hcju/isabelleps huggingface dataset:
proper_itE: fixes it :: "('f \<Rightarrow> bool) \<Rightarrow> ('g \<Rightarrow> 'f \<Rightarrow> 'f) \<Rightarrow> 'f \<Rightarrow> 'f"   and it' :: "('h \<Rightarrow> bool) \<Rightarrow> ('g \<Rightarrow> 'h \<Rightarrow> 'h) \<Rightarrow> 'h \<Rightarrow> 'h" assumes "proper_it it it'" obtains l :: "'g list" where "it = foldli l" and "it' = foldli l"
iIN_leD: fixes x :: "nat"   and n :: "nat"   and d :: "nat" assumes "x \<in> [n\<dots>,d]" shows "x \<le> n + d"
digraph_isomorphism_def: fixes G :: "('e, 'f) pre_digraph"   and hom :: "('e, 'f, 'g, 'h) digraph_isomorphism" shows "pre_digraph.digraph_isomorphism G hom \<equiv> wf_digraph G \<and> inj_on (iso_verts hom) (verts G) \<and> inj_on (iso_arcs hom) (arcs G) \<and> (\<forall>a\<in>arcs G. iso_verts hom (tail G a) = iso_tail hom (iso_arcs hom a) \<and> iso_verts hom (head G a) = iso_head hom (iso_arcs hom a))"
fps_of_ratfps_def: shows "fps_of_ratfps \<equiv> map_fun Rep_ratfps id (\<lambda>x. fps_of_poly (numerator x) / fps_of_poly (denominator x))"
mve1: shows "make_vert_equiv 1 = mat1 2"
phis'_aux_keys_super: fixes phis :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping"   and g :: "'g"   and v :: "'var"   and ns :: "'node set" shows "Mapping.keys phis \<subseteq> Mapping.keys (phis'_aux g v ns phis)"
can_step_def: fixes lstep :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'a"   and s :: "'b" shows "system.can_step lstep l s \<equiv> \<exists>s'. lstep l s s'"
lin_Ls_Bs: shows "lin_indpt_list (Ls @ Bs)"
results_in_mono_1: fixes t1 :: "'t"   and t2 :: "'t"   and s :: "'sum" assumes "t1 \<le> t2" shows "results_in t1 s \<le> results_in t2 s"
inf_commute: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.semilattice_inf inf less_eq less" shows "inf x y = inf y x"
borel_measurable_Max_finite: fixes n :: "nat"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "0 < n"   and "card A = n"   and "\<forall>a\<in>A. f a \<in> borel_measurable M" shows "(\<lambda>w. MAX a\<in>A. f a w) \<in> borel_measurable M"
acc_ss: fixes q :: "'Q" assumes "acc q \<noteq> {}" shows "q \<in> V"
Lower_is_closed: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set" assumes "weak_partial_order L"   and "A \<subseteq> carrier L" shows "is_closed\<^bsub>L\<^esub> (Lower L A)"
simps: fixes a :: "db" shows "val a = ((\<exists>n. a = DBNum n) \<or> a = DBff \<or> a = DBtt \<or> (\<exists>e. a = DBAbsN e) \<or> (\<exists>e. a = DBAbsV e))"
distinct_1: fixes x11 :: "'a"   and x12 :: "nat"   and x13 :: "'a interval"   and x21 :: "'a"   and x22 :: "nat"   and x23 :: "'a interval" shows "INS x11 x12 x13 \<noteq> DEL x21 x22 x23"
sorted_rev_nth_mono: fixes xs :: "'a list"   and i :: "nat"   and j :: "nat" assumes "sorted (rev xs)"   and "i \<le> j"   and "j < length xs" shows "xs ! j \<le> xs ! i"
labelAtOut_def: fixes v :: "nat"   and p :: "(form, nat) out_port" shows "task.labelAtOut v p = subst (inst v) (predicate.freshen (id v) (task.labelsOut (nodeOf v) p))"
abs_eq: fixes x :: "dflt_size word" shows "uint32_of_uint (Abs_uint x) = Abs_uint32 (ucast x)"
zero_below_val_Zp: fixes x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "x \<in> carrier Zp"   and "eint (int n) \<le> val_Zp x" shows "x n = 0"
