hcju/isabelleps huggingface dataset:
exec_step_ind_Return_NonStatic: fixes D :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "jvm_method"   and P :: "jvm_prog"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and h :: "heap"   and stk\<^sub>0 :: "val list"   and loc\<^sub>0 :: "val list"   and pc :: "nat"   and ics :: "init_call_status"   and stk' :: "val list"   and loc' :: "val list"   and C' :: "char list"   and m' :: "char list"   and pc' :: "nat"   and ics' :: "init_call_status"   and frs' :: "frame list"   and sh :: "sheap" assumes "(D, NonStatic, Ts, T, m) = method P C\<^sub>0 M\<^sub>0"   and "M\<^sub>0 \<noteq> clinit" shows "exec_step_ind (StepI Return) P h stk\<^sub>0 loc\<^sub>0 C\<^sub>0 M\<^sub>0 pc ics ((stk', loc', C', m', pc', ics') # frs') sh (None, h, (hd stk\<^sub>0 # drop (length Ts + 1) stk', loc', C', m', Suc pc', ics') # frs', sh)"
distinct_49: fixes x51 :: "('d, 'f) ODE"   and x52 :: "('d, 'e, 'f) formula"   and x8 :: "('d, 'e, 'f) hp" shows "EvolveODE x51 x52 \<noteq> x8**"
tab_simps_2: shows "src rs.tab = src (tab\<^sub>0 (r \<star> s))"
B: fixes u :: "complex" assumes "u \<in> cball z r" shows "cmod (f u) \<le> B"
integral_combine_division_topdown: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and \<D> :: "'a set set" assumes "f integrable_on S"   and "\<D> division_of S" shows "integral S f = (\<Sum>i\<in>\<D>. integral i f)"
normalize_length_le: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list" assumes "ring R" shows "length (ring.normalize R p) \<le> length p"
causal_network_def: fixes history :: "nat \<Rightarrow> 'c event list"   and msg_id :: "'c \<Rightarrow> 'd" shows "causal_network history msg_id \<equiv> network history msg_id \<and> causal_network_axioms history"
dense: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.dense_order less_eq less"   and "less x y" shows "\<exists>z. less x z \<and> less z y"
assms_4: shows "r'' \<notin> RID\<^sub>G s\<^sub>2"
maxR_range: fixes x :: "'a"   and y :: "'a" shows "MaxR_empty.maxR x y \<in> {x, y}"
eq2: shows "snd \<circ> spp_of = rep_list"
continuous_within_subset: fixes x :: "'b"   and s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and t :: "'b set" assumes "continuous (at x within s) f"   and "t \<subseteq> s" shows "continuous (at x within t) f"
env_delete_below_arg: fixes x :: "'a"   and m :: "'a \<Rightarrow> 'b" shows "env_delete x m \<sqsubseteq> m"
remove_neighbor_pres_dis_invar: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and u :: "nat"   and fo :: "nat \<times> nat \<Rightarrow> 'a"   and lo :: "nat \<Rightarrow> nat"   and no :: "nat \<Rightarrow> nat set"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and n :: "nat \<Rightarrow> nat set"   and v :: "nat" assumes "discharge_invar c s t u fo lo no f l n"   and "(u, v) \<notin> Network.adm_edges c f l" shows "discharge_invar c s t u fo lo no f l (n(u := n u - {v}))"
elims: fixes x :: "'a"   and xa :: "'a list"   and y :: "'a list set" assumes "insert_in_list_at_arbitrary_pos x xa = y" obtains xb :: "'a" where "x = xb__" and "xa = []" and "y = {[xb__]}"   | xb :: "'a" and ya :: "'a" and ys :: "'a list" where "x = xb" and "xa = ya # ys" and "y = insert (xb # ya # ys) ((#) ya ` insert_in_list_at_arbitrary_pos xb ys)"
