hcju/isabelleps huggingface dataset:
to_nat_on_from_nat_into_infinite: fixes A :: "'c set"   and n :: "nat" assumes "countable A"   and "infinite A" shows "to_nat_on A (from_nat_into A n) = n"
roofed_idem: fixes G :: "('k, 'l) graph_scheme"   and B :: "'k set"   and S :: "'k set" shows "roofed_gen G B (roofed_gen G B S) = roofed_gen G B S"
sec_id_hash_def: fixes n :: "nat"   and m :: "nat" shows "sec_id_hash n m \<equiv> insert (Spy, \<langle>n, Hash (ID n (Sec_PubKey m))\<rangle>) (sec_id_pubk_more n m)"
NTCatCod_def: fixes \<eta> :: "('g, 'd, 'h, 'e, 'i, 'f, 'j) NatTrans_scheme" shows "NTCatCod \<eta> \<equiv> CatCod (NTCod \<eta>)"
cart_product_singleton_right: fixes N :: "'a multiset"   and b :: "'a set" shows "{#m \<in># N \<times># {#b#}. fst m \<in> snd m#} = {#n \<in># N. n \<in> b#} \<times># {#b#}"
Inf_islimpt: fixes S :: "real set" assumes "bdd_below S"   and "Inf S \<notin> S"   and "S \<noteq> {}" shows "Inf S islimpt S"
lessThan_Suc_shift: fixes g :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "prod g {..<Suc n} = g 0 * (\<Prod>i<n. g (Suc i))"
injects_1: fixes cfun :: "Value \<rightarrow> Value"   and cfun' :: "Value \<rightarrow> Value" shows "(Fn\<cdot>cfun = Fn\<cdot>cfun') = (cfun = cfun')"
strict_iff_order: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" shows "(a \<prec> b) = (a \<preceq> b \<and> a \<noteq> b)"
cases: fixes x :: "tree" obtains "x = Leaf"   | l :: "tree" and r :: "tree" where "x = Branching l r"
Cons_eq_filterD: fixes x :: "'b"   and xs :: "'b list"   and P :: "'b \<Rightarrow> bool"   and ys :: "'b list" assumes "x # xs = filter P ys" shows "\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
is_on_stack_impl_def: fixes SBIP :: "'c list \<times> nat list \<times> ('c \<Rightarrow> node_state option) \<times> (nat \<times> 'c set) list"   and v :: "'c" shows "GS.is_on_stack_impl SBIP v \<equiv> case GS.I SBIP v of None \<Rightarrow> False | Some (STACK x) \<Rightarrow> True | Some DONE \<Rightarrow> False"
init_comp_ty_PrimT: fixes pt :: "prim_ty" shows "init_comp_ty (PrimT pt) = Skip"
prj_in_hom_4: shows "\<guillemotleft>gfg.Prj\<^sub>1 : Chn (g \<star> f \<star> g) \<rightarrow>\<^sub>C g.apex\<guillemotright>"
scaleR_diff_left: fixes a :: "real"   and b :: "real"   and x :: "'c" shows "(a - b) *\<^sub>R x = a *\<^sub>R x - b *\<^sub>R x"
rw: shows "basis_left_null_space A = basis_null_space (Finite_Cartesian_Product.transpose A)"
strict_sorted_list_of_set: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "class.linorder less_eq less" shows "linorder.strict_sorted less (linorder.sorted_list_of_set less_eq A)"
create_opt_sol_in_LP_4: fixes A :: "rat mat"   and b :: "rat vec"   and c :: "rat vec"   and m :: "nat"   and n :: "nat"   and X :: "(nat, rat) mapping"   and x :: "rat vec"   and y :: "rat vec"   and i :: "nat" assumes "rat_LP A b c m n"   and "create_optimal_solutions A b c = Inr X"   and "(x, y) = split_n_m_x (dim_vec c) (dim_vec b) X"   and "i < dim_vec y" shows "0 \<le> y $ i"
eq_B: fixes x :: "vote" shows "(x = B) = (x \<noteq> A)"
SeqSkip: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate" shows "\<Gamma>\<turnstile> (Seq Skip c\<^sub>2, s) \<rightarrow> (c\<^sub>2, s)"
