hcju/isabelleps huggingface dataset:
state_qbit_is_lincomb: fixes n :: "nat" shows "state_qbit n = {module.lincomb (module_cpx_vec (2 ^ n)) a A |a A. finite A \<and> A \<subseteq> unit_vectors (2 ^ n) \<and> a \<in> A \<rightarrow> UNIV \<and> \<parallel>module.lincomb (module_cpx_vec (2 ^ n)) a A\<parallel> = 1}"
subgroupE_2: fixes H :: "'a set" assumes "subgroup H (G\<lparr>carrier := H\<rparr>)" shows "H \<noteq> {}"
rref_imp_upper_triagular: fixes A :: "(('b, 'c) vec, 'c) vec" assumes "reduced_row_echelon_form A" shows "upper_triangular A"
DONE_eq_Done: fixes a :: "'f"   and b :: "'f option" shows "(DONE a = Resumption.resumption.Done b) = (b = Some a)"
simps_2: fixes x :: "'a" shows "vwalk_arcs [x] = []"
not_inf_Fault: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and x :: "'c" shows "\<not> \<Gamma>\<turnstile> (c, Fault x) \<rightarrow> \<dots>(\<infinity>)"
wlp_Loop1: fixes P :: "'a \<Rightarrow> real"   and body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "unitary P"   and "nearly_healthy (wlp body)" shows "wlp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip) P = gfp_exp (\<lambda>Q s. \<guillemotleft> G \<guillemotright> s * wlp body Q s + \<guillemotleft> \<N> G \<guillemotright> s * P s)"
coeffs_1_eq: shows "coeffs 1 = [1::'a]"
Vfrom_in_bundle_3: fixes i :: "V"   and j :: "V"   and A :: "V"   and B :: "V" assumes "i \<in>\<^sub>\<circ> j"   and "A = A"   and "B = B" shows "Vfrom B j = Vfrom B (rank j)"
cone_iff: fixes S :: "'b set" assumes "S \<noteq> {}" shows "cone S = ((0::'b) \<in> S \<and> (\<forall>c>0. (*\<^sub>R) c ` S = S))"
cast: shows "e = \<lparr>C\<rparr>e\<^sub>0"
vec_carrier_vec: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and m :: "nat" shows "(Matrix.vec n f \<in> carrier_vec m) = (n = m)"
slice2_ddep_slice2: fixes nx :: "'node SDG_node"   and n :: "'node SDG_node"   and V :: "'var"   and n' :: "'node SDG_node" assumes "nx \<in> sum_SDG_slice2 n"   and "n s-V\<rightarrow>\<^sub>d\<^sub>d n'" shows "nx \<in> sum_SDG_slice2 n'"
cod_char: fixes f :: "'a" shows "Left_a.cod f = (if Left_a.arr f then cod f else null)"
update_arg_wf_tuples_1: fixes nhip :: "nat" shows "update_arg_wf (0, unk, val, Suc 0, nhip)"
is_ceq_def: fixes uu :: "'b itself" shows "is_ceq uu = (ID CEQ('b) \<noteq> None)"
surjective: fixes r :: "'a view_scheme" shows "r = \<lparr>ext = ext r, lan = lan r, own = own r, \<dots> = more r\<rparr>"
commuting_imp_normal2: assumes "carrier G \<subseteq> A <#> B"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> B\<rbrakk> \<Longrightarrow> x \<otimes> y = y \<otimes> x" shows "B \<lhd> G"
simps_5: fixes v :: "'b"   and l :: "value list"   and v' :: "'b"   and l' :: "value list" shows "gexp_same_structure (In v l) (In v' l') = (v = v' \<and> l = l')"
writes_not_ok: fixes h :: "'a"   and f :: "('a, 'b, 'c) prog"   and S :: "('a, 'b, 'd) prog set"   and h' :: "'a" assumes "\<not> h \<turnstile> ok f" shows "writes S f h h'"
assms: shows "\<forall>x\<in>fv\<^sub>s\<^sub>e\<^sub>t M. (\<exists>f. \<theta> x = Fun f []) \<or> (\<exists>y. \<theta> x = Var y)"
SFAccRedsNone: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and D :: "char list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool" assumes "\<nexists>b t. P \<turnstile> C has F,b:t in D" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D},s,b\<rangle> \<rightarrow>* \<langle>THROW NoSuchFieldError,s,False\<rangle>"
