hcju/isabelleps huggingface dataset:
eqd_suf1: fixes x :: "'b list"   and y :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "x \<cdot> y = u \<cdot> v"   and "\<^bold>|x\<^bold>| \<le> \<^bold>|u\<^bold>|" shows "y\<^sup><\<inverse>v  \<cdot> v = y"
rel_fun_assoc_weak: fixes \<rho> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool"   and \<xi> :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd" assumes "class.rel_monoid \<rho> \<xi>" shows "rel_magma.times_rel_fun \<rho> (rel_magma.times_rel_fun \<rho> f g) h = rel_magma.times_rel_fun \<rho> f (rel_magma.times_rel_fun \<rho> g h)"
methdMembr_qmdecl_simp: fixes sig :: "sig"   and old :: "qtname \<times> methd" shows "methdMembr (qmdecl sig old) = method sig old"
superset_imp_sort_leq: fixes cs :: "(String.literal \<times> String.literal) set"   and s2 :: "String.literal set"   and s1 :: "String.literal set" assumes "wf_subclass_loc cs"   and "sort_ex cs s2"   and "s2 \<subseteq> s1" shows "sort_leq cs s1 s2"
mult_mat_vec_hom: fixes hom :: "'a \<Rightarrow> 'b"   and A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and v :: "'a Matrix.vec" assumes "semiring_hom hom"   and "A \<in> carrier_mat nr n"   and "v \<in> carrier_vec n" shows "map_vec hom (A *\<^sub>v v) = map_mat hom A *\<^sub>v map_vec hom v"
less_eq_assn_def: fixes a :: "assn"   and b :: "assn" shows "a \<le> b \<equiv> a = a \<and>\<^sub>A b"
norm1_def: fixes v :: "('b, 'c) Finite_Cartesian_Product.vec" shows "norm1 v = (\<Sum>i\<in>UNIV. norm (v $h i))"
some_gcd_ddvd_gcd: fixes a :: "'a"   and b :: "'a" shows "some_gcd a b ddvd gcd a b"
def_of_tl: fixes y :: "'a list"   and a :: "'a"   and x :: "'a list" shows "y = a # x \<longrightarrow> tl y = x"
transpD: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "transp r"   and "r x y"   and "r y z" shows "r x z"
simps_2: fixes g1 :: "'d \<Rightarrow> 'c"   and g2 :: "'d \<Rightarrow> 'd"   and a :: "'d" shows "stl (unfold_stream g1 g2 a) = unfold_stream g1 g2 (g2 a)"
spr_simObs_def: fixes envObs :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "SPRViewSingle.spr_simObs envObs \<equiv> \<lambda>a. envObs a \<circ> ODList.hd"
R2E: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "f \<in> \<R>\<^sup>2" obtains r :: "recf" where "recfn 2 r" and "Partial_Recursive.total r" and "f = (\<lambda>x\<^sub>1 x\<^sub>2. eval r [x\<^sub>1, x\<^sub>2])"
inverse_arrowsE: fixes f :: "'b"   and g :: "'b" assumes "\<phi>.B.inverse_arrows f g" obtains "\<phi>.B.ide (g \<cdot>\<^sub>B f)" and "\<phi>.B.ide (f \<cdot>\<^sub>B g)"
Merge_defined: fixes A :: "transaction\<^sub>m set" assumes "finite A"   and "A \<noteq> {}"   and "\<forall>a\<in>A. \<forall>b\<in>A. hash_transaction a = hash_transaction b" shows "transaction.Merge A \<noteq> None"
singleton_sup: fixes default :: "unit \<Rightarrow> 'a"   and A :: "'a pred"   and B :: "'a pred" shows "singleton default (A \<squnion> B) = (if A = \<bottom> then singleton default B else if B = \<bottom> then singleton default A else if singleton default A = singleton default B then singleton default A else default ())"
eq_iff_3: fixes fmla :: "fmla"   and fmlaa :: "fmla" shows "(Robinson_Arithmetic.neg fmla = Robinson_Arithmetic.neg fmlaa) = (fmla = fmlaa)"
get_proc_return: fixes a :: "'b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node"   and Q' :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and f' :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option" assumes "lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a"   and "knd a = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'" shows "lift_get_proc get_proc Main (src a) = p"