k_invariant_step_new_aux: fixes X :: "'rep list"   and x :: "'rep"   and ec :: "'rep"   and ec' :: "'rep"   and Y :: "'mt" assumes "set X \<subseteq> set (k_succs x)"   and "k_isNode x"   and "k_isNode ec"   and "simAbs ec' \<in> simAbs ` set X"   and "simAbs ec = simAbs x" shows "\<exists>r. lookup tOps (foldr (transUpdate x) X Y) (ec, simObs a ec') = Some r \<and> simAbs r \<in> simAbs ` set (k_succs ec) \<and> simObs a r = simObs a ec'"
setcompr_eq_image: fixes f :: "'c \<Rightarrow> 'b"   and P :: "'c \<Rightarrow> bool" shows "{f x |x. P x} = f ` {x. P x}"
list_all2_append2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and zs :: "'b list" shows "list_all2 P xs (ys @ zs) = (\<exists>us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and> list_all2 P us ys \<and> list_all2 P vs zs)"
trans: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "prefix a b"   and "prefix b c" shows "prefix a c"
real_of_nat_less_numeral_iff: fixes n :: "nat"   and w :: "num" shows "(real n < numeral w) = (n < numeral w)"
fresh_Pair: fixes a :: "atom"   and x :: "'c"   and y :: "'d" shows "a \<sharp> (x, y) = (a \<sharp> x \<and> a \<sharp> y)"
INF_emeasure_decseq': fixes A :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "\<And>i. A i \<in> sets M"   and "decseq A"   and "\<exists>i. emeasure M (A i) \<noteq> \<infinity>" shows "(INF n. emeasure M (A n)) = emeasure M (\<Inter> (range A))"
lookup_lift_poly_fun: fixes f :: "('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "lookup (lift_poly_fun f p) v = (lookup (f (proj_poly (component_of_term v) p)) (pp_of_term v) when component_of_term v \<in> keys (vectorize_poly p))"
simps_1: fixes x1 :: "'a"   and x2 :: "'b"   and y1 :: "'a"   and y2 :: "'b" shows "(Monoid_ap x1 x2 = Monoid_ap y1 y2) = (x1 = y1 \<and> x2 = y2)"
ground_SMP_disjointI: fixes A :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and B :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and C :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set" assumes "fv\<^sub>s\<^sub>e\<^sub>t (set A) \<inter> fv\<^sub>s\<^sub>e\<^sub>t (set B) = {}"   and "finite_SMP_representation arity Ana \<Gamma> A"   and "finite_SMP_representation arity Ana \<Gamma> B"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s C"   and "\<forall>t\<in>set A. \<forall>s\<in>set B. \<Gamma> t = \<Gamma> s \<and> mgu t s \<noteq> None \<longrightarrow> intruder_synth' public arity {} t \<and> intruder_synth' public arity {} s \<or> (\<exists>u\<in>C. is_wt_instance_of_cond \<Gamma> t u) \<and> (\<exists>u\<in>C. is_wt_instance_of_cond \<Gamma> s u)" shows "{t \<in> SMP (set A). fv t = {}} \<inter> {t \<in> SMP (set B). fv t = {}} \<subseteq> {t \<cdot> \<delta> |t \<delta>. t \<in> C \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> fv (t \<cdot> \<delta>) = {}} \<union> {m. {} \<turnstile>\<^sub>c m}"
wprepare_loop_start_in_middle_2_Oc: fixes m :: "nat"   and a :: "nat"   and aa :: "nat"   and listaa :: "nat list"   and b :: "cell list"   and list :: "cell list" assumes "wprepare_goto_start_pos m (a # aa # listaa) (b, Oc # list)" shows "wprepare_loop_start_in_middle m (a # aa # listaa) (Oc # b, list)"
nat_pow_closed: fixes x :: "'c \<Rightarrow> 'a"   and n :: "nat" assumes "x \<in> carrier F" shows "x [^]\<^bsub>F\<^esub> n \<in> carrier F"
