hcju/isabelleps huggingface dataset:
connectedin_def: fixes X :: "'c topology"   and S :: "'c set" shows "connectedin X S \<equiv> S \<subseteq> topspace X \<and> connected_space (subtopology X S)"
ign_subst_res: fixes P :: "'b assn"   and w :: "vals" shows "P\<down>\<leftarrow>w = P\<down>"
ring_axioms: shows "class.ring (+) (0::'a) (-) uminus (*)"
f10: fixes F :: "nat \<Rightarrow> 'a poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'a"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'a" assumes "primitive_remainder_sequence F n \<delta> f k \<beta>" shows "f 1 \<noteq> (0::'a)"
to_tree_order_child: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and nodes :: "(_) object_ptr list"   and parent :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and child :: "(_) node_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.to_tree_order ptr \<rightarrow>\<^sub>r nodes"   and "h \<turnstile> Shadow_DOM.get_child_nodes parent \<rightarrow>\<^sub>r children"   and "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child \<noteq> ptr"   and "child \<in> set children"   and "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child \<in> set nodes" shows "parent \<in> set nodes"
ND0aux1: fixes x :: "'a"   and y :: "'a"   and b :: "('a, 'b) Combinators list" assumes "DenyAllFromTo x y \<in> set b" shows "x \<in> set (net_list_aux b)"
induct: fixes P :: "iface \<Rightarrow> iface \<Rightarrow> bool"   and a0 :: "iface"   and a1 :: "iface" assumes "\<And>uu_. P (Iface []) (Iface uu_)"   and "\<And>v va. P (Iface (v # va)) (Iface [])"   and "\<And>a as b bs. (a = b \<Longrightarrow> P (Iface as) (Iface bs)) \<Longrightarrow> P (Iface (a # as)) (Iface (b # bs))" shows "P a0 a1"
admissible_kbo_axioms: shows "admissible_kbo w w0 pr_strict' pr_weak' least scf"
partn_lst_imp_partn_lst_VWF_eq: fixes r :: "('a \<times> 'a) set"   and U :: "'a set"   and \<alpha> :: "V list"   and n :: "nat"   and \<beta> :: "V" assumes "Partitions.partn_lst r U \<alpha> n"   and "ordertype U r = \<beta>"   and "small U"   and "wf r"   and "trans r"   and "total_on U r" shows "partn_lst_VWF \<beta> \<alpha> n"
exec': fixes ts :: "'t list"   and f :: "'op" assumes "\<forall>s\<in>set ts. inv' s"   and "length ts = arity f" shows "hom (exec' f ts) = exec f (map hom ts)"
singular_relboundary_chain_map: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and X' :: "'c topology"   and g :: "'b \<Rightarrow> 'c"   and T :: "'c set" assumes "singular_relboundary p X S c"   and "continuous_map X X' g"   and "g ` S \<subseteq> T" shows "singular_relboundary p X' T (chain_map p g c)"
Int_insert_right: fixes A :: "'c set"   and a :: "'c"   and B :: "'c set" shows "A \<inter> insert a B = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"
upset_to_downset: shows "(`) \<partial> \<circ> \<up> = \<down> \<circ> \<partial>"
add_zmset_diff_bothsides: fixes a :: "'b"   and M :: "'b zmultiset"   and A :: "'b zmultiset" shows "add_zmset a M - add_zmset a A = M - A"
is_FunE: fixes t :: "('a, 'b) Term.term" assumes "is_Fun t" obtains f :: "'a" and ts :: "('a, 'b) Term.term list" where "t = Fun f ts"
infinite_growing: fixes X :: "'a set" assumes "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> \<exists>y\<in>X. x \<prec> y" shows "infinite X"
subseqs_distinctD: fixes ys :: "'b list"   and xs :: "'b list" assumes "ys \<in> set (subseqs xs)"   and "distinct xs" shows "distinct ys"
assms_3: shows "u \<in> ball a r \<inter> T"
ell_field_def: fixes R :: "('c, 'd) ring_scheme" shows "ell_field R \<equiv> field R \<and> ell_field_axioms R"