idem_normalize: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and top :: "'a"   and bot :: "'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "bounded_quasi_semilattice f top bot normalize" shows "f a a = normalize a"
empty_sb_sims: fixes ts\<^sub>s\<^sub>b :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and ts\<^sub>h :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m\<^sub>h :: "nat \<Rightarrow> nat"   and \<S>\<^sub>h :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option" assumes "\<forall>i p is xs sb \<D> \<O> \<R>. i < length ts\<^sub>s\<^sub>b \<longrightarrow> ts\<^sub>s\<^sub>b ! i = (p, is, xs, sb, \<D>, \<O>, \<R>) \<longrightarrow> sb = []"   and "ts\<^sub>s\<^sub>b \<sim>\<^sub>h ts\<^sub>h "   and "(ts\<^sub>h, m\<^sub>h, \<S>\<^sub>h) \<sim> (ts, m, \<S>)" shows "ts\<^sub>s\<^sub>b \<sim>\<^sub>d ts  \<and> m\<^sub>h = m \<and> length ts\<^sub>s\<^sub>b = length ts"
bisim_inv_def: shows "bisim_inv \<equiv> (\<forall>s1 ta1 s1' s2 t. t \<turnstile> s1 \<approx> s2 \<longrightarrow> t \<turnstile> s1 -1-ta1\<rightarrow> s1' \<longrightarrow> (\<exists>s2'. t \<turnstile> s1' \<approx> s2')) \<and> (\<forall>s2 ta2 s2' s1 t. t \<turnstile> s1 \<approx> s2 \<longrightarrow> t \<turnstile> s2 -2-ta2\<rightarrow> s2' \<longrightarrow> (\<exists>s1'. t \<turnstile> s1' \<approx> s2'))"
image_add_atLeastAtMost: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and k :: "'b"   and i :: "'b"   and j :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "plus k ` ord.atLeastAtMost less_eq i j = ord.atLeastAtMost less_eq (plus i k) (plus j k)"
hom_one: fixes Ra :: "('e, 'f) ring_scheme"   and Sa :: "('g, 'h) ring_scheme"   and ha :: "'e \<Rightarrow> 'g"   and h :: "'i \<Rightarrow> 'j"   and R :: "('i, 'k) ring_scheme"   and S :: "('j, 'l) ring_scheme" assumes "ring_hom_ring Ra Sa ha"   and "h \<in> ring_hom R S" shows "h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>"
Diff_Int: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" shows "A - B \<inter> C = A - B \<union> (A - C)"
cnj_add: fixes x :: "quat"   and y :: "quat" shows "Quaternions.cnj (x + y) = Quaternions.cnj x + Quaternions.cnj y"
box01_nonempty: shows "box (0::'a) One \<noteq> {}"
seq_bot_right: fixes c :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice) ; \<bottom> \<sqsubseteq> c"
map_update: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and k :: "nat"   and y :: "'d" shows "map f (xs[k := y]) = (map f xs)[k := f y]"
chamber_system_def: shows "folding_g.\<C> \<equiv> Collect folding_g.chamber"
sorted_poly_list_rel_empty_l: fixes s' :: "(char list multiset \<times> int) multiset"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and T :: "('a \<times> char list multiset) set" shows "(([], s') \<in> sorted_poly_list_rel_wrt S T) = (s' = {#})"
learn_consI: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "environment \<psi> U s"   and "\<And>f. f \<in> U \<Longrightarrow> \<exists>i. \<psi> i = f \<and> (\<exists>n\<^sub>0. \<forall>n\<ge>n\<^sub>0. s (f \<triangleright> n) \<down>= i)"   and "\<And>f n. f \<in> U \<Longrightarrow> \<forall>k\<le>n. \<psi> (the (s (f \<triangleright> n))) k = f k" shows "learn_cons \<psi> U s"