CHO_trans_def: fixes A :: "('d, 'e, 'f, 'g) CHOAlgorithm_scheme"   and HOs :: "nat \<Rightarrow> 'd \<Rightarrow> 'd set"   and SHOs :: "nat \<Rightarrow> 'd \<Rightarrow> 'd set"   and coord :: "nat \<Rightarrow> 'd \<Rightarrow> 'd" shows "CHO_trans A HOs SHOs coord = {((r, st), r', st') |r r' st st'. r' = Suc r \<and> CSHOnextConfig A r st (HOs r) (SHOs r) (coord r) st'}"
rejects_trivial: fixes \<F> :: "nat set set"   and S :: "nat set"   and M :: "nat set"   and F :: "nat set" assumes "rejects \<F> S M"   and "thin_set \<F>"   and "init_segment F S"   and "F \<in> \<F>" shows "False"
integrable_on_lborel: fixes f :: "'b \<Rightarrow> 'c" assumes "integrable lborel f" shows "f integrable_on UNIV"
reachable_in_verts_1: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v" shows "u \<in> verts H"
dvd_mult_unit_iff: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b dvd (1::'a)" shows "(a dvd c * b) = (a dvd c)"
1: shows "to_fun (lin_part (T\<^bsub>a\<^esub> f)) b = fa \<oplus> f'a \<otimes> b"
mult_ac_2: fixes a :: "'b"   and b :: "'b" shows "a * b = b * a"
intros_1: fixes arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and \<zeta> :: "('a, 'b) hd" shows "arity.wary arity_sym arity_var (Hd \<zeta>)"
ivp_sols_def: fixes f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and U :: "'b \<Rightarrow> real set"   and S :: "'b set"   and t\<^sub>0 :: "real"   and s :: "'b" shows "Sols f U S t\<^sub>0 s = {X \<in> U s \<rightarrow> S.  D X = (\<lambda>t. f t (X t)) on U s \<and> X t\<^sub>0 = s \<and> t\<^sub>0 \<in> U s}"
coprime_commute: fixes p :: "'a"   and q :: "'a" shows "coprime p q = coprime q p"
right_plus_below_circ: fixes x :: "'a" shows "x\<^sup>\<star> * x \<le> x\<^sup>\<star>"
matrix_match_condn_1: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zer :: "'b"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inver :: "'b \<Rightarrow> 'b"   and A1 :: "'b list list"   and A2 :: "'b list list"   and B1 :: "'b list list"   and B2 :: "'b list list"   and i :: "nat"   and j :: "nat" assumes "plus_mult id f zer g inver"   and "plus_mult.matrix_match A1 A2 B1 B2 \<and> i < mult.row_length A1 * mult.row_length B1 \<and> j < length A2 * length B2" shows "mult.Tensor f (mat_multI zer g f (mult.row_length A1) A1 A2) (mat_multI zer g f (mult.row_length B1) B1 B2) ! j ! i = f (plus_mult.scalar_product f zer g (row A1 (i div mult.row_length B1)) (col A2 (j div length B2))) (plus_mult.scalar_product f zer g (row B1 (i mod mult.row_length B1)) (col B2 (j mod length B2)))"
Diff_idemp: fixes A :: "'a set"   and B :: "'a set" shows "A - B - B = A - B"
merge_eq_appendI: fixes xs :: "'a list"   and ys :: "'a list" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set ys\<rbrakk> \<Longrightarrow> compare cmp x y \<noteq> Greater" shows "local.merge xs ys = xs @ ys"
some_gcd_dvd_2: fixes d :: "'a"   and a :: "'a"   and b :: "'a" shows "(d dvd some_gcd a b) = (d dvd gcd a b)"
infsuff_open: fixes r :: "'a llist" assumes "r \<in> A\<^sup>\<star>" shows "infsuff A r open"
The_on_SomeI: fixes a :: "'c"   and U :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "a \<in> U"   and "P a"   and "\<And>x. \<lbrakk>x \<in> U; P x\<rbrakk> \<Longrightarrow> x = a" obtains x :: "'c" where "(THE x on U. P x) = Some x" and "P x"
WHILEIT_def: shows "WHILEIT \<equiv> \<lambda>I b f. REC\<^sub>T (WHILEI_body bind return I b f)"
simps: fixes p :: "real poly"   and q :: "real poly" shows "smods_ext p q = (if p = 0 then [] else if p mod q \<noteq> 0 then p # smods_ext q (- (p mod q)) else p # smods_ext q (pderiv q))"
ipath_sim1: fixes r :: "nat \<Rightarrow> 'u" assumes "ipath E r" shows "ipath G'.E (f \<circ> r)"
