hcju/isabelleps huggingface dataset:
simps: fixes C :: "'a literal set"   and \<sigma> :: "('a \<times> 'a trm) list" shows "mset_cl (C, \<sigma>) = {#mset_lit (subst_lit x \<sigma>). x \<in># mset_set C#}"
rel_G_neg_distr_cond_eq: fixes tytok :: "('t \<times> 'u \<times> 'v \<times> 'w \<times> 'x \<times> 'y \<times> 'z) itself" shows "rel_G_neg_distr_cond (=) (=) (=) (=) (=) (=) (=) (=) tytok"
PGC_strong: shows "\<forall>c x. \<lfloor>PPA x \<^bold>\<rightarrow> RightTo x FWB\<rfloor>\<^sub>c"
is_le_rel_distinct: fixes rel :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and x :: "'t"   and y :: "'t" assumes "ord_term_lin.is_le_rel rel"   and "rel x y"   and "x \<noteq> y" shows "x \<prec>\<^sub>t y"
simps_4: fixes uv :: "typ"   and v :: "nat" shows "eta_reducible (Abs uv (Bv v)) = eta_reducible (Bv v)"
select_convs_5: fixes types :: "tf list"   and funcs :: "nat list"   and tab :: "nat option"   and mem :: "nat option"   and globs :: "nat list"   and more :: "'a" shows "inst.globs \<lparr>types = types, funcs = funcs, tab = tab, mem = mem, globs = globs, \<dots> = more\<rparr> = globs"
bal_upd: fixes t :: "('c \<times> 'd) tree234"   and x :: "'c"   and y :: "'d" assumes "bal t" shows "bal (tree\<^sub>i (upd x y t)) \<and> height (upd x y t) = height t"
keys_add: fixes f :: "'b \<Rightarrow>\<^sub>0 'c"   and g :: "'b \<Rightarrow>\<^sub>0 'c" shows "Poly_Mapping.keys (f + g) \<subseteq> Poly_Mapping.keys f \<union> Poly_Mapping.keys g"
bounded_cbox: fixes a :: "'c"   and b :: "'c" shows "bounded (cbox a b)"
card_of_Func_Plus: fixes A :: "'c set"   and B :: "'d set"   and C :: "'e set" shows "|Func (A <+> B) C| =o |Func A C \<times> Func B C|"
finalConflictingState: fixes state :: "State"   and F0 :: "Literal list list" assumes "\<not> applicableBacktrack state F0"   and "formulaFalse F0 (elements (getM state))" shows "decisions (getM state) = []"
pseudo_divmod_2: fixes g :: "'b poly"   and f :: "'b poly"   and q :: "'b poly"   and r :: "'b poly" assumes "g \<noteq> 0"   and "pseudo_divmod f g = (q, r)" shows "r = 0 \<or> degree r < degree g"
church_rosser_var: fixes y :: "'a"   and x :: "'a" assumes "y \<cdot> x\<^sup>\<star> \<le> x\<^sup>\<star> \<cdot> y\<^sup>\<star>" shows "(x + y)\<^sup>\<star> = x\<^sup>\<star> \<cdot> y\<^sup>\<star>"
assms: fixes pi :: "(name \<times> name) list" assumes "(pi \<bullet> xvec) \<sharp>* c"   and "distinctPerm pi"   and "set pi \<subseteq> set xvec \<times> set (pi \<bullet> xvec)" shows "thesis"
convex_translation: fixes S :: "'a set"   and a :: "'a" assumes "convex S" shows "convex ((+) a ` S)"
ccw_system4_axioms: shows "ccw_system4 ccw S"
simps: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" shows "gt_sub a1 a2 = (\<exists>t s. a1 = t \<and> a2 = s \<and> is_App t \<and> (fun t \<ge>\<^sub>t s \<or> arg t \<ge>\<^sub>t s))"
simpUpdateI: fixes e1 :: "'v"   and e2 :: "'v" assumes "\<not> P (nP e1) (nP e2)"   and "\<not> P \<bottom> (nP e2)" shows "\<not> P ((nP(i := \<bottom>)) e1) (nP e2)"
2: shows "fst (sub2_wit [] fs) = map (\<lambda>i. map (\<mu> i) [0..<i]) [0..<length fs]"
unpower_eval: fixes v :: "nat"   and \<phi> :: "atom fm"   and L :: "real list" shows "eval (unpower v \<phi>) L = eval \<phi> L"