subterms_subset_set: fixes M :: "('a, 'b) Term.term set"   and t :: "('a, 'b) Term.term" assumes "M \<subseteq> subterms t" shows "subterms\<^sub>s\<^sub>e\<^sub>t M \<subseteq> subterms t"
diff: fixes x :: "char list multiset" shows "poly_of_vars x - poly_of_vars (remdups_mset x) \<in> More_Modules.ideal polynomial_bool"
zip_map_option2: fixes x :: "'c option"   and g :: "'e \<Rightarrow> 'd"   and y :: "'e option" shows "zip_option x (map_option g y) = map_option (apsnd g) (zip_option x y)"
eventually_ge_at_top: fixes c :: "'b" shows "eventually ((\<le>) c) at_top"
sorted_merge_distinct: fixes xs :: "'c list"   and ys :: "'c list" assumes "sorted xs"   and "sorted ys" shows "sorted (merge_distinct xs ys)"
sat_eq_reduc_sat: fixes N :: "'f set" shows "saturated N = reduc_saturated N"
finsum_def: fixes G :: "('b, 'c) ring_scheme" shows "finsum G = finprod (add_monoid G)"
K_def: shows "K \<equiv> \<lambda>(s\<^sub>1, o\<^sub>1). \<K> s\<^sub>1 \<bind> (\<lambda>s\<^sub>2. map_pmf (Pair s\<^sub>2) (\<O> s\<^sub>2))"
pair_loopfree_digraph_subdivide: fixes e :: "'a \<times> 'a"   and w :: "'a" assumes "e \<in> parcs G"   and "w \<notin> pverts G" shows "pair_loopfree_digraph (subdivide G e w)"
atms_of_negs: fixes AA :: "'b multiset" shows "atms_of (negs AA) = set_mset AA"
fun_eq_iff: fixes f :: "'p \<Rightarrow> 'q"   and g :: "'p \<Rightarrow> 'q" shows "(f = g) = (\<forall>x. f x = g x)"
item_nonterminal_finished_item: fixes \<alpha> :: "symbol list" shows "item_nonterminal (finished_item \<alpha>) = \<SS>"
simps_2: fixes E :: "exp"   and Es :: "exp list" shows "vars_list (E # Es) = vars E \<union> vars_list Es"
i'_1: shows "p = (\<^bold>\<diamond> Nom i')"
f_foldl_abs_alt_def: fixes \<phi> :: "'b ltl"   and w :: "'b set list" shows "af_G_abs.f_foldl_abs (Abs \<phi>) w = foldl af_G_abs.f_abs (Abs \<phi>) w"
collinear_closed_segment: fixes a :: "'a"   and b :: "'a" shows "collinear (closed_segment a b)"
all_same_arities_def: fixes F1 :: "'l \<Rightarrow> 'm fundef option"   and F2 :: "'l \<Rightarrow> 'n fundef option" shows "all_same_arities F1 F2 \<equiv> \<forall>f. rel_option (\<lambda>fd gd. arity fd = arity gd) (F1 f) (F2 f)"
opp_chamberD1: fixes C :: "'a set" assumes "C \<in> f \<turnstile> folding.\<C>" shows "(THE D. D \<in> folding.\<C> - f \<turnstile> folding.\<C> \<and> f ` D = C) \<in> folding.\<C> - f \<turnstile> folding.\<C>"
trunc_bound_pdevs_def: fixes p :: "nat"   and x :: "'a pdevs" shows "trunc_bound_pdevs p x = (trunc_pdevs p x, tdev' p (trunc_err_pdevs p x))"
distinct_3: shows "hp_Idle \<noteq> hp_InitMark"
PR2_def: shows "PR2 = prod_list (map AB [3..<Suc i])"
mset_ran_mem: fixes r :: "int set"   and i :: "int"   and a :: "int \<Rightarrow> 'a" assumes "finite r"   and "i \<in> r" shows "a i \<in># mset_ran a r"
snoc_Cons: fixes xs :: "'b list"   and x :: "'b"   and y :: "'a"   and ys :: "'a list" assumes "as = xs @ [x]"   and "bs = y # ys" shows "P"
nb_def: fixes b :: "contour"   and c :: "nat" shows "nb b c = Abs_contour (c # Rep_contour b)"
parts_subset_iff: fixes G :: "msg set"   and H :: "msg set" shows "(parts G \<subseteq> parts H) = (G \<subseteq> parts H)"
