hcju/isabelleps huggingface dataset:
are_all_operators_applicable_def: fixes s :: "'b \<Rightarrow> bool option"   and ops :: "'b strips_operator list" shows "are_all_operators_applicable s ops \<equiv> list_all (is_operator_applicable_in s) ops"
trivial_limit_def: fixes F :: "'a filter" shows "(F = bot) = (\<forall>\<^sub>F x in F. False)"
Theorem3_derivAux: fixes p :: "L"   and G :: "Var \<Rightarrow> L"   and c :: "IMP"   and H :: "Var \<Rightarrow> L"   and X :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and q :: "L" assumes "(p, G, c, H) \<in> HS" shows "Theorem3derivProp X p G c H q"
no_change_1_1: fixes c :: "nat \<Rightarrow>f value option"   and r :: "nat \<Rightarrow>f value option"   and t :: "(String.literal \<times> value list) list" assumes "obtains 1 c drinks2 1 r t" shows "c = r"
Lab: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and c :: "stmt"   and l :: "jump"   and Q :: "'b assn" assumes "G,A\<turnstile>{Normal P} .c. {abupd (absorb l) .; Q}" shows "G,A\<turnstile>{Normal P} .l\<bullet> c. {Q}"
less_1: fixes X :: "'a nmultiset"   and Y :: "'a nmultiset"   and Z :: "'a nmultiset" assumes "Max {|X|, |Y|, |Z|} < Max {|Xa__|, |Ya__|, |Za__|}"   and "X < Y"   and "Y < Z" shows "X < Z"
freshBoundDerivative_2: fixes P :: "pi"   and a :: "subject"   and x :: "name"   and P' :: "pi"   and y :: "name" assumes "P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P'"   and "y \<sharp> P"   and "y \<noteq> x" shows "y \<sharp> P'"
simps_1: fixes f :: "typ \<Rightarrow> typ"   and a :: "String.literal"   and T :: "typ" shows "map_types f (Ct a T) = Ct a (f T)"
coeff_principal: fixes t :: "'a"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "t \<in> carrier S"   and "pol_coeff S (n, f)"   and "\<forall>j\<le>n. f j \<in> S \<diamondsuit>\<^sub>p t" shows "\<exists>f'. pol_coeff S (n, f') \<and> (\<forall>j\<le>n. f j = t \<cdot>\<^sub>r\<^bsub>S\<^esub> f' j)"
trans: fixes b :: "'a \<times> 'b"   and a :: "'a \<times> 'b"   and c :: "'a \<times> 'b" assumes "less_eq_prod b a"   and "less_eq_prod c b" shows "less_eq_prod c a"
step_8: shows "InvariantWatchesEl (getF state') (getWatch1 state') (getWatch2 state')"
less_eq_seq_def: fixes xs :: "'b seq"   and ys :: "'b seq" shows "(xs \<le> ys) = (xs = ys \<or> xs < ys)"
Next_geq_not_member: fixes S :: "'a set"   and s :: "'a" assumes "finite S"   and "Next S \<le> s" shows "s \<notin> S"
Next_2: fixes cstate :: "call \<times> (nat \<Rightarrow> contour option) \<times> ((nat \<times> char list) \<times> contour \<Rightarrow> d option) \<times> contour"   and cstate_a :: "call \<times> (nat \<Rightarrow> 'c option) \<times> ((nat \<times> char list) \<times> 'c \<Rightarrow> 'c \<aproc> set) \<times> 'c" assumes "|cstate| \<lessapprox> cstate_a" shows "|evalC\<cdot>(Discr cstate)| \<lessapprox> \<aC>\<cdot>(Discr cstate_a)"
abs_conv_abscissa_cmult: fixes c :: "'a"   and f :: "'a fds" assumes "c \<noteq> (0::'a)" shows "abs_conv_abscissa (fds_const c * f) = abs_conv_abscissa f"
simps_1: fixes N :: "bool stream measure"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat" shows "delta_predict N der matur 0 = delta_price N der matur 0"