arr_dom_iff_arr: fixes f :: "'a" shows "Right.arr (Right.dom f) = Right.arr f"
angle_nonneg: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "0 \<le> angle a b c"
G_is_faithful: shows "faithful_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G"
varsSubsetFormula: fixes F :: "Literal list list"   and F' :: "Literal list list" assumes "\<forall>c. c el F \<longrightarrow> c el F'" shows "vars F \<subseteq> vars F'"
component_of_term_splus: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and t :: "'b"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.component_of_term pair_of_term (term_powerprod.splus pair_of_term term_of_pair t v) = term_powerprod.component_of_term pair_of_term v"
leg_simps_3: shows "C.arr \<pi>.cod.leg1"
simps_2: fixes x2 :: "nat"   and y2 :: "nat" shows "(Indet x2 = Indet y2) = (x2 = y2)"
var_app_typesE: fixes e :: "nat \<Rightarrow> LambdaType.type"   and i :: "nat"   and ts :: "dB list"   and T :: "LambdaType.type" assumes "e \<turnstile> Var i \<degree>\<degree> ts : T" obtains Ts :: "LambdaType.type list" where "e \<turnstile> Var i : Ts \<Rrightarrow> T" and "e \<tturnstile> ts : Ts"
set_integral_sum: fixes B :: "'a set"   and M :: "'b measure"   and A :: "'b set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "finite B"   and "\<And>x. x \<in> B \<Longrightarrow> set_integrable M A (f x)" shows "\<integral>y\<in>A. (\<Sum>x\<in>B. f x y)\<partial>M = (\<Sum>x\<in>B. set_lebesgue_integral M A (f x))"
vector_eq: fixes x :: "'b"   and y :: "'b" shows "(x = y) = (x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x)"
set_of_rem: fixes t :: "tree" assumes "t \<noteq> Tip" shows "set_of t = set_of (rem t) \<union> {rm t}"
truncate_up_def: fixes prec :: "nat"   and x :: "real" shows "truncate_up prec x = round_up (int prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor>) x"
cbox_interval: fixes a :: "'a"   and b :: "'a" shows "cbox a b = {a..b}"
r_ifeq_else_diverg_def: shows "r_ifeq_else_diverg \<equiv> Cn 3 r_add [recf.Id 3 2, Mn 3 (Cn 4 r_add [recf.Id 4 0, Cn 4 r_eq [recf.Id 4 1, recf.Id 4 2]])]"
ccNeighbors_mono: fixes G :: "CoCalls"   and G' :: "CoCalls"   and x :: "Vars.var" assumes "G \<sqsubseteq> G'" shows "ccNeighbors x G \<subseteq> ccNeighbors x G'"
cone_hull_eq: fixes S :: "'c set" shows "(cone hull S = S) = cone S"
cSup_least: fixes X :: "'a multiset set"   and z :: "'a multiset" assumes "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> x \<subseteq># z" shows "Sup X \<subseteq># z"
map_leD: fixes m :: "'a \<Rightarrow> 'b option"   and m' :: "'a \<Rightarrow> 'b option"   and x :: "'a"   and v :: "'b" assumes "m \<subseteq>\<^sub>m m'"   and "m x = Some v" shows "m' x = Some v"
Suc: shows "emeasure M {w \<in> space M. stake n w = stake n x} \<noteq> 0"
eq: shows "x *\<^sub>R p = u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c"
less_infI2: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and b :: "'a"   and x :: "'a"   and a :: "'a" assumes "class.semilattice_inf inf less_eq less"   and "less b x" shows "less (inf a b) x"
unit_SE_def: fixes e :: "'e" shows "unit\<^sub>S\<^sub>E e = (\<lambda>\<sigma>. Some (e, \<sigma>))"
b: shows "braun \<langle>l, x, r\<rangle>"
