hcju/isabelleps huggingface dataset:
elims_2: fixes x :: "atom list \<times> 'a"   and xa :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xb :: "'a \<Rightarrow> atom set"   and xc :: "perm"   and xd :: "atom list \<times> 'a" assumes "x \<approx>lst xa xb xc xd" obtains bs :: "atom list" and xe :: "'a" and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and f :: "'a \<Rightarrow> atom set" and p :: "perm" and cs :: "atom list" and y :: "'a" where "x = (bs, xe)" and "xa = R" and "xb = f" and "xc = p" and "xd = (cs, y)" and "f xe - set bs = f y - set cs \<and> (f xe - set bs) \<sharp>* p \<and> R (p \<bullet> xe) y \<and> p \<bullet> bs = cs"
tinres_nonempty_1: fixes r :: "cell list" assumes "tinres r []"   and "r \<noteq> []" shows "hd r = Bk"
pow_order_eq_1: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c" assumes "Group.group G"   and "a \<in> carrier G" shows "a [^]\<^bsub>G\<^esub> order G = \<one>\<^bsub>G\<^esub>"
project_coord_aform_def: fixes sctn :: "'a sctn"   and cxs :: "'a \<times> 'a pdevs" shows "project_coord_aform sctn cxs = (project_coord (fst cxs) (normal sctn) (pstn sctn), project_coord_pdevs sctn (snd cxs))"
simps_2: fixes w :: "'a \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and scf :: "'a \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and f :: "'a"   and ts :: "('a, 'b) Term.term list" shows "weight_fun.weight w w0 scf (Fun f ts) = (let n = length ts; scff = scf (f, n) in w (f, n) + sum_list (map2 (\<lambda>ti i. weight_fun.weight w w0 scf ti * scff i) ts [0..<n]))"
mset_factors_imp_dvd: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and F :: "'b multiset"   and x :: "'b"   and f :: "'b" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.mset_factors times one zero F x"   and "f \<in># F" shows "dvd.dvd times f x"
ereal_distrib_minus_left: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal" assumes "a \<noteq> \<infinity> \<or> b \<noteq> \<infinity>"   and "a \<noteq> - \<infinity> \<or> b \<noteq> - \<infinity>"   and "\<bar>c\<bar> \<noteq> \<infinity>" shows "c * (a - b) = c * a - c * b"
h: shows "h \<in> Ar\<^bsub>set_cat U\<^esub>"
bij_betw_imageI: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "inj_on f A"   and "\<P> f A = B" shows "bij_betw f A B"
ilist_Previous_Suc: fixes f :: "nat \<Rightarrow> 'a"   and init :: "'a"   and n :: "nat" shows "f\<^bsup>\<leftarrow> init\<^esup> Suc n = f n"
cl_inv'I: fixes l :: "('a, 'b) rbt"   and r :: "('a, 'b) rbt"   and c :: "color"   and k :: "'a"   and v :: "'b" assumes "cl_inv l"   and "cl_inv r" shows "cl_inv' (Node l c k v r)"
Polys_mono: fixes X :: "'c set"   and Y :: "'c set" assumes "X \<subseteq> Y" shows "P[X] \<subseteq> P[Y]"
aezfun_transfer: fixes a :: "('b, 'a) aezfun"   and b :: "('b, 'a) aezfun" assumes "aezfun a = aezfun b" shows "a = b"
simps: fixes a :: "expr" shows "lc a = ((\<exists>xT. a = \<langle>xT\<rangle>) \<or> (\<exists>b. a = b) \<or> (\<exists>e1 e2. a = e1 ? e2 \<and> lc e1 \<and> lc e2) \<or> (\<exists>e1 e2. a = e1 \<cdot> e2 \<and> lc e1 \<and> lc e2) \<or> (\<exists>T X e. a = \<Lambda>\<langle>T\<rangle> e \<and> (\<forall>x. (x, T) |\<notin>| X \<longrightarrow> lc (open0_Var (x, T) e))))"