open_continuous_vimage: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and B :: "'b set" assumes "open s"   and "continuous_on s f"   and "open B" shows "open (f -` B \<inter> s)"
set_decode_inverse: fixes n :: "nat" shows "set_encode (set_decode n) = n"
no_pd_path: fixes x :: "'n"   and y :: "'n" assumes "x \<in> local.nodes"   and "\<not> y pd\<rightarrow> x" obtains \<pi> :: "nat \<Rightarrow> 'n" and n :: "nat" where "is_path \<pi>" and "\<pi> 0 = x" and "\<pi> n = local.return" and "\<forall>k\<le>n. \<pi> k \<noteq> y"
cnj_upper_circle_01: shows "cnj ` upper_circle_01 = lower_circle_01"
resDerivative: fixes P :: "pi"   and Q :: "pi"   and a :: "subject"   and x :: "name"   and Rel :: "(pi \<times> pi) set"   and Rel' :: "(pi \<times> pi) set"   and y :: "name" assumes "derivative P Q a x Rel"   and "\<And>P Q x. (P, Q) \<in> Rel \<Longrightarrow> (<\<nu>x>P, <\<nu>x>Q) \<in> Rel'"   and "eqvt Rel" shows "derivative (<\<nu>y>P) (<\<nu>y>Q) a x Rel'"
linear_homeomorphic_image: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "linear f"   and "inj f" shows "S homeomorphic f ` S"
subcls: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog" assumes "G\<turnstile>C\<preceq>\<^sub>C D" shows "G\<turnstile>Class D\<succ>Class C"
SL_def: fixes p :: "real"   and A :: "'a set" shows "random_skip_list.SL p A = Pi_pmf A 0 (\<lambda>_. geometric_pmf p)"
k_prop: shows "\<forall>y<k. (b - c) * y < y\<^sup>2"
bounded_antilinear_intro: fixes f :: "'d \<Rightarrow> 'e"   and K :: "real" assumes "\<And>x y. f (x + y) = f x + f y"   and "\<And>r x. f (r *\<^sub>C x) = cnj r *\<^sub>C f x"   and "\<And>x. norm (f x) \<le> norm x * K" shows "bounded_antilinear f"
stable_translation_length_eq_additive_strength: fixes f :: "'b \<Rightarrow> 'b"   and xi :: "'b Gromov_completion" assumes "isometry f"   and "Gromov_extension f xi = xi" shows "stable_translation_length f = \<bar>additive_strength f xi\<bar>"
induct: fixes P :: "'b list \<Rightarrow> bool"   and a0 :: "'b list" assumes "P []"   and "\<And>a. P [a]"   and "\<And>a b as. P (b # as) \<Longrightarrow> P (a # b # as)" shows "P a0"
that_2: assumes "\<exists>a b. a \<in> carrier R \<and> b \<in> carrier R \<and> a \<otimes> b \<in> I \<and> a \<notin> I \<and> b \<notin> I" shows "thesis"
inject_2: fixes x2 :: "'d"   and x1 :: "('d, 'e, 'f) boundOutput"   and y2 :: "'d"   and y1 :: "('d, 'e, 'f) boundOutput" shows "(ROut x2 x1 = ROut y2 y1) = (x2 = y2 \<and> x1 = y1)"
tinfinite_ex: fixes V :: "Preliminaries.var set"   and \<sigma> :: "'tp" assumes "tinfinite V" shows "\<exists>x\<in>V. tpOfV x = \<sigma>"
ereal_less_divide_pos: fixes x :: "ereal"   and y :: "ereal"   and z :: "ereal" assumes "0 < x"   and "x \<noteq> \<infinity>" shows "(y < z / x) = (x * y < z)"
sum_list_prod_map_aezfun_apply: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> ('a, 'b) aezfun"   and xys :: "('c \<times> 'd) list"   and a :: "'b" shows "aezfun (\<Sum>(x, y)\<leftarrow>xys. f x y) a = (\<Sum>(x, y)\<leftarrow>xys. aezfun (f x y) a)"