star_1r: fixes x :: "'a" shows "x\<^sup>\<dagger> \<cdot> x \<le> x\<^sup>\<dagger>"
denot_implies_rel: fixes v :: "val"   and e :: "exp"   and \<rho> :: "(nat \<times> val) list" assumes "v \<in> E e \<rho>" shows "\<rho> \<turnstile> e \<Rightarrow> v"
induct: fixes x1 :: "Syntax.term list"   and x2 :: "Syntax.term"   and x3 :: "mode"   and x4 :: "nat"   and x5 :: "Syntax.term list"   and x6 :: "Syntax.term"   and P :: "Syntax.term list \<Rightarrow> Syntax.term \<Rightarrow> mode \<Rightarrow> nat \<Rightarrow> Syntax.term list \<Rightarrow> Syntax.term \<Rightarrow> bool" assumes "\<lless>x1, x2\<ggreater> x3\<rightarrow>x4 \<lless>x5, x6\<ggreater>"   and "\<And>\<pi> e uu_. P \<pi> e uu_ 0 \<pi> e"   and "\<And>\<pi>\<^sub>1 e\<^sub>1 m i \<pi>\<^sub>2 e\<^sub>2 \<pi>\<^sub>3 e\<^sub>3. \<lbrakk>\<lless>\<pi>\<^sub>1, e\<^sub>1\<ggreater> m\<rightarrow>i \<lless>\<pi>\<^sub>2, e\<^sub>2\<ggreater>; P \<pi>\<^sub>1 e\<^sub>1 m i \<pi>\<^sub>2 e\<^sub>2; \<lless>\<pi>\<^sub>2, e\<^sub>2\<ggreater> m\<rightarrow> \<lless>\<pi>\<^sub>3, e\<^sub>3\<ggreater>\<rbrakk> \<Longrightarrow> P \<pi>\<^sub>1 e\<^sub>1 m (i + 1) \<pi>\<^sub>3 e\<^sub>3" shows "P x1 x2 x3 x4 x5 x6"
pell_valuation_nth_solution: fixes n :: "nat" shows "pell_valuation (case nth_solution n of (x, y) \<Rightarrow> (int x, int y)) = pell_valuation (case fund_sol of (x, y) \<Rightarrow> (int x, int y)) ^ n"
Suc_funpow: fixes n :: "nat" shows "Suc ^^ n = (+) n"
simps_2: fixes f :: "'c" shows "\<lbrace>\<^bold>\<langle>f\<^bold>\<rangle>\<rbrace> = E f"
cl_invI: fixes l :: "('a, 'b) rbt"   and r :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b" assumes "cl_inv l"   and "cl_inv r" shows "cl_inv (Node l B k v r)"
order_iff_strict: fixes b :: "'c"   and a :: "'c" shows "(b \<le> a) = (b < a \<or> a = b)"
nat_numeral_diff_1: fixes v :: "num" shows "numeral v - 1 = nat (numeral v - 1)"
left: fixes c21' :: "'a list" assumes "c1 = c1' @ qs' # c21'"   and "c2' = c21' @ qs # c2" shows "thesis"
isCont_tendsto_compose: fixes l :: "'a"   and g :: "'a \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'a"   and F :: "'c filter" assumes "isCont g l"   and "(f \<longlongrightarrow> l) F" shows "((\<lambda>x. g (f x)) \<longlongrightarrow> g l) F"
Abs_process_inverse: fixes y :: "('e list \<times> 'e set) set \<times> 'e list set" assumes "y \<in> process_set" shows "Rep_process (Abs_process y) = y"
simps: fixes xs :: "'d list"   and a :: "'d list" shows "cyclic1 xs a = (a = rotate1 xs)"
distinct_57: fixes x4 :: "nat"   and x6 :: "nat" shows "Key x4 \<noteq> Passwd x6"
DOWN_UP: fixes x :: "hf" shows "DOWN (UP x) = x"
wf_Un: fixes r :: "('c \<times> 'c) set"   and s :: "('c \<times> 'c) set" assumes "wf r"   and "wf s"   and "Domain r \<inter> Range s = {}" shows "wf (r \<union> s)"
gs_dim: fixes x :: "complex vec" assumes "x \<in> set gs" shows "dim_vec x = d"
simps_29: fixes \<C> :: "t_context"   and t :: "t"   and tp_sx :: "(tp \<times> sx) option"   and a :: "nat"   and off :: "nat"   and ts :: "checker_type" shows "check_single \<C> (Load t tp_sx a off) ts = (if memory \<C> \<noteq> None \<and> load_store_t_bounds a (option_projl tp_sx) t then type_update ts [TSome T_i32] (Type [t]) else Bot)"
