: "complex" assumes "z \<in> S" shows "f z = exp (g z)"
power_increasing_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and b :: "'b"   and x :: "nat"   and y :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one b" shows "less_eq (power.power one times b x) (power.power one times b y) = (x \<le> y)"
S1_def: fixes msgs :: "'a \<times> 'a"   and out1 :: "unit" shows "S1 msgs out1 = (let (m0, m1) = msgs in sample_uniform (order \<G>) \<bind> (\<lambda>a. sample_uniform (order \<G>) \<bind> (\<lambda>b. sample_uniform (order \<G>) \<bind> (\<lambda>c. return_spmf (msgs, \<^bold>g [^] a, \<^bold>g [^] b, \<^bold>g [^] c, \<^bold>g [^] (a * b))))))"
interp_alt_factorization_def: shows "interp_alt_factorization = (\<lambda>(a, cs). smult a (\<Prod>(c, n)\<leftarrow>cs. [:1::'a, - c:] ^ Suc n))"
x: shows "x = v \<^sub>v* polys_to_mat ts ps"
indRelRT_def: shows "indRelRT \<equiv> \<lambda>TRel. _Collect (xa, x) (indRelRTp (\<lambda>x xa. (x, xa) \<in> TRel) xa x)"
rsum_component_le: fixes p :: "('a \<times> 'a set) set"   and a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and g :: "'a \<Rightarrow> 'b" assumes "p tagged_division_of cbox a b"   and "\<And>x. x \<in> cbox a b \<Longrightarrow> f x \<bullet> i \<le> g x \<bullet> i" shows "(\<Sum>(x, K)\<in>p. content K *\<^sub>R f x) \<bullet> i \<le> (\<Sum>(x, K)\<in>p. content K *\<^sub>R g x) \<bullet> i"
equivpI: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "reflp R"   and "symp R"   and "transp R" shows "equivp R"
simps: fixes v :: "'a val" shows "the_Val (Val v) = v"
gcs_leq_fun_1: fixes s :: "'a \<Rightarrow> 'b"   and t :: "'a \<Rightarrow> 'b" shows "gcs s t \<le> s"
HA_def: shows "HA \<equiv> fst \<circ> Rep_status"
V_def: fixes X :: "'d set"   and v :: "'d \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'d" assumes "Regions X v n not_in_X" shows "Regions.V X \<equiv> {v. \<forall>x\<in>X. (0::'e) \<le> v x}"
singleton_eq_iff: fixes a :: "V"   and b :: "V" shows "(ZFC_in_HOL.set {a} = ZFC_in_HOL.set {b}) = (a = b)"
fps_cos_def: fixes c :: "'b" shows "fps_cos c = Abs_fps (\<lambda>n. if even n then (- (1::'b)) ^ (n div 2) * c ^ n / of_nat (fact n) else (0::'b))"
vU: shows "adjuster n w us \<in> local.span (set us)"
eq_iff: fixes a :: "'a"   and b :: "'a" shows "(a = b) = (a \<le> b \<and> b \<le> a)"
subsetCE: fixes A :: "'d set"   and B :: "'d set"   and c :: "'d" assumes "A \<subseteq> B" obtains "c \<notin> A"   | "c \<in> B"
weak_reduction_simulation_impl_OCom: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel"   and "\<forall>S T. (SourceTerm S, TargetTerm T) \<in> Rel \<longrightarrow> (\<lbrakk>S\<rbrakk>, T) \<in> TRel\<^sup>*"   and "weak_reduction_simulation Rel (STCal Source Target)" shows "operational_complete (TRel\<^sup>*)"
mr: shows "applied_rule_rev Cp x p = \<lfloor>DenyAllFromTo a b\<rfloor>"
sum_singleton: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and f :: "'c \<Rightarrow> 'b"   and x :: "'c" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.sum plus zero f {x} = f x"
map_to_set_dom: fixes m :: "'c \<Rightarrow> 'd option" shows "dom m = fst ` map_to_set m"
strict_prefix_def: fixes xs :: "'b list"   and ys :: "'b list" shows "strict_prefix xs ys = (prefix xs ys \<and> xs \<noteq> ys)"
dg_op_dg_op_dg: shows "op_dg (op_dg \<CC>) = \<CC>"
configuration_token_squats: fixes x :: "nat"   and q :: "'b"   and n :: "nat"   and y :: "nat" assumes "x \<in> configuration q n"   and "y \<in> configuration q n" shows "token_squats x = token_squats y"