Nil_is_map_conv: fixes f :: "'h \<Rightarrow> 'g"   and xs :: "'h list" shows "([] = map f xs) = (xs = [])"
gen_iapath_rev_path: fixes V :: "'a set"   and v :: "'a"   and p :: "('a \<times> 'a) list"   and u :: "'a" shows "gen_iapath V v (rev_path p) u = gen_iapath V u p v"
nat_pow_closed: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e"   and n :: "nat" assumes "Group.monoid G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> n \<in> carrier G"
foldr_Nil: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" shows "foldr f [] = id"
csidi: shows "s\<inverse> O d\<inverse> \<subseteq> d\<inverse>"
fresh_permute_left: fixes a :: "atom"   and p :: "perm"   and x :: "'a" shows "a \<sharp> p \<bullet> x = - p \<bullet> a \<sharp> x"
inv_swap_id: fixes a :: "'b"   and b :: "'b" shows "inv (Fun.swap a b id) = Fun.swap a b id"
Bloch_lemma: fixes f :: "complex \<Rightarrow> complex"   and a :: "complex"   and r :: "real" assumes "f holomorphic_on cball a r"   and "0 < r"   and "\<And>z. z \<in> ball a r \<Longrightarrow> cmod (deriv f z) \<le> 2 * cmod (deriv f a)" shows "ball (f a) ((3 - 2 * sqrt 2) * r * cmod (deriv f a)) \<subseteq> f ` ball a r"
complex_conjugate_root: fixes p :: "complex poly"   and c :: "complex" assumes "set (coeffs p) \<subseteq> \<real>"   and "poly p c = 0" shows "poly p (cnj c) = 0"
in_angle_two_sides: fixes B :: "'p"   and A :: "'p"   and P :: "'p"   and C :: "'p" assumes "\<not> Col B A P"   and "\<not> Col B C P"   and "P InAngle A B C" shows "P B TS A C"
derivation_ge_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and i :: "nat" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "derivation_ge D i = (\<forall>d\<in>set D. i \<le> fst d)"
has_derivative_unique: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and x :: "'c"   and F' :: "'c \<Rightarrow> 'd" assumes "(f has_derivative F) (at x)"   and "(f has_derivative F') (at x)" shows "F = F'"
assms: shows "\<lbrakk>M; map (\<lambda>d. \<forall>X\<langle>\<or>\<noteq>: [(pair (t, s), pair (snd d))]\<rangle>\<^sub>s\<^sub>t) (filter (\<lambda>d. d \<notin> set Di) D)\<rbrakk>\<^sub>d \<I>"
lzip_lappend: fixes xs :: "'c llist"   and us :: "'d llist"   and ys :: "'c llist"   and vs :: "'d llist" assumes "llength xs = llength us" shows "lzip (lappend xs ys) (lappend us vs) = lappend (lzip xs us) (lzip ys vs)"
the_elem: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and f' :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "f \<in> A \<rightarrow>\<^sub>E B"   and "f' \<in> A \<rightarrow>\<^sub>E B"   and "(\<lambda>b. {x \<in> A. f x = b}) ` B - {{}} = (\<lambda>b. {x \<in> A. f' x = b}) ` B - {{}}"   and "x \<in> A" shows "the_elem (f ` {xa \<in> A. f' xa = f' x}) = f x"
assms_2: fixes t :: "complex" assumes "t \<in> closed_segment 0 u" shows "cmod (poly f t) \<le> C"
run_ask_nondet: fixes ask :: "('f \<Rightarrow> 'd) \<Rightarrow> 'd"   and f :: "'f \<Rightarrow> ('e, 'd) nondetT" shows "run_nondet (ask_nondet ask f) = ask (\<lambda>r. run_nondet (f r))"
axioms_2: fixes pi1 :: "(name \<times> name) list"   and pi2 :: "(name \<times> name) list"   and x :: "'a" shows "(pi1 @ pi2) \<bullet> x = pi1 \<bullet> pi2 \<bullet> x"