scaleR_eq_0_iff: fixes a :: "real"   and x :: "'b" shows "(a *\<^sub>R x = (0::'b)) = (a = 0 \<or> x = (0::'b))"
CONS_surj: fixes \<AA> :: "'i"   and idx :: "'n" assumes "Length \<AA> \<noteq> 0"   and "#\<^sub>V \<AA> = idx" shows "\<exists>x \<BB>. \<AA> = CONS x \<BB> \<and> #\<^sub>V \<BB> = idx \<and> size x = idx"
Fpow_def: fixes A :: "'b set" shows "Fpow A \<equiv> {X. X \<subseteq> A \<and> finite X}"
multiset_nonemptyE: fixes A :: "'b multiset" assumes "A \<noteq> {#}" obtains x :: "'b" where "x \<in># A"
ccloses_elim: fixes a :: "coname"   and B :: "ty"   and \<Delta> :: "(coname \<times> ty) list"   and \<theta>_c :: "(coname \<times> name \<times> trm) list" assumes "(a, B) \<in> set \<Delta>"   and "\<theta>_c ccloses \<Delta>" shows "\<exists>x P. \<theta>_c cmaps a to Some (x, P) \<and> x:P \<in> \<parallel>(B)\<parallel>"
a_notin_sb: shows "a \<notin> outstanding_refs is_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"
set_mset_empty: shows "set_mset (\<Empt>) = {}"
cert: shows "cert_ok c (length ins) \<top> \<bottom> A"
in_obs_ids_is_kth_obs: fixes i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n" assumes "i \<in> obs_ids \<pi>" obtains k :: "nat" where "is_kth_obs \<pi> k i"
assms_3: shows "\<forall>x\<in>U\<^sub>2. f x \<in> U"
Diff_single_insert: fixes A :: "'b set"   and x :: "'b"   and B :: "'b set" assumes "A - {x} \<subseteq> B" shows "A \<subseteq> insert x B"
leg0_composite_2: shows "HHfgh.dom.leg0 = fgh.\<pi>.leg0 \<cdot> fgh.Prj\<^sub>0"
IH: fixes A :: "'a list" assumes "set A \<subseteq> set B"   and "distinct A"   and "distinct B" shows "length A \<le> length B"
tr_par_preserves_typing_cond: fixes A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and A' :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list" assumes "par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t A Sec"   and "typing_cond\<^sub>s\<^sub>s\<^sub>t (unlabel A)"   and "A' \<in> set (tr\<^sub>p\<^sub>c A [])" shows "typing_cond (unlabel A')"
map\<^sub>0_def: fixes a :: "'b" shows "G.map\<^sub>0 a \<equiv> src\<^sub>D (G a)"
ge: shows "p \<ge>p poly_subst (\<lambda>w. poly_of (if w = v then PSum [PNum (1::'a), PVar v] else PVar w)) p"
resultant_const_2: fixes p :: "'b poly"   and a :: "'b" shows "resultant p [:a:] = a ^ degree p"
has_density_emeasure_space': fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "has_density M N f" shows "emeasure (density N f) (space (density N f)) = integral\<^sup>N N f"
Subring_one_ring_one: fixes S :: "('a, 'd) Ring_scheme" assumes "Subring R S" shows "1\<^sub>r\<^bsub>S\<^esub> = 1\<^sub>r"
IP: fixes \<sigma> :: "'\<sigma>"   and k :: "'u"   and it :: "'u set"   and v :: "'v" assumes "c \<sigma>"   and "k \<in> it"   and "\<alpha> m k = Some v"   and "it \<subseteq> dom (\<alpha> m)"   and "I it \<sigma>" shows "I (it - {k}) (f (k, v) \<sigma>)"
mem_ordinal_number_equiv: fixes D :: "'c Order"   and X :: "'c Order" assumes "Worder D"   and "X \<in> ordinal_number D" shows "ord_equiv X D"
iIN_iTILL_subset: fixes n' :: "nat"   and d' :: "nat"   and n :: "nat" assumes "n' + d' \<le> n" shows "[n'\<dots>,d'] \<subseteq> [\<dots>n]"
add_less_same_cancel2: fixes a :: "'a multiset"   and b :: "'a multiset" shows "(a + b \<subset># b) = (a \<subset># {#})"
prj_simps_4: shows "C.arr src_src.prj\<^sub>0"
path_trans: fixes l1 :: "'loc"   and l2 :: "'loc"   and xs :: "('loc \<times> 'sum \<times> 'loc) list"   and l3 :: "'loc"   and ys :: "('loc \<times> 'sum \<times> 'loc) list" assumes "path l1 l2 xs"   and "path l2 l3 ys" shows "path l1 l3 (xs @ ys)"