steps_preserves_terminations: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and cs :: "('d, 'e, 'f) com list"   and css :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list"   and s :: "('d, 'f) xstate"   and cs' :: "('d, 'e, 'f) com list"   and css' :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list"   and t :: "('d, 'f) xstate" assumes "\<Gamma>\<turnstile> (cs, css, s) \<rightarrow>\<^sup>* (cs', css', t)"   and "\<Gamma>\<turnstile>cs,css \<Down> s" shows "\<Gamma>\<turnstile>cs',css' \<Down> t"
step_2_main_dims_main: fixes n :: "nat"   and j :: "nat"   and A :: "'b mat" shows "dim_row (??.Jordan_Normal_Form_Existence.step_2_main n j A) = dim_row A \<and> dim_col (??.Jordan_Normal_Form_Existence.step_2_main n j A) = dim_col A"
\<beta>_boundedness_gt: fixes m :: "int"   and x :: "'a" assumes "m \<le> int (k x)"   and "x \<in> X" shows "\<exists>U\<subseteq>\<R>. \<Union> U = {u \<in> V. real_of_int m < u x}"
init_pnet_fst_sr_netgmap: fixes s :: "'s net_state"   and p :: "net_tree"   and i :: "nat" assumes "s \<in> init (pnet np p)"   and "i \<in> net_ips s"   and "wf_net_tree p" shows "the (fst (netgmap sr s) i) \<in> (fst \<circ> sr) ` init (np i)"
assms_1: shows "m1 f|` S \<sqsubseteq> m2 f|` S"
restrict_index_in: fixes i :: "nat"   and X :: "'b option list"   and I :: "nat set" assumes "i < length X"   and "i \<in> I" shows "restrict I X ! i = X ! i"
that: fixes Q :: "'c multiset \<Rightarrow> 'a" assumes "Q \<in> carrier L"   and "\<not> index_free Q j" shows "thesis"
intro: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b" assumes "monad return bind"   and "monad_fail_axioms bind fail" shows "monad_fail return bind fail"
valid_Tree\<^sub>\<alpha>_Act\<^sub>\<alpha>: fixes P :: "'state"   and \<alpha> :: "'act"   and t\<^sub>\<alpha> :: "('b, 'pred, 'act) Tree\<^sub>\<alpha>" shows "valid_Tree\<^sub>\<alpha> P (Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha>) = (\<exists>\<alpha>'' t\<^sub>\<alpha>' P'. Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha> = Act\<^sub>\<alpha> \<alpha>'' t\<^sub>\<alpha>' \<and> P \<rightarrow> \<langle>\<alpha>'',P'\<rangle> \<and> valid_Tree\<^sub>\<alpha> P' t\<^sub>\<alpha>')"
axioms_1: fixes \<Omega> :: "'c set"   and M :: "'c set set" assumes "sigma_algebra \<Omega> M" shows "algebra \<Omega> M"
osig_tvsT_subset_SortsT: fixes T :: "typ" shows "snd ` tvsT T \<subseteq> SortsT T"
empty_not_add_mset: fixes a :: "'b"   and A :: "'b multiset" shows "{#} \<noteq> add_mset a A"
iso_def: fixes f :: "'d" shows "D.iso f \<equiv> \<exists>g. D.inverse_arrows f g"
alt_seq_limit: fixes p :: "int"   and s :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "padic_integers p"   and "padic_integers.is_Zp_cauchy p s" shows "res_lim (alt_seq s) = res_lim s"