nth_zip: fixes i :: "nat"   and xs :: "'b list"   and ys :: "'c list" assumes "i < length xs"   and "i < length ys" shows "zip xs ys ! i = (xs ! i, ys ! i)"
size_list_simp_2: fixes xs :: "'b list" shows "length xs = (if xs = [] then 0 else Suc (length (tl xs)))"
assoc'_simps_1: fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "src\<^sub>C f = trg\<^sub>C g"   and "src\<^sub>C g = trg\<^sub>C h" shows "C.arr \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, h]"
dens: shows "dens_ctxt_\<alpha> ([], vs @ vs', \<Gamma>, CReal 1) \<turnstile>\<^sub>d b \<Rightarrow> (\<lambda>\<rho> x. ennreal (eval_cexpr f \<rho> x))"
awalk_empty_ends: fixes u :: "'a"   and v :: "'a" assumes "G.awalk u [] v" shows "u = v"
simps: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_erase2 ires rs (l, r) = (\<exists>ln rn. l = Bk # Oc # ires \<and> tl r = Bk \<up> ln @ Bk # Bk # Oc \<up> Suc rs @ Bk \<up> rn)"
compositionality: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and list :: "'c list" shows "map f (map g list) = map (f \<circ> g) list"
simps_2: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" shows "wt (App s t) = wt s - \<delta> + wt t"
g2_i: shows "gs2.gso i = fs.gs.gso (i - 1) - map of_int_hom.vec_hom fs ! (i - 1) \<bullet> gs2.gso (i - 1) / \<parallel>gs2.gso (i - 1)\<parallel>\<^sup>2 \<cdot>\<^sub>v gs2.gso (i - 1)"
start_def: fixes A :: "'d \<times> 'e \<times> 'f" shows "start A = fst A"
nth_prime_eq_iff: fixes m :: "nat"   and n :: "nat" shows "(nth_prime m = nth_prime n) = (m = n)"
coeff_sum: fixes p :: "'b \<Rightarrow> 'a Polynomial.poly"   and A :: "'b set"   and i :: "nat" shows "poly.coeff (sum p A) i = (\<Sum>x\<in>A. poly.coeff (p x) i)"
list_set_rel_sv: fixes R :: "('a \<times> 'b) set" assumes "single_valued R" shows "single_valued (\<langle>R\<rangle>list_wset_rel)"
path_between_odds: fixes v :: "'v"   and v' :: "'v" assumes "odd (degree v G)"   and "odd (degree v' G)"   and "finite E"   and "v \<noteq> v'"   and "num_of_odd_nodes G = 2" shows "\<exists>ps. is_trail v ps v'"
inj_on_def: fixes f :: "'a \<Rightarrow> 'c"   and A :: "'a set" shows "inj_on f A = (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
preserves_arr: fixes f :: "'c" assumes "Cop.arr f" shows "S.arr (Y a f)"
ifte_thms_2: fixes e1 :: "'r"   and e2 :: "'r" shows "If FF then e1 else e2 = e2"
sgn_1_pos: fixes a :: "'b" shows "(sgn a = (1::'b)) = ((0::'b) < a)"
take_upt: fixes i :: "nat"   and m :: "nat"   and n :: "nat" assumes "i + m \<le> n" shows "take m [i..<n] = [i..<i + m]"
csmall_nstep_Rec: fixes P :: "'prog"   and \<sigma> :: "'state"   and n :: "nat" shows "csmall_nstep P \<sigma> (Suc n) = {(\<sigma>2, coll). \<exists>\<sigma>1 coll1 coll2. (\<sigma>1, coll1) \<in> csmall_nstep P \<sigma> n \<and> (\<sigma>2, coll2) \<in> csmall P \<sigma>1 \<and> combine coll1 coll2 = coll}"
ProblemIkTpartG_GEModel_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"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool"   and eintT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and eintF :: "('a, 'c) G.efsym \<Rightarrow> univ list \<Rightarrow> univ"   and eintP :: "('b, 'c) epsym \<Rightarrow> univ list \<Rightarrow> bool" shows "ProblemIkTpartG_GEModel wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl eintT eintF eintP \<equiv> ProblemIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl \<and> CM.Model (ProblemIkTpartG.GE_wtFsym wtFsym resOf protCl) (ProblemIkTpartG.GE_wtPsym wtPsym prot protFw) (ProblemIkTpartG.GE_arOf arOf) (ProblemIkTpartG.GE_resOf resOf) (ProblemIkTpartG.GE_parOf parOf) (ProblemIkTpartG.gPB wtFsym arOf resOf \<Phi> prot protFw protCl) eintT eintF eintP"