keys_deg_le_sect_pp_aux: fixes xs :: "'x list"   and n :: "nat" shows "keys (deg_le_sect_pp_aux xs n) = {t. deg_pp t \<le> n \<and> keys_pp t \<subseteq> set xs}"
P: fixes t :: "real" assumes "t \<in> {0..1}" shows "P (\<lambda>x. h (t, x))"
l_create_element_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_create_element_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr known_ptrs type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel set_tag_name set_tag_name_locs set_disconnected_nodes set_disconnected_nodes_locs create_element get_shadow_root get_shadow_root_locs get_tag_name get_tag_name_locs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
add_lub: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x + y \<le> z) = (x \<le> z \<and> y \<le> z)"
intro: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "ThinChamberComplex X"   and "ChamberComplexFolding X f" shows "ThinChamberComplexFolding X f"
hyp4: shows "fst (snd (foldl Gauss_Jordan_column_k_PA (mat (1::'a), 0, A) [0..<Suc k])) \<le> nrows A"
these_image_Some_eq: fixes A :: "'c set" shows "Option.these (Some ` A) = A"
fun_Compl_def: fixes A :: "'b \<Rightarrow> 'c" shows "- A = (\<lambda>x. - A x)"
dsj_EH_1: fixes A :: "fmla"   and H :: "fmla set"   and C :: "fmla"   and B :: "fmla" assumes "insert A H \<turnstile> C"   and "insert B H \<turnstile> C" shows "insert (A OR B) H \<turnstile> C"
conc_fun_fail_iff_2: fixes R :: "('c \<times> 'd) set"   and S :: "'d nres" shows "(FAIL = \<Down> R S) = (S = FAIL)"
top_greatest: fixes less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and top :: "'a::type"   and a :: "'a::type" assumes "class.order_top (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (top::'a::type)" shows "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (a::'a::type) (top::'a::type)"
sum_with_def: fixes pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and f :: "'c \<Rightarrow> 'b"   and S :: "'c set" shows "sum_with pls z f S = (if \<exists>C. f ` S \<subseteq> C \<and> comm_monoid_add_ow C pls z then Finite_Set.fold (pls \<circ> f) z S else z)"
i_le3: shows "pick I i < card {i. i < dim_row A \<and> i \<in> UNIV}"
UNIV_atMost: shows "UNIV = Abs_atMost ` {..<LENGTH('c)}"
dim_insert: fixes x :: "'b"   and S :: "'b set" shows "dim (insert x S) = (if x \<in> span S then dim S else dim S + 1)"
prems_2: shows "I (u \<lhd> \<sigma>) (v \<lhd> \<sigma>)"
head_arev: fixes a :: "'a \<times> 'a" assumes "a \<in> arcs (with_proj G)" shows "head (with_proj G) (rev_G a) = tail (with_proj G) a"
degree_reflect_poly_eq: fixes p :: "'a poly" assumes "coeff p 0 \<noteq> (0::'a)" shows "degree (reflect_poly p) = degree p"