eq_refl: fixes x :: "'a list"   and y :: "'a list" assumes "x = y" shows "y \<le>lex x"
assms: shows "nop_v11 A (2\<^sub>\<nat>) f"
integral_Dirichlet_kernel: fixes n :: "nat" shows "integral\<^sup>L (lebesgue_on {- pi..pi}) (Dirichlet_kernel n) = pi"
lex_ord_def: fixes ko :: "'o"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> order option"   and xs :: "'x"   and ys :: "'y" shows "lex_ord ko f xs ys = lex_ord_raw ko f (list_of_oalistx xs) (list_of_oalisty ys)"
IH: assumes "0 < n"   and "n < \<^bold>|w\<^bold>|" shows "Lyndon_rec w n"
prems_2: shows "P,h \<turnstile>\<^sub>s sh \<surd>"
minimum_spanning_forest_def: fixes f :: "'a"   and g :: "'a" shows "minimum_spanning_forest f g \<equiv> spanning_forest f g \<and> (\<forall>u. spanning_forest u g \<longrightarrow> sum_class.sum (f \<sqinter> g) \<le> sum_class.sum (u \<sqinter> g))"
minus_pairs_def: fixes A :: "('e \<times> 'e) set"   and B :: "('e \<times> 'e) set" shows "A -\<^sub>p B = A - (B \<union> prod.swap ` B)"
comp_left_subdist_inf: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<sqinter> z \<le> x \<sqinter> z \<sqinter> (y \<sqinter> z)"
invertible_right_inverse2: fixes u :: "'a"   and v :: "'a" assumes "additive.invertible u"   and "u \<in> R"   and "v \<in> R" shows "u + (- u + v) = v"
induced_automorphism_simplex_image_1: fixes C :: "'a set"   and x :: "'a set" assumes "C \<in> f \<turnstile> folding_g.\<C>"   and "order.greater_eq C x" shows "\
 ` x = g ` x"
encode_kind_def: fixes e :: "nat" shows "encode_kind e \<equiv> if e = 0 then 0 else if e = 1 then 1 else pdec1 e"
cont2cont_split_pair: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'i"   and g :: "'g \<Rightarrow> 'h \<Rightarrow> 'j" assumes "cont f"   and "\<And>x. cont (f x)"   and "cont g"   and "\<And>x. cont (g x)" shows "cont (\<lambda>(a, b). (f a b, g a b))"
lookup_none_notin_dom: fixes \<rho> :: "(nat \<times> exp) list"   and x :: "nat" assumes "lookup \<rho> x = None" shows "x \<notin> sdom \<rho>"
fst_conv_simps: fixes xa :: "'a \<times> 'b" shows "fst xa = (case xa of (x2aa, x1aa) \<Rightarrow> x2aa)"
min_avg: fixes m :: "nat"   and T :: "nat \<Rightarrow> nat" assumes "LoadBalancing m" shows "m * T (min\<^sub>k T m) \<le> sum T {1..m}"
hyps_1: shows "s\<turnstile> l reachable_from (s@@k)"
eq: shows "Collect ((\<lambda>v. v = signs ! i) \<circ> consistent_sign_vec qs) \<inter> set (sorted_list_of_set {x. poly p x = 0}) = {x. poly p x = 0 \<and> consistent_sign_vec qs x = signs ! i}"
res_def: fixes ts :: "traffic" shows "res ts \<equiv> fst (snd (Rep_traffic ts))"
less_neg_neg: 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 a :: "'b" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less a (uminus a) = less a zero"
lfp_dual_var: fixes f :: "'a \<Rightarrow> 'a" shows "\<partial> (lfp f) = gfp (\<partial>\<^sub>F f)"
take_zip: fixes n :: "nat"   and xs :: "'b list"   and ys :: "'c list" shows "take n (zip xs ys) = zip (take n xs) (take n ys)"
degree_slist_eq_SucD: fixes xs :: "(nat, 'd) slist"   and n :: "nat" assumes "degree_slist xs = Suc n" shows "pdevs_apply (Pdevs xs) n \<noteq> (0::'d)"
assms: shows "(P, Q) \<in> Rel\<^sup>*"
comp_arr_inv: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "B.VV.inverse_arrows f g" shows "B.VV.comp f g = B.VV.dom g"
