hcju/isabelleps huggingface dataset:
thmD13_part1: fixes k :: "nat"   and u :: "nat" assumes "items_le k (\<J> k 0) = Gen (paths_le k (\<P> k 0))"   and "k \<le> length Doc" shows "items_le k (\<J> k u) = Gen (paths_le k (\<P> k u)) \<and> \<T> k u = \<Z> k u"
indep_vars_compose: fixes M' :: "'e \<Rightarrow> 'f measure"   and X :: "'e \<Rightarrow> 'a \<Rightarrow> 'f"   and I :: "'e set"   and Y :: "'e \<Rightarrow> 'f \<Rightarrow> 'g"   and N :: "'e \<Rightarrow> 'g measure" assumes "indep_vars M' X I"   and "\<And>i. i \<in> I \<Longrightarrow> Y i \<in> M' i \<rightarrow>\<^sub>M N i" shows "indep_vars N (\<lambda>i. Y i \<circ> X i) I"
no\<phi>_def: fixes istate :: "'a"   and step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and \<phi> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and T :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and K :: "'a \<Rightarrow> bool" shows "BD_Security.no\<phi> istate step \<phi> T K \<equiv> \<forall>s a ou s'. BD_Security.reachNT istate step T s \<and> K s \<and> step s a = (ou, s') \<longrightarrow> \<not> \<phi> (Trans s a ou s')"
ACI_norm_alt: fixes r :: "'c rexp" shows "ACI_norm_alt r = \<guillemotleft>r\<guillemotright>"
cong: fixes nfv :: "nat"   and fv :: "nat set"   and sat :: "'c trace \<Rightarrow> 'd list \<Rightarrow> nat \<Rightarrow> bool"   and progress :: "'c prefix \<Rightarrow> nat" shows "progress.M nfv fv sat progress \<equiv> progress.M nfv fv sat progress"
inf_sup_aci_1: fixes inf :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and sup :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and x :: "'k"   and y :: "'k" assumes "class.lattice inf less_eq less sup" shows "inf x y = inf y x"
assms_2: fixes a :: "'a"   and b :: "'b" assumes "a \<in> set1_FGl x"   and "b \<in> set1_FGl y"   and "L1 a b" shows "L1' a b"
positive_proj_k: fixes k :: "nat" shows "positive (proj_k k)"
fg_local_def: fixes A :: "('f, 'g, 'h, 'i) CHOAlgorithm_scheme"   and p :: "'f"   and HO :: "'f set"   and crd :: "'f"   and config :: "('g, 'f, 'h, 'j) config_scheme"   and config' :: "('g, 'f, 'h, 'j) config_scheme" shows "fg_local A p HO crd config config' \<equiv> sent config p = UNIV \<and> dom (rcvd config p) = HO \<and> (\<exists>s. CnextState A (round config p) p (state config p) (rcvd config p) crd s \<and> config' = config\<lparr>round := (round config)(p := Suc (round config p)), state := (state config)(p := s), sent := (sent config)(p := {}), rcvd := (rcvd config)(p := Map.empty)\<rparr>)"
finite_subset: fixes A :: "'g set"   and B :: "'g set" assumes "A \<subseteq> B"   and "finite B" shows "finite A"
GSiKnormGSi: shows "\<GG> ! (i + 1) \<inter> K \<lhd> G\<lparr>carrier := \<GG> ! (i + 1)\<rparr>"
oInv_bounds: fixes a :: "ordinal" assumes "F 0 \<le> a" shows "F (oInv F a) \<le> a \<and> a < F (oSuc (oInv F a))"
rel_interior_rel_interior: fixes S :: "'a set" assumes "convex S" shows "rel_interior (rel_interior S) = rel_interior S"
unit_char_2: fixes a :: "'c" assumes "C.obj a" shows "D.iso (F.unit a)"
fold_graph_OR: fixes \<Phi> :: "'b pltl set" assumes "finite \<Phi>" shows "fold_graph Or_ltlp false\<^sub>p \<Phi> (OR \<Phi>)"
gamma_i_component_smooth: shows "\<forall>x\<in>s. (\<lambda>x. \<gamma> x \<bullet> i) differentiable at x"
assoc: fixes a :: "'f list"   and b :: "'f list"   and c :: "'f list" shows "(a @ b) @ c = a @ b @ c"
wrapS_unwrapS_id': fixes f :: "nat discr Box \<rightarrow> 'a"   and n :: "nat discr Box" assumes "f\<cdot>\<bottom> = \<bottom>" shows "unwrapS\<cdot>f !! n = f\<cdot>n"