Compl_NegPf: fixes var :: "'d set"   and trm :: "'e set"   and fmla :: "'f set"   and Var :: "'d \<Rightarrow> 'e"   and FvarsT :: "'e \<Rightarrow> 'd set"   and substT :: "'e \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'e"   and Fvars :: "'f \<Rightarrow> 'd set"   and subst :: "'f \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'f"   and eql :: "'e \<Rightarrow> 'e \<Rightarrow> 'f"   and cnj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and imp :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and all :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and exi :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and fls :: "'f"   and dsj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and num :: "'e set"   and enc :: "'f \<Rightarrow> 'e"   and prv :: "'f \<Rightarrow> bool"   and bprv :: "'f \<Rightarrow> bool"   and P :: "'f"   and isTrue :: "'f \<Rightarrow> bool"   and Pf :: "'f"   and n :: "'e"   and \<phi> :: "'f" assumes "Minimal_Truth_Soundness_HBL1iff_Compl_Pf_Compl_NegPf var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num enc prv bprv P isTrue Pf"   and "n \<in> num"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "let PPf = \<lambda>t1 t2. Generic_Syntax.psubst var Var FvarsT Fvars subst Pf [(t1, Generic_Syntax.Variable var 11), (t2, Generic_Syntax.Variable var 10)] in isTrue (Syntax_with_Connectives_False.neg imp fls (PPf n (enc \<phi>))) \<longrightarrow> bprv (Syntax_with_Connectives_False.neg imp fls (PPf n (enc \<phi>)))"
bset_quot_map: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Abs :: "'e \<Rightarrow> 'f"   and Rep :: "'f \<Rightarrow> 'e"   and T :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "Quotient R Abs Rep T" shows "Quotient (rel_bset R) (map_bset Abs) (map_bset Rep) (rel_bset T)"
pw_ele_iff: fixes m :: "('c + 'd) nres"   and n :: "('c + 'd) nres" shows "(m \<le> n) = (nofail n \<longrightarrow> nofail m \<and> (\<forall>e. inres m (Inl e) \<longrightarrow> inres n (Inl e)) \<and> (\<forall>x. inres m (Inr x) \<longrightarrow> inres n (Inr x)))"
simps_2: fixes b :: "nat"   and p :: "String.literal"   and \<phi> :: "formula"   and \<psi> :: "formula" shows "fvi b (formula.Let p \<phi> \<psi>) = fvi b \<psi>"
emb_wit: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist" assumes "chain R\<^sup>+\<^sup>+ xs" shows "emb xs (wit R xs)"
AMin_mem: fixes A :: "ant set"   and z :: "int" assumes "A \<subseteq> LBset (ant z)"   and "A \<noteq> {}" shows "AMin A \<in> A"
set_to_map_dom: fixes S :: "('a \<times> 'b) set" shows "dom (set_to_map S) = fst ` S"
\<Z>_\<beta>: fixes \<beta> :: "V" assumes "\<beta> = \<alpha>" shows "\<Z> \<beta>"
not_in_keys_iff_lookup_eq_zero: fixes k :: "'a"   and f :: "'a \<Rightarrow>\<^sub>0 'b" shows "(k \<notin> keys f) = (lookup f k = (0::'b))"
bool_cast: shows "G\<turnstile>PrimT Boolean\<preceq>? T"
less_aexp_trans: fixes x :: "'b aexp"   and y :: "'b aexp"   and z :: "'b aexp" assumes "x < y"   and "y < z" shows "x < z"
delete_map_eqE: fixes i :: "nat"   and g :: "nat \<Rightarrow> 'c"   and g' :: "nat \<Rightarrow> 'c" assumes "delete_map i g = delete_map i g'" obtains x :: "'c" where "g = g'(i := x)"
add_commute: fixes a :: "'b \<times> 'c"   and b :: "'b \<times> 'c" shows "sup_prod a b = sup_prod b a"
dM2: shows "M2 \<in> carrier_mat d2 d2"
cr_view_def: shows "cr_view \<equiv> \<lambda>x y. x = from_view y"