multpw_cancel: fixes ns :: "('a \<times> 'a) set"   and X :: "'a multiset"   and Z :: "'a multiset"   and Y :: "'a multiset" assumes "refl ns"   and "trans ns" shows "((X + Z, Y + Z) \<in> multpw ns) = ((X, Y) \<in> multpw ns)"
image_subset_iff_subset_vimage: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and B :: "'c set" shows "(f ` A \<subseteq> B) = (A \<subseteq> f -` B)"
iso_verts_head: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism"   and e :: "'d" assumes "H.digraph_isomorphism hom"   and "e \<in> arcs (app_iso hom G)" shows "iso_head hom (iso_arcs hom e) = iso_verts hom (head (app_iso hom G) e)"
unitl_eq: fixes e :: "'a"   and x :: "'a"   and y :: "'a" assumes "e \<in> \<xi>"   and "\<rho> x e y" shows "x = y"
simps: fixes x :: "'a ext0"   and y :: "'a ext0" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
WbisT_transT: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and s' :: "'state" assumes "c \<approx>wT d"   and "s \<approx> t"   and "(c, s) \<rightarrow>t s'" obtains t' :: "'state" where "(d, t) \<rightarrow>*t t'" and "s' \<approx> t'"
card_insert_disjoint: fixes A :: "'g set"   and x :: "'g" assumes "finite A"   and "x \<notin> A" shows "card (insert x A) = Suc (card A)"
rT_is_acc_type: fixes G :: "prog"   and P :: "pname"   and sig :: "sig"   and m :: "mhead" assumes "wf_mhead G P sig m" shows "is_acc_type G P (resTy m)"
path_is_trancl: fixes E :: "('b \<times> 'b) set"   and u :: "'b"   and l :: "'b list"   and v :: "'b" assumes "path E u l v"   and "l \<noteq> []" shows "(u, v) \<in> E\<^sup>+"
simps_2: fixes a :: "'a"   and b :: "'a" shows "un_MSSCons2 (mssalternate a b) = mssalternate b a"
Barbara: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A Q B"   and "B Q C" shows "A Q C"
dom_simp_2: shows "partial_magma.dom (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} {False, True} (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g)) quiver.One = quiver.One"
cases: fixes x :: "pseqp" obtains "x = PAodv"   | "x = PPkt"   | "x = PNewPkt"   | "x = PRreq"   | "x = PRrep"   | "x = PRerr"
b: fixes i :: "Quotient_Int.int" assumes "P i" shows "P (i + 1)"
set_set_to_list: fixes xs :: "'a set" assumes "finite xs" shows "set (set_to_list xs) = xs"
sorted_on_distinct_set_unique: fixes xs :: "'a list"   and ys :: "'a list" assumes "sorted_on r xs"   and "distinct xs"   and "sorted_on r ys"   and "distinct ys"   and "set xs = set ys" shows "xs = ys"
pre_buf: shows "wf_mbuf2' \<sigma> P V j n R \<phi> \<psi> buf"
preserves_section_retraction: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and e :: "'a"   and m :: "'a" assumes "functor A B F"   and "partial_magma.ide A (A e m)" shows "partial_magma.ide B (B (F e) (F m))"
listset_ConsI: fixes y :: "'b"   and x :: "'b set"   and ys' :: "'b list"   and xs :: "'b set list"   and ys :: "'b list" assumes "y \<in> x"   and "ys' \<in> listset xs"   and "ys = y # ys'" shows "ys \<in> listset (x # xs)"
WT: shows "\<I>' \<turnstile>g gpv \<surd>"
chain_boundary_boundary_alt: fixes p :: "nat"   and X :: "'c topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "singular_chain (Suc p) X c" shows "chain_boundary p (chain_boundary (Suc p) c) = 0"
assms_1: shows "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b"
itrev_Nil: fixes xs :: "'b list" shows "itrev xs [] = rev xs"