IZ_asubst: fixes i :: "int"   and ks :: "int list"   and a :: "atom"   and xs :: "int list" shows "I\<^sub>Z (asubst i ks a) xs = I\<^sub>Z a ((i + \<langle>ks,xs\<rangle>) # xs)"
R22_1: shows "pmf (sds R22) a = 1 / 2"
fib_wrap_def: shows "fib_wrap \<equiv> wrapS\<cdot>fib_work"
dom12_d_U: fixes x :: "('a \<times> 'a set) set" shows "mrel_c_lattice.d x = 1\<^sub>\<sigma> \<inter> (x \<cdot> 1\<^sub>\<pi> \<parallel> Multirelations.U)"
ed_ou_witness_pow: fixes X :: "ed_ou_witness set" shows "X \<in> set ` set (subseqs [ed_ou_witness.a, ed_ou_witness.b, ed_ou_witness.c, d, e])"
measurable_cong: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and M' :: "'e measure" assumes "\<And>w. w \<in> space M \<Longrightarrow> f w = g w" shows "(f \<in> M \<rightarrow>\<^sub>M M') = (g \<in> M \<rightarrow>\<^sub>M M')"
enat_of_option_simps_2: fixes x :: "nat option" shows "(Common.enat_of_option x = \<infinity>) = (x = None)"
SemFalse: fixes b :: "BExpr"   and s :: "Var \<Rightarrow> Val"   and c2 :: "IMP"   and n :: "nat"   and t :: "Var \<Rightarrow> Val"   and c1 :: "IMP" assumes "\<not> evalB b s"   and " s , c2 \<rightarrow>\<^sub>n  t " shows " s , Iff b c1 c2 \<rightarrow>\<^sub>n + 1  t "
homeomorphic_maps_def: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" shows "homeomorphic_maps X Y f g \<equiv> continuous_map X Y f \<and> continuous_map Y X g \<and> (\<forall>x\<in>topspace X. g (f x) = x) \<and> (\<forall>y\<in>topspace Y. f (g y) = y)"
smult_vec_def: fixes a :: "'b"   and v :: "'b Matrix.vec" shows "a \<cdot>\<^sub>v v \<equiv> Matrix.vec (dim_vec v) (\<lambda>i. a * v $v i)"
Abs_Exp_inverse: fixes y :: "freeExp set" assumes "y \<in> Exp" shows "Rep_Exp (Abs_Exp y) = y"
residuated_iff_conjugate: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" shows "residuated_pair f g = conjugation_pair f (\<lambda>x. - g (- x))"
cv_part: fixes X :: "'a set"   and n :: "nat"   and i :: "nat"   and a :: "V set" assumes "\<forall>x\<in>[X]\<^bsup>n\<^esup>. f (cv x) = i"   and "X \<subseteq> U"   and "a \<in> [cv X]\<^bsup>n\<^esup>" shows "f a = i"
divide_nonpos_neg: 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 less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq x zero"   and "less y zero" shows "less_eq zero (divide x y)"
abs_eq: fixes xa :: "'c regular \<times> 'c dense Filters.filter"   and x :: "'c regular \<times> 'c dense Filters.filter" assumes "eq_onp (\<lambda>x. x \<in> stone_phi.pairs) xa xa"   and "eq_onp (\<lambda>x. x \<in> stone_phi.pairs) x x" shows "Abs_stone_phi_pair xa \<sqinter> Abs_stone_phi_pair x = Abs_stone_phi_pair (stone_phi.pairs_inf xa x)"
diff_single_trivial: fixes x :: "'a"   and M :: "'a multiset" assumes "x \<notin># M" shows "M - {#x#} = M"
cbb: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and e :: "('b \<times> 'b) set"   and m :: "('b \<times> 'b) set"   and b :: "('b \<times> 'b) set"   and ov :: "('b \<times> 'b) set"   and d :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and f :: "('b \<times> 'b) set" assumes "class.arelations meets \<I> e m b ov d s f" shows "b O b \<subseteq> b"