simps_132: fixes f1 :: "char list \<Rightarrow> 'b"   and f2 :: "48 word \<Rightarrow> 'b"   and f3 :: "48 word \<Rightarrow> 'b"   and f4 :: "16 word \<Rightarrow> 'b"   and f5 :: "16 word \<Rightarrow> 'b"   and f6 :: "16 word \<Rightarrow> 'b"   and f7 :: "32 prefix_match \<Rightarrow> 'b"   and f8 :: "32 prefix_match \<Rightarrow> 'b"   and f9 :: "8 word \<Rightarrow> 'b"   and f10 :: "16 word \<Rightarrow> 16 word \<Rightarrow> 'b"   and f11 :: "16 word \<Rightarrow> 16 word \<Rightarrow> 'b"   and x111 :: "16 word"   and x112 :: "16 word" shows "(case L4Dst x111 x112 of IngressPort x \<Rightarrow> f1 x | EtherSrc x \<Rightarrow> f2 x | EtherDst x \<Rightarrow> f3 x | EtherType x \<Rightarrow> f4 x | VlanId x \<Rightarrow> f5 x | VlanPriority x \<Rightarrow> f6 x | IPv4Src x \<Rightarrow> f7 x | IPv4Dst x \<Rightarrow> f8 x | IPv4Proto x \<Rightarrow> f9 x | L4Src x xa \<Rightarrow> f10 x xa | L4Dst x xa \<Rightarrow> f11 x xa) = f11 x111 x112"
subset_Pow_Union: fixes A :: "'a set set" shows "A \<subseteq> Pow (\<Union> A)"
is_pure_assn_pure: fixes P :: "bool" shows "is_pure_assn (\<up> P)"
Union_eq: fixes A :: "'d set set" shows "\<Union> A = {x. \<exists>B\<in>A. x \<in> B}"
mgu_wf_trm: fixes s :: "('fun, 'b) Term.term"   and t :: "('fun, 'b) Term.term"   and \<sigma> :: "'b \<Rightarrow> ('fun, 'b) Term.term"   and v :: "'b" assumes "mgu s t = Some \<sigma>"   and "wf\<^sub>t\<^sub>r\<^sub>m s"   and "wf\<^sub>t\<^sub>r\<^sub>m t" shows "wf\<^sub>t\<^sub>r\<^sub>m (\<sigma> v)"
vector_add_component: fixes x :: "('a, 'b) Finite_Cartesian_Product.vec"   and y :: "('a, 'b) Finite_Cartesian_Product.vec"   and i :: "'b" shows "(x + y) $h i = x $h i + y $h i"
minus: shows "(R ===> R ===> R) (arith_ops_record.minus ff_ops) (-)"
these_def: fixes A :: "'d option set" shows "Option.these A = the ` {x \<in> A. x \<noteq> None}"
distinct_27: fixes x3 :: "nat"   and x61 :: "msg"   and x62 :: "msg" shows "Nonce x3 \<noteq> \<lbrace>x61, x62\<rbrace>"
fmdom'_restrict_set_precise: fixes A :: "'c set"   and m :: "('c, 'd) fmap" shows "fmdom' (fmrestrict_set A m) = fmdom' m \<inter> A"
atLeastatMost_psubset_iff: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset"   and d :: "'c multiset" shows "(subset_mset.atLeastAtMost a b \<subset> subset_mset.atLeastAtMost c d) = ((\<not> a \<subseteq># b \<or> c \<subseteq># a \<and> b \<subseteq># d \<and> (c \<subset># a \<or> b \<subset># d)) \<and> c \<subseteq># d)"
neighbourhood_base_of_def: fixes P :: "'c set \<Rightarrow> bool"   and X :: "'c topology" shows "neighbourhood_base_of P X \<equiv> \<forall>x\<in>topspace X. neighbourhood_base_at x P X"
dehomogenize_homogenize_id: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "x \<notin> indets p" shows "dehomogenize x (homogenize x p) = p"
assms_2: shows "x \<longlonglongrightarrow> x0"
Un_Image: fixes R :: "('c \<times> 'b) set"   and S :: "('c \<times> 'b) set"   and A :: "'c set" shows "(R \<union> S) `` A = R `` A \<union> S `` A"
minus_vec_def: shows "(-) \<equiv> \<lambda>x y. \<chi>i. x $ i - y $ i"
im_sum_list_prod: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'b"   and xys :: "('d \<times> 'e) list" assumes "VectorSpaceHom smult V smult' T"   and "set (map (\<lambda>(x, y). f x y) xys) \<subseteq> V" shows "T (\<Sum>(x, y)\<leftarrow>xys. f x y) = (\<Sum>(x, y)\<leftarrow>xys. T (f x y))"
r_update_rightmost_eq_r: fixes f :: "'b \<Rightarrow> 'b"   and t :: "'b tree" shows "r (update_rightmost f t) = r t"
fun_upd_apply: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d"   and y :: "'c"   and z :: "'d" shows "(f(x := y)) z = (if z = x then y else f z)"