interaction_any_bounded_converter_start_2: fixes "consider" :: "'i \<Rightarrow> bool"   and gpv :: "('j, 'i, 'k) gpv"   and n :: "enat"   and m :: "enat" assumes "interaction_bounded_by consider gpv n"   and "n \<le> m" shows "interaction_bounded_by consider gpv m"
gbinomial_neg: fixes a :: "int"   and k :: "nat" assumes "a < 0" shows "a gchoose k = (- 1) ^ k * int (k + nat (- a) - 1 choose k)"
cc_lt_c: shows "\<forall>C'\<in>CC. C' < C"
closed_orbitI: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and t :: "real"   and t' :: "real"   and y :: "'b" assumes "auto_ll_on_open f X"   and "t \<noteq> t'"   and "t \<in> auto_ll_on_open.existence_ivl0 f X y"   and "t' \<in> auto_ll_on_open.existence_ivl0 f X y"   and "auto_ll_on_open.flow0 f X y t = auto_ll_on_open.flow0 f X y t'" shows "auto_ll_on_open.closed_orbit f X y"
span_mem: fixes E :: "complex vec set"   and u :: "complex vec" assumes "E \<subseteq> carrier_vec d"   and "u \<in> E" shows "u \<in> local.span E"
LIMSEQ_imp_Cauchy: fixes X :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "X \<longlonglongrightarrow> x" shows "Cauchy X"
fds_divisor_sigma: fixes x :: "'a" shows "fds (divisor_sigma x) = fds_zeta * fds_shift x fds_zeta"
intT_def: shows "intT \<equiv> ProblemIkTpart_TEModel.intT tpD tpFD (\<lambda>\<sigma>. D) eintF"
wp_imp_wlp: fixes \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "wp \<pi> c Q s" shows "wlp \<pi> c Q s"
q_def: shows "q \<equiv> if ty_q = ty_q then CARD('q) else undefined"
assms: shows "tmf_ivl_cs prec n A a f = Some cs"
nonzero_mult_divide_mult_cancel_left2: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.field divide inverse times one plus zero minus uminus"   and "c \<noteq> zero" shows "divide (times c a) (times b c) = divide a b"
subs: shows "interp D' \<subseteq> \<Union> (production ` CC)"
unfold: shows "nats = 0 && smap\<cdot>(\<Lambda> x. 1 + x)\<cdot>nats"
facetrel_def: fixes y :: "'b set"   and x :: "'b set" shows "y \<lhd> x \<equiv> \<exists>v. v \<notin> y \<and> x = insert v y"
OAlist_tc_lex_ord_def: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> order option"   and xs :: "('a, 'b) oalist_tc"   and ys :: "('a, 'c) oalist_tc" shows "OAlist_tc_lex_ord f xs ys = lex_ord_pair_tc f (list_of_oalist_tc xs) (list_of_oalist_tc ys)"
not_two_sides_id: fixes P :: "'p"   and Q :: "'p"   and A :: "'p" shows "\<not> P Q TS A A"
s2BC: fixes l :: "hf" assumes "l \<^bold>\<in> succ k2" shows "Builds B C s2 l"
ordLess_irreflexive: fixes r :: "('a \<times> 'a) set" shows "(r, r) \<notin> ordLess"
map\<^sub>0_simps_2: fixes a :: "'b" assumes "B.obj a" shows "src\<^sub>C (\<tau>.G.map\<^sub>0 a) = \<tau>.G.map\<^sub>0 a"
indep: shows "indep_sets E (\<Union> (I ` J))"
sg8: shows "\<not> t mod cycleLength (nC i) mem schedule (nC j)"
proj_Eps: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and X :: "'a set" assumes "equiv A r"   and "X \<in> A // r" shows "proj r (SOME x. x \<in> X) = X"
prems_3: fixes x :: "'a"   and it :: "'a set"   and \<sigma> :: "'b" assumes "x \<in> it"   and "it \<subseteq> set (x__ # S)"   and "I it \<sigma>" shows "I (it - {x}) (f \<sigma> x)"
cmult_Nil: fixes c :: "'a" shows "c %* [] = []"
assms_1: fixes x :: "'a" assumes "x \<in> carrier T" shows "f x \<in> carrier S"