add_decreasing2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq c zero"   and "less_eq a b" shows "less_eq (plus a c) b"
assms_2: shows "s\<^sub>1 \<leadsto> s\<^sub>2'"
a: assumes "k < j" shows "(X \<in> set (inner G T i (k + 1) j)) = (\<exists>l\<ge>k + 1. l < j \<and> X \<in> set (match_prods G (T (i, l)) (T (i + l, j - l))))"
nullhomotopic_through_contractible: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and U :: "'d set" assumes "continuous_on S f"   and "f ` S \<subseteq> T"   and "continuous_on T g"   and "g ` T \<subseteq> U"   and "contractible T" obtains c :: "'d" where "homotopic_with_canon (\<lambda>h. True) S U (g \<circ> f) (\<lambda>x. c)"
a_l_coset_subset_G: fixes H :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "x <+ H \<subseteq> carrier G"
less_top: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and top :: "'a"   and a :: "'a" assumes "class.order_top less_eq less top" shows "(a \<noteq> top) = less a top"
goodPar_def: fixes P :: "('d, 'e, 'varSort, 'var, 'f) param" shows "goodPar P \<equiv> Ball (termsOf P) good \<and> Ball (absOf P) goodAbs \<and> Ball (envsOf P) goodEnv"
take_all_iff: fixes n :: "nat"   and xs :: "'g list" shows "(take n xs = xs) = (length xs \<le> n)"
pt_set_eq_ineq: fixes pi :: "('b \<times> 'b) list"   and P :: "'a \<Rightarrow> bool" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "{pi \<bullet> x |x. P x} = {x. P (rev pi \<bullet> x)}"
mtf2_def: fixes n :: "nat"   and x :: "'c"   and xs :: "'c list" shows "mtf2 n x xs = (if x \<in> set xs then swaps [index xs x - n..<index xs x] xs else xs)"
empty_iff: fixes c :: "'ap" shows "(c \<in> {}) = False"
image_filter_cong: fixes M :: "'b multiset"   and P :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>C. \<lbrakk>C \<in># M; P C\<rbrakk> \<Longrightarrow> f C = g C" shows "image_mset f (filter_mset P M) = image_mset g (filter_mset P M)"
ereal_max: fixes x :: "real"   and y :: "real" shows "ereal (max x y) = max (ereal x) (ereal y)"
orbit_apply_power: fixes f :: "'a perm"   and n :: "nat"   and a :: "'a" shows "orbit f ((f ^ n) \<langle>$\<rangle> a) = orbit f a"
insert_def: shows "RBT.insert \<equiv> map_fun id (map_fun id (map_fun rbt.impl_of RBT)) rbt_insert"
ordinal_util_imp_rat_prefs: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set"   and u :: "'a \<Rightarrow> real" assumes "ordinal_utility carrier relation u" shows "rational_preference carrier relation"
commute: fixes a :: "'b"   and b :: "'b" shows "a \<squnion> b = b \<squnion> a"
disjoint_family_Suc: fixes A :: "nat \<Rightarrow> 'b set" assumes "\<And>n. A n \<subseteq> A (Suc n)" shows "disjoint_family (\<lambda>i. A (Suc i) - A i)"
is_glb_def: fixes s :: "val option"   and s1 :: "val option"   and s2 :: "val option" shows "is_glb s s1 s2 \<equiv> is_lb s s1 s2 \<and> (\<forall>s'. is_lb s' s1 s2 \<longrightarrow> s' \<preceq> s)"
mult_rel_aux_def: fixes r :: "'a"   and s :: "'a"   and r' :: "'a"   and s' :: "'a" shows "mult_rel_aux r s r' s' \<equiv> (r \<cdot> r') / (s \<cdot> s')"
assm: shows "N_Chop(i,i1,i2) \<and> N_Chop(i2,i3,i4)"
enabled_def: fixes r :: "rule"   and s :: "fmla fset \<times> fmla fset" shows "enabled r s \<equiv> \<exists>sl. eff' r s = Some sl"
assms: shows "\<exists>U. U \<in> sets (stream_space borel) \<and> U \<inter> range X = {r}"
L4_2c: fixes \<sigma> :: "nat \<Rightarrow> 'a node"   and \<mu> :: "'a ltlr"   and \<eta> :: "'a ltlr" assumes "ipath gba.E \<sigma>"   and "\<mu> R\<^sub>r \<eta> \<in> old (\<sigma> 0)" shows "\<forall>i. \<eta> \<in> old (\<sigma> i) \<or> (\<exists>j<i. \<mu> \<in> old (\<sigma> j))"