select_convs_4: fixes orderA :: "('b, 'c) gorder_scheme"   and orderB :: "('a, 'd) gorder_scheme"   and lower :: "'b \<Rightarrow> 'a"   and upper :: "'a \<Rightarrow> 'b"   and more :: "'e" shows "\<pi>\<^sub>*\<^bsub>\<lparr>orderA = orderA, orderB = orderB, lower = lower, upper = upper, \<dots> = more\<rparr>\<^esub> = upper"
17: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<squnion> z \<sqinter> y = (x \<squnion> z) \<sqinter> y"
Eadd_prop: shows "E' \<union> Eadd = E"
1: shows "\<guillemotleft>\<phi> : k \<star> p\<^sub>1 \<Rightarrow> h \<star> p\<^sub>0\<guillemotright>"
eval_minus: fixes F :: "'a"   and G :: "'a" shows "eval (F - G) = (\<lambda>x. eval F x - eval G x)"
normalize_ratfps_def: fixes x :: "'a ratfps" shows "normalize x = (if x = 0 then 0 else ratfps_of_poly (monom (1::'a) (ratfps_subdegree x)))"
Kcknows_3: shows "toSet aec = agent_abs a t"
suminf_def: fixes f :: "nat \<Rightarrow> 'a" shows "suminf f = (THE s. f sums s)"
zero_less_one: fixes one :: "'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.zero_less_one one zero less_eq less" shows "less zero one"
unsynlr_inf: fixes x :: "('a, 'b) synlr"   and y :: "('a, 'b) synlr" shows "unsynlr (inf x y) = unsynlr x \<inter> unsynlr y"
inverse_nonpositive_iff_nonpositive: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (inverse a) zero = less_eq a zero"
vertical_chop_add2: fixes v :: "view"   and x :: "nat"   and y :: "nat" shows "|lan v| = x + y \<longrightarrow> (\<exists>u w. v=u--w \<and> |lan u| = x \<and> |lan w| = y)"
narrow1_acom: fixes C2 :: "'a acom"   and C1 :: "'a acom" assumes "C2 \<le> C1" shows "C2 \<le> C1 \<triangle> C2"
Nat2nat_SucNat: fixes N :: "ZF" assumes "Elem N HOLZF.Nat" shows "Nat2nat (SucNat N) = Suc (Nat2nat N)"
sep_substate_def: fixes x :: "'a"   and y :: "'a" shows "x \<preceq> y \<equiv> \<exists>z. x ## z \<and> x + z = y"
simps_19: fixes f1 :: "bool \<Rightarrow> nat \<Rightarrow> nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and f2 :: "bool \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b"   and f3 :: "'c \<Rightarrow> 'c \<Rightarrow> instr list \<Rightarrow> 'b"   and f4 :: "nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and x31 :: "'c"   and x32 :: "'c"   and x33 :: "instr list" shows "(case Prog\<^sub>s\<^sub>b x31 x32 x33 of Write\<^sub>s\<^sub>b x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Read\<^sub>s\<^sub>b x xa xb xc \<Rightarrow> f2 x xa xb xc | Prog\<^sub>s\<^sub>b x xa xb \<Rightarrow> f3 x xa xb | Ghost\<^sub>s\<^sub>b x xa xb xc \<Rightarrow> f4 x xa xb xc) = f3 x31 x32 x33"
borel_measurable_ereal: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
req_free_theorem: fixes xs :: "'a list"   and ys :: "'a list" shows "(case xs @ ys of [] \<Rightarrow> Zero | x # xa \<Rightarrow> One) = (if (case xs of [] \<Rightarrow> Zero | x # xa \<Rightarrow> One) = One \<or> (case ys of [] \<Rightarrow> Zero | x # xa \<Rightarrow> One) = One then One else Zero)"
pt_set_bij: fixes pi :: "('e \<times> 'e) list"   and x :: "'d"   and X :: "'d set" assumes "pt TYPE('d) TYPE('e)"   and "at TYPE('e)" shows "(pi \<bullet> x \<in> pi \<bullet> X) = (x \<in> X)"
isnpolyh_polybound0: fixes p :: "poly"   and n :: "nat" assumes "isnpolyh p (Suc n)" shows "polybound0 p"
wt_subst: fixes \<rho> :: "'v \<Rightarrow> ('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "wary_subst \<rho>"   and "wary s" shows "local.kbo_std.wt (subst \<rho> s) = local.kbo_std.wt s + local.kbo_std.extra_wt \<rho> s"