elims: fixes x :: "nat"   and y :: "nat" assumes "local.Dube_aux x = y" obtains j :: "nat" where "x = j" and "y = (if j + 2 < n then 2 + (local.Dube_aux (j + 1) choose 2) + (\<Sum>i = j + 3..n - 1. local.Dube_aux i choose Suc (i - j)) else if j + 2 = n then d\<^sup>2 + 2 * d else 2 * d)"
path_connected_UNION: fixes A :: "'a set"   and S :: "'a \<Rightarrow> 'b set"   and z :: "'b" assumes "\<And>i. i \<in> A \<Longrightarrow> path_connected (S i)"   and "\<And>i. i \<in> A \<Longrightarrow> z \<in> S i" shows "path_connected (\<Union> (S ` A))"
exhaust: fixes y :: "'b vertex" obtains x1 :: "'b" where "y = Vertex x1"   | x21 :: "'b" and x22 :: "'b" where "y = Edge x21 x22"
ik\<^sub>s\<^sub>tD': fixes t :: "('c, 'd) Term.term"   and S :: "('c, 'd) strand_step list" assumes "t \<in> ik\<^sub>s\<^sub>t S" shows "t \<in> trms\<^sub>s\<^sub>t S"
combine_def: shows "combine \<equiv> map_fun id (map_fun rep (map_fun rep Mapping)) (\<lambda>f m1 m2 x. combine_options f (m1 x) (m2 x))"
absorb2: fixes a :: "'c"   and b :: "'c" assumes "a \<le> b" shows "sup a b = b"
subseq_drop_many: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "xs \<preceq> ys" shows "xs \<preceq> zs @ ys"
reachable_ss_V: fixes s :: "nat" assumes "s \<in> h.cf.V" shows "h.cf.reachableNodes s \<subseteq> h.cf.V"
fix_least_below: fixes F :: "'b \<rightarrow> 'b"   and x :: "'b" assumes "F\<cdot>x \<sqsubseteq> x" shows "fix\<cdot>F \<sqsubseteq> x"
RedNewFail: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and E :: "char list \<Rightarrow> ty option"   and C :: "char list"   and l :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" assumes "new_Addr h = None" shows "P,E \<turnstile> \<langle>new C,(h, l)\<rangle> \<rightarrow> \<langle>Throw (addr_of_sys_xcpt OutOfMemory, [OutOfMemory]),(h, l)\<rangle>"
xor_ac_1: fixes x :: "'e"   and y :: "'e"   and z :: "'e" assumes "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "(x \<oplus> y) \<oplus> z = x \<oplus> y \<oplus> z"
open_typedef_to_Quotient: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and P :: "'b \<Rightarrow> bool"   and T :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "type_definition Rep Abs {x. P x}"   and "T \<equiv> \<lambda>x y. x = Rep y" shows "Quotient (eq_onp P) Abs Rep T"
ts_inv_okI: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and I :: "'a \<Rightarrow> 'd option" assumes "\<And>t. (ts t = None) = (I t = None)" shows "ts_inv_ok ts I"
mset_ran_by_sum: fixes a :: "int \<Rightarrow> 'b"   and r :: "int set" shows "mset_ran a r = (\<Sum>i\<in>r. {#a i#})"
whileLoop_unroll: fixes C :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'c) set \<times> bool"   and r :: "'d" shows "whileLoop C   B r = condition (C r)   (B r >>= whileLoop C   B)   (return r)"
n2_type: fixes t1 :: "'b bro"   and h :: "nat"   and t2 :: "'b bro"   and a :: "'b" shows "(t1 \<in> Bp h \<and> t2 \<in> T h \<longrightarrow> n2 t1 a t2 \<in> Bp (Suc h)) \<and> (t1 \<in> T h \<and> t2 \<in> Bp h \<longrightarrow> n2 t1 a t2 \<in> Bp (Suc h))"
finite_imp_Sup_less: fixes X :: "'a set"   and x :: "'a"   and a :: "'a" assumes "finite X"   and "x \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> x < a" shows "\<Squnion> X < a"
HaddP_unique: fixes w :: "tm"   and x :: "tm"   and y :: "tm"   and y' :: "tm" shows "{HaddP w x y, HaddP w x y'} \<turnstile> y' EQ y"