d1': shows "(sterm_sem I \<theta>1 has_derivative frechet J \<theta>2 (fst \<nu>)) (at (fst \<nu>))"
set_less_eq_conv_set_less: fixes A :: "'a set"   and B :: "'a set" shows "(A \<sqsubseteq> B) = (A = B \<or> A \<sqsubset> B)"
mul_def: fixes r :: "('a \<times> 'a) set" shows "mul r = {(M, N). \<exists>I J K. M = I + K \<and> N = I + J \<and> set_mset K \<subseteq> r \<down>m J \<and> J \<noteq> {#}}"
assms_4: shows "isUnitClause bClause bLiteral (elements p)"
iT_Plus_neg_cut_greater2: fixes k :: "nat"   and t :: "nat"   and I :: "nat set" assumes "k \<le> t" shows "(I \<oplus>- k) \<down>> (t - k) = I \<down>> t \<oplus>- k"
diff_diff_eq2: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.group_add minus plus zero uminus" shows "minus a (minus b c) = minus (plus a c) b"
R18_c: shows "pmf (sds R18) c = pmf (sds R9) c"
distributed_distr_eq_density: fixes M :: "'a measure"   and N :: "'b measure"   and X :: "'a \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> ennreal" assumes "distributed M N X f" shows "distr M N X = density N f"
butlast_snoc: fixes xs :: "'c list"   and x :: "'c" shows "butlast (xs @ [x]) = xs"
target_def: fixes t :: "'d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd" shows "target t = snd (snd t)"
Root_CTRL_Init_def: shows "Root_CTRL_Init = ''CarAudioSystem''"
inflstsE: fixes x :: "'b llist"   and A :: "'b set" assumes "x \<in> A\<^sup>\<omega>" obtains "x \<in> A\<^sup>\<infinity>" and "x \<notin> UNIV\<^sup>\<star>"
S_nonneg: fixes x :: "real" shows "0 \<le> S x"
sep_conj_sep_emptyI: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "class.sep_algebra plus zero sep_disj"   and "P h" shows "sep_algebra.sep_conj plus sep_disj P (sep_algebra.sep_empty zero) h"
24: fixes x :: "'a"   and y :: "'a" shows "- - x \<squnion> - (- x \<sqinter> y) = - (- x \<sqinter> y)"
finite_well_order_on_ordIso: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and r' :: "('a \<times> 'a) set" assumes "finite A"   and "well_order_on A r"   and "well_order_on A r'" shows "r =o r'"
step_1_similar: fixes A :: "'c mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "similar_mat (??.Jordan_Normal_Form_Existence.step_1 A) A"
subgroupE_4: fixes H :: "'a set"   and a :: "'a"   and b :: "'a" assumes "subgroup H (add_monoid G)"   and "a \<in> H"   and "b \<in> H" shows "a \<oplus> b \<in> H"
plus_adds_ppI_2: fixes t :: "'a"   and v :: "'t"   and s :: "'a" assumes "t adds\<^sub>p v"   and "s adds\<^sub>p v \<ominus> t" shows "t + s adds\<^sub>p v"
is_expansion_aux_expands_to: fixes f :: "real \<Rightarrow> real"   and xs :: "('a \<times> real) msllist"   and g :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" assumes "(f expands_to MS xs g) basis" shows "is_expansion_aux xs f basis"
insert_absorb: fixes a :: "'c"   and A :: "'c set" assumes "a \<in> A" shows "Set.insert a A = A"
bindS_def: shows "bindS = (\<Lambda> x g. lconcat\<cdot>(lmap\<cdot>g\<cdot>x))"
homeomorphic_Borsukian_eq: fixes S :: "'b set"   and T :: "'c set" assumes "S homeomorphic T" shows "Borsukian S = Borsukian T"
WP1: fixes a :: "('a, 'b) Combinators"   and list :: "('a, 'b) Combinators list" assumes "a \<notin> set list" shows "a \<notin> set (removeShadowRules2 list)"