exhaust: fixes y :: "'c stream" obtains x1 :: "'c" and x2 :: "'c stream" where "y = x1 ## x2"
group_hom_axioms_def: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme"   and h :: "'e \<Rightarrow> 'g" shows "group_hom_axioms G H h \<equiv> h \<in> hom G H"
Extend_commute_unsafe: fixes ord :: "WS1S_Prelim.order"   and ord' :: "WS1S_Prelim.order"   and j :: "nat"   and i :: "nat"   and \<AA> :: "interp"   and P1 :: "nat fset"   and P2 :: "nat fset" assumes "ord \<noteq> ord'" shows "Extend ord j (Extend ord' i \<AA> P1) P2 = Extend ord' i (Extend ord j \<AA> P2) P1"
restrict_relI: fixes x :: "'b"   and y :: "'b"   and R :: "('b \<times> 'b) set"   and A :: "'b set" assumes "(x, y) \<in> R"   and "x \<in> A"   and "y \<in> A" shows "(x, y) \<in> restrict_rel A R"
dg_prod_vunion_Arr_in_Arr: fixes J :: "V"   and K :: "V"   and b :: "V"   and \<AA> :: "V \<Rightarrow> V"   and c :: "V" assumes "J \<inter>\<^sub>\<circ> K = []\<^sub>\<circ>"   and "b \<in>\<^sub>\<circ> dg_prod J \<AA>\<lparr>Arr\<rparr>"   and "c \<in>\<^sub>\<circ> dg_prod K \<AA>\<lparr>Arr\<rparr>" shows "b \<union>\<^sub>\<circ> c \<in>\<^sub>\<circ> dg_prod (J \<union>\<^sub>\<circ> K) \<AA>\<lparr>Arr\<rparr>"
ideD_1: fixes a :: "'b" assumes "hhom\<^sub>D.ide a" shows "hhom\<^sub>D.arr a"
rga_ops_rem_penultimate: fixes xs :: "('a \<times> 'a option) list"   and i1 :: "'a"   and r1 :: "'a option"   and i2 :: "'a"   and r2 :: "'a option" assumes "rga_ops (xs @ [(i1, r1), (i2, r2)])"   and "\<And>r. r2 = Some r \<Longrightarrow> r \<noteq> i1" shows "rga_ops (xs @ [(i2, r2)])"
assigned_nodes_slot_is_slot: fixes h :: "(_) heap"   and slot :: "(_) element_ptr" assumes "h \<turnstile> ok assigned_nodes slot" shows "h \<turnstile> get_tag_name slot \<rightarrow>\<^sub>r ''slot''"
S: shows "summable (\<lambda>n. Sigma_Algebra.measure M (A n))"
fun_upd_eqvt: fixes p :: "perm"   and f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and y :: "'e" shows "p \<bullet> f(x := y) = (p \<bullet> f)(p \<bullet> x := p \<bullet> y)"
coeff_minus: fixes lp1 :: "linear_poly"   and lp2 :: "linear_poly"   and var :: "nat" shows "coeff (lp1 - lp2) var = coeff lp1 var - coeff lp2 var"
that: fixes u :: "'a list"   and a :: "'a"   and v :: "'a stream" assumes "w = u @- a ## v"   and "fold (count cs) u k = k"   and "(cs ! k) a" shows "thesis"
results_gpv_inline: fixes x :: "'a"   and s' :: "('b, 'c, 'd, 'e) converter"   and \<I>' :: "('d, 'e) \<I>"   and gpv :: "('a, 'b, 'c) gpv"   and s :: "('b, 'c, 'd, 'e) converter"   and \<I> :: "('b, 'c) \<I>" assumes "(x, s') \<in> results_gpv \<I>' (inline run_converter gpv s)"   and "\<I> \<turnstile>g gpv \<surd>"   and "plossless_converter \<I> \<I>' s \<and> \<I>, \<I>' \<turnstile>\<^sub>C s \<surd>" shows "x \<in> results_gpv \<I> gpv \<and> plossless_converter \<I> \<I>' s' \<and> \<I>, \<I>' \<turnstile>\<^sub>C s' \<surd>"
coef_gt_0: fixes A :: "'v pvar \<Rightarrow> zhmultiset"   and s :: "('s, 'v) tm"   and i :: "nat" assumes "legal_zpassign A" shows "0 < eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (coef s i))"
pref_eq: fixes X :: "'b \<Rightarrow> 'c"   and Y :: "'b \<Rightarrow> 'c" assumes "|~ X = Y" shows "X = Y"