satisfies_lderiv0: fixes \<AA> :: "'i"   and a :: "'a"   and x :: "'x" assumes "wf0 (#\<^sub>V \<AA>) a"   and "#\<^sub>V \<AA> = size x"   and "lformula0 a" shows "(\<AA> \<Turnstile> lderiv0 x a) = (CONS x \<AA> \<Turnstile>\<^sub>0 a)"
homeomorphism_image2: fixes S :: "'a set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" assumes "homeomorphism S T f g" shows "g ` T = S"
arr_char: fixes Obj :: "'b set"   and Hom :: "'b \<Rightarrow> 'b \<Rightarrow> 'c set"   and Id :: "'b \<Rightarrow> 'c"   and Comp :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "('b, 'c) Maps.arr" assumes "concrete_category Obj Hom Id Comp" shows "partial_magma.arr (concrete_category.COMP Obj Hom Comp) f = (f \<noteq> Maps.Null \<and> concrete_category.Dom f \<in> Obj \<and> Maps.Cod f \<in> Obj \<and> Maps.Map f \<in> Hom (concrete_category.Dom f) (Maps.Cod f))"
config_config_distinct: fixes A :: "('c list \<Rightarrow> 'd) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> (nat \<times> nat list) \<times> 'd)"   and init :: "'c list"   and qs :: "'c list" shows "distinct (fst (config A init qs)) = distinct init"
chine_hcomp_props_5: fixes \<mu> :: "'a arrow_of_spans_data"   and \<nu> :: "'a arrow_of_spans_data" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<nu> = trg \<mu>" shows "\<p>\<^sub>0[Leg0 (Cod \<nu>), Leg1 (Cod \<mu>)] \<cdot> chine_hcomp \<nu> \<mu> = Chn \<mu> \<cdot> \<p>\<^sub>0[Leg0 (Dom \<nu>), Leg1 (Dom \<mu>)]"
remove_elt_pairs_map: fixes ys :: "('c \<times> 'd) list"   and x :: "'c" assumes "Lists_Ex.strict_sorted (map fst ys)" shows "map_of_alist (remove_elt_pairs x ys) = delete_map x (map_of_alist ys)"
Ctxt_validn_def: fixes n :: "nat"   and G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set" shows " \<Turnstile>\<^sub>n G   = (\<forall>m\<le>n. \<forall>A. A \<in> G \<longrightarrow>  \<Turnstile>\<^sub>m Call : A )"
closedin_continuous_map_preimage: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and C :: "'b set" assumes "continuous_map X Y f"   and "closedin Y C" shows "closedin X {x \<in> topspace X. f x \<in> C}"
\<F>\<G>_semantics': fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<F>\<G> \<phi> w = subformulas\<^sub>\<nu> \<phi> \<inter> {\<psi>. w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>)}"
csum_mono1: fixes p1 :: "('d \<times> 'd) set"   and r1 :: "('e \<times> 'e) set"   and q :: "('f \<times> 'f) set" assumes "p1 \<le>o r1" shows "p1 +c q \<le>o r1 +c q"
card_geq_2_iff_contains_2_elems: fixes A :: "'a set" shows "(2 \<le> card A) = (finite A \<and> (\<exists>x y. x \<noteq> y \<and> x \<in> A \<and> y \<in> A))"
inf_aci_1: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.semilattice_inf inf less_eq less" shows "inf x y = inf y x"
dg_SemiCAT_components_2: fixes \<alpha> :: "V" shows "dg_SemiCAT \<alpha>\<lparr>Arr\<rparr> = all_smcfs \<alpha>"
iso_inv_iso: fixes f :: "'a" assumes "L'.C.iso f" shows "L'.C.iso (L'.C.inv f)"
SD_singleton_left: fixes x :: "'a"   and q :: "'a pmf" assumes "x \<in> carrier"   and "q \<in> lotteries_on carrier" shows "SD le (return_pmf x) q = (\<forall>y\<in>set_pmf q. le x y)"
max_coeff_bound_right: fixes a :: "nat list"   and b :: "nat list"   and xs :: "nat list"   and ys :: "nat list" assumes "hlde a b"   and "(xs, ys) \<in> Linear_Diophantine_Equations.Minimal_Solutions a b" shows "\<forall>x\<in>set xs. x \<le> maxne0 ys b"