refl: fixes n :: "'node"   and S :: "'node set" assumes "valid_node n"   and "n \<in> S" shows "n \<in> backward_slice S"
frame_miracle: fixes x :: "'c \<Longrightarrow> 'b" shows "{x}:[false] = false"
card_insert_le_m1: fixes n :: "nat"   and y :: "'a set"   and x :: "'a" assumes "0 < n"   and "card y \<le> n - 1" shows "card (x \<triangleright> y) \<le> n"
le_cong_r: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "weak_partial_order L"   and "x \<sqsubseteq>\<^bsub>L\<^esub> y"   and "y .=\<^bsub>L\<^esub> z"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<sqsubseteq>\<^bsub>L\<^esub> z"
fzero_def: shows "fzero = COERCE(udom\<cdot>'c, 'b\<cdot>'c)\<cdot>zeroU"
lossless_resourceD: fixes \<I> :: "('c, 'd) \<I>"   and res :: "('c, 'd) resource"   and a :: "'c" assumes "lossless_resource \<I> res"   and "a \<in> outs_\<I> \<I>" shows "lossless_spmf (run_resource res a) \<and> (\<forall>(x, res')\<in>set_spmf (run_resource res a). lossless_resource \<I> res')"
atLeast_subset_iff: fixes x :: "'c"   and y :: "'c" shows "({x..} \<subseteq> {y..}) = (y \<le> x)"
assms_2: shows "fmrel R\<^sup>+\<^sup>+ xm ym"
abs_summable_on_Sigma_iff: fixes A :: "'d set"   and B :: "'d \<Rightarrow> 'e set"   and f :: "'d \<times> 'e \<Rightarrow> 'f" assumes "countable A"   and "\<And>x. x \<in> A \<Longrightarrow> countable (B x)" shows "(f abs_summable_on Sigma A B) = ((\<forall>x\<in>A. (\<lambda>y. f (x, y)) abs_summable_on B x) \<and> (\<lambda>x. \<Sum>\<^sub>ay\<in>B x. norm (f (x, y))) abs_summable_on A)"
simps_1: shows "constant0 \<B> = \<langle>(''bool'', \<B>)\<rangle>"
arr_def: fixes f :: "'b" shows "C'.arr f \<equiv> C'.domains f \<noteq> {} \<or> C'.codomains f \<noteq> {}"
SimplicialComplexIsomorphism_axioms_def: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'c" shows "SimplicialComplexIsomorphism_axioms X f \<equiv> inj_on f (\<Union> X)"
spmf_spmf_of_pmf: fixes p :: "'c pmf"   and x :: "'c" shows "spmf (spmf_of_pmf p) x = pmf p x"
vec_of_basis_enum_divide: fixes \<psi> :: "'a"   and \<phi> :: "'a" shows "vec_of_basis_enum (\<psi> / \<phi>) = vec_of_list [vec_of_basis_enum \<psi> $ 0 / vec_of_basis_enum \<phi> $ 0]"
A_ub3: fixes t :: "'b tree"   and a :: "'b" assumes "bst t" shows "S34.A a t \<le> log (root 3 4) (real (size1 t)) + 1"
bij_lens_def: fixes x :: "'a \<Longrightarrow> 'b" shows "bij_lens x \<equiv> weak_lens x \<and> bij_lens_axioms x"
eq_implies_lang: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" assumes "\<phi> \<sim> \<psi>" shows "\<phi> \<sim>\<^sub>L \<psi>"
transitive_acyclic_irreflexive: fixes x :: "'a" assumes "x \<sqinter> x \<le> x" shows "(x \<sqinter> top \<le> - top) = (x \<le> - top)"
takeWhile_dropWhile_id: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "takeWhile P xs @ dropWhile P xs = xs"
set_less_eq_aux_finite_total2: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B" shows "A \<sqsubseteq>' B \<or> B \<sqsubseteq>' A"
filter_is_subset: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "set (filter P xs) \<subseteq> set xs"
f: fixes x :: "'a" shows "lfp.mono_body (\<lambda>f. F f x)"
bind_contT_def: shows "Monad_Overloading.bind = bind_cont"
Select1_def: fixes d :: "d data" shows "Select1 d = Sel1 (d !P! 1)"