bounded_def: fixes step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list"   and n :: "nat" shows "bounded step n = (\<forall>p<n. \<forall>\<tau>. \<forall>(q, \<tau>')\<in>set (step p \<tau>). q < n)"
finite_less_ub: fixes f :: "nat \<Rightarrow> nat"   and u :: "nat" assumes "\<And>n. n \<le> f n" shows "finite {n. f n \<le> u}"
Sym: fixes R :: "pi"   and S :: "pi" shows "(R \<parallel> S, S \<parallel> R) \<in> Rel"
rel_fun_comp1: fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and S1 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f"   and R2 :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and S2 :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and h :: "'g \<Rightarrow> 'h"   and R' :: "'c \<Rightarrow> 'g \<Rightarrow> bool" assumes "rel_fun R1 S1 f g"   and "rel_fun R2 S2 g h"   and "R' = R1 OO R2" shows "rel_fun R' (S1 OO S2) f h"
degeneralize_language: fixes A :: "('c, 'd) dgba" shows "DBA.language (degeneralize A) = DGBA.language A"
openE: fixes S :: "'c set"   and x :: "'c" assumes "open S"   and "x \<in> S" obtains e :: "real" where "0 < e" and "ball x e \<subseteq> S"
leq_zero_imp: fixes x :: "'a" assumes "(x, zero) \<in> r" shows "x = zero"
assms: shows "\<tau>trsys.silent_move (init_fin t) init_fin_\<tau>move ((s, x), m) ((s', x'), m')"
simps_4: fixes uu :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and uv :: "'d" shows "map_match_tac uu uv MatchAny = TernaryValue TernaryTrue"
FAssReds2: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and v :: "val"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>Val v\<bullet>F{D} := e,s,b\<rangle> \<rightarrow>* \<langle>Val v\<bullet>F{D} := e',s',b'\<rangle>"
dim_row_transfer_rule: fixes A :: "'b mat"   and A' :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" assumes "Mod_Type_Connect.HMA_M A A'" shows "dim_row A = CARD('d)"
remdups_adj_singleton: fixes xs :: "'g list"   and x :: "'g" assumes "remdups_adj xs = [x]" shows "xs = replicate (length xs) x"
intro: fixes I :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "I \<noteq> R"   and "\<And>\<aa>. \<lbrakk>lideal \<aa> R addition multiplication zero unit; \<aa> \<noteq> R; I \<subseteq> \<aa>\<rbrakk> \<Longrightarrow> I = \<aa>" shows "max_lideal_axioms I R addition multiplication zero unit"
size_big_union_sum: fixes M :: "'a multiset multiset" shows "size (\<Sum>\<^sub># M) = \<Sum>\<^sub># (image_mset size M)"
maxsimpchainD_maxsimp: fixes xs :: "'a set list"   and x :: "'a set" assumes "folding_g.gallery xs"   and "x \<in> set xs" shows "folding_g.chamber x"
finite_dimensional_vector_space_with_overloaded: shows "finite_dimensional_vector_space = finite_dimensional_vector_space_with (+) (0::'e) (-) uminus"
inv_ide: fixes a :: "'d" assumes "D.ide a" shows "D.inv a = a"
simps_2: fixes e :: "('a, 'b) Expression"   and f :: "'b"   and d :: "('a, 'b) Expression" shows "sub e in f E@ d = f E@ sub e in d"
not_Base_subcls_Ext: fixes R :: "bool" assumes "tprg\<turnstile>Base\<prec>\<^sub>C Ext" shows "R"
simps_8: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Text of Cext \<Rightarrow> f1 | Text \<Rightarrow> f2 | Fext \<Rightarrow> f3) = f2"
dom_inv: fixes f :: "'c \<times> 'c" assumes "C.VV.iso f" shows "C.VV.dom (C.VV.inv f) = C.VV.cod f"