prodag_carrier: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "carrier (a\<Pi>\<^bsub>I\<^esub> A) = carr_prodag I A"
dsjcnj: fixes C :: "fmla"   and A :: "fmla"   and B :: "fmla" shows "C OR A IMP (C IMP B) IMP A OR B \<in> boolean_axioms"
wf_table_iff: fixes n :: "nat"   and A :: "nat set"   and Q :: "'a option list \<Rightarrow> bool"   and X :: "'a option list set" shows "wf_table n A Q X = (\<forall>x. (x \<in> X) = (Q x \<and> wf_tuple n A x))"
elims: fixes x :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and xa :: "'c"   and xb :: "'b rule list"   and xc :: "state"   and y :: "state" assumes "approximating_bigstep_fun x xa xb xc = y" obtains \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)" and p :: "'c" and s :: "state" where "x = \<gamma>'__" and "xa = p__" and "xb = []" and "xc = s" and "y = s"   | \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)" and p :: "'c" and v :: "'b rule" and va :: "'b rule list" and X :: "final_decision" where "x = \<gamma>''__" and "xa = pa__" and "xb = v # va" and "xc = Decision X" and "y = Decision X"   | \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)" and p :: "'c" and m :: "'b match_expr" and a :: "action" and rs :: "'b rule list" where "x = \<gamma>" and "xa = p" and "xb = Rule m a # rs" and "xc = Undecided" and "y = (if \<not> matches \<gamma> m a p then approximating_bigstep_fun \<gamma> p rs Undecided else case a of Accept \<Rightarrow> Decision FinalAllow | Drop \<Rightarrow> Decision FinalDeny | Log \<Rightarrow> approximating_bigstep_fun \<gamma> p rs Undecided | Reject \<Rightarrow> Decision FinalDeny | Empty \<Rightarrow> approximating_bigstep_fun \<gamma> p rs Undecided)"
Atm_ZObisT: fixes indis :: "('a \<times> 'a) set"   and aval :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and atm :: "'b"   and tval :: "'c \<Rightarrow> 'a \<Rightarrow> bool" assumes "PL_Indis indis"   and "PL_Indis.compatAtm aval indis atm" shows "(Atm atm, Atm atm) \<in> PL_Indis.ZObisT tval aval indis"
diff_Suc_diff_eq1: fixes k :: "nat"   and j :: "nat"   and i :: "nat" assumes "k \<le> j" shows "i - Suc (j - k) = i + k - Suc j"
empty_def: shows "Leftist_Heap.empty = \<langle>\<rangle>"
one_unique: fixes G :: "('e, 'f) Group_scheme"   and a :: "'e"   and x :: "'e" assumes "Group G"   and "a \<in> carrier G"   and "x \<in> carrier G"   and "x \<cdot>\<^bsub>G\<^esub> a = x" shows "a = \<one>\<^bsub>G\<^esub>"
compact_path_image: fixes g :: "real \<Rightarrow> 'b" assumes "path g" shows "compact (path_image g)"
unx_some_lemmas_4: fixes u :: "'a"   and y :: "name" assumes "unconst u = Some y" shows "unfree u = None"
assms_2: fixes t1 :: "'a trm"   and t2 :: "'a trm" assumes "is_Const t1" shows "P (Eq t1 t2)"
fps_unit_factor_mult': fixes f :: "'a fps"   and g :: "'a fps" assumes "fps_nth f (subdegree f) * fps_nth g (subdegree g) \<noteq> (0::'a)" shows "unit_factor (f * g) = unit_factor f * unit_factor g"
arrI: fixes f :: "'a arrow_of_spans_data"   and a :: "'a arrow_of_spans_data"   and b :: "'a arrow_of_spans_data" assumes "V.in_hom f a b" shows "V.arr f"
sol': shows "(b, es) = solve_factored_ratfps p' ds"