narrow_Boolean: fixes G :: "prog"   and T :: "ty" assumes "G\<turnstile>PrimT Boolean\<succ>T" shows "T = PrimT Boolean"
get_subsets_def: fixes data :: "rat mat \<times> nat list list \<times> rat list list" shows "get_subsets data = fst (snd data)"
FAssRed1: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and F :: "char list"   and D :: "char list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>F{D} := e\<^sub>2,s\<rangle> \<rightarrow> \<langle>e'\<bullet>F{D} := e\<^sub>2,s'\<rangle>"
count_empty: fixes a :: "'d" shows "count {#} a = 0"
equal_length2: fixes l :: "tm list"   and l' :: "tm list" assumes "equal_length l l'" shows "length2 l = length2 l'"
run_def: fixes s\<^sub>0 :: "'b"   and lstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'b" shows "run s\<^sub>0 lstep s \<equiv> df_system s\<^sub>0 lstep \<and> run_axioms s\<^sub>0 lstep s"
is_interval_existence_ivl: fixes T :: "real set"   and f :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and X :: "'d set"   and t0 :: "real"   and x0 :: "'d" assumes "ll_on_open_it T f X" shows "is_interval (ll_on_open.existence_ivl T f X t0 x0)"
adjoint_pair_antipar_1: fixes f :: "'a"   and g :: "'a" assumes "adjoint_pair f g" shows "ide f"
preserves_ide: fixes a :: "'d" assumes "Adj.Cop.ide a" shows "Adj.Cop.ide (Adj.Cop.map a)"
indep: shows "gs.lin_indpt_list (map of_int_hom.vec_hom fs)"
simps: fixes P1 :: "'a \<Rightarrow> bool"   and P2 :: "'b \<Rightarrow> bool"   and a :: "'a + 'b" shows "pred_sum P1 P2 a = ((\<exists>aa. a = Inl aa \<and> P1 aa) \<or> (\<exists>b. a = Inr b \<and> P2 b))"
map_add_dom_app_simps_2: fixes m :: "'a"   and l1 :: "'a \<Rightarrow> 'b option"   and l2 :: "'a \<Rightarrow> 'b option" assumes "m \<notin> dom l1" shows "(l2 \<Oplus> l1) m = l2 m"
Diag_TensorE_3: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Diag (t \<^bold>\<otimes> u)" shows "Diag t"
preserves_ide: fixes a :: "'b" assumes "D.ide a" shows "C.ide (F.right_map a)"
fps_inverse_mult_divring: fixes f :: "'a fps"   and g :: "'a fps" shows "inverse (f * g) = inverse g * inverse f"
Lim_def: fixes f :: "'b \<Rightarrow> ('a, 'b) node set" shows "Lim f \<equiv> \<Union> {z. \<exists>x. z = Push_Node (Inl x) ` f x}"
bind_put: fixes s :: "'s"   and m :: "'m"   and f :: "'a \<times> 'w list \<Rightarrow> 'm" shows "bind (put s m) f = put s (bind m f)"
map_poly_inj_comm_ring_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "map_poly_inj_comm_ring_hom hom \<equiv> inj_comm_ring_hom hom"
elims: fixes x :: "('b \<times> 'c) list"   and xa :: "'c"   and y :: "('b \<times> 'c) list \<times> ('b \<times> 'c) list" assumes "linear_split x xa = y" obtains xs :: "('b \<times> 'c) list" and x :: "'c" where "xa__ = xs" and "xa = x" and "y = linear_split_help xs x []"
facet_of_polyhedron_explicit: fixes F :: "'b set set"   and S :: "'b set"   and a :: "'b set \<Rightarrow> 'b"   and b :: "'b set \<Rightarrow> real"   and C :: "'b set" assumes "finite F"   and "S = affine hull S \<inter> \<Inter> F"   and "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> (0::'b) \<and> h = {x. a h \<bullet> x \<le> b h}"   and "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter> F'" shows "(C facet_of S) = (\<exists>h. h \<in> F \<and> C = S \<inter> {x. a h \<bullet> x = b h})"