gb_rename_ecnv_def: fixes ecnv :: "('d, 'e) gb_graph_rec_scheme \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'b"   and G :: "('d, 'e) gb_graph_rec_scheme" shows "gb_rename_ecnv ecnv f G \<equiv> \<lparr>gbg_F = {f ` A |A. A \<in> gbg_F G}, \<dots> = ecnv G\<rparr>"
elims: fixes x :: "'a simplex_state"   and y :: "'a simplex_checkpoint" assumes "checkpoint_simplex x = y" obtains cs :: "QDelta ns_constraint list" and asi_tv :: "('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list" and s :: "('a, QDelta) state" where "x = Simplex_State (cs, asi_tv, s)" and "y = Simplex_Checkpoint (checkpoint_s s)"
incl_in_def: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "incl_in a b = (ide a \<and> ide b \<and> local.set a \<subseteq> local.set b)"
bernoulli'_fps_def: shows "bernoulli'_fps = fps_X / (1 - fps_exp (- (1::'b)))"
algebraic_int_fact: fixes n :: "nat" shows "algebraic_int (fact n)"
map_raw_cong: fixes xs :: "('a \<times> 'b) list \<times> 'c"   and f :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'd"   and g :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'd" assumes "\<And>kv. kv \<in> set (fst xs) \<Longrightarrow> f kv = g kv" shows "map_raw f xs = map_raw g xs"
mrw: shows "mrw_values P vs0 (obs @ [ob]) (ad, al) = \<lfloor>(v, b)\<rfloor>"
content_const: fixes c :: "'a" shows "content [:c:] = normalize c"
det_fst_Gauss_Jordan_in_ij_PA_eq_fst_Gauss_Jordan_in_ij_det_P: fixes A :: "(('b, 'c) vec, 'c) vec"   and i :: "'c"   and j :: "'c"   and P :: "(('b, 'c) vec, 'c) vec" shows "fst (Gauss_Jordan_in_ij_det_P A i j) * det P = det (fst (Gauss_Jordan_in_ij_PA (P, A) i j))"
bbind_leftID: fixes a :: "'b"   and f :: "'b \<rightarrow> 'a" shows "up\<cdot>a >>= f = f\<cdot>a"
value_flow: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" assumes "flow \<Delta> f"   and "\<And>y. edge \<Delta> (source \<Delta>) y = (y = x)" shows "value_flow \<Delta> f = f (source \<Delta>, x)"
x_2: shows "infinite {n. birkhoff_sum f n x \<in> {real n * expectation f - 1..real n * expectation f}}"
spanning_treeE: fixes H :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" assumes "spanning_tree H G" obtains "tree H \<and> spanning H G"
Rep_mod_ring_cases: fixes y :: "int" assumes "y \<in> {0..<int CARD('b)}" obtains x :: "'b mod_ring" where "y = Rep_mod_ring x"
iT_Min_2: fixes n :: "nat" shows "iMin [\<dots>n] = 0"
iTILL_div_Min: fixes n :: "nat"   and k :: "nat" shows "iMin ([\<dots>n] \<oslash> k) = 0"
mantissa_nonneg_iff: fixes x :: "Float.float" shows "(0 \<le> mantissa x) = (0 \<le> x)"
multi_winkerp: fixes k :: "nat" shows "\<beta> \<squnion> k \<otimes> \<alpha> = \<beta>"
fv_ik_subset_fv_st: fixes S :: "('d, 'c) strand_step list" shows "fv\<^sub>s\<^sub>e\<^sub>t (ik\<^sub>s\<^sub>t S) \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S"
delete_point_strong_block_not_in: fixes p :: "'a"   and bl :: "'a set" assumes "p \<in> bl" shows "bl \<notin># str_del_point_blocks p"
ffold_fimage: fixes g :: "'a \<Rightarrow> 'b"   and A :: "'a fset"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c" assumes "inj_on g (fset A)" shows "ffold f z (g |`| A) = ffold (f \<circ> g) z A"