add_mset_lt_le_lt: fixes a :: "'b"   and b :: "'b"   and A :: "'b multiset"   and B :: "'b multiset" assumes "a \<le> b"   and "A < B" shows "add_mset a A < add_mset b B"
simps_1: fixes do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and st :: "'a" shows "GM_sec_model.\<delta> do (Idle st) = {Idle st} \<union> {State (do st u' c') u' c' |u' c'. True}"
Un_Diff_cancel2: fixes B :: "'g set"   and A :: "'g set" shows "B - A \<union> A = B \<union> A"
Infinitesimal_scaleR2: fixes x :: "'a star"   and a :: "real" assumes "x \<in> Infinitesimal" shows "a *\<^sub>R x \<in> Infinitesimal"
alist_product_def: fixes xs :: "('a \<times> 'b) list"   and ys :: "('c \<times> 'd) list" shows "local.alist_product xs ys = concat (map (\<lambda>(a, b). map (\<lambda>(c, d). ((a, c), f a b c d)) ys) xs)"
Not2I: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set" shows "F \<triangleright> \<Gamma> \<turnstile> \<^bold>\<not> (\<^bold>\<not> F)"
y_gt_x: shows "y \<succeq> x \<and> (x, y) \<notin> relation"
k_invariantTOD: fixes ec :: "'rep"   and ec' :: "'rep"   and A :: "('ma, 'mt) AlgState"   and obs :: "'obs" assumes "k_isNode ec"   and "k_isNode ec'"   and "simAbs ec' = simAbs ec"   and "k_invariant A" shows "lookup tOps (aTrans A) (ec, obs) = lookup tOps (aTrans A) (ec', obs)"
name_swap: fixes a :: "name"   and b :: "name"   and x :: "'d" shows "[(a, b)] \<bullet> x = [(b, a)] \<bullet> x"
sig_trd_red_rtrancl: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b" shows "(sig_red (\<prec>\<^sub>t) (\<preceq>) (set bs))\<^sup>*\<^sup>* p (sig_trd bs p)"
restrict_untouched_id: fixes x :: "'b"   and t :: "'b ifex"   and val :: "bool" assumes "x \<notin> ifex_var_set t" shows "restrict t x val = t"
subset_antisym: fixes A :: "'a set"   and B :: "'a set" assumes "order.greater_eq B A"   and "order.greater_eq A B" shows "A = B"
\<nu>_in_hom_2: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c"   and u :: "'c"   and w :: "'c"   and \<theta> :: "'c"   and \<omega> :: "'c"   and \<nu> :: "'c" assumes "uw\<theta>\<omega>\<nu> V H \<a> \<i> src trg r \<rho> f g u w \<theta> \<omega> \<nu>" shows "partial_magma.in_hom V \<nu> (partial_magma.dom V \<omega>) (H g w)"
a1: shows "\<forall>s. P s \<longrightarrow> P' (s(v := e s))"
hyps_5: assumes "cs' \<noteq> []"   and "upd_cs cs' as = []"   and "\<forall>xs ys. as = xs @ ys \<and> ys \<noteq> [] \<longrightarrow> upd_cs cs' xs \<noteq> []" shows "last as \<in> get_return_edges (last cs')"
cases: fixes a1 :: "'a list"   and a2 :: "'a list"   and a3 :: "'a list" assumes "appending a1 a2 a3" obtains ys :: "'a list" where "a1 = []" and "a2 = ys__" and "a3 = ys__"   | xs :: "'a list" and ys :: "'a list" and zs :: "'a list" and x :: "'a" where "a1 = x # xs" and "a2 = ys" and "a3 = x # zs" and "appending xs ys zs"
SUP_bot_conv_2: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and B :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(bot = Sup (B ` A)) = (\<forall>x\<in>A. B x = bot)"
int_f_eq_y: shows "integral\<^sup>N M f = Sup (integral\<^sup>N M ` G)"
FAssThrow1: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and F :: "char list"   and D :: "char list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
funion_fempty_right: fixes A :: "'c fset" shows "A |\<union>| {||} = A"
CollectI: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "(P::'b \<Rightarrow> bool) (a::'b)" shows "(a::'b) \<in> {x::'b. (P::'b \<Rightarrow> bool) x}"