ta_upd_proj_simps_4: fixes ta :: "('b, 'a, 'c, 'd, 'e, 'f) thread_action"   and obs :: "'f" shows "\<lbrace>ta_update_obs ta obs\<rbrace>\<^bsub>c\<^esub> = \<lbrace>ta\<rbrace>\<^bsub>c\<^esub>"
id: shows "of_int_poly H = of_int_poly F div of_int_poly G"
assms_3: shows "wf\<^sub>t\<^sub>r\<^sub>m t"
subspaceI: fixes S :: "'b poly set" assumes "0 \<in> S"   and "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x + y \<in> S"   and "\<And>c x. x \<in> S \<Longrightarrow> smult c x \<in> S" shows "poly_vs.subspace S"
inject_4: fixes x41 :: "char list list"   and x42 :: "nat"   and y41 :: "char list list"   and y42 :: "nat" shows "(StepT x41 x42 = StepT y41 y42) = (x41 = y41 \<and> x42 = y42)"
assms: shows "G \<subseteq> dgrad_p_set d m"
terminating_path_iff: fixes x :: "'a" shows "terminating_path x = (backward_terminating_path x \<and> forward_terminating_path x)"
None_in_Inf: fixes A :: "'b option set" assumes "None \<in> A" shows "\<Sqinter> A = None"
suminf_le: fixes f :: "nat \<Rightarrow> 'c"   and g :: "nat \<Rightarrow> 'c" assumes "\<And>n. f n \<le> g n"   and "summable f"   and "summable g" shows "suminf f \<le> suminf g"
P2_a: fixes x :: "'a"   and y :: "'a" assumes "x l\<rightarrow> y = (1::'a)"   and "y l\<rightarrow> x = (1::'a)" shows "x = y"
hext_trans: fixes h :: "heap"   and h' :: "heap"   and h'' :: "heap" assumes "h \<unlhd> h'"   and "h' \<unlhd> h''" shows "h \<unlhd> h''"
primepow_odd_imp_primepow: fixes n :: "nat" assumes "primepow_odd n" shows "primepow n"
Cycle_def: fixes L :: "'c list set" shows "Cycle L = {u @ w |u w. w @ u \<in> L}"
reinterpret: fixes t1 :: "t"   and v :: "v"   and t2 :: "t" assumes "types_agree t1 v" shows "\<lparr>[$C v, $Cvtop t2 Reinterpret t1 None]\<rparr> \<leadsto> \<lparr>[$C wasm_deserialise (bits v) t2]\<rparr>"
convert_term_simps_3: fixes t :: "'d"   and u :: "'d" shows "convert_term (app t u) = app (convert_term t) (convert_term u)"
dghm_ArrMap_app_in_HomCod_Arr: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>" shows "\<HH>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"
Cons_2: shows "generalized_sfw (a # as) p = Some (r1, d1)"
rewrite_negated_src_ports_def: fixes m :: "'b common_primitive match_expr" shows "rewrite_negated_src_ports m \<equiv> rewrite_negated_primitives (is_Src_Ports, src_ports_sel) Src_Ports l4_ports_negate_one m"
infty_plus_def: fixes pl :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and a :: "'b infty"   and b :: "'c infty" shows "infty_plus pl a b \<equiv> case (a, b) of (infty.Infty, ba) \<Rightarrow> infty.Infty | (Num a, infty.Infty) \<Rightarrow> infty.Infty | (Num a, Num b) \<Rightarrow> Num (pl a b)"
preserves_reflects_arr: fixes f :: "'a" shows "S.arr (S.src f) = S.arr f"
assms_1: shows "naturally_isomorphic (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F G"
states_Power_dfa: fixes M :: "'b nfa" assumes "nfa M" shows "dfa.states (nfa.Power_dfa M) = HF ` nfa.epsclo M ` Pow (nfa.states M)"
cPar1B: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi"   and C :: "'a" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'"   and "x \<sharp> Q"   and "\<And>C. F C P a x P'" shows "F C (P \<parallel> Q) a x (P' \<parallel> Q)"