that: fixes g :: "'a" assumes "g \<in> carrier G"   and "H \<subseteq> g <# (P #> inv g)" shows "thesis"
ev_iff_sdrop: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" shows "ev P \<omega> = (\<exists>m. P (sdrop m \<omega>))"
def_scheme: fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and a1 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1"   and a2 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2" shows "f self a1 a2 \<equiv> \<lambda>\<tau>. SOME res. let res = \<lambda>_. res in if \<tau> \<Turnstile> \<delta> self \<and> \<tau> \<Turnstile> \<upsilon> a1 \<and> \<tau> \<Turnstile> \<upsilon> a2 then \<tau> \<Turnstile> PRE self a1 a2 \<and> \<tau> \<Turnstile> POST self a1 a2 res else \<tau> \<Turnstile> res \<triangleq> invalid"
fsmultD: fixes a :: "'f"   and v :: "'v" shows "a \<sharp>\<cdot> v = (a \<delta>\<delta> (0::'g)) \<cdot> v"
abs_eq: shows "0 = Abs_poly_mapping (\<lambda>k. 0::'b)"
open_subset: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T"   and "open S" shows "openin (top_of_set T) S"
assms: shows "\<Psi> \<simeq> \<Psi>'"
configuration_Min_in: fixes q :: "'a ltl_prop_equiv_quotient"   and n :: "nat" assumes "configuration q n \<noteq> {}" shows "Min (configuration q n) \<in> configuration q n"
invert_side_of_triangle_2: fixes h :: "('a, 'a set) Maps.arr"   and f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr" assumes "Maps.arr h"   and "f \<odot> g = h"   and "Maps.iso g" shows "Maps.seq h (Maps.inv g) \<and> f = h \<odot> Maps.inv g"
assms_3: shows "path_image \<gamma> \<subseteq> s"
integral_cos_Z: fixes n :: "real" assumes "n \<in> \<int>" shows "LINT x|lebesgue_on {- pi..pi}. cos (x * n) = (if n = 0 then 2 * pi else 0)"
assms: shows "v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> w"
Invoke_handlers: fixes P :: "'a prog"   and C :: "char list"   and pc :: "nat"   and xt :: "ex_table"   and pc' :: "nat"   and d' :: "nat"   and n :: "char list"   and M :: "nat" assumes "match_ex_table P C pc xt = \<lfloor>(pc', d')\<rfloor>" shows "\<exists>(f, t, D, h, d)\<in>set (relevant_entries P (Invoke n M) pc xt). P \<turnstile> C \<preceq>\<^sup>* D \<and> pc \<in> {f..<t} \<and> pc' = h \<and> d' = d"
cases: fixes a1 :: "'a \<rightharpoonup>\<^sub>f 'b \<rightharpoonup>\<^sub>f 'c"   and a2 :: "'a"   and a3 :: "'b"   and a4 :: "'a"   and a5 :: "'c" assumes "unique_closest_attribute a1 a2 a3 a4 a5" obtains attributes :: "'a \<rightharpoonup>\<^sub>f 'b \<rightharpoonup>\<^sub>f 'c" and \<C> :: "'a" and attr :: "'b" and \<D> :: "'a" and \<tau> :: "'c" where "a1 = attributes" and "a2 = \<C>" and "a3 = attr" and "a4 = \<D>" and "a5 = \<tau>" and "closest_attribute attributes \<C> attr \<D> \<tau>" and "\<not> closest_attribute_not_unique attributes \<C> attr \<D> \<tau>"
buffon_altdef: fixes l :: "real"   and d :: "real" shows "buffon l d = uniform_measure lborel ({- d / 2..d / 2} \<times> {- pi..pi}) \<bind> (\<lambda>(x, \<phi>). return (count_space UNIV) (let a = x - l / 2 * sin \<phi>; b = x + l / 2 * sin \<phi> in min a b + d / 2 \<le> 0 \<and> 0 \<le> max a b + d / 2 \<or> min a b - d / 2 \<le> 0 \<and> 0 \<le> max a b - d / 2))"