ab_specI: fixes ab :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> nat \<times> 'h \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list" assumes "\<And>gs bs ns data. ns \<noteq> [] \<Longrightarrow> set (ab gs bs ns data) = set bs \<union> set ns"   and "\<And>gs bs data. ab gs bs [] data = bs" shows "ab_spec ab"
med01iai_def: fixes t :: "l1_state" shows "med01iai t \<equiv> \<lparr>signals = signalsInit t\<rparr>"
check_cond_conv: fixes x :: "nat list \<times> nat"   and y :: "nat list \<times> nat"   and b :: "nat list"   and a :: "nat list" assumes "(x, y) \<in> set (alls2 (Max (set b)) (Max (set a)) a b)" shows "check_cond a b (fst x, fst y) = (static_bounds a b (fst x) (fst y) \<and> a \<bullet> fst x = b \<bullet> fst y \<and> boundr_impl a b (fst x) (fst y) \<and> suffs (cond1 b (fst y)) a x \<and> suffs (cond2 a b) b y)"
jfinish_eq: shows "jumpF_pathfinish (part_circlepath z r st tt) z0 = jumpF h (at_left tt)"
find_Some_iff: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and x :: "'a" shows "(List.find P xs = Some x) = (\<exists>i<length xs. P (xs ! i) \<and> x = xs ! i \<and> (\<forall>j<i. \<not> P (xs ! j)))"
le_less_trans: fixes x :: "'b list"   and y :: "'b list"   and z :: "'b list" assumes "prefix x y"   and "strict_prefix y z" shows "strict_prefix x z"
echelon_form_of_upt_k_invertible: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and A :: "(('d, 'f) vec, 'e) vec"   and k :: "nat" assumes "is_bezout_ext bezout" shows "\<exists>P. invertible P \<and> P ** A = echelon_form_of_upt_k A k bezout"
intros_4: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "F, \<Gamma> \<Rightarrow> \<Delta>" shows "\<Gamma> \<Rightarrow> \<^bold>\<not> F, \<Delta>"
gen_ne: fixes B :: "nat"   and n :: "nat" shows "gen B n \<noteq> []"
sup_commute: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "sup_prod x y = sup_prod y x"
fps_lr_inverse_unique_ring1_2: fixes f :: "'a fps"   and g :: "'a fps" assumes "f * g = 1"   and "fps_nth g 0 * fps_nth f 0 = (1::'a)" shows "fps_right_inverse f (fps_nth g 0) = g"
axioms_1: fixes I :: "'c set"   and R :: "('c, 'd) ring_scheme" assumes "ideal I R" shows "additive_subgroup I R"
rotateR_in_trav: fixes t :: "('c, 'd) tree" shows "in_traverse (rotateR t) = in_traverse t"
add_mono_thms_linordered_field_1: fixes i :: "'d"   and j :: "'d"   and k :: "'d"   and l :: "'d" assumes "i < j \<and> k = l" shows "i + k < j + l"
elims_3: fixes atom_ordering :: "('a \<times> 'a) set"   and x :: "'a Literal"   and xa :: "'a Literal" assumes "propositional_atoms atom_ordering"   and "\<not> propositional_atoms.literal_ordering atom_ordering x xa" obtains L1 :: "'a Literal" and L2 :: "'a Literal" where "x = L1" and "xa = L2" and "(atom L1, atom L2) \<notin> atom_ordering"
U: shows "\<forall>v\<in>\<V>. U v \<in> \<U> \<and> v = {x \<in> topspace X. f x \<in> U v}"
unwind_dec_secure: fixes \<Delta>0 :: "'state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool"   and \<Delta>s :: "('state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool) set"   and "next" :: "('state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool) \<Rightarrow> ('state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool) set" assumes "\<Delta>0 \<in> \<Delta>s"   and "\<And>vl vl1. B vl vl1 \<Longrightarrow> \<Delta>0 istate vl istate vl1"   and "\<And>\<Delta>. \<Delta> \<in> \<Delta>s \<Longrightarrow> next \<Delta> \<subseteq> \<Delta>s \<and> unwind_to \<Delta> (next \<Delta>)" shows "secure"
has_terminal_arrows_from_functor_1: fixes x :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.ide x" shows "terminal_arrow_from_functor (\<cdot>) (\<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>]) \<Delta>.map (G x) x (Adj.\<epsilon> x)"