Ints_odd_nonzero: fixes a :: "'a" assumes "a \<in> \<int>" shows "(1::'a) + a + a \<noteq> (0::'a)"
41: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> (x \<sqinter> y) = bot"
simps_2: fixes b :: "nat list"   and ys :: "nat list"   and x :: "nat"   and xs :: "nat list"   and s :: "nat" shows "cond1 b ys (x # xs) s = (s \<le> b \<bullet> ys \<and> x \<le> maxne0_impl ys b)"
neg1: fixes A :: "'d"   and k :: "'d"   and a :: "'d" assumes "A \<equiv> k + a" shows "- A \<equiv> - k + - a"
ub_def: shows "ub = (\<lambda>x. x powr (1 - \<alpha>) / \<alpha> + c2 * x powr - \<alpha> + \<bar>Ra x\<bar>)"
ih_14: shows "InvariantNoDecisionsWhenUnit (getF state') (getM state') (currentLevel (getM state'))"
card_Un_disjoint: fixes A :: "'g set"   and B :: "'g set" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "card (A \<union> B) = card A + card B"
Ifh: fixes t :: "real"   and s :: "real" assumes "dist x y \<le> t"   and "t \<le> s"   and "s \<le> dist x z" shows "dist (f s) (f t) = s - t"
decides_subsets_subset: fixes \<F> :: "nat set set"   and M :: "nat set"   and N :: "nat set" assumes "decides_subsets \<F> M"   and "N \<subseteq> M" shows "decides_subsets \<F> N"
comm_ring_isom_def: fixes hom :: "'c \<Rightarrow> 'd" shows "comm_ring_isom hom \<equiv> ring_isom hom"
add_diff_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "c + (b - a) = c + b - a"
lift_pre_var: fixes x :: "'b \<Longrightarrow> 'c" shows "\<lceil>x\<rceil>\<^sub>< = $x"
extf_compat_list: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and y :: "('b, 'a) tm"   and x :: "('b, 'a) tm"   and gt :: "('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool"   and f :: "'b"   and xs :: "('b, 'a) tm list"   and xs' :: "('b, 'a) tm list" assumes "rpo_basis ground_heads_var gt_sym arity_sym arity_var extf"   and "y \<noteq> x"   and "gt y x" shows "extf f gt (xs @ y # xs') (xs @ x # xs')"
filterlim_inverse_at_infinity: shows "filterlim inverse at_infinity (at (0::'c))"
app\<^sub>i_Putstatic: fixes C :: "char list"   and F :: "char list"   and D :: "char list"   and P :: "'a prog"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and T :: "ty"   and ST :: "ty list"   and LT :: "ty err list" shows "app\<^sub>i (Putstatic C F D, P, pc, mxs, T\<^sub>r, T # ST, LT) = (\<exists>T\<^sub>f. P \<turnstile> C sees F,Static:T\<^sub>f in D \<and> subtype P T T\<^sub>f)"
lift_Def_High: fixes V :: "'f"   and H :: "'f set"   and E :: "'g"   and Def :: "'g \<Rightarrow> 'f set"   and X :: "'g"   and L :: "'f set" assumes "V \<in> H" shows "(Node E, V) \<in> lift_Def_set Def E X H L"
iT_Minus_mono: fixes A :: "nat set"   and B :: "nat set"   and k :: "nat" assumes "A \<subseteq> B" shows "k \<ominus> A \<subseteq> k \<ominus> B"
Solution_imp_cond_D: fixes x :: "nat list"   and y :: "nat list" assumes "(x, y) \<in> local.Solutions" shows "cond_D x y"
noStuck_startn: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"   and "t \<noteq> Stuck" shows "s \<noteq> Stuck"
rt_fresherI2: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dip \<in> kD rt1"   and "dip \<in> kD rt2"   and "nsqn rt1 dip = nsqn rt2 dip"   and "the (dhops rt2 dip) \<le> the (dhops rt1 dip)" shows "rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2"
v11_def: fixes r :: "V" shows "v11 r \<equiv> vsv r \<and> v11_axioms r"
discr0_MtransC_indis: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "('b, 'c) com"   and s :: "'a"   and c' :: "('b, 'c) com"   and s' :: "'a" assumes "PL_Indis indis"   and "PL_Indis.discr0 tval aval indis c"   and "PL.mustT tval aval c s"   and "PL.MtransC tval aval (c, s) (c', s')" shows "(s, s') \<in> indis"