hyps: assumes "\<forall>as' a as''. asx = as' @ a # as'' \<longrightarrow> \<not> sourcenode a -{V}a # as''\<rightarrow>\<^sub>d\<^sub>d n'"   and "asx \<noteq> [] \<longrightarrow> n' = last (targetnodes asx)" shows "(V, asx, asx) \<in> dependent_live_vars n'"
l_inv: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b" assumes "Group.group G"   and "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
prems_2: fixes i :: "nat"   and j :: "nat" assumes "i \<in> {..Suc n}"   and "j \<in> {..Suc n}"   and "i \<noteq> j" shows "I i <+> I j = carrier R"
phis'_fst: fixes g :: "('b, 'a, 'c) gen_cfg_wf"   and n :: "'b"   and v :: "'a \<times> 'b \<times> Def"   and vs :: "('a \<times> 'b \<times> Def) list"   and v' :: "'a \<times> 'b \<times> Def" assumes "gen_wf.phis' g (n, v) = Some vs"   and "v' \<in> set vs" shows "gen_wf_var g v' = gen_wf_var g v"
higher_differentiable_on_subset: fixes T :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat"   and S :: "'b set" assumes "higher_differentiable_on T f n"   and "S \<subseteq> T" shows "higher_differentiable_on S f n"
trivial_combine_imp_independent: fixes Us :: "'a list" assumes "set Us \<subseteq> carrier R"   and "\<And>Ks. \<lbrakk>set Ks \<subseteq> K; combine Ks Us = \<zero>\<rbrakk> \<Longrightarrow> set (take (length Us) Ks) \<subseteq> {\<zero>}" shows "local.independent K Us"
sds_impossibility_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" shows "sds_impossibility agents alts sds \<equiv> (anonymous_sds agents alts sds \<and> neutral_sds agents alts sds) \<and> sd_efficient_sds agents alts sds \<and> strategyproof_sds agents alts sds \<and> sds_impossibility_axioms agents alts"
assms_4: shows "c / 2 + d + 1 \<le> c"
untilEN: fixes n :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'cmp"   and \<gamma>' :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool" assumes "\<not> (\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>)"   and "eval c t t' n (\<gamma>' \<UU>\<^sub>b \<gamma>)" shows "\<exists>n'\<ge>n. eval c t t' n' \<gamma> \<and> (\<forall>n''\<ge>n. n'' < n' \<longrightarrow> eval c t t' n'' \<gamma>')"
submanifold_atlasI: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and S :: "'c set"   and c :: "('c, 'd) chart" assumes "submanifold charts k S"   and "c \<in> c_manifold.atlas charts k" shows "restrict_chart S c \<in> c_manifold.atlas (manifold.charts_submanifold charts S) k"
set_remove_nth: fixes xs :: "'b list"   and j :: "nat" assumes "distinct xs"   and "j < length xs" shows "set (remove_nth j xs) = set xs - {xs ! j}"
set_child_nodes_get_child_nodes: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and h' :: "(_) heap" assumes "type_wf h"   and "known_ptr ptr"   and "h \<turnstile> set_child_nodes ptr children \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"
simps: fixes n :: "char list"   and params :: "(variable \<times> PDDL_STRIPS_Semantics.type) list"   and pre :: "PDDL_STRIPS_Semantics.term atom formula"   and eff :: "PDDL_STRIPS_Semantics.term ast_effect" shows "wf_action_schema (Action_Schema n params pre eff) = (let tyt = ty_term (map_of params) constT in distinct (map fst params) \<and> wf_fmla tyt pre \<and> wf_effect tyt eff)"
Spy_see_shrK: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> orb" shows "(Key (shrK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