simps_1: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and a :: "('a, 'b) rbt"   and y :: "'a"   and s :: "'b"   and lt :: "('a, 'b) rbt"   and z :: "'a"   and v :: "'b"   and rt :: "('a, 'b) rbt" shows "ord.rbt_del_from_right less x a y s (Branch B lt z v rt) = balance_right a y s (ord.rbt_del less x (Branch B lt z v rt))"
assms_1: shows "p\<^sub>1 = Point x\<^sub>1 y\<^sub>1"
Mismatch_1: fixes P :: "pi"   and Rs :: "residual"   and a :: "name"   and c :: "name" assumes "P \<Longrightarrow>\<^sub>l Rs"   and "a \<noteq> c" shows "[a\<noteq>c]P \<Longrightarrow>\<^sub>l Rs"
finsum_Un_disjoint: fixes A :: "'e set"   and B :: "'e set"   and g :: "'e \<Rightarrow> 'c" assumes "finite A"   and "finite B"   and "A \<inter> B = {}"   and "g \<in> A \<rightarrow> carrier M"   and "g \<in> B \<rightarrow> carrier M" shows "finsum M g (A \<union> B) = finsum M g A \<oplus>\<^bsub>M\<^esub> finsum M g B"
InitState_def: shows "InitState = fst \<circ> snd \<circ> Rep_seqauto"
simps_2: fixes queue :: "'a list"   and t :: "'a"   and b :: "bool" shows "Round_Robin.enqueue_new_thread queue (ThreadExists t b) = queue"
removeKeyList_subset: fixes ws :: "'a list"   and ps :: "('a \<times> 'b) list" shows "set (removeKeyList ws ps) \<subseteq> set ps"
cod_dom: fixes f :: "'a" assumes "L'.C.arr f" shows "L'.C.cod (L'.C.dom f) = L'.C.dom f"
bi_unique_Union_r: fixes T :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and K :: "'e set set"   and xr :: "'f set" assumes "bi_unique T"   and "\<Union> K \<subseteq> Collect (Domainp T)"   and "rel_set T (\<Union> K) xr" shows "\<Union> ({(x, y). rel_set T x y} `` K) = xr"
emeasure_PiM_Collect_single: fixes i :: "'i"   and A :: "'a set" assumes "i \<in> I"   and "A \<in> M.events i" shows "emeasure (Pi\<^sub>M I M) {x \<in> space (Pi\<^sub>M I M). x i \<in> A} = emeasure (M i) A"
prop_atoms_entailment_inter: fixes \<phi> :: "'a ltln"   and P :: "'a ltln set"   and \<A> :: "'a ltln set" assumes "prop_atoms \<phi> \<subseteq> P" shows "(\<A> \<inter> P) \<Turnstile>\<^sub>P \<phi> = \<A> \<Turnstile>\<^sub>P \<phi>"
that_1: shows "linear_on S T (*\<^sub>R) (*\<^sub>R) f"
map_def: fixes f :: "'a" shows "I.map f = (if arr f then \<I> else null)"
divide_dimvec_def: fixes x :: "('a, 'b) dimvec"   and y :: "('a, 'b) dimvec" shows "x / y = x \<cdot> inverse y"
finite_dom_rbt_lookup: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "('c, 'd) rbt" assumes "class.linorder less_eq less" shows "finite (dom (ord.rbt_lookup less t))"
kdt: shows "k dvd \<lfloor>Inum (b' # bs) t\<rfloor>"
c_tl_aux3: fixes u :: "nat" shows "c_tl u = sgn1 (c_len u - 1) * (c_pair (c_len u - 2) (c_snd (c_snd (u - 1))) + 1)"
comp_converter_id_left: fixes conv :: "('p, 'q, 'r, 's) converter" shows "1\<^sub>C \<odot> conv = conv"
SCallInitThrow: fixes P :: "(char list list \<times> char list exp) prog"   and ps :: "char list exp list"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and vs :: "val list"   and h\<^sub>1 :: "heap"   and l\<^sub>1 :: "sfields"   and sh\<^sub>1 :: "sheap"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp"   and D :: "char list"   and a :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>ps,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>map Val vs,(h\<^sub>1, l\<^sub>1, sh\<^sub>1)\<rangle>"   and "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = (pns, body) in D"   and "\<nexists>sfs. sh\<^sub>1 D = \<lfloor>(sfs, Done)\<rfloor>"   and "M \<noteq> clinit"   and "P \<turnstile> \<langle>INIT D ([D],False) \<leftarrow> unit,(h\<^sub>1, l\<^sub>1, sh\<^sub>1)\<rangle> \<Rightarrow> \<langle>throw a,s'\<rangle>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(ps),s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw a,s'\<rangle>"