llistE: fixes y :: "'b llist" obtains "y = LNil"   | x21 :: "'b" and x22 :: "'b llist" where "y = x21 ## x22"
regular_conv_closed: fixes x :: "'a" assumes "regular x" shows "regular (x\<^sup>T)"
del_point_block_count: fixes p :: "'a" shows "size (del_point_blocks p) = size \<B>"
scaleR_left_diff_distrib: fixes a :: "real"   and b :: "real"   and x :: "'a" shows "(a - b) *\<^sub>R x = a *\<^sub>R x - b *\<^sub>R x"
keys_monom_multE: fixes v :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'b"   and t :: "'x \<Rightarrow>\<^sub>0 nat"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "v \<in> keys (local.punit.monom_mult c t p)" obtains u :: "'x \<Rightarrow>\<^sub>0 nat" where "u \<in> keys p" and "v = local.punit.splus t u"
prems_1: shows "a \<in> set (insert_spec (x # xs) (e, i))"
allAllocationsFinite: fixes N :: "'a set"   and G :: "'b set" assumes "finite N"   and "finite G" shows "finite (allAllocations N G)"
2: shows "bprv (all yy' (imp (cnj (NN \<langle>\<phi>\<rangle> \<langle>neg \<phi>\<rangle>) (NN \<langle>\<phi>\<rangle> (Var yy'))) (eql \<langle>neg \<phi>\<rangle> (Var yy'))))"
TriSumA_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" shows "A B C TriSumA D E F \<equiv> \<exists>G H I. A B C B C A SumA G H I \<and> G H I C A B SumA D E F"
discrete_topology_unique_alt: fixes U :: "'c set"   and X :: "'c topology" shows "(discrete_topology U = X) = (topspace X \<subseteq> U \<and> (\<forall>x\<in>U. openin X {x}))"
belowI: fixes d :: "'e defl"   and f :: "'e \<rightarrow> 'e" assumes "\<And>x. cast\<cdot>d\<cdot>x = x \<Longrightarrow> f\<cdot>x = x" shows "cast\<cdot>d \<sqsubseteq> f"
dgrad_p_set_le_crit_pair_zero: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "dgrad_p_set_le d {fst (crit_pair p 0)} {p}"
E16: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<diamond>\<box>F \<longrightarrow> \<diamond>F"
coprime_2: fixes c1 :: "complex poly"   and c2 :: "complex poly"   and m :: "nat"   and n :: "nat" assumes "\<not> coprime c1 c2"   and "(c1, m) \<in> set cs"   and "(c2, n) \<in> set cs" shows "m = n"
subcls1_wfD: fixes C :: "char list"   and D :: "char list"   and P :: "'b prog"   and wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool" assumes "P \<turnstile> C \<prec>\<^sup>1 D"   and "wf_prog wf_md P" shows "D \<noteq> C \<and> (D, C) \<notin> (subcls1 P)\<^sup>+"
rep_eq: fixes x :: "('a, 'b) pp"   and xa :: "('a, 'b) pp" shows "pp.mapping_of (lcs x xa) = lcs (pp.mapping_of x) (pp.mapping_of xa)"
ltake_LCons: fixes n :: "enat"   and x :: "'a"   and xs :: "'a llist" shows "ltake n (LCons x xs) = (case n of 0 \<Rightarrow> LNil | eSuc n' \<Rightarrow> LCons x (ltake n' xs))"
compI_3: fixes f :: "'a"   and g :: "'a" assumes "f \<in> Ar"   and "g \<in> Ar"   and "Cod f = Dom g" shows "Cod (g \<bullet> f) = Cod g"
family_subset_Un_sub: fixes C :: "'c set set"   and B :: "'c set" assumes "\<forall>A\<in>C. A \<subseteq> B" shows "\<Union> C \<subseteq> B"
assms_1: shows "\<Gamma>(a \<mapsto> \<sigma>) \<turnstile> M : \<tau>"