chine_assoc'_props_4: shows "Prj\<^sub>0 \<cdot> chine_assoc' = Prj\<^sub>0\<^sub>0"
assms_2: fixes x :: "'fun" shows "rel_option (\<lambda>ar fd. arity fd = ar) (get_arity x) (F x)"
psimps: fixes data :: "nat \<times> nat \<times> 'd"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" assumes "gb_schema_aux_dom (data, bs, ps)" shows "gb_schema_aux data bs ps = (if ps = [] then gs @ bs else let sps = sel gs bs ps (snd data); ps0 = ps -- sps; aux = compl gs bs ps0 sps (snd data); remcomps = fst data - count_const_lt_components (fst aux) in if remcomps = 0 then full_gb (gs @ bs) else let (hs, data') = add_indices aux (snd data) in gb_schema_aux (remcomps, data') (ab gs bs hs data') (ap gs bs ps0 hs data'))"
graph_more_rel_sv: fixes Rm :: "('a \<times> 'b) set"   and Rv :: "('c \<times> 'd) set"   and Rw :: "('e \<times> 'f) set" assumes "single_valued Rm"   and "single_valued Rv"   and "single_valued Rw" shows "single_valued (\<langle>Rm, Rv, Rw\<rangle>graph_more_rel)"
Stone_Weierstrass_uniform_limit: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "compact S"   and "continuous_on S f" obtains g :: "nat \<Rightarrow> 'b \<Rightarrow> 'c" where "uniform_limit S g f sequentially" and "\<And>n. polynomial_function (g n)"
execute_upd_1: fixes i :: "nat"   and h :: "heap"   and a :: "'a array"   and x :: "'a" assumes "i < Array.length h a" shows "execute (Array.upd i x a) h = Some (a, Array.update a i x h)"
tanh_real_altdef: fixes x :: "real" shows "tanh x = (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))"
init: fixes m0 :: "'c nres"   and X :: "'c set"   and x :: "'c"   and cond :: "'c \<Rightarrow> bool"   and step :: "'c \<Rightarrow> 'c nres" assumes "m0 = RES X"   and "x \<in> X" shows "rwof m0 cond step x"
poly_pos: fixes p :: "real poly" shows "(\<forall>x. 0 < poly p x) = (p \<noteq> 0 \<and> poly_inf p = 1 \<and> count_roots p = 0)"
wt: shows "P,E,h \<turnstile> {V:T; e} : T'"
sets_2: fixes x :: "'a" assumes "x \<in> A" shows "{x} \<in> sets N"
divide_prod_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x / (y * z) = x / y / z"
chop_assoc2: fixes i :: "nat_int"   and i1 :: "nat_int"   and i2 :: "nat_int"   and i3 :: "nat_int"   and i4 :: "nat_int" shows "N_Chop(i,i1,i2) \<and> N_Chop(i1,i3,i4) \<longrightarrow> N_Chop(i,i3,i4 \<squnion> i2) \<and> N_Chop(i4 \<squnion> i2,i4,i2)"
not_list_ex_equals_list_all_not: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "(\<not> list_ex P xs) = list_all (\<lambda>x. \<not> P x) xs"
LIM_zero: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and F :: "'a filter" assumes "(f \<longlongrightarrow> l) F" shows "((\<lambda>x. f x - l) \<longlongrightarrow> (0::'b)) F"
env_C_restr_upd: fixes r :: "C"   and \<rho> :: "'a \<Rightarrow> C \<rightarrow> 'b"   and x :: "'a"   and v :: "C \<rightarrow> 'b" shows "\<rho>(x := v)|\<^sup>\<circ>\<^bsub>r\<^esub> = (\<rho>|\<^sup>\<circ>\<^bsub>r\<^esub>)(x := v|\<^bsub>r\<^esub>)"
poly_of_vec_scalar_mult: fixes b :: "'a poly"   and n :: "nat"   and a :: "'a" assumes "degree b < n" shows "poly_of_vec (a \<cdot>\<^sub>v vec_of_poly_n b n) = smult a b"