positively_invariant_restrict_dom: fixes M :: "'a set" shows "positively_invariant M = positively_invariant (M \<inter> X)"
stop_rel: fixes mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat" shows "(\<langle>Stop, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>C\<rangle>\<^sub>A, \<langle>Stop, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2"
pi_foldri: fixes l :: "'b list" shows "proper_it (foldri l) (foldri l)"
S5n_kripke: fixes M :: "('e, 'f, 'g) KripkeStructure" assumes "S5n M" shows "kripke M"
simps: fixes ord :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "'b option"   and a2 :: "'c option" shows "ord_option ord a1 a2 = ((\<exists>x. a1 = None \<and> a2 = x) \<or> (\<exists>x y. a1 = Some x \<and> a2 = Some y \<and> ord x y))"
WT_gpv_PauseI: fixes out :: "'d"   and \<Gamma> :: "('d, 'e) \<I>"   and c :: "('f, 'd, 'e) rpv" assumes "out \<in> outs_\<I> \<Gamma>"   and "\<And>input. input \<in> responses_\<I> \<Gamma> out \<Longrightarrow> \<Gamma> \<turnstile>g c input \<surd>" shows "\<Gamma> \<turnstile>g Generative_Probabilistic_Value.Pause out c \<surd>"
bezout_matrix_works1: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and a :: "'c"   and b :: "'c"   and A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and j :: "'d" assumes "is_bezout_ext bezout"   and "a \<noteq> b" shows "(bezout_matrix A a b j bezout ** A) $h a $h j = snd (snd (snd (snd (bezout (A $h a $h j) (A $h b $h j)))))"
simps_5: fixes v :: "'b formula"   and va :: "'b formula" shows "is_cnf (v \<^bold>\<or> va) = is_disj (v \<^bold>\<or> va)"
orthogonal_imp_lin_indpt_list: fixes gs :: "'a vec list" assumes "orthogonal gs"   and "set gs \<subseteq> carrier_vec n" shows "lin_indpt_list gs"
plossless_exec_gpv: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and s :: "('a, 'b) resource" assumes "plossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>s x. \<lbrakk>x \<in> outs_\<I> \<I>; lossless_resource \<I> s \<and> \<I> \<turnstile>res s \<surd>\<rbrakk> \<Longrightarrow> lossless_spmf (run_resource s x)"   and "lossless_resource \<I> s \<and> \<I> \<turnstile>res s \<surd>" shows "lossless_spmf (exec_gpv run_resource gpv s)"
sub: fixes \<Gamma>\<^sub>1 :: "'Var \<Rightarrow> Sec option"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>\<^sub>1' :: "'Var \<Rightarrow> Sec option"   and \<Gamma>\<^sub>2 :: "'Var \<Rightarrow> Sec option"   and \<Gamma>\<^sub>2' :: "'Var \<Rightarrow> Sec option" assumes "\<turnstile> \<Gamma>\<^sub>1 {c} \<Gamma>\<^sub>1'"   and "\<Gamma>\<^sub>2 \<sqsubseteq>\<^sub>c \<Gamma>\<^sub>1"   and "\<Gamma>\<^sub>1' \<sqsubseteq>\<^sub>c \<Gamma>\<^sub>2'" shows "\<turnstile> \<Gamma>\<^sub>2 {c} \<Gamma>\<^sub>2'"
RETURN_SPEC_conv: fixes r :: "'b" shows "RETURN r = SPEC (\<lambda>x. x = r)"
groupcastT: fixes \<xi> :: "'e"   and l :: "'h"   and s\<^sub>i\<^sub>p\<^sub>s :: "'e \<Rightarrow> nat set"   and s\<^sub>m\<^sub>s\<^sub>g :: "'e \<Rightarrow> 'f"   and p :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" shows "((\<xi>, {l}groupcast(s\<^sub>i\<^sub>p\<^sub>s, s\<^sub>m\<^sub>s\<^sub>g) . p), groupcast (s\<^sub>i\<^sub>p\<^sub>s \<xi>) (s\<^sub>m\<^sub>s\<^sub>g \<xi>), \<xi>, p) \<in> seqp_sos \<Gamma>"
iaN: fixes p :: "nat"   and a :: "(real \<times> real pdevs) list"   and b :: "(real \<times> real pdevs) list" shows "inner_aforms' p a b \<noteq> None"
dom_eq_singleton_conv: fixes f :: "'a \<Rightarrow> 'b option"   and x :: "'a" shows "(dom f = {x}) = (\<exists>v. f = [x \<mapsto> v])"
runit_simps_3: fixes f :: "'a" assumes "ide f" shows "trg \<r>[f] = trg f"