obtains_recognises: fixes s :: "nat"   and c :: "nat \<Rightarrow>f value option"   and e :: "((nat \<times> nat) \<times> transition) fset"   and s' :: "nat"   and r :: "nat \<Rightarrow>f value option"   and t :: "(String.literal \<times> value list) list" assumes "obtains s c e s' r t" shows "recognises_execution e s' r t"
map_tailrec_def: fixes f :: "'e \<Rightarrow> 'd"   and as :: "'e list" shows "map_tailrec f as = rev (map_tailrec_rev f as [])"
except_Diff_singleton: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a" shows "except p (keys p - {t}) = monomial (lookup p t) t"
fun_upd_triv: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d" shows "f(x := f x) = f"
infsetsum_Sigma: fixes f :: "'c \<times> 'd \<Rightarrow> 'e"   and A :: "'c set"   and B :: "'c \<Rightarrow> 'd set" assumes "f abs_summable_on Sigma A B" shows "infsetsum f (Sigma A B) = (\<Sum>\<^sub>ax\<in>A. \<Sum>\<^sub>ay\<in>B x. f (x, y))"
children_ordered: fixes lt :: "dag"   and p :: "ref"   and rt :: "dag"   and var :: "ref \<Rightarrow> nat" assumes "ordered (Node lt p rt) var" shows "ordered lt var \<and> ordered rt var"
obj_at_field_on_heap_def: fixes P :: "'g \<Rightarrow> bool"   and r :: "'g"   and f :: "'e" shows "obj_at_field_on_heap P r f \<equiv> \<lambda>s. case map_option obj_fields (sys_heap s r) of None \<Rightarrow> False | Some fs \<Rightarrow> case fs f of None \<Rightarrow> True | Some r' \<Rightarrow> P r'"
norm_pres_extensions_def: fixes E :: "'b set"   and p :: "'b \<Rightarrow> real"   and F :: "'b set"   and f :: "'b \<Rightarrow> real" shows "norm_pres_extensions E p F f = {graph H h |H h. linearform H h \<and> Subspace.subspace H E \<and> Subspace.subspace F H \<and> graph F f \<subseteq> graph H h \<and> (\<forall>x\<in>H. h x \<le> p x)}"
no_await_trans: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "no_await c"   and "\<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" shows "no_await c'"
abs_eq: fixes xa :: "'e \<Longrightarrow> 'f"   and x :: "'f \<Rightarrow> 'g" shows "(xa \<natural> mk\<^sub>e x) = (\<forall>b b'. x (b' \<triangleleft>\<^bsub>xa\<^esub> b) = x b)"
fimage_inter_dom_2: fixes m :: "('d, 'c) fmap"   and A :: "'d fset" shows "fmimage m (A |\<inter>| fmdom m) = fmimage m A"
distinct_1: fixes x21 :: "'c"   and x22 :: "'c Tree"   and x23 :: "'c Tree" shows "E \<noteq> T x21 x22 x23"
closed_Compl: fixes "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "class.topological_space open"   and "open S" shows "topological_space.closed open (- S)"
trancl_converse: fixes r :: "('c \<times> 'c) set" shows "(r\<inverse>)\<^sup>+ = (r\<^sup>+)\<inverse>"
map_def: fixes f :: "'a" shows "\<chi>.A.map f = (if \<chi>.J.arr f then a else \<chi>.C.null)"
set_mode_get_disconnected_nodes: fixes ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) document_ptr" shows "\<forall>w\<in>set_mode_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')"
Phase1or2ReadThen_HInv3_hasRead: fixes s :: "state"   and pp :: "Proc"   and dd :: "Disk"   and qq :: "Proc"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "\<not> hasRead s pp dd qq"   and "Phase1or2ReadThen s s' p d q"   and "pp \<noteq> p \<or> qq \<noteq> q \<or> dd \<noteq> d" shows "\<not> hasRead s' pp dd qq"
llength_eq_infty_conv_lfinite: fixes xs :: "'b llist" shows "(llength xs = \<infinity>) = linfinite xs"