simps: fixes t1 :: "('b \<times> nat) tree"   and t2 :: "('b \<times> nat) tree" shows "balanced1 t1 t2 = (10 * (size_wbt t2 + 1) \<le> 25 * (size_wbt t1 + 1))"
roots_bound: fixes f :: "nat \<Rightarrow> int" assumes "f \<in> carrier (UP R)"   and "f \<noteq> \<zero>\<^bsub>UP R\<^esub>"   and "finite (carrier R)" shows "finite {a \<in> carrier R. eval R R id a f = \<zero>\<^bsub>R\<^esub>} \<and> card {a \<in> carrier R. eval R R id a f = \<zero>\<^bsub>R\<^esub>} \<le> deg R f"
eq: fixes i :: "'b"   and a :: "real" shows "{x \<in> S. f x \<bullet> i < a} = {x \<in> S. f x \<in> {y. y \<bullet> i < a}}"
ideD_2: fixes a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "ide a" shows "local.dom a = a"
return_time_exivl: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and P :: "'b set"   and x :: "'b" assumes "auto_ll_on_open f X"   and "auto_ll_on_open.returns_to f X P x"   and "closed P" shows "auto_ll_on_open.return_time f X P x \<in> auto_ll_on_open.existence_ivl0 f X x"
Sup_sup_pres: fixes f :: "'c \<Rightarrow> 'd" assumes "Sup_pres f" shows "sup_pres f"
complement_connected_component_unions: fixes S :: "'c set"   and x :: "'c" shows "S - connected_component_set S x = \<Union> (connected_component_set S ` S - {connected_component_set S x})"
normal_series_prefix_closed: fixes i :: "nat" assumes "i \<le> length \<GG>"   and "0 < i" shows "normal_series (G\<lparr>carrier := \<GG> ! (i - 1)\<rparr>) (take i \<GG>)"
conj_disj_distribs_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<^bold>\<sqinter> (y \<^bold>\<squnion> z) = x \<^bold>\<sqinter> y \<^bold>\<squnion> x \<^bold>\<sqinter> z"
a_set_tag_name_locs_def: fixes element_ptr :: "(_) element_ptr" shows "l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_tag_name_locs element_ptr \<equiv> all_args (put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr tag_name_update)"
mk_inter_def: shows "mk_inter \<equiv> (\<inter>)"
assms: shows "do_app (refs s, ffi s) op0 (rev vs) = Some ((refs', ffi'), res)"
beukers_aux_integrable1: shows "set_integrable lborel (({0<..<1} \<times> {0<..<1}) \<times> {0<..<1}) (\<lambda>((x, y), z). P x * P y / (1 - (1 - x * y) * z))"
gfp_F_def: fixes ds :: "'b set" shows "gfp_F ds = map_prod id undual (gfp (F ds))"
intros_6: fixes n'' :: "'node SDG_node"   and p :: "'pname"   and n' :: "'node SDG_node"   and n :: "'node SDG_node" assumes "n'' s-p\<rightarrow>\<^bsub>sum\<^esub> n'"   and "n' \<in> sum_SDG_slice2 n" shows "n'' \<in> sum_SDG_slice2 n"
submanifold_atlasE: fixes c :: "('a, 'b) chart" assumes "c \<in> sub.atlas" shows "c \<in> src.atlas"
LeftDerivation_breakdown: fixes u :: "symbol list"   and v :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and w :: "symbol list" assumes "LeftDerivation (u @ v) D w" shows "\<exists>n w1 w2. w = w1 @ w2 \<and> LeftDerivation u (take n D) w1 \<and> derivation_ge (drop n D) (length w1) \<and> LeftDerivation v (derivation_shift (drop n D) (length w1) 0) w2"
simps_12: fixes \<nu> :: "'b"   and v :: "'b local.term"   and va :: "'b local.term" shows "\<^bold>\<langle>\<nu>\<^bold>\<rangle> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> v \<^bold>\<star> va = \<^bold>\<langle>\<nu>\<^bold>\<rangle> \<^bold>\<star> v \<^bold>\<star> va"