hd_in_set: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "hd xs \<in> set xs"
newP': shows "P' \<in> carrier_mat (n - Suc 0) (n - Suc 0)"
klift_prop1: fixes f :: "'f \<Rightarrow> 'e set"   and g :: "'d \<Rightarrow> 'f set" shows "bd\<^sub>\<F> (bd\<^sub>\<F> f \<circ> g) = bd\<^sub>\<F> f \<circ> bd\<^sub>\<F> g"
equivalent_implies_exist_matrix': fixes A :: "(('a, 'b) vec, 'c) vec"   and B :: "(('a, 'b) vec, 'c) vec" assumes "equivalent_matrices A B" shows "\<exists>X Y X' Y' f. linear (*s) (*s) f \<and> matrix' X Y f = A \<and> matrix' X' Y' f = B \<and> is_basis (set_of_vector X) \<and> is_basis (set_of_vector Y) \<and> is_basis (set_of_vector X') \<and> is_basis (set_of_vector Y')"
singleton: fixes x :: "'a::type" shows "\<Sqinter>\<^sub>f\<^sub>i\<^sub>n {x::'a::type} = x"
mult_right_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq zero c" shows "less_eq (times a c) (times b c)"
split_beta: fixes f :: "'l \<Rightarrow> 'm \<Rightarrow> 'k"   and prod :: "'l \<times> 'm" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (fst prod) (snd prod)"
rel_interior_convex_hull_union: fixes I :: "'a set"   and S :: "'a \<Rightarrow> 'b set" assumes "finite I"   and "\<forall>i\<in>I. convex (S i) \<and> S i \<noteq> {}" shows "rel_interior (convex hull \<Union> (S ` I)) = {\<Sum>i\<in>I. c i *\<^sub>R s i |c s. (\<forall>i\<in>I. 0 < c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> rel_interior (S i))}"
beukers_nn_integral1_swap: fixes r :: "nat"   and s :: "nat" shows "beukers_nn_integral1 r s = beukers_nn_integral1 s r"
C_p: shows "C \<in> Liminf_llist Gs - sr.Rf (Liminf_llist Gs)"
fsubset_refl: fixes A :: "'d fset" shows "A |\<subseteq>| A"
no_inf_from_last: shows "\<forall>y. \<not> llast Sts \<leadsto>\<^sub>w y"
zp: shows "\<forall>bs. wf_bs bs p \<longrightarrow> Ipoly bs p = (0::'a)"
lessThan_Suc: fixes g :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "sum g {..<Suc n} = sum g {..<n} + g n"
iso_is_section: fixes f :: "'s" assumes "local.iso f" shows "section f"
wf_measure_triple: shows "wf (measure size <*lex*> SNa_Redu <*lex*> SNa_Redu)"
below_sctn_def: fixes X :: "'b set"   and sctn :: "'b sctn" shows "below_sctn X sctn = SPEC (\<lambda>b. b \<longrightarrow> X \<subseteq> below_halfspace sctn)"
AssertionStatImp_def: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b" shows "\<Psi> \<hookrightarrow> \<Psi>' \<equiv> \<forall>\<Phi>. \<Psi> \<turnstile> \<Phi> \<longrightarrow> \<Psi>' \<turnstile> \<Phi>"
image_mset_add_mset: fixes f :: "'l \<Rightarrow> 'k"   and a :: "'l"   and M :: "'l multiset" shows "image_mset f (add_mset a M) = add_mset (f a) (image_mset f M)"
is_Guard_def: fixes c :: "('d, 'e, 'f) com" shows "is_Guard c = (case c of Guard f g c' \<Rightarrow> True | _ \<Rightarrow> False)"
count_def: fixes t :: "('c, 'd) FingerTree" shows "count t \<equiv> FingerTreeStruc.count (Rep_FingerTree t)"