nn_integral_fst: fixes f :: "'a \<times> 's \<Rightarrow> ennreal"   and M1 :: "'a measure" assumes "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M count_space \<Omega>)" shows "\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space \<Omega> \<partial>M1 = integral\<^sup>N (M1 \<Otimes>\<^sub>M count_space \<Omega>) f"
T_vrestr_same_emeasure_2: fixes A :: "'a set"   and n :: "nat" assumes "A \<in> events" shows "emeasure M ((Tinv ^^ n) --` A) = emeasure M A"
between_less: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubset> y" shows "x \<sqsubset> between x y \<and> between x y \<sqsubset> y"
npd0: shows "n pd\<rightarrow> \<pi> 0"
set_remdups: fixes xs :: "'g list" shows "set (remdups xs) = set xs"
sp_lift: fixes n :: "nat"   and p1 :: "('c, 'd, 'fun, 'op, 'opinl) Inca.instr list"   and \<Sigma>1 :: "Unboxed.type option list"   and p2 :: "('c, 'd, 'fun, 'op, 'opinl, 'opubx, 'e, 'f) Ubx.instr list"   and \<Sigma>2 :: "Unboxed.type option list"   and F :: "'fun \<Rightarrow> 'g fundef option" assumes "lift n p1 \<Sigma>1 = Ok (p2, \<Sigma>2)"   and "\<And>x. rel_option (\<lambda>ar fd. arity fd = ar) (get_arity x) (F x)" shows "Subx.sp F p2 \<Sigma>1 = Ok \<Sigma>2"
assms: shows "(\<lambda>b. (\<Sum>k = a..b. f (real k)) - F (real b) - T (real b)) \<longlonglongrightarrow> C"
monic_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "monic p = monom_mult ((1::'c) / lc p) (0::'a) p"
8: shows "injective (w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T)"
ceiling_ge_round: fixes x :: "'a" shows "round x \<le> \<lceil>x\<rceil>"
fixp_preserves_mono2: fixes F :: "('c \<Rightarrow> 'd \<Rightarrow> 'e llist) \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e llist"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e llist"   and ord :: "'c \<times> 'd \<Rightarrow> 'c \<times> 'd \<Rightarrow> bool" assumes "\<And>x. mono_llist (\<lambda>f. case x of (x, xa) \<Rightarrow> F (curry f) x xa)"   and "f \<equiv> curry (llist.fixp_fun (\<lambda>f (x, y). F (curry f) x y))"   and "\<And>f. monotone ord (\<sqsubseteq>) (\<lambda>(x, y). f x y) \<Longrightarrow> monotone ord (\<sqsubseteq>) (\<lambda>(x, y). F f x y)" shows "monotone ord (\<sqsubseteq>) (\<lambda>(x, y). f x y)"
r: assumes "a \<in> carrier G" shows "\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as a"
subset_entailed: fixes N2 :: "'b literal multiset set"   and N1 :: "'b literal multiset set" assumes "N2 \<subseteq> N1" shows "N1 \<TTurnstile>e N2"
distinct_butlast: fixes xs :: "'c list" assumes "distinct xs" shows "distinct (butlast xs)"
sumI': fixes u :: "'b"   and U :: "'b set"   and v :: "'b"   and V :: "'b set" assumes "u \<in> U"   and "v \<in> V" shows "u + v \<in> U + V"
pnet_init_net_ips_net_tree_ips: fixes s :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and p :: "net_tree" assumes "s \<in> init (pnet np p)" shows "net_ips s = net_tree_ips p"
distinct_3: fixes x1 :: "'a"   and x3 :: "'b" shows "Oldf x1 \<noteq> Wit x3"
intros_3: fixes \<Gamma> :: "binding list"   and t\<^sub>1 :: "trm"   and T\<^sub>1\<^sub>1 :: "POPLmarkRecord.type"   and T\<^sub>1\<^sub>2 :: "POPLmarkRecord.type"   and t\<^sub>2 :: "trm" assumes "\<Gamma> \<turnstile> t\<^sub>1 : T\<^sub>1\<^sub>1 \<rightarrow> T\<^sub>1\<^sub>2"   and "\<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>1\<^sub>1" shows "\<Gamma> \<turnstile> t\<^sub>1 \<bullet> t\<^sub>2 : T\<^sub>1\<^sub>2"