simps_1: fixes name :: "name" shows "frees (Svar name) = {|name|}"
convex_def: fixes S :: "'a vec set" shows "convex S = (convex_hull S = S)"
space_completion: fixes M :: "'a measure" shows "space (completion M) = space M"
trail_def: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c" shows "pre_digraph.trail G u p v \<equiv> pre_digraph.awalk G u p v \<and> distinct p"
seq_mono_right: fixes c\<^sub>0 :: "'b::refinement_lattice"   and c\<^sub>1 :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice" assumes "(c\<^sub>0::'b::refinement_lattice) \<sqsubseteq> (c\<^sub>1::'b::refinement_lattice)" shows "(d::'b::refinement_lattice) ; (c\<^sub>0::'b::refinement_lattice) \<sqsubseteq> d ; (c\<^sub>1::'b::refinement_lattice)"
prems_2: shows "eqvt_at cCCexp_sumC body"
finfun_rec_upd: fixes a :: "'b"   and f :: "('b, 'a) finfun"   and a' :: "'b"   and b' :: "'a" shows "finfun_rec (\<lambda>b. b) (\<lambda>a'. If (a = a')) (finfun_update f a' b') = (if a = a' then b' else finfun_rec (\<lambda>b. b) (\<lambda>a'. If (a = a')) f)"
S_edges_def: shows "ES.S_edges = all_edges ES.S_verts"
restrict_preds_to_vars_def: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and P :: "'d set"   and V :: "'a set" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars" shows "sifum_lang_no_dma.restrict_preds_to_vars bexp_vars P V \<equiv> {e \<in> P. bexp_vars e \<subseteq> V}"
\<R>\<^sub>3_sym: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option" shows "sym (\<R>\<^sub>3 \<Gamma>)"
d_delta_lnexp_cf2_nonpos: fixes x :: "real" shows "diff_delta_lnexp_cf2 x \<le> 0"
pff: fixes I :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and x :: "char list"   and n :: "nat" shows "support (\<lambda>l. I (l(x := n))) \<subseteq> support I"
simps_22: fixes x1 :: "'g option" shows "outputs (Resumption.resumption.Done x1) = {}"
simps_2: fixes x :: "'c"   and xs :: "'c list"   and ys :: "'c list" shows "(x # xs) @ ys = x # xs @ ys"
transp_on_subset: fixes A :: "'a set"   and B :: "'a set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "A \<subseteq> B"   and "transp_on P B" shows "transp_on P A"
x1_gt: fixes x :: "'a" assumes "x \<in> set l1" shows "x < x1"
nn_integral_zero': fixes f :: "'c \<Rightarrow> ennreal"   and M :: "'c measure" assumes "AE x in M. f x = 0" shows "integral\<^sup>N M f = 0"
fst_vimage_eq_Times: fixes S :: "'a set" shows "fst -` S = S \<times> UNIV"
Hash_EphK: fixes X :: "msg"   and X' :: "ephkey" shows "Hash X \<noteq> EphK X'"
iMOD_singleton_subset_conv: fixes r :: "nat"   and m :: "nat"   and a :: "nat" shows "([ r, mod m ] \<subseteq> {a}) = (r = a \<and> m = 0)"
insert: fixes A :: "'c set"   and x :: "'c"   and g :: "'c \<Rightarrow> 'd" assumes "finite A"   and "x \<notin> A" shows "sum g (insert x A) = g x + sum g A"
wfp_to_wffp: fixes sl :: "('St, 'U, 'C) state list"   and i :: "nat"   and \<pi> :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream" assumes "sl = map fst (stake i \<pi>)"   and "0 < i"   and "wfp UNIV \<pi>" shows "wffp sl \<and> (\<forall>j<length sl. fst (\<pi> !! j) = sl ! j) \<and> stateOf \<pi> = hd sl"
domain_eq: shows "Domainp (pcr_poly_mapping (=) (=)) = (\<lambda>f. finite {x. f x \<noteq> (0::'c)})"