unfold_congs_7: fixes r :: "'a t_context_scheme"   and r' :: "'a t_context_scheme"   and v' :: "t list list"   and f :: "t list list \<Rightarrow> t list list"   and f' :: "t list list \<Rightarrow> t list list" assumes "r = r'"   and "label r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "label_update f r = label_update f' r'"
not_outside_2: fixes x :: "'a"   and y :: "'a" assumes "of_weak_ranking (weak_ranking_aux carrier) x y" shows "y \<in> carrier"
Joints_msum_aforms: fixes xs :: "(real \<times> real pdevs) list"   and d :: "nat"   and ys :: "(real \<times> real pdevs) list" assumes "degree_aforms xs \<le> d"   and "degree_aforms ys \<le> d" shows "Joints (msum_aforms d xs ys) = {map2 (+) a b |a b. a \<in> Joints xs \<and> b \<in> Joints ys}"
wg_equiv: fixes H :: "'b program"   and F :: "'b program"   and X :: "'b program set" shows "(H \<in> wg F X) = (F component_of H \<longrightarrow> H \<in> X)"
openin_open_eq: fixes s :: "'b set"   and t :: "'b set" assumes "open s" shows "openin (top_of_set s) t = (open t \<and> t \<subseteq> s)"
not_vset_not_disj_eff_prod_dom_diff: fixes a :: "('c, bool) fmap \<times> ('c, bool) fmap"   and PROB :: "(('c, bool) fmap \<times> ('c, bool) fmap) set"   and vs :: "'c set" assumes "a \<in> PROB"   and "\<not> varset_action a vs" shows "fmdom' (snd a) \<inter> (prob_dom PROB - vs) \<noteq> {}"
path_endpoints_2: fixes g :: "'b ugraph"   and u :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b" assumes "path g u p v"   and "p \<noteq> []" shows "v \<in> nodes g"
aadd_commute: fixes x :: "ant"   and y :: "ant" shows "x + y = y + x"
typ_ok_sig_Ty: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and n :: "String.literal"   and Ts :: "typ list" assumes "typ_ok_sig \<Sigma> (Ty n Ts)" shows "list_all (typ_ok_sig \<Sigma>) Ts"
dim_row_mat_1: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'c" shows "dim_row (mat nr nc f) = nr"
plus_vinsert: fixes x :: "V"   and z :: "V"   and y :: "V" shows "x + vinsert z y = vinsert (x + z) (x + y)"
NTRS_3: shows "((s, ce), e, s', ce') \<in> ntrs fg"
negligible_imp_measure0: fixes S :: "'c set" assumes "negligible S" shows "Sigma_Algebra.measure lebesgue S = 0"
problem_plan_bound_S_bound: fixes PROB :: "(('b, bool) fmap \<times> ('b, bool) fmap) set"   and vs :: "'b set"   and lss :: "('b, bool) fmap list" assumes "finite PROB"   and "top_sorted_abs (\<lambda>x y. y \<in> state_successors (prob_proj PROB vs) x) lss"   and "set lss = valid_states (prob_proj PROB vs)" shows "problem_plan_bound PROB \<le> Sup {S vs lss PROB s' |s'. s' \<in> valid_states (prob_proj PROB vs)}"
Mf: shows "f \<in> N \<rightarrow>\<^sub>M subprob_algebra R"
sorted_nth_mono: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and i :: "nat"   and j :: "nat" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs"   and "i \<le> j"   and "j < length xs" shows "less_eq (xs ! i) (xs ! j)"
min_lvar_not_in_bounds_Some: fixes min_lvar_not_in_bounds :: "('a, 'b) state \<Rightarrow> nat option"   and s :: "('a, 'b) state"   and x\<^sub>i :: "nat" assumes "MinLVarNotInBounds min_lvar_not_in_bounds" shows "min_lvar_not_in_bounds s = Some x\<^sub>i \<longrightarrow> \<not> in_bounds x\<^sub>i \<langle>\<V> s\<rangle> (\<B> s)"
sup_state_opt_any_None: fixes P :: "'a prog"   and s :: "(ty list \<times> ty err list) option" shows "P \<turnstile> s \<le>' None = (s = None)"