substAbs_vsubstAbs_compose2: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and Z :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var"   and y1 :: "'var" assumes "goodAbs A"   and "good Z"   and "ys \<noteq> zs \<or> y \<noteq> z"   and "fresh ys y Z" shows "A $[y1 // y]_ys $[Z / z]_zs = A $[Z / z]_zs $[Var ys y1 #[Z / z]_zs / y]_ys"
abs_def: fixes R :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and Abs :: "'g set \<Rightarrow> 'h"   and Rep :: "'h \<Rightarrow> 'g set"   and x :: "'g" assumes "quot_type R Abs Rep" shows "quot_type.abs R Abs x = Abs (Collect (R x))"
assms: fixes S :: "'a set"   and T :: "'a set" assumes "connected S"   and "connected T"   and "U = S \<union> T"   and "closedin (top_of_set U) S"   and "closedin (top_of_set U) T" shows "connected (S \<inter> T)"
mpoly_monom_0_eq_Const: fixes c :: "'b" shows "MPoly_Type.monom 0 c = Const c"
poly_times_exp_tendsto_zero: fixes b :: "'a"   and k :: "nat" assumes "norm b < 1" shows "(\<lambda>x. of_nat x ^ k * b ^ x) \<longlonglongrightarrow> (0::'a)"
strict_trans2: fixes b :: "'e word"   and a :: "'e word"   and c :: "'e word" assumes "b <s a"   and "c \<le>s b" shows "c <s a"
wpt_def: fixes c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "wp\<^sub>t c Q = (\<lambda>s. \<exists>t. (c, s) \<Rightarrow> t \<and> Q t)"
ct_suffixE: fixes xs :: "ct list"   and ys :: "ct list" assumes "ct_suffix xs ys" obtains as :: "ct list" and zs :: "ct list" where "ys = as @ zs" and "ct_list_eq zs xs"
well_defI: fixes prog :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "healthy (wp prog)"   and "nearly_healthy (wlp prog)"   and "wp_under_wlp prog"   and "sub_distrib_pconj prog"   and "sublinear (wp prog)"   and "bd_cts (wp prog)" shows "well_def prog"
absorb1: fixes a :: "'a word"   and b :: "'a word" assumes "a \<le>s b" shows "signed.min a b = a"
left_neutral: fixes a :: "('a \<times> ('b \<times> 'a) list) set" shows "t_prod t_one a = a"
simps_2: fixes h :: "'b \<times> 'a"   and t :: "('b \<times> 'a) list"   and l :: "'b" shows "(h # t)\<down>l = (if fst h = l then Some (snd h) else t\<down>l)"
HRP_graph_aux_def: shows "HRP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>HRP_aux s x x' k. x1 = (x, x') \<and> x2 = SyntaxN.Ex s (SyntaxN.Ex k (SeqHRP x x' (Var s) (Var k))) \<and> atom s \<sharp> (x, x', k) \<and> atom k \<sharp> (x, x'))"
IH: shows "delete\<cdot>a\<cdot>xs = filter\<cdot>(\<Lambda> x. neq\<cdot>a\<cdot>x)\<cdot>xs"
dim_def: shows "dim = (LEAST n. \<exists>A. finite A \<and> card A = n \<and> A \<subseteq> carrier V \<and> gen_set A)"
mem_existence_ivl_iv_defined_1: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl0 x0" shows "0 \<in> UNIV"
simps_2: shows "local.map (J.MkIde True) = a1"
pr0: shows "\<guillemotleft>pr0 (cod f) (cod g) : local.prod (cod f) (cod g) \<rightarrow> cod g\<guillemotright>"
principalideal_axioms_def: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" shows "principalideal_axioms I R \<equiv> \<exists>i\<in>carrier R. I = Idl\<^bsub>R\<^esub> {i}"
prefix_length_le: fixes xs :: "'e list"   and ys :: "'e list" assumes "prefix xs ys" shows "length xs \<le> length ys"
added_ids_Deliver_Create_diff_collapse: fixes e :: "'b"   and e' :: "'b"   and i :: "'c"   and j :: "'c" assumes "e \<noteq> e'" shows "added_ids [Deliver (i, Create j e)] e' = []"
hta_ensure_idx_f_correct: shows "hashedTa (hta_ensure_idx_f H)"