vec_deriv_i_comp_cont: shows "continuous_on {0..1} (\<lambda>x. vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1}))"
eventually_imp_iff: fixes P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(\<forall>\<^sub>F x in \<U>. P x \<longrightarrow> Q x) = (eventually P \<U> \<longrightarrow> eventually Q \<U>)"
assms_3: shows "fps_nth G 0 \<noteq> (0::'a)"
1: shows "Poly_Mapping.range r0 \<subseteq> punit_dgrad_max_set d"
wf_unknown_match_tac_def: fixes \<alpha> :: "action \<Rightarrow> 'b \<Rightarrow> bool" shows "wf_unknown_match_tac \<alpha> \<equiv> \<alpha> Drop = \<alpha> Reject"
Sync_\<tau>execI: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and V :: "nat"   and e' :: "'addr expr1" assumes "\<tau>exec_move ci P t e h s s'" shows "\<tau>exec_move ci P t (sync\<^bsub>V\<^esub> (e) e') h s s'"
schroeder_4_p: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x * y \<le> - z) = (z * y\<^sup>T \<le> - x)"
SN_onI: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "\<And>f. \<lbrakk>f 0 \<in> A; \<forall>i. (f i, f (Suc i)) \<in> r\<rbrakk> \<Longrightarrow> False" shows "SN_on r A"
mem_Collect_eq: fixes a :: "'a"   and P :: "'a \<Rightarrow> bool" shows "s2p (Collect P) a = P a"
bind_def2: fixes m :: "'f\<cdot>'e"   and k :: "'f \<rightarrow> 'd\<cdot>'e" shows "m \<bind> k = join\<cdot>(fmap\<cdot>k\<cdot>m)"
case_split: fixes P :: "bool" obtains "P"   | "\<not> P"
nres_rel_def: fixes R :: "('a \<times> 'c) set" shows "\<langle>R\<rangle>nres_rel \<equiv> {(c, a). c \<le> \<Down> R a}"
ptd_lookup_unchanged_2: fixes va :: "word32"   and ptp :: "word32"   and s1 :: "('b, 'c) sparc_state_scheme"   and addr :: "word36"   and val :: "word8" shows "ptd_lookup va ptp (mem s1) 2 = ptd_lookup va ptp ((mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))) 2"
prems: shows "m' \<in> set (map (\<lambda>spt. MatchAnd (Match (C spt)) ms) (f as))"
bij_betw_iff_bijections: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set" shows "bij_betw f A B = (\<exists>g. (\<forall>x\<in>A. f x \<in> B \<and> g (f x) = x) \<and> (\<forall>y\<in>B. g y \<in> A \<and> f (g y) = y))"
eq: shows "h\<^sub>0 ++ h\<^sub>1 = h\<^sub>0' ++ h\<^sub>1'"
d: fixes p :: "'b"   and q :: "'b" shows "((p, q) \<in> d) = (\<exists>k l u v. k \<parallel> l \<and> l \<parallel> p \<and> p \<parallel> u \<and> u \<parallel> v \<and> k \<parallel> q \<and> q \<parallel> v)"
assms_1: shows "\<not> i < Suc (length x + length y)"
absorb2: fixes b :: "'b \<times> 'c"   and a :: "'b \<times> 'c" assumes "less_eq_prod b a" shows "inf_prod a b = b"
subs_1: shows "{Inl x |x. x \<in> SIGT t1} \<subseteq> {Inl x |x. x \<in> SIGT (Times t1 t2)}"
connectedin_topspace: fixes X :: "'a topology" shows "connectedin X (topspace X) = connected_space X"
star_star_absorb: shows "top \<sqinter> top \<sqinter> top = top \<sqinter> top"
times_preal_def: fixes x :: "preal"   and y :: "preal" shows "x * y \<equiv> preal_plus x y"
inf1D1: fixes A :: "'f \<Rightarrow> bool"   and B :: "'f \<Rightarrow> bool"   and x :: "'f" assumes "inf A B x" shows "A x"
simps_2: fixes x :: "char list"   and e :: "aexp"   and P :: "'a" shows "strip (x ::= e {P}) = x ::= e"