surj1: shows "f ` {a \<in> A. P a} \<subseteq> {b \<in> B. Q b}"
zero_le_power2: 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 a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq zero (power.power one times a 2)"
emeasure_compl: fixes s :: "'b set"   and M :: "'b measure" assumes "s \<in> sets M"   and "emeasure M s \<noteq> \<infinity>" shows "emeasure M (space M - s) = emeasure M (space M) - emeasure M s"
left_commute: fixes b :: "'c multiset"   and a :: "'c multiset"   and c :: "'c multiset" shows "b \<inter># (a \<inter># c) = a \<inter># (b \<inter># c)"
deterministic_union: fixes S :: "('a \<times> 'b) set"   and R :: "('a \<times> 'b) set" assumes "fst ` S \<inter> fst ` R = {}"   and "deterministic S"   and "deterministic R" shows "deterministic (S \<union> R)"
Ints_eq_abs_less1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> \<int>"   and "y \<in> \<int>" shows "(x = y) = (\<bar>x - y\<bar> < (1::'a))"
sums_unique: fixes f :: "nat \<Rightarrow> 'a"   and s :: "'a" assumes "f sums s" shows "s = suminf f"
eq: shows "{a \<in> A. max 0 (f a) \<noteq> 0 \<and> g a \<noteq> 0} = {a \<in> A. 0 < f a \<and> g a \<noteq> 0}"
tab_simps_1: shows "seq (r \<star> \<phi>) \<rho>"
weight_spmf_le_1: fixes p :: "'h spmf" shows "weight_spmf p \<le> 1"
supp_perm_perm_eq: fixes x :: "'b"   and p :: "perm"   and q :: "perm" assumes "\<forall>a\<in>supp x. p \<bullet> a = q \<bullet> a" shows "p \<bullet> x = q \<bullet> x"
triangle_in_hom_2: shows "\<guillemotleft>(g \<star> \<xi>) \<cdot> \<a>[g, f, g] \<cdot> (\<zeta> \<star> g) : trg g \<star> g \<Rightarrow> g \<star> src g\<guillemotright>"
induct: fixes P :: "'k \<Rightarrow> ('a, 'k) aformula \<Rightarrow> bool"   and a0 :: "'k"   and a1 :: "('a, 'k) aformula" assumes "\<And>k \<phi> \<psi>. \<lbrakk>P k \<phi>; P k \<psi>\<rbrakk> \<Longrightarrow> P k (FAnd \<phi> \<psi>)"   and "\<And>k v. P k (FBool v)"   and "\<And>k v. P k (FBase v)"   and "\<And>k v. P k (FNot v)"   and "\<And>k v va. P k (FOr v va)"   and "\<And>k v va. P k (FEx v va)"   and "\<And>k v va. P k (FAll v va)" shows "P a0 a1"
acceptD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and m :: "'c match_expr" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = [Rule m Accept]"   and "matches \<gamma> m p"   and "s = Undecided" shows "t = Decision FinalAllow"
at_least_delta: fixes x :: "real" assumes "x \<le> \<delta>"   and "local.t_stop \<le> x" shows "ego.q x = ego2.s (x - \<delta>)"
linear_pat_cases: fixes t :: "term" assumes "linear t" obtains s :: "name" where "t = Free s"   | name :: "name" and args :: "term list" where "linears args" and "t = name $$ args"
preserves_comp: fixes g :: "'b"   and f :: "'b" assumes "D.seq g f" shows "src\<^sub>D (g \<cdot>\<^sub>D f) = src\<^sub>D g \<cdot>\<^sub>D src\<^sub>D f"
not_empty_rootsE: fixes p :: "'a list list" assumes "p \<in> carrier (poly_ring (poly_ring R))"   and "UP.roots p \<noteq> {#}" obtains a :: "'a list" where "a \<in> carrier (poly_ring R)" and "a \<in># UP.roots p" and "[\<one>\<^bsub>poly_ring R\<^esub>, \<ominus>\<^bsub>poly_ring R\<^esub> a] \<in> carrier (poly_ring (poly_ring R))" and "[\<one>\<^bsub>poly_ring R\<^esub>, \<ominus>\<^bsub>poly_ring R\<^esub> a] pdivides\<^bsub>poly_ring R\<^esub> p"
mult_left_cancel: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_no_zero_divisors_cancel plus zero times"   and "c \<noteq> zero" shows "(times c a = times c b) = (a = b)"
simps_2: fixes n :: "nat" shows "boundslt n F = True"
simps_4: fixes c :: "int"   and e :: "MIR.num" shows "\<zeta> (NEq (CN 0 c e)) = c"