trans_wf_iff: fixes r :: "('a \<times> 'a) set" assumes "trans r" shows "wf r = (\<forall>a. wf (Restr r (r\<inverse> `` {a})))"
map: shows "(E(p' \<mapsto> T', pns' [\<mapsto>] Ts')) (pns' ! n) = \<lfloor>Ts' ! n\<rfloor>"
conv_radius_cong: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in sequentially. norm (f x) = norm (g x)" shows "conv_radius f = conv_radius g"
assms_2: fixes J :: "'b set"   and A :: "'b \<Rightarrow> 'a set" assumes "J \<noteq> {}"   and "J \<subseteq> I"   and "finite J"   and "\<forall>j\<in>J. A j \<in> F j" shows "prob (\<Inter> (A ` J)) = (\<Prod>j\<in>J. prob (A j))"
TS_yy'_1: fixes x :: "nat"   and y :: "nat"   and h :: "nat list"   and h0 :: "nat list" assumes "x \<noteq> y"   and "\<exists>hs. h = [x, y] @ hs" shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) [y, y] = 1"
add_neg_numeral_special_9: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.neg_numeral minus plus zero uminus" shows "plus (uminus one) (uminus one) = uminus (numeral.numeral one plus (num.Bit0 num.One))"
signed_eq_0_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and w :: "'c word" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(ring_1.signed one plus zero minus w = zero) = (w = 0)"
heap_eq_ListI1: fixes p :: "ref"   and h :: "ref \<Rightarrow> ref"   and ps :: "ref list"   and g :: "ref \<Rightarrow> ref" assumes "List p h ps"   and "\<forall>x\<in>set ps. g x = h x" shows "List p g ps"
connected_component_of_maximal: fixes X :: "'b topology"   and S :: "'b set"   and x :: "'b" assumes "connectedin X S"   and "x \<in> S" shows "S \<subseteq> connected_component_of_set X x"
simps_14: fixes v :: "Formula.formula"   and va :: "\<I>"   and vb :: "Formula.formula" shows "remove_neg (formula.Since v va vb) = formula.Since v va vb"
real_distribution_distr: fixes M :: "'b measure"   and X :: "'b \<Rightarrow> real" assumes "prob_space M"   and "X \<in> borel_measurable M" shows "real_distribution (distr M borel X)"
ZObis_finished_discr_R: fixes c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd" assumes "proper c"   and "finished c"   and "c \<approx>01 d" shows "discr d"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "comm_ring_hom hom" shows "idom_hom hom"
Pi_UNIV: fixes A :: "'e set" shows "A \<rightarrow> UNIV = UNIV"
holomorphic_fun_eq_on_connected: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and g :: "complex \<Rightarrow> complex"   and z :: "complex"   and w :: "complex" assumes "f holomorphic_on S"   and "g holomorphic_on S"   and "open S"   and "connected S"   and "\<And>n. (deriv ^^ n) f z = (deriv ^^ n) g z"   and "z \<in> S"   and "w \<in> S" shows "f w = g w"
rel_fun_Sup_distl: fixes \<rho> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and G :: "('c \<Rightarrow> 'd) set" shows "rel_magma.times_rel_fun \<rho> f (\<Squnion>G) = \<Squnion>{rel_magma.times_rel_fun \<rho> f g |g. g \<in> G}"
Entry_Exit_edge: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Exit :: "'g" assumes "CFGExit sourcenode targetnode kind valid_edge Entry Exit" shows "\<exists>a. valid_edge a \<and> sourcenode a = Entry \<and> targetnode a = Exit \<and> kind a = (\<lambda>s. False)\<^sub>\<surd>"
CARD_valid_states: fixes PROB :: "(('c, bool) fmap \<times> ('c, bool) fmap) set" assumes "finite PROB" shows "card (valid_states PROB) = 2 ^ card (prob_dom PROB)"
cm_preserves_impl_safe: fixes c0 :: "('loc, 't) Propagate.configuration"   and c1 :: "('loc, 't) Propagate.configuration"   and loc :: "'loc"   and t :: "'t"   and n :: "int" assumes "impl_safe c0"   and "next_change_multiplicity' c0 c1 loc t n" shows "impl_safe c1"