Ap_reduce1: fixes x :: "comb"   and y :: "comb"   and z :: "comb" assumes "x \<rightarrow> y" shows "x \<bullet> z \<rightarrow> y \<bullet> z"
lang_inorm: fixes \<Sigma> :: "'a list"   and n :: "nat"   and r :: "'a atom rexp" assumes "M2L_Equivalence_Checking.wf \<Sigma> n r" shows "lang \<Sigma> n (inorm r) = lang \<Sigma> n r"
LID\<^sub>LI_3: fixes s :: "('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and \<tau> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and e :: "('b, 'a, 'c) expr"   and r :: "'a" assumes "s = (\<sigma>, \<tau>, e)"   and "r \<in> LID\<^sub>E e" shows "r \<in> LID\<^sub>L s"
ide_char: fixes a :: "'j" shows "J.ide a = (J.arr a \<and> J.dom a = a \<and> J.cod a = a)"
that: fixes c :: "'a"   and d :: "'a" assumes "\<not> P (cbox c d)"   and "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i \<le> c \<bullet> i \<and> c \<bullet> i \<le> d \<bullet> i \<and> d \<bullet> i \<le> b \<bullet> i \<and> 2 * (d \<bullet> i - c \<bullet> i) \<le> b \<bullet> i - a \<bullet> i" shows "thesis"
name_list_avoiding: fixes xvec :: "name list"   and c :: "'c" obtains pi :: "(name \<times> name) list" where "(pi \<bullet> xvec) \<sharp>* c" and "distinctPerm pi" and "set pi \<subseteq> set xvec \<times> set (pi \<bullet> xvec)"
concat_eq_Nil_conv: fixes xss :: "'a list list" shows "(concat xss = []) = (\<forall>xs\<in>set xss. xs = [])"
two_not_zero: 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" assumes "class.ell_field divide inverse times one plus zero minus uminus" shows "numeral.numeral one plus (num.Bit0 num.One) \<noteq> zero"
kcomp_def: fixes F :: "'b \<Rightarrow> 'd set"   and G :: "'d \<Rightarrow> 'c set" shows "F ; G = \<mu> \<circ> \<P> G \<circ> F"
wf_prog_def2: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog" shows "wf_prog wf_md P = (wf_syscls P \<and> (\<forall>C rest. class P C = \<lfloor>rest\<rfloor> \<longrightarrow> wf_cdecl wf_md P (C, rest)) \<and> distinct_fst (classes P))"
in_keys_iff: fixes x :: "'b"   and s :: "'b \<Rightarrow>\<^sub>0 'c" shows "(x \<in> Poly_Mapping.keys s) = (poly_mapping.lookup s x \<noteq> (0::'c))"
prj_simps_10: shows "C.arr Prj\<^sub>1"
axioms_1: fixes Cn :: "'b set \<Rightarrow> 'b set"   and contraction :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "AGMC_S Cn contraction true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "AGM_Contraction Cn contraction"
is_scc_connected: fixes E :: "('a \<times> 'a) set"   and U :: "'a set"   and x :: "'a"   and y :: "'a" assumes "is_scc E U"   and "x \<in> U"   and "y \<in> U" shows "(x, y) \<in> E\<^sup>*"
power_down_eq_zero_iff: fixes prec :: "nat"   and b :: "real"   and n :: "nat" shows "(power_down prec b n = 0) = (b = 0 \<and> n \<noteq> 0)"
rank_take_cols_leq: fixes R :: "'a mat"   and nc :: "nat"   and ls :: "nat list" assumes "R \<in> carrier_mat n nc" shows "rank (take_cols R ls) \<le> rank R"
content_ball_conv_unit_ball: fixes r :: "real"   and c :: "'b" assumes "0 \<le> r" shows "content (ball c r) = r ^ DIM('b) * content (ball (0::'b) 1)"
Cons_shuffles_subset2: fixes y :: "'g"   and xs :: "'g list"   and ys :: "'g list" shows "(#) y ` shuffles xs ys \<subseteq> shuffles xs (y # ys)"
take_bit_mask: fixes m :: "nat"   and n :: "nat" shows "take_bit m (mask n) = mask (min m n)"
bchoice: fixes S :: "'d set"   and Q :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<forall>x\<in>S. \<exists>y. Q x y" shows "\<exists>f. \<forall>x\<in>S. Q x (f x)"
close_world_extensive: fixes M :: "object atom formula set" shows "M \<subseteq> close_world M"