le_supE: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<le> a \<sqinter> b" obtains "x \<le> a" and "x \<le> b"
eq_mk_fun_iff: fixes T :: "Unary_PCF.type"   and Ts :: "Unary_PCF.type list" shows "(T = Ts \<rightarrow>\<rightarrow> \<B>) = (Ts = dest_fun T)"
eq_1: fixes m :: "nat" assumes "coprime m n" shows "\<chi> m = 1"
dbl_simps_3: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl plus one = numeral.numeral one plus (num.Bit0 num.One)"
emsapss_encode_help7_def: fixes maskedDB :: "bit list"   and H :: "bit list"   and emBits :: "nat" shows "emsapss_encode_help7 maskedDB H emBits = emsapss_encode_help8 (maskedDB_zero maskedDB emBits) H"
ex_reg: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<forall>x. P x \<longrightarrow> Q x"   and "Ex P" shows "Ex Q"
sublens'_def: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e" shows "X \<subseteq>\<^sub>L' Y = (\<forall>s\<^sub>1 s\<^sub>2 s\<^sub>3. s\<^sub>1 \<triangleleft>\<^bsub>Y\<^esub> s\<^sub>2 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>3 = s\<^sub>1 \<triangleleft>\<^bsub>Y\<^esub> (s\<^sub>2 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>3))"
countable_approach: fixes x :: "ereal" assumes "x \<noteq> - \<infinity>" shows "\<exists>f. incseq f \<and> (\<forall>i. f i < x) \<and> f \<longlonglongrightarrow> x"
countable_insert: fixes A :: "'f set"   and a :: "'f" assumes "countable A" shows "countable (insert a A)"
ennreal_zero_less_divide: fixes a :: "ennreal"   and b :: "ennreal" shows "(0 < a / b) = (0 < a \<and> b < top)"
mod_mult_cong_left: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and c :: "'b"   and b :: "'b"   and a :: "'b"   and d :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo (modulo c (times b a)) d a = unique_euclidean_semiring.cong modulo c d a"
assms_1: shows "\<forall>as s. s \<in> valid_states PROB \<and> as \<in> valid_plans PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < f PROB)"
continuous_on_mult_const: fixes s :: "'d set"   and c :: "'d" shows "continuous_on s ((*) c)"
word_ops_nth_size: fixes n :: "nat"   and x :: "'b word"   and y :: "'b word" assumes "n < size x" shows "(x OR y) !! n = (x !! n \<or> y !! n) \<and> (x AND y) !! n = (x !! n \<and> y !! n) \<and> (x XOR y) !! n = (x !! n \<noteq> y !! n) \<and> NOT x !! n = (\<not> x !! n)"
decrnum: fixes t :: "Cooper.num"   and x :: "int"   and bs :: "int list" assumes "numbound0 t" shows "Inum (x # bs) t = Inum bs (decrnum t)"
branch_instr_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (branch_instr instr s1)) \<and> t2 = snd (fst (branch_instr instr s2))" shows "low_equal t1 t2"
m_def: shows "m = max_list_non_empty (map (\<lambda>i. \<bar>coeff p i\<bar>) [0..<n])"
mta_bisim_conv: fixes bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and t :: "'a"   and ta1 :: "('f, 'a, 'b, 'c, 'g, 'h) thread_action"   and t' :: "'a"   and ta2 :: "('f, 'a, 'd, 'e, 'g, 'h) thread_action" shows "FWbisimulation_base.mta_bisim bisim (t, ta1) (t', ta2) = (t = t' \<and> ta_bisim bisim ta1 ta2)"
lim_ln_over_ln1: shows "(\<lambda>n. ln (real n) / ln (real (Suc n))) \<longlonglongrightarrow> 1"