integral_lborel_f: fixes f :: "(nat \<Rightarrow> real) \<Rightarrow> ennreal"   and n :: "nat" assumes "f \<in> borel_measurable (lborel_f (Suc n))" shows "integral\<^sup>N (lborel_f (Suc n)) f = \<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x(n := y)) \<partial>lborel_f n \<partial>lborel"
empty_worklists_vacant_to: fixes c :: "('loc, 't) configuration"   and t :: "'t" assumes "\<forall>loc. c_work c loc = {#}\<^sub>z" shows "worklists_vacant_to c t"
bind_UNION: fixes A :: "'g set"   and f :: "'g \<Rightarrow> 'f set" shows "Set.bind A f = \<Union> (f ` A)"
lhs_dot_rewrite: fixes p :: "real poly"   and signs :: "rat list list"   and I :: "nat list"   and qs :: "real poly list" assumes "p \<noteq> 0" shows "vec_of_list (mtx_row signs I) \<bullet> w_vec p qs signs = (\<Sum>s\<leftarrow>signs. z I s * rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec_copr qs x = s})))"
Posix_elims_2: fixes s :: "'b list"   and v :: "'b val" assumes "s \<in> One \<rightarrow> v" obtains "s = []" and "v = Void"
smap_less_trans: fixes m1 :: "'a \<Rightarrow> 'b set"   and m2 :: "'a \<Rightarrow> 'b set"   and m3 :: "'a \<Rightarrow> 'b set" assumes "m1 \<subseteq>. m2"   and "m2 \<subseteq>. m3" shows "m1 \<subseteq>. m3"
functor_is_transformation: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah" assumes "functor A B F" shows "natural_transformation A B F F F"
row_mat_of_row_fun: fixes i :: "nat"   and nr :: "nat"   and f :: "nat \<Rightarrow> 'a vec"   and nc :: "nat" assumes "i < nr"   and "dim_vec (f i) = nc" shows "row (mat\<^sub>r nr nc f) i = f i"
comp_arr_mkPoint: fixes f :: "'s"   and t :: "'s" assumes "arr f"   and "t \<in> Dom f" shows "f \<cdot> mkPoint (local.dom f) t = mkPoint (cod f) (Fun f t)"
lookup_operator_eq_name: fixes name :: "char list"   and name' :: "char list"   and pres :: "(nat \<times> nat) list"   and effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer :: "nat" assumes "lookup_operator name = Some (name', pres, effs, layer)" shows "name = name'"
in_Vset_rank: fixes a :: "hf" shows "a \<^bold>\<in> Vset (succ (rank a))"
refl: fixes x :: "'a" assumes "x \<in> A" shows "boa x x"
comp_comp_fun_commute: fixes f :: "'k \<Rightarrow> 'l \<Rightarrow> 'l"   and g :: "'m \<Rightarrow> 'k" assumes "comp_fun_commute f" shows "comp_fun_commute (f \<circ> g)"
distinct_25: fixes x2 :: "nat"   and x81 :: "'b poly"   and x82 :: "nat"   and x83 :: "'b poly" shows "Bound x2 \<noteq> CN x81 x82 x83"
prevVertex_def: fixes f :: "face"   and v :: "nat" shows "f\<^bsup>-1\<^esup> \<bullet> v \<equiv> let vs = vertices f in nextElem (rev vs) (last vs) v"
ext_inject: fixes states :: "hf set"   and init :: "hf set"   and final :: "hf set"   and nxt :: "hf \<Rightarrow> 'b \<Rightarrow> hf set"   and eps :: "(hf \<times> hf) set"   and more :: "'c"   and states' :: "hf set"   and init' :: "hf set"   and final' :: "hf set"   and nxt' :: "hf \<Rightarrow> 'b \<Rightarrow> hf set"   and eps' :: "(hf \<times> hf) set"   and more' :: "'c" shows "(\<lparr>nfa.states = states, init = init, final = final, nxt = nxt, eps = eps, \<dots> = more\<rparr> = \<lparr>nfa.states = states', init = init', final = final', nxt = nxt', eps = eps', \<dots> = more'\<rparr>) = (states = states' \<and> init = init' \<and> final = final' \<and> nxt = nxt' \<and> eps = eps' \<and> more = more')"
rangeI: fixes f :: "'c::type \<Rightarrow> 'b::type"   and x :: "'c::type" shows "(f::'c::type \<Rightarrow> 'b::type) (x::'c::type) \<in> range f"