external_from_internal_invariant: fixes S :: "('e, 'f) spec"   and P :: "'e set"   and Q :: "'f set" assumes "reach S \<subseteq> P"   and "obs S ` P \<subseteq> Q" shows "oreach S \<subseteq> Q"
bot_coreflexive_def: shows "bot \<equiv> Abs_coreflexive bot"
cycle_free_def: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat" shows "cycle_free m n \<equiv> \<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> (\<forall>j\<le>n. len m i j (rem_cycles i j xs) \<le> len m i j xs) \<and> (0::'b) \<le> len m i i xs"
fresh_psubst_Ctxt: fixes X :: "W.tvar"   and \<theta> :: "(W.tvar \<times> ty) list"   and \<Gamma> :: "(W.var \<times> tyS) list" assumes "X \<sharp> \<theta>"   and "X \<sharp> \<Gamma>" shows "X \<sharp> \<theta><\<Gamma>>"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>0 'b"   and xa :: "'a \<Rightarrow>\<^sub>0 'b" shows "lookup (x + xa) = (\<lambda>k. lookup x k + lookup xa k)"
var_update_out: fixes x :: "'c \<Longrightarrow> 'b"   and A :: "'a"   and A' :: "'b"   and v :: "'c" shows "put\<^bsub>$x\<acute>\<^esub> (A, A') v = (A, put\<^bsub>x\<^esub> A' v)"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and msg :: "msg" shows "P (case msg of Rreq x xa xb xc xd xe xf xg xh \<Rightarrow> f1 x xa xb xc xd xe xf xg xh | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = (\<not> ((\<exists>x11 x12 x13 x14 x15 x16 x17 x18 x19. msg = Rreq x11 x12 x13 x14 x15 x16 x17 x18 x19 \<and> \<not> P (f1 x11 x12 x13 x14 x15 x16 x17 x18 x19)) \<or> (\<exists>x21 x22 x23 x24 x25. msg = Rrep x21 x22 x23 x24 x25 \<and> \<not> P (f2 x21 x22 x23 x24 x25)) \<or> (\<exists>x31 x32. msg = Rerr x31 x32 \<and> \<not> P (f3 x31 x32)) \<or> (\<exists>x41 x42. msg = Newpkt x41 x42 \<and> \<not> P (f4 x41 x42)) \<or> (\<exists>x51 x52 x53. msg = Pkt x51 x52 x53 \<and> \<not> P (f5 x51 x52 x53))))"
surj_card_le: fixes A :: "'a set"   and B :: "'b set"   and f :: "'a \<Rightarrow> 'b" assumes "finite A"   and "B \<subseteq> f ` A" shows "card B \<le> card A"
parts_def: shows "parts \<equiv> \<lambda>H. {x. partsp (\<lambda>x. x \<in> H) x}"
cod_comp: fixes g :: "('obj, 'obj) concrete_category.arr"   and f :: "('obj, 'obj) concrete_category.arr" assumes "seq g f" shows "cod (g \<cdot> f) = cod g"
m_assoc: fixes x :: "'a vec"   and y :: "'a vec"   and z :: "'a vec" assumes "x \<in> carrier_vec n"   and "y \<in> carrier_vec n"   and "z \<in> carrier_vec n" shows "x + y + z = x + (y + z)"
hyps_5: assumes "a \<sharp> s1" shows "a \<sharp> s2"
x0_hb_bound5: fixes x :: "real"   and i :: "nat" assumes "x\<^sub>1 \<le> x"   and "i < k" shows "bs ! i + (hs ! i) x / x \<le> bs ! i * 3 / 2"
append_Cons_nth_left: fixes i :: "nat"   and xs :: "'c list"   and u :: "'c"   and ys :: "'c list" assumes "i < length xs" shows "(xs @ u # ys) ! i = xs ! i"
simps_9: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and v :: "'a"   and va :: "value list" shows "local.less_gexp_aux (Eq e1 e2) (In v va) = True"
zero_le_mult_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less_eq zero (times a b) = (less_eq zero a \<and> less_eq zero b \<or> less_eq a zero \<and> less_eq b zero)"