abs_pi: shows "\<bar>arg x\<bar> \<le> pi"
relcompp_witness_1: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'a"   and y :: "'c" assumes "(A OO B) x y" shows "A x (relcompp_witness A B (x, y))"
apply_updates_foldr: fixes u :: "(nat \<times> vname aexp) list"   and old :: "vname \<Rightarrow> value option" shows "apply_updates u old = foldr (\<lambda>h r. r(fst h $:= aval (snd h) old)) (rev u)"
not_mem_set_dropWhileD: fixes x :: "'a"   and P :: "'a \<Rightarrow> bool"   and xs :: "'a list" assumes "x \<notin> set (dropWhile P xs)"   and "x \<in> set xs" shows "P x"
saturated_upto_def: fixes N :: "'a literal multiset set" shows "saturated_upto N = (inferences_from (N - Rf N) \<subseteq> Ri N)"
rep_eq: fixes x :: "'b fract"   and xa :: "'b fract" shows "(x \<le> xa) = (fst (rep_fract x) * snd (rep_fract xa) * (snd (rep_fract x) * snd (rep_fract xa)) \<le> fst (rep_fract xa) * snd (rep_fract x) * (snd (rep_fract x) * snd (rep_fract xa)))"
split_list_unique_prefix: fixes x :: "'c"   and xs :: "'c list" assumes "x \<in> set xs" shows "\<exists>pre suf. xs = pre @ x # suf \<and> (\<forall>y\<in>set pre. x \<noteq> y)"
not_squarefreeE: fixes n :: "'b" assumes "\<not> squarefree n" obtains x :: "'b" where "x\<^sup>2 dvd n" and "\<not> x dvd (1::'b)"
simps_2: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" shows "binrelchain P [x] = True"
poly_inv_def: fixes p :: "('c monom \<times> 'd) list" shows "poly_inv p \<equiv> (\<forall>c\<in>snd ` set p. c \<noteq> (0::'d)) \<and> distinct (map fst p)"
countable_def: fixes S :: "'b set" shows "countable S = (\<exists>f. inj_on f S)"
encR: fixes S :: "'procS" shows "indRelEQp (SourceTerm S) (TargetTerm (\<lbrakk>S\<rbrakk>))"
dgrad_set_le_subset: fixes S :: "'e set"   and T :: "'e set"   and d :: "'e \<Rightarrow> nat" assumes "S \<subseteq> T" shows "dgrad_set_le d S T"
transpos_mem: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and l :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "i \<noteq> j"   and "l \<le> n" shows "(\<tau>\<^bsub>i j\<^esub>) l \<le> n"
I_def: shows "I = (\<lambda>i. if i < k then i else Suc i)"
low_mds_eq_def: fixes mds :: "Mode \<Rightarrow> 'Var\<^sub>A set"   and mem\<^sub>1 :: "'Var\<^sub>A \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var\<^sub>A \<Rightarrow> 'Val" shows "abs.low_mds_eq mds mem\<^sub>1 mem\<^sub>2 \<equiv> \<forall>x. dma\<^sub>A mem\<^sub>1 x = Low \<and> (x \<in> \<C>\<^sub>A \<or> x \<notin> mds AsmNoReadOrWrite) \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x"
hyps_7: shows "x\<^sub>3 = l\<^sup>2 - x\<^sub>1 - x\<^sub>2"
agree_sub: fixes A :: "('a + 'a) set"   and B :: "('a + 'a) set"   and \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and \<omega> :: "(real, 'a) vec \<times> (real, 'a) vec" assumes "A \<subseteq> B"   and "Vagree \<nu> \<omega> B" shows "Vagree \<nu> \<omega> A"
snapshot_state_unchanged: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and p :: "'p" assumes "c \<turnstile> ev \<mapsto> c'"   and "ps c p \<noteq> None" shows "ps c p = ps c' p"