vbrelation_Limit_vsubset_VsetI: fixes \<alpha> :: "V" assumes "Limit \<alpha>"   and "\<D>\<^sub>\<circ> a \<subseteq>\<^sub>\<circ> Vset \<alpha>"   and "\<R>\<^sub>\<circ> a \<subseteq>\<^sub>\<circ> Vset \<alpha>" shows "a \<subseteq>\<^sub>\<circ> Vset \<alpha>"
UnI2: fixes c :: "'d"   and B :: "'d set"   and A :: "'d set" assumes "c \<in> B" shows "c \<in> A \<union> B"
assms: shows "\<forall>i\<in>I. i < length T"
simps_1: fixes p :: "assertion" shows "bot_ass (Rib p) = Ribbon p"
right_diff_distrib: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec"   and z :: "(real, 3) vec" shows "cross3 x (y - z) = cross3 x y - cross3 x z"
perm_power_inverse: fixes f :: "'a perm"   and n :: "nat" shows "inverse f ^ n = inverse (f ^ n)"
tiny_dghm_in_Vset: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF> \<in>\<^sub>\<circ> Vset \<alpha>"
fmap_fplus: fixes f :: "'g \<rightarrow> 'e"   and a :: "'g\<cdot>'f"   and b :: "'g\<cdot>'f" shows "fmap\<cdot>f\<cdot>(fplus\<cdot>a\<cdot>b) = fplus\<cdot>(fmap\<cdot>f\<cdot>a)\<cdot>(fmap\<cdot>f\<cdot>b)"
assms: shows "(x = (0::'a)) = (pdevs_apply X n = (0::'a))"
nested_sequence_unique: fixes f :: "nat \<Rightarrow> real"   and g :: "nat \<Rightarrow> real" assumes "\<forall>n. f n \<le> f (Suc n)"   and "\<forall>n. g (Suc n) \<le> g n"   and "\<forall>n. f n \<le> g n"   and "(\<lambda>n. f n - g n) \<longlonglongrightarrow> 0" shows "\<exists>l. ((\<forall>n. f n \<le> l) \<and> f \<longlonglongrightarrow> l) \<and> (\<forall>n. l \<le> g n) \<and> g \<longlonglongrightarrow> l"
comp_ide_self: fixes a :: "'b" assumes "C'.ide a" shows "C' a a = a"
evenE: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_parity plus minus zero times one divide modulo"   and "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" obtains b :: "'a" where "a = times (numeral.numeral one plus (num.Bit0 num.One)) b"
right_unique_alt_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "right_unique R = (R\<inverse>\<inverse> OO R \<le> (=))"
assms: shows "classes_above P C \<inter> classes_changed P P' = {}"
ipset_from_cidr_base_wellforemd: fixes l :: "nat"   and base :: "'a word" assumes "mask (LENGTH('a) - l) && base = 0" shows "ipset_from_cidr base l = {base..base || mask (LENGTH('a) - l)}"
2: shows "(\<lambda>k. ln (ln (real (primorial' k)))) \<sim>[sequentially] (\<lambda>x. ln (real x))"
integrable_bounded_linear: fixes T :: "'b \<Rightarrow> 'c"   and M :: "'d measure"   and f :: "'d \<Rightarrow> 'b" assumes "bounded_linear T"   and "integrable M f" shows "integrable M (\<lambda>x. T (f x))"
rat_precision_pos: fixes x :: "nat"   and y :: "nat"   and n :: "nat" assumes "0 \<le> x"   and "0 < y"   and "2 * x < y" shows "0 < rat_precision n (int x) (int y)"
Smith_normal_form_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "Smith_normal_form A = ((\<forall>a b. mod_type_class.to_nat a = mod_type_class.to_nat b \<and> mod_type_class.to_nat a + 1 < nrows A \<and> mod_type_class.to_nat b + 1 < ncols A \<longrightarrow> A $h a $h b dvd A $h (a + (1::'d)) $h (b + (1::'c))) \<and> isDiagonal A)"
30: shows "subst (instInpP Pinp 0 (instInp (encF N) (Var xx))) \<langle>\<phi>\<rangle> xx = instInpP Pinp 0 (instInp (encF N) \<langle>\<phi>\<rangle>)"
to_nat_plus_one_less_card': fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and a :: "'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "plus a one \<noteq> zero" shows "mod_type.to_nat Rep a + 1 < CARD('a)"
H: shows "set zs \<subseteq> range p \<union> range q"
r'': fixes msrFn :: "'a \<Rightarrow> nat"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "\<forall>x. (\<forall>y. msrFn y < msrFn x \<longrightarrow> P y) \<longrightarrow> P x" shows "P a"
keys_frag_of: fixes a :: "'c" shows "Poly_Mapping.keys (frag_of a) = {a}"