twoNetsDistinct_def: fixes a :: "'b set set"   and b :: "'c set set"   and c :: "'b set set"   and d :: "'c set set" shows "twoNetsDistinct a b c d = (netsDistinct a c \<or> netsDistinct b d)"
subset_wens_single_finite: fixes W :: "'b set set"   and act :: "('b \<times> 'b) set"   and B :: "'b set"   and k :: "nat" assumes "W \<subseteq> wens_single_finite act B ` {..k}"   and "single_valued act"   and "W \<noteq> {}" shows "\<exists>m. \<Union> W = wens_single_finite act B m"
discI_1: fixes llist :: "'a llist" assumes "llist = LNil" shows "lnull llist"
eq_alt: shows "(=) = BNF_Def.Grp UNIV id"
psubset_finite_imp_set_less_aux: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subset> B"   and "finite B" shows "A \<sqsubset>' B"
compI_1: fixes f :: "'a"   and g :: "'a" assumes "f \<in> Ar"   and "g \<in> Ar"   and "Cod f = Dom g" shows "g \<bullet> f \<in> Ar"
cr_set_dlist_def: shows "cr_set_dlist \<equiv> \<lambda>x y. x = list_of_dlist y"
finite_state_imp_restriction_finite_2: fixes s :: "'b \<Rightarrow> (('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr) option"   and r :: "'b" assumes "finite (LID\<^sub>G s)" shows "finite (LID\<^sub>G (s(r := None)))"
set_less_trans_set_less_eq: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A \<sqsubset>' B"   and "B \<sqsubseteq>' C" shows "A \<sqsubset>' C"
g_2: shows "eval (Pr n f g) (Suc y # xs) = eval g (y # hd (trace n f g xs y) # xs)"
sp_instr_no_jump: fixes F :: "'q \<Rightarrow> 'r fundef option"   and instr :: "('s, 't, 'q, 'e, 'f, 'g, 'u, 'v) instr"   and \<Sigma> :: "Unboxed.type option list"   and type :: "Unboxed.type option list" assumes "sp_instr F instr \<Sigma> = Ok type" shows "\<not> is_jump instr"
pnsubst_correct: fixes l :: "'a list"   and P1 :: "pol"   and M :: "mon"   and P2 :: "pol"   and n :: "nat"   and P3 :: "pol" assumes "in_carrier l"   and "pnsubst P1 M P2 n = Some P3"   and "Imon l M = Ipol l P2" shows "Ipol l P1 = Ipol l P3"
assms: fixes x :: "'a"   and y :: "'b" shows "ovalid (P x y) (B x y) Q"
get_elements_by_tag_name_result_in_tree_order: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and name :: "char list"   and results :: "(_) element_ptr list"   and to :: "(_) object_ptr list"   and result :: "(_) element_ptr" assumes "h \<turnstile> get_elements_by_tag_name ptr name \<rightarrow>\<^sub>r results"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r to"   and "result \<in> set results" shows "cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^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 result \<in> set to"
simps_1: fixes s :: "'c list" shows "Partial_Cost_Model.steps s [] [] = s"
cases: fixes a :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "mthr.\<tau>diverge a" obtains s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" where "a = s" and "\<tau>mredT s s'" and "mthr.\<tau>diverge s'"
mult_oner: fixes x :: "('a \<times> 'a) set" shows "x O Id = x"
flattening_lemma: fixes \<Pi> :: "'c strips_problem"   and \<pi> :: "'c strips_operator list list" assumes "is_valid_problem_strips \<Pi>"   and "\<forall>ops\<in>set \<pi>. \<exists>op. ops = [op]"   and "STRIPS_Semantics.is_parallel_solution_for_problem \<Pi> \<pi>" shows "STRIPS_Semantics.is_serial_solution_for_problem \<Pi> (concat \<pi>)"
wf_interp_for_formula_FEXISTS: fixes I :: "(nat + nat set) list"   and \<phi> :: "'a formula"   and w :: "'a list" assumes "wf_formula (length I) (FEXISTS \<phi>)" shows "wf_interp_for_formula (w, I) (FEXISTS \<phi>) = (\<forall>P. finite P \<longrightarrow> wf_interp_for_formula (w, Inr P # I) \<phi>)"