has_fields_mono_lem: fixes D :: "char list"   and C :: "char list"   and P :: "'a prog"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list" assumes "P \<turnstile> D \<preceq>\<^sup>* C"   and "P \<turnstile> C has_fields FDTs" shows "\<exists>pre. P \<turnstile> D has_fields pre @ FDTs \<and> dom (map_of pre) \<inter> dom (map_of FDTs) = {}"
fin_2: fixes i :: "nat" shows "g i \<noteq> \<infinity>"
suminf_sum: fixes I :: "'a set"   and f :: "'a \<Rightarrow> nat \<Rightarrow> 'b" assumes "\<And>i. i \<in> I \<Longrightarrow> summable (f i)" shows "(\<Sum>n. \<Sum>i\<in>I. f i n) = (\<Sum>i\<in>I. \<Sum>n. f i n)"
merge: shows "merge I J -` A \<inter> space (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) \<in> sets (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M)"
mono: shows "mono (sum (\<lambda>i. (cmod (x i))\<^sup>2))"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "inv_on_right_moving (as, lm) (s, l, r) ires = (\<exists>lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and> ml + mr = m \<and> (if lm1 = [] then l = Oc \<up> ml @ Bk # Bk # ires else l = Oc \<up> ml @ [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and> (r = Oc \<up> mr @ [Bk] @ <lm2> @ Bk \<up> rn \<or> r = Oc \<up> mr \<and> lm2 = []))"
symetric: fixes P :: "pi"   and Q :: "pi" assumes "P \<approx>\<^sup>s Q" shows "Q \<approx>\<^sup>s P"
c: shows "\<not> other.s \<delta> - ego.q \<delta> \<le> delayed_safe_distance.snd_safe_distance"
abs_eq: shows "1 = Abs_blinop 1\<^sub>L"
not_greater_Greatest: fixes P :: "'b \<Rightarrow> bool"   and k :: "'b" assumes "(GREATEST x. P x) < k" shows "\<not> P k"
antisym: fixes s1 :: "'a"   and s2 :: "'a" assumes "s1 \<preceq> s2 \<and> s2 \<preceq> s1" shows "s1 = s2"
apfst_def: fixes f :: "'a \<Rightarrow> 'c" shows "apfst f = map_prod f id"
sum_nonneg: 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 A :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq zero (f x)" shows "less_eq zero (comm_monoid_add.sum plus zero f A)"
atLeastLessThan_empty: fixes b :: "'a"   and a :: "'a" assumes "b \<le> a" shows "{a..<b} = {}"
top_on_sup: fixes o1 :: "'b st option"   and X :: "char list set"   and o2 :: "'b st option" assumes "top_on\<^sub>o o1 X"   and "top_on\<^sub>o o2 X" shows "top_on\<^sub>o (o1 \<squnion> o2) X"
adds_pp_iff: fixes s :: "('b, 'c) pp"   and t :: "('b, 'c) pp" shows "(s adds t) = (pp.mapping_of s adds pp.mapping_of t)"
fmdrop_set_null: fixes m :: "('d, 'e) fmap" shows "fmdrop_set {} m = m"
strategy_extends_backwards_VVpstar: fixes v0 :: "'a"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "v0 \<in> G'.VV p**"   and "G'.strategy p \<sigma>"   and "\<And>w. v0 \<rightarrow>\<^bsub>G'\<^esub> w \<Longrightarrow> G'.winning_strategy p \<sigma> w" shows "G'.winning_strategy p \<sigma> v0"
l2_inv5_trans: shows "{l2_inv5} TS.trans l2 {> l2_inv5}"
gmft_correct2: fixes t :: "('c, 'd) FingerTreeStruc" assumes "ft_invar t" shows "gmft t = sum_list (map snd (toList t))"
dim_gen_eigenspace_def: fixes A :: "'b mat"   and ev :: "'b"   and k :: "nat" shows "dim_gen_eigenspace A ev k = kernel_dim (char_matrix A ev ^\<^sub>m k)"
solvable_subgroup: fixes H :: "'a set" assumes "subgroup H G"   and "solvable G" shows "solvable_seq G H"
disjoint_from_second_factor: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "CM"   and "P x y \<and> \<not> O x (y \<otimes> z)" shows "\<not> O x z"
Span_base_incl: fixes K :: "'c set"   and Us :: "'c list" assumes "subfield K S"   and "set Us \<subseteq> carrier S" shows "set Us \<subseteq> S.Span K Us"
field_char_0_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "field_char_0_hom hom \<equiv> field_hom hom"
inner_Basis_sum: fixes I :: "'b set"   and x :: "'b \<Rightarrow> 'a"   and v :: "'a" assumes "\<And>i. i \<in> I \<Longrightarrow> x i \<in> S" shows "local.inner_Basis (sum x I) v = (\<Sum>i\<in>I. local.inner_Basis (x i) v)"
wpt_Seq: fixes c\<^sub>1 :: "com"   and c\<^sub>2 :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "wp\<^sub>3\<^sub>' (c\<^sub>1;; c\<^sub>2) Q = wp\<^sub>3\<^sub>' c\<^sub>1 (wp\<^sub>3\<^sub>' c\<^sub>2 Q)"
cut: fixes G :: "('a, 'b) form list"   and A :: "('a, 'b) form"   and B :: "('a, 'b) form" assumes "G \<turnstile> A"   and "A # G \<turnstile> B" shows "G \<turnstile> B"
result: shows "closure_invariant ([([], init r, init s)], [], {(post (init r), post (init s))}) = Some (ws, ps, N)"
SUP_cong: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c \<Rightarrow> 'd"   and D :: "'c \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd" assumes "A = B"   and "\<And>x. s2p B x \<Longrightarrow> C x = D x" shows "Sup (\<P> C A) = Sup (\<P> D B)"
form_split_def: fixes a :: "nat"   and b :: "nat"   and c :: "nat"   and d :: "nat"   and i :: "nat" shows "form_split a b c d i \<equiv> a \<le> c \<and> (i = 0 \<and> a < b \<and> b < c \<and> c < d \<or> i = 1 \<and> a < c \<and> c < b \<and> b < d \<or> i = 2 \<and> a < c \<and> c < d \<and> d < b \<or> i = 3 \<and> a = c \<and> b \<noteq> d)"
SUP_upper2: 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 i :: "'c"   and A :: "'c set"   and u :: "'b"   and f :: "'c \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A"   and "less_eq u (f i)" shows "less_eq u (Sup (f ` A))"
eq_refl: fixes x :: "'a"   and y :: "'a" assumes "x = y" shows "leq1 x y"
simps_233: fixes f :: "'c \<Rightarrow> 'b"   and x111 :: "'c ltlc"   and x112 :: "'c ltlc" shows "map_ltlc f (x111 U\<^sub>c x112) = map_ltlc f x111 U\<^sub>c map_ltlc f x112"
simps_1: fixes t1 :: "'b trm"   and t2 :: "'b trm" shows "lhs (t1 \<cdot> t2) = t1"
lexordp_def: shows "char.lexordp \<equiv> lfp (\<lambda>p x1 x2. (\<exists>y ys. x1 = [] \<and> x2 = y # ys) \<or> (\<exists>x y xs ys. x1 = x # xs \<and> x2 = y # ys \<and> less_char x y) \<or> (\<exists>x y xs ys. x1 = x # xs \<and> x2 = y # ys \<and> \<not> less_char x y \<and> \<not> less_char y x \<and> p xs ys))"
simps_1: fixes X :: "tyvrs"   and U :: "ty"   and Y :: "tyvrs"   and T :: "ty" shows "TVarB X U[Y \<mapsto> T]\<^sub>b = TVarB X (U[Y \<mapsto> T]\<^sub>\<tau>)"
direct_decompE: fixes A :: "'a set"   and ss :: "'a set list"   and a :: "'a" assumes "direct_decomp A ss"   and "a \<in> A" obtains qs :: "'a list" where "qs \<in> listset ss" and "a = sum_list qs"
not_eq_order_implies_strict: fixes a :: "'b"   and b :: "'b" assumes "a \<noteq> b"   and "order.greater_eq b a" shows "order.greater b a"
list_constr_def: fixes L :: "nat list"   and n :: "nat" shows "list_constr L n \<equiv> list_all (\<lambda>x. x < n) L"