tame12o_def: fixes g :: "graph" shows "tame12o g = (\<forall>v\<in>\<V> g. except g v \<noteq> 0 \<and> degree g v = 6 \<longrightarrow> vertextype g v = (5, 0, 1))"
mk_eqcl_bound: fixes xs :: "nat option list"   and zs :: "nat list"   and x :: "nat"   and ii :: "nat"   and T :: "bool list list" assumes "\<And>x k. \<lbrakk>x \<in> set xs; x = Some k\<rbrakk> \<Longrightarrow> k < length zs"   and "x \<in> set (fst (mk_eqcl xs zs ii T))" shows "x < length (snd (mk_eqcl xs zs ii T))"
Can_red_1: fixes u :: "'c local.term" assumes "Ide u" shows "Can (u\<^bold>\<down>)"
frag_extend_add: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow>\<^sub>0 int"   and a :: "'c \<Rightarrow>\<^sub>0 int"   and b :: "'c \<Rightarrow>\<^sub>0 int" shows "frag_extend f (a + b) = frag_extend f a + frag_extend f b"
Suc: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" shows "alternate f g (Suc k) = (if even k then f else g) \<circ> alternate f g k"
fim: shows "f ` sphere a r \<subseteq> sphere 0 1"
conga_diff1: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "A B C CongA A' B' C'" shows "A \<noteq> B"
bl_set_size: fixes y :: "'a" assumes "y \<in># mset_set (\<V> - {x})" shows "size (filter_mset ((\<in>) y) (filter_mset ((\<in>) x) \<B>)) = nat \<Lambda>"
assms: shows "ref = None \<or> ref = Some r \<and> r \<in> set list"
simps_1: fixes d :: "'dyn" shows "Subx.norm_unboxed (OpDyn d) = d"
execn_dynCall_Normal_elim: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and init :: "'e \<Rightarrow> 'e"   and p :: "'e \<Rightarrow> 'd"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>dynCall init p return c,Normal s\<rangle> =n\<Rightarrow> t" obtains "\<Gamma>\<turnstile> \<langle>call init (p s) return c,Normal s\<rangle> =n\<Rightarrow> t"
simps_4: fixes a :: "name"   and x :: "'a"   and b :: "name" assumes "a \<sharp> x"   and "b \<sharp> x" shows "[(a, b)] \<bullet> x = x"
linear_diff: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and x :: "('a, 'b) vec"   and y :: "('a, 'b) vec" assumes "Vector_Spaces.linear (*s) (*s) f" shows "f (x - y) = f x - f y"
intros_2: fixes x :: "'a"   and r :: "('a \<times> 'a) set"   and xs :: "'a list" assumes "x \<in> Field r"   and "\<forall>y\<in>set xs. (x, y) \<in> r"   and "sorted_on r xs" shows "sorted_on r (x # xs)"
ring_iso_def: fixes R :: "('g, 'i) ring_scheme"   and S :: "('h, 'j) ring_scheme" shows "ring_iso R S = {h \<in> ring_hom R S. bij_betw h (carrier R) (carrier S)}"
units_power_order_eq_one: fixes a :: "'a mod_ring poly" assumes "finite (Units R)"   and "a \<in> Units R" shows "a [^]\<^bsub>R\<^esub> card (Units R) = \<one>\<^bsub>R\<^esub>"
Uvariation_trans: fixes \<omega> :: "variable \<Rightarrow> real"   and \<nu> :: "variable \<Rightarrow> real"   and U :: "variable set"   and \<mu> :: "variable \<Rightarrow> real"   and V :: "variable set" assumes "Uvariation \<omega> \<nu> U"   and "Uvariation \<nu> \<mu> V" shows "Uvariation \<omega> \<mu> (U \<union> V)"
one_subst_aux: fixes N :: "lam"   and N' :: "lam"   and x :: "name"   and M :: "lam" assumes " N \<longrightarrow>\<^sub>1 N'" shows " M[x::=N] \<longrightarrow>\<^sub>